Transcript Thin-Walled Column Design Considering Local, Distortional

34mm
8mm
64mm
Euler (torsional)
t=0.7mm
Thin-Walled Column Design Considering
Local, Distortional and Euler Buckling
Distortional
Local
Ben Schafer
Asst. Professor
Johns Hopkins University
Euler (flexural)
Overview
•
•
•
•
•
•
Introduction
Elastic Buckling
Ultimate Strength
Column Design Methods
Performance of Methods
Conclusion
Introduction
500
34mm
8mm
450
64mm
Euler (torsional)
400
t=0.7mm
buckling stress (MPa)
350
300
250
Distortional
Euler (flexural)
200
150
Local
100
50
0
10
100
1000
half-wavelength (mm)
10000
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)
(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 Local 1.02
0.92
0.96
1.09
Distortional
st.dev.
0.15
0.08
0.07
0.06
0.16
(fcr)true
(fcr)true
(fcr)true
(fcr)true
(fcr)true
Commercial Drywall Studs
avg.
1.38
1.07
0.93
1.00
0.81
(fcr)element
(fcr)interact
(fcr)Schafer
(fcr)Hancock
(fcr)AISI
0.09
0.05
0.02
0.07
0.26
All Data st.dev.
avg.
1.34
1.03
0.93
0.96
0.79
AISI Manual C's st.dev.
avg.
1.33
1.01
0.93
0.99
0.81
0.13
0.06
0.05
0.06
0.33
st.dev.
0.13
0.07
0.05
0.03
0.26
max
1.49
1.15
1.07
1.08
1.45
AISI Manual Z's
avg.
1.39
1.04
0.92
0.92
0.41
min
0.96
0.78
0.81
0.83
0.18
st.dev.
0.03
0.04
0.03
0.06
0.18
count
149
149
89
89
89
Schafer
(1997)
Members buckling
avg. stress from
1.16 finite strip1.02
0.92
0.96
1.09
(fcr)true
= local
or distoritonal
analysis
st.dev. stress of0.15
0.08
0.07 1-3
0.06
0.16
(fcr)element = minimum local buckling
the web, flange
and lip via Eq.'s
Commercial
Drywall Studs
avg. stress using
1.38 the semi-empirical
1.07
0.93(Eq.'s 4-6)1.00
0.81
(fcr)interact
= minimum
local buckling
equations
st.dev.
0.097-15
0.05
0.02
0.07
0.26
(fcr)Schafer = distortional buckling
stress via Eq.'s
Manual C's
avg.stress via Lau
1.33 and Hancock
1.01(1987)
0.93
0.99
0.81
(fcr)HancockAISI
= distortional
buckling
0.07(1996) from
0.05
0.26
(fcr)AISI = buckling stress for anst.dev.
edge stiffened0.13
element via AISI
Desmond et 0.03
al. (1981)
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
1 For a =wide
variety of cold-formed steel lipped channels, zees and racks
(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)
*For members
with slender
small
flanges the Lau and Hancock (1987) approach
(fcr)Schafer
= distortional
bucklingwebs
stressand
via Eq.'s
7-15
(fcr)Hancock
= distortional
buckling
via Lau
and Hancock
conservatively
converges
to a stress
buckling
stress
of zero (1987)
(these members are ignored in the
(fcr)AISI = buckling stress for an edge stiffened element via AISI (1996) from Desmond et al. (1981)
Elastic Buckling Comparisons1
*
summary statistics given above)
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)
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
Experiments on Distortional Buckling Failures
High Strength Rack Columns (U. of Sydney)
not predicted by AISI Spec.
1
channel
rack
rack+lip
hat
channel+ web st.
strength (Fu/F y) or (Pu /Py )
0.8
0.6
(a)
(b)
(d)
(c)
(e)
0.4
Eq. 16
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
.5
4
.5
distortional slenderness (Fy /Fcr) or (Py /Pcr)
4.5
5
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)
Effective Width Methods
A eff 
 beff t
b eff 
f cr  f cr 
f
 1  0.22
 0.673 , elseb eff  b .(17)
for




b
f  f 
f cr

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
f cr is the local buckling stress
* 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)
Direct Strength Methods
Local
.4
.4
Pn 
 Pcr   Pcr 
P
 1  0.15
 
 for
 0.776 , else Pn = P . (19)

P 
P
P
Pcr

 


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.
Pcr is the critical elastic local buckling load (A
f crg
)
Distortional
.6
.6

 Pcrd   Pcrd 
Pn 
 
 for P  0.561 , else Pn = P. (16)
 1  0.25
 P   P 
Pcrd
P 






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)
Performance of Effective Width Methods
(for subset of tests on lipped Zees)
120
L = 610 mm
L = 1220 mm
100
100
80
80
Pu (kn)
Pu (kn)
120
60
experiment
A1=AISI(1996)
local by B1
distortional by B1
B1
40
0
40
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
20
0
20
40
d (mm)
60
20
60
0
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
0
20
40
d (mm)
60
Performance of Direct Strength Methods
(for subset of tests on lipped Zees)
120
L = 610 mm
L = 1220 mm
100
100
80
80
Pu (kn)
Pu (kn)
120
60
experiment
A1=AISI(1996)
local by B3
distortional by B3
B3
40
0
40
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
20
0
20
40
d (mm)
60
20
60
0
h ~ 200 mm
b ~ 75 mm
t = 1.5 mm
0
20
40
d (mm)
60
Overall Performance (Direct Strength)
Solution C3
1.6
1.4
1.2
local buckling controlled
strength P u/P ne
distortional buckling cont rolled
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
.5
slenderness of controlling mode (Pn/Pcr)
7
8
Conclusions
• Must consider local, distortional, and Euler modes, but closedform 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