Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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Section B4.2 Uniformly compressed elements with an
edge stiffener (see Figure 7.9d)
S 1:28
E
f
s
32:77 Eq: B4-1
Case I (w=t 4 S=3) Flange fully effective without
stiffener
Not applicable since
w
t
>
S
3
:
Case II (S=3 < w=t < S) Flange fully effective with
I
s
5 I
a
I
a2
399t
4
w=t
S
ÿ
k
u
4
r
!
3
5:322 10
ÿ3
in:
4
Eq: B4:2-4
n2 0:5
Case III (w=t 5 S) Flange not fully effective
I
a3
t
4
115
w=t
S
5
2:058 10
ÿ3
in:
4
Eq: B4:2-11
n3 0:333
Since
w
t
> S; then
I
a
I
a3
2:058 10
ÿ3
in:
4
n n3 0:333
Calculate buckling coef®cient (k) and stiffener-reduced
effective width (d
s
):
k
a
5:25 ÿ 5
D
w
4:21 > 4:0 Eq: B4:2-8
Chapter 7
218
Hence,
k
a
4:0
C
2
I
s
I
a
0:207 < 1:0 Eq: B4:2-5
C
1
2 ÿ C
2
1:793 Eq: B4:2-6
k C
n
2
k
a
ÿ k
u
k
u
2:543 Eq: B4:2-7
d
s
C
2
d
0
s
0:091 in: Eq: B4:2-9
Section B2.1(a) Effective width of ¯ange element 3 for
strength (see Figure 7.9d)
l
1:052
k
p
w
t
f
E
r
1:129 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:713 Eq: B2:1-3
Since
l > 0:673; then b
f
rw 1:875 in: Eq: B2:1-2
F. Stub Column Strength (P
no
)
Section C4
A
e
tb
w
2b
f
4u 2d
s
0:439 in:
2
P
no
A
e
F
y
19:74 kips Eq: C4-1
(Note that P
n
15:75 kips < P
no
19:74 kips.)
G. Design Axial Strength (P
d
)
LRFD f
c
0:85
P
d
f
c
P
n
13:39 kips
ASD O
c
1:80
P
d
P
n
O
c
8:75 kips
Compression Members
219
REFERENCES
7.1 Hancock, G. J. and Roos, O., Flexural-Torsional Buck-
ling of Storage Rack Columns, Eighth International
Specialty Conference on Cold-Formed Steel Struc-
tures, St Louis, MO, November 1986.
7.2 Rack Manufacturers Institute, Speci®cation for the
Design, Testing and Utilization of Industrial Steel
Storage Racks, Part I, June 1997 edition, Charlotte,
NC.
7.3 Peko
È
z, T. B. and Sumer, O., Design Provision for Cold-
Formed Steel Columns and Beam-Columns, Final
report submitted to AISI, Cornell University, Septem-
ber 1992.
7.4 Popovic, D., Hancock, G. J., and Rasmussen, K. J. R.,
Axial compression tests of DuraGal angles, Journal of
Structural Engineering, ASCE, Vol. 125, No. 5, 1999,
pp. 515±523.
7.5 Rasmussen, K. J. R. and Hancock, G. J., The ¯exural
behaviour of ®xed-ended channel section columns,
Thin-walled Structures, Vol. 17, No. 1, 1993, pp. 45±
63.
7.6 Young, B. and Rasmussen, K. J. R., Compression
Tests of Fixed-ended and Pin-ended Cold-formed
Plain Channels, Research Report R714, School of
Civil and Mining Engineering, University of Sydney,
September 1995.
7.7 Young, B. and Rasmussen, K. J. R., Tests of ®xed-
ended plain channel columns, Journal of Structural
Engineering, ASCE, Vol. 124, No. 2, 1998, pp. 131±
139.
7.8 Young, B. and Rasmussen, K. J. R., Design of lipped
channel columns, Journal of Structural Engineering,
ASCE, Vol. 124, No. 2, 1998, pp. 140±148.
Chapter 7
220
8
Members in Combined Axial Load
and Bending
8.1 COMBINED AXIAL COMPRESSIVE LOAD
AND BENDING: GENERAL
Structural members which are subjected to simultaneous
bending and compression are commonly called beam-
columns. The bending generally results from one of three
sources, depending upon the application of the beam-
column. These sources are shown in Figure 8.1a and are
described as follows:
(i) Eccentric axial load of the type usually encoun-
tered in columns where the line of action of the
axial force is eccentric from the centroid. A
typical application is the columns of industrial
steel storage racks where the beams apply an
eccentric load to the columns as a consequence of
the nature of the connections. In the studs in
steel-framed housing, particular end connections
221
at the top and bottom plates may produce
eccentric loading.
(ii) Distributed transverse loading normally result-
ing from wind forces on sheeting attached to the
structural member. A typical application is the
purlins in the end spans of industrial buildings
where the end wall applies axial load to the
purlin and the sheeting applies a distributed
load. Another application is the stud walls of
steel-framed houses where lateral load from
wind and axial load from upper stories or the
roof can occur simultaneously.
(iii) End moments, shears, and axial forces which
occur in members of rigid jointed structures
such as portal frames, where the members are
rigidly connected together using bolted joints.
The combination of concentric axial force and bending
moment at a cross section can be converted to an equivalent
eccentric axial force as shown in Figure 8.1b. The eccen-
tricity of this axial force de®nes three different situations.
For the singly symmetric section shown in Figure 8.1b,
these are
(i) Eccentric compression with bending in the plane
of symmetry when e
x
6 0 and e
y
0
(ii) Eccentric compression with bending about the
axis of symmetry when e
x
0 and e
y
6 0
(iii) Biaxial bending when e
x
6 0 and e
y
6 0
In the AISI Speci®cation, Section C5 speci®es the accepta-
ble combinations of action effects (force, moment) for
members subject to combined bending and compression.
Section C5 uses the design capacity for beams given in
Section C3 as well as those for columns given in Section C4.
The resistance factors (f
b
; f
c
) and safety factors (O
b
; O
c
) for
bending and compression remain the same as when the
bending or compression is considered alone.
Chapter 8
222
8.2 INTERACTION EQUATIONS FOR
COMBINED AXIAL COMPRESSIVE LOAD
AND BENDING
The rules governing combined actions are based on the use
of linear interaction equations expressed in terms of axial
force and moment. Two sets of equations are provided in
the AISI Speci®cation, one set for ASD and one set for
FIGURE 8.1 Combined compression and bending: (a) member
loads; (b) cross-section loads.
Members in Combined Axial Load and Bending
223
LRFD. The set for ASD is given in Section C5.2.1 of the
AISI Speci®cation as
When O
c
P=P
n
4 0:15,
O
c
P
P
no
O
b
M
x
M
nx
O
b
M
y
M
ny
4 1:0 8:1
When O
c
P=P
n
> 0:15,
a
O
c
P
P
n
O
b
C
mx
M
x
M
nx
a
x
O
b
C
my
M
y
M
ny
a
y
4 1:0 8:2
b
O
c
P
P
no
O
b
M
x
M
nx
O
b
M
y
M
ny
4 1:0 8:3
where P nominal axial load
M
x
; M
y
nominal moments with respect to the centroidal
axes of the effective section determined for the
required axial strength alone
P
n
nominal axial strength determined in accordance with
Section C4
P
no
nominal axial strength determined in accordance
with Section C4 with F
n
F
y
(stub column strength)
M
nx
; M
ny
nominal ¯exural strengths about the centroidal
axes determined in accordance with Section C3
a
x
; a
y
moment ampli®cation factors as discussed below
C
mx
; C
my
coef®cients for unequal end moments as
discussed below
The set for LRFD is given in Section C5.2.2.
When P
u
=f
c
P
n
4 0:15,
P
u
f
c
P
n
M
ux
f
b
M
nx
M
uy
f
b
M
ny
4 1:0 8:4
Chapter 8
224
When P
u
=f
c
P
n
> 0:15,
a
P
u
f
c
P
n
C
mx
M
ux
f
b
M
nx
a
x
C
my
M
uy
f
b
M
ny
a
y
4 1:0 8:5
b
P
u
f
c
P
no
M
ux
f
b
M
nx
M
uy
f
b
M
ny
4 1:0 8:6
where P
u
required axial strength
M
ux
; M
uy
required ¯exural strengths with respect to the
centroidal axes of the effective section deter-
mined for the required axial strength alone
P
n
nominal axial strength determined in accordance with
Section C4
P
no
nominal axial strength determined in accordance
with Section C4 with F
n
F
y
(stub column strength)
M
bx
; M
by
nominal ¯exural strength about the centroidal
x- and y-axes, respectively, determined in accor-
dance with Section C3
C
mx
; C
my
coef®cients for unequal end moments as
discussed below
a
x
; a
y
moment ampli®cation factors as discussed below
The linear interaction equations expressed as Eqs. (8.1) to
(8.3) for ASD or Eqs. (8.4) to (8.6) for LRFD allow for the
limit states of in-plane section strength failure, in-plane
member strength including in-plane buckling, and out-of-
plane member strength including lateral buckling.
A detailed discussion of these limit states is given in
Ref. 8.1.
The coef®cients C
mx
; C
my
allow for unequal end
moments in beam-columns. They are set out for members
free to sway in Figure 8.2a and members braced against
sway in Figure 8.2b.
Members in Combined Axial Load and Bending
225
The moment ampli®cation factors for ASD are de®ned
by
a
x
1 ÿ
O
c
P
P
Ex
8:7
a
y
1 ÿ
O
c
P
P
Ey
8:8
where
P
Ex
p
2
EI
x
K
x
L
x
2
8:9
FIGURE 8.2 C
m
factors: (a) free to sway; (b) braced against sway.
Chapter 8
226
and
P
Ey
p
2
EI
y
K
y
L
y
2
8:10
The moment ampli®cation factors for the LRFD method are
de®ned by
a
x
1 ÿ
P
u
P
Ex
8:11
a
y
1 ÿ
P
u
P
Ey
8:12
where P
Ex
, P
Ey
are given by Eqs. (8.9) and (8.10).
The values of I
x
, I
y
are the second moments of area of
the full, unreduced cross section about the x-, y-axis,
respectively. The values of K
x
L
x
, K
y
L
y
are the effective
lengths for buckling about the x-, y-axis, respectively.
They may be greater than the member length for members
in sway frames. The moment ampli®cation factors (a
x
; a
y
)
increase the ®rst-order elastic moments (M
x
; M
y
) to second-
order elastic moments (M
x
=a
x
; M
y
=a
y
). The values of C
mx
=a
nx
and C
my
=a
ny
should never be less than 1.0 since the second-
order moment must always be greater than or equal to the
®rst-order moment.
8.3 SINGLY SYMMETRIC SECTIONS UNDER
COMBINED AXIAL COMPRESSIVE LOAD
AND BENDING
The behavior of singly symmetric sections depends on
whether the sections are bent in the plane of symmetry,
about the axis of symmetry, or both.
8.3.1 Sections Bent in a Plane of Symmetry
Singly symmetric sections bent in the plane of symmetry
may fail by either of two ways:
Members in Combined Axial Load and Bending
227