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

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transversely and longitudinally demonstrates the

Bauschinger effect (Ref. 2.7). In Figure 2.6b, long-

itudinal specimens demonstrate a higher yield in

tension than compression, whereas in Figure 2.6c,

FIGURE 2.6 Effects of cold work on stress-strain characteristics:

(a) effect of strain hardening and strain aging; (b) longitudinal

specimen; (c) transverse specimen.

Chapter 2

transverse specimens of the same steel with the

same degree of cold work demonstrate a higher

yield in compression than in tension.

4. Generally, the larger the ratio of the ultimate

tensile strength to the yield strength, the larger

is the effect of strain hardening during cold work.

5. Aging of a steel occurs if it is held at ambient

temperature for several weeks or for a much

shorter period at a higher temperature. As demon-

strated in Figure 2.6a, the effect of aging on a cold-

worked steel is to

a. Increase the yield strength and the ultimate

tensile strength

b. Decrease the ductility of the steel

c. Restore or partially restore the sharp yielding

characteristic

Highly worked steels, such as the corners of cold-formed

tubes which have characteristics such as shown in Figure

2.3b, do not return to a sharp yielding characteristic. Some

steels, such as the cold-reduced killed steel described in

Ref. 2.6, may not demonstrate aging.

2.5 CORNER PROPERTIES OF COLD-

FORMED SECTIONS

As a consequence of the characteristics of cold-worked steel

described in Section 2.4, the forming process of cold-formed

sections will result in an enhancement of the yield strength

and ultimate tensile strength in the corners (bends) as

demonstrated in Figure 1.15 for a square hollow section.

For sections undergoing cold-forming as demonstrated in

Figure 1.10, Section A7.2 of the AISI Speci®cation provides

formulae to calculate the enhanced yield strength F

 of

the corners. These formulae were derived from Ref. 2.6. A

brief summary of these formulae and their theoretical and

experimental bases follows.

Materials and Cold Work of Forming

The effective stress±effective strain s-E characteristic

of the steel in the plastic part of the stress-strain curve is

assumed to be described by the power function

s  kE

2:6

where k is the strength coef®cient and n is the strain-

hardening exponent. The effective stress s is de®ned by

the Von Mises distortion energy yield criterion (Ref. 2.8),

and the effective strain E is similarly de®ned. In Ref. 2.9, k

and n are derived from tests on steels of the types described

in Section 2.4 to be

k  2:80F

ÿ 1:55F

2:7

n  0:255

ÿ 0:120 2:8

where F

is the virgin yield point and F

is the virgin

ultimate tensile strength.

The equations for the analytical corner models were

derived with the following assumptions:

1. The strain in the longitudinal direction during

cold-forming is negligible, and hence a condition

of plane strain exists.

2. The strain in the circumferential direction is equal

in magnitude and opposite in sign to that in the

radial direction on the basis of the volume

constancy concept for large plastic deformations.

3. The effective stress-effective strain curve given by

Eq. (2.6) can be integrated over the area of a

strained corner.

The resulting equation for the tensile yield strength of the

corner is



R=t

2:9

Chapter 2

where



2:10

b  0:045 ÿ 1:315n 2:11

m  0:803n 2:12

and R is the inside corner radius. Equations (2.11) and

(2.12) were derived empirically from tests.

Substituting Eqs. (2.7) and (2.8) into Eqs. (2.10) and

(2.11) and eliminating b produce

 3:69

ÿ 0:819

ÿ1:79 2:13

which is Eq. A7.2-3 in the AISI Speci®cation. Substituting

Eq. (2.8) into Eq. (2.12) produces

m  0:192

ÿ 0:068 2:14

which is Eq. A7.2-4 in the AISI Speci®cation. From Eqs.

(2.9), (2.13), and (2.14) it can be shown that steels with a

large F

ratio have more ability to strain-harden than

those with a small ratio. Also, the smaller the radius-to-

thickness ratio R=t, the larger is the enhancement of the

yield point. The formulae were calibrated experimentally

by the choice of b and m in Eqs. (2.11) and (2.12) for

(a) R=t < 7

(b) F

 1:2



The formulae are not applicable outside this range.

The average yield stress of a section including the

effect of cold work in the corners can be simply computed

using the sum of the products of the areas of the ¯ats and

corners with their respective yield strengths. These are

Materials and Cold Work of Forming

taken as the virgin yield point F

 and the enhanced

yield point F

, respectively, as set out in the AISI Speci-

®cation.

2.6 FRACTURE TOUGHNESS

In some instances, structures which have been designed

according to proper static design sections fail by rapid

fracture in spite of a signi®cant safety factor and without

violating the design loading. The cause of this type of

failure is the presence of cracks, often as a result of

imperfect welding or caused by fatigue, corrosion, pitting,

etc. The reason for the rapid fracture is that the applied

stress and crack dimensions reach critical values (related to

the ``fracture toughness'' of the material) at which instabil-

ity leads to crack extension which may even accelerate to

crack propagation at the speed of an elastic wave in the

propagating medium. Thus there are three factors which

come into play: the applied load, the crack dimension, and

the fracture toughness of the material. The discipline of

``fracture mechanics'' is formulating the scienti®c and engi-

neering aspects of conditions that deal with cracked struc-

tures. A brief introduction to fracture mechanics can be

found in Refs. 2.10 and 2.11.

It is clear that fracture toughness relates to the

amount of plastic work which is needed for crack extension.

However, crack extension also relates to the elastic energy

released from the loaded body as the volume in the vicinity

of the cracked surfaces elastically relaxes. The energy

components involved therefore are positive, plastic work

done by crack extension, and negative, elastic energy

release. Once critical conditions are reached, the negative

term dominates and the structure undergoes sudden fail-

ure by a rapid fracture.

There are two factors in cold-formed materials which

reduce their ability to resist rapid fracture. The ®rst is the

reduced material ductility, described in Section 2.3, which

Chapter 2

reduces the energy absorbed in plastically deforming and

fracturing at the tip of a crack, and hence its fracture

toughness. The second is the increased residual stresses

in cold-formed sections, produced by the forming processes,

which increases the energy released as a crack propagates.

Little research appears to have been performed to date on

the fracture toughness of cold-formed sections other than

that discussed below, probably as a result of the fact that

cold-formed sections are not frequently welded. However,

where connections between cold-formed sections contain

welds, designers should be aware of the greater potential

for rapid fracture. More research is required in this area.

Investigations of cold-formed tubes with wall thickness

greater than 0.8 in. have recently been performed in

Japan (Refs. 2.12, 2.13).

A study of the fracture toughness of G550 sheet steels

in tension has been performed and described by Rogers

and Hancock (Ref. 2.14). The Mode 1 critical stress inten-

sity factor for 0.42-mm (0.016 in.) and 0.60-mm (0.024 in.)

G550 (80 ksi yield) sheet steels was determined for the

cold-rolled steel in three directions of the plane of the

sheet. Single notch test specimens with fatigue cracks

were loaded in tension to determine the resistance of

G550 sheet steels to failure by unstable fracture in the

elastic deformation range. The critical stress intensity

factors obtained by testing were then used in a ®nite

element study to determine the risk of unstable fractures

in the elastic deformation zone for a range of different

G550 sheet steel structural models. It was determined that

the perforated coupon (Ref. 2.5) and bolted connection

specimens were not at risk of fracture in the elastic

zone. A parametric study of perforated sections revealed

that the probability of failure by unstable fracture

increases, especially for specimens loaded in the transverse

direction, as G550 sheet steel specimens increase in width,

as cracks propagate, and for specimens loaded in a localized

manner.

Materials and Cold Work of Forming

REFERENCES

2.1 American Society for Testing Materials, Standard

Test Methods and De®nition for Mechanical Testing

of Steel Products, ASTM A370-95, 1995.

2.2 Dhalla, A. K. and Winter, G., Steel ductility measure-

ments, Journal of the Structural Division, ASCE, Vol.

100, No. ST2, Feb. 1974, pp. 427±444.

2.3 Dhalla, A. K. and Winter, G., Suggested steel ductility

requirements, Journal of the Structural Division,

ASCE, Vol. 100, No. ST2, Feb. 1974, pp. 445±462.

2.4 Standards Australia, Steel Sheet and Strip-Hot-

Dipped Zinc-Coated or Aluminium Zinc-Coated, AS

1397±1993, 1993.

2.5 Rogers, C. A. and Hancock, G. J., Ductility of G550

sheet steels in tension, Journal of Structural Engi-

neering, ASCE, Vol. 123, No. 12, 1997, pp. 1586±1594.

2.6 Chajes, A., Britvec, S. J., and Winter, G., Effects of

cold-straining on structural sheet steels, Journal of

the Structural Division, ASCE, Vol. 89, No. ST2, April

1963, pp. 1±32.

2.7 Abel, A., Historical perspectives and some of the main

features of the Bauschinger effect, Materials Forum,

Vol. 10, No. 1, First Quarter, 1987.

2.8 Nadai, A., Theory of Flow and Fracture in Solids, Vol.

1, McGraw-Hill, New York, 1950.

2.9 Karren, K. W., Corner properties of cold-formed steel

shapes, Journal of the Structural Division, ASCE,

Vol. 93, No. ST1, Feb. 1967, pp. 401±432.

2.10 Chipper®eld, C. G., Fracture Mechanics, Metals

Australasia, March 1980, pp. 14±16.

2.11 Abel, A., To live with cracks, an introduction to

concepts of fracture mechanics, Australian Welding

Journal, March=April 1977, pp. 7±9.

2.12 Toyoda, M., Hagiwara, Y., Kagawa, H., and Nakano,

Y., Deformability of Cold Formed Heavy Gauge RHS:

Deformations and Fracture of Columns under Mono-

Chapter 2

tonic and Cyclic Bending Load, Tubular Structures,

V, E & F.N. Spon, 1993, pp. 143±150.

2.13 Kikukawa, S., Okamoto, H., Sakae, K., Nakamura,

H., and Akiyama, H., Deformability of Heavy Gauge

RHS: Experimental Investigation, Tubular Struc-

tures V, E. & F.N. Spon, 1993, pp. 151±160.

2.14 Rogers, C. A. and Hancock, G. J., Fracture toughness

of G550 sheet steels subjected to tension, Journal of

Constructional Steel Research, Vol. 57, No. 1, 2000,

pp. 71±89.

Materials and Cold Work of Forming

Buckling Modes of Thin-Walled

Members in Compression and

Bending

3.1 INTRODUCTION TO THE FINITE STRIP

METHOD

The ®nite strip method of buckling analysis of thin-walled

sections is a very ef®cient tool for investigating the buck-

ling behavior of cold-formed members in compression and

bending. The buckling modes which are calculated by the

analysis can be drawn by using computer graphics, and

consequently it is a useful method for demonstrating the

different modes of buckling of thin-walled members. It is

the purpose of this chapter to demonstrate the ®nite strip

buckling analysis to describe generally the different modes

of buckling of cold-formed members in compression and

bending. Although this description is not central to the

application of the design methods described in later chap-

ters, it facilitates an understanding of these methods. In

addition, the ®nite strip method of analysis can be used to