Effect of lateral impact loads on failure of pressurized pipelines supported by foundation.

193

Effectoflateralimpactloadsonfailureofpressurizedpipelinessupportedbyfoundation

CSNg1ÃandWQShen21

JPKennyLtd(formerlyUniversityofPortsmouth),Staines,UK2

DepartmentofMechanicalandDesignEngineering,UniversityofPortsmouth,Portsmouth,UK

Themanuscriptwasreceivedon1March2006andwasacceptedafterrevisionforpublicationon23June2006.DOI:10.1243/0954408JPME97

Abstract:Thedynamicinelasticresponseandfailurepredictionofpipelinesunderlateralmassimpactatthemid-spanpositionarepresentedinthisarticle.Atotalof52impacttestswerecarriedoutonseamlessmildsteelpipeswithdifferentinternalpressurelevelsandfoundationsupports.Criticalinitialimpactenergieswereobtainedforeachgroupofpipespecimenswithdifferentinternalpressurelevelsandfoundationsupports.

Numericalsimulationsoftheexperimentaltestswerecarriedoutusingthree-dimensionaldynamicnon-linear niteelementanalysis,wherebothgeometricalandmaterialnon-linear-itieswereconsidered.Goodagreementswereachievedbetweentheimpacttestsresultsandthenumericalpredictions.

Ithasbeenfoundthatthein uencesoftheinternalpressureandthefoundationonthecriti-calinitialimpactenergyaresigni cant.Thecriticalinitialimpactenergyismuchlesssensitivetointernalpressureforthepipeswithafoundationthanthosewithoutafoundation.Keywords:pipeline,foundation, niteelementanalysis,impact,pressure

1INTRODUCTION

Researchonpipelineshasbeencarriedoutforafewdecades.Muchworkhasrecentlybeendoneinanefforttorationalizethedesignofpipelines.Theimpactresponseofsteelpipelinessubjectedtoimpactloadinghasbeenstudiedinmanyarticles[1–8].

Leisetal.[1]showedtheirworksofquantifyingtheeffectsofdentfeatureson awinitiationandgrowthinhigh-pressurepipelines.Itwasreportedthatthefourfactorsthathaveastrongin uenceontheseve-rityofthemechanicaldamageofthepipelinesaresoilsupport,linepressurelevel,linetensionorcompression,andthetype,magnitude,andtheorientationoftheimpactload.

Thomasetal.[2]conductedaseriesofexperimen-taltestsonsimplysupportedtubesunderacentralload.Threedifferentmodesofdeformations,i.e.localdeformation,localandglobaldeformations,andstructuralcollapse,havebeenidenti ed.

Ã

Correspondingauthor:JPKennyLtd,ThamesPlaza,Chertsey

Lane,StainesTW183DT,UK.email:ncs1018@

Jonesetal.[3]successfullyconductedsevenseriesofimpacttestsoncoldworkedseamlessmildsteelpipes,whichhaddifferentspans,outsidediameters,andthicknesses.Thepipespecimensweresubjectedtoanimpactloadingatdifferentlocationswiththeimpactvelocitiesrangingupto14m/s.

Atheoreticalrigid-perfectlyanalysisforpredictingthequasi-staticresponseofpipelinesthatarefullyclampedacrossaspanandsubjectedtoalateralimpactloadinghasbeendevelopedinreference[4].Thistheoreticalpredictionachievedagoodagree-mentwiththeexperimentalresultsinreference[3].JonesandBirch[5]presentedsomeexperimentaldata,whichwererecordedfromtheir54impacttestsonpressurizedandunpressurizedmildsteelpipes.Allthesepipesweresubjectedtoimpactload-ingfromarigidwedge-shapedindenter,whichhada atimpactface.Moreover,thepipeswerefullyclampedacrossaspanandimpactedatthemid-andone-quarter-spanpositions.Theyrevealedtwomajorfailuremodesoccurringatthepipes,i.e.localfailureandglobalfailure.

Morerecently,anexperimentalstudyonthefail-ureoffullyclampedsteelpipeswasreportedin

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reference[6].Atotalof226impacttestswereconductedonthedifferentgeometriesandinternalpressurelevelpipespecimens,whichwerefullyclampedatbothendsandimpactedtransverselybywedge-shapedindentersatdifferentpositions.

Athree-dimensionaldynamicnon-linear niteelementanalysis(FEA)waspresentedinreference[7]inordertosimulatetheactualdynamicinelasticresponseofallthepipesthatweresubjectedtoimpactloading,asreportedinreference[6].Agoodagreementwasobtainedbetweenthenumericalpre-dictionsandthecorrespondingexperimentalresultsforboththecriticalinitialimpactenergyandthemaximumpermanenttransversedisplacementswhenfailureoccurred.

Thetheoreticalanalysisinreference[4]hasbeenfurtherdevelopedtopredictthequasi-staticresponseandtheonsetoffailureforboththepress-urizedandunpressurizedpipelines[8].Themaxi-mumstrainfailurecriterion,theeffectofmaterialstrainratesensitivity,andanempiricalformulaforestimatingtheplastichingelengthwereusedintheanalysis.Goodagreementswereachievedbetweenthetheoreticalpredictionsandthecorrespondingexperimentalresultsinreference[6]forboththecriticalinitialimpactenergyandthemaximumpermanenttransversedeformationofthepipe.

Somefoundationspossiblysupportthepipelinesinthepracticalworkingenvironment,e.g.oil-trans-portationpipelineslyingonseabed.Theeffectof

thefoundationonthedynamicresponsesandfailureofpipelineswhensubjectedtoanaccidentimpactmaybesigni cantandhasnotbeeninvestigatedinthepreviouspublishedarticles.Therefore,aseriesofcarefullyinstrumentedtestsandnumericalsimu-lationsonpressurizedandunpressurizedpipessup-portedbydifferentfoundationswereconductedintheDepartmentofMechanicalandDesignEngineer-ingattheUniversityofPortsmouthforpredictingthefailureofpipelinesundersevereimpactloading.

22.1

EXPERIMENTALDETAILS

Testfacilitiesandinstrumentation

Aschematicdiagramoftheconnectionofinstru-mentsandequipmentsforcarryingouttheexper-imentalworkisshowninFig.1.Apairoftimesensorswas xedatthedrophammerrig,wheretheinitialvelocityoftheindenteratthetimeofimpactingonthetopsurfaceofthepipespecimencouldbepredictedandrecordedusingadigitaltimerandaspeciallydesignedelectroniccircuit.ThearrangementfortheunpressurizedtestisshowninFig.2(a).Steelplatessurroundingthepressurizedpipespecimenswereusedinthepress-urizedtest,asshowninFig.2(b),inordertostoppossibledebrisbeingprojectedduringtheimpact

test.

Fig.1Aschematicdiagramshowingtheconnectionofthetestinstrumentsandequipments

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Fig.2(a)Thearrangementfortheunpressurizedtestswithafoundationsupportand(b)steel

platessurroundingthepipespecimenwithafoundationsupportinthepressurizedtest

2.22.2.1

MaterialstestingPipespecimens

Allthespecimensweremadeofcolddrawnseamlessmildsteelpipes,havinganoutsidediameterD¼57mm,awallthicknessH¼1.6mm,andaspanofpipeline2L¼570mm,respectively.Thespan2LwastentimestheoutsidepipediameterD,consideringthecharacteristicsofpracticalpipe-lines[3].

Staticuniaxialtensiletests[9,10]werecarriedoutforthecolddrawnseamlessmildsteelpipeÃ.Auniax-ialtruestress–truestrainrelation[11]forthepipespecimenswasobtainedwiththeassumptionofdoublelinearrelationship,wherestaticyieldstresssy¼772.6MPa,statictruerupturestresssrü1080.5MPa,andstatictruerupturestrain1Ãr¼67.6percentwiththeassumedYoung’smodulusE¼208GPa.

2.2.2Foundations

Thereweretwokindsofsoil,i.e.sandandkaolin,usedasthefoundationforthepipelines.Lightbrown neandmediumdrysandwasusedasafoun-dationtosupportthepipespecimensinthetest

Ã

Fourtensiletestswereconductedandtheaveragevaluesofthe

mechanicalpropertieswereused.

seriesA,ALP,AMP,andAHP,andasandfoundationdensityof1548kg/m3wasusedinthesetests.

Plateloadingtests[12,13]werecarriedoutonthesandfoundationinordertodeterminetheverticaldeformationandstrengthcharacteristicsofthesandinsitubyassessingtheforceandamountofpenetrationwithtimewhenarigidplatewasmadetopenetratethesand.Atypicaltestresultwasused,inwhichtheinitialtangentmodulusis7.14MN/m2andtheassumedPoisson’sratiois0.2[14].

Kaolin,alsoknownasChinaClay,isanaturalmaterialthatconsistsofvariableproportionsofvar-iousminerals.ThecommerciallyavailablekaolincalledPolwhiteTMB[15]wasusedasthefoundationsupportinthetestseriesB,BLP,BMP,andBHP.‘Light’manualcompactiontests[16,17]werecarriedout,wherethemoistkaolin,whichisamix-tureofkaolinpowderanddistilledwater,waspackedmorecloselytogether,therebyincreasingthedrydensityofthekaolin.Theobjectiveofthistestwasto ndtherelationshipbetweenthecom-pacteddrydensityandthemoisturecontentofthekaolin.Amaximumdrydensityof1.45Mg/m3attheoptimummoisturecontentof27.9percentwasnoted.

Triaxialcompressiontests[18,19]werecarriedoutonthespecimenstakenfromthecompactiontests.Thismethodincludedthedeterminationoftheundrainedstrengthofaspecimenofcohesive

soil

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whenitwassubjectedtoaconstantcon ningpressureandstrain-controlledaxialloading,wherenochangeintotalmoisturecontentwasallowed.Themaximumstrengthofthekaolinoccurredatthemoisturecontentof24.5percent.Atypicalresultforthesoilspecimenwithamoisturecontentof47.77percentwasusedto ndoutthepropertiesofthepresentkaolinfoundation.Thevaluefortheinitialtangentmoduluswascalculatedtobe4.4MN/m2.Poisson’sratioof0.5wasusedinthiscaseandwassubjecttothevolumeofthekaolinfoundationremainingconstantaftertheimpactloading[14].

2.3Methodology

Thisexperimentwasconcernedwiththebehaviourofaconsiderablylonguniformgaspipelineundertheactionofsingletransverseloadingactingatthetopsurface.Thepipespecimenswerefullyclampedacrossaspanof2Latbothends,aswellaswithandwithoutafoundationsupportalongtheunderneathofthepipe,asshowninFig.3aandb,respectively.Thepipesweredesignedtofreelymovealongthelongitudinaldirectionatthesupports.Thepipesweresubjectedtoalateralimpactloadingatthemid-spaninthedirectionperpendiculartothepipeaxisbyarigidwedge-shapedindenter,whichhadacylindricalimpactsurfaceofradius2.65mmatthetipandamassofG¼80kgwasusedforallthetests.Notethatthelengthoftheindenterinthedirectionperpendiculartothepipeaxiswaslongerthanthehalfcirclelengthofthepipe.

Inordertoexaminethein uenceoffoundationsoncriticalinitialimpactenergy,threedifferent

majorgroupsoffoundationsupportforthepipespecimenswereconducted.TheyweregroupsA,B,andCforthesandfoundation,kaolinfoundationandnon-foundation,supports,respectively.Assump-tionsweremadesuchthatthefoundationprovidedaperfectlinesupportforthepipe.Alinesupportdoesnotdiffersigni cantlyfromasupporthaving 308contactwiththepipe,asshowninFig.4[1].

Nitrogengaswasusedtoachievetheinternalpressureinthepipes,ratherthanusingrelativelyincompressiblewater,assuggestedinreference[5].Thiswasduetoadecreaseinasmallamountinpipevolumethatwouldcauseanincredibleincreaseinpressureactingontheinnersurfaceofthepipe,ifwaterwasused.Inordertoexaminethein uenceofpressuresonthecriticalinitialimpactenergy,threedifferentvaluesofinternalpressurewereused,i.e.63bars(su¼0.136sy),95bars(su¼0.207sy)and125bars(su¼0.272sy).Thesewereinadditiontotheimpacttestswithoutinternalpressureinsidethepipe.Internalpressuresbeforeandafterimpacttestswererecordedforthepurposeof ndingoutthedifferencesbetweenthem.

2.4Impacttestsresults

Atotalof52identicalpipespecimensfrom12groupsofdifferentinternalpressuresandfoundationsup-portswereconducted.Theinitialimpactvelocityrangedupto7.70m/sandcausedpermanentinelas-ticdeformationforlowervelocitiesandfailureoccur-ringneartheendsupportsforhighervelocities,whichwerestudiedinreferences[3,5,6].

ThedatarecordedonthepipespecimensfromgroupsAtoCthroughouttheimpacttestsarelistedinTables1to3forthesandfoundation,kaolinfoun-dation,andnon-foundation,respectively.Thevaluesofharethedropheightoftheindenter;poandp1theinternalpressuresbeforeandaftertheimpacttests;Votheinitialimpactvelocityoftheindenterjust

2

beforestrikingthepipespecimen;Ek¼GVo/2denotestheinitialimpactenergy;PmaxandPmeanarethemaximumandaverageimpactloadings;nthenumberofreboundsoftheindenterafter

striking

Fig.3

Afullyclampedpipesubjectedtoanimpactloadingatthemid-spanposition:(a)withafoundationsupportand(b)withoutafoundation

support

Fig.4

Foundationsupports(drawingsarenottoscale):(a)perfectlinesupportand(b) 308curvesupport

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Table1Experimentalresultsofthepipespecimenssupportedbyasandfoundation(C,largecrack;I,intact;J,smallcrackorjustonsetfailure)

Specimenhsp1Vo

EkPmaxPmeanDmaxDminWfoWf1WbCf

lA

Failurecode(mm)upoy

(bar)(bar)(m/s)(J)(kN)(kN)n(mm)(mm)(mm)(mm)(mm)(mm)(mm)modeA133110007.70236937.06––86.869.75138.35134.6591.10–39.06CA224110006.62174833.2122.92–81.4821.9676.2674.9141.2247.1313.30CA312000005.27110834.1728.88–73.9232.5738.3737.8813.9417.93–IA418110005.77133133.2123.80–78.0327.0555.9354.4125.9830.319.32JA516110005.44118239.6025.48–76.1130.1946.3945.8619.5824.010.94IA617110005.51121138.0026.59475.9530.2945.5545.3518.8422.772.16IA717610005.58124348.5424.74276.5929.1550.2549.4122.4025.309.30JALP116000.1366305.43117843.4326.96071.3136.1543.7041.3122.85–5.70CALP214000.13663635.06102240.8837.08365.7442.7127.5626.6713.2717.52–IALP315000.1366305.25109946.9531.58068.4539.1734.8033.0816.9722.44–JProc.ALP414500.13663645.14105541.5235.19266.5541.6829.9829.3714.6620.723.25IALP514750.13663655.19107640.2436.85266.4441.6129.2027.7813.8118.966.17IIMechEAMP114500.2079505.15106140.2427.81069.2638.9238.1538.2720.0728.729.67JAMP213500.20795944.9597845.3537.20264.6843.3726.2925.4912.6616.935.31IAMP314000.2079505.03101246.9534.15066.2942.3429.6329.3814.9717.586.47JVol.AMP413750.2079505.0099938.0026.60068.7039.2137.5636.7519.7729.1410.15JAMP513650.2079504.9899142.7929.21067.8940.0333.9333.5516.9622.657.69J220PartAHP114000.2721251255.07102541.2037.44463.9045.1627.3826.3815.5422.327.99IAHP215000.27212505.25110144.3932.99066.9841.8533.3732.9918.2224.466.86JE:AHP314500.27212505.11104341.8425.97068.4139.7640.1640.2222.9224.2911.61CJ.AHP414000.27212505.06102340.2429.57066.4341.8834.6034.4419.4820.838.13JProcessAHP5

1350

0.272

125

125

4.96

984

40.56

32.40

4

64.54

44.79

30.37

31.88

18.16

22.55

7.75

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Proc.IMechETable2Experimentalresultsofthepipespecimenssupportedbyakaolinfoundation

Specimenhwspop1Vo

EkPmaxPmeanDmaxDminWfoWf1WbCf

lA

FailureVol.code(mm)(%)uy

(bar)(bar)(m/s)(J)(kN)(kN)n(mm)(mm)(mm)(mm)(mm)(mm)(mm)mode220B1181049.65000––35.45–377.1528.6050.7750.4422.3723.887.16JB2175052.160005.59124932.8927.31176.1829.9645.7345.0818.6920.162.60IPartB3178049.380005.62126342.1625.93376.6629.0648.7048.3420.7624.798.34JE:J.BLP1145049.870.13663635.12104838.3231.54267.2440.4733.2331.7216.7017.225.09IProcessBLP2147048.890.13663635.16106339.6035.72366.5041.6229.7628.3014.3817.658.35IBLP3152050.050.13663635.26110438.6431.72068.0539.9834.8032.3017.7821.709.72IBLP4156048.780.13663635.33113340.8834.90267.5240.5632.4632.3116.0220.297.52IBLP5162048.650.1366305.41116938.9626.42071.7935.9144.2543.9723.1630.1512.16JMechanicalBMP1145048.580.20795955.12104939.2834.35465.6942.8830.5430.4416.4219.867.47IBMP2150048.430.20795955.21108441.2039.20665.0243.3127.6527.0213.9617.794.45IBMP3157048.620.2079505.34113740.8832.57067.6240.4934.9134.1418.4025.045.84JEngineeringBHP1150046.390.272125125––44.39–463.8145.3226.7626.6515.0818.865.06IBHP2155047.840.2721251245.28111140.2435.87464.8744.4130.9730.3918.3822.735.96IBHP3160047.970.27212505.421171–26.13070.5137.5944.8243.5925.4127.596.22CBHP4

1560

48.48

0.272

125

125

5.34

1140

42.16

35.06

64.97

44.07

32.52

31.57

19.59

25.30

8.12

I

Table3Experimentalresultsofthepipespecimenswithoutafoundationsupport

SpecimenhsppVEPPDDWWW1lcode(mm)uy

(bar)o(bar)1(m/s)o

(J)k(kN)max(kN)meann(mm)max(mm)min(mm)fo(mm)f1(mm)b(mm)A

(mm)B

FailuremodeC118500005.72130544.0328.69575.6930.6145.4845.9119.093.64–IC219000005.79133849.1327.81576.3829.5048.1248.2020.627.67–JC318750005.76132439.2428.11175.9729.8147.1046.5119.915.75–ICLP118000.1366305.66127844.3530.33070.5037.3842.1341.4222.518.444.87JCLP217500.13663635.59124445.9435.59568.0240.0134.9534.5617.966.192.13ICMP116000.20795955.37115044.6731.39567.0141.6236.6436.8221.267.302.50ICMP217000.20795955.52121746.5834.15667.1241.2835.6436.4619.927.412.29ICMP318000.207950––35.73–071.9536.0648.0448.7527.1012.802.37CJPME97CMP417500.2079505.46119045.9431.69369.0838.5737.5538.0319.126.602.14JCHP117000.27212505.52121345.3031.46068.3140.0838.5638.9621.645.88–CCHP216500.27212505.45118543.0728.73068.9039.1041.2541.3923.358.64–C#CHP316000.27212505.36114641.4827.83068.9139.3741.1841.1023.558.02–CIMechECHP415000.2721250––42.11–067.4140.8735.8835.2819.755.91–JCHP514000.2721251255.03100746.9040.85563.3145.2224.6524.6312.873.861.19ICHP6

1450

0.272

125

125

5.14

1052

46.26

39.50

6

63.86

45.03

26.63

26.68

14.66

4.32

0.79

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thepipespecimen;DmaxandDminthemaximumandminimumdiametersofthedeformedpipecross-section;WbthetotaldisplacementofpipebottomsurfaceattheimpactlocationCfthemaximumper-manentdeformationoffoundationattheimpactpoint;andlAthetotalrelativedisplacementofpointsAatbothsidesofthesupportsasshowninFig.3(a).ThevaluesofwinTable2arethemoisturecontentofsoilfoundation.InTable3,lBisthetotalrelativedisplacementofthepipeslidingoutoftheclampsatboththesupports.

ThevaluesofthetotalmaximumpermanenttransverseplasticdeformationWfofthepipespeci-menslocatedattheimpactpointwererecordedinTables1to3.Thesevaluesweremeasuredaccord-ingtothesupportpositionacrossthespan2LbeforeandafterunclampingfromtherigasWfoandWf1,respectively.However,therewasnotmuchdifferencebetweenthesetwovalues.Speci-menBLP3showedthemaximumdeviationof7.18percent(i.e.2.50mm),whereasspecimenCHP5showedtheminimumdeviationof0.08percent(i.e.0.02mm).

Threefailuremodeswerecarefullyobservedinthepresentimpacttests,asidenti edinreference[6].FailuremodeIcorrespondstothefailurewiththeshearslidingorindentationattheimpactpoint.Thephenomenonofbucklingatthebottomsurfaceofthepipenearthesupportsisassociatedwithfail-uremodeII.FailuremodeIIIrelatestothetensiletearingonthetopsurfaceofthepipenearthesupports.

ThesethreefailuremodesareshowninFig.5andrecordedinTables1to3foreachgroupofthepipespecimens,wheresymbolIdenotesthatthepiperemainsintact(modesIandII);Jindicatesasmallcrackorjusttheonsetoffailureoccurringatthepipenearthesupports(allthreefailuremodes);andCrelatestothefailureofpipewithalargecrack,i.e.crack.3mm.Thecriticalinitialimpactenergyisde nedastheaveragevalueofthesmallest

energythatcausedpipefractureandthelargestenergythatdidnot.Noneofthepipeswasobservedfracturedunderneaththeindenter,asthecylindricalimpactsurfacewasused,insteadofa atimpactsurfaceatthetipoftheindenter.

TheresultsfromtheimpacttestsshowedthatonlyfailuremodeIIIcausedalossofthepipesintegrity,andthisoccurrednearthesupportsasreportedinreference[6].WiththeexceptionofspecimensA1,A2,A4,andBHP3,itwasfoundinthepresentexper-imentalstudythatfailuremodeIIImostlyonlyoccurredononesideofthepipespecimens.Besidesthis,thisfailuremodejustoccurredontheinletsideofthenitrogengasonthepressurizedpipespeci-mens.FailuremodesIandIIoccurredinalltheimpactedpipespecimens.

Onacarefulinspectionofthedeformedpipespecimens,itwasfoundthatthemiddleaxiallineofthepipespecimenwascreatedbynearlytwostraightlinesthatconnectedattheimpactpoint,asreportedinreference[6].Thisphenomenonwasusedinthetheoreticalanalysesofreferences[4,8].

33.1

FINITEELEMENTFAILUREPREDICTIONFEAmodelling

Fig.5

Threemajorfailuremodesoccurringatthepipe:(a)shearslidingattheimpactpoint(modeI)and(b)bucklingonthebottomsurface(modeII)andtensiletearingonthetopsurface(modeIII)

LUSASV13.4isageneralFEApackage,whichcannotdirectlyestimatethefailureofthepipelinesunderimpactloading.Therefore,FEAcalculationswerecarriedoutforthemostpipespecimensusedintheimpacttestsinordertocomparewiththeexper-imentalresultsforassessingtheaccuracyofthenumericalanalysis.Arangeofpracticalimportantconditions,includinginternalpressure,foundationsupport,andpipedimensions,weresimulatedasintheexperimentsinsection2.Furthermore,boththegeometricalandthematerialnon-linearitieswereconsideredwiththeintentionofsimulatingtheactualdeformationpatternand ndingoutthestressandstraindistributionsintheareasofinterest,wherematerialruptureoccurredbecauseofthehighstrainconcentrationinthepotentialfailureareas.Asaresultofthesymmetricalnature,onlyone-quarterofthestructureswasexamined.ThepipeswithH¼1.6mmshouldbedividedintofourlayersalongthewallthickness[7].ThelengthoftheruptureareanearthesupportandthecontactareasbetweentheindenterandthepipeinthepipelongitudinaldirectionshouldnotbelessthanthepipemeanradiusR[7].

Non-linearelasto-plasticisotropicmaterialprop-ertieswithstrainhardeningwereusedforallthemildsteelpipesinthepresentsimulations.Thesewereconstructedaccordingtothevon

Mises

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criterionforthecurrentmodel,wherethenon-linearhardeningwasspeci edinsteadoftheyieldstress.Linearelasticisotropicmaterialpropertieswereassignedtothefoundationsinthepresentmodelling.Datawereobtaineddirectlyfromtheplateloadingtestandthetriaxialcompressiontestforthesandandkaolinfoundations,respectively.

Thefoundation,indenter,andpipewerecon-structedusingthethree-dimensionalSolidConti-nuumExplicitDynamicsElements(HX8E)[20].Asuitabletimestepautomaticallycalculatedbytheprogramwasusedtoeliminatetheunconvergenterror,asthistimeintegrationschemewascon-ditionallystable.

Thefullyclampedpipeswithandwithoutafoun-dationsupportweresubjectedtoanimpactloadingatthemid-span,whereitwasfreetomovealongthelongitudinalaxis.Inordertostartadynamicanalysis,theknowledgeoftheinitialconditionsoftheproblemwascompulsory,i.e.theinitialvelocityoftheindenterinthepresentstudy.Theinitialvel-ocityshouldonlybeappliedtothe rstloadcase.Asthecompressednitrogengaswasusedtopres-surizethepipespecimensintheimpacttests,thepressureinsidethepipeswasalmostunchangedaftertheimpact,forthepipelinesremainingintact.Inordertomodeltheimpactloadactingonthepressurizedpipe,aninternalconstantpressurewasassignedperpendiculartothelocalfaceoftheelementsattheinnersurfaceofthepipebyusingtheelementfaceload[20].Thefaceloadsforexplicitdynamicselementsbeingconstantshouldbe

highlighted,i.e.theaverageoftheinputnodalpress-ures.Astheinitialvelocityshouldonlybeappliedtothe rstloadcase,boththeinitialvelocityandtheinternalpressureactedsimultaneouslyinthepipes,i.e.attime¼0s.

Slidelinewasusedtomodelthenon-linearcontactandseparationbetweenthestructures[20].Forsim-pli cation,three-dimensionalgeneralslidingwith-outfrictionslidelinewasusedinordertoignorethefrictionbetweenthestructuresforalltheFEAmodelsinthepresentnumericalstudy.

Geometricalnon-linearitywasappliedforalltheFEAthroughtheEulerianformulation[20].Themainadvantageofthisformulationwasthatthestressandstrainmeasuresnaturallydescribedthematerialresponse.

3.2FEAresults

Someofthesigni cantFEAresultsarerecordedinTables4to6forthepipessupportedbysandandkaolinfoundationsandwithoutafoundationsup-port,respectively.Theseresultscorrespondtothetotalresponsetimetfwhenthevelocityoftheinden-terreachedorwasnearto0m/s.

ThevaluesofWfandWginTables4to6arethetotalmaximumpermanenttransversedisplacementandtheglobaldeformationofthepipesattheimpactlocation,respectively.ThelocaldeformationofthepipeWlatthesameimpactlocationisthedifferencebetweentheWfandWg[4,8].

Table4FEAresultsforthepipessupportedbyasandfoundation

SpecimencodeA1A2A3A4A5A6A7ALP1ALP2ALP3ALP4ALP5AMP1AMP2AMP3AMP4AMP5AHP1AHP2AHP3AHP4AHP5

tf(ms)15.7215.2213.8814.5614.1314.1214.3212.5512.3812.3712.3712.3711.9011.7111.8111.8111.9111.6411.8211.9411.7511.76

Wf(mm)70.2758.2642.5948.3544.4445.2246.0239.9036.2138.0837.0037.4836.0934.1734.9334.6634.4734.7536.5335.1834.6833.77

Wg(mm)45.7834.7721.5926.2523.0323.6424.3022.1219.1220.5919.7820.1320.0318.4719.0818.8518.6919.6621.1620.0119.6518.86

Wl(mm)24.4923.4821.0022.1021.4221.5821.7217.7817.0917.4917.2317.3516.0615.7015.8415.8115.7715.1015.3715.1715.0214.91

Cf(mm)19.3215.9710.0612.2910.9811.5311.2616.1913.6014.9514.1814.7215.3713.9914.5114.7414.3715.7817.2116.3815.8315.36

1max(%)9.637.324.345.524.744.835.084.944.514.704.614.785.054.594.824.674.755.415.955.555.355.03

lA(mm)12.939.135.296.605.735.905.996.255.295.725.415.595.765.195.415.365.325.756.315.915.775.51

lB(mm)11.167.624.225.354.614.764.824.834.034.404.174.304.403.944.124.104.044.434.914.624.454.25

lp(mm)23.1723.0620.6221.5521.7121.7321.3411.9411.1011.4111.4111.5110.7810.5110.6910.6010.6310.3310.3510.3410.4410.19

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Table5FEAresultsforthepipessupportedbyakaolinfoundation

SpecimencodeB1B2B3BLP1BLP2BLP3BLP4BLP5BMP1BMP2BMP3BHP1BHP2BHP3BHP4

tf(ms)–13.9113.9112.0612.0612.2612.3512.3411.8011.8912.08–11.8212.0111.92

Wf(mm)–45.4645.7736.4536.8337.8038.5039.3035.4936.3337.61–36.4437.7937.00

Wg(mm)–24.2124.4719.3919.7020.4721.0421.7219.5720.2521.28–21.0622.1921.54

Wl(mm)–21.2521.3017.0617.1217.3317.4517.5715.9216.0816.33–15.3815.6015.47

Cf(mm)–13.2013.3214.0514.2714.8515.2915.8415.0115.5416.38–16.7817.7117.18

1max(%)–5.245.314.534.644.885.085.305.135.395.76–6.186.636.38

lA(mm)–6.246.315.435.535.795.976.195.635.866.23–6.276.696.45

lB(mm)–4.924.994.054.134.334.484.664.204.384.67–4.755.094.89

lp(mm)–20.0219.9510.9411.0011.0611.1911.0410.3810.4910.65–10.0610.2510.10

44.1

DISCUSSION

Experimentalresults

Figures6(a)to(c)showthepresentexperimentalresultsonthevariationofthedimensionlessmaxi-mumpermanentdeformationf¼Wf=Hwiththedimensionlessinitialimpactenergyl¼Ek=PcHforthepipespecimenssupportedbyasandfoundation,kaolinfoundation,andnon-foundationsupport,respectively.Figures6(a)and(c)showthatthemaxi-mumpermanenttransversedeformationfortheunpressurizedpipespecimensAandCwasapproxi-matelyproportionaltotheinitialimpactenergy,regardlessofthedifferentfailuremodes.Asimilarresultwasalsoreportedinreferences[3,5,6]thatthemaximumpermanenttransversedeformation

fortheunpressurizedpipespecimenssubjectedtoanimpactloadingatdifferentlocationsvariedalmostlinearlywiththeinitialimpactenergyoftheindenter.

However,thislinearrelationshipwasnolongertrueforthepressurizedpipespecimenssubjectedtoanimpactloadingatthemid-spanlocationastheplotsbegantoscatter[5,6].Thescatteroftheplotsbecomesmoresigni cantforthecurrentpressurizedpipespecimenswithandwithoutafoundationsup-port,asshowninFig.6.ThescatterresultsdisplayedinFigs6(c)withoutafoundationsupportaresimilartothoseinFigs6(a)and(b)withafoundationsupport,whichindicatesthatthefoundationsupportisposs-iblynotthecauseforthescatterresults.Thisrevealsthatthedynamicresponseofpressurizedpipelinesbecomingunstableisanomalous.

Table6FEAresultsforthepipeswithoutafoundationsupport

SpecimencodeC1C2C3CLP1CLP2CMP1CMP2CMP3CMP4CHP1CHP2CHP3CHP4CHP5CHP6

tf(ms)16.1616.4316.3415.1714.9814.3514.81–14.6314.9414.6614.28–13.5313.72

Wf(mm)51.3952.4451.9947.0446.0242.3044.44

–43.5844.2643.2441.93

–37.5538.97

Wg(mm)30.7631.6831.2829.5228.5926.3528.23

–27.4828.9328.0126.85

–22.9624.21

Wl(mm)20.6420.7620.7117.5217.4315.9516.20

–16.1015.3315.2315.08

–14.6014.76

1max(%)7.728.007.889.148.728.469.32–8.9710.239.799.24–7.438.01

lA(mm)8.839.179.029.088.728.038.76

–8.479.168.798.34

–6.877.33

lB(mm)6.937.227.106.916.616.066.69

–6.437.086.766.36

–5.135.51

lp(mm)18.7818.8718.8111.3411.3910.7410.85

–10.8510.1710.1610.18

–10.0610.06

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Fig.6

Wf=Hversusl:(a)specimenssupportedbyasandfoundation;(b)specimenssupportedbyakaolinfoundation;and(c)specimenswithoutafoundation

support

AHP.Thereasonforthisinstabilityandanomalousphenomenonwaspossiblyduetotheexistenceoftheinternalpressure.Thisphenomenonhasnotbeenreportedfortheunpressurizedpipelines.Thecauseisstillunclear.Furtherexplorationintothisproblemisveryimportant,becausemostoftheprac-ticaloilandgaspipelinesintheindustrieshavehighinternalpressuresandareverylongfortransportingoilorgas,especiallywhenlyingontheseabed.

Theamountofenergyabsorbedbythefoundationwasestimatedanditwasfoundthatitwasverysmallwhencomparedwiththeinitialimpactenergy.Forexample,theenergiesabsorbedbythesandfoun-dationwere4.53,4.88,3.91,and7.08JinthetestsA3,ALP4,AMP2,andAHP5,respectively.

Figure7showsthecomparisonsofthevariationoffversuslforthepresentpipespecimensA,B,andCwithspecimensEinreference[3],specimensC12andC13inreference[6],andthosespecimenswith-outinternalpressureinreference[5].Thecommonfeaturesofallthespecimensaretheimpactlocation,e.g.mid-spanimpacts,andasimilarratioof2L/D,e.g.10forspecimensA,B,C,andEandthoseunpres-surizedpipespecimensand9.5and10.6forspeci-mensC12andC13.Moreover,nofractureoccurredforthespecimensshowninFig.7.ThedifferenceofratioD/Hforallthespecimenswassmall,e.g.35.6forspecimensA,B,C,C12,andC13;30forspecimenE;and35.3fortheunpressurizedspecimens.Allplotsexceptthosethatcorrespondingtothecurrentspeci-mensA,B,andCgatheredaroundastraightline.TherestraintatsupportwasdifferentforthecurrentspecimensA,B,andCfromtheremainingspecimensinFig.7.AlthoughafullyclampedsupportwasemployedforallspecimensinFig.7,themovementintheaxialdirectionforthepresentspecimensA,B,andCwasallowed.Thiswasthemainreasonforthesigni cantincreaseoffforspecimensA,

B,

Onewouldexpectthatthehigh-pressurizedspeci-menAHP5wouldhavealowerde ectionthanthelow-pressurizedspecimenALP2becauseofthelessimpactenergyandhigherinternalpressurerecordedinTable1.However,specimenALP2hadaWf1¼26.67mmwithanimpactenergyEk¼1022J,whereasAHP5hadaWf1¼31.88mmandEk¼984J.Thisindividualresultwasnottheonlyonethatshowedanobviousanomalousbehaviour.Therewasatendencyexistingbetweenthetwogroups,forexample,fromTable1,whereanaveragevalueofWf1¼31.64mmwithanaverageimpactenergyEk¼1086Jforthelowerpressurizedspeci-menALPcomparedwithWf1¼33.18mmandEk¼1035Jforthehigherpressurizedspecimen

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Fig.7

Wf=Hversusl:(S,A,W),presentspecimensA,B,andC,respectively;(Â,4),specimensC12andC13[6];(Ã),unpressurizedpipespecimens[5];and(þ),specimenE[3]

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Effectoflateralimpactloads203

andCandhadnoregardtothesupportgivenbythefoundationsofpipespecimensAandB,whichmayhavereducedtheglobaldeformationofthepipeline.Thevariationsoflcversustheinternalpressure(usingsu=sy)forthecurrenttestresultsareillustratedinFigure8,togetherwiththetestresultsreportedinreference[5].Inadditiontotheexistenceofthefoun-dationsupporttothepresentspecimensAandB,anotherdifferencebetweenthecurrenttestswiththoseinreference[5]wasthelongitudinalrestrictionsatthesupports,asmentionedpreviously.Itwaspointedoutinreference[5]thatthefullylongitudinalrestrictionatthesupportwouldlowerthecriticalinitialimpactenergystrikeagainstthemid-spanoftheunpressurizedpipeline.Inordertoinvestigatetheirprediction,impacttestswerecarriedoutonthepresentspecimenthatwaswithoutanyfoundationsupportunderneathandhadthesamesupportcon-ditionsasthecurrentspecimensAandB.AsprovedinFig.8,alargedifferenceoflcappearedatthebegin-ningoftheplotsforthecurrentspecimenCandthespecimensreportedinreference[5]whensu=sy¼0.Furthermore,thevalueoflcforthepresentspecimenCwasmuchlowerthanpredictedandthespecimensreportedinreference[5].

Figure8showsthatthevariationcharacteristicsoflcwithsu=sywerequitedifferentforthepipeswithandwithoutafoundationsupportaswellasamem-braneforce,wherelcdecreasedwiththeincreaseintheinternalpressureforallthefourseriesoftestsexceptforgroupsAandBwithhigherinternalpressure.However,thecriticalinitialimpactenergywasmuchlesssensitivetotheinternalpressureforthepresentpipeswithoutmembraneforcethanthosewithmembraneforcereportedinreference[5].Itwasalsoshownthatthedifferenceofthelcbetweenthetwodifferentfoundationsupportsslightlyincreasedasthepressureincreased.

Thepipewouldbeweakerduetotheexistenceofthecircumferentialstresses,resultingfromtheinternalpressure,asshowninFig.8.Thecircumfer-entialstressesofthepipeneartheendsupportscouldcausethepipetofractureeasilyduringtheimpactloadingatthemid-spanposition.ChenandShen[6]discussedthatmostoftheimpactenergycouldbecontributedtotheglobaldeformationandcausespiperupturingnearthesupports,asthelocaldeformationofthepressurizedpipewassup-pressedbytheinternalpressure.Inotherwords,lesscriticalinitialimpactenergywouldberequiredtofracturethepressurizedpipeneartheendsup-portsthanthatfortheunpressurizedpipe.Theyalsopointedoutthatthiscouldbeanotherreasonwhythecriticalfailureimpactenergiesforthepress-urizedpipesliebelowthanthosefortheunpressur-izedpipes.

Experimentalresultsreportedinreference[5]revealedthatlcwasinverselyproportionaltotheinternalpressure,asshowninFig.8.Incontrast,itwasfoundinthecurrentresults,forthepipeswiththefoundationsupportandwithoutthemembraneforce,thatlcinitiallydecreasedslightlyasthepressureincreaseduptothepressurelevelofsu=sy¼0:21.Then,lcstartedtoincreaseslightlyasthepressureincreased.Inotherwords,thehigherinternalpressure(heresu=sy.0.21)wouldstrengthenthestructure.Although,lcwasinverselyproportionaltotheinternalpressureforthepresentspecimensC,thisrelationshipwasonlyuptothepressurelevelofsu=sy¼0:21,itthendecreasedmoresigni cantly.Asaresult,thecriticalpointforallthepresentpipes,withandwithoutafoundationsupport,wassu=sy¼0:21,atwhichthecharacter-isticsofthepipeswereunexpectedlychangedunderinternalhighpressure.Furtherresearchmightbedoneinorderto ndoutthereasonfortheseanomalousphenomena.

Thein uencesoftheinternalpressureloadingandthefoundationsupportonthecriticalinitialimpactenergytranspiredtobeverysigni cant.Hence,thepreviousresearchonthepipelineswithoutafoun-dationsupportmaynolongerbevalidfortheactualpipelineswithafoundationsupport,evenmoreseriously,thesecouldleadtoawrongesti-mationofthefailureofpipelines.

4.2FEAresults

Fig.8

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lcversussu=sy

Thedynamicnon-linearresponseofthepipessub-jectedtoalateralimpactloadingatthemid-spanpositionwasobtained.ResultsobtainedfromtheFEAwerecomparedwiththeimpacttestresultsforassessingtheaccuracyofthenumericalanalysisas

follows.

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Table7ComparisonbetweentheFEAresultsandtheimpacttestsresults

FEAresult

SpecimencodeA1A2A3A4A5A6A7ALP1ALP2ALP3ALP4ALP5AMP1AMP2AMP3AMP4AMP5AHP1AHP2AHP3AHP4AHP5B1B2B3BLP1BLP2BLP3BLP4BLP5BMP1BMP2BMP3BHP1BHP2BHP3BHP4C1C2C3CLP1CLP2CMP1CMP2CMP3CMP4CHP1CHP2CHP3CHP4CHP5CHP6

po(bar)0000000636363636395959595951251251251251250006363636363959595125125125125000636395959595125125125125125125

Ek(J)236917481108133111821211124311781022109910551076106197810129999911025110110431023984–1249126310481063110411331169104910841137–1111117111401305133813241278124411501217

–1190121311851146

–10071052

Wf(mm)70.2758.2642.5948.3544.4445.2246.0239.9036.2138.0837.0037.4836.0934.1734.9334.6634.4734.7536.5335.1834.6833.77–45.4645.7736.4536.8337.8038.5039.3035.4936.3337.61–36.4437.7937.0051.3952.4451.9947.0446.0242.3044.44

–43.5844.2643.2441.93

–37.5538.97

Cf(mm)19.3215.9710.0612.2910.9811.5311.2616.1913.6014.9514.1814.7215.3713.9914.5114.7414.3715.7817.2116.3815.8315.36–13.2013.3214.0514.2714.8515.2915.8415.0115.5416.38–16.7817.7117.18

–––––––––––––––

lA(mm)12.939.135.296.605.735.905.996.255.295.725.415.595.765.195.415.365.325.756.315.915.775.51–6.246.315.435.535.795.976.195.635.866.23–6.276.696.458.839.179.029.088.728.038.76–8.479.168.798.34–6.877.33

lB(mm)11.167.624.225.354.614.764.824.834.034.404.174.304.403.944.124.104.044.434.914.624.454.25–4.924.994.054.134.334.484.664.204.384.67–4.755.094.896.937.227.106.916.616.066.69–6.437.086.766.36–5.135.51

Wf(mm)134.6574.9137.8854.4145.8645.3549.4141.3126.6733.0829.3727.7838.2725.4929.3836.7533.5526.3832.9940.2234.4431.8850.4445.0848.3431.7228.3032.3032.3143.9730.4427.0234.1426.6530.3943.5931.5745.9148.2046.5141.4234.5636.8236.4648.7538.0338.9641.3941.1035.2824.6326.68

ExperimentalresultCf(mm)–47.1317.9330.3124.0122.7725.30–17.5222.4420.7218.9628.7216.9317.5829.1422.6522.3224.4624.2920.8322.5523.8820.1624.7917.2217.6521.7020.2930.1519.8617.7925.0418.8622.7327.5925.30

–––––––––––––––

lA(mm)39.0613.30

–9.320.942.169.305.70––3.256.179.675.316.4710.157.697.996.8611.618.137.757.162.608.345.098.359.727.5212.167.474.455.845.065.966.228.123.647.675.758.446.197.307.4112.806.605.888.648.025.913.864.32

lB(mm)––––––––––––––––––––––––––––––––––––––––4.872.132.502.292.372.14––––1.190.79

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Itwasassumedthattheimpactloadingandtheinternalpressurewouldacttogethertothepipesimultaneously.Twosolidsteelbungswereusedintheinvestigationinordertopreventthedistortionofthepipewithintheclamps.Thepressureinthepipesputforceontothebungsandtriedtoelongatethepipes.TheeffectofthepressureactingonthebungswasneglectedinthepresentFEA.

Someofthesigni cantnumericalresultsarerecordedinTable7andcomparedwiththeimpacttestsresults.Thecorrespondingvaluesfromtheimpacttestsarealsopresentedinthetableforcom-parison.Asmentionedinsection2,thepipeswerefreetomovealongthelongitudinalaxisduringtheimpacttests.Therewerestillsomerestrictionsonthepipeswithintheclampsduringtheimpacttests.Linearelasticisotropicmaterialpropertieswereassignedtothefoundationsinthenumericalmodel-ling.Therefore,thefoundationsreactedelasticallyandthedeformationofthefoundationwouldhaverecoveredduringtheimpactloading.BecauseofthisthevaluesofCfintheFEpredictionswerealwayslessthantheexperimentaltestsresults,asshowninTable7.

ShenandChen[7]proposedanempiricalformulafordeterminingthecriticalmeanplastichingelengthaslpc¼2Rl1.ThevaluesoflpobtainedfromtheFEAasrecordedinTables4to6werecomparedwiththevalueoflpc,whichisequalto47.31mm.Itwasfoundthatthevaluesoflpforthepresentspecimensweremuchlowerthantheempiricalvalueoflpc,i.e.,50percent.Thiswasduetotheeffectofthesup-portconditionasthepresentspecimenswerefreetomoveintheaxialdirection.Foracertainimpactenergyactinginsideapipe,aplasticenergydensityofthepipenearasupportincreaseswiththedecreasesintheplastichingelength,whichcausesthepipebecomeweaker.Thisiswhythevaluesoflcforthepresentunpressurizedpipeswerelowerthanthosefortheunpressurizedpipereportedinreference[5],asshowninFig.8.5

CONCLUSION

deformationpro leofthestructuresinthethree-dimensionaldynamicnon-linearFEAwereobtained.Thein uencesofthecircumferentialstressbecauseoftheinternalpressure,foundationsupport,andthemembraneforceonthethresholdoffailureforthepipelinesweresigni cant.Hence,thepre-viousresearchonthepipelineswithoutafoundationsupportmaynolongerbevalidfortheactualpipelineswithafoundationsupport.

Furtherexperimentaltestsneedtobecarriedoutinordertoobtaintheplasticbehavioursoffoun-dations,suchastheinitialcohesion,theinitialand nalfrictionangles,andthelimitingplasticstrain.Moreunderstandingofthephenomenawillbegainedfromtheproposedtheoreticalworkinthefuture,especiallytofurtherdevelopthetheoreticalstudyproposedinreference[8].Thein uencesofinternalpressureloading,foundationsupport,membraneforces,andthepropertiesofthematerialsinordertopredicttheinelasticresponseandtheonsetoffailureforthepipelinesshouldbeconsidered,asshouldtheroleoftheindenteronthedynamicresponseandthegeometricalscalinglaw.

REFERENCES

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3Jones,N.,Birch,S.E.,Birch,R.S.,Zhu,L.,andBrown,M.Anexperimentstudyonthelateralimpactoffullyclampedmildsteelpipes.Proc.InstnMech.Engrs,PartE:J.ProcessMechanicalEngineering,1992,206,111–127.

4Jones,N.andShen,W.Q.Atheoreticalstudyofthelat-eralimpactoffullyclampedpipelines.Proc.InstnMech.Engrs,PartE:J.ProcessMechanicalEngineering,1992,206,129–146.

5Jones,N.andBirch,R.S.In uenceofinternalpressureontheimpactbehaviorofsteelpipelines.Trans.ASMEJ.Press.VesselTechnol.,1996,118,464–471.

6Chen,K.S.andShen,W.Q.Furtherexperimentalstudyonthefailureoffullyclampedsteelpipes.Int.J.ImpactEng.,1998,21(3),177–202.

7Shen,W.Q.andChen,K.S.Aninvestigationontheimpactperformanceofpipelines.Int.J.Crashworthiness,1998,3(2),191–209.

Experimentaldatarecordedfromatotalof52impacttestsontheseamlessmildsteelpipeswithandwith-outafoundationsupportarepresentedinthisarticle.Thepipespecimenswithdifferentinternalpressureloadingsthatproducecircumferentialstressesupto0.272sywereconducted.

Numericalsimulationsoftheexperimentaltestswereconducted.Thesenumericalsimulationswerecarriedoutthroughathree-dimensionaldynamicnon-linearFEA,whereboththegeometricalandmaterialnon-linearitieswereconsidered.Thedetailsofthestressandstraindistributionsandthe

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8Shen,W.Q.andShu,D.W.Atheoreticalanalysisonthefailureofunpressurizedandpressurizedpipelines.Proc.InstnMech.Engrs,PartE:J.ProcessMechanicalEngineering,2002,216,151–165.

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10John,V.Testingofmaterials,1992(Macmillan,Hamp-shireandLondon).

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straincurveforaductilemetal.Nucl.Eng.Des.,1993,140,153–158.

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Siteinvestigation,2ndedition,1995,pp.500–506(Blackwell,Oxford).

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HlAlBlplpcLnpp1poPcPmaxPmeanRtfVowWbWfWf1WfoWgWlf

APPENDIXNotationCfDDmaxDminEEkGh

maximumpermanentdeformationoffoundationatimpactlocationoutsidediameterofpipe

maximumdiameterofdeformedpipeatimpactlocation

minimumdiameterofdeformedpipeatimpactlocation

Young’smodulusofpipe

initialimpactenergy,1=2GVo2massofindenter

freedropheightofindenter

wallthicknessofpipe

totalrelativehorizontaldisplacementofpointsAshowninFig.3(a)

totalrelativehorizontaldisplacementofpipeslidingoutoftheclampsmeanplastichingelengthcriticalvalueoflp

half-distancebetweentwosupportsofpipe

numberofreboundsofindenterinternalpressure

internalpressureafterimpacttestinternalpressurebeforeimpactteststaticplasticcollapseloadforapipewithoutafoundation

maximumvalueofconcentratedimpactforce

averagevalueofconcentratedimpactforce

meanradiusofpipe

totalresponsetimein niteelementanalysiswhenthemotionceasesinitialimpactvelocity

moisturecontentofsoilfoundation

totaldisplacementofpipebottomsurfaceattheimpactlocation,WfoþDmin2Dtotalmaximumpermanenttransverseplasticdisplacementofpipeattheimpactlocation,WlþWg

valueofWfafterunclampingvalueofWfbeforeunclamping

globaldeformationofpipeatimpactlocation

localdeformationofpipeatimpactlocation

dimensionlessvalueofWf,Wf/Hmaximumaverageaxialplasticstrainacrosspipewallthickness

staticuniaxialtruerupturestraindimensionlessvalueofEk,Ek/PcHcriticalvalueofl

dimensionlesscriticalmeanlengthofplastichinge,lpc/2R

staticuniaxialtruerupturestressofpipestaticuniaxialyieldstressofpipe

circumferentialstressinpressurizedpipe,pR/H

1max

Ã1rllcl1Ãsrsysu

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