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Thin-Walled Structures 44 (2006) 969985

Aluminum alloy tubular columnsPart II: Parametric study and designusing direct strength method

Ji-Hua Zhu, Ben YoungDepartment of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

Received 23 February 2006; received in revised form 11 August 2006; accepted 18 August 2006Available online 18 October 2006

Abstract

A parametric study of aluminum alloy columns of square and rectangular hollow sections was performed using nite element analysis(FEA). The columns were compressed between xed ends. The parametric study included 120 columns with and without transverse weldsat the ends of the columns. An accurate and reliable nite element model was used for the parametric study. Design approaches foraluminum alloy tubular columns with and without transverse welds were proposed. Column strengths predicted by the FEA werecompared with the design strengths calculated using the current American, Australian/New Zealand and European specications foraluminum structures. In addition, the direct strength method (DSM), which was developed for cold-formed carbon steel members, wasused in this study for aluminum alloy columns. The design strengths calculated using the DSM were compared with the numerical results.Furthermore, design rules modied from the DSM were proposed. It is shown that the proposed design rules accurately predicted theultimate strengths of aluminum welded and non-welded columns. The reliability of the current and proposed design rules was evaluatedusing reliability analysis.r 2006 Elsevier Ltd. All rights reserved.

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

1. Introduction

Finite element analysis (FEA) has been widely used instructural design. Compared with physical experiments,FEA is relatively inexpensive and time efcient, especiallywhen a parametric study of cross-section geometry isinvolved. In addition, FEA is more convenient forinvestigation involving geometric imperfections of struc-tural members, whereas this could be difcult to investigatethrough physical tests. Although FEA is a useful and

powerful tool for structural analysis and design, it isimportant to obtain an accurate and reliable nite elementmodel (FEM) prior to a parametric study of FEA to becarried out.

Aluminum members are being used increasingly instructural applications. The current American AluminumDesign Manual [1], Australian/New Zealand Standard [2]and European Code [3] for aluminum structures provide

design rules for compression members. The design rules inthe three specications follow the element approach thatseparate the cross-section into elements for design. Onedisadvantage of the element approach is that the computa-tion procedure becomes more tedious for members havingcomplex cross-sections. This disadvantage can be overcomeby a new design method called the direct strength method(DSM) that was developed by Schafer and Peko z [4]. TheDSM was developed for cold-formed steel structures. Thetest 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 geo-metric limits [57]. The DSM has been adopted by theNorth American Specication (NAS) [8,9] for cold-formedsteel structures. In this study, the DSM was used for thedesign of aluminum alloy columns.

There are some advantages in using aluminum as astructural material, such as high strength-to-weight ratio,lightness, corrosion resistance and ease of production.However, one disadvantage is that heat-treated aluminumalloys could suffer loss of strength in a localized region

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www.elsevier.com/locate/tws

0263-8231/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.doi: 10.1016/j.tws.2006.08.012

Corresponding author. Tel.: +852 2859 2674; fax: +852 25595337.E-mail address: [email protected] (B. Young).
http://www.elsevier.com/locate/twshttp://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.tws.2006.08.012mailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_10/dx.doi.org/10.1016/j.tws.2006.08.012http://www.elsevier.com/locate/tws

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when welding is involved, and this is known as heat-affected zone (HAZ) softening. Previous research [10,11]indicated that welds have signicant effect on columnstrength. The test program presented by Zhu and Young[12] showed that transverse welds at the ends of thecolumns reduce the column strength for nearly 45%. Thecurrent American Aluminum Design Manual [1], Austra-lian/New Zealand Standard [2] and European Code [3] foraluminum structures provide design rules for structuralmembers containing transverse welds.

An accurate and reliable non-linear FEM for aluminumcolumns with and without transverse welds at the ends of the columns has been developed by Zhu and Young [13].The purpose of this paper is rstly to investigate thebehavior and design of aluminum columns using aparametric study of FEA. The developed FEM [13] is usedfor a parametric study of cross-section geometries.Secondly, the design rules for aluminum welded and non-welded columns are proposed based on the DSM. Thecolumn strengths predicted by the FEA were compared

with the design strengths calculated using the AmericanAluminum Design Manual (AA), Australian/New ZealandStandard (AS/NZS) and European Code (EC9) foraluminum structures, as well as the direct strength method(DSM) and proposed design rules. In addition, reliabilityanalysis was performed to assess the reliability of thesedesign rules.

2. Summary of test program

The test program presented by Zhu and Young [12]provided experimental ultimate loads and failure modes of aluminum alloy square and rectangular hollow sections(RHS) compressed between xed ends. The test specimenswere fabricated by extrusion using 6063-T5 and 6061-T6heat-treated aluminum alloys. The test program included29 columns with both ends transversely welded toaluminum end plates (welded columns), and 12 columnswithout welding of end plates (non-welded columns).The non-welded and welded material properties of the

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Nomenclature

A gross cross-section areaB overall width of SHS and RHSCOV coefcient of variation

DL dead loadE Youngs modulusFEA nite element analysisFEM nite element modelF m mean value of fabrication factor f y material yield strength f y-nw non-welded material yield strength f y-w welded material yield strengthH overall depth of SHS and RHSk c coefcient in the AS/NZS StandardL length of specimenLL live loadl e column effective lengthM m mean value of material factorP AA unfactored design strength for American Alu-

minum Design ManualP AS/NZS unfactored design strength for Australian/New

Zealand StandardP cre critical elastic buckling load in exural buckling,

p2EA /( l e/r )2

P crl critical elastic local column buckling loadP DSM column design strength calculated using the

direct strength methodP DSM-NW non-welded column design strength calcu-

lated using the modied direct strength

methodP DSM-W1 welded column design strength calculatedusing the modied direct strength method (rstapproach)

P DSM-W2 welded column design strength calculatedusing the modied direct strength method(second approach)

P EC9 unfactored design strength for Eurocode 9P Exp experimental ultimate load of column

P FEA ultimate load predicted by FEA for parametricstudyP m mean value of tested-to-predicted load ratioP ne nominal axial strength for exural bucklingP n l nominal axial strength for local bucklingP u column strengthP y yield strength of the section ( f y A)P y-nw yield strength of the section calculated using the

non-welded material properties ( f y-nw A)P y-w yield strength of the section calculated using the

welded material properties ( f y-w A )r radius of gyration of gross cross-section about

the minor y-axis of bucklingt thickness of sectionV F coefcient of variation of fabrication factorV M coefcient of variation of material factorV P coefcient of variation of tested-to-predicted

load ratiob reliability indexl c non-dimensional slenderness for exural buck-

ling ( ffiffiffiffiffiffiffiffiffiP y=P crep ; ffiffiffiffiffiffiffiP y nw =P crep or ffiffiffiffiP y w=P crep )l l non-dimensional slenderness for interaction of local and exural buckling ( ffiffiffiffiffiP ne =P crlp )r c local buckling coefcient specied in the EC9Code

r haz heat-affected zone (HAZ) softening factor spe-cied in the EC9 Codef resistance factors 0.2 static 0.2% proof stress.

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specimens were determined by longitudinal tensile coupontests. Initial overall and local geometric imperfections weremeasured prior to testing. The specimens were testedbetween xed ends at various column lengths ranged from300 to 3000 mm. The column tests were performed usingdisplacement control. The details of the test program were

reported in Zhu and Young [12].

3. Finite element modeling

The nite element program ABAQUS [14] version 6.5was used in the analysis for the simulation of aluminumalloy xed-ended columns tested by Zhu and Young [12].An accurate and reliable non-linear FEM for aluminumwelded and non-welded columns has been presented byZhu and Young [13]. The development of the FEM isdetailed in Zhu and Young [13]. In the FEM, the measuredcross-section dimensions, material properties and initial

geometric imperfections of the test specimens weremodeled. The xed-ended boundary condition was mod-eled by restraining all the degrees of freedom of the nodesat both ends of the column, except for the translationaldegree of freedom in the axial direction at one end of thecolumn. The nodes other than the two ends were free totranslate and rotate in any directions. The material non-linearity was included in the FEM by specifying the truevalues of stresses and strains. The plasticity of the materialwas simulated by a mathematical model, known as theincremental plasticity model, in which the true stresses andtrue plastic strains were calculated in accordance withABAQUS [14]. The geometric imperfections were includedin the FEM by using the eigenvalue analyses. Thedisplacement control loading method was used in theFEA that was identical to the loading method used in thecolumn tests. The S4R general-purpose shell elements wereused in the FEM. The size of the nite element mesh of 10 10mm 2 (length width) was used in the modeling of the non-welded columns. The welded columns weremodeled by dividing the columns into different portionsalong the column length. Therefore, the HAZ softening atboth ends of the columns were simulated. The size of thenite element mesh for the welded columns is detailed inZhu and Young [13].

4. Parametric study

The FEM closely predicted the experimental ultimateloads and failure modes of the tested aluminum columns aspresented by Zhu and Young [13]. Hence, the model wasused for an extensive parametric study. The parametricstudy included 120 specimens that consisted of 24 series, asshown in Table 1 . Each series contained 5 specimens withcolumn lengths of 500, 1200, 2000, 2700 and 3500 mm. Thespecimens were labelled such that the type of aluminumalloy, cross-section dimensions, welding condition andcolumn length could be identied, as shown in Tables 25 .

For example, the label T5-A-NW-L500 denes thefollowing specimen:

The rst letter indicates the type of material of thespecimen, where T5 refers to the aluminum alloy6063-T5, and T6 refers to the aluminum alloy 6061-T6.

The second part of the label indicates the cross-sectionshape of the specimen, where A refers to a squarehollow section (SHS) with nominal cross-section dimen-sion of 60 60 1.5 mm 2. Table 1 shows the cross-section dimensions of each series using the nomenclaturedened in Fig. 1 .

The following part of the label NW indicates thewelding condition of the specimen, where the letterNW refers to the non-welded column, and the letterW 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 columnlength and the following digits are the nominal length of the specimen in millimeters (500 mm).

The material properties of the specimens of 6063-T5alloy investigated in the parametric study are identical tothe material properties of Series N-R2 in the experimentalprogram for the welded and non-welded material, whereasthe material properties of the specimens of 6061-T6 alloy

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Table 1Cross-section dimensions of the series for parametric study

Series Type of material

DepthH (mm)

Width B (mm)

Thickness t(mm)

(H 2t=t

T5-A-NW 6063-T5 60 60 1.5 38.0T5-B-NW 6063-T5 60 60 3.0 18.0T5-C-NW 6063-T5 120 60 1.5 78.0T5-D-NW 6063-T5 120 60 3.0 38.0T5-E-NW 6063-T5 180 60 1.5 118.0T5-F-NW 6063-T5 180 60 3.0 58.0

T6-A-NW 6061-T6 60 60 1.5 38.0T6-B-NW 6061-T6 60 60 3.0 18.0T6-C-NW 6061-T6 120 60 1.5 78.0T6-D-NW 6061-T6 120 60 3.0 38.0T6-E-NW 6061-T6 180 60 1.5 118.0T6-F-NW 6061-T6 180 60 3.0 58.0

T5-A-W 6063-T5 60 60 1.5 38.0T5-B-W 6063-T5 60 60 3.0 18.0T5-C-W 6063-T5 120 60 1.5 78.0

T5-D-W 6063-T5 120 60 3.0 38.0T5-E-W 6063-T5 180 60 1.5 118.0T5-F-W 6063-T5 180 60 3.0 58.0

T6-A-W 6061-T6 60 60 1.5 38.0T6-B-W 6061-T6 60 60 3.0 18.0T6-C-W 6061-T6 120 60 1.5 78.0T6-D-W 6061-T6 120 60 3.0 38.0T6-E-W 6061-T6 180 60 1.5 118.0T6-F-W 6061-T6 180 60 3.0 58.0

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

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are identical to the material properties of Series H-R2 inthe experimental program for the welded and non-weldedmaterial, as detailed in Zhu and Young [12,13]. The localimperfection magnitude was 16% of the section thicknesswhich is equal to the mean value of the measured localimperfection magnitudes of the tested specimens. Theoverall imperfection magnitude was 1/2000 of the columnlength. The size of the nite element mesh was kept at

10 10mm 2 (length width) for the non-welded columns.The welded columns were modeled with 25mm HAZextension at both ends of the columns. The nite elementmesh dimension was approximately 8.3 10mm 2 (length

width) for the HAZ regions at both ends of the weldedcolumns, and 10 10mm 2 (length width) for the mainbody of the columns. The column strengths ( P FEA ) obtainedfrom the parametric study are shown in Tables 25 .

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Table 2Comparison of FEA and design strengths for non-welded columns of aluminum alloy 6063-T5

Specimen FEA Comparison

P FEA (kN) P FEA /P AA P FEA /P AS/NZS P FEA /P EC9 P FEA /P DSM P FEA /P DSM-NW

T5-A-NW-L500 48.3 0.92 0.92 0.95 0.91 0.90T5-A-NW-L1200 48.1 0.91 0.91 1.00 0.93 0.93T5-A-NW-L2000 47.5 0.90 0.90 1.07 0.96 0.96T5-A-NW-L2700 46.6 0.92 0.92 1.19 0.96 0.98T5-A-NW-L3500 38.4 0.88 0.88 1.25 1.01 1.05Mean, P m 0.91 0.91 1.09 0.96 0.96COV, V P 0.019 0.019 0.116 0.039 0.111Reliability index, b 2.57 2.35 2.29 2.73 2.65T5-B-NW-L500 125.4 0.97 1.09 0.99 0.98 0.98T5-B-NW-L1200 120.7 0.98 1.05 1.01 1.01 1.01T5-B-NW-L2000 113.6 1.04 1.04 1.07 1.02 1.02T5-B-NW-L2700 94.7 0.97 0.97 1.09 1.09 1.09T5-B-NW-L3500 75.4 0.93 0.93 1.20 1.04 1.04Mean, P m 0.98 1.02 1.07 1.03 1.03COV, V P 0.039 0.060 0.078 0.061 0.039Reliability index, b 2.83 2.65 2.51 3.04 3.04T5-C-NW-L500 63.2 1.21 1.21 1.16 1.18 1.05T5-C-NW-L1200 62.7 1.20 1.20 1.19 1.21 1.08T5-C-NW-L2000 61.9 1.25 1.25 1.24 1.29 1.16T5-C-NW-L2700 56.0 1.38 1.38 1.20 1.28 1.17T5-C-NW-L3500 40.5 1.19 1.19 0.98 1.06 1.00Mean, P m 1.25 1.25 1.16 1.20 1.09COV, V P 0.063 0.063 0.089 0.074 0.068Reliability index, b 3.67 3.47 2.72 3.43 3.09T5-D-NW-L500 164.0 0.95 1.01 0.99 0.96 0.96T5-D-NW-L1200 155.3 0.90 0.95 0.99 0.94 0.95T5-D-NW-L2000 152.1 0.89 0.93 1.05 1.00 1.02T5-D-NW-L2700 125.5 0.82 0.82 0.98 0.99 1.02T5-D-NW-L3500 114.6 0.85 0.85 1.13 1.03 1.03Mean, P m 0.88 0.91 1.03 0.98 1.00COV, V P 0.057 0.084 0.062 0.036 0.037Reliability index, b 2.29 2.06 2.45 2.86 2.92T5-E-NW-L500 62.1 1.20 1.20 1.12 1.15 0.95T5-E-NW-L1200 61.3 1.18 1.18 1.13 1.17 0.97T5-E-NW-L2000 58.3 1.49 1.49 1.12 1.19 1.00T5-E-NW-L2700 51.8 1.62 1.62 1.04 1.15 0.98T5-E-NW-L3500 41.7 1.55 1.55 0.90 1.06 0.92Mean, P m 1.41 1.41 1.06 1.14 0.96COV, V P 0.146 0.146 0.090 0.045 0.033Reliability index, b 3.21 3.05 2.40 3.46 2.78T5-F-NW-L500 178.9 1.05 1.11 1.02 1.01 0.95T5-F-NW-L1200 174.2 1.02 1.08 1.04 1.02 0.96T5-F-NW-L2000 162.8 0.95 1.01 1.03 1.02 0.98T5-F-NW-L2700 146.2 0.91 0.91 1.00 1.01 0.98T5-F-NW-L3500 124.9 0.92 0.92 1.00 0.99 0.98Mean, P m 0.97 1.01 1.02 1.01 0.97COV, V P 0.061 0.088 0.016 0.010 0.012Reliability index, b 2.66 2.42 2.61 3.06 2.88

Note : 1 kip 4.45kN.


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5. Design approaches

5.1. Current design rules for aluminum structures

The American Aluminum Design Manual (AA) [1],Australian/New Zealand Standard (AS/NZS) [2] andEuropean Code (EC9) [3] for aluminum structures providedesign rules for aluminum columns with and withouttransverse welds. The design rules in the AA Specication

for calculating the design strengths of non-welded alumi-num columns are based on the Euler column strength. Theinelastic column curve, based on the tangent modulus, iswell approximated by a straight line using bucklingconstants [11]. The buckling constants were obtained fromTables 3.33 and 3.34 of Part I-B of the AA Specication.Local buckling stress for the section as a whole was theweighted average local buckling stress for the individualelements, based on gross section properties, while the local

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Table 3Comparison of FEA and design strengths for non-welded columns of aluminum alloy 6061-T6


P FEA (kN) P FEA /P AA P FEA /P AS/NZS P FEA /P EC9 P FEA /P DSM P FEA /P DSM-NW

T6-A-NW-L500 48.3 0.97 0.97 0.99 0.93 0.89

T6-A-NW-L1200 48.1 0.96 0.96 1.04 0.96 0.93T6-A-NW-L2000 47.5 0.95 0.95 1.14 0.99 0.97T6-A-NW-L2700 46.6 0.86 0.86 1.23 1.06 1.06T6-A-NW-L3500 38.4 0.91 0.91 1.20 0.99 1.02Mean, P m 0.93 0.93 1.12 0.99 0.97COV, V P 0.049 0.049 0.093 0.049 0.071Reliability index, b 2.56 2.35 2.56 2.80 2.61T6-B-NW-L500 181.1 0.96 1.08 0.98 0.98 0.98T6-B-NW-L1200 177.6 1.01 1.06 1.05 1.06 1.06T6-B-NW-L2000 169.0 1.11 1.11 1.21 1.23 1.23T6-B-NW-L2700 135.5 1.04 1.04 1.32 1.27 1.27T6-B-NW-L3500 79.4 0.96 0.96 1.15 1.10 1.10Mean, P m 1.02 1.05 1.14 1.13 1.13COV, V P 0.062 0.053 0.115 0.107 0.107Reliability index, b 2.85 2.83 2.46 2.86 2.86T6-C-NW-L500 76.0 1.18 1.18 1.12 1.11 0.96T6-C-NW-L1200 69.9 1.08 1.08 1.07 1.07 0.93T6-C-NW-L2000 63.9 1.26 1.26 1.04 1.09 0.96T6-C-NW-L2700 51.0 1.23 1.23 0.91 0.99 0.89T6-C-NW-L3500 41.5 1.19 1.19 0.87 0.99 0.91Mean, P m 1.19 1.19 1.00 1.05 0.93COV, V P 0.057 0.057 0.109 0.054 0.031Reliability index, b 3.53 3.33 2.04 3.04 2.64T6-D-NW-L500 220.9 0.98 1.03 1.00 1.00 0.98T6-D-NW-L1200 207.8 0.92 0.97 1.00 0.99 0.98T6-D-NW-L2000 205.6 0.91 0.96 1.11 1.10 1.10T6-D-NW-L2700 152.8 0.73 0.73 0.99 0.95 0.96T6-D-NW-L3500 125.3 0.85 0.85 1.11 0.99 1.02Mean, P m 0.88 0.91 1.04 1.00 1.01COV, V P 0.105 0.131 0.058 0.056 0.055Reliability index, b 1.98 1.77 2.53 2.84 2.85T6-E-NW-L500 79.3 1.24 1.24 1.14 1.16 0.92T6-E-NW-L1200 77.4 1.38 1.38 1.15 1.18 0.94T6-E-NW-L2000 74.4 1.86 1.86 1.16 1.25 1.01T6-E-NW-L2700 60.8 1.86 1.86 1.01 1.15 0.96T6-E-NW-L3500 45.8 1.67 1.67 0.84 1.04 0.89Mean, P m 1.24 1.24 1.14 1.15 0.95COV, V P 0.178 0.178 0.133 0.063 0.048Reliability index, b 3.24 3.10 2.07 3.36 2.63T6-F-NW-L500 243.1 1.08 1.15 1.05 1.07 0.98T6-F-NW-L1200 237.4 1.06 1.12 1.08 1.10 1.01T6-F-NW-L2000 213.0 1.07 1.07 1.05 1.09 1.02T6-F-NW-L2700 171.8 1.05 1.05 0.95 1.01 0.96T6-F-NW-L3500 131.7 0.96 0.96 0.91 0.94 0.92Mean, P m 1.04 1.07 1.01 1.04 0.98COV, V P 0.178 0.178 0.133 0.063 0.040Reliability index, b 3.05 2.79 2.29 2.94 2.81



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buckling stress for each element is weighted in accordancewith the ratio of the area of the element to the total areaof the section [1]. Effects of elastic local buckling oncolumn strength were determined based on Clause 4.7.4of Part I-B of the AA Specication. The design rules inthe AS/NZS Standard for calculating the design strengthsof non-welded aluminum columns are generally identicalto those in the AA Specication, except that the AS/NZS

Standard reduces the yield load of the column usinga parameter k c which is not included in the AASpecication. The EC9 Code adopts the Perry curve forcolumn design, and values of the imperfection factorsare listed in Table 5.6 of the Code. The effects of localbuckling on column strength are considered by replacingthe true section with an effective section. The effectivecross-section is obtained by employing a local buckling

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Table 4Comparison of FEA and design strengths for welded columns of aluminum alloy 6063-T5


P FEA (kN) P FEA /P AA P FEA /P AS/NZS P FEA /P EC9 P FEA /P DSM P FEA /P DSM-W1 P FEA /P DSM-W2

T5-A-W-L500 30.6 1.24 1.39 1.10 0.58 0.90 0.88T5-A-W-L1200 30.2 1.22 1.37 1.14 0.60 0.92 0.90T5-A-W-L2000 29.9 1.21 1.35 1.22 0.64 0.99 0.97T5-A-W-L2700 28.4 1.15 1.29 1.29 0.68 1.05 1.02T5-A-W-L3500 24.5 0.99 1.11 1.38 0.71 1.08 1.04Mean, P m 1.16 1.30 1.23 0.64 0.99 0.96COV, V P 0.087 0.087 0.092 0.086 0.080 0.073Reliability index, b 3.16 3.41 2.92 0.89 2.60 2.54T5-B-W-L500 105.5 2.19 2.45 1.38 0.83 1.27 1.12T5-B-W-L1200 99.1 2.06 2.30 1.38 0.83 1.26 1.10T5-B-W-L2000 90.3 1.88 2.10 1.42 0.87 1.29 1.12T5-B-W-L2700 67.6 1.40 1.57 1.30 0.78 1.13 0.97T5-B-W-L3500 55.9 1.16 1.30 1.48 0.84 1.18 1.00Mean, P m 1.74 1.95 1.39 0.83 1.23 1.06COV, V P 0.253 0.253 0.048 0.040 0.057 0.067Reliability index, b 2.73 2.87 3.81 2.11 3.66 2.99T5-C-W-L500 35.8 0.96 1.07 1.20 0.67 0.93 0.91T5-C-W-L1200 35.2 0.94 1.05 1.22 0.68 0.94 0.92T5-C-W-L2000 35.6 0.95 1.07 1.30 0.74 1.02 1.00T5-C-W-L2700 32.1 0.86 0.96 1.24 0.73 1.01 0.99T5-C-W-L3500 31.7 0.93 0.95 1.36 0.83 1.15 1.11Mean, P m 0.93 1.02 1.27 0.73 1.01 0.99COV, V P 0.044 0.059 0.051 0.090 0.087 0.081Reliability index, b 2.58 2.68 3.40 1.38 2.63 2.59T5-D-W-L500 117.1 1.59 1.78 1.25 0.68 1.08 1.06T5-D-W-L1200 112.5 1.53 1.71 1.26 0.68 1.08 1.05T5-D-W-L2000 111.6 1.52 1.70 1.35 0.73 1.16 1.13T5-D-W-L2700 98.0 1.33 1.49 1.33 0.71 1.12 1.09T5-D-W-L3500 84.1 1.14 1.28 1.42 0.75 1.12 1.08Mean, P m 1.42 1.59 1.32 0.71 1.11 1.08COV, V P 0.129 0.129 0.053 0.045 0.029 0.026Reliability index, b 3.43 3.64 3.55 1.45 3.43 3.33T5-E-W-L500 44.1 1.16 1.16 1.45 0.82 1.05 1.03T5-E-W-L1200 43.4 1.14 1.14 1.47 0.83 1.06 1.04T5-E-W-L2000 42.4 1.11 1.11 1.49 0.87 1.11 1.09T5-E-W-L2700 41.0 1.28 1.28 1.50 0.91 1.17 1.14T5-E-W-L3500 37.5 1.39 1.39 1.46 0.95 1.21 1.18Mean, P m 1.22 1.22 1.47 0.88 1.12 1.10COV, V P 0.097 0.097 0.013 0.064 0.063 0.057Reliability index, b 3.23 3.05 4.26 2.22 3.24 3.19T5-F-W-L500 131.1 1.33 1.49 1.32 0.74 1.09 1.07T5-F-W-L1200 131.8 1.33 1.49 1.38 0.77 1.13 1.11T5-F-W-L2000 126.3 1.28 1.43 1.40 0.79 1.17 1.14T5-F-W-L2700 115.8 1.17 1.31 1.39 0.80 1.17 1.14T5-F-W-L3500 104.8 1.06 1.19 1.45 0.83 1.22 1.18Mean, P m 1.23 1.38 1.39 0.79 1.16 1.13COV, V P 0.095 0.095 0.032 0.045 0.041 0.035Reliability index, b 3.32 3.55 3.91 1.87 3.54 3.46



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coefcient r c to reduce the thickness of the element in thesection.

The strength of aluminum column with transverse welds(welded column) depends on the location and number of welds [1]. For SHS and RHS with transverse welds at theends of the columns only, the design equations given by theAA and AS/NZS specications are identical to the design

equations of non-welded columns. However, the designstrength of welded columns is limited by the yield strengthof the welded material. The EC9 Code uses a factor r haz toconsider the weakening effects of welding on columnstrength, and r haz is equal to 0.60 and 0.50 for the 6000Series alloys of T5 and T6 conditions, respectively, asshown in Table 5.2 of the Code.

ARTICLE IN PRESS

Table 5Comparison of FEA and design strengths for welded columns of aluminum alloy 6061-T6


P FEA (kN) P FEA /P AA P FEA /P AS/NZS P FEA /P EC9 P FEA /P DSM P FEA /P DSM-W1 P FEA /P DSM-W2

T6-A-W-L500 40.5 1.16 1.30 1.39 0.60 0.89 0.90T6-A-W-L1200 40.0 1.15 1.29 1.44 0.62 0.93 0.93T6-A-W-L2000 39.4 1.13 1.27 1.57 0.70 1.04 1.03T6-A-W-L2700 35.5 1.02 1.14 1.65 0.74 1.09 1.07T6-A-W-L3500 29.4 0.85 0.95 1.79 0.78 1.15 1.11Mean, P m 1.06 1.19 1.57 0.69 1.02 1.01COV, V P 0.126 0.150 0.103 0.114 0.113 0.089Reliability index, b 2.48 2.69 3.71 1.06 2.51 2.61T6-B-W-L500 126.5 1.87 2.09 1.40 0.68 1.05 1.03T6-B-W-L1200 123.3 1.82 2.04 1.48 0.73 1.11 1.06T6-B-W-L2000 110.6 1.63 1.83 1.60 0.80 1.18 1.08T6-B-W-L2700 82.7 1.22 1.37 1.62 0.78 1.10 0.96T6-B-W-L3500 66.4 0.98 1.10 1.93 0.92 1.22 1.04Mean, P m 1.50 1.68 1.60 0.78 1.13 1.03COV, V P 0.258 0.258 0.126 0.112 0.058 0.045Reliability index, b 2.37 2.50 3.51 1.52 3.29 3.02T6-C-W-L500 47.1 0.95 1.00 1.52 0.69 0.92 0.93T6-C-W-L1200 45.6 0.92 0.97 1.53 0.70 0.93 0.94T6-C-W-L2000 46.8 0.94 1.00 1.66 0.80 1.06 1.06T6-C-W-L2700 39.7 0.96 0.96 1.52 0.77 1.03 1.01T6-C-W-L3500 37.6 1.08 1.08 1.65 0.89 1.19 1.15Mean, P m 0.97 1.00 1.58 0.77 1.03 1.02COV, V P 0.064 0.047 0.047 0.108 0.105 0.091Reliability index, b 2.65 2.67 4.35 1.49 2.54 2.63T6-D-W-L500 146.0 1.41 1.58 1.41 0.66 1.01 1.02T6-D-W-L1200 146.9 1.42 1.59 1.50 0.70 1.07 1.07T6-D-W-L2000 140.5 1.36 1.52 1.59 0.75 1.15 1.14T6-D-W-L2700 109.7 1.06 1.19 1.47 0.68 1.03 1.01T6-D-W-L3500 101.7 0.98 1.10 1.83 0.80 1.19 1.15Mean, P m 1.25 1.40 1.56 0.72 1.09 1.08COV, V P 0.167 0.167 0.106 0.080 0.069 0.059Reliability index, b 2.62 2.81 3.65 1.35 3.08 3.11T6-E-W-L500 56.2 1.14 1.14 1.78 0.82 1.01 1.02T6-E-W-L1200 55.8 1.13 1.13 1.82 0.85 1.05 1.05T6-E-W-L2000 54.1 1.36 1.36 1.85 0.91 1.12 1.12T6-E-W-L2700 51.0 1.56 1.56 1.83 0.96 1.19 1.17T6-E-W-L3500 42.2 1.53 1.53 1.65 0.96 1.18 1.15Mean, P m 1.34 1.34 1.79 0.90 1.11 1.10COV, V P 0.205 0.205 0.080 0.072 0.071 0.059Reliability index, b 2.99 2.84 4.91 2.28 3.13 3.21T6-F-W-L500 179.3 1.29 1.44 1.65 0.79 1.13 1.13T6-F-W-L1200 175.4 1.26 1.41 1.69 0.81 1.15 1.16T6-F-W-L2000 159.5 1.15 1.28 1.66 0.82 1.16 1.15T6-F-W-L2700 142.8 1.03 1.15 1.65 0.84 1.18 1.17T6-F-W-L3500 122.9 0.89 0.99 1.74 0.88 1.24 1.20Mean, P m 1.12 1.26 1.68 0.83 1.17 1.16COV, V P 0.147 0.150 0.023 0.042 0.037 0.022Reliability index, b 2.48 2.65 4.80 2.09 3.62 3.66

Note : 1kip 4.45kN.


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5.2. Direct strength method for cold-formed steel structures

The direct strength method, which was developed forcold-formed carbon steel members, is based on the sameunderlying empirical assumption as the effective widthmethod: ultimate strength is a function of elastic bucklingand yielding of the material [7]. The direct strength methodhas been proposed by Schafer and Peko z [4] for laterallybraced exural members undergoing local or distortionalbuckling. Subsequently, the method has been developed forconcentrically loaded pin-ended cold-formed steel columnsundergoing local, distortional, or overall buckling [5,6],which allows for interaction of local and overall bucklingas well as distortional buckling. The SHS and RHS areinvestigated in this study, and distortional buckling does

not occur on these sections. Therefore, distortionalbuckling is not considered in this study. As summarizedin the NAS [8,9] for cold-formed steel structures, thecolumn design rules of the direct strength method thatconsidered the local and overall exural buckling areshown in Eqs. (1)(3). The values of 0.15 and 0.4 are thecoefcient and exponent of the direct strength equation,respectively, that were calibrated against test data of concentrically loaded pin-ended cold-formed steel columnsfor certain cross-sections and geometric limits:

P DSM min P ne ; P nl , (1)

P ne 0:658 l

2c

P y for l cp 1:5;0:877l 2c P y for l c4 1:5;8>:

(2)

P nl P ne for l l p 0:776 ;

1 0:15 P crlP ne 0:4 P crlP ne

0:4P ne for l l 4 0:776 ;

8>:

(3)

where P y f y A ; l c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP y=P crep ; l l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP ne =P crlp .A is the gross cross-section area, f y is the material yieldstrength which is the static 0.2% proof stress ( s 0.2 ) usingthe non-welded material properties in this paper, P cre is thep2EA /( l e/r )2, critical elastic buckling load in exural

buckling for SHS and RHS columns, P crl is the Criticalelastic local column buckling load, E is the Youngsmodulus, l e is the column effective length, r is the radius of gyration of gross cross-section about the minor y-axis of buckling.

The nominal axial strengths ( P DSM ) are calculated for

the two cases, as shown in Eqs. (2) and (3), respectively,where P ne refers to the nominal axial strength for exuralbuckling, and P nl refers to the nominal axial strength forlocal buckling as well as interaction of local and overallbuckling. The nominal axial strength, P DSM , is theminimum of P ne and P nl , as shown in Eq. (1). In calculatingthe axial strengths, the critical elastic local buckling load(P crl ) of the cross-section was obtained from a rationalelastic nite strip buckling analysis [15].

5.3. Proposed design rules for aluminum alloy non-welded columns

It should be noted that the direct strength method wasdeveloped based on open sections, such as simple lippedchannel, lipped channel with web stiffeners, Zed section,hat section and rack upright section. In this study, squareand RHSs are investigated. Therefore, the direct strengthmethod for cold-formed carbon steel members wasmodied for aluminum alloy columns. The proposeddesign equations for aluminum alloy SHS and RHScolumns without transverse welds at the ends of thecolumns (non-welded columns) are as follows:

P DSM NW min P ne ; P nl , (4)

P ne 0:658 l

2c P y nw for l cp 1:5;

0:877l 2c P y nw for l c4 1:5;

8>:

(5)


1 0:15 P crlP ne 0:3 P cr l P ne

0:3P ne for l l 4 0:713 ;

8>:

(6)

where P y-nw f y-nw A; l c ffiffiffiffiffiffiP y nw =P crep ; and l l ffiffiffiffiffiffiffiffiffiffiffiffiP ne =P crlp ; and f y-nw is the non-welded material yieldstrength.The design equations were veried against the numerical

results obtained from the parametric study as presented inthis paper, and the test results reported by Zhu and Young[12]. The proposed design Eqs. (4)(6) for aluminum non-welded columns require only small modications to thecurrent direct strength method for cold-formed steelmembers. In Eq. (3), the value of the exponent 0.4 wasmodied to 0.3, and the non-dimensional slenderness ( l l)has been adjusted to 0.713 for a smooth transition of theelastic and inelastic buckling loads as shown in Eq. (6). Asa result, the reliability index ( b0) of 2.86 was obtained forthe proposed design rules, which is closer to the targetvalue of 2.5 compared with the reliability index ( b0) of 3.07

ARTICLE IN PRESS

B

t H

Fig. 1. Denition of symbols.


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for the current direct strength method, as shown in Fig. 2 .The load combination of 1.2 DL (dead load)+1.6 LL (liveload) was used in calculating the reliability index asspecied in the AA Specication [1]. The resistance factor(f ) of 0.85 was used in the calculation. The reliabilityanalysis is detailed in Section 6 of this paper. Fig. 2 showsthe comparison of FEA and experimental results againstthe current and modied direct strength curves plottedfrom Eqs. (3) and (6), respectively, for the non-weldedcolumns. Generally, the results predicted using the currentand modied direct strength methods compared reasonablywell with the FEA and experimental results. However, themodied direct strength method provided a reliabilityindex closer to the target value compared with the current

direct strength method.

5.4. Proposed design rules for aluminum alloy welded columns

Two design approaches were proposed for aluminumcolumns with transverse welds at both ends of the columns(welded columns). The two design approaches were alsomodied from the current direct strength method as well asthe current nominal axial strength ( P ne ) equations forexural buckling. The proposed design rules were cali-brated with the welded column strengths obtained from the

parametric study presented in this paper, as well as the testresults reported by Zhu and Young [12]. The rst approachadopts the non-welded material properties in calculating thewelded column strength. The proposed design equationsare shown in Eqs. (7)(9). It should be noted that the P creand P crl are obtained based on the non-welded materialproperties:

P DSM W1 min P ne ; P nl , (7)

P ne 0:65 0:7l

2c P y nw for l cp 1:56;

0:664

l2c

P y nw for l c4 1:56;

8>:

(8)


0:88 1 0:15 P crlP ne 0:3 P crlP ne

0:3P ne for l l 4 0:535 ;8>: (9)where P y-nw f y-nw A; l c ffiffiffiffiffiP y nw =P crep ; and l l ffiffiffiffiffiffiffiffiffiffiffiP ne =P crlp ; and f y-nw is the non-welded material yieldstrength.

Figs. 3(a) and (b) show the comparison of FEA andexperimental results against the results predicted by theproposed design rules for exural buckling and interactionof local and exural buckling, respectively. The proposeddesign Eqs. (7)(9) for aluminum welded columns weremodied based on the proposed design Eqs. (4)(6) foraluminum non-welded columns. In Eq. (5), the value of thecoefcients 0.658 and 0.877 were modied to 0.7 and 0.664,respectively, as shown in Eq. (8). In addition, a coefcientof 0.65 was multiplied to one of the equations, and the non-dimensional slenderness ( l c) has been adjusted to 1.56 for asmooth transition of the elastic and inelastic bucklingloads. In Eq. (6), a coefcient of 0.88 was multiplied to oneof the equations, as shown in Eq. (9). Similarly, the non-dimensional slenderness ( l l) has been adjusted to 0.535 fora smooth transition of the elastic and inelastic bucklingloads.

ARTICLE IN PRESS

0

0.5

1

1.5

0 1 2 2.5 3 3.5 4

l

P u

/ P n e

Eqn. (3) DSM

Eqn. (6) Modified DSM

Non-Welded FEA data

Non-Welded experimental data

0.5 1.5

( = 0.85; = 2.86)

( = 0.85; = 3.07)

Fig. 2. Comparison of FEA and experimental data with direct strengthcurves for non-welded columns.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 3 4

c

P u

/ P

y - n w

Eqn. (8)Welded FEA dataWelded experimental data

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 4

l

P u

/ P

n e


2.5 3.5

3.5

(a)

(b)

Fig. 3. Comparison of FEA and experimental data with proposed designrules ( P DSM-W1 ) using non-welded material properties for welded columns:(a) exural buckling and (b) interaction of local and exural buckling.


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The second approach adopts the welded material proper-ties in calculating the welded column strength. Theproposed design equations are shown in Eqs. (10)(12),where the P cre and P crl are obtained based on the weldedmaterial properties:

P DSM W2 min P ne ; P nl , (10)

P ne 2 0:4l

2c P y w for l cp 1:0;

0:8l 2c P y w for l c4 1:0;

8>:

(11)

P nl P ne for l lp 0:445 ;

0:81 1 0:15 P crlP ne 0:3 P crlP ne

0:3P ne for l l4 0:445 ;

8>: (12)

where P y-w f y-w A ; l c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP y w=P crep ; l l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP ne =P crlp , and f y-w is the welded material yield strength.The results obtained from the proposed design equations

were compared with the FEA and experimental results of welded columns, as shown in Figs. 4(a) and (b) . Theproposed design Eqs. (10)(12) for aluminum weldedcolumns were also modied based on the proposeddesign Eqs. (4)(6) for aluminum non-welded columns.However, the welded material properties were used in these

equations. In Eq. (5), the value of the coefcients 0.658 and0.877 were modied to 0.4 and 0.8, respectively, and acoefcient of 2 was multiplied to one of the equations, asshown in Eq. (11). The non-dimensional slenderness ( l c)has been adjusted to 1.0. In Eq. (6), a coefcient of 0.81was multiplied to one of the equations, and the non-

dimensional slenderness ( l l) has been adjusted to 0.445, asshown in Eq. (12).

6. Reliability analysis

The reliability of the design rules for aluminum columnsis evaluated using reliability analysis. Reliability analysis isdetailed in the AA Specication [1], and the ratio of dead(DL) to live (LL) load of 0.2 was used in the analysis. Ingeneral, a target reliability index of 2.5 for aluminum alloycolumns as a lower limit is recommended by the AASpecication [1]. If the reliability index is greater than orequal to 2.5 ( bX 2:5), then the design is considered to bereliable. The AA and AS/NZS specications providedifferent resistance factors ( f ) for compression memberswith different failure modes. The resistance factor varieswith the slenderness parameter for exural buckling failuremode. The resistance factor is a constant and equal to 0.85for local buckling or interaction of local and overallbuckling failure mode. The observed failure modes of thecolumns in this study included local buckling, exuralbuckling, interaction of local and overall buckling, andfailure in the HAZ. Hence, the resistance factor of thecolumns given by the AA and AS/NZS specicationsranged from 0.73 to 0.95. In calculating the reliability

indices of the AA and AS/NZS design rules, the resistancefactor for the columns is chosen to be equal to 0.85 for allfailure modes in this study. The EC9 Code provides aconstant resistance factor of 1/1.1 0.91 for compressionmembers, which is used in the reliability analysis. Thereliability of the direct strength method and proposeddesign rules for aluminum columns is also evaluated, andthe resistance factor is equal to 0.85.

The load combination of 1.2 DL+1.6 LL is used in theanalysis for the AA Specication, the direct strengthmethod and the proposed design rules. The load combina-tions of 1.25 DL+1.5 LL and 1.35 DL+1.5 LL are used inthe analysis for AS/NZS and EC9 specications, respec-tively. The statistical parameters M m , F m , V M , and V F arethe mean values and coefcients of variation (COV) of material and fabrication factors. These values are obtainedfrom Section 9 of Part I-B of the AA Specication [1],where M m 1.10, F m 1.00, V M 0.06, and V F 0.05.The statistical parameters P m and V P are the mean valueand the COV of test-to-predicted load ratios, respectively.

7. Comparison of numerical and experimental results withdesign predictions

The nominal axial strengths (unfactored design strengths)predicted by the AA Specication ( P AA ), AS/NZS Standard

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0

1

2

3

4

0 1 2

c

P u

/ P

y - w


0

0.5

1

1.5

0 1 2 3 4

l

P u

/ P

n e

Eqn. (12)Welded FEA data

Welded experimental data

0.5 1.5 2.5 3.5

1.50.5

(a)

(b)

Fig. 4. Comparison of FEA and experimental data with proposed designrules ( P DSM-W2 ) using welded material properties for welded columns: (a)exural buckling and (b) interaction of local and exural buckling.


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(P AS/NZS ), and European Code ( P EC9 ) for aluminumstructures, as well as the direct strength method ( P DSM )and proposed design equations ( P DSM-NW , P DSM-W1, P DSM-W2 ) are compared with the column strengths obtained fromthe parametric study ( P FEA ) and experimental program(P Exp ) [12], as shown in Tables 27 . The statistical

parameters P m and V P which are the mean value andCOV of FEA and experimental-to-predicted load ratios of each series of specimens are shown in Tables 27 .Reliability indices ( b) of the design rules for each seriesof specimens are also shown in Tables 27 . The FEAresults are also compared with the column design curvesobtained from the design rules, as shown in Figs. 528 .

The design strengths are calculated using the materialproperties for each series of specimens, as reported by Zhuand Young [1213], and the 0.2% proof stress ( s 0U2) wasused as the corresponding yield stress. The design strengthsof non-welded column specimens are calculated using thenon-welded material properties. In calculating the designstrengths of welded columns, P AA and P AS/NZS arecalculated using the welded material properties, as speciedin the AA and AS/NZS specications. Whereas P EC9 arecalculated using the non-welded material properties asrequired by the EC9 Code. The design strengths of theproposed design equations P DSM-W1 are calculated usingthe non-welded material properties, while P DSM-W2 arecalculated using the welded material properties, as detailedin Section 5.4 Proposed design rules for aluminum alloywelded columns of the paper. The column specimens weredesigned as concentrically loaded compression members,and the effective length ( l e) was taken as one-half of the

column length ( L ).For the non-welded columns, it is shown that the column

strengths predicted by the AA and AS/NZS specicationsare quite close. It should be noted that the AA and AS/NZS predictions are unconservative for some specimens, asshown in Tables 2 and 3 . Reliability index as low as 1.98

and 1.77 were obtained for column Series T6-D-NW, withthe corresponding load ratios P FEA /P AA and P FEA /P AS/NZS of 0.88 and 0.91, respectively. However, the AA andAS/NZS predictions are generally conservative or lessunconservative for short specimens. Table 6 indicates theAA and AS/NZS specications accurately predicted the

column strengths obtained from the experimental programwith column length ranged from 300 to 1650 mm. Thedesign strengths calculated using the EC9 Code aregenerally conservative for the non-welded columns. How-ever, the design strengths calculated using the EC9Code are unconservative for a few specimens with longcolumn lengths, such as specimens T6-C-NW-L3500 andT6-E-NW-L3500 with the P FEA /P EC9 load ratio of 0.87 and 0.84, respectively. The reliability index for thedesign rules of the EC9 Code are mostly less than the targetvalue of 2.5, where seven series among twelve series are lessthan the target value, as shown in Tables 2 and 3 . This isdue to the large resistance factor of 0.91 has been used inthe EC9 Code. The design strengths P DSM predicted by thecurrent direct strength method are generally accurate andreliable for all the non-welded column series, as shownin Tables 2, 3 and 6 . The mean values of the load ratioP FEA /P DSM for each series of FEA specimens ranged from0.96 to 1.20, with the corresponding COV ranged from0.010 to 0.107. In terms of the experimental program, themean value of the load ratio P Exp /P DSM is 1.10, with thecorresponding COV of 0.063, as shown in Table 6 . It isshown that the current direct strength method could besuccessfully used in the design of aluminum non-weldedcolumns of square and RHSs. However, for all the non-

welded column series of FEA and test specimens, thereliability indices of the current direct strength method aregreater than the target value of 2.5, which ranged from 2.73to 3.46. Thus, the design equations shown in Section 5.3Proposed design rules for aluminum alloy non-weldedcolumns of this paper have been proposed that require

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Table 6Comparison of test strengths with design strengths for non-welded columns

Specimen Experimental Comparison

P Exp (kN) P Exp /P AA P Exp /P AS/NZS P Exp /P EC9 P Exp /P DSM P Exp /P DSM-NW

N-S1-L300 34.1 1.08 1.08 1.13 1.09 1.09N-S1-L1000 33.7 1.06 1.06 1.20 1.14 1.14N-S1-L1650 33.6 1.08 1.08 1.34 1.26 1.28N-R1-L300 42.3 1.08 1.08 1.02 1.05 0.94N-R1-L1000 41.7 1.03 1.03 1.07 1.09 0.98N-R2-L300 147.9 1.03 1.10 1.08 0.99 1.03N-R2-L1000 145.8 1.01 1.08 1.14 1.04 1.08H-R1-L300 53.3 1.08 1.08 1.06 1.09 0.95H-R1-L1000 51.6 1.06 1.06 1.10 1.12 0.99H-R2-L300 209.2 1.06 1.14 1.14 1.10 1.10H-R2-L1000 202.4 1.04 1.12 1.21 1.15 1.16Mean, P m 1.06 1.08 1.14 1.10 1.07COV, V P 0.023 0.027 0.078 0.063 0.096Reliability index, b 3.24 3.13 2.88 3.30 3.00

Note : 1kip 4.45kN.


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Table 7Comparison of test strengths with design strengths for welded columns

Specimen Experimental Comparison

P Exp (kN) P Exp /P AA P Exp /P AS/NZS P Exp /P EC9 P Exp /P DSM P Exp /P DSM-W1 P Exp /P DSM-W2

N-S1-W-L300 18.8 1.45 1.62 1.15 0.60 0.99 0.97N-S1-W-L1000 19.2 1.47 1.65 1.24 0.62 1.01 0.99N-S1-W-L1650 19.8 1.55 1.73 1.45 0.66 1.16 1.12N-S1-W-L2350 18.4 1.41 1.59 1.59 0.65 1.27 1.22N-S1-W-L3000 15.2 1.16 1.30 1.72 0.57 1.28 1.23Mean, P m 1.41 1.58 1.43 0.62 1.14 1.10COV, V P 0.105 0.104 0.166 0.059 0.121 0.112Reliability index, b 3.69 3.93 2.73 0.83 2.77 2.74N-R1-W-L300 26.4 0.88 0.95 1.16 0.65 0.92 0.93N-R1-W-L1000 27.7 0.94 1.01 1.30 0.70 1.01 1.02N-R1-W-L1650 28.5 0.94 1.02 1.38 0.74 1.14 1.13N-R1-W-L2350 25.1 0.92 0.92 1.39 0.68 1.16 1.14N-R1-W-L3000 23.2 0.93 0.93 1.45 0.67 1.28 1.25Mean, P m 0.92 0.96 1.33 0.69 1.10 1.09COV, V P 0.025 0.047 0.082 0.050 0.127 0.113Reliability index, b 2.62 2.51 3.32 1.28 2.59 2.70N-R2-W-L300 101.0 1.77 1.98 1.29 0.68 1.07 1.05N-R2-W-L1000 89.7 1.61 1.76 1.23 0.61 1.04 1.01N-R2-W-L1650 85.4 1.53 1.68 1.30 0.60 1.09 1.06N-R2-W-L2350 74.3 1.35 1.47 1.45 0.57 1.13 1.08N-R2-W-L3000 60.4 1.09 1.19 1.60 0.51 1.19 1.07Mean, P m 1.47 1.62 1.37 0.59 1.10 1.05COV, V P 0.177 0.185 0.110 0.104 0.053 0.026Reliability index, b 3.00 3.04 3.16 0.55 3.26 3.22H-R1-W-L300 37.5 1.02 1.03 1.65 0.76 1.04 1.29H-R1-W-L1000 37.9 1.04 1.04 1.76 0.79 1.12 1.37H-R1-W-L1650 37.7 1.04 1.04 1.87 0.81 1.26 1.48H-R1-W-L2350 30.3 1.06 1.06 1.72 0.69 1.24 1.37H-R1-W-L3000 23.8 1.00 1.00 1.68 1.12 1.24 1.27Mean, P m 1.03 1.03 1.74 0.83 1.18 1.36COV, V P 0.022 0.021 0.049 0.199 0.081 0.061Reliability index, b 3.13 2.92 4.74 1.31 3.30 4.06H-R2-W-L300 118.0 1.48 1.66 1.38 0.62 1.00 1.30H-R2-W-L1000 139.3 1.80 1.96 1.76 0.74 1.25 1.58H-R2-W-L1650 119.4 1.54 1.68 1.75 0.67 1.23 1.49H-R2-W-L2350 95.2 1.23 1.33 1.94 0.57 1.23 1.38H-R2-W-L3000 75.4 0.98 1.06 2.22 1.39 1.38 1.32Mean, P m 1.41 1.54 1.81 0.80 1.22 1.41COV, V P 0.222 0.226 0.169 0.422 0.112 0.084Reliability index, b 2.48 2.54 3.37 0.65 3.07 3.95


0

20

40

60

80

0 1500 2000 2500 3000

C o l u m n s t r e n g t

h , P

u ( k N )

Non-welded FEA

P DSM

Flexural bucklingP AA P AS/NZS

Effective length, l e (mm)

500 1000

P EC9

P DSM-NW

Fig. 5. Comparison of FEA and design column strengths for SeriesT5-A-NW.

0

20

40

60

80

100

120

140160

180

0 500 1000 1500 2000 2500 3000

C o

l u m n s t r e n g t

h , P

u ( k N )

Non-welded FEA

P EC9

P DSM

P DSM-NW

Flexural buckling

PAA P AS/NZS


Fig. 6. Comparison of FEA and design column strengths for SeriesT5-B-NW.


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only small modications to the current direct strengthmethod and obtaining closer reliability index to thetarget value. The load ratios of the FEA ( P FEA ) andexperimental ( P Exp ) column strengths to the designstrengths ( P DSM-NW ) calculated using the proposed designequations for non-welded columns are shown in Tables 2, 3and 6 . The mean values of load ratios P FEA /P DSM-NW andP Exp /P DSM-NW ranged from 0.93 to 1.13, with thecorresponding COV ranged from 0.012 to 0.111 for all

the non-welded column series. It should be noted that thereliability indices of the proposed design rules for eachcolumn series are generally closer to the target value of 2.5compared with the current direct strength method, whichranged from 2.61 to 3.09. The non-welded column designcurves predicted by the AA, AS/NZS and EC9 specica-tions, as well as the current and modied direct strengthmethods are shown in Figs. 516 for each non-weldedcolumn series.

ARTICLE IN PRESS

0

20

40

60

80

0 500 1000 2000 3000

C o l u m n

s t r e n g

t h , P u

( k N ) Non-welded FEA

P EC9

Flexural buckling

P AA P AS/NZS

Effective length, le (mm)1500 2500

P DSM-NW

P DSM

Fig. 7. Comparison of FEA and design column strengths for SeriesT5-C-NW.

0

40

80

120

160

200

240

0 500 1000 1500 2000 2500 3000

C o


h , P

u ( k N )

Non-welded FEA

Flexural buckling

P AA P AS/NZS


P EC9

P DSM

P DSM-NW

Fig. 8. Comparison of FEA and design column strengths for SeriesT5-D-NW.

0

20

40

60

80

0 500 1000 1500 2000 2500 3000

C o l u m n s t r e n g

t h , P u

( k N )

Non-welded FEA

P DSM

Flexural buckling

P AA P AS/NZS

P EC9

P DSM-NW

Effective length,l

e (mm)

Fig. 9. Comparison of FEA and design column strengths for SeriesT5-E-NW.

0

50

100

150

200

250

0 500 1000 2000 2500 3000


t h , P u

( k N )

Non-welded FEA

P DSM

Flexural buckling

P AA P AS/NZS

P EC9

P DSM-NW

Effective length, l e (mm)1500

Fig. 10. Comparison of FEA and design column strengths for SeriesT5-F-NW.

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000


t h , P u

( k N )

Non-welded FEA

P DSM-NW

Flexural bucklingP AA P AS/NZS

P DSM

P EC9


Fig. 11. Comparison of FEA and design column strengths for SeriesT6-A-NW.

0

50

100

150

200

250

0 500 1000 1500 2000 2500 3000


t h , P u


P DSM-NW

Flexural buckling

P AAP AS/NZS

P DSM

P EC9

Effective length,l

e (mm)

Fig. 12. Comparison of FEA and design column strengths for SeriesT6-B-NW.


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For the welded columns, it is shown that the designstrengths calculated using the AA specication ( P AA ) aregenerally quite conservative, as shown in Tables 4, 5 and 7 .For the column series of FEA specimens, the mean valuesof the load ratio P FEA /P AA ranged from 0.93 to 1.74, withthe corresponding COV ranged from 0.044 to 0.258, andthe reliability index ranged from 2.37 to 3.43. For the

column series of test specimens, the mean values of the loadratio P Exp /P AA ranged from 0.92 to 1.47, with thecorresponding COV ranged from 0.022 to 0.222, and thereliability index ranged from 2.48 to 3.69. The designstrengths calculated using the AS/NZS Standard aregenerally more conservative than the predictions givenby the AA Specication, as shown in Tables 4, 5 and 7 .

ARTICLE IN PRESS

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000

C o l u m n

s t r e n g

t h , P u


P EC9

P DSM

P DSM-NW

Flexural buckling

P AA P AS/NZS


Fig. 13. Comparison of FEA and design column strengths for SeriesT6-C-NW.

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500 3000


h , P

u ( k N ) Non-welded FEA

P EC9

P DSM

P DSM-NW

Flexural buckling

P AA P AS/NZS


Fig. 14. Comparison of FEA and design column strengths for SeriesT6-D-NW.

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000


t h , P u

( k N )

Non-welded FEA

P DSM

P EC9

P DSM-NW

Flexural buckling

P AA P AS/NZS


Fig. 15. Comparison of FEA and design column strengths for SeriesT6-E-NW.

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500 3000


h , P

u ( k N )

Non-welded FEA

P DSM

P EC9

P DSM-NW

Flexural buckling

P AA P AS/NZS


Fig. 16. Comparison of FEA and design column strengths for SeriesT6-F-NW.

0

20

40

60

0 1000 1500 2000 2500 3000


t h , P u

( k N )

Welded FEA

Flexural buckling

P DSM

P DSM-W2

P DSM-W1

P EC9

P AA P AS/NZS

Effective length, l e (mm)500

Fig. 17. Comparison of FEA and design column strengths for SeriesT5-A-W.

0

30

60

90

120

150

0 1000 1500 2000 2500 3000


t h , P u

( k N ) Welded FEA

Flexural buckling

P DSM

P DSM-W2

P DSM-W1

P EC9P AA P AS/NZS

500Effective length, l e (mm)

Fig. 18. Comparison of FEA and design column strengths for SeriesT5-B-W.


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The design strength calculated using the EC9 Code forwelded columns are also quite conservative. The meanvalues of the load ratio P FEA /P EC9 ranged from 1.23 to1.79 for each column series of FEA specimens, and themean values of the load ratio P Exp /P EC9 ranged from 1.33to 1.81 for each column series of test specimens. Thecorresponding COV for the load ratios P FEA /P EC9 andP Exp /P EC9 ranged from 0.013 to 0.169, and the reliability

index ranged from 2.73 to 4.91. It is also shown that thecurrent direct strength method is not suitable for the designof aluminum welded columns. The mean values of the loadratio P FEA /P DSM ranged from 0.64 to 0.90, with thecorresponding COV ranged from 0.040 to 0.114, and thereliability index ranged from 0.89 to 2.28 for each columnseries of FEA specimens, as shown in Tables 4 and 5 .The mean values of the load ratio P Exp /P DSM ranged from

ARTICLE IN PRESS

0

20

40

60

80

0 500 1000 1500 2000 2500 3000

C o

l u m n s t r e n g

t h , P u

( k N )

Welded FEA

Flexural bucklingP DSM

P DSM-W2

P DSM-W1

P EC9

P AA

Effective length, l e ( mm)

P AS/NZS

Fig. 19. Comparison of FEA and design column strengths for SeriesT5-C-W.

0

40

80

120

160

200

0 500 1000 1500 2000 2500 3000

C o

l u m n s t r e n g

t h , P u

( k N ) Welded FEA

Flexural buckling

P DSM

P DSM-W2

PDSM-W1

P EC9P AA P AS/NZS


Fig. 20. Comparison of FEA and design column strengths for SeriesT5-D-W.

0

20

40

60

0 1000 1500 2000 2500 3000

C o

l u m n s t r e n g

t h , P u

( k N )

Welded FEAFlexural buckling

P DSMP DSM-W2

P DSM-W1P EC9

P AA P AS/NZS

500


Fig. 21. Comparison of FEA and design column strengths for SeriesT5-E-W.

0

50

100

150

200

0 1000 1500 2000 2500 3000


t h , P u

( k N ) Welded FEA

Flexural buckling

P DSM

P DSM-W1P DSM-W2

P EC9

P AA P AS/NZS


Fig. 22. Comparison of FEA and design column strengths for SeriesT5-F-W.

0

20

40

60

80

0 500 1000 1500 2000 2500 3000


t h , P u (

k N )


P DSM

P DSM-W2

P DSM-W1

P EC9

P AA P AS/NZS


Fig. 23. Comparison of FEA and design column strengths for SeriesT6-A-W.

0

50

100

150

200

0 500 1000 1500 2000 2500 3000


t h , P u

( k N )


P DSM

P DSM-W2

P DSM-W1

P EC9

P AA P AS/NZS

Effective length, le (mm)

Fig. 24. Comparison of FEA and design column strengths for SeriesT6-B-W.


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0.59 to 0.83, with the corresponding COV ranged from0.050 to 0.422, and the reliability index ranged from 0.55 to1.31 for each column series of test specimens, as shown inTable 7 .

Two design approaches have been proposed for alumi-num columns with transverse welds at both ends of thecolumns (welded columns). The rst approach ( P DSM-W1 )adopts the non-welded material properties in calculating thewelded column strength as detailed in Eqs. (7)(9), whereas

the second approach ( P DSM-W2 ) adopts the welded materialproperties in the calculation, as described in Eqs. (10)(12).The design strengths ( P DSM-W1 and P DSM-W2 ) calculatedusing the two proposed design rules for welded columns arein good agreement with the FEA and experimental results,as shown in Tables 4, 5 and 7 . For the rst design approachthat uses the non-welded material properties, the meanvalues of the load ratios P FEA /P DSM-W1 and P Exp /P DSM-W1ranged from 0.99 to 1.23, with the corresponding COVranged from 0.029 to 0.127. The reliability indices for thisdesign rules ( P DSM-W1 ) are greater than the target value of 2.5 that ranged from 2.51 to 3.66 for each series of specimens. Fig. 3(a) shows the comparison of the weldedcolumn results that failed by exural buckling against theresults obtained from the proposed Eq. (8), whereas

Fig. 3(b) shows the comparison of all the welded columnresults against the results obtained from the proposedEq. (9). The design strengths calculated using the seconddesign approach ( P DSM-W2 ) that uses the welded materialproperties are generally conservative, as shown in Tables 4,5 and 7 . The mean values of the load ratio P FEA /P DSM-W2ranged from 0.96 to 1.16, with the corresponding COVranged from 0.022 to 0.091. The mean values of the loadratio P Exp /P DSM-W2 ranged from 1.05 to 1.41, withthe corresponding COV ranged from 0.026 to 0.113. Thereliability indices for P DSM-W2 are also greater than thetarget value of 2.5 that ranged from 2.54 to 4.06 for allwelded column series. Fig. 4(a) shows the comparison of the welded column results that failed by exural bucklingwith the results obtained from the proposed Eq. (11).Fig. 4(b) shows the comparison of all the welded columnresults against the results obtained from the proposedEq. (12). The welded column design curves predicted by theAA, AS/NZS and EC9 specications, as well as the currentdirect strength method and the two proposed design rulesare shown in Figs. 1728 for each welded column series. Itis shown that the two proposed design rules can be used forthe design of aluminum columns of square and RHSs withtransverse welds at the ends of the columns. Generally, it isconsidered that the non-welded material properties couldbe obtained easily compared with the welded material

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0

20

40

60

80

0 1000 1500 2000 2500 3000


t h , P u

( k N )

Welded FEA

Flexural buckling

P DSM

P DSM-W2

P DSM-W1P EC9

P AA P AS/NZS


Fig. 25. Comparison of FEA and design column strengths for SeriesT6-C-W.

0

50

100

150

200

250

0 500 1000 1500 2000 2500 3000


t h , P u

( k N ) Welded FEAFlexural buckling

P DSM

P DSM-W2

P DSM-W1

P EC9

P AA P AS/NZS

Effective length, le (mm)

Fig. 26. Comparison of FEA and design column strengths for SeriesT6-D-W.

0

20

40

60

80

0 1000 1500 2000 2500 3000

C o


h , P

u ( k N )

Welded FEA

Flexural buckling

P DSM

P DSM-W2P DSM-W1

P EC9

P AA P AS/NZS

500

Effective length,l

e (mm)

Fig. 27. Comparison of FEA and design column strengths for SeriesT6-E-W.

0

50

100

150

200

250

0 500 1000 1500 2000 2500 3000

C o

l u m n

s t r e n g t

h , P u

( k N ) Welded FEA

Flexural buckling

P DSM

P DSM-W2

P DSM-W1P EC9

P AA P AS/NZS


Fig. 28. Comparison of FEA and design column strengths for SeriesT6-F-W.


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properties. The designer may choose either one of theproposed design rules for the welded columns based on theavailable material properties.

8. Conclusions

This paper presents a parametric study of aluminumalloy square and RHS columns using FEA. A non-linearFEM was used in the parametric study that has beenveried against experimental results. The research pro-gram contained aluminum columns of 6063-T5 and 6061-T6 heat-treated aluminum alloys with and withouttransverse welds at both ends of the columns. Theparametric study included 24 column series with differentcross-section geometry and type of aluminum alloy. Eachseries contained 5 specimens with the column lengthsranged from 500 to 3500mm. The column strengthsobtained from experimental and numerical investigations

were compared with the design strengths calculated usingthe current American, Australian/New Zealand andEuropean specications for aluminum structures. Thecolumn strengths were also compared with the designstrengths calculated using the direct strength method thatwas developed for cold-formed steel members. Based onthe available data obtained from experimental andnumerical investigations, design rules modied from thedirect strength method were proposed for aluminumcolumns with and without transverse welds at the endsof the columns. Reliability analysis was performed toevaluate the reliability of the design rules. It is shown thatthe design strengths calculated using the proposed designrules are in good agreement with the test and numericalresults. It is also shown that the proposed design rules arereliable.

References

[1] AA. Aluminum design manual. Washington, DC: The AluminumAssociation; 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 structuresPart 1-1: Generalrulesgeneral 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 steelmembers using numerical elastic buckling solutions. In: Proceedingsof the 14th international specialty conference on cold-formed steelstructures, University of Missouri-Rolla, Rolla, Mo 1998. p. 6976.

[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-walledcolumns. J Struct Eng 2002;128(3):28999.

[7] Schafer BW. Progress on the direct strength method. In: Proceedingof 16th international specialty conference on cold-formed steel

structures, Orlando, FL, 2002. p. 64762.[8] North American Specication for the design of cold-formed steel

structural members. Washington, DC: American Iron and SteelInstitute, 2001.

[9] Supplement to the North American Specication for the design of cold-formed steel structural member. Washington, DC: AmericanIron and Steel Institute; 2004.

[10] Mazzolani FM. Aluminum alloy structures. 2nd ed. London: E & FNSpon; 1995.

[11] Sharp ML. Behaviour and design of aluminum structures. New York:McGraw-Hill; 1993.

[12] Zhu JH, Young B. Tests and design of aluminum alloy compressionmembers. J Struct Eng 2006;132(7):1096107.

[13] Zhu JH, Young B. Aluminum alloy tubular columnsPart I: Finiteelement modeling and test verication. Thin-Walled Struct 2006, in

press, doi:10.1016/j.tws.2006.08.011 .[14] ABAQUS Analysis Users Manual, Version 6.5. ABAQUS, Inc., 2004.[15] Papangelis JP, Hancock GJ. Computer analysis of thin-walled

structural members. Comput Struct 1995;56(1):15776.

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