22陈On the Impact force and Energy Transformation during Ship.docx
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22陈On the Impact force and Energy Transformation during Ship.docx
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22陈OntheImpactforceandEnergyTransformationduringShip
OntheImpactforceandEnergyTransformationduringShip-BridgeCollisions
LiliWang1,,LimingYang1,ChanggangTang1,ZhongweiZhang1,GuoyuChen2andZonglinLU3
1,MechanicsandMaterialScienceResearchCenter,NingboUniversity,Ningbo,315211China;
2,ShanghaiMarineSteelandStructureResearchInstitute,Shanghai,201204,China;
3,DepartmentofBridgeEngineering,TongjiUniversity,Shanghai,200092,China
ABSTRACT:
Takingship-bridgecollisionasanexample,howthewavepropagationandthedynamicbehaviorofmaterialsinfluencetheimpactforceandenergytransformationarestudiedfromtheimpactdynamicsview-point.Byusinganewflexible,energy-dissipatingcrashworthydevicedevelopedbytheauthors,boththeimpactforceandenergytransformedcanbemarkedlyreduced,whichismuchconductivetoprotectbothbridgeandship.
Keywords:
Impactforce;energytransformation;ship-bridgecollision;wavepropagation;highstrainrate
1.INTRODUCTION
Inthestudiesofdynamicresponseofstructuresandmaterialsunderimpactloading,twodynamiceffects,i.e.theinertiaeffectandthestrainrateeffect,shouldbetakenintoconsideration.Thestudyoftheformerresultsinanalysesofwavepropagationinvariousforms,eitherexplicitlyorimplicitly;andthestudyofthelatterhaspromotedtheresearchofallkindsofnonlinearrate-dependentconstitutiverelationsandfailurecriteria.
Correspondingly,thosetwodynamiceffectsshouldbetakenintoaccountintheanalysesofimpactforceanddynamicenergyabsorption/transformationofstructuresunderimpactloading,suchasthosetakeplaceduringship-bridgecollisions.
Infact,consideringatypicalstructurewithacharacterizedscaleofLsunderatransientlyappliedloadingwithacharacterizeddurationTL,ifthevelocityofwavepropagatingwithinthestructureischaracterizedbyCw,thenthedynamicresponsetimeofthestructurecanbecharacterizedbyTW(=Ls/Cw).NotethatthecharacterizedwavevelocityCwisstronglydependentonthedynamicphysical-mechanicalpropertiesofmaterials.Obviously,whenthedimensionlesstimeT*(=TL/TW)<1orintheorderof100(calledtheearlyresponseofstructures);thewavepropagationwilldominatethedynamicresponseofstructures.Insuchsituations,theimpactforceandenergyabsorptionofstructurescouldnotbecorrectlyanalyzedwithouttakingintoaccountthestresswavepropagationandthestrainratedependentbehaviorofmaterials.
Onthecontrary,ifthedimensionlesstimeT*(=TL/TW)>>1(calledthelateresponseofstructures),andthentheanalysisofthedetailedprocessofwavepropagationisnolongerneeded.
Inthepresentpaper,theattentionismainlyfocusedontheearlyresponseofstructures.Takingship-bridgecollisionasatypicalexample,howthewaveeffectandthedynamicbehaviorofmaterialsinfluencetheimpactforceandtheenergyabsorption/transformationwillbestudied.
Finally,atypicalengineeringexampleofship-bridgecollisionisanalyzedanddiscussed.
2.IMPACTFORCE
Accidentsduetoship-bridgecollisionhavetakenplaceoftensincebridgesappeared,andbecomemoreseriouswithincreasinghugeshipandlargebridge[1-4].Moreattentionshouldbepaidtothetechniquestoavoidtheincreasinglyseriousaccidentsresultedfromship-bridgecollisions.
Forbothship-designersandbridge-designers,thekeyshouldbeknownishowtheimpactforceintheprocessofship-bridgecollisioncanbedeterminedscientifically.
InChina,likeinothercountries,twotypicalformulasarecurrentlyrecommendedtoestimatetheimpactforceduringship-bridgecollision:
oneisbasedonthemomentumprinciple(orimpulseprinciple),whichisusuallyexpressedas[5]
;
(1)
andanotherisbasedonthekineticenergyprinciple,whichisusuallyexpressedas[6]
(2)
whereFistheimpactcompressiveforce(MN),Wandmtheweight(MN)andthemassofshiprespectively,vtheshipimpactvelocity(m/s),ttheimpactduration,αtheimpactanglebetweentheshipnavigationdirectionandthebridgepiersurface,C1andC2theelasticflexibility(m/MN)oftheshipandthebridgepierrespectively,γ(s/m1/2)thereducedcoefficientofkineticenergy,consideringthatthekineticenergyofshipisnottotallyabsorbedbybridge.
Wheneqn
(1)isadopted,themostdifficultishowtocorrectlydeterminetheimpactdurationtindifferentprocessofship-bridgecollision.Inpractice,designershavetousesomeempiricaldata,whicharelackoftheoreticalorexperimentalevidenceandconsequentlymayleadtolargeerrors.Thus,moredesignersprefertoadopteqn
(2).
Theoretically,eqn
(1)andeqn
(2)areinherentlyinterrelated.IftherelativedisplacementbetweentheshipandbridgepierduringcollisionisdenotedbyU,thentheaveragetineqn
(1)canbecalculatedfromUdividedbytheimpactvelocityv,namely,t=U/v,andconsequentlyeqn
(1)canbere-writtenas
.(1a)
Thephysicalmeaningofeqn(1a)isthattheworkdonebytheimpactforceisequaltothekineticenergyofship,anotherexpressionofkineticenergyprinciple.Furthermore,foranelasticsystem,thedisplacementUresultedfromanimpactforceFisproportionaltotheimpactforce,U=CF,whereCistheelasticflexibility(thereciprocaloftherigidityK)ofelasticsystem,thuseqn
(1)canbefurtherre-writtenas
.(1b)
Takingintoaccounttheinfluenceofimpactangleαinthecaseofobliquecollision,consideringonlyaβ(=γg1/2)partofkineticenergyofshipbeingabsorbedbythebridge,andtakingCasthesumoftheflexibilityC1ofshipandtheflexibilityC2ofbridge,C=C1+C2,eqn(1b)exactlycoincideswitheqn
(2).
However,ascanbeseenintheaboveanalysis,botheqn
(1)andeqn
(2),aswellasothersimilarequationsrecommendedinmostexistingcodesorguidespecifications[7,8],areallbasedonsimplified,quasi-static,elasticanalyses,takingshipandbridgerespectivelyasawholebody,withoutanydynamicconsiderationofwavepropagationandrate-dependentplasticdeformationforbothshipandbridge.
Althoughthespeedofashipwhenitpassesthroughabridgeisgenerallyrequestedtobedecreasedtoseveralmeterspersecond,itshouldbeemphasizedthatcollisionbetweenashipandabridgeisacomplicateddynamicprocess.Infact,theshipdimensionisnowasbigasintheorderof102m,theshipmassisaslargeasintheorderof106-108kg,andthusthekineticenergyofashipisaslargeasintheorderof102MJ,sotheship-bridgecollisionisadynamicprocesswithhugeenergyexchangetakingplaceinashortduration(intermsof102millisecondtofewsecond),particularlywhentheearlyresponseofshipandbridgeareconcerned.Thereforeitshouldbesolvedbyanimpactdynamicsapproach,notbyaquasi-staticapproachastraditionallyused.
Fromaviewbasedonwavepropagationtheory[9,10],thedynamicimpactforceatthecollisioninterfaceiscomplexlydeterminedbythecoupledprocessofwavepropagationwithinbothofshipandbridgeunderagiveninitial-boundaryconditions.However,thespecificprocessofwavepropagationwithinship/bridgeisfurtherdependentontheirstructuraldetailsandtheconstitutivebehaviorofthematerialused.
Infact,inthecaseofone-dimensional(1D)elastic-plasticstresswavepropagation,itiswell-knownthatthegoverningequationsareconstitutedbythefollowingequations[10],i.e.themotionequation(momentumconservationequation)
(3)
thecontinuityequation(massconservationequation)
(4)
andtheconstitutiveequationofmaterialathighstrainrate
(5)
whereρ0denotesthedensity,σthestress(definedaspositivefortensilecase),εthestrain(definedaspositivefortensilecase),andvtheparticlevelocity(definedaspositivewhentheparticlemovesinthexaxis).
Bymeansofthewell-knowncharacteristicsmethod[10],theabovepartialdifferentialequationsareequivalenttotwosetsofordinarydifferentialequations,eachsetconsistingofacharacteristicsequationandacorrespondingcompatibilityconditionalongthecharacteristics:
(6)
or
(7)
wherethepositivesignisforrightwardwaves,thenegativesignforleftwardwaves,Cwthewavevelocityalongthecharacteristics,
(8)
andρ0Cwiscalled“waveimpedance”.
Forrightwardsimplewaves,satisfyingtheboundaryconditionofconstantvimpact,thesolutioncanbeobtainedfromeqns(6)and(7)as[10]:
ifσ≥σy;(9a)
ifσ≤σy,(9b)
whereσyistheyieldstress,Cwetheelasticwavevelocity,Cwe=(E/ρ0)1/2whichisaconstant,andEtheYoung’smodulus.
NoticingF=-σA,whereAistheareaofcollisioninterface,eqn(9b)canbere-writtenas
inelasticcases.(9c)
Therefore,evenintheelasticcases,ascanbeseenbycomparingthedynamiceqn(9c)andthequasi-staticeqn(1b),althoughtheimpactforceFisidenticallyproportionaltotheimpactvelocityvinbothequations,anessentialdifferenceexistsbetweenthosetwoequations.Inthequasi-staticanalysis(eqn1b),FisproportionaltothesquarerootofthetotalmassmandstructuralrigidityK(=1/C)ofship,whileinthedynamicanalysis(eqn9c),Fisproportionaltothegeneralizedwaveimpedanceρ0CweA(orthesquarerootofthedensityρ0andelasticmodulusKofmaterial).Itmeanswhentheearlyresponseofship-bridgecollisionisconcerned,thematerial-dependentwaveimpedanceplaysadominantrole,ratherthanthetotalmassandrigidityofship.Itisnotdifficulttounderstand,onlywiththewavepropagatingforthandback
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