基于高等分析的钢结构设计——材料建模与应变极限

HighPerformanceStructures:BuildingStructuresandMaterials—Article

SteelDesignbyAdvancedAnalysis:MaterialModelingandStrainLimits

LeroyGardner⇑,XiangYun,AndreasFieber,LorenzoMacorini

DepartmentofCivilandEnvironmentalEngineering,ImperialCollegeLondon,LondonSW72AZ,UKarticleinfoabstract

Structural analysis of steel frames is typically performed using beam elements. Since these elements are unable to explicitly capture the local buckling behavior of steel cross-sections, traditional steel design specifications use the concept of cross-section classification to determine the extent to which the strength and deformation capacity of a cross-section are affected by local buckling. The use of plastic design methods are restricted to Class 1 cross-sections, which possess sufficient rotation capacity for plastic hinges to develop and a collapse mechanism to form. Local buckling prevents the development of plastic hinges with such rotation capacity for cross-sections of higher classes and, unless computation-ally demanding shell elements are used, elastic analysis is required. However, this article demonstrates that local buckling can be mimicked effectively in beam elements by incorporating the continuous strength method (CSM) strain limits into the analysis. Furthermore, by performing an advanced analysis that accounts for both geometric and material nonlinearities, no additional design checks are required. The positive influence of the strain hardening observed in stocky cross-sections can also be harnessed, provided a suitably accurate stress–strain relationship is adopted; a quad-linear material model for hot-rolled steels is described for this purpose. The CSM strain limits allow cross-sections of all slender-ness to be analyzed in a consistent advanced analysis framework and to benefit from the appropriate level of load redistribution. The proposed approach is applied herein to individual members, continuous beams, and frames, and is shown to bring significant benefits in terms of accuracy and consistency over current steel design specifications.

Ó 2019 THE AUTHORS. Published by Elsevier LTD on behalf of Chinese Academy of Engineering and Higher Education Press Limited Company. This is an open access article under the CC BY-NC-ND license

Articlehistory:Received31July2018Revised10September2018Accepted12November2018Availableonline26February2019Keywords:AdvancedanalysisContinuousstrengthmethodLocalbucklingMaterialmodelingStrainlimits1.IntroductionThemostrecenteditionsofmanyinternationalstructuraldesigncodespermittheuseofgeometricallyandmateriallynon-linearanalysis,alsoknownasadvancedanalysis,forthedesignofsteelstructuresconsistingofcompactcross-sections.ExamplesincludeSection5ofEN1993-1-1[1],AppendixDofAS4100[2],andAppendix1ofAISC360[3].Theanalysisistypicallyperformedusingthefiniteelement(FE)method,andstructuralmembersarecommonlyrepresentedusingbeamelements.Thebenefitsofadvancedanalysisdesignarewidelyrecognized[4–10].Comparedwiththetraditionalapproachtostructuraldesign,wherebythestructuralanalysisisfollowedbyindividualmemberandcross-sectionchecks,inadvancedanalysis,globalframeandmemberinstabilities(i.e.,P-DandP-d)arecapturedandtheneedforsubse-quentmemberchecksiseliminated.However,sincebeam⇑Correspondingauthor.E-mailaddress:Leroy.gardner@imperial.ac.uk(L.Gardner).elementscannotcapturelocalbuckling,plasticanalysismethodsarecurrentlylimitedtostockycross-sectionsthatpossesssuffi-cientrotationalcapacityforplastichingestodevelopandacol-lapsemechanismtoform.Toavoidusingcomplexshellelementsforaglobalframeanal-ysis,structuraldesigncodesgenerallyaccountforlocalbucklingbyprovidinglimitationsoncross-sectioncapacityanddeformabilitybyintroducingdiscretecross-sectionclassesbasedonplatewidth-to-thicknessratios.However,thisapproachcreatesartificial‘‘steps”inthecapacitypredictionsofstructuralframes.Forexam-ple,aClass1cross-sectionisassumedtohaveinfiniterotationcapacity,whileaClass2cross-sectionisassumedtohavezero.Inthispaper,amoreconsistentdesignapproachisproposed,wherebystrainlimitsareusedtomimiclocalbucklinginanadvancedstructuralanalysisusingbeamelements.Thestrainlim-itsarebasedonthecontinuousstrengthmethod(CSM)[11,12]andcontrolthedegreeofplasticredistributionthatoccurswithinthestructure,takingdueaccountofthelocalslendernessofthecon-stituentcross-sections.Hence,structureswithcross-sectionsof244L.Gardneretal./Engineering5(2019)243–249anyclasscanbetreatedusingbeamelementsinthesameconsis-tentmanner.Tocapturethestress–straincharacteristicsofstruc-turalcarbonsteelsaccurately,aquad-linearmaterialmodelhasbeendeveloped[13]andincorporatedintotheCSMframeworkforthedesignofhot-rolledsteelcross-sectionsunderdifferentloadingconditions[14,15];thismaterialmodelisemployedherein.Intheproposedapproachtodesignbyadvancedanalysis,thestructuralresistanceistakenasthelowerofeither①thepeakloadfactorreachedduringtheanalysisor②theloadfactoratwhichtheCSMstrainlimitisfirstreached.Notethat,asrecom-mendedinRef.[16],connectionswillgenerallyhavetobecheckedseparately.Thispaperoutlinestheframeworkrequiredtoincludestrainlimitsindesignbyadvancedanalysis.Thequad-linearmaterialmodelforhot-rolledsteelisdescribedandworkedexamplesillus-tratethebenefitsandgeneralapplicabilityoftheproposedmethod.Itshouldbenotedthatthefollowingdiscussionislimitedtothein-planebehaviorofindividualmembers,continuousbeams,andframesconsistingofhot-rolledsteelI-shapedsectionsundermajoraxisbending.2.MaterialmodelingAtypicalengineeringstress–straincurveforhot-rolledstruc-turalcarbonsteelsubjectedtoquasi-statictensileloadisshowninFig.1.Intheelasticrange,theslopeislinearandisdefinedbythemodulusofelasticity,orYoung’smodulusE,whichistakenas210000NÁmmÀ2accordingtoEN1993-1-1.Afterreachingtheyieldstressfy,withthecorrespondingstrainofey=fy/E,thestraincontinuestoincreasealongayieldplateauwithoutanyapparentincreaseinstress.Whenthestrainreachesthestrain-hardeningstrain,esh,strainhardeninginitiatesandthestressbeginstoincreaseagainwithincreasingstrain.Finally,thestressreachesamaximumvalueattheultimatetensilestrengthfuandthecorre-spondingultimatestraineu.Beyondthispoint,neckingandeven-tuallyfractureoccur.Thestress–strainrelationshipforstructuralcarbonsteelsisoftenidealizedbyanelastic-perfectlyplasticmodel,andthissim-plifiedmodelgenerallyformsthebasisofthecurrentdesignprovi-sionsinEN1993-1-1.However,thismodelfailstocapturethestrain-hardeningcharacteristicsofthematerial,whichareimpor-tantforadvancedcomputationalanddesignmethodssuchastheCSM,particularlyforstockycross-sections.Thus,aquad-linearFig.1.Atypicalengineeringstress–straincurveforhot-rolledstructuralcarbonsteel.E:Young’smodulus;fy:yieldstress;ey:straincorrespondstofy;fu:ultimatetensilestrength;eu:ultimatestraincorrespondstofu;esh:strain-hardeningstrain.Fig.2.Quad-linearmaterialmodelforhot-rolledstructuralcarbonsteel.materialmodel,asillustratedinFig.2anddescribedbyEq.(1)[13],hasbeenproposedtorepresentaccuratelytheyieldplateauandstrain-hardeningbehaviorofhot-rolledstructuralcarbonsteels,baseduponandcalibratedagainstalargedatasetoftensilecoupontestresults.Twomaterialcoefficients(C1andC2)areusedinEq.(1):C1definesa‘‘cut-off”straintoavoidover-predictionsofmaterialstrengthandisalsoincludedintheCSMbasecurve,asdescribedinSection3.2;andC2isemployedinEq.(2)todeterminethestrain-hardeningslopeEsh.ThesetwocoefficientsmaybedeterminedrespectivelyfromEqs.(3)and(4),whichareexpressedintermsofthestrain-hardeningstraineshandtheultimatestraineu.8>>>>Eefore ey>>forey>f>yþEshðeÀeshÞforesh< C1euð1Þ>>>:ffuÀfC1Ceu1euþeuÀC1euðeÀC1euÞforC1eu0:68,whicharereferredtoasClass4cross-sectionsinEN1993-1-1.NotethatinEq.(7),twoupperboundshavebeenplacedonthepredictedCSMstrainratioecsm/ey;thefirstlimitXdefinesthelevelofplasticFig.6.CSMbasecurve:acontinuousrelationshipbetweencross-sectionslender-ness󰀁kpanditsdeformationcapacityecsm/ey.246L.Gardneretal./Engineering5(2019)243–249deformationthatisdeemedtolerableatultimatelimitstateonagivenprojectwitharecommendedvalueof15,whilethesecondlimitofC1eu,whereC1isamaterialcoefficientdescribedinSec-tion2andeuisthematerialultimatestrainwhichmaybedeter-minedusingEq.(6),definesa‘‘cut-off”straintoavoidover-predictionsofmaterialstrength.AlsonotethatbothEqs.(7)and(8)passthroughtheidentifiedtransitionpoint,thatis,(0.68,1)inFig.6,thusensuringcompatibilitybetweentheCSMbasecurvesfornon-slenderandslendercross-sections.e󰀂󰀃csm0:25eecsmfor󰀁y¼C1󰀁kkp 0:68ð7Þpeeu3:6buty minX;eye01csm¼@1À0:2221ey󰀁1:05Ak1:05for󰀁kp>0:68ð8Þp󰀁kpTheaccuracyoftheCSMstrainlimitsusedinadvancedanalysisdependsontheabilitytoaccuratelydeterminethecross-sectionslenderness.Thecross-sectionslendernessisadimensionlessparameterthatquantifiessusceptibilitytolocalinstabilityandisdefinedbyEq.(9),wherefyistheyieldstressandrcristheelasticcriticalbucklingstress.sffiffiffiffiffiffi󰀁kfffiyp¼rcrð9ÞVariousmethodsareavailabletocalculatetheelasticcriticalbucklingstress.Standardplatetheory,utilizingbucklingcoeffi-cientsksuchasthosepresentedinEN1993-1-5[20],issimpletouseyetignoresanyelementinteractionbyassumingsimplysupportedboundaryconditionsalongtheadjoinedplateedges.TheelasticcriticalbucklingstresscanthenbecalculatedusingEq.(10),wheretandbaretherelevantplatethicknessandwidth,visthePoisson’sratio;theplateelementwiththelowestbucklingstressisthenusedtodefinethecross-sectionslendernessthroughEq.(9).rp2E󰀂󰀃2cr¼kt12ð1Àm2Þbð10ÞAlternatively,approximateexpressionscalibratedagainstnumericalresults[21,22]maybeusedtodeterminetheelasticcrit-icalbucklingstressofthefullcross-section.Plateinteractionisimplicitlyaccountedforintheseexpressions,whichincreasestheiraccuracy.TheexpressionsinRef.[21]onlycoverpurecompressionandpuremajor/minoraxisbending,whereasinstructuralframes,cross-sectionsaretypicallysubjectedtoacombinationofcompres-sionandbending;theexpressionsdevelopedinRef.[22]addressthislimitationandcovercompression,bendingandcombinedcompressionplusbending.Numericaltools,suchasthecon-strainedandunconstrainedfinitestripmethod(CUFSM)[23],canalsobeusedtodeterminetheelasticcriticalbucklingstressofthefullcross-section;CUFSMisemployedinthepresentstudy.3.3.Strain-averagingapproachExperiments[24,25]haveshownthatthemaximumin-planebendingresistanceofamemberisgreaterunderamomentgradi-entthanunderuniformbending.Theincreaseincapacityarisesfromtherestraintaffordedtothecriticalsection(i.e.,thatisundermaximumbendingmoment)bythelowerstressedadjacentregions,thusdelayingtheonsetoflocalbucklinginboththeelasticandplasticregimes.Amomentgradientisofcoursepresentinmostpracticalapplications,yetEN1993-1-1andtheCSMbasecurvedonotconsidertheeffectsoflocalmomentgradientsonthelocalstabilityofcross-sections.Itisproposedtoexploitthebeneficialinfluenceofmomentgra-dientsonlocalstabilitybyapplyingtheCSMstrainlimittoastrainextractedfromtheadvancedanalysisthatisaveragedoverafinitememberlength,asopposedtoconsideringsimplythemosthighlystressed(orstrained)cross-section,asiscustomary.Sincelocalbucklingrequiresafinitelengthofmemberoverwhichtodevelop—thatis,thelocalbucklinghalf-wavelengthLb—thisrepre-sentsasuitabledistanceoverwhichtoaveragethestrains.ThisapproachisinlinewiththeobservationsofLayandGalambos[26],whodemonstratedexperimentallythatinelasticlocalbuck-lingcommencesonlywhenyieldinghasextendedtoafinitelengthofmemberrelatedtothelocalbucklinghalf-wavelength.Byapply-ingtheCSMstrainlimittoanaveragedstrain,itisthereforepossi-bletodirectlyaccountforthemagnitudeofthelocalmomentgradient;underuniformmoment,theaveragingapproachpredictsthecross-sectioncapacity,whileforincreasingmomentgradients,thepeakmomentcapacityincreasesaccordingly.Thestrain-averagingapproachalsoreducesthesensitivityofthestrengthpre-dictionstothemeshdensityemployedintheFEmodel.4.Implementation4.1.GeneralInthissection,theproposedadvancedanalysisisappliedtoaseriesofindividualmembers,aswellascontinuousbeamsandaframe,inordertoillustratehowtheCSMstrainlimitsareabletomimiclocalbuckling,controlthelevelofinelasticredistributionofforcesandmoments,andcapturethebeneficialeffectsofmomentgradients.Forcomparison,thecapacitypredictionsaccordingtoEN1993-1-1arealsoshown.Inallcases,calibratedshellFEmodelsfeaturinglocalgeometricimperfectionswereemployedtoprovidebenchmarkresults.TheshellmodelsweredevelopedinAbaqus[27]andsolvedusingthemodifiedRiksmethod[28].Thequad-linearmaterialmodeldescribedinSection2wasadoptedinthesimulations.4.2.BeamsThenormalizedbendingcapacityatthecriticalcross-sectionofaseriesofbeamswithcross-sectionsofdifferentlocalslendernessanalyzedunderfour-pointbendingandasacantileverisshowninFig.7,whereMmaxisthemaximumbendingmoment,Mplistheplasticmomentcapacity,andMelistheelasticmomentcapacity.ItmaybeseenthatapplicationoftheCSMstrainlimitstothebeamelementmodelsyieldssafe-sidedpredictionsoftheshellFEmodelresultsacrosstherangeofcross-sectionslenderness.AccordingtoEN1993-1-1,thereisanartificial‘‘step”inbendingcapacityfromClass2to3cross-sections,whichcanbeeliminatedusingthemod-ifiedelastic–plasticbendingcapacity[29]toaccountforpartialyieldinginClass3cross-sections.ForClass4cross-sections,thestrainlimitspredictsimilarcapacitiestoEurocode3butavoidthecumbersomecalculationproceduresassociatedwiththeeffec-tivewidthmethod.TheshellFEmodelscaptureanincreaseinbendingcapacityof5%–10%acrosstherangeofcross-sectionslen-dernessfromtheuniformmomentcasetothecantilevercase,duetothebeneficialinfluenceofthemomentgradient.Similarly,theproposedadvancedanalysisusingbeamelementsaccuratelypre-dictsanincreaseof2%–7%duetothemomentgradient,whichisaccountedforthroughthestrain-averagingapproach.Ontheotherhand,EN1993-1-1doesnotcapturethiseffectanddoesnotdistin-guishbetweenthetwocases,butpredictsasinglebendingcapac-ityforeachcross-section.L.Gardneretal./Engineering5(2019)243–249247Fig.7.NormalizedbendingcapacitypredictionsfromtheshellFEmodel,proposedapproach(i.e.,beamFEmodelwithstrainlimits),andEN1993-1-1forbeamsunderuniformmomentandamomentgradientwithvaryingcross-sectionslenderness.EC3:EN1993-1-1.4.3.ContinuousbeamsAtanisolatedmemberlevel,thereisnoscopetoallowformomentredistribution.Inindeterminatestructures,however,suchascontinuousbeams,thebenefitsoftheproposedmethodcanbecomeevenmoresignificant.Inadditiontothelocalcross-sectionstrengthandtheeffectsofmomentgradients,thecollapseloadofanindeterminatesystemdependsonthelevelofinelasticforceandmomentredistribution.Fig.8showstheultimateloadfactoratcollapse,au,normalizedbytheloadfactoratwhichthefirstyieldoccurs,ael,forshellFEmodelsofaseriesofcontinuousbeamsunderpointloadsPmultipliedbyloadfactora,alongwiththecapacitypredictionsfromtheproposedmethodofdesignbyadvancedanalysisandEurocode3.EN1993-1-1onlypermitstheuseofplasticanalysismethods,whichareabletocapturetheredistributionofforcesandmomentsasthematerialyields,forClass1cross-sections.Ontheotherhand,Class2cross-sectionsmustbeanalyzedelasticallyandthusbenefitfromnoredistribu-tion,eventhoughsomeyieldingoccursbeforetheirplasticmomentcapacityisreached.Theresultingstepsinresistancepre-dictionsclearlydonotreflecttheshellFEresultsandareanoverlysimplisticrepresentationofreality.Theuseofaconsistentadvancedanalysis,inwhichtheCSMstrainlimitsdefinethecross-sectiondependentlevelofpermissibleredistribution,isshowntoprovidesignificantlymoreaccuratecapacityprediction.Theaveragecapacitypredictionforthe46analyzedcontinuousbeamsaccordingtoEN1993-1-1is0.810oftheshellFEresults,withacoefficientofvariation(COV)of5.3%.Incomparison,theproposedmethodofdesignbyadvancedanalysispredictsameancapacityof0.924oftheshellFEresults,withaCOVof4.9%.4.4.FramesConsideringstructuralbehaviorattheframelevel,aslightlymodifiedversionof‘‘Testframe2,”asreportedbyAveryandMahendran[30],isusedtoillustratethefullbenefitsofthepro-poseddesignmethod.Thesinglebayportalframecomprisedasin-gleClass3cross-sectionthroughout.ComparedwiththeoriginalFig.8.NormalizedbendingcapacitypredictionsfromtheshellFEmodel,proposedapproach(i.e.,beamFEmodelwithstrainlimits),andEN1993-1-1fortwo-spancontinuousbeamswithmid-spanpointloadsandvaryingcross-sectionslenderness.248L.Gardneretal./Engineering5(2019)243–249Fig.9.Normalizedload–strainresponseforaframewithaClass3cross-section.P:loadoftheframe;Mep:theelastic-plasticmoment.testframe,thematerialmodelwasadjustedtothequad-linearmaterialmodeldescribedearlierwithanelasticmodulusE=200000MPa,ayieldstressfy=360MPa,andanultimatestressfu=510MPa,matchingthemeasuredflangematerialpropertiesof‘‘Testframe2.”TheglobalswayimperfectionwasbasedontherecommendationsofEN1993-1-1,whilethememberimper-fectionsweretakenasl/1000inthemostunfavorabledirection,wherelisthememberlength;bothP-DandP-deffectsarethere-foredirectlyaccountedforintheadvancedanalysis.Theloadversusnormalizedstrain(averagedoverLbfromthecriticalcross-section)responsesoftheshellandbeamFEmodelsofthesinglebayportalframeareshowninFig.9.ThemeasuredtestcollapseloadoftheframewasP=135.0kN.IntheshellFEmodel,someyieldingisobservedatthebaseoftherightcolumnpriortolocalbucklingandultimatelycollapseoftheframeataloadofP=133.9kN.Incontrast,thebeamFEmodelcannotcapturethelocalbucklingatthebaseofthecolumnandreachesaglobalswayfailureatapeakloadofP=156.0kNwhenallowingforunlimitedstrains—clearlyanunsafesituation,sincetheshellFEcollapseloadisover-predictedby16%.Toensuresafecapacitypredictions,suitablecross-sectionandmemberdesignchecksarerequiredbyEN1993-1-1followinganelasticanalysisofthestructure;failureispredictedataloadofP=108.9kN.Thisisconservativeby20%comparedwiththeshellFEmodel,sincenoaccountorbenefitistakenfromthespreadofplasticity.AllowingforpartialplastificationinClass3cross-sectionsusingalineartransitionfromtheplastictotheelasticbendingmomentcapacitythroughtheelastic-plasticmomentMep[29]increasesthefailureloadpredictiontoP=116.9kN.TheproposedmethodofdesignbyadvancedanalysisdefinesfailureoftheframeastheloadlevelatwhichtheCSMstrainlimitof1.35eyforamodeledcross-sectionwith󰀁kp=0.626isreached(whichoccursfirstatthebaseoftheright-handcolumn).ThisoccursataloadlevelofP=127.0kN,whichisonly6%shyoftheshellFEcollapseload.ItshouldbeemphasizedagainthatthestrainsconsideredintheaveragingprocedurearetheextremefibercompressiveflangestrainsofallelementsfullycontainedwithinLb.Thedescribedframeexampleillustratesboththeeaseofappli-cationoftheproposedapproachandtheenhancementsindesignefficiencythatcanbeachieved.5.ConclusionsBeamfiniteelementsarecommonlyusedtoanalyzesteelstruc-tures.Plasticanalysismethodsarelimitedtocompactcross-sectionsthatpossesssufficientrotationalcapacityforplastichingestodevelopandrotatetoformaplasticcollapsemechanism.Anewmethodofdesignbyadvancedanalysisispresent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ResearchHigh Performance Structures: Building Structures and Materials—Article基于高等分析的钢结构设计——材料建模与应变极限Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UKLeroy Gardner *，Xiang Yun，Andreas Fieber，Lorenzo Macorinia r t i c l e i n f oArticle history:Received 31 July 2018Revised 10 September 2018Accepted 12 November 2018Available online 26 February 2019摘要我们对于钢框架的结构分析通常通过梁单元来进行。然而，由于该类单元无法确切地捕捉钢材截面的局部屈曲行为，因此，传统的钢结构设计规范采用截面分类的概念来确定截面强度以及变形能力受材料局部屈曲影响的程度。而塑性设计方法的使用仅限于 1 级截面，其具有足够的转动能力以形成塑性铰并引发倒塌机制。在更高级截面中，局部屈曲阻止了具有这种转动能力的塑性铰的形成，除非出于计算需求而使用壳单元，否则我们需要对材料进行弹性分析。然而，本文证明了通过将连续强度法（CSM）及其应变极限纳入分析，可以在梁单元中有效地模拟局部屈曲。此外，通过进行几何非线性和材料非线性的高等分析，可确保无需进行额外的设计检查。如果采用适当而精确的应力 - 应变关系，我们在较粗截面中观察到的应变硬化所带来的积极影响亦可以得到有效应用；为此，我们在文详尽地描述了一个用于热轧钢的四元线性材料模型。对于一致的高等分析框架中任意细长比截面的分析问题，CSM 应变极限分析法均适用，同时还可以从荷载重新分配水平的优化中受益。本文所提出的方法可用于单个构件、连续梁单元及相关框架结构，并且在精度与一致性等方面与当前钢结构设计规范相比，本方法具有显著优势。

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关键词高等分析连续强度法局部屈曲材料建模应变极限1.引言在许多最新版本的国际结构设计规范中，几何非线性与材料非线性分析的应用均得到了批准，这些方法也被称作是高等分析，常用于由紧凑截面组成的钢结构设计。比如，EN 1993-1-1 [1]的第5节、AS 4100的附录D [2]以及AISC 360的附录1 [3]等。在常规操作中，我们主要使用有限元（FE）法来进行分析，并且通常使用梁单元来表示其结构构成。目前，高等分析设计所带来的益处被人们广泛认可[4−10]。与传统的结构设计方法相比，高等分析首先进行结构性分析，并辅以单个构件的分析与截面检查，通过对整体框架与单个构件结构不稳* Corresponding author. E-mail address: Leroy.gardner@imperial.ac.uk (L. Gardner) 定性（如P-Δ和P-δ）的捕捉，从而免去了对后面单个构件进行检查的需求。然而，由于梁单元无法对局部屈曲进行捕捉，因此塑性分析法目前仅限用于具有足够转动能力的较粗截面，以便形成塑性铰并引发倒塌机制。为了避免在整体框架分析中使用复杂的壳单元，结构设计规范通常通过引入基于板材宽厚比的离散截面等级以对截面的承载力和变形能力做出，并以此来解决局部屈曲问题。然而，在框架结构的承载能力的测算中，这种方法往往人为地添加了额外的“步骤”。比如，人们常假设1级截面具有无限转动能力，而2级截面的转动能力为0。在本文中，我们将提出一种更具一致性的设计方272Author name et al. / Engineering 2(2016) xxx–xxx案，即在用梁单元进行的高级结构分析中，利用应变极限来模拟局部屈曲。基于连续强度法（CSM）[11,12]的应变极限控制了结构内发生的塑性变形程度，同时其也受材料截面局部的细长比的影响。因此，我们可以采用相同的模式，通过梁单元来处理任意截面类型的结构问题。为了准确捕捉结构碳钢的应力-应变特性，我们开发了一种四元线性材料模型[13]，并将其纳入CSM分析框架中，用于不同荷载条件下的热轧钢截面的结构设计[14,15]。该材料模型即在此处发挥作用。在我们所提出的基于高等分析的设计方法中，结构阻力取决于分析过程中达到的荷载系数峰值中的较低者，或首先达到CSM应变极限的荷载系数。请注意参考文献[16]中给出的诸项建议，其连接通常必须单独检查。本文概述了通过高等分析将应变极限引入设计所需的研究框架。描述了热轧钢的四元线性材料模型，并通过实例说明了该方法的优点和普遍适用性。应当注意的是，下文的讨论仅限于单个构件、连续梁以及由热轧钢工字型截面组成的框架在主轴弯曲下的平面内行为。2.材料建模在准静态拉伸荷载作用下热轧结构碳钢的典型工程应力-应变曲线如图1所示。在弹性范围内，其关系是线性的，并且由弹性模量或杨氏模量E定义，根据EN 1993-1-1，E取值为210 000 N·mm–2。在达到屈服应力fy之后，在相应的应变条件εy = fy/E之下，应变沿屈服平台继续增加，而应力没有任何显著增长。而当应变达到应变硬化εsh状态时，应力开始随着应变的增加而再次开始增加。最终，应力在极限拉伸强度fu和相应的极限应图1. 热轧结构碳钢的典型工程应力-应变曲线。变εu处达到最大值，如若超过这一点，材料将会出现颈缩现象并最终断裂。结构碳钢的应力-应变关系通常通过弹塑性体模型来理想化，且该简化模型是构成EN 1993-1-1中现行设计规范的基础。然而，该模型无法捕获材料的应变硬化特征，而这对于诸如CSM的高等计算与设计方法，尤其是对于较粗截面的材料而言，具有重要意义。因此我们提出了四元线性材料模型，如图2所示并由方程（1）表述[13]。该模型已经经过基于拉伸试样测试结果的大数据集校准，可准确地表示热轧结构碳钢的屈服平台和应变硬化行为。在方程（1）中我们使用了C1与C2两个材料系数：其中，C1定义了一个“截止”应变，以免对材料强度做出过度预测，且如3.2节所述，该设定也被包含在了CSM的基础曲线当中；而C2则在方程（2）中决定了应变硬化斜率Esh。上述两个系数可由方程（3）与方程（4）确定，分别用应变硬化应变εsh和极限应变εu表示。（1）（2）（3）（4） εsh和εu的值可由方程（5）和方程（6）分别预测得到。因此，四元线性材料模型只需三个基础参数：E、fy和图2. 热轧结构碳钢的四元线性材料模型。Author name et al. / Engineering 2(2016) xxx–xxx273fu。近来该模型已被纳入CSM分析框架，用于热轧钢的（以及其他截面水平的各类型屈曲）。因此，这些现象往截面设计[14,15]，并显示出比EN 1993-1-1设计规范所要求的更高的精度。 （5） （6）应力-应变曲线的典型对比[17–19]与通过使用我们开发的四元线性材料模型所得结果[方程（1）]以及由方程（2）～（6）预测的结果，参见图3至图5。结果表明，该模型对于不同的钢种均兼容，其结果具有准确性与一致性。关于此问题，Yun和Gardner[13]在其研究中进行了进一步比较与讨论。3. 基于高等分析的设计3.1. 引言为解释钢结构中的复杂现象，我们通常应该考虑几何非线性以及材料非线性的问题。在传统意义上，我们必须进行适当的设计用以检查任何未包含在结构分析中的极限状态。例如，线性（一阶）分析不能捕获局部屈曲，因此我们需要额外进行相应的局部屈曲检查。而高等分析通过将种种极限状态纳入分析本身减少了所需的设计检查次数。理想情况下，我们可以消除所有的设计检查，并且结构系统的承载力仅由分析过程中所能达到的荷载系数峰值定义。然而，尽管在高等分析中通常会考虑到框架与局部构件的不稳定性（即P-Δ和P-δ）效应，但是仅通过梁单元无法对更复杂的现象进行捕捉，如局部屈曲图3. 数据曲线的比较。四元线性材料模型与由Yun等[17]测试的S235钢的应力-应变实验往被排除在分析之外，并大致通过截面分类以及相应的检查来对其进行考量。本研究提出的设计方法打破了这种局限，我们通过采用CSM应变极限，以模拟梁单元中的局部屈曲现象及其影响。因此，我们可以在包括所有四级截面的结构中运用梁单元进行相似的高等非线性分析，因为应变极限以合理的方式控制着局部屈曲的程度、塑性的扩展以及塑性变形。这与现行的设计规范中逐步处理的方式形成了鲜明对比，其中，在1级截面的情况下允许进行完全塑性再分配，而在其他级别的截面中，则无法进行再分配。3.2. 连续强度法与应变极限CSM是一种基于变形的设计方法，该方法将截面材料的细长比与变形能力联系起来[11]。CSM基础曲线并非将截面分为不同的离散组，而是定义了截面可以承受的应变峰值。利用该应变极限，将从所采用的材料模型上获得的相应应力分布（见第2节）在截面面积上进行图4. 应变实验数据曲线的比较。四元线性材料模型与由Chan和Cardner[18]测试的S355钢的应力-图5. 验数据曲线的比较。四元线性材料模型与由Wang等[19]测试的S460钢的应力-应变实274Author name et al. / Engineering 2(2016) xxx–xxx积分，即可得出截面承载力。在高等分析中引入CSM应变极限，可以使我们能够直接计算CSM截面的承载力，因为在分析的每个荷载增量处都要进行数值积分运算。如图6所示，CSM基础曲线给出了截面细长比λp与其变形能力之间的连续关系。后者以标准化形式被定义为εcsm/εy，其中εcsm是截面可以承受的最大压缩应变，εy是屈服应变。该基础曲线被分为两部分：方程（7）适用于λp ≤ 0.68的非细长截面，在EN 1993-1-1中被称为1~3级截面；方程（8）适用于λp > 0.68的细长截面，在EN 1993-1-1中被称为4级截面。请注意，在方程（7）中，我们预测的CSM应变比εcsm/εy有两个上限，第一个上限Ω定义了塑性变形的水平，对于建议值为15的给定项目在极限状态下被认为是可容忍的；第二个极限是C1εu，其中C1是第2节中所描述的材料系数，εu是可以使用方程（6）确定的材料极限应变。另外，我们定义了一个“截止”应变，以避免对材料强度做出过度预测。还需注意的是，方程（7）与方程（8）均需经过一个确定的临界点，即图6中的（0.68, 1），从而确保CSM基础曲线对于非细长截面与细长截面之间的兼容性。（ 7） （8）应用于高等分析的CSM应变极限的准确度取决于准确确定截面细长比的能力。截面细长比是一个无量纲参数，可对局部不稳定性的敏感性进行量化，并由方程（9）定义，式中fy是屈服应力，σcr是弹性临界屈曲应力。 （ 9）图6. CSM基础曲线：截面细长比λp与其变形能力εcsm/εy之间的连续关系。有多种方法可用于计算弹性临界屈曲应力。标准板材理论易于使用，利用了诸如EN 1993-1-5 [20]中所提出的屈曲系数k，但通过对沿相邻板材边缘的简支边界条件进行假设，往往会忽略任意单元间的相互作用。弹性临界屈曲应力可通过方程（10）算得，其中t和b是相关板材的厚度与宽度，v是泊松比；然后我们使用具有最低屈曲应力的板材单元通过方程（9）来确定截面的细长比。 （10）或者，我们还可以通过经数值结果[21,22]校准的近似表达式来确定整个截面的弹性临界屈曲应力。这些表达式含蓄地解释了板材的相互作用，同时也提高了其准确度。然而，表达式[21]仅仅包括了纯压缩与纯主轴/短轴弯曲的情况。在结构框架中，截面往往要承受压弯组合作用，而参考文献[22]中提出的表达式则解决了这一，并对压缩、弯曲以及压弯组合影响等不同情况均兼容。约束和无约束有限条法（CUFSM）[23]等数值工具也可用于整个截面的弹性临界屈曲应力的测算；在本研究中我们采用CUFSM法。3.3. 应变平均法实验[24,25]表明，在弯矩梯度作用下，单个构件的最大面内抗弯承载力大于均匀弯矩情况下的对应值。该抗弯承载力的增加源于低应力相邻区域对临界截面（即在最大弯矩情况下）所施加的约束，从而，延迟了弹性与塑性状态下局部屈曲的发生。在大多数的实际应用中，弯矩梯度当然存在，但EN 1993-1-1设计规范以及CSM基础曲线均没有考虑局部的弯矩梯度对截面局部稳定性的影响。我们建议利用弯矩梯度对局部稳定性产生的有利影响，将CSM应变极限应用于提取自高等分析且在构件有限长度上均匀分布的应变中，而不是按照惯例简单地考虑最高应力（或应变）的截面。由于局部屈曲需要在构件的有限长度上形成——也就是说局部屈曲的半波长Lb——这代表了一个可将应变均分的合适距离。这种方法与Lay和Galambos [26]的观察结果一致，他们通过实验证明，只有当屈服已延伸到与局部屈曲半波长相关的构件的有限长度时，截面才会开始出现非弹性局部屈曲。因此，通过将CSM应变极限应用于平均应变，可直接计算局部弯矩梯度的大小；在均匀弯矩下，可用平均法预测截面承载力，而对于增大的弯矩梯度，弯矩的承Author name et al. / Engineering 2(2016) xxx–xxx275载力峰值也相应增大。应变平均法也降低了强度预测对FE模型中网格密度的敏感性。释3级截面中的部分屈服。对于4级截面，经由应变极限预测的承载力与欧洲设计规范3的结果相似，但避免了与有效宽度法相关的繁琐计算过程。由于弯矩梯度的积极影响，从等弯矩情况到悬臂梁弯矩情况，壳体FE模型在截面细长比范围内的抗弯承载力增加了5%~10%。同样，我们所提出的使用梁单元进行的高等分析也准确预测了由于弯矩梯度而导致的2%~7%的增长，此结果是通过应变平均法计算而得的。EN 1993-1-1没有捕捉到这一效应，也没有对这两种情况做出区分，而是仅仅预测了每个截面的单一抗弯承载力。4.3. 连续梁单元在孤立的构件上无法进行弯矩的再分配。然而在不确定的结构当中，如连续梁单元，我们提出的方法所带来的益处可能会更加显著。除去局部截面强度和弯矩梯度的影响，不确定系统的极限荷载取决于非弹性力以及弯矩重分布的水平。图8显示了倒塌时的极限荷载系数αu，该系数是由第一次屈服时的荷载系数αel标准化而得，而αel又是由荷载系数α乘以点荷载P条件下的系列连续梁单元中的壳体FE模型而求得。此外，我们结合高等分析以及欧洲设计规范3对所提出的方法进行了承载力预测。EN 1993-1-1只允许使用塑性分析方法，对于1级截面，该方法能够有效捕捉材料屈服时的力与弯矩重分布。然而，对于2级截面则必须进行弹性分析，这样即使在达到塑性弯矩承载力之前出现一定程度屈服，2级截面也不会产生弯矩重分布。在此基础上的阻力预测过程显然不能反映壳体FE的结果，而是对现实的过度简4. 应用4.1. 一般情况在本节中，我们所提出的高等分析将被应用于一系列的构件以及连续的梁单元和框架结构中，以说明CSM应变极限是如何模拟局部屈曲、控制力和弯矩的非弹性再分配水平，以及捕捉弯矩梯度所带来的积极影响。为方便比较，本节中还展现了基于EN 1993-1-1设计规范的承载力预测方法。我们在各种情况下都使用了具有局部几何缺陷的校准壳体有限元模型来提供基准结果。该壳体模型利用Abaqus软件开发[27]，并通过改进的Riks方法进行求解[28]。在此模拟中，我们引入了在第2节中描述的四元线性材料模型。4.2. 梁单元在四点弯曲和悬臂梁作用下，我们对具有不同局部细长比截面的一系列梁单元在临界截面处的标准化抗弯承载力的分析如图7所示。其中Mmax是最大弯矩值，Mpl是塑性抗弯承载力，而Mel是弹性抗弯承载力。可以看出，将CSM应变极限应用于梁单元模型，可以在截面细长比范围内对壳体FE模型的结果进行安全侧预测。根据EN 1993-1-1的设计规范，在2级至3级截面的抗弯承载力之间存在人为额外添加的“步骤”，而其可以利用经改进的弹塑性抗弯承载力来消除[29]，这恰好可以解图7. 我们所提出的壳体FE模型的标准化抗弯承载力预测方法（即具有应变极限的梁FE模型）和EN1993-1-1设计规范中不同截面细长比的均匀弯矩及弯矩梯度下的梁单元预测法。EC3: EN 1993-1-1。276Author name et al. / Engineering 2(2016) xxx–xxx单化表示。在一致的高等分析方法应用中，CSM应变极限定义了允许进行再分布的截面相关水平，这保证了我们所提供的荷载预测的准确性。根据EN 1993-1-1，46个连续梁单元的平均荷载预测值为壳体FE预测结果的0.810，变异系数（COV）为5.3%。相比之下，由我们所提出的高等分析设计法得到的平均荷载预测值为壳体FE预测结果的0.924，COV为4.9%。4.4. 框架根据Avery和Mahendran[30]的报告，在对框架水平的结构性能进行考量时，我们对“测试框架2”进行了略微修改，以说明前文所提出的设计方法的全部优点。单跨门式钢架由一个3级截面构成。与原试验框架相比，我们将材料模型调整为前文所述的四元线性材料模型，其弹性模量为E = 200 000 MPa，屈服应力fy = 360 MPa，极限应力fu = 510 MPa，这与“测试框架2”的翼缘材料特性一致。根据EN 1993-1-1设计规范，框架存在整体摆动缺陷。我们对构件缺陷在最不利方向上取值为l/1000，其中l是构件长度。P-Δ以及P-δ效应亦均被直接纳入高等分析的考量中。单跨门式钢架的壳体和梁单元FE模型的荷载与标准化应变（其临界截面均值超过Lb）响应如图9所示。测得的框架极限荷载为P = 135.0 kN。在壳体FE模型中，我们在局部屈曲之前于右柱底部观察到屈服现象，最终，该框架在P = 133.9 kN的荷载下倒塌。相比之下，梁单元的FE模型无法捕捉柱体底部的局部屈曲，并在允许无限应变的条件下最终在峰值荷载P = 156.0 kN时出现整体摇摆并最终被破坏——这显然是一种不安全的解决方案，因为壳体FE模型的承载力被明显过度预测多达16%。为确保安全的承载力预测，EN 1993-1-1要求对截面与构件进行适当的设计检查，然后再对结构进行弹性分析；预计结构在荷载P = 108.9 kN时会发生破坏。与利用壳体FE模型进行的预测相比，我们将预测值调低20%，因为在预测中我们没有考虑塑性扩展的影响或益处。如若考虑3级截面的局部塑性，由于通过弹塑性弯矩Mep [29]可以对从塑性到弹性弯矩承载力的线性过渡进行有效利用，因此我们将结构破坏时施加的荷载预测值增加到P = 116.9 kN。我们提出的基于高等分析的设计方法将框架的破坏定义为：对于细长比λp = 0.626的模型截面，当CSM应变极限为1.35εy时，会引发框架倒塌（首先发生在右侧柱的底部）。这种情况发生在荷载P=127.0 kN时，且仅比壳体FE模型预测的极限荷载低6%。应再次强调的是，我们考虑的应变是均值低于Lb的所有单元的翼缘最外层纤维的压缩应变。所述框架示例说明了我们所提出方法的易用性以及增强的设计效率。5. 结论梁单元FE模型常用于钢结构分析。塑性分析方法仅限于具有足够旋转能力的致密截面，以便形成塑性铰并旋转引发塑性倒塌机制。本文提出了一种新的基于高等分析的设计方法，通过应用经CSM法获得的与截面相关图8. 我们所提出的壳体FE模型的标准化抗弯承载力预测法（即具有应变极限的梁FE模型）和 EN 1993-1-1设计规范中具有中跨点荷载和变截面细长比的两跨连续梁的梁单元预测法。Author name et al. / Engineering 2(2016) xxx–xxx277图9. 3级截面的框架标准化荷载-应变响应。P：框架的荷载；Mep：弹塑性弯矩。的应变极限来模拟局部屈曲。该方法能够捕获局部弯矩梯度的积极影响并且预测力与弯矩重分布的实际水平。结果表明，我们所提出的方法比现行的欧洲设计规范3更为准确，因此，通过高等分析法，我们在提高钢结构设计的准确性与效率的道路上迈出了重要一步。Compliance with ethics guidelinesLeroy Gardner, Xiang Yun, Andreas Fieber, and Loren-zo Macorini declare that they have no conflict of interest or financial conflicts to disclose.References[1] EN 1993-1-1: Eurocode 3—Design of steel structures—Part 1-1: General rules and rules for buildings. 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