Contents

Preface

Chapter1.DesignatServiceabilityLimitState(SLS)1.1.Nomenclature1.2.Bendingbehaviorofreinforcedconcretebeams–qualitativeanalysis1.3.Backgroundontheconceptoflimitlaws1.4.LimitlawsforsteelandconcreteatServiceabilityLimitState1.5.Pivotsnotionandequivalentstressdiagram1.6.Dimensionlesscoefficients1.7.Equilibriumandresolutionmethodology1.8.CaseofpivotAforarectangularsection1.9.CaseofpivotBforarectangularsection1.10.Examples–bendingofreinforcedconcretebeamswithrectangularcross-section1.11.ReinforcedconcretebeamswithT-cross-section

Chapter2.VerificationatServiceabilityLimitState(SLS)2.1.Verificationofagivencross-section–controldesign2.2.Cross-sectionwithcontinuouslyvaryingdepth2.3.Composedbendingwithcombinedaxialforces2.4.DeflectionatServiceabilityLimitState

Chapter3.ConceptsfortheDesignatUltimateLimitState(ULS)3.1.Introductiontoultimatelimitstate3.2.Postfailureanalysis

3.3.Constitutivelawsforsteelandconcrete

Chapter4.Bending-CurvatureatUltimateLimitState(ULS)4.1.Onthebilinearapproximationofthemoment-curvaturerelationshipofreinforcedconcretebeams4.2.Postfailureofreinforcedconcretebeamswiththeinitialbilinearmoment-curvatureconstitutivelaw4.3.Bendingmoment-curvaturerelationshipforbucklingandpostbucklingofreinforcedconcretecolumns

Appendix1:Cardano’sMethodA1.1.IntroductionA1.2.Rootsofacubicfunction–methodofresolutionA1.3.Rootsofacubicfunction–synthesisA1.4.Rootsofaquarticfunction–principleofresolution

Appendix2:SteelReinforcementTable

Bibliography

Index

Firstpublished2013inGreatBritainandtheUnitedStatesbyISTELtdandJohnWiley&Sons,Inc.

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Preface

The authors have written two books on the theoretical and practical design of reinforcedconcrete beams, columns and frame structures. This book, entitled Reinforced ConcreteBeams,ColumnsandFrames–MechanicsandDesign,dealswiththefundamentalaspectsofthemechanicsanddesignofreinforcedconcreteingeneral,bothrelatedtotheServiceabilityLimitState(SLS)andtheUltimateLimitState(ULS).Therelatedbook,entitledReinforcedConcreteBeams,ColumnsandFrames–SectionandSlenderMemberAnalysis,dealswithmoreadvancedULSaspects,alongwithinstabilityandsecond-orderanalysisaspects.Thetwobooks are complementary, and, indeed, could have been presented together in one book.However,forpracticalreasons,ithasprovedmoreconvenienttopresentthematerialintwoseparatebookswiththesameprefaceinbothtitles.The books are based on an analytical approach for designing usual reinforced concrete

structuralelements,compatiblewithmostinternationaldesignrules,includingforinstancetheEuropeandesignrulesEurocode2 forreinforcedconcretestructures.Thepresentationshavetried to distinguish between what belongs to the philosophy of structural design of suchstructural elements (related to strength ofmaterials arguments) and the design rules aspectsassociatedwithspecificcharacteristicdata(for thematerialor the loadingparameters).TheEurocode2designrulesareusedinmostoftheexamplesofapplicationsinthebooks.Evenso, older international rules, as well as national rules such as the old French rules BAEL(“Béton Armé aux Etats Limites”, or Reinforced Concrete Limit State in English) willsometimesbementioned,atleastforhistoricalreasons.Whatever the design rules considered, the fundamental concept of Limit State will be

detailed,andmorespecifically,theServiceabilityLimitState(SLS)andUltimateLimitState(ULS)bothinbendingandincompressionwillbeinvestigated.The books are devoted mainly to the bending (flexural) behavior of reinforced concrete

elements,includinggeometricalnonlineareffects(secondbook).However,twomajoraspectsofreinforcedconcretedesignarenottreated.Theseareshearforceeffectsandthecalculationof crack width as dealt with in theCrackOpening Limit State inEurocode 2. The latterrepresentsamajornewcontributionascomparedtosomeolderEuropeanrulessuchasBAEL.ThereadersarereferredtotheverygoodmonographsdevotedtothegeneralpresentationofEurocode2fortheseadditionalparts(seeforinstance[CAL05];[DES05];[MOS07];[EUR08];[PAI09];[PER09];[ROU09a];[ROU09b];[THO09];[PER10];[SIE10];[PAU11]).Wewould also like topoint out that the calculationof crackwidths, evenunder a simple

loading configuration, such as uniform tension loading, still remains a difficult topic.Moreover, theauthorsareevenconvinced thatmeaningfuleffortsshouldbeaddressed in thefuture,forfacilitatingthetransferofknowledgefromtheoreticalresearchinfractureordamagemechanics,toapplied,practicaldesignrules.Inconnectionwiththis,cohesivecrackmodelswere introduced in the1970s to investigate thecrackopening inmodeIof failure[HIL76],

whereas non-local damage mechanics models were developed in the 1980s for efficientcomputationsofdamagesofteningmaterials[PIJ87].Bothappeartobelongtothefamiliesofnonlocalmodelswhich contain an internal length, for the control of the postfailure process[PLA93].Non-localdamagemechanicsisnowwidelyusedintheresearchcommunityforthestudyofreinforcedconcretestructures(seeforinstance[BAZ03];[MAZ09]).Theauthorsofthesebookshavealsoconductedsomeresearchinthisfieldtobetterunderstandthefailureofsomesimplereinforcedconcretestructuralelements(researchatINSAofRennes,UniversityofRennesI,UniversityofSouthBrittanyorUniversityofOslo–seeforinstance[CHA05];[CHA06];[CHA07];[CHA08];[CHA09];[CHA10];[CHA11];[CHA12]).However,theengineeringcommunityhasnotyetnecessarilyintegratedtheseresultsintothedesignprocessorevenintotherules.Thegapbetweentheresearchactivityandtheengineeringmethodologyisprobablytoolargeatpresent,andresearcherswillprobablyhavesomeresponsibilityinthefuturetomaketheirresultsmoretractabletotheengineeringcommunity.Withrespecttothesebooks,someverysimpleconceptsofnon-localmechanicswillbepresentedwhennecessary.However, thebooksaremainlydevoted to thedesignofa reinforcedconcretestructureatagiven limit state, the cracking evolution problem often being considered as a secondaryproblem.Wehavechosentoconcentrateoureffortsonthebendingdesignbasedonthepivotconcept,atboththeServiceabilityLimitState(SLS)andtheUltimateLimitState(ULS).Thelastpartofthesecondbookdealswiththedesignofcolumnsagainstbuckling,andhowtotakeinto account second-order effectswill be presented for stability design. In particular, someengineeringapproachespracticedbyengineerswillbedetailed, to replaceefficiently,whenpossible,thenonlinearevolutionproblemassociatedwithmicrocrackingandfailure.Thebooks are aimedat bothundergraduate andgraduate (Licence andmaster) students in

civilengineering,engineersandteachersinthefieldofreinforcedconcretedesign.Inaddition,researchers and PhD students can find something of interest in the books, including thepresentationonelementaryapplicationsofnon-localdamageorplasticitymechanicsappliedto theultimatebendingof reinforced concretebeams (and columns).Wehope that thebasicideas presented in the books can contribute to stimulating the links between the researchcommunity in this field (computational modeling and structural analysis) and the designcommunity with practical structural cases. The principles of Limit State design will beintroducedanddevelopedfirst,bothat theServiceabilityLimitState(SLS)andtheUltimateLimit State (ULS), illustrated by some detailed examples to illustrate the introducedmethodology.Olderbooks(seeforinstance[HOG51];[BAK56];[SAR68];[ROB74];[PAR75];[FUE

78];[LEO78];[ALB81];[LEN81];[BAI83];[GYO88];[WAL90];[PAU92];[MAC97])have been used in some section of the books for establishing familiar and well-knownequationsonsectiondesign(inparticularequationsbasedonthesimplifiedrectangularstress-straindiagramforconcreteincompression).Inparticular,theauthorswanttoacknowledgethevery exhaustive work of Professor Robinson, at Ecole Nationale des Ponts et Chaussées,whosereinforcedconcreteteachingbookpublishedin1974canstillbeconsideredasamainreferencewithmoderninsightsintoreinforcedconcretedesign[ROB74].Wehavealsobeen

inspiredbythemorerecentandveryexhaustiveworksofProfessorThonier(seeforinstance[THO09]),alsoatEcoleNationaledesPontsetChaussées.The current book, Reinforced Concrete Beams, Columns and Frames – Mechanics and

Design,isorganizedasfollows.Chapters1and2dealwiththeServiceabilityLimitState,forboththedesignandcross-sectionverification.TheFrenchschoolofreinforcedconcretedesignhavecommonlyusedtheconceptof“Pivot”,whichisrelatedtothelimitbehaviorofthecross-sectionwith respect to steel and concretematerial characteristics. The Pivot A (where thesteelmaterialcharacteristicscontrolthebehaviorofthecross-sectionattheLimitState),andPivotB(wheretheconcretematerialcharacteristicscontrolthebehaviorofthecross-sectionat theLimitState) concepts are introducedwith theServiceabilityLimitState inChapter 1.Chapter 1 is mainly focused on the design aspects, whereas Chapter 2 deals with theverification of the reinforced concrete sectionwith both the bending and the normal forceseffects.Thegeneraltheorypresentedinthesefirsttwochaptersisvalidforarbitraryshapesofreinforcedconcretecross-sectionsincludingforinstancerectangular,triangular,trapezoidalorT-cross-sections. Chapter 2 ends with the presentation of a cubic equation for thedetermination of the neutral axis in the general loading configuration, including the normalforce effects. This elegant equation is also known as the cubic equation of the Frenchreinforced concrete design rules dating from 1906 (“Circulaire du 20Octobre 1906”) (andreportedinthebookbyMagny,1914[MAG14])orthosedatingfrom1934(“Règlementsdesmarchésdel’étatde1934”–alsoinFrench),alsorecentlyreportedbyProfessorThonierforT-cross-sections[THO09].Finally, thetensionstiffeningphenomenonis introducedintermsofanonlinearbendingmoment-curvatureconstitutivelawandsomeverificationexamplesaregiventoillustratethetheoreticalresultsobtainedinthefundamentalparts.Chapters 3 and 4 focus on the fundamental aspects of theUltimateLimit State.Chapter 3

startswithabriefintroductiontotheconceptoftheUltimateLimitStateforthebendingofareinforcedconcretebeam.Theneedtousesomenon-localtheorytocorrectlymodelthepost-failurebehaviorofreinforcedconcretestructuralelementsisshowninthepresenceofglobalcurvature softening. The material characteristics of the steel and concrete allowed byEurocode2arelisted,andcomparedwitheachother.Itispossibletoderiveanalyticallythenormalforcesandtheresultantbendingmomentinthecompressionblockforeachconsideredconcrete law, including the parabolic-rectangle constitutive law, the simplified rectangularconstitutive law, the bilinear constitutive law or Sargin’s nonlinear constitutive law. ThesepreliminarieswillbeusedlaterforthedesignofreinforcedconcretesectionsatUltimateLimitState. Chapter 4 discusses some possible bending moments – curvature law of typicalreinforcedconcretesections.Thesecross-sectionalbehaviorscanbededucedfromthelocalcharacteristics of the steel and concrete constituents. The relevancy of a bilinearapproximationforthemoment-curvatureconstitutivelawisdiscussed,withpossibletractableanalyticalresultsforengineeringpurposes.Chapter4concludeswithsomebucklingandpost-buckling results obtained for a reinforced concrete column modeled with a simplifiednonlinear bending-curvature constitutive law. It is shown that reinforced concrete columnstypicallybehavelikeimperfection-sensitivestructuralsystems.

Therelatedbook,ReinforcedConcreteBeams,ColumnsandFrames–SectionandSlenderMember Analysis, is organized as follows. The advanced design of general reinforcedconcretesectionsistreatedinChapter1.Thereinforcedconcretesectioncanbeoptimizedfora given loading (in terms of minimization of the steel quantity for instance), with someconstrainedequations.AlsodiscussedishowtheServiceabilityandtheUltimateLimitStatescanbecompared,dependingonthematerialandloadingfeaturesoftheproblem.Adesignofthe cross-section in biaxial bending is also proposed. More generally in this chapter, thereinforced concrete section is designed for various constitutive laws for the behavior ofconcrete and steel, includingpossible steelhardening,withpossible analytical solutions forthe optimized design. Some design examples are included for the various solicitationsincludingsimplebending,bendingcombinedwithnormalforcesorbi-axialbending.Thelastpart of Chapter 1 discusses the possible use ofmoment-normal forces interaction diagramsavailable in international codes, and some new possible improvements of these simplifieddiagrams.Chapter 2 is devoted to general aspects of instability of second-order effects in slender

compression members, and in frames that include such members. For such cases, it isnecessary to consider second-order load effects in the analysis anddesign.The concepts ofbraced,unbracedandpartiallybracedsystemsaswellasassociatedmomentformulationsarepresented,andtheusefuldistinctionbetweenlocalandglobalsecond-ordereffectsdiscussed.Thegeneralprinciplesofanalysisanddesignof individual reinforcedconcretecolumnsandframesystemsarereviewedinordertoprovideageneralunderstandingoftheproblemarea.This includes a presentation and discussion of fundamental concepts and theory behindapproximateanalysisanddesignmethodstoprovideareasonablecompletebasisforrelevantanalysisanddesignrequirementsasgiveninexistingdesignrules,suchasinEurocode2.Thisalso includes a discussion of the applicability of equivalent elastic analysis as anapproximation to nonlinear analyses (accounting for both material and geometric nonlineareffects).Local andglobal slenderness limits, allowing second-ordereffects tobeneglected,are presented and discussed. Chapter 3 deals with approximate analysis methods used forefficientandpracticalelasticstabilitycalculations,andsecond-orderelasticswayandmomentcalculations.Includedinthischapteraredifferentmethodsforcomputingeffectivelengths,andmethodsemployingthewidelyusedeffectivelengthconceptinframeanalysis.Basicconceptsareexplainedandsimpleandmorecomplexengineeringexamplesareincludedtoprovideabetterunderstandingofthemethods.This book and the first chapter of the related book were mainly written by Charles

Casandjian, Noël Challamel and Christophe Lanos, whereas Jostein Hellesland mostlycontributedtothefinaltwochaptersofReinforcedConcreteBeams,ColumnsandFrames–SectionandSlenderMemberAnalysis.Finally, an appendix is provided that gives further developments on the theoretical

background of Cardano’s method, useful for the resolution of a cubic equation, oftenencountered in the designing of reinforced concrete sections at both Serviceability andUltimateLimitStates.Anappendixgivingatableofsteeldiameters isalsoprovidedfor the

quickandefficientselectionofreinforcementsizesindesigncalculations.

CharlesCASANDJIAN,NoëlCHALLAMEL,

ChristopheLANOSandJosteinHELLESLAND

December2012

Chapter1

DesignatServiceabilityLimitState(SLS)

1.1.Nomenclature1.1.1.Conventionwiththenormalvectororientation

Thenormalvector ischosen tobeoriented toward theexternalpartof theconsideredbody.The usual conventions of mechanics of continuous media are chosen, leading to a positivestressfortensionandanegativestressforcompression.

Figure1.1.Definitionofthenormalunit

1.1.2.VectorialnotationAsoppositetothenotationusedforfigures,wherevectorsarerepresentedwithanarrow,inthe text, vectors are denoted by bold characters and its components have normal non-boldcharacters.Asanexample,wewillhave“M=Mxi+Myj+Mzk”.

1.1.3.PartoftheconservedreferencesectionTheconservedreferencepartofthebeamusedforthecalculationofgeneralizedstressinuseofthestatictheoremsisthe“right”part.

1.1.4.FrameFigure1.2.Orientationoftheframeforageneralsection

1.1.5.Compressionstressσc,supinthemostcompressedfiber

Itisadmittedthattheneutralaxisislocatedinsidethecross-section,thusdelimitingatensionzoneandacompressedzone.Thislastassumptionofaneutralaxisinsidethecross-sectionisno more valid when considering additional meaningful normal forces. Typically, under apositive moment (in span), the tension zone is located under the neutral axis, and thecompressionzoneabovetheneutralaxis,asshowninFigure1.3.Obviously,inthepresenceofa negative bending moment, the tension zone and the compression zone are permuted withrespect to the notation of Figure 1.3. The neutral axis as shown in Figure 1.3 allows theintroductionoftheconceptofextremalcompressedfiber,definedfromthemostdistantparalleltotheneutralaxisbelongingtothecross-section.Themostcompressedfiberinconcreteisbydefinitionthefiberassociatedwiththeminimalcompressivestressinalgebraicvalue,denotedbyσc,sup.

Figure1.3.Conceptofextremalcompressedfiber

1.2.Bendingbehaviorofreinforcedconcretebeams–qualitativeanalysis1.2.1.Frameworkofthestudy

1.2.1.1.ConstitutivelawofconcreteTheconstitutivelawofconcreteisastrongunsymmetricallawintensionandincompression,bothfromthestrengthandthepostfailureresponse,whichischaracterizedbyitsductility(seeFigure1.4).As a natural choice, the subscript c refers to concretewhereas the subscript sreferstothesteelmaterial.Weadoptbyσc,mintheextremestressatthecompressionpeak.

Figure1.4.Unsymmetricalresponseofconcreteinuniaxialtensionandcompression

1.2.1.2.BeamtheoryinsimplebendingInthissection,reinforcedconcretebeamsinsimplebendingarestudied(withoutaxialforces),composedofarectangularcross-sectionwithatotalheightdenotedbyhandawidthb.Thissection is reinforced by some steel reinforcement working in tension with a cross-section

denoted by As1 and by steel reinforcement working in compression with a cross-sectiondenotedbyAs2.Thecenterofgravityofthetensilereinforcementisatadistancedoftheupperfiber,andoneof thecompressionreinforcements isatadistanced'of theupper fiberof thecross-section.

Figure1.5.Geometricalparametersofthereinforcedcross-section;tensileandcompressionsteelreinforcement

In this case, again, it is implicitly accepted that the bending solicitation corresponds to apositive moment (the lower fibers are in tension with this convention, typically in span).Designingundernegativemoment (typically at support, for instance) is formally feasiblebypermutingthebehaviorofthecross-section.Furthermore,wecanintroducethestrainεs1asthestrainofthetensilereinforcementwiththelargesttensilestressσs1andwiththereinforcementareaAs1.

1.2.2.Classificationofcross-sectionalbehaviorThree kinds of reinforced concrete beam responses can be distinguished, depending on thesteelreinforcementdensity(Figure1.6).Theseresponsesareexplicitlydetailedinthespaceofthebendingmomentwithrespecttothestressinthemostcompressedfiberinconcrete.

Figure1.6.Bendingbehaviorofreinforcedconcretebeamswithrespecttothesteelreinforcementdensity

F:brittleresponse,whichappearswhenthebeamdesigndoesnotrespecttheconditionofnon-brittleness.A: beam with low steel reinforcement density, characterized by a global ductile response.Failureisinducedbyalargedrawingofthetensilesteelreinforcement.Asdiscussedbelow,theletterAreferstoPivotA.B: beam with high steel reinforcement density, characterized by the breaking up of thecompressedpartof theupperpartconcrete.The letterB refers to thebehaviorclassifiedasPivotB.Inthefollowing,brittlereinforcedconcretebeamsoftypeFwillnotbeinvestigated.

1.2.3.Parameterizationoftheresponsecurvesbythestressσs1ofthemoststressedtensilereinforcement

In Figure 1.7, the response curves are parameterized by the stress σs1 of themost stressedtensilereinforcement.WhenreadingFigure1.7inthesenseofincreasingthebendingmoment(thesolicitation),thestressvalueofthetensilesteelreinforcementalsoincreases.Thestressinthecompressedpartof theconcrete(typicallyin theupperpartof thesectionforpositivebending moment in span) is limited by a material characteristic value fcs defined in theNationalAnnexestotheEurocode2-EC2[EUR04](EC27.2.1).Thismateriallimitationisdefined in the rules for reducing the longitudinalcrack level in the tensilepartof thecross-section,forreducingthemicrocackingphenomenaofconcreteinthecompressedpartandforlimitingthetime-dependentinducedeffects(includingthespecificcreepeffect).

Figure1.7.Bendingsectionalbehaviorofreinforcedconcretebeamswithrespecttothesteelreinforcementdensity;theresponsecurvesareparameterizedbythestresslevelinthetensilesteelreinforcement

Forthesamebendingmoment,thesharetakenbytensilereinforcementismoreimportantincaseA.Consequently,thesharetakenbytheconcreteislessimportant;thisgeneratestherefore,inthismaterial,alowernormalstress.Thisexplainstherelativepositionoftwocurves.

1.2.4.Comparisonofσs1ofthetensilereinforcementforagivenstressinthemostcompressedconcrete

fiberσc,supInthissection,thephysicsofthereinforcedconcretebeambehaviorisdiscussedwithrespectto the steel reinforcement density. For a given value of the stress in the most compressedconcretefiber,bendingmomentsforthetwodifferenttypesofreinforcedconcretebeamsAandBaredifferent(Figure1.8).As thereinforcedconcretebeamof typeAhasa lowdensityoftensilesteelreinforcement,thefollowinginequalityholds:MB>MA.Thestress in themostcompressed fiber inconcrete isbeing fixed; the linear strainof the

upper fibers is the same for the two kinds of reinforced concrete beams. As a result, it isnecessarythatthecompressedpartinthesectionoftypeBhastobelargerthantheoneinthesectionoftypeA,inordertofulfillthemomentinequality.ThelineardistributionofstrainsandtheassociatedstressinthetensilereinforcementofsectionoftypeAare then larger than thesectionoftypeB(seeFigure1.8).

Figure1.8.Bendingbehaviorofreinforcedconcretebeamswithrespecttothesteeltensilereinforcementdensity;sectionsoftypeAandtypeB

Reciprocally, if thestressvalue in the tensilesteel reinforcement (in the lowerpartof thereinforcedsection)isthesameforbothtypesofthesection,thesectionoftypeAwillhavealower stress value (in absolute value) in the upper part of the compressed concrete incomparisontothesectionoftypeB(seeFigure1.9).

Figure1.9.Stressinthecompressionconcreteblock;sectionsoftypeAandtypeB

1.2.5.ComparisonofthebendingmomentsToreducethenon-elasticstrainlevel(existenceofpermanentstrainforastresshistoryendedby a vanishing stress value, for instance) leading to some possible large cracks or somepossible structural member strains, the steel reinforcement tensile stress is limited to amaterialcharacteristicvaluefssthatisalsodefinedintheNationalAnnexestoEurocode2–

EC2(EC27.2.2).For designing the structural members at the Serviceability Limit State (SLS), we should

satisfythefollowingtwosimultaneousinequalitiesthathavetobealgebraicallyfulfilled(seealsoFigure1.10or1.11):

[1.1]For reinforced concrete beams of kindA, slightly reinforced (or with a low tensile steel

reinforcementdensity),theSLSiscontrolledbythedrawingofthetensilesteelreinforcement.ThebendingmomentattheSLSisthencalculatedfor:

[1.2]

Figure1.10.IntroductiontotheconceptsofpivotsAandBattheserviceabilitylimitstate(SLS);inaPivotBsection,stresslimitationinconcreteoccursbeforestresslimitationinsteel

Figure1.11.IntroductiontotheconceptsofpivotsAandBattheserviceabilitylimitstate(SLS)

For reinforced concrete beams of kindB, highly reinforced (or with a high tensile steelreinforcementdensity), theSLS is controlledby the compression in concrete, considered asunacceptablebytherules.ThebendingmomentattheSLSisthencalculatedfor:

[1.3]

1.3.Backgroundontheconceptoflimitlaws1.3.1.Limitlawformaterialbehavior

1.3.1.1.DefinitionAlimitlaw,foragivenmaterial,isdefinedfromtheoverallpossibleconfigurationsreachedbythismaterialandiscompatiblewithagivenstatecriterion.Alimitlaw,initsessence,isdifferent from a constitutive law; the constitutive law implicitly contains all the successivetemporalconfigurationsbeforereachingthelimitstate.Therefore,intheconceptoflimitlaw,thenotionofthestressorstrainpathisreplacedbythenotionofthelimitstate,whichisthesetofconfigurationscompatiblewithagivenstatecriterion.

1.3.1.2.Applicationtotherectangularparabolicdiagram,alimitlawoftheultimatelimitstateInthecaseofaparabola–rectanglediagramappliedtoconcretemodeling(oneofthepossiblelimit lawsforconcretemodelingat theultimate limitstate(ULS)), if thesectionreaches theconcretelimitlaw,thentheBernoulliassumptionleadstoanonlinearstressdistributionalongthecross-section,asshownbythestressdistributionidentifiedbynumber5inFigure1.11.

InFigure 1.12, for the solicited bendingmomentsM1<M2<M3<M4<M5 =Mu,act, thestressdistributionalong thecross-sectioncannotbe rangedasaparabola–rectangle law, forthebendingmomentM1–M4; for these solicitations, the stress responsecurvesare transitorycurvesbetweenthelinearstateandthelimitstateoftheparabolicrectangularlimitstate.

Figure1.12.Illustrationoftheconceptoflimitlaw–stressevolutioninsidethecross-section;Ultimatelimitstateparabolic-rectangularstress–strainrelationship

However,forthebendingmomentdenotedbyM5(ultimatelimitactingbendingmoment),thelimitstateisreached,andthestresslimitdiagramisknown.

1.3.2.Exampleoflimitlawsinphysics,caseofthetransistor

Figure1.13.Electronicexampleforlimitlawsinphysics:thetransistor

Figure1.13 shows the setting of a polarized transistor,which is oftenmet in the field ofelectronicsdesigns.Thecharacteristicsof the transistoraregiven inFigure1.13.The linearpartofFigure1.14withasofteningslopecontainsthelocusofthestaticpointofthiselectronicsystem.Infact,theworkingpointislocatedonthisloadingline.

Figure1.14.Limitlawsinelectronicsdesign

Thissofteningstraightlineisclearlyalimitlaw:thepathfromtheorigintoreachthislimitlawisnotknownapriori,andonlythelimitstateisofinterest.

1.3.3.DesignofreinforcedconcretebeamsinbendingatthestressServiceabilityLimitState

Figure1.15.Limitlawatthestressserviceabilitylimitstate(SLS);stressdistributionalongthecross-section

InthecaseofthelimitbehaviorofconcreteattheSLS,asectionatitslimitstatewillhavealinear distribution of stress in the compressed part of the section (as a consequence ofBernoullikinematicsandelasticitybehaviorofeachmaterialat theSLS), thecontributionofconcrete in tension being neglected as a fundamental assumption. This “limit” behavior isvisualizedinFigure1.15bythelinearstressdistributionindicatedbynumber5.FortheexternalmomentsM1,M2,M3,M4andM5withM1<M2<M3<M4<M5=Mser,act,

the stress distribution is theoretically not necessarily linear along the cross-section, for themoments fromM1–M4; such curves are not knownapriori. Only for themomentM5 (limitmomentat theSLS), thestressdistributionatSLS isknownand is linear.However, for thisspecific limit state, it canbe also admitted in this case that the limit case (characterizedbylinearelasticity)isalsoreachedforthetransitorystates.

1.4.LimitlawsforsteelandconcreteatServiceabilityLimitState

1.4.1.Concreteatthecross-sectionalSLSConcrete, at the cross-sectional SLS, is modeled by a linear elastic constitutive law incompression, with a stress limitation denoted by fcs. The strength in tension is neglected(Figure1.16)anditsYoung’smodulusisdenotedbyEc.

Figure1.16.Limitlawofconcreteatserviceabilitylimitstate

1.4.2.Steelatthecross-sectionalSLSSteel at the cross-sectionalSLS is alsomodeledbya linear elastic constitutive law,with atension limit denoted by fss (Figure 1.17). A symmetrical behavior in tension and incompression is also accepted for modeling the behavior of the steel reinforcement, whichmeans that the potential buckling of the steel reinforcement in the presence of a crushingphenomena in concrete is generally neglected, except in so far as the spacing of transversereinforcement.Asaresult,forthecompressionsteelreinforcement,stressintheabsolutevalueisalsolimitedbythesamevaluefss.Young’smodulusofthesteelreinforcementisdenotedbyEs.

Figure1.17.Limitlawofsteelreinforcementatserviceabilitylimitstate

1.4.3.EquivalentmaterialcoefficientTheequivalencecoefficient(betweensteelandconcrete)isbydefinitiontheratiobetweenthesteelmodulusandtheconcretemodulusαe=Es/Ec.Thisratiodependsontheshort-termorthelong-term analysis considered at the SLS. As the Young’s modulus is decreasing with theeffectsof time, theequivalence ratio tends to increasewith theeffectsof time.This ratio isalso sometimes denoted by the equivalence coefficient “n” in the earlier textbooks onreinforcedconcretedesign.Atypicalorderofmagnitudeforthisratioisαe=15.

1.5.Pivotsnotionandequivalentstressdiagram

1.5.1.FrameandneutralaxisFigure1.18.Neutralaxisinageneralsection

LetusconsiderabeaminwhichthedirectorthonormalframeO,x,y,andzisdefined.Thebeam is solicitated by an external screwgiven at theSLS.The frame is in conformitywithsection1.1.3.or1.1.4.Under theactionof theexternal forces,aneutralaxiscanbedefined(Figure 1.18). It is admitted that the direction of the neutral axis is already known. In theoppositecase,somesymmetricalconsiderationshavetobeaddedinthereasoningorwecanproceedbyiterations.Theneutralaxisseparatesthetensionandthecompressionpartsofthesection.Eachofthe“n”steelbarsofthereinforcedsectionwillbeidentifiedbythesubscriptqwith

q [1;n].LetPqbethevectorwhoseoriginisdenotedbyOandwithanextremitydefinedatthecenterofgravityoftheqthreinforcementbar.Theorientedanglebetweentheaxiszandtheneutralaxisisdenotedbyω.Letdq be the distance between the center of gravity of theqth reinforcement bar and the

paralleltotheneutralaxisassignedtopassthroughthemostcompressedfiber.Amongthe“n”

steelbarsof the reinforcedsection, themost tensionedreinforcementbar is identifiedby itssubscriptr and its cross-sectional areaAsr with r [1;n]. The effective height “dr” of thecross-section can now be defined as the distance between the center of gravity of themosttensioned reinforcement bar, and the parallel to the neutral axis passing through the mostcompressedfiber(Figure1.19).

Figure1.19.Definitionofthenotionofeffectiveheightdr

1.5.2.Conservationofplaneityofacross-sectionTheconservationlawofplaneityofacross-sectionduringthedeformationprocessallowsustorelateduetosomegeometricalproperties,thestrainofeachconstituentofthecross-section,thecompressedpartof theconcreteandeachsteel reinforcementbar.Theordinateaxis“u”canbedefinedfromastraightlineperpendiculartotheneutralaxis,andorientedfromthelesscompressed to themostcompressedconcrete fibers.Theoriginof thisaxis is locatedat thebasisofthisaxis,positionedfromtheprojectionoftheoriginpointOonthisline.Thestressfieldinthissectionisafunctionofthisparameter“u”.Thefollowingscalarvariablecanbeintroducedfromuq=Pq·(jcosω+ksinω),wherethepointdenotesthescalarproduct(dotproduct)betweentwovectors(Figure1.20).

Figure1.20.Parameterizationofthestressfieldwiththevariableu

Theneutral axis is identifiedby thevalueur+ (1–α)dr for thevariableu,whereα is adimensionless distance characterizing the position of the neutral axis. α is a dimensionless

parameternormalizedbythetotalheightof thecross-section.Themostcompressedconcretefiberisassociatedwiththeprojectionvalueu0forthevariableu;theequalityu0=dq+uqisexactwhatever theconsideredsubscriptq [1;n].The strainεsq in each reinforcement barandthestrainεc(u)inconcretearerelatedwitheachotherbytherelationshipinducedbytheplaneityconservationofthecross-section:

[1.4]

1.5.3.PlaneityconservationlawintermofstressFromequation[1.4],the“limit”lawsassociatedwitheachmaterial(namelyelasticityforeachmaterialconstituents),at theSLS,allowus toobtainarelationshipbetweenthestress in thesteelreinforcementbarsandthestressinthecompressedconcretepartalongthecross-section.Thefollowingequalitiesfinallyholdbetweenthestressintheqreinforcementbarsσsqandthestressdistributioninthecompressedpartofconcreteσc(u):

[1.5]Bysettingktheproportionalityfactorofthislastequality(proportionaltothecurvature),a

seriesofaffinerelationshipsforthestressvalueinboththetensileandthecompressionsteelreinforcement bars, as for the compressed part of concrete is obtained,which can be againwrittenas:

[1.6]Thecalculatedstressinthemostcompressedfiberofconcreteisgivenby:

[1.7]Itisworthmentioningthatthecharacteristiclimitvaluesinboththesteelandconcretepartof

the composite section are defined in a algebraicmanner, with themechanics of continuousmedia convention, that is fss is positive and fcs is negative. At the limit state formally, thefollowingequalitiescanbeviewedtobevalidonlyinthecase(σsr=fssorσc(u0)=fcs (seesection1.2.5).Wecanalsoconsiderthatthelimitstateisalwaysreachedinelasticity,atleastattheSLS.WefinallyobtaininPivotA,withσsr=fss:

[1.8a]

[1.8b]InPivotB,thestressrelationshipsareobtainedfromσc(u0)=fcs:

[1.9a]

[1.9b]

1.5.4.IntroductiontopivotconceptsFigure1.21.DefinitionofPivotA;αe=Es/Ec

Designing by h the distance between the two extremal parallels to the neutral axis (h istypically the height of the cross-section), the series of equalities, equation [1.8], can bepresentedas:

[1.10a]

[1.10b]This systemofequationscanbe interpretedas theanalytical expressionof aplanewhose

traceinthespace{u,x}isthestraightlineappearinginFigure1.21.Thislinecrossesthefixedpointcorrespondingtothemosttensionedsteelreinforcementwithanequivalenttensilestressequaltofss/αe.ThispointiscalledPivotA(Figure1.21).Theseriesofequalities,equation[1.9],canbewrittenas:

[1.11a]

[1.11b]Again,thissystemofequationscaninterpretedastheanalyticalexpressionofaplanewhose

traceinthespace{u,x}isthestraightlineappearinginFigure1.22.Thislinecrossesthefixedpoint,correspondingtothemostcompressedfiberwithanequivalentcompressionstressequaltofcs.ThispointiscalledPivotB.

Figure1.22.DefinitionofPivotB;αe=Es/Ec

Thesefixedpointsplaytheroleofaturningpointorpivotpoint,aroundwhichtheequivalentstress lines turn,anddefine theprofileofequivalentstressalongthecross-section,which isthestressfor theconcretepart incompression,and theequivalentorhomogenizedstressforthesteelreinforcementbars,typicallythestressdividedbytheequivalentcoefficientαe.

1.5.5.PivotrulesThe planeity conservation assumption at SLS, which is in fact a fundamental kinematicsassumption, leads to an equivalent linear stress diagram that contains one of the two Pivotpoints:

–thepoint“PivotA”thatcorrespondstoanequivalentstressfss/αeofthemosttensionedsteelreinforcementbars;–thepoint“PivotB”thatcorrespondstoacompressionstressfcsinthemostcompressedfiberinconcrete.

Figure1.23.Workingzoneofthecompositecross-sectioninPivotAandPivotB

Thisstraightlineexactlycorrespondstothecompressionstressfieldinthecompressedpartofconcrete,whereas thestress in thesteel reinforcementbars, in tensionor incompression,arededucedfromtheequivalentstresslinebyintroducingtheequivalencecoefficient1/αe.Atthecross-sectionalSLS,thediagramofequivalentstressesshouldcrossoneofthetwopivotpoints (Pivot A and Pivot B) and should also respect the limit stress requirement for theextremalpointattheotherpivotpoint(seeFigure1.23).ThelimitcasebetweenthetwopivotsisreferredasPivotAB,andischaracterizedbythelimitvaluefortherelativepositionoftheneutralaxisαABdefinedfromtheupperfiberofthecross-sectionasαAB=αefcs/(αefcs–fss).

1.6.Dimensionlesscoefficients1.6.1.Goal

Tocharacterizethestaticequilibriumequationsofthecross-section,itisnecessarytointegrateboth the stress field and themoment in the compressed part of concrete, defined at a givenpointofthecross-section.Thiscalculationwillleadtothedeterminationoftheinternalforcesscrewapplied to concrete.The total internal screwwill thenbededuced from the concretescrew,byadding the tensileand thecompressionsteel reinforcement screwevaluatedat thesamepoint.Thecalculationof internalandexternalscrews,bothwithequilibriumequationswillbewritten,asmuchaspossible,inadimensionlessformat,toavoiddimensionconfusionand try to give themore general presentation.Dimensionless numbers are also important toclassifythesectionalbehavior,initsmoregeneralformulation.

1.6.2.Totalheightofthecross-sectionAsindicatedinPart1.5.4,thetotalheightofthecross-sectionisdenotedbyh(histhedistancebetween twoparallels to theneutralaxispassing through theextremityof thecross-section).ForarectangularoraT-beam,hhasitsusualdefinitionandistheheightofthecross-section.

1.6.3.RelativepositionoftheneutralaxisInsection1.5.2,thepositionoftheneutralaxiswasidentifiedfromthefollowingequationu=ur+(1–α)dr,whereαistherelativeheightoftheneutralaxis.Eveniftheunderstandingofthis relative position is quite intuitive as explained in Figure1.24, a rigorousmathematicaldefinition can be given based on geometrical arguments. The height parameter dr can bedefinedfromthedistancebetweentwolinesparallel to theneutralaxis,onepassingthroughthemostcompressedfiberandtheothercrossingthemosttensionedsteelreinforcement.Thedistance between themost compressed fiber and the neutral axis is denoted byαdr (Figure1.24).

Figure1.24.Definitionoftherelativeheightoftheneutralaxis

1.6.4.ShapefillingcoefficientFigure1.25.Definitionoftheshapefillingcoefficientψ

Thescrewof internal forcesapplied toconcrete iscalculated for thenormal forceNc, byintegrationofthecompressionstressoverthedomainofthecompressedconcretedenotedbyDc:

[1.12]Theresultantcanbewritteninthefollowingform:Nc=Nci.Itisworthmentioningthatthe

stressfunctionσc(u)isacontinuousfunctionwithinthecompressiondomainDc.Byvirtueofthemeanvaluetheorem,itcanberigorouslydeducedthatthereexistsacharacteristicvariableu*belongingtothedomainofvariationofu,suchas:

[1.13]The integral insuchadefinition represents theareaof thecompressiondomainDc, which

canbedenotedbyAc.TheshapefillingcoefficientΨcannowbedefinedfromthismeanvalue:

[1.14]The meaning of the shape filling coefficient can be more easily understood from

considerationsofthevolumeshighlightedinFigure1.25,andtheintrinsicrelationshipbetween

theactualstressvolumeandtheequivalentuniformstressvolume.Theshapefillingcoefficientis theratiobetween thevolumeofacylinderwithabasisequal toDcandaheightequal toσ(u*)andthevolumeofanothercylinderofthesamebasisbutwithheightequaltofcs.Itcanbe remarked that thevolumeof a cylinderwith abasis equal toDc andwithheightσ(u*)isequaltoNc.Thestressσ(u*)iscalledthemeanstress.

1.6.5.Dimensionlessformulationforthepositionofthecenterofpressure

Figure1.26.Notionofcenterofpressure

TheinternalforceoftheinternalscrewincludingthenormalforceinconcreteNchasbeenevaluatedpreviously;theinternalmomentneedsnowtobecalculatedforcompleteequilibriumdetermination of the cross-section. By designingDc, the concrete part in compression, thescrewofresultantmomentMccalculated,forinstance,attheoriginoftheframewillbegivenby:

[1.15]Suchanexpressionoftheresultantmoment,evaluatedattheoriginoftheframe,dependson

thestressfunctionσc(y,z)thatpossessesaconstantsign,intheconcretedomainincompressionDc.Thisstressfunctionhaslowerandupperbounds,thusalsoincludingthevectornormofyj+ zk that parameterizes the point of the studied compression domain. The conditions ofapplicationofthesecondtheoremofthegeneralizedmeanvaluearesatisfied,andthenwecanconcludetheexistenceofavectorG=ξ2yj+ξ2zkwhoseextremityisinsidethecompressiondomainDc(Figure1.26),andsuchthat:

[1.16]The origin of the frame for calculation of this screw can be chosen to coincidewith the

centerofgravityofthemosttensionedsteelreinforcementbar(Asr).Bydividingthedifferentlength(associatedwiththelevelarmoftheequivalenttorquecalculation)ξ2yandξ2zbydr,thedimensionlessparametersδgyandδgzappearas:

[1.17]These size parameters are called the dimensionless coefficient of the center of pressure.

Using some symmetry or mechanical considerations, it is possible to calculate thesedimensionless parameters by engineering rules. However, in the more general case, forunsymmetricalcross-section,amathematicalintegralcalculationisoftenneeded.

1.7.Equilibriumandresolutionmethodology

1.7.1.EquilibriumequationsThescrewofinternalforcescanbedecomposedintotwointernalscrews,onerelatedtothereinforcement steel bars and the other to the concrete domain in compression. Whenconsideringonlytheinternalscrewrelatedtothereinforcementsteelbars,bothintensionandin compression, the normal force resultant Ns and the internal torque resultant Ms arecalculatedas:

[1.18]The cross-sectionalmechanics equilibrium in bending, at SLS, can then bewritten in the

followingform(Figure1.27):

[1.19]Themomentequilibriumequationcanbealsorewrittenas:

[1.20]Thenormalforceequilibriumequationissimplyreducedtothefollowingscalarequation:

[1.21]

Figure1.27.Equilibriumofageneralcompositesection;calculationofthescrewofinternalaction

1.7.2.Discussionontheresolutionofequationswithrespecttothenumberofunknowns

Ithasbeenshown,fromtheapplicationoftheconservationprincipleofplaneitycoupledwiththenotionsoflimitlawsandpivotconcepts,thatthestressdistributioninsidethecross-sectioncan be considered as a discretized function of the following characteristic parameters

,fssorfcs.Theequilibriumequationsgiveasystemcomposedoftwovectorialequations(threescalarequations)inthegeneralcase: andNact=Ns+Nc.Themathematicalproblem, inconformitywith thespecificationsat theSLS, isawell-posedproblem only if the number of unknowns is also equal to three in the general case. Theseunknown variables are of different types and are typically of geometrical nature: steelreinforcementarea,positionoftheneutralaxis,etc.–Generalcase: threelinearlyindependentscalarequations;

(twomomentscalarequationsandonenormalforceresultantequation).–Symmetricalcase: twolinearlyindependentscalarequations;

(onemomentscalarequationandonenormalforceresultantequation).

For symmetrical sections, with symmetrical steel reinforcement, and with symmetricalloading(inthiscaseωisknown),thethree-parameterproblemisreducedtoatwo-parameterproblem associated with two equations: one scalar moment equation and one resultant

equation. Different engineering problems can be envisaged, for reinforced concrete design,whichcanbeillustratedthroughthefollowingexamples:

– Design of a unique unknown steel reinforcement area (in tension or in compression)whosecenterofgravityisknown.Oneof thesteelreinforcementareaAsibeingunknown(in tensionor incompression), two

additionalparametershavetobechosen,intheunsymmetricalcase,suchastherelativeheightoftheneutralaxisαandtheneutralaxisdirection(angleω).Wehavethentosolveaproblemof three equations with three unknowns Asi,ω and α. For a symmetrical section, the twounknownparameterscanbethesteelreinforcementareaandtherelativepositionoftheneutralaxisα.

–Designofauniqueunknownsteelreinforcementarea(intensionorincompression)withagivenneutralaxis(relatedtothevariableωortothevariableα).Oneof thesteelreinforcementareaAsibeingunknown(in tensionor incompression), two

additionalparametershave tobe chosen, in theunsymmetrical case, suchas the two spatialcoordinatesoftheconsideredsteelreinforcement.Forasymmetricalsection,thetwounknownparameterscanbethesteelreinforcementareaandtheordinateofthecenterofgravityofthesteelreinforcement.Acurrentcaseisthecaseofasymmetricalsection,wheretheneutralaxisis fixed at theboundaryofPivotA andPivotB, to avoid thedesignof the cross-section atPivotB(inorder to limit thesteel reinforcementquantity). Insuchacase, the twounknownparameters canbe the areasof the tensionandcompression steel reinforcement, for agivenfixedlocationofthesesteelreinforcements.

1.7.3ReducedmomentsFromtheequilibriummomentequation,wehaveMact=Ms+ [ξ2z.j-dr (1-α.δgy).k].ψ.Ac.fcs,which,bydividingeachmemberbythemomentquantity-dr.Ac.fcs/f(α)wheref(α)isafunctionofthedimensionlessparameterα,canbeformulatedas:

[1.22]From this dimensionless equilibrium equation, the dimensionless reducedmoment can be

introduced:

[1.23]withμact,ser is the acting reducedmoment andμres,ser is the reduced concrete contribution

momentThefunctionf(α)canbechosensuchasthereducedactingmomentwhichdoesnotcontain

explicitly the variable α; it is quite efficient to take f(α) = dr.Ac/(dr2.b) where b is acharacteristiclengthofthecross-section.Forinstance,forarectangularsectionsolicitedinitssymmetryplane,thenaturalchoicef(α)=αleadstothefollowingactingreducedmoment:

[1.24]andthereducedconcretecontributionmomentiswrittenas:

[1.25]Anotherexamplecanillustratetheroleplayedbythefunctionf(α)tohighlighttheroleofthe

dimensionless parameters. Let us take the example of a T-section, solicited in its plan ofsymmetry.Itispossibletochoosethefunctionf(α)issuchaway:

[1.26]leadingto:

[1.27]Thereducedbendingmomentisthenformulatedas:

[1.28]Anotherchoiceforthefunctionf(α)inthecaseoftheT-sectionwouldbebasedon:

[1.29]With this alternative of the function f(α), the reduced bendingmoment is normalizedwith

respecttothewidthoftheconcreteslab:

[1.30]Thereducedbendingmomentisthenformulatedas:

[1.31]In the symmetrical loading casewith symmetrical cross-section,we finally obtain, for the

reducedmomentequation:

[1.32]Inthisrelationship,itisconvenienttodefinethephysicalmeaningofeachterm:

Thetotalmomentequilibriumequationissimplyobtainedbythesumofeachcontributionof

thesteelandtheconcretepartofthereinforcedconcretesection:

[1.33]

1.7.4.CaseofarectangularsectionFigure1.28.Characteristicsofarectangularreinforcedconcretesection

UsingthenotationsofFigure1.28,thetwoequilibriumequationsbothforthenormalforceandthebendingmoment(onlytwoequationsforsymmetricalreasons)arewrittenas:

[1.34]

[1.35]

1.8.CaseofpivotAforarectangularsection1.8.1.Studiedsection

Thecaseofarectangularcross-sectionreinforcedbybothtensile(areaAs1)andcompressive(area As2) steel reinforcements symmetrically disposed along the median axis y is nowconsidered. The origin of the frame is located at the center of gravity of the tensile steelreinforcement.Thegeometricalcharacteristicsofthecross-sectionaredenotedbyb,h,dandd′andareillustratedinFigure1.28.TheconcreteandsteelmaterialstresslimitsaregivenattheSLS by fss for the steel reinforcement, and fcs for the concrete, the equivalence coefficientbeing classically denoted by αe. The acting solicitation is evaluated at the pointO and ischaracterized by two components, the acting bendingmomentMact =Mactk and the normalforce resultant Nact = Nact.i. The unknown of the problem are, respectively, the tensilereinforcementareaAs1andtherelativeheightoftheneutralaxisα.

1.8.2.ShapefillingcoefficientFigure1.29.StressdiagraminthehomogenizedsectionofthereinforcedconcretesectionforPivotA

Thevariableαvariesintheinterval ,whichdependsonboththeelasticandthestresslimitvaluesattheSLS.Stressesinsidethecompositecross-sectionarerelatedbytheequalities:

[1.36]Thestressdiagram in thehomogenizedsection is shown inFigure1.29.Bydefinition, the

normalforce in thecompressedpartofconcrete iscalculatedasNc=αψ.b.d fcs.,which isalsoequivalentaftercalculationtoNc=0,5.σc,sup.α.b.d.Theshapefillingcoefficientψ thensimplyexpressedby:

[1.37]

1.8.3.Dimensionlesscoefficientrelatedtothecenterofpressure

AttheSLS,thepressurecenterofthecompressedpartofconcreteisatadistanceαd/3ofthemostcompressedfiber(theupperfiberofconcrete), leadingtothedimensionlesscoefficientδg=1/3(Figure1.30).

Figure1.30.Positionofthecenterofpressureattheserviceabilitylimitstate;PivotA

1.8.4.EquationsformulationAthird-orderpolynomialequationoftheunknownneutralaxispositionhastobesolvedattheSLS.Thispolynomialequationisobtainedfromwritingtheequilibriumequationsinbendingmomentandnormalforces,coupledwiththekinematicsassumptionsofthecross-sectionandtheelasticbehaviorofthesteelandconcreteconstituentsofthecompositecross-section:

[1.38]

[1.39]Thestressinthecompressionsteelreinforcementσs2alsodependsontheunknownα.

[1.40]Itmeansthatonlythesecondbendingmoment,equation[1.39],containsonesingleunknown

αoutofthetwounknownsofthedesignproblem,respectively,thetensilereinforcementareaAs1andtherelativeheightoftheneutralaxisα.AsthefillingshapecoefficientΨisgivenby:

[1.41]andbysettingthesectionequilibriumM=Mser,equationoftheequilibriumbendingmomentfinallyleadsto:

[1.42]Werecognizeathird-orderpolynomialequation:

[1.43]Someotherdimensionlesscoefficientscanbeintroducedas:

[1.44]Wenotethatallthesedimensionlesscoefficientsδ’,ρ’csandμactarepositivenumbers.The

third-orderpolynomialequationcanfinallybewrittenas:

[1.45]Algebraically, this third-order equationcaneasilybe solvedusingCardano’smethod (see

Appendix1).

1.8.5.ResolutionFollowingCardano’smethod, such a third-order polynomial equation can be presented in acanonicalformatusingcanonicalparameters:

[1.46]Bothcanonicalparameterspandq arenegativenumbers.The solutionα can be extracted

from Cardano’s formula as one of the three real equations, whose solution of interest, forphysicalreasons,is:

[1.47]whichcanbeequivalentlywrittenas:

[1.48]If there is no compression steel reinforcement (As2 = 0), the neutral axis equation to be

solvedisstillathird-orderpolynomialequationthatisnowsimplifiedin:

[1.49]Thecanonicalparametersinthisparticularcase(As2=0)arealsosimplified:

[1.50]Intheabsenceofcompressionsteelreinforcement(As2=0),thesolutionforthepositionof

neutralaxisαinPivotAisthencalculatedas:

[1.51]Oncetheneutralaxispositionαiscalculated,theaxialforceequilibriumequationgivesthe

tensilesteelarea:

[1.52]Intheabsenceofcompressionsteelreinforcement(As2=0),thetensilesteelareainPivotA

isgivenby:

[1.53]ThestressSLShasbeenthensolvedinPivotA.

1.9.CaseofpivotBforarectangularsection1.9.1.Studiedsection

Thecaseofarectangularcross-sectionreinforcedbybothtensile(areaAs1)andcompressive(areaAs2)steelreinforcementssymmetricallydisposedalongthemedianaxisyisconsidered,butnowthesection is ruledbyPivotB.Thegeometricalcharacteristicsof thecross-sectionarealsodenotedbyb,h,dandd'andare illustrated inFigure1.28.Theconcreteand steelmaterial stress limits are given at theSLSby fss for the steel reinforcement, and fcs for theconcrete,theequivalencecoefficientbeingclassicallydenotedbyαe.TheactingsolicitationisevaluatedatthepointO,andischaracterizedbytwocomponents,theactingbendingmomentMact=Mactk and thenormal force resultantNact=Nact.i.Theunknownof theproblemare,respectively,thetensilereinforcementareaAs1andtherelativeheightoftheneutralaxisα.

1.9.2.ShapefillingcoefficientInacaseofsimplebendingsolicitation(withoutcombinedaxialforces),thevariableαvariesintheinterval ,whichdependsonboththeelasticandthestresslimitvaluesattheSLS.Incaseofbendingwithcombinedaxialforces,theneutralaxiscanbelocatedoutsidethecross-section.However, in the simplebendingcase, theneutral axis is inside the cross-section, which gives the upper bound variation of the dimensionless coefficient α. Bydefinition,thenormalforceinthecompressionpartNciscalculatedfrom:

[1.54]As the stress diagram is linear at the SLS, it can be easily deduced that the filling shape

coefficientΨinPivotBisequaltoΨ=1/2,asNc=0,5.αfcs.b.d.

1.9.3.Dimensionlesscoefficientrelatedtothecenterofpressure

AttheSLS,thepressurecenterofthecompressedpartofconcreteisatadistanceαd/3ofthemostcompressedfiber(theupperfiberofconcrete),leadingtothedimensionlesscoefficientofδg=1/3(Figure1.31).

Figure1.31.Positionofthecenterofpressureattheserviceabilitylimitstate;PivotB

1.9.4.EquationsformulationInthemoregeneralcase,atPivotB,athird-orderpolynomialequationoftheunknownneutralaxispositionαhasalso tobesolvedat theSLS.Thispolynomialequation isobtainedfromwriting theequilibriumequations inmomentandnormal forces,coupledwith thekinematicsassumptionsofthecross-sectionandtheelasticbehaviorofthesteelandconcreteconstituentsofthecompositecross-section:

[1.55]

[1.56]Thestressinthecompressionsteelreinforcementσs2alsodependsontheunknownα.

[1.57]It isworthmentioning that the stress expression in the compression steel reinforcement is

differentinPivotAandPivotB.InsertingthevalueoftheshapefillingcoefficientΨatPivotB, ,andtheequilibriummomentequationisfinallygivenby:

[1.58]This equation is equivalent to the third-order polynomial equation, which allows us to

calculatealgebraicallythepositionoftheneutralaxisatPivotB:

[1.59]Thedimensionlesscoefficientscanbeintroducedas:

[1.60]Wenotethatallthesedimensionlesscoefficientsδ′,p′csandμactarealsopositivenumbers.

Thethird-orderpolynomialequationcanfinallybewrittenas:

[1.61]In this case, again, this third-order equation can be easily algebraically solved using

Cardano’smethod(seeAppendix1).Itcanbeoutlinedthatthethird-orderequationinPivotBisdifferentfromthethird-orderequationinPivotA,andthe tworesolutionsare indeedtwodifferentmathematicalproblems.

1.9.5.ResolutionFollowing Cardano’s method, such a third-order polynomial equation can be presented incanonicalformatusingthecanonicalparameters:

[1.62]Bothcanonicalparameterspandq aregenerallynegativenumbers.The solutionα canbe

extractedfromCardano’sformulaas:

[1.63]Ifthereisnocompressionsteelreinforcement(As2=0),theneutralaxisequationtobesolved

isnowasecond-orderpolynomialequationwhichcanbewrittenas:

[1.64]whosesolutioniswellknowninreinforcedconcretedesigns:

[1.65]Once theneutral axispositionα has been calculated, the axial force equilibrium equation

givesthetensilesteelarea:

[1.66]Intheabsenceofcompressionsteelreinforcement(As2=0),thetensilesteelareainPivotB

isgivenby:

[1.67]ThestressSLShasbeenthensolvedinPivotB.

1.9.6.SynthesisIn Pivot A, the calculation of the position of the neutral axis at the stress SLS needs theresolution of a third-order polynomial equation,whatever compression steel reinforcementsare incorporated in the cross-section (As2= 0 orAs2≠0). In Pivot B, the calculation of thepositionoftheneutralaxisatthestressSLSneedstheresolutionofapolynomialthird-orderpolynomial equation, when compression steel reinforcement are incorporated in the cross-section(As2≠0).However,onlyinPivotB,thethird-orderequationdegeneratestoasecond-orderpolynomialequationinabsenceofcompressionsteelreinforcement(As2=0).

1.10.Examples–bendingofreinforcedconcretebeamswithrectangularcross-

section

1.10.1.AdesignproblematSLS–exercise

1.10.1.1.StructuralproblemFigure1.32.Simplysupportedreinforcedconcretebeam

Asafirstapplicationofthepresentedtheory,wewouldliketodesignthereinforcementofagiven beam with a given rectangular cross-section (see Figures 1.32 and 1.33). This is asimplysupportedbeamoflengthL=6m.

Figure1.33.Characteristicsoftherectangularcross-section

ThegeometricalcharacteristicsaregiveninFigure1.33anddetailedbelow:

[1.68]Forengineeringdesign,theorderofmagnitudeforboththepositionofthecenterofgravity

ofthetensilesteelreinforcementandthecompressionsteelreinforcementisusuallytakenas:[1.69]

whichisalmostsatisfiedinourpresentcase.

1.10.1.2.MaterialparametersThe reinforced concrete section is composed of B500B steel bars and a C25/30 type ofconcrete. In the nomenclature of steel classificationB500B, the firstB stands for steel forreinforcedconcrete.Thenextthreedigitsrepresentthespecifiedcharacteristicvalueofupperyieldstrengthfyk=500MPa.ThelastsymbolBstandsforductilityclass.InthenomenclatureofconcreteclassificationC25/30,thefirstCstandsforconcrete.Thenextdigitsarerelatedto

characteristic strengths. The first characteristic compression strength is the cylindercharacteristicstrengthfck=25MPa,whereasthesecondcharacteristiccompressionstrengthisthecubiccharacteristicstrength.Hence,wecalculatethelimitstressesattheSLSas:

[1.70]Theequivalencecoefficientisgivenforthisproblemas:αe=15.ThesteelYoung’smodulus

isequaltoEs=200,000MPa.TherecouldbesomediscussionsonthecalculationoftheconcreteYoungmodulusEc.From

EC2(Eurocode2)[EUR04],wehave(inMPa):

[1.71]whereφ is the creep coefficient (time-dependent effects) and the subscriptm represents themeanvalueoftheconsideredvariable.Theorderofmagnitudeofφtypicallyvariesbetween1and 2. The formula in this equation is not homogeneous from a dimensional point of view,except for the numbers 22,000, 10 and 8 can be viewed as specific stress values given inmegapascals(MPa).WecalculateforthemeanvalueoftheconcreteYoungmodulus:

[1.72]Foraspecificconcrete,thecalculationoftheequivalentcoefficientαedependsonthecreep

coefficientφ.Foragivenvalueof theequivalentcoefficientαe, thecreepcoefficientcanbecalculatedas:

[1.73]Itmeans, in thisexample, that thevalueofαe=15corresponds toacreepcoefficientφ of

approximately1.361.Thedeterminationoftheequivalencecoefficientisstilldebated,actually(seealso[THO09]or[PAI09],forinstance),andthetypicalvalueofαe=15,whichwastheusualvalueemployedintheoldFrenchrulesBAEL91,whichisstillrelevanttousewiththenewEC2rules.More generally, the creep coefficient depends on the type of concrete and on the type of

analysis,namelytheshort-termorthelong-termanalyses.

1.10.1.3.LoadingparametersThesimplysupportedreinforcedconcretebeamisloadedbysomeuniformdistributedloads:

–permanentloads(withouttheownweight):g1=24.75kN/m;–variableloads:q=20kN/m.

Thevolumetricdensityofthereinforcedconcretebeamcanbetakenas =25kN/m3.Thepermanentloadsarethencalculatedasthesumofg1andtheownweightg2:

[1.74]ThecombinationofactionattheSLSisusuallyoneforthepermanentloadandoneforthe

variableload,leadingtotheserviceabilitydistributedload:

[1.75]Thecriticalsectionofthisproblemistheonewherethebendingmomentismaximum,thatis

atmid-span,leadingtothebendingsolicitationfordesigningattheSLS:

[1.76]

1.10.1.4.SteelreinforcementatSLSTheexerciceconsistsofdesigningthesteelreinforcementofthereinforcedconcretebeamattheStressSLSforthissolicitationMser=225kN.m,andforahighersolicitation:

–Mser=225kN.m;–Mser=405kN.m.WefirststartfromthefirstsolicitationMser=225kN.m.

1.10.2.ResolutioninPivotA–Mser=225kN.mAspresentedinsection1.5.5,thelimitvalueoftheneutralaxispositionattheboundaryofthetwopivots,PivotAandPivotB,isgivenfrom:

[1.77]Thebendingmomentsolicitationat theboundaryofPivotAandPivotBcanbecalculated

from:

[1.78]Itisalsopossibletointroducethedimensionlessreducedmomentattheboundarybetween

PivotAandPivotBas:

[1.79]Itcanbeeasilychecked, for thissolicitation, thatPivotArules thebehaviorof thecross-

sectionatSLSas:

[1.80]Then, the dimensionless position of the neutral axis given by parameter α should be the

solutionofthefollowingthird-orderpolynomialequation:

[1.81]Thecanonicalparametersneededfortheresolutionofthisthird-orderpolynomialequation

byCardano’smethodarethencalculated:

[1.82]ThesolutionforthepositionofneutralaxisαinPivotA,forthiscrosssection,iscalculated

as:

[1.83]Finally,thetensilesteelreinforcementareaisobtainedfromequilibriumofthenormalforce

componentas:

[1.84]Thetensilesteelreinforcementareaof9.843cm2isneededtoverifythestressSLSofthis

cross-sectionforthebendingsolicitationMser=225kN.m.UsingtheabacusofAppendix2,wecanchoose5ϕ16 (10.053cm2),whichhas tobe anupperboundof theminimum tensilesteel area of 9.843 cm2. An example of steel reinforcement location is presented in Figure1.34.

Figure1.34.5ϕ16tensilesteelreinforcementforMser=225kN.matserviceabilitylimitstate

1.10.3.ResolutioninPivotB–Mser=405kN.mBycomparingthereducedmomentμserwiththereducedfrontiermomentbetweenPivotAandPivot B,μAB = 0.1584,we see that for this solicitationMser = 405 kN.m, the sectionwillbehaveinPivotB:

[1.85]In Pivot B, and without additional compression steel reinforcement, the position of the

neutralaxisisobtainedfromresolutionofasecond-orderpolynomialequationwhoserootofinterestis:

[1.86]Finally,thetensilesteelreinforcementareaisobtainedfromequilibriumofthenormalforce

component

[1.87]Thetensilesteelreinforcementareaof39.34cm2isneededtoverifythestressSLSofthis

cross-sectionforthebendingsolicitationMser=405kN.m.UsingtheabacusofAppendix2,wecanchoose5ϕ32 (40.212cm2),whichhas tobe anupperboundof theminimum tensilesteelareaof39.34cm2.ItcanbeseenthatforasolicitationMser=405kN.mlessthantwicetheothersolicitation

Mser=225kN.m,thesteelreinforcementquantityhasbeenmultipliedbyapproximatelyfour(from approximately 10 cm2 to 40 cm2). Even if physically possible, it is generally not

interesting todesign a section atPivotB for economical reasons.This canbe explainedbyFigure 1.35, where the variation of the steel reinforcement quantity measured by ρs1 iscomparedinPivotAandPivotBwithoutcompressionsteelreinforcement(As2=0).

Figure1.35.Evolutionofthedimensionlesssteelquantityρs1withrespecttothesolicitationexpressedwiththereducedmomentμser

InPivotA,wehavethefollowingrelationshipsforthecalculationofthedimensionlesssteelreinforcementareaρs1:

[1.88]InPivotB,wehavethefollowingrelationshipsforthecalculationofthedimensionlesssteel

reinforcementarea:

[1.89]Figure 1.35 shows the change of the slope of the steel area variationwith respect to the

solicitationrepresentedbythereducedbendingmoment.ToavoidadesigninPivotBwithtoomuchsteelreinforcement,itissuggestedtoaddsome

compression steel reinforcement and to design the section at the boundary between the twopivots:PivotAB.

1.10.4.ResolutioninpivotABFigure1.36.Decompositionofthecross-sectionintotwoparts;DesignatPivotAB

WechoosetoaddsomeadditionalcompressionsteelreinforcementwithasectionareaAs2,locatedatadistanced′fromtheupperfiberofthecross-section(d’≈0.1h;inthepresentcase,we choosed’ = 5 cm). In order to keep the design problem as awell-posedmathematicalproblem,wehave to fixanadditionalparameterbyadding theunknownof thecompressionsteel reinforcement area. Following the previous analysis, it has been shown that we haveinteresttofixthepositionoftheneutralaxisinordertobehaveinPivotA.Itischosentofixthepositionoftheneutralaxissuchas:

[1.90]The section is decomposed into parts as shown in Figure 1.36, one section with the

compressedpartoftheconcreteandonlythetensilereinforcement,whereastheothersectionis composed of some tensile and compression steel reinforcement. The total area of tensilesteelreinforcementisequalto:

[1.91]Thecalculationof the tensile steel reinforcement in the firstpartof thecross-section (see

Figure1.36)isobtainedfromthenormalforceequilibriumequation,havinginmindthatthiscross-sectionissolicitatedbyafictitiousbendingmomentequaltoMAB:

[1.92]Considering the second part of the cross-section (see also Figure 1.36), the stress in the

compressionsteelreinforcementiscalculatedfromthecompatibilityofstraininthedeformedcross-section:

[1.93]The moment equilibrium equation written at the center of gravity of the tensile steel

reinforcementinduces:

[1.94]Thenormalforceequilibriumequationgivesthetensionsteelreinforcement:

[1.95]Wefinallyobtainforthiscross-sectiondesignedatPivotABbyaddingsomecompression

steelreinforcement:As1=12.96+4.79=17.75cm2andAs2=10.87cm2.Wecantake6ϕ20(18.85 cm2) for the tensile steel reinforcement, and 2ϕ20 + 1ϕ25 (11.19 cm2) for thecompression steel reinforcement. The total steel quantity for this alternative less consumingdesignis30.04cm2(withAs2≠0)comparedto40.21cm2obtainedbefore inPivotBwithoutthe use of compression steel reinforcement (with As2 = 0). In this case, adding somecompression steel reinforcement allows us to gain approximately 25% of the total steelreinforcement.Infact,theinterestofaddingcompressionsteelreinforcementcanbeclearlyseeninFigure

1.37forsufficientlylargebendingsolicitationμser≥μAB. In thecasewherecompressionsteelreinforcementsareaddedtotheupperpartofthecross-sectionandthesectionisdesignedatPivotAB,thetotaldimensionlesssteelareaρsisgivenby:

[1.96]AsshownbyFigure1.37,addingsomecompressionsteelreinforcementinthiscasemakes

the variation of the steel quantity linearwith respect to the bending solicitation,whereas adesign at Pivot B without additional compression steel reinforcement leads to a stronglynonlinearcurve,whichmayoverestimatesignificantlythesteelquantityneededatSLS.

Figure1.37.Evolutionofthetotaldimensionlesssteelquantityρswithrespecttothesolicitationexpressedwiththereducedmomentμser

1.10.5.Designofareinforcedconcretesection,anoptimizationproblem

In fact, designing the section at Pivot AB at SLS may lead to the optimized solution withrespect to the total steel quantity used in the reinforced section. It can be shown that theposition of the neutral axis (see Chapter 2 on the so-called “static moment equation”) isobtainedfromasecond-orderpolynomialequationwrittenas:

[1.97]wherethedimensionlesscoefficientδ’=d′/dhasbeenused.Wecanintroduce thechangeofvariable:

[1.98]It is easy to check thatρS=ρs1+ρs2 and tan2 θ= ρs2/ρs1. The optimization consists of

finding theoptimizedvalueθ so that theSLScapacity reducedmomentμ ismaximized foragivenquantityofsteelreinforcementρs.ThisoptimizationproblemisillustratedinFigure1.38fortheproblemwithgeometricalparametersdefinedinFigure1.33.

Figure1.38.Optimizationofthereducedmomentμserwithrespecttothesteelreinforcementratio;ρs2/ρs2=tan2θforagivenquantityρs {0.1;0.2;0.3;0.4;0.5};δ’=0.078125;B500BsteelbarsandC25/30typeofconcrete

Theoptimizationproblemcanbeexplicitlyformulatedfromtheanalyticalexpressionoftheneutralaxispositionand the reducedmomentparameters.Thepositionof theneutralaxis isthenexpressedwithrespecttothenewoptimizationvariables:

[1.99]ThereducedmomentinPivotA,forα(ρs,θ)≤αAB,isgivenby:

[1.100]ThereducedmomentinPivotB,forα(ρs,θ)≥αAB,isgivenby:

[1.101]AsshownbyFigure1.38, theoptimizationproblemisasingularoptimizationproblemand

theoptimizationparameterisassociatedwithPivotAB,leadingtothetranscendentalequation:

[1.102]whichcanbeanalyticallysolvedfrom:

[1.103]Forμser≤μAB=0.1584,thatisforaPivotAsectionwithonlytensilesteelreinforcement,

thebestsolutionisobtainedforθ=0,thatis,thereisnoneedtoaddsomecompressionsteelreinforcementinthereinforcedconcretesection.However,itcanbeshownthatdesigningthesectionatSLSattheboundarybetweenPivotA

andPivotBforμser≥μABisnotnecessarilythebestsolutionforalltherangeofparameters.

TheoptimizationproblemcoincideswiththePivotABdesignintheproblemhandledinFigure1.38,basedonB500B steel bars and theC25/30 type of concrete, andwithδ′ = 0.078125.However,forthesamematerialsbutchangingthevalueofthegeometricalratioδ′upto0.15,itcanbeseeninFigures1.39and1.40thattheoptimizedsolutionisdifferentfromtheoptimizedsolutionofPivotAB.Forinstance,thisgeometricalratioδ′=0.15correspondstothereinforcedsectionbasedon

thefollowingparameters:

[1.104]

Figure1.39.Optimizationofthereducedmomentμserwithrespecttothesteelreinforcementratio;ρs2/ρs1=tan2θforagivenquantityρs {0.1;0.2;0.3;0.4;0.5};δ′=0.15;B500BsteelbarsandC25/30typeofconcrete

Figure1.40.Optimizationofthereducedmomentμserwithrespecttothesteelreinforcementratio;ρs2/ρs1=tan2θforagivenquantityρs {0.1;0.15;0.2};δ′=0.15;B500BsteelbarsandC25/30typeofconcrete

AsanexamplerelatedtothesensitivityanalysisshownbyFigures1.39and1.40,wegivebelowanexampleofanreinforcedconcretesectioninPivotBwithatotalsteelquantitylesserthan the one associatedwith the design in PivotAB. For this solicitationμser = 0.183, thedesignatPivotABleadsto:

[1.105]whereas theoptimizedsolution, for thesamesolicitationμser=0.183 isgiven inPivotB,

for:

[1.106]In this case, the optimized solution allows the gain of more than 5% of steel area with

respecttothedesignatPivotAB.

1.10.6.GeneraldesignatServiceabilityLimitStatewithtensileandcompressionsteelreinforcements

In this section, the design of a reinforced concrete section at SLS with both tensile andcompressionsteelreinforcementsispresented.ThesectionisnotnecessarilydesignedatPivotAB,asdeveloped insection1.10.4. In thegeneralcase, thesection isdecomposed into twoparts(seeFigure1.41).

Figure1.41.Decompositionofthecross-sectionintotwoparts;generaldesign

Itisassumedtowritethetotalbendingsolicitationintotwoterms:

[1.107]whereξisafreedimensionlessparameterthatfixestheneutralaxisposition.Forinstance,iftheneutralaxispositionisfixedatα=αAB,thentheparameterξisequaltoξAB=MAB/Mser.Butthislastcaseisofcourseaparticularcase.Thedesignofthefirstfictitioussectionwithonlytensilesteelreinforcementsisfirstpresented(solicitationM1=ξMser).Foragivenvalueofthedimensionlesscoefficientξ,thesteelquantityAs1,1iscalculatedforthereducedmoment

. If the calculation of the dimensionless neutral axis position α1 iscomputedfromthecubicequation:

[1.108]leadingtotheroot:

[1.109]Ifµser,1≥µAB,thecalculationofthedimensionlessneutralaxispositionα1iscomputedfrom

thesecond-orderpolynomialequation:

[1.110]leadingtothesolution:

[1.111]Inbothcases,inPivotAorPivotB,thetensilesteelareaofsectionAs1,1isobtainedfrom:

[1.112]Nowconsidering the second fictitious sectionwith some tensile steel reinforcement (area

As1,2),compressionsteelreinforcements(areaAs2),withoutanyconcrete incompressionandsolicitedbythemomentM2=(1–ξ)Mser.Ifα1correspondstoPivotA,thenthestressesinthetensileandcompressionsteelreinforcementsarecalculatedfrom:

[1.113]Thebendingmomentequilibriumequationcanbewrittenatthecenterofthegravityofeach

steelreinforcement:

[1.114]Itisworthmentioningthatacombinationofeachoftheseequationsisinfactequivalentto

thenormalforceequilibriumequationAs1,2σs1+As2σs2=0.Thetensileandcompressionsteelareacanbefinallyobtainedfrom:

[1.115]Ifα1correspondstoPivotB,thenthestressesinthesteelreinforcementsareobtainedfrom:

[1.116]Also,usingthebendingmomentequilibriumequation[1.114], thesteelreinforcementareas

canbecalculatedinPivotBfrom:

[1.117]

Using theseequations, itwouldbepossible tocompute the total steel areaAs1+As2 withrespecttothedimensionlessfreeparameterξItisalsopossibletooptimizethedesignofthereinforcedconcretesectionwithrespecttothetotalsteelareabystudyingtheevolutionofthetotalsteelareaAs1+As2with respect to thedimensionlesspositionof theneutralaxisα1. InPivotA,thedimensionlesscompressionsteelareaρs2isobtainedfrom:

[1.118]andthetotaldimensionlesssteelareaρsisthenequalto:

[1.119]InPivotB,thedimensionlesscompressionsteelareaρs2isobtainedfrom:

[1.120]andthetotaldimensionlesssteelareaρsisthenequalto:

[1.121]

Figure1.42.Evolutionofthesteelquantitywithrespecttothedimensionlesspositionoftheneutralaxis;B500BsteelbarsandC25/30typeofconcrete;µser=0.183;δ′{0.078125;0.15}

Theoptimizationproblemconsistsoffindingthesmalleststeelquantityratioρswithrespecttoα, for a given reducedmomentµser = 0.183. It is clearly shown in Figure 1.42 that theoptimization problem is sensible regarding the value of the parameter δ′ = d′/d. For theparameter δ′ = 0.078125 (problem analyzed in Figure 1.38), it is confirmed that theoptimizationsolutionisthesolutionassociatedwiththedesignatPivotAB,whereasforδ′=0.15(problemanalyzedinFigures1.39and1.40),theoptimizedsolutionisdifferentfromthe

solutionattheboundarybetweenPivotAandPivotB.Theoptimalsolutionαopt=0.416>αAB=0.36isobtainedinthiscaseforthebranchruledbyPivotB,fromthequarticequation:

[1.122]whereρsisgivenbyequation[1.121]inPivotB.Forthisvalueofδ′=0.15,wefindαopt≥αABandα=αoptisthesolutionoftheconstrainedoptimizationproblem.Ontheotherhand,ifthe solution of the quartic equation [1.122] was less than αAB, then the optimized solutionwouldbetheoneofPivotAB,asalreadymentionedforthecasewithδ′=0.078125.

1.11.ReinforcedconcretebeamswithT-cross-section

1.11.1.IntroductionAT-cross-sectionisanalyzedwherebothcompression(withareaAs2)andtensile(withareaAs1)steelbarsreinforcethecompositecross-section(seeFigure1.43).Thegeometryofthecross-sectionischaracterizedbythedifferentlengthparametersb,bw,h,

h0,d andd′ whereb is thewidth of the concrete slab,h0 is the depth of the flange (slab)thickness,and thewidthof theweb isdenotedbybw.Thepositionof theneutral axis is asusual,characterizedbyαdfromtheupperfiberofthecross-section.

Figure1.43.GeometryofaT-cross-section

If thedepthof thecompressionblock iswithin the flangedportionof thebeam, that is the

neutralaxisdepthαdislessthantheflange(slab)thicknessh0,measuredfromthetopoftheslab(αd<h0),thenthesectioncanbecalculatedasan“equivalent”rectangularcross-sectionwiththewidthequaltob(astensileconcretecontributionisneglectedintheanalysis).Then,wefindagaintheconfigurationpreviouslyinvestigated,forthedesignofreinforcedconcretebeamswithrectangularcross-sectionattheSLS.However,whenthedepthofthecompressionblockislargerthantheflange(slab)thickness,theneutralaxisislocatedintheweboftheT-cross-section, and the calculation has to be based on the T-cross-section calculation, asdetailedinthissection.The concrete and steel material stress limits are given at the SLS by fss for the steel

reinforcement,andfcsfortheconcrete,theequivalencecoefficientbeingclassicallydenotedbyαe.TheactingsolicitationisevaluatedatpointOandischaracterizedbytwocomponents,theactingmomentMact=MactkandthenormalforceresultantNact=Nact.i.Theunknownsoftheproblem are, respectively, the tensile reinforcement areaAs1 and the relative height of theneutralaxisα.Itisassumedthattheneutralaxisislocatedinsidetheweb,thatish0≤αd≤h.TosimplifythetheoreticalstudyoftheT-cross-sectionanalysis,thesummationoftheforces

resistant screw is induced by a decomposition of the total cross-section within piecewiserectangular cross-sections, without forgetting the steel reinforcements, both in tension andcompression.

1.11.2.Decompositionofthecross-sectionThecross-sectionSisdecomposedintothreerectangularparts:tworectangularparts,S1 thatcanbeadded,andS2thatcanbesubtracted,plustheadditionalsteelreinforcementS3thathasto be added to the total crosssection (see Figure 1.44). These subsections have the sameneutralaxisandthesamecurvatureforcompatibilityreasons.

Figure1.44.DecompositionoftheT-cross-section

ThecharacteristicsofeachsubdomainaregiveninTable1.1.

Table1.1.CharacterizationofeachsubdomainoftheT-cross-section

Thecenterofpressureofthecompressionblockinconcreteisatadistanceαd/3ofthemostcompressedfiberofthecross-section,thatistheupperconcretefiber,andthenδg=1/3.Asfortherectangularcross-section,theshapefillingcoefficientΨhasbeenintroducedinTable1.1. The normal force in the compressed part of concrete can be expressed through thisdimensionlesscoefficient:

[1.123]InTable1.1,thedimensionlessshapefillingcoefficientΨdependsontheconsideredpivot,

PivotAorPivotB.

[1.124]InTable1.1, thedimensionless parameterΨ′ denotes another shape filling coefficient that

alsodependsontheconsideredpivot,PivotAorPivotB:

[1.125]

1.11.3.CaseofpivotAforaT-cross-section

1.11.3.1.EquationsformulationBy settingM =Mser, the bending moment equilibrium consideration leads to a nonlinearequation of the dimensionless relative depth α, similar to what has been found for therectangularcross-sectioninsection1.8.4as:

[1.126]

or,equivalently,byusingthedimensionlesscoefficients:

[1.127]Athird-orderpolynomialequationisfinallyobtainedforthedeterminationoftheneutralaxis

position,characterizedbyα,foragivencross-section:

[1.128]wherethedimensionlesscoefficientδ=d′/dhasbeenused.

1.11.3.2.Resolution–PivotACardano’smethodcanbeusedfortheresolutionofthisthird-orderpolynomialequation(seeAppendix1).Thecanonicalparametersare:

[1.129]Typically, for the physical parameters associated with reinforced concrete design

application,thethird-orderpolynomialequationinαhasthreeroots,whichtheoneofinterestinourstudyis:

[1.130]Once the neutral axis position α is numerically calculated, the normal force equilibrium

equation gives the area of tensile steel reinforcement as found in section 1.8.5 for therectangularcross-section:

[1.131]DifferentspecificcasescanbededucedfromthisgeneralequationvalidforT-cross-section.

InthecaseofsimplebendingNact=0,thesteelareaequationisreducedto:

[1.132]In thecaseofsimplebendingNact=0,andwithonly tensilesteel reinforcement (As2=0),

thisequationissimplifiedinto:

[1.133]Finally,inthecaseofsimplebendingNact=0,withonlytensilesteelreinforcement(As2=0),

andforrectangularcross-section(bw=b),wefindagainwhathasbeenfoundinsections1.8and1.9:

[1.134]

1.11.4.CaseofpivotBforaT-cross-section

1.11.4.1.EquationsformulationBy settingM =Mser, the bending moment equilibrium consideration leads to a nonlinearequation of the dimensionless relative depth α, similar to what has been found for therectangularcross-sectioninsection1.9.4as:

[1.135]or,equivalently,byusingthedimensionlesscoefficients:

[1.136]Athird-orderpolynomialequationisfinallyobtainedforthedeterminationoftheneutralaxis

position,characterizedbyα,foragivencross-section:

[1.137]whosesolutionisalsogivenbyCardano’smethod(seesection1.11.3).

1.11.4.2.ResolutionOnce the neutral axis position α is numerically calculated, the normal force equilibriumequation gives the area of tensile steel reinforcement as found in section 1.9.5 for therectangularcross-section:

[1.138]Thestressesinthetensileandinthecompressionsteelreinforcementsareobtainedfromthe

equivalentstressdiagram:

[1.139]It isworthmentioning that a third-orderpolynomial equation isgenerallyobtained for the

determinationofthepositionoftheneutralaxisatPivotB,asinthecaseofPivotA(evenifthethird-order equations in both pivots are not the same). However, in the specific case of arectangularcross-section(bw=b)withoutcompressionsteelreinforcement(As2=0),thelastterm a3 vanishes in the third-order equation and the third-order equation degeneratesmathematicallyintoasecond-orderequation:

[1.140]Werecognizethesecond-orderequationforthedeterminationof thepositionoftheneutral

axis at Pivot B in the case of a rectangular crosssection without compression steelreinforcement.However,inthecaseofaT-cross-sectional,withb≠bw,athird-orderequationis obtained for the determination of the position of neutral axis α, even in a case withoutcompressionsteelreinforcement,whichisanotabledifferencewiththecaseofarectangularcross-section.Finally,inthecaseofsimplebendingNact=0,withonlytensilesteelreinforcement(As2=0),

andforarectangularcross-section(bw=b),wefindagain,asinthecaseofPivotA:

[1.141]

1.11.5.Example–designofreinforcedconcretebeamscomposedofT-cross-section

1.11.5.1.DataofthedesignproblemThegeometrical characteristics of theT-cross-section aregiven inFigure1.45 and detailedbelow:

[1.142]Thereinforcedconcretesectioniscomposedofsteelbarswithfyk=300MPaandC25/30

typeofconcrete.Hence,wecalculatethelimitstressesattheSLSas:

[1.143]Theequivalencecoefficientisgivenforthisproblemas:αe=15.ThesteelYoung’smodulus

isequaltoEs=200,000MPa.Thesolicitationisasimplebendingsolicitationcharacterizedatthecenterofgravityofthe

tensile steel reinforcement byMser = 0.49MN.m andNser = 0MN. In simple bending, thesolicitationwouldhavebeenthesameinanotherpointofthecross-section.

Figure1.45.SteelreinforcementdesignofareinforcedconcreteT-cross-sectionatSLS;lengthsgiveninmeters

It is assumed that there is no compression steel reinforcement (As2=0m2). The exerciseconsistsofdesigningthetensilereinforcementofthisT-cross-sectionatthestressSLS.

1.11.5.2.ResolutionAspresentedinChapter5.5,thelimitvalueoftheneutralaxispositionattheboundaryofthetwopivots,PivotAandPivotB,isgivenfrom:

[1.144]Thebendingmomentsolicitationat theboundaryofPivotAandPivotBcanbecalculated

from:

[1.145]Hence, the T-cross-section has to be calculatedwith the PivotA rule. The dimensionless

neutral axis position is the solutionof a third-order equationgivenby equation[1.128] andwrittenwiththenumericalapplicationas:

[1.146]ThethreesolutionscanbecomputedfromCardano’smethodaccordingtoAppendix1–see

equation[A1.23].

[1.147]Onlyα2=0.2720isofinterestforphysicalreasons.Furthermore,wecancheckthatforthis

valueoftheneutralaxis,theT-cross-sectiondoesnotbehaveasarectangularcross-sectionas:

[1.148]In otherwords, it is confirmed that the neutral axis is located in theweb of theT-cross-

section,whichjustifiesthecalculationwiththewebpartoftheT-cross-section.InPivotA,thestressinthetensilesteelreinforcementσs1isequaltotheserviceabilitylimitstressfssandthesteelareaAs1canbecalculatedfromequation[1.132]:

[1.149]Wefinallyobtainforthiscross-sectiondesignedatPivotA,As1=24.30cm2.Wecan take

3ϕ32(As1=24.13cm2)forthetensilesteelreinforcement(seeAppendix2).But5φ25 (As1=24.54cm2)would probably be safer, even if the first solution based on3ϕ32 could also beused, if theanticipatedvalueofd, thedistancefromthecenterofgravityof the tensilesteel

reinforcementtotheupperfiberofthecross-section,isalittlebitlargerthan0.92m.

Chapter2

VerificationatServiceabilityLimitState(SLS)

2.1.Verificationofagivencross-section–controldesign

2.1.1.PositionoftheneutralaxisFigure2.1.Positionoftheneutralaxisatserviceabilitylimitstate

In this section, we assume a given reinforced concrete cross-section at its serviceabilitylimit state (SLS), with a given area of tensile steel reinforcementAs1 and a given area ofcompression steel reinforcement As2. To check the correctness of the reinforced concretedesign,thestressesineachpartofthecross-sectionhavetobecalculatedandcomparedtotheserviceabilitylimitstresses.For a given reinforced concrete cross-section, the position of the neutral axis can be

calculated at theSLS (seeFigure2.1). It is found that this position does not depend on thesolicitationlevel,atleastinsimplebendingwithoutanormalforce.Thisfundamentalpropertyis not true at the ultimate limit state (ULS), where the position of the neutral axis, in fact,dependsonthesolicitationlevel.Theneutralaxisisusedtodefinethez-axisalongwhichtheframe axis is defined (see Figure 2.1). The upper part of the cross-section is assumed incompression,which corresponds to the positivemoment in span. The axis origin is chosenarbitrarilyalongthisneutralaxis.Theframeiscompletedbythex-axis,perpendiculartotheplane of the cross-section, and oriented towards the external part of the considered body(conventionofpositivetraction).Intheplaneofthecross-section,theframeiscompletedbyaverticaly-axisorientedtowardthecompressedpartofconcrete.

2.1.2.Equationofstaticmomentsforthedeterminationofthepositionofneutralaxis

Starting for theconservation lawofplaneity, the strain relationship inboth the steeland theconcretepartislinearlyrelatedthrough:

[2.1]Theequivalentstressequationsarededucedfromthestrainrelationshipbyintroducingthe

elasticityconstitutivelawinthesteelσs=Esεsandtheconcretepartσs=Esεsofthecross-section,andusingtheequivalencecoefficientαe=Es/Ec,as:

[2.2]ThenormalforceinthecompressedpartofconcreteDc,denotedbyNc,iscalculatedthrough

anintegraloperatoras:

[2.3]whereScisthestaticmomentcalculatedwithrespecttotheneutralaxisofthecompressedareaofconcreteDc.ItisalsopossibletocalculatethenormalforcesNs1andNs2inboththetensilesteelreinforcementofareaAs1andcompressionsteelreinforcementofareaAs2:

[2.4]

[2.5]where,withrespecttotheneutralaxisOz,Ss1isthestaticmomentof“αe”timestheareaofthetensilesteelreinforcementwithanareaAs1andSs2isthestaticmomentof“αe”timestheareaof the compression steel reinforcement with an area As2. The normal force equilibriumequationattheSLSthenleadstoastaticmomentrelationship:

[2.6]In the case of simple bending without additional normal force Nact= 0, this equation is

reducedtothestaticmomentequation:

[2.7]Theconceptofahomogenisedsectionmaybeintroducedatthisstage,byaffectationaunity

coefficientontheconcretepartandbytheequivalentYoung’smoduluscoefficientαe,thesteelreinforcements both in tension and compression. It isworthmentioning that this notion of ahomogenizedsectionnolongerholdsattheULSwherethenonlinearbehaviorsofthesteelandtheconcreteparts arenot compatible ingeneralwithelasticity.Afterhaving introduced thisconcept of a homogenized section, the meaning of the static moment equation can now be

discussed. This equation expresses the fact that the sum of the homogenized staticmomentswithrespecttotheneutralaxisisvanishinginareinforcedconcretesectionattheSLS.Wecanalso say that theneutral axiscrosses thecenterofgravityof thehomogenizedcross-section,neglecting of course the tensile part of concrete. The static moment equation allows thedeterminationoftheneutralaxisposition,foragivenreinforcedconcretecross-section.Thisstaticmomentequationisasecond-orderequationforacross-sectionwithpiecewiseconstantwidth,includingforinstancearectangularcross-sectionorT-cross-sections.

2.1.3.Stresscalculation–generalcaseOnce theneutralaxispositionhasbeencalculated thanks to thestaticmomentequation, thestressesineachpartofthereinforcedcross-sectioncanbecalculated.Themomentequilibriumequationhastonowbeused:

[2.8]where the resultant moment with respect to the compression block of concrete isMc, theresultant moment with respect to the tensile steel reinforcements isMs1 and the resultantmomentwithrespect to thecompressionsteel reinforcement isMs2.Themoment resultant inthecompressionpartofconcrete,denotedbyMc,iscalculatedwithrespecttotheneutralaxisas:

[2.9]whereIcisthequadraticmomentwithrespecttotheneutralaxisofthecompressionblockofconcreteDc. The resultant moment in the tensile and the compression steel reinforcements,denotedbyMs1andMs2,canalsobecalculatedwithrespecttotheneutralaxisas:

[2.10]

[2.11]whereIs1isthequadraticmomentwithrespecttotheneutralaxisofthetensilereinforcementhomogenizedbytheequivalencecoefficientαeandIs2isthequadraticmomentwithrespecttotheneutralaxisofthecompressionreinforcementhomogenizedbytheequivalencecoefficientαe. These quadratic moments are the exact quadratic moments of the homogenized steelreinforcements,assuming that theprincipalquadraticmomentofeachsteelcross-sectioncanbe neglected (only the transport term is meaningful in application of Huygens’s formulae).

Going back to the equilibrium moment equation, the maximum compression stress in thecompressionblockcannowbecalculatedfrom:

[2.12]where I is the quadraticmoment of the homogenized section (neglecting the tensile part ofconcrete) with respect to the neutral axis. The stresses in the steel reinforcement andcompressionblockarefinallyobtainedfromtheequivalentlineardiagraminthehomogenizedsection:

[2.13]Thisequationcaneasilybecompared to theusual equationvalid in linearelasticity fora

homogeneouscross-sectionσ=±Mv/I.However,inthiscase,asamaindifferencewithlinearelasticity,thecontributionofconcreteintensionhasbeenneglected.

2.1.4.Rectangularcross-section–verificationofagivencross-section

2.1.4.1.Rectangularcross-section–stressequationsFigure2.2.Positionoftheneutralaxisatserviceabilitylimitstate;rectangularcross-section

Therectangularcross-sectionhasthegeometricalcharacteristicsdenotedbyb,h,dandd′,andareillustratedinFigure2.2.Theequationy=αdisthedistancefromtheneutralaxistotheupper compression fiber of the cross-section,whereasy =αd is the unknownof the designproblemthatcanbedeterminedusingthestaticmomentequation:

[2.14]This second-order equation can be easily solved, and it leads to the root of the physical

interest,as:

[2.15]The stresses are then calculated from equation [2.13], where the equivalent moment

quadraticfortherectangularcross-sectionisequalto:

[2.16]

2.1.4.2.Example–rectangularcross-sectionAsanexample,therectangularcross-sectionisreinforcedbytensilesteelreinforcementwithareaAs1=12.57cm2(4ϕ20)andbycompressionsteelreinforcementwithareaAs2=1.57cm2

(2ϕ10).Thecross-sectionhasthefollowingcharacteristics:b=25cm;h=40cm;d=35cm;

d′=3.5cm.ThereinforcedconcretesectioniscomposedofB500BsteelbarsandtheC25/30typeofconcrete.Theequivalencecoefficientisgivenforthisproblemas:αe=15.ThebeamissolicitatedinsimplebendingwithMser=80kN m.Theexerciseconsistsofcontrollingthedesignofthisreinforcedrectangularcross-sectionatthestressSLS.Thestaticmomentequation[2.14],inthiscase,givesthesecond-orderpolynomialequation:

[2.17]The calculation of the quadraticmoment of the homogenized section is given in equation

[1.138]anddetailedforthespecificstudiedcross-section:

[2.18]Thestressesarefinallycalculatedineachpartofthereinforcedconcretesection:

[2.19]It can be checked that this reinforced concrete cross-section is correctly designed with

respecttotheSLS.

2.1.5.T-cross-section–verificationofagivencross-section

2.1.5.1.T-cross-sectionorrectangularcross-section?Figure2.3.Positionoftheneutralaxisatserviceabilitylimitstate;T-cross-section

AT-cross-sectionisanalyzedwhereboththecompression(withareaAs2)andtensile(withareaAs1)steelbarsreinforcethecompositecross-section.Thegeometryofthecross-sectionischaracterizedby thedifferent lengthparametersb,bw,h,h0,d andd′. b is thewidth of theconcrete slab and h0 is the depth of the flange (slab) thickness. The width of the web isdenotedbybw.Thepositionof theneutralaxis isasusualcharacterizedbyy=αd from theupperfiberofthecross-section.If thedepthof thecompressionblock iswithin the flangedportionof thebeam, that is the

neutralaxisdepthαdislessthantheflange(slab)thicknessh0,andmeasuredfromthetopoftheslab(y=αd<h0),thenthesectioncanbecalculatedasan“equivalent”rectangularcross-sectionwiththewidthequaltob(astensileconcretecontributionisneglectedintheanalysis).Then, we find again the configuration previously investigated for the design of reinforcedconcretebeamswitharectangularcross-sectionat theSLS.However,whenthedepthof thecompressionblockislargerthantheflange(slab)thickness,theneutralaxisislocatedintheweboftheT-cross-section(y=αd>h0)andthecalculationhastobebasedontheT-cross-sectioncalculation.Thetransitorycasebetweenthesetwokindsofbehaviorisobtainedfory=αd=h0(seeFigure2.3).To evaluate the type of configuration (rectangular or T-cross-section calculation), it is

requiredtoassumearectangularcross-sectionandseewhentherectangularassumptiony=αd<h0 is fulfilled. The position of the neutral axis for a rectangular cross-section is alreadygiveninequation[2.14]andtheconditionoftherectangularassumptionforthecalculationisthenwrittenas:

[2.20]whichisequivalentto:

[2.21]Thefollowingcompressionandtensilestaticmomentscanbeintroducedas:

[2.22]Hence,ifthecondition isfulfilled,thenthedepthofthecompressionblockiswithin

theflangedportionofthebeam,thatistheneutralaxisdepthαdislessthantheflange(slab)thicknessh0(y=αd<h0).Thesectioncanbecalculatedasan“equivalent”rectangularcross-sectionwiththewidthequaltob.Onthecontrary,ifthecondition isfulfilled,thenthedepthof the compressionblock is larger than the flange (slab) thickness, the neutral axis islocatedintheweboftheT-cross-section(y=αd>h0)andthecalculationisbasedontheT-cross-sectioncalculation,asdetailedbelow.

[2.23]Itisusefultonotethattheseconditionsareindependentofthesolicitationanddependonly

onthepropertiesofthecross-section.

2.1.5.2.Calculationofthepositionofneutralaxis–T-cross-sectionIt is assumed that theneutral axis is located inside theweb, that ish0 ≤αd ≤h.The staticmoment of the compression block Sc is obtained as the sum of the static moment of thecompressionpartofthewebbw.y.y/2,withoneoftheflanges(b–bw).h0.(y–h0/2):

[2.24]Thestaticmomentequationisthenwrittenas:

[2.25]whosepositivesolutionis:

[2.26]Oncethepositionyoftheneutralaxisisobtained,thequadraticmomentofthehomogenized

sectioniscalculatedas:

[2.27]The stresses are then calculated for themost compression concrete fiber, the tensile steel

reinforcement and the compression steel reinforcement from equation [2.13], which stillremainsvalid.Nowbyusingthedimensionlesscoefficients:

[2.28]thepositionoftheneutralaxisexpressedindimensionlessformatisgivenby:

[2.29]Thecaseoftherectangularcross-sectionissimplydeducedbyputtingbw=b

[2.30]When the reinforcedconcrete section isdesignedwithoutcompressionsteel reinforcement

(ρs2=0),thecaseoftherectangularcross-sectionbw=bleadsto:

[2.31]

2.1.6.Example–verificationofareinforcedT-cross-section

Thecross-sectioncharacteristicsoftheT-cross-sectionarethecharacteristicsgiveninFigure2.1.Asanexample,therectangularcross-sectionisreinforcedbytensilesteelreinforcementwithareaAs1=24.54cm2 (5ϕ25)andbycompressionsteel reinforcementwithareaAs2=0cm2(nocompressionsteelreinforcement).Thecross-sectionhasthefollowingcharacteristics:b=0.8m;d=0.92m;d′=0.04m;bw=0.3m;andh0=0.2m.ThereinforcedconcretesectioniscomposedofB300Bsteelbars(fyk=300MPa)andC25/30typeofconcrete(fck=25MPa).Theequivalencecoefficientisgivenforthisproblemas:αe=15.ThebeamissolicitatedinsimplebendingwithMser=0.49MN m.TheexerciseconsistsofcontrollingthedesignofthisreinforcedT-cross-sectionatthestressSLS.The following compression and tensile static moments can be calculated from equation

[2.22]:

[2.32]

Thetwostaticmomentscanbecomparedandweobtain ,whichmeansthatthedepthofthecompressionblockislargerthantheflange(slab)thickness:theneutralaxisislocatedintheweboftheT-cross-section(y=αd>h0)andthecalculationisbasedontheT-cross-sectioncalculation.Thestaticmomentequation[2.25]inthiscasegivesthesecond-orderpolynomialequationwithrespecttotheunknownpositionoftheneutralaxismeasureswithy=αd:

[2.33]whosepositive root isy=0.251m,which is indeedgreater thanh0= 0.2m.The quadraticmomentofthehomogenizedsectionwithrespecttotheneutralaxisisequalto:

[2.34]The stress values in both the tensile steel reinforcement and the compression block in

concretearethencalculatedas:

[2.35]

[2.36]It can be checked that this reinforced concrete cross-section is correctly designed with

respecttotheSLS.ThisreinforcedconcretesectionhasbeenclearlydesignedatPivotA.

2.1.7.Determinationofthemaximumresistingmoment

Ithasbeenshownabovehowtodesignareinforcedconcretesectionbydeterminingthesteelquantityforagivensolicitationinsimplebending,accordingtotheSLS.WehavealsoshownhowtocheckthatagivenreinforcedconcretesectionverifiesthestressSLS.Inthissection,we present the methodology to compute the maximum solicitation that a given reinforcedconcretesectioncansupportattheSLS.For a given reinforced concrete section, the neutral axis position expressed with the

dimensionlessparameterαcanbedeterminedbyusingthemomentstaticequationleadingtothe solution given by equation [2.29]. The parameter α has to be compared to

to determine the type of pivot that should pilot the reinforced concretesectionattheSLS.Ifα<αAB(PivotA),theresistantmomentattheSLSiscalculatedfrom:

[2.37]Ifα>αAB(PivotB),theresitantmomentattheSLSiscalculatedfrom:

[2.38]

2.2.Cross-sectionwithcontinuouslyvaryingdepth

2.2.1.Triangularortrapezoidalcross-sectionEvenifrectangularorT-cross-sectionshaveawideareaofapplicationsincivilengineeringorbuilding,somespecificengineeringproblemswithcontinuousvaryingdepthcross-sectionscanbemet,especiallyinbridgeengineering,withprestressedconcretecaissons,forinstance(seeFigure2.4).

Figure2.4.Differentkindsofcross-sectionwithrectangular,triangularortrapezoidalcross-sections

In these cases, theuseof theT-cross-section analogy is not alwayspossible and an exactcalculationwithanintegrationovertheprismaticcross-sectionisneeded.Inthissection,wewillstudyareinforcedconcretetriangularcross-section(seeFigure2.5),designedatitsSLS.Inacertainsense, thiscaseenglobessomeother trapezoidalvariationcross-section,at leastfromamethodologypointofview.

Figure2.5.Reinforcedconcretetriangularcross-sectionwithtensilesteelreinforcement

The triangular cross-section is reinforced by some tensile steel reinforcement (with areaAs1).Thegeometryofthecross-sectionischaracterizedbythedifferentlengthparametersb0,h

andd,whereb0 is thewidth of the upper part triangle base andh is the total depth of thetriangle.Thepositionof theneutralaxis isasusual,characterizedbyy=αd fromtheupperfiberofthecross-section.Attheneutralaxislevel,thewidthofthecross-sectionisequalto:

[2.39]

2.2.2.Equilibriumequations–normalforceresultantTheequilibriumequationsareexpressedbothforthenormalforcecomponentandthemomentatthecenterofgravityofthetensilesteelreinforcementas:

[2.40]Inlinearelasticity,atSLS,thecompressionstressintheconcreteblocklinearlyevolvesas:

[2.41]The width of the cross-section at the fiber parameterized by the ordinate y′ in the

compressionblock(y′=0attheneutralaxis–seeFigure2.5)isgivenby:

[2.42]Thenormalforceintheconcreteblockcanthenbecalculatedas:

[2.43]Thedoubleintegralcanbeeasilyconvertedintoasingleintegralwiththesliceintegration

technique:

[2.44]Thenormalforceintheconcretecompressionblockcanthenbecalculatedas:

[2.45]Thenormalforceequilibriumequationisthenwrittenas:

[2.46]Itwouldalsobepossibletohaveusedageometricalproofforthisnormalforceexpression.

In fact, the calculation of the double integral for the determination of the normal force inconcreteisequivalenttothecalculationofthevolumeofapentahedroncomposedofaregularpentahedron and two symmetrical tetrahedral (see Figure 2.6). The total volume of thepentahedronVcanbedecomposedasthesumoftwoothervolumesV=V3+2V4,withV3asthevolumeoftheregularpentahedronandV4asthevolumeofeachtetrahedron.ThevolumeV3oftheregularpentahedronwithawidthb(α)anddepthαdcanbecalculatedas:

[2.47]ThevolumeV4ofeachtetrahedronisequaltotheproductofthebaseareamultiplyingathird

ofthedepth,whichisexpressedas:

[2.48]ItcaneasilybeverifiedthatNc=V3+2V4(orequivalentlyNc=V1−2V2withthenotation

ofFigure2.6).

2.2.3.Equilibriumequations–bendingresultantmoment

Thebendingmomentintheconcreteblockcanthenbecalculatedatthecenterofgravityofthetensilesteelreinforcementas:

[2.49]Thedoubleintegralcanbeeasilyconvertedintoasingleintegralwiththesliceintegration

technique:

[2.50]Wefinallyobtainfromthisintegralequation:

[2.51]Thisresultcanalsobeshownfromgeometricalarguments,asdetailedinFigure2.6.

Figure2.6.Linearstressdiagramintheconcretecompressionblockofthetriangularcross-section–serviceabilitylimitstate

Thecalculationofthebendingmomentcanalsobebasedongeometricalarguments:

[2.52]whered3 andd4 are the distance from the center of gravity of eachvolume to the center ofgravityofthetensilesteelreinforcement.Itcanbecheckedthatequation[2.52]isequivalenttothemathematicalexpressionofequation[2.51].

2.2.4.CaseofpivotAforatriangularcross-sectionAtPivotAB(boundarybetweenPivotAandPivotB),wehavethecharacteristicvalues:

[2.53]For μ ≤μAB , the section has to be designed with respect to Pivot A. At Pivot A, the

compressionstressinconcreteinthemostcompressedfiberisrelatedtothetensilestressinthetensilesteelreinforcementas:

[2.54]Insertingequation[2.54]intoequation[2.51]leadstothedimensionlessreducedmomentμ,

whichisexpressedas:

[2.55]Equation[2.55]isafourth-orderpolynomialequationinαandiswrittenas:

[2.56]

2.2.5.CaseofpivotBforatriangularcross-sectionForμ≥μAB,thesectionhastobedesignedwithrespecttoPivotB.AtPivotB,thecompressionstress in concrete is equal to the limit compressive stress at SLS σc,sup = fcs, and thedimensionlessreducedmomentμisdirectlyobtainedfromequation[2.51]as:

[2.57]Equation[2.57]isathird-orderpolynomialequationinα,andiswrittenas:

[2.58]Itisshownthatintheparticularcaseofatriangularreinforcedconcretesection,theorderof

theneutralaxispositionpolynomialequation ishigher inPivotA than inPivotB, as for therectangular cross-section without compression steel reinforcement. Furthermore, it isinteresting to note that the order of the polynomial equation is 1ο higher for the triangularsectionthanitisfortherectangularsection.

2.2.6.Staticmomentequationforatriangularcross-section

Foragiven reinforcedconcrete triangularcross-section, thedeterminationof thepositionoftheneutralaxiscanbeobtainedfromthestaticmomentequation,whichissimplywrittenas:

[2.59]As a result, a third-order equation is obtained for the determinationof the positionof the

neutralaxis:

[2.60]Inthiscase,again,theorderofthepolynomialequationissuedofthestaticmomentequation

isonedegreehigherforthetriangularsectionthanitisfortherectangularsection.

2.2.7.Designexampleofatriangularcross-sectionThecross-sectioncharacteristicsofthetriangularcross-sectionareshowninFigure1.43.Thecross-sectionhasthefollowingcharacteristics:b0=0.8m;d=0.9×h.ThereinforcedconcretesectioniscomposedofB500Bsteelbars(fyk=500MPa)andtheC30/37typeofconcrete(fck=30MPa). The equivalence coefficient is given for this problem as: αe = 15. The beam issolicitated in simple bendingwithMser= 1MN m. The exercise consists of controlling thedesignofthisreinforcedtriangularcross-sectionatthestressSLS.Asthetriangularcross-sectionisanequilateraltriangle,thedepthhisrelatedtothewidthb0

as:

[2.61]ThecriticalparametersαABandμABarefirstcalculatedfromequation[2.53]as:

[2.62]Wethencalculatethereducedmomentμ,whichiscomparedtothecriticalreducedmoment

μAB:

[2.63]Asμ≥μAB,thesectionhastobedesignedwithrespecttoPivotB.Thedeterminationofthe

positionofneutralaxisneedstosolveathird-orderequationinαgivenbyequation[2.58],andnumericallypresentedinthefollowingformat:

[2.64]WeuseCardano’smethod tocompute the rootof thiscubic.Thecanonicalparametersare

calculatedfrom:

[2.65]Itcanbecheckedthat4p3+27q2=56.3963>0,andthecubicadmitsonlyonerealsolution

(seeAppendix1):

[2.66]ThetensilesteelreinforcementareaAs1isthendeducedfromtheconsiderationofthenormal

forceequilibriumequation:

[2.67]AsthesectionisdesignedunderPivotB,thestressesoftheextremalfibersaregivenby:

[2.68]Theareaoftensilesteelreinforcementisthencalculatedas:

[2.69]Wefinallyobtainforthiscross-sectiondesignedatPivotB,As1=68.07cm2.Wecanconsider

4ϕ25+4ϕ40(As1=69.9cm2)forthetensilesteelreinforcement(seeAppendix2).

2.3.Composedbendingwithcombinedaxialforces

2.3.1.Steelreinforcementdesignforagivenreinforcedconcretesection

Equationsof sections1.8,1.9and1.11 (Chapter1)arevalidboth in simplebendingand incomposed bending with combined axial forces. The bending moment and normal forcescomponentsaregivenatthecenterofgravityofthetensilesteelreinforcement(Mact,Nact)(seeFigure2.7forrectangularcross-sectionandFigure2.8forT-cross-section).

Figure2.7.Designofarectangularreinforcedconcretecross-sectionatserviceabilitylimitstateforcomposedbendingstatewithcombinedaxialforces

When dealingwith the design of the steel reinforcement quantity for a given solicitation,dependingon thepivotnatureof the reinforcedconcrete section, theequationsgiven for theposition of the neutral axis are different in PivotA and PivotB. In PivotA, for a T-cross-section (the rectangularcross-section isconsideredasaparticularcaseofaT-cross-sectionwithbw=b), the determination of the position of the neutral axis needs to solve the cubicequation [1.102], and the tensile steel reinforcement can then be calculated in composedbending from equation [1.105]. In Pivot B, for a T-cross-section, the determination of thepositionoftheneutralaxisalsoneedstosolveacubicequationgivenbyequation[1.111],andthe tensile steel reinforcement can then be calculated in composed bending from equation[1.112].ThelimitcasebetweenthetwopivotsisreferredtoaspivotAB,andischaracterizedbythelimitvaluefortherelativepositionoftheneutralaxisαABdefinedfromtheupperfiberofthecross-sectionasαAB=αefcs/(αefcs–fss).

Figure2.8.DesignofareinforcedconcreteT-cross-sectionatserviceabilitylimitstateforcomposedbendingstatewithcombinedaxialforces

2.3.2.Determinationofthepositionoftheneutralaxis–simplebending

As shown in section 2.1, Chapter 2, the calculation of the stresses for a given reinforcedcompositecross-sectionisadifferentproblem,whichisassociatedwiththedeterminationofthe position of neutral axis inside the cross-sectionwith the given steel reinforcement. Forsimplebendingsolicitation,withoutnormalforce(Nser=0), thisproblemissolvedwith theso-calledstaticmomentequation,whichleadstoasecond-orderpolynomialequationgivenbyequation[2.25]foraT-cross-section.Again,therectangularcross-sectioncanbeviewedasaparticularcaseofaT-cross-sectionwithbw=b.Thestressesineachpartofthecross-sectionare thencalculated from theequivalent linear stressdiagramgivenbyequation[2.13].NotethatthedeterminationofthepositionofneutralaxisinthiscasedoesnotrequirethespecificnotionofPivotAorPivotB.Forsimplebendingsolicitation,withoutnormalforce(Nser=0),thepositionoftheneutralaxisissummarizedbelowforaT-cross-sectionas:

[2.70]

withthesteelreinforcementratiodefinedfrom and .Intheparticularcaseofarectangularcross-section,thisequationissimplifiedas:

[2.71]

2.3.3.Determinationofthepositionoftheneutralaxis–composedbendingwithnormalforcesolicitation

2.3.3.1.Rectangularcross-sectionThe acting solicitation evaluated at the center of gravity of the tensile steel reinforcement,leadingtotheequilibriumequation:

[2.72]Thestaticmomentequation[2.6]can stillbeused for thenormal forcecomponent, and is

detailedbelowforthecaseoftherectangularcross-section:

[2.73]The bending moment equation evaluated at the center of gravity of the tensile steel

reinforcementcanbedevelopedas:

[2.74]Forarectangularcross-section,wefinallyobtaintheratiobetweenthebendingmomentand

normalforce:

[2.75]Asinthecaseofasimplebendingmoment,thisequationisindependentonthepivotnature

ofthecross-section.Foraloadingwithconstanteccentricityeact,wehave:

[2.76]Inthiscase,theneutralaxispositionshouldbethesolutionofacubic,whichisindependent

ofthesolicitationexceptthroughtheeccentricityparametereact:

[2.77]ThiscubicequationcanthenbesolvedusingCardano’smethod.Suchacubicequationcan

be found in some other textbooks such as in [WAL 90] for rectangular cross-sections or in[ROB 74] for both rectangular and T-cross-sections [ROB 74]. This equation can be

understoodasthegeneralizationofthestaticmomentequationforcomposedbendingwiththecombinednormalforceeffect.ThisequationisalsoknownasthecubicequationoftheFrenchreinforcedconcretedesign

rules,datingfrom1906(Circulairedu20Octobre1906)(andreportedinthebookofMagny[MAG14])ortheFrenchreinforcedconcretedesignrulesdatingfrom1934(règlementsdesmarchésde l’état de1934 – in French),whichwas recently reported in [THO09], and ispresentedinthefollowingcanonicalformatforrectangularreinforcedconcretecross-sections:

[2.78]It can be easily shown that equations [2.78] and [2.77] are equivalent. We note that the

simplebendingcasecanbededucedasymptoticallywithaninfiniteeccentricityleadingto:

[2.79]whosesolutionhasbeenalreadygiven inequation[2.30].Thecubicequation[2.77] canbesimplifiedincaseoftensilesteelreinforcementwithoutcompressionsteelreinforcementAs2=0as:

[2.80]

2.3.3.2.T-cross-sectionThe staticmoment equation [2.6] can still be used for the normal force component, and isdetailedbelowforthecaseofaT-cross-section:

[2.81]The bending moment equation evaluated at the center of gravity of the tensile steel

reinforcementcanbedevelopedas:

[2.82]Thepositionof the neutral axis givenby the dimensionless parameterα can be computed

fromthefractionalequationforaloadingwithconstanteccentricityeact:

[2.83]Wealsoobtainacubicequationwithrespecttotheparameterα:

[2.84]which,again,canbesolvedusingCardano’smethod.ThisequationisalsoknownasthecubicequationoftheFrenchreinforcedconcretedesign

rulesdatingfrom1906(Circulairedu20Octobre1906)(seealso[MAG14])ortheFrenchreinforcedconcretedesignrulesdatingfrom1934(règlementsdesmarchésdel’étatde1934–inFrench),whichwasrecentlyreportedin[THO09]forT-cross-sections,andispresentedinthefollowingcanonicalformat:

[2.85]Itcanbeeasilyshownthatequations[2.84]and[2.85]areequivalent.Itisworthmentioning

thatequation[2.85]ispresenteddirectlyincanonicalformatsothatCardano’smethodcanbedirectlyappliedwithoutvariabletransformation.Wenote that thesimplebendingcasecanbededucedasymptoticallyfromequation[2.84],

withaninfiniteeccentricityleadingto:

[2.86]whosesolutionhasbeenalreadygiveninequation[2.26].

2.3.4.Exercisesforcomposedbendingwithnormalforcesolicitation

2.3.4.1.Designofaspecificrectangularcross-section

2.3.4.1.1.DataoftheproblemArectangular cross-sectionof a reinforcedconcretebeam is solicited in composedbendingwithnormal forces.Theexternal screwelements aregivenat thepointG of the rectangularcross-section (see Figure 2.9). The pointG is located at the center of gravity of the fullrectangularcross-section(seeFigure2.9).Thedataoftheproblemaregivenbelow:

[2.87]

Figure2.9.Designofareinforcedconcreterectangularcross-sectionatserviceabilitylimitstateforcomposedbendingstatewithcombinedaxialforces;solicitationdefinedatthecenterofgravityofthecross-section

Theexerciseconsistsofdesigningthetensilesteelreinforcementofthisreinforcedconcretebeam solicitated in composed bending with normal compression. In the second part of theproblem,itissuggestedtocontrolthedesignofagivenreinforcedconcretesectionforagivensolicitationincomposedbendingwithnormalcompression,orequivalentlyforagivenloadingeccentricity (see Figure 2.10). For an easier resolution of the problem, the design isdecomposedintodifferentstepsthatcanbehelpfulforoptimizingthedesignofthisreinforced

cross-sectionatSLS.

Figure2.10.Interactiondiagraminthenormalforce–bendingmomentspace

1)Determine the screw reduction elements (momentMEd and normal forceNEd) of theexternalscrewforcesevaluatedattheframeorigin.2)Inthisquestion,thecharacterizationofthepivotAB is investigated.It isrecalledthatthepivotABisdefinedattheboundaryofPivotAandPivotBfromastressattheupperfiber equal to fcs and a stress at the center of gravity of the tensile steel reinforcement(lowerfiber)equalto fss.GivethevalueαABof therelativedepthof this limitcase.Byconsidering thatMAB is the associatedmoment of this limit case, calculate this bendingmomentandconcludeonthetypeofpivotthatcontrolsthedesignofthisrectangularcross-sectionwiththissolicitation.3)Givetheanalyticalexpressionof thenonlinearequationfor thispivotwithrespect tothedimensionlesspositionoftheneutralaxis,andsolvethispolynomialequationwiththegivendataoftheproblem,especiallywiththeexternalsolicitationgivenby{MEd,NEd}.4) Under the action of the external screw {MEd, NEd}, calculate the stress level in theupperfiberofconcreteσc,sup,thestressinthetensilesteelreinforcementσs1andthesteelareaAs1ofthesteelreinforcementneededattheSLS.Thiswillconcludethedesignofthereinforcedconcretesectionforthegivensolicitation.5)Forthefollowing,wewouldliketocontrolthedesignofagivenreinforcedconcretesection,withthetensilesteelreinforcementcomposedof6HA14(withAs1=9.236cm2).Undertheactionoftheexternalscrew{MEd,NEd},calculatethestresslevelintheupperfiber of concrete σc,sup, the stress in the tensile steel reinforcement σs1 and check thedesignofthisreinforcedconcretesectionattheSLS.

2.3.4.1.2.Resolution

1)TheexternalscrewforcesevaluatedattheoriginpointOarecalculatedfromthescrewforcesgivenatthepointGfromthetransportrule:

[2.88]ItcanbecheckedthatMEd≥MG,whichisphysicallyconsistentwithrespecttotheapplied

load.Theequivalenteccentricityoftheexternalscrewforcesisalsocalculatedfrom:

[2.89]2)ThelimitcaseatPivotABischaracterizedbythelimitvalues:

[2.90]ItiseasytocheckthatMEd≥MABandthesectionbehavesatPivotBattheSLS.3) At Pivot B, the dimensionless position α of the neutral axis is expressed indimensionlessformatwith:

[2.91]4)ThetensilesteelreinforcementareaneededattheSLSisthencalculatedfrom:

[2.92]5)Fortheverificationofthereinforcedconcretecross-section,acubicequationhastobesolvedfrom:

[2.93]Cardano’smethodisusedtosolvethiscubicequation.

[2.94]WeuseCardano’smethod tocompute the rootof thiscubic.Thecanonicalparametersare

calculatedfrom:

[2.95]Itcanbecheckedthat4p3+27q2=−1363.73<0,andthecubicequationadmitsthreereal

solutions (seeAppendix1), calculated from the following termArc cos ,andleadsto:

[2.96]Thethirdsolutionisthephysicalsolutionoftheproblemα=0.42567.Thestressinthemost

compressedfiberofconcreteiscalculatedfrom:

[2.97]

Itiseasytocheckthat and Clearly,thesectioniscorrectlydesignedattheSLS,anditsbehaviorisclosetotheoneinPivotB.

2.3.4.2.DesignofaspecificT-cross-section

2.3.4.2.1.DataoftheproblemAT-cross-sectionofareinforcedconcretebeamissolicitedincomposedbendingwithnormalforces.TheexternalscrewelementsaregivenatthepointOchosenatthecenterofgravityof

thetensilesteelreinforcement(seeFigure2.8).Thedataoftheproblemaregivenbelow:

[2.98]Theexerciseconsistsofdesigningthetensilesteelreinforcementofthisreinforcedconcrete

beamsolicitatedincomposedbendingwithnormalcompression.ThecalculationofthetensilesteelareaneedstodetermineanalyticallyandnumericallythefunctionMres,s(α),whereMres,sis the resistantmomentof the cross-section atSLS, evaluated at the center of gravityof thetensile steel reinforcement and α is as usual the dimensionless depth of the neutral axisposition.ItcanbeshownthatthefunctionMres,s(α)iscontinuousinthedomain[0,h/d]andisexactly

knownatleastineachsub-intervalasapiecewisefunction.Thetotaldomainforthevariationoftheneutralaxispositionαissubdividedintosub-domainsconnectedatsomecharacteristicpoints associatedwith amaterialor ageometricaldiscontinuity.These characteristicvaluescanbeclassifiedinaformalsuite(α0,α1,α2,…,αn)of[0,h/d]withα0=0<α1<α2<…<αn=h/dsuchthatforeachi {1,2,…,n},thereexistsanapplicationMbi(α),thatiscontinuousin[αi-1,αi]andverifies:

EachfunctionMbi(α)canbecalleda“branch”ofMres,s(α),whichdiffers fromeachotherfrom the discontinuous nature of the material or the geometrical characteristics of thereinforced cross-section. The choice of the relevant branchwith respect to the design of agivenreinforcedconcretesectioncanbedonefromthecomparisonoftheboundarymomentswithrespecttotheserviceabilitymomentMact,s,leadingtoauniquenonlinearequationtobesolvedforthedeterminationofthepositionofneutralaxisα.Foraneasier resolutionof theproblem, thedesign isdecomposed intodifferentsteps that

canbehelpfulforoptimizingthedesignofthisreinforcedcross-sectionatSLS.1)Letusfixεc,sup=0(orequivalentlyα=0).Determineforthisparticularpositionoftheneutralaxis,theexpressionoftheresistantmomentM0withrespecttothevariablesoftheproblem.2)WearenowinterestedincalculatingtheresistantmomentMtatSLS,whichcorrespondsto the position of the neutral axis located at the distanceh0 from the upper fiber of thecross-section (boundary between the flanged portion of the beam and the web). After

explainingthepivotassociatedwiththislimitcase,itissuggested,tofindanalytically,thisbendingmomentMt.3)Inthisquestion,thecharacterizationofthepivotAB is investigated.It isrecalledthatthepivotABisdefinedattheboundaryofPivotAandPivotBfromastressattheupperfiber equal to fcs and a stress at the center of gravity of the tensile steel reinforcement(lowerfiber)equalto fss.GivethevalueαABof therelativedepthof this limitcase.Byconsidering thatMAB is the associatedmoment of this limit case, calculate this bendingmoment and specify how the section can be calculated for this limit case (rectangularcross-sectionorT-cross-sectioncalculations).4)Afterhavingclassifieddifferentcharacteristiccases,analyzethetypeofpivotandthetype of calculation (rectangular cross-section or T-cross-section calculations),which isneededforthedesignofthisT-cross-sectionatSLS.Givetheanalyticalexpressionofthenonlinear equation for this pivot with respect to the dimensionless position of theneutralaxisα.5)Undertheactionoftheexternalscrew{MO,NO},calculatethestresslevelintheupperfiberofconcreteσc,sup,thestressinthetensilesteelreinforcementσs1andthesteelareaAs1ofthesteelreinforcementneededattheSLS.

2.3.4.2.2.Resolution1) In this case, the neutral axis is located at the upper fiber of the reinforced concretesection,α=0,meaningthatthecross-sectionisfullyincompressionorintension.AtPivotA,thelowersteelreinforcementsofareaAs1areintensionwithastrainvalueεs1=fss/Es.Byneglectingthetensilepartoftheconcrete,thebendingmomentM0issimplycalculatedfrom:

[2.99]Thisnegative signof thebendingmoment is linked to the tensionbehaviorof theupper

steelreinforcementofareaAs2.Thereducedmoment isequaltoμ0=−2.148×10−4.2)Thelimitcaseat theboundarybetweentherectangularcross-sectionandtheT-cross-

section is obtained from . It means that thiscaseistypicallyruledbyPivotAforarectangularcross-section:

[2.100]

Wecalculate,inthiscase,thereducedmoment .3)Asαt=h0/d<αAB ,thecalculationoftheresistantmomentatPivotABisnecessarilyassociatedwiththeT-cross-sectionformulaeas:

[2.101]

Wecalculate,forPivotAB,thereducedmomentequalto .4)Forthesolicitation(M0=0.49MN m;N0=0.49MN),wehaveM0 [Mt;MAB]andthen thesection iscontrolledbyPivotAwithaT-cross-sectionalcalculation.ThecubicequationforthedeterminationofthepositionoftheneutralaxisatPivotAforaT-cross-sectionisgivenbyequation[1.102]withthecoefficientsnumericallycalculatedas:

[2.102]Cardano’smethodisused(seeAppendix1),withthecanonicalparameterscalculatedas:

[2.103]5) At Pivot A, the stress in the tensile steel reinforcement is equal to the steelserviceability stress limit σs1 = fss. The steel area is obtained from the normal forceequilibriumequation:

[2.104]WefinallyobtainforthisT-cross-sectiondesignedatPivotA,As1=1.44cm2.Wecandesign

thecross-sectionwith3ϕ8(1.508cm2).Inthiscase,asshowninFigure2.11,thesectionbehavesinPivotAandisquitefarfromthe

PivotAB.Equationsofeachbranchcanbeanalyticallygiven.Forinstance,theexpressionofthe reducedmomentbranch inPivotAwhen thecross-sectionbehavesas rectangular cross-

sectionisgivenby:

[2.105]ThebranchinPivotAbutwithacompleteT-cross-sectionalbehaviorisgivenby:

[2.106]Finally,thebranchinPivotBbutwithacompleteT-cross-sectionalbehaviorisgivenby:

[2.107]The continuous branches shown in Figure 2.11 are specific of the cross-section. For the

solicitationinvestigatedintheproblem,thecross-sectionalbehaviorismaterializedbyapointwiththecoordinates(α,μ)=(0.1998;0.1088).

Figure2.11.Momentversusthepositionofneutralaxisforeachbranch;designofareinforcedconcreteT-cross-section

2.4.DeflectionatServiceabilityLimitState

2.4.1.Effectofcrackonthebendingcurvaturerelationship

2.4.1.1.TensionstiffeningphenomenonAsreportedinEurocode2,theappearanceandutilityofthestructuremaybeimpairedwhenthecalculatedsagofabeam,slaborcantileversubjectedtoquasi-permanentloadsexceedsabeam’s span/250. The sag is relative to the supports.Deflections that can damage adjacentparts of the structure should be limited. For the deflection after construction, a beam’sspan/500isnormallyanappropriatelimitforquasi-permanentloads.ThedeflectioncontrolatSLSisthenformulatedas:

[2.108]where forstructureswhereadjacentpartsshouldnotbeaffectedbythedeflectionofthebeam(withaspan lengthequal toL)or in theothercases.Thedeflectionsarecomputedfromanappliedloadwithanonlinearbending-curvatureconstitutivelawthattakesintoaccountboth the reductionofstiffnessdue to thecrackingprocess in the tensionpartofconcreteandthetime-dependentphenomenasuchascreepthatsoftenstheapparentstiffnessofthereinforcedconcretebeam.Considerthebendingbehaviorofareinforcedconcretebeam,asshowninFigure2.12.

Figure2.12.Moment-curvaturerelationshipatserviceabilitylimitstateforthecalculationofdeflectionincludingtensionstiffeningphenomenon

Thelawtocalculatethereductionofstiffnessisthefollowing:

[2.109]whereκisthecurvature,Misthebendingmoment,EIIisthebendingstiffnessoftheuncrackedbeam,andEIIIisthebendingstiffnessofthecrackedbeam.Forphysicalreasonslinkedtothemicro-crackingprocess,thestiffnessofthedamagedbeamissmallerthanthestiffnessofthe

undamagedbeamEIII≤EII.Notethatthislawcanalsobeformulatedas:

[2.110]The contribution of the cracking part of concrete in the global stiffness of the reinforced

concretebeamiscalledthetensionstiffeningeffect.InEurocode2, thedistributioncoefficient isanempiricalcoefficient that isassumedtobe

dependentonthebendingvariableas:

[2.111]Intheaboveequation,ζ=0correspondstotheuncrackedsectionandζ=1correspondsto

thecrackedsection.β1isacoefficientthattakesintoaccountthetime-dependenteffects:β1=1forshort-termanalysesandβ1=1/2forlong-termanalyses(typicallyincaseofpermanentorquasi-permanentloading).Infact,ζcouldbeinterpretedasacross-sectionalbendingdamageparameter,evenif thelawisessentiallyahardeninglaw.Mcr is thecrackingmoment that isassociatedwithastressvalueinthelowerfiberofconcreteσc,infequaltotheuniaxialtensilestrengthofconcretefctmintheuncrackedsection.Withrespecttodeflectioncontrol,thelong-termanalysisisthemostunfavorable,andcreep

effectshavetobetakenintoaccountwiththecreepcoefficientφgiven inequation[1.71] toderive the effective Young’s modulus of concrete . Shrinkage effectsshouldalsobeaddedforanexhaustivedesign.

2.4.1.2.Uncrackedsection

2.4.1.2.1.Uncrackedsection–rectangularcross-sectionConsidera rectangularcross-sectionas shown inFigure1.28.This section is theuncrackedsection,meaning thatwedo take into account the tensionpart of concrete.We calculate theparametersoftheuncrackedhomogenizedsectionasfollows.Thepositionoftheneutralaxisisobtainedfromtheupperfiberofthecross-sectionas:

[2.112]The equivalent stiffness of the uncracked section is then EII = Ec,eff × II. The cracking

momentcanbeexplicitlycalculatedfrom:

[2.113]Wenotethatthiscriticalmomentdoesnotdependonthesolicitation,butonlydependsonthe

materialandgeometricalparametersofthereinforcedconcretecross-section.

2.4.1.2.2.Uncrackedsection–T-cross-sectionForaT-cross-section,theformulavalidfortherectangularcross-sectioncanbegeneralizedasfollowing,whentheneutralaxisoftheuncrackedsectionislocatedintheweb:

[2.114]The equivalent stiffness of the uncracked section is then EII = Ec,eff × II. The cracking

momentcanbeexplicitlycalculatedfrom:

[2.115]

2.4.1.3.Crackedsection

2.4.1.3.1.Crackedsection–rectangularcross-sectionInthecrackedsectionwithrectangularcross-section,weusetheequationof“staticmoment”toderivetheneutralaxispositionfromequations[2.15]and[2.16]thatarerewrittenas:

[2.116]Theinertiaofthehomogenizedcrackedsectioniscalculatedfrom:

[2.117]TheequivalentstiffnessofthecrackedsectionisthenEIII=Ec,eff×III.Wealsonotethatthe

parameters of the cracked sectiondonot dependon the solicitation, but dependonlyon thematerialandgeometricalparametersofthereinforcedconcretecross-section.

2.4.1.3.2.Crackedsection–T-cross-sectionInthecrackedsectionwithT-cross-section,weusetheequationof“staticmoment”toderivetheneutralaxispositionfromequations[2.26]and[2.27]thatarerewrittenhereas:

[2.118]Theinertiaofthehomogenizedcrackedsectioniscalculatedfrom:

[2.119]TheequivalentstiffnessofthecrackedsectionisthenEIII=Ec,eff×III.

2.4.1.4.Onthenonlinearbending-curvatureconstitutivelawatSLSInthepost-crackedsection,anonlinearbendingcurvatureconstitutivelawappearsthatcanbepresentedasadamagelaw.

[2.120]Thenonlinearpost-crackedcurvature-bendingmomentiscontinuousforβ1=1.

2.4.2.SimplysupportedreinforcedconcretebeamAsimplysupportedreinforcedconcretebeamisstudied,loadedbysomeuniformdistributedloadattheSLSdenotedbypser.ThebeamhasalengthL.Thebendingmomentisparabolicinthisstaticallydeterminatesystem,andisgiven,forx [0;L],by:

[2.121]It is assumed that the reinforcement is homogeneous along the beam. For symmetrical

reasons,twosymmetricalpartsofthebeamoflengthLIcanbehighlightedwithanuncrackedbehavior(seeFigure2.13):

[2.122]ThelengthLIcorrespondstoabendingmomentthatisequaltothecriticalmoment:

[2.123]The smallest solution is of interest, with respect to the notation of Figure 2.13, which is

writtenas:

[2.124]

Figure2.13.Simplysupportedreinforcedconcretebeamloadedbyuniformloads;definitionofcrackedanduncrackedportionofthebeam

Forthecrackedpartofthebeam,namelyforx [LI;L−LI],adetailedstructuralanalysisisneededfortheintegrationofthenonlinearconstitutivelawalongthebeam.

2.4.3.Calculationofdeflection–safeapproachThe safer approach (approach in security)would consist of neglecting the tension stiffeningeffectofconcrete.Inthissimplifiedapproach,thecrackedsectionisassumedalongtheentirebeam:

[2.125]Inthiscase,thissimplifiedapproachleadstothedeflectionofthecrackedbeam:

[2.126]Thisapproach isequivalent toneglecting the tensionstiffeningeffectand toneglecting the

uncrackedcontributionofthebeamaroundthesupports.Ifthisinequalityischecked,itisnotnecessarilytorefinetheapproachestocontrolthedeflection.

2.4.4.Calculationofdeflection–amorerefinedapproach;tensionstiffeningneglected

Amorerefinedapproachwouldbebasedonthefollowingconstitutivelaw:

[2.127]The uncracked contribution of the beam around the supports is taken into account, but the

tensionstiffeningeffectinthecrackedportionofthebeamisneglected,andapproximatedbythe fully cracked section behavior. For homogeneous reinforcement along the beam, thisengineering approach is in fact equivalent to a piecewise homogeneous beam with twodifferentstiffnessesEIIandEIII.Inthiscase,thedeflectioninthebeamcanbecalculatedfromtheintegrationofsecond-orderdifferentialequationsgivenby:

[2.128]For a symmetrical reason, onlyhalf of thebeam is considered in the study.Theboundary

conditionsattheconnectionx=LIexpressthecontinuityofthedeflectionw(x)andtheslopeanglew′(x):

[2.129]Thedeflectionsineachpartofthebeamarefinallycalculatedas:

[2.130]Thedeflectionatmid-spanf=w(L/2)ispresentedinthefollowingformat:

[2.131]Asanticipated, thedeflectionwith thismodel issmaller than thedeflectionof thecracked

sectionf≤fII.Itisalsoeasytocheckthecorrectnessoftheseformulasforthelimitcases:

[2.132]Itisquiteinterestingtooutlinethatthedeflectionfinequation[2.131]isanonlinearfunction

oftheloadpserduetothenonlineardependenceofLIwithrespecttopser,asgivenbyequation[2.124].Clearly,thiscalculationisnotbasedonelasticconstitutivelaws.Withthismodel,thedeflection,Δf,hastobesubtractedfromthedeflectionfIIofthecracked

beam,whichcanbefinallypresentedas:

[2.133]

2.4.5.Calculationofdeflection–amorerefinedapproach;tensionstiffeningincluded

It is possible to take into account the tension stiffening effect with the nonlinear bending-curvatureconstitutivelaw.Amorerefinedapproachwouldbebasedonthefollowingconstitutivelaw:

[2.134]Inthiscase,thedeflectioninthebeamcanbecalculatedfromtheintegrationofsecond-order

differentialequationsgivenby:

[2.135]Integrating each second-order differential equation, and introducing the same boundary

conditionsasinequation[2.129],leadstothedeflectionsolution:

and

[2.136]Thedeflectionatmid-spanf=w(L/2)canbepresentedinthefollowingformat:

[2.137]Finally, when the tension stiffening phenomenon is taken into account, the deflection

correctionΔfhas tobesubtractedfromthedeflection fIIof thecrackedbeam,whichcanbefinallypresentedas:

[2.138]This result coincideswith the deflection calculation presented in theworked examples of

EC2[EUR08a].Of course, in the most general configuration, including, for instance, variable steel

reinforcementalongthebeam,ananalyticalintegrationisnotalwaysfeasible,andanumericalintegrationhastobeperformed.

2.4.6.ApproximatedapproachAnotherengineeringapproachpresentedintheliteratureconsistsofexpressingthedeflectionasacombinationofthedeflectionofthecrackedandtheuncrackedsectionas:

[2.139]Applying this engineering approach to the present simply supported beam leads to the

correctiondeflection:

[2.140]The two approaches (the exact one with and without tension stiffening effect) and the

engineeringapproacharecomparedinFigure2.14.Itcanbeseenthattheengineeringapproachgives a safer result than the exact one including the stiffening effect, as the correctiondeflectionissmallerfortheengineeringapproachthanfortheexactone.Itisalsoshownthatthe tension stiffening effect is significant with respect to the deflection correction termneglectingthisphenomenon.

Figure2.14.Effectofthetensionstiffeningphenomenononthedeflectionbehaviorofthecrackedbeam;β1=1

2.4.7.Calculationofdeflection–astructuralexample

2.4.7.1.DataoftheproblemWeconsiderareinforcedconcretebeamwitharectangularcross-section.AsshowninFigure2.13,thesimplysupportedbeamisloadedbysomeuniformserviceabilityloadpser.Thedataoftheproblemarethefollowing:

[2.141]Thisreinforcedconcretesectionhasnocompressionsteelreinforcement,As2=0.Giventhecreepcoefficientφ=2,theequivalentcoefficientαecanbecalculated.

[2.142]where themeanvalueof theconcreteYoung’smodulushasbeengiven for thisC25/30 typeconcreteinequation[1.72].ThequestionistochecktheSLSforthedeflectioncontrol.

2.4.7.2.CharacteristicsoftheuncrackedsectionFromequation[2.112],wecalculatethepositionoftheneutralaxisoftheuncrackedsectionyIandtheassociatedhomogenizedinertiaI1as:

[2.143]Thecrackingmomentcanbeexplicitlycalculatedfrom:

[2.144]Wecalculatetheloadratioas:

[2.145]The length of the uncracked part of the beam, denoted byLI, is calculated from equation

[1.247]as:

[2.146]

Thestiffnessoftheuncrackedsectionisobtainedfrom:

[2.147]

2.4.7.3.CharacteristicsofthecrackedsectionThecharacteristicsofthecrackedsectionaregivenbyequations[2.116]and[2.117]andaredetailedbelow:

[2.148]Thestiffnessofthecrackedsectionisobtainedfrom:

[2.149]

2.4.7.4.CalculationofdeflectionWecomputethecharacteristicsdeflectionfI,fIIand from:

[2.150]

It canbe seen that and thedeflectioncontrolneeds to take into account the tensionstiffeningeffecttochecktheadmissiblemaximumdeflection.From equation [2.137], the correction deflection induced by the tension stiffening

phenomenoniscalculatedforβ=1as:

[2.151]Wecalculatethetotaldeflectionincludingthetensionstiffeningeffectas:

[2.152]We conclude that this beam is correctly designedwith respect to the deflection control at

SLS.Notethattheapproximatedengineeringformula(whichissafewithrespecttotheexactoneincludingrigorouslythetensionstiffeningeffect)wouldleadinthiscaseto:

[2.153]In this case, the deflection SLS would have been possibly checked with this simplified

engineeringformula.

Chapter3

ConceptsfortheDesignatUltimateLimitState(ULS)

3.1.Introductiontoultimatelimitstate3.1.1.Yielddesign

Thepurposeofdesigningareinforcedconcretestructureistoachieveacceptableprobabilitiesthatthisstructureunderagivenloadwillnotreachalimitstate,namelytheserviceabilitylimitstate(SLS)ortheultimatelimitstate(ULS).Inthischapter, thestrengthofULSispresentedfor general in-plane bending loading including the normal force coupling. The ULS ischaracterizedbytherequirementthatthestructuremustbeabletowithstand,withanadequatefactorofsafetyagainstcollapse,toensurethesafetyofboththebuilding’soccupantsandthestructureitself.Thepossibilityofbucklingphenomenonorotherkindsofstructuralinstabilitieshasformallyalsotobetakenintoaccount.However,inthischapter,weonlyfocusonsectionalULSwithoutconsideringpossiblestructuralinstabilities.TheULSdesigncanbebasedontheconcept of yield design, where the maximum capacity of a given structure under a givenloadingsystemcanbeevaluatedfromthestrengthofitsconstituents(forareinforcedconcretestructure,thestrengthofthesteelreinforcement,thestrengthoftheconcreteandthestrengthoftheinterfacebetweentheseconstituents).AsgreatlyanalyzedbyTimoshenko[TIM83],YuandZhang[YU96],Heyman[HEY99]or

Salençon([SAL90],[SAL02]or[SAL06]),Galileo’sdialogs(1638)areoftenconsideredasa fundamental pioneering studyof the strength ofmaterials at the beam scale [GAL02].AsdetailedbySalençon[SAL06]:“ThemaintopictreatedintheFirstDayandtheSecondDayofGalileo’sDialogsistheresistancethatsolidsoffertofracturewithspecialconsiderationtoprismsandcylinderssubmitted to axial tensile loading or to “transverse”, i.e. bending forces.Without anyconsideration to the deformation of the considered solid before fracture, Galileo’sreasoning implicitly introduces the concept of a Continuum within which coherenceforces do act in order to maintain the filaments, fibres or any constituent particlestogether: onemay say thatGalileo opens theway to the concept of stress, whichwassettled explicitly some 200 years later. Having recognised that coherence forces andgravity forces in a solid are not related in the same way to its geometric scale, heperformswhat can be considered as the first striking example of dimensional analysis

withapplicationtosimilarity.Initscelebratedanalysisoftheresistanceofacantileverbeamsubmitted tobending,Galileogivesa firstattempt toderiving theresistanceofawhole solid submitted to some kind of loading from the resistance of its constituentmaterialdeterminedfromanothertest.”

Galileo’scantileverbeam(showninFigure3.1)canbeclassifiedasastructuralparadigmasthisstructuralmodelcontainssomestressgradientalongthebeam,allowingfortheso-calledlocalizationeffectthatwewilldetaillater.

Figure3.1.ThecantileverbeamstudiedbyGalileo(1638)

“ThisapproachmustalsobeassociatedwithCoulomb’scelebratedessay(1773)asoneofmilestonesofthetheoryofyielddesign,implicitlyevidencingthedualitybetweenforcesand virtual velocity fields, which is presently implemented in the virtual rate of workmethod.The theory of yield design is the fundamental root of theUltimateLimit Statedesignnowintroducedininternationalcodesforcivilengineering”.

3.1.2.Applicationofyielddesigntothecantileverbeam

3.1.2.1.Acantileverbeamwithgeneralinelasticbending-curvatureconstitutivelawThehomogeneouscantileverbeamoflengthLisloadedbyaverticalconcentratedloadPatitsend (Figure 3.2). We recognize Galileo’s cantilever beam previously solved by Galileohimself (1564–1642) using equilibrium, strength and dimensional arguments. The cantileverbeam loadedby a concentrated force canbeviewedas a typical caseofplasticbeamwithnon-constant bending moment. The beam is assumed to behave with a complex nonlinearbending-curvatureconstitutivelaw,includingpossibleirreversiblephenomena.Thenonlinearbending-curvatureconstitutivelawisconsideredasinFigure3.3.Thebending

strengthcriterionissimplygivenbythefollowinginequality:

[3.1]

whereMpistheyieldmoment(strengthcapacityofthecross-section).

Figure3.2.Thecantileverbeam

The reasoning that we will present here includes all kinds of structural applications(reinforced concrete design, steel structural design, timber engineering, etc.). Whenconsideringreinforcedconcretedesign, thestrengthcapacityof thecross-sectiondependsonthe strength of the steel reinforcement and the one of the concrete in compression. Thereasoning is independent of the constitutive law of the material and only depends on thestrength parameter Mp, which is the upper bending moment value. The reasoning isindependentofthehistorypathinthebending-curvaturespace.

Figure3.3.PossiblenonlinearbendingM-curvatureκlawforareinforcedconcretemember

Figure 3.4 also shows other possible bending-curvature constitutive law included in theyielddesignreasoning,whichisquiteindependentofthepathusedtoreachthestrengthlimitMp.

Figure3.4.Possiblenonlinearbending-curvaturelawforareinforcedconcretemember

3.1.2.2.StaticapproachbyinsideIn the case of the cantilever beam, the structural system is statically determinate, and thebendingmoment is then calculatedwithout any difficulties as a linearly decreasing functionassociatedwithaconstantshearforce:

[3.2]Introducingthebendingmomentexpressioninthestrengthcriterionleadstotheoptimization

problem:

[3.3]whichisalsoequivalenttothefollowinginequality:

[3.4]P+istheyieldloadorthemaximumcapacityofthebeamwithrespecttoitslocalsectional

strengthconstituentdependentonMp.Thisapproach iscalledastaticapproach,andgivesalowerboundofthevalueP+.However,fromequation[3.3],asthemomentdistributionistheexact one of the problem for this statically determinate problem, an upper bound of themaximumcapacityofthebeamcanalsobeobtained:

[3.5]Thisisalsocalledstaticapproachbyoutside(asdetailedbySalençon[SAL02]).

3.1.2.3.KinematicapproachbyoutsideThesameresultcanbeobtainedfromdirectapplicationoftheprincipleofvirtualwork:

[3.6]where[δθ] denotes the jumpof anyquantityδθ at thepoint of abscissa si. The fundamentalreasoningofthekinematicapproachofyielddesignisbasedonthesurroundingoftheinternal

workwithrespecttothestrengthcapacityofthebeam[SAL02]:

[3.7]Thesimplestvirtualdisplacementfieldistherigidmechanismdefinedbyoneconventionally

positivekinematicparameterδθ0:

[3.8]Theinequalityequation[3.6]isthenreducedto:

[3.9]The present reasoning can be generalized to geometrically nonlinear problems (see, for

instance,[CHA09a]).

3.1.2.4.DiscussiononthemeaningofthecapacityloadInthissimplestaticallydeterminatecase,bothstaticandkinematicapproachesgivethesamecapacityloadofthecantileverbeam:

[3.10]Note that no constitutive law arguments have been introduced in the reasoning. Only the

ultimate limit moment denoted by Mp has been used. The yield design theory gives usinformationabout themaximumstrengthcapacityof thiscantileverbeaminbending,withoutansweringthecomplexproblemofwhatwouldhappenbeyondthisstrengthlimit.Itmeansthatdesigningthiscantileverbeamwithrespecttothisstrengthcapacitywouldbesafe,whateverthe bending constitutive law considered at the sectional level. The yield design theory alsodoesnotanswer,generallyspeaking,whetherthestrengthcapacitywillbereachedduringthedeformationprocess.Onlyinthecaseofperfectplasticity(nohardening/softeningprocesses)withassociatedflowrule,wecanrelatethecapacityloadwiththelimit loadoftheelastic–plasticevolutionproblem.

3.1.3.Inelastic(plasticityorcontinuumdamagemechanics)bending-curvatureconstitutivelaw

3.1.3.1.Softeningplasticitymoment-curvatureconstitutivelawFigure3.5.Softeningplasticityconstitutivelawwithplasticcurvatureκp

In thissection,wewillpresent twocommonlyusedinelasticconstitutivelaws,namelytheplasticityandthedamageconstitutivelaws.Thesetwoconstitutivelawscanbeviewedastwoparticular cases of the more general laws introduced in Figure 3.4. The bending-curvatureelastoplastic model represented in Figure 3.5 is a standard plasticity model with negativehardening (softening). Softening here means that the strength decreased during the inelasticprocess.Using thedecompositionof the totalcurvatureκ intoanelasticpartandaplasticpart, the

moment-elasticcurvaturerelationgives:

[3.11]where theslopeof thebendingmoment-curvaturecurve is initiallyequal toEIandκp is theplasticcurvature(seeFigure3.5).Theplasticityyieldfunctionfisgivenby:

[3.12]whereM*isanadditionalmomentvariablethataccountsfor theloadinghistory.Theplasticcurvaturerate isobtainedusingthenormalityrule:

[3.13]Theplasticmultiplier mustsatisfythecomplementaryconditions:

[3.14]Equation[3.14]canbeviewedastheso-calledoptimalityKuhn–Tuckerconditions(see,for

instance, [JIR 02] for a discussion on Kuhn–Tucker conditions for inelastic analysis). Formonotonic loading with positive or negative bending moments, the plastic multiplier λ issimplyrelatedtotheabsolutevalueoftheplasticcurvatureλ=|κp|.Thehardening/softeningbeinglinear,thefollowingrelationholdsforthe“local”case:

[3.15]According to the signof theplasticmodulusk,we canget a softening fork =k− < 0 (as

showninFigure3.5),orahardeningfork=k+>0.Note that the plasticity softening branch is obtained by combining equation [3.11] with

equation [3.12] leading to the linear relationship between the bending moment and thecurvature,whichisexpressedinthecaseofsofteningplasticityas:

[3.16]Theslopeofthesofteningpartofthebending-curvaturerelationshipisthenequalto:

[3.17]Wewill showlater thatsucha“local”bending-curvatureconstitutive law leads to theso-

called Wood’s paradox where the propagation of the plasticity inelastic zone cannot holdbeyondthepeakload.Toavoid“local”snapback,thefollowingconditionshouldbeverified:

[3.18]

3.1.3.2.Softeningdamagemoment-curvatureconstitutivelawThecontinuumdamagemechanics(CDM)moment-elasticcurvaturerelationiswrittenas:

[3.19]where the slopeof thebendingmoment-curvaturecurve is initiallyequal toEIandD is thecross-sectionaldamageparameterthatevolvesbetween0and1(seeFigure3.6).Thedamageyieldfunctionfisgivenby:

[3.20]whereκ*isanadditionalcurvaturevariablethataccountsfortheloadinghistory.Thedamagerate isobtainedusingthenormalityrule:

[3.21]Theplasticmultiplier mustsatisfythecomplementaryconditions:

[3.22]Thehardening/softeningbeinglinear,thefollowingrelationholdsforthe“local”case:

[3.23]Thisdamagelawinthesofteningpartisequivalenttoequation[3.16].

Figure3.6.SofteningdamageconstitutivelawwithdamagevariableD

Wewillalsoshowlaterthatsucha“local”bending-curvatureconstitutivelawleadstotheso-calledWood’s paradoxwhere the propagation of the damage inelastic zone cannot holdbeyondthepeakload.Asfarasnoelasticunloadingappearsduringtheevolutionproblem,thetwoplasticityand

damagemechanicsmodelspresentedabovearemathematicallyequivalent.However, inbothcases, these kind of softeningmodels (plasticity orCDMmodels) cannot be used in such alocalformat,asthestructuralproblemtobesolved,evenforthesimplecantilevercase,isill-posed,astheinelasticlocalizationzonecannotpropagateinbothcases.Whateverthesofteninginelasticmodelconsidered,itcanbeshownthatthebeamcannotcollapsewithsuchkindsofmodels,asdetailedinthenextsection.

3.2.Postfailureanalysis3.2.1.Historicalperspective

Thissectiondealswiththepropagationoflocalizationinhardening–softeningplasticitybeams.The propagation of plasticity along a bending beam is studied for a piecewise hardening–softening moment-curvature relationship. Historically, moment-curvature relationships withsofteningbranchwerefirstintroducedforreinforcedconcretebeams[WOO68].Wood[WOO68] did point out some specific difficulties occurring during the solution of the evolutionproblemforplasticsofteningmodels.Moreprecisely,hehighlighted the impossibilityof theplasticsofteningbeamtoflow,iftheplasticcurvatureisassumedtobeacontinuousfunctioninspace, a phenomenon sometimes called Wood’s paradox. Hardening–softening plasticity

modelsmayconcernawideclassofstructuralmechanicsproblems.Suchsimplifiedmodelscan be useful for the fundamental understanding of bending of structural members at theirultimate state (reinforced concrete members, timber beams, composite members, etc.). Theplasticbucklingoftubesinbendingcanbealsomodeledwithahardening–softeningmoment-curvature relationship.Thebending responseof thin-walledmembers can also experience asofteningphenomenoninducedbythelocalbucklingphenomenon(seealso[CHA10b]).Wood’s paradox is met for local softening moment-curvature relationship. A non-local

(gradient) moment-curvature constitutive relation was introduced by Challamel [CHA 03]associated with some scale effects for controlling the postfailure of softening beams. Non-localmodelsatthebeamscaleabandontheclassicalassumptionoflocality,andadmitthatthebendingmomentdependsnotonlyon thestatevariables(curvature,plasticcurvature)at thatpoint.Non-local inelasticmodels(damageorplasticitymodels)weresuccessfullyusedasalocalizationlimiterwitharegularizationeffectonsofteningstructuralresponsein the1980s.Thenon-localcharacteroftheconstitutivelaw,generallyintroducedthroughaninternallength,is restricted to the loading function (damage loading functionorplasticity loading function).Pijaudier-CabotandBažant[PIJ87]firstelaboratedanon-localdamagetheory,basedontheintroductionofthenon-localityinthedamageloadingfunction.Thistheoryhastheadvantageofleavingtheinitialelasticbehaviorunaffected,andtocontrolthelocalizationprocessinthepost-peakregime.Itisworthmentioningthatthisideawasalreadyusedbeforetomodelshearbands.Gradientplasticitymodels(alsocalledexplicitgradientplasticitymodels)andintegralplasticitymodelsmaybedistinguished.In thecaseofexplicitgradientplasticitymodels, theplasticityloadingfunctiondependsontheplasticstrainanditsderivative,whereasforintegralplasticitymodels,theplasticityloadingfunctionisexpressedfromanintegraloperatoroftheplasticstrain.Moreover,itcanbeshown,asinthecaseofnon-localelasticmodels([ERI83],seealso[ELI12]), thatsomerelevant integralplasticitymodelscanbecast inadifferentialform [ENG 03]. These models are called implicit gradient plasticity models, but can beviewedasparticularcasesofintegralplasticitymodelswithspecificweightfunctionsdefinedasGreen’sfunctionofthedifferentialoperator[ENG03].More recently, an implicit gradient plasticitymodelwas used at the beam scale to solve

Wood’sparadoxinbeamswithmomentgradientandwithouthardeningrange([CHA08b]or[CHA 08c]). The localization phenomenon is controlled by a non-local softening plasticitymodel,basedonacombinationofthelocalandthenon-localplasticvariables(assuggestedby Vermeer and Brinkgreve [VER 94], see also [JIR 02]). It has been shown on simplestructural examples that the softeningevolutionproblemwaswellposedwith thisnon-localconstitutivelaw.Inparticular, theuniquenessof theevolutionproblemisclearlyobtainedinthepresenceofgradientmoment,typicallyforacantileverbeamsolicitedbyaverticalforce.Note that this uniqueness result of the evolution problem would not be obtained forhomogeneousstructureswithconstantgeneralizedstress(constantmoment)(see[DEB92]forgradientplasticitymodels,ormorerecentlyforthenon-localbeamproblem[CHA08c]).Thesamekindof results (lossofuniquenesswithuniformstateof stress)hasalsobeen recentlynoticed by Benallal and Marigo [BEN 07] for damage problems. Introduction of some

heterogeneities can restore the uniqueness property for these non-local damage problems[CHA 09b]. Most of the presented theoretical results deal with softening media withouthardeningrange.Hence,uptonow,veryfewresultsareavailableforhardening–softeningnon-local plasticity media, even if this configuration is of fundamental importance from anengineeringpointofview.

3.2.2.Wood’sparadoxTheaxialandtransversalcoordinatesaredenotedbyxandy,respectively,andthetransversedeflection isdenotedbyw.The stiffnessof thealreadycrackedbeam isdenotedbyEI (thetensionstiffeningeffectisneglectedwithrespecttothespecificreinforcedconcretebehavior).Attheendofthebeam,thedisplacementv=w(L)ofconcentratedforcePisusedtocontroltheloadingprocess.Thelocalmoment-curvaturerelationship(M,κ)consideredisbilinearwithalinear elastic part and a linear softeningpart (Figure3.5).Thecurvatureκ is related to thedisplacementfieldthankstothekinematicsκ=–w”.Thismodelisfirstconsideredinalocalform,that isstandardplasticitymodelwithnegativehardening.Thenon-localextensionwillbeinvestigatedlaterinthechapter.Mpisthelimitelasticmoment,andκY is thelimitelasticcurvature, related throughMp/κY=EI. Inpractice, thecurvaturecannot increase indefinitelyandislimitedbyκu(theultimateadmissiblecurvature).However,suchlimitationisnottakenintoaccountinthepresentstudy.Themaximumbendingmomentoccursatx=0,wherethebeamisclamped.Plasticrotation

startsassoonas thebendingmomentreaches theplasticbendingmomentMp.ThemaximumelasticdisplacementatthebeamendvYandthecorrespondingloadPYaregivenby:

[3.24]FordisplacementvsmallerthanvY(v≤vY),thebeamremainselasticandthedeflectioncan

be calculatedusing the elasticity solution.Forv greater thanvY (v ≥vY), the plastic regimestartsand thebeamcanbesplit intoanelasticandaplasticdomain.Thesizeof theplasticdomain is denoted by l0 ≤L (see Figure 3.2).We can introduce the notation of for thesofteningdomainand forthehardeningdomain.Consideringonlythesofteningprobleminthissection,thegoverningequationsintheplasticdomainare:

[3.25]wherew−denotesthedeflectionintheplasticregion.Theelasticadjacentdomainisgovernedby:

[3.26]w+isthedeflectionintheelasticregion.Theboundaryconditionscanbesummarizedas:

[3.27]Thedeflectionw(x)andtherotationw′(x)mustbecontinuousfunctionsofx(inparticularat

theintersectionoftheelasticandtheplasticdomains).Enforcingthatκp isalsoacontinuousfunctionofx(κp(l0−)=0),leadsto:

[3.28]ThisadditionalassumptiongivesWood’sparadox.Theunloadingelasticsolutionistheonly

possible solutionof the local softeningproblem, if the plastic curvature is assumed to be acontinuous function in space (Figure 3.7). This paradox can also be interpreted as theappearance of plastic curvature increments localized into one single section, leading to thephysicallynon-reasonablephenomenonoffailurewithzerodissipation.Thisparadoxiswelldocumentedintheliterature([WOO68],[BAZ76],[ROY01],[BAZ03]and[CHA05d]).Apossible way to overcome Wood’s paradox is to introduce a non-local plastic softeningconstitutivelaw(see[CHA08b],[CHA08c]).

Figure3.7.Wood’sparadox–localsofteningplasticitymodels

Exactlythesamereasoningcanholdforthelocaldamagesofteningbeamproblem(withtheconstitutive law shown in Figure 3.6, for instance), which is also associated withWood’sparadox. For this kind of constitutive law, a non-local damage softening also allows us tosolvethisapparentparadox.

3.2.3.Non-localhardening/softeningconstitutivelaw,avariationalprinciple

Fortheimplicitgradientplasticitymodel,thenon-localplasticcurvature isdefinedasthesolutionofthedifferentialequation:

[3.29]Therefore, acharacteristic length lc is introduced in thedefinitionof thenon-local plastic

curvature .AsshownbyEringen[ERI83]fornon-localelasticity,thisdifferentialequationclearly shows that the non-local plastic curvature is a spatial weighted average of thevariableκp.Thisspatialweightedaverageiscalculatedontheplasticdomain:

[3.30]where the weighting function G(x, y) is Green’s function of the differential system withappropriate boundary conditions. The non-local hardening/softening constitutive law ofmodulus k (k = k+ for hardening evolutions, k = k− for softening evolutions) including theassociated boundary conditions can be obtained from a variational principle, as alreadyobtained in the rate form for gradient plasticity [MUH 91]. The extension to non-localplasticity is inspired by the micromorphic approach recently developed for elastic andinelasticmedia, in a consistent thermodynamic framework [FOR09]. The following energyfunctionalcanbechosen:

[3.31]whereζ is a dimensionlessparameter that appears in thehardening/softening evolution law.Following a classical procedure, also used for explicit gradient plasticitymodels (see, forinstance,[MUH91]),theoveralldomaincanbedividedintoaplasticdomainandanelasticdomain.Thefirstvariationofthisenergyfunctionalleadstotheextremalcondition:

[3.32]Moreover, followingaGreen-type identityassociatedwith the self-adjointpropertyof the

regularizedoperator for relevantboundaryconditions, andaccording to thedefinitionof thenon-localplasticcurvature,thefollowingidentityholds:

[3.33]Therefore, usingMδκp = |M |δλ, the first variation of the energy functional can be also

simplifiedas:

[3.34]withtheassociatedconstitutivelawfortheelastic,andthenon-localhardening/softeninglaw:

[3.35]Thefollowingintegrationbypartcanbeconsideredforthedeflection:

[3.36]Theextremalconditionleadstotheequilibriumequationandtheyieldfunction:

[3.37]withthenaturalboundaryconditions:

[3.38]Thehigh-orderboundaryconditionsof thenon-localplasticitymodelare includedin these

equations,andareappliedattheboundaryoftheelastoplasticdomain.Consideringthehigherorderboundaryconditionsattheelastoplasticboundaryhastheadvantagetobevariationallyand physicallymotivated. In this case, the non-local plastic variable is calculated over theplasticdomain(seeequation[3.30]as formost integral-basednon-localplasticitymodels–seealsoPolizzottoetal.[POL98]forthethermodynamicsbackgroundofintegral-basednon-localplasticitymodels).Forinstance,auniformplasticvariableintheplasticdomainwouldlead to a non-local variable that is identical.The introduction of the higher order boundaryconditionsatthephysicalboundaryofthesolidwouldleadtodifferentresults,asdetailedin[CHA 10b]. Note that the non-local plastic curvature does not necessarily vanish at theboundaryoftheelastoplasticdomain,whereastheplasticcurvatureisacontinuousvariableofthespatialcoordinateandvanishesattheboundarybetweentheelasticandtheplasticdomain.Note that a different functional was considered in [CHA 08b] leading to the same

constitutivebehaviorwiththesameboundaryconditions[CHA10b]:

[3.39]Thenon-localplasticconstitutivelawappearingfromthevariationalprincipleisbasedona

combinationofthelocalplasticcurvatureandthenon-localplasticcurvature.

[3.40]Such a combination of local and non-local plastic variables was initially proposed by

VermeerandBrinkgreve[VER94]forsofteningevolutions.Inthepresentcase,thismodelcanalsobewrittenintheformofadifferentialequation:

[3.41]In the case of the cantilever beam, it is worth mentioning that the non-local differential

formatlookslikeagradientplasticitymodel:

[3.42]However, the boundary conditions written in equation [3.38] for the non-local plastic

curvaturesaredifferent from theboundaryconditionsof theusualgradientplasticitymodelsdealing with only the derivative of the plastic curvatures. Equation [3.41] is the plasticitygeneralization of themixed elastic constitutive law investigated for a one-dimensional non-localelasticbar([CHA09d];seealso[AIF03]ormorerecently[AIF11]and[ELI12]):

[3.43]where σ and ε are the uniaxial stress and the uniaxial strain, respectively. Equation [3.43]givessatisfactoryresultsforadispersivewaveequationoflatticemodels.The sign of ζ controls the well-posedness of the plasticity evolution problem for both

hardeningandsofteningbehaviors(asshownin[CHA08c]).Typically,ζcanbeunderstoodasaregularizationparameter.Forhardeningevolutions,ζ+hastobepositive,leadingtothenon-localhardeningconstitutivelaw:

[3.44]Thismodelcomprisesthepurelynon-localplasticsofteningmodel(a=0),andthegradient

plasticitymodelforhardeningevolution(lc=0).Thedifferential formatequation[3.44]hasbeenalreadyusedinthepastinstructuralengineeringforsomespecificapplications:

[3.45]Interestingly, the moment-curvature (M, κ) constitutive model equation [3.45] has been

proposed for applications in composite beamswith imperfect connections between the twoelements (such as steel-concrete composite structures, timber-concrete elements and layeredwood systemswith interlayer slip) ([GIR07], [CHA11a], [CHA11b]).Note the similaritywith thenon-localbendingconstitutive law recently studied for theelasticproblems ([CHA08a],[ZHA10]).AsrecentlyshownbydiPaolaetal.[DIP09],modelsofelasticfoundationcan also involve somenon-locality. In fact, the soil–structure interactionmodel ofReissner[REI58] isalsobasedon thedifferentialequation[3.33],wherepandy are the foundationreactionandthedeflection,respectively(seealso[CHA10c]).Ontheopposite,forsofteningevolutions,ζ−hastotakenegativevalues[CHA08c],leading

tothenon-localsofteningconstitutivelaw:

[3.46]Arelevantchoiceoftenassumedinthesofteningconstitutivebehavioristoassumethatais

equal to lc (ζ− = −1) (see also [CHA 08c]). In the following, a local hardening moment-curvature relationship is incorporated in the model, leading to a well-posed evolutionproblem.Infact,itisnotnecessarytointroducesomenon-localityinthehardeningrangefromamathematical point of view. However, it is also possible to introduce some non-localityduring hardening, to introduce some scale effects in the hardening range. For the non-localhardeningplasticitymodel,M*isrelatedtothecombinednon-localplasticcurvaturevariablethroughthelinearmodel:

[3.47]Introducing the combined non-local plastic curvature into the loading function leads to a

differentialequation:

[3.48]Thegeneralsolutionofthisdifferentialequationiswrittenas:

[3.49]withtheboundaryconditionsobtainedfromthevariationalprinciple:

[3.50]ThenonlinearsystemofthreeequationswiththreeunknownsA,Band isfinallyobtained:

[3.51]Thefollowingdimensionlessparametersmaybeintroducedas:

[3.52]leadingtothelocalizationrelation:

[3.53]The plastic zone ξ versus the loading parameterp β is shown in Figure 3.8 and is

parameterizedbythedimensionlessparameterζ.

Figure3.8.Evolutionoftheplasticzoneξversustheloadingparameterβ–non-localhardeningplasticitymodel:ζ>0

Thegradientplasticitymodel(inthehardeningrange)isrecoveredfromthisrelationshipasanasymptoticlaw(lc→0inequation[3.53]):

[3.54]The width of the plastic zone associated with the non-local models is larger than the

referencewidthofthelocalmodel,forζlargerthanunity,whereasthiswidthissmallerthantheoneofthelocalmodelforζsmallerthanunity.Thelocalhardeningplasticzonerelationisobtainedbysettingζ=1.

3.2.4.Non-localsofteningconstitutivelaw:applicationtothecantileverbeam

Forthenon-localsofteningplasticitymodel,M*isrelatedto thecombinednon-localplasticcurvaturevariable throughthelinearmodel(see,forinstance,equation[3.46]):

[3.55]Introducingthecombinednon-localplasticcurvature(witha=lc) intotheloadingfunction

leadstoalineardifferentialequation:

[3.56]Thegeneralsolutionofthisdifferentialequationiswrittenas:

[3.57]withtheboundaryconditionsobtainedfromthevariationalprinciple:

[3.58]ThenonlinearsystemofthreeequationswiththreeunknownsA,Band isfinallyobtained:

[3.59]Thefollowingdimensionlessparametersmaybeintroducedas:

[3.60]andtheload-plasticzonerelationshipisfinallywrittenas:

[3.61]Inotherwords,Wood’sparadoxisovercomeforthenon-localsofteningcantilevercaseand

uniquenessprevailsforthesofteningevolutionconsideredinthechapter.Figure3.9showstheevolutionoftheplasticzoneξintermsofthepositivedimensionlessparameter|β|.

Theparameter|β|variesbetween0andtendstowardaninfinitevaluewhenPtendstowardzero.Moreover,thesizeoftheplasticzonetendstowardanasymptoticvalueforlargevaluesof|β|(andsufficientlysmallvaluesofP).ξ0=πisthelimitingvalueofthemaximumwidthofthelocalizationzone.Theplasticzoneevolvesfromatransitoryregimetowardamaterial(orsection)scalethatdoesnotdependanymoreontheloadingrange.Theresultsrevealthat theevolutiontendstowardoneuniquesolutionwithafiniteenergydissipationthatdependsonlyonthecharacteristiclength.Themaximumwidthofthelocalizationzone directlydependsonthecharacteristiclengthofthenon-localmodelviatherelation (for thecantileverbeam). The determination of the characteristic length lc (or the maximum width of thelocalization zone ) is related to the question of the finite-length hinge model, a centralquestionofthepresentnon-localmodel.Wood[WOO68],inspiredbytheworksofBarnardand Johnson [BAR 65], suggested the term of discontinuity length.Many papers have beenpublishedontheexperimentalortheoreticalinvestigationofsuchalength[WOO68,BAZ76,BAZ 87, PAU 92, BAZ 03 LEE 09] for reinforced concrete beams. It is generallyacknowledgedthatthevalueoflc(orthemaximumlocalizationzone )mustberelatedtothedepthofthecross-sectionh.Therigidbodymoment-rotationmechanismdetailedin[DAN08]or[HAS09]maybeusedtocalibratethischaracteristiclengthforreinforcedconcretebeams.Therefore,itisrecommendedthatthemaximumwidthofthelocalizationzone ischosenintheorderofmagnitudeofthedepthofthecross-sectionh.Thisimpliesforthecantileverbeamthat the characteristic length lc is in the order ofmagnitude of h/π. This theoretical aspectcertainlymeritssomefurtherinvestigations.However,theexistenceofthefinitesizefractureprocesszoneleadstothespecificstructuralsizeeffect.

Figure3.9.Evolutionoftheplasticzoneξversustheloadingparameterβ–non-localhardeningplasticitymodel:ζ>0

The deflection in the plastic zone is obtained by integrating twice the elasticcurvature:

[3.62]Thedeflectionintheelasticzone isderivedfromthecontinuityconditiongivenby

equation[3.27]:

[3.63]Figure 3.10 shows the resolution ofWood’s paradox with the non-local softening plastic

model.Afterthepeakload,thesofteningplasticitypropagatesalongthebeam,leadingtotheglobalsofteningphenomenon.Theinfluenceofthecharacteristiclength,whichcontrolsthepropagationofthelocalization

zone,isshowninFigure3.11.

Figure3.10.Responseoftheelastoplasticnon-localsofteningbeam;

Figure3.11.InfluenceofthecharacteristiclengthonthesofteningresponseoftheelastoplasticbeamP/PYversusv/vY;

Theglobalductility increasesas thestiffness ratio |EI/k| increases,or the length ratio lc/Lincreases.EvolutionoftheplasticzoneduringtheplasticsofteningprocessisshowninFigure3.12.

Figure3.12.Evolutionoftheplasticzoneduringthesofteningprocess;deflectionofthecantileverbeam;

3.2.5.Someotherstructuralcases–thesimplysupportedbeam

Another simple statically determinate model is the simply supported beam under centralconcentrated load (Figure 3.13). We will show that this case can be analyzed from thecantilevercase.Morespecifically,theplasticzoneparameterizedbythelengthl0isrelatedto

theloadingparameterthroughasimilarlaw.Forsymmetricalreasons,thebendingmomentissymmetricalwithrespecttothecentralaxis(bendingmomentispositiveinthiscase):

[3.64]

Figure3.13.Simplysupportedbeam

Usingsymmetricalconsiderations, it is sufficient toanalyzehalfa structure, leading to thedifferentialequationforthenon-localplasticcurvature:

[3.65]Inthiscase,andusingsymmetricalarguments,theboundaryconditionsarewrittenas:

[3.66]Letusintroducethechangeofvariable:

[3.67]Thenewdifferentialsystemisobtained:

[3.68]Werecognizethecantileverproblemwiththenewvariablesintroducedinequation[3.67].

The load-plastic zone relationship is given by equation [3.61] corrected with the newvariables:

[3.69]

Figure3.9obtainedforthecantileverbeamisstillvalidforthesimplysupportedbeam(withthenewnotationofequation[3.67]).However, in the caseof a simply supportedbeam, theboundary conditions dealing with the displacement function differ from the one of thecantilevercase:

[3.70]Thismeansthatthesolutionofthecantilevercasecanbeusedfortherotationfunction,but

notforthedisplacementfunction.Therotationintheplasticzoneisobtainedfrom:

[3.71]The rotation in the elastic zone is derived from the continuity of the rotation along the

elastoplasticboundary:

[3.72]Thedeflectionintheelasticdomaincanbewrittenas:

[3.73]

3.2.6.Postfailureofreinforcedconcretebeamsunderdistributedlateralload

The failure process of the cantilever beam solicited by its own weight (or under uniformdistributedlateralload)canalsobestudied(Figure3.14).In this case, the bendingmoment varies nomore as a linear function, but as a parabolic

function:

[3.74]Thedifferentialequationforthenon-localplasticcurvatureisnowwrittenas:

[3.75]Thesolutionofsuchadifferentialequationisobtainedfrom:

[3.76]with the three constants (A, B,l0) identified from the three boundary conditions given byequation[3.68].

Figure3.14.Cantileverbeamunderdistributedload

Thefollowingdimensionlessparametersmaybeintroducedas:

[3.77]andtheload-plasticzonerelationshipisfinallywrittenas(seeFigure3.16):

[3.78]Another simple statically determinatemodel is the simply supported beam under uniform

distributed lateral load (Figure 3.15).We will show that this case is not analogous to thecantilever case under distributed load. For symmetrical reasons, the bending moment issymmetricwithrespecttothecentralaxis(bendingmomentispositiveinthiscase):

[3.79]

Figure3.15.Simplysupportedbeamunderdistributedload

Usingsymmetricalconsiderations, it is sufficient toanalyzehalfa structure, leading to thedifferentialequationforthenon-localplasticcurvature:

[3.80]Thefollowingchangeofvariablecanbeadopted:

[3.81]Thenewdifferentialsystemisobtained:

[3.82]Thissystemcannotbecastintheformgivenbyequation[3.78],andthereisnoequivalence

between thecantilevercaseand the simply supportedbeam incaseofuniform loading.Theload-plasticzonerelationshipisfinallywrittenas(seeFigure3.16):

[3.83]

Figure3.16.Evolutionoftheplasticzoneξ(or )versustheloadingparameterβ(or )–beamunderdistributedload–lc*=0.1

Tosummarize, thenumericaldescriptionofthepostfailureofreinforcedconcretemembersneedstheuseofnon-localmechanicstheory(inthischapter,non-localplasticitymechanicshasbeenused).However,thesameargumentswouldbeneededforthebendinganalysisofdamagebeams, where non-local damage mechanics models are needed in presence of softening inordertoavoidWood’sparadox(see,forinstance,[CHA10a]).Thebendingdamagelawcanbegeneralizedinanon-localformatbycorrectingthedamage

loadingfunction,assuggestedbyPijaudier-CabotandBažant[PIJ87]:

[3.84]wherethenon-localvariableisthenon-localcurvature-drivenvariable .Butotherkindsofvariationallybaseddamagemodelscanalsobeused,suchas[CHA10a]:

[3.85]where isthenon-localdamageenergyreleaserateatthecross-sectionallevel.However,inthefollowing,wewillbemainlyconcernedwith theyieldcharacterizationof thereinforcedconcrete cross-section before softening, and non-locality in this case is not necessary to beintroduced.

3.3.Constitutivelawsforsteelandconcrete3.3.1.Steelbehavior

3.3.1.1.DesignvaluesforsteelatUltimateLimitStateThe design yield strength of reinforcement fyd also denoted by fsu, is related to the

characteristic yield strength of reinforcement fyk through the partial factor γs for reinforcingsteelas:

[3.86]γs is equal to1.15, except for combinationswith accidental actions (for combinationswithaccidentalaction,γsisequaltounity).Inthefollowing,wewillmainlyusethevalueγs=1.15.Asusual(seealsotheSLS),theYoung’smodulusofsteelEsistakenequalto200,000MPa.Asanexample,letusconsiderareinforcedconcretesectioncomposedofB500Bsteelbars.

In the nomenclature of steel classificationB500B, the firstB stands for steel for reinforcedconcrete. The next three digits represent the specified characteristic value of upper yieldstrengthfyk=500MPa.ThelastsymbolBstandsforductilityclass.

3.3.1.2.PerfectelastoplasticityTwopossibilitiesareofferedtomodelthesteelbehaviorinEurocode2.Thefirstpossibilityisbasedonperfectplasticity(hardeningneglected–seeFigure3.17).

Figure3.17.Perfectelastoplasticitydiagramforthesteelreinforcements

EveniftheoldFrenchrules,suchasBétonArméauxEtatsLimites(BAEL)dated1983andmodifiedin1999(orsomeotherinternationalrulessuchastheACIdesignrules)introducedalimitationofthetensilestrain(typicallyεud=10‰),thenewrules–Eurocode2donotlimitthestraininthesteelreinforcement.Thesteelinthereinforcementismodeledwithaperfectplasticityrulewithoutanylimitationinstrain.ItalsomeansthatthepivotAconceptdoesnotexist anymore with such amodeling of the steel reinforcement. Formonotonic loading, thestress–strainrelationshipofthesteelbehavioristhengivenby:

[3.87]wherefsu,alsodenotedbyfyd,isthedesignyieldstrengthofreinforcement.

3.3.1.3.HardeningelastoplasticityFigure3.18.Elastoplasticitywithlinearhardeningforthesteelreinforcements

ItiscorrecttotakeintoaccountthesteelhardeninginEurocode2(seeFigure3.18).Inthiscase,thecharacteristicultimatelimitstrainisboundedεuk(typicallyεuk=50‰).Thedesignultimate limit strain iscalculated from thecharacteristic strainof reinforcementatmaximumloadthroughapartialcoefficient:

[3.88]Formonotonicloading,thestress–strainlawcanbewrittenasapiecewiselinearlaw:

[3.89]wherevalueoftheparameters(m,m’)dependsonthebranchconsidered.Theseparameterscanbeexpressedby:

[3.90]Onlythestraininsidetheinterval|εs|≤εudisallowed.Theplasticityhardeningparameters

aredefinedbelow:

[3.91]Thelinearbrancheshavebeencalculatedfromcontinuitywiththeelasticbranches,withthe

followingcharacteristicpoints:

[3.92]Thecoefficientkdependsonthesteelductility,andisgiveninEurocode2as:

[3.93]Notethatwedonotfocusoncyclingbehaviorwithpossibleunloading–reloading,andthen,

the typeof hardening (isotropic, kinematic ormixedhardening) is not of importance for thedesignofthereinforcedconcretesection,atleastinthestaticrange.Withthisconstitutivelaw,thePivotA isdefinedfor themaximumdesignultimatestrainin

thetensilesteelreinforcementas:

[3.94]AsaresultusingEurocode2,PivotAislesspresentthanPivotB,duetotheallowedlarge

ductilityofsteel,asignificantchangewith respect to theolderEuropeanrules, for instance.Typically,εud=45‰isallowedinEurocode2,whereasεud=10‰wasthelimitedvalueintheoldFrenchrules(BétonArméauxEtatsLimitesBAEL83whichwasmodifiedin1999).It can be interesting also to have some other material characteristics for the steel

reinforcement, suchas thedensityρ=7,850kg/m3, or the thermicdilatationcoefficientα =10−5K−1(thedilatationcoefficientforthesteelandtheconcretepartofthereinforcedconcretesectionarealmostidentical).

3.3.1.4.NumericalapplicationAs an example, for steel reinforcement bars made of B500B steel bars, the steel materialparametersattheULSare:

[3.95]

3.3.2.Concretebehavior

3.3.2.1.DesignvaluesforconcreteatUltimateLimitStateConcretehascomplexunsymmetricalmaterial responses in tensionandincompression,bothfromthestrengthandtheductilityproperties(seeFigure3.19).

Figure3.19.Unsymmetricalresponseofconcreteinuniaxialtensionandcompression

AsfortheSLS,thetensionstrengthofconcretewillbeneglectedatULS.Thedesigncompressionyieldstrengthofconcretefcd,alsodenotedbyfcu,isrelatedtothe

characteristicyieldstrengthofconcretefckthroughthepartialfactorγcforconcreteas:

[3.96]γc is equal to 1.5, except for the combination with accidental actions combinations (for

combination with accidental action, γs is equal to 1.2). αcc is a reducing coefficient forincludingthetime-dependentcrackingprocessatlargestressvalues.Typically,thiscoefficientαccvariesbetween0.8and1,andisavailableinthenationalAnnexesofEurocode2foreachcountry.This coefficient was initially included to take into account the possible creep failure

phenomenon (tertiary creep) for large stress values, typically for compressive stress valueshigherthan85%oftheinstantaneouscompressivestrength.This time-dependentphenomenonhasbeenexperimentallystudied indetailsbyRüsch[RUS60],and recentlymodeledwithasimplecreepdamagemodel[CHA05a].Itisshownin[CHA05a]or[CHA05c]thatthecreepfailure phenomenon is associated with the existence of a limit point in the stress–straindiagram at a very slow loading rate, and then creep failure is interpreted as a loss ofequilibriumsolutionassociatedwiththeconstitutivelaw.IntheFrenchNationalAnnexes(eachannexhasdifferentrules),thiscoefficientisequalto

αcc=1inmostdesigncases(forbuildingdesign,forinstance)except,forinstance,inbridgedesign,orforprestresseddesign.Forbridgedesign,thevalueαcc=0.85isrecommended.Forthenextapplicationspresentedinthischapter,wewillfocusonbuildingapplications,andthenαcc=1.Historicallyspeaking,theoldFrenchrulesBétonArméauxEtatsLimitesBAEL83,gaveαcc=0.85(thisvalueαcc=0.85alsoappearsintheAmericanrulesACIforreinforcedconcrete structure design, or in the CEB-FIP model code for concrete structures –ComitéEuro-International duBéton/Fédération Internationale de laPrécontrainte). Accordingly,the transition rules fromBAEL toEC2 have changed significantly the calculation of design

strength for building applications (15%), at least for the French National Annexes. Oneargumentwouldbethatthepartialfactorγcalreadyincludedthesetime-dependenteffects.The strength tensionparameters are alsogiven, even if theyarenotused for the sectional

designatULS(asthetensioncontributionofconcreteisneglected).

[3.97]wherefctk ,005(seealsoTable3.1)isthetensilestrength(5%lowertensilestrength).Asanexample, letusconsidera reinforcedconcrete section,composedofaC30/37 type

concrete,thefirstCstandsforconcrete.Thenextdigitsarerelatedtocharacteristicstrengths.The first characteristic compression strength is the cylinder characteristic strength fck = 30MPa, whereas the second characteristic compression strength is the cubic characteristicstrength.Wecalculate:

[3.98]It can be interesting to also have some other material characteristics for the reinforced

concrete material, such as the volumetric weight γ = 25 kN/m3, or the thermic dilatationcoefficientα = 10−5K−1 (the dilatation coefficient for the steel and the concrete part of thereinforcedconcretesectionarealmostidentical,asoutlinedin3.3.1inthesteelchapter).TheconstitutivelawsavailableinEurocode2formodelingconcretebehaviorattheULSare

nowpresented.Onlythemonotonicbehaviorismentioned.

3.3.2.2.Parabola–rectangleconstitutivelawTheparabola–rectangleconstitutivelawiscomposedofanonlinearparabolicpartforsmallcompressivestrainandaconstantstresspartforlargerstrains(seeFigure3.20).

Figure3.20.Parabolicrectangulardiagramforconcrete;typically,εc2=−2‰andεcu2=−3.5‰forfck≤50MPa

For fck ≤ 50 MPa, the nonlinear constitutive law is mathematically described by thefollowingequations:

[3.99]Thematerialparametersaredefined inTable3.2.Note thatEurocode2allows theuseof

someothernonlinearconstitutive laws forhighconcrete strength fck≥55MPa,withpowerstrainfunctiondifferentfromn=2,asconsideredfortheparabola–rectanglelaw.

3.3.2.3.RectangularsimplifiedconstitutivelawForsimplifyingdesignpurposes,amuchsimplerconstitutivelawcalledrectangularsimplifiedconstitutivelawcanbeusedinEurocode2,insteadoftheparabola–rectanglelaw.BothlawswerealreadyavailableintheoldFrenchrulesBétonArméauxEtatsLimitesBAEL83.Thislawissimilartoarigidplasticconstitutivelawthatactsonlyonareducedpartλyof

thecompressionblock,namelyalongaportionλy=λαdfromtheupper(compressed)fiberofthecross-section(seeFigure3.21).(λ=0.8forfck≤50MPainEurocode2).Forcomparison,the0.8factorappearingforthesizeofthecompressionblockischangedto0.85forordinaryconcreteintheAmericanACIdesignrules(see,forinstance,[MAC97]).

Figure3.21.Rectangularsimplifieddiagramforconcrete;strainandstressevolutionalongthecross-section

The fundamental assumptions of the so-called simplified rectangular constitutive law canalsoberepresentedinthestress–straindiagramsforbothPivotA(maximumstraincapacityofthe tensile steel reinforcement) and Pivot B (maximum strain capacity of the compressionconcretefiber)–seeFigure3.22.

Figure3.22.Rectangularsimplifieddiagramforconcrete;stress–strainrepresentation

Inthestress–straindiagram,thestressisvanishinguptoastrainvalueequalto0.2εc,sup. InpivotB,asthestraincapacityoftheconcreteisfullyreached,εc,supisequalto–3.5‰andthentheminimumstrainvaluewithzerostressvalueisequalto0.2×3.5‰=0.7‰.However,inPivotA,asεc,supislargerthan–3.5‰,thisminimumstrainvaluewilldependontheconcretestrainoftheupperfiber(seeFigure3.22).ThisconstitutivelawisdifferentinPivotAorPivotB,andismathematicallydescribedby

thefollowingsingleequation:

[3.100]whereεcsup=εcu2=–3.5‰inPivotB,andεcsup=−αεud/(1–α)inPivotA.

3.3.2.4.BilinearconstitutivelawThebilinearconstitutivelawiscomposedofalinearelasticpartforsmallcompressivestrainand a constant stress part for larger strains (see Figure 3.23). This kind of model can betypically classified as an elastic-perfectly plastic constitutive law, even if for concrete, thisductileclassificationshouldbeusedwithprecaution.

Figure3.23.Bilinearconstitutivelawforconcrete;typically,εc3=–1.75‰andεcu3=–3.5‰forfck≤50MPa

Thisconstitutivelawismathematicallydescribedbythefollowingequations:

[3.101]Thematerial parameters are defined inTable3.2. This law is simpler than the parabolic

law,asitavoidsanypiecewisenonlinearstress–strain,butonlypiecewiselinearstress–strainlaws. The bilinear law can be considered as a good compromise between the parabola–rectangleconstitutivelawandtherectangularsimplifiedconstitutivelaw.

3.3.2.5.Sargin’sconstitutivelawAccordingtoEurocode2,itisalsopossibletouseanonlinearstress–strainconstitutivelawintroduced by Sargin [SAR 68] – see also [SAR 69], and sometimes called “Sargin’sparabola”insometextbooks(seealso[SAR71];alsocitedin[OTT05]).Sargin’slawcanbeusedfor thestructuralanalysiswithrespect tothebucklingdesign.EvenifEurocode2doesnot explicitlymention the reference of Sargin, thementioned nonlinear stress–strain law isreferredtotheCEB90,whichisbasedonSargin’slaw.Thisnonlinearstress–strainconstitutivelawcanbewrittenas(seealso[THO09]):

[3.102]withεcu1=–3.5‰forconcreteclasslowerthanC55.ThecharacteristicsstrainvariablesofSargin’slawaregiveninTable3.1.

Table3.1.ConcretematerialparametersofSargin’slawatEurocode2forfck≤50MPa

Thislawisparameterizedbyonedimensionlesscoefficientk,whichcontrolstheductilityoftheconcrete.Itiseasytocheckthat:

[3.103]The parameter k can be theoretically identified from the initial slope of the undamaged

Young’smodulus,accordingto:

[3.104]TheEurocode2rulessuggestedacorrectionofthislawintheevaluationofthesecond-order

effectsforstabilitydesign,leadingtoasmallerYoung’smodulusassociatedwithlargerstrainanddeflectionvaluesinasaferdesign:

[3.105]Sargin’s law is represented in Figure 3.24 for a concrete, associatedwith a k parameter

equalto2.5(ktypicallyvariesbetween1.3and2.6).

Figure3.24.Sargin’slawforaC30concretewithk=2.5;comparisonwiththeparabolic-rectangularstress-strainrelationship

Thethreecurves(Sargin’slaw,bilinearconstitutivelawandparabola–rectangleconstitutivelaws)arecomparedinFigure3.24.ItisdifficulttodistinguishtheparabolicpartandSargin’slawwiththeseparameters,atleastinthehardeningpartofthenonlinearconstitutivelaw.Moregenerally,Sargin’slaw[SAR68]canbepresentedinthefollowingform:

[3.106]Thenonlinearequation[3.102]ofEurocode2isobviouslyaparticularcaseofSargin’slaw

[SAR68]withα=0.Some other nonlinear laws are available in the literature such as the law of Desayi and

Krishnan [DES64] suggested in theoldFrench rulesBétonArméauxEtatsLimites for thestructuralanalyses(seealso[ROB74]),andobtainedfromSargin’slawbysettingα=1andk=2:

[3.107]Unfortunately, this last nonlinear lawhas nodegree of adjustment of theYoung’smodulus

identification.Infact,withsuchalaw[DES64],wenecessarilyhave:

[3.108]whichcanbequiteunrealistic.

Anotherclassicallawistheparaboliclawoftenusedinreinforcedconcretedesign,whichissimplyobtainedasaparticularcasewithα=0andk=2:

[3.109]Wehaveonlypresented a fewengineering laws that are recommended inEurocode2.Of

course,somemorerefinedmodelshavebeenpublishedintheliteratureforconcretemodeling.CDMmodelshavebeenfoundtobeverypowerfulwithrespecttothesereinforcedconcretestructuralproblems.TheCDMmodelofMazars([MAZ86],[MAZ96],seealso[MAZ09],for instance)canbecited, for instance, foragoodrepresentationof thestrongasymmetryofconcrete in the tensionand in thecompressiondomain.One loading function isused for thecompression behavior and another loading function is used for the tension behavior. Someother tensorialCDMmodels havebeendeveloped in the literature (see, for instance, [HAL96],[MUR97],[CHA05b],[LEM05]and[MUR12]);themodelof[CHA05b]presentstheadvantagetodescribethestrongunsymmetricalbehaviorintensionandincompressionwithasingledamageloadingfunction.We also note that the constitutive laws available in Eurocode 2 have no softening parts,

excepteventuallyinSargin’slawthatismainlyusedforstabilitydesigns.Hence,forpracticalbendingdesign,thesofteningphenomenonfrombothlocalandglobalstatevariablesdoesnotappearandnon-localitymechanicsisnotneeded,infact,intheevolutionproblemforthestrainrange of interest in the design process. However, the post-cracking process of the bendingbeamneedstheuseofnon-localmechanicsasalreadypresentedinthischapter.

3.3.2.6.Synthesis–materialparametersforconcreteThematerialparameters for theconcreteat theULSare summarized inTable3.2, basedonEurocode2rules.

Table3.2.GeneralconcretematerialparametersatEurocode2forfck≤50MPa

3.3.3.DimensionlessparametersatULSA rectangular reinforced concrete cross-section is considered, with both tensile and

compressionsteel reinforcement, respectively, in the lowerpartandupperpartof thecross-section(seeFigure3.25).

Figure3.25.Rectangularcross-sectionwithcompressionandtensilesteelreinforcement

Figure3.26.Strainprofileatultimatelimitstateaccordingtotheplanecross-sectionassumption(Navier–Bernoulliassumption)

According to Navier–Bernoulli assumption (plane cross-section assumption), the strain islinearly varying along the cross-section, as shownby the geometrical relationship inFigure3.26.UsingthenotationsofFigure3.25,thetwoequilibriumequationsbothforthenormalforce

andthebendingmoment(onlytwoequationsforsymmetricalreasons)arewrittenwithrespecttothecenterofgravityofthetensilesteelreinforcementas:

[3.110]

[3.111]wherethedimensionlessparametersψandμaredefinedfrom:

[3.112]NcandMcare,respectively,thenormalforcecomponentofthecompressionblockofconcreteandthemomentcomponentofthiscompressionblock.Thedimensionlessparametersψandμcanbealsointroducedinanintegralformatas:

[3.113]wherethestressinconcreteσcisgenerallyanonlinearfunctionofthestrainεc,whichisitselfalineardecreasingfunctionoftheverticalaxis,asaresultofthekinematics.Theoriginofthey-axisisheretakenatthecenterofgravityofthetensilesteelreinforcement.ThecoefficientsofthislinearfunctiondependonthePivotconsidered:

[3.114]εcui is the maximum strain capacity of concrete in compression for each constitutive law(parabola–rectangle,simplifiedrectangular,bilinearorSargin’slaw).AsfortheSLS,itispossibletodefinethepivotrulesattheULS:–thepoint“pivotA”,whichcorrespondstothemaximumstraincapacityεudof themosttensionedsteelreinforcementbars;–thepoint“pivotB”,whichcorrespondstothemaximumstraincapacityεcui inthemostcompressedfiberinconcrete.

Figure3.27.Workingzoneofthecompositecross-sectioninPivotAandPivotBatULS–strainprofile

At the cross-sectional ULS, the strain diagram should cross one of the two pivot points(PivotA and PivotB) and should also respect the limit strain requirement for the extremalpointattheotherpivotpoint(seeFigure3.27).ThelimitcasebetweenthetwopivotsisreferredtoaspivotAB,andischaracterizedbythe

limitvaluefortherelativepositionoftheneutralaxisαABdefinedfromtheupperfiberofthecross-sectionas:

[3.115]PivotAB sometimes is often referred to as the balance section, because at the ULS, the

concreteandtensionedsteelreinforcementreachtheirultimatelimitstrainsatthesametime.εcui,whichtakesnegativevalues,isexplicitlygiveninTable3.2,accordingtotheEurocode

2rules.AthirdcaseisPivotC,whereallpartsofthesectionisincompression,eventhelowerpart

of thecross-section.PivotC can alsobeunderstood as a particular caseofPivotB, as thesectioniscontrolledbythemaximumstraincapacityofconcreteincompression.TheboundarybetweenPivotBandPivotC,alsodenotedbyPivotBC,isobtainedfor:

[3.116]Figure 3.28 shows the nonlinear stress profile along the cross-section, at PivotB, to be

integratedtocalculatethenormalforceandbendingmomentcomponentsinconcreteatULS.

Figure3.28.StressprofileatUltimateLimitState–PivotB

3.3.4.Calculationoftheconcreteresultantfortherectangularsimplifieddiagram

According toFigure3.21, thenormal forceandbendingmomentcomponents inducedby theconcretecompressionblockarecalculatedas:

[3.117]WenotethattheseexpressionsareindependentonthePivotconsidered(validforbothPivot

AandPivotB).The dimensionless parameters ψ and μ are then easily identified for this simplified

constitutivelaw:

[3.118]

3.3.5.Calculationoftheconcreteresultantforthebilineardiagram

3.3.5.1.Introduction,PivotA1,PivotA2andPivotBItisusefultodistinguishPivotA,characterizedbyεs=εudandPivotB,characterizedbyεc,sup=εcu3,forthebilinearconcreteconstitutivelaw.However,PivotAitselfcanbedecomposedintoPivotA1 (where the compression block has only linear elastic behavior) and PivotA2(where the compression block has an elastic and a “perfectly plastic” part). Figure 3.29represents the stress in the compression block at PivotAB, for a concretemodeling by thebilinearconstitutivelaw.

Figure3.29.Calculationofthenormalforceandbendingmomentcomponentsinthecompressionblockofconcrete

3.3.5.2.PivotA1–bilineardiagramPivotA1 is a part of PivotA,meaning that themaximum strain capacity of the tensile steelreinforcement is reached εs = εud, but the concrete part in compression has only an elasticbehavior:

[3.119]Inthiscase,weonlyhavetointegratethelinearlyincreasingpartoftheconcretestressalong

thecross-section,andtheproblemistheelasticone.Thebendingmomentiscalculatedfromequation[3.113]as:

[3.120]leadingtothedimensionlessreducedmomentcoefficientμ:

[3.121]Theshapecoefficientψisequalto:

[3.122]

3.3.5.3.PivotA2–bilineardiagramPivotA2 isalsoapartofPivotA,meaning that themaximumstraincapacityof the tensile

steelreinforcementisreachedεs=εud,buttheconcretepartincompressionhasnowanelasticandaconstantstresscomponent:

[3.123]Weneedtodecomposethestressvariationintoalinearlydecreasingoneandaconstantone

(see Figure 3.29).αcd is the distance from the upper fiber to the fiber associatedwith thetransitionfromtheelasticparttotheconstantpart(orplasticpart)ofthecompressionblock.ApplyingThalesgeometricalrelationship,weobtainforthischaracteristicparameter:

[3.124]Thebendingmomentiscalculatedfromequation[3.113]as:

[3.125]leadingtothedimensionlessreducedmomentcoefficientμ:

[3.126]

Thereducedmomentcanalsobesimplifiedas:

[3.127]Theshapecoefficientψisequalto:

[3.128]

3.3.5.4.PivotB–bilineardiagramIn PivotB, the maximum strain capacity of the upper fiber of concrete in compression isreachedεc,sup=εcu3suchas:

[3.129]Wealsoneedtodecomposethestressvariationintoalinearlydecreasingoneandaconstant

one(seeFigure3.29).αcdisthedistancefromtheupperfibertothefiberassociatedwiththetransitionfromtheelasticparttotheconstantpart(orplasticpart)ofthecompressionblock.ApplyingThalesgeometricalrelationship,weobtainforthischaracteristicparameter:

[3.130]The bending moment expression is the same as in equation [3.125], leading to the

dimensionlessreducedmomentcoefficientμ:

[3.131]Thereducedmomentcanalsobesimplifiedas:

[3.132]Theshapecoefficientψisequalto:

[3.133]

3.3.5.5.PivotC–bilineardiagram

PivotC is reached when the overall section is in compression, which is mathematicallyequivalentto:

[3.134]Inthiscase,thelowerfiberisincompressionanditsstrainisequalto:

[3.135]whereαc also defined the fiber associated with the transition from the elastic part to theconstantpart(orplasticpart)ofthecompressionblock.Thebendingmomentiscalculatedfromequation[3.113]as:

[3.136]Assumingthatthelowerfiberisintheelasticrange,thereducedmomentisfinallycalculated

as:

[3.137]Thereducedmomentcanalsobesimplifiedas:

[3.138]Theshapecoefficientψisequalto:

[3.139]Theshapecoefficientψcanbealsosimplifiedas:

[3.140]ItiseasytocheckthecontinuityofPivotBandCattheboundarybetweenthetwobehaviors.

3.3.6.Calculationoftheconcreteresultantfortheparabola–rectanglediagram

3.3.6.1.PivotA1–parabola–rectanglelawPivotA1 is a part of PivotA,meaning that themaximum strain capacity of the tensile steelreinforcement is reachedεs=εud,but theconcretepart incompressionhasonlyaparabolicbehavior(thestrengthcapacityofconcretehasnotbeenreached):

[3.141]Thebendingmomentandthenormalforcesintheconcretecompressionblockareobtained

byintegrationofthenonlinearstress–strainrelationshipfromequation[3.113]presentedas:

[3.142]where thestress inconcreteσc isnowanonlinear functionof thestrainεc,which is itselfalineardecreasingfunctionoftheverticalaxis,asaresultofthekinematics.Theoriginofthez-axis is here taken at the location of the neutral axis, in order tomake the calculationmucheasier.ThecoefficientsofthislinearfunctiondependonthePivotconsidered:

[3.143]Thedimensionlessnormalforcecomponentintheconcretecompressionblockiscalculated

fromtheshapecoefficientψ:

[3.144]InpivotA1,thereducedmomentiscalculatedfromtheintegralequation:

[3.145]

3.3.6.2.PivotA2–parabola–rectanglelawPivotA2 isalsoapartofPivotA,meaning that themaximumstraincapacityof the tensile

steel reinforcement is reached εs = εud, but the concrete part in compression has now aparabolicandaconstantstresscomponent:

[3.146]We need to decompose the stress variation into a parabolic one and a constant one (as

alreadysuggestedforthebilinearconstitutivelaw).αcdisthedistancefromtheupperfibertothefiberassociatedwiththetransitionfromtheelasticparttotheconstantpart(orplasticpart)of the compression block. Applying Thales geometrical relationship, we obtain for thischaracteristicparameter:

[3.147]Thebendingmomentiscalculatedfromequation[3.113]as:

[3.148]wherethefirsttermistheassociatedwiththeparabolicpartoftheconstitutivelaw,whereasthe second term is related to the maximum capacity strength fcu. After some lengthy

manipulations,wefinallyobtainforthedimensionlessreducedmomentcoefficientμ:

[3.149]Theshapecoefficientψcanbeshowntobeequalto:

[3.150]

3.3.6.3.PivotB–parabola–rectanglelawIn PivotB, the maximum strain capacity of the upper fiber of concrete in compression isreachedεc,sup=εcu3suchas:

[3.151]We also need to decompose the stress variation into parabolic stress variation and

rectangular stress variation.αcd is the distance from the upper fiber to the fiber associatedwiththetransitionfromtheelasticparttotheconstantpart(orplasticpart)ofthecompressionblock. Applying Thales geometrical relationship, we also obtain for this characteristicparameter:

[3.152]Thebendingmomentiscalculatedfromequation[3.113]as:

[3.153]leadingtothedimensionlessreducedmomentcoefficientμ:

[3.154]Theshapecoefficientψcanbeshowntobeequalto:

[3.155]

3.3.6.4.PivotC–parabola–rectanglelawForPivotC,thedimensionlessreducedmomentcoefficientμiscalculatedas:

[3.156]Theshapecoefficientψcanbeshowntobeequalto:

[3.157]

3.3.7.CalculationoftheconcreteresultantforthelawofDesayiandKrishnan

3.3.7.1.LawofDesayiandKrishnan–introductionSargin’slawisdefinedinonesingleequation,andthereisnoneedtoseparatePivotA1andPivotA2,asintroducedfortheparabola–rectanglelaworthebilinearlaw.Inthissectionweuse the particular version of Sargin’s law given by equation [3.107], which is the law ofDesayiandKrishnan[DES64].ThisparticularcaseofSargin’slawwithα=1andk=2leadstoaneasieranalyticaldeterminationof thedimensionlessconcrete resultantparameters (seealso[ROB74]).ThemoregeneralintegrationresultsvalidforSargin’slawintheothercasescanbeobtainedfromtheintegrationresultspresentedin[THO09].Thebendingmomentandthenormalforcesintheconcretecompressionblockareobtained

byintegrationofthenonlinearstress–strainrelationshipfromequation[3.113]presentedas:

[3.158]where thestress inconcreteσc isnowanonlinear functionof thestrainεc,which is itselfa

lineardecreasingfunctionoftheverticalaxis,asaresultofthekinematics.Theoriginofthez-axisisheretakenatthelocationoftheneutralaxis,inordertomakethe

calculation much easier. The coefficients of this linear function depend on the Pivotconsidered:

[3.159]

3.3.7.2.PivotA–lawofDesayiandKrishnanThedimensionless normal force component in the concrete compressionblock is calculatedfromtheshapecoefficientψ:

[3.160]whichcanbeeasilyintegratedleadingto:

[3.161]InpivotA,thereducedmomentiscalculatedfromtheintegralequation:

[3.162]whichisalsoequivalentto:

[3.163]

3.3.7.3.PivotB–lawofDesayiandKrishnanThedimensionless normal force component in the concrete compressionblock is calculatedfromtheshapecoefficientψ:

[3.164]whichcanbeeasilyintegratedleadingto:

[3.165]whichisalso thevaluepresentedbyCarreiraandChu[CAR86]whopresentedthecaseofPivotBwithsomeadditionaltensioneffectsforthecrackedconcreteintension.InpivotB,thereducedmomentiscalculatedfromtheintegralequation:

[3.166]whichisalsoequivalentto:

[3.167]InPivotB,theseresultsarealsocoincidentwiththeresultsofCarreiraandChu[CAR86],

whocalculatedthebendingmomentwithrespecttotheneutralaxis,whereasthemomentofthecompressionblock ispresentedherewithrespect to thecenterofgravityof the tensilesteelreinforcement.

3.3.7.4.PivotC–lawofDesayiandKrishnanThedimensionless normal force component in the concrete compressionblock is calculatedfromtheshapecoefficientψ:

[3.168]whichcanbeeasilyintegratedleadingto:

[3.169]Thereducedmomentμcisgivenby:

[3.170]Theintegralcalculationsaregivenin[THO09]forSargin’slawgiveninequation[3.102],

whichispossibletousefornonlinearstructuralanalysisaccordingtoEurocode2.

3.3.8.CalculationoftheconcreteresultantforSargin’slawofEurocode2

3.3.8.1.Sargin’slawofEurocode2–introductionSargin’s law inEurocode2 is aparticular caseofSargin’s law [SAR68]withα = 0.TheintegrationresultsvalidforSargin’slawarealsopresentedin[THO09].Thebendingmomentandthenormalforcesintheconcretecompressionblockareobtained

byintegrationofthenonlinearstress–strainrelationshipfromequation[3.113]presentedas:

[3.171]where thestress inconcreteσc isnowanonlinear functionof thestrainεc,which is itselfalineardecreasingfunctionoftheverticalaxis,asaresultofthekinematics.Theoriginofthez-axis is here taken at the location of the neutral axis, in order tomake the calculationmucheasier. The coefficients of this linear function depend on the Pivot considered, as given byequation[3.159].

3.3.8.2.PivotA–Sargin’slawofEurocode2Thedimensionless normal force component in the concrete compressionblock is calculatedfromtheshapecoefficientψ:

[3.172]whichcanbeeasilyintegratedleadingto:

[3.173]InpivotA,thereducedmomentiscalculatedfromtheintegralequation:

[3.174]whichisalsoequivalentto:

[3.175]

3.3.8.3.PivotB–Sargin’slawofEurocode2Thedimensionless normal force component in the concrete compressionblock is calculatedfromtheshapecoefficientψ:

[3.176]whichcanbeeasilyintegrated,leadingto:

[3.177]Typically,fork=2.5,theshapecoefficientisequalto:

[3.178]Inthislastcase,foraC30typeconcrete,theshapecoefficientfactorisequalto:

[3.179]TheshapecoefficientψB=0.787isanintermediatevaluebetweenψB=0.75obtainedfor

thebilineardiagramandψB=0.8obtainedfortherectangularsimplifieddiagram.InpivotB,thereducedmomentiscalculatedfromtheintegralequation:

[3.180]whichcanbedevelopedasfollowing:

[3.181]Typically,fork=2.5,thereducedmomentisequalto:

[3.182]

3.3.8.4.PivotC–Sargin’slawofEurocode2Thedimensionless normal force component in the concrete compressionblock is calculatedfromtheshapecoefficientψ:

[3.183]whichcanbeeasilyintegrated,leadingto:

[3.184]Thereducedmomentμcisgivenby:

[3.185]

3.3.9.OntheuseofthereducedmomentparameterIn absence of additional steel compression reinforcement, the neutral axis position iscalculatedfromthenonlinearequation:

[3.186]which isalwaysasecond-orderequationfor therectangularsimplifiedconstitutive law,andstill remains a second-order equation inPivotB for the constitutive lawpresented indetail(bilinear, parabola–rectangle or Sargin’s law). Only the law of Desayi and Krishnan ispresentedinTable3.3, for theclarityofpresentation,but theparametersofSargin’s lawforEC2canalsobededucedfromtheformulapresentedinsection3.3.2.5devotedtothislaw.ThedimensionlessparametersofinterestarepresentedforeachmodelinTable3.3(seealso

[THO09]forthepresentationoftheseparameters).For a reinforced cross-section without additional compression steel reinforcement, the

dimensionlesssteelareainthecaseofsimplebending(withoutaxialforce)canbegiveninadimensionlessformatas:

[3.187]Thetensilesteelreinforcementisassumedinplasticity,withatensilestressequaltofsu.Itis

showninFigure3.30, that thefourmodelsare in factveryclose.Therectangularsimplifiedlaw gives the lower steel area (unsafe design), whereas Sargin’s nonlinear law gives thelargeststeelarea,orthesaferdesign(bothSargin’slawforEC2,orDesayiandKrishnanlawwhichisalsoaparticularcaseofSargin’slaw).Boththebilinearandtheparabola–rectangleconstitutivelawsleadtoverycloseresults.

Figure3.30.Comparisonofthesteelreinforcementareaforeachmodel

Table3.3.ComparisonofthecharacteristicsoftheconstitutivelawsinpivotBatultimatelimitstateforfck≤50MPa

Theseresultsobtainedfromasecond-orderpolynomialequationforthedeterminationoftheposition of the neutral axis, for a given actingmoment, are based on PivotB. However, inPivotA, the nonlinear equation of the neutral axis position can be a cubic equation for thebilinearconstitutive law,a fourth-orderequation for theparabola–rectangleconstitutive lawandatranscendalequationforSargin’sconstitutivelaw.

Chapter4

Bending-CurvatureatUltimateLimitState(ULS)

4.1.Onthebilinearapproximationofthemoment-curvaturerelationshipof

reinforcedconcretebeams4.1.1.Phenomenologicalapproach

Thebehaviorofareinforcedconcretebeaminbendinguptotheultimatelimitstate(ULS)iscomposed of different complex inelastic stages (cracking of concrete in tension, plasticityphenomena in steel, crackingof concrete incompression), as shown inFigure4.1 (see also[MAC 97]). First, we start from a phenomenological approach describing qualitatively themainphenomenainvolvedat theULSofareinforcedconcretebeam.Second,somebending-curvature’smodels are analyzed based on both the local constituent behaviors and a globalsimplifiedbending-curvatureconstitutivelaw.Figures4.1and4.2showtwotypicalresponsesofreinforcedconcretebeamswithdifferent

steelreinforcements.Bothcurvesarelinearintheinitialstages.Initially,forthelowloadinglevel,thebeamcanbeconsideredasuncracked.Thestrainsatthisstageareverysmall,andthe stress distribution is essentially linear. The moment-curvature diagram at this stage isessentially linear. When the stress of the lower fiber reaches the tensile strength of theconcrete, cracking occurs at the lower part of the reinforced concrete beam, for positivebendingmoment. After cracking, the tensile force in the concrete is transferred to the steelreinforcement.Theresultisthatthelowerpartofaconcretesectionisinvolvedinresistanceto the acting moment: a decrease in the beam stiffness is observed. However, it isexperimentally observed that the beam stiffness is slightly larger than the stiffness totallyneglectingtheconcreteintension.Thisphenomenon,alreadydescribedinChapter2,iscalledthetensionstiffeningeffect.Althoughpotentiallysignificantforthedeflectioncalculationattheserviceabilitylimitstate(SLS),thetensionstiffeningeffectwillbeneglectedinthefollowingfortheULSdesign.Whenneglectingtheconcretebehaviorintension,thestressdistributionintheconcreteisstilllinearatthisstage.

Figure4.1.Typicalexperimentalmoment-curvaturediagramforathebendingofaductilereinforcedconcretebeam–bilinearapproximation–lightlyreinforcedsections

Thebehaviorofthesectionaftercrackingisdependentmainlyonthesteelcontent(seealso[PAR 75]). For lightly reinforced sections, after a certain loading level, the tensile steelreinforcement reaches the yield point and enters into the plasticity range. Even during thisplasticity stageof the steel reinforcement, the concrete in compression can still behave in alinear range.This step is associatedwith a drastic change of slope in the bendingcurvaturediagram. Once yielding occurs in the tensile steel reinforcement, the curvature increasesrapidlywithasmallincreaseinmoment(seealso[MAC97]).Finally,thebeammainlyfailsduetocrushingoftheconcreteatthetopofthebeam,meaningthatPivotBmostlycontrolstheULSofthebendingbehaviorofreinforcedconcretebeamsforhigh-ductilitysteel(evenifthetensionsteelreinforcementsareintheplasticitystage).However,PivotAfailurecanalsobepotentially observedwith a strain of the tensile steel reinforcement,which has reached theultimatestrainofsteelintension.PivotA-typefailureleadstohigherductilitiesthanPivotB-typefailure,asalsomentionedin[PAR75].

Figure4.2.Typicalexperimentalmoment-curvaturediagramforathebendingofaquasibrittlereinforcedconcretebeam–heavilyreinforcedsections

Inheavilyreinforcedconcretesections,thebending-curvatureresponseishighlynonlinear,

duetothenonlinearityofthestress–strainresponseinducedbythemicro-crackingphenomena.Aquasi-brittleresponseisthenobservedunlesstheconcreteisconfinedbyclosedstirrupsatclose centers. If the concrete is not confined, the concrete crushes at a relatively smallcurvature before the steel yields leading to the softening behavior in the bending-curvaturediagram (see Figure 4.2). To ensure ductility behavior, it can be required to design thereinforcedconcretebeamswithasteelcontentlessthanthebalanceddesignvalueleadingtoquasi-brittleresponses.ThenatureoffailureatULSdependsonthedesignofthereinforcedconcretesection.Inthe

following,thecharacteristicmaterialparametersassociatedwithdifferentkindofbehaviorsatfailurewillbeanalyzed.Asoutlinedin[PAR75]or[MAC97],althoughconcreteisnotaductilematerial(concrete

istypicallyaquasi-brittlematerial),reinforcedconcretebeamscanexhibitlargeductilitiesinbendingasshowninFigure4.1.Typically,thecurvatureatULScanbelargerthanfivetosixtimes the curvature at SLS, due to the abrupt change of slope of the bending-curvatureconstitutivelawaftertheyieldingofthetensilereinforcement.Abilinearapproximationof thiscomplexnonlinearbending-curvatureconstitutive lawcan

model the different nonlinear stages with sufficient accuracy (see also the bilinearapproximation in Figure 4.1). The tension stiffening effect is neglected for the bendingmodelingofreinforcedconcretebeamsatULS.

4.1.2.Moment-curvaturerelationshipforconcrete–briefoverview

Themoment-curvaturerelationshipscanbeobtainedfromthelocalconstitutivelawsforboththe concrete and the steel part, integrating over the cross-section.Most of the results in theliteraturearefocusedonthenumericalinvestigationofthebending-curvaturerelationshipfordifferentamountofaxialforces.Forinstance,Pfrangetal.[PFR64]numericallyderivedthemoment-curvature relationship of reinforced concrete sections for different axial load level.TheconcretemodelingwasbasedonthelawofHognestad[HOG51].Hognestad’slaw[HOG51] is composedof the usual parabolic lawup to themaximumcompressive strength and alinearly decreasing (softening) branch up to the ultimate strain failure. Hognestad’s law isgivenby[HOG51]:

[4.1]TheassumptionsusedbyPfrangetal.[PFR64]arequiteusual:thetensionpartofconcrete

wasneglected.Strainhardening inplasticitywas alsoneglected for the steel reinforcement.

Somebending-curvature relationship andnormal force-bendingmoment interactiondiagramswere numerically obtained for symmetrically reinforced concrete beams. The sections haveshownalargeamountofductilityatlowaxialloadlevel(axialloadincompression).Astheloadlevelincreased,ductilitydecreasedmarkedly.Furthermore,thereinforcedconcretebeambecamestifferwithincreasingaxialloaduptoacriticalloadcalledthebalanceload.Since the paper of Pfrang et al. [PFR 64], many papers have been published on the

theoreticalmoment-curvature determination in reinforced concrete design (see, for instance,[CAR86];ormorerecentlythepaperscoauthoredbyProfessorChandrasekaran[CHA09e],[CHA10d],[CHA11d]).CarreiraandChu[CAR86]alsoneglectthestrainhardeninginthesteelreinforcements.The

concretelawincompressionwasmodeledbythefollowingnonlinearstress–strainlaw:

[4.2]EvenifthelawofCarreiraandChucannotbecastintheframeworkofSargin’slawinthe

general case [CAR86],we recognize the lawofDesayi andKrishnan [DES64] forβ = 2,whichisitselfaparticularcaseofSargin’slaw[SAR71].CarreiraandChutakeintoaccountsome contribution of concrete in tension, with a model expressed by a similar homotheticnonlinear law normalized by the tension strength and expressed with respect to thedimensionlesstensionstrain[CAR86].Chandrasekaran et al. [CHA 11d] derived the analytical bending-moment curvature

relationship with the effect of normal force. Strain hardening was neglected for the steelreinforcement, and a limitation of the tensile strain to 10%o was taken into account. Theconcrete was modeled with a parabola–rectangle constitutive law in compression (and asusualthetensionpartinconcreteisneglected).

4.1.3.Analyticalmoment-curvaturerelationshipforconcrete

4.1.3.1.GeneralformulationThe bending-curvature relationship is studied in this part assuming an elastic and perfect-plastic constitutive law for the steel reinforcement (σs1 = fsu at ULS), and a bilinearconstitutivelawforconcrete(σc,sup=fcuatULS).Thecontributionofconcrete in tension isneglected, which is a reasonable assumption for ULS design. An unsymmetrical reinforcedconcretesectionisanalyzed(seeFigure4.3),withtensilesteelreinforcementofsteelareaAs1,but without compression steel reinforcement As2 = 0. The presence of the additionalcompression steel reinforcement does not change fundamentally the complexity of themathematicalproblem.

Figure4.3.Rectangularcross-sectionatUltimateLimitState

The bending-curvature can be obtained in closed-form solution for this problem,with thebilinear constitutive law for the concrete in compression.Weoutline that bending-curvatureconstitutivelawshavealsobeenanalyticallyinvestigatedbyChandrasekaranetal.[CHA11d]for parabola–rectangle constitutive laws, leading to some more complex calculations. Thedetailed equations of the nonlinear bending-curvature constitutive law of a rectangularreinforced concrete section with a parabola–rectangle constitutive law for concrete can bealso available in closed-form solutions for this problem, even if the solutions are moredifficulttomanipulate[CHA11d].AsalreadydetailedbothatSLSandULS,theNavier–Bernoullikinematicsassumptionsare

writtenas:

[4.3]whereκisthecurvatureandyisthedistanceoftheconsideredfibertotheneutralaxis(y=0definesthepositionoftheneutralaxis).Aswe are only presenting results for the case of a simple bendingbehavior (N = 0), the

equilibriumequationsare:

[4.4]

4.1.3.2.ElasticityforboththesteelandtheconcretepartsForsufficientlysmallbendingmoments, thebehavior is linear.Wefirstassume thatboth theconcreteandthesteelareintheelasticityrange(butconcreteintensionisneglectedatULSasalreadyspecified):

[4.5]whereEcandEsare,respectively,theYoung’smodulusofconcreteandsteel.Introducingthe

concrete constitutive law in the bending equilibrium equation, and using the kinematicrelationship between the strain in concrete and the curvature leads to the bending-curvatureequationinthelinearrange:

[4.6]The positionα of the neutral axis is still an unknown of the problem. The normal force

equilibriumequationleadstothenecessaryadditionalequationforresolutionoftheproblem.

[4.7]Note that in the reasoning, up to now, the steel constitutive law has not been used in the

bending-curvature equationpresented in equation[4.6], or in the axial equilibrium equationpresentedinequation[4.7].Now introducing theelasticconstitutive lawof steelwith thecurvaturedependenceof the

straininthetensilereinforcementgivesasecond-orderequationfortheneutralaxis,whichisnothingmorethanthe“staticmomentequation”:

[4.8]whichcanbealsowritteninamoreclassicalequationas:

[4.9]Intheelasticrange,theneutralaxispositiondoesnotdependonthesolicitation(asalready

showninthepartdevotedtotheSLS),andisequalto:

[4.10]The elastic moment-curvature constitutive law, neglecting the tension stiffening effect

(crackedsection),isthenequalto:

[4.11]TheequivalentstiffnessEIeqwasalsodenotedbyEcIII inChapter2,andis theequivalent

stiffnessofthehomogeneouscrackedsection.Fromthisquasi-elasticstate(thesectionisassumedtobefullycrackedandtensionbehavior

is neglected), by increasing the curvature, the section will reach the elastic domain of theconcreteorthesteelpartofthesection.

4.1.3.3.PivotA1–linearbehaviorofconcreteWe first assume that the inelastic part of the section is reached for the tensile steel

reinforcement,whereas theconcretepart incompressionof thesectionremains in itselasticrange.ThiscanalsobereferredtoasPivotA1.Thiskindofbehavioristypicalofalowsteelreinforcementratioandisassociatedwithveryhigh-ductilitybehavior.ItispossibletodefineacriticalcurvatureκYassociatedwithastrainεs1inthetensilesteel

reinforcement,equaltothemaximumelasticstrainεsu,suchas:

[4.12]Thedimensionlesscriticalcurvaturecanbeeasilydefinedas:

[4.13]Thereducedmomentparameterscanbealsointroducedas:

[4.14]In the elastic range (neglecting tension stiffening effect), the bendingcurvature constitutive

lawcanbepresentedwithdimensionlessvariablesas:

[4.15]Oncethetensilesteelreinforcementshaveyielded(σs1=fsu),thepositionoftheneutralaxis

is no more constant but depends on the bending moment solicitation. The normal forceequilibriumequationgivesinfact:

[4.16]Thepositionoftheneutralaxisgivenbythedimensionlessparameterαisthenproportional

totheinverseofthesquarerootofthecurvature:

[4.17]Replacing theneutral axisvariable as anonlinear functionof the curvature in thebending

momentequilibriumequationleadstothenonlinearbendingmoment-curvatureconstitutivelawas:

[4.18]The bendingmoment-curvature constitutive law can be also presented in a dimensionless

formatas:

[4.19]whereκu is theultimate curvature associatedwithPivotA1.TheULSatPivotA is definedfromtheultimatestraincondition:

[4.20]The calculation of the ultimate curvature κu can be achieved from the resolution of this

ultimate strain condition, which is a nonlinear equation expressed with respect to thecurvature:

[4.21]Thisnonlinearequationisinfactasecond-orderpolynomialequationofthedimensionless

curvature:

[4.22]The ultimate curvature is then obtained as the solution of this secondorder polynomial

equation:

[4.23]Among these twosolutions,onlyoneverifies theconstraint condition inducedbyequation

[4.21],andwrittenas:

[4.24]Wefinallycalculatetheultimatecurvatureforthisreinforcedconcretesectionasthelargest

solutionofthesecond-orderpolynomialequation:

[4.25]

Figure4.4.Bendingmoment-curvaturerelationshipofareinforcedconcretebeam–failurewithPivotA1–linearbehaviorofconcrete;C30-37fortheconcrete;B500Bforthesteel;

It has been shown that the bendingmoment-curvature response of the reinforced concretebeam is nonlinear in the post-yield regime, as shown by equation [4.19]. It is howeverpossible to linearize this nonlinear function around the yield point, and the linearapproximationiswrittenas:

[4.26]As shown in Figure 4.4, this linear approximation solution is not very accurate for a

sufficiently largecurvature, as canbe the case for thisductile section inPivotA (low steelreinforcement ratio). The asymptotic value of the dimensionless bending moment is quiteaccurateinfactashighlightedinFigure4.4:

[4.27]Thissolution(PivotAfailurewithonlyelasticbehaviorofconcrete,i.e.PivotA1)isvalid

aslongasthestrainoftheupperfiberhasnotreachedtheelasticconcretestraindenotedbyεc3,whichismathematicallyexpressedbytheinequality:

[4.28]Thisinequalityisalsoequivalenttothefollowinginequality:

[4.29]This solution is in fact equivalent to consider that the steel reinforcement ratio has to be

lowerthanacriticalratiogivenby:

[4.30]WecalculateforaC30-37typeconcreteandforaB500Bsteelthesteelreinforcementratio

limitas:

[4.31]Thenumerical resultspresented inFigure4.4show thebendingcurvatureconstitutive law

for a PivotA1 failure responsewith a reinforcement density equal to which is

lowerthantheonecomputedabove andassociatedwiththetransitiontoPivotA2.

4.1.3.4.PivotA2–bilinearbehaviorofconcreteItishoweverpossibletohaveaPivotA-typefailuregovernedbythemaximumcapacitystraininthetensilesteelreinforcementwithsomenonlinearbehaviorsoftheconcrete.ThisbehaviorcanbealsoreferredasPivotA2.Forthiskindofcross-section,typicallyfor:

[4.32]thesectionbehaveslikeinPivotA1(elasticbehavioroftheconcrete)uptoacriticalcurvatureandthenthesectionwillfailinPivotA2(nonlinearbehaviorofconcrete).

[4.33]Inthiscase,theequilibriumequations[4.4]arewrittenas:

[4.34]Asusual,αdisthedistancefromtheupperfiberofthecross-sectiontotheneutralaxis,and

αcdisthedistancefromtheupperfiberofthecrosssectiontothefiberinconcreteassociatedwiththemaximumelasticstrain(seeFigure3.29).Theintegralequationscanbeeasilydevelopedas:

[4.35]associatedwiththestraincompatibilityequationgivenby:

[4.36]Thedimensionless neutral axis parameters canbe expressedwith respect to the curvature

variablesfromthenormalforceequilibriumequationas:

[4.37]Introducingtheseneutralaxisparametersinthebendingmomentequilibriumequationleads

tothereducedmomentnonlinearfunctionofthecurvature:

[4.38]whereκu is theultimatecurvatureassociatedwithPivotA2.TheULSatPivotA2 isdefinedfromtheultimatestraincondition:

[4.39]The calculation of the ultimate curvature κu can be achieved from the resolution of this

ultimatestraincondition,whichisalinearequationexpressedwithrespecttothecurvature:

[4.40]This solution (PivotA2 failure) is valid as long as the strain of the upper fiber has not

reachedtheultimateconcretestraindenotedbyεcu3,whichismathematicallyexpressedbytheinequality:

[4.41]This solution is in fact equivalent to consider that the steel reinforcement ratio has to be

lowerthanacriticalratiogivenby:

[4.42]ThisupperboundisclearlyassociatedwiththePivotABboundary.Forhigherreinforcement

steel density, the beam failswith PivotB,meaning that the strain capacity in the section iscontrolledbythestraincapacityinthecompressionconcreteblock.WecalculateforaC30-37typeconcreteandforaB500Bsteelthesteelreinforcementratio

limitas:

[4.43]Atthecriticalsteelreinforcementratio:

[4.44]ThebeamfailsatPivotAB, that is theultimatestate ischaracterizedbyboththemaximum

strain capacity of the tensile steel reinforcement and the maximum strain capacity of theconcreteblock in compression.At theULSassociatedwith this critical steel reinforcement,weclearlyhave:

[4.45]Figure4.5.showsthetypicalresponseofareinforcedconcretesectionthatiscontrolledby

PivotA2 failure, for low steel reinforcement density ratio. The two stages of PivotA1 andPivotA2areclearlyseenwiththeinitialyieldingof thetensilesteelreinforcement,andthen

the contributionof themicro-cracking effect in the compressionblock, even if the failure isstillaPivotAfailurecontrolledbythetensilestraincapacityinthesteelreinforcement

Figure4.5.Bendingmoment-curvaturerelationshipofareinforcedconcretebeam–failurewithPivotA2–bilinearbehaviorofconcrete;C30-37fortheconcrete;B500Bforthesteel;

4.1.3.5.PivotB–elastoplasticbehaviorofthetensilesteelreinforcementWewillnowstudythecaseofaPivotB-typeoffailure,forthesteelreinforcementratiolargerthanthecharacteristicdensity:

[4.46]For this kind of reinforced concrete cross-section, the section behaves like in Pivot A1

(elasticbehavioroftheconcrete)uptoacriticalcurvatureandthenthesectionwillbehaveinPivotB(failureatPivotBwithplasticityinthetensilesteelreinforcement).

[4.47]ThePivotB branch is detailed below. The reducedmoment equation is still valid, if the

tensilesteelreinforcementisintheelasticityrange,andisgivenagainbelow:

[4.48]whereκuistheultimatecurvatureassociatedwithPivotB.TheULSatPivotBisdefinedfromtheultimatestraincondition:

[4.49]The calculation of the ultimate curvature κu can be achieved from the resolution of this

ultimatestraincondition,whichisalinearequationexpressedwithrespecttothecurvature:

[4.50]Figure4.6showsatypicalfailureatPivotB,whichclearlyshowsthelowerductilityofthe

responsewithafailureatPivotB(failureincompression),ascomparedtothefailureatPivotA(failureintensioninthetensilesteelreinforcement).

Figure4.6.Bendingmoment-curvaturerelationshipofareinforcedconcretebeam–failurewithpivotB–bilinearbehaviorofconcrete;C30-37fortheconcrete;B500Bforthesteel;

In thiscase,with theparametersof interest, theductility thatcanbemeasuredas the ratiobetween the ultimate curvature and the elastic curvature is approximately three, which issignificantlydifferentfromtheductilityobservedinPivotA.Furthermore,itisalsoobservedthat the asymptotic approximation of the reducedmoment suggested for pivotA is nomorerelevantinthiscase.

4.13.6.PivotB–elasticbehaviorofthetensilesteelreinforcement

Forveryhigh steel reinforcementdensity, the concrete still remains in the linear range.Theequilibriumequationsarenowwrittenas:

[4.51]Introducingtherelationshipbetweentheneutralaxisparameters(seealsoFigure3.29):

[4.52]the normal force equilibrium equation gives the neutral axis position with respect to thecurvatureas:

[4.53]The other parameterαc,which defines the limit of the elastic domain in the compression

concreteblockalongthecross-section,isdeducedas:

[4.54]Thebendingmoment-curvature isobtainedusing thebendingmomentequilibriumequation

thatgivesanonlinearfunctionas:

[4.55]ItispossibletodefineanewcriticalcurvatureκYassociatedwithastrainεc,supintheupper

fiberofthecross-section,equaltothemaximumelasticstrainεc3inthecompressionblockofconcrete,suchas:

[4.56]Thedimensionlesscriticalcurvaturecanbeeasilydefinedas:

[4.57]Thereducedmomentparameterscanbealsointroducedas:

[4.58]Thereducedmomentcanberewrittenas:

[4.59]

whereκuistheultimatecurvatureassociatedwithPivotB.TheULSatPivotBisdefinedfromtheultimatestraincondition:

[4.60]The calculation of the ultimate curvature κu can be achieved from the resolution of this

ultimate strain condition, which is a nonlinear equation expressed with respect to thecurvature:

[4.61]The ultimate curvature is then obtained as the positive solution of this second-order

polynomialequation,whichis:

[4.62]Thissolution(PivotBfailurewithonlyelasticbehaviorofthetensilesteelreinforcement)is

validaslongasthestraininthetensilesteelreinforcementεs1hasnotreachedtheelasticsteelstraindenotedbyεsu,whichismathematicallyexpressedbytheinequality:

[4.63]This solution is in fact equivalent to consider that the steel reinforcement ratio has to be

largerthanacriticalratiogivenby:

[4.64]WecalculateforaC30-37typeconcreteandforaB500Bsteelthesteelreinforcementratio

limitas:

[4.65]Thenumerical resultspresented inFigure4.7 show thebending-curvatureconstitutive law

for a PivotB failure response with a reinforcement density equal to which islargerthantheonecomputedabove andassociatedwiththetransitiontoPivotB.

Figure4.7.Bendingmoment-curvaturerelationshipofareinforcedconcretebeam–failurewithpivotB–bilinearbehaviorofconcrete;C30-37fortheconcrete;B500Bforthesteel;

4.1.3.7.Synthesisofthebending-curvatureconstitutivelawsEach kind of reinforced cross-section is associated with a specific bending-curvatureconstitutivelaw.Table4.1summarizesthemainresultsontheductilityformulaedependingonthesteelcontentofthereinforcedconcretecross-section.

Table4.1.Ductility withrespecttothedimensionlesssteelreinforcementquantity

Thecalculationofthedimensionlesselasticcurvature iscontrolledbythesteelcontentρ′s1inthereinforcedconcretesection,andthesteelandconcretestraincapacitiesas:

[4.66]Inacertainsense,thiscalculationislinkedtothedeterminationofPivotAorPivotBatSLS.

WenotethatthedeterminationofthepivotnatureatULScanbedifferentfromthepivotnatureat SLS, the elastic SLS can be controlled, for instance, by the steel strain capacity at SLSassociated accompaniedwith aPivotB failure (ULS).These inequalities conditions can beexpresseddirectlywithrespecttothesteelcontentas:

[4.67]

4.1.3.8.DuctilityofthereinforcedconcretesectionFigure 4.8 shows the decrease in ductility with the steel reinforcement density, and thetransition from Pivot A failure to Pivot B failure. The ductility is very sensitive to thereinforcementdesign,anditisnotrecommendedtohaveheavilyreinforcedsections,toavoidquasi-brittleorbrittleresponsesofthereinforcedbeams.Inpractice,toavoidthequasibrittleresponsesofthereinforcedconcretebeams,itisrequiredthatthesteelcontentisboundedbyacriticalratioρb,calculatedas:

[4.68]

Figure4.8.Theductilityκu/κYstronglydependsonthesteelcontentρ′s1;C30-37fortheconcrete;B500Bforthesteel;αe=17.5

This condition expressed the need of inelastic strains in the tensile steel reinforcement to

guarantyaminimumamountofductility.ThestrengthcapacityevolutionwithrespecttothesteelcontentisstudiedinFigures4.9and

4.10.The curveshavebeendrawndue to the analytical expressionsof the reducedmomentpresented in this chapter for each kind of behavior, and piecing together each solution.Theinelastic strength reservemeasured by the dimensionless factorµu /µY is close to unity inPivotAandinPivotBatULSuptothecriticalpointwherethesectionbehavesinPivotABatSLS,wheretheinelasticstrengthreservesignificantlyincreases.Oncethesectionhasreacheditsbehaviorcharacterizedbytheelasticbehaviorofthetensilesteelreinforcement,thestrengthcapacityisstableandisnomoresensitivetothesteelcontent.Moregenerally,astheductilityκu / κY decreases with the steel content ρ′s1, the strength capacity of the reinforced cross-section measured by the ultimate reduced moment µu monotonically increases (see Figure4.10).However, a singular point is clearly exhibited in Figure 4.10 for a steel content ρ′s1equal to the critical ratio ρb associated with the elasticity behavior of the tensile steelreinforcementatULS.Thispoint isclassifiedasthebalancefailure(see,for instance,[PAR75]).ThispointislocatedinPivotBwherethesectioniscontrolledbythestraincapacityinconcreteatULS,andisassociatedwithachangeofbehaviorofthetensilesteelreinforcementfromtheplateaurangeinplasticitytotheelasticityrange.ItisshowninFigure4.10thattheultimatereducedmomentincreasesalmostlinearlyupto

thepointcorrespondingtothebalancefailureρ′s1≤ρb.Inthecompressionfailureregionforρ′s1≥ρb,theincreaseinmomentwithsteelareaisextremelysmallbecauseboththesteelstressandtheleverarmdecreasewithincreaseinthesteelcontentforthisregime.Hence,thereisachangeofslopeinthediagramofthereducedmomentwithrespecttothesteelcontentatthisspecificpointand there isonlya littleadditionalstrength tobegainedbyan increase in thesteelarea(seeFigure4.10).

Figure4.9.Theinelasticstrengthreserveµu/µystronglydependsonthesteelcontentρ′s1;C30-37fortheconcrete;B500Bforthesteel;αe=17.5

Figure4.10.Thestrengthcapacityµustronglydependsonthesteelcontentρ′s1;C30-37fortheconcrete;B500Bforthesteel;αe=17.5

4.1.3.9.Conditionofnon-fragilityForductilityreasons,ithasbeenshownthatitisrequiredforthesteelcontentρ′s1 tobelessthanacriticalsteelcontentρbcorrespondingtothebalancefailurepoint.Itisalsorequiredtostipulate a minimum reinforcement ratio that should always be exceeded to avoid a brittleresponse of the reinforced concrete beam. This condition is called the condition of non-fragility.Thisisbecauseifthereinforcementratioistoosmall,thecalculatedflexuralstrengthof the reinforced concrete sectionMu becomes lower than the bendingmoment required tocrack the sectionMcr (seeChapter 2), and on cracking, failure becomes sudden and brittle.Thisconditionismathematicallyexpressedby:

[4.69]It can also be a requirement that the elasticmoment of the cracked sectionMY has to be

largerthanthecriticalmomentoftheuncrackedsectionMcr:

[4.70]Asshowninthischapter,thesetwolastinequalitiesareverycloseinPivotA,whichisthe

Pivotconcernedforlowsteeldensity.Fortheunsymmetricalreinforcedconcretesectionwithonlytensilesteelreinforcement,thisinequalityinPivotA(orinPivotBwithanSLSinPivotA)isequivalentto:

[4.71]Thenonlinearinequalityhasthentobesolvedas:

[4.72]-Figure4.11showstheevolutionoftheminimumsteelreinforcementρ′s1withrespecttothe

tensionstrengthratiofctm/fsu,andthiscurvewasobtainedfromthenon-linearfunctiondefinedby[4.72].

Figure4.11.Conditionofnon-fragilityexpressedwiththeminimumsteelreinforcementρ′s1withrespecttothetensionstrengthratiofctm/fsu;αe=17.5;d/h=0.9

An approximated formula can be obtained by assuming that the critical moment isindependentofthesteelcontentandisexpressedby:

[4.73]Hence,forsufficientlysmallvaluesofthesteelcontent,thelowerboundisobtainedas:

[4.74]Thislowerboundiscalculatedbelowfordifferentgeometricald/hratiosas:

[4.75]ItisshowninFigure4.11that thelinearapproximationwiththeproportionalitycoefficient

0.21correspondingtod/h=0.9isrelevantonlyforsmallsteelcontent.However,forlargersteel content, the proportionality coefficient 0.23 is more accurate, due to the weaknonlinearityof thecurve.Thiscoefficient0.23 is, in fact, thecoefficient thatcorresponds tod/h=0.85.Thesamecoefficient0.23isusedasacorrectioncoefficientfortheconditionofnon-fragilityoftheoldFrenchrulesBétonArméauxEtatsLimites:

[4.76]The condition of non-fragility of the old French rules Béton Armé aux Etats Limites is

normalizedwithrespecttotheuniaxialtensilelimitstrengthfyk:

[4.77]

4.1.4.Amodelbasedonthebilinearmoment-curvatureapproximation

Thebending-curvaturerelationshipisstronglydependentonthesteelcontentandthematerialconstitutive laws. The normalized bendingcurvature relationship of each studied bending-curvaturelawcanbepresentedinFigure4.12.

Figure4.12.Normalizedbending-curvatureconstitutivelawasafunctionofthesteelcontentρ′s1;C30-37fortheconcrete;B500Bforthesteel;αe=17.5

Each curve, numerically obtained with bilinear constitutive laws for both the steelreinforcementandthecompressionblock,canbereasonablyapproximatedbyaglobalbilinearconstitutivelaw(atthebeamscale),withalinearelasticstageandalinearhardeningbending-curvature constitutive law. Such kinds of bilinear bending-curvature constitutive laws havealsobeenobtainedintheliteraturewithsomeotherconstitutivelawsfortheconcrete.ParkandPaulay[PAR75](withaHognestadconstitutivelawfortheconcretepart),CarreiraandChu[CAR86](withthegeneralizationofSargin’slawfortheconcretepart)orChandrasekaranetal. [CHA09e] (withaparabola–rectangleconstitutive law for theconcretepart) also foundthat the bending moment-curvature relationship is close to a bilinear approximation (withhardeningrange).Furthermore,Chandrasekaranetal.gavetheexactanalyticalexpressionsofthe bending-curvature constitutive law with a rectangle-parabolic constitutive law for theconcrete([CHA09e],[CHA10d]and[CHA11d]).Thiskindofbilinearbendingmoment-curvatureconstitutivelawisalsousedbyChallamel

[CHA03]forsteelsections,and,infact,manyengineeringapplicationsareconcernedbysuchamathematicalbilinearmodeling.Thebilinearapproximationcanbemodeledbyaplasticitymodelatthebeamscalewithalinearhardeninglaw(seeFigure4.13).

Figure4.13.Normalizedbending-curvatureconstitutivelaw;Mp=MYandm=Mu/MY=µu/µY

Foramonotonicloading,thebilinearconstitutivelawisgivenby:

[4.78]TheparametersoftheconstitutivelawκY,µY,κuandµuaregivenforeachsteelcontentand

foreachbehaviorofthereinforcedconcretecrosssection(see,forinstance,Table4.1 for theidentificationofeachparameterfromthelocalsteelandconcreteconstitutivelawsbasedonbilinearapproximations).Thelocalmoment-curvaturerelationship(M,κ)consideredisbilinearwithalinearelastic

partandalinearsofteningpart(Figure4.13).Thismodel isfirstconsideredinalocalform,that is standard plasticity model with negative hardening. The non-local extension will beinvestigated later in this chapter.Mp is the limit elasticmoment, andκY is the limit elasticcurvature, related throughMp / χY = EIeq = EI, which is the equivalent stiffness of thehomogeneouscrackedsectionThe elastoplastic model represented in Figure 4.13 is a standard plasticity model with

positive plasticity hardening, associatedwith the yield function f given by equation [3.12],with the plasticity flow rule given by equation [3.13]. The hardening being linear, thefollowingrelationholdsforthe“local”case:

[4.79]Usingequation[3.16],thehardeningbranchofthebendingmomentcurvatureconstitutivelaw

canbepresentedinthefollowingform:

[4.80]Thehardeningmoduluscanthenbecalculatedfrom:

[4.81]Theplasticityhardeningmodulusκ+ associatedwith themacroscopicbending responseof

the reinforced concrete beam depends on the material constituent of the cross-section, andespeciallyontheductilityofthereinforcedcross-section,whichisverysensitivetothesteelcontent (see Figure 4.14). Figure 4.14 shows the dependence of the equivalent hardeningmodulus of an unsymmetrical reinforced concrete section, calculated from the bilinearconstitutive law for concrete at the local state, and an elasticperfect plasticity law for thetensilesteelreinforcement.

Figure4.14.Evolutionoftheequivalenthardeningmodulusk+/EIwithrespecttothesteelcontentρ′s1foranunsymmetricalreinforcedconcretesectionwithbilinearresponseforconcrete

4.2.Postfailureofreinforcedconcretebeamswiththeinitialbilinearmoment-

curvatureconstitutivelaw4.2.1.Elastic-hardeningconstitutivelaw

In this chapter, the bending response of a reinforced concrete cantilever beam will betheoreticallyinvestigateduptothefailure.Thecantilever isshowninFigure3.2,asalreadyinvestigatedwithanelastic-softeningresponse.Followingthetheoreticaldevelopmentsforthejustificationofthemacroscopicbendingresponsefromthelocalsteelandconcreteconstitutivelaws, an elastic-hardening response will be first considered. This reinforced concretecantileverproblemhasalsobeen investigatedbyParkandPaulay [PAR75]withabilinearbending-curvature constitutive law associated with positive hardening (see, for instance,Figure 4.13). This problem, with elastic-hardening bending-curvature constitutive law, hasbeenrecentlystudiedbyChallameletal.[CHA10b]forthecantileverstructuralcase,andbyChandrasekaran et al. [CHA 09e] for fixed–fixed beams or simply supported beams underconcentratedanddistributedloading.The cantilever beam problem is a statically determinate problemwith a bendingmoment

evolutiongivenbyequation[3.2].Formonotonicincreasingloading,thebeambehavesintheelastic range, forP smaller thanPY=Mp /L. In thehardening range, the load is increasingsuchas:

[4.82]Forincreasingvalueoftheloadoutsidetheelasticdomain,theplasticregimestartsandthe

beamcanbesplit intoanelasticandaplasticdomain(seealsoFigure3.2).Thesizeof theplasticzoneisdenotedbyl0+.Intheplasticzone,theplasticcurvatureislinearlyincreasing:

[4.83]The continuity of the plastic curvature at the elastic–plastic interface leads to the plastic

zone-loadrelationship:

[4.84]Note that the propagation of the local hardening process zone is equivalent to the linear

relationshipbetweenpreviouslyuseddimensionlessparameters:

[4.85]The displacement field in the plastic zone is obtained using the boundary conditions

(clampedbeam):

[4.86]The displacement in the elastic zone is obtained by enforcing the continuity of the

displacementandtherotationattheinterface:

[4.87]Theload-deflectionrelationshipisfinallydeducedinthehardeningrangefrom:

[4.88]Figure4.15showsthehardeningrangeforthecantileverbeam.Itcanbeeasilycheckedfrom

equation[4.88]thattheload-deflectionrelationshipisnotlinear,evenforthelocalhardeningconstitutivebehaviorconsideredinthisparagraph(seealsoFigure4.15).

Figure4.15.Load-deflectionrelationshipforthecantileverreinforcedconcretebeamwithabilinearbending-curvatureconstitutivelaw;m=5/4;EI/k+=11

Aremarkable result is that theplasticcurvaturedistributiondependson thehardening lawbutthepropagationlaw(equation[4.84]orequation[4.85])doesnotdependonthemodelofthehardeninglaw.Infact,whateverthehardeningmodel(evenincaseofnonlinearhardening),thesameequalityisvalid:

[4.89]Thetotalcurvaturealongthebeamiscalculatedfrom:

[4.90]The curvature increases drastically inside the inelastic plasticity zone that occurs in an

equivalent“plastichinge”inthevicinityofthecriticalsection(seeFigure4.16).

Figure4.16.Evolutionofthetotalcurvaturealongthebeamm=5/4;EI/k+=11

Usingequation[4.81]with thesectionalparametersm=5/4andEI/k+=11,wecalculateκu/κY=4,which is thecurvaturevalueobtained in theclampedsectionwith theassociatedloadingparameter.Themaximumsizeoftheplasticlengthinthehardeningprocessisobtainedfrom:

[4.91]Thevalueofmthatcontrolsthestrengthcapacityreservebeyondtheelasticmomentdepends

on the design of the reinforced concrete section and the considered Pivot. In the presentreasoning,thephenomenonofincreaseinsteelforcesduetoinclineddiagonaltensioncracksisneglected.Thisadditionalphenomenoncanbetakenintoaccountbysomecorrectionofthelinearly decreasing curvature (see [PAR 75], [PAU 92]). Diagonal tension cracks in theplasticityzoneincreasetheavailableplasticrotationbyspreadingthezoneofyieldingalongthemember(tensionshifteffect).

4.2.2.PlastichingeapproachAsimplifiedplastichingeapproachcanbeused,associatedwithanequivalent lengthof theplastic hingeoverwhich the plastic curvature is considered to be constant and equal to the

meanvalueoftheexactcurvaturefield

[4.92]Theequivalentplasticrotationθpcanbecalculatedas:

[4.93]wherelp=lo+isthelengthofthe“equivalent”plastichinge,whichisloaddependentaccordingtoequation[4.84]. It is clearly seen from equations[4.92] and [4.93] that the length of the“equivalent”plastichinge,themeanvalueoftheplasticcurvatureandtheplasticrotationareall dependent on the loading parameter, which has been also reported by [CHA 09a] orChallamel et al. [CHA 10b] for the bending of elastoplastic hardening beams. Lee andFilippou[LEE09]alsousedasimilarload-dependenthingemodeltocomputethepost-failureresponseofreinforcedconcretebeam-columns.The equivalent load-dependent hingemodel can be computed from the elastic differential

equationswiththefollowingboundaryconditions:

[4.94]Thismodelisinfactequivalenttoaloading-dependentbending-rotationconstitutivelawfor

theconcentratedhinge,obtainedfromequation[4.93]as:

[4.95]The“simplified”nonlinearload-deflectionrelationship,associatedwiththeload-dependent

hingemodel,isobtainedas:

[4.96]Thisdeflection-load relationshipcanbealsoexpressedwith respect to thevariablehinge

lengthas:

[4.97]Thisrelationshipisvalidintheplasticitybranch,andespeciallyattheULS:

[4.98]Amoresimplifiedapproachwouldbebasedonthehingebending-rotationconstitutivelaw

withaconstantglobalhardeningmoduluscalculatedfromthemaximumsizeofthehardeningplasticityzoneatULSas:

[4.99]The“simplified”nonlinearload-deflectionrelationship,associatedwithaload-independent

hingemodel,isobtainedas:

[4.100]AttheULS,equation[4.98]isalsovalid,andbothequivalenthingemethodscoincide.Althoughnotexact,theapproximationsbasedontheequivalenthingemethodsprovidegood

engineering results, especially theonebasedon the loading-dependenthingemodel analysis(seeFigure4.17).Thehingemodelbasedon a load-independenthardeningmoment-rotationconstitutive law leads to the linear curve in Figure 4.17, and leads to relevant resultsespeciallyinthevicinityoftheULS(forκ=κuatthesectionbasis).

Figure4.17.Comparisonoftheexactbilinearmomentcurvaturemodelwiththehingemodels;m=5/4;El/k+=11

There is a large discussion in the literature on the identification of the finite hinge length(see,forinstance,[PAR75],[BAZ03],[MEN01],[BAE08],[ZHA12]and[BAR12]).Asmentioned by Park and Paulay [PAR 75]: “The difference between the various empiricalexpressions emphasizes that the rotation capacity of plastic hinges in reinforced concretememberscanonlybeapproximatedatpresent.Mostresearchisneededtoclearthedifferencesbetweenthevariousempiricalexpressions”.AsshowninFigure4.18basedonthebending-curvatureconstitutivelawobtainedfromthe

bilinearconstitutivelaw,theplastichingelengthstronglydependsonthesteelcontent,andthemodeoffailure.

Figure4.18.Evolutionofthefiniteplasticityhingelengthlpu/Lwithrespecttothesteel

content foranunsymmetricalreinforcedconcretesectionwithbilinearresponseforconcrete;L/d=5;C30-37fortheconcrete;B500Bforthesteel;αe=17.5

ParkandPaulay[PAR75]presentedvariousempiricalexpressionsforthecorrelationofthisfinitehingelength,basedonexperimentalresults(see,forinstance,[BAK56],[SAW65]and[MAT67]).Baker[BAK56]definedanempiricalruleexpressedby:

[4.101]where z is the distance of the critical section to the point of contraflexure; z = L for thecantileverbeamstudiedinthischapter.κ1,κ2andκ3areempiricalcoefficientsthatdependonthesteelandconcreteusedinthesection.ForL/d=5,wetypicallyobtainlp/d=0.83withthismodel.Sawyer[SAW65]usedalinearregressionlawwithrespecttodandzas:

[4.102]ForL/d= 5,we typically obtain lp/d= 0.63with thismodel.Mattock [MAT67] used a

slightlydifferentcorrelationexpression:

[4.103]ForL/d=5,wetypicallyobtainlp/d=0.75withthismodel.Thesethreemodelsbasedon

experimental results are compared in Figure 4.18 with the theoretical model based on thestraincapacityofthecross-sectiondefinedbyEurocode2rules.Inthetheoreticalanalysis,thephenomenonof increase in steel forcesdue to inclineddiagonal tensioncracks isneglected,and the post-failure analysis is not included. It can be observed in Figure 4.18 that theempiricalcorrelationexpressionsmainlygivehigherplasticitylengththanthetheoreticalone

in Pivot A (or for tension failure), whereas the correlation length can underesti-mate thetheoreticalplasticitylengthinPivotB(orforcompressionfailure).ItisalsoshowninFigure4.18thatthetheoreticalplasticityhingelengthissensitivetothe

steelcontentandisgloballyanincreasingfunctionofthesteelcontent.PriestleyandPark[PRI87],orPaulayandPriestley [PAU92]suggestedanotherempirical lawfor theplastichingelength,whichissensitivetothesteelcontentandcanbepresentedas:

[4.104]where lp,0 and γ1 are two fitting parameters. Priestley and Park [PRI 87], or Paulay andPriestley[PAU92]suggestedthevaluelp,0=0.08LwhereasEurocode8,forinstance,givestheclosevaluelp,0=0.10L.AccordingtoFigure4.18,anew

simpleproposalcouldbebasedon:

[4.105]wherethefittingparameterγischosentobeequaltoγ=2.37,foraC30-37typeconcrete,witha B500B for steel, in order to enforce the square root approximation function (equation

[4.103])topassthroughthebalancedfailurepointdefinedby Equation[4.105]isaparticularcaseofequation[4.104]wheretheconstantparameterlp,0hasbeenassumedtovanish.Figure4.18showsthatthenewproposalgivesthecorrecttendencyforthematchingofthetheoreticalresultsbasedonthetheoreticalstraincapacitysection.Finally, as outlined by Park and Paulay [PAU 75], the plastic hingewithin the span of a

symmetrically loadedbeamwillhavea totalequivalent lengthtwice the lengthfoundfor thecantilever(seealso[CHA08a]).

4.2.3.Elastic-hardeningconstitutivelawandlocalsofteningcollapse:Wood’sparadox

IthasbeenshownhowthereinforcedconcretebeamcanbedesignedwithrespecttotheULS,which is a strain-controlledULS,compatiblewith theallowablebehaviorof thebeamwithrespecttothespecificities.However,therulesincludingEurocode2donotquestionthebehaviorbeyondthisultimate

limit.Wewilltrytounderstandnowwhathappensbeyondthepeak,orduringtheinitiationofcollapse.Thetri-linearbendingmomentcurvaturediagramisnowconsideredwithasofteningbranchassociatedwiththefailurebehavior(seeFigure4.19).Itisassumedthatthesofteningbranchstartsattheultimatecurvatureκubutthereasoningcan

begeneralizedbyconsideringastartingbranchatahighervalue.Thehardening/softeningruleisnowgiveninthefollowingform:

[4.106]Mpisthelimitelasticmoment,andκYisthelimitelasticcurvature,relatedthroughMp/κY=

EI.m is the ratio between themaximummoment reached during positive hardening and thelimit elastic moment (m is necessarily greater than unity), and κc is the plastic curvaturereachedbeforetheinitializationofthesofteningprocess.Thehardeningmodulusκ+ispositivewhereasthesofteningmodulusκ-isnegative.Wefirststudiedthe“local”softeningprocessinthesensethatthesofteningconstitutivelaw

isexpressedonlywithrespecttothelocalplasticcurvature.

Figure4.19.Elastic–plastichardening–softeningmoment-curvaturelaw

Theplasticsofteningresponsemaystartonce the loadP reaches themaximumvaluemPY.

Enforcingthatκpisacontinuousfunctionofx

[4.107]This additional assumption gives the newWood’s paradox for hardening–softening local

constitutive relationship. The unloading elastic solution is the only possible solution of thelocalsofteningproblem(Figure4.20).In this case again, the paradox can be interpreted as the appearing of plastic curvature

incrementslocalizedintoonesinglesection,leadingtophysicallynoreasonablephenomenonoffailurewithzerodissipation.

Figure4.20.Wood’sparadox-localhardening/softeningplasticitymodels;m=5/4;EI/κ+=11

4.2.4.Elastic-hardeningconstitutivelawandnon-locallocalsofteningcollapse

Wood’sparadoxforthehardening–softeningbeamcanbesolvedbyusinganon-localsofteningmoment-curvaturelaw,asinthecaseoftheelastic-softeningbeams.Oncethebendingmomentintheclampedsection|M(x=0)|=PLreachestheyieldstrengthmMp,thesofteningzonecanpropagate from theclampedsection,whereasunloading isobserved in thehardeningplasticzone and in the elastic zone. The local hardening and nonlocal softening constitutiverelationshiparegivenby:

[4.108]Theproblemofthecontinuityrequirementbetweenhardeningandsofteningconstitutivelaws

(a local lawandanon-local law)willbe implicitlysolvedby thefact that the lengthof thesofteningzoneattheyieldstrengthmMpwillvanishaswewillsee(asfortheelastic-softeningproblem). Furthermore, the non-local plastic variable is integrated over the active plasticdomain, that is the softening zone, as the hardening zone is in unloading during this finalprocess.Asintroducedinequation[3.55],thenon-localsofteningmeasureisalsoequivalentto:

[4.109]

By considering the yield function in the softening area, the linear differential equation isobtainedinthesofteningdomain:

[4.110]withtheboundaryconditionsassociatedwiththehigherorderboundaryconditionsofthenon-localmodelandthecontinuityrequirementoftheplasticcurvature:

[4.111]Thegeneralsolutionofthedifferentialequation[4.110]iswrittenas:

[4.112]

ThenonlinearsystemofthreeequationswiththreeunknownsA,Band isfinallyobtained:

[4.113]Thefollowingdimensionlessparametersmaybeintroducedas:

[4.114]leadingtothelocalizationrelationofequation[3.61].Aremarkableresultisthattheplasticdiffusioninthesofteningrangedoesnotdependonthe

hardening range. In other words, the hardening modulus (or the material history in thehardeningdomain)doesnotaffect the localizationprocess, fromaqualitativepointofview.

The deflection in the plastic zone is obtained by integrating twice the elasticcurvature:

[4.115]

The deflection in the elastic zone is derived from the kinematics continuitycondition, along the active softening–passive hardening zone, whereas the plastic curvaturedistributionisconstantintheunloadingphase:

[4.116]

The deflection in the elastic zone is derived from the kinematics continuityconditionalongthepassivehardening-elasticinterface:

[4.117]Generallyspeaking, theplasticzonegrowsin thehardeningrangeuntil themaximumload,

thenamorelocalizedsofteningzonearisesfromtheclampedsectionandcontrolsthemodeofcollapse.Theglobalsofteningisthenobservedafterthehardeningbehavior(Figure4.21)andWood’sparadoxissolvedwithinthenon-localplasticityconstitutivelaw.

Figure4.21.Responseoftheelastoplastichardening–non-localsofteningbeam;EI/k-=−5;lc/L=0.1;m=5/4;EI/k+=11

l0+relatedtothehardeningdomainisthelengthofthehardeningplasticityzonethatpropagatesalongthebeamwithoutanymateriallimits,whereasthesofteninglocalizationzone,denotedbyl0–, is increasing during the softening process, until a finite length, which depends on thematerial-sectionmodel.Ofcourse,thesetwolocalizationzonesarestronglydifferent(l0+≠l0-

). For the non-localmodels studied in this chapter, the plastic variable is integrated on anactiveplasticdomain.Inparticular,duringthesofteningprocess,thenon-localplasticvariableisintegratedonthelocalizationlengthl0–,eveniftheplasticityzoneisgenerallymuchlarger(l0+>l0–).Infact,thehardeningandthesofteningplasticzonesaregivenby:

[4.118]wherethehardeningzonehasbeencalculatedfromthelocalhardeningmodel.Therefore,thesofteningplasticzoneisnecessarilysmallerthanthehardeningplasticzoneforasufficientlysmallcharacteristiclength,thatis:

[4.119]It is not excluded, however, the softening localization zone is influenced by the overall

plasticity phenomena in the precracked reinforced concrete beam, and the simple non-localmodelpresentedinthischapterhasitsownlimits.Thecalibrationofthecharacteristiclengthlc, which includes material and cross-sectional properties, should certainly merit furtherinvestigations. The research on the global modeling of the collapse of reinforced concretebeamsisstillopen,andwouldmeritfurtherstudies,especiallyincludingthecouplingmodeofcollapsebetweentheshearandthebendingmodes.

4.3.Bendingmoment-curvaturerelationshipforbucklingandpostbucklingofreinforced

concretecolumns4.3.1.Acontinuumdamagemechanics-basedmoment

curvaturerelationshipThebucklingandpostbucklingbehaviorsofreinforcedconcretesofteningcolumnscomposedof quasi-brittle materials such as concrete and composite materials are of interest in thissection.Suchmembersmustbedesignedtakingintoaccountthesecond-ordereffectsproducedby the axial loads on the deformed member. The main international rules, including, forinstance,theEuropeancodeforreinforcedconcretedesign,Eurocode2[EUR04],arebasedon this concept, even though theymay allow second-order effects to be neglectedwhen thecolumnslendernessdoesnotexceedcertainlimits.Suchlimitshavebeeninvestigated(e.g.[HEL05a],[HEL08]and[MAR05])andincluded

inmostreinforcedconcretecodes.Variousempiricalortheoreticalbasedmethodshavebeenpublished in the past, which all introduced some necessary realistic imperfections throughadditional load eccentricities ([ROB75], [BAZ03]). This is typical and necessary, from atheoreticalstabilitypointofview, fornonlinearstructuralsystems thatbelong to thefieldofimperfectionsensitivesystems.Therehavebeennumeroustextbooksdevotedtotheelasticorinelasticbucklingofcolumns(see,forinstance,[LHE76]or[BAZ03]),butthelinkbetweentheimperfectionsensitivityofthestructuralmodelsandthebucklingphenomenonofreinforcedconcretecolumnswasnotclearlyhighlighted in theauthors’pointofview,at leastnot fromsimplephysicallybasedmodels.In such analyses, it is necessary to include some nonlinearity in the bending-curvature

constitutive law.Suchnonlinearitiesmay lead tounstablepost-bifurcationandpost-bucklingbranchesinthebucklingproblem.Asaresult,andratherwellknown,thebucklingofsofteningsystemsmay lead to the imperfection sensitivityphenomenon.This studywill focuson suchproblems based on continuum damage mechanics (CDM) models that may specifically beappliedinaqualitativesensetoreinforcedconcretesections.Theproblemhandledinthisstudyisnotsodifferentfromtheelasticproblemofasoftening

column, as alreadynumerically investigatedby [ODE70], [MON74], [YU82], [HAS85],[KOU87],[LEW87],[WAN96],[VIR04],[BRO07]or[LAC08].Theseauthorsstudiedthepost-bucklingof columnsmodeledwithanonlinearbending-curvatureconstitutive law,withdifferentpotentialapplicationsincludingsteelorcompositestructures.Amajorobservationbythese authors is that the local stiffness softening of the constitutive law (associated with adecrease in the secant stiffness) may induce some global softening of the post-buckling

behavior, that is the axial load must decrease with increasing deflection during the post-buckling range in order to maintain equilibrium. This kind of postbuckling behavior is, asmentioned,typicallyresponsibleforimperfectionsensitivityphenomena.The continuous clamped-free cantilever column considered is shown in Figure 4.22. A

similar problem has been recently studied by Krauberger et al. [KRA 11] for reinforcedconcretecolumnswithoutanyimperfection.However,inthepresenceofsofteninginducedbythe micro-cracking in concrete, the buckling load of the perfect system may overestimatesignificantly the limit load of the imperfect column.An imperfection analysis is needed, aspreviously shown for the single-degree-of-freedom softening structural system. Theimperfectionis,asbefore,introducedthroughanadditionaleccentricityoftheaxialload.

Figure4.22.Eccentricallyloaded,continuouscantilevercolumn,anddefinitionofparametersandvariablesusedinthebucklingandpostbucklingstudyofsofteningreinforcedconcretecolumns

Figure4.23.CDMconstitutivelawoftheinelasticspring

Thebending-curvature relationship is assumed tobe inelastic andbehaveswith adamage

lawillustratedinFigure4.23anddefinedby:

[4.120]whereM is the bending moment, κ is the curvature of the cross-section, EI is the initialstiffnessof thespringandD is thedamagevariable thatmeasures the integrityof thecross-section(seealso[CHA07]forasimilardamagebendingconstitutivelaw).Devolvesfrom0for a virgin crosssection to 1 for the totally broken cross-section. As seen in the figure,plasticityeffectsareneglected,asthereisnoremainingrotationatcompletedunloading.Thedamageloadingfunctionispostulatedas:

[4.121]whereκisthecurvature,θistherotationofthecross-sectionandsisthecurvilinearabscissa(coordinatealongthedeformedaxis).ThecurvatureκYistheonlymaterialparameter,andisassociatedwiththemaximummomentcapacityofthecross-section.Theirreversibledamageconstitutivelawincludingtheloading–unloadingconditionscanbewrittenas:

[4.122]Equation[4.122]canbeviewedastheso-calledoptimalityKuhn–Tuckerconditionswhere

theLagrangemultiplierisequaltothedamagerate(see,forinstance,[JIR02]foradiscussiononKuhn–Tuckerconditionsforinelasticanalysis).First,monotonicloadingbehaviorwillbeconsidered,without anyunloadingphenomena.Unloadingwill bediscussed in a subsequentsectionofthechapter.

4.3.2.Governingequationsoftheproblemandnumericalresolution

Applicationoftheprincipleofvirtualworkintheso-calledgeometricallyexactconfigurationfortheinextensiblesofteningcolumnleadsto:

[4.123]wheree0istheeccentricityoftheappliedaxialloadatthecolumntop.Anintegrationbypartgivesthedifferentialequation:

[4.124]andthenaturalandessentialboundaryconditions:

[4.125]

Fortheclamped-freecolumnstudiedinthischapter,theseboundaryconditionsbecome:

[4.125]Incaseofmonotonicloadinginthehardeningregime,uniquenessprevails.Hill’suniqueness

criterion[HIL58]atthebeamscale(seealso[CHA07])canbeapplied:

[4.127]For this loading range, the column behaves as a nonlinear elastic columnwith a damage

loadingfunctiongivenbyf(κ,D)fromequation[4.121].Inthemonotonicloadingcase,f (κ,D)=0,implyingD=|κ|/(2κY),thenonlinearconstitutivelawisassumedinthesameform:

[4.128]Thisistypicallyahyperelasticconstitutivelawwhosepotentialπ0isgivenby:

[4.129]By substituting the damage parameter into equation [4.124], the nonlinear differential

equation governing the hardening (increasing moment) path of the nonlinear material andgeometricalproblemisthengivenby:

[4.130]The general solution of such a nonlinear differential equation is difficult to achieve

analytically.Withthedimensionlessparameters:

[4.131]where typicallyhasanorderofmagnitudeof0.05or0.1forreinforcedconcretecolumns,thenonlinearboundaryvalueproblemforthehardeningpathcanbeexpressedbythenonlineardifferentialequationandboundaryconditionsgivenby:

[4.132]For a curvature dθ/ds equal to κY (or κ* = κ*Y), equation [4.132] shows that the tangent

modulusvanishes,andthedifferentialequationabovebecomessingular,leadingtothelossofwellposednessofthestructuralproblem.InordertoproceedforcurvaturesbeyondκY, it isnecessary to introduce non-local consideration in a region of the column instead of localsectionalconsiderationsalone.Thiswillbedealtwithindetaillater.It is theoretically possible to express the above nonlinear differential equation as a first-

orderdifferentialequation.Multiplyingthenonlineardifferentialequationbythecurvatureandintegratingonetimeleadstothefollowingnonlinearfirst-orderdifferentialequation:

[4.133]This isacubicequationwith respect to thecurvature. Itcanbe theoreticallysolvedusing

Cardano’s method, but then the post-buckling behavior investigated from the first-ordernonlinear differential equation still needs some numerical computations. In the paper ofHaslach[HAS85],basedonacubicmoment-curvatureconstitutivelaw,aquarticequationinthe curvature was obtained by integration, leading to an easier analytical solution of thecurvature,evenifthefinalintegrationprocesswasnumericallyperformed.Fortheparabolicconstitutivelawadoptedinthischapter,adifferentapproachisrequired.

In the hardening range, that is for k<κY, the nonlinear boundary value problem (equation[4.132])can,inordertofacilitateacomputationalformat,berewritteninanexplicitformwithrespecttothesecondderivative:

[4.134]Therotationαatthecolumnendwiththeappliedloadistakenasthecontrolparameter.Simplificationsoftheaboveformulations,inthegeometricallyexactframework,willoften

givesufficientlyaccurateresultsforcaseswithsmallrotations.Forthisreason,wewillderivesome approximate solutions for the so-called “second-order analysis” of the consideredproblem.Introducing(sinθ≈θandcosθ≈1)inthedifferentialequations[4.134],theycanbewrittenas:

[4.135]

wherederivativesarenowwithrespecttotheabscissa(longitudinalaxis)x(ratherthanwithrespecttothecurvilinears).Figure4.24showsnumericalload-rotationresultsobtainedfortheunstable,softeningpost-

bifurcationbranchesof theperfect system(withno loadeccentricity).Theabscissaα in thefigure is the rotationat the topof the cantilever column.Resultsbasedon thegeometricallyexactandonthesecond-orderapproximatedanalysiswerefoundtobeidenticalwithinthreedecimals.Inthefigure,thisdifferenceisnotvisible.Similarcorrespondencebetweenthetwoanalyses was found for load-rotation results for imperfect columns, for a range of loadeccentricities [0;0.01].SuchresultsarepresentedinFigure4.25.Hence,thesecond-orderanalysis is sufficient formost engineering applications, including the analysis anddesign ofreinforcedconcretecolumns.

Figure4.24.Bifurcationdiagramfortheperfectsofteningbucklingsystemshowingunstablebrancheswithasingularpoint; =0.05and =0.1

Figure4.25.Eccentricityeffectonthepre-andpost-bucklingresponseofthesofteningcolumn; =0.1

Inthenumericalcomputationofthesefigures, waschosentobeequaltoeither0.05or0.1.This range isof anorderofmagnitude that is typical for reinforcedconcrete columns.Withthesematerialparameters,thecurvaturecorrespondingtothesectionmomentcapacity,κ(0)=κY, is reached for a column end rotation of approximatelyαY ≈ 0.0228 in the case of =

0.05,andforαY≈0.0455inthecaseof =0.1.Thus,inbothcases,αYisapproximatelyequal

to half of the curvature ,αY ≈ /2. This is an interesting result. It can be checked in asimplified manner based on an assumed linear moment distribution along the cantilevercolumn.Thisgives

[4.136]whereM0=M(0)isthemomentatthecolumnbase.Then,fromequation[4.136]:

[4.137]which,asanapproximation,confirmstheresultsabovefromthesecond-orderanalysis.Forthecontinuouscolumn,theunstablepost-bifurcationbranches(Figure4.24)canbeseen

tohaveasingularbifurcationpoint,aphenomenonalsonoticedforthesofteningsingle-degree-of-freedomsystem.Figure4.25reflectsthestrongloadsensitivityoftheimperfectcolumnwithrespecttotheloadeccentricityatthecolumnend.Thelimitpointsoftheimperfectcolumn,aswellasthecompletesoftening(descending)post-bucklingbranchesshowninthefigures,areallobtainedforsectioncurvaturesthatareinthehardening(ascending)rangeofthemoment-curvaturelaw,thatisforκ≤κY.Inengineeringdesigncontexts,itisoftenthelimitloadsthatare of prime interest. Since these are reached prior to or at k = κY, as shown above, thepresented analyses (limited to k ≤ κY) are sufficient. However, if the response beyond thisvalueisofinterest,whichitoftenis,anon-localCDMapproachcanbeused,aswewillsee

later,inthepostfailurerangeκ≥κY.

4.3.3.Second-orderanalysis–someanalyticalarguments

In an effort to derive imperfection sensitivity results, in an analytical form, an asymptoticmethodbasedona linearizationofan integral formulationof theboundaryvalueproblem issuggestedinthissection,whichstillislimitedtoκ≤κY.Anumericalcomparisonoftheexactandthelinearizedsolutionwillbepresentedinasecondstep.Smallrotationsareassumedinordertosimplifythenonlineardifferentialequation[4.130]

as:

[4.138]where the prime derivative is now expressedwith respect to the spatial abscissa x (in thelengthdirection).Anintegrationofequation[4.138]leadstothedifferentialequation:

[4.139]The constantC1 can be identified from the boundary conditions at the loaded end of the

column:

[4.140]equation[4.140]canthenbewrittenas:

[4.141]which is a quadratic nonlinear differential equation, fromwhich the roots (curvatures) canreadilybeextracted.Thesmallestrootisimportantforthehardeningmoment-curvatureregimestudiedinthissection.Thus,

[4.142]Notethattheexpressionunderthesquarerootispositiveinthecurvaturehardeningrange.

Thiscanbearguedphysically,butalsomathematicallyasshownbelow.

[4.143]

The mathematical problem can be simplified by introducing the variables A and B asfollows:

[4.144]Bymultiplyingeachterminthisnonlineardifferentialequationbytherotation,suchthat:

[4.145]and then integrating this equation, the nonlinear first-order differential equation below isobtained.

[4.146]TheconstantC1 iseasily identifiedfromthe introductionof theboundaryconditionsat the

clampedsection:

[4.147]Wefinallyobtainthefollowingnonlineardifferentialequation:

[4.148]thatcanbepresentedinanintegralformatby:

[4.149]Byincorporatingthedimensionlessparameters:

[4.150]theintegralequation[4.149]canthenbeexpressedindimensionlessformatby:

[4.151]

Inanintegralformat,equation[4.151]givestheexactload-displacementrelationshipoftheimperfectsofteningcolumnbasedonthesecond-orderanalysis.Itisnowinvestigatedtoseeifitispossibletodevelopasuitableapproximateimperfection

sensitivelawbasedonthesecond-orderanalysis,integral-formulationofequation[4.151].Forsufficiently small deflection valuesw* << 1, an asymptotic expansion of equation [4.151]yields:

[4.152]Fromequation[4.152],thetopdefectionw*(1)canbecalculatedandexpressedby:

[4.153]fromwhichthedimensionlessloadpcanbesolvedforandexpressedby:

[4.154]Fromthisequation,thecritical(bifurcation)loadoftheperfectcolumnbecomesp=p0(for=0).Theanalysisabovegavep0=2,whichclearlyonlyrepresentsanapproximationofthe

correctbuckling loadof theclamped-free linearlyelasticcolumn,pE=π2/4.Thedifferencebetweenp0andpEisduetotheexpansionapproximation.Fortheperfectproblem =0,thepost-bifurcationbranchistypicallyanunstablebranch.Fortheimperfectproblem,with ≠0,thedeflectionatthelimitloadscanbecalculatedfrom:

[4.155]whichforthelargerrootgives:

[4.156]By substituting this expression into equation [4.154], the imperfection sensitive rule is

obtainedas:

[4.157]The inaccuracy impliedbyp0 = 2will be corrected in a simplemanner in the following,

simplybytakingp0equaltothecorrectvaluepE=π2/4.Thus,withp0=pE=π2/4, thefinalimperfectionsensitiverulebecomes:

[4.158]Other closely related imperfection sensitivity rules can be derived by means of the

Rayleigh–Ritz approach or the Galerkin method based on only one-term sinusoidalapproximatefunction.AnapproximationformulacanalsobederivedfromtheapplicationofRayleigh–Ritzmethod,startingfromRayleigh’squotient:

[4.159]Notethattheenergytermπ0definedinequation[4.159]isconvexforcurvatureκlesserthan

κY, which is the case considered for the initial postbuckling behavior considered here.Introducingtheinitialbucklingmode

[4.160]inRayleigh’squotientleadstotheupperbound:

[4.161]AnotherupperboundcanbeobtainedfromGalerkin’smethodbasedontheweakformofthe

differentialequations[4.141]writtenas:

[4.162]Usingagain thecomparison functionequation[4.160] inequation[4.162] leads to a close

upperbound:

[4.163]

Thetwobounds,Rayleigh’sboundandGalerkin’sbound,areverycloseas:

[4.164]Moretermsintheuseoftheseapproximatedvariationallybasedmethodswouldbeprobably

needed for a more accurate description of the analytical post-buckling response of thesoftening column (see, for instance, [LAC08] or [YU12] for the use ofGalerkin’smethodwithseveralcomparisonfunctions).Itissufficientatthisstagetocomparethestructureofthepostbucklingequationissuedofthevariationallybasedmethodswiththepreviousasymptoticoneforacorrectunderstandingofthephysicalproblem.

Figure4.26.Comparisonoftheexactnumericalimperfectionresultsfromthenonlinearboundaryvalueproblem,andtheasymptoticanalyticalrule(equation[4.158])

Figure4.26comparesnumericalresultsfromtheexactimperfectionsensitivityruletothosefromthecorrectedasymptoticimperfectionsensitivityrule(equation[4.158])givenbythefulllines.Theresultsforboth =0.05and =0.1areinverygoodagreement.Forsufficientlysmalleccentricities,equation[4.158]canbesimplifiedtothefollowingrule:

[4.165]whichis identical toKoiter’s½powerlaw(see[KOI45]or[BAZ03]).Itreflectsaratherextremesensitivityofthelimitloadontheeccentricities,ortheimperfection,thatisgenerally

foundinasymmetricalstructuralsystems,aspointedoutbyBazantandCedolin[BAZ03].Itisremarkable that this strong imperfection sensitivity appears in the present damage problem,whichisinitiallyasymmetricalstructuralproblem,whereasharpangularbifurcationbranchintersectioncharacterizesthepost-bucklingsystem.

4.3.4.Postfailureofthenon-localcontinuumdamagemechanicscolumn

Asfarasthecalculationofthelimitpointcalculationisconcerned,alocalbendingcurvatureconstitutive law (applicable at each local section) is sufficient.However, once the bendingmoment at the clamped,maximummoment sectionhas reached thebendingmoment capacityMY(attheapexofthemoment-curvaturerelationship),thelocallawisadequateforcalculatingunloading(decreasingP)fordecreasingrotations(andcurvatures),butnotforthecalculationof continued unloading associated with increasing induced rotations (or deflections). Thisphenomenonisdiscussedinsomedetailinthissection.This limitation of a local section model may be explained physically. Consider first the

columnstateatwhichtheclamped,maximummomentsectionhasjustreachedthemomentMY(capacity)andcorrespondingcurvatureκ=κYforanaxialloadP=PY.Forsimplereference,thisstateinthischapterwillbedenotedbythe“localmodelinglimitstate”(LMLS).UnloadingfromPYforanimposedincreasingcolumnrotationwillrequirethecurvaturetoincreaseattheclamped section and its moment to decrease, since κ > κr. A reduced moment at the basesection requires (from equilibrium) themoments and curvatures at all other sections of thecolumntodecreasealonganunloading(elastic)moment-curvaturecurve(similartothatshownin Figure 4.2). Considering that an increased curvature at the base section, which may beconsideredasalocalizeddamagezoneofzero(infinitesimal,or“vanishing”)length,willnotcontribute to an increasedcolumn rotation,while at the same time thecurvature (unloading)reductionsatalloftheothersectionswillcallforareducedrotation,itisclearthatunloadingforincreasingrotationsisimpossible.Therefore,localmodelingmethodsarelimitedtoκ≤κY.This problem is also encountered in material nonlinear finite element analysis ([BAZ 76],[HEL81],[BAZ03]).ThisphenomenonissometimesreferredtoasWood’sparadox[WOO68],andimpliesthat

thesoftening,damagelocalizationzonevanishesinlocalmodelingmethods,leadingtothezerodissipation phenomenon ([WOO68], [BAZ76], [BAZ03], [CHA05d], [CHA08b], [CHA08c]).Wood’sparadoxcanbemathematicallyproveninthispost-bucklingproblem,byassuminga

softening damage zone (or damage localization zone) over a length adjacent to the clampedsection of size l0, where x [0;l0]. Inside this zone, the constraint κ>κY implies a positivederivative (curvaturegradient),ensuringan increasingcurvature.Fromequation[4.135], thecurvatureinsidethesofteningzonecanbegivenby:

[4.166]

where =l0/L.Thecontinuityofthecurvatureatthesofteningdamageboundaryimpliesthat:

[4.167]Asthecurvatureκisincreasinginsidethesofteningdamagezone,thecurvatureκshouldbe

necessarilylowerthanthecharacteristiccurvatureκY,whichisincontradictiontothefactthatthecurvatureshouldbelargerthanthecharacteristiccurvatureκYfromtheadoptedconstitutivelawforthesofteningbranch.Asareferenceforlater,theLMLS,definedbyP=PYandthelocallimitrotationfunctionθY

(s*)correspondingtoκ(0)=κYandM(0)=MY,canbecalculatedforsmallrotations,whichforthiscasebecomes:

[4.168]ThedamagefunctionD(x)atthisstateisdenotedbyDY(x).Itcanbeobtainedfromequation

[4.121]forf(θ,D)=0.Thisleadstotheproportionalequation:

[4.169]whichrelatesthedamagefunctionDY(x*)andthepositivecurvatureκY(x*)It is of interest to study what unloading implies within the local model framework. As

mentioned,atthelimit,damagelocalizationvanishestoa“zone”consistingofasinglesectioninsuchmodels.Consequently,unloadingfromthisstateimplieselasticunloadingalongsecantsstiffnesses defined byEI(x) =EI0 [1 −DY (x)]. The resulting unloading analysis thereforebecomes equivalent to an analysis of a non-uniform column with varying (non-uniform)stiffness.

[4.170]TypicalnumericalresultsarepresentedinFigure4.27foraperfectcolumnandanimperfect

column.Thefigureshowsthehysteresislooplinkedtotheloadingandunloadingphase.The

hysteresisisduetothedamagedissipationassociatedwiththedamageconstitutivelaw.Inbothcases,unloading implies rotation reversals (after theκ=κY curvature state). For the perfectcolumn,unloadingfromthepost-bifurcationbranchfollowsahorizontallinetoθ=0,andthenalongtheordinatetop=0.

Figure4.27.Lod-rotationresponseincludingunloadingbehaviorofcolumnsmodeledwithlocaldamagelaw(curvaturereversalattheclampedsectionfromκ*= =0.1);Wood’sparadoxfoundinthepostbucklingrange

Inorder to extend the analysisvaliditybeyondκ=κY at the base (clamped) section, it isnecessary to develop a non-local formulation of the boundary value problem. This iscontemplated in thispartwith theuseofnon-localdamagemechanics forbending-curvatureconstitutive law, as shown, for instance, by Challamel [CHA 10a]. We start from thegeneralizedversionoftheprincipleofvirtualworkforgradientdamagesystemsdefinedby:

[4.171]

Where isanon-localcurvature,lcisanadditionalinternallength,specificofthenon-localdamagemodelneededforthesofteningpropagation(causedbymicro-cracking),andl0istheactiveregionofthedamagesofteningprocess,heredenotedasthedamagelocalizationzone(seealso[CHA10a]incaseofnon-localhardening–softeningplasticity).λhasthedimensionof a moment variable. The presentation of the weak formulation as considered in equation[4.171]mayrigorouslyjustifytheuseofhigherorderboundaryconditionsassociatedwiththehigherorderdamagemodel.Foramoredetailedpresentation,thereaderisreferredtovariouspapersforthegeneralizedprincipleofvirtualworkappliedtonon-localorgradientdamagesystems (see, for instance, [FRE 96], [LOR99], [FOR09], [PHA10a], [PHA10b], [NGU11a]). An integration by parts of equation [4.171] gives the following coupled system of

differentialequations:

[4.172]The secondequationabove is similar toEringen’sdifferential equation [ERI83]used for

non-local elasticmodels, and later usedbyPeerlingsetal. [PEE96] for non-local damagemodels.Notethatthepresentmodelexpressedbyequation[4.172]cannotbederivedfromasingle energy functional in this case, even if alternative variationally consistent non-localdamagemodelsalsocanbeusedasshownbyChallamel [CHA10a], for instance.Thenon-

localcurvature ,whichcontrolsthenon-localdamageprocess,iscalculatedovertheactivepart l0 of the damage process, as also suggested by Challamel et al. [CHA 10a] fromvariational arguments or Nguyen et al. [NGU 11b] from micromechanics and numericalarguments.Thenaturalandessentialboundaryconditionsaregivenby:

[4.173]Thenon-localdamageloadingfunction,validforD≥1/2,ischosenas:

[4.174]The damage loading function is expressed with respect to the non-local curvature-driven

parameter.Thismodeltypicallybelongstotheclassofstrain-basednon-localdamagemodelsdevelopedbyPijaudier-CabotandBažant[PIJ87],wherethenon-localoperatorinthisworkischosenfromEringen’skernel,asadopted,forinstance,byPeerlingsetal.[PEE96].Inthepresentanalysis,theimplicationof inthedamagelocalizedzoneisthatthenon-local

curvature can be equivalently replaced by the damage variableD. Inserting the damagevariable in the implicit differential equation [4.172] dealing with the non-local curvaturevariableleadstotheequivalentdamageloadingfunctiondefinedby:

[4.175]We recognize in this case that this is a typical gradient damagemodel where the second

derivativeofthedamagevariableappearsintheloadingfunction(see,forinstance,[FRE96];see more recently [PHA 10b], [NGU 11a]). Therefore, in the present problem, there is anequivalencebetweenthenon-localintegralstrain(curvature)-drivendamagemodel(equation[4.174]) and the gradient damage model (equation [4.175]). Note that the gradient of thedamagevariablevanishesattheboundary(x=l0)oftheactivedamagezonewiththepresentmodel.

Inviewoftheabove,themultipoint(three-point)nonlinearboundaryvalueproblemcannowbepresentedwithsecond-orderanalysisassumptionsby

[4.176]Withtheboundaryconditions:

[4.177]Therotationandthemomentvariablesarecontinuousvariablesalongthecolumn.Notethat

theequivalencebetweenthenon-localintegralstrain(curvature)-drivendamagemodelandthegradient damage model both concern the loading function and the higher order boundaryconditionsatthesoftening,damagelocalizationinterfaceasreflectedby:

[4.178]Itisfoundnumericallythatthegradientofthedamageparameterisnotvanishingatthefree-

endboundaryofthecolumn,aswegenerallyhaveD′(L)=D′Y(L)=D′Y(L)≠0.This (three-point) nonlinear boundary value problem can be numerically solved with the

Matlab® program bvp4c, using a first-order vectorial differential equation with fourcomponents(θ,dθ/dx,D,dD/dx).Asbefore,solutionsareobtained for specified (controlledor imposed) rotationsat thecolumn’s free

end(α).Thenon-localproblemdependsontheadditionallengthscaleparameter thatcontrols

theextensionofthelocalizationzonel0andthepostfailureregime(seealso[CHA03]orlater[CHA10b]fornon-localplasticitybeammodels,seealso[CHA09b]or[CHA10a]fornon-localdamagebeammodels).Thedeterminationofthecharacteristiclengthlc(orthemaximumlength of the localization zone l0) is related to the question of the finitelength hingemodel,whichisacentralquestionofthepresentnon-localmodel.Wood[WOO68]inspiredbytheworksofBarnardandJohnson[BAR65]suggestedthetermdiscontinuitylength.Manyworkshave been published on the experimental or theoretical investigation of such a length forreinforced concrete beams [BAZ 03]. It is generally acknowledged that the value of lc(affectingthemaximumlocalizationzonel0)mustberelatedtothedepthofthecross-sectionh.Therefore, it is recommended that themaximum lengthof the localization zone l0 should bechosentobeintheorderofthemagnitudeofthecross-sectiondepthh([BAZ87],[BAZ03]).

In thepresentnumerical calculations, theorderofmagnitudeof is chosen equal to0.1.Thedamagelocalizationzonepropagates(increases)withincreasingvaluesofthecontrolparameterα.Thesimulationwasarbitrarilystoppedforavalueofαgiving ≈0.3.Figure4.28showscolumnsofteningresultsobtainedwiththenon-localdamagemodelforκ>κY.Itisobservedthatthelocalandnon-localanalysisregimeshaveacommontangentatthetransitionbetweenthetworegimes.

Figure4.28.Pre-andpost-bucklingresponseaccordingtolocaldamagelawforκ<κY,andcontinuedcolumnsofteningforincreasingrotations(duetoincreasingcurvaturesinthedamagelocalizationzone)aspredictedbythenon-localdamagelaw;

Some further studies would probably be needed in the future to investigate the possiblecoexistence of multiple solutions for this propagating problem, and more specifically thebifurcationfromthisfundamentalsolution,forlargerotations.Inthischapter,wehavemainlybeenconcernedaboutshowingthepossibilityofmodelingthepost-buckling,softeningcolumnresponsewithanon-localCDMmodelforcurvaturesinthesofteningmoment-curvaturerange.Theneedofintroducingnon-localityintheproblemformulationhasbeendiscussedforthe

post-buckling response for increasing curvatures in the softening range. A local bending-curvatureconstitutivelawleadstotheunloadingWood’sparadox,aphenomenonwellknownforthebendingoflocalsofteningbeams(seeChapter3).Byincludingsomenon-localityinthemoment-curvature constitutive law, the propagation phenomenon of damage localization hasbeen solved. However, non-locality needs not to be necessary, as far as the limit loadcalculationsareconcerned,astheseareobtainedforlocalcurvaturesinthehardeningregime.

Appendix1

Cardano’sMethod

A1.1.IntroductionThisappendixgivesthemathematicalmethodtosolvetherootsofapolynomialofdegree3,calledacubicequation.Someresultsinthissectioncanbefound,forinstance,in[ART04].Asausefulextension,wealsogivethemethodologytodeterminetherootsofapolynomial

ofdegree4,calledaquarticequation.Therootsofacubicequation, like thoseofaquarticequation,canbe foundalgebraically. It canbeshown that thisproperty isnomorevalid, ingeneral,foraquinticequation(equationofthefifthorder)orequationsofhigherdegrees.ThisisknownastheAbel–Ruffinitheorem,whichwasfirstpublishedin1824.ReferringtotheFrenchdictionaryLeRobert,thecompletemethodforthegeneralresolution

ofacubic isprobablydue toTartaglia(NiccoloFontana,1500–1557,alsocalledTartaglia)fromhisworksconcludedin1537,basedonthefirstapproachofGerolamoCardano(1501–1576).In1539,TartagliarevealedhismethodtoCardanoontheconditionthatCardanowouldnever reveal it. Some years later,Cardano learned about Ferro’s priorwork and publishedFerro’smethodinhisbookArsmagnasivederegulisalgebraicis,linerunus in1545.Theseworks,whichareproducedbytheItalianmathematicsschool,arealsobasedon:

– Rafael Bombelli (1526–1572) was the one who finally managed to address theresolutionofpolynomialequationswithimaginarynumbers.– Lodovico Ferrari (1522–1565), as Cardano’s student, gave the solution of a quarticequation,whichwaspublished inonechapterofArsmagnasivede regulisalgebraicis,linerunuswrittenbyCardanoin1545.–ScipionedelFerro(1465–1526)firstdiscoveredthemethodtosolvethecanonicalformofacubicequation(x3+px+q=0),thefirststeptowardthemoregeneralmethodofacubicequation.Inthefollowing,weusethemathematicalfunctionsgn(x)forthesignfunctionofarealx,and

wealsouse:

[A1.1]

A1.2.Rootsofacubicfunction–methodofresolution

A1.2.1.CanonicalformWeconsiderthecubicequationwithrealcoefficients:

[A1.2]Eachtermcanbedividedbythefirstcoefficient,leadingto:

[A1.3]Thiscubicequationcanbefactorizedas:

[A1.4]whichisknownasthecanonicalform:

[A1.5]Thiscanonicalequationissolvedfromtheintroductionoftwonumbersyandzsuchthatx=

y+zarerootsofthecubicequationf(x)=0,byimposingsomeconstraintequalities:

[A1.6]Thefollowingchangeofvariablecanbechosenas:

[A1.7]KnowingthesumandtheproductofYandZ,thesenumbersarenecessarilytherootsU1and

U2 of the quadratic equation: . IfU1 andU2 are known, then y and z arecalculatedfrom(e2kipU1)1/3and(e2kipU2)1/3,whichshouldbeassociatedbyapairsuch thatthe product yz is a real number. We can distinguish several possible cases using the

discriminantconcept,dependingonthesignof orequivalently,dependingonthesignof4p3+27q2.

A1.2.2Resolution–onerealandtwocomplexrootsCase4p3+27q2>0(onerealandtwocomplexconjugaterootsforf(x)=0).

Thiscaseincludesthecasep=0.Inthiscase,bothU1andU2arerealnumbers:

[A1.8]Tohavetheproductyzasarealnumber, thepossiblecouples(y,z) (orequivalently (z,y))

arethen:

[A1.9]wherejdenotesthecomplexnumberthatisthecubicrootofunity.Thesolutionsinxarethen:

[A1.10]In reinforcedconcretedesign,weareonlyconcernedwithrealsolutions,and thenonlyx1

willbeofinterest,whichfinallyleadstotherootoftheinitialcubicequation[A1.2],as:

[A1.11]

Inthespecificcasep=0,thisrealrootissimplyreducedto

A1.2.3.Resolution–tworealrootsCase4p3+27q2=0(onerealandonedoublerealrootsforf(x)=0).

In this case,U1 andU2 are real numberswith . The product yzbeingreal,thepossiblecouples(y,z)(orequivalently(z,y))aregivenby:

[A1.12]where j denotes the complex number that is the cubic root of unity. Using the fundamentalproperty1+j+j2=0,thesolutionsinxaregivenby:

[A1.13]Therealrootsoftheinitialcubicequationg=0in“t”(equation[A1.2])arethen:

[A1.14]

A1.2.4.Resolution–threerealroots

In this case,U1 andU2 are conjugate imaginary numbers. It can be checked that ify is a

cubicrootofthecomplexnumberU1,then istheimaginaryconjugatenumberofy;andx=y+zisarealnumber.Practically,weusethefactthatanecessaryandsufficientconditionfortwopolynomialequationstohavethesamerootsisthatthecoefficientsofthesepolynomialequationsareproportional.Weusetheequality4cos3a-3cosa-cos3a=0whichisalwaystrue.The unknown y= cos a is a root of4y3 - 3 y - cos 3a= 0.We are looking for theconditions to have both equations x3+ px + q = 0 and 4 y3 - 3 y - cos 3a = 0 withproportionalcoefficients.Inthiscase,ifxisarootofthefirstcubic,kxwouldbetherootofthesecondcubic,withkasaproportionalcoefficient,leadingtotheequivalenceprinciple:

[A1.15]Thesetwoequationsareequivalenttotheconditions:

[A1.16]

Theeliminationofkgives: whichshouldbecomprisedbetween–1and

+1,leadingto with-4p3>0.Werecognizetheconditionthatthediscriminant isnegative, that is4p3+27q2≤0. In thiscase,andfromequation[A1.16],wehavetheinverserelationship:

[A1.17]Astherootsofthecubicequation4y3-3y-cos3a=0arey=cosa,therootsofthecubic

equationx3+px+q=0are1/kproportionaltothepreviousones(withx1<x2<x3):

[A1.18]

Therootsoftheinitialcubicequation[A1.2]g(t)=0arethen(witht1<t2<t3):

[A1.19]

A1.3.Rootsofacubicfunction–synthesis

A1.3.1.SummaryofCardano’smethodConsideringthecubicequationnowexpressedintermsoftheunknownαthatisrelatedtothedimensionlessneutralaxispositioninreinforcedconcretedesign:

[A1.20]Theparameterspandqcanbeintroducedas:

[A1.21]If4p3+27q2>0,theuniquerealsolutionofthecubicequationisobtainedfrom:

[A1.22]If4p3+27q2<0,thethreerealsolutionsaregivenby:

[A1.23]

A1.3.2.Resolutionofacubicequation–exampleWegiveherea smallexample to illustrateourpurpose.Letusconsider the followingcubicequation:

[A1.24]Thecoefficients(a,b,c,d)areidentifiedfromequation[A1.20]as:

[A1.25]Wecalculatenowpandqfordeterminingthenatureofthesolutions:

[A1.26]Wecalculatethediscriminantas:

[A1.27]Hence,wehavethreerealsolutionsforthiscubicequation.Itcanberelevanttocomputethe

followingnumberfortherootcalculation:

[A1.28]Wethencomputethethreerootsofthecubicfromequation[A1.23]as:

[A1.29]Ofcourse,itiseasytocheckthatα3−2α2−α+2=(α+1)(α−1)(α−2)

A1.4.Rootsofaquarticfunction–principleofresolution

Wenowconsiderthequarticequationwithrealcoefficients:

[A1.30]Itcanbepostulatedthatthequarticcorrespondstothebeginningofthesquareofasecond-

orderpolynomialequationlike:

[A1.31]whereyisarealnumber.Forthefollowing,wewillassumethat:

[A1.32]yischosensuchthatthesecondtrinomeoff(x),constitutedofthethreelasttermsoff(x),couldbeconsideredinasquareformat.Itisthennecessarythat:

[A1.33]Werecognizeacubicequationexpressedwiththeunknowny:

[A1.34]whichcanbesolvedwiththepreviousCardano’scubicmethod.Lety1,y2andy3bethethreerootsofthiscubicequation.Theparametersαandβwillbechosenas:

[A1.35]Oncethecubicrootyiscalculatedy=y1,thequarticfunctionf(x)hasthefollowingform:

[A1.36]Then,thedeterminationoftherootsofthequarticfunctionisreducedtothedeterminationof

therootsoftwoquadraticfunctions.

Appendix2

SteelReinforcementTableTableA2.1.AbacusofthesteelareaAsincm2foreachbardiameterΦ(mm)

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IndexBBendingmomentandnormalforces,Bending-curvaturelaw,Bilinearbehaviorofconcrete,Bucklingandpost-buckling,

CCantilever,Collapse,Compositecrosssection,Conditionofnon-fragility,Continuumdamagemechanics,Cubicequation,

DDeflectioncalculation,Dimensionlessparameters,Ductility,

EEccentricity,Elasticbehavior,

FFundamentalanalysis,

IImperfectionsensitivelaw,

LLimitload,

NNeutralaxis,Non-localmechanics,

OOptimization,

PParabola-rectanglediagram,PivotA,PivotB,PivotC,Plastichingelength,Plasticity,Post-failure,

RRectangularcrosssection,Reducedmoments,Reinforcedconcretedesign,

SSargin’slaw,Serviceabilitylimitstate,Simplifiedapproach,Simplifieddiagram,Softening,Steelreinforcement,

TT-cross-section,Tensionstiffeningphenomenon,Triangularcrosssection,

UUltimatelimitstate,

WWood’sparadox,

Y

Yielddesign,