Yu W., LaBoube R.A. Cold-Formed Steel Design

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SPACING OF CONNECTIONS IN COMPRESSION ELEMENTS 305

The requirement of item 2 is based on the following Euler

formula for column buckling:

(KL/r)

(8.83)

by substituting σ

= 1.67f

, K = 0.6, L = s, and r =

√

12. This provision is conservative because the length

is taken as the center distance instead of the clear distance

between connections, and the coefﬁcient K is taken as 0.6

instead of 0.5, which is the theoretical value for a column

with ﬁxed end supports.

The requirement of item 3 is to ensure the spacing of

connections close enough to prevent the possible buckling

of unstiffened elements.

Additional information can be found in Refs. 8.60–8.62

and 8.92–8.94.

Example 8.9 Use the ASD method to determine the

required spacing of resistance spot welds for the compres-

sion member made of two channels and two sheets

(0.105 in. in thickness), as shown in Fig. 8.48. Assume

that the member will carry an axial load of 45 kips based

on the yield point of 33 ksi and an unbraced length

of 14 ft.

SOLUTION. Using a general rule, the following sectional

properties for the combined section can be computed:

A = 3.686 in.

= 26.04 in.

= 32.30 in.

= 2.65 in. r

= 2.96 in.

The spacing of spot welds connecting the steel sheets to

channel sections should be determined on the basis of the

following considerations:

1. Required Spacing Based on Shear Strength. Even

though the primary function of a compression member

is to carry an axial load, as a general practice, built-up

compression members should be capable of resisting

Figure 8.48 Example 8.8

a shear force of 2% of the applied axial load, that is,

V = 0.02(45) = 0.9kip

If the shear force is applied in the y direction, then

the longitudinal shear stress in line a–a is

vt =

Since

s(vt) = s





= 2 ×shear strength per spot

then

s =

2 × shear strength per spot × I

2(2.10)(26.04)

0.9(9 ×0.105)(3.0 + 0.105/2)

= 42.1in.

In the above calculation, the shear strength of resis-

tance spot welds is obtained from Eq. (8.39a) using a

safety factor of 2.50.

If the shear force is applied in the x direction, then

the shear stress is

vt =

and

s =

2 × shear strength per spot ×I

2(2.10)(32.30)

0.9[(6.0 × 0.105 × 3.0625) + (2 ×1.385

×0.105 × 3.808)]

= 49.6in.

2. Required Spacing Based on Column Buckling of Indi-

vidual Steel Sheets Subjected to Compression. Based

on the AISI requirements, the maximum spacing of

welds is

s = 1.16t



in which

45.0

3.686

= 12.2ksi

Then

s = 1.16(0.105)



29, 500

12.2

= 6.0in.

3. Required Spacing Based on Possible Buckling of

Unstiffened Elements

s = 3w = 3 × 0.75 = 2.25 in.

306 8 CONNECTIONS

However, based on item 3 of the AISI requirements,

0.75

0.105

= 7.14

0.50



= 0.50



29, 500

= 14.95

Since w/t < 0.50



E/F

, the required spacing deter-

mined above need not be less than the following

value:

1.11t



= 1.11(0.105)



29, 500

= 3.48in.

Comparing the required spacings computed in items

1, 2, and 3, a spacing of 3.5 in. may be used for the

built-up section.

If the LRFD method is used in design, the shear

force applied to the member should be computed by

using the factored loads and the design shear strength

should be determined by φP

CHAPTER 9

Shear Diaphragms and Roof

Structures

9.1 GENERAL REMARKS

A large number of research projects conducted throughout

the world have concentrated on the investigation of the

structural behavior not only of individual cold-formed

steel components but also of various structural systems.

Shear diaphragms and roof structures (including purlin roof

systems, folded-plate, and hyperbolic paraboloid roofs) are

some examples of the structural roof systems that have been

studied.

As a result of the successful studies of shear diaphragms

and roof structures accompanied by the development of new

steel products and fabrication techniques, the application

of steel structural assemblies in building construction has

increased rapidly.

In this chapter the research work and the design methods

for the use of shear diaphragms and roof structures are

brieﬂy discussed. For details, the reader is referred to the

related references.

9.2 STEEL SHEAR DIAPHRAGMS

9.2.1 Introduction

In building construction it has been a common practice to

provide a separate bracing system to resist horizontal loads

due to wind load, blast force, or earthquake. However,

steel ﬂoor and roof panels (Fig. 1.11), with or without

concrete ﬁll, are capable of resisting horizontal loads in

addition to the beam strength for gravity loads if they

are adequately interconnected to each other and to the

supporting frame.

1.6,9.1,9.2

The effective use of steel ﬂoor

and roof panels can therefore eliminate separate bracing

systems and result in a reduction of building costs (Fig. 9.1).

Figure 9.1 Shear diaphragms.

For the same reason, wall panels can provide not only

enclosure surfaces and support normal loads but also

diaphragm action in their own planes.

In addition to the utilization of diaphragm action, steel

panels used in ﬂoor, roof, and wall construction can be

used to prevent the lateral buckling of beams and the

overall buckling of columns.

4.115–4.120

Previous studies

made by Winter have shown that even relatively ﬂexible

diaphragm systems can provide sufﬁcient horizontal support

to prevent the lateral buckling of beams in ﬂoor and roof

construction.

4.111

The load-carrying capacities of columns

can also be increased considerably if they are continuously

braced with steel diaphragms.

4.116

9.2.2 Research on Shear Diaphragms

Because the structural performance of steel diaphragms

usually depends on the sectional conﬁguration of panels,

the type and arrangement of connections, the strength and

thickness of the material, span length, loading function, and

concrete ﬁll, the mathematical analysis of shear diaphragms

is complex. At the present time, the shear strength and the

stiffness of diaphragm panels can be determined either by

tests or by analytical procedures.

Since 1947 numerous diaphragm tests of cold-formed

steel panels have been conducted and evaluated by a

number of researchers and engineers. The diaphragm tests

conducted in the United States during the period 1947–1960

were summarized by Nilson in Ref. 9.2. Those tests

were primarily sponsored by individual companies for the

purpose of developing design data for the diaphragm action

of their speciﬁc panel products. The total thickness of the

panels tested generally range from 0.04 to 0.108 in. (1 to

2.8 mm). Design information based on those tests has been

made available from individual companies producing such

panels.

307

308 9 SHEAR DIAPHRAGMS AND ROOF STRUCTURES

In 1962 a research project was initiated at Cornell

University under the sponsorship of the AISI to study the

performance of shear diaphragms constructed of corrugated

and ribbed deck sections of thinner materials, from 0.017 to

0.034 in. (0.4 to 0.9 mm) in total thickness. The results of

diaphragm tests conducted by Luttrell and Apparao under

the direction of George Winter were summarized in Refs.

9.3, 9.4, and 9.5. Recommendations on the design and

testing of shear diaphragms were presented in the AISI

publication, ‘’Design of Light Gage Steel Diaphragms,”

which was issued by the institute in 1967.

9.6

Since 1967 additional experimental and analytical studies

of steel shear diaphragms have been conducted throughout

the world. In the United States, research projects on

this subject have been performed by Nilson, Ammar,

and Atrek,

9.7–9.9

Luttrell, Ellifritt, and Huang,

9.10–9.13

Easley and McFarland,

9.14–9.16

Miller,

9.17

Libove, Wu,

and Hussain,

9.18–9.21

Chern and Jorgenson,

9.22

Liedtke

and Sherman,

9.23

Fisher, Johnson, and LaBoube,

9.24–9.26

Jankowski and Sherman,

9.90

Heagler,

9.91

Luttrell,

9.92

and

others. The research programs that have been carried out

in Canada include the work of Ha, Chockalingam, Fazio,

and El-Hakim

9.27–9.30

and Abdel-Sayed.

9.31

In Europe, the primary research projects on steel shear

diaphragms have been conducted by Bryan, Davies, and

Lawson.

9.32–9.38

The utilization of the shear diaphragm

action of steel panels in framed buildings has been well

illustrated in Davies and Bryan’s book on stressed skin

diaphragm design.

9.39

In addition, studies of tall buildings

using diaphragms were reported by El-Dakhakhni in Refs.

9.40 and 9.41.

More recently, shear diaphragms have been studied

by Caccese, Elgaaly, and Chen,

9.96

Kian and Pekoz,

9.97

Miller and Pekoz,

9.98

Easterling and Porter,

9.99

Serrette and

Ogunfunmi,

9.100

Smith and Vance,

9.101

Elgaaly and Liu,

9.102

Lucas, Al-Bermani, and Kitipornchai,

9.103,9.104

Elgaaly,

9.105

Lease and Easterling,

9.110

and others. References 1.269,

9.106 and 9.111 provide additional design information on

the design and use of shear diaphragms.

In addition to the shear diaphragm tests mentioned

above, lateral shear tests of steel buildings and tests

of gabled frames with covering sheathing have been

performed by Bryan and El-Dakhakhni.

4.113,4.114

Recent

studies and design criteria for cold-formed steel framed

shear walls will be discussed in Chapter 13. The structural

behavior of columns and beams continuously braced

by diaphragms has also been studied by Pincus, Fisher,

Errera, Apparao, Celebi, Pekoz, Winter, Rockey, Nethercot,

Trahair, Wikstrom, and others.

4.115–4.120,4.135,4.136

recently, experimental work by Wang et al.

9.112

and

analytical research by Schafer and Hiriyur

9.113

have

extended the state-of-the-art. This subject is discussed

further in Section 9.3.

In order to understand the structural behavior of shear

diaphragms, the shear strength and the stiffness of steel

diaphragms are brieﬂy discussed in subsequent sections.

9.2.2.1 Shear Strength of Steel Diaphragms Results of

previous tests indicate that the shear strength per foot of

steel diaphragm is usually affected by the panel conﬁgura-

tion, the panel span and purlin or girt spacing, the mate-

rial thickness and strength, acoustic perforations, types and

arrangements of fasteners, and concrete ﬁll, if any.

Panel Conﬁguration. The height of panels has consid-

erable effect on the shear strength of the diaphragm if a

continuous ﬂat-plate element is not provided. The deeper

proﬁle is more ﬂexible than are shallower sections. There-

fore the distortion of the panel, in particular near the ends,

is more pronounced for deeper proﬁles. On the other hand,

for panels with a continuous ﬂat plate connected to the

supporting frame, the panel height has little or no effect on

the shear strength of the diaphragm.

With regard to the effect of the sheet width within

a panel, wider sheets are generally stronger and stiffer

because there are fewer side laps.

Panel Span and Purlin Spacing. Shorter span panels

could provide a somewhat larger shear strength than longer

span panels, but the results of tests indicate that the failure

load is not particularly sensitive to changes in span.

The shear strength of panels is increased by a reduction of

purlin spacing; the effect is more pronounced in the thinner

panels.

Material Thickness and Strength. If a continuous ﬂat

plate is welded directly to the supporting frame, the failure

load is nearly proportional to the thickness of the mate-

rial. However, for systems with a formed panel, the shear

is transmitted from the support beams to the plane of

the shear-resisting element by the vertical ribs of the

panels. The shear strength of such a diaphragm may be

increased by an increase in material thickness, but not

linearly.

When steels with different material properties are used,

the inﬂuence of the material properties on diaphragm

strength should be determined by tests or analytical proce-

dures.

Acoustic Perforations. The presence of acoustic perfora-

tions may slightly increase the deﬂection of the system and

decrease the shear strength.

STEEL SHEAR DIAPHRAGMS 309

Types and Arrangement of Fasteners. The shear

strength of steel diaphragms is affected not only by the

types of fasteners (welds, bolts, sheet metal screws, and

others) but also by their arrangement and spacing. The

shear strength of the connection depends to a considerable

degree on the conﬁguration of the surrounding metal.

Previous studies indicate that if the fasteners are small

in size or few in number, failure may result from shearing

or separation of the fasteners or by localized bearing or

tearing of the surrounding material. If a sufﬁcient number

of fasteners are closely spaced, the panel may fail by elastic

buckling, which produces diagonal waves across the entire

diaphragm.

The shear strength will be increased considerably by

the addition of intermediate side lap fasteners and end

connections.

Concrete Fill. Steel panels with a concrete ﬁll provide a

much more rigid and effective diaphragm. The stiffening

effect of the ﬁll depends on the thickness, strength, and

density of the ﬁll and the bond between the ﬁll and the

panels.

The effect of lightweight concrete ﬁll on shear

diaphragms has been studied by Luttrell.

9.11

It was found

that even though this type of concrete may have a very low

compressive strength of 100–200 psi (0.7–1.4 MPa), it can

signiﬁcantly improve the diaphragm performance. The most

noticeable inﬂuence is the increase in shear stiffness. The

shear strength can also be increased, but to a lesser extent.

Insulation. Based on the results of full-scale diaphragm

tests, Lease and Easterling

9.110

concluded that the presence

of insulation does not reduce the shear strength of the

diaphragm. Insulation thicknesses of up to 6

in. were

studied.

9.2.2.2 Stiffness of Shear Diaphragms In the use of

shear diaphragms, deﬂection depending upon the stiffness

of the shear diaphragms is often a major design criterion.

Methods for predicting the deﬂection of cold-formed steel

panels used as diaphragms have been developed on the basis

of the speciﬁc panels tested. In general, the total deﬂection

of the diaphragm system without concrete ﬁll is found to

be a combination of the following factors

9.2,9.39

1. Deﬂection due to ﬂexural stress

2. Deﬂection due to shear stress

3. Deﬂection due to seam slip

4. Deﬂection due to local distortion of panels and rela-

tive movement between perimeter beams and panels

at end connections

The deﬂection due to ﬂexural stress can be determined

by the conventional formula using the moment of inertia

due to perimeter beams and neglecting the inﬂuence of the

diaphragm acting as a web of a plate girder.

For simplicity the combined deﬂections due to shear

stress, seam slip, and local distortion can be determined

from the results of diaphragm tests. If shear transfer devices

are provided, the deﬂection due to relative movement

between marginal beams and shear web will be negligible.

The above discussion is based on the test results of shear

diaphragms without concrete ﬁll. The use of concrete ﬁll

will increase the stiffness of shear diaphragms considerably,

as discussed in the preceding section. When the advantage

of concrete ﬁll is utilized in design, the designer should

consult individual companies or local building codes for

design recommendations.

9.2.3 Tests of Steel Shear Diaphragms

In general, shear diaphragms are tested for each proﬁle or

pattern on a reasonable maximum span which is normally

used to support vertical loads. The test frame and connec-

tions should be selected properly to simulate actual building

construction if possible. Usually the mechanical proper-

ties of the steel used for the fabrication of the test panels

should be similar to the speciﬁed values. If a substantially

different steel is used, the test ultimate shear strength may

be corrected on the basis of Ref. 9.114.

During the past, cantilever, two-bay, and three-bay steel

test frames have been generally used. Another possible test

method is to apply compression forces at corners along a

diagonal. Nilson has shown that the single-panel cantilever

test will yield the same shear strength per foot as the three-

bay frame and that the deﬂection of an equivalent three-bay

frame can be computed accurately on the basis of the single-

panel test. It is obvious that the use of a cantilever test is

economical, particularly for long-span panels. References

9.42–9.44 and 9.114 contain the test procedure and the

method of evaluation of the test results.

The test frame used for the cantilever test is shown

in Fig. 9.2a, and Fig. 9.2b shows the cantilever beam

diaphragm test.

The three-bay simple beam test frame is shown in

Fig. 9.3a, and Fig. 9.3b shows the test setup for a simple

beam diaphragm test.

The test results can be evaluated on the basis of the

average values obtained from the testing of two identical

specimens if the deviation from the average value does

not exceed 10%. Otherwise the testing of a third identical

specimen is required by Refs. 9.42–9.44. The average of the

two lower values obtained from the tests is regarded as the

result of this series of tests. According to Ref. 9.43, if the

310 9 SHEAR DIAPHRAGMS AND ROOF STRUCTURES

Figure 9.2 (a) Plan of cantilever test frame.

9.6

(b) Cantilever beam diaphragm test.

9.1

frame has a stiffness equal to or less than 2% of that of the

total diaphragm assembly, no adjustment of test results for

frame resistance need be made. Otherwise, the test results

should be adjusted to compensate for frame resistance.

The ultimate shear strength S

in pounds per foot can be

determined from

(9.1)

where (P

ult

)

avg

= average value of maximum jack loads

from either cantilever or simple beam

tests, lb

b = depth of beam indicated in Figs. 9.2a

and 9.3a, ft

The computed ultimate shear strength divided by the

proper load factor gives the allowable design shear S

des

pounds per linear foot. (See Fig. 9.4 for the tested ultimate

shear strength of standard corrugated steel diaphragms.)

According to Ref. 9.42, the shear stiffness G



is to be

determined on the basis of an applied load of 0.4(P

ult

)

avg

for use in deﬂection determination.

∗

For the evaluation of

shear stiffness, the measured deﬂections at the free end of

∗

Reference 9.44 suggests that the shear stiffness G



is to be determined

on the basis of a reference level of 0.33(P

ult

)

avg

. If the selected load

level is beyond the proportional limit, use a reduced value less than the

proportional limit.

STEEL SHEAR DIAPHRAGMS 311

Figure 9.3 (a) Plan of simple beam test frame.

9.6

(b) Simple beam diaphragm test.

9.1

the cantilever beam or at one-third the span length of the

simple beam for each loading increment can be corrected

by the following equations if the support movements are to

be taken into account:

 =



−



+ D

)



for cantilever tests

(9.2)

+ D

− D

) for simple beam tests (9.3)

where D

, D

,andD

are the measured deﬂections

at locations indicated in Figs. 9.2a and 9.3a and a/b is

the ratio of the diaphragm dimensions. The load-deﬂection

curve can then be plotted on the basis of the corrected test

results.

The shear deﬂection for the load of 0.4(P

ult

)

avg

can be

computed from



= 



− 



(9.4)

where 



= shear deﬂection for load of 0.4P

ult



= average value of deﬂections obtained from

load–deﬂection curves for load of 0.4P

ult



= computed bending deﬂection

312 9 SHEAR DIAPHRAGMS AND ROOF STRUCTURES

Figure 9.4 Tested ultimate shear strength of standard 2

-in.

corrugated galvanized steel diaphragms.

9.42

In the computation of 



the following equations may

be used for cantilever beams. The bending deﬂection at the

free end is



(12)

3EI

(9.5)

For simple beam tests, the bending deﬂection at one-third

the span length is



5Pa

(12)

6EI

(9.6)

In Eqs. (9.5) and (9.6),

P = 0.4(P

ult

)

avg

,lb

E = modulus of elasticity of steel, 29.5 × 10

psi (203

GPa, 2070 GPa)

I = moment of inertia considering only perimeter

members of test frame,

= Ab

(12)

/ 2, in.

A = sectional area of perimeter members CD and GE

in Figs. 9.2a and 9.3a,in.

a, b = dimensions of test frame shown in Figs. 9.2a and

9.3a, ft

Finally, the shear stiffness G



of the diaphragm can be

computed as



P/b



0.4P max







(9.7)

The shear stiffness varies with the panel conﬁguration

and the length of the diaphragm. For standard corrugated

sheets, the shear stiffness for any length may be computed

by Eq. (9.8) as developed by Luttrell

9.5

if the constant K

can be established from the available test data on the same

proﬁle:

Figure 9.5 Cross section of corrugated sheets.

9.42



[2(1 +μ)g]/p + K

/(Lt)

(9.8)

where G



= shear stiffness, lb/in.

E = modulus of elasticity of steel, = 29.5 × 10

psi (203 GPa, 2070 GPa)

t = uncoated thickness of corrugated panel, in.

μ = Poisson’s ratio, = 0.3

p = corrugation pitch, in. (Fig. 9.5)

g = girth of one complete corrugation, in.

(Fig. 9.5)

L = length of panels from center to center of end

fasteners, measured parallel to corrugations,

in.

= constant depending on diaphragm cross

section and end-fastener spacing, in.

Knowing G



from the tested sheets, the constant K

can

be computed as





−

2(1 + μ)g



(Lt)

(9.9)

Figure 9.6 shows graphically the tested shear stiffness for

0.0198-in. (0.5-mm-) thick standard corrugated diaphragms.

Based on a review of diaphragm test methods, AISI

developed the Test Standard for Cantilever Test Method

for Cold-Formed Steel Diaphragms

9.114

. According to Ref.

9.114, the test frame stiffness may inﬂuence the test results

and therefore adjustments to compensate for frame resis-

tance and stiffness are provided.

The nominal diaphragm web shear strength, S

,which

is the load per unit length across the full frame test, is

calculated by

(9.10)

where

⎧

⎪

⎨

⎪

⎩

max

− P

max



max

− 0.02



max

> 0.02 (9.10a)

max

≤ 0.02 (9.10b)

STEEL SHEAR DIAPHRAGMS 313

Figure 9.6 Tested shear stiffness for 2

-in. standard corrugated steel diaphragms. Thickness

of panels = 0.0198 in.

9.42

max

= 0.4 P

max

G' =

Figure 9.7 Typical load–net deﬂection curve.

9.114

where P

max

= maximum applied load P to test frame

= load P from testing of bare frame at

deﬂection equal to deﬂection for load of

max

for strength

b = depth of diaphragm test frame and

dimension parallel with load, P

Equation (9.10) adjusts the test load for the contribution

of the frame stiffness.

For design the nominal diaphragm web strength is

reduced by either a safety factor or resistance factor as

described in Section D5 of AISI S100

1.345

. See Section

9.2.5 in this volume.

According to Ref. 9.114, the shear stiffness G’ is deter-

mined at the test load P

=0.4P

max

as illustrated by Fig. 9.7.

The shear stiffness is computed by Eq. (9.11),









(9.11)

314 9 SHEAR DIAPHRAGMS AND ROOF STRUCTURES

where

⎧

⎪

⎨

⎪

⎩

0.4P

max

− 0.4P

max



0.4P

max

− 0.02



0.4P

max

> 0.02 (9.11a)

0.4P

max

0.4P

max

≤ 0.02 (9.11b)

where P

= load P at which diaphragm stiffness shall be

determined

= load P from testing of bare test frame at

deﬂection equal to deﬂection for load of

0.4 P

max

for stiffness



= net shear deﬂection of diaphragm test at load

0.4 P

max

The shear net deﬂection illustrated by Fig. 9.8 may be

determined from either diagonal measurements or rect-

angular measurements. Both measurement methods are

acceptable. However, the rectangular method is the histor-

ical method. Common existing analytical methods

9.11, 9.115

were veriﬁed using the rectangular method.

When using diagonal measurements the net shear

deﬂection, 

, is computed as follows:



(



)

√

+ b

(9.12)

where 

1,2

= displacement measured at points 1 or 2

(see Fig. 9.8)

a = length of diaphragm test frame

b = depth of diaphragm test frame

When using the rectangular displacements (Fig. 9.9), the

net shear deﬂection is computed by Eq. (9.13),



= 

−





(



+ 

)



(9.13)

where 1,2,3,4 is the displacement measured at points 1,

2, 3, or 4 (see Fig. 9.9).

The number of tests and the conditions for acceptance of

test results are described in Ref. 9.114.

9.2.4 Analytical Methods for Determining Shear

Strength and Stiffness of Shear Diaphragms

In Section 9.2.3 the test method to be used for establishing

the shear strength and stiffness of shear diaphragms was

discussed. During the past two decades, several analytical

methods have been developed for computing the shear

strength and the stiffness of diaphragms. The following ﬁve

methods are commonly used:

Length, a

Beam

Bearing

reaction

Tension

reaction

Steel deck

panel

Load, P

Depth, b

Legend:

Displacement device mounted on top

flange of test frame. Arrow indicates

direction of

ositive readin

Wire

Figure 9.8 Diagonal displacement measurements.

9.114

1. Steel Deck Institute (SDI) method

9.45,9.106, 9.111

2. Tri-Service method

9.46

3. European recommendations

9.47

4. Nonlinear ﬁnite-element analysis

9.9

5. Simpliﬁed diaphragm analysis

9.36

6. Metal Construction Association (MCA) method

9.115

For details, the reader is referred to the referenced docu-

ments and publications.

With regard to the use of the European recommenda-

tions, a design guide was prepared by Bryan and Davies

in 1981.

9.48

This publication contains design tables and

worked examples which are useful for calculating the

strength and stiffness of steel roof decks when acting as

diaphragms.

In addition to the above publications, general design rules

on steel shear diaphragms can also be found in Ref. 9.116.

9.2.5 Design of Shear Diaphragms

Steel shear diaphragms can be designed as web elements

of horizontal analogous plate girders with the perimeter

framing members acting as the ﬂanges. The primary stress

in the web is shear stress, and in the ﬂanges the primary

stresses are axial stresses due to bending applied to the plate

girder.

It has been a general design practice to determine the

required sections of the panels and supporting beams based