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Marine Techn ology , Vo l. 38 , No. 2 , Apr i l 2001, pp. 9 2-94

A n A p p r o x i m a t e M e t h o d f o r C r o s s C u r v e s o f C a r g o V e s s e l s

H 0 s e y i n Y , I m a z 1 a n d M e s u t G l n e r 1

In th i s s tudy , a fo rmula i s presented to es t imate c ross curves o f cargo vesse l s and to pred i c t s ta t i ca l

s tab i l it y a t the pre l im inary des ign s tage o f the vesse l . The pred i c t i ve techn ique i s obta ined by regression

anal ys i s o f sys temat i ca l l y var i ed cargo ves se l ser i es data . In order to ach ieve th is procedu re, some ca rgo

vesse l fo rms are gene rated us ing Ser i es -60. The m athema t i ca l model i n th i s pred i c t ive techn ique i s

cons t ruc ted as a func ti on o f des ign param eters such as Length , beam, d epth , dra ft , and b lock coefficient.

The pred i c t i on method

developed in

th i s work can a l so be used to determ ine the e f fec t o f spec if i c hu ll fo rm

pa r am e t e r s and the load condit ions on s tab i l i ty o f cargo vesse l s . The presen t method i s app l ied to a cargo

vesse l and then the resu lts o f the ac tua l sh ip are com pared w i th those o f regress ion va lues .

I n t r o d u c t i o n

THE analytical, statistical, and empirical formulations for

ships have generally been based on theoretical or experimen-

tal information. However, the naval architect needs mostly

some practical techniques to obtain appropriate designs and

some indication of an acceptable level of performance in as-

suring vessel safety at the prelimin ary design stage. There-

fore, the stab ility of ships can be considere d crucially impor-

tant for the nav al architect. With in this context, approximate

expressions for the ship stability computations can be ob-

tained by analyzing a large amount of computer calculations.

The development of ship stability methods has had a long

period of evolution. During that period, man y studies have

been carri ed out on various aspects of the subject. For ex-

ample, the Prohaska method [1] has been used in stability

calculations for a long time. There are some papers concern-

ing the application of regression analysis for calculating the

stabil ity of ships, such as Campa nile and Cassella [2], Ku-

pras [3], and Yflmaz and Kii kner [4]. Camp anil e and Cas sella

have in vesti gate d the reducti on of form stabili ty for Series-60

ship forms in waves and have established some graphs for

the practical calculation of the righting moment arm. Kupras

has modeled KN-~ values based on design parameters for

cargo ships at certain heeling angles. Ydmaz and Ktikner

used a similar method as shown in this study to develop a

mathematical model for computation of cross curves of fish-

ing vessels at the prelimi nary design stage.

The objective of this paper is therefore to derive a m ath-

ematical model for est ima ting cross curves of cargo vessels by

use of regression analysis. The math ematica l model based on

hull form parameters is established for cross curves. There-

fore, the sys temati cally varied hull series [5] has been chosen

for the application of regression analysis to obtain an ap-

proximate e xpression for the stabi lity characteristics of cargo

vessels.

M o d e l i n g o f c r o s s c u r v e s K N - O ) v a l u e s

The prediction of ship stability at the prelimin ary design

stage is very difficult since the statical stability of a ship

cannot be easily predicted without knowing the ship's hull

geometry and her center of gravity. Therefore it is intended

1 Assistant professors, Depar tment of Naval Architecture, Yfldlz

Technical Universi ty, 80750 Beoiktao, Istanbul , Turkey.

Manuscript received at SNAME headquarters May 24, 2000.

that at the initial stage of design, stability cross curves ( K N -

T), which are a funct ion of ship draft, are proposed to be

estimated by usin g approximate statistical methods. As can

be seen from the geometrical featu res of an in clined ship

given in Fig. 1, the righting momen t arm G Z ) value is cal-

culated as follows in terms of KN value:

G Z = K N - K G sin~ (1)

A number of cargo vessels have been derived from Series-

60 for the r egressio n modeling of cross curves. The lengt hs of

those vessels are kept as a constant and their actual cross

curves are used in the regression analysis method. A total of

72 different hull forms are generated by in cremen ting the

L / B ratio by 0.5 where it provides a sufficient number of

sample sta bility data for reliable statistical evaluation. For

each L / B ratio, the B / T value is varied by 0.25 increments ,

and then for each B / T rati o the block coefficient (CB) is in-

cremented by 0.01. All calculations are considered up to the

main deck.

The pa rame ter rang e of gene rated vessels is set as follows:

5.0 < L / B < 7.5

2.25 < B / T < 3.0

0.60 < CB < 0.80

1.3 < D / T C < 4.0

where D / T c is the depth-draft ratio.

The actua l cross-curve computations of the vessels are per-

formed by a well-established stability software. The results

obtained from the implem entati on of the software are used to

establish approximated s tability expression.

Since the (KN-~) values have been modeled in terms of

design parameters, it is necessary to use the values of KN in

the nondime nsiona l form. Hence, the KN values should be

nondimens ionalized by the molded beam B, which is one of

the m ost effective design param eter s i n a vessel's cross-curve

calculations. At the beginn ing of the regression modeling, the

(KN-~) curve has been fitted for the fifth-degree polynomial

by us ing polyn omial regression as follows:

5

K N / B = E a Cpi (2)

i 1

where (I) is the heel angle in ra dia ns a nd ai is the polyno mial

coefficient which is determined by the application of the

least -squa res method. I n order to determ ine ai coefficients as

a function of design parameters, the following multiple linear

regression model can be used:

92 APRIL 2001 0025-3316/01/3802-0092500.31/0 MARINE TECHNOLOGY

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T a b l e 1 L i s t o f r e r e s s i o n c o e f f i c i e n t s f o r N c a l c u l a t i o n

W A

Fig. 1 Geometric representat,on of a heeled vessel

5 5

a i= E E b d X j ( 3 )

i - 1 j=O

w h e r e X d i s t h e j t h d e s i g n p a r a m e t e r a n d b u i s t h e c o ef fi -

c i e n t o f t h e j t h d e s i g n p a r a m e t e r f o r t h e i t h c o e f f i c i e n t o f t h e

p o l y n o m i a l e x p r e s s i o n , a ~. T h e s e l e c t io n o f d e s i g n p a r a m e t e r s

( Xj, j = 1 , 2 , . , 5 ) c a n b e a c h i e v e d b y s y s t e m a t i c t r i a l i n w h i c h

a p a r a m e t e r o r c o m b i n a t i o n o f s o m e p a r a m e t e r s s h o u ld g i v e

t h e h i g h e s t p o s s i b l e c o r r e l a t i o n w i t h t h e c o m p u t e d r i g h t i n g

a r m c u r v e d e f i n e d b y K N - q ) v a l u e s . T h e s e l e c t e d p a r a m e t e r s

u s e d i n t h i s s t u d y a r e g i v e n i n t h e f o l l o w i n g fo r m s :

X o = l

X 1 = B / T

X2 CB

X a = B / T ) 2

X 4 CB) 2

X 5 = B / T ) C B

I t i s a s s u m e d t h a t t h e d r a f t i s c h a n g e d a t 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 , t h e b u c o e f f i c i e n t s g i v e n i n e q u a t i o n ( 3 )

h a v e b e e n d e t e r m i n e d a t e a c h l o a d i n g c o n d it i o n .

T h e o t h e r d e s i g n p a r a m e t e r v a r i a t i o n s w i t h t h e l o a d in g

c o n d i t i o n h a v e b e e n c o n s i d e r e d a s c o n s t a n t s i n c e i t i s d if f i c u lt

t o c a r r y o u t t h e c a l c u l a t i o n s a t t h e i n i t i a l d e s i g n s t a g e . T h u s ,

t h e v a r i a t i o n s o f b u c o e f fi c i e n ts a r e d e f i n e d w i t h t h e f r e e -

b o a r d v a r i a t i o n

D / T c )

o n ly . F o r t h e i n c l u s i o n o f lo a d i n g

v a r i a t i o n , t h e f o l l o w i n g t h i r d - d e g r e e p o l y n o m i a l r e g r e s s i o n

w i t h a m a x i m u m e r r o r o f 2 % w a s u s e d:

5 5 3

= EE Ec, , , 4 )

z=l j - O k - 1

w h e r e 5 i s

D / T C .

U s i n g t h e r e g r e s s i o n c o e f f i c i e n t s ( cu ~) g i v e n i n T a b l e 1 , t h e

r i g h t i n g a r m c u r v e c a n b e e a si l y c o m p u t e d f o r a g i v e n s e t of

d e s i g n p a r a m e t e r s .

A p p l i ca t i o n o f t h e m e t h o d

T h e p r e s e n t e d r e g r e s s i o n m o d e l h a s b e e n a p p l i e d t o a c a r g o

v e s s e l , w h o s e p r i n c i p a l d i m e n s i o n s a r e :

a2

a3

o .

b]o -2.328721

bH 2.717601

bl2 -5.640604

bi3

bt4

bl5

b2o

b21

b~2

b23

b24

b25

b3o

b31

b32

b33

b34

b35

b4o

b4~

b42

a4 b43

}344

b45

bso

bsl

b52

a5 b53

b~4

4.109446

-3 745978

5,965310

b55

-1.458433

1.346525

-2.055787

c,p3

0.135208

-0.133742

0 217996

-0 463889 0 637674 -O 226011 0.022080

2.428064 -2.320000 0.790000 -0.090000

1 09 36 91 -1.270000 0.414975 -0. 035 071

22.570978 -27 .50 251 8 9.052024 -0.848615

-19.2 96043 24.898382 -8.951266 0.900041

43.692879 -5 4.3 868 90 19 .79 78 15 -2.0050-04

3.255165 -4 121551 1. 44 52 87 -0.141919

-15.864207 20 892283 -8.408362 0.933074

-9.679033 10 687704 -3.125419 0.249320

-33.921013 43 51 33 51 -15.438095 1 539757

34.550896 -45 926285 17 31 176 6 -1.791367

-112.551231 138026321 -48,579960 4.767433

-6.027348 7 621548 -2.745575 0.275166

41.359619 -54 768772 21.547258 -2.304114

22.077457 -23 .55 197 3 6.477677 -0 495855

17.957562 -25 852377 10 .4 46 90 3 -1.123475

-24.510056 33. 53 934 9 -13.263817 1.409822

-12743425806.314568 43 502827

-4.184539

4 386174 -5.559719 2.062754 -0.211117

-41.033024 52 90 082 6 -20.119068 2.099168

-18.532402 19 273975 -5.063540 0 373116

-3.007130 5.270205 -2.515895 0.291860

6.157536 -8 639531 3.559559 -0 386909

-32.953545 38 .810 867 -12 963937 1.230697

-1 119601 1. 42 27 59 -0.542013 0.056536

13.310734 -16 780735 6. 22 03 91 -0.638607

5.2 263 68 -5.3 2409 2 1.34 630 6 -0.095566

L e n g t h ( L ) : 1 2 0 ( m )

B r e a d t h ( B ) : 2 0 ( m )

D e p t h m o l d e d ( D ) : 1 2 ( m )

D e s i g n d r a f t ( T ) : 8 ( m )

C o m p u t a t i o n d r a f t (T c ) : 8 ( m )

B l o c k c o e f f i c i e n t

C B )

: 0 .70

T h e c o m p u t a t i o n p r o c e d u r e f o r t h e r i g h t i n g a r m c u r v e ( K N -

q~) o f t h e p r o p o s e d c a r g o v e s s e l s h o u l d b e a s f o l lo w s :

1 . C a l c u l a t e r e g r e s s i o n p a r a m e t e r s ,

X i

X o = I

X 1 = B / T

= 2 . 5

x 2 = c B = 0 . 7 0

X : ~ = B / T ) 2

= 6 . 2 5

X 4 = C B ) 2

= 0 . 4 9

X s = B / T ) C B = 1 . 7 5

5 = D / T , )

= 1 . 5

2 . C a l c u l a t e b u c o e f f i c i e n t s

bu = 0 + C i j l ~ -}- CU2 5 2 Jr- C 13 83

( i = 1 , 2 , .. , 5 j = 1 , 2 , , 5 )

b l o = - 2 . 3 2 8 7 2 1 + 4 . 1 0 9 4 4 6 * ( 1 . 5 )

- 1 . 4 5 8 4 3 3 * ( 1 . 5 ) 2 + 0 . 1 3 5 2 0 8 * ( 1 . 5 ) ~

a, , bd , cu~ -

reg re ss ion coef f i c i en ts o f KN-qb

B = m o l d e d b r e a d t h

B M -

t r a n s v e r s e m e t a c e n t r ic r a d i u s

C B

= b lock coef f i c i en t

C p

= l o n g i t u d i n a l p r i s m a t i c c o e f f i c i e n t

D = d e p t h

N o m e n c l a t u r e

G M ~

m e t a c e n t r i c h e i g h t

G Z

= r i g h t i n g m o m e n t a r m

K B =

h e i g h t o f c e n t e r o f b u o y a n c y

K G = v e r t i c a l c e n t e r o f g r a v i t y

K N ~ f o r m s t a b i li t y l e v e r i n k e e l

L = l e n g t h o f b e t w e e n p e r p e n d i c u l a r s

T = d e s i g n d r a f t

T c

= c a l c u l a t i n g d r a f t

= D / To ,

r a t i o

• - h e e l a n g l e

A P R I L 2 0 01 M A R I N E T E C H N O L O G Y 9 3

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3/3

51o = 1.01030

bll = -0.32306

bl~ = -0.58242 b t a = 0.05861

b14 = 0.41 972 bl.~ = 0.00160

b.~o = -1 .1 79 82 b,~ - 0.948 81

b2., = -0 .1 09 26 b~.~ = -0 .1 54 24

b~4 = -0 .2 95 47 b.,.~ = 0.161 78

bao = 1.80998 ba~ = -1. 432 92

b:~ = 1.27342 baa = 0.15611

b34 = -0.08 859 has = -0.3 4924

b4o = -1.10720 b41 = 0.71352

542 = -1. 078 2 b4~ = -0. 024 72

b44 =

0.13501

b45 =

0.244862

bso = 0.22244 bs1 = -0. 098 57

bs, = 0.24750 b~3 = -0.01418

b~ 4 = -0. 019 78 bs.~ = -0. 053 11

3. Calculate the a ,

a, = bio X o + bi l X~

coeff ic ients

bi2 X 2 b,:~ X3 bi4 X4 bi5 X5

a l = b l o X o + b l l X 1 + b l 2 X ~ + b 1 3 X 3 + b 1 4 X 4 + b l ~ X ~

at = 1.010301 - 0.3230642 * (2. 5)- 0.5824 233 (0 .70)

+ 0 .058617 23*(6 .25)+ 0 .4197221 (0 .49)

+ 0.001608094 *(1.75)

= 0.3697794

a l = 0.3697794

The res t of the a~ coeff ic ients have b een o btained in the same

way:

a2 = 0.2900721

a 4 = 0.26 1939 1

a 3 = -0.55 9783 6

as = -0.04202612

4. Calculate KN values

KN = (a~(l) * + a2~ e + aa ~ al + a 4~ 4 + as4)51B

KN = (0.369 7794 ¢P + 0.2900 721 cI)2 - 0.55 9783 6 q)a

+ 0.2619391 4) 4 - 0.042026 12 ~s) 20

whe re q~ is in rad ian s.

K N - O C H A R T

L = 1 2 0 m , B = 2 0 m . , D = 1 2 m , T = 8 m , T c = 8 m , C B = O 7 0

7 7

5

2

1 ' 1 : 0 ' 2 0 ~ 3 0 ' 4 0 ' 5 0 ' 6 0 - ' 7 0 ' 8 0

0 ' ' '

¢ degree) j

F i g

2 Comparison of KN-q~ values

K N - T ¢

8

6 .

2 •

0

I l l ~ l r l ,

3 4

n

5

., 10

~. 20

* 30

0

40 °

~. 50°

= 60

7o °

-

actual

r e g r e s .

5 6 7 8 9 10

T~ [m]

F i g

3 Comparison of KN-Tvalues

The ac tua l and sugges ted methods of comput ing resul ts

of (KN-dp) values for the sampl e cargo vessel are show n in

Fig. 2.

For o the r loading condi t ions , the KN va lues have been

computed by changing the dra f t va lues. The KN- T va lues of

the ac tua l and sugges ted methods of comput ing resul ts for

different heeling angles have been plotted in Fig. 3.

C o n c l u s i o n s

The present method and the computa t iona l tool g iven in

this pape r can be successfully applied to the calculation of a

cargo vessel 's cross curves at the p reli minar y design stage.

The accuracy of the met hod is based on the hull form par am-

eters as des cr ibed herein. I t can be seen from the compu ta-

t iona l r esul ts tha t the pr esent meth od g ives ve ry h igh accu-

r a c y f o r S e r i e s - 60 a nd f o r o the r d i f f e r e n t c a r go hu l l

geomet r ies as well , and the level of error is acceptable . I f the

approximate

K G

value is know n for a cargo vessel , the sta ti-

cal stabil i ty calculation of the ves sel can be executed at t he

pre l iminary des ign s tage .

One of the o the r im por tan t r esul ts of th is s tudy is tha t i t

provides the des igner wi th appropr i a te des ign pa ramet e rs for

a cargo vessel f rom th e p oint of view of vessel stabil i ty.

R e f e r e n c e s

1. P rohaska, C. W., Intluence of Ship Form on Transv erse Stability,

TINA,

1951.

2. Cam panile, A. and Cassella, P., For m Stabi lity Reduction Amon g

Waves for Series-60 Hulls,

Oceazz Enginee ring,

Vol. 16, 1989, pp. 431-

462.

3. Kupras, L. K., Optimization Method and Paramet ric Study in Pre-

contracted Ship Design, International Shipbuilding Progress, Vol. 23,

1976, pp. 138-155.

4. Yllma, H. and Ktikner, A., Evalua tion of Cross Curves of Cargo

Vessels in Prel iminary Design Stage, Ocean Engbzeering, Vol. 26, No. 10,

1999, pp. 979-990.

5. Todd, F.H ., Some Fur the r Experim ents on Single-Screw Mer-

chant Ship Form--60 Series,

Transactions.

SNAME, Vol. 1, 1953, pp.

516-589.

6. Yllmaz, H., The Determi nation of Practical Stability Criteria De-

pending Upon Ship Design Parameters, MS thesis (in Turkish), Ylldlz

Technical University, Ist anbul, 1994.

94 APRIL 2001 MARINE TECHNOLOGY