A closed form assessment of the capsizal probability - The si Factor

TECHNICAL UNiVERSITY OF DE]MARK DEPARTMENT OF OCEAN ENGINEERING

WEGEMT Workshopon Damage tabiity of

_ShiPs

scii

Copenhagen, 20 October 1995

Scheepshydromcj

Archef

Recent Developments, Trends and Proposals on _{Meketweg 2: 2628 CD O&ft} Damage Stability Criteria 015 - 75873 Fax: 015 781838

A CLOSED-FORM ASSESSMENT OF 1JiF CAPSIZAL PROBABILITY

- TILE s FACTOR

Maciej Pawlowski

ABSTRACT. A review is presented of the past effOrts to developan estimate for the factor s-the most dicult and controversial part of s-the new subdivision regulations based on s-the probabi-listic concept. This may facilitate our present efforts in developing a sound assessment, consistent with the results of lmge stability model tests A new methodis put forward, based on the theo-ry underlying numerical sirmilations of diniiged ship behaviour in natural conditions, promising a high accuracy of prediction for the factor s.

INTRODUCTION

The calculation of the factor sis definitely the weal est part of the new tegulations, both for pas-senger ships [1] and dry cargo ships [21 alike. Though the methods for the s factor in the two in-struments are' not identical, the differences are not substantial. The s factor for dry cargo ships evolved in the late 1980s from the method developed about twentyyears earlier for passenger ships. The knowledge on damage survivability of ships, however, did not increase during that time - in fact - there was lack of knowledge for whatever progress It was mainly decided at 1MO to abandon the idea of the effective freeboard, not very handy in practical apphcations and uncertain as to its correctness, and base the whole calculation on the GZ curve. Thus, in practice, the two methods are equally deficient and reflect the lack of relevant knowledge in_{the are&} None the less, these new probabilistic regulations provide much higher standards of safety than the SOLAS Convention The point is that none of the existing ro-ro passenger ships have been built according to these new regulations.

fiLE ORIGINAL s FACTOR

The inadequacy of the orinal method is clearly illustrated in [1] (in Fig.. 4.20 on page 94), high-hghting large and inconsistent discrepancies between the results of model experiments, camed out separately in the UK [3] and the USA [4], for two different ro-ro passenger ferries which indi-cates failure to generalise correctly the results of model experiments The critical sea state, cha-racterised by the significant wave height H3, was considered as a function of the damge stability parameter GM1F/B:

J1

_{= H(GM1P/B)}

which is simply not the case, where: G'I - the metacentnc height flooded

0.6

0.4

02

0.0

Fe - the effective freeboard. after damage [1] B - breadth of the ship

The above relationship was treated as determinate, even though some degree of randomness is undoubtedlypresent especially in natural sea conditions and because of non-linear behaviour of a d2mged ship in waves. The probability s that a ship with a given vahte of the dmge stability parameter GMJF/B will not capsize after mge is equal to the probability that the cntical significant wave height related to this parameter is not exceeded Therefore the probability s for _a given particular case flooded can be derived from the sea state distribution at the moment of colli-sion F = F(H3), with a combination of the above damage stability criterion (the boundaiy stability curve), as a composite function of GMjFe'B:

s=F[H5(GMP/B)}

A graph of this probability is shown in Fig. 1, obtaiñed using data given in [1, 3]. A simple but adequate approximation of this functiòn is

s =

1.7(GMF/B)"6

=

(xix,)"6

(.1)

for s(0, 1), where x stands for the damage stability parameter n metres and x1 0.0416 mis a value of r yielding s = 1. 1MO, however, approximates this próbability with a large under-estima-tion, as shown in Fig. 1, using the equation:

s =

4.9(GM1F/B)"2 =

(xIx1)"2

(2)

fors(0,1). Ifsislessthan0 6, whíchhappensx(GM1F)islessthan0 015m, thenIMØ

requires in regulation 5 thais is taken as zero [1], reflecting thus the lack of confidence in the.

Figure 1. The probability of collision survival s: a - based on model tests and sea state distribution, b - based on a comparative method [6], c - approximation adcpted by 1MO

X=GM1Fe/B

O 0.01 0.02 0.03 0.04 [mJ

1.0

s

0.8

GMJAJ GMJCTJ

GM=

LBPB'

B2 (4)

quality of prediction for sm2fl s fàctors, and admitting the pöor adequacy of the current lmge stability parameter for a measure of ship's ability to resist capsng at a given sea state In this

range, the s factor could be equally well approximated using liñear interpolation between s = 0.6

forx= 0.015m, ands= I forx=Ó.04m.

SOME MODIFICATIONS

The dmge stability parameter can be easily epressed in terms of parameters related to the GZ curve. It may be observed that GMjFe/B 1/2 GZd, i.e. the dmge stability parameter equals roughly half the righting arm at an angle of heel at which the deck edge becomes submerged. Hence, the following is obtained from Eq. 1:

s =

(GZdIGZ1)"6

(3)

for SE(0, 1), where GZ1 0.083 m is a value of GZd yielding s = 1. According to the latest tests [5-7] the value of GZ1 should be preferably O i m To facilitate applications m all cases of floo-ding. including end compartments, and to conform with the definition of the effective freeboard Fe, provided in [1], GZd should be taken as the maximum value of the GZ curve within the range extending:fromtheangleofequilirium 6eupto:

(2J3)6,,.,taken as an assumed angle of deck edge immersion, where 6,,,,is the anglé

at which GZ,, occurs, or

- the angle at which weathertight openings immerse, or (2/3)6» where is the angle of flooding, or

22 degrees ( arc tan. 0.4), whichever is less.

Such an estimatic)n should be adopted for the s factor for dry cargo ships to adhere the method developed for passenger ships Unfortunately, 1MO has employed another estimation in [2], based on heunstic assumptions, using GZ with a combination of other parameters associated with ship stability. It is worth noting that GZd by definition is less than GZ.

NEW SURGE OF RESEARCE

Following the tragic capsize of the "Herald ofFree Enterprise" in March 1987, realising the lack of a reliable dnie stability criterion, the UK Department of Transport initiated an extensive Ro-Ro Passenger Ferry Safety Research Programme, comprising a number of studies into damage

survivability of these ships, including model tests inwaves [5-9] and numerical simnnbtions [10-'li]. The main oljective of the above experiments was to determine the standard of residual

stabi-lity needed to enable a ro-ro passenger ferry to survive flooding and avoid rapid capsize in reali-stic sea going conditions. Unfortunately, this primary objective has only been partly achieved. The results obtained, whether through model tests or numerical cimnlMion, are fully valid only for the particular ships investigated. Whether they can be extended to other ships iS a matter of spe-culation, although a new attempt to generalise them has been proposed inthe form of a relation-ship between the ratio Hi/Fe and the non-dimensional metacentric height flooded G?f, defiìied as

where: - volume displacement of the vessel together with the flooded water CB - block coefficient of the vessel to th damaged waterline

- draught of the vessel in the flooded condition

LBP - length between perpendiculars

with the other parameters as defined earlier. The non-dimensional metacentric height flooded is derived from the equation:

GM

GM-

f

BM,.

where BM1is the metacentric radius in the flooded condition, and this ratio should be used rathet than Eq. 4. However, this type of boundaiy stability curve is still far from representing all vesséls

ø.SGM and Fe by nature do not reflect the effect of different stnictural arrangements on the

vehicle deck and above that can be used for enhancing dmge survivability of existing ro-ro pas-senger ships. For this purpose, the. definition of the effective freeboard would have to be much

more intricate. -,

Damage stability is a complex problem invOlving a large number of variables and with results only

from one or two given shapes it is dicult to identifr the siguiflcant parameters governing the problem The matter can be now much more extensively investigated using numerical simu1atioiis [10-11], of the same accuracy as model tests. Full benefit from these results, however, will not achieved until there is a consistent logical theory which will enable the results to be generalised o other ship forms, sizes, and subdivision arrangement. Such theory, at the moment, does not exit. The objective is then to define the unique boundary stability curve, valid for all vessels. Such a curve is badly needed by regulatory bodies, designets and industiyto know hw existing ship can be modified to make them safer for the public.

However, it is worth telling loudly that not much can be done about existing ro-ro passenger shi,s we cannot rebuild these ships extensively to meet the high standards of safety in the new regu lations [1]. Many of them do not meet even the SOLAS 90 criteria which are rather lenient. These ships, as a rule, were poorly designed with no reserve of buoyancy above the subdivision (vehicle) deck and with excessively close bulkhead spacing below this deck, being misled by the illogical aspect of the SOLAS Convention Subdivision of that kind results inevitably in very lovi indices of subdivision [12]. Such ships are simply unacceptable:and, in practice, their safety can be upgraded only to a limited extent.

On the other hand, new standards of safety can be easily implemented on new ro-ro designs [13-16]. The fact that ro-ro ships can be as safe, in terms of subdivision indices as other ship types without destroying their operational features is not yet widely recognised. Fortunately, this fact

has been eventually recognised by 1MO who now make no distinction between ro-ro and conven-tional passenger ships, and the same applies to dry cargo ships.

Nevertheless, establishing promptly the right boundary stability curve,, valid for all ships and forns of subdivision, is of prime importance. Otherwise, we may allow for developing regulations that' are unreasonable, more stringent than necessary which can even destroy in future the ro-ro con-cept. Another important aspect is that whatever we do now can be totally ignored ormay be irre-levant to 1MO activities for many years to come, including harmonisation of probabilistic subdivi-sion regulations fòr all ships. Our efforts to solve these problems ration'lly can thus be wasted, with the detriment for the travelling public.

PROPERTIES OF BOUNDARY STABILiTY CURV

For any ship, with a given loading condition and compartment flooded, the critical sea state the ship can withstand, characterised by the significant wave height H3, cannot be determmed unique-ly, and this fact is widely acknowledged nowadays. This is not because of some inaccuracies of

model experimentS or numerical simulations, nor because of insucient time öf duration of test runs, but simply because of the random nature of the critical sea state, characterised by certain distribution Hence, any boundaty stability curvt is a fuzzy curve rather than distinct, indicating mean values of the critical sea states and surrounded by a confidence leveL Therefore, to find out distnbution of critical sea states (and its mean or median value first of ail), you have to repeat many times the same case of flooding ax the sanie sea state but with different initial conditions. Tó arrive at a boundary curve, you cannot do just one run. The random nature of critical sea states comes mainly from the non-linear roll motion in irregular waves.

The probability s'that a ship with a given Iöading condition and compartment flooded will not capsize after dimge is equal to the mew! probability that the critical significant wave height related to this case is not exceeded:

S

-

$H F(Ii)f(H) dH5

(5)

where: F(H3) - cumulative distribution functiôn of sea states at the moment of collision f(H5) - probability density function of critical sea states for the ship with a given

loading condition and compartment flooded

ecause for moderate and higher critical sea states F(H) h a mll rate of change, as can be seen in [1] (in Fig. 4.21 on page 95), whereas' for low critical sea states (when c1amged stability is deficient) the range of variation of critical sea states is narrow, by virtue of the mean value

theo-rem Eq. 5 yields:

S =F(Hsm,)

(6)

that is, in practice, the s factor can be calculated as if the critical sea states were of binary nature, cut off at the mean value.

Averaging the s factor. The calculation of the probability s s would therefore be relatively simple if the mean ritical sea' state H3, was determinate for each compartment group. How-ever, this quantity is not determinate becaise it dependS on such tandom quantities as the loading

condition (draught T, trim t, .metacentric height GM and permeability 4u) at the moment of

colli-sion, and vertical extent of flooding H. Therefore, in order to ohtain the composite probability s for all possible combinations of p, H,. T, t and Gli4, it is necessary to average sfor each compart-ment group with respect to these random variables. This follows from the formula for the entire probability. Hence:

s. = E(s)

= L

J L J

sfu, T, t, GM) dp dT dt dGM

(7)

where the probability s = s(u, T, t, GM) is itself a function of the four random quantities, averaged previously, for ships with horizontal subdivision above the waterline, with respect to the vertical extent of flooding H that is of discrete character.

As can be seen, to find the s, factor for each compartment group it is necessary to know the joint distribution density functionfCu, T, t, GM) which can only be derived from statistical data, and which in practice is virtually impossible to obtain. Such a distribution might also be related to the ship type and possibly to the ship's route, but again the understandable lack of data would prevent these variables from being considered.

Remembering that the method is aimed at arriving at an assumed rather than the actual probability of survival, the averaging procedure may be largely simplified by accepting draught and vertical extent of flooding as the only random variables and assuming the others to be determinateeither as constants or as functions of draught. Hence, Eq. 7 for the s factor reduces then to the

follow-ing:

S1 =JT s(flf(fldr

(8)

where s(1) is the probability s = s(u, T, t, GM) as a function of the ship draught only, obtained by averaging the s fàctor (for each compartment group and draught) relative to different descrete vertical extents of flooding, if any, andf(7) is the marginal distribution density of draughts at the moment of collision. Hence, if several watertight decks are fitted above the waterline in question, with the heights 11k above the baseline (k= 1,2, ...), the factor s(7) for each compartment group

and draught is given by:

s(T)

=

_s(HkXvk

-

vk_1) (9)

where: s(14) - s factor calculated for each compartment group and intact draught, assuming vertical extent of flooding up to a height 14

Vk - reduction factor which represents the probability that the spaces above the hori-zontal subdivisionataheightHwillnotbeflooded, with v0=O, and vk i for the last (heighest) deck.

Obviously, if there is no horizontal subdivision within a given compartment group (k = I only), then there is only one vertical extent of flooding, usually from the base line or double bottom up-wards without liinit and naturally there is no need for any averagmg. In such a case, s(7) equals simply the one and only s factor calculated for the subject compartment group with a given intact ship draught and for the one extent of flooding,

The reduction factor y is currently being developed at 1MO. It will be based on bow heights sta-tistics and will correspond to the cumulative distribution function of bow heights [17].

Having calculated the s(7) factor for each compartment group and intact draught; the s1 factor is the mean value of s(7) relative to draught that can be obtained from Eq. 8 by applying numerical integration:

s1

=wjs(r)

(10)

where: n - number of draughts used for calculating s and

{} - weighting factors, depending on distribution density of draughts at the moment of collision and the number of draughts used.

The present calculation of the s1 factor for dry cargo ships [2J is based on two draughts only. Be.. cause of the fact, however, that

- flooding information for the master is supposed to be carned out for a widerange of draughts, and

- there are problems ai-ising with determinption of the KG _{value lithe calculàtion of the} index is based on two draughts only

a minimum of three draughts should be used for these caltuiations. It is noteworthy that thiee draughts of five are employed in [1] for calculating the A index for passenger ships, although these ships have much smaller draught variation than cargo ships. Three draughts seem to be

entirely sufficient for practical applications.

The weighting factors for passenger ships were originally derived assuming a triangular

distribu-tion of draughts at a range between d0 and d3, vanishing at the ends and assuming a. maximum

va-lue at d2, as is explained in [1]. These factors represent the relative frequency of a ship operating at a given draùght at the moment of collision and can be derived from the damage statistics. They

may depend on the type and. category of ship.

A NIW METhOD FOR TILE s FACTOR

As can be seen, of prime importance for the determination of the s. factor, and thus for the whole method, is the basic expression for the s factor, given by Eq.. 5, that requires the critical sea state

Hsm _{to be known for each damage case, denoted further down simply by H3. However, to get}

this basic factor, we wish to avoid determination of H3 with the aid of model tests or numerical simulations, as they are not suited and intended for rOutine àpplications. For this purpose, we have to find out a simple but meaningful procedure, verified by the previous results and preferably based on the theory underlying numerical simulations [10]. We hope that in spite of the complexi-ty of the problem, this goal can be achieved.

Main Observations. The m2in observation which can be made from the previous tests results is that the damaged ship in waves behaves quasi-staticly when _{it reaches a point of no returnbut} reaching this point and the tme to reach this point are determined by the dynamics of the ship. Qwsi-static behaviour of the ship at the point ofno return denotes lack of roll motion, which is marginal at this position of the ship, while the other ship motions such as sway, heave and possibly pitch (in case of flooded compartments other than those at mid-ships) remain in practice unaffected by the quasi-static heel of the ship. Heave_{is then particularly dominant.}

The three other motions in principle can be regarded as linear even though they can be of large amplitudes, in particular heave and pitch, and despite the large variation of the waterplane area with draught and trim, typical for modern ferries. The linear part is dominant in these motions. What is more important in this case is that we are going to disregard here possible couplings between heave and roll and heave and pitch.. Above couplings can develop for asymmetrical floodings with a high position of the centre of gravity äbove the waterline, typical for short sea ferries, and for compartments flooded at the end parts of the ship. By doing so, we err on the side of safety and can then make_{use mainly of the results for symmetrical mid-ships. floodings.}

Having analysed the theoretical model for the _{dynamic behaviour of a damaged ship in natural} waves [IO], and exaimned a number of the results from numerical simulations of the ship behavi-our for different loading conditions, sea states and various internal configurations, _{the follOwing} observations can be made:

The point of no return occurs when the ship has reached the angle of heel 6,,,, at which maximum of the GZ curve occurs. This angle, at a large majority of cases, is less then. i O degrees. Reference is made here to the GZ curve calculated traditionally, using the constant

displacement method and. allowing for free floociii of the vehicle deck when the deck edge

is submerged.

The amount of water on deck at this point can be predicted from stability calculations for the ship at another flooding scenario, in which the ship is damaged only below the vehicle deck but with sorne amount of water on the (undamaged) deck inside the upper (intact) part of the ship. The critical amount of water on deck is such that the ship assumes the angle of 1911 (angle of equilibrium) 6 that equals the angle 6, determined previously.

Of crucial importance for the seeking damaged stability criterion, capable of generalising damage stability model tests results, are the following quantities occurring at the point of no

return:

- the elevation of waxer on deck above the sea level, h, and

- the depth of the deck edge below sea level, f, measured at the centre of damage at

the inner shell of wing spaces, if any.

The space above the vehicle deck at the other scenario is enclosed by the undamaged deck and undamaged ship sides above the deck, with the damage extending from below the deck down-wards. For the ship with a side casing or wing tanks, the space is enclosed by the outer shell ber yond the flooded part of the double hull, and by the inner shellin way of the flooded part of the

wing spaces. Consequently, the depth of deck edge is understood then as draught of deck edge measured at the inner shell of the casing (wing tanks) in the middle of clmage

Due tó the drnamic action of waves, the flooded water accumulates on deck causing the ship to heel and, whén the deck edge becomes submerged, the water continues to elevate above sea levl until it reaches á height h, depending on sea state, at which inflow and outflow rate of water tbrbugh the opening is balanced maintaining the amount of trapped water on deck coistant ovei some period of time as if the upper part of the ship were intact. The ship assumes then aheel

angle at which the heeling moment due to the accumulated water on deck is balanced by the restoring moment. This quasistatic heel angle determines in turn the mean roll angle about whiçh the ship oscillates-4t cannot, therefore, be greaterthan the angle 6m

An immediate practical conclusion can be drawn from these observationany measures increa-sing the GZ and/or decreasing the heeling moment due to water accumulated on deck are bene-ficial to ship safety. These are first of all:

buoyant chambers made on the vehicle deck along the ship sides by sealing off the space between the flanges of side girders and the side itself. These chambers should be parti-cularly effective and recommended as they do the both things and, in addition, do not restrict the stowage space.

down-flooding arrangements which counteract the accumulation of water on deck and, if properly designed, can largely reduce or even eliminate this phenomenoa

There are many other solutions that are feasible for existing roro ships such as sponsons, trans-verse removable bulkheads, partial bulkheads, to mention only few, but above two are ro-ro friten-dly and at the same time very effective. However, the use of passive down-flooding arrangements on existing ships is not an easy task and requires special attentionthey have to befitted in such a way to be put into operation exclusivelywithin the extent of the compartment (group) flooded

which can be known only after flooding. Theréfore, they have to be combined with other

enhan-cing devices or be of a special design.

A Rational Boundary Stability Curve. It can be expected that a generalised damage stability criterion (boundary stability curve), valid for any dimge scenario and subdivision configuration,

can be presented in a non-dimensional form:

h/H3.=f(f/H)

(11)

which can be deduced from the considerations regarding water accumulation on deck [18], where H is the mean critical sea state, charaçterised by significant wave height, found with the aid of

physical or numerical experiments for any drnge case, characterised by h andf, as described above. This type of relationship can be influenced mainly by one parameterthe ratio TO/TZ, where T0 is modal period of the critia1 sea state and T is the natural heave period.

The natural heave period T is proportional to (TC)° where T and C are ship draught and vertical prismatic coefficient for the shipin flooded condition. In practice, however, T is con-stant for a given ship, the same for intact and flooded condition. This is particularly true fór ships with large B11 ratios and the V-type stations, typical for ferries and a majority of dry cargo ships. On the other hand, the modal period T0 is itself a function of H for the critical sea state, which in turn is a complex function of cbmged stability and subdivision arrangement. Hence, for the end product to be simple, the boundary stability curve, in the form given by Eq. 11, should'be necessa-rily obtained keeping constant values of the ratio rji;, taken as a parameter. Such curves should be universal, valid for any ship of any size, any type of subdivision, loading condition and compar-tment floöded

Practical Applications. This type of boundaxy curves, although novel, would be very simple in practical applications. Having calculated for a given compartment group the elevation of flooded water on deck h and draught of deck edge at the openingf a straight line with a slope of h/f can be drawn on a graph presen*ing the boundary curves, passing through the origin. The straight line intersects a boirnWy curve, with a given value of raño TJT, at a certain point with given values of hIH and flHr Themeanciitical sea state is given then as

H = h/(h/H5) = fI(fIH)

for given T(JTZ (12)

As T,, is itself a function of sea state, the true value of H5 is determined by a point of intersection between a curve ¡I = ¿15(T0), defined by the above equation for different values of the parameter

T0, and a curve T,, = T,,(H5), a charäteristic curve known for the giveñ sea spectrum. In natural sea conditions, the elevation of water above the sea level, h, is a random quantity, therefore the critical sea state is random as well, and what we use here is the mean critical sea

state H5.

Other ship types. The proposed procedure has so far focused on ro-ro vessels but the approach adopted could also be easily applied to conventional ships, if the effect of water shipping on the weather deck, unprotected by the ship's sides, is regarded as equivalent to the mechanism of water accumulating on the vehicle deck. In such a case, the method should be formally applied to the ship as if her sides were extended vertically above the upper deck thus reducing this deck to a ro-ro type.

CONCLUSIONS

The damaged stability criterion proposed here seems to be well founded as it is based on comj,ari-son of some real quantities derived from a physical model. There is, therefore, hope for a tational cnterion that could be delivered to 1MOm a short time scale Equation 11 can be quickly estb-lished (and verified) if only the new quantities h andf are provided for the dmge cases which have been previously investigated. For this purpose, existing software for damage stability calcu-lation has to be modified slightly so as to be able to calculate them.. Nobody before regárded them ofinterest for ship safety.

ACKNOWLEDGEMENT

The author is indebted to Dr. Dracos Vassalos who made the necessary arrangements for award-ing him by Strathclyde University a six month Visitaward-ing Fellowship at the Department of Ship and Marine Technology.

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(j1,

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