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Numerical investigation and design of perforated aluminium alloy SHS and RHS columns
Engineering Structures 199 (2019) 109591
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Numerical investigation and design of perforated aluminium alloy SHS and RHS columns
T
Ran Fenga,b, , Jiarui Liua ⁎
a b
School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen 518055, China State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China
ARTICLE INFO
ABSTRACT
Keywords: Aluminium alloy Numerical investigation Parametric study Perforated column Rectangular hollow section (RHS) Square hollow section (SHS)
This paper describes a numerical investigation on perforated aluminium alloy square and rectangular hollow section (SHS and RHS) columns by the finite-element analysis (FEA). The non-linear finite-element model (FEM) was developed by considering both geometric and material non-linearities. The initial local and overall geometric imperfections of aluminium alloy columns were incorporated in the FEM. The non-linear FEM was verified against the corresponding experimental results, which was further used for an extensive parametric study that consisted of 594 specimens with different cross-section dimensions, overall lengths, as well as diameters, numbers and locations of circular openings. The ultimate strengths of the columns obtained from FEA were employed to evaluate the current design specifications for aluminium alloy and cold-formed steel structures. It is demonstrated that American Specification (AA) and limit state design (LSD) in Australian/New Zealand Standard (AS/NZS) are somewhat unconservative, whereas allowable stress design (ASD) in Australian/ New Zealand Standard (AS/NZS) and Chinese Code are quite conservative, while European Code (EC9) is generally appropriate for imperforated aluminium alloy columns. In addition, North American Specification (NAS) and direct strength method (DSM) for cold-formed steel structural members with openings are quite conservative for aluminium alloy columns with circular openings. The design equations proposed based on the design rules of EC9 were verified to be accurate for perforated aluminium alloy SHS and RHS columns under axial compression.
1. Introduction Aluminium alloy has been widely used in structural applications such as curtain walls, space structures and bridges owing to its material advantages of high strength-to-weight ratio, excellent corrosion resistance and ease of manufacture. Currently, structural members are often perforated with one or more openings to facilitate the construction assembly, building service and equipment installation. These prepunched openings usually weaken the sectional dimensions and destroy the geometric continuity of structural members, which result in the deterioration of the elastic stiffness and ultimate strengths of structural members. The shape, size, number and location of openings have a certain degree of influences on the behaviour of perforated structural members. Finite-element analysis (FEA) is a useful and powerful tool that has been widely used in structural analysis and design. Compared to the physical experiments, FEA is a relatively time-saving and convenient method, especially for the structural members with various cross-
⁎
section geometries. In order to find an accurate and stable finite-element model (FEM) for the parametric study, it is mandatory to verify the FEM against the corresponding experimental results. A large number of researches were performed on perforated coldformed steel compression members. In 2001, Shanmugam and Dhanalakshmi [1] studied the ultimate compressive strengths of perforated cold-formed steel channel stub columns with different plate slenderness ratios, opening shapes and sizes by the FEA. The web plate slenderness ratio and opening area ratio were found to be two main variables on the ultimate compressive strengths. In 2007, Moen and Schafer [2] carried out a series of compression tests on 24 cold-formed steel short and intermediate-length columns with and without slotted web holes. The relationship between the elastic buckling and test response of perforated cold-formed steel columns was investigated. It was found that the post-peak response and ductility were influenced by the slotted holes, the cross-section type and length of the columns. For the short columns with higher hole-to-web width ratio, the slotted holes influenced the post-peak response and reduced the ductility of cold-
Corresponding author at: School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen 518055, China. E-mail address: [email protected] (R. Feng).
https://doi.org/10.1016/j.engstruct.2019.109591 Received 30 January 2019; Received in revised form 23 August 2019; Accepted 25 August 2019 Available online 04 September 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Structures 199 (2019) 109591
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Nomenclature
PDSM2
Notation Aeff B COV d D E fy L Le n Nb,Rd Nc,Rd PAA PA/N-1 PA/N-2 PCC PDSM1
PDSM3
Effective area Outer width of SHS and RHS Coefficient of variation Diameter of circular opening Outer depth of SHS and RHS Young’s modulus Yield stress (0.2% tensile proof stress) Overall length of aluminium alloy column Effective length of aluminium alloy column Number of opening Flexural buckling load Local squashing load Design strength obtained from design formulae of American Specification Design strength obtained from design formulae of AS/NZS 1664.1: 1997 Design strength obtained from design formulae of AS/NZS 1664.2: 1997 Design strength obtained from design formulae of GB 50429-2007 Design strength obtained from 1st design criteria proposed
PEC PEXP PFEA PNAS Ppd Pu r S Sh t ε εf εpl ln κ ν σ σtrue σu χ
formed steel columns. For the short columns with lower hole-to-web width ratio and intermediate-length columns, the slotted holes had a small influence on the post-peak response and ductility. Moreover, the existence of slotted holes reduced the ultimate strengths of cold-formed steel columns. In 2012, Yao and Rasmussen [3] researched the effects of different shapes, sizes and spacings of perforations on the inelastic stress distributions, load transfers and failure modes of perforated simply-supported plates and perforated C-section columns in compression. It was found that the perforations changed the stress distributions within the members obviously and triggered the distinct failure modes of inward and outward buckles at the nodal lines of local buckles. In 2013, Kulatunga and Macdonald [4] investigated the influence of perforation positions on the ultimate compressive strengths of cold-formed steel columns with lipped channel cross-sections through the FEA. The ultimate compressive strengths were found to be affected by perforation positions. And the perforation positions near the ends of the columns have a greater weakening effect than the perforations at other positions. Furthermore, Kulatunga et al. [5] also researched the influence of perforation shapes on the buckling behaviour of cold-formed steel columns with lipped channel cross-sections by the FEA. It was found that the ultimate compressive strengths varied greatly with the presence of perforations, which decreased with the increase of the length of perforations. However, limited researches were conducted on perforated
by Moen and Schafer Design strength obtained from 2nd design criteria proposed by Moen and Schafer Design strength obtained from 3rd design criteria proposed by Moen and Schafer Design strength obtained from design formulae of EC9 Ultimate strength obtained from experiment Ultimate strength obtained from FEA Design strength obtained from design formulae of NAS Design strength obtained from proposed design equations Ultimate strength Radius of gyration of SHS and RHS Total area of surface Net area of surface Thickness of SHS and RHS Strain Elongation after fracture Logarithmic plastic strain Reduction factor to allow for weakening effect of welding Poisson’s ratio Stress True stress Ultimate tensile stress Reduction factor for relevant buckling mode
aluminium alloy structural members. In 2010, the experimental and numerical investigations were carried out by Zhou and Young [6] on aluminium alloy square hollow sections (SHSs) with circular holes in the webs subjected to web crippling. The diameter of circular holes was found to be the main factor affecting the web crippling behaviour. In 2015, Feng and Young [7] experimentally investigated the compressive behaviour of aluminium alloy SHS stub columns with circular openings. The ultimate compressive strengths obtained from the tests were used to assess the current design rules. It was found that the current design rules for perforated cold-formed steel members were not suitable for perforated aluminium alloy SHS stub columns. Nevertheless, there is no design rule applicable to perforated aluminium alloy columns. It is worth noting that North American Specification (NAS) [8] provides design rules for cold-formed steel members containing holes. The main objective of this study is to derive the design guidelines for perforated aluminium alloy square and rectangular hollow section (SHS and RHS) columns under axial compression based on the numerical investigation. Hence, this paper mainly focuses on the development and verification of the FEM, which was used to carry out the parametric study and propose the design equations for perforated aluminium alloy SHS and RHS columns. An accurate and reliable nonlinear FEM was developed in this study for aluminium alloy SHS and RHS columns with and without circular openings. The FEM was validated by comparing with the corresponding experimental results,
Table 1 Initial geometric imperfections and material properties of aluminium alloy columns in experimental investigation. Section (D × B × t)
SHS50 × 50 × 1 SHS80 × 80 × 2 RHS120 × 80 × 3 SHS100 × 100 × 3 RHS120 × 60 × 2 RHS150 × 100 × 2.5
Max local geometric imperfection
Max overall geometric imperfection
Δ/t
δ/L
0.36 0.082 0.03 0.12 0.07 0.048
1/1028 1/984 1/1181 1/1897 1/1050 1/1920
Aluminium alloy
6061-T6 6063-T5
2
Material property E (GPa)
σ0.2 (MPa)
σu (MPa)
εf (%)
68.0 68.2 66.8 69.3 69.6 66.7
222.19 207.61 203.75 184.30 191.27 167.55
237.22 219.23 226.23 209.07 211.72 183.91
8.02 10.60 12.08 13.13 15.14 12.06
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(a) 6061-T6
(b) 6063-T5
Fig. 1. Engineering and true stress-strain curves of high strength aluminium alloy type 6061-T6 and normal strength aluminium alloy type 6063-T5.
Table 2 Comparison of ultimate strengths of aluminium alloy columns between experiments and FEA. Specimen
Experimental result
H-S50 × 50 × 1-L150 H-S50 × 50 × 1-L150-d19.2-n1 H-S50 × 50 × 1-L1000-d19.2-n6 H-S80 × 80 × 2-L240 H-S80 × 80 × 2-L240-d30.4-n1 H-S80 × 80 × 2-L800 H-S80 × 80 × 2-L800-d30.4-n3 H-S80 × 80 × 2-L1600 H-S80 × 80 × 2-L1600-d30.4-n6 H-R120 × 80 × 3-L360 H-R120 × 80 × 3-L360-d46.4-n1 N-S100 × 100 × 3-L300 N-S100 × 100 × 3-L300-d37.6-n1 N-S100 × 100 × 3-L1000-d37.6-n3 N-S100 × 100 × 3-L2000 N-R120 × 60 × 2-L360 N-R120 × 60 × 2-L360-d46.4-n1 Mean COV
FEA result
Comparison
Failure mode
Ultimate strength PEXP (kN)
Failure mode
Ultimate strength PFEA (kN)
PEXP/PFEA
L L L L L L L L+F L L L L L L L+F L L
32.4 27.5 25.6 95.6 82.9 90.5 82.8 82.4 62.4 180.0 158.0 166.7 149.5 135.5 163.5 87.2 72.9
L L L L L L L L+F L L L L L L L+F L L
31.5 27.7 23.1 96.0 81.0 90.1 75.1 78.5 61.7 179.5 163.0 157.3 145.9 146.1 167.8 78.9 73.9
1.03 0.99 1.11 1.00 1.02 1.00 1.10 1.05 1.02 1.00 0.97 1.06 1.02 0.93 0.97 1.10 0.99 1.02 0.050
Note: L = Local buckling; F = Flexural buckling.
ultimate strengths of the columns predicted by the FEA are compared with the design strengths calculated using the design rules of NAS [8] and derived by Moen and Schafer [9] for cold-formed steel columns with holes by substituting the material properties of aluminium alloys. The design equations are proposed for perforated aluminium alloy SHS and RHS columns based on the current design rules.
Table 3 Sensitivity analysis of mesh density for aluminium alloy SHS 80 × 80 × 2 with different column lengths. Specimen
Mesh density (mm)
No. of element
Running time (s)
PFEA (kN)
PEXP/PFEA
H-S80 × 80 × 2-L240
3 3.5 4 4 5 6 5 8 10 12
8,640 6,348 4,800 16,000 10,240 6,916 20,480 8,000 5,120 3,724
86 63 56 379 226 156 345 154 84 61
96.3 96.0 91.9 91.8 90.1 85.3 80.5 79.4 78.5 75.9
0.99 1.00 1.04 0.99 1.00 1.06 1.02 1.04 1.05 1.09
H-S80 × 80 × 2-L800 H-S80 × 80 × 2-L1600
2. Summary of experimental investigation The experimental investigation presented by Feng et al. [10] provided the ultimate strengths and failure modes of aluminium alloy SHS and RHS columns with and without circular openings compressed between pin-ends. The test specimens included six different cross-section dimensions with different column lengths ranged from 150 to 3000 mm, which were fabricated by extrusion of SHS and RHS using heat-treated aluminium alloys of 6061-T6 and 6063-T5. The material properties of high strength aluminium alloy type 6061-T6 and normal strength aluminium alloy type 6063-T5 were determined by longitudinal tensile coupon tests, which include the initial Young’s modulus (E), the 0.2% tensile proof stress, the ultimate tensile stress (σu) and the elongation after fracture (εf), as summarized in Table 1. The full stress-
which was then used for the parametric study of perforated aluminium alloy SHS and RHS columns with different geometrical dimensions, as well as diameters, numbers and locations of circular openings. The 3
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formation of the heat-affected zone. The pin-ended bearing was employed at both ends of the special aluminium alloy sleeves for pin-ended compression tests. The identical loading device including the special fixed-ended bearing and pin-ended bearing as that used in test series II were employed in test series III for pin-ended compression tests. The same testing procedure as that used in test series II was employed in test series III for pin-ended compression tests. All specimens failed by either local buckling or flexural buckling. Initial local and overall geometric imperfections about the minor axis of the specimens were measured prior to testing. It is worth noting that five types of geometric imperfections on the minor-axis global buckling mode, major-axis global buckling mode, pure torsional buckling mode, local buckling mode and distortional buckling mode were investigated by Kechidi et al. [11] on single and built-up sections by using the Modal Imperfection Decomposition (MID) method, which is particularly useful for the open sections with various patterns of geometric imperfections. However, the SHSs and RHSs investigated in this study are doubly symmetric closed sections, which are normally subjected to the minor-axis global buckling and local buckling only. Hence, the conventional hand-measurement approach was adopted in the experimental investigation to obtain the initial geometric imperfections. The maximum local and overall geometric imperfections of the specimens with six different cross-section dimensions are also shown in Table 1. The measured specimen dimensions, test rig and procedure are detailed in Feng et al. [10]. The ultimate strengths and failure modes obtained from test series III are summarized in Table 2, which were carefully simulated in the corresponding numerical analysis in this study. 3. Finite-element modelling 3.1. General The finite-element program ABAQUS [12] was used in the numerical analysis for the simulation of aluminium alloy SHS and RHS pinended columns tested by Feng et al. [10]. The material properties, measured geometric dimensions, initial local and overall geometric imperfections of test specimens were all incorporated in the FEM. Residual stress caused by heat treatment was not considered in the FEM since it has little influence on the load-carrying capacities of extruded aluminium alloy profiles [13]. The finite-element simulation consisted of two steps. The first step is an elastic buckling analysis (linear perturbation analysis in the ABAQUS library), which was performed to determine the buckling modes (Eigen-mode) of aluminium alloy columns by the Eigen-value analysis. The second step is a load-displacement non-linear analysis, which was performed to obtain the ultimate strengths and failure modes of aluminium alloy columns by using the (*RIKS) method available in the ABAQUS library. The non-linear geometric parameter (*NLGEOM) was activated in the (*RIKS) method to consider the large displacement in the numerical simulation. Furthermore, the initial geometric imperfections and material non-linearities were all taken into account in the (*RIKS) method.
Fig. 2. Finite-element mesh of the specimen H-S80 × 80 × 2-L800-d30.4-n3.
strain curves of aluminium alloy SHS and RHS tubes with six different cross-section dimensions are clearly shown in Fig. 1. There are three series of compression tests. Test series I included 5 columns with both ends welded to aluminium end plates. The specimens compressed between pin-ended bearings were allowed to rotate about the minor axis only. The axial compression force was applied to the specimens through displacement control by using a servo-controlled hydraulic testing machine. It was found that all specimens failed by material yielding at the heat-affected zone (HAZ). Test series Ⅱ included 35 columns with both ends reinforced by carbon fibre-reinforced polymer (CFRP) at the heat-affected zone resulted from the welding. The similar testing procedure was employed by using the special fixedended bearing and pin-ended bearing to conduct pin-ended compression tests. It was found that some specimens also failed by material yielding at the heat-affected zone (HAZ) even if the affected area has been reinforced by CFRP, which means the material strength of aluminium alloy is significantly deteriorated by the welding. Test series III included 17 columns with both ends fixed by a special aluminium alloy sleeve instead of welding, which was designed to preclude the
3.2. Type of element and finite-element mesh For the modelling of thin-walled metal structures, the shell element is one of the most appropriate elements. A four-node doubly curved shell element with reduced integration (S4R) was used in the FEM, which has six degrees of freedom per node and could provide accurate solutions based on the previous research performed by Ellobody and Young [14]. In order to obtain the optimum finite-element mesh size, the convergence studies were carried out with the sensitivity analysis results of mesh density shown in Table 3 for aluminium alloy SHS 80 × 80 × 2 with different column lengths. It was found that the size of the finiteelement mesh of approximately 3.5 × 3.5 mm, 5 × 5 mm and 4
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(b) Overall geometric imperfection
(a) Local geometric imperfection
Fig. 3. Eigen-mode 1 of the specimen H-S80 × 80 × 2-L800-d30.4-n3.
10 × 10 mm (length by width) in the modelling of aluminium alloy stub, intermediate-length and slender columns, respectively, could achieve accurate results with minimum computational time. In order to obtain the reasonable finite-element mesh around the circular openings, a straight line with an angle of 45° from the y-direction was used for partition. Furthermore, the shell element with quadrilateral shape and structured meshing technique were adopted in the FEM, which could achieve higher accuracy with less computational cost. Hence, the fine finite-element meshes were used around the circular openings for all specimens, as shown in Fig. 2 for the specimen H-S80 × 80 × 2-L800d30.4-n3.
true curves of high strength aluminium alloy type 6061-T6 and normal strength aluminium alloy type 6063-T5 are clearly plotted in Fig. 1a and 1b, respectively, where the true stress and logarithmic plastic strain were obtained from the measured stress (σ) and strain (ε) by the equations as specified in ABAQUS standard user’s manual [12] as follows: true
pl ln
=
(1)
(1 + )
= ln(1 + )
true / E
(2)
3.5. Initial geometric imperfections
3.3. Boundary condition and loading method
The initial local and overall geometric imperfections were included in the FEM. The initial geometric imperfections in the FEM were applied by an elastic buckling analysis, which is known as linear perturbation analysis. The main purpose of the linear perturbation analysis was to establish the possible buckling modes of aluminium alloy columns, which were used to simulate the initial geometric imperfections of aluminium alloy columns for non-linear analysis. The measured initial local and overall geometric imperfections of aluminium alloy columns were superimposed with the lowest buckling modes (Eigen-mode 1), which were obtained from the linear perturbation analysis as shown in Fig. 3. The lowest local and overall buckling modes were normalized to 1.0 by multiplying the magnitudes of the measured initial local and overall geometric imperfections. The developed FEM was used for test verification.
The reference points were assumed at both ends of the columns. Then, the rigid constraints were established between two ends of the cross-sections and the reference points. The pin-ended boundary condition was simulated by restraining all translational degrees of freedom and rotational degrees of freedom along the x- and z-directions of the reference points at both ends of the columns, except for the translational degree of freedom along the axial direction at one end of the columns, as shown in Fig. 2 for the specimen H-S80 × 80 × 2-L800d30.4-n3. The loading method in terms of displacement control, which was employed in the experimental work, was also used in FEM. The displacement was applied in increments by using the (*STATIC) method available in the ABAQUS library. The compressive axial load was applied by means of displacement to the reference point at one end of the columns. The non-linear geometric parameter (*NLGEOM) was activated in the FEM to consider the large displacement in the numerical simulation.
3.6. Test verification The developed FEM was verified against the experimental results of test series III. A total of 17 aluminium alloy columns with different cross-section dimensions, overall lengths, as well as diameters and numbers of circular openings were analyzed. The measured dimensions and material properties of aluminium alloy columns were used in the modelling. The ultimate strengths, axial load versus axial shortening curves and deformed shapes based on different failure modes of aluminium alloy columns were all investigated. The comparison of the ultimate strengths obtained from the experiments (PEXP) and FEA (PFEA) is shown in Table 2. The mean value of the ultimate strength ratio of experiment-to-FEA is 1.02 with the corresponding coefficient of variation (COV) of 0.050. It is found that the ultimate strengths obtained
3.4. Material properties In the linear elastic stage of FEM, the material properties of aluminium alloys include the density, initial Young’s modulus and Poisson’s ratio. The initial part of the non-linear stress-strain curve represents the elastic property up to the proportional limit stress with measured initial Young’s modulus (E), and Poisson’s ratio (ν) equals 0.3. In the non-linear plastic stage of FEM, the material plasticity or nonlinearity of aluminium alloy was presented by the incremental plasticity model [12], which includes the true stress (σtrue) and logarithmic plastic strain (εpl ln). The full stress-strain curves including both engineering and 5
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(a) H-S50×50×1-L150
(b) H-S50×50×1-L150-d19.2-n1
(c) H-S50×50×1-L1000-d19.2-n6
(d) H-S80×80×2-L240
(e) H-S80×80×2-L240-d30.4-n1
(f) H-S80×80×2-L800
(g) H-S80×80×2-L800-d30.4-n3
(h) H-S80×80×2-L1600
Fig. 4. Comparison of axial load versus axial shortening curves for aluminium alloy columns between experiments and FEA.
6
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(i) H-S80×80×2-L1600-d30.4-n6
(j) H-R120×80×3-L360
(k) H-R120×80×3-L360-d46.4-n1
(l) N-S100×100×3-L300
(m) N-S100×100×3-L300-d37.6-n1
(o) N-S100×100×3-L2000
(n) N-S100×100×3-L1000-d37.6-n3
(p) N-R120×60×2-L360 Fig. 4. (continued)
7
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(a) H-S50×50×1-L150
(b) H-S50×50×1-L150-d19.2-n1
(c) H-S50×50×1-L1000-d19.2-n6
(d) H-S80×80×2-L240
(e) H-S80×80×2-L240-d30.4-n1
(f) H-S80×80×2-L800
Fig. 5. Comparison of deformed shapes of aluminium alloy columns between experiments and FEA.
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(g) H-S80×80×2-L800-d30.4-n3
(h) H-S80×80×2-L1600
(i) H-S80×80×2-L1600-d30.4-n6
(j) H-R120×80×3-L360
(k) H-R120×80×3-L360-d46.4-n1
(l) N-S100×100×3-L300 Fig. 5. (continued)
9
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Fig. 5. (continued)
10
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Table 4 Parameter design of aluminium alloy columns in parametric study. Cross-section (D × B)
Outer depth D (mm)
Outer width B (mm)
Thickness t (mm)
Overall length L
Diameter of opening d
Number of opening n
Location of opening x
SHS 120 × 120 RHS 200 × 120 RHS 300 × 120 SHS 200 × 200 RHS 300 × 200 SHS 300 × 300
120 200 300 200 300 300
120 120 120 200 200 300
2, 4, 6
3D, 15D, 30D
0, 0.2D, 0.5D, 0.8D
1, 3, 6
L/2, L/3, L/4
from the FEA are in good agreement with the experimental results, which may attribute to the accurate measurements of dimensions and material properties of aluminium alloy columns. The comparison of the axial load versus axial shortening curves between the experiments and FEA is plotted in Fig. 4. It is found that a good agreement between the experimental and numerical results is generally obtained. Some discrepancies lie in the initial stiffness and post-ultimate range of the deformation curves, which may attribute to the possible inconsistency of initial geometric imperfections included in the experiments and FEA. Three different failure modes such as local buckling failure, flexural buckling failure and the combination of local and flexural buckling failure were observed in the experimental investigation, which were also verified by the FEA. The comparison of the failure modes between the experiments and FEA is shown in Fig. 5. It is found that the failure modes observed in the experiments are closely simulated by the FEA, except for the specimens N-S100 × 100 × 3L300 and N-R120 × 60 × 2-L360-d46.4-n1. These two specimens failed by material yielding at the end of the columns in the experiments, which are quite different from the failure modes observed in the FEA. This may attribute to the pronounced end effects on the failure of stub columns, which are unavailable in the FEA. Based on the comparison between experimental and FEA results, it is demonstrated that the developed FEM accurately predicted the structural behaviour of aluminium alloy columns in test series III.
Table 5 Comparison of ultimate strengths of aluminium alloy SHS 120 × 120 × 2 with different diameters and locations of circular openings. Specimen
PFEA (kN)
Reduction
S120 × 120 × 2-L360 S120 × 120 × 2-L360-d24-n1 S120 × 120 × 2-L360-d60-n1 S120 × 120 × 2-L360-d96-n1 S120 × 120 × 2-L1800 S120 × 120 × 2-L1800-d24-n1-1/2 S120 × 120 × 2-L1800-d24-n1-1/3 S120 × 120 × 2-L1800-d24-n1-1/4 S120 × 120 × 2-L1800-d60-n1-1/2 S120 × 120 × 2-L1800-d60-n1-1/3 S120 × 120 × 2-L1800-d60-n1-1/4 S120 × 120 × 2-L1800-d96-n1-1/2 S120 × 120 × 2-L1800-d96-n1-1/3 S120 × 120 × 2-L1800-d96-n1-1/4 S120 × 120 × 2-L3600 S120 × 120 × 2-L3600-d24-n1-1/2 S120 × 120 × 2-L3600-d24-n1-1/3 S120 × 120 × 2-L3600-d24-n1-1/4 S120 × 120 × 2-L3600-d60-n1-1/2 S120 × 120 × 2-L3600-d60-n1-1/3 S120 × 120 × 2-L3600-d60-n1-1/4 S120 × 120 × 2-L3600-d96-n1-1/2 S120 × 120 × 2-L3600-d96-n1-1/3 S120 × 120 × 2-L3600-d96-n1-1/4
102.6 99.0 94.5 89.7 96.9 90.0 97.3 91.8 90.5 90.5 88.9 77.1 76.6 76.7 77.4 77.4 77.5 77.1 74.8 74.3 74.9 72.1 71.5 71.6
— 3.51% 7.89% 12.57% — 7.05% −0.45% 5.25% 6.55% 6.53% 8.26% 20.36% 20.88% 20.86% — 0.00% −0.12% 0.32% 3.28% 4.01% 3.22% 6.77% 7.54% 7.43%
Fig. 6. Comparison of Mises stress cloud charts for aluminium alloy columns with and without circular openings.
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ratio (Le/r) of the columns, the opening size ratio (d/D), number (n) and location (x) of the openings are the main geometric parameters investigated in the parametric study. The similar labelling system as that adopted in the experiments was also used in the parametric study. The specimens were labelled based on the cross-section dimensions and overall length of the column, as well as the diameter, number and location of circular openings. For example, the label “S120 × 120 × 2-L1800-d24-n1-1/2” defined an aluminium alloy column with the cross-section of SHS 120 × 120 × 2 and the overall length of 1800 mm, while one circular opening with the diameter of 24 mm was positioned at the middle length of the column. The material properties of high strength aluminium alloy type 6061-T6 obtained from tensile coupon tests conducted by Feng et al. [10] were used in the parametric study. The magnitude of local geometric imperfection was set as 11.8% of the wall thickness in the parametric study, which was equal to the mean value of the maximum measured local geometric imperfections. Furthermore, the magnitude of overall geometric imperfection was adopted as 1/1600 and 1/3000 of the column length for intermediate-length and slender columns, respectively, in the parametric study, which was equal to the mean value of the measured overall geometric imperfections. The size of the finiteelement mesh was kept as 3.5 × 3.5 mm, 5 × 5 mm and 10 × 10 mm (length by width) for the modelling of aluminium alloy stub, intermediate-length and slender columns, respectively, in the parametric study. The ultimate strengths of aluminium alloy SHS and RHS columns with and without circular openings are compared in the parametric study. It is found that the opening reduced the ultimate strengths of the columns, as shown in Table 5, which may attribute to the existence of the opening that weakened the section area of the columns and resulted in the stress concentrations around the opening, as shown in Fig. 6. Therefore, the perforated aluminium alloy columns usually failed at the location of openings, except for the columns with large slenderness ratio. The ultimate strengths of perforated aluminium alloy columns decreased up to 12.57%, 20.88% and 7.54% with the increase of the diameter of circular openings up to 0.8D for aluminium alloy stub, intermediate-length and slender columns, respectively, as shown in Table 5. The axial load versus axial shortening curves are also shown in Fig. 7 for perforated aluminium alloy columns with the cross-section of SHS 120 × 120 × 2 and different diameters of circular openings. Furthermore, the large diameter of circular openings changed the failure modes of aluminium alloy columns, especially for the intermediatelength and slender columns, which also decreased the ultimate strengths of aluminium alloy columns sharply, as shown in Fig. 8. The decrease of the depth-to-thickness ratio of aluminium alloy columns significantly enhanced the ultimate strengths, as shown in Fig. 9 for the specimens with the cross-sections of SHS 120 × 120 and different depth-to-thickness ratios. For aluminium alloy columns with large slenderness ratio, the decrease of the depth-to-thickness ratio also changed the failure modes of aluminium alloy columns from local buckling failure to flexural buckling failure or the combination of local and flexural buckling failure, as shown in Fig. 10. The number of openings has a certain degree of influence on the ultimate strengths of aluminium alloy columns. Table 6 shows the comparison of the ultimate strengths of perforated aluminium alloy columns with the cross-sections of SHS 120 × 120 and different numbers of openings. The ultimate strengths of perforated aluminium alloy columns decreased up to 25.08% and 19.12% with the increase of the number of openings up to 3 and 6 for aluminium alloy intermediatelength and slender columns, respectively, as shown in Table 6. And the decrease of the ultimate strengths further increased with the increase of the diameter of circular openings. The stress distributions along the longitudinal direction of perforated aluminium alloy columns were also influenced by the number of openings. The stress concentrations occurred around the openings by applying axial loads to the perforated aluminium alloy columns, as shown in Fig. 11.
(a) S120×120×2-L360
(b) S120×120×2-L1800
(c) S120×120×2-L3600 Fig. 7. Comparison of axial load versus axial shortening curves for aluminium alloy SHS 120 × 120 × 2 with different diameters of circular openings.
4. Parametric study The FEM was verified to accurately simulate the strengths and behaviour of aluminium alloy columns in test series III as presented by Feng et al. [10]. Therefore, the validated FEM was used for an extensive parametric study. The parametric study was performed on 594 specimens that consisted of six different cross-section dimensions with three different wall thicknesses (t) and three different overall lengths (L). The multiple circular openings with four different diameters (d), three different numbers (n) and three different locations (x) were introduced in the specimens at two opposite sides of the cross-sections along the longitudinal direction of the columns, as summarized in Table 4. Hence, the depth-to-thickness ratio (D/t) of the cross-sections, the slenderness 12
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(a) d=0 mm
(b) d=60 mm
(c) d=96 mm
Fig. 8. Comparison of failure modes for aluminium alloy S120 × 120 × 2-L1800 with different diameters of circular openings.
column strength using the gross cross-section area. The influence of material non-linearity was considered by introducing the tangent modulus theory, and the Euler column curve approximated a hyperbola. The inelastic column curve based on the tangent modulus can be well approximated by a straight line using the buckling constants [20], which can be obtained from Tables 3.3-3 and 3.3-4 of Part I-B of AA [15]. And the influence of elastic local buckling on the column strengths was determined based on Clause 4.7.4 of Part I-B of AA [15]. The design rules of EC9 [16] for calculating the design strengths of imperforated aluminium alloy columns were derived based on the Perry curve. The effect of local buckling on the strengths of aluminium alloy columns was taken into account by replacing the cross-section with the effective cross-section, from which the thickness of the cross-section was reduced by multiplying the local buckling coefficient. Two different design methods including the limit state design (LSD) method and allowable stress design (ASD) method were adopted in AS/NZS [17,18] to calculate the strengths of aluminium alloy columns. It is worth noting that the LSD method in AS/NZS [17] is generally identical to the design rules of AA [15], except for the resistance coefficient and capacity factor introduced in the LSD method. The Perry curve was also adopted in Chinese Code [19] for calculating the design strengths of imperforated aluminium alloy columns. The stability factor of global buckling was calculated by using the Perry-Roberson equation, while the effect of local buckling was considered by using the effective thickness method. It should be noted that the aforementioned design specifications do not have design rules for perforated aluminium alloy members. Nevertheless, North American Specification (NAS) [8] has design rules for cold-formed steel members with openings, which were derived based on the effective width method (EWM). Furthermore, the design rules of NAS [8] were extended by the direct strength method (DSM) proposed by Moen and Schafer [9] for calculating the design strengths of cold-formed steel members with openings. Therefore, the design rules for cold-formed steel members with openings were used in this study to evaluate the strengths of aluminium alloy columns with circular openings by substituting the material properties of aluminium alloys.
Fig. 9. Comparison of ultimate strengths for aluminium alloy SHS 120 × 120 with different depth-to-thickness ratios.
Table 5 shows the comparison of the ultimate strengths of perforated aluminium alloy columns with different locations of the opening. It is demonstrated that the location of the opening has an insignificant influence on the ultimate strengths of perforated aluminium alloy columns with similar decrement for both intermediate-length and slender columns. Whereas, it has a certain degree of influence on the deformations of perforated aluminium alloy columns since the location of the opening changed the stress distributions along the longitudinal direction of perforated aluminium alloy columns, which resulted in the maximum deformations of perforated aluminium alloy columns occurred at the location of the opening, as shown in Fig. 12. 5. Design rules of aluminium alloy members 5.1. Current design rules American Specification (AA) [15], European Code (EC9) [16], Australian/New Zealand Standard (AS/NZS) [17,18] and Chinese Code [19] for the design of aluminium alloy structures provide design rules for imperforated aluminium alloy columns. It is worth noting that the design rules of AA [15] for calculating the design strengths of imperforated aluminium alloy columns were derived based on the Euler
5.2. Comparison of test and FEA strengths with design strengths The ultimate strengths (Pu) of imperforated aluminium alloy columns obtained from the tests and FEA are compared with the design strengths (PAA, PEC, PA/Z1, PA/Z2 and PCC) calculated using the design 13
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(a) t=2 mm
(b) t=6 mm
Fig. 10. Comparison of failure modes for aluminium alloy S200 × 200-L6000-d100 with different depth-to-thickness ratios.
RHSs under axial compression. The ultimate strengths (Pu) of perforated aluminium alloy columns obtained from the tests and FEA are compared with the design strengths (PNAS, PDSM1, PDSM2 and PDSM3) calculated using the design rules of NAS [8] and proposed by Moen and Schafer [9] for cold-formed steel members with openings, respectively, as shown in Fig. 14. The mean values of Pu/PNAS, Pu/PDSM1, Pu/PDSM2 and Pu/PDSM3 are 1.23, 1.65, 1.26 and 1.31, with the corresponding COVs of 0.411, 0.518, 0.304 and 0.303, respectively, as shown in Table 8. It can be seen from the comparison that the current design rules of cold-formed steel members with openings are all quite conservative for aluminium alloy columns with circular openings, in which the design rules derived from DSM1 are most conservative with the largest scatter of predictions.
Table 6 Comparison of ultimate strengths of aluminium alloy SHS 120 × 120 × 2 with different diameters and numbers of circular openings. Specimen
PFEA (kN)
Reduction
S120 × 120 × 2-L1800 S120 × 120 × 2-L1800-d24-n1-1/2 S120 × 120 × 2-L1800-d60-n1-1/2 S120 × 120 × 2-L1800-d96-n1-1/2 S120 × 120 × 2-L1800-d24-n3 S120 × 120 × 2-L1800-d60-n3 S120 × 120 × 2-L1800-d96-n3 S120 × 120 × 2-L3600 S120 × 120 × 2-L3600-d24-n1-1/2 S120 × 120 × 2-L3600-d60-n1-1/2 S120 × 120 × 2-L3600-d96-n1-1/2 S120 × 120 × 2-L3600-d24-n3 S120 × 120 × 2-L3600-d60-n3 S120 × 120 × 2-L3600-d96-n3 S120 × 120 × 2-L3600-d24-n6 S120 × 120 × 2-L3600-d60-n6 S120 × 120 × 2-L3600-d96-n6
96.9 90.0 90.5 77.1 88.2 88.1 72.6 77.4 77.4 74.8 72.1 74.9 72.3 65.1 71.8 69.3 62.6
— 7.05% 6.55% 20.36% 8.98% 9.08% 25.08% — 0.00% 3.28% 6.77% 3.22% 6.59% 15.89% 7.24% 10.47% 19.12%
5.3. Proposed design rules for aluminium alloy columns with circular openings It is concluded from the evaluation of current design specifications that the design rules of EC9 [16] are generally appropriate for imperforated aluminium alloy columns. And it can be seen from the numerical results that the diameter and number of circular openings significantly affected the ultimate strengths of perforated aluminium alloy columns. Therefore, the design equations are newly proposed for perforated aluminium alloy columns based on the design rules of EC9 [16] for imperforated aluminium alloy columns by modifying the flexural buckling load (Nb,Rd) and local squashing load (Nc,Rd) through introducing the ratio of net area-to-total area of surface (Sh/S) and the correction factor as follows:
rules of AA [15], EC9 [16], LSD and ASD in AS/NZS [17,18] and Chinese Code [19], respectively, as shown in Fig. 13. The mean values of Pu/PAA, Pu/PEC, Pu/PA/Z1, Pu/PA/Z2 and Pu/PCC are 0.74, 1.14, 0.88, 1.41 and 1.55, with the corresponding COVs of 0.238, 0.149, 0.282, 0.493 and 0.756, respectively, as shown in Table 7. It can be seen from the comparison that the design rules of AA [15] and LSD in AS/NZS [17] are somewhat unconservative, whereas the design rules of ASD in AS/ NZS [18] and Chinese Code [19] are quite conservative with the largest value of COV. The design rules of EC9 [16] are somewhat conservative with the smallest value of COV. Therefore, the design rules of EC9 [16] are generally appropriate for imperforated aluminium alloy SHSs and
Ppd = min(Nb, Rd, Nc, Rd )
(3)
For the flexural buckling failure, the design resistances of perforated aluminium alloy columns can be calculated as follows:
14
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(a) n=0
(b) n=1
(c) n=3
Fig. 11. Comparison of failure modes for aluminium alloy S200 × 200 × 2-L6000-d100 with different numbers of openings.
(a) p=1/2
(b) p=1/3
(c) p=1/4
Fig. 12. Influence of locations of opening on deformations of perforated aluminium alloy columns.
Nb, Rd = 1.037 Aeff fy
Sh S
obtained from the tests and FEA are also compared with the proposed design strengths (Ppd) calculated using the proposed design equations, as shown in Fig. 15. It is demonstrated from the comparison that the proposed design strengths are in good agreement with the test and FEA strengths with the mean value (Pu/Ppd) of 1.01 and the corresponding COV of 0.098, as shown in Table 8. Therefore, the proposed design equations were verified to be accurate for aluminium alloy SHS and RHS columns with circular openings under axial compression.
10.94
(4)
where κ is a reduction factor to allow for the weakening effects of welding, χ is a reduction factor for relevant buckling mode, Aeff is the effective area, fy is the yield stress (0.2% tensile proof stress for aluminium alloy), Sh is the net area of surface, and S is the total area of surface. For the local squashing failure, the design resistances of perforated aluminium alloy columns can be calculated as follows:
Nc , Rd = 1.037A eff f y
Sh S
6. Conclusions
10.94
This paper presents a numerical investigation on perforated aluminium alloy SHS and RHS columns by the FEA. The geometric and material non-linearities of aluminium alloy columns were included in the FEM. The non-linear FEM was verified against the corresponding experimental results, which was used to carry out an extensive parametric study. The parametric study consisted of 594 specimens with different
(5)
where Aeff is the effective area, fy is the yield stress (0.2% tensile proof stress for aluminium alloy), Sh is the net area of surface, and S is the total area of surface. The ultimate strengths (Pu) of perforated aluminium alloy columns
15
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(a) American Specification (AA) [15]
(b) European Code (EC9) [16]
(c) LSD in AS/NZS [17]
(d) ASD in AS/NZS [18]
(e) Chinese Code [19] Fig. 13. Comparison of test and FEA strengths with design strengths for aluminium alloy columns.
16
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Table 7 Comparison of test and FEA strengths with design strengths for imperforated aluminium alloy columns. Specimen (A total of 62 SHS and RHS columns)
Comparison Pu/PAA
Pu/PEC
Pu/PA/Z1
Pu/PA/Z2
Pu/PCC
Mean COV
0.74 0.238
1.14 0.149
0.88 0.282
1.41 0.493
1.55 0.756
Table 8 Comparison of test and FEA strengths with design strengths for perforated aluminium alloy columns. Specimen (A total of 552 SHS and RHS columns)
Comparison Pu/PNAS
Pu/PDSM1
Pu/PDSM2
Pu/PDSM3
Pu/Ppd
Mean COV
1.23 0.411
1.65 0.518
1.26 0.304
1.31 0.303
1.01 0.098
depth-to-thickness ratio also changed the failure modes of aluminium alloy columns from local buckling failure to flexural buckling failure or the combination of local and flexural buckling failure. (3) It is shown from the comparison that AA and LSD in AS/NZS are somewhat unconservative, whereas ASD in AS/NZS and Chinese Code are quite conservative, while EC9 is generally appropriate for imperforated aluminium alloy columns. On the other hand, it is also demonstrated from the comparison that NAS and DSM for coldformed steel members with openings are all quite conservative for aluminium alloy columns with circular openings. (4) The design equations are proposed in this study based on the design rules of EC9, which were verified to be accurate for perforated aluminium alloy SHS and RHS columns under axial compression.
cross-section dimensions, overall lengths, as well as diameters, numbers and locations of circular openings. Some conclusions are drawn from the FEA as follows: (1) The ultimate strengths of aluminium alloy columns were significantly weakened by the introduction of the openings, especially the increase of the diameter and number of circular openings, whereas the location of the opening has little influence on the ultimate strengths of perforated aluminium alloy columns. (2) The decrease of the depth-to-thickness ratio of aluminium alloy columns significantly enhanced the ultimate strengths. For aluminium alloy columns with large slenderness ratio, the decrease of the
(a) NAS [8]
(b) DSM1
(c) DSM2
(d) DSM3
Fig. 14. Comparison of test and FEA strengths with design strengths for perforated aluminium alloy columns.
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Civil Engineering, Shenzhen Durability Center for Civil Engineering (Shenzhen University, Grant No. GDDCE 18-5). References [1] Shanmugam NE, Dhanalakshmi M. Design for openings in cold-formed steel channel stub columns. Thin-Walled Struct 2001;39(12):961–81. [2] Moen CD, Schafer BW. Experiments on cold-formed steel columns with holes. ThinWalled Struct 2008;46(10):1164–82. [3] Yao Z, Rasmussen KJR. Inelastic local buckling behaviour of perforated plates and sections under compression. Thin-Walled Struct 2012;61(6):49–70. [4] Kulatunga MP, Macdonald M. Investigation of cold-formed steel structural members with perforations of different arrangements subjected to compression loading. Thin-Walled Struct 2013;67:78–87. [5] Kulatunga MP, Macdonald M, Rhodes J, Harrison DK. Load capacity of cold-formed column members of lipped channel cross-section with perforations subjected to compression loading—Part I: FE simulation and test results. Thin-Walled Struct 2014;80:1–12. [6] Zhou F, Young B. Web crippling of aluminium tubes with perforated webs. Eng Struct 2010;32(5):1397–410. [7] Feng R, Young B. Experimental investigation of aluminium alloy stub columns with circular openings. J Struct Eng, ASCE 2015;141(11):04015031. [8] NAS. North American Specification for the Design of Cold-Formed Steel Structural Members. American Iron and Steel Institute, AISI S100-16, Washington, D.C., USA; 2016. [9] Moen CD, Schafer BW. Direct strength method for design of cold-formed steel columns with holes. J Struct Eng, ASCE 2011;137(5):559–70. [10] Feng R, Zhu W, Wan HY, Chen AY, Chen Y. Tests of perforated aluminium alloy SHSs and RHSs under axial compression. Thin-Walled Struct 2018;130:194–212. [11] Kechidi S, Fratamico DC, Castro JM, Bourahla N, Schafer BW. Numerical study on the behavior and design of screw connected built-up CFS chord studs. In: Proceedings of the Annual Stability Conference. Structural Stability Research Council, San Antonio, USA, 2017; p. 1–25. [12] ABAQUS Analysis User’s Manual, Version 6.14. ABAQUS, Inc., 2014. [13] Mazzolani FM. Aluminium alloy structures. 2nd ed. London: E & FN Spon; 1995. [14] Ellobody E, Young B. Structural performance of cold-formed high strength stainless steel columns. J Constr Steel Res 2005;61(12):1631–49. [15] AA. Aluminum Design Manual. The Aluminum Association, Washington, D.C., USA, 2005. [16] EC9. Eurocode 9: Design of Aluminium Structures—Part 1-1: General Structural Rules. European Committee for Standardization, EN 1999-1-1:2007, CEN. Brussels, Belgium, 2007. [17] AS/NZS. Aluminium Structures Part 1: Limit State Design, Australian/New Zealand Standard, AS/NZS 1664.1: 1997. Standard Australia, Sydney, Australia, 1997. [18] AS/NZS. Aluminium Structures Part 2: Allowable Stress Design, Australian/New Zealand Standard, AS/NZS 1664.2: 1997. Standard Australia, Sydney, Australia, 1997. [19] Chinese Code. Code for Design of Aluminium Structures. GB 50429-2007, Beijing, China, 2007. [in Chinese]. [20] Sharp ML. Behaviour and Design of Aluminium Alloy Structure. New York: McGraw-Hill; 1993.
Fig. 15. Comparison of test and FEA strengths with proposed design strengths for perforated aluminium alloy columns.
Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements The authors are grateful for the financial support from National Natural Science Foundation of China (Grant No. 51528803), Natural Science Foundation of Guangdong Province of China (Grant No. 2018A030313208), State Key Laboratory of Subtropical Building Science (South China University of Technology, Grant No. 2018ZA02), and Guangdong Provincial Key Laboratory of Durability for Marine
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Numerical investigation and design of perforated aluminium alloy SHS and RHS columns
T
Ran Fenga,b, , Jiarui Liua ⁎
a b
School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen 518055, China State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China
ARTICLE INFO
ABSTRACT
Keywords: Aluminium alloy Numerical investigation Parametric study Perforated column Rectangular hollow section (RHS) Square hollow section (SHS)
This paper describes a numerical investigation on perforated aluminium alloy square and rectangular hollow section (SHS and RHS) columns by the finite-element analysis (FEA). The non-linear finite-element model (FEM) was developed by considering both geometric and material non-linearities. The initial local and overall geometric imperfections of aluminium alloy columns were incorporated in the FEM. The non-linear FEM was verified against the corresponding experimental results, which was further used for an extensive parametric study that consisted of 594 specimens with different cross-section dimensions, overall lengths, as well as diameters, numbers and locations of circular openings. The ultimate strengths of the columns obtained from FEA were employed to evaluate the current design specifications for aluminium alloy and cold-formed steel structures. It is demonstrated that American Specification (AA) and limit state design (LSD) in Australian/New Zealand Standard (AS/NZS) are somewhat unconservative, whereas allowable stress design (ASD) in Australian/ New Zealand Standard (AS/NZS) and Chinese Code are quite conservative, while European Code (EC9) is generally appropriate for imperforated aluminium alloy columns. In addition, North American Specification (NAS) and direct strength method (DSM) for cold-formed steel structural members with openings are quite conservative for aluminium alloy columns with circular openings. The design equations proposed based on the design rules of EC9 were verified to be accurate for perforated aluminium alloy SHS and RHS columns under axial compression.
1. Introduction Aluminium alloy has been widely used in structural applications such as curtain walls, space structures and bridges owing to its material advantages of high strength-to-weight ratio, excellent corrosion resistance and ease of manufacture. Currently, structural members are often perforated with one or more openings to facilitate the construction assembly, building service and equipment installation. These prepunched openings usually weaken the sectional dimensions and destroy the geometric continuity of structural members, which result in the deterioration of the elastic stiffness and ultimate strengths of structural members. The shape, size, number and location of openings have a certain degree of influences on the behaviour of perforated structural members. Finite-element analysis (FEA) is a useful and powerful tool that has been widely used in structural analysis and design. Compared to the physical experiments, FEA is a relatively time-saving and convenient method, especially for the structural members with various cross-
⁎
section geometries. In order to find an accurate and stable finite-element model (FEM) for the parametric study, it is mandatory to verify the FEM against the corresponding experimental results. A large number of researches were performed on perforated coldformed steel compression members. In 2001, Shanmugam and Dhanalakshmi [1] studied the ultimate compressive strengths of perforated cold-formed steel channel stub columns with different plate slenderness ratios, opening shapes and sizes by the FEA. The web plate slenderness ratio and opening area ratio were found to be two main variables on the ultimate compressive strengths. In 2007, Moen and Schafer [2] carried out a series of compression tests on 24 cold-formed steel short and intermediate-length columns with and without slotted web holes. The relationship between the elastic buckling and test response of perforated cold-formed steel columns was investigated. It was found that the post-peak response and ductility were influenced by the slotted holes, the cross-section type and length of the columns. For the short columns with higher hole-to-web width ratio, the slotted holes influenced the post-peak response and reduced the ductility of cold-
Corresponding author at: School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen 518055, China. E-mail address: [email protected] (R. Feng).
https://doi.org/10.1016/j.engstruct.2019.109591 Received 30 January 2019; Received in revised form 23 August 2019; Accepted 25 August 2019 Available online 04 September 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature
PDSM2
Notation Aeff B COV d D E fy L Le n Nb,Rd Nc,Rd PAA PA/N-1 PA/N-2 PCC PDSM1
PDSM3
Effective area Outer width of SHS and RHS Coefficient of variation Diameter of circular opening Outer depth of SHS and RHS Young’s modulus Yield stress (0.2% tensile proof stress) Overall length of aluminium alloy column Effective length of aluminium alloy column Number of opening Flexural buckling load Local squashing load Design strength obtained from design formulae of American Specification Design strength obtained from design formulae of AS/NZS 1664.1: 1997 Design strength obtained from design formulae of AS/NZS 1664.2: 1997 Design strength obtained from design formulae of GB 50429-2007 Design strength obtained from 1st design criteria proposed
PEC PEXP PFEA PNAS Ppd Pu r S Sh t ε εf εpl ln κ ν σ σtrue σu χ
formed steel columns. For the short columns with lower hole-to-web width ratio and intermediate-length columns, the slotted holes had a small influence on the post-peak response and ductility. Moreover, the existence of slotted holes reduced the ultimate strengths of cold-formed steel columns. In 2012, Yao and Rasmussen [3] researched the effects of different shapes, sizes and spacings of perforations on the inelastic stress distributions, load transfers and failure modes of perforated simply-supported plates and perforated C-section columns in compression. It was found that the perforations changed the stress distributions within the members obviously and triggered the distinct failure modes of inward and outward buckles at the nodal lines of local buckles. In 2013, Kulatunga and Macdonald [4] investigated the influence of perforation positions on the ultimate compressive strengths of cold-formed steel columns with lipped channel cross-sections through the FEA. The ultimate compressive strengths were found to be affected by perforation positions. And the perforation positions near the ends of the columns have a greater weakening effect than the perforations at other positions. Furthermore, Kulatunga et al. [5] also researched the influence of perforation shapes on the buckling behaviour of cold-formed steel columns with lipped channel cross-sections by the FEA. It was found that the ultimate compressive strengths varied greatly with the presence of perforations, which decreased with the increase of the length of perforations. However, limited researches were conducted on perforated
by Moen and Schafer Design strength obtained from 2nd design criteria proposed by Moen and Schafer Design strength obtained from 3rd design criteria proposed by Moen and Schafer Design strength obtained from design formulae of EC9 Ultimate strength obtained from experiment Ultimate strength obtained from FEA Design strength obtained from design formulae of NAS Design strength obtained from proposed design equations Ultimate strength Radius of gyration of SHS and RHS Total area of surface Net area of surface Thickness of SHS and RHS Strain Elongation after fracture Logarithmic plastic strain Reduction factor to allow for weakening effect of welding Poisson’s ratio Stress True stress Ultimate tensile stress Reduction factor for relevant buckling mode
aluminium alloy structural members. In 2010, the experimental and numerical investigations were carried out by Zhou and Young [6] on aluminium alloy square hollow sections (SHSs) with circular holes in the webs subjected to web crippling. The diameter of circular holes was found to be the main factor affecting the web crippling behaviour. In 2015, Feng and Young [7] experimentally investigated the compressive behaviour of aluminium alloy SHS stub columns with circular openings. The ultimate compressive strengths obtained from the tests were used to assess the current design rules. It was found that the current design rules for perforated cold-formed steel members were not suitable for perforated aluminium alloy SHS stub columns. Nevertheless, there is no design rule applicable to perforated aluminium alloy columns. It is worth noting that North American Specification (NAS) [8] provides design rules for cold-formed steel members containing holes. The main objective of this study is to derive the design guidelines for perforated aluminium alloy square and rectangular hollow section (SHS and RHS) columns under axial compression based on the numerical investigation. Hence, this paper mainly focuses on the development and verification of the FEM, which was used to carry out the parametric study and propose the design equations for perforated aluminium alloy SHS and RHS columns. An accurate and reliable nonlinear FEM was developed in this study for aluminium alloy SHS and RHS columns with and without circular openings. The FEM was validated by comparing with the corresponding experimental results,
Table 1 Initial geometric imperfections and material properties of aluminium alloy columns in experimental investigation. Section (D × B × t)
SHS50 × 50 × 1 SHS80 × 80 × 2 RHS120 × 80 × 3 SHS100 × 100 × 3 RHS120 × 60 × 2 RHS150 × 100 × 2.5
Max local geometric imperfection
Max overall geometric imperfection
Δ/t
δ/L
0.36 0.082 0.03 0.12 0.07 0.048
1/1028 1/984 1/1181 1/1897 1/1050 1/1920
Aluminium alloy
6061-T6 6063-T5
2
Material property E (GPa)
σ0.2 (MPa)
σu (MPa)
εf (%)
68.0 68.2 66.8 69.3 69.6 66.7
222.19 207.61 203.75 184.30 191.27 167.55
237.22 219.23 226.23 209.07 211.72 183.91
8.02 10.60 12.08 13.13 15.14 12.06
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(a) 6061-T6
(b) 6063-T5
Fig. 1. Engineering and true stress-strain curves of high strength aluminium alloy type 6061-T6 and normal strength aluminium alloy type 6063-T5.
Table 2 Comparison of ultimate strengths of aluminium alloy columns between experiments and FEA. Specimen
Experimental result
H-S50 × 50 × 1-L150 H-S50 × 50 × 1-L150-d19.2-n1 H-S50 × 50 × 1-L1000-d19.2-n6 H-S80 × 80 × 2-L240 H-S80 × 80 × 2-L240-d30.4-n1 H-S80 × 80 × 2-L800 H-S80 × 80 × 2-L800-d30.4-n3 H-S80 × 80 × 2-L1600 H-S80 × 80 × 2-L1600-d30.4-n6 H-R120 × 80 × 3-L360 H-R120 × 80 × 3-L360-d46.4-n1 N-S100 × 100 × 3-L300 N-S100 × 100 × 3-L300-d37.6-n1 N-S100 × 100 × 3-L1000-d37.6-n3 N-S100 × 100 × 3-L2000 N-R120 × 60 × 2-L360 N-R120 × 60 × 2-L360-d46.4-n1 Mean COV
FEA result
Comparison
Failure mode
Ultimate strength PEXP (kN)
Failure mode
Ultimate strength PFEA (kN)
PEXP/PFEA
L L L L L L L L+F L L L L L L L+F L L
32.4 27.5 25.6 95.6 82.9 90.5 82.8 82.4 62.4 180.0 158.0 166.7 149.5 135.5 163.5 87.2 72.9
L L L L L L L L+F L L L L L L L+F L L
31.5 27.7 23.1 96.0 81.0 90.1 75.1 78.5 61.7 179.5 163.0 157.3 145.9 146.1 167.8 78.9 73.9
1.03 0.99 1.11 1.00 1.02 1.00 1.10 1.05 1.02 1.00 0.97 1.06 1.02 0.93 0.97 1.10 0.99 1.02 0.050
Note: L = Local buckling; F = Flexural buckling.
ultimate strengths of the columns predicted by the FEA are compared with the design strengths calculated using the design rules of NAS [8] and derived by Moen and Schafer [9] for cold-formed steel columns with holes by substituting the material properties of aluminium alloys. The design equations are proposed for perforated aluminium alloy SHS and RHS columns based on the current design rules.
Table 3 Sensitivity analysis of mesh density for aluminium alloy SHS 80 × 80 × 2 with different column lengths. Specimen
Mesh density (mm)
No. of element
Running time (s)
PFEA (kN)
PEXP/PFEA
H-S80 × 80 × 2-L240
3 3.5 4 4 5 6 5 8 10 12
8,640 6,348 4,800 16,000 10,240 6,916 20,480 8,000 5,120 3,724
86 63 56 379 226 156 345 154 84 61
96.3 96.0 91.9 91.8 90.1 85.3 80.5 79.4 78.5 75.9
0.99 1.00 1.04 0.99 1.00 1.06 1.02 1.04 1.05 1.09
H-S80 × 80 × 2-L800 H-S80 × 80 × 2-L1600
2. Summary of experimental investigation The experimental investigation presented by Feng et al. [10] provided the ultimate strengths and failure modes of aluminium alloy SHS and RHS columns with and without circular openings compressed between pin-ends. The test specimens included six different cross-section dimensions with different column lengths ranged from 150 to 3000 mm, which were fabricated by extrusion of SHS and RHS using heat-treated aluminium alloys of 6061-T6 and 6063-T5. The material properties of high strength aluminium alloy type 6061-T6 and normal strength aluminium alloy type 6063-T5 were determined by longitudinal tensile coupon tests, which include the initial Young’s modulus (E), the 0.2% tensile proof stress, the ultimate tensile stress (σu) and the elongation after fracture (εf), as summarized in Table 1. The full stress-
which was then used for the parametric study of perforated aluminium alloy SHS and RHS columns with different geometrical dimensions, as well as diameters, numbers and locations of circular openings. The 3
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formation of the heat-affected zone. The pin-ended bearing was employed at both ends of the special aluminium alloy sleeves for pin-ended compression tests. The identical loading device including the special fixed-ended bearing and pin-ended bearing as that used in test series II were employed in test series III for pin-ended compression tests. The same testing procedure as that used in test series II was employed in test series III for pin-ended compression tests. All specimens failed by either local buckling or flexural buckling. Initial local and overall geometric imperfections about the minor axis of the specimens were measured prior to testing. It is worth noting that five types of geometric imperfections on the minor-axis global buckling mode, major-axis global buckling mode, pure torsional buckling mode, local buckling mode and distortional buckling mode were investigated by Kechidi et al. [11] on single and built-up sections by using the Modal Imperfection Decomposition (MID) method, which is particularly useful for the open sections with various patterns of geometric imperfections. However, the SHSs and RHSs investigated in this study are doubly symmetric closed sections, which are normally subjected to the minor-axis global buckling and local buckling only. Hence, the conventional hand-measurement approach was adopted in the experimental investigation to obtain the initial geometric imperfections. The maximum local and overall geometric imperfections of the specimens with six different cross-section dimensions are also shown in Table 1. The measured specimen dimensions, test rig and procedure are detailed in Feng et al. [10]. The ultimate strengths and failure modes obtained from test series III are summarized in Table 2, which were carefully simulated in the corresponding numerical analysis in this study. 3. Finite-element modelling 3.1. General The finite-element program ABAQUS [12] was used in the numerical analysis for the simulation of aluminium alloy SHS and RHS pinended columns tested by Feng et al. [10]. The material properties, measured geometric dimensions, initial local and overall geometric imperfections of test specimens were all incorporated in the FEM. Residual stress caused by heat treatment was not considered in the FEM since it has little influence on the load-carrying capacities of extruded aluminium alloy profiles [13]. The finite-element simulation consisted of two steps. The first step is an elastic buckling analysis (linear perturbation analysis in the ABAQUS library), which was performed to determine the buckling modes (Eigen-mode) of aluminium alloy columns by the Eigen-value analysis. The second step is a load-displacement non-linear analysis, which was performed to obtain the ultimate strengths and failure modes of aluminium alloy columns by using the (*RIKS) method available in the ABAQUS library. The non-linear geometric parameter (*NLGEOM) was activated in the (*RIKS) method to consider the large displacement in the numerical simulation. Furthermore, the initial geometric imperfections and material non-linearities were all taken into account in the (*RIKS) method.
Fig. 2. Finite-element mesh of the specimen H-S80 × 80 × 2-L800-d30.4-n3.
strain curves of aluminium alloy SHS and RHS tubes with six different cross-section dimensions are clearly shown in Fig. 1. There are three series of compression tests. Test series I included 5 columns with both ends welded to aluminium end plates. The specimens compressed between pin-ended bearings were allowed to rotate about the minor axis only. The axial compression force was applied to the specimens through displacement control by using a servo-controlled hydraulic testing machine. It was found that all specimens failed by material yielding at the heat-affected zone (HAZ). Test series Ⅱ included 35 columns with both ends reinforced by carbon fibre-reinforced polymer (CFRP) at the heat-affected zone resulted from the welding. The similar testing procedure was employed by using the special fixedended bearing and pin-ended bearing to conduct pin-ended compression tests. It was found that some specimens also failed by material yielding at the heat-affected zone (HAZ) even if the affected area has been reinforced by CFRP, which means the material strength of aluminium alloy is significantly deteriorated by the welding. Test series III included 17 columns with both ends fixed by a special aluminium alloy sleeve instead of welding, which was designed to preclude the
3.2. Type of element and finite-element mesh For the modelling of thin-walled metal structures, the shell element is one of the most appropriate elements. A four-node doubly curved shell element with reduced integration (S4R) was used in the FEM, which has six degrees of freedom per node and could provide accurate solutions based on the previous research performed by Ellobody and Young [14]. In order to obtain the optimum finite-element mesh size, the convergence studies were carried out with the sensitivity analysis results of mesh density shown in Table 3 for aluminium alloy SHS 80 × 80 × 2 with different column lengths. It was found that the size of the finiteelement mesh of approximately 3.5 × 3.5 mm, 5 × 5 mm and 4
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(b) Overall geometric imperfection
(a) Local geometric imperfection
Fig. 3. Eigen-mode 1 of the specimen H-S80 × 80 × 2-L800-d30.4-n3.
10 × 10 mm (length by width) in the modelling of aluminium alloy stub, intermediate-length and slender columns, respectively, could achieve accurate results with minimum computational time. In order to obtain the reasonable finite-element mesh around the circular openings, a straight line with an angle of 45° from the y-direction was used for partition. Furthermore, the shell element with quadrilateral shape and structured meshing technique were adopted in the FEM, which could achieve higher accuracy with less computational cost. Hence, the fine finite-element meshes were used around the circular openings for all specimens, as shown in Fig. 2 for the specimen H-S80 × 80 × 2-L800d30.4-n3.
true curves of high strength aluminium alloy type 6061-T6 and normal strength aluminium alloy type 6063-T5 are clearly plotted in Fig. 1a and 1b, respectively, where the true stress and logarithmic plastic strain were obtained from the measured stress (σ) and strain (ε) by the equations as specified in ABAQUS standard user’s manual [12] as follows: true
pl ln
=
(1)
(1 + )
= ln(1 + )
true / E
(2)
3.5. Initial geometric imperfections
3.3. Boundary condition and loading method
The initial local and overall geometric imperfections were included in the FEM. The initial geometric imperfections in the FEM were applied by an elastic buckling analysis, which is known as linear perturbation analysis. The main purpose of the linear perturbation analysis was to establish the possible buckling modes of aluminium alloy columns, which were used to simulate the initial geometric imperfections of aluminium alloy columns for non-linear analysis. The measured initial local and overall geometric imperfections of aluminium alloy columns were superimposed with the lowest buckling modes (Eigen-mode 1), which were obtained from the linear perturbation analysis as shown in Fig. 3. The lowest local and overall buckling modes were normalized to 1.0 by multiplying the magnitudes of the measured initial local and overall geometric imperfections. The developed FEM was used for test verification.
The reference points were assumed at both ends of the columns. Then, the rigid constraints were established between two ends of the cross-sections and the reference points. The pin-ended boundary condition was simulated by restraining all translational degrees of freedom and rotational degrees of freedom along the x- and z-directions of the reference points at both ends of the columns, except for the translational degree of freedom along the axial direction at one end of the columns, as shown in Fig. 2 for the specimen H-S80 × 80 × 2-L800d30.4-n3. The loading method in terms of displacement control, which was employed in the experimental work, was also used in FEM. The displacement was applied in increments by using the (*STATIC) method available in the ABAQUS library. The compressive axial load was applied by means of displacement to the reference point at one end of the columns. The non-linear geometric parameter (*NLGEOM) was activated in the FEM to consider the large displacement in the numerical simulation.
3.6. Test verification The developed FEM was verified against the experimental results of test series III. A total of 17 aluminium alloy columns with different cross-section dimensions, overall lengths, as well as diameters and numbers of circular openings were analyzed. The measured dimensions and material properties of aluminium alloy columns were used in the modelling. The ultimate strengths, axial load versus axial shortening curves and deformed shapes based on different failure modes of aluminium alloy columns were all investigated. The comparison of the ultimate strengths obtained from the experiments (PEXP) and FEA (PFEA) is shown in Table 2. The mean value of the ultimate strength ratio of experiment-to-FEA is 1.02 with the corresponding coefficient of variation (COV) of 0.050. It is found that the ultimate strengths obtained
3.4. Material properties In the linear elastic stage of FEM, the material properties of aluminium alloys include the density, initial Young’s modulus and Poisson’s ratio. The initial part of the non-linear stress-strain curve represents the elastic property up to the proportional limit stress with measured initial Young’s modulus (E), and Poisson’s ratio (ν) equals 0.3. In the non-linear plastic stage of FEM, the material plasticity or nonlinearity of aluminium alloy was presented by the incremental plasticity model [12], which includes the true stress (σtrue) and logarithmic plastic strain (εpl ln). The full stress-strain curves including both engineering and 5
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(a) H-S50×50×1-L150
(b) H-S50×50×1-L150-d19.2-n1
(c) H-S50×50×1-L1000-d19.2-n6
(d) H-S80×80×2-L240
(e) H-S80×80×2-L240-d30.4-n1
(f) H-S80×80×2-L800
(g) H-S80×80×2-L800-d30.4-n3
(h) H-S80×80×2-L1600
Fig. 4. Comparison of axial load versus axial shortening curves for aluminium alloy columns between experiments and FEA.
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(i) H-S80×80×2-L1600-d30.4-n6
(j) H-R120×80×3-L360
(k) H-R120×80×3-L360-d46.4-n1
(l) N-S100×100×3-L300
(m) N-S100×100×3-L300-d37.6-n1
(o) N-S100×100×3-L2000
(n) N-S100×100×3-L1000-d37.6-n3
(p) N-R120×60×2-L360 Fig. 4. (continued)
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(a) H-S50×50×1-L150
(b) H-S50×50×1-L150-d19.2-n1
(c) H-S50×50×1-L1000-d19.2-n6
(d) H-S80×80×2-L240
(e) H-S80×80×2-L240-d30.4-n1
(f) H-S80×80×2-L800
Fig. 5. Comparison of deformed shapes of aluminium alloy columns between experiments and FEA.
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(g) H-S80×80×2-L800-d30.4-n3
(h) H-S80×80×2-L1600
(i) H-S80×80×2-L1600-d30.4-n6
(j) H-R120×80×3-L360
(k) H-R120×80×3-L360-d46.4-n1
(l) N-S100×100×3-L300 Fig. 5. (continued)
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Fig. 5. (continued)
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Table 4 Parameter design of aluminium alloy columns in parametric study. Cross-section (D × B)
Outer depth D (mm)
Outer width B (mm)
Thickness t (mm)
Overall length L
Diameter of opening d
Number of opening n
Location of opening x
SHS 120 × 120 RHS 200 × 120 RHS 300 × 120 SHS 200 × 200 RHS 300 × 200 SHS 300 × 300
120 200 300 200 300 300
120 120 120 200 200 300
2, 4, 6
3D, 15D, 30D
0, 0.2D, 0.5D, 0.8D
1, 3, 6
L/2, L/3, L/4
from the FEA are in good agreement with the experimental results, which may attribute to the accurate measurements of dimensions and material properties of aluminium alloy columns. The comparison of the axial load versus axial shortening curves between the experiments and FEA is plotted in Fig. 4. It is found that a good agreement between the experimental and numerical results is generally obtained. Some discrepancies lie in the initial stiffness and post-ultimate range of the deformation curves, which may attribute to the possible inconsistency of initial geometric imperfections included in the experiments and FEA. Three different failure modes such as local buckling failure, flexural buckling failure and the combination of local and flexural buckling failure were observed in the experimental investigation, which were also verified by the FEA. The comparison of the failure modes between the experiments and FEA is shown in Fig. 5. It is found that the failure modes observed in the experiments are closely simulated by the FEA, except for the specimens N-S100 × 100 × 3L300 and N-R120 × 60 × 2-L360-d46.4-n1. These two specimens failed by material yielding at the end of the columns in the experiments, which are quite different from the failure modes observed in the FEA. This may attribute to the pronounced end effects on the failure of stub columns, which are unavailable in the FEA. Based on the comparison between experimental and FEA results, it is demonstrated that the developed FEM accurately predicted the structural behaviour of aluminium alloy columns in test series III.
Table 5 Comparison of ultimate strengths of aluminium alloy SHS 120 × 120 × 2 with different diameters and locations of circular openings. Specimen
PFEA (kN)
Reduction
S120 × 120 × 2-L360 S120 × 120 × 2-L360-d24-n1 S120 × 120 × 2-L360-d60-n1 S120 × 120 × 2-L360-d96-n1 S120 × 120 × 2-L1800 S120 × 120 × 2-L1800-d24-n1-1/2 S120 × 120 × 2-L1800-d24-n1-1/3 S120 × 120 × 2-L1800-d24-n1-1/4 S120 × 120 × 2-L1800-d60-n1-1/2 S120 × 120 × 2-L1800-d60-n1-1/3 S120 × 120 × 2-L1800-d60-n1-1/4 S120 × 120 × 2-L1800-d96-n1-1/2 S120 × 120 × 2-L1800-d96-n1-1/3 S120 × 120 × 2-L1800-d96-n1-1/4 S120 × 120 × 2-L3600 S120 × 120 × 2-L3600-d24-n1-1/2 S120 × 120 × 2-L3600-d24-n1-1/3 S120 × 120 × 2-L3600-d24-n1-1/4 S120 × 120 × 2-L3600-d60-n1-1/2 S120 × 120 × 2-L3600-d60-n1-1/3 S120 × 120 × 2-L3600-d60-n1-1/4 S120 × 120 × 2-L3600-d96-n1-1/2 S120 × 120 × 2-L3600-d96-n1-1/3 S120 × 120 × 2-L3600-d96-n1-1/4
102.6 99.0 94.5 89.7 96.9 90.0 97.3 91.8 90.5 90.5 88.9 77.1 76.6 76.7 77.4 77.4 77.5 77.1 74.8 74.3 74.9 72.1 71.5 71.6
— 3.51% 7.89% 12.57% — 7.05% −0.45% 5.25% 6.55% 6.53% 8.26% 20.36% 20.88% 20.86% — 0.00% −0.12% 0.32% 3.28% 4.01% 3.22% 6.77% 7.54% 7.43%
Fig. 6. Comparison of Mises stress cloud charts for aluminium alloy columns with and without circular openings.
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ratio (Le/r) of the columns, the opening size ratio (d/D), number (n) and location (x) of the openings are the main geometric parameters investigated in the parametric study. The similar labelling system as that adopted in the experiments was also used in the parametric study. The specimens were labelled based on the cross-section dimensions and overall length of the column, as well as the diameter, number and location of circular openings. For example, the label “S120 × 120 × 2-L1800-d24-n1-1/2” defined an aluminium alloy column with the cross-section of SHS 120 × 120 × 2 and the overall length of 1800 mm, while one circular opening with the diameter of 24 mm was positioned at the middle length of the column. The material properties of high strength aluminium alloy type 6061-T6 obtained from tensile coupon tests conducted by Feng et al. [10] were used in the parametric study. The magnitude of local geometric imperfection was set as 11.8% of the wall thickness in the parametric study, which was equal to the mean value of the maximum measured local geometric imperfections. Furthermore, the magnitude of overall geometric imperfection was adopted as 1/1600 and 1/3000 of the column length for intermediate-length and slender columns, respectively, in the parametric study, which was equal to the mean value of the measured overall geometric imperfections. The size of the finiteelement mesh was kept as 3.5 × 3.5 mm, 5 × 5 mm and 10 × 10 mm (length by width) for the modelling of aluminium alloy stub, intermediate-length and slender columns, respectively, in the parametric study. The ultimate strengths of aluminium alloy SHS and RHS columns with and without circular openings are compared in the parametric study. It is found that the opening reduced the ultimate strengths of the columns, as shown in Table 5, which may attribute to the existence of the opening that weakened the section area of the columns and resulted in the stress concentrations around the opening, as shown in Fig. 6. Therefore, the perforated aluminium alloy columns usually failed at the location of openings, except for the columns with large slenderness ratio. The ultimate strengths of perforated aluminium alloy columns decreased up to 12.57%, 20.88% and 7.54% with the increase of the diameter of circular openings up to 0.8D for aluminium alloy stub, intermediate-length and slender columns, respectively, as shown in Table 5. The axial load versus axial shortening curves are also shown in Fig. 7 for perforated aluminium alloy columns with the cross-section of SHS 120 × 120 × 2 and different diameters of circular openings. Furthermore, the large diameter of circular openings changed the failure modes of aluminium alloy columns, especially for the intermediatelength and slender columns, which also decreased the ultimate strengths of aluminium alloy columns sharply, as shown in Fig. 8. The decrease of the depth-to-thickness ratio of aluminium alloy columns significantly enhanced the ultimate strengths, as shown in Fig. 9 for the specimens with the cross-sections of SHS 120 × 120 and different depth-to-thickness ratios. For aluminium alloy columns with large slenderness ratio, the decrease of the depth-to-thickness ratio also changed the failure modes of aluminium alloy columns from local buckling failure to flexural buckling failure or the combination of local and flexural buckling failure, as shown in Fig. 10. The number of openings has a certain degree of influence on the ultimate strengths of aluminium alloy columns. Table 6 shows the comparison of the ultimate strengths of perforated aluminium alloy columns with the cross-sections of SHS 120 × 120 and different numbers of openings. The ultimate strengths of perforated aluminium alloy columns decreased up to 25.08% and 19.12% with the increase of the number of openings up to 3 and 6 for aluminium alloy intermediatelength and slender columns, respectively, as shown in Table 6. And the decrease of the ultimate strengths further increased with the increase of the diameter of circular openings. The stress distributions along the longitudinal direction of perforated aluminium alloy columns were also influenced by the number of openings. The stress concentrations occurred around the openings by applying axial loads to the perforated aluminium alloy columns, as shown in Fig. 11.
(a) S120×120×2-L360
(b) S120×120×2-L1800
(c) S120×120×2-L3600 Fig. 7. Comparison of axial load versus axial shortening curves for aluminium alloy SHS 120 × 120 × 2 with different diameters of circular openings.
4. Parametric study The FEM was verified to accurately simulate the strengths and behaviour of aluminium alloy columns in test series III as presented by Feng et al. [10]. Therefore, the validated FEM was used for an extensive parametric study. The parametric study was performed on 594 specimens that consisted of six different cross-section dimensions with three different wall thicknesses (t) and three different overall lengths (L). The multiple circular openings with four different diameters (d), three different numbers (n) and three different locations (x) were introduced in the specimens at two opposite sides of the cross-sections along the longitudinal direction of the columns, as summarized in Table 4. Hence, the depth-to-thickness ratio (D/t) of the cross-sections, the slenderness 12
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(a) d=0 mm
(b) d=60 mm
(c) d=96 mm
Fig. 8. Comparison of failure modes for aluminium alloy S120 × 120 × 2-L1800 with different diameters of circular openings.
column strength using the gross cross-section area. The influence of material non-linearity was considered by introducing the tangent modulus theory, and the Euler column curve approximated a hyperbola. The inelastic column curve based on the tangent modulus can be well approximated by a straight line using the buckling constants [20], which can be obtained from Tables 3.3-3 and 3.3-4 of Part I-B of AA [15]. And the influence of elastic local buckling on the column strengths was determined based on Clause 4.7.4 of Part I-B of AA [15]. The design rules of EC9 [16] for calculating the design strengths of imperforated aluminium alloy columns were derived based on the Perry curve. The effect of local buckling on the strengths of aluminium alloy columns was taken into account by replacing the cross-section with the effective cross-section, from which the thickness of the cross-section was reduced by multiplying the local buckling coefficient. Two different design methods including the limit state design (LSD) method and allowable stress design (ASD) method were adopted in AS/NZS [17,18] to calculate the strengths of aluminium alloy columns. It is worth noting that the LSD method in AS/NZS [17] is generally identical to the design rules of AA [15], except for the resistance coefficient and capacity factor introduced in the LSD method. The Perry curve was also adopted in Chinese Code [19] for calculating the design strengths of imperforated aluminium alloy columns. The stability factor of global buckling was calculated by using the Perry-Roberson equation, while the effect of local buckling was considered by using the effective thickness method. It should be noted that the aforementioned design specifications do not have design rules for perforated aluminium alloy members. Nevertheless, North American Specification (NAS) [8] has design rules for cold-formed steel members with openings, which were derived based on the effective width method (EWM). Furthermore, the design rules of NAS [8] were extended by the direct strength method (DSM) proposed by Moen and Schafer [9] for calculating the design strengths of cold-formed steel members with openings. Therefore, the design rules for cold-formed steel members with openings were used in this study to evaluate the strengths of aluminium alloy columns with circular openings by substituting the material properties of aluminium alloys.
Fig. 9. Comparison of ultimate strengths for aluminium alloy SHS 120 × 120 with different depth-to-thickness ratios.
Table 5 shows the comparison of the ultimate strengths of perforated aluminium alloy columns with different locations of the opening. It is demonstrated that the location of the opening has an insignificant influence on the ultimate strengths of perforated aluminium alloy columns with similar decrement for both intermediate-length and slender columns. Whereas, it has a certain degree of influence on the deformations of perforated aluminium alloy columns since the location of the opening changed the stress distributions along the longitudinal direction of perforated aluminium alloy columns, which resulted in the maximum deformations of perforated aluminium alloy columns occurred at the location of the opening, as shown in Fig. 12. 5. Design rules of aluminium alloy members 5.1. Current design rules American Specification (AA) [15], European Code (EC9) [16], Australian/New Zealand Standard (AS/NZS) [17,18] and Chinese Code [19] for the design of aluminium alloy structures provide design rules for imperforated aluminium alloy columns. It is worth noting that the design rules of AA [15] for calculating the design strengths of imperforated aluminium alloy columns were derived based on the Euler
5.2. Comparison of test and FEA strengths with design strengths The ultimate strengths (Pu) of imperforated aluminium alloy columns obtained from the tests and FEA are compared with the design strengths (PAA, PEC, PA/Z1, PA/Z2 and PCC) calculated using the design 13
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(a) t=2 mm
(b) t=6 mm
Fig. 10. Comparison of failure modes for aluminium alloy S200 × 200-L6000-d100 with different depth-to-thickness ratios.
RHSs under axial compression. The ultimate strengths (Pu) of perforated aluminium alloy columns obtained from the tests and FEA are compared with the design strengths (PNAS, PDSM1, PDSM2 and PDSM3) calculated using the design rules of NAS [8] and proposed by Moen and Schafer [9] for cold-formed steel members with openings, respectively, as shown in Fig. 14. The mean values of Pu/PNAS, Pu/PDSM1, Pu/PDSM2 and Pu/PDSM3 are 1.23, 1.65, 1.26 and 1.31, with the corresponding COVs of 0.411, 0.518, 0.304 and 0.303, respectively, as shown in Table 8. It can be seen from the comparison that the current design rules of cold-formed steel members with openings are all quite conservative for aluminium alloy columns with circular openings, in which the design rules derived from DSM1 are most conservative with the largest scatter of predictions.
Table 6 Comparison of ultimate strengths of aluminium alloy SHS 120 × 120 × 2 with different diameters and numbers of circular openings. Specimen
PFEA (kN)
Reduction
S120 × 120 × 2-L1800 S120 × 120 × 2-L1800-d24-n1-1/2 S120 × 120 × 2-L1800-d60-n1-1/2 S120 × 120 × 2-L1800-d96-n1-1/2 S120 × 120 × 2-L1800-d24-n3 S120 × 120 × 2-L1800-d60-n3 S120 × 120 × 2-L1800-d96-n3 S120 × 120 × 2-L3600 S120 × 120 × 2-L3600-d24-n1-1/2 S120 × 120 × 2-L3600-d60-n1-1/2 S120 × 120 × 2-L3600-d96-n1-1/2 S120 × 120 × 2-L3600-d24-n3 S120 × 120 × 2-L3600-d60-n3 S120 × 120 × 2-L3600-d96-n3 S120 × 120 × 2-L3600-d24-n6 S120 × 120 × 2-L3600-d60-n6 S120 × 120 × 2-L3600-d96-n6
96.9 90.0 90.5 77.1 88.2 88.1 72.6 77.4 77.4 74.8 72.1 74.9 72.3 65.1 71.8 69.3 62.6
— 7.05% 6.55% 20.36% 8.98% 9.08% 25.08% — 0.00% 3.28% 6.77% 3.22% 6.59% 15.89% 7.24% 10.47% 19.12%
5.3. Proposed design rules for aluminium alloy columns with circular openings It is concluded from the evaluation of current design specifications that the design rules of EC9 [16] are generally appropriate for imperforated aluminium alloy columns. And it can be seen from the numerical results that the diameter and number of circular openings significantly affected the ultimate strengths of perforated aluminium alloy columns. Therefore, the design equations are newly proposed for perforated aluminium alloy columns based on the design rules of EC9 [16] for imperforated aluminium alloy columns by modifying the flexural buckling load (Nb,Rd) and local squashing load (Nc,Rd) through introducing the ratio of net area-to-total area of surface (Sh/S) and the correction factor as follows:
rules of AA [15], EC9 [16], LSD and ASD in AS/NZS [17,18] and Chinese Code [19], respectively, as shown in Fig. 13. The mean values of Pu/PAA, Pu/PEC, Pu/PA/Z1, Pu/PA/Z2 and Pu/PCC are 0.74, 1.14, 0.88, 1.41 and 1.55, with the corresponding COVs of 0.238, 0.149, 0.282, 0.493 and 0.756, respectively, as shown in Table 7. It can be seen from the comparison that the design rules of AA [15] and LSD in AS/NZS [17] are somewhat unconservative, whereas the design rules of ASD in AS/ NZS [18] and Chinese Code [19] are quite conservative with the largest value of COV. The design rules of EC9 [16] are somewhat conservative with the smallest value of COV. Therefore, the design rules of EC9 [16] are generally appropriate for imperforated aluminium alloy SHSs and
Ppd = min(Nb, Rd, Nc, Rd )
(3)
For the flexural buckling failure, the design resistances of perforated aluminium alloy columns can be calculated as follows:
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(a) n=0
(b) n=1
(c) n=3
Fig. 11. Comparison of failure modes for aluminium alloy S200 × 200 × 2-L6000-d100 with different numbers of openings.
(a) p=1/2
(b) p=1/3
(c) p=1/4
Fig. 12. Influence of locations of opening on deformations of perforated aluminium alloy columns.
Nb, Rd = 1.037 Aeff fy
Sh S
obtained from the tests and FEA are also compared with the proposed design strengths (Ppd) calculated using the proposed design equations, as shown in Fig. 15. It is demonstrated from the comparison that the proposed design strengths are in good agreement with the test and FEA strengths with the mean value (Pu/Ppd) of 1.01 and the corresponding COV of 0.098, as shown in Table 8. Therefore, the proposed design equations were verified to be accurate for aluminium alloy SHS and RHS columns with circular openings under axial compression.
10.94
(4)
where κ is a reduction factor to allow for the weakening effects of welding, χ is a reduction factor for relevant buckling mode, Aeff is the effective area, fy is the yield stress (0.2% tensile proof stress for aluminium alloy), Sh is the net area of surface, and S is the total area of surface. For the local squashing failure, the design resistances of perforated aluminium alloy columns can be calculated as follows:
Nc , Rd = 1.037A eff f y
Sh S
6. Conclusions
10.94
This paper presents a numerical investigation on perforated aluminium alloy SHS and RHS columns by the FEA. The geometric and material non-linearities of aluminium alloy columns were included in the FEM. The non-linear FEM was verified against the corresponding experimental results, which was used to carry out an extensive parametric study. The parametric study consisted of 594 specimens with different
(5)
where Aeff is the effective area, fy is the yield stress (0.2% tensile proof stress for aluminium alloy), Sh is the net area of surface, and S is the total area of surface. The ultimate strengths (Pu) of perforated aluminium alloy columns
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Engineering Structures 199 (2019) 109591
R. Feng and J. Liu
(a) American Specification (AA) [15]
(b) European Code (EC9) [16]
(c) LSD in AS/NZS [17]
(d) ASD in AS/NZS [18]
(e) Chinese Code [19] Fig. 13. Comparison of test and FEA strengths with design strengths for aluminium alloy columns.
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Engineering Structures 199 (2019) 109591
R. Feng and J. Liu
Table 7 Comparison of test and FEA strengths with design strengths for imperforated aluminium alloy columns. Specimen (A total of 62 SHS and RHS columns)
Comparison Pu/PAA
Pu/PEC
Pu/PA/Z1
Pu/PA/Z2
Pu/PCC
Mean COV
0.74 0.238
1.14 0.149
0.88 0.282
1.41 0.493
1.55 0.756
Table 8 Comparison of test and FEA strengths with design strengths for perforated aluminium alloy columns. Specimen (A total of 552 SHS and RHS columns)
Comparison Pu/PNAS
Pu/PDSM1
Pu/PDSM2
Pu/PDSM3
Pu/Ppd
Mean COV
1.23 0.411
1.65 0.518
1.26 0.304
1.31 0.303
1.01 0.098
depth-to-thickness ratio also changed the failure modes of aluminium alloy columns from local buckling failure to flexural buckling failure or the combination of local and flexural buckling failure. (3) It is shown from the comparison that AA and LSD in AS/NZS are somewhat unconservative, whereas ASD in AS/NZS and Chinese Code are quite conservative, while EC9 is generally appropriate for imperforated aluminium alloy columns. On the other hand, it is also demonstrated from the comparison that NAS and DSM for coldformed steel members with openings are all quite conservative for aluminium alloy columns with circular openings. (4) The design equations are proposed in this study based on the design rules of EC9, which were verified to be accurate for perforated aluminium alloy SHS and RHS columns under axial compression.
cross-section dimensions, overall lengths, as well as diameters, numbers and locations of circular openings. Some conclusions are drawn from the FEA as follows: (1) The ultimate strengths of aluminium alloy columns were significantly weakened by the introduction of the openings, especially the increase of the diameter and number of circular openings, whereas the location of the opening has little influence on the ultimate strengths of perforated aluminium alloy columns. (2) The decrease of the depth-to-thickness ratio of aluminium alloy columns significantly enhanced the ultimate strengths. For aluminium alloy columns with large slenderness ratio, the decrease of the
(a) NAS [8]
(b) DSM1
(c) DSM2
(d) DSM3
Fig. 14. Comparison of test and FEA strengths with design strengths for perforated aluminium alloy columns.
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Engineering Structures 199 (2019) 109591
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Civil Engineering, Shenzhen Durability Center for Civil Engineering (Shenzhen University, Grant No. GDDCE 18-5). References [1] Shanmugam NE, Dhanalakshmi M. Design for openings in cold-formed steel channel stub columns. Thin-Walled Struct 2001;39(12):961–81. [2] Moen CD, Schafer BW. Experiments on cold-formed steel columns with holes. ThinWalled Struct 2008;46(10):1164–82. [3] Yao Z, Rasmussen KJR. Inelastic local buckling behaviour of perforated plates and sections under compression. Thin-Walled Struct 2012;61(6):49–70. [4] Kulatunga MP, Macdonald M. Investigation of cold-formed steel structural members with perforations of different arrangements subjected to compression loading. Thin-Walled Struct 2013;67:78–87. [5] Kulatunga MP, Macdonald M, Rhodes J, Harrison DK. Load capacity of cold-formed column members of lipped channel cross-section with perforations subjected to compression loading—Part I: FE simulation and test results. Thin-Walled Struct 2014;80:1–12. [6] Zhou F, Young B. Web crippling of aluminium tubes with perforated webs. Eng Struct 2010;32(5):1397–410. [7] Feng R, Young B. Experimental investigation of aluminium alloy stub columns with circular openings. J Struct Eng, ASCE 2015;141(11):04015031. [8] NAS. North American Specification for the Design of Cold-Formed Steel Structural Members. American Iron and Steel Institute, AISI S100-16, Washington, D.C., USA; 2016. [9] Moen CD, Schafer BW. Direct strength method for design of cold-formed steel columns with holes. J Struct Eng, ASCE 2011;137(5):559–70. [10] Feng R, Zhu W, Wan HY, Chen AY, Chen Y. Tests of perforated aluminium alloy SHSs and RHSs under axial compression. Thin-Walled Struct 2018;130:194–212. [11] Kechidi S, Fratamico DC, Castro JM, Bourahla N, Schafer BW. Numerical study on the behavior and design of screw connected built-up CFS chord studs. In: Proceedings of the Annual Stability Conference. Structural Stability Research Council, San Antonio, USA, 2017; p. 1–25. [12] ABAQUS Analysis User’s Manual, Version 6.14. ABAQUS, Inc., 2014. [13] Mazzolani FM. Aluminium alloy structures. 2nd ed. London: E & FN Spon; 1995. [14] Ellobody E, Young B. Structural performance of cold-formed high strength stainless steel columns. J Constr Steel Res 2005;61(12):1631–49. [15] AA. Aluminum Design Manual. The Aluminum Association, Washington, D.C., USA, 2005. [16] EC9. Eurocode 9: Design of Aluminium Structures—Part 1-1: General Structural Rules. European Committee for Standardization, EN 1999-1-1:2007, CEN. Brussels, Belgium, 2007. [17] AS/NZS. Aluminium Structures Part 1: Limit State Design, Australian/New Zealand Standard, AS/NZS 1664.1: 1997. Standard Australia, Sydney, Australia, 1997. [18] AS/NZS. Aluminium Structures Part 2: Allowable Stress Design, Australian/New Zealand Standard, AS/NZS 1664.2: 1997. Standard Australia, Sydney, Australia, 1997. [19] Chinese Code. Code for Design of Aluminium Structures. GB 50429-2007, Beijing, China, 2007. [in Chinese]. [20] Sharp ML. Behaviour and Design of Aluminium Alloy Structure. New York: McGraw-Hill; 1993.
Fig. 15. Comparison of test and FEA strengths with proposed design strengths for perforated aluminium alloy columns.
Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements The authors are grateful for the financial support from National Natural Science Foundation of China (Grant No. 51528803), Natural Science Foundation of Guangdong Province of China (Grant No. 2018A030313208), State Key Laboratory of Subtropical Building Science (South China University of Technology, Grant No. 2018ZA02), and Guangdong Provincial Key Laboratory of Durability for Marine
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