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1. 34mm
64mm
8mm
t=0.7mm
Local
Distortional
Euler (torsional)
Euler (flexural)
Thin-Walled Column Design Considering
Local, Distortional and Euler Buckling
Ben Schafer
Asst. Professor
Johns Hopkins University

2. Overview
• Introduction
• Elastic Buckling
• Ultimate Strength
• Column Design Methods
• Performance of Methods
• Conclusion

3. Introduction
0
50
100
150
200
250
300
350
400
450
500
10 100 1000 10000
half-wavelength (mm)
bucklingstress(MPa)
34mm
64mm
8mm
t=0.7mm
Local
Distortional
Euler (torsional)
Euler (flexural)

4. Elastic Buckling Prediction
• Numerical Methods
– finite element, finite strip (www.ce.jhu.edu/bschafer)
• Hand Methods (for use in a traditional Specification)
– Local Buckling
• Element methods, e.g. k=4
• Semi-empirical methods that include element interaction
– Distortional Buckling
• Proposed (Schafer) method, rotational stiffness at web/flange
juncture
• Hancock’s method
• AISI (k for Edge Stiffened Elements per Spec. section B4.2)

5. Elastic Buckling Comparisons1
(fcr)element (fcr)interact (fcr)Schafer (fcr)Hancock (fcr)AISI
All Data avg. 1.34 1.03 0.93 0.96 0.79
st.dev. 0.13 0.06 0.05 0.06 0.33
max 1.49 1.15 1.07 1.08 1.45
min 0.96 0.78 0.81 0.83 0.18
count 149 149 89 89 89
Schafer (1997) Members avg. 1.16 1.02 0.92 0.96 1.09
st.dev. 0.15 0.08 0.07 0.06 0.16
Commercial Drywall Studs avg. 1.38 1.07 0.93 1.00 0.81
st.dev. 0.09 0.05 0.02 0.07 0.26
AISI Manual C's avg. 1.33 1.01 0.93 0.99 0.81
st.dev. 0.13 0.07 0.05 0.03 0.26
AISI Manual Z's avg. 1.39 1.04 0.92 0.92 0.41
st.dev. 0.03 0.04 0.03 0.06 0.18
(fcr)true = local or distoritonal buckling stress from finite strip analysis
(fcr)element = minimum local buckling stress of the web, flange and lip via Eq.'s 1-3
(fcr)interact = minimum local buckling stress using the semi-empirical equations (Eq.'s 4-6)
(fcr)Schafer = distortional buckling stress via Eq.'s 7-15
(fcr)Hancock = distortional buckling stress via Lau and Hancock (1987)
(fcr)AISI = buckling stress for an edge stiffened element via AISI (1996) from Desmond et al. (1981)
Local Distortional
(fcr)true (fcr)true (fcr)true (fcr)true (fcr)true
(fcr)element (fcr)interact (fcr)Schafer (fcr)Hancock (fcr)AISI
All Data avg. 1.34 1.03 0.93 0.96 0.79
st.dev. 0.13 0.06 0.05 0.06 0.33
max 1.49 1.15 1.07 1.08 1.45
min 0.96 0.78 0.81 0.83 0.18
count 149 149 89 89 89
Schafer (1997) Members avg. 1.16 1.02 0.92 0.96 1.09
st.dev. 0.15 0.08 0.07 0.06 0.16
Commercial Drywall Studs avg. 1.38 1.07 0.93 1.00 0.81
st.dev. 0.09 0.05 0.02 0.07 0.26
AISI Manual C's avg. 1.33 1.01 0.93 0.99 0.81
st.dev. 0.13 0.07 0.05 0.03 0.26
AISI Manual Z's avg. 1.39 1.04 0.92 0.92 0.41
st.dev. 0.03 0.04 0.03 0.06 0.18
(fcr)true = local or distoritonal buckling stress from finite strip analysis
(fcr)element = minimum local buckling stress of the web, flange and lip via Eq.'s 1-3
(fcr)interact = minimum local buckling stress using the semi-empirical equations (Eq.'s 4-6)
(fcr)Schafer = distortional buckling stress via Eq.'s 7-15
(fcr)Hancock = distortional buckling stress via Lau and Hancock (1987)
(fcr)AISI = buckling stress for an edge stiffened element via AISI (1996) from Desmond et al. (1981)
*
*For members with slender webs and small flanges the Lau and Hancock (1987) approach
conservatively converges to a buckling stress of zero (these members are ignored in the
summary statistics given above)
1
For a wide variety of cold-formed steel lipped channels, zees and racks

6. Ultimate Strength
• Numerical Studies (nonlinear FEA)
– Analysis of isolated flanges
– Parametric studies on lipped channels
• Existing Experimental Data
(pin ended, concentrically loaded columns)
– 100+ tests on lipped channels
– 80+ tests on lipped zees
– 40 tests on rack columns (variety of stiffeners)

7. Numerical Studies on Ultimate Strength
• Primarily focused on differences in the behavior and in
the failure mechanisms associated with local and
distortional buckling.
• Key findings in this area:
– distortional buckling may control the failure mechanism even
when the elastic distortional buckling stress (fcrd
) is higher than
the elastic local buckling stress (fcrl)
– distortional failures have lower post-buckling capacity and
higher imperfection sensitivity than local failures

8. Experiments on Distortional Buckling Failures
High Strength Rack Columns (U. of Sydney)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.2
0.4
0.6
0.8
1
channel
rack
rack+lip
hat
channel+web st.
distortional slenderness (Fy/Fcr)
.5
or (Py/Pcr)
.5
strength(Fu/Fy)or(Pu/Py)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.2
0.4
0.6
0.8
1
channel
rack
rack+lip
hat
channel+web st.
distortional slenderness (Fy/Fcr)
.5
or (Py/Pcr)
.5
strength(Fu/Fy)or(Pu/Py)
(a) (b) (c)
(d)
(e)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.2
0.4
0.6
0.8
1
channel
rack
rack+lip
hat
channel+web st.
distortional slenderness (Fy/Fcr)
.5
or (Py/Pcr)
.5
strength(Fu/Fy)or(Pu/Py)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.2
0.4
0.6
0.8
1
channel
rack
rack+lip
hat
channel+web st.
distortional slenderness (Fy/Fcr)
.5
or (Py/Pcr)
.5
strength(Fu/Fy)or(Pu/Py)
(a) (b) (c)
(d)
(e)
Eq. 16
not predicted by AISI Spec.

9. Considered Design Methods
• Effective Width Methods
– AISI Design Specification (1996)
– Element by element effective width approach,
local and distortional buckling treated separately
• Direct Strength Methods
– Hand solutions for member elastic buckling
– Numerical solutions (finite strip) for elastic buckling
• Varying levels of interaction amongst the failure
modes considered (see paper for full discussion)

10. Effective Width Methods
















−=
f
f
f
f
22.01
b
b crcreff 
for 673.0
f
f
cr
>

, else bbeff =.(17)
where: beff is the effective width of an element with gross width b
f is the yield stress (f = fy) when interaction with other modes is
not considered, otherwise f is the limiting stress of a mode
interacting with local buckling
crf is the local buckling stress
∑= tbA effeff
*
* for Euler (long column) interaction f=Fcr of the column curve used in AISC Spec.
(the notation for f is Fn in the AISI Spec. but the same column curve is employed)

11. Direct Strength Methods
4.
cr
4.
crn
P
P
P
P
15.01
P
P




















−= 
for 776.0
P
P
cr
>

, else Pn = P . (19)
where: Pn is the nominal capacity
P is the squash load (P = Py = Agfy) except when interaction with
other modes is considered, then P = Agf, where f is the
limiting stress of the interacting mode.
crP is the critical elastic local buckling load (Ag crf )
6.
crd
6.
crdn
P
P
P
P
25.01
P
P


























−= for 561.0
P
P
crd
> , else Pn = P. (16)
where: Pn is the nominal capacity in distortional buckling
P is the squash load (P = Py = Agfy) when interaction with other
modes is not considered, otherwise P = Agf, where f is the
limiting stress of a mode that may interact
Pcrd is the critical elastic distortional buckling load (Agfcrd)
Local
Distortional

12. Performance of Effective Width Methods
(for subset of tests on lipped Zees)
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
experiment
A1=AISI(1996)
local by B1
distortional by B1
B1
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
L = 610 mm
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
L = 1220 mm
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
experiment
A1=AISI(1996)
local by B1
distortional by B1
B1
0 20 40 60
0
20
40
60
80
100
120
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
experiment
A1=AISI(1996)
local by B1
distortional by B1
B1
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
L = 610 mm
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
L = 1220 mm
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm

13. Performance of Direct Strength Methods
(for subset of tests on lipped Zees)
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
experiment
A1=AISI(1996)
local by B3
distortional by B3
B3
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
L = 610 mm
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
L = 1220 mm
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
experiment
A1=AISI(1996)
local by B3
distortional by B3
B3
0 20 40 60
0
20
40
60
80
100
120
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
experiment
A1=AISI(1996)
local by B3
distortional by B3
B3
0 20 40 60
0
20
40
60
80
100
120
d (mm)
Pu(kn)
L = 610 mm
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
L = 1220 mm
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm

14. Overall Performance (Direct Strength)
0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
local buckling controlled
distortional buckling controlled
slenderness of controlling mode (Pn/Pcr)
.5
strengthPu/Pne
Solution C3
0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
local buckling controlled
distortional buckling controlled
slenderness of controlling mode (Pn/Pcr)
.5
strengthPu/Pne
Solution C3

15. Conclusions
• Must consider local, distortional, and Euler modes, but closed-
form and numerical methods are accurate and available
• Current effective width based design methods ignore local
web/flange interaction and distortional buckling, leading to
systematic error and inflexibility in dealing with new shapes
• A direct strength method using separate column curves for
local and distortional buckling:
– provides a consistent and accurate treatment of the relevant buckling modes
– avoids lengthy effective width calculations
– demonstrates the effectiveness of directly using numerical elastic buckling solutions
– opens the door to rational analysis methods
– provides greater potential innovation in cold-formed shapes