Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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Shear, Progress Report No. 1 of Phase I of
CSSBI=IRAP Project, Department of Civil Engineer-
ing, University of Waterloo, Waterloo, Canada, 1995.
6.13 Eiler, M. R., LaBoube, R. A., and Yu, W. W., Behavior of
Web Elements with Openings Subjected to Linearly
Varying Shear, Final Report, Civil Engineering Series
97-5, Cold-Formed Steel Series, Department of Civil
Engineering, University of Missouri-Rolla, 1997.
6.14 Langan, J. E., LaBoube, R. A., and Yu, W. W., Struc-
tural Behavior of Perforated Web Elements of Cold-
Formed Steel Flexural Members Subjected to Web
Crippling and a Combination of Web Crippling and
Bending, Final Report, Civil Engineering Series, 94-
3, Cold-Formed Steel Series, Department of Civil
Engineering, University of Missouri-Rolla, 1994.
6.15 Uphoff, C. A., Structural Behavior of Circular Holes
in Web Elements of Cold-Formed Steel Flexural
Members Subjected to Web Crippling for End-One-
Flange Loading, thesis presented to the faculty of the
University of Missouri-Rolla in partial ful®llment for
the degree Master of Science, 1996.
6.16 Deshmukh, S. U., Behavior of Cold-Formed Steel Web
Elements with Web Openings Subjected to Web Crip-
pling and a Combination of Bending and Web Crip-
pling for Interior-One-Flange Loading, thesis
presented to the faculty of the University of
Missouri-Rolla in partial ful®llment for the degree
Master of Science, 1996.
Chapter 6
188
7
Compression Members
7.1 GENERAL
The design of cold-formed members in compression is
generally more complex than conventional hot-rolled steel
members. This is a result of the additional modes of buck-
ling deformation which commonly occur in thin-walled
structural members. These are
(a) Local buckling and post±local buckling of stif-
fened and unstiffened compression elements, as
described in Chapter 4
(b) Flexural, torsional, and torsional-¯exural modes
of buckling of the whole compression member as
shown in Figure 1.13b.
The design for (a) determines the nominal stub column
axial strength (P
no
) described in Section 7.3. The design for
(b) gives the nominal axial strength (P
n
) given in Section
7.4. The nominal axial strength is de®ned in Section C4,
Concentrically Loaded Compression Members, of the AISI
Speci®cation. The elastic buckling stress for ¯exural,
189
torsional, and torsional-¯exural buckling of members in
compression is discussed in Section 7.2.
Section C4 of the AISI Speci®cation includes rules
which enable the designer to account for these additional
buckling modes as well as the normal failure modes of
¯exural buckling and yielding commonly considered for
doubly symmetric hot-rolled compression members as
described in the AISC Speci®cation (Ref. 1.1).
7.2 ELASTIC MEMBER BUCKLING
The elastic buckling load for ¯exural, torsional, and
torsional-¯exural buckling of a thin-walled member of
general cross section, as shown in Figure 7.1a, was derived
by Timoshenko and is explained in detail in Timoshenko
and Gere (Ref. 3.6) and Trahair (Ref. 5.2). For a member of
length L subjected to uniform compression and restrained
at its ends by simple supports which prevent lateral dis-
placement of the section perpendicular to its longitudinal
axis, as well as twisting rotation, the elastic critical load
(P
e
) is given by solution of the following three simultaneous
equations in the displacement amplitudes A
1
, A
2
in the x-,
y-directions, respectively, and twist angle A
3
as follows:
P
ey
ÿ P
e
A
1
ÿ P
e
y
0
A
3
0 7:1
P
ex
ÿ P
e
A
2
P
e
x
0
A
3
0 7:2
ÿP
e
y
0
A
1
P
e
x
0
A
2
r
2
0
P
t
ÿ P
e
A
3
0 7:3
where
P
ex
p
2
EI
x
L
2
7:4
P
ey
p
2
EI
y
L
2
7:5
P
t
GJ
r
2
0
1
p
2
EC
w
GJL
2
7:6
r
2
0
I
x
I
y
A
x
2
0
y
2
0
7:7
Chapter 7
190
The terms x
0
and y
0
are the shear center coordinates shown
in Figure 7.1a. The modes of buckling for ¯exure about both
principal axes and twist angle are sinusoidal with ampli-
tudes of A
1
, A
2
, and A
3
, respectively. The buckling stresses
FIGURE 7.1 Thin-walled sections: (a) general cross-section;
(b) section types.
Compression Members
191
s
ex
, s
ey
, and s
t
corresponding to P
ex
, P
ey
, and P
t
, respec-
tively, are simply derived from Eqs. (7.4), (7.5) and (7.6) by
dividing by the gross area.
Two solutions to Eqs. (7.1)±(7.3) exist. Either the
column does not buckle and A
1
A
2
A
3
0, or the
column buckles and the determinant of the matrix of
coef®cients of A
1
, A
2
, and A
3
in Eqs. (7.1)±(7.3) is zero. In
this case the resulting cubic equation to be solved for the
critical load (P
e
)is
P
3
e
r
2
0
ÿ x
2
0
ÿ y
2
0
ÿP
2
e
P
ex
P
ey
P
t
r
2
0
ÿP
ey
x
2
0
P
ex
y
2
0
P
e
r
2
0
P
ex
P
ey
P
ey
P
t
P
t
P
ex
ÿ P
ex
P
ey
P
t
r
2
0
0 7:8
For general nonsymmetric sections of the type shown in
Figure 7.1b, the solution of Eq. (7.8) in its general form is
necessary. Section C4.3, Nonsymmetric Sections, of the
AISI Speci®cation allows the elastic buckling stress of a
nonsymmetric section to be determined by a rational analy-
sis, such as the solution of Eq. (7.8), or by testing.
For singly symmetric sections of the type shown in
Figure 7.1b, for which x
0
or y
0
is zero, then simpler solu-
tions exist. In the case of the x-axis as the axis of symmetry,
then y
0
is zero; hence
P
e
1
P
ey
7:9
P
e
2;3
P
ext
P
ex
P
t
P
ex
P
t
2
ÿ 4P
ex
P
t
1 ÿx
0
=r
0
2
q
21 ÿx
0
=r
0
2
7:10
Equation (7.9) gives the ¯exural buckling load about the
y-axis, and the smaller of the two values computed using
Eq. (7.10) gives the torsional-¯exural buckling load.
Chapter 7
192
Equation (7.9) for the unlipped channel section in
Figure 3.2 produces identical results with the curve
through C in Figure 3.3 for column lengths greater than
40 in. where local buckling effects have no in¯uence. Simi-
larly, evaluation of Eq. (7.10) for the channel section in
Figure 3.2 produces identical results with the curve
through D in Figure 3.3 for column lengths greater than
40 in. In Figures 3.6, 3.7, and 3.9, the curves shown as the
dashed line labeled ``Timoshenko ¯exural-torsional buck-
ling formula'' were calculated using the lower root of Eq.
(7.10).
For doubly symmetric or point symmetric sections
where x
0
and y
0
are both zero, the three solutions of Eq.
(7.8) are simply
P
e
1
P
ex
7:11
P
e
2
P
ey
7:12
P
e
3
P
t
7:13
For doubly symmetric sections, closed cross sections, and
any other sections that can be shown not to be subject to
torsional or torsional-¯exural buckling, Section C4.1 of the
AISI Speci®cation uses the lesser P
e
derived from Eqs.
(7.11) and (7.12) divided by the gross area (A) to give the
¯exural buckling stress (F
e
) (Eq. C4.1-1 of the AISI Speci-
®cation). For singly symmetric sections, where torsional-
¯exural buckling can also occur, Section C4.2 of the AISI
Speci®cation uses P
e
derived from Eq. (7.10) divided by A to
give F
e
. The lesser value of F
e
from Section C4.1 and from
Section C4.2 must be used for the design of singly
symmetric sections. For point symmetric sections, the
AISI Speci®cation uses the lesser of P
t
derived from Eq.
(7.13) divided by the gross area A and F
e
from Section C4.1
for ¯exural buckling alone.
In Eqs. (7.4)±(7.6), L is the unbraced length between
the simply supported ends of the column. However, in Eqs.
(7.4)±(7.6) L has been replaced by K
x
L
x
, K
y
L
y
, and K
t
L
t
,
Compression Members
193
respectively, in Section C3.1.2 of the AISI Speci®cation.
These terms are the effective lengths about the x-, y-, and z-
axes, respectively, as de®ned in Section C3.1.2. The use of
different lengths in the computation of the torsional-¯ex-
ural buckling load by Eqs. (7.8) or (7.10) is not theoretically
justi®able. However, it usually produces a conservative
estimate of the torsional-¯exural buckling load of columns
which have different ¯exural and torsional effective
lengths. This procedure has been justi®ed experimentally
for the uprights of steel storage rack columns as described
in Ref. 7.1.
In Section 6 of the Rack Manufacturers Institute
Speci®cation (Ref. 7.2), effective length factors are speci®ed
for ¯exural buckling in the direction perpendicular to the
upright frame (typically K
x
1:7 for racking not braced
against sidesway), for ¯exural buckling in the plane of the
upright frame (typically K
y
1:0), and for torsional buck-
ling (typically K
t
0:8), provided twisting of the upright is
prevented at the brace points.
7.3 STUB COLUMN AXIAL STRENGTH
The strength of short-length columns compressed between
rigid end platens (P
no
) is governed mainly by the yield
strength of the material and the slenderness of the plate
elements forming the cross section. For sections with stocky
plate elements, the strength is simply equal to the squash
load given by the product of the full section area (A) and the
yield stress (F
y
). However, for sections with slender plate
elements, the section capacity in compression (often called
the stub column strength) is less than the squash load as a
result of local buckling.
In Section C4 of the AISI Speci®cation, the stub
column strength is de®ned by
P
no
A
e
F
y
7:14
where A
e
is the effective area of the section at the yield
stress F
y
computed by summing the effective areas of all
Chapter 7
194
the individual elements forming the cross section, as shown
for the unlipped channel section in Figure 7.2. The effective
areas of the individual elements forming the cross section
are computed from the effective width formula [Eq. (4.9)]
for each element in isolation, ignoring any interaction
which may occur between different elements of the cross
section. This approach appears to produce reasonable esti-
mates of the stub column axial strength of thin-walled
sections.
7.4 LONG COLUMN AXIAL STRENGTH
The column design curve in the AISI Speci®cation is based
upon the American Institute of Steel Construction LRFD
Speci®cation curve (Ref. 1.1). The formula for the critical
stress (F
n
) on a column is given in Section C4 of the AISI
Speci®cation as
F
n
0:658
l
2
c
F
y
for l
c
4 1:5 7:15
F
n
0:877
l
2
c
F
y
for l
c
> 1:5 7:16
FIGURE 7.2 Effective section for post±locally buckled channel.
Compression Members
195
where
l
c
F
y
F
e
s
7:17
The elastic buckling stress (F
e
) is the least of the ¯exural,
torsional, or torsional-¯exural buckling stresses deter-
mined in accordance with Sections C4.1 to C4.3 of the
AISI Speci®cation and discussed in Section 7.2 of this
book. The resulting column design curve is shown in
Figure 7.3, where it is compared with the 1991 AISI
Speci®cation column design curve. The design curve was
lowered in the 1996 Speci®cation to allow for the effect of
geometric imperfection (Ref. 7.3).
To compute the nominal axial strength (P
n
) the critical
stress (F
n
) is multiplied by the effective area (A
e
):
P
n
A
e
F
n
7:18
FIGURE 7.3 Compression member design curves.
Chapter 7
196
The area A
e
is computed at the stress F
n
and not the yield
stress as for the nominal stub column strength given by Eq.
(7.14). The use of the effective area computed at F
n
rather
than F
y
allows for the fact that the section may not be fully
stressed when the critical load in compression is reached
FIGURE 7.4 Pin-ended angle tests (Popovic, Hancock, and
Rasmussen): (a) 0.1 in. thick; (b) 0.15 in. thick.
Compression Members
197