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Address Carreswmimm t.:

NQval.% S@?ms Command.%xn*, Ship StFx:hue Cormntk=

Miky .%M Command{/.s cccw Guard .Hfid<~&G!,s,(<;.>.!/E!2)

hfimbe AdmhistmihjWidl@Rm, L?c X5$90

bhtd .%&6 (koloyd Sl@QyAme#icml Eh53u d Sflipphg

&gfip&@

An InteragencyAdvisory~ornm]tiesDedicatedtoImpmvingtheStructureofShips

The Ship Structure Cmwnittee has beer,Iconcerned tiit.hthe various parameters that influence the hLt.Llfkxi”b:ilityof ships. An earlier study, which was reported on in shipStructure Committee Report 249, reviewed different skip .J~.brat.ior!prediction methods and evaluated the effects at hull sti~’fr,es:+variati[m on vibratory response. The resu~l:.s ~KXhlCl?C~modeshapes and frequencies induced by w;Ivesand the prupe..ll.e:rsinthree specific ships.

SSC-288

FINALREPORT

on

ProjectSR-1239

“RationalLimitof”HullFlexibility”

TELSEFFECTSOFVARYINGSHIPHULLPROPORTIONSANDHULLMATERIALSQNHULLFLEXIBILITY,BENDINGAND

VIBRATORYSTRESSES

iiy

P.Y.Chang

Hydronautics,Inc.

under

Eeparh’ErltofThll+3titimUnitedStatesCoastGuardContractNo.DOT-CG-61906-A

-.-,

U..S.CoastGuardHeadquartersWastii:ngton;D.C.

1979

TechnicalIteportDocumentationPage

r===~’”’”- 3. Rec,ptent’sCntologNq,r_——-——~—.4. T,!le qnd5ubt,tle

.——. —

THEEFFECTSOFVARYINGSHIPHULL

T

5. RvportDote

PROPORTIONSANDHULLMATERIALSONHULLFLEXIBILITY, April1979—----13END1NGANDVIBRATORYSTRESSES 6. Pcrforrn~n3Orgnnizatio.Code

-—.—l——8. PerformingOrgaoizotionReportNo.7. Author(s) PinYuChang Report7715-1

I9. Ptrforrn,r,gOrgoniz”.t,onNom.md Addre<s

.—— —.10 warkUnlfNo.fTRAIS)1~,h,ngton,,c,o,go

Hydronautics,Inc.Laurel 11, ComrocI o. GrontNe.

Maryland20810 nnTm-- 61qofi-A13, Typcof ReportondPvr,odCovered

-.—...—-12. SponsoringAgencyNameandAdd,es*

ShipStructureCommittee FinalReportU.S.CoastGuardHeadquarters —.Offi:eofMerchantMarineSafety 14, SponsoringAgencyCOJ.

!&M —-15. Stipplemen+o,y NOI.S

L-m=---Theeffectofvaryingshipproportionsandhull

materialsonhuqlflexibilityandontheconcomitantbendingandvibratorystressesforanorecarrier,atanker,containership,anda generalcargoshipisevaluated.

Withtheflexibilityoftheship’shullrepre-sentedbythenaturalfrequencyoftheshipassociatedwiththetwo-nodeshape,a potentiallyusefulrelationbetweer~theflex:6~litYandbendingmomenthasbeenestablished’.

Ananalysisindicatesthatforwardspeedaffectshydrodynamicdampingandforcesaswellashullflexibility,andtheremayexistanoptimalflexibilityforeveryship,butthereisnotnecessarilya limittotheflexibility.

I 17. KeyWords 18. D~stribut,on$?oiement

DocumentavailabletothepublicthroughtheNationalTechnicalInformationService,Springfield,Virginia22161

19. SecurityClo.sif. (ol thisreport) 20. SecurityClassIf. (ofthispoge)

lJNCLASSIFIFD UNCLASSIFIED ~FormDOTF 1700,7(8-72) Reproductionofcomplctcd page authorized

L

,:.

SHIPSTRUCTURECOMMITTEETheSHIPSTRUCTURECOMMITTEEisconstitutedtoprosecutea research,

programtoimprovethehullstructuresofshipsandothermarinestructuresbyanextensionofknowledgepertainingtodesign,materialsandmethodsofconstruction.

RADMH.H.Bell (Chairman) Mr.M.PitkinChief,OfficeofMerchantMarine AssistantAdministratorfor

Safety CommercialDevelopmentU.S.CoastGuardHeadquarters Marit?meAdministration

Mr. l?.M.Palermo Mr.R.B.KrahlAssistantforStructures Chief,BranchofMarineOilandNavalShipEngineeringCenter GasOperationsNavalSeaSystemsCommand U.S.GeologicalSurvey

Mr.W.N.Hannan Mr.C. J. Whit@StOneVicePresident ChiefEngineerAmericanBureauofShipping MilitarySealiftCommand

I.CDRT.H.Robinson,U.S.CoastGuard(Secretary)

SHIPSTRUCTURESUBCOMMITTEETheSHIPSTRUCTURESUBCOMMITTEEactsfortheShipStructure

Committeeontechnicalmattersbyprovidingtechnicalcoordinationforthedeterminationofgoalsandobjectivesoftheprogram,andbyevaluatingandinterpretingtheresultsintermsofstructuraldesign,constructionandoperation.

U.S.COASTGUARD

Cdr.J.C.CardLcdrS.H.DavisCaptC.B.GlassDr.W.C.Dietz

NAVALSEASYSTEMS

Mr.R.ChiuMr.R.JohnsonMr.G.SorkinMr.J.B.O’Brien

COMMAND

(ContractsAdmin.)

MARITIMEADMINISTRATIONMr.l?.J.DashnawMr.N.O.HammerMr. l?’.SeiboldMr.M.Touma

NATIONALACADEMYOFSCIENCESSHIPRESEARCHCOMMITTEE

Mr.O.H.Oakley- LiaisonMr.R.W.Rumke- Liaison

SOCIETYOFNAVALARCHITECTS&MARINEENGINEERS

ljr.A.B.Stavovy- Liaison

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U.S.MERCHANT

MILITARYSEALIFTCOMMANDMr.T.W.ChapmanMr.A.B.StavovyMr.D.SteinMr.J.Torresen

AMERICANBURXAUOFSHIPPING

Dr.H.Y.JanMr.D.LiuMr.T..L.SternMr.S.G.Stiansen(Chairman)

U.S.GEOLOGICALSURVEYMr.R.GiangerelliMr.J.Gregory

INTERNATIONALSHIPSTRUCTURESCONGRESSProf.J.H.Evans- Liaison

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U.S.COASTGUARDACADEMYCaptW.C.Nolan- LiaisonU.S.NAVALACADEMYDr.R.Battacharyya- Liaison

MARINEACADEMYDr.Chin-BeaKim- Liaison

-iii--..

METRICCONVERSIONFACTORS

AgproximMeConversionstoMet!icMeasures Appro~imateConversionsfromMetiicMeasuresWh@nYouKnDw Mulliplyby

LENGTH

ToFindSvmb0!whenYmJMnow Mukiplyby la Find $rmbal

LENGTH millm!aers 0.04cent,mews 0.4reelers 3.3IIMms >.1kilmnclers 0.6

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inches .2.5feet 30yards 0.9mile, 1.6

Jmcmmkm

cmzIi#,7dha

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zquarecentifremr5 0.16quarometers 1.2squarokiIcmctws 0,4heclaws(I0,00Un12) 2.5

wuareinclms 6.5sq,,rnfeet 0.09sq,,,,eyards 0.8squaremiles 2.6.Ic,Gs 0.4

Id,<1 MASS(weight]MASS{weight)

Qmnls 0.035kilograms 2.2tonnes[1000kg! 1.1

m ricespoundsd,orttails

orlbounces 29po,w?ds 0.65shorl!ons 0.9(2000lb!

Ozib

VOLUMEVOLUME”tl 0!F.tq!g.tI?yd3

m!tI

milliliters 0.03Iiiers 2.1Iilws 1.06Iite,s 0.26cubicmeters 35cubicnwers 1.3

teaspoms 5mblespmms 15IEu,dOUllC*S 30cups 0.24pints 0.47q,arls 0,95galIons 3,8cub,cfeet 0.03cubicyards 0.76

rmll,lit.rsmillilitersm,ll,l,wrsIitwsI,!W5IitersIilcrscubicmetermchicn>eters

15pTbspfl02cplql

TEMPERATURE[esacl}Celsius 9/5(thenTEMPERATURE{exact~ “c

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1’ ,1 r f I , 1 1 ? 1 , 1-:: -20 0 3.0 40 60 00 tOD

37 Qc

TABLEOFCONTENTS

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

PageINTRODUCTION

1.1 Objectives....................................... 11.2 Smaryof Findi~gs.............................. 1

THESTATE-OF-THE-ARTOFSHIP-VIBMTIONANALYSIS....... 4

2.1 ShipVibration-A HydroelasticProblem.......... 42.2 ProblemAreasinExistingSeawayResponseAnalysis.5

A HYDROELASTICFORMULATIONOFTHESHIP-VIBRATIONPROBLEM...............................................

3.1 ExistingMethods..,..............................3.2 ComparisonAmongtheExistingMethods...........3.3”TheEffectofForwardSpeedonShipMotions......

METHODOLOGY...........................................

4.1 SelectionofSeaSpectra.........................4.2 EquationsofMotion.............................4.3 ThePredictionofDamping........................4.4 DeterminationoftheEffectsofShip

ProportiononHullFlexibility...................4.5 DeterminationofMaximumWaveLoads..............4.6 EffectsofHullMaterials........................

SELECTIONOFREPRESENTATIVESHIPSFORANALYSIS........

5.1 GreatLakesOreCarri”erSTEWARTJ.CART..........5.2 264,000DWTTanker...............................5,3 “C4’’GeneralCargoVessel........................5.4 “C6”and“C8”Containerships.....................

COITPUTATIONRESULTS...................................

DISCUSSIONSOFTHEMETHODOLOGYANDRESULTS............

7.1 Methodology......................................7.2 Results..........................................

CONCLUSIONS,APPLICATIONSANDRECOMMENDATIONS.........

8.1 Conclusions.......................................8.2 Applications.....................................8.3 Recommendations..................................

REFERENCES.................................................

10

101214

16

161618

191932

35

:;4242

55

56

5656

60

60

;:

67

-v-

Figure1

Figure2

Figure3

Fig~re4

Figure5

Figure6

Figure7

Figure8

Figure9

Figure10

Figure11

Figure12

Figure13

Figure14

Figure15

Figure16

Figure17

Figure18

Figure19

LIST

EffectofHullBendingMoment

OFFIGURES

FlexibilityontheVertical

SignificantWaveHeightvs.PeakEnergyFrequency.

TotalDampingasa functionofF~oucleNumber.

DampingCoefficientsUsedByVariousInvestigators.

EffectofL/BonDeflectionforGreatLakesOreCarriers.

EffectofL/BonBendingStressforGreatLakesOreCarriers.

EffectofL/BonVerticalBendingMomentforGreatLakesOreCarriers.

EffectofL/BonLateralandTorsionalMomentfor .GreatLakesOreCarriers.

EffectofL2/BI~onDeflectionforGreatLakesOreCarriers.

.EffectofL2/BI*onBendingStressforGreatLakesOreCarriers.

EffectofforGreat

EffectofLakesOre

Effectof

Effectof

Effectof

L2/BI~onLateralandTorsionalMomentLakesOreCarriers.

L2/BI~-onVerticalMomentforGreatCarriers.

L/Bon

L/Bon

L/BonTankVessels.

EffectofL/Bon

DeflectionforTankVessels.

BendingStressforTankVessels.

VerticalBendingMomentfor

TorsionalandLateralBendingMomentforTankVessels,

EffectofL2/BI~onDeflectionforTankVessels.

EffectofL2/BI~onBendingStressforTankVessels.

EffectofL2/BItonVerticalMomentforTankVessels.

-vi-

Figure20

Figure21

Figure22

Figure23

Figure24

Figure25

Figure26

Figure27

Figure28

Figure29

Figure30

Figure31

Figure32

Figure33

Figure34

Figure35

EffectofL2/BI~onLateralandTorsionalMomentforTankVessels.

EffectofL/BonDeflectionforCargoShips.

EffectsofL/BonVerticalBendingStressforCargoShips.

EffectofL/BonVerticalBendingMomentforCargoShips.

EffectofL/BonTorsionalandLateralBendingMomentforCargoShips.

EffectofL2/BI~onDeflectionforCargoShips.

EffectofL2/BI~onVerticalBendingStressforCargoShips.

EffectofL2/BI~onVerticalBendingMomentforCargoShips.

EffectofL2/BI:-onTorsionalandLateralMomentforCargoShips.

EffectsofB/TonVerticalBendingMomentforCargoShips.

EffectofFullLoad

EffectofFullLoad

Effectof

L/BonDeflectionforContainerships,Condition.

L/BonBendingStressforContainerships,Condition.

L/BonVerticalBendin~MomentforContainerships,FullLoadCondit;on.

EffectofL/BonTorsionalandLateralMomentforContainerships,FullLoadCondition.

1EffectofL2/BITonDeflectionforContainerships,FullLoadCondition.

1EffectofL2/BITonBendingStressforContainerships,FullLoadCondition.

-vii-

—..

ligure36

Figure37

Figure38

Figure39

Figure40

Figure41

Figure42

Figure43

Figure44

Figure45

Figure46

EffectofL2/BI~onVerticalBendingMomentforContainerships,FullLoadCondition.

1

EffectofL2/BI:onLateralandTorsionalMomentforContainerships,FullLoadCondition.

EffectofWaveHeightandHeadingonSeaLoadsof’GreatLakesOreCarrier,FullLoadCondition.

EffectofWaveHeightandHeadingonSeaLoadsofTanker,FullLoadCondition,

EffectofWaveHeightandHeadingonSeaLoadsofGreatLakesOreCarrier,BallastCondition.

EffectofWaveHeightandHeadingonSeaLoadsofTanker,BallastCondition.

EffectofWaveHeightandHeadingonSeaLoadsofCargoShip,FullLoadCondition.

EffectofWaveHeightandHeadingonSeaLoadsofCargoShip,BallastCondition.

EffectofWaveHeightandHeadingonSeaLoadsofC6Containership,FullLoadCondition.

EffectofWaveHeightandHeadingonSeaLoadsofC8 ,.Containership,FullLoadCondition.

EffectofShipProportionsontheHullFlexibility(RepresentedbytheTwo-NodeFrequency).

,..-vlll-

LISTOFTABLES

1.

2.

3.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

ApproximateMeasuresofCorrelationBetweenTheoryandExperimentforHeadSeas.

ApproximateMeasuresofCorrelationBetweenTheoryandExperimentforFollowingSeas.

ApproximateMeasuresofCorrelationBetweenTheoryandExperimentforBowSeas(Headings120to1500).

ApproximateMeasuresofCorrelationBetweenTheoryandExperimentforQuarteringSeas(Headings30to60°).

ProposedVariationinDimensionsofGreatLakesVessel“StewartJ.Cort”.

PropertiesofGreatLakesOreCarrier“StewartJ.Cort”. FullLoadCondition.

PropertiesofGreatLakesOreCarrier“StewartJ.Cort”. BallastCondition.

VariationofProportionsandResponsesofGreatLakesOreCarriers.FullLoadCondition.

VariationofProportionsLakesOreCarriers.Ba”

ProposedVariationsinD264,000DWTTanker.

andResponsesofGreatlastCondition.

mensionsof

Propertiesof264,000DWT“TIO”Tanker.FullLoadCondition.

Propertiesof264,000DWT“TIO”Tanker.BallastCondition.

VariationofProportionsandResponsesofTankVessels.FullLoadCondition.

VariationofProportionsandResponsesofTankVessels.BallastCondition.

ProposedVariationsinDimensionsofC4-S-69b.GeneralCargoVessel.

PropertiesofC4-S-69bGeneralCargoVessel.FullLoadCondition.

-ix-

LISTOFTABLES(cONT.)

17.

18.

19.

m.

21.

22.

23.

24.

PropertiesofC4-S-69bCargoVessel.BallastCondition.

VariationofProportionsofGeneralCargoVessels.FullLoadCondition.

VariationofProportionsandResponsesofGeneralCargoVessels.BallastCondition.

ProposedVariationsinDimensionsof“C6”and“C8”FamilyofContainerships.

PropertiesofC6-S-85aContainership.FullLoadCondition.

PropertiesofC8-S-85dContainership.FullLoadCondition.

VariationofShipProportionsandResponseforContainerships.FullLoadCondition.

RelationBetweenShipProportionsandHullFlexibilityfortheTwo-NodeFrequency.

-x-

LISTOFSYMBOLSANDABBREVIATIONS

Thesymbolsandabbreviationsdefinedinthetextaftertheequationsmay “ ..

A

AIYai’ ai’ ai

A I~l,Y.si‘ sl

notberepeatedhere,

Shearareaofshipsection.

Area,momentofinertia,anddistancefromtheneutralaxisoftheithaluminummember.

“thsteelArea,mom~ntofinertiaofthe1member,andthedistancebetweenitscenterofgravityandtheneutralaxisofthewholeshipsection.

B

BM

B%

BMT

BMV

c

CB

c c C3,C5o’ s“

D

Defltn

E

G

gGM

I,11

Beamof

Bending

Lateral

shipsection.

momentamplitude.

bendingmomentamplitude.

Torsionalbendingmomentamplitude,

Verticalbendingmomentamplitude(wave+stillwater).

Totaldampingcoefficientofshipsectionassociatedwithverticalmotion.

Blockcoefficient

Dampingcoefficients/lengthasdefinedaftertheequationsinthetext.

Depthofship.

Shipmaximumdeflection.

Modulusofelasticity.

Shearmodulusofelasticity.

Accelerationofgravity.

Shipmetacentricheight.

Momentofinertiaofshipsection.

.Xi-

IIo’ my+<I

L

M

msma

N

P

SM

T

t

u

v

w

W’)fi

x

a

A

Aa

Massrotarymomentofinertiaofshipsection/length.

Equivalentmomentofinertia.

Lengthofship.

Bendingmoment

Massofship/length.

Addedmassofwater/length.

Hydrodynamicdampingcoefficientofshipsection.

Axialforce.

Sectionmodulus

Draftofship,

Time.

Forwardspeed

Shearforce.

“Displacement,bodymotion.

w’==”~x,w=

ofship.

ofship.

includingdeflectionandrigid

awm

Coordinatealongthelongitudinalcenterline.

Headingangleofship.

Displacementinlongtons.

Addeddisplacementduetotheaddedmassofwater.Foreandaftattitude,includestrimandelasticslope.Wavesurfaceelevationrelativetostillwater.

-xii-

me

Un

=m~+ma

Poisson’sratio.

Densityofwater.

Verticalbendingstress.

Naturalfrequencyofship.

Firstortwo-nodefrequencyofship.

Encounterfrequency.

Frequencyofnthmode.

-xili-

-1-

1.0 INTRODUCTION

1.1 Objectives

Shipboardvibrationhasbeenamajorproblemforshipbuildersandoperators.Vibratorystressesadverselyaffectshipstructuresandequipment,reducefatiguelifeofa ship,andimpaircrewoperations.Atthistimetherearenogenerallyacceptedlimitingstandardsorcorrespondingdesignproceduresforassessinghullvibration,dueinpa~ttothelackofunderstandingoftherela-tionshipbetweenshipproportionsandhullvibration.Accordingly,theobjectiveofthisstudyi~todeterminetheeffectsofshipproportionsonhullflexibilityandtoestablishsuitablecriteriaforhull-vibrationlimits,suchasa limittothehullflexibility.

1.2 SummaryofFindings

Themethodologyadoptedforthisstudyisbasedontwoassump-tions.First,itisgenerallybelievedthattheexistingmethodsfordeterminingtheseawayloadsareadequate.Secondly,itisbelievedthatshipswithmoreflexibilityare~.nferiortostiffershipswithrespecttohullvibration.Thesetwoassumptionsaregenerallyacceptedandarebasedonreliableinformation.Forexample,in1970,Salvesen,TuckandFaltinsenpublishedtheirpaperonsealoads(l),whereinthecomparisonbetweentheanalyti-calandexperimentalresultsaregenerallyquitegood.

Theoretically,forthesamesealoads,moreflexibleshipsaregenerallysubjectedtohigherstress.Forthisreason,amoreflexibleshipis,indeed,inferiortoa stiffership.However,studyresultsreportedhereindifferconsiderablyfromthesetwoassumptions. First,manyshortcomingshavebeenfoundintheexistingmethodsofanalysisandthecorre$pendingerrorsindicateexistingmethodologymaybeinadequateforsomeproblems.Secondly,resultsindicatetheflexibilityoftheship’shullisnotnecessarilyanundesirableproperty.A moreflexibleshipcanactuallybesaferthana stiffership.Forthesereasons,a limittoflexibilityhasnotbeenestablished.Fromtheresultsobtainedinthisstudy,theinvestigatorstendtobelievethatthereexistsanoptimalflexibilityforeveryship,butthereisnotnecessarilya limi~totheflexibility.Thisconclusionwillbediscussedindetailinthefollowingsectionsofthisreport.

Theprimarystudyobjectiveofdeterminingtheeffectsofvariationsofshipproportionsonhullflexibilityandvibratoryresponsesforfourshiptypes,havebeenachieved.Theshipproportionsaredefinedbytwonondimensionalparameters:The

-.2-

length-beamratio,L/BandL2/BI~.Theeffectsofthedepth,D,areincludedinthemomentofinertia,I. Theeffectsofthebeam-draftratio,B/T,werefoundtobenegligible.

Theflexibilityoftheship’shullisrepresentedinthisreportbythenaturalfrequencyoftheshipassociatedwi~hthetwo-nodemodeshape.AnimportantandusefulrelationbetweentheflexibilityandbendingmomenthasbeenestablishedinFigur@1.

Becauseoftheshortcomingsoftheexistingmethodsofana-lysis,thequalitativevaluesofthesecurvesaremoreimportantthanthequantitativevalues.Untilthesequantitativevaluesareconfirmedbymorereliableinputdataandstudymethodology,theresultspresentedareconsideredtentative..

Inadditiontostudyingtheeffectsoftheshipproportions,thestudyalsoachieveda broadergoalofbetterunderstandingof.theresponsesofshipsina seaway.Itisclearthatamoreac-curatemethodforship-vibrationanalysisisrequiredandcanbedevelopedwithinthestate-of-the-artofthecurrenttheoriesofhydrodynamicsandstructuralmechanics.Forthisreasona re-viewoftheexistingtheoriesandrecommendationsfornewmethodo-logiesareincludedinthisreport.

Duringthecourseofthestudy,theeffectofshipspeedondampingwasa subjectofmajorconcernandcorrespondinginvesti-gation.A ten~ativeanalysisindicatesthatforwardspeedhaseffectsonhydrodynamicdampingandforcesaswellashullflexi-bility.

-3-

z’>m

.

0 0 0● ✎ ✎m 04

0

001 Xu

-4-

2.0 THESTATE-OF-THE-ARTOFSHIP-VIBRATIONANALYSIS

Theexcitationforceofwavesontheshiphullisdeterminedbyuseofseakeepingtheoriesinwhichrigidhullsareassumed,asinReferences1,2,and3. Inspiteoftheconsiderableeffortspentin-thelastdecadetoimprovetheseakeepingtheories,theresultsobtainedwiththevariousimprovedmethodsstilldiffersomewhatfromtestresultsofrigidmodels.TablesI,II,III,andIVfromReference4 indicatetheerrorofthevariousmethods.

2.1 ShipVibration-A HydroelasticProblem

Allshiphullsae flexibletosomedegree.Loadsonflexiblehullsdifferfr~mloadsonrigidhulls.Theoretically,completelyflexibleshiphullswillbehavedifferentlythanrigidhullsandwillresponddirectlytothewavesurfaceconfiguration.Inpracticetherearenocompletelyrigidorflexibleships.Betweenthesetwoextremesjtheaccuracyoftherigid-hullsea-keepingtheorydecreaseswiththeincreaseinhullflexibility.

Inrecentyears,shipshavebeenbuiltwithincreasedhullflexibilityanditis,therefore,necessarytoimprovetherigidhullseakeepingtheory.Theshiphullisanelasticbodyandtheseawayresponseproblemandtheshipvibrationproblemareasinglehydroelasticproblem.

Theshortcomingsofexistingmethodsforship-vibrationanalysiswererecognizedbyKline,Reference5,whereinhecon-sideredthemosturgentproblemtobetheaccuratedeterminationofdampingandthedevelopmentofa hydroelasticsolutionforshipvibration.SomehydroelasticeffectswereconsideredinthemethoddevelopedbyGoodman,Reference6. Hismethodisbasedontheassumptionofzeropitchandheave.Althoughitistruethatheaveandpitchofa rigidshipamongregularwavesofshortwavelengthwithrespecttothehulllengtharequitesmall,theseshipmotionsmaystillbeimportantsinceCheshipisnotper-fectlyrigid.

Itisunderstoodthatclassificationsocietiesaregenerallyusingtherigid-bodyapproachinthecalculationofthehydrody-namicloads.Forexample,currentpracticeatABS,Reference8,is~ousetherigid-bodyapproachinthecalculationofthehydro-dynamicloads,andtotakeintoconsiderationthehullflexibilityinthevibrationanalysis.Thisapproachisnota true“hydro-elastic”formulationoftheproblem,sincethecouplingeffectisneglected.

-5-

TheeffectsofforwardspeedhavebeenrecognizedbyHoffman,Reference7,tobequiteimportantinhisinvestigationwithmodelexperiments.Thesubjectisdiscussedlaterina separatesectionofthisreport.IEisinterestingtonoteherethesizeablediscrepanciesbetweenGoodman’stheoreticalresultsandtheex-perimentalresults.Hoffmanwasabletoexplainsomeofthedis-crepancies.Fromtheequationsofmotiongiveninthefollowingsection,itcanbeshownthatGoodmanneglectedsomeimportantterms,whichmayexplainthediscrepancies..

2.2 ProblemAreasinExistingSeawayResponseAnalysis

Incomparisonwithresultsfromrigid-modelexperimentstherigid-shipseakeeping--methodisnotentirelyaccurate.TheerrorsshowninTables1:-.4 areinadditiontotheerrorsduetotheflexibilityoftheshiphullandthesumoftheerrorsmaybesignificant.

Despitegreatprogressinthepredictionoftheseawayloadsofrigid-shiphullsinrecentyears,twosourcesoferrorremaintobecorrected.First,striptheoriesare,ingeneral,validonlyforthemid-bodyoftheshiphull.Thetheoryisnotvalidforthehullendsanderrorstendtoincreasetowardtheends.Sincetheeffectoftheforwardspeedisproportionaltothechangesofhydrodynamiccoefficients,withgreatchangestowardtheends,theaccumulatederrorscanbesignificant.Inrecentyears,effortshavebeenmadetoimprovetheaccuracyoftheaddedmassanddampingcoefficients.A promisingapproachistheuseoffinite-elementmethodswhereinalltypesofhullcross-sectionscanbeconsidered.

Theeffectsofforwardspeedareanotherunsettledarea.Salvesen,Reference1,hasindicatedthattheforward-speedtermsintheequationsofmotiondevelopedbyvariousinvestigatorsdiffergreatly.Froma briefreviewofthevariousversionsofthefarward-speedeffects,Salvese~’sversionappearstobeacceptable.However~additionalstudiesandcomparisonsshouldbemadetoidentifytheimportanceofvarioustermsi-ntheanalysisofforward-speedeffects.

-6-

TABLE1

ApproximateMeasuresofCorrelationBetweenTheoryandExperimentfarHeadSeas

PercentErrorMidshipMidship

Source Froude Pifch Heave VerticalVertical RelativeNumber Moment Shear BowMotion

Baitis,etal .13-.2 5-1o I0-20 - . .(1974)

CoxandGerzina .22 5-1o 5-15,-(1975)

5-10.30 10-15 5-15 - . 5-30.37 20 10-30 - 5-30

BaitisandWermter..15 10 . .(1972) .46 :: 20 .

Flokstra(1974 .22 10 - . ..245 10- 10 10 20 1O-I5.27 . 10 . .

WahabandVink .15 5 10 15 15(1975) .245 Is 25 15 20 25

Journee(1976) .15 10 20 . .●20 10 25.25 10 25.30 10 20 . .

Kaplan,etal .25-.30 10-15 - 30 20(1974)

Kim(1975) .25 . 10 30

Loukakis(1975) .15 10 10 ..20 15 10.25 15 10 ..30 15 10

.09-.14 - 10

Salvesen,etal 5 5 - .(1970) ::5 20 10

●I5 10 10

Oosterveldand ●3-.4 - . 10van f)ossanen(1975)

-7.

TABLE2

ApproximateMeasuresofCorrelation BetweenThmry and ExperimentforFollowing Seas

Percent Error

Source

Baitis andWermter(1972)

Journee(1976)

Kaplan,etal(1974)

Kim (1975)

WahabandVink(1975)

FroudeNumber

0.150.46

0.150.200.250.30

0.25-0.30

0.25

0.15

Pitch

10150

10201515

15

.

5

Heave

:;

5101015

.

HldshipVerticalMamcnt

60

25

25

MidshipVertical

Shear

80

15

I00

-3-

Source

Baitls an<Wermter(1972)

Salvesen,et al

Flokstra.(1974)

FujiiandIkegami(1975)

Kaplan,et al(1974

WahabandVink(1975)

TABLE3

ApproximateMeasuresof Correlation EetweenTheory and Experimentfor BowSeas (Headings 120 to 150°)

FroudeNumber

0.150.460.15

0.15

0.245

0.195

b.2540.303.25_0.30

0.150-245

IO-1510-6010

10

20

15

1010-30

Heave

5-1o10-2010

30

25

.

20=o

Roll

10-5025-6o50

.

15

.

20

A MidshiD Moments_Vertical

“

.

15

15

20

4040

3::50

-

.

15

25

30-50

20-4020-40

2025

‘orsiona

.

.

20

40

30-50

20-9020-90

3020

MidshipVerticalShear

.

15

30

40-9040-90

5;:100

CB

.486

.4S6

.4B!5

.80

.598

.6994

.56

.56

.80

.598

Gfl/B

I2%1FL&A

5

3.6

4.1

2.55.0

5.03.6

CB= Block Coefficient

GM= Metacentric Height

B = Breadth

TABLE4

Source

Ba;tis andWermter(1972)

Salve5enJet al(1970)

Flokstra(1974)

Fujii andI kcgami(1975)

Kaplanetal(1974)

KIm(1974)

WahabandVink(1975)

ApproximateMeasure$of Correlation BetweenTheory and Experimentfor Quartering Seas (Headings ~0 to 600)

FroudeNumber

0.15

0.15

0.245

0.195

0.25-0.30

0.25-0.30

0.25

0.15o+2115

Pitch

10

10

15

S-20

1010-15

CB= Block Coefficient

+eave

10

.

15

1s-20

.

.

Percent Error

Roll

10

90

20-3s

9030

50-10(

30-40

~ Midshio Moments~

.

15

10

20-2s

;:

20-40

2020-40

20

“25

20-80

30-IOC20-j’O

30-40

3;:50

Torsional

20

.

30-40

10-504!3-90

30-90

5f60

MidshipVcrtica

shear

.

30

60-8060-80

40-100

10050-100

CB

0.486

0.80

0.595

0.6994

0.560.56

0.56

0.800.593

Gtl/B

12%

5

3.6

4.1

2.55.0

2.5

5.03.6

GM= Metacentric Height

B = Breadth

-1o-

3.0 A HYDROELASTICFORMULATIONOFTHESHIP-VIBRATIONPROBLEM

3.1 ExistingMethods

Theexistingship-vibrationmethodscanbeexplaine~mostconvenientlybytheequationsofmotionusedbyvariousin-vestigators;

3.1.1Thefourth-mderequation:

wherew isthekwistheC isthep isthe

F(x,t)istheA istheI isthe10isthe

‘kw-(~:+lo)ti’’+&+~:=F(xt’“(1deflectionrestoringforcetotaldampingcoefficientshipmassplusaddedmassverticalexcitationforceperunitlengthshearareaoftheshipsectionmomentofinertiaoftheshipsectionmassmomentofinertiaoftheshipsection

. aw‘r==

w’=%. ::

w,,,,_ a~w.——@x’

Thisequation,withslightvari i ns,hasbeenusedbymanyinvestigators,includingNoonan~!yo ,Kline(s),McGoldrickando~hers.

3.1.2Thesecond-orderequations,obtainedfromReference8:

‘[EIzx,’]’- ImyXS- C5&5 +K3A3G(x3’--x5) = o

PX3 + c3i3 + k3x3+ [EIz>~s’]”- [I~yX5 + C5~51’ = F3(X,E)

(2)

-11-

X3X5

C5, C3

is theverticalexcitationforceisthemassrotarymomentofLnertia/lengthistherestoringforceistheverticalshearareais themomentofinertiaistheshipmassplusaddedmassistheverticaldeflection

ax~istherotation,X5= —ax

arethetotaldampingcoefficientsassociatedwiththelongitudinalrotationandverticalmotionoftheshipsection,respectively.

Notetihatintheabovetwoequationstheloadontheshiphullisnota functionofthedeflectionofthehull.Thehydro-dynamicforcesarepartiallyincludedinthetermsassociatedwiththeaddedmassandthehydrodynamicdampingcoefficients.Mostofthehydrodynamicforcesduetotheforward-speedeffectshavebeenignored.

3.1.3Thefirst-ordereauations- Thefollowinpeauationswereusedinthisstudy: “

[

W’=e+

gf=~

M’=v+

VIQ mswwhere

PO+ 10:+ Co~ (3)

+ Cs++ F(w,C,x,t)

w,Q,M,Varethedeflection,slope,bendingmoment,andshearresponsesofthehull,

P istheaxialforce10isthemassrotarymomentof

CoandCsarethedampingcoefficientsassociatedwiththerotation

respectively

inertia/lengthperunitlengthandverticalmotions

ofthes isthemA isthe

F(w,L,x,t)isthe

shipsectionshipmass/lengthshearareaverticalexcitationforce

.12.

Noteintheaboveequationthattheloadontheshiphullisa functionofthehulldeflection.Thedifferencesbetweenequation(3)andequations(1)and(2)areexplainedfurtherinthefollowingparagraphs.

3.2 Cornp=isonAmongtheExistingMethods

Forsimplebeams,Equations(l),(2),and(3)canbere-writtenassimilarfourth-orderequationsinthefollowingmanner:

EIw””+ C&+ uwi-kw= F(x,t) (4)

E’lxa””+ C;~i-@?3+ kqxa= F3(x,t) (5)

EIw’”~+ C~t+ msw = F(w,C,x,t) (6)

Intheaboveequationssheardeflectionandrotaryinertiahavebeenneglectedforcomparativepurposes.Notethatmsisthemassof theshiponly,whileu isthemassoftheshipandtheaddedmass.SimilarlyC~istheinternalshipdamping,andC isCheinternalshipdampingplusthehydrodynamicdamping.

Equations(4)and(5)havebeengenerallyacceptedbynavalarchitects.Theseequationsaccountfortheeffectofthesur-roundingwaterontheaddedmassandhydrodynamicdampinginadditiontothestructuraldamping.Thisconceptisnotentirelycorrect.InEquation(6),thetermstotheleftoftheequalsigndonotincludeanyconsiderationofsurroundingwater.Thisimpliesthattheshipismovingasanelasticbodyandisexcitedbythesurroundingwater,andthattheexcitationforceisafunctionofthewavesandthedeflectionoftheship.PhysicallyLhisconceptismorerealistic.Mathematically$itshouldleadtoa morereliablesolutionoftheshipvibraticmandshipmotionproblems.Thisisexplainedfurtherinthefollowingsection.

3.2.1Wave-ExcitationForces-Wave-excitationforcesarestillanunsettledsubjectamongseakeepinginvestigatorsandvariousformulationsarecurrentlyinuse(seeReference1). A fulldis-cussionoftherelativemeritsoftheseversionsisbeyondthescopeofthis-study.

Frombasicfluidmechanicstheorytheexcitationforceofthesurroundingwatercanbeexpressedas:

( )F(w,~,x,t)= - & ma# (w-C)-N~(w-C)- PgB(w-L)(7)

D_a a—— —.Dt at %

-13-

wherem~ istheaddedmass/length

U istheforwardspeedN isthehydrodynamicdampingcoefficientw isthedeflectionoftheshipc isthewatersurfaceB isthebeam

Thisexpressionsimplystatesthattheexcitationconsistsoftheinertiaforce(firstterm),thedampingforce(secondterm),andtherestoringforce.Alloftheseforcecomponentsarefunctionsoftherelativepositionbetweenthewatersurfaceandtheshipsection.

Forrigid–shiphullstheverticaldisplacementofa sectioncanbeexpressedas

w = ~3 - x?15 (8)where

rISandr15areheavingandpitchingdisplacements,respectively.

SubstitutingEquation(7)intoEquation(3),andcombiningthesefourfirst-orderequations,wecanobtaina fourth-orderequation.Thisfourth-orderequationistoocomplicatedforcomparisonwiththeexistingmethods.Forconvenience,thesheardeflection,therotaryinertia,thero~ary damping,axialforce,etc.areneglectedandEquations(4),(5),(6)areusedforcom-parison.

SubstitutingEquation(7)intoEquation(6),wehave

EIw’’’’+(cs+ N - Um’a)~- 2Uma;’+ maU2w”

- U(N- Uma’)w’+ PgBw+ (m~+ ma)ti (9)

= mat+ (N-Uma’)5- 2Uma~’- U(N- l.Jma)~’+ pgB~where amam’=—a ax

NotethatEquation(7)isjustoneofthemanyexpressionsforthewave-excitationforces.However,differentexpressionsforwave-excitingforcesresultindifferentandlesscompletetermsforforward-speedeffects.Infact}theexcitationforcesfrommodernseakeepingtheorybasedonincidentanddiffractionwavepotentialsaredifferentfromthoseinEquation(7).ThesignificantfactoristheabsenceinprevioustheoriesofsomeimportanttermswhichappearinEquation(9).Thosetermsare

-14-

discussedinthefollowingsection.

3.2.2TheEffectsofForwardSpeed- Ifthesheardeflectionandotherpropertiesareincluded,theaboveexpressionbecomesmuchmorecomplicated.Thefollowingconclusionscanbedrawn:

(1)Theforwardspeedaffect-s(1)thehy~~d~~~~~~&~~gand(2)thestiffnessofthe”ship.hasbeengenerallyignoredbyshipvibrationinvestigators.

(2) Someofthetermsrelatingtodampingduetotheforward-speedeffectshavebeenignoredbymanyvibrationin-vestigators. Thesignificanceofthisomissioniscon-sideredinthefollowingsection.

(3)Thetermsignoredbytheship-vibrationinvestigatorshaveproventobeimport-antbyinvestigatorsconcernedwithflow-inducedvibrationofpipesandrodsasshowninReferences12,13,14,and15.

3.3 TheEffectofForwardSpeedonShipMotions

Theeffectofforwardspeedonshipmotionshasbeenofparticularconcernduringthestudyandthesubjecthasbeencon-sideredinsomedepthtosupportthemethodologyadopted.In-dependentstructuralanalysesofoceanthermalenergy(OTEC)cold-waterpipes,reportedinReference12,providessomeinsightintotheeffectsofwaterflow.Sincethecold-waterpipeproblemalsousesa setofequationsofmotionssimilartoEquation(3),theeffectsoftheinternalwaterflowareequivalenttotheeffects,!oftheforwardspeedoftheships.Incomparingthecold-waterpipesolutionwiththemethodsusedbyvariousship-vibrationinvestigators,itisevidentthatsomeimport-afittermshavebeenignoredintheship-vibration,problem.

FromEquation(9),theforwardspeedhasthreetypesofeffec~sontheresponsesoftheshipina seaway:

3.3.1Effectsondamping- TheeffectsondampingareshowninthefollowingtermswiththespeedU:

Dampingforce= (Cs+N -Uma’)J- 2Uma+’

ThemethodsrecommendedbyGoodman(Reference6),andusedbyHoffman(Reference7)andKline(References4 and5),haveignoredthesecondterm.ThistermisalsoneglectedintheABSmethod,(Reference8).

3.3.2EffectsontheHullStiffness- ThetermsmaU2w”andU(N-Uma’)w’havetheeffectofchangingthenaturalfrequencies

-15-

andthevibrationresponses.ThefirsttermmaU2w”canactuallycausetheresonancevibrationofa pipeconveyingfluidorsolidrodsinparallelflow.Thesetwotermsareentirelyignoredintheusualship-vibrationanalysis.ThesecondtermU(N-m’a)w’oritsequivalentdoesexistintheseakeepin~theorybySalvesen,Reference1,andothers.

3.3.3EffectsonWaveLoads- Physically,alltermsassociatedwithforwardspeedgeneratecertainforcesupontheship’shull.Mathematically,thetermsortheirequivalentsontherightoftheequalsignofEquation(g),aredefinedasthewaveloads,forcomparisonwiththeexistingmethods.

Thetermsmat+ (N- LJma’)~+ PgBLareeXactlYthesameasGoodman’ssolution(Reference6), Theterms,2Uma~’,U(N- Uma)~’,havebeenignored.

Againjitisnecessarytonotethata differentversionoftheexcitationwillresultina setofdifferenteffects.However,allversionsofexistingmethodologydoindicatethatmanytermshavebeenignored.

-16-

4.0 METHODOLOGY

Becauseofthelimitedscopeofthisstudy,projectcal-culationsofthevibrationresponsewerecarriedoutusingexistingmethods.ThewaveloadsandhydrodynamiccoefficientswerecalculatedbytheprogramMIT5DdevelopedbytheMassachusettsInstituteofTechnology.Usingthedataobtainedfromthispro-gram,thevibrationoftheshipwascalculatedbytheprogramBEAMRESPONSE,Reference19,withmodificationsforhandlingdampedvibrations,asshownindetailinReference21.

4.1 SelectionofSeaSpectra

Figure2 showstheassumedvariationofwavepeakenergyfrequencywithsignificantwaveheightforoceanandGreatLakeswaves. ThelowercurveisforrepresentativeoceanwavesandtheuppercurveisforwavesinLakeSuperior.TheoceanwavesarerepresentedbytheBretschneiderspectrumhavinga peakenergyfrequency10percentgreaterthanthewellknownPierson-Moskowitzspectrum.TheBretschneiderspectrumisprobablymorerepresen-tativethanthePierson-Moskowitzforalloceanlocationsandisofmore’interestforthepresentstudysincethehigherfrequenciesofwaveenergywillproducelargerspringingstresses.TheGreatLakeswavesarerepresentedbytheJonswapspectrumwhichisbasedonanalysisofavailablewavespectraldata,withemphasisontheLakeSuperiordatafromReference16.

Forthebaselineshipsandthevariations,theresponsesfordifferentwaveheightsanddifferentheadingswerecalculatedbytheseakeepingprogram.Theconditionsassociatedwiththemaximumwave-inducedbendingmomentwerethenadoptedforthevibrationanalysis.Seesection4.5.

4.2 EquationsofMotion

Asindicatedearlier,theusualanalyticalmethodsforestimatingshipvibrationarenotentirelysatisfactory.Errorsmaybeintroducedintheusualassumptionsofrigidhullsforseawayloadestimatesandflexiblehullsforvibrationanalysis.Becauseofthelimitedscopeofthisstudy,however,theforwardspeedeffectsdiscussedearlierhaveonlybeenpartiallyaccountedforasindicatedinFigure3.

TheconstantparametersinEquation(3)aredefinedasfollows:

E = 30 x 106psi= 1.9286x 106tons/ft2G=E/2(l+v)v = 0.3P = O (noaxialforce)

0x

2.52.01.5

1.00.80—0.700.60—0.500.40—0.30—0.25—0.20—

0“’5LL-L11111

LAKESWAVES

I! 1.522.52.4 5678910 15 202530 4050

SIGNIFICANTWAVEHEIGHTIN FEET

Figure2 - SignificantWaveHeightVS.PeakEnergyFrequency.

0.08

0.06

0.04

0.02

I

0 0

I0.1 0.2

Figure3 - TotalDampingas.aFunctionof FroudeNumber

-18-

Co=oC isdefinedinFigure3 (Seenextsection.)m~ isreplacedbyms+ maF isthesealoadcalculatedbytheprogramMIT5D

4,3 ThePredictionofDamping

Thestate-of-the-artofestimatingofvibrationdampingofshipswasreviewedbyWoolmanin1965,References17and18. Heconcludedthat“informationisabundantbutinadequateinpredictingresponsesoftheshiphullatresonantconditions.”Sincethen,thissituationhaschangedverylittle.

Severalpertinentshortcomingsandlimitationsintheexistingmethodsformeasurementandcomputationofthedampingcoefficients,whichhavenotbeenconsideredbyWoolman,arediscussedinthefollowingparagraphs.

Manymeasuringandcomputingmethodstreattheshipasasingledampedmass-springsystem.Theresults,evenifaccurate,providethetotaldampingoftheship.Whilesuchdataareabundantandreadilyavailable,theyarenotadequateforshipvibrationanalysis.

Themeasurementsobtainedindampingexperimentsprovideonlythetotalresponseduetocertaincontrolled-excitations.Itisgenerallyunderstoodthatthetotaldampingconsistsofatleastthreebasiccomponents,i.e.,hydrodyn~ic,cargo,andstructuraldamping,andthatthesecomponentsarefunctionsoffrequency.Inordertoidentifyanddeterminethesecomponents,andtheeffectsofdifferentfrequencies,anexperimentalprogrammustincludemethodsfordifferentiatingthesecomponents.Littleefforthasbeenmadeinthisdirectioninpastexperiments.Forexample,methodsforthispurposearenotconsideredinReferences17and18.

Equations(1),(2),and(3)indicatethatthemagnitudeaswellasthedistributionofdampingcoefficientsarerequired.Noneoftheexistingexperimentaldataandcomputationmethodscanbeusedtodeterminethedistributionofthedampingforce.Thisisanobviousshortcominginthemethodology,resultinginerrorsinthedeterminationofvibrationresponses.

Theforward-speedeffects,havebeenrecognizedasquiteimportant,Reference1; Inthepast,nodampingexperimentshavebeenconductedtodeterminetheforward-speedeffectsondamping.Hoffman,Reference7,hascalculatedthedifferencebetweentheexperimentalresultsandtheresultsbyGoo~an~smethod!Re-.ference6,andheindicatedtheimportanceoftheforward-speed‘effects.However,heattributedthesedifferencestothedamping

-19-

alone.SinceGoodman’ssolutionalsoignorestheforward-speedeffectsontheexcitationforceandthestiffnessofthehull,theactualforward-speedeffectsondampingarestillunknown.

ThecurrentindeterminatestatusofdampingisconsideredinFigure4. Variousinvestigatorsuseentirelydifferentvaluesofthedampingcoefficient.Notethatalmost,ifnotall,oftheseexperimentaldataweremeasuredwiththeshipsbeingstationary.

4.4 DeterminationoftheEffectsofShipProportionsonHullFlexibility

Theflexibilityofanystructurecanbedefinedasthede-formationofthestruc~ureata givenlocationproducedbyaunitgeneralizedforce,suchasa deflectionduetoa unitforce,androtationduetounitmoment,etc.Thisdefinitionisnotconvenientforshipsanditsmeaningistoovagueforthedesigners.A betterdefinitionisthetwo-nodefrequency.Itcanbeshownthatshipswithsmallvaluesofvibrationfrequencyrespondtounitforcewithrelativelygreatdeformation.Forthisreason

Flexibility-& (lo)

Thedeflectionduetoa standardwaveofunitwaveheighthasalsobeenusedasameasureofflexibility.However,theresultis-notsatisfactorybecauseotherfactorssuchasheadinganglesandweightdistributionhavenotbeenstandardized.

Theeffectsofvaryinghullproportionshavebeenexaminedusingnon-dimensionalhullgeometryratiossuchasL/B,B/T,L2/BI~,L/D,B/D,etC.Theeffectsoftheseonbendingmoment,deflection,stress,andnaturalfrequencyofthehullhavebeenplottedinFigures5 through37. Fromthestudyresults,theeffectsofB/Twerefoundtobequitesmall.Sincethedepthoftheship,D, doesnotaffectthehydrodynamicforceorthehydrodynamiccoefficients,theeffectofvariationofD canbeincludedinthestructuralmo-mentofinertiaoftheship,I.4.5 DeterminationofMaximumWaveLoads

4.5,1MaximumWaveLoadsforHighEnergyWaves- Inthestudy,theshipmotionsresponsesina seawaywerecomputedfora com-pleterangeofheadinganglesanddifferentsignificantwaveheightstodeterminetherelationbetweentheverticalbendingmomentandthewaveheights.SomeoftheresultsareshowninFigures38to45. Themaximumwaveheightfortheoceangoingshipsislimitedto25feetatawavefrequencyof0.5rad/sec

-u.w

12345b79 9 10II12,13!41514!7la1920FR39ULNC?INHz

Fig.4-DampingCoefficientsUsedbyVariousInvestigators

— FUILLOAD—— BALLAS?40 f

30 2WAVE +

20

.:’”+ *“-

‘o g“”___

0 7 a 9 10 11 12 13L/B

Fig.6 - EffectofL/B,onBendingStressforGreatLakesOreCarriers

#.o

}

— FULLLOAD----~*~~~~

,

WA\ +2,0

STILLWATER

o7 8 9 10 11 12 13

L/8

Fiq.5 - EffectofL/B onDeflectionforGreatLakesOreCarriers

.ILLLOAOBALLAsr

5?31NGiWG

WAVf+SllLLWAnl

- ~>

1~

“ ~,— -=+,<RING,NG---- ‘I -----

I ----— .——.

7 n v L/B ‘0 II 17 la

Fig.7 - EffectofL/BonVerticalBendingMonientforGreatLakesOreCarriers

Fig.

16,0

— FWLLOAD--- BALLAST

12.0

8.0

10PSIONA1

4.0— - ----

? 8 ? 10 ,11 12 13L/B

8-Effectof1/8onLateralandTorsionalMomentforGreatLakesOreCarriers

_ FULLLOAD—— MLLAST

la

nr-—

L2/BX*

~rg. la - EffectofL2/B1?onBendingStressforGreatLakesOreCarriers

— SALMSr

8.0

4.0 /

4,0

2.0

0m 603 m em VWJ lWQ I!ml

L2/dFig.9 - EffectofL2/BI%on DeflectionforGreat

LakesOreCarriers

-2 iL /BI

Fig.11- EffectofL2/BI~onLateralandTorsionalMomentforGreatLakesOreCarriers

,—FULLLOAD---OALIAST

40

30

’10

10.

0 5m &m 7m am ?W 10W llml

L2fBIt

Fig.12- EffectofL2/BI*onVert~ca?MomentforGreatLakesOreCarriers

60~ I I I 1—FLYLLOAD ,,----BALLAST

50

40

_ ---—,30

WAVE +.>

20 Y31uWATER1

10I ____ .--- .-”- .

04 5 4 7L/B

Fig..14-,EffectofL/Bon8endtngStressforTankVessels

— FLILLOAD---- BALL/J~

4.0

5 b 7L/B

Fig.13- EffectofL/BOnDeflectionforTankVessels

60, I I IE=++--+‘--:”-40 1 1

Fig.15- EffectofL/13anverticalBendin,gMomentforTankVessels

4.0

‘WLL LOAD‘--- kALLAS1

3.0

TOP.HONAL2.0

T.o

.L -

1r04 I I I

EffectofL/B onTorsionalandLateralBendingMomentforTankVessels

4— FULLLOAD4,0 ‘----BALLAST

k~ 3.0-zoE:H 2.0

STILLWATER

----- __5PRtNGtNG

Fig.17- EffectofL2/BI%onDeflectionforTankVessels

#l

1‘sol-=+------x 40~

3 — fUkLLOAD>.

---- BALLAS,

o-, 30

~il.,18==EffectofL2/131%onBendingStressforTankYessels

o~d200 303 4W 503

L2/d

Fig.19-EffectofLz/BI~onVerticalMomentforTankVessels

I-LmI

50

{0

30

20

10

02

/-

20-EffectofL2/BI*onLateralandTorsionalMomentforTankVessels

50FILLLOAD

—= BALLAST40

5 30 I I I I~

WAVf+’STILLWAIEM——

: .——.—.——— —&: 20 I I

I

++=--++ -,———_— ________o

b

EffectsofStressfor

7L/B

1/3onVert”CargoShips

B

calBending

t-i-

.1 10.2

I5P~lNGlNG

———————.——————— ————— ———I I I 15 6 7 a

1/0

Fig.21- EffectofL/BonDeflectionforCargoShips

6.0- FLU LOAD

k

~ t,

“--% _—— %ALLASTWAVE~ \

: 5“0 I0. STILLWATER ------ l’,— —.— ——————___. 1

4.0

3.0

2:0”

1.0

——_ —— ___ ——. .1 I 1 (5 6 7 6

1/0

Fig.23- EffectofL/BonVerticalBendingMomentforCargoShips

FJ

\

\4\\II

“(du-PvaJ

wn

/

—,

..

-14-0+vu-l: .?q-rWw)

I

IrJN

.m

I

,/

—0. -. =...”

u

1334HI NOll>lU3a

—1

.r--0!=2

—

—

—

\.

“mvor-PLam>!a

I.

laml

( H - sNO1) t_Olx 1N3WOW1VNO15301lNYldlNOl$(M- sNO1) #Jl xlNgWOWlV131k71NVX41N91S 1s1NISS381S u-

—— __

—

—

—

—

—

—

—

—

—

.

—

\

\

\

I

1$XNISS3US

It+m

.’= ‘k.-..,”5

10

8

d

4

2

Fig.32-

L/B

EffectofL/BonMomentforConta-LoadCondition

t

VerticalBendingnerships,Full&

“m

Fig.34-

500 m

L2/d

EffectofL2/BI%onDeflectionforContainerships,FullLoadCondition

4.0

I3.0

LA?EPAL

2.0

1.0

h ? nL{B

Fig.33- EffectofL/BonTorsionalandLateralMomentforContainersFullLoadCondition

40

30

20

10 SPRINGING

o4c0 503 hw

Fig.35-

L2/B16

EffectofL2/BI%onBendingStressforContainerships,FullLoadCondition

ps.I-LuI

WAVE+5T!LLWA7E

\

/

SPRINGtNG

1s /I I403

Fig.36-

5WJ 6WL2/B?*

Effectof12/731%onV~rticalBendingMomentforContainerships,FullLoadCondition

~TEML

4.0 -

3.0

2.0—

I .0

4m w 620L2/BIi

Fig,37- EffectofL2/131%cmLateralandTorsionalMomentforContainerships,FullLoadCondition

-29-

FWNCIPALCHA~ClEklSTlC5OFM/VSTMARTJ. CORT ILO? - 9ss.5’ IBnT

t

. lc4.6,. *Q, I

.-7.

15 20 25

SIGNIFICANTWAVEHEIGHTINFEET

Fig.,38.-EffsctafwaveHeightandHeadinaon’SeaLoadsofGreatLakesOreCa~rier,FullLoadCondition15

I IPRINCIPA1CHAk4CTERlSTiC5OF

M/v STEWARTJ. CORT

SIGNIFICANTWAVEHElGIiTF+FEET

Fig.40- EffectofWaveHeightandHeadingonSeaLoadsofGreatLakesOreCarrier,BallastCondition

SIGNlFICANTWAVEHEIGHTINFfET

Fig.39- EffectofWaveHeightandHeadingonSeaLoadsofTanker,FullLoadCondition

10 20 30SIGNIFICANTWAVEHEIGHTIN FEFT

Fig.41- EffectofWaveHeightandHeadingonSeaLoadsofTanker,BallastCondition

..——— ___

C4-S-S%CMGOSHIP11? - 544.5’

- W—: - 45.5fl_

t = 32*~A -3’2M310N5v - 23KNOTS

i-

SIGNtFKANTWAVEHEIGHTIN FEET

Fig42- EffectofWaveHeightandHeadingonSeaLoadsofCargoShip,FullLoadCondition

ICA.S-&CONTAlNEk5HlP

ILB? - A25, I ..-.1 - “& I /,1ao-D ● S3’ M5°T . 2’.T~ —

10 20 30

SIGNIFICANTWAVEHEIGHTIN -ET

Fig.44- EffectofWaveHeightandHeadingonSeaLoadsofC6Container-ship,FullLoadCcndition

-30-

1

0

Fig.43HeadingBallast

SIGNIFICANTWAVEHEIGHTINFEET

- EffectofWaveHeightandHeadingonSeaLoadsofCargoShip,Condition

51GN1F1CANTWAVE}IEIGHTIN FEET

Fig.45- EffectofWaveHeight-andHeadingonSeaLoadsofC8Container-shipyFullLoad.C.cmditicn

-31-

and20.5feetforGreakLakesships.Withintheselimitsofseastatesjthewaveloadsassociatedwiththemaximumverticalbendingmomentwereadoptedforthevibrationanalysis.4.5.2WaveLoadsforSpringingCondition- Forthespringingcondition,thetwo-nodefrequencyof.theshipwasfirstcalculated,Usingthetwo-nodefrequency,Ml,thewavefrequenciesandhead-ingswhichcouldcausespringingweredeterminedfromtherelation:

(11)

wherea istheheadingangle,beginningwithzerodegreescorres-pondingtofollowingseas.

Amongallthesetsofpeakenergywavefrequenciesandhead-ingangles,‘asetofwavesandheadingsassociatedwiththemaxi-mumrigid-hullsignificantbendingmomentwasdetermined.Thesealoadsforthissetofheadingsandwaveswereadoptedforthespringinganalysis.4.5.3ApproximateMethodforDeterminingtheFlexibleHullBend-ingMoment- Theoreticallythemaximumvibrationbendingmomentcanonlybedeterminedbycalculatingthebendingmomentassociatedwiththeentirewavespectrum,a taskbeyondthescopeofthispro-ject.Inviewofthemanyuncertaintiesinexistingvibrationtheory,anabsolutemaximum.isnotofinterest.Relativemaximawithintheaccuracyoftheexistingtheorycanbeobtainedbythefollowingapproximatemethod:LetBMi,BMrbetheflexibleshipbendingmomentinirregular

waveandunitregularwaves,‘“Ri’‘Rr bethebendingmomentsinirregularwaveandunit

regularwavesforthesameshipassumedtoberigid.,Forthesameregularwaveload,theflexiblehullbending

moment,BMr’andtherigidhullbendingmomentBMRr,canbecal-culated.Sincethesealoadinirregularwavescanberegardedasa combinationofmanvregularwaveloads,theratiobetweentherigidhullbendingmome~t,~MRi,andthemoment,BMi,forthesameirregularwaveapproximatelyasfollows:

‘“Ri_ BMRr— —BMi ~

flexible.hullbendingload,canbedetermined

(12)

Inhigh-energywaves,thedeflectionoftheflexiblehullissmallincomparisonwiththerigid-hullmotions.Thedifferencesbetweentheseakeepingandflexiblehullbendingmomentsaresmall.Forthiscase,theaboveequationisquitegood.Inthespringingcondition,thedeflectionoftheflexiblehullmaybemuchgreater

-32-

thantherigid-hullmotions.Inthatcase,theaboveequationmayinducesomeerrors.Itwasindicatedinpreviousdiscussionthatthestate-of-the-artisinaccurateforthespringingconditionunlesstheneglectedhydroelasticeffectsaretakenintocon-sideration.Intheabsenceofmoreaccuratemethodsforanalyzingspringing,theaboveequationcanbeusedforestimatingtheapproximatespringingmoment.

AccordingtoGoodman,Reference6,thetwo-nodemodevibra-tionpredominatesatandaroundthetwo-nodenaturalfrequency.Ifthisistrue,theerrorsduetoequation(12)shouldbesmalleveninthespringingcondition.

Equation(12)wasusedtocalculatethe.verticalbendingmoment”fortheflexibleshipforboththewavebendingcaseandthespringingcase.Thebendingmomentinirregularwavesfortherigidship(B~i)wascalculatedusingtheMIT5D.Thispro-gramwasalsoused-tocalculateBMRr,thewavebendingmomentinunitregularwavesfortherigidship.Thebendingmomentfortheflexibleshipinregularwaves,BMr,wascalculatedusingthemodifiedBEAMRESPONSEprogram.

Forthewavebendingeaselthesignificantwaveheightwastakenas25ft.fortheocean-goingshipsand20.5ft.forGreatLakesships.ForthespringingcasethewaveheightwaschosentocorrespondtotheheadingthatgavethemaximumbendingmomentforUel,theencounterfrequency,equaltoml,thenatural2-nodedhullfrequency;whereMelcorrespondstothepeakfrequencyofthewave,spectrum.

InthecurvesinFigures5 through37,thestill-waterbend-ingmomentwasaddedtothewavebendingmoment,calculatedasdescribedabove,toobtainEMv,butwasnotaddedtotheplottedspringingbendingmoment.

4.6 EffectsofHullMaterials

Theeffectsofhullmaterialsonthehullflexibilitywereconsideredin.thestudyinthefollowingmanner:

(1)High-StrengthSteelClassificationsociehiesusuallyallowcertain

reductionsinthescantlingsoftheshipstructureifhigh-strengthsteelisused.Thisreductioninscantlingswillreducethemomentofinertiaoftheshipsectionwitha correspondingincreaseinthehullflexibility.Usingthetwo-nodefrequencyastheparameterforhullflexibility,theincreaseinflexibilitycanbedeterminedfromtherelationship

-33”

(13)

whereI~ andIsarethemomentofinertiaofthecross-sectionwithandwithouthigh-strengthsteel.

Accordingly,theeffectsofthehigh-strengthsteelcanbeaccountedforbyproperlyusingthevalueofthemomentofinertia.

(2) Aluminum

Sincethemodulusofelasticityofaluminumislessthanthatofsteel,boththemomentofinertiaandthemodulusofelasticitymustbetakenintocon-siderationasintheexpression

(3)

whereandu‘a s

IaandIsEaandEs

(11,)

arethetwo-nodefrequenciesforaluminumandsteel,respectively;arethemomentsofinertiaforaluminumandsteel,respectively;arethemoduliiofelasticityforaluminumandsteel,respectively.

Thus,theeffectsofusingal~inumaidsteelcanbetakenintoconsiderationbyevaluatingtheproductofcross-sectionmomentofinertiaandthemodulusofelasticity.

CompusiteMaterialsHullswithmildsteelandhigherstrengthsteels

canbereadilycomparedsincethemoduliiofelasticityofthesetwomaterialsarethesame.

Forshipsconstructedofbothmildsteelandaluminum,theproblemismorecomplicated.Forthiscase$theconceptofequivalentmomentofinertiamustbeused.LettingA i,

aAsibethecross-sectionalareas

ofthealuminuman steelmembers;yai,ysithedistancefromthecenterofgravityofthoseareastotheneutralaxisoftheshipcross-section,theequivalentmomentofinertiaisdefinedas

-34-

N

x( EI*=. A “y~i2+Isi si)‘$ (& Aaiyai2+ Iai

)i=l s(15)

whereIsi,Iaiarethemomentsofinertiaofeachstructuralmemberaboutitsowncenterofgravity;

N,M arethenumbersofthesteelandaluminummembersinthecrosssection.

Theeffectofthealuminumstructureisincludedintheequivalentmomentofinertia.Asaspecialcasewhentheentirehullismadeofa“luminum,Equation(15)reducesto

Ex( )

EI*= ~ Aaiyai2+ Iai = # Ia

s s(16)

5.0 SELECTIONOFREPRESENTATIVE’SHIPSFORANALYSIS

Thefollowingfourvesselswereselectedasvehiclesforconductingthehullflexibilitystudy:

(1) GreatLakesorecarrierSTEWARTJ.CORT.(2) 264,000dwtU.S.flagtankvessel,designatedTIO-S-10lb.(3) C6-S-85aandC8-S-85dfamilyofcontainerships.(4) C4-S-69bgeneralcargovesselofMICHIGANclass.

Characteristicsoftheabovevessels,andtheproposedpara-metricvariationsindimensions,areconsideredinthefollowingparagraphs.,

Eachofthevesselswasstudiedforonefullloadandonerepresentativeballastcondition,Effectofdimensionalvariationsonfullloadservicespeedwasignored.Foreachsetofparametricvariationsofa givenparentvessel,onevalueeachoffullloadandballastspeeds,correspondingtotheparentvesselcharacter-istics,wasassumed.

Asihdicatedearlier,therequiredevaluationoftheeffectsofchangesindepthandstructuralmaterialswasobtainedbyappropriatevariationinmomentofinertia.

5.1 GreatLakesOreCar.rie~STEWARTJ.CORT

ThematrixshowninTable5waspreparedassumingconstantvaluesofbreadth,B,anddraft,T. Theseassumptionsreflectrealisticlimitsfortheforeseeablefuture,reflectinglockdimensionsandoperatingdraftconstraints.The1,000ftoverallleng~hreflectsexistingmaximumpermissiblelengthfortransitofthePoeLocks.Itisunderstood,however,thatthisconstraintmayberelaxedtopermitlengthincreasesofabout100ft.

Accordingly,a five-shipparallelbodyseriesbasedonthepresentCORT,withlengthincreasesto1,300ftoverallandlengthreductionsto800ftoverall,wasinvestigated.Itwasassumedthatthesechangesindimensionswouldbeaccomplishedbysimpleadditionandsubtractionof parallelmid-body,forconstantbreadthanddraft.ThefullloadservicespeedoftheCORTwasassumedconstantfortheseriesanda higherservicespeedwasassumedforthelighterballastdraft.

VesselssimilartotheCORThavebeenbuilttothesameoveralllengthandbreadthconstraints,butwithincreaseddepthtoobtainthehighercubiccapacityrequiredforcoaltransport.ThemostrecentvesselbuiltforthisserviceistheBELLERIVER,BayShipbuildingHullNo.716,withD = 56ft. Accordinglyjtheseriesincludestwovaluesofdepth,withD = 49ftfortheshortervesselsandD = 56ftforthelongervessels.The1,000ft

TABLE5.-PROPOSEDVARIATIONINDIMENS1ONSOF

GREATLAKESVESSEL“STEWARTJ.CORT”

Length,overall,ft. 800

Length,B.P.,ft.,L

k

788.5Breadth,rnld.,ft.,BDepth,mid.,ft.,D F

Draft,fullload,keel,ft.,T LDisplacement,mid.,f.w.,1.tons57,834

I

CB 0.907

L/~ 7.538

16.092L/DforD = 49.0forD = 56.0 1-

2.133B/DforD = 49.0forD=56.O

NTt-

900 1000 1200(BasicDesign)

888.5 988.5 1188.5104.6049.056.027.83

65,917 74,000 90,166

0,918 0.926 0.939

8.494 9.450 11.362

18.133 20.173 24.25517.652 21.223

2.133 2.1331.866 1.866

3.758

1300

1288.5>

>>

98,249

0.943

12.318

23.009

1.866

CLo-l

-37-

TABLE6-PROPERTIESOFGREATLAKESORECARRIER“STEWARTJ.CORT”FULLLOADCONDITION

LBP= 988.5’B = 104.6’D = 49’T = 27.83’A = 74472tons

Slc’r:l[ltdl’iiilF’E:R”T:ll$i

o+o*o+0+0+()+()+o+(;,*()+

-38-

TABLE7- PROPERTIESOFGRJZATLAKESORECARRIER“STEWARTJ.CORT”BALLASTCONDITION

LBP= 988.5’B = 104.6’D = 49’T = 20.75’A = 54840tons

0+(}.()+()+()*,,.,.,,+o●

0.0+o+

TABLE8 - VARIATIONOFPROPORTIONSANDRESPONSESOFGREATLAKESORECARRIERS- FULLLOADCONDITION,,

B% X10-3B~x 10-3B% x 10-’ ~2

(deg~ees)(#t)Defltn

(to:,) (f:’)(ton-ft)(ton-ft)‘b

B/T (ton-ft)(ft) LIB B1~‘1 (radi%lsee)(ftiYsec)(psi)

60 788.5 3.763957,48616,680 894 463 45.6 1.22757.5274 522.27 3.0588 21.44121,726

180 429 0.5989 10,425

60 888.5 3.763966,95016,680 904 570 40.0 1.72108.4943 644.10 2.5940 21,480

180 550 0.7980 13,068

75 988.5 3.763074,47216,680 1049 696 41.6 2.18949.4499 821.97 2,1077 25,502

180 799 1.6923 19,422

7’5 1188.53.763685,51025,950 1550 1030 70.4 3.3450.11.36231063.98 1.5230 24,220

180 1840 5.4902 28,752

75 1288.53.764990,26733,360 2099 1206 95.4 3.402012.30071172.7 1.4252 29,150

180 2844 8.8349 39,508

LmTABLE9 - VARIATIONOFPROPORTIONSANDRESPONSESOF GREATLAKESORECARRIERS- BALLASTCONDITION

r , 1 1 r 1 I 1 IB% x 10-3BEIL~10-9B% x10-’ ~z

(deg~ees)(:t)Defltn

Iz/T (to:s) (f:’)(ton-ft)(ton-ft)(ton-ft)(ft) L/B @

I I I I I [ 4 1 r I60 788.54.978845,64516,680 707 346 34.0 0.95967.5274522.27

i 180 127 0.174460 888.55.042449,85016,680 860 394 38,9 1.72008.4943664.10180 220 0.295060 988.55.042454,83916,680I 1087 483 45.6’2,28799.4499821.97180 307 0.696360 1188.54.995067,10025,9501640 ?89 66.5 3.19511.36231063.98180 840 2.45060 1288.54.988074,54933,3602062 1042 83.3 3.316912,30071172.7

I180 1260 3.9141

‘b(rad;h/see)(ft~sec)(psi)I3.1412 21.93917,241

3,0872.6400 20,970

5,3642.1495 26,419

7,4561.8210 25,620

13,1221.9263 28,641

17,620

.40-

parentwasstudiedforbothvaluesofdepth,thusprovidingfora six-shipseries.

Characteristicsoftheproposedseriesatfull-loaddraftaresummarizedinTable5. TherangeofvaluesofL/BindicateproportionsthatextendwellbeyondcurrentpracticeontheGreatLakes.ThevaluesofL/b,from16toover24,exceedoceangoinglimitsandextendfromcurrentGreatLakeslimitingvaluesof20towellbeyondcurrentdesignpractice.ValuesofB/TareconstantandreflecttheexistingvaluesofthisratioforvesselsdesignedtotransitthePoeLocks.

SectionalpropertiesusedintheanalysisaresummarizedinTables6 and ? andassumedspeedsareincludedwiththeresponsedata,Tables8and9.

5.2 264,000DWTTanker

TheproposedmatrixofsystematicdimensionalvariationsfortheTIO-S-10lbtankerisshowninTable10.Draftanddisplacementvaluesarespecifictothefull-loadcondition.

Thematrixwaspreparedassumingconstantvaluesofdisplace- Amentanddraft.A systematicvariationintheratioL/B,-wasassumed,thusprovidingforcorrespondingvariationsin:hesignifi-cantratiosL/D,B’/D.Thisapproachdiffersfromtheal~arnativesnormallyexaminedbytheshipdesignerinthattheowner’~re-quirementsgenerallyincludedefinedvaluesofdeadweightanddraftrestriction.Lightshipanddeadweightvalueswillvarywithproportions.Howevertheconstantdisplacementseriesisareasonableapproximationtothedesigner’sconstantdeadweightapproachanditprovidesa practicalbasisforstudyanalysis.

ThefivedesignpointsindicatedinthetablewerefurtherexaminedbyanalysiswiththeHYDRONAUTICS’conceptdesigncom-puterprogram,toobtainrealisticvaluesofdepthfortheassumedvariationsinhulldimensionsandproportions.Thiscomputeranalysisalsoprovidedthenecessaryweightinformationtosupportpreparationofa systematicvariationofweightcurvesforthefourvariationsoftheparent.

Thecomputerdesignanalysisindicatedthatrequireddepthvariationswouldbesmall,approximately1.25ftfromthebasevalueof86ft. Further,fora constantservicespeedandCB,powerrequirementsvariedonlyapproximately800hp. Accordingly,itwasconsideredreasonabletoholdtheoriginalvaluesofD andCBconstantforbasicvariationinparameters.

1 ..

TABLE10—PROPOSEDVARIATIONSINDIMENS1ONS

OF264,000DWT TANKER

~o~in~lL/Bv~l~.e

Item 5 5.5

Length,B.P,,ft.,LBreadth,mid,,ft.,BDepth,mid.,ft.,D

*

Draft,fullload,mid,ft.,T4Displacement,mld.,1.tons 4

CB I 0.8425I 0.8425

L/B 4.9664 5.&712

L/DforD = 86.00 11.2558 11.8140forD = 76.00

B/DforD = 86.00 2.2664 2:1593forD = 76.00

6.5(BasicDesign)

1060,00 1105.001.78.00 170.75

—86.00

+=

“ 76.00–67.0625—304573

0.84247I 0.8425I

5.9551 I 6.4714

12.3256 12.848813.9474 14.5395

2.0698 1.98552.3421 2.2467

2.6542 2.5461B/T 2.9064 2.7691I I 1 I

*Valueof76ft.isinadequatetoobtainfreeboarddraftof67ft.,hencethisseriesisofacademicinterestonly.

1147.00164.50

0.8425

6.9726 I13.337215.0921

1.91282.1645

2.4529I

-42-

ToobtainshipcharacteristicswithhighvaluesoftheratioL/D,a reducedvalueofD = 76ftwasarbitrarilyassumedforthethreelongestdesigns.Thisthree-points“eriesisacademicinthatthe76ftdepthisinadequatetoobtainthefreeboarddraftof67ft.

SectionalpropertiesusedintheanalysisaresummarizedinTables11and12andassumedspeedsareincludedintheresponsedata,Tables.13and14.

5.3 “C4~’Gener’alCarpoVessel

AnexistingU.S.flaggeneralcargovessel,designatedC4-S-69b,wasselectedforstudy.TheMICHIGANofthisclasshasbeenthesubjectofearlierstudiesunderShipSturctureCommitteesponsorship.TheproposedmatrixofsystematicdimensionalvariationsisshowninTable15.Draftanddisplacementvaluesarespecifictothefullloadcondition.

Thematrixwaspreparedassumingconstantvaluesofdisplace-ment,draft,andblockcoefficient,CB. A systematicvariationintheratioL/Bwasassumed,thusprovidingforcorrespondingvaria-tionsinthesignificantratiosL/DandB/D.Itisrecognizedthatthetwolongestvessels,withhighestvaluesofL/BandlowestvaluesofB/D,mayhavemarginalstabilitycharacteristics.Howeve~,theserieswasretainedforstudysincetheintentwastoinvestigatesystematicallytheeffectofvaryingshippropor-tions.

SectionalpropertiesusedintheanalysisaresummarizedinTablesI-6and17andassumedspeedsareincludedintheresponsedata,Tables18and19.

5.4 “C6”and“C8”Containerships

ExistingC6-S-85aandC8-S-85dcontainershipswereselectedforthestudy.Theoriginal“C6”designwascompletedin1968andhasbeeninservicesincethattimefortwoU.S.flagoperators.Recently,a groupoftheoriginal“C6”vesselswasconvertedbylengthening144ftand,subsequently,newconstructionofthislatterconfiguration,designatedC8-S-85d,wasinitiatedandiscurrentlyunderconstruction.Designandcharacteristicsdataforbothconfigurationsiscurrentlyavailable.Further,opera-tionaldataexistsforthe“C6”designandwillbecomeavailableforthelonger“c8”design.Accordingly,selectionofthesevesselsforstudyprovidedpointsin”ashipserieswhereinanalyticalstudiescanberelatedtoactualdesignandoperatingexperience.

..__,._.

--- _+ +++

-.

-, -,,..,,.:++=0?,.;?..3

-. .--.’..+--

L5

IIIIIiIIII

nozuHHHoz

+ --1- .w

“’44-

TABLE12 PROPERTIESOF264,000DWT“TIO”TANKER,BALLASTCONDITION

LBP= 1060’B = 178’D = 86’T = 35.925’A = 144392tons

MOD.of: ELAS. M(). INE!?rI,4

HOI’.JXAS. [T)}l SHEARAREA(ft’)

(J. 7● 300JOL)E+00d. 1.06.}10.)!:”!-01”u. 1.39,~i<O)E”i-r.)1fJ. 1..39’~,?(lM”Fol

I

TABLE13-YHRIATIONOFPROPORTIONSANDRESPONSESOF TANKVESSELS- FULL LOADCONDITION

I I —

‘b(psi)1~?

~+ (radk/see)(ftlvsec)(deg~ees)(#t) B/T (to:. ) ( f~’ j --1--B%x 10-3B%. 10-3(ton-ft)(ton-ft)

4404 1659

mq.10-

1

Defltn:ton-ft) (ft) LIB

1 50 I968I 2.905I302,850]129,450 252.4412.537 \25.637422,839I18050 1016 2.769303,010123,010

+=

1721‘4750 185011024993 2055844

26,500II 180 II I [0.4520i I

60 106CI 2,653303,110118,220180

28,355I=129,200

30,937

I 50 11105\ 2.546I303,120I113,2005042I 2250 389.892.050 ,,

18050 1147 2.451304,020109,260

900 I 1 1

439.891.9773 “5053I 2459180 I I 0.6516

TABLE14 - VARIATIONOF PROPORTIONSANDRESPONSESOFTANKVESSELS- BALLASTCONDITION

1 1I ‘%x10-’WLX10-’‘%x10-3D&(deg~ees)(:t} BIT (to:. ) (f:’) (ton-ft)(ton-ft)(ton-ft) LIB

L2—1BIJ

‘b(i-adk/see)(ft!sec)(psi)

252.443.231 28.71225,03311,714I 45 969 5.699143,870129,4504827 6!33 185.7 1.64824.9664

180 2258 0.770445 1016 5.430144,100123,0104920 802 195,0 1.86005.4712180 1995 0.7490

296.822.7602I I 26,300,

I10,064

I 45 ]1060I 5.205I144,392I118:220I 5054I 933 I192.6i2.189I5.9550340.422.8953t I28,701

I I I 1814I I 10.7863I389.8S45 1105 4.993144,200113,2005230 950 169,0 2.57006.4714

180 2010 I0.9360I 50 jl147~ 4.810I143,859~109,260] 5477I 916 ]137.9I3.1490I6.9726439.852.582 I I33,532I 180 II I I ] 2436I I I ~14,916

TABLE15

PROPOSEDVARIATIONSINDIMENSIONSOFC4-S-69b

GENEFUILCARGOVESSEL

Item

Length,“B.P.,ft.,L.

Breadth,mid.,ft.,B.

Depth,mid.,ft.,D.

Draft,freeboard,mid,ft.,T.

Displacement,rnld.,l-tons

CB

L/B

L/DforD = 45.5

B/DforD = 45.5

B/T

NominalL/B I5.1 5.8 7.5 8.5

(Ba~~~Design)

475 509 544 579 614

94 87.5 82 77 72.51

[1 45.6 ‘ *

[ 32.0/ I I I

*[

1 22500 ‘ , *

0.5512

5.0532

10.4396

2.0659

3.0519

0.5526

5.8171

11.1868

1.9231

2.8409

0.5517 0.5520

6.6341 7.5195

11.956012.7253

1.8022 1.6923

2.6623 2.5000

0.5528

I8.469013.4945

1.5934

2.3539

., ..,.,= ,.

-47”

TABLE16-PROPERTIESOFC4-S-69bGENERALCARGOVESSELFULLLOADCONDITION

LBP= 544’B = 82’D = 45.6’T = 321A = 22643tons

SECTIoiiPROPEH1”lES

SEC~IONLE}IGTH(ft)I 6.850000E+012 5.400000E+OI3 5.4(IOOOOE+OI4 5.400000E+01q 5.40L)OOOE+OI6 5.400000E+OI7 5.400000E+UI8 5.400000E+019 ki.400000Ei-Lll10 -7.4JUOOOE+OI

s.Ec-rrOl~ ELAS.FDN(ton fft’)

I (.).2 5.52f3000E-013 I .215000E+OL)4 I .[iJbwx)E+(JJb 2.293000E+UU6 2.3430UUE+W“1 2.2”750L)(JE+Wd 1.90900()!E+W

‘9 I .l”7!Jux)E+w10 2.2300WE-01

?JOD.OFELAS.(tom/ft’)

1.92ti600E+06] .(,)2H~,~oE+06I .92tj6(loE+06I .~2~600E+06I .928690E+061.928600E+061.Y28G90EW61.9,28600E+06I .928600E+06I .Y2H600E+06

ROT.EJ-AS.FDN

().L).o.0.().o.(.).L).o.0.

M()./~y;r?TIA

3.6.33:)OQE+035.867909E+037. oo!loo:)lE+o.36.Y33900E+0.37.3.3300.1E+036.9]-/~~,JS+037.933U0’.IE+O.36.466U(XIE+0.34..?33OOOE+O.?2.doouoJE+03

SHE&~,)AREA

4.30000:IE+O05.970(.)!lJE+o(36.95000JE+O07.35000JE+O07.07dooul:+oL)6.469tiWJE+O06.64000\)E+oo7.55[)00.)E+OO7. 110000E+OO4.”74L)UWZ+O0

/JASS~EIJSITY(Con-sec’lft’)

3.877000E-017.589000E-011.477100E+O02.513200E+C)03.243500E+(I03..424.3OOE+OO3.160900E+O01.t317FlooF+ooI.074400E+O06.575000E–Q1

POISSON”RATI()

3.0000OOE-013.0090(IOE-31.3.0000OOE-013.0000OOE-013.0000OOE-013.00(.)OOOE-013.0000OOE-013.0000OOE-013.0000OOE-013.000(IOOE-01

-48-

TABLElY-PROPERTIESOFC4-S-69bCARGOVESSELBALLASTCONDITION

LBP= 544’B = 82’D = 45.5’T = 19.5’A = 12004tons

SECTIONPROPER~’IES

SECrION LEh!GTH(ft)

“1 6.850000E+OI2 5.400000E+OI”3 5.4ooo(loE+dl4 5.4QOOOOE+015 5.400000E+OI6 5.400000E+OI7 5.4:IOOOOEWI8 5.4000WE+019 5.4UOOOOE+OI

l(.) ‘/.43(JOOUE+til

SECTION ELAS. FDN(tcm/ft’)

1 ().2 5.520000E-013 I.215000E+O~4 1.LldbouuE+uL)~ 2.2$300(IE+OL)o 2.343000E+O07 2.2”/5oooE+oLl8 1.909000E+O!J‘J I.17LC)OOE+OO”10 2.2300UOE-01

!JO13.OFELA5.(run f~t’)

1 ~9286QOE+061.928630E+06I .92H600E+G61.928600E+(161.92H6C)OE+061.928600E+061.928600E+061.Y28600E+061.928600E+061.928600E+06

ROT.ELAS.FDN

0.0.().o.0.cl.(1.o*o.0.

M(I. INEVTIA,,ZL,,.1~~)3.6333D3E+035.2(57303E+937. O(X)!)()’)E+026.Y.3.3i)03E+037.33300.)E+036.91700]5+037.(.)3300’JE+036.46hU(X)E+0.34. .3?I33O(IE+O32.doouodE+03

SHE~l?2AREA\.-~)

4*300009E+O05.9700(L)E+O06.Y5000JE+O07.350W3E+O07.07dclo3E+oo6.’’l6ouooE+ocl6.640(XIJE+O07.55UOOJE+(XI7. IIOOOOE+OO4.740(XWE+O0

),f~ss9E]J5.ITY(tOn-sec2/fc2)

3.8771900E-017.5a9000E-olI.4771OOE+OO2.513200E+O03.243500E+O03.424300E+O03.159YOOE+O01.817600E+O01.074400E+O06.575000E-01

POISSON”F!ATI()

3.0000OOE-013.00C)OOOE-01.3.000300E-013.000moE-ol3.0000OOE-013.00U0OOE-013.0000OOE-013.0000OOE-013*OOOOOOE-013.0000COE-01

TAELE18-VARIATI,QNQFPROPORTIONSANDRESPONSESOF GENERALCARGOVESSELS- FLILLLOP,DCONDI

B% xlfj-~(degr~es)(f:) BIT (to:,) (f:“) (ton-ft)

I [ I I I30 475 2.93722,6418406 488180 16S30 509 2.73422,.6427850 495180 12445 544..52.56322,6437400 489180 10345 579 2.40622,6146850 485180 9945 614 2.26622,5856433 .458180 99

1 I 1 I 1 1B~x 10-’B~x 10-3Defltn ~2

(ton-ft)(ton-ft)(ft) lJB ~z (rad;;n/see)(ftfI I 1 I I I

133 18 0.67635.0532250,6.6.767 38.0.2181

181 21 0.78105.8171314.5 6.3120.1980

203 23 0.91966.6400.389.8 5.7900.1829

198 27 1.06217.5195478.5 5.2930.1895

176 30 1.23418.4689590.6 4.72120.2521 /

TABLE19- VARIATIONOFPROPORTIONSANDRESPONSESOF GENERALCARGOVESSELS- BALLASTCONDI

BMVX10-3 B~x il)-3B?~~o-3~efltn L2(deg~ees)(f:) (to\s)(ft=’)BIT (ton-ft)(ton-ft) (ton-ft) (fe) LIB ~z {rad~~n]sec)(ftYse

15 475 4.825 12,003 8406 542 689 34.6 0.7068 5.0532 250.6 9.1986 44.75180 46 0.060130 509 4.491 12,003 7850 481 1025 33.4 0.7352 5.8171 314.5 8.0602

180 30 0.520180 544.5 4.209 12,004 7400 448 1317 34.2 0.7925 6.6400 389.8 7.3272180 29 0.0538180 579 3.952 12,003 6850 432 1480 48,6 o.a960 7.5195 478.5 6.7435180 20 0.0904180 614 3.722 12,003 6433 426 1516 85.8 1.0810 8.4689 590.6 6.5350180 19.5 0.0403

-50-

ThematrixofshipcharacteristicsselectedforstudyisshowninTable2!3.Thematrixwaspreparedassumingconstantvaluesoffullloaddraftanddepth.Theexisting“C6”and“C8”parallelmid-bodyseriesvesselswereselectedforthefirsttwodesignpoints;Thethirddesignpointisa furtherparallelmid-bodyextensiontoa lengthof875ft. There-sultingcharacteristicsofthisdesignpointarecurrentlyofacademicinterestinthatthevalueofL/D= 16.5exceedsclassificationsocietylimits.

A fourth“designpointwasobtainedbyincreasingthebreadthofthe“c8”designto106ft,correspondingtothenominaladditionoftworowsof8 ftwidthcontainers.The106ftbreadthalso.correspondstoexistingPanamaCanalconstraints.

Containershipstendtooperatewithcargoaboardinbothoutboundandreturnvoyagesanddraftisnearconstantfortheoperatingconditionsofinterest.Thisconclusionhasbeenverifiedthroughdiscussionwiththeoperatorsofthe“C6”and“c8”vessels.Accordingly,draftandservicespeedwereheldconstantfortheseries.

SectionalpropertiesusedintheanalysisaresummarizedinTables21and22 andassumedspeedisincludedintheresponsedata,Table23.

-. ,,,

TABLE20PROPOSEDVARIATIONSINDIMENSIONSOF“C6”AND“c8”FAMILYOFCONT.AINERSHIPS

Length,B.P.,ft.,L 625 769 769 875Breadth,mid,,ft.,B 90 90 106 90Depth,mid.,ft.,D 53 53 53 53

Draft,scantling,mid.,ft.,T 33 33 33 33Displacement,mid,,1. tons 30300 42100 49585 50770

Blockcoefficient,CB 0.5713 0.6452 0.6452 0.6838

L/B 6,9444 8.54444 7.2547 9.7222L/D 11.7925 14.5094 14.509416.5094B/T 2.7272 2.7272 3.2121 2.7272B/D 1.6981 1.6981 2.0000 2.0000Notes (1) (2)

AI

Notes: (1)BasicC6-S-85a design.(2)Lengthened“C6’Tdesign,currentlyunderconstruction,designated

C8-S-85d.

-52-

TABLE21-PROPERTIESOFC6-S-85aCONTAINERSHIPFULLLOADCONDITION

LBP= 625’B = 90’D = 53’T = 32.7’A = 29880tOnS

sEc’l”IoldP1/0PEill”IE5

!?()”i.ELLS.“FI)!d

0.u.rJ.(j.0.L).ti.u.LJ.u.

-53-

TABLE22- PROPERTIESOFC8-S-85dCONTAINERSHIPFULLLOADCONDITION

LBP= 769’B = 90’D = 53’T = 33!A = 42163tons

l!oi-. ELAs. I%)N POI%OIJIMi”IO

TABLE23 - VARIATIONOF PROPORTIONSANDRESPONSEFORCONTAINERSHIPS- FULLLOADCONDITION

B?x10-3B% X10-9B1-$.10-3S (rad;~n/see)(deg;ees)(i%

Defltn ‘bB/T (to:,) (f;’) (tori-fc)(ton-ft)(ton-ft)(ft) LIB (ft~sec)(psi)

180 625 2.754 29,87813,750 778 209 37.6 0.98726.944 400.8 5.617 38.84624,465

180 221 0.2902 6,943

180 769 3.212 49,65018,585 710 360 27.0 1.00007.254 477.8 3.711 18,940

180 129 0.3022 6,577 l!n-P

180 769 2.727 42,163I

15,780~ 850 470 44.0 1.40997.847 586.2 3.792 23,270

180 395 0.6620 10,814

180 875 2.727 49,80015,780~ 1071 742.3 45.8 0.81979.722 759.0 3.276 38.00129,320

180 i 148.4 0.2747 4,062

,-

-55-

6.0.COMPUTATIONRESULTS

Theseawayresonseso~thefBurparentshipswerecomputedfor12headings, 1?forO to180 in15 increments,andforthreesignificantwaveheights.ResultsareincludedinFigures38to45. Forthevariationsoftheparentship,computationswere~ad~onlyforthoseheadingswherehighvaluesofresponseswereanticipated.Forexample,ifthemaximumwavemomentoccurredat45a@adlngfortheparentship,thencomputationsweremadefor30°,4!i~,60aand180°forthevariations.Headseascaseswerecalculatedforallparentsandvariationstoassessthepossibilityofspringing.

-Calculatedresultsaresummarizedinthefollowingtablesandfigures.

ParentShip Tables -~TEklARTJ.CORT[GreatLakesOreCarrier) 8,9 5-12

‘LTIO}[Tanker 13,14 13-20“C4”GeneralCargo 18,19 21- 39

‘~LC6/C8’4’Containerships 23 30-37ItshouldbenotedthatshipdisplacementsinTables5,10,

15and20maydifferfromvaluesgiveninTables8,9,13,14,18,19,20.and23. Valuesinthelattertablesreflectactualloadingconditionsstudiedandincludeinaccuraciesinherentintheiterativeprocedureforbalancingtheshipona wave.

-56-

7.0 DISCUSSIONOFMETHODOLOGYANDRISULTS

7.1 Methodology

Theresponsesofthefourtypesofshipsina seawayhavebeencalculatedusingthebestmethodscurrentlyavailable.TheeffectsofshipproportionshavebeenobtainedandplottedinFigures1,5 through37and46,andsummarizedinTables17through24.

Theoretically,boththedeflectionandbendingmomentareaffectedbythefollowingfourfactors:

(1)Hullproportions,(2)Ratiobetweenshipencounterfrequencyandthetwo-node

frequency,(3)Ratiobetweenwavelengthandshiplength,(4)Headingangleandwaveheight.

Unlesstheotherfactorsareconstrainedtosmall.variations,theeffectsofvariationofproportionsalonecannotbeshownexplicitly.Insuchcases,however,theresponsesmaybebeyondtherangeofinterest.Inanycase,theeffectsofthevariationofshippro-portionsonthemaximumresponsesisofgreatestinterest.Toobtainthisinformationalltheabovefourfactorsmust‘bevariedinamultiple-dimensionspace.

Becauseofthelimitedscopeofthisstudy,thisapproachisnotfeasible.Asa compromise,theproblemhasbeenseparatedintotwophases.

First,theconditionsassociatedwiththemaximumbendingmomentoftherigidshipweresearchedbyusingtheseakeepingprogram.ThenthesealoadsattheseconditionswereusedasinputtothevibrationanalsisasdescribedinSection4.5.

iAsnoted

earlier,thisapproacisonlyapproximatebecausethemaximumsea-wayloadsfromtheseakeepingprogramarenotentirelyvalidbe-causeoftheflexibilityoftheship’shull.

7.2 Results

TheresultsobtainedfromtheabovemethodsareingenerallygoodagreementwiththerequirementsofABS.Therelationsobtainedinthestudy,asshowninFigures1 and46,maybeusefulforde-signpurposes.

Forpresentationoftheresults,variousrelationsbetweentheshipproportionsandtheresponsesweretried.Onlythree

.

10.0

5.0

[

1P=

L = LBPI= MOMENTOFINERTIAATMIDSHIPD = DEPTHB = BEAMT = DRAFT

o CARGOSHIPS● ORECARRIERSo CONTAINERSFIIPS~ TANKERS

o 2 3

SHIPPROPORTIONFACTOR,PX I&

4

,

FIGURE46-EFFECTOFSHIPPROPORTIONSON THEHULLFLEXIBILITY(REPRESENTEDBYTHETWO-NODEFREQUENCY)

TABLE24-RELATIONBETldEENSHIPPROPORTIONSANDHULLFLEXIBILITYFORTHETWO-I!ODEFREQUENCY

~ = (A+iYbB’L)L k = 0.0097143

SHIP (:t) (A) (t~ns) f’;$q Tig:; ,(~,, CB ,

CargoShtp,FullLoad . 94.0 475.0 22,641 6.7670 4.880 0.01751 0.5465 0,0228782.0 544.5 22,643 5.7900 4.890 0.01964 0.5470 0.0213372.5 614.0 22,585 4.7212 4.580 0.02182 0.5482 0.01875

CargoShip,Ballast 94.0 475.0 12,003 9.1986 5.419 0.4830 0.0359982.0 544.5 12,004 7.327 4.480 0.4830 0.0281972.5 614.0 12,003 6.535 4.260 0.4844 0.02551

Tanker,FullLoad 194.9 968.0 302,850 2.537 44.040 0.00818 0.8425 0.00753178.0 1060.0 303,114 2.347 49.930 0.00889 0.8424 0.00815164.5 1147.0 304,017 1.977 50.530 0.00926 0.8425 0.00789

Tanker,Ballast 194.9 968.0 143,870 2.537 48.270 0.7804 0.01179178.0 1060.0 144,392 2.895 50.540 0.7832 0.01192164.5 1147.0 143,859 2.582 54.770 0.7803 0.01259

OreCarrier,FullLoad 104.6 788.5 57,486 3.058 8.939 0.9070 0.00849104.6 988.5 74,472 2.107 10.493 0.9260 0.00618104.6 1288.5 90,267 1.425 20.990 0.9430 0.00742

OreCarrier,Ballast 104.6 788.5 45,645 3.141 7.074 0.6960 0.00862104.6 988.5 54,639 2.149 10.870 0.6646 0.00883104.6 1288.5 74,549 1,926 20.620 0.6956 0.00942

ContainerShip,FullLoad 90.0 625.0 29,878 5.617 7.786 0.01920 0.5690 0.0215390.0 769.0 49,650 3.710 7,100 0,01697 0.6459 0.01040106.0 769.0 42,163 3.790 8,500 0.01840 0.6458 0.01146

-59-

non-dimensionalparameterscanproducemeaningfulrelations,namelyL/B,L2/BI~BM/(A+A’)L.Theeffectsoftheseparametersarediscussedintiefollowingparagraphs.

Basedinthecalculatedresults,thefollowingobservationscanbemaderegardingtheeffectsofvariationofshipproportions:

(1)

(2)

(3)

(4)

(5)

(6)

NoobviouseffectsofB/Tonanyresponsecanbefoundforconstantdisplacementofa givenship,

AllresponsesareaffectedbytheL/BratioandbyL2/BI~.However,increaseoftheseproportionsdoesnotnecessarilyincreasetheresponse,especiallywhenL/Bissmall.

Theeffecc’sofallshipproportionsonthemaximumvertical vibration bendingmomentcanbeobtainedfromFigure46andthefollowingsimplerelations:

BMV= 0.004(A+A’)LuI(17)

A’= 0.0097143CBB2L

Foranyshipwithgivenproportionsthetwo-nodefre-quencycanbeobtainedfromFigure46. Withml,themaximumverticalvibrationbendingmomentcanbecal-culatedfromtheaboveequation.

Themaximumverticalbendingmomentwascalculatedfora significantoceanwaveheightof25feetand20.5feetfortheGreatLakes.

Theaboveequationsindicatethatthebendingmomentisproportionaltodisplacement,addeddisplacementduetothewater,blockcoefficient,squareofthebeam,squareofthelength,andthetwo-nodefrequency.

Sincethehullflexibilityisinverselyproportionaltothefrequency,thebendingmomentfora givenshipandloadingconditiondecreaseswithincreaseinflexibility.

Ofparticularinterestaretheresultsobtainedforthe.GreatLakesorecarrierSTEWARTJ.CORTgiveninFigures5through7. Resultsshowthatresponsesforthespringingconditionfora 5 foot-highsignificantwavecanbehigherthantheresponsesfortheshipona 20foot-highsig-nificantwave.

-60.-

8.0 CONCLUSIONS,APPLICATIONS,ANDRECOMMENDATIONS

8.1 Conclusions

Fromthepreviousdiscussionsandtheresultsofthevibrationcalculationsforthefourshiptypes,thefollowingconclusionscanbemade:

8.1.1HullFlexibility- Inthepast,manyshipboardvibrationproblemshavebeenattributedtothetrendtowardincreasinghullflexibility.Forthisreason,itfollowedthata criterionforalimittohullflexibilitywasneeded.However,fromEheresultsshowninFigure1 and46,itcanbeconcludedthata specificlimittohullflexibility,withparticularrespecttototalbendingmoment,maynotbenecessaryforthefollowingreasons:

(1)

(2)

Eventhoughthecalculationsandresultsreportedhereinaresubjecttothelimitationsdiscussedearlier,therelationshowingthebendingmomentfora givenshipandloadingconditiondecreasingwithincreaseinhullflexi-bilityisvalid(Figure1). Obviously,a completelyflexi-bleshipinwavescannotbesubjectedtoanybendingmo-ment.Forexample,navalarchitectshaveseriouslypro-posedhingedships,toreducebendingmomentbyincreas-inghullflexibility.

Almostallshipboardvibrationproblemsarelocalpro-blems.Thefollowingareoftenmentionedexamples: .

Propulsionsystemproblemsoccurbecauseofhullflexibilityandcorrespondinghull-shafting-bearingsysteminteractions.Theshiphullmustprovideafounda~ionstiffenoughfortheshaftingsystemandmachinery.Thisproblemcanbesolvedbyreinforcingtheportionofhullinvolved.Machinerycompartmentsoflargevesselsaregenerallylocatedwellaft.Inthesecases,theafterone-fourthorone-fifthofthehulllengthcanbereinforcedtothedesirabledegree.Thisonlyaffectsthehullflexibilityslightly.Inthiscase,averyflexiblehull,withpropersupportofthemachineryandshaftingsystem,canstillbeacceptable.

ForspecialshipssuchasLNGcarriers,hulldeforma-tionscancauseproblemsinwayofLNGcontainment.Thedegreeofflexibilitythatcanbetoleratedgenerally,orlocally,dependsuponthenatureofthecontainmentsystem,includingthechoiceofindepen-denttankversusintegratedcontainmentsystems.

. HullFlexibilitycanbea causeofhullvibrationswhichcausehabitabilityproblemsinpersonnelspaces.Againthisisa localproblemwhichcanbesolvedlocallyandisnotnecessarilyrelatedtohullflexi-bility.

(3)Withintheaccuracyoftheexistingshipvibrationandseakeepingtheories,t-heverticalbendingmomentseemstobedecreasingwithdecreaseintheshiphulltwo-nodefrequency.Thisimpliesthatbendingmomentdecreaseswithincreaseinhullflexibilitywhenspringingisnota factor.

8.1.2Methodology- Theshipmotionandshipvibrationproblemisessentiallya hydroelasticproblem.Theexistingmethodsbasedoncombiningrigid-shipseakeepingtheoriesandflexible-shipvibra-tiontheoriesmayleadtounacceptableerrorsforflexibleships,whichisthegeneralcase.Withrigidhullgirders,thevibrationproblemreducestoallowuseoftheexistingseakeepingtheories.

Becauseoftheuncertaintiesinthedampingandtheforward-speedeffects,andthehydroelasticeffectsontheshipresponse,theexistingmethodsofanalysis,includingtheoneusedinthisstudy,arenotadequateforspringingcalculations.

8.1.3SeakeepingTheories

(1)

(2)

(3)

(.4)

Theassumptionofrigid-shiphullsinherentinallexistingseakeepingtheoriesisnotvalidforlargeshipsbecauseoftheeffectsofthehullflexibility.

Evenforrelativelysmall,stiffshipstheexistingsea-keepingtheoriesdonotproperlyaccountfortheforward-speedeffectsandthehydrodynamiccoefficientstowardtheendsoftheship.

Inhigh-energywaves,wherethetwo-nodefrequencyoftheshipismuchhigherthantheencounterfrequency,thesea-keepingtheoriestendtooverestimatetheseawayloads.

Inlow-energywaves,wherethetwo-nodefrequencyoftheshipiscloseto,orcoincideswith,theen;ounterfre-quency,theseakeepingtheoriestendtooverestimateorunderestimatethesealoads.

8.1.4VibrationTheories

(1) Sincea hydroelasticformulationofthevibrationtheoryisbeyondthescopeofthisproject,allequationsofmotionsinthisreportaretentativeandshouldnotbeuseddirectly.Thecompletesetofequationsofmotion

-62-

hasnotbeenformulated.Forthisreason,theexac~expressionsforthetermsassociatedwiEhtheforwardspeedhavenotbeenformulated.Atthistime,itisonlycertainthatsomeimportanttermsforforward-speedeffectshavebeenignoredintheexistingvibrationtheoriesandtheexactexpressionsofthesetermshavenotbeenes.dab-lished.

(2)Theforwa~d-speedeffectsonthevibrationresponseareimportant.However,differentinvestigatorsstillusedifferenttermsfortheforward-speedeffects.Itisevidentthatmanysignificantforward-speedeffectshavebeenignoredintheexistingshipvibrationtheories.

(3)Forwardspeedhasthefollowingimportanteffec~sonthevibrationcharacteristicsofships:

● Naturalfrequenciesoftheshiphull.. Lineardampingoftheverticalmotionoftheship

section.. Rotarydampingoftheshipsection.. Hydrodynamicexcitationforceupontheship’shull.

Notethatinmanyexistingship-vibrationmethodsonlytheeffectsonlineardampinghavebeenconsidered.

8.1.5-Damping- Thedampingcoefficientsandaddedmassfromtheseakeepingprogramsare,ingeneral,quiteaccurateformostoftheship’shull.However,thisaccuracydecreasestowardtheends.

Structuraldampingofshipsisstillanunsettledsubjectbecauseofthelackofreliabledata.Methodsusedforfull-scaledampingtestsaregenerallyinadequate.

?63-

8.2 Applications

ThemajorstudyresultsthatmayhavefutureapplicationarepresentedinFigures1 and46. AnanalyticalexpressionofthelineshowninFigure1 isgiveninEquation(17).ThelinegiveninFigure46canberepresentedbytheexpression:

ml = 218,000 1+\AL3(l+*)[l+21(18)

wheremomentofinertiaoftheshipinft4displacement,longtonsLBP,feetbreadth,feetdepth,feetdraft,feettwo-nodefrequency

TheeffectsofvaryingshipproportionscanbeevaluatedbyusingEquatiors(17)and(18),asillustratedinthefollowingexample:

ThetankerUNIVERSEIRELANDhasbeenselectedforthisexamplecalculationsincesomeexperimentalandcalculatedvibrationdatahasbeenpublishedinReferenc~20.

TheUNIVERSEIRELANDisa 326,000DWTtankvesselwiththefollowingprincipalcharacteristics:

LBP 1075.94’B 174.83’D 104.99TT 81.417’CB 0.86A 375.811longtonsI 216;483ftb-

8.2.1EstimationofTwo-NodeFrequency-canbeestimatedbysubstitutingtheship(18). FromEquation(18)wehave:

Thetwo-nodefrequencyproportionsintoEquation

l.1)1= 2.8724rad/sec

.64.

ThevalueMlcalculatedbytheAmericanBureauofShippingis3.09rad/secforthefull-loadcondition(Reference20).Thedifferenceisabout7%. SincetheABScomputerprogramhasbeenvalidatedbymanyfull-scalemeasurements,thevalueof3.09rad/secshouldbeacceptedasquiteaccurate.

8.2.2EffectofBreadthVariation- Toevaluatetheeffectofvaryingshipbreadth,allparameterswiththeexceptionofdraftwereheldconstantinthisexample.Thedraftwasallowedtochangetomaintaindisplacementconstant.Insucha caseEquation(18)reducesto:

4.68815al=r (1+ 3.51267132X 10-5)(0.943694+0.0019505B)

Ifnowbreadthisincreasedfrom174.83’to200’,theU1valuebecomes

ml = 2.61754radlsec

Thisvalueofthetwo-nodefrequencyislessthanthe2.872rad/seccalculatedforthe174.83~breadthship.Accordingly,thewidership~smoreflexible.

Variationinothershipparameterssuchaslength,depth,draftanddisplacementcanbeevaluatedina similarmannerusingEqua-tion(18).

8.2.3EffectofMaterialChanges- Ifhigh-strengthsteelisused,theclassificationsocietiesusuallyallowcertainreductionsinthesectionmodulus.Ifallothershipparametersareheldconstant,theeffectonhullflexibilitycanbereadilycalculated.

Itisassumedthattheoriginaldeckandbottomplatingofthetankerwasconstructedofmildsteel.Ifa high-strengthsteelwithOy= 34,000psi,au= 66,000psiwassubstitutedforthebottomanddeckplating,thenthesectionmoduluscanbereducedto

s~ts= 70900(34000+ 44000)x SM

= 0.908974SM

FromEquation(18)wehave

ul,hts= 0.95340ml

-65-

Forothermaterialsorconsiderationsofmaterials,theme~hodofequivalentmomentofinertiaisgiveninSection4.6.Followingthecalculationoftheequivalentmomentofinertia,Equation(18)shouldbeusedtoobtainthetwo-nodefrequency.

8.2.4EstimationofTotalVerticalBendingMoment- Themaximumverticalbendingmomentthata shipmayencountercanbecalcu-latedbyusingEquation(17)orFigure1. Fortheexampletanker

A’= 274,745

BMV= 0.004 x 650556 X 2.8724 x 1075.94= 8,042,251ft-ton

Theabovebendingm,omentappearstobeexcessive.However,asindicatedthroughoutthisreport,thebendingmomentpredictedbyEquation(17)orFigure1maybetooconservativebecauseofthemanyuncertaintiesinvolvedintheexistingtheory.Thedevelopmentofa moreaccuratetheoryasrecommendedinSection8.3,wouldresultinappropriatemodificationofEquation(17).

8.3 Recommendations

Thescopeofthestudyreporthereinwasnecessarilylimited.Accordingly,thefollowingspecificareasofinvestigationarerecommendedforfuturestudies:

8.3.1Developmentofa ComputerProgramBasedona HydroelasticFormulation-Allelementsforthedevelopmentofa computerpro-gramforstudyofshipvibrationsbasedona hydroelasticformu-lationareavailable,Thepotentialbenefitstobederivedbysucha programincludethefollowing:

(1)Thecomputerprogramcouldbeusedtoevaluatethevariousmethodsavailableforinvestigatingtheeffectsofforwardspeedonnaturalfrequencies,damping,andtheexcitationforceontheship.Thebestmethodcouldthenbeselectedforfutureuse.

(2)Theprogramcouldbeusedtodeterminetheerrorintro-ducedbyrigid-bodyseakeepi.ngtheories.

(3) Dampingexperimentsshouldbeguidedbytheory,andavailabilityofa moreaccuratevibrationtheorywillimproveresults.Forexample,existenceofsuchatheorywouldpermittheisolationandverificationofthecomponentsofdampingforces.

(4) Thecomputerprogramcouldbeusedtoverifythere-lationsgiveninFigures1 and45.

-66-

8.3.2GeneralizationoftheAnalyticalApproachtoHullFlexibility-Forthepurposeofthisstudy,fourspecificshipswereselectedforanalysis.Theshipvibrationproblemiscom-plexandthevibrationanalysisiscostly.Inordertoproperlydeterminethegeneralrelationshipsbetweenshipproportionsandvibrationresponses,.withinreasonablelimitsoftimeandbudget,thefollowingapproachisrecommendedfor,futurestudies:

(.1)Allstructuralandhydrodynamiccoefficientsintheequationsofmotionsshouldbetreatedasfunctionsofshipproportionsinappropriateexpressionsratherthenassimplenumericalvalues.

(2)By definingthehullgeometrybysimplebutrealisticmathematicalexpressions,thesolutionforvibrationresponsecouldbeexpressedintermsoftheshippro-portions.

-67-

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

N.Salvesen,E.O.Tuck,andO.Faltinsen,“ShipMotionsandSeaLoads”,TransactionsSNAME,1970.

B.V..Korvin-KroukovskyandW.R.Jacobs,“PitchingandHeav-ingMotionsofa ShipinRegularWaves”,TransactionsofSNAME,Vol.65,1957.

E.V.Lewis,“TheMotionsofShipsinWaves”,PrincipalsofNavalArchitecture,ChapterIX,SNAME,1967.

ltlthAmericanTowingTankConference,ReportofSeakeepingCommittee,August1977.

R. Kline,“SpringingandHydroelasticProblemsofLargeShips”,August26-29,1975,SNAME.

R.A.Goodman,“Wave-ExcitedMainHullVibrationinLargeTankersandBulkCarriers”,RINA,1971.

D.Hoffman,etal,“ExperimentalandTheoreticalEvaluation,ofSpringingonA GreatLakesBulkCarrier”,AD-776861,July1973.

S.G.Stiansen,A.Mansour,andY.N.Chen,“DynamicResponseofLargeGreatLakesBulkCarrierstoWave-ExcitedLoads”,TransactionsofSNAME,1977.

H.Chin,“SolutionoftheBoundary-ValueProblemsofPotentialEquationbyInvariantEmbedding”,Dissertation,UniversityofCalifornia,Berkeley,1975;ProgramPOT3.

E.F.Noonan,“DesignConsiderationsforShipboardVibration”,PresentedattheFebruary17,1970meetingoftheNewYorkSectionofSNAME.

R.T.McGoldrick,“ShipVibration”,DTNSRDCReport1451,December1960.

P.Y.Chang,“StructuralAnalysisofColdWaterPipesforOceanThermalEnergyConversionPowerPlants”,HYDRONAUTICSTechnicalReport7676-1,May1977.

S,S.Chen,“Parallel-Flow-InductedVibrationofFuelRods”,NuclearEngineeringandDesign,18(1972),253-278.

14.

15

16.

17.

18.

19.

20.

21.

R.W. Gregory and M.P. Paidoussis, “Unstable Oscillation ofTubular Cantilevers Conveying Fluid”, I Theory, Proceedingsof Royal Society A 293, 512-542, 1966.

T.B. Benjamin< “Dynamics of A System of Articulate PipesConveying Fluid,’’Part I and II, Proceedings of the RoyalSociety A261, 457-499, 1961.

J. Ploeg, “Wave Climate Study - Great Lakes and Gulf ofSt. Laurence” , SNAME T and R Bulletin No. 2-17, May 1971.

W.E, Woolman, “A Review of the ‘State-of-the-Art’ of Vibra-tion Damping Associated with Ship Hull Structures”, TR No. 1,for NDTMB, February 1965.

W.E. Woolman, “Analysis of Methods for Computing and Mea-suring Vibration Damping Characteristics of Ship HullStructures” , TR No. 2 for NDTM8, June 1965.

W.D. Pilkey and P.Y. Chang, “BEAM: A Program for the Static,Stability, and Dynamic Response of Beams,” Structural MechanicsSoftware Series I, Part I , PP. 347-376, University Pressof Virginia, 1977.

S.G. Stiansen and A.E. Mansour, “Ship Primary Strength Basedon Statistical Data Analysis,” Transactions, SNAMF 1975.

R.A. Barr and P.Y. Chang, “Methods for and Examples ofDynamic Load and Stress Analysis of OTEC Cold Water PipeDesigns,” HYDRONAUTICS, Incorporated, Technical Report7825-2, Volume II, Appendix B, November 1978.

SHIP RESEARCH COMMITTEEIlar’itim Transportation Research IIc!ai”d

Nationa? fl.cadefiyof Sciences-National Research Council

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Tt,e ~lii;, ileseevch Committee has teci]nica’1cognizmce of it:y Ship Stricture Lomn:ttee’s researcn pl-)cjram:

SHIP STRUCTURE CONMI”rI”EEPUBL ICATIOMS