1803586192_1999999096_McTaggart_&_de_Kat_-_SNAME_2000_-_Capsize_risk

8/12/2019 1803586192_1999999096_McTaggart_&_de_Kat_-_SNAME_2000_-_Capsize_risk

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Capsize Risk of Intact Frigates in Irregular Seas

Kevin McTaggart (M), Defence Research Establishment Atlantic, Dartmouth, NovaScotiaJan Otto de Kat (M), Maritime Research Institute Netherlands, Wageningen, TheNetherlands

ABSTRACT

Risk analysis provides a rational method for safe design and operation of complex engineeringsystems. This paper describes the application of capsize risk analysis to the safe design andoperation of intact naval frigates. Input probabilistic descriptions of wave conditions andship operating conditions are combined with time domain simulations of ship motions to

determine capsize risk of an intact ship. The maximum roll angle of a ship in an irregularseaway is dependent on the seaway realization, and this effect is included in the risk analysis.Sample calculations for a naval frigate demonstrate application of the risk analysis procedure.Risk analysis results can be used to develop simplified design guidelines for given ship types.In addition, operator guidelines can be developed for minimizing capsize risk.

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NOMENCLATURE

Ajk infinite frequency added massai amplitude of wave componentaX Gumbel distribution scale parameterX estimate ofaX

bX biased estimate ofaX

B ship beamBjk frequency independent dampingBjk(e) frequency dependent dampingbX Gumbel distribution location parameterX estimate ofbXC ship capsizeCDx local cross-flow drag coefficientCjk frequency independent stiffnessCV P vertical prismatic coefficientD durationDp propeller diameterDs simulation duration

Fjk(t) radiation force componentFFK Froude-Krylov force on shipFH quasi-steady transverse hull forceFH,NL nonlinear component ofFHF n Froude numberFPR propeller thrustFX(X) cumulative distribution function ofXFx, Fy, Fz translational forces on shipGMfluid corrected metacentric heightg gravitational accelerationHs significant wave heightHs,max maximum Hs for given wave period

H/ nominal wave steepnessIij ship rotational inertia componentJ(t) propeller advance coefficientKjk(t) retardation functionKxx hull moment coefficientKT propeller thrust coefficientKG height of CG above baselineL ship life or length[M] ship generalized (6 6) mass matrixMx, My, Mz rotational moments on shipMx,H roll moment from hull lift and dragm0 ship massNC number of capsizes

Np propeller RPMNs number of simulationsNw number of wave componentsNX number of discretized values ofX{n} generalized normal vectorP(CD) probability of capsize duringD

P(CD) estimate of P(CD)P(C|X) probability of capsize givenX

p pressurep, q, r ship rotational velocity components

pX(Xi) probability of occurrence ofXipX|Y(Xi|Y) conditional probability ofXiQX(X) exceedence probability ofXQX|Y(X|Y) conditional exceedence probabilityS(t) instantaneous wetted surface of shipS(i) wave spectral density

T local draftTmid midships draftTp peak wave periodTz zero-crossing periodT natural roll periodt timetp propeller thrust deduction coefficientts trim by sternU, V ship horizontal velocity componentsUa mean wind speeduG, vG, wG, velocity at ship centre of gravityVs nominal ship speedv local transverse velocity

X random variableXi discretized value of random variableXx, y, z ship-based axes{x(t)} ship acceleration vectorxe, ye, ze earth-based axesxk(t) displacement for modek wave slope anglemax maximum absolute value of incident sea direction relative to ship Eulers constant (0.5772 . . .)i discretized wave frequency intervali wave component random phaset local transverse wave orbital velocity

water elevation ship pitch anglei wave component wave numberX mean value of variableXX standard deviation of variableXI incident wave velocity potential roll anglemax,D max roll angle for durationDmax,L max roll angle for ship lifeLmax,i max roll angle of ranki wave frequencye wave encounter frequencyi

frequency of wave componenti

INTRODUCTION

In recent years, a consensus has emerged amongengineers that risk analysis is the most rational ap-proach for safe design and operation of complex sys-tems. Blockley and contributors [1] give a compre-hensive overview of risk analysis and its applicationto engineering systems. For application to ships,

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Kobylinski [2] gives an example of capsize risk analy-sis, while Mansour et al. [3] discuss structural designusing risk analysis.

For stability of warships, risk analysis offers analternative to the rules developed approximately 40years ago by Sarchin and Goldberg [4], which con-

tinue to be used by several navies. Although therules of Sarchin and Goldberg have resulted in verysafe operation for naval warships, there are sev-eral reasons for moving toward new risk-based ap-proaches. Sarchin and Goldbergs stability crite-ria were developed based on experience with shipsin World War II. Modern ships have considerablydifferent hull forms for which the older criteria arenot necessarily valid. For example, modern warshipshave wide transom sterns, making them more vul-nerable to loss of static stability in following seas.For novel hull forms such as trimarans, lack of op-erational experience makes risk analysis the only vi-

able approach available for developing safe designs.Another important reason for using risk analysis isthat it provides information regarding what com-binations of speed and heading should be avoidedto minimize capsize risk. In the past, lack of ac-curate simulation tools and large computational re-quirements have been the major deterrents for appli-cation of risk analysis to ship capsize. Fortunately,time domain models of ship capsize now appear tobe sufficiently accurate and fast.

This paper describes ongoing work to developrational risk analysis procedures for the stability ofintact naval frigates. This work is an extension of

earlier work presented by McTaggart [5, 6]. Thepresent research is being conducted in parallel withCanadas ongoing participation in the CooperativeResearch Navies Dynamic Stability Project, whichhas been described by de Kat [7] and de Kat et al.[8].

NUMERICAL SIMULATION MODEL

The numerical model FREDYN has been devel-oped to simulate the large amplitude motions of asteered ship in severe seas and wind. The model con-

sists of a non-linear strip theory approach, where lin-ear and non-linear potential flow forces are combinedwith maneuvering and viscous drag forces. The non-potential force contributions are of a nonlinear na-ture and based on (semi)empirical models.

The derivation of the equations of motions isbased on the conservation of linear and angular mo-mentum. These are given in principle in the inertial(earth-fixed) reference system, defined by the sys-tem of axes (xe, ye, ze). Eulers method is applied

for deriving the equations of motion in terms of arotating, ship-fixed coordinate system. The equa-tions of motion are given by:

[M] {x(t)} =

FxFyFzMxMyMz

m0(wGq

vGr)

m0(uGr wGp)m0(vGp uGq)(Izz,0 Iyy,0)qr(Ixx,0 Izz,0)pr(Iyy,0 Ixx,0)pq

(1)

The matrix [M] is the generalized (6 6) mass ma-trix of the intact ship and{x(t)} is the accelerationvector at the center of gravity; p, qandr representthe rotational velocities for roll, pitch and yaw, re-

spectively. The summation signs in the right handside represent the sum of all force and moment con-tributions, which result from:

Froude-Krylov force (nonlinear) Wave radiation (linear) Diffraction (linear) Viscous and maneuvering forces (nonlinear) Propeller thrust and hull resistance (nonlinear)

Appendages rudders, skeg, active fins (nonlin-

ear)

When present: wind or internal fluid (nonlin-ear)

Large angles are retained in the matrices for trans-formation between the ship-fixed and the earth-fixedcoordinate system. The combination of the inte-grated hydrostatic and dynamic wave pressures rep-resents the total Froude-Krylov force (or moment),given by the vector:

{FFK

} = S(t) p{n} dS (2)

where{n} is the generalized normal vector and

p = I

t + gz (3)

S(t) represents the instantaneous wetted hull surfacein the presence of the undisturbed, incoming wavewith velocity potential I. Linear wave theory isused to describe the sea surface and wave kinematics.

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In the case of irregular waves, the model makes use oflinear superposition of sinusoidal components withrandom phasing.

Linear, 3D, transfer functions are used in the de-termination of the diffraction forces, and the waveradiation forces are based on linear retardation func-

tions and convolution integrals (including forwardspeed terms). The potential, non-lifting hydrody-namic force terms are given by:

Fjk(t) = Ajk xk(t) Bjk xk(t) Cjk xk(t)

+

t0

Kjk(t )xk()d (4)

whereFjk(t) is the contribution to the j mode forcefrom motion in the k-mode, Ajk is the infinite fre-quency added mass, Bjk is the frequency indepen-dent damping, Cjk is the frequency independentstiffness, Kjk(t) is the retardation function, xk(t)

is the time dependent ship displacement for mode k,xk(t) is the time dependent ship velocity, and xk(t)is the time dependent ship acceleration.

The retardation functions can be obtained asfollows in terms of the 3D hydrodynamic coefficients:

Kjk(t) = 2

0

Bjk(e) Bjk

cos(t)de(5)

where e is wave encounter frequency. Viscous ef-fects include roll damping due to hull and bilge keels,wave-induced drag due to wave orbital velocities,and non-linear maneuvering forces with empiricallydetermined coefficients. The quasi-steady hull forces

resulting from the motions in the horizontal planeconsist of a linear and non-linear part:

FH(t) = FH,L(U,V,r,; t)

+FH,NL(v(x, t), T(x, t),

CDx(U(t), x),U,V,r, t) (6)

whereUandVare the time-dependent longitudinaland transverse velocity components, is the pitchangle,v(x, t) is the local transverse velocity at (sec-tional) location x, T(x, t) is the local draft, CDx isthe local cross-flow drag coefficient, and t is the lo-cal wave orbital velocity in transverse direction. The

roll moment resulting from lift and hull drag forcesis given by:

Mx,H(t) = FH,Ly(t) + FH,NL(t)y(x, t)

+Kur U(t)r + KupU(t)p

+Kpp|p|p (7)The propeller thrust is estimated by:

FPR(t) = FPR[Np, tp, KT(J(t), Np, Dp)] (8)

whereNp is the propeller RPM (assumed constant),tpis the thrust deduction coefficient at the propeller,andKTis the propeller thrust coefficient. The pro-peller thrust coefficient KT is a function of the in-stantaneous advance coefficientJ(t), propeller RPMNp, and propeller diameter Dp.

The hull resistance is based on the calm watercharacteristics as a function of instantaneous speedand sinkage. Appendage forces are estimated by us-ing wing theory in the case of a rudder or active fin,and by using pressure drag in the case of a skeg.Interaction between rudder and hull is considered,and the instantaneous angle of attack depends onthe ship motions, rudder angle, propeller race andlocal wave orbital velocities. The equations of mo-tion are solved in the time domain using a 4th orderRunge-Kutta scheme.

Wind forces are modelled using standard semi-empirical methods. McTaggart and Savage [9] pro-

vide wind tunnel results for a naval frigate which canbe used in simulations.

Simulation of Random Seaway

The numerical model can simulate motions of aship in a regular or random long-crested seaway. Tosimulate a long-crested random seaway with a givenwave spectrum S(), a random phase method isused (see Chakrabarti [10]). Using the conventionthat waves propagate along the earth-fixed xe axis,the instantaneous water elevation is then given by:

(xe, t) =Nwi=1

ai sin(i ixe + i) (9)

where Nw is the number of discretized wave com-ponents, ai is the amplitude of component i, i isthe component frequency, i is the component wavenumber, and i is a randomly generated componentphase between 0 and 2. Assuming small amplitudelinear wave theory, the amplitude and wave numberof the wave components are as follows:

ai = 2S(i) i (10)

i = 2i

g (11)

where i is the discretized interval of wave com-ponenti.

For a realistic representation of a random sea-way, at least 20 discretized wave components mustbe used. The random phasesi are determined us-ing a random number generator, and are determinedby an input integer seed number. It will be shown

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later that maximum roll angle in a randomly gen-erated seaway can be very dependent on the inputseed number for the seaway.

PHYSICS OF CAPSIZE FOR NAVAL

FRIGATES

Critical wave and operational conditions maycause a frigate to capsize according to the follow-ing mechanisms:

Static loss of stability Dynamic loss of stability Broaching

Static Loss of Stability

This refers to the quasi-static loss of transverse

stability (associated with an excessive righting armreduction) in the wave crest. This mode occurs typ-ically in regular or irregular following to stern quar-tering waves with low encounter frequencies. Theship can capsize when it experiences temporarily acritically reduced (possibly negative) righting armfor a sufficient amount of time, while the wave crestovertakes the ship slowly and the ship is surging orsurf-riding periodically. For this mode of capsize tooccur in irregular waves, one encountered wave ofcritical length and steepness is sufficient to cause thesudden catastrophic event. Experimental evidenceof this for a frigate hull form has been observed by

de Kat and Thomas [11].

Dynamic Loss of Stability

A frigate can lose stability dynamically in con-junction with extreme rolling motions and lack ofrighting energy under a variety of conditions. Thiscapsize mode may be associated with the phenom-ena described below.

Dynamic Rolling: This mode of motion occursat forward speed in stern quartering seas, whichcan be of regular or irregular nature. Here all

six degrees of freedom are coupled, where inaddition to roll, surge, sway and yaw can ex-hibit large amplitude fluctuations. The motionis characterised by asymmetric rolling: the shiprolls heavily to the leeward side in phase withthe wave crest (approximately) amidships androlls back to the windward side in the wavetrough, albeit with a shorter half-period andsmaller amplitude. Due to the associated surg-ing behaviour, the ship spends more time on the

wave crest than in the trough, resulting in a pe-riodic but asymmetric reduction on the crestand restoring (in the trough) of the rightingarm. The actual roll period may exceed thenatural roll period significantly. In the case of acapsize, the roll motion typically builds up over

a number of wave encounters to a critical level,and the ship will usually capsize to leeward. Ex-perimental evidence of this for a frigate can befound in de Kat and Thomas [11, 12].

Parametric Excitation: Parametric excitationresults from the time-varying roll restoringcharacteristics of a ship typically found in lon-gitudinal waves. The periodic changes in staticrighting arm during the repeated passage of awave crest followed by the trough can causelarge amplitude roll motions, which occur atapproximately the natural roll period and si-

multaneously at twice the wave encounter pe-riod. The wavelength must be of the order ofthe ship length. In such circumstances, para-metric rolling - also referred to as low cycle res-onance - can result in capsizing. It can occurin regular and irregular waves. It has been ob-served in head seas, but parametric excitationin astern seas is typically more critical in termsof capsizing. In particular, when a ship travelsat the mean group speed in following seas, para-metric excitation can occur during the passage(in a regular fashion) of a wave group with asufficient number of encountered waves of criti-

cal height and length (de Kat [13]). Ships witha wide transom stern may be more prone toparametric rolling; typically this is not a criti-cal capsize mode for frigates.

Resonant Excitation: In principle large ampli-tude roll motions can result when a ship is ex-cited at or close to its natural roll frequency.Roll resonance conditions are determined by thecombination ofGZcurve characteristics, weightdistribution, roll damping, heading angle (e.g.,beam seas), ship speed, wavelength and height.

Impact Excitation: Steep, breaking waves can

cause severe roll motions and may overwhelm avessel. The impact due to a breaking wave thathits a vessel from the side will affect the shipdynamics and may cause extreme rolling andcapsizing. Possible damage to deck structuresand subsequent water ingress are not consideredhere. This capsize mode is relevant especially tosmaller vessels in steep seas.

Bifurcation: In laboratory conditions, the roll

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response may jump from one steady state toanother steady-state condition at the same fre-quency following a sudden disturbance. Bi-furcation is usually associated with large am-plitude motions. The experimental studies onthese and other types of non-linear systems be-

haviour have so far been restricted to regu-lar waves. Double period bifurcation in long( = 2L), high waves has been observed for afrigate by de Kat and Thomas [11, 12].

Broaching

Broaching is related to the loss of course keep-ing in waves. A variety of broaching modes exist inregular and irregular waves:

Successive overtaking waves while the ship istravelling at low speed;

Low frequency, large amplitude yaw motions;

Broaching caused by a single wave.The first mode may occur in steep following seas

at low ship speed, where the ship is gradually forcedto a beam sea condition during the passage of sev-eral steep waves. The other modes occur at higherspeed, typically at a Froude number F n > 0.3. Thethird mode is usually characterised by quasi-steadysurf-riding at wave phase speed and steadily increas-ing yaw angle; this broaching mode has been alsobeen observed for frigates in combination with bowsubmergence during surf-riding. Experimental evi-dence of this for a frigate can be found in de Katand Thomas [11, 12].

Other Factors

For steep wave conditions most likely to causecapsize, the wind direction will typically be approxi-mately collinear with the waves. Consequently, windwill not strongly influence wave-induced capsizing inastern seas. In beam waves, however, it can be moreimportant. Furthermore, capsizing may be associ-ated with a combination of modes, e.g., static loss ofstability after a surf-riding and a high speed broach.

PREDICTION OF SHIP CAPSIZE RISK

The complex nature of ship capsize means thata risk analysis must consider all conditions encoun-tered, and that the analysis cannot be restricted toa limited range of headings (e.g. beam seas) or toonly very large wave heights. The present approach

considers the possibility of ship capsize for all en-countered seaways, headings, and ship speeds. For aship in a seaway of duration D (e.g. one hour), theprobability of capsize P(CD) is:

P(CD) =

NVs

i=1N

j=1NHs

k=1NTp

l=1 pVs(Vsi)p(j) pHs,Tp(Hsk, Tpl) P(CD|Vs, , H s, Tp) (12)

whereVs is ship speed, is the relative wave head-ing, Hs is significant wave height, and Tp is peakwave period. Each independent variable X in theabove equation has been discretized into NX dif-ferent values. The last term of Equation (12) de-notes a conditional probability given a set of oper-ational and seaway conditions. Similar approacheshave been presented by Kobylinski [2] and Dahle and

Myrhaug [14, 15]. An important assumption of theabove equation is that desired ship speed and head-ing are independent of wave conditions. This as-sumption is conservative because ship operators willalter speed and course to reduce capsize risk.

Equation (12) does not explicitly include windeffects, which are considered to be of secondary im-portance relative to waves for naval frigates; how-ever, wind effects can be included in the risk analysisby considering wind conditions to be dependent onwave conditions. A reasonable approximation is toassume that winds are collinear with waves. Datafrom Bales et al. [16] suggest that mean wind speed

can be approximated as a linear function of signifi-cant wave height. The following approximation hasbeen derived for the North Atlantic:

Ua = Hs 1.823/s + 3.45 m/s (13)

where Ua is mean wind speed at an elevation of19.5 m.

Once the probability of capsize for duration Dhas been computed using Equation (12), the associ-ated annual probability of capsize can be computedas follows:

P(Cannual) = 1 [1 P(CD)]1year/D

(14)

The application of Equation (12) for predic-tion of capsize risk requires a suitable methodto determine capsize risk for given conditionsP(CD|Vs, , H s, Tp). For a FREDYN simulation ofa ship in irregular waves, a random phase approachis used to generate a wave realization. The waverealization is dependent upon an integer seed num-ber provided as input for generation of random wave

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phases. The occurrence of capsize can be highly de-pendent on the input seed number. Upon initialconsideration, it would seem appropriate to deter-mine P(CD|Vs, , H s, Tp) usingNs simulations withdifferent seed numbers as follows:

P(CD|Vs, , H s, Tp) = NC

Ns (15)

where NC is the number of simulations for whichcapsize occurs. The main disadvantage of this ap-proach is that extrapolation cannot be used to esti-mate capsize risk in cases with no observed capsizes.A more useful approach is to apply a statistical fitto the maximum roll angles max,D from Ns sim-ulations of duration D. In the present discussion,variableXdenotes the maximum roll angle in a sea-way with given duration and conditions. McTaggart[17, 18] has shown that a Gumbel distribution is ap-propriate for modelling the maximum roll angle in a

one hour seaway. Properties of the Gumbel distribu-tion are described in various texts, including Mad-sen et al. [19] and Thoft-Christensen and Baker [20].The cumulative distribution function of the Gumbeldistribution is as follows:

FX(X) = exp

exp

bX X

aX

(16)

whereaX andbX are scale and location parameters,withbX being the 36.8th percentile ofX. The meanX and standard deviation X of the Gumbel dis-tribution are related to the distribution parameters

as follows:

X = bX+ aX (17)

X = aX

6(18)

where is Eulers constant (0.5772. . .).Various methods are available for determin-

ing Gumbel distribution parameters, including themethod of moments and a least squares fit proce-dure. The method of moments relates the distribu-tion parameters to the computed mean and standarddeviation for the variable X. The second method

uses a least squares linear fit between the variableXor its transformation (e.g. ln X) and the cumula-tive distribution function FX(X) or its transforma-tion (e.g. ln [ ln(FX(X))]). A significant advantageof the least squares method is that it can be appliedto a limited probability range of greatest interest.When simulating ship motions using the programFREDYN, a simulation will terminate when the shiproll angle exceeds a specified maximum value of 90degrees; thus, a Gumbel least squares fit provides a

suitable approach for handling the limited range ofvalid maximum roll angles.

For a random variable having a Gumbel distri-bution, a linear relationship exists between X andln [ ln(FX(X))]. The Gumbel parametersaX andbX can be determined by a least squares linear fit

minimizing the error in eitherXor ln [ ln(FX(X))].For predicting ship capsize risk, McTaggart [18]shows that it is preferable to minimize the error inln [ ln(FX(X))], which can be done using the fol-lowing equation:

ln [ ln(FX(X))] = (bX X)aX

(19)

When determining the least squares fit parametersfrom Ns simulations in a given seaway, the simu-lated maximum roll angles can be ranked in ascend-ing order to assign cumulative distribution values asfollows:

FX(Xi) = iNs+ 1

(20)

whereXiis the sorted maximum roll angle for givenconditions with rank i in ascending order. The lin-ear regression fit has a slope of1/abX and an in-tercept of bX/abX , where a

bX is a biased estimate of

aX , and bX is an estimate ofbX . Monte Carlo sim-ulation indicates that the least squares estimate bXhas negligible bias, and that a corrected estimate aXcan be obtained as follows:

aX = abX

1 0.64Ns

(21)

Once the Gumbel parameters have been determinedusing a least squares fit for a given seaway, the as-sociated probability of exceeding a specified capsizeroll angle (e.g., 90 degrees) can be easily estimatedusing Equation (16).

To reduce required computing time, it is usefulto use simulations of a shorter duration Ds to esti-mate statistics of maximum roll for a longer seawaydurationD. The following relationship can then beused:

FX(X(D)) = [FX(X(Ds))]D/Ds (22)

Simulation results indicate that simulation durationsof 30 minutes give good estimates of maximum rollstatistics for seaway durations of one hour. If a max-imum value from simulation durationDshas a Gum-bel distribution, then the maximum value for a sea-way duration D will have the following properties:

X(D) = X(Ds) (23)

X(D) = X(Ds) +

6

ln

D

Ds

X(Ds) (24)

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Figure 1 shows simulation results and fittedGumbel distributions for a frigate at a speed of 10knots, heading of 75 degrees, peak wave period of12.4 s, and wave height of 9.5 m. The wind iscollinear with the waves and has a steady speed ofof 40 knots, as given by Equation (13). The results

indicate that a Gumbel distribution provides a verygood fit to the conditional exceedence probabilityof maximum roll angle. An added benefit of usinga fitted distribution for each seaway is that Equa-tion (12) can be revised to give exceedence proba-bilities for all maximum roll angles in all seaways asfollows:

Qmax,D(max,D) =

NVsi=1

Nj=1

NHsk=1

NTpl=1

pVs(Vsi)p(j)

pHs,Tz(Hsk, Tpl)

Qmax,D |Vs,,Hs,Tp(max,D|Vs, , H s, Tp) (25)

Of major practical importance is the requirednumber of simulations for predicting capsize risk ingiven conditions. McTaggart [18] indicates that 10simulations of 30 minute duration can provide suffi-cient data for estimating hourly capsize probabilityin given conditions. Subsequent work examining avery wide range of conditions has indicated that alarger number of simulations (e.g., 20) should be per-formed if the hourly probability of capsize is greaterthan 0.001. In some cases with hourly capsize prob-

abilities greater than 0.001, the actual distributionof maximum roll angle can deviate significantly froma Gumbel distribution. To overcome this problem,the Gumbel fit can be limited to an upper rangeof maximum roll angles (e.g., the upper 30 degreerange) to ensure that the fitted distribution providesa good fit in the range of greatest interest for shipcapsize. For consistent results, the limited range ofroll angles should include at least 5 values from sim-ulations. Figure 2 gives examples of Gumbel fits toall data points and to the upper 30 degree range ofmaximum roll angles.

Estimates of capsize risk using fitted distribu-

tions have associated errors related to sample size(number of simulations). For estimates of GumbelparametersaXand bXobtained using a least squaresfit procedure, their variances have been estimatedusing Monte Carlo simulation:

2(aX) 1.45Ns

a2X (26)

2(bX) 1.16Ns

a2X (27)

The following equation can then be used to estimatethe sample size error of predicted capsize risk forgiven conditions:

2(P(CD|Vs, , H s, Tp)

P(CD|Vs, , H s, Tp)aX 2

2

(aX)

+

P(CD|Vs, , H s, Tp)

bX

22(bX) (28)

The following summation then provides an estimateof sample size error of predicted capsize risk for allconditions:

2(P(CD)) =

NVsi=1

Nj=1

NHsk=1

NTpl=1

p2Vs(Vsi)

p2(j)p

2Hs,Tp(Hsk, Tpl)

2(P(CD|Vs, , H s, Tp)) (29)It should be emphasized that Equations 26 to 29 pro-vide approximate estimates to the sample size errorin predicted capsize risk. The actual error in pre-dicted capsize error will likely be larger because theactual distribution of maximum roll angle deviatesfrom a Gumbel distribution.

To minimize the computational time for capsizerisk analysis, time domain simulations are only per-formed for conditions in which hourly capsize riskis non-negligible (> 106). For a given combina-

tion of ship speed Vs, heading , and peak waveperiod Tp, maximum roll statistics are computed forthe maximum significant wave height Hs given Tp.Additional time domain simulations are performedat lower significant wave heights until the predictedhourly capsize risk is below 106. For small waveheights, maximum roll statistics are assumed to beproportional toHsusing values from the lowest waveheight for time domain simulations.

SOURCES OF WAVE STATISTICS

The present risk analysis procedure requires the

joint distribution of significant wave height and peakwave period as input. Bales et al. [16, 21] havepublished wave scattergrams developed using theSOWM wave hindcast model. BMT Global WaveStatistics [22] is another commonly used data sourcefor wave scattergrams.

The present study originally used wave data forArea 15 (western portion of North Atlantic) fromBMT Global Wave Statistics; however, the resultingFREDYN simulations often experienced numerical

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0 30 60 90Hourly Maximum Roll max

0.01

0.1

1.0

Q(max)

Simulations......................................................... Fitted Gumbel

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................

Figure 1: Roll Exceedence Probability for Frigate in One Hour Seaway, Vs = 10 knots, = 75 degrees,Tp = 16.4 s, Hs = 9.5 m, Ua = 40 knots

0 30 60 90

Hourly Maximum Roll max

0.0001

0.001

0.01

0.1

1.0

Q(max)

Simulations............. ............. ....... Gumbel fit to all data points

......................................................... Gumbel fit to upper range

............. ............. ............. ............. ............. ............. ............. ..........................

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.. .............

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...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 2: Roll Exceedence Probability for Frigate in One Hour Seaway, Vs = 10 knots, = 45 degrees, Tp= 13.9 s, Hs = 9.5 m, Ua = 44 knots

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instabilities, and predicted capsize risks appeared tobe unrealistically high. It was postulated that theseproblems were caused by unrealistically high wavesteepnesses in the BMT data, which are based pri-marily on visual observations of wave conditions.

To address concerns regarding wave steepnesses

arising from BMT Global Wave Statistics, compar-isons were made with data from Bales et al. andBuckley [23]. Note that BMT uses zero-crossingwave period when presenting wave data while Baleset al. and Buckley use peak wave period. Whencomparing the different data sources, the followingequation based on a Bretschneider spectrum can beused:

Tp = 1.408Tz (30)

The introduction of nominal wave steepness

H/be-

low also facilitates comparison of data from the dif-

ferent data sources:

H/ = 2 Hsg T2p

(31)

Using numerous wave buoy observations, Buckleyhas developed limiting envelopes of significant waveheight versus peak wave period. Buckleys analy-

sis indicates that nominal wave steepnessH/ hasa maximum possible value of 0.049. Figure 3 givesmaximum nominal wave steepness versus peak waveperiod using using data from BMT, Bales et al.and Buckleys limiting envelope for the Northern

Hemisphere. The nominal wave steepnesses basedon the BMT data are much steeper than Buckleyslimiting envelope. In contrast, the North Atlanticdata from Bales give steepnesses which are consis-tent with Buckleys envelope, with the exception ofa single observation ofHs = 8.5 m and Tp = 9.7 s.For the present capsize risk analysis study, the BalesNorth Atlantic scattergram is used, with the Hs =8.5 m,Tp= 9.7 s observation changed toHs= 6.5 m,Tp = 9.7 s. Table 1 gives the adapted wave scatter-gram used for risk analysis in this paper.

EXAMPLE RISK ASSESSMENT FORHALIFAX CLASS FRIGATE

A risk assessment has been performed to eval-uate the hourly capsize probability for a CanadianHalifax class frigate (also know as Canadian PatrolFrigate) operating throughout the year in the NorthAtlantic. Table 2 gives dimensions for the frigate.The ship is assumed to capsize when the roll angleexceeds 70 degrees.

Table 2: Main Particulars for Halifax Class Frigate

Length,L 124.5 m

Beam,B 14.7 m

Midships draft, Tmid

4.64 m

Trim by stern, ts 0.0 m

Displacement, 4077 tonnesVertical centre of gravity, K G 6.44 m

Metacentric height,GMfluid 1.224 m

(corrected for free-surface)

Natural roll period, T 11.7 s

Input Probability Distributions

The ship has discretized calm water speeds of10, 17, and 24 knots, with a distribution as shownin Figure 4. Figure 5 shows the distribution of rela-tive wave headings, which are assumed to be evenlydistributed with a discrete interval of 15 degrees.The assumption of evenly distributed relative waveheadings is likely conservative because a ship willtypically alter course in severe conditions to reducethe risk of capsize. For numerical simulations, head-ings of 3.75 and 176.25 degrees are used rather thanthe nominal headings of 0 and 180 degrees, whichactually represent ranges of 0-7.5 and 172.5-180 de-grees respectively.

Annual wave climate statistics for the North At-lantic are taken from Bales [16, 21], as discussed inthe previous section. Figure 6 shows the distribu-tion of significant wave heights and Figure 7 showsthe distribution of nominal wave steepnesses.

Simulations of Motions for Relevant

Conditions

The first stage of the risk analysis procedure in-volved simulations of ship motions for relevant com-binations of ship speed, heading, significant waveheight, and peak wave period. The following guide-lines determined the number of 30 minute simula-tions for each condition:

The maximum wave height given peak wave pe-riod for simulations was based on the maximumheight of observed waves,

Ten initial simulations were performed for agiven condition. If the predicted hourly capsize

10


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0 10 20 30Peak Wave Period Tp (m)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

MaxNominal

WaveSteepnessH/max

......................................................... Bales North Atlantic

..... BMT Area 15............. ............. ....... Buckley Northern Hemisphere

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...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Figure 3: Maximum Nominal Wave Steepness Versus Peak Wave Period

Table 1: North Atlantic Wave Scattergram Adapted from Bales et al. for Capsize Risk Analysis

Wave Period Tp (s)Hs (m) 3.2 4.5 6.3 7.5 8.5 9.7 10.9 12.4 13.9 15.0 16.4 18.0 20.0 22.5 25.7

0.5 541 341 5488 5622 3892 4861 2667 2612 1180 1321 592 213 52 41.5 289 3929 6631 5475 5361 3627 3428 1617 1707 945 569 164 102.5 29 2150 5501 5695 3829 2968 1598 1267 780 421 124 33.5 5 822 4582 3873 3393 1457 1098 623 350 97 64.5 13 613 3115 3381 1293 1008 556 316 75 125.5 22 631 3014 1263 993 467 308 48 56.5 2 27 1336 1103 934 429 287 407.5 229 722 778 347 230 28 68.5 16 218 599 310 200 31 79.5 2 26 187 316 163 20 2 1

10.5 1 41 202 106 21 111.5 7 67 95 15 112.5 13.5 57.5 17.513.5 13.5 57.5 17.514.5 13 13 1.5 0.515.5 13 13 1.5 0.516.5 2.5 2.517.5 2.5 2.518.5 0.519.5 0.5

Total number of observations: 130,339

11


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0 5 10 15 20 25

Ship Speed Vs (degrees)

0.0

0.1

0.2

0.3

0.4

0.5

SpeedProbability

pVs(Vs)

Figure 4: Probability Mass Function for Discretized Ship Speeds

0 30 60 90 120 150 180

Ship Heading (degrees)

0.00

0.02

0.04

0.06

0.08

0.10

HeadingProbability

p()

Figure 5: Probability Mass Function for Discretized Ship Headings

12


13/24


14/24

probability exceeded 0.001, then the number ofsimulations was increased to 50.

For a given combination of ship speed, head-ing, and peak wave period, computations wereperformed for the largest wave height, followed

by computations for sequentially smaller waveheights. If the hourly capsize risk for a givenwave height was below 106, then linear re-sponse was assumed for smaller wave heights,with no time domain simulations being re-quired.

Each irregular seaway was modelled by aBretschneider spectrum, with uni-directional wavesbeing assumed. Winds were assumed to be steadyand collinear with waves, with wind speed given byEquation (13).

The motion simulations were performed on a 300MHz Dell Pentium II personal computer, which runsFREDYN approximately 15 times faster than realtime. The total required CPU time was approxi-mately 15 days.

Predicted Capsize Risk

Figure 8 shows hourly and annual exceedenceprobability versus roll angle. The annual exceedenceprobabilities are based on the hourly values with theassumption that the ship is at sea 33 percent of thetime. Assuming uniformly distributed ship headingsand that capsize occurs when maximum roll angle

exceeds 70 degrees, the hourly capsize probability is2107, with a corresponding annual capsize prob-ability of 6 104.

Figures 9 to 11 show conditional probabilitiesof hourly capsize given speed and heading. Whenall headings are considered, Figure 9 suggests thatthe safest ship speed is 17 knots. At the lowest shipspeed of 10 knots, the ship is likely prone to lossof steering control in severe seas and subsequentbroaching. The highest speed of 24 knots makesthe ship vulnerable to loss of static stability stabil-ity and surf-riding. Figure 10 indicates that cap-size risk is negligible for headings of 135 degrees and

greater. This finding is consistent with good sea-manship practice, which is to head into waves atmoderate speed when conditions are severe. Fig-ure 11 indicates that a speed of 24 knots and headingof 75 degrees is the most dangerous speed-headingcombination for the present ship, although otherspeed-heading combinations also have relatively highrisk levels.

Table 3 gives conditional hourly capsize proba-bilities given significant wave height and peak wave

period. Note that the highest capsize probabilitiesare along the limiting wave heights for each waveperiod. Figures 12 to 14 give conditional probabil-ities of significant wave height, peak wave period,and nominal wave steepness given capsize. The con-ditional probabilities in each of these three figures

have a sum of unity. Figure 12 indicates that signif-icant wave heights of 9.5 m and larger present thegreatest capsize risk. Similarly Figure 14 suggeststhat a nominal wave steepnesses of 0.04 and is mostlikely to be the cause of capsize.

Closer Examination of Most Likely

Conditions to Cause

One of the most useful benefits of risk analysisis that it indicates how likely a given combination ofship speed, heading, and wave conditions is to causecapsize. This likelihood is given by the following

conditional probability given capsize:

P(Vs, , T p, Hs|CD) =pVs(Vs)p();pHs,Tp(Hs, Tp)

P(CD|Vs, , T p, Hs)P(CD)

(32)

The most likely conditions given capsize can indicatewhat conditions should be avoided by a ship oper-ator. From a research perspective, the most likelyconditions given capsize indicate where effort shouldbe directed to ensure accuracy of capsize prediction

methods. Table 4 gives the ten most likely sets ofconditions given capsize for the example frigate. Theconditions in this table account for 78 percent of thetotal predicted capsize risk of the ship. The meanand standard deviations of hourly maximum roll arebased on Gumbel distributions fitted to the upper 30degree range of maximum roll angles from 30 minutesimulations. Where necessary, the range of maxi-mum roll angles has been expanded to include atleast ten data points from simulations.

Figure 15 gives plots of hourly exceedence prob-ability and maximum wave slope versus maximumroll angle for the four most likely conditions given

capsize. The fitted Gumbel distributions give goodagreement with maximum roll angle from simula-tions for the upper range

The wide range of maximum roll angles shownin Figure 15 prompted further investigation into whythe range of maximum roll angles can vary so greatlyfor simulations of given ship speed, heading, peakwave period, and significant wave height. It wasfound that maximum roll angle can be very depen-dent upon the maximum absolute wave slope angle

14


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0 30 60 90

Maximum Roll Angle max (deg)

107

106

105

104

0.001

0.01

0.1

1.0

ExceedenceProbability

Q(max)

......................................................... One hour

............. ............. ....... Annual

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Figure 8: Hourly and Annual Exceedence Probabilities for Maximum Roll

0 5 10 15 20 25

Ship SpeedVs (degrees)

108

107

106

105

ConditionalCapsize

ProbabilityPC|Vs(C|Vs)

Figure 9: Conditional Hourly Probability of Capsize Given Speed

15


16/24

0 30 60 90 120 150 180

Ship Heading (degrees)

109

108

107

106

105

ConditionalCapsize

ProbabilityPC|(C|)

Figure 10: Conditional Hourly Probability of Capsize Given Heading

0 30 60 90 120 150 180

Ship Heading(degrees)

109

108

107

106

105

104

ConditionalCapsize

ProbabilityPC|Vs,(C|Vs, )

10 knots

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Figure 11: Conditional Hourly Probability of Capsize Given Speed and Heading

16


17/24

Table 3: Conditional Hourly Probability of Capsize Given Significant Wave Height and Peak Wave Period

Conditional probabilities for wave height and period combinations

Cell values are capsize probability in seaway given Hs and Tp

Blank cells wave-speed combination

Cell value 0 indicates < 1E-9

Hs | Wave Period Tp (s)

(m) | 3.2 4.5 6.3 7.5 8.5 9.7 10.9 12.4 13.9 15.0 16.4 18.0 20.0 22.5 25.7

0.5 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1.5 | 0 0 0 0 0 0 0 0 0 0 0 0 0

2.5 | 0 0 0 0 0 0 0 0 0 0 0 0

3.5 | 0 0 0 0 0 0 0 0 0 0 0

4.5 | 1E-9 0 0 0 0 0 0 0 0 0

5.5 | 0 0 0 0 0 0 0 0 0

6.5 | 3E-4 2E-6 0 0 0 0 0 0

7.5 | 2E-9 0 0 0 0 0 0

8.5 | 8E-7 0 0 0 0 0 0

9.5 | 0.008 7E-9 5E-9 0 0 0 0 0

10.5 | 0.001 2E-8 0 0 0 011.5 | 4E-6 5E-9 4E-9 0 3E-9

12.5 | 7E-8 2E-6 0

13.5 | 4E-6 1E-6 0

14.5 | 9E-6 3E-9 6E-7 6E-6

15.5 | 3E-4 4E-5 4E-7 6E-6

16.5 | 2E-4 7E-5

17.5 | 5E-4 3E-4

18.5 | 0.001

19.5 | 0.005

4.5 6.5 8.5 10.5 12.5 14.5 16.5 18.5

Significant Wave Height Hs (m)

0.0

0.2

0.4

0.6

ConditionalWave

HeightProbability

pHs|C(Hs|C)

Figure 12: Conditional Probability of Significant Wave Height Given Capsize

17


18/24

9.7 10.9 12.4 13.9 15.0 16.4 18.0 20.0 22.5 25.7

Peak Wave Period Tp (s)

0.0

0.2

0.4

0.6

0.8

ConditionalWave

PeriodProbability

pTp|C(Tp|C)

Figure 13: Conditional Probability of Peak Wave Period Given Capsize

0.00 0.01 0.02 0.03 0.04 0.05

Nominal Wave SteepnessH/0.0

0.1

0.2

0.3

0.4

0.5

0.6

ConditionalWave

SteepnessProbability

pH/|C(H/|C)

Figure 14: Conditional Probability of Nominal Wave Steepness Given Capsize

18


19/24

Vs = 10 knots, = 75 degrees, Tp = 12.4 s, Hs = 9.5 m, Ua = 40 knots

0 30 60 90


0.01

0.1

1.0

Q(max)

..........................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................

0 30 60 90

Maximum Roll max (deg)

10

20

30

max(deg)


0 30 60 90


0.01

0.1

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Figure 15: Hourly Exceedence Probability and Maximum Wave Slope Angle Versus Maximum Roll Anglefor Most Likely Conditions Given Capsize

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Table 4: Most Likely Conditions Given Capsize

P(Vs, , T p, Hs|C) Vs Tp Hs P(C|Vs, , T p, Hs) Hourly max roll (deg)(knots) (deg) (s) (m) Hourly Mean

0.181 10 75 12.4 9.5 0.0587 43.3 15.40.166 10 45 12.4 9.5 0.0539 42.3 15.3

0.125 10 90 12.4 9.5 0.0405 45.8 11.9

0.106 10 60 12.4 9.5 0.0343 37.3 15.1

0.076 24 75 18.0 15.5 0.0380 50.1 9.5

0.036 10 45 13.9 10.5 0.0234 23.7 18.8

0.031 24 75 25.7 19.5 0.4040 69.8 3.1

0.023 24 60 18.0 15.5 0.0115 46.8 7.7

0.020 24 90 25.7 19.5 0.2614 68.8 2.5

0.017 24 60 20.0 17.5 0.0443 56.7 6.8

encountered during a simulation. Figure 15 indi-cates that a very large maximum roll angle is usuallyassociated with a large wave slope angle. The mostinteresting aspect of Figure 15 is the very high sensi-tivity of maximum roll angle to wave slope at highervalues ofmax. This high sensitivity indicates thatspecial attention should be given to accurate mod-elling of steep wave conditions. Unfortunately, thepresent FREDYN numerical model uses simple su-

perposition of linear waves to simulate a randomseaway. For real ocean waves, wave deformationswhich occur at large amplitudes will cause higher lo-cal wave slopes, likely causing increased capsize risk.One encouraging aspect of the results in Figure 15 isthat the wave slope angles never exceed 30 degrees,the breaking limit for a fifth-order Stokes wave (seeSarpkaya and Isaacson [24]).

APPLICATION TO SHIP DESIGN AND

OPERATION

The presented risk analysis procedure is suitablefor application to both ship design and operation.The general nature of the procedure means that itcan be used if reliable input data and an accuratetime domain ship motion model are available.

Direct Application to Design

The presented risk analysis procedure can beused to determine whether a given ship meets a pre-

scribed acceptable level of safety against intact cap-size. Table 5, citing data from Reference 25, givessome typical risks which can be used to select anappropriate target risk level. It appears that an ac-ceptable annual risk level for ship capsize would beapproximately 104. Using Equation (14), the cor-responding acceptable hourly capsize risk is 4108for a ship at sea 30 percent of the time.

Table 5: Annual Probability of Early Death (fromReid [25])

Cause Annual Risk (106)Air travel 9

Coal mining 210

Construction work 210

Road accidents 300

All causes, man aged 30 1000

The risk analysis presented in the previous sec-tion raises the question of how the dependence ofcapsize on ship heading should be considered in de-sign. The assumption of uniform heading distribu-tion for a ship in all conditions is overly conservativebecause a ship captain will go to safer headings un-der severe conditions; however, there is still a pos-sibility that a ship will be operated at dangerous

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headings in severe waves. One possible approach isto specify one acceptable risk level that is based ona uniform distribution of headings, and a second ac-ceptable acceptable risk level that is based on theship maintaining a limited range of headings. Forexample, it could be prescribed that a ship must

meet both of the following requirements:

1. For uniform distribution of headings, hourlycapsize risk must be less that 105,

2. For headings within 30 degrees of head seas,hourly capsize risk must be less than 108

It should be emphasized that the risk levels givenin the above criteria are merely initial estimates ofsuitable values.

A more rigorous approach to treating the influ-

ence of operator action on capsize risk is to con-sider ship speed and heading to be conditional uponwave conditions. For example, the following equa-tion treats distributions of ship speed and headingto be conditional upon significant wave height:

P(CD) =

NVsi=1

Nj=1

NHsk=1

NTpl=1

pVs|Hs(Vsi|Hsk)

p|Hs(j |Hsk)pHs,Tp(Hsk, Tpl) P(CD|Vs, , H s, Tp) (33)

Ongoing work examining ship operational pro-files will provide data for determining the condi-tional probability distributions pVs|Hs(Vs|Hs) and

p|Hs(|Hs).As indicated previously, approximately 15 days

of computer CPU time was required to perform thenumerical simulations for the frigate risk example.For practical application, it is very desirable to re-duce the elapsed computing time to approximatelyone day. Faster computer processors will continueto reduce the time required for time domain simula-tions of ship motions. Processors currently availablefor desktop computers are approximately three times

as fast as the processor used in the present calcula-tions. Parallel processing offers great potential forreducing required time for a risk analysis. Desktopcomputers with four processors are becoming com-monly available, and could likely reduce the timerequired for a risk analysis by a factor of approxi-mately three. Considering all of the above factors,it is likely that the required elapsed computer timefor a capsize risk analysis could be reduced to ap-proximately two days within the near future.

Modifications to Existing Ships

Risk analysis can assist with making decisionsregarding modifications to existing ships. For war-ships, modifications are often proposed which wouldincrease the height of the centre of gravity. Theinfluence of proposed modifications on ship stabil-ity could be examined by comparative risk analysis.Results from risk analysis can also be used to iden-tify modifications which could possibly make a shipsafer. For example, the roll exceedence probabilitiesgiven in Figure 8 for the Halifax class suggest thatthe capsize risk level could be reduced by a factor ofapproximately ten by increasing the angle of down-flooding from 70 to 90 degrees.

Development of Simple Design Criteria

Due to the complexity of quantitative risk anal-ysis, it is important to have simple design criteria

as an alternative. The simple design criteria mustensure that the resulting ship will achieve a targetlevel of safety. For naval frigates, de Kat et al. [8]have proposed the following simple criteria:

1. The righting lever (GZ()) in calm water mustremain positive up to an angle of at least 90degrees,

2. The positive area under the righting lever curvemust exceed a specified value dependent on ver-tical prismatic coefficient CV P. For CV P 0.55, the required positive area is 1.00 mrad.For C

V P 0.70, the required positive area is

0.67 mrad. For intermediate values of CV P,linear interpolation should be used.

Simple design criteria such as those given above areintended for a specified range of ship parameters.Application of risk analysis to a number of repre-sentative ships can ensure that the simple designcriteria give adequate levels of safety.

Operator Guidance

Risk analysis results can provide very useful in-formation to ship operators regarding which combi-

nations of speed and heading present the greatestrisk of capsize. Alman et al. [26] discuss develop-ment of operational guidelines based on time domainsimulations. The risk analysis for the Halifax classfrigate indicates that an operator can reduce capsizerisk to negligible levels by steering a heading within45 degrees of head seas.

Work is currently underway to provide real-timeguidance to ship operators for avoidance of danger-ous combinations of speed and heading. Proposed

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systems would combine current sea state informa-tion with a database of predicted ship motions. Fig-ure 16 gives an example of output from a guidancesystem for a severe seaway with significant waveheight of 9.5 m and peak wave period of 12.4 s. Theshaded areas in Figure 16 represent capsize risk lev-

els predicted for the Halifax class frigate. Despitethe severity of the seaway, the operator still has awide range of safe heading-speed combinations.

CONCLUSIONS

A procedure has been developed for determin-ing capsize risk of intact ships in irregular seaways.The availability of a fast and accurate program forpredicting ship motions makes the procedure viablefor practical application. The maximum roll angleof a ship in an irregular seaway is very dependent onwave process realization. This dependence is mod-

elled using a Gumbel fit to maximum roll angles fromseveral different wave process realizations.

Sample calculations for a naval frigate demon-strate the application of the method. The requiredcomputational time of approximately 15 days couldlikely be shortened considerably using faster proces-sors and parallel processing.

Results from risk analysis can be used directlyto determine whether a ship meets a target level ofsafety against capsize. Other applications includedevelopment of simplified design guidelines for givenship types and provision of operational guidance foravoiding conditions likely to cause capsize.

REFERENCES

[1] D. Blockley, ed., Engineering Safety, McGraw-Hill, London, 1992.

[2] L.K. Kobylinski, Rational Approach to ShipSafety Requirements, in Second InternationalConference on Marine Technology (Szczecin,Poland, 1997).

[3] A.E. Mansour, P.H. Wirsching, M.D. Lucket,A.M. Plumpton, and Y.H. Lin, StructuralSafety of Ships,Transactions, Society of NavalArchitects and Marine Engineers105 (1997).

[4] T.H. Sarchin and L.L. Goldberg, Stabilityand Buoyancy Criteria for U.S. Naval SurfaceShips, Transactions, Society of Naval Archi-tects and Marine Engineers70, 418458 (1962).

[5] K.A. McTaggart, Risk Analysis of Intact ShipCapsizing, in United States Coast Guard Ves-

sel Stability Symposium 93(New London, Con-necticut, March 1993).

[6] K.A. McTaggart, Capsize Risk Prediction In-cluding Wind Effects, in Proceedings of theThird Canadian Marine Hydrodynamics and

Structures Conference (Dartmouth, Nova Sco-tia, August 1995), pp. 115124.

[7] J.O. de Kat, The Development of Surviv-ability Criteria Using Numerical Simulations,in United States Coast Guard Vessel Stabil-ity Symposium 93(New London, Connecticut,March 1993).

[8] J.O. de Kat, R. Brouwer, K. McTaggart, andL. Thomas, Intact Ship Survivability in Ex-treme Waves: Criteria from a Research andNavy Perspective, in STAB 94, Fifth Inter-

national Conference on Stability of Ships andOcean Vehicles(Melbourne, Florida, 1994).

[9] K. McTaggart and M. Savage, Wind HeelingLoads on a Naval Frigate, in STAB 94, FifthInternational Conference on Stability of Shipsand Ocean Vehicles(Melbourne, Florida, 1994).

[10] S.K.Chakrabarti, Hydrodynamics of OffshoreStructures, Springer-Verlag, 1987.

[11] J.O. de Kat and W.L. Thomas III, ExtremeRolling, Broaching, and Capsizing Model

Tests and Simulations of a Steered Ship inWaves, in 22nd Symposium on Naval Hydro-dynamics(Washington, August 1998).

[12] J.O. de Kat and W.L. Thomas III, Broach-ing and Capsize Model Tests for Validation ofNumerical Ship Motion Predictions, in FourthInternational Stability Workshop (St. Johns,Newfoundland, September 1998).

[13] J.O. de Kat, Irregular Waves and Their In-fluence on Extreme Ship Motions, in Twenti-eth Symposium on Naval Hydrodynamics(Santa

Barbara, 1994).

[14] E.A. Dahle and D. Myrhaug, Risk AnalysisApplied to Capsize of Smaller Vessels in Break-ing Waves, Transactions, Royal Institution ofNaval Architects136, 237252 (1994).

[15] E.A. Dahle and D. Myrhaug, Capsize Risk ofFishing Vessels, Schiffstechnik (Ship Technol-ogy Research) 43, 164171 (1996).

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[16] S.L. Bales, W.T. Lee, and J.M. Voelker,Standardized Wave and Wind Environ-ments for NATO Operational Areas, ReportDTNSRDC/SPD-0919-01, DTNSRDC, July1981.

[17] K.A. McTaggart, Ship Capsize Risk in a Sea-way Using Time Domain Simulations and Fit-ted Gumbel Distributions, in 18th Interna-tional Conference on Offshore Mechanics andArctic Engineering - OMAE99 (St. Johns,Newfoundland, July 1999).

[18] K.A. McTaggart, Ship Capsize Risk in a Sea-way Using Fitted Distributions To Roll Max-ima, Transactions of the ASME, Journal ofOffshore Mechanics and Arctic Engineering(ac-cepted for publication).

[19] H.O. Madsen, S. Krenk, and N.C. Lind,Meth-

ods of Structural Safety, Prentice-Hall, Engle-wood Cliffs, New Jersey, 1986.

[20] P. Thoft-Christensen and M. J. Baker, Struc-tural Reliability and Its Applications, Springer-Verlag, 1982.

[21] S.L. Bales, Development and Application of aDeep Water Hindcast Wave and Wind Clima-tology, in Royal Institute of Naval ArchitectsWave and Wind Climate World Wide Sympo-sium(1984).

[22] British Maritime Technology Limited, Global

Wave Statistics, Unwin Brothers, London,1986.

[23] W.H. Buckley, Stability Criteria: Develop-ment of a First Principles Methodology, inSTAB 94, Fifth International Conference onStability of Ships and Ocean Vehicles (Mel-bourne, Florida, 1994), Vol. 3.

[24] T. Sarpkaya and M. Isaacson, Mechanics ofWave Forces on Offshore Structures, Van Nos-trand Reinhold, 1981.

[25] S.G. Reid, Acceptable Risk, in EngineeringSafety, edited by D. Blockley (McGraw-Hill,London, 1992).

[26] P.R. Alman, P.V. Minnick, R. Sheinberg, andW.L. Thomas III, Dynamic Capsize Vulnera-bility: Reducing the Hidden Operational Risk,Transactions, Society of Naval Architects andMarine Engineers107 (1999).

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