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

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GENERAL DESIGN CONSIDERATIONS OF COLD-FORMED STEEL CONSTRUCTION 25

Figure 1.33 Example 1.1.

The accuracy of the linear method for computing the

properties of a given section depends on the thickness of the

steel sheet to be used and the conﬁguration of the section.

For the thicknesses of steel sheets generally used in cold-

formed steel construction, the error in the moment of inertia

determined by the linear method is usually negligible,

particularly for relatively deep sections made of thin mate-

rial. For example, as indicated in Table 1.3, the expected

Table 1.3 Expected Error in I

Channel Thickness of Expected

Section Material (in.) in I

(%)

A 0.50 3.3

0.25 0.7

0.10 0.1

B 0.50 0.6

0.25 0.15

0.10 0.02

Note: 1in. = 25.4 mm.

errors in the computed moment of inertia of the two arbi-

trarily chosen channel sections as shown in Fig. 1.34 are

less than 1% if the material is

in. or thinner.

For cylindrical tubes, the error in the computed moment

of inertia about the axis passing through the center of the

tube determined by the linear method varies with the ratio of

mean diameter to wall thickness, D/t; the smaller the ratio,

the larger the error. The expected errors in the moment of

inertia are approximately 2.7 and 0.2% for D/t ratios of 6

and 20, respectively, if the wall thickness is

in. Errors

smaller than the above values are expected for materials

thinner than

in.

The Direct Strength Method Design Guide

1.383

indicates

that the use of midline dimensions ignoring the corner is

adequate for analysis unless the corner radius is larger than

10 times the thickness.

1.6.10 Tests for Special Cases

In Section 1.1 it was indicated that in cold-formed

steel construction unusual sectional conﬁgurations can

be economically produced by cold-forming operations.

However, from the point of view of structural design, the

analysis and design of such unusual members may be very

complex and difﬁcult. In many cases it may be found

that their safe load-carrying capacity or deﬂection cannot

be calculated on the basis of the design criteria presently

included in the North American speciﬁcation.

1.345

For

this case the North American Speciﬁcation permits their

structural performance to be determined by load tests

conducted by an independent testing laboratory or by a

manufacturer’s laboratory. It is not the intent of the North

American provision, however, to s ubstitute load tests for

design calculations.

Figure 1.34 Sections used for studying the accuracy of the linear method.

26 1 INTRODUCTION

A detailed discussion on the method of testing is beyond

the scope of this book. However, when tests are found

necessary to determine structural strength or stiffness of

cold-formed sections and a ssemblies, Chapter F of the

North American speciﬁcation

1.345

and Part VI of the AISI

Design Manual

1.349

should be used for the e valuation of

test results and the determination of allowable load-carrying

capacities. In some cases the test results may be evaluated

by an experienced laboratory or consultant.

1.6.11 Cold Work of Forming

It is well known that the mechanical properties of steel

are a ffected by cold work of forming. The North American

speciﬁcation permits utilizing the increase in yield stress

from a cold-forming operation subjected to certain limita-

tions. Sections 2.7 and 2.8 discuss the inﬂuence of cold

work on the mechanical properties of steel and the utiliza-

tion of the cold work of forming, respectively.

1.7 ECONOMIC DESIGN AND OPTIMUM

PROPERTIES

The basic objective of economic design is to achieve

the least expensive construction that satisﬁes the design

requirements. One of the conditions required for the low

cost of the erected structure is that the weight of the

material be kept to a minimum, which is associated with

the maximum structural efﬁciency.

It has been shown by numerous investigators that for

a given loading system the maximum efﬁciency can be

obtained when the member strengths for all the possible

modes of failure are the same.

In practice, such ideal conditions may not be obtained

easily because of unavoidable limitations, such as pres-

elected shapes and speciﬁc dimensional limitations.

However, it can be shown that in some cases there may be

a possible mode of failure or limit state that will result in

a maximum efﬁciency within the practical limitations.

The efﬁciency of the use of high-strength steel depends

on the type of mode of failure. Under certain conditions,

such as long columns having large slenderness ratios, the

failure is usually limited by overall elastic buckling. For

this case the use of high-strength s teel may not result in

an economic design because the performance of structural

members under the above-mentioned conditions will be the

same for different grades of steel. For this reason the use

of high-strength s teel for these cases may not be justiﬁed

as far as the overall cost is concerned.

In any event the general aim should always be to utilize

the full potential strength of the steel that can be used in

fabrication by designing the detail outline of the section

for maximum structural efﬁciency. Flexibility of the cold-

forming process to produce an endless variety of shapes is

ideal for this purpose.

1.225,1.247,1.402–1.406

CHAPTER 2

Materials Used in Cold-Formed

Steel Construction

2.1 GENERAL REMARKS

Because material properties play an important role in the

performance of structural members, it is important to be

familiar with the mechanical properties of the steel sheets,

strip, plates, or ﬂat bars generally used in cold-formed

steel construction before designing this type of steel struc-

tural member. In addition, since mechanical properties are

greatly affected by temperature, special attention must be

given by the designer for extreme conditions below −30

◦

(−34

◦

C) and above 200

◦

F(93

◦

C).

Sixteen steels are speciﬁed in the current edition

of the North American Speciﬁcation

1.345

for struc-

tural applications. These steels are identiﬁed in ASTM

standards for sheet material as SS or, in the case of

high-strength, low-alloy steels, as HSLAS or HSLAS-F

steels:

ASTM A36, Carbon Structural Steel

ASTM A242, High-Strength Low-Alloy Structural Steel

ASTM A283, Low and Intermediate Te nsile Strength

Carbon and Steel Plates

ASTM A500, Cold-Formed Welded and Seamless Carbon

Steel Structural Tubing in Round and Shapes

ASTM A529, High-Strength Carbon-Manganese Steel of

Structural Quality

ASTM A572, High-Strength Low-Alloy Columbium-

Vanadium Structural Steel

ASTM A588, High-Strength Low-Alloy Structural Steel

with 50 ksi (345 MPa) Minimum Yield Point to 4 in.

(100 mm) Thick

ASTM A606, Steel, Sheet and Strip, High-Strength,

Low-Alloy, Hot-Rolled and Cold-Rolled, with Improved

Atmospheric Corrosion Resistance

ASTM A653, Steel Sheet, Zinc-Coated (Galvanized) or

Zinc-Iron Alloy-Coated (Galvannealed) by the Hot-Dip

Process

ASTM A792, Steel Sheet, 55% Aluminum-Zinc Alloy-

Coated by the Hot-Dip Process

ASTM A847, Cold-Formed Welded and Seamless High-

Strength, Low-Alloy Structural Tubing with Improved

Atmospheric Corrosion Resistance

ASTM A875, Steel Sheet, Zinc-5% Aluminum Alloy-

Coated by the Hot-Dip Process

ASTM A1003, Steel Sheet, Carbon, Metallic- and

Nonmetallic-Coated for Cold-Formed Framing Members

ASTM A1008, Steel, Sheet; Cold-Rolled, Carbon, Struc-

tural, High-Strength Low-Alloy, High-Strength Low-

Alloy with Improved Formability, Solution Hardened,

and Bake Hardenable

ASTM A1011, Steel, Sheet and Strip, Hot-Rolled, Carbon,

Structural, High-Strength Low-Alloy and High-Strength

Low-Alloy with Improved Formability

ASTM A1039, Steel, Sheet, Hot-Rolled, Carbon, Commer-

cial and Structural, Produced by the Twin-Roll Casting

Process

See Table 2.1 for the mechanical properties of these

16 steels.

In addition to the above-listed steels, other steel sheet,

strip, or plate can also be used for structural purposes

provided such material conforms to the chemical and

mechanical requirements of one of the listed speciﬁcations

or other published speciﬁcation that establishes its proper-

ties and suitability

1.345

for the type of application. Speciﬁca-

tion Section A2.2 includes additional speciﬁc requirements

for using other steels.

From a structural standpoint, the most important proper-

ties of steel are as follows:

1. Yield stress

2. Tensile strength

3. Stress–strain characteristics

4. Modulus of elasticity, tangent modulus, and shear

modulus

5. Ductility

6. Weldability

7. Fatigue strength

8. Toughness

In addition, formability and durability are also impor-

tant properties for thin-walled cold-formed steel structural

members.

Table 2.1 Mechanical Properties of Steels Referred to in Section A2.1 of the AISI North American Speciﬁcation

1.345,1.349

Minimum Minimum

Minimum Yield Tensile Elongation (%)

ASTM Thickness Stress F

Strength F

in 2 in. Gage

Steel Designation Designation (in.) (ksi) (ksi) F

Length

Carbon structural steel A36 — 36 58–80 1.61 23

High-strength, low-alloy

structural steel

A242

and under 50 70 1.40 21

to 1

46 67 1.46 21

Low- and intermediate-

tensile-strength carbon

steel plates

A283

A — 24 45–60 1.88 30

B — 27 50–65 1.85 28

C — 30 55–75 1.83 25

D — 33 60–80 1.82 23

Cold-formed welded and

seamless carbon steel

structural tubing in rounds

and shapes

A500

Round tubing

A — 33 45 1.36 25

B — 42 58 1.38 23

C — 46 62 1.35 21

D — 36 58 1.61 23

Shaped tubing

A — 39 45 1.15 25

B — 46 58 1.26 23

C — 50 62 1.24 21

D — 36 58 1.61 23

High-strength

carbon–manganese steel

A529 Gr. 50 — 50 70–100 1.40 21

55 — 55 70–100 1.27 20

High-strength, low-alloy

columbium–vanadium

steels of structural quality

A572 Gr. 42 — 42 60 1.43 24

50 — 50 65 1.30 21

55 — 55 70 1.27 20

60 — 60 75 1.25 18

65 — 65 80 1.23 17

High-strength, low-alloy

structural steel with 50 ksi

minimum yield point

A588 4 in. and under 50 70 1.40 21

Hot-rolled and cold-rolled

high-strength, low-alloy

steel sheet and strip with

improved corrosion

resistance

A606

Hot rolled as — 50 70 1.40 22

rolled

Hot rolled — 45 65 1.44 22

annealed or

normalized

Cold rolled — 45 65 1.44 22

Zinc-coated or zinc–iron

alloy-coated steel sheet

A653 SS

33 — 33 45 1.36 20

37 — 37 52 1.41 18

40 — 40 55 1.38 16

50 Class 1 — 50 65 1.30 12

50 Class 3 — 50 70 1.40 12

50 Class 4 — 50 60 1.20 12

55 — 55 70 1.27 11

HSLAS

40 — 40 50 1.25 22

50 — 50 60 1.20 20

55 Class 1 — 55 70 1.27 16

55 Class 2 — 55 65 1.18 18

60 — 60 70 1.17 16

70 — 70 80 1.14 12

80 — 80 90 1.13 10

HSLAS-F

40 — 40 50 1.25 24

50 — 50 60 1.20 22

55 Class 1 — 55 70 1.27 18

55 Class 2 — 55 65 1.18 20

60 — 60 70 1.17 18

70 — 70 80 1.14 14

80 — 80 90 1.13 12

55% aluminum–zinc

alloy–coated steel sheet by

the hot-dip process

A792 SS

Gr. 33 — 33 45 1.36 20

37 — 37 52 1.41 18

40 — 40 55 1.38 16

50 Class 1 — 50 65 1.30 12

50 Class 4 — 50 60 1.20 12

Cold-formed welded and

seamless high-strength,

low-alloy structural tubing

with improved

atmospheric corrosion

resistance

A847 — 50 70 1.40 19

(continuous)

Table 2.1 (continued )

Minimum Minimum

Minimum Yield Tensile Elongation (%)

ASTM Thickness Stress F

Strength F

in 2 in. Gage

Steel Designation Designation (in.) (ksi) (ksi) F

Length

Zinc–5% aluminum

alloy–coated steel sheet by

the hot-dip process

A875 SS

Gr. 33 — 33 45 1.36 20

37 — 37 52 1.41 18

40 — 40 55 1.38 16

50 Class 1 — 50 65 1.30 12

50 Class 3 — 50 70 1.40 12

HSLAS

Gr. 50 — 50 60 1.20 20

60 — 60 70 1.17 16

70 — 70 80 1.14 12

80 — 80 90 1.13 10

HSLAS-F

Gr. 50 — 50 60 1.20 22

60 — 60 70 1.17 18

70 — 70 80 1.14 14

80 — 80 90 1.13 12

Metallic- and

nonmetallic-coated carbon

steel sheet

A1003 ST

Gr. 33 H — 33 See note 1.08 10

37 H — 37 See note 1.08 10

40 H — 40 See note 1.08 10

50 H — 50 See note 1.08 10

Cold-rolled steel sheet,

carbon structural,

high-strength, low-alloy

and high-strength,

low-alloy with improved

formability

A1008

SS:

Gr. 25 — 25 42 1.68 26

30 — 30 45 1.50 24

33 Types 1 & 2 — 33 48 1.45 22

40 Types 1 & 2 — 40 52 1.30 20

HSLAS:

Gr. 45 Class 1 — 45 60 1.33 22

45 Class 2 — 45 55 1.22 22

50 Class 1 — 50 65 1.30 20

50 Class 2 — 50 60 1.20 20

55 Class 1 — 55 70 1.27 18

55 Class 2 — 55 65 1.18 18

60 Class 1 — 60 75 1.25 16

60 Class 2 — 60 70 1.17 16

65 Class 1 — 65 80 1.23 15

65 Class 2 — 65 75 1.15 15

70 Class 1 — 70 85 1.21 14

70 Class 2 — 70 80 1.14 14

HSLAS-F:

Gr. 50 — 50 60 1.20 22

60 — 60 70 1.17 18

70 — 70 80 1.14 16

80 — 80 90 1.13 14

Hot-rolled steel sheet and

strip, carbon, structural,

high-strength, low-alloy

with improved corrosion

resistance

A1011

SS:

Gr. 30 — 30 49 1.63 21-25

33 — 33 52 1.58 18-23

36 Type 1 — 36 53 1.47 17-22

36 Type 2 — 36 58–80 1.61 16-21

40 — 40 55 1.38 15-21

45 — 45 60 1.33 13-19

50 — 50 65 1.30 11-17

55 — 55 70 1.27 9-15

HSLAS:

Gr. 45 Class 1 — 45 60 1.33 23-25

45 Class 2 — 45 55 1.22 23-25

50 Class 1 — 50 65 1.30 20-22

50 Class 2 — 50 60 1.20 20-22

55 Class 1 — 55 70 1.27 18-20

55 Class 2 — 55 65 1.18 18-20

60 Class 1 — 60 75 1.25 16-18

60 Class 2 — 60 70 1.17 16-18

65 Class 1 — 65 80 1.23 14-16

65 Class 2 — 65 75 1.15 14-16

70 Class 1 — 70 85 1.21 12-14

70 Class 2 — 70 80 1.14 12-14

(continuous)

Table 2.1 (continued )

Minimum Minimum

Minimum Yield Tensile Elongation (%)

ASTM Thickness Stress F

Strength F

in 2 in. Gage

Steel Designation Designation (in.) (ksi) (ksi) F

Length

HSLAS-F:

Gr. 50 — 50 60 1.20 22-24

60 — 60 70 1.17 20-22

70 — 70 80 1.14 18-20

80 — 80 90 1.13 16-18

Hot-rolled, carbon,

commercial and structural

steel sheet

A1039

Gr. 40 — 40 55 1.38 15-20

50 — 50 65 1.30 11-16

55 — 55 70 1.27 9-14

60 — 60 70 1.17 8-13

70 — 70 80 1.14 7-12

80 — 80 80 1.13 6-11

Notes:

1. The tabulated values are based on ASTM standards.

2.1

2. 1 in. = 25.4 mm; 1 ksi = 6.9 MPa = 70.3 kg/cm

3. Structural Grade 80 of A653, A875, and A1008 steel and Grade 80 of A792 are allowed in the North American speciﬁcation under special conditions. For these grades,

= 80 ksi, F

= 82 ksi, and elongations are unspeciﬁed. See North American speciﬁcation for the reduction of yield stress and tensile strength.

4. For Type H of A1003 steel, the minimum tensile strength is not speciﬁed. The ratio of tensile strength to yield stress shall not be less than 1.08. Type L of A1003

steel is allowed in the North American Speciﬁcation under special conditions.

5. For A1011 steel, the speciﬁed minimum elongation in 2 in. of gage length varies with the thickness of steel sheet and strip.

6. For A1039 steel, the larger speciﬁed minimum elongation is for the thickness under 0.078–0.064 in. The smaller speciﬁed minimum elongation is for the thickness

under 0.064–0.027 in. For Grades 55 and higher that do not meet the requirement of 10% elongation, Section A2.3.2 of the North American Speciﬁcation shall be used.

YIELD STRESS, TENSILE STRENGTH, AND STRESS–STRAIN CURVE 33

2.2 YIELD STRESS, TENSILE STRENGTH,

AND STRESS–STRAIN CURVE

2.2.1 Yield Stress F

and Stress–Strain Curve

The strength of cold-formed steel structural members

depends on the yield point or yield strength of steel,

except in connections and in those cases where elastic

local buckling or overall buckling is critical. In the 2007

edition of the North American Speciﬁcation and in this

book, the generic term yield stress is used to denote either

yield point or yield strength. As indicated in Table 2.1,

the yield stresses of steels listed in the North American

speciﬁcation range from 24 to 80 ksi (165 to 552 MPa or

1687 to 5624 kg/cm

There are two general types of stress–strain curves,

as shown in Fig. 2.1. One is of the sharp-yielding type

(Fig. 2.1a) and the other is of the gradual-yielding type

(Fig. 2.1b). Steels produced by hot rolling are usually sharp

yielding. For this type of steel, the yield stress is deﬁned

by the level at which the stress–strain curve becomes

horizontal. Steels that are cold reduced or otherwise cold

worked show gradual yielding. For gradual-yielding steel,

the stress–strain curve is rounded out at the “knee” and the

yield stress is determined by either the offset method or the

strain-underload method.

2.2,2.3

Figure 2.1 Stress–strain curves of carbon steel sheet or stri p:

(a) sharp yielding; (b) gradual yielding.

Figure 2.2 Determination of yield stress for gradual-yielding

steel: (a) offset method; (b) strain-underload method.

In the offset method, the yield stress is the stress corre-

sponding to the intersection of the stress–strain curve and

a line parallel to the initial straight-line portion offset by a

speciﬁed strain. The offset is usually speciﬁed as 0.2%, as

shown in Fig. 2.2a. This method is often used for research

work and for mill tests of stainless and alloy steels. In the

strain-underload method, the yield stress is the stress corre-

sponding to a speciﬁed elongation or extension under load.

The speciﬁed total elongation is usually 0.5%, as shown in

Fig. 2.2b. This method is often used for mill tests of sheet

or strip carbon and low-alloy steels. In many cases, the

yield stresses determined by these two methods are similar.

2.2.2 Tensile Strength

The tensile strength of s teel sheets or strip used for cold-

formed steel sections has little direct relationship to the

design of such members. The load-carrying capacities of

cold-formed steel ﬂexural and compression members are

usually limited by yield stress or buckling stresses that are

less than the yield stress of steel, particularly for those

compression elements having relatively large ﬂat-width

ratios and for compression members having relatively large

34 2 MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION

slenderness ratios. The exceptions are tension members and

connections, the strength of which depends not only on the

yield stress but also on the tensile strength of the material.

For this reason, in the design of tension members and

connections where stress concentration may occur and the

consideration of ultimate strength in the design is essential,

the North American Speciﬁcation includes special design

provisions to ensure that adequate safety is provided for the

ultimate strengths of tension members and connections. As

indicated in Table 2.1, the minimum tensile strengths of the

steels listed in the North American Speciﬁcation range from

42 to 100 ksi (290 to 690 MPa or 2953 to 7030 kg/cm

). The

ratios of tensile strength to yield stress, F

, range from

1.08 to 1.88. Previous studies indicated that the effects of

cold work on cold-formed steel members depend largely

upon the spread between tensile strength a nd yield stress of

the virgin material.

2.3 MODULUS OF ELASTICITY, TANGENT

MODULUS, AND SHEAR MODULUS

2.3.1 Modulus of Elasticity E

The strength of members that fail by buckling depends

not only on the yield stress but also on the modulus of

elasticity E and the tangent modulus E

. The modulus

of elasticity is deﬁned by the slope of the initial straight

portion of the stress–strain curve. The measured values of

E on the basis of the standard methods

2.4,2.5

usually range

from 29,000 to 30,000 ksi (200 to 207 GPa or 2.0 × 10

to 2.1 × 10

kg/cm

2.76,2.77

A value of 29,500 ksi (203 GPa

or 2.07 × 10

kg/cm

) has been used by AISI in its speciﬁ-

cations for design purposes since 1946

2.78

and is retained in

the North American speciﬁcation.

1.345

This value is slightly

higher than 29,000 ksi (200 GPa or 2.0 × 10

kg/cm

)

currently used in the AISC speciﬁcation.

1.411

2.3.2 Tangent Modulus E

The tangent modulus is deﬁned by the slope of the

stress–strain curve at any point, as shown in Fig. 2.1b.

For sharp yielding, E

= E up to the yield stress, but with

gradual yielding, E

= E only up to the proportional limit.

Once the s tress exceeds the proportional limit, the tangent

modulus E

becomes progressively smaller than the initial

modulus of elasticity. For this reason, for moderate slender-

ness the sharp-yielding steels have larger buckling strengths

than gradual-yielding steels. Various buckling provisions

of the North American Speciﬁcation have been written for

gradual-yielding steels whose proportional limit is usually

not lower than about 70% of the speciﬁed minimum yield

stress.

2.3.3 Shear Modulus G

By deﬁnition, shear modulus G is the ratio between the

shear stress and the shear strain. It is the slope of the

straight-line portion of the shear stress–strain curve. Based

on the theory of elasticity, the shear modulus can be

computed by the following equation

2.52

G =

2(1 + μ)

(2.1)

where E is the tensile modulus of elasticity and μ is

Poisson’s ratio. By using E = 29,500 ksi (203 GPa or

2.07 × 10

kg/cm

)andμ = 0.3 for steel in the elastic

range, the value of shear modulus G is taken as 11,300 ksi

(78 GPa or 794 × 10

kg/cm

) in the North American Spec-

iﬁcation. This G value is used for computing the torsional

buckling stress for the design of beams, columns, and wall

studs.

2.4 DUCTILITY

Ductility is deﬁned as the extent to which a material can

sustain plastic deformation without rupture. It is not only

required in the forming process but also needed for plastic

redistribution of stress in members and connections, where

stress concentration would occur.

Ductility can be measured by (1) a tension test, (2) a

bend test, or (3) a notch test. The permanent elongation of

a tensile test specimen is widely used as the indication of

ductility. As s hown in Table 2.1, for the customary range in

thickness of steel sheet, strip, or plate used for cold-formed

steel structural members, the minimum elongation in 2 in.

(50.8 mm) of gage length varies from 10 to 30%.

The ductility criteria and performance of low-ductility

steels for cold-formed steel members and connections have

been studied by Dhalla, Winter, and Errera at Cornell

University.

2.6–2.9

It was found that the ductility measure-

ment in a standard tension test includes (1) local ductility

and (2) uniform ductility. Local ductility is designated

as the localized elongation at the eventual fracture zone.

Uniform ductility is the ability of a tension coupon to

undergo sizable plastic deformations along its entire length

prior to necking. This study also revealed that for the

different ductility steels investigated the elongation in 2 in.

(50.8 mm) of gage length did not correlate satisfactorily

with either the local or the uniform ductility of the material.

In order to be able to redistribute the stresses in the plastic

range to avoid premature brittle fracture and to achieve

full net-section strength in a tension member with stress

concentrations, it was suggested that (1) the minimum local

elongation in

in. (12.7 mm) of gage length of a stan-

dard tension coupon including the neck be at least 20%;

(2) the minimum uniform elongation in 3 in. (76.2 mm)