Reinforced Concrete Beam - California State University .

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Mechanics of MaterialsReinforced Concrete Beam jkmConcrete Beam2Concrete Beam We will examine a concrete beam in bendingP2P2 A concrete beam is what we call acomposite beam It is made of two materials: concreteand steel Concrete is also a composite jkm1

Concrete Beam3Concrete Beam Test LoadingL1PPP2LL1P2P222VPL1 P22M Maximum Moment is in the center region jkmConcrete Beam 4Concrete StrengthWe determine the compressive strength ofconcrete with a concrete cylinder testCylinders are typically 6 inches in diameter12 inches longWe want the stress-strain results so wemeasure load and change in length jkm2

Concrete Beam5Concrete Cylinder Test jkmConcrete Beam6Concrete Stress Strain The concrete stress-strain diagram is not linearf ’cstressf 'c2strain jkm3

Concrete Beam 7CompositeWhen we examine a beam we generally need avalue of E to use in the analysisDo we use E for concrete or steel?We will usually pretend that the beam is madeof only one material, usually concreteWe Transform the steel to an equivalentamount of concrete How much concrete do we substitute forthe steel? jkmConcrete Beam 8Modular RatioWhich is stiffer, steel or concrete?Es for steel is about 29,000 ksiEc for concrete is about 3 to 6,000 ksi, and willbe different for every batch of concreteWe get Ec from the cylinder testsLet’s define n to be the modular ration EsEC It tells you how much stiffer steel is comparedto concrete jkm4

Concrete Beam9Modulus of Concrete-Ec The concrete stress-strain diagram is not linearf ’cstressf 'c2Do a regressionthrough thesepointsEcstrain Ec is the slope of the stress-strain curve upto about half the strength of the concreteConcrete Beam jkm10Steel is Stiffer Since the steel is stiffer than the concrete,when we replace the steel with concrete, wewill need more concrete to make itequivalent How much more? We will multiply the amount of steel by n toget the equivalent amount of concreteAs ' n As As is the total cross sectional area of thesteel for all the bars jkm5

Concrete Beam11Transformed Section When we replace the steel with equivalentconcrete, we have effectively transformedeverything to concrete We call the resulting all concrete beam theTransformed Section Let’s examine the process . . . jkmConcrete Beam12Equivalent Steel Replace As by As’This concretenow models thesteelAs total areaof steel(n 1) ASAs’ nAs totalAs’ (n-1) As afterfilling “holes” jkm6

Concrete Beam13Neutral Axis The NA passes through the centroid, thebalance pointNA(n 1) ASAssume thisheight is small* The moment of the area above theNA is the same as the moment of thearea below the NA (both the realconcrete and the equivalent concrete)Concrete Beam jkm14Transformed Section We want to find the Itr of our transformedsection about the NA We first need to locatethe NANA(n 1) AS The section is composed of three parts The part above the NA The portion below the NA And the transformed area, (n-1)As jkm7

Concrete Beam15Neutral Axis Locate the NAbdNActrhBelow NATransformed steelAbove NA(n 1) AS cbctr tr 2 h ctr (n 1) A d c s tr b h ctr 2 Solve for ctr This locates the NA jkmConcrete Beam16Itr of the Transformed SectionParallel axis theorem Find ItrI I cg Ad 2bbc 3 b h ctr 2I tr tr (n 1) A s d ctr 333dNA(n 1) AS Solve for ItrctrhTop,compressionconcreteSteeltransformedinto concreteBottom,Tensionconcrete jkm8

Concrete Beam17Cracks What happens to the tension concrete asthe load get higher and higher? It starts to crack, first at the bottom As the loadincreases, theNAcracks progress nAShigher and higher Finally, all the concrete in tension will crack Once the tension concrete cracks, it jkmcannot hold tension stressesConcrete Beam18Cracked Section We call the resulting section with all theconcrete below the NA cracked, the crackedsection Now we effectively haveonly two remainingareas, because theconcrete below the NA iscrackedNAnA S We must find the new location of the NA jkm9

Concrete Beam19Cracked Section NA Since there is less concrete toward the bottom,the NA will move upb Calculate the newcentroid to locate the NA cbccr cr 2 nA s d ccr ccrdNAhnA S Solve for ccr This locates the NA jkmConcrete Beam20Icr of the Cracked Section Now we can find the cracked moment ofinertiabParallel axis theoremI I cg Ad 2bccr 32I cr nA s d ccr 3Top,compressionconcretedNAccrhnA SSteeltransformedinto concrete Solve for Icr, the moment of Inertia ofthe cracked section jkm10

Concrete Beam21Load the Beam We have two values of the moment of InertiaI, Itr and Icr Which do we use for the beam bendingtest? At the start (before cracking) we use Itr At a load corresponding to steel yielding,we can use Icr for the beam In between I for the beam varies from Itr to Icras tension cracks propagate upward and alongthe length of the beam jkmConcrete Beam22Nonlinear Beam The response of the beam to the load isnonlinear in two ways The stress-strain curve for concrete isnonlinear, E changes The section itself is nonlinear, since the Ichanges as the load increases jkm11

Concrete Beam23Concrete Stress What is the stress in the beam at cracking?b Stress in the concrete will be conc MyI Which I do we use? we use ItrctrydNAh(n 1) AS c MctrI tr jkmConcrete Beam24Steel Stress What is the steel stress up to cracking?b Since our transformedbeam is made of concrete,ctrdwe must multiply thehANstress we calculated inthe steel by n(n 1) A we use ItrS steel nM d ctr I tr jkm12

Concrete Beam Flexural FailureHow will the beam fail?b2 possibilities for flexure: The concrete on the topcrushes before the steel dyields (brittle)NA The steel yields beforeconcrete crushes (ductile)The concrete will fail in compressionat a concrete strain of 0.003-0.004.The steel will yield at a steel strain offy/Es or a steel stress of fy.ccr25hnA S jkmConcrete Beam26Cracking of the Concrete in Tension As the load is applied to the beam, thetension stress at the bottom of the beamincreases This is the approximate cracking stress forconcrete in tension Here is the Bending stress equation for thetensile stress in the concrete at the bottomof the beam.Why use Itr? jkm13

Concrete Beam27Cracking of the Concrete in Tensionb Use these equations:dNActrhynA S Find the moment Mcr that will cause theconcrete to start cracking. Then find the load, Pcr, that will cause thisP LTry to locate this pointmoment.M crcr22M crPcr L11on your graph of load vsdeflection as a change inslope jkmConcrete BeamYielding of the Steel Rebarb Here is the bending stressequation for the steel rebardwith y 68,000psiNAy28ccrhnA S Find the moment My and the load Py that willcause the steel to yieldMy Py Py L122M yL1If you had steel yielding, try tolocate this point on your graphof load vs deflection. jkm14

Concrete BeamYielding of the Steel Rebarb Next, calculate the stress inthe concrete at the top ofthe beam at the load that dycauses the steel to yield NA29ccrhnA S Determine also the ratio fc-yield /f’c, to seeif it is less than approximately 0.7, an upperbound for “roughly” linear concrete stresses Was the concrete behaving linearly? Ifnot, more accurate equations can be used. jkmConcrete Beam30Ultimate Failure of the Concrete Here is the nonlinear stress distribution inthe compression concrete due to bending It is shaped like the concrete stress-straincurvebhdccrASaEquivalent RectangularStress Distributionassumes constantstress over smallerarea To make it easier to model the nonlinearresponse of the concrete, a simplified stressdiagram is used jkm15

Concrete Beam31Ultimate Failure of the Concrete Here is the resulting force diagrambccrhdCad T Asfya2This is a couple For equilibrium:C T jkmConcrete Beam32Ultimate Failure of the Concrete Here is the resulting force diagrambccrhdCaT As fy Moment ofd a2This is a couplea M ult T d fy2 the couple: a As fy can be replaced by fult to0.85f 'C ba account for strain hardeningM ult A sf y d 2 f 1.25f jkmulty16

Concrete Beam33Ultimate Failure of the Concrete Once this ultimate moment for the beam isfound, calculate the load, Pult, that wouldcause this momenta P LM ult ult 122M ultPult L1M ult T d 2 a M ult A sf y d 2 a As fy0.85f 'C b This is the load that would cause theconcrete to crush, usually after the steelyields jkmConcrete Beam34Other Types of Ultimate Failure Although failure of the concrete usuallyoccurs instead of ultimate fracture of thesteel, other final failure modes may occurinstead This could include concrete bond failurealong the plane of the rebar Shear failure of the concrete may also occur How did your beam ultimately fail? jkm17

Shear FailureConcrete Beam35 The beam may fail due to excessive shearstresses (i.e., diagonal tension), if the shearcapacity of the beam is less than itsflexural capacity The average shear stress at which adiagonal tension crack forms depends on thepresence of flexural stresses There is a lot of scatter in the data thatdescribes the maximum shear stress that aconcrete beam can withstand jkmShear FailureConcrete Beam36 Here is a range of average shear stresses that canbe used to estimate shear failure : The average shear stress that concrete can withstand can be approximated by this rangeFor most cases and especially regions of highshear and moment, flexural cracks will form first,reducing shear stress at diagonal cracking to minIn regions of high shear and small moment (e.g.,simple supports and p.i.), then "web shear" cracksmay form at a stress level given by max jkm18

Shear FailureConcrete Beam37 Here is the approximate shear stress equation: With the range of defined by our equations, this defines a range of internal shear force, Vconcrete,that should represent the maximum shear forcethe beam can withstand.PV represents an estimate for the maximum loadthat would cause the concrete to fail in shear.This failure will occur only if another mode offailure has not occurred prior to reaching this load.You will have a range of PV based upon the range ofVconcrete calculated above. jkmShear FailureConcrete Beam38 You will have a range of PV based upon the min and maxPV 2Vconcrete Shear failures are more likely as the amount of steel rebar increases.For shear failure, cracking and sudden failureshould be shown on the load-deflection plots. jkm19

Concrete Beam39EndConcreteBeam jkm20

The concrete on the top crushes before the steel yields (brittle) The steel yields before concrete crushes (ductile) The concrete will fail in compression at a concrete strain of 0.003-0.004. The steel will yield at a steel strain of fy/Es or a steel stress of fy. N A ccr h b d nAS Concrete Beam 26 jkm Cracking of the Concrete in Tension

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