Cyclic steady states of a thick-walled reactor with stress...
Transcript of Cyclic steady states of a thick-walled reactor with stress...
* Dr inż. GrzegorzWidłak, Prof. dr hab. inż.Andrzej P. Zieliński, Instytut Konstrukcji Maszyn,WydziałMechaniczny,PolitechnikaKrakowska.
GRZEGORZWIDŁAK,ANDRZEJP.ZIELIŃSKI*
CYCLICSTEADYSTATESOFATHICK-WALLEDREACTORWITHSTRESSCONCENTRATORS
CYKLICZNESTANYuSTALONEWGRubOśCIENNYMREAKTORZEZKONCENTRATORAMINAPRężEŃ
A b s t r a c t
Thepaperdealswithelastic-plasticstressstatesinvicinityofaradialcross-boreinathick-walledreactorloadedbyvariableinternalpressureandatemperaturegradient.Inthedescriptionofmaterial,thelinearPragermodelofhardeninghasbeenapplied.Themechanismofdevelopmentofreverseplastificationineachloadcyclehasbeenobserved.Thetermaleffectispresentedonanexampleofthesteadystressstates.Thetransientstates(startingandclosingworkperiodsofthereactor)havebeeninvestigatedinthePhDthesisofthefirstauthor
Keywords: thick-walled reactor, radial cross-bore, shakedown, thermal effects, finite element method
S t r e s z c z e n i e
Wartykulebadanosprężysto-plastycznestanynaprężeńwpobliżuotworugrobościennegore-aktoraobciążonegozmiennymciśnieniemwewnętrznymiróżnicątemperatur.Wopisiemate-riałustosowanoliniowymodelwzmocnieniaPragera.Obserwowanomechanizmrozwojuprze-ciwzwrotnegouplastycznieniawystępującyprzykażdymcykluobciążenia.Efektytermicznepokazanonaprzykładziestanówustalonych.Stanynieustalone(rozruchiwygaszaniereaktora)byłybadanewramachrozprawydoktorskiejpierwszegoautora.
Słowa kluczowe: reaktor grubościenny, otwór promienny, przystosowanie plastyczne, efekty termiczne, metoda elementów skończonych
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1. Introduction
High-pressure, thick-walled cylindrical vessels, which are used in the petrochemicalandotherprocessindustryareusuallysubjecttocyclicmechanicalandthermalloads.Suchvessels very often have small radial holes in theirwalls, which are necessary formediatransmissionandattachmentofinstrumentation.Theseholesarestrongstressconcentrationsourcesandintheirvicinityplasticeffectscanoftenoccur.Therefore,itisessentialtodesignthevesselsagainstcyclicplasticityfailuremechanisms,whichcanresultindevelopmentofcracksdangerousforthewholestructure.ThecracksofthiskindwereobservedinoneofpolyethylenereactorsinthepetroleumrefineryinPoland[1,2].Theyappearednearasmallholefordischargingthereactorfromthefinalproduct(Fig.1).Thepurposeofthepresentworkis,therefore,toexaminecloselyandcomprehensivelythisimportantindustrialproblemwithrespecttolocalplasticityeffectsduringcyclicoperatingofthereactor.
a) b)
Fig.1.Reactorusedinthepetroleumrefinery:a)generalview,b)objectofinterest–radialhole anditsvicinity
Rys.1.Reaktorstosowanywrafineriiropynaftowej:a)widokogólny,b)okolicepromieniowegootworu–przedmiotbadań
2. Cyclic response of thick-walled reactor subject to pressure load
Inthepresentsectionamechanismofdevelopmentofreverseplastificationsisillustrated.Asimpleassessmentofthelinearkinematichardeningruleisalsopresentedinthecontextoftheratchettingevaluation.
2.1.Materialmodel
Therate-independentplasticitymodelusedinthisstudyisassumedtoexhibitkinematichardeningwiththeHuber–vonMisesyieldcondition[3].Therefore,theequivalentstresscanbedefinedas
1/23 ( ) : ( ) ,2e σ = − −
S a S a (1)
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whereSisthedeviatoricstresstensorandadenotesthebackstressdeviator,representingcurrentcenteroftheyieldsurface.Theyieldfunction
0ef Y= σ − (2)
givesyieldconditionf=0,whichmustholdthroughouttheplasticresponse.ThesizeoftheyieldsurfaceisdenotedbyY0,whichremainsconstantinkinematichardeningmodels.Therearemanyformulationsofcoupledkinematicmodels[4].Pragerproposed[5]thesimplestpossibleformofevolutionoftheyieldsurfaceduringplasticstrainingbylineartranslationinthestressspace:
a pl= 2
3C εε (3)
Thismodelisverypopularinengineeringcomputations,becauseitneedsonlyoneplasticparameterC.Despitethefactthatthismodelcouldreasonablyrepresenttheshapeofsomemonotonic stress-strain curves, it fails to produce ratchetting in uniaxial tests.Moreover,rapid prediction of shakedown in the initial cycles ofmultiaxial loading does notmatchexperimentalresults.Moreappropriatemodelswereinvestigatedin[6].Nevertheless,inthefirstapproachtotheproblem,itseemstobereasonabletousethissimplemodel inordertodemonstratemechanismsresponsibleforthedevelopmentofcyclicplasticity.InfurthercalculationsthevaluesofY0 =275MPaandC=2000MPawereassumed.
2.2.FEmodel
In reactors, the pressure load and thermal gradients usually have cyclic character,thereforeitisessentialtodevelopaFEMmodel(Ansys),whichisappropriateforaccurateand effective cyclic plasticity simulations [7]. Estimation and investigation of the robustintegrationalgorithmofconstitutiveequations,whichishereinutilized,canbefoundin[8].
Ithasbeenassumedthattheinvestigatedreactorhasthefollowingdimensions(compareFig.2):internalradiusa=150mm,externalradiusisdefinedbyratioa/b=0,7andradialholediameterratioisd/a=0,05.Theradialcrossholeconfigurationhastwoplanesofsymmetry,soonlyaquarter-modelsarerequired(seeFig.3).Thex-Rplanehasthefollowingsymmetryconditions
u 0
00
R
x
θ
θ
θ
=τ =τ =
(4)
andthex–θplane
u 0
00
x
xR
xθ
=τ =τ =
(5)
where uθ ,uxarecorrespondingdirectdisplacementsand Rθτ , xθτ , xRτ , xθτ representshearstresses.
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Fig.2.Generalviewofinvestigatedreactormodel
Rys.2.Ogólnywidokmodelubadanegoreaktora
Fig.3.Reducedcomputationalmodelandboundaryconditions;globalandlocal
coordinatesystems
Rys.3.Modelobliczeniowyijegowarunkibrzegowe;globalnyilokalnyukład
współrzędnych
Dimensionsofthesegmentaresuitablychosensoasnottodisturbthelocalstressfield.Apressureloadpisappliedontheinternalsurfaceofthemainboreaswellasontheradialholesurface.Theclosed-endboundaryconditionisdefinedbyapplicationofuniformaxialtensionattheendofthecylinder
( )( )
2
2
/
1 /b
a bp p
a b=
− (6)
The presented segment is discretized with 13200 higher-order finite elements. Everyelement is defined by 20 nodes having three translational degrees of freedom per node.Aspecificvalueofthepressureload,loadingmanneraswellasthermalboundaryspecificationaregiveninthesuccessivesections,whichdescribeparticularanalyses.
2.3.Mechanismofdevelopmentofreverseplastification
Radialholesinthethick-walledvesselstructure,havenegligibleinfluenceonreductionoftheglobalload-carryingcapacity.Nevertheless,theyarestrongstressconcentrationsources,causingsignificantdecreaseofelasticlimitloadsandrapidplastificationnearthehole[6].
Inordertodemonstratecharacteristicmechanismwhichisresponsibleforthedevelopmentofcyclicplasticity,loadingandunloadingprocesshasbeensimulated.Forthegivenpressurevaluesupto3p0(p0=32MPa–isthereactorelasticcapacity),thematerialeffortstateandredistributionoftheprincipalstressesareobserved.Themaximumeffortpointisidentified
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in the subsequent stages. In Fig. 4 and Fig. 5 a position of this point is presented. It islocatedontheedgebetweentheradialholesurfaceandthex-Rsymmetryplane.Therefore,dimensionlessshift /z cζ = uniquelydefinesitslocation.Theaxiallocalcoordinateishereindenotedbyz,whereascisthethicknessofreactorwall.
Withtheincreaseofpressure,aproportionaldecreaseoftheminorstressesisobserved.Theminorstressdirectioncoincideswiththeradiallocalcoordinate.Aslightdecrease,inorder to fulfill theconsistencycondition,of themajor (hoopdirection)andmiddle (axialdirection)stressesisnoticed.Suchatendencyisobservedinallthesubsequentmaximumeffortpoints,inwhichtheprincipaldirectionsareillustratedinthefiguresbelow.
Anessentialredistributionofprincipaldirectionsisnoticedduringtheunloadingprocess(Fig.5).Justbelowthereliefpressurep/p0=2,acharacteristicstressstateoccurs,inwhichtheinitiallypositivemajor(hoop)stresschangestobecompressiveanditsvalueisequalinthismomenttotheaxialstressmagnitude.Therefore,intheplanedeterminedbytheaxialdirectionandthetangenttothecircumferentialdirectioninthispoint,theshearvanishandthenormalstressesinalldirectionsarethesame(localhydrostaticstate).
Furthermore, with the progressive relief, a proportional decrease of compression intheradialprincipaldirectionisobserved.Eventually,itreachesazerovalueandtheradialdirectioncouldbeconsideredasamajorprincipaldirection.Astrongcompressionof theplastifiedzone,occurringinthehoopdirection,contributestotheconsiderablechangeoftheorderofprincipalstresses.Thefinalstressstate,resultsinreverseplastificationandtheactiveprocesscausesfurthershiftofthemaximumeffortpoint.Incompressibilityoftheplasticflowproducesalsointhiscaseconsiderableenlargementofcompressionintheaxialdirection.
Fig.4.Stressesinprincipaldirectionatthemaximumeffortpoint (p0=32MPa), /z cζ = , z–reactorthickness
Rys.4.Naprężeniegłównewpunkciemaksymalnegonatężenia (p0=32MPa)–obciążenie, /z cζ = , z–grubośćreaktora
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Fig.5.Stressesinprincipaldirectionatthemaximumeffortpoint(po=32MPa),
/z cζ = ,z–reactorthickness
Rys.5.Naprężeniagłównewpunkciemaksymalnegowytężenia(po=32MPa)–odciążenie,
/z cζ = ,z–grubośćreaktora
2.4.Inadaptationrange
When reactor loads exceed elastic shakedown limit, phenomenon of cyclic plasticityoccurs,whichisacauseoffatiguefailureinvery-low-cycleandlow-cycleregimes.Inadditiontoalternatingplasticity,aratchettingresponseisoftenpresent,accountingforaccelerationoffatiguedamageoractingasafailuremechanismitself.Although,plasticstrainsdevelopedinthefirstcyclearerelativelylow,theycanbeaccumulatedinapredominantdirectionduringsubsequentcycles.Theaccumulationofplasticstrainsisoneaspectofcyclicplasticity.Theother, appearing in the fatiguedamage form, is characterizedbymicro- andmacro-crackpropagationprocessandfinal fracture.However, there isastrong interactionbetween thedamageprocesses.Crackinitiationandpropagationarestronglyaffectedbytheaccumulatedplasticstrains.
Operatinginaninadaptationrangeisadmittedbysomedesigncodes.Moreover,whenthenumberofcyclesissolowthatthefatiguedamageisprevented,asitisinthecaseofreactors,emphasisshouldbeputoncheckofprogressivedeformation.Thisrequiresapplicationofappropriatemodelsofratchetting,whichcanresultinfailureitself.
Inthepressurevesselwithahole,translationofthepointofmax.effort(Fig.4,Fig.5)fadeswiththecyclingandthispointsettlesintheinitiallocationcorrespondingtothefirstloading.In thecaseofmodellingof theratchettingmode,magnitudeofplasticstrainschangeandprogressivedeformationisobservedinthesurroundingregions,asithasbeendemonstratedin[6].Inthefollowing,plasticstraincomponentsarepresentedforthemaximumeffortpoint,asafunctionofthenumberofcycles(Fig.6).ForthePragerhardeningmodel,therearenosignificantchanges in theplasticstrainhistory.After initial, relatively largeplastification,areverseplasticityisobservedduetostrongcompressiveeffectsafterunloading(1stcycle).A slight difference between magnitudes of plastic strain increments during loading andunloadingisnoticeableonlyforthefirstthreecycles.Further,anintegralofeverycomponentof plastic strain tensor is practically equal to zero over each individual cycle.Therefore,thelinearPragermodelshouldnotbeusedforrepresentationofratchettinginthisspecificstructre.
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Fig.6.Plasticstrains(localholecoordinates)inducedby100MPapressureloading–Pragerhardeningrule(a/b =0,7;d/a =0,01)
Rys.6.Odkształceniaplastyczne(współrzędnelokalneotworu)wywołanetętniącymciśnieniem 100MPa–modelwzmocnieniaPragera(a/b =0,7;d/a =0,01)
Asithasbeennoticedabove,thecasewithlinearhardeningmodelshowsslightincrementsofplasticstrains,whichisrathercausedbyaredistributioneffectandexpireswiththefurthercycling(Fig.7).
Fig.7.Hysteresisloop–Pragerhardeningrule
Rys.7.Pętlahisterezy–modelwzmocnieniaPragera
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3. Influence of thermal gradients at steady states
uptonowonly themechanical loading (pressure)hasbeenconsidered.However, thehighpressurereactorsareusuallysubjecttointernalpressureandincreasedtemperatures[9].Inthepresentsection,thesteadythermaleffectsaretakenintoaccount.Thethermalgradientcausesdevelopmentofthermalstresses,moreover, thehightemperaturescansignificantlychange thematerial properties. The study concerns examination of reactors subjected todifferentpressureandthermalloadrelationsduringsteadystatesoftheloading-unloadingcyclicprocess.
Inordertoevaluatetemperaturesinthereactormodel,whichissubjectedtostabilizedconditions,thewell-knownhomogeneousequationofsteadystateheattransferisutilized.Once the temperaturefield isknown, resulting thermal strains and stresses are calculated[10]. For detailed investigation of the thermal effects, different combinations of pressureloadsandassociatedsteadytemperaturestateshavebeenchosen.Themechanicalloadingsareappliedintheformofpressurewhichtakesvalues:60MPa,80MPaand100MPa.Ontheinternalreactorsurfaces(Fig.8)theconvectionboundaryconditionsareestablishedwith
theconvectioncoefficient 2
W2000m Kih = .Internaltemperaturestakevaluesfrom 1 50 CT = °
to 6 300 CT = ° each 50 CT∆ = ° . The external reactor surface has the ambient boundary
conditionswiththeconvectioncoefficient 0 2
W20m K
h = andtemperature 0 20 CT = ° .
Amaterialmodel used in calculations corresponds to a typical boiler steelDIN13Cr-
Mo44.Theyieldstressforthissteelis 020 300MPaY = forthetemperature 0 20 CT = ° and
decreasesto 0300 235MPaY = inthetemperature max 300 CT = ° .Thefirstanalysisconcernsdeterminationoftemperaturedistributionwiththespecified
boundaryconditions.With theaboveassumedvaluesof theconvectioncoefficients, therearenosignificantthermalgradientsinthesteadystate.Forexample,inthecaseofprocessed
medium temperature 300 CiT = ° and ambient temperature 0 20 CT = ° the difference ofinternalandexternaltemperaturesofthevesselisequal 11,2 Ct∆ = ° (Fig.9).
Inthecaseofsteadythermaleffectsdifferentdistributionsofthestressandstrainstatesinthevesselresultmainlyfromthepressurechanges.Presenceofstrongconcentrationcausesthatevenforthelowestpressurethelocalplasticstrainoccurs.Thepressureincreaseresultsinregulareffortvariationuptothehighestanalyzedtemperaturesinwhichaconsiderableyieldstressreductionisobservedandasignificantshiftingofthemaximumeffortpointintheanalyzedstructure.The largest loadconfigurationresults inplasticstrainswhichhaveaglobalrange(Fig.10).
A difference of the local equivalent plastic strains between the loaded and unloadedstates ∆ε ε εeq
pleqpl load
eqpl unloadmax = − (7)
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Fig.8.Acomputationalmodel–thermal
boundaryconditions
Rys.8.Modelobliczeniowy–termicznewarunkibrzegowe
Fig.9.Steadystatetemperaturedistribution (Ti =300°C,T0 =20°C;tmin=287°C,
tmax=298°C)
Rys.9.ustalonystanrozkładutemperatury (Ti =300°C,T0 =20°C;tmin=287°C,
tmax=298°C)
has been chosen as a measure of reverse plastification of the structure. This differenceincreaseswithincreasingpressurebutalmostdoesnotchangewithtemperatures.Onlythehighestinternaltemperaturecausesitsconsiderablereductionbecauseofstronggrowthoftheplasticstrainzoneinthevicinityofthehole,whichresultsindecreaseofcompressiveeffectsafterunloading(Fig.11).
Fig.10.Equivalentplasticstrainsafterloading
Rys.10.Zastępczeodkształcenieplastyczne poobciążeniu
Fig.11.Equivalentplasticstraindifference(loading–unloading)
Rys.11.Różnicaodkształceńplastycznych(obciążenie–odciążenie)
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4. Conclusions
The investigationofmaterialmodelswhich are intended to represent cyclic plasticity(theArmstrong-FrederickandChabochemodels)proved[6]thattheratchettingrateinthereactorstructureissignificantattheinitialprocessstageandfurtheriskeptconstant,whicheventually leads to the failuredue to incrementalplasticcollapse.Theconstitutivemodelconsideredhere(Pragerhardeningrule)isnotintendedtodefinecyclicplasticity.AplasticmodulusC (Eqn(3)),whichisthesameforloadingandunloading(noeffectofmeanstress)results,therefore,inthestresscycles,whichformaclosedhysteresisloop.
The small thermal effects noted above change to large values in starting and closingworkperiodsofreactors(stronglyunsteadystates).Theseeffectsandtheirinfluenceonlocalelastic-plasticresponsehavebeeninvestigatedindetailandarepresentedin[6].
R e f e r e n c e s
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[2] Z i e l i ń s k i A.P., Ł a c z e k S., R y ś J. Optimization of Thick-Walled High-Pressure Vessels with Holes with Respect to Ductile Fracture, Proceedings of the 6thWorldCongressofStructuralandMultidisciplinaryOptimization,RiodeJaneiro2005.
[3] Ż y c z k o w s k i M.,Combined Loadings in the Theory of Plasticity,PWN,Warszawa1981.
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[5] P r a g e r W., A new method of analyzing stresses and strains in work hardening plastic solids,JournalofAppliedMechanics,23,1956.
[6]W i d ł a k G., Localshakedown analysis of a thick-walled reactor subject to mechanical and thermal loads., PhD thesis supervised byA.P. Zieliński, Cracowuniversity ofTechnology,Cracow2010.
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[9]C a m i l l e r i D., M a c k e n z i e D., Shakedown of a thick cylinder with a radial crosshole,ASMEPressureVesselsandPipingDivision,2006.
[10] W i d ł a k G., Z i e l i ń s k i A.P., Local shakedown analysis of reactors subject to pressure and thermal loads,Proceedingsofthe8thWorldCongressonComputationalMechanics,Venice2008.