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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 P1 P2 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 Pr Pc Mrx Mcx Mry Mcy

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Axial Capacity Pc

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Axial Capacity Pc Fe: Elastic Buckling Stress corresponding to the controlling mode of failure (flexural, torsional or flexural torsional) Theory of Elastic Stability (Timoshenko & Gere 1961) Flexural Buckling Torsional 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**

Free to rotate and translate Fixed on top Free to rotate Fixed on bottom Fixed on bottom Fixed on bottom

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**Effective Length of Columns**

A B Ig Lg Ic Lc 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 ,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 Pc LRFD

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Axial Capacity Pc ASD

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**Moment Capacity Mcx or Mcy**

REMEMBER TO CHECK FOR NON-COMPACT SHAPES

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**Moment Capacity Mcx or Mcy**

REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

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**Moment Capacity Mcx or Mcy**

LRFD ASD

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Demand

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Axial Demand Pr LRFD ASD factored service

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Demand

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**Second Order Effects & Moment Amplification**

y P W x=L/2 = d x=L/2 = Mo + Pd = wL2/8 + Pd additional moment causes additional deflection

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**Second Order Effects & Moment Amplification**

Consider Mmax = Mo + PD 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-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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**Moment Amplification Method**

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

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**Derivation of Moment Amplification**

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**Derivation of Moment Amplification**

Moment Curvature P M 2nd 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**

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 Mnt = Maximum 1st order moment assuming no sidesway occurs Mlt = Maximum 1st order moment caused by sidesway B1 = Amplification factor for moments in member with no sidesway B2 = 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 Frames Pr = required axial compressive strength = Pu for LRFD = Pa for ASD Pr 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 Frames Cm coefficient accounts for the shape of the moment diagram

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**Braced Frames Cm For Braced & NO TRANSVERSE LOADS**

M1: Absolute smallest End Moment M2: Absolute largest End Moment

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Braced Frames Cm For Braced & NO TRANSVERSE LOADS COSERVATIVELY Cm= 1

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Unbraced Frames Eq. C2-1a Mnt = Maximum 1st order moment assuming no sidesway occurs Mlt = Maximum 1st order moment caused by sidesway B1 = Amplification factor for moments in member with no sidesway B2 = 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 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

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**Used when shape is known e.g. check of adequacy **

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 K2 = Unbraced length factor corresponding to the unbraced condition L = Story Height Rm = 0.85 for unbraced frames DH = drift of story under consideration SH = sum of all horizontal forces causing DH

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