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Buckling behaviorThin-walled structuresDirect strength method

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tures and the Direct Strength Method (DSM) on thin-walled structures. Following reliability analysis, the

tio, betthrougin str

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buckling behaviors of re exposed aluminum alloy columns,

design of aluminum alloy columns design considering the stressstrain relationship of aluminum alloys at elevated temperatures.Manganiello et al. [7] evaluated the inelasticexural behavior of alu-minum alloy structures through numerical method and proposed amethod for the ultimate strength of the rotational capacity of across-section in bending. Maljaars et al. [8] studied local bucklingof compressed aluminum alloy at elevated temperatures through

accuracy of the design rules in the current specications. Theseuckling behaviors

gn codes fminum alloy structural members, such as EN1999-1-2 (ECAmerican Aluminum Design Manual (AA) [15], AustraliaZealand Standard [16], and Chinese Design SpecicatioAluminum Structures (GB50429) [17]. To make full use of struc-tural material, the cross-section of an aluminum alloy member isusually made up of thin-walled plates. These design codes followthe element approach to calculate the buckling strength of eachthin-walled element considering effects of local bucking. In designof thin-walled steel structures, the effective section is usuallydetermined through the effective width method [18]. While the

Corresponding author. Fax: +86 531 88392843.E-mail address: pjwang@sdu.edu.cn (P. Wang).

Engineering Structures 95 (2015) 127137

Contents lists availab

g

lseMaljaars et al. [5] found that EN1999-1-2 [6] didnot give anaccurateprediction for exural buckling strength of re exposed aluminumcolumns. A new design method was proposed for the re resistance

researches greatly advanced the mechanism of bof extruded aluminum alloy columns.

Many countries have already published desihttp://dx.doi.org/10.1016/j.engstruct.2015.03.0640141-0296/ 2015 Elsevier Ltd. All rights reserved.or alu-9) [6],n/Newns foret al. [3] quantied the response and failure of 5083-H116 and6082-T6 aluminumplates under compression loadwhile being sub-jected to a re. Rasmussen and Rondal [4] proposed a column curveto predict the strengths of the extruded aluminum alloy columnfailed at exural buckling. Based on the FEM parametric studies on

deformation based continuous method gave more accurate predic-tion for the ultimate strength. Wang et al. [13] carried out tests onthe columns of 6082-T6 circular tubes. Zhu and Young [14] pre-sented tests results of aluminum alloy circular hollow section col-umns with and without transverse welds and assessed the1. Introduction

For its high strength-to-weight raand exural manufacture procedurealloy members are being widely usedSummers et al. [2] performed a seriAA5083H116 and AA6061T651 aand developed empirical laws fordesign strength predicted by current design specications were found to be generally conservative,whereas DSM offered more accurate results.

2015 Elsevier Ltd. All rights reserved.

ter corrosion resistanceh extrusion, aluminumuctural applications [1].niaxial tension tests onmulated re exposurel yield strength. Fogle

tests. Adeoti et al. [9] presented a column curve for extruded mem-bers made of 6082-T6 aluminum alloy. Yuan et al. [10] investigatedthe local buckling and postbuckling strengths of aluminum alloy I-section stub columns under axial compression. Their researchresults showed that current design codes were conservative to pre-dict ultimate strength of aluminum alloy columns. Su et al. [11,12]carriedout a series of stub-columntests onbox sections and two ser-ies of experiments on aluminum alloy hollow section beams. TheAluminum alloy columnIrregular shaped cross section

nd the potential buckling failure mode at a given length. Tested ultimate strengths were compared withthose predicted by the current American, European and Chinese specications on aluminum alloy struc-Buckling behaviors of section aluminucompression

Mei Liu, Lulu Zhang, Peijun Wang , Yicun ChangSchool of Civil Engineering, Shandong University, Jinan, Shandong Province 250061, Chi

a r t i c l e i n f o

Article history:Received 30 January 2015Revised 27 March 2015Accepted 28 March 2015Available online 10 April 2015

Keywords:

a b s t r a c t

Thin-walled columns withoffering high strength-to-wpaper, thin-walled alumincally to investigate the bucelement model (FEM) wasEffects of plate thickness o

Engineerin

journal homepage: www.ealloy columns under axial

section are used widely as columns in aluminum alloy framed structures,ht ratios and convenience in connection with maintaining walls. In thisalloy columns with section were studied experimentally and numeri-g behavior and to assess the accuracy of current design methods. A niteeloped and used to perform parametric studies after being veried by tests.lastic buckling stress was studied using nite strip method (FSM) and to

le at ScienceDirect

Structures

vier .com/ locate /engstruct

aluminum alloy members usually have complex shape cross sec-tions, the effective width method appears tedious because it needsiterations for the effective width dependent on stress distributionacross the section. At this kind of circumstance, the effective thick-ness method [19] is more feasible.

Schafer and Pekz [20] developed DSM for predicting the ulti-mate strength of thin-walled steel structural members. The DSMhad been adopted by AISI [21,22] now. The design equations ofDSM was proposed by curve tting the test data and FEA resultson open section thin-walled structural members such as channel,lipped channel with web stiffeners, Z-section, hat section and rackupright section. Unlike the traditional design method uses theeffective section, DSM uses whole section to calculate the ultimatestrength, which provides rational analysis procedure for irregularshaped section and allows section optimization. Zhu and Young[23,24] found that the modied DSM could be used in the designof square hollow section (SHS) and rectangular hollow section(RHS) aluminum alloy columns. Aluminum alloy extruded mem-bers usually have complex sections to include as many functionsas possible. However, the applicability of DSM in aluminum alloymember with irregular shaped section has not been investigatedyet.

This paper presented experimental and numerical investigationon the buckling behaviors of aluminum alloy columns withshape cross-section under axial compression. The structural com-ponent with the studied cross section is usually used as columnsin an aluminum alloy framed structure, as shown in Fig. 1(a). TheFiber Reinforced Plastic (FRP) wall can be easily xed in the chan-nel of the section, as shown in Fig. 1(b).The FSM software CUFSM

[25] was used to illustrate effects of plate thickness on the bucklingstrength and the potential buckling mode at a given length. TheFEA software ABAQUS [26] was used to obtain the ultimatestrength of the member considering effects of initial geometricimperfection and the elasticplastic properties of aluminum alloy.Current design codes were assessed through the comparison of FEAresults with predictions by AA [15], EC9 [6], GB50429 [17] andDSM [21,22], as well as AISI by substituting the material propertiesof steel with those of aluminum alloy.

aluminum alloy columns. (b) Connection of the aluminum alloy column with FRP.

f stress)

0.2% proof stressf 0:2 (MPa)

Ultimate strengthf u (MPa)

Ultimatestrain eu (%)

Parametern

Fig. 2. Stressstrain relationships.

128 M. Liu et al. / Engineering Structures 95 (2015) 127137Fig. 1. Application of aluminum alloy column with section. (a) Layout of the

Table 1Material properties.

Specimens Area A(mm2)

Length L(mm)

Elastic modulusE (GPa)

0.1% proof 0:1 (MPaT01a 113.46 399.6 67.07 187.2T01b 110.25 400.8 70.06 200.5Mean value 111.86 400.2 68.57 193.9193.1 233.2 9.7 22.0207.6 235.4 7.9 19.9200.4 234.3 8.8 20.9

2. Experimental studies

2.1. Material properties

Material properties of the specimens were determined by ten-sile coupon tests. The mean value of the elastic modulus and yieldstrength of the aluminum alloy were 68.57 GPa and 200.4 MPa,respectively. The stressstrain relationship of aluminum alloywas described by RambergOsgood expression [27],

e rE 0:002 r

f 0:2

n1

where E and f0.2 were elastic modulus and nominal yield strength

n ln 2ln f 0:2f 0:1

2

The ultimate tensile strain of the tested aluminum alloy wasonly about 8.8%, which indicated that the aluminum alloy mem-bers might encounter ruptures where the tensile strain was rela-tive high. The stressstrain relationship obtained from coupontests was shown in Fig. 2.

2.2. Column test

The test specimens were fabricated by extrusion using 6063-T5aluminum alloy. Two column lengths including 350 mm and190 mm were studied. The section dimension was shown in Fig. 3.It had one symmetric axis and was made up of four cantilever andone stiffened plates.

7 specimens were tested to investigate the inuence of columnlength, direction of global buckling and initial geometric imperfec-tion on buckling behaviors. Since the short tested specimens allwould fail at local buckling but not interaction of local and globalbuckling without initial imperfection, two levels of initial lack ofstraightness, 0.70 mm (l/500) and 5.00 mm (l/70), were exertedthrough rolling machine before tests. The initial geometric imper-fection was listed in Table 2.

A servo-controlled hydraulic testing machine was used to applycompressive force by displacement control at a constant rate of0.1 mm/ min. The ends of the specimens were milled at and2 kN was applied before recording in order to ensure full contactwith the hydraulic machine bearing plates. Two LVDTs were used

Fig. 3. Column section dimension.

agn

.7

.7

.0

.0

.0

.7

s

M. Liu et al. / Engineering Structures 95 (2015) 127137 129(stress at 0.2% plastic strain), respectively. The index n was usedto describe the shape of the inelastic portion of the stressstraindiagram, as listed in Table 1. The value of n was calculated by

Table 2Comparisons of buckling strengths obtained from tests and FEA.

Specimen Length L (mm) Direction of initial deection M

C01 350 Y 0C02 350 X 0C03 350 Y 5C04 350 X 5C05 350 Y+ 5C06 350 X 0C07 190

Note: X stands for axis of symmetry; Y stands for axis of non-symmetric axis; + andFig. 4. Test arrto measure the axial displacement and the horizontal deectionof the specimens, as shown in Fig. 4. The applied load and LVDTsreadings were recorded at 5 second intervals during tests.

itude of initial deection (mm) Test FEA ComparisonPEXP PFEA PEXP/PFEA

60.48 60.24 1.0068.78 64.32 1.0742.56 46.24 0.9243.50 47.46 0.9239.54 43.53 0.9065.28 64.32 1.0166.02 61.91 1.07

Mean 0.99COV 0.072

tand for the positive and negative direction of the axis, respectively.angement.

2.3. Test results

2.3.1. Failure modesDeformations of the specimens after tests were shown in Fig. 5.

Since the section had only one symmetric axis, the specimensfailed at exuraltorsional buckling when it buckled around thesymmetric axis; and failed at exural buckling when it buckledaround the non-symmetric axis. For the 350 mm specimens, twofailure modes were observed during tests: (1) exural buckling,

as shown in Fig. 5(a), (c) and (e); and (2) exuraltorsional buck-ling, as shown in Fig. 5(b) and (d). The 190 mm specimens failedat local buckling, as shown in Fig. 5(f) and (g).

The initial geometric imperfection would affect the failuremode of the column. The specimens with initial deection towardx axis (non-symmetric axis) failed at exuraltorsional bucklingaround the symmetric axis, as shown in Fig. 5(b) and (d). Whilethe specimens with initial deection toward y axis (symmetricaxis) failed at exural buckling around the non-symmetric axis,

130 M. Liu et al. / Engineering Structures 95 (2015) 127137Fig. 5. Deformation of test specimens. (a) Deformation of specimen C01. (b) DeformatioC04. (e) Deformation of specimen C05. (f) Deformation of specimen C06. (g) Deformation of specimen C02. (c) Deformation of specimen C03. (d) Deformation of specimenn of specimen C07.

the increase in axial displacement at rst. The axial load kept

and column end. Hard contact was dened to prevent the penetra-tion of the column into the rigid plates. The tangential behaviorwas dened by friction formulation. The friction coefcient wasset as 0.47 [28]. In order to obtain a stable numerical solution, apseudo-dynamic was adopted with a default dissipated energyfraction of 2 104 that are suitable for most applications. FEMis shown in Fig. 8.

Structures 95 (2015) 127137 131increasing till it reached the vertex of the loadaxial displacementcurve, where global buckling occurred, as shown in Fig. 6(ac).

3. Finite element model and verication

3.1. Finite element model

The FEA software ABAQUS [26] was used to predict the bucklingbehavior and ultimate strength of the aluminum alloy column. S4R,a 4-node reduced integration shell element, with 5 mm 5 mmmesh size was used after mesh sensitivity analysis consideringaccuracy, CPU time and memory. The loadaxial displacementcurves of the FEM with different mesh size were shown in Fig. 7.Three curves almost coincided with each other indicated that meshsize with 5 mm 5 mmwas accurate enough to predict the behav-ior of the studied column. The general purpose shell element S4Rgave robust and accurate solutions and allowed transverse sheardeformations. The classical metal plasticity model was applied. Itused standard Mises yield surfaces with associated plastic ow.Perfect plasticity and isotropic hardening denitions were bothavailable. It was simple and adequate for common applicationsincluding crash analyses, metal forming, and general collapse stud-ies. The normal stressstrain data obtained through tensile testswas converted to true stress (Cauchy stress) and true strain (loga-rithmic strain). Poissons ratio of aluminum alloy was assumed tobe 0.3.

Meshed by C3D8R, 8-node linear reduced integrated structuralbrick element, the rigid plates were used to apply axial load tocolumn through contact interaction between test machine andcolumn end. The mesh size was selected to be10 mm 10 mm 10 mm. The rigid plates were made of steelwith Youngs modulus of 2.05 105 MPa and behaved elastically.as shown in Fig. 5(a), (c) and (e). Because of the smaller ultimatetensile strain of aluminum alloy, cracks were observed at the ten-sion side of specimen C04 and C05, as shown in Fig. 5(d) and (e).Specimen C07 failed at local buckling nally because of its shortlength, as shown in Fig. 5(g).

2.3.2. Ultimate strengthsThe magnitude of the initial imperfection had signicant effects

on the ultimate strength of the column, as listed in Table 2. The0.7 mm initial lack of straightness was found appropriate to triggerglobal buckling. While the initial lack of straightness with 5.00 mmwas too large, thus the buckling strength reduced a lot due to theP-4 effect.

Length of specimen C03, C04 and C05 was 350 mm and themagnitudes of initial imperfection was 5.0 mm. Measured ultimatestrengths were 42.56 kN, 43.50 kN and 39.54 kN, respectively. Thedifference from the average value were very small, which were1.8%, 4.0% and 5.8%, respectively. However, specimen C03 andC05 failed at exural buckling and specimen C04 failed at exu-raltorsional buckling, which showed that the failure mode hadno effects on the exural buckling strength.

The average ultimate strength of specimen C01, C02 and C06with 0.7 mm initial deection was 64.58 kN, which was muchgreater than the average value of specimen C03, C04 and C05 as41.80 kN.

2.3.3. Loadaxial displacement curvesLoadaxial displacement curves obtained from tests were

shown in Fig. 6. The axial compression load increased linearly with

M. Liu et al. / EngineeringOne rigid plate was xed and the other could only move axiallytoward the column to apply the axial compressive load. Contactalgorithm was dened between the surface of the rigid platesFig. 6. Loadaxial displacement curves obtained from tests and FEA. (a) Loadaxialdisplacement of specimen C01, C02 and C06 with 0.7 mm initial imperfection. (b)

Loadaxial displacement of specimen C03, C04 and C05 with 5 mm initialimperfection. (c) Loadaxial displacement of specimen C07 without initialimperfection.

tructures 95 (2015) 127137Fig. 7. Mesh sensitivity analysis.132 M. Liu et al. / Engineering SThe initial imperfection of the column followed a half-sinecurve with the maximum deection at mid-span with magnitudeof 0.7 mm. The model was adjusted by modifying the coordinatesof nodes.

3.2. Effect of residual stress

The residual stress of extruded aluminum alloy members isusually lower than 20 MPa [29]. The distribution of residual stressacross the studied section was shown in Fig. 9(a). The maximummagnitude was 15.7 MPa. Stress distribution across the sectionand deformation shape of the member with and without residualstress were nearly the same, as shown in Fig. 9(b) and 9(c). Theaxial loadaxial displacement curves coincided with each otherbefore reaching the ultimate strength, as shown in Fig. 9(d). Theultimate strength was 59.06 kN with residual stress and 60.48 kNwithout residual stress. The difference was only 2.40%, whichshowed very little inuence from residual stress. Residual stresswas ignored in the following FEM parametric studies.

3.3. Model verication

The measured material properties in Table 1 were adopted inFEM. The comparisons of ultimate loads obtained from tests and

Fig. 8. FEM of the aluminum alloy column.

Fig. 9. Effect of residual stress. (a) The distribution of initial residual stress. (b)Stress distribution of the column without residual stress. (c) Stress distribution ofthe column with residual stress. (d) Axial loadaxial displacement curve.

FEA were listed in Table 2. The mean value of Test/FEA was 0.99.The associated coefcient of variation was only 0.07.

There existed differences between the loadaxial displacementcurves obtained from FEA and tests, as shown in Fig. 6. However,the ultimate loads and the corresponding axial displacementswhen the columns buckled globally were almost the same.Furthermore, the loadaxial displacement curves obtained fromFEA clearly showed the turning points on the ascending branchwhich reected the reduction of the axial stiffness due to the localbuckling.

Deformation of specimen C01 and C07 at global buckling failureobtained from tests and FEA were shown in Fig. 10. The local buck-ling and global buckling deformation agreed very well with eachother.

Based on the comparisons in ultimate strengths, loadaxial dis-placement curves and deformations obtained from FEA and tests, itwas concluded that the presented FEM could accurately predict thebehavior of aluminum alloy columns with section subject toaxial compression. The validated FEM was used thereafter in theparametric studies to assess current design codes for aluminumalloy columns.

4. Section optimization

M. Liu et al. / Engineering Structures 95 (2015) 127137 133Fig. 10. Comparison of buckling modes obtained from tests and FEA. (a) SpecimenC01 failed at the exural buckling. (b) Specimen C07 failed at local buckling.The section was made up of 6 plates, as shown in Fig. 11.FSM software CUFSM was used to obtain the elastic buckling stressof the column with different plate thickness and column length.

Effects of plate thickness on elastic buckling stress were shownin Fig. 12. It was seen that the thickness of plate t2, t3 and t6 had noinuences on global buckling stress, as shown inFig. 12(b), (c) and (f). With the increase in t1 thickness, the columnglobal buckling stress increased. However, the increments werevery small, as shown in Fig. 12(a). Except for small thickness, theincrease in the thickness of t4 and t5, did not lead to thecorresponding increase in global buckling stress, as shown inFig. 12(d) and (e). That was, for the column section studied, itwas not an effective way to improve the global buckling stressby increasing the plate thickness.

While for short columns less than 100 mm, t1 thickness greatlyaffected the buckling stress and bucking modes, as shown inFig. 12(a). When t1 was 0.2 mm or 0.3 mm, it only occurred localbucking. When t1 was 0.4 mm or 0.5 mm, it occurred both localand distortional buckling; however, the local buckling dominatedthe column failure. When t1 was 0.6 mm or 0.7 mm, it occurredboth local and distortional buckling and the distortional bucklingdominated the column failure. Under this circumstance, the buck-ling capacity of the column could not be improved by increasing t1thickness. When t1 was 0.7 mm, the local buckling stress exceededaluminum alloy yield stress. That was, no local buckling occurredand the whole plate was effective.

Plate t2 was a cantilever plate and was vulnerable to local buck-ling. When t2 was less than 0.9 mm, short columns with lengthbeing less than 100 mm failed at local buckling, as shown inFig. 12(b). When t2 was greater than 1.0 mm, the column failedat distortional buckling. The increase in t2 thickness did notincrease the local or distortional buckling stress of the column.

Plate t3 was a stiffened plate in the section. When t3 wasgreater than 0.5 mm, the increase in t3 thickness did not lead tothe increase of the critical buckling stress accordingly, as shownin Fig. 12(c).

Plate t4 was a cantilever plate and was vulnerable to distor-tional buckling. The critical elastic distortional buckling stresswas signicantly inuenced by t4 thickness. With the increase int4 thickness, the critical elastic distortional buckling stressincreased. And the column behavior was controlled by localFig. 11. Plates division.

buckling when t4 was 1.5 mm. Furthermore, it was not efcient toincrease t4 thickness when it was greater than 1.5 mm since thecritical elastic distortional buckling stress had exceeded the yieldstress of aluminum alloy, as shown in Fig. 12(d).

Pate t5 was a stiffened plate and its local buckling stress keptconstant when t5 increased from 0.5 mm to 2.0 mm. However,the distortional buckling stress increased along with the increasein t5 thickness, as shown in Fig. 12(e). The elastic local bucklingstress was less than the elastic distortional buckling stress whent5 was greater than 1.5 mm. That was, increase in t5 thicknesshad no signicance under this circumstance.

The existence of plate t6 did not increase the buckling stress ofthe column, as shown in Fig. 12(f). Plate t6 was not a structural partin the section. It was to satisfy the leak proof requirement.

5. Current design codes

5.1. Current design codes for aluminum alloy column

EC9 [6], AA [15] and GB50429 [17] provide design rules for alu-minum columns with traditional section shapes. AISI [21,22] for

of t1

134 M. Liu et al. / Engineering Structures 95 (2015) 127137Fig. 12. Effects of plate thickness on elastic buckling stress of the column. (a) Effects

elastic buckling stress (t1 = 0.6 mm, thickness of other plates are 1.5 mm). (c) Effects ofEffects of t4 on elastic buckling stress (t1 = 0.6 mm, thickness of other plates are 1.5 mm)1.5 mm). (f) Effects of t6 on elastic buckling stress (t1 = 0.6 mm, thickness of other plateon elastic buckling stress (thickness of other plates are 1.5 mm). (b) Effects of t2 on

t3 on elastic buckling stress (t1 = 0.6 mm, thickness of other plates are 1.5 mm). (d). (e) Effects of t5 on elastic buckling stress (t1 = 0.6 mm, thickness of other plates ares are 1.5 mm).

cold-formed steel structures was also used to obtain the ultimatestrength of thin-walled aluminum alloy column by substitutingthe material properties of steel with those of aluminum alloy.

The ultimate strength of the column under axial compressiveload is the minimum strength of global buckling, local bucklingand interaction of local and global buckling in AA [15]. Globalbuckling strength is calculated based on Euler formula. Local buck-ling is the summation of the local buckling strength of each plate.The interaction of local and global buckling is calculated when thestrength of local buckling is lower than that of global buckling. Thedesign strength is reduced when considering the effect of localbuckling in AA Specication.

EC9 and GB50429 adopt Perry curve for the design of axial com-pressive column. The stability factor of global buckling is

Table 3Comparisons of ultimate strengths obtained from design codes and FEA.

Comparison PFEAPAA

PFEAPAISI

PFEAPEC9

PFEAPGB

PFEAPDSM

Mean 1.21 1.18 1.47 1.62 1.13COV 0.103 0.083 0.101 0.145 0.107

M. Liu et al. / Engineering Structures 95 (2015) 127137 135Fig. 13. Comparisons of results by FEA and design codes. (a) Comparison of ultimate streand FEA. (c) Comparison of ultimate strength obtained by EC9 and FEA. (d) Comparisostrength obtained by DSM and FEA.ngth obtained by AA and FEA. (b) Comparison of ultimate strength obtained by AISIn of ultimate strength obtained by GB50429 and FEA. (e) Comparison of ultimate

calculmetho

minumDS

calculbers.

Pne

5.2. Pa

t4: 2.0 mm;

5.3. Comparison of ultimate strengths

Table 3 gave the mean value and Coefcient of Variation (COV)of the ratios of FEA results and design codes predictions.Comparisons of ultimate strengths by FEA and design codes wereshown in Figs. 1316. Results of EC9 were very close to GB50429since they both considered initial imperfections based on Perrycurve. Ratios of PFEA/PEC9 and PFEA/PGB were 1.47 and 1.62 withthe corresponding COV of 0.101 and 0.145, respectively. GB50429considered a bigger imperfection, which leaded to the ultimatestrength being the most conservative among these design meth-ods. AA predictions were also conservative with PFEA/PAA of 1.21and COV of 0.103. Results by AISI were close to DSM and were rela-tively economic with PFEA/PAISI and PFEA/PDSM of 1.18 and 1.13 andthe corresponding COV of 0.083 and 0.107, respectively.

Fig. 13 and 14 compared FEA and design codes calculations foraluminum alloy column with section. From Fig. 13 and 14, itwas seen that current design codes were all conservative to predictthe ultimate strength. The column design curves were divided into3 parts in AA. The curves of EC9 and GB50429 agreed well witheach other. And GB50429 were the most conservative prediction.

DSM exural buckling curve agreed well with that obtainedfrom FEA, as shown in Fig. 15. Fig. 16 showed the strength ratio dis-tribution, and the majority of PFEA/PDSM was between 1.05 and 1.15.

Fig. 15. Comparisons of results by FEA and DSM.

136 truc t5: 2.0 mm t6: 1.0 mm.

(2) group II: (30 specimens) t1: 0.8, 1.2, 1.5 mm; t2 and t3:0.6, 0.8, 1.0, 1.2, 1.4, 1.5, 1.6, 1.8, 2.0, 2.2,

2.4 mm; t4:2.0 mm; t5:2.0 mm; t6:1.0 mm.

(3) group III: (30 specimens) t1:0.8 mm; t2 and t3: 1.5 mm; t4: 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0 mm; t5: 1.0, 2.0, 3.0 mm; t6: 1.0 mm.

(4) group IV: (30 specimens) t1: 0.8 mm; t2 and t3: 1.0, 1.5, 2.0 mm;The veried FEMwas used to obtain the ultimate strength of thecolumns with various parameters. The obtained ultimate strengthswere compared with those calculated by design codes.

The 0.2% proof stress of the aluminum alloy obtained from testswas adopted in FEM, which was 200.4 MPa. In order to eliminatethe effects of local buckling at column ends, the Multi-PointsConstraint (MPC) was used to simulate the xed boundary condi-tion in FEM. The Beam type MPC was used ensuring a rigid beamconnection in order to constrain the displacement and rotation ofeach slave nodes to those of the control point. The displacementload was applied at the control point.

The parametric studies consisted of 615 specimens including 5lengths and 123 sections with different plate thickness. Studiedcolumn lengths were 600, 1200, 1800, 2400, 3000 mm, respec-tively. The section dimension remained unchanged while platethickness varied. The column lengths and plates thicknesses werecarefully selected to ensure global buckling occur.

Four groups of sections were studied and plate thicknesses ineach group were:

(1) group I: (33 specimens) t1: 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5 mm; t2 and t3: 1.0, 1.5, 2.0 mm;Pne is the global buckling strength of the column; Py is then yielding strength; kc is dimensionless slenderness ratio.

rameter studieswheresectiorule of DSM for global buckling strength is

0:658k2c

Py for kc 6 1:5

0:877k2c

Py for kc > 1:5

8>: 3their interactions are all taken into consideration. The columndesignis also used to predict the ultimate strength of the alu-alloy column with section in this paper.

M [21,22] assumes that the full section is effective whenating the ultimate strength of thin-walled structural mem-Global buckling, local buckling, distortional buckling andsiders the effect of local buckling using effective width method,whichated using PerryRoberson formula. Effective thicknessd is applied to consider the effect of local buckling. AISI con-

M. Liu et al. / Engineering S t4: 2.0 mm; t5: 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0 mm; t6: 1.0 mm.Fig. 14. Comparisons of ultimate strengths by FEA and design codes (t1 = 0.9 mm;t2 = 2.0 mm; t3 = 1.0 mm; t4 = 2.0 mm; t5 = 1.0 mm; t6 = 1.0 mm).

tures 95 (2015) 127137Therefore, it was concluded that DSM exural buckling curve pre-dicted the ultimate strength more accurately for aluminum alloycolumn with section under axial compressive load.

References

[1] Mazzolani FM. Competing issues for aluminium alloys in structuralengineering. Progr Struct Eng Mater 2004;6(2):18596.

[2] Summers PT, Case SW, Lattimer BY. Residual mechanical properties ofaluminum alloys AA5083-H116 and AA6061-T651 after re. Eng Struct2014;76(1):4961.

[3] Fogle EJ, Lattimer BY, Feih S, Kandare E, Mouritz AP, Case Scott W. Compressionload failure of aluminum plates due to re. Eng Struct 2012;34(1):15562.

[4] Rasmussen KJR, Rondal J. Strength curves for aluminum alloy columns. EngStruct 2000;22(11):150517.

M. Liu et al. / Engineering Structures 95 (2015) 127137 1376. Conclusions and suggestions

This paper presented experimental and numerical investiga-tions on the behaviors of section aluminum alloy columnsunder axial compression. Seven axial compression tests were car-ried out. The ultimate strengths and failure modes were reported.Columns with length of 190 mm failed at local buckling; and thosewith length of 350 mm failed at global buckling. A FEMwere devel-oped and veried by tests in three aspects including ultimate loads,loadaxial displacement curves and deformations.

Effects of plate thickness and column length on the elastic buck-ling stress were investigated by FSM. The thickness of plate t2, t3and t6 had no signicant inuences on global buckling stress.With the increase in t1 thickness, the column global buckling stressincreased slightly. Except for small thickness, the increase in t4 andt5 thickness did not increase the global buckling stress.

Parametric studies on 615 specimens were presented usingFEM. Ultimate strengths obtained by FEA were compared withthose calculated by AA, AISI, EC9, GB50429, as well as DSM.Results of EC9 were close to GB50429 because they both consid-ered initial imperfections based on Perry curve. GB50429 consid-ered imperfection greater than EC9, which leaded to the mostconservative predictions. AA was also a little conservative.

Fig. 16. Strength ratio distribution.Results of AISI were close to DSM and they both agreed well withFEA results. DSM exural buckling equation which considered thegross area could predict the ultimate strength for section col-umn under axial compression more accurately and conveniently.

Acknowledgements

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Buckling behaviors of section aluminum alloy columns under axial compression1 Introduction2 Experimental studies2.1 Material properties2.2 Column test2.3 Test results2.3.1 Failure modes2.3.2 Ultimate strengths2.3.3 Loadaxial displacement curves

3 Finite element model and verification3.1 Finite element model3.2 Effect of residual stress3.3 Model verification

4 Section optimization5 Current design codes5.1 Current design codes for aluminum alloy column5.2 Parameter studies5.3 Comparison of ultimate strengths

6 Conclusions and suggestionsAcknowledgementsReferences