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Journal of Constructional Steel Research 63 (2007) 1590–1602www.elsevier.com/locate/jcsr

Compression tests of welded section columns undergoingbuckling interaction

Young Bong Kwona,∗, Nak Gu Kim b, G.J. Hancock c

a Department of Civil Engineering, Yeungnam University, Gyongsan, 712-749, Republic of Koreab Yeungnam University, Gyongsan, 712-749, Republic of Korea

c Faculty of Engineering, University of Sydney, NSW 2006, Australia

Received 1 September 2006; accepted 31 January 2007

Abstract

This paper describes a series of compression tests performed on welded H-section and channel section columns fabricated from a mild steel

plate of thickness 6.0 mm with nominal yield stress of 240 MPa. The ultimate strength and performance of the compression members undergoing

nonlinear interaction between local and overall buckling were investigated experimentally and theoretically. The compression tests indicated that

the interaction between local and overall buckling had a significant negative effect on the ultimate strength of the thin-walled welded steel section

columns. The Direct Strength Method (DSM), which was newly developed and adopted as an alternative to the effective width method for the

design of cold-formed steel sections recently by NAS (AISI, 2004), was calibrated by using the test results for application to welded steel sections.

This paper confirms that the Direct Strength Method can properly predict the ultimate strength of welded section columns when local buckling

and flexural buckling occur simultaneously or nearly simultaneously.c 2007 Elsevier Ltd. All rights reserved.

Keywords: Direct strength method; Effective width method; Interaction between local and overall buckling; Ultimate strength; Performance; Welded sections

1. Introduction

The compression and flexural members of hot-rolled

shapes and welded sections fabricated from hot-rolled plate

will normally buckle in local and flexural/flexural–torsional

buckling mode [1], while the cold-formed steel sections will

buckle in the distortional mode in addition to those modes [2].

However, whenever the local or distortional buckling stress

is lower than the overall buckling stress, the interaction

between local or distortional and overall buckling may occur

and have a significant effect on the performance of thesections [3–5]. Since the interaction between local and overall

buckling generally deteriorates the overall column strength,

it is necessary to account for the negative effect of buckling

interaction in the conservative prediction of the ultimate

strength of columns.

∗ Corresponding address: Department of Civil and Environmental Engineer-ing, 214-1 Daedong, 712-749 Gyongsan-si, Gyongbuk-do, Republic of Korea.Tel.: +82 53 810 2418, +82 11 802 2418; fax: +82 53 810 4622.

E-mail address: [email protected] (Y.B. Kwon).

The ultimate strength of compression members, which are

composed of thin plate elements, is dependent on both the

width–thickness ratio of the plate elements and the slenderness

ratio of the columns. When the local buckling stress is lower

than the overall buckling stress or both types of buckling

occur nearly simultaneously, local buckling may negatively

affect the column strength. Because the local buckling mode

has a post-buckling strength reserve, it has generally been

considered in the design strength through the effective width

concept [6–8]. However, since the computation of the effective

width can be tedious and complicated, some countries such asJapan and Korea, do not account for the post-local-buckling

strength reserve in the design strength of compression members

to maintain simplicity. Instead of using the effective width

concept, the overall column strength is simply reduced by

the ratio of the design strength based on the local buckling

stress to the yield stress in the Korean Highway Bridge Design

Specifications [9].

The Direct Strength Method (DSM) has been developed

by Schafer and Pekoz [10] and studied further by many

researchers [11–13]. The method has been developed to

0143-974X/$ - see front matter c

2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jcsr.2007.01.011

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Y.B. Kwon et al. / Journal of Constructional Steel Research 63 (2007) 1590–1602 1591

Notation

Abbreviation

ASD Allowable stress design

DSM Direct strength method

EWM Effective width method

LRFD Load and resistance factor design

Symbols

A Cross sectional area

b Clear width of plate element

b f Flange width

d Web depth

F crl Elastic local buckling stress

F cro Elastic flexural/flexural–torsional buckling stress

F ne Overall column strength based on the overall

buckling stress

f nl Design limiting stress accounting for interaction

of local and overall buckling

F l Nominal strength based on the local buckling

stress

F max Ultimate strength determined in test

F y Yield stress

l Length of specimen

Pcrl Elastic local buckling load (=F crl × A)

Pl Resulting limiting load (=F l × A)

Pnl Design limiting strength (=F nl × A)

Pne Compression member design strength (=F ne× A)

Qa Form factor for stiffened element

Qs Form factor for unstiffened elementr Radius of gyration

t f Flange thickness

t w Web thickness

λ Slenderness ratio factor (=√ F ne /F crl ,√

Pne /Pcrl )

overcome the weak points of the effective width method

(EWM), which has been used in thin-walled steel section design

for over 60 years. These weak points are the complication in

accurate computation of effective width and the difficulty in

consideration of elements interacting in isolation. Also, as cold-formed steel sections become more complex with additional

edge and intermediate stiffeners, calculations of effective width

can become very complex. The interaction between local or

distortional and overall buckling for cold-formed steel sections

has been studied, and simple design formulas have already been

presented. In 2004, the Direct Strength Method was adopted as

an alternative to the effective width method, which has been

used for thin section design by NAS Supplement 1 (AISI,

2004) [14] and the Australian/New Zealand Cold-Formed Steel

Structures Standard AS/NZS 4600 [15]. However, there has

been less research concerning the application of this method

to hot-rolled and welded sections.

This paper aims to develop the direct strength formulas

of design for welded section compression members. The

application of the Direct Strength Method to mild steel

welded sections was studied experimentally and theoretically.

A series of compression tests was performed on welded H-

sections and C-sections fabricated from mild steel plates of

thickness 6.0 mm with nominal yield stress 240 MPa todevelop the direct strength formulas for the welded steel section

compressive members undergoing interaction between local

and flexural/flexural–torsional buckling. Nonlinear analyses

of the sections tested have also been conducted to compare

their results with the test results. The direct strength formulas

of design were compared with the effective width method

(EWM) and the allowable stress design method (ASD), which is

currently used in Korea. The direct strength formulas proposed

have been proven accurate and efficient to predict the ultimate

strength of columns when local buckling and overall buckling

occur simultaneously or nearly simultaneously. The results of

David and Hancock [16] and Rasmussen and Hancock [17] are

also included in the paper for comparison.

2. Test sections

2.1. General

A series of compression tests were performed on mild steel

H-sections and C-sections of thickness 6.0 mm with nominal

yield and ultimate stresses 240 MPa and 400 MPa, respectively,

which were fabricated by the continuous fillet welding of

effective width on both sides of the flange–web joint. The

size of fillet welds was determined as 6.0 mm according to

the AISC specifications [7]. The geometry of the welded H-section and the C-section tested are shown in Fig. 1. The

dimensions of the test specimens were optimized in order to

ensure that the local buckling stress was slightly lower than

the flexural/flexural–torsional buckling stress and consequently,

the interaction between local and overall buckling occurred

before the ultimate load was reached. The cross sections tested

were optimized by using a finite strip analysis program THIN-

WALL [18], repeatedly. The width–thickness ratios of the webs

and the flanges of the test sections were selected, such that

the elastic local buckling stress of the section was lower than

half the yield stress ( F y /2) and significant post-local-buckling

strength reserve was displayed before the ultimate load, which

is mainly dependent on the overall buckling strength, was,reached.

The limiting width–thickness ratio for Class 3 cross sections

is expressed as d /t w ≤ 42

235/F y for the web and

b/t f ≤ 14

235/F y for the flange of the sections according

to Eurocode3 [8]. Since the width–thickness ratios of the webs

or the flanges of sections tested are larger than the slenderness

limit values, the sections selected can be classified as Class 4,

where the effective width must be used to account for the local

buckling effect. The dimensions of the H-sections and the C-

sections tested are summarized in Table 1. The thickness of all

the specimens tested was 6.0 mm, and the dimensions of the

webs and flanges of the sections were adjusted to ensure the

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1592 Y.B. Kwon et al. / Journal of Constructional Steel Research 63 (2007) 1590–1602

(a) H-section. (b) C-section.

Fig. 1. Test sections.

Table 1

Dimensions of test sections

Specimens b f (mm) d (mm) t f (mm) t w (mm) d /t w b/t f l (mm) k ∗l/r A (mm2)

H-1 100.0 400.0 6.0 6.0 66.7 7.8 1400.0 58.59 3600.0

H-2 160.0 550.0 6.0 6.0 91.7 12.8 2200.0 54.95 5220.0

H-3 140.0 500.0 6.0 6.0 83.3 11.2 2400.0 63.49 4680.0

C-1 153.0 550.0 6.0 6.0 91.7 24.5 2200.0 55.37 5136.0

C-2 103.0 500.0 6.0 6.0 83.3 16.0 2200.0 53.97 4236.0

C-3 303.0 150.0 6.0 6.0 25.0 49.5 2200.0 22.89 4536.0

k ∗ = 0.7.

interaction between local buckling and overall buckling of the

test sections.

2.2. Determination of overall length of specimens

For the columns tested, the effect of the interaction betweenlocal and overall buckling was the focus of the tests. In order to

determine the optimum lengths of the test sections, the local

and overall buckling stresses of the specimens needed to be

computed accurately. The elastic buckling stresses of the H-

2 and C-2 specimens, subjected to uniform compression, are

illustrated in Fig. 2(a) and 2(b) as the buckling stress versus

buckle half-wavelength, respectively. The lengths of the test

specimens were determined by using the elastic buckling stress

versus buckle half-wavelength curves obtained by the program

THIN-WALL [18]. To ensure that the overall buckling stress

is higher than the local buckling stress and that there exists

a significant post-local-buckling strength reserve before the

final collapse of the specimens, the test column lengths were

determined. The lengths of test specimens should be in the

hatched range shown in Fig. 2(a) and (b), which are longer than

the half-wavelengths where the mixed mode of local and overall

buckling occurs (point B) and shorter than the half-wavelengths

(point D) where the overall buckling stresses are equal to the

local buckling stresses. To avoid the failure due to only the local

buckling without buckling interaction, the overall lengths of the

test specimens were determined to be in range from 1400 to

2400 mm.

The local buckling stress minima (point A) occurred in the

curves at the half-wavelengths of 480 mm and 500 mm and the

local buckling stresses of 118.7 MPa and 106.5 MPa for the

C-2 specimen and the H-2 specimen, respectively. The mixed

mode buckling between local and overall buckling occurred

at the half-wavelength of 1000 mm and 1200 mm for the C-

2 specimen and the H-2 specimen, respectively. The overall

buckling modes of the H-2 specimen of 1540 mm in length

(point C) was the flexural buckling mode in slight interactionwith the local buckling mode, as shown in Fig. 2(b). Since

the flexural buckling load about the unsymmetrical axis ( x-

axis) of the C-2 section was larger than the flexural–torsional

buckling load about the symmetrical axis ( y-axis), the overall

buckling mode at the length of 1540 mm was the flexural

mode about x -axis in slight interaction with the local buckling

rather than the flexural–torsional mode about y -axis, as shown

in Fig. 2(a). A noticeable interaction between overall buckling

and local buckling was generally observed for the H-section

and C-section columns at lengths between points B and D in

Fig. 2(a) and (b). Since the unloaded and the loaded end of the

test columns were a fixed and a hinged boundary condition,respectively, the effective length of the columns was taken

as 0.7 L. The slenderness ratios (k L/r ) of the test specimens

ranged from 22.9 to 63.5.

2.3. Numerical analysis results

The material and geometrical nonlinear analysis of the

specimens selected for the compression test was conducted

using the program LUSAS [19], to investigate the ultimate

strength and the structural behavior. To study the effect of

buckling interaction between local and overall modes on the

ultimate strength of the columns, an elastic buckling analysis

was conducted first to find out the local buckling mode.

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(a) C-2 section. (b) H-2 section.

Fig. 2. Typical buckling stress versus half-wavelength curves.

The first several local buckling loads and modes were nearly

similar. Since the buckling interaction seemed to be quite

sensitive to the imperfections, the sensitivity of the magnitude

of imperfections was studied. The load versus shortening curvesof the H-2 section obtained by the nonlinear analysis using the

LUSAS with the different initial imperfections which were the

first local buckling mode multiplied by the magnification factor

0.1, 0.01 and 0.001, respectively, are compared in Fig. 3(a) and

3(b). In the analysis, a triggering load was applied laterally

at the column center to cause the interaction between local

buckling and overall buckling. The ultimate loads of the H-2

section computed with different magnitudes of imperfections

were nearly similar in the case of 0.1% triggering load applied

as shown in Fig. 3(a). The ultimate loads obtained with 0.1%

triggering load were slightly higher than those obtained with

0.5% triggering load. The ultimate loads and the structuralbehavior after the peak load became slightly different in the

case of 0.5% triggering load applied as shown in Fig. 3(b). In

consideration of the effect of the initial imperfections and the

triggering load, the initial imperfections could be assumed as

the first local buckling mode multiplied by the factor of 0.01

and the triggering load could be taken as 0.1% of the vertical

reference load in the further analysis.

The 4-node shell element (QTS4) was used for the numerical

modeling of the section, and the loaded end was assumed as

a hinged boundary condition with the vertical direction of the

loaded end to be free to move to allow the vertical application of

the load. The uniform displacement control technique was used

at the loaded end to make a similar boundary condition to thecompression test. The bottom end of the column was assumed

as a fixed boundary condition as the test condition. The average

yield stress obtained from the tensile tests of the coupons which

were cut from the flange and web of the welded sections was

approximately 260 MPa, which was slightly higher than the

nominal yield stress of 240 MPa. The effect of the magnitude

of the yield stress on the ultimate strengths of the C-2 and

H-2 sections was not negligible as illustrated in Fig. 3(c).

Therefore, the yield stress of the material was taken as 260 MPa

in the further analysis of the specimens. Young’s modulus was

assumed to be 2.0 × 105 MPa and Poisson’s ratio was taken

as 0.3. The stress–strain relation of the material was assumed

elastic–perfectly plastic neglecting the strain-hardening and the

von Mises yield criterion was applied for the plasticity theory

of the material.

The ultimate stresses of the sections obtained by thenonlinear analysis are summarized in Table 2. The elastic

local and overall buckling stresses, and the local buckle half-

wavelength obtained by the program THIN-WALL [18] are also

given in the table for comparison. The elastic overall buckling

modes obtained by THIN-WALL were in the interaction

mode between local buckling and flexural/flexural–torsional

buckling. Since the interaction between local buckling and

flexural/flexural–torsional buckling negatively affected the

column buckling stress, Euler buckling stresses F E were

expected to be higher than the overall buckling stresses

computed by THIN-WALL. The elastic local buckling stresses

were lower than the maximum stresses and overall buckling

stresses of the sections. Since the elastic overall buckling

stresses of test columns were larger than the elastic local

buckling stresses by the difference between 69.6 MPa for H-

1 and 112.9 MPa for C-2, it was supposed that most of test

columns might fail in the mixed mode of local and overall

buckling.

3. Compression tests

3.1. General

The H-sections and the C-sections listed in Table 1 were

chosen for the pseudo-static compression test, considering the

maximum loading capacity and the dimension of the swivel

head of the testing machine. The steel grade of the test sections

was SM400 structural steel to KS D3515 [20] (equivalent to

ASTM A36 Steel), of which the nominal yield stress and the

ultimate tensile stress were 240 MPa and 400 MPa, respectively.

The end plates of thickness 30 mm were welded to both ends of

the specimens to minimize eccentric loading and to prevent the

local failure of the specimen ends. The loaded end boundary

condition was free about the x- and y-direction rotations and

movable in the vertical direction, and the bottom end boundary

condition of the specimens was fully fixed.The concentric compression test was conducted by using

a 3000 kN MTS testing machine. Downward loading at

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Table 2

Maximum and buckling stress of sections

Specimen Maximum stress

F max (MPa)

Elastic local buckling

stress F crl (MPa)

Elastic overall buckling

stress F cro (MPa)

Euler buckling stress

F E (MPa)

Local buckle half-wavelength

l (mm)

H-1 213.5 197.9 267.5 575.0 380

H-2 172.2 106.5 194.3 653.7 500

H-3 184.2 120.9 233.8 489.6 460

C-1 136.9 97.9 185.7 643.8 550

C-2 177.6 118.7 231.6 621.1 480

C-3 151.7 77.1 159.8 433.6 550

F cro : Elastic buckling stress computed by THIN-WALL.

(a) Effect of initial imperfections (0.1% triggering load) for H-2

section.

(b) Effect of initial imperfections (0.5% triggering load) for H-2

section.

(c) Effect of yield stress for H-2 and C-2 sections.

Fig. 3. Effect of initial imperfections and yield stress.

(a) Locations of gauges

& LVDT.

(b) Test configuration.

Fig. 4. Test set-up (H-section).

0.01 mm/s was controlled by the displacement control method.

The vertical displacement was obtained from the machine

directly and the horizontal displacements were measured by

displacement transducers, which were attached at the center and

the quarter points of the test specimens. Since the local buckling

occurred in three half-waves, six strain gauges were attached at

the centers of each local buckling half-wave expected, as shownFig. 4(a). Typical compression test configurations for the H-

section are shown in Fig. 4(a) and (b). Since the C-section was

assumed to buckle in the flexural mode about the minor axis,

its test set-up was prepared to be quite similar to that of the

H-section.

3.2. Test section behavior

Axial load versus displacement relations of H-sections and

C-sections tested are shown in Fig. 5(a) and 5(b), respectively.

The displacements in Fig. 5(a) and (b) were the axial shortening

of the test columns measured by the LVDT, which was located

at the top of the specimen. As shown in Fig. 5(a) and (b),

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(a) H-sections. (b) C-sections.

Fig. 5. Axial load versus displacement curves.

(a) Side view. (b) Front view.

Fig. 6. Buckled shape of H-section.

the columns displayed a stable post-local-buckling behavior but

some of the C-section did not show a stable behavior after the

ultimate loads. As the load was increased, the column shortened

elastically and gradually at the beginning of the test. At or

near the elastic local buckling load, local buckling occurred as

expected, first at the web of the section and then propagated

to the flanges of the H-sections and C-sections, except the

C-3 section which had a very large width–thickness ratio in

the flange. For the H-2 sections, the test local buckling stress

was much higher than that computed numerically. For the C-1 section, the local buckling occurred at a load 71% lower

than that expected. The premature local buckling was due to

the concentration of the load applied. Other H-sections and C-

sections buckled at the load near the theoretical local buckling

load within a range of approximately 10%. After the local

buckling load was reached, a significant post-local-buckling

strength reserve was observed before flexural buckling about

the minor axis for the H-section and about the unsymmetrical

axis for the C-section occurred, respectively.

A significant buckling mode interaction between local

buckling and flexural buckling was observed before the

maximum load was reached. After the peak load, the horizontal

displacement continued to increase slowly as the load was

decreased. Typically, the buckled shapes of the H-2-1 section

and C-2-1 section are shown in Fig. 6(a), 6(b) and Fig. 7(a),

7(b), respectively. All the buckled shapes of the test sections

agreed well with those of the numerical analysis. The numerical

analysis results are also shown in Figs. 6(a), (b) and 7(a), (b)

for comparison of the buckled shapes. The buckled shape of the

H-section was a mixed mode of local buckling in three short

half-waves and flexural buckling in a long half-wave about the

minor axis. This mixed mode was similar to that obtained by thenumerical analysis, as shown in Fig. 6(a) and (b). The overall

buckling mode of the C-section was the flexural buckling about

the unsymmetrical axis, and the flange and the web buckled in

the local mode as shown in Fig. 7(a) and (b) additionally. The

mode interaction between local buckling and flexural buckling

observed during testing was quite similar to that obtained by a

nonlinear analysis as shown in Fig. 7(a) and (b).

3.3. Test results

The experimental local buckling stresses and the ultimate

stresses of the test sections are summarized in Table 3 and

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(a) Side view. (b) Front view.

Fig. 7. Buckled shape of C-section.

Table 3Ultimate stress and local buckling stress of specimens

Specimen Ultimate stress ( F max) Local buckling stress ( F cr )

Test (MPa) Analysis (MPa) Test/Analysis Test (MPa) Analysis (MPa) Test/Analysis

H-1-1 198.7 213.5 0.93 185.2 197.9 0.93

H-2-1 170.9 172.2 0.99 143.7 106.5 1.35

H-2-2 182.0 172.2 1.06 155.2 106.5 1.46

H-3-1 158.1 184.2 0.86 121.8 120.9 1.01

H-3-2 165.6 184.2 0.90 128.2 120.9 1.06

C-1-1 120.0 136.9 0.88 66.7 97.9 0.68

C-1-2 106.4 136.9 0.78 73.8 97.9 0.75

C-2-1 155.1 177.6 0.87 128.9 118.7 1.09

C-3-1 116.4 151.7 0.77 86.7 77.1 1.12

compared with the numerical analysis results obtained by the

program THIN-WALL [18] and LUSAS [19], respectively. The

test ultimate stresses were generally lower than the analysis

results except for the H-2-2 specimen. As shown in Table 3,

the test ultimate stresses of the C-1-2 and the C-3-1 were

not in good agreement with those obtained by the numerical

analysis. The comparison of the local buckling stresses of

the test specimens showed a different trend from the ultimate

stresses. The experimental elastic local buckling stresses of

the C-1-1 and the C-1-2 specimens were lower than the

theoretical elastic local buckling stresses by 25%–32% due

to the local collapse of the web, which was caused by the

concentration of load. The local buckling stress of the C-1sections was very much lower than the yield stress and the

translation of the effective centroid of the buckled section

caused the premature failure of the web before the interaction

between local buckling and overall buckling. Except for the H-

2-2 section, the experimental ultimate stresses of the columns

tested were slightly lower than the numerical values because

of the adverse effect of the interaction between local buckling

and overall buckling of the specimens. For the C-3-1 section,

the test ultimate stress was lower than the analysis result

by 23.0%, while the test ultimate stress was higher than the

analysis result by 6.0% for the H-2-2 section. However, the

difference between test ultimate stress and local buckling stress

was 29.7 MPa and 26.8 MPa, respectively. All the specimens

displayed more or less significant post-local-buckling strength

reserve up to the ultimate load which was determined by the

minor axis flexural buckling load. The differences between the

experimental ultimate stresses and the local buckling stresses

ranged from 13.5 MPa for the H-1-1 section to 53.3 MPa for

the C-1-1 section.

Since the local buckling stress of the column tested cannot

be determined easily, the experimental buckling stress was

taken as the stress at the location where the deterioration of

stiffness on the load–displacement curves commenced. The

local buckling stress was determined by two different methods.

The first set of buckling stresses was determined by examiningthe change in the slope of the load–displacement curves. The

second set of buckling stresses was determined by plotting

the stress versus the square of the strain and subsequently

fitting a line through the test results in the post-buckling region.

The intersection of the fitted line and load or stress axis was

assumed to be the experimental buckling stress [21]. Since

the change in the overall flexural stiffness was subtle and the

intersection point was difficult to choose, the average value of

the two methods was taken as the experimental local buckling

stress in Table 3. In cases where the slope change of the

load–displacement curves was not clear and the local buckling

stress was difficult to decide, the stress–strain curves, which

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Fig. 8. Stress versus strain curves (H-section).

were obtained by the strain gauges attached, were additionally

used to determine the local buckling stress. The local buckling

stresses of the H-2 sections were higher than those obtained

by the numerical analysis by approximately 40.0%. The C-1

sections displayed a local buckling load that was approximately

28.0% lower than that of the numerical analysis. However, the

local buckling stresses of all the other sections agreed well with

those obtained by the numerical analysis.

The typical axial stress versus strain curves for H-sections

are shown in Fig. 8. The stress–strain curves for H-sections

tested are similar to one another. At the intersection point

where the slope changed, local buckling commenced at the

web of the columns. Before the ultimate load, a significant

post-local-buckling strength reserve was observed on the

load–displacement and stress–strain curves except for H-1-1

section, whose width–thickness ratio was comparatively lower

than those of other specimens. In Fig. 8, the stress was theaverage stress obtained by dividing the applied load by the

unreduced gross section area, and the strain was the average

strain, which was calculated from the gauge values measured

at the column center. As shown in Fig. 8, the local buckling

started at the strain of approximate 0.001. In some cases of

H-2-2 and H-3-1 sections, the local buckling stress could be

decided from the stress–strain curves more easily than the

load–displacement curves. A significant post-local-buckling

strength reserve similar to that in the load–displacement curves

was observed. The H-2-2 and H-3-2 sections displayed a

continuous increase of strain after the ultimate stress. However,

the H-2-1 and H-3-1 sections showed a slightly unstable stress-softening behavior.

4. Column design methods

4.1. Korean highway bridge design specifications

The design compressive strength defined in terms of

allowable stress in KHBDS [9] (Korean Highway Bridge

Design Specifications, 2005) and KS D 3515 [20] can be

modified in terms of nominal strength as

f nl =

f ne ×

F l

F y(1)

where f nl = design axial column strength accounting for the

interaction of local buckling and overall buckling; f ne = overall

column strength not considering the local buckling strength;

F l = nominal strength based on the local buckling stress;

F y = nominal yield stress of the material. The nominal

strengths are obtained by multiplying the allowable stresses

by the factor of safety of 1.7, which is basically used inthe specifications. Eq. (1) incorporates the concept that the

compressive strength of columns is based on the overall

buckling stress and is reduced by the ratio of the nominal

strength based on the local buckling stress to the yield

stress (F l /F y ) in order to consider the local buckling of the

composing elements.

Whenever the nominal stress based on the local buckling

stress is higher than the yield stress of the section, the strength

formula can predict reasonable design strengths of the section.

However, if the nominal local buckling stress is lower than the

yield stress, Eq. (1) is liable to produce unreasonable strengths.

In the case where the overall column strength and the design

local strength are lower than the yield stress, whether the overallcolumn strength is larger than the design local buckling strength

or not, the design axial column strength computed by Eq. (1) is

extremely low. Consequently, Eq. (1) cannot reasonably include

the negative effect of buckling mode interaction between local

and overall buckling.

In the cases where the local buckling stress is larger than

the flexural/flexural–torsional buckling stress, the strength of

the column should be determined by the overall buckling

stress. However, if the local buckling stress is smaller than the

overall buckling stress, the post-local-buckling strength reserve

should be considered in the design strength to some extent.

To incorporate the negative effect of mode interaction for thecompression members undergoing interaction of local buckling

and overall buckling, the strength formula in Eq. (1) should be

revised properly.

The design column strength ( Pnl ) can be obtained by

multiplying the axial column strength ( f nl ) by the unreduced

cross section area ( A). The resulting alternative formulation in

Eq. (1) in terms of force is

Pnl = Pne ×Pl

P y

(2)

where pnl = design column strength accounting for interaction

of local buckling and overall buckling AF ne

, Pl =

resulting

limiting load AF l , and P y = yield load AF y .

4.2. Effective width method

The NAS (AISI 2001) [6] and Eurocode 3 [8] have used

the same effective width formula to include the post-local-

buckling strength reserve in the design strength of compression

members, where the width–thickness ratio is larger than the

limit values specified. Therefore, even if the overall column

strength formulas are slightly different, the design strengths

of the compression members computed according to the

NAS and the EC3 are quite similar. The AISC LRFD [7]

provides different buckling coefficient K values, and the

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calculation method of the effective width is also different from

other specifications mentioned. Since the section is generally

composed of stiffened and unstiffened elements, the design

procedure according to the AISC LRFD is as follows: the

reduction factor Qs for the unstiffened elements with adequate

K values specified should be calculated first, and then the

reduction factor Q a for the stiffened elements is computed withthe assumed stress level in the iterative method. The NAS and

the EC3 adopted the same effective width formula for both

stiffened and unstiffened elements for simplicity. The formula

was originally proposed for the stiffened element and produces

more conservative values than that for the unstiffened element.

Consequently, the AISC LRFD may produce more or less

higher design strengths than the NAS and EC3 in general. The

buckling coefficient K values 4.0 and 0.43 are used to calculate

the effective widths of the stiffened and unstiffened elements

in isolation, respectively. The unreduced gross area is used to

compute the limiting load in the AISC LRFD rather than the

effective area which is used in the NAS and EC3.

4.3. Direct strength method

The effective width method (EWM), which can consider

the post-buckling strength reserve into the design, may need

iterative calculations to compute the effective width of the

composing element separately. As the section shape becomes

more complicated, the accurate computation of the effective

width becomes more difficult and the separation of each

element of complicated sections becomes unreasonable. To

overcome these problems, the direct strength method (DSM)

for cold-formed steel sections has been developed by Shafer

and Pekoz [10] and further studied by Hancock et al. [2,12].The method incorporates the empirical formulae and the elastic

local or distortional buckling stress obtained by the rigorous

buckling analysis or reliable strength formulas. The application

of the direct strength method to a welded section was studied

recently by Kwon et al. [13].

The direct strength formulas considering the interaction

between local buckling and overall buckling for cold-formed

steel sections, which have been proposed by Shafer and

Pekoz [10], are given by

for λ ≤ 0.776

f nl

= F ne . (3a)

For λ > 0.776

f nl =

1− 0.15

F crl

F ne

0.4

F crl

F ne

0.4

F ne (3b)

where λ = √ F ne /F crl ; f nl = limiting stress accounting for

local buckling and overall buckling (MPa) ; F crl = elastic

local buckling stress (MPa) ; F ne = overall column strength

based on the overall failure mode (MPa), which is determined

from the minimum of the elastic flexural, torsional, and

flexural–torsional buckling stresses. The overall column

strength F ne can be calculated from Eqs. (C4-2) and (C4-

3) of the NAS (AISI) [6] or Eqs. (6.47), (6.48) and (6.49)

in the Eurocode 3 [8] or the equations in Table 3.3.2 of the

KHBDS [9]. The elastic local buckling stress F crl can be

computed by the rigorous Finite Element Method (FEM) or

the Finite Strip Method (FSM). The exponent 0.4 was used

instead of 0.5, which was used in Von Karman [22] and Winter

formulae [23], for the effective width of elements to reflect a

higher post-local-buckling strength reserve for the unreducedgross section than for an element in isolation. Eqs. (3a) and

(3b) were recently adopted as an alternative design method to

the EWM by the NAS [14] and Australian Standards [15]. The

DSM is much simpler to apply than the EWM since it uses

gross section properties rather than effective section properties.

It has already been proven quite accurate in comparison with

the EWM for cold-formed steel sections [2].

4.4. Proposed direct strength equations

The direct strength formula considering the interaction

between local buckling and overall buckling can be reasonably

used for welded sections and hot-rolled sections. However,the strength formula should be modified to account for the

different characteristics between cold-formed and welded steel

sections. First, the effect of interaction between local buckling

and overall buckling on the strength of the welded compressive

members might be less significant, and the post-local-buckling

strength reserve is smaller than that for the cold-formed steel

section since the width–thickness ratios of commonly used

welded sections are comparatively smaller than those of cold-

formed steel sections. Secondly, unlipped welded channel

sections are commonly used as compression members, while

lipped channel sections are common for cold-formed steel

sections. However, in the case of single symmetrical sections

such as the unlipped C-section, of which the local bucklingstress was very much lower than the yield stress, the transition

of the effective centroid of the locally buckled section might

cause a premature failure. To account for these phenomena,

the modified Winter formula [23] was proposed rather than

Eqs. (3a) and (3b). When the exponent 0.5, as used in Winter

formulas, is used, the coefficient 0.15 in Eq. (3b) is adopted

rather than 0.22 in Winter formulas, which reflects a higher

post-local-buckling strength reserve in the inelastic buckling

range of a material for the intermediate length column. The

equations for the limiting stress f nl considering the interaction

between local buckling and overall buckling for the welded

section are given byfor λ ≤ 0.816

f nl = F ne . (4a)

For λ > 0.816

f nl =

1− 0.15

F crl

F ne

0.5

F crl

F ne

0.5

F ne (4b)

where

λ =

F ne /F crl . (4c)

The results predicted by the strength formulas proposed

are compared with tests results of the welded H-sections and

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Fig. 9. Comparison between DSM curves and test results.

the C-sections, and Eqs. (3a) and (3b) in Fig. 9. As shown

in Fig. 9, Eqs. (3a) and (3b) predict reasonable strengths for

the columns undergoing the interaction between local buckling

and overall buckling in comparison with the test results,

as long as the slenderness ratio factor is small. However,when the slenderness ratio factor λ becomes larger than 1.5,

Eq. (3b) produces very optimistic strengths in comparison with

the test results of channel sections. The strength curve equation

(4b) proposed can predict slightly higher design strengths

than Eq. (3b) when the slenderness ratio factor λ remains

between 0.776 and 1.0. However, when the slenderness ratio

factor λ becomes larger than 1.0, Eq. (4b) produces slightly

lower values than Eq. (3b). The difference in the strengths

predicted by Eqs. (3b) and (4b) becomes more significant

as the slenderness ratio factor becomes large. The ultimate

strengths of C-sections tested are under the design strength

curves of Eq. (4b) proposed. However, Eq. (3b) predicts theultimate strengths of C-sections too optimistically. As shown

in Fig. 9, the proposed design curves have a less satisfactory

fit to the ultimate strength of H-sections than Eq. (3b), but

predict a less optimistic ultimate strength for C-sections than

Eq. (3b) does. Since the single symmetric C-section generally

shows a premature failure due to the translation of the effective

centroid of the locally buckled section, the design strength

curves is forced to predict somewhat optimistically according

to the slenderness of the plate elements. Consequently, it can

be concluded that the proposed ultimate strength curves of

Eqs. (4a) and (4b) can predict more reliable ultimate strengths

for the welded section columns than Eqs. (3a) and (3b) when

local buckling and overall buckling occur simultaneously ornearly simultaneously.

4.5. Comparison with test results

To compare the DSM with the EWM directly, the strength

equations should be expressed in terms of load. If the

distortional buckling does not occur, the design strength of

compression members can be taken as the nominal strength

Pnl , which is obtained by multiplying the limiting stress ( f nl )

in Eqs. (4a) and (4b) by the full unreduced section area ( A).

Therefore, the alternative formulations of Eqs. (4a) and (4b) in

terms of load are given by

Fig. 10. Comparison between DSM (based on NAS) and test results.

for λ ≤ 0.816

Pnl = Pne . (5a)

For λ > 0.816

Pnl =

1− 0.15

Pcrl

Pne

0.5

Pcrl

Pne

0.5

Pne (5b)

where λ = √ Pne /Pcrl , Pcrl = elastic local buckling load

AF crl , pne = nominal column design strength AF ne .

The Direct Strength Method (DSM) curves based on the

overall column strength formula provided by the NAS (AISI)

and Eurocode 3 are compared with the test results in Fig. 10

and Fig. 11, respectively. Since the nominal column strengths

computed by the NAS and EC3 column strength formula are

slightly different, the test results were generalized by the two

different design strengths and compared. The test results of thewelded I-section and channel section columns with the nominal

yield stress of 350 MPa, which were executed previously at

the University of Sydney [16,17], are also included in Figs. 10

and 11. For most of the columns, the DSM curve predicts the

ultimate strengths fairly conservatively. However, in the case of

very slender unlipped channel sections, the ultimate strengths

are predicted optimistically by the DSM curve. The DSM curve

predicts the ultimate strengths too optimistically in comparison

with the test results of very slender channel sections, which

were executed by Rasmussen and Hancock [17]. However, it

can be concluded that the ultimate strengths of welded H-

sections and C-sections, which are not too slender, can be

predicted fairly reasonably by the DSM.The column design strengths, which are calculated

according to the Effective Width Method (EWM), the NAS [6]

and the EC3 [8], and the DSM proposed have been summarized

and compared with test results in Table 4. Since the web of

the C-1-1 and C-1-2 sections failed prematurely in the local

mode before interaction between local buckling and overall

buckling, the test results of the C-1-1 and C-1-2 sections

were not compatible with the design strengths predicted by

the specifications. For the H-sections tested, the ratio of the

maximum test strength to NAS ranged from 1.33 to 1.62, the

ratio of the maximum test strength to EC3 from 1.43 to 1.83

and the ratio of the maximum test strength to DSM ranged from

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Table 4

Comparison of tests and design strengths

Specimen Design strengths (kN) Test ultimate strength (kN) Test/Design

NAS EC3 AISC DSM NAS EC3 AISC DSM

H-1-1 513.0 444.0 575.0 550.2 715.3 1.39 1.61 1.24 1.30

H-2-1 585.7 519.6 766.9 566.2 892.1 1.52 1.72 1.16 1.58

H-2-2 585.7 519.6 766.9 566.2 950.0 1.62 1.83 1.24 1.68H-3-1 584.8 519.0 682.7 564.3 739.9 1.27 1.43 1.08 1.31

H-3-2 584.8 519.0 682.7 564.3 775.0 1.33 1.49 1.14 1.37

C-1-1 497.6 447.9 596.9 529.7 502.9 1.01 1.12 0.84 0.95

C-1-2 497.6 447.9 596.9 529.7 444.1 0.89 0.99 0.74 0.84

C-2-1 504.6 442.8 632.6 505.5 697.8 1.38 1.58 1.10 1.38

C-3-1 323.1 334.3 251.9 400.8 519.9 1.61 1.56 2.06 1.30

Fig. 11. Comparison between DSM (based on EC3) and test results.

1.30 to 1.68. The ratio of the test ultimate strength to the AISC

ranged from 1.08 to 1.24. For the C-sections tested, the ratios of

the ultimate test strength to NAS were 1.38 and 1.61, the ratios

of the ultimate test strength to EC3 1.56 and 1.58 and the ratiosof the maximum test strength to DSM were 1.30–1.38. The

ratios of the test ultimate strength to the AISC were 1.10 and

2.06. For H-sections and C-sections, the difference between the

test ultimate strength and design strength was fairly constant.

However, for the C-section, the difference for AISC was highly

variable in magnitude. The DSM and EWM predicted the

ultimate strength for the H-sections fairly conservatively. In the

case of the C-sections, the predictions of the ultimate strength

by the specifications and the DSM were fairly conservative.

However, it can also be concluded that the ultimate strength

of the channel sections such as the C-3 specimen which had

exceptionally slender flange was predicted too conservatively

by the EWM but fairly by the DSM. Generally speaking, the test

results for H-sections and C-sections validate the DSM clearly.

The comparison of predictions for the ultimate strength of

the sections among the design specifications and the DSM

proposed is summarized in Table 5. As shown in Table 5, since

the provisions to take account of the local buckling stress on the

column strength are not adequate and the post-local-buckling

strength is not considered at all, the KHBDS [9] cannot closely

predict the ultimate strength of the slender welded section

columns undergoing interaction between local buckling and

flexural/flexural–torsion buckling. The average ratio of KHBDS

to DSM was 0.19. Even if the width–thickness of the test

sections is beyond the limit prescribed in the specification, the

Table 5

Comparison between design methods

Specimen NAS/DSM EC3/DSM AISC/DSM KHBDS/DSM

H-1 0.93 0.81 1.05 0.25

H-2 1.03 0.92 1.35 0.19

H-3 1.04 0.92 1.21 0.19C-1 0.94 0.85 1.13 0.20

C-2 1.00 0.88 1.25 0.21

C-3 0.81 0.83 0.63 0.10

Average 0.97 0.88 1.15 0.19

Standard deviation 0.07 0.04 0.15 0.03

predictions of the ultimate strength by the KHBDS are absurdly

lower than the test ultimate strengths for the sections which

may have the buckling interaction between local buckling and

overall buckling.

The slight difference between NAS, EC3 or AISC and DSM

might result from the magnitude of post-local-buckling strength

considered in the specifications and the effect of interactionbetween the elements composing the whole cross section.

Unlike the EWM, which estimates the effective width of each

element in isolation, the DSM handles the whole section at

a one time. Thus, the difficulty in computing the effective

width of the each element in isolation for the EWM has been

exempted in the DSM. The design strengths predicted by the

DSM proposed agree quite closely with those estimated by

the NAS. The ratio of the design strengths predicted by the

NAS to those obtained according to the DSM ranged from

0.81 to 1.04 with the average 0.97 and the standard deviation

0.07, as shown in Table 5. The C-3 section has an exceptional

width–thickness ratio of the flange, which is four times the

buckling limit of 14.0. In comparison with EC3, the DSM

estimates the design strength higher than the EC3 by an average

of 12%. The prediction by the EC3 is slightly more conservative

than that by the NAS except for the C-3 specimen. The ultimate

strengths predicted by the AISC are higher than those predicted

by the DSM except for the C-3 by an average of 15%. The

fact that the AISC uses a different effective width formula

between stiffened and unstiffened elements made the difference

in the predictions. The difference in the ultimate strengths

predicted by the specifications is mainly the result of the single,

effective width formula for both stiffened and unstiffened

elements adopted by the NAS and the EC3 with the advantage

of simplicity. Since the formula for the unstiffened element

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has more post-local-buckling strength reserve and produces a

higher value than that for the stiffened element, the NAS and

EC3 predict the ultimate strength less optimistically than the

AISC.

From the comparison of the ultimate strengths predicted by

the design specifications based on the effective width concept

and the DSM, it can be concluded that the design ultimatestrength of welded sections can reasonably be predicted by

the DSM. The DSM proposed can be used as an alternative

method to the conventional EWM, which has been used for the

welded section columns. However, in the case of the welded

sections such as a lipped channel section, which is liable to

undergo distortional buckling, even if the sections are not

of common shape, the distortional limiting strength should

also be considered to predict the overall column strength.

Therefore, the DSM should determine the ultimate strength for

the interaction between local buckling and overall buckling,

and distortional buckling and overall buckling and then take the

lesser of the two as the column strength. The DSM should also

be carefully applied to very slender single symmetric sectionssuch as a channel section. In that case, if the shift of the centroid

of the effective area relative to the center of gravity of the

gross section is not prevented by constructional arrangements,

the possible shift of the centroid and the resulting additional

moment should be determined, and the beam–column strength

formula for the DSM should be used. The beam–column

strength formula for the DSM is under preparation at the

moment.

5. Conclusions

The experimental study for the application of the directstrength method (DSM) to the thin-walled welded section

columns undergoing interaction between local buckling and

flexural/flexural–torsional buckling was conducted. The local

buckling, which occurs prior to the overall column buckling

and has a significant post-local-buckling strength reserve,

deteriorates the overall column strength to some extent. This

phenomenon should be considered appropriately to predict

conservatively and accurately the ultimate strength of the

welded sections as well as the cold-formed sections.

The direct strength formulas, which were adopted for the

design of the cold-formed steel sections, have been modified

to take into account the effect of the interaction between local

buckling and overall buckling and post-local-buckling strengthreserve for the welded sections. The member strengths from

the direct strength formulas proposed were compared with the

test results of welded H-sections and C-sections fabricated

from the hot-rolled steel plates with nominal yield stresses

of 240 and 350 MPa. The ultimate strengths predicted by

the direct strength method were also compared with those

estimated by the effective width method adopted in the NAS,

AISC and EC3. The comparison of the ultimate strengths

predicted by the design specifications and the test results has

proven the reliability of the direct strength method. The direct

strength formulas proposed have been proven easier to apply

and accurate to predict the ultimate strength of the columns

that undergo the interaction between local buckling and overall

buckling and fail in the mixed mode between local buckling and

flexural–torsional buckling. However, detailed provisions as to

the width–thickness ratio limit of single symmetric sections,

such as a channel section, should be prepared immediately. The

proposed direct strength formulas should be further verified and

calibrated against the test results of various kinds of sections forpractical use.

Acknowledgements

This research was supported by the Korea Bridge Design

and Engineering Center under the sponsorship of 2004

Development of Core Construction Technology Project of

Korea Ministry of Construction and Transportation.

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