structural stability column strength

8/2/2019 Structural Stability Column Strength

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CEE 6554, Column Strength 3

Column Strength: A History



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Double Modulus Theory

y

Key assumption: axial force remains constant at buckling


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y

Key assumption: axial force remains constant at buckling

0MPy int =

MintP =Pr



tensilecompr PP =

0PP tensilecompr =

0dAdA 22c

01

1c

0=

0dAEdAE 22c

01

1c

0 t=

( ) ( ) 0dAyzEdAyzE 22c

01

1c

0 t =

0dAzyEdAzyE 22c

01

1c

0t=

0dAzEdAzE 22c

01

1c

0t=

or

Assuming axial force remainsconstant

z1 z2


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0dAzEdAzE 22c

01

1c

0t =

0EQQE 21t = Locate Neutral Axis

For a rectangular section

2/bcdzbzdAzQ 21111c

01

1c

01===

2/bcdAzQ 2222c

02==

t

2

2

1

E

E

c

c=

For other sections ??

z1 z2



dAzdAzM 222c

011

1c

0int+=

( ) ( ) dAzyEzdAzyzE 222c

011t

1c

0=

( )dAzEdAzEy 222c0211c

0t +=

( )21t IEIEy +=

( )IEy r=

y

0MPy int =

MintP =Pr

( ) 0IEyPy r =+

er

2

r

2

r PE

E

L

IEP =

=

I

IEIEE 21tr

+=


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Tangent Modulus Theory

Same as double modulus theory, except we assume that thecompressive strains continue to increase throughout the cross-sectionat buckling

everywhereEt =

( ) 0IEyPy t =+

r2

t

2

t PL

IEP 0.5Py

(5)

Axial Strength Pn Pn based on KL(3) Pn based on L (no K)

(6)

(1) Includes first-order analysis with amplifiers

(2) Minimum notional load of 0.002Yi is required for gravity load only combinations

(3) K = 1 allowed if sidesway amplifier B2 = 2/1 < 1.1

(4) Out-of-plumbness o / L = 0.002 may be used in lieu of notional load(5) b = 4 (Pu/Py) (1 Pu/Py)

(6) If Pu < 0.01PeL of if a member out-of-straightness of 0.001L or the equivalentnotional loading is included, Pn = QPy (LRFD)


Relationship between notional loads and out-of-plumbness


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Example Beam-Columns


Appropriate nominal attributes for distributed plasticityanalysis

A sinusoidal or parabolic out-of-straightness with a maximum amplitude of o =

L/1000, where L is the unsupported length in the plane of bending.

An out-of-plumbness of o = L/500, the maximum tolerance specified in the AISC

(2005) Code of Standard Practice.

The Lehigh (Galambos and Ketter 1959) residual stress pattern.

An elastic-perfectly plastic material stress-strain response.


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Lehigh Residual Stress Pattern


Nominal strength curves by distributed plasticityanalysis versus AISC (2005) ELM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

HL/Mp

P/Py

AISC (2005) Effective Length

Distributed plasticity analysis, major-axis

Distributed plasticity analysis, minor axis


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structural stability column strength

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