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

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STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND DESIGN CRITERIA 75

with Eqs. (3.41)–(3.44) for uniformly compressed stiffened

elements with f

substituted for f, h/t substituted for w/t,

and the k value computed above. From Eq. (3.44),

λ =



1.052

√









1.052

√

17.304



(78.21)



31.25

29,500

= 0.644

Since λ<0.673, b

= h = 4.693 in. Because h

5.00/6.50 < 4andψ>0.236, Eqs. (3.55a)and(3.55b)

are used to compute b

and b

as follows:

3 + ψ

4.693

3 + 0.704

= 1.267 in.

= 2.347 in.

+ b

= 1.267 + 2.347 = 3.614 in.

Since b

+ b

> 2.754 in., the compression portion of the

web is fully effective.

3.5.2 Unstiffened Compression Elements

3.5.2.1 Unstiffened Elements under Uniform

Compression

Yielding. An unstiffened compression element, such as

the ﬂange of the I-shaped column shown in Fig. 3.32a,

may fail in yielding if the column is short and its w/t ratio

is less than a certain value. It may buckle as shown in

Fig. 3.32b at a predictable unit stress, which may be less

than the yield stress, when its w/t ratio exceeds that limit.

Local Buckling. The elastic critical local buckling stress

for a uniformly compressed plate can also be determined

by Eq. (3.16), which gives

kπ

12(1 − μ)

(w/t)

where E = modulus of elasticity

μ = Poisson’s ratio

w/t = ﬂat width–thickness ratio

k = constant depending upon conditions of edge

support and aspect ratio a/w

Figure 3.32 Local buckling of unstiffened compression elements.

1.6

76 3 STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA

Figure 3.33 Buckling coefﬁcient for rectangular plates simply

supported along three sides with one unloaded edge free.

3.7

For a long rectangular plate simply supported along three

sides, with one unloaded free edge as shown in Fig. 3.33,

k = 0.425. However, when the restraining effect of the web

is considered, k may be taken as 0.5 for the design of an

unstiffened compression ﬂange.

If the steel exhibits sharp yielding and an unstiffened

compression element is ideally plane, the element will

buckle at the critical stress determined by Eq. (3.16) with

the upper limit of F

(Fig. 3.34). However, such ideal

conditions may not exist, and an element with a moderate

w/t ratio may buckle below the theoretical elastic buckling

stress.

On the basis of experimental evidence a straight line B

is drawn in Fig. 3.34 representing those s tresses at which

sudden and pronounced buckling occurred in the tests. The

1980 edition of the AISI Speciﬁcation considered that the

Figure 3.34 Maximum stress for unstiffened compression

elements.

1.161

Figure 3.35 Correlation between test data on unstiffened

compression elements and predicted maximum stress.

1.161,3.13

upper limit of such buckling is at w/t = 63.3/



and

the endpoint of the line is at w/t = 144/

√

3.68∗

this region the element will buckle inelastically. If w/t ≤

63.3/



, the element will fail by yielding, represented by

horizontal line A.

Additional e xperimental and analytical investigations on

the local buckling of unstiffened compression elements in

the elastic range have been conducted by Kalyanaraman,

Pekoz, and Winter.

3.8,3.69–3.71

These studies considered the

effects of initial imperfection and rotational edge restraint

on the local buckling of compression elements. By using the

procedure outlined in Ref. 3.8, a more realistic value of the

local buckling coefﬁcient can be calculated for compression

elements of cold-formed steel members.

Figure 3.35 shows the correlation between some test data

and the predicted maximum stresses.

Postbuckling Strength. When the w/t ratio of an unstiff-

ened element exceeds about 25, the element distorts more

gradually at a stress about equal to the theoretical local

buckling stress (curve D in Fig. 3.34) and returns to its

original shape upon unloading because the buckling stress

is considerably below the yield stress. Sizable waving can

∗

When the yield stress of steel is less than 33 ksi (228 MPa or 2320

kg/cm

), the endpoint of line B is w/t = 25.

STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND DESIGN CRITERIA 77

occur without permanent set being caused by the additional

stress due to distortion. Such compression elements show

a considerable postbuckling strength.

Based upon the tests made on cold-formed steel sections

having unstiffened compression ﬂanges, the following

equation has been derived by Winter for the effective

width of unstiffened compression elements, for which the

postbuckling strength has been considered

3.13

b = 0.8t



max



1 − 0.202







max



(3.59)

where f

max

is the stress in the unstiffened compression

element at the supported edge (Fig. 3.36). Curve E in

Fig. 3.34 is based on Eq. (3.59) and represents the ultimate

strength of the element, which is considerably larger than

the elastic buckling stress.

Figure 3.36 Effective width of unstiffened compression element.

Based on a selected local buckling coefﬁcient of k = 0.5,

Eq. (3.59) can be generalized as follows:

b = 1.13t



max



1 − 0.286







max



(3.60)

where k is the local buckling coefﬁcient for unstiffened

compression elements. Figure 3.37 shows a comparison

between Eq. (3.36) for stiffened elements and Eq. (3.60)

for unstiffened elements.

Equation (3.59) can also be written in terms of

max

= 1.19



max



1 − 0.3



max



(3.61)

where f

is the e lastic local buckling stress determined by

Eq. (3.16) with a value of k = 0.5. The above equation

is practically identical to the empirical formula derived by

Kalyanaraman et al. on the basis of some additional results

of tests.

3.70

Based on Eq. (3.61), the reduction factor ρ for the effec-

tive width design of unstiffened elements can be determined

as follows:

ρ =

1.19(1 − 0.3/λ)

(3.62)

where λ is deﬁned in Eq. (3.44).

Figure 3.37 Comparison of generalized equations for stiffened and unstiffened compression

elements.

78 3 STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA

Prior to 1986, it had been a general practice to design

cold-formed steel members with unstiffened ﬂanges by

using the allowable stress design approach. The effective

width equation was not used in the AISI Speciﬁcation

due to lack of extensive experimental veriﬁcation and the

concern for excessive out-of-plane distortions at service

loads.

In the 1970s, the applicability of the effective width

concept to unstiffened elements under uniform compres-

sion was studied in detail by Kalyanaraman, Pekoz, and

Winter.

3.69–3.71

The evaluation of the test data using

k = 0.43 is presented and summarized by Pekoz in

Ref. 3.17, which shows that Eq. (3.43) gives a conser-

vative lower bound to the test results of unstiffened

compression elements. In addition to the strength determi-

nation, the same study also investigated the out-of-plane

deformations in unstiffened e lements. The results of

theoretical calculations and test results on sections having

unstiffened elements with w/t = 60 are presented in

Ref. 3.17. It was found that the maximum amplitude of

the out-of-plane deformations at failure can be twice the

thickness as the w/t ratio approaches 60. However, the

deformations are signiﬁcantly less at service loads.

Based on the above reasons and justiﬁcations, the

following provisions were included for the ﬁrst time in

Section B3.1 of the 1986 AISI Speciﬁcation for the design

of uniformly compressed unstiffened elements. The same

approach was used in the 1996 Speciﬁcation and is retained

in the North American speciﬁcation:

(a) Strength Determination. The effective w idths b of

unstiffened compression elements with uniform

compression are determined in accordance with

Eqs. (3.41)–(3.44) with the exception that k is taken a s

0.43 and w is as deﬁned in Section 3.2. See Fig. 3.5.

(b) Serviceability Determination. The effective widths

used in determining serviceability are calculated

in accordance with Eqs. (3.45)–(3.47) except that

k = 0.43.

Example 3.5 Determine the critical buckling stress and

critical buckling load for the thin sheet simply supported at

three edges and one edge free, as shown in Fig. 3.38. Use

U.S. customary unit.

Figure 3.38 Example 3.5.

SOLUTION

1. The critical buckling stress of the unstiffened

compression element based on Eq. (3.16) is

kπ

12(1 − μ

)(w/t)

0.425π

(29.5 × 10

)

12(1 − 0.3

)(4/0.105)

= 7.808 ksi

In the above calculation, the value of k = 0.425 is

slightly conservative (see Fig. 3.33).

2. The critical buckling load is

= Af

= 4(0.105)(7.808) = 3.279 kips

Example 3.6 Calculate the effective width of the

compression ﬂange of the channel section (Fig. 3.39) to be

used as a beam. Use F

= 33 ksi. Assume that the beam

web is fully effective and that lateral bracing is adequately

provided. Use U.S. customary unit.

SOLUTION. Because the compression ﬂange of the given

channel is a uniformly compressed unstiffened element

which is supported at only one edge parallel to the direction

of the stress, the effective width of the ﬂange for strength

determination can be computed by using Eqs. (3.41)–(3.44)

with k = 0.43.

According to Eq. (3.44), the slenderness factor λ for

f = F

λ =



1.052

√











1.052

√

0.43



1.8463

0.06





29,500

= 1.651

Figure 3.39 Example 3.6.

STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND DESIGN CRITERIA 79

Figure 3.40 Unstiffened lip subjected to stress gradient.

Since λ>0.673, use Eqs. (3.42) and (3.43) to calculate the

effective width b as follows:

b = ρw =



1 − 0.22/λ





1 − 0.22/1.651

1.651



(1.8463) = 0.97 in.

3.5.2.2 Unstiffened Elements with Stress Gradient In

concentrically loaded compression members and in ﬂexural

members where the unstiffened compression element is

parallel to the neutral axis, the stress distribution is uniform

before buckling. However, in some cases, such as the lips

of the beam section s hown in Fig. 3.40, which are turned

in or out and are perpendicular to the neutral axis, the

compression stress is not uniform but varies in proportion

to the distance from the neutral axis.

An exact determination of the buckling condition of such

elements is complex. When the stress distribution in the lip

varies from zero to the maximum, the buckling coefﬁcient

k may be obtained from Fig. 3.41.

3.7

The local buckling of

unstiffened elements under nonuniform compression was

discussed by Kalyanaraman and Jayabalan in Ref. 3.169.

In Section B3.2 of earlier editions of the AISI

Speciﬁcation,

1.314,1.336

the effective widths of unstiffened

compression elements and edge stiffeners with stress

gradient were treated as uniformly compressed elements

Figure 3.41 Buckling coefﬁcient for unstiffened compression

elements subjected to nonuniform stress.

3.7

with the stress f to be the maximum compressive stress in

the element. This conservative design a pproach was found

to be adequate by Rogers and Schuster on the basis of the

comparisons made with the available test data.

3.170

In the early 2000s, additional investigations on the

unstiffened elements with stress gradient were carried

out by Yiu and Pekoz at Cornell University

3.207,3.208

and by Bambach and Rasmussen at the University of

Sydney.

3.209–3.216

These s tudies included plain channels

bending about the minor axis, so that the unstiffened

elements are under a stress gradient with one longitudinal

edge in compression and the other longitudinal edge in

tension. According to the studies of the University of

Sydney, the effects of the stress distribution in unstiffened

elements on the effective width are shown in Fig. 3.42.

1.346

It can be seen that the effective width of an unstiffened

element increases as the stress at the supported edge

changes from compression to tension.

Subsequently, in 2004, new design provisions were added

in Section B 3.2 of the North American Speciﬁcation for

determining the buckling coefﬁcient k , the reduction factor

ρ, and the effective width b of the unstiffened elements and

edge stiffeners with stress gradient.

1.343

These provisions

can be used not only for unstiffened elements under a

stress gradient with both longitudinal edges in compression

but also for unstiffened elements under a stress gradient

with one longitudinal edge in compression and the other

longitudinal edge in tension. The following excerpts are

adapted from Section B3.2 of the 2007 edition of the North

American Speciﬁcation:

B3.2 Unstiffened Elements and Edge Stiffeners with

Stress Gradient

The following notation shall apply in Section B3.2 of the

speciﬁcation:

b = effective width measured from the supported

edge, determined in accordance with

Eq. (3.41)–Eq. (3.44) with f equal to f

and

with k and ρ being determined in accordance with

this section

= overall width of unstiffened element of unstiffened

C-section member as deﬁned in Fig. 3.45

80 3 STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA

Figure 3.42 Reduction factor ρ vs. slenderness factor λ for unstiffened elements with stress

gradient.

1.346

Figure 3.43 Unstiffened elements under stress gradient, both longitudinal edge in

compression.

1.345

, f

= stresses shown in Figs. 3.43, 3.44, and 3.45

calculated on the basis of the gross section, where

and f

are both compression, f

≥ f

= overall depth of unstiffened C-section member as

deﬁned in Fig. 3.45

k = plate buckling coefﬁcient deﬁned in this section or,

otherwise, as deﬁned in Section 3.5.1.1

t = thickness of element

w = ﬂat width of unstiffened element, where w/t ≤ 60

ψ =|f

| (absolute value) (3.63)

λ = slenderness factor deﬁned in Section 3.5.1.1 with

f = f

ρ = reduction factor deﬁned in this section or,

otherwise, as deﬁned in Section 3.5.1.1

(a) Strength Determination. The effective width, b,ofan

unstiffened element under stress gradient shall be determined

in accordance with Section 3.5.1.1 with f equal to f

and the

plate buckling coefﬁcient, k , determined in accordance with

Section B3.2 of the speciﬁcation, unless otherwise noted. For

the cases where f

is in compression and f

is in tension,

ρ in Section 3.5.1.1 shall be determined in accordance with

this section.

1. When both f

and f

are in compression (Fig. 3.43), the

plate buckling coefﬁcient shall be calculated in accordance

STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND DESIGN CRITERIA 81

Figure 3.44 Unstiffened elements under stress gradient, one longitudinal edge in compression

and the other longitudinal edge in tension.

1.345

Figure 3.45 Unstiffened elements of C -section under stress gradient for alternative methods.

1.345

with either Eq. 3.64 or Eq. 3.65 as follows: If the stress

decreases toward the unsupported edge (Fig. 3.43a),

k =

0.578

ψ +0.34

(3.64)

If the stress increases toward the unsupported edge

(Fig. 3.43b),

k = 0.57 − 0.21ψ + 0.07ψ

(3.65)

2. When f

is in compression and f

in tension (Fig. 3.44),

the reduction factor and plate buckling coefﬁcient shall be

calculated as follows:

i. If the unsupported edge is in compression (Fig. 3.44a):

ρ =

⎧

⎪

⎨

⎪

⎩

1whenλ ≤ 0.673(1 + ψ)

(1 + ψ)

{

1 − [0.22(1 +ψ)]/λ

}

when λ>0.673(1 +ψ)

(3.66)

k = 0.57 + 0.21ψ + 0.07ψ

(3.67)

ii. If the supported edge is in compression (Fig. 3.44b),

for ψ<1

ρ =

⎧

⎪

⎨

⎪

⎩

1whenλ ≤ 0.673

(1 − ψ)



1 −

0.22



+ ψ

when λ>0.673

(3.68)

k = 1.70 + 5ψ + 17.1ψ

(3.69)

and for ψ ≥ 1

ρ = 1

The effective width, b, of the unstiffened elements of

an unstiffened C-section member shall be permitted t o

be determined using the following alternative methods, as

applicable:

1. Alternative 1 for unstiffened C-sections: When the unsup-

ported edge is in compression and the supported edge is in

tension (Fig. 3.45a),

b =



w when λ ≤ 0.856 (3.70)

ρw when λ>0.856 (3.71)

where

ρ =

0.925

√

(3.72)

k = 0.145





+ 1.256 (3.73)

0.1 ≤

≤ 1.0

2. Alternative 2 for unstiffened C-sections: When the

supported edge is in compression and the unsupported edge

is in tension (Fig. 3.45b), the effective width is determined

in accordance with Section 3.5.1.2.

In calculating the effective section modulus S

in Section

4.2.2 or S

in Section 4.2.3.3, the extreme compression ﬁber in

Figs. 3.43b,3.44a, and 3.45a shall be taken as the edge of the

effective section closer to the unsupported edge. In calculating

82 3 STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA

the effective section modulus S

in Section 4.2.2, the extreme

tension ﬁber in Figs. 3.44b and 3.45b shall be taken as the

edge of the effective section closer to the unsupported edge.

(b) Serviceability Determination. The effective width b

used

in determining serviceability shall be calculated in accordance

with speciﬁcation Section B3.2(a), except that f

and f

are

substituted for f

and f

as shown in Figs. 3.43, 3.44, and 3.45,

respectively, based on the gross section at the load for which

serviceability is determined.

The applications of the above design provisions are

illustrated in Examples 3.7 and 4.2.

3.5.3 Uniformly Compressed Elements with Stiffeners

3.5.3.1 Uniformly Compressed E lements with a Simple

Lip Edge Stiffener An edge stiffener is used to provide

a c ontinuous support along a longitudinal edge of the

compression ﬂange to improve the buckling s tress. Even

though in most cases the edge stiffener takes the form

of a simple lip (Fig. 3.46), other types of stiffeners, as

shown in Fig. 3.47, can also be used for cold-formed steel

members.

3.78,3.173

In order to provide the necessary support for the compres-

sion element, the edge stiffener must possess sufﬁcient

rigidity. Otherwise it may buckle or displace perpendicular

to the plane of the element to be stiffened.

Both theoretical and experimental investigations on the

local stability of ﬂanges stiffened by lips and bulbs have

been conducted in the past.

3.75–3.82,3.217–3.220

The design

requirements included in the 1986 and 1996 editions of the

AISI Speciﬁcation for uniformly compressed elements with

Figure 3.46 Edge stiffener.

Figure 3.47 Edge stiffeners other than simple lip.

3.78

an edge stiffener are based on the analytical and exper-

imental investigations on adequately stiffened elements,

partially stiffened elements, and unstiffened elements

conducted by Desmond, Pekoz, and Winter

3.75,3.76

with

additional studies carried out by Pekoz and Cohen.

3.17

Those design provisions were developed on the basis of

the critical buckling criterion and the ultimate-strength

criterion. In this design approach, the design requirements

recognize that the needed stiffener rigidity depends on

the width-to-thickness ratio of the plate element being

stiffened. The interaction between the plate element and the

edge stiffener is compensated for in the design equations

for the plate buckling coefﬁcient, the reduced effective

width of a simple lip edge stiffener, and the reduced area

of other stiffened shapes. Because a discontinuity exists

in the 1996 AISI design provisions, in 2001, Dinovitzer’s

expressions were adopted in the ﬁrst edition of the North

American Speciﬁcation for determining the constant “n”

to eliminate the discontinuity.

3.221

In 2007, the design provisions of the ﬁrst edition of

the North American Speciﬁcation were revised to limit the

design equations for applying only to simple lip edge stiff-

eners due to the fact that previous equations for complex

lip stiffeners were found to be unconservative, in compar-

ison with the nonlinear ﬁnite-element analysis conducted

by Schafer, Sarawit, and Pekoz.

3.222

According to Section B4 of the 2007 edition of the

North American Speciﬁcation, the effective width of the

uniformly compressed elements with a simple lip edge

stiffener can be calculated by the following equations.

For other stiffener shapes, the design of member strength

may be handled by the direct-strength method provided in

Appendix 1 of the North American Speciﬁcation:

(a) Strength Determination. For w/t ≤ 0.328S

= 0 (no edge stiffener needed) (3.74)

b = w (3.75)

= b

w (see Figure 3.48) (3.76)

= d



For w/t > 0.328 S





(b)(R

) (see Figure3.48) (3.77)

= b − b

(see Figure 3.48) (3.78)

= d



) (3.79)

where

S = 1.28



(3.80)

STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND DESIGN CRITERIA 83

Figure 3.48 Elements with simple lip edge stiffener.

1.345

w is the ﬂat dimension deﬁned in Fig. 3.48; t is the

thickness of the section; I

, the adequate moment of inertia

of the stiffener so that each component element will behave

as a stiffened element, is deﬁned as

= 399t



w/t

− 0.328



≤ t



115

w/t

+ 5



(3.81)

b is the effective design width; b

, b

are the portions

of effective design width as deﬁned in Fig. 3.48; d

the reduced effective width of the stiffener as deﬁned

in Fig. 3.48 and used in computing overall effective

sectional properties; d



is the effective width of the stiff-

ener calculated in accordance with Section 3.5.2.2 (see

Fig. 3.48); and

) =

≤ 1 (3.82)

where I

is the moment of inertia of the full section

of stiffener about its own centroidal axis parallel to the

element to be stiffened. For edge stiffeners, the round corner

between the stiffener and element to be stiffened is not

considered as a part of the stiffener:

t sin

θ) (3.83)

See Fig. 3.48 for deﬁnitions of other dimensional variables.

The effective width b in Eqs. (3.77) and (3.78) shall be

calculated in accordance with Section 3.5.1.1 with the plate

buckling coefﬁcient k as given in Table 3.5, where

n =



0.582 −

w/t



≥

(3.84)

(b) Serviceability Determination. The effective width b

used in determining serviceability shall be calculated as

in item (a), except that f

is substituted for f, where f

Table 3.5 Determination of Plate Buckling

Coefﬁcient k

Simple Lip Edge Stiffener (140

◦

≥θ ≥40

◦

)

D/w ≤ 0.25 0.25 < D/w ≤ 0.8

3.57 (R

)

+ 0.43 ≤ 4 (4.82 – 5D/w)(R

)

+ 0.43 ≤ 4

84 3 STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA

Figure 3.49 Stress distribution in edge-stiffened ﬂange.

3.17

the computed compressive stress in the effective section at

the load for which serviceability is determined.

According to Ref. 3.17, the distribution of longitudinal

stresses in a compression ﬂange with an edge stiffener is

shown in Fig. 3.49 for three cases.

The design criteria are intended to account for the

inability of the edge stiffener to prevent distortional buck-

ling by reducing the local buckling coefﬁcient k for

calculating the effective design width of the compression

element. Because the empirical equations were derived on

the basis of the tests of back-to-back sections with strong

restraint against web buckling, past research demonstrated

that these AISI design equations may provide unconser-

vative strength predictions for laterally braced beams and

columns with edge-stiffened ﬂanges when the distortional

buckling mode of the compression ﬂange is critical.

3.168

Additional discussions of distortional buckling are given in

Section 4.2.4 for beams and Section 5.6 for columns.

Recent compression tests c onducted by Young and

Hancock on channels with inclined simple edge stiffeners

using a yield stress of 450 MPa (65.3 ksi or 4588 kg/cm

)

indicated that the design strength predicted by the North

American Speciﬁcation are conservative for all channels

with outward and inward edge stiffeners, when the ﬂange

w/t ratios are between 20 and 30, but are slightly

unconservative for channels with the ﬂange w/t ratios

between 40 and 50, except for channels with inward

edge stiffeners.

3.219

For channels having ﬂange w/t ratio