# Beam-Columns.

## Presentation on theme: "Beam-Columns."— Presentation transcript:

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 P1 P2 C E A D F B

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

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

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

REQUIRED CAPACITY Pr Pc Mrx Mcx Mry Mcy

Axial Capacity Pc

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

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

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:

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

Axial Capacity Pc LRFD

Axial Capacity Pc ASD

Moment Capacity Mcx or Mcy
REMEMBER TO CHECK FOR NON-COMPACT SHAPES

Moment Capacity Mcx or Mcy
REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

Moment Capacity Mcx or Mcy
LRFD ASD

Demand

Axial Demand Pr LRFD ASD factored service

Demand

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

Second Order Effects & Moment Amplification
Consider Mmax = Mo + PD additional moment causes additional deflection

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

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

Derivation of Moment Amplification

Derivation of Moment Amplification
Moment Curvature P M 2nd order nonhomogeneous DE

Derivation of Moment Amplification
Boundary Conditions Solution

Derivation of Moment Amplification
Solve for B

Derivation of Moment Amplification
Deflected Shape

Derivation of Moment Amplification
Mo(x) Amplification Factor

Braced vs. Unbraced Frames
Eq. C2-1a

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

Braced Frames

Braced Frames

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

Braced Frames a = 1 for LRFD = 1.6 for ASD

Braced Frames Cm coefficient accounts for the shape of the moment diagram

Braced Frames Cm For Braced & NO TRANSVERSE LOADS
M1: Absolute smallest End Moment M2: Absolute largest End Moment

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

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

Unbraced Frames

Unbraced Frames

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

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

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