Experimental and Analytical Study of SHS Aluminium Members under Uniform Compression ^†

Abstract

The aim of this work was to study the ultimate behaviour of box-shaped aluminium members subjected to uniform compression. Eight stub column tests have been carried out at the University of Salerno. In particular, four box sections made of 6060 aluminium alloys with different width-to-thickness ratios have been investigated. The results have been reported in terms of the maximum compressive resistance and corresponding deformation capacity. Subsequently, the experimental results have been compared with those obtained by two accurate methodologies: (1) a theoretical procedure based on the deformation theory of plasticity (J2); and (2) an extension of the effective thickness method (ETM) provided by Annex L of the Eurocode 9 draft. These approaches take into account the local buckling effects within the elastic-plastic region, the strain-hardening be-haviour of the aluminium material, and the interaction between the plate elements constituting the cross-section. Finally, a comparison between new methodologies and current design rules has been presented.

1. Introduction

The utilization of aluminium alloys is experiencing significant growth in the realm of structural engineering applications, particularly in the context of long-span roof systems, movable bridges, and prefabricated frameworks. Aluminium possesses highly enticing characteristics, including a remarkable strength-to-weight ratio, exceptional resistance against corrosion, flexibility in accommodating varied cross-section shapes, an advantageous life-cycle cost, and expedient fabrication processes [1,2,3].

Aluminium alloys also exhibit structural disadvantages: high initial cost, high deformability resulting from an elastic modulus one-third that of steel, increased local and global stability issues, high sensitivity to thermal variations, and reduced strength due to heat-treated zones in welded areas. Similar to all metallic elements, aluminium alloy elements are sensitive to instability phenomena. One of the main stability problems for metal alloys is local buckling, which impacts the ultimate behavior of compressed portions of the cross-section and subsequently influences the ultimate resistance and plastic deformation capacity. This factor becomes even more significant for aluminium alloys given their lower elastic modulus. Local buckling is influenced by the slenderness of the cross-section, with buckling occurring in either the elastic or plastic range, depending on the ratio of width to thickness of the plate elements composing the member section. The European code, EN 1999-1-1 [4], establishes a classification method for cross-sections that considers the slenderness of the individual plate elements and the conventional elastic limit, $f_{0.2}$.

The current approaches for metal members are based on an elastic-perfectly plastic analysis performed on an effective cross-section. However, this approach [5] neglects two main aspects: (1) the real behaviour of the material, which is relevant for aluminium members because of their continuous strain-hardening behaviour [3]; and (2) the interaction between the plate elements constituting the cross-sections.

Many research studies have been dedicated to enhancing the accuracy in forecasting the ultimate response of aluminium alloy members. Numerous stub column tests have been performed on different cross-sectional shapes in order to acquire additional insights into the impact of local buckling phenomena on the ultimate resistance. Faella et al. [6] examined sections of a square hollow section (SHS) and a rectangular hollow section (RHS), providing an empirical formulation for computing the strain that corresponds to the complete development of local buckling. Su et al. [7] undertook an exhaustive experimental campaign on SHS and RHS sections fabricated from aluminium alloys 6061-T6 and 6063-T5, both with and without internal stiffeners.

In the study of Moen et al. [8], they investigated the behavior of aluminium alloy beams subjected to both uniform and non-uniform bending. They performed three- and four-point bending tests on beams made of aluminium alloys 6082-T4, 6082-T6, and 7108-T7. They found that the rotation capacity of these beams is influenced by the strain-hardening characteristics of the aluminium material and the magnitude of the moment gradient. In a more recent study, Wang et al. [9] performed experiments on extruded I-shaped beams made of 6000 series aluminium alloy. These beams were tested both with and without intermediate stiffeners under concentrated loads. Furthermore, Piluso et al. [10], Montuori et al. [11,12], and Pisapia [13,14] have proposed mathematical equations to calculate the ultimate flexural strength and rotation capacity of box-shaped (SHS and RHS) and I-shaped aluminium beams.

Following the research provided by Nastri et al. [15,16], a variation of the effective thickness method (ETM), which is currently utilized by Eurocode 9 [4], has been formulated to calculate the ultimate resistance of aluminium beams. This revised edition of ETM is currently presented in Annex L of the revised version of Eurocode 9 [17].

In recent years, there has been a significant advancement in the understanding of the inelastic response of aluminium compression members. However, there still exists a significant deficiency in experimental data compared to steel members. In order to address this issue, the University of Salerno conducted eight stub column tests on square hollow section (SHS) members. These members were made of 6060 aluminium alloys and varied in terms of their width-to-thickness ratios. Subsequently, the experimental results were obtained in terms of maximum compression resistance and maximum strains. Then, two accurate methodologies were presented: (1) a theoretical procedure based on the deformation theory of plasticity (J2) [18,19,20]; and (2) an extension of the effective thickness method (ETM) provided by Annex L of the Eurocode 9 draft [4]. These approaches consider the local buckling effects in the elastic-plastic range, the strain-hardening behaviour of aluminium material, and the interaction between the plate elements constituting the cross-section. Finally, the experimental results presented in this work with those provided in [6,7] have been compared with the theoretical results obtained by means of the previous methodologies and current design provisions.

2. Experimental Campaign

2.1. Geometric and Mechanical Properties

The aluminium members analysed in the current experimental tests are extruded beams made of EN-AW 6060 T66 with the SHS section.

Four types of section are considered for both stub and bending tests, characterized by a nominal width of 40, 60, 80, or 100 mm and with the same nominal thickness equal to 2 mm. According to Figure 1, the nominal and measured geometrical properties of specimens are reported in

Table 1. Nominal and measured geometric properties of tested specimens.

Section	${\mathit{B}}_{1.\mathbf{n}\mathbf{o}\mathbf{m}}$ [mm]	${\mathit{B}}_{2.\mathbf{n}\mathbf{o}\mathbf{m}}$ [mm]	${\mathit{t}}_{\mathbf{n}\mathbf{o}\mathbf{m}}$ [mm]	${\mathit{A}}_{\mathbf{n}\mathbf{o}\mathbf{m}}$ [mm2]	${\mathit{B}}_1$ [mm]	${\mathit{B}}_2$ [mm]	${\mathit{t}}_1$ [mm]	${\mathit{t}}_2$ [mm]	${\mathit{t}}_3$ [mm]	${\mathit{t}}_4$ [mm]	$\mathit{A}$ [mm2]
SHS40	40.00	40.00	2.00	304	40.18	40.11	1.99	2.20	2.09	2.00	315.26
SHS60	60.00	60.00	2.00	464	60.31	60.40	2.00	2.16	2.00	2.09	480.92
SHS80	80.00	80.00	2.00	624	80.17	80.06	2.00	1.98	1.90	1.89	607.40
SHS100	100.00	100.00	2.00	784	100.30	100.30	2.24	2.04	2.12	2.37	860.40

.

Standard tensile tests have been performed on specimens cut from each type of section according to UNI-EN-ISO 6892-1-1 [21]. In particular, the specimens ${\mathrm{P}}_{\mathrm{I}}$ were cut from each plate constituting the section according to the shape shown in Figure 2. The geometric dimensions are reported in

Table 2. Geometric properties of tensile coupons.

Section	${\mathit{S}}_0$ [mm]	$\mathit{S}$ [mm]	${\mathit{L}}_{\mathit{c}}$ [mm]	${\mathit{L}}_{\mathit{s}}$ [mm]	${\mathit{L}}_{\mathit{t}\mathit{o}\mathit{t}}$ [mm]
SHS40	15	30	80	90	275
SHS60	15	30	80	90	275
SHS80	20	40	100	100	320
SHS100	20	40	100	100	320

.

The monotonic tests have been performed under displacement control according to Method A2 as described in UNI-EN-ISO 6892-1-1 [21]. The experimental results for each plate constituting each tested section are reported in

Table 3. Mechanical properties of tested tensile coupons.

$\mathbf{S}\mathbf{p}\mathbf{e}\mathbf{c}\mathbf{i}\mathbf{m}\mathbf{e}\mathbf{n}$		${\mathit{E}}_{\mathit{e}\mathit{x}\mathit{p}}$ [mPa]	${\mathit{f}}_{0.1}$ [mPa]	${\mathit{f}}_{0.2}$ [mPa]	${\mathit{f}}_{\mathit{u}}$ [mPa]	${\mathit{\epsilon }}_{\mathit{u}}$ [-]	${\mathit{\epsilon }}_{\mathit{r}}$ [-]	${\mathit{n}}_{\mathit{i}}$ [-]	${\mathit{n}}_{\mathit{p}}$ [-]
SHS 40	P1	73054	171.00	181.75	207.85	0.055	0.075	11.34	22.96
	P2	75767	176.70	182.59	203.03	0.046	0.062	21.14	28.22
	P3	71465	160.83	174.03	202.17	0.053	0.074	8.78	20.35
	P4	66793	154.92	166.92	192.36	0.057	0.072	9.29	23.53
SHS 60	P1	65669	210.11	227.32	252.65	0.054	0.064	8.80	30.38
	P2	61254	207.09	215.00	238.83	0.065	0.088	18.49	33.09
	P3	65277	224.68	231.39	252.22	0.065	0.082	23.55	38.62
	P4	72674	213.20	226.59	253.58	0.069	0.095	11.38	31.90
SHS 80	P1	69665	210.82	224.17	246.02	0.039	0.064	11.29	35.70
	P2	75975	199.00	211.00	253.41	0.061	0.068	11.84	18.66
	P3	65174	226.46	236.92	255.41	0.042	0.050	15.35	39.65
	P4	63341	217.45	231.12	253.71	0.070	0.091	11.37	38.07
SHS 100	P1	71564	168.65	174.75	199.10	0.060	0.088	19.51	26.39
	P2	70044	165.38	175.61	201.80	0.060	0.098	11.55	24.83
	P3	66906	140.02	152.86	187.38	0.080	0.103	7.90	17.64
	P4	69317	155.32	169.36	195.79	0.060	0.085	8.00	22.51

. In particular, the following measured properties are provided:

• The experimental Young’s modulus $\left. [E_{exp}\right]$
• The stress at a residual strain of 0.1% $\left. [f_{0.1}\right]$
• The stress at a residual strain of 0.2% $\left. [f_{0.2}\right]$
• The engineering maximum stress $\left. [f_u\right]$
• The engineering strain corresponding to the maximum stress $\left. [{\epsilon }_u\right]$
• The ultimate strain $\left. [{\epsilon }_r\right]$
• The Ramberg–Osgood coefficient $\left. [n\right]$ given by:

$$n_i=\frac{ln2}{ln\left(\frac{f_{0.2}}{f_{0.1}}\right)}\hspace{1em}\hspace{1em}\hspace{1em}{n}_p=\frac{ln(0.002/{\epsilon }_{0.u})}{ln\left(\frac{f_{0.2}}{f_u}\right)}$$

(1)

$n_i$ and $n_p$ are, respectively, the R-O coefficients in the inelastic and plastic region and ${\epsilon }_{0.u}$ is the residual strain corresponding to the maximum stress $f_u$, it is equal to ${\epsilon }_{0.u}={\epsilon }_u-0.002$. The Ramberg–Osgood law is given by:

$$\epsilon =\frac{\sigma }{E}+0.002{\left(\frac{\sigma }{f_{0.2}}\right)}^n$$

(2)

2.2. Stub Column Tests

Stub column tests have been carried out to evaluate the maximum load $N_{u.exp}$ and corresponding displacement ${\delta }_{u.exp}$ of SHS aluminium members. The test setup is shown in Figure 3. The compression tests were provided by means of a Schenck Hydropuls S56 testing machine (maximum load 630 kN, piston stroke ± 125 mm). For each member, a minimum of two stub tests were tested. The loading protocol is based on the application of two different test speeds, equal for each specimen: an initial low speed $v_i$ equal to 0.42 mm/min up to post-elastic level and, successively, a final speed $v_f$ equal to 1.20 mm/min. The speed change was set for each specimen according to the theoretical yield displacement ${\delta }_{0.2}$.

The axial displacements have been measured by three inductive transducers (LDT). The mean value of the three measures was considered. Almost all the specimens have failed due to pure local buckling, except the tests referring to SHS100 (test a, test b) and SHS80 (test a) where the instability phenomenon occurred, preliminarily, of a single plate due to the geometrical imperfections of sections and as can also be depicted by observing the shapes of $N-\delta$ curves provided in Figure 4. The results are shown in

Table 4. Results of stub column tests.

Specimen	Test	$\mathit{a}$ [mm]	${\mathit{N}}_{\mathit{u}.\mathit{e}\mathit{x}\mathit{p}}$ [kN]	${\stackrel{-}{\mathit{N}}}_{\mathit{u}}$ [-]	${\mathit{\delta }}_{\mathit{u}.\mathit{e}\mathit{x}\mathit{p}}$ [mm]	${\stackrel{-}{\mathit{\delta }}}_{\mathit{u}}\left({\stackrel{-}{\mathit{\epsilon }}}_{\mathit{u}}\right)$ [-]
SHS40	a	120.05	58.30	1.05	0.87	2.96
	b	120.12	57.53	1.04	0.86	2.92
	c	130.43	55.55	1.01	0.51	1.60
SHS60	a	181.10	92.26	0.85	0.59	0.96
	b	180.80	85.62	0.79	0.61	0.99
SHS80	a	240.12	79.50	0.58	0.60	0.76
	b	240.32	87.18	0.63	0.64	0.81
SHS100	a	300.10	90.22	0.62	0.62	0.58
	b	299.00	80.85	0.55	0.61	0.75

, where the specimen height $a$, the ultimate resistance $N_{u.exp}$ its non-dimensional value ${\stackrel{-}{N}}_u$, the maximum displacement ${\delta }_{u.exp}$, and its non-dimensional value ${\stackrel{-}{\delta }}_u$ are provided. The non-dimensional values are expressed as:

$${\stackrel{-}{N}}_u=\frac{N_{u.exp}}{A·f_{0.2}}\hspace{1em}\hspace{1em}\hspace{1em}{\stackrel{-}{\delta }}_u=\frac{{\delta }_{u.exp}}{{\delta }_{0.2}}$$

(3)

where $A$ represents the section area of each section, while ${\delta }_{0.2}$ corresponds to the displacement according to the conventional strain ${\epsilon }_0=f_{0.2}/E$. Consequently, considering that $\delta =\epsilon ·a$, ${\stackrel{-}{\delta }}_u$ coincides with the normalised buckling strain ${\stackrel{-}{\epsilon }}_u$:

$${\stackrel{-}{\epsilon }}_u=\frac{{\epsilon }_u}{{\epsilon }_0}$$

(4)

3. New Theoretical Procedures

3.1. Theoretical Approach Based on J2 Theory

Recently, a new theoretical approach has been developed to evaluate the ultimate compression load of aluminium members under uniform compression by considering the local buckling effects in the elastic-plastic region. This procedure is based on the J2 deformation theory of plasticity. Starting from the J2 theory [18], the plate stability equation is derived in the elastic-plastic range by including the variability of the Poisson’s ratio depending on the stress level as follows [20]:

$$C_1\frac{{\partial }^{4}w}{\partial x^4}+2C_3\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+C_5\frac{{\partial }^{4}w}{\partial y^4}=-\frac{N}{D_s}\frac{{\partial }^{2}w}{\partial x^2}$$

(5)

where $D_s$ represents the secant flexural stiffness of the plate:

$$D_s=\frac{E_s{t}^3}{12(1-{\nu }^2)}$$

(6)

and the coefficients $C_i$ accounts for the nonlinear behaviour of the material:

$$\begin{gathered} C_1=1-\frac{{\left(2-\nu \right)}^2}{4H(1-{\nu }^2)}\left(1-\frac{E_t}{E_s}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{C}_5=1-\frac{{\left(1-2\nu \right)}^2}{4H(1-{\nu }^2)}\left(1-\frac{E_t}{E_s}\right) \\ {C}_3=1+\frac{\left(2-\nu \right)\left(1-2\nu \right)}{4H(1-{\nu }^2)}\left(1-\frac{E_t}{E_s}\right)\hspace{1em}\hspace{1em}\hspace{1em}H=1+\frac{{\left(1-2\nu \right)}^2}{4(1-{\nu }^2)}\left(1-\frac{E_t}{E_s}\right) \end{gathered}$$

(7)

where $E_s$ and $E_t$ indicates the secant and tangent modulus, respectively, and $\nu$ is the Poisson’s ratio, and according to Gerard and Wildhorn [22], it can be expressed as:

$$\nu ={\nu }_p-\left({\nu }_p-{\nu }_e\right)\frac{E_s}{E}=\frac{1}{2}-\left(\frac{1}{2}-{\nu }_e\right)\frac{E_s}{E}$$

(8)

The solution of Equation (5) can be defined in the Levy’s form, as reported in [18]. The final solution is expressed as:

$$w\left(x,y\right)=\left(A_{1}cosh\alpha y+A_{2}sinh\alpha y+A_{3}cos\beta y+A_{4}sin\beta y\right)sinkx$$

(9)

where:

$$\alpha =\sqrt{\frac{C_3{k}^2}{C_5}+\sqrt{{\left(\frac{C_3}{C_5}\right)}^2{k}^4-k^2\left(k^2\frac{C_1}{C_5}-\frac{N}{D_s{C}_5}\right)}}\hspace{1em}\hspace{1em}\hspace{1em}\beta =\sqrt{-\frac{C_3{k}^2}{C_5}+\sqrt{{\left(\frac{C_3}{C_5}\right)}^2{k}^4-k^2\left(k^2\frac{C_1}{C_5}-\frac{N}{D_s{C}_5}\right)}}$$

(10)

In order to investigate the interactive local buckling of box-shaped sections, the Equation (9) can be applied to the plate elements constituting the member sections. In particular, according to Figure 5, eight integration constants have to be derived to get:

$$\begin{gathered} \mathrm{Plate}1:\left(A_1^{\left(1\right)}cosh{\alpha }_1{y}_1+A_2^{\left(1\right)}sinh{\alpha }_1{y}_1+A_3^{\left(1\right)}cos{\beta }_1{y}_1+A_4^{\left(1\right)}sin{\beta }_1{y}_1\right)sinkx \\ \mathrm{Plate}2:\left(A_1^{\left(2\right)}cosh{\alpha }_2{y}_2+A_2^{\left(2\right)}sinh{\alpha }_2{y}_2+A_3^{\left(2\right)}cos{\beta }_2{y}_2+A_4^{\left(2\right)}sin{\beta }_2{y}_2\right)sinkx \end{gathered}$$

(11)

The following boundary conditions can be applied:

$$\begin{gathered} {(1)\left.\frac{\partial w_1}{\partial y_1}\right|}_{y_1=0}=0\hspace{1em}\hspace{1em}\hspace{1em}(2){\left.R_{x.1}^{*}\right|}_{y_1=0}=0 \\ (3){\left.\frac{\partial w_2}{\partial y_2}\right|}_{y_2=0}=0\hspace{1em}\hspace{1em}\hspace{1em}(4){\left.R_{x.2}^{*}\right|}_{y_2=0}=0 \\ (5){\left.w_1\right|}_{y_1=b_1}=0\hspace{1em}\hspace{1em}\hspace{1em}(6){\left.w_2\right|}_{y_2={-b}_2} \\ (7){\left.\frac{\partial w_1}{\partial y_1}\right|}_{y_1=b_1}={\left.\frac{\partial w_2}{\partial y_2}\right|}_{y_2={-b}_2}\hspace{1em}\hspace{1em}\hspace{1em}(8){\left.M_y\right|}_{y_1=b_1}={\left.M_y\right|}_{y_2=-b_2} \end{gathered}$$

(12)

where $R_{x.i}^{*}$ are the equivalent shear actions while $M_y$ represents the bending moments. By substituting Equation (9) into the relations provided in Equation (12), a system of four equations is obtained and it can be expressed in matrix form:

$$\left. [\begin{array}{cccc}cosh{\alpha }_1{b}_1& cos{\beta }_1{b}_1& 0& 0\\ 0& 0& cosh{\alpha }_2{b}_2& cos{\beta }_2{b}_2\\ {\alpha }_{1}sinh{\alpha }_1{b}_1& -{\beta }_{1}sin{\beta }_1{b}_1& {\alpha }_{2}sinh{\alpha }_2{b}_2& {-\beta }_{2}sin{\beta }_2{b}_2\\ D_s^{\left(1\right)}{\alpha }_1^{2}cosh{\alpha }_1{b}_1& -D_s^{\left(1\right)}{\beta }_1^{2}cos{\beta }_1{b}_1& -D_s^{\left(2\right)}{\alpha }_2^{2}cosh{\alpha }_2{b}_2& {D_s^{\left(2\right)}\beta }_2^{2}cos{\beta }_2{b}_2\end{array}\right]\left\{\begin{array}{c}{A}_1^{\left(1\right)}\\ A_3^{\left(1\right)}\\ A_1^{\left(2\right)}\\ A_3^{\left(2\right)}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right\}$$

(13)

A trivial solution $\underset{\_}{A}=0$ can be neglected. A non-trivial result is obtained by imposing the determinant of the coefficient matrix is equal to zero. The buckling stress corresponding to the solution of Equation (13) can be determined by means of a numerical procedure that works to increase the axial stress values in the plate elements until satisfying Equation (13).

3.2. Extension of Effective Thickness Method (ETM)

In prEN1999-1-1 Annex L [17], an extension of the effective thickness method (ETM) that incorporates buckling in the elastic-plastic range is introduced.

This approach is aimed at enhancing the accuracy in assessing the nonlinear response of aluminium structural elements subjected to uniform compression and non-uniform bending. It represents a more advanced version of the conventional effective thickness method typically used for considering the effects of elastic local buckling in class 4 sections. The refined ETM relies on a strain-dependent analysis of the effective thickness that accounts for local buckling in the elastic-plastic region and the interaction between the plate components forming the section of the member.

The slenderness parameter of the plate elements constituting the section is evaluated as a function of the strain level $\epsilon$ [18] as:

$$\frac{\beta }{{ϵ}_0}=17.54\eta \frac{b}{t}\sqrt{\frac{\epsilon }{\mu \zeta }}$$

(14)

In accordance with Eurocode 9, $\beta$ denotes the slenderness parameter, which is a function of the ratio between the width $b$ and thickness $t$ of the plate. ${ϵ}_0$ is a non-dimensional parameter determined by the yield strength of the aluminium material and it is equal to $\sqrt{250/f_{0.2}}$. On the other hand, $k$ represents the buckling factor, which considers the edge-restraining conditions and the stress distribution along the loaded edges. The factors $\eta$ and $\mu$ are computed as:

$$\eta =\frac{2}{\sqrt{k}}\hspace{1em}\hspace{1em}\hspace{1em}\mu =\frac{\xi E}{E_s}\frac{1}{1-{\nu }^2}$$

(15)

$\zeta$ is a correction factor accounting for interactive buckling and it is equal to $\zeta =k/k_0$ where the buckling factor $k$ accounts for interactive buckling and $k_0$ is a buckling factor for an isolated plate element. In the case of plate elements, acting as a flange, connected to webs on both edges:

$$\zeta =1.75-\frac{0.45b_2/b_1}{0.15+b_2/b_1}-0.02275{\left(b_2/b_1\right)}^3\ge 1$$

(16)

here according to, $b_2=B_2-t_f$ and $b_1=B_1-t_w$. Equation (16) is derived from the expression of k given by BS5950-5 [23] considering that, in this case, $k_0=4$.

The coefficient $\xi$ accounts for the nonlinear behaviour of the material, i.e., local buckling in the elastic-plastic range, according to the following relation:

$$\xi =\frac{E_s}{E}\left. [\frac{n-8}{8}+\frac{8}{n}\sqrt{\frac{E_t}{E_s}}\right]$$

(17)

The application of the effective thickness method requires a procedure under displacement control. For increasing values of the axial displacement δ, the corresponding average strain $\epsilon =\delta /a$ is determined. Therefore, the slenderness parameter is given by Equation (14) and it increases for increasing values of ε. Then, the effective thickness can be obtained for all the plate elements and the effective cross-section area $A_{eff}$ can be computed. The axial force corresponding to δ is computed as ${N=\sigma A}_{eff}$ where $\sigma$ is the stress level corresponding to $\epsilon$, evaluated according to the Ramberg–Osgood model. So, the axial force-displacement curve can be provided. The maximum value of this curve is the ultimate buckling load.

4. Comparison with Available Experimental Results

A comparison between the experimental results with those derived by European provisions and theoretical procedures has been provided. In particular, according to EN1999-1-1 [4], the following relation is used to predict the maximum compression resistance:

$$N_{u.EC9}=\frac{f_{0.2}{A}_{eff}}{{\gamma }_{M1}}$$

(18)

Obviously, in the sections belonging to first, second, or third class, $A_{eff}$ coincides with the whole cross-section area, while in the case of the fourth class, $A_{eff}$ is reduced due to local buckling effects. The safety factor ${\gamma }_{M1}$ is assumed equal to 1.00.

In addition to the experimental data provided in Section 2, the comparison has been performed by considering the experimental results provided by Faella et al. [6] and Su et al. [7]. A summary of the comparison between the experimental values with theoretic values obtained by EN1999-1-1 and the theoretical procedures is reported in

Table 5. Comparison between experimental and theoretical values.

Section	Design Code EN 1999-1-1	Deformation Theoretical Procedure (DTP)		Effective Thickness Method (ETM)
SHS, RHS	$\frac{N_{u.EC9}}{N_{u.exp}}$	$\frac{N_{u.DTP}}{N_{u.exp}}$	$\frac{{\stackrel{-}{\epsilon }}_{u.DTP}}{{\stackrel{-}{\epsilon }}_{u.exp}}$	$\frac{N_{u.ETM}}{N_{u.exp}}$	$\frac{{\stackrel{-}{\epsilon }}_{u.ETM}}{{\stackrel{-}{\epsilon }}_{u.exp}}$
Mean [μ]	0.93	1.02	0.99	0.96	0.81
$\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left[\mathsf{\sigma }\right]$	0.09	0.09	0.18	0.07	0.22

.

It is immediately observed that in the prediction of the maximum resistance the accuracy is very high. In fact, the mean value of the $N_{u.DTP}/N_{u.exp}$ ratios is equal to 1.02. Moreover, an important advantage of the new methodologies is related to the possibility to estimate the non-dimensional strains corresponding to the maximum compression load. Also, regarding the prediction of non-dimensional strains ${\stackrel{-}{\epsilon }}_u$, the values provided by the DTP procedure are very close to the experimental values. In fact, the mean value of the ${\stackrel{-}{\epsilon }}_{u.DTP}/{\stackrel{-}{\epsilon }}_{u.exp}$ ratios is equal to 0.99 with a standard deviation of 0.18. However, the great advantage of the ETM procedure is that it allows for the determination of a continuous curve between the axial load and the non-dimensional strain by also including the softening branch due to the post-buckling behaviour. This phenomenon occurs when the gradual decrease in the effective thickness is no longer counterbalanced by the corresponding increase in stress caused by the rise in strain magnitude.

For comprehensive references, all numerical values can be found in Appendix A.

5. Conclusions

In this study, the local buckling phenomenon of aluminium members subjected to uniform compression was analyzed within the elastic-plastic region. Specifically, new experimental tests were conducted on square hollow sections (SHS) made of 6060 aluminium alloys. Subsequently, novel theoretical approaches were presented. The first one is based on the application of the J2 deformation theory of plasticity. The second method is an extension of the effective thickness method (ETM) outlined in Annex L of prEN1999-1-1. To assess the accuracy of these new methodologies, a comparison was made between the experimental results of the stub column tests and the values obtained using the European provisions and the new methods. The results revealed that the European rules underestimate the actual behavior of aluminium members subjected to uniform compression. On the other hand, the new methods demonstrate a high level of accuracy. Specifically, both the fully theoretical approach and the effective thickness method were investigated by comparing the predicted buckling resistance with the results from available experimental tests. The obtained outcomes demonstrated that the average ratio of $N_{u.th}/N_{u.exp}$ between the theoretical and experimental values of the buckling resistance is 1.02, with a standard deviation of 0.09. When applying the effective thickness method, the same ratio has an average value of 0.96 with a standard deviation of 0.07. Consequently, despite its simplicity, the effective thickness method provides satisfactory results for code provisions. Finally, these methodologies also offer the advantage of estimating the non-dimensional strains corresponding to the maximum compression loads.