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3 Flexural Analysis/Design of Beam3.BENDING OF HOMOGENEOUS BEAMSREINFORCED CONCRETE BEAM BEHAVIORS G OOF TENSIONS O REINFORCEDO CREC.C BEAMSSDESIGNDESIGN AIDSPRACTICAL CONSIDERATIONS IN DESIGNREC. BEAMS WITH TEN. AND COMP. REBART BEAMS447.327Theory of Reinforced Concrete and Lab. ISpring 2008

3. Flexural Analysis/Design of BeamTypical StructuresTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamTypical StructuresTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamTypical StructuresTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamBENDING OF HOMOGENEOUS BEAMSConcrete is homogeneous?Reinforced Concrete is homogeneous?The fundamental principles in the design and analysis ofreinforced concrete are the same as those of homogeneousstructural material.Two componentsInternal forcesat any cross sectionTheory of Reinforced Concrete and Lab I.normal to the section - flexuretangential to the section - shearSpring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORBasic Assumptions in Flexural Design1. A cross section that was plane before loading remainsplane under loadF unit strain in a beam above and below the neutralaxis are proportional to the distance from that axis;strain distribution is linear G Bernoulli’s hypothesis( not true for deep beams)Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORBasic Assumptions in Flexural Design2. Concrete is assumed to fail in compression,when εc εcu (limit state) 00.0030033. Stress-strain relationshipp of reinforcement is assumed tobe elastoplastic (elastic-perfectly plastic).G Strain hardening effect is neglected4. Tensile strength of concrete is neglected for calculationof flexural strength.strengthTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORBasic Assumptions in Flexural Designcf.)f ) Typicall s-s curve off hhomogeneous materiallproportionalnot proportionalTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORBasic Assumptions in Flexural Design5. Compressive stress-strain relationship for concretemay be assumed to be any shape (rectangular(rectangular,trapezoidal, parabolic, etc) that results in anacceptableppredictionpof strength.gG Equivalent rectangular stress distributionTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORBehavior of RC beam under increasing loadTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORMomeentBehavior of RC beam under increasing loadεMφ y EICurvatureTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORElastic uncracked Sectionnot sIn elastic range(n-1)AsfcfsEsF fs εc εs f c nff cEcEsEcTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORExample 3.1 (SI unit)A rectangular beamAs 1,520 mm2fcu 27 MPa (cylinder strength)fr 3.5 MPa (modulus of rupture)fy 400 MPaCalculate the stresses caused by abending moment M 60 kNkN·mmTheory of Reinforced Concrete and Lab I.250600650(unit: mm)D25Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORSolution250Es2.0 105n 7.847 84 8Ec 8,500 3 f cu600650(n 1) As 7 1,520 10, 640 mm 2transformed area of rebarsD25transformed sectionTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORSolutionAssuming the uncracked sectionsection,250neutral axis600650bh 2 (n 1) As ddAQx ydA 2y A dA bh (n 1) As 342 mm3bh h I x y 2 dA y bh12 2 D25transformed sectionTheory of Reinforced Concrete and Lab I. (d y ) 2 (n 1) As 6, 477 106 mm 4Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORSolutionCompressive stress of concreteat the top fiber250Mfc y 3.17 MPaIxTension stress of concreteat the bottom fiberMf ct (h y ) 2.85 MPa IxfrAssumption of uncracked, transformed section is justified !!Theory of Reinforced Concrete and Lab I.600650D25Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORSolutionStress in the tensile steel250Mf s n (d y ) 19.1219 12 MPaMPIxCompare fc and fs with fcu andfy respectively!!!600650D25Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORElastic Cracked SectionThis situation is under service load state1f f ct f c f ck fs f yr2Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORElastic Cracked SectionTo determine neutral axisb(kd )Theory of Reinforced Concrete and Lab I.kd nAs (d kd ) 02(1)Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORElastic Cracked SectionTension & Comp. forcefcC b((kdkd )2Theory of Reinforced Concrete and Lab I.T As f s(2)Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORElastic Cracked SectionBending moment about CM T ( jd ) As f s ( jd )(3)Mfs As ( jd )(4)FBending moment about TfcM C ( jd ) b(kd )( jd )222MMfc Fkjbd 2Theory of Reinforced Concrete and Lab I.(5)(6)How to get k and j ?Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORElastic Cracked SectionDefiningρ AsbdAs ρbdThen,Substitute (7) into (1) and solve for kb(kd )k ( ρn) 2 2 ρn ρncf.)f)(7)kd nAs (d kd ) 02(8)jdd d kd / 3F j 1 k3Theory of Reinforced Concrete and Lab I.See Handout #3#3-33Table A.6Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORExample 3.2 (Quiz)The beam of Example 3.1 is subjected to a bendingmoment M 120 kN·m (rather than 60 kN·m as previously).Calculate the relevant properties and stress right away!!Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORFlexural Strengthα f avf ck(9)fav ave. compressivestress on the area bcC αf ck bcbTheory of Reinforced Concrete and Lab I.( )(10)Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORFlexural Strengthα0.72decrease by 0.04 for every 7 MPa0.56fck 28 MPa28 MPa fck 56 MPafck 56 MPaβ0.425fck 28 MPadecrease by 0.025 for every 7 MPa 28 MPa fck 56 MPa0.325fck 56 MPaTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORFlexural StrengthThis values apply to compression zone with other crosssectional shapes (circular, triangular, etc)However, the analysis of those shapes becomes complex.Note that to compute the flexural strength of the sectionsection, itis not necessary to know exact shape of the compressionstress block. Only need to know C and its location.These two quantities are expressed in α and β .Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORFlexural StrengthG The higher compressive strength, the more brittle.Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORFlexural StrengthTension failure (εu 0.003, fs fy)EquilibriumC Tαf ck bc As f s(11)Bending momentorM Tz As f s (d βc)(12)M Cz α f ck bc(d β c)(13)Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORTension failure (εu 0.003, fs fy)Neutral axis at steel yielding, fs fyFrom Eq.(11)ρf y dc αf ck b αf ckAs f y(14)Nominal bending momentρ f yd M n As f y ( d β c ) ρ bdf y d β αfck ρ fy ρ fy 2 2 ρ f y bd 1 β0 59 ρ f y bd 1 0.59 (15)α f ck f ck Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORFlexural StrengthCompression failure (εu 0.003, fs fy)Hook’ss lawHookf s ε s Es(16)from strain diagramf s ε u Esd cc(17)Equilibriumd cα f ck bc As f s Asε u EscTheory of Reinforced Concrete and Lab I.(18)Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORCompression failure (εu 0.003, fs fy)Solving the quadratic for cαf ckk bc 2 Asε u Es c Asε u Es d 0(19)c (20)fs (21)N i l bendingNominalb di momenttMn Theory of Reinforced Concrete and Lab I.(22)Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORFlexural StrengthBalanced reinforcement ratio ρbThe amount of reinforcement necessary for beam fail toby crushing of concrete at the same load causing thesteel to yield; (εu 0.003,0.003, fs fy) ρ ρb lightly reinforced, tension failure, ductile ρ ρb balanced, tension/comp. failure ρ ρb heavilyy reinforced, compressionpfailure, brittleTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORBalanced reinforcement ratio ρbBalanced conditionfs f yεy Substitute Eq. (23) into Eq.(17)c εuεu ε yρb εuf y εu ε yTheory of Reinforced Concrete and Lab I.(23)Esf s ε u Esd ccdSubstitute Eq. (24) into Eq.(11)αf ckfy(24)αf ck bc As f s(25)Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORExample 3.3 (SI unit)A rectangular beamAs 1,520 mm2fcu 27 MPa (cylinder strength)fr 3.5 MPa (modulus of rupture)fy 400 MPaCalculate the nominal moment Mnat which the beam will fail.failTheory of Reinforced Concrete and Lab I.250600650(unit: mm)D25Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORSolutionCheck whether this beam fail in tension or compressionAs1 5201,520ρ 0.0101bd (250)(600)ρb α f ckfyεu(0.72)(27)0.003 0.0292 ρ4000.003 0.002εu ε yF The beam will fail in tension by yielding of the steelTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamREINFORCED CONCRETE BEAM BEHAVIORSolutionUsing Eq. (15) for tension failureρ fy M n ρ f y bd 1 0.59 f ck 2(0 0101)(400) (0.0101)(400) (0.0101)(400)(250)(600) 1 0.59 27 332 kN m2Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSKorea’s design method is Ultimate Strength Design.called as Limit States Design in the US and Europe1. Proportioning for adequate strength2. Checking the serviceability:deflections/crack width compared against limiting valuesTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSEquivalent Rectangular Stress Distributionis called as Whitney’s Block (Handout #3-1)What if the actual stress block is replaced by anequivalent rectangular stress block for compression zone.ActualTheory of Reinforced Concrete and Lab I.EquivalentSpring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSEquivalent Rectangular Stress DistributionGo toP.25Cactual αf ck bc γf ck ab Cequi.Fγ αcaαγ β1a βc2Ga β1cγdependson α,βGa β1cβ1 2 βTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSEquivalent Rectangular Stress Distributionfck, MPa 2835424956 1 2 β0.850.8010.7520.7030.654γ α/ β10.8570.8490.8510.8530.856- γ is essentially independent of fck.- β1 0.850 85 - 0.007 (0 007 (fck -28)28)Theory of Reinforced Concrete and Lab I.andd0.650 65 β1 0.850 85Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSEquivalent Rectangular Stress Distribution-KCI 6.2.1(6) allows other shapes for the concrete stressblock to be used in the calculations as longg as theyy resultin good agreement with test results.-KCI6.2.1(5) makes a further simplification. Tensilestresses in the concrete may be neglected in the cals.G Contribution of tensile stresses of the concrete belowN.A. is very small.Concrete Stress Distribution in Ultimate Strength Design”Design“ConcreteHandout #3-2by HognestadTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSBalanced Strain Conditionsteel strain is exactly equal to εy andconcrete simultaneouslyy reaches εu 0.003Eq. (24)c εuεu ε ydEquilibrium C TAs f y ρbbd f y 0.85 f ck ab 0.85 f ck β1cbTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSBalanced Strain ConditionBalanced reinforcement ratiof ck cf ck ε uρb 0.85β1 0.85β1f ydf y εu ε y(26)Apply εu 0.003 and Es 200,000 MPaf ckρb 0.85β1fyTheory of Reinforced Concrete and Lab I.f ck 600 0.85β1fyf y 600 f y0 003 0.003Es0.003(27)Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSUnderreinforced BeamsIn actual practice, ρ should be below ρb for the reasons,1. Exactly ρ ρb, then concrete reaches the comp. strain limit and steelreaches its yield stressstress.2. Material properties are never known precisely.3 Strain hardening can cause compressive failure3.failure, although ρ may besomewhat less than ρb.4. The actual steel area provided, will always be equal to or larger thanrequired based on ρ.5. The extra ductility provided by low ρ provides warning prior to failure.Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSKCI Code Provisions for Underreinforced BeamBy the way, how to guarantee underreinforced beams?F KCI Code provides,(1) The minimum tensile reinforcement strain allowed atnominal strengthg in the designg of beam.(2) Strength reduction factors that may depend on thetensile strain at nominal strength.gNote Both limitations are based on the net tensile strain εtof the rebar farthest from the compression face at thedepth dt.(d dt)Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSKCI Code Provisions for Underreinforced Beam(1) For nonprestressed flexural members and members with factoredaxial compressive load less than 0.1fckAg, εt shall not be less than 0.004((KCI 6.2.2(5))( ))Substitute dt for d and εt for εyεuc dFεu ε yf ck ε uρb 0.85β1f y εu ε yFdt cεt εucf ck ε uρ 0.85β1f y εu εtThe reinforcement ratio toproduce a selected εtTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSKCI Code Provisions for Underreinforced BeamMaximum reinforcement ratio (KCI 2007)ρ max 0.85β1f ck ε ufεu 0.85β1 ckf y εu εtf y ε u 0.004Cf.) (prior to KCI 2007& ACI 2002)ρ max 0.75ρb(29)(28)F εt 0.00376 at ρ 0.75ρb for fy 400 MPa.F 0.00376 0.004 ,i.e., KCI 2007 is slightly conservative.Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSKCI Code Provisions for Underreinforced Beamεu 0.003εu 0.003εu 0.003cdtεt 0.002c0 0030.003 0.600dt 0.003 0.002εt 0.004c0 0030.003 0.429dt 0.003 0.004CompressioncontrolledmemberTheory of Reinforced Concrete and Lab I.Minimumnet strainfor flexuralmemberεt 0.005c0 0030.003 0.375dt 0.003 0.005TensioncontrolledmemberSpring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSKCI Code Provisions for Underreinforced ontrolledΦ 0.85SpiralOtherΦ 0.700 70Φ 0.70 (εt - 0.002) 0.85Φ 0.65 (εt - 0.002)) 200/3/Φ 0.65εt 0.002Theory of Reinforced Concrete and Lab I.εt 0.005Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSKCI Code Provisions for Underreinforced BeamNominal flexural strengtha M n As f y d 2 As f ya 0.85 f ck bTheory of Reinforced Concrete and Lab I.(30)(31)Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSExample 3.4 (the same as Ex.3.3)A rectangular beamAs 1,520 mm2fcu 27 MPa (cylinder strength)fr 3.5 MPa (modulus of rupture)fy 400 MPaCalculate the nominal strength Mnusing the equivalent stress block.Theory of Reinforced Concrete and Lab I.250600650(unit: mm)D25Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSSolutionMaximum reinforcement ratioρ maxf ckkεu 0.85β1f y ε u 0.004As(27)0.003 (0.85)(0.85) 00.0209(0 85)(0 85)0209 00.01010101 (400) 0.003 0.004bdF This beam is underreinforced (tension controlled)and will fail yielding of the steelTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSSolutionDepth of stress blockAs f y(1520)(400)a 106 mm0.85 f ck b (0.85)(27)(250)Nominal Strengtha 106 M n As f y d (1520)(400) 600 333 kN m2 2 CComparethisthi withith ExampleEl 3.33 3 !!!Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSNominal flexural strength (Alternative)Eq(31) can be written w.r.t. ρa ρ f yd0.85 f ck(32)Nominal flexural strengthρ f yd M n ( ρ bd ) f y d 1.7 f ck AsTheory of Reinforced Concrete and Lab I.ρ fy 2 ρ f y bd 1 0.59 fck (33)a/2Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSNominal flexural strength (Alternative)simplified expression of Eq.(33)M n Rbd 2where,,R ρ f y (1 0.590 59(34)ρ fyf ck)(35)flexural resistance factor R depends on1. reinforcement ratio2. material properties (Handout #3-3)Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSDesign flexural strengthKCI Code Provisionsaφ M n φ As f y (d )2 φρ f y bd (1 0.590 592 φ Rbd 2Theory of Reinforced Concrete and Lab I.ρ fyf ck)(36)(37)Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSKCI Code Provisions for Underreinforced rolledΦ 0.85SpiralOtherΦ 0.700 70Φ 0.70 (εt - 0.002) 0.85Φ 0.65 (εt - 0.002)) 200/3/Φ 0.65εt 0.002Theory of Reinforced Concrete and Lab I.εt 0.005Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSExample 3.4 (continued)Calculate the design moment capacity for the beamanalyzed in Example 3.4Hint net tensile strain should be known.Theory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSSolutiona106c 125 mmβ1 0.85dt c600 125 0.003 0.0114 0.005F εt εu125cFstrength reduction factor is 0.85!!!Design strengthTheory of Reinforced Concrete and Lab I.φ M n (0.85)(333) 283 kN mSpring 2008

3. Flexural Analysis/Design of BeamDESIGN OF TENSION REINFORCED REC. BEAMSMinimum Reinforcement Ratio; very lightly reinforced beams will also fail without warningwarning.so, lower limit is required.Rectangular cross sectionAs.min0.15 f ck bdfy(38)ProofEquating the cracking moment to the flexural strength, based underthe assumptions,assumptions h 1.1dh 1 1d and internal lever arm 0.95d0 95dTheory of Reinforced Concrete and Lab I.Spring 2008

3. Flexural Analysis/Design of BeamDES

3. Flexural Analysis/Design of Beam3. Flexural Analysis/Design of Beam REINFORCED CONCRETE BEAM BEHAVIORREINFORCED CONCRETE BEAM BEHAVIOR Flexural Strength This values apply to compression zone with other cross sectional shapes (circular, triangular, etc) However, the analysis of those shapes becomes complex.