* 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

158

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.

160

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

161

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

162

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.

163

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)

165

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)

166

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

[1]R y ś J., Strength analysis in the region of a hole in high pressure vessel,Proceedingsof the 8th InternationalResearchExpertConference “Trends in theDevelopment ofMachineryandAssociatedTechnologyTMT”,Neum,2004.

[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.

[4]C h a b o c h e J.L., Time-independent constitutive theories for cyclic plasticity,InternationalJournalofPlasticity,2,1985.

[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.

[7] Z i e l i ń s k i A.P., W i d ł a k G., Local shakedown analysis in regions of holes in high pressure vessels, Proceedings of the IX International Conference onComputationalPlasticity,COMPLASIX,barcelona,2007.

[8]W i d ł a k G., Radial return method applied in thick-walled cylinder analysis,JournalofTheoreticalandAppliedMechanics,48,2010.

[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.