7
A Probabilistic Method on Ship Damages H. Sayyaadi, Assistant Professor M. T. Tavakoli, Research Assistant Mechanical Engineering D ept. Sharif University of Technology Azadi Avenue, Tehran IRAN [email protected]  Abstract Marine accidents especially when considering ship accidents and crashes are addressed here in this paper. In order to iden tify damage stability in ships and marine vessels, there are basically two main approaches including: deterministic methods, and probabilistic methods. A probabilistic method is used here in this paper and because of probabilistic calculus; statistical data is needed to identify models, methods, etc. IMO data bas e is used to do analysis and present research results. Probabilistic modeling has been performed by introducing three individual factors which are as follows: a) the probability that one or mo re than one ship compartments to be flooded (there is no any more longitudinal andlor horizontal subdivision in it), b) the probability of not damaging longitudinal subdivision (if there will be some), c) the probability of not damaging horizontal subdivision (if there will be some). In this research work focu s is on the f irst and second factors that are functions of the location, the length and the width of compartment(s). In order to ease the proposed method and also to generalize the results, non-dimensional damage location and non-dimensional damage length and non-dimensional damage penetration are introduced. Referring o IMO data base, the bi-linear functions are used to describe non-dimensional damage length and non -dimensional location. Completion of these methods and models will lead us to a new formulation for probability of the flooded compartments in ships and vessels and the results will be illustrated to prove the validity of the method. I. INTRODUCTION Marine accidents especially when considering ship accidents and crashes are addressed here in this paper. From T itanic sank due to iceberg hit in North Pacific till now, naval architects, marine engineers and marine industries researchers all over the globe have done too many efforts to analyze, predic t, and prevent ship accidents and crashes. In order to identify damage stability in ships and marine vessels, there are basically wo main approaches including: deterministic methods, and probabilistic me thods . A probabilistic method is used here in this paper to do analysis and maki ng results. The basic conceptual thing in probabilistic study of ship damages is Attained Subdivision Index, which are nominated by letter A. It is including two m [email protected]  0-7803-8669-8/04/ 20.00 02004 IEEE. 205 parts that the first part is used for determination of the dama ge probability of one compartment, a nd the second part is used for determination of the survival probability of the ship while the compartment s flooded. In 1960 ship damage probabilistic studies were founded by Wendel [I] and Densis d id also some researches n this field [2]. In this method the studies of damage prob ability are concentrated on three individual factors, which are nominated by p, v, and r. p factor represents the probability that one or more than one ship compartments o be flooded (there is no any more longitudinal and/or horizontal subdivision n the shi p), r factor shows probability of not damaging longitudinal subdivision (if there will be some)., and finally v factor is the probability of not damaging horizontal subdivision (if there will be some). The main procedure of th e method was introduce d n par t B-I of SOLAS for the first time [3]. In SOLAS it is assumed that the probability of damage is increased from stern to middle of the ship and it has no any variation from amidship s to the stem. SLF (Stability and Load lines and on Fishing vessels safety) researchers also did some additional studies on ship damages and probability [4]. In their results the damage probability of the ship is increasing along ship from stern to stem. It can be inferred from SOLAS and SLF results hat their methodologies and data bases are based on some traditional knowledge of general cargo carriers and thus needed to be updated for modern s hips and navigator systems. Recently Lutzen dev eloped new distributions or damage length and damage location along ship [5]. n this new proposal damage probability is assumed to be unvaried along ship and it is a function of the ship compartments length only. In this paper damage probability s assumed to be functions of compartments length and compartments loca tion. For this new distribution some recent data base are in use and these data are gathered from some new ship accidents. II. MODELING OF DAMA GE PROBABILI TY OF SHIPS A Damage Probability In damage probability studies Attained Subdivision Index is a very important factor and can be shown by the following formula:

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A

Probabilistic Method

on

Ship Damages

H.

Sayyaadi, Assistant Professor M. T. Tavakoli, Research Assistant

Mechanical Engineering Dept. Sharif University of Technology

Azadi Avenue, Tehran

IRAN

[email protected] 

Abstract Marine accidents especially when

considering ship accidents and crashes are addressed

here in this paper. In order to iden tify damage stability

in ships and marine vessels, there are basically two

main approaches including: deterministic methods, and

probabilistic methods.

A

probabilistic method is used

here in this paper and because of probabilistic calculus;

statistical data is needed to identify models, methods,

etc.

IMO

data base is used to do analysis and present

research results. Probabilistic modeling has been

performed by introducing three individual factors which

are as follows: a) the probability that one or more than

one ship compartments to be flooded (there is no any

more longitudinal andlor horizontal subdivision in it), b)

the probability of not damaging longitudinal subdivision

(if there will be some), c) the probability of not damaging

horizontal subdivision (if there will be some). In this

research work focus is on the f irst and second factors

that are functions of the location, the length and the

width of compartment(s). In order

to

ease the proposed

method and also to generalize the results,

non-dimensionaldamage location and non-dimensional

damage length and non-dimensional damage

penetration are introduced. Referring o

IMO

data base,

the bi-linear functions are used to describe

non-dimens ional damage length and non-dimensional

location. Completion of these methods and models will

lead us to a new formulation for probability of the

flooded compartments in ships and vessels and the

results will be illustrated to prove the validity of the

method.

I. INTRODUCTION

Marine accidents especially when considering ship

accidents and crashes are addressed here in this paper.

From Titanic sank due to iceberg hit in North Pacific till now,

naval architects, marine engineers and marine industries

researchers all over the globe have done too many efforts

to

analyze, predict, and prevent ship accidents and crashes.

In order

to

identify damage stability in ships and marine

vessels, there are basically

wo

main approaches including:

deterministic methods, and probabilistic methods. A

probabilistic method is used here in this paper

to do

analysis

and making results. The basic conceptual thing

in

probabilistic study of ship damages is Attained Subdivision

Index, which are nominated by letter A.

It

is including

two

m

[email protected] 

0-7803-8669-8/04/ 20.00 02004

IEEE.

205

parts that the first part is used for determination of the

damage probability of one compartment, and the second part

is used for determination of the survival probability of the ship

while the compartment s flooded.

In

1960

ship damage probabilistic studies were founded

by Wendel

[ I ]

and Densis did also some researches n this

field [2]. In this method the studies of damage probability are

concentrated on three individual factors, which are

nominated by p, v, and r. p factor represents the

probability that one or more than one ship compartments o

be flooded (there is no any more longitudinal and/or

horizontal subdivision n the ship), r factor shows probability

of not damaging longitudinal subdivision (if there will be

some)., and finally v factor is the probability of not

damaging horizontal subdivision (if there will be some).

The main procedure of the method was introduced n part

B-I

of SOLAS for the first time

[3].

In SOLAS it is assumed

that the probability of damage is increased from stern

to

middle of the ship and it has no any variation from amidships

to the stem. SLF (Stability and Load lines and on Fishing

vessels safety) researchers also did some additional studies

on ship damages and probability

[4].

In their results the

damage probability of the ship is increasing along ship from

stern

to

stem. It can be inferred from SOLAS and SLF

results hat their methodologiesand data bases are based on

some traditional knowledge of general cargo carriers and

thus needed to be updated for modern ships and navigator

systems.

Recently Lutzen developed new distributions or damage

length and damage location along ship

[5]. n

this new

proposal damage probability is assumed

to

be unvaried

along ship and it is a function of the ship compartments

length only. In this paper damage probability s assumed

to

be functions of compartments length and compartments

location. For this new distribution some recent data base

are in use and these data are gathered from some new ship

accidents.

II.

MODELING OF DAMAGE PROBABILITYOF SHIPS

A Damage

Probability

In damage probability studies Attained Subdivision Index is

a very important factor and can be shown by the following

formula:

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Damaaed

ComD.

Fig.

1.

Damage probability of ship compartments [7]

“p factor” represents the probability that one or more

than one ship compartments

to

be flooded (there is no any

more longitudinal andlor horizontal subdivision n the ship), “r

factor” shows probability of not damaging longitudinal

subdivision (if there will be some)., “v factor” is the probability

of not damaging horizontal subdivision (if there will be some),

and i is the compartment number.

All damages which open single compartments of length

Li re represented in Fig.

by points in triangles with the

base

Li

Triangles with the base

Li L j

where

= i

+

1)

enclose points corresponding

to

damages opening either

compartment “i”, or compartment

“j”,

or both of them.

Correspondingly, the points in the parallelogram ”ij”

represent damages which open both the compartments ’7”

and

“j”

Damage location “x” and damage length

”y”

are random

variables. Their distribution density f x , y ) can be derived

from the damage statistics. The meaning of

f x , y )

s as

follows (see Fig. 2): the total volume between the x-y plane

and the surface given by f x , y )equals one and represents

the probability that there is damage (this has been assumed

to

be certain).

Fig.

2.

Damage probability including probability density

function [7]

Fig.

3.

Damage location and damage length

The volume above a triangle corresponding

to

damage

which opens a compartment represents the probability that

this compartment s opened. In a similar manner or all areas

in the x-y plane which correspond

to

the opening of

compartments or group of compartments, here are volumes

which represent the probability that the considered

compartments or group

of

compartments are opened.

The probability that a compartment or a group of adjacent

compartments is opened is expressed by the “p factor” as

calculated according

to

the following formula:

_ _

f x,y)

is the probability density function and is related

to

the

non-dimensional damage location and non-dimensional

damage length.

Rgferring

to

the Fig.

non-dimensional damage location

(( x = x L

, which is the ratio of the damaged distance from

stern

to

fie overall length), and non-dimensional damage

length ((

y

= y / L ),

which is the ratio of the damaged length

to

the overall length), can be considered two independent

parameters.

B

Non dimensional dam age location

SOLAS

and SLF proposed linear functions for

non-dimensional damage location, referring

to

the accidents

reported for some traditional cargo carrier

[3, 41. These

formulas are as follows:

SOLAS:

0.4+1.6;

for x10.5

1.2

for

;>

0.5

a ( x )

=

SLF:

a ;)

= 0.6+0.8; (2.4)

Fig. 4 shows density function according

to

the

non-dimensional damage location of new data base.

Fig.

4. 

Statistical data base

of

non-dimensional damage location

[5]

2 6

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D i ” t t o n ensi@

d a r ~ n - d i i e n s i o ~ l

mqe ccltion

Fig. 5. Distributiondensity for the non-dimensionaldamage

location

[5]

Error

0

2 0.4

Fig.

6.

Distribution density for the non-dimensionaldamage

location using different functions [SI

Tab.

1.

Mean square error of different unctions for damage

Location

1.11 1.51

2.80 4.08

1.59

Bi-linear Linear Weibull

SOLAS

Uniform

Bi-linear

Weibull

SOLAS

Fig. 5 illustrates distribution functions using SLF and

SOLAS formulas, while new and old data bases are

represented. Fig.

show some new distribution functions,

which are Bi-linear function, Weibull function, Uniform

function, and Linear function. In this figure SOLAS

distribution function is also depicted.

In Tab. 1 mean square error of each distribution

functions are listed, and it shows that the best distribution

function is the Bi-linear.

Error

E

0.62

0.51 1.05

Fig. 8. Distribution density for the non-dimensionaldamage

length [5]

Fig. 9.

Distribution density for the non-dimensionaldamage

length using different unctions

[6]

C Non-dimensionalDamage Length

SOLAS proposed linear function for non-dimensional

damage location, referring o the accidents reported or some

traditional cargo carrier

[3].

hese formulas are as follows:

Fig.

shows density function according

to

the

non-dimensional damage length of new data base. Fig.

8

illustrates distribution functions using

SOLAS

formula, while

new and old data bases are represented.

Fig. 9 show some new distribution functions, which are

Bi-linear function, and Weibull function. In this figure

SOLAS distribution unction is also depicted.

In Tab. 2 mean square error of each distribution unctions are

listed, and

it

shows that the best distribution function is the

Wiebull but for sake of difficulty here the Bi-linear function is

in use.

Fig.

7.

Statistical data base for non-dimensional damage length [5]

207

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D. New Proposal for

“ p

factor”

According to the previous discussion, a new proposal is

suggested here for “p factor”. In this paper Bi-linear

functions are used for both non-dimensionaldamage length

and non-dimensional damage location. Lutzen proposed

uniform distribution function for non-dimensional damage

location and Bi-linear function for non-dimensional damage

length

[5].

Lutzen

Proposal

Bi-linear function is used for non-dimensional damage

location as follows:

- -

_ _ (2.6)

a 2 X ) = a l l x + a Z 2 or

X > X k

That

x

is a knuckle point between

U x)

and

u 2 x)

Parameters a l l

uI2

U and

u2,

have to satisfy

conditions and function has to be continuous at the knuckle

point.

0.455 1.154 0.708 -0.768 1.583

1 0

1

0

0

6 a x)dx = 1

(2.7)

Similar to the non-dimensional damage location, a Bi-linear

function is introduced here for non-dimensional damage

length as follows:

That y , is a knuckle point between b,

X)

and b,

X)

.

Parameters

b,, b,, , b,, ,

and

b,,

have

to

satisfy

conditions and function has to be continuous at the knuckle

point.

b y ) d y

= 1

In Tab. 3  and Tab.

parameters of non-dimensional

damage location and non-dimensional damage length are

listed.

In

these tables the first rows are based

on

the

proposed Bi-linear function and

the

second

rows

are based

on Lutzen.

Tab. 3. Parameters or Bi-linear function of damage location

Tab.

4.

Parameters or Bi-linear function of damage length

E

New Proposal for

“r

factor”

Till now the effect of longitudinal bulkhead on damage

probability was not considered. In this section modeling will

be completed while considering this effect. “r factor” is

determined by the following formula:

(2.10)

That

=f ; , v , i )

s probability density function and is

related to the non-dimensional damage location,

non-dimensional damage length, and non-dimensional

penetration. “r factor” actually shows probability of not

damaging ongitudinal subdivision. As the vessel is assumed

to be symmetrical t, should be expected that

b 1

B 2

r = r - = - ) = 1

and

(2.11)

The probability of damaging only a wing tank of the breadth

b, p.r can be written as

(2.12)

Where

.Jb

is equal to

b/15B

and

c ik)

s the

conditional probability function

of

the non- imensional

penetration, given the non-dimensional damage length. The

corresponding conditional probability distribution function is

defined by:

(2.13)

The function c ~

)

depend on non-dimensional damage

length

so

that: 1

Now, the expression for p.r can be written:

(2.14)

(2.15)

Fig. I O . Distribution density for the non-dimensional

penetration

[5]

208

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1.2

,

1

0 8

-

r

0 6 -

0.4

~

0 2

Fig:11. Density distribution of non-dimensional penetration

And the probability r that the penetration is less than the

breadth of the wing tank b can be written as:

Where

G

is

defined by

(2.16)

(2.17)

When the integration is carried out, the position of the

knuckle point, Jk, in the expression for the non-dimensional

damage length must be considered. The G function can be

expressed:

J I J ,

(2.18)

(2.19)

Where J is min Jp,J).

Proposal or the C(z) function

Data from the old database and the updated database is

shown in Fig. I O . A linear function has been used for

describing the non-dimensional penetration or the c(z)

function.

As

only

5

of the non-dimensionalpenetrationsare

greater than 0.5, the maximum value will be taken as

0.5.

The proposal is fitted to the data by the least-squared error

method for the cumulative distribution function requiring

that all parameters must be described by fractions. The

function of c(z) can thus be determined as

And the corresponding probability distribution unction as

0 4

0 0.1 0.2 0.3 0.4 0.5 0 6

b lB

Fig. 12. Comparison of the r factor for the

SOLAS

and the

new proposal

The functions c(z) can be seen in Fig. 11 with the results rom

the database.

In the SOLAS, the reduction factor shall be determined by

following formulas:

b

r =

~ . F . ~ + * ] + o . I f i r b50.2

(2.22)

for

- ->0 .2

J+O.03

+ b + 0 . 3 6 B

In Fig. 12 the r-factor s shown as a functionof the relation

b/B for both the current regulation and the new proposal.

Ill.

CASE STUDIES

In this research work three different cases are studied. In

the first case a vessel 120 meters in length and 20 meters in

width having two same compartments without any

longitudinal bulkheads s considered. n Fig. 13 results based

on the proposed method and Lutzen method are illustrated

together. It can be concluded that based on the proposed

method, damage probability or the aft. compartment s less

than that of for. compartment, while in Lutzen both are the

same and equal. It is obvious that because of ship traveling n

forward direction damage probability in for. compartment

should be greater than aft. compartment, which are

consistence with the statistical data base depicted in Fig.4.  

In the second case, a vessel same in dimensions to the

first one, but having four same compartments without any

longitudinal bulkheads s considered. In Fig. 14 results based

on the proposed method and Lutzen method are illustrated

together. Based on the Lutzen damage probabilityfor the aft.

and for. compartmentsare the same and damage probability

for the two middle compartments are less and the same,

while in the proposed method damage probability or these

four compartments are different and close

to

the statistical

data base.

- 1

-2

C z)=- -12.z +16.z) for 01290.5

5

(2.21)

209

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0

30

60

90 12

Location

0.2 -

Fig. 13. Damage probability of a ship with two compartments

(only “p factor”)

Fig. 14. Damage probability of a ship with four compartments

(only “p factor”)

In the third case, a vessel 300 meters in length and 60

meters in width that is illustrated in Fig. 15 is studied. This

vessel has 11 compartments aligned along vessel.

Compartments at for. and aft. are shorter in length and are

15

meters each, wile the other compartment are longer and

30 meters in length. Compartments alignments are depicted

in Fig. 15. Damage probability of the vessel’s compartments

without considering longitudinal bulkheads can be seen in

Fig. 16. In this case

“ p

factor” is activated only and Lutzen

and proposed method are compared together. In Fig.

17

“r

factor” is considered only and Lutzen and the proposed

method are compared together. In forward compartments

non damaging probability of longitudinal bulkheads are the

same as Lutzen and less than that of afterward

compartments. In fig.18 both “p factor” and “r factor” are

considered or this vessel. The proposed method and Lutzen

are compared together and it can be seen that although both

methods have enough consistency with the statistical data

base of Fig. 4, but it can be concluded from the results of

case 1 and

2

that the proposed method better satisfies the

statistical data base.

Fig. 15. Compartments distribution along an actual vessel

004

p 003

0.02

1 ,

I

0 30 60 90 120 150 180 210 240 270 300

Location

Fig. 16. Damage probability of the actual ship (only “p

factor“)

Fig.

17.

Non damage probability of the actual ship with

longitudinal bulkheads (only “p factor”)

0.06

P*r

0.03

0 02

1

30

60

90 120 150 180 210  240 270 300

Location

Fig. 18. Damage probability of the actual ship (“p factor” and

“r factor”)

210

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IV. CONCLUSION cknowledgments

Marine accidents especially when considering ship

accidents and crashes are addressed here in this paper. In

order

to

identify damage stability n ships and marine vessels,

there are basically two main approaches including:

deterministic methods, and probabilistic methods. A

probabilistic method is used here in this paper and because

of probabilistic calculus; statistical data is needed

o

identify

models, methods, etc. In this method the studies of damage

probability are concentrated on three individual factors,

which are nominated by p, v, and r. “p factor” represents he

probability hat one or more than one ship compartments

o

be flooded (there is no any more longitudinal and/or

horizontalsubdivision in he ship), “r factor” shows probability

of not damaging longitudinal subdivision, and finally “v factor”

is the probability of not damaging horizontal subdivision.

Modeling has been carried

out

using some new statistical

data bases, which are including all types of vessels accidents

all over globe.

It

should be emphasized here again that in the method

non-dimensional damage location and non-dimensional

damage length both have been considered and Bi-linear

functions have been used for modeling them. Results show

that the proposed method has reasonable compatibility with

the statistical damage data bases and experimental results.

Authors acknowledge Sharif University of Technology or

his support. Authors also acknowledge Mrs. Lutzen for her

cooperation.

REFERENCES

[I]. K. Wendel, “Die Wahrschenulichkiet des Ueberstehens

von Verletzungen”, 1960.

[2]. M.

S.

Denis,

“ A

Note on the Probabilistic Method of

assessing Survivability o Collision Damage”, 1962.

31.

SOLAS,”Subdivision and damage stability of cargo

ships”, Part B- I of chapter

I

1990

[4]. SLF

43/3/2,

Development of revised

SOLAS”

Report of

the

SDS

Working Group, part A and B [Chapter 11-I],

2000.

[5].

M. Lutzen, “Damage Distributions”, Technical Report

2-22-D-01-3, EU-Project GRDI

-1

999

-1

0721

HARDER, 2001.

[6]. M. Lutzen, “Ship Collision Damage,” Ph.D. thesis

Technical University of Denmark, 2001.

[7]. MCA Maritime and Coastal Agency, “Subdivision and

Damage Stability of Cargo Ships of 80m in Length

and Over”, 1999.

-211