bending tests to determine slenderness limits for cold-formed circular hollow sections

Journal of Constructional Steel Research 58 (2002) 1407–1430www.elsevier.com/locate/jcsr

Bending tests to determine slenderness limitsfor cold-formed circular hollow sections

M. Elchalakani, X.L. Zhao∗, R. GrzebietaDepartment of Civil Engineering, Monash University, PO Box 60, Victoria 3800, Australia

Received 23 July 2001; received in revised form 7 December 2001; accepted 10 December 2001

Abstract

There are significant differences in slenderness limits recommended in various codes forcircular hollow sections (CHS) under bending as there have been little experimental studies.In this paper an attempt is made to establish more accurate slenderness limits for cold-formedcircular hollow sections. This paper describes a series of bending tests to examine the influenceof section slenderness on the inelastic bending properties of cold-formed CHS. Twelve bendingtests were performed up to failure on different sizes of CHS with diameter-to-thickness ratiod/t ranging from 37 to 122. This range ofd/t was obtained by machining as-received cold-formed circular hollow sections grade C350L0. The test results are compared with other experi-mental data and the design rules given in various steel specifications. The slenderness limitswere established to define Class 1 (compact), 2, 3 (non-compact) and 4 (slender). These limitswere based on modifications of criteria for rotation capacity commonly used for steel struc-tures. A design curve was developed and recommended for the design of cold-formed CHSunder pure bending. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Buckling; Slenderness limits; Flexural strength; Cold-formed; Circular tubes

1. Introduction

In recent years cold-formed steel hollow sections have become more popular asa construction member and consequently attracted a lot of research efforts to deter-mine their structural properties. Cold-formed tubular sections are manufactured inAustralia to meet the quality of AS 1163 [1] while in the US they are manufactured

∗ Corresponding author. Tel.:+61-3-9905-4972; fax:+61-3-9905-4944.E-mail address: [email protected] (X.L. Zhao).

0143-974X/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0143 -974X(01)00106-7

1408 M. Elchalakani et al. / Journal of Constructional Steel Research 58 (2002) 1407–1430

Nomenclature

d Outside diameter of CHSdh Outside horizontal radius of ovalised tubedhi Inside horizontal radius of ovalised tubedv Outside vertical radius of ovalised tubedvi Inside vertical radius of ovalised tubeeu Percentage elongation at fractureEn Young’s modulus of elasticity (200 GPa)E0 Measured initial Young’s modulusET Tangent modulusI Second moment of areaIm Second moment of area based on measured dimensionsL Beam length under constant momentLm Machined lengthLf Free deformation lengthM Applied momentMp Full plastic moment of the cross sectionMpt Plastic moment based on measured dimensions and material

propertiesMy First yield momentMo Ovalisation momentMu Ultimate moment obtained in a testR1 Rotation capacity defined in eq. (1)R2 Rotation capacity defined in eq. (2)S Plastic section modulusSm Plastic section modulus based on measured dimensionsSo Plastic section modulus of an ovalized tubeSF Shape factort Thickness of CHSZ Elastic section modulusZm Measured elastic modulusZe Effective section modulusa Non-dimensional local buckling parameterap Plastic buckling parameterac2 Buckling parameter to define Class 2 sectionsay Yield buckling parameterb Ratio of inside diameter to outside diameter of CHS� Curvature�pt Curvature corresponds to Mpt

ls Section slenderness defined in AS 4100lp Plastic slenderness limit defined AS 4100

1409M. Elchalakani et al. / Journal of Constructional Steel Research 58 (2002) 1407–1430

ly Yield slenderness limit defined in AS 4100lc2 Section slenderness limit to define Class 2 sections�y Yield stress�yn�yt Nominal tensile yield stress and measured tensile yield stress�un�ut Nominal ultimate tensile strength and measured ultimate tensile

strengthq Relative angle of rotationqmax Inelastic rotation at the plastic hingequ Inelastic rotation defined in Fig. 1qy Rotation corresponds to My

Fig. 1. The classical definition for rotation capacity based on normalised moment-rotation relationships[16–18].

to the quality of ASTM-A500 [2]. Cold-formed steel framed houses become costeffective compared to those constructed from timber particularly in North America[3]. Cold-formed tubular members are currently used more often in modern steelconstruction worldwide mainly as space trusses and frames. The Colonial Stadiumin Melbourne used 6000 tons of structural steel mainly cold-formed circular tubes[4]. The Sydney Olympic Aquatic Centre utilised 13 km of tubular sections, whilethe construction of the Year 2000 Sydney Olympic Stadium required 12000 tons ofstructural steel work mainly cold-formed tubular members [5]. In addition, cold-formed tubular sections are used in other structures, such as: the construction of steel


communication towers and truss bridges; crane booms; lightening poles and highway sign supports [6]. An interesting state-of-the art on cold-formed steel structureshas been presented by Hancock [7] and more recently by Rondal [8]. A summaryof the research performed on cold-formed steel structures in Australia was presentedin Hancock et al. [9].

Local buckling may affect the bending behaviour of circular tubular sections. Steeldesign specifications define different classes of cross sections depending on the pointat which local buckling occurs during bending [10]. In a number of specifications,such as AS 4100 [11] and AISC-LRFD [12] sections can be classified as compact,noncompact or slender. In other specifications such as Eurocode [13], CAN 16.1[14] and BS 5950 [15] sections are classified as Class 1, 2, 3 or 4. Fig. 1 showsthe normalised moment-rotation relationship for compact, noncompact and slendersections based on a rotation capacity defined as [16,17,18]

R1 � qu /qy�1 (1)

where qu/qy is a dimensionless inelastic rotation defined in Fig. 1 and qy is the yieldrotation corresponds to the moment causes yield in the outermost fibres (My). Class4 or slender section fails by elastic buckling before the attainment of My. Class 4circular section was studied experimentally and theoretically by Donnell [19], Otsukaand Koga [20,21] and more recently by Elchalakani et al. [22]. Class 3 circularsection buckles inelastically at a moment between My and the fully plastic momentMp. A rotation capacity R1�1 measured at My is required for Class 3 circular section.A Class 2 circular section can obtain a moment of Mp but can not sustain this momentfor adequate inelastic rotation (R1�1 at Mp). Class 2 and 3 circular sections aregrouped together as noncompact sections in AS4100 [11] and AISC-LRFD [12].These noncompact sections were studied by Murray and Bilston [23] and Ju andKyriakides [24]. Class 1 circular section can maintain Mp for sufficiently largerotations (R1�3 at Mp) to allow for moment redistribution in the plastic design ofa framed structure. A slightly larger value of R�4, based on normalised curvaturesrather than rotations, is recommended by Korol and Huboda [25] to define a compactsection. Class 1 circular section was studied experimentally and theoretically byBrazier [26], Mamalis [27], Kyriakides and Shaw [28,29,30] and Wierzbicki andSinmao [31]. Class 1 circular section received much research efforts as it comprisesthe most commonly used d/t ratios in inland and offshore construction. Table 1 showsthe large differences in section slenderness limits (particularly for the yield limit ly)specified in a number of steel specifications. A wider range of codes is comparedin Rondal et al. [32].

A related paper [33] dealt with compact cold-formed circular section, and a plasticslenderness limit to define this class was obtained. The current paper mostly dealswith noncompact and slender sections, while compact section is briefly reviewed. Aseries of bending tests are described to examine the influence of section slendernesson the rotation capacity of cold-formed CHS. Twelve bending tests were performedup to failure on different sizes of CHS with d/t ranging from 37 to 122 (or ls rangingfrom 60 to 200). The large range of d/t was obtained by machining cold-formedcircular hollow sections grade C350L0. The test results are compared with the design


Table 1Cross section classification

AS 4100 [11] AISC-LRFD Eurocode 3 [13] & NZS 3404 [42] Proposed[12] CIDECT [32]

(1) (2) (3) (4) (5)

ls�50 ls�50 ls�47 Class 1 ls�50 Class 1 ls�60 Class 1 ls�60Compact Compact Compact50 � ls�120 50 � ls�250 47 � ls�66 Class 50 � ls�65 60 � ls�88 60 � ls�140Non-

Non-compact Non-compact 2 Class 2 Class 2 compact66 � ls�84 Class 65 � ls�170 88 � ls�1403 Class 3 Class 3

ls�120 ls�250 ls�84 Class 4 ls�170 Class 4 ls�140 Class 4 ls�140Slender Slender Slender

rules given in various codes. The newly obtained slenderness limits and design modelare discussed towards the end of the paper.

2. Previous relevant bending tests of CHS

Sherman [34] reported the results of 53 four-point bending tests including cold-formed (electric-resistance welded), fabricated (submerged or manual arc welded)and hot-formed seamless cylinders. These tests form the database used to derivemost of the current design rules (see Fig. 2). Sherman’s own testing program [35]

Fig. 2. Design rules for CHS and available test results.


included 21 constant-moment tests with d/t ranging between 17 and 90 (ls=19 to149) and 8 cantilever tests with d/t =28 to 72 (ls =39 to 78). A total of 16 specimenshad ls�57, ie. noncompact and slender sections (according to [12]). In these tests,the loads were applied and reacted through gravity load simulators to prevent anyaxial load due to chord shortening. The buckling failures occurred away from theload fixtures that would prevent ovalisation. In the constant-moment tests the length-to-diameter ratio was L/d=4. Fig. 2 summarizes the test results obtained in constant-moment [34] and pure bending [33], whereas the cantilever test results [35] are notshown. The ultimate strength (Ze/Z) is plotted against the slenderness parameter (ls),where Ze is the effective section modulus, and Z is the elastic section modulus. Aconversion factor of Ze/Z =1.273Mu/Mp was used to convert the test data to AS 4100[11] format. The main conclusions drawn by the present authors from the 53 tests[34] are:

� The bending strength for hot-formed seamless pipes are generally larger than thecorresponding strength for fabricated and cold-formed pipes.

� The plastic moment Mp was reached for fabricated cylinders made of ASTM-A36steel with average au/�y =1.6 [35]. In a number of the tests (with ls�57) Mp couldnot be achieved, which was attributed to lack of strain hardening in the materialused to fabricate the cylinders with average �u/�y =1.4 [35]. Similar conclusionswere drawn for cold-formed CHS with �u/�y =1.23 [33] where ovalisation wasalso considered as part of the reasons.

� The section yield slenderness limit of circular tubes is ly= 250. This later valueis the same specified as that in the AISC-LRFD [12] specification.

� Tubes with ls�50 reached the plastic moment Mp and have adequate rotationcapacity which is in the range of 4�R2�25.5 (where R2 is defined in eq. (2)).

� Tubes with section slenderness in the range of 50�ls�57 reached Mp, but havecomparatively smaller rotation capacity in the range of 2.4�R2�4.5. The slender-ness limit lp=57 to define a compact section is specified in the AISC-LRFD [12].A smaller value of ls=36 is recommended for plastic design in [12].

� Machined specimens tested by Schilling [36] with ls�90 were able to achievethe plastic moment in Schilling’ tests [36]. The machined length-to-diameter ratiowas Lm/d=1.3. This was due to the relatively small Lm/d, which restricted ovalis-ation, but not due to the machining operation itself.

� The non-dimensional buckling parameter ac2=8.2 (lc2 =97) can be interpreted(based on R1�1 at My) from the test results to define Class 2 section for fabricatedcylinders tested in [35].

� The classical definition of rotation capacity defined by Korol and Huboda [25]for tubular sections is not suitable for CHS. The rotation capacity of CHS wasredefined as

R2 �qmax

qy

�1 (2)

qmax is the rotation corresponding to the formation of plastic hinge [33] and qy is


the rotation corresponding to the first yield moment (My). The section slendernessls [11] and non-dimensional buckling parameter a [34] are shown below:

ls � �dt�.� sy

250� (3)

a ��Esy�

� dt �

(4)

where d is the outside diameter, t is the tube wall thickness, �y is the yield stressand E is the elastic modulus.

3. Material properties

The tensile coupons were prepared and tested according to the Australian StandardAS 1391 [37] to determine the initial Young’s modulus (E0), the yield stress (syt),the ultimate tensile strength (sut) and the percentage elongation (eu) at fracture. Thetensile specimens were tested in a 500 kN capacity Baldwin Universal TestingMachine. The measured values of E0, syt (0.2% proof stress), sut and eu are shownin Table 2. It can be seen in Table 2 that the average ratio of the measured sut/syt

=1.25, the average percentage elongation eu=26% and the average yield stresssyt=404 MPa. These later values are larger than the minimum specified to meet theductility requirements in AS/NZS 4600 [38] of sut/syt =1.08 and eu=10%. Also, these

Table 2Tensile coupons test results

Specimenno. E0GPa �ytMPa �utMPa �ut/�yt �yt/�yn �ut/�un E0/En eu%(1) (2) (3) (4) (5) (6) (7) (8) (9)

TB1 190.9 408 510 1.25 1.17 1.19 0.95 26TB2 190.9 408 510 1.25 1.17 1.19 0.95 27TB3 190.9 408 510 1.25 1.17 1.19 0.95 28TB4 190.9 408 510 1.25 1.17 1.19 0.95 29TB5 212.3 410 501 1.22 1.17 1.17 1.06 30TB6 191.2 404 505 1.25 1.15 1.17 0.96 19TB7 191.2 404 505 1.25 1.15 1.17 0.96 19TB8 191.2 404 505 1.25 1.15 1.17 0.96 19TB9 199.8 365 469 1.28 1.04 1.09 1.00 34TB10 212.3 410 501 1.22 1.17 1.17 1.06 30TB11 217.9 412 502 1.22 1.18 1.17 1.09 28TB12 191.2 404 505 1.25 1.15 1.17 0.96 19Mean 197.6 404 503 1.25 1.15 1.17 0.99 26COV 0.05 0.03 0.02 0.01 0.03 0.02 0.05 0.21


values meet a number of key plastic design requirements in AS 4100 [11], such assut/syt =1.2, eu=15% and sy � 450 MPa.

4. Pure bending test specimens and procedure

4.1. Specimens

A total of 12 specimens were tested in the pure bending rig. The steel sectionsused in the preparation of the specimens were cold-formed circular hollow sections(CHS) grade C350L0 with nominal yield stress of 350 MPa produced by PalmerTube Mills in Australia. Table 3 shows the measured dimensions of the specimens.The range of section slenderness examined in this paper is 59.9�ls�198.9 (d/t=36.4to 121.9). AS4100 [11] and the recently obtained plastic slenderness limit in [33]were used to define the classification of the specimens. Although the compact behav-iour was studied in Elchalakani et al. [33], one compact section B11 was selectedto examine the effect of machining on the inelastic bending properties of cold-formedCHS. Six specimens (B5, B7 to B10 and B12) were chosen to represent the noncom-pact behaviour with 60 � ls�120. Five specimens (B1 to B4 and B6) were chosento represent the slender behaviour with ls �120. The calculated elastic stiffness(E0Im), plastic moment capacity (Mpt) and plastic curvature (�pt) are listed in Table4. The measured material properties are used in calculating Mpt which in turn wasused to determine �pt=Mpt/E0Im.

The necessary d/t ratios required for the investigation were obtained by machiningall the as-received CHS to the required thickness. The total length of the specimenwas 1500 mm long, while the middle machined length was Lm=600 mm. SpecimensB1 to B4 were filled with plaster from the ends to avoid premature instability at thetransition from the as-received section to the machined section. The tube was insertedvertically in a double jacket mould container filled with plaster. The internal jacketwas about 30 mm in diameter to pervert air entrapment. A 400 mm length gap wasleft in the middle of the specimen unfilled. The free deformation length for specimensB1 to B4 is Lf=400 mm, while Lf=600 mm for B5 to B12. This makes the minimalfree deformation length-to-diameter ratio equals Lf/d=3.62 which occurs at B4 (seeTable 3). This later value is favourably more than the recommended minimal valueof Lf/d=2 [39] to eliminate the effect of load fixtures on the bending properties. Thisallows the formation of local buckles without end effects and the development offull inelastic rotation of the cross section.

4.2. Test procedure

A unique pure bending rig was used to test the bending specimens. This rig wasdesigned and fully commissioned at Monash University [39]. The advantage of thisrig is its ability to apply a pure bending moment over the middle span of the testspecimen without inducing significant axial or shear forces. Fig. 3 shows schematicof the pure bending rig and its key components. The machined length (Lm), bending


Tab

le3

Mea

sure

ddi

men

sion

sof

spec

imen

s

Spec

imen

no.

Nom

inal

size

dx

tA

vera

gem

easu

red

dim

ensi

ons

Lf/d

d/t

Sect

ion

Buc

klin

g(m

mx

mm

)sl

ende

rnes

sl s

coef

ficie

nta

d m(m

m)

t m(m

m)

Are

a,A

m(m

m2)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

B1

114.

3x

3.2

110.

11.

1037

6.83

3.63

100.

116

3.3

4.7

B2

114.

3x

3.2

109.

91.

0034

2.26

3.64

109.

917

9.4

4.3

B3

114.

3x

3.2

109.

70.

9030

7.75

3.65

121.

919

8.9

3.8

B4

114.

3x

3.2

110.

41.

2542

8.80

3.62

88.3

144.

15.

3B

510

1.6

x2.

698

.61.

7051

7.72

6.09

58.0

95.1

8.9

B6

101.

6x

2.6

98.8

1.20

368.

096.

0782

.313

3.1

5.7

B7

101.

6x

2.6

99.2

1.40

430.

326.

0570

.911

4.5

6.7

B8

101.

6x

2.6

99.6

1.60

492.

806.

0262

.310

0.6

7.6

B9

101.

6x

2.6

100.

01.

8055

5.53

6.00

55.6

81.1

9.9

B10

101.

6x

3.2

99.8

2.30

704.

796.

0143

.471

.211

.9B

1188

.9x

3.2

87.3

2.40

640.

396.

8736

.459

.914

.5B

1210

1.6

x2.

610

0.6

2.10

650.

105.

9647

.977

.49.

9


Tab

le4

Mec

hani

cal

prop

ertie

sof

spec

imen

s

Spec

imen

Seco

ndE

last

icse

ctio

nPl

astic

Yie

ldm

omen

tO

valin

gPl

astic

mom

entq y

k yt

My/

E0I m

k pt

Mp

t/E0I m

Shap

eno

.m

omen

tof

mod

ulus

,se

ctio

nM

y=Z

m.�

ym

omen

tM

oM

pt(

kN.m

)(D

eg.)

(105

mm

�1)

(105

mm

�1)

fact

orar

ea,

I m(1

06Z

m(1

03m

m3)

mod

ulus

,S m

(kN

.m)

(kN

.m)

S m/Z

m

mm

4)

(103

mm

3)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

B1

0.56

10.1

713

.07

4.15

4.95

5.33

1.78

3.88

4.99

1.29

B2

0.51

9.23

11.8

63.

774.

504.

841.

783.

895.

001.

28B

30.

468.

3010

.65

3.39

4.04

4.35

1.79

3.90

5.00

1.28

B4

0.64

11.5

714

.89

4.72

5.64

6.08

1.78

3.87

4.98

1.29

B5

0.61

12.3

315

.96

5.06

6.08

5.83

1.80

3.92

4.52

1.29

B6

0.44

8.87

11.4

33.

584.

295.

321.

964.

286.

341.

29B

70.

5110

.37

13.3

94.

195.

024.

891.

954.

264.

971.

29B

80.

5911

.88

15.3

74.

805.

775.

611.

954.

244.

961.

29B

90.

6713

.40

17.3

64.

895.

886.

341.

683.

654.

731.

30B

100.

8416

.79

21.8

76.

888.

327.

981.

773.

874.

491.

30B

110.

5813

.23

17.3

05.

456.

617.

131.

994.

335.

671.

31B

120.

7915

.68

20.3

86.

347.

648.

351.

934.

205.

541.

30


Fig. 3. Pure bending rig.

moment and shear force diagrams, and relative angle of rotation (q) of a specimenare shown in Fig. 4.

Adequate modelling of the boundary conditions is of prime importance in theanalysis of buckling problems. Therefore, the specimen was carefully mounted onthe two load application wheels and positioned using saddle clamps that were fabri-cated to suit each tube size. The intention of these clamps was to provide full bearingof the load onto the specimen thus reducing the localised stress raiser particularlyat the inside 40 mm diameter loading pins. The specimen was positioned horizontallyon the saddles to have its weld seam levelled with its centroid. The moment wasapplied to the specimen using a hydraulic pump connected to two hydraulic jacksthrough a manifold. The presence of the manifold ensured that the load applied byboth jacks was approximately the same. The jacks are connected to the load appli-cation wheels on either side of the test specimen. Extension of the jacks causes theload application wheels to rotate opposite to each other and hence apply a bendingmoment to the specimen via four load application pins.

In order to determine the moment-curvature (M-�) curve, it was necessary to meas-ure the change of six key angles and jack loads. Four inclinometers were attachedto the rig, one on each side of the load application wheels and one on each side ofthe hydraulic jacks. In addition, two inclinometers where attached directly (by amagnet) to the top side of the specimen to measure the relative angle of rotation.The applied bending moment was determined from the measured angular rotationsand forces from the load cells attached to the jacks. The inclinometers were calibratedusing a Divided Head set to a 30 degree range and 2.0 kN intervals. The load cellswere calibrated using a Mohr and Federhaff Universal testing machine set to a 10kN range and 1.0 kN intervals. The curvatures were determined from the incli-nometers (attached directly to the ends of the specimen) then were used in plotting


Fig. 4. Bending moment and shear force diagrams.

M-� curves. The curvature was determined as �=q/(LAB�a) [see Fig. 4 for definitionof these terms]. The load cells and inclinometers were connected to a standard datalogger through an amplifier. The sampling rate was carried out at one second inter-vals. The test was interrupted for a few minutes at about 2 to 5 degrees relativerotation to allow for measurements and photographs.

Ovalisation of the cross section was measured using a newly constructed device.The device comprises a frame and a linear displacement measuring device (stringpot). A centre punch was used to locate (at mid span) two positioning holes approxi-mately 0.5 mm in diameter in the tube to allow attaching the string pot and the frameto the tube (top and bottom). The drilling holes have no influence on the mechanismforming as shown in Fig. 5 (a). The string pot was calibrated using a standardmicrometer before testing. The ovalisation measuring device was capable of measur-ing the distortions occuring only in the plane of bending (see Fig. 3). Initial ovalis-ation before the start of the test was measured at eight locations along the machinedlength, and found to be small and of the order of ±0.2 mm.


Fig. 5. Compact CHS under pure bending. (a) B11 at q=40° ls=59.9 (smooth kink); (b) normalisedmoment-curvature response for compact sections.

5. Test results

5.1. Compact sections

Elchalakani et al. [33] examined compact cold-formed CHS under pure bending.Consequently, only the effect of machining on inelastic bending properties of com-


pact CHS is examined in the following. Columns 4, 5 and 6 of Table 5 show theultimate moment obtained in the tests normalised using the yield moment My, theovalisation moment Mo and the full plastic moment Mpt, respectively.

My � Zm·syt (5)

Mo � So·syt �43·(d2

v·dh�d2vi·dhi)·syt (6)

Mpt � Sm·syt (7)

where Zm and Sm are the measured elastic and plastic section moduli of the tube,respectively. �yt is the measured yield stress. So is the plastic section modulus of anovalised tube, dh=0.55d and dv=0.45d are the external horizontal (normal to planeof bending) and vertical (in the plane of bending) radii of an ovalised tube, respect-ively. These values are based on 10% uniform ovalisation measured in [33]. Theinternal horizontal and vertical radii are dhi =(dh�t) and dvi =(dv�t), respectively.Column 7 of Table 5 summarises the rotation capacity determined from.eq. (2) Onespecimen B11 with ls=59.9 (machined to thickness required) was tested and com-pared to the as-received specimen BC2 tested in [33]. B11 did not achieve Mpt asfor BC2 (see Table 5). Fig. 5a shows a plastic hinge formed in B11 at q=40° bendingrotation. This smooth kink was the typical failure mode obtained for the compacttubes tested in [33]. Fig. 5b shows the normalised moment (M/Mpt) versus normalised

Table 5Pure bending test results

Specimen Ultimate Maximum Normalised ultimate moment Rotation capacityno. moment Mu rotation qmax R2=(qmax�qy)/qy

(kN.m) (Deg.)

Mu/My Mu/Mo Mu/Mpt

(1) (2) (3) (4) (5) (6) (7)

B1 3.89 1.79 0.94 0.79 0.73 0.01B2 3.67 1.71 0.97 0.82 0.76 �0.04B3 3.42 1.83 1.01 0.85 0.79 0.02B4 4.53 1.71 0.96 0.80 0.75 �0.04B5 5.78 3.77 1.14 0.95 0.99 1.10B6 4.33 3.35 1.21 1.01 0.81 0.71B7 4.92 3.84 1.17 0.98 1.01 0.97B8 5.38 3.29 1.12 0.93 0.96 0.69B9 5.35 3.60 1.09 0.91 0.84 1.15B10 8.89 4.90 1.29 1.07 1.11 1.76B11 5.67 11.23 1.04 0.86 0.80 4.65B12 7.50 8.05 1.18 0.98 0.90 3.18Mean -- -- 1.093 0.913 0.871 --Coefficient -- -- 0.1013 0.0980 0.1396 --of variation


curvature (�/�pt) curves for B11 and BC2. The normalised strength Mu/Mpt=0.8 and0.9 for B11 and BC2, respectively. The rotation capacity (as given in eq. (2)) R2=4.65and 5.57 for B11 and BC2, respectively. It appears that machining reduced the nor-malised strength by 11% and the rotation capacity by 16.5%. Based on this limitedcomparison, the design recommendations made in the following section may be con-sidered as favourably conservative.

5.2. Noncompact sections

Fig. 6a shows a plastic hinge formed in B5 at q=15° bending rotation. This non-symmetric buckling mode was the typical failure mode obtained for the noncompactCHS. Fig. 6b shows the normalised moment (M/Mpt) versus normalised curvature(�/�pt) curves for B5, B7 to B10 and B12. It is seen that the peak is relatively sharperthan that obtained for compact sections in Fig. 5b. In general, after the formationof the plastic hinge the radial deformation was somewhat confined to the region ofthe plastic mechanism and the load carrying capacity decreased. Some ovalisationdeformations extended beyond the plastic hinge region. The collapse of the noncom-pact sections was relatively rapid and occurred at smaller rotations when comparedwith compact sections. A lock-up in the mechanism can be identified for B7 by asecond peak on the unloading curve in Fig. 6 of �/�pt of 8. Similar phenomenon forSHS was observed in [36]. Plastic mechanism analysis used by Grzebieta [40] andKawata et al. [41] for the analysis of the so-called Yoshimura patterns (diamondindentations) observed for cylinders under axial load may be useful in modelling thenon-symmetric mode observed for noncompact CHS.

5.3. Slender sections

Fig. 7a shows a plastic hinge formed in B2 at q=8° bending rotation. Sectionswith ls�140 all failed by elastic buckling and formed the non-symmetric mode inFig. 7a. This failure mode is similar to the one obtained for noncompact sections inFig. 6a, however more folds around the circumference were found for slender sec-tions. Fig. 7b shows the normalised moment (M/Mpt) versus normalised curvature(�/�pt) curves for B1 to B4 and B6. B6 has ls=133.1 which is classified in AS 4100[11] as slender section. It is seen that the moment-curvature response is very similarto those for noncompact sections shown in Fig. 6b. This emphasis the suitability ofthe proposed ly=140 in Section 6 of the present paper. In general, the peak in Fig.7 is very sharp when compared with curves obtained for noncompact sections in Fig.6b except for B6. The collapse of the slender sections was very rapid and occurred atapproximately q=qy.

5.4. Ovalisation

Fig. 8 shows the average measured ovalisation at the ultimate moment Mu in thepresent and previous tests [33]. The ovalisation is defined as %�dv=[100(d�dv)/d],where dv is the reduced diameter of the cross section in the plane of bending due


Fig. 6. Noncompact CHS under pure bending. (a) B5 at q=15° ls=95.1 (non-symmetric mode); (b)normalised moment-curvature response for noncompact sections.

to ovalisation and d is the initial outside diameter of the CHS. The average measured%�dv was larger for smaller d/t ratios and of the order of 10, 5, 1 for compact [33],noncompact and slender sections, respectively. The accumulation of ovalisation wasvery rapid with increasing bending rotations for specimens with larger d/t. It maybe concluded that the instability of compact and noncompact CHS is controlled byflattening (ovalisation) when compared to slender sections. Similar conclusions weregiven in Otsuka and Koga [20,21]. Two types of ovalisation were observed in thetests; i.e. before and after the formation of the plastic hinge. Different analytical


Fig. 7. Slender CHS under pure bending. (a) B2 at q=8° ls=179.4 (non-symmetric mode); (b) normalisedmoment-curvature response for slender sections.

treatments of these two types of ovalisation are discussed in Wierzbicki and Sin-mao [31].

6. Slenderness limits

It was concluded in Elchalakani et al. [33] that the as-received compact CHS didnot achieve Mpt due to the combined effect of low strain hardening capacity measuredby an average value of �u/�y=1.23 and ovalisation. It is currently unknown how the


Fig. 8. Average measured ovalisation in the tests for compact [33], noncompact and slender CHS (thispaper).

strain hardening of a CHS interacts with the ovalisation prior to the formation ofthe plastic hinge. Previous research [23] showed that the instability of a CHS in theelastic–plastic range is controlled by the tangent modulus (ET) and a graphical pro-cedure was developed to determine the buckling strength. The intersection point ofthe stress–strain curve (��e) and ET�e curve (both curves plotted on the same graph)was used to determine the critical buckling stress under bending. However ovalis-ation was not accounted for in this graphical procedure.

In our previous tests [33], strain measurements indicated that the cross sectionfully yielded, but the tube has taken an oval shape before the formation of the localbuckling (smooth kink mode). This oval section has smaller plastic section modulus(So) compared to the corresponding one for the initial undistorted circulrrar section(Sm). Therefore it was necessary to modify the classical definition for rotationcapacity by using eq. (2)where it is implied that a compact section shall haveadequate rotation capacity at My instead of Mp recommended in [16,17,18]. Therotation capacities R2 for B1 to B12 are listed in column 7 of Table 5. These valuesand those obtained in [33,35] are plotted against the non-dimensional buckling para-meter (a) in Fig. 9. The plastic slenderness limit ap=14 (lp=60), based on a sharptransition in the rotation capacity was obtained in [33]. The rotation capacity (R2)at the transition point (ap=14) is about 5.57 in Fig. 9.


Fig. 9. Normalised inelastic rotation versus non-dimensional buckling parameter.

A modified criterion is adopted to define Class 2 section based on R1 at My insteadof Mp recommended in [16–18]. It is seen in Fig. 9 that, CHS with a8.93 satisfythis requirement. This limit of ac2=8.93 can be converted to the section slendernesslimit (lc2) where E is expressed in MPa:

lc2 � �dt�limit

·� sy

250� � �E /sy

ac2�·� sy

250� �E

ac2·250(8)

After substituting the average measured value of E0=1.98×105 MPa (see Table 2)in eq. (8) lc2=88 is obtained.

Fig. 2 and Table 5 show that all sections with ls�140 achieved My in the tests.Note that a value of Ze/Z=1 in Fig. 2 corresponds to Mu/My=1 in the tests. Therefore,it appears that a satisfactory yield slenderness limit ly=140 is suitable for cold-for-med CHS. It is interesting to see a similar transition in constant-moment tests [34]in Fig. 2 at about ls=140. The newly obtained limits in this paper for lp=60, lc2=88and ly=140 are compared with the slenderness limits in the current steel specifi-cations in Table 1. It seems that these new limits are somewhat larger than the currentones, except for the yield limit specified in AISC-LRFD [12].

7. Design recommendations

The classification of compact, non-compact and slender sections are based on theslenderness limits derived in the previous section. The nominal moment capacity Ms

can be calculated in the same format as given in AS4100 [11], i.e.


Ms � fy·Ze (9)

where fy is the yield stress of CHS and Ze is the effective section modulus givenbelow.

7.1. Compact sections

Since the full plastic moment Mpt was not reached in the present tests or previoustests for compact CHS [33], and the ovalisation moment Mo shown in eq. (6) gavegood prediction of the ultimate moment capacity for compact sections, it is proposedin this paper that Ze be taken as the plastic section modulus of an ovalised tube So

with 10% uniform ovalisation, i.e.

Ze � So �43·(d2

v·dh�d2vi·dhi) (10)

where dh=0.55d, dv=0.45d, dhi =(dh� t) and dvi =(dv�t).A ratio of Ze/Z is often used in comparing different codes as shown in Fig. 2

where Z is the elastic section modulus of CHS which is given by

Z �p32

·d3(1�b4) (11)

in which,

b �d�2·t

d� 1�

2(d / t)

(12)

The ratio of Ze/Z can be written in terms of (d/t), i.e.

Ze

Z�

43·�0.1113·�d

t�3

��0.45·dt�1�2

·�0.55·dt�1��

p32

·�dt�3

·[1�b4]

(13)

The ratio of Ze/Z is plotted in Fig. 10 against d/t for compact CHS sections withd/t ranging from 5 to 50. It can be seen that a lower bound of Ze/Z of 1.2 is obtained.Therefore eq. (10) can be simplified to:

Ze � 1.2·Z (14)

where Z is given in eq. (11).

7.2. Noncompact sections

The proposed effective section modulus Ze for noncompact sections includingClass 2 and 3 sections with section slenderness in the range of 60 � ls�140 is

Ze � Z ��ly�ls

ly�lp�(Zc�Z)� (15)


Fig. 10. Ze/Z versus d/t.

where Zc=1.2Z, which is the effective section modulus of a compact section (definedin eq. (14)), ls is defined in eq. (3), lp=60 and ly=140.

7.3. Slender sections

For sections which satisfy ls � 140, the effective section modulus shall be calcu-lated as the lesser of the values determined in eqs. (16) and (17).

Ze � Z�ly

ls

(16)

Ze � Z �2ly

ls�2

(17)

where ly=140. Eq. (16) was found to give lesser values than those predicted usingeq. (17) for ls�355. A recommended design curve makes use of eqs. (14)–(16) isplotted and shown in Fig. 2 together with other design rules. It is seen that therecommended design curve initially lies lower, at smaller ls, but it lies higher forlarger ls in comparison with the AS 4100 [11] design curve. It is also observed thatthe Eurocode 3 [13] design curve which has a sudden drop from Ze/Z=1.273 toZe/Z=1.0 at ls=65.8 have reasonable predictions (particularly for ls�60) when com-pared with the test results for the machined CHS.

8. Conclusions

The following conclusions are made based on the test results described in thispaper:


� The plastic moment capacity Mpt of CHS can not be reached for machined or as-received compact CHS due to the combined effect of low strain hardening ofcold-formed CHS and ovalization. Therefore it was necessary to modify the exist-ing criteria to define new slenderness limits for cold-formed CHS.

� The plastic slenderness limit of lp=60 to define a compact section which wasobtained for cold-formed CHS in a related paper [33] is confirmed from theadditional tests performed in this paper. This limit is slightly larger than lp=50which was recommended by Sherman [34] for plastic design and is specified inAS 4100 [11].

� A new section slenderness limit of lc2=88 was obtained to define Class 2 cold-formed CHS. This value was obtained based on a rotation capacity R2�1 at My.This value is slightly smaller than lc2=97 (ac2=8.2) interpreted from Sherman[35] tests.

� A new section yield slenderness limit ly=140 was obtained to define slender cold-formed CHS. This limit is considerably lower than ly=250 that obtained by Sher-man [34] and currently specified in the AISC-LRFD [12]. The reason for thisdiscrepancy is the limited amount of test results available in the slenderness rangeof ls=100 to 300.

� A new design curve, makes use of the newly obtained slenderness limits, consist-ent with AS 4100 [11] format is derived and recommended for design of cold-formed CHS.

Acknowledgements

The writers are grateful to the Australian Research Council and Monash Universityfor their financial assistance for the project. Thanks to Professor Don Sherman forproviding the data on fabricated cylinders. Thanks are given to Palmer Tube Mills forproviding the steel tubes. The experiments were carried out in the Civil EngineeringLaboratory at Monash University and the technical assistance of Mr Graham Rundleand Mr Geoff Doddrell is gratefully acknowledged.

References

[1] Standards Association of Australia. Structural steel hollow sections. AS 1163, Sydney, Australia,1991.

[2] ASTM. Cold-formed welded and seamless carbon steel structural tubing in rounds and shapes.ASTM-A500, Philadelphia, Pa, 1984.

[3] Galambos TV. Recent research and design developments in steel and composite steel-concrete struc-tures in USA. Journal of Construction Seel Research 2000;55:289–303.

[4] Sheldon M, Sheldon R. Colonial stadium-a highly innovative multi purpose stadium. AustralianAISC 1999;33(4):15–27.

[5] McDonald D. Stadium Australia, The Year 2000 Olympic Stadium. Australian AISC1999;33(4):3–14.

[6] Dean BK, Bennett JD. Roof top communication towers. In: Grundy P, Holgate A, Wong B, editors.Tubular Structures VI, Melbourne, Australia, December 1994;14–16:41-44.


[7] Hancock GJ. Light gauge construction. Progress in Structural Engineering and Materials1997;1(1):25–30.

[8] Rondal J. Cold-formed steel members and structures general report. Journal of Construction SteelResearch 2000;55:155–8.

[9] Hancock GJ, Zhao XL, Wilkinson T. Future directions in cold-formed tubular structures. The PaulGrundy Symposium, Department of Civil Engineering, Monash University. November 2001:17–24.

[10] Wilkinson T, Hancock GJ. Tests to examine compact web slenderness of cold-formed RHS. Journalof Structural Engineering, ASCE 1998;124(10):1166–74.

[11] Standards Association of Australia. SAA steel structures code. AS4100, Sydney, Australia, 1998.[12] AISC-LRFD. Load and resistance factor design specification for structural steel buildings. American

Institute of Steel Construction Inc., Chicago, December 27, 1999.[13] Eurocode 3. Design of Steel Structures, Part 1.1, General rules and rules for buildings. ENV 1993

1-1, February 1992.[14] CAN-SCA S16.1. Steel structures for buildings. Limit States Design, Canadian Standards Associ-

ations, 1989.[15] British Standards Institution. BS 5950, Part 1, Structural use of steelwork in building, 1990.[16] Kemp AR. Factors affecting the rotation capacity of plastically designed members. The Structural

Engineer 1986;64B(2):28–35.[17] Kemp AR. Strength of pipes continuous over a series of saddle supports. Journal of Constructional

Steel Research 1990;15:233–48.[18] Kemp AR. Applications of research on ductility of structures. Canadian Journal for Civil Engineering

1992;19:86–96.[19] Donnell LH. A new theory for the buckling of thin cylinder under axial compression and bending.

ASME Transactions 1934;56:795–806.[20] Otsuka H, Koga T. Buckling of circular cylindrical shell under beam-like bending, 1st report, experi-

ment. Transactions of Japan Society for Aerospace Science 1997;41(133):38–45.[21] Otsuka H, Koga T. Buckling of circular cylindrical shell under beam-like bending, 2nd report, buck-

ling stress of cylindrical shells. Transactions of Japan Society for Aerospace Science1997;41(133):144–51.

[22] Elchalakani M, Zhao XL, Grzebieta R. A closed-form solution for elastic buckling of thin-walledunstiffened circular cylinders under pure flexure. 1st International Conference on Steel and Com-posite Structures, Pusan, Korea, Techno-Press 2001:267–274.

[23] Murray NW, Bilston P. Local buckling of thin-walled pipes being bent in the plastic range. Thin-Walled structures 1992;14:411–34.

[24] Ju GT, Kyriakides S. Bifurcation and localisation instabilities in cylindrical shells under bending.International Journal of Solids and Structures, II-Predictions 1992;29:1143–71.

[25] Korol RM, Huboda J. Plastic behaviour of hollow structural sections. J Structural Division ASCE1972;98:1007–23.

[26] Brazier LG. On the flexural of thin cylindrical shells and other “ thin sections” . Proc. Royal SocietyLondon (Series A) 1927;116:104–14.

[27] Mamalis AG, Manolacas DE, Baldoukas AK, Viefelahn GL. Deformation characteristics of crash-worthy thin-walled steel tubes subjected to bending, part C. International Journal of MechanicalSciences, Proceedings of Institute of Mechanical Engineers 1989;203:411–7.

[28] Kyriakides S, Shaw PK. Response and stability of elastic-plastic circular pipes under combinedbending and external pressure. International Journal of Solids and Structures 1982;18:957–73.

[29] Kyriakides S, Shaw PK. Inelastic buckling of tubes under cyclic bending. Journal of Pressure VesselTechnology 1987;109:169–78.

[30] Shaw PK, Kyriakides S. Inelastic analysis of thin-walled tubes under cyclic bending. Journal ofsolids and structures 1985;21(11):1073–100.

[31] Wierzbicki T, Sinmao MV. A simplified model for brazier effect in plastic bending of cylindricaltubes. International Journal of Pressure Vessels and Piping 1997;71:19–28.

[32] Rondal J, Wurker KG, Dutta D, Wardenier J, Yeomans N. Structural stability of hollow sections,CIDECT series on construction with hollow steel sections. Verlag, TUV, Rheinland, Germany,reprinted, 1996.


[33] Elchalakani M, Zhao XL, Grzebieta R. Plastic slenderness limit for cold-formed circular hollowsections plastic. Australian Journal of Structural Engineering, Institutions of Engineers Australia2001;3(3):1–16.

[34] Sherman DR. Inelastic flexural buckling of cylinders, “steel structures” . In: Pavlovic MN, editor.Recent Research Advance and Their Applications to Design, Proceedings of the invited papers forthe International Conference. Budva, Yugoslavia, 29 September–1 October, 1986:339–357.

[35] Sherman DR. Bending capacity of fabricated pipe at end connections. The University of WisconsinMilwaukee research report, December 1987.

[36] Schilling GS. Buckling strength of circular tubes. Journal of Structural Division, ASCE1965;91:325–48.

[37] Standards Association of Australia. “Methods for tensile testing of metals” . AS 1391, Sydney, Aus-tralia, 1991.

[38] Standards Association of Australia. “Cold-formed steel structures. Australian/New Zealand StandardAS/NZS 4600, Sydney, Australia, 1996.

[39] Cimpoeru SJ. The modelling of collapse during roll-over of bus frames consisting of square thin-walled tubes. PhD Thesis, Monash University, Australia, 1992.

[40] Grzebieta RH. On the equilibrium approach for predicting the crush response of thin-walled mildsteel structures. PhD Thesis, Monash University, Melbourne, 1990.

[41] Kawata K, Miyamoto I, Minamitani, R. Plastic collapse in dynamic axial compression of thin-walledcircular cylinder. Proceeding of the International Symposium on Intense Dynamic and its Effects,Beijing, China, 1986:1–5.

[42] New Zealand Standard. Steel Structures Standard, NZS 3404, Part 1, Wellington, New Zealand,1997.

bending tests to determine slenderness limits for cold-formed circular hollow sections

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