Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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and bending is covered in Section C3.3. The design of webs
for web crippling under point loads and reactions is
governed by Section C3.4, and combined bending and web
crippling is prescribed in Section C3.5. The design of
transverse stiffeners at bearing points and as intermediate
stiffeners is set out in Section B6.
6.2 WEBS IN SHEAR
6.2.1 Shear Buckling
The mode of buckling in shear is shown in Figure 6.1a. This
mode of buckling applies for a long plate which is simply
supported at the top and bottom edges. Unlike local buck-
ling in compression, the nodal lines are not perpendicular
to the direction of loading and parallel with the loaded
edges of the plate. Consequently, the basic mode of buckling
is altered if the plate is of ®nite length. The buckling stress
for shear buckling is given by Eq. (4.1), with the value of
the local buckling coef®cient k selected to allow for the
shear loading and boundary conditions. The theoretical
values of k as a function of the aspect ratio a=b of the
plate are shown in Figure 6.1b. As the plate is shortened,
the number of local buckles is reduced and the value of k is
increased from 5.34 for a very long plate to 9.34 for a square
plate. An approximate formula for k as a function of a=b is
shown by the solid line in Figure 6.1b and given by
k 5:34
4
a=b
2
6:1
Equation (6.1) is a parabolic approximation to the garland
curve in Figure 6.1b.
For plates with intermediate transverse stiffeners
along the web, the effect of the stiffeners is to constrain
the shear buckling mode between stiffeners to be the same
as a plate of length equal to the distance between the
stiffeners. Consequently, the value of a in Eq. (6.1) is
Chapter 6
168
taken as the stiffener spacing and an increased value of the
local buckling coef®cient over that for a long unstiffened
plate can be used for webs with transverse stiffeners.
As explained in Ref. 6.1, plates in shear are unlikely to
have a substantial postbuckling reserve as for plates in
compression unless the top and bottom edges are held
FIGURE 6.1 Simply supported plate in shear: (a) buckled mode in
shear for long narrow plate; (b) buckling coef®cient (k) versus
aspect ratio (a=b).
Webs
169
straight. Since cold-formed members usually only have thin
¯anges, it is unlikely that the postbuckling strength in
shear can be mobilized in the web of a cold-formed member.
Hence, the elastic critical stress is taken as the limiting
stress in Section C3.2 of the AISI Speci®cation. Section
C3.2 uses Eq. (4.1) to give a nominal shear strength V
n
of
V
n
0:905Ek
v
h=t
2
ht
0:905Ek
v
t
3
h
6:2
where h is web depth, which is taken as the depth of the ¯at
portion of the web measured parallel to the web, and k
v
is
taken as the parabolic approximation to k given by Eq.
(6.1). When the stiffener spacing is greater than the web
depth such that a=h > 1:0, then b in Eq. (6.1) is replaced by
h. When the stiffener spacing is less than the web depth
such that a=h 1:0, then the plate is taken as a plate of
length h with a ¯at width ratio of a=t. Hence, w=t in Eq.
(4.1) is replaced by a=t, and a=b in Eq. (6.1) is replaced by
h=a . The resulting equation for k
v
to be used with Eq. (6.2)
is
k
v
4:0
5:34
a=h
2
6:3
For plates without stiffeners, k
v
is simply taken as 5.34.
These formulae for k
v
are speci®ed in Section C3.2.
6.2.2 Shear Yielding
A stocky web (small h=t) subject to shear will yield in shear
at an average stress of approximately F
y
=
3
p
, as given by
the von Mises' yield criterion (Ref. 6.2). The nominal shear
capacity for yield V
n
is therefore
V
n
0:60F
y
ht 6:4
The value of 0.60, which is higher than 1=
3
p
, is consistent
with a reduced factor of safety normally assumed for shear
yielding in allowable stress design standards such as the
Chapter 6
170
AISC Speci®cation (Ref. 1.1). Figure 6.2 shows the failure
stresses for a web in shear, including yield as given by Eq.
(6.4) and buckling as given by Eq. (6.2). In the region where
shear and buckling interact, the failure stress is given by
the geometric mean of the buckling stress and 0.8 the
yield stress in shear (Ref. 6.3). The resulting equation for
the nominal shear strength V
n
is given by Eq. (6.5a) and is
shown in Figure 6.2 for a range of yield stresses:
V
n
0:64t
2
Ek
v
F
y
q
6:5a
This equation applies in the range of web slenderness:
0:96
Ek
v
F
y
s
h
t
1:415
Ek
v
F
y
s
6:5b
The resistance factors f
v
and safety factors O
v
are 1.0
and 1.5, respectively, for Eq. (6.4), and 0.90 and 1.67,
respectively, for Eqs. (6.2) and (6.5).
FIGURE 6.2 Nominal shear strength V
n
of a web expressed as
average shear stress.
Webs
171
6.3 WEBS IN BENDING
The elastic critical stress of a web in bending is given in Eq.
(4.1) with k 23:9, as shown as Case 5 in Figure 4.1. As
described in Section 4.4, the web in bending has a substan-
tial postbuckling reserve. Consequently, the design proce-
dure for webs in bending should take account of the
postbuckling reserve of strength. Two basic design methods
have been described in Ref. 6.4. Both methods result in
approximately the same design strength but require differ-
ent methods of computation.
The ®rst procedure extends the concept of effective
width to include an effective width for the compression
component of the web as shown in Figure 6.3a. Conse-
quently the effective section involves removal of portions of
both the compression ¯ange and web if each is suf®ciently
slender. The effective width formulae for b
1
and b
2
in the
web, as shown in Figure 6.3a, have been proposed in Ref.
6.4 and con®rmed by tests. They are described in Section
4.4 and demonstrated in Example 4.6.3. The disadvantage
of the method is the complexity of the calculation, including
the determination of the centroid of the effective section.
The 1996 AISI Speci®cation uses this design approach.
An alternative procedure has also been proposed in
Ref. 6.4 where only the ineffective component of the
compression ¯ange has been removed and the full web
depth is considered as shown in Figure 6.3b. In this case,
a limiting stress F
bw
in the web in bending is speci®ed.
The method is simpler to use than the effective web
approach since it is not iterative. It was selected for the
1980 edition of the AISI Speci®cation as well as the British
Standard BS 5950 (Part 5) (Ref. 1.5). The resulting formu-
lae for the maximum permissible stress in a web in bending
are Eqs. (6.6) and (6.7).
For sections with stiffened compression ¯anges:
F
bw
1:21 ÿ 0:00034
h
t
F
y
q
F
y
6:6
Chapter 6
172
For sections with unstiffened compression ¯anges:
F
bw
1:26 ÿ 0:00051
h
t
F
y
q
F
y
6:7
Both formulae have been con®rmed by the testing reported
in Ref. 6.4. The resulting design curves are plotted in
FIGURE 6.3 Effective width models for thin-walled sections in
bending: (a) effective compression ¯ange and web; (b) effective
compression ¯ange and full web.
Webs
173
Figure 6.4, where they are compared with the elastic local
buckling stress for a web in bending. The limiting stress
F
bw
has been used in the purlin design model of Rousch and
Hancock (Ref. 5.18), described in Section 5.3.2.
6.4 WEBS IN COMBINED BENDING AND
SHEAR
The combination of shear stress and bending stress in a
web further reduces web capacity. The degree of reduction
depends on whether the web is stiffened. The interaction
formulae for combined bending and shear are given in
Section C3.3 of the AISI Speci®cation.
For an unstiffened web the interaction equation is a
circular formula as shown in Figure 6.5a. This interaction
formula is based upon an approximation to the theoretical
interaction of local buckling resulting from shear and
bending as derived by Timoshenko and Gere (Ref. 3.6). As
applied in the AISI Speci®cation, the interaction is based
on the nominal ¯exural strength M
nxo
, which includes
FIGURE 6.4 Limiting stress on a web in bending.
Chapter 6
174
post±local buckling in bending. Hence, the justi®cation for
the use of the circular formula is actually empirical as
con®rmed by the testing reported in Ref. 6.5.
For a stiffened web, the interaction between shear and
bending is not as severe, probably as a result of a greater
FIGURE 6.5 Combined bending and shear in webs: (a) unstif-
fened webs; (b) webs with transverse stiffeners.
Webs
175
postbuckling capacity in the combined shear and bending
buckling mode. Consequently, a linear relationship, as
shown in Figure 6.5b with a larger design domain, is
used to limit the design actions under a combination of
shear and bending.
In Figures 6.5a and b, the LRFD and ASD formulae in
the AISI Speci®cation are shown.
6.5 WEB STIFFENERS
The design sections for the web stiffener requirements
speci®ed in Section B6 are based on those introduced in
the 1980 AISI Speci®cation. The section for transverse
stiffeners (B6.1) has been designed to prevent end crushing
of transverse stiffeners [Eq. (B6.1-1)] and column-type
buckling of the web stiffeners. It is based on the tests
described in Ref. 6.6. The resistance factor f
c
and safety
factor O
c
for transverse stiffeners are the same as for
compression members.
The section for shear stiffeners (B6.2) is based mainly
upon similar sections in the AISC Speci®cation for the
design of plate girders (Ref. 6.7), although the detailed
equations were con®rmed from the tests reported in
Ref. 6.6.
6.6 WEB CRIPPLING
6.6.1 Edge Loading Alone
In the design of cold-formed sections, it is not always
possible to provide load-bearing stiffeners at points of
concentrated edge loading. Consequently, a set of rules is
given in Section C3.4 of the AISI Speci®cation for the
design against web crippling under concentrated edge
load in the manner shown in Figure 6.6. The equations in
the AISI Speci®cation have been empirically based on tests.
The nominal web crippling strength P
n
has been found to
Chapter 6
176
be a function of the following parameters as shown in
Figure 6.7. These are:
(a) The nature of the restraint to web rotation
provided by the ¯ange and adjacent webs as
shown in Figure 6.7a. The back-to-back channel
beam has a higher restraint to web rotation at the
top and bottom, and hence a higher load capacity
than a single channel. Similarly, the channel with
a stiffened, or partially stiffened, compression
¯ange has a higher restraint to web rotation
than the channel with an unstiffened ¯ange,
and hence a higher web-bearing capacity.
(b) The length of the bearing N shown in Figure
6.7b and its proximity to the end of the section
(de®ned by c). In addition the proximity of other
opposed loads, de®ned by e in Figure 6.7b, is also
important. A limiting value of c=h of 1.5, where h
is the web ¯at width, is used to distinguish
between end loads and interior loads. Similarly
a limiting value of e=h 1:5 is used to distinguish
between opposed loads and nonopposed loads.
FIGURE 6.6 Web crippling of an open section.
Webs
177