beam-columns aisc summary
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Beam-Columns
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Members Under Combined ForcesMost beams and columns are subjected to some degree of both bending and axial loade.g. Statically Indeterminate Structures
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Interaction Formulas for Combined Forcese.g. LRFD If more than one resistance is involved consider interaction
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Basis for Interaction FormulasTension/Compression & Single Axis BendingTension/Compression & Biaxial BendingQuite conservative when compared to actual ultimate strengthsespecially for wide flange shapes with bending about minor axis
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AISC Interaction Formula CHAPTER HAISC Curver = required strengthc = available strength
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REQUIRED CAPACITYPr PcMrx McxMry Mcy
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Axial Capacity Pc
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Axial Capacity PcElastic Buckling Stress corresponding to the controlling mode of failure (flexural, torsional or flexural torsional)Fe:Theory of Elastic Stability (Timoshenko & Gere 1961)Flexural BucklingTorsional Buckling2-axis of symmetryFlexural Torsional Buckling1 axis of symmetryFlexural Torsional BucklingNo axis of symmetryAISC EqtnE4-4AISC EqtnE4-5AISC EqtnE4-6
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Effective Length FactorFixed on bottomFree to rotate and translateFixed on bottomFixed on topFixed on bottomFree to rotate
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Effective Length of ColumnsAssumptionsAll columns under consideration reach buckling Simultaneously
All joints are rigid
Consider members lying in the plane of buckling
All members have constant ADefine:
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Effective Length of ColumnsUse alignment charts (Structural Stability Research Council SSRC) LRFD Commentary Figure C-C2.2 p 16.1-241,242Connections to foundations(a) HingeG is infinite - Use G=10(b) Fixed G=0 - Use G=1.0
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Axial Capacity PcLRFD
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Axial Capacity PcASD
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Moment Capacity Mcx or McyREMEMBER TO CHECK FOR NON-COMPACT SHAPES
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Moment Capacity Mcx or McyREMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE
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Moment Capacity Mcx or McyLRFDASD
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Demand
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Axial Demand PrLRFDASDfactoredservice
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Demand
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Second Order Effects & Moment AmplificationWPymax @ x=L/2 = dMmax @ x=L/2 = Mo + Pd = wL2/8 + Pdadditional moment causes additional deflection
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Second Order Effects & Moment AmplificationConsiderMmax = Mo + PD
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Second Order Effects & Moment AmplificationTotal Deflection cannot be Found DirectlyAdditional Moment Because of Deformed ShapeFirst Order Analysis Undeformed Shape - No secondary moments
Second Order Analysis (P-d and P-D) Calculates Total deflections and secondary moments Iterative numerical techniques Not practical for manual calculations Implemented with computer programs
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Design CodesAISC Permits
Second Order Analysis
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Moment Amplification MethodCompute moments from 1st order analysisMultiply by amplification factor
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Derivation of Moment Amplification
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Derivation of Moment AmplificationMoment CurvatureMP2nd order nonhomogeneous DE
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Derivation of Moment AmplificationBoundary ConditionsSolution
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Derivation of Moment AmplificationSolve for B
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Derivation of Moment AmplificationDeflected Shape
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Derivation of Moment AmplificationMomentMo(x)Amplification Factor
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Braced vs. Unbraced FramesEq. C2-1a
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Braced vs. Unbraced FramesEq. C2-1aMnt = Maximum 1st order moment assuming no sidesway occursMlt = Maximum 1st order moment caused by sideswayB1 = Amplification factor for moments in member with no sideswayB2 = Amplification factor for moments in member resulting from sidesway
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Braced Frames
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Braced Frames
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Braced FramesPr = required axial compressive strength = Pu for LRFD = Pa for ASDPr has a contribution from the PD effect and is given by
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Braced Frames a = 1 for LRFD = 1.6 for ASD
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Braced FramesCm coefficient accounts for the shape of the moment diagram
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Braced FramesCm For Braced & NO TRANSVERSE LOADSM1: Absolute smallest End MomentM2: Absolute largest End Moment
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Braced FramesCm For Braced & NO TRANSVERSE LOADSCOSERVATIVELY Cm= 1
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Unbraced FramesEq. C2-1aMnt = Maximum 1st order moment assuming no sidesway occursMlt = Maximum 1st order moment caused by sideswayB1 = Amplification factor for moments in member with no sideswayB2 = Amplification factor for moments in member resulting from sidesway
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Unbraced Frames
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Unbraced Frames
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Unbraced Framesa= 1.00 for LRFD= 1.60 for ASD = sum of required load capacities for all columns in the story under consideration= sum of the Euler loads for all columns in the story under consideration
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Unbraced Frames Used when shape is knowne.g. check of adequacyUsed when shape is NOT knowne.g. design of members
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Unbraced Frames I = Moment of inertia about axis of bendingK2 = Unbraced length factor corresponding to the unbraced conditionL = Story HeightRm = 0.85 for unbraced framesDH = drift of story under considerationSH = sum of all horizontal forces causing DH
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