屈服线理论.docx
- 文档编号:5376732
- 上传时间:2022-12-15
- 格式:DOCX
- 页数:16
- 大小:326.60KB
屈服线理论.docx
《屈服线理论.docx》由会员分享,可在线阅读,更多相关《屈服线理论.docx(16页珍藏版)》请在冰豆网上搜索。
屈服线理论
8YieldLineTheory
8.1Theessentials
YieldLineDesignisawell-foundedmethodofdesigningreinforcedconcreteslabs,andsimilartypesofelements.ItusesYieldLineTheorytoinvestigatefailuremechanismsattheultimatelimitstate.Thetheoryisbasedontheprinciplethat:
workdoneinyieldlinesrotating=workdoneinloadsmoving
YieldLineDesigniseasytograspbutthereareseveralfundamentalprinciplesthatneedtobeunderstood.YieldLineDesignisaplasticmethod:
itisdifferentfrom‘normal’elasticmethods.
8.1.1AshorthistoryofYieldLineTheory
YieldLineTheoryasweknowittodaywaspioneeredinthe1940sbytheDanishengineerandresearcherKWJohansen.
Asearlyas1922,theRussian,AIngerslevpresentedapapertotheInstitutionofStructuralEngineersinLondononthecollapsemodesofrectangularslabs.AuthorssuchasRHWood,LLJones,ASawczukandTJaeger,RPark,KOKemp,CTMorley,MKwiecinskiandmanyothers,consolidatedandextendedJohansen’soriginalworksothatnowthevalidityofthetheoryiswellestablishedmakingYieldLineTheoryaformidableinternationaldesigntool.
Inthe1960s70sand80sasignificantamountoftheoreticalworkontheapplicationofYieldLineTheorytoslabsandslab-beamstructureswascarriedoutaroundtheworldandwaswidelyreported.
Tosupportthiswork,extensivetestingwasundertakentoprovethevalidityofthetheory.ExcellentagreementwasobtainedbetweenthetheoreticalandexperimentalYieldLinepatternsandtheultimateloads.
Thedifferencesbetweenthetheoryandtestsweresmallandmainlyontheconservativeside.Inthetestswhererestraintwasintroducedtosimulatecontinuousconstruction,theultimateloadsreachedatfailureweresignificantlygreaterthantheloadspredictedbythetheoryduetothebeneficialeffectofmembraneforces.
8.1.2Whatisayieldline?
Ayieldlineisacrackinareinforcedconcreteslabacrosswhichthereinforcingbarshaveyieldedandalongwhichplasticrotationoccurs.
8.1.3WhatisYieldLineTheory?
YieldLineTheoryisanultimateloadanalysis.Itestablisheseitherthemomentsinanelement(e.g.aloadedslab)atthepointoffailureortheloadatwhichanelementwillfail.Itmaybeappliedtomanytypesofslab,bothwithandwithoutbeams.ConsiderthecaseofasquareslabsimplysupportedonfoursidesasillustratedbyFigure8.1.1.Thisslabissubjectedtoauniformlydistributedload,whichgraduallyincreasesuntilcollapseoccurs.
Initially,atserviceload,theresponseoftheslabiselasticwiththemaximumsteelstressanddeflectionoccurringatthecentreoftheslab.Atthisstage,itispossiblethatsomehairlinecrackingwilloccuronthesoffitwheretheflexuraltensilecapacityoftheconcretehasbeenexceededatmidspan.
Figure8.1.1Onsetofyieldingofbottomreinforcementatpointofmaximumdeflectioninasimplysupportedtwo-wayslab
Increasingtheloadhastenstheformationofthesehairlinecracks,Increasingtheloadfurtherwillincreasethesizeofthecracksfurtherandinduceyieldingofthereinforcement,initiatingtheformationoflargecracksemanatingfromthepointofmaximumdeflection.Onincreasingtheloadyetfurther,thesecracksmigratetothefreeedgesoftheslabatwhichtimeallthetensilereinforcementpassingthroughayieldlineyields.
Atthisultimatelimitstate,theslabfails.AsillustratedbyFigure8.1.2,theslabisdividedintorigidplaneregionsA,B,CandD.Yieldlinesformtheboundariesbetweentherigidregions,andtheseregions,ineffect,rotateabouttheyieldlines.Theregionsalsopivotabouttheiraxesofrotation,whichusuallyliealonglinesofsupport,causingsupportedloadstomove.Itisatthisjuncturethattheworkdissipatedbythehingesintheyieldlinesrotatingisequatedtoworkexpendedbyloadsontheregionsmoving.ThisisYieldLineTheory.
Underthistheory,elasticdeformationsareignored;allthedeformationsareassumedtobeconcentratedintheyieldlinesand,forconvenience,themaximumdeformationisgiventhevalueofunity.
ForamoredetailedappraisalofthesituationatcornerseeSection?
Figure8.1.2Theformationofamechanisminasimplysupportedtwo-wayslabwiththebottomsteelhavingyieldedalongtheyieldlines
8.1.4Whatisayieldlinepattern?
Whenaslabisloadedtofailure,yieldlinesforminthemosthighlystressedareasandthesedevelopintocontinuousplastichinges.Asdescribedabove,theseplastichingesdevelopintoamechanismformingayieldlinepattern.
Yieldlinesdividetheslabupintoindividualregions,whichpivotabouttheiraxesofrotation.YieldlinesandaxesofrotationconformtorulesgiveninTable8.1.1,whichhelpwiththeidentificationofvalidpatternsandtheYieldLinesolution.
Table8.1.1Rulesforyieldlinepatterns
•Axesofrotationgenerallyliealonglinesofsupportandpassalongsideanycolumns.
•Yieldlinesarestraight.
•Yieldlinesbetweenadjacentrigidregionsmustpassthroughthepointofintersectionoftheaxesofrotationofthoseregions.
•Yieldlinesmustendataslabboundary.
•Continuoussupportsrepelandsimplesupportsattractpositiveorsaggingyieldlines.
8.1.5WhatisaYieldLinesolution?
Intheory,theremaybeseveralpossiblevalidyieldlinepatternsthatcouldapplytoaparticularconfigurationofaslabandloading.However,thereisoneyieldlinepatternthatgivesthehighestmomentsorleastloadatfailure.Thisisknownastheyieldlinesolution.
Thedesignerhasseveralwaysofdeterminingthecriticalpatternandensuringsafe
design:
•Fromfirstprinciples,e.g.byusingTheWorkMethod
•Usingformulaeforstandardsituations.
Itwillbenotedthatvalidyieldlinepatternsgiveresultsthatareeithercorrectortheoreticallyunsafe.These‘upperboundsolutions’candetersomedesignersbut,asdiscussedlater,thistheoreticalawkwardnessiseasilyovercomebytestingdifferentpatternsandbymakingsuitableallowances(see10%rulelater).
8.1.6Howdoyouselectrelevantyieldlinepatterns?
Ayieldlinepatternisderivedmainlyfromthepositionoftheaxesofrotation,(i.e.thelinesofsupport)andbyensuringthattheyieldlinesthemselvesarestraight,gothroughtheintersectionofaxesofrotationandendattheslabboundary,i.e.conformtotherulesinTable8.1.1.SomesimpleexamplesareshowninFigure8.1.3Consideringaslabtobeapieceofpastrylaidoversupportsmayhelpdesignerstovisualiseappropriateyieldlinepatterns.
Figure8.1.3SimpleYieldLinepatterns
Theaimofinvestigatingyieldlinepatternsistofindtheonepatternthatgivesthecriticalmoment(thehighestmomentortheleastloadcapacity).However,anexhaustivesearchisrarelynecessaryandselectingafewsimpleandobviouspatternsisgenerallysufficientastheirsolutionsarewithinafewpercentoftheperfectlycorrectsolution.
8.1.7Whatisafanmechanism?
Slabssubjectedtoheavyconcentratedloadsmayfailbyaso-calledfanmechanism,withpositiveYieldLinesradiatingfromtheloadandanegativecircularYieldLinecentredunderthepointload.ThismechanismisshowninFigure8.1.4.Itisrareforthisformoffailuretobecriticalbutnonethelessacheckisadvisedwherelargeconcentratedloadsarepresentorforinstanceinflatslabswheretheslabissupportedoncolumns.
Figure8.1.4Fancollapsepatternforaheavyconcentratedloadontoareinforcedslab
Themechanismforaslabsupportedbyacolumnisthesameshapebutwiththepositiveandnegativeyieldlinesreversed.
8.1.8WhatistheWorkMethod?
TheWorkMethod(orvirtualWorkMethod)ofanalysisisthemostpopular(andmosteasy)wayofapplyingYieldLinetheoryfromfirstprinciples.Indeed,manyexperiencedusersofYieldLinetheoryofdesignchoosetousetheWorkmethodbecauseitissoveryeasy.Thefundamentalprincipleisthatworkdoneinternallyandexternallymustbalance.
Inotherwords,atfailure,theexpenditureofexternalenergyinducedbytheloadontheslabmustbeequaltotheinternalenergydissipatedwithintheyieldlines.Inotherwords:
Externalenergy
=
Internalenergy
byloadsmoving
=
dissipatedbyrotationsaboutyieldlines
Expended
=
Dissipated
E
=
D
Σ(Νxδ)forallregions
=
Σ(mxlxθ)forallregions
where
N=load(s)actingwithinaparticularregion[kN]
δ=theverticaldisplacementoftheload(s)Noneachregionexpressedasafractionofunity[m]
m=themomentinormomentofresistanceoftheslabpermetrerun[kNm/m]
l=thelengthofyieldlineoritsprojectedlengthontotheaxisofrotationforthatregion[m]
θ=therotationoftheregionaboutitsaxisofrotation[m/m]Bywayofillustration,considertheslabshowninFigure8.1.2.Figure8.1.5showsanaxonometricviewofthistwo-waysimplysupportedslabthathasfailedduetoauniformlydistributedload.Notethat:
•ThetriangularregionsA,B,CandDhaveallrotatedabouttheirlinesofsupport.
•Theloadsontheregionshavemovedverticallyandrotationhastakenplaceabouttheyieldlinesandsupports.
•Theuniformlydistributedloadoneachoftheseregionswillhavemovedonaverage1/3ofthemaximumdeflection.
Therotationoftheregionsabouttheyieldlinescanberesolvedintorotationabouttheprincipalaxesofrotation,andtherebymeasuredwithrespecttothelocationandsizeofthemaximumdeflection.
Figure8.1.5Deformedshapeatfailure
This,fundamentally,isthe‘WorkMethod’.AnyslabcanbeanalysedbyusingtheprincipleofE=D.Somejudgementisrequiredtovisualiseandchecklikelyfailurepatternsbutabsoluteaccuracyisrarelynecessaryandallow
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 屈服 理论
![提示](https://static.bdocx.com/images/bang_tan.gif)