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.
Trang 1Ngoc Hieu Pham, Cao Hung Pham ⁎ , Kim J.R Rasmussen
School of Civil Engineering, The University of Sydney, NSW 2006, Australia.
a b s t r a c t
a r t i c l e i n f o
Article history:
Received 13 November 2020
Accepted 4 January 2021
Available online xxxx
Keywords:
Cold-rolled aluminium sections
Global buckling
Distortional buckling
Experimental investigation
Finite element modelling
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 commer-cially available channel sections with two thicknesses and various lengths were tested at the University of Syd-ney Mechanical properties of aluminium alloy 5052 material were reported on the basis of tests of tensile and compression coupons cut longitudinally from bothflat 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 detailsfinite element (FE) models developed using the com-mercial ABAQUS software package to simulate the behaviour and ultimate member buckling capacities of cold-rolled 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 lay-ing the foundation for undertaklay-ing parametric studies and proposlay-ing new design rules for the global and distor-tional buckling capacities of these types of cold-rolled aluminium alloy channel sections in the subsequent work
© 2021 Elsevier Ltd All rights reserved
1 Introduction
The use of aluminium alloys in structural members has seen signi
fi-cant 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
prod-ucts recently produced by BlueScope Permalite [2] and have proven to
be more cost-effective as compared to extruded sections A large
num-ber 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
alu-minium members Faella et al [5] studied rectangular and square hollow
aluminium beams and proposed buckling formulae to determine the
in-elastic buckling moments of these beams Zhu and Young [6] conducted
a total of 30 tests on thin-walled square hollow section aluminium
beams for a range of sectional slendernesses, observing failure by yield-ing or local bucklyield-ing 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 sec-tion 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 toflexural-torsional buckling
For research studies on numerical modelling,finite element soft-ware packages have been widely used to simulate and analyse the be-haviour of structural members Well-developedfinite element models can accurately predict the behaviour of actual structures, resulting in savings of the time and cost of conducting physical experiments How-ever, 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
⁎ 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.
Trang 2Su et al [7] to study local, yielding and tensile fracture of rectangular
hollow beams Numerical investigations offlexural-torsional or
interac-tion buckling modes of aluminium open secinterac-tion beams remain scarce
Roll-formed aluminium alloy sections in the form of thin-walled
sec-tions are prone to local, distortional and global buckling instabilities
This paper presents an experimental program recently performed at
the University of Sydney to investigate theflexural-torsional buckling
of cold-rolled aluminium alloy 5052 channel beams under 4-point
bending including the interaction offlexural-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
2 Selection of section geometries and specimen lengths
The cross-sections used for the tests were chosen from the
“Roll-Formed 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
non-dimensional slenderness of a section is defined as follows:
λl¼
ffiffiffiffiffiffiffiffiffi
f0 :2
fcr
s
ð1Þ
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]
devel-oped at the University of Sydney The elastic buckling stresses and
slen-dernesses for bending are reported in Pham [9] The cross-sections
selected for this study are C10030, C25025 and C40030 channels,
cho-sen to provide low, intermediate and high slendernesses, respectively
The nominal dimensions of each section are given inTable 1using the
nomenclature shown inFig 1
The specimen lengths were selected to captureflexural-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:
M0¼ CbArol
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
foyfoz
q
ð2Þ
where Cbis the coefficient accounting for moment distribution in the
laterally unbraced segment; A is the area of the full cross-section; rolis
the polar radius of gyration of the cross-section about the shear centre;
foy, fozare respectively the elastic buckling stresses of an axially loaded
compression member forflexural buckling about the y-axis and for
tor-sional buckling
The elastic global buckling analyses were conducted for simple
sup-port boundary conditions at both ends with free warping
displace-ments Accordingly, the effective length factors taken were taken as
1.0 for the x-, y- and z- axes The selected specimen lengths are given
inTable 2
3 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 inFig 2 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 mea-surement 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 twoflanges were captured through lines (1), (2), (8) and (9) Full details of the laser scan-ner 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 inFig 3where (d1, d2) are the web andflange de-formations, and (G1, G2, G3) are the bow, camber and twist, respectively The mean values of sectional imperfection components (d1, d2) for each
Table 1
Nominal cross-section dimensions with the internal corner radius of 5 mm.
mm
D mm
B mm
L mm
Fig 1 Nomenclature for channel section
Table 2 Bending specimen lengths.
3.0 4.0
4.0 5.0
5.0 6.0
Trang 3type of cross-sections and global ones (G1, G2, G3) for all channels are
summarised inTables 3 and 4, respectively
4 Mechanical properties of aluminium alloy 5052 material and
re-sidual stresses in coll-rolled sections
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
Struc-tures 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
cor-ner 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
of theflat 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 proper-ties of the corners which were expected to be affected by the roll-forming process
Fig 3 Locations of measurement lines and imperfection components.
Table 3 Sectional imperfection components.
mm
d 2 -flg1 mm
d 2 -flg2 mm
Fig 2 Imperfection measuring rig.
Trang 4The key mechanical properties of aluminium alloy 5052 material in
tension and compression are summarised inTables 5 and 6, where
σ0.2is the 0.2% proof stress;σuis the ultimate tensile strength; E0is
the Young’s modulus; εfis 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 5shows that the yield and ultimate
strengths of the corner coupons only increased by approximately 10%
as compared to those of theflat 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
cross-sections were also measured thoroughly using the sectioning technique
where two strain-gauges were attached to both sides of specimens
be-fore 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]
Subse-quently, the residual stress distributions around the cross-sections were
incorporated into the numericalfinite 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
straighten-ing process durstraighten-ing coupon teststraighten-ing led to the re-introduction of bendstraighten-ing
residual stresses The membrane residual stresses were introduced into
thefinite element models using “Predefined Fields/Initial conditions,
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 mem-brane residual stress component was therefore ignored in the develop-ment of FE models in this study
Table 4
Global imperfection components.
Table 5
Tensile material properties [ 12 ].
Coupon Load Stroke
mm/strain
E 0
GPa
σ 0.01
MPa
σ 0.2
MPa
σ u
MPa n
-ε u
%
ε f
%
Table 6
Compression material properties [ 12 ].
mm/min
t mm
b mm
L mm
E 0
GPa
σ 0.01
MPa
σ 0.2
MPa
ε 0.01
%
ε 0.2
%
n
Fig 4 Experimental set-up and actual test
Trang 55 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
de-sign 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
load-ing configuration is shown inFig 4
A dual-actuator loading system comprising a vertical and lateral
ac-tuator was set up to apply upward vertical loads through the shear
cen-tre of each beam during testing The vertical actuator was bolted to a
trolley driven by the lateral actuator, as shown inFig 5 As
flexural-torsional buckling occurred, the lateral displacement signal from a
Linear Variable Displacement Transducer (LVDT) attached to the middle
of the channel web adjacent to the loading point was sent to the control-ler 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 ap-plied 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 vary-ing moment in the spans between the loadvary-ing frames and the pinned supports The two loading frames were connected to the web of the
Fig 5 Dual-actuator loading system and mechanism of testing
Fig 6 Tapered washers between the loading frames and the web of the specimen
Trang 6specimen by four brackets as shown inFig 6 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 inFig 6
At the two end supports, two restraint boxes consisting of two
perpendicular shafts were specifically designed to allow the
speci-men to freely rotate about the vertical and horizontal axes at the
same longitudinal point, as shown inFig 7 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
dif-ferentially, 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 inFig 8 For the interface between the
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 inFig 8, thus minimising the en-hancement 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 inFig 9 The ver-tical angle section shown inFigs 8 and 9was used to clamp the last row of bolts to avoid distortion at each end section due to unbalanced shearflow
For instrumentation, transducers were attached at mid-span, load-ing 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 trans-ducers (LC1, LC2 and LC3) were attached to the aluminium frame
Fig 7 Restraint box at the end support and bearings allowing rotations in horizontal and vertical planes
Fig 8 Connection between connecting plates and the web of specimen
Trang 7mounted on the specimen at mid-span, as shown inFig 10a This
spe-cially designed frame was connected at the twoflange-web corners of
the beam specimen (seeFig 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 globalflexural and torsional deformations Four
verti-cal transducers (G1 to G4) were attached to two stable columnsfixed to
the strongfloor and connected to two corners of the transducer frame
through four steel strings to measure global deformations at mid-span
(seeFig 10a) Three inclinometers were attached to the web at the
load-ing points and to the LVDT frame at mid-span to measure global twist
(seeFig 10b)
The specimen nomenclature and test matrix information are given
de-fined 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 1 of several nominally identical
specimens
6 Development of nonlinearfinite element models 6.1 Material property assignment
The material properties and full stress-strain curves of the cold-rolled 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 theflat and corner coupon tests were subsequently converted into true stresses and true plastic strains calculated using the following standard equations:
εtp¼ ln 1 þ εð Þ−σt
whereσ and ε are the measured engineering stress and strain, re-spectively The true stress and true plastic strain were obtained from Huynh et al [12]
Fig 9 Mechanism of tapered washers
Fig 10 LVDT and inclinometer set-up at mid-span
Trang 8The assignment of material properties in thefinite 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 fromflat coupon tests were assigned to the
flat parts of the cross-section from the neutral axis to the top of the
sec-tion, whereas compression properties were assigned to theflat parts of
the cross-section below the neutral axis
6.2 Modelling loading and boundary conditions
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
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 inFig 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 forfinite 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 num-ber of S4R shell elements was selected using a mesh convergence analy-sis 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 inFig 13
It was found that the ultimate load converged asymptotically with increas-ing 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
6.4 Incorporation of initial geometric imperfections intofinite element models
The procedure for incorporating the measured geometric imper-fections 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
Table 7
Specimens in bending tests.
C10030-2.0m-2 C10030-2.0m-3 C10030-3.0m-1 C10030-3.0m-2 C10030-4.0m-1 C10030-4.0m-2
C25025-2.5m-2 C25025-4.0m-2 C25025-4.0m-3 C25025-5.0m-1 C25025-5.0m-2 C25025-5.0m-3
C40030-4.0m-2 C40030-5.0m-1 C40030-5.0m-2 C40030-6.0m-1 C40030-6.0m-2
Fig 11 Material property assignment for bending specimens
Fig 12 Finite element model and simplified boundary conditions
Trang 9cross-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
inputfile (*.INP) was exported from the original model in ABAQUS
software [16], and the coordinates of all nodes in this inputfile
were reproduced using a MATLAB code This code superimposed
im-perfections 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
inter-mediate nodes between scanning lines
f xð Þ ¼X∞
n ¼1
Knsinnπx
L ðwhere 0 < x < LÞ ð5Þ
Kn¼2LZL
0
f xð Þ sinnπxL dx nð ¼ 1; 2; 3; …:∞Þ ð6Þ The number of series terms was chosen between 20 and 35 depend-ing on the specimen length.Fig 14shows an example of the results for specimen C10030-2.0m-1 where the actual measuring and Fourier ex-pression curves at line 5 are plotted for comparison.Fig 15shows the
Fig 13 Mesh size density converge study for C10030-2m specimen
Fig 14 Actual measuring and Fourier expression curves at measuring line 5 of specimen C10030-2.0m-1 [ 11 ]
Fig 15 Perfect and imperfect C10030-2.0m-1section for model calibration
Trang 10globally perfectfinite element model after extruding the end
cross-section and the imperfect one after incorporating geometric
imperfec-tions for C10030-1 specimen with 2000 mm length
7 Experimental behaviour and test vsfinite 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
load-ing points and at the end supports At the two loadload-ing points, the
verti-cal loads were transferred to the shear centre of the channel by
adjusting the horizontal position of the loading frames as shown in
Figs 4 and 5 At the two end supports, the distances between the
spec-imen web and the vertical shafts of the restraint boxes were also
ad-justed to ensure the reactions passed through the shear centre of the
channel specimen, again by monitoring and eliminating initial twist
ro-tation by adjusting the bolted connections between the connecting bars
and the web of the specimen (seeFig 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
ef-fect 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 signi
fi-cantly 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 inTable 8 All test
points are graphically reproduced inFigs 16,17and18for C10030,
C25025 and C40030 section beams, respectively Thefigures include
the elastic local, distortional and global buckling moments as well as
the failure modes, where FT and D-FT stand forflexural-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
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
neg-ative direction of twist were found to be significantly lower than those with positive twist direction as given inTable 8, because a negative twist induced additional compression of the lip stiffener on the com-pressed side of the cross-section causing a localised inelastic buckle of the stiffener to form Conversely, a positive twist reduced the compres-sion of the lip stiffener and delayed the formation of a localised buckle of the stiffener The direction of initial crookedness and twist had signi fi-cant 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 di-rection was sudden, almost snap-through and occurred due to inelastic local buckling of the lip stiffener This conclusion is clearly demon-strated inFig 20, and was also reported by previous investigators [23,24] for the lateral-torsional buckling of cold-formed steel channel
Table 8
Test and Finite Element (FE) modelling results for channel beams.
Specimens Test and FE results
P exp (kN) M exp (kNm) Failure mode P FE (kN) P exp /P FE
Mean 1.027 Std Dev 0.016 (*): The tests failed in the positive direction (clockwise).
Fig 16 Test results for C10030 specimens
Fig 17 Test results for C25025 specimens