Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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with caution when there are more than eight purlin lines. If
the top ¯anges of adjacent purlins or sets of purlins face in
opposite directions, an anchorage is required to resist only
the net downslope component.
Anchorage force calculations for a Z-purlin supported
roof system are illustrated in Section 10.5.3.
10.4.2 C-Purlin Supported Systems
The anchorage requirements for C-purlins are much less
than those for Z-purlins since bending is about a principal
axis. The eccentricity caused by the load not acting through
the shear center requires a resisting force of about 5% of
the load parallel to the web of the C-section. AISI Speci®ca-
tion Supplement No. 1 Section D3.2.1(a), C-Sections, gives
the anchorage force for a C-section-supported roof system
with all compression ¯anges facing in the same direction
(Eq. D3.2.1-1) as
P
L
0:05a cos y ÿ sin yW 10:19
with a 1 for purlins facing in the upslope direction and
ÿ1 for purlins facing in the downslope direction. A
FIGURE 10.19 Anchorage force vs. Panel stiffness: Eq. (10.10)
and stiffness model results.
Chapter 10
328
positive value of P
L
means that anchorage is required to
prevent movement of the purlin ¯anges in the upslope
direction, and a negative value means that anchorage is
required to prevent movement in the downslope direction.
As with Z-purlin systems, if the top ¯anges of adjacent
C-purlin lines face in opposite directions, the anchorage
need only resist the downslope component of the gravity
load.
10.5 EXAMPLES
10.5.1 Computation of R-value from Base
Tests
Problem
Determine the R-value relationship for the gravity loading
base test data shown. The tests were conducted using Z-
sections with nominal thicknesses of 0.060 in. and 0.095 in.
and nominal yield stress of 55 ksi. The span length was 22 ft
9 in., and intermediate lateral braces were installed at the
third points. The total supported load (w
ts
) is equal to the
sum of the applied load (w) and the weight of the sheathing
and purlins (w
d
). The maximum applied moment is M
ts
.
Solution
A. Summary of Test Loadings
Test
Max. applied
load
Deck
weight
Purlin
weight w
d
Total
load, w
ts
M
ts
no. w (lb=ft) (lb=ft) (lb=ft) (lb=ft) (lb=ft) (K-in.)
1 91.79 4.0 3.10 7.10 98.9 76.8
2 86.29 4.0 3.14 7.14 93.4 72.5
3 81.84 4.0 3.10 7.10 88.9 69.0
4 186.4 4.0 5.05 9.05 195.4 151.7
5 189.1 4.0 5.01 9.01 198.1 153.8
6 184.5 4.0 4.91 8.91 193.4 150.2
Roof and Wall Systems
329
B. Reduction Factor Data
Test
no. t (in.)
S
eff
(in.)
F
y
(ksi)
M
nt
S
eff
F
y
(K-in.)
M
ts
(K-in.)
R
t
(%)
1 0.059 1.88 60.0 112.8 76.8 68.1
2 0.059 1.90 59.2 112.5 72.5 64.5
3 0.060 1.92 57.3 110.0 69.0 62.8
Average 111.8 65.1
4 0.097 3.38 68.4 231.2 151.7 65.6
5 0.096 3.38 67.1 226.8 153.8 67.8
6 0.097 3.30 66.5 219.4 150.2 68.5
Average 225.8 67.3
s
max
one standard deviation of the modi®cation
factors of the thickest purlins tested 0.0148
s
min
one standard deviation of the modi®cation
factors of the thinnest purlins tested
0.0271
R
t max
0:673 ÿ 0:0148 0:658
R
t min
0:651 ÿ 0:0271 0:624
M
nt
max
225:8 kip-in.
M
nt
min
111:8 kip-in.
C. Reduction Factor Relation
Equation (10.4):
R
0:658 ÿ 0:624
225:8 ÿ 111:8
M
n
ÿ 111:80:624 1:0
0:298M
n
ÿ 111:8
1000
0:624
The variation of R with purlin strength for the stand-
ing seam roof system tested is shown in Figure 10.20.
(Figure 10.20 shows the R-value line sloping upward to
the right. For some standing seam roof systems, the
line will slope downward to the right.)
Chapter 10
330
D. Application
For a purlin having the same depth, ¯ange width, edge
stiffener slope, and material speci®cation as those used
in the above example and with a nominal ¯exural
strength M
n
S
e
F
y
135 in.-kips, the reduction
factor is
R
0:298135 ÿ 111:77
1000
0:624 0:631
The positive moment design strength:
1. LRFD
f
b
0:90
fM
n
f
b
RS
e
F
y
0:90 0:631 135 76:7 kip-in:
2. ASD
O
b
1:67
M
RS
e
F
y
O
b
0:631 135
1:67
51:0 kip-in:
FIGURE 10.20 Reduction factor vs. nominal moment strength.
Roof and Wall Systems
331
10.5.2 Continuous Lapped Z-Section Purlin
Problem
Determine the maximum uplift and gravity service loads
for a three-span continuous purlin line using the Z-section
purlins (8Z060) shown in Figure 4.14. The spans are 25 ft
with interior lap lengths of 5 ft (3-ft end spans and 2-ft
interior span) as shown in Figure 10.21. The purlin lines
are 5 ft on center, and the roof sheathing is screw-fastened
to the top ¯ange of the purlins. The weight of the sheath-
ing, insulation, and purlin is equivalent to 13.5 pounds per
linear foot (lb=ft).
The following section, equation, and table numbers
refer to those in the AISI Speci®cation.
F
y
55 ksi E 29;500 ksi H 8 in. y
cg
4 in.
Solution
A. Full Section Properties
Area (A), second moment of area about centroidal axis
perpendicular to web (I
x
), second moment of area about
FIGURE 10.21 Three-span continuous lapped Z-purlin.
Chapter 10
332
centroidal axis parallel to the web (I
y
), and section
modulus of compression ¯ange (wide) for full section
(S
f
) accounting for rounded corners (from THIN-WALL
(Ref. 11.3)):
A 0:8876 in:
2
I
x
8:858 in:
4
I
y
1:714 in:
4
I
yc
I
y
2
0:857 in.
4
S
f
I
x
y
cg
2:215 in:
3
B. Design Load on Continuous Lapped Purlin under
Uplift Loading
1. Strength for Bending Alone Braced by Sheathing
Section C3.1.3 Beams Having One Flange
Through-fastened to Sheathing
a. Uplift Loading: Continuous Lapped Z-Section
LRFD
R 0:70
S
e
1:618 in:
3
(see Example 4.6.4)
M
n
RS
e
F
y
62:29 kip-in. 5:191 kip-ft
f
b
0:90 (LRFD)
f
b
M
n
4:672 kip-ft (Eq. C3.1.4-1)
LRFD design load per unit length (see moment
diagram in Figure 10.21 w 120 lb=ft, where
maximum moment 5.715 kip-ft).
w
u
(LRFD)
f
b
M
n
120
5:715
98:1 lb/ft (LRFD)
ASD
O
b
1:67 (ASD)
M
n
O
b
3:108 kip-ft
Roof and Wall Systems
333
ASD design load per unit length (see moment
diagram in Figure 10.21 at w 120 lb=ft,
where maximum moment 5.715 kip-ft):
w (ASD)
M
n
O
b
120
5:715
65:3 lb/ft (ASD)
2. Strength for Shear Alone
Section C3.2 Strength for Shear Only
k
v
5:34 for an unreinforced web
h H ÿ 2R t8 ÿ20:20 0:06
7:48 in:
h
t
7:48
0:060
124:7
Ek
v
F
y
s
29;500 5:34
55
r
53:51
Since
h
t
> 1:415
Ek
v
F
y
s
75:73
then
V
n
0:905Ek
v
t
3
h
4:12 kips (Eq. C3.2-3)
f
v
0:90 (LRFD)
O
v
1:67 (ASD)
Maximum V at end of lap in end span 1.468 kips
when w 120 lb=ft
Chapter 10
334
LRFD
Assuming that bending alone controls
w
u
(LRFD) 98:1 lb/ft B:1; then
V
u
98:1
120
1:468 1:20 kips
V
u
f
v
V
n
1:20
0:90 4:12
0:324 < 1:0
Hence, shear alone is satis®ed for LRFD under
uplift loading.
ASD
Assuming that bending alone controls
w ASD65:3lb=ft (B.1), then
V
65:3
120
1:468 0:80 kips
V
V
n
=O
v
0:80
4:12=1:67
0:324 < 1:0
Hence, shear alone is satis®ed for ASD under
uplift loading.
3. Strength for Bending at End of Lap
Section C3.1.1 Nominal Section Strength
S
e
1:618 in:
3
at F
y
M
nxo
S
e
F
y
(see Example 4.6.4)
1:618 55 Eq. C3.1.1-1)
89:0 kip-in: 7:42 kip-ft
f
b
0:95 LRFD
O
b
1:67 ASD
Maximum M at end of lap in interior span
5.452 kip-ft when w 120 lb=ft.
Roof and Wall Systems
335
LRFD
Again assuming that bending alone controls with
w
u
(LRFD) 98.1 lb=ft (B.1), then
M
u
98:1
120
5:452 4:457 kip-ft
M
u
f
b
M
nxo
4:457
0:95 7:42
0:632 < 1:0
ASD
Again assuming that bending alone controls with
w (ASD) 65.3 lb=ft (B.1), then
M
65:3
120
5:452 2:97 kip-ft
M
M
nxo
=Q
b
2:97
7:42=1:67
0:668 < 1:0
4. Strength for Combined Bending and Shear
Section C3.5 Combined Bending and Shear
LRFD
For M
u
and V
u
based on bending alone (B.1) with
w
u
(LRFD) 98.1 lb=ft. Maximum V at end of lap
in interior span 1.260 kips when w 120 lb=ft.
Hence
V
u
f
v
V
n
98:1=120 1:260
0:90 4:12
0:278
M
u
f
b
M
nxo
2
V
u
f
v
V
n
2
0:632
2
0:278
2
0:477 < 1:0
Hence, combined bending and shear are satis®ed
for LRFD at end of lap in interior span. Checking
end of lap in end span produces a lower (safer)
result due to signi®cantly lower moment.
Chapter 10
336
ASD
For M and V based on bending alone (B.1) with w
(ASD) 65.3 lb=ft:
M
M
nxo
=O
b
2
V
V
n
=O
v
2
0:668
2
0:278
2
0:524 < 1:0
Hence, combined bending and shear are satis®ed
for ASD at end of lap in interior span. Checking
end of lap in end span produces a lower result due
to signi®cantly lower moment.
C. Design Load on Continuous Lapped Purlin under
Gravity Loading
1. Strength for Bending with Lateral Buckling
Section C3.1.2 Lateral Buckling Strength
The Center for Cold-Formed Steel Structures
Bulletin (Vol. 1, No. 2, August 1992) suggests
using a distance from in¯ection point to end of
lap, as unbraced length for gravity load with
C
b
1:75. Check interior span since it is more
highly loaded in negative moment region.
L distance from end of lap to in¯ection point
as shown in Figure 10.21
73.1 in. based on statics using Figure
10.21 with zero moment at in¯ection
point
C
b
1:75
M
e
p
2
EC
b
dI
yc
2L
2
326:8 kip-in: 27:2 kip-ft
Eq. C3.1.2-16
Critical Moment (M
c
)
M
y
S
f
F
y
121:8 kip-in. 10:15 kip-ft
Eq. C3.1.2-5
Roof and Wall Systems
337