Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification
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By comparison, the value of r in Eq. (5.21) for a C-
section is zero if the section remains untwisted. Hence, for
m
t
to be zero, the ®rst term in Eq. (5.21) (prying force term)
must balance with the third and fourth terms in Eq. (5.21).
That is, the torque from the prying forces is required to
counterbalance the torque from the eccentricity of the uplift
force in the center of the ¯ange from the shear center.
Consequently, the prying forces are much greater for C-
sections, than they are for Z-sections before structural
instability of the unrestrained ¯ange occurs.
5.3.2 Stability Considerations
When studying the lateral stability of C- and Z-section
¯exural members restrained by sheathing, two basically
different models of the buckling mode have been developed.
FIGURE 5.12 C-section twisting deformations: (a) distorted (D)
section model; (b) undistorted (U) section model.
Chapter 5
148
These are the distorted section model (D model) shown in
Figures 5.12a and 5.13a for C- and Z-sections, respectively,
and the undistorted section model (U model) shown in
Figures 5.12b and 5.13b for C- and Z-sections, respectively.
The D model is similar to the behavior of sheeted purlins as
observed in tests. The U model is based on the well-known
thin-walled beam theories of Timoshenko (Ref. 3.6) and
Vlasov (Ref. 5.14), so the stability can be computed by using
the conventional stability theory of thin-walled beams with
undistorted cross sections.
The D model has required the development of a new
stability model to account for the section distortion which is
not accounted for in the Timoshenko and Vlasov theories.
In the D model, the unrestrained ¯ange is usually in
compression for purlins subjected to wind uplift. Hence,
FIGURE 5.13 Z-section twisting deformations: (a) distorted (D)
section model; (b) undistorted (U) section model.
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the compression ¯ange will become unstable if the lateral
restraint provided through the web is insuf®cient to
prevent column buckling of the unrestrained ¯ange. The
usual model for cases where the torsional restraint
provided by the sheathing is substantial is to consider the
unrestrained compression ¯ange as a compression member
supported by a continuous elastic restraint of stiffness K as
shown in Figure 5.14b. The elastic restraint of stiffness K
shown in Figure 5.14b consists of the stiffness of the web
and the purlin-sheeting connection, and the ¯exural resis-
tance of the sheathing.
A design method for simply supported C-sections with-
out bridging and subjected to uniformly distributed load
was developed at Cornell University and is described in
detail in Ref. 5.15. The method has been extended recently
to continuous lapped C- and Z-sections, with and without
FIGURE 5.14 D model of C-section under uplift: (a) de¯ection;
(b) models.
Chapter 5
150
bridging, by Rousch and Hancock (Refs. 5.16±5.18). A brief
summary of the method follows.
As shown in Figure 5.14a, the de¯ected con®guration
is assumed to be the sum of a torsion stage, which includes
distortion of the section as a result of ¯exure of the web,
and a vertical bending stage. It is assumed that the torsion
stage is resisted by a ¯ange element restrained by an
elastic spring of stiffness K as shown in Figure 5.14b.
FIGURE 5.15 Test of C-section±sheathing combination to deter-
mine spring stiffness K.
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The ¯ange element is assumed to be subjected to a distrib-
uted lateral load wz per unit length and a distributed
axial force pz per unit length as shown in Figure 5.16a. It
de¯ects u in its own plane as shown in Figure 5.16b. The
torsional rotation, and hence the torsional stiffness GJ
f
,
of the ¯ange element is ignored.
The spring stiffness K depends upon the
(a) Flexibility of the web
(b) Flexibility of the connection between the tension
¯ange and the sheathing
(c) Flexibility of the sheathing
A test of a short length of purlin through-fastened to
sheathing as shown in Figure 5.15 is required to assess
this total ¯exibility and, hence, K .
The ¯ange element is then analyzed as a beam-column
as shown in Figure 5.16. In Ref. 5.15, the de¯ected shape
u of the beam-column is assumed to be a sine curve, and
an energy analysis of the system is performed, including
FIGURE 5.16 Idealized beam-column in D model: (a) forces and
restraint on ¯ange element; (b) plan of assumed de¯ected shape.
Chapter 5
152
the ¯exural strain energy of the ¯ange element, the strain
energy of the elastic spring, the potential energy of the
lateral loads wz, and the potential energy of the axial
loads pz. In Refs 5.16±5.18, a ®nite element beam-
column model is used to determine the de¯ected shape u
of the unrestrained ¯ange.
The failure criteria involve comparing the maximum
stress at the ¯ange-web junction with the local buckling
failure stress F
bw
, and the maximum stress at the ¯ange
lip junction with the distortional buckling failure stress.
5.4 BRACING
As described in Section 1.7.4, bracing of beams can be used
to increase the torsional-¯exural buckling load. In addition,
non-doubly-symmetric sections, such as C- and Z-sections,
twist or de¯ect laterally under load as a consequence of the
loading which is not located through the shear center for C-
sections or not in a principal plane for Z-sections. Conse-
quently, bracing can be used to minimize lateral and
torsional deformations and to transmit forces and torques
to supporting members.
For C- and Z-sections used as beams, two basic situa-
tions exist as speci®ed in Section D.3.2 of the AISI Speci-
®cation.
(a) When the top ¯ange is connected to deck or
sheathing in such a way that the sheathing effec-
tively restrains lateral de¯ection of the connected
¯ange as described in Section D3.2.1
(b) When neither ¯ange is so connected and bracing
members are used to support the member as
described in Section D3.2.2
A full discussion of bracing requirements for metal roof and
wall systems is given in Section 10.4.
The formulae for the design of braces speci®ed in
Section D3.2.2 are based on an analysis of the torques
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and lateral forces induced in a C- or Z-section, respectively,
assuming that the braces completely resist these forces. For
the simple case of a concentrated load acting along the line
of the web of a C-section as shown in Figure 5.17a, the
brace connected to the C-section must resist a torque equal
to the concentrated force times its distance from the shear
center. For the case of a vertical force acting along the line
FIGURE 5.17 Theoretical bracing forces at a point of concen-
trated load: (a) C-section forces; (b) Z-section forces.
Chapter 5
154
of the web of a Z-section as shown in Figure 5.17b, the brace
must resist the load tending to cause the section to de¯ect
perpendicular to the web. Section D3.2.2 also speci®es how
the bracing forces should be calculated when the loads are
not located at the brace points or are uniformly distributed
loads.
5.5 INELASTIC RESERVE CAPACITY
5.5.1 Sections with Flat Elements
As described in Section 1.7.10, a set of rules to allow for the
inelastic reserve capacity of ¯exural members is included in
Section C3.1.1(b) of the AISI Speci®cation. These rules are
based on the theory and tests in Ref. 5.19.
Cold-formed sections are generally not compact as are
the majority of hot-rolled sections and hence are not
capable of participating in plastic design. However, the
neutral axis of cold-formed sections may not be at mid-
depth, and initial yielding may take place in the tension
¯ange. Also, since the web and ¯ange thicknesses are the
same for cold-formed sections, the ratio of web area to total
area is larger than for hot-rolled sections. The inelastic
reserve capacity of cold-formed sections may therefore be
greater than that for doubly symmetric hot-rolled sections.
Consequently an allowance for inelastic reserve capacity
may produce substantial gains in capacity for cold-formed
sections even if the maximum strain in the section is
limited in keeping with the reduced ductility of cold-
formed steel sections.
Tests on cold-formed hat sections were reported in Ref.
5.19 and are shown in Figure 5.18a where the ratio C
y
of
the maximum compressive strain to the yield strain has
been plotted against the compression ¯ange slenderness for
a range of section slenderness values. An approximate
lower bound to these tests is shown, which is the basis of
Section C3.1.1(b) in the AISI Speci®cation. For sections
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with stiffened compression ¯anges with slenderness less
than 1:11=
F
y
=E
p
, a maximum value of compression ¯ange
strain equal to three times the yield strain C
y
3:0 is
allowed. For sections with stiffened compression ¯anges
FIGURE 5.18 Stresses and strains for assessing inelastic reserve
capacity: (a) compression strain factor for stiffened compression
elements; (b) stress distribution for tension and compression
¯ange yielded; (c) stress distribution for tension ¯ange not
yielded.
Chapter 5
156
with slenderness greater than 1:28=
F
y
=E
p
, the maximum
compression ¯ange strain is the yield strain C
y
1:0. For
intermediate slenderness values, linear interpolation
should be used.
From these values of C
y
, the stress and strain distri-
butions in Figures 5.18b and 5.18c can be determined. The
stress distributions assume an ideally elastic-plastic ma-
terial. The ultimate moment M
u
causing a maximum
compression strain of C
y
e
y
can then be calculated from
the stress distributions. Section C3.1.1(b) of the AISI
Speci®cation limits the nominal ¯exural strength M
n
to
1:25S
e
F
y
. This limit requires that conditions (1)±(5) of
Section C3.1.1(b) be satis®ed. These limits ensure that
other modes of failure such as torsional-¯exural buckling
or web failure do not occur before the ultimate moment is
reached. For sections with unstiffened compression ¯anges
and multiple stiffened segments, Section C3.1.1(b) of the
AISI Speci®cation requires that C
y
1.
5.5.2 Cylindrical Tubular Members
Section C6.1 of the AISI Speci®cation gives design rules for
cylindrical tubular members in bending and includes the
inelastic reserve capacity for compact sections for which
M
n
1:25F
y
S
f
.
The effective section modulus S
e
nondimensionalized
with respect to the full elastic section modulus S
f
has
been plotted in Figure 5.19 versus the section slenderness
l
s
based on the outside diameter-to-thickness ratio D=t
and the yield stress F
y
. A test database provided by
Sherman (Ref. 5.20) for constant moment and cantilever
tests on electric resistance welds (ERW) and fabricated
tubes has also been plotted on the ®gure. The most signi®-
cant feature of the AISI Speci®cation is that the full-section
plastic moment capacity can be used for compact section
tubes.
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