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    Analysis

1184

    ;JournalofZhejiangUniversitySCIENCEA

    ;ISSN1009-3095(Print);ISSN1862-1775(Online)

    ;wwwzjueducn~zus;wwwspringerlinkcom

    ;E-mail:jZUS@zjueduca

    ;Xuetal/JZhejiangUnivSciA20089(9):1184-1192

    ;Analysisofstructuralresponseunderblastloadsusing ;thecoupledSPH.FEMapproach

    ;JunxiangXU,XilaLIU

    ;(DepartmentofCivilEngineering,ShanghaiJiaoTongUniversity,Shanghai200240,Chin

    a)

    ;E-mail:xujunx@msn.corn

    ;ReceivedNov.25,2007;revisionacceptedFeb.25,2008 ;Abstract:Anumericalmodelusingthecoupledsmoothedpaniclehydrodynamicsfinite

    elementmethod(SPHFEM)approach

    ;ispresentedforanalysisofstructuresunderblastloads.Theanalysesontwonumericalcases

    ,oneforfreefieldexplosiveandthe

    ;otherforstructuralresponseunderblastloads,areperformedtomodelthewholeprocessesf

    romthepropagationofthepressure

    ;wavetotheresponseofstructures.Basedonthesimulation,itisconcludedthatthismodelca

    nbeusedforreasonablyaccurate

    ;explosiveanalysisofstructures.Theresultinginformationwouldbevaluableforprotectin

    gstructuresunderblastloads.

    ;Keywords:Smoothedparticlehydrodynamics(SPH),Finiteelementmethod(FEM),Rein

    forcedconcretestructure,Explosion

    ;doi:10.1631/jzus.A0720080Documentcode:ACLCnumber:TU312 ;INTRODUCTION

    ;Theresponseofstructuressubjectedtoblast

    ;loadsisacomplicatedprocess.Manyresearcheson ;thisfieldhavebeenreported(LuandJiang,2001;

    ;Luccionieta1.,2004;Dueta1.,2006).Amongthem, ;thenumericalsimulationhasbeenpaidextensive

    ;attention.

    ;Inthetraditionalfiniteelementmethod(FEM),

    ;thenumericalmeshmovesanddeformswiththe

    ;ph;yrsicalmateria1.Freesurfaceandmaterialinterfaces ;arelocatedatelementboundariesandhencecanbe

    ;maintainedthroughoutthecalculation.Therefore, ;FEMiswellsuitedforsolidmateria1.However.dueto

    ;largedeformationandchuckofsoilnearthecharges. ;thecomputationaldifficultyforFEMresultedfrom ;meshdistortionwilloccurinexplosiveanalysis. ;SmoothedparticlehydrodynamicsfSPH)isa

    ;meshless,particlemethodofLagrangiannature(Liuet ;a1..2003).Ithasanaturaladvantageovermeshed

     ;methodsforproblemsinwhichtopologicallydiscon

    ;tinuousdeformationssuchascollapse,fracture,and ;ProjectsupposedbyNationalBasicResearchProgram(973)of ;ChinafNo.2002CB412709,andtheNationalNaturalScienceFoun. ;dationofChina(No.50378054)

    ;fragmentationoccur.SPHnotonlyovercomesthe ;aforementionedgridtanglingproblemOfFEMunder ;extremedeformations.butalsocaneasilytrackthe ;materialboundaryandhistorydependentbehavior. ;However,accurateSPHsimulationsrequireuniformly ;sizedparticlesthroughouttheSPHregion.Henceif ;highaccuracyissoughtinnumericalsimulations.1arge ;runtimecanbeaproblemfCenturyDynamicInc., ;l998).Ontheotherhand.theimplementationof ;boundaryconditionsinparticlemethodsisnotso ;clearlydefinedasitiswithmeshedmethods. ;AcombinationOfFEMandSPHwillbeespe

    ;ciallyeffectiveformodelingtheprocessinvolvingthe ;formationofcratersbytheexplosion,thepropagation ;ofpressureandtheresponseofstructures.Thebasic ;ideaistocoupleSPHwithFEMelementsthroughthe ;definitionofinteractionbetweenthem.Inthepresent ;study.SPHisusedtomodelexplosionchargesandthe ;nearfieldwithlargedeformationresultedfromblast ;loads.whileFEMelementsareusedtomodelthefar ;mediumincludingthebuilding.

    ;InSection2.thecoupledSPHFEMapproachis

    ;employedtoexplosiveanalysisoffreefield.Simula. ;tionresultsarecomparedwithempiricalequationsto ;validatethematerialmodelandthecoupledmethod. ;

    ;Xueta1./JZhefiangUnivSciA20089(9).‟1184-1192

    ;Then,basedontheobtainedresults,theresponseofthe ;structuresubjectedtosubsurfaceblastisstudiedand ;finallysomeconclusionsaredrawn.Theexplicithy

    ;drocodesAutodynisusedforthecalculations. ;COUPLEDSPHFEMAPPROACH

    ;SPHbasis

;TheSPHtechniquewasfirstdevelopedinas

    ;trophysicsbyLucyfl977)andGingoldandMona

    ;ghan(1977).Inearlyl990s,itwasappliedtothe ;simulationofelasticityandfragmentationinsolids.In ;thismethod.thestateofasystemisrepresentedby ;arbitrarilydistributedparticles.whichcarryallthe ;computationalinformationincludingmassandve

    ;locityaswellaselasticmodulus.Usingtheconceptof ;kernelandparticleapproximation,thediscretized ;SPHequationsofmotionfornumericalsimulationof ;blastanalysisaregivenasfollows.

    ;Conservationequationofmass:

    ;(=N.(1)

    ;Conservationequationofmomentum:

    ;ca,fl,)

    ;Conservationequationofenergy

    ;(一暑(

    (3),Pisthedensity,misthemass,l,isthe ;InEqs.(1)

    ;spatialvelocity,Eistheenergy,isthestresstensor,

    ;xiisthepositionvectorforparticlei,.[andarethe ;spaceindices,AUistheoperatorincludingkernel ;functionandcanbewrittenas

    ;=

    ;xfxJa

    ;

    ;wherer/jisthedistancefromparticleitoparticlej,W ;isthekernelfunction.

    ;1185

    ;CouplingalgorithmofSPHwithFEM

    ;TomakethemostofadvantagesofbothSPHand ;FEM.thecouplingbetweenSPHandFEMisneeded. ;Ingeneral,thecouplingalgorithmisbasedona ;master-slavealgorithmtoaccountforthecontact ;interactionofFEandSPHparticles(deVuystelal

    ;2005).Thecontactalgorithmwilldetectwhenapar- ;ticlepenetratesthesegmentsofthesurfaceoftheFE. ;Thecontactthicknessindicatesthedistanceaway ;fromacontactfacewherephysicalcontactisestab

    ;lished,eneveranodeentersthisdetectionregion ;itisrepelledbyaforcethatisafunctionofthepre

    ;dictedpenetration.

    ;Inthepresentstudy.becausethenearfield(SPH ;region)andfarfield(FEregion)arethesamesoil ;medium,SPHparticlesareioinedwithFEsasasingle

;nodeinthecalculation.Theywillremainexactlyto

    ;getherthroughoutthecalculationunlesstheioincon

    ;ditionisremoved(Fig.1,.Aninterfaceconstraint ;betweenFEandSPHregionsisreducedtostandard ;FEshapefunctionsandposestheKroneckerdelta ;prOpertyasfollows:

    ;=0?..),

    ;where.isthemodifiedcoupledFEandSPHfunc

    ;tionforthenodef,ffaceiStheinterfacebetweenthe ;FEandSPHregions.

    ;ItisimportantthattheorderoftheFEinterpola

    ;tionischosentobethesameasthatofthekernel ;function.Inaddition,nearfield,theextremedefor

    ;mationregion,canbeestimatedthroughempirical ;functionsorthetrialcomputationinadvance. ;,‟<<+

    ;IlIlll

    ;Fig.1CouplingofSPHtoFEM

    ;SolutionoftwoSPHmaterials

    ;Thesoilsurroundingtheexplosivechargewhere ;severedeformationoccursandthechargearemod- ;eledusingSPHregion.Itisunnecessarytodefinea ;contactbetweenSPHpartswhentwoSPHmaterials ;,?,????

    ;,???J

    ;r????c??l

    ;

    ;1186Xuetal/JZhejiangUnivSciA20089f9J.1184?1192 ;existinthesamemode1.Thecontactbetweentwolatedby ;SPHmaterialswilloccurwhenparticlesofthefirst ;p

    ;ti

    ;a

    ;c

    ;rt

    ;les

    ;w

    ;ofoIh

    ;illbe

    ;e

    ;l

    ;rpan

    ;ocated

    ;s.

;inlhesphennuefpar_

    ;D:

    ;Agp]

    ;,ticlesofotherparts.一厶了,

    ;=

    ;D1(pp)n,

    ;C0NSTITUTIVEM0DEL

    ;Soilmodel

    ;Accordingtothebehaviorofsoil,theDrucker

    ;Pragemodel(Druckereta1.,1952)isemployedfor ;analysisofstructures.Itsyieldfunctionisdescribed ;aS

    ;F=aI1+?2=k,

    ;whereI1isthefirstinvariantofthestresstensor:J2is ;thesecondinvariantofthedeviatoricstresstensor: ;istheangleoffriction;kisthecohesion.Modelpa

    ;rametersofandkarecomputedbyusingthe ;MohrCoulombstrengthparametersofthefriction ;angleandcohesionc,respectively.Theycanbe ;expressedas

    ;,l=+D+,

    ;=

    ;[()+(o-1)+(o-2)]/6,

    ;243sin=——=================

    ;,

    ;~

    ;/2x/‟3x(9-sin2)

    ;6?cos09

    ;~

    ;/.2.....x..../...-3.....n......(....9......-........s....i...n.....2.‟‟..‟(„.p‟—)

    ;(5)

    ;(6)

    ;(7)

    ;(8)

    ;Materialmodelforconcrete

    ;Thedamageofconcretecanbecausedbyblast ;loads.Inaddition,thestrainratehasaninfluenceon ;thebehaviorofconcrete.TheRHTmodeldeveloped ;byRiede1.HiermaierandThomainErnstMachIn

    ;stitute(Riedeleta1.,l999)isadoptedforthepresent ;dynamicanalysis.Themodelcanbesplitintofour ;components,namelythefailuresurface,theelastic ;limitsurface.theresidualfailuresurfaceanddamage ;mode1.Thepartforstrengthmodelcanbefoundin

;(Riedeleta1.,1999;Wangeta1.,2005).

    ;Indamagemodel,additionalplasticstrainingof ;themateriallcadstodamageandstrengthreduction ;followingthehardeningphase.Damageisaccumu

    ;(10)

    ;wheree.plisthecurrentplasticstrain,DI,D2are ;damageconstants,P.isthepressurenormalizedby ;thecompressivestrength,Ps.allisthestrainrate ;function.andmistheminimumstraintoreach ;failure.

    ;Thepost--damagefailuresurfaceistheninter-- ;polatedby

    ;=

    ;(1D)|lu.+Dl,

    ;andthepostdamageshearmodulusisinterpolatedby ;6fm=(1D)q+DGl,(12)

    ;whereanddIla1arethecurrentfailuresurface ;andresidualfailuresurface,respectively.Ginitiaiand ;GresiduaIareinitialandresidualshearmoduli,respec

    ;tively.

    ;Forthevolumetriccompaction,theHerrmann ;r1969)‟sP—amodelisusedastheequationofstatefor ;concretemateria1.

    ;Highexplosives

    ;Inordertoaccuratelyattainthelocationandthe ;magnitudeofexplosiveload,andtoavoidtheshort

    ;comingofsimplificationwheretheexplosiveloadis ;generallyconsideredasaloadingfunction.blast ;analysisisneeded.WeappliedtheJonesWilkinsLee

    ;(JWL)equationofstatetohighexplosiveproposedby ;Leeeta1.r1968)assumingidealstationarydetonation. ;Theequationofstateis

    ;P二【,B,/,J

    ;whereVisthespecificvolume;Eisspecificenergy;A, ;B,Ri,R2and09areconstantsofvariouscommon ;explosives.

    ;

    ;NUMERICALINVESTIGAT10N

    ;Xueta1./JZhejiangUnivSciA20089(9):11841192

    ;Tovalidatetheparametersusedinmaterial ;modelsandthecoupledSPHFEMapproach.theblast

    ;analysisforfreefieldisfirstlyperformedandthe ;resultiscomparedwithempiricalequations.Onbasis ;ofthevalidation,thestructuralperformanceunder

    ;subsurfaceblastisreproduced.andsomehelpful ;conclusionsareaddressed.

    ;Freefieldblastanalysis

    ;Pillarlikeexplosivecharge,33.2kg,isembedded ;at3.0mdeepintheinfinitesoilmedium.Itissim

    ;plifiedtoanaxissymmetricalproblem.Theexplosive ;chargeandtheextremedeformationsoilmediumare ;modeledbySPHregion.whileFEMelementsare ;usedtomodelthefarsoilandthestructure(Wanget ;a1..2005).Transmissionboundariesareapplied. ;Mechanicalpropertiesusedforbothsoilandcharge ;areshownin1‟ablelandTable2.respectively.In

    ;Table1.PCJistheinitialCJpressure:V0Disthe

    ;CJdetonationvelocity.

    ;Table1Parametersforexplosiongeneration ;ParameterlueParameterlue

    ;A(GPa)3.720.3

    ;B(GPa)3.53/90(kg/m)1.63~10

    ;R4.15VOD(m/s)6950

    ;R,0.95PCJ(GPa)28

    ;Table2Mechanicalpropertiesofsoil

    ;ParameterValueParameterValue

    ;P(kN/m)18.0G(MPa)0.0036

    ;E(GPa)0.0l(.)15.8

    ;C(GPa)0.04

    ;Fig.2showsvariationsofthepeakpressureand ;thepeakvelocityatdifferentpointsofthesoilme

    ;dium.Itcanbeseenthatthepeakpressureandthe ;Deakvelocityfa11rapidlywiththeincreaseofdistance ;fromcharge.Thepeakpressureandthepeakvelocity ;bythepresentedSPHFEMapproachareconsistent

    ;withthosebycorrespondingempiricalequationsfLu ;e,a1..2005).

    ;Theevolutionoftheshapeofthecrateratdifferent ;time(t=-2.5.9.6and39.0ms)isshowninFig.3.Itcan ;beseenthatthecraterequablyexpandsoutwardatthe ;beginningofexplosion,andthattheinitialshapeof ;cratergraduallyturnstobecircularandthenis ;Distanceawayfromthecharge(m

    ;1187

    ;Distanceawayfromthecharge(m)

    ;(b)

    ;Fig.2Variationsofthe(a)peakpressureand(b) ;peakvelocityatdifferentlocations

    ;changedbyreflectionwaveaftertheimpulsewave ;reachesthefreesurface.Itagreeswellwiththegen

    ;eralrecognitionofexplosives.

    ;Responseofstructuressubjectedtosubsurface ;blast

    ;Theresponseofathreestorey,threebayrein

    ;forcedconcreteflame.withtheheightOfl1.8mand ;thewidthof14.0m.issimulatedundersubsurface ;blast.ThenumericalmodelandlocationOftarget ;pointsareshowninFig.4.Asitcanbeseeninthis ;figure,theexplosivechargeof400kgOfTNTis ;embeddedat1.25mdeep.Thehorizontaldistance ;fromthefrontofthebuildingtothechargeis2.5m. ;Theflamecolumnsdimensionsarethesamefrom ;groundtoupperstory,andallbeamshavethesame ;dimensionsandreinforcementlayout.Mechanical ;propertiesofthesoilandthechargearethesameas ;theformerexample.Sinceitisimpracticableina ;simulationforthecompletestructuretotakeinto ;accountallthereinforcementdetails.thetensile ;strengthofconcreteisincreasedtoresisttension ;stresses(Luccionieta1..2004).Propertiesofrein

    ;forcedconcretearelistedin1ble3.

    ;(_)I??0.IQu0?/Iu】五0_100

    ;

    ;1188Xuetal/JZhejiangUnivSciA20089(9):11841192

    ;Ca)

    ;Pressure(kPa)Table3Mechanicalpropertiesforreinforcedconcrete

    ;(b)

    ;(c)

    ;4726E+06

    ;4253E+06

    ;378lE+06

    ;3308E+06

    ;2836EH6

    ;2363E+06

    ;l890E+06

    ;l4l8E+06

    ;9452E+05

    ;4726E05

    ;0O00E+O0

    ;Pressure(kPa)

    ;992lE+06

    ;8929E+06

;793706

    ;6.945E+06

    ;5953E+06

    ;496lE+06

    ;3969E+06

    ;2976E+06

    ;1984E+06

    ;992lE+05

    ;r

    ;3302E0l

    ;Pressure(kPa)

    ;2973E+07

    ;2642E+O7

    ;23l2E+07

    ;I982E+O7

    ;1652E+07

    ;l32lE+07

    ;9909E+06

    ;6606E+06

    ;3303E+06

    ;9999E0l

    ;Fig.3Formationofcraterinfreefield.(a)t=2.5ms;

    ;(b)f=9.6ms;(c)f=39.0ms

    ;C

    ;Fig.4NumericalmodelusingcoupledSPH..FEM ;approachandlocationoftargetpoints ;model

    ;ParameterValueParameterValue ;(MPa)35Dl0.04

    ;(MPa)10D21.0

    ;1.60.01

    ;Jv0.61Pe(MPa)2.33×10

    ;“l0.036Ps(MPa)6.0x10

    ;“,0.032G…iial(Mpa)1.67x10

    ;Q2.00.61GIdul(Mpa)2.17x10

    ;1.6Epl1.93x10

    ;M0.61

    ;1.Propagationofpressurewaveinstructural ;foundation

    ;Rightafterthedetonation,apressurewaveis ;generatedinthesoilmedium.Theoverpressurehisto.

    ;desofselectedtargetsareshowninFig.5.Itcanbe

    ;seenthat,theoverpressuredecreasesgraduallywith

    ;400

;,

    ;300

    ;200

    ;

     ;100

    ;O

    ;

    ;\

    ;0

    ;0

    ;020406080100120

    ;Time(ms)

    ;Fig?5Pressure-timehistoryatdifferenttargetsinsoil ;medium?(a)Target7;(b)Target9;(c)Target10 ;)0.Ij??0.I

    ;

    ;Xueta1./JZhejiangUnivSciA20089I9):11841192

     ;theincreasingdistance.Hence,oneofthemosteffec

    ;tivemeanstoprotectstructuresistoincreasethedis- ;tancebetweenapotentialbombandthebuildingtobe ;protected

    ;TheaccelerationhistoriesatTarget8inFig.4 ;areshowninFig.6.Itisobviousthatthehorizontal ;accelerationislargerthantheverticalone.Forthe ;verticalacceleration.thenegativeaccelerationis ;smallerthanthepositiveone.Thisindicatesthatthe ;accelerationinthesoilissensitivetoafreesurface. ;Itshouldbenotedalsothatthedurationofshock ;waveinducedbyblastloadsisabout50msinthe ;presentcase.However,itisgenerallyknownthatthe ;durationofearthquakeisabout10s.Inthesense.the ;hazardfromtheearthquakeissevererthanthatfrom ;explosion.

    ;0

    ;30

    ;300

    ;200

    ;l00

    ;0

    ;

    ;l00

    ;-

    ;200

    ;0

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