Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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Section C4.2
F
e1
s
ey
28:46 ksi
F
e2
s
ex
s
t
ÿ
s
ex
s
t
2
ÿ 41 ÿx
0
=r
0
2
s
ex
s
t
q
21 ÿx
0
=r
0
2
35:60 ksi Eq: C4:2-1
F
e
lesser of F
e1
and F
e2
28:46 ksi
l
c
F
y
F
e
s
1:125 Eq: C4-4
Since
l
c
< 1:5; then F
n
0:658
l
2
c
F
y
21:2 ksi
Eq: C4-2
C. Effective Area at Critical Stress (F
n
)
Section B2 applied to web of channel
k 4 f F
n
l
1:052
k
p
h
t
f
E
r
0:62
Eq: B2:1-4
Since
l < 0:673; then b
w
h 5:5in: Eq: B2:1-1
Section B3.1 applied to ¯anges of channel
k 0:43
l
1:052
k
p
w
t
f
E
r
0:602
Eq: B2:1-4
Since
l < 0:673; then b
f
w 1:75 in: Eq: B2:1-1
Chapter 7
208
D. Nominal Axial Strength (P
n
)
Section C4
A
e
tb
w
2b
f
2u1:199 in:
2
P
n
A
e
F
n
25:41 kips
Eq: C4-1
Centroid of Effective Section Under Axial Force Alone
x
cea
t
2
b
w
2ut R ÿ c2b
f
t R
b
f
2
t
A
e
0:455 in:
E. Effective Area at Yield Stress (F
y
)
Section B2 applied to web of channel
k 4 f F
y
l
1:052
k
p
h
t
f
E
r
0:808 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:90 Eq: B2:1-3
Since
l > 0:673; then b
w
rh 4:952 in: Eq: B2:1-2
Section B3.1 applied to ¯anges of channel
k 0:43
l
1:052
k
p
w
t
f
E
r
0:785 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:917 Eq: B2:1-3
Since
l > 0:673; then b
f
rw 1:605 in: Eq: B2:1-2
Compression Members
209
F. Nominal Section Stub Column Strength (P
no
)
Note that P
no
is required for Example 8.5.1, Combined
Loading.
Section C4
A
e
tb
w
2b
f
2u1:094 in:
2
P
no
A
e
F
y
39:38 kips Eq: C4-1
(Note that P
n
25:41 kips < P
no
39:38 kips.)
G. Design Axial Strength (P
d
)
LRFD f
c
0:85
P
d
f
c
P
n
21:60 kips
ASD O
c
1:80
P
d
P
n
O
c
14:34 kips
7.6.3 Lipped Channel Column
Problem
Determine the nominal axial strength (P
n
) for the channel
section of length 80 in. shown in Figure 7.9, assuming the
channel is loaded concentrically through the centroid of the
effective section and the effective lengths in ¯exure and
torsion are based on lateral and torsional restraints in the
plane of symmetry at mid-height. The nominal yield stress
(F
y
) is 45 ksi. The following section numbers refer to the
AISI Speci®cation.
Solution
K
x
L
x
80 in: K
y
L
y
40 in: K
t
L
t
40 in: F
y
45 ksi
B 3in: H 4in: D 0:625 in: R 0:125 in:
t 0:060 in:
Chapter 7
210
A. Section Properties of Full Section
Corner radius r R
t
2
0:155 in:
Length of arc u 1:57 r 0:243 in:
Distance of centroid c 0:637 r 0:099 in:
from center of radius
FIGURE 7.9 Lipped channel section: (a) actual section; (b) idea-
lised section; (c) line element model; (d) effective widths.
Compression Members
211
w B ÿ 2R t2:63 in:
h H ÿ 2R t3:63 in:
d D ÿR t0:44 in:
A t2w h 4u 2d0:645 in:
2
I
x
2t 2u
h
2
c
2
w
H
2
ÿ
t
2
2
"
d
H
2
ÿR tÿ
d
2
2
#
t
h
3
12
1:812 in:
4
x
c
t
2
h 2ut R ÿ c2uB ÿ R ÿ t c
2w
B
2
2dBÿ
t
2
t
A
1:124 in:
I
y
t
2
2
h 2ut R ÿ c
2
"
2uB ÿ R ÿ t c
2
2w
B
2
2
2dBÿ
t
2
2
#
t
2t
w
3
12
ÿ Ax
2
c
0:792 in:
4
J
At
2
3
0:774 10
ÿ3
in:
4
Shear Center and Warping Constant
a H ÿ t 3:94 in: b B ÿ t 2:94 in:
c D ÿ
t
2
0:595 in:
m
s
b3a
2
b c6a
2
ÿ 8c
2
a
3
6a
2
b c8c
2
ÿ 12ac 6a
2
1:493 in:
x
0
ÿ m
s
ÿ
t
2
ÿ x
c
ÿ2:587 in:
Chapter 7
212
C
w
a
2
b
2
t
12
2a
3
b 3a
2
b
2
48c
4
112bc
3
8ac
3
48abc
2
12a
2
c
2
12a
2
bc 6a
3
c
6a
2
b a 2c
3
ÿ 24ac
2
8
>
<
>
:
9
>
=
>
;
2:951 in:
6
r
y
I
y
A
r
1:109 in: r
x
I
x
A
r
1:677 in:
r
0
I
x
I
y
A
x
2
0
r
3:28 in: Eq: C3:1:2-13
B. Critical Stress (F
n
) According to Section C4
Use Section C3.1.2 to compute s
ex
, s
ey
, and s
t
:
s
ex
p
2
E
K
x
L
x
=r
x
2
127:9 ksi Eq: C3:1:2-8
s
ey
p
2
E
K
y
L
y
=r
y
2
223:6 ksi Eq: C3:1:2-9
s
t
GJ1 p
2
EC
w
=GJK
t
L
t
2
Ar
2
0
78:89 ksi
Eq: C3:1:2-10
Section C4.2
F
e1
s
ey
223:6 ksi
F
e2
s
ex
s
t
ÿ
s
ex
s
t
2
ÿ 41 ÿx
0
=r
0
2
s
ex
s
t
q
21 ÿx
0
=r
0
2
54:13 ksi Eq: C4:2-1
F
e
lesser of F
e
1
and F
e2
54:13 ksi
l
c
F
y
F
e
s
0:912 Eq: C4-4
Since l
c
< 1:5, then
F
n
0:658
l
2
c
F
y
31:78 ksi Eq: C4-2
Compression Members
213
C. Effective Area at Critical Stress (F
n
) as Shown in
Figure 7.9d
Section B2 applied to web of channel
k 4 f F
n
l
1:052
k
p
h
t
f
E
r
1:044 Eq: B2:1-4
r
1 ÿ 0:22
l
0:756 Eq: B2:1-3
Since l > 0:673, then
b
w
rh 2:744 in: Eq: B2:1-2
Section B3.1 applied to lips of channel
d 0:44 in: k
u
0:43
l
1:052
k
u
p
d
t
f
E
r
0:386 Eq: B2:1-4
Since l < 0:673, then
d
0
s
d 0:44 in: Eq: B2:1-1
I
s
d
3
t
12
0:426 10
ÿ3
in:
4
Element 3 Flange ¯at
w 2:63 in:
w
t
43:83 this value must not exceed 60
Section B1:1a
Section B4.2 Uniformly compressed elements with an
edge stiffener (see Figure 7.9d)
S 1:28
E
f
s
39:00 Eq: B4-1
Chapter 7
214
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
2:608 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
1:740 10
ÿ3
in:
4
Eq: B4:2-11
n3 0:333
Since
w
t
5 S; then I
a
I
a3
1:740 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
Hence
k
a
4:0
C
2
I
s
I
a
0:245 < 1:0 Eq: B4:2-5
C
1
2 ÿ C
2
1:755 Eq: B4:2-6
k C
n
2
k
a
ÿ k
u
k
u
2:664 Eq: B4:2-7
d
s
C
2
d
0
s
0:108 in:
Compression Members
215
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
0:927 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:823 Eq: B2:1-3
Since l > 0:673, then
b
f
rw 2:164 in: Eq: B2:1-2
D. Nominal Axial Strength (P
n
)
Section C4
A
e
tb
w
2b
f
4u 2d
s
0:496 in:
2
P
n
A
e
F
n
15:75 kips
Eq: C4-1
Centroid of effective section under axial force alone
x
1
t
2
b
w
2ut R ÿ c2uB ÿ R ÿ t c
1:542 in:
2
x
2
2C
1
b
f
2
R t C
1
b
f
4
2C
2
b
f
2
R t w ÿ C
2
b
f
4
2d
s
B ÿ
t
2
6:368 in:
2
x
cea
x
1
x
2
t
A
e
0:958 in:
Chapter 7
216
E. Effective Area at Yield Stress (F
y
)
Section B2 applied to web of channel
k 4 f F
y
l
1:052
k
p
h
t
f
E
r
1:243 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:662 Eq: B2:1-3
Since l > 0:673, then
b
w
rw 2:404 in: Eq: B2:1-2
Section B3.1 applied to lips of channel
d 0:44 in: k
u
0:43
l
1:052
k
u
p
d
t
f
E
r
0:459 Eq: B2:1-4
Since l < 0:673, then
d
0
s
d 0:44 in: Eq: B2:1-1
I
s
d
3
t
12
0:426 10
ÿ3
in:
4
Element 3 Flange ¯at
w 2:63 in:
w
t
43:83 this value must not exceed 60
Clause B1:1a
Compression Members
217