STRUCTURAL ANALYSIS OF LONGITUDINALLY FRAMED ...SSC-225 STRUCTURAL ANALYSIS OF LONGITUDINALLY FRAMED...
Transcript of STRUCTURAL ANALYSIS OF LONGITUDINALLY FRAMED ...SSC-225 STRUCTURAL ANALYSIS OF LONGITUDINALLY FRAMED...
-
SSC-225
STRUCTURAL ANALYSIS OFLONGITUDINALLY FRAMED SHIPS
This document has been approvedfor public release and sale; its
distribution is unlimited.
SHIPSTRUCTURE COMMITTEE*
1972
-
SHIP STRUCTURE COMMITTEEAN INTERAGENCY ADVISORY
COMMITTEE I) EOICATEO TO IMPROVINGTHE STRUCTURE OF SHIPS
ADDRE5S cORRESPONDENCE TO:
sEci7ETAt7Y
?,I{IPSIRUCT(IHF (: [)MMI TTEFi!.S. c
-
SSC-225
FinalReport
on
ProjectSR-196,“ComputerDesignofLongitudinallyFramedShips”
tothe
ShipStructureCommittee
STRUCTURALANALYSISOFLONGITUDINALLYFRAMEDSHIPS
by
R.Nielson,P.Y.Chang,andL.C.DeschampsCOM/CODECorporation
under
DepartmentoftheNavyNavalShipEngineeringCenterContractNo.NOO024-70-C-5219
Thisdocumenthasbeenapp~ovedforpublic?eleaseandsale;itsdistributionisunlimited.
U.S.CoastGuardHeadquartersWashington,D.C.
1972
-
ABSTRACT
Thetechniqueoffiniteelementshasbroughtabouta neweratothefieldofstructuralanalysisofshipstructures.Theapplicationofthistechnique,however,is limitedbythecostandcapacityofthecomputer.Straightforwardapplicationsofthe finiteelementmethodtothewholeortoa majorportionoftheshiphavesofarbeeninac-curateandtooexpensivefordesignpurposes.
Themethodpresentedcombinestheadvantagesofthefiniteelementtechniqueandtheuncouplingbycoordinatetransformation.A finemeshmaynowbeusedtoproducemoreaccurateboundaryconditions.Theun-couplingtransformationsalsoreducethecomputertimetoaboutone-t.enthofthatbyothermethods.Thecriticalassumptionsandthebasictheorieshavebeen verifiedwithexperimentaltestresultsfromthetanker“JOHNA.MCCONE.”
Thisreportdiscussesthreecomputerprograms;oneforthelongitu-dinalstrengthanalysis,onefortransversestrengthanalysis,andonefarthelocalstabilitycheckofthestructure.Theprogramsthemselvesappearinsubsequentreports.
ii
-
CONTENTS
Page
INTRODUCTION.. . . . . . . , . . . . . . . . . . . . . . . . . 1
LONGITUDINALSTRENGTH. . . . . . . . . . . . . . . . . . . ...6
TRANSVERSESTRENGTH. . . . . . . . . . . . . . . . . . . . ...10
CORRELATIONOFTHEORETICALSTRESSESWITHSTRAINGAUGEEXPERIMENTS.. . . . . . . . . . . . . . . . . . . . . . .20
REFERENCES.. . . . . . . . . . . . . . . . . . . . . . . . . .26
APPENDICES
A.
B.
c.
D.
E.
F.
G.
H.
I.
LONGITUDINALSTRENGTH,MATHEMATICALDEVELOPMENT. , . . . . 28
TRANSVERSESTRENGTH,MATHEMATICALDEVELOPMENT. . . . . . . 31
STIFFNESSOFLONGITUDINAL.. . . . . . . . . . . . . . . .38
INFLUENCECOEFFICIENTSANDDEFLECTIONSOFPRIMETRANSVERSEMEMBERS.. . . . . . . . . . . . . . . . . . . .41
EFFECTOFLONGITUDINALNOTCOINCIDENTWITHNODALPOINT. .43
SIMILARITYOFTRANSVERSES. . . . . . . . . . . . . . . . .46
APPLICATIONSOFPROGRAMTOORECARRIERSANDCONTAINERSHIPS.. . . . . . . . . . . . . . . . . . . . .50
ANALYSISOFPARTSOFTHEHULL. . . . . . . . . . . . ...54
STABILITYANALYSIS.. . . . . . . . . . . . . . . . . . . .58
iii
—— —— —
-
SHIPSTRUCTURECOMMITTEETheSHIPSTRUCTURECOMMITTEEisconstitutedtoprosecutea research
programtoimprovethehullstructuresofshipsbyan extensionofknowledgepertainingtodesign,materialsandmethodsoffabrication.
RADMW.F.Rea,III, USCG,ChairmanChief,OfficeofMerchantMarineSafety
U.S.CoastGuardHeadquarters
Capt.J.E.Rasmussen,USN Mr.E.S.DillonHead,ShipSystemsEngineering Chief
andDesignDepartment OfficeofShipConstructionNavalShipEngineeringCenter MaritimeAdministrationNavalShipSystemsCommand
Capt.L.L.Jackson,USNMr.K.Morland,VicePresident MaintenanceandRepairOfficerAmericanBureauofShipping MilitarySealiftCommand
SHIPSTRUCTURESUBCOMMITTEE
TheSHIPSTRUCTURESUBCOMMITTEEactsfortheShipStructureCommitteeontechnicalmattersbyprovidingtechnicalcoordinationforthedeterminationofgoalsandobjectivesoftheprogram,andbyevaluatingandinterpretingthere-sultsintermsofshipstructuraldesign,constructionandoperation.
NAVALSHIPENGINEERINGCENTER OFFICEOFNAVALRESEARCH
Mr.P.M.Palermo- Chairman Mr.J.M.Crowley-MemberMr.J.B.O’Brien- ContractAdministratorDr.W.G.Rauch- AlternateMr.G.Sorkin-MemberMr.H.S.Sayre-AlternateMr.I.Fioriti-Alternate
U.S.COASTGUARD
LCDRC.S.Loosmore,USCG- SecretaryCAPTC.R.Thompson,USCG- MemberCDRJ.W.Kime,USCG- AlternateCDRJ.L.Coburn,USCG- Alternate
MARITIMEADMINISTRATION
Mr.F.Dashnaw-MemberMr.A.Maillar-MemberMr.R.Falls-AlternateMr.R.F.Coombs- Alternate
MILITARYSEALIFTCOMMAND
Mr.R.R.Askren-MemberLTJGE. T. Powers,USNR- MemberAMERICANBUREAUOFSHIPPING
Mr.S.G.Stiansen-MemberMr.F.J.Crum-Member
NAVALSHIPRESEARCH& DEVELOPMENTCENTER
Mr.A.B.Stavovy-Alternate
NATIONALACADEMYOFSCIENCES-ShipResearchCommittee
Mr.R.W.Rumke,LiaisonProf.R.A.Yagle,Liaison
SOCIETYOFNAVALARCHITECTS&MARINEENGINEERS
Mr.T.M.Buermann,Liaison
BRITISHNAVYSTAFF
Dr.V.Flint,LiaisonCDRP.H.H.Ablett,RCNC,Liaison
WELDINGRESEARCHCOUNCIL
Mr.K.H.Koopman,LiaisonMr.C.Larson,Liaisoniv
-
INTRODUCTION
Prelude
Theshiphullisa complexstructuresubjecttothemultiplestaticanddynamicloadingsimposedbyitsmass,itscontents,andthetime-dependentforcesofthesea.A satisfactoryprocedureofstructuraldesignrequiresa completeknowledgeoftheloadingsaswellasa methodofaccuratestructuralanalysis.Whilethisultimategoalmaynotbereachedforsometime,improvementshavebeenmadeinbothareas,l~2J3Themostnotabledevelopmentsinstructuralmechanicsasappliedtoshipshavebeenthetehniquesoffiniteelements4*5andthetheoryofgrillages.6*7~8~8 Thecomplexityofa ship’shullsuggeststhattheversatilefiniteelementtechniquewouldbetheidealanalysistoolifthecomputertimecanbeheld?8wntowithina reasonablelimitforanacceptabledegreeofaccuracy.In thelightofthisproblem,then,thisreportpresentsa newapproachtotheanalysisoflongitudinallyframedships.Itstheoreticalfoundationisbaseduponthefollowingthreeobservations:
1. Resultsfromfull-scaleshiptestsconfirmthattheshiphullofmoderatesizebehavescloselyasa simplebeamwithsheardeflect~on.Thetrendofmodernshipbuilding,however,hasbeentowardincreasinglylargershiphulls,andthefuturemayrequirerefinementstothiselemen-tarymethodofanalysistoincludethepossibleeffectsofexpandedbeam/lengthandbeam/depthratios.
2. Paststructuralfailuresoflargetankersrevealedprincipalareasofdamageattheintersectionsoftheprimelongitudinal(longitu-dinalbulkheads,sideshells,deeplongitudinalgirders)andtransverse
{memberstheoil-tightandswashbulkheads,andthedeeptransversewebframes).1 Thelinesofbucklingoftenshowedthecharacteristicfeaturesofdeformationunderexcessiveshearloadsandindicatethattheshearloadsattheintersectionsoftheseprimemembersmustbere gnizedasimportantfactorsinthestructuralanalysesoftheseships.t?
3. Thetheoreticalnavalarchitecthaslongrecognizedtheflowpatternofloadtransferenceamongthestructuralmembersofthelongi-tudinallyframedshipasfollows:Loadsfromtheplatearetransferredtothelongitudinalframes,thentotheprimetransversemembers(i.e.,bulkheadsandwebframes)andfinallytotheprimelongitudinalmembers.However,thisflowpatternhasneverbeenconsideredinthecalculationofthelongitudinalstrengthofships,andonlyver recentlyhasItbeenconsideredinthetransversestrengthanalysis.351#AS thesizeOftankersincreases,thisflowpatternassumesgreaterimportance.
A BRIEFREVIEWOFTHESTATUSOFSHIPSTRUCTURALANALYSIS
Theconventionalapproachfora ship’sstructuralanalysiscanbedividedintothreestages:First,theshiphullistreatedasa thin-wallsimplebeamtodeterminetheprimaryorlongitudinalstrength.
-
2
ThevalidityofthismodelingtechniquehasbeenverifiedbyVasta’sinvestigations13forshipsof50,000tonsorsmaller.Butfortheshiphulltobehaveasa thin-wallsimplebeam,thetransversemembersmustbestrongenoughtomaintainanessentiallyconstanthullcross-section.
Thesecondstageisthetransversestrengthanalysisforwhichseveralapproacheshavebeenused.Oneistotreatthetransversememberasanindependent,reinforcedtwo-dimensionalspaceframelorelasticbodylowiththeshells,longitudinalbulkheadsandcentralgirdersmodeledasconcentratedsprings,butwiththeeffectofthesmallerlongitudinalmembersneglected.Toreqainmoreofthecouplingeffects,anothermethodhasbeentotreatthestiffenedpanelsofbulk-heads,decks,bottomandsideshellsasorthotropicplates,grillagesortwoorthree-dimensionalspaceframeswithcalculationsconfinedtooneholdonly.Morerecently,thetechniqueoffiniteelementshasbeenappliedwherebytheentirehullora portion“thereofismodeledasathree-dimensionalstructure.Theresultingsolutionsthenprovidetheboundaryconditionsfora moredetailedanalysisofthetransversememb-er understudy.10!5~3!15
Finally,theunstiffenedplatepanelsaretreatedasisotropicplatestodeterminethetertiarystresses.
Althoughtheconventionallongitudinalstrengthanalysishasprovedtobeadequateforshipsofmoderatesize,evidencesuqgeststhata moreelaboratemethodisneededforverylargeships.Asa three-dimensionalfloatingstructure,theshipissub,jecttonotonlyverticalbending,butalsogirthbendingandcompression,horizontalbending,andtwisting.Forshipswithsmallbeam/lengthandbeam/depthratios,theonlyimportantfactorsaretheverticalbendingandthegirthcompression.Allotherfactorsmaybeneglected,exceptfortwistinginshipswithlargedeckopenings.Forverylargevesselsthesefactorsmaybesignificant.
,ontheotherhand,allofThisreportgoesonestepbeyond
existingmethodstoincludetheeffectsofdeformationofthetransverseandsheardistributionbetweentheprimelongitudinalhullgirdersandtransversemembers.Theeffectsoftwistingandhorizontalbendingareleftforfutureinvestigations.
Forthetransversestrength,theframeanalysisIssimple,butthereisnouniquewaytodeterminethestiffnessandspanofeachmember.Themethodtodeterminethespringfactorsisstillanartratherthanascience,anddifferentinvestigatorscanobtainverydifferentresultsforthesamestructureevenwhenusingthesamecomputerprogram.Thedeterminationofboundaryconditionshasbeena subjectofmuchdiscussionbutremainsunsettled.Thebestwaytoavoiddifficultyistotakealargerportionofthestructureintoconsideration.
Theprincipleofsuper-positionisvalidonlyiftheboundarycondi-tionsadoptedineachstageofthecalculationsareexactlydefinable;thissituation,however,isgenerallyimpossiblesincetheboundarycon-ditionsofanyregionarefunctionsofboththeship’sgeometryandtheloadsactinguponit.
—_ — —- ._._
-
3
THEAPPLICATIONOFTHETECHNIQUEOFFINITEELEMENTSTOTHEANALYSISOFSHIPSTRUCTURES
Thefiniteelementmethod(FEFl).hasbeenusedeffectivelyformanyyearsbytheaerospaceandcivilengineers.Infact,itconstitutestheonlypracticalmethodfortheanalysisofcomplexstructuresandasappliedtoshipstructuresinrecentyearshasproducedgoodresults.3,F,~~,l~
AlthoughthebasictheoryofFEhliswellknowntoengineers,itisimportanttoreviewtheaccuracyofthistechnique,whichdependsuponthefollowingfourfactors:
1. Thediscretizationoftherealstructure.Thecontinuousstruc-turemustbeidealizedIntodiscreteelements.Theconsequenceofapproxi-matinga continuumofinfinitedegreesoffreedomwitha fiodeloffiiitedegreesoffreedomisthecliscretizationerrorwhichisoftenmeasuyedbyhowcloselyassumeddisplacementfunctionscanrepresentthetruedisplacements.
2. Thetypesofelements.Nanytypesofelementshavebeendevelopedfordifferent.purposes.Thetv~eofelementforwhichtheassumeddisplace-ment-Functions’satisfyallcom~atibilityconditionsattheboundaries&ftheelementiscalledconforming.Sincesuchfunctionsaredifficulttodevelopforsometypesofelements,functionswhichsatisfyonlyportionsofthecompatibilityconditionsmayhavetobeused.Theseelementsarethennon-conforming.Thedifferencebetweenthetwotypesisthatasthesizeoftheelementapproacheszero,thesequenceofapproximateSo-lutionsconvergestotheexactsolutionfortheconformingelementbutmaycon-vergetoanincorrectvalueorevendivergefornon-conformingelements.Althoughtheconformingelementsdonotnecessarilyyieldbetterresultsina verycoarsemesh,duetootherapproximationsinvolved,theyarepreferablewheneverpossibleforanalysisinfinermeshes.
3. Thenumberofelementsandtheroundingerror.Foranalysesusingconformingelements,thediscretizationerrormayber-educedbyusinga finermesh;~.e.,increasingthenumberofelements.Howevergasthenumberofdegreesoffreedomincreases,anotherkindoferrorbeginstogrow.Thecomputerrecognizesonlya certainnumberofdigitsofanynumericalvalue;and,consequently,round-offerrorscanaccumu-lateandbecomeverylargeattheendofa computation.Sincethiserrorincreaseswiththenumberofdegreesof-Freedcm,a finermeshmayevenproducea greaterneterrordependinguponthemethodsofcmnputa-tionandthecomputer.Thiserrormaybereducedwithanimprovedcom-putationalprocedureandwithdoubleprecision,buta limitwillalwaysexistwheretheincreaseinroundingerrorislargerthanthedecreaseinthediscretizationerror.
4. Theaccuracyofboundaryconditions.Toreducerounci~ngerrorsandthecomputertimeexpense,thecommonapproachistousea macromeshforthewholestructure;fora shipthemacromeshmayconsistofelementsaslargeasa basketballcourt.Thesolutionsfromthismacromeshanalysisarethenusedasboundaryconditionsfortheanalysisofa stillsmallerregionusinga micromesh,andsoon.
——— — —-——
-
4
Unfortunately,theaccuracyofthedetailanalysiscanneverbebetterthantheaccuracyoftheboundaryconditions.Iftheresultsfromthemacromeshanalysisarequestionable,thesolutionsfromthemicroanalysisarealsosuspect.Theuseofthemacromeshofnon-conformingelementspromisesresultsthatare,atbest,veryroughapproximations.
PROPOSEDMODIFICATIONSTOTHEFINITEELEMENTMETHODOFANALYSIS
Toreducethediscretizationerrorsatboundaryconditions,themethodpresentedinthisreportemploysa muchfinermeshforthethree-dimensionalanalysis.Theproblemsoftheround-offerrors,thecom-puterexpense,andofthelimitedcomputercapacityarealleviatedbythecoordinatetransformationtechniqueintroducedinAppendixB.
OUTLINEOFTHENEWAPPROACHANDITSBASICASSUMPTIONS
Thefollowingapproachincludesbotha longitudinalanda trans-versestrengthanalysis.
A. Longitudinalstrengthanalysis.
1. Thelongitudinalbulkheadswithdeckandbottomplateofthecentraltankandthesideshellswithdeckandbottomplateofthewingtanksaredefinedasprimelongitudinal.Theprimelongitudinalandthetransversesbehaveassimpleshearbeams.Vasta13hasverifiedthisassumptionforthenot-so-largeships,andnoevidenceindicatesthatthisassumptionisinvalidforthelargetankers,
2. Theprimelongitudinalareassumedtobesimplysupportedatbothends.Thesimplysupportedendisthesameasthefreeendiftheshearforceiszero.Sincetheexternalloadsareself-balanced,theshearforcesattheendshouldbesmall,andthesumofshearforcesoftheseprimelongi-tudinalisactuallyequaltozero.
3. Thetransversemembersareassumedto befree at bothends.4. Theshearforcesinthedeckandbottomplatingaresmall
relativetothelongitudinalstressesneartheintersectionsofthelongitudinalbulkheads.Thisassumptionhasbeenverifiedbyexperimentina 90,000tontanker.14
5. ThetransversesareinturnsupportedbytheprimeIongitudinals.Thisloadtransferpatternindicatesthattheprimelongitudinalareacteduponbythereactionsofthetransversesonly.Thisisanimprovementoverthecon-ventionalmethodwhichimpliesthattheexternalloadsareactedupontheprimelongitudinaldirectly.
—— _.
-
c2
B.
c,
6. Theeffectoflocalorsecondarydeformationsofthetrans-versesonthelongitudinalstressisnegligible.Thisisthebasicassumptionfortheconventionallongitudinalstrengthanalysis.However,theprimarydeformationsofthetransversememberbetweentheshellsandlongitudinalbulkheadsarenotneglected.
Transversestrengthanalysis.
1. Alllongitudinalareassumedtobesimilar,orofpropor-tionalstiffness.Thisimpliesthatthemomentofinertiachangesbythesameratioalongthelengthforalllongitudinal.Thisistrueformostships.
2. Alltransversemembersareassumedtobesimilar,orofproportionalstiffness.Thisassumptionisa necessaryapproximation.Theerrorcausedbythisassumptionissmallintermsoftheactual‘reactionsactinguponthetransverses.
3. Theexternalloadsareactingupontheplateandtransmittedtothelongitudinalsupportedbythetransverses.Loadsmaybedistributedandneednotbeconvertedtoconcentratedforcesatelementnodalpoints.
4. Theexternalloadsarearbitraryinsofarastheyaresym-metricaboutthecenterplaneoftheship.Unsymmetricload-ingsystemscanbetreatedasthesumofa symmetricsystemandanantisymmetricsystem.Thepresentcomputerprogramisapplicabletobothloadingsystems.
5. Thelongitudinalbeamelementsareassumedtobesimplysupportedatbothendsoftheship.Sincetheexternalloadsoftheshipareself-balanced,thisassumptionisthesameastheconventionalfree-freecondition.Theymaybesimplysupportedorfixedforpartialanalysisdependingupontheloadingconditionsandthenatureofthehullstructure.
6. Theeffectofthetorsionalrigidityofthelongitudinalisneglectedfortworeasons.Thiseffectisnegligibleforalllongitudinalwithopencrosssection,andthein-planetwistingatthenodalpointscannotbeaccommodatedbytheplanefiniteelementtheory.
7. Thebendingstiffnessesoftheplateelementsareneglected.Kendrick15verifiedthisstandbyshowingthatbendingstiff-nesshasvjrtuallynoeffectonin-planestress.
Thestabilitycheck
ThestabilitycheckformulasasgiveninAppendixIareinterpo-latedfromestablishedcriteriaintheliterature16*17Y18forthesimplysupportedplate.Whilethismodelingassumptionisnotexactforthewebplateofthetransversemembers,itdoesprovidea goodupperboundfordesignpurposes.
-
...- -------–– “-
6
LONGITUDINALSTRENGTHOFLARGESHIPS
Formanyyears,thelongitudinalstrengthofshipshasbeencal-culatedbythesimplebeamtheory.Recently,attemptshavebeenmadetoapplythreeimensionalfiniteelementanalysestpthewholeshipstructure.lO*1~ Inadditiontoprovidingthelongitudinalstrength,thisanalysiscanalsoprovideinformationabouttheverticalshearloadsuponthelongitudinalbulkheadsandsideshellsandalsoboundaryconditionsforlocalanalysis.Duetocomputerexpenseandthelimitofthenumberofelementsavailable,onlycoarsemeshanaTyseshavebeenpossible.
In recentyears,anexhasbeenevelopedbyKanlelI
~fil~;n;lcomputerprogramfortankeranalysisThelongitudinalstressescalculated
byDAISY1showonlyslightdeviati~nsfromthelinearstressdistributionexceptatlocationswherethebendingmomentissmall.Thiscoarsemeshanalysis,however,doesinvolvesomeerrorontheidealizationofboththeloadingandthebalancingthemodel.
fbructurewhereadditionalforcesarerequiredfor
AnanalysissimilartoDAISYwasperformedontheESSONOR!dAYusingtheprogramSESAM-69.19Theresultsindicatefairlylargediscrepanciesbetweenthemeasuredandcalculateddeflections.Resultsobtainedfromsimplebeamtheory,ontheotherhand,haveprovedtocorrelatequitewellwithfull-scaleexperiments,althoughtheseexperimentswerecon-ductedonmuchsmallershipsthantheESSONORWAY.
Authoritiesgenerallyagreethatthelongitudinalstrengthstandardsadoptedbythesocietiesusingthesimplebeamtheoryarequiteadequateevenforsupertankers.Forthesereasons,then,a coarsemeshfiniteelementanalysisforthelongitudinalstrengthisreallyunnecessary.
A simpleandaccuratemethodtocalculatethedistributionoftheshearloadbetweenthelongitudinalbulkheadsandthesideshells,how-ever,isneeded.Thiscanbedonebytreatingthehullasa grillageconsistingoffourprimelongitudinalmembers(thesideshellsandthelongitudinalbulkheads)andtheprimetransversemembers(transversebulkheads-andwebframes).(SeeFig.2-l).Thetransversemembersincludeportionsofthedeckandbottomasflanges.Theshellmembersincludeportionsoftheconnecteddeckandbottomplatingasflanges.Similarly,portionsofthebottomanddeckareascribedasflangesforthelongitudinalbulkheadmembers.Thememberdefinitionsshouldbesuchthatthetotalmomentofinertiaofthehullisexactlyequaltothatderivedintheconventionalmanner.
Theprimelongitudinalareassumedtobesimplysupportedatbothends.Thesimplysupportedendconditionisthesameasthatforthefreeendiftheshearforceiszero.Sincetheexternalloadsareself-balanced,theshearforcesattheendsshouldbesmall,andthesumofshearforcesofthesefourlongitudinalmembersshouldactuallybezero.Thetransversemembersareassumedtobefreeatbothends.
-
7
Theshearforcesinthedeckandbottomplatingareassumedsmallrelativetothelongitudinalstressesneartheintersectionsofthelongitudinalbulkheads.Theexternalloadsareassumedtobeactingupontheplateandtransmittedtotheprimetransversemembersthroughthelongitudinal.
Forsymmetricalloadsthisnewapproachisidenticaltothecon-ventionalsimplebeamtheoryiftheprimetransversememberscanbetreatedasperfectlyrigid.Butthismethodismoreusefulbecausetheshearloadsontheprimememberscanbecalculatedaccurately.Inaddition,thestressduetotransversebendingcanalsobecalculated.
TheformulationofthismethodisgivenindetailinAppendixA.
A computerprogramhasalsobeenprepared.AsillustratedinAppendixA,thismethodisonlyslightlymorecomplicatedthanthatoftheconventionalsimplebeambutstillrequiresonlya fewsecondsofthecomputertimeforthecalculations.
Table2-1providesa samplecomparisonofthelongitudinaldeckstressesderivedfromboththeconventionalsim~lebeammethodandfromthenewgrillageapproachwhichfurtherdeterminestherelativeshariroftheloadsupportbythesideshellsandthelongitudinalbulkhead.Thevalidationofthegrillagemethodhereactuallygivesevidenceoftheapproximationsmadebytheconventionalmethod.
Thedistributionofverticalshearforcebetweenthelongitudinal:bulkheadsandthesideshellisplotedinpercentageinFigure2-2.~ItispointedoutherethatRobertshastreatedthecargoportionofthetankerasa grillagefortheshearloadspriortothispaperand
9
hasdeviseda formulaForthispurpose.Theprincipleofhis-methodissimilartothatofthenewapproach,althoughsomedifferencesarenotable.ThesediscrepanciesareduetothefactthattheRoberts’formulaexcludestheeffectsofthepositionoftheloadingrelativetothecentralplaneandofthestiffnessesofthetransverses.Resultsfroma longitudinalanalysisofthe“JOHNA.MCCONE”indicatetheimportanceofthesetwofactorsandhencedonotconformwithRoberts’simpleformula.
SeveralcomputerprogramssuchasSTRESSandsTRUDLcanbeade-quatelyapplied,buttheyaremoredifficultandexpensivetousethanwhatisintroducedinthisreport.Theproposedgrillageanalysisistailoredforthelongitudinalstrengthcalculationsandincludesdefor-mationsduetoshearaswellasbendingofthedeepprimarymembers.Themethodisbasedonthetechniqueoftransfermatrices,andhence,theresultsshouldbethesameasthoseobtainedbya frameanalysisexceptthatthecomputertimeshouldbesignificantlyreduced.
-
PercentageofShearLoadOnSideSheli IntersectIon -m
A.co.
m
-
9
Table2-1.RelativeLongitudinalStressonDeck.kg/mm2
FrameNo. By BynewGrillageMethodConventional NearSide NearLongitudinalMethod Shell Bulkhead
97 ... --- ---99 0.59 0.57101 0.87 0.88103 1.14 1.15105 1.40 1.43107 1.64 1.75109 1.86 2.03111 2.02 2.20113 2,13 2.30115 2.18 2.37117 2.18 2.37119 2.13 2.28121 2.02 2.1!7123 1,86 2.02125 1.65 1.78127 1.39 1.51129 1.07 1.16131 0.70 0.74133 0.27 0.30135 -0.20 -0.22137 -!0.74 -0.85139 -1.23 -1.34141 -1.57 -1.70143 -1.83 -1.94145 -1.83 -1.98147 -1.75 -1,89149 -1.53 -1.66151 -1.17 -1.23
-0.66 -0.68’53 ..-–,..-..___.,----
0.660.917.201.451.641.801.952.082.122.122.101.971.811.621.341.040.690.26-0.19-0.69-1.20-1.54-1.71-1.80-1.73-1.49-1.17-0.68—-. .
-
10
TRANSVERSESTRENGTHOFLARGESHIPS
Theconventionaltwoorthree-dimensionalanalysesoftransversestrengthoftenassumesa pre-deformedstateofthestructure,wherethesupportingforcesuponthetransversesofthehullhavebeeneitherneglectedorroughlyapproximated.Mostanalystssimplytreattheeffectoftheship’shullasseveralrigidorspringsupports.Thespringconstants,however,aremoreartfullyderivedthanpreciselydevelopedfromthehullstructureasa contiguoussystem.A three-dimensionalfiniteelementsolutionfortheseboundaryconditionshasalsoprovedinaccuratesincea coarsemeshmustbeused.
Themethodpresentedhereissimilartothethree-dimensionalfiniteelementtechniquesinuse,buta muchfinermeshisgeneratedtoimproveaccuracy,andtheuncouplingviacoordinatetransformationssimplifiesthenumericalcomputationandthusreducesthecomputertime.Theship’shullismodeledasa three-dimensionalelasticbodyconsisti-ng ofbeamelementsrepresentingthelongitudinalandplateandbarelementsrepresentingthetransverses(Fig.3-l).Thenodalpointsattheboundariesandthetransversesarelocatedontheintersectionswiththelongitudinalwhereverpossible.(TheeffectoflongitudinalthatdonotcoincidewithanynodalpointisaccountedforbythemethodintroducedinAppendixE.) Thelongitudinalaresimplysupportedatbothendsandthetransversesarerestrainedfromhorizontalmovementalongthecenterlinebecauseofsymmetryandarealsosupportedbyanartificialsupportatthebottomofthelongitudinalbulkhead(Fig.3-2).
Thethree-dimensionalcoupledstructureofthetransverseanalysisrequiresasinputthesupportingshearforcesgeneratedwithinthehullgirderbytheexternalloadingconditions.Theexternalloadingcondi-tions,then,areusedtocomputethesecondarydeflectionsofthelongi-tudinalmembersandtheelasticdeformationsoftransverses.Boththedeflectionsoftheprimelongitudinal(equaltotherigidbodymotionsofthetransverses)andthesupportingshearforcesareavailabledirectlyfromtoelongitudinalstrengthanalysis.(Fig.3-3).Theshearforcesuponthetransversesareactuallythechangesinlongitudinalshearandmaybeapplieddirectlytothetransversemembersasexternalloads.Manyanalysessimplyneglecttheseforcesinthetransversemodelandallowtheresultingforceloadinginbalancetobecorrectedbythedevelopmentofconcentratedreactionforcesatthetransverseboundarysupports,asillustratedinFigure3-2.
Thepointsupportattheintersectionofthebottomplateandthelongitudinalbulkheadisnotnecessarywhenthehullistreatedasathree-dimensionalstructure,butisnecessaryforthefinaltwodimen-sionalanalysiswhenthesupportsofthelongitudinalarereplacedbyboundaryforces.Ifthereactionsofthelonqitudinalsuponthetrans-versesarebalancedexactlybythesupportingforces(longitudinalsheardrop)ofthelongitudinalbulkheadsandsideshells,thenforcesattheimaginarytransverseboundarysupportsshouldbezero.Butsincethelocaldeformationsofthetransversesarenotconsideredinthelongi-tudinalstrengthcalculations,thisidealconditionmaynotbecompletelysatisfied.Theresultingdiscrepancies,however,shouldbesmallerthanthosedevelopedwiththeconventionaltreatmentofthesesupportforces.
-
11
Fig.3-1.StructuralModelfortheTransverseStrengthAnalysisofShips
Fig.3-2.TransverseBoundaryConditionsforSymmetricalLoading
/\,
—— ——— . —
-
12
Fig.3-3a.PrimaryDeflectionsandSupportingForces(FromLongitudinalStrengthAnalysis)
Fig.3-3b.PrimaryDeflectionsSuper-ImposedWithSecondaryDeflections(FromTransverseStrengthAnalysis)
BythemethodpresentedinAppendixB,thisthreedimensionalsystemismathematicallyuncoupledintoa setofequivalenttwodimen-sionaltransversemembers,eachloadedwithtransformedforcesandsupportedbytransformedspringelementswhichrepresenttheeffectsofthelongitudinal.Sinceboththesetransformedforcesandspringconstantscanbecomputeddirectly,theresultingquasidisplacementsofthetransverseboundariescanbecalculated.Uponre-couplingthesystem,thesedisplacementsprovidetheactualforcesexertedbythelongitudinaluponthetransverses.Withthesereactionforcesknown,thestresseswithinthetransversemembersmaybecomputedusingaconventionaltwodimensionalfiniteelementanalysis.SeeFigure3-4.
Thefeasibilityoftheuncouplingtechniquedependsupontheac-ceptanceofcertainassumptionswhichrenderthemathematicsmoretractable.First,themethodassumesthatalllongitudinalaresimilar;thisisa gondapproximationformostlargeships,particularlywithinthemid-bodysection.Secondly,alltransversemembers(webframes,oil-tightandswashbulkheads)aretreatedasbeingofproportionalstiffness.!ihilethislattermodelingtechniquemaynotappearveryexact,a 100percenterrorina givenconstantofproportionalitywillproduceonlyaverysmallpercentageerror(perhaps0.5percentmaximum)intheforcereactionsatthetransverseboundaries.Infact,twosetsofcalculatflonsweremadeforthesamestructureunderthesameloadingconditionbutwith
—. —— .
-
13
/
++-’
-
14
Thepropertiesofthismodelarelistedasfollows:
Length 80‘Depth 60‘Width 60‘
LongitudinalmembersNo.1,11 Ix = I.y= 0.22ft.4A= 0.22ft.z
LongitudinalmembersNo.4,8 Ix= Iy= 1.1ft.4Ad.lft.2
LongitudinalmembersNo.2,3,5,6,7,Ix= Iy= 0.11ft.49,10 A= 0.11ft.z
WhereA =
Ix,(Thevaluesof
attachedplate.)
Thicknessandsideplate 0.02ft.
crosssectionarea
IymomentofinertiaaboutX,Y- axis.
cross-sectionalareaandmomentofinertiaincludethe
ofthedeck.bottom.
Thicknessofthewebandbulkhead 0.04ft.
Cross-sectionalareaoftheflangeofwebframes 0.4ft.2
Forthissimpleexample,itisnotnecessarytousethelongi-tudinalstrengthprogram.Thebendingmomentcanbecalculatedbythesimplebeammethod.Theshearforceactingu~onthetransversesarejustthesumoftheexternalloadsatthebottom,5,000kips.Thissumisdividedevenlytothetwosides.Bysimplebeamtheory,thisloadmaybeidealizedtoconcentratedloads,750kipsattheintersectionwithlongitudinalNo.5 and7,and1,000kipsatNo.6.
Bytheuncouplingandrecouplingprocedure,theboundaryforcesactinguponthetransversemembersarecalculatedasindicatedinTable3-2.Notethattheboundaryforces,i.e.,thereactionsfromthelongitudinal,arequitedifferentfromtheexternalloadsatthesamenodepoints.Themaximumdifferenceismorethan18percent.Usingtheseboundaryforces,thestressesinsidethetrans-versescanbecalculatedbytheseparatetwo-dimensionalanalysis.Theresultsofthestresseswithinthewebframesobtainedfromthetransversestrenathprogramarecomparedwiththosebythethree-dimensionanalysisbyEASE/CDCinFigure3-9.
-
15
Table3-1.DominantBoundaryForcesduetoTwoDifferentStiffnessesforOil-TightBulkheads.
Longitudinal Component StiffnessFactor StiffnessFactorp=5.6(actual) p=2.64
1-1011-22
2324252627282951525354
YYYYYYYYYxxxx
69610-50560-14.12*69610-505605047500046004200-47470-43130-36330-32550
69610-50560-9.09*69610-505605047500046004200-47470-43130-36330-32550
55 x -28960 -2896056 x -26620 -2662957 x -26620 -2662!758 x -24600 -2460059 x -24600 -2460060 x -24600 -24600
Forloadingcondition5 , “JOHNA.MCCONE”*Valuesareinsignificant
— .-—
-
I16
Y
//m, /h—-B-+
L.2A,, D. B. 60’
Fig.3-5B. ExternalLoadsontheTransversesP = 1000kipsLoadsatBottomofSwashBHD= 2000kips
t +Y
Fig.3-5A,SimpleBoxGirder
I /
H
ng10117.5P 6 .5P‘Wp
/ 1/“x/’7-
1/ !,‘YW777’–+X,’i j//
6’ {
///
~ /// /’ /
Fig.3-6.OneQuarterofBoxGirderWithOneSwashBulkheadandOneWebFrame
.
-
17
8 9 10 11
3
w
4837
33 31f4
36
35
1An’
o indjcatesbeamelementnumber4136 indicatesmembraneelementnumber36
/’”
Fig.3-7.PartIofBeamandTriangularElementsforSampleBoxGirderAnalysis
-
18
8
7
19
39
1 20 21 222~
26r23 ‘6,7 ‘% 2%14 ““-“/ T12 281317 .-. 29 “-30. A1516m26r‘ u-”%“0%12’Y38
37
36
3
@ indicatesbeamelementnumber4la indicatesmembraneelementnumber
Fig.3-8.PartII of BeamandTriangularforSampleBoxGirderAnalysis
12
Elements
-
19
Table3-2.BoundaryForcesonTransversesinkips
No.of WebFrame1 Swashi3HD WebFrame2
longls. Fx Fy Fx Fy Fx Fy
1
234.5.6.78.9.
10.11
0--
1.0
1.6
20.8
-997.5
-500.
0.4
-8.0
-.7
-.4
0.
-506.0
-1021.
-1022.
0.
763.6
1184.
761.7
-27.4
-1.0
.3
.7
0.-
-1.3
-2.1
-26.5
.1003.
-500.
-.7
!3.8
.9
.5
0.
-989.8
-1968.
-1967.
0.
1480.
1745.
1482
36.
-1.4
-.3
-1.0
Fig.3-9.NormalStressesonTransverseNo.1.
0.-
1.0
1.6
20.8
-997.5
-500.
.4
-7.9
-.7
-.4
0.
-506.
-1021.
-1022.
-0.
763.6
1184.
761.7
27.4-1.0
—— —- —
-
20
CORRELATIONOFTHEORETICALSTRESSESWITHSTRAINGAUGEEXPERIMENTS
Solutionsobtainedbythemethodspresentedinthispaperhavebeencomparedwithstraingaugereadingsfromthefullscaleexperimentsofa200,000tontanker.TheloadingconditionsareindicatedinFig.4-1.
Comparisonsbetweenexperimentalandanalyticalresultsareoftendifficulttomakebecausethetheoreticalapproachesarebaseduponidealizedconditionswhileactualexperimentsreflectthereal,imper-fectstructure.Thefullscaletestswereconductedtwoyearsago,andsomeofthedataneededforclosercomparisonsisnolongeravailable.Forexample,thereisnorecordofthewaterheadforthe100percentfulltankloadingconditions.Thetankcapacitiesshowninthedrawingsare,ingeneral,largerthanthoserecordedduringtheexperiments,Sincethedeckofthisparticulartankeratthelongitudinalbulkheadsisabout1.15metershigherthanattheedges,thereissomeupwardpressureactinguponthedeckwhenthewingtankis100percentfull.Themagnitudeofthispressurecanbedeterminedonlyiftheactualwaterheadisknown.Thispossibleupwardpressureonthedeckisnotconsideredintheanalysis,eventhoughtheeffectofthispressurecanbequitegreat.
PartofthecalculatedresultsareplottedinFiqure4-2through4-6.Duetothediscretizationerror,thestressofoneelementattheboundaryis,ingeneral,notthesameasthestressofanotherelementatthesameboundary.Forsomelocationsthisdiscontinuityissmall,asshowninelementsbetweenColumn18and19,Figure4-2.Forsomelocationsofgreatstressconcentrations,thisdiscontinuitymaybelarge;thenormalstressesintheelementsbetweenRow19and20reveallargediscontinuitiesbetweentheelementsattheboundariesalongColumn12andColumn13,andindicatesthatsmallerelementsaredesir-ableforthisarea.Sincethestressdistributionmustbecontinuous,thecommonpracticeistodeterminetheaveragevalueattheboundariestoproducea continuousdistribution,asillustratedinFigure4-2.Figure4-7showsthefiniteelementmeshused.
Withfewexceptions,thecorrelationsbetweenthecomputedandmeasuredstressesareverygood.Insomecases,thediscrepancybetweenthetwogaugesata givenlocationisgreaterthanthecomputedresult.Furthermore,thecomputedresultsaregenerallycloserthanthosecom-putedbyothermethods,
Thelargediscrepanciesintheupperpartofthewebframeandthedeckbeammaybeduetotheupwardpressureofthetanksduetoawaterheadabovethedecks. Thispressurehasnotbeentakenintoconsidera-tionbecauseoflackofdata.Also,a finermeshforlocationsnearthebracketsandcornersmaybenecessaryformoreaccurateresults.
Becauseofa limitationinthepresentinputsubroutine,theelementsgeneratedaroundthecornersarenotexactlythesameasexistingintherealstructure,particularlyatthewingtankcornernearthelongitudinalbulkhead.Thishastheeffectofincreasingthestressconcentrationattheselocationsasisindicatedbytheresults.
— — —.
-
21
FU1lLoadCondition: draft = 62’ - 4.75”, ‘trim = 0’ - O“
W,600 19,9
-
o .5 1 kgslmn2
— stress dlstrlbuticmby the present m!thod
4 stfess ffum strain gaugeo stress fromstrain gauge
on oth@rside of the plateOnevalue Is plotted if kthreadingsare closed
Fiq.4-3.NormalStressesonWebFrameNo.127forLoadCondition5
—.,. . .0 .5 1 kgSlm2
— stress distributionbythe presefitm!thnd
+ stressesfromstrain gaugereadingsononcsik of the plate
o stressesfrrm strain gaugeread$ngson the oth@rside of the plate
\ I. .
..
Fig.4-4.NormalStressesonWebFrameNo.127forLoadCondition5
-
Fig.4-5.NormalStressesonWebFrameNo.127forLoadCondition6
r.~ “--i kgdstressesdistributionby
the ormcntwthod
—
Fig.4-6.NormalStressesonWebFrameNo.127forLoadCondition6
-
24
FiniteElementGridDefinitionforTransverseFrame
cohmn number1 231, 5!s70 9 10 11 12 13 14 1$ 16 17 18 19 20 21 22 23 24 25 26 27
,!
w+/.-:.1,
~1<
4--’-+1--+-
11,’
~.,--!-1!,,
,A-.L -
~--
+=+
+-7---1
1:~
1:,(
*--+-
-.-
y--
+--
* .—T—
1:#-=++-7-;-q-,..+.
I
%
.-4-
-l-i--~-
! 1 I,— ,—:=—. .-{__:_+’—- _T --- ‘1,, 1, !,
~ ,’,’!1. .- .— , ,. , ..-. .
:,, 1
II 1 I I1,!1--- l“l--”!”’- ,, !
I—. I_.—— ‘—.F-..}-}I
‘c&mnn-%
r ... . —.–-
— . . .. ‘-. ..
. -.
.. . . .—7. .
,_ \..
.
— —. — —.
.-— — . ..
---- --. .—
;-.,.-.21I
. . ., ..2 0
19
. .. . ..-. — k6.“. 15
14
—
— ,–. +.**,- g—.—. -- ‘-- 3-1-
.- ..., ..”—— . . -...—-....J ~
... “ .,,..-- — .- .—.I
,1,-.l-.. *1-.. .—— .-!. -.— ---- -B
I 1,., !:. -1’-
....”.,. ._ .
F?g.4-7.ElementMeshfortheSampleCalculations
-
25
Thecomputertimeditionwas129centralincludesinput/outPut)
requiredtodotheanalysisforoneloadingcorl-processorseconds(1,044systemseconds,whichontheControlDataCorporation’s6600computer.
Thetransverse-analysismodelincluded29transverses(699quadrilateralandtriangularplateandbarelements)and!35Ionqitudinals.Thismodelwouldthenbeequivalenttoonecomprisingofabout23,000finiteelements.
Muchoftheeffortrequiredfordatapreparationisconductedauto-maticallyinsidethecomputerprogramwhichreceivesonlyaminimumamountofinputtodefinethegeometryofthestructureandtheIoadlngs.Thisfeaturenotonlyreducesthetimeneededfordatapreparation,butalsoeliminatesmanyofthepossibleinputerrors.Furthermore,eachoftherequiredinputdatacardsischeckedbytheprogramforpossibleerrors.Thecomputationsarestoppedautomat~callyupondetectionofanyerrorandappropriatediagnosticstatementsareprintedoutfortheengineer.
Themostdifficultpartoftheinputistheloadingdefinition,fortheexternalloadsmustbeaccuratelydistributedontothelongitudinal.Thisprogramdoesallowtheuser-toinputtheseforcesingreatdetail,andnoidealizationoftheloadingisnecessary.Inputpreparationfora transversestrengthanalysisrequiresabouttwotothreeman-weeks,dependinguponthecomplexityoftheloadingcondition.!.luchofthemanualeffortsfordefiningtheseToadinqscouldwellbegeneratedbya specialroutineadaptedtothepresentcomputerprogram.Sucha routinewouldthenrequireonlya verygeneraldescriptionoftheloadsinvolved;theroutinethenwoulddevelopthedetailneededfortheanalysis.
Sincetheanalysisconsumesrelativelylittlecomputerexpenseandproducesquiteaccuratestresssolutionsnevertheless,thisnewtechniquecouldbeincorporatedwithina truedes{gnprogram.Todate,a full-scalestressanalysishasbeenreservedforfinalstructuralchecking~urposesonly.
.
-
26
REFERENCES
1.
2.
3.
4.
5.
6.
i’.
8.
9.
10.
11.
12.
13.
14.
15.
16.
EgilAbrahamsen“RecentDevelopmentsinthePracticalPhilosophyofShipStructuralDesign.”Trans.SNAME,1967
EgilAbrahamsen“StructuralDesignAnalysisofLargeShips”Trans.SNAME1969
W.J.Roberts,“StrengthofLargeTankers”,No.55Lloyd’sRegisterofShipping,London,England,January1970
0.C.ZienkiewiczandY.K.Cheuna“TheFiniteElementMethodinStructuralandContinuumMechanic:”Lir.lited,London-NewYork,1967
IvanHolandandKolbelnBell“Finite
G.Vedeler,“GrillageBeamsGrondahl& Son,Oslo,1945
M.Hetenyi,“BeamsonElastPress,1946
RichardNielsen,Jr.,andF
inShips
McGraw-HillPublishingCo.
ElementMethods”Tapir,1969
andSimilarStructures”
c Foundation”TheUniversityofMichgan
nnC.Michelsen,“GriliageStructureAnalysisThroughApplicationoftheLaplaceTransformation”Trans.SNAME,1965
P.Y.Chang“ASimpleMethodforElasticAnalysisofGrillages”JournalofShipResearch,Vol.12,No.2,June1968
H.A.Kamel,U.Birchler,D.Liu,J.W.McKinley,&W. R.Reid‘AnAutomatedApproachtoShipStructureAnalysis”Trans.SNAME1969
A BulletinPublishedbyLloyd’sRegisterofShipping.No.21,1969
MasahiroMori,“OnTransverseStrengthofOilTankers”,HiroshimaTechnicalInstitute,Jan.1968
JohnVasta“LessonsLearnedfromFullScaleShipStructuralTests”Trans.SNAMEVol.66.pp.165-203,1958
EivoldM.Q.Roren“TransverseStrengthofTankers-FiniteElementApplications”
S.Kendrick,
TheinWah,“ADepartmentof
EuropeanShipbuilding,Part1,No.3,PartII, No.4, 1968“TheStructuralDesignofSupertankers”RINA,1970
GuidefortheAnalysisofShipStructures”,U.S.Commerce,1960
-
27
17. M.SteinandR.W.Fralich,“CriticalShearStressofInfinitelySimplySupportedPlatewithTransversestiffeners”,NACATN1851,1949
18. J.H.JohnsonandR.G.Noel,“CriticalBendingStressforFlatRectangularPlates”JournalofAero.Science,Vol.20,p.535,August,1953
19. P.O.Araldsen,G.Holtsmark,andE.M.O.Roren,“AnalysisofOilTankerbySESAM-69”.Presentedatseminaronthepracticalapplicationofthefiniteelementmethod,Trondheim,Norway,Jan. 1971.
20. J.R.Paulling,Jr.,“TheAnalysisofComplexShipStructuresbytheFiniteElementTechnique”JournalofShipResearch,Dec.1964,pp1-14
21. G.O.ThomasandJ.H.Ma“UserManualfortheFiniteElementProgramofStructuralAnalysis”NSRDCReport2712,April1958.
-
0DL
Wi(Q
28
APPENDIXA: LONGITUDINALSTRENGTHOFLARGESHIPS
AbbreviationsandNomenclature
Thewidthoftheship’shullThedepthoftheship’shullThelengthoftheshiu’shullThereactionbetweentheathtransverseandtheith~rimelongitudinalmemberThewidthofthewingtankHalfwidthofthecentraltankTheinfluencecoefficientoftheuthtransversewhilethetransverseissumosecitobesim~lysurmortedatbothendsDeflectionoftheathtransverseattheintersectionwiththelongitudinalbulkheads(i=2)orsideshells(i=l)Thedeflectionofthetransverseattheintersectionsofthelongitudinalbulkheadssub.iectedtotheqivenuniformloadqw,qcwhenthistransverseissim~lysu~~ortedatbothends.UniformloadontheUthtransverseinthewingtankandcentraltankrespectively.Totalloaduponthe uthtransverse.
—
-
29
GRILLAGEANALYSISFORLONGITUDINALSTRENGTH—.. ——
Considerthetransversemembers(transversebulkheadsandwebframes)asshortdeepbeamsacteduponbythesymmetricalloadingsystemasshowninthefollowingfigure:
FigureA-1
Letk~jbetheinfluencecoefficientsoftheuthtransverse,andd: bethedeflectionati duetoexternalloadsattheathtransverse.Then,
W2(za)-W,(ZU) = $ - $2 R; (A-1)
SolvingforR; ,L [$R; ‘aa +W,(za)-W2(ZJI,’22
a andd;can‘here’22 beobtainedbythebeamtheory.Sincetheload-ingissymmetrical,R;canbecalculatedfromthefollowing:
or
a b
‘7 = Jqw(x)dx + iqc(x)dx
o-?’
a.‘1- Qa - R; (A-2)
—
-
30
Treatingtheprimelongitudinalmembersasshearbeams,the
influencecoefficientsassociatedwiththeintersectionsofthe
transversememberscanbeobtainedfrombeamtheory.LetA andQ(3B~B betheinfluencecoefficientsforthesideshellandlongitud-
inalbulkheadsrespectively,Thus,
Combiningequations(A-l),(A-2),(A-3),and(A-4),
(A-3),..
(A-4)
(A-5)
Thereactionsbetweenthelongitudinalbulkheads,R;, can
thenbesolvedfromequation(A-5).WithR; known,R~canbeob-
tainedfromequation(A-2).Withbothofthesereactionsknown,
thebendingmomentsanddeflectionsofthelongitudinalandtrans-
versememberscanbecalculatedwithbeamtheory.
Sinceboththecross-sectionsoftheprimelongitudinaland
ofthetransversesmaynotbeuniformalongtheirrespectivelengths,
themethodoftransfermatricesisa moreconvenientmeansofcalcu-
latingtheinfluencecoefficients.
-
31
APPENDIXB:TRANSVERSESTRENGTHASA PLAINSTRESSPROBLEM
U,v
X,Y
v
E
s
n,nx Yfx,fy
L,B
ki
[~a%]
cijC;jx
v
F
Pi
Yxa,-fya
x,y
z
di diXQ’ya
AijAijXa’ya
~ij
AbbreviationsandNomenclature
Displacementinthex,y-direction
Concentratedforcesinthex,y-direction
Poisson’sratio
Young’smodulus
Coordinatealonga boundary
Componentsofa unitnormalona boundary
Componentsofboundaryforcesinthex,y-direction
Differentialoperators
Theitheigenvalue
DiagonalmatrixwitheigenvaluesAiasdiagonalelements
ElementsoftheunitarymatrixCElementsofthetransposematrixofC
Elasticityconstant
Displacementvector
Boundaryforcevector
Stiffnessfactorfortheithtransverse
Stiffnessfactorforthe ath longitudinalinthex,y-direction
Coordinatesinthetransverseplane
Coordinatealongthelengthoftheship
Deflectionoftheathlongitudinalattheintersectionwiththe i-thtransverseinthex,y-directionsduetoexternallyappliedloads
Influencecoefficientsfortheuthlongitudinalassociatedwiththeintersectionsofthei-thandthej-thtransversesinthex,y-directions
Influencecoefficientsoftheonelongitudinalthatisusedforthestandard.
-
THETRANSVERSESOFTANKERS:
Thetransversesoftankersmaybetreatedastwo-dimensionalelasticbodieswiththeboundaryS asshownbythesolidlinesinFigureB-1below.
s~—...——..,.. Ior–”(-JIp-—.——_a. WebFrame
rL ob. SwashBHD
——-—.._ -—.—.. ..—1I
c. Oil-TightBHDFigureB-1.TypicalTransverses
LetB beanoperatorrelatingthe boundarydeformationV ofa trans-versetotheboundaryforcesF,andL beanoperatorgoverningthedeform-ationwithintheboundaryofthetransverses,thenthedeformationVmustsatisfythefollowingequations.
LV=OBV=F attheboundarySV=v attheboundaryS
OR:
[ 1[1[1uUu Uv = oLvuLvv V Oi ‘1[/[1’11’12 u . ‘x’21’22 v ‘y
(B-1)(B-2)(B-3)
(B-4)
at S (B-5)
(B-6)at S
-
33
ThetransversesareacteduponbythereactionforcesfromtheIongitudinals;theseboundaryforcesareappliedalongthedeck,bottom,shell,andlongitudinalbulkheadseams.Usuallywithintheparallelmid-bodyofa tanker,thereisaone-to-onecorrespondenceoflongitudinalintersectingthetransversesalongtheseboundarylinesforalltransverseswithinthemid-bodysection.Hence,theboundarylinesofalltransversesarethesame.Theonlydifferencebetweendifferenttransversesistheirstiffness.
Assumethatthestiffnessesofthesetransversesdifferbya scalar. .factor.If B istheoperatorforonewebframe,L1B1canbeexpressedas
For
and
The
Bi= PiB (B-7)
Li= PiL
wherePiisa scal”arfactor
theithtransverse,equations(B-4)and(B-5)reduceto
(L Luv’)Pi Ui” ‘O”Uu .:L Lvv Pi Vi O, vu ,~‘ /’/
. .\[B “ i’)i 11 ’12 ‘i ‘1 elf+
‘’21 ’22,lpi ‘1, ,f~r~
LB-8)
onS
equationsaboveimplythatfortheboundaryforceF,
1U’=—u,P. Vi=;v , (B-9)1 i
whereU,Varetheboundarydisplacementsofa givenstandardtransversewherethestiffnessisknownprecisely.Equations(B-9)arenotexactlytrue,buttheerrorscausedbytheirusearenegligible.
-
34
LONGITUDINALASCONTINUOUSBEAMS:Letdi bethedeflectioninthex-directionofthe ~thXa 1ong-.thitudinalatitsintersectionwiththeI transverse.Theactual
deflection,U1 atz=z., ofthislongitudinalcanbeexpressed1bythefollow~ng:
(B-1O)
. . .thwhereA~~ istheinfluencecoefficientfortheI longitudinal,.thandX: isthesupportingforceoftheJ transverse.
SincealltheIongitudinalsareassumedtobesimilarinbending..stiffness,AIJcanbeexpressedinthefollowingway:
(E-11)
. .whereyxa isa scalarfactorandAIJistheinfluencecoefficient
fora,givenstandardlongitudinal.Combiningwithequation(B--11),
(B-1O)canbereducedto
n
TAijXj= Yxa(di - u:)aJ= ya
Similarlyfordeflectionsinthey-direction,
n
(B-12)
(B-13)
-
35
Forequilibriumandcompatibilityoftheintersections,the
followingisa necessarycondition:
Combiningequations(B-12)and(B-15),
n
z.. .
B1lAijP.uJ+ B12A1JPjvJJ = yxu[d~a- Ui] (B-1,6)j=l
Theaboverevealsa couplingrelationshipbetweentheboundary
displacementsofdifferenttransverses.Let
+andmultiplyequation(B-16)by Pi :
n
z,2-$i“ -mB1~P~AijPfij+ B p.AJp VJj j
.Yxa[p~di “Xa- til](B-18)
jzl
. .Since[P~AIJP$issymmetrical,thereexistsa unitarymatrixC
suchthat
CtC= I and CtPAPC = [-A..] (B19)
. .—
-
36
Let fii= Cij~j,andmultiplyequation(B-18)byC~j:
‘llai;’ + ‘12Aii’=‘XU[,c;jP*:a-:i] (B-20)(sumonj)
Similarly,
HenceatS = Sa,
[
’11
’21
’12
’221
(B-21)
(B-22)
where.lct Pidj
‘Xa Ai ij j xu (sumonj)
—if .&t ~}djya Aiij j ya (sumonj)
Similartransformationsreducethesetofequations(B-4)
forthetransversestothefollowing:
Forhonmgeneousboundaryconditions,theboundaryrestraints
reducetozero:
II
-
37
Fromequations(B-22),(B-23),and(B-24),thisisaplainstressproblemforanequivalenttwo-dimensionalelasticbodytowhichtheboundaryforces~~,,?~u, (u=1,.....mwherem isthenumber
oflongitudinal)areapplied.Thisbodyhastheboundaryconstraints
asdefinedbyequation(B-24)andissupportedbya setofconcentrated
springsatS = SUwhichhasspringconstantsequaltoyx,/Ai,Yyu/Ai.
Thisproblemcanbesolveddirectlybya two-dimensionalfiniteelement
approach.
Letthenumberoftransversesben andthenumberoflongitud-
inalbem. Leteachtransverseincludek-degreesoffreedom.This
isa problemof2nk-degreesoffreedomusingthisnewmethod.By
treatingtransversesassuper-elements,theproblemisreducedtoonly
n problems,eachofk-degreesoffreedom.
After~i iia’ a arecalculated,therealdisplacementsonthe
boundarycanbeobtainedbythereversetransformations:
(B-25)
WithUi and V1 theboundaryforcescanbecalculatedfroma a’equations(B-12)and(B-13’),Withtheseboundaryforcesknown,the
realdisplacementsandstressesofanytransversecanbecalculated
bya standardtwo-dimensionalfiniteelementmethod.Thefinite20elementprogramusedassubroutinewasdevelopedbyPauling,and
extendedbyThomasandMa.21
-
38
APPENDIXC - THESTIFFNESSOFTHELONGITUDIN,ALS
Thedeflectionoflongitudinal.
Theloaduponeachlongitudinalisdefinedasthatloadacting
ontheareasupportedbythelongitudinal.Formostpracticalpurposes,
theloadswithinoneframespacemaybeassumeduniformwithsufficient
accuracy.
LetqiabetheloadsandI. bethestiffnessoftheUthlongitudinallainithspacing,andlet W,G,M,Vi bethedeflection,slope,bending
.thmoment,andshearforceattheintersectionwiththe1 transverse.Thenfrombeamtheory.
w
o
M
v
1
2 31 ‘Zi+l- ‘1+1 ‘j+lm-m
z?o 1 ‘j+l *
o 0 1 Zi
!o 0 010 0 00”
4‘i+lzi+l~
qj+lz~+l6EI
2‘i+l
‘i+lT
‘i+]zi+~
1
ii-l,U
wherezi+listhespacing,qiistheuniformload.
u istheindexfortheu‘hlongitudinal.
Omittingtheindexawe have
s.1+1= Li+lSi
—.
w
o
M
v
1.—
(c-l)
,a
-
39
Therefore,
s L~+1Ln-----LISOn+l=
or s = LSOn+l
Sincethelongitudinalissimplysupported,
W.= Wn+l= M.= Mn+l= O
and
or
‘12
‘1
’14
’32 ’34L -11[1B ’15=-V ’35‘14L35-‘15L34
‘o= L12L34- ‘14L32
V.= ‘15L32- ‘12L35‘12L34- ‘14L32
(c-2)
(C-3)
(c-4)
(c-5)
LetLi= LiLi-l----L, then
Wi= L~2Qo+ L~4Vo+ Li15 (C-6)
UsingthenotationinChapterII
d = (L~2Goxia + L;4V0+ Lj5)a (c-7)
Theindicesu andx indicatethatalltheaboveequationsaredealing
withtheUthlongitudinalinthex-direction.
Theinfluencecoefficients.
Lettheqi+lbezeroandinsertthefollowingpointmatrixbetween
Li+,andLi.
. ..—
-
I
—
1
0
0
0
0
wehave
sn+l=LSO
where
40
0000
1 000
0100
001-1
0001
(C-8)
-J.
(c-9)
L=L Li+lLpLi----L1n+lLn‘--- (c-lo)
Fromequation(c-5)wehaveO.,Vo,andfromequation(C-6)wehave
(C-n)
wherethe indexi indicates
ofthe~h longitudinal.
Notethat
Aaxij= A;;
Thus,onlytheupper
thedeflectionati duetoa unitloadatj
halfofthematrix
alllongitudinalaresimilar,onlyoneora
becomputed.Ingeneral,dxia,dyiamustbenalunlesstheexternalloadsarethesame.
(C-12)
mustbecalculated,andsince
fewtypicalA~~andA~~need
calculatedforeachlongitudi-
1
-
41
APPENDIXD: INFLUENCECOEFFICIENTSANDDEFLECTIONOFTHEPRIMETRANSVERSEMEMBERS
.thLetIibethemomentofInertiaandAithewebareaoftheT
sectionofthesimplysupportedshearbeamasillustratedinFigureD-1.
Theinfluencecoefficient,Bij,isdefinedasthedeflectionati dueto
a unitloadorloadsatj.
FigureD-1
Deflectionduetouniformloads.
LettheloadbeqlinO-1,q2in1-2.Bylinesolution
‘\w
e
M
v
1\/
/
1 -a
o 1
0 0
0 0
0 0
$
\
2 ~4 a2%$ (#. +&,) ‘1(24E11—-q)
a2’~ ~
-qla3~
.
1 a -qlaz2
0 1. -qla
o 0 1
0
‘\w
e
M
v
1
,/
(D-1)
o
-
42
Or S1=J so
BychangingtheindicesS2= ~2S1
Combining(D-1)and(D-2)wehaveS2= ~zJ so= ~so
Theboundaryconditionsareo =M”=G2=V2=0
Fromequation(D-3)
[ 1[1 [ 1
’22’24 ‘o 25=’42‘4 ‘o ’45
101“ = Li2‘o+ ’14v + ’15
W2= L~29°+ L;4VO+ L;5
InfluenceCoefficients
(D-1)
(D-2)
(D-3)
(D-4)
(D-5)
(D-6)
Inderivinginfluencecoefficients,thetransfermatricesL1andL2
arethesameasgivenaboveexceptthattheelementsassociatedwiththe
loadsvanish.In additiona pointmatrixisaddedatthelocationoftheunitload.Thepointmatrixis
/ \
[
1 0000
01 ,0 0 0-
00100
0 001-1
00001
,/
-
43
APPENDIXE,-THEEFFECTOFLONGITUDINALNOTATTHENODALPOINTS
In anyfiniteelementanalysis,theterminalsofanyelementmustbelocatedatthenodalpoints.Forthisreason,themeshforthetransverses
shouldcontainallintersectionswiththelongitudinalasnodalpoints.
Thisrequirement,however,putsa greatrestrictiononthediscretization
ofthetransversesand thereforemaybeundesirableforotherpurposes.
Thisappendixinvestigatestheeffectofthelongitudinallocatedonone
edgeoftheelements.
Triangularelements
/
FigureE-l
Fora constantstresstriangularelementthedisplacementislinear;
therefore,au2+bu3
‘P= a-l-b(E-1)
av2+bv3‘P‘ a-i-b
Anyforceactingat(XPYP)canbereplacedbytwoequivalentforcesat
2,and3.
,.- —!. -.. ———--——” ~,--
-
Similarly
44
‘2‘*+*3 =+i A
=~Y aY‘2 a+b p’y3=~p
(E-2)
(E-3)
Replacethislongitudinalby
perequation(1)ofAppendix
‘i2=
I
‘i3=
twoimaginaryIongitudinalsat2 and3 as
B:
d 2Xxi2 - aij‘j2
(E-4)
d 3xxi3 - aij‘j3
Theforceanddisplacementcomponentsofthesetwoimaginarylongi-
tudinalshouldbecompatibleandequivalenttothoseofthereallongi-
tudinalonlyif
Thisequationissatisfiedif
Idxi2= dxi3=dxip=~11 =LS12 a+b‘ 3 a+b(E-5)
(E-6)
Equation(E-6)impliesthatonelongitudinallocatedinoneedgeof
thetriangularelementcanbereplacedbytwoimaginarylongitudinalat
thenodalpointsofthisedgeifthestiffnessandtheloadforthesetwo
imaginaryIongitudinalsareproportionaltothedistanceratiosofthe
twonodalpointsandthelocationoftheactuallongitudinal.I ~Sthemomentofinertiaofthislongitudinallocatedatp,and12and13arethe
respectiveequivalentmomentsofinertiaatthenodalpoints2 and3.
-
45
Forothertypesofelementswithlinearstresses,thedisplacement
orientsarenon-linear.Thecompatibilityconditionmaynotbesatis-
Sandtheerrorinvolvedisequivalenttothatinducedbyreplacing
elementwithtwoormoretriangularelements.Butforallpractical
>ses,thiserrorisnegligible,andassuchthismethodofdetermin-
?quivalentlongitudinalisappliedtoallothertypesofelements.
-
44
L a=— x. .‘Jk-~... --- _-[ E-.2----.._.———.. .. ...
46
APPENDIXF: THESIMILARITYOFTRANSVERSES
ThetheoryintroducedinChapter3 assumesthatalltransverse
members(webframes,oil-tightandswashbulkheads)aresimilarin
stiffness:morespecifically,theinfluencecoefficientsofonetransverse
aredirectlyproportionaltothecorrespondingcoefficientsofany
othertransverse.
Withoutcausingtoomuchdifficulty,thetheoryassumesthatatthe
veryleastallwebframeswithinthemid-bodysectionareidentical.
Sincethesimilarityprincipalrequirestheuseofonetypeoftransverse
asthestandardagainstwhichthestiffnessesofothersmaybemeasured,
thewebframeisselectedasthisstandardsinceitisalsoperhapsone
ofthemostcriticalmemberswithintheshipstructure.
I
FigureF-1. TransverseMembers
1SVJ4M+BULWHEAO I
Therelativestiffnessfactor(theproportionalityconstant)
beobtainedbycomparingthedeflectionsofthebulkheadtothose
webframewhenbothmembersareacteduponbya unitloadapplied
may
ofthe
atthe
-
47
uppercornerasillustratedbelow.
I
FigureF-2
Mathematicallythestiffnessfactor,Pb,ofthebulkheadis
expressedas
(F-1)
whered.,’andd.,aretherespectivedeflectionsatu ofthewebframeu u
andthebulkhead.
Asconcludedfromtheexperimentsconducted
boththewebframesandbulkheadsmaybemodeled
verylittlebendingdeflection:
‘(1=Gif
whereAfandAbarethetotalshearareasofthe
by
as
Mori17and~oberts14,
shearbeamsexperiencing
(F-2)
frameandbulkheadre-
spectively,andx isthedistanceoftheunitloadfromthesupport.
Bysubstitution,thestiffnessfactormaybeexpressedas‘b
‘b‘~(F-3)
. ...—.. -—?----— .--—. -.....—- .—....-—.—,—., —.
-
48
Theaboveprovidesanapproximatesolutionforthestiffnessfactor,
whichperhapscouldbemoreaccuratelyresolvedusinga finiteelement
analysis.
Atthispointthequestionarisesastothevalidityofapplying
thissamestiffnessfactortootherpositionsofthebulkhead,forexample,
location6 ofFigureF-2.Naturallysomeerrorwilloccurandtheextent
ofthisinaccuracymustbeestablished.‘hlongitudinal,andLetA~j,betheinfluencecoefficientofthea
letaibetheinfluenceoftheithtransverseattheintersectionofthis
longitudinal(seeFigureF-3below).Thereaction,Ri,andtheactualde-
flection,Wi,oftheintersectionmaybeexpressedasi
Wi=L” Ria (F-4)
forthetransverse,and
Wi= d;-A~jRj (F-5)
forthelongitudinal,d;isthedeflectionofthelongitudinalunderex-
ternalloadsbuttreatedasa simplysupportedbeamwithnosupportby
thetransverses.
~FigureF-3
-
49
CombiningequationsF-4andF-5,thematrixequationyields
(A+ ~)R=D (F-6)
where~ isa diagonalmatrix.
Sincethetransverseisassumedtobemuchstifferthanthelongi-.
tudinal(A~j>>L;) theresolvedreactionsmaybeexpressedas
R+[I- n;l(-l)n-’(A-lI)n]A-lD (F-7)Let~bL~betheerrorintheinfluencecoefficientofthemth
transverse,a bulkhead. .thThemaximumerrorinRI(the1 transverse)may
befoundtobe‘t
~%1Ei=+t)A-lLm +1im a j (F-8)j=l ‘t= Numberofthetransverses
In termsofordersofmagnitude,theratioofRiandEiisapproximately+ bA-lLmima” Sincethisexpressioncomparesthestiffnessofthelongi-
tudinalwitha muchgreaterstiffnessofthebulkhead,(A~~L~)isesti-
matedtobelessthanhalfofonepercentfora largetanker.Thusa
onehundredpercenterrorinthestiffnessfactorforthebulkheadwould
produceanerrorlessthanonepercent.Thisconclusionhasbeenvalidatedbytheanalysisofthetanker,
“JOHNA.MCCONE”,thestiffnessfactoroftheoil-tightbulkheadswasdeli-
beratelyincreasedby100percent;thischangeproduceda maximumerror
withintheresultingboundaryforcesoflessthan0.5percent.
-
50
APPENDIXG -APPLICATIONOFTHISPROGRAMFORORECARRIERSANDCONTAINERSHIPS
Fortheanalysisoforecarriersorcontainershipswithsymmetrical
loads,nomodificationsarenecessaryforthetransversestrengthcalcula-
tions.
Thelongitudinalorprimarystrengthcalculations,however,canaccom-
modatesymmetricalloadingsonly.FortheStressesduetounsymmetricalloads
(thehorizontal-bendingandthetwistingofthehull),additionalinvesti-
gationsarenecessarytodeterminethesignificanceoftheseeffects.The
theoreticalsolutionsforthesestressesmaybeapproachedinthefollowing
manner:
BasicAssumption
Thedeformationoftheship’sstructureissufficientlysmallsuchthat
thestressesduetoverticalbending,horizontalbending,andtwistingcan
becalculatedseparately.
HorizontalBending
Thehorizontalbendingcanbecalculatedsimilarlyastheverticalbend-
ing.Theonlydifferenceistheloads.
Twistin~
Forthetwistingstresses,thehullistreatedasanopenthinwallbeam
withbracesasshowninFigureG-1.
FigureG-1.
-
51
Thecrosssectionbetweenthebracesmaybeassumedasconstant.Bytransfermatrix,thestatevariablesbetweentwostationswithoutloadscanbewrittenas
or
where
/ \
$$‘B
‘t1
/
si+l
I \L11’LIZ’L133L141o’21’’22’’23’’24’0’31’’32’’33’’34’0—— L41’’42>’43’L44YO+1 00001 /= LiSi
i
@- thetwistingangle
IJ- thederivativeofthetwistingangle
MB-thebimoment
‘t- thetwistingmoment
Lnj’i=l- 4,j=l- 4 aregiveninTableG-1
62=)
C@- t;etorsionalrigidityCm- thewarpingrigidity
(G-1)
i
—- .-——. -.-. -— —-———— ———.—— --.—-—-—-- ---——-, ,--—,-_.. —.
-
52
TableG-1.TheTransferMatrix(Lij)
F 9sin~(x-ai) (l-cosB(x-ai)) -13(x-ai)+sin13(X-ai)
1 - B CUB2 Cti??
cos13(x-ai) sinB(x-ai) +l-cosB(x-ai)o CUB C*B2
- BCwsin(x-ai)o
cosf3(x-ai) Sit113(X-ai)f3
o o’ 0 1*
Thetransfermatrixfora concentratedtwistingmoment,Mt,is
10000
01000
00100
0 0 0 1 -Mt00001
,Bythemethodofl,,,esolution,thegTobalmatrixisL.
(G-2)
where
sn+l=Ln,Ln-l..GLoSo(G-3)
sn+L=LSO
Theeffectofthebracescanbetakenasredundant.LetZibethetotal.thshearforceacrosstiemiddlesectionoftheT braceasindicatedin
FigureG-2,ThevaluesofZicanbecalculatedbyequationG-3,
-
53
Figure
wheredi- deformationatthecutout
G-2.
oftheith
6 = 1,ifi=j,otherwisezeroij
aij - deformationattheithcutoutdueto
6,- .thdeformationattheJ cutoutduetoJ
‘i j anddicanbecalculatedwiththeshiphullasa thinwallbeam
thetransfer
(G-3)
braceduetoexternalloads
.tha unitload.attheJ cutout.ththedeformationoftheJ brace.
matrixmethodbytreating
withoutbraces. Bicanbecalculatedbyshearbeamtheory;hence,equationG-’3maybesolveddirectly.WithZiknown,therealdeformationsandstressesofthehullcanbecalculated
bythetransfermatrixmethod.Thesestressesshouldbeaddedtothe
stressesduetoverticalandhorizontalbending.Thesignificanceof
theseadditionalstressesdependsuponseveralfactors:themagnit~deofthe
possibleskewloadsuponthehull;thedimensionsoftheopeningsrelative
tothedimensionsoftheship;andthedesignofthebraces.
Fromtheabovetheory,stressconcentrationswilloccuratthe
cornersoftheopenings.Thus,a finiteelementanalysisofthisportion
ofthestructuremayberequired.
-
.-—. — “--- —.— .. ——. — —.
54
APPENDIXH: ANALYSISOFPARTOFTHE
Thegeneraltheoryintroducedbythe
beappliedforanalysisofpartofthehu’
HULL
uncouplingtechniquecan
1. A partalhullmodelis
desirableforseveralreasons.Firstofall,wearenorms
interestedonlyinthemiddlebodyoftheship.Secondly,
sectionreducestheamountofinputdataandcomputertime
ly
thesmaller
Andthirdly,
theresultsmaybeaccurateenoughforthedesignpurposes.
Themagnitudeofanyerrordependsupontheloaddistribution,the
geometryoftheshipstructure,andtheportionofstructuretakeninto
consideration.Therelationshipsbetweentheseparameterscanbebriefly
describedasfollows:
TheLoadDistribution
Fora structureconsistingofa finitenumberofdiscreteelements,
theexternalloadsaresharedbyalltheelementssothattheequilibrium
andcompatibilityconditionswithinandbetweentheelementscanbe
satisfiedeverywhere.Thestifferelementsorthestiffersubstructureswill
sharemoreloadthantheweakerones.In general,thesharesofloadsortheterminalforcesfortheelementsaredifficulttodeterminewithouta
completeanalysisofthewholestructure.However,somespecialloaddis-
tributionsmayberesolvedwithenoughaccuracy.
Considera ship-likecompositeboxgirderwithequallyspacedand
identicaltransverses(FigureH-l).Iftheself-balancedexternalloadis
uniformalongthelength,thenalltransversessharethesameamountof
loadregardlessofthestiffnessoftheIongitudinals.In thiscase,theconventionaltwo-dimensionalanalysisforonetransversewillgeneratethe
sameresultsasthosefromthethree-dimensionalanalysisforthewhole
girder.
-
55
SupposetheloadisnotuniformbutinsomeperiodicpatternasindicatedinFigureH-2.Thesharingoftheloadwilltakeplaceonlyamongthememberswithinseveraltransverses.
i===
7DfIl
—
.—
mL..------.T -- —m ===4—— —.$’
FigureH-1. UniformShip-LikeGirderSubjectedtoUniformLoad
FigureH-2.PeriodicLoad
TheGeometryoftheStructure
If thetransversesofthecompositeboxgirderareof proportionalstiffnessandiftheyarearrangedina regularpatternwhichcorresponds
toa similarpatternoftheloaddistribution,ananalysismaybeconfined
tothisportionofthehullwithaccurateresultstobeexpected.
Althoughshipsareusuallydesignedwitha definitepatternof
transverses,theloaddistributionsrarelyfollowaccordingly.Hence,some
errormaylikelyevolvefroma partialanalysis.But’inthelightofa
full-sizeanalysis,whereerrorsmaybeintroducedinroundingoff(a
greaternumberofdegreesoffreedom}and/orindiscretizingbya coarser
mesh,thepartialanalysismaystillbepreferable.
—..——. ..— .....—. ..—— —.. - . -——. .—-.. -—. —-—. ..— .-..— ——....— -—
-
56
TheErrorInvolvedinthePartialAnalysis
AsindicatedinChapterII, therigidbodymovementofthetransversesisthedeflectionoftheshiphullandcanbeobtained
bytreatingtheshiphullasa grillagesubjectedtoa setofline
forcesalongthetransverses,LetA~jbetheinfluencecoefficientsof
theprimelongitudinaltreatedasa simplebeam;then
voi=A;jYj (H-1).thwhereYjisthedifferenceofshearforceatthelocationoftheJ
transverse.Since-theshiphullhasthesamelengthandthesame
fixitiesasthelongitudinal,A~jareapproximatelyequaltoaij/F,
whereF isa scalarfactoranda..lJisthebasicinfluencecoefficient.Consideringthelastterminequation(B-34)wehave
- $Y”a.. “[email protected] Oj=~ 1
where{~ij}={aij~l
Similarlya-Bu ~ rBXijoj‘Ei
(H-2)
(H-3)
Notethat theexpressionattheleftoftheequalsigndoesnotinvolvethelengthoftheportionoftheshipnorthefixitiesofthe
longitudinalwewanttoanalyze.Thus,ifwetreatthisportionoftheshipandallthelongitudinalassimplya“beamof
thesamefixities,thefirsttwotermsofequation
thesame,andthethirdtermisalnmtthesameas
analysis.Theonlysignificantdifferencebetween
thislengthwith
(B-34)and(B-33)are
theseintheglobal
theglobal
andthepartialanalysisisthelastterm.Considerequation
thisseriesrepresentsthecouplingeffectofthedeformation
analys
(B-31)
ofthe
s
-
57
transverses.In a globalanalysis,thecouplinginvolvesalltransverses.In a partialanalysis,thiscouplinginvolvesonlythosetransverseswithinthisportionoftheship.Forsomeloaddistributionsthe
couplingdoesnotinvolvealltransverses;therefore,a partialanalysis
isasgoodasa globalanalysis,
identicaltransversesanduniform
thefirsttwoinequation(B-33);
Forexample,thespecialcaseof
loadyieldsalltermsnegligibleexcept
thus,thefullanalysisreducesto
theconventionaltwo-dimensionalanalysis.
Ingeneral,thepartialanalysisaspresentedherehasneglectedthecouplingeffectsofthoseexcludedportionsofthestructure.The
significanceoftheseeffectsincreaseswiththestiffnessofthe
connectingelements(theIongitudinals).Theresultsfroma partial
analysisofthe“JOHNA.MCCONE”,whichincludesonlyholdsno.1,2,3,4,
haveprovidedgoodcorrelationswiththeexperimentalmeasurements.
-
.
t=
d=
b=
v=a’
s
D=c
‘s =
.——” —. ____ ——-.—.————
58
APPENDIXI : STABILITYCHECK
NomenclaturethicknessoftheplatespacingoftheverticalstiffnessmomentofinertiaofthestiffenersmodulusofelasticitydepthofthewebPoisson’sratioshearstressbendingstresscompressivestressfactorofsafety
‘cB.e---- ..-“.——.Ub
1Whiletheanalyticalmethodsdevelopedinthisreportare
basedupontheassumptionthatthestructureiseverywherestable,
structuralfailuresinlargetankersoftenrevealthe characteristicsof shearbuckling.
—
-
59
Numerouscontributionshavebeenpublishedonthesubjectof
stabilityofstiffenedplatesandwebsofdeepgirders.TheinWah23
haspresenteda verythoroughreviewofthevariousmethods,allof
whicharequitecomplicatedandbaseduponassumptionsthatarenot
strictlyvalidforrealstructures,especiallytankers.Fortunately
fordesignpurposes,though,thesetheoriesdoprovideameansfor
determiningtheupperboundsofthein-planeforces.
CriterionforBucklingShearLoads
ThefollowingequationswereinterpolatedfromSteinandFrallch.24
= ksEm2(t/b)2‘s,cr —
12(1-V2)
where ks= 5.3+5 (b/d)2g
and g = 21#v2)(b/d)3
t3d
(1-1)
> b/d 5 (1-2)
5 b/d 5 (1-3)
CriterionforBucklingBendingStress
JohnsonandNoel25haveprovidedthefollowingcriticalbending
stressfora simplysupportedplate.
‘b,cr= 23,9T2E(t/b)2 (1A)12(1~2)
CriterionforBucklingCompressiveStress
aCcr=41T2E(t/b)2#12(1-112) (1-5)
.—-.
-
60
CriteriaforPlateBucklingUnderShearBendingandCompressiveStresses
Thefollowingequationprovidestheupperboundlimitforstability.
Wherethisrelationshipexceedsunity,bucklingislikelytooccur.
(-2 +(”b-.-)2 + (g)*= l/Fs, (1-6)● 3 c,cr
WhereFSisa factorofsafety.
FortranIVComputerProgramforDeterminingBucklingStability
20 wIE FR5STABLE.STABLE,STABLE30 ,---- . - . ,.-. - -4050607080909295100110120130140150160170180190%00E!lo220230240250260270.280290300310?25327330340345
3503ZQ mm
mr.wl.l\3,Luu, C..b101 FORMAT(//” THICKNESSWIDTHSHEARCOMPRESSIVEBENDINGSTRESS
1 SPACINGANDMoMENTOFINERTIAOFSTIFFENERS‘//)100 FORMAT()102 FORMAT(/ ‘ THISAREGNEEDSREINFORCEMENT‘ /1103 FORMAT(/ “THISAREAisSTABLEUNDERTHESE STRESSES‘ /)10 READ(5.100)T.B.SJSC.SB.XI,D
IF.(T.LT.O.) GOTO60104 FORMATc7E10.4)
WRITE(6,101)WRITE(6.104)T,B,S,SC,SB,D,XIY=T/BA=l.-G*GAl=E*3.14159**2/12./A*Y*YIF(XI.EQ.O.) GOTO20X=Ej/DGA=2.*xI*A/T**3/DGO TO30
20 GA=O.30 SK=5.3+5.*X*X*GA
P=S/SK/AlQ=sc/4./olR=SB/23.9/AlC=P*P+Q*Q+R*RIF(C.GT.1)WRITE(6,103)GOTO 10
40 WRITE C6,102)P=SK*AlQ=4.*A1
GOTO40
R=23.9*AIWRITE(6,104)S,SC,SBWRITEC6,105)WRITE(6s104)P,Q,R
GOTO 10105 FORMATC/wTHECRITICALSHEAR COM ANDBENDINGSTRESSESAREc/)
60 STOP
—.—
-
UNCLASSIFIEDSeci, titV classification
DOCUMENTCONTROLDATA-R&D,st.curItYclassificationof title,bodyol., b.tract :#r,dindcxirlfi.mnotarionmu.! ~C~nt=ed WIICnrf)eOVC~allrWOr( is ~ffi~s:fied)
ORIGINATING ACTIVITY (Corporate.mzthor) 2a. RE!50RT 5ZCUR1TY CL A5S8Fl CA T10N
COM/CODECorporation UNCLASSIFIEDAlexandria,Virginia .?b.GROUP
REPOWT TITLE
StructuralAnalysisofLongitudinallyFramedShips
DESCRIPTIVE NOTES(TYPC ofreportandinclusivedafes)
AU THOR(5I (Fi
-
UNCLASSIFIEDSecurityClassification
14KEY WORDS
GPO 939.038
LINK A
ROLE WTLINK B
ROLE WT
UNCLASSIFIED(PAGE’2) SecurityClassification
LINK C
ROLE W’t
—
—
-
f
SHIP RESEARCH COMMITTEEMaritime Transportation Research Board
National Academy of Sciences-National Research Council
‘The Ship Research Committee has technical cognizance of the inter-agencyShip Structure Committee’s research program:
PROF. R. A. YAGLE, Chairman, tioj’. of Naval Amlzikec?t?.we, Liriuersit!{ of Mich2ym
OR. H. N. ABRAMSON, DirwLo?, I’)e pt. of Mzh. fkiencc?s, SO-uihz!!est Reseurch. fnstitutg?
DR. W. D. DOTY, lieseach [consultant, U. S. steel Corporation
PROF. J. E. GOLDBERG, ,scbo,! of Cioil Eng;neeringjPUFdUe hivmsitj
Advisory Group 11, “Ship Structural Design” prepared the pl”ojectprospectus and evaluated the proposals for this project.
-
SSC-213,
SSC-214,
SSC-215,
SSC-216,
SSC-217,
SSC-218,
SSC-219,
SSC-220,
SSC-?21 ,
SSC-222,
SSC-223,
SSC-224 ,