â _ 0 Ű۔۱Ùۧ ÙŰȘۧۚ â Reinforced Concrete Deep Beams â 2025 Beams: Beams Reinforced Concrete Deep Beams Edited by PROFESSOR F K KONG Professor of Structural Engineering Department Civil Engineering University Newcastle upon Tyne Published in the United States America by Van Nostrand Reinhold Contents Preface Contributors 1 concrete deep beams F KONGâ andâ M CHEMROUK Notation 1 1 Introduction 1 2 Historyâ andâ development 1 3 Currentâ designâ practice 1 4 CIRIAâ Guideâ 2 1 CIRIAâ Guideâ Ă«Simpleâ RulesĂ 1 CIRIAâ Guideâ Ă«Supplementaryâ RulesĂ 1 5 Draftâ Eurocodeâ andâ CEB FIPâ Modelâ Code 1 Flexuralâ strength:â simplyâ supportedâ deepâ beams 1 Flexuralâ strength:â continuousâ deepâ beams 1 Shearâ strengthâ andâ webâ reinforcement 1 6 ACIâ Buildingâ Codeâ 318ñ83â (revisedâ 1986) 1 Flexuralâ strength 1 Shearâ strength 1 7 Canadianâ Codeâ CAN3 A23 M84 1 Shearâ strength References 2 Strength and behaviour beams M D KOTSOVOS Notation 2 Introduction 2 Currentâ conceptsâ forâ beamâ design 2 Effectâ ofâ transverseâ stresses 2 Flexuralâ capacity 2 Shearâ capacity 2 Compressiveâ forceâ pathâ concept 2 Deepâ beamâ behaviourâ atâ ultimateâ limitâ state 2 Causesâ ofâ failure 2 Archâ andâ tieâ action 2 Effectâ ofâ transverseâ reinforcement 2 Designâ implications 2 Modelling © 2002 Taylor & Francis Books, Inc CONTENTS xii 2 Designâ method 2 Verificationâ ofâ designâ method References 3 beams with web openings S P RAY Notation 3 Introduction 3 Factorsâ influencingâ behaviour 3 Generalâ behaviourâ inâ shearâ failureâ (underâ two pointâ loading) 3 Beamâ withâ rectangularâ webâ openings 3 Beamâ withâ circularâ webâ openings 3 Flexuralâ cracks 3 Generalâ behaviourâ inâ shearâ failureâ (underâ four Effectâ ofâ webâ opening 3 Effectsâ ofâ mainâ andâ webâ reinforcements 3 Diagonalâ modeâ ofâ shearâ failureâ load 3 8 Definitions 3 9 Criterionâ ofâ failureâ andâ strengthâ theory 3 10 Ultimateâ shearâ strength 3 1Evaluationâ ofâ webâ openingâ parameter 3 2 Ultimateâ shearâ strength 3 11 Simplifiedâ designâ expression 3 12Ultimateâ strengthâ inâ flexur 3 13Simplifiedâ expressionâ forâ flexuralâ strengt 3 14Extensionâ ofâ theoryâ ofâ ultimateâ shearâ strengthâ ofâ beam toâ four pointâ loading 3 15Extensionâ forâ uniformlyâ distributedâ loading 3 16 Recommendationsâ forâ designâ ofâ beamsâ forâ shearâ andâ flexure 3 17 Recommendationsâ forâ leverâ armâ (Z) 3 18 Designâ example References 4 Continuous beams D ROGOWSKY 4 1 Introduction 4 2 Distinguishingâ behaviourâ ofâ continuousâ deepâ beams 4 1 Previousâ tests 4 2 Continuousâ deepâ beamsâ vs â continuousâ shallowâ beams 4 3 Continuousâ deepâ beamsâ vs â simpleâ spanâ deepâ beams 4 3 Capacityâ predictionsâ byâ variousâ methods 4 1 Elasticâ analysis 4 2 Finiteâ elementâ analysis © CONTENTS xiii 4 ACI318 4 Kong,â Robinsâ andâ Sharp 4 Trussâ models 4 Trussâ modelsâ forâ continuousâ deepâ beams 4 Designâ ofâ continuousâ deepâ beams 4 Designâ example 4 Summary References 5 Flanged beams H SOLANKIâ andâ A GOGATE Notation 5 Introduction 5 Reviewâ ofâ currentâ knowledge 5 Modesâ ofâ failure 5 Modeâ ofâ failureâ 1:â flexural shearâ failure 5 Modeâ ofâ failureâ 2:â flexural shear compressionâ failure 5 Modeâ ofâ failureâ 3:â diagonalâ splittingâ failure 5 Modeâ ofâ failureâ 4:â splittingâ withâ compressionâ failure 5 Analysis 5 ACIâ Buildingâ Code 5 CIRIAâ Guideâ 2 5 Methodâ ofâ Tanerâ et al 5 Methodâ ofâ Reganâ andâ Hamadi 5 Methodâ ofâ Subediâ forâ flangedâ beamsâ with webâ stiffeners 5 Designâ exampleâ 1:â Beam panelâ P311â (Tanerâ et , â 1977) 5 Designâ exampleâ 2:â ACIâ Code References 6 under top bottom loading A R CUSENS Notation 6 Introduction 6 Earlyâ testsâ onâ deepâ beamsâ underâ topâ andâ bottomâ loading 6 Testsâ atâ Leedsâ University 6 Descriptionâ ofâ testâ specimens 6 Crackâ patterns 6 6Crackâ width 6 7 Designâ approaches 6 1 Americanâ Concreteâ Institute 6 2 Sch ̧ttĂsâ equations 6 3 CIRIAâ Guide © CONTENTS xiv 6 Top loadedâ wall beams 6 Bottom Combinedâ topâ andâ bottomâ loading 6 11 Summaryâ andâ recommendations References 7 Shear strength predictionĂłsoftened truss model S T MAUâ andâ T C HSU Notation 7 Introduction 7 Modellingâ ofâ deepâ beams 7 Shearâ element 7 Effectiveâ transverseâ compression 7 Softenedâ trussâ model 7 Fundamentalâ assumptions 7 Stressâ transformationâ (equilibrium) 7 Strainâ transformationâ (compatibility) 7 Materialâ laws 7 Solutionâ algorithm 7 Accuracy 7 Parametricâ study 7 span to heightâ ratio 7 Longitudinalâ reinforcement 7 Transverseâ reinforcement 7 Explicitâ shearâ strengthâ equation 7 Derivationâ ofâ equation 7 Calibration 7 Conclusions References 8 predictionĂłplastic method M W BRAESTRUP Notation 8 Introduction 8 Plasticityâ theory 8 Limitâ analysis 8 Rigid,â perfectlyâ plasticâ model 8 Structuralâ concreteâ planeâ elements 8 Concreteâ modelling 8 Reinforcementâ modelling 8 Yieldâ lines 8 Shearâ strengthâ ofâ deepâ beams 8 Lowerâ boundâ analysis © CONTENTS xv 8 Upperâ boundâ analysis 8 Experimentalâ evidence 8 Shearâ reinforcement 8 Conclusion References 9 Finite element analysis Y CHEUNGâ andâ H CHAN Notation 9 Introduction 9 Conceptâ ofâ finiteâ elementâ method 9 Triangularâ planeâ stressâ elements 9 Rectangularâ planeâ stressâ elements 9 Isoparametricâ quadrilaterals 9 Equivalentâ loadâ vector 9 Stiffnessâ matrixâ ofâ rectangleâ withâ sidesâ 2aâ2b 9 Elasticâ stressâ distributionâ inâ deepâ beamâ byâ finite elementâ method 9 Finiteâ elementâ modelâ forâ crackedâ reinforcedâ concrete 9 Modellingâ ofâ reinforcingâ steelâ bars 9 Pointâ elementâ orâ linkageâ element 9 Discreteâ crackingâ model 9 Smearedâ crackingâ model 9 Modellingâ ofâ constitutiveâ relationshipâ ofâ concrete 9 12 Constitutiveâ relationshipâ ofâ steelâ bars 9 13 Crackingâ inâ concreteâ andâ yieldingâ inâ steel 9 14 Stiffnessâ ofâ crackedâ element 9 15 Solutionâ procedure 9 Incrementâ procedure 9 Iterativeâ procedure 9 Mixedâ procedure 9 Flowâ chartâ ofâ non linearâ analysisâ procedure 9 16Exampleâ ofâ non linearâ analysisâ ofâ reinforcedâ concret deepâ beams References 10 Stability slender KONGâ andâ H H A WONG Notation 10 Introduction 10 Slenderâ deepâ beamâ behaviour 10 Elasticâ behaviour 10 Ultimateâ loadâ behaviour © CONTENTS xvi 10 Current design methodsĂłCIRIA Guide (1977) 10 CIRIA Simple Rules 10 Supplementary Appendix C: Single Panel Method 10 Two The equivalent column method 10 Theoretical background 10 analysis columns: graphical improved graphical methods 10 analytical beams: column 6Deep beam buckling comparison test result 10 Concluding remarks Reference ۧÙÙÙۯ۳۩ ۧÙÙ ŰŻÙÙŰ© Ù ŰŹŰ§ÙŰ§Ù PDF ۧÙÙÙۧÙÙ ÙŰłÙ ŰźŰ§Ű” ŰšÙŰȘŰš ÙÙŰȘŰÙ ÙÙ Ù ŰŹŰ§Ù ÙÙÙ ŰŁŰŰŻ Ù۱ÙŰč ÙۧÙÙ ŰčÙÙŰ© ۚۯ۱ۧ۳۩ ÙŰȘŰ”Ù ÙÙ ÙŰȘŰÙÙÙ Ű§ÙÙ ÙŰŽŰąŰȘ ۧÙÙ ŰźŰȘÙÙŰ© ÙۧÙŰŁŰšÙÙŰ© ۧÙŰłÙÙÙŰ© ÙۧÙŰźŰŻÙ ÙŰ© ÙۧÙŰ·Ű±Ù ÙۧÙŰŹŰłÙ۱ ÙۧÙŰŁÙÙŰ§Ù ÙۧÙÙ Ű·Ű§Ű±Ű§ŰȘ ÙۧÙÙ ÙۧÙŰŠ ÙŰŽŰšÙۧŰȘ Ű„Ù ŰŻŰ§ŰŻ Ù ÙŰ§Ù Ű§Ùێ۱ۚ ÙÙ Ű۷ۧŰȘ ۶۟ ۧÙÙ ÙŰ§Ù Ű§ÙŰ”Ű±Ù Ű§ÙŰ”ŰÙ Ű§ÙŰȘÙÙÙŰ© ÙÙ ŰčۧÙŰŹŰ© ÙۧÙ۳ۯÙŰŻ ÙÙ۰ÙÙ Ù ŰŽŰ§Ű±ÙŰč ۧÙŰ±Ù ÙۧÙŰ„ŰŽŰ±Ű§Ù ŰčÙ Ù Ù۰ۧ ŰŁŰ«Ùۧۥ ÙŰȘ۱۩ ۧ۳ŰȘÙ Ű±Ű§Ű±Ùۧ Ù۰ۧ Ùۧ ÙŰŹÙŰČ Ű۔۱ ۧÙŰčÙÙ ŰšŰŁÙÙ Ű§ÙÙ 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