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LOCAL AND DISTORTIONAL BUCKLING OF COLD-FORMED
STEEL MEMBERS WITH EDGE STIFFENED FLANGES
B.W. Schafer1 & T. Peköz2
1
Simpson Gumpertz & Heger, Arlington, MA 02474, USA
2
Professor, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853,
USA
ABSTRACT
Cold-formed steel members with edge stiffened flanges have three important buckling phenomena:
local, distortional, and global. Current North American specification (e.g., AISI) methods do not
explicitly treat the distortional mode or account for interaction of elements in local buckling. Hand
methods are presented for proper estimation of the critical buckling stress of compression and flexural
members in both the local and distortional mode. Post-buckling behavior of edge stiffened flanges is
examined. Phenomena unique to the distortional mode include: reduced post-buckling capacity and
heightened imperfection sensitivity. A design method for strength prediction, based on the unified
effective width approach, is proposed.
KEYWORDS
local buckling, distortional buckling, cold-formed steel design, imperfection sensitivity
INTRODUCTION
80
Buckling of cold-formed steel members with 1.33"
0.33"
edge stiffened flanges may be generally 70 Torsional
2.5"
characterized as occurring in one of three 60
t =0.0284"
modes: local, distortional, or global (e.g.,
buckling stress (ksi)
50
torsional, flexural etc.). The finite strip 40
analysis shown in Figure 1 illustrates these
30 Local
three buckling modes for a member under pure Distortional
Flexural
compression. 20
10
The local mode repeats at short wavelengths, 0
generally involves only rotation at element 0.1 1.0 10.0
half-wavelength (in)
100.0 1000.0
junctures (i.e., the elements appear “pinned” at
each fold line) and hand methods in current Figure 1 Finite Strip Analysis of a
use typically ignore any interaction amongst Member in Compression
the elements. The distortional mode repeats at
wavelengths from short to long depending on geometry and loading, generally involves rotation and
translation of multiple elements (but not the entire cross-section) and hand methods for prediction are
relatively cumbersome. The global mode repeats at long wavelengths (often only one half-wavelength
1
occurs in a given member) generally involves rotation and/or translation of the entire cross-section and
hand methods for prediction are considered classical examples of buckling phenomena.
ELASTIC BUCKLING PREDICTION BY HAND CALCULATION
Local Buckling
In design local buckling is typically treated by ignoring any interaction that exists between elements
(e.g., the flange and the web). Each element is treated independently and classic plate buckling
solutions based on isolated simply supported plates are generally employed. The result of such an
approach is that each element is predicted to buckle at a different stress. This approach shall be termed
the “element model”.
Numerical methods, such as the finite strip method, or finite element method may be used to determine
the local buckling stress of an entire member; however, for design purposes hand methods are
desirable. In order to better predict the actual local buckling stress, but not result to numerical
solutions a second class of local buckling solutions is introduced: the “semi-empirical interaction
model”. These local buckling solutions account for the interaction between two elements, but not the
entire member. The solutions are approximations to finite strip analyses of two isolated members, e.g.,
the flange and the web. In general, it is found that the minimum local buckling stress of any two
attached elements is a reasonable approximation of the entire member local buckling stress.
Table 1 Proposed Models for Local Buckling Prediction
ELEMENT MODEL
Flange: ( fcr)f k 4
Web: ( fcr)w
k 0.5 3 4 2 4 b h
web web
2
k k lip b d
2
Lip: ( fcr)l
for 0 lip 11
. k lip 14lip 0.25lip 0.425
. 2
for 11 lip 2 k lip 13lip 655lip 131lip 80
. 3
. 2
SEMI-EMPIRICAL INTERACTION MODEL
.
Flange/Lip: ( fcr)fl k 8.55 lip 1107 d b 159 lip 3.95 d b 4
.
2
for lip 1 and d b 0.6
Flange/Web1: ( fcr)fw .
k 1125 min 4 , 0.5 3 4 2 4 b h
web web 2
Flange/Web2: ( fcr)fw
k 2 b h 4b h
0.4 2
if h b 1
2 h b 4 if h b 1
0.2
1
local buckling of the flange and web when the web is under flexure
2
local buckling of the flange and web when the web is in pure compression.
2
Local buckling solutions for both the element model and the semi-empirical interaction model are
given in Table 1. For an edge stiffened member (e.g., a lipped channel) h = web height, b = flange
width, and d = lip length. All of the k values are in terms of the critical buckling stress of the flange,
where:
2D
f cr k 2 .
bt
Several of the elements are subjected to a stress gradient, which is defined in terms of ,
f f2
1 .
f1
Where, f1 and f2 are defined as the stresses at the opposite edges of the element. For the web, f1 is at the
web/compression flange juncture. For the lip, f1 is at the lip/compression flange juncture. Compression
stresses are positive.
Distortional Buckling
Prediction of distortional buckling by hand methods is markedly more involved than local buckling.
The key issue is related to the rotational restraint at the juncture of the flange and the web. Consider an
isolated edge stiffened flange in which the restraint at
the web/flange juncture is first modeled as simply 15
supported, then fixed – finite strip results of such a Fixed Support
Simple Support
model are given in Figure 2. The plate buckling
coefficient and hence the buckling stress are 10
markedly more sensitive to rotational restraint for
k
distortional buckling than for local buckling. Hence,
distortional buckling requires careful treatment of the 5 Local
rotational restraint at the web/flange juncture and Distortional
local buckling in most cases does not.
0
0 1 10 100
Closed-form prediction of the distortional buckling
stress is based on the rotational restraint at the
Figure 2 Importance of Rotational Restraint
web/flange juncture. Consider a typical cross-section
in an Edge Stiffened Flange
as shown in Figure 3 and the definition of the
rotational stiffness. The rotational stiffness may be expanded as a summation of elastic and stress
dependent geometric stiffness terms with contributions from both the flange and the web,
M=k
k k f k w k
e
f kw g
.
1 Buckling ensues when the elastic stiffness at the web/flange juncture is eroded by
1
the geometric stiffness, i.e.,
k 0 .
Writing the stress dependent portion of the geometric stiffness explicitly,
~
~
k kfe kwe f kfg kwg 0 .
Figure 3 k Therefore, the buckling stress ( f ) is
kfe kwe
f ~ ~ .
kfg kwg
Complete expressions for the stiffness terms for members in flexure and compression are given in
Table 2. The expressions for flexure are derived in Schafer and Peköz (1999). The stiffness terms for
the flange are completed in the classic manner – assuming the flange acts as a column undergoing
3
flexural-torsional buckling with springs along one edge. The expressions for the web stiffness are
derived by truncating the solution for the buckling of a single finite strip.
Table 2 Proposed Model for Distortional Buckling Prediction
4
PROPOSED MODEL FOR DISTORTIONAL BUCKLING PREDICTION
kfe kwe
f ed ~ ~
kfg kwg
L min Lcr , Lm
Flange Rotational Stiffness:
4
I2
EI xf xo hx 2 ECwf E xyf xo hx 2 GJ f
2
kf
L
L
e
I yf
2 2
2 I xyf I
~
k f
g
A f x o hx
L I
yf
2 yo xo hx xyf
I
yf
hx yo I xf I yf
2 2
Flexural Member: Critical Length and Web Rotational Stiffness
I
1
4 4 h 1 2 2
I xyf 4h4
4
Lcr x o hx x o hx
2 2
Cwf
t3 xf 720
I yf
Et 3 3 2 19h 4 h 3
k we
h L 60 L 240
12 1 2
2 2
453601 62160 L 448 2 h 53 31 4
ht 2
web web
~ h L
kwg 2 4
13440 L L
4 28 2 420
h h
Compression Member: Critical Length and Web Rotational Stiffness
I
1
6 4 h 1 2 2
I xyf
4
x o hx x o hx
2 2
Lcr Cwf
t3 xf
I yf
2
th
3
Et 3 ~
k we k wg
6h 1 2 L 15
E = Modulus of Elasticity Af, Ixf, Iyf, Cwf, Jf = Section properties of the
G = Shear Modulus compression flange (flange and edge
= Poisson’s Ratio stiffener) about x, y axes respectively,
t = plate thickness where the x, y axes are located at the
h = web depth centroid of flange with x-axis parallel
= (f1- f2)/f1 stress gradient in the web with flat portion of the flange
Lm = Distance between restraints which limit xo = x distance from the flange/web junction
rotation of the flange about the to the centroid of the flange.
flange/web junction hx = x distance from the centroid of the flange
to the shear center of the flange
5
The distortional buckling methods proposed for compression members by Lau and Hancock (1987)
and flexural members Hancock (1995) and Hancock (1997) perform in a manner similar to the
proposed method except in the cases where the geometric stiffness of the web is “driving” the
distortional buckling solution (e.g., distortional buckling in which essentially the flange is
restraining the web from buckling). The explicit treatment of the role of the elastic and geometric
rotational stiffness at the web/flange juncture and the expressions for the web’s contribution to the
rotationanl stiffness are unique to the method presented here.
Verification
In order to verify the proposed buckling models a parametric study of members in either flexure or
compression is performed. The geometry of the studied members is summarized in Table 3 and the
results are given in Table 4. The results are determined by comparison to finite strip analysis. For
calculation of the local buckling moment or load (M or P) the minimum buckling stress of the
elements is used to compare to the finite strip solution.
For local buckling prediction use of the minimum element buckling stress for the entire member
(element model) is quite conservative. Use of the semi-empirical interaction model that accounts
for any two attached elements is generally a reasonable local buckling predictor. For distortional
buckling prediction the proposed method is a reasonable predictor, but not without error. For cases
with slender webs the proposed distortional buckling solution correctly converges to the web local
buckling stress, Hancock’s method conservatively converges to zero buckling stress.
Table 3. Geometry of Members used for Verifcation*
h/b h/t b/t d/t
max min max min max min max min count
Schafer (1997) Members 3.0 1.0 90 30 90 30 15.0 2.5 32
Commercial Drywall Studs 4.6 1.2 318 48 70 39 16.9 9.5 15
AISI Manual C's 7.8 0.9 232 20 66 15 13.8 3.2 73
AISI Manual Z's 4.2 1.7 199 32 55 18 20.3 5.1 50
7.8 0.9 318 20 90 15 20.3 2.5 170
* for members in flexure only Schafer (1997) members are studied
Table 4. Performance of Elastic Buckling Methods*
Distortional
Local Buckling Buckling
Element Model Interaction Model Proposed Method
Mpredicted /Mlocal Mpredicted /Mlocal Mpredicted /Mdist.
Average 0.74 0.90 0.95
St. Dev. 0.12 0.05 0.08
Ppredicted /Plocal Ppredicted /Plocal Ppredicted /Pdist.
Average 0.75 0.97 1.07
St. Dev. 0.13 0.06 0.05
* finite strip analysis does not always have a minimum for both local and distortional buckling, comparisons
are only made for those cases in which finite strip analysis revealed a minimum in the appropriate mode.
6
POST-BUCKLING BEHAVIOR
To investigate the post-buckling behavior in the local and distortional modes, nonlinear FEM
analysis of isolated flanges is completed using ABAQUS (HKS 1995). The boundary conditions
and the elements used to model the flange are shown in
Figure 4. The material model is elastic-plastic with strain
hardening. Initial imperfections in the local and distortional
d
"Roller" Supp ort
ne
i
ra
DOF 2,3 restrained
st
mode are superposed to form the initial imperfect
re
6
2-
F
geometry. A longitudinal through thickness flexural 4
O
D
d,
xe
residual stress of 30% fy is also modeled. D.O.F. 1
Fi
2 5
3
The geometry of the members investigated is summarized
in Table 5. The thickness is 1mm and fy = 345MPa. It is "Pin" Sup port
6
observed that the final failure mechanism is consistent with DOF 1-3 restrained
the distortional mode even in cases when the distortional
buckling stress is higher than the local buckling stress. Figure 4 Isolated Flange
Consider Figure 5, which shows the final failure (fixed at flange/web juncture)
mechanism for all the members studied. Based solely on elastic buckling one would expect the
local mode to control in all cases in which (fcr)local / (fcr)dist. < 1 – as the figure shows, this is not the
Table 5. Edge Stiffened Flanges 2
Distortional Mechanism
Distortional Mechanism + Local Yielding
1.8
Pcr,local Mixed ~ Mechanism Depends on Imp.
Local Mechanism + Distortional Yielding
1.6
b/t d/t Pcr,dist Local Mechanism
1.4
25 4.00 - 19.0 90 1.82 - 0.25
f
1.2
6.25 - 12.5 45 1.94 - 0.96 cr local
50 5.00 - 25.0 90 1.58 - 0.27 f
cr dist.
1
0.8
6.25 - 25.0 45 1.76 - 0.51
0.6
75 6.25 - 37.5 90 1.34 - 0.18
0.4
6.25 - 37.5 45 1.73 - 0.35
0.2
100 6.25 - 50.0 90 1.40 - 0.14
6.25 - 50.0 45 1.75 - 0.23 0
0 0.5 1 1.5 2 2.5
fy f
cr local
Figure 5 Failure Mechanism
case.
Finite element analysis also reveals that the post-buckling capacity in the distortional mode is less
than that in the local mode. Consider Figure 6, for the same slenderness values the distortional
failures exhibit a lower ultimate strength. Similar loss in strength is experimentally observed and
summarized in Hancock et al. (1994). Note for Figure 6, (fcr)mechanism is the buckling stress, either
local or distortional, that corresponds to the final failure mechanism. As shown in Figure 5
(fcr)mechanism is not equal to the minimum of (fcr)local and (fcr)dist..
Geometric imperfections are modeled as a superposition of the local and distortional mode. The
magnitude of the imperfection is selected based on the statistical summary provided in Schafer and
Peköz (1998). The error bars in Figure 6 demonstrate the range of strengths predicted for
imperfections varying over the central 50% portion of expected imperfection magnitudes. The
greater the error bars, the greater the imperfection sensitivity. The percent difference in the strength
over the central 50% portion of expected imperfection magnitudes is used as a measure of
imperfection sensitivity:
f u 25%imp. f u 75%imp.
100% .
2 f u 25% imp. f u 75% imp.
1
7
A contour plot of this imperfection sensitivity statistic is shown in Figure 7. Stocky members prone
to failure in the distortional mode have the greatest sensitivity. In general, distortional failures are
more sensitive to initial imperfections than local failures. Areas of imperfection sensitivity risk are
2
1 1.8
1.6 15%
0.8 1.4 10%
P HIGH MEDIUM
1.2
f
5%
f u 0.6 cr local 15%
fy
f
cr dist.
1
0.8
0.4
Winter's Curve
Local Buckling Failures
0.6 LOW
Distortional Buckling Failures 10% 0%
0.4
0.2 error bars indicate the range of strengths
observed between imperfection magnitudes of
25 and 75 % probability of exceedance
0.2 MEDIUM 5%
0 0
0 0.5 1 1.5 2 2.5
0 1 2 3 fy f
f
cr mechanism
fy cr mechanism
Figure 7 Imperfection Sensitivity
Figure 6 Failure Strength
assigned.
INTEGRATING DISTORTIONAL BUCKLING INTO DESIGN
The current North American specification approach for the capacity of a member involves
determining an effective area or section modulus to account for local buckling. The reduction is
based on an empirical correction to the work of von Kármán et al. (1932) completed by Winter
(1947). The extension of this approach to all members of the cross-section is based on the unified
approach of Peköz (1987). The resulting effective section is used to (1) calculate the capacity due
to local buckling alone and (2) determine the reduced section properties for use in global buckling
modes in order to account for interaction between local and global modes.
When considering distortional buckling in design one must consider whether distortional buckling
should be treated in a manner similar to local buckling, global buckling, or in an entirely new way.
If distortional buckling is a separate failure mode it may be treated as such (e.g., the method of
Hancock et al. 1996). If distortional buckling can interact with global modes then an effective
width type of approach that accounts for local and distortional buckling would be appropriate – this
is the method currently suggested. Further, the results of the previous section show a direct
competition between local and distortional buckling that must be accounted for.
If distortional buckling is considered then the critical buckling stress of an element (flange, web or
lip) is no longer solely dependent on local buckling, as is currently assumed in most specifications.
In order to properly integrate distortional buckling, reduced post-buckling capacity in the
distortional mode and the ability of the distortional mode to control the failure mechanism even
when at a higher buckling stress than the local mode must be incorporated. Consider defining the
critical buckling stress of the element as:
f min f
cr cr local , Rd f cr dist .
The slenderness (for an applied stress equal to the yield stress) is:
f y f cr
For strength, if the reduced distortional mode governs, then effective width would be:
beff b where 1 0.22 / /
For Rd < 1 this method provides an additional reduction on the post-buckling capacity. Further, the
method also allows the distortional mode to control in situations when the distortional buckling
stress is greater than the local buckling stress. Thus, Rd provides a framework for solving the
8
problem of predicting the failure mode and reducing the post-buckling capacity in the distortional
mode. The selected form for Rd based on Figure 5 and 6 and the experimental results of Hancock et
al. (1994) is:
117
0.3 where d f y f cr dist . .
.
Rd min1,
d 1
If numerical methods (finite strip analysis) are not used to determine the critical buckling stress in
the local and distortional modes, then the models proposed herein are suggested for use.
The above approach was examined for the strength capacity of laterally braced flexural members.
Experimental data of Cohen (1987), Desmond (1981), Ellifritt et al. (1997), LaBoube and Yu
(1978), Moreyra (1993), Rogers (1995), Schardt and Schrade (1982), Schuster (1992), Shan et al.
(1994), and Willis and Wallace (1990) on laterally braced lipped channel and Z sections was
gathered and examined – see Schafer and Peköz (1999). Using the proposed hand methods for
calculation of the local and distortional buckling stress (for local buckling the interaction model is
used) a test to predicted ratio of 1.07 with a standard deviation of 0.08 was determined for the 190
experiments. In addition to properly accounting for distortional buckling individual cases are
observed where including the local buckling interaction yields markedly better results. For
example, local buckling initiated by long lips (long edge stiffeners) and local buckling with highly
slender webs and compact flanges are examples where including the interaction is observed to
improve the strength prediction markedly.
Currently work is underway to investigate similar approaches for compression members and also to
investigate the possibility of directly using finite strip analysis results on the entire member instead
of the current element by element approach.
CONCLUSIONS
Cold-formed steel members with edge stiffened flanges have three important buckling phenomena:
local, distortional, and global. Current North American specification methods do not explicitly treat
the distortional mode or account for interaction in local buckling. Distortional buckling deserves
special attention because it has the ability to control the final failure mechanism and is also
observed to have lower post-buckling capacity and higher imperfection sensitivity than local
buckling. New hand methods are developed to properly estimate the critical buckling stress in both
the local and distortional mode. A design method for strength prediction, based on the unified
effective width approach, is discussed. The design method uses the new expressions for prediction
of the local and distortional buckling stress. Proper incorporation of the distortional buckling
phenomena is imperative for accurate strength prediction of cold-formed steel members.
ACKNOWLEDGEMENT
The sponsorship of the American Iron and Steel Institute in conducting this research is gratefully
acknowledged.
APPENDIX I. REFERENCES
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American Iron and Steel Institute. Washington, D.C.
Cohen, J. M. (1987). Local Buckling Behavior of Plate Elements, Department of Structural Engineering Report,
Cornell University, Ithaca, New York.
Desmond T.P., Peköz, T. and Winter, G. (1981). "Edge Stiffeners for Thin-Walled Members." Journal of the
Structural Division, ASCE, February 1981.
Elhouar, S., Murray, T.M. (1985) “Adequacy of Proposed AISI Effective Width Specification Provisions for Z- and C-
Purlin Design.” Fears Structural Engineering Laboratory, FSEL/MBMA 85-04, University of Oklahoma,
Norman, Oklahoma.
Ellifritt, D., Glover, B., Hren, J. (1997) “Distortional Buckling of Channels and Zees Not Attached to Sheathing.”
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Hancock, G.J. (1995). “Design for Distortional Buckling of Flexural Members.” Proceedings of the Third International
Conference on Steel and Aluminum Structures, Istanbul, Turkey.
9
Hancock, G.J. (1997). “Design for Distortional Buckling of Flexural Members.” Thin-Walled Structures, 27(1).
Hancock, G.J., Kwon, Y.B., Bernard, E.S. (1994) “Strength Design Curves for Thin-Walled Sections Undergoing
Distortional Buckling.” Journal of Constructional Steel Research, 31(2-3), 169-186.
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Schafer, B.W., Peköz, T.P. (1998). “Computational Modeling of Cold-Formed Steel: Characterizing Geometric
Imperfections and Residual Stresses.” Journal of Constructional Steel Research, 47(3), 193-210.
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ASCE Journal of Structural Engineering, 125(2), 118-127.
Schardt, R. Schrade, W. (1982). “Kaltprofil-Pfetten.” Institut Für Statik, Technische Hochschule Darmstadt, Bericht
Nr. 1, Darmstadt.
Schuster, R.M. (1992). “Testing of Perforated C-Stud Sections in Bending.” Report for the Canadian Sheet Steel
Building Institute, University of Waterloo, Waterloo Ontario.
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the Combination of Bending and Shear.” Civil Engineering Study Structural Series, 94-2, Department of Civil
Engineering, University of Missouri-Rolla, Rolla, Missouri.
von Kármán, T., Sechler, E.E., Donnell, L.H. (1932). “The Strength of Thin Plates In Compression.” Transactions of
the ASME, 54, 53-57.
Willis, C.T., Wallace, B. (1990). “Behavior of Cold-Formed Steel Purlins under Gravity Loading.” Journal of
Structural Engineering, ASCE. 116(8).
Winter, G., (1947) “Strength of Thin Steel Compression Flanges.” Transactions of ASCE, Paper No. 2305, Trans., 112,
1.
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