Numerical simulation of aluminum alloy 6082-T6 columns failing by overall buckling

Abstract

Aluminum alloys have found increasing use in space structures, mainly because of their high strength-to-weight ratio and satisfactory corrosion resistance. The work presented in this article is focused on finite element simulation of the failure behavior of aluminum alloy members subjected to an axial compression load and failing in overall buckling mode. The column members were fabricated by extrusion using the relatively new heat-treated aluminum alloy 6082-T6. An accurate finite element model has been developed for the simulation of aluminum alloy columns. The developed finite element model was verified against experimental results and demonstrated that it is capable of accurately predicting the deformed shapes and the ultimate loads of the tested columns. A parametric study was carried out on the stability of aluminum alloy column members failing by overall buckling. A fitting column curve was proposed for aluminum alloy design; it was somewhat more conservative than the column curve from current Chinese code GB50429.

Keywords

aluminum alloy 6082-T6 buckling circular hollow section column finite element analysis initial geometrical imperfection

Introduction

Aluminum alloy is an established primary structural material in transportation applications, and it has been increasingly applied in space structures, curtain walls, and other structural applications because of its satisfactory corrosion resistance, high strength-to-weight ratio, good formability, and aesthetics (Kissell and Ferry, 2002; Mazzolani, 1995, 2012; Shen et al., 2007; Shi et al., 2013; Spyrakos and Ermopoulos, 2005). 6000 series aluminum alloys are widely used in structural applications because of their particularly favorable combinations of such properties. Among 6000 series alloys, 6082 alloy is relatively new and provides the most favorable combination of properties such as strength after heat treatment, corrosion resistance, machining properties, and weldability (Kissell and Ferry, 2002). Compared to the widely used and classic 6061 alloy, 6082 alloy provides higher strength (10%–18% higher for allowable tensile strengths provided in EC9), better general corrosion resistance, and other approximately equivalent properties such as density and extrudability (Eurocode 9, 2007; Kissell and Ferry, 2002). It has already been widely used in Europe and is included in European Standard (EC9) (Eurocode 9, 2007); its application in the United States has been increasing, and it has been added into the latest edition of the American Aluminum Design Manual (2010a) (Aluminum Association, 2005, 2010b).

Stability is one of the main concerns regarding aluminum alloy members, because Young’s modulus for aluminum alloy is only about one-third of that for carbon steel (Mazzolani, 1995). An important failure mode of aluminum alloy columns is flexural buckling under axial compression. Significant advances in the study of stability of aluminum alloy structural column members have been gained through persistent experimental and analytical investigations as summarized in Mazzolani (1995), Sharp (1993), and Rasmussen and Rondal (2000). Guo (2006) carried out experimental and numerical investigations on aluminum alloy 6061-T6 extruded profiles for several cross-sectional types under axial compression. Zhu and Young (2006, 2008) and Su et al. (2014) conducted experimental and numerical investigations on aluminum alloy columns with circular, square, and rectangular hollow sections extruded from alloys 6063-T5 and 6061-T6. Compared with the extensive studies on the popular 6061 alloy members, there has been only sparse previous research on the behavior of 6082 alloy members under compression. A few experimental tests on 6082 alloy columns with circular hollow sections (CHSs) and H-shaped sections were conducted in the 1970s under the auspices of the European Convention for Constructional Steelwork (ECCS) (Bernard et al., 1973; Klöppel and Barsch, 1971, 1973; Sowa, 1971). Wang et al. (2013) conducted an experimental study on material constitutive relations of alloy 6082-T6. Zhai et al. (2011) and Fan et al. (2013) carried out an experimental investigation of alloy 6082-T6 columns with rectangular, square, and H-shaped cross sections.

With respect to the study of aluminum alloy structural members, physical experiments are valuable, but expensive and time-consuming, especially when extensive analysis is needed. Finite element (FE) simulation is a powerful complement to physical experimentation, because it can provide accurate, inexpensive, and time-efficient analysis, especially when a parametric study is required. Before FE simulation can be used to conduct an investigation; however, it is necessary to develop an accurate and reliable FE model with accuracy verified against available experimental results, for example, the work on aluminum alloy structural members conducted by Zhu and Young (2006, 2008), and the work on stainless steel structural members conducted by Ashraf et al. (2006), Ellobody (2007), Ellobody and Young (2005), and Gardner and Nethercot (2004).

The purpose of this study was to develop an accurate FE model employing the FE software package ABAQUS v6.12 to predict the behavior and load capacity of aluminum columns failing by overall buckling, and then to carry out a parametric study on the stability of the columns. Comparisons were carried out to determine the optimal element type, balanced element size, and suitable simulation of column ends. The developed FE model was verified against an experimental program on a total of 73 columns made of heat-treated aluminum alloy 6082-T6.

Summary of column test program

The experiments were performed on a total of 73 column specimens, including 15 circular tubes (Wang et al., 2015), 17 H-section profiles, 15 box-section profiles, 16 equal-leg angles, and 10 unequal-leg angles. The specimens were fabricated by extrusion using 6082-T6 heat-treated aluminum alloy with a nominal yield stress ( $σ_{0.2}$ , 0.2% proof stress) of about 260 MPa, a tensile strength of about 310 MPa, and an elongation greater than 10%, as specified in (GB 50429-2007, 2007). The slenderness ratio ( $λ$ ) of a column specimen is defined by equation (1), where L_e is the effective length and r is the radius of gyration of the cross section. The regularized slenderness ratio ( $\bar{λ}$ ) of a column specimen is defined by equation (2), in which $σ_{E}$ is the elastic Euler buckling stress and E is the Young’s modulus. The values of $σ_{0.2}$ and E were obtained from material tests

λ = \frac{L_{e}}{r}

(1)

\bar{λ} = \sqrt{\frac{σ_{0.2}}{σ_{E}}} = \frac{λ}{π} \sqrt{\frac{σ_{0.2}}{E}}

(2)

Local buckling may affect the load-bearing capacity of a column member when the width-to-thickness ratio of cross-sectional plate is large enough. In this study, only column members failing by overall buckling were concerned. For the column specimens with circular section, rectangular section, and H-section, the width-to-thickness ratios or radius-to-thickness ratios (for circular sections) were less than the maximum ratios for wholly effective plates in compression according to standard GB 50429-2007 (2007), which means local buckling would not occur in these column specimens. For some column specimens with angle sections, the width-to-thickness ratios exceeded the maximum values for wholly effective compressive plates given in GB 50429-2007 (2007). However, the calculated critical stresses for local buckling of these angle column specimens were greater than the experimental critical stresses of the column specimens. This indicated that overall buckling occurred before local buckling and local buckling had no effect on the load-bearing capacities of the column specimens in this study. Our experimental study was limited to the column specimens in which local buckling would not occur before overall buckling.

The end device used to perform the column tests is depicted in Figure 1. The single-knife-edge steel bearings shown in Figure 1(e) were used to provide rotational capability at the ends of the aluminum specimens with H- and box-sections, while the double-knife-edge steel bearings shown in Figure 1(f) were used for the specimens with CHSs. High-strength pattern stone was used to reliably connect the aluminum specimen to the steel end plates (Figure 2). The nominal material properties of the aluminum circular tubes and the steel end plates are given in Table 1 (GB 50429-2007, 2007; Steinhardt, 1971). Some angle-section specimens failed by flexural-torsional buckling and the other column specimens failed by overall flexural buckling, as shown in Figure 3. Local buckling did not occur before load capacities were reached because of the relative small width-to-thickness ratios of the specimen plates. The load-carrying capacities obtained from the experiments are shown in Tables 2 to 4.

Figure 1.

Column test set-up: (a) H-section specimen, (b) box-section specimen, (c) circular tube specimen, (d) angle-section specimen, (e) single-knife-edge end bearing, and (f) double-knife-edge end bearing.

Figure 2.

Connection with pattern stone in steel case: (a) H-section specimen, (b) box-section specimen, (c) circular tube specimen, and (d) angle-section specimen.

Table 1.

Nominal material properties of column specimens and end plates.

Aluminum alloy (circular tubes)	E (MPa)	v	σ _0.2 (MPa)	σ_u (MPa)	n
	70,000	0.3	260	310	26
Steel (end plates)	E (MPa)	v	σ_y (MPa)	σ_u (MPa)	−
	206,000	0.3	355	600	−

Figure 3.

Failure of column specimens: (a) H-section specimen, (b) box-section specimen, (c) circular tube specimen, (d) flexural-torsional buckling of angle specimen, and (e) flexural buckling of angle specimen.

Table 2.

Experimental results of H-section specimens, box-section specimens, and circular-hollow-section specimens.

No.	Section	Label	L (mm)	L_e (mm)	$λ$	E (GPa)	f _0.2 (kN)	$\bar{λ}$	P _exp (kN)	σ _exp (MPa)	ϕ _exp	Failing mode
Experimental results of H-section specimens
1	H120 × 60 × 5 × 7	H1-A	1042.0	1066.0	78.11	72.03	314.23	1.642	148.73	109.79	0.3494	F
2	H120 × 80 × 6 × 8	H2-A	1056.0	1080.0	57.02	70.60	307.99	1.199	311.78	167.61	0.5442	F
3		H3-A	966.0	990.0	52.14	70.60	307.99	1.096	351.95	187.54	0.6089	F
4		H3-B	966.0	990.0	52.38	70.60	307.99	1.101	375.58	204.56	0.6642	F
5		H3-C	966.0	990.0	52.42	70.60	307.99	1.102	435.42	238.88	0.7756	F
6	H100 × 70 × 6 × 8	H4-A	1034.0	1058.0	63.08	71.46	322.86	1.350	240.52	150.33	0.4656	F
7		H4-B	1030.5	1054.5	62.50	71.46	322.86	1.337	262.38	163.28	0.5057	F
8		H4-C	1038.5	1062.5	63.12	71.46	322.86	1.350	263.5	164.26	0.5088	F
9	H100 × 50 × 5 × 7	H5-A	1030.0	1054.0	91.99	69.36	319.54	1.987	89.85	80.06	0.2505	F
10		H5-B	1018.0	1042.0	91.10	69.36	319.54	1.968	94.58	85.04	0.2661	F
11		H5-C	1040.0	1064.0	92.90	69.36	319.54	2.007	93.64	83.94	0.2627	F
12		H6-A	1216.0	1240.0	108.74	69.36	319.54	2.349	66.87	59.93	0.1876	F
13		H6-B	1215.0	1239.0	108.87	69.36	319.54	2.352	67.5	60.72	0.1900	F
14		H7-A	1517.0	1541.0	135.44	69.36	319.54	2.926	44.36	39.70	0.1243	F
15		H7-B	1517.0	1541.0	135.18	69.36	319.54	2.921	44.83	40.35	0.1263	F
16		H8-A	1817.0	1841.0	161.46	69.36	319.54	3.488	33.41	30.11	0.0942	F
17		H8-B	1818.0	1842.0	162.19	69.36	319.54	3.504	30.23	27.07	0.0847	F
No.	Section	Label	L (mm)	L_e (mm)	$λ$	E (GPa)	f _0.2 (kN)	$\bar{λ}$	P _exp (kN)	σ _exp (MPa)	ϕ _exp	Failing mode
Experimental results of box-section specimens
1	□80 × 4	□1-A	1032.0	1056.0	34.13	66.30	279.45	0.705	294.77	247.14	0.8844	F
2		□1-B	1037.0	1061.0	34.33	66.30	279.45	0.709	289.47	255.91	0.9158	F
3		□1-C	1032.0	1056.0	34.19	66.30	279.45	0.706	295.17	256.97	0.9195	F
4	□80 × 40 × 4	□2-A	1524.5	1548.5	97.26	68.18	254.21	1.890	63.8	77.46	0.3047	F
5		□2-B	1522.0	1546.0	96.93	68.18	254.21	1.884	68.89	82.63	0.3250	F
6		□2-C	1525.9	1549.9	96.87	68.18	254.21	1.883	63.08	77.20	0.3037	F
7		□4-A	659.8	683.8	42.86	68.18	254.21	0.833	176.62	213.19	0.8387	F
8		□4-B	659.8	683.8	42.90	68.18	254.21	0.834	164.58	197.18	0.7757	F
9	□100 × 50 × 4	□3-A	1524.0	1548.0	76.38	67.68	281.49	1.568	120.69	115.36	0.4098	F
10		□3-B	1525.9	1549.9	76.19	67.68	281.49	1.564	127.66	118.61	0.4214	F
11		□3-C	1527.5	1551.5	76.43	67.68	281.49	1.569	130.07	123.70	0.4395	F
12	□80 × 40 × 6	□5-A	416.0	440.0	28.93	70.36	305.13	0.606	353.46	272.78	0.8940	F
13		□5-B	416.0	440.0	28.93	70.36	305.13	0.606	375.99	290.17	0.9510	F
14		□6-A	316.0	340.0	22.36	70.36	305.13	0.469	389.44	300.55	0.9850	F
15		□6-B	316.0	340.0	22.36	70.36	305.13	0.469	390.34	301.24	0.9873	F
No.	Section	Label	L (mm)	L_e (mm)	λ	E (GPa)	f _0.2 (kN)	$\bar{λ}$	P _exp (kN)	σ _exp (MPa)	ϕ _exp	Failing mode
Experimental results of circular-hollow-section specimens
1	O89×6.5	O89-L500-A	500.5	598.5	20.44	67.96	327.01	0.451	508.71	311.34	0.9521	F
2		O89-L500-B	500.0	598.0	20.44	67.96	327.01	0.451	467.49	286.69	0.8767	F
3		O89-L500-C	500.0	598.0	20.42	67.96	327.01	0.451	471.93	288.83	0.8832	F
4	O89×6.5	O89-L800-A	800.0	898.0	30.69	67.96	327.01	0.678	444.26	272.44	0.8331	F
5		O89-L800-B	800.0	898.0	30.69	67.96	327.01	0.678	429.38	263.32	0.8052	F
6		O89-L800-C	799.0	897.0	30.66	67.96	327.01	0.677	473.67	290.48	0.8883	F
7	O89×6.5	O89-L1200-A	1200.5	1298.5	44.38	67.96	327.01	0.980	416.24	255.26	0.7806	F
8		O89-L1200-B	1200.0	1298.0	44.36	67.96	327.01	0.979	378.68	232.23	0.7102	F
9		O89-L1200-C	1200.5	1298.5	44.38	67.96	327.01	0.980	398.53	244.40	0.7474	F
10	O89×6.5	O89-L1650-A	1649.0	1747.0	59.86	67.96	327.01	1.322	239.67	146.84	0.4490	F
11		O89-L1650-B	1650.0	1748.0	59.87	67.96	327.01	1.322	269.84	165.70	0.5067	F
12		O89-L1650-C	1650.0	1748.0	59.83	67.96	327.01	1.321	265.55	163.00	0.4984	F
13	O76×3.0	O76-L1700-A	1700.0	1798.0	69.89	74.77	240.95	1.263	74.17	115.69	0.4801	F
14		O76-L1700-B	1700.0	1798.0	69.96	74.77	240.95	1.264	68.54	107.74	0.4471	F
15		O76-L1700-C	1699.0	1797.0	69.78	74.77	240.95	1.261	72.23	111.35	0.4621	F

F represents flexural buckling.

Table 3.

Experimental results of the specimens with equal-leg angle sections.

No.	Section	Label	L (mm)	L_e (mm)	$λ_{x}$	$λ_{y ω}$	E (GPa)	f _0.2 (kN)	$\bar{λ}$	P _exp (kN)	σ _exp (MPa)	ϕ _exp	Failing mode
1	L110×8	L110-800A	799.6	897.6	39.38	68.45	71.33	323.55	1.468	221.46	135.45	0.4186	TF
2		L110-800B	799.2	897.2	39.39	68.74	71.33	323.55	1.474	224.31	137.97	0.4264	TF
3		L110-1200A	1199.8	1297.8	57.00	72.46	71.33	323.55	1.553	203.11	124.45	0.3846	TF
4		L110-1200B	1199.8	1297.8	56.96	71.72	71.33	323.55	1.538	210.22	127.31	0.3935	TF
5		L110-1800A	1799.4	1897.4	83.20	76.14	71.33	323.55	1.784	142.27	86.08	0.2661	F
6		L110-1800B	1799.2	1897.2	83.32	76.74	71.33	323.55	1.786	123.03	75.29	0.2327	F
7	L90×10	L90-400A	400.1	497.7	26.39	40.04	71.35	290.21	0.813	427.97	257.19	0.8862	TF
8		L90-400B	399.6	497.4	26.38	39.87	71.35	290.21	0.809	466.36	279.33	0.9625	TF
9		L90-800A	799.1	897.6	47.60	47.49	71.35	290.21	0.966	348.08	209.41	0.7216	TF
10		L90-800B	799.6	897.2	47.57	47.34	71.35	290.21	0.966	383.3	229.81	0.7919	TF
11	L90×8	L90-1200A	1199.4	1297.4	69.67	62.43	71.35	290.21	1.414	159.7	120.93	0.4167	F
12		L90-1200B	1199.1	1297.1	69.18	62.20	71.35	290.21	1.404	141.29	105.27	0.3627	F
13		L90-1500A	1499.5	1597.5	85.64	65.56	71.35	290.21	1.739	111.02	83.51	0.2878	F
14		L90-1500B	1499.6	1597.6	85.53	65.08	71.35	290.21	1.736	102.67	76.39	0.2632	F
15	L70×4	L70-1500A	1498.3	1596.3	110.75	98.01	69.16	279.14	2.240	24.31	46.11	0.1652	F
16		L70-1500B	1499.6	1597.6	110.75	98.07	69.16	279.14	2.240	26.3	49.85	0.1786	F

F represents flexural buckling and TF represents torsional-flexural buckling.

Table 4.

Experimental results of the specimens with unequal-leg angle sections.

No.	Section	Label	L (mm)	L_e (mm)	$λ_{x}$	$λ_{y}$	$λ_{ω}$	E (GPa)	f _0.2 (kN)	$λ_{xy ω}$	$\bar{λ}$	P _exp (kN)	σ _exp (MPa)	ϕ _exp	Failing mode
1	L70×45×8	L70×45-1000A	1016.9	1114.9	115.38	47.71	36.78	69.26	306.00	116.82	2.472	47.82	57.27	0.1872	TF
2		L70×45-1000B	1010.4	1108.4	115.06	47.38	36.66	69.26	306.00	116.52	2.465	48.88	58.31	0.1906	TF
3		L70×45-1500A	1499.3	1597.3	166.24	68.46	37.47	69.26	306.00	167.25	3.539	25.7	31.20	0.1020	TF
4		L70×45-1500B	1499.5	1597.5	165.47	68.32	37.13	69.26	306.00	166.46	3.522	24.8	29.70	0.0970	TF
5	L100×70×8	L100×70-800A	799.3	897.3	59.40	25.99	52.67	71.35	290.21	67.18	1.364	192.77	154.14	0.5311	TF
6		L100×70-800B	799.2	897.2	59.12	25.87	52.55	71.35	290.21	66.91	1.358	197.54	156.24	0.5384	TF
7		L100×70-1000A	1000.1	1098.1	72.40	31.68	53.47	71.35	290.21	77.73	1.578	119.33	94.39	0.3252	TF
8		L100×70-1000B	1000.0	1098.0	72.36	31.67	53.54	71.35	290.21	77.72	1.578	141.44	111.94	0.3857	TF
9		L100×70-1500A	1499.8	1597.8	105.25	46.11	54.45	71.35	290.21	108.16	2.196	84.52	66.82	0.2302	TF
10		L100×70-1500B	1499.4	1597.4	105.30	46.11	54.34	71.35	290.21	108.20	2.196	78.9	62.30	0.2147	TF

F represents flexural buckling and TF represents torsional-flexural buckling.

Initial eccentricity at column mid-length

The initial eccentricity at mid-length cross section, the sum of initial crookedness, and initial loading eccentricity at ends affect the load capacity of a column member, because the initial crookedness and initial loading eccentricity at ends both cause bending moment at mid-length cross section (Figure 4(a)). In some aluminum alloy structural design specifications (Aluminum Association, 2010; AS/NZS 1664.1:1997, 1997; Eurocode 9, 2007; Mazzolani, 1995), an initial sinusoidal curvature, with a magnitude of L/1000, which accounts for the initial crookedness and initial loading eccentricity at ends, is the basis for computation of the instability curves of extruded aluminum profiles.

Figure 4.

Loading eccentricities at mid-length of column specimens: (a) initial eccentricity e_0x at mid-length, (b) H-section specimen, (c) box-section specimen, (d) circular hollow section specimen, and (e) angle section specimen.

According to the Chinese fabrication specification (GB 5237.1-2008, 2008), the allowable deviation from straightness of ordinary grade extruded aluminum alloy profiles, with the diameters of circumscribed circles greater than 38 mm, is 1.5L/1000; according to the European standard (EN 12020-2:2008, 2008) for extruded precision profiles in alloys 6060 and 6063, the allowable deviation from straightness is 0.7 mm for L ≤ 1000 mm and 1.3 mm for 1000 mm < L≤ 2000 mm; according to the American standard (ANSI H35.2(M):2009, 2009), the allowable deviation from straightness for extruded profiles with circumscribing circle diameters greater than 40 mm and thickness greater than 2.5 mm is L/1000.

The initial eccentricities at mid-length of the column specimens in this study were measured by means of the strain gauges at mid-length cross sections. The schematic diagram for H-section is shown in Figure 4(b). Considering the cross section basically obeys plane section assumption when the load was low at the beginning of the test, bending moments at mid-length cross section of column specimen are expressed as follows

M_{x} = F e_{y}

(3)

M_{y} = F e_{x}

(4)

where M_x and M_y are the bending moments about major axis x and minor axis y, F is the axial load, e_x and e_y are the eccentricity components in the two directions. The eccentricities e_x and e_y consist of three components

e_{y} = e_{ey} + v_{0 y} + D_{y}

(5)

e_{x} = e_{ex} + v_{0 x} + \frac{D_{x 1} + D_{x 2}}{2}

(6)

where e_ex and e_ey are the components in x- and y-directions of load eccentricity at mid-length cross section, respectively; v_ex and v_ey are the components in x- and y-directions of initial crookedness at mid-length, respectively; D_x1 and D_x2 are the deflections of two H-section flanges at mid-length, respectively; D_y is the deflection about major axis x of mid-length cross section.

The stresses on the four flange tips of mid-length cross section can be expressed in terms of the moments M_x, M_y, and the axial force F

σ_{neg_1} = E_{0} ε_{neg_1} = - \frac{M_{x} d_{g}}{2 I_{x}} - \frac{M_{y} h}{2 I_{y}} - \frac{F}{A}

(7)

σ_{pos_1} = E_{0} ε_{pos_1} = - \frac{M_{x} d_{g}}{2 I_{x}} + \frac{M_{y} h}{2 I_{y}} - \frac{F}{A}

(8)

σ_{neg_2} = E_{0} ε_{neg_2} = \frac{M_{x} d_{g}}{2 I_{x}} - \frac{M_{y} h}{2 I_{y}} - \frac{F}{A}

(9)

σ_{pos_2} = E_{0} ε_{pos_2} = \frac{M_{x} d_{g}}{2 I_{x}} + \frac{M_{y} h}{2 I_{y}} - \frac{F}{A}

(10)

where $E_{0}$ is the initial elastic modulus of material; $σ_{neg_1}$ and $ε_{pos_1}$ are the stress and strain on the compressive tip of one flange, respectively; $σ_{pos_1}$ and $ε_{pos_1}$ are the stress and strain on the tensile tip of one flange, respectively; $σ_{neg_2}$ and $ε_{neg_2}$ are the stress and strain on the compressive tip of the other flange, respectively; $σ_{pos_2}$ and $ε_{pos_2}$ are the stress and strain on the tensile tip of the other flange, respectively; I_x is the moment of inertia of cross section about major axis x; I_y is the moment of inertia of cross section about minor axis y; d_g is the distance between two strain gauges on flange; h is the cross-sectional height.

Combining equations (7) and (8) gives

E_{0} (ε_{pos_1} - ε_{neg_1}) = \frac{M_{y} d_{g}}{I_{y}}

(11)

Combining equations (9) and (10) gives

E_{0} (ε_{pos_2} - ε_{neg_2}) = \frac{M_{y} d_{g}}{I_{y}}

(12)

Combining equations (11) and (12) gives

M_{y} = \frac{E_{0} I_{y} (ε_{pos_1} - ε_{neg_1} + ε_{pos_2} - ε_{neg_2})}{2 d_{g}}

(13)

Summing up equations (7) to (10) obtains

F = - \frac{E_{0} A}{4} \times (ε_{neg_1} + ε_{pos_1} + ε_{neg_2} + ε_{pos_2})

(14)

Substituting equations (13) and (14) into equations (4) and (6) gets the initial loading eccentricity about the minor axis x at mid-length of H-section specimen

\begin{matrix} e_{0 x} = e_{ex} + v_{0 x} \\ = \frac{2 I_{y} (ε_{neg_1} - ε_{pos_1} + ε_{neg_2} - ε_{pos_2})}{d_{g} A (ε_{neg_1} + ε_{pos_1} + ε_{neg_2} + ε_{pos_2})} - \frac{D_{x 1} + D_{x 2}}{2} \end{matrix}

(15)

Due to the length limit of the article, only the derivation for H-section is detailed. The initial loading eccentricity at mid-length of rectangular, round, and angle-section specimens is given in equations (16) to (18), respectively, where Ds are the displacements at the measuring points and εs are the strains at the measuring points, as shown in Figure 4(c) to (e). All the parameters on the right-hand side of equations (16) to (18) can be measured in the test. For angle specimens, it is very challenging to accurately estimate the initial eccentricity at mid-length, because possible torsional buckling is difficult to be measured. To solve this problem, equation (18) is employed to calculate the initial eccentricity at mid-length for angle specimen when the applied load is very low and the deformation of the specimen is negligible

e_{0 x} = \frac{2 I_{y} (ε_{neg_1} - ε_{pos_1} + ε_{neg_2} - ε_{pos_2})}{d_{g} A (ε_{neg_1} + ε_{pos_1} + ε_{neg_2} + ε_{pos_2})} - D_{x}

(16)

e_{0 x} = \frac{4 I_{y} (ε_{x 1} - ε_{x 2})}{DA (ε_{x 1} + ε_{x 2} + ε_{y 1} + ε_{y 2})} - \frac{D_{x 1} + D_{x 2}}{2}

(17)

e_{0 x} = \frac{M_{y}}{F} = \frac{E I_{y} (ε_{3} - ε_{1})}{F d_{g 1}} e_{0 y} = \frac{M_{x}}{F} = \frac{E I_{x} (ε_{4} - ε_{6})}{F d_{g 2}}

(18)

The calculated initial eccentricities of some specimens in Figure 5 demonstrate that the calculated results at low load level basically remain constant and can be treated as initial eccentricities at mid-length of the column specimens. The initial eccentricities at mid-length of a total of 73 column specimens are shown in Figure 6. Initial eccentricities of 14 specimens (about 30% of the measured specimens) were larger than L/1000, initial eccentricities of 6 specimens were larger than 1.5L/1000, and initial eccentricities of 4 specimens were larger than 1.8L/1000.

Figure 5.

Calculated initial eccentricities at mid-length of some column specimens: (a) Specimen □4-2 and (b) Specimen O89-1-B.

Figure 6.

Calculated initial eccentricities of 73 column specimens.

Mechanical properties

A total of 117 tensile tests were carried out to investigate the material properties of domestic aluminum alloy 6082-T6 extruded profiles. The tensile coupons were cut from extruded profiles with four types of cross section, as shown in Table 5. The tensile test devices for plate and spherical coupons are shown in Figure 7. The obtained stress–strain curves are shown in Figure 8 and the statistical parameters of mechanical properties of all the tensile coupons are listed in Table 6.

Table 5.

Sampling summary of tensile coupons.

Group No.	Sampling profile	Group No.	Sampling profile	Group No.	Sampling profile
1	□80×80×4	14	H100×70×6×8 (flange)	27	∟70×45×6
2	□80×80×6	15	H120×60×5×7 (web)	28	∟70×45×8
3	□100×100×6	16	H120×60×5×7 (flange)	29	∟100×70×6
4	□100×100×8	17	H100×80×6×8 (web)	30	∟100×70×8
5	□80×40×4	18	H100×80×6×8 (flange)	31	Ø76×3.0
6	□80×40×6	19	∟70×70×4	32	Ø76×4.5
7	□100×50×4	20	∟70×70×6	33	Ø76×5.5
8	□100×50×6	21	∟70×70×8	34	Ø89×4.0
9	H80×50×4×6 (web)	22	∟90×90×6	35	Ø89×5.0
10	H80×50×4×6 (flange)	23	∟90×90×10	36	Ø89×6.5
11	H100×50×5×7 (web)	24	∟110×110×8	37	Ø114×4.0
12	H100×50×5×7 (flange)	25	∟110×110×12	38	Ø114×5.5
13	H100×70×6×8 (web)	26	∟70×45×4	39	Ø114×7.0

Figure 7.

Tensile testing device: (a) plate coupon and (b) spherical coupon.

Figure 8.

Stress–strain curves of all tensile coupons: (a) full range and (b) initial parts.

Table 6.

Statistical parameters of mechanical properties of all the coupons.

	E (GPa)	f _0.1 (MPa)	f _0.2 (MPa)	f_u (MPa)	A_t (%)	$υ$	f _0.2/f_0.1
Mean ( $μ$ )	70.519	298.7	304.8	332.1	13.81	0.31	1.02
Standard deviation ( $σ$ )	3.007	25.4	24.7	24.9	2.21	0.05	0.03
COV	0.043	0.085	0.081	0.075	0.160	0.170	0.028
$μ - 1.645 σ$	−	257.0	264.3	291.3	−	−	−

COV: coefficient of variation.

The Ramberg–Osgood parameters n were calculated using n = ln2/ln(σ_0.2/σ_0.1) (Mazzolani, 1995) where σ_0.1 is the 0.1% proof stress and σ_0.2 is the 0.2% proof stress. The Ramberg–Osgood curve matched the measured stress–strain curves well within the strain range [0, 1%] (Wang et al., 2015), so the verified Ramberg–Osgood expression will be used in the follow-up FE simulation.

Development of FE model

General comments

The commercial FE software package ABAQUS was used to produce the simulations of the aluminum circular tubular pin-ended columns summarized in section “Summary of column test program.” Measured dimensions, measured initial out-of-straightness, and material properties of the test specimens were used in building the FE model; residual stresses were not included in this model because in extruded aluminum alloy profiles, regardless of the type of the heat treatment, residual stresses typically have very small values, and for all practical purposes, these stresses have a negligible effect on load-carrying capacity (Mazzolani, 2012). The simulation was performed in two steps: (1) introduction of geometric imperfection and (2) load–displacement nonlinear analysis.

Element type and wall discretization

Two kinds of analysis, linear elastic-buckling analysis, and nonlinear load–displacement analysis were performed to select element type and element number in the thickness direction. Linear elastic-buckling analysis was carried out to analyze the effect of element type and element number in the thickness direction on the buckling eigenvalue. Nonlinear analysis was performed to analyze the effect of element number in the thickness direction on column strength.

Linear elastic-buckling analysis

ABAQUS has an extensive element library that provides a powerful set of tools for solving a broad range of problems; continuum solid and shell element packages are particularly appropriate for accurately modeling metal profiles. ABAQUS/Standard provides more than 20 types of three-dimensional (3D) solid elements and 13 types of 3D shell elements, each with its own particular advantages, so several common and suitable element types were selected for comparison to find an optimal choice for aluminum alloy CHS columns. The solid element types involved included C3D8 (eight-node linear brick), C3D8R (eight-node linear brick, reduced integration with hourglass control), C3D8I (eight-node linear brick, incompatible modes), C3D20 (20-node quadratic brick), and C3D20R (20-node quadratic brick, reduced integration); selected shell element types included S4 (four-node general-purpose shell, finite membrane strains) and S4R (four-node general-purpose shell, reduced integration with hourglass control, finite membrane strains).

The element number chosen for solid element types or integration point number for shell element types in the thickness direction might possibly affect the simulation results, so element numbers 1 and 2 for solid element types and choices of 5 (default) and 9 integration points for shell element types in the thickness direction were also compared. In all, 18 combinational scenarios of element types and element numbers (or integration point numbers) in the thickness direction were swept, with the results shown in Table 7. Labels, such as “C3D8R-E-T1,” indicate the element type and wall discretization using the following descriptions:

The first part of the label indicates the element type. As mentioned above, the element types considered included C3D8, C3D8R, C3D8I, C3D20, C3D20R, S4, and S4R.

The second part of the label “E,” if it appears, indicates that enhanced hourglass control was adopted for linear reduced integration element types, such as C3D8R and S4R.

The third part of the label indicates the wall discretization. The letter “T” refers to the discretization in the thickness direction, and the following digit is the number of solid elements or the number of integration points of shell elements in the thickness direction.

Table 7.

Labels of the combination scenarios of element types and wall discretization.

Solid element					Shell element
No.	Label	Element type	Element number	Hourglass control	No.	Label	Element type	IP number	Hourglass control
1	C3D8-T1	C3D8	1	−	13	S4-T5	S4	5	−
2	C3D8R-T1	C3D8R	1	Normal	14	S4R-T5	S4R	5	Normal
3	C3D8R-E-T1	C3D8R	1	Enhanced	15	S4R-E-T5	S4R	5	Enhanced
4	C3D8I-T1	C3D8I	1	−	16	S4-T9	S4	9	−
5	C3D20-T1	C3D20	1	−	17	S4R-T9	S4R	9	Normal
6	C3D20R-T1	C3D20R	1	−	18	S4R-E-T9	S4R	9	Enhanced
7	C3D8-T2	C3D8	2	−
8	C3D8R-T2	C3D8R	2	Normal
9	C3D8R-E-T2	C3D8R	2	Enhanced
10	C3D8I-T2	C3D8I	2	−
11	C3D20-T2	C3D20	2	−
12	C3D20R-T2	C3D20R	2	−

IP: integration point.

Element number and integration number are in the thickness direction.

To consider the effect of diameter-to-thickness ratio (D/t), comparisons of a variety of element types and a variety of meshing techniques in the thickness direction were based on a total of eight cross-sectional geometries, as shown in Table 8. All tube cases had a slenderness ratio ( $\bar{λ}$ ) of 50 and an outer diameter of 100 mm, and the wall thickness ranged from 2.17 to 30 mm. The tube cases were labeled in such a way that dimensions of the cross sections could be easily identified. For example, the label “C-100×2.17” defines a circular tube case with a cross-sectional dimension of 100×2.17 mm².

Table 8.

CHS tube geometries of the cases for the comparisons of element types and wall discretization.

Label of tube case	Dimension, D×t×L (mm)	D/t	Slenderness ratio $λ$	Regularized slenderness ratio $\bar{λ}$
C-100×2.17	100×2.17×1730	46.08	50	0.97
C-100×3	100×3×1716	33.33	50	0.97
C-100×5	100×5×1682	20.00	50	0.97
C-100×10	100×10×1601	10.00	50	0.97
C-100×15	100×15×1526	6.67	50	0.97
C-100×20	100×20×1458	5.00	50	0.97
C-100×25	100×25×1398	4.00	50	0.97
C-100×30	100×30×1346	3.33	50	0.97

The tubes were discretized using 30 elements in the circumferential direction, and the global element size in the longitudinal direction was 10 mm. Figure 9 shows the different meshing techniques on the cross section. A 3D view of the FE model of the tube is shown in Figure 10. In the FE model, one reference point was created at the center of the cross section at each end and coupled with the corresponding end surface. The material properties used were the nominal values listed in Table 1. Linear elastic-buckling analysis of the circular tube cases shown in Table 7 was carried out in ABAQUS/Standard by sweeping the 18 combinational scenarios of element type and thickness meshing shown in Table 7.

Figure 9.

Cross-sectional discretization of circular tube examples: (a) solid element type (one element), b) solid element type (two elements), and (c) shell element type (one element, with five or nine integration points).

Figure 10.

A FE model of circular tube examples.

Theoretically, the first-order buckling eigenvalues ( $P_{cr}$ ) should be equal to the Euler load ( $P_{E}$ ), as calculated by $P_{E} = (π^{2} EA) / λ^{2}$ . The ratios $P_{cr} / P_{E}$ , varying with different combination scenarios of element types and wall meshing, are shown in Figures 11 and 12 for both solid elements and shell elements.

Figure 11.

P_cr/P_E versus element type and number of solid elements or shell element integration points in the thickness direction.

Figure 12.

Variation P_cr/P_E with wall thickness using shell element.

For solid element types, the comparison shown in Figure 11 indicates the following:

When element type C3D8 and one element in the thickness direction (label C3D8-T1) were used, the ratio $P_{cr} / P_{E}$ varied considerably, from 0.98 (t = 2.17 mm) to 1.11 (t = 30 mm), with a maximum discrepancy of 13.5%. Increasing the element number in the thickness direction (label C3D8-T2) produced a nearly constant ratio $P_{cr} / P_{E}$ for various thickness values, with a maximum discrepancy of 0.23%.

Using element type C3D8R and one element in the thickness direction (label C3D8R-T1) produced an unacceptable result for the circular tube case EC-100×2.17 with the smallest thickness, because a numerical problem, an hour-glassing phenomenon for linear reduced integration elements, occurred. For other circular tube cases, the ratio $P_{cr} / P_{E}$ varied from 0.97 when t = 3 mm to 0.94 when t = 30 mm, with a maximum discrepancy of 3.7%. Adopting enhanced hour-glassing control (label C3D8R-E-T1) or increasing the element number in the thickness direction (labels C3D8R-T2 and C3D8R-E-T2) produced a nearly constant ratio $P_{cr} / P_{E}$ for various thickness values, with a maximum discrepancy of 0.95%.

Using element types C3D8I, C3D20, and C3D20R resulted in a $P_{cr} / P_{E}$ value that remained virtually constant at 0.975, 0.987, and 0.987, with maximum discrepancies of 0.70%, 0.42%, and 0.37%, respectively. Increasing the element number in the thickness direction did not change the results for different thickness values.

For shell element types, the comparisons in Figure 11 indicate that varying the number of integration points in the thickness direction from 5 to 9 did not affect the results, but the ratio $P_{cr} / P_{E}$ varied significantly with wall thickness, as shown in Figures 11 and 12. When the thickness t was less than 10 mm, the difference between the ratio $P_{cr} / P_{E}$ for that value and that with t = 2.17 mm remained less than 0.01. The ABAQUS Inc. (2012) manual points out that the shell element type can be used for a circular tube when the ratio of curvature radius-to-thickness ratio (R_m/t) is greater than 10. In this case, for a CHS with a diameter of 100 mm, the thickness t should be less than 4.76 mm. However, circular tubes with R_m/t values smaller than 10 are very common in practical applications, so the shell element type was not selected for FE simulation of the CHS column in this work. After an extensive comparison of element types and numbers, solid element type C3D8I was ultimately chosen for use in developing an FE model with high accuracy and efficiency.

Nonlinear buckling analysis

After performing linear elastic-buckling analysis of the effect of element type on the buckling eigenvalue, nonlinear analysis was carried out to study the effect of solid element number in the thickness direction on the column curve; this curve represents the relationship between column strength (P_u) and regularized slenderness ratios ( $\bar{λ}$ ). For this study, a cross-sectional geometry of Ø100×30 (D×t, mm) was selected. The regularized slenderness ratio ( $\bar{λ}$ ) varied from 0.3 to 3.0. Based on the results from the previous section, solid element type C3D8I was selected to discretize the FE model. The global element size in the longitudinal direction was 10 mm, and the element number in the circumferential direction was 30. The wall thickness was discretized using the element numbers 1 and 3.

The nonlinear analysis took into account both material nonlinearity and geometric nonlinearity. The adopted material properties were the nominal values as shown in Table 1. As required in ABAQUS, the true nonlinear stress–strain relationship of the material was adopted to describe the material nonlinearity. The true nonlinear stress–strain relationship was obtained by a simple conversion of the engineering stress–strain relationship (ABAQUS Inc., 2012), generally described by the Ramberg–Osgood expression; both the conversion and Ramberg–Osgood expression are described in Section “Material properties” To represent geometric nonlinearity, the initial out-of-straightness adopted the first-order buckling mode obtained from the linear elastic-buckling analysis, with a magnitude of L/1000.

The column curves of obtained column strength (P_u) versus regularized slenderness ratio ( $\bar{λ}$ ) in Figure 13 indicate that increasing the element number for wall thickness indeed did not affect the simulation results. Consequently, solid element type C3D8I and one element in the thickness direction were selected for the FE model in this study.

Figure 13.

Effect of wall thickness element number on the column curve (cross section Ø100×30).

FE size in the longitudinal and circumferential directions

The effect of element size on the column curve in the longitudinal and circumferential directions ( $P_{u} - \bar{λ}$ ) was investigated using nonlinear analysis of the four CHS geometries Ø100×3, Ø100×10, Ø100×20, and Ø100×30 (D×t, in mm). For each cross section, five global element sizes (5, 10, 15, 20, and 25, in mm) in the longitudinal and circumferential directions were considered, and the wall thickness was meshed with one element consistent with the results of Section “Element type and wall discretization”.

The column curves of all considered cross-sectional geometries and element sizes in Figure 14 demonstrate that column strength P_u increases when element size decreases. When the global element size is less than 10 mm, the ratios $P_{u_10 mm} / P_{u_5 mm}$ remain less than 3%, reflecting convergence. A global size of 10 mm was therefore selected for discretization of the FE model.

Figure 14.

Column curves for four circular hollow sections using five element sizes (one element in thickness direction): (a) cross section Ø100×3, (b) cross section Ø100×10, (c) cross section Ø100×20, and (d) cross section Ø100×30.

For the specimens with H-, box-, and angle sections, the buckling strength from FE simulation might vary with element sizes and types. To study the element effect on the buckling strength, some thick-walled and short specimens with a slenderness ratio of 0.45 were simulated with various combinations of element sizes (5, 10, 15, 20, 25, and 30 mm) and types (solid element: C3D8I, shell elements: S4 and S4R). The simulation results in Table 9 indicate that (1) the differences of the buckling strengths are less than 3% with 5 mm versus 10 mm elements, (2) the results with shell elements are very different from those with solid elements, (3) the effect of increasing the number of solid elements on the simulation results is negligible, and (4) the conclusions obtained for H-, box-, and angle-section specimens are consistent with that from the circular tube specimens detailed above.

Table 9.

Comparisons of buckling strengths with various combinations of element sizes and types for H-, box-, and angle-section specimens.

Specimen	Item	Element size (mm, element type C3D8I)						Element type (size: 10 mm)
		5	10	15	20	25	30	C3D8I (two in wall)	S4	S4R
H80×80×6×8	P_u	468.8	468.4	485.7	484.0	474.7	474.6	468.5	484.6	481.7
	Diff. (%)	0.00	−0.11	3.59	3.23	1.26	1.23	−0.07	3.37	2.73
□80×40×6	P_u	313.1	315.6	316.8	319.8	320.6	321.5	316.4	301.9	304.7
	Diff. (%)	0.00	0.792	1.183	2.135	2.388	2.663	1.049	−3.577	−2.677
L65×40×10	P_u	236.8	237.0	237.0	216.7	222.1	220.0	236.8	230.4	230.5
	Diff. (%)	0.000	0.046	0.082	−8.482	−6.209	−7.091	0.000	−2.730	−2.652

Material properties

The nonlinear analysis in ABAQUS/Standard requires input values for Young’s modulus (E), Poisson’s ratio (v), and the relationship between true stress ( $σ_{true}$ ) and true plastic strain ( $ε_{true}^{pl}$ ) of the material. In uniaxial tension/compression testing, the obtained stress and strain data are generally expressed as engineering stress ( $σ_{eng}$ ) and engineering strain ( $ε_{eng}$ ). For isotropic material, the engineering stress and strain can be converted into true stress and true plastic strain (ABAQUS Inc., 2012) using

σ_{true} = σ_{eng} (1 + ε_{eng})

(19)

ε_{true}^{pl} = \ln (1 + ε_{eng}) - \frac{σ_{true}}{E}

(20)

For aluminum alloy, the engineering stress–strain relationship generally adopts the Ramberg–Osgood expression (De Martino et al., 1990; Ramberg and Osgood, 1943) given below, in which the material properties E, $σ_{0.2}$ , and the exponent n may be obtained from material tests

ε = \frac{σ}{E} + 0.002 {(\frac{σ}{σ_{0.2}})}^{n}

(21)

End simulation

Because the experimental end device was complex (Figure 15), the five end models shown in Figure 16 were compared to find a suitable solution. The dashed line A represents the top surface of the pattern stone, and the dashed line B represents the position of the knife edge T1 or B1 about which the column specimen would bend. If there was a steel plate at the tube end (e.g. Models 1, 2, and 3), the surface of the tube end was tied to the corresponding surface of the steel plate, and the reference point (RF-T or RF-B) located at the hinge in Figure 16 was coupled with the other surface of the steel plate. If there was no steel plate in the end simulation (e.g. Models 4 and 5), the reference point (RF-T or RF-B) was directly coupled to the surface of the tube end.

Figure 15.

Specimen end.

Figure 16.

Five simplified simulations of the tube end: (a) Model 1, (b) Model 2, (c) Model 3, (d) Model 4, and (e) Model 5.

Considering that the shorter the specimen the greater the influence of the modeling method on the simulation results, a very short circular tube with a nominal dimension of 89×6.5×492 (diameter×thickness×length, in mm) was selected to analyze effects produced by the end-modeling method. Elastic-buckling analysis was performed on the five FE models for a unit load applied vertically at the reference point RF-T. The five first-order buckling modes obtained are shown in Figure 17 and the five first-order buckling eigenvalues obtained (P_cr) are listed in Table 10. Theoretically, the first eigenvalue (P_cr) equals the Euler load (P_E), because the tube had pinned ends and the first buckling mode was that of overall buckling. Actually, the discrepancies between the eigenvalue P_cr and the Euler load P_E were very small; the ratios P_cr/F_E of the five FE models ranged from 0.97 to 1.03. Although Models 2 and 3 have no discrepancies, Model 5 with a discrepancy of 0.01 was selected for the FE model of the circular tube columns in this article because of the relatively easy modeling offered by that model for FE simulations.

Figure 17.

First-order buckling modes obtained using five models to simulate the tube end: (a) Model 1, (b) Model 2, (c) Model 3, (d) Model 4, and (e) Model 5.

Table 10.

First-order buckling eigenvalues P_cr of the five end models.

	End plate thickness (mm)	Tube length (mm)	L_e (mm)	P_cr (kN)	Euler load, P_E (kN)	P_cr/P_E
Model 1	40	452	590	2.94×10³	2.86×10³	1.03
Model 2	20	492	590	2.87×10³	2.86×10³	1.00
Model 3	49	492	590	2.87×10³	2.86×10³	1.00
Model 4	0	590	590	2.78×10³	2.86×10³	0.97
Model 5	0	492	590	2.83×10³	2.86×10³	0.99

Test verification

After a comprehensive study of several principal factors affecting the FE simulation, the FE model of aluminum alloy column members was developed and used to perform load–displacement nonlinear buckling analysis. The developed FE model was verified with respect to the experimental results summarized in section “Summary of column test program” to check the model’s reliability.

Because the complete crookedness was measured only for circular and angle-section specimens and the crookedness of the H- and box-section specimens was measured only at mid-length, the initial geometrical imperfection models used for test verification were the completely measured initial crookedness for circular and angle-section specimens and the first buckling modes with measured initial crookedness at mid-length for the H- and box-section specimens. The loading locations on the end sections of all specimens were adjusted according to the derived initial eccentricities and the measured initial crookedness at mid-length.

The failure modes predicted by the FE model were flexural buckling, in agreement with experimental results, as shown in Figures 18 and 19. As an example, Figure 20 shows the comparison of the load–deflection curves obtained from both the experiment and the FE simulation for specimen ○76-L1700-A. A comparison between the ultimate loads predicted by the FE analysis and the experimental ultimate load is shown in Table 10, in which the experimental and simulated ultimate loads (P_exp and P_FE) were normalized with respect to the yield load (P_0.2) calculated by equations (22) to (24). It can be seen that the FE simulation results and experimental results from most specimens are in good agreement

φ_{Exp} = \frac{P_{Exp}}{P_{0.2}}

(22)

φ_{FE} = \frac{P_{FE}}{P_{0.2}}

(23)

P_{0.2} = σ_{0.2} A

(24)

Figure 18.

Comparison of experimental and FE deformed shapes for specimen ○89-L500-C.

Figure 19.

Comparison of experimental and FE deformed shapes for specimen ○76-L1700-A.

Figure 20.

Comparison of experimental and FE load–deflection curves for specimen ○76-L1700-A.

Parametric study

Possible factors related to the stability coefficient of a column member failing by overall buckling mainly include material nonlinearity, geometric imperfection, cross-sectional type and dimensions, and so on. A parametric study was carried out to investigate the effects of such factors on the column stability coefficient. The investigation in this article was limited to those aluminum alloy columns in which local buckling did not occur before overall buckling.

Initial imperfection

Initial imperfections of a metal column member include both geometric imperfection and material imperfection; the former refers mainly to initial crookedness at mid-length and initial loading eccentricity at the ends, while the latter refers mainly to residual stress that is typically so small in extruded aluminum alloy profiles that its effect on column stability can be ignored (Mazzolani, 1995).

Initial crookedness

The first buckling mode of axially loaded member was used to represent the initial geometrical imperfection of column member. CHS Φ100×10 and angle section L100×65×10 were chosen for study of initial crookedness on the column curve with respect to flexural buckling and torsional-flexural buckling, respectively, as shown in Figure 21(a) and (b). It can be seen that the stability coefficient decreases with increased initial mid-length crookedness. Such initial crookedness has a larger influence on the stability coefficient of the column member with a regularized slenderness ratio of about 1.1.

Figure 21.

Influence of initial crookedness on column curve: (a) flexural buckling and (b) torsional-flexural buckling.

For asymmetric cross sections such as angle sections, the direction of initial crookedness may also affect the stability coefficient. Angle section L100×65×10 was studied with the results shown in Figure 22 that indicate that the initial crookedness direction for asymmetric cross section affects the stability coefficient, especially when the regularized slenderness ratio is near 1. It can also be seen that the stability coefficient with initial crookedness creating an angle point in compression is higher than that with initial crookedness creating an angle point in tension.

Figure 22.

Influence of initial crookedness direction on column curve.

Initial loading eccentricity

Section H100×80 × 6×10 was chosen to study the effect of initial loading eccentricity on column curve, as shown in Figure 23. It can be seen that increasing the loading eccentricity at column ends decreases the column curve, especially when the regularized slenderness ratio is near 1.

Figure 23.

Influence of initial eccentricity on column curve.

Material properties

According to tangent modulus theory, the column buckling stress is given by

σ_{c} = \frac{π^{2} E_{t, m}}{λ^{2}}

(25)

The stability coefficient can then be written as follows

φ = \frac{σ_{c}}{f_{0.2}} = \frac{π^{2} E_{t, m}}{λ^{2} f_{0.2}} = \frac{E_{t, m}}{E} \frac{1}{{\bar{λ}}^{2}}

(26)

For aluminum alloy, the tangent modulus can be obtained from the Ramberg–Osgood expression

E_{t, m} = \frac{d σ}{d ε} = \frac{E}{1 + 0.002 n (E σ^{n - 1} / {f_{0.2}}^{n})}

(27)

By substituting equation (27) into equation (26), the stability coefficient can be obtained as follows

{\bar{λ}}^{2} = \frac{1}{φ + nk φ^{n}}

(28)

where

k = \frac{1}{500} \frac{E}{f_{0.2}}

(29)

Ramberg–Osgood constitutive curves with varying values of exponent n are shown in Figure 24(a). Taking cross section Φ100×10 as an example, the influence of n on column curves according to tangent modulus theory is shown in Figure 24(b) to (d). It can be seen that when n increases, the column curve decreases, and this effect is enhanced when ratio k increases. A similar conclusion can be drawn for column curves using FE nonlinear simulation (Figure 25), in which the initial crookedness at mid-length is L/1000. It is also seen that when n is larger than 20, the effect of n on column curve becomes very small.

Figure 24.

Influence of n on column curve (tangent modulus theory): (a) influence of n on Ramberg–Osgood curve, (b) column curves when k = 0.54, (c) column curves when k = 1.0, and (d) boundaries when k = 0.54 and k = 0.1.

Figure 25.

Influence of n on column curve (nonlinear finite element simulation): (a) when k = 0.54 and b) when k = 1.0.

Cross-sectional dimensions

The influence of dimensions on column curves was investigated for four cross-sectional types, including an H-section, a box-section, an angle section, and a CHS with the results shown in Figures 26 to 29, respectively. It can be seen that the influence of cross-sectional dimensions on column curve is negligible for the four cross sections when local buckling does not occur.

Figure 26.

Influence of dimensions on column curves for H-section: (a) height-to-width ratio and (b) width-to-thickness ratio.

Figure 27.

Influence of dimensions on column curves for box-section: (a) height-to-width ratio and (b) width-to-thickness ratio.

Figure 28.

Influence of dimensions on column curves for circular hollow section.

Figure 29.

Influence of dimensions on column curves for angle section.

Proposed column curve

Because crookedness tolerance of aluminum alloy extruded profiles is 1.5L/1000 (according to GB 5237.1-2008, 2008) and initial loading eccentricity at member ends may exist, four cases representing different material properties or crookedness at column mid-length were considered and compared. As shown in Figure 30, column curve expressions were derived by fitting Perry curves to the lowest stability coefficients. The fitting column curves were compared with the column curves consistent with EC9 and GB50429. It can be seen that a fitting curve using initial crookedness L/1000 and representative material properties from GB50429 (Figure 30(a)) is the closest to both the EC9 curve and the GB50429 curve. Increasing initial crookedness at mid-length will decrease the fitting curve, as shown in Figure 30(b) and (d). The fitting column curve using initial crookedness L/1000 and statistical material properties from experimental results (Figure 30(c)) is higher than the EC9 and GB50429 curves. This is because the measured elastic modulus E, the 0.2% proof stress, and the Ramberg–Osgood exponents n were a little higher than the representative values given by GB50429, which can increase the buckling strength of the column. The fitting column curve in Figure 30(d), using initial crookedness 1.8L/1000 and the measured material properties, is proposed to predict overall buckling strength for column design. The $α$ and ${\bar{λ}}_{0}$ values of the fitting column curve expressions are listed in Table 13.

Figure 30.

Fitting column curve for 6082-T6 aluminum alloy with different material properties and initial crookedness at mid-length: (a) L/1000 and representative properties, (b) 1.8L/1000 and representative properties, (c) L/1000 and experimental properties, and (d) 1.8L/1000 and experimental properties.

Table 11.

Comparison between test and simulation results for H-section specimens and box-section specimens.

No.	Cross section	Specimen label	L_e (mm)	$λ$	$\bar{λ}$	Test			Simulation			Diff (%)
						P _exp (kN)	ϕ _exp	Failing mode	P _FE (kN)	ϕ _FE	Failing mode
Comparison between test and simulation results for H-section specimens
1	H120×60×5×7	H1-A	1066.0	78.11	1.642	148.7	0.3494	F	155.5	0.3654	F	4.6
2	H120×80×6×8	H2-A	1080.0	57.02	1.199	311.8	0.5442	F	330.3	0.5765	F	5.9
3		H3-A	990.0	52.14	1.096	352.0	0.6089	F	403.1	0.6974	F	14.5
4		H3-B	990.0	52.38	1.101	375.6	0.6642	F	376.6	0.6660	F	0.3
5		H3-C	990.0	52.42	1.102	435.4	0.7756	F	418.5	0.7454	F	−3.9
6	H100×70×6×8	H4-A	1058.0	63.08	1.350	240.5	0.4656	F	264.1	0.5113	F	9.8
7		H4-B	1054.5	62.50	1.337	262.4	0.5057	F	262.7	0.5064	F	0.1
8		H4-C	1062.5	63.12	1.350	263.5	0.5088	F	271.3	0.5239	F	3.0
9	H100×50×5×7	H5-A	1054.0	91.99	1.987	89.9	0.2505	F	90.6	0.2525	F	0.8
10		H5-B	1042.0	91.10	1.968	94.6	0.2661	F	89.0	0.2505	F	−5.9
11		H5-C	1064.0	92.90	2.007	93.6	0.2627	F	78.9	0.2214	F	−15.7
12		H6-A	1240.0	108.74	2.349	66.9	0.1876	F	61.5	0.1725	F	−8.1
13		H6-B	1239.0	108.87	2.352	67.5	0.1900	F	62.0	0.1746	F	−8.1
14		H7-A	1541.0	135.44	2.926	44.4	0.1243	F	40.7	0.1141	F	−8.3
15		H7-B	1541.0	135.18	2.921	44.8	0.1263	F	41.3	0.1165	F	−7.8
16		H8-A	1841.0	161.46	3.488	33.4	0.0942	F	28.8	0.0813	F	−13.8
17		H8-B	1842.0	162.19	3.504	30.2	0.0847	F	28.2	0.0791	F	−6.6
No.	Cross section	Specimen label	L_e (mm)	$λ$	$\bar{λ}$	Test			Simulation			Diff (%)
						P _exp (kN)	ϕ _exp	Failing mode	P _FE (kN)	ϕ _FE	Failing mode
Comparison between test and simulation results for box-section specimens
1	□80×4	□1-A	1056.0	34.13	0.705	294.8	0.8844	F	307.7	0.9231	F	4.4
2		□1-B	1061.0	34.33	0.709	289.5	0.9158	F	292.8	0.9263	F	1.1
3		□1-C	1056.0	34.19	0.706	295.2	0.9195	F	288.2	0.8979	F	−2.4
4	□80×40×4	□2-A	1548.5	97.26	1.89	63.8	0.3047	F	56.0	0.2677	F	−12.2
5		□2-B	1546.0	96.93	1.884	68.9	0.3250	F	59.0	0.2781	F	−14.4
6		□2-C	1549.9	96.87	1.883	63.1	0.3037	F	57.2	0.2751	F	−9.4
7		□4-A	683.8	42.86	0.833	176.6	0.8387	F	181.5	0.8620	F	2.8
8		□4-B	683.8	42.90	0.834	164.6	0.7757	F	170.4	0.8030	F	3.5
9	□100×50×4	□3-A	1548.0	76.38	1.568	120.7	0.4098	F	116.2	0.3945	F	−3.7
10		□3-B	1549.9	76.19	1.564	127.7	0.4214	F	121.6	0.4014	F	−4.8
11		□3-C	1551.5	76.43	1.569	130.1	0.4395	F	116.3	0.3929	F	−10.6
12	□80×40×6	□5-A	440.0	28.93	0.606	353.5	0.8940	F	350.7	0.8869	F	−0.8
13		□5-B	440.0	28.93	0.606	376.0	0.9510	F	398.1	1.0068	F	5.9
14		□6-A	340.0	22.36	0.469	389.4	0.9850	F	423.7	1.0715	F	8.8
15		□6-B	340.0	22.36	0.469	390.3	0.9873	F	388.0	0.9812	F	−0.6

Table 12.

Comparison between test and simulation results for circular-hollow-section specimens, equal-leg L-section specimens, and unequal-leg L-section specimens.

No.	Cross section	Specimen label	L_e (mm)	$λ$	$\bar{λ}$	Test			Simulation		Diff (%)
						P _exp (kN)	ϕ _exp	Failing mode	P _FE (kN)	Failing mode
Comparison between test and simulation results for circular-hollow-section specimens
1	O89×6.5	O89-L500-A	598.5	20.44	0.451	508.7	0.9521	F	529	F	4.0
2		O89-L500-B	598.0	20.44	0.451	467.5	0.8767	F	516	F	10.4
3		O89-L500-C	598.0	20.42	0.451	471.9	0.8832	F	432	F	−8.5
4	O89×6.5	O89-L800-A	898.0	30.69	0.678	444.3	0.8331	F	467	F	5.1
5		O89-L800-B	898.0	30.69	0.678	429.4	0.8052	F	407	F	−5.2
6		O89-L800-C	897.0	30.66	0.677	473.7	0.8883	F	499	F	5.3
7	O89×6.5	O89-L1200-A	1298.5	44.38	0.980	416.2	0.7806	F	389	F	−6.5
8		O89-L1200-B	1298.0	44.36	0.979	378.7	0.7102	F	352	F	−7.1
9		O89-L1200-C	1298.5	44.38	0.980	398.5	0.7474	F	370	F	−7.2
10	O89×6.5	O89-L1650-A	1747.0	59.86	1.322	239.7	0.4490	F	234	F	−2.4
11		O89-L1650-B	1748.0	59.87	1.322	269.8	0.5067	F	274	F	1.6
12		O89-L1650-C	1748.0	59.83	1.321	265.6	0.4984	F	281	F	5.8
13	O76×3.0	O76-L1700-A	1798.0	69.89	1.263	74.2	0.4801	F	79	F	6.5
14		O76-L1700-B	1798.0	69.96	1.264	68.5	0.4471	F	68	F	−0.7
15		O76-L1700-C	1797.0	69.78	1.261	72.2	0.4621	F	79	F	9.4
No.	Cross section	Specimen label	L_e (mm)	$\bar{λ}$	Test			Simulation			Diff. (%)
					P _exp (kN)	ϕ _exp	Failing mode	P _FE (kN)	Failing mode
Comparison between test and simulation results for equal-leg L-section specimens
1	L110×8	L110-800A	897.6	1.468	221.46	0.4186	TF	248.24	TF		12.1
2		L110-800B	897.2	1.474	224.31	0.4264	TF	249.54	TF		11.2
3		L110-1200A	1297.8	1.553	203.11	0.3846	TF	217.45	TF		7.1
4		L110-1200B	1297.8	1.538	210.22	0.3935	TF	227.51	TF		8.2
5		L110-1800A	1897.4	1.784	142.27	0.2661	F	135.86	F		−4.5
6		L110-1800B	1897.2	1.786	123.03	0.2327	F	139.95	F		13.8
7	L90×10	L90-400A	497.7	0.813	427.97	0.8862	TF	453.22	TF		5.9
8		L90-400B	497.4	0.809	466.36	0.9625	TF	455.31	TF		−2.4
9		L90-800A	897.6	0.966	348.08	0.7216	TF	305.63	TF		−12.2
10		L90-800B	897.2	0.966	383.3	0.7919	TF	361.78	TF		−5.6
11	L90×8	L90-1200A	1297.4	1.414	159.7	0.4167	F	147.87	F		−7.4
12		L90-1200B	1297.1	1.404	141.29	0.3627	F	149.89	F		6.1
13		L90-1500A	1597.5	1.739	111.02	0.2878	F	111.22	F		0.2
14		L90-1500B	1597.6	1.736	102.67	0.2632	F	96.92	F		−5.6
15	L70×4	L70-1500A	1596.3	2.240	24.31	0.1652	F	26.08	F		7.3
16		L70-1500B	1597.6	2.240	26.3	0.1786	F	26.60	F		1.1
No.	Cross section	Specimen label	L_e (mm)	$\bar{λ}$	Test			Simulation			Diff. (%)
					P _exp (kN)	ϕ _exp	Failing mode	P _FE (kN)	Failing mode
Comparison between test and simulation results for unequal-leg L-section specimens
1	L70×45×8	L70×45-1000A	1114.9	2.473	47.8	0.1872	TF	42.0	TF		−12.1
2		L70×45-1000B	1108.4	2.466	48.9	0.1906	TF	42.5	TF		−13.1
3		L70×45-1500A	1597.3	3.539	25.7	0.1020	TF	24.4	TF		−5.1
4		L70×45-1500B	1597.5	3.522	24.8	0.0970	TF	23.5	TF		−5.2
5	L100×70×8	L100×70-800A	897.3	1.392	192.8	0.5311	TF	205.7	TF		6.7
6		L100×70-800B	897.2	1.387	197.5	0.5384	TF	206.7	TF		4.7
7		L100×70-1000A	1098.1	1.59	119.3	0.3252	TF	140.2	TF		17.5
8		L100×70-1000B	1098	1.59	141.4	0.3857	TF	145.5	TF		2.9
9		L100×70-1500A	1597.8	2.198	84.5	0.2302	TF	73.2	TF		−13.4
10		L100×70-1500B	1597.4	2.199	78.9	0.2147	TF	75.4	TF		−4.4

F denotes overall flexural buckling and TF indicates torsional-flexural buckling.

Table 13.

$α$ and ${\bar{λ}}_{0}$ values of fitting column curves.

Material properties	Representative values from GB50429-2007		Statistical values of measurements
Crookedness at mid-length	0.001L	0.0018L	0.001L	0.0018L
Parameter $α$	0.20	0.27	0.17	0.26
Parameter ${\bar{λ}}_{0}$	0.19	0.11	0.16	0.17

Figure 31 shows the comparison of the experimental stability coefficients, the column curves from various codes, and the fitting column curve using initial crookedness of 0.0018L and the measured material properties. Stability coefficients from the proposed curve, tests, and various codes are given in Table 14. Most of the experimental stability coefficients of a total of 73 specimens are higher than the fitting column curve, indicating that the fitting column curve is lower and therefore more conservative than the column curves from AA, AS/NZS, EC9, and GB50429.

Figure 31.

Comparison of the proposed column curve, test results, and column curves from codes.

Table 14.

Comparison of stability coefficients from the proposed curve, tests, and various codes.

Cross section	No.	$\bar{λ}$	Test	AA (2010)		AS/NZS (1997)		EC9 (2007)		GB 50429-2007 (2007)		Fitting curve
			$φ$	$φ$	Diff. (%)	$φ$	Diff. (%)	$φ$	Diff. (%)	$φ$	Diff. (%)	$φ$	Diff. (%)
H-section	1	1.642	0.349	0.315	−9.8	0.371	6.2	0.318	−9.1	0.319	−8.7	0.308	−11.8
	2	1.199	0.544	0.576	5.8	0.677	24.5	0.526	−3.4	0.531	−2.5	0.505	−7.1
	3	1.096	0.609	0.609	0.0	0.716	17.6	0.592	−2.8	0.598	−1.8	0.568	−6.7
	4	1.101	0.664	0.607	−8.6	0.714	7.5	0.589	−11.4	0.595	−10.5	0.565	−14.9
	5	1.102	0.776	0.607	−21.8	0.714	−8.0	0.588	−24.2	0.594	−23.4	0.565	−27.2
	6	1.350	0.466	0.466	0.2	0.549	17.8	0.441	−5.4	0.444	−4.7	0.425	−8.9
	7	1.337	0.506	0.476	−6.0	0.559	10.6	0.447	−11.5	0.451	−10.9	0.431	−14.8
	8	1.350	0.509	0.466	−8.3	0.549	7.8	0.441	−13.4	0.444	−12.8	0.425	−16.6
	9	1.987	0.251	0.215	−14.1	0.253	1.1	0.225	−10.0	0.226	−9.7	0.220	−12.5
	10	1.968	0.266	0.219	−17.5	0.258	−3.0	0.229	−13.8	0.230	−13.5	0.223	−16.0
	11	2.007	0.263	0.211	−19.7	0.248	−5.5	0.221	−15.7	0.222	−15.5	0.216	−18.0
	12	2.349	0.188	0.154	−17.9	0.181	−3.4	0.165	−12.0	0.165	−11.8	0.161	−14.1
	13	2.352	0.190	0.154	−19.1	0.181	−4.9	0.165	−13.3	0.165	−13.1	0.161	−15.2
	14	2.926	0.124	0.099	−20.1	0.117	−6.0	0.109	−12.5	0.109	−12.4	0.107	−13.9
	15	2.921	0.126	0.100	−21.1	0.117	−7.2	0.109	−13.6	0.109	−13.5	0.107	−15.0
	16	3.488	0.094	0.070	−25.8	0.082	−12.7	0.078	−17.7	0.078	−17.6	0.076	−18.8
	17	3.504	0.085	0.069	−18.3	0.081	−3.8	0.077	−9.3	0.077	−9.2	0.076	−11.0
Box-section	18	0.705	0.884	0.734	−17.0	0.864	−2.4	0.829	−6.2	0.840	−5.0	0.811	−8.3
	19	0.709	0.916	0.733	−20.0	0.862	−5.9	0.827	−9.6	0.838	−8.5	0.809	−11.7
	20	0.706	0.920	0.734	−20.2	0.863	−6.1	0.829	−9.9	0.839	−8.7	0.810	−11.9
	21	1.890	0.305	0.238	−21.9	0.280	−8.1	0.247	−18.9	0.248	−18.7	0.240	−21.2
	22	1.884	0.325	0.239	−26.3	0.282	−13.3	0.248	−23.5	0.249	−23.3	0.242	−25.6
	23	1.883	0.304	0.240	−21.1	0.282	−7.1	0.249	−18.1	0.250	−17.8	0.242	−20.4
	24	0.833	0.839	0.693	−17.4	0.815	−2.8	0.763	−9.1	0.773	−7.9	0.739	−12.0
	25	0.834	0.776	0.693	−10.7	0.815	5.1	0.762	−1.8	0.772	−0.5	0.738	−4.9
	26	1.568	0.410	0.346	−15.6	0.407	−0.7	0.344	−16.0	0.346	−15.6	0.333	−18.8
	27	1.564	0.421	0.347	−17.5	0.409	−3.0	0.346	−18.0	0.347	−17.6	0.334	−20.6
	28	1.569	0.440	0.345	−21.4	0.406	−7.6	0.344	−21.8	0.345	−21.4	0.333	−24.4
	29	0.606	0.894	0.766	−14.3	0.901	0.8	0.871	−2.6	0.881	−1.4	0.858	−4.0
	30	0.606	0.951	0.766	−19.5	0.901	−5.3	0.871	−8.5	0.881	−7.3	0.858	−9.8
	31	0.469	0.985	0.810	−17.8	0.953	−3.3	0.915	−7.1	0.926	−6.0	0.911	−7.5
	32	0.469	0.987	0.810	−18.0	0.953	−3.5	0.915	−7.3	0.926	−6.2	0.911	−7.7
Circular hollow section	33	0.451	0.952	0.815	−14.3	0.959	0.8	0.920	−3.3	0.931	−2.2	0.918	−3.6
	34	0.451	0.877	0.815	−7.0	0.959	9.4	0.920	5.0	0.931	6.2	0.918	4.6
	35	0.451	0.883	0.815	−7.7	0.959	8.6	0.920	4.2	0.931	5.4	0.918	3.9
	36	0.678	0.833	0.743	−10.8	0.874	4.9	0.841	1.0	0.852	2.3	0.825	−1.0
	37	0.678	0.805	0.743	−7.8	0.874	8.5	0.841	4.5	0.852	5.8	0.825	2.4
	38	0.677	0.888	0.743	−16.4	0.874	−1.6	0.842	−5.2	0.853	−4.0	0.825	−7.1
	39	0.980	0.781	0.646	−17.3	0.760	−2.7	0.670	−14.2	0.678	−13.2	0.644	−17.5
	40	0.979	0.710	0.646	−9.0	0.760	7.1	0.670	−5.6	0.678	−4.5	0.645	−9.2
	41	0.980	0.747	0.646	−13.6	0.760	1.7	0.670	−10.4	0.678	−9.3	0.644	−13.8
	42	1.322	0.449	0.486	8.3	0.572	27.4	0.455	1.4	0.459	2.1	0.438	−2.4
	43	1.322	0.507	0.486	−4.0	0.572	12.9	0.455	−10.1	0.459	−9.5	0.438	−13.5
	44	1.321	0.498	0.487	−2.3	0.573	15.0	0.456	−8.5	0.459	−7.9	0.439	−11.9
	45	1.263	0.480	0.533	11.0	0.627	30.6	0.488	1.6	0.492	2.4	0.469	−2.2
	46	1.264	0.447	0.532	19.0	0.626	40.0	0.487	9.0	0.491	9.9	0.469	4.9
	47	1.261	0.462	0.535	15.7	0.629	36.1	0.489	5.8	0.493	6.7	0.470	1.8
Mean				−	−11.9%	−	3.7%	−	−8.6%	−	−7.8%	−	−11.0%
Max.				−	19.0%	−	40.0%	−	9.0%	−	9.9%	−	4.9%
Min.				−	−26.3%	−	−13.3%	−	−24.2%	−	−23.4%	−	−27.2%
Diff. >0				−	6	−	23	−	8	−	8	−	5

After comparing the measurements with the tolerance limits given in the Chinese fabrication standard, a representative value of the initial crookedness, 1.8L/1000, is recommended. One should note that this value was only based on the measurement results of a small group of specimens. To determine a more appropriate value for aluminum column design, further studies are still needed to investigate the initial geometric imperfection of domestic aluminum extruded profiles.

Conclusion

This article presents a comprehensive exploration of FE simulation of pinned-ended aluminum alloy members which fail by overall buckling under axial compression using the FE software package ABAQUS. By comparing simulations with a variety of element types, mesh sizes, and end-simulation models, an advanced nonlinear FE model incorporating initial geometric imperfections and material nonlinearity of the 6082-T6 heat-treated aluminum alloy was developed. Performance of this model was verified against an experimental program of 73 column tests on aluminum alloy 6082-T6 extruded profiles. The verification demonstrated that the developed FE model can accurately predict experimental ultimate loads and failure modes for aluminum alloy columns. A parametric study was carried out on the stability of aluminum alloy column members failing by overall buckling. A fitting column curve was proposed for aluminum alloy design; it was somewhat more conservative than the column curve from current Chinese code GB50429. Further investigation is still needed in future to determine a more appropriate representative value of the initial geometrical imperfection of an aluminum column member for design purpose.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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