PREDICTION METHODS FOR CAPSIZING UNDER DEAD
SHIP CONDITION AND OBTAINED SAFETY LEVEL- FINAL REPORT OF SCAPE COMMITTEE (PART 4) Yoshitaka OGAWA*, Naoya UMEDA**, Daeng PAROKA**
Harukuni TAGUCHI*, Shigesuke ISHIDA*, Akihiko MATSUDA***,
Hirotada HASHIMOTO**and Gabriele BULIAN*** National Maritime Research Institute, Tokyo, JAPAN** Department of Naval Architecture and Ocean Engineering, Osaka University, Osaka, JAPAN*** National Research Institute of Fisheries Engineering, Kamisu, JAPANABSTRACTRegarding capsizing of ships under dead shipcondition, this paper reports experimental, numericaland analytical studies conducted by the SCAPE
committee, together with critical review of theoreticalprogress on this phenomenon. Here capsizing
probability under dead ship condition is calculated witha piece-wise linear approach. Effects of statisticalcorrelation between wind and waves are examined. Thisapproach is extended to the cases of water on deck andcargo shift. Further, experimental and theoreticaltechniques for relevant coefficients are examined. Basedon these outcomes, it also proposes a methodology fordirect assessment as a candidate for the new generation
criteria at the IMO.
KEY WORDS: Dead Ship Condition; Intact Stability
Code; Beam Winds and Waves; Capsizing Probability,
Piece-Wise Linear Approach, Water on Deck
INTRODUCTION
Stability under dead ship condition is one of threemajor capsizing scenarios in terms of performance-based
criteria, which is requested to be developed for the revisionof the Intact Stability Code (IS Code) by 2010 at theInternational Maritime Organization (IMO). It was expected
to be a probabilistic stability assessment based on physics
by utilizing first principle tools. This is owing to the factthat probabilistic approach can be linked with a risk analysis
and dynamic-based approach enables us to deal with new
ship-types without experience.
Therefore, it is important to establish a method forevaluating capsizing probability under dead ship condition
because its safety level could be a base for other capsizing

scenarios. In case of the dead ship condition, under which
the main propulsion plant, boilers and auxiliaries are not in
operation owing to absence of power, a shipmaster cannot
take any operational countermeasure. This means that
operational factors are not relevant to safety level evaluation.
Although some operational actions such as high-speed
running in following waves can decrease safety level
against capsizing, they can be avoided by operators if thesafety level under dead ship condition is ensured. In otherwords, if perfect operation is taken, the safety level during a
ship’s life-cycle could be comparable to that under deadship condition.
For the dead ship condition, the weather criterion iscurrently implemented in the IS Code and provides asemi-empirical criterion for preventing capsizing in beamwind and waves. The dead ship condition, however, does
not always mean beam wind and waves. If a ship has alongitudinally asymmetric hull form, ship may drift toleeward with a certain heading angle. Therefore, it isimportant to estimate the drifting attitude of a ship and toevaluate its effect on capsizing probability. It is also widely
known that empirical estimation of effective wave slope
coefficient in the weather criterion is often difficult to be
applied to new ship-types such as a RoPax ferry, a large
passenger ship and so forth. Therefore, the IMO (2006)
recently allows us to use model experiments for thispurpose although it is not always feasible. A simplified
prediction method is still desirable if accurate enough.
Based on the above situation, the authors presented amethodology for calculating capsizing probability of a ship
under dead ship condition. In this present methodology,
capsizing probability under dead ship condition is
calculated by means of piece-wise linear approach.
Through this paper, we summarize this methodology andits application of this methodology to the new generation
criteria at the IMO.

First, a prediction method of the annual capsizing
probability in beam seas based on a piece-wise linear
approach was presented.
Second, the methodology is verified with the Monte
Carlo simulation and is extended to the case with cargo shift,down-flooding and water on deck. The effect of several
important factors, such as drifting attitude, drifting velocity,hydrodynamic coefficient estimation and statisticalcorrelation between wind and waves are investigated.
Third, safety level in terms of capsizing probability wasassessed by means of the present approach for passenger
and cargo ships as well as fishing vessels.
HISTORICAL REVIEW
Several methods were developed for estimating
capsizing probability in stationary irregular waves. One is
the numerical experiment known as the Monte Carlosimulation, in which the expected number of capsizing isestimated by repeating numerical simulation in time domain
with different wave realizations (McTaggart and de Kat
2000). If a reliable mathematical model is available, thismethod seems to be easily applicable. However, since
capsizing probability of an existing ship is small, thenumber of numerical runs could be prohibitively large. In
addition, statistical variation of outcomes is not suitable forregulatory purposes. Sheinberg et al. (2006) attempted to
improve this approach by focusing on critical local waves.
Others are theoretical ones, in which capsizing probability
is obtained as the product of the probability of dangerous
condition and the conditional probability of capsizing. Somemethods utilize stationary linear or non-linear theory for theformer and non-linear time-domain simulation ofdeterministic process for the latter (Umeda et al. 1992; Shenand Huang 2000; Bulian and Francescutto, 2004). Shengand Huang (2000) use the Markov process for probability
density of roll and use the deterministic simulation in the
final stage. Bulian and Francescutto (2004) use a simplified
stochastic linearization technique, with a Poisson modellingof the capsize event. Their main drawback is an assumption
for the final deterministic process. Although some studies(Jiang and Troesch, 2000; Falzarno et al., 2001) for
extending the Melnikov approach into random seaways
were carried out, no direct relationship with capsizing
probability was not provided. Avoiding such a drawback,
Belenky (1993) proposed a piece-wise linear approach
where both probabilities are analytically calculated byapproximating the restoring arm curve with piece-wiselinear ones. Because of its fully analytical scheme, thepiece-wise linear approach has been utilized for practical
purpose as the most promising methodology. An approachthat has a series of similarities with the piece-wise linear
method, but that is based on a simplified statisticallinearization technique, has been developed by Bulian andFrancescutto (2004, 2006)
For a ship in irregular beam seas, Belenky derived exact
formulae based on a piece-wise linear approach without
numerical results. Then he extended his method to the casein both beam winds and waves (Belenky 1994) andprovided numerical results with his simplified method(Belenky 1995). Here the formulae relating to the windeffect were not published in detail. Recently, Iskandar et al.
(2001) applied Belenky’s simplified method to estimatedannual capsizing probability of Indonesian Ro-Ro ferry
accidents. Others (Umeda et al. 2002; Munif et al. 2004,
Francescutto et al., 2004) also applied that to estimate thecapsizing probability of fishing vessels and a large
passenger ship.
Although Belenky’s piece-wise linear method is widely
utilized, some unsolved problems remain. Firstly, complete
formulae for covering the wind effect have not been
published. Secondly, numerical results of the exact
formulae have not yet been reported for evaluating theaccuracy of the simplified formula. Thirdly, effects ofseveral elements such as wind have not been examined.
Therefore, Paroka et al. (2006) examined these
problems with deriving the detailed formulae for the case ofbeam winds and waves and calculating with both exact andsimplified formulae for a car carrier, which has a large
windage area.
CAPSIZING PROBABILITY BY MEANS OF THE
PIECE-WISE LINEAR APPROACH
The methodology adopted by the SCAPE committee forcalculating capsizing probability under dead ship conditionis based on the piece-wise linear approach proposed by
Belenky(1993). This is partly because it rigorously takes
acount of nonlinear restoring moment and its efficient
calculation load enables us to obtain annual capsizing
probability by integrating many combinations of wave
height and wave period in long-term wave statistics. As
mentioned above, Belenky’s formulation for beam waves(Belenky, 1993) and its extension to beam wind and waves
case (Belenky, 1994) had not been complete enough so that
Paroka et al. (2006) corrected them. Therefore, in this paper,
the upgraded formulation is overviewed.
kfi(φ)

Steady Wind heeling arm-φv -φCR-φm0
φm0

φCR φv

φ

Fig. 1 Piece-wise linear approximation of righting arm.
The capsizing probability here is calculated with thefollowing nonlinear and uncoupled equation of relative or
absolute rolling angle of a ship under a stochastic wave
excitation and steady wind moment. Usually the ship
motion in beam seas is modeled with equations of coupledmotions in sway and roll with wave radiation forces anddiffraction forces. Watanabe (1938), however, proposed an
one-degree of freedom equation of roll angle, φ , as follows(I xx + A44 )φ&& + B44φ& + WGZ (φ ) = M wind (t ) + M wave (t )(1)where Ixx is momentum of inertia of the ship, A44 is thehydrodynamic coefficient of added inertia, W is the shipdisplacement, B44 is the hydrodynamic coefficient of roll
damping, GZ( φ ) is the righting arm. Mwind (t) is the wind

induced moment consisting of the steady and fluctuating
wind moment and Mwave (t) is the wave exciting moment
based on the Froude-Krylov assumption. This is because theroll diffraction moment and roll radiation moment due tosway can cancel out when the wavelength is sufficiently
longer than the ship breadth. (Tasai, 1969)
The uncoupled equation of the absolute roll motion (1)can be rewritten by dividing by the virtual moment ofinertia as follows:

φ&& + 2αφ& + ω 02 kf i (φ ) = ω 02 (m wind (t ) + m wave (t ))(2).

where kfi( φ )=GZ( φ )/GM, α: roll damping coefficient,ω0 : the natural roll frequency and kf1( φ ) : thenon-dimensional righting arm, mwind (t) : the wind induced
moment consisting of the steady and fluctuating wind
moment and mwave (t) : the wave exciting moment based on
the Froude-Krylov assumption.
As shown in Fig. 1, the restoring moment coefficient inequation (2) is approximated with continuous piece-wisedlines in leeward and windward conditions as follows:

0 < φ < φ m0 range1⎫⎧ kf 0φ
kf i (φ ) = ⎨⎬⎩kf 1 (φ v − φ ) φ m0 < φ range2⎭(3)

P (H 1 / 3 , T01 , Ws , T ) =Pl PT (φ > φm 0 orφ < −φ m 0 )PA ( A > 0; φ > φ m 0 ) +Pw PT (φ > φ m 0 orφ < −φm 0 )PA ( A < 0; φ < −φ m 0 )(6).
Here Pl and PW denote the probability of up-rossing towardleeward and that of down–crossing toward windward.
Under the assumption of Poisson process, PT indicates theprobability of at least one up-crossing or down-crossing at
the border between the first and second range, φ m0 or−φ m0 . PA is the probability of diverging behavior of
absolute value of the roll angle in the second range
including the angle of vanishing stability.
These can be determined as following formulas:
PT (φ > φ m0 orφ < −φ m0 ) = 1 − exp(− (u l + u w )T )Pl =ulul + u w

(8)

uw
Pw =
ul + u w∞

PA ( A > 0;φ > φm0 ) = ∫ f ( A)dA
0

and

(7)

(9)

PA ( A < 0 ;φ < −φm0 ) = ∫ f ( A)dA
0

−∞

kf 0φ
− φ m0 < φ < 0 range1⎫⎧
kf i (φ ) = ⎨(4).
⎬⎩kf 1 (− φ v − φ ) φ < −φ m0 range2⎭
Here, kf0 and kf1 are the slope of the range 1 and range 2,
respectively. φ m0 and −φ m0 are the border between therange 1 and the range 2 in leeward and windward,
respectively. Effects of dividing methods of the righting
arm curve into piece-wise lines and the case using therelative roll angle can be found in Paroka et al. (2006).
Since Equation (2) is linear within each range, it can beanalytically solved without any problems. The bordercondition, however, should be satisfied. Therefore,
capsizing occurs when the roll angle up-croses, φm0, or
down-crosses, -φm0, and then the absolute value of roll angle
increases further. Therefore, capsizing probability can becalculated as the product of the outcrosing probability in therange 1 and the conditional probability of divergence of theabsolute value of roll angle in the range 2. Belenky (1993)
presented the following formula for the probability that aship capsizes when the duration, T, passes in stationary
waves.

P ( H 1 / 3 , T01 , Ws , T ) = 2 PT (φ > φ m 0 ) PA ( A > 0; φ > φ m 0 )(5)
Where H1/3 : the significant wave height, T01: the mean
wave period, WS: the mean wind velocity and T: duration.
Since this formula could exceed 1, Umeda et al. (2004)
proposed the following formula.

where ul and uW denote the expected number of up-crossing
and down-crossing, respectively. f(A) denotes theprobability density function of the coefficient A,, which is afunction of three random variables, namely the initial
angular velocity of roll motion, φ&1 , the initial forced roll, p1,
and the initial forced angular velocity, p&1 , in the second
range.
The probability density function of the A coefficient canbe determined from a joint probability density function ofthese three random variables, which can be calculated byusing the three-dimensional Gaussian probability density(Price and Bishop, 1974).Then, the probability density
function of the coefficient A in leeward can be described asfollows:f ( A) =∞ ∞1∫ ∫ f φ&1 , p& 1 A,φ&1 , p1 , p1 dφ&1 dp1
λ 2 − λ1 − ∞ 0

(

(

) )

(10)

That in windward can be described as follows:f ( A) =∞ 0
1∫ ∫ f φ&1 , p& 1 A,φ&1 , p1 , p1 dφ&1 dp1
λ 2 − λ1 − ∞ − ∞

(

(

) )

(11)

where
λ1,2 = −α ± ω0 2 kf1 + α 2(12).

Furthermore, Belenky (1993) simplified Equations (10)and (11) by using the fact that roll resonance in the second
range does not exist because of negative restoring slope. As
a result, f(A) can be evaluated with a single integral in placeof a double integral.
For evaluating risk of capsizing, it is necessary to
calculate annual capsizing probability with sea stateVERIFICATION AND EXTENSIONS OF PIECE-WISE
LINEAR APPROACH

In case of a ship has large angle of vanishing stability, theobtained capsizing probability could be very small but theship could capsize owing to cargo shift or down flooding
when the roll angle is larger than her critical angle. In thiscase, the calculation method mentioned above cannot bedirectly applied because angle of vanishing stability cannot
be determined.
Therefore,. Paroka & Umeda. (2006a) extend thepiece-wise linear seas approach to the case having thecritical roll angle. Here the probability of roll angle
exceeding the critical value is defined as the probability ofroll angle of the second range exceeds the critical value
when the roll motion up-crosses or down-crosses the border
between the first and the second ranges of the piece-wiselinear righting arm curve. This can be written as follows:

Comparison with the Monte Carlo simulation

P(H 1 / 3 ,T01 ,Ws ,T ) =statistics of operational water area. The formula of theannual capsizing probability, Pan, can be found in Iskandar
et al. (2000) as follows:()(

Pan = 1 − 1 − P* (T )365×24×3600 / T )

(13)

where
∞∞

P (T ) = ∫ ∫ f (H 1/ 3 , T01 ) ⋅ P(H1/ 3 , T01 , Ws , T )dH1/ 3 dT01 .
*00

Here Ws is assumed to be fully correlated with H1/3

For verifying the piece-wise linear approach, Paroka &Umeda (2006b) calculated capsizing probability of a large
passenger ship in stationary beam wind and waves andcompared it with the Monte Carlo simulation. First, they
confirmed that stochastic scattering of the Monte Carlosimulation results is comparable to its confidence interval
estimated with the assumption of binomial distribution.
Second, it was that the piece-wise linear approach coincides
with the Monte Carlo simulation in moderate sea states,
such as the mean wind velocity of 28 m/s, within theconfidence interval but underestimates it in more severewind velocity as shown in Fig.2. The results of thisunderestimation are that the assumption of the Poissonprocess for outcrossing is not appropriate when the sea stateis extremely severe. In addition, it is impossible for theMonte Carlo simulation to accurately quantify the capsizing
probability in lower wind velocity because the probability is
too small. In conclusion, it is not so easy to verify thepiece-wise linear approach but it seems not to beinappropriate to use the piece-wise linear approach for themean wind velocity of 26 m/s, which the weather criterion
deals with, but there is a room for further improvement.
Wind Velocity (m/s)
28.0

28.5

29.0

29.5

30.0

30.5

31.0

Capsizing probability

1.0E+00

1.0E-01

1.0E-02

1.0E-03

Numerical Experiment
+Err
-Err
Present Method

1.0E-04

Figure 2. Comparison of capsizing probability of thelarge passenger ship between the piece-wise linear
approach and the error tolerance of Monte Carlosimulation. (Paroka & Umeda, 2006b)
Effect of critical angle for down-flooding or cargo shift

Pl PT (φ > φ m0 orφ < −φ m0 )PCR (φ > φ CR ;φ > φ m0 ) + (14)Pw PT (φ > φ m0 orφ < −φ m0 )PCR (φ < −φ CR ;φ < −φ m0 )where PCR is the conditional probability of roll angle
exceeds the critical value, φCR .
The conditional probability of roll angle exceeds thecritical value is calculated with assuming that effect of
exciting moment on the roll motion in the second range isnegligibly small. This assumption is based on the fact that
resonance is impossible in the second range owing to its
negative stiffness. Therefore the roll motion in this range
depends only on the initial condition, which is the same as
the roll motion at the end of the first range. The solution of
the roll motion equation in the second range can be written
as follows:

φ2 (t ) = A exp(λ1t ) + B exp(λ2t ) + (φv − φD )

(15)

where

λ1, 2 = −α ± − α + ω 02 kf1
φD =

(16)

m0

ω 02 kf1
1
A=
φ& − λ {φ − (φ − φ )}(λ1 − λ2 ) 02 2 m0 v D
1B=
λ {φ − (φ − φ )} − φ&02
(λ1 − λ2 ) 1 m0 v D

(17)

[

]

(18)

[

]

(19)

Here φ&02 indicates the initial angular velocity of roll motionin the second range. m0 indicates the non-dimensional
steady wind moment. In order to investigate whether or not
the roll angle exceeds the critical value, the following
iteration procedure is proposed. Firstly the initial angularvelocity of roll motion in the second range with negative A
coefficient in the equation (18) is arbitrarily choose.
Secondly the time of the angular velocity of roll motion
becomes zero is calculated as follows:tm =⎛ Bλ2 ⎞
1⎟ln⎜⎜ −(λ1 − λ2 ) ⎝ Aλ1 ⎟⎠(20)Then the roll angle with the time, tm , can be calculated
by means of the equation (15). If this roll angle is smaller()

P φ&02 > φ&CR =∞2⎧⎪ φ&02 ⎫⎪

&
exp⎨
⎬ dφ
2πD ∫
⎪⎩ 2 D ⎪⎭
q& &
φ

CR

q&

02

(21)

where φ&CR indicate the critical value of the angularvelocity in the second range. The probability of roll angle
exceeding the critical value in a certain exposure time canbe calculated by means of the equation (14) with theconditional probability of roll angle exceeds the criticalvalue is replaced by the equation (21).
Numerical results of the car carrier for several criticalangles ranging from 55 degrees to 75 degrees are shown in
Fig. 3 (Paroka et al., 2006c). The probability of dangerous
condition increases when the critical angle decreases. Thisis because the roll angle can exceed the critical value even if
the roll motion in the second range does not diverge. The
difference disappears when the unstable heel angle owing tosteady wind moment is smaller than the critical value. Whenthe unstable heel angle is smaller than the critical value, theroll angle can exceed the critical value only if the rollmotion in the second range diverges.

trapped on deck have been published (e.g. Belenky, 2003)
however its effect to capsizing probability has not been
investigated.
Paroka & Umeda (2006c) extended the piece-wise linear
approach to the case with trapped water on deck. Here theamount of trapped water was estimated with modelexperiments (Matsuda et al., 2005) by measuring heel andsinkage of the model in irregular beam waves. Water
ingresses and egresses through the bulwark top or freeing
ports and the balance of these can be determined duringb
the experiment. As a result, it was found out that the trappedwater on deck could increase the capsizing probability forthis ship type as shown in Fig. 4. This is because the trappedwater can significantly reduce the mean restoring arm up to
the angle of immersion of the bulwark top, which isapproximated with piece-wise linear curves as a function of
stationary sea state..
As a next step, Paroka & Umeda (2007) incorporated a
numerical model for describing water ingress and egresswith a hydraulic model into the piece-wise linear approach.
As a result, capsizing probability with water on deck can becalculated without a model experiment. Here the dynamic
effect of trapped water on deck such as change in addedmass, damping coefficient and exciting moment are ignored
for simplicity’s means.
GM without Water on Deck (m)
0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1

Long-term Capsizing Probability

than the critical one, the initial angular velocity is increased
and we repeat these processes. When the maximum rollangle has been larger than the critical value, the iteration is
stopped and the initial angular velocity is set as the criticalangular velocity. The probability of roll angle exceeds thecritical value then can be calculated as the same as theprobability of the initial angular velocity of roll motion
higher than the critical angular velocity. The probability of
the initial angular velocity larger than the critical angularvelocity can be calculated as follows:

1E-5

1E-10

1E-15

1E-20

Without Water on Deck
With Water on Deck

1E-25

1E-30

Fig. 4 Long-term capsizing probability with and withoutwater on deck for the purse seiner operating off Kyusyu.
(Paroka & Umeda, 2006c)
Effect of drifting attitude and velocity

Fig. 3 Dangerous probabilities of the car carrier for different
critical roll angles (Paroka & Umeda., 2006a).
Effect of water on deck
For smaller ships such as fishing vessels, the effect oftrapped water on deck should be taken into account. This isbecause these ships have smaller freeboard but higher
bulwark comparing with their ship size. As a result, whenshipping water occurs, water can be easily trapped bybulwark. Although a certain amount of water can flow out
through freeing ports, the balance between ingress andegress can be trapped on deck. The righting arm candecrease because of the water trapped on deck. Manyexperimental and numerical works about effect of water

The dead ship condition does not always mean beamwind and waves. If a ship has a longitudinally asymmetrichull form, ship may drift to leeward with a certain heading
angle. Therefore, it is important to estimate the drifting
attitude of a ship and to evaluate its effect on capsizing
probability.
X
U

Wind

x
χ
Y

β
y

0
ψ

Wave
y

φ

Y
0'

z
Z

Fig. 5 Co-ordinate systems

(m + m y )v& + (m + mx )ur − m y l yφ&& = YH + YA + YW

(23)(I z + J z )ψ&& = N H + N A + NW
(I x + J x )φ&& − m y l y v& − mx l x ur + 2μφ& + W ⋅ GMφ

(24)
(25)

= KH + KA
whereu,v ; ship speed in x-axis and y-axis,
ψ , φ ; yaw and roll angle,
m ; ship mass,
W ; displacement,
GM ; metacentric height,
mx, my; added mass in x-axis and y-axis,
Ix, Iz; moment of inertia around x-axis and z-axis,
Jx, Jz ; added moment of inertia around x-axis and z-axis,
lx, ly; vertical position of centre of mx,and my
X, Y; external forces in x-axis and y-axis
N, K; yaw and roll moments

Here the subscript H, A and W means hydrodynamic
force/moment, wind force/moment and wave-induced
drifting force/moment.
In this model, a mathematical model for hydrodynamic
forces under low speed manoeuvring motion (Yoshimura,1988), an empirical method for wind forces (e.g. Fujiwara,
2001), and a potential theory for wave-induced drifting
forces (Kashiwagi, 2003) are used.
The Newton method was used to identify equilibria of theequations. As an initial input, the equilibria in the beamwind and waves from starboard and those from port side arecalculated. And then, equilibria for different angle between
the wind and waves, χ , are traced. The calculation aredone in two ways: the first one is that χ changes from 0
degrees to 180 degrees (forward) with the wind direction
fixed, and the second is that χ changes from 180 degrees
to 0 degrees (backward). Stability of equilibrium isevaluated by calculating the eigenvalues of the Jacobean
matrix in the locally linearised system around equilibrium.
The example of results is shown in Fig 6.
These figures indicate that when the relative angle
between wind and waves is 0 degrees, the ship is drifting
leeward in beam wind and wave. Having the relative angle
between wind and waves larger from 0 degrees, the heading
angle gradually increases from the beam wind.
Ogawa et al.(2006b) examined the present method
through the comparison with measured drifting motions. Fig.
7 shows the sway velocity in wind and waves as a functionof wind speed. In this calculation, the significant wave
height and mean wave period were of the same as in theexperiments (H1/3 = 9.5 m, T01 = 10.4 sec). Only the windspeed was varied in the present calculation. It is found thatheading angle (degrees)

(22)

330
300
270
240
210
180
150
120

forward(stable)
forward(unstable)
backward(stable)
backward(unstable)

90
60
30
0
0

30

60

90

120

150

180

angle between wind and waves (degrees)Fig. 6 Heading angle of fixed points as a function of relative
angle between wind and waves (wind speed is 26 m/s, wave
amplitude is 2.85m and wave length ratio is 2.94). (Umeda
et al., 2006b)
Drift speed due to wind & wave
Cal.( wind only)
Cal. (wave only(H1/3=9.5m, T01=12.4sec))Cal.(wind&wave(H1/3=9.5m, T01=12.4sec))Exp.(wind only)Exp.(wave only(H1/3=9.5m, T01=12.4sec))Exp.(wind&wave(H1/3=9.5m, T01=12.4sec))Cal.(Beaufort)

3.5
3
2.5
Drift speed (m/s)(m + m x )u& − (m + m y )vr = X H + X A + X W

360

2
1.5
1
0.5
0
0

5

10

15

20

25

30

35

Wind speed (m/s)

Fig. 7 Relation between the wind speed and the drift speed
in wind and waves. (Ogawa et al., 2006b)
0

20

40

60

80

100

1.00E+00

capsizing probability (case1)

For estimating drifting velocity and attitude of a ship in
regular waves and steady wind, Umeda et al. (2006b,
2007a) proposed a method for identifying equilibria of asurge-sway-yaw-roll model and for examining its local
stability. Equilibria can be estimated by solving 4-DOF
non-linear equations in the following co-ordinate systems
shown in Fig. 5;

the computed drift speed is close to the measured drift speed,
although the drift speed in beam waves only (wind speed =0 in Fig. 7) was underestimated. It is confirmed that present
method is useful for the quantitative computation of thedrift speed under dead ship condition.

1.00E-04
1.00E-08
1.00E-12
1.00E-16
1.00E-20
1.00E-24
1.00E-28
1.00E-32
1.00E-36

angle between wind and waves (degree)

Fig.8 Capsizing probabilities under two coexisting drifting
conditions as functions of relative angle between wind andwaves. (Umeda et al., 2006b)
An example of capsizing probabilities under two
coexisting stable steady drifting states are shown in Fig 8.
Umeda et al. (2006b, 2007a) Here the mean wind velocity is
26 m/s and the duration is one hour. The fully developed
wind waves under this wind velocity are assumed. In thiscalculation, the effect of relative angle to wind is modelled
as the change in lateral windage area, the effect of relative
angle to waves is done as the change in effective wave slope
coefficient and the effect of drifting velocity is done as thechange in the exciting frequency. In addition, the heelinglever due to drift are estimated with the model tests by
Taguchi et al. (2005). It is confirmed that the mostdangerous condition for this subject ship under dead shipcondition is one in beam wind and waves, i.e. χ is 0
degrees. It is clarified that drifting motion obviously
depends on underwater and above-water ship geometry.

simultaneous equation for determining two dimensionalflows, however, a further simplified method is expected to
be developed. For this purpose, Sato et al. (2007) carriedout a series of model tests by means of two dimensionalmodels covering large breadth to draft ratios. They
confirmed that measured effective wave slope coefficient
could be much smaller than that from a strip theory in caseof very large breadth to draft ratios. This fact coincides withthe experiments by Ikeda et al. (1992) for a small
high-speed craft.

Effect of hydrodynamic coefficient estimationroll amplitude (deg)

25
20
15
10
5
0

0.7

0.8

0.9

1

1.1

1.2

1.3

tuning factor
Fig. 9 Roll amplitude of the RoPax ferry in regular beam
waves measured with different constraining methods(Umeda et al., 2006a).
30
25roll amplitude (deg)It is also widely known that empirical estimation ofeffective wave slope coefficient in the weather criterion isoften difficult to be applied to modern ship-types such as alarge passenger ship, a RoPax ferry and so forth. Based on
this background, the IMO allows us to estimate the effective
wave slope and damping coefficients as well as heelinglever owing to beam wind by model experiments with theinterim guidelines (2006).
Taguchi et al. (2005) and Ishida et al. (2006) carried out
almost full set of model tests for a RoPax ferry and verified
the interim guidelines as the alternative assessment of theweather criterion. It was clarified that the evaluation of
weather criterion considerably differs that with the
combination of tests. Umeda et al. (2006a) applied thismodel test approach to the Ro-Pax ferry for further
investigating its feasibility. Based on the interim guidelines,
the model experiments were executed with three different
constraints: a guide rope method, a looped wire system anda mechanical guide method. The guide rope method meansthat the model was controlled by guide ropes. The looped
wire system means that, by utilizing a looped wire attached
to a towing carriage, the yaw motion of the model is fixed
but the sway, heave and roll motions are allowed. The
mechanical guide is the combination of a sub-carriage, aheaving rod and gimbals. Among them, as shown in Fig.9,
the looped wire system is less accurate because natural rollperiod and damping was slightly changed owing to a looped
wire. In case the yaw angle increases with larger wave
steepness, the guide rope method has inherent difficulty forkeeping a beam wave condition. The guideline allows thethree different methods for determining the roll angle fromthe experiment: the direct method, the three step method
and the parameter identification. As shown in Fig 10, thedifference between the three methods is negligibly small. In
conclusion, the interim guidelines can guarantee reasonableaccuracy if appropriate constraint is used.
When we calculate capsizing probability with thepiece-wise linear approach, the same difficulty exists. Thisis because fluid-dynamic coefficients such as effective wave
slope coefficient should be separately estimated in advance.
Therefore it is desirable to execute model experiments
following the IMO interim guidelines. Utilizing model tests,
however, could be not always practical owing to cost andtime. Therefore, it is desirable to develop an empirical or atheoretical method to accurately predict obtained resultsfrom the model test.
It is well known that a strip theory can explain theeffective wave slope coefficient in general. (Mizuno, 1973)
Since a strip theory requires numerical solution of

Guide Rope
Looped Wire System
M echanical Guide
NM RI (Guide Rope)

s=1/60

25.22

26.56

26.18

three step methodparameter identificatin

20
15
10
5
0
direct method

Fig. 10 Comparison of estimated roll amplitude by meansof three different estimation methods. (Umeda et al., 2006a)
Umeda & Tsukamoto (2007c) investigated several
possible simplified versions of theoretical methods andfinally proposed a formula to calculate the effective wave
slope coefficient of the equivalent rectangular sections with
the Froude-Krylov component on its own. Here the local
breath and area of the rectangular section should be thesame as those of original transverse ship sections. As shownin Table 1, this simplified formula agrees well with theexperiment and a strip theory for existing ships having large
windage areas probably because the diffraction component
can almost cancel out the radiation owing to sway and the
local breadth and area are responsible for restoring moment.
Furthermore, they indicate that difference in the effective
wave slope coefficient between the present formula andexperiments results in only small effects on the estimationof the one-hour capsizing probability in stationary beam

wind and waves as shown in Fig.11. Therefore, it isrecommended to use this simplified formula for a regulatory
purpose.
Table.1 Comparison of effective wave slope coefficients
under resonant condition. Here IMO means the formula in
the IMO weather criterion and FK indicates the
Froude-Krylov prediction. (Umeda & Tsukamoto, 2007c)

19

21

23

25

27

29

∞∞∞
P* (T ) = ∫ ∫ ∫ f (H 1 / 3 ,T01 ,Ws )000⋅ P(H 1 / 3 ,T01 ,Ws ,T )dH 1 / 3 dT01 dW s

(27)

Ogawa et al. (2006a) calculated the annual capsizing
probability of the RO-PAX ferry. Fig. 12 shows her rightingarm (GZ) curves. The limiting KG of this ship is governed by theweather criterion so that it was used as a loading condition for thesafety assessment. The approximated line of the righting arm curve,of which GM, the area under the GZ curve and the angle ofvanishing stability are not changed, is also shown in Fig. 12.

Fig. 13 shows the capsizing probabilities within one year
as a function of wind speed. In addition to the capsizing
probability by means of the present method, two kinds ofcapsizing probability are calculated by means of two otherwave scatter diagrams. One is the wave scatter diagram
normalized for each mean wind velocity (Normalized) andthe other is the wave scatter diagram without correlation ofwind (No correlation). The wave scatter diagram without
correlation with wind was conducted by summing up thepresent scatter diagram of wave height, wave period andwind speed. Then the capsizing probabilities using two
kinds of diagrams is defined as follows:

31
33
35
Wind Velocity [m/ｓ]1.0E-01
Probability

1.0E-03
1.0E-05(Normalized)

1.0E-07
1.0E-09∞∞
P* (T ,W s ) = ∫ ∫ f N (H 1 / 3 ,T01 ; W s )1.0E-11
1.0E-13

IMO

1.0E-15

Strip theory

1.0E-17
1.0E-19

0 0⋅ P(H 1 / 3 ,T01 ,W s ,T )dH 1 / 3 dT01

EXP
simplified Froude-Krylov

Fig.11 The effect of the effective wave slope coefficientestimation on the one-hour capsizing probability of the
RoPax ferry in stationary beam wind and waves. (Umeda etal., 2007a)
Effect of correlation between wind and waves
For the provision of the criteria based on
performance-based approaches, the capsizing probability
should be assessed preferably with long-term sea statestatistics. Ogawa et al. (2006a,2007) constructed a scatter
diagram of wave height, wave period and wind speed bymeans of the wind and wave database developed fromhindcast calculation of wind-wave interactions andexamined the effect of the correlation of winds with waves
on the long-term capsizing probability. Table 2 shows
examples of the scatter diagram of wave height, wave
period and wind speed for a water area near Minami Daito
Islands in the Northern Pacific. It is found that theprobability of large wave height and long wave period
increases as the wind speed increases.
The long-term probability of capsizing of a ship within
N years can be described as follows:
PN = 1 − ⎛⎜ 1 − P* (T )⎞⎟
⎝
⎠

where

(N × 365 × 24 × 3600 ) / T

(26)

(28)

(No correlation)∞∞
P* (T ,W s ) = ∫ ∫ f NC (H 1 / 3 ,T01 )0 0⋅ P(H 1 / 3 ,T01 ,W s ,T )dH 1 / 3 dT01

(29)

where fN(H1/3, T01; Ws) denotes the normalized wave scatter
diagram and fNC(H1/3, T01) denotes the wave scatter diagram
without correlation with wind. For comparison sake, theshort-term probability in the constant wind and waves isalso shown in Fig. 13.
It is found that capsizing probability of the RO-PAX
ferry complying with the weather criterion is adequately
small if the statistical correlation between wind and waves
is taken into account. This probability does not depend on
the mean wind velocity because the occurrence probabilityof wind is taken into account in the calculation. The
capsizing probability with the normalized scatter diagram islarger than the present method when the mean wind velocity
is larger 16 m/s. This indicates that occurrence probabilityof such strong wind is not so large in this water area during
a year. The capsizing probability with the no correlation
scatter diagram is larger than the present method. This isprobably because unrealistic combinations of wind andwaves are included in the calculation.
The capsizing probability taken from the stationary sea
state is larger than the present method because theoccurrence probability of such sea state is not large during ayear in this water area. On the other hand, the capsizing
probability from the stationary sea state of the mean windvelocity of 26 m/s is much smaller than the present method.
This suggests that the occurrence probability of sea stateshould be taken into account when we estimate the annual
risk from the short-term capsizing probability.

Capsizing probability (T=3600sec.)
0

5

10

15

20

25

30

35

0.0

Ws=2(m/s) T(sec.)
Hw(m)311.2510.7510.259.759.258.758.257.757.256.756.255.755.254.754.253.753.252.752.251.751.250.750.250-

4-

5-

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

Ws=20(m/s) T(sec.)
Hw(m)5615.250
14.75014.25013.75013.25012.75012.25011.75011.25010.75010.250
9.750
9.250
8.750
8.250
7.750
7.250
6.750
6.250
5.750
5.250
4.750
4.250
3.750
3.250
2.750
2.250
1.750

Ws=40(m/s) T(sec.)
Hw(m)8915.250
14.75014.25013.75013.25012.75012.25011.75011.25010.75010.250
9.750
9.250
8.750
8.250
7.750
7.250

67891011120
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0.000297 0.000224
000
0 0.000414 0.000405
00
0 0.000176 0.000589 0.001021 0.000847 3.91E-06
0
0 0.000887 0.001245 0.002002 0.002682 5.05E-05
0 1.1E-05 0.001218 0.004456 0.004191 0.001685 0.000424
0 0.000378 0.001439 0.001877 0.000579 9.28E-05 1.9E-05
0 0.000101 4.15E-05 3.73E-05 3.91E-06 1.86E-07 1.86E-07
0
0
0
0
0
0
0

130
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

140
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

150
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

789101112131415160
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 3.73E-07 3.17E-06
0
0
0
0
0
0
0
0
0 3.35E-06 1.86E-07
0
0
0
0
0
0
0
0
0 1.12E-06
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 7.83E-06 4.84E-06
0
0
0
0
0
0
0
0 6.71E-06 8.24E-05
0
0
0
0
0
0
0
0
0 1.58E-05 0.00011 7.45E-07 3.91E-06
0
0
0
0
0
0 2.61E-06 7.32E-05 6.17E-05 5.78E-06 3.5E-05
00000 1.3E-06 1.01E-05 0.000168 4.84E-06 6.15E-06 1.3E-05
0000
0 9.69E-06 4.45E-05 0.000351
0
0
0
0
0
0
0 2.24E-06 4.49E-05 0.000201 9.82E-05 1.49E-06
0
0
0
0
0
0 4.47E-06 7.84E-05 0.000212 0.000119 7.27E-06 1.45E-05
0000
0 5.33E-05 0.000301 0.000302 0.000128 4.1E-06 3.82E-05
00
0
0 5.03E-06 0.000106 0.00066 1.29E-05 6.28E-05 8.57E-06 6.15E-06
00
0
0 3.78E-05 9.69E-05 0.000248
0 3.73E-06 1.86E-07
00000 5.29E-05 7.81E-05 2.5E-05
0 5.59E-07 4.84E-06
00
0
0 2.96E-05 1.9E-05 3.63E-05
0
0
0
0
0
0
0
0 1.58E-05 2.83E-05 3.06E-05
0
0
0
0
0
0
0
0
0 2.24E-06
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

100
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

110
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

12130
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 7.45E-07
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

141516170 1.86E-07
000
0 1.49E-06
0 9.32E-07 5.59E-07
00
0 4.66E-06
00
0
0 9.32E-06
00
0
0 9.5E-06
00
0 5.03E-06 3.17E-06
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

1.4
Original GZ (kg=10.63m)
Present method

1.2

GZ(m)

1
0.8
0.6
0.4
0.2
0
0

10

20

30

40

50

60

70

80

Angle of Heel(deg.)

Fig. 12 Righting arm curves of the RO-PAX ferry. (Ogawa
et al., 2006a)
The capsizing probability calculation in this paper
except for this section assumes that wind is fully correlated
wave height based on the WMO’s reference data (Umeda etal., 1992). It will be a future task to compare thismethodology with the capsizing probability calculated
taking account of the statistical correlation between wind
and waves. Then the final conclusion whether the statisticalcorrelation should be considered for the performance-based
criteria or not will be discussed.

-5.0
log10(probability)

Table.2 Samples of wave height –period diagram in eachwind speed (Autumn, 1994-2003, Ws=2, 20, 40(m/s)).
(Ogawa et al., 2006a)

-10.0
-15.0
-20.0
-25.0
-30.0
-35.0

Present method
Beaufort(short term)
Normalized
No correlation-40.0

Wind speed(m/s)

Fig. 13 Capsizing probability in one year of the Ro-PAX
ferry under dead ship condition. (Ogawa et al., 2006a)
ASSESSMENT OF SAFETY LEVEL IN TERMS OF THE
PIECE-WISE LINEAR APPROACH
Passenger and cargo ships
Umeda et al. (2007a, 2007b) attempted to evaluate thesafety levels of four passenger and cargo ships by utilizing
the above mentioned capsizing probability calculation.
Their principal dimensions of these ships are shown in
Table 3. The method for calculating capsizing probability
here is the piece-wise linear approach with down-flooding
angle taken into account. The mean wind velocity is
assumed to be fully correlated with the wave height and theeffect of drifting attitude and velocity is ignored. The
restoring arm is approximated to keep the metacentricheight, the angle of vanishing stability and the maximum
restoring arm unchanged.
The calculation results of the annual capsizing
probabilities under the designed conditions in the North
Atlantic are shown in Fig. 14. Here the annual capsizing
probabilities of a LPS (large passenger ship) and acontainership under their designed conditions are smaller
than 10-8 so that the associated risks are assumed to be
negligibly small. In particular, the safety against capsizing
in beam wind and waves for the containership is sufficientalthough her designed metacentric height is small. This isbecause the large freeboard of the containership results insignificant increase of the slope of righting arm curve atlarger heel angle. On the other hand, larger probabilities are
obtained for the Ro-Pax and the PCC (pure car carrier). In
particular, the Ro-Pax has large annual capsizing probability,
although her number of persons onboard is not small,because her natural roll period is small so that resonant
probability increases in ocean waves.
The Ro-Pax ferry investigated here is operated not in the
North Atlantic but in Japan’s limited greater coastal area,
which is water area within about 100 miles from theJapanese coast. (Watanabe and Ogawa, 2000) In addition,
her navigation time ranges only from 6 to 12 hours so that
she can cancel her voyage if bad weather such as a typhoon
is expected. Therefore the calculation of capsizing
probabilities of the case in Japan’s limited greater coastal
area and the case in Japan’s limited greater coastal area with
operational limitation to avoid the significant wave heightgreater than 10 meters are executed and compared with thecase in the North Atlantic, as shown in Fig. 15. Here theaero- and hydrodynamic coefficients were estimated withexperimental data.
The comparison indicates that the effect of water areas is
not so large because worst combinations of significant wave

height and mean wave period exist also in Japan’s limited
greater coastal area. This seems to be reasonable because
typhoons are included in the wave statistics in Japan’s
limited greater coastal area.
Table.3 Principal dimensions of ships under designed
conditions. Here W: displacement, Tφ: natural roll period,
N: number of persons onboard. (Umeda et al., 2007b).
Container
PCC
RO/PAX
Items
LPS
Ship
W (t)
53,010109,225
27,216
14,983
Lpp (m)B (m)d (m)GM (m)

242.25
36.0
8.4
1.58

283.8

192.0

42.8
14.0

Units

new
135 GTold 135
GT

80 GT

39 GT

Lpp

Meters

38.50

34.50

29.00

23.00

BS

Meters

8.10

7.60

6.80

5.90

d

Meters

2.851

2.65

2.25

1.77

FBS

Meters

0.485

0.496

0.484

0.396

Δ

Tons

480.98431.07276.61145.70GM

Meters

1.95

1.65

1.57

1.79

6.06

Tφ

Seconds

5.80

5.87

5.20

3.886

Tz

Seconds

3.413

3.435

3.246

2.218

1.41

hB

Meters

1.50

1.45

1.35

1.30

Af

Meters2

7.05

4.75

3.40

2.60

25.0

8.2

1.06

Items

170.0

32.3

1.25

Table. 4 Principal dimension of the subject ships at fullload departure condition. Here FBS: freeboard, TZ: natural
heave period, hB: height of bulwark and Af: area of freeing
port. (Paroka & Umeda, 2007)

T (s)

23.0

30.3

22.0

17.9

N

4332

33

25

370
0 .8

1.0E+00

0 .7

PROBABILITY

1.0E-04

4.98E-05

3.66E-05

Freeboard (m)0 .6

1.0E-02

0 .5
0 .4
0 .3

N ew 13 5 G T
O ld 13 5 G T
80 G T
39 G T0 .2
0 .1

1.0E-06

0 .0
0 .0

0 .2

0 .4

0 .6

1.0E-10
RO/PAX

1 .0

1 .2

1 .4

1 .6

1 .8

2 .0

LPS

CONTAINER SHIP

PCC

Fig. 16 Marginal lines of purse seiners for the annual
capsizing probability of 10-6. (Paroka & Umeda, 2007)

Fig. 14 Annual Capsizing Probability for passenger andcargo ships in the North Atlantic. (Umeda et al., 2007b)

0.8
0.7
0.6

3.66E-05
1.00E-05

Freeboard (m)1.00E+00

PROBABILITY

0 .8

M e tace ntric H e ight (m )1.0E-08

1.23E-05

0.5
0.4

New 135 GT
Old 135 GT
80 GT
39 GT

0.3
0.2
0.1

1.00E-10

2.92E-12

0.0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

M etacentric Height (m)1.00E-15

1.00E-20
NORTH ATLANTIC

JPN(Limited greater coastal area）

JPN(Limited greater coastal
area+Operational limitation）

Fig. 15 Effect of operational conditions on annual capsizing
probability for the RO/PAX. (Umeda et al., 2007b)
On the other hand, the significant reduction in thecapsizing probability is realized by the operational
limitation. This explains that the Ro-Pax ferry is safely
operated in actual situations with the aid of the operational
limitation to avoid bad weather.
Fishing vessels

Fig. 17 Marginal lines of purse seiners for the IMO
weather criterion for fishing vessels. (Paroka & Umeda,
2007)
Paroka & Umeda (2007) estimated the safety levels offour Japanese purse seiners, whose principal dimensions arelisted in Table 4, as one of typical fishing vessels types. Byutilizing the piece-wise linear approach with water trappedon deck taken into account, combinations of metacentricheight and freeboard realizing the annual capsizing
probability is 10-6 are identified as shown in Fig. 16. The
water areas used here are actual fishing grounds within theJapanese EEZ for these vessels so that wave statistics areobtained from the NMRI database and actual fish catchrecords (Ma et al., 2004) with assumption that wind velocity
is fully correlated with the wave height. This diagram
indicates that the designed full departure condition for each
vessel has safety level above 10-6. Combinations ofmetacentric height and freeboard for marginally complying
with IMO weather criterion for fishing vessels and thewater-on-deck criterion in the recommendation of theTorremolinos Convention are shown in Figs.17-18,
respectively. The comparison among these results
demonstrates that the weather criterion for fishing vessels
on its own is not sufficient for ensuring safety of thesefishing vessels because of trapped water on deck but thewater-on-deck criterion in the Torremolinos Convention iscomparable to the current probability-based calculation.

2.

3.

calculated probability coincides with the Mote Carlosimulation results within its confidence interval for thesea state relevant to the weather criterion
The calculation method was extended to cover effectsof flooding angle or cargo shift angle, water on deck,hydrodynamic coefficient estimation, drifting attitudeand velocity and the correlation between wind andwaves.
By utilizing the above extended methods, safety levelsof passenger and cargo ships as well as fishing vessels
are calculated as annual capsizing probability, andshow that current designs ensures reasonable levels ofsafety against intact capsizing under good operation
practice.

ACKNOWLEDGEMENTS
0 .8
0 .7

Freeboard (m)

0 .6
0 .5
0 .4
0 .3
0 .2

New 135 GT
O ld 1 3 5 G T
80 GT
39 GT

0 .1
0 .0
0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

1 .8

2 .0

M e t a c e n t r ic H e ig h t ( m )Fig. 18 Marginal lines of purse seiners for thewater-on-deck criterion in the IMO Torremolinos
Convention. (Paroka & Umeda, 2007)
RECOMMENDATION FOR IMO Performance-BASEDCRITERIA
As shown in this paper, it is feasible to calculate annual
capsizing probability of a given ship in beam wind andwaves by utilizing the piece-wise linear approach presented
here. Since a beam wind and wave condition is moredangerous than other drifting attitude, safety level ofstability under dead ship condition at least as a relative
measure can be estimated with this approach. It isnoteworthy here the calculation based on this approach is
not time-consuming so that it can be used as aperformance-based criterion. In case a simpler approach isalso required for a vulnerability criterion, as an existing
criterion for stability under dead ship condition the weather
criterion can be recommended with refinement with thepresent achievements such as effective wave slope
coefficient and roll damping.

CONCLUSIONS
The activity of the SCAPE committee for stability underdead ship condition provides the following conclusions:
1. A piece-wise linear approach for calculating capsizing
probability in beam and waves is presented. The

A part of the present study was carried out in cooperation
with the Japan Ship Technology Research Association
through the part of the Japanese project for the stability
safety (SPL project) that is supported by the Nippon
Foundation. This research was also supported by a Grant-in
Aid for Scientific Research of the Japan Society for
Promotion of Science (No. 18360415). The authors thankDrs. Y. Sato and M. Ueno, Misses E. Maeda and I.
Tsukamoto, Messrs. J. Ueda, S. Koga, H. Sawada and H.
Takahashi for their contribution to the works described inthis paper.
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