Beam-Columns

Members Under Combined ForcesMost beams and columns are subjected to some degree of both bending and axial loade.g. Statically Indeterminate Structures

Interaction Formulas for Combined Forcese.g. LRFD If more than one resistance is involved consider interaction

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

AISC Interaction Formula CHAPTER HAISC Curver = required strengthc = available strength

REQUIRED CAPACITYPr PcMrx McxMry Mcy

Axial Capacity Pc

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

Effective Length FactorFixed on bottomFree to rotate and translateFixed on bottomFixed on topFixed on bottomFree to rotate

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:

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

Axial Capacity PcLRFD

Axial Capacity PcASD

Moment Capacity Mcx or McyREMEMBER TO CHECK FOR NON-COMPACT SHAPES

Moment Capacity Mcx or McyREMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

Moment Capacity Mcx or McyLRFDASD

Demand

Axial Demand PrLRFDASDfactoredservice

Demand

Second Order Effects & Moment AmplificationWPymax @ x=L/2 = dMmax @ x=L/2 = Mo + Pd = wL2/8 + Pdadditional moment causes additional deflection

Second Order Effects & Moment AmplificationConsiderMmax = Mo + PD

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

Design CodesAISC Permits

Second Order Analysis

or

Moment Amplification MethodCompute moments from 1st order analysisMultiply by amplification factor

Derivation of Moment Amplification

Derivation of Moment AmplificationMoment CurvatureMP2nd order nonhomogeneous DE

Derivation of Moment AmplificationBoundary ConditionsSolution

Derivation of Moment AmplificationSolve for B

Derivation of Moment AmplificationDeflected Shape

Derivation of Moment AmplificationMomentMo(x)Amplification Factor

Braced vs. Unbraced FramesEq. C2-1a

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

Braced Frames

Braced Frames

Braced FramesPr = required axial compressive strength = Pu for LRFD = Pa for ASDPr has a contribution from the PD effect and is given by

Braced Frames a = 1 for LRFD = 1.6 for ASD

Braced FramesCm coefficient accounts for the shape of the moment diagram

Braced FramesCm For Braced & NO TRANSVERSE LOADSM1: Absolute smallest End MomentM2: Absolute largest End Moment

Braced FramesCm For Braced & NO TRANSVERSE LOADSCOSERVATIVELY Cm= 1

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

Unbraced Frames

Unbraced Frames

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

Unbraced Frames Used when shape is knowne.g. check of adequacyUsed when shape is NOT knowne.g. design of members

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