Numerical investigation and design of aluminum alloy circular hollow section columns

ARTICLE IN PRESS Thin-Walled Structures 46 (2008) 1437– 1449

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Numerical investigation and design of aluminum alloy circular hollow section columns Ji-Hua Zhu a, Ben Young b, a b

School of Civil Engineering, Shenzhen University, Shenzhen 518060, PR China Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

a r t i c l e in fo

abstract

Article history: Received 19 October 2007 Accepted 10 March 2008 Available online 5 May 2008

This paper presents a numerical investigation of aluminum alloy circular hollow section non-welded and welded columns using finite element analysis. A non-linear finite element model was developed and verified against fixed-ended column tests. The column specimens were extruded from heat-treated aluminum alloys of 6063-T5 and 6061-T6, and the ends of the columns were transversely welded to aluminum end plates for the welded columns. The non-welded columns without welding of end plates were also investigated. The welded columns were modeled by dividing the column into different portions along the column length, so that the heat-affected zone softening at both ends of the welded columns was included in the simulation. The initial local geometric imperfections of the columns were measured in this study. Geometric and material non-linearities were incorporated in the finite element model. The verified finite element model was used for a parametric study of fixed-ended aluminum alloy circular hollow section columns. A comparison of the column strengths predicted by the finite element analysis and the design strengths calculated using the current American, Australian/New Zealand and European specifications for aluminum structures was presented. The column strengths were also compared with the design strengths predicted by the direct strength method, which was developed for cold-formed carbon steel members. Design rules were proposed for aluminum alloy circular hollow section columns with transverse welds at the ends of the columns. Reliability analysis was performed to evaluate the reliability of the design rules. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Aluminum alloys Buckling Circular hollow section Column Design Finite element analysis Heat-affected zone Parametric study Transverse welds

1. Introduction Aluminum members are being used increasingly in structural applications. The current American Aluminum Design Manual [1], Australian/New Zealand Standard [2] and European Code [3] for aluminum structures provide design rules for compression members. Schafer and Peko¨z [4] developed a new design method called the direct strength method (DSM) for cold-formed steel structures. The test data used in the development of column design for DSM were based on concentrically loaded pin-ended cold-formed steel column for certain cross sections and geometric limits [5,6]. The DSM has been adopted by the North American Specification [7,8] for cold-formed steel structures. Zhu and Young [9,10] showed that the DSM with some modification can be used in the design of aluminum alloy square hollow section (SHS) and rectangular hollow section (RHS) columns. For the purpose of obtaining accordant design rules of different cross-sections, the DSM was used in this study for the design of aluminum alloy circular hollow section (CHS) columns.

 Corresponding author. Tel.: +852 2859 2674; fax: +852 2559 5337.

E-mail address: [email protected] (B. Young). 0263-8231/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2008.03.006

One disadvantage in using aluminum as a structural material is that heat-treated aluminum alloys could suffer loss of strength in a localized region when welding is involved, and this is known as heat-affected zone (HAZ) softening. Previous research [11,12] indicated that welds have significant effect on column strength. The test program presented by Zhu and Young [13] showed that transverse welds at the ends of the CHS columns reduce the column strength for nearly 46%. In addition, it was also shown that the design rules in the current American, Australian/New Zealand and European specifications are generally quite conservative for aluminum alloy welded columns of circular hollow sections [13]. Hence, it is necessary to obtain accurate design rules for aluminum alloy columns containing transverse welds. Finite element analysis (FEA) has been widely used in structural design. Compared with physical experiments, FEA is relatively inexpensive and time efficient, especially when a parametric study of cross-section geometry is involved. In addition, FEA is more convenient for investigation involving geometric imperfections of structural members, whereas this could be difficult to investigate through physical tests. Although FEA is a useful and powerful tool for structural analysis and design, it is important to obtain an accurate and reliable finite element model (FEM) prior to a parametric study of FEA to be

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Nomenclature

PFEA15

A COV D DL E e FEA FEM Fm fy kc L LL le Mm P PAA

PFEA20

PAS/NZS Pcre Pcrl PDSM PDSM-W PEC9 PExp PFEA

gross cross-section area coefficient of variation overall diameter of CHS dead load Young’s modulus axial shortening finite element analysis finite element model mean value of fabrication factor material yield strength coefficient in the AS/NZS Standard length of specimen live load column effective length mean value of material factor axial load unfactored design strength for American Aluminum Design Manual unfactored design strength for Australian/New Zealand Standard critical elastic buckling load in flexural buckling, p2EA/ (le/r)2 critical elastic local column buckling load column strength calculated using the direct strength method welded column strength calculated using the proposed design rules unfactored design strength for Eurocode 9 experimental ultimate load of column ultimate load predicted by FEA

carried out. A non-linear FEM for aluminum columns of SHS and RHS with and without transverse welds has been developed by Zhu and Young [9]. In this study, a non-linear FEM for aluminum columns of CHS was developed and verified against experimental results. The purpose of this paper is firstly to investigate the behavior and design of aluminum CHS columns using non-linear FEA. The verified FEM is used for a parametric study of cross-section geometries. Secondly, the current DSM is used for the design of aluminum non-welded and welded columns of CHS. Thirdly, design rules for aluminum welded columns of CHS are proposed based on the current DSM. The column strengths predicted by the FEA were compared with the design strengths calculated using the American Aluminum Design Manual (AA), Australian/New Zealand Standard (AS/NZS) and European Code (EC9) for aluminum structures, as well as the DSM and proposed design rules. Lastly, reliability analysis was performed to assess the reliability of these design rules.

Pm Pne Pnl Pu Py r t VF VM VP z a1 a2 b ef lc ll rc rhaz f s0.2 su

ultimate load predicted by FEA using 15 mm heataffected zone extension for welded column ultimate load predicted by FEA using 20 mm heataffected zone extension for welded column mean value of tested-to-predicted load ratio nominal axial strength for flexural buckling nominal axial strength for local buckling column strength yield strength of the section (fy A) radius of gyration of gross cross-section about the minor y-axis of buckling thickness of section coefficient of variation of fabrication factor coefficient of variation of material factor coefficient of variation of tested-to-predicted load ratio longitudinal coordinates factor in the proposed design equation due to welding (1.3(D/t)0.19) factor in the proposed design equation due to welding (0.6(x/L)0.12) reliability index elongation (tensile strain) at fracture non-dimensional slenderness for flexural buckling pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð P y =Pcre Þ non-dimensional slenderness for ffi interaction of local pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and flexural buckling ð P ne =Pcrl Þ local buckling coefficient specified in the Eurocode 9 heat-affected zone (HAZ) softening factor specified in the Eurocode 9 resistance factor static 0.2% proof stress static ultimate tensile strength

ended CHS columns without the welding of end plates. In this paper, the term ‘‘welded column’’ refers to a specimen with transverse welds at the ends of the column, whereas the term ‘‘non-welded column’’ refers to a specimen without transverse welds. The testing conditions of the non-welded and welded columns are identical, other than the absence of welding in the non-welded columns. The experimental program included four test series with different cross-section geometry and type of aluminum alloy, as shown in Table 1 using the symbols illustrated in Fig. 1. The measured cross-section dimensions of each specimen are detailed in Zhu and Young [13]. The specimens were tested between fixed ends at various column lengths ranged from 300–3000 mm. The test rig and operation are also detailed in Zhu and Young [13]. The experimental ultimate loads (PExp) and failure modes observed at ultimate loads obtained from the non-welded and welded column tests are shown in Tables 2–6. The test specimens were labeled such that the type of aluminum alloy, test series, welding condition and specimen length could be easily identified, as shown in Tables 2–6. For example, the label

2. Summary of test program 2.1. Column tests and material properties Experimental results of aluminum alloy circular hollow sections compressed between fixed ends have been reported by Zhu and Young [13]. The test specimens were fabricated by extrusion using 6063-T5 and 6061-T6 heat-treated aluminum alloys. The test program included 21 fixed-ended CHS columns with both ends welded to aluminum end plates, and eight fixed-

Table 1 Nominal specimen dimension of test series Test series

Type of material

Dimension, D  t (mm)

N-C1 N-C2 H-C1 H-C2

6063-T5 6063-T5 6061-T6 6061-T6

50  1.6 50  3.0 50  1.6 50  3.0

Note: 1 in. ¼ 25.4 mm.

ARTICLE IN PRESS J.-H. Zhu, B. Young / Thin-Walled Structures 46 (2008) 1437–1449

‘‘H-C2-W-L1000’’ defines the following specimen: the first letter indicates the type of material of the specimen, where ‘‘H’’ refers to the high strength aluminum alloy 6061-T6; and ‘‘N’’ refers to the normal strength aluminum alloy 6063-T5. The second part of the label indicates the cross-section shape of the specimen, where ‘‘C2’’ refers to a circular hollow section with nominal cross-section dimension of 50  3.0 mm. The cross-section dimensions for other

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sections are shown in Table 1. The third part of the label indicates the welding condition. If a specimen has transverse welds to the aluminum end plates, then ‘‘W’’ indicates a welded column specimen; if a specimen was tested without welding to the end plates, then the letter ‘‘NW’’ indicates a non-welded column specimen. The last part of the label ‘‘L1000’’ indicates the length of the specimen, where the letter ‘‘L’’ refers to the column length and the following digits are the nominal length of the specimen in millimeters (1000 mm). The non-welded and welded material properties for each series of specimens were determined by longitudinal tensile coupon tests as detailed in Zhu and Young [13]. The measured material properties obtained from the coupon tests are summarized in Table 7. 2.2. Measured local and overall geometric imperfections

t

D Fig. 1. Definition of symbols.

Table 2 Comparison of test and FEA results for non-welded columns Specimen

Experimental

FEA

Comparison

PExp (kN) Failure mode PFEA (kN) Failure mode PExp/PFEA N-C1-NW-L300 48.5 N-C1-NW-L1000 45.9 N-C2-NW-L300 102.4 N-C2-NW-L1000 86.1 H-C1-NW-L300 75.9 H-C1-NW-L1000 71.7 H-C2-NW-L300 129.6 H-C2-NW-L1000 119.6

Y F Y Y Y Y Y F

46.3 43.6 87.1 76.0 69.6 65.6 124.3 117.1

Y F Y F Y F Y F

1.05 1.05 1.18 1.13 1.09 1.09 1.04 1.02

Mean COV

1.08 0.048

Note: 1 kip ¼ 4.45 kN; F, flexural buckling; Y, yielding.

In this study, the initial local geometric imperfections were measured on four CHS specimens in length of 300 mm. The specimens were cut from those specimens belonged to the same batch of specimens as the column tests. Hence, the measured local geometric imperfections are considered nearly the same order of imperfections as the column specimens. A Mitutoyo co-ordinate Measuring Machine (CMM) with an accuracy of 0.001 mm was used to measure the initial local geometric imperfections. The measurements were taken at the longitudinal quarter lines A–A, B–B, C–C and D–D of each specimen as shown in Fig. 2. Readings were taken at regular intervals of 2 mm along the specimen length. The measured local imperfection profiles for Series N-C1 is shown in Fig. 2. The vertical axis is plotted against the normalized location along specimen length and the horizontal axis is plotted against the measured local geometric imperfections. The maximum measured local geometric imperfections were 6.3%, 2.0%, 10.4% and 16.0% of the section thickness for Series N-C1, N-C2, H-C1 and H-C2, respectively. Initial overall geometric imperfections were measured on all specimens prior to testing, except for the short specimens of 300 mm in length, as detailed in Zhu and Young [13]. The maximum measured overall geometric imperfections at midlength were 1/1732, 1/1432, 1/562 and 1/854 of the specimen length for Series N-C1, N-C2, H-C1 and H-C2, respectively.

3. Finite element modeling An accurate and reliable non-linear FEM for aluminum nonwelded and welded columns of square and rectangular hollow sections has been developed using the finite element program ABAQUS [14], as presented by Zhu and Young [9]. In this study, the FEM was used for the simulation of aluminum alloy circular hollow section columns tested by Zhu and Young [13]. The development of the FEM is detailed in Zhu and Young [9]. In the

Table 3 Comparison of test and FEA results for welded columns of Series N-C1 Specimen

N-C1-W-L300 N-C1-W-L1000 N-C1-W-L1650 N-C1-W-L2350 N-C1-W-L3000

Experimental

FEA

Comparison

PExp (kN)

Failure mode

PFEA20 (kN)

PFEA15 (kN)

Failure mode

PExp/PFEA20

PExp/PFEA15

35.9 30.3 27.9 21.4 17.3

HAZ HAZ F F F

29.0 28.0 25.1 19.6 14.8

32.4 31.3 27.3 21.0 15.4

HAZ HAZ F F F

1.24 1.08 1.11 1.09 1.17

1.11 0.97 1.02 1.02 1.13

Mean COV

1.14 0.057

1.05 0.063

Note: 1 kip ¼ 4.45 kN; F, flexural buckling; HAZ, failure in the HAZ.

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Table 4 Comparison of test and FEA results for welded columns of Series N-C2 Specimen

Experimental

N-C2-W-L300 N-C2-W-L1000 N-C2-W-L1650 N-C2-W-L2350 N-C2-W-L3000

FEA

Comparison

PExp (kN)

Failure mode

PFEA20 (kN)

PFEA15 (kN)

Failure mode

PExp/PFEA20

PExp/PFEA15

69.4 65.1 48.6 35.3 28.3

HAZ HAZ F F F

71.1 61.9 45.5 34.8 27.5

77.2 64.2 46.9 37.1 28.4

HAZ HAZ F F F

0.98 1.05 1.07 1.02 1.03

0.90 1.01 1.04 0.95 1.00

Mean COV

1.03 0.034

0.98 0.056

Note: 1 kip ¼ 4.45 kN; F, flexural buckling; HAZ, failure in the HAZ.

Table 5 Comparison of test and FEA results for welded columns of Series H-C1 Specimen

Experimental

H-C1-W-L300 H-C1-W-L1000 H-C1-W-L1650 H-C1-W-L2350 H-C1-W-L3000

FEA

Comparison

PExp (kN)

Failure mode

PFEA20 (kN)

PFEA15 (kN)

Failure mode

PExp/PFEA20

PExp/PFEA15

47.2 39.0 36.8 25.3 17.0

HAZ HAZ F F F

44.1 34.0 29.3 21.8 15.7

45.2 38.6 32.3 23.3 16.3

HAZ HAZ F F F

1.07 1.15 1.25 1.16 1.08

1.04 1.01 1.14 1.08 1.04

Mean COV

1.14 0.064

1.06 0.047

Note: 1 kip ¼ 4.45 kN; F, flexural buckling; HAZ, failure in the HAZ.

Table 6 Comparison of test and FEA results for welded columns of Series H-C2 Specimen

Experimental

H-C2-W-L300 H-C2-W-L1000 H-C2-W-L1650 H-C2-W-L2350 H-C2-W-L3000

FEA

Comparison

PExp (kN)

Failure mode

PFEA20 (kN)

PFEA15 (kN)

Failure mode

PExp/PFEA20

PExp/PFEA15

88.0 84.5 66.9 44.6 33.0

HAZ HAZ F F F

79.6 80.1 59.4 44.6 32.7

90.7 89.0 61.6 47.2 33.2

HAZ HAZ F F F

1.11 1.05 1.13 1.00 1.01

0.97 0.95 1.09 0.95 0.99

Mean COV

1.06 0.053

0.99 0.058

Note: 1 kip ¼ 4.45 kN; F, flexural buckling; HAZ, failure in the HAZ.

Table 7 Measured non-welded and welded material properties of tensile coupons

Outside

1.0

E (GPa)

s0.2 (MPa)

su (MPa)

ef (%)

N-C1-W N-C1-NW N-C2-W N-C2-NW H-C1-W H-C1-NW H-C2-W H-C2-NW

73.0 66.7 71.6 67.1 72.6 67.1 71.7 70.2

71.3 194.6 75.3 185.9 92.5 286.7 94.3 278.9

120.9 214.4 109.7 207.7 148.3 310.1 161.2 284.3

9.9 10.0 6.9 10.4 10.7 10.7 10.9 11.7

Note: 1 ksi ¼ 6.89 MPa; NW, non-welded tensile coupon; W, welded tensile coupon.

FEM, the measured cross-section dimensions, material properties and initial geometric imperfections of the test specimens were modeled. The fixed-ended boundary condition was modeled by

Line A-A Line B-B Line C-C Line D-D

0.8 Location, z/L

Specimen

Inside

0.6 D C

A 0.4

B

0.2 C

A 0.0 -0.2

B -0.1

0.0 Imperfection (mm)

0.1

0.2

Fig. 2. Measured initial local geometric imperfection profiles for Series N-C1 specimen of 300 mm in length.

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140 Test Axial load, P (kN)

120 100 80

FEA

60 40 20 0 0

1

2 3 4 Axial shortening, e (mm)

6

5

Fig. 3. Comparison of experimental and FEA axial load-shortening curves for Specimen H-C2-NW-L1000.

25 20 Axial load, P (kN)

restraining all the degrees of freedom of the nodes at both ends of the column, 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 material non-linearity was included in the FEM by specifying the true values of stresses and strains. The plasticity of the material was simulated by a mathematical model, known as the incremental plasticity model, in which the true stresses and true plastic strains were calculated in accordance with ABAQUS [14]. The geometric imperfections were included in the FEM by using the Eigenvalue analyses. The displacement control loading method was used in the FEA that was identical to the loading method used in the column tests. The S4R general-purpose shell elements were used in the FEM. The size of the finite element mesh of 5  5 mm (length by width) was used in the modeling of the non-welded and welded columns of CHS. The welded columns were modeled by dividing the columns into different portions along the column length. Therefore, the HAZ softening at both ends of the columns were simulated. The welded columns were separated into three parts, the HAZ regions at both ends of the columns, and the main body of the columns that are not affected by welding. In this study, different length of HAZ extension at both ends of the welded columns were considered, that are equal to 20 and 15 mm. The material properties obtained from the nonwelded and welded tensile coupon tests were used for the main body and the HAZ regions of the welded columns, respectively.

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15 FEA20

10

FEA15 Test

5 4. Test verification

0 It is necessary to verify the FEM. The developed FEM was verified against the experimental results. For the non-welded columns, the ultimate loads and failure modes predicted by the FEA are compared with the experimental results as shown in Table 2. It is shown that the ultimate loads (PFEA) obtained from the FEA are generally in good agreement with the experimental ultimate loads (PExp). The ultimate loads predicted by the FEA are slightly lower than the experimental ultimate loads. The mean value of the experimental-to-FEA ultimate load ratio is 1.08 with the corresponding coefficient of variation (COV) of 0.048 for the nonwelded columns, as shown in Table 2. For the welded columns, both the ultimate loads predicted by the FEA using the HAZ extension of 20 mm (PFEA20) and 15 mm (PFEA15) are compared with the experimental results as shown in Tables 3–6 for Series N-C1, N-C2, H-C1 and H-C2, respectively. It is shown that the PFEA15 are in better agreement with the experimental ultimate loads compared with the PFEA20. The mean values of the experimental-to-FEA ultimate load ratio (PExp/PFEA20) are 1.14, 1.03, 1.14, and 1.06 with the corresponding COV of 0.057, 0.034, 0.064 and 0.053 for Series N-C1, N-C2, H-C1 and H-C2, respectively. The mean values of the load ratio PExp/PFEA15 are 1.05, 0.98, 1.06 and 0.99 with the corresponding COV of 0.063, 0.056, 0.047 and 0.058 for Series N-C1, N-C2, H-C1 and H-C2, respectively. The failure modes at ultimate load obtained from the tests and FEA for each specimen are also shown in Tables 2–6. The observed failure modes included yielding (Y), flexural buckling (F), and failure in the HAZ. The failure modes predicted by the FEA are identical to those observed in the tests, except for the specimens N-C2-NW-L1000 and H-C1-NW-L1000. Fig. 3 shows the comparison of the load-shortening curves obtained from the test and predicted by the FEA for the non-welded specimen H-C2-NW-L1000. It is shown that the FEA curve follows the experimental curve closely, except that the loads predicted by the FEA are slightly lower than the experimental loads in the postultimate range. Fig. 4 also shows the load-shortening curves for

0

1

2

3

4

5

6

Axial shortening, e (mm) Fig. 4. Comparison of experimental and FEA axial load-shortening curves for Specimen N-C1-W-L2350.

the welded specimen N-C1-W-L2350. The load-shortening curves predicted by the FEA using the HAZ extension of 20 and 15 mm are shown in Fig. 4. In addition, Fig. 5(a) shows the photograph of specimen N-C1-W-L2350 after the ultimate load has reached. The specimen failed in flexural buckling. Fig. 5(b) shows the deformed shape of the specimen predicted by the FEA after the ultimate load. The resemblance of Fig. 5(a) and (b) demonstrates the reliability of the FEA predictions.

5. Parametric study The FEM closely predicted the experimental ultimate loads and failure modes of the tested aluminum circular hollow section columns, as shown in Tables 2–6. Hence, the model was used for an extensive parametric study. The parametric study included 80 specimens that consisted of 16 series, as shown in Table 8. Each series contained five specimens with column lengths of 500, 1200, 2000, 2700 and 3500 mm. The specimens were labeled such that the type of aluminum alloy, section thickness, welding condition and column length could be identified, as shown in Tables 9–12. For example, the label ‘‘T5-t0.4-NW-L500’’ defines the following specimen:

 The first letter indicates the type of material of the specimen, 

where ‘‘T5’’ refers to the aluminum alloy 6063-T5, and ‘‘T6’’ refers to the aluminum alloy 6061-T6. The second part of the label ‘‘t0.4’’ indicates the section thickness of the specimen, where the letter ‘‘t’’ refers to the section thickness and the following digits are the thickness of the CHS in millimeters (0.4 mm). Table 8 shows the

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cross-section dimensions of each series using the nomenclature defined in Fig. 1. The following part of the label ‘‘NW’’ indicates the welding condition of the specimen, where the letter ‘‘NW’’ refers to the non-welded column, and the letter ‘‘W’’ refers to the welded column. The last 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 millimeters (500 mm).

The material properties of the specimens of 6063-T5 alloy investigated in the parametric study are identical to the material properties of Series N-C1 in the experimental program for the non-welded and welded material, whereas the material properties of the specimens of 6061-T6 alloy are identical to the material properties of Series H-C1 in the experimental program for the non-welded and welded material, as detailed in Zhu and Young [13]. The local imperfection magnitude was 10% of the section thickness, and the overall imperfection magnitude was 1/2000 of the column length used in the parametric study. The size of the finite element mesh was kept at 5  5 mm (length by width) for the non-welded and welded columns. The welded columns were modeled with 15 mm HAZ extension at both ends of the columns. The column strengths (PFEA) obtained from the parametric study are shown in Tables 9–12.

6. Design approaches 6.1. Current design rules for aluminum structures

Fig. 5. Comparison of experimental and FEA deformed shapes for Specimen N-C1W-L2350: (a) experimental and (b) FEA.

The American Aluminum Design Manual (AA) [1], Australian/ New Zealand Standard (AS/NZS) [2] and European Code (EC9) [3] for aluminum structures provide design rules for aluminum columns with and without transverse welds. The design rules in the AA Specification for calculating the design strengths of nonwelded aluminum columns are based on the Euler column strength. The inelastic column curve, based on the tangent modulus, is well approximated by a straight line using buckling constants [12]. The buckling constants were obtained from Tables 3.3-3 and 3.3-4 of Part I-B of the AA Specification. Local buckling strength can be calculated using an empirical

Table 8 Cross-section dimensions of the series for parametric study Series

Type of material

Diameter, D (mm)

Thickness, t (mm)

Area, A (mm2)

T5-t0.4-NW T5-t0.6-NW T5-t1.0-NW T5-t2.0-NW T6-t0.4-NW T6-t0.6-NW T6-t1.0-NW T6-t2.0-NW T5-t0.4-W T5-t0.6-W T5-t1.0-W T5-t2.0-W T6-t0.4-W T6-t0.6-W T6-t1.0-W T6-t2.0-W

6063-T5 6063-T5 6063-T5 6063-T5 6061-T6 6061-T6 6061-T6 6061-T6 6063-T5 6063-T5 6063-T5 6063-T5 6061-T6 6061-T6 6061-T6 6061-T6

75.4 75.6 76.0 77.0 75.4 75.6 76.0 77.0 75.4 75.6 76.0 77.0 75.4 75.6 76.0 77.0

0.4 0.6 1.0 2.0 0.4 0.6 1.0 2.0 0.4 0.6 1.0 2.0 0.4 0.6 1.0 2.0

94.2 141.4 235.6 471.2 94.2 141.4 235.6 471.2 94.2 141.4 235.6 471.2 94.2 141.4 235.6 471.2

Note: 1 in. ¼ 25.4 mm; NW, non-welded column series; W, welded column series.

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Table 9 Comparison of FEA and design strengths for non-welded columns of aluminum alloy 6063-T5

Table 10 Comparison of FEA and design strengths for non-welded columns of aluminum alloy 6061-T6

Specimen

Specimen

FEA

Comparison

PFEA (kN)

PFEA/PAA

PFEA/PAS/NZS

PFEA/PEC9

PFEA/PDSM

T5-t0.4-NW-L500 T5-t0.4-NW-L1200 T5-t0.4-NW-L2000 T5-t0.4-NW-L2700 T5-t0.4-NW-L3500

16.5 15.9 15.3 13.9 11.5

1.10 1.05 1.02 0.96 0.91

1.10 1.05 1.02 0.96 0.91

1.27 1.27 1.31 1.31 1.29

0.98 0.97 1.01 1.05 1.08

T6-t0.4-NW-L500 T6-t0.4-NW-L1200 T6-t0.4-NW-L2000 T6-t0.4-NW-L2700 T6-t0.4-NW-L3500

Mean, Pm COV, VP Reliability index, b

– – –

1.01 0.073 2.72

1.01 0.073 2.52

1.29 0.014 3.67

1.02 0.045 2.96

Mean, Pm COV, VP Reliability index, b

T5-t0.6-NW-L500 T5-t0.6-NW-L1200 T5-t0.6-NW-L2000 T5-t0.6-NW-L2700 T5-t0.6-NW-L3500

26.3 25.4 24.2 21.5 17.4

1.07 1.03 1.01 0.99 0.91

1.07 1.03 1.01 0.99 0.91

1.16 1.18 1.21 1.20 1.20

0.97 0.99 1.05 1.08 1.08

T6-t0.6-NW-L500 T6-t0.6-NW-L1200 T6-t0.6-NW-L2000 T6-t0.6-NW-L2700 T6-t0.6-NW-L3500

Mean, Pm COV, VP Reliability index, b

– – –

1.00 0.059 2.81

1.00 0.059 2.60

1.19 0.016 3.31

1.03 0.052 2.98

Mean, Pm COV, VP Reliability index, b

T5-t1.0-NW-L500 T5-t1.0-NW-L1200 T5-t1.0-NW-L2000 T5-t1.0-NW-L2700 T5-t1.0-NW-L3500

45.4 44.1 41.3 36.3 29.0

1.02 0.99 1.03 1.00 0.91

1.11 1.08 1.03 1.00 0.91

1.02 1.05 1.08 1.09 1.13

1.00 1.02 1.07 1.09 1.09

T6-t1.0-NW-L500 T6-t1.0-NW-L1200 T6-t1.0-NW-L2000 T6-t1.0-NW-L2700 T6-t1.0-NW-L3500

Mean, Pm COV, VP Reliability index, b

– – –

0.99 0.047 2.84

1.03 0.074 2.60

1.07 0.036 2.78

1.06 0.038 3.16

Mean, Pm COV, VP Reliability index, b

T5-t2.0-NW-L500 T5-t2.0-NW-L1200 T5-t2.0-NW-L2000 T5-t2.0-NW-L2700 T5-t2.0-NW-L3500

92.1 89.1 83.2 73.1 58.1

1.00 1.00 1.04 1.01 0.91

1.13 1.09 1.04 1.01 0.91

1.02 1.04 1.06 1.08 1.12

1.02 1.03 1.08 1.10 1.09

Mean, Pm COV, VP Reliability index, b

– – –

0.99 0.048 2.85

1.03 0.079 2.59

1.06 0.037 2.73

1.06 0.034 3.21

formula which was first developed by Clark and Rolf [15]. The design rules in the AS/NZS Standard for calculating the design strengths of non-welded aluminum columns are generally identical to those in the AA Specification, except that the AS/NZS Standard reduces the yield load of the column using a parameter kc which is not included in the AA Specification. The EC9 Code adopts the Perry curve for column design, and values of the imperfection factors are listed in Table 5.6 of the Code. The effects of local buckling on column strength are considered by replacing the true section with an effective section. The effective cross-section is obtained by employing a local buckling coefficient rc to reduce the thickness of the element in the section. The strength of aluminum column with transverse welds (welded column) depends on the location and number of welds [1]. For CHS columns with transverse welds at the ends only, the design equations given by the AA and AS/NZS specifications are identical to the design equations of non-welded columns. However, the design strength of CHS welded columns is calculated using the welded mechanical properties and the buckling constants obtained from Table 3.3-3 of Part I-B of the AA Specification regardless of temper before welding. The EC9 Code uses a factor rhaz to consider the weakening effects of welding on column strength, and rhaz is equal to 0.60 and 0.50 for the 6000 Series alloys of T5 and T6 conditions, respectively, as shown in Table 5.2 of the Code.

FEA

Comparison

PFEA (kN)

PFEA/PAA

PFEA/PAS/NZS

PFEA/PEC9

PFEA/PDSM

1.11 1.07 1.03 0.94 0.91

1.11 1.07 1.03 0.94 0.91

1.42 1.43 1.50 1.47 1.30

1.06 1.08 1.16 1.15 1.06

1.01 0.084 2.66

1.01 0.084 2.46

1.42 0.052 3.88

1.10 0.044 3.31

1.09 1.05 1.03 0.98 0.92

1.09 1.05 1.03 0.98 0.92

1.31 1.34 1.40 1.38 1.24

1.01 1.03 1.11 1.15 1.07

1.01 0.064 2.82

1.01 0.064 2.61

1.33 0.049 3.63

1.07 0.053 3.14

1.03 1.00 1.06 0.99 0.97

1.10 1.07 1.06 0.99 0.97

1.15 1.19 1.25 1.26 1.24

1.00 1.04 1.15 1.17 1.13

– – –

1.01 0.034 3.00

1.04 0.051 2.80

1.22 0.039 3.31

1.10 0.067 3.12

T6-CHS2.0-NW-L500 T6-CHS2.0-NW-L1200 T6-CHS2.0-NW-L2000 T6-CHS2.0-NW-L2700 T6-CHS2.0-NW-L3500

134.7 130.5 121.2 99.1 66.7

1.00 1.01 1.07 1.00 0.93

1.12 1.08 1.07 1.00 0.93

1.02 1.06 1.14 1.19 1.15

1.01 1.06 1.16 1.17 1.09

Mean, Pm COV, VP Reliability index, b

– – –

1.00 0.050 2.87

1.04 0.072 2.66

1.11 0.064 2.75

1.10 0.062 3.17

23.3 22.5 21.6 18.6 13.0 – – – 37.9 36.7 34.8 29.0 19.8 – – – 66.2 64.3 59.9 49.2 34.8

6.2. DSM for aluminum alloy non-welded columns The DSM has been proposed by Schafer and Peko¨z [4] for laterally braced flexural members undergoing local or distortional buckling. Subsequently, the method has been developed for concentrically loaded pin-ended cold-formed steel columns undergoing local, distortional, or overall buckling [5,6], which allows for interaction of local and overall buckling as well as interaction of distortional and overall buckling. Zhu and Young [10] reported that the modified DSM can be used for the design of aluminum alloy columns of SHS and RHS. The CHS is investigated in this study. As summarized in the North American Specification (NAS) [7,8] for cold-formed steel structures, the column design rules of the DSM 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 limits: PDSM ¼ minðP ne ; P nl Þ

Pne

 8 l2c > > > 0:658 P y < ! ¼ 0:877 > > Py > : l2c

(1) for

lc p1:5

for

lc 41:5

(2)

ARTICLE IN PRESS 1444

J.-H. Zhu, B. Young / Thin-Walled Structures 46 (2008) 1437–1449

Table 11 Comparison of FEA and design strengths for welded columns of aluminum alloy 6063-T5

Table 12 Comparison of FEA and design strengths for welded columns of aluminum alloy 6061-T6

Specimen

Specimen

T5-t0.4-W-L500 T5-t0.4-W-L1200 T5-t0.4-W-L2000 T5-t0.4-W-L2700 T5-t0.4-W-L3500

FEA

Comparison

PFEA (kN)

PFEA/ PAA

PFEA/ PAS/NZS

PFEA/ PEC9

PFEA/ PDSM

PFEA/ PDSM-W

7.9 7.8 8.0 7.6 7.3

1.27 1.26 1.28 1.30 1.36

1.32 1.31 1.33 1.30 1.36

1.11 1.15 1.24 1.29 1.44

0.47 0.48 0.53 0.57 0.68

1.06 1.07 1.10 1.16 1.34

FEA

Comparison

PFEA (kN)

PFEA/ PAA

PFEA/ PAS/NZS

PFEA/ PEC9

PFEA/ PDSM

PFEA/ PDSM-W

T6-t0.4-W-L500 T6-t0.4-W-L1200 T6-t0.4-W-L2000 T6-t0.4-W-L2700 T6-t0.4-W-L3500

10.3 10.3 10.3 9.7 9.2

1.32 1.32 1.32 1.31 1.35

1.32 1.32 1.32 1.31 1.35

1.37 1.45 1.56 1.65 1.91

0.47 0.49 0.55 0.60 0.75

1.07 1.10 1.15 1.22 1.46

Mean, Pm COV, VP Reliability index, b

– – –

1.29 0.028 4.10

1.32 0.018 4.02

1.24 0.104 2.86

0.54 0.161 0.21

1.15 0.101 2.98

Mean, Pm COV, VP Reliability index, b

– – –

1.32 0.010 4.26

1.32 0.010 4.04

1.59 0.132 3.41

0.57 0.194 0.32

1.20 0.131 2.84

T5-t0.6-W-L500 T5-t0.6-W-L1200 T5-t0.6-W-L2000 T5-t0.6-W-L2700 T5-t0.6-W-L3500

13.2 12.5 12.7 12.4 11.7

1.34 1.27 1.36 1.41 1.42

1.47 1.40 1.42 1.41 1.42

1.07 1.06 1.16 1.24 1.41

0.49 0.48 0.55 0.62 0.73

1.03 1.00 1.07 1.17 1.32

T6-t0.6-W-L500 T6-t0.6-W-L1200 T6-t0.6-W-L2000 T6-t0.6-W-L2700 T6-t0.6-W-L3500

17.7 17.6 16.5 15.6 13.4

1.32 1.32 1.37 1.39 1.41

1.43 1.42 1.41 1.39 1.41

1.27 1.33 1.44 1.57 1.85

0.47 0.49 0.53 0.62 0.73

0.99 1.02 1.02 1.15 1.32

Mean, Pm COV, VP Reliability index, b

– – –

1.36 0.046 4.19

1.42 0.020 4.34

1.19 0.120 2.55

0.57 0.180 0.34

1.12 0.115 2.75

Mean, Pm COV, VP Reliability index, b

– – –

1.36 0.030 4.32

1.41 0.010 4.33

1.50 0.155 2.97

0.57 0.186 0.30

1.10 0.124 2.62

T5-t1.0-W-L500 T5-t1.0-W-L1200 T5-t1.0-W-L2000 T5-t1.0-W-L2700 T5-t1.0-W-L3500

25.2 25.7 25.4 23.8 21.3

1.50 1.53 1.62 1.62 1.57

1.68 1.72 1.70 1.62 1.57

1.04 1.11 1.19 1.27 1.43

0.56 0.60 0.66 0.72 0.80

1.06 1.12 1.17 1.22 1.31

T6-t1.0-W-L500 T6-t1.0-W-L1200 T6-t1.0-W-L2000 T6-t1.0-W-L2700 T6-t1.0-W-L3500

32.8 33.0 32.5 29.3 25.1

1.51 1.51 1.61 1.57 1.50

1.68 1.69 1.66 1.57 1.50

1.25 1.33 1.46 1.58 1.85

0.49 0.54 0.62 0.69 0.83

0.95 1.01 1.10 1.18 1.35

Mean, Pm COV, VP Reliability index, b

– – –

1.57 0.034 4.90

1.66 0.037 4.89

1.21 0.124 2.58

0.66 0.143 0.86

1.18 0.081 3.27

Mean, Pm COV, VP Reliability index, b

– – –

1.54 0.032 4.84

1.62 0.051 4.68

1.50 0.157 2.94

0.63 0.209 0.57

1.12 0.141 2.51

T5-t2.0-W-L500 T5-t2.0-W-L1200 T5-t2.0-W-L2000 T5-t2.0-W-L2700 T5-t2.0-W-L3500

63.2 63.1 62.9 54.4 46.5

1.88 1.88 2.00 1.85 1.71

2.11 2.10 2.10 1.85 1.71

1.16 1.22 1.34 1.34 1.49

0.70 0.73 0.82 0.82 0.87

1.17 1.21 1.27 1.22 1.26

T6-t2.0-W-L500 T6-t2.0-W-L1200 T6-t2.0-W-L2000 T6-t2.0-W-L2700 T6-t2.0-W-L3500

82.9 82.7 82.5 66.8 54.1

1.90 1.90 2.05 1.80 1.60

2.13 2.13 2.12 1.80 1.60

1.30 1.39 1.59 1.63 1.88

0.62 0.67 0.79 0.79 0.88

1.05 1.11 1.23 1.18 1.28

Mean, Pm COV, VP Reliability index, b

– – –

1.86 0.056 5.41

1.97 0.093 4.91

1.31 0.096 3.13

0.79 0.089 1.67

1.23 0.031 3.85

Mean, Pm COV, VP Reliability index, b

– – –

1.85 0.090 4.89

1.95 0.125 4.38

1.56 0.146 3.18

0.75 0.137 1.27

1.17 0.078 3.28

P nl

8 Pne > > <"

  #  Pcrl 0:4 P crl 0:4 ¼ 1  0:15 P ne > > : Pne Pne

for

ll p0:776

for

ll 40:776

(3)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Py ¼ fy A; lc ¼ P y =P cre ; ll ¼ P ne =Pcrl ; A, gross crosssection area; fy, material yield strength which is the static 0.2% proof stress (s0.2) using the non-welded material properties in this paper; Pcre, p2EA/(le/r)2, critical elastic buckling load in flexural buckling for CHS columns; Pcrl, critical elastic local column buckling load; E, Young’s modulus; le, column effective length; and r, radius of gyration of gross cross-section. 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 section was obtained from a rational elastic finite strip buckling analysis [16]. Fig. 6 shows the local buckling (cross-section distortion) of CHS generated from the finite strip analysis. Fig. 7(a) and (b) show the comparison of FEA and experimental results against the direct strength curves plotted from Eqs. (2) and (3), respectively, for the non-welded columns. The unfactored design strengths (PDSM) calculated using the DSM are compared with the

numerical and test results of aluminum non-welded columns of CHS, as shown in Tables 9, 10 and 13.

6.3. Proposed design equation based on DSM for aluminum alloy welded columns Design equation was proposed based on the current DSM for aluminum alloy CHS columns with transverse welds at both ends of the columns (welded columns). The research reported by Zhu and Young [17] indicated that the effects of transverse welds on aluminum stub column strengths are varied with the overall diameter-tothickness ratio of the CHS. The HAZ softening factor proposed by Zhu and Young [17] is used in this study. The unfactored design strengths (PDSM-W) is obtained based on the column strengths (PDSM) calculated using Eqs. (1)–(3) and multiplied by the factors a1 and a2 due to welding, as shown in Eq. (4). The proposed design equation was calibrated with the welded column strengths obtained from the parametric study presented in this study, as well as the test results reported by Zhu and Young [13]. The unfactored design strengths (PDSM-W) are compared with the numerical and test results of aluminum welded columns of CHS, as shown in Tables 11, 12 and 14: PDSM-W ¼ a1 a2 P DSM

(4) 0.19

0.12

where a1 ¼ 1.3(D/t) and a2 ¼ 0.6(30/L) ; PDSM-W, welded column strength; PDSM, column strength calculated using

ARTICLE IN PRESS J.-H. Zhu, B. Young / Thin-Walled Structures 46 (2008) 1437–1449

1445

Table 13 Comparison of test strengths with design strengths for non-welded columns Specimen

Experimental

Comparison

PExp (kN)

PExp/PAA

PExp/PAS/NZS

PExp/PEC9

PExp/PDSM

N-C1-NW-L300 N-C1-NW-L1000 N-C2-NW-L300 N-C2-NW-L1000 H-C1-NW-L300 H-C1-NW-L1000 H-C2-NW-L300 H-C2-NW-L1000

48.5 45.9 102.4 86.1 75.9 71.7 129.6 119.6

1.04 1.08 1.20 1.10 1.11 1.16 1.04 1.06

1.16 1.12 1.37 1.16 1.25 1.17 1.16 1.07

1.07 1.11 1.24 1.14 1.13 1.20 1.06 1.10

1.05 1.11 1.24 1.14 1.13 1.22 1.05 1.12

Mean, Pm COV, VP Reliability index, b

– – –

1.10 0.052 3.31

1.18 0.077 3.26

1.13 0.055 3.25

1.13 0.062 3.38

Fig. 6. Local buckling (cross-section distortion) of CHS generated from finite strip analysis. Table 14 Comparison of test results and design rules for welded columns

1.5

Specimen

Eqn. (2) – DSM Non-Welded FEA data Non-Welded experimental data

Test

Comparison

PExp (kN)

PExp/ PAA

PExp/ PAS/NZS

PExp/ PEC9

PExp/ PDSM

PExp/ PDSM-W

N-C1-W-L300 N-C1-W-L1000 N-C1-W-L1650 N-C1-W-L2350 N-C1-W-L3000

35.9 30.3 27.9 21.4 17.3

2.15 1.86 1.87 1.61 1.43

2.41 2.02 1.86 1.61 1.43

1.33 1.22 1.32 1.46 1.69

0.79 0.73 0.80 0.81 0.93

1.29 1.19 1.23 1.19 1.33

Mean, Pm COV, VP Reliability index, b

– – –

1.78 0.156 3.84

1.87 0.203 3.24

1.40 0.129 3.03

0.81 0.089 1.79

1.24 0.049 3.80

N-C2-W-L300 N-C2-W-L1000 N-C2-W-L1650 N-C2-W-L2350 N-C2-W-L3000

69.4 65.1 48.6 35.3 28.3

2.06 1.99 1.65 1.35 1.22

2.30 2.16 1.65 1.35 1.22

1.40 1.44 1.28 1.33 1.56

0.84 0.87 0.77 0.75 0.87

1.20 1.24 1.04 0.97 1.09

Mean, Pm COV, VP Reliability index, b

– – –

1.65 0.226 2.84

1.74 0.276 2.44

1.40 0.077 3.59

0.82 0.065 1.95

1.11 0.101 2.86

H-C1-W-L300 H-C1-W-L1000 H-C1-W-L1650 H-C1-W-L2350 H-C1-W-L3000

47.2 39.0 36.8 25.3 17.0

2.18 1.88 1.97 1.52 1.17

2.45 2.03 1.97 1.52 1.17

1.43 1.35 1.68 1.85 1.91

0.71 0.67 0.83 0.87 0.94

1.15 1.09 1.27 1.28 1.35

Mean, Pm COV, VP Reliability index, b

– – –

1.74 0.228 2.95

1.83 0.270 2.60

1.64 0.151 3.30

0.80 0.141 1.47

1.23 0.084 3.41

H-C2-W-L300 H-C2-W-L1000 H-C2-W-L1650 H-C2-W-L2350 H-C2-W-L3000

88.0 84.5 66.9 44.6 33.0

2.09 2.10 1.85 1.42 1.21

2.34 2.24 1.85 1.42 1.21

1.44 1.56 1.63 1.77 1.97

0.72 0.79 0.82 0.84 1.00

1.02 1.13 1.10 1.08 1.25

Mean, Pm COV, VP Reliability index, b

– – –

1.73 0.232 2.90

1.81 0.274 2.55

1.68 0.123 3.69

0.83 0.125 1.67

1.12 0.076 3.12

Pu / Py

1

0.5

0 0

0.5

1

1.5

2

2.5

3

3.5

4

λc 1.5 Eqn. (3) – DSM Non-Welded FEA data Non-Welded experimental data Pu / Pne

1

0.5

0 0

0.5

1

1.5

2

λl Fig. 7. Comparison of FEA and experimental data with direct strength method (PDSM) for non-welded columns: (a) flexural buckling and (b) interaction of local and flexural buckling.

Eqs. (1)–(3); D, overall diameter of CHS; L, column length in mm, but LX1200; and t ¼ thickness.

7. Comparison of numerical and experimental results with design predictions The nominal axial strengths (unfactored design strengths) predicted by the AA Specification (PAA), AS/NZS Standard (PAS/NZS), and European Code (PEC9) for aluminum structures, as well as the

current DSM (PDSM) and proposed design equation (PDSM-W) are compared with the column strengths obtained from the parametric study (PFEA) and experimental program (PExp) [13], as shown in Tables 9–14. The statistical parameters Pm and VP which are the mean value and coefficient of variation (COV) of FEA and experimental-to-predicted load ratios of each series of specimens are shown in Tables 9–14. Reliability indices (b) of the design rules for each series of specimens are also shown in Tables 9–14.

ARTICLE IN PRESS J.-H. Zhu, B. Young / Thin-Walled Structures 46 (2008) 1437–1449

The FEA results are also compared with the column design curves obtained from the design rules, as shown in Figs. 8–23. The design strengths are calculated using the material properties for each series of specimens, as shown in Table 7, and the 0.2% proof stress (s0.2) was used as the corresponding yield stress. The design strengths of non-welded column specimens are calculated using the non-welded material properties. In calculating the design strengths of welded columns, PAA and PAS/NZS are calculated using the welded material properties, as specified in the AA and AS/NZS specifications. Whereas PEC9 are calculated using the non-welded material properties as required by the EC9 Code. The design strengths (PDSM-W) of the proposed design Eq. (4) are calculated using the non-welded material properties. The fixed-ended

120 PAA Column strength, Pu (kN)

1446

Non-welded FEA

100

Flexural buckling

80 60

PAS/NZS PDSM

40

PEC9

20 0 0

500

1500

1000

2500

2000

3000

Effective length, le (mm)

20

Fig. 11. Comparison of FEA and design column strengths for Series T5-t2.0-NW.

Flexural buckling

15

30 Non-welded FEA

PAA

10

PAS/NZS PDSM

5 PEC9 0 0

500

1000 1500 2000 Effective length, le (mm)

2500

3000

Column strength, Pu (kN)

Column strength, Pu (kN)

Non-welded FEA

20

Flexural buckling

PAA

10

PAS/NZS PDSM PEC9

0 500

0

Fig. 8. Comparison of FEA and design column strengths for Series T5-t0.4-NW.

1000 1500 2000 Effective length, le (mm)

2500

3000

Fig. 12. Comparison of FEA and design column strengths for Series T6-t0.4-NW.

30 50

Flexural buckling

Non-welded FEA Column strength, Pu (kN)

Column strength, Pu (kN)

Non-welded FEA

20 PAA

PAS/NZS PDSM

10

PEC9 0 0

500

1500 2000 1000 Effective length, le (mm)

2500

3000

40

Flexural buckling

30 PAA

20

PAS/NZS PDSM

10

PEC9 0 0

500

Fig. 9. Comparison of FEA and design column strengths for Series T5-t0.6-NW.

1000 1500 2000 Effective length, le (mm)

2500

3000

Fig. 13. Comparison of FEA and design column strengths for Series T6-t0.6-NW.

80

PAA

Non-welded FEA

50

Column strength, Pu (kN)

Column strength, Pu (kN)

60

Flexural buckling

40 30

PAS/NZS

20 PDSM

10

PEC9

Non-welded FEA

70

Flexural buckling

60 50

PAA

40 30

PAS/NZS

20

PDSM PEC9

10 0

0 0

500

1000 2000 1500 Effective length, le (mm)

2500

3000

Fig. 10. Comparison of FEA and design column strengths for Series T5-t1.0-NW.

0

500

1500 2000 1000 Effective length, le (mm)

2500

3000

Fig. 14. Comparison of FEA and design column strengths for Series T6-t1.0-NW.

ARTICLE IN PRESS J.-H. Zhu, B. Young / Thin-Walled Structures 46 (2008) 1437–1449

120 Non-welded FEA

140

Column strength, Pu (kN)

Column strength, Pu (kN)

160

1447

Flexural buckling

120 100

PAA PAS/NZS

80

PDSM

60 40

PEC9

20 0

Flexural buckling

80 60 40 20 PAA

0 0

500

1000

1500

2000

2500

3000

0

PAS/NZS PEC9 500

Effective length, le (mm) Fig. 15. Comparison of FEA and design column strengths for Series T6-t2.0-NW.

Welded FEA

PDSM

100

PDSM-W

1000 1500 2000 Effective length, le (mm)

2500

3000

Fig. 19. Comparison of FEA and design column strengths for Series T5-t2.0-W.

Welded FEA

30

Flexural buckling

15

Column strength, Pu (kN)

Column strength, Pu (kN)

20

PDSM

10

5 PAA P AS/NZS PEC9

0

500

0

PDSM-W

1500 2000 1000 Effective length, le (mm)

2500

3000

Welded FEA

PDSM

Flexural buckling 20

10

PAA PAS/NZS PEC9

0 0

500

1000

PDSM-W 1500

2000

2500

3000

Effective length, le (mm) Fig. 16. Comparison of FEA and design column strengths for Series T5-t0.4-W. Fig. 20. Comparison of FEA and design column strengths for Series T6-t0.4-W.

30 50

20

Column strength, Pu (kN)

Column strength, Pu (kN)

Welded FEA Flexural buckling PDSM 10 PAA

PAS/NZS

PEC9

PDSM-W

0 0

500

2000 1000 1500 Effective length, le (mm)

2500

30 20 10 PAA

0 0

60 Column strength, Pu (kN)

Flexural buckling

3000

Fig. 17. Comparison of FEA and design column strengths for Series T5-t0.6-W.

Welded FEA

PDSM 40

PAS/NZS 500

PEC9

PDSM-W

2000 1000 1500 Effective length, le (mm)

2500

3000

Fig. 21. Comparison of FEA and design column strengths for Series T6-t0.6-W.

Welded FEA

PDSM

50

Flexural buckling

40 30 20 10 PAA

0 0

PAS/NZS

PEC9

500

1000 2000 1500 Effective length, le (mm)

PDSM-W 2500

3000

Fig. 18. Comparison of FEA and design column strengths for Series T5-t1.0-W.

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 [18]. The reliability of the design rules for aluminum columns is evaluated using reliability analysis. Reliability analysis is detailed in the AA Specification [1], and the ratio of dead (DL) to live (LL) load of 0.2 was used in the analysis. In general, a target reliability index of 2.5 for aluminum alloy columns as a lower limit is recommended by the AA Specification [1]. If the reliability index is greater than or equal to 2.5 (bX2.5), then the design is considered to be reliable. The AA and AS/NZS specifications provide different

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J.-H. Zhu, B. Young / Thin-Walled Structures 46 (2008) 1437–1449

Column strength, Pu (kN)

80 Welded FEA

PDSM

70 60

Flexural buckling

50 40 30 20 10

PAA

0 0

PAS/NZS P EC9 PDSM-W 500

1000 1500 2000 Effective length, le (mm)

2500

3000

Fig. 22. Comparison of FEA and design column strengths for Series T6-t1.0-W.

Column strength, Pu (kN)

160 Welded FEA

PDSM

140 120

Flexural buckling

100 80 60 40 20

PAA

0 0

PAS/NZS 500

PEC9

PDSM-W

1500 2000 1000 Effective length, le (mm)

2500

3000

Fig. 23. Comparison of FEA and design column strengths for Series T6-t2.0-W.

resistance factors (f) for compression members with different failure modes. The resistance factor varies with the slenderness parameter for flexural buckling failure mode. The resistance factor is a constant and equal to 0.85 for local buckling or interaction of local and overall buckling failure mode. The observed failure modes of the columns in this study included local buckling, flexural buckling, interaction of local and overall buckling, and failure in the HAZ. Hence, the resistance factor of the columns given by the AA and AS/NZS specifications ranged from 0.76 to 0.95. In calculating the reliability indices of the AA and AS/NZS design rules, the resistance factor for the columns is chosen to be equal to 0.85 for all failure modes in this study. The EC9 Code provides a constant resistance factor of 1/1.1 ¼ 0.91 for compression members, which is used in the reliability analysis. The reliability of the DSM and proposed design rule for aluminum columns is also evaluated, and the resistance factor is equal to 0.85. The load combination of 1.2DL+1.6LL is used in the analysis for the AA Specification, the DSM and the proposed design rule. The load combinations of 1.25DL+1.5LL and 1.35DL+1.5LL are used in the analysis for AS/NZS and EC9 specifications, respectively. The statistical parameters Mm, Fm, VM, and VF are the mean values and coefficients of variation (COV) of material and fabrication factors. These values are obtained from Section 9 of Part I-B of the AA Specification [1], where Mm ¼ 1.10, Fm ¼ 1.00, VM ¼ 0.06, and VF ¼ 0.05. The statistical parameters Pm and VP are the mean value and the coefficient of variation of test-to-predicted load ratios, respectively. For the non-welded columns, it is shown that the column strengths predicted by the AA and AS/NZS specifications are quite close with the numerical and experimental results, except for the

long columns that are slightly unconservative, as shown in Tables 9 and 10. The mean values of load ratios PFEA/PAA and PExp/PAA ranged from 0.99 to 1.10, with the corresponding COV ranged from 0.034 to 0.084, and the reliability index ranged from 2.66 to 3.31 for all the non-welded column series. The mean values of load ratios PFEA/PAS/NZS and PExp/PAS/NZS ranged from 1.00 to 1.18, with the corresponding COV ranged from 0.051 to 0.084, and the reliability index ranged from 2.46 to 3.26 for all the non-welded column series. The design strengths calculated using the EC9 Code are generally more conservative compared with the predictions given by the AA and AS/NZS specifications. The mean values of load ratios PFEA/PEC9 and PExp/PEC9 ranged from 1.06 to 1.42, with the corresponding COV ranged from 0.014 to 0.064, and the reliability index ranged from 2.73 to 3.88 for all the non-welded column series. The design strengths PDSM predicted by the current DSM are generally conservative and reliable for all the nonwelded column series, as shown in Tables 9, 10 and 13. The mean values of the load ratio PFEA/PDSM for each series of FEA specimens ranged from 1.02 to 1.10, with the corresponding COV ranged from 0.034 to 0.067, and the reliability index ranged from 2.96 to 3.31. In terms of the test results, the mean value of the load ratio PExp/ PDSM is 1.13, with the corresponding COV of 0.062, and the reliability index of 3.38, as shown in Table 13. It is shown that the current DSM could be successfully used in the design of aluminum non-welded columns of circular hollow sections. The non-welded column design curves predicted by the AA, AS/NZS and EC9 specifications, as well as the DSM are shown in Figs. 8–15 for each non-welded column series. For the welded columns, it is shown that the design strengths calculated using the AA specification (PAA) are quite conservative, as shown in Tables 11, 12 and 14. For the column series of FEA specimens, the mean values of the load ratio PFEA/PAA ranged from 1.29 to 1.86, with the corresponding COV ranged from 0.010 to 0.090, and the reliability index ranged from 4.10 to 5.41. For the column series of test specimens, the mean values of the load ratio PExp/PAA ranged from 1.65 to 1.78, with the corresponding COV ranged from 0.156 to 0.232, and the reliability index ranged from 2.84 to 3.84. The design strengths calculated using the AS/NZS Standard are more conservative than the predictions given by the AA Specification, as shown in Tables 11, 12 and 14. The design strength calculated using the EC9 Code for welded columns are also quite conservative, but generally less conservative than the predictions by the AA and AS/NZS specifications. The mean values of the load ratio PFEA/PEC9 ranged from 1.19 to 1.59 for each column series of FEA specimens, and the mean values of the load ratio PExp/PEC9 ranged from 1.40 to 1.68 for each column series of test specimens. The corresponding COV for the load ratios PFEA/PEC9 and PExp/PEC9 ranged from 0.077 to 0.157, and the reliability index ranged from 2.55 to 3.69. It is also shown that the current DSM is not suitable for the design of aluminum CHS welded columns. The mean values of the load ratio PFEA/PDSM ranged from 0.54 to 0.79, with the corresponding COV ranged from 0.089 to 0.209, and the reliability index ranged from 0.21 to 1.67 for each column series of FEA specimens, as shown in Tables 11 and 12. The mean values of the load ratio PExp/PDSM ranged from 0.80 to 0.83, with the corresponding COV ranged from 0.065 to 0.141, and the reliability index ranged from 1.47 to 1.95 for each column series of test specimens, as shown in Table 14. The design equation based on the DSM for aluminum alloy CHS columns with transverse welds at both ends of the columns (welded columns) is shown in Eq. (4) of the paper. The unfactored design strengths (PDSM-W) calculated using the proposed design rules are generally conservative for the welded columns, as shown in Tables 11, 12 and 14. The mean values of the load ratios PFEA/PDSM-W and PExp/PDSM-W ranged from 1.10 to 1.24, with the corresponding COV ranged from 0.031 to 0.141. The reliability indices for the design

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rules (PDSM-W) are greater than the target value of 2.5 that ranged from 2.51 to 3.85 for each series of specimens. The welded column design curves predicted by the AA, AS/NZS and EC9 specifications, as well as the current DSM and the proposed design rules are shown in Figs. 16–23 for each welded column series. It is shown that the proposed design equation based on the DSM can be used for the design of aluminum columns of circular hollow sections with transverse welds at the ends of the columns.

8. Conclusions Numerical investigation and design of aluminum alloy circular hollow section columns have been presented in this paper. A nonlinear FEM incorporating geometric imperfections and material non-linearities was developed. The FEM was verified against experimental results. The column specimens were fabricated by extrusion using heat-treated aluminum alloy of 6061-T6 and 6063-T5. The ultimate loads and failure modes predicted by the FEM are in good agreement with the experimental results. A parametric study was performed using the verified FEM that included 80 specimens with the column lengths ranged from 500 to 3500 mm. The column strengths obtained from the experimental and numerical investigations were compared with the design strengths calculated using the current American, Australian/New Zealand and European specifications for aluminum structures, as well as the current DSM that was developed for cold-formed steel members. It is shown that the design strengths predicted by the current DSM are generally conservative for the aluminum non-welded columns of circular hollow sections. Design equation was proposed based on the current DSM for aluminum alloy circular hollow section columns with transverse welds at the ends of the columns. It is also shown that the design strengths calculated using the proposed design rules are generally conservative for the aluminum welded columns. The reliability of the current and proposed design rules was evaluated using reliability analysis. It is shown that the proposed design rules are reliable. It is recommended of using the proposed design rules in the design of aluminum columns of circular hollow section containing transverse welds at the ends of the columns.

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Acknowledgement The research work described in this paper was supported by a grant from The University of Hong Kong under the Seed Funding Programme for Basic Research. References [1] AA. Aluminum design manual. Washington, DC: The Aluminum Association; 2005. [2] AS/NZS. Aluminum structures. Part 1: Limit state design. Australian/New Zealand Standard AS/NZS 1664.1:1997. Sydney, Australia: Standards Australia; 1997. [3] EC9. Eurocode 9: Design of aluminum structures—Part 1-1: General rules—general rules and rules for buildings. DD ENV 1999-1-1:2000. Final draft October 2000. European Committee for Standardization, 2000. [4] Schafer BW, Peko¨z T. Direct strength prediction of cold-formed steel members using numerical elastic buckling solutions. In: Proceeding of 14th international specialty conference on cold-formed steel structures. University of Missouri-Rolla, Rolla, MO, 1998. p. 69–76. [5] Schafer BW. Distortional buckling of cold-formed steel columns. August final report to the American Iron and Steel Institute, Washington, DC, 2000. [6] Schafer BW. Local, distortional, and Euler buckling of thin-walled columns. J Struct Eng 2002;128(3):289–99. [7] North American Specification for the design of cold-formed steel structural members. American Iron and Steel Institute, Washington, DC, 2001. [8] Supplement to the North American Specification for the design of coldformed steel structural member. American Iron and Steel Institute, Washington, DC, 2004. [9] Zhu JH, Young B. Aluminum alloy tubular columns—part I: finite element modeling and test verification. Thin-Walled Struct 2006;44(9):961–8. [10] Zhu JH, Young B. Aluminum alloy tubular columns—part II: parametric study and design using direct strength method. Thin-Walled Struct 2006;44(9): 969–85. [11] Mazzolani FM. Aluminum alloy structures. 2nd ed. London: E&FN Spon; 1995. [12] Sharp ML. Behaviour and design of aluminum structures. New York: McGrawHill; 1993. [13] Zhu JH, Young B. Experimental investigation of aluminum alloy circular hollow section columns. Eng Struct 2006;28(2):207–15. [14] ABAQUS analysis user’s manual, version 6.5. ABAQUS Inc., 2004. [15] Clark JW, Rolf RL. Design of aluminum tubular columns. J Struct Div ASCE 1964;90:259. [16] Papangelis JP, Hancock GJ. Computer analysis of thin-walled structural members. Comput Struct 1995;56(1):157–76. [17] Zhu JH, Young B. Effects of transverse welds on aluminum alloy columns. Thin-Walled Struct 2007;45(3):321–9. [18] Young B, Rasmussen KJR. Tests of fixed-ended plain channel columns. J Struct Eng 1998;124(2):131–9.