structural stability column strength
TRANSCRIPT
-
8/2/2019 Structural Stability Column Strength
1/19
-
8/2/2019 Structural Stability Column Strength
2/19
CEE 6554, Column Strength 3
Column Strength: A History
CEE 6554, Column Strength 4
-
8/2/2019 Structural Stability Column Strength
3/19
CEE 6554, Column Strength 5
CEE 6554, Column Strength 6
-
8/2/2019 Structural Stability Column Strength
4/19
CEE 6554, Column Strength 7
CEE 6554, Column Strength 8
Double Modulus Theory
y
Key assumption: axial force remains constant at buckling
-
8/2/2019 Structural Stability Column Strength
5/19
CEE 6554, Column Strength 9
Double Modulus Theory
y
Key assumption: axial force remains constant at buckling
0MPy int =
MintP =Pr
CEE 6554, Column Strength 10
Double Modulus Theory
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
-
8/2/2019 Structural Stability Column Strength
6/19
CEE 6554, Column Strength 11
Double Modulus Theory
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
CEE 6554, Column Strength 12
Double Modulus Theory
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
+=
-
8/2/2019 Structural Stability Column Strength
7/19
CEE 6554, Column Strength 13
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)
CEE 6554, Column Strength 32
Relationship between notional loads and out-of-plumbness
-
8/2/2019 Structural Stability Column Strength
17/19
CEE 6554, Column Strength 33
Example Beam-Columns
CEE 6554, Column Strength 34
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.
-
8/2/2019 Structural Stability Column Strength
18/19
CEE 6554, Column Strength 35
Lehigh Residual Stress Pattern
CEE 6554, Column Strength 36
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
-
8/2/2019 Structural Stability Column Strength
19/19