Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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FIGURE 3.8 Flange distortional buckle of lipped channel
columnÐSection 2.
Chapter 3
68
the additional lip stiffeners will be detrimental to the
strength of the section where torsional-¯exural buckling
controls the design. Additional information on the distor-
tional mode of buckling, including design formulas, is given
in Refs. 3.7, 3.8, and 3.9.
3.2.3 Lipped Channel (Fixed Ended)
The spline ®nite strip method of buckling analysis (Ref. 3.4)
uses spline functions in the longitudinal direction in place
of the single half-sine wave over the length of the section as
demonstrated in Figure 3.1b. The advantage of the spline
®nite strip analysis is that boundary conditions other than
simple supports can be investigated. However, the spline
®nite strip analysis uses a considerably greater number of
degrees of freedom and takes much longer to solve a given
problem than the semianalytical ®nite strip method.
To investigate the effect of ®xed-ended boundary
conditions on the distortional buckling of a simple lipped
FIGURE 3.9 Section 3ÐBuckling stress versus half-wavelength
for concentric compression.
Buckling Modes of Thin-Walled Members
69
channel, both types of ®nite strip analysis were performed
for the channel section in Figure 3.10. The results of
buckling analyses of the lipped channel section are shown
in Figure 3.10, where BFINST denotes the semianalytical
®nite strip analysis and BFPLATE denotes the spline ®nite
strip analysis. The BFINST analysis gives the elastic local
s
le
and elastic distortional s
de
buckling stresses at given
half-wavelengths, whereas the BFPLATE analysis gives
the actual buckling stress of a given length of section
between ends that may be ®xed. The abscissa in Figure
3.10 for the BFINST analysis indicates one half-wave
whereas for the BFPLATE analysis the abscissa indicates
the real column length of the section. The different marks
indicate local (L), distortional (D), and overall buckling
(FT), respectively. It can be observed in Figure 3.10 that
where the sections buckle in multiple distortional half-
waves the change in the buckling stress in the distor-
tional mode with increasing section length reduces as
the effect of the end conditions reduces, and eventually
FIGURE 3.10 Finite strip buckling analyses.
Chapter 3
70
approaches the value produced by the semianalytical
®nite strip method.
3.3 PURLIN SECTION STUDY
3.3.1 Channel Section
To demonstrate the different ways in which a channel
purlin may buckle when subjected to major axis bending
moment as shown in Figure 3.11, a ®nite strip buckling
analysis of a channel purlin of depth 6 in., ¯ange width
2.5 in., lip size 0.55 in., and plate thickness 0.060 in. was
performed and the results are shown in Figure 3.12. This
graph is similar to that for the lip-stiffened channel in
Figure 3.6 except for the shape of the buckling modes and
their nomenclature.
FIGURE 3.11 Channel section purlin subjected to major axis
bending moment.
Buckling Modes of Thin-Walled Members
71
FIGURE 3.12 Channel section purlin buckling stress versus half-
wavelength for major axis bending.
FIGURE 3.13 Flange distortional buckle of Z-section purlin.
Chapter 3
72
The ®rst minimum point (A) is again a local buckling
mode, which now involves only the web, the compression
¯ange, and its lip stiffener. The second minimum point (B)
is associated with a mode of buckling where the compres-
sion ¯ange and lip rotate about the ¯ange web junction
with some elastic restraint to rotation provided by the web.
This mode of buckling is called a ¯ange distortional buck-
ling mode and is shown in a test specimen of a Z-section in
Figure 3.13. The value of the buckling stress is highly
dependent upon the size of the lip stiffener.
At long wavelengths (point C) where the purlin is
unrestrained, a torsional-¯exural buckle occurs which is
often called a lateral buckle. However, if the tension ¯ange
is subjected to a torsional restraint such as may be provided
by sheeting screw fastened to the tension ¯ange, then a
lateral distortional buckle will occur at a minimum half-
wavelength of approximately 160 in., as shown at point D
in Figure 3.12. The value of the minimum buckling stress
and its half-wavelength depend upon the degree of
torsional restraint provided to the tension ¯ange.
3.3.2 Z-Section
A similar study to that of the channel section in bending
was performed for the two Z-sections in Figure 3.14. The
®rst section contains a lip stiffener perpendicular to the
¯ange, and the second has a lip stiffener located at an angle
of 45
to the ¯ange. In Figure 3.14, the buckling stresses
have been computed for buckle half-wavelengths up to
40 in., so only the local and distortional buckling modes
have been investigated.
As for the channel section study described above, the
distortional buckling stresses are signi®cantly lower than
the local buckling stresses for both Z-sections. For the lip
stiffener turned at 45
to the ¯ange, the distortional buckling
stress is reduced by 19% compared with the value for the lip
Buckling Modes of Thin-Walled Members
73
stiffener perpendicular to the ¯ange, thus indicating a
potential failure mode of purlins with sloping lip stiffeners.
3.4 HOLLOW FLANGE BEAM IN BENDING
The hollow ¯ange beam section in Figure 3.15 was investi-
gated by a semianalytical ®nite strip buckling analysis. Two
sections have been analyzed to demonstrate the effect of
the electric resistance weld (ERW) on the section buckling
behavior. These are the section with closed ¯anges (HBS1)
and the section with open ¯anges (HBS2). Figure 3.15
shows graphs of buckling stress versus buckle half-wave-
lengths for the two sections subjected to pure bending about
their major principal axes so that their top ¯anges are in
compression and their bottom ¯anges are in tension as in a
conventional beam. The buckling stress is the value of the
stress in the compression ¯ange farthest away from the
bending axis when the section undergoes elastic buckling.
FIGURE 3.14 Z-section purlinÐbuckling stress versus half-wave-
length for bending about a horizontal axis.
Chapter 3
74
At short half-wavelengths (2±10 in. in Figure 3.15),
the effect of welding the ¯ange to the abutting web clearly
demonstrates the changed buckling mode from local buck-
ling in the unattached ¯ange for HBS2 to local buckling of
the top ¯ange at a higher stress for HBS1.
At long half-wavelengths (80±400 in. in Figure 3.15),
the increased torsional rigidity of the ¯anges has in-
creased the buckling stress, with the increase exceeding
100% for lengths greater than 200 in. The mode of buckling
at a half-wavelength of 200 in. for the open section (HBS2)
is a conventional torsional-¯exural buckle, of the type
described in Ref. 3.6. The torsional-¯exural buckling
mode of the open HBS2 section involves longitudinal
displacements of the cross-section (called warping displace-
ments) such that the longitudinal displacements at the free
edges of the strips are different from the longitudinal
displacements of the web at the points where the free
edges abut the web.
FIGURE 3.15 Hollow ¯ange beamÐbuckling stress versus half-
wavelength for major axis bending.
Buckling Modes of Thin-Walled Members
75
The mode of buckling at 200 in. for the section with
closed ¯anges (HBS1) shows a new type of buckling mode
not previously described for sections of this type. It involves
lateral bending of the two ¯anges, one more than the other,
with the ¯anges substantially untwisted as a result of their
increased torsional rigidity. The web is distorted as a result
of the relative movement of the ¯anges. The mode is called
lateral distortional buckling and was ®rst reported for
sections of this type in Ref. 3.10. It has a substantially
increased buckling stress value over that of the torsional-
¯exural buckling of the open section HBS2. It is not valid to
compute the torsional-¯exural buckling capacity for HBS1
using Ref. 3.6 because this would produce erroneous
results, as shown by the dashed line in Figure 3.15. The
differential warping displacements where the free edges of
the ¯ange abut the web in the open HBS2 section are
eliminated by the welds in the closed HBS1 section. Conse-
quently, the enhanced lateral distortional buckling stress of
the HBS1 section is obtained entirely as a result of the
welds. An analytical formula for computing lateral distor-
tional buckling is given in Ref. 3.11.
REFERENCES
3.1 Cheung, Y. K., Finite Strip Method in Structural
Analysis, Pergamon Press, New York, 1976.
3.2 Przmieniecki, J. S. D., Finite element structural
analysis of local instability, Journal of the American
Institute of Aeronautics and Astronautics, Vol. 11, No.
1, 1973.
3.3 Plank, R. J. and Wittrick, W. H., Buckling under
combined loading of thin, ¯at-walled structures by a
complex ®nite strip method, International Journal
For Numerical Methods in Engineering, Vol. 8, No.
2, 1974, pp. 323±329.
3.4 Lau, S. C. R. and Hancock, G. J., Buckling of thin ¯at-
walled structures by a spline ®nite strip method,
Chapter 3
76
Thin-Walled Structures, 1986, Vol. 4, No. 4, pp. 269±
294.
3.5 Hancock, G. J., Local, distortional and lateral buck-
ling of I-beams, Journal of the Structural Division,
ASCE, Vol. 104, No. ST11, Nov. 1978, pp. 1787±1798.
3.6 Timoshenko, S. P. and Gere, J. M., Theory of Elastic
Stability, McGraw-Hill, New York, 1959.
3.7 Hancock, G. J., Distortional buckling of steel storage
rack columns, Journal of Structural Engineering,
ASCE, Vol. 111, No. 12, Dec. 1985.
3.8 Lau, S. C. W. and Hancock, G. J., Distortional buck-
ling formulas for channel columns, Journal of Struc-
tural Engineering, ASCE, Vol. 113, No. 5, May 1987.
3.9 Lau, S. C. W. and Hancock, G. J. Distortional Buck-
ling Tests of Cold-Formed Channel Sections, Proceed-
ings, Ninth International Specialty Conference on
Cold-Formed Steel Structures, St. Louis, Missouri,
Nov. 1988.
3.10 Centre for Advanced Structural Engineering, Distor-
tional Buckling of Hollow Flange Beam Sections,
Investigation Report S704, University of Sydney,
Feb. 1989.
3.11 Pi, Y. K. and Trahair, N. S., Lateral-distortional
buckling of hollow ¯ange beams, Journal of Struc-
tural Engineering, ASCE, Vol. 123, No. 6, 1997, pp.
695±702.
Buckling Modes of Thin-Walled Members
77