Design of cold-formed steel oval hollow section columns

Journal of Constructional Steel Research 71 (2012) 26–37

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Journal of Constructional Steel Research

Design of cold-formed steel oval hollow section columns Ji-Hua Zhu a, Ben Young b,⁎ a b

Shenzhen Key Lab on Durability of Civil Engineering, College of Civil Engineering, Shenzhen University, Shenzhen, PR China Dept. of Civil Engineering, The University of Hong Kong, Pokfulam road, Hong Kong

a r t i c l e

i n f o

Article history: Received 10 August 2011 Accepted 29 November 2011 Available online 28 December 2011 Keywords: Buckling Cold-formed steel Column Direct strength method Finite element analysis Oval section

a b s t r a c t This paper presents the numerical simulation and design of cold-formed steel oval hollow section columns. An accurate finite element model was developed to simulate the fixed-ended column tests of oval hollow sections. The material non-linearities obtained from tensile coupon tests as well as the initial local and overall geometric imperfections were incorporated in the finite element model. Convergence study was performed to obtain the optimized mesh size. A parametric study consisted of 100 columns was conducted using the verified numerical model. The failure modes of material yielding, local buckling and flexural buckling as well as interaction of local and flexural buckling were found in this study. The experimental column strengths and numerical results predicted by the parametric study were compared with the design strengths calculated using the current North American, Australian/New Zealand and European specifications for cold-formed steel structures. In addition, the direct strength method, which was developed for cold-formed steel members for certain cross-sections but not cover oval hollow sections, was used in this study. The reliability of these design rules was evaluated using reliability analysis. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Finite element analysis (FEA) is a powerful tool that can be employed to a wide range of applications, such as the numerical simulation of steel and aluminium structures [1]. The finite element approach provides many advantages over physical experiments, especially when a parametric study of cross-section geometry is involved. However, it is necessary to verify the finite element model (FEM) against experimental results to ensure an accurate and reliable model. The FEA should be capable to predict the ultimate loads and failure modes of steel structural members. Oval hollow section (OHS) is one of the new sections used in steel structure constructions. Typical steel sections such as I-section, channel section, circular hollow section and rectangular hollow section are used in various structures. However, new sections are being developed with the requirements on both the efficiency and aesthetics. The geometry of the OHS combined with its aesthetic appearance makes it an exciting choice for architects. Besides, it has the advantage of different major and minor axes properties. Gardner and Ministro [2] reported some applications of oval hollow sections in structural engineering projects. In the previous research, different definition of oval hollow section has been used. The elliptical geometry, which is commonly referred as oval hollow section, has been investigated by a number of

⁎ Corresponding author. Tel.: + 852 2859 2674; fax: + 852 2559 5337. E-mail addresses: [email protected] (J.-H. Zhu), [email protected] (B. Young). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.11.013

researchers. Bortolotti et al. [3] investigated welded joints of elliptical hollow sections. Gardner and Chan [4] proposed slenderness parameters and cross-section classification limits for elliptical hollow sections. The flexural buckling behaviour was also investigated by Chan and Gardner [5,6]. Theofanous et al. [7] investigated stainless steel elliptical section compression members. Silvestre [8] investigated the buckling behaviour of elliptical cylindrical shells and tubes under compression. Research on oval hollow section has been extended to concrete-filled members, as presented by Zhao and Packer [9] and Yang et al. [10]. It should be noted that the aforementioned oval hollow sections were focused on elliptical hollow sections, which is different to the oval hollow sections presented in this study. A series of fixed-ended column tests on cold-formed steel oval hollow sections has been conducted and presented by Zhu and Young [11]. The oval hollow section investigated in this study is composed of two flat web plates and two semi-circle flanges, as shown in Fig. 1. The oval hollow sections were cold-rolled at room temperature and then the electric resistance welding was used to close the hollow sections. The test program included 28 columns compressed between fixed ends. The observed failure modes included material yielding, flexural buckling and local buckling in the flat web plates as well as interaction between local and flexural buckling. Local buckling of the curved flanges of semi-circle was not observed since the crosssections were relatively compact in the test program. Following the experimental investigation, a numerical investigation using FEA is performed and presented in this paper with particular emphasis on slender sections, where local buckling was observed in the curved flanges. It should be noted that the semi-circle element with W/t

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

27

Nomenclature A COV D DL E e FEA FEM Fm fy h L le ley LL Mm P PAS/NZS PEC3 PExp PFEA PNAS Pm Pu ry t VF VM VP W β κ λmax σ σ0.2 σtrue σu ε εf pl εtrue ϕ

Gross cross-section area Coefficient of variation Overall depth of section Dead load Young's modulus Axial shortening Finite element analysis Finite element model Mean value of fabrication factor Material yield strength Clear depth of flat portion of section Column length Column effective length Effective length for buckling about the minor y-axis Live load Mean value of material factor Axial load Unfactored design strength for Australian/New Zealand Standard Unfactored design strength for Eurocode 3 Experimental ultimate load of column Ultimate load predicted by FEA Unfactored design strength for North American Specification Mean value of tested-to-predicted load ratio Column strength Radius of gyration of full unreduced cross-section about the y-axis Thickness of section Coefficient of variation of fabrication factor Coefficient of variation of material factor Coefficient of variation of tested-to-predicted load ratio Overall width of section Reliability index Plate buckling coefficient Nominal maximum slenderness ratio Stress Static 0.2% proof stress True stress Static ultimate tensile strength Strain Elongation (tensile strain) at fracture True plastic strain and Resistance factor.

ratio less than 40 should be considered as fully effective according to the Australian/New Zealand Standard for cold-formed steel structures [12]. Therefore, the expressions of compact and slender sections were used to identify the oval hollow section in this study. The objective of this study is firstly to develop an advanced non-linear FEM for the investigation on the strengths and behaviour of cold-formed steel oval hollow section columns; secondly, to compare the test strengths with the design strengths predicted using the current North American [13,14], Australian/New Zealand [12] and European [15–17] specifications for cold-formed steel structures with certain assumptions on the calculation of effective width. The direct strength method, which has been adopted by the North American Specification for cold-formed steel structures as an alternative procedure, was also used in this study; and lastly, to examine the reliability of these design rules using reliability analysis. Finite element program ABAQUS [18] was used to perform the numerical analysis. Initial geometric

y x D t

h

W Fig. 1. Definition of symbols.

imperfections and material non-linearity were included in the model. The finite element model was verified against the column test results conducted by Zhu and Young [11]. 2. Summary of test program The test program presented by Zhu and Young [11] provided experimental ultimate loads and failure modes of cold-formed steel oval hollow sections compressed between fixed ends. The experimental program consisted of 28 specimens of four test series with different cross-section sizes and steel grades, as shown in Table 1 using the symbols illustrated in Fig. 1. The series A, B, C and D refer to the test specimens with nominal cross-section dimensions of 120×48×2.0, 115×38×2.0, 42×21×2.8 and 30×15×1.6 mm, respectively. The measured cross-section dimensions of each specimen are detailed in Zhu and Young [11]. Table 2 shows the material properties of each series of specimens obtained from tensile coupon tests in the flat portion (web) of the specimen, as shown in Zhu and Young [11]. The specimens were tested between fixed ends at various column lengths ranged from 90 to 3000 mm, and the nominal maximum slenderness ratio (λmax) for each test series ranged from 75.5 to 131.6. The longest specimen lengths produced ley/ry ratios of 75.5, 94.5, 97.5 and 131.6 for Series A, B, C and D, respectively, where ley is the

Table 1 Oval hollow section column test series. Test series

Dimension, D× W× t (mm)

h/t

A B C D

120×48×2.0 115×38×2.0 42×21×2.8 30×15×1.6

36.0 38.5 7.5 9.4

Note: 1 in. = 25.4 mm.

Table 2 Measured material properties of tensile coupons in the web plates. Coupon

E (GPa)

σ0.2 (MPa)

σu (MPa)

εf (%)

A B C D

202 202 200 199

359 359 443 432

403 387 456 454

18 27 20 21

28

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

Table 3 Comparison of test strengths with FEA strengths of different mesh densities for Series A. Specimen

A360 A360# A600 A1200 A1200# A1800 A2400 A3000 Mean, Pm COV, VP

Test

Comparison

PExp (kN)

Failure mode

PExp/PFEA1

PExp/PFEA2

PExp/PFEA3

Failure mode

181.2 185.9 196.0 190.3 188.5 183.9 173.1 157.7 – –

L L L F F F F F – –

1.00 1.03 1.07 1.05 1.04 1.04 1.04 1.08 1.04 0.023

1.00 1.03 1.06 1.05 1.06 1.04 1.05 1.08 1.04 0.024

1.00 1.03 1.07 1.05 1.06 1.05 1.04 1.08 1.04 0.025

L L L L L F L+F L+F – –

effective length for buckling about the minor y-axis and ry is the radius of gyration of full unreduced cross-section about the y-axis. The test rig and operation are also detailed in Zhu and Young [11]. Initial overall geometric imperfections were measured for all specimens prior to testing, except for short specimens of less than 360 mm in length, as detailed in Zhu and Young [11]. The experimental ultimate loads (PExp) and failure modes observed at ultimate loads are shown in Tables 3–6 for Series A, B, C and D, respectively. The test specimens were labeled such that the test series and specimen length could be easily identified, as shown in Tables 3–6. For example, the label “A360#” defines the following specimen: the first letter indicates the specimen dimension and test series, where “A” refers to an OHS with nominal cross-section dimension of 120 × 48 × 2.0 mm for overall depth (D), overall width (W) and thickness (t) of the section, respectively. The cross-section dimensions for other sections are shown in Table 1. The following digits in the label indicate the nominal specimen length in millimeters (360 mm); and if a test was repeated, then the symbol “#” indicates the repeated test. 3. Development of finite element model The finite element program ABAQUS [18] version 6.9 was used in the analysis for the simulation of cold-formed steel OHS columns tested by Zhu and Young [11]. The measured geometry, initial geometric imperfections, and material non-linearity of the test specimens were included in the finite element model. The model was based on the centerline dimensions of the cross-sections. The simulation consisted of two steps. The first step is the Eigenvalue analysis to determine the buckling modes, whereas the second step is the load–displacement nonlinear analysis. The finite element model described in this section followed the same approach as detailed in Yan and Young [19] for cold-formed steel columns. Shell element is one of the most appropriate types of element for modeling thin-walled metal structures. The 4-noded doubly curved

B345 B600 B1200 B1800 B2400 B3000 Mean, Pm COV, VP

Comparison

C126 C300 C600 C900 C900# C1200 C1500 Mean, Pm COV, VP

Comparison

FEA

Failure mode

PFEA (kN)

PExp/PFEA

Failure mode

144.1 122.3 111.0 96.8 92.8 64.6 51.5 – –

Y Y F F F F F – –

126.4 114.8 108.9 92.7 95.2 74.4 52.7 – –

1.14 1.07 1.02 1.04 0.97 0.87 0.98 1.01 0.083

Y F F F F F F – –

Note: 1 kip = 4.45 kN; F = Flexural buckling; Y = Yielding.

shell elements with reduced integration S4R were used in the model. The S4R element has six degrees of freedom per node and shown to provide accurate results from previous research as described in Yan and Young [19]. The finite element mesh dimension is important on the calculation efficiency and simulation accuracy. Hence, a convergence study was performed on Series A test specimens with three different mesh dimensions of FEA1 (3.6 × 3.6 mm, length by width), FEA2 (5 × 5 mm) and FEA3 (8 × 8 mm), as shown in Table 3. The three mesh dimensions separated the cross-section into 80, 56 and 36 elements, respectively. The ultimate loads and failure modes of each specimen obtained from the three mesh densities are shown in Table 3 for Series A. It is shown that the experimental-toFEA ultimate load ratios PExp/PFEA1, PExp/PFEA2 and PExp/PFEA3 are very close for each specimen, with the same mean value of 1.04, and the corresponding coefficient of variation (COV) of 0.023, 0.024 and 0.025, respectively. By considering calculation efficiency, the FEA2 (5× 5 mm) is a better choice for Series A. The finite element mesh dimensions of 5 × 5 mm, 2.9 × 2.9 mm and 2.4 × 2.4 mm were used that separated the cross-sections into 56, 36 and 30 elements for Series B, C and D, respectively. The fixed-ended boundary condition was simulated by restraining all the degrees of freedom of the nodes at both ends, except for the translational degree of freedom in the axial direction at one end of the column. The nodes other than the two ends were free to translate and rotate in any directions. The displacement control loading method, which is identical to that used in the column tests, was used in the finite element model. Compressive axial load was applied to the column by specifying an axial displacement to the nodes at one end of the column. The material properties obtained from the flat tensile coupon tests were used in the numerical modeling of the respective test series. In the linear analysis stage of the simulation, the material properties of the columns were defined by the density, initial Young's modulus and Poisson's ratio only. In the non-linear analysis stage, material non-linearity or “plasticity” was included in the FEM using a mathematical model known as the incremental plasticity model [18], in pl which true stresses (σtrue) and true plastic strains (εtrue ) were

Specimen

Test PExp (kN)

Failure mode

PFEA (kN)

PExp/PFEA

Failure mode

D90 D300 D600 D900 D900# D1200 D1500 Mean, Pm COV, VP

58.4 51.1 38.9 30.4 28.4 19.0 12.7 – –

Y Y F F F F F – –

50.3 47.2 43.8 32.1 31.7 19.5 13.2 – –

1.16 1.08 0.89 0.95 0.90 0.97 0.97 0.99 0.101

Y F F F F F F – –

FEA

PExp (kN)

Failure mode

PFEA (kN)

PExp/PFEA

Failure mode

163.3 156.2 146.6 131.6 118.8 103.5 – –

L L L+F L+F L+F F – –

151.1 146.6 146.6 139.2 123.8 97.3 – –

1.08 1.07 1.00 0.95 0.96 1.06 1.02 0.053

L L L+F L+F L+F L+F – –

Note: 1 kip = 4.45 kN; F = Flexural buckling; L = Local buckling.

Test PExp (kN)

Table 6 Comparison of test strengths with FEA strengths for Series D.

Table 4 Comparison of test strengths with FEA strengths for Series B. Test

Specimen

FEA

Note: 1 kip = 4.45 kN; F = Flexural buckling; L = Local buckling.

Specimen

Table 5 Comparison of test strengths with FEA strengths for Series C.

Comparison

Note: 1 kip = 4.45 kN; F = Flexural buckling; Y = Yielding.

FEA

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

4. Test verification

Axial Load, P (kN)

200 Test FEA

160 120 80 40 0 0

1

2

3

4

Axial shortening, e (mm) Fig. 3. Axial load versus shortening of specimen A360.

60

Axial Load, P (kN)

specified. The true stresses and true plastic strains were obtained from the static engineering stresses (σ) and strains (ε) using σtrue = σ(1 + ε), pl and εtrue = ln(1 + ε) − σtrue/E, as specified in ABAQUS [18], where E is the initial Young's modulus of the static engineering stress–strain curve. The incremental plasticity model required only a range of the true stress–strain curve from the point corresponding to the last value of the linear range of the static engineering stress–strain curve to the ultimate point of the true stress–strain curve. Fig. 2 shows the stress– strain curves of Series B. Both initial local and overall geometric imperfections were incorporated in the model. Superposition of local buckling mode and overall buckling mode with the identified magnitudes was carried out. These buckling modes were obtained by Eigenvalue analysis of the columns with very high value of width-to-thickness ratio and very low value of width-to-thickness ratio to ensure local and overall buckling occurs respectively. Only the lowest buckling mode (Eigenmode 1) is used in the Eigenvalue analysis. All the buckling modes predicted by ABAQUS Eigenvalue analysis are generalized to 1.0. Therefore, the buckling modes were factored by the measured magnitudes of the initial local and overall geometric imperfections. The measured magnitudes of the initial overall geometric imperfections of each specimen are detailed in Zhu and Young [11]. The local geometric imperfection was not measured in the test program and hence the magnitude of 10% of plate thickness was used in the FEM as suggested by Chan and Gardner [20].

29

45 30 15

Test FEA

0 0

The developed finite element model was verified against the experimental results. The ultimate loads and failure modes predicted by the FEA are compared with the experimental results as shown in Tables 3–6, for Series A, B, C and D. It is shown that the ultimate loads (PFEA) obtained from the FEA are in good agreement with the experimental ultimate loads (PExp). The mean values of the experimental-to-FEA ultimate load ratio are 1.04, 1.02, 1.01 and 0.99, with the corresponding coefficient of variation (COV) of 0.024, 0.053, 0.083 and 0.101 for Series A, B, C and D, respectively. The failure modes at ultimate load obtained from the tests and FEA for each specimen are also shown in Tables 3–6. The failure modes included material yielding (Y), local buckling (L) and flexural buckling (F) as well as interaction of local and flexural buckling (L + F). The failure modes predicted by the FEA are generally in good agreement with those observed in the tests, except for a few specimens. For example, the failure mode of flexural buckling was observed in the test for specimen B3000, whereas the failure mode predicted by the FEA was interaction of local and flexural buckling. Figs. 3 and 4 show the comparison of the load-shortening curves obtained from the tests and predicted by the FEA for the specimens A360 and C1500, respectively. It is shown that the FEA curves followed the experimental curves closely. Fig. 5(a) shows the photograph of test specimen

1

2

3

Axial shortening, e (mm) Fig. 4. Axial load versus shortening of specimen C1500.

B1200 immediately after the ultimate load has reached. The specimen failed in interaction of local and flexural buckling. Fig. 5(b) shows the deformed shape of the specimen predicted by the FEA right after the ultimate load. The resemblance of Fig. 5(a) and (b) demonstrates the reliability of the FEA predictions.

a) Test

b) FEA

Stress σ, (MPa)

500 400 300 200 Test curve Engineering curve True curve

100 0 0

5

10

15

20

25

30

Strain ε, (%) Fig. 2. Modeling of material plasticity for Series B.

Fig. 5. Comparison of experimental and FEA deformed shapes for specimen B1200.

30

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

5. Parametric study

Table 8 Comparison of FEA strengths with design strengths.

The FEM closely predicted the experimental ultimate loads and failure modes of the cold-formed steel OHS columns conducted by Zhu and Young [11]. Hence, the model was used for an extensive parametric study. The parametric study included 100 specimens that consisted of 20 series, as shown in Table 7. Each series contained 5 specimens with column lengths of 500, 1200, 2000, 2700 and 3500 mm. The specimens were labeled such that the cross-section dimensions and column length could be identified, as shown in Table 8. For example, the label “W60T2L500” defines the following specimen: • The first part of the label “W60T2” indicates the cross-section dimension, where “W60” refers to the section overall width of 60 mm, and “T2” refers to the specimen thickness of 2 mm. Table 7 shows the cross-section dimensions of each series using the nomenclature defined in Fig. 1. It should be noted that the overall depth of all sections is 300 mm. • The second part of the label “L500” indicates the length of the column, where the letter “L” refers to the column length and the following digits are the nominal length of the specimen in millimetres (500 mm). The material properties of the specimens in the parametric study are identical to the material properties of Series B in the experimental program [11]. The local imperfection magnitude was 10% of the section thickness and the overall imperfection magnitude was 1/1500 of the column length. The size of the finite element mesh was 10 × 10 mm (length by width) since the specimen cross-section dimensions are much larger than the specimens in the experimental program. The column strengths (PFEA) obtained from the parametric study are shown in Table 8. 6. Design approaches 6.1. Current design rules The current North American Specification [13,14], Australian/New Zealand Standard [12] and European Codes [15–17] provide design rules for cold-formed steel structures. However, these specifications do not cover design rules of the oval hollow sections investigated in this study. Therefore, certain assumptions on the calculation of effective width have been made in the calculation of the column strengths in this study, as described in Zhu and Young [11]. The OHS Table 7 Cross-section dimensions of each series for parametric study. Series

W60T2 W60T2.4 W60T3 W60T4 W60T10 W75T1.9 W75T2.2 W75T2.8 W75T3.8 W75T10 W100T1.7 W100T2 W100T2.5 W100T3.3 W100T10 W150T1.3 W150T1.5 W150T1.8 W150T2.5 W150T10

Depth

Width

Thickness

D (mm)

W (mm)

t (mm)

300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

60 60 60 60 60 75 75 75 75 75 100 100 100 100 100 150 150 150 150 150

2 2.4 3 4 10 1.9 2.2 2.8 3.8 10 1.7 2 2.5 3.3 10 1.3 1.5 1.8 2.5 10

h/W

h/t

W/t

4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1

120.0 100.0 80.0 60.0 24.0 118.4 102.3 80.4 59.2 22.5 117.6 100.0 80.0 60.6 20.0 115.4 100.0 83.3 60.0 15.0

30.0 25.0 20.0 15.0 6.0 39.5 34.1 26.8 19.7 7.5 58.8 50.0 40.0 30.3 10.0 115.4 100.0 83.3 60.0 15.0

Specimen

W60T2L500 W60T2L1200 W60T2L2000 W60T2L2700 W60T2L3500 W60T2.4L500 W60T2.4L1200 W60T2.4L2000 W60T2.4L2700 W60T2.4L3500 W60T3L500 W60T3L1200 W60T3L2000 W60T3L2700 W60T3L3500 W60T4L500 W60T4L1200 W60T4L2000 W60T4L2700 W60T4L3500 W60T10L500 W60T10L1200 W60T10L2000 W60T10L2700 W60T10L3500 W75T1.9L500 W75T1.9L1200 W75T1.9L2000 W75T1.9L2700 W75T1.9L3500 W75T2.2L500 W75T2.2L1200 W75T2.2L2000 W75T2.2L2700 W75T2.2L3500 W75T2.8L500 W75T2.8L1200 W75T2.8L2000 W75T2.8L2700 W75T2.8L3500 W75T3.8L500 W75T3.8L1200 W75T3.8L2000 W75T3.8L2700 W75T3.8L3500 W75T10L500 W75T10L1200 W75T10L2000 W75T10L2700 W75T10L3500 W100T1.7L500 W100T1.7L1200 W100T1.7L2000 W100T1.7L2700 W100T1.7L3500 W100T2L500 W100T2L1200 W100T2L2000 W100T2L2700 W100T2L3500 W100T2.5L500 W100T2.5L1200 W100T2.5L2000 W100T2.5L2700 W100T2.5L3500 W100T3.3L500 W100T3.3L1200 W100T3.3L2000 W100T3.3L2700 W100T3.3L3500 W100T10L500 W100T10L1200 W100T10L2000 W100T10L2700

FEA

Comparison

PFEA (kN)

PFEA/PNAS

PFEA/PAS/NZS

PFEA/PEC3

PFEA/PDSM

223.4 205.0 202.3 178.3 171.7 299.9 273.3 267.1 237.7 227.3 429.4 395.9 380.7 355.7 326.5 700.6 658.5 626.3 598.5 562.8 2350.6 2279.6 2133.6 1994.4 1826.2 237.5 231.1 220.6 212.8 182.1 286.0 284.8 266.2 263.1 245.7 415.5 406.2 387.8 373.2 348.9 678.9 648.1 621.9 603.4 579.7 2433.9 2380.1 2266.4 2147.8 2016.8 228.3 226.3 222.4 216.8 208.8 283.7 282.4 278.1 271.8 261.3 385.9 384.1 376.7 369.5 356.4 578.0 574.0 560.6 550.9 535.7 2514.8 2487.9 2422.6 2338.4

0.88 0.83 0.86 0.81 0.85 0.91 0.85 0.87 0.83 0.87 0.94 0.89 0.90 0.89 0.89 1.00 0.96 0.96 0.97 1.00 0.99 0.98 0.98 0.99 1.02 0.89 0.88 0.87 0.88 0.80 0.88 0.89 0.86 0.88 0.88 0.91 0.90 0.89 0.89 0.89 0.96 0.93 0.92 0.93 0.95 0.99 0.99 0.98 0.98 1.00 0.83 0.83 0.83 0.83 0.83 0.83 0.84 0.84 0.84 0.84 0.84 0.85 0.85 0.85 0.86 0.87 0.87 0.87 0.88 0.89 0.98 0.98 0.98 0.98

0.88 0.83 0.86 0.81 0.85 0.91 0.85 0.87 0.83 0.87 0.94 0.89 0.90 0.89 0.89 1.00 0.96 0.96 0.97 1.00 0.99 0.98 0.98 0.99 1.02 0.89 0.88 0.87 0.88 0.80 0.88 0.89 0.86 0.88 0.88 0.91 0.90 0.89 0.89 0.89 0.96 0.93 0.92 0.93 0.95 0.99 0.99 0.98 0.98 1.00 0.83 0.83 0.83 0.83 0.83 0.83 0.84 0.84 0.84 0.84 0.84 0.85 0.85 0.85 0.86 0.87 0.87 0.87 0.88 0.89 0.98 0.98 0.98 0.98

0.87 0.81 0.86 0.81 0.87 0.90 0.83 0.88 0.84 0.90 0.93 0.87 0.91 0.93 0.96 0.99 0.95 0.99 1.04 1.12 0.98 1.00 1.04 1.10 1.20 0.88 0.86 0.86 0.88 0.81 0.87 0.86 0.85 0.90 0.91 0.90 0.88 0.89 0.92 0.94 0.95 0.91 0.94 0.98 1.03 0.99 0.99 1.02 1.06 1.12 0.82 0.81 0.82 0.83 0.85 0.82 0.82 0.83 0.85 0.87 0.84 0.83 0.84 0.87 0.89 0.86 0.86 0.87 0.90 0.93 0.98 0.97 1.00 1.02

0.98 0.91 0.94 0.87 0.90 0.97 0.90 0.91 0.86 0.88 0.96 0.90 0.90 0.89 0.88 0.97 0.93 0.92 0.93 0.95 0.99 0.98 0.98 0.99 1.02 1.06 1.04 1.02 1.02 0.92 0.99 1.00 0.96 0.98 0.97 0.97 0.96 0.94 0.94 0.92 0.97 0.93 0.92 0.93 0.94 0.99 0.99 0.98 0.98 1.00 1.11 1.10 1.10 1.10 1.09 1.05 1.05 1.05 1.05 1.04 0.98 0.98 0.98 0.98 0.98 0.93 0.93 0.92 0.93 0.93 0.98 0.98 0.98 0.98

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37 Table 8 (continued) Comparison

PFEA (kN)

PFEA/PNAS

PFEA/PAS/NZS

PFEA/PEC3

PFEA/PDSM

2234.0 211.5 208.5 206.6 204.3 203.5 255.9 253.3 252.1 249.4 240.0 325.3 322.7 318.0 313.9 307.8 499.5 482.1 479.5 473.5 466.8 2788.1 2755.4 2694.9 2638.7 2566.9 – – –

0.98 0.79 0.78 0.78 0.78 0.80 0.80 0.80 0.80 0.81 0.79 0.82 0.82 0.82 0.82 0.82 0.85 0.82 0.82 0.83 0.83 1.01 1.00 0.99 0.99 0.98 0.89 0.076 2.23

0.98 0.79 0.78 0.78 0.78 0.80 0.80 0.80 0.80 0.81 0.79 0.82 0.82 0.82 0.82 0.82 0.85 0.82 0.82 0.83 0.83 1.01 1.00 0.99 0.99 0.98 0.89 0.076 2.03

1.06 0.78 0.77 0.76 0.78 0.80 0.80 0.79 0.79 0.80 0.80 0.82 0.81 0.80 0.81 0.83 0.84 0.81 0.81 0.83 0.85 1.01 1.00 0.98 1.00 1.02 0.90 0.100 1.79

0.98 1.28 1.26 1.26 1.26 1.28 1.22 1.21 1.21 1.21 1.18 1.14 1.14 1.13 1.13 1.12 1.02 0.99 1.00 0.99 0.99 1.01 1.00 0.99 0.99 0.98 1.01 0.097 2.90

1.5

PFEA/Pne and PExp /Pne

W100T10L3500 W150T1.3L500 W150T1.3L1200 W150T1.3L2000 W150T1.3L2700 W150T1.3L3500 W150T1.5L500 W150T1.5L1200 W150T1.5L2000 W150T1.5L2700 W150T1.5L3500 W150T1.8L500 W150T1.8L1200 W150T1.8L2000 W150T1.8L2700 W150T1.8L3500 W150T2.5L500 W150T2.5L1200 W150T2.5L2000 W150T2.5L2700 W150T2.5L3500 W150T10L500 W150T10L1200 W150T10L2000 W150T10L2700 W150T10L3500 Mean, Pm COV, VP Reliability index, β

a) Flexural buckling FEA

Eqn (2) Tests FEA

1

0.5

0 0

1

2

3

4

λc

b) Local buckling 1.5

PFEA/Pne and PExp /Pne

Specimen

31

Eqn (3) Tests FEA

1

0.5

0 0

1

2

3

4

λl investigated in this study was considered as two flat web plates and two semi-circle flanges. The flat plates are supported by the two semi-circle flanges and assumed to be stiffened elements. Hence, the plate buckling coefficient k of the flat plates is taken as 4.0. The NAS [13,14], AS/NZS [12] and EC3 [15–17] specifications do not have design rules for effective width of curved plate, except for cylindrical tubular member. However, the local buckling resistance of a curved plate, such as semi-circle, is greater than a flat plat. Local buckling was not observed in the curved portions of the OHS for all the test specimens. In this study, the semi-circle flanges are considered as fully effective. The calculation procedure of NAS, AS/NZS and EC3 specifications are detailed in Zhu and Young [11]. It should be noted that the Automotive Steel Design Manual [21] published by the American Iron and Steel Institute, although it was

Fig. 7. Comparison of FEA and experimental data with design rules (PDSM) for columns: (a) flexural buckling and (b) local buckling.

not developed for structural members, includes design rules for local instability of sections with curved and straight elements. However, it has been demonstrated that the design strengths predicted using the ASDM [21] are generally unconservative for the coldformed steel OHS columns, as concluded by Zhu and Young [11]. Therefore, the ASDM Specification was not used in this study. 6.2. Direct strength method The direct strength method, which was developed for cold-formed steel members, is based on the same underlying empirical

a) Buckling on semi-circle flanges

Buckled shape

Original shape

b) Buckling on flat webs

Buckled shape

Original shape

Fig. 6. Oval hollow section buckling shapes generated from finite strip analysis.

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

Table 9 Comparison of test strengths with design strengths.

A360 A360# A600 A1200 A1200# A1800 A2400 A3000 B345 B600 B1200 B1800 B2400 B3000 C126 C300 C600 C900 C900# C1200 C1500 D90 D300 D600 D900 D900# D1200 D1500 Mean, Pm COV, VP Reliability index, β

Test

Comparison

PExp (kN)

PExp/PNAS

PExp/PAS/NZS

PExp/PEC3

PExp/PDSM

181.2 185.9 196.0 190.3 188.5 183.9 173.1 157.7 163.3 156.2 146.6 131.6 118.8 103.5 144.1 122.3 111.0 96.8 92.8 64.6 51.5 58.4 51.1 38.9 30.4 28.4 19.0 12.7 – – –

0.96 0.95 1.03 1.05 1.06 1.10 1.14 1.19 0.97 0.95 0.97 0.96 1.01 1.08 1.18 1.06 1.04 1.10 1.02 0.90 0.96 1.16 1.09 0.97 1.01 0.96 0.99 0.99 1.03 0.073 2.83

0.96 0.95 1.03 1.05 1.06 1.10 1.14 1.19 0.97 0.95 0.97 0.96 1.01 1.08 1.18 1.06 1.04 1.10 1.02 0.90 0.96 1.16 1.09 0.97 1.01 0.96 0.99 0.99 1.03 0.073 2.63

0.95 0.94 1.01 1.10 1.11 1.21 1.34 1.49 0.96 0.95 1.04 1.11 1.26 1.41 1.17 1.09 1.16 1.31 1.24 1.16 1.27 1.16 1.12 1.13 1.30 1.23 1.24 1.17 1.17 0.117 2.70

0.92 0.92 0.99 1.01 1.01 1.06 1.13 1.20 0.93 0.91 0.91 0.92 0.99 1.09 1.18 1.06 1.04 1.10 1.03 0.91 1.00 1.16 1.09 0.97 1.03 0.97 1.01 1.03 1.02 0.081 3.00

Column strength, Pu (kN)

Specimen

160 Tests 120

PDSM PEC3

80

PNAS 40

PAS/NZS

0 0

500

1000

1500

2000

Effective length, le (mm) Fig. 10. Fixed-ended column curves for Series C.

60

Column strength, Pu (kN)

32

PNAS

50

Tests

PDSM PAS/NZS

40 30

PEC3

20 10 0

0

500

1000

1500

2000

Effective length, le (mm) Fig. 11. Fixed-ended column curves for Series D.

Column strength, Pu (kN)

250

PDSM

Tests

200 150

PNAS

100

PAS/NZS

50

PEC3

0 0

500

1000

1500

2000

2500

3000

Effective length, le (mm) Fig. 8. Fixed-ended column curves for Series A.

assumption as the effective width method: ultimate strength is a function of elastic buckling and yielding of the material [22]. The direct strength method has been proposed by Schafer and Peköz [23] for laterally braced flexural members undergoing local and distortional buckling. Subsequently, the method has been developed for concentrically loaded pin-ended cold-formed steel columns undergoing local, distortional and overall buckling [24,25]. In this study, distortional buckling is not considered since the oval hollow section is a closed section. As summarized in the North American Specification (NAS) [13,14] for cold-formed steel structures, the column design rules of the direct strength method that considered the local and overall flexural buckling are shown in Eqs. (1)–(3). The values of 0.15 and 0.4 are the coefficient and exponent of the direct strength equation, respectively, that were calibrated against test data of concentrically loaded pin-ended cold-formed steel columns for certain cross-sections and geometric

400 Tests

Column strength, Pu (kN)

Column strength, Pu (kN)

250

PDSM

200 150 100

PNAS 50

PAS/NZS PEC3

0 0

500

1000

1500

2000

Effective length, le (mm) Fig. 9. Fixed-ended column curves for Series B.

2500

3000

FEA

PEC3 320 240 160

PNAS

PAS/NZS PDSM

80 0

0

500

1000

1500

Effective length, le (mm) Fig. 12. Fixed-ended column curves for Series W60T2.

2000

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

3000

FEA

PEC3

Column strength, Pu (kN)

Column strength, Pu (kN)

500 400 300 200

PNAS PAS/NZS

PDSM

100 0

0

500

1000

1500

PDSM

2000 1500

PNAS

PEC3

500 0

2000

0

P DSM ¼ minðP ne ; P nl Þ

P nl ¼

: 1−0:15

ð1Þ for λc ≤1:5

ð2Þ

for λc > 1:5

for λl ≤0:776  0:4  0:4 P crl P crl P ne for λl > 0:776 P ne P ne

ð3Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Py = fy A λc ¼ P y =P cre ; λl ¼ P ne =P crl . A Gross cross-section area;

fy

Pcrl E le ry

400

FEA

PDSM

600

Column strength, Pu (kN)

Column strength, Pu (kN)

2000

The nominal axial strengths (PDSM) are calculated for the two cases, as shown in Eqs. (2) and (3), respectively, where Pne refers to the nominal axial strength for flexural buckling, and Pnl refers to the nominal axial strength for local buckling as well as interaction of local and overall buckling. The nominal axial strength, PDSM, is the minimum of Pne and Pnl, as shown in Eq. (1). In calculating the axial strengths, the critical elastic local buckling load (Pcrl) of the cross-

500 400 300

PNAS PAS/NZS

200

PEC3 100

300

200

PNAS

500

1000

1500

PAS/NZS

100

0 0

FEA

PEC3

2000

PDSM

0

Effective length, le (mm)

500

1000

1500

2000

Effective length ,le (mm)

Fig. 14. Fixed-ended column curves for Series W60T3.

Fig. 17. Fixed-ended column curves for Series W75T1.9.

1000

500

FEA

PDSM

Column strength, Pu (kN)

Column strength, Pu (kN)

1500

Material yield strength which is the static 0.2% proof stress (σ0.2); π 2EA/(le/ry) 2, critical elastic buckling load in flexural buckling for OHS columns; Critical elastic local buckling load; Young's modulus; Column effective length; Radius of gyration of gross cross-section about the minor y-axis of buckling.

Pcre

700

800 600 400

PNAS

PAS/NZS PEC3

200 0

1000

Fig. 16. Fixed-ended column curves for Series W60T10.

where

0

500

Effective length, le (mm)

limits. It should be noted that the direct strength method does not cover OHS.

8 P ne <

PAS/NZS

1000

Fig. 13. Fixed-ended column curves for Series W60T2.4.

 8
λ2 > < 0:658 c P y   ¼ 0:877 > : Py λ2c

FEA

2500

Effective length, le (mm)

P ne

33

0

500

1000

1500

Effective length, le (mm) Fig. 15. Fixed-ended column curves for Series W60T4.

2000

PEC3

FEA

400 300 200

PNAS PAS/NZS PDSM

100 0

0

500

1000

1500

Effective length, le (mm) Fig. 18. Fixed-ended column curves for Series W75T2.2.

2000

34

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

400 FEA

PEC3

600

Column strength, Pu (kN)

Column strength, Pu (kN)

700

500 400 300

PNAS PAS/NZS

200

PDSM

100 0

PNAS

PAS/NZS

PEC3

FEA

320 240 160

PDSM

80 0

0

500

1000

1500

0

2000

500

1000

1500

2000

Effective length, le (mm)

Effective length, le (mm) Fig. 19. Fixed-ended column curves for Series W75T2.8.

Fig. 22. Fixed-ended column curves for Series W100T1.7.

section was obtained from a rational elastic finite strip buckling analysis [26]. Fig. 6 shows the section buckling shapes obtained from the analysis. Fig. 7 shows the comparison of FEA and experimental results against the direct strength curve plotted from Eqs. (2) and (3). The unfactored design strengths PDSM calculated using the direct strength method are compared with the numerical and test results, as shown in Tables 8 and 9, respectively.

design of cold-formed steel structural members. The ratio of dead (DL) to live (LL) loads is assumed as 0.2 (DL/LL = 0.2) in this analysis, as suggested by the NAS [13]. In general, a target reliability index of 2.5 is recommended as a lower limit [13]. If the reliability index is greater than 2.5 (β > 2.5), then the design is considered to be reliable. In calculating the reliability indices of the design rules, resistance factors of 0.85, 0.85 and 0.91 were used for the NAS, AS/NZS and EC3 specifications, as required by the relevant specifications, respectively. The resistance factor of 0.80 was used for the DSM, as specified by the NAS [13] for sections not being considered as pre-qualified sections, such as the OHS in this study. The load combinations of 1.2DL + 1.6LL, 1.25DL + 1.5LL and 1.35DL + 1.5LL are used in the analysis for the NAS, AS/NZS and EC3 specifications, respectively. The statistical parameters Mm, Fm, VM,

7. Reliability analysis The reliability of the design rules for cold-formed steel oval hollow section columns is evaluated using reliability analysis. Reliability analysis is detailed in the commentary on NAS Specification [14] for

500

FEA

PDSM

Column strength, Pu (kN)

Column strength, Pu (kN)

1000 800 600

PNAS PAS/NZS

400

PEC3 200 0

PNAS PAS/NZS PEC3 400 300 200

500

1000

1500

PDSM

100 0

0

2000

0

Fig. 20. Fixed-ended column curves for Series W75T3.8.

Column strength, Pu (kN)

Column strength, Pu (kN)

2500 2000

PDSM

1000

PEC3

500 0

0

500

1000

1500

2000

600

FEA

PAS/NZS

1000

Fig. 23. Fixed-ended column curves for Series W100T2.

3000

PNAS

500

Effective length, le (mm)

Effective length, le (mm)

1500

FEA

1500

Effective length, le (mm) Fig. 21. Fixed-ended column curves for Series W75T10.

2000

FEA

PDSM

500 400 300

PNAS PAS/NZS PEC3

200 100 0

0

500

1000

1500

Effective length, le (mm) Fig. 24. Fixed-ended column curves for Series W100T2.5.

2000

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

400

PDSM

FEA

Column strength, Pu (kN)

Column strength, Pu (kN)

900 750 600 450

PNAS

PAS/NZS

PEC3

300 150 0 0

500

1000

1500

PEC3

PAS/NZS

FEA

350 300 250 200 150 100

PDSM

50 0

2000

PNAS

35

0

500

Effective length, le (mm)

1000

1500

2000

Effective length, le (mm)

Fig. 25. Fixed-ended column curves for Series W100T3.3.

Fig. 28. Fixed-ended column curves for Series W150T1.5.

and VF are the mean values and coefficients of variation (COV) of material and fabrication factors. These values are obtained from Table F1 of the NAS Specification [13], where Mm = 1.10, Fm = 1.00, VM = 0.10, and VF = 0.05.

Tables 8 and 9. Reliability indices (β) of the design rules are also shown in the two tables. The experimental and FEA results are also compared with the column design curves obtained from the design rules, as shown in Figs. 8–31. The design strengths of specimens from the experimental program were calculated using the material properties for each series of specimens, as shown in Table 2, and the 0.2% proof stress (σ0.2) was used as the corresponding yield stress. The design strengths of specimens from the parametric study were calculated using the material properties of Series B, which have been also used in the finite element modeling. The fixed-ended column specimens were designed as concentrically loaded compression members, and the effective length (le) was taken as one-half of the column length (L), as recommended by Young and Rasmussen [27]. It is shown in Table 8 that the design strengths predicted by the NAS, AS/NZS and EC3 specifications are generally unconservative for the numerical results, with the mean load ratios PFEA/PNAS, PFEA/PAS/NZS

8. Comparison of numerical and experimental results with design predictions The nominal column strengths (unfactored design strengths) predicted by the NAS Specification (PNAS), AS/NZS Standard (PAS/NZS), European Code (PEC3) and the direct strength method (PDSM) are compared with the column strengths obtained from the parametric study (PFEA) and experimental investigation (PExp), as shown in Tables 8 and 9, respectively. The statistical parameters Pm and VP which are the mean value and coefficient of variation (COV) of FEA and Experimental-to-predicted strength ratios are shown in

500

FEA

Column strength, Pu (kN)

Column strength, Pu (kN)

3000 2500 2000

PNAS

1500

PDSM

PAS/NZS PEC3

1000 500 0 0

500

1000

1500

300 200

PDSM

100

0

Fig. 26. Fixed-ended column curves for Series W100T10.

FEA

300 250 200 150 100

PDSM

50 0 0

500

1000

1500

2000

700

PEC3

PAS/NZS

1000

Fig. 29. Fixed-ended column curves for Series W150T1.8.

Column strength, Pu (kN)

Column strength, Pu (kN)

PNAS

500

Effective length, le (mm)

Effective length, le (mm)

350

FEA

400

0

2000

PEC3

PAS/NZS

PNAS

1500

Effective length, le (mm) Fig. 27. Fixed-ended column curves for Series W150T1.3.

2000

FEA

600 500 400

PNAS PAS/NZS

PEC3 PDSM

300 200 100 0

0

500

1000

1500

Effective length, le (mm) Fig. 30. Fixed-ended column curves for Series W150T2.5.

2000

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37

Column strength, Pu (kN)

3500

FEA 2800 2100

PNAS

PAS/NZS

PDSM

PEC3

1400 700 0 0

500

1000

1500

2000

Effective length, le (mm) Fig. 31. Fixed-ended column curves for Series W150T10.

and PFEA/PEC3 of 0.89, 0.89 and 0.90, and the corresponding coefficients of variation (COV) of 0.076, 0.076 and 0.100, respectively. The DSM design strengths are generally conservative with the mean load ratio PFEA/PDSM of 1.01 and the corresponding COV of 0.097. The reliability indices of the design rules provided by the NAS, AS/NZS and EC3 specifications for the numerical results are less than the target value of 2.5, which are 2.23, 2.03 and 1.79, respectively. The reliability index of 2.90 was obtained for the DSM, which is greater than the target value, as shown in Table 8. It should be noted that the design strengths predicted by the NAS and AS/NZS specifications are identical. It is shown that the design strengths predicted by the NAS, AS/NZS and EC3 specifications are generally conservative for the experimental results, with the mean load ratios PFEA/PNAS, PFEA/PAS/NZS and PFEA/PEC3 of 1.03, 1.03 and 1.17, and the corresponding COV of 0.073, 0.073 and 0.117, respectively. This is due to the fact that the oval hollow sections are compact in the experimental investigation. The EC3 design strengths are more conservative compared to the NAS and AS/NZS predictions. It can be seen that the DSM design strengths are generally conservative with the load ratio PExp/PDSM of 1.02, and the corresponding COV of 0.081. The reliability indices of the design rules provided by the NAS, AS/NZS, EC3 and DSM specifications for the experimental results are greater than the target value having the values of 2.83, 2.63, 2.70 and 3.00, respectively. It should be noted that local buckling in the semi-circle flanges of the oval hollow sections was not observed for the columns in the experimental investigation, whereas this is not the case for the slender sections in the parametric study and local buckling was observed in the semi-circle flanges. Fig. 32 shows the average ratios of design strengths obtained from different specifications over numerical column strengths against W/t for each series of specimens as shown in Table 8. It can be seen that the current design rules provided by the NAS, AS/NZS and EC3 specifications are unconservative. It is also shown that the direct strength method is generally conservative. Therefore, it is demonstrated that the design rules of the current NAS, AS/NZS and EC3 specification for cold-formed steel structures can be used for the design of compact oval hollow sections by adopting the two assumptions that firstly the flat plates are supported by the two semi-circle flanges as stiffened elements with the plate buckling coefficient k equal to 4.0; and secondly, the semi-circle flanges are fully effective, as described in Zhu and Young [11]. It is also demonstrated that the direct strength method is reliable and can be used for the design of both compact and slender oval hollow section columns. 9. Conclusions This paper presents the design of cold-formed steel oval hollow section columns. A parametric study of oval hollow section columns using finite element analysis was performed. A non-linear finite element model was developed and used in the parametric study. The

Design strength / column strength

36

1.4 1.2 1 0.8 0.6 0.4

NAS and AS/NZS EC3 DSM

0.2 0 0

20

40

60

80

100

120

140

W/t Fig. 32. Average ratios of design strengths obtained from different specifications over numerical column strengths against W/t.

parametric study included 20 column series with different crosssection sizes. Each column series contained 5 specimens with the column lengths ranged from 500 to 3500 mm. The column strengths obtained from experimental and numerical investigations were compared with the design strengths calculated using the current North American, Australian/New Zealand and European specifications for cold-formed steel structures. The column strengths were also compared with the design strengths calculated using the direct strength method. Reliability analysis was performed to evaluate the reliability of the design rules. It is demonstrated that the column design rules in the current North American, Australian/New Zealand and European specifications for cold-formed steel structures can be used for compact oval hollow sections. It is also demonstrated that the direct strength method is conservative and reliable for the design of both compact and slender oval hollow section columns. Acknowledgments The research work described in this paper was supported by the Chinese National Natural Science Foundation (Project Nos. 50808126 and 51078237) and a grant from The University of Hong Kong under the seed funding program for basic research. References [1] Zhu JH, Young B. Design of aluminum alloy flexural members using direct strength method. Journal of Structural Engineering – ASCE 2009;135(5):558–66. [2] Gardner L, Ministro A. Structural steel oval hollow sections. Struct Eng 2005;83 (21):32–6. [3] Bortolotti E, Jaspart JP, Pietrapertosa C, Nicaud G, Petitjean PD, Grimault JP. Testing and modelling of welded joints between elliptical hollow sections. Proceedings of the 10th International Symposium on Tubular Structures. Madrid; 2003. p. 259–66. [4] Gardner L, Chan TM. Cross-section classification of elliptical hollow sections. Steel Compos Struct 2007;7(3):185–200. [5] Chan TM, Gardner L. Bending strength of hot-rolled elliptical hollow sections. J Constr Steel Res 2008;64(9):971–86. [6] Chan TM, Gardner L. Flexural Buckling of Elliptical Hollow Section Columns. J Struct Eng ASCE 2009;135(5):546–57. [7] Theofanous M, Chan TM, Gardner L. Structural response of stainless steel oval hollow section compression members. Eng Struct 2009;31(4):922–34. [8] Silvestre N. Buckling behaviour of elliptical cylindrical shells and tubes under compression. IntJ Solids Struct 2008;45(16):4427–47. [9] Zhao XL, Packer JA. Tests and design of concrete-filled elliptical hollow section stub columns. Thin Walled Struct 2009;47(6):617–28. [10] Yang H, Lam D, Gardner L. Testing and analysis of concrete-filled elliptical hollow sections. Eng Struct 2008;30(12):3771–81. [11] Zhu JH, Young B. Cold-formed-steel oval hollow sections under axial compression. J Struct Eng ASCE 2011;137(7):719–27. [12] AS/NZS. Cold-formed steel structures.” AS/NZS No. 4600. Sydney: Standards Australia; 2005. [13] NAS. North American Specification for the Design of Cold-Formed Steel Structural Members. Washington, DC: American Iron and Steel Institute; 2007. [14] NAS. Commentary on North American Specification for the Design of Cold-Formed Steel Structural Members. Washington, DC: American Iron and Steel Institute; 2007.

J.-H. Zhu, B. Young / Journal of Constructional Steel Research 71 (2012) 26–37 [15] EC3. Design of steel structures — Part 1–1: General rules and rules for buildings. EN 1993-1-1. Brussels: European Committee for Standardization; 2005. [16] EC3. Design of steel structures — Part 1–3: General rules — Supplementary rules for cold-formed members and sheeting. EN 1993-1-3. Brussels: European Committee for Standardization; 2006. [17] EC3. Design of steel structures — Part 1–5: Plated structural elements. EN 1993-15. Brussels: European Committee for Standardization; 2006. [18] ABAQUS. analysis user's manual, Version 6.9. Dassault Systemes; 2009. [19] Yan J, Young B. Numerical investigation of channel columns with complex stiffeners-part I: tests verification. Thin Walled Struct 2004;42(6):883–93. [20] Chan TM, Gardner L. Compressive Resistance of Hot-Rolled Elliptical Hollow Sections. Eng Struct 2008;30(2):522–32. [21] AISI. Automotive Steel Design Manual - Revision 6.1. Southfield, MI: Automotive Applications Committee, American Iron and Steel Institute; 2002.

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[22] Schafer BW. Progress on the direct strength method. Proceeding of 16th International specialty conference on cold-formed steel structures. Orlando, Florida; 2002. p. 647–62. [23] Schafer BW, Peköz T. Direct strength prediction of cold-formed steel members using numerical elastic buckling solutions. Proceedings of the 14th Int. specialty conference on cold-formed steel structures. Rolla, Mo: University of MissouriRolla; 1998. p. 69–76. [24] Schafer BW. Distortional buckling of cold-formed steel columns. Washington, DC: August Final Report to the American Iron and Steel Institute; 2000. [25] Schafer BW. Local, distortional, and Euler buckling of thin-walled columns. J Struct Eng ASCE 2002;128(3):289–99. [26] Papangelis JP, Hancock GJ. Computer analysis of thin walled structural members. Comput Struct 1995;56(1):157–76. [27] Young B, Rasmussen KJR. Design of lipped channel columns.”. J Struct Eng ASCE 1998;124(2):140–8.