R E P O R T 4 of 4
UNIVERSITY O F S T R A T H C L Y D E
MARINE SAFETY AGENCY
Marine Safety Agency Spring Place 105 Commercial Road
Southampton S015 l E G Tel: 0703 329100
SUMMARY
INTRODUCTION
AIMS OF THE PROJECT
APPROACH ADOPTED MATHEMATICAL MODELLING PARA>fETRIC INVESTIGATION DISCUSSION CONCLUSIONS ACKNOWLEDGEMENT REFERENCES
SUMMARY
This i n t e r i m r e p o r t d e t a i l s t h e t h e o r e t i c a l r e s u l t s achieved so f a r a t the U n i v e r s i t y o f S t r a t h c l y d e , based on a n a l y s i s o f t h e dynamic behaviour o f a damaged s h i p i n r e a l i s t i c e n v i r o n m e n t a l c o n d i t i o n s . - A t h r e e - d e g r e e s - o f - freedom m a t h e m a t i c a l model was used, a c c o u n t i n g f o r coupled heave, sway and r o l l w h i l e a l l o w i n g f o r water i n g r e s s . M o d e l l i n g o f the water i n g r e s s was based on a s i m p l e f o r m u l a o f water f l o w t h r o u g h an opening when t h e p r e s s u r e head i s kno^.Ti, w i t h the water f l o w c o e f f i c i e n t c a l i b r a t e d f r o m model experiments c a r r i e d out i n Phase I o f the Ro-Ro r e s e a r c h programme.
E x c e l l e n t agreement between e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s p r o v i d e d s u f f i c i e n t j u s t i f i c a t i o n f o r embarking on a s m a l l - s c a l e p a r a m e t r i c i n v e s t i g a t i o n which a l l o w e d l i m i t i n g curves o f s t a b i l i t y t o be d e r i v e d and r e l a t i o n s h i p s t o be e s t a b l i s h e d between s h i p design and e n v i r o n m e n t a l parameters and l i m i t i n g s t a b i l i t y parameters.
There i s a s t r o n g f e e l i n g among r e s e a r c h e r s a t S t r a t h c l y d e t h a t the development o f s u r v i v a l c r i t e r i a f o r damaged passenger s h i p s i s w i t h i n reach.
1. INTRODUCTION
The t r a g i c a c c i d e n t t o t h e HERALD OF FREE ENTERPRISE was t h e s t r o n -g e s t reminder y e t o f t h e v u l n e r a b i l i t y o f Ro-Ro passen-ger f e r r i e s . I n the wake o f t h i s d i s a s t e r t h e Marine D i r e c t o r a t e o f t h e UK Depart-ment o f T r a n s p o r t i n i t i a t e d a c o - o r d i n a t e d r e s e a r c h programme ( t h e Ro-Ro F e r r y S a f e t y Research Programme) aimed a t d e v e l o p i n g more soundly based s u r v i v a l c r i t e r i a f o r damaged passenger s h i p s w i t h a view t o enhancing t h e s a f e t y o f these v e s s e l s .
The programme comprised two phases.
Phase I addressed t h e r e s i d u a l s t a b i l i t y o f e x i s t i n g v e s s e l s and t h e key reasons behind c a p s i z e s . To t h i s end a s e r i e s o f model e x p e r i -ments was c a r r i e d o u t i n v o l v i n g two d i f f e r e n t models: t h e f i r s t , a t r i p l e - s c r e w Ro-Ro f e r r y w i t h s p l i t c a s i n g ( l ) , and t h e second, a t w i n - s c r e w Ro-Ro f e r r y w i t h a c e n t r e c a s i n g ( 2 ) . I n a d d i t i o n , t h e o r e t i c a l s t u d i e s were u n d e r t a k e n i n t o t h e p r a c t i c a l b e n e f i t s and p e n a l t i e s o f i n t r o d u c i n g a number o f d e v i c e s f o r i m p r o v i n g t h e r e s i d u a l s t a b i l i t y o f e x i s t i n g Ro-Ro's.
Phase I I o f t h i s r e s e a r c h programme was s e t up w i t h s e v e r a l o b j e c t i v e s i n mind. These were:
a) To c o n f i r m the f i n d i n g s i n r e s p e c t o f zones o f p r o b a b l e c a p s i z e .
b) To c a r r y o u t damaged model t e s t s , i n which t h e enhancing devices assessed i n Phase I would be modelled t o d e t e r m i n e the improve-ment i n s u r v i v a b i l i t y a c h i e v e d i n r e a l i s t i c sea-going c o n d i t i o n s .
c ) To c o n f i r m t h a t damage i n t h e amidships r e g i o n i s l i k e l y t o lead t o t h e most onerous s i t u a t i o n i n r e s p e c t o f t h e p r o b a b i l i t y o f c a p s i z e .
d) To undertake t h e o r e t i c a l s t u d i e s i n t o t h e n a t u r e o f the c a p s i z e phenomenon, w i t h a view t o e x t r a p o l a t i n g t h e model t e s t r e s u l t s t o Ro-Ro passenger s h i p s o f d i f f e r e n t s i z e s and p r o p o r t i o n s . I d e a l l y , t h i s would i n c l u d e t h e e f f e c t o f f i t t i n g s i d e buoyancy
and by r e s t r i c t i n g t h e f l o o d w a t e r on the v e h i c l e deck by the use o f t r a n s v e r s e and l o n g i t u d i n a l d i v i s i o n s .
S t r a t h c l y d e U n i v e r s i t y ' s Marine Technology Centre (SMTC) was one o f t h r e e o r g a n i s a t i o n s t o undertake a t h e o r e t i c a l i n v e s t i g a t i o n , and the r e s u l t s o f t h i s t o d a t e are presented i n t h i s i n t e r i m r e p o r t .
The r e p o r t begins by s t a t i n g t h e aims o f the i n v e s t i g a t i o n b e f o r e d e s c r i b i n g the approach adopted i n the r e s e a r c h work and t h e m a t h e m a t i c a l model used. Using a t i m e domain s i m u l a t i o n computer program based on t h i s model and s e l e c t i n g t h e t r i p l e - s c r e w f e r r y used by BMT ( l ) f o r model e x p e r i m e n t s , a p a r a m e t r i c s t u d y was t h e n u n d e r t a k e n which addressed a number o f r e a l i s t i c damage s c e n a r i o s and c o n d i t i o n s as w e l l as v a r i a t i o n s i n s e l e c t i v e s h i p d e s i g n and e n v i r o n m e n t a l parameters, t o d e r i v e u s e f u l r e l a t i o n s between these and l i m i t i n g s t a b i l i t y parameters. On the b a s i s of the r e s u l t s o b t a i n e d a framework f o r d e v e l o p i n g r e a l i s t i c s u r v i v a l c r i t e r i a has emerged.
2. AIMS OF THE PROJECT
The main aim of t h e r e s e a r c h work i s t o f o r m u l a t e and propose new s u r v i v a l c r i t e r i a f o r Ro-Ro passenger s h i p s . D e r i v i n g from the r e q u i r e m e n t s set by t h e Marine D i r e c t o r a t e o f the Department o f T r a n s p o r t , the s p e c i f i c aims o f t h i s work are as f o l l o w s :
a) To develop a dynamic t h e o r e t i c a l c a p s i z e model which can p r e d i c t the minimum s t a b i l i t y needed t o p r e v e n t c a p s i z i n g i n a g i v e n sea s t a t e .
b) To v a l i d a t e SMTC's motion s i m u l a t i o n program t h r o u g h comparisons w i t h r e s u l t s f r o m model e x p e r i m e n t s from Phase I o f the Ro-Ro Research Programme, and f r o m Phase I I when r e s u l t s a r c a v a i l a b l e .
c) To use t h i s s o f t w a r e t o e s t a b l i s h l i m i t i n g s t a b i l i t y parameters by computing the v e s s e l b e h a v i o u r i n r e p r e s e n t a t i v e "severe" wave and wind e n v i r o n m e n t s , a l l o w i n g f o r water i n g r e s s .
d ) To e s t a b l i s h m a t h e m a t i c a l r e l a t i o n s h i p s between l i m i t i n g s t a b i -l i t y parameters and s h i p d e s i g n and e n v i r o n m e n t a -l p a r a m e t e r s .
3- APPROACH ADOPTED
To achieve t h e aims o u t l i n e d above i t was d e c i d e d t o address t h e f o l l o w i n g t a s k s :
a ) V a l i d a t i o n o f The M a t h e m a t i c a l Model
The e x i s t i n g m a t h e m a t i c a l model was developed i n t h e time domain -and used t o examine t h e sway, heave and r o l l b e h a v i o u r o f t h e s h i p t o g e t h e r w i t h s i n k a g e , s t a t i c h e e l and t r i m , w h i l e a l l o w i n g f o r p r o -g r e s s i v e f l o o d i n -g t o t a k e p l a c e . Water i n -g r e s s and a c c u m u l a t i o n o f w a t e r were a l s o i n c l u d e d i n t h e model. D>Tiamic e x c i t a t i o n was modelled by t a k i n g i n t o account wave and wind e f f e c t s .
The m a t h e m a t i c a l model which had a l r e a d y been used i n o t h e r r e s e a r c h (3) was v a l i d a t e d f o r t h i s p a r t i c u l a r p r o j e c t t o ensure t h a t t h e s o f t w a r e a t SMTC p r e d i c t s measured b e h a v i o u r o f t h e s h i p a c c u r a t e l y and c o n s i s t e n t l y . Where a p p r o p r i a t e , c o r r e l a t i o n s t u d i e s w i t h e x p e r i m e n t a l r e s u l t s were u n d e r t a k e n . For t h i s purpose use was made o f model experiments w h i c h had been c a r r i e d o u t by BMT ( l ) .
b ) S i m u l a t i o n S t u d i e s
A s y s t e m a t i c p a r a m e t r i c i n v e s t i g a t i o n was u n d e r t a k e n t o e s t a b l i s h r e l a t i o n s h i p s between e n v i r o n m e n t a l and s h i p d e s i g n parameters and s t a b i l i t y parameters. For t h i s purpose s i m u l a t i o n s t u d i e s were c a r r i e d i n two s t a g e s . I n t h e f i r s t stage they were done t o e s t a b -l i s h these r e -l a t i o n s h i p s f o r t h e s h i p t e s t e d by BMT. T h i s was a c h i e v e d by u s i n g a range o f sea s t a t e s , o f damage f r e e b o a r d s and o f l o a d i n g c o n d i t i o n s . A t t h e second stage t h e s t u d i e s addressed t h e e f f e c t o f d i f f e r e n t s h i p p a r a m e t e r s , e s p e c i a l l y beam, on t h e damage s t a b i l i t y o f t h e s h i p . To c a r r y o u t t h i s r e s e a r c h t h e p a r e n t form was m o d i f i e d by u s i n g d i f f e r e n t l e n g t h / b r e a d t h (L / B) r a t i o s t o c r e a t e
a f a m i l y o f f i v e forms which were s u p p l i e d by t h e Department o f T r a n s p o r t . The b e h a v i o u r o f each form f o r a f i x e d f r e e b o a r d was then i n v e s t i g a t e d f o r v a r i o u s sea s t a t e s and l o a d i n g c o n d i t i o n s .
c ) L i m i t i n g Curves o f S t a b i l i t y
Based on the r e s u l t s d e r i v e d from t h e work i n ( a ) and ( b ) , l i m i t i n g curves o f s t a b i l i t y were e s t a b l i s h e d i n v o l v i n g s h i p design and e n v i r o n m e n t a l parameters a g a i n s t l i m i t i n g s t a b i l i t y p a r a m e t e r s , on the b a s i s o f which s u r v i v a l c r i t e r i a c o u l d be d e v e l o p e d .
4- MATHEMATICAL MODELLING
4.1 COORDINATE SYSTEM
Three c o o r d i n a t e systems a r e used. The f i r s t , oxyz, i s used t o d e f i n e the s h i p h u l l f o r m , F i g u r e l a . I t i s l o c a t e d amidships a t t h e i n t e r s e c t i o n betweeen t h e k e e l and t h e c e n t r e l i n e , and i s f i x e d t o the s h i p .
The second system, O X Y Z , i s used t o c a l c u l a t e t h e underwater G c c G
volume o f the s h i p , and i s a l s o l o c a t e d amidships a t t h e i n t e r s e c t i o n between the s t i l l w a t e r p l a n e and the s h i p ' s c e n t r e l i n e , and i s f i x e d to the e a r t h . F i g u r e l b .
The t h i r d c o o r d i n a t e system i s l o c a t e d a t t h e c e n t r e o f g r a v i t y , G, and i s f i x e d t o the e a r t h (OgeXgeYgeZge). This t h i r d system i s used to measure the s h i p m o t i o n s . Since i t i s assumed t h a t t h e s h i p r o t a t e s about t h e c e n t r e o f g r a v i t y , a l l r o t a t i o n a l m o t i o n s , e x c i t a t i o n and r e s t o r i n g moments a r e c a l c u l a t e d assuming t h e c e n t r e o f g r a v i t y t o be f i x e d t o t h e e a r t h . F i g u r e l b . A n t i - c l o c k w i s e r o l l m o t i o n , upward heave i n z d i r e c t i o n and s t a r b o a r d sway i n y. a r e d e f i n e d as p o s i t i v e m o t i o n s .
4.2 DYNAMIC ANALYSIS
I n o r d e r t o d e v e l o p a r e a l i s t i c model t o i n v e s t i g a t e t h e s t a b i l i t y and c a p s i z i n g phenomena o f t h e s h i p i n a dynamic sense, t h e approach adopted and t h e p a r a m e t e r s t a k e n i n t o a c c o u n t i n t h e m a t h e m a t i c a l model a r e e x p l a i n e d below.
a ) Time S i m u l a t i o n Approach
Damage and f l o o d i n g a r e c o n t i n u o u s phenomena which may l e a d t o d i f f e r e n t r e s u l t s depending on such parameters as l o c a t i o n and e x t e n t of damage, e n v i r o n m e n t , e t c . I n v e s t i g a t i n g t h e b e h a v i o u r o f s h i p s i n d i f f e r e n t c o n d i t i o n s would c e r t a i n l y h e l p t o p r o v i d e a b e t t e r u n d e r s t a n d i n g o f t h e c a p s i z i n g phenomenon and t o develop a r e a l i s t i c damage s t a b i l i t y assessment p r o c e d u r e .
I n o r d e r t o take i n t o a c c o u n t n o n - l i n e a r i t i e s , changes i n e x c i t a t i o n f o r c e s , p r o g r e s s i v e f l o o d i n g and responses o f t h e s h i p i n t i m e , i t was necessary t o adopt a t i m e s i m u l a t i o n approach. The t i m e simu-l a t i o n process s t a r t s f r o m t h e b e g i n n i n g o f f simu-l o o d i n g , when 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 a r e known. At each t i m e s t e p d i f -f e r e n t parameters, such as t h e amount o -f w a t e r -f l o w i n g i n , h e e l , d i s p l a c e m e n t , e x c i t a t i o n f o r c e s and responses o f t h e s h i p a r e examined i n d e t a i l . T h i s process c o n t i n u e s u n t i l e i t h e r c a p s i z i n g occurs o r t h e t o t a l t i m e a l l o w e d f o r s i m u l a t i o n i s used up.
b ) M o t i o n s
As i s common knowledge, t h e s t a t i c and dynamic s t a b i l i t y o f a s h i p depend on i t s h e e l i n g o r i t s r o l l i n g m o t i o n . T h i s h e e l o r r o l l angle i s i t s e l f a c r i t e r i o n which i s taken i n t o account by i n t a c t and damage s h i p s t a b i l i t y assessments. However, i n a r e a l environment t h e r e a r e o t h e r m o t i o n s which a l s o a f f e c t t h e s h i p ' s s t a b i l i t y and r o l l m o t i o n d i r e c t l y o r i n d i r e c t l y .
The s t r o n g hydrodynamic c o u p l i n g o f sway and t h e n o n l i n e a r h y d r o -s t a t i c c o u p l i n g o f heave, b o t h -s i g n i f i c a n t l y change t h e underwater volume o f t h e s h i p i n r o l l , and so b o t h m o t i o n s were i n c l u d e d i n t h e time s i m u l a t i o n model. P r o g r e s s i v e t r i m ( w h i c h i s due t o w a t e r f l o o
-d i n g i n ) was a l s o i n c l u -d e -d i n t h e c a l c u l a t i o n s i n c e i t can a f f e c t water i n g r e s s and s h i p m o t i o n s .
The coupled sway, heave and r o l l motions t o g e t h e r w i t h i n s t a n t a n e o u s t r i m can be w r i t t e n as shown below.
S u b s c r i p t s i and j (X.,A.^) i n d i c a t e t h e mode o f m o t i o n . i , j = 2 denotes sway m o t i o n .
i > j = 3 denotes heave m o t i o n . i,j=4 denotes r o l l m o t i o n .
(M22+A22) X2"+ B22 X2* + A23 X 3 - + B23 X3' + A24 X4" + B24 X4' =
P2wind + P2wave
A32 X2" + B32 X2' + (A33 4-M33) X3" 4- B33' X3' + RES3(X4,e,X3,t) +
A34 X4" + B34 X4' = F3wave+F3wod
A42 X2" + B42 X2' + A43 X3" 4- B43 X3' + (A44+I44) X4'' + B44 X4' +
RES4 (X4,9 ,X3, t) = M4 wind+M4 wave+^Awod
Where: M. . i J A. . i j B. . i j RES. 1 F M i w i n d , i w i n d F. M. xwave, iwave F M iwod, iwod : Mass o f Ship : Added mass : Damping C o e f f i c i e n t
: R e s t o r i n g Forces and moment i n c l u d i n g t h e r e s t o r i n g c o u p l i n g
: Mass Moment o f I n e r t i a
: Wind e x c i t a t i o n f o r c e s and moment : Wave e x c i t a t i o n f o r c e s and moment
: E x c i t a t i o n f o r c e s and moment due t o w a t e r on deck : I n s t a n t a n e o u s t r i m .
4-3 HYDRODYNAMIC FORCES
a) Wave E x c i t a t i o n F o r c e s and Moment
Wave e x c i t a t i o n f o r c e s and moment ( F r o u d e - K r y l o v and D i f f r a c t i o n ) a r e c a l c u l a t e d by a t w o - d i m e n s i o n a l d i f f r a c t i o n program w i t h s e c t i o n a l f o r c e s computed by u s i n g t h e Frank C l o s e - F i t Method ( 4 ) , (5)- The s e c t i o n a l f o r c e s a r e t h e n i n t e g r e a t e d a l o n g the s h i p ' s l e n g t h t o f i n d the t o t a l f o r c e s and moment. For r e g u l a r waves, t h e maximum f o r c e s and moment f o r each mode o f m o t i o n a r e t r a n s f o r m e d i n t o t i m e -dependent f o r c e f u n c t i o n s as shown; Fw(0 = Fw cos(cot+8) Mw(t)= Mw cos(©t+s) Where 0) : Wave frequency 8 : Phase angle
Fw : Maximum amplitude of wave excitation force Mw : Maximum amplitude of wave excitation moment
b) I r r e g u l a r Waves
To r e p r e s e n t t h e N o r t h Sea wave e n v i r o n m e n t , a JONSWAP spectrum was used ( 6 ) . I n o r d e r t o c r e a t e a f o r c e spectrum f o r a g i v e n sea s t a t e and f r e q u e n c y range f o r each mode o f m o t i o n , e x c i t a t i o n f o r c e s p e r u n i t wave a m p l i t u d e a r e c a l c u l a t e d u s i n g the same method as was used f o r t h e c a l c u l a t i o n o f t h e e x c i t a t i o n f o r c e s f o r r e g u l a r waves as d e s c r i b e d i n t h e l a s t s e c t i o n . Wave and f o r c e s p e c t r a f o r a g i v e n sea s t a t e and f r e q u e n c y range a r e then t r a n s f o r m e d t o random f o r c e and wave r e a l i s a t i o n s by u s i n g s t a n d a r d F o u r i e r t r a n s f o r m a t i o n s (6).
c ) Hydrodynamic C o e f f i c i e n t s
The m o t i o n - i n d u c e d f o r c e s (added mass and damping t e r m s ) a r e a g a i n c a l c u l a t e d by u s i n g t h e Frank C l o s e - F i t method ( 5 ) , ( 6 ) . T h i s approach does n o t c o n s i d e r v i s c o u s e f f e c t s and i s based on s m a l l a m p l i t u d e s . These l i m i t a t i o n s may a f f e c t s h i p m o t i o n s s i g n i f i c a n t l y , e s p e c i a l l y t h e r o l l m o t i o n . However, n o n - l i n e a r i t i e s i n t h e r o l l
m o t i o n s t i l l remained t o be s o l v e d t h e o r e t i c a l l y , as t h e y a r e s i g n i f i c a n t l y a f f e c t e d by f l u i d v i s c o s i t y . Ikeda's s e m i - e m p i r i c a l r o l l damping c a l c u l a t i o n method was t h e r e f o r e used, as t h i s i n c l u d e s v i s c o u s e f f e c t s t o g e t h e r w i t h l i f t i n g , wave damping and o t h e r c o n t r i b u t i o n s , ( 7 ) .
d ) Mass o f t h e S h i p
The mass o f t h e s h i p changes i n s t a n t a n e o u s l y d u r i n g f l o o d i n g and so does t h e mass moment o f i n e r t i a . Both e f f e c t s a r e c o n s i d e r e d a t each time s t e p .
e) Wind H e e l i n g Moment
A beam wind i s assumed w i t h t h e wind h e e l i n g moment c a l c u l a t e d a c c o r d i n g t o t h e weather c r i t e r i o n ( 8 ) .
4.4 RESTORING FORCES AND MOMENT
The r e s t o r i n g f o r c e s and moment a r e b a s i c a l l y o f a h y d r o s t a t i c n a t u r e as t h e y a r e r e l a t e d t o t h e underwater volume o f t h e s h i p . I n order to take i n t o account t h e n o n - l i n e a r i t y i n t h e r e s t o r i n g f o r c e s and moment which appear as a r e s u l t o f l a r g e a m p l i t u d e m o t i o n s , they a r e c a l c u l a t e d i n s t a n t a n e o u s l y a t each t i m e s t e p . T h i s i s done by c a l c u l a t i n g the u n d e r w a t e r volume o f t h e s h i p up t o t h e f r e e s u r f a c e at each time s t e p and by t a k i n g i n t o account t h e i n s t a n t a n e o u s r o l l and heave m o t i o n s . Free s u r f a c e i s d e f i n e d by t a k i n g i n t o account the calm water l e v e l . T h i s n o n - l i n e a r r e s t o r i n g can be f o r m u l a t e d as shown:
RES-HEAVE(t,X3,e,X4) g [A(t,X3,e,X4) -A(to)]
RES-ROLL(t,X3,e,X4) g A(t,X3,e,X4) [TCGs-TCB(t,X3,e,X4)]
Where: A(t,X3,e,X4) A(to) TCB(t,X3,e,X4) TCGs g : Instantaneous displacement
: Initial displacement at time = tg
: Instantaneous transverse centre of buoyancy : Transverse centre of gravity of the ship : Gravitational acceleration
4.5 MODELLING THE DAMAGE SCENARIOS
a ) Added Weight Method
I n t h i s approach w a t e r i s added t o t h e compartment which was assumed t o be f l o o d e d . Using the i n t a c t h y d r o s t a t i c v a l u e s o f t h e s h i p , s i n k a g e , h e e l and t r i m can be c a l c u l a t e d f o r each time s t e p when e x t r a water i s added. T h i s method c o u l d be used t o model t h e i n t e r mediate stages o f f l o o d i n g as w e l l as the f l o o d i n g o f the c o m p a r t -ments above the w a t e r l i n e .
b) Combination of Time Dependent Added Weight Method and Accumulation of Water
Entrapped water on deck poses s t a b i l i t y problems and c o n t r i b u t e s s u b s t a n t i a l l y t o c a p s i z i n g , e s p e c i a l l y on a l a r g e deck such as t h a t found on Ro-Ro s h i p s . Water on deck c o u l d induce b o t h s t a t i c and dynamic e f f e c t s ( 9 ) , (10).
S l o s h i n g and w a t e r a c c u m u l a t i o n can have dynamic e f f e c t s when the e x c i t a t i o n f r e q u e n c y i s c l o s e t o the n a t u r a l f r e q u e n c y o f t h e w a t e r i n the t a n k . A l t h o u g h t h i s i s p o s s i b l e , i t i s d i f f i c u l t t o meet r e s o n a n t c o n d i t i o n s i n the case o f f e r r i e s s i n c e t h e i r r o l l i n g f r e -quency i s g e n e r a l l y v e r y low. I n o r d e r t o have the same f r e q u e n c y , the water depth i n the t a n k must be v e r y low, i n which case t h e e f f e c t o f s l o s h i n g w i l l n a t u r a l l y be s m a l l . On the o t h e r hand, p r o g r e s s i v e f l o o d i n g would n o t a l l o w resonance t o b u i l d up as the water c o n t i n u o u s l y changes. The s t a t i c e f f e c t r e s u l t s when w a t e r i n a v e r y l a r g e compartment f l o w s t o the c o r n e r o f the deck, c a u s i n g s t a t i c h e e l . As a r e s u l t , the e f f e c t i v e b r e a d t h o f the tank would be v e r y s m a l l and w a t e r d e p t h h i g h , so t h a t the n a t u r a l f r e q u e n c y o f t h e water i n the tank - which depends on depth and b r e a d t h - would be v e r y h i g h . This would d r a s t i c a l l y reduce the p o s s i b i l i t y o f w a t e r s l o s h i n g . A l i m i t e d p a r a m e t r i c s t u d y showed t h a t the dynamic p r e s s u r e due t o s l o s h i n g i s v e r y s m a l l compared w i t h the h y d r o s t a t i c p r e s s u r e . I t i s , however, suggested t h a t more i n v e s t i g a t i o n s s h o u l d be c a r r i e d out t o i d e n t i f y the p r e c i s e e f f e c t .
Because o f the d i f f i c u l t i e s i n m o d e l l i n g s l o s h i n g c o r r e c t l y , and t h e f a c t t h a t i t i s the s t a t i c e f f e c t o f the water which i s d o m i n a n t , t h c
e f f e c t o f the a c c u m u l a t i n g w a t e r was i n c l u d e d i n t h e t i m e s i m u l a t i o n by t a k i n g i n t o account t h e i n s t a n t a n e o u s amount o f w a t e r on deck, r o l l a n g l e and t r i m .
The f o r m u l a t i o n o f t h e e f f e c t o f water on deck i s as f o l l o w s :
I n s t a n t a n e o u s amount o f w a t e r :
I t was assumed t h a t t h e s h i p r o t a t e s around t h e i n i t i a l c e n t r e o f g r a v i t y , and hence t h e i n s t a n t a n e o u s s t a t i c h e e l i n g moment would be:
ADW(t)= ADW(t-At)+AW(At)
I n s t a n t a n e o u s s t a t i c f o r c e t o s i n k the s h i p :
SSF(t)= g ADW(t)
SRM(t,X4,e)=g ADW(t) [(TCG)s-tcg(t,X4,e)]
I n s t a n t a n e o u s t r i m moment: STM(t,X4,G)=g ADW(t) [(LCG)s-lcg(t,X4,e)] Where: ADW(t) icg(t,X4,e) tcg(t,X4,e) SSF(t)
Longitudinal centre of gravity of water on deck Transverse centre of gravity of water on deck Instantaneous static sinkage force due to water Instantaneous amount of water on deck
on deck
SRM(t,X4,e) Instantaneous static heeling moment due to the water on deck
Instantaneous static trimming moment due to the water on deck
STM(t,X4,e)
c ) M o d e l l i n g The Water I n g r e s s
Water i n g r e s s can be modelled i n two d i f f e r e n t ways, each o p t i o n s e r v i n g a d i f f e r e n t purpose.
i ) O p t i o n 1
T h i s o p t i o n i s based on p r e d e f i n e d parameters, which a r e r e l a t e d t o f l o o d i n g and a r e k e p t c o n s t a n t d u r i n g t h e s i m u l a t i o n . Damage l o c a t i o n and f l o o d i n g f o r each damaged compartment i s d e f i n e d s e p a r a t e l y by p r o v i d i n g i n f o r m a t i o n about t h e i n i t i a l amount o f w a t e r i n t h e tank b e f o r e t h e s i m u l a t i o n began, t h e t o t a l amount o f w a t e r assumed t o be f l o o d i n g i n , t h e f l o w r a t e , time s t e p and t o t a l simu-l a t i o n t i m e . The i n s t a n t a n e o u s amount o f water and i t s e n s u i n g e f f e c t a r e then c a l c u l a t e d by u s i n g these parameters. T h i s a r r a n g e -ment a f f o r d s t o t a l c o n t r o l o f t h e mode o f f l o o d i n g and p r o v i d e s t h e o p p o r t u n i t y f o r e x t e n s i v e a n a l y s e s over a wide range o f p a r a m e t e r s .
i i ) O p t i o n 2
T h i s o p t i o n i s based on t h e c a l c u l a t i o n o f t h e r e l a t i v e p o s i t i o n between t h e wave e l e v a t i o n and t h e damage l o c a t i o n . T h i s i n s t a n -taneous r e l a t i v e p o s i t i o n i s c a l c u l a t e d by t a k i n g i n t o account t h e i n s t a n t a n e o u s wave e l e v a t i o n and s h i p m o t i o n s . T h i s o p t i o n p r o v i d e s a more r e a l i s t i c m o d e l l i n g o f p r o g r e s s i v e f l o o d i n g o f v e s s e l compart-ments, e s p e c i a l l y f o r t h e decks above t h e w a t e r l i n e , as water i n g r e s s depends on wave h e i g h t and s h i p motions.
Due t o t h e v a r y i n g n a t u r e o f w a t e r l e v e l a t t h e damage h o l e , t h e r a t e o f water e n t r y has t o be m o d e l l e d t o accommodate b o t h f l o w t h r o u g h an o r i f i c e and f l o w over a n o t c h . The mode o f f l o w i s i n f l u e n c e d l a r g e l y by t h e p r e s s u r e head and t h e area o f t h e damage h o l e . The s t a t i c p r e s s u r e head i s m o d i f i e d by dynamic e f f e c t s , edge e f f e c t , wave p r o f i l e , e t c . The emphasis here, however, i s m a i n l y on t h e h y d r o s t a t i c e f f e c t i n c l u d i n g edge e f f e c t , wave d i r e c t i o n and l o c a t i o n o f damage. F o r m u l a t i o n s f o r d i f f e r e n t damage c o n d i t i o n s a r e shown below.
I n t h e p r e s e n t i n v e s t i g a t i o n . O p t i o n 2 was used, f o r t h e reasons g i v e n above.
Flow o f water t h r o u g h a damage-hole below t h e water l i n e : Fi-^ 2 a
U =
K^|2JÊ
Flow r a t e :
Q = U A
Where:
U = V e l o c i t y o f t h e w a t e r
H = D i s t a n c e between w a t e r . l e v e l and c e n t r e o f damage h o l e ( F i g . 2 a ) K = Flow c o e f f i c i e n t
A = a r e a o f damaged h o l e
Flow o f water t h r o u g h a damage-hole above the w a t e r l i n e F i g 2b
Flow r a t e : Q = U A
U = K ^ / 7 H
The K v a l u e changes depending on t h e shape o f t h e damaged a r e a , l o c a t i o n o f damage, f l o w d i r e c t i o n , e t c . I n the case o f f l o o d i n g o f a s h i p compartment above t h e w a t e r l i n e , t h i s v a l u e changes s i g n i -f i c a n t l y depending on t h e wave d i r e c t i o n . On t h e o t h e r hand, the amount o f water t h a t may f l o w o u t due t o r o l l m o t i o n must be c o n s i d e r e d and t h i s o u t f l o w can be r e p r e s e n t e d by m o d i f y i n g K. D e t a i l s o f the m a t h e m a t i c a l model used can be found i n (3).
5- PARAMETRIC INVESTIGATION
a ) S e l e c t i o n of the Ship
For t h e v a l i d a t i o n o f t h e computer programme and f o r t h e p a r a m e t r i c s t u d y an e x i s t i n g s h i p was chosen, one which had been used by BMT f o r model e x p e r i m e n t s . T h i s s h i p works m a i n l y between t h e UK and t h e C o n t i n e n t and i s s i m i l a r t o t h e HERALD OF FREE ENTERPRISE. Her main p a r t i c u l a r s a r e g i v e n i n Table I , w h i l e the body p l a n , p r o f i l e and g e n e r a l deck arrangement a r e shown i n F i g u r e s 3 and 4.
b) P a r t i c u l a r s of Damage and Flooded Compartments
The damaged compartment was s i t u a t e d around amidships such t h a t t h e r e was s i n k a g e o n l y , w i t h no s i g n i f i c a n t t r i m and, f o l l o w i n g g i v e n spe-c i f i spe-c a t i o n s , t h e opening had a t r a p e z o i d a l f o r m . I t s t a r t e d a t t h e double bottom and extended t o t h e t o p o f t h e model s u p e r s t r u c t u r e ( F i g u r e 4 ) . I t s s i d e s s l o p e a t 15° t o the v e r t i c a l and i t s w i d t h a t the w a t e r l i n e , W^, i s g i v e n by t h e Department o f T r a n s p o r t r e g u l a t i o n s as f o l l o w s :
Wj^ = (0.03LS + 3.0)m o r 11 m e t r e s , whichever i s l e s s .
TABLE 1 - MAIN PARTICULARS OF SHIP
L Q^ ( o v e r a l l l e n g t h ) : 131.9m Lgp ( L e n g t h between p e r p e n d i c u l a r s ) : 126.1m B (moulded b r e a d t h ) : 22.7m 1 ( d e s i g n d r a u g h t ) : 5.7m D^j (Depth t o bulkhead d e c k ) : 7.3m ( d i s p l a c e m e n t : 8S07 tonnes C^ ( B l o c k c o e f f i c i e n t ) : 0.53 p ( P r i s m a t i c C o e f f i c i e n t ) : O.56 C^ ( M i d s h i p C o e f f i c i e n t ) : 0.943 Design KG : 10.10m Number o f Passengers : 1326 Number o f Cars : 350
By u s i n g t h e above d e f i n i t i o n o f damage, t h e p o s s i b i l i t y was taken i n t o account o f t h e f l o o d i n g o f compartments b o t h below and above t h e bulkhead deck.
The same compartment l e n g t h was used as t h a t s e l e c t e d by BMT i n t h e i r t e s t s f o r each damaged f r e e b o a r d ( F ) , Table I I . T h e r e f o r e t h e damaged d r a u g h t changed.
TABLE I I - RANGE OF LENGTHS OF DAMAGE) COMPARTMENT
Freeboard (m) 0.18 0.25 0.5 0.75 1.0 Compartment Length (m) 27.7 25.7 21.1 15.0 7.0
For t h e p a r a m e t r i c i n v e s t i g a t i o n t h e compartment l e n g t h was f i x e d a t 27.7 m e t r e s , which o r i g i n a l l y p r o v i d e d O.I8 metres f r e e b o a r d ( 7 . 1 2 metres d r a u g h t ) . However, i n o r d e r t o i n v e s t i g a t e t h e e f f e c t o f f r e e b o a r d , t h e depth t o b u l k h e a d deck (Dbd) o f t h e s h i p was m o d i f i e d t o o b t a i n t h e r e q u i r e d f r e e b o a r d and these v a l u e s can be found i n Table I I I .
TABLE I I I - RANGE OF DEPTH TO BULKHEAD DECK Freeboard (m) 0.18 0.25 0.5 0.75 1.0
Depth t o Bulkhead Deck (m)
7.3 7.48 7.72 7.97 8.2 c ) Sea S t a t e s
In o r d e r t o c r e a t e randon sea s t a t e s t h e JONSWAP spectrum w h i c h was a l s o used f o r model e x p e r i m e n t s i s c o n s i d e r e d . JONSWAP s p e c t r a w i t h r e p e a t p e r i o d o f t w e n t y m i n u t e s i n f u l l s c a l e a r e used. The peak enhancement f a c t o r i s t a k e n a t 3.3 and t h e sea s t a t e s a r e d e f i n e d a c c o r d i n g t o t h e North-Sea wave s t a t i s t i c p r o v i d e d by BMT ( l ) . The modal p e r i o d s , as w e l l as t h e sea s t a t e s used f o r t h i s s t u d y , a r e g i v e n i n Table I V .
TABLE I V - SEA STATES AND CORRESPONDING MODAL PERIODS
S i g n i f i c a n t Wave H e i g h t Modal P e r i o d H (m) T ( s e c ) O O 0.0 - 0.99 4.5 1.0 - 1,99 5.5 2.0 - 2.99 6.0 3.0 - 3.99 6.25 4.0 - 4.99 6.5 d) Loading Conditions
Since t h e compartment l e n g t h f o r each f r e e b o a r d changes, and hence the KM v a l u e s a l s o change, t h e l o a d i n g c o n d i t i o n i s a d j u s t e d f o r comparison purposes i n o r d e r t o o b t a i n t h e r e q u i r e d GM v a l u e s , as shown i n Table V. I n a l l cases damage GMs were used u n l e s s s p e c i f i c a l l y s t a t e d o t h e r w i s e .
TABLE V - KG AND GM VALUES REQUIRED FOR COMPARISON PURPOSES ( I n i t i a l Draught - 6.12m) KG(m) GM(m) 11.52 0.1 11.12 0.5 10.8 0.93 10.05 1.56 9.3 2.31 8.27 3.35
The damaged draught was f i x e d a t 7.12ra f o r t h e p a r a m e t r i c i n v e s t i -g a t i o n . Hence, t h e KG d i d n o t have t o be chan-ged t o p r o v i d e t h e same GM f o r d i f f e r e n t f r e e b o a r d s (Depth t o bulkhead deck i s m o d i f i e d ) . I n o r d e r t o see t h e e f f e c t o f d i f f e r e n t l o a d i n g c o n d i t i o n s , t h e r e f o r e , the v a l u e s shown i n Table V I a r e used.
TABLE VI - KG AND GM VALUES REQUIRED FOR THE PARAMETRIC INVESTIGATION KG(m) GM(m) GM factor 12.15 0.1 0.0058 11.75 0.5 0.0293 10.5 1.75 0.1026 8.9 3.35 0.196
The GM factor is a non-dimensional parameter, used by BMT (Fig 139, (1)) and is calculated by using the following expression:
GM factor = 10 GM ^ B
5.2 STATICAL STABILITY CALCULATIONS
The s t a t i c a l s t a b i l i t y curves f o r t h e i n t a c t and damaged v e s s e l and f o r d i f f e r e n t l o a d i n g c o n d i t i o n s were c a l c u l a t e d and t h e r e s u l t s were compared w i t h BMT's measured and c a l c u l a t e d s t a t i c a l s t a b i l i t y c u r v e s .
a ) I n t a c t Ship
The s t a t i c a l s t a b i l i t y o f t h e i n t a c t s h i p w i t h 2.45m GM (KG = 9.27m) was c a l c u l a t e d . As shown i n F i g u r e 5, t h e s h i p has v e r y good r e s t o r i n g a b i l i t y which i s h e l p e d c o n s i d e r a b l y by t h e s u p e r s t r u c t u r e w i t h a v a n i s h i n g a n g l e o f over 1 0 0 ° . S t r a t h c l y d e U n i v e r s i t y ' s r e s u l t s a r e i n good agreement w i t h BMT's e x p e r i m e n t a l and c a l c u l a t e d r e s u l t s .
b) Damaged Ship
The s t a t i c a l s t a b i l i t y c u r v e was c a l c u l a t e d o f t h e damaged s h i p w i t h 1.66m i n t a c t GM as g i v e n i n t h e BMT Report ( F i g . 27 ( l ) ) . The damaged compartment l e n g t h was 27.7m, p r o v i d i n g o n l y 0.076m damaged f r e e b o a r d . As shown i n F i g u r e 6, t h e s h i p has good r e s t o r i n g a b i l i t y d e s p i t e t h e v e r y l o n g f l o o d e d compartment, and t h e r e s u l t s f r o m a l l the methods used a r e i n good agreement.
5.3 CALIBRATION OF STRATHCLYDE•S WATER INGRESS MODEL
As i n d i c a t e d i n S e c t i o n 4, t h e i n g r e s s o f water depends on c e r t a i n parameters such as wave h e i g h t , wave d i r e c t i o n , l o c a t i o n o f damage and e x t e n t o f damage. As BMT's e x p e r i m e n t a l r e s u l t s show, t h e wave d i r e c t i o n c o n s i d e r a b l y a f f e c t s t h e w a t e r i n g r e s s , and thus a l s o t h e f l o o d i n g r e s u l t s . S i n c e t h e S t r a t h c l y d e model i s based on c a l c u -l a t i n g t h e i n s t a n t a n e o u s p r e s s u r e head a t the damage o p e n i n g , t h e water i n g r e s s model cannot take i n t o account the e f f e c t o f wave d i r e c t i o n . F u r t h e r m o r e , t h e f l o w o f water o u t o f t h e damaged compartment i s a n o t h e r c o m p l i c a t e d problem which cannot be m o d e l l e d . I n o r d e r t o i n c l u d e these e f f e c t s i n t h e water i n g r e s s model used a t S t r a t h c l y d e , t h e m a t h e m a t i c a l model was c a l i b r a t e d by u s i n g e x p e r i -m e n t a l r e s u l t s . T h i s c a l i b r a t i o n was achieved by -m o d i f y i n g t h e edge e f f e c t c o e f f i c i e n t K, which i s n o r m a l l y taken i n t h e r e g i o n o f 0.55 to 0.58. The c o r r e l a t e d K v a l u e f o r t h e wave i n t o the damage o p e n i n g was found t o be around O.4 t o O.45 ( F i g u r e 7),. a t which range b o t h c a p s i z i n g and n o n - c a p s i z i n g r e s u l t s a r e very much i n accordance w i t h the e x p e r i m e n t a l outcome. The K v a l u e s f o r the wave away f r o m t h e damage opening were found t o bc between 0.2 and 0.25 ( F i g u r e 8 ) . Tiiese c o r r e l a t e d v a l u e s a r c o n l y a p p r o x i m a t i o n s which p r o v i d e
a c c e p t a b l e r e s u l t s i n comparison w i t h t h e e x p e r i m e n t a l r e s u l t s . I n o r d e r t o e s t a b l i s h a c c u r a t e water i n g r e s s and w a t e r o u t f l o w , t h e r e f o r e , f u r t h e r s t u d i e s must be c a r r i e d o u t , i n c l u d i n g an e x p e r i -m e n t a l v e r i f i c a t i o n .
F o l l o w i n g the above, e x c e l l e n t agreement was d e m o n s t r a t e d between t h e t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s over t h e whole range o f t h e parameters used. A t t h i s p o i n t a f u l l p a r a m e t r i c i n v e s t i g a t i o n was u n d e r t a k e n , c l o s e l y f o l l o w i n g BMT's model e x p e r i m e n t s , and t h e r e s u l t s are presented n e x t .
5.4 DYNAMIC DAMAGE STABILITY COMPUTATIONS
a) The Damage S c e n a r i o
B e f o r e t h e s t a r t o f t h e s i m u l a t i o n i t was assumed t h a t t h e damaged compartment below the b u l k h e a d deck i s f l o o d e d up t o the l e v e l o f t h e e x t e r n a l w a t e r l i n e w h i l e t h e r e i s a damage h o l e a t the s i d e and above the bulkhead deck ( F i g u r e 4 ) . As s i m u l a t i o n began, t h e damaged s h i p was assumed t o be moving under t h e e f f e c t o f random waves and t h e i n s t a n t a n e o u s water i n g r e s s was modelled by t a k i n g i n t o account t h e wave h e i g h t and s h i p m o t i o n s which were a l s o e s t i m a t e d a t each time s t e p . Runs were c a r r i e d o u t f o r d i f f e r e n t KG's ( T a b l e 6) and f r e e b o a r d s w h i l e t h e sea s t a t e used i n the c a l c u l a t i o n s was i n c r e a s e d to a l i m i t a t which the s h i p c a p s i z e d s y s t e m a t i c a l l y . Sample r e s u l t s of t h e heave, r o l l , w a t e r i n g r e s s and t r a n s v e r s e c e n t r e o f w a t e r on deck are presented i n F i g u r e s 9 t o 43.
b) A n a l y s i s o f the R e s u l t s
A n a l y s i s was c a r r i e d out w i t h the f o l l o w i n g t h r e e damage f r e e b o a r d s :
i ) Freeboard = 0.18m
Since the f r e e b o a r d i s v e r y s m a l l , no m a t t e r what wave h e i g h t and GM, even t h e s m a l l e s t r o l l m o t i o n causes water i n g r e s s i n t o t h e v e h i c l e deck. At v e r y low GM, r e s u l t s depend s i g n i f i c a n t l y on t h e i n i t i a l
amount o f water on t h i s deck and t h e d i r e c t i o n o f t h e f i r s t r o l l c y c l e , which depends i n t u r n on t h e r e l a t i v e p o s i t i o n o f t h e s h i p t o the wave. As even t h e s m a l l e s t amount o f w a t e r on t h e deck causes a s t a t i c h e e l , t h e s h i p may s u r v i v e i f she h e e l s away from damage, r e s u l t i n g i n an i n c r e a s e d f r e e b o a r d ( F i g u r e 9 ) . I f t h e wave h e i g h t i s n o t s u f f i c i e n t t o e x c i t e t h e s h i p i n t o l a r g e r o l l o r t o cause w a t e r i n g r e s s , t h e s h i p w i l l be s a f e . However, i f t h e s h i p i n c l i n e s
towards the damaged s i d e she l o s e s her s m a l l damaged f r e e b o a r d i m m e d i a t e l y , and t h e v e h i c l e deck s t a r t s f l o o d i n g . I n t h i s case cap-s i z i n g becomecap-s i n e v i t a b l e . When t h e wave h e i g h t i cap-s cap-s m a l l t h e w a t e r on deck seems t o be a d o m i n a t i n g f a c t o r i n t h e s h i p ' s b e h a v i o u r .
As F i g u r e s l l a and l i b show, t h e time h i s t o r i e s o f r o l l and w a t e r i n g r e s s have i d e n t i c a l t r e n d s which i n i t i a l l y v a r y l i n e a r l y w i t h t i m e , b u t as t h e w a t e r on t h e deck i n c r e a s e s , t h e i n c l i n a t i o n a l l o w s f o r a l a r g e r area o f t h e damaged opening t o immerse and t h e w a t e r f l o w r a t e i n c r e a s e s e x p o n e n t i a l l y , c a u s i n g i n c l i n a t i o n t o i n c r e a s e a t the same r a t e . I n e v i t a b l y t h e s h i p c a p s i z e s w i t h i n a v e r y s h o r t t i m e . T h i s mode o f c a p s i z e appears t o be r e a s o n a b l e f o r o t h e r con-d i t i o n s as w e l l . F o r 0.1m GM, i t i s con-d i f f i c u l t t o i con-d e n t i f y a c r i t i c a l Hs/F r a t i o and hence t o d e r i v e a l i m i t i n g curve o f s t a b i l i t y . I n f a c t , t h e s h i p c o u l d c a p s i z e i n any sea s t a t e as she has a l m o s t zero r e s t o r i n g a b i l i t y .
Even i f t h e GM were i n c r e a s e d f u r t h e r t o O.Sm a s i m i l a r problem would s t i l l e x i s t , b u t t o a l e s s e r degree, which a l l o w s f o r a c l e a r e r p i c -t u r e -t o emerge w i -t h r e g a r d -t o l i m i -t i n g curves o f s -t a b i l i -t y ( c a p s i z e ) .
When GM i s i n c r e a s e d t o l.JSm f o r t h e damaged c o n d i t i o n , t h e p e r -formance o f t h e s h i p i s improved c o n s i d e r a b l y and the wave h e i g h t then becomes an i m p o r t a n t parameter i n water i n g r e s s and s h i p m o t i o n s , and thus on c a p s i z i n g . As t h e s h i p does n o t i n c l i n e w i t h a s m a l l amount o f w a t e r on deck a s m a l l wave h e i g h t does n o t cause any w a t e r i n g r e s s or even r o l l m o t i o n . Even heave m o t i o n i s t o o s m a l l t o cause any water i n g r e s s . As a r e s u l t , t h e c r i t i c a l Hs/F r a t i o i n c r e a s e s c o n s i d e r a b l y , i n d i c a t i n g t h a t t h c s h i p w i l l be s a f e even i n the case o f t h e h i g h e r waves.
The t r e n d c o n t i n u e s w i t h i n c r e a s i n g GM as can be seen from F i g u r e 30 f o r a GM o f 3.35m. T h i s v e r y h i g h GM, which i s d i f f i c u l t t o a c h i e v e i n f e r r i e s , i n c r e a s e s t h e s a f e Hs/F r a t i o up t o 2.5, which g i v e s a maximum wave h e i g h t a t which t h e s h i p i s s a f e a g a i n s t c a p s i z i n g a t around 0.5ra. For b o t h GMs, 1.75m and 3.35m, c a p s i z i n g and non-c a p s i z i n g boundaries non-can be d e f i n e d v e r y e a s i l y .
I f t h e r e s u l t s o f a l l t h e runs a r e p l o t t e d , t h e e f f e c t s o f l o a d i n g c o n d i t i o n , GM, wave h e i g h t and f r e e b o a r d can be seen v e r y c l e a r l y ( F i g u r e 1 6 ) . O b v i o u s l y , as GM i n c r e a s e s t h e chance o f s u r v i v i n g damage i n c r e a s e s c o n s i d e r a b l y , b u t t h i s may l e a d t o an i m p r a c t i c a l l o a d i n g c o n d i t i o n w h i c h cannot be achieved v e r y e a s i l y i n t h e case o f f e r r i e s . On t h e o t h e r hand, t h e p o s i t i v e e f f e c t o f GM can be d r a s -t i c a l l y reduced i f -t h e damaged f r e e b o a r d i s s m a l l , as i -t can cause t h e v e h i c l e deck t o f l o o d e a s i l y even a t modest wave h e i g h t s . These r e s u l t s suggest t h a t e x i s t i n g f r e e b o a r d l i m i t s must be i n c r e a s e d s i g n i f i c a n t l y t o p r e v e n t w a t e r i n g r e s s , and hence c a p s i z i n g , i n even modest sea c o n d i t i o n s .
i i ) Freeboard = 0.75m
At t h i s f r e e b o a r d t h e r e i s , i n g e n e r a l , l i t t l e danger o f c a p s i z i n g even a t v e r y low-GMs, as l o n g as t h e Hs/F r a t i o i s l e s s than 1.0. A g a i n , when Hs/F i s around 1 and GM i s v e r y low, t h e d i r e c t i o n o f t h e i n i t i a l r o l l c y c l e and t h e amount o f w a t e r on t h e deck determine whether t h e s h i p c a p s i z e s o r n o t . When GM i s i n c r e a s e d t o moderate v a l u e s t h e Hs/F r a t i o i m p r o v e s . At GM = 0.94m, t h e s h i p c a p s i z e s when Hs/F i s c l o s e t o 2. The s h i p ' s b e h a v i o u r i s s t i l l dominated by the water i n g r e s s as t h e s h i p r o l l s w i t h o n l y s m a l l a m p l i t u d e s ( F i g u r e s 17a, 17b) w i t h h e r i n c l i n a t i o n f o l l o w i n g a t r e n d almost i d e n t i c a l w i t h t h a t o f t h e w a t e r i n g r e s s .
When GM i s i n c r e a s e d t o 1.785m t h e Hs/F boundary v a l u e i n c r e a s e s t o around 2.5. which means a 2.0m s i g n i f i c a n t wave h e i g h t . At t h i s sea s t a t e t h e s h i p r o l l s t o o n l y 5° t o 6 ° , and does n o t i n c l i n e t o one s i d e unless the amount o f water on deck i n c r e a s e s c o n s i d e r a b l y . However, when the w a t e r reaches an amount t h a t exceeds t h e wave e x c i t a t i o n f o r c e , t h e s h i p e i t h e r c a p s i z e s i f t h e i n c l i n a t i o n i s
towards the damaged s i d e o r s u r v i v e s i f t h e i n c l i n a t i o n i s away from the damaged s i d e . There i s , t h e r e f o r e , a s m a l l Hs/F range w i t h i n which the s h i p may e i t h e r s u r v i v e o r c a p s i z e .
However, d e s p i t e a c o n s i d e r a b l e i n c r e a s e i n GM (2.25m), t h e boundary v a l u e which s e p a r a t e s t h e c a p s i z e and non-capsize zones i n c r e a s e s o n l y s l i g h t l y . I n t h i s c o n d i t i o n , a l t h o u g h t h e amount o f w a t e r t h a t can cause i n c l i n a t i o n i n c r e a s e s , t h e s h i p r o l l s w i t h l a r g e r a m p l i -tudes. More s u r p r i s i n g l y , when GM i s i n c r e a s e d t o 3.35m, t h e boundary Hs/F v a l u e decreases t o around 2. The main reason f o r t h i s b e h a v i o u r i s p r o b a b l y t h a t , due t o t h e v e r y h i g h GM t h e s h i p ' s r o l l e x c i t a t i o n moment f o r t h e same wave h e i g h t i n c r e a s e s ( F i g u r e 1 9 ) , and t h i s causes an i n c r e a s e i n t h e maximum s h i p r o l l a m p l i t u d e (around 10'^). A l t h o u g h t h i s i n c r e a s e i n r o l l motion i s n o t i m p o r t a n t i n terms o f t h e s h i p ' s dynamics, i t i n c r e a s e s t h e w a t e r i n g r e s s by a l l o w i n g a l a r g e r damaged area t o immerse below t h e w a t e r s u r f a c e .
As shown i n F i g u r e 22, t h e boundary r a t i o o f Hs/F changes depending on GM, w i t h Hs/F l y i n g between 1 and 2.5, c o r r e s p o n d i n g t o 0.75m and 1.75m s i g n i f i c a n t wave h e i g h t s , r e s p e c t i v e l y . T h i s proves t h a t as w e l l as damage f r e e b o a r d , adequate GM i s v e r y i m p o r t a n t i n o r d e r t o p r e v e n t c a p s i z i n g .
i i i ) Freeboard = 1.Om
R e s u l t s f o r f r e e b o a r d o f l.Om a r e s i m i l a r t o those f o r 0.75m. At low GMs t h e r e i s no c l e a r boundary v a l u e o f Hs/F, i m p l y i n g t h a t t h e s h i p i s prone t o c a p s i z i n g i n any sea c o n d i t i o n ( F i g u r e s 23 t o 2 5 ) . As H S / F i n c r e a s e s f u r t h e r the s h i p appears t o be c a p s i z i n g systema-t i c a l l y . When GM i s i n c r e a s e d systema-t o 1.75m systema-the boundary v a l u e o f Hs/F reaches around 2.0, a t which p o i n t t h e s h i p again r o l l s w i t h s m a l l a m p l i t u d e s b u t an average o f 0.75m heave a l l o w s more water t o f l o w i n , ( F i g u r e 2 7 a ) .
Again a t 3-35m GM, t h e boundary v a l u e o f Hs/F reduces t o around 1.75 f o r the same reason as i s i n d i c a t e d i n ( i i ) ( F i s u r o 2 8 ) . T h i s de-crease can be observed i n F i g u r e 20. which shows t h c r e s u l t s f o r l.Om f r e e b o a r d .
i v ) E f f e c t o f Freeboard on t h e L i m i t i n g Curves o f S t a b i i t y
As shown i n F i g u r e 30, t h e f r e e b o a r d a f f e c t s t h e boundary curves between c a p s i z e and n o n - c a p s i z e zones. I f the damaged f r e e b o a r d i s s m a l l , the p o s s i b i l i t y o f w a t e r i n g r e s s i s v e r y h i g h even i f the s h i p r o l l s or i n c l i n e s t o v e r y s m a l l a n g l e s . T h i s o b v i o u s l y reduces t h e boundary o f the n o n - c a p s i z e zone c o n s i d e r a b l y . E s p e c i a l l y a t low GMs t h e s h i p can f l o o d above the bulkhead deck i n almost calm water. However, when GM i s i n c r e a s e d , the chances o f s u r v i v i n g i n c r e a s e as t h e boundary curve o f t h e non-capsize zone i n c r e a s e s l i n e a r l y w i t h GM. U n f o r t u n a t e l y i n r e a l terms, even a t u n r e a l i s t i c a l l y h i g h GMs, t h e maximum s i g n i f i c a n t wave h e i g h t t h a t the s h i p can s u r v i v e i s as l i t t l e as 0.45ni.
T h i s dangerous c o n d i t i o n improves o n l y when t h e f r e e b o a r d i s i n c r e a s e d c o n s i d e r a b l y . For 0.75m and lm f r e e b o a r d s , t h e r e s u l t s are c l o s e t o each o t h e r i n terms o f Hs/F r a t i o s . A l t h o u g h the lm f r e e -board has a s l i g h t l y l o w e r Hs/F boundary curve compared w i t h 0.75m, t h e lm f r e e b o a r d case can s u r v i v e a g a i n s t h i g h e r waves than the c o r r e s p o n d i n g 0.75ni f r e b o a r d . S u r p r i s i n g l y , the boundary curves f o r b o t h 0.75ni and l.Om s t a r t t o decrease a t v e r y h i g h GM v a l u e s . This change c o u l d be due t o e i t h e r l a r g e r motion a t v e r y h i g h GM values or i n a c c u r a t e m o d e l l i n g o f t h e water f l o w i n and o u t a t v e r y h i g h GMs.
I n g e n e r a l i t can be s a i d t h a t a t h i g h damaged f r e e b o a r d s (0.75-lm) and a t average GMs, the s h i p w i l l s u r v i v e i f the Hs/F r a t i o i s l e s s than 1.5-2.0. However, as the damaged f r e e b o a r d decreases, the damaged s h i p ' s chances o f s u r v i v i n g decrease even i n v e r y moderate c o n d i t i o n s as f l o o d i n g o f t h e v e h i c l e deck depends s u b s t a n t i a l l y on the damaged f r e e b o a r d .
v ) P r e s e n t a t i o n o f O v e r a l l R e s u l t s and Comparison w i t h Experiments I f t h e r e s u l t s f o r a l l t h e f r e e b o a r d s c o n s i d e r e d i n t h e p a r a m e t r i c s t u d y are p l o t t e d i n one graph, the average boundary curve between n o n - c a p s i z e and c a p s i z e zones can be d e r i v e d f o r a].] c o n d i t i o n s , as shown i n F i g u r e 31. Three main zones can be i d e n t i f i e d . F i r s t l y , a non-capsize zone, i n which the s h i p s u r v i v e s s y s t e m a t i c a l l y - and t h i s can he seen c l e a r l y a t moderate and hi^h CM v a l u e s : secondlv a
c a p s i z e zone, i n w h i c h t h e s h i p c a p s i z e s s y s t e m a t i c a l l y , and t h i s can be seen i n the whole range o f GM; t h i r d l y , a c r i t i c a l zone i n w h i c h the s h i p can e i t h e r c a p s i z e o r s u r v i v e , depending on s p e c i f i c i n s t a n t a n e o u s c o n d i t i o n s . There i s , t h e r e f o r e , no d e f i n i t e boundary l i n e between c a p s i z e and n o n - c a p s i z e zones.
Since t h e r e i s no d e f i n i t e boundary c u r v e , an average boundary c u r v e w i t h i n t h e c r i t i c a l zone can be drawn and t h i s r e s u l t i s shown i n F i g u r e 3 2 . When t h e c u r v e d e r i v e d from t h e s i m u l a t i o n r e s u l t s i s compared w i t h t h e average boundary c u r v e d e r i v e d from BMT's model e x p e r i m e n t s , an e x c e p t i o n a l l y good agreement can be seen up t o 0.125 of the n o n - d i m e n s i o n a l p a r a m e t e r w h i c h c o r r e s p o n d s t o 2m GM. Beyond t h i s p o i n t , w h i l e e x p e r i m e n t a l r e s u l t s tend t o i n c r e a s e e x p o n e n t i a l l y ( t h e r e i s no d a t a a v a i l a b l e beyond 2.41m GM), t h e s i m u l a t i o n r e s u l t s i n c r e a s e l i n e a r l y up t o O.136 (GM = 2.41m), then decrease s l i g h t l y when t h e GM f a c t o r i s i n c r e a s e d f u r t h e r t o O.I96 (GM = 3•35m). I t m i g h t be u s e f u l i f e x p e r i m e n t s were c a r r i e d o u t f o r h i g h e r GMs t o v e r i f y t h e t h e o r e t i c a l r e s u l t s , o r o t h e r w i s e .
T h i s change o f t r e n d w i t h t h e t i m e s i m u l a t i o n curve d e r i v e s f r o m t h e l a r g e r a m p l i t u d e s w h i c h cause more damaged area t o immerse, t h u s , more water t o f l o w i n . A l t h o u g h t h e s h i p has a v e r y good r e s t o r i n g c a p a b i l i t y , a few degrees o f i n c r e a s e i n r o l l motion i n c r e a s e s c o n s i -d e r a b l y t h e amount o f w a t e r f l o w i n g i n , thus r e -d u c i n g t h e Hs/F r a t i o a t which t h e s h i p c a p s i z e s a t h i g h GMs.
The d i f f e r e n c e between e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s a t v e r y h i g h GMs c o u l d a l s o be due t o the f a c t t h a t t h e s i m u l a t i o n r e s u l t s used an edge e f f e c t c o e f f i c i e n t which was d e t e r m i n e d f r o m t h e e x p e r i m e n t a l r e s u l t s , w i t h t h c l a t t e r c o v e r i n g o n l y m o d e r a t e l y h i g h GMs. I t i s t h e r e f o r e recommended t h a t f u r t h e r s t u d i e s on water i n g r e s s - w i t h e x p e r i m e n t a l v e r i f i c a t i o n a r e e s s e n t i a l and s h o u l d bc undertaken as soon as p o s s i b l e .
EFFECT OF THE AMOUNT OF WATER ON DECK
O b v i o u s l y a s h i p would c a p s i z e o r s i n k o n l y i f a s u f f i c i e n t amount o f water f l o o d e d i n , o r as a d i r e c t r e s u l t o f wave e x c i t a t i o n . As shown by the r e s u l t s o b t a i n e d , t h e djmamic e f f e c t o f waves on the r o l l i n g m o t i o n o f the s h i p i s s m a l l i n a random wave e n v i r o n m e n t . The o n l y dominant e f f e c t r e m a i n i n g , t h e r e f o r e , has t o be t h e amount o f water on the v e h i c l e deck. The p a r a m e t r i c s t u d y i n d i c a t 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 s h i p between the amount o f w a t e r on t h e deck which can c a p s i z e the s h i p and t h e r e s i d u a l GM ( F i g u r e 3 3 ) . The approximate amount o f water which can c a p s i z e the s h i p changes f r o m 10^ t o 3 0 ^ o f the d i s p l a c e m e n t depending on the GM v a l u e . The c u r v e i n F i g u r e 33, which i s based on t h e average values o f d i f f e r e n t r u n s , i s f o r t h e s p e c i f i c damage s c e n a r i o used and t h i s curve may change i f a n o t h e r damage s c e n a r i o i s c o n s i d e r e d .
THE EFFECT OF HULL FORMS ON THE CAPSIZING OF A DAMAGED SHIP
I n o r d e r t o 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 form and 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 between l i m i t i n g s t a b i l i t y parameters and s h i p d e s i g n parameters, a s e r i e s o f p a r a m e t r i c i n v e s t i g a t i o n s was c a r r i e d o u t . For t h i s purpose the p a r e n t h u l l form was m o d i f i e d by u s i n g d i f f e r e n t L / B r a t i o s which were chosen by SMTC and the Department o f T r a n s p o r t . These r a t i o s are o b t a i n e d by changing the b r e a d t h o f the s h i p . New h u l l forms a r e c r e a t e d by keeping t h e d i s p l a c e m e n t , l e n g t h and d r a u g h t c o n s t a n t even though d u r i n g the g e n e r a t i o n o f the new forms the draught had t o be a d j u s t e d s l i g h t l y t o o b t a i n t h e same d i s p l a c e m e n t . The changes i n the s h i p parameters and body forms a r e g i v e n i n Table V I I and a r e shown i n F i g u r e s 34 t o 37.
TABLE V I I - FAMILY OF SHIPS USED FOR THE PARAMETRIC INVESTIGATION Parameter Ship 1 L/B 6.5 B(m.) 19.4 Cb 0.625 T 5.62 Ship 2 Main 6.0 5.5 21. 22.7 0.581 0.525 5.58 5.62 S h i p 3 S h i p 4 5.0 4.5 25.22 28.02 0.481 0.441 5.62 5.52
U s i n g the same damage s c e n a r i o as a l r e a d y d e s c r i b e d , a p a r a m e t r i c i n v e s t i g a t i o n was c a r r i e d o u t f o r GMs e q u a l t o 0.1m, 0.5m, 1.75m and 3.3511 w i t h the damaged f r e e b o a r d f i x e d a t 0.75m.
For h i g h L / B r a t i o s , 6.0 and 6.5, t h e m o t i o n s and c a p s i z i n g b e h a v i o u r a r e s i m i l a r t o those o b t a i n e d f o r t h e p a r e n t f o r m , e x c e p t a t GM o f 3.35m. At t h i s v a l u e , when t h e Hs/F r a t i o i s below 1.8, t h e s h i p s
s u r v i v e a t a l l t i m e s , b u t as the Hs/F i s i n c r e a s e d beyond 1.8, s h i p s are s y s t e m a t i c a l l y l o s t ( F i g u r e s 38 and 3 9 ) . T h i s , however, i s n o t due t o c a p s i z i n g b u t t o s i n k i n g . A s h i p w i t h h i g h L / B i s v e r y s t i f f b u t o f f u l l underwater f o r m (Cb = O.58I and 0.625, r e s p e c t i v e l y ) . I n i t i a l l y t h e amount o f water on deck i s n o t q u i t e s u f f i c i e n t t o c r e a t e a s t a t i c h e e l as t h e f u l l e r u n d e r w a t e r f o r m r e s i s t s t h e h e e l i n g moment c r e a t e d by the w a t e r . I n a d d i t i o n , as a r e s u l t o f r o l l m o t i o n the water accumulates on b o t h s i d e s o f the c e n t r e l i n e . As the amount of w a t e r i n c r e a s e s the t r a n s v e r s e c e n t r e o f water on the deck approaches t h e c e n t r e l i n e and because o f the s u b s t a n t i a l amount o f water on the deck the s h i p e v e n t u a l l y s i n k s r a t h e r than c a p s i z e s .
At low L / B r a t i o s ( 4 . 5 and 5 - 0 ) , s h i p s can appear t o c a p s i z e v e r y e a s i l y even a t h i g h GMs, but e q u a l l y t h e chance o f s u r v i v a l i s h i g h as shown i n F i g u r e s 40 and 41. T h i s c o n t r a d i c t o r y s t a t e m e n t i m p l i e s t h a t the c r i t i c a l boundary zone i s v e r y wide a t h i g h GM v a l u e s . The main reason i s t h a t w a t e r on deck can v e r y e a s i l y cause a s t a t i c h e e l as the s h i p s have a w i d e r beam and v e r y s l e n d e r u n d e r w a t e r f o r m (Cb = 0.441 and 0.481, r e s p e c t i v e l y ) . These s h i p s , t h e r e f o r e , have t o i n
r e s i s t the h e e l i n g moment o f the water on deck. I f the s h i p i n c l i n e s towards the damaged s i d e c a p s i z e i s i n e v i t a b l e as the damaged f r e e b o a r d w i l l be reduced, b u t i f the s h i p i n c l i n e s away from the damage she w i l l s u r v i v e as the damaged f r e e b o a r d i n c r e a s e s .
From the boundary c u r v e s o f the non-capsize zones i t can be seen t h a t a l l boundary curves f o r a l l L / B r a t i o s have s i m i l a r t r e n d s and Hs/F v a l u e s f o r the same GM values ( F i g u r e 4 2 ) . However, a l t h o u g h the average boundary curves are s i m i l a r , a t h i g h GM v a l u e s , s h i p s w i t h l o w e r L / B r a t i o s and slimmer underwater forms have a w i d e r c r i t i c a l boundary zone which i n c r e a s e s the chance o f c a p s i z i n g a t l o w e r Hs/F r a t i o s ( F i g u r e s 40 and 4 1 ) . H u l l forms w i t h h i g h L / B r a t i o s and hence narrower c r i t i c a l zones a t h i g h GMs appear, t h e r e f o r e , t o be s l i g h t l y s a f e r .
On the o t h e r hand, i f the boundary curves f o r each L / B r a t i o a r e r e -drawn a g a i n s t l o a d i n g c o n d i t i o n (KG), i t can be seen t h a t as L / B de-creases the s h i p ' s f l e x i b i l i t y o f l o a d i n g improves c o n s i d e r a b l y w i t h the maximum a l l o w a b l e KG i n c r e a s i n g t o as h i g h as l6m ( F i g u r e 4 3 ) . I t has t o be n o t e d , however, t h a t the r e s u l t i n g l o a d i n g c o n d i t i o n s are unacceptable. For i n s t a n c e , the K G / T o f the s h i p w i t h L / B = 6 . 5 equals 1.0 when GM i s 3.35m w h i l e f o r the maximum a l l o w a b l e KG, K G / T equals 1.6. These v a l u e s are p r a c t i c a l l y i m p o s s i b l e f o r f e r r i e s which have a v e r y h i g h s u p e r s t r u c t u r e and c a r r y a l l cargo on upper decks. These r e s u l t s suggest t h a t the i d e a l s h i p a t t h i s d i s -placement must have a s l i g h t l y l o w e r L / B b u t a f u l l e r h u l l f o r m .
I t i s , o f course, p o s s i b l e t o have v e r y h i g h L / B r a t i o s - as f o r example F2 ( 6 . 4 ) , b u t t h i s v e s s e l a l s o has a h i g h B / T r a t i o which improves both s t a b i l i t y and l o a d i n g f l e x i b i l i t y .
As expected, s h i p d e s i g n parameters and s t a b i l i t y a r e c l o s e l y r e l a t e d to each o t h e r . A l l the main s h i p parameters, t h e r e f o r e , such as L / B . B/T, KG/T and Cb must be c o n s i d e r e d i n p u r s u i n g an optimum d e s i g n which has improved s a f e t y .
DISCUSSION
The r e s u l t s o f t h i s s t u d y show t h a t when t h e r e i s damage above t h e bulkhead deck the s u r v i v a b i l i t y o f a damaged s h i p depends m a i n l y on the f o l l o w i n g p a r a m e t e r s : damaged f r e e b o a r d , wave h e i g h t and m e t a c e n t r i c h e i g h t (GM o r l o a d i n g c o n d i t i o n ) .
Waves have two main e f f e c t s on a damaged s h i p : they i n d u c e motions and they exacerbate f l o o d i n g . Random waves, however, do n o t cause a s i g n i f i c a n t r o l l m o t i o n - w i t h t h e s h i p r o l l i n g r a r e l y t o t e n degrees. Resonant c o n d i t i o n s a r e a l s o u n l i k e l y t o b u i l d up due t o the randomness o f t h e sea s t a t e . The heave m o t i o n appears t o be s l i g h t l y more s e r i o u s when (2"-Ha/Hs) i s a p p r o x i m a t e l y 1, b u t a g a i n t h i s i s n o t a cause f o r c o n c e r n (Ha i s t h e average heave a m p l i t u d e ) . The most i m p o r t a n t e f f e c t o f waves i s t h e f l o o d i n g o f the v e h i c l e deck. The r a t e o f w a t e r i n g r e s s i n t o t h e v e h i c l e deck depends on how much o f the damage o p e n i n g i s immersed i n t h e water. As the wave h e i g h t i n c r e a s e s , t h e r e f o r e , t h e p o s s i b i l i t y o f immersion o f t h e damaged area becomes g r e a t e r . A l t h o u g h a s m a l l r o l l m o t i o n does n o t cause any dj-namical p r o b l e m , even t h e s m a l l e s t r o l l angle towards t h e damaged s i d e causes more w a t e r t o f l o w i n t o t h e v e h i c l e deck as i t reduces the damaged f r e e b o a r d . The e f f e c t o f heave i n r e d u c i n g t h e damaged f r e e b o a r d i s as i m p o r t a n t as the r o l l a n g l e .
F o l l o w i n g an e x a m i n a t i o n o f t h e e f f e c t s o f wave h e i g h t and r o l l and heave motions on w a t e r i n g r e s s , i t become^ c l e a r t h a t the damaged f r e e b o a r d i s the key p a r a m e t e r f o r p r e v e n t i n g water f l o o d i n g t h e v e h i c l e deck. As F i g u r e 31 shows, on average, i f the Hs/F r a t i o i s l e s s than 1, f l o o d i n g o f t h e v e h i c l e deck i s u n l i k e l y u n l e s s b o t h f r e e b o a r d and GM a r e v e r y s m a l l . The 0.076m damaged f r e e b o a r d r e q u i r e d by the r u l e s i s t h e r e f o r e u n r e a l i s t i c and u n a c c e p t a b l e , as the v e h i c l e deck w i l l be f l o o d e d i f the v e s s e l i n c l i n e s towards t h e damaged s i d e o f the s h i p by as l i t t l e as h a l f a degree. I n p r a c t i c e , sevei-al degrees of i n c l i n a t i o n a r c q u i t e l i k e l y as t h e r e may be a s h i f t o f cargo due t o c o l l i s i o n o r d u r i n g passenger e v a c u a t i o n . T h i s v e r y smal] f r e e b o a r d a l s o c r e a t e s a c o n t r a d i c t i o n between the STAB
amendment, the area under the r e s t o r i n g l e v e r curve should be a t l e a s t 0.015m rad measured f r o m the a n g l e o f e q u i l i b r i u m t o the l e s s e r o f :
- The angle a t which p r o g r e s s i v e f l o o d i n g occurs - 22° from the u p r i g h t f o r one-compartment f l o o d i n g
- 27° from the u p r i g h t f o r f l o o d i n g o f two or more compartments
and the r e s i d u a l r i g h t i n g l e v e r , (GZ) w i t h i n a 15° range from e q u i l i b r i u m must be n o t l e s s than 0.1m.
However, i n the case o f an 0.076m f r e e b o a r d , p r o g r e s s i v e f l o o d i n g w i l l occur i f the s h i p i n c l i n e s t o one or two degrees. I n t h i s c o n d i t i o n the s h i p must have around lm GM t o comply w i t h the minimum GZ v a l u e r e q u i r e m e n t . S u r p r i s i n g l y , t h i s b r i n g a n o t h e r c o n t r a d i c t i o n to l i g h t as the minimum GM r e q u i r e d by the r u l e s i s as l i t t l e as 0.05m. These c o n t r a d i c t i o n s show t h a t r u l e s d e r i v e d from s t a t i c a l s t a b i l i t y are f a r f r o m s a t i s f y i n g the minimum d>-namic s t a b i l i t y r e q u i r e m e n t s as the consequences d e r i v i n g f r o m the v e s s e l ' s dynamic b e h a v i o u r are i g n o r e d c o m p l e t e l y . The h i g h l i g h t e d c o n t r a d i c t i o n s suggest t h a t a l l r e g u l a t i o n s , i n c l u d i n g the STAB '90 amendments, must be re-examined and a t l e a s t rendered c o n s i s t e n t .
At low GMs, even a s m a l l amount o f w a t e r on deck can cause s t a t i c h e e l . The s h i p must t h e r e f o r e have s u f f i c i e n t f r e e b o a r d t o reduce the chances o f f l o o d i n g . As GM i n c r e a s e s , the chances i n c r e a s e o f the s h i p s u r v i v i n g a t g r e a t e r wave h e i g h t s ( F i g u r e 3 1 ) . However, f o r low f r e e b o a r d (say 0.1m) the a c t u a l maximum wave h e i g h t t h a t can be s u r v i v e d by a s h i p w i t h a l a r g e GM, may be as l i t t l e as 0.3m. I n o r d e r t o e s t a b l i s h a minimum r e q u i r e d f r e e b o a r d , t h e r e f o r e , the w o r s t s e r v i c e a b l e sea c o n d i t i o n s must be used, t o a l l o w a s h i p t o have a reasonable chance o f s u r v i v i n g i n moderate sea c o n d i t i o n s .
As can be seen, GM i s a n o t h e r key parameter f o r d e t e r m i n i n g the sea-s t a t e sea-s a t which a sea-s h i p can sea-s u r v i v e . Asea-s GM determinesea-s the r e sea-s t o r i n g a b i l i t y o f a s h i p , the c r i t i c a l amount o f water on deck t h a t w i l l cause thc s h i p to c a p s i z e , depends on thc GM ( F i g u r e 3 3 ) . As shown
