Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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Effective Distance from
Element length, top ®ber,
no., L
i
y
i
L
i
y
i
L
i
y
2
i
i (in.) (in.) (in.
2
) (in.
3
)
1 1.130 3.532 3.992 14.101
2 0.973 3.914 3.809 14.910
3 5.260 3.970 20.882 82.902
4 7.260 2.000 14.520 29.040
5 2.575 0.030 0.077 0.002
6 0.487 0.086 0.042 0.004
P
6
i1
L
i
17:685 in:
P
6
i1
L
i
y
i
43:323 in:
2
P
6
i1
L
i
y
i
2
140:959 in:
3
Determine distance y
cg
of neutral axis from top ®ber
y
cg
P
6
i1
L
i
y
i
P
6
i1
L
i
2:450 in:
I
Ly
2
P
6
i1
L
i
y
2
cg
106:127 in:
3
I
0
x
P
6
i1
L
i
y
2
i
I
0
1
I
0
4
ÿ I
Ly
2
42:834 in:
3
I
x
I
0
x
t 2:570 in:
4
B. Check Web (Section B2.3)
f
1
y
cg
ÿR t
f
y
cg
46:22 ksi
f
2
ÿh R tÿy
cg
f
y
cg
ÿ27:87 ksi
c
f
2
f
1
ÿ0:603 Eq: B2:3-5
Chapter 4
98
k 4 21 ÿ c
3
21 ÿ c15:442Eq: B2:3-4
l
1:052
k
p
h
t
f
1
E
r
0:641 Eq: B2:1-4
b
e
b h 3:63 in: since l < 0:673
b
1
b
e
3 ÿ c
1:008 in: Eq: B2:3-1
b
2
b
e
2
1:815 in: since c < ÿ0:236
Eqs: B2:3-2 and B2:3-3
b
cw
y
cg
ÿR t2:265 in:
b
cw
is the compression portion of the web
b
1
b
2
2:823 in:
Since b
1
b
2
> b
cw
; b
e
2:265 in: Since b
e
b
cw
, the web is
fully effective and no iteration is required.
C. Calculation of effective section modulus (S
e
)
for compression ¯ange stressed to yield
S
e
I
x
y
cg
1:049 in:
3
D. Calculation of nominal ¯exural strength (M
n
)
and design ¯exural strength (M
d
)
M
n
S
e
F
y
52:457 kip-in: Eq: C3:1:1-1
LRFD f
b
0:95
M
d
f
b
M
n
49:834 kip-in:
ASD O
b
1:67
M
d
M
n
O
b
34:411 kip-in:
Compression Elements
99
4.6.2 Hat Section in Bending with Intermediate
Stiffener in Compression Flange
Problem
Determine the design positive bending moment about a
horizontal axis of the hat section in Figure 4.10 when an
intermediate stiffener is added to the center of the compres-
sion ¯ange as shown in Figure 4.11. The following section
numbers refer to the AISI Speci®cation:
D 4:0in: B 9:0in: t 0:060 in: R 0:125 in:
L
f
0:75 in: B
f
3:0in: d
s
0:30 in:
F
y
50 ksi E 29;500 ksi
Solution
I. Properties of 90
corner (Elements 2 and 6)
Corner radius r R
t
2
0:155 in:
Length of arc u 1:57r 0:243 in:
Distance of centroid from c 0:637r 0:099 in:
center of radius
I
0
of corner about its own centroidal axis is negligible
II. Nominal ¯exural strength (M
n
) (Section C3.1.1)
Based on Initiation of Yielding [Section C3.1.1(a)]
A. Computation of I
x
assuming web is fully effective
Assume f F
y
in the top ®bers of the section.
Element 4: Webs
h D ÿ 2R t3:63 in:
h
t
60:5 this value must not exceed 200
Section B1:2a
Chapter 4
100
Element 5: Compression Flange
b
o
B ÿ 2R t8:63 in:
b
o
t
144 this value must not exceed 500
Section B1:1a2
Section B4.1(a) Uniformly compressed elements with
one intermediate stiffener (see Figure 4.11)
FIGURE 4.11 Hat section ¯ange with intermediate stiffener: (a)
cross section of compression ¯ange; (b) line element model of
¯ange; (c) stiffener section; (d) line element model of stiffener.
Compression Elements
101
S 1:28
E
f
s
31:091 3S 93:273 Eq: B4-1
Case III b
o
=t > 3S: Flange not fully effective
I
a
t
4
128b
o
=t
S
ÿ 285
0:0040 in:
4
Eq: B4:1-6
Full section properties of stiffener
L
i
is the effective length of all elements of type i, and y
i
is the distance from the top ®ber.
Corner properties of stiffener
Corner radius r R
t
2
0:155 in:
Length of arc u 1:57r 0:243 in:
Distance of centroid from c 0:637r 0:099 in:
center radius
L
8
2uy
8
R tÿc
L
9
2d
s
y
9
R t
d
s
2
I
0
9
2d
3
s
12
0:0045 in:
3
L
10
2uy
10
R td
s
c
Element Effective Distance from
no., length, top ®ber,
iL
i
y
i
L
i
y
i
L
i
y
2
i
(in.) (in.) (in.
2
) (in.
3
)
8 0.487 0.086 0.042 0.0036
9 0.600 0.335 0.201 0.067
10 0.487 0.584 0.284 0.166
Chapter 4
102
P
10
i8
L
i
1:573 in:
P
10
i8
L
i
y
i
0:527 in:
2
P
10
i8
L
i
y
2
i
0:237 in:
3
y
cg
P
10
i8
L
i
y
i
P
10
i8
L
i
0:335 in:
I
Ly
2
P
10
i8
L
i
y
2
cg
0:177 in:
3
I
0
s
P
10
i8
L
i
y
i
2
I
0
9
ÿ I
Ly
2
0:065 in:
3
I
s
I
0
s
t 0:0039 in:
4
A
0
s
P
10
i8
L
i
t 0:094; in:
2
Reduced area of stiffener (A
s
)
A
s
I
s
I
a
A
0
s
0:092 in:
2
Eq: B4:1-8
L
s
A
s
t
1:535 in:
Continuing with Element 5
k 3
I
s
I
a
1=3
1 3:98 4:00 Eq: B4:1-7
w
B ÿ 2R tÿ2R ÿ 2R t
2
4:005 in:
w
t
67
Compression Elements
103
l
1:052
k
p
w
t
f
E
r
1:450 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:585 Eq: B2:1-3
b rw 2:343 in: since l > 0:673
Element 1: Lip
L
u
L
f
ÿR t0:565 in:
Element 3: Flange
b
f
B
f
ÿ 2R t2:63 in:
L
i
is the effective length of all elements of type i and y
i
is the distance from the top ®ber:
L
1
2L
u
y
1
D ÿR tÿ
L
u
2
I
0
1
2L
3
u
12
0:03 in:
3
L
2
4uy
2
D ÿR tc
L
3
2b
f
y
3
D ÿ
t
2
L
4
2hy
4
R t
h
2
I
0
4
2h
3
12
7:972 in:
3
L
5
2wy
5
t
2
L
6
2uy
6
R tÿc
L
7
L
s
y
7
R t
d
s
2
I
0
7
I
0
s
Effective Distance from
Element length, top ®ber,
no., L
i
y
i
L
i
y
i
L
i
y
2
i
i (in.) (in.) (in.
2
) (in.
3
)
1 1.130 3.533 3.992 14.101
2 0.973 3.914 3.810 14.910
3 5.260 3.970 20.882 82.902
4 7.260 2.000 14.520 29.040
5 4.686 0.030 0.141 4.218
6 0.487 0.086 0.042 3.622
7 1.535 0.335 0.514 0.173
Chapter 4
104
P
7
i1
L
i
21:331 in:
P
7
i1
L
i
y
i
43:9in:
2
P
7
i1
L
i
y
2
i
141:133 in:
3
Determine distance y
cg
of neutral axis from top ®ber:
y
cg
P
7
i1
L
i
y
i
P
7
i1
L
i
2:058 in:
I
Ly
2
P
7
i1
L
i
y
2
cg
90:348 in:
3
I
0
x
P
7
i1
L
i
y
2
i
I
0
1
I
0
4
ÿ I
Ly
2
58:787 in:
3
I
x
I
0
x
t 3:527 in:
4
B. Check Web (Section B2.3)
f
1
y
cg
ÿR t
f
y
cg
45:51 ksi
f
2
ÿh R tÿy
cg
f
y
cg
ÿ42:69 ksi
c
f
2
f
1
ÿ0:938 Eq: B2:3-5
k 4 21 ÿ c
3
21 ÿ c22:434 Eq: B2:3-4
l
1:052
k
p
h
t
f
1
E
r
0:528 Eq: B2:1-4
b
e
3:63 in: since l < 0:673
b
1
b
e
3 ÿ c
0:922 in: Eq: B2:3-1
Compression Elements
105
b
2
b
e
2
1:815 in: since c < ÿ0:236
Eqs: B2:3-2 and B2:3-3
b
cw
y
cg
ÿR t1:873 in:
b
cw
is the compression portion of the web
b
1
b
2
2:737 in:
Since b
1
b
2
> b
cw
then b
e
2:737 in:
C. Calculation of effective section modulus (S
e
)
for compression ¯ange stressed to yield
S
e
I
x
y
cg
1:714 in:
3
D. Calculation of nominal ¯exural strength (M
n
)
and design ¯exural strength (M
d
)
M
n
S
e
F
y
85:69 kip-in: Eq: C3:1:1-1
LRFD f
b
0:95
M
d
f
b
M
n
81:41 kip-in:
ASD O
b
1:67
M
d
M
n
O
b
51:31 kip-in:
4.6.3 C-Section Purlin in Bending
Problem
Determine the design positive bending moment about the
horizontal axis for the 8C0.060 purlin section shown in
Figure 4.12. The yield stress of the material is 55 ksi.
Assume the element lies along its centerline and eliminate
thickness effects. The section modulus should be computed
Chapter 4
106
assuming the section is fully stressed f F
y
. The follow-
ing section numbers refer to the AISI Speci®cation.
H 8:0in: B 2:75 in: t 0:060 in: R 0:1875 in:
D 0:625 in: F
y
55 ksi E 29;500 ksi
Solution
I. Properties of 90
corner (Elements 2 and 6)
Corner radius r R
t
2
0:218 in:
length of arc u 1:57r 0:341 in:
Distance of centroidal c 0:637r 0:139 in:
from center of radius
I
0
of corner about its own I
0
c
0:149r
3
0:0015 in:
3
centroidal axis
II. Nominal ¯exural strength (M
n
) (Section C3.1.1)
Based on Initiation of Yielding [Section C3.1.1(a)]
FIGURE 4.12 Purlin section with elements: (a) cross section; (b)
line element model.
Compression Elements
107