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Lecture 7-FlexureJune 16,2003CVEN 444Lecture GoalsDoubly Reinforced beamsT Beams and L Beams Pan JoistAnalysis of Flanged SectionFloor systems with slabs and beams are placed in monolithic pour.Slab acts as a top flange to the beam;T-beams,and Inverted L(Spandrel)Beams.Analysis of Flanged SectionsPositive and Negative Moment Regions in a T-beamAnalysis of Flanged SectionsIf the neutral axis falls within the slab depth analyze the beam as a rectangular beam,otherwise as a T-beam.Analysis of Flanged SectionsEffective Flange WidthPortions near the webs are more highly stressed than areas away from the web.Analysis of Flanged SectionsEffective width (beff)beff is width that is stressed uniformly to give the same compression force actually developed in compression zone of width b(actual)ACI Code Provisions for Estimating beffFrom ACI 318,Section 8.10.2T Beam Flange:ACI Code Provisions for Estimating beffFrom ACI 318,Section 8.10.3Inverted L Shape FlangeACI Code Provisions for Estimating beffFrom ACI 318,Section 8.10Isolated T-BeamsVarious Possible Geometries of T-BeamsSingle TeeTwin TeeBoxAnalysis of T-BeamCase 1:Same as rectangular section Steel is yielding under reinforcedCheckAnalysis of T-BeamCase 1:EquilibriumAnalysis of T-BeamCase 1:ConfirmAnalysis of T-BeamCase 1:Calculate MnAnalysis of T-BeamCase 2:Assume steel yields Analysis of T-BeamCase 2:Assume steel yields The flanges are considered to be equivalent compression steel.Analysis of T-BeamCase 2:EquilibriumAnalysis of T-BeamCase 2:ConfirmAnalysis of T-BeamCase 2:ConfirmAnalysis of T-BeamCase 2:Calculate nominal momentsAnalysis of T-BeamsThe definition of Mn1 and Mn2 for the T-Beam are given as:Analysis of T-BeamsThe ultimate moment Mu for the T-Beam are given as:For a T-Beam with the tension steel yielded using a function c/dLimitations on Reinforcement for Flange BeamsLower LimitsnFlange in compressionLimitations on Reinforcement for Flange BeamsLower LimitsnFlange in tensionLimitations on Reinforcement for Flange BeamsLower LimitsnIf As(provided)4/3 As(reqd)based on analysisthen As(min)is not required(i.e.)fMn 4/3Mu for As(provided)See ACI 10.5.3Example-T-BeamFind Mn and Mu for T-Beam.beff=54 in.hf=3 in.b=7 ft.d =16.5 in.As=8.5 in2 fy=50 ksi fc=3 ksi bw=12 in L=18 ftExample of L-BeamConfirm beffExample-T-BeamCompute the equivalent c value and check the strain in the steel,esSteel will yield in the tension zone.Example-T-BeamCompute the reinforcement r and check to make sure it is greater than rminSection works for minimum reinforcement.Example-T-BeamCompute w and check that the c value is greater than hfAnalysis the beam as a T-beam.Example-T-BeamCompute w and check that the c value is greater than hfCompute aExample-T-BeamCompute nominal moment componentsExample-T-BeamCompute nominal momentCompute ultimate momentExample of L-BeamDetermine the effective b for the spandrel beam and do the analysis.Use 4#9 bars and find the ultimate moment capacity.fy=50 ksi,fc=3 ksiExample of L-BeamCompute beffExample of L-BeamCompute beffExample of L-BeamThe value beff and AsExample-L-BeamCompute the equivalent c value and check the strain in the steel,esSteel will yield in the tension zone.Example-L-BeamCompute the reinforcement r and check to make sure it is greater than rminSection works for minimum reinforcement.Example-L-BeamCompute w and check that the c value is greater than hfAnalysis the beam as a Singly reinforced beam.False!Example-L-BeamCompute aExample-L-BeamCompute nominal momentExample-L-BeamCompute ultimate momentPan Joist Floor SystemsView of Pan Joist Slab from BelowWalter P.Moore&Assoc.Pan Joist Floor SystemsView of Double Skip Joist Slab from BelowWalter P.Moore&Assoc.Pan Joist Floor SystemsPlacing Reinforcement for a Pan Joist SlabWalter P.Moore&Assoc.Pan Joist Floor SystemsGeneral framing layout of the pan joist system.Pan Joist Floor SystemsPouring a Pan Joist SlabWalter P.Moore&Assoc.Pan Joist Floor SystemsDefinition:The type of slab is also called a ribbed slab.It consists of a floor slab,usually 2-4 in.thick,supported by reinforced concrete ribs.The ribs are usually tapered and uniformly spaced at distances that do not exceed 30 in.The ribs are supported on girders that rest on columns.In some ribbed slabs,the space between ribs may be filled with permanent fillers to provide a horizontal slab soffit.One-Way Joist ConstructionMacGregor,Fig.10-28Definition:Pan joist floor systems are series of closely spaced cast-in-place T-beams or joists used for long-span floors with relatively light loads.Typically removable metal forms(fillers or pans)are used to form joists.One-Way Joist ConstructionDetails of ribbed floor with removable steel pans.Ribbed-floor cross sections.One-Way Joist ConstructionThe design of a ribbed floor with steel pan forms and average weight of the floor.One-Way Joist ConstructionThe design of a ribbed floor with steel pan forms and average weight of the floor.One-Way Joist ConstructionJoist DetailsPan Joist Floor SystemsACI Requirements for Joist Construction(Sec.8.11,ACI 318-02)nSlabs and ribs must be cast monolithically.nRibs must be spaced consistentlynRibs may not be less than 4 inches in widthPan Joist Floor SystemsACI Requirements for Joist Construction(cont.)(Sec.8.11.2,ACI 318-02)nDepth of ribs may not be more than 3.5 times the minimum rib widthnClear spacing between ribs shall not exceed 30 inches.*Ribbed slabs not meeting these requirements are designed as slabs and beams.*Pan Joist Floor SystemsSlab Thickness n(ACI Sec.8.11.6.1)t 2 in.for joints formed with 20 in.wide panst 2.5 in.for joints formed with 30 in.wide pans(1/12 distance)Pan Joist Floor SystemsSlab Thickness(cont.)nBuilding codes give minimum fire resistance rating:1-hour fire rating:in.cover,3”-3.5”slab thickness2-hour fire rating:1 in.cover,4.5”slab thicknessPan Joist Floor SystemsStandard Removable Form Dimensions nNote the shapesPan Joist Floor SystemsStandard Removable Form Dimensions nStandard Widths:20 in.&30 in.(measured at bottom of ribs)nStandard Depths:6,8,10,12,14,16 or 20 in.Pan Joist Floor SystemsStandard Removable Form Dimensions(cont.)nEnd Forms:one end is closed(built-in)to form the supporting beamnTapered End Forms:provide additional shear capacity at ends of joists by tapering ends to increase rib width.Pan Joist SlabsStandard Pan Joist Form DimensionsRef.CECO Concrete Construction CatalogPan Joist SlabsStandard Pan Joist Form DimensionsRef.CECO Concrete Construction CatalogPan Joist Floor SystemsLaying Out Pan Joist FloorsnRib/slab thicknesswGoverned by strength,fire rating,available spacenOverall depth and rib thicknesswGoverned by deflections and shearPan Joist Floor SystemsLaying Out Pan Joist Floors(cont.)nTypically no stirrups are used in joistsnReducing Forming Costs:wUse constant joist depth for entire floorwUse same depth for joists and beams(not always possible)Pan Joist Floor SystemsDistribution RibsnPlaced perpendicular to joists*nSpans 30 ft.:Provided at third-pointsnAt least one continuous#4 bar is provided at top and bottom of distribution rib.*Note:not required by ACI Code,but typically used in constructionMember DepthACI provides minimum member depth and slab thickness requirements that can be used without a deflection calculation(Sec.9.5 ACI 318)nUseful for selecting preliminary member sizesMember DepthACI 318-Table 9.5a:nMin.thickness,h(for beams or ribbed one-way slab)wFor beams with one end continuous:L/18.5wFor beams with both ends continuous:L/21wL is span length in inchesnTable 9.5a usually gives a depth too shallow for design,but should be checked as a minimum.Member DepthACI 318-99:Table 9.5aMember DepthRule of Thumb:nhb(in.)L(ft.)nEx.)30 ft.span-hb 30 in.nMay be a little large,but okay as a start to calc.DLAnother Rule of Thumb:nwDL(web below slab)15%(wSDL+wLL)Note:For design,start with maximum moment for beam to finalize depth.Select b as a function of dnb (0.45 to 0.65)(d)
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