On the probability of ship capsizing

On

the

Probability of Ship Capsizing

M. A. Shama, B. Sc. Ph. D.)

Summary

The variabilities of the main factors affectug the heelinq and righting moments are enined. ilecause of the random variation of these factors, the reserve of dynamical stability is

treated as a

stochast phenonrauon. The

mathe-matical model repr?sonting the random variation of the relevant parameters is assumed to follow the normal probability density function. This assumption is made because of lite absence of sufficient

statistical data and also to simplify the

subsequent calculations. The risk of

capsizing is calculated using the coef-ficients of variation and the results are presented graphically.

It is concluded that since the reserve o dynamical stability has a direct in-fluence on ship safety, its variability

should be carefully examined with

particular emphasis on the risk of

capsiz-ing.

rrtrod oction

Dynamical stability of ships is measur-ed by the area under the statical stability curve. The latter is normally obtained

from the cross curves of stability. Various methods are available for the

calculation of these curves [1, 2, 3, 41.

These methods, however, are based on the assumption that inertia forces and

hyd rodynarnic pressure are neglected.

Threfore, experimental and theoretical methods 15.61 are proposed to determine ship stab:ity among waves. The effects

of the hoqiri

and sagging conditions are indthter JI [7J. The effect of ship

SpeOd is exar:ned in (8(.

Because of the random variation of the main parameters affecting the shape and area under the statical stability curve, thn characteristics of the latter should be treated as random variables.

Con-seq.uontly, the reserve of dynamical

stability should be associated with the risk of capsizing, since the external forces

acting ori a ship among waves are

ra'dom in nature.

In this

paper, the variability of the

mein parameters affecting the reserve of dynamical stability is discussed. Particular emphasis being placed on the

calcula-tion of the risk of capsizing. Because

)Assoc. Prof., Marine Engineering Dept., Fac, of

Eng., Basrah University, Iraq.

670 Schi! & Halen, SMM-Sonderausgabe, September 1976

of the lack at adequate statistical data to establish the mathematical niodel re-presenting the random variation of the relevant parameters associated with the reserve of dynamical stability, a normal probability function is assumed.

Dynamical stability

Dynamical stability is generally defined by the

work done by the

righting

moment in incli'ìing a ship through an angle of heel E). It is, th'refore, equal

to the area under the statical stability

curve [9(, Hence

o e

Mp ₌ J Mr dO

= Î

_{J CZ} dO,

o o

(1.1)

where: MD =" dynamical sLability,

Mit righting moment,

ship displacement,

CZ = righting arm.

This definition, however, does not Lake into account inertia, hydrodynamic and

friction forces. The calculations are,

therefore, based on quasi-static

condi-tions. The errors inherent in this

assump-tion

has not yet been fully identified

(I 01.

Reserve of dynamical stability Io c'er to ensure adequate dynamical

stability, the following condition must be satisfied:

SD = MD - Mit > O

where

Sp reserve of dynamical stability, see fig. 1,

M11 =

work done by an

_arbitrary

heeling moment.

The angle of dynamical equilibrium, Od, is determined from the following condition, see fig. 1:

t1,l t1, a.'

l)ct University

of Technology J

Hyromethanics

Laboratory

tjbrary

Mekelweg 2 - 2628 CD Deift The Netherlands °'-ne. 3115 78673 -Fax: 31 5 781836 ML) = Mii (2.2) The limiting value of a heeling mo-ment independent of the angle of heel O

codld be determined as showit in

fig.

1. In this case, the reserve of clyimamical stability vanishes.

Probability of capsizing

The probability of cdpizing, R, can

be calculated from the probability density function, p.d.f., of the reserve of dynamical

stability as follows: o R = P (SI) s _{O) = J p} (S11) dSp,

00

(3.1) where p (SI)) = p.d.f. of Sp. The p.d.f.

of Sp can be determined

from the p.d.f. of both MD and M, which could be determinen from tile nature of

their variabilities.

Variables nl Mfl and 1D

Since M1 depends largely on tile wind and sea conditions, it can be treated as

a stochastic phenomenon. However, in the absence of sufficient statistical data, M11 may be assumed to follow the normal

p.d.f.; i.e.

(x_;i

2 a (4.1)

where:

and o

are

the mean and

variance of X, X = M11.

Since MD depends totally on the shape of the statical stability curve, its bility must be deduced from the varia-bilities of tile main factors affecting the stalical stability curve. These factors are:

initial stability, GM0, ii maximum righting arm, GZ,,,

11L angle of vanishing stalility,

O.

-

variability of GM0

'l'ho variability of GM,, results freni the variabilities of KM,, and KG, since

GM0 KM0 - KG (4.2)

KG is a variable quantily given by

1 n KG _{¿t,,, 'KG,, + .' Wt Z} A

_i=

(2.1) _{Px (x)} -/

i

2i'o,

Fig.i (4.3)

However, the variability of KG

de-pends ort ship type and size. For cer-tain types of ships, such as oil tankers, KG may be treatcU as a deterministic quantity for the particular loading

con-dition.

For other types

of ships, KG must be treated as a random variable.

The variability of KM0 depends on ship size, form, geometry, sea condition, hull stiffness [II] and speed of vessel [81. On

a wave crest and in a following sea [71,

KM is seriously affected. Therefore, it is possible to treat KM as a random vari-a b le.

In the absence of sufficient statistical data on the variabilities of both KG and KM0, it may be sufficient to assume that they both follow the nonnal p.d.f. Since these two variables are statistically in-dependent random variables, GM0 also follows a normal p.d.i. [12] having the following prlicu1ars: GM0 =r _{KM0 - KG} (4.4) 02

=o

±e

GM,, KM,) KG (4.5) where :

=

mean value of X,(XGM0, KM,) ami KG) =

variance of X, (XGM, KM,) and KG)

It should be mentioned here that al-though GM, cannot he used alone as a general measure of ship stability, a posi-tive value must be maintained, and in-itial stability should be measured by the coefficient of stability, i.e. GM0. ii - variabilities of CZ,0, O.,.

The variabilities of GZ, and O result from:

a - accuracy of the method used

for

calculating the cross curves of sta-i) ility,

h - the inevitable trim associated with the different angles of heel,

e - the effect of waves,

d -- errors resulting from the presence, or absence, of watertight structures

above the muni deck,

e - errors

with the presence of appendages,

f

treating the ship as a rigid body,

Fig. 2

the reserve of dynamical stability does not vary from - cc to + cc, Therefore, a trunc.ted p.d.f. should be used in cal-culating the risk of capsizing. The effect

of truncation could be taken into

ac-count as follows [12]:

f(x)

=

p(x)/E

(4.6)

where (x) = truncated p.d.f., and

X0 X

E

f

Px (z) dx

-

_{J p}

(z) dx

-

ce

(4.7) The range of variation is, therefore, given by

XL.<X<XU (4.8)

where X = any random variable, KG, KM, ... etc.

U and L stand for upper and lower respectively.

5. Risk et capsizing

In order to calculate the risk of cap-sizing, R, the p.d.f. of both M and M should be known apriori. For the

par-ticular case when both M and M

fol-low the normal p.d.f., Si) will also fol-low the normal p.d.f. by virtue of sta-tistical independence.

The calculation of R could he greatly simplified by using the coefficients of variation of M0 and M11 as follows [13]:

u = °u and u =°M ¡M11 I) ii 1-fence, R is given by R =P (SI) 0)

-V 2 r where t

w

-)/ 2 . +

i

F= M0 / Mn> 1.0,

SI) =

MDMH, see fig. 2,

w

Jexp.

cc

72, = 02)1 + _0\I

1) II H

M1) and = mean values of M11 and

M11 respectively,

= variance of x, (x = Mj, Mii and S0).

The standard deviations of MD and Mii could be approximately calculated es

follows:

(range of variation of z) / 6, = M11, M11.

The effect on R of variation of u, y

and F is slìow,i in figs. 3, 4. lt can be seen that when:

u = 0.10, y

0.10 and F

= 1.4,

l00

It is evident now that the risk of cap-sizing is greatly influenced by the shape of the statical stability curve. The latter is normally obtained frein the cross curves of stability. Therefore, improving the accuracy of calculating these curves cannot be overemphasised.

lt should he mentioned here that the

area under the statical stability curve

does not give a correct measure of dy-namical stability, as

it does not take

into account such factors as inertia cud l)ydrodynamic forces, among several othr factors. However, the risk of capsizing

based on the arca under the

statical

stability curve could be used for quali-faUve measures as well as for comparing riifferent designs.

(5.2)

Schiff&Hafen. SMM-Sondcrausgabe, September 1976 371

here

L0 =

light ship displacement, KG0 = height of C.G. of o from

base line. lt could be de-termined by au inclining experiment,

W weight of item i,

Z = height of C.G. of W from

bose line,

n = total number of weight

items.

The carriage of deck cargoes, as well is the presence of partially filled tanks, 'ave a marked influence on the

varia-ility of KG.

g - neglecting

dynamic and hydro-dynamic effects,

h - neglecting wind effects,

i - neglecting effect of ship speed,

j -

neglecting friction forces,

k - the inaccuracy of calculating KG. However, because of the lack of ad-equate statistical data needed to estab-lish the matematical model representing the random variation of the relevant para-meters affecting MD, the latter is

as-sumed to follow the normal p.d.f. lt should be mentioned here that the variability of each parameter affecting

t2

dt 2

6. Concluding remarks

Although there are major assumptions used in this investigation, the following

IOdiO _{conclusions are generally valid:}

I. _{The probability of ship capsizing is} greatly influenced by the variabilities of initial stability and tIre shape of the statical stability curve.

2. Deficient initial stability has an ad-verse effect on initial and dynamical stability.

Die Kapzitat des Forschungs- und Eut-wicklungszen trums der Perki na-Motoren-Gruppe in Peterborough!England ist jetzt erheblich erweitert worden. Für umge-rechnet über 20 Mio. DM wurden neue Einrichtungen und Anlagen, die der Die-selniotoren-Entwicklung dienen, in

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872 _{Schiff & Hafen, SMM-Sonderausgabe, September 1976}

Stability

criteria should be trealed

stochastically as the main parameters affecting ship capsizing are stochastic in nature.

Much work is needed to determine

the effects of inertia, friction and

hydrodynamic forces on dynamical

stability.

Statistical data on the variabilities of the relevant parameters affecting the

reserve of dynamical stability are needed.

Ferner wurden drei besondere Prüf-räume errichtet, die groß genug sind, um komplette Fahrzeuge oder Geräte noii Motoren bis zu 350 PS aufnehmen zu können. In diesen Testkainrnern können Temperatur und Luftfeuchtigkeit kontrol-liert eingestellt werden. Daneben gibt es seit langem eine Kältekamrner, in der Dieselmotoren bei extrem nied rigen Tem-peraturen getestet werden. Damit hat Perkins jetzt die Möglichkeit, alle Um-welt- und Klimabedingungen _unter de-nen Perkins-Motoren irgendwo auf der Welt zuro Einsatz kommen können, zu

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7. _Relerences

[Il M. A. Shama, "A method for co'iutating ship stability curves", Shipbuilding and Shipping

ACcord, August, 1968.

-2] M. A, Shama. "A Computer programme for

ship stability curves", Shiptruilding and Ship-ping Record, May, 1969.

1 31 K. Hoppe, "A method for the determination

ot cross curves of stability', Schiff & Halen.

Vol. 10, 1953.

1 4] C. Prohaska, "Influence of ship form on

transverse stability", RINA, Oct., 1951.

1 5] C. Boden and R. Hallidsy. "Computation of the transverse stability at a ship in a

long-itudirisi seaway", RINA, Jan.. 1964.

16] J. R. Psulling. "The transverse stability of

a ship in a longitudinal seaway", Journal of

Ship Research, March, 1961.

17] R. A. Norrby, '"Stability, problems of coastal

vessels". International Shpbuilding Progress,

Vol. It, No. 121, 1264.

1 81 A. M. Ferguson and J. F. C. Cann, "The effect of forward mnlion on the transverso stability of a displac'rnent vessel", LESS.,

Jan., 1970.

191 V. Semeyonov - T,'.'ri-Shsnsky, "Statics

and dynamics of t.h Ship", Peace

Pub-lishers, Moscow, U.S

[101 C. Kuo and Y. Odabasi, "Alternative

ap-proachea to ship and ocean vehicle

sta-bility criteria", The Nvsl Architect, July,

1974.

[ill M. A. Shama. "An investigation into ship

hull _{girder deflection", Bull of th Fac, of}

Eng. Alexandria University, 1973.

P. L. Meyer. "Introductory probability and

statistical applications", Addison - Wesely,

Reading, Masschuselts, 1966.

t.1. A. Shams, "The risk of tosing stability",

Shipping World and Shipbuilder, Oct., 1075.

Drehzahlen bis zu 4500/min erforscht werden, Das umfassende

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Die Gesamt-Investition von über 20 Mio, DM attf dem Gebiet Forschung und Entwicklung soll es Perkins ermöglichen, die weltweit immer strenger werdenden Urnweltschutzgesetze zu. erfüllen. Zur Zeit richten sich die nordamroerikanischen Gesetze hauptsächlich gegen gasförmige Schadstoffe, während die Gesetzgebung de r Eu rop ¿lise hen Gemeinschaft die Rauchbekämpfung zum Hauptziel hat. In naher Zukunft werden aber auch in der EG Gesetze über gasförmige Schadstoffe erlassen. «'C'o

HI

4 6 20 12 1.4 16 ₂₀ o_s lO 00' 05 'C 001 e Fig. 3 _{Fig. 4}