subdivision

Subdivisionandstability

REPORT

KaiyeHu

CollegeofShipbuildingEngineering

HarbinEngineeringUniversity,China,2007

CONTENTS

uction.................................................................................2

lframeworkofnewregulations...............................................2

2.1OverallIndexofSubdivision...........................................................2

2.2Localindicesofsubdivision...........................................................3

2.3RequiredSubdivisionIndex...........................................................4

hodtocalculatepi..............................................................4

3.1Probabilityoffloodingspacesforshipswithtransversubdivisiononly.....4

3.2Examples................................................................................7

hodtocalculatesi............................................................12

4.1Example................................................................................18

References...................................................................................22

1

uction

Theprobabilisticconceptofshipsubdivisionisbadontheprobabilisticasssment

ofstabilityforamultitudeofpossibledamagesprovidingtheprobabilityforashipto

isionisunderstoodasthepartitioningofa

ship’sinternalvolumeintowatertightcompartmentsinordertolimitthequantityof

ntssuchas

collision,,

however,mayalsoentertheshipduetomanyotherreasonssuchasstructuralfailure,

negligence,internalpipingfailure,firefighting,ntrolledfloodingoccurs,

then,withoutadequatesubdivision,the

shiptypicallyinvolveslossoflife,lossofcargo,andpossiblereleaofquantitiesof

eriousconquencesof

floodingcanbesubstantiallyreducedifadequatesubdivision,throughwell-placed

watertightbulkhead,isprovidedinthedesignofships.

InternationalMaritimeOrganization(IMO)providesmanyregulationsforjudgingthe

ship’dominantIMOtreatydocumentaffectingshipsubdivisionand

damagestabilityistheInternationalConventionfortheSafetyofLifeatSea(1974

SOLAS).Ithassofarbeenratifiedby131countriesandappliestomorethan98%of

wordmerchantshippingtonnage.

lframeworkofnewregulations

Asisknown,manyfactorsaffectingthefinalconquencesofshiphulldamageare

sreason

probabilityofcollisionsurvival(indexofsubdivision)istakenasameasureofship

safetyinthedamagedcondition-measureofmeritofaship're

twoindicesthatcanbepostulatedforjudgingtheeffectivenessofaship'ssubdivision:

zOverall(global)indexofsubdivision,reflectingtheaveragedegreeofsubdivision

forthewholeshipthatdenotesamean(conventional)probabilityofsurvivalforthe

wholeshipincaofaccidentalflooding.

zLocalindicesofsubdivisionoftwokindsof(notinuintheregulationsyet),

reflectingthedegreeofsubdivisionforindividualpartsoftheship,whichdenote

meanprobabilitiesofsurvivaleitherforthecasoffloodinginwhichagiven(wing,

ifany)compartmentisflooded,or-inwhichagiventransverbulkheadis

sreasonthediscussionisfocudonone-compartmentindicesof

merexpresstheso-called

one-compartmentstandardonthegroundoftheprobabilisticapproach;the

latter-thetwo-compartmentstandardsoregardedbythepractitioners.

2.1OverallIndexofSubdivision

Theactualdegreeofsubdivisionprovidedforashipisdeterminedbytheprobabilityof

dexshouldbe

:

RA>(1)

2

Theattainedsubdivisionindex"A"shallbecalculatedfortheshipbythefollowing

formula:

ii

spAΣ=forIi∈(2)

where:

"i"reprentachcompartmentorgroupofcompartmentsunderconsideration,

"p

i

"accountsfortheprobabilitythatonlythecompartmentorgroupofcompartments

underconsiderationmaybeflooded,disregardinganyhorizontalsubdivision,

"s

i

"accountsfortheprobabilityofsurvivalafterfloodingthecompartmentorgroupof

compartmentsunderconsideration,includingtheeffectsofanyhorizontalsubdivision.

“I”isthetofallfeasiblecasofflooding,comprisingsinglecompartmentsand

groupsofadjacentcompartments.

BecautheattainedsubdivisionindexAistheentireprobability,therefore

1=Σ

i

p(3)

thatis,thesumofprobabilitiesofallcasoffloodingequals1.

Thefactorsp

i

aregenerallynotaproblemfromthetheoreticalpointofviewasthey

tpurpoitis

nec

otherextremeisthefactors

i

cannotbederivedusingstatisticalmethodsonly,asit

accountsforstabilityandshipdynamicinthedamagedcondition.

2.2Localindicesofsubdivision

Theuoftheglobalmeasureofmeritaloneisnotsufficientforregulatorypurpo.

Twodifferentshipswiththesameoverallindexofsubdivisionareobviouslyofequal

overallsafetywithrespecttoflooding,althoughtheshipsmayhavequitedifferent

actualcapabilitiesforwithstandinghulldamageinsomepartsoftheirlengththatisnot

neutralforthequalityofsubdivision.

Topreventsuchanunsatisfactorysituation,thebasicglobalmeasureofmerit

(subdivisionindexA)shouldbesupplementbyitslocalcounterpartstheshowhowthe

probabilityofsurvivalisdistributedalongtheship’alindicesof

subdivisionwillindicatedirectlyifanypartoftheshipisleftwithunacceptable

vulnerabilitytoflooding.

Thistypeofrequirementisbestdonebytheuofso-called(partial)indicesof

subdivisionoftwokinds,givenbytheequations:

i

ii

jp

sp

A

∑

∑

=fornj,....2,1=,(4)

∑

∑

==

+

i

ii

jjjp

sp

AB

1,

for1,....2,1−=nj.(5)

TheA

j

quantitiesreprentone-compartmentindicesandB

j

two-compartmentindices

merreflectstheuniformityofsubdivisionalongtheshiplength,

3

whereasthelatterareameasureoftheship’sabilitytosurvivedamageinwayofa

bulkhead(leasttwoadjacentcompartmentsareflooded).Theyarealso

referredto,particularlyattheIMOcircles,astheminordamageindices,orthe

bulkheadindices.

2.3RequiredSubdivisionIndex

Thedegreeofsubdivisionprovidedforashipisdeterminedbyarequiredvalueofthe

probabilityofcollisionsurvival,

beobtainedonthebasisofnumericalvaluesoftheattainedindicesofsubdivision

laadoptedbytheIMOforpasngershipsisas

follows:

1500)4(

1000

1

++

−=

NL

R(6)

whereN=N

1

+N

2

isanassumednumberofpersonsonboardtheship,N

1

isthe

numberofpersonsforwhomlife-boatsareprovided,N

2

isatotalnumberofpersons

onboardtheshippermittedtocarryinexcessofN

1

,andListhesubdivisionlengthof

theshipinmeters.

Fordrycargoships,IMOadoptedinitiallythesameequationasforpasngerships,

butwiththeomissionofN,modifyingitsubquentlyto,arriving

finallyattheformula:

3/1)001.0(LR=

3/1)0009.0002.0(LR+=(7)

hodtocalculatep

i

3.1Probabilityoffloodingspacesforshipswithtransversubdivisiononly

Toanalysthesimplestcaofsubdivision,itisonlynecessarytoconsiderthe

locationandlengthofdamageinthelongitudinaldirection,assumingnolongitudinal

andhorizontalwatertightstructuraldivisionxist.

WiththenondimensionaldamagelocationLx/=ξandnondimensionaldamage

lengthLl/=λ,asdefinedinFigure1,allpossibledamagescanbereprentedby

pointsinanisoscelestriangle,showalsointhisfigure,whobaandheightare

iangleisthusthedomainofthetwo-dimensionalrandomvariable

(λξ,).ThelengthLisunderstoodasasubdivisionlengthoftheship.

4

Figure1:DomainsG

i

correspondingtodamagesopeningsinglecompartment

(triangles)andgroupsofadjacentcompartment(parallelograms)

Sincethelocationandsizeofthedamagesarerandomvariables,itisnotpossibleto

r,theprobabilityoffloodinga

spacecanbedeterminediftheprobabilityofoccurrenceofcertaindamageisknown,

babilityoffloodingaspace,

boundedbyundamagedwatertightstructuraldivisions,equalstheoccurrenceofall

,inmathematicalterms

∫=

i

G

ii

fdGp(8)

inwhichthejointdistributiondensityfisintegratedoverthedomainG

i

enclosing

zonesinthetrianglereferredtoabove,correspondingtodamagesopeningthespace

sityfunctionfistermedinprobabilitytheoryasthe

probabilitydensityfunction(pdf).Itdenoteshowthetotallikelihoodoftheshipbeing

damaged,equalto1,isdistributedoverthewholedamagedomain.

AlldamagesthatopensinglecompartmentsoflengtharereprentedinFigure

1bypointsintriangleswith;thusthetrianglesarethedomainsudfor

i

l

LlJ

ii

/=

5

spointsinany

parallelogramreprentalldamagessimultaneouslyopeningthecompartments

,theparallelograms

arethedomainsudforcalculatingtheprobabilitiesofsimultaneousfloodingof

groupsofcompartments.

Thefollowingrelationshipscanbedirectlyinferredfromequation(8)andFigure1

fortheprobabilityoffloodinganarbitrarynumberofadjacentcompartments:

forcompartmentstakenbypairs:

etcpppp

pppp

i

i

,

3223

2112

−−=

−−=

(9)

forcompartmentstakenbygroupsofthree:

etcppppp

ppppp

i

i

,

33423234

22312123

+−−=

+−−=

(10)

forcompartmentstakenbygroupsoffour:

etcppppp

ppppp

i

i

,

343452342345

232341231234

+−−=

+−−=

(11)

wheretheindicesassignedtopindicatetheprobabilityoffloodingasingle

compartmentwhonondimensionallengthJcorrespondstothatofagroupof

rwords,thefactorattributableto

acompartmentgroupisobtainedbyacombinationoffactorscorrespondingto

adehereoflinearityofintegration.

i

p

i

p

Itisclearfromequation(8)andFigure1thatforanydistributiondensityfatthe

range

max

,0Jthefactorisalwayspositiveforasinglecompartmentandapair

ofadjacentcompartments,whereisthemaximumnondimensionaldamage

oupoftheormoreadjacentcompartmentsthefactorsmayequal

zeroifthenondimensionallengthofsuchagroupwithouttheoutermost

rword,thefactor,ifthe

wholedomaincorrespondingtosuchagroupliesabove

i

p

max

J

max

J0=

i

p

i

G

maz

J=λ,wherethe

densityfunctionfvanishes.

Ascanbeenfromequation(9-11),inordertocalculatethefactorforagroup

ofadjacentcompartments,itissufficienttoknowdetailedformulatefordetermination

oftheprobabilitypoffloodingasinglearbitrarycompartment,asthefactoris

insuchformulate,the

i

p

i

p

6

two-dimensionaldensityfunction),(λξfisindispensableandthisfollowsdirectly

fromequation(8).Thisdensityfunctioncanbederivedfromdamagestatistics.

3.2Examples

ernow

probabilityoffloodingatransvercompartment(ofanylengthandlocationalongthe

ship’slength)aca,theexpression

forthefactorcanbeobtainedstraightaway,byintegrationofthedensityfunction

foverthetriangledomain,correspondingtoagivencompartment,asshownin

Figure1.

i

p

i

p

i

G

First,considerauniformdensityfunctionf=constattheentiredomainG(thebig

triangleinFigure1).

Theintegraloffovertheentiredomainequals1,thatis

1=⋅Gf(12)

fromFigure1,weknowthattheareaoftheentiredomain5.011

2

1

=⋅⋅=G,sof=2,in

suchaca,equation(8)yieldsimmediatelythefollowingexpressionforthe

factor:

i

p

222JGdGfdGp

i

GG

i

ii

====∫∫(13)

/=

Letustrytofindnowmarginaldistributions,whicharenormallyobtainedfromthe

redistributionsobtainedbydisregardingtheotherrandom

variable(s),i.e,ematicalterms,

theyareobtainedbycalculatingtheelementaryareaorvolumerelatedtoonevariable.

Hence,inourcathemarginaldistributionofdamagelengthisgivenby

)1(2)(

2/1

2/

λξλ

λ

λ

−==∫−

fdf(14)

whereasthemarginaldistributionofdamagelocationisgivenby

⎪

⎪

⎩

⎪

⎪

⎨

⎧

∈−=

∈=

=

∫

∫

−ξ

ξ

ξξλ

ξξλ

ξ

22

0

2

0

1,5.044

5.0,04

)(

fd

fd

f(15)

7

Figure2:Marginaldistributionsincaofauniformpdfattheentiredomain

ascanbeen,thedensityfunctionfisintegratedoverahorizontalcross-ction

whileinthecondca,

marginaldistributionsare

een,theyfollowtheshapeofthedomainperceived

fromtherespectiveaxis,withamultiplicationfactorequalto2.

era

uniformdistributiondensityfinsideastripofwidth,his

strip(i.e.,for

max

J

max

J>λ)thedensityfunctionvanishes(i.e.,0=f).Whereasf=const

gure3,wecangettheareaofthestrip(trapezoid):

2

maxmaxmax

max

2

1

)

2

11

(JJJ

J

G−=⋅

+−

=(16)

considertheequation(12),thedensityfunctionis:

12

maxmax

)

2

1

(−−=JJf(17)

8

Figure3:Uniformdistributionatpartofthedomain

obviously,f>2becauthedensityfunctionisdistributednowoverasmallerareathan

tillf=const,therefore,whereisactiveareaofthe

ii

Gfp⋅=

i

G

,)(

2

1

2

1

,

2

1

2

max

2

max

2

otherwiJJJG

JJifJG

i

i

−−=

<=

(18)

thereforethefactorisgivenby

i

p

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

−

−

≤

−

=

otherwi

J

y

JJfor

J

y

p

i

1

2

12

1

2

max

max

max

2

(19)

whereisanormalizedlengthofthecompartment.

max

/JJy=

Themarginaldistributionofdamagelengthisgivenby

max

,0)1()(Jforff∈−⋅=λλλ(20)

wherefisgivenbyequation(17).

Hereisanexampleaboutmarginaldistribution,considerauniformprobability

densityfunctionf=constforatwo-dimensionaldamage,whonondimensional

lengthλ

max

.IfthenondimensionallengthequalsJ,assumeJ

max

=et

9

themarginaldistributionsofthenon-dimensionaldamagelengthanddamagelocation.

Followingisthesolution.

Fromequation(17),wecanget

2

maxmax2

1

1

JJ

f

−

==4.15282(21)

forthemarginaldistributions∫−−⋅==2

1

2

)1()(

λ

λ

λξλffdf,andbecauJ

max

=0.28,

so

max

)1(15282.4)(Jandf<−=λλλ(22)

Wecanalsoget:

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎨

⎧

−∈−=

−∈⋅=

∈⋅⋅=

=

∫

∫

∫

−)1(2

0

max

0

maxmaxmax

2

0

max

]1,

2

1

1[,)1(2

)

2

1

1,

2

1

(

)

2

1

,0[2

)(max

ξ

ξ

ξξλ

ξλ

ξξλ

ξ

Jffd

JJJffd

Jffd

fJ

(23)

Figure4:marginaldistribution1

10

Figure5:marginaldistribution2

Figure4andFigure5aregraphsofthemarginaldistributionsbaontheequation

(22)andequation(23).

Abouttheoverallandlocalindicesofsubdivision,wehaveanexample:Suppoa

shiphastransverbulkheadsatthefollowingpointsξ:0.05,0.20,0.35,0.50,0.65,

0.80,torsp

i

ands

i

fortheship(J

max

=0.24)areasfollows:

No.12345678

p

i

0.01270.04450.06230.08020.08910.08910.08910.0358

single

comprt.

s

i

11111111

p

i

0.02800.05160.06870.08440.08580.08580.0673

pairsof

comprt.

s

i

10.950.900.800.760.901

p

i

0.00240.00350.00460.00510.00510.0049

groups

oftriple

s

i

0.1500000.12

Table1:Thefactorsp

i

ands

i

forthisship

wecancheckwhethertheprobabilitiesformacompleteprobabilityfromtable1,the

resultsareshowninTable2.

Singlecompartment.∑1i

p=0.5028∑1ii

sp=0.5028

Pairsofcompartments.∑2i

p=0.4716∑2ii

sp=0.416098

Groupsoftriple∑3i

p=0.0256∑3ii

sp=0.00948

Sumup:∑∑i

p=1∑ii

sp=0.919846

Table2:Results

11

becau=1,sothisisacompleteprobability.∑i

p

FromTable2,wecangettheattainedindexofsubdivision:

A=∑ii

sp=0.919846(24)

WecanfindthemostvulnerablepartofthisshipfromTable3,badonthe

equation(4):

i

ii

jp

sp

A

∑

∑

=fornj,....2,1=.

FromTable3,wecanfindwhenn=5,hasthesmallestvalue,thispartisthe

mostvulnerablepartoftheship.

j

A

WecanalsofindthemostvulnerablebulkheadfromTable4,badontheequation

(5):∑

∑

==

+

i

ii

jjjp

sp

AB

1,

for1,....2,1−=nj.

n12345678

ii

sp0.012700.044530.062340.080160.089060.089060.089060.03582

i

p0.012700.044530.062340.080160.089060.089060.089060.3582

j

A

0.95210.93710.89840.8502

0.8096

0.84200.92900.96006

Table3:values

j

A

n1234567

ii

sp0.027990.048950.061830.067570.065270.077290.06734

i

p0.027990.051530.068700.084460.085880.085880.06734

j

B

0.93210.85770.80510.7182

0.6798

0.81300.9410

Table4:values

j

B

whenn=5,hasthesmallestvalue,sothispartisthemostvulnerablebulkhead.

j

B

hodtocalculates

i

Thes

i

factorisdifficulttoget,becauwecan‘tgets

i

factorfromstatisticalmethods.

ProfessorPawlowskidevelopedaveryufulmethodtocalculates

i

factor:TheStatic

EquivalencyMethod(SEM),nowI’llintroducethismethod.

TheStaticEquivalencyMethod(SEM)isbadonanumberofinsightsand

esumedthatitistheaccumulationofwateronthevehicledeckthat

12

caustheshiptocapsize,ratherthanthebasicdamagedstabilityintheflooded

conditionasmeasuredbytheSOLAScriteria.

Therequiredcapsizevolume(orweight)ofwaterondeckisassumedtobethat

whichwouldcautheshiptololltoitsangleofmaximumGZinthefloodedcondition.

Anyadditionalheelwiththisvolumeondeck,oranyadditionalvolumeatthesame

heelangle,llberesistedbyasmaller

,theshipwillinevitablycapsize.

Thedepthofwaterondeckatthecriticalconditioncorrespondstoanelevation

abovethemeanexternalalevel,andthilevationcan,inturn,becorrelatedwith

cwaveenergyis,ineffect,transformedinto

potentialenergy.

Theprocesscanbetreatedquasi-statically,asthetimeframesassociatedwith

ceffectsdo

needtobeaccountedforbothinthestabilitycalculationapproachandinthe

correlationofwaterelevationandwaveheight.

Thestabilitycalculationsneededtopredictcapsizewatervolumecanbe

und

arerelativelyeasytounderstandbutdifficulttoapply,whileothersaresimpletou

report,I’llintroducethreemethods.

a):Elevatedwater

Figure6:Stabilityofadamagedro-roveslwithariofwateronthecardeck

Theelevationofwaterabovealevelatthecriticalpointforcapsizecanbeeasily

visualizedasafunctionofheelangleφ,usingthelostbuoyancyorconstant

eenfromFigure6,therightingmomentisproduced

stoneiscreatedbyweightoftheship,passingthroughits

centerofgravityG,balancedbybuoyancyforceDwhichpassthroughthecenterof

13

ercoupleiscreatedbyweightoftheelevatedwaterondeck,

balancedbyachangeinbuoyancy

duetosinkageoftheshipdT,whichisappliedatthecenterofflotationofthe

damagedwaterline(thecentroidofthewaterplanewithoutthepartoccupied

bythefloodedwater).Hence,therightingmomentisgivenasfollows:

el

p

el

C

WLD

F

MlpGZD

elel

=−⋅(25)

Where:

D–displacement(weight)oftheship

∇–volumedisplacementoftheship.

GZ–rightingarmcalculatedbytheconstantdisplacementmethod,allowingforfree

floodingofthevehicledeck;

el

p–weightofwaterelevatedabovealevel=;

el

gv

el

l–heelingleverduetoelevatedwaterondeck,equaltothehorizontaldistance

betweenthecenterofflotationandcenterofgravityofelevated

water.

WLD

F

el

C

Theadditionalamountofwaterondeck,elevatedabovealevel(shadingin

Figure6)issuchthattheresultantrightingmomentM=ng

Equation(25)throughoutbythedensityofawater,thefollowingisobtained:

GZlv

elel

⋅∇=(26)

whereisvolumeofwaterelevatedabovealevel.

el

v

Equation(26)iquation

venheelangle,

therighthandsideoftheequation,GZ⋅∇,isknownfromroutinestability

calculations,whichallowandthushtobefoundprecilybyaniterative

ecalculationsthesinkageoftheshipdTisneededthatcanbe

foundfromachangeinvolumedisplacement:

el

v

WLdel

AvdT/=(27)

whereisthemeanwaterplanearea,excludingthepartoccupiedbythewater.

WLd

A

b)Addedwater

14

Thecalculationprocedurefortheheelingmomentcanbesomewhatmodifiedby

takingintoaccounttheentireamountofwaterondeckabovethewaterline

(beforesinkage).Theheelingmomentinsuchacaiscreatedbyweightofthe

additionalwaterondeck,abovethewaterline(darkandlightgreyinFigure

7),passingthroughitscentreofgravity,balancedbychangeofbuoyancydueto

ngeofbuoyancyisappliedatthecentroidofthe

undamagedlayer(includingthepartoccupiedbythefloodedwaterondeck,denoted

bylightgrey,butwithoutthedamagedpartbelowthedeck)cutoffbywaterlinesWL

and

0

WL

ad

p

0

WL

ad

C

ad

F

0

WL

Figure7:Alternativewayofheelingmomentcalculation

AsthesinkagedTistypicallyverysmall,thecentroidcoincideswiththecentre

clearthatislargerthanbytheamountofwatercontainedbetweenthetwo

waterlines,theircentresofgravityandarenotthesame,thelocationsof

heless,sinkageof

sonisthatweaddazero

forceconsistingofweightofwaterondeckandchangeofbuoyancycontained

ad

F

WL

F

ad

p

el

p

ad

C

el

C

WL

F

WLd

F

15

treason

0

WL

adadelel

lplp=,and,

whereisweightoftheadditionalwaterondeckabovethewaterline,is

theheelingleverduetotheadditionalwaterondeck,equaltothehorizontaldistance

betweenthecentreofflotationofwaterline(undamagedabovethecar

deck)andthecentreofgravityoftheadditionalwaterondeck,measured

perpendicularlytothesameaxisofrotationasbefore.

adadelel

lvlv=

ad

p

0

WL

ad

l

WL

F

0

WL

ad

C

Conquently,alltheexpressionsfortheheelingmomentbadontheelevated

waterintheforegoingequationscanbereplacedbythobadontheadditional

kagedTisgivenbytheequation:

WLad

AvdT/=(28)

whereisthedamagedwaterplanearea,includingthepartoccupiedbythe

waterondeck.

WL

A

Thesolutionofequation(26)venheelangleφthe

right-handsideofequation(26)-GZ⋅∇isknown,forwhichthewaterlineis

alsoknown,includingstaticcharacteristicssuchasdraught,trim,thearea

(or)lastquantitieshavetobedeterminedforthe

shipwiththeundamaged(ordamaged)ther

hand,theleft-handsideofequation(26)-isafunctionof

0

WL

WLd

A

WL

A

WLd

F

elel

lv

h

′

-theelevationof

iredwaterelevationis

determinedwhenequation(

0

WL

h

′

GZ⋅∇)issatisfied,andthatcanbeeasilyfound

y,thewaterelevatedwithrespecttoalevelh,ofprime

importanceforthedamagedsafety,isfoundfromasimplerelation:,

wherethesinkageoftheshipisdefinedbyequation(27)or(28).Intheequations

(or)isknownfunctionof

dThh−

′

=

el

v

ad

v

h

′

.

c)Totalwater

Mostfrequently,however,thecalculationsareperformedbyuofvarioussoftware,

atprentmainlybyNAPA,wherespaceonthevehicledeckistreatedlikean

undamagedtank,aca,theheeling

momentiscreatedbyweightofthetotalamountwaterondeck(darkandlight

greyinFigure8),passingthroughitscentreofgravity.

3

p

3

C

16

Figure8:Calculationofheelingmomentbytypicalsoftware

Theentireweightoftheship,includingweightofwaterondeck,isbalancedbythe

entirebuoyancyforcebelowtheactualwaterplane,includingbuoyancy

givenbytheundamagedvehicledeck(lightgreyinFigure8)Theentirebuoyancy

forcepassthroughthecentreofbuoyancyfortheshipdamagedonlybelow

,theresultingmoment(restoring)isgivenby:

33

pDD+=

3

B

3333

lpGZDM−=(29)

whereistherightingarmforafreelyfloatingship,withtheundamagedvehicle

deck,isweightoftotalwaterondeck,andheelingleverduetowaterondeck,

equaltothehorizontaldistancebetweencentreofgravityoftheintactshipGand

centreofgravityofwaterondeck,measuredperpendicularlytotheaxisof

shipatequilibrium,theaboveequationyields

3

GZ

3

p

3

l

3

C

3333

GZVlv=(30)

whereistotalvolumeofwaterondeck,andisvolumedisplacementoftheship

withtheundamagedvehicledeck.

3

v

3

V

Theoutcomeofthecalculationsforgivenamountofwaterondeckistheangleof

lollφ,trim,waterhead,freeboardattheopeningattheheelangle(depthofthe

deckedge),volumeoftheelevatedwater,gvolumeofelevatedwater

h

f

el

v

17

thesinkageoftheshipdTcanbefoundfromequation(27).Next,runningsoftware

again,thetraditionalrightingleverGZcanbefound,correspondingtothedamaged

waterplane(beforeparallelsinkage)withfreefloodedwateronthevehicledeck,

asshowinFigure6.

0

WL

4.1Example

Hereisanexampleofabox-shapedro-rovesl,wecanfindit’sfactorsandthe

(mean)ndimensions

forthisbox-shapedro-roveslareasfollows:

LengthL...............................................................143.00m

BeamB................................................................27.50m

HeightH8.00m

DraughtT5.75m

BlockcoefficientC

B

...........................................................1

KG(=z

G

)...............................................................12.00m

suppothattheshipwithamidshipscompartmentfloodedoflengthl=16,5mand

withpermeability1=μ.Assumethatthefloodedcompartmentbelowthecardeck

hasnodoublesidesandnodoublebottom,whereasthevehicledeckextendsover

bovethecardeckhasnosidecasings(thisshipis

prentedinFigure9).

f

h

δT

N

F

WL

B

WL

WL

0

Z

C

ad

D

p

ad

h'

N

G

Figure9:Stabilityoftheshipwithariofwateronthecardeck

First,

rightingarmGZforabox-shapeveslisgivenbytheequation:

GZ=r

0

m(φ)–asinφ (31)

18

inequation(31),r

0

istheinitialmetacentricradius,a=z

G

–z

B

isaheightofthecentre

ofgravityoverthecentreofbuoyancyatanuprightpositionoftheship,andthe

functionm(φ),reprentingthestabilityofform,isgivenasfollows:

m(φ)=(1+½tan2φ)sinφfor φ≤φ

1

,(32)

m(φ)=6t(cosφ–tsinφ)–8t3/2cos2φ/√sin2φfor φ∈〈φ

1

,φ

2

)(33)

wheret=F/B,Fisthefreeboard,tanφ

1

=2t,tanφ

2

=H2/(2BF).

Whentheshipflooded,wecangetthedamageddraught:

dam

T

m

l

L

LT

T

dam

5.6

5.16143

143*75.5*

=

−

=

−

=(34)

so,theinitialmetacentricradiusr

0

is:

m

T

B

r

dam

696.9

5.6*12

5.27

*12

22

0

===(35)

and

m

T

zdam

B

25.3

2

==(36)

mzza

BG

75.825.312=−=−=(37)

wecanalsogett,φ

1

andφ

2

fromthefollowingequation:

05455.0

5.27

5.1

===

B

F

t(38)

()0

1

23.62arctan=⋅=tφ(39)

0

2

2

8.37)

2

arctan(==

BF

H

φ(40)

knowingtheinitialmetacentricradiusr

0

,a,φ

1

andφ

2

,fromequation(31)-equation

(33),wecandrawit’sGZcurve,showninFigure10.

19

Figure10:GZcurve

Fromthiscure,wecanfindtheangle

max

φ,whereamaximumGZoccurs,and

vanishangle:,.Wecanalsofindthemaximumrightingarm:

m.

0

max

75.6=φ081.9=

van

φ

014.0

max

=GZ

Thewidthofthecardeckabovethewateris:

398.26

tan

2

==

α

BF

bm(41)

inequation(41),

180

max

πφ

α

⋅

=istheradianof

max

φ.

so,thewidthofthecardeckbelowthewateris:

102.1=−=bBcm(42)

themergedfreeboardis:

125.3tan=⋅=αbFe

merged

m(43)

wecancalculatethedraughtofthedeckedgerelativetoWL

0

:

130.0sin

0

=⋅=αcdm(44)

theareaofWL

0

is:

36.3521

cos0

=

⋅−⋅

=

α

blBL

A

WL

m2(45)

thecentroidofWL

0

fromtheship’ssideis:

777.13

cos)(

)2/(2/2

=

⋅⋅−⋅

−⋅⋅−⋅

=

′

αblBL

bBblBL

F

WL

m(46)

thecentroidofWL

0

fromtheverticallineNN(showninFigure9)is:

20

762.13tan

0

=⋅−

′

=αdFF

WLWL

m(47)

thewidthofWL

0

abovethecardeckis:

110.1cos/==

′αccm(48)

thecentroidofWL

0

abovethecardeckrelativetonormalNNis:

540.0tan

2

1

0

=⋅−

′

⋅=

′αdcF

c

m(49)

so,thewidthofthefreesurfaceofaddedwateris:

183.5/

0

=⋅

′

=

′′

ddccm(50)

tocalculatethevolumeoftheaddedwater,wecanuthefollowingequation:

99.213)(

2

)(

0

=−⋅

′′

+

′

⋅

=dd

ccL

V

ad

m3(51)

wecangettheheightofthecentroidofaddedwateraboveWL

0

:

289.0

3

2

0=

−

⋅

′′

+

′

′′

+

′

=

′

dd

cc

cc

h

ad

m(52)

andthecentroidofaddedwaterrelativetonormalNN:

743.1

0

0=

′

⋅

+

′

=

c

ad

ad

F

d

dh

Fm(53)

theheelingmomentduetotheaddedwateris:

712.2571)(=−⋅=

adWLadad

FFVMnm(54)

so,thesinkageoftheshipafterdamageis

061.0/

0

==

WLad

AVTδm(55)

theelevatedwaterheightis:

415.0)(

0

=−−=Tddhδm(56)

themeancriticalastateis:

4.3)

085.0

(3.1

1

==

h

H

s

m(57)

thefactorscangetfromthefollowingequation:

990.0)0148.06301.24095.27494.0(3/123=++−=xxxs(58)

inequation(58),.4/

s

Hx=

21

References

ski,M:calUniversityofGdansk,

2004

skiMandVassalosD:RiskCharacterizationoftheRequiredIndexRintheNew

dingsofthe8thInternationalShipStability

Workshop,Istanbul,October2005.

ationalMaritimeOrganization:SubdivisionandStability,PartBofChapterⅡ-1in

theInternationalConventionfortheSafetyofLifeatSea,SOLAS1974,IMO,London.

ck,ski,:FloodingProtectionofRo-RoFerries,Pha

ortCanada,March1998.

22