第3章 受弯构件正截面受弯承载力

Chapter3CalculationofFlexuralCapacity3.1Generaldetailingrequirementsforbeamsandslabs3.2Normal-sectionflexuralbehaviorofflexuralmembers（受弯构件正截面的受弯性能）3.3Calculationprincipleforflexuralcapacity（正截面承载力计算原理）3.4Calculationforflexuralcapacity

ofsinglyreinforcedrectangularbeams(单筋矩形截面梁正截面受弯承载力计算）3.5Calculationforflexuralcapacity

ofdoublyreinforcedrectangularbeams(双筋矩形截面梁正截面受弯承载力计算）3.6Calculationforflexuralcapacity

ofT-sectionbeams

(T形截面梁正截面受弯承载力计算）StressedsteelbarsDistributionsteelbars3.1Generaldetailingrequirementsforbeamsandslabs

1、Cross-sectiontypesbeamsandslabs2、Dimensionofbeams（1）Fitmodulus：Dimensionisgenerallymultipleof50mm(millimeter)。（2)Depth-span

rationforbeams：Accordingtoexperience（Detailingmanual构造手册）（3)Depth-widthrationforbeams：Rectangularsectionh/b=2-3.53.1.1Typesanddimensionofcrosssection3、Dimensionforcast-in-situslab（1）Minimumsectionthickness（Table3.1）（2）Depth-spanration3.1.2Selectionofmaterialsandgeneraldetailing1、ConcreteGradeC20~C30，GenerallylessthanC402、ReinforcementsGradeandordinarydiameterinbeams1）LongitudinalstressedreinforcementsGrade：Diameter：10mm2）Erectionsteelreinforcements(stirrup-supportbars,tiesteelbars)(架立钢筋）RelatedtoSpans。L<4m，d8mm；L＝4-6m，d10mm；L6m，d12mmd1

30mm1.5dd2

25mmd3.ReinforcementsGradeandordinarydiameterinslabsGrade：HPB300，HRB335，HRB400d8mm；Spacing≤200mm2）Distributionreinforcementsd：6mm、8mm；Spacing≤250mmStressedsteelbarsDistributionsteelbars1）Longitudinalstressedreinforcements(bendingreinforcements)5、Concretecover(混凝土保护层厚度）（Seealsoappendix4-4）4、Longitudinal-stressedReinforcementratio(纵向钢筋的配筋率h0hbh0-EffectiveDepthbh0-Effetivearea3.2Normal-sectionflexuralbehaviorofflexuralmembers（受弯构件正截面的受弯性能）1、Experimentforunder-reinforcedbeams(实验研究）M图V图pph0hb？DeformationPropertyFailureforunder-reinforcedbeams3.2.1、Threestressedstagesinnormal-sectionflexureofunder-reinforcedbeams(适筋梁正截面受弯的三个受力阶段）2、Threestressedstagesinthenormal-sectionflexureofunder-reinforcedbeams（1）StageⅠ——ElasticStageStageⅠ—ThestressandstrainofConcreteincompressivezoneislinear.Ⅰa(第一阶段末）

-Thestressattheextremetensilefiberreachesthetensilestrength(orstrainreachestheultimatetensilestrain.)（2）StageⅡ——StageofworkingwithcracksStageⅡ

：StartsfromcrackingtotensilereinforcementsyieldingⅡa(第二阶段末）

:Thestressesintensilereinforcementsreachtheyieldstrength.（3）StageⅢ——Failurestage阶段StageⅢ

：Startsfromyieldingofreinforcementstocrushingofconcreteattheendofcompressivezone.Ⅲa(第三阶段末）:Thestrainattheextremeouterfiberincompressivezonereachestheultimatecompressivestrain.3.2.2Threetypesoffailuresinnormal-sectionflexure（1）Under-reinforcedfailure：First，tensilereinforcementsyield,thentheconcreteincompressivezonecrushes.Thefailureisductile.（2）Scarce-reinforcedfailure：Oncetheconcretecrackinginthetensilezone,thereinforcementsyield。Thefailureisbrittle.（3）Over-reinforcedfailure：Beforethetensilereinforcementsyield,theconcretecrushesincompressivezone.Thefailureisbrittle.Itisavoidedbystipulatingtheminimumreinforcementratio.Itisavoidedbythecalculationofload-bearingcapacityItisavoidedbystipulatingthemaximumdepthofcompressivezone.3.3CalculationPrincipleofFlexuralCapacity3.3.1FundamentalAssumptionsforcalculation（1）Plane-sectionalassumption：AplanesectionbeforebendingremainsplaneafterbendingThemeanstraindistributeslinearlyoverthedepthofsectioncu（2）Thetensilestrengthintheconcretemaybeneglected.MufyMufy（3）Stress-strainrelationforconcreteandsteelfyMuMufyfc3.3.2-3DesignformulaforflexuralcapacityMufy1、Thestressdistributionforunder-reinforcedbeamsreachingtheload-bearingcapacity2、DerivationofformulasEquilibriumofforcesX＝0EquilibriumofmomentofforcesM＝01fcTs＝fyAsCMuxh0asbMuCxcTs＝fyAsfcx=1xcRelativedepthofcompressionzoneDeformedformofformulah0-x/2Deformedformofformulas3.3.4-5Accommodatedconditionforformulas(公式的适用条件）Theyieldingoftensilereinforcementsandthecrushingofconcreteoccuratthesametime.cuyThedepthofcompressionzoneRelativelimitdepthofcompressionzonexcbBalanced-reinforcedFailure（2）Howtoavoidover-reinforcedfailureAvoidover-reinforcedfailurecuxcbyForover-reinforcedfailure即Forunder-reinforcedfailure即（3）Avoidscare-reinforcedfailure：Conclusion：Formulasandaccommodatedconditions3.4Calculationofflexuralcapacityforsinglyrectangular-sectionbeams(单筋矩形截面梁的承载力计算）3.4.1Basiccalculationformulasandtheiraccommodatedconditions3.4.2Twotypesofproblemsinthecalculationofflexuralcapacity1、DefinitionforsinglyrectangularsectionThereinforcementisembeddedinthetensilezoneinrectangularsection.3.4.1Fundamentalcalculationformulasandtheiraccommodatedconditions2、Calculationformulasandtheiraccommodatedconditions1.Section-DesigningProcedure3.4.2Twotypesofproblemsinthecalculationofflexuralcapacity（2）Selectionformaterials（3）Calculatetheareaofreinforcements（1）EstimatesectiondimensionConcrete：C20－C30Reinforcements：CalculationformulasCalculationprocedure（4）SelectreinforcementsSelectreinforcementsProblemsLiveloadontheflooris2.0KN/m2UnitweightofRC25KN/m3Unitweightofmortar20KN/m324024030602100Arectangularbeamsupportsafloorslabwhichis100mmthickwith20mm-thicklime-mortarplasteronitsbottomand20mm-thickmixed-mortarlevelingblanketonthetop.Designthebeam.（1）计算简图；3300（2）荷载统计（估算截面尺寸）；3300gkqk截面尺寸：400X200例题图（3）内力计算及组合（4）材料选择荷载组合（4）计算受压区高度计算公式（5）计算受拉钢筋面积选择钢筋2.Section-AnalysisProcedureCalculationformulas（1）Checkthereinforcementratio（2）Calculatex（3）Ifxxb,thencalculatingtheMubasedontheformula（4）Ifx>xb,thentakex＝xbto

calculateMubasedontheformula

3.5Calculationofflexuralcapacity

fordoublyreinforcedrectangularbeams(双筋矩形截面梁的承载力计算）3.4.2Basiccalculationformulasandtheiraccommodatedconditions3.4.2Twotypesofproblemsinthecalculationofsectionalload-bearingcapacity-DesignandAnalysis3.4.1Introduction1.DefinitionofdoublyreinforcedrectangularbeamsThereinforcementsareembeddedinbothcompressiveandtensilezonePP3.4.1Introduction2.ReasontoadoptdoublyreinforcedbeamsThebeamisover-reinforcedwhendesignedbysinglyreinforcedsection.(2)Thebeamispossiblysubjectedtopositiveandnegativemoments(1)Thestressdistributionoversectionwhenbeamreachesitsload-bearingcapacityh0asash3.5.2Calculationformulaandaccommodatedconditions2.Calculationformulah0asashh0asashxch0cuAccordingtotheplane-sectionalassumptionGenerally,when,compressivereinforcementsyield.3Accommodatedconditions★Avoidover-reinforcedfailure：★

Avoidscare-reinforcedfailure：★

Thecompressivereinforcementscanyield：ConclusionFormulaAccommodatedconditions(1)Thefirsttypeofdesignproblem——Knowingbandh，fc，fy，fy，M，DetermineAsandAsDesignProcedure1）Setting＝b；2）AccordingtoEq.(2）,determinetheareaofcompressivereinforcements.3）AccordingtoEq.(1）,determinetheareaoftensilereinforcements.4)Checktheminimumreinforcementratio1.Designsection3.5.3DesignversusAnalysisexampleh0fyAsfcsAsasas1）AccordingtoEq.(2)or(4)todetermine；2)If2as/h0

b3)If<2as/h0,4)If>

bCalculationformula则假定受压区合力作用点在受压钢筋重心处，并以受压区合力点为求矩中心求矩，得到计算公式，根据公式求解纵向受拉钢筋截面面积；(2)ThesecondtypeofdesignproblemThendeterminetheAsbasedonEq.(1)；Thecompressivereinforcementscannotyieldthetensilereinforcementscannotyield.Thatsuggeststhatthecompressivereinforcementsareinadequate.Knowingb,h，fc，fy，fy，MandAs，DetermineAsExample(2)If2as/h0

b2.AnalysisofsectionKnowingb×h，fc，fy，fy，AsandAs，DetermineMu（1）AccordingtoEq.(1)todetermine；ThendetermineMubasedonEq.(3);Thensubstituting＝bintoEq.(2)tocalculateMu.CalculationFormulaThendetermineMubasedonEq.(2)；(3)If<

2as/h0(4)If>

bExample3.6Calculationofflexuralcapacity

forT-sectionbeam3.6.2Formulaandtheiraccommodatedconditions1、TwotypesofT-sectionbeamThefirsttypeofT-sectionalbeam(RectangularT-beam)floorslabofcast-in-situReinforcedconcrete

3.6.1Introduction（TypesofT-sectionbeams）ThesecondtypeofT-sectionalbeam(TrueT-beam)2FormulaandaccommodatedconditionsforthefirsttypeofT-beamAccommodatedconditions3FormulaandaccommodatedconditionsforthesecondtypeofT-beamAccommodatedconditions4StressblockcompletelycoversthewholeflangeKnowingb,h,fc,M,determineAs(2)AccordingtoEq.(4）or（6）tocompute(3)AccordingtoEq.(3）or（5）todeterminetheAs(1)DeterminethetypesofT-beam1.DesignSectionCalculationformulas4.6.3DesignVersusAnalysisIf>b，substituting＝b代入intoEq.(4）or（6）tocomputeMuCalculationformulas2.AnalysisofSectionKnowingb,h,fc,As,determineMu(1)DeterminethetypesofT-beam(2)AccordingtoEq.(3）or（5）toCalculate(3)AccordingtoEq.(4）or（6）tocomputeMu计算公式变形Formulaofflexuralcapacityforsinglyrectangularbeams(单筋梁正截面承载力计算公式）双筋梁正截面承载力计算公式T形截面梁正截面承载力计算公式Sectionsofbeamsandslabs普通纵向受力钢筋宜采用HRB400、HRB500、HRBF400、HRBF500钢筋；也可采用HRB335、HRBF335、HPB300和RRB400钢筋；普通箍筋宜采用HRB400、HRBF400、HRB500、HRBF500钢筋；也可采用HRB335、HRBF335和HPB300钢筋。HRBF细晶粒热轧带肋钢筋-Hot-rolledribbedfinebarsRRB-余热处理带肋钢筋RemainedHeat-treatedRibbedBars1）Fitmodulus：Dimensionisgenerallymultipleof50mm(millimeter)。2)Height-span

rationforbeams：Accordingtoexperience（Detailingmanual构造手册）3）Height-widthrationforbeams：Rectangularsectionh/b=2-3.5Calculationformulash0has(1)DetermineasAlayerreinforcementsTwolayerreinforcements(2)Calculateandchecktheaccommodatedconditions(3)Calculatetheareaofreinforcementsandcheckthereinforcementratio(4)Chooseanddrawthesectionreinforcementpictures(Satisfythedetailingrequirements）（Othercalculationmethod）ReinforcementsGradeandordinarydiameterinbeams1）LongitudinalstressedreinforcementsDiameter：10mm2）Erectionsteelreinforcements(stirrup-support