RESEARCHOpenAccess - PDF document

RESEARCHOpenAccess FEM-basedshakedownanalysisofhardening structures PhúTìnhPh
m 1* andManfredStaat 2 *Correspondence: phamphutinh@ yahoo.com 1 FacultyofCivilEngineering,Hanoi ArchitecturalUniversity,NguyenTrai Street,ThanhXuânDistrict,Hanoi, Vietnam Fulllistofauthorinformationis availableattheendofthearticle Abstract Thispaperdevelopsanewfiniteelementmethod(FEM)-basedupperbound algorithmforlimitandshakedownanalysisofhardeningstructuresbyadirect plasticitymethod.Thehardeningmodelisasimpletwo-surfacemodelofplasticity withafixedboundingsurface.Theinitialyieldsurfacecantranslateinsidethe boundingsurface,anditisboundedbyoneofthetwoequivalentconditions:(1)it alwaysstaysinsidetheboundingsurfaceor(2)itscentrecannotmoveoutsidethe back-stresssurface.Thealgorithmgivesaneffectivetooltoanalyzetheproblems withaveryhighnumberofdegreeoffreedom.Ournumericalresultsareveryclose totheanalyticalsolutionsandnumericalsolutionsinliterature. Keywords: Ratchetting;Kinematichardening;Two-surfaceplasticity;Shakedown;FEM Background Shakedownanalysisforhardeningstructure shasbeeninvestigated bymanyresearchers. Amonghardeningmodels,theisotropichardeni nglawisgenerallynotreasonableinsitua- tionswherestructuresaresub jectedtocyclicloadingbecauseitdoesnotaccountforthe Bauschingereffectandrejectsthepossibilityo fincrementalplasticity.Theunboundedkine- matichardeningmodelhasalreadybeenintrod ucedtheoreticallybyMelan[1]andlaterby Prager[2].Applicationsofthismodelhaveb eeninvestigatedbyMaier[3]andPonter[4]. Theunboundedkinematichardeningmodelcannotestimatetheplasticcollapseandalsoin- crementalplasticitybutonlylow-cyclefatigue,whilelow-cyclefatiguelimitwiththeki- nematicalhardeningmodelseemsnottobeessentiallydifferentfromtheperfectly plasticmodel,cf.GokhfeldandCherniavsky[5]andSteinandHuang[6]. plasticitywithafixedboundingsurfaceisachievedwhichappearstobemostbasic, suitableandsimpleforshakedownanalysis.Applicationofboundedkinematicharden- ingmodelwasintroducedtheoreticallyandnumericallybyWeichertandGroß-Weege [7]whousedthegeneralizedstandardmaterialmodel(GSM).TheyusedAiry'sstress functiontosatisfytheequilibriumconditionsintheinteriorofthestructuresfulfilled. Shakedowntheoremsforboundedlinearandnonlinearkinematichardeninghavebeen proposedbyBodovilléanddeSaxcé[8],Pham[9,10]andNguyen[11]. Numericalinvestigationsforboundedkinematichardeningusingbasicreduction techniquehavebeenintroducedbyStaatandHeitzer[12,13]andSteinandZhang [14].Bythelowerboundapproach,itpermitstoavoidthenondifferentiabilityofthe ©2014Ph
mandStaat;licenseeSpringer.ThisisanopenaccessarticledistributedunderthetermsoftheCreativeCommons AttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproductioninany medium,providedtheoriginalworkisproperlycited. Ph
mandStaat AsiaPacificJournalonComputationalEngineering 2014, 1 :4 http://www.apjcen.com/content/1/1/4 objectivefunction,whichmustberegularizedviainternaldissipationenergyandthereisnoincompressibilityconstraintinnonlinearprogrammingproblem,butthisap-proachsuffersfromnonlinearinequalityconstraints.Acompanyoflowerboundalgorithmistheupperboundalgorithm,whichisbasedonKoitertheorem.Forperfectlyplasticstructures,theupperboundalgorithmhasbeenestab-lishedbyYanandNguyenDang[15,16]andYanetal.[17].Themajornumericalobstacleinthisapproachisthesingularpropertyofplasticdissipationfunction.Dealingwiththisdiffi-culty,theresearchersreplacedtheoriginaldissipationfunction,whereisaverysmallnumber.Thistechniqueisalsousedinouralgorithm.Byusingthestaticapproachandthecriterionofthemean,NguyenDangandKönig[18]showedthattheshakedownsolutioncanbeobtainedbyamaximizationoraminimizationproblem.TheyieldcriterionofthemeanwasfurtherappliedinpracticalcomputationsbydisplacementmethodandequilibriumfiniteelementbyNguyenDangandPalgen[19].Averyefficientprimal-dualalgorithm,whichcanderivelowerandupperboundsim-ultaneouslyofshakedownlimitloadfactorforcomplicatedstructures,hasbeenintro-ducedbyVu,YanandNguyenDang[20-22]andVu[23].Intheseworks,dualrelationshipbetweenupperboundandlowerboundforshakedownanalysisofperfectlyplasticstructureshasbeenproven.Theoreticallyspeaking,primal-dualalgorithmhelpstofindaveryaccuratesolutionofshakedownanalysisproblem.Whileusingthefiniteelementmethod(FEM)forlimitandshakedownanalysis,thestressmethodcanbeused,butthismethodisrestrictedsinceforcertainstructures,itisverydifficulttofindappropriatestressfunction,sothedisplacementmethodispre-ferredtomakethenumericalapproachasgeneralaspossible.Forthestructureswithhardeningmaterial,itisdifficulttoprovetherelationshipbetweenupperboundandlowerboundbecauseofthecomplicationoftheobjectivefunction.Fur-thermore,inthestaticapproach,itisdifficulttopresentalternatinglimitandratchetinglimitseparately.Inthispaper,wehavepresentedaFEM-basedupperboundalgorithmforshake-downanalysisofboundedkinematichardeningstructureswithvonMisesyieldcriterion.Bythedirectplasticitymethods,shakedownanalysisisanonlinearprogrammingproblem.Thepresentalgorithmcandealwithcomplicatedrealisticstructureswhicharemodelledby3D,20-nodeelementswithhugenumberofdegreeoffreedom.Twonumericalexamplesarein-cludedtovalidatethealgorithmandtostudytheinfluenceofhardeningeffect.BoundedkinematichardeningmodelForkinematichardeningmodel,theinitialyieldsurfacecantranslateinthemulti-axialstressspace,withoutchangingitsshapeandsize.Ifthetranslationisunlimited,orinotherwords,theultimatestrengthofmaterialisinfinite,wehaveunboundedmodel(Figure1).Thismodelisinadequatetopredicttheplasticcollapse(bothincrementalandinstantaneous)ofstructure.Itcanonlydescribethealternatingplasticitymode.TheinitialyieldsurfaceforvonMisesmaterialisdefinedasbelowThesubsequentsurfaceisdefinedasisthebackstress.Ifhardeningisunbounded,isinfinite.mandStaatAsiaPacificJournalonComputationalEngineering:4Page2of13http://www.apjcen.com/content/1/1/4 Formorerealisticmaterial,yieldstress  y mustbeboundedbyultimatestrength  u . Asimpletwo-surfacemodelisusedtomodeltheboundedhardening.Thesubsequent yieldsurfacemayormaynottouchthefixedboundingsurface;seeFigure2.Thisis satisfiedbyoneofthetwofollowingconditions: 1.Centreofsubsequentyieldsurfacecannotmoveoutsidetheback-stresssurface. Thisisexpressedby F
½

u Š
y  2 : ð 3 Þ 2.Subsequentyieldsurfacealwaysstaysinsideboundingsurface.Thisisexpressedby F
½   2 u : ð 4 Þ Figure1 Unboundedkinematichardeningmodel. Figure2 Asimpletwo-surfaceplasticitywithfixedboundingsurface. Ph
mandStaat AsiaPacificJournalonComputationalEngineering 2014, 1 :4Page3of13 http://www.apjcen.com/content/1/1/4 Intheprecedingconditions,Equations3and4,equalitiesoccurwhenthesubsequentsurfacetouchesboundingsurface.Wehaveproventhatboundingconditions(3)and(4)areexactlyequivalent.SeedetailinthestudyofPhamandStaat[24].ShakedownformulationbasedonKoiter'stheoremProblemestablishmentUpperboundsolutionofshakedownloadmultiplieristhesolutionofaconstrainednonlinearprogrammingproblemdVdtðÞ¼ ðÞþwheretotalplasticenergydissipationinthestructureisasfollows: dVdt ThefirsttermintherighthandsideofEquation6isplasticenergydissipationofper-fectplasticitymaterial,andthesecondtermishardeningeffect.Evidently,if,wehaveidealplasticmaterial.Constraint(5b)isthedefinitionofplasticstrainaccumulation.Theplasticstrainratemaynotnecessarilybecompatible,butmustbecompatible.Thisisexpressedbyconstraints(5d)and(5e).Constraint(5c)istheincompressibilitycondition,and(5g)isthenormalizedcondition.ProblemdiscretizationBasedonFEM,wholestructureisdiscretizedintofiniteelementswithNG=Gaussianpoints,whereisnumberofGaussianpointsineachelement.Iftheloaddomainisconvex,itissufficienttocheckifshakedownwillhappenatallverticesof.Sotheloaddomaincanbediscretizedintofinitenumberofload=1,,andistotalnumberofverticesof.Bythesedis-cretizations,theshakedownanalysisisreducedtocheckingshakedownconditionsatallGaussianpointsandallloadvertices,insteadofcheckingforwholemandStaatAsiaPacificJournalonComputationalEngineering:4Page4of13http://www.apjcen.com/content/1/1/4 structureandentireloaddomain.Then,numericalformofEquation5isasfollows:  denotestheshakedownmultiplierinboundedlinearlykinematichardening.isthestrainvectorcorrespondingtoloadvertexatGaussianpointisthefictitiouselasticstressvectorcorrespondingtoloadvertexatGaussianisthenodaldisplacementvector,isthedeformationmatrixandisthesmallnumbertoavoidsingularity.andaresquarematrices,expressedinEquation9:111 2 1 111000111000111000000000000000000000Forthesakeofsimplicity,wedefinesomenewplasticstrain,fictitiouselastic,deformationmatrix,respectivelyasThenEquation(7)becomes  tosolveproblem(),usingpenaltyfunctionmethodforconstraints(b)andc),combinedwithLagrangemultipliermethodforconstraint(d).PenaltymandStaatAsiaPacificJournalonComputationalEngineering:4Page5of13http://www.apjcen.com/content/1/1/4 functionandLagrangefunctionareexpressedinEquationsandrespectively. c2Xmk¼1eTikDMeikþ !) c2Xmk¼1eTikDMeikþ !)AlgorithmStep1:Choosestartingpoint:displacementandstrainvectorsandsuchthatthenormalizedcondition(11d)issatisfied:Step2:Calculate)fromcurrentvaluesofmandStaatAsiaPacificJournalonComputationalEngineering:4Page6of13http://www.apjcen.com/content/1/1/4 Q i ¼ I i þ X m k ¼ 1 ~ M  1 ik N ik ð 21 Þ ~ M ik   X m k ¼ 1 e T ik X m k ¼ 1 e ik þ
2 s I ik þ cb ik D M ! ð 22 Þ N ik  a  e T ik e ik þ
2 q þ cb ik \f I ik ð 23 Þ b ik ¼  X m k ¼ 1 e T ik X m k ¼ 1 e ik þ
2 s  e T ik e ik þ
2 q ð 24 Þ  1 ¼ a X m k ¼ 1 e ik  e T ik e ik þ
2 q þ e ik  X m k ¼ 1 e T ik X m k ¼ 1 e ik þ
2 s þ c D M e ik b ik þ c X m k ¼ 1 e ik  ^ B i u ! b ik  cb ik ^ B i d u 1 ð 25 Þ  2 ¼ t ik  c ^ B i d u 2  ð 26 Þ  þ d  ðÞ¼ 1  X NG i ¼ 1 X m k ¼ 1 t T ik e ik þ d e ik ðÞ 1  X NG i ¼ 1 X m k ¼ 1 t T ik d e ik ðÞ 2 ¼  X NG i ¼ 1 X m k ¼ 1 t T ik d e ik ðÞ 1 X NG i ¼ 1 X m k ¼ 1 t T ik d e ik ðÞ 2 ð 27 Þ Step3:Performalinesearchtofind  u suchthat  u ¼ F P u þ  d u ; e þ  d e ðÞ
min ð 28 Þ Updatedisplacement u ,plasticstrain e ik u ¼ u þ  u d u a ðÞ e ik ¼ e ik þ  u d e ik b ðÞ ð 29 Þ Step4:Checkconvergencecriteria:iftheyareallsatisfied,thenstop;otherwisegoto step2. Figure3 Continuousbeam. Ph
mandStaat AsiaPacificJournalonComputationalEngineering 2014, 1 :4Page7of13 http://www.apjcen.com/content/1/1/4 Resultsanddiscussions Twoexamplesarereported.Tocomparetheresultsonshakedownlimitforperfectly plasticmaterialswithotherresearches,wechoose  u =  y .Toinvestigatetheeffectof boundedhardening,wechoose  y  u  y .When  u  2  y ,wehaveunboundedkine- matichardeningmodel. Continuousbeam ThecontinuoussteelbeamisdescribedinFigure3subjectedtouniformdistributed loads: p 1 and P 2 varyindependentlyinthedomain: p 1 \b [1.2,2], p 2 \b [0,1].Theload domainisdescribedinFigure4. ThematerialmechanicalpropertiesareYoung'smodulus, E =1.8 \t 10 5 N/mm 2 ;yield stress,  y =100N/mm 2 ;ultimatestrength,  u =1.35  y andPoisson'sratio, \b =0.3.Bythe symmetryoftheproblem,onlyhalfofthestruct ureisdiscretizedinto589elements,8-node quadrangle,Figure5.Thestructureisconsideredasaplanestressproblem.Numericallimit andshakedownanalysisforthisstructuremade ofperfectlyplasticmaterialwerepresented inGarceaetal.[25]andTranetal.[26]. Table1showstheresultsoflimitandshakedownanalysis.Presentresultsarecloseto othersinliterature. InteractiondiagramofshakedownloadmultiplierisplottedinFigure6.Inthisstruc- ture,when p 2 isnotverylarge,thestructurefailsinratchetingmode,andbenefitof hardeningisquiteclear. Cylindricalpipeundercomplexloading Thisclosed-endpipeisinvestigatedforperfectlyplasticmaterialinVu[23]using primal-dualshakedownalgorithm.Thestructureissubjectedtobending M b and Figure4 Loaddomainforexample4.1. Figure5 FEMmesh. Ph
mandStaat AsiaPacificJournalonComputationalEngineering 2014, 1 :4Page8of13 http://www.apjcen.com/content/1/1/4 torsion M t moments,internalpressure p andaxialtension T .Materialpropertiesare Young'smodulus, E =2.1 \t 10 5 N/mm 2 ;yieldstress,  y =160N/mm 2 ;ultimatestrength,  u =1.25  y andPoisson'sratio, \b =0.3.Using20-node3Delementstomodelwhole structurewiththedimensions:length L =2,700mm,meanradius r =300mmand thickness h =60mm,seeFigure7. Theanalyticalsolutionsofplasticcollapselimitforcylindricalpipeundercomplex loadingcanbecitedfromVu[23]. Purebendingcapacity: M b lim ¼ 4  y hr 2 þ h 2 12 \f ¼ 3647 : 52 \t 10 6 Nmm : ð 30 Þ Puretorsioncapacity: M t lim ¼ 2  3 p \t r 2 h  y ¼ 3134 : 24 \t 10 6 Nmm : ð 31 Þ Puretensioncapacity: T lim ¼ 2 \t rh  y ¼ 18095573 : 6N : ð 32 Þ Pureinternalpressurecapacity: p lim ¼  y h r ¼ 32N = mm 2 ð 33 Þ Table1Comparisonofplasticlimitcollapseandshakedownresults LimitShakedown Author[ p 1 , p 2 ]=[2.0,0.0][ p 1 , p 2 ]=[0.0,1.0][ p 1 , p 2 ]=[1.2,1.0][ p 1 , p 2 ]=[2.0,1.0] p 1
1 : 2 ; 2 ½ p 2
0 ; 1 ½ Garceaetal.[ 25 ]3.2808.7185.4673.2803.244 Tranetal.[ 26 ]3.4029.1925.7203.3883.377 Present(perfectly plastic) 3.3008.7445.5003.3003.264 Present(kin. hardening) 4.45511.8047.4254.4554.406 Figure6 Interactiondiagramforshakedownboundsofcontinuousbeam. Theresultsare notnormalized. Ph
mandStaat AsiaPacificJournalonComputationalEngineering 2014, 1 :4Page9of13 http://www.apjcen.com/content/1/1/4 andthenormalizedloadmultiplierwhenbending,internalpressureandtensionare combinedisasfollows: m ¼  4  3 n 2
q 2 cos n
 2 n x  4  3 n 2
q \t 2 2 6 4 3 7 5 ; ð 34 Þ where m ¼ M = M b lim n \n ¼ p = p lim n x ¼ T = T lim : 8 : ð 35 Þ Iftheaxialtensionforcecomesfromonlyinternalpressureonclosedends,then n x = n \n /2,andformula(35)canberewrittenas m ¼  4  3 n 2
q 2 ð 36 Þ FEanalysisisfulfilledforstructuresubjectedtocombinedinternalpressure p and bending M b .ResultsarepresentedinTable2,normalizedbypurebendingcapacityin Figure7 FEMmeshofcylindricalpipe. Table2Limitandshakedownloadmultipliersofcylindricalpipesubjectedtointernal pressureandbending LoadcombinationElastic factor Limitfactor (perfectly plastic) Shakedown factor(perfectly plastic) Shakedown factor(bounded hardening) Shakedownfactor (unbounded hardening) 0.0p_1.0M0.73381.00120.73380.73380.7338 0.2p_1.0M0.72280.98700.72970.73100.7304 0.4p_1.0M0.70110.94780.72280.72360.7231 0.6p_1.0M0.65700.89140.71310.71320.7134 0.8p_1.0M0.60230.82670.70110.70130.7014 1.0p_1.0M0.55090.76080.66670.68550.6853 1.0p_0.8M0.61680.85400.76960.81280.8127 1.0p_0.6M0.69210.95560.89060.98940.9894 1.0p_0.4M0.77271.05461.01791.23181.2346 1.0p_0.2M0.84861.13061.12041.38741.5506 1.0p_0.0M0.90191.15891.15861.44821.8091 Ph
mandStaat AsiaPacificJournalonComputationalEngineering 2014, 1 :4Page10of13 http://www.apjcen.com/content/1/1/4 formula(28)andpureinternalpressureinformula(33).Limitanalysisisimplemented for  u /  y =1.0tobecomparedtoformula(36),andinteractiondiagramisplottedin Figure8.Shakedownanalysiswithandwithouthardeningeffectisimplementedforthe loaddomain: p \b [0,1]; M b \b [  1,1].InteractiondiagramisplottedinFigure9. Figure8showsthatthepresentresultsoflimitanalysisfor  u /  y =1areclosetoana- lyticalsolutions.Figure9showsthatthehardeningeffectiscleariftheappliedmoment islessthan0.5 M b lim .If  u  2  y ,boundedhardeningmodelbecomesunbounded,and shakedownlimitofstructurecannotexceedtwotimesofelasticlimit. Conclusions Thepaperdevelopedanewupperboundalgorithmforshakedownanalysisofelastic plastic-boundedlinearlykinematichardeningstructures.Thisisanefficienttoolforprac- ticalcomputation,especiallyforcomplicatedstructuressubjecttomechanicalloads. Figure8 Interactiondiagramforlimitbounds. Comparisonbetweenanalyticalandnumericalsolutions. Figure9 Interactiondiagramforelasticandshakedownbounds,normalizedbypureplastic collapselimits, M b lim and p lim. Ph
mandStaat AsiaPacificJournalonComputationalEngineering 2014, 1 :4Page11of13 http://www.apjcen.com/content/1/1/4 Theproposedalgorithmgivesresultsthatareclosetotheresultsinliteratures.Ifitleadstoperfectlyplasticmaterial;if,itleadstounboundedkinematichardeningmaterial;otherwise,,wehaveboundedkinematichardeningmaterial.denoterespectivelyelasticlimit,shakedownlimitforelasticper-fectlyplasticandshakedownlimitforboundedkinematichardeningmaterial,respect-ively,then: Intheprecedingexpression,theleftequalityoccursifthesubsequentyieldsurfacetranslatesinsidetheboundingsurface,themiddleequalityoccursifthesubsequentyieldsurfacefixedontheboundingsurfaceandthelastequalityoccurswhenyieldsur-facetranslatesunboundedly.Ifthestructureshakesdowninalternatingplasticitymode,thenthereisnodifferencebetweenperfectlyplasticandkinematichardeningmodels.CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.AuthordetailsFacultyofCivilEngineering,HanoiArchitecturalUniversity,NguyenTraiStreet,ThanhXuânDistrict,Hanoi,Vietnam.FacultyofMedicalEngineeringandTechnomathematics,AachenUniversityofAppliedScience,JülichCampus,Heinrich-Mußmann-Str.1Jülich52428,Germany.Received:15August2013Accepted:30December2013Published:29April20141.MelanE(1938)ZurPlastizitätdesräumlichenKontinuums.Ing-Arch8:1162.PragerW(1956)Anewmethodofanalyzingstressandstraininworkhardeningplasticsolids.JApplMechASME3.MaierGA(1973)Shakedownmatrixtheoryallowingforworkhardeningandsecond-ordergeometriceffects.In:SawczukA(ed)Foundationsofplasticity.Springer,North-Holland,Amsterdam,pp4174.PonterARS(1975)Ageneralshakedowntheoremforelasticplasticbodieswithworkhardening.In:Proc.SMiRT-3,paperL5/25.GokhfeldDA,CherniavskyOF(1980)Limitanalysisofstructuresatthermalcycling.Sijthoff&Noordhoff,The6.SteinE,HuangYJ(1995)Shakedownforsystemsofkinematichardeningmaterials.In:WeichertD,DorozS,MrózZ(ed)Inelasticbehaviourofstructuresundervariablesloads.KluwerAcademicPublishers,Springer,Netherlands,pp337.WeichertD,Groß-WeegeJ(1988)Thenumericalassessmentofelastic-plasticsheetsundervariablemechanicalandthermalloadsusingasimplifiedtwo-surfaceyieldcondition.IntJMechSci30:7578.BodovilléG,deSaxcéG(2001)Plasticitywithnon-linearkinematichardeningmodellingandshakedownanalysisbythebipotentialapproach.EurJMechA/Solids20:999.PhamDC(2005)Shakedownstaticandkinematictheoremsforelastic-plasticlimitedlinearkinematic-hardeningsolids.EurJMechA/Solids24:3510.PhamDC(2007)Shakedowntheoryforelasticplastickinematichardeningbodies.IntJPlast23:124011.NguyenQS(2003)Onshakedownanalysisinhardening.JMechPhysSolids51:10112.StaatM,HeitzerM(2002)Therestrictedinfluenceofkinematichardeningonshakedownloads.ProceedingsofWCCMV,5thWorldCongressonComputationalMechanics,Vienna,Austria.http://opus.bibliothek.fh-aachen.de/13.StaatM,HeitzerM(ed)(2003)NumericalMethodsforLimitandShakedownanalysisDeterministicandProbabilisticProblems.NICSeries,vol15.JohnvonNeumannInstituteforComputing,Jülich.http://webarchiv.14.SteinE,ZhangG(1992)Theoreticalandnumericalshakedownanalysisforkinematichardeningmaterials.In:OwenDRJ,OñateE,HintonE(ed)Proc.3rdInt.Conf.onComputationalPlasticity(COMPLAS3),CIMNE.PineridgePress,Barcelona,Spain,pp115.YanAM,NguyenDangH(2000)Directfiniteelementkinematicalapproachesinlimitandshakedownanalysisofshellsandelbows.In:InelasticAnalysisofStructuresundervariableLoads,TheoryandEngineeringApplications.KluwerAcademicPublishers,Springer,Netherlands,pp23316.YanAM,NguyenDangH(2001)Kinematicalshakedownanalysiswithtemperature-dependentyieldstress.IntJNumMechEngng50:1415mandStaatAsiaPacificJournalonComputationalEngineering:4Page12of13http://www.apjcen.com/content/1/1/4 17.YanAM,KhoiVD,NguyenDangH(2003)Kinematicalformulationoflimitandshakedownanalysis.In:Numerical MethodsforLimitandShakedownAnalysis – DeterministicandProbabilisticProblems.NICSeries,vol15.Johnvon NeumannInstituteforComputing,Jülich.http://webarchiv.fz-juelich.de/nic-series/volume15/volume15.html 18.NguyenDangH,KönigJ(1976)Afiniteelementformulationforshakedownproblemsusingayieldcriterionof themean.CompApplMechEng1(Nr.2):179 – 182 19.NguyenDangH,PalgenL(1980-81)Shakedownanalysisbydisplacementmethodandequilibriumfiniteelement. TransacCSME6(Nr.1):32 – 39 20.VuDK,YanAM,NguyenDangH(2003)Adualformfordiscretizedkinematicformulationinshakedownanalysis. IntJSolidsStruct41(1):267 – 277 21.VuDK,YanAM,NguyenDangH(2004)Aprimal-dualalgorithmforshakedownanalysisofstructures.Comp MethodsApplMechEng(Elsevier)193(42 – 44):4663 – 4674 22.YanAM,VuDK,NguyenDangH(2004)Dualinkinematicalapproachesoflimitandshakedownanalysisof structures.In:DavidY(ed)Complimentarily,dualityandsymmetryinnonlinearmechanics,vol6.Gao,Kluwer AcademicPublishers,Springer,Netherlands,pp127 – 148 23.VuDK(2001)DualLimitandShakedownanalysisofstructures.PhDThesis.UniversitédeLiège,Belgium 24.Ph
mPT,StaatM(2013)Anupperboundalgorithmforlimitandshakedownanalysisofboundedlinearly kinematichardeningbodies.In:DeSaxcéGetal.(ed)DirectMethods.Springer,Netherlands 25.GarceaG,ArmentanoG,PetroloS,CasciaroR(2005)Finiteelementshakedownoftwo-dimensionalstructures.Int JNumerMechEngng63:1174 – 1202 26.TranTN,LiuGR,NguyenXH,NguyenTT(2010)Anedge-basedsmoothedfiniteelementmethodforprimal-dual shakedownanalysisofstructures.IntJNumerEngng82:917 – 938 doi:10.1186/2196-1166-1-4 Citethisarticleas: Ph
mandStaat: FEM-basedshakedownanalysisofhardeningstructures. AsiaPacificJournalon ComputationalEngineering 2014 1 :4. Submit your manuscript to a journal and bene“ t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the “ eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Ph
mandStaat AsiaPacificJournalonComputationalEngineering 2014, 1 :4Page13of13 http://www.apjcen.com/content/1/1/4