Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification

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can provide torsional restraint to the purlin. These stiff-

nesses can be incorporated in the ®nite element analysis

described above to simulate the effect of sheathing.

A ®nite element analysis has been performed for the

four-span continuous lapped purlin shown in Figure 5.6a

FIGURE 5.5 Sheathing stiffness: (a) plan of sheathing; (b) sheath-

ing shear stiffness (k

); (c) sheathing ¯exural stiffness (k

Chapter 5

138

by treating it as a doubly symmetric beam. It is restrained

by bracing as well as sheathing on the top ¯ange. The

sheathing was assumed to provide a diaphragm shear

stiffness of the type shown in Figure 5.5b, but no ¯exural

stiffness of the type shown in Figure 5.5c. The loading was

located at the level of the sheathing. The bending moment

diagram is shown in Figure 5.6b, and the resulting buck-

FIGURE 5.6 BMD and buckling modes for half-beam: (a) element

subdivision; (b) BMD; (c) buckling modes.

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139

ling modes for both inward and outward loading are shown

in Figure 5.6c. In the case of outward loading, buckling

occurs mainly in the end span as a result of its unrest-

rained compression ¯ange. However, for inward loading,

buckling occurs mainly at the ®rst interior support where

there is a large negative moment and an unrestrained

compression ¯ange. The buckling loads for both inward

and outward loading are approximately the same.

To analyze problems of this type in practice without

recourse to a ®nite element analysis, the C

factor has been

FIGURE 5.7 Bending moment distributions: (a) positive moment

(or negative) alone; (b) positive and negative moments.

Chapter 5

140

included in Section C3.1.2 as in Eqs. (5.8) and (5.15). The

factor is



12:5M

max

2:5M

max

 3M

 4M

 3M

5:16

where M

max

, M

, and M

are the absolute values of

moment shown in Figures 5.7a and b. Equation (5.16) is

Equation (C3.1.2-11) of the AISI Speci®cation and was

developed by Kirby and Nethercot (Ref. 5.7).

5.2.3 Bending Strength Design Equations

The interaction of yielding and lateral buckling has not

been thoroughly investigated for cold-formed sections.

Consequently, the design approach adopted in the AISI

Speci®cation is based on the Johnson parabola but factored

by 10=9 to re¯ect partial plasti®cation of the section in

bending (Ref. 5.8).

The set of beam design equations in Section C3.1.2 of

the AISI Speci®cation is



for M

 2:78M

5:17

1:11M

1 ÿ

10M

36M



for 2:78M

> M

> 0:56M

5:18

for M

 0:56M

5:19

They have been con®rmed by research on beam-columns in

Refs 5.9 and 5.10. The Eurocode 3 Part 1.3 (Ref. 1.7) beam

design curve is also shown in Figure 5.8 for comparison. A

detailed discussion of the lateral buckling strengths of

unsheathed cold-formed beams is given in Ref. 5.11.

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141

5.3 BASIC BEHAVIOR OF C- AND Z-SECTION

FLEXURAL MEMBERS

5.3.1 Linear Response of C- and Z-Sections

5.3.1.1 General

Two basic section shapes are normally used for ¯exural

members. They are Z-sections, which are point-symmetric,

and C-sections, which are singly symmetric. For wind

uplift, loading is usually perpendicular to the sheathing

and parallel with the webs of the C- and Z-sections. If there

is no lateral or torsional restraint from the sheathing, Z-

sections will move vertically and de¯ect laterally as a result

of the inclined principal axes, while C-sections will move

vertically and twist as a result of the eccentricity of the load

from the shear center. If there is full lateral and torsional

restraint, both C- and Z-sections will bend in a plane

parallel with their webs, and simple bending theory based

on the section modulus computed for an axis perpendicular

FIGURE 5.8 Comparison of beam design curves.

Chapter 5

142

to the web can be used to compute the bending stress and

shear ¯ow distributions in the section. If the sections have

the same thickness, ¯ange width, web depth, and lip width,

then the shear ¯ow distributions are approximately the

same for both sections.

5.3.1.2 Sections with Lateral Restraint Only

In order to investigate the basic linear behavior of C- and

Z-sections when full lateral restraint is applied at the top

¯ange with no torsional restraint, simply supported C- and

Z-sections subjected to uplift along the line of the web and

laterally restrained at the top ¯ange have been analyzed

with a linear elastic matrix displacement analysis which

includes thin-walled tension (Refs. 5.12 and 5.13). The

models used in the analysis are shown in Figure 5.9,

where the sections have been subdivided into 20 elements

and a lateral restraint is provided at each of the 19 internal

nodes at the points where the uplift forces are applied. The

uplift forces are statically equivalent to a uniformly distrib-

uted force w of 113 lb=in. The longitudinal stress distribu-

tions at the central cross sections are shown in Figure 5.10

and are almost identical for C- and Z-sections. The stress

distributions show that the top ¯ange remains in uniform

stress, whereas the bottom ¯ange undergoes transverse

bending in addition to compression from the vertical

component of bending. If plain C- and Z-sections were

compared, the stress distributions would be identical. The

lateral displacement at point C on the bottom ¯ange of

the sections in Figure 5.9 was computed to be 2.5 in. for

the C-section and 2.4 in. for the Z-section. It can be

concluded that C- and Z-sections with lateral restraint

only from sheathing attached to one ¯ange tend to behave

similarly, whereas C- and Z-sections without restraint

behave differently.

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143

FIGURE 5.9 Simply supported beam models: (a) C-section;

(b) Z-section.

Chapter 5

144

5.3.1.3 Sections with Lateral and Torsional

Restraint

The magnitude of the torsional restraint will govern the

degree to which the unrestrained ¯ange moves laterally for

both C- and Z-sections. For C- and Z-sections under the

wind uplift and restrained both laterally and torsionally by

sheathing, deformation will tend to be as shown in Figure

5.11a. The forces per unit length acting on the purlins as

shown in Figure 5.11b are q, representing uplift, p, repre-

senting prying, and r, representing lateral restraint at the

top ¯ange. The prying forces can be converted into a

statically equivalent torque pb

=2 as shown in Figure

5.11c. If the line of action of the uplift force q and lateral

FIGURE 5.10 Longitudinal stress distributions at central cross

section of beam models: (a) C-section, (b) Z-section.

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145

restraint force r are moved to act through the shear center

for both sections, then a statically equivalent torque m



results, as shown in Figure 5.11d, and is given by Eq. (5.20)

for a Z-section and by Eq. (5.21) for a C-section:





5:20





ÿ qd

5:21

The pb

=2 term is the torque resulting from the prying

force. The rb

=2 term is the torque resulting from the

eccentricity of the restraining force from the shear center.

The qb

=2 term is the torque resulting from the eccentricity

of the line of the screw fastener, presumed at the center of

the ¯ange, from the web centerline. The term qd

in Eq.

(5.21) is the torque resulting from the eccentricity of the

shear center of the C-section from the web centerline. Since

the forces in Figure 5.11d act through the shear center of

each section, the value of m

must be zero if the section does

not twist. Hence, the different components of Eqs. (5.20)

and (5.21) must produce a zero net result for m

for

untwisted sections.

For the Z-section, the value of r in Eq. (5.20) is given

by Eq. (5.22) for a section which is untwisted and braced

laterally:

r 

ÿI

5:22

Using Eqs. (5.20) and (5.22) and assuming that the value of

ÿI

is equal to b

, where the x- and y-axes are as

shown in Figure 5.11b, so that the sign of I

is negative, then

the second and third terms in Eq. (5.20) add to zero. Hence,

the ®rst term in Eq. (5.20) (prying force term) is zero if m

zero. Consequently, since it can be shown that the second and

third terms in Eq. (5.20) cancel approximately for conven-

tional Z-sections with the screw fastener in the center of the

¯ange, the prying force will be small for Z-sections before

structural instability of the unrestrained ¯ange occurs.

Chapter 5

146

FIGURE 5.11 C- and Z-section statics: (a) deformation under

uplift; (b) forces acting on purlins (a); (c) static equivalent of (b);

(d) static equivalent of (c) with forces through shear center (S).

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147