The paper presents a series of fourpoint bending tests investigating global buckling of coldrolled aluminium alloy channel beam members. These types of sections have been commercially fabricated in Australia using a rollforming process as distinct from the usual extrusion process. A total of twenty specimens of three commercially available channel sections with two thicknesses and various lengths were tested at the University of Sydney. Mechanical properties of aluminium alloy 5052 material were reported on the basis of tests of tensile and compression coupons cut longitudinally fromboth flat and corner parts of the crosssections. Prior to the testing of beams, initial geometric imperfections of each specimen were measured using a laser scanning method. A dualactuator test rig was specially designed to maintain the load vertically applied through the shear centre of the channel section throughout the test. Flexuraltorsional and distortional buckling modes were observed including interaction of thesemodes. The paper also details finite element (FE)models developed using the commercial ABAQUS software package to simulate the behaviour and ultimate member buckling capacities of coldrolled aluminiumalloy channel beams. The FE models with the incorporation of measured properties and actual imperfections are verified against the experimental results indicating good agreements. The test results are then compared with the design strength predictions from current specifications. The calibratedmodel results are laying the foundation for undertaking parametric studies and proposing new design rules for the global and distortional buckling capacities of these types of coldrolled aluminium alloy channel sections in the subsequent work.
Journal of Constructional Steel Research 179 (2021) 106521 Contents lists available at ScienceDirect Journal of Constructional Steel Research Global buckling capacity of cold-rolled aluminium alloy channel section beams Ngoc Hieu Pham, Cao Hung Pham ⁎, Kim J.R Rasmussen School of Civil Engineering, The University of Sydney, NSW 2006, Australia a r t i c l e i n f o Article history: Received 13 November 2020 Accepted January 2021 Available online xxxx Keywords: Cold-rolled aluminium sections Global buckling Distortional buckling Experimental investigation Finite element modelling a b s t r a c t The paper presents a series of four-point bending tests investigating global buckling of cold-rolled aluminium alloy channel beam members These types of sections have been commercially fabricated in Australia using a roll-forming process as distinct from the usual extrusion process A total of twenty specimens of three commercially available channel sections with two thicknesses and various lengths were tested at the University of Sydney Mechanical properties of aluminium alloy 5052 material were reported on the basis of tests of tensile and compression coupons cut longitudinally from both flat and corner parts of the cross-sections Prior to the testing of beams, initial geometric imperfections of each specimen were measured using a laser scanning method A dual-actuator test rig was specially designed to maintain the load vertically applied through the shear centre of the channel section throughout the test Flexural-torsional and distortional buckling modes were observed including interaction of these modes The paper also details finite element (FE) models developed using the commercial ABAQUS software package to simulate the behaviour and ultimate member buckling capacities of coldrolled aluminium alloy channel beams The FE models with the incorporation of measured properties and actual imperfections are verified against the experimental results indicating good agreements The test results are then compared with the design strength predictions from current specifications The calibrated model results are laying the foundation for undertaking parametric studies and proposing new design rules for the global and distortional buckling capacities of these types of cold-rolled aluminium alloy channel sections in the subsequent work © 2021 Elsevier Ltd All rights reserved Introduction The use of aluminium alloys in structural members has seen significant growth in recent times in the construction sector [1] Extrusion is the conventional method used to produce aluminium alloy sections, whereas cold-rolled aluminium alloy sections are new Australian products recently produced by BlueScope Permalite [2] and have proven to be more cost-effective as compared to extruded sections A large number of research studies on cold-formed carbon steel and stainless steel structures have been available in the literature, whereas research on cold-rolled aluminium alloy structural members is scarce Early studies of local buckling of aluminium structural sections were presented by Clark and Rolf [3,4] for tubular, I-section and channel aluminium members Faella et al [5] studied rectangular and square hollow aluminium beams and proposed buckling formulae to determine the inelastic buckling moments of these beams Zhu and Young [6] conducted a total of 30 tests on thin-walled square hollow section aluminium ⁎ Corresponding author E-mail addresses: npha0499@alumni.sydney.edu.au (N.H Pham), caohung.pham@sydney.edu.au (C.H Pham), kim.rasmussen@sydney.edu.au (K.J.R Rasmussen) https://doi.org/10.1016/j.jcsr.2021.106521 0143-974X/© 2021 Elsevier Ltd All rights reserved beams for a range of sectional slendernesses, observing failure by yielding or local buckling The test strengths and results from a parametric study were compared with the predictions from the American, Australian/New Zealand and European specifications for aluminium structures Predictions using the Direct Strength Method (DSM) for cold-formed steel members were also included in the comparison New design rules were subsequently proposed for aluminium alloy square hollow section beams based on the DSM framework Su et al [7] presented a series of 29 tests on square and rectangular hollow section beams in which the failure modes of the specimens were local buckling, yielding or tensile fracture These hollow section beams had high torsional rigidity and were not susceptible to flexural-torsional buckling For research studies on numerical modelling, finite element software packages have been widely used to simulate and analyse the behaviour of structural members Well-developed finite element models can accurately predict the behaviour of actual structures, resulting in savings of the time and cost of conducting physical experiments However, FE models require careful validation against experimental results to ensure reliability Zhu and Young [6] developed a parametric study to investigate yielding and local buckling of square hollow beams for a range of slenderness values Similar FE models were developed by N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Su et al [7] to study local, yielding and tensile fracture of rectangular hollow beams Numerical investigations of flexural-torsional or interaction buckling modes of aluminium open section beams remain scarce Roll-formed aluminium alloy sections in the form of thin-walled sections are prone to local, distortional and global buckling instabilities This paper presents an experimental program recently performed at the University of Sydney to investigate the flexural-torsional buckling of cold-rolled aluminium alloy 5052 channel beams under 4-point bending including the interaction of flexural-torsional buckling with distortional buckling Subsequently, detailed FE models are developed to simulate the global buckling capacities of cold-rolled channel beams, and validated against the experimental results The test ultimate strengths are then compared with predictions of current Australian, American and European specifications and conclusions are drawn about the accuracy of these specifications Selection of section geometries and specimen lengths The cross-sections used for the tests were chosen from the “RollFormed Aluminium Purlins Solutions” catalogue published by BlueScope Permalite [2] The selection of the cross-sections was based on the sectional slenderness varying from low to high The nondimensional slenderness of a section is defined as follows: s f 0:2 l ẳ f cr 1ị The term (f0.2) is the 0.2% proof stress or equivalent yield stress of the aluminium alloy and the term (fcr) is the elastic local or distortional buckling stress determined by the program THIN-WALL-2 [8] developed at the University of Sydney The elastic buckling stresses and slendernesses for bending are reported in Pham [9] The cross-sections selected for this study are C10030, C25025 and C40030 channels, chosen to provide low, intermediate and high slendernesses, respectively The nominal dimensions of each section are given in Table using the nomenclature shown in Fig The specimen lengths were selected to capture flexural-torsional buckling and interactions between sectional and global modes using elastic global buckling analyses The elastic global buckling moment was determined according to AS/NSZ 4600:2018 [10] as follows: M0 ¼ C b Arol qffiffiffiffiffiffiffiffiffiffiffiffiffiffi f oy f oz Fig Nomenclature for channel section Initial geometric imperfection measurement Actual initial geometric imperfections were measured prior to testing using a laser scanner method The imperfection measuring rig included two high precision bars attached to a rigid frame, and a trolley running along the bars as shown in Fig The laser devices were attached to the trolley to measure distances to the surface of the specimens while the trolley ran along the bars at a constant speed A total of nine measurement lines were located around the channel cross-sections (see Fig 3) The global imperfections were measured through lines (3), (4), (6) and (7) The local imperfection of the web was captured through line (5), whilst the distortional imperfections of the two flanges were captured through lines (1), (2), (8) and (9) Full details of the laser scanner method and measured data are reported in Pham [9] and Pham et al [11] Measured imperfections can be classified into sectional and global components as shown in Fig where (d1, d2) are the web and flange deformations, and (G1, G2, G3) are the bow, camber and twist, respectively The mean values of sectional imperfection components (d1, d2) for each ð2Þ where Cb is the coefficient accounting for moment distribution in the laterally unbraced segment; A is the area of the full cross-section; rol is the polar radius of gyration of the cross-section about the shear centre; foy, foz are respectively the elastic buckling stresses of an axially loaded compression member for flexural buckling about the y-axis and for torsional buckling The elastic global buckling analyses were conducted for simple support boundary conditions at both ends with free warping displacements Accordingly, the effective length factors taken were taken as 1.0 for the x-, y- and z- axes The selected specimen lengths are given in Table Table Bending specimen lengths Table Nominal cross-section dimensions with the internal corner radius of mm Sections t mm D mm B mm L mm C10030 C25025 C40030 3.0 2.5 3.0 105 255 400 60.5 76.0 125.5 16.0 25.5 30.0 Sections Specimen lengths (m) C10030 2.0 3.0 4.0 C25025 2.5 4.0 5.0 C40030 4.0 5.0 6.0 N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Fig Imperfection measuring rig type of cross-sections and global ones (G1, G2, G3) for all channels are summarised in Tables and 4, respectively Table Sectional imperfection components Parameters Mechanical properties of aluminium alloy 5052 material and residual stresses in coll-rolled sections C10030 C25025 The mechanical properties of the aluminium alloy 5052 material used for both experimental and numerical investigations were obtained from coupon tests performed in the J.W Roderick Materials and Structures Laboratory at the University of Sydney and are fully reported in Huynh et al [12] Coupon tensile and compression tests were conducted for the three studied C10030, C25025 and C40030 sections Flat and corner coupons were taken from the respective parts of the sections and tested in their longitudinal direction Flat coupons were tested in both tension and compression whereas corner coupons were only tested in tension tests due to the relatively small corner radius The dimensions C40030 Mean St dev Mean St dev Mean St dev d1 mm d2-flg1 mm d2-flg2 mm 0.284 0.078 0.739 0.212 1.366 0.597 0.962 0.367 1.328 0.276 1.577 0.471 0.928 0.361 1.391 0.309 1.664 0.669 of the flat coupons conformed to Australian Standard AS1391 [13] for tensile testing, featuring a width of 12.5 mm and a gauge length of 50 mm The corner coupon tests were conducted to determine the properties of the corners which were expected to be affected by the rollforming process Fig Locations of measurement lines and imperfection components N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Table Global imperfection components Parameters Channel sections Mean St dev G1/L G2/L G3/L (deg/m) 1/1580 1/1973 1/4236 1/5766 0.599 0.607 Table Tensile material properties [12] Coupon Load Stroke mm/strain E0 GPa σ0.01 MPa σ0.2 MPa σu MPa n - εu % εf % C10030_Flange C10030_Web C10030_Corner 0.4 0.4 0.4 69.7 69.6 70.5 174 173 192 220 220 244 267 267 286 12.7 12.5 12.5 4.7 5.1 3.3 6.0 6.4 4.7 C25025_Lip C25025_Flange C25025_Web C25025_Corner 0.4 0.4 0.4 0.4 69.9 69.7 69.8 70.5 170 172 174 190 217 219 221 240 257 269 271 286 12.3 12.4 12.5 12.8 5.9 6.1 6.9 5.8 7.7 7.5 7.9 6.8 C40030_Lip C40030_Flange C40030_Web C40030_Corner 0.4 0.4 0.4 0.4 69.3 69.4 69.5 70.4 174 170 175 192 219 219 223 244 268 270 272 284 12.4 11.8 12.3 12.5 4.5 5.6 6.4 4.2 6.0 7.6 8.2 6.1 The key mechanical properties of aluminium alloy 5052 material in tension and compression are summarised in Tables and 6, where σ0.2 is the 0.2% proof stress; σu is the ultimate tensile strength; E0 is the Young’s modulus; εf is the uniform elongation corresponding the total elongation after fracture; and ‘n’ is the Ramberg-Osgood index It was found that the cold-forming process had a negligible impact on the material properties Table shows that the yield and ultimate strengths of the corner coupons only increased by approximately 10% as compared to those of the flat coupons The elongation of the corner coupons also decreased insignificantly The compression properties are in close agreement with the tensile properties Residual stresses throughout the cold-rolled aluminium crosssections were also measured thoroughly using the sectioning technique where two strain-gauges were attached to both sides of specimens before cutting the plates into strips Full details of the results for the three sections investigated in this study are given in Huynh et al [12] Subsequently, the residual stress distributions around the cross-sections were incorporated into the numerical finite element (FE) models developed by Huynh et al [14,15] The bending residual stresses were included in the stress-strain curve obtained from the coupon tests as the straightening process during coupon testing led to the re-introduction of bending residual stresses The membrane residual stresses were introduced into the finite element models using “Predefined Fields/Initial conditions, Fig Experimental set-up and actual test type = stress” function in the ABAQUS software [16] It was found in Huynh et al [15] that the membrane residual stress had an insignificant effect on the behaviour and strength of the channel columns The membrane residual stress component was therefore ignored in the development of FE models in this study Table Compression material properties [12] Coupon Strain Rate mm/min t mm b mm L mm E0 GPa σ0.01 MPa σ0.2 MPa ε0.01 % ε0.2 % n - C10030_LC_1 C10030_LC_2 C10030_LC_3 0.1 0.1 0.1 8.97 9.00 8.90 25.13 25.10 25.03 93.18 93.00 93.17 Mean 69.41 69.58 69.76 69.59 170 169 176 172 225 222 228 225 0.255 0.254 0.263 0.257 0.525 0.520 0.528 0.524 10.6 11.0 11.6 11.1 C25025_LC_1 C25025_LC_2 C25025_LC_3 0.1 0.1 0.1 7.27 7.33 7.34 25.07 25.11 25.11 93.20 93.21 93.10 Mean 69.38 69.34 69.24 69.32 180 178 177 178 228 223 223 225 0.270 0.266 0.265 0.267 0.528 0.522 0.522 0.524 12.7 13.3 12.9 13.0 C40030_LC_1 C40030_LC_2 C40030_LC_3 0.1 0.1 0.1 8.74 8.69 8.70 25.11 24.58 25.00 93.14 93.09 93.12 Mean 69.68 69.67 69.57 69.64 171 175 174 173 228 227 226 227 0.298 0.261 0.217 0.259 0.533 0.529 0.524 0.529 10.4 11.5 11.5 11.1 N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Fig Dual-actuator loading system and mechanism of testing Linear Variable Displacement Transducer (LVDT) attached to the middle of the channel web adjacent to the loading point was sent to the controller The lateral actuator would instantly adjust the lateral position of the vertical actuator through signals received from the controller to ensure that the applied loads always acted vertically The loading frames were shaped as parallelograms with pin connections at the four corners to allow the specimen to rotate freely about the shear centre The load applied by the vertical actuator was equally transferred through the spreader beam to two loading frames, thus producing a state of uniform bending in the specimen between the loading frames and linearly varying moment in the spans between the loading frames and the pinned supports The two loading frames were connected to the web of the Test rig design and set-up The experimental program consisted of a total of 20 channel beam tests of three different cross-sections with various lengths The basic design of the test rig was developed by Niu et al [17,18] A diagram of the test set-up and overview photograph for the common four-point loading configuration is shown in Fig A dual-actuator loading system comprising a vertical and lateral actuator was set up to apply upward vertical loads through the shear centre of each beam during testing The vertical actuator was bolted to a trolley driven by the lateral actuator, as shown in Fig As flexuraltorsional buckling occurred, the lateral displacement signal from a Fig Tapered washers between the loading frames and the web of the specimen N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Fig Restraint box at the end support and bearings allowing rotations in horizontal and vertical planes connecting bars and the web of the specimen, similar specially designed tapered washers to those used to connect the loading frames were also attached to the two sides of the web and the three connecting bars as shown in Fig 8, thus minimising the enhancement of torsional rigidity of the specimen while undergoing flexural-torsional buckling A secondary purpose of the tapered washers was to allow for differential rotation of the connecting bars and twist of the cross-section as demonstrated in Fig The vertical angle section shown in Figs and was used to clamp the last row of bolts to avoid distortion at each end section due to unbalanced shear flow For instrumentation, transducers were attached at mid-span, loading points and ends of the specimen to monitor the local and overall displacements Four transducers (L1, L2, R1 and R2) were attached to the two ends of the specimen to measure end-shortening Three transducers (LC1, LC2 and LC3) were attached to the aluminium frame specimen by four brackets as shown in Fig To minimise the clamped contact between these brackets and the web, which would otherwise enhance the torsional rigidity and the strength of the beam, a series of specially tapered washers were attached to the brackets and both sides of the web as shown in Fig At the two end supports, two restraint boxes consisting of two perpendicular shafts were specifically designed to allow the specimen to freely rotate about the vertical and horizontal axes at the same longitudinal point, as shown in Fig The vertical shaft was connected to the web of the specimen using three connecting solid bars These bars were welded to three separate bearings attached to the vertical shaft, allowing them to rotate independently and differentially, and enabling the specimen to twist at the connection point while preventing twist rotation at the support point The three connecting bars were bolted to the web of the specimen by two rows of bolts, as shown in Fig For the interface between the Fig Connection between connecting plates and the web of specimen N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Fig Mechanism of tapered washers Development of nonlinear finite element models mounted on the specimen at mid-span, as shown in Fig 10a This specially designed frame was connected at the two flange-web corners of the beam specimen (see Fig 10a) and followed the specimen during global buckling It enabled the transducers mounted on the frame to capture local and distortional buckling deformations without these being affected by global flexural and torsional deformations Four vertical transducers (G1 to G4) were attached to two stable columns fixed to the strong floor and connected to two corners of the transducer frame through four steel strings to measure global deformations at mid-span (see Fig 10a) Three inclinometers were attached to the web at the loading points and to the LVDT frame at mid-span to measure global twist (see Fig 10b) The specimen nomenclature and test matrix information are given in Table where, for example, the C10030-2m-1 specimen label is defined as follows: (i) “C10030” indicates a channel section of 100mm depth and 3.0mm thickness; (ii) “2.0 m” is the length of the specimen; (iii) “1” indicates test number of several nominally identical specimens 6.1 Material property assignment The material properties and full stress-strain curves of the coldrolled aluminium alloy 5052 sections were obtained from coupon tests, as described in Huynh et al [12] and summarised in previous sections The measured stresses and strains from the flat and corner coupon tests were subsequently converted into true stresses and true plastic strains calculated using the following standard equations: σ t ¼ σ ð1 ỵ ị ẳ ln ỵ ị 3ị t E ð4Þ where σ and ε are the measured engineering stress and strain, respectively The true stress and true plastic strain were obtained from Huynh et al [12] Fig 10 LVDT and inclinometer set-up at mid-span N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 6.2 Modelling loading and boundary conditions Table Specimens in bending tests Cross-sections Specimen labels C10030 C10030-2.0m-1 C10030-2.0m-2 C10030-2.0m-3 C10030-3.0m-1 C10030-3.0m-2 C10030-4.0m-1 C10030-4.0m-2 C25025 C25025-2.5m-1 C25025-2.5m-2 C25025-4.0m-2 C25025-4.0m-3 C25025-5.0m-1 C25025-5.0m-2 C25025-5.0m-3 C40030 C40030-4.0m-1 C40030-4.0m-2 C40030-5.0m-1 C40030-5.0m-2 C40030-6.0m-1 C40030-6.0m-2 In order to accurately simulate the ultimate strengths and failure mechanism of the cold-rolled aluminium channel members subject to four-point bending, finite element (FE) models were created using the Finite Element Method (FEM) program ABAQUS [16], as shown in Fig 12 In the models, the web of each end section was connected by rigid links to three reference points which were located on the vertical shear centre plane of the channel section and restrained in the x-axis and y-axis, while the longitudinal z-axis displacement was free Hence, the end boundary conditions were modelled as simple supports with free warping displacements at the two end cross-sections At the loading points, rigid links were used to connect the web of the specimen to the shear centre location of the cross-section Two vertical loads were then applied at the shear centre of the cross-section at loading point as also shown in Fig 12 6.3 Element type and mesh density The four-node reduced integration shell element, S4R, having three translational and three rotational degrees of freedom at each node was chosen from the ABAQUS element library to model the specimens The S4R shell element accounts for finite membrane strains and arbitrarily large rotations and is suitable for large strain analyses and geometrically nonlinear problems To ensure the reliability of the FE results, the number of S4R shell elements was selected using a mesh convergence analysis Also, previous experience by Pham and co-authors [19–22] in developing reliable numerical FE models was drawn upon An FE model of the C10030 section with 2.0 m length was used to check for convergence with variable mesh sizes of 20×20 mm, 12×12 mm, 10×10 mm, 8×8 mm and 5×5 mm The graphs of the ultimate loads, “CPU time” and “axial shortening” versus mesh size are shown in Fig 13 It was found that the ultimate load converged asymptotically with increasing mesh density Mesh refinement beyond 10×10 mm resulted in only a small gain in accuracy, while the differences in ultimate load between 10×10 mm and 8×8 mm or 5×5 mm mesh sizes were less than 0.2% Also, a mesh size of 10×10 mm was more computationally efficient than 8×8 mm or 5×5 mm mesh sizes Similarly, differences in end shortening at the peak load for a 10×10 mm mesh compared to 8×8 mm or 5×5 mm mesh sizes were less than 0.5% Therefore, the mesh size of 10×10 mm was deemed to be adequate and used for all further investigated models Fig 11 Material property assignment for bending specimens The assignment of material properties in the finite element (FE) models depended on the region of the cross-section as illustrated in Fig 11 The tensile corner properties were assigned to all corner regions Tensile properties obtained from flat coupon tests were assigned to the flat parts of the cross-section from the neutral axis to the top of the section, whereas compression properties were assigned to the flat parts of the cross-section below the neutral axis 6.4 Incorporation of initial geometric imperfections into finite element models The procedure for incorporating the measured geometric imperfections into the FE models is described in Pham et al [11] The end cross-section of an actual specimen was manually traced on paper and subsequently converted into CAD format The end Fig 12 Finite element model and simplified boundary conditions N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Fig 13 Mesh size density converge study for C10030-2m specimen Fig 14 Actual measuring and Fourier expression curves at measuring line of specimen C10030-2.0m-1 [11] cross-section thus obtained was used as the original cross-section After being imported into ABAQUS by using the “File>Import/ Sketch” command, the original cross-section was extruded to create the perfectly straight but cross-sectionally imperfect specimen An input file (*.INP) was exported from the original model in ABAQUS software [16], and the coordinates of all nodes in this input file were reproduced using a MATLAB code This code superimposed imperfections using the Fourier series curves onto the coordinates of nodes longitudinally along scanning lines as given in Eqs (5) and (6) and interpolated transversely between the co-ordinates of intermediate nodes between scanning lines f xị ẳ X n¼1 Kn ¼ L Z L nπx ðwhere < x < LÞ L ð5Þ nπx dxðn ¼ 1; 2; 3; …:∞Þ f ðxÞ sin L ð6Þ K n sin The number of series terms was chosen between 20 and 35 depending on the specimen length Fig 14 shows an example of the results for specimen C10030-2.0m-1 where the actual measuring and Fourier expression curves at line are plotted for comparison Fig 15 shows the Fig 15 Perfect and imperfect C10030-2.0m-1section for model calibration N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 globally perfect finite element model after extruding the end crosssection and the imperfect one after incorporating geometric imperfections for C10030-1 specimen with 2000 mm length Experimental behaviour and test vs finite element results Before testing, to ensure the specimens were loaded to produce pure bending between the loading points, care was taken to apply the load vertically upward through the shear centre of the specimen at the loading points and at the end supports At the two loading points, the vertical loads were transferred to the shear centre of the channel by adjusting the horizontal position of the loading frames as shown in Figs and At the two end supports, the distances between the specimen web and the vertical shafts of the restraint boxes were also adjusted to ensure the reactions passed through the shear centre of the channel specimen, again by monitoring and eliminating initial twist rotation by adjusting the bolted connections between the connecting bars and the web of the specimen (see Fig 8) During the testing process, as the C10030 specimens were relatively compact beams with high local and distortional buckling stresses, the flexural-torsional buckling mode was observed with no discernible effect of sectional buckling Similarly, for the C25025 specimens, flexural-torsional buckling with no evidence of sectional buckling was observed in all tests with lengths ranging from 2.5 m to 5.0 m, since the local and distortional buckling stresses of this section were significantly higher than the elastic global buckling stresses in this range For the C40030 specimens, flexural-torsional buckling occurred in long beams, while distortional-global interaction buckling was observed in the shorter beams C40030-4.0m due to the relatively low distortional buckling stresses of this section The ultimate experimental and Finite Element (FE) results for all channel beams investigated in this study are shown in Table All test points are graphically reproduced in Figs 16, 17 and 18 for C10030, C25025 and C40030 section beams, respectively The figures include the elastic local, distortional and global buckling moments as well as the failure modes, where FT and D-FT stand for flexural-torsional and distortional-flexural torsional interaction bucklings, respectively It can be seen that the ultimate strengths of the long specimens are close to the global buckling curve, whereas the strengths of the C40030-4.0m Fig 16 Test results for C10030 specimens beams are considerably lower than the global buckling curve as a result of distortional-global interaction buckling It is interesting to note that the twist rotation may occur in either the positive (clockwise) or negative (anti-clockwise) direction as shown in Fig 19 for C25025-4.0m The ultimate strengths of specimens with negative direction of twist were found to be significantly lower than those with positive twist direction as given in Table 8, because a negative twist induced additional compression of the lip stiffener on the compressed side of the cross-section causing a localised inelastic buckle of the stiffener to form Conversely, a positive twist reduced the compression of the lip stiffener and delayed the formation of a localised buckle of the stiffener The direction of initial crookedness and twist had significant impact on the direction of twist and the post-ultimate response The failure process in the positive twist direction occurred gradually with a low rate of load shedding, whereas the failure in the negative direction was sudden, almost snap-through and occurred due to inelastic local buckling of the lip stiffener This conclusion is clearly demonstrated in Fig 20, and was also reported by previous investigators [23,24] for the lateral-torsional buckling of cold-formed steel channel Table Test and Finite Element (FE) modelling results for channel beams Specimens Test and FE results Pexp (kN) Mexp (kNm) Failure mode PFE (kN) Pexp/PFE C10030-2.0m-1 C10030-2.0m-2 C10030-2.0m-3 C10030-3.0m-1 C10030-3.0m-2 C10030-4.0m-1 C10030-4.0m-2 15.110* 10.140 9.981 3.762 3.479 2.270 2.090 4.646 3.118 3.069 1.626 1.505 1.266 1.165 FT FT FT FT FT FT FT 14.71 9.93 9.95 3.69 3.49 2.21 2.00 1.027 1.021 1.003 1.019 0.997 1.027 1.045 C25025-2.5m-1 C25025-2.5m-2 C25025-4.0m-2 C25025-4.0m-3 C25025-5.0m-1 C25025-5.0m-2 C25025-5.0m-3 24.730* 24.630* 8.340* 7.210 8.040* 4.861 4.768 7.604 7.574 4.650 4.020 4.482 2.709 2.659 FT FT FT FT FT FT FT 24.32 24.21 8.20 6.93 7.78 4.65 4.79 1.017 1.017 1.017 1.040 1.033 1.045 0.996 C40030-4.0m-1 C40030-4.0m-2 C40030-5.0m-1 C40030-5.0m-2 C40030-6.0m-1 C40030-6.0m-2 36.579 36.231 31.025* 29.093 17.824* 16.993* 20.393 20.198 17.299 16.218 14.390 13.719 D-FT D-FT FT FT FT FT 35.07 35.00 30.30 27.78 16.87 16.62 Mean Std Dev 1.043 1.035 1.024 1.047 1.056 1.022 1.027 0.016 (*): The tests failed in the positive direction (clockwise) Fig 17 Test results for C25025 specimens 10 N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Fig 20 Load vs lateral displacement curves of C25025-4.0m specimens in negative (blue) and positive (red) directions (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig 18 Test results for C40030 specimens Fig 23 shows the corresponding failure mode shapes of the test and FE model of the larger C40030-4.0m-2 specimen for which distortionalglobal interaction buckling was observed It is evident in Fig 23c that the bottom flange deforms approximately mm in the distortional mode before reaching the peak load The load-twist rotation curves at mid-length for the test and the FE model are also in good agreement as shown in Fig 23d It is concluded that the nonlinear FE simulation can accurately predict the ultimate strengths and buckling failure modes of cold-rolled aluminium channel beams in the four-point bending configuration The reliable and calibrated FE models can be used for subsequent parametric studies to further extend the experimental data in order to propose a new set of design guides for cold-rolled aluminium beams Note that as shown in Fig 20, when buckling occurred in the positive twist direction, the applied load increased after the onset of global flexural-torsional buckling and produced ultimate strengths above the elastic flexural-torsional buckling moment, as shown in Figs 16–18 Figs 21 and 22 show a comparison between the load vs mid-length deformation responses observed in the tests and FE analyses of specimens C10030-2.0m-3 and C25025-4.0m-3, respectively, both failing in the negative twist direction Flexural-torsional buckling was observed in both tests and good agreements were obtained for the load-lateral displacement and load-twist rotation curves as shown in Figs 21c,d and 22c,d Fig 19 Failure directions of specimens (a) C25025-4.0m-3 and (b) C25025-4.0m-2 11 N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Fig 21 Experimental and FE analysis results for specimen C10030-2.0m-3 Fig 22 Experimental and FE analysis results for specimen C25025-4.0m-3 12 N.H Pham, C.H Pham and K.J.R Rasmussen Journal of Constructional Steel Research 179 (2021) 106521 Fig 23 Experimental and FE analysis results for specimen C40030-4.0m-2 sections subject to global buckling and global-distortional interaction buckling ϕF rb ¼ ϕy ðF ec Þ1=3 ðF cr Þ2=3 ð11Þ where Fec is the elastic global buckling stress of the beam; Fcr is the element local buckling stress given in Clause 4.7.1 [25]; ϕ, ϕy are capacity (resistance) factors Comparison of test results with design predictions from current specifications 8.1 Australian/New Zealand Standard AS/NZS 1664.1 Clause 3.4.12 of the Australian Standard AS/NZS 1664.1 [25] provides design formulae for aluminium alloy beams The design bending capacity is determined as the limit state design stress (FL) multiplied by the elastic section modulus (Wel), (Eq (7)) MAS=NZS ¼ F L W el 8.2 American Aluminum Design Manual The member strength provisions for aluminium alloy beams are contained in Chapter F of the Aluminum Design Manual [26] The nominal flexural strength of an aluminium alloy beam is the least of the available limit state strengths for yielding, rupture, local buckling, lateral-torsional buckling and interaction of local and lateral-torsional buckling (Eqs (12)–(18)) Yielding and rupture: À Á Yielding : Mnp ¼ 1:5St F ty , 1:5Sc F cy ð12Þ ð7Þ The limit state stress is FL is determined as follows: For Lb =r y ≤S1 : F L ẳ F cy 8ị For S1 < Lb =ry < S2 : F L ¼ Bc − For Lb =r y ≥S2 : FL ¼ Dc Lb 1:2r y π E 2 Lb 1:2r y 9ị Rupture : Mnu ẳ ZF tu =kt 10ị 13ị Local buckling: Weighted average method : Mnlb ¼ F c If =ccf ỵ F b Iw =ccw where S1, S2 are slenderness limits; Lb is the length of the beam between bracing points; ry is the radius of gyration of about an axis parallel to the web; B c, Dc are the buckling constants (Table 3.3 of the Standard) The effect of local buckling on the member strength is accounted for in Clause 4.7 [25] when the local buckling stress is less than the global buckling stress The member strength determined using Eqs (8)–(10) cannot exceed the combined elastic local-global buckling stress determined using Eq (11) as follows: Direct strength method : M nlb ẳ F b Sxc 14ị 15ị Lateral-torsional buckling: Inelastic buckling λ