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Beam-Columns

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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

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

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Basis for Interaction Formulas Tension/Compression & Single Axis Bending Tension/Compression & Biaxial Bending Quite conservative when compared to actual ultimate strengths especially for wide flange shapes with bending about minor axis

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AISC Interaction Formula – CHAPTER H AISC Curve r = required strength c = available strength

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

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Axial Capacity P c

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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

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Effective Length Factor Fixed on bottom Free to rotate and translate Fixed on bottom Fixed on top Fixed on bottom Free to rotate

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Effective Length of Columns A B I g L g I c L c Assumptions All columns under consideration reach buckling Simultaneously All joints are rigid Consider members lying in the plane of buckling All members have constant A Define:

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Effective Length of Columns Use alignment charts (Structural Stability Research Council SSRC) LRFD Commentary Figure C-C2.2 p 16.1-241,242 Connections to foundations (a) Hinge G is infinite - Use G=10 (b) Fixed G=0 - Use G=1.0

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Axial Capacity P c LRFD

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Axial Capacity P c ASD

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

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

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Moment Capacity M cx or M cy LRFDASD

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Demand

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Axial Demand P r LRFDASD factoredservice

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Demand

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Second Order Effects & Moment Amplification W P P M y y max @ x=L/2 = M max @ x=L/2 = P wL 2 /8 + P additional moment causes additional deflection

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Second Order Effects & Moment Amplification Consider M max = P additional moment causes additional deflection

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Second Order Effects & Moment Amplification Total Deflection cannot be Found Directly Additional Moment Because of Deformed Shape First Order Analysis Undeformed Shape - No secondary moments Second Order Analysis (P- and P- ) Calculates Total deflections and secondary moments Iterative numerical techniques Not practical for manual calculations Implemented with computer programs

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

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Derivation of Moment Amplification

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Moment Curvature M P 2 nd order nonhomogeneous DE

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Derivation of Moment Amplification Boundary Conditions Solution

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Derivation of Moment Amplification Solve for B

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Derivation of Moment Amplification Deflected Shape

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Derivation of Moment Amplification Moment Mo(x)Mo(x) Amplification Factor

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Braced vs. Unbraced Frames Eq. C2-1a

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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

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Braced Frames

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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

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Braced Frames a = 1 for LRFD = 1.6 for ASD

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

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

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

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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

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Unbraced Frames

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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

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Unbraced Frames Used when shape is known e.g. check of adequacy Used when shape is NOT known e.g. design of members

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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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