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B

P1

C

D

P2

E

F

Members Under Combined ForcesMost beams and columns are subjected to some degree of both bending and axial load

e.g. Statically Indeterminate Structures

Axial Capacity Pc

Elastic Buckling Stress corresponding to the controlling mode of failure (flexural, torsional or flexural torsional)

Fe:

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

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

y

M

Second Order Effects & Moment AmplificationP

W

ymax @ x=L/2 = d

Mmax @ x=L/2 = Mo + Pd = wL2/8 + Pd

additional moment causes additional deflection

additional moment causes additional deflection

Second Order Effects & Moment AmplificationConsider

Mmax = Mo + PD

Design Codes

AISC Permits

Second Order Analysis

or

Moment Amplification Method

Compute moments from 1st order analysis

Multiply by 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

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

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

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

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

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

Homework

- 6.2-1
- 6.2-2
- 6.5-2
- 6.5-6
- 6.6-1

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