Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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Case I w=tS=3 Flange fully effective without
stiffener
Not applicable w=t > S=3
Case II S=3 < w=t < S Flange fully effective
with I
s
I
a
k
u
0:43
I
a
2
399 t
4
w=t
S
ÿ
k
u
4
r
!
3
7:708 10
ÿ3
in:
4
Eq: B4:2-4
Case III w=t S Flange not fully effective
I
a
3
t
4
115
w=t
S
5
2:259 10
ÿ3
in:
4
Eq: B4:2-11
n
3
0:333
Since w=t S, then
I
a
I
a
3
2:259 10
ÿ3
in:
4
n n
3
0:333
Calculate buckling coef®cient k and compression
stiffener reduced effective width d
s
:
k
a
5:25 ÿ 5
D
w
4:10 > 4:0 Eq:B4:2-8
Hence use
k
a
4:0
C
2
I
s
I
a
0:143 < 1:0 Eq: B4:2-5
C
1
2 ÿ C
2
1:857 Eq: B4:2-6
k C
n
2
k
a
ÿ k
u
k
u
2:296 Eq: B4:2-7
d
s
C
d
d
0
s
0:068 in: Eq: B4:2-9
Chapter 4
118
Section B2.1 Effective width of ¯ange element 3 for
strength
l
1:052
k
p
w
t
f
E
r
1:308 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:636 Eq: B2:1-3
b rw 1:665 in: since l > 0:673 Eq: B2:1-2
Calculate second moment of area of effective section
L
i
is the effective length of all elements of type i, and y
i
is the distance from the top ®ber:
L
1
d
s
y
1
R
t
2
1 ÿ cos y I
0
1
d
3
s
sin
2
y
12
d
s
2
sin y
t
2
1:55410
ÿ5
in:
3
L
2
uy
2
R tÿc
1
I
0
2
I
0
c1
1:22710
ÿ4
in:
3
L
3
by
3
t
2
I
0
3
0in:
3
L
4
uy
4
R tÿcI
0
4
I
0
c
1:81310
ÿ3
in:
3
L
5
hy
5
R t
h
2
I
0
5
h
3
12
34:876 in:
3
L
6
uy
6
H ÿR tcI
0
6
I
0
c
1:81310
ÿ3
in:
3
L
7
wy
7
H ÿ
t
2
I
0
7
0in:
3
L
8
u
1
y
8
H ÿR tc
1
I
0
8
I
0
c1
1:22710
ÿ4
in:
3
L
9
dy
9
H ÿ
R t
2
I
0
9
d
3
12
sin
2
y
ÿt
2
1 ÿ cos y
ÿd
2
sin y
ÿt
2
5:36610
ÿ3
in:
3
Compression Elements
119
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 0.068 0.138 0.009 0.001
2 0.201 0.058 0.012 0.001
3 1.665 0.030 0.050 0.001
4 0.361 0.113 0.041 0.005
5 7.48 4.00 29.92 119.68
6 0.361 7.887 2.848 22.459
7 2.619 7.97 20.872 166.346
8 0.201 7.942 1.594 12.660
9 0.479 7.704 3.689 28.419
P
9
i1
L
i
13:434 in:
P
9
i1
L
i
y
i
59:034 in:
2
P
9
i1
L
i
y
2
i
349:572 in:
3
P
9
i1
I
0
i
34:885 in:
3
Determine distance y
cg
of neutral axis from top ®ber:
y
cg
P
9
i1
L
i
y
i
P
9
i1
L
i
4:394 in:
I
Ly
2
P
9
i1
L
i
y
2
cg
259:411 in:
3
I
0
x
P
9
i1
L
i
y
2
i
P
9
i1
I
0
i
ÿ I
Ly
2
125:046 in:
3
I
x
I
0
x
t 7:503 in:
4
S
ex
I
x
y
cg
1:707 in:
3
Chapter 4
120
B. Check Web (Section B2.3)
f
1
y
cg
ÿR t
f
y
cg
51:75 ksi
f
2
ÿh R tÿy
cg
f
y
cg
ÿ41:88 ksi
c
f
2
f
1
ÿ0:809 Eq: B2:3-5
k 4 21 ÿ c
3
21 ÿ c19:464 Eq: B2:3-4
l
1:052
k
p
h
t
f
1
E
r
1:245 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:661 Eq: B2:1-3
b
e
rh 4:946 in: since l > 0:673 Eq: B2:1-2
b
1
b
e
3 ÿ c
1:298 in: Eq: B2:3-1
b
2
b
e
2
2:473 in: since c < ÿ0:236 Eq: B2:3-2
b
c
y
cg
ÿR t4:134 in:
b
1
b
2
3:772 in:<b
c
4:134 in:
Web is not fully effective in this case since b
1
b
2
< b
c
.
C. Iterate on web effective width
Try a value of the centroid position and iterate until
convergence:
y
0
cg
4:50 in:
D. Check Web (Section B2.3)
f
1
y
cg
ÿR t
f
y
cg
51:82 ksi
f
2
ÿh R tÿy
cg
f
y
cg
ÿ39:6 ksi
c
f
2
f
1
ÿ0:764 Eq: B2:3-5
Compression Elements
121
k 4 21 ÿ c
3
21 ÿ c18:509 Eq: B2:3-4
l
1:052
k
p
h
t
f
1
E
r
1:278 Eq: B2:1-4
r
1 ÿ 0:22=l
l
0:648 Eq: B2:1-3
b
e
rh 4:846 in: since l > 0:673 Eq: B2:1-2
b
1
b
e
3 ÿ c
1:287 in: Eq: B2:3-2
b
2
b
e
2
2:423 in: since c < ÿ0:236 Eq: B2:3-2
Break into three elements as shown in Figure 4.14.
Elements 5 and 10 are the top and bottom effective portions
of the compression zone, and Element 11 is the web tension
zone which is fully effective
L
5
b
1
y
5
R t
b
1
2
I
0
5
b
3
1
12
0:178 in:
3
L
10
b
2
y
10
y
cg
ÿ
b
2
2
I
0
10
b
3
2
12
1:186 in:
3
L
11
H ÿ y
cg
y
11
H ÿR t I
0
11
L
11
3
12
ÿR tÿ
L
11
2
2:834 in:
3
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 0.68 0.138 0.009 0.001
2 0.201 0.058 0.012 0.001
3 1.665 0.030 0.050 0.001
(continued)
Chapter 4
122
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
)
4 0.361 0.113 0.041 0.005
5 1.287 0.904 1.164 1.052
6 0.361 7.887 2.848 22.459
7 2.619 7.97 20.872 166.346
8 0.201 7.942 1.594 12.660
9 0.479 7.704 3.689 28.419
10 2.423 3.288 7.968 26.203
11 3.240 6.120 19.829 121.352
P
11
i1
L
i
12:905 in:
P
11
i1
L
i
y
i
58:075 in:
2
P
11
i1
L
i
y
2
i
378:499 in:
3
P
11
i1
I
0
i
4:207 in:
3
Determine distance y
cg
of neutral axis from top ®ber:
y
cg
P
11
i1
L
i
y
i
P
11
i1
L
i
4:50 in:
The computed centroid position y
cg
matches with the
assumed value:
I
Ly
2
P
11
i1
L
i
y
2
cg
261:347 in:
3
I
0
x
P
11
i1
L
i
y
2
i
P
11
i1
I
0
i
ÿ I
Ly
2
121:359 in:
3
I
x
I
0
x
t 7:282 in:
4
Compression Elements
123
E. Calculation of effective section modulus (S
ex
for compression ¯ange stressed to yield
S
ex
I
x
y
cg
1:618 in:
3
The resulting effective section modulus S
ex
allows for the reduction in the effective area of both
the web and ¯ange due to their slenderness and is
the value when the section is yielded at the
extreme compression ®ber.
F. Calculation of nominal ¯exural strength (M
n
)
and design ¯exural strength (M
d
)
M
n
S
e
F
y
88:99 kip-in: Eq: C3:1:1-1
LRFD f
b
0:95
M
d
f
b
M
n
84:54 kip-in:
ASD O
b
1:67
M
d
M
n
O
b
53:929 kip-in:
REFERENCES
4.1 Bleich, F., Buckling Strength of Metal Structures,
McGraw-Hill, New York, 1952.
4.2 Bulson, P. S., The Stability of Flat Plates, Chatto and
Windus, London, 1970.
4.3 Allen, H. G. and Bulson, P., Background to Buckling,
McGraw-Hill, New York, 1980.
4.4 Von Karman, T., Sechler, E. E., and Donnell, L. H.,
The strength of thin plates in compression, transac-
tions ASME, Vol. 54, MP 54-5, 1932.
4.5 Winter, G., Strength of thin steel compression
¯anges, Transactions, ASCE, Vol. 112, Paper No.
2305, 1947, pp. 527±576.
4.6 Winter, G., Thin-Walled StructuresÐTheoretical
Solutions and Test Results, Preliminary Publications
Chapter 4
124
of the Eighth Congress, IABSE, 1968, pp. 101±112.
4.7 Kalyanaraman, V., Peko
È
z, T., and Winter, G., Unstif-
fened compression elements, Journal of the Struc-
tural Division, ASCE, Vol. 103, No. ST9, Sept. 1977,
pp. 1833±1848.
4.8 Weng, C. C. and Peko
È
z, T. B., Subultimate Behavior of
Uniformly Compressed Stiffened Plate Elements,
Research Report, Cornell University, Ithaca, NY,
1986.
4.9 Ortiz-Colberg, R. and Peko
È
z, T. B., Load Carrying
Capacity of Perforated Cold-Formed Steel Columns,
Research Report No. 81±12, Cornell University,
Ithaca, NY, 1981.
4.10 Peko
È
z, T., Development of a Uni®ed Approach to the
Design of Cold-Formed Steel Members, American
Iron and Steel Institute, Research Report CF87-1,
March 1987.
4.11 Desmond, T. P., Peko
È
z, T., and Winter, G., Edge
stiffeners for thin-walled members, Journal of Struc-
tural Engineering, ASCE, 1981, 1-7(2), p. 329±353.
4.12 Desmond, T. P., Peko
È
z, T., and Winter, G., Intermedi-
ate stiffeners for thin-walled members, Journal of
Structural Engineering, ASCE, 1981, 107(4), pp.
627±648.
4.13 Schafer, B. and Peko
È
z, T., Cold-formed steel members
with multiple longitudinal intermediate stiffeners,
Journal of Structural Engineering, ASCE, 1998, 124
(10), pp. 1175±1181.
Compression Elements
125
5
Flexural Members
5.1 GENERAL
The design of ¯exural members for strength is usually
governed by one of the following limit states:
(a) Yielding of the most heavily loaded cross section
in bending including local and postlocal buckling
of the thin plates forming the cross section, or
(b) Elastic or inelastic buckling of the whole beam in a
torsional-¯exural (commonly called lateral) mode,
or
(c) Yielding and=or buckling of the web subjected to
shear, or combined shear and bending, or
(d) Yielding and=or buckling of the web subjected to
bearing (web crippling) or combined bearing and
bending
The design for (a) determines the nominal section strength
M
n
given by Eq. (5.1).
M
n
S
e
F
y
5:1
127