UBC Theses and Dissertations

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UBC Theses and Dissertations

Dynamic response of ship structures to impact loads Asadi, Ghasem Vaez-Zadeh 1989

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DYNAMIC RESPONSE OF SHIP STRUCTURES TO IMPACT LOADS By GHASEM VAEZ-ZADEH ASADI B . S c , S h i r a z U n i v e r s i t y , I r a n , 1974 M . E n g . , N o r t h C a r o l i n a S t a t e U n i v e r s i t y , 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRIT ISH COLUMBIA © GHASEM VAEZ-ZADEH ASADI May 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Date J u t j / ^ , I f g q DE-6 (2/88) ABSTRACT I n t h i s s t u d y t h e dynamic r e s p o n s e o f a s h i p s t r u c t u r e t o i m p a c t l o a d s i s i n v e s t i g a t e d . The s h i p m o t i o n i s f u l l y t h r e e - d i m e n s i o n a l and t h e s h i p s t r u c t u r e i s m o d e l e d as a t h r e e - d i m e n s i o n a l e l a s t i c beam. F i n i t e e l e m e n t methods a r e u s e d t o d i g i t i z e t h e e q u a t i o n s o f m o t i o n o f t h e s y s t e m . The f o r c e s , o n t h e s h i p a r e i n t e r a c t i v e w i t h t h e s h i p m o t i o n and p o s i t i o n so t h a t a f u l l dynamic a n a l y s i s i s e s s e n t i a l . Two m a i n p r o b l e m s a r e c o n s i d e r e d : i ) E s t i m a t i o n o f h u l l damage when a s h i p c o l l i d e s w i t h a n o t h e r s h i p , f l o a t i n g s t r u c t u r e o r f i x e d i n s t a l l a t i o n . A p a r t i c u l a r a s p e c t o f t h i s a n a l y s i s w h i c h h a s n o t p r e v i o u s l y b e e n e x a m i n e d a n a l y t i c a l l y i n v o l v e s e s t i m a t i n g damage t o t h e b o t t o m o f s h i p when i t r u n s a g r o u n d . D e p e n d i n g on t h e n a t u r e o f t h e g r o u n d t h e s h i p may be p i e r c e d and s i g n i f i c a n t amounts o f s t e e l may be t o r n , o r t h e s h i p may r i d e o v e r a s a n d b a r w i t h o u t t e a r i n g b u t w i t h n o t i c e a b l e d e n t i n g a n d b e n d i n g . I n s u c h g r o u n d i n g s t u d i e s i t h a s b e e n n e c e s s a r y t o i n t r o d u c e c e r t a i n s t r e n g t h c o e f f i c i e n t s , r e a l i s t i c v a l u e s o f w h i c h h a v e n o t b e e n d e t e r m i n e d , b u t f o r w h i c h s e n s i b l e e s t i m a t e s h a v e b e e n made. The r e s u l t s o f a n u m e r i c a l s t u d y i n t o g r o u n d i n g and c o l l i s i o n damage i l l u s t r a t e c l e a r l y t h a t s h i p s p e e d i s t h e m a j o r v a r i a b l e i n t h e damage p r o c e s s . I n p a r t i c u l a r t h e e f f e c t o f s u b s e q u e n t a n g u l a r m o t i o n s i n c u r r e d d u r i n g a h i g h s p e e d c o l l i s i o n c a n c a u s e s e c o n d a r y b u t a l s o s i g n i f i c a n t c o l l i s i o n s f u r t h e r a f t . I t i s b e l i e v e d t h a t t h e s e a s p e c t s o f c o l l i s i o n and g r o u n d i n g , and t h e r e l a t e d p r o b l e m s a s s o c i a t e d w i t h c o l l i s i o n w h i l s t m a n e u v e r i n g , h a v e n o t b e e n i n v e s t i g a t e d p r e v i o u s l y . i i ) B e n d i n g s t r e s s e s i n d u c e d i n i c e - b r e a k i n g s h i p s d u r i n g o p e r a t i o n i n i c e . I n t h i s s e c o n d c l a s s o f p r o b l e m s two modes o f o p e r a t i o n s a r e c o n s i d e r e d ; c o n t i n u o u s o p e r a t i o n i n l e v e l i c e w i t h o u t l o s s o f s p e e d , and h i g h s p e e d ramming o f i c e r i d g e s i n w h i c h t h e s h i p i s b r o u g h t t o r e s t . I n t h e c o n t i n u o u s i c e b r e a k i n g mode, t h e i m p u l s e l o a d s a r e r e l a t i v e l y l o w b u t p e r i o d i c . The p e r i o d o f t h e i m p u l s e l o a d s v a r i e s l i n e a r l y w i t h s h i p s p e e d and a l s o depends on t h e h a r d n e s s and t h i c k n e s s o f t h e i c e . S i n c e t h e s h i p i s a n e l a s t i c s y s t e m w i t h n a t u r a l f r e q u e n c i e s o f t h e same o r d e r as i m p a c t f r e q u e n c y , some i n t e r e s t i n g r e s p o n s e c o n d i t i o n s have b e e n i d e n t i f i e d l e a d i n g t o l a r g e f l e x u r a l b e n d i n g s t r e s s e s i n t h e s h i p . I n t h e ramming m o d e , ' two r e s p o n s e s t a t e s a r e o f i m p o r t a n c e . , The i n i t i a l i m p u l s e a t t h e bow o f t h e s h i p , when c o n t a c t i s f i r s t made, c a u s e s t h e s h i p t o r e s p o n d p r i m a r i l y i n i t s f i r s t f l e x u r a l mode w i t h p o s s i b l y l a r g e b e n d i n g s t r e s s e s d e v e l o p i n g d u r i n g t h e f i r s t s e c o n d a f t e r i m p a c t . The s h i p t h e n r i d e s o n t o t h e i c e i n a " b e a c h i n g mode " c a u s i n g l a r g e q u a s i - s t a t i c b e n d i n g s t r e s s e s i n t h e h u l l w h i c h r e a c h a p e a k a f t e r f i v e s e c o n d s o r s o . B o t h o f t h e s e p e a k b e n d i n g s i t u a t i o n s h a v e b e e n i n v e s t i g a t e d and t h e i r dependence on s p e e d , h u l l " s t i f f n e s s , bow a n g l e , and s h i p s p e e d h a s b e e n e s t a b l i s h e d . I n t h e p a s t few y e a r s some d a t a o b t a i n e d f r o m s h i p s o p e r a t i n g i n t h e B e a u f o r t s e a h a s b e e n r e l e a s e d , b o t h f o r c o n t i n u o u s i c e - b r e a k i n g and f o r ramming . Whenever p o s s i b l e t h o s e d a t a h a v e b e e n compared w i t h t h e r e s u l t s p r e d i c t e d b y t h e n u m e r i c a l method d e v e l o p e d h e r e . The a g r e e m e n t i s shown t o be v e r y g o o d . i i i TABLE OF CONTENTS Page A b s t r a c t i i L i s t o f T a b l e s i x L i s t o f F i g u r e s x N o m e n c l a t u r e x v A c k n o w l e d g e m e n t x x i v 1 - I N T R O D U C T I O N : 1 1 . 1 - Impact L o a d s d u r i n g s h i p c o l l i s i o n . 1 1 . 1 . 1 - L i t e r a t u r e R e v i e w . 4 1 . 2 - I c e B r e a k i n g S h i p s and Impact L o a d s . 8 1 . 2 . 1 - L i t e r a t u r e R e v i e w . 9 1 . 3 - I n t r o d u c t i o n t o T h i s Work. 16 1 . 3 . 1 - C o l l i s i o n o f A S h i p and an E x t e r n a l O b j e c t . 17 1 . 3 . 2 - C o n t i n u o u s I c e B r e a k i n g Mode. 18 1 . 3 . 3 - Ramming o f H i g h P r e s s u r e I c e R i d g e s . 18 2- D E R I V A T I O N OF E Q U A T I O N S OF M O T I O N . 20 2 . 1 - G e n e r a l F o r m u l a t i o n . 20 2 . 2 - D e v e l o p m e n t o f S t r u c t u r a l P a r a m e t e r s f o r an E l a s t i c Beam. 24 2 . 2 . 1 - Shape F u n c t i o n . 24 2 . 2 . 2 - Mass and S t i f f n e s s M a t r i x . 26 2 . 2 . 3 - N o d a l V a l u e s o f t h e A p p l i e d L o a d s on t h e e l e m e n t . 29 i - W e i g h t o f t h e E l e m e n t . 30 i v i i - Buoyancy F o r c e . 30 i i i - R e s t o r i n g To rque C a u s e d by H e e l i n g . 32 i v - The O t h e r E x t e r n a l F o r c e s . 35 2 . 3 - D e v e l o p m e n t o f t h e D i s c r e t i z e d E q u a t i o n o f M o t i o n o f t h e F l o a t i n g Beam E l e m e n t . 36 2 . 4 - Deve lopment o f t h e G l o b a l E q u a t i o n o f M o t i o n o f t h e F l o a t i n g Beam. ' 37 3 . ICE BREAKING PROCEDURE. •  41 3 . 1 - I n t r o d u c t i o n . 41 3 . 2 - D e f i n i t i o n o f C o o r d i n a t e s . 41 3 . 3 - C o n t i n u o u s I c e B r e a k i n g Mode. 42 3 . 3 . 1 - A n a l y s i s o f t h e C o n t a c t F o r c e . 43 3 . 3 . 2 - E q u a t i o n s o f M o t i o n o f t h e S h i p . 49 3 . 4 - Ramming o f L a r g e P r e s s u r e R i d g e s . 52 3 . 4 . 1 - A n a l y s i s o f t h e Ramming P r o c e d u r e . 52 3 . 4 . 2 - R e s p o n s e o f t h e S h i p S t r u c t u r e Due t o t h e I n i t i a l I m p u l s e . 54 3 . 4 . 2 . 1 - E n e r g y L o s s I n C o l l i s i o n . 59 3 . 4 . 3 - R e s p o n s e o f t h e S h i p S t r u c t u r e D u r i n g t h e B e a c h i n g P e r i o d . 60 3 . 4 . 3 . 1 - B e a c h i n g M o t i o n Over A F i x e d P o i n t o f t h e bow. 60 3 . 4 . 3 . 2 - B e a c h i n g M o t i o n Over a F i x e d P o i n t on t h e I c e R i d g e . 65 3 . 5 - D e t e r m i n a t i o n o f C o n t a c t A r e a Be tween t h e I c e S h e e t and t h e Bow o f t h e I c e - B r e a k i n g S h i p . 68 3 . 6 - D e t e r m i n a t i o n o f t h e U n i t V e c t o r I n t h e D i r e c t i o n o f t h e V e l o c i t y T a n g e n t t o t h e P l a n e . 71 3 . 7 - D e t e r m i n a t i o n o f t h e Maximum B e n d i n g Moment A l o n g a Beam E l e m e n t . 73 v 3 . 8 - A M e t h o d o f D e t e r m i n i n g t h e C o n s t a n t s f o r P r o p o r t i o n a l Damping . 75 4 . - COLLISION OF THE SHIP WITH AN EXTERNAL OBJECT 79 4 . 1 - I n t r o d u c t i o n . 79 4 . 2 - D e f i n i t i o n o f t h e C o o r d i n a t e s . 79 4 . 3 - E q u a t i o n o f M o t i o n o f The S h i p . 80 4 . 3 . 1 - Head on and S h o u l d e r C o l l i s i o n . 81 4 . 3 . 2 - S i d e C o l l i s i o n D u r i n g A S h i p Maneuver o r G r o u n d i n g . 84 4 . 4 - G e n e r a t e d C o l l i s i o n F o r c e . 85 4 . 4 . 1 - G e n e r a t e d F o r c e I n M a j o r C o l l i s i o n . 86 4 . 4 . 2 - G e n e r a t e d F o r c e I n M i n o r C o l l i s i o n and M o d i f i c a t i o n F o r M a j o r c o l l i s i o n . 87 4 . 4 . 3 - G e n e r a t e d F o r c e When H u l l I s C u t by a Sharp O b j e c t . 89 4 . 5 - D e t e r m i n a t i o n o f The P e n e t r a t i o n o f t h e E x t e r n a l O b j e c t I n t h e H u l l and C a l c u l a t i o n o f t h e C r o s s - s e c t i o n o f Damaged P a r t . 91 4 . 6 - C a l c u l a t i o n o f F o r c e s i n T h r e e P r i n c i p a l D i r e c t i o n s . 95 4 . 7 - D e t a i l o f t h e S o l u t i o n o f t h e E q u a t i o n o f M o t i o n . 96 5 - SOME NUMERICAL SIMULATIONS AND DISCUSSION. 101 5 . 1 - I c e B r e a k i n g P r o c e d u r e . : 102 5 . 1 . 1 - I n t r o d u c t i o n . 102 5 . 1 . 2 - C o n t i n u o u s I c e B r e a k i n g Mode. 107 5 . 1 . 3 - Ramming Heavy I c e . 110 ' 5 . 1 . 3 . 1 - R e s p o n s e o f t h e S h i p t o t h e I n i t i a l I m p u l s e . I l l 5 . 1 . 3 . 2 - B e a c h i n g p e r i o d . 112 v i 5 . 1 . 3 . 2 . 1 - B e a c h i n g on a F i x e d C o n t a c t P o i n t on t h e Bow. 112 5 . 1 . 3 . 2 . 2 - B e a c h i n g on a m o v i n g C o n t a c t P o i n t on the Bow. 115 5 . 1 . 4 - C o m p a r i s o n o f T h e o r y w i t h F i e l d D a t a . 115 5 . 2 - E s t i m a t i o n o f t h e Damage D u r i n g c o l l i s i o n . 143 5 . 2 . 1 - I n t r o d u c t i o n . 143 5 . 2 . 2 - C o l l i d i n g D u r i n g F o r w a r d M o t i o n . 144 5 . 2 . 2 . 1 - C o l l i s i o n o f S h i p W i t h a Smooth R o c k . 144 5 . 2 . 2 . 2 - C o l l i s i o n o f S h i p W i t h a Sharp Edge R o c k . 147 5 . 2 . 3 - C o l l i s i o n o f t h e S h i p D u r i n g M a n e u v e r i n g . 149 5 . 2 . 3 . 1 - C o l l i s i o n o f S h i p w i t h a Smooth R o c k . 150 5 . 2 . 3 . 2 - C o l l i s i o n o f S h i p W i t h a Sharp Edge R o c k . 151 5 . 2 . 4 - R e l a t i o n B e t w e e n Volume o f Damaged S t r u c t u r a l E l e m e n t s and D i s s i p a t e d K i n e t i c E n e r g y D u r i n g C o l l i s i o n . 152 5 . 2 . 5 - C l o s u r e • 152 6 - SUMMARY O F R E S U L T S AND C O N C L U S I O N . 169 6 . 1 - I c e B r e a k i n g S h i p . 169 6 . 1 . 1 - C o n t i n u o u s I c e B r e a k i n g 169 6 . 1 . 2 - Ramming Mode. 171 6 . 2 - S h i p C o l l i s i o n . 173 6 . 3 - Recommendat ions F o r F u t u r e W o r k s . 175 R e f e r e n c e s 176 APPENDICES A - N o d a l V a l u e s o f t h e G e n e r a t e d L o a d s . 180 v i i A . l - N o d a l v a l u e s o f t h e C o n c e n t r a t e d L o a d . 181 A . 2 - N o d a l V a l u e o f t h e D i s t r i b u t e d L o a d . 182 A . 3 - N o d a l v a l u e s o f t h e C o n c e n t r a t e d L o a d on a Two D i m e n s i o n a l Beam E l e m e n t 184 B- U n c o u p l i n g t h e E q u a t i o n s o f M o t i o n . 185 C- R e l a t i o n B e t w e e n The H o r i z o n t a l and v e r t i c a l F o r c e on t h e Bow. 187 D- P e n e t r a t i o n I n t o t h e H u l l . : 189 v i i i LIST OF TABLES T a b l e Page 5 . 1 - S p e c i f i c a t i o n o f t h e ' S t a n d a r d I c e - B r e a k e r ' a t t h e s t a t i o n s . 105 5 . 2 - F r e q u e n c i e s o f t h e m o t i o n o f t h e i c e b r e a k e r . 106 5 . 3 - S p e c i f i c a t i o n s o f t h e s h i p i m m e d i a t e l y b e f o r e and a f t e r c o l l i s i o n t o t h e i c e . I l l 5 . 4 - S p e c i f i c a t i o n o f t h e ' S t a n d a r d T a n k e r S h i p ' a t t h e s t a t i o n s . 143 i x LIST OF FIGURES F i g u r e Page 2 . 1 - The beam e l e m e n t w i t h t e n d e g r e e s o f f r e e d o m . 25 2 . 2 - T y p i c a l c r o s s - s e c t i o n a r e a o f a beam e l e m e n t . 35 3 . 1 - C o o r d i n a t e a x i s and t h e i c e b r e a k i n g s h i p . , 42 3 . 2 - D i s t r i b u t i o n o f t h e i c e l o a d on t h e s t e m . 47 3 . 3 - A two d i m e n s i o n a l beam model 53 3 . 4 - I c e b r e a k i n g s h i p and t h e i c e r i d g e w i t h c o r r e s p o n d i n g beam m o d e l 61 3 . 5 - The f l e x i b l e beam e l e m e n t s l i d i n g o v e r an i n c l i n e d s u r f a c e . 66 3 . 6 - T y p i c a l c r o s s - s e c t i o n a r e a a t t h e bow. 68 3 . 7 - P r i n t o f t h e bow on t h e i c e s h e e t . 69 A 3 . 8 - A p l a t e w i t h u n i t n o r m a l v e c t o r n and v e l o c i t y u n i t v e c t o r is . 72 4 . 1 - Sys tems o f body and g l o b a l c o o r d i n a t e s . 80 4 . 2 - C r o s s - s e c t i o n a r e a o f t h e beam e l e m e n t and p o s i t i o n o f t h e e x t e r n a l o b j e c t . 92 4 . 3 - Damaged on t h e c r o s s - s e c t i o n a r e a o f t h e beam e l e m e n t . 94 4 . 4 - The n o d a l d i s p l a c e m e n t a l o n g t h e beam m o d e l . 99 5 . 1 - The s t a n d a r d s h i p and t h e s e l e c t e d s t a t i o n s . 103 5 . 2 - a ) Mass d i s t r i b u t i o n , b ) B u o y a n c y d i s t r i b u t i o n a l o n g t h e i c e - b r e a k i n g s h i p . 104 5 . 3 - S c h e m a t i c v i e w o f t h e t h r e e d i m e n s i o n a l beam mode l o f t h e s h i p 105 5 . 4 - a) V e r t i c a l component o f t h e i c e f o r c e ; b) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 0 . 5 m / s . 117 5 . 4 - V a r i a t i o n o f , c)maximum b e n d i n g moment d) R o l l i n g a n g l e when s h i p v e l o c i t y i s 0 . 5 m / s . 118 x 5 . 5 - a) V e r t i c a l component o f t h e i c e f o r c e and b ) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 0 . 7 m/s 119 5 . 5 - V a r i a t i o n o f , c)maximum b e n d i n g moment d) R o l l i n g a n g l e when s h i p v e l o c i t y i s 0 . 7 m / s . 120 5 . 6 - a ) V e r t i c a l component o f t h e i c e f o r c e ; b ) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 1 .4 m/s 121 5 . 6 - V a r i a t i o n o f , c ) maximum b e n d i n g moment; d) R o l l i n g a n g l e , when s h i p v e l o c i t y i s 1 . 4 m / s . 122 5 . 7 - a) V e r t i c a l component o f t h e i c e f o r c e ; b ) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 1 . 6 m / s . 123 5 . 7 -. V a r i a t i o n o f , c ) maximum b e n d i n g moment; d) R o l l i n g a n g l e when s h i p v e l o c i t y i s 1 . 6 m / s . 124 5 . 8 - a) V e r t i c a l component o f t h e i c e f o r c e ; b ) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 1 . 8 m / s . 125 5 . 8 - V a r i a t i o n o f , c ) maximum b e n d i n g moment; d) R o l l i n g a n g l e when s h i p v e l o c i t y i s 1 .8 m / s . 126 5 . 9 - a) V e r t i c a l component o f t h e i c e f o r c e ; b ) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 3 .0 m / s . 127 5 . 9 - V a r i a t i o n o f , c ) maximum b e n d i n g moment; d) R o l l i n g a n g l e when s h i p v e l o c i t y i s 3 .0 m/s 128 5 . 1 0 - a ) V e r t i c a l component o f t h e i c e f o r c e , b ) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 5 . 0 m / s . 129 5 . 1 0 - V a r i a t i o n o f , c ) maximum b e n d i n g moment, d) R o l l i n g a n g l e when s h i p v e l o c i t y i s 5 . 0 m / s . 130 5 . 1 1 - a ) V e r t i c a l component o f t h e i c e f o r c e , b ) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 6 . 5 m / s . 131 x i 5 . 1 1 - V a r i a t i o n o f , c ) maximum b e n d i n g moment, d) R o l l i n g a n g l e when s h i p v e l o c i t y i s 6 . 5 m/s 132 5 . 1 2 - a) V e r t i c a l component o f t h e i c e f o r c e , b ) G e n e r a t e d t o r q u e on t h e s h i p when s h i p v e l o c i t y i s 7 . 0 m / s . 133 5 . 1 2 - V a r i a t i o n o f , c ) maximum b e n d i n g moment, d) R o l l i n g a n g l e when s h i p v e l o c i t y i s 7 . 0 m/s 134 5 . 1 3 - a ) P e a k o f maximum b e n d i n g moment, b) Peak o f r o l l i n g a n g l e v e r s e s s h i p v e l o c i t y f o r d i f f e r e n t i c e t h i c k n e s s 135 5 . 1 4 - R e l a t i o n b e t w e e n f r e q u e n c y o f t h e i c e l o a d and the s h i p v e l o c i t y . 136 5 . 1 5 - I c e - b r e a k i n g power i n d i f f e r e n t i c e f i e l d . 136 5 . 1 6 - Mode s h a p e s o f t h e s h i p m o t i o n d u r i n g t h e ramming p e r i o d . 137 5 . 1 7 - Maximum o f b e n d i n g moment a l o n g t h e s h i p and c o n t a c t f o r c e on t h e bow f o r ramming o f t h e i c e r i d g e w i t h a f i x e d c o n t a c t p o i n t on t h e bow. 138 5 . 1 8 - Maximum b e n d i n g moment f o r d i f f e r e n t ramming v e l o c i t i e s 139 5 . 1 9 - Maximum o f c o n t a c t f o r c e s f o r d i f f e r e n t ramming v e l o c i t i e s 139 5 . 2 0 - Maximum o f , a ) b e n d i n g moment, b) c o n t a c t f o r c e f o r t h r e e d i f f e r e n t s t i f f n e s s o f t h e h u l l w i t h ramming v e l o c i t y o f 5 m/s 140 5 . 2 1 - Maximum o f , a ) b e n d i n g moment, b) c o n t a c t f o r c e f o r ramming o f t h e i c e r i d g e w i t h a m o v i n g c o n t a c t p o i n t on the bow. 141 5 . 2 2 - C o l l e c t e d d a t a f r o m Kigoriak w i t h ramming v e l o c i t y o f 4 . 9 m / s . 142 5 . 2 3 - a) M a s s , b ) Buoyancy d i s t r i b u t i o n a l o n g t h e ' S t a n d a r d S h i p ' . 154 5 . 2 4 - E x t e n t o f damage due t o f o r w a r d c o l l i s i o n w i t h o u t r u p t u r e o f t h e s t a n d a r d h u l l , f o r s h i p v e l o c i t i e s o f 1 , 3 , 5 and 7 m/s 155 x i i 5 . 2 5 - G e n e r a t e d f o r c e d u r i n g t h e f o r w a r d c o l l i s i o n w i t h o u t r u p t u r e o f t h e s t a n d a r d h u l l f o r s h i p v e l o c i t i e s o f 1 , 3 , 5 , and 7 m / s . 156 5 . 2 6 - E x t e n t o f damage due t o f o r w a r d c o l l i s i o n w i t h o u t r u p t u r e o f t h e r e i n f o r c e d h u l l f o r s h i p v e l o c i t i e s o f 1 , 3 , 5 , and 7 m / s . 157 5 . 2 7 - G e n e r a t e d f o r c e d u r i n g t h e f o r w a r d c o l l i s i o n w i t h o u t r u p t u r e o f t h e r e i n f o r c e d h u l l f o r s h i p v e l o c i t i e s o f 1 , 3 , 5 , and 7 m / s . 158 5 . 2 8 - E x t e n t o f damage due t o f o r w a r d c o l l i s i o n w i t h r u p t u r e o f t h e s t a n d a r d h u l l . f o r s h i p v e l o c i t i e s o f 1 , 3 , 5 , and 7 m / s . 159 5 . 2 9 - G e n e r a t e d f o r c e d u r i n g t h e f o r w a r d c o l l i s i o n w i t h r u p t u r e o f t h e s t a n d a r d h u l l f o r s h i p v e l o c i t i e s o f 1 , 3 , 5 , and 7 m / s . 160 5 . 3 0 - E x t e n t o f damage due t o f o r w a r d c o l l i s i o n w i t h r u p t u r e o f t h e r e i n f o r c e d h u l l f o r s h i p v e l o c i t i e s o f 1 , 3 , 5 , and 7 m / s . 161 5 . 3 1 - G e n e r a t e d f o r c e d u r i n g t h e f o r w a r d c o l l i s i o n w i t h r u p t u r e o f t h e r e i n f o r c e d h u l l f o r s h i p v e l o c i t i e s o f 1 , 3 , 5 , and 7 m / s . 162 5 . 3 2 - V a r i a t i o n o f r o l l i n g a n g l e when s h i p c o l l i d e s w i t h t h e r o c k a t s p e e d o f 7 m / s . 163 5 . 3 3 - G e n e r a t e d f o r c e and e x t e n t o f damage, d u r i n g t h e s i d e w i s e c o l l i s i o n t h r o u g h t h e s h i p maneuver w i t h o u t r u p t u r e o f t h e s t a n d a r d h u l l 164 5 . 3 4 - G e n e r a t e d f o r c e and e x t e n t o f damage, d u r i n g t h e s i d e w i s e c o l l i s i o n t h r o u g h t h e s h i p maneuver w i t h o u t r u p t u r e o f t h e r e i n f o r c e d h u l l . 165 5 . 3 5 - G e n e r a t e d f o r c e and e x t e n t o f damage, d u r i n g t h e s i d e w i s e c o l l i s i o n t h r o u g h t h e s h i p maneuver w i t h r u p t u r e o f t h e s t a n d a r d h u l l 166 5 . 3 6 - G e n e r a t e d f o r c e and e x t e n t o f damage, d u r i n g t h e s i d e w i s e c o l l i s i o n t h r o u g h t h e s h i p maneuver w i t h r u p t u r e o f t h e r e i n f o r c e d h u l l 167 5 . 3 7 - V a r i a t i o n o f r o l l i n g a n g l e o f t h e s h i p due t o s i d e w i s e c o l l i s i o n . 168 5 . 3 8 - R e l a t i o n b e t w e e n vo lume o f t h e damaged s t r u c t u r a l e l e m e n t s and l o s s o f k i n e t i c e n e r g y o f t h e s h i p . 168 x i i i A . l - The c o n c e n t r a t e d l a t e r a l f o r c e on t h e beam e l e m e n t . 181 A . 2 - D i s t r i b u t e d f o r c e on a beam e l e m e n t . 182 C. l - The c o n t a c t f o r c e and i t s Components . 187 D. l - C r o s s - s e c t i o n o f damaged a r e a when s h i p i s m o v i n g t o t h e r i g h t . 192 D.2 - C r o s s - s e c t i o n o f damaged a r e a when s h i p i s m o v i n g t o t h e l e f t . 193 x i v NOMENCLATURE c r o s s - s e c t i o n o f t h e e l e m e n t . a m p l i t u d e o f t h e i m p a c t l o a d on t h e i c e - b r e a k e r , p a r t s o f t h e s h i p c r o s s - s e c t i o n . g e o m e t r i c c o n t a c t a r e a b e t w e e n bow and t h e i c e s h e e t c r o s s - s e c t i o n o f t h e damaged m a t e r i a l . u n d e r w a t e r c r o s s - s e c t i o n a r e a o f t h e beam e l e m e n t . c r o s s - s e c t i o n o f t h e damaged p a r t i n x , y , and z - d i r e c t i o n . s o l i d c r o s s - s e c t i o n o f damaged m a t e r i a l , p a r a m e t e r s . c o n s t a n t p a r a m e t e r i n t h e e q u a t i o n o f c u t t i n g f o r c e , added mass i n t h e l a t e r a l m o t i o n , added mass o f t h e s h i p i n s u r g e , b r e a t h o f t h e c l a m p e d beam, p a r a m e t e r s . b u o y a n c y f o r c e a l o n g t h e beam e l e m e n t , c o n s t a n t p a r t o f t h e b u o y a n c y f o r c e , a body f o r c e d i s t r i b u t i o n . a b o d y f o r c e d i s t r i b u t i o n e x c l u d i n g t h e i n e r t i a , d i s t a n c e o f m e t a c e n t r i c p o i n t and c e n t e r o f b u o y a n c y a v e r a g e v a l u e o f t h e b r e a t h o f t h e s t a t i o n , t h e v a r i a b l e p a r t o f t h e b u o y a n c y f o r c e . t h e b r e a d t h o f t h e beam e l e m e n t , p r o p o r t i o n a l r a t i o i n { f } = c [N ] (QJ } x v s u r f a c e r e d u c t i o n c o e f f i c i e n t , shape c o e f f i c i e n t . damaged e x t e n t i n a s p e c i f i e d d i r e c t i o n . l a t e r a l d e f l e c t i o n o f t h e beam. e l e m e n t o f v e c t o r {d}. modu le o f e l a s t i c i t y . i n i t i a l k i n e t i c e n e r g y o f t h e s h i p . k i n e t i c e n e r g y o f t h e s h i p due t o t h e s u r g e immediate a f t e r c o l l i s i o n . k i n e t i c e n e r g y a f t e r c o l l i s i o n . e n e r g y o f d e f o r m a t i o n . k i n e t i c e n e r g y o f t h e s h i p due t o i t s l a t e r a l m o t i o n a b s o r b e d k i n e t i c e n e r g y d u r i n g a damage g e n e r a t i o n , c r u s h i n g f o r c e n o r m a l t o t h e bow. t o t a l c o n t a c t f o r c e , b e n d i n g and c u r l i n g f o r c e , c u t t i n g f o r c e o f a p l a t e . c o n s t a n t p a r a m e t e r i n t h e e q u a t i o n o f c u t t i n g f o r c e , f o r c e w h i c h g e n e r a t e s damage i n d i r e c t i o n D. f r i c t i o n f o r c e v e c t o r , l a t e r a l f o r c e on t h e s h i p . u l t i m a t e l o a d w h i c h an i c e s h e e t c a n c a r r y , n o r m a l component o f c r u s h i n g f o r c e v e c t o r , components o f t h e e x t e r n a l f o r c e , components o f f o r c e i n l o c a l c o o r d i n a t e s . F + F c B x v i module o f r i g i d i t y , g r a v i t y a c c e l e r a t i o n . d i s t a n c e b e t w e e n m e t a c e n t r i c p o i n t and c e n t e r o f g r a v i t y . t r a n s v e r s e m e t a c e n t r i c h e i g h t o f t h e e l e m e n t , components o f d i s t r i b u t e d f o r c e i n y and z - d i r e c t i o n , t h e a v e r a g e v a l u e o f t h e d e p t h o f t h e s t a t i o n , t h i c k n e s s o f t h e c l a m p e d beam, t h i c k n e s s o f i c e s h e e t . t r a n s v e r s e moment o f i n e r t i a o f t h e e l e m e n t . moments o f i n e r t i a o f t h e c r o s s - s e c t i o n o f t h e e l e m e n t w i t h r e s p e c t t o t h e y and z a x e s r e s p e c t i v e l y . h o r i z o n t a l , v e r t i c a l , and n o r m a l i m p u l s e . p o l a r moment o f i n e r t i a o f t h e e l e m e n t a r o u n d x - a x i s . c o n s t a n t s i n t h e e n e r g y - v o l u m e e q u a t i o n . s t i f f n e s s f o r t h e i - t h mode i n g e n e r a l i z e d c o o r d i n a t e . f o r c e c o n s t a n t i n t h r e e d i r e c t i o n s x , y , a n d z . h a l f o f t h e c l a m p e d beam l e n g t h . s h i p l e n g t h . l e n g t h o f e l e m e n t . two g i v e n c o o r d i n a t e s a l o n g t h e e l e m e n t , shape p a r a m e t e r o f t h e s t a t i o n , t o t a l mass o f t h e s h i p . mass i n t h e i - t h mode i n g e n e r a l i z e d c o o r d i n a t e . b e n d i n g moment i n a beam. number o f n o d e s . u n i t v e c t o r n o r m a l t o t h e bow. x v i i n , n , n d i r e c t i o n c o s i n e s o f v e c t o r n. x y z O-XYZ f i x e d c o o r d i n a t e s y s t e m . A A A A . o - x y z b o d y c o o r d i n a t e s y s t e m , p number o f c y c l e s . Pc c o l l a p s e l o a d o f a f u l l y c l a m p e d beam. P i c e b r e a k i n g power . P^ , P , P z p e n e t r a t i o n o f t h e e x t e r n a l o b j e c t i n t h e h u l l i n t h r e e d i r e c t i o n s o f x , y , and z . f o r c e i n g e n e r a l i z e d c o o r d i n a t e . R r e s t o r i n g t o r q u e . S a r e a o f t h e h u l l u n d e r f o r c e . s s u r f a c e doma in o f i n t e g r a t i o n . T p l a t e t h i c k n e s s . t t i m e . A t u n i t v e c t o r t a n g e n t t o t h e c o n t a c t a r e a and i n d i r e c t i o n o f m o t i o n . t h i c k n e s s o f t h e f u l l y c l a m p e d beam. T a s u r f a c e t r a c t i o n o v e r a p a r t o f t h e b o u n d a r y o f t h e b o d y . T p r o p e l l e r s t h r u s t f o r c e . r A t , t , t d i r e c t i o n c o s i n e s o f v e c t o r t . x y z u ( x , t ) o r u l a t e r a l d e f l e c t i o n o f t h e beam. u ( x , t ) v e r t i c a l component o f u ( x , t ) . u , . . . u n o d a l d i s p l a c e m e n t s . 1 1 0 v vo lume domain o f i n t e g r a t i o n . V vo lume o f t h e damaged m a t e r i a l . is s h i p v e l o c i t y . x v i i i bow v e l o c i t y v e c t o r . s i d e w i s e v e l o c i t y o f a f t . s i d e w i s e v e l o c i t y o f s t e m . i n i t i a l s h i p v e l o c i t y . v e l o c i t y o f t h e N - t h n o d e . components o f v e l o c i t y v e c t o r V . d i r e c t i o n c o s i n e s o f v e l o c i t y v e c t o r V . w e i g h t o f t h e e l e m e n t . w e i g h t p e r u n i t l e n g t h o f t h e beam e l e m e n t , c o o r d i n a t e o f a p o i n t . v e l o c i t y and a c c e l e r a t i o n o f t h e s h i p . m e a s u r e d a m p l i t u d e . c o o r d i n a t e s o f t h e e x t e r n a l o b j e c t . d i s t a n c e b e t w e e n t h e i - t h and t h e f i r s t n o d e . v a r i a b l e s . v e l o c i t y and a c c e l e r a t i o n o f t h e N - t h n o d e , c o o r d i n a t e s o f t h e c e n t e r o f t h e s t a t i o n c r o s s - s e c t i o n , c o o r d i n a t e s o f p o i n t s on t h e h u l l , maximum o f Z o c o o r d i n a t e o f t h e p o i n t o f a p p l i c a t i o n o f i c e f o r c e . T [T ] {d } a v e c t o r t h a t a l l o f i t s e l e m e n t s a r e z e r o b u t t h e ( 2 N + l ) - t h w h i c h i s e q u a l t o u n i t y . a v e c t o r t h a t a l l o f i t s e l e m e n t s a r e z e r o b u t t h e f o u r components w h i c h a r e r e l a t e d t o t h e K - t h e l e m e n t . v e c t o r o f d i s t r i b u t e d a p p l i e d f o r c e . x i x {f} d i s s i p a t i o n f o r c e p e r u n i t v o l u m e . {S-} c o n c e n t r a t e d f o r c e v e c t o r . <N> a l i n e a r shape f u n c t i o n . <N> shape f u n c t i o n o f t h e two d i m e n s i o n a l beam e l e m e n t . {Q} v e c t o r o f t h e n o d a l v a l u e s o f t h e a p p l i e d l o a d on t h e w h o l e beam. {q} v e c t o r o f n o d a l v a l u e s o f t h e t o t a l f o r c e s on t h e e l e m e n t . (Q) v e c t o r o f n o d a l v a l u e s o f t h e f o r c e s on two d i m e n s i o n a l beam. {q} n o d a l v a l u e s o f b u o y a n c y f o r c e . B {q} v e c t o r o f n o d a l v a l u e s o f t h e c o n s t a n t p a r t o f t h e B C b u o y a n c y f o r c e . { q ) B v v e c t o r o f n o d a l v a l u e s o f t h e v a r i a b l e p a r t o f t h e b u o y a n c y f o r c e . (q) n o d a l v a l u e s o f t h e e x t e r n a l f o r c e . e x {q } n o d a l v a l u e s o f t h e r e s t o r i n g t o r q u e . R {q} n o d a l w e i g h t o f t h e e l e m e n t . w e i (T) s u r f a c e t r a c t i o n v e c t o r . {U } i n i t i a l v a l u e s o f n o d a l v e l o c i t i e s , o {QJ} , {JJ} , {U} n o d a l d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n , {u } d i s p l a c e m e n t f i e l d m a t r i x . {U} , {U} , { i i } v e c t o r s o f n o d a l d e f l e c t i o n , v e l o c i t y and a c c e l e r a t i o n i n g l o b a l c o o r d i n a t e s . {u} n o d a l d i s p l a c e m e n t o f t h e e l e m e n t w h i c h i s i n c o n t a c t . {U } , {U} , { i i } v e c t o r o f n o d a l d i s p l a c e m e n t . v e l o c i t y , and a c c e l e r a t i o n o f t h e two d i m e n s i o n a l beam. x x { $ } , } , { § } d e f l e c t i o n , v e l o c i t y and a c c e l e r a t i o n v e c t o r i n g e n e r a l i z e d c o o r d i n a t e s . {$} , { * } , { £ } d e f l e c t i o n , v e l o c i t y and a c c e l e r a t i o n v e c t o r i n g e n e r a l i z e d c o o r d i n a t e s f o r two d i m e n s i o n a l beam. az [A] = ——- [N ] [a ] added mass m a t r i x [B] = [ L ] [ N ] [C] damping m a t r i x o f t h e two d i m e n s i o n a l beam, [c ] t h e e l e m e n t damping m a t r i x . [C] t h e g l o b a l damping m a t r i x . [C] ^ d i a g o n a l i z e d damping m a t r i x . [D] c o n s t i t u t i v e l a w m a t r i x . [K] t h e g l o b a l s t i f f n e s s m a t r i x . [k] t h e s t i f f n e s s m a t r i x o f t h e e l e m e n t . [k ] [k ]+[k ] + [k ] [K] s t i f f n e s s m a t r i x o f t h e two d i m e n s i o n a l beam, [k] t h e g e n e r a t e d s t i f f n e s s c a u s e d b y b u o y a n c y . B A [ K ] p d i a g o n a l i z e d s t i f f n e s s m a t r i x o f two d i m e n s i o n a l beam. [ K ] p d i a g o n a l i z e d s t i f f n e s s m a t r i x . [k ] ' t h e s t i f f n e s s m a t r i x g e n e r a t e d b y r o t a t i o n o f t h e R e l e m e n t i n t h e w a t e r . [L] d i f f e r e n t i a l o p e r a t i o n . [M] mass m a t r i x o f t h e two d i m e n s i o n a l beam. [M] t h e g l o b a l mass m a t r i x , [m] = [m ] + [m ] 2 [m] mass m a t r i x o f t h e e l e m e n t r e l a t e d t o i t s i n e r t i a . x x i [m] mass m a t r i x o f t h e e l e m e n t r e l a t e d t o t h e r o t a t o r y i n e r t i a . [M] d i a g o n a l i z e d mass m a t r i x o f a two d i m e n s i o n a l beam. [ M ] p d i a g o n a l i z e d mass m a t r i x . [N] i n t e r p o l a t i o n m a t r i x . [N ] i n t e r p o l a t i o n m a t r i x e v a l u a t e d a t a s p e c i f i e d p o i n t . T [N] t r a n s p o s e o f [ N ] . T [N ] q t r a n s p o s e o f [N ] . [T ] m o d a l m a t r i x f o r two d i m e n s i o n a l beam. [T ] m o d a l m a t r i x . [T ] T t r a n s p o s e o f [T ] . [T 1 e l e m e n t t r a n s f e r m a t r i x . 1 ' s a p a r a m e t e r i n [c ] = a [m] + /3[k] a bow a n g l e . /3 damping r a t i o d e f i n e d i n [a ] d = 0 [D]{e }. (3 c o n s t a n t i n J o n e s ' f o r m u l a . 7 yaw a n g l e . AE l o s s o f k i n e t i c e n e r g y . S ( x - x ) d e l t a f u n c t i o n a t x = x . o o S ( t ) d e l t a f u n c t i o n a t t= 0 . e . s t a i n t h r o u g h t h e b o d y . T) damping c o e f f i c i e n t . r\. , T) . damping c o e f f i c i e n t o f t h e i - t h e and j - t h mode. L i 9 t o t a l a n g l e o f i c e s h e e t . o 0 a n g l e o f i c e wedge. h e l l a n g l e . x x i i K -Z / I A r a t i o b e t w e e n v e r t i c a l and h o r i z o n t a l i m p u l s e . A . =di / m i 1 ' is c o e f f i c i e n t o f f r i c t i o n b e t w e e n i c e and t h e s h i p . £, £, £ n o d a l d e f l e c t i o n , v e l o c i t y and a c c e l e r a t i o n i n g e n e r a l i z e d c o o r d i n a t e s . v =x / £ p d e n s i t y o f t h e b o d y . p s p e c i f i c mass o f t h e s e a w a t e r . w a c r u s h i n g s t r e n g t h o f i c e . a^ c r u s h i n g s t r e n g t h o f i c e when s h i p v e l o c i t y i s 1 m / s . a y i e l d s t r e s s o f t h e s t r u c t u r a l m a t e r i a l . 0 J r p e r i o d o f m o t i o n o f t h e p o i n t o f a p p l i c a t i o n o f c o n t a c t f o r c e . T s t r e s s t h r o u g h t h e b o d y . 1 i 4>^, <f>2 s h e a r d e f o r m a t i o n p a r a m e t e r s . x = y / * V> ,ip ,i> r a t i o b e t w e e n t h e g e o m e t r i c and s o l i d c r o s s - s e c t i o n s , x y z w . , w . n a t u r a l f r e q u e n c i e s . a>d f r e q u e n c y o f damped o s c i l l a t i o n . V vo lume o f t h e e l e m e n t . {5u} i n c r e m e n t o f d e f l e c t i o n . {SQJ} i n c r e m e n t o f n o d a l d i s p l a c e m e n t . [e] t h e s t r a i n f i e l d m a t r i x . [a] a s t r e s s m a t r i x . [ a ] d v i s c o u s damping s t r e s s t e n s o r . x x i i i AKNOWLEDGEMENT The a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e g r a t i t u d e and a p p r e c i a t i o n t o h i s s u p e r v i s o r P r o f e s s o r H. Vaughan f o r h i s c o n t i n u e d g u i d a n c e , e n c o u r a g e m e n t , a s s i s t a n c e , and s u p p o r t w i t h o u t w h i c h t h i s work w o u l d h a v e b e e n i m p o s s i b l e . P a r t i c u l a r l y l i k e t o a c k n o w l e d g e my w i f e Mahnaz f o r h e r u n f a i l i n g e n t h u s i a s m and p a t i e n c e d u r i n g t h e l a s t f i v e y e a r s and my c h i l d r e n M o s t a f a , M o r t e z a and A t e f e h f o r many weekends and e v e n i n g s t h e y s p e n t w i t h o u t me. x x i v To My W i f e , My M o t h e r and Memory o f My F a t h e r x x v 1 1- INTRODUCTION Impulsive loads a c t i n g on the h u l l of a ship can only confound the ope r a t i o n and performance of that ship. Even i f the ship i s designed to withstand impacts the o v e r a l l e f f e c t s are never advantageous, o f t e n d e l e t e r i o u s and sometimes c a t a s t r o p h i c . The spectrum of s t r u c t u r a l damage incl u d e s abrasion, p l a t e dimpling, f r a c t u r e , t e a r i n g , l o c a l s t r u c t u r a l c o l l a p s e and o v e r a l l p l a s t i c h i n g i n g . In a d d i t i o n the ship may founder or los e s t a b i l i t y . The source of impact forces i s e q u a l l y broad. The ship may c o l l i d e w i t h a f i x e d i n s t a l l a t i o n or w i t h another ship. I t may run aground. I t may be subjected to very heavy wave loads. In the case of A r c t i c Ships i t may have been d e l i b e r a t e l y headed i n t o heavy i c e . In t h i s study we consider two scenarios; i ) the ship has c o l l i d e d w i t h an object r e s u l t i n g i n some s t r u c t u r a l c o l l a p s e and rupture. i i ) the s h i p , i n t h i s case an ice- b r e a k e r , i s operating i n heavy i c e i n c l u d i n g ramming. In the f i r s t case we are concerned w i t h being able to estimate the extent of the s t r u c t u r a l damage. In the second case we need to know the magnitude of the peak bending s t r e s s e s and impact fo r c e s on the h u l l . 1 . 1 - Impact Loads During Ship C o l l i s i o n Assessment of the damage which may occur during c o l l i s i o n of a ship w i t h another object i s becoming an important s a f e t y c o n s i d e r a t i o n i n the design of ship s t r u c t u r e s , p a r t i c u l a r l y f o r ships which may have to 2 t r a n s p o r t h a z a r d o u s o r d a n g e r o u s c a r g o e s . I n s p i t e o f modern n a v i g a t i o n s y s t e m s and p i l o t a w a r e n e s s , t h e r e a r e o v e r 1000 s i g n i f i c a n t a c c i d e n t s e a c h y e a r i n t h e U n i t e d S t a t e ' s w a t e r a l o n e [ 1 ] . The s t a t i s t i c s c o l l e c t e d b y M i n o r s k y e t a l [2] show t h a t , f o r t h e p e r i o d 1970 t o 1975 , f o r e a c h 1000 s h i p s i n o p e r a t i o n , an a v e r a g e o f 4 t o 5 s h i p s i n t h e w e i g h t r a n g e f r o m 6000 t o 60000 t o n s were i n v o l v e d i n g r o u n d i n g i n c i d e n t s . One t h i r d o f t h e s e g r o u n d i n g s were w r i t t e n o f f as t o t a l l o s s e s . K i n k e a d [3] showed t h a t , a c c o r d i n g t o t h e s t a t i s t i c s f r o m IMCO, i n t h e p e r i o d 1968 t o 1980 t h e r e were 239 g r o u n d i n g i n c i d e n t s i n v o l v i n g t a n k e r s . S t a t i s t i c s f r o m t h e N o r w e g i a n c l a s s i f i c a t i o n s o c i e t y , N o r s k e V e r i f a s , showed t h a t g r o u n d i n g i n c i d e n t s a c c o u n t e d f o r 46 p e r c e n t o f a l l s h i p a c c i d e n t s and c a s u a l t i e s i n t h e p e r i o d 1970 t o 1978 . O b v i o u s l y r e p a i r c o s t s and c a r g o s p i l l a g e l o s s e s a r e c o n s i d e r a b l e . I n a d d i t i o n , a r u p t u r e i n t h e h u l l o f an o i l t a n k e r o r L i q u i d N a t u r a l Gas (LNG) c a r r i e r c a n c a u s e m a j o r e n v i r o n m e n t a l p o l l u t i o n . O t h e r v e r y h a z a r d o u s s i t u a t i o n s a r i s e when an a c c i d e n t o c c u r s t o a s h i p w i t h a n u c l e a r power p l a n t , o r when a s h i p i s c a r r y i n g s p e n t n u c l e a r f u e l s o r d a n g e r o u s c h e m i c a l s . The e f f e c t s o f h u l l r u p t u r e c o u l d t h e n be c a t a s t r o p h i c . The number o f LNG c a r r i e r s w h i c h u s e c o a s t a l r o u t e s , o r p l a c e s t h a t a r e i c e - b o u n d i s i n c r e a s i n g , h e n c e , t h e p r o b a b i l i t y o f g r o u n d i n g i s a l s o i n c r e a s i n g . I t i s d e s i r a b l e t o be a b l e t o p r e d i c t a s a f e o p e r a t i n g s p e e d f o r s u c h s h i p s , c o n s i d e r e d as a s p e e d t h a t i n t h e c a s e o f g r o u n d i n g o r s l i t t i n g on a s h a r p i c e r e e f , l i m i t s t h e e x t e n t o f damage t o t h e s h i p h u l l and o u t e r s h e l l w h i l e l e a v i n g t h e i n n e r p r o t e c t e d i t e m s undamaged. I n v e s t i g a t o r s who h a v e s t u d i e d t h e s h i p c o l l i s i o n p r o b l e m n o r m a l l y t a l k a b o u t M i n o r and M a j o r C o l l i s i o n s . T h e r e i s no e x a c t d e f i n i t i o n on 3 t h e c l a s s i f i c a t i o n o f c o l l i s i o n s b u t b r o a d l y t h e y a r e c l a s s i f i e d a s : i ) M i n o r C o l l i s i o n s - c o l l i s i o n s i n w h i c h t h e s h i p h u l l i s d e n t e d w i t h o u t r u p t u r e , and p e n e t r a t i o n o f t h e h u l l i s r e s i s t e d by l a r g e i n p l a n e f o r c e s . These f o r c e s a r e g e n e r a t e d b y p l a s t i c d e f o r m a t i o n o f t h e s t r u c t u r a l e l e m e n t s s u c h as h u l l p l a t i n g o r s t i f f e n e r s . i i ) M a j o r C o l l i s i o n s - c o l l i s i o n s i n w h i c h s i g n i f i c a n t d i s t o r t i o n and r u p t u r e o f s t r u c t u r a l e l e m e n t s o c c u r . T h e r e a r e o t h e r t y p e s o f c o l l i s i o n s w h i c h a r e n o t i n c l u d e d i n t h i s c a t e g o r y , f o r i n s t a n c e , r u p t u r i n g o f t h e h u l l w i t h a s h a r p o b j e c t w h i l e a s i g n i f i c a n t amount o f s t r u c t u r a l m a t e r i a l i s n o t d e f o r m e d . A s h i p c o l l i s i o n u s u a l l y i n v o l v e s d e f l e c t i o n , b u c k l i n g and f r a c t u r e o f many s t r u c t u r a l members w h i c h a r e g e n e r a l l y n o n l i n e a r p r o c e s s e s . C o n s e q u e n t l y s h i p c o l l i s i o n p r o b l e m s a r e v e r y d i f f i c u l t t o a n a l y z e t h e o r e t i c a l l y o r e v e n e x p e r i m e n t a l l y . The c o m p l e x i t y o f t h e p r o b l e m i s f u r t h e r compounded b y many p o s s i b l e c o l l i s i o n s c e n a r i o s . Some p a r a m e t e r s i n t h e s h i p c o l l i s i o n p r o b l e m a r e : t y p e o f s h i p o r s h i p s w h i c h a r e i n v o l v e d , s p e e d o f t h e v e s s e l o r v e s s e l s b e f o r e c o l l i s i o n , d i r e c t i o n o f i m p a c t , and t h e p o s i t i o n o f i m p a c t on t h e s h i p h u l l . I n t h e f o l l o w i n g s e c t i o n some o f t h e p r e v i o u s i n v e s t i g a t i o n s on s h i p c o l l i s i o n s a r e b r i e f l y r e v i e w e d . 4 1 . 1 . 1 - L i t e r a t u r e R e v i e w M i n o r s k y [4] was t h e f i r s t p e r s o n t o a n a l y z e t h e s h i p s t r u c t u r a l damage c a u s e d when two s h i p s c o l l i d e . He p u b l i s h e d h i s s e m i - e m p i r i c a l a r t i c l e i n 1959 . He p r o v i d e d t h e f i r s t and s t a n d a r d method o f e s t i m a t i n g g r o s s damage s u f f e r e d b y two c o l l i d i n g s h i p s i n o b l i q u e i m p a c t . He u s e d t h e p r i n c i p a l o f c o n s e r v a t i o n o f momentum f o r r i g i d b o d i e s t o f i n d t h e k i n e t i c e n e r g y l o s s d u r i n g a c o l l i s i o n b e t w e e n two v e s s e l s . M i n o r s k y i n v e s t i g a t e d d a t a f r o m t h e r e c o r d s o f 26 a c t u a l s h i p c o l l i s i o n s and c o n c l u d e d t h a t t h e r e i s a l i n e a r r e l a t i o n b e t w e e n t h e t o t a l l o s s o f k i n e t i c e n e r g y and t h e vo lume o f t h e damaged s t r u c t u r a l e l e m e n t s . M i n o r s k y ' s o r i g i n a l e q u a t i o n was s u c c e s s f u l i n p r e d i c t i n g t h e damage i n c u r r e d b y s h i p s i n m a j o r c o l l i s i o n b u t was n o t a p p r o p r i a t e f o r i n v e s t i g a t i n g m i n o r c o l l i s i o n i n c i d e n t s . W o i s i n [5] i n 1971 and A k i t a and K i t a m u r a [6] i n 1972 v e r i f i e d M i n o r s k y ' s f o r m u l a u s i n g s m a l l and m o d e r a t e s c a l e m o d e l s i n t h e i r t e s t s . S i n c e M i n o r s k y ' s o r i g i n a l w o r k , many a r t i c l e s h a v e b e e n p u b l i s h e d on t h e c o l l i s i o n r e s i s t a n c e o f s h i p s . I n two l i t e r a t u r e r e v i e w s J o n e s [ 7 , 8 ] , h a s r e f e r e n c e d o v e r 190 a r t i c l e s w h i c h d e a l e i t h e r t h e o r e t i c a l l y o r e x p e r i m e n t a l l y w i t h t h e s h i p c o l l i s i o n p r o b l e m . I n s p i t e o f t h e e x t e n s i v e l i t e r a t u r e on s h i p c o l l i s i o n s , v e r y few a r t i c l e s have b e e n p u b l i s h e d on t h e g r o u n d i n g p r o b l e m [ 7 , 8 , 9 ] . A s h o r t r e v i e w o f work done on g r o u n d i n g i s now p r e s e n t e d . I n 1928 C o k e r [10] i n v e s t i g a t e d t h e s t r u c t u r a l f a i l u r e o f t h e s t r a n d e d c a r g o v e s s e l Lochmonar and n o t i c e d t h a t f a i l u r e h a d i n i t i a t e d a t a d i s c o n t i n u i t y i n t h e h u l l p l a t i n g . C o k e r , t h r o u g h an e x p e r i m e n t a l p h o t o e l a s t i c e x p e r i m e n t , showed t h a t s t r e s s c o n c e n t r a t i o n e f f e c t s a r e 5 v e r y i m p o r t a n t i n s h e l l s t r e n g t h and recommended t h a t a r e a s o f s t r e s s c o n c e n t r a t i o n s h o u l d be a v o i d e d by a p p r o p r i a t e new d e s i g n p r o c e d u r e s . I n 1945 , Thomson [11] s t u d i e d f a i l u r e o f d e c k p l a t i n g o f s t a n d a r d v e s s e l s and c a l c u l a t e d t h e s t r e s s i n t h e d e c k p l a t i n g f o r s e v e n c a s e s o f g r o u n d i n g s . I n 1 9 7 8 , K i t a m u r a e t a l , [12] i n v e s t i g a t e d t h e damage t o t h e bow o f a s h i p w h i c h s t r u c k a r o c k t h a t p e n e t r a t e d t h e b o t t o m s t r u c t u r e . From a s e r i e s o f e x p e r i m e n t s w h i c h t h e y p e r f o r m e d t o s i m u l a t e t h e g r o u n d i n g p r o c e s s , K i t a m u r a e t a l o b s e r v e d t h a t , t h e c r u s h e d members s u c h as o u t e r and i n n e r b o t t o m p l a t i n g , l o n g i t u d i n a l s and g i r d e r s , c o l l a p s e d a p p r o x i m a t e l y a t 80 p e r c e n t o f t h e a s s o c i a t e d y i e l d l o a d . B a s e d on t h e i r own e x p e r i m e n t s t h e y e s t i m a t e d t h a t a r o c k w o u l d p e n e t r a t e 0 . 2 t o 0 . 5 t h e l e n g t h o f a f u l l y l a d e n 1 0 0 , 0 0 0 t o n n e s d o u b l e b o t t o m o i l c a r r i e r when t r a v e l l i n g a t o p e r a t i n g s p e e d . Ueda e t a l [13] t e s t e d a 1 / 7 s c a l e mode l o f a d o u b l e b o t t o m s h i p s t r a n d i n g on a c y l i n d r i c a l shape r o c k and compared t h e r e s u l t s w i t h a t h e o r e t i c a l a n a l y s i s w h i c h t h e y d e v e l o p e d . They f o u n d good agreement b e t w e e n t h e i r t h e o r e t i c a l a n a l y s i s and t h e c o r r e s p o n d i n g e x p e r i m e n t a l r e s u l t s . I n 1977 , Vaughan [14] p o i n t e d o u t t h a t M i n o r s k y ' s [4] a p p r o x i m a t e p r o c e d u r e was d e v e l o p e d f o r m a j o r s h i p t o s h i p c o l l i s i o n s i n v o l v i n g l a r g e v o l u m e s o f d i s t o r t i o n o f t h e s h i p s t r u c t u r e w h e r e a s i n g r o u n d i n g i n c i d e n t s t h e r e may be r e l a t i v e l y l i t t l e vo lume d i s t o r t i o n b u t s i g n i f i c a n t p l a t e t e a r i n g . S i n c e M i n o r s k y ' s method was t h e r e f o r e n o t s u i t a b l e f o r i n v e s t i g a t i n g g r o u n d i n g damage, Vaughan d e v e l o p e d a s i m p l e s e m i - e m p i r i c a l p r o c e d u r e t o e s t i m a t e t h e damage when s h i p s r u n a g r o u n d . I n 1978 , Vaughan [15] u s e d d i m e n s i o n a l a n a l y s i s t o i n v e s t i g a t e t h e 6 t e a r i n g s t r e n g t h o f p l a t e s . He showed t h a t t h e work done f o r t e a r i n g and b e n d i n g a p l a t e c o n s i s t s o f two p a r t s ; a p a r t r e l a t i n g t o t h e vo lume o f d i s t o r t e d p l a t e , as i n M i n o r s k y ' s work [ 4 ] , and a p a r t p r o p o r t i o n a l t o t h e t o t a l f r a c t u r e d a r e a o f t h e t o r n p l a t e . I n 1979 , Vaughan [9] s t u d i e d t h e p l a t e t e a r i n g p r o b l e m more c l o s e l y . He p e r f o r m e d a s e r i e s o f d rop t e s t s on m i l d s t e e l p l a t e s o f v a r i o u s t h i c k n e s s u s i n g s h a r p r i g i d wedges . The work done by t h e d r o p p e d wedge was e q u a t e d t o t h e w o r k t o t e a r and b e n d t h e p l a t e . T e a r i n g work was i s o l a t e d and h e n c e e v a l u a t e d . These t e s t s l e d t o s c a l i n g l a w s w h i c h were a p p l i e d t o t h e g r o u n d i n g p r o b l e m f o r s h i p s . I n 1 9 8 1 , t h e damage e x t e n t o f a g r o u n d e d LNG c a r r i e r was e x a m i n e d b y P o u d r e t , e t a l [ 1 6 ] . T h e i r a r t i c l e d e s c r i b e d t h e damaged s u f f e r e d by 3 a f u l l y l o a d e d 1 3 0 , 0 0 0 m LNG c a r r i e r a f t e r g r o u n d i n g a t s p e e d on a r o c k . I n t h a t a r t i c l e V a u g h a n ' s e q u a t i o n and M i n o r s k y ' s p r e d i c t i o n method were a p p l i e d t o t h e c a s e . I t was shown t h a t t h e a d a p t a t i o n g i v e n b y Vaughan was a d e q u a t e t o a n a l y z e s u c h an a c c i d e n t and i t was c o n c l u d e d g e n e r a l l y t h a t , i n i n c i d e n t s i n v o l v i n g s i g n i f i c a n t p l a t e t e a r i n g , M i n o r s k y ' s method u n d e r e s t i m a t e s t h e e n e r g y a b s o r b e d b y t h e c r u s h e d s t r u c t u r e . I n 1 9 7 9 , t h e f i n i t e e l e m e n t p r o g r a m ADINA was u s e d by O r s e r o , e t a l [17] t o s t u d y t h e c o l l i s i o n o f a b a r g e w i t h t h e r i g i d p i e r o f a b r i d g e . F o r t h e p a r t i c u l a r p r o b l e m w h i c h t h e y c o n s i d e r e d 25 p e r c e n t o f t h e i n i t i a l k i n e t i c e n e r g y o f t h e b a r g e was t r a n s f e r r e d t o t h e b a r g e a n g u l a r r o t a t i o n . P e t t e r s o n and J o h n s o n [ 1 8 , 1 9 ] have s t u d i e d damage o f v a r i o u s m o b i l e o f f s h o r e u n i t s s u b j e c t e d t o c o l l i s i o n l o a d s , b y means o f a s i m p l i f i e d n u m e r i c a l a p p r o a c h . They c o n s i d e r e d t h e l o c a l damage ( d e n t ) o f t u b e s , 7 b u c k l i n g o f b r a c i n g s and t h e d e v e l o p m e n t o f p l a s t i c mechan isms i n o r d e r t o e s t i m a t e t h e g l o b a l damage. I n 1 9 8 3 , J o n e s [8] gave an a n a l y t i c a l v e r i f i c a t i o n f o r M i n o r s k y ' s e m p i r i c a l m e t h o d . He showed t h a t f o r m i n o r c o l l i s i o n s i t i s n e c e s s a r y t o s p e c i f y t h e r a t i o o f t h e e x t e n t o f damage on t h e e l e m e n t , t o t h e l e n g t h o f t h e e l e m e n t . He a l s o showed t h a t m i n o r c o l l i s i o n i n c i d e n t s , w h i c h M i n o r s k y ' s f o r m u l a d i d n o t c o v e r , c o u l d be d e a l t w i t h b y p r o p e r s e l e c t i o n o f t h i s r a t i o . I n 1 9 8 3 , an e x p e r i m e n t a l s t u d y was done b y H a g i w a r a , e t a l [20] on t h e s t r e n g t h o f s h i p s t r u c t u r e s . They p r o p o s e d t h a t h u l l s t r e n g t h c o u l d be e v a l u a t e d b y summing up t h e p r o d u c t s o f e f f e c t i v e b r e a d t h , t h e p l a t e t h i c k n e s s and m a t e r i a l y i e l d i n g s t r e s s f o r a l l members i n t h e c o l l a p s i n g f r a m e s p a c e . The r e s u l t s were i n good ag reement w i t h t h e t h e o r e t i c a l w o r k s done b y J o n e s [ 2 1 , 8 , 7 ] where he r e l a t e d t h e s l o p e o f v o l u m e - e n e r g y l i n e o f M i n o r s k y ' s method w i t h t h e y i e l d s t r e n g t h o f damaged s t r u c t u r a l members. 8 1.2- I c e B r e a k i n g S h i p s and Impact L o a d s E x p l o r a t i o n o f t h e a r c t i c b e g a n i n t h e s i x t e e n t h c e n t u r y w i t h t h e a i m o f f i n d i n g a n o r t h e r n p a s s a g e f r o m Europe t o A s i a . M a r t i n F r o b i s h e r made a v o y a g e t o B a f f i n I s l a n d i n 1576 and w i t h i n 40 y e a r s , D a v i s S t r a i t , B a f f i n Bay and Hudson Bay were d i s c o v e r e d . S e v e r a l e x p e d i t i o n s were made t o t h e a r c t i c d u r i n g t h e n i n e t e e n t h c e n t u r y b y B r i t i s h e x p l o r e r s and a s e r i e s o f maps o f t h e a r c t i c i s l a n d s were p r e p a r e d . D u r i n g 1 9 0 3 - 1 9 0 6 , R o a l d Amundsen o f Norway made t h e f i r s t t r i p t h r o u g h t h e N o r t h West P a s s a g e . E a r l y i n t h e t w e n t i e t h c e n t u r y t h e C a n a d i a n Government c o m m i s s i o n e d e x t e n s i v e a r c t i c v o y a g e w i t h s h i p s s u c h as t h e Arctic [ 2 2 ] . A f t e r t h e s e c o n d w o r l d war C a p t a i n L a r s e n s a i l e d t h e R o y a l C a n a d i a n M o u n t e d P o l i c e (RCMP) v e s s e l , St. Roch, f r o m V a n c o u v e r t o H a l i f a x and b a c k . T h i s was t h e f i r s t v e s s e l t o t r a v e l t h r o u g h t h e N o r t h West P a s s a g e i n b o t h d i r e c t i o n s . I n 1969 t h e i c e - s t r e n g t h e n e d t a n k e r Manhattan made a v o y a g e t h r o u g h t h e N o r t h West P a s s a g e w h i c h s t a n d s as a m i l e - s t o n e i n a r c t i c m a r i n e t r a n s p o r t a t i o n [ 2 2 ] . I n N o r t h A m e r i c a we see t h e d e v e l o p m e n t • o f t h e C a n a d i a n A r c t i c S h i p p i n g P o l l u t i o n P r e v e n t i o n R e g u l a t i o n s (CASPPR) i n 1972 . I n r e c e n t y e a r s , t h e c o n t i n u i n g s e a r c h f o r h y d r o c a r b o n r e s o u r c e s i n t h e A r c t i c h a s l e d t o d e v e l o p m e n t and p r o d u c t i o n o f s p e c i a l f a c i l i t i e s f o r o p e r a t i o n i n i c e - i n f e s t e d w a t e r s and much work h a s b e e n done t o s o l v e t h e p r o b l e m s c a u s e d by t h e h o s t i l e c o n d i t i o n o f t h e a r c t i c m a r i n e e n v i r o n m e n t . I c e b r e a k e r s a r e one v a l u a b l e component o f t h e A r c t i c o i l 9 i n d u s t r y . I c e b r e a k e r s t r a d i t i o n a l l y a r e u s e d t o open a c h a n n e l t h r o u g h t h e i c e so t h a t o t h e r s e l f - p r o p e l l e d v e s s e l s c a n maneuver and make way. The o p e r a t i o n a l s t r e n g t h o f an i c e - b r e a k e r i s c a t e g o r i z e d b y t h e i c e - c l a s s i f i c a t i o n i n d e x . A c c o r d i n g t o t h e C a n a d i a n A r c t i c S h i p p i n g P o l l u t i o n P r e v e n t i o n R e g u l a t i o n (CASPPR) , t h e c l a s s number i n d i c a t e s t h e t h i c k n e s s ( i n f t ) o f u n i f o r m l e v e l i c e i n w h i c h t h e i c e b r e a k e r c a n o p e r a t e w i t h o u t l o s s o f s p e e d and w i t h o u t s u s t a i n i n g s t r u c t u r a l damage, a t a u n i f o r m s p e e d o f 3 k n o t s . E v e n t h o u g h l e v e l I c e o f c o n s t a n t t h i c k n e s s i s s e l d o m , i f e v e r e n c o u n t e r e d i n t h e a r c t i c , t h e c o n c e p t o f l e v e l i c e o f u n i f o r m t h i c k n e s s and s t r e n g t h i s a u s e f u l one as i t e n a b l e s r e l a t i v e p e r f o r m a n c e s t a n d a r d s o f d i f f e r e n t s h i p s t o be made. When a n i c e - b r e a k e r o p e r a t e s i n i c e e q u a l i n t h i c k n e s s t o t h e i c e i n d e x o f t h e v e s s e l i t i s a b l e t o p r o c e e d i n c o n t i n u o u s i c e - b r e a k i n g mode a t c o n s t a n t s p e e d . When i t r e a c h e s t h i c k i c e , o r an i c e r i d g e , i t i s n e c e s s a r y f o r t h e s h i p t o ram t h e i c e i f i t i s t o p r o c e e d . I n t h e ramming mode t h e s h i p p r o g r e s s e s b y r e p e a t e d l y c h a r g i n g t h e i c e . F o r e a c h ramming p r o c e s s t h e s h i p r e v e r s e s f r o m t h e i c e and t h e n r u n s t o w a r d i t a t s p e e d . I n t h e f o l l o w i n g s e c t i o n we r e v i e w some p r e v i o u s p u b l i s h e d w o r k s r e l a t i n g t o i c e - b r e a k i n g s h i p s 1.2.1- L i t e r a t u r e R e v i e w Many a r t i c l e s have b e e n p u b l i s h e d on t h e c h a r a c t e r i s t i c s o f i c e , i n t e r a c t i o n o f i c e and s h i p s , and t h e b e h a v i o r o f i c e - b r e a k i n g s h i p s . I n 1 9 0 0 , R u n e b e r g [23] s t u d i e d t h e i c e b r e a k i n g p r o b l e m and d e v e l o p e d a method t o i n v e s t i g a t e c o n t i n u o u s mode i c e - b r e a k i n g . He i s 10 c r e d i t e d as t h e f i r s t p e r s o n t o examine t h e i c e b r e a k i n g p r o c e s s . L a t e r o n , i n 1958 V i n a g r a d o v [ 2 4 ] , i n 1961 M i l a n o [ 2 6 ] , and i n 1965 W h i t e [25] i n v e s t i g a t e d t h e i c e b r e a k i n g p r o b l e m and gave some e q u a t i o n s f o r i c e f o r c e s on t h e s h i p s t e m . They u s e d methods w h i c h were v e r y s i m i l a r t o t h o s e o f R u n e b e r g . N e i t h e r o f t h e f o r e g o i n g methods c o n s i d e r e d s h i p s p e e d . They assumed t h a t s u f f i c i e n t f o r c e i s d e v e l o p e d b y t h e s h i p p r o p e l l e r t h r u s t t o c a u s e i c e f a i l u r e , so t h e s h i p p r o c e e d s a t some v e r y l o w s p e e d t h r o u g h t h e i c e . I n 1 9 3 8 , S h i m a n s k y [27] d e v e l o p e d an e q u a t i o n f o r t h e r e s i s t a n c e o f i c e b r e a k e r s i n t h e c o n t i n u o u s mode p r o c e s s . By means o f a s e m i - e m p i r i c a l method he d e v e l o p e d s e v e r a l p a r a m e t e r s f o r i c e b r e a k e r s w h i c h he t e r m e d "conditional Ice quality. sdandaridA.''. H i s e q u a t i o n s c o n t a i n p a r a m e t e r s w h i c h must be d e t e r m i n e d f r o m f u l l s c a l e d a t a . I n 1 9 6 5 , W h i t e [25] i n v e s t i g a t e d t h e ramming p r o c e s s and d e v e l o p e d an e q u a t i o n f o r t h e downward f o r c e on t h e i c e i n i t i a t e d b y t h e s h i p , as a f u n c t i o n o f v e l o c i t y . He compared t h e d e c e l e r a t i o n o f t h e s h i p p r e d i c t e d b y h i s m a t h e m a t i c a l mode l w i t h t h o s e o b t a i n e d d u r i n g s t r u c t u r a l t e s t s on t h e USCGC Westwind, w h i c h were r e p o r t e d by Waterman [ 2 8 ] . I n 1962 , K o r z h a v i n [29] p u b l i s h e d r e s u l t s o f h i s e x p e r i m e n t a l i n d e n t o r t e s t s t o d e t e r m i n e t h e i c e s t r e n g t h . He showed t h e v a r i a t i o n o f i c e s t r e n g t h due t o t h e i n d e n t o r d i m e n s i o n , f o r m , c o n t a c t a r e a b e t w e e n t h e i c e and t h e i n d e n t o r and t h e s t r a i n r a t e . The s u g g e s t i o n s made by h i m w e r e l a t e r u s e d b y d i f f e r e n t i n v e s t i g a t o r s s u c h as N o b l e [32] and B e r n a r d [ 3 0 ] . I n 1 9 6 8 , K a s h t e l j a n , e t a l [31] u s e d a s e m i - e m p i r i c a l method w h i c h was f i r s t i n t r o d u c e d b y S h i m a n s k y f o r t h e c o n t i n u o u s mode. They a l s o 11 i n c l u d e d t h e e f f e c t s o f b u o y a n c y f o r c e s and v e l o c i t y d e p e n d e n t f o r c e s . A c c o r d i n g t o t h e i r e q u a t i o n , t o t a l i c e r e s i s t a n c e i s t h e summat ion o f r e s i s t a n c e due t o b r e a k i n g t h e i c e , r e s i s t a n c e due t o f o r c e s c o n n e c t e d t o t h e w e i g h t s u c h as s u b m e r s i o n and r o t a t i o n o f b r o k e n i c e , change o f p o s i t i o n o f i c e b r e a k e r and w a t e r f r i c t i o n , and t h e r e s i s t a n c e due t o p a s s i n g t h r o u g h t h e b r o k e n i c e p i e c e s . K a s h t e l j a n , e t a l , s t a t e d t h a t t h e c o e f f i c i e n t s o f t h e i r e q u a t i o n were d e t e r m i n e d f r o m t h e f u l l s c a l e and m o d e l t e s t s on t h e s h i p Ermak, t h e r e f o r e i t s t r i c t l y a p p l i e d t o t h a t s h i p . However t h e y d i d i n t r o d u c e some f a c t o r s i n t h e i r e q u a t i o n r e l a t i n g t o t h e shape o f t h e h u l l , so t h a t t h e i r e q u a t i o n c o u l d be m o d i f i e d and a p p l i e d t o o t h e r i c e b r e a k e r s . L e w i s and Edwards [32] i n 1970 , i n v e s t i g a t e d t h e c o n t i n u o u s and ramming mode o f i c e b r e a k e r s . F o r c o n t i n u o u s i c e b r e a k i n g r e s i s t a n c e t h e y i n t r o d u c e d a s e m i - e m p i r i c a l e q u a t i o n w h i c h has o n l y t h r e e t e r m s . The f i r s t t e r m a c c o u n t s f o r r e s i s t a n c e a t t r i b u t e d t o b r e a k i n g t h e i c e , t h e s e c o n d t e r m a c c o u n t s f o r r e s i s t a n c e f o r c e s a s s o c i a t e d w i t h b u o y a n c y , and f i n a l l y t h e t h i r d t e r m i s f o r a l l r e s i s t a n c e f o r c e s a t t r i b u t e d t o momentum i n t e r c h a n g e b e t w e e n t h e s h i p and t h e b r o k e n i c e . The a u t h o r s m e n t i o n e d t h a t t h e a p p l i c a b i l i t y o f t h e i r s u g g e s t e d e q u a t i o n was c a r r i e d o u t b y p e r f o r m i n g r e g r e s s i o n a n a l y s i s o f a l l mode l and f u l l - s c a l e i c e r e s i s t a n c e t e s t d a t a a v a i l a b l e a t t h e t i m e . L e w i s and Edwards i n t r o d u c e d a s i m p l e e q u a t i o n f o r t h e maximum v e r t i c a l l o a d w h i c h c a n be s u p p o r t e d b y an i c e p l a t e o f a s p e c i f i e d t h i c k n e s s and s t r e n g t h . They compared t h e i r e q u a t i o n w i t h t h e f u l l s c a l e t e s t o b s e r v a t i o n o f USCGC Northwind. W i t h a s p e c i f i e d p a r a m e t r i c m a g n i t u d e w h i c h t h e y i n t r o d u c e d i n t h e i r e q u a t i o n , t h e y c o u l d f i n d good ag reement b e t w e e n t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s . 12 I n 1 9 7 9 , N o b l e , e t a l [33] i n t r o d u c e d a m a t h e m a t i c a l m o d e l f o r s h i p and i c e i n t e r a c t i o n . They c o n s i d e r e d t h e c o l l i s i o n b e t w e e n a m o v i n g i c e f l o e and a r u n n i n g r i g i d s h i p . They u s e d L e w i s and Edwards [32] i c e s t r e n g t h m o d e l and s o l v e d t h e e q u a t i o n o f m o t i o n o f t h e s h i p u n d e r i m p a c t l o a d due t o i c e c o l l i s i o n . The a u t h o r s compared t h e p r e d i c t i o n s o f t h e i r m a t h e m a t i c a l mode l w i t h r e s u l t s o b t a i n e d f r o m t h e s e l f p r o p e l l e d 1 / 3 6 s c a l e m o d e l o f t h e CCGS Louise S. St. Laurent w h i c h was t e s t e d b y A r c t e c C a n a d a . They a l s o compared t h e i r r e s u l t s w i t h f u l l s c a l e a c c e l e r a t i o n d a t a on Louis w h i c h was g a t h e r e d i n 1977 . I n 1 9 8 1 , J o h a n s s o n , e t a l [34] p u b l i s h e d an a r t i c l e i n w h i c h t h e y i n t r o d u c e d t h e i d e a o f d e s i g n i n g and b u i l d i n g an i c e b r e a k i n g t a n k e r w h i c h w o u l d h a v e t h e c a p a b i l i t y o f t r a n s p o r t a t i o n o i l o u t o f t h e C a n a d i a n a r c t i c . I n t h e i r p a p e r t h e y s u g g e s t e d a s p o o n s h a p e d bow t o m a x i m i z e t h e i c e - b r e a k i n g c a p a c i t y i n t h i c k i c e . I n 1983 a f u l l s c a l e d a t a c o l l e c t i o n was made by Ghoneim and K e i n o n e n [35] on Kigoriak, C a n m a r ' s c l a s s 3 i c e b r e a k e r , w h i l s t ramming h e a v y i c e r i d g e s . The f u l l - s c a l e t e s t s showed t h a t t h e bow o f t h e s h i p c o u l d be damaged i n u n l i m i t e d o p e r a t i o n o f t h e s h i p i n m u l t i - y e a r i c e . They a l s o n o t i c e d t h a t t h e l o n g i t u d i n a l s t r e n g t h o f Kigoriak w o u l d be p o t e n t i a l l y c r i t i c a l when o p e r a t i n g i n m u l t i - y e a r i c e . I n 1984 Vaughan [36] c o n s i d e r e d t h e e x t r e m e c a s e i n ramming p r o c e s s . He s t u d i e d t h e c a s e when t h e i c e b r e a k e r t r i e s t o b r e a k up l a r g e p r e s s u r e i c e r i d g e s many t i m e s t h i c k e r t h a n t h e i c e - c l a s s t h i c k n e s s f o r w h i c h i t i s c l a s s i f i e d . I n s u c h c a s e s t h e s h i p must ram t h e i c e r i d g e many t i m e s . D u r i n g s u b s e q u e n t rams t h e bow s l i d e s o v e r t h e r i d g e w i t h o u t f u r t h e r c r u s h o f t h e i c e and t h e s h i p r i d e s up t h e i c e r i d g e w h i c h c a u s e s an u n u s u a l l y h i g h c o n t a c t f o r c e on t h e bow. The h i g h 13 c o n t a c t l o a d on t h e bow r e g i o n c a u s e s a h i g h b e n d i n g s t r e s s a l o n g t h e s h i p . V a u g h a n , t h r o u g h a t h e o r e t i c a l and d i m e n s i o n a l a n a l y s i s i n v e s t i g a t e d t h e m o t i o n o f a r i g i d s h i p i n t h e ramming p r o c e d u r e . He d e r i v e d a s e r i e s o f e q u a t i o n s and i n t r o d u c e d d e s i g n c u r v e s f o r p r e d i c t i n g t h e u p p e r bounds o f t h e s e c t i o n modu lus and s h e a r a r e a s o f i c e - b r e a k i n g s h i p s . He a l s o gave f o r m u l a s f o r p r e d i c t i n g t h e maximum c o n t a c t f o r c e , and maximum b e n d i n g moment i n t h e s h i p . He showed t h a t w h i l e t h e maximum c o n t a c t f o r c e o c c u r s a t t h e end o f t h e ramming p r o c e s s , t h e maximum b e n d i n g moment o c c u r s a t some e a r l i e r t i m e . Vaughan c o n c l u d e d t h a t , s i n c e t h e maximum b e n d i n g moment and maximum c o n t a c t f o r c e were n o t h a p p e n i n g s i m u l t a n e o u s l y t h e r e was a n e e d f o r a dynamic a n a l y s i s o f t h e c o m p l e t e m o t i o n . I n 1984 , p r o g r a m B A F F I N , a compute r p r o g r a m b a s e d on m o d a l a n a l y s i s o f t h e s h i p s t r u c t u r e , was i n t r o d u c e d by D a l e y [ 3 7 ] . I n t h i s p r o g r a m t h e s h i p i s c o n s i d e r e d as a two d i m e n s i o n a l e l a s t i c beam. I n p u t t o t h e p r o g r a m p r e f e r a b l y , a r e t h e mode shapes o f t h e s h i p m o t i o n s . These mode s h a p e s c o u l d be p r e d e t e r m i n e d u s i n g a n o t h e r f i n i t e e l e m e n t p r o g r a m . B A F F I N c o u l d be u s e d t o s t u d y t h e c o n t i n u o u s i c e - b r e a k i n g mode o f an i c e b r e a k e r . A g a i n i n 1984 , a n u m e r i c a l a n a l y s i s o f t h e s h i p - i c e i n t e r a c t i o n p r o b l e m was p u b l i s h e d b y M a t s u i s h i , e t a l [ 3 8 ] . The a u t h o r s d e v e l o p e d a f i n i t e e l e m e n t method t o a n a l y z e t h e c o n t i n u o u s i c e b r e a k i n g mode p r o c e d u r e . They c o n s i d e r e d a 2 0 0 , 0 0 0 dwt t a n k e r o f l e n g t h 360 m e t e r s . The s h i p was m o d e l e d as a two d i m e n s i o n a l beam. I n t h e i r f i n a l r e s u l t s t h e y i g n o r e d t h e h y d r o d y n a m i c s and s t r u c t u r a l damping e f f e c t . I n t h e same y e a r , 1984 , Ghone im, e t a l [39] p u b l i s h e d t h e r e s u l t s 14 o f a s e r i e s o f f u l l - s c a l e ramming t e s t s c o n d u c t e d i n t h e B e a u f o r t s e a , on Canmar Kigoriak and Robert LeMeur, s h i p s w i t h 90 m WL and 6800 t o n e s d i s p l a c e m e n t . S e l e c t e d d a t a f r o m s e v e r a l h u n d r e d ramming t e s t s a t speeds r a n g i n g f r o m 3 t o 1 2 . 5 k n o t s were p r e s e n t e d . The a u t h o r s compared t h e d a t a w i t h t h e p r e d i c t i o n o f a f i n i t e e l e m e n t p r o g r a m . The mode l f o r Kigoriak was composed o f 32 beam e l e m e n t s w h i l e Robert LeMeur was m o d e l e d as an a s s e m b l a g e o f 34 e l e m e n t s . The Kigoriak mode l was u n d e r i m p a c t l o a d s w i t h p e a k s o f 45 and 17 MN w h i l e Robert LeMeur was r u n u n d e r a 8 MN i m p u l s e l o a d . Good ag reement b e t w e e n t h e m e a s u r e d v a l u e s and t h e p r e d i c t e d q u a n t i t i e s f r o m t h e f i n i t e e l e m e n t p r o g r a m showed t h a t t h e f i n i t e e l e m e n t m o d e l i n g was a d e q u a t e f o r d e t e r m i n i n g t h e g l o b a l d y n a m i c h u l l r e s p o n s e . I n 1986 Vaughan [40] composed a m a t h e m a t i c a l a n a l y s i s t o s t u d y t h e ramming o f a n i c e r i d g e b y a f l e x i b l e s h i p . The i c e r i d g e and c o l l i s i o n h a d t h e same c o n d i t i o n s as t h o s e i n t r o d u c e d i n [35] b y t h e same a u t h o r . B u t i n t h i s a r t i c l e t h e f l e x i b i l i t y o f t h e s h i p was c o n s i d e r e d . I n o r d e r t o s i m u l a t e t h e dynamic c h a r a c t e r i s t i c s o f t h e r e a l s h i p , Vaughan c o n s i d e r e d t h e s h i p as a u n i f o r m beam w i t h t h e same f u n d a m e n t a l n a t u r a l f r e q u e n c y and i n e r t i a as t h a t o f t h e r e a l s h i p . W i t h t h e s e a s s u m p t i o n s c e r t a i n r e s p o n s e s o f t h e r e a l s h i p and t h e u n i f o r m beam w o u l d be e q u a l , n a m e l y , t h e e n e r g y l o s t , t h e f o r c e a t f i r s t i m p a c t , t h e r a t i o o f t h e b e a c h i n g p e r i o d t o t h e p e r i o d o f f r e e f l e x u r a l v i b r a t i o n , and t h e r a t i o o f f l e x u r a l s t r e s s e s t o t h e b e a c h i n g s t r e s s e s . As t h e f l e x u r a l m o t i o n s o f t h e s h i p were s u c h t h a t t h e s t e r n was f r e e b u t t h e bow was c o n s t r a i n e d t o r e m a i n i n c o n t a c t w i t h t h e i c e , t h e beam mode l was s e l e c t e d as a p i n n e d - f r e e beam w i t h t h e p i n n e d end c o n s t r a i n e d t o move a l o n g a s p e c i f i e d p a t h d u r i n g t h e b e a c h i n g m o t i o n . He a l s o made t h e 15 a s s u m p t i o n t h a t t h e s h i p r e s p o n d e d e l a s t i c a l l y i n i t s f i r s t mode o f v i b r a t i o n , t o t h e i m p u l s e a t c o n t a c t as w e l l as t o t h e b e a c h i n g l o a d . T h r o u g h t h i s a n a l y t i c a l m e t h o d , Vaughan c o n c l u d e d t h a t t h e f l e x u r a l r e s p o n s e due t o t h e i n i t i a l i m p u l s e c a n l e a d t o b e n d i n g s t r e s s o f t h e same o r d e r o f m a g n i t u d e as t h o s e p r o d u c e d s u b s e q u e n t l y d u r i n g t h e b e a c h i n g m o t i o n . A l s o t h e f l e x i b i l i t y o f t h e s h i p r e d u c e d t h e i m p u l s e a t t h e bow b y 36%, a r e s u l t w h i c h was i n a good a g r e e m e n t w i t h t h e M a t s u i s h i , e t a l [38] f i n i t e e l e m e n t a n a l y s i s . F i n a l l y t h e a u t h o r c a l c u l a t e d t h e maximum b e n d i n g moment f o r Kigoriak and compared i t t o t h e f u l l - s c a l e d a t a p u b l i s h e d b y G h o n e i m , e t a l [ 3 9 ] . Good ag reement b e t w e e n t h e c a l c u l a t e d v a l u e s and c o l l e c t e d d a t a showed t h e a c c u r a c y o f t h e d e v e l o p e d e q u a t i o n s i n t h e g l o b a l a n a l y s i s o f t h e f l e x i b l e s h i p . I n 1986 , t h e r e s u l t s o f a f u l l - s c a l e t e s t o f 'USCGC Polar Sea' was p u b l i s h e d b y Cowper , e l a l [ 4 1 ] . 'USCGC Polar Sea' i s an i c e - b r e a k i n g s h i p o f 1 0 7 . 3 m WL w i t h d i s p l a c e m e n t o f 1 3 , 1 9 0 t o n s . T e s t s were c a r r i e d o u t i n t h e A l a s k a n B e a u f o r t Sea b e t w e e n September 19 , 1985 and O c t o b e r 1 3 , 1985 . The o v e r a l l o b j e c t i v e o f t h e t e s t s was t o m e a s u r e , r e c o r d and a n a l y z e t h e g l o b a l i c e - h u l l i n t e r a c t i o n s t r e s s e s i n m u l t i - y e a r i c e . Maximum b e n d i n g s t r e s s and i t s l o c a t i o n , and t h e v e r t i c a l component o f t h e i c e f o r c e on t h e bow d u r i n g t h e ramming p r o c e s s were o b t a i n e d . The r e s u l t s were compared w i t h t h o s e p u b l i s h e d b y Ghone im , e t a l [39] f o r Kigoriak. Some d i f f e r e n c e s b e t w e e n two t e s t s were r e c o g n i z e d , i . e . , t h e i m p a c t d u r a t i o n f o r Polar Sea i s a p p r o x i m a t e l y t w i c e t h a t m e a s u r e d f o r Kigoriak . A l s o t h e maximum b e n d i n g moment on t h e Polar Sea o c c u r r e d f u r t h e r f o r w a r d t h a n on t h e Kigoriak. The a u t h o r s m e n t i o n e d t h a t t h e d i s p l a c e m e n t o f Polar sea was a p p r o x i m a t e l y 1 .7 t i m e s t h a t o f Kigoriak, and bow s l o p e s o f t h e v e s s e l s were d i f f e r e n t , f a c t o r s w h i c h c o u l d 16 a c c o u n t f o r some o f t h e o b s e r v e d d i f f e r e n c e s . 1.3- Introduction to This Work E v e n t h o u g h t h e two t y p e s o f p r o b l e m s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n s a r e g e n e r a l l y d i f f e r e n t f r o m e a c h o t h e r , i n b o t h c a s e s t h e r e e x i s t a s t r u c t u r e w h i c h i s a c t e d upon b y an i m p a c t l o a d . T h e r e f o r e , i n e a c h c a s e , dynamic r e s p o n s e o f t h e s t r u c t u r e t o t h e i m p a c t l o a d s c o u l d be u s e d t o a n a l y z e t h e s y s t e m . I n t h i s w o r k , t h e s h i p i s m o d e l e d as a f l o a t i n g beam. The beam i s t h e n a n a l y z e d b y t h e f i n i t e e l e m e n t method u s i n g t h e f o l l o w i n g i d e a l i z a t i o n s : e a c h e l e m e n t o f t h e beam i s assumed t o h a v e c o n s t a n t f l e x u r a l a n d t o r s i o n a l r i g i d i t y and c o n s t a n t c r o s s - s e c t i o n a r e a , e a c h node o f t h e beam h a s 6 d e g r e e s o f f r e e d o m , t h e beam i s l o n g i t u d i n a l l y v e r y r i g i d , and s p e c i f i c a t i o n o f e a c h e l e m e n t i n t h e mode l i s e q u a l t o t h e a v e r a g e v a l u e o f t h a t f a c t o r on t h e c o r r e s p o n d i n g s t a t i o n a l o n g t h e r e a l s h i p . G l o b a l mass and s t i f f n e s s m a t r i c e s o f t h e beam a r e d e v e l o p e d . W e i g h t a n d b u o y a n c y f o r c e s a r e t r e a t e d as d i s t r i b u t e d f o r c e s a l o n g t h e s h i p . The n o d a l v a l u e s f o r t h e s e f o r c e s a r e e v a l u a t e d . B u o y a n c y f o r c e s a l o n g t h e s h i p a c t l i k e an e l a s t i c f o u n d a t i o n f o r t h e beam. T h i s phenomenon n o r m a l l y makes t h e s h i p s t i f f e r . Added masses o f t h e s h i p i n l o n g i t u d i n a l and l a t e r a l m o t i o n s a r e c o n s i d e r e d . S t r u c t u r a l damping i s assumed t o be n e g l i g i b l e compared t o h y d r o d y n a m i c d a m p i n g . Damping c o e f f i c i e n t s o f t h e f l o a t i n g beam a r e d e v e l o p e d t h r o u g h a p r o p o r t i o n a l m a t r i x . E a c h c a s e o f t h e i m p a c t l o a d , d e p e n d i n g on t h e n a t u r e o f t h e f o r c e i s s t u d i e d s e p a r a t e l y . 17 1 . 3 . 1 - C o l l i s i o n o f A S h i p and an E x t e r n a l O b j e c t The g e n e r a t e d damage on t h e h u l l due t o c o l l i d i n g w i t h a n o t h e r o b j e c t i s e s t i m a t e d . C o m b i n a t i o n o f M i n o r s k y ' s [4] g e n e r a l e q u a t i o n and i t s m o d i f i c a t i o n b y J o n e s [8] w i t h V a u g h a n ' s e q u a t i o n [9] f o r p l a t e t e a r i n g f o r c e , i s u s e d t o c a l c u l a t e t h e c o l l i s i o n f o r c e and damage e x t e n t . I t i s shown t h a t , t h e g e n e r a t e d c o l l i s i o n f o r c e i n e a c h d i r e c t i o n i s p r o p o r t i o n a l t o t h e s t r u c t u r a l s p e c i f i c a t i o n o f t h e h u l l and c r o s s - s e c t i o n o f t h e s t r u c t u r a l damage i n t h a t s p e c i f i c d i r e c t i o n . The e q u a t i o n s o f m o t i o n o f t h e w h o l e s h i p a r e d e v e l o p e d b y means o f f i n i t e e l e m e n t m e t h o d s . These e q u a t i o n s a r e s o l v e d , i n c o n j u n c t i o n w i t h t h e i n i t i a l c o n d i t i o n s o f t h e s h i p m o t i o n . I n i t i a l c o n d i t i o n s depend on t h e s c e n a r i o o f t h e c o l l i s i o n , s u c h as s h o u l d e r o r h e a d on c o l l i s i o n i n n o r m a l n a v i g a t i o n o r s i d e c o l l i s i o n d u r i n g s h i p m a n e u v e r i n g . S o l u t i o n o f t h e e q u a t i o n s o f m o t i o n f o r e a c h t i m e s t e p i n d i c a t e s t h e p o s i t i o n o f t h e s h i p a t t h e end o f t h a t t i m e i n t e r v a l . H e n c e , i t i s p o s s i b l e t o c a l c u l a t e t h e i n t e r f a c e o f t h e v e s s e l w i t h t h e e x t e r n a l o b j e c t a t t h e end o f e a c h t i m e s t e p . T h i s c a n be u s e d t o d e t e r m i n e t h e f o r c e on t h e s h i p w h i c h i n t u r n d e t e r m i n e s t h e s h i p m o t i o n d u r i n g t h e n e x t t i m e i n c r e m e n t . T h i s p r o c e s s w o u l d be r e p e a t e d t i l l t h e end o f t h e c o l l i s i o n . The end o f c o l l i s i o n i s m o n i t o r e d b y c h e c k i n g t h e v e l o c i t y and p o s i t i o n o f t h e s h i p . When t h e s h i p comes t o r e s t o r p a s s e s t h e e x t e r n a l o b j e c t t h e c o l l i s i o n i s c o n s i d e r e d f i n i s h e d . I n e a c h t i m e s t e p t h e damaged vo lume i s e s t i m a t e d , c o n s e q u e n t l y a t t h e end o f c o l l i s i o n t h e g l o b a l damage i s known. 18 1 . 3 . 2 - C o n t i n u o u s I c e B r e a k i n g Mode I n t h e c o n t i n u o u s i c e - b r e a k i n g mode, t h e i m p a c t l o a d on t h e h u l l i s due t o t h e c r u s h i n g and b e n d i n g f a i l u r e o f t h e l e v e l i c e . When an i c e - b r e a k i n g s h i p i s r u n n i n g w i t h c o n s t a n t s p e e d i n t o l e v e l i c e , c o n t a c t a r e a b e t w e e n t h e i c e and h u l l g r a d u a l l y i n c r e a s e s , and c a u s e s an i n c r e a s e i n t h e c o n t a c t f o r c e s . B e n d i n g s t r e n g t h o f t h e i c e s h e e t d e t e r m i n e s t h e l i m i t f o r t h e s e f o r c e s and when t h e l i m i t i n g v a l u e s a r e r e a c h e d t h e i c e f a i l s . A t t h i s t i m e t h e c o n t a c t f o r c e d r o p s t o z e r o and r e m a i n s so w h i l e t h e s h i p t r a v e l s a d i s t a n c e e q u a l t o t h e l e n g t h o f t h e b r o k e n p a r t s o f i c e . A t t h e end o f t h i s d i s t a n c e t h e s h i p makes t h e n e x t c o n t a c t w i t h a r e l a t i v e l y b i g p i e c e o f l e v e l i c e , and t h e f o r m e r p r o c e s s i s r e p e a t e d . K o r z h a v i n ' s [29] mode l i s s e l e c t e d t o c a l c u l a t e t h e c o n t a c t f o r c e and l e n g t h o f t h e b r o k e n i c e p i e c e s . The e q u a t i o n s u g g e s t e d by L e w i s and Edward [32] i s a l s o u s e d f o r t h e maximum c o n t a c t f o r c e . As i t i s e x p e c t e d , f r e q u e n c y o f t h e i c e l o a d h a s a m a j o r e f f e c t on t h e r e s p o n s e o f t h e s h i p s t r u c t u r e . I n t h i s s t u d y r e l a t i o n b e t w e e n t h e f r e q u e n c y o f t h e c o n t a c t l o a d and s h i p v e l o c i t y and i t s e f f e c t on t h e s h i p m o t i o n and maximum b e n d i n g moment i n d u c e d i n t h e h u l l i s i n v e s t i g a t e d . 1 . 3 . 3 - Ramming o f H i g h P r e s s u r e I c e R i d g e s I n a d d i t i o n t o t h e c o n t i n u o u s i c e - b r e a k i n g mode m e n t i o n e d above we a l s o c o n s i d e r e d t h e e x t r e m e c a s e when a s h i p rams h e a v y i c e and f a i l s t o b r e a k i t . The s h i p t h e n s l i d e s upwards on t h e I c e and e v e n t u a l l y comes 19 t o r e s t b e f o r e s l i d i n g b a c k w a r d s . The ramming o p e r a t i o n i s t h e most s e v e r e i n t e r m s o f s t r e s s e s i n d u c e d i n t h e s h i p so a k n o w l e d g e o f t h e p e a k b e n d i n g s t r e s s e s and i m p a c t f o r c e s a r e e s s e n t i a l f o r d e s i g n p u r p o s e s . 20 2 - DERIVATION OF EQUATIONS OF MOTION 2 . 1 — G e n e r a l F o r m u l a t i o n The v i r t u a l w o r k method i s u s e d t o d e r i v e t h e e q u a t i o n s o f m o t i o n o f a d e f o r m a b l e b o d y . E q u a t i o n ( 3 . 3 ) on page 114 o f [ 4 2 ] , shows t h a t , t h e v i r t u a l w o r k o f a b o d y f o r c e and a s u r f a c e t r a c t i o n on a d e f o r m a b l e b o d y a r e r e l a t e d t o t h e s t r e s s and s t r a i n t h r o u g h t h e body as f o l l o w s : B. S u . d v + l l T . S u . d s = l l T . . 8e . . dv v V w h e r e ; B^ i s a b o d y f o r c e d i s t r i b u t i o n t h r o u g h o u t t h e b o d y . T^ i s a s u r f a c e t r a c t i o n o v e r a p a r t o f t h e b o u n d a r y o f t h e b o d y . r „ i s t h e s t r e s s t h r o u g h t h e b o d y . e _ i s t h e s t r a i n t h r o u g h t h e b o d y . I n e r t i a o f t h e body i n d i r e c t i o n ( i ) i s a p a r t o f t h e b o d y f o r c e B^, t h e r e f o r e , u s i n g D ' a l e m b e r t p r i n c i p a l i t c a n be w r i t t e n i n t h e f o r m o f : B.= B. - p l i 2 du d t 2 where p i s t h e d e n s i t y o f t h e b o d y . S u b s t i t u t i n g f o r B^ i n t h e above e q u a t i o n we w o u l d h a v e : B. 5 u . d v l l 2 du d t 5 u . d v + 2 1 T . S u . d s l l T . . 6e. . dv 21 The above e q u a t i o n c o u l d be w r i t t e n i n t h r e e p r i n c i p a l d i r e c t i o n s , and c o m b i n e d i n t h e v e c t o r f o r m . The v e c t o r f o r m o f t h e above e q u a t i o n i s shown i n page 644 o f [42] a s : {Su} {b }dv p{6u} {ii} dv + {Su} {T )ds = [Se] [ a ] d v C o n s i d e r i n g t h e damping e f f e c t and m o d i f y i n g t h e above e q u a t i o n , i t w o u l d be w r i t t e n as : (Su) (b }dv - {Su} I f } dv v p[S\i) {ii} dv + {Su} {T} ds = T d [Se] [a] dv [Se] [<r]dv (a) w h e r e , W , . = d i s s { 5u } { f } dv T d [ 5 e ] [ a ] d v i s t h e v i r t u a l d i s s i p a t e d w o r k . {f} i s t h e d i s s i p a t i o n f o r c e p e r u n i t vo lume a n d . d [a] i s a v i s c o u s damping s t r e s s t e n s o r . I n o r d e r t o d i s c r e t i z e t h e body i n t o s m a l l f i n i t e e l e m e n t s , t h e d i s p l a c e m e n t o f any p o i n t i n t h e b o d y , {u}, must be e x p r e s s e d i n t e r m o f n o d a l d i s p l a c e m e n t {U} , and an i n t e r p o l a t i o n f u n c t i o n [ N ] . Thus we have {u} = [N] {U} (b) where {U} i s a n o d a l d i s p l a c e m e n t , and i s a f u n c t i o n o f t i m e , and [N] i s t h e i n t e r p o l a t i o n f u n c t i o n m a t r i x . E a c h member o f [N] i s a f u n c t i o n o f 22 p o s i t i o n w h i c h s a t i s f i e s t h e p r e s c r i b e d g e o m e t r i c b o u n d a r y c o n d i t i o n s f o r t h e e l e m e n t . The s t r a i n f i e l d m a t r i x [e] i s r e l a t e d t o t h e d i s p l a c e m e n t f i e l d {u} b y a s u i t a b l e d i f f e r e n t i a l o p e r a t i o n [ L ] , d e v e l o p e d w i t h g e o m e t r i c c o n s i d e r a t i o n , i . e . , [e] = [L ] {u} Hence [e] - [L] [N] (V) =[B]{(U} ( c ) where [B] - [L] [N] The c o n s t i t u t i v e l a w m a t r i x [D] r e l a t e s t h e s t r a i n f i e l d m a t r i x and t h e s t r e s s m a t r i x as f o l l o w ; [a] - [D] U ) = [D] [B] (U) The v i s c o u s d i s s i p a t i o n f o r c e ( f ) i s assumed t o be p r o p o r t i o n a l t o t h e v e l o c i t y , w h i l e t h e v i s c o u s dampen ing m a t r i x [ c r ] d i s assumed t o be p r o p o r t i o n a l t o [D] {e} as s u g g e s t e d i n [ 4 2 ] . T h e s e a s s u m p t i o n s t h e n e n a b l e t o w r i t e : { f }= c {u} = c [N] {U} and [af - 0 [D] {€) = p [D] [B] {U} where c and ft axe c o n s t a n t o f p r o p o r t i o n a l i t y . S u b s t i t u t i n g f o r {u}, {e}, [<*] > [ ° ] d a n d ( f ) f r o m t h e above r e l a t i o n s i n e q u a t i o n (a) we h a v e , T T {5QJ} [N] {b} dv -T T c [ 5 U ] [ N ] [ N ] { U } d v v 23 T T p [SU] [B] [D] [B] {U} dv T T p {SU}[N] [N] {U} dv v T T {5U) [N] {T} ds T T {5U} [B] [D] [B] W) dv v o r T {5U} [N] {b} dv - c [N] [N] dv {QJ} v V p [B] [D] [B] dv {U} p [N] [N] dv (i)) T >, T [N] (T) ds = [B] [D] [B] dv {U} As t h e above r e l a t i o n must be t r u e f o r any v a l u e o f {5U}, t h e n t h e e q u a t i o n o f m o t i o n c o u l d be w r i t t e n a s : ' [m] (U) + [c] (IU} +[k] {(U) ={q} ( 2 . 1 ) w h e r e ; Mass M a t r i x Damping M a t r i x [m] = c = p [N] [N] dv c [N] [N] dv + ( 2 . 2 ) P [B] [D] [B] dv S t i f f n e s s M a t r i x [ k ] - [B] [D] [B] dv ( 2 . 3 ) N o d a l v a l u e s o f t h e a p p l i e d l o a d s a r e , 24 {q} = T [N] {b} dv + T [N] {T} ds ( 2 . 4 ) v C o m p a r i n g t h e damping m a t r i x [c] w i t h t h e mass m a t r i x [m] and t h e s t i f f n e s s m a t r i x [ k ] , shows t h a t t h e f i r s t i n t e g r a l f o r [c ] i s p r o p o r t i o n a l t o [m] , w h i l e t h e s e c o n d one i s p r o p o r t i o n a l t o [ k ] . Thus [c ] = a [m] + 0 [k] w h i c h i s t h e R a y l e i g h p r o p o r t i o n a l damping m a t r i x . P r o p o r t i o n a l damping p e r m i t s u n c o u p l i n g o f t h e e q u a t i o n s o f m o t i o n . A l s o , t h e v i b r a t i o n mode shapes i n t h e damped s y s t e m w i l l be s i m i l a r t o t h o s e i n t h e undamped s y s t e m . 2 . 2 - D e v e l o p m e n t o f S t r u c t u r a l P a r a m e t e r s f o r an E l a s t i c Beam 2 . 2 . 1 - Shape F u n c t i o n I n t h i s s t u d y a s h i p i s m o d e l e d as a f l o a t i n g beam, t h e r e f o r e e a c h e l e m e n t i s an e l a s t i c beam w i t h a u n i f o r m c r o s s - s e c t i o n a r e a . I t s l a t e r a l d i s p l a c e m e n t s a r e u and m i n t h e y and z d i r e c t i o n s r e s p e c t i v e l y . The beam i s assumed t o be a x i a l l y r i g i d and i t s a x i a l m o t i o n i s c o n s i d e r e d s e p a r a t e l y . The beam e l e m e n t s w i t h i t s n o d a l d i s p l a c e m e n t s i s i l l u s t r a t e d i n F i g . ( 2 . 1 ) . The d i s p l a c e m e n t s o f t h e end p o i n t s a l o n g t h e y and z a x i s o f t h e l o c a l c o o r d i n a t e s y s t e m a r e r e s p e c t i v e l y u^ and f o r t h e end (1) and u , and U-. f o r end (2) , w h i l e r o t a t i o n s o f t h e end p o i n t s a r o u n d t h e s e o / a x i s a r e u„, u and u c f o r end (1) and u , u_ and u . f o r end ( 2 ) . 25 10 7 8 z Fig. 2.1- The Beam Element With Ten Degrees of Freedom. Since the beam element i s axially r i g i d , there i s no relative motion along the x axis. Therefore the equation of axial motion of the element i s uncoupled from those of the lateral motions and is considered separately. This decreases by two the number of degrees of freedom of the beam element leaving a total of ten for each element. A shape function for a beam element with 12 degrees of freedom is introduced i n page 293 of [42]. The special shape function for an element with ten degrees of freedom, which is shown in Fig. (2.1), is deduced from [42] to be: 26 1 - 3 i / 2 + 2u 3 0 2 3 l - 3 i / + 2i/ (-1/ +2»/2 - i / 3 )£ 0 0 [N] T 0 0 0 0 2 3 (i/ - i / )* 2 3 (-1/ +1/ ) £ 0 The n o n - d i m e n s i o n a l p a r a m e t e r s u s e d a r e : 2 .2 .2 - Mass and S t i f f n e s s Matr ix The most g e n e r a l f o r m o f t h e mass m a t r i x f o r a beam e l e m e n t i n v o l v e s t w e l v e d e g r e e s o f f r e e d o m and i s g i v e n on page 294 o f [ 4 3 ] . I f t h e a x i a l m o t i o n i s d i s r e g a r d e d , as i s t h e c a s e h e r e , t h e n t h e mass m a t r i x r e d u c e s t o a t e n b y t e n m a t r i x . I n p a r t i c u l a r i t c a n be r e p r e s e n t e d as t h e sum o f two s u c h m a t r i c e s [m]^ and [ m ] 2 > where [ni]^ a c c o u n t s f o r t h e i n e r t i a and [m] 2 a c c o u n t s f o r t h e r o t a r y i n e r t i a w h i c h i s i m p o r t a n t when c o n s i d e r i n g t h e h i g h e r modes o f v i b r a t i o n o f t h e s h i p . [m] i =p A I 13 35 0 0 0 111 210 9 70 0 0 0 131 420 A n d f 6 I 2 [m] =p A £ 2 5 A r 0 0 0 1 2 lOAl 6Iz 5A£2 0 0 0 Iz 10A£ 13 35 0 1U "210 0 4TT 0 9_ 70 0 131 420 0 J x 3A 0 0 0 0 J x 6A 0 0 6 l z 5A£2 0 Iy "10A£ 0 0 6Iy 5A£2 0 Iy "10A£ 0 0 0 0 0 0 0 105 0 0 131 '420 0 '140 0 2Iy 15A 0 0 Iy 10A£ 0 Iy 3 OA 0 105 131 420 0 0 0 13 35 0 0 0 i n 140 " 210 13 35 0 111 210 0 2Iz 15A I z '10A£ 0 0 0 Iz  "30A "10A£ 61 z 5 At 0 0 0 I z J x 3A 0 0 6Iy 5A£2 0 Iy 10 Al " 0 0 105 105 2Iy 15A 0 21 = 15A) i s t h e c r o s s - s e c t i o n o f t h e e l e m e n t i s t h e l e n g t h o f e l e m e n t . 28 J i s t h e p o l a r moment o f i n e r t i a o f t h e e l e m e n t a r o u n d t h e x - a x i s w h i c h r e p r e s e n t s t h e t o r s i o n a l i n e r t i a o f t h e e l e m e n t . I & I a r e t h e moments o f i n e r t i a o f t h e c r o s s - s e c t i o n o f t h e e l e m e n t y z w i t h r e s p e c t t o t h e y and z a x e s r e s p e c t i v e l y . As t h e beam e l e m e n t i s f l o a t i n g on t h e w a t e r , t h e c o r r e s p o n d i n g a d d e d mass s h o u l d be c o n s i d e r e d i n i t s m o t i o n . Added mass o f a beam e l e m e n t c a n be r e p r e s e n t e d as a f r e q u e n c y d e p e n d e n t m a t r i x t o c o v e r t h e e f f e c t o f a d d e d mass i n l a t e r a l , v e r t i c a l and r o t a t i o n a l m o t i o n o f t h e beam. Added mass e f f e c t s f o r t r a n s i e n t and i m p u l s e m o t i o n a r e d i f f i c u l t t o c a l c u l a t e and i n t h i s t h e s i s we h a v e t a k e n h a r m o n i c m o t i o n v a l u e s . The e f f e c t o f s h a l l o w w a t e r a r e a l s o i m p o r t a n t and i s a n a r e a o f o n g o i n g r e s e a r c h i n s h i p h y d r o d y n a m i c s . The v a l u e s h e r e a r e f o r deep w a t e r . The mass m a t r i x i s now r e p r e s e n t e d a s : [ m ] = [ I + a ] [ [ m ] + [ m ] ] where [a ] i s t h e added mass m a t r i x r e f e r r e d t o above and [ I ] i s t h e H i d e n t i t y m a t r i x . The s t i f f n e s s m a t r i x f o r a beam e l e m e n t w i t h 12 d e g r e e s o f f r e e d o m i s d e v e l o p e d i n page 79 o f [ 4 3 ] . The e f f e c t o f s h e a r d e f o r m a t i o n i s i n c l u d e d t h r o u g h two f a c t o r s <f>^ and <j>^. He re we i g n o r e t h e e f f e c t s o f s h e a r on d e f o r m a t i o n and t a k e b o t h <f> and <j>^ t o be z e r o . The s y m m e t r i c s t i f f n e s s m a t r i x o f t h e beam e l e m e n t w i t h t e n d e g r e e s o f f r e e d o m t h e n r e d u c e s t o : 29 f 12EI2 [k ] = 0 12EIy e 0 0 GJx I 0 6EIy e 0 4EIy i 6EIz e 0 0 0 12EIz I3 0 0 0 0 - 12EIy 0 6EIy e e 0 0 GJx 'I 0 0 6EIy e 0 2EIz I 6EIz e 0 0 0 4 E I ; I - 6 E I 2 £ 2 0 0 0 2 E I 2 12EI = 0 0 0 - 6 E I z 12E I j 0 — x 6EIy 4EIy 0 0 4EI = w h i l e E and G a r e t h e modu lus o f e l a s t i c i t y and r i g i d i t y r e s p e c t i v e l y . 2.2.3- N o d a l V a l u e s o f t h e A p p l i e d L o a d s on t h e E l e m e n t The s h i p i s m o d e l e d a s - a f l o a t i n g beam. The w h o l e s h i p and t h e r e f o r e e a c h e l e m e n t o f t h e mode l w o u l d be u n d e r t h e e f f e c t o f t h e f o l l o w i n g f o r c e s : i - W e i g h t o f t h e e l e m e n t as a body f o r c e . i i - B u o y a n c y f o r c e as a s u r f a c e t r a c t i o n . i i i - R e s t o r i n g t o r q u e c a u s e d b y h e e l i n g i v - O t h e r e x t e r n a l f o r c e s 30 i - W e i g h t o f t h e E l e m e n t A l o n g e a c h e l e m e n t , w i t h a good a p p r o x i m a t i o n , w e i g h t c a n be c o n s i d e r e d as a u n i f o r m l y d i s t r i b u t e d body f o r c e i n , t h e v e r t i c a l , y - d i r e c t i o n . N o d a l v a l u e s o f t h i s f o r c e i s c a l c u l a t e d t h r o u g h E q u . ( 2 . 4 ) a s : S u b s t i t u t i n g t h e i n t r o d u c e d shape f u n c t i o n f o r [ N ] , w h i l e t a k i n g K = 0 b e c a u s e o f t h e symmetry o f t h e d i s t r i b u t i o n o f w e i g h t i n z - d i r e c t i o n , t h e n o d a l w e i g h t s a r e : 0 0 f 6 0 0 0 I {q} = W { - 6 0 0 0 I w e i i i - B u o y a n c y F o r c e The b u o y a n c y f o r c e on t h e beam e l e m e n t i s r e p r e s e n t e d a s : B ( x ) = p g I (A - Br u ( x , t ) ) w w w h e r e , P S w i s t h e s p e c i f i c w e i g h t o f t h e s e a w a t e r . A i s t h e u n d e r w a t e r c r o s s - s e c t i o n a r e a o f w t h e beam e l e m e n t . 31 Br i s t h e b r e a d t h o f t h e beam e l e m e n t and i s c o n s t a n t e x c e p t n e a r t h e bow, when t h e a v e r a g e f o r t h e e l e m e n t i s t a k e n u ( x , t ) i s t h e l a t e r a l d e f l e c t i o n o f t h e beam i n t h e v e r t i c a l y - d i r e c t i o n . B = p g A £ i s c o n s i d e r e d as t h e c o n s t a n t p a r t o f t h e b u o y a n c y f o r c e . B e c a u s e o f t h e u n i f o r m c r o s s - s e c t i o n o f t h e beam e l e m e n t , i t i s assumed t h a t t h e c o n s t a n t p a r t o f t h e b u o y a n c y f o r c e i s u n i f o r m l y d i s t r i b u t e d a l o n g t h e e l e m e n t . The n o d a l v a l u e s o f t h e c o n s t a n t p a r t a r e c a l c u l a t e d s i m i l a r t o t h o s e o f t h e w e i g h t and t h e y a r e : r e i 0 0 0 {qlx, = P g A • DC w w 6 0 0 0 £ The s e c o n d p a r t o f t h e b u o y a n c y f o r c e i s , B - p g Br I a ( x , t ) V w The b u o y a n c y f o r c e a c t s i n t h e v e r t i c a l d i r e c t i o n and i s a f u n c t i o n o f t i m e and p o s i t i o n . I t c a n be r e p r e s e n t e d i n m a t r i x n o t a t i o n a s : B = p g Br £ <1 0> {u} V w S u b s t i t u t i n g f o r {u} f r o m r e l a t i o n (b) i n S e c t i o n (2.1) we w o u l d h a v e : B = p g Br £<1 0>[N] {tU} V w To c a l c u l a t e t h e n o d a l v a l u e s o f t h i s p a r t o f t h e b u o y a n c y , B i s 32 s u b s t i t u t e d i n E q u . ( 2 . 4 ) t o g i v e ; B [ N ] T -I V | - dx 0 B dx v w h i c h c a n be w r i t t e n a s : l C l } B V = > „ g B l 1 1 0 )• <1 0>[N] {U}dx ' 1 0 ' g Br £ [ N ] T . 0 1 . [N]dx {QJ} 0 F i n a l l y i t c a n be w r i t t e n as ; <q>B V = [ k ] B { U ) w h e r e , B w ' 1 . 0 ' g B r £ [ N ] T [N] dx . 0 1 , 0 The n o d a l v a l u e s o f t h e t o t a l b u o y a n c y f o r c e a r e f o u n d b y summat ion and a r e : <q)B= < q ) B c - [k]B{U} i i i - R e s t o r i n g T o r q u e Caused by H e e l i n g When t h e e l e m e n t r o t a t e s t h r o u g h a n g l e 6 a b o u t i t s a x i s t h e r e s t o r i n g t o r q u e i s p r o p o r t i o n a l t o t h e r o t a t i o n a l a n g l e 6 and i s ; 33 R W GM 8 where W i s t h e w e i g h t o f t h e e l e m e n t GM i s t h e t r a n s v e r s e m e t a c e n t r i c h e i g h t 9 i s t h e a n g l e o f h e e l The h e e l a n g l e 9 i s l i n e a r l y r e l a t e d t o t h e r o t a t i o n s u and u , 3 8 t h e two ends o f t h e beam. T h a t i s , t w i s t o f any p o i n t on t h e beam e l e m e n t c o u l d be w r i t t e n i n t e r m s o f t h e r o t a t i o n o f t h e end p o i n t s and i t s p o s i t i o n a l o n g t h e beam e l e m e n t . H e e l a n g l e 9 c a n be shown i n te rms o f t h e n o d a l p o s i t i o n o f t h e beam e l e m e n t {U}, and t h e p o s i t i o n o f t h e p o i n t a l o n g t h e e l e m e n t v , a s : 9 = <N > {U} where <N> = <0 0 l-i/ 0 0 0 0 0 0> T h e r e f o r e t h e r e s t o r i n g t o r q u e R c o u l d be w r i t t e n a s ; -W GM <N> {U} To f i n d t h e n o d a l v a l u e s o f R t h e e q u a t i o n ( 2 . 4 ) i s u s e d a s , <N> R dx W GM <N> <N> dx {(U) w h i c h c o u l d be w r i t t e n a s ; {q}= -W GM I <N> <N> &v {QJ} = - [ k ] R { U ] where [k] i s t h e h y d r o s t a t i c s r e s t o r i n g s t i f f n e s s g i v e n b y : > R 34 [ k ] = I W GM L J R '0 0 0 0 0 0 0 0 0 0' 0 0 0 0 0 0 0 0 0 0 0 0 1 / 3 0 0 0 0 1 / 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 / 6 0 0 0 0 1 /3 0 0 0 0 0 0 0 0 0 0 0 0 [o 0 0 0 0 0 0 0 0 A c c o r d i n g t o t h e r e l a t i o n g i v e n on Page 70 o f [45] t h e t r a n s v e r s e m e t a c e n t r e M i s l o c a t e d above t h e c e n t e r o f b u o y a n c y B an amount ; — I T BM = — (d) where V = A x I i s t h e u n d e r w a t e r vo lume o f t h e e l e m e n t and I i s t h e w T t r a n s v e r s e moment o f i n e r t i a o f t h e w a t e r p l a n e a r e a o f t h e e l e m e n t g i v e n b y : 1 = Z-T 3 y 3 dx 0 A l o n g e a c h beam e l e m e n t t h e v a l u e o f y i s c o n s t a n t and i s e q u a l t o B r / 2 . T h e r e f o r e , I = Z- ( ^ ) 3 I T 3 2 I f r o m t h e above e q u a t i o n i s s u b s t i t u t e d i n t o Eq (d) and t h e v o l u m e o f d i s p l a c e m e n t V i s r e p l a c e d b y (A x I) . The m e t a c e n t r i c h e i g h t w BM i s t h e n : — B r 3 I B r 3 12 A I 12 A 35 fi P o s i t i o n o f the center o f |M buoyancy and the center o f g r a v i t y f o r each s t a t i o n of the ship depends upon the geometry o f the s t a t i o n and the load ing c o n d i t i o n o f the s h i p . As F i g . (2.2) shows, when p o s i t i o n of the centers o f buoyancy and g r a v i t y are r e s p e c t i v e l y , at B and G, the d i s tance between the g r a v i t y center , G, and the transverse metacentric p o i n t , M, i s : K F i g . 2.2- Typical Cross-Section Area of a Beam Element. GM - (KB + BM) - KG i v - The Other E x t e r n a l Forces Other ex terna l forces such as impact loads are t r e a t e d l i k e the surface t r a c t i o n s . Depending on the nature of the a p p l i e d forces they may be e i t h e r d i s t r i b u t e d or concentrated f o r c e s . The nodal values of the e x t e r n a l forces are developed through the equat ion (2 .4 ) , and are shown as {q} . Therefore the nodal values o f the t o t a l forces on the ^ Ex f l o a t i n g beam element i s : ( q J - - { q ) w e i + , q ) B c - [ k ] B ,U> - [ k ] a + {q}^ 36 2 .3- Development of the D i s c r e t i z e d Equation of Motion of the F l o a t i n g  Beam Element The e x p r e s s i o n s f o r mass m a t r i x , s t i f f n e s s m a t r i x and t h e n o d a l v a l u e s o f t h e a p p l i e d l o a d s on t h e beam e l e m e n t , w h i c h were i n t r o d u c e d i n t h e p r e v i o u s s e c t i o n s , a r e s u b s t i t u t e d i n e q u a t i o n ( 2 . 1 ) . The new f o r m o f e q u a t i o n i s t h e n : [m] { 0 5 + [ c ] {U} + [k ] {U}= - [ k ] ( U ) - [ k ] {OJ} +{q} +{q} _{q} B R EX BC Wei o r [m] { 0 } + [c ] {U} + [k] {U> - {q} + {q} - { q l hi A EC Wei Where [k ]= [k] + [k] +[k] B R I t i s n o t i c e d t h a t t h e b u o y a n c y f o r c e i n c r e a s e s t h e a p p a r e n t s t i f f n e s s f o r t h e beam. As t h e m a g n i t u d e o f a l l o f t h e components o f l q ) B C a n c l a r e c o n s t a n t and on t h e w h o l e s h i p t h e y a r e i n e q u i l i b r i u m , so t h e y h a v e no e f f e c t on t h e d y n a m i c s o f t h e s y s t e m and c a n be n e g l e c t e d i n t h e e q u a t i o n s o f m o t i o n . Then t h e e q u a t i o n o f m o t i o n b e c o m e s : [m] {(0) + [c ] {U} + [k] {U} = {q} ( 2 . 5 ) E X T h i s e q u a t i o n i s d e v e l o p e d w i t h r e s p e c t t o t h e l o c a l c o o r d i n a t e o f t h e e l e m e n t . A s was m e n t i o n e d e a r l i e r , t h e beam e l e m e n t h a s t e n d e g r e e s o f f r e e d o m , so t h e m a t r i c e s i n e q u a t i o n ( 2 . 5 ) a r e o f o r d e r t e n b y t e n . 37 2 . 4 - D e v e l o p m e n t o f t h e G l o b a l E q u a t i o n o f M o t i o n o f t h e F l o a t i n g Beam The s h i p i s d i v i d e d i n t o n s t a t i o n s , and e a c h s t a t i o n i s c o n s i d e r e d as a beam e l e m e n t . P h y s i c a l c h a r a c t e r i s t i c s o f e a c h s t a t i o n a r e t a k e n as t h o s e o f t h e beam e l e m e n t s as i n p u t d a t a . T h e s e p r o p e r t i e s a r e , t h e a v e r a g e v a l u e s o f t h e c r o s s - s e c t i o n a r e a , b r e a d t h , d e p t h , and A A moment o f i n e r t i a w i t h r e s p e c t t o t h e b o t h v e r t i c a l ( y ) and l a t e r a l ( z ) a x i s , and t h e mass and l e n g t h o f e a c h e l e m e n t . The mass and s t i f f n e s s m a t r i c e s f o r a l l o f t h e e l e m e n t s a r e t h e n d e v e l o p e d . N o d a l v a l u e s o f t h e e x t e r n a l f o r c e on e a c h e l e m e n t a r e a l s o c a l c u l a t e d f r o m t h e s e p a r a m e t e r s , and t h e g l o b a l e q u a t i o n o f m o t i o n o f t h e s h i p i s g e n e r a t e d . E q u a t i o n o f m o t i o n o f t h e w h o l e beam i s d e v e l o p e d i n t h e g l o b a l c o o r d i n a t e s y s t e m ( X , Y , Z ) , b y a s s e m b l i n g t h e d i s c r e t e e l e m e n t s . D i s p l a c e m e n t o f e a c h node i s m e a s u r e d w i t h r e s p e c t t o t h e g l o b a l c o o r d i n a t e s y s t e m . A s s e m b l a g e i s b a s e d on t h e r e q u i r e m e n t t h a t , t h e d i s p l a c e m e n t a t a node s h a r e d b y two e l e m e n t s must be t h e same f o r b o t h o f t h o s e e l e m e n t s . I n page 300 o f [ 4 4 ] , i t i s shown t h a t b y means o f a s e r i e s o f t r a n s f e r m a t r i c e s [T] , t h e c o o r d i n a t e s o f t h e e l e m e n t s a r e t r a n s f e r r e d 1 s f r o m l o c a l t o t h e g l o b a l s y s t e m o f r e f e r e n c e . S p e c i f i c a l l y t h e r e l a t i o n b e t w e e n t h e l o c a l c o o r d i n a t e o f t h e s—th e l e m e n t and t h e g l o b a l c o o r d i n a t e i s : {U ) s = [ T ] g {U} where {U} i s t h e c o o r d i n a t e o f t h e s - t h e l e m e n t , {U} i s t h e g l o b a l c o o r d i n a t e o f t h e n o d e s , and [T] i s a r e c t a n g u l a r m a t r i x c o r r e s p o n d i n g t o t h e s - t h e l e m e n t . The e l e m e n t s o f e v e r y row o f [T] a r e a l l z e r o , 38 w i t h e x c e p t i o n o f one e l e m e n t i n e a c h row w h i c h i s e q u a l t o u n i t y . The p o s i t i o n o f t h e u n i t e l e m e n t i n e v e r y row o f [T] i s s u c h t h a t t h e above r e l a t i o n r e p r e s e n t s an i d e n t i t y . By means o f s u c h t r a n s f e r m a t r i c e s g l o b a l m a s s , ' s t i f f n e s s and damping m a t r i c e s o f t h e w h o l e beam a r e d e v e l o p e d a s : n T [M]= I [ T ] s [ m ] s [ T ] s s= 1 [K] - I [ T ] g [ k ] s [ T ] s s = i [C] = I [ T ] g [ C ] s [ T ] s s = i The n o d a l v a l u e s o f t h e a p p l i e d l o a d on t h e w h o l e beam w o u l d b e : S = 1 T where [T] i s t h e t r a n s p o s e o f t h e t r a n s f e r m a t r i x [T] . s s The a s s e m b l e d g l o b a l mass m a t r i x i s shown s c h e m a t i c a l l y b e l o w . I t i s n o t i c e d t h a t t h e e l e m e n t mass m a t r i x [m ] and [m 1 c o r r e s p o n d t o i i+1 r two a d j a c e n t e l e m e n t s I and l+l o v e r l a p w i t h a (5x5) p o r t i o n o f e a c h m a t r i x w h i c h r e l a t e s t o t h e same n o d e . G l o b a l s t i f f n e s s m a t r i c e s a r e a s s e m b l e d s i m i l a r l y f r o m t h e e l e m e n t s t i f f n e s s m a t r i c e s . 39 1 o 1 o [M] = m The a p p l i e d l o a d s a r e a s s e m b l e d i n t h e same way and t h e g l o b a l f o r c e v e c t o r i s o f t h e f o r m shown b e l o w . The e l e m e n t n o d a l f o r c e s o v e r l a p w i t h f i v e e l e m e n t s t o make t h e g e n e r a l n o d a l f o r c e s . f ri. ( Q ) f \ • A v / q I n t h e g l o b a l c o o r d i n a t e , t h e e q u a t i o n o f m o t i o n o f t h e f l o a t i n g beam w h i c h i s d e d u c e d f r o m t h e e q u a t i o n o f m o t i o n o f a l l o f i t s e l e m e n t s i s : [M]{U} + [C] {U} +[K] (U ) = {Q } ( 2 . 6 ) 40 O b v i o u s l y t h e g l o b a l damping m a t r i x [C] w o u l d be p r o p o r t i o n a l t o t h e g l o b a l mass and s t i f f n e s s m a t r i c e s , as t h i s p r o p e r t y h o l d s f o r e a c h e l e m e n t . O r d e r o f m a t r i c e s i n e q u a t i o n ( 2 . 6 ) i s ( 5 ( n + l ) b y 5 ( n + l ) ) where n i s t h e number o f d i s c r e t e e l e m e n t s a l o n g t h e beam, and v e c t o r s {U} and {Q} a r e o f o r d e r 5 ( n + l ) . E v e r y d i s c r e t e e l e m e n t i s c o n s i d e r e d as a f r e e - f r e e beam. An i m p l i c a t i o n i s t h a t t h e c o m p l e t e beam i s c a p a b l e o f r i g i d - b o d y m o t i o n as w e l l as t h e e l a s t i c m o t i o n . 41 3. ICE BREAKING PROCEDURE 3 . 1 - I n t r o d u c t i o n I c e b r e a k e r s a r e o f t e n u s e d t o a s s i s t o t h e r v e s s e l s t h r o u g h i c e c o v e r e d w a t e r s . I n t h i s r o l e t h e y open a c h a n n e l t h r o u g h t h e i c e t h r o u g h w h i c h t h e o t h e r v e s s e l s f o l l o w . These v e s s e l s may t h e m s e l v e s be i c e s t r e n g t h e n e d and c a p a b l e o f t r a n s f e r r i n g c a r g o i n c o n d i t i o n s w h i c h w o u l d be h a z a r d o u s t o s t a n d a r d s h i p s . O p e r a t i o n o f a n i c e b r e a k e r t h r o u g h i c e may be c a t e g o r i z e d b y two d i f f e r e n t modes: t h e c o n t i n u o u s i c e b r e a k i n g and t h e ramming mode. I n t h e f o l l o w i n g s e c t i o n s b o t h o p e r a t i o n a l modes a r e c o n s i d e r e d , f i r s t l y c o n t i n u o u s i c e b r e a k i n g and s e c o n d l y ramming o f h e a v y i c e r i d g e s . I n e a c h c a s e an a n a l y s i s i s p r e s e n t e d . The f o l l o w i n g a s s u m p t i o n s a r e made f o r c o n t i n u o u s i c e b r e a k i n g mode. i ) I m p a c t b e t w e e n t h e s h i p and i c e i s i n e l a s t i c , t h a t i s , t h e s h i p a l w a y s m a i n t a i n s c o n t a c t w i t h t h e i c e . i i ) ^ C u r v a t u r e o v e r t h e c o n t a c t a r e a i s n e g l e c t e d , i i i ) F l e x u r a l s t r e n g t h and c r u s h i n g s t r e n g t h o f t h e i c e i s assumed c o n s t a n t . 3 .2— D e f i n i t i o n o f C o o r d i n a t e s The g l o b a l c o o r d i n a t e s y s t e m O-XYZ i s t h e r e f e r e n c e f rame f i x e d w i t h r e s p e c t t o t h e e a r t h . X Z - p l a n e i s t h e s t i l l w a t e r p l a n e . When t h e s h i p makes f i r s t c o n t a c t w i t h t h e i c e , a t t= 0 , t h e s h i p i s a l o n g t h e 42 X - a x i s , the s t ern i s at the o r i g i n and the bow i s i n the p o s i t i v e X reg ion . Y - a x i s i s v e r t i c a l p o s i t i v e upward and Z - a x i s i s normal to the XY-plane . The axis c o n f i g u r a t i o n i s shown i n F i g . (3 .1 ) . Y F i g . 3 .1- Coordinate Axis and the Ice-Braking Ship. 3.3- Continuous Ice Breaking Mode We cons ider the case i n which the i ce breaking sh ip proceeds through a l e v e l i ce f i e l d with constant speed. In t h i s ana lys i s the model introduced by Lewis and Edward. [32] for f a i l u r e of i ce sheet i s s e l ec t ed . According to t h i s model, as the ship proceeds in to the i c e , i t exerts an i n c r e a s i n g force on the i ce sheet. The exerted force crushes the i ce i n the immediate v i c i n i t y of the bow, whi le i t generates long r a d i a l cracks s t a r t i n g from the contact p o i n t . Advance of the ship into the i ce causes the generated cracks to propagate and subsequently a number of wedges are broken out of the ice sheet. Each i ce wedge i s broken away from the sheet by the c i r c u m f e r e n t i a l cracks generated on 43 i t . The g e n e r a t e d f o r c e d u r i n g t h e i c e c r u s h i n g i s p r o p o r t i o n a l t o t h e a r e a o f c o n t a c t b e t w e e n t h e s h i p and t h e i c e . P r o g r e s s o f t h e s h i p t h r o u g h t h e i c e i n c r e a s e s t h i s c o n t a c t a r e a w h i c h i n t u r n c a u s e s an i n c r e a s e i n t h e m a g n i t u d e o f t h e c o n t a c t f o r c e s . T h r o u g h o u t t h e c o n t i n u o u s i c e b r e a k i n g p r o c e s s t h e v e r t i c a l component o f t h e c o n t a c t f o r c e c a u s e s f l e x u r e o f e a c h o f t h e wedges . The m a g n i t u d e o f t h e b e n d i n g moment i n e a c h i c e wedge i n c r e a s e s w i t h t h e g r o w t h o f t h e c o n t a c t f o r c e . F i n a l l y a t a c e r t a i n l i m i t o f t h e l o a d , t h e i c e wedges c o l l a p s e due t o t h e i n d u c e d b e n d i n g moment, and a s e r i e s o f c i r c u m f e r e n t i a l c r a c k s w o u l d be g e n e r a t e d a l o n g t h e wedges [ 3 2 ] . The l e n g t h o f t h e b r o k e n wedges i n t h i s s t a g e depends on t h e t h i c k n e s s o f * i c e s h e e t as w e l l as t h e s t r e n g t h o f t h e i c e and g e o m e t r y o f c o n t a c t b e t w e e n t h e i c e and t h e i c e b r e a k e r . When t h e i c e s h e e t s c o l l a p s e an o p e n w a t e r s p a c e i s g e n e r a t e d i n f r o n t o f t h e s h i p . I d e a l l y t h e s h i p w o u l d h a v e a f r e e l a t e r a l m o t i o n w h i l e i t i s r u n n i n g i n t h i s open a r e a b u t i n p r a c t i c e t h e r e i s o f t e n c o n s i d e r a b l e r e s i s t a n c e due t o j amming o f t h e s h i p i n t h e c h a n n e l b y b r o k e n i c e b i t s . We do n o t c o n s i d e r t h i s p r o b l e m h e r e b u t assume t h a t t h e s h i p r u n s t h i s d i s t a n c e w i t h c o n s t a n t s p e e d . A t t h e end o f t h e o p e n i n g , when i t r e a c h e s t h e u n b r o k e n i c e t h e l o a d i n g c y c l e i s r e p e a t e d . I n t h e f o l l o w i n g s e c t i o n an a n a l y s i s f o r t h i s p r o c e s s i s i n t r o d u c e d . 3 . 3 . 1 - A n a l y s i s o f t h e C o n t a c t F o r c e The c o n t a c t f o r c e i s a p p l i e d o v e r a l o c a l i z e d a r e a n e a r t h e bow. I n 44 t h i s a n a l y s i s t h e c o n t a c t f o r c e i s t r e a t e d as a c o n c e n t r a t e d l o a d . The p o i n t o f a p p l i c a t i o n o f t h e c o n t a c t f o r c e moves s l i g h t l y a f t d u r i n g t h e c o n t a c t p e r i o d . The c o n t a c t f o r c e m a i n l y depends on t h e s t r e n g t h o f t h e i c e and c o n t a c t a r e a b e t w e e n t h e s h i p and t h e i c e s h e e t . K o r z h a v i n [29] i n t r o d u c e d t h e f o l l o w i n g r e l a t i o n f o r t h e m a g n i t u d e o f t h e c r u s h i n g f o r c e . F = C C a A 1 2 C w h e r e , F i s t h e c r u s h i n g f o r c e n o r m a l t o t h e bow. A i s t h e g e o m e t r i c c o n t a c t a r e a b e t w e e n t h e bow and t h e i c e c ° s h e e t , c a l c u l a t i o n o f t h e c o n t a c t a r e a i s r e p r e s e n t e d i n S e c t i o n ( 3 . 5 ) . C i s a r e d u c t i o n c o e f f i c i e n t w h i c h r e d u c e s t h e g e o m e t r i c c o n t a c t l b a r e a t o t h e e f f e c t i v e c o n t a c t a r e a . K o r z h a v i n [ 2 9 ] , showed t h a t t h e m a g n i t u d e o f t h i s c o e f f i c i e n t i s b e t w e e n 0 . 4 t o 0 . 6 . C z i s t h e shape f a c t o r d e p e n d e n t s upon t h e shape o f t h e bow a r e a . K o r z h a v i n [29] recommended t h e c o r r e s p o n d i n g v a l u e s f o r . d i f f e r e n t s h a p e s o f t h e bow. a i s t h e c r a s h i n g s t r e n g t h o f t h e i c e . K o r z h a v i n [29] p r o p o s e d t h a t t h e i c e s t r e n g t h depends on t h e s t r a i n r a t e i n t h e c r u s h i n g p r o c e s s and he c o n c l u d e d t h a t t h e s t r e n g t h o f i c e v a r i e s w i t h t h e i n d e n t e r v e l o c i t y a s : -1/ 3 a = a. i s w h i l e 45 a. i s t h e c r a s h i n g s t r e n g t h o f t h e i c e when i t i s b r o k e n by a s h i p r u n n i n g w i t h t h e v e l o c i t y o f 1 m / s e c . is i s t h e v e l o c i t y o f t h e s h i p . The c r u s h i n g f o r c e n o r m a l t o t h e bow c o u l d be r e p r e s e n t e d i n v e c t o r n o t a t i o n a s : F = - F • n n A w h e r e , n i s t h e u n i t v e c t o r n o r m a l t o t h e c o n t a c t a r e a and i n t e r m o f i t s ^ d i r e c t i o n a l c o s i n e i s : n = n i + n j + n k x y z The f r i c t i o n f o r c e F i s t a n g e n t t o t h e h u l l i n t h e e f f e c t i v e c o n t a c t a r e a , and i s r e l a t e d t o t h e c o n t a c t f o r c e b y t h e c o e f f i c i e n t o f f r i c t i o n a s : A F = - n F t f Where , p i s t h e c o e f f i c i e n t o f f r i c t i o n b e t w e e n i c e and t h e bow. A t i s a u n i t v e c t o r , t a n g e n t t o t h e c o n t a c t a r e a and i n t h e d i r e c t i o n o f t h e m o t i o n o f t h e bow w i t h r e s p e c t t o t h e i c e . T h i s v e c t o r i s r e p r e s e n t e d a s : A A A A t = t i + t i + t k x y z where t , t , and t t h e a r e t h e d i r e c t i o n c o s i n e s o f t h e t a n g e n t u n i t x y z A A A A v e c t o r t . R e l a t i o n b e t w e e n t , n and V t h e v e l o c i t y v e c t o r o f t h e bow i s d e r i v e d i n s e c t i o n ( 3 . 6 ) . The t o t a l c o n t a c t f o r c e F on t h e s h i p w o u l d be t h e summat ion o f t h e 46 n o r m a l and f r i c t i o n a l f o r c e w h i c h i s shown a s : A A A A A A F = - F (n i +n j + n k) - n F ( t i + t J + t k) x y z x y z The v e r t i c a l component o f t h e c r u s h i n g f o r c e , F =- (n + \x t ) F y y y i s n o r m a l t o t h e i c e s h e e t . Whenever t h e m a g n i t u d e o f t h i s f o r c e e x c e e d s t h e maximum a l l o w a b l e v a l u e i t c a u s e s t h e f l e x u r e f a i l u r e o f t h e i c e . T h e r e f o r e i t i s n e e d e d t o compare t h e m a g n i t u d e o f t h e v e r t i c a l f o r c e , F y , w i t h t h e maximum a l l o w a b l e f o r c e f o r t h e s p e c i f i e d i c e f i e l d . T h i s c o m p a r i s o n i s done a t e a c h i n s t a n t d u r i n g t h e i n t e g r a t i o n p r o c e d u r e . The u l t i m a t e l o a d t h a t a s p e c i f i e d i c e s h e e t c a n c a r r y b e f o r e i t f a i l s b e c a u s e o f f l e x u r e , depends upon many v a r i a b l e s s u c h as i c e s t r e n g t h , l o a d i n g a r e a , and i c e t h i c k n e s s . A f o r m u l a p r o p o s e d b y L e w i s and Edward [32] i s : „ . . , Go • a t h 2 t . 0 . / o n s F m a x = 1 A 5 ~ 0 3 — t a n ( — > ( 3 ' 1 } w h e r e , 0^ i s t h e t o t a l a n g l e o f t h e i c e s h e e t , w h i c h i s e q u a l t o 180 d e g . f o r a s e m i - i n f i n i t e s h e e t . 0 i s t h e wedge a n g l e . h i s t h e t h i c k n e s s o f t h e i c e s h e e t . L e w i s and Edward [32] , a l s o m e n t i o n t h a t f u l l s c a l e o b s e r v a t i o n s h a v e p r e d i c t e d v a l u e s o f 0 q = 270 d e g . and 0 = 6 7 . 5 d e g . f o r s e a i c e . The c r u s h i n g f o r c e i s d i s t r i b u t e d a l o n g t h e s tem on t h e w a t e r l e v e l as shown i n F i g . ( 3 . 2 ) . The p o s i t i o n o f t h e p o i n t o f a p p l i c a t i o n o f t h e r e s u l t a n t f o r c e c o u l d v a r y on t h e s tem d e p e n d i n g on t h e c o n d i t i o n and 47 thickness of the ice sheet. In i d e a l c o n d i t i o n s , when the ice has uniform s trength and thickness,, the B r e s u l t a n t force could be app l i ed on the bow along the axis of the ship F i g . 3 .2 - Distribution of the and i n the symmetric plane of the Ice Load on the Stem. s h i p . In a r e a l i c e f i e l d the i ce thickness i s not constant and ne i ther i s i ce s t rength , so the p o s i t i o n of the a p p l i c a t i o n of the r e s u l t a n t force might not be i n the plane of symmetry of the s h i p . V a r i a t i o n of the p o s i t i o n of t h i s po int on the bow of the ship has e f f e c t on the generated torque and r o l l i n g motion and s t a b i l i t y of the s h i p . The f o l l o w i n g model i s introduced to consider the e f f e c t of change of the p o s i t i o n of t h i s p o i n t . I t i s assumed that the thickness of the i ce sheet versus the beam of the ship v a r i e s i n a way that the Z-coordinate of the contact po int could be w r i t t e n as: Z -= Z S in (27rt/r) o ' Where, Z , i s the maximum value o f Z . o T i s the p e r i o d of v a r i a t i o n which depends on the v e l o c i t y of the ship as w e l l as i t s beam. The p e r i o d r could be the time requ ired for the ship to t r a v e l a d is tance equal to the beam of the sh ip . That i s , r - Br / <s Then the l a t e r a l p o s i t i o n of the contact po int on the stem i s : 48 — 2 7 T U Z - Z S i n t ) 0 B r I t i s assumed t h a t t h e maximum v a r i a t i o n o f Z q i s e q u a l t o t h e d i s t a n c e o f t h e c e n t r o i d o f t h e q u a r t e r c i r c l e w i t h r a d i u s o f t h e h a l f o f t h e beam . T h a t i s , - = 4 _ Br 2Br 37T 2 37T T h e r e f o r e , „ 2Br . 2 7 T W Z = —=— S i n (—5— t ) o 3?r Br T h e r e i s e q u a l p o s s i b i l i t y f o r t h i s p o s i t i o n t o be on e i t h e r t h e r i g h t o r l e f t s i d e o f t h e s h i p . T h i s c o u l d be c o n s i d e r e d i f , i n e a c h s u b s e q u e n t c r u s h i n g p e r i o d t h e p o s i t i o n o f t h e c o n t a c t f o r c e s h i f t s f r o m one s i d e o f t h e c e n t e r l i n e o f t h e s h i p t o t h e o t h e r s i d e . T h i s change w o u l d be r e g a r d e d t h r o u g h a f a c t o r o f ( - 1 ) J , where j i s t h e number o f i c e c r u s h i n g p e r i o d s . T h e r e f o r e t h e l a t e r a l p o s i t i o n o f t h e c o n t a c t p o i n t c o u l d be shown a s : z » - < - » s i " E> < 3 ' 2 > The c r u s h i n g f o r c e and t h e p o i n t o f i t s a p p l i c a t i o n on t h e s h i p i s d e t e r m i n e d a c c o r d i n g t o t h e above p r o c e d u r e . I n o r d e r t o s o l v e t h e n o d a l e q u a t i o n o f m o t i o n o f t h e s h i p , t h e n o d a l v a l u e s o f t h e c o n t a c t f o r c e a r e d e t e r m i n e d a c c o r d i n g t o t h e method i n t r o d u c e d i n A p p e n d i x ' A ' . The m a g n i t u d e o f Z q w h i c h i s c a l c u l a t e d a c c o r d i n g t o t h e E q u . ( 3 . 2 ) i n c o n j u n c t i o n w i t h Y q e q u a l t o z e r o ( as t h e f o r c e i s a p p l i e d on t h e w a t e r l e v e l ) a r e u s e d t o c a l c u l a t e t h e n o d a l v a l u e s o f t h e c o n t a c t f o r c e . 49 3 . 3 . 2 - Equations of Motion of the Ship The beam m o d e l f o r t h e s h i p w h i c h i s i n t r o d u c e d i n C h a p t e r 2 , i s u s e d t o a n a l y z e t h e i c e b r e a k i n g p r o c e d u r e i n t h e c o n t i n u o u s mode. The i c e b r e a k i n g s h i p i s d i v i d e d i n t o n s t a t i o n s . E a c h s t a t i o n i s c o n s i d e r e d as a beam e l e m e n t w i t h t w e l v e d e g r e e s o f f r e e d o m . N o r m a l l y , d u r i n g t h e c o n t i n u o u s i c e b r e a k i n g mode, t h e s h i p p r o c e e d s w i t h c o n s t a n t s p e e d u n d e r t h e e f f e c t o f t h e i c e p r e s s u r e and t h e t h r u s t f o r c e o f t h e p r o p e l l e r s . H e n c e , t h e m o d e l beam i s assumed t o be m o v i n g w i t h c o n s t a n t s p e e d a l o n g t h e l o n g i t u d i n a l a x i s . E q u a t i o n s o f t h e l a t e r a l m o t i o n o f t h e s h i p as i s shown i n E q u . ( 2 . 6 ) i s : [M ] {U }+ [C ]{U } + [K ] {U } = {Q } ( 3 . 3 ) The mass m a t r i x [M] , t h e s t i f f n e s s m a t r i x [ K ] , and t h e p r o p o r t i o n a l damping m a t r i x [C] a r e d e v e l o p e d i n C h a p t e r 2 . A method f o r d e t e r m i n a t i o n o f t h e damping m a t r i c [C] i s r e p r e s e n t e d i n S e c t i o n 3 . 8 . The i n i t i a l c o n d i t i o n s f o r t h e e q u a t i o n s o f m o t i o n a r e t h e d i s p l a c e m e n t v e c t o r {U } and t h e v e l o c i t y v e c t o r {U } w h i c h a r e e q u a l t o z e r o i m m e d i a t e l y b e f o r e t h e f i r s t c o n t a c t o f t h e s h i p w i t h t h e i c e , a t t - 0 . I n o r d e r t o f i n d t h e m a g n i t u d e o f t h e c r u s h i n g f o r c e on t h e bow and t h e maximum b e n d i n g moment i n d u c e d a l o n g t h e s h i p , t h e e q u a t i o n s o f m o t i o n o f t h e s h i p a r e t o be s o l v e d . A s l o n g as t h e m a g n i t u d e o f t he v e r t i c a l component o f t h e c o n t a c t f o r c e i s l e s s t h a n t h e maximum a l l o w a b l e v a l u e r e p r e s e n t e d i n E q u . ( 3 . 1 ) , t h e s h i p w o u l d be m o v i n g u n d e r t h e e f f e c t o f t h e i c e f o r c e . D u r i n g t h i s p e r i o d t h e c o n t a c t p o i n t 50 w o u l d be m o v i n g a l o n g t h e bow w i t h a r e l a t i v e s p e e d e q u a l t o t h a t o f t h e s h i p . D u r i n g t h e i n t e g r a t i o n p r o c e d u r e f o r e a c h t i m e i n c r e m e n t A t , t h e p o s i t i o n o f t h e c o n t a c t p o i n t on t h e s h i p i s d e t e r m i n e d , t h e s t a t i o n a l o n g t h e s h i p ( o r t h e beam e l e m e n t on t h e mode l ) w h i c h i s i n c o n t a c t w i t h t h e i c e i s i n d i c a t e d and t h e n o d a l v a l u e s o f t h e c o n t a c t f o r c e s a r e d e v e l o p e d . E q u . ( 3 . 3 ) i s a s e t o f c o u p l e d e q u a t i o n s , t h e s e e q u a t i o n s a r e u n c o u p l e d when t r a n s f e r r e d t o t h e n o r m a l c o o r d i n a t e s }. The p r o c e d u r e r e p r e s e n t e d i n a p p e n d i x ' B ' i s f o l l o w e d t o u n c o u p l e t h e e q u a t i o n s a s : £. + (a + p to.) £.+ w 2 £. — ( 3 . 4 ) l E a c h e q u a t i o n o f t h e g r o u p ( 3 . 4 ) i s s o l v e d b y t h e f i n i t e d i f f e r e n c e method f o r a t i m e s t e p A t . A t t h e end o f e a c h t i m e i n t e r v a l t h e new p o s i t i o n o f t h e n o d e s , v e l o c i t y o f e a c h node and t h e p o s i t i o n o f t h e s h i p w i t h r e s p e c t t o t h e i c e s h e e t , a r e d e t e r m i n e d , and t h e g e n e r a t e d c o n t a c t f o r c e i s e s t i m a t e d . The v e r t i c a l component o f t h e c o n t a c t f o r c e i s c o m p a r e d w i t h t h e u l t i m a t e s t r e n g t h o f t h e i c e s h e e t . I n t h e c a s e t h a t t h e f o r c e i s l e s s t h a n t h e l i m i t i n t r o d u c e d i n E q u . ( 3 . 1 ) t h e i n t e g r a t i o n p r o c e s s c o n t i n u e s . A n d when t h i s f o r c e e x c e e d s t h e maximum f o r c e i n d i c a t e d by E q u . ( 3 . 1 ) t h e i c e s h e e t i s c o n s i d e r e d b r o k e n . A c c o r d i n g t o t h e s e m i - e m p i r i c a l d e r i v a t i o n w h i c h i s done by K o r z h a v i n [ 4 7 ] , t h e b e n d i n g f a i l u r e happens a t a d i s t a n c e o f 3 t o 6 t i m e s t h e t h i c k n e s s o f t h e i c e s h e e t . T h i s b e n d i n g f a i l u r e o f t h e i c e s h e e t , opens a s h o r t c h a n n e l i n f r o n t o f t h e s h i p . 51 W h i l e t r a n s i t i n g t h i s open w a t e r t h e E q u . ( 3 . 4 ) a r e homogeneous c o r r e s p o n d i n g t o a f r e e l y damped v i b r a t i n g s y s t e m . When t h e s h i p h a s t r a v e r s e d t h e open c h a n n e l c o n t a c t w i t h t h e i c e i s r e - e s t a b l i s h e d and t h e l o a d i n g c y c l e i s r e p e a t e d . When t h e s h i p i s i n c o n t a c t w i t h i c e t h e l a t e r a l c o n t a c t f o r c e F L i s : A A F - - F [ ( n +/it ) j + (n +ftt )k] L y y z z The m a g n i t u d e o f t h i s f o r c e i s d e t e r m i n e d d u r i n g t h e i n t e g r a t i o n p r o c e s s . R e s o n a n c e r e s p o n s e c o u l d be e x p e c t e d i f t h e f r e q u e n c y o f v a r i a t i o n o f t h e m a g n i t u d e o f t h i s l o a d i s c l o s e t o t h a t o f t h e s h i p s t r u c t u r e . The v e l o c i t y o f t h e s h i p i n a s p e c i f i e d i c e f i e l d i s an i n d e p e n d e n t v a r i a b l e t h a t c a n c o n t r o l t h e f r e q u e n c y o f t h e i c e l o a d on t h e h u l l . The maximum i n d u c e d b e n d i n g moment a l o n g t h e s h i p c a n be e v a l u a t e d t h r o u g h - o u t t h e i c e b r e a k i n g p r o c e d u r e . F o r t h i s c a l c u l a t i o n a t e a c h t i m e s t e p A t , m a g n i t u d e o f t h e b e n d i n g moment i n e a c h beam e l e m e n t i s d e t e r m i n e d t h e n t h e s e v a l u e s f o r a l l o f t h e e l e m e n t s a r e compared t o f i n d t h e maximum b e n d i n g moment a l o n g t h e beam. The method o f c a l c u l a t i n g t h e maximum b e n d i n g moment i n e a c h beam e l e m e n t i s r e p r e s e n t e d i n s e c t i o n ( 3 . 7 ) . 52 3 . 4 - Ramming o f L a r g e P r e s s u r e R i d g e s A n i c e - b r e a k i n g s h i p , i n a d d i t i o n t o t h e n o r m a l i c e b r e a k i n g p r o c e d u r e s j u s t d i s c u s s e d , may be r e q u i r e d t o b r e a k up l a r g e p r e s s u r e r i d g e s many t i m e s t h i c k e r t h a n t h a t f o r w h i c h i t i s c l a s s i f i e d . I n s u c h c a s e s t h e s h i p a t t e m p t s t o b r e a k t h e r i d g e b y ramming t h e i c e a t s p e e d , t o p r o d u c e u n u s u a l l y h i g h c o n t a c t f o r c e s on t h e i c e . I n t h i s c a s e t h e m a g n i t u d e o f t h e c o n t a c t f o r c e and i n d u c e d b e n d i n g moment a l o n g t h e s h i p , a r e v e r y much h i g h e r t h a n t h e c o r r e s p o n d i n g v a l u e s g e n e r a t e d i n t h e r e g u l a r i c e - b r e a k i n g p r o c e d u r e . F o r m u l a t i o n o f t h e ramming mode i n t h i s s t u d y , i s done w i t h two more a s s u m p t i o n s as f o l l o w e d : i ) The " i c e r i d g e i s c o n s i d e r e d f i x e d and c o m p l e t e l y r i g i d . i i ) I m m e d i a t e l y b e f o r e c o l l i s i o n t h e l a t e r a l m o t i o n o f t h e s h i p i s n e g l i g i b l e . 3 . 4 . 1 — A n a l y s i s o f t h e Ramming P r o c e d u r e I m m e d i a t e l y b e f o r e c o l l i s i o n , t h e v e l o c i t y o f t h e s h i p t o w a r d s t h e ' i c e r i d g e i s 4 S q . I m m e d i a t e l y a f t e r c o l l i s i o n t h e v e l o c i t y d e c r e a s e s t o is and t h e h u l l o b t a i n s a l a t e r a l v e l o c i t y . D u r i n g t h e i n e l a s t i c c o l l i s i o n some p a r t o f t h e k i n e t i c e n e r g y i s l o s t . A f t e r t h e f i r s t i m p a c t t h e bow o f t h e s h i p s l i d e s upwards on t h e i c e ( b e a c h i n g m o t i o n ) . • D u r i n g t h e b e a c h i n g p e r i o d t h e s h i p d e c e l e r a t e s and f i n a l l y comes t o t h e r e s t b e f o r e s l i d i n g b a c k o f f t h e i c e . The m a i n p a r a m e t e r s w h i c h a f f e c t t h e m o t i o n o f t h e i c e b r e a k e r d u r i n g t h e b e a c h i n g p e r i o d a r e m a i n l y t h e s t r u c t u r a l s t i f f n e s s o f t h e 53 s h i p , the i n i t i a l v e l o c i t y of the ship immediately before c o l l i s i o n and the bow angle . We consider the two dimensional problem of the ship ramming symmetrical ly into an i ce r i d g e . A two dimensional beam model i s shown i n F i g . ( 3 . 3 ) and the equation of v e r t i c a l motion of the beam i s : [M ](U ) + [C ]{U } + [K ](U > - { § } . . ( 3 . 5 ) where, [M ], [C ], and [K ] are matrices of order 2(n+l) x 2(n+l) and {U },{U }, and (u } are vectors of order 2(n+l) , while n i s the number o f the beam elements considered along the model. 4)0 j)©-4) ®; u u e 0® 4) 6 ( 2 n + 2 ) F i g . 3 .3- A Two Dimensional Beam Model. In order to study the motion of the ship i n the ramming per iod through the Equ. ( 3 . 5 ) i n a d d i t i o n to the s t r u c t u r a l parameters of the s h i p , the i n i t i a l condi t ions of the motion as w e l l as the condi t ions of the force vec tor {Q} are needed to be known. In t h i s study each ram i s considered to be composed of two consequent stages, f i r s t , an impulse due to the c o l l i s i o n exc i tes the l a t e r a l motion of the s h i p , then a beaching motion br ings the ship to a stop wi th a cons iderable t r i m . Therefore , the system state immediately a f t e r impulse defines the i n i t i a l condi t ions for the beaching motion. We 54 now examine t h e s e two s t a g e s i n o r d e r . 3 . 4 . 2 - R e s p o n s e o f t h e S h i p S t r u c t u r e Due t o t h e I n i t i a l I m p u l s e G e n e r a t i o n o f w a t e r waves i s an i m p o r t a n t s o u r c e o f damping o f f l o a t i n g o b j e c t s . The i c e b r e a k e r s h i p w o u l d n o t have any l a t e r a l m o t i o n i n t h e v e r y s h o r t d u r a t i o n o f t h e f i r s t i m p u l s e . T h e r e f o r e , no w a t e r wave c o u l d be g e n e r a t e d b y t h e h u l l o f t h e s h i p . O b v i o u s l y t h e m o t i o n w o u l d n o t be damped d u r i n g t h i s s t a g e and t h e damping c o e f f i c i e n t does n o t a p p e a r i n t h e e q u a t i o n o f m o t i o n o f t h e s h i p . I n t h i s p e r i o d t h e e q u a t i o n o f m o t i o n i s : [M ]{U } + [K ]{U } = {Q } ( 3 . 6 ) The e f f e c t o f t h e f i r s t c o l l i s i o n o f t h e i c e b r e a k e r and t h e i c e r i d g e c o u l d be m o d e l e d b y an i m p u l s e l o a d on t h e s h i p . The v e r t i c a l component o f t h e t h e i m p u l s e l o a d i s shown as A - S ( t ) , where § ( t ) i s t h e u n i t impulse f u n c t i o n and A , i s t h e unknown m a g n i t u d e o f t h e i m p u l s e w h i c h s h o u l d be i n v e s t i g a t e d i n t h i s s t a g e . When t h e N - t h node o f t h e s h i p makes f i r s t c o n t a c t w i t h t h e i c e r i d g e , t h e f o r c e v e c t o r {Q} i s : IQ } = {d } A - f i ( t ) where {d} i s a v e c t o r o f d i m e n s i o n 2 ( n + l ) , a l l e l e m e n t s o f {d} a r e z e r o b u t t h e ( 2 N + l ) - t h e l e m e n t w h i c h i s e q u a l t o u n i t y . T h e r e f o r e t h e e q u a t i o n o f m o t i o n ( 3 . 6 ) c o u l d be w r i t t e n a s : A [M ] {U } + [K ]{U } = {d } A - S ( t ) ( 3 . 7 ) 55 The s e t o f c o u p l e d E q u . ( 3 . 7 ) i s t r a n s f e r r e d t o t h e n o r m a l c o o r d i n a t e s {ip }, f o l l o w i n g t h e same method i n t r o d u c e d i n A p p e n d i x ' B ' . I n t h e n o r m a l c o o r d i n a t e s t h e u n c o u p l e d e q u a t i o n s a r e : w h e r e , [M ] p W ) + [K ]p{V> > = (d } A - S ( t ) ( 3 . 8 ) [M ] p = [T ] [M ] [T ] [K ] p =" [T ] [ K ] [T ] T _ (d } - [T ][d ] -1 A n d [T ] i s a m a t r i x whose co lumns a r e t h e e i g e n - v e c t o r s o f t h e [M] [ K ] . E a c h e q u a t i o n i n t h e u n c o u p l e d s e t ( 3 . 8 ) may be i n n o r m a l i z e d f o r m a s : A d I + w 2 £ = — 5 ( t ) ( 3 . 9 ) i i i m k. where , w = — -i m S o l u t i o n o f t h e E q u . ( 3 . 9 ) i s : = 1 , A d i i w m i i . 5 ( t ) S i n oo. ( t - t ) d t = S i n w t i to m i i i The n o d a l v e l o c i t y , u s i n g n o r m a l c o - o r d i n a t e s i s : i A d i t = Cos co t i m i The v e l o c i t y o f t h e nodes a t t=0 i s 56 w h e r e , A = i m i T h e r e f o r e , t h e i n i t i a l v e l o c i t y v e c t o r i n t h e n o r m a l c o o r d i n a t e i s : 2n+2 The p h y s i c a l n o d a l v e l o c i t y i s : {U} = [T ](V> ) w h i c h c a n be r e p r e s e n t e d a s : 2n+2 ( 3 . 1 0 ) The i n e l a s t i c c o l l i s i o n o f t h e bow w i t h t h e i c e i m p l i e s t h a t t h e v e l o c i t y o f t h e bow must be i n t h e d i r e c t i o n p a r a l l e l t o t h e bow a n g l e . H e n c e : 18 = is C o t a N S u b s t i t u t i n g f o r <s f r o m E q u . ( 3 . 1 0 ) t h e above r e l a t i o n becomes : is = (A A ) C o t a ( 3 . 1 1 ) 2N + 1 where is i s t h e v e l o c i t y o f t h e s h i p i m m e d i a t e l y a f t e r c o l l i s i o n . a i s t h e bow a n g l e . The h o r i z o n t a l i m p u l s e c a u s e s a change i n t h e m a g n i t u d e o f t h e h o r i z o n t a l momentum o f t h e s h i p w h i c h c a n be shown t o b e : 57 M (1 +a ) ( A S -AS ) = 3 ( 3 . 1 2 ) s o' x M i s t h e t o t a l mass o f t h e s h i p . 3 i s t h e h o r i z o n t a l i m p u l s e on t h e s h i p . X a i s t h e added mass o f t h e s h i p i n s u r g e . s A s m e n t i o n e d e a r l i e r , added mass e f f e c t s d u r i n g i m p u l s e a r e d i f f i c u l t t o c a l c u l a t e . The v a l u e t a k e n h e r e i s t h e h a r m o n i c m o t i o n v a l u e and i s r e l a t i v e l y s m a l l , b e i n g e q u a l t o 0 . 1 . The v e r t i c a l and h o r i z o n t a l i m p u l s e a r e r e l a t e d t o e a c h o t h e r t h r o u g h t h e c o e f f i c i e n t o f f r i c t i o n and t h e bow a n g l e a s : os. , ~ , , S i n a + u Cos a 3 = -A o where A = x y Cos a - n S i n a w h i l e , n i s t h e c o e f f i c i e n t o f f r i c t i o n b e t w e e n t h e bow and t h e i c e r i d g e . 3 i s t h e v e r t i c a l i m p u l s e . y ( d e r i v a t i o n o f t h e above r e l a t i o n i s r e p r e s e n t e d i n S e c t i o n ( 3 . 8 ) ) The m a g n i t u d e o f t h e v e r t i c a l i m p u l s e 3 i s : y r°° 3 = A <5(t) d t = A y o T h e r e f o r e t h e h o r i z o n t a l i m p u l s e w o u l d b e : 3 = - A A X S u b s t i t u t i n g f o r 3 i n t o E q u . ( 3 . 1 2 ) , t h e r e l a t i o n b e t w e e n t h e change o f t h e h o r i z o n t a l momentum o f t h e s h i p and t h e v e r t i c a l i m p u l s e w o u l d b e : M (1 +a ) (AS -AS ) = - A A ( 3 . 1 3 ) s 0 M a g n i t u d e o f t h e v e l o c i t y o f t h e s h i p i m m e d i a t e l y a f t e r c o l l i s i o n , AS, i s s u b s t i t u t e d i n t o E q u . ( 3 . 1 3 ) f r o m E q u . ( 3 . 1 1 ) . The e q u a t i o n i s s o l v e d 58 f o r t h e unknown v a l u e o f t h e i m p u l s e , A a s : i s A = ( 3 . 1 4 ) A C o t a + 2N+1 M (1 + a ) E q u . ( 3 . 1 4 ) shows a l i n e a r r e l a t i o n b e t w e e n t h e i n i t i a l i m p u l s e and t h e ramming v e l o c i t y . I t a l s o shows t h a t t h e i n i t i a l i m p u l s e i s r e l a t e d t o t h e mass and bow a n g l e o f t h e s h i p as w e l l as t h e s t i f f n e s s o f t h e h u l l s t r u c t u r e ( t h r o u g h t h e p a r a m e t e r ^ 2 N + 1 ) a n d l n a l o w e r e x t e n t t o t h e c o e f f i c i e n t o f f r i c t i o n . S u b s t i t u t i n g f o r A f r o m E q u . ( 3 . 1 4 ) i n t o E q u . ( 3 . 1 0 ) , t h e l a t e r a l v e l o c i t y v e c t o r o f t h e w h o l e beam i m m e d i a t e l y a f t e r c o l l i s i o n i s d e t e r m i n e d a s : AS {U } A C o t a + 2N+1 M (1 + a ) 2n+2 ( 3 . 1 5 ) The v e l o c i t y o f t h e s h i p i m m e d i a t e l y a f t e r c o l l i s i o n i s c a l c u l a t e d b y s u b s t i t u t i n g A f r o m E q u . ( 3 . 1 4 ) i n t o E q u . ( 3 . 1 1 ) t o g i v e : AS = AS A C o t a 2N + 1 A C o t a + 2N+1 M (1 + a ) T h i s e q u a t i o n shows t h a t f o r a s p e c i f i e d i c e b r e a k e r t h e r a t i o o f AS s h i p v e l o c i t y i m m e d i a t e l y a f t e r c o l l i s i o n t o t h e ramming v e l o c i t y , AS ' i s c o n s t a n t and depends on t h e s t r u c t u r a l p r o p e r t i e s o f t h e s h i p as shown. The l a t e r a l v e l o c i t y and p o s i t i o n o f t h e s h i p a t t h e s t a r t o f b e a c h i n g s t a g e a r e r e q u i r e d as i n i t i a l c o n d i t i o n s . The i n i t i a l v e l o c i t y 59 i s i n t r o d u c e d i n E q u . ( 3 . 1 5 ) and t h e i n i t i a l p o s i t i o n o f t h e nodes a r e z e r o , b e c a u s e i n i t i a l l y t h e s h i p l i e s a l o n g t h e X - a x i s . 3 . 4 . 2 . 1 - E n e r g y L o s s I n C o l l i s i o n A n i n e l a s t i c c o l l i s i o n i s assumed b e t w e e n t h e i c e r i d g e and t h e bow o f t h e s h i p . T h i s i s i n ag reement w i t h t h e l i m i t e d f i e l d d a t a a v a i l a b l e ; t h a t r e b o u n d does n o t o c c u r . C o n s e q u e n t l y a l o s s i n k i n e t i c e n e r g y i s e x p e c t e d d u r i n g t h e i m p u l s e s t a g e . T h i s l o s s o f e n e r g y i s d e p e n d e n t upon t h e s t i f f n e s s o f t h e s h i p . H e n c e , t h e e f f e c t o f t h e s t i f f n e s s o f t h e s h i p on t h e e n e r g y l o s s due t o t h e f i r s t i m p u l s e c o u l d l e a d t o t h e p r a c t i c a l l y v a l u a b l e i n f o r m a t i o n . T h i s f a c t i s i n v e s t i g a t e d i n C h a p t e r 5 , where m o t i o n o f a ' S t a n d a r d I c e B r e a k e r ' i s s t u d i e d . The i n i t i a l k i n e t i c e n e r g y o f t h e s h i p i s : Er ~j~H ( 1 + a s K ( 3- 1 6 ) I m m e d i a t e l y a f t e r c o l l i s i o n t h e k i n e t i c e n e r g y o f t h e s h i p w o u l d be summat ion o f t h e k i n e t i c e n e r g y due t o t h e s h i p v e l o c i t y AS, and t h e l a t e r a l v e l o c i t y {U }. T h i s k i n e t i c e n e r g y c a n be w r i t t e n a s : E = E + E ( 3 . 1 7 ) 2 2 L w h e r e , i s k i n e t i c e n e r g y due t o s h i p f o r w a r d m o t i o n a s : I \- M (1 + a ) A S 2 2 2 s and E^ i s k i n e t i c e n e r g y due t o i t s l a t e r a l m o t i o n a s : 1 • T E L = ~~T {U 1 [M ] {U } T h e r e f o r e t h e k i n e t i c e n e r g y l o s s o f t h e w h o l e s h i p d u r i n g t h e f i r s t c o l l i s i o n AE i s : 60 AE = E - E l 2 3 .4 .3 - Response of the Ship Structure During the Beaching Per iod I m m e d i a t e l y a f t e r t h e f i r s t c o l l i s i o n b e t w e e n t h e i c e r i d g e and t h e s h i p , t h e bow o f t h e s h i p s t a r t s t o s l i d e upward o v e r t h e i c e . The e x e r t e d r e a c t i o n f o r c e on t h e s h i p d u r i n g t h i s s t a g e c a u s e s f l e x u r e o f t h e h u l l as w e l l as d e c e l e r a t i o n o f t h e m o t i o n u n t i l t h e s h i p comes t o a r e s t . I n t h i s s t u d y , two d i f f e r e n t a p p r o a c h e s a r e t a k e n t o a n a l y z e t h i s s t a g e o f m o t i o n . i ) I n t h e f i r s t a p p r o a c h , i t i s assumed t h a t , d u r i n g t h e b e a c h i n g p e r i o d a f i x e d p o i n t on t h e bow i s i n c o n t a c t w i t h t h e i c e . T h i s a s s u m p t i o n i s v a l i d when t h e v a r i a t i o n o f t h e p o s i t i o n o f t h e c o n t a c t p o i n t on t h e bow i s v e r y s m a l l compared t o t h e l e n g t h o f t h e s h i p . W i t h t h i s a s s u m p t i o n a f i x e d ' p o i n t on t h e beam m o d e l w o u l d be assumed t o s l i d e on an i n c l i n e d s u r f a c e w i t h t h e i n c l i n a t i o n e q u a l t o t h e bow a n g l e o f t h e s h i p . i i ) V a r i a t i o n o f p o s i t i o n o f t h e c o n t a c t p o i n t on t h e bow i s c o n s i d e r e d , w h i l e i t i s assumed f i x e d w i t h r e s p e c t t o t h e i c e r i d g e . E a c h o f t h e above m e n t i o n e d c a s e s a r e d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n s . The r e s u l t s c o n c l u d e d f r o m e a c h o f t h e s e two methods a r e c o m p a r e d i n t h e n e x t c h a p t e r . 3 . 4 . 3 . 1 - Beaching Motion Over A Fixed Point of the Bow E v e n t h o u g h t h e p o s i t i o n o f t h e c o n t a c t p o i n t may v a r y s l i g h t l y a l o n g t h e bow, i t i s assumed t h a t a f i x e d p o i n t on t h e bow i s i n c o n t a c t w i t h t h e i c e t h r o u g h o u t t h e b e a c h i n g s t a g e . T h i s p o i n t i s d e n o t e d by 61 N-th node. With t h i s assumption, the v e l o c i t y of the ship and the l a t e r a l v e l o c i t y of the N-th node are r e l a t e d through the bow angle as: X - Y Cot a N where X i s the v e l o c i t y of the ship and Y i s the l a t e r a l v e l o c i t y of the N-th node. The above r e l a t i o n i s v a l i d at a l l time dur ing the beaching stage, therefore i t could be d i f f e r e n t i a t e d with respect to time to get the s i m i l a r r e l a t i o n between the a c c e l e r a t i o n s : X - Y Cot Q (3.18) The c o n f i g u r a t i o n of the ship and the i ce r idge i n conjunct ion with the corresponding beam model i s represented i n F i g . (3 .4 ) . y F i g . 3 . 4 - The Ice-Breaking Ship and the Ice Ridge With the Corresponding Beam Model. Newton's Law a p p l i e d to ship i n the h o r i z o n t a l d i r e c t i o n g ives: 62 T + F = M (1 + a )X r X s w h e r e , T i s t h e t h r u s t f o r c e o f t h e p r o p e l l e r s . r F^ i s t h e h o r i z o n t a l component o f t h e c o n t a c t f o r c e . a i s t h e added mass o f t h e s h i p i n s u r g e . D u r i n g t h e ramming m o t i o n t h e m a g n i t u d e o f t h e p r o p e l l e r t h r u s t i s n o r m a l l y n e g l i g i b l e compared t o t h e c o n t a c t f o r c e s . T h e r e f o r e t h e above e q u a t i o n c o u l d be s i m p l i f i e d a s : F = M (1 + a ) X ( 3 . 1 9 ) X s The v e r t i c a l and h o r i z o n t a l components o f t h e c o n t a c t f o r c e a r e r e l a t e d t h r o u g h t h e p a r a m e t e r A. w h i c h i s i n t r o d u c e d i n t h e f o r m e r s e c t i o n . T h e n , F A F X Y T h e r e f o r e E q u . ( 3 . 1 9 ) c o u l d be w r i t t e n a s : F ^_ M (1 + a ) X ( 3 . 2 0 ) i A s S u b s t i t u t i n g f o r X f r o m E q u . ( 3 . 1 8 ) t h e v e r t i c a l f o r c e e x e r t e d on t h e s h i p i s r e l a t e d t o t h e l a t e r a l a c c e l e r a t i o n o f t h e c o n t a c t p o i n t Y N a s : F ^ - M (1 + a ) Y C o t a ( 3 . 2 1 ) Y A s N The c o n t a c t f o r c e i s a p p l i e d a t t h e N - t h node o f t h e beam, t h e r e f o r e t h e o n l y n o n - z e r o e l e m e n t o f t h e f o r c e v e c t o r (Q ) , i n t h e e q u a t i o n o f m o t i o n ( 3 . 5 ) , i s i t s (2N + l ) - t h e l e m e n t w h i c h c o r r e s p o n d s t o t h e N - t h n o d e . H e n c e , t h e f o r c e v e c t o r (Q)may be r e p r e s e n t e d a s : {QJ= (d) F y 63 where {d } i s a 2 ( n + l ) d i m e n s i o n v e c t o r , h a v i n g a l l i t s e l e m e n t s z e r o e x c e p t f o r d = 1 . 2N + 1 S u b s t i t u t i n g f o r F y f r o m E q u . ( 3 . 2 1 ) , t h e above e q u a t i o n becomes a s : {Q } Y " (1+a ) Y C o t o {d } ( 3 . 2 2 ) — A s N w i t h t h i s f o r c e v e c t o r , t h e e q u a t i o n o f l a t e r a l m o t i o n o f t h e s h i p i s : [M ]{U } + [C ]{U } + [K ] (U } = - (1+a ) Y C o t a {d } ( 3 . 2 3 ) - - - - - - A s N The r i g h t s i d e o f t h e above e q u a t i o n i s r e l a t e d t o t h e l a t e r a l a c c e l e r a t i o n o f t h e N - t h n o d e , Y , t h e r e f o r e i t w o u l d be p o s s i b l e t o s h i f t i t t o t h e l e f t s i d e o f t h e e q u a t i o n and combine i t w i t h t h e f i r s t t e r m w h i c h r e p r e s e n t s t h e i n e r t i a o f t h e s y s t e m . Hence t h e e q u a t i o n o f m o t i o n w o u l d be shown a s : A [M ]{U } + [C ]{U } + [K ]{U } = 0 ( 3 . 2 4 ) These a r e t h e e q u a t i o n s o f m o t i o n o f t h e s h i p - b e a m - m o d e l d u r i n g t h e b e a c h i n g p h a s e . They may be i n t e g r a t e d w i t h r e s p e c t t o t i m e t o d e t e r m i n e t h e d i s p l a c e m e n t s , v e l o c i t i e s and f o r c e s i n t h e s h i p . C l e a r l y t h e E q u . ( 3 . 2 4 ) i s t h e e q u a t i o n o f f r e e v i b r a t i o n o f a l i n e a r s y s t e m . Any n u m e r i c a l method c o u l d be u s e d t o s o l v e t h e above e q u a t i o n i n c o n j u n c t i o n w i t h t h e i n i t i a l c o n d i t i o n s w h i c h a r e c a l c u l a t e d i n t h e i m p u l s e s t a g e . I n E q u . ( 3 . 2 4 ) t h e p r o p o r t i o n a l damping m a t r i x [C ] i s c o n s i d e r e d a s : 64 [C ] = a [M ] + /3 [K ] where t h e method o f s e l e c t i o n o f p a r a m e t e r s a and (3 a r e r e p r e s e n t e d i n S e c t i o n 3 . 8 . The mode s u p e r p o s i t i o n method i s u s e d t o s o l v e t h e E q u . ( 3 . 2 4 ) . To u n c o u p l e t h e e q u a t i o n o f m o t i o n i t i s t r a n s f e r r e d t o t h e n o r m a l c o o r d i n a t e s a s i s shown i n a p p e n d i x ' B ' . The n o r m a l i z e d u n c o u p l e d n o d a l e q u a t i o n s o f m o t i o n a r e : A . A £ + (a +/3 to 2 ) £ + w 2 £ = 0 ( 3 . 2 5 ) i i i i i A A A 2 where to = k / m i i i The g e n e r a l s o l u t i o n f o r e a c h o f t h e above e q u a t i o n s i s g i v e n on page 25 o f [ 4 7 ] . C o n s i d e r i n g z e r o d i s p l a c e m e n t a t t=0 , t h e s o l u t i o n o f t h e s e e q u a t i o n s i s : £.(0) £ ( t ) = [exp (- rj co t ) ] ( — - S i n co t ) ( 3 . 2 6 ) i i i CO d i d i A A / A 2 / 2 Where , r? = ( a + fl w ) / 2u and co = V 1-rj w i i i d i i i The i n i t i a l v e l o c i t y v e c t o r {T/>} i s c a l c u l a t e d f r o m t h e i n i t i a l n o d a l v e l o c i t y v e c t o r {U} a s : A {U } = [T ]{# } The c a l c u l a t e d v a l u e s o f t h e d i s p l a c e m e n t i n t h e n o r m a l c o o r d i n a t e s £ ( t ) a r e t r a n s f e r r e d t o t h e n o d a l c o o r d i n a t e (U) t h r o u g h t h e r e l a t i o n : i A {U } = [T ](V. ) ( 3 . 2 7 ) A where [T ] i s a m a t r i x whose co lumns a r e t h e e i g e n - v e c t o r s o f t h e m a t r i x 65 A - 1 [M ] [K ] . The i n t e g r a t i o n p r o c e s s s t o p s when t h e m a g n i t u d e o f t h e v e l o c i t y o f t h e s h i p i s z e r o . D u r i n g t h e i n t e g r a t i o n p r o c e s s t h e b e n d i n g moment f o r e a c h beam e l e m e n t i s c a l c u l a t e d , as i s e x p l a i n e d i n S e c t i o n ( 3 . 7 ) . The m a g n i t u d e and l o c a t i o n o f t h e maximum b e n d i n g moment a r e t h e n f o u n d . 3 . 4 . 3 . 2 - B e a c h i n g M o t i o n Over a F i x e d P o i n t On t h e I c e R i d g e I n t h i s a p p r o a c h i t i s assumed t h a t w h i l e t h e bow i s s l i d i n g o v e r a f i x e d p o i n t o f t h e h i g h p r e s s u r e i c e r i d g e , t h e c o n t a c t p o i n t moves a l o n g t h e bow. T h i s mode l h a s an a d v a n t a g e t h a t t h e N - t h n o d e , w h i c h makes t h e f i r s t c o n t a c t w i t h t h e i c e , i s n o t r e s t r i c t e d t o r e m a i n on t h e i c e r i d g e d u r i n g t h e b e a c h i n g s t a g e . W h i l e t h r o u g h - o u t t h i s p e r i o d , t h e r e w o u l d a l w a y s be a c o n t a c t p o i n t b e t w e e n t h e bow and i c e r i d g e . The i n t e g r a t i o n method u s e d i n t h e f o r m e r s e c t i o n c a n n o t be u s e d i n t h i s case . , b e c a u s e t h e v a r i a t i o n o f t h e c o n t a c t p o i n t a l o n g t h e bow makes a v a r i a b l e mass m a t r i x . D u r i n g t h e b e a c h i n g s t a g e t h e c o n t a c t p o i n t h a s a l o c a l c o o r d i n a t e u a l o n g t h e K - t h e l e m e n t . The beam e l e m e n t u n d e r c o n t a c t c o u l d be c o n s i d e r e d as a f l e x i b l e beam s l i d i n g on an i n c l i n e d s u r f a c e . F i g . ( 3 . 5 ) shows t h i s c o n f i g u r a t i o n . A s i m i l a r r e l a t i o n r e p r e s e n t e d i n E q u . ( 3 . 1 8 ) i s v a l i d b e t w e e n t h e a c c e l e r a t i o n o f t h e s h i p and t h e l a t e r a l a c c e l e r a t i o n o f t h e c o n t a c t p o i n t a s : X = Y C o t a ( 3 . 2 8 ) 66 F i g . 3 .5- The Flexible Beam Element Sliding Over An Inclined Surface. D e f l e c t i o n of any po int along the element which i s under contact i s r e l a t e d to the displacements of i t s end nodes through the shape func t ion <N> as: Y - <N >{u } (3.29) where {u } i s the nodal displacement of the K - t h element which i s under contact . D i f f e r e n t i a t i n g Equ.(3.29) twice with respect to time g ives: Y= <N >{u } (3.30) S u b s t i t u t i n g for Y from Eq (3.30) in to Eq (3.28) , the a c c e l e r a t i o n of the ship i s : X - <N >{u } Cot a (3.31) Cons ider ing the Equ. (3.20) which r e l a t e s the l a t e r a l force F^and the a c c e l e r a t i o n of the ship as: F Y r H ( 1 + V * ( 3 - 2 0 ) and s u b s t i t u t i n g for X from Equ. (3 .31 ) , the l a t e r a l force on the ship 67 i s : F ^ - <N >{u } (1 + a ) C o t a ( 3 . 3 2 ) y A s The n o d a l v a l u e s o f t h e l a t e r a l f o r c e F y , a c t i n g on t h e s h i p a r e r e p r e s e n t e d b y {Q}. The f o r c e v e c t o r {Q} i s : {Q } = {d } F _ _ y where a l l e l e m e n t s o f t h e v e c t o r {d } a r e z e r o e x c e p t t h o s e f o u r w h i c h a r e r e l a t e d t o t h e K - e l e m e n t . The m a g n i t u d e o f t h e s e f o u r t e r m s depend on t h e p o s i t i o n o f t h e c o n t a c t p o i n t a l o n g t h e e l e m e n t . The method o f c a l c u l a t i o n o f t h e n o d a l v a l u e s o f t h e c o n t a c t f o r c e i s r e p r e s e n t e d i n A p p e n d i x ' A ' . The e q u a t i o n o f l a t e r a l m o t i o n o f t h e s h i p w o u l d b e : A [M ] {U } + [C ] {U } + [K ]{U } - {d } F ( 3 . 3 3 ) The s i m u l t a n e o u s s o l u t i o n o f Eqs ( 3 . 3 2 ) and ( 3 . 3 3 ) l e a d s t o t h e m a g n i t u d e o f t h e d i s p l a c e m e n t , v e l o c i t y , and a c c e l e r a t i o n o f t h e s h i p d u r i n g t h e b e a c h i n g s t a g e . T h r o u g h o u t t h e i n t e g r a t i o n p r o c e d u r e the m a g n i t u d e o f t h e maximum b e n d i n g moment a l o n g t h e s h i p i s a l s o c a l c u l a t e d . The f o l l o w i n g p r o c e d u r e i s u s e d t o s o l v e t h e e q u a t i o n s o f m o t i o n . F o r t h e f i r s t t i m e i n t e r v a l A t , i t i s assumed t h a t t h e c o n t a c t f o r c e on t h e s h i p i s z e r o , E q u . ( 3 . 3 3 ) i s s o l v e d t o f i n d t h e n o d a l v a l u e s o f t he l a t e r a l a c c e l e r a t i o n o f t h e s h i p . The m a g n i t u d e o f t h e n o d a l a c c e l e r a t i o n o f t h e K - t h e l e m e n t a r e s u b s t i t u t e d i n t o E q u . ( 3 . 3 2 ) t o c a l c u l a t e t h e c o n t a c t f o r c e . The c a l c u l a t e d f o r c e i s t a k e n as t h e a p p l i e d l o a d on t h e s y s t e m d u r i n g t h e n e x t t i m e i n t e r v a l . The 'Newmark j3 M e t h o d ' w i t h i t s p a r a m e t e r e q u a l t o 0 . 2 5 , i s f o u n d t o be t h e most 68 e f f i c i e n t method of s o l u t i o n for Equ . (3 .33 ) . At each step, the p o s i t i o n of contact po in t on the bow and the v e l o c i t y of the ship are monitored. The p o s i t i o n of the contact po int determines the magnitude of the shape func t ion <N>. The i n t e g r a t i o n procedure stops whenever the v e l o c i t y of the ship i s zero . In each time step A t , the maximum tending moment along the ship i s determined by the same method which i s introduced i n the former s e c t i o n . 3 .5- Determination of Contact Area Between the Ice Sheet and the Bow of  the Ice-Breaking Ship As mentioned, i n the continuous ice breaking mode contact area between the bow and the i ce should be determined to c a l c u l a t e the contact f o r c e . In t h i s s ec t i on the method used to c a l c u l a t e t h i s area i s presented. A F i g . 3 .6- Typical Cross-Section Area At the Bow. In general the cross s ec t ion of the ice breaking ship along i t s bow, as shown i n F i g (3 .6) , could be approximated by: Y — (a Zm+ b) where the parameters of a, b , and m would be s p e c i f i e d for each ves se l 69 i n a way t h a t t h e above e q u a t i o n g i v e s t h e b e s t a p p r o x i m a t i o n o f t h e r e a l c r o s s s e c t i o n o f t h e s h i p . When t h e s h i p c r u s h e s t h e i c e , t h e f o o t p r i n t o f i t s bow on t h e edge o f t h e i c e s h e e t w o u l d be s i m i l a r t o t h e c o n f i g u r a t i o n shown i n F i g . ( 3 . 7 ) . p r o j e c t i o n o f t h e c o n t a c t a r e a on t h e v e r t i c a l p l a n e i s d e f i n e d by t h e a r e a s u r r o u n d e d by t h e c u r v e o f Y = f ( Z ) and t h e u p p e r edge o f t h e i c e s h e e t . T h i s a r e a i s c a l c u l a t e d a s : F i g . 3 . 7 - 'Print' of the Bow on the Ice Sheet. B / z A - 2 l m+ 1 B / 2 (aZ +b) dZ - 2 ( — - — , Z + b Z ) m + 1 o ( 3 . 3 4 ) 70 C o n s i d e r i n g t h e e x t e n t s o f t h e c r u s h e d p a r t s as shown i n F i g . ( 3 . 7 ) we w o u l d h a v e : Q — a = — and b = C ( B / 2 ) m S u b s t i t u t i n g f o r a and b i n E q u . ( 3 . 3 4 ) , t h e c r o s s s e c t i o n a r e a i s c a l c u l a t e d a s : ni ~ — A = —r B C l m + 1 I t i s a good a p p r o x i m a t i o n t o assume t h e t h e c a l c u l a t e d a r e a A^ i s t h e p r o j e c t i o n o f t h e p l a n e c o n t a c t a r e a A on t h e v e r t i c a l s u r f a c e . T h e r e f o r e t h e c o n t a c t a r e a c o u l d be r e l a t e d t o t h e c r o s s s e c t i o n a r e a A ^ t h r o u g h t h e bow a n g l e a a s : A A S i n a A n d t h e c o n t a c t a r e a i s : S i n a From g e o m e t r y o f t h e bow, i t i s n o t i c e d t h a t , t h e v e r t i c a l p e n e t r a t i o n o f t h e bow i n t h e i c e , C, i s r e l a t e d t o t h e h o r i z o n t a l p e n e t r a t i o n , D, t h r o u g h t h e bow a n g l e a s : > C = D Tan a ( 3 . 3 6 ) A l s o f r o m t h e g e o m e t r y o f t h e bow t h e r e l a t i o n b e t w e e n t h e h o r i z o n t a l p e n e t r a t i o n o f bow i n t h e i c e , D, and b r e a d t h o f t h e engaged p a r t c o u l d be a p p r o x i m a t e d a s : 71 D = a B o r B = ( - r - ) ( 3 . 3 7 ) a and p f o r e a c h v e s s e l a r e s e l e c t e d i n a way t h a t t h e above r e l a t i o n g i v e s a good a p p r o x i m a t i o n o f t h e r e a l bow. S u b s t i t u t i n g f o r B and C i n t o t h e E q u . ( 3 . 3 5 ) f r o m E q u . ( 3 . 3 6 ) and E q u . ( 3 . 3 7 ) we h a v e : _ < i + p ) / p . m D A = ( 3 . 3 8 ) k S i n a where k = ( a ) 1 / p E q u . ( 3 . 3 8 ) r e l a t e s t h e g e o m e t r i c c o n t a c t a r e a b e t w e e n t h e i c e - b r e a k i n g s h i p and t h e i c e s h e e t t o t h e p e n e t r a t i o n o f t h e bow i n t o t h e i c e . 3 . 6 - D e t e r m i n a t i o n o f t h e U n i t V e c t o r I n t h e D i r e c t i o n o f t h e V e l o c i t y  T a n g e n t t o t h e P l a n e I n t h e c o n t i n u o u s i c e b r e a k i n g mode t h e t a n g e n t v e c t o r t o t h e bow i s u s e d t o c a l c u l a t e t h e f r i c t i o n f o r c e a n t t h e n t h e t o t a l i c e f o r c e . I n t h i s s e c t i o n t h e method s e l e c t e d t o c a l c u l a t e t h e t a n g e n t v e c t o r a t e a c h i n s t a n t i s p r e s e n t e d . A •ft P l a n e P w h i c h i s d e f i n e d b y t h e n o r m a l u n i t v e c t o r n i s m o v i n g w i t h v e l o c i t y V . The v e l o c i t y v e c t o r c o u l d be r e p r e s e n t e d a s : A A A V = v i + v j + v k x y z A U n i t v e c t o r t i s a l o n g t h e p r o j e c t i o n o f t h e v e l o c i t y v e c t o r V on t h e p l a n e P * . The f o l l o w i n g p r o c e d u r e i s s e l e c t e d t o f i n d t h i s v e c t o r . The u n i t v e c t o r o f t h e v e l o c i t y i s : A A A A x y z 72 v v where AS • , AS •= — / 2 ~ 2 ~ 2 V Z - 2 2~ 2 v v + v + v v v + v + v x y z x y z V z AS ' z / 2 , 2 . V V + V + 2 V x y z The c r o s s p r o d u c t o f t h e u n i t v e l o c i t y v e c t o r s AS and t h e u n i t A n o r m a l v e c t o r n g i v e s a u n i t v e c t o r w h i c h l i e s i n t h e p l a n e P. T h i s v e c t o r w o u l d be n o r m a l t o t h e p l a n e w h i c h i s i n t r o d u c e d by t h e u n i t A A v e c t o r s n and AS. t h i s c o n f i g u r a t i o n i s shown i n F i g . ( 3 . 8 ) . A A F i g . 3 . 8 - A Plane With Unit Normal Vector n And Unit Velocity Vector v . C r o s s p r o d u c t o f AS and f i i s : A A A A A AS x n •= (AS n - A s n ) i + ( A s n - A s n ) J + ( A s n - A s n ) / c y z z y z x x z x y y x A A A A and t h e c r o s s p r o d u c t o f n and (AS x n ) i s a u n i t v e c t o r t . T h i s v e c t o r l i e s on t h e p l a n e P and i s i n t h e d i r e c t i o n o f t h e component o f v e l o c i t y t a n g e n t t o t h e p l a n e . T h i s v e c t o r i s d e t e r m i n e d as f o l l o w e d : 73 A A w h e r e , t = n x (« x a ) = t i + t ; " + t k x y z 2 2 x x y x z y x y z x z 2 2 t = « n + < 5 ( l - « f t f l - < > a A y y z y x z y z x y x 2 2 t = « ft + « n . - 4 j n n - i s a n . z z x z y x z x y z y 3.7- D e t e r m i n a t i o n o f t h e maximum B e n d i n g Moment A l o n g A Beam E l e m e n t The b e n d i n g moment a l o n g a beam i n m a t r i x n o t a t i o n i s : M z M = E a x i 2 0 0 I Y z ( 3 . 3 9 ) {I} V e c t o r r e p r e s e n t t h e d i s p l a c e m e n t o f a p o i n t on t h e beam e l e m e n t . T h i s d i s p l a c e m e n t i s r e l a t e d t o t h e d i s p l a c e m e n t o f t h e ends o f t h e beam b y : - [N ] {u) w h e r e , t h e shape f u n c t i o n [N ] f o r a beam e l e m e n t i s t h e one w h i c h i s i n t r o d u c e d i n C h a p t e r 2 , and {u} r e p r e s e n t s t h e d i s p l a c e m e n t o f t h e end p o i n t s o f t h e beam e l e m e n t . S u b s t i t u t i n g f o r -| ^ j" f r o m t h e above r e l a t i o n i n t o E q u . ( 3 . 3 9 ) we h a v e : 74 M z M = E I Z 0 0 I I — , ] < » > S e c o n d d e r i v a t i v e o f t h e shape f u n c t i o n [N] w i t h r e s p e c t t o x i s w r i t t e n i n t e r m o f t h e n o n - d i m e n s i o n a l c o o r d i n a t e v a s : [A ] = [ a x N T h e r e f o r e t h e m a t r i x [A ] i s : [A ] f - 6 +12i/ 0 0 0 ( - 4 +6i/)£ 6 - 1 2 i / 0 0 0 ( - 2 +6i/)« £2 L d s J o - 6 +12i/ 0 (4 -6i/)« 0 0 6 - 1 2 i / 0 (2 -6 J / ) £ 0 ( 3 . 4 0 ) f M 1 " I 0 z • = E z M V y 0 I y J H e n c e , t h e b e n d i n g moment a l o n g t h e beam e l e m e n t w o u l d b e : [A ] {u } A s l o n g as t h e n o d a l p o s i t i o n v e c t o r {u } f o r t h e e l e m e n t i s known, t h e b e n d i n g moment a t any p o i n t o f t h e beam e l e m e n t c o u l d be c a l c u l a t e d . Any s e l e c t e d p o i n t on t h e beam e l e m e n t i s i d e n t i f i e d b y i t s n o n - d i m e n s i o n a l c o o r d i n a t e v = x /£, w i t h t h i s c o o r d i n a t e t h e m a t r i x [A] i s e v a l u a t e d and b e n d i n g moment a t t h e p o i n t i s c a l c u l a t e d . D u r i n g c o n t i n u o u s i c e b r e a k i n g and d u r i n g a c o l l i s i o n , t h e c o n t a c t f o r c e s a r e n o t a p p l i e d as p o i n t l o a d s a t t h e nodes o f t h e beam. I n some 75 c a s e s t h e s e f o r c e s a r e d i s t r i b u t e d a l o n g a f i n i t e l e n g t h o f t h e beam e l e m e n t . The f i n i t e e l e m e n t r e s u l t s a r e t h e n n o t e x a c t and we c a l c u l a t e t h e b e n d i n g moments a t t h e Gauss p o i n t s on t h e e l e m e n t . On p 399 o f [48] i t i s shown t h a t Gauss p o i n t s on a beam e l e m e n t a r e l o c a t e d a t v = 0 . 2 1 1 5 and 0 . 7 8 8 5 . Page 236 o f t h e same r e f e r e n c e a l s o shows t h a t t h e m a g n i t u d e o f t h e b e n d i n g moment a t t h e Gauss p o i n t s a l o n g a u n i f o r m l y l o a d e d beam a r e e x a c t . 3.8- A M e t h o d o f D e t e r m i n i n g t h e C o n s t a n t s f o r P r o p o r t i o n a l Damping When t h e s h i p moves i n w a t e r , waves a r e g e n e r a t e d and p r o p a g a t e d . Wave p r o p a g a t i o n d i s s i p a t e s some p a r t o f t h e k i n e t i c e n e r g y o f t h e s y s t e m . T h i s e n e r g y d i s s i p a t i o n i s t r e a t e d as h y d r o d y n a m i c d a m p i n g . I n a d d i t i o n t h e f l e x u r a l m o t i o n o f t h e s t r u c t u r a l e l e m e n t s d i s s i p a t e some e n e r g y t h r o u g h m a t e r i a l d a m p i n g . M a t e r i a l damping i s v e r y much s m a l l e r t h a n h y d r o d y n a m i c d a m p i n g , so i s n e g l e c t e d . W i t h t h i s a s s u m p t i o n t h e damping r a t i o rj^ . w o u l d depend upon t h e m o t i o n o f t h e s h i p i n t h e w a t e r e i t h e r as t h e r i g i d - b o d y o r e l a s t i c m o t i o n . Then damping r a t i o rj^, f o r 2 t h e 4 . - t h mode o f m o t i o n , i s r e l a t e d t o t h e damping c o e f f i c i e n t a + B a s : 2 2 n .w.= a + B to. I t i s n o t i c e d t h a t i f t h e m a g n i t u d e o f t h e damping r a t i o r\ ^  i s known f o r two d i f f e r e n t f r e q u e n c i e s , t h e n t h e two unknowns a and 8 c a n be c a l c u l a t e d . I f t h e m a g n i t u d e s o f r>. f o r two d i s t i n c t n o n - z e r o 76 f r e q u e n c i e s o f t h e s y s t e m w . and w„ a r e known t h e n we h a v e : 2n.<a.=a + Bu>. i t 1 -2 VfUk = a + B W h i c h g i v e s t h e v a l u e s o f a and B as a = ( 3 . 4 1 ) P - 2 2 I f we know damping r a t i o s f o r two d i s t i n c t f r e q u e n c i e s E q u . ( 3 . 4 1 ) c a n be u s e d t o c a l c u l a t e t h e p a r a m e t e r s a and B. Damping r a t i o s c a n be d e t e r m i n e d t h r o u g h t h e e x p e r i m e n t w i t h a s h i p m o d e l . I n t h i s s t u d y t h e f o l l o w i n g m o d e l i s u s e d t o d e t e r m i n e t h e m a g n i t u d e o f t h e damping r a t i o s . The f i r s t and t h e s e c o n d n o n - z e r o e i g e n - v a l u e s o f [M ] [K ] a r e r e l a t e d t o two o s c i l l a t i n g r i g i d b o d y m o t i o n s o f t h e s h i p s u c h as h e a v e and p i t c h . T h e s e two f r e q u e n c i e s a r e r e p r e s e n t e d b y » i and u>^ and t h e i r c o r r e s p o n d i n g damping r a t i o s a r e rj and r\ r e s p e c t i v e l y . M a g n i t u d e o f i s d e t e r m i n e d t h r o u g h a l o g a r i t h m i c d e c r e m e n t a n a l y s i s . T h i s method i s shown i n page 27 o f [ 4 7 ] . I n t h i s m e t h o d , i f t h e a m p l i t u d e o f t h e m o t i o n i s m e a s u r e d b e f o r e and a f t e r a c e r t a i n number o f f r e q u e n c i e s , t h e damping r a t i o c o u l d be c a l c u l a t e d t h r o u g h t h e e q u a t i o n : - 1 1 I n o 1 = X p 77 Where , t] i s t h e damping r a t i o . X and X a r e t h e i n i t i a l and f i n a l m e a s u r e d a m p l i t u d e s o f m o t i o n , o P F p i s t h e number o f o s c i l l a t i o n s b e t w e e n t h e two measu rements . M a g n i t u d e o f damping r a t i o ij^ r e l a t e d t o t h e s e c o n d f r e q u e n c y o^has t h e f o l l o w i n g r e s t r i c t i o n s . i ) I n e q u a t i o n o f m o t i o n s u c h as ( 3 . 4 ) damping c o e f f i c i e n t s h o u l d a l w a y s be p o s i t i v e . i i ) The e q u a t i o n s o f m o t i o n r e p r e s e n t e d i n ( 3 . 4 ) , a r e t h o s e o f a l l modes o f m o t i o n . T h a t i s , i t g o v e r n s t h e m o t i o n w i t h any f r e q u e n c y i n c l u d i n g t h e c a s e o f z e r o f r e q u e n c i e s s u c h as yaw and sway . I n t h e c a s e o f z e r o f r e q u e n c y t h e damping c o e f f i c i e n t s h o u l d s t i l l be p o s i t i v e . T h e s e two r e s t r i c t i o n s i m p l y : a + B co2. > 0 and a > 0 l 2 a and ft a r e s u b s t i t u t e d f r o m E q u . ( 3 . 4 1 ) i n t o a + B to ^  > 0 and we h a v e : 2 w w ( w n - w n ) ( w n - to n )to. 1 2 1 '2 2 1 + l ' l 2 2 L 2 2 2 2 to - to to - to 1 2 1 2 As t h e f r e q u e n c i e s i n c r e a s e s i n o r d e r o f to < to <to < <to t h e n n 1 2 3 n 2 2 to — to < 0 and t h e above r e l a t i o n c a n be w r i t t e n a s : 1 2 2 1 2 1 '2 2*1 4 v 1 ' l 2 ' 2 o r 78 2 2 2 2 co (co - co.) 77 < co (co - co.) n 2 1 L' ' Z 1 2 L ' l Z 2 M a g n i t u d e o f (w - co.) i s n e g a t i v e t h e n we h a v e : .2 2 W ( W - W . ) 1 2 t r? > n f o r i. > 2 2 co . 2 2. l 2 (w - co.) T h i s r e l a t i o n i s v a l i d f o r a l l v a l u e s o f I > 2 and i t i s n o t i c e d 2 2 2 2 t h a t t h e m a g n i t u d e o f (co^- UK)/(oo^-co/) i s a l w a y s s m a l l e r t h a n u n i t y . T h e r e f o r e t h e above r e l a t i o n c o u l d be w r i t t e n a s : co ri > —- V ( 3 . 4 2 ) 2 CO 1 V ' 2 and f r o m t h e r e s t r i c t i o n o f a > 0 we h a v e : ( W 1 n2 " W2 " l } - > 0 w h i c h i m p l i e s : 2 2 co - co l 2 co V < —- V ( 3 . 4 3 ) 2 CO 1 V ' 1 C o m b i n i n g r e l a t i o n s ( 3 . 4 2 ) and ( 3 . 4 3 ) we h a v e : co co -IT V -zr ^ ( 3 - 4 4 ) CO 1 2 CO 1 1 2 R e s t r i c t i o n ( 3 . 4 4 ) h e l p s t o d e c i d e a b o u t t h e v a l u e o f t h e damp ing r a t i o o f t h e s e c o n d v i b r a t i n g mode. Then b y E q u . ( 3 . 4 1 ) m a g n i t u d e o f a and 8 are c a l c u l a t e d . 79 4- COLLISION OF THE SHIP WITH AN EXTERNAL OBJECT 4 . 1 - I n t r o d u c t i o n When a s h i p r o u t e p a s s e s t h r o u g h s h a l l o w w a t e r s o r c l o s e t o o f f s h o r e s t r u c t u r e s , t h e r e i s a r i s k o f g r o u n d i n g o r c o l l i d i n g w i t h t h e e x t e r n a l o b j e c t s . The c o l l i s i o n c a n c a u s e e i t h e r d e n t i n g o f t h e s h i p ' s h u l l as a m i n o r c o l l i s i o n o r t e a r i n g and b e n d i n g o f t h e h u l l p l a t i n g as a m a j o r c o l l i s i o n . The e x t e n t o f damage depends on t h e s t r u c t u r a l c h a r a c t e r i s t i c s o f t h e s h i p , t h e a p p r o a c h v e l o c i t y , the' l o c a t i o n o f t h e i m p a c t on t h e h u l l and t h e n a t u r e o f t h e s t r u c k o b j e c t . S t r u c t u r a l c h a r a c t e r i s t i c s o f t h e s h i p , o r t he h u l l s t r e n g t h has a m a j o r e f f e c t on t h e m a g n i t u d e o f t h e e x t e r n a l f o r c e , w h i l e t h e i n i t i a l p o s i t i o n and v e l o c i t y d e t e r m i n e t h e i n i t i a l c o n d i t i o n s o f t h e s u b s e q u e n t m o t i o n o f t h e s h i p d u r i n g and a f t e r c o l l i s i o n . I n t h i s s t u d y t h e e x t e r n a l o b j e c t i s c o n s i d e r e d f i x e d w i t h r e s p e c t t o t h e e a r t h and c o m p l e t e l y r i g i d . 4 . 2 - D e f i n i t i o n o f t h e C o o r d i n a t e s A A A A The body c o o r d i n a t e s y s t e m o - x y z i s f i x e d w i t h r e s p e c t t o the A s h i p . The o r i g i n i s a t t h e s t e r n o f t h e s h i p , x - a x i s i s a l o n g t h e s h i p A and p o s i t i v e t o w a r d bow, y - a x i s i s v e r t i c a l and p o i n t i n g u p w a r d , and A z - a x i s i s s i d e w i s e n o r m a l t o t h e s h i p . The g l o b a l c o o r d i n a t e s y s t e m O-XYZ i s a s y s t e m o f r e f e r e n c e s f i x e d w i t h r e s p e c t t o t h e e a r t h . I m m e d i a t e l y b e f o r e c o l l i s i o n b o t h o f t he c o o r d i n a t e s y s t e m s c o i n c i d e . F i g . ( 4 . 1 ) shows t h e two s y s t e m s . 80 F i g . 4 .1 - Systems of Body and Global Coordinates. The equation of motion of the ship i s developed and so lved with respect to the i n e r t i a l system 0-XYZ, while damage to the h u l l i s A A A A measured i n the l o c a l coordinates o-x y z . The coordinates of the ex terna l object are measured with respect to the 0-XYZ coordinates and would be represented as X^ ,Y^ and . 4 .3- Equation of Motion of The Ship The whole ship i s d i v i d e d in to n s t a t i o n s , each s t a t i o n i s cons idered as a beam element with twelve degrees of freedom as descr ibed i n Chapter 2. Immediately before c o l l i s i o n a l l of the nodes on the beam are a long the X - a x i s . The mass and s t i f f n e s s matrices for each element are developed. Then the g l o b a l mass and s t i f f n e s s matrices are assembled i n accordance wi th the method descr ibed i n Chapter 2. The p r o p o r t i o n a l damping matrix 81 i s g e n e r a t e d by t h e a p p r o p r i a t e p r o p o r t i o n a l p a r a m e t e r s . These p a r a m e t e r s a r e d e t e r m i n e d a c c o r d i n g t o t h e method d e s c r i b e d i n s e c t i o n ( 3 . 8 ) . E q u a t i o n o f l a t e r a l m o t i o n o f t h e w h o l e s h i p i s d e v e l o p e d s i m i l a r t o t h e E q u . ( 2 . 6 ) . T h i s s e t o f e q u a t i o n s i n c o n j u n c t i o n w i t h t h e e q u a t i o n o f a x i a l m o t i o n d e s c r i b e t h e t o t a l e q u a t i o n o f m o t i o n o f t h e s h i p . The n o d a l v a l u e s o f t h e e x t e r n a l f o r c e s a r e c a l c u l a t e d and e q u a t i o n s o f m o t i o n a r e s o l v e d f o r e a c h s e l e c t e d t i m e s t e p d t . Two d i f f e r e n t c a s e s o f c o l l i s i o n s a r e c o n s i d e r e d : i ) h e a d o n and s h o u l d e r c o l l i s i o n . i i ) s i d e c o l l i s i o n d u r i n g m a n e u v e r i n g . I n e a c h c a s e t h e g e n e r a t e d l o a d s and t h e m o t i o n o f t h e s h i p i s s t u d i e d s e p a r a t e l y . 4 . 3 . 1 - Head on and S h o u l d e r C o l l i s i o n A h e a d - o n c o l l i s i o n i s s a i d t o o c c u r when t h e i m p a c t f o r c e i s i n t h e v e r t i c a l p l a n e o f symmetry o f t h e s h i p . T h e r e i s t h e n no s i d e w i s e m o t i o n and t h e p r o b l e m i s t w o - d i m e n s i o n a l . A s h o u l d e r c o l l i s i o n i s t h r e e d i m e n s i o n a l and o c c u r s when t h e i m p a c t f o r c e i n t h e bow r e g i o n , h a s ' a l a t e r a l component i n a d d i t i o n t o t h e components i n t h e v e r t i c a l p l a n e o f symmet ry . D u r i n g t h e s h o u l d e r c o l l i s i o n , t h e a x i a l component o f t h e e x e r t e d f o r c e d e c e l e r a t e s t h e s h i p , w h i l e t h e l a t e r a l components o f t h e l o a d c a u s e t h e s h i p t o d e f l e c t , t w i s t and move a s i d e . The t o t a l e q u a t i o n s o f m o t i o n o f t h e s h i p a r e : 82 [M]{U } + [C] {U } + [K] {U } = {Q } M ( l + a ) X = T - F ( 4 . 1 ) s r . x where t h e f i r s t e q u a t i o n i s t h a t o f t h e l a t e r a l m o t i o n w h i c h i s d e v e l o p e d i n C h a p t e r 2 and t h e s e c o n d one i s t h e e q u a t i o n o f a x i a l m o t i o n o f t h e s h i p . M i s t h e t o t a l mass o f t h e s h i p . T i s t h e p r o p e l l e r t h r u s t f o r c e . r F i s t h e component o f g e n e r a t e d c o l l i s i o n f o r c e a l o n g t h e s h i p . X i s t h e s h i p a x i a l a c c e l e r a t i o n , a i s a d d e d mass o f t h e s h i p i n s u r g e s The m a g n i t u d e o f F and t h e n o d a l v a l u e s o f {Q } a r e n o t known i n g e n e r a l b e f o r e h a n d . They a r e g e n e r a t e d t h r o u g h t h e c o l l i s i o n w h i c h w o u l d b e s i m u l a t e d t h r o u g h t h e i n t e g r a t i o n p r o c e s s . A t e a c h i n s t a n t o f c o l l i s i o n t h e p o s i t i o n o f t h e c o l l i s i o n p o i n t w i t h r e s p e c t t o t h e s h i p i s d e t e r m i n e d b y c o m p a r i n g t h e c o o r d i n a t e s o f t h e b a r r i e r w i t h t h o s e o f t h e nodes a l o n g t h e s h i p . A c o m p a r i s o n o f t h e X - c o o r d i n a t e s shows w h i c h s t a t i o n (beam e l e m e n t ) i s i n v o l v e d i n c o l l i s i o n , w h i l e c o m p a r i n g and w i t h t h e c r o s s s e c t i o n o f t h e s h i p i n d i c a t e s t h e m a g n i t u d e o f p e n e t r a t i o n o f t h e e x t e r n a l o b j e c t i n t h e h u l l . The above c o m p a r i s o n a r e made a t e a c h t i m e i n c r e m e n t o f i n t e g r a t i o n o f t h e e q u a t i o n s o f m o t i o n . To b e g i n t h e i n t e g r a t i o n , a t t=0 f o r a s h o r t t i m e i n t e r v a l d t a f t e r c o l l i s i o n , ( d t i s i n t h e o r d e r o f 0 . 0 5 s e c . ) i t i s assumed t h a t no f o r c e i s a p p l i e d on t h e s y s t e m , t h a t i s , F = 0 and a l l e l e m e n t s o f X t h e v e c t o r {Q } a r e z e r o . The s e t o f E q u . ( 4 . 1 ) a r e s o l v e d f o r a t i m e s t e p d t i n c o n j u n c t i o n w i t h t h e i n i t i a l c o n d i t i o n s {U} = 0 and X=« 83 where A S q i s t h e v e l o c i t y o f t h e s h i p j u s t b e f o r e c o l l i s i o n . D e t a i l s f o r t h e s o l u t i o n o f t h e e q u a t i o n s o f m o t i o n a r e d e s c r i b e d i n S e c t i o n ( 4 . 7 ) . O b v i o u s l y a t t h e end o f t i m e s t e p d t , t h e r e w o u l d n o t be any change i n t h e d e f l e c t i o n {U}, b u t t h e s h i p has moved a d i s t a n c e o f P w i t h r e s p e c t t o t h e e x t e r n a l o b j e c t . A t t h i s s t a g e p e n e t r a t i o n o f t h e e x t e r n a l o b j e c t i n t h e h u l l i s c a l c u l a t e d . (The p r o c e d u r e u s e d t o c a l c u l a t e the p e n e t r a t i o n i n t h e h u l l f r o m t h e c o o r d i n a t e s o f t h e s h i p and e x t e r n a l o b j e c t i s e x p l a i n e d i n S e c t i o n ( 4 . 5 ) and A p p e n d i x ' D ' ) . When we know t h e damage e x t e n t i n t h e h u l l , t h e m a g n i t u d e o f t h e g e n e r a t e d f o r c e i n e a c h d i r e c t i o n i s e s t i m a t e d b y c o n s i d e r i n g t h e s t r u c t u r a l c h a r a c t e r o f t h e s h i p a r o u n d t h e damaged a r e a . The method o f c a l c u l a t i o n o f t h e f o r c e i s e x p l a i n e d i n S e c t i o n ( 4 . 4 ) . A t t h i s s t a g e we know t h e damage e x t e n t and g e n e r a t e d f o r c e s i n t h r e e p r i n c i p a l d i r e c t i o n s . Now t h e n o d a l v a l u e s o f t h e l a t e r a l f o r c e s a r e c a l c u l a t e d a c c o r d i n g t o t h e E q u . ( 2 . 4 ) . D e t a i l s f o r c a l c u l a t i n g t h e n o d a l v a l u e s o f t h e g e n e r a t e d l o a d a r e shown i n A p p e n d i x ' A ' . W i t h t h e s e m a g n i t u d e s o f t h e f o r c e s , t h e e q u a t i o n s o f m o t i o n ( 4 . 1 ) a r e s o l v e d w i t h t h e same i n i t i a l v a l u e s o f {U}=0 and X= u f o r a n o t h e r t i m e s t e p d t . A t t h e end o f t h e o s e c o n d t i m e p e r i o d t h e new p o s i t i o n o f t h e s h i p i s c a l c u l a t e d , p e n e t r a t i o n o f t h e e x t e r n a l o b j e c t i n t h e h u l l i s e s t i m a t e d and t h e g e n e r a t e d f o r c e i n e a c h d i r e c t i o n i s e v a l u a t e d . From t h i s i n s t a n t t h e r e w o u l d be some n o n - z e r o v a l u e s f o r {U} and X w h i c h a r e u s e d as i n i t i a l v a l u e s f o r s o l u t i o n o f e q u a t i o n o f m o t i o n i n t h e s u b s e q u e n t t i m e s t e p s . T h i s p r o c e d u r e c o n t i n u e s u n t i l t h e c o l l i s i o n i s c o m p l e t e d . The end o f t h e c o l l i s i o n i s d e t e r m i n e d as f o l l o w s : i ) The s h i p a x i a l v e l o c i t y X r e a c h e s z e r o , w h i c h means t h a t t h e s h i p h a s b e e n b r o u g h t t o r e s t . 84 i i ) S h i p m i g h t be f o r c e d a s i d e a f t e r a c e r t a i n t i m e i n a s h o u l d e r c o l l i s i o n . D u r i n g t h e f r e e m o t i o n o f t h e s h i p , h e r p o s i t i o n i s f o l l o w e d t o c h e c k i f t h e s t e r n comes i n t o c o n t a c t w i t h t h e b a r r i e r , p o s s i b l y l e a d i n g t o f u r t h e r damage. When t h e s t e r n has p a s s e d by t h e b a r r i e r t h e i n t e g r a t i o n i s s t o p p e d . T h r o u g h o u t t h e i n t e g r a t i o n p r o c e s s , t h e p o s i t i o n and v e l o c i t y o f a l l o f t h e nodes a l o n g t h e s h i p a r e d e t e r m i n e d . A l s o p e n e t r a t i o n o f t h e e x t e r n a l o b j e c t i n t o t h e h u l l i s f o l l o w e d , and a t t h e end o f t h e i n t e g r a t i o n we have an e s t i m a t e f o r t h e o v e r a l l damage e x t e n t . The r o l l i n g m o t i o n o f t h e s h i p i s c h e c k e d f o r s t a b i l i t y d u r i n g t h e c o l l i s i o n . I t i s a l s o p o s s i b l e t o l o o k a t t h e d e f l e c t i o n o f t h e w h o l e s h i p and c h e c k t h e maximum b e n d i n g moment a l o n g t h e h u l l . 4 .3 .2- Side C o l l i s i o n During A Ship Maneuver o r G r o u n d i n g D u r i n g t h e maneuver o f a s h i p , i t m i g h t h i t a n e x t e r n a l o b j e c t s i d e w i s e . A s t r o n g wave m i g h t p u s h a s h i p , o r i n some c a s e s l i f t i t and d r o p i t on a r o c k . I n a l l o f t h e s e c a s e s a l a t e r a l f o r c e w o u l d be e x e r t e d on t h e s h i p w h i c h may d e n t o r t e a r t h e h u l l . The l a t e r a l f o r c e on t h e s h i p c o u l d be e i t h e r a c o n c e n t r a t e d o r a d i s t r i b u t e d f o r c e . A c o n c e n t r a t e d f o r c e m i g h t o c c u r when t h e s h i p s t r i k e s a s i n g l e r o c k w h e r e a s a d i s t r i b u t e d l o a d m i g h t o c c u r when t h e s h i p c o l l i d e s w i t h a s e r i e s o f r e e f s o r becomes g r o u n d e d on a s h o a l . I n t h e c a s e o f s i d e c o l l i s i o n , t h e p o s i t i o n o f t h e c o l l i s i o n p o i n t a l o n g t h e s h i p i s f o l l o w e d . The s t a t i o n ( s ) (beam e l e m e n t ( s ) ) w h i c h a r e u n d e r c o n t a c t a r e i n d i c a t e d and t h e n o d a l v a l u e s o f t h e f o r c e s a r e c a l c u l a t e d t h r o u g h E q u . ( 2 . 4 ) . I f t h e c o n t a c t f o r c e d u r i n g t h e i n i t i a l i 85 c o l l i s i o n e x t e n d s o v e r a c o n s i d e r a b l e a r e a o f t h e h u l l , t h e n t h e nodes r e c e i v i n g i n i t i a l i m p a c t f o r c e s must be i d e n t i f i e d . T h i s w o u l d be t h e c a s e , f o r e x a m p l e , when a s h i p r u n s a g r o u n d . The n o d a l v a l u e s o f t h e c o l l i s i o n f o r c e on t h e w h o l e s h i p a r e d e t e r m i n e d . E q u a t i o n s o f m o t i o n a r e e x a c t l y t h e same as E q u . (4.1), b u t t h e i n i t i a l c o n d i t i o n s f o r s i d e c o l l i s i o n a r e d i f f e r e n t f r o m t h o s e o f h e a d on c o l l i s i o n . I n t h i s c a s e , i n i t i a l l a t e r a l v e l o c i t y o f t h e s h i p i s n o t z e r o . The i n i t i a l v a l u e s f o r t h e v e l o c i t y v e c t o r {U} a r e d e t e r m i n e d f r o m t h e i n i t i a l l a t e r a l v e l o c i t y o f t h e w h o l e s h i p . D e t a i l s f o r c a l c u l a t i o n s o f {U} i s p r e s e n t e d i n S e c t i o n (4.7). The p r o c e d u r e u s e d t o s o l v e t h e e q u a t i o n s o f m o t i o n . i s t h e same as f o r h e a d - o n c o l l i s i o n . Damage e x t e n t , p o s i t i o n , v e l o c i t y , and s t a b i l i t y o f t h e s h i p a r e c a l c u l a t e d i n t h e same way as was done f o r t h e h e a d - o n c o l l i s i o n . Some t i m e s i t happens t h a t t h e a x i a l v e l o c i t y o f t h e s h i p becomes z e r o d u r i n g a s i d e a c c i d e n t w h i l e t h e s h i p i s s t i l l m o v i n g l a t e r a l l y and t h e damage i s e x t e n d i n g . I n s u c h c a s e s , t h e p o s i t i o n o f t h e e x t e r n a l o b j e c t i n t h e h u l l i s c h e c k e d f o r z e r o a x i a l f o r c e w h i c h m i g h t be g e n e r a t e d . The end o f c o l l i s i o n i n t h i s c a s e i s d e t e r m i n e d i n two d i f f e r e n t c a s e s a s ; i ) when t h e s h i p comes t o r e s t b y l o s s o f e n e r g y i n a c c i d e n t . i i ) when t h e s h i p p a s s e s o v e r t h e b a r r i e r . 4 . 4 - G e n e r a t e d C o l l i s i o n F o r c e I n t h i s w o r k , t h e damage on a s h i p h u l l g e n e r a t e d d u r i n g a c o l l i s i o n i s c l a s s i f i e d i n t h r e e d i f f e r e n t c a t e g o r i e s as f o l l o w e d : 86 i ) M a j o r damage i n w h i c h t h e h u l l t e a r s and a s i g n i f i c a n t amount o f t h e s t r u c t u r a l m a t e r i a l s a r e d i s t o r t e d , i i ) M i n o r damage i n w h i c h t h e h u l l i s d e n t e d w i t h o u t any r u p t u r e , i i i ) C o l l i s i o n s i n w h i c h t h e h u l l p l a t e t e a r s b u t t h e r e i s no s i g n i f i c a n t d i s t o r t i o n o f t h e i n t e r n a l s t r u c t u r e . The f o r c e s w h i c h c a u s e e a c h o f t h e above m e n t i o n e d s c e n a r i o s a r e i n t e r a c t i v e and a r e d e v e l o p e d d u r i n g t h e c o l l i s i o n . The m a g n i t u d e s o f t h e c o l l i s i o n f o r c e s i n e a c h d i r e c t i o n depends on t h e d e s i g n o f t h e s t r u c t u r e and t h e e x t e n t o f damage i n t h a t s p e c i f i c d i r e c t i o n . E a c h o f t h e above s i t u a t i o n s i s c o n s i d e r e d s e p a r a t e l y and a n a l y z e d i n t h e f o l l o w i n g s e c t i o n s . 4 . 4 . 1 - G e n e r a t e d F o r c e I n M a j o r C o l l i s i o n The damage g e n e r a t e d when two s h i p s a r e i n c o l l i s i o n was f i r s t s t u d i e d b y M i n o r s k y i n 1959 [ 4 ] . He u s e d t h e p r i n c i p l e s o f c o n s e r v a t i o n o f momentum t o e s t i m a t e t h e l o s s o f k i n e t i c e n e r g y o f t h e s h i p s i n v o l v e d . He a l s o i n v e s t i g a t e d a s e r i e s o f a c t u a l a c c i d e n t s i n w h i c h t h e s h i p h u l l s were damaged and showed t h a t t h e r e e x i s t a l i n e a r r e l a t i o n b e t w e e n t h e a b s o r b e d k i n e t i c e n e r g y i n c o l l i s i o n and t h e vo lume o f damaged s t r u c t u r e . H i s s e m i - e m p i r i c a l e q u a t i o n c o u l d s u c c e s s f u l l y p r e d i c t and d e s c r i b e m a j o r c o l l i s i o n s , b u t was n o t t h a t s u c c e s s f u l i n p r e d i c t i n g t h e damage e x t e n t i n m i n o r c o l l i s i o n s . I n t h i s w o r k , M i n o r s k y ' s method i s u s e d t o f i n d t h e g e n e r a t e d f o r c e d u r i n g a m a j o r c o l l i s i o n . M i n o r s k y ' s r e l a t i o n b e t w e e n t h e vo lume o f damaged s t r u c t u r e and t h e a b s o r b e d e n e r g y i s : 87 E = K V + K ( 4 . 2 ) s o ' where E^ i s t h e a b s o r b e d e n e r g y i n c o l l i s i o n . V i s t h e vo lume o f damaged m a t e r i a l . K and K ^ a r e two c o n s t a n t s o f e q u a t i o n The v o l u m e o f t h e damaged m a t e r i a l c o u l d be w r i t t e n a s : V = A D ( 4 . 3 ) Where D i s t h e damage e x t e n t i n a s p e c i f i c d i r e c t i o n and A d i s t h e c r o s s - s e c t i o n o f t h e damaged m a t e r i a l p e r p e n d i c u l a r t o t h a t s p e c i f i e d d i r e c t i o n . S u b s t i t u t i n g f o r V f r o m E q u . ( 4 . 3 ) i n t o E q u . ( 4 . 2 ) , we h a v e : E = K A , D + k s d o To f i n d t h e f o r c e w h i c h c a u s e s t h e damage i n d i r e c t i o n o f D, Es i s d e r i v a t e d w i t h r e s p e c t t o D a s : dE F = . S = K A ( 4 . 4 ) D d D d v ' When we know t h e s t r u c t u r a l p r o p e r t y K i n a s p e c i f i e d d i r e c t i o n and t h e c r o s s - s e c t i o n o f t h e damaged p a r t n o r m a l t o t h a t d i r e c t i o n , t h e g e n e r a t e d f o r c e c a n be c a l c u l a t e d . 4 . 4 . 2 - G e n e r a t e d F o r c e I n M i n o r C o l l i s i o n and M o d i f i c a t i o n F o r M a j o r C o l l i s i o n As m e n t i o n e d e a r l i e r , M i n o r s k y ' s s e m i - e m p i r i c a l e q u a t i o n i s n o t a b l e t o p r e d i c t damage c a u s e d i n m i n o r c o l l i s i o n s . J o n e s [8] has i n v e s t i g a t e d m i n o r c o l l i s i o n s and h a s m o d i f i e d t h e M i n o r s k y ' s e q u a t i o n . He a l s o i n t r o d u c e d a r e l a t i o n b e t w e e n t h e p r o p o r t i o n a l i t y c o n s t a n t k o f 88 M i n o r s k y ' s e q u a t i o n and t h e y i e l d s t r e n g t h o f t h e s t r u c t u r a l m a t e r i a l s . I n h i s m o d i f i c a t i o n , he assumed t h a t d u r i n g t h e c o l l i s i o n t h e membrane f o r c e s i n t h e e l e m e n t s were d o m i n a n t and c o n t r o l t h e d e f o r m a t i o n and f r a c t u r e o f t h e members. He c o n c l u d e d t h a t i n m i n o r c o l l i s i o n t h e r a t i o o f d e f l e c t i o n t o t h e s p a n o f t h e e l e m e n t must be c o n s i d e r e d . J o n e s [21] c o n s i d e r e d a r i g i d p e r f e c t l y p l a s t i c beam o f l e n g t h 2L w i t h f u l l y c l a m p e d s u p p o r t s a t t h e e n d s . A c o n c e n t r a t e d l o a d P was a p p l i e d a t t h e m i d s p a n o f t h e beam. The e n e r g y a b s o r b e d b y t h e beam i n t h e p l a s t i c d e f o r m a t i o n was c a l c u l a t e d . The a b s o r b e d e n e r g y w h i c h he c a l c u l a t e d c o u l d be w r i t t e n a s : Df 2 E = 2 a V ( — ) ( 4 . 5 ) S 0 2L ^ ' where a i s t h e y i e l d s t r e s s o f t h e m a t e r i a l , o J V i s t h e vo lume o f damaged m a t e r i a l . D i s t h e l a t e r a l d e f l e c t I o n o f t h e beam, f 2L i s t h e l e n g t h o f t h e e l e m e n t . J o n e s p l o t t e d e n e r g y v e r s e s vo lume u s i n g h i s e q u a t i o n ( 4 . 5 ) f o r d i f f e r e n t v a l u e s o f — ^ . He w a s • t h u s a b l e t o show t h a t f o r D / 2 L < 1 / 3 h i s e q u a t i o n c a n c o v e r l o w e n e r g y c o l l i s i o n s on M i n o r s k y ' s g r a p h , and f o r D ^ / 2 L =1 /3 h i s e q u a t i o n p r e d i c t s damage e s t i m a t e s f o r m a j o r c o l l i s i o n s w h i c h a r e s i m i l a r t o t h o s e p r e d i c t e d b y M i n o r s k y ' s e q u a t i o n . To f i n d t h e m a g n i t u d e o f t h e g e n e r a t e d f o r c e d u r i n g t h e a c c i d e n t , t h e v a l u e o f V f r o m E q u . ( 4 . 3 ) i s s u b s t i t u t e d i n E q u . ( 4 . 5 ) , t h e n E s i s d i f f e r e n t i a t e d w i t h r e s p e c t t o D we t h u s o b t a i n : Df 2 F = 6 a ( — ) A ( 4 . 6 ) D 0 2L d F o r m a j o r c o l l i s i o n s w i t h Df / 2 L =1 /3 we h a v e : 89 ( 4 . 7 ) C o m p a r i n g E q u a t i o n s ( 4 . 7 ) and ( 4 . 4 ) , we see t h a t t h e p r o p o r t i o n a l i t y 2 c o n s t a n t K = a . 3 o The m a g n i t u d e o f t h e c o l l i s i o n f o r c e s i n m i n o r c o l l i s i o n s i s e s t i m a t e d b y u s i n g E q u . ( 4 . 6 ) w i t h 0 < D / 2 L < 1 / 3 . 4 . 4 . 3 - G e n e r a t e d F o r c e When H u l l I s Cut by a S h a r p O b j e c t When a s h i p r u n s o v e r a s h a r p r e e f t h e g e n e r a t e d damage i s n o t v o l u m e t r i c . T h e r e i s o f t e n no s i g n i f i c a n t amount o f d i s t o r t e d i n t e r n a l s t r u c t u r e b u t i n s t e a d a s i g n i f i c a n t amount o f b o t t o m p l a t e r u p t u r e and t e a r i n g . I n s u c h c a s e s , M i n o r s k y ' s e q u a t i o n i s n o t a p p r o p r i a t e f o r p r e d i c t i n g t h e t y p e o f damage i n c u r e d . Vaughan [ 1 4 , 1 5 ] c o n s i d e r e d t h i s p r o b l e m and showed t h a t i n s u c h c a s e s , some p a r t o f t h e k i n e t i c e n e r g y i s a b s o r b e d b y d i s t o r t i o n o f t h e s t r u c t u r a l members and some o t h e r p a r t by t e a r i n g p r o c e d u r e . The e n e r g y a b s o r b e d b y d i s t o r t i o n o f t h e s t r u c t u r a l members was p r o p o r t i o n a l t o t h e v o l u m e o f t h e damaged m a t e r i a l s w h i l e t h e e n e r g y o f t e a r i n g was p r o p o r t i o n a l t o t h e f r a c t u r e d a r e a o f t h e t o r n p l a t e . T h e r e f o r e t h e t o t a l f o r c e on t h e h u l l w h i c h c a u s e s t e a r i n g and d e n t i n g o f t h e p l a t e c o u l d be w r i t t e n a s : F = F + F D C B where F^ i s t h e c u t t i n g f o r c e and F^ i s t h e b e n d i n g and c u r l i n g f o r c e on t h e m e t a l l i c s t r u c t u r e . Vaughan [9] s t u d i e d t h e t e a r i n g f o r c e o f m i l d s t e e l p l a t e s t h r o u g h 90 a s e r i e s o f e x p e r i m e n t s . He i s o l a t e d t h e t e a r i n g f o r c e i n h i s e x p e r i m e n t s and showed t h a t t h e t e a r i n g f o r c e was r e l a t e d t o t h e t h i c k n e s s o f t h e p l a t e a s : a F = F T c c Where T i s t h e p l a t e t h i c k n e s s F and a a r e two c o n s t a n t s c — N He a l s o showed t h a t f o r m i l d s t e e l F = 5500 and a =1 .5 where T i s c m e a s u r e d i n mm. I n o r d e r t o e v a l u a t e t h e c u r l i n g f o r c e F g , and i t s r e l a t i o n w i t h t h e p r o p e r t i e s o f t h e s t r u c t u r a l members, J o n e s ' a p p r o a c h i s f o l l o w e d . I f t h e r e i s a n o t c h i n t h e f i x e d - f i x e d member w h i c h J o n e s i n t r o d u c e d i n [ 2 1 ] , t h e n t h e r e w o u l d be no membrane f o r c e a t t h e t o r n p o i n t . T h i s c a u s e s b e n d i n g w i t h o u t any r e s i s t a n c e f r o m t h e i n p l a n e f o r c e s . U s i n g t h e E q u . 23a o f [21] w i t h t h e v a l u e o f B= -—^ f o r no membrane f o r c e 2 2a B H c o n d i t i o n we w o u l d h a v e : — - 1 w h i l e P = — -P= C 2 L where B i s t h e b r e a d t h o f t h e e l e m e n t . H i s t h e t h i c k n e s s o f t h e e l e m e n t , o Then t h e e n e r g y o f d e f o r m a t i o n w h i c h i s t h e work o f t h e b e n d i n g f o r c e F i s : B E = B " D - 2 D D F dD =2 a B H D / 2L = 4 a B H L — = 2 a V ^ - i B f o r o 2L 2L o 2L 2L Where V i s t h e vo lume o f d e f o r m e d e l e m e n t . S u b s t i t u t i n g f o r V f r o m E q u . ( 4 . 3 ) i n t h e above e q u a t i o n t h e a b s o r b e d e n e r g y w o u l d b e : 91 E . 2 a A J L 5£ B 0 d 2L 2L T a k i n g t h e d e r i v a t i v e o f w i t h r e s p e c t t o D f w i l l l e a d t o t h e f o r c e o f b e n d i n g i n d i r e c t i o n o f . t h e n : F = — 4 a A B dD 0 d 2L 2L f D i s t o r t i o n e n e r g y i n t h i s c a s e i s r e l a t e d t o t h e v o l u m e o f t h e damaged m a t e r i a l s , so t h a t M i n o r s k y ' s r e l a t i o n i s a p p l i c a b l e , and as J o n e s c o n c l u d e d , w i t h D / 2L = 1 / 3 , t h e M i n o r s k y ' s mode l i s r e p r o d u c e d . Then t h e above e q u a t i o n w i t h Df / 2L = 1 / 3 c o i n c i d e s w i t h M i n o r s k y ' s e q u a t i o n . U s u a l l y t h e d e f o r m a t i o n e x t e n t D i s b i g compared t o t h e t h i c k n e s s o f t o r n p l a t e , t h a t i s , D > H , t h e r e f o r e Df / 2L >H / 2 L , when D f 0 0 f 1 2L , H 0 / -L. c a n be t a k e n as 1 / 3 and we w o u l d h a v e : F = a A ( 4 . 8 ) B 9 0 d V ' T h e r e f o r e t h e t o t a l f o r c e i n a s p e c i f i e d d i r e c t i o n D^can be w r i t t e n a s : _ a ••• F w = F c T + — V d < 4- 9> 4 . 5 - D e t e r m i n a t i o n o f The P e n e t r a t i o n o f t h e E x t e r n a l O b j e c t i n t h e H u l l and C a l c u l a t i o n o f t h e C r o s s - s e c t i o n o f Damaged P a r t To c a l c u l a t e t h e p e n e t r a t i o n o f t h e e x t e r n a l o b j e c t i n t o t h e h u l l a t t h e c o l l i s i o n p o i n t t h e p r o f i l e o f t h e c r o s s - s e c t i o n o f t h e s h i p and i t s p o s i t i o n w i t h r e s p e c t t o t h e e x t e r n a l o b j e c t i s u s e d . F o r t h e c a s e t h a t t h e p r o f i l e o f t h e c r o s s - s e c t i o n o f t h e s h i p i s n o t a v a i l a b l e , t h e 92 fo l l owing approximation i s used to define the p r o f i l e of the c r o s s - s e c t i o n and c a l c u l a t e penetrat ion of the ex terna l object in to the h u l l . When the p o s i t i o n of the contact po in t along the ship i s known the s t a t i o n (beam element) invo lved i n the c o l l i s i o n i s c l e a r l y i d e n t i f i e d . Y H P — z F i g . 4 . 2 - Cross-Section Area of the Beam Element and Position of the External Object. A schematic view of the c r o s s - s e c t i o n of the beam element, which i s i n contact wi th the ex terna l objec t , and i t s p o s i t i o n with respect to the coordinates system i s shown i n F i g . (4 .2 ) . This c r o s s - s e c t i o n i s approximated by: y - H [ ( 2Z m 1 ] and y = H (- — ) m 1 ] f o r - z > 0 for z < 0 93 A A A A where y and z a r e m e a s u r e d w i t h r e s p e c t t o t h e l o c a l o - x y z r e f e r e n c e s y s t e m . H i s t h e a v e r a g e v a l u e , o f t h e d e p t h o f t h e s t a t i o n . B r i s t h e a v e r a g e v a l u e o f t h e b r e a d t h o f t h e s t a t i o n . m i s a shape p a r a m e t e r w h i c h v a r i e s f r o m e l e m e n t t o e l e m e n t . The m a g n i t u d e o f m depends on t h e p o s i t i o n o f t h e s t a t i o n (beam e l e m e n t ) a l o n g t h e s h i p . F o r example (m >10) i s u s e d f o r t h e m i d s h i p s t a t i o n s w h i l e t h e s m a l l e r v a l u e s a r e u s e d f o r s t a t i o n s w h i c h a r e c l o s e t o e i t h e r end o f t h e s h i p . C o o r d i n a t e s o f t h e e x t e r n a l o b j e c t w i t h r e s p e c t t o t h e l o c a l s y s t e m o f r e f e r e n c e a r e c a l c u l a t e d a t e a c h i n s t a n t b y a t r a n s f o r m a t i o n m a t r i x . R o t a t i o n and t r a n s l a t i o n o f t h e l o c a l c o o r d i n a t e w i t h r e s p e c t t o t h e i n e r t i a l c o o r d i n a t e s y s t e m a r e c o n s i d e r e d i n t h i s t r a n s f o r m a t i o n a s : z b c o s 9 s i n 9 - s i n 9 c o s 9 b N Z , - Z b N w h e r e , 6 i s t h e a n g l e o f r o t a t i o n o f t h e s t a t i o n a r o u n d t h e x - a x i s ( r o l l i n g ) . A Y and Z d e f i n e p o s i t i o n o f p o i n t o , c e n t e r o f t h e c r o s s - s e c t i o n , N N i n t h e i n e r t i a l f r a m e . A P o s i t i o n o f p o i n t o i s c a l c u l a t e d t h r o u g h t h e n o d a l c o o r d i n a t e s o f t h e beam e l e m e n t and t h e shape f u n c t i o n [N] w h i c h i s e v a l u a t e d a t t h e c o l l i s i o n p o i n t a l o n g t h e e l e m e n t . T h a t i s ; Y N ' = [ N Q ] { U } 94 where {U} i s the nodal p o s i t i o n of the element. z F i g . 4 . 3 - Damaged on the Cross-Section Area of the Beam Element. To f i n d the extent of damage exerted on the s t a t i o n , coordinates of. two po ints on the outer s h e l l of the s e c t i o n , as A and B shown i n F i g . ( 4 . 3 ) , should b e . c a l c u l a t e d . These coordinated are determined through the equation of, the c r o s s - s e c t i o n as: when z, £ 0 b 2z m y L = H [ ( ^ ) - i ] - i y — j — Br , n b . m and Z l - _ ( ! + _ ) And when z, < 0 D 2z m and z — L — _1 5i ( i + ^ >» 2 U H ; For c a l c u l a t i o n of the penetra t ion of the ex terna l object in to the h u l l depending on the r e l a t i v e motion of the ship and p o s i t i o n of the e x t e r n a l b a r r i e r four d i f f e r e n t cases could be cons idered. These four 95 c a s e s a r e d e s c r i b e d i n A p p e n d i x ' D ' . I n e a c h c a s e P and P t h e y z A A p e n e t r a t i o n i n z and y d i r e c t i o n r e s p e c t i v e l y and A^ t h e c r o s s - s e c t i o n A o f t h e damaged vo lume n o r m a l t o x - a x i s a r e c a l c u l a t e d . 4 . 6 - C a l c u l a t i o n o f F o r c e s i n T h r e e P r i n c i p a l D i r e c t i o n s I n t h e f o r m e r s e c t i o n s we n o t i c e d t h a t t h e g e n e r a t e d f o r c e i n e a c h d i r e c t i o n depends on t h e c r o s s - s e c t i o n a r e a o f t h e damaged m a t e r i a l s n o r m a l t o t h a t s p e c i f i e d d i r e c t i o n and t h e s t r u c t u r a l s p e c i f i c a t i o n o f t h e s h i p . F o l l o w i n g t h e p r o c e d u r e shown i n A p p e n d i x ' D ' t h e c r o s s - s e c t i o n a l a r e a o f t h e damaged s t r u c t u r e i n t h e x - d i r e c t i o n , a l o n g t h e s h i p a x i s , A i s c a l c u l a t e d . The c r o s s - s e c t i o n a l a r e a o f t h e g e n e r a t e d damage i n t h e y A and z d i r e c t i o n a r e , r e s p e c t i v e l y : A = P P and A = P P y z x z y x where P and P a r e d e s c r i b e d i n S e c t i o n ( 4 ^ 5 ) . y z A c e r t a i n p o r t i o n o f t h e s e s u r f a c e s i s a c t u a l l y f i l l e d w i t h s t r u c t u r a l members. The r a t i o b e t w e e n t h e s o l i d c r o s s - s e c t i o n and t h e g e o m e t r i c c r o s s - s e c t i o n i s p r o v i d e d by t h e d a t a f r o m t h e s h i p s t r u c t u r e . Then t h e s o l i d c r o s s - s e c t i o n a l a r e a w o u l d b e : A = V A A = V > A A = - 0 A x s x x y s y y z s z z where \b , ib and \1> a r e t h e r a t i o s b e t w e e n t h e g e o m e t r i c and s o l i d x y z a c r o s s - s e c t i o n a l a r e a s . T h e r e f o r e t h e g e n e r a t e d f o r c e s i n t h e l o c a l c o o r d i n a t e s y s t e m w o u l d b e : F = K A , F = K A , and F = K A x x s y y s z z s 96 The components o f t h e g e n e r a t e d f o r c e a l o n g t h e a x e s o f t h e g l o b a l c o o r d i n a t e 0 - X Y Z a r e c a l c u l a t e d t h r o u g h t h e t r a n s f o r m a t i o n : ( F ) C O S 7 0 • s i n 7 0 1 s m 7 0 C O S 7 z where 7 i s t h e a n g l e o f r o t a t i o n o f t h e s h i p a b o u t t h e v e r t i c a l a x i s ( y a w ) . The a n g l e o f r o t a t i o n a b o u t t h e h o r i z o n t a l a x i s ( p i t c h ) i s c o n s i d e r e d t o be s m a l l i n t h e above c a l c u l a t i o n s . 4.7- D e t a i l o f t h e S o l u t i o n o f t h e E q u a t i o n o f M o t i o n The g l o b a l e q u a t i o n o f m o t i o n o f t h e beam ( s h i p ) i s r e p r e s e n t e d i n E q u . ( 4 . 1 ) as : [M ]{U} + [C ] {U } + [K ] {U } = {Q} M ( l + a ) X = T - F s r x The i n i t i a l c o n d i t i o n s f o r t h e e q u a t i o n s a r e X = « , X = 0 , {U } = {0 } and {U }= {U } 0 0 As m e n t i o n e d , t h e beam i s c o n s i d e r e d a x i a l l y r i g i d , t h e n t h e e q u a t i o n o f a x i a l m o t i o n o f t h e s h i p i s u n c o u p l e d w i t h t h e s e t o f i t s l a t e r a l e q u a t i o n s o f m o t i o n . The e q u a t i o n s o f l a t e r a l m o t i o n a r e c o u p l e d . D u r i n g t h e c o l l i s i o n , t h e p r o p e l l e r t h r u s t f o r c e T i s assumed t o r be i n b a l a n c e w i t h t h e h y d r o d y n a m i c r e s i s t a n c e and s m a l l compared t o t h e a x i a l component o f t h e c o l l i s i o n f o r c e . T h e r e f o r e i t s e f f e c t i n t h e s h i p 97 m o t i o n c a n be n e g l e c t e d . Then t h e a x i a l e q u a t i o n o f m o t i o n c a n be w r i t t e n a s : M (1+ a )X = - F ( 4 . 1 0 ) S X A c c e l e r a t i o n o f t h e s h i p d u r i n g e a c h t i m e s t e p i s c a l c u l a t e d t h r o u g h t h e E q u . ( 4 . 1 0 ) . T h i s v a l u e o f a c c e l e r a t i o n i n c o n j u n c t i o n w i t h t h e f o l l o w i n g e q u a t i o n s AS = AS + X d t and X = X + A S d t + X d t o o o a r e u s e d t o f i n d t h e v e l o c i t y AS and p o s i t i o n X , a t t h e end o f e a c h t i m e i n t e r v a l d t . T h e s e v a l u e s a r e t a k e n as t h e i n i t i a l v a l u e s f o r n e x t t i m e i n t e r v a l . To c a l c u l a t e p o s i t i o n o f t h e s h i p t h e s e t o f e q u a t i o n s o f l a t e r a l m o t i o n a r e s i m u l t a n e o u s l y s o l v e d i n t h e same t i m e i n t e r v a l . These e q u a t i o n s a r e u n c o u p l e d b y t r a n s f e r r i n g t o t h e g e n e r a l i z e d c o o r d i n a t e s }. The method o f u n c o u p l i n g t h e s e e q u a t i o n s i s p r e s e n t e d i n page 194 o f [ 4 6 ] . A c c o r d i n g t o t h a t p r o c e d u r e t h e e q u a t i o n o f l a t e r a l m o t i o n o f t h e s h i p i s u n c o u p l e d . ( I n A p p e n d i x ' C t h e u n c o u p l i n g p r o c e d u r e i s s h o w n ) . The n o r m a l i z e d u n c o u p l e d e q u a t i o n i n g e n e r a l i z e d c o o r d i n a t e s w o u l d b e : •• . • q. £. + <<* + B w 2 ) £. + co2 £. = — — ( 4 . 1 1 ) •t 2 k. where = — - i s t h e i - t h n a t u r a l f r e q u e n c y o f t h e w h o l e s y s t e m . The method o f d e t e r m i n i n g t h e m a g n i t u d e o f a and 0 are d e s c r i b e d i n s e c t i o n ( 4 . 8 ) . The damping r a t i o r/^ i s d e f i n e d t h r o u g h t h e r e l a t i o n 2 2 i i . u . = a + i3 o. 98 and t h e e q u a t i o n o f m o t i o n c a n be w r i t t e n a s : £. + 2 rj. w. £. + ooZ. £. = ( 4 . 1 2 ) m. The above e q u a t i o n i s t h e s t a n d a r d s e c o n d o r d e r e q u a t i o n o f m o t i o n o f a f o r c e d v i b r a t i n g s y s t e m . As l o n g as we have t h e m a g n i t u d e o f t h e e x e r t e d f o r c e q^ we c a n f i n d t h e d i s p l a c e m e n t £^. I n t h e c o l l i s i o n p r o b l e m t h e e x e r t e d f o r c e on t h e s y s t e m i s g e n e r a t e d t h r o u g h t h e c o l l i s i o n and must be d e v e l o p e d t h r o u g h t h e i n t e g r a t i o n p r o c e s s . I n t e g r a t i o n i s done f o r a c e r t a i n t i m e i n t e r v a l d t . A t t h e end o f e a c h t i m e s t e p t h e l o c a t i o n o f t h e s h i p r e l a t i v e t o t h e e x t e r n a l o b j e c t and t h e m a g n i t u d e o f t h e c o n t a c t f o r c e i s c a l c u l a t e d . T h i s v a l u e i s t a k e n as t h e f o r c e on t h e s y s t e m d u r i n g t h e n e x t t i m e s t e p . A s e c o n d o r d e r f i n i t e d i f f e r e n c e method o f i n t e g r a t i o n i s u s e d t o s o l v e e a c h o f t h e d i f f e r e n t i a l E q s . o f ( 4 . 1 2 ) . The i n i t i a l c o n d i t i o n s a r e t h e m a g n i t u d e o f d i s p l a c e m e n t £^  and t h e v e l o c i t y £^  i n t h e g e n e r a l i z e d c o o r d i n a t e s . The method w h i c h p r e s e n t e d i n page 108 o f [47] i s f o l l o w e d t o s o l v e t h e d i f f e r e n t i a l e q u a t i o n s . F o r a h e a d on c o l l i s i o n , t h e i n i t i a l l a t e r a l d i s p l a c e m e n t and v e l o c i t y o f t h e s h i p a r e z e r o . T h e r e f o r e a t t=0 t h e d i s p l a c e m e n t v e c t o r (U) and t h e v e l o c i t y v e c t o r {U} a r e z e r o , t h e n i t i s c o n c l u d e d t h a t a t t=0 , } and } a r e z e r o v e c t o r s w h i c h i m p l i e s t h e v a l u e s o f £^ and £. a r e z e r o f o r a l l o f t h e modes ( d i f f e r e n t v a l u e s o f I ) . F o r o b l i q u e c o l l i s i o n s t h e n o d a l v a l u e s o f t h e l a t e r a l v e l o c i t y j u s t b e f o r e c o l l i s i o n s h o u l d be c a l c u l a t e d f r o m t h e g l o b a l v e l o c i t y o f t h e s h i p . T h e s e v a l u e s w o u l d be c o n s i d e r e d as t h e i n i t i a l v e l o c i t y v e c t o r {U}. The v e c t o r } i s t h e n c a l c u l a t e d b y m u l t i p l y i n g {U } w i t h 99 the inverse of transformation matrix [T ] ( Transform matrix [T] i s introduced i n appendix ' B ' ) . As an example, i f magnitude of the l a t e r a l v e l o c i t i e s at the front and a f t of the ship are known, the nodal v e l o c i t i e s are c a l c u l a t e d according to the fo l lowing procedure. C o n f i g u r a t i o n of the ship and the se l ec ted nodes along i t are schemat ica l ly shown i n F i g . (4 .4 ) . V e l o c i t y of node number 1 i n the z - d i r e c t i o n normal to the ship axis i s « and that of the l a s t node a f t (number n) normal to the ship ax is i s « . Therefore we have : f r t At t - 0 u = AS 2 a f t and U = 49 n+1 f r t F i g . 4.4 The Nodal Displacements Along the Beam Model. I n i t i a l l y , the l a t e r a l v e l o c i t y of the ship i n v e r t i c a l d i r e c t i o n i s zero; that i s : u(5l-4) •= 0 and u ( 5 £ ) •= 0 for I - l , n 100 A l s o , t h e r o l l i n g v e l o c i t y o f t h e s h i p a t t = 0 i s z e r o ; t h a t i s : u ( 5 i - 2 ) = 0 f o r I = l , n I n t h i s c a s e as t h e s h i p i s m o v i n g l a t e r a l l y , a c e n t e r o f r o t a t i o n f o r t h e w h o l e s h i p on t h e w a t e r l e v e l ( X - Z p l a n e ) c o u l d be c o n s i d e r e d and t h e a n g u l a r v e l o c i t y o f t h e s h i p i s c a l c u l a t e d . T h i s g i v e s t h e m a g n i t u d e o f r o t a t i o n a l v e l o c i t i e s o f t h e nodes a r o u n d t h e y - a x i s . T h e n , AS - AS . c. » . frt aft u ( 5 t - 1)= -s where L g i s t h e l e n g t h o f t h e s h i p . And f i n a l l y t h e n o d a l v a l u e s o f t h e v e l o c i t i e s i n z d i r e c t i o n a r e c a l c u l a t e d a s : AS - AS / a • o \ . ->r f r t aft U ( 5 - L - 3 ) = AS + X . = aft t L S where X^ i s t h e p o s i t i o n o f t h e i - t h node a l o n g t h e s h i p . I n t h i s way t h e w h o l e v e l o c i t y v e c t o r i s c a l c u l a t e d w h i c h i n c o n j u n c t i o n w i t h t h e i n i t i a l p o s i t i o n v e c t o r o f t h e s h i p c a n be u s e d as t h e i n i t i a l c o n d i t i o n s f o r t h e i n t e g r a t i o n p r o c e s s . 101 5- SOME NUMERICAL SIMULATIONS AND DISCUSSION I n t h i s c h a p t e r some a p p l i c a t i o n o f t h e methods i n t r o d u c e d i n t h e f o r m e r c h a p t e r s a r e d e m o n s t r a t e d . The r e l i a b i l i t y o f t h e s e s i m u l a t i o n s i s t e s t e d by c o m p a r i n g t h e c a l c u l a t e d r e s u l t s w i t h t h e c o l l e c t e d d a t a whenever i t i s a v a i l a b l e . I c e b r e a k i n g p r o c e d u r e i s s t u d i e d u n d e r two d i s t i n c t s i t u a t i o n s as c o n t i n u o u s and ramming modes. I n t h e c o n t i n u o u s i c e b r e a k i n g p r o c e d u r e , t h e m o t i o n o f a ' S t a n d a r d I c e B r e a k e r ' i s a n a l y z e d . I t i s p o s s i b l e t o s p e c i f y a s a f e l i m i t i n g v e l o c i t y f o r t h e s h i p , b y e x a m i n i n g t h e v a r i a t i o n o f maximum b e n d i n g moment a l o n g t h e s h i p and i t s r e l a t i o n t o t h e s h i p v e l o c i t y , D u r i n g t h e ramming mode, f o r d i f f e r e n t v a l u e s o f t h e ramming v e l o c i t i e s , t h e maximum b e n d i n g moment a l o n g t h e ' S t a n d a r d I c e B r e a k e r ' i s c a l c u l a t e d A l s o i n e a c h c a s e , v a r i a t i o n o f c o n t a c t f o r c e b e t w e e n t h e i c e r i d g e and t h e i c e b r e a k e r i s e s t i m a t e d . S t r u c t u r a l s p e c i f i c a t i o n o f t h e s h i p l e a d s t o t h e j a l l o w a b l e b e n d i n g moment and c o n t a c t f o r c e on t h e h u l l . By c o m p a r i n g t h e s e v a l u e s t o t h e c a l c u l a t e d v a l u e s o f t h e maximum b e n d i n g moment and t h e c o n t a c t f o r c e , t h e maximum a l l o w a b l e ramming s p e e d o f a s p e c i f i e d i c e b r e a k e r c o u l d be d e c i d e d . The p r e s e n t a n a l y s i s c o u l d a l s o be u s e d i n t h e s t a g e o f d e s i g n o f t h e i c e - b r e a k e r t o d e t e r m i n e t h e m a i n s t r u c t u r a l p r o p e r t i e s o f t h e s h i p . F o r t h e s h i p c o l l i s i o n p r o b l e m , t h e damage e x t e n t on t h e h u l l o f a ' S t a n d a r d T a n k e r S h i p ' i s d e t e r m i n e d . I t i s shown how t h e s u g g e s t e d a p p r o a c h c a n be u s e d t o f i n d t h e s a f e r e g i o n i n t h e s h i p , i f t h e r e i s a n y , d u r i n g an a c c i d e n t . P a t t e r n s o f t h e g e n e r a t e d damage a r e e s t i m a t e d f o r c o l l i s i o n o f t h e s h i p w i t h two d i f f e r e n t t y p e s o f e x t e r n a l o b j e c t s . The f i r s t o b j e c t i s assumed t o g e n e r a t e o n l y d e n t s on t h e h u l l d u r i n g t h e c o l l i s i o n w h i l e t h e s e c o n d one c a u s e s d e n t s and r u p t u r e o f t h e h u l l . The e f f e c t o f s h i p v e l o c i t y on t h e e x t e n t o f g e n e r a t e d damage i s a l s o 102 s t u d i e d . T h i s c a n be u s e d t o d e t e r m i n e a maximum a l l o w a b l e s a f e s p e e d f o r any s p e c i f i e d s h i p t h r o u g h c o a s t a l r o u t e s and s h a l l o w w a t e r s . 5.1- Ice Breaking Procedure 5 .1 .1 - In troduct ion As m e n t i o n e d above we c o n s i d e r c o n t i n u o u s i c e - b r e a k i n g and ramming . I n e a c h c a s e t h e maximum b e n d i n g moment a l o n g t h e s h i p i s c a l c u l a t e d and p l o t t e d a g a i n s t t i m e . The e f f e c t o f t h e s h i p v e l o c i t y and i c e t h i c k n e s s on t h e maximum b e n d i n g moment and r o l l i n g a n g l e o f t h e s h i p i n t h e c o n t i n u o u s i c e b r e a k i n g mode i s i n v e s t i g a t e d . The r e l a t i o n b e t w e e n f r e q u e n c y o f t h e g e n e r a t e d i c e l o a d and v e l o c i t y o f t h e s h i p i s p r e s e n t e d f o r i c e f i e l d s o f d i f f e r e n t t h i c k n e s s e s , and t h e e f f e c t o f t h i s f r e q u e n c y on t h e r e s p o n s e o f t h e s h i p s t r u c t u r e i s e x a m i n e d . A s i s t o be e x p e c t e d i t i s f o u n d t h a t when t h e f r e q u e n c y o f t h e i c e l o a d i s c l o s e t o a n a t u r a l f r e q u e n c y o f t h e s h i p - m o t i o n , t h e l e v e l o f maximum b e n d i n g moment a n d / o r r o l l i n g a n g l e o f t h e s h i p g r o w s . The maximum a t t a i n a b l e power o f a n i c e b r e a k e r s e t s a l i m i t on t h e v e l o c i t y d u r i n g i c e b r e a k i n g . The power n e e d e d t o b r e a k s h e e t - i c e a t c o n s t a n t s p e e d i s c a l c u l a t e d b y m u l t i p l y i n g t h e a x i a l component o f t h e i c e f o r c e and t h e s h i p v e l o c i t y . I n t h e ramming mode, t h e e f f e c t o f s h i p v e l o c i t y on t h e p e a k i m p a c t l o a d and t h e r e s p o n s e o f t h e s h i p s t r u c t u r e i s i n v e s t i g a t e d . The m a g n i t u d e s o f t h e maximum b e n d i n g moment a l o n g t h e i c e b r e a k e r and t h e c o n t a c t f o r c e on t h e bow d u r i n g t h e ramming p r o c e d u r e a r e c a l c u l a t e d and p l o t t e d v e r s u s t i m e . The e f f e c t o f s h i p s t r u c t u r a l s t i f f n e s s on t h e maximum b e n d i n g moment and c o n t a c t f o r c e i s a l s o s t u d i e d . I n t h e n u m e r i c a l example w h i c h i s i n t r o d u c e d i n t h i s p a r t , an i c e - b r e a k i n g s h i p o f l e n g t h 90 m e t e r s and mass 6800 m e t r i c t o n n e s i s s e l e c t e d 103 as the 'Standard Ice B r e a k e r ' . The bow angle of t h i s ship i s assumed to be 20 degrees. 1 2 3 4 5 6 7 8 9 10 11 F i g . 5 .1- The Standard Ship and the Selected Stations. Ten d i s t i n c t s ta t ions are se l ec ted along t h i s s h i p . F i g . ( 5 . 1 ) schemat ica l ly shows the s ta t ions and node numbers along the s h i p . Geometrical p r o p e r t i e s o f each s t a t i o n are represented i n Table (5 .1 ) . P lo t s o f mass and buoyancy d i s t r i b u t i o n are represented i n F i g s . (5.2a) and (5 .2b) . The "Standard I ce-breaker" has l ength , mass, bow angle , and midship dimensions s i m i l a r to those f o r the Kigoriak g iven i n [53]. Dimensions of the other sec t ions a long the ship and moment o f i n e r t i a o f each s e c t i o n were est imated. i Fundamental frequency o f t h i s ship i s equal to that of Kigoriak. I i i T h i s sh ip i s modeled as a beam composed o f ten uniform elements. Each element i s assumed to have constant f l e x u r a l and t o r s i o n a l r i g i d i t y , breadth, j d r a f t and cross s e c t i o n area equal to the average | i dimension of i t s corresponding s t a t i o n along the ship which i s shown i n j Table (5 .1 ) . F i g . (5.3) represents a schematic view of the beam model of the ship and i t s nodal freedom. 104 250 Fig. 5.2 - a) Mass distribution, b) Buoyancy distribution along the 'Standard Ice-Breaker' 105 r STATION STATION STATION STATION STATION MOMENT OF INERTIA No. LENGTH BREADTH DRAFT SHAPE FCT. I I meter meter meter y 4 z 4 m m 1 20 12 4 5 4 6 5 2 10 12 4 5 4 7 6 3 10 13 5 5 6 8 7 4 10 14 7 4 7 10 9 5 10 16 8 5 7 12 10 6 6 16 8 5 7 12 10 7 6 16 8 5 7 12 10 8 6 16 6 2 6 10 9 9 6 16 5 0 4 8 8 10 6 12 4 0 3 7 6 Table 5 .1 - Specification of the 'Standard Ice-Breaker' at the Stations. 514 F i g . 5 .3- Schematic View of the Three Dimensional Beam Model of the Ship. Added mass o f t h i s sh ip i n surge i s taken as a g - 0.1 i n agreement with the va lue est imated by the r e l a t i o n suggested i n p . 47 [51]. And i n the l a t e r a l motion as a - 0.75 i n agreement with the e s t imat ion done through the method represented i n page 722 [45]. Added mass o f t h i s sh ip i n sidewise motion accord ing to the r e l a t i o n i n [51] i s a l so c a l c u l a t e d and i s equal to 0.6 reasonably c lose to 0.75 f o r v e r t i c a l motion. Data f o r the computer program comes from these s p e c i f i c a t i o n s of the s h i p . Information from the dimension, weight, and m a t e r i a l p r o p e r t i e s o f the ship s t r u c t u r e , enable the 106 [M ] and [K ] m a t r i c e s t o be d e v e l o p e d . N a t u r a l f r e q u e n c i e s o f m o t i o n o f t h e s h i p a r e d e t e r m i n e d t h r o u g h t h e c a l c u l a t i o n s o f t h e e i g e n - v a l u e s o f t h e m a t r i x - l [[M] [k] ] . The f i r s t t w e l v e n a t u r a l f r e q u e n c i e s o f t h e s h i p m o t i o n a r e r e p r e s e n t e d i n T a b l e ( 5 . 2 ) . The f i r s t f i v e o f t h e s e f r e q u e n c i e s a r e r e l a t e d t o t h e r i g i d b o d y m o t i o n w h i l e t h e r e s t o f t h e f r e q u e n c i e s a r e r e l a t e d t o t h e e l a s t i c d e f l e c t i o n o f t h e s h i p . FREQUENCY No . C Y C L E / S E C 1 0 . 0 RELATED TO SWAY 2 0 . 0 RELATED TO YAW 3 0 . 1 5 4 RELATED TO HEAVE 4 0 . 1 8 6 RELATED TO PITCH 5 0 . 4 2 6 RELATED TO ROLL 6 1 .638 1 s t STRUCTURAL FREQ. ( v e r t i c a l v i b . ) 7 1 . 7 5 1 2 n d STRUCTURAL FREQ. ( l a t e r a l v i b . ) 8 4 . 3 0 7 3 r d STRUCTURAL FREQ. ( t o r s i o n a l v i b . ) 9 4 . 6 6 1 : 10 8 . 2 4 2 : 11 8 . 8 5 0 : 12 1 3 . 5 9 0 : T a b l e 5 . 2 - Frequencies of the Motion of the Ice Breaker. A p r o p o r t i o n a l damping c o e f f i c i e n t [C] i s d e v e l o p e d t h r o u g h t h e e q u a t i o n : [C] = a [M]+ B [K] where c o n s t a n t s a and B are c a l c u l a t e d b y t h e p r o c e d u r e shown i n S e c t i o n 3 . 8 . . I n E q u . ( 3 . 4 1 ) t h e c o e f f i c i e n t s o f damping i n h e a v e and p i t c h a r e c o n s i d e r e d t o be 0 . 0 2 and 0 . 0 2 2 r e s p e c t i v e l y . These v a l u e s compared w e l l w i t h t h o s e o b t a i n e d f r o m t h e c u r v e s r e p r e s e n t e d i n p . 299 o f [52] f o r damping c o e f f i c i e n t s . A n d t h e c a l c u l a t e d damping r a t i o f o r t h e f i r s t f l e x u r a l mode i s 0 . 1 6 . The s t r e n g t h o f t h e i c e f i e l d i s t a k e n t o be 15 MPa d u r i n g t h e c o n t i n u o u s i c e b r e a k i n g p r o c e d u r e . T h i s m a g n i t u d e i s t h e maximum v a l u e q u o t e d b y Ghoneim 107 and K e i n o n e n [34] f o r w i n t e r i c e . The c o e f f i c i e n t o f f r i c t i o n i n t h e i c e b r e a k i n g p r o c e d u r e i s t a k e n e q u a l t o 0 . 1 . 5 . 1 . 2 - C o n t i n u o u s I c e B r e a k i n g Mode The s t r e n g t h and t h i c k n e s s o f t h e i c e , have c o n s i d e r a b l e e f f e c t on t h e r e s p o n s e o f t h e s h i p , as w e l l as on t h e i n d u c e d b e n d i n g moment a l o n g i t . The v e l o c i t y o f t h e s h i p c o m b i n e d w i t h t h e p r o p e r t i e s o f t h e i c e f i e l d d e t e r m i n e t h e f r e q u e n c y o f t h e g e n e r a t e d i c e l o a d . The mode l f o r d i s t r i b u t i o n o f t h e i c e f o r c e on t h e bow a t t h e w a t e r l i n e i s i n t r o d u c e d i n S e c t i o n ( 3 . 3 . 1 ) . A c c o r d i n g t o t h i s m o d e l , t h e p o i n t o f a c t i o n o f t h e c o n t a c t f o r c e moves on t h e bow a l o n g t h e w a t e r l e v e l . D i r e c t i o n o f t h e l a t e r a l f o r c e c h a n g e s i n e a c h s e q u e n c e o f c o n t a c t o f t h e s h i p and i c e . T h i s makes f r e q u e n c y o f t h e l a t e r a l component o f t h e c o n t a c t f o r c e and g e n e r a t e d t o r q u e on t h e s h i p , h a l f o f t h e f r e q u e n c y o f t h e v e r t i c a l component o f t h e c o n t a c t f o r c e . T h i s f a c t i s shown on t h e p l o t s o f t h e v e r t i c a l f o r c e and g e n e r a t e d t o r q u e on t h e s h i p . M o t i o n o f t h e ' S t a n d a r d I c e B r e a k e r ' i s s t u d i e d i n t h e i c e f i e l d s o f d i f f e r e n t t h i c k n e s s e s f o r d i f f e r e n t s h i p v e l o c i t i e s . T h i s makes i t p o s s i b l e t o i n v e s t i g a t e t h e e f f e c t o f i c e t h i c k n e s s and t h e s h i p v e l o c i t y on t h e s t r u c t u r a l r e s p o n s e o f t h e s h i p . The c a l c u l a t e d r e s u l t s a r e p r e s e n t e d i n g r a p h s w h i c h a r e i n t r o d u c e d as f o l l o w s : F i r s t , an i c e f i e l d w i t h t h e a v e r a g e t h i c k n e s s o f 0 . 7 m e t e r i s s e l e c t e d , and s h i p s p e e d i s v a r i e d i n t h i s f i e l d . F o r e a c h v a l u e o f t h e s h i p v e l o c i t y , v a r i a t i o n o f t h e v e r t i c a l component o f t h e c o n t a c t f o r c e , g e n e r a t e d t o r q u e on t h e s h i p , maximum b e n d i n g moment a l o n g t h e s h i p and i t s r o l l i n g a n g l e i s e x a m i n e d . These v a l u e s a r e p r e s e n t e d T h r o u g h F i g s . ( 5 . 4 a ) t o ( 5 . 1 2 d ) f o r t he s h i p v e l o c i t i e s f r o m 0 . 5 t o 7 m / s . V e r t i c a l component o f t h e g e n e r a t e d c o n t a c t f o r c e s a r e shown i n F i g s . ( 5 . 4 a ) t o ( 5 . 1 2 a ) f o r d i f f e r e n t s h i p v e l o c i t i e s . These f i g u r e s show t h e i c e 108 f o r c e i s a r e p e a t e d i m p u l s e f o r c e s w i t h a f r e q u e n c y w h i c h l i n e a r l y i n c r e a s e s w i t h s h i p s p e e d . F i g s . ( 5 . 4 b ) t o ( 5 . 1 2 b ) p r e s e n t t h e g e n e r a t e d t o r q u e on t h e s h i p , . t h e s e f i g u r e s show t h a t , t h e f r e q u e n c y o f t h e g e n e r a t e d t o r q u e i s h a l f o f t h a t o f t h e v e r t i c a l f o r c e i n e a c h c a s e . As m e n t i o n e d , t h i s i s e x p l a i n e d when we c o n s i d e r t h e v a r i a t i o n i n t h e d i r e c t i o n o f l a t e r a l f o r c e a t e a c h i c e b r e a k i n g s e q u e n c e . The e f f e c t o f t h e f r e q u e n c y o f i c e l o a d on t h e maximum o f b e n d i n g moment i n d u c e d a l o n g t h e s h i p i s shown t h r o u g h F i g s . ( 5 . 4 c ) t o ( 5 . 1 2 c ) . These f i g u r e s p r e s e n t v a r i a t i o n o f maximum b e n d i n g moment a l o n g t h e s h i p i n two v e r t i c a l and h o r i z o n t a l p l a n e s f o r d i f f e r e n t s h i p v e l o c i t i e s . These f i g u r e s show t h a t a t t h e l o w e r s p e e d s , when t h e i m p u l s e p e r i o d i s much g r e a t e r t h a n t h e f l e x u r a l p e r i o d o f t h e s h i p s t r u c t u r e , maximum b e n d i n g moment i s v a r y i n g w i t h t h e f u n d a m e n t a l f r e q u e n c y o f t h e s h i p h u l l as a damped f r e e v i b r a t i n g s y s t e m . A l s o p e a k o f maximum b e n d i n g moment c o i n c i d e s w i t h t h e p e a k o f t h e i c e i m p u l s e . I t i s a l s o n o t i c e d t h a t whenever f r e q u e n c y o f t h e i c e i m p u l s e i s c l o s e t o any o f t h e n a t u r a l f r e q u e n c i e s o f t h e s h i p m o t i o n , maximum b e n d i n g moment i n c r e a s e s c o m p a r e d t o t h e o t h e r c a s e s . F i g s . ( 5 . 4 d ) t o ( 5 . 1 2 d ) show v a r i a t i o n o f t h e r o l l i n g a n g l e o f t h e s h i p d u r i n g t h e i c e b r e a k i n g p r o c e d u r e w i t h d i f f e r e n t v e l o c i t i e s . These f i g u r e s show t h a t t h e r o l l i n g a n g l e does n o t e x c e e d 5 d e g r e e s . A n i n c r e a s e i n t h e r o l l i n g a n g l e i s n o t i c e d when t h e f r e q u e n c y o f t h e g e n e r a t e d t o r q u e on t h e s h i p i s c l o s e t o t h e t h i r d n a t u r a l f r e q u e n c y o f t h e s h i p ( 0 . 4 H z . ) The above s t u d y on t h e i c e b r e a k e r i s r e p e a t e d f o r t h e i c e f i e l d s o f d i f f e r e n t t h i c k n e s s e s . P e a k s o f maximum b e n d i n g moment i n d u c e d a l o n g t h e i c e b r e a k e r and r o l l i n g a n g l e o f t h e s h i p a r e c a l c u l a t e d i n e a c h c a s e . V a r i a t i o n o f p e a k maximum b e n d i n g moment a l o n g t h e s h i p , and maximum r o l l i n g a n g l e a r e 109 p r e s e n t e d i n F i g . ( 5 . 1 3 a ) , and ( 5 . 1 3 b ) r e s p e c t i v e l y f o r d i f f e r e n t s h i p s p e e d s . A s m e n t i o n e d , i c e f i e l d t h i c k n e s s and s h i p v e l o c i t y d e t e r m i n e t h e f r e q u e n c y o f t h e g e n e r a t e d c o n t a c t f o r c e d u r i n g t h e c o n t i n u o u s i c e b r e a k i n g mode. F r e q u e n c y o f c o n t a c t l o a d , f o r d i f f e r e n t i c e t h i c k n e s s e s , i s p l o t t e d v e r s u s s h i p v e l o c i t y i n F i g . ( 5 . 1 4 ) . The e f f e c t o f i c e - b r e a k e r s p e e d on peak maximum b e n d i n g moment and i t s r e l a t i o n t o t h e s h i p s t r u c t u r a l f r e q u e n c i e s a r e shown i n F i g . ( 5 . 1 3 a ) and F i g . ( 5 . 1 4 ) . These f i g u r e s show t h a t , f o r e a c h i c e f i e l d , when i c e l o a d f r e q u e n c y i s b e l o w t h e f u n d a m e n t a l f r e q u e n c y o f t h e s h i p , v a r i a t i o n s i n p e a k maximum b e n d i n g moment a r e s m a l l , a l t h o u g h a s l i g h t i n c r e a s e does o c c u r whenever t h e l o a d f r e q u e n c y c o i n c i d e s w i t h a s h i p r i g i d - b o d y - m o t i o n . We c o n c l u d e t h a t t h e p e a k o f maximum b e n d i n g moment v a r i e s s l i g h t l y w i t h s h i p v e l o c i t y as l o n g as t h e s h i p i s n o t a c t u a t e d b y a l o a d w i t h a f r e q u e n c y c l o s e t o one o f i t s s t r u c t u r a l n a t u r a l f r e q u e n c i e s , i n w h i c h c a s e i t i n c r e a s e s d r a m a t i c a l l y . C o n c l u s i o n i s t h a t , an i c e b r e a k e r w o u l d be a b l e t o p r o c e e d i n t h e t h i c k i c e f i e l d s w i t h h i g h e r v e l o c i t i e s when f u n d a m e n t a l f r e q u e n c y o f t h e s h i p s t r u c t u r e i s f a r f r o m t h e f r e q u e n c i e s o f i t s r i g i d body m o t i o n . T h a t i s , t h e s t i f f e r i c e - b r e a k e r s a r e c a p a b l e o f b r e a k i n g t h e t h i c k e r i c e f i e l d s w i t h h i g h e r s p e e d s . U s u a l l y h i g h e r s p e e d s a r e n o t a t t a i n a b l e i n t h i c k i c e f i e l d s b e c a u s e o f l i m i t a t i o n s on t h e power o f t h e s h i p . The e x t r a power n e e d e d f o r i c e b r e a k i n g p r o c e d u r e i s c a l c u l a t e d b y c o n s i d e r i n g t h e a x i a l component o f t h e c o n t a c t f o r c e and t h e s p e e d o f t h e s h i p . V a r i a t i o n o f t h e i c e b r e a k i n g power v e r s u s t h e v e l o c i t y o f t h e s h i p i s p l o t t e d i n F i g . ( 5 . 1 5 ) . From t h i s p l o t , i t i s c o n c l u d e d t h a t t h e i c e b r e a k i n g power o f t h e ' s t a n d a r d I c e B r e a k e r ' i s r e l a t e d t o i t s v e l o c i t y and t h i c k n e s s o f i c e f i e l d a s : 110 1.64 1.13 P = 760 h A S w where P i s t h e i c e b r e a k i n g power i n h p . h i s t h e a v e r a g e t h i c k n e s s o f i c e f i e l d i n m e t e r s , is i s t h e s h i p v e l o c i t y i n m / s . V a r i a t i o n o f r o l l i n g a n g l e o f t h e s h i p w i t h r e s p e c t t o v e l o c i t y i s p l o t t e d i n F i g ( 5 . 1 3 b ) f o r d i f f e r e n t i c e t h i c k n e s s e s . From t h i s f i g u r e and F i g . ( 5 . 1 4 ) we see t h a t when f r e q u e n c y o f t h e g e n e r a t e d v e r t i c a l f o r c e on t h e bow i s e q u a l t o , o r d o u b l e t h e r o l l i n g f r e q u e n c y o f s h i p ( 0 . 4 2 6 H z . ) , p e a k o f r o l l i n g a n g l e i s i n c r e a s e d . To h a v e a s t a b l e i c e b r e a k i n g p r o c e s s t h e s e f r e q u e n c i e s s h o u l d be a v o i d e d . 5 . 1 . 3 - Ramming Heavy I c e F i r s t l y we examine t h e r e s p o n s e o f t h e s h i p t o t h e i n i t i a l i m p a c t . I n p a r t i c u l a r we examine t h e change i n s h i p v e l o c i t y , g e n e r a t e d f l e x u r a l r e s p o n s e and t h e e n e r g y l o s s . The n o d a l v e l o c i t i e s i m m e d i a t e l y a f t e r c o l l i s i o n a r e u s e d as i n i t i a l c o n d i t i o n s f o r t h e b e a c h i n g m o t i o n . The two d i f f e r e n t a p p r o a c h e s f o r s i m u l a t i o n o f t h e b e a c h i n g p e r i o d d e s c r i b e d i n C h a p t e r 3 a r e e x a m i n e d . The b e n d i n g moment a l o n g t h e s h i p and t h e c o n t a c t f o r c e on t h e bow a r e d e t e r m i n e d f r o m b o t h methods and c o m p a r e d . 5 . 1 . 3 . 1 - R e s p o n s e o f t h e S h i p t o t h e I n i t i a l I m p u l s e T a b l e ( 5 . 3 ) shows t h e ramming v e l o c i t y («^) , v e l o c i t y o f t h e s h i p i m m e d i a t e l y a f t e r c o l l i s i o n ( A S ) , k i n e t i c e n e r g y o f t h e s u r g e o f t h e s h i p b e f o r e and a f t e r c o l l i s i o n ( E ^ and ( E 2 ) , k i n e t i c e n e r g y o f t h e l a t e r a l m o t i o n o f t h e s h i p i m m e d i a t e l y a f t e r c o l l i s i o n (E ) and e n e r g y l o s s i n t h e i n e l a s t i c I l l c o l l i s i o n ( A E ) . I n t h e l a s t two c a s e s t h e f l e x u r a l r i g i d i t y o f t h e s h i p s a r e f i v e t i m e s and h a l f o f t h a t o f t h e ' S t a n d a r d I c e B r e a k e r ' r e s p e c t i v e l y . We see t h a t t h e p e r c e n t a g e o f t h e i n i t i a l k i n e t i c e n e r g y w h i c h i s l o s t i n t h e c o l l i s i o n i s i n d e p e n d e n t o f t h e ramming v e l o c i t y , b u t depends on t h e e l a s t i c i t y o f t h e s h i p . T a b l e ( 5 . 3 ) , shows t h a t t h e more f l e x i b l e s h i p l o s e s l e s s k i n e t i c e n e r g y i n c o l l i d i n g w i t h h a r d i c e . T h a t i s , t h e more r i g i d s h i p h a s a h a r d e r c o l l i s i o n w i t h more e n e r g y l o s s . The v e l o c i t y o f t h e s h i p i m m e d i a t e l y a f t e r c o l l i s i o n i s shown t o be h i g h e r f o r t h e more f l e x i b l e s h i p . T h e r e f o r e , a t t h i s s t a g e i t c o u l d be c o n c l u d e d t h a t l o w e r l e v e l o f i m p u l s e w o u l d be g e n e r a t e d on t h e bow o f t h e more f l e x i b l e s h i p s w h i c h means s o f t e r and t h e r e f o r e s a f e r c o l l i s i o n . A S 0 A S E l E 2 E L AE %AE m /s m /s MJ MJ MJ MJ 2 1 .938 14 98 14 07 0 . 4 6 0 . 4 5 3 . 0 0 5 3 2 . 9 0 8 33 71 31 67 1 .03 1 . 0 1 3 . 0 0 5 4 3 . 8 7 7 59 93 56 30 1 . 8 3 1 . 8 0 3 . 0 0 5 5 4 . 8 4 6 93 64 87 96 2 . 8 6 2 . 8 1 3 . 0 0 5 6^ 5 . 8 1 5 134 84 126 67 4 . 1 2 4 . 0 5 3 . 0 0 5 51* 4 . 7 6 2 93 64 84 92 4 . 3 5 4 . 3 7 4 . 6 6 5 5** 4 . 8 7 3 93 64 88 96 2 . 3 3 2 . 3 4 2 . 5 0 7 T a b l e 5 . 3 - Specifications of the Ship Immediately Before and After Collision to the Ice. w h e r e ; A S Q i s ramming v e l o c i t y . A S i s s h i p v e l o c i t y i m m e d i a t e l y a f t e r c o l l i s i o n E^ i s i n i t i a l k i n e t i c e n e r g y o f t h e s h i p . E z i s k i n e t i c e n e r g y o f t h e s h i p i n s u r g e . E^ i s k i n e t i c e n e r g y o f t h e l a t e r a l m o t i o n o f t h e s h i p i m m e d i a t e l y a f t e r c o l l i s i o n . AE i s e n e r g y l o s s i n t h e i n e l a s t i c c o l l i s i o n . 112 %AE i s p e r c e n t o f t h e i n i t i a l e n e r g y w h i c h i s l o s t i n c o l l i s i o n . * f l e x u r a l r i g i d i t y i s f i v e t i m e s o f t h a t o f t h e ' s t a n d a r d I c e B r e a k e r ' . ** f l e x u r a l r i g i d i t y i s h a l f o f t h a t o f t h e ' S t a n d a r d I c e B r e a k e r ' . 5 . 1 . 3 . 2 - B e a c h i n g p e r i o d 5 . 1 . 3 . 2 . 1 - B e a c h i n g on a F i x e d C o n t a c t P o i n t on t h e Bow The e q u a t i o n o f m o t i o n o f t h e s h i p d u r i n g t h e b e a c h i n g p e r i o d a s s u m i n g a f i x e d c o n t a c t p o i n t on t h e bow i s g i v e n b y E q . ( 3 . 2 4 ) . E i g e n - v e c t o r s o f t h e A - 1 m a t r i x [ [M ] [K ]] r e p r e s e n t mode s h a p e s o f t h e s h i p m o t i o n d u r i n g t h e b e a c h i n g p e r i o d . F i g . ( 5 . 1 6 ) shows t h e p l o t o f t h e f i r s t e i g h t e i g e n - v e c t o r s , t h e f i r s t two r e p r e s e n t t h e r i g i d body m o t i o n s o f t h e s h i p w h i l e t h e l a s t s i x a r e r e l a t e d t o s u b s e q u e n t modes o f e l a s t i c m o t i o n o f t h e s h i p . The mode shapes o f e l a s t i c m o t i o n a r e t h o s e o f a p i n n e d - f r e e beam, i n ag reement w i t h t h e a s s u m p t i o n t h a t t h e bow r e m a i n s i n c o n t a c t w i t h t h e i c e t h r o u g h - o u t t h e b e a c h i n g p e r i o d . V a r i a t i o n o f maximum b e n d i n g moment a l o n g t h e s h i p , and t h e c o n t a c t f o r c e on t h e bow d u r i n g t h e b e a c h i n g m o t i o n a r e r e p r e s e n t e d i n F i g . ( 5 . 1 7 ) . E a c h p l o t shows t h e c o r r e s p o n d i n g v a l u e s f o r d i f f e r e n t ramming v e l o c i t i e s . The b e n d i n g moment and t h e c o n t a c t f o r c e on t h e bow a r e f l u c t u a t i n g w i t h a f r e q u e n c y e q u a l t o t h a t o f t h e f u n d a m e n t a l f l e x u r a l f r e q u e n c y o f t h e s h i p ( 1 . 6 H z ) . F i g . ( 5 . 1 7 a ) shows t h a t t h e b e n d i n g moment g e n e r a t e d due t o t h e i n i t i a l i m p u l s e i s h i g h e r t h a n t h e b e n d i n g moment a t t h e end o f t h e b e a c h i n g p e r i o d . I t a l s o shows t h a t t h e r a t i o b e t w e e n t h e m a g n i t u d e o f maximum b e n d i n g moment i n d u c e d b y t h e i n i t i a l i m p u l s e and t h a t a t t h e end 113 o f t h e b e a c h i n g p e r i o d , i s c o n s t a n t , i n d e p e n d e n t o f t h e ramming v e l o c i t y . M a g n i t u d e o f t h e i n i t i a l i m p u l s e i s c a l c u l a t e d i n E q u . ( 3 . 1 4 ) , r e s p o n s e o f t h e f l o a t i n g e l a s t i c beam ( s h i p ) t o t h i s i m p u l s e i s s t u d i e d as r e p r e s e n t e d i n S e c t i o n ( 3 . 4 ) . The v e r t i c a l component o f t h e n o d a l f o r c e a t t h e end node i s t a k e n as t h e v e r t i c a l component o f t h e c o n t a c t f o r c e . T h i s f o r c e c a l c u l a t e d i m m e d i a t e l y a f t e r c o l l i s i o n shows t h e i n i t i a l c o n t a c t f o r c e . The c o n t a c t f o r c e on t h e bow i s p r e s e n t e d i n F i g . ( 5 . 1 7 b ) . I t shows t h a t t h e i m p a c t f o r c e on t h e bow due t o t h e i n i t i a l i m p u l s e i s h i g h e r t h a n t h e c o n t a c t f o r c e d u r i n g t h e b e a c h i n g p e r i o d . A change o f s i g n o f F i n d i c a t e t h a t t h e s h i p h a s b o u n c e d . The maximum b e n d i n g moment i n d u c e d i n t h e s h i p f o r t h e d i f f e r e n t ramming v e l o c i t i e s a r e p l o t t e d i n F i g . ( 5 . 1 8 ) . I t i s r e a l i z e d t h a t , t h e r e i s a l i n e a r r e l a t i o n b e t w e e n t h e maximum b e n d i n g moment and t h e ramming v e l o c i t y . The maximum c o n t a c t f o r c e on t h e bow p r e d i c t e d b y p r e s e n t m e t h o d , i s p l o t t e d i n F i g . ( 5 . 1 9 ) as a f u n c t i o n o f ramming v e l o c i t y . Vaughan [35] i n t r o d u c e d t h e maximum v a l u e o f t h e c o n t a c t f o r c e t h r o u g h t h e e q u a t i o n : _0 . 9 1 . 2 F = 0 . 6 3 M u m o and J o h a n s s o n e t a l [33] p r o p o s e d t h e f o l l o w i n g r e l a t i o n f o r maximum o f c o n t a c t f o r c e . 0 . 9 F = M « m o I n b o t h o f t h e above e q u a t i o n s 19 i s t h e ramming v e l o c i t y i n m e t e r p e r s e c o n d o and M i s t h e d i s p l a c e m e n t o f t h e s h i p i n t h o u s a n d s o f t o n n e s . They h a v e b o t h shown t h a t F i s i n s e n s i t i v e t o bow a n g l e a f o r a b e t w e e n 20 and 30 d e g s . , m w h i c h c o v e r s most modern i c e - b r e a k e r bow f o r m s . I n F i g . ( 5 . 1 9 ) t h e s e two p r e d i c t i o n i s compared w i t h t h e v a l u e s c a l c u l a t e d b y t h e p r e s e n t a p p r o a c h . A good ag reement i s n o t i c e d b e t w e e n t h e r e s u l t s o f t h e p r e s e n t w o r k and t h e o t h e r two m e t h o d s . 114 Two s p e c i a l i c e b r e a k e r s w h i c h a r e i n t r o d u c e d i n t h e f o r m e r s e c t i o n a r e a l s o s t u d i e d d u r i n g t h e b e a c h i n g m o t i o n . To compare t h e maximum b e n d i n g moment a l o n g t h e h u l l and c o n t a c t f o r c e on t h e bow o f t h e s h i p , v a r i a t i o n o f t h e s e v a l u e s a r e shown i n F i g . ( 5 . 2 0 a ) and ( 5 . 2 0 b ) . These f i g u r e s a r e r e l a t e d t o t h e ramming v e l o c i t y o f 5 m/s f o r t h e s h i p w i t h t h r e e d i f f e r e n t s t i f f n e s s . F i g . ( 5 . 2 0 a ) shows t h a t d u r i n g t h e b e a c h i n g m o t i o n maximum b e n d i n g moment a p p r o a c h e s t o t h e same m a g n i t u d e i n t h r e e d i f f e r e n t h u l l s , w h i l e t h e b e n d i n g moment i n d u c e d i n t h e s t i f f e r s h i p due t o t h e i n i t i a l i m p u l s e i s h i g h e r t h a n t h e t h a t o f t h e o t h e r s h i p s . F i g . ( 5 . 2 0 b ) r e p r e s e n t s v a r i a t i o n o f t h e c o n t a c t f o r c e on t h e bow d u r i n g t h e ramming p e r i o d . I t shows t h a t t h e i n i t i a l c o n t a c t f o r c e i n c r e a s e s w i t h an i n c r e a s e i n t h e s t i f f n e s s o f t h e h u l l , b u t d u r i n g t h e b e a c h i n g p e r i o d m a g n i t u d e o f c o n t a c t f o r c e a p p r o a c h e s t o a common v a l u e f o r t h e s h i p s w i t h s t a n d a r d and h a l f s t i f f h u l l s . Though t h e c o n t a c t f o r c e i n t h e c a s e o f r i g i d s h i p shows an i n c r e a s e d u r i n g t h e p r o c e s s . I t c a n be c o n c l u d e d t h a t , a b i g g e r c o l l i s i o n f o r c e i s p r e d i c t e d i n t h e c a s e o f r i g i d s h i p s , b u t as l o n g as t h e bow a r e a i n t h e s e s h i p s a r e s t r o n g enough t o s t a n d t h e i n i t i a l i m p a c t , t h e h u l l o f t h e more r i g i d s h i p s r e m a i n s a f e r d u r i n g t h e ramming mode. 5 . 1 . 3 . 2 . 2 - B e a c h i n g on a m o v i n g C o n t a c t P o i n t on t h e Bow I n t h e s e c o n d a p p r o a c h t h e s h i p i s a l l o w e d t o move r e l a t i v e t o t h e c o n t a c t p o i n t . The e q u a t i o n s o f m o t i o n f o r b e a c h i n g m o t i o n , ( 4 . 3 2 ) and ( 4 . 3 3 ) a r e s o l v e d s i m u l t a n e o u s l y b y 'Newmark B M e t h o d ' ( T h i s method o f n u m e r i c a l i n t e g r a t i o n i s d e s c r i b e d i n pp 9 0 - 9 5 o f [ 4 9 ] ) . The maximum v a l u e s o f t h e b e n d i n g moment a l o n g t h e s h i p and t h e c o n t a c t f o r c e on t h e bow d u r i n g t h e b e a c h i n g p e r i o d a r e a g a i n d e t e r m i n e d and shown i n F i g s . ( 5 . 2 1 ) . The m a i n d i f f e r e n c e s i n t h e r e s u l t s o b t a i n e d b y a l l o w i n g t h e s h i p t o move 115 r e l a t i v e t o t h e c o n t a c t p o i n t a r e : i ) The b e a c h i n g p e r i o d i s a b o u t 8% s h o r t e r . i i ) The f i n a l b e n d i n g moment a p p r o a c h e s a c e r t a i n v a l u e i n d e p e n d e n t o f t h e i n i t i a l ramming v e l o c i t y . T h i s i n d i c a t e s t h a t t h e r e d u c e d moment arm a s s o c i a t e d w i t h t h e " m o v i n g c o n t a c t p o i n t " r e d u c e s t h e maximum moment d u r i n g b e a c h i n g , as m i g h t be e x p e c t e d . However t h e maximum f l e x u r a l moments w h i c h o c c u r v e r y e a r l y i n t h e m o t i o n , a r e t h e same f o r b o t h m e t h o d s . i i i ) Maximum b e n d i n g moments i n d u c e d b y t h e i m p u l s e a r e e s t i m a t e d t o be 9% l o w e r . V a r i a t i o n o f t h e maximum b e n d i n g moment v e r s u s t h e ramming v e l o c i t y p r e d i c t e d b y e a c h o f t h e methods a r e compared i n F i g . ( 5 . 1 8 ) . I t shows t h a t i n b o t h c a s e s t h e r e i s a l i n e a r r e l a t i o n b e t w e e n t h e maximum b e n d i n g moment and t h e ramming v e l o c i t y and t h a t f o r any p a r t i c u l a r v e l o c i t y , t h e " m o v i n g p o i n t " method g i v e s a l o w e r b e n d i n g moment. 5 . 1 . 4 - C o m p a r i s o n o f T h e o r y W i t h F i e l d D a t a Ghoneim e t a l , [ 3 8 ] , i n 1984 p u b l i s h e d some r e s u l t s f r o m a s e r i e s o f f u l l - s c a l e ramming t e s t s on Kigoriak, a s h i p w i t h 90 m WL and t o t a l mass o f 6800 t o n n e s ( t h e same l e n g t h and mass as o u r ' S t a n d a r d I c e - B r e a k e r ' ) . The m a j o r o b j e c t i v e o f t h e i r t e s t s was t o e s t i m a t e t h e g l o b a l i c e f o r c e w h i c h o c c u r s d u r i n g t h e ramming p r o c e s s and t o e v a l u a t e t h e h u l l r e s p o n s e . Thus t h e y were a b l e t o e s t a b l i s h t h e r e l a t i o n s h i p b e t w e e n maximum i c e f o r c e , and s h i p v e l o c i t y . They c a r r i e d o u t a s e r i e s o f ramming t e s t w i t h Kigoriak a t speeds o f 1 . 5 t o 6 . 5 m /s i n h e a v y m u l t i - y e a r i c e . They m e a s u r e d t h e s h i p r e s p o n s e and e s t a b l i s h e d t h e v e l o c i t y dependency o f ramming f o r c e s . The g r a p h s shown i n F i g s . ( 5 . 2 2 ) (a ) and (b) a r e s a m p l e s o f t h e i r p u b l i s h e d r e s u l t s . These f i g u r e s r e s p e c t i v e l y show v a r i a t i o n o f t h e b e n d i n g moment i n Kigoriak and t h e 116 i c e f o r c e on h e r bow d u r i n g t h e ramming p r o c e d u r e when she rams t h e r i d g e w i t h v e l o c i t y o f 4 . 9 m / s . The t h e o r e t i c a l r e s u l t s were o b t a i n e d f o r a c o e f f i c i e n t o f f r i c t i o n o f 0 . 1 . The a c t u a l v a l u e o f t h e f r i c t i o n f o r c e on Kigoriak i s o f c o u r s e unknown. However , i t h a s b e e n shown i n [36] t h a t f r i c t i o n does n o t i n f l u e n c e t h e maximum f o r c e and b e n d i n g moment s i g n i f i c a n t l y . T h i s s t u d y a l s o shows t h a t t h e c o e f f i c i e n t o f f r i c t i o n h a s m i n o r i n f l u e n c e on t h e b e n d i n g moment. T h i s c a n be n o t i c e d i n E q u . ( 3 . 2 3 ) where any v a r i a t i o n o f f r i c t i o n c o e f f i c i e n t c a n s l i g h t l y change t h e m a g n i t u d e o f one e l e m e n t i n t h e a p p a r e n t mass m a t r i x . C o m p a r i s o n o f t h e s e d a t a w i t h t h o s e o b t a i n e d f r o m t h e p r e s e n t method , r e p r e s e n t e d i n F i g s . ( 5 . 1 7 ) and ( 5 . 2 1 ) , shows a v e r y good a g r e e m e n t . I n p a r t i c u l a r : i ) The b e a c h i n g p e r i o d m e a s u r e d i n t h e f i e l d i s s l i g h t l y more t h a n f i v e s e c o n d s and t h a t p r e d i c t e d b y t h i s work i s a l m o s t 5 . 5 s e c . i i ) Maximum c o n t a c t f o r c e due t o t h e i n i t i a l i m p u l s e i n b o t h t h e p r e s e n t method a n d t e s t measurements a r e 2 5 . 5 MN and i s a l w a y s b i g g e r t h a n t h e f o r c e d u r i n g b e a c h i n g p e r i o d . i i i ) P e a k b e n d i n g moment i n t h e s h i p , i n b o t h t h e measurements and t h e p r e s e n t m e t h o d , i s a b o u t 550 MN-m and o c c u r s a l m o s t h a l f a s e c o n d a f t e r t h e f i r s t c o n t a c t . The g e n e r a l f o r m s o f t h e c u r v e s f o r c o n t a c t f o r c e and b e n d i n g moment as m e a s u r e d on t h e Kigoriak d u r i n g t h e ramming t e s t s ( F i g . ( 5 . 2 2 a ) and ( 5 . 2 2 b ) ) a r e v e r y s i m i l a r t o t h o s e o b t a i n e d u s i n g t h e n u m e r i c a l a n a l y s i s ( F i g s . ( 5 . 1 7 ) and ( 5 . 2 1 ) ) . N o t e t h a t t h e t i m e datum f o r t h e Kigoriak t e s t i s d i f f e r e n t t o t h e one u s e d h e r e i n F i g 5 . 2 2 ( b ) . 117 10 8 . Fig. 5.4- a) Vertical component of the ice force; b) Generated torque on the ship when ship velocity is 0.5 m/s. 118 119 F i g . 5 .5- a) Vertical component of the ice force and b) Generated torque on the ship when ship velocity is 0.7 m/s. F i g . 5.5- Variation of, c)maximum bending moment d) Rolling angle, when ship velocity is 0.7 m/s. 121 F i g . 5 .6- a) Vertical component of the ice force; b) Generated torque on the ship when ship velocity is 1.4 m/s. 122 z L L I X o z o z o z U J CD 150 100 50 -50 -100 0 (c) o 2 J o -4 (d) IN VERT. PLANE IN H0RZ. PLANE 10 TIME 12 ( S E C . ) T -14 16 i e 10 12 14 TIME ( S E C . ) T " 20 F i g . 5 .6- Variation of, c) maximum bending moment; d) Rolling angle, when ship velocity is 1A m/s, 123 10 e . o or o 6 . u o o 3 or o 0 0 (a) L 15 10 5 . -5 . -ID -15 (b) 10 TIME 12 14 ( S E C . ) 1^ 16 i 6 10 i 12 14 16 TIME ( S E C . ) 18 20 16 20 F i g . 5.7- a) Vertical component of the ice force; b) Generated torque on the ship when ship velocity is l.b m/s, F i g . 5.7- Variation of, c) maximum bending moment; d) Rolling angle when ship velocity is 1.6 m/s. 125 10 6 . UJ o or o 6 . o 4 . 2 . (a) 10 12 i 14 16 TIME ( S E C . ) I 18 I 20 or o -10 -15 (b) r ~ — 1 >" 10 12 14 16 18 20 2 4 6 8 TIME ( S E C . ) F i g . 5.8- a) Vertical component of the ice force; b) Generated torque on the ship when ship velocity is 1.8 m/s. J 150 IN VERT. PLANE IN HORZ. PLANE 100 J F i g . 5.8- Variation of, c) maximum bending moment; d) Rolling angle when ship velocity is 1.8 m/s. 127 10 o CC O o c_> 0 2 4 6 (a) • y — 1 r 8 10 12 14 16 18 20 TIME (SEC.) E I Z 3 a . o 15 10 -5 -10 -15 (b) - i —i 1 1 1 1 r B 10 12 14 16 18 20 TIME (SEC.) F i e 5.9- &) Vertical component of the ice force; -b) Generated torque on the ship when ship velocity is 3.0 m/s 128 129 10 or O z o u B . 2 . (a) i \-u-LX-t r 0 2 4 6 I i —r 6 10 12 14 16 IB 20 TIME (SEC.) E I 3 a o -10 (b) 8 10 12 TIME (SEC. ) 16 IB 20 Fig. 5.10- a) Vertical component of the ice force, b) Generated torque on the ship when ship velocity is 5.0 m/s, 150 IN VERT . PLANE • - • IN HORZ. PLANE 100 J 0 2 4 6 8 10 12 14 16 16 20 0 2 4 6 8 10 12 14 16 18 20 ( d ) TIME ( S E C . ) Fig. 5.10- Variation of, c) maximum bending moment, d) Rolling angle when ship velocity is 5.0 m/s. 131 o or o z o o 10 8 . 2 . (a) 1 i r 0 2 4 6 'I H I 1 1 ! 1" 1 1 1 1 \ 8 10 12 14 16 18 20 TIME (SEC.) E I z X or o 15 10 . 5 . -5 . -10 . -15 (b) TIME (SEC.) Fig. 5.11- a) Vertical component of the ice force, b) Generated torque on the ship when ship velocity is 6.5 m/s. 132 F i g . 5 .11- Variation of, c) maximum bending moment, d) Rolling angle when ship velocity is 6.5 m/s. 133 10 U I U a. O O U 0 0 (a) 14 T 16 TIME ( S E C . ) 18 20 => a. O 15 10 . 0 . -5 . -10 -15 (b) TIME ( S E C . ) F i g . 5.12- a) Vertical component of the ice force, b) Generated torque on the ship when ship velocity is 7.0 m/s. 0 . -150 0 ( C ) II | H I | | | | | | IN VERT. PLANE IN HORZ. PLANE ~T~ 16 10 12 TIME ( S E C . ) 14 16 20 F i g . 5.12- Variation of, c) maximum bending moment, d) Rolling angle, when ship velocity is 7.0 m/s a—a ICE THICK. 0 9 m ICE THICK. 0 8 m ICE THICK. 0 7 m ICE THICK. * 0 6 in 0 0 ICE THICK. m 0 4 m ICE THICK. m 0 3 Rl •Q' 3-w O 9 V 0 (a) 2 3 4 5 ICE BREAKER VELOCITY (m/S) ICE BREAKER VELOCITY (mis) 5.13- a) Peak of maximum bending moment, b) Peak of rolling angl verses ship velocity for different ice thickness. 136 3.5 2.5 a o o 1.5 J ICE THICK. - 0.9m • ---O ICE THICK. - 0.8 m *-•-» ICE THICK. - 0.7 m ICE THICK. - 0.6 m ICE THICK. - 0.4 III ICE THICK. -0.3m 0.5 r 5 6 7 8 ICE BREAKER VELOCITY (m/s) F i g . 5.14- . R e l a t i o n b e t w e e n i c e l o a d frequency and the ship velocity. 7000 6000 5000 J= 4000 o o. 3000 2000 1000 J 2 3 4 ICE BREAKER VELOCITY (m/s) F i g . 5.15- Ice-breaking power in different ice field. 137 138 10 -25 . 0 1 2 3 4 5 6 (b) TIME (SEC.) F i g . 5.17- Maximum of a) bending moment along the ship, b) contact force on the bow for ramming of the ice ridge considering fixed contact point on the bow. 139 o X C3 o z 600 500 J 400 J 300 200 J 100 t> a RAMMING ON Q-- -E RAMMING ON FIXED POINT MOVING POINT o 1 l \ 0 1 2 3 4 5 6 7 RAMMING VELOCITY (m/S) F i g . 5.18- Maximum bending moment for different ramming velocities. 40 z u o or o o u , 3 Z *—< X 35 30 J 25 J 20 J 15 10 J a — * FROM PRESENT METHOD VAUGHAN' S PREDICTION o-j<> JOHANSSON'S PREDICTION 0 1 2 3 4 5 6 7 RAMMING VELOCITY (n/s) F i g . 5.19- Maximum of contact forces for different ramming velocities. 140 F i g . 5.20- Maximum of, a)bending moment, b)contact force, for three different stiffness of the hull with ramming velocity of 5 m/s. 400 . 0 . RAMMING V E L . RAMMING V E L . RAMMING V E L . RAMMING V E L . RAMMING V E L . 6 m/s 5 m/s 4 m/s 3 m/s 2 m/s 1 1 2 3 TIME ( S E C . ) (b) TIME ( S E C . ) 5.21- Maximum of, a)bending moment, b) contact force, for ramming of the ice ridge with a moving contact point on the'bow. 142 z z — it 2 "2*1 -b -II (b) we! 2i i i 3i 4> S. TIME Typica l bow force t ime history F i g . 5.22- Collected data from Kigoriak with ramming velocity of 4.9 m/s. a) Maximum bending moment, b) contact force on the bow. 143 5.2- Es t imat ion of the Damage During c o l l i s i o n 5 .2 .1 - In troduct ion I n t h i s s e c t i o n , t h e method i n t r o d u c e d i n C h a p t e r 4 , i s u s e d t o e s t i m a t e t h e e x t e n t o f damage g e n e r a t e d on t h e h u l l o f a s h i p d u r i n g c o l l i s i o n w i t h an e x t e r n a l o b j e c t . The e f f e c t o f h u l l s t r e n g t h and s h i p v e l o c i t y on t h e damage e x t e n t i s i n v e s t i g a t e d , and t h e v a r i a t i o n o f c o l l i s i o n f o r c e s d u r i n g t h e c o l l i s i o n a r e e s t i m a t e d . N u m e r i c a l d a t a i s o b t a i n e d f o r a ' S t a n d a r d T a n k e r S h i p ' , a t a n k e r o f l e n g t h 330 m e t e r s and mass 260 t h o u s a n d t o n n e s . Ten s t a t i o n s a r e c o n s i d e r e d a l o n g i t , t h e same as shown i n F i g . ( 5 . 1 ) . S p e c i f i c a t i o n o f t h i s s h i p a t e a c h s t a t i o n i s r e p r e s e n t e d i n T a b l e ( 5 . 4 ) , mass and b u o y a n c y d i s t r i b u t i o n o f t h e s h i p a r e shown i n F i g . ( 5 . 2 3 ) , and h u l l p l a t e t h i c k n e s s i n t h e r e g i o n o f c o l l i s i o n i s s e l e c t e d t o be 15 mm. STATION STATION STATION STATION STATION MOMENT OF INERTIA No . LENGTH BREADTH DRAFT SHAPE FCT I I m e t e r m e t e r m e t e r y 4 m z 4 m 1 33 32 22 5 500 360 2 33 38 22 7 600 400 3 33 42 22 8 800 600 4 33 42 22 8 800 600 5 33 42 22 8 800 600 6 33 42 22 8 800 600 7 33 42 22 8 800 600 8 33 42 22 8 800 600 9 33 41 22 7 600 400 10 33 40 22 5 600 400 Table 5.4- Specification of the 'Standard Tanker Ship' at the Stations. 144 Two g e n e r a l c a s e s o f a c c i d e n t s a r e c o n s i d e r e d , a - The s h i p c o l l i d e s w i t h an e x t e r n a l o b j e c t d u r i n g n o r m a l f o r w a r d m o t i o n . b - C o l l i s i o n o c c u r s d u r i n g m a n e u v e r i n g . I n e a c h o f t h e above c a s e s , we c o n s i d e r two s c e n a r i o s . i - The s t r u c k o b j e c t d e n t s t h e h u l l w i t h o u t r u p t u r e . i i - R u p t u r e o f t h e h u l l o c c u r s w i t h s u b s e q u e n t p l a t e t e a r i n g . 5 . 2 . 2 - C o l l i d i n g D u r i n g F o r w a r d M o t i o n To i l l u s t r a t e t h e method a s p e c i f i c i n c i d e n t i s c o n s i d e r e d . The ' S t a n d a r d T a n k e r S h i p ' i s assumed t o be r u n n i n g s t r a i g h t a h e a d when i t s t r i k e s a r o c k . The r o c k i s l o c a t e d 15 m e t e r s b e l o w w a t e r l e v e l and 17 m e t e r t o t h e s i d e o f t h e s h i p a x i s . Y i e l d s t r e n g t h f o r s t r u c t u r a l members i s t a k e n as 280 M P a . , and r a t i o b e t w e e n t h e s o l i d c r o s s - s e c t i o n and t h e g e o m e t r i c c r o s s - s e c t i o n f o r t h e s t r u c t u r e o f t h e h u l l i n t h e c o l l i s i o n r e g i o n , i n t h r e e d i r e c t i o n s x , y , and z , a r e c o n s i d e r e d t o be 0 . 0 5 , 0 . 0 4 , and 0 . 0 4 r e s p e c t i v e l y . I t i s a l s o assumed t h a t t h e t h r u s t f o r c e o f t h e p r o p e l l e r s i s e q u a l t o t h e w a t e r r e s i s t a n c e d u r i n g t h e c o l l i s i o n . 5 . 2 . 2 . 1 - C o l l i s i o n o f S h i p and a Smooth R o c k F i g . ( 5 . 2 4 ) r e p r e s e n t s t h e damage g e n e r a t e d a l o n g t h e s t a n d a r d s h i p f o r d i f f e r e n t v e l o c i t i e s b e f o r e c o l l i s i o n . I t i s n o t i c e d t h a t when t h e i n i t i a l v e l o c i t y i s 1 m / s , t h e damaged vo lume i s l i m i t e d t o a r e g i o n c l o s e t o t h e f r o n t o f t h e s h i p and a s m a l l e r p a r t c l o s e t o t h e m i d s h i p . 145 The e x p l a n a t i o n i s as f o l l o w s ; t h e i n i t i a l c o l l i s i o n p u s h e s t h e bow away f r o m t h e r o c k , and t h i s g e n e r a t e s a r o t a t i o n a l m o t i o n f o r t h e s h i p . The s h i p c o n t i n u e s a l o n g i t s r o u t e b u t t h e c o m b i n a t i o n o f r o t a t i o n and t r a n s l a t i o n , b r i n g s o t h e r p a r t s o f t h e h u l l i n c o n t a c t w i t h t h e r o c k . F i g . ( 5 . 2 4 a ) shows t h e e x t e n t o f l a t e r a l damage does n o t e x c e e d s 1 m e t e r . Thus a l i m i t e d d e n t o f t h e h u l l c o u l d be e x p e c t e d i f t h i s s h i p i s i n v o l v e d i n a n a c c i d e n t w i t h v e l o c i t y o f 1 m / s . F i g . ( 5 . 2 4 a ) a l s o shows t h a t t h e s h i p comes t o s t o p a f t e r t h e s e c o n d c o l l i s i o n . F i g . ( 5 . 2 4 b ) shows t h e h u l l damage f o r a s p e e d o f 3 m / s . The l a t e r a l damage e x c e e d s 1 m e t e r and a b i g g e r r e g i o n a l o n g t h e s h i p i s d e n t e d . I t i s a l s o n o t i c e d t h a t t h e g e n e r a t e d damage c a u s e d d u r i n g t h e s e c o n d c o l l i s i o n i s c o n s i d e r a b l e . F i g s . ( 5 . 2 4 ) c and d , r e s p e c t i v e l y r e p r e s e n t t h e damage e x t e n t on t h e s h i p , when i t s v e l o c i t y b e f o r e c o l l i s i o n i s 5 and 7 m / s . The l a t e r a l damage e x c e e d s 2 m e t e r s and a l o n g e r d i s t a n c e a l o n g t h e s h i p i s d e n t e d . I t i s a l s o s e e n t h a t i n t h e s e two c a s e s , t h e s h i p does n o t s e p a r a t e f r o m t h e r o c k i n b e t w e e n t h e c o l l i s i o n , e v e n t h o u g h t h e e x t e n t o f damage i s d e c r e a s e d i n t h e a r e a c l o s e t o t h e m i d s h i p . V a r i a t i o n o f c o l l i s i o n f o r c e s f o r e a c h o f t h e m e n t i o n e d c a s e s a r e shown i n F i g s . ( 5 . 2 5 ) . C o l l i s i o n f o r c e when t h e i n i t i a l s h i p v e l o c i t y i s 1 m / s i s r e p r e s e n t e d I n F i g . ( 5 . 2 5 a ) , i t shows t h a t t h e f i r s t c o l l i s i o n i n t h e bow a r e a t a k e s n e a r l y 90 s e c o n d s . D u r i n g t h i s t i m e i n t e r v a l , s h i p v e l o c i t y d e c r e a s e s a l m o s t t o z e r o . F i g . ( 5 . 2 5 b ) shows v a r i a t i o n o f t h e c o l l i s i o n f o r c e v e r s u s t i m e when t h e i n i t i a l s h i p v e l o c i t y i s 3 m / s . I t i s n o t i c e d t h a t t h e d u r a t i o n o f t h e f i r s t c o l l i s i o n i s d e c r e a s e d t o 40 s e c o n d s and t h e t i m e i n t e r v a l b e t w e e n two a d j a c e n t c o l l i s i o n s i s d e c r e a s e d t o 35 s e c o n d s , much s h o r t e r 146 t h a n t h e p r e v i o u s c a s e . The p e a k o f t h e c o l l i s i o n f o r c e i s i n c r e a s e d t o 20 MN. w h i c h i s more t h a n t h r e e t i m e s t h e f o r c e i n t h e f i r s t c a s e . F i g . ( 5 . 2 5 c ) shows v a r i a t i o n o f c o l l i s i o n f o r c e when t h e s h i p s p e e d i s i n c r e a s e d t o 5 m / s . The t o t a l d u r a t i o n o f c o l l i s i o n i s d e c r e a s e d t o 56 s e c . and t h e p e a k c o l l i s i o n f o r c e i s i n c r e a s e d t o 30 MN. F i g . ( 5 . 2 5 d ) shows v a r i a t i o n o f t h e c o l l i s i o n f o r c e s v e r s u s t i m e f o r a s p e e d o f 7 m / s . The t o t a l d u r a t i o n o f t h e c o l l i s i o n i s d e c r e a s e d t o 36 s e c . and p e a k o f t h e c o l l i s i o n f o r c e i s i n c r e a s e d t o 38 MN. The s h o r t c o l l i s i o n p e r i o d s and t h e g r e a t e r c o l l i s i o n f o r c e s w h i c h a r e g e n e r a t e d a t h i g h e r v e l o c i t i e s , c o u l d c a u s e s e r i o u s damage t o s e n s i t i v e o r h a z a r d o u s c a r g o s . By r e p e a t i n g t h e s e t y p e s o f c o l l i s i o n s t u d i e s b u t w i t h t h e i n i t i a l i m p a c t p o i n t l o c a t e d a t d i f f e r e n t p a r t s o f t h e h u l l , i t w o u l d be p o s s i b l e t o e s t a b l i s h t h e s a f e s t p o s i t i o n i n any p a r t i c u l a r s h i p f o r s t o r i n g h a z a r d o u s c a r g o e s . To s t u d y t h e e f f e c t o f s t r u c t u r a l c o n f i g u r a t i o n o f t h e h u l l on t h e damage e x t e n t , i t i s assumed t h a t t h e s h i p s t r u c t u r e i s r e i n f o r c e d s u c h t h a t , r a t i o b e t w e e n t h e s o l i d c r o s s - s e c t i o n and t h e g e o m e t r i c c r o s s - s e c t i o n a r e d o u b l e d , t h a t i s V = 0 . 1 , and ij> = ib = 0 . 0 8 . The x y z same c o l l i s i o n s c e n a r i o i s a c c e p t e d f o r t h i s s h i p . The e x t e n t o f g e n e r a t e d damage t o t h e h u l l f o r e a c h s h i p v e l o c i t y i s p l o t t e d i n F i g . ( 5 . 2 6 ) . T h i s f i g u r e shows t h a t t h e g e n e r a t e d damage a t e a c h s h i p s p e e d i s a l m o s t s i m i l a r t o t h e damage on t h e s t a n d a r d h u l l t h o u g h a s m a l l d e c r e e s i s n o t i c e d i n t h e damage e x t e n t o f e a c h c a s e . C o l l i s i o n f o r c e s f o r e a c h o f t h e s e c a s e s a r e c a l c u l a t e d and r e p r e s e n t e d i n F i g . ( 5 . 2 7 ) . F i g . ( 5 . 2 7 a ) shows v a r i a t i o n o f t h e c o l l i s i o n f o r c e when i n i t i a l v e l o c i t y o f t h e s h i p i s 1 m / s . I t i s s e e n 147 t h a t t h e i n i t i a l c o l l i s i o n t a k e s a p p r o x i m a t e l y 50 s e c o n d s and t h e t i m e i n t e r v a l b e t w e e n t h e f i r s t and s e c o n d c o l l i s i o n i s a l m o s t 7 . 5 m i n u t e s . T h i s t i m e i n t e r v a l c o u l d be u s e d t o c o n t r o l t h e s h i p m o t i o n and a v o i d t h e s e c o n d c o l l i s i o n . The maximum c o l l i s i o n f o r c e i n t h i s c a s e i s a l m o s t 6 MN. C o l l i s i o n f o r c e s f o r s h i p v e l o c i t i e s o f 3 , 5 , and 7 m / s , a r e a l s o r e p r e s e n t e d i n F i g . ( 5 . 2 7 ) . I t shows d u r a t i o n o f t h e c o l l i s i o n i s g e t t i n g s h o r t e r and s h o r t e r w i t h an i n c r e a s e i n t h e i n i t i a l v e l o c i t y , w h i l e m a g n i t u d e o f t h e c o l l i s i o n f o r c e i n c r e a s e s w i t h t h e i n i t i a l v e l o c i t y . The damage s u s t a i n e d i n a c o l l i s i o n b y t h e s t a n d a r d s h i p i s now c o m p a r e d w i t h t h a t s u s t a i n e d b y t h e r e i n f o r c e d s h i p . F i g s . ( 5 . 2 4 ) and ( 5 . 2 6 ) show t h a t f o r t h e same i n i t i a l v e l o c i t i e s t h e damage e x t e n t t o t h e r e i n f o r c e d s h i p i s n o t much l e s s t h a n t h a t o f t h e n o r m a l s h i p . C o m p a r i n g t h e c o l l i s i o n f o r c e s i n F i g s . ( 5 . 2 5 ) and ( 5 . 2 7 ) show a b o u t 100% i n c r e a s e o f t h e p e a k o f c o l l i s i o n f o r c e g e n e r a t e d i n e a c h s i m i l a r c a s e on t h e r e i n f o r c e d h u l l . T h i s i n c r e a s e i n t h e c o l l i s i o n f o r c e c o u l d c a u s e a r i s k o f damage on t h e s h i p c a r g o . A n i n t e r e s t i n g e f f e c t o f t h e h a r d e r h u l l c o u l d be n o t i c e d as b o u n c i n g m o t i o n o f t h e s h i p e s p e c i a l l y when t h e she c o l l i d e s t o t h e r o c k w i t h r e l a t i v e l y h i g h v e l o c i t y . F o r e x a m p l e , i n F i g . ( 5 . 2 6 c ) two m i n o r c o l l i s i o n s i s s e e n a t t h e m i d s h i p a r e a . C l e a r l y i t r e p r e s e n t t h e b o u n c i n g m o t i o n o f t h e s h i p w i t h r e s p e c t t o t h e r o c k . 5 .2 .2 .2 - C o l l i s i o n of Ship with a Sharp Rock We now c o n s i d e r what happens when a s h i p c o l l i d e s w i t h a s h a r p r o c k c a u s i n g r u p t u r e o f t h e h u l l . As m e n t i o n e d i n C h a p t e r 4 , s t r e n g t h o f t h e 148 t o r n s t r u c t u r a l member i s e x p e c t e d t o be l e s s t h a n t h a t o f a d e n t e d one . As i n t h e p r e v i o u s s e c t i o n we compare t h e damage t o t h e ' S t a n d a r d T a n k e r S h i p ' w i t h t h a t o f t h e ' S t r e n g t h e n e d S h i p ' i n t h e same c o l l i s i o n s c e n a r i o . The damage e x t e n t f o r a c o l l i s i o n s p e e d o f 1 m / s e c . i s shown i n F i g . ( 5 . 2 8 a ) t o be s l i g h t l y i n e x c e s s o f 1 m e t e r . The s h i p comes t o r e s t a t t h e end o f t h e f i r s t c o l l i s i o n . I n F i g . ( 5 . 2 8 ) b t o d , damaged r e g i o n s a r e shown f o r i n i t i a l s h i p s p e e d s o f 3 , 5 , and 7 m / s , i n s e q u e n c e . I t i s s e e n t h a t t h e l e n g t h o f t h e damage a l o n g t h e s h i p i s a l m o s t c o n s t a n t w h i l e t h e l a t e r a l damage i n c r e a s e s w i t h v e l o c i t y . The c o r r e s p o n d i n g c o l l i s i o n f o r c e s a r e shown i n F i g . ( 5 . 2 9 ) . A c o m p a r i s o n o f t h e c o l l i s i o n f o r c e s t h r o u g h F i g s . ( 5 . 2 7 ) and ( 5 . 2 9 ) show t h a t when r u p t u r e o c c u r s t h e p e a k c o n t a c t f o r c e i s l e s s t h a n when t h e r e i s no r u p t u r e and t h a t t h e l o a d i s r e l a t i v e l y u n i f o r m d u r i n g t h e c o l l i s i o n t i m e w h e r e a s , w i t h o u t r u p t u r e , c o l l i s i o n f o r c e s a r e s h a r p . N e x t we c o n s i d e r t h e s h i p w i t h r e i n f o r c e d b o t t o m p l a t i n g . The g e n e r a t e d damage f o r d i f f e r e n t v a l u e s o f t h e s h i p v e l o c i t i e s a r e c a l c u l a t e d a n d shown i n F i g . ( 5 . 3 0 ) and c o r r e s p o n d i n g c o l l i s i o n f o r c e s a r e shown i n F i g . ( 5 . 3 1 ) . A c o m p a r i s o n o f F i g s . ( 5 . 2 8 ) and ( 5 . 3 0 ) show t h a t t h e r e i n f o r c e d s h i p i s damaged a l m o s t as much as t h e s t a n d a r d s h i p . T h a t i s i f t h e h u l l t e a r s i n an a c c i d e n t t h e n p l a t e t h i c k n e s s h a s m i n o r e f f e c t on t h e s i z e o f damage. R o l l i n g a n g l e o f t h e s h i p i n e a c h o f t h e above m e n t i o n e d c a s e s i s c a l c u l a t e d . I t i s n o t i c e d t h a t i n a l l o f t h e s e c a s e s t h e s h i p i s v e r y s t a b l e and a n g l e o f r o l l i n g does n o t e x c e e d s 1 . 5 d e g r e e . As an e x a m p l e , v a r i a t i o n o f t h e r o l l i n g a n g l e f o r t h e r e i n f o r c e d s h i p c o l l i d i n g w i t h 149 t h e r o u n d r o c k w i t h v e l o c i t y o f 7 m/s i s r e p r e s e n t e d i n F i g . ( 5 . 3 2 ) . T h i s c a s e h a s t h e most s e v e r e c o l l i s i o n and i t i s n o t i c e d t h a t t h e r o l l i n g a n g l e v a r i e s v e r y s l o w and does n o t e x c e e d 1 . 5 d e g r e e . 5 . 2 . 3 - C o l l i s i o n o f t h e S h i p D u r i n g M a n e u v e r i n g We now c o n s i d e r what happens when a s h i p i s m a n e u v e r i n g (has r o t a t i o n a l m o t i o n ) and c o l l i d e s w i t h a f i x e d o b j e c t . The f o r w a r d v e l o c i t y o f t h e s h i p i s t a k e n as 2 m / s e c , and t h e f o r e and a f t speeds n o r m a l t o t h e p a t h a r e 2 m / s e c . and 0 . 5 m / s e c r e s p e c t i v e l y . I t i s assumed t h a t a r o c k i s l o c a t e d 15 m e t e r s u n d e r w a t e r and t h a t c o n t a c t i s made o v e r 20 m e t e r s o f t h e s h i p h u l l w i t h c e n t e r o f c o l l i s i o n 50 m e t e r s f r o m t h e f o r e p e r p e n d i c u l a r . A g a i n two t y p e s o f c o l l i s i o n s a r e c o n s i d e r e d ; w i t h o u t and w i t h r u p t u r e . R e s u l t s o f t h e c a s e s u n d e r c o n s i d e r a t i o n a r e compared i n o r d e r t o g a i n some i n s i g h t i n t o t h i s c o m p l e x and p r e v i o u s l y u n e x p l o r e d p r o b l e m . As a r e s u l t o f t h e i m p a c t , t h e s h i p s t a r t s t o move away f r o m t h e r o c k , and c a u s e s a d e c r e a s e i n d e p t h o f t h e g e n e r a t e d damage a l o n g t h e s h i p . S t r e n g t h o f t h e g e n e r a t e d i m p u l s e depends on d i f f e r e n t p a r a m e t e r s s u c h as t h e c o l l i s i o n s c e n a r i o , t y p e o f c o l l i s i o n , mass and t h e s t r u c t u r a l s p e c i f i c a t i o n o f t h e s h i p . I n t h e f o l l o w i n g s e c t i o n t h e s e p o i n t s a r e e x a m i n e d f o r a number o f c a s e s . The p r e s e n t a p p r o a c h makes i t p o s s i b l e t o f o l l o w r o l l i n g a n g l e o f t h e s h i p d u r i n g and a f t e r e a c h c o l l i s i o n . F o r e a c h o f t h e s p e c i f i e d c o l l i s i o n c a s e s on t h e ' S t a n d a r d T a n k e r S h i p ' and ' S t r e n g t h e n e d S h i p ' v a r i a t i o n o f r o l l i n g a n g l e i s r e p r e s e n t e d . 150 5 . 2 . 3 . 1 - C o l l i s i o n o f t h e S h i p W i t h a Smooth R o c k F i r s t , c o l l i s i o n o f t h e s t a n d a r d s h i p i s s t u d i e d . C o l l i s i o n f o r c e v e r s u s t i m e i s p l o t t e d i n F i g . ( 5 . 3 3 a ) , w h i c h shows t h a t t h e f i r s t i m p a c t t a k e s l e s s t h a n 0 . 4 s e c o n d and i t s p e a k e x c e e d s 70 MN b u t a f t e r t h e f i r s t i m p u l s e t h e c o n t a c t l o a d d r o p s t o a v e r y l o w l e v e l . A c o m p a r i s o n o f t h i s f i g u r e w i t h F i g . ( 5 . 3 3 b ) , w h i c h r e p r e s e n t s t h e e x t e n t s o f g e n e r a t e d damage v e r s u s t i m e , shows t h a t t h e s h i p s t a r t s t o move away f r o m t h e r o c k a f t e r t h e f i r s t i m p u l s e . I t i s a l s o n o t i c e d t h a t a f t e r t h e f i r s t i m p u l s i v e f o r c e t h e h u l l d e n t s f o r 0 . 5 m e t e r s i d e w i s e w h i l e t h e v e r t i c a l e x t e n t o f t h e d e n t e x c e e d s 1 m e t e r . The g e n e r a t e d damage a l o n g t h e s h i p i s e s t i m a t e d t o be 2 3 . 4 8 m e t e r s . We now c o n s i d e r t h e r e i n f o r c e d h u l l . The i m p a c t l o a d g e n e r a t e d on t h e h u l l d u r i n g t h i s c o l l i s i o n i s p l o t t e d i n F i g . ( 5 . 3 4 a ) . T h i s f i g u r e shows t h a t t h e d u r a t i o n o f t h e i m p a c t l o a d i s d e c r e a s e d t o l e s s t h a n 0 . 3 s e c o n d and i t s maximum v a l u e e x c e e d s 100 MN. T h i s i s an i n c r e a s e o f 30%. E x t e n t o f l a t e r a l damage f o r t h i s s h i p i s r e p r e s e n t e d i n F i g . ( 5 . 3 4 b ) , i t i s n o t i c e d t h a t t h e t h e l a t e r a l damage e x t e n t i s s m a l l e r t h a n on t h e n o r m a l s t r e n g t h h u l l . I n t h i s c a s e , d e n t e d a r e a w o u l d h a v e a l e n g t h o f 2 2 . 7 m e t e r s . I t a p p e a r s t h a t s t r e n g t h e n i n g t h e h u l l h a r d l y a f f e c t s t h e e x t e n t o f damage b u t i t i n c r e a s e s t h e i m p u l s e on t h e s h i p as p r e v i o u s l y o b s e r v e d i n t h e f o r w a r d c o l l i s i o n s t u d i e s . 151 5 . 2 . 3 . 2 - C o l l i s i o n o f t h e S h i p w i t h A S h a r p Edge R o c k The same c o l l i s i o n s c e n a r i o i s t a k e n as b e f o r e b u t now i t i s assumed t h a t t h e h u l l t e a r s t h r o u g h - o u t t h e c o l l i s i o n . V a r i a t i o n o f t h e i m p a c t l o a d f o r t h e s t a n d a r d s h i p i s shown i n F i g . ( 5 . 3 5 a ) . I t i s n o t i c e d t h a t t h e f i r s t i m p a c t t a k e s a l m o s t 0 . 6 s e c o n d ; 50% l o n g e r t h a n t h e p e r i o d o f i m p a c t i n t h e " no r u p t u r e " c o l l i s i o n . P e a k o f t h e i m p a c t l o a d i s l e s s t h a n 45 MN w h i c h shows a d e c r e a s e o f 35% w i t h r e s p e c t t o t h e " no r u p t u r e " c o l l i s i o n . L e n g t h o f t h e damage a r e a a l o n g t h e s h i p i s e s t i m a t e d t o be 2 3 . 7 3 m e t e r s . E x t e n t o f t h e l a t e r a l damage on t h e h u l l i s r e p r e s e n t e d i n F i g ( 5 . 3 5 b ) w h i c h shows t h a t t h e h o r i z o n t a l damage e x c e e d s 0 . 6 m e t e r and v e r t i c a l damage i s 1 .7 m e t e r , g r e a t e r t h a n when t h e r e i s no r u p t u r e . F i n a l l y t h e r e i n f o r c e d s h i p i s c o n s i d e r e d . F i g . ( 5 . 3 6 a ) shows v a r i a t i o n o f t h e i m p a c t l o a d d u r i n g t h e c o l l i s i o n . I t i s n o t i c e d t h a t t h e f i r s t i m p u l s e t a k e s l e s s t h a n 0 . 5 s e c o n d s and i t s maximum v a l u e i s 65 MN. L e n g t h o f t h e damaged a r e a a l o n g t h e s h i p i s e s t i m a t e d t o be 2 3 . 6 1 m e t e r s . F i g . ( 4 . 3 6 b ) shows t h e e x t e n t o f l a t e r a l damage. I t shows a g a i n t h a t t h e s t r e n g t h e n e d h u l l i n c r e a s e s t h e p e a k i m p u l s e l o a d and does n o t h e l p s i g n i f i c a n t l y t o r e d u c e t h e damage e x t e n t . V a r i a t i o n o f t h e r o l l i n g a n g l e i n e a c h c o l l i s i o n d u r i n g t h e maneuver o f t h e s h i p i s shown i n F i g . ( 5 . 3 7 ) , f o r a n i n t e r v a l o f 7 s e c o n d s a f t e r t h e f i r s t c o n t a c t . The r o l l i n g a n g l e i s l i m i t e d t o 4 d e g r e e s a n d d e c r e a s e s i n t h e damping p r o c e d u r e . T h i s shows a s t a b l e s h i p d u r i n g t h e s e s e r i e s o f a c c i d e n t s . 152 5.2 .4- R e l a t i o n Between Volume of Damaged S t r u c t u r a l Elements and  D i s s i p a t e d K i n e t i c Energy During C o l l i s i o n I n e a c h o f t h e c o l l i s i o n c a s e s e x a m i n e d d u r i n g t h e p r e c e d i n g s e c t i o n s t h e v o l u m e o f t h e damaged s t r u c t u r a l e l e m e n t s o f t h e ' S t a n d a r d T a n k e r S h i p ' were c a l c u l a t e d b y a s s u m i n g a r a t i o b e t w e e n t h e s o l i d and g e o m e t r i c vo lume i n t h e damaged r e g i o n . C o m p a r i n g k i n e t i c e n e r g y o f t h e s h i p i m m e d i a t e l y b e f o r e and a t t h e end o f t h e c o l l i s i o n , l o s s o f k i n e t i c e n e r g y d u r i n g e a c h c o l l i s i o n i s c a l c u l a t e d . F o r e a c h c a s e t h e v o l u m e o f damaged m a t e r i a l and t h e c o r r e s p o n d i n g l o s s o f k i n e t i c e n e r g y i s p l o t t e d i n F i g . ( 5 . 3 8 ) . I t i s n o t i c e d t h a t f o r e a c h s p e c i f i e d s h i p t h e r e i s a l i n e a r r e l a t i o n b e t w e e n t h e d i s s i p a t e d e n e r g y and t h e c o r r e s p o n d i n g damaged v o l u m e . G r a p h o f F i g . ( 5 . 3 8 ) shows t h a t r e s u l t s o f t h e p r e s e n t a p p r o a c h a g r e e w i t h M i n o r s k y ' s [4] method a n d t h e m o d i f i c a t i o n s made b y J o n e s [ 8 ] . M i n o r s k y [ 4 ] , showed t h a t t h e r e i s a l i n e a r r e l a t i o n b e t w e e n t h e v o l u m e o f damaged m a t e r i a l and d i s s i p a t e d k i n e t i c e n e r g y i n t h e c o l l i s i o n . A n d J o n e s [8] shows t h a t t h e s l o p e o f t h e V o l u m e - E n e r g y l i n e depends on t h e damage e x t e n t i n e a c h c o l l i s i o n . 5 .2 .5 - Change i n Yaw During Grounding The t h e o r y h a s b e e n d e v e l o p e d a s s u m i n g t h a t t h e change i n a n g u l a r m o t i o n (yaw) i s s m a l l w h i l s t t h e i m p a c t f o r c e s a r e a c t i n g . I n t h e c a s e shown i n F i g . ( 5 . 2 6 d ) , t h e most s e v e r e c a s e c o n s i d e r e d , i t h a s b e e n n e c e s s a r y t o f o l l o w t h e a n g u l a r m o t i o n d u r i n g t h e w h o l e c o l l i s i o n p r o c e s s . I n t h i s c a s e t h e a n g u l a r r o t a t i o n d u r i n g c o l l i s i o n was f o u n d t o 153 be l e s s t h a n 3 d e g r e e s . I t i s p o s s i b l e t h a t a n a n g u l a r c o l l i s i o n d u r i n g m a n e u v e r i n g c o u l d r e s u l t i n l a r g e r r o t a t i o n s and i t i s t h e n n e c e s s a r y t o c h e c k t h e m a g n i t u d e o f t h e r o t a t i o n s . 5 . 2 . 6 - E f f e c t o f G r o u n d i n g Damage on S t a b i l i t y When a s h i p g r o u n d s and s u f f e r s b o t t o m damage t h e damage p r o c e s s o c c u r s i n a t i m e i n t e r v a l l e s s t h a n one m i n u t e , v e r y o f t e n d u r i n g a t e n - s e c o n d i n t e r v a l . On t h e o t h e r h a n d t h e t i m e t a k e n f o r t h e c a r g o ( o i l ) t o s p i l l t o any g r e a t e x t e n t i s s e v e r a l h o u r s and i n some c a s e s s e v e r a l d a y s . T h e r e f o r e i t h a s n o t b e e n n e c e s s a r y t o c o n s i d e r l o s s o r change i n s t a b i l i t y o f t h e s h i p due t o c h a n g i n g c a r g o , d u r i n g t h e g r o u n d i n g p r o c e s s . The e f f e c t o f t h e c o n t a c t f o r c e on s t a b i l i t y i s , o f c o u r s e , i n c l u d e d . I t i s f u r t h e r assumed t h a t d u r i n g g r o u n d i n g t h e l o s s i n b u o y a n c y due t o t h e r e d u c e d h u l l a r e a i n c o n t a c t w i t h t h e f l u i d , i s n e g l i g i b l e . 5 . 2 . 7 - C l o s u r e I n t h e above s e c t i o n s i t was shown how t o e s t i m a t e t h e e x t e n t o f damage t o t h e h u l l o f a s h i p when i t i s i n c o l l i s i o n w i t h a submerged o b j e c t . C o n s e q u e n t l y we a r e a b l e t o d e t e r m i n e t h o s e r e g i o n s w h i c h r e m a i n undamaged d u r i n g a c o l l i s i o n , n a m e l y t h e s a f e r e g i o n o f t h e s h i p . T h i s s t u d y c o u l d be c o n d u c t e d on any s h i p t o f i n d t h e s a f e o p e r a t i n g s p e e d t h r o u g h a h a z a r d o u s a r e a . A l s o t h e p r e s e n t e d method c o u l d be u s e d i n t h e d e s i g n s t a g e o f t h e s h i p t o d e t e r m i n e t h e p r o p e r s t r u c t u r a l c o n f i g u r a t i o n o f t h e h u l l , i n o r d e r t o m i n i m i z e t h e damage on t h e s h i p and t h e c a r g o s h o u l d a c o l l i s i o n o c c u r . 154 2000 1750 J Fig. 5.23- a) Mass, b) Buoyancy distribution along the 'Standard Tanker Ship'. 155 X O X Id L3 < 2 . 0 (a) F . P . 4 (b) F P -4 2 . ( C ) F . P . 4 (d) o F . P . V E R T I C . DAMAGE HORIZO. DAMAGE A —r— 1 I 100 150 200 SHIP LENGTH (M) 250 300 A . P . V E R T I C . DAMAGE HORIZO. DAMAGE 150 SHIP LENGTH 300 A . P . V E R T I C . DAMAGE HORIZO: DAMAGE 100 150 SHIP LENGTH 200 (M) 250 300 A . P. V E R T I C . DAMAGE HORIZO. DAMAGE 100 150 SHIP LENGTH 200 (M) 250 I— 300 A . P. F i g . 5.24- Extent of damage due to forward collision without rupture of the standard hull. a) Velocity - 1 m/s, b) Velocity - 3 m/s c) Velocity - 5 m/s, d) Velocity - 7 m/s. 10 (a) - i 1 1 1 1 1 1 1 r - ^ 0 I0O 100 300 ' 0 0 900 <00 TOO BOO 9 0 0 t | » C ( i C C . ) - «o JO 29 3D 39 t i n t ( 9 C C . ) F i g . 5.25- Generated force during the forward collision without rupture of the standard hull. a) Velocity - 2 m/s, b) Velocity - 3 m/s c) Velocity - 5 m/s, d) Velocity - 7 m/s. 157 Ixl ' *- ' X 13 < X U o UJ < X < A > F . P . 4 (b) F . P . 4 ( C ) F . P . 4 2 . <a> F . P . 4-i—; — i — 100 150 200 SHIP LENGTH (M) I 100 150 200 SHIP LENGTH (M) A. I 100 150 200 SHIP LENGTH (M) VERTIC . DAMAGE H0RIZ0. DAMAGE 250 300 A. P. VERTIC . DAMAGE HORIZO. DAMAGE 250 I 300 A . P VERTIC . DAMAGE HORIZO. DAMAGE 300 A . P . VERTICAL DAMAGE HORIZONTAL DAMAGE r 100 150 SHIP LENGTH 300 A . P . F i g . 5.26- Extent of damage due to forward collision without rupture of the reinforced hull, a) Velocity - 1 m/s, b) Velocity - 3 m/s c) Velocity - 5 m/s, d) Velocity - 7 m/s. (StC.) F i g . 5.27- Generated force during the forward collision without rupture of the reinforced hull. a) Velocity - 1 m/s, b) Velocity - 3 m/s c) Velocity - 5 m/s, d) Velocity - 7 m/s. 00 159 LU < U l I— X 13 < X o < (3 ( a ) (b) (c) 4 F i g . (d) 5 . 2 8 -F . P . F . P . F . P . VERTIC . DAMAGE HORIZO. DAMAGE 100 150 200 SHIP LENGTH (M) 250 150 200 SHIP LENGTH (M) 250 100 150 200 SHIP LENGTH (M) 300 A . P . VERTIC. DAMAGE HORIZO. DAMAGE 300 A . P . VERTIC. DAMAGE HORIZO. DAMAGE 300 A . P . VERT IC . DAMAGE HORIZO. DAMAGE i 5 0 200 250 300 F . P . SHIP LENGTH (M) A . P . Extent of damage due to forward collision with rupture of the standard hull. a) Velocity - 1 m/s, b) Velocity - 3 m/s c) Velocity - 5 m/s, d) Velocity - 7 m/s. 10 i « 20 7 J JO 3 ) 40 ti 0 J 10 15 JO * » 30 J J T I K I ( S E C . ) (d ) M N t ( 5 t C . ) Generated force during the forward collision with rupture of the standard hull. a) Velocity - 1 m/s, b) Velocity - 3 m/s c) Velocity - 5 m/s, d) Velocity - 7 m/s. 4 VERTIC. DAMAGE HORIZO. DAMAGE » ) F . P . 4 F . P . 4 1) F . P . ^ I I 1 50 100 150 200 SHIP LENGTH (M) 250 300 A . P . VERTIC . DAMAGE HORIZO. DAMAGE 150 200 SHIP LENGTH (M) 250 300 A . P ; VERTIC . DAMAGE HORIZO. DAMAGE 1 200 150 SHIP LENGTH (M) 250 300 A . P . VERTIC. DAMAGE HORIZO. DAMAGE ~ l — 200 150 SHIP LENGTH (M) 250 300 A . P . 30- Extent of damage due to forward collision with rupture the reinforced hull. a) Velocity - 1 m/s, b) Velocity - 3 m/s c) Velocity - 5 m/s, d) Velocity - 7 m/s. 40 — R — 2 0 —R— 3 0 —R— 4 0 5 0 TINE ( S E C . ) to I TO _ JO ( S E C . ) F i g . 5.31- Generated force during the forward collision with rupture the reinforced hull. a) Velocity - 1 m/s, b) Velocity - 3 m/s c) Velocity - 5 m/s, d) Velocity - 7 m/s. 163 5 . 3 2 - Variation of rolling angle when ship collides with at speed of 7 m/s. the rock 164 o a . o u. 140 120 . ~ 100 60 . 60 . 40 20 (a) COLLISION TIME ( S E C . ) I 1 I I I I I I I 0 0 .25 0 . 5 0 .75 1 1.25 1.5 1.75 2 2 .25 2 .5 X U J 2 . 5 2 . 1 .5 1 . 0 .5 . VERTICAL DAMAGE HORIZONTAL DAMAGE 0 ' 0 .25 0 . 5 0 .75 1 (b) COLLISION TIME ( S E C . ) F i g . 5 . 3 3 - a) Generated force, b) extent of damage, due to the sidewise collision through the ship maneuver without rupture of the standard hull. COLLISION TIME ( S E C . ) (b) VERTICAL DAMAGE HORIZONTAL DAMAGE 0 .2 0 .4 .0 .6 0 .8 1 COLLISION TIME ( S E C . ) 5.34- a) Generated force, b) extent of damage, due to the sidewise collision through the ship maneuver without rupture of the reinforced hull. 166 o a . o 140 120 100 . 80 . 60 40 . 20 . (a) 1 1.5 2 2 . 5 3 COLLISION TIME ( S E C . ) 2 . 5 2 . 1.5 0 .5 . - VERTICAL DAMAGE •• HORIZONTAL DAMAGE 3.5 F i g . 5.35- a) Generated force, b) extent of damage, due to the sidewise collision through the ship maneuver with rupture of the standard hull. 167 F i g . 5.36- a) Generated force, b) extent of damage, due to the sidewise collision through the ship maneuver with rupture of the reinforced hull. 168. 3 . 2 . Ol o _ l C3 O I X • 1 . •2 . - 3 . Fig 5.37- Variation of rolling angle of the ship due to sidewise collision. 10 or Ul *— 111 03 => O o > o « « a A * NORMAL HULL DENT ONLY B---ID REINF. HULL DENT ONLY NORMAL HULL DENT 4 TEAR * - - - * R E I N F . HULL DENT & TEAR 1000 2000 3000 DISSIPATED ENERGY (MJ) 4000 F i g . 5.38- Relation between volume of the damaged structural elements and loss of kinetic energy of the ship. 169 6- SUMMARY OF RESULTS AND CONCLUSION 6 . 1 - I c e B e a k i n g S h i p I c e b r e a k i n g h a s b e e n i n v e s t i g a t e d f o r b o t h t h e c o n t i n u o u s and ramming modes . The f i n i t e e l e m e n t p r o g r a m d e v e l o p e d e n a b l e s t h e g l o b a l v a l u e s o f t h e b e n d i n g moment a l o n g t h e s h i p and t h e c o n t a c t f o r c e on t h e bow t o be e v a l u a t e d as f u n c t i o n s o f t i m e . S t r u c t u r a l s p e c i f i c a t i o n o f t h e s h i p l e a d s t o t h e a l l o w a b l e b e n d i n g moment and c o n t a c t f o r c e on t h e bow. The p r e s e n t method c a n be u s e d t o f i n d t h e g l o b a l s t r u c t u r a l s p e c i f i c a t i o n o f t h e s h i p and t h e i r r e l a t i o n t o t h e o p e r a t i n g s p e e d . 6 . 1 . 1 - C o n t i n u o u s I c e B r e a k i n g A m o d e l i s d e v e l o p e d t o a n a l y z e t h e e f f e c t o f i c e t h i c k n e s s and i c e s t r e n g t h on t h e b e h a v i o u r o f t h e s h i p d u r i n g c o n t i n u o u s i c e b r e a k i n g . The s e m i - e m p i r i c a l a n a l y s e s o f K o r z h a v i n [29] and L e w i s & Edward [32] a r e e m p l o y e d i n t h i s w o r k . T h i s s t u d y shows t h a t t h e i c e f o r c e i s a r e p e a t e d i m p u l s e w i t h a f r e q u e n c y w h i c h i n c r e a s e s l i n e a r l y w i t h s h i p s p e e d . a n d a l s o depends on t h e i c e t h i c k n e s s and i c e s t r e n g t h . V a r i a t i o n o f t h e maximum b e n d i n g moment a l o n g t h e s h i p i s i n v e s t i g a t e d f o r d i f f e r e n t v a l u e s o f s h i p v e l o c i t y i n i c e f i e l d s o f d i f f e r e n t t h i c k n e s s e s . I t i s shown t h a t a t t h e l o w e r s p e e d s , when t h e i m p u l s e p e r i o d i s much g r e a t e r t h a n t h e f l e x u r a l p e r i o d o f t h e s h i p s t r u c t u r e , i n t h e t i m e p e r i o d b e t w e e n two s u b s e q u e n t i m p u l s e s b e n d i n g moment i s v a r y i n g w i t h t h e f u n d a m e n t a l f r e q u e n c y o f t h e s h i p h u l l as a 170 damped f r e e v i b r a t i n g s y s t e m . A l s o p e a k o f maximum b e n d i n g moment c o i n c i d e s w i t h t h e p e a k o f t h e i c e i m p u l s e . T h i s s t u d y shows t h a t , as i s t o be e x p e c t e d , when t h e f r e q u e n c y o f t h e i c e l o a d i s c l o s e t o a n a t u r a l f r e q u e n c y o f t h e s h i p - m o t i o n , t h e p e a k o f maximum b e n d i n g moment a n d / o r r o l l i n g a n g l e o f t h e s h i p g r o w s . B u t as l o n g as t h i s f r e q u e n c y i s b e l o w t h e f u n d a m e n t a l f r e q u e n c y o f t h e s h i p , v a r i a t i o n s i n p e a k maximum b e n d i n g moment a r e s m a l l , a l t h o u g h a s l i g h t i n c r e a s e does o c c u r whenever t h e l o a d f r e q u e n c y c o i n c i d e s w i t h a s h i p r i g i d - b o d y - m o t i o n . We c o n c l u d e t h a t t h e p e a k b e n d i n g moment v a r i e s s l i g h t l y w i t h s h i p v e l o c i t y as l o n g as t h e s h i p i s n o t e x c i t e d b y a l o a d w i t h a f r e q u e n c y c l o s e t o one o f i t s s t r u c t u r a l n a t u r a l f r e q u e n c i e s , i n w h i c h c a s e p e a k o f maximum b e n d i n g moment a l o n g t h e s h i p i n c r e a s e s d r a m a t i c a l l y . We c o n c l u d e t h a t , an i c e b r e a k e r w o u l d be a b l e t o p r o c e e d i n t h i c k i c e f i e l d s w i t h h i g h e r v e l o c i t i e s when t h e f u n d a m e n t a l f r e q u e n c y o f t h e s h i p s t r u c t u r e i s f a r f r o m t h e f u n d a m e n t a l h y d r o d y n a m i c f r e q u e n c i e s i n h e a v e and p i t c h . T h a t i s , c o n s i d e r i n g t h e b e n d i n g s t r e s s i n t h e s h i p , t h e s t i f f e r i c e - b r e a k e r s a r e c a p a b l e o f b r e a k i n g t h i c k e r i c e f i e l d s w i t h h i g h e r ' s p e e d s . U s u a l l y h i g h e r s p e e d s a r e n o t a t t a i n a b l e i n t h i c k i c e f i e l d s b e c a u s e o f l i m i t a t i o n s on t h e power o f t h e s h i p . The e x t r a power n e e d e d f o r i c e b r e a k i n g p r o c e d u r e v e r s u s t h e s h i p v e l o c i t y i s c a l c u l a t e d . A r e l a t i o n s h i p b e t w e e n t h e i c e b r e a k i n g power o f t h e ' s t a n d a r d I c e B r e a k e r ' , t h i c k n e s s o f i c e f i e l d and s h i p v e l o c i t y i s i n t r o d u c e d . A l l t h e s e make i t p o s s i b l e t o d e t e r m i n e t h e s h i p o p e r a t i n g s p e e d . I n t h i s w o r k v a r i a t i o n o f r o l l i n g a n g l e o f t h e s h i p i s s t u d i e d t o f i n d a r e l a t i o n s h i p b e t w e e n t h i s a n g l e and t h e s h i p s p e e d . I t i s n o t i c e d 171 t h a t when t h e f r e q u e n c y o f t h e g e n e r a t e d f o r c e on t h e bow i s e q u a l t o o r d o u b l e t h e r o l l i n g f r e q u e n c y o f t h e s h i p t h e n p e a k r o l l i n g a n g l e i s i n c r e a s e d . 6 . 1 . 2 - Ramming Mode ' The p r e s e n t s i m u l a t i o n o f t h e ramming mode c o u l d be a p p l i e d t o any i c e - b r e a k i n g s h i p t o c a l c u l a t e t h e g l o b a l ramming f o r c e and dynamic r e s p o n s e o f t h e s h i p . T h i s work g i v e s t h e r e l a t i o n s h i p b e t w e e n t h e g e n e r a t e d c o n t a c t f o r c e on t h e bow a n d maximum b e n d i n g moment a l o n g t h e s h i p w i t h m a s s , bow a n g l e and s t i f f n e s s o f t h e h u l l . V a r i a t i o n o f maximum b e n d i n g moment a l o n g t h e s h i p , and t h e c o n t a c t f o r c e on t h e bow d u r i n g t h e b e a c h i n g m o t i o n a r e c a l c u l a t e d f o r d i f f e r e n t ramming v e l o c i t i e s . I t i s shown t h a t t h e b e n d i n g moment and t h e c o n t a c t f o r c e on t h e bow a r e f l u c t u a t i n g w i t h a f r e q u e n c y e q u a l t o t h a t o f t h e f u n d a m e n t a l f r e q u e n c y o f t h e s h i p . A l s o i t i s n o t i c e d t h a t t h e b e n d i n g moment g e n e r a t e d due t o t h e i n i t i a l i m p u l s e i s h i g h e r t h a n t h e b e n d i n g moment a t t h e end o f t h e b e a c h i n g p e r i o d . V a r i a t i o n o f t h e c o n t a c t f o r c e shows t h a t t h e i m p a c t f o r c e on t h e bow due t o t h e i n i t i a l i m p u l s e i s h i g h e r t h a n t h e c o n t a c t f o r c e d u r i n g t h e b e a c h i n g p e r i o d . The e f f e c t o f s h i p v e l o c i t y on t h e p e a k i m p a c t l o a d and t h e maximum b e n d i n g moment i n .the s h i p i s i n v e s t i g a t e d . A c o m p a r i s o n b e t w e e n the c a l c u l a t e d maximum c o n t a c t f o r c e w i t h p r o p o s e d f o r m u l a s b y Vaughan [35] and J o h a n s s o n e t a l [33] i s p e r f o r m e d w h i c h shows a good ag reement b e t w e e n t h e p r e s e n t c a l c u l a t i o n s and t h e i r p r e d i c t i o n s . I n t h i s s t u d y t h e e f f e c t o f t h e s t i f f n e s s o f t h e h u l l on t h e s h i p s t r u c t u r a l r e s p o n s e t o t h e i m p a c t l o a d o f t h e i c e i s i n v e s t i g a t e d . I t i s 172 shown that the more f l e x i b l e ship loses l e s s k i n e t i c energy i n c o l l i d i n g w i t h hard i c e . That i s , the more r i g i d ship has a harder c o l l i s i o n w i t h more energy l o s s . We conclude that higher i n i t i a l impulse would be generated on the bow of the more r i g i d s h i p s . I t i s a l s o n o t i c e d that while the bending moment induced i n the s t i f f e r ship due to the i n i t i a l impulse i s higher than the that of the other s h i p s , during the beaching motion maximum bending moment approaches to the same magnitude i n the h u l l s w i t h d i f f e r e n t s t i f f n e s s . V a r i a t i o n of the contact f o r c e on the bow during the ramming p e r i o d shows that the i n i t i a l contact f o r c e increases w i t h an increase i n the s t i f f n e s s of the h u l l , but during the beaching p e r i o d magnitude of contact f o r c e approaches to a common value. Though the contact f o r c e i n the case of r i g i d ship shows an increase during the process. I t can be concluded t h a t , a bigger c o l l i s i o n f o r c e is. p r e d i c t e d i n the case of r i g i d s h i p s , but as long as the bow area i n these ships i s strong enough to stand the i n i t i a l impact, the h u l l s of the more r i g i d ships remain s a f e r during the ramming mode. 173 6 . 2 - S h i p C o l l i s i o n I n t h i s w o r k c o l l i s i o n o f a s h i p w i t h an e x t e r n a l o b j e c t i s s t u d i e d t o d e v e l o p a method f o r e s t i m a t i n g t h e e x t e n t o f damage g e n e r a t e d i n an a c c i d e n t . A s h i p a c c i d e n t c o u l d h a p p e n e i t h e r when t h e s h i p i s r u n n i n g t h r o u g h i t s n o r m a l r o u t e s o r d u r i n g t h e m a n e u v e r i n g . The c o n s e q u e n c e o f a s h i p c o l l i s i o n may i n v o l v e r u p t u r e o r s i g n i f i c a n t c r u s h i n g o f t h e s h i p s t r u c t u r e . To d e v e l o p a mode l f o r p r e d i c t i n g t h e g e n e r a t e d f o r c e i n a c o l l i s i o n t h e a n a l y t i c a l w o r k s r e p r e s e n t e d b y M i n o r s k y [4] , J o n e s [21] and Vaughan [9] h a v e b e e n e m p l o y e d i n t h i s w o r k . The g e n e r a t e d c o l l i s i o n f o r c e depends on t h e s t r u c t u r a l p r o p e r t i e s o f t h e s h i p and t h e c o l l i s i o n s c e n a r i o . T h i s w o r k c o v e r s a w i d e v a r i e t y o f p o s s i b l e c o l l i s i o n s as i s e x p l a i n e d i n C h a p t e r 4 . O b v i o u s l y t h e e x t e n t o f damage i s r e l a t e d t o t h e s h i p v e l o c i t y . To i n v e s t i g a t e t h i s c o n n e c t i o n , c o l l i s i o n o f a ' S t a n d a r d T a n k e r S h i p ' w i t h a n e x t e r n a l o b j e c t i s s i m u l a t e d f o r d i f f e r e n t v a l u e s o f t h e s h i p s p e e d . I t i s shown t h a t a m o d e r a t e - i n c r e a s e i n t h e v e l o c i t y o f t h e s h i p c o u l d d r a m a t i c a l l y i n c r e a s e t h e damage e x t e n t . A p p l y i n g t h i s a p p r o a c h t o a s h i p makes i t p o s s i b l e t o i n d i c a t e t h e maximum a l l o w a b l e s a f e s p e e d f o r any s h i p t h r o u g h c o a s t a l r o u t e s and s h a l l o w w a t e r i f t h e damage e x t e n t i s s u p p o s e d t o be i n a c e r t a i n l i m i t . To i n v e s t i g a t e t h e e f f e c t o f h u l l r u p t u r e on damage e x t e n t , two s i m i l a r c o l l i s i o n s s c e n a r i o s a r e c o m p a r e d . I n t h e f i r s t c o l l i s i o n the h u l l o n l y d e n t s , and i n t h e s e c o n d one i t t e a r s and d e n t s . T h i s c o m p a r i s o n shows t h a t when r u p t u r e o c c u r s t h e p e a k c o l l i s i o n f o r c e i s s m a l l e r t h a n when t h e r e i s no r u p t u r e and t h e c o l l i s i o n f o r c e i s 174 r e l a t i v e l y u n i f o r m d u r i n g t h e c o l l i s i o n t i m e , w h e r e a s i n c o l l i s i o n w i t h o u t a r u p t u r e on t h e h u l l c o l l i s i o n f o r c e s a r e s h a r p . A l s o c o m p a r i s o n o f t h e damage e x t e n t i n t h e s e two c a s e s shows t h a t i n c o l l i s i o n s w i t h o u t r u p t u r e t h e g e n e r a t e d damage i s a l m o s t h a l f o f t h e c a s e w i t h r u p t u r e and d e n t . The e f f e c t o f h u l l s t r e n g t h on t h e b e h a v i o r o f t h e s h i p d u r i n g a c o l l i s i o n i s a l s o s t u d i e d b y c o m p a r i n g t h e ' S t a n d a r d T a n k e r S h i p ' and a s h i p o f t h e same s i z e and d i s p l a c e m e n t b u t w i t h a h u l l t w i c e as s t r o n g . I t i s n o t i c e d t h a t s t r e n g t h e n i n g t h e h u l l i s e f f e c t i v e as l o n g as t h e c o l l i s i o n does n o t c a u s e r u p t u r e o f t h e h u l l . I t i s a l s o n o t i c e d when t h e h u l l t e a r s i n a c o l l i s i o n t h e g e n e r a t e d f o r c e on t h e s t r e n g t h e n e d h u l l i s a l m o s t two t i m e s t h a t o f t h e n o r m a l h u l l w h i l e t h e g e n e r a t e d damage i s n o t d e c r e a s e d i n t h e same o r d e r . By r e p e a t i n g t h e s e t y p e s o f c o l l i s i o n s t u d i e s b u t w i t h t h e i n i t i a l i m p a c t p o i n t l o c a t e d a t d i f f e r e n t p a r t s o f t h e h u l l , i t w o u l d be p o s s i b l e t o e s t a b l i s h t h e s a f e s t p o s i t i o n i n any p a r t i c u l a r s h i p f o r s t o r i n g h a z a r d o u s c a r g o e s . 175 6 . 3 - Recommendat ions F o r F u t u r e Work C o l l i s i o n o f t h e s h i p w i t h an e x t e r n a l o b j e c t c a u s i n g t h e h u l l t o r u p t u r e w i l l change t h e f l o a t a t i o n o f t h e s h i p . T h i s e f f e c t i s n o t c o n s i d e r e d i n t h e p r e s e n t w o r k . I n f u t u r e w o r k t h e e f f e c t o f c h a n g e s i n b u o y a n c y o f t h e r u p t u r e d s h i p c a n be c o n s i d e r e d . I t s h o u l d be m e n t i o n e d t h a t t h e p r e s e n t work s t u d i e s t h e s h i p m o t i o n d u r i n g a c o l l i s i o n p e r i o d o f a b o u t 1 m i n u t e w h i l e t h e t i m e f o r t i l t i n g o f a damaged t a n k e r s h i p t o 25 d e g r e e s i s r e c o r d e d t o be a b o u t 1 . 5 h o u r s [ 5 0 ] . T h i s shows t h e p r e s e n t work c o v e r s a v e r y s h o r t p o r t i o n o f t h e f l o a t a t i o n p e r i o d . C o n s i d e r i n g t h e change o f b u o y a n c y makes i t p o s s i b l e t o s i m u l a t e t h e t o t a l p e r i o d o f a c c i d e n t . When t h e s h i p h u l l t e a r s i n c o l l i s i o n , e s p e c i a l l y when t h e e x t e n t o f l a t e r a l damage i s c o n s i d e r a b l e , t h e g l o b a l s t i f f n e s s o f t h e s h i p c h a n g e s . T h i s v a r i a t i o n o f t h e h u l l s t i f f n e s s c a n be c o n s i d e r e d i n t h e f u t u r e w o r k s t o s i m u l a t e t h e c o l l i s i o n more a c c u r a t e l y . I t s h o u l d be m e n t i o n e d t h a t i n t h e p r e s e n t e d s i m u l a t i o n t h e e x t e n t o f l a t e r a l damage i s n o t more t h a n 2 m e t e r s f o r a t a n k e r s h i p o f 40 m e t e r s beam. 176 REFERENCES 1 . R o b e s o n , D . E J r . , H a f t k a , R . T . , and S u n d k v i s t , K . E . , " P o t e n t i a l o f O p t i m a l S h i p S t r u c t u r e R e d e s i g n f o r M i n o r C o l l i s i o n s , " J o u r n a l o f S h i p R e s e a r c h V o l . 31 N o . l , M a r c h 1978 , 5 3 - 5 9 . 2 . M i n o r s k y , V . U . , P a r k e r , C .W . , and G o t i m e r , J . C . , " S h i p A c c i d e n t S t u d i e s , " Symp. on S a f e t y o f N u c l e a r S h i p s , S p o n s o r e d b y OECD N u c l e a r E n e r g y A g e n c y , Hamburg, Dec . ( 1 9 7 7 ) . 3 . K i n k e a d , A . N . "Some C o l l i s i o n and G r o u n d i n g C o n s i d e r a t i o n f o r R e f r i g e r a t e d G a s C a r r i e r s , " M a r i n and O f f s h o r e S a f e t y , E l s e v i e r 1984 , E d i t e d b y : P . A . F r i e z e , R . C . M c G r e g o r and I . E . W i n k l e . 4 . M i n o r s k y , V . U . , " A n A n a l y s i s o f S h i p C o l l i s i o n w i t h R e f e r e n c e t o P r o t e c t i o n o f N u c l e a r Power P l a n t s , " J . S h i p R e s e a r c h , 3 , 1 - 4 , 1 9 5 9 . 5 . W o i s i n , G. (1971) " S c h i f f b a u l i c h e F o r s c h u n g s a r b e i t e n f u r d i e S i c h e r h e i t K e r n e n e r g i e g e t r i e b e n e r H a n d e l s s c h i f f e , " J a h r b u c h d e r S c h i f f b a u t e c h n i s c h e n G e s e l l s c h a f t , 6 5 , 2 2 5 - 2 6 3 . 6 . A k i t a , Y . and K i t a m u r a , K. ( 1 9 7 2 ) , "A S t u d y on C o l l i s i o n b y an E l a s t i c Stem t o a S i d e S t r u c t u r e o f S h i p s , " J . S o c . N a v a l A r c h . J a p a n , 1 3 1 , 3 0 7 - 3 1 7 . 7 . J o n e s , N . , "A l i t e r a t u r e S u r v e y on t h e C o l l i s i o n And G r o u n d i n g P r o t e c t i o n o f S h i p s , " S h i p S t r u c t u r e C o m i t y R e p o r t S S C - 2 8 3 ( 1 9 7 4 ) . 8 . J o n e s , N. " S t r u c t u r a l A s p e c t o f S h i p C o l l i s i o n , " S t r u c t u r a l C r a s h w o r t h i n e s s , B u t t e r w o t h s , 1 9 8 3 . E d i t o r : J o n e s and W i e r z b i c k i . 9 . V a u g h a n , H . , " T h e T e a r i n g and C u t t i n g o f M i l d S t e e l P l a t e W i t h A p p l i c a t i o n t o S h i p G r o u n d i n g Damage," M e c h a n i c a l B e h a v i o r o f M a t e r i a l , V o l . 3 , ICM 3 C a m b r i d g e , E n g l a n d , A u g u s t 1979 . 10 . C o k e r , E . G . , " S t r e s s e s i n t h e H u l l s o f S t a n d a r d V e s s e l s , " T r a n s . I n s t . Nav . A r c h . 7 0 , 1 4 4 - 1 5 1 ( 1 9 2 8 ) . 1 1 . Thomson, W., "Some C a s e s o f F a i l u r e o f Deck P l a t i n g Under C o m p r e s s i v e S t r e s s e s Due t o S t r a n d i n g , " T r a n s . I n s t . N a v a l A r c h . 87 , 7 1 - 8 0 ( 1 9 4 5 ) . 1 2 . K i t a m u r a , K . , Okumoto, Y . , and S h i b u e , T . , "On t h e M o d e l T e s t s o f D o u b l e B o t t o m S t r e n g t h f o r S t r a n d i n g , " ( i n J a p a n e s e ) , J o u r n a l o f S o c i e t y o f N a v a l A r c h i t e c t s o f J a p a n , 143 , 3 6 9 - 3 7 9 , ( 1 9 7 8 ) . 1 3 . U e d a , Y . , K i t a m u r a , K . , Okumoto, Y . , Y o s h i d a , Y . and K a t a y a m a , M . , " U l t i m a t e S t r e n g t h o f t h e D o u b l e B o t t o m f o r S t r a n d i n g on a R o c k , " ( i n J a p a n e s e ) . J o u r n a l o f S o c i e t y o f N a v a l A r c h i t e c t s o f J a p a n , 143 , 3 5 7 - 3 6 8 ( 1 9 7 8 ) . 177 14 . V a u g h a n , H . , "Damage t o S h i p Due t o C o l l i s i o n and G r o u n d i n g , " Det N o r s k e V e r i t a s T e c h . R e p . 7 7 - 3 4 5 ( 1 9 7 7 ) . 1 5 . 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J . , " M a t r i x and F i n i t e E l e m e n t D i s p l a c e m e n t A n a l y s i s o f S t r u c t u r e s , " O x f o r d E n g i n e e r i n g S c i e n c e S e r i e s , 1984 . 4 9 . W a r b u r t o n G . B . , "The D y n a m i c a l B e h a v i o u r o f S t r u c t u r e s , " Pergamon p r e s s , O x f o r d , 1 9 7 6 5 0 . P r a t y u s h S e n , and C. K o n s t a n t i n i d i s , "A T ime S i m u l a t i o n A p p r o a c h t o A s s e s s m e n t o f Damage S u r v i v a b i l i t y o f RO/RO C a r g o S h i p s , " SNAME T r a n s a c t i o n s , V o l . 9 5 , 1987 , PP 3 3 7 - 3 5 5 . 5 1 . T u r g u t S a r p k a y a , and M i c h a e l I s a a c s o n , " M e c h a n i c s o f Wave F o r c e s on O f f s h o r e S t r u c t u r e s , " V a n N o s t r a n d R e i n h o l d Company, 1 9 8 1 . 5 2 . J . N . Newman, " M a r i n e H y d r o d y n a m i c s , " The M a s s a c h u s e t t s I n s t i t u t e o f T h e c h n o l o g y , 1977 . 5 3 . K i n K . , T u e - F e e , and A r n o J . K e i n o n e n , " F u l l - S c a l e M a n e u v e r i n g T e s t i n L e v e l I c e o f Canmar K i g o r i a k And R o b e r t L e M e u r , " M a r i n e T e c h n o l o g y , V o l . 2 3 , No. 2 , A p r i l 1986 , P P . 1 3 1 - 1 3 8 . 180 APPENDIX 'A' N o d a l V a l u e s o f t h e G e n e r a t e d L o a d s I n g e n e r a l , t h e c o l l i s i o n f o r c e s c o u l d e i t h e r be d i s t r i b u t e d o r c o n c e n t r a t e d . A l o a d a p p l i e d on t h e h u l l a l o n g a s h o r t d i s t a n c e , c o m p a r e d t o t h e l e n g t h o f t h e s h i p , i s c o n s i d e r e d a c o n c e n t r a t e d f o r c e , t h o u g h i t m i g h t be d i s t r i b u t e d o v e r a f i n i t e a r e a . C o n c e n t r a t e d l o a d s a r e g e n e r a t e d on t h e bow o f an i c e - b r e a k i n g s h i p o r when a s h i p r u n s o v e r a s h a r p r e e f o r when i t c o l l i d e s w i t h a s i n g l e r o c k . The d i s t r i b u t e d l o a d s a r e a p p l i e d t o a l o n g l e n g t h o f t h e h u l l . The l o a d s g e n e r a t e d d u r i n g a g r o u n d i n g i n c i d e n t , o r i n c o l l i s i o n w i t h a b i g o b j e c t i n m a n e u v e r i n g m o t i o n a r e e x a m p l e s o f t h e s e d i s t r i b u t e d l o a d s . As i s m e n t i o n e d i n C h a p t e r 2 , t h e e x t e r n a l l o a d s on a beam e l e m e n t a r e t h e t y p e o f s u r f a c e t r a c t i o n s , and t h e i r n o d a l v a l u e s c a n be c a l c u l a t e d b y : where s i s t h e a r e a o f t h e h u l l w h i c h i s a c t i v e l y l o a d e d . On a s h i p , n o r m a l l y , t h e f o r c e s a r e a p p l i e d on a p o r t i o n o f t h e h u l l . When t h e d i s t r i b u t i o n o f t h e f o r c e p e r u n i t l e n g t h o f t h e s h i p i s known, t h e above s u r f a c e i n t e g r a l w o u l d e a s i l y be c o n v e r t e d t o a l i n e i n t e g r a l as f o l l o w e d : T {q} = [N ] {F } ds s T {q} = [N ] {F } dx ( A . l ) 181 where {F} i s f o r c e p e r u n i t l e n g t h . E q u . ( A . l ) i s u s e d t o d e v e l o p t h e n o d a l v a l u e s o f t h e g e n e r a t e d l o a d on t h e e l e m e n t i n t h e f o l l o w i n g p a g e s . A . l - N o d a l v a l u e s o f t h e C o n c e n t r a t e d L o a d The beam e l e m e n t w i t h a c o n c e n t r a t e d l a t e r a l l o a d , a p p l i e d a t a c e r t a i n p o i n t on i t , i s c o n s i d e r e d . The c o n f i g u r a t i o n i s shown i n F i g . ( A . l ) . The l a t e r a l 3 -Z « F y F i g . A . l - The Concentrated Lateral p o i n t f o r c e F i s a p p l i e d a t d i s t a n c e Force on the Beam Element. X q f r o m t h e end number (1) and i n v e c t o r f o r m i s r e p r e s e n t e d b y : {F 5 ( x - x ) o w h e r e S ( x - x ) i s d e l t a f u n c t i o n a t x - x . o o T h e r e f o r e b y means o f E q u . ( B . l ) . t h e n o d a l v a l u e s o f t h e f o r c e i s : T T. [N ] (F } 6"(x - x ) dx - [N ] {F } 1 J0 0 0 where [N ] i s t h e t r a n s p o s e o f shape f u n c t i o n [N] e v a l u a t e d a t t h e p o i n t x - X q . The shape f u n c t i o n [N] w h i c h i s i n t r o d u c e d i n C h a p t e r 2 i s e v a l u a t e d a t v = x /I. Then t h e n o d a l v a l u e s o f t h e c o n c e n t r a t e d l o a d 182 {q} -'(1 - 3«/ +2 v ) F o o y (1 - 3i/2 +2 » / ) F 0 0 z -ll-u )z F - (1 -»/ ) y F o o 0 y 2 0 z j/ )£ F ' o z ( i/ - 2i/ 2 + J / 3 ) £ F 0 0 0 y (3^2- 2t / 3 )F 0 .0 y (3i, 2- 2 ^ 3 ) F 0 0 z - i / z F -»/ y F 0 0 y 0-7 0 z i / J )£F 0 z (-i/Z + i / 3 )£F 0 0 y w h e r e Z q and y Q r e p r e s e n t t h e c o o r d i n a t e s o f t h e p o i n t o f a p p l i c a t i o n o f t h e l o a d . The g l o b a l f o r c e v e c t o r i s p r o d u c e d b y a s s e m b l i n g o f t h e e l e m e n t a l f o r c e v e c t o r s i n a c c o r d a n c e w i t h t h e method shown i n C h a p t e r 2. A . 2 - Nodal Value of the D i s t r i b u t e d Load When t h e c o l l i s i o n f o r c e on t h e s h i p i s d i s t r i b u t e d a l o n g a c o n s i d e r a b l e e x t e n t o f t h e h u l l t h e f o r c e d i s t r i b u t i o n s h o u l d be c o n s i d e r e d . I n t h i s c a s e t h e s t a t i o n s on t h e s h i p (beam e l e m e n t s on t h e m o d e l ) w h i c h a r e u n d e r c o n t a c t a r e i n d i c a t e d . The n o d a l v a l u e s o f t h e f o r c e on e a c h e l e m e n t a r e c a l c u l a t e d a c c o r d i n g t o t h e f o l l o w i n g p r o c e d u r e . F i g . A . 2 - Distributed Force On a Beam Element. 183 The f o r c e on e a c h e l e m e n t i s a p p r o x i m a t e d as a u n i f o r m d i s t r i b u t e d l o a d o f m a g n i t u d e G. T h i s u n i f o r m l o a d w o u l d have two components o f G y and G u n i f o r m l y d i s t r i b u t e d a l o n g t h e e l e m e n t as r e p r e s e n t e d i n z F i g . ( A . 2 ) . The n o d a l v a l u e s o f t h e d i s t r i b u t e d l o a d a r e c a l c u l a t e d t h r o u g h t h e E q u . ( A . l ) as f o l l o w e d : {q} = [N ] {G } dx = lz/l [N ] {G } di / I and I d e f i n e t h e p o s i t i o n o f t h e l o a d on t h e e l e m e n t . 1 2 r S u b s t i t u t i n g t h e c o r r e s p o n d i n g v a l u e o f shape f u n c t i o n , [N ] i n t h e above e q u a t i o n and t a k i n g t h e i n t e g r a l , t h e n o d a l v a l u e s o f t h e d i s t r i b u t e d l o a d w o u l d b e : lq}= I (v- i / + 0 . 5 i / ) G y (v- l / 2 + 0 . 5 i / ) G z - (1/-0.5 i / 2 ) z G - ( i / - 0 . 5 i / 2 ) y G 0 y J 0 z ( - 0 . 5 i / 2 +2 /3 v3- 0 . 2 5 i / ) £ G z ( - 0 . 5 i / +2 /3 v3- 0 . 2 5 v*)tG , . y (1/ - 0 . 5 i / ) G y (1/ - 0 . 5 i / ) G z - 0 . 5 z J/ 2G - 0 . 5 y i / 2 G 0 y J 0 z ( 1 / 3 i / 3 - 0 . 2 5 i / ) £ G z ( 1 / 3 v3 - 0 . 2 5 i/*)£G v = £ i / £ The g l o b a l f o r c e v e c t o r i s p r o d u c e d b y a s s e m b l i n g e l e m e n t a l f o r c e v e c t o r s i n a c c o r d a n c e w i t h t h e method i n t r o d u c e d i n C h a p t e r 2 . 184 A . 3 - N o d a l v a l u e s o f t h e C o n c e n t r a t e d L o a d on a Two D i m e n s i o n a l Beam E l e m e n t Shape f u n c t i o n f o r a beam e l e m e n t w h i c h d e f l e c t s i n a p l a n e may be d e d u c e d f r o m t h e one i n t r o d u c e d i n C h a p t e r 2 a s : f l - 3i / + 2v <N > = -3 \ O -2vZ+v3)l , 2 3 iv - 2v The n o d a l v a l u e s o f a c o n c e n t r a t e d f o r c e !F a p p l i e d on t h e e l e m e n t a r e c a l c u l a t e d t h r o u g h t h e f o l l o w i n g r e l a t i o n . <N > ^ S ( x - x ) dx = ^ <N > • 0 0 where <N > i s t h e t r a n s p o s e o f shape f u n c t i o n <N > e v a l u a t e d a t x = T h e r e f o r e , t h e n o d a l f o r c e {q } i s : (q } = ? f 1 - 3v2 + 2»3 ( i / -2v2+v*)l . 0 0 0 o 2 3 3i / -2 i^ 0 0 (- v2  K K o a } The g l o b a l f o r c e v e c t o r i s g e n e r a t e d f r o m an a s s e m b l a g e o f t h e f o r c e v e c t o r s f o r a l l e l e m e n t s . O b v i o u s l y when t h e r e i s o n l y one p o i n t f o r c e on t h e w h o l e beam, a l l e l e m e n t s o f t h e g l o b a l f o r c e v e c t o r w o u l d be z e r o e x c e p t t h e f o u r t e rms c o r r e s p o n d i n g t o t h e e l e m e n t w h i c h i s u n d e r t h e f o r c e . The above e q u a t i o n r e p r e s e n t t h e r e l a t i o n o f t h e s e t e r m s and t h e p o s i t i o n o f t h e c o n c e n t r a t e d f o r c e . 185 APPENDIX 'B' U n c o u p l i n g t h e E q u a t i o n s o f M o t i o n A s e t o f l i n e a r and c o u p l e d e q u a t i o n s s u c h a s , [M]{U} + [C]{U} + [K] {U) = {Q} c a n be u n c o u p l e d b y t r a n s f o r m i n g t o t h e g e n e r a l i z e d c o o r d i n a t e s . I n g e n e r a l i z e d c o o r d i n a t e s t h e e q u a t i o n s w o u l d b e : [M ] p { * } + [C ] p { * } + [K ] p { * } = (Q } ( B . l ) T where [M ] = [T ] [M ] [T ] T [K ] p - [T ] [K ] [T ] T [C ] p = [T ] [C ] [T ] T - 1 {Q ) = [T ] { Q } and } = [T] {U} M a t r i c e s [M ] p , [K ] p a n d [C ] a r e t h e d i a g o n a l i z e d m a t r i c e s and m a t r i x [T] i s s u c h t h a t a l l o f i t s co lumns a r e t h e e i g e n - v e c t o r s o f t h e - 1 T m a t r i x [ [M ] [K ] ] ; [T ] i s t h e t r a n s p o s e o f [ T ] . When we h a v e p r o p o r t i o n a l damping , i . e . [C ] i s p r o p o r t i o n a l t o t h e mass and s t i f f n e s s m a t r i c e s s u c h a s : [C ] = a [M ] + B [K ] t h i s p r o p o r t i o n a l i t y i s s t i l l v a l i d b e t w e e n t h e d i a g o n a l i z e d damping m a t r i x [C ] and d i a g o n a l i z e d mass and s t i f f n e s s m a t r i c e s . Then : [C ] p = a [M ] p + B [K ] p Then E q u . ( B . l ) c a n be w r i t t e n a s : 186 [M ] p { * } + [a [M ] p + B [K ] p ] { * } + [K ] p { * } = {Q } ( B . 2 ) E a c h u n c o u p l e d e q u a t i o n i n t h e s e t o f E q u . ( B . 2 ) c a n be r e p r e s e n t e d a s : m. £. + (a m. + B k . ) £. + k . £. = q . n o r m a l i z i n g t h e above e q u a t i o n : q. £. + (a + Bco\) h + J . h = - ± - ( B . 3 ) k. 2 i where a>. = i s t h e i - t h n a t u r a l f r e q u e n c y o f t h e w h o l e s y s t e m . I The damping r a t i o r\^ i s d e f i n e d t h r o u g h t h e r e l a t i o n 2 r>. w. = a + / ) w 2 t h e n n o d a l u n c o u p l e d e q u a t i o n o f m o t i o n c a n be w r i t t e n a s : q. £ + 2 r, oo £ + co2 £ = — — ( B . 4 ) •t i L t i, -t m. 187 APPENDIX X ' R e l a t i o n Be tween The H o r i z o n t a l and V e r t i c a l A p p l i e d F o r c e on t h e Bow D u r i n g t h e ramming mode when t h e s h i p i s i n c o n t a c t w i t h t h e i c e , t h e c o n t a c t f o r c e F may be r e s o l v e d e i t h e r i n t o i t s t a n g e n t i a l and n o r m a l c o m p o n e n t s , F and F , o r i n t o i t s h o r i z o n t a l and v e r t i c a l £ - n components F and F . A s shown i n F i g . ( C . l ) c l e a r l y t h e y a r e r e l a t e d b y : F i g . C . l - Contact Force and Its Components. F =-F S i n a - F Cos a x n f F - F Cos a - F S i n a y n f ( C . l ) ( C 2 ) w h e r e , a i s t h e bow a n g l e . S i n c e , F - u F , where u i s t h e c o e f f i c i e n t o f f r i c t i o n . The above f n e q u a t i o n s become: 188 F = - F ( S i n a + fi Cos a) x n F = F (Cos a - (j, S i n a) y n ( C 3 ) ( C 4 ) D i v i d i n g F b y F we t h e n g e t : x y ( S i n a + jit Cos a ) (Cos a — /J, S i n a) T h a t i s , F A F y where A = ( S i n a + ix Cos a ) (Cos a — n S i n a) O b v i o u s l y t h e same r e l a t i o n i s v a l i d b e t w e e n t h e components o f t h e h o r i z o n t a l and v e r t i c a l i m p u l s e . When t h e Eqs ( C . 3 ) and ( C . 4 ) a r e i n t e g r a t e d o v e r t i m e , t , ( w i t h c o n s t a n t bow a n g l e , a), t h e r e s u l t w o u l d b e : I = — I ( S i n a + u, Cos a) x n I = I (Cos a - u S i n a) y n ( C 5 ) ( C . 6 ) Where , I y. F d t , I = y F d t and y I = F d t a r e t h e t h e h o r i z o n t a l , v e r t i c a l and n o r m a l i m p u l s e s on t h e bow. I t i s n o t i c e d f r o m Eqs ( C . 5 ) and ( C . 6 ) t h a t : I A I 189 APPENDIX 'D' P e n e t r a t i o n I n t o t h e H u l l P e n e t r a t i o n o f t h e e x t e r n a l o b j e c t i n t h e s h i p h u l l depends upon t h e p o s i t i o n and d i r e c t i o n o f m o t i o n o f t h e s h i p . F o u r d i f f e r e n t c a s e s c o u l d be c o n s i d e r e d as f o l l o w e d : i ) The s h i p i s mov ing t o t h e r i g h t t o w a r d t h e b a r r i e r and t h e e x t e r n a l o b j e c t i s s t i l l i n t h e r i g h t hand s i d e o f s h i p (z >0) F i g . ( D . l a ) shows p e n e t r a t i o n s P and P f o r t h i s c a s e a s : y z A A P = z - z, Z L b and c r o s s - s e c t i o n a r e a o f t h e damaged vo lume i s : A = A - A X 1 2 where A and A a r e c a l c u l a t e d a s : 1 2 A = l y dz = H 1 1 dz = H 2(m+l) B A m+1 A m+1 A A [ ( 2 z ) - (2z ) ] - H (z - z ) L b L b A A And A = y • (z - z ) 2 ' ^ b 1 L b 190 i i ) The s h i p i s m o v i n g t o t h e r i g h t t o w a r d t h e b a r r i e r and e x t e r n a l A o b j e c t i s on i t s l e f t hand s i d e ( z < 0 ) F i g . ( D . l b ) shows p e n e t r a t i o n s P and P f o r t h i s c a s e a s : y z A A * - i y , - y> and y ' y L • ' b 1 The damaged c r o s s - s e c t i o n a r e a i s c a l c u l a t e d a s , Z J + I Zb" A = A - A where A and A a r e : x 1 2 1 2 A = l L „ L A A y dz = ,2z m H [ ( £ - ) - 1 ] dz + 2z H [ ( - - g ) - 1 ] dz -H m+i A m+i A A 2(m+l )B • [ ( - 2 z ) + <-2z ) ]+ H (z + z ) m L b L b And A 2 - | y b | - ( | z b | + | z j ) i i i ) The s h i p i s m o v i n g t o t h e l e f t t o w a r d t h e b a r r i e r and t h e e x t e r n a l A o b j e c t i s on i t s t h e l e f t hand s i d e (z < 0 ) F i g . ( D . 2 a ) s h o w s p e n e t r a t i o n s P and P f o r t h i s c a s e a s : y z P y - | y L - y j and P z - | Z l - z j and t h e damaged c r o s s - s e c t i o n a r e a i s c a l c u l a t e d a s : A = IA I - IA I Where A and A a r e : x 1 l 1 1 2 1 1 2 191 z z r. b „ b A A A = l y dz = 2z m H [ ( ~ ) - 1 ] dz = H A m+i A m+i A A 2(m+l )B t - ( - 2 z ) + ( - 2 z ) ]+ H (z - z ) m L b L b And A 2 = l y b l - ( l V zh\ ) i v ) The s h i p i s m o v i n g t o t h e l e f t t o w a r d t h e b a r r i e r and t h e e x t e r n a l A o b j e c t i s on i t s r i g h t h a n d s i d e ( z > 0 ) F i g . (D .2b ) shows p e n e t r a t i o n s P and P f o r t h i s c a s e a s : y z A A p = i y - y I and P = z + z y , J L J b' Z L b And t h e damaged c r o s s - s e c t i o n a r e a i s c a l c u l a t e d a s : A = IA I - IA I where A and A a r e : x 1 l 1 1 2 ' 1 2 A = l b ,.0 A A y dz = H [ < - g - ) - 1 ] dz + fy i l l A H [ ( - g ) - 1 ] dz -— z — z H m+i A m+i A A 2(m+l )B • [ (2z ) + ( - 2 z ) ] - H ( z + z ) m b L L b A A A A n d A = | y | • ( z + z ) 2 ' - ' b 1 L b 192 y z (a ) (b ) y A . z F i g . D. l - c 7 r o s s - S e c t i o n o f Damaged Area When Ship Is Moving to the Right. a) The Barrier Is on the Right Side of the Ship, a) The Barrier Is on the Left Side of the Ship. 193 y z (b) F i g . T).2-Cross-Section of Damaged Area When Ship Is Moving to the Left. a) The Barrier Is on the Left Side of the Ship. b) The Barrier Is on the Right Side of the Ship. 

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