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8/17/2019 Sayyaadi & Tavakoli - A Probabilistic Method on Ship Damages
http://slidepdf.com/reader/full/sayyaadi-tavakoli-a-probabilistic-method-on-ship-damages 1/7
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
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
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.
1
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.
3
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.
6
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.
7
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.
4
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.
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