Safety for damaged vessels as probability of non-capsizing in follwing seas

SAFETY FOR DAMAGED

_{VESSELS AS PROBABILITY}

OF NON-CAPSIZING IN

_{FOLLOWING SEAS}

Accident reccxds refer many causes leading to possible capsizing and among them the dangerous

condition when a ship sailing in a longitudinal seaway is subject to the action of damage floodwater.

Nowadays there are neither international statutccy regulations ncr' reliable guidelines to facilitate the

selection of data f- modelling flooding scenarios and

to evaluate the level of risk resulting from damage.

So far, flooding mechanism simulated in available

damage stabllity _{computer píogams is essentially}

based on a static approach and searches fcr static

equiliium both at intermediate and at final stages of

flooding. Such a physical modelling is only a rou approximation of the real phenomenon since it is

sufficient to consider the intrinsic phase lag between

restcxing fc*ces and oscillations to remind that a damaged ship, also in calm water condition, cannot be

considered in instantaneous hy&ostatic equiliium. It has been verified that the static approaches usually

applied to assess damage stabllity [16,181, induding

the 1MO probabllistic one, cannot explain the most p&t

of capsizing accidents.

Damaged ship suvivablity is a very difficult ta'get

to reach also because of the unsuffident knowledge

Giorgio Trincas ()

ABSTRACT

() lnstttute of Naval Architecture,Unlverslty_{of Trieste, vi Yalerto tO, 31127 TSIESTE, Ithty}

637

-TECHpJJsCIg

_UNIVERSffET

.aboratorium voor

$cheepshyd orne hanica

Archlef

Mekeiweg Z 2628 CD

_D&ft

1e O1Ö. 7887

- Faxa O18-181e3

Analysis of damage stality di.ring intermediate stages of flooding ¡s a very imp«tant item to fcresee dangerous situations which _{may indice ship capsizing. The damage scenario is} simulated through a simpüfied deterministic model which reflects the dynamics of the ship. A time-domain simulation fct- computing ship motions coupled with sioshing effects ¡n damaged compartments is developed to monitci the histcy of _{ansient flooding. Variations of static and} hy&odynamic characteristics of the ship and f1oodiater as well are taken into _{account at each}

small step in time.

To determine the non-capsizing probability _{duing the flooding elapsed time of}

a ro-ro passenger ferry, initially at rest in a following_{sea, the Bayes' fccmula fcc the entre probability is}

applied by computing the probability _{distribution of a wave goup excitation and the conditional}

probability of the dangerous event.

MODELLING OF FLOODING IN A SEAWAY

of the

dynamics and hyd-odynamics of flooding

phenomenon. The effect of floodwater on the transient

response of

the ship has not been

investigated

extensively up to now. Flooding transients can play a &amatic role in damaged stability since it is sufficient

one excessive excu-sion to have the system faiftre.

Mcceover, su-vivaI capability ¡n the event of damage

should not be assessed through _{p&re deterministic}

approach, but from the statistics of the shipresponses obtained from

a time

simulation of fully coupled

equations of motion. The necessity of introducing statistics derives from the randomness and uncertainty

of meaningful ship _{parameters depending also} n,

operative conditions (centre of g-avity, freeboard), of

extent and position of damage, of three-dimensional permeability distribution, of sea state,

of floo<ater

dynamics, and of initial kinematic conditions.

Because of the complex and stochastic _{natue of a} damage scenario, in the benning we prefer to model

the phenomenon by

means of a

simulatíon that

desc-ibes flooding merely ttrough fluid mechanisms

based on fcronomia methods accccdng to Bernoulli's

the<xy of flow [12] and is indusive of the effects of sioshing in the damaged compalments. The flooding evolution is descibed if-i-ough _{a series of small time} steps. The hyd-aulics of the mechanism operates ¡n

ui

we rcy

u uiø Ui SdLIC cdcualO(ìS Kl intermediate stages of flooding, but without searching lcr eqilibiium positions. At each point of time

hy&ostatic and hy&odynamic chac1eristics of the instantaneous wetted strface of the ship are computed

ori the g'ound of the values of volume, heel, and im at previous time step. Such a mechanism easily permits to continuously monitor the inemental

variation in ship motions during the course of flooding.

The study presented here is pail of a research

prog-amme on dama e stability desianed to be

developed in prog-essive stages, the final goal being both the improvement of existing rules lcr capsizing prevention and the development of guidelines that can help the desicner to limit the risk of exceeding some roll ane considered dangerous for survivability. The

time-domain flooding simulation could be a useful tool for different design purposes. lt should be used to

determine the ship behaviour at each damage case in combination with a probabilistic assessment of the various eventual cases of accident, to predict the

probable time until the vessel sinks or until a heel angle is reached at which the lifesaving gear is useless. lt could be of use not only to assess the safety implications of watertight door closure, but also to determine the optimum subdivision scheme for a given

level of safety. Finally, also the operative and safety

procedures could be previewed by time-based

calculations lcr realistic conditions.

MATHEMATICAL MODEL

In _{order to investigate how the motions} of a

damaged ship, particularly roll, build up and which ship p&ametei-s are the most important, a general formulation of the ship motion problem in the time-domain is considered to be representative of the physical event. t has to indude theories of

seakeeping, computational fluid dynamics to take into account floodwater effects, and methods of random phenomena in a seaway. Damage flooding is a typical ansient phenomenon so that it can be considered as

an initiak'alue problem. At initial time the ship in upright position is assumed at rest and subject to the disturbance of a regular wave whose profile along the free sirface simulates a stochastic longitudinal sea. The improvement of stich a mathematical model is

deemed as preliminary to necessary experimental

studies on models because numerical simulations

638

-aiiow

i'ge

_{satstics and preiminary}

_concfuons

about capsizing phenomenon that should be tuned in

a future experimental study.

ConU'arily to other time domain approaches [18, 19].

our dynamic model does not envisage time evolution

as a seçience of states of hyd-ostatic equilibl-ium. In _a

frst study by Francescutto and Trincas 17], a linear

mechanical _{model was assumed to desoribe the}

motions in heave, roll and pltch. The mass of the ship

was considered slowly varying in time under the

added weight of floodwater. The equations of motion

were considered only implicitly couple-c though

hydostatic terms. The results were not promising

enough because nonlinear terms of mechanical origin representing coupling were not introduced. Moreover, hyäostatic forces and moments were computedon the

gound of a simplifying approach considering super-position of independent effects.

ln the light of these considerations, a time-based

simulation of damage flooding has been developed

where all the time-varying restoring ¡excitation forces and moments, the ship hy&odynamic coefficients and floodwater forces are computed with reference to the

instantaneous hull, regardless of ship equilibiium

being. The ship is assumed to be a

six-deg-ee-of-freedom rigid body system in unbounded rionviscid

fluid. As pure loss of stability in longitudinal seas is one of the most frequent modes of capsizing, priority has been given to the mathematical model desa'ibing

the dynamics of a _{side damaged ship at rest} in following waves.

Two sets of right handed coordinate systems are

used to desa'ibe the motions of the ship (Fig. 1). The frst is an inertial frame 00x0y0z0 with the origin fixed

on the undisturbed surface, where the x0-y0 plane corresponds to the calm water level. The positive

directions of the x0. _Yo z0

axes are fward, to

starboard and downward respectively. The next is a body-fixed reference system Oxyz with its reference

point at the centre of g-avity of the ship. As large

motions are forecastable and every rotation is wanted to be independent of the other angles, the so-called Euledan angles - , O, ¡n that soq.ience - are

introduced to define the rotation of one system relative

to the other.

Although the simulation results do not depend on the choice of the coordinate system, in a time-domain simulation it is suitable to perform hy&ostatic and

hydi-odynamic calculations in the ship-fixed system of

'ie1odty and position the ship as-ts at each pnt of

time. Kinematics ves

_{the ret'_'vship between}

a splacement vedc x defined in ship-fixed system

and the displacement vecicx

x,

_{esaibed in}

the inertial system

x [TIx0

where [TI is the 3x3 ansfamath.c aU-ix

cosy cose coscsinesin -

cossin8cos +

sin cosQ sinç.sin

[T] - snícosG

sinsin6sin

cos1.co$o

[

-sine cosOsinò

Notice that the ans1aton

coc*dinate systems

s not cOfl«ed because

the ship is assumed at zero speed Then, f the vectc(s

V(u,v,w) and (p,q,r) rep-esent t'e Lnear and angular velocities in the instantaneous _{'-fixed system, the} ansfcmations between the inertiti nnd ship axes are

givep by

Y(u,v,w) = [TI

The f cced motions r (i - 1, 2 _{6) desa-ibed in the}

instantaneous ship-fixed reference system are ven respectively by the follong set of linearized

second-order diflerential equations based on Newton's law of

dynamics: m (ü + qw - rv) - X

m (" + ru - pw) - y

m (

+ pv - qu) - z

+

- fy)qr - K

+ (I- l)rp - M

U

+ (1w- l))pq

N

where m s the ship mass, and I,,

_{, l e the}

p-inc.ipal moments of inertia _{independent of the}

motions. The right hand sides of the equations

rep-esent the components of the excitation force and

moment in X, y, z directions ard around them,

respectively. They depend On the time history of the

ship motion.

639

The set of differential equations can be written as a system of 12 coupled frst-ordec diflerential equations. They are solved by means of a Runge-Kutta-Merson

integatìon scheme where time intervals of diuferent inteqation step may be selected. Thus, small steps are used in ansients of relatively high frequency

response and large steps beyond these ansients.

The advantage of such a mathematical model comes from the fact that il allows all the external forces to be compi.rted separately in terms of the combination

of the ship condition and the sea state. The total

external force in whichever mode of motion is the linear superposition of various conibutors. In this

study the following set of forces (and moments) has been taken into account:

weight and hy&ostatic forces of the intact ship hydodynamic forces p-opertional to the velocity and

acceleration of the ship

excitation forces caused by the incident wave and

diffraction effects

- static forces of floodwater

sioshing

forces due to

the interaction between

floodwater and ship motions.

As the hydodynamic forces a-e expensive to compute at each frequency of oscillation, the

frequency domain coefficients can be calculated

off-line and stored fcc fute interpolations, so giving the

momentary forces at each time step. Two subsequent

20 interpolations by splines are used, the

fist

in

nodulus- and phase-frequency spaces, and the second fcc each ansverse section in daft space.

The ship exciting fluid actions are diven by the

natixal roll frequency As hy&odynamic coefficients

are frequency dependent owing to memory effect and

since c cannot be the frequency in

_{upight ship}

position, it is assumed to be the one depending on tf-c' time-varying arm GZ iii]

o, t) -

2g t) d

where is the roll amplitude at p-evious step and is the Ûansverse gadíus.

The wave exciting forces and moments, and the hydodynamic coefficients as well are computed by means of a seakeeping p-ogam based on the

two-dimensional linear potential theory inclusive of the Frank close-fit-method [171. The nonlinear damping effects in roll are computed according to lkeda's

method [io]. App-opiate kinematic relationships are

(p,q.r) = [S]

-sin8 O

i

[SI = cose sin cos O

cosO cost -n

O_

sin.sin O cos

cossin

cosG.co$

used to _{ansfer the fluid actions from the water plane}

to the cene of rotation of _{the ship [3]. The stip-they} e<.iatíons of moon are decoupled into one set fc _the

lontucinal motions and a second set _{desa-ibing the}

lateral motions. Since _{the shape cf the huH}

'ansverse

sections can be hìly asymmeic in the pesence of large _{motions, this}

decoupling could be incc*-rect

Nevertheless, st1ing

_{from the fact}

_{that the}

low-freq.iency folloMng sea motions are dominated by the

hy&odynamic coupling given in _{limman-Newman} relationship [20), some _{fundamental effect Fke the}

complete coupling of _{sngfe modes of motions}

and the

exact evaluation of the hy&odynamic coefficients can be diegarded. _{Mcíeover, when the}

instantaneous

position of the ship notably differs from is mean

position, motions will be largely determined _{by the}

qjasì-static fcrces and moments due to the waves.

Thus, it results of _{paramount imp1ance to calculate}

them exactly by _{considering the}

time-dependent

underwater geomey relative _{to the wave pofile. The} wave disturbance has to be referred to the ship-fixed cocxdinate system taking _{into account the relative} motions between the _{at-rest water surface and}

the aacent ship surface

(x,t) = 0s [k (x

_{+ -}

_f

_{- oit]}

6efae develop4ng mxe complicate _{and exact} models, only the effect _{of static-static coupling}

has

been inodi.iced by

_{considering the second order} derivatives of heave force, roll _{moment, and pitch}

moment for heave-roll, roJl-pdch, _{and pitch-heave}

couping [4]. In reality, _{a pseudo-coupling íninsically} exists because _{of the mutual}

influence between vertical and lateral _{motions dije to the variation of}

hull geomey.

The exduing forces _{include also the forces}

exerted

on the ship by the floodwater which are nonlinearly

dependent on the ship motions. This is

_{a hiher}

reason to model damaged ship

_{moons by a}

time-domain simulation. _{In order to derive the}

forces and moments acting on the damaged compartment _{lt is} necessary to compute the displacement, velodty and

acceleration components _{at fls reference point through}

a transformation p-ocedtre considering the vecicdal

distance from the ship centre of rotation. The fluid actions due to flooater _{motion ere then transferred}

to the ship centre of _{gavity by means of}

an inverse

transformation and used_{to solve the motion}

equations at the next time_{step in the simulation pocedure}

640

COUPLING BETWEEN _SLOSHING AND SHIP MOTIONS

Eamage fioong is a case of

_{ack long where}

shic2 motion is affected by

the floodwater n-ction. The cLCem of determining the

dynanc effects due to the

fk-water movement under _{ship osdllations remains}

one- of the actual çxoblems of

seakeeping studies. The

mamtude of

sloshing _{forces and}

moments is

cosderably affected by the

_{compartment geometry} anC baffles, the quantlty of _{moving floodwater,}

and the

amoí1ude and frequency _{of lts motion.}

Ali of them

mair'1Y depend on the fill _level.

Ti-eorY and experiments _{show that}

oqessive

fboo-Ç of

a ship's _compartment

partially filled

berefl 10 and 90 percent of

_{ils depth can cause}

a

resc<flt motion

of _{floodwater which .-ay} match

resor'ant ship motions. The _{relation for the lowest}

resc,aflt liquid period in roll_{versus tank filling level for}

recgular tanks,

_{that correlates}

well enough with

expe mental results [2], is given_as

2rt

-(ng/b) -tanh (nh/b) wher h is the liquid filling _{heicht and b is the}

hreadth

of the 'ompartment.

Th _{excitation periods of}

ship motions have to be

comp'ed with the

_{resonant &oshing periods}

of the damajd compartments. _{When periods overlap}

or the

forsir period is about the _{natu-al period of flocótater,}

maxirr,.im amplitudes of these motions will result in maxirrr,im slosl-iing forces _{as a result of ampificated} couplir< between heave and roll _{motions 1141. When}

the lcì"ing period is away

_{from either the}

natural

perfod 'f roll or of pitch, there is

very lfttle liquid -otion

th sloshing _{loads do noi}

need nonlinear

matherraiiC2l models.

When the compartments _are

deeply filled, for

exdtatíMl periods suffidently _{arger than (her natual}

period, particularly in roll motion, that is. Mien

damaged compartments a-e undergoing _non-resonant

low amAitUde motions, the _{linear equations ven by}

Ahrams('fl [i) or the equation of motion of the liquid

free

_{jface derived from}

Lag-ange's equation (151

can be of use to derive the _{slostiing forces}

on the

flooded compartment. If _{the motions are large} it is necessol y to _{introduce nonlinear}

free _{surf ace} conditionO When the fill level is _{small and it does exist} strong syna-onizetion, the Glimm's _{method appears to} be (ho bc'st tool to solve the _{slostiing p-oblem [5,151.}

The evaluate the interaction between sioshing n

a damaged comp1ment and ship motions when

syncfronisation is expected, a simplified model of the coupled problem has been set up whereby the sway, heave, and ro motions of the ship &ive the sloshing

progam which in turn determines the dynamic components of the floodwater action

to be

then combined with the other exciting fcrces in the next time

step. Ecc the time belng, ox study makes use of the

20 simulation developed by Eguchi and Niho [6].

They fccmulate the accelerations of the fluid elements in the damaged comp1ment as

where v and are the velocity components relative to

the compartment-fixed coccdinate system. A finite-difference integ'ation scheme is used to derive the

accelerations in the momentum equations of the fluid

as the body fccces per unit mass.

Other linear and nonlinear techniques have been developed to compute the sloshing effects in different situations. A complete p&ameUic analysis is deemed

necessary in ccder to evaluate the ship responses when modelling the sloshing mechanism in alternative

ways and to analyse sloshing fccces when variatons in subdivision arrangement are studied.

PROBABILITY OF NON-CAPSIZING

Like every deterministic model, also oiis built to simulate motions of a damaged ship has pocc ability to deal with uncertainties. Since variables such as

loading, seaway, position and extent of damage, ail of them associated to uncertainty, heavily affect ship motions, it ¡s impalant to consider theù probebilislic

features. Among the others, a widely used procedure fcc dealing with uncertainty while using deterministic

models s to define different damage scenarios and to

rerun the deterministic model fcc each one of them.

Instead of computing the probability of capsizing

within some diiation of time under certain operative

conditions and in given sea regions, safety could be better managed by the designer in terms of probability of non-occurrence to exceed some extreme roll angie

either derived from some rule

and consaint

cc

believed dangerous. In ccder to provide adequate

safety with respect to design situations, the product of

641

-the probability of occurrence of certain enviocimental

conditions times the probability of exceeding the

dangerous angle must be near to zero. The measure of survival capability is still a probability, namely, the one's complement of dangerous event

P(S) = i - P(D)

As exeme rolling can be the result of cumulative

build up of roll due to a sequence of waves, acccrding to Bayes' fccmula the risk of dangerous event cari be

given as

P(D) P(A) P(DlA) = P(AD)

j

'j

also using the axiom of total probabilities.

The first term P(Aj) takes into account the

environmental condition of the numerical experiment.

if the presence of a wave goup is assumed with a

number of successive waves exceeding a given

maximum amplitude Hmax, the hypothesis of event Aj

can be given as

P(A) = P(j) pi1(1

-where p is the probability of the simulated wave and

the parameter i is given by the fcrmula [9]

=

[t-

e 2(Hmax/Hs)2]l

Here H represents the significant wave heigi'it of the - wave g'oup.

In the case of damage accident the designer can be

interested in assessing risk due to rolling in very

shcct-term situations leading to dangerous oscillation up to

capsizing.

That may occur

if the sea spectium

becomes so narrow that it can be associated to waves whose profile resembles a slowly varying wave -oup. Thus, the sequences of waves will be modelled by means of a wave data basis with given probability

disthbutions of heights and periods using the

narrow-band wave specti-um concept. This one is desibed

by a dominant frequency , a mean zero'-upaossing

period T, and an average wave length L, all of them defined in the domain of the number of waves in terms

of the specti-al moments.

n the time-domain analysis the amplitude expected value of the highest wave and the other characteristics

of a given sea state have to be obtained fcc the required simulation time T, that is, fcc the number of cycles N T3/Tz. The probability of an individual

peak amplitude of N independent waves to exceed

a tteshold level

tc

a certain sea specum

of veriance m0 can be rnode1led

as a

stationery Gaussian stochastic pxocess,

desibed

by the

1heetical Raegl distribution

p -

>

-

1-

eo]N

where fc a narrow frequency band ocean wave the most robabte wave amplitude in N cydes is related to the rms wave amplitude by the fmula 1131

4'fl3X

/;:; [(In N)h/2

+ 0.28.86 (In N)1/2]

The second term P(DIA) of the fcmula fc* the total p-obabililies repesents the condItional xobablIity of

the occuence of exceeding the dangerous roll angle

under the hypothesis of event A. t is derived as the ones complement of the cumulative pobabllity

distribution of the standard deviation of roll in the

(Hmx,T0) plane up to the dangerous angle.

The wave pofile is simulated as a regular wave

whose wave amplitude is the most pxobabfe maximum

cc*responding to the selected energy spectrum peak.

It is whtlen as

(x.t) = cas {k0x - t +

where x is the location of each ansvecse section with

respect to the wave taking into

account the ship

relative motion whilst S1 is a random phase angle

chosen to follow an unifcm disÜbution of p-obability

within the range (0,2t) by a random number

generatc. When the peak frequency o is considered,

the wave number k0 cccresponds to the wave length assumed to have a maximum steepness glven by the

relation [91

L = Hmax/(O.I5l - 0.0072 T0)

A NUMERICAL SIMULATION

The p-obabílity of damage suvivabffity has been computed assuming to run the experiment n a sea desaibed by the JONSWAP spectrum when using N1h Sea wave data 181. Instead of reíesenting the

shc1-term sea by ils single wave-energy spectrum and

then deriving random wave amplitudes whose finite

sum re-oduces the sea, a family of wave spectra has

been selected f c- the most p-obable peak periods and

642

covering a meaningful range of significant wave

helìts. Thus, each envi-onmental event has been

reçx-esented as a shal-term wave condItion whose

kinematics are derived from the value of the selected

peak frequency and rms height. A number of 50

random phase angles unifc-mly distributed has been assumed f c* each wave of simulation. The conditional obabilities of dangerous events have been weighted

with the obabilities of ocaxrence fc* each peak period. Therefe, the risk of dangerous event can be desaibed as

P(D) =[ w1P(A1).P(DlA) ]/ w

where m s the number of sea speca of the family and

WI is the obabiIity of occurrence of a (Hm&x,To) pai,

that is, the fraction of the total number of observations fcr each T0 to the total sum of observations fc ail the recc-deced periods.

The architecture of the computer code fc the motion simulation of a damaged ship requires a pevious

stcx'ing in a suitably arranged data base of the results

poduced off-line by the modules relative to the variables that are dependent on a few parameters only

(hydostatics, hy&odynamic coefficients, wave fcces). During the time-domain simulation the system of

differential equations fcr fcecasting ship motions are integ-ated at each time step to give the effective hull

geome'y with respect to the

instantaneous wave p'ofile. The &iving frequency is then derived and the actual values of hydostatics and hy&odynamics are

interpolated after scrtíng.

The simulation runs on the VAX 8820 computer at the University of Trieste. The mean time required to compute the ship motion in a seaway with flc-oòwater in damage compartments is about half a second of

CPU time fcx' every second of real-time motion.

Here the investigation is discussed only fcx' one

experiment. A oss-channel ro-co passenger ferry has been used to simulate the flooding phenomenon in a stochastic sea. Principal particulars of the ship

together with a small scale body plan and layout of

compartments are shoít in Fig. 2. The ship resuhs not to comply with the 1MO probabilistic damage stability

regulations [21], achieving a s,x'vival probability of

58.2%, while the required subdivision index amounts

to 70.2%. The operative and damage conditions,

considered as a numerical example, are desanbed in

Fig. 3 together with the responses in heave, roll, and

'by MO [231 as the limiting ane not to be exceeded duing intermediate asymmetical stages fcr flooding

two compartments, was considered as the dangerous

roll ande. The simulation time was assumed to be

equai to 10 mint.rtes, whh the ship at rest (4) = 4) = 0) at

time t =0. The flooded compartments were considered to be partially filled (-15%) when the flooding started.

The summary of the results of the numerical simulation is ven in the follving table.

lt results a s.rvival capability P(S) = 0.986 fc the

considered operative condition, sea state, position and

extent of damage. The ship oscillates very iregul'1y in heave and roll

as a result

of the presence of nonlinearities at large amplitudes and of heave-roll

coupling. The maxima crespond to the states when natixal frequency of floodwater in roll is at near the

excitation ä-iving frequency (Fig. 4). Differently from

the preliminary conclusions by Petey who studied the same phenomenon in beam seas [16), here the floodwater is not effective as a roll damper. On the conary, ¡t appears that flooding mechanism was

exciting heave and roll motions when there was a arge flow of flodwater on and off the compartment. Another impalant peculiarity is that the re.iction of

righting levers resulted to be mxe relevant than fci the intact ship under equivaient conditions, particularly in the presence of resonance between ship &iving excitation and fiooóNater motion.

CONCLUSIONS

The results of the numerical time-domain simulation of the damaged ship motions are to be analysed only

from a qualitative point of view. Apart from the futtxe

improvement of existing modules and the inoduction

of lacking ones, they heavily depend on the initial

conditions of ship motion and real seaway parameters. So, as it is evident to every naval architect who has not

- 643

a blind faith in mathematics, also this model has to be

experimentally validated with respect to its physical

consistence. The present state of the art does not allow to derive simple and handsome .iidelines useful du-ing

the desii process

as far as the implications of damage survivability on subdivision

and hull fccm are concerned. Befcce reaching such a

target,

a deep and intensive seening of different

approaches must be perfc'med by testing their validity and reliability 1cc ro.itine use. Fcc the time being, our

goal is to g-adually build up an appropriate model which can be useful to the desiier as a qualitative

tool to perfccm parameic analyses. As far as accuracy

of predictions is concerned, nobody can &eam to reach an exact solution in the near future also because time simulations of nonlinear dynamic systems cannot up to now result in reliable quantitative conclusions.

In the opinion of the authcr, a direct probabilistic

approach is the better method to predict rare events like capsizing also in the case of flooding. In any case, improvement in understanding dynamic stability and updating of existing rules fa- damage stability are to be

explained ¡n terms of the -aditional naval architecture.

To this purpose, and also in crder to investigate the

parame-ic influence of hull fa-m, size, and subdivision arrangement on damage stability in a seaway, the

time-domain modelling approach seems to be

promising and advisable. It could be induded in a

future stochastic optimization design. Last but not

east, the time-domain approach is to be preferred to

obtain a sample disibution fcc capsizing margin to

combine with 1MO marginal densities fcc extent and location of damage. That could be the way to define in a dynamic sense the "s-factce, that is, the probability of non-sinking cc non-capsizing when defining the MC)

Attained Subdivision lndex.

Acknowlegment

The authcc is indebted to his colleagues profi. Alberto

Francescutto and Radoslav Nabergoj fcc the fruitful

discussions in the framewcrk of a common research

and fa- their help ¡n developing some software.

SpectrurJ 1 2 3 4 5 6 7 T0 6.0 7.5 9.0 10.5 12.0 13.5 15.0 m0 0.118 0.207 0.385 0.551 0.639 0.601 0.512 L 28.5 14.7 64.4 87.6 114.4 144.8 178.8 N 140 112 93 80 70 62 56 w 0.090 0.210 0.260 0.170 0.100 0.060 0.030 P(A) 0.016 0.035 0.107 0.105 0.103 0243 0.244 P(DIA,) 0.012 0.031 0.139 0.178 0.222 0 200 0.165

Ref erences

¡ _{Abramson, H.N.}

The Dynamic Behavior of Liquids in Moving Containers, NAASP- 106, _1966.

Bass, QL., Bowles, E.8. and_{Ccx, PA., Liquid Dynamic} Loais in LNG Car _{Tanks, Trans. SNAME, Vol.}

88,

l980,pp. 103-126.

Bishop, RED., Price, WO., On the Use of Equilibrium Axes and By Axes in the Dynamics of a Rigid Ship, Journal of Mechanical _{Enoineeríng Science, Vol. 23,} 1981 pp. 23-256.

Blecki, W., Ship Safety In Connection with Parametric Resonance of (he Roll, _{nt, $hiob, Prresa, Vol. 25,} 1978, pp. 36-53.

5 _{Dlllingham, J., Motion Studies of a Vessel with Water}

on Deck, Çlarjne Technolcç',', Vol.18, _{1981 pp.38-50.}

Eguchi, T., Niho, 0., A numerical _{and experimental}

study of sloshing problems, Hydrosoft Vol. 2, No. i

1989, pp. 27-36.

Fronutto, A., Trincas, G., Ship _{Motions by} Time-8as Dam

_{Mlling, Prc. IX Simpozi)}

Tecrija I

Praksa Brtradnje, Dubrovnik, _{1990, pp. 299-309.}

Haver, S., Wave Climate of _{Northern Norway,} cJJi an Rearçh, 1985, Vol. 7, No. 2, pp. 85-92.

Hben, N., Wills, JAB.,_{Environmental Data for High}

Risk Areas Relating _{to Ship Stability Assessment,} Prec. STAB86, Gdonsk, _{pp. 279-290.}

ike, Y., Himeno, Y., Tanaka, N., _{A Pr1iction Methi} for Ship Roiling Damping, Dep. of Naval Archiiecture, University of Osaka, Report _{O5, 1978.}

li. Kastner, S., Stability ol Ships and _{Safety from}

Capsizin, Prc. Safety at Sea, WEMI. _{I 977,pp. 95-98.} 12. Le Conte, J.N., l-tydrauiiçs,_{McGrw-Hiii, New York.}

9.

LiL

ccozocoo

r0jL.___

:-,

_zir.cco

ccZ00000000 000Cc _{0 0000CC 000 0000000}

Fig. 2 : S hip main particulars and layout or comparirnents

644

-3. Longuet-Hiins, _{MS. On the S(etls(icaì}

distributIon

of the _{heights of sea}

waves, Journai of Marine

Vol. II, 1952,_{pp. 245-266.}

Mikells, N.E., Miller, J.K., Taylor, KV., Siothing In Partially Fi11i Liquid Tanks end ((s Effect on Ship'

Motions: Numerical _{Simulations and}

ExperImental Verification, RINA Spring_{Mtings,1984, Paper No.?,}

pp.1-Il.

Pet&.', F., Numerical Calculation of Forces and

Moments due to Fluid Motions _{in Tanks end Damaj} Compartmen(s, Prec. STAB'86, Gdensk, pp. 77-82. Pete'i', F., [mittlung der _{Kentersicnerheit lecker} Schiffe im Seng, _{chiffstechnik, Ed. 35, Heft 4,}

1988,pp. 155-172.

I?. Salvesen, N., Tuck, ED., Faitinsen, O., Ship _Motions and Sea Loads, Trans. SNAME, Vo. 78, 1970, pp. 250-28 7.

18. Sen, P., _{Konstantinidis, C., A Tine}

Simulation Approh to he Assessment of _{Dam Srvivabili(y of} RO/RO Carge Ships, Trans. _{SNAME, VoI. 95, 1987,} pp. 337-355.

19 Spouge,J.R., The _{Technical investition of (he} Sinking

of the Ro-Ro Ferry

European-Gateway, Trans. RINA, 1986, pp. 49-72.

Timman, R., Newman, _{J.N The CoupI}

Damping Coefficients _{of Symmetric Ships, Journal}

of $hi essarch, Vol. 5, No. 4, 1962.

TrIncas, G., Applicabilità _{dei criteri 1MO} probabili-stici al caso di falla di ro-ro passenger ferries,

TecnIca Italiana, Vol. Lii, No.4, ¡989.

MOüAssembly, 1973, R8solutionA.265

_(Viii).

MO-Paper MSC/Circular 484, 1988. MAIN DIM4SIONS LOA 146.00 ru

k

= 14000m Lpp =138.60m T .... B = 1840m

4L

D

= 1285m

LU LU 12.5 10.0 7.5 5.0 25 0.0 (J

-2.

-5.0

-j

-J

o

-7.5 -70.0 -12.5 1.5 12 0. 9 0.6 0.3 O

-0.3

-0.6

-0.9

- 1.2 - 1.5 EXPERIMENT CONDF!ONS J q V

I

-

645

--1

I

J time (s)

Ship Dala Side Dama-ge Wave Data

Displacement = 9550t Length =6.00m Waveampliwde =2.48.5m

Draught. = 5.8.5 ni Height = 300m

Wave fruency = 0.734

Trim

_{= 000m}

Z-bowm =4.00m Hedingang1e 0.00deg

KG = 7.81 ni Z-top =7.5ni Phase lag = 7.25 deg

Speed = 0.Ok.nots X-centre = 122.2-5m Nurnberof cycles = 70

0 10 20 _{30 40 50 60 70 80 90 100 flO 120}

10 20 _{30 40 50} 60 70 80 90 100 110 120

0 10 _{20 30 40 50 60} 70 80 90 00 110 120

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h/b

Fig. 4 : Syncronization been driving frequency and resonant liquid

frequency in ecrns o(comparnentriIling level.

646

-12 11 10 TR 9

8

7

6

5

4

3

2

1 t J f