Beam-Columns

Members Under Combined Forces Most beams and columns are subjected to some degree of both bending and axial load e.g. Statically Indeterminate Structures P1P1 P2P2 C E A D F B

Interaction Formula REQUIRED CAPACITY P r P c M rx M cx Mry Mcy

Axial Capacity P c

Elastic Buckling Stress corresponding to the controlling mode of failure (flexural, torsional or flexural torsional) Fe:Fe: Theory of Elastic Stability (Timoshenko & Gere 1961) Flexural BucklingTorsional Buckling 2-axis of symmetry Flexural Torsional Buckling 1 axis of symmetry Flexural Torsional Buckling No axis of symmetry AISC Eqtn E4-4 AISC Eqtn E4-5 AISC Eqtn E4-6

Axial Capacity P c LRFD

Axial Capacity P c ASD

Moment Capacities

Moment Capacity M cx or M cy REMEMBER TO CHECK FOR NON- COMPACT SHAPES

Moment Capacity M cx or M cy REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

Moment Capacity M cx or M cy LRFDASD

Axial Demand

Axial Demand P r LRFDASD factoredservice

Demand

Second Order Effects & Moment Amplification W P P M y y x=L/2 =  M x=L/2 =    P  wL 2 /8 + P  additional moment causes additional deflection

Second Order Effects & Moment Amplification Consider M max =    P  additional moment causes additional deflection

Design Codes AISC Permits Second Order Analysis or Moment Amplification Method Compute moments from 1 st order analysis Multiply by amplification factor

Braced vs. Unbraced Frames Eq. C2-1a

Braced vs. Unbraced Frames Eq. C2-1a M nt = Maximum 1 st order moment assuming no sidesway occurs M lt = Maximum 1 st order moment caused by sidesway B 1 = Amplification factor for moments in member with no sidesway B 2 = Amplification factor for moments in member resulting from sidesway

Braced Frames P r = required axial compressive strength = P u for LRFD = P a for ASD P r has a contribution from the P  effect and is given by

Braced Frames a = 1 for LRFD = 1.6 for ASD

Braced Frames C m coefficient accounts for the shape of the moment diagram

Braced Frames C m For Braced & NO TRANSVERSE LOADS M 1 : Absolute smallest End Moment M 2 : Absolute largest End Moment

Braced Frames C m For Braced & NO TRANSVERSE LOADS COSERVATIVELY C m = 1

Unbraced Frames Eq. C2-1a M nt = Maximum 1 st order moment assuming no sidesway occurs M lt = Maximum 1 st order moment caused by sidesway B 1 = Amplification factor for moments in member with no sidesway B 2 = Amplification factor for moments in member resulting from sidesway

Unbraced Frames

a= 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 known e.g. check of adequacy Used when shape is NOT known e.g. design of members

Unbraced Frames I = Moment of inertia about axis of bending K 2 = Unbraced length factor corresponding to the unbraced condition L = Story Height R m = 0.85 for unbraced frames  H = drift of story under consideration  H = sum of all horizontal forces causing  H

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