Preliminary Study of Interactive Local Buckling for Aluminium Z-Section

Abstract

In this study, a theoretical investigation is conducted on the local buckling resistance of aluminium Z-sections subjected to uniform compression. A method is developed based on the $J_2$ deformation theory of plasticity (DTP) to calculate the critical buckling load within the elastic–plastic range. The deformation theory of plasticity relies on the assumption that the strain state is uniquely defined by the stress state. Consequently, it serves as a specific path-independent non-linear constitutive model. The study commences with the elastoplastic differential equation for a single compressed plate. By incorporating the boundary conditions and the interaction between plate elements, the interactive buckling load is determined. An example is provided to illustrate the incremental nature of the numerical procedure. Additionally, numerical analyses are performed to examine the impact of the strain-hardening properties of aluminium alloys on local buckling resistance. In the final stage, the theoretical results are compared with those found in existing scientific literature. This comparison serves to evaluate the accuracy of the DTP procedure.

1. Introduction

Aluminium alloy structural members, especially those employed in civil and offshore applications, have garnered considerable attention due to their superior corrosion resistance and high strength-to-weight ratio [1,2,3]. These attributes position aluminium alloys as a viable alternative to conventional steel, particularly in environments where steel’s propensity for corrosion could compromise structural integrity [1].

Over recent decades, the research community has demonstrated an escalating interest in the application of aluminium alloys for seismic design. Numerous studies have explored various structural applications where aluminium is used to fabricate dissipative elements, such as shear walls and seismic links [4,5,6,7]. Consequently, there has been a pronounced emphasis on examining the cyclic behaviour of these materials, as documented in [8,9,10,11], where the dissipative capacity of various aluminium alloys from the 5000, 6000 and 7000 series has been scrutinised through both experimental tests and numerical models. A notable advancement in the field of aluminium alloy structures is the inclusion of a section on seismic design in the latest version of Eurocode 8 [12], marking the inaugural instance of aluminium structures being specifically addressed in seismic design codes.

When employed as a structural material, aluminium is subject to the same issues as steel members, specifically, the instability phenomena that can manifest at different levels and in both the elastic and plastic range. These phenomena are of particular significance in aluminium structures due to their lower Young’s modulus compared to steel. This paper probes the effect of local buckling on the ultimate response of aluminium elements under uniform compression. The instability issues at the section level are generally perceived to be influenced by factors such as local slenderness, resulting from width-to-thickness ratios, mechanical properties of the aluminium alloys and constraints between the plate elements comprising the cross-sections.

It is widely acknowledged that current design provisions [13,14,15] propose conservative analytical methods for evaluating the resistance to local instability of aluminium profiles. This is attributed to the fact that these methods often mirror those used for steel structures but fail to adequately account for the distinct material behaviour. As a result, numerous research efforts have been directed towards proposing advanced methodologies to estimate the actual behaviour of aluminium sections subjected to local instability phenomena.

Drawing on experimental tests documented in scientific literature [16,17,18,19,20], a variety of analytical methods have been proposed for determining the maximum resistance of aluminium components, factoring in local buckling effects. These methods encompass the continuous strength method (CSM) [21,22], the direct strength method (DSM) [23,24,25] and the numerical slenderness approach for designing intricate aluminium sections [26]. Both Chinese and Australian codes currently integrate these approaches [14,15]. Furthermore, a substantial volume of research has been devoted to formulating empirical equations using finite element parametric analyses to predict the ultimate flexural strength and rotational capacity of aluminium beams subjected to non-uniform bending [27,28,29,30,31]. Some of the findings obtained have informed the definition of the current Annex L present in the recent version of EN 1999-1-1 [13], where empirical mathematical formulations and an extended version of the effective thickness method (ETM) [32] have been proposed to predict the inelastic response of aluminium members under uniform compression or non-uniform bending moment. It is useful to underline that, in the present study, the main reference code adopted is the new version of EN 1999-1-1 [13].

In this work, an initial theoretical study is conducted on the aluminium Z-section to evaluate the local buckling resistance under uniform compression. It is important to underline that the aforementioned topic represents a significant gap in the scientific literature, as few experimental tests have been conducted since 1945. The only more recent work related to the subject concerns a study of the torsional and flexural–torsional stability for Z-shaped steel beams [33].

In the present study, attention is focused on the influence of the strain-hardening behaviour and the plates’ interaction on the local buckling resistance of aluminium Z-sections. Specifically, a theoretical procedure grounded in the deformation theory of plasticity (DTP) [34,35,36] is developed to estimate the critical buckling load in the elastic–plastic range. This approach has previously been applied to other cross-sections (box-shaped sections, channels and angles), and the results are presented in [18,37,38]. Commencing from the elastoplastic differential equation of a single plate in compression, the interactive buckling load is obtained by imposing the boundary conditions and considering the interaction between the plate elements constituting the cross-section. It is important to underline that the main advantage of this procedure, compared to the current methodologies available in scientific and technical literature, is its consideration of the strain-hardening behaviour of aluminium alloys and the interaction between the plate elements.

Subsequently, an example case is presented, and numerical analyses are performed to assess the influence of the strain-hardening behaviour of aluminium alloys on the local buckling resistance. Finally, despite the limited number of stub tests on aluminium Z-profiles, the experimental results presented in scientific literature are compiled and compared with the corresponding theoretical values derived from the application of the DTP approach.

2. Theoretical Remarks

Recently, advancements have been made in the theory of plate stability within the elastic-plastic region, taking into account the variability in the Poisson ratio depending on in the stress state [39]. In accordance with the J2 deformation theory of plasticity, the differential equation for buckling of a single plate subjected to uniform compression (Figure 1) is defined as follows:

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

(1)

where $D_s$ is the secant flexural stiffness, and its expression is equal to

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

(2)

and $E_s$ represents the secant modulus; $t$ is the plate thickness; and $\nu$ is the Poisson ratio in the elastic–plastic range according to Gerard and Wildhorn’s expression [40]:

$$\nu =0.50-\left(0.50-{\nu }_e\right)\frac{E_s}{E}$$

(3)

where $E$ is the elastic modulus of the material, and ${\nu }_e$ is the Poisson ratio in the elastic range (${\nu }_e=0.30$ for steel and aluminium alloys).

Regarding the coefficients $C_i$, Equation (4) provides the expressions presented in [39]:

$$\begin{array}{cc}{C}_1=1-\frac{{\left(2-\nu \right)}^2}{4V\left(1-{\nu }^2\right)}\left(1-\frac{E_t}{E_s}\right)& C_5=1\frac{{\left(1-2\nu \right)}^2}{4V(1-{\nu }^2)}\left(1-\frac{E_t}{E_s}\right)\\ C_3=1+\frac{\left(2-\nu \right)\left(1-2\nu \right)}{4V(1-{\nu }^2)}\left(1-\frac{E_t}{E_s}\right)& V=1+\frac{{\left(1-2\nu \right)}^2}{4(1-{\nu }^2)}\left(1-\frac{E_t}{E_s}\right)\end{array}$$

(4)

with $E_t$ as the tangent modulus.

It is immediately observed that in the elastic range, i.e., $E_s=E_t=E$, the coefficients $C_i$ are equal to 1, and consequently, Equation (1) becomes the well-known De Saint Venant differential equation [41]. Instead, in the pure plastic region, i.e., $\nu =0.50$, the expressions (4) become

$$C_1=\frac{1}{4}-\frac{3}{4}\frac{E_t}{E_s}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{C}_3=C_5=V=1$$

(5)

The expression of the $C_1$ coefficient was derived by Elbridge Z. Stowell and presented in [42]. As indicated in [39], the solution of Equation (1) can be obtained in the Levy form, and it is given by

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

(6)

where $A_i$ are the integration constants, and they depend on the boundary conditions of the plate’s edges, while the expressions of factors $\alpha$,$\beta$ and $k$ are expressed as [39]

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

(7)

with $k=m\pi /a$. The parameter $m$ is the number of half-waves along the x-direction at the buckling, and $a$ is the length of the plate according to Figure 1.

3. DTP Procedure for Z-Sections

This section provides the theoretical procedure based on the deformation theory of plasticity (DTP) [34] to evaluate the local buckling resistance of aluminium Z-sections in the elastic–plastic range.

3.1. Theoretical Derivation

According to Figure 2, the generic Z-shaped cross-section is composed of three plates (plate 1, plate 2 and plate 3). The interactive local buckling of the whole section in either the elastic or the plastic range can be evaluated by considering the displacement function $w(x,y)$, provided in Equation (8), for the three plate elements:

$$w^{\left(i\right)}\left(x,y_i\right)=\left(A_1^{\left(i\right)}cosh{\alpha }_i{y}_i+A_2^{\left(i\right)}sinh{\alpha }_i{y}_i+A_3^{\left(i\right)}cos{\beta }_i{y}_i+A_4^{\left(i\right)}sin{\beta }_i{y}_i\right)sinkx$$

(8)

with $i=\mathrm{1,2},3$.

Therefore, 12 integration constants must be determined, 4 for each plate.

The integration constants can be derived by accounting for the boundary conditions, which can be kinematic conditions (i.e., related to displacements and rotations) and/or static conditions (i.e., related to internal actions). In this case, the boundary conditions along the common edges, i.e., the edge between plate 1 and plate 2 (for $y_1=0$ and $y_2=b_2$) and the edge between plate 2 and plate 3 (for $y_3=0$ and $y_2=-b_2$), provide six kinematic conditions, which are expressed as

$$\begin{array}{cc}{\left.w^{\left(1\right)}\right|}_{y_1=0}=0& {\left.w^{\left(2\right)}\right|}_{y_2=b_2}=0\\ {\left.w^{\left(2\right)}\right|}_{y_2={-b}_2}=0& {\left.w^{\left(3\right)}\right|}_{y_3=0}=0\\ {\left.{\phi }^{\left(1\right)}\right|}_{y_1=0}={\left.{\phi }^{\left(2\right)}\right|}_{y_2=-b_2}& {\left.{\phi }^{\left(3\right)}\right|}_{y_3=0}={\left.{\phi }^{\left(2\right)}\right|}_{y_2=b_2}\end{array}$$

(9)

where ${\phi }^{\left(i\right)}$ is the rotation function of the $i$th plate. The six static boundary conditions along the common edges, and the free edges of plates 1 and 3 (for $y_1=-b_1$ and $y_3=b_3$), are given by

$$\begin{array}{cc}{\left.M^{\left(1\right)}\right|}_{y_1=0}={\left.M^{\left(2\right)}\right|}_{y_2={-b}_2}& {\left.M^{\left(3\right)}\right|}_{y_3=0}={\left.M^{\left(2\right)}\right|}_{y_2=b_2}\\ {\left.M^{\left(1\right)}\right|}_{y_1={-b}_1}=0& {\left.M^{\left(3\right)}\right|}_{y_3=b_3}=0\\ {\left.R^{\left(1\right)}\right|}_{y_1={-b}_1}=0& {\left.R^{\left(3\right)}\right|}_{y_3=b_3}=0\end{array}$$

(10)

where $M^{\left(i\right)}$ and $R^{\left(i\right)}$ are, respectively, the bending moment and the equivalent shear action of the ith plate. The expressions of $M^{\left(i\right)}$ and $R^{\left(i\right)}$ in the elastic–plastic range are derived in [37], and they are given by

$$M^{\left(i\right)}=-D_s^{\left(i\right)}\left[C_5\frac{{\partial }^{2}w}{\partial y_i^2}+(\nu +C_3-1)\frac{{\partial }^{2}w}{\partial x^2}\right]{ R}^{\left(i\right)}=-D_s^{\left(i\right)}\left[C_5\frac{{\partial }^{3}w}{\partial y_i^3}+(C_3+1-\nu )\frac{{\partial }^{3}w}{\partial x^2\partial y_i}\right]$$

(11)

By substituting Equation (8) into Equations (9) and (10) and by neglecting the trivial solution $sinkx=0$, a system of twelve equations can be determined (see Appendix A), but it can be simplified in ten equations through some mathematical steps and expressed in the matrix form:

$$\underset{\_}{\underset{\_}{\mathbf{K}}}\underset{\_}{\mathbf{A}}=\underset{\_}{0}$$

(12)

where $\underset{\_}{\mathbf{A}}$ indicates the vector of the unknown integration constants

$$\underset{\_}{\mathbf{A}}={\left\{\begin{array}{cccccccccc}{A}_1^{\left(1\right)}& A_2^{\left(1\right)}& A_4^{\left(1\right)}& A_1^{\left(2\right)}& A_2^{\left(2\right)}& A_3^{\left(2\right)}& A_4^{\left(2\right)}& A_1^{\left(3\right)}& A_2^{\left(3\right)}& A_4^{\left(3\right)}\end{array}\right\}}^T$$

(13)

and $\underset{\_}{\underset{\_}{\mathbf{K}}}$ is the coefficient matrix, which can be expressed as

$$\underset{\_}{\underset{\_}{\mathbf{K}}}=\left[\begin{array}{cccccccccc}0& 0& 0& cosh{\alpha }_2{b}_2& -sinh{\alpha }_2{b}_2& cos{\beta }_2{b}_2& -sin{\beta }_2{b}_2& 0& 0& 0\\ 0& 0& 0& cosh{\alpha }_2{b}_2& sinh{\alpha }_2{b}_2& cos{\beta }_2{b}_2& sin{\beta }_2{b}_2& 0& 0& 0\\ 0& {\alpha }_1& {\beta }_1& {\alpha }_{2}sinh{\alpha }_2{b}_2& -{\alpha }_{2}cosh{\alpha }_2{b}_2& -{\beta }_{2}sin{\beta }_2{b}_2& -{\beta }_{2}cos{\beta }_2{b}_2& 0& 0& 0\\ 0& 0& 0& {-\alpha }_{2}sinh{\alpha }_2{b}_2& -{\alpha }_{2}cosh{\alpha }_2{b}_2& {\beta }_{2}sin{\beta }_2{b}_2& -{\beta }_{2}cos{\beta }_2{b}_2& 0& {\alpha }_3& {\beta }_3\\ D_s^{\left(1\right)}\left({\alpha }_1^2+{\beta }_1^2\right)& 0& 0& -D_s^{\left(2\right)}{\alpha }_2^{2}cosh{\alpha }_2{b}_2& D_s^{\left(2\right)}{\alpha }_2^{2}sinh{\alpha }_2{b}_2& {D_s^{\left(2\right)}\beta }_2^{2}cos{\beta }_2{b}_2& {{-D}_s^{\left(2\right)}\beta }_2^{2}sin{\beta }_2{b}_2& 0& 0& 0\\ 0& 0& 0& -D_s^{\left(2\right)}{\alpha }_2^{2}cosh{\alpha }_2{b}_2& -D_s^{\left(2\right)}{\alpha }_2^{2}sinh{\alpha }_2{b}_2& {D_s^{\left(2\right)}\beta }_2^{2}cos{\beta }_2{b}_2& {D_s^{\left(2\right)}\beta }_2^{2}sin{\beta }_2{b}_2& D_s^{\left(3\right)}\left({\alpha }_3^2+{\beta }_3^2\right)& 0& 0\\ {\mathsf{\Phi }}_{\alpha .c}^{\left(1\right)}+{\mathsf{\Phi }}_{\beta .c}^{\left(1\right)}& -{\mathsf{\Phi }}_{\alpha .s}^{\left(1\right)}& {\mathsf{\Phi }}_{\beta .s}^{\left(1\right)}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\mathsf{\Phi }}_{\alpha .c}^{\left(3\right)}+{\mathsf{\Phi }}_{\beta .c}^{\left(3\right)}& {\mathsf{\Phi }}_{\alpha .s}^{\left(3\right)}& -{\mathsf{\Phi }}_{\beta .s}^{\left(3\right)}\\ {\mathsf{\Psi }}_{\beta .s}^{\left(1\right)}-{\mathsf{\Psi }}_{\alpha .s}^{\left(1\right)}& {\mathsf{\Psi }}_{\alpha .c}^{\left(1\right)}& -{\mathsf{\Psi }}_{\beta .c}^{\left(1\right)}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\mathsf{\Psi }}_{\alpha .s}^{\left(3\right)}-{\mathsf{\Psi }}_{\beta .s}^{\left(3\right)}& {\mathsf{\Psi }}_{\alpha .c}^{\left(3\right)}& -{\mathsf{\Psi }}_{\beta .c}^{\left(3\right)}\end{array}\right]$$

(14)

where

$$\begin{array}{cc}{\mathsf{\Phi }}_{\alpha .c}^{\left(i\right)}={[C}_5{\alpha }_i^2-(\nu +C_3-1)k^2]cosh{\alpha }_i{b}_i& {\mathsf{\Phi }}_{\alpha .s}^{\left(i\right)}={[C}_5{\alpha }_i^2-(\nu +C_3-1)k^2]sinh{\alpha }_i{b}_i\\ {\mathsf{\Phi }}_{\beta .c}^{\left(i\right)}=[C_5{\beta }_i^2+\left(\nu +C_3-1\right)k^2]cos{\beta }_i{b}_i& {\mathsf{\Phi }}_{\beta .s}^{\left(i\right)}=[C_5{\beta }_i^2+\left(\nu +C_3-1\right)k^2]sin{\beta }_i{b}_i\\ {\mathsf{\Psi }}_{\alpha .c}^{\left(i\right)}=[C_5{\alpha }_i^3-{\alpha }_i\left(C_3+1-\nu \right)k^2]cosh{\alpha }_i{b}_i& {\mathsf{\Psi }}_{\alpha .s}^{\left(i\right)}=[C_5{\alpha }_i^3-{\alpha }_i\left(C_3+1-\nu \right)k^2]sinh{\alpha }_i{b}_i\\ {\mathsf{\Psi }}_{\beta .c}^{\left(i\right)}=[C_5{\beta }_i^3+{\beta }_i\left(C_3+1-\nu \right)k^2]cos{\beta }_i{b}_i& {\mathsf{\Psi }}_{\beta .s}^{\left(i\right)}=[C_5{\beta }_i^3+{\beta }_i\left(C_3+1-\nu \right)k^2]sin{\beta }_i{b}_i\end{array}$$

(15)

for $i=1$ and 3. The expressions of the plastic coefficients $C_i$ and the factors ${\alpha }_i,{\beta }_i$ are provided in Equations (4) and (7), respectively.

A non-trivial solution can be computed by imposing that the determinant of the coefficient matrix is equal to zero, i.e., $\left|\underset{\_}{\underset{\_}{\mathbf{K}}}\right|=0$, which represents the buckling condition in the elastic–plastic range. It is useful to note that the parameters ${\alpha }_i,{\beta }_i$ and $D_s^{\left(i\right)}$ are dependent on the stress level. Consequently, in order to compute the buckling stress, an incremental procedure is adopted by computing $\left|\underset{\_}{\underset{\_}{\mathbf{K}}}\right|$ for increasing values of the axial stress in the plate elements until the determinant is equal to zero. For the sake of completeness, all mathematical steps to determine the system of twelve equations previously mentioned are reported in Appendix A. Moreover, in order to clarify the theoretical procedure illustrated previously, the following flowchart is shown in Figure 3.

3.2. Case Study

As described in Section 3.1, the determination of the buckling load cannot be computed in a closed form; it is necessary to apply a numerical procedure.

In this section, an example of the application of the theoretical procedure is illustrated. The numerical case was developed through a simple spreadsheet [43]. In particular, with reference to Figure 4, the Z-section is characterised by $B=106 \mathrm{m}\mathrm{m}$, $H=112 \mathrm{m}\mathrm{m}$ and $t_f=t_w=12 \mathrm{m}\mathrm{m}$. Moreover, the Z-section is made of the EN AW 6082-T6 aluminium alloy. It is useful to underline that the aluminium material was modelled by means of the Ramberg–Osgood law, as suggested by EN 1999-1-1 [13]:

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

(16)

where $f_{0.2}$ represents the conventional yield stress, and $n$ is the well-known strain-hardening coefficient.

With reference to the line model shown in Figure 2, the main geometric properties are computed as

$$b_1\left(b_3\right)=B-\frac{t_w}{2}{ b}_2=H-t_f t_f=t_1\left(t_3\right) t_w=t_2$$

(17)

and they are reported in

Table 1. Geometric and mechanical properties.

${\mathit{b}}_1={\mathit{b}}_3$ [mm]	${\mathit{b}}_2$ [mm]	${\mathit{t}}_1={\mathit{t}}_2={\mathit{t}}_3$ [mm]	$\mathit{a}$ [mm]
100.00	50.00	12.00	300.00
$E$ [MPa]	$f_{0.2}$ [MPa]	$f_u$ [MPa]	$n_{exp.}$ [-]
70,000	260	310	25

. Moreover, according to EN 1999-1-1 [13], the nominal values of the mechanical properties of the EN AW 6082-T6 alloy are indicated in Table 1.

According to the previous section, an incremental procedure is carried out for increasing values of the stress $\sigma$ uniformly applied to each plate element until the determinant of the matrix $\underset{\_}{\underset{\_}{\mathbf{K}}}$ provided in Equation (14) is equal to zero. Operatively, for each plate, $N$ is equal to $N=\sigma t_i$, where $t_i$ is the thickness of the $i$th plate.

Moreover, it is useful to underline that the previous procedure is repeated by varying the number of half-waves $m$ along the x-direction. In particular, the value of $m$ is varied between 1 and 6. Obviously, critical stress is defined by considering the minimum value among those obtained by varying the $m$ parameter. In this case, the minimum value of critical stress is reached for $m=2$.

Table 2. Mechanical properties corresponding to the occurrence of buckling.

$\mathit{\nu }$ [-]	${\mathit{E}}_{\mathit{s}}$ [MPa]	${\mathit{E}}_{\mathit{t}}$ [MPa]
0.41	30016.35	2130.08
$C_1$ [-]	$C_3$ [-]	$C_5$ [-]
0.30	1.08	0.99

and

Table 3. Main parameters of each plate corresponding to the occurrence of buckling.

$\mathbf{P}\mathbf{l}\mathbf{a}\mathbf{t}\mathbf{e}$	$\mathit{k}$ [mm−1]	${\mathit{D}}_{\mathit{s}}^{\left(\mathit{i}\right)}$ [Nmm]	${\mathit{\alpha }}_{\mathit{i}}$ [mm−1]	${\mathit{\beta }}_{\mathit{i}}$ [mm−1]
1	0.021	5,217,675.285	0.034	0.014
2	0.021	5,217,675.285	0.034	0.014
3	0.021	5,217,675.285	0.034	0.014

provide the plastic mechanical properties and the main parameters corresponding to the occurrence of the buckling condition in the elastic–plastic range. The matrix $\underset{\_}{\underset{\_}{\mathbf{K}}}$ corresponding to critical stress is shown in

Table 4. Matrix of coefficients $\underset{\_}{\underset{\_}{\mathbf{K}}}$ corresponding to the occurrence of buckling.

$\mathbf{Matrix} \underset{\_}{\underset{\_}{\mathbf{K}}}$
0.00	0.00	0.00	2.80	−2.61	7.72 × 10−1	−6.35 × 10−1	0.00	0.00	0.00
0.00	0.00	0.00	2.80	2.61	7.72 × 10−1	6.35 × 10−1	0.00	0.00	0.00
0.00	3.38 × 10−2	1.38 × 10−2	8.83 × 10−2	−9.46 × 10−2	−8.75 × 10−3	−1.06 × 10−2	0.00	0.00	0.00
0.00	0.00	0.00	−8.83 × 10−2	−9.46 × 10−2	8.75 × 10−3	−1.06 × 10−2	0.00	3.38 × 10−2	1.38 × 10−2
6.94 × 103	0.00	0.00	−1.67 × 10−4	1.56 × 104	7.64 × 102	−6.28 × 102	0.00	0.00	0.00
0.00	0.00	0.00	−1.67 × 10−4	−1.56 × 104	7.64 × 102	6.28 × 102	6.94 × 103	0.00	0.00
1.35 × 10−2	−1.34 × 10−2	3.95 × 10−24	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.00	0.00	0.00	0.00	0.00	0.00	0.00	1.35 × 10−2	1.34 × 10−2	−3.95 × 10−4
−1.87 × 10−4	2.00 × 10−4	−2.43 × 10−6	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.00	0.00	0.00	0.00	0.00	0.00	0.00	1.87 × 10−4	2.00 × 10−4	−2.43 × 10−6

.

Figure 5 reports the value of the determinant $\left|\underset{\_}{\underset{\_}{\mathbf{K}}}\right|$ as a function of stress applied to the plate elements of the Z-section. Critical stress is obtained as the intersection between the determinant function and the horizontal axis. Figure 6 shows the buckling shape of the Z-section for $m=2$ due to critical stress ${\sigma }_{cr}=268.89 \mathrm{M}\mathrm{P}\mathrm{a}$.

Finally, it is worthwhile highlighting that the numerical computation is performed through a simple spreadsheet, although it is based on an incremental approach.

3.3. Numerical Analyses

In order to evaluate the influence of the width-to-thickness ratio of the plate elements and the strain-hardening coefficient $n$ on the buckling stress in the elastic–plastic range, numerical analyses were performed. In particular, with reference to Figure 2, given the plates’ widths ($b_1=b_3=100 \mathrm{m}\mathrm{m}$ and $b_2=50 \mathrm{m}\mathrm{m}$), the plates’ thicknesses were varied from 1.00 mm to 20.00 mm ($5\le b/t\le 100)$. Moreover, five aluminium alloys were considered according to EN 1999-1-1 [13]. They are characterised by different yields and ultimate stresses ($f_{0.2}$ and $f_u$, respectively) and strain-hardening coefficients $\left(n\right)$. Their properties are summarised in

Table 5. Properties of aluminium alloys according to EN 1999-1-1 [13].

$\mathbf{A}\mathbf{l}\mathbf{l}\mathbf{o}\mathbf{y}$	${\mathit{f}}_{0.2}$ [MPa]	${\mathit{f}}_{\mathit{u}}$ [MPa]	$\mathit{n}$ [-]
5754-H111	80	180	6
6063-T6	160	195	16
6061-T6	240	290	23
6082-T6	260	310	25
7020-T6	290	350	23

.

The results of the numerical analyses are shown in Figure 7. It is immediately observed that for values of $b/t$ greater than 40, the buckling stress occurs in the elastic range for each analysed alloy. Conversely, for $b/t\le 40$, the buckling stress can occur in the post-elastic range depending on the combination of yield stress with the strain-hardening coefficient. In particular, the 5754-H111 aluminium alloy, characterised by the smallest value of the R-O coefficient, exhibits a strongly non-linear behaviour starting from relatively high slenderness values ($b/t>40$). Additionally, the instability law trend with respect to slenderness differs from that of other aluminium alloys. In fact, other aluminium alloys are characterised by the same elastic behaviour up to $b/t\cong 25$. In addition, it is interesting to observe that the corresponding value of $b/t$ at which the transition occurs between elastic and plastic buckling depends on the value of the coefficient $n$. Specifically, when the R-O coefficient ($n$) increases, the corresponding value of $b/t$ decreases.

4. Comparison with Stub Column Tests

To evaluate the accuracy of the DTP procedure, the theoretical approach was applied to compute the buckling resistance of the stub column tests available in the scientific literature. As is well known from the scientific literature [17,18], stub column tests are used to evaluate the resistance to local buckling phenomena of thin-walled profiles. A representative scheme of the stub column test is shown in Figure 8. Unfortunately, in the technical literature, only few tests for Z-shaped aluminium profiles are available. In this study, reference is made to the experimental tests performed by George et al. [44,45,46]. The experimental procedures involved different aluminium cross-sections, subject to all possible instability phenomena (local and global). Attention was focused only on profiles subject to local instability. Consequently, only experimental data referring to Z-section profiles with a height-to-web width ratio not exceeding 4 were considered. According to Figure 4, the main experimental data and results, provided in [44,45,46], are presented in

Table 6. Geometric and mechanical properties of specimens with test and numerical results [44,45,46].

$\mathbf{S}\mathbf{p}\mathbf{e}\mathbf{c}\mathbf{i}\mathbf{m}\mathbf{e}\mathbf{n}$	$\mathbf{R}\mathbf{e}\mathbf{f}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{n}\mathbf{c}\mathbf{e}$	$\mathit{B}$ [mm]	$\mathit{H}$ [mm]	${\mathit{t}}_{\mathit{f}}$ [mm]	${\mathit{t}}_{\mathit{w}}$ [mm]	$\mathit{a}$ [mm]	$\mathit{E}$ [MPa]	${\mathit{f}}_{0.2}$ [MPa]	$\mathit{n}$ [-]
1a	G.J. Heimerl and A. Roy [44]	30.23	47.75	3.18	3.05	155.70	72395	540	24
1b		29.97	47.50	3.18	3.05	154.94	72395	540	24
1c		30.23	47.60	3.23	3.05	155.19	72395	540	24
2a		32.77	47.60	3.23	3.05	178.05	72395	540	24
2b		32.51	47.50	3.18	3.05	177.55	72395	540	24
2c		32.77	47.60	3.23	3.05	177.04	72395	540	24
1a	G.J. Heimerl and D. Fay [45]	28.52	47.55	3.07	3.12	154.94	72395	495	25
1b		28.27	47.80	3.07	3.12	154.94	72395	495	25
1c		28.27	47.60	3.10	3.12	154.43	72395	495	25
2a		30.56	48.11	3.10	3.12	165.10	72395	495	25
2b		30.56	47.19	3.02	3.12	165.10	72395	495	25
2c		30.56	47.85	3.10	3.12	162.56	72395	495	25
3a		32.59	48.01	3.05	3.12	176.78	72395	495	25
3b		32.49	47.80	3.07	3.02	175.26	72395	495	25
3c		32.84	47.80	3.07	3.12	175.26	72395	495	25
1a	G.J. Heimerl and D. Niles [46]	28.09	47.40	3.25	3.20	101.60	72395	370	30
1b		28.37	47.45	3.28	3.23	101.60	72395	370	30
1c		27.97	47.45	3.28	3.07	101.60	72395	370	30
2a		30.38	47.40	3.25	3.20	114.30	72395	370	30
2b		30.02	47.55	3.33	3.10	114.30	72395	370	30
2c		30.10	47.65	3.25	3.18	114.30	72395	370	30
6b		32.00	62.69	3.15	3.05	256.54	72395	370	30
10a		32.00	75.69	3.05	3.05	315.98	72395	370	30

. Operatively, to perform a comparison with the theoretical procedure, numerical analysis is carried out by considering the geometric and mechanical properties presented in Table 6.

The critical buckling load in the elastic–plastic range is computed by defining the matrix of the coefficients $\underset{\_}{\underset{\_}{\mathbf{K}}}$ according to Equation (14) and by increasing the stress level $\sigma$ until the determinant of the $\underset{\_}{\underset{\_}{\mathbf{K}}}$ matrix is equal to zero (as described in Section 3.2). The analyses have to be repeated for different values of the parameter $m$, looking for the one leading to the minimum value of ultimate resistance. The analyses are carried out by means of a simple script (without a specific toolbox) developed in a MATLAB environment (R2022b) [47], where the input geometry refers to the thickness midline of the section, as depicted in Figure 2. The MATLAB script is reported in Appendix B.

Figure 9 provides a comparison between the theoretical buckling load ($N_{u.th})$ and the corresponding experimental results ($N_{u.exp})$ presented in [44,45,46]. In particular, the maximum value between $b_1/t_1$ and $b_2/t_2$ (Figure 2) is reported on the $x-axis$, while the $N_{u.th}/N_{u.exp }$ ratio is provided on the $y-axis$. Although the number of tests is limited, the accuracy of the theoretical approach is quite satisfactory. This is testified by the mean value $\left(\mathsf{\mu }\right)$ of the $N_{u.th}/N_{u.exp }$ ratio being equal to 0.94, with a standard deviation of $\mathsf{\sigma }=0.05$. The minimum and maximum values of the ratio $N_{u.th}/N_{u.exp }$ are equal to 0.86 and 1.02, respectively. The same comparison is performed by considering the current rules provided in EN 1999-1-1. In particular, the local buckling resistance ($N_{u.EC9 })$ is evaluated by means of new version of Eurocode 9, by considering the partial safety factor being equal to 1.00. It is immediately recognised that Eurocode 9 is more conservative than the DTP procedure. In fact, the mean value $\left(\mathsf{\mu }\right)$ of the $N_{u.th}/N_{u.exp }$ ratio is equal to 0.79, with a standard deviation of $\mathsf{\sigma }=0.03$. A further comparison between the experimental and numerical results was obtained in terms of the percentage error, calculated as

$$r_{\mathrm{E}\mathrm{C}9}=100\left|\frac{N_{u.\mathrm{E}\mathrm{C}9}-N_{u.exp}}{N_{u.exp}}\right|\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{r}_{\mathrm{t}\mathrm{h}}=100\left|\frac{N_{u.\mathrm{t}\mathrm{h}}-N_{u.exp}}{N_{u.exp}}\right|$$

(18)

For the sake of completeness, the values of experimental and numerical results ($N_{u.exp },N_{u.th}$, $N_{u.\mathrm{E}\mathrm{C}9}$) are shown in

Table 7. Comparison between the theoretical and experimental values.

$\mathbf{S}\mathbf{p}\mathbf{e}\mathbf{c}\mathbf{i}\mathbf{m}\mathbf{e}\mathbf{n}$	$\mathbf{R}\mathbf{e}\mathbf{f}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{n}\mathbf{c}\mathbf{e}$	${\mathit{N}}_{\mathit{u}.\mathit{e}\mathit{x}\mathit{p}}$ [kN]	${\mathit{N}}_{\mathit{u}.\mathbf{E}\mathbf{C}9}$ [kN]	$\frac{{\mathit{N}}_{\mathit{u}.\mathbf{E}\mathbf{C}9}}{{\mathit{N}}_{\mathit{u}.\mathit{e}\mathit{x}\mathit{p}}}$	${\mathit{r}}_{\mathbf{E}\mathbf{C}9}$ [%]	${\mathit{N}}_{\mathit{u}.\mathit{t}\mathit{h}}$ [kN]	$\frac{{\mathit{N}}_{\mathit{u}.\mathit{t}\mathit{h}}}{{\mathit{N}}_{\mathit{u}.\mathit{e}\mathit{x}\mathit{p}}}$	${\mathit{r}}_{\mathit{t}\mathit{h}}$ [%]
1a	G.J. Heimerl and A. Roy [44]	161.65	136.68	0.85	15.45	164.38	1.02	1.68
1b		162.40	136.15	0.84	16.16	163.71	1.01	0.81
1c		165.03	138.14	0.84	16.30	166.73	1.01	1.03
2a		168.36	139.17	0.83	17.34	168.97	1.00	0.36
2b		164.75	137.15	0.83	16.75	165.63	1.01	0.54
2c		167.67	139.17	0.83	16.99	169.01	1.01	0.80
1a	G.J. Heimerl and D. Fay [45]	155.66	124.77	0.80	19.85	147.41	0.95	5.30
1b		155.06	125.04	0.81	19.36	147.43	0.95	4.92
1c		154.34	125.49	0.81	18.69	148.20	0.96	3.97
2a		159.23	127.25	0.80	20.08	151.62	0.95	4.78
2b		156.16	123.48	0.79	20.93	146.62	0.94	6.11
2c		159.06	126.86	0.80	20.24	151.31	0.95	4.87
3a		158.62	126.21	0.80	20.43	150.16	0.95	5.33
3b		159.39	124.36	0.78	21.98	149.55	0.94	6.17
3c		162.53	126.78	0.78	21.99	151.55	0.93	6.75
1a	G.J. Heimerl and D. Niles [46]	135.98	103.07	0.76	24.20	118.01	0.87	13.21
1b		138.28	104.35	0.75	24.54	119.60	0.86	13.51
1c		131.21	101.56	0.77	22.60	116.47	0.89	11.24
2a		140.84	104.29	0.74	25.95	122.15	0.87	13.27
2b		136.65	104.56	0.77	23.48	122.21	0.89	10.57
2c		137.04	104.03	0.76	24.09	121.59	0.89	11.27
6b		146.77	116.19	0.79	20.84	135.02	0.92	8.01
10a		158.07	118.15	0.75	25.26	142.33	0.90	9.96
Mean [$\mathsf{\mu }$]				0.79	20.59		0.94	6.28
St. Dev. [$\mathsf{\sigma }$]				0.03	3.14		0.05	4.39

.

5. Conclusions

The inelastic response of aluminium Z-sections under uniform compression was examined from a theoretical perspective. Specifically, a theoretical procedure was developed to estimate the local buckling resistance of Z-profiles, taking into account the strain-hardening behaviour of the aluminium material and the interaction between the plate elements that constitute the cross-sections.

Following this, a numerical example was presented for a generic member composed of the EN AW 6082-T6 alloy, characterised by an identical width-to-thickness ratio for each plate in the Z-section $(b/t=8.33)$. Numerical analyses were conducted, with variations in the mechanical properties of aluminium alloys according to EN 1999-1-1 and the plates’ slenderness in the Z-sections ($5\le b/t\le 100$).

In addition, a comparison was put forward between the experimental results provided by the stub column tests—as presented in the scientific literature—and those derived from application of the DTP procedure. According to this theoretical study, it is possible to formulate the following remarks:

• The occurrence of buckling stress is strongly influenced by the width-to-thickness ratio $b/t$. As expected, for a value of $b/t<40$, the critical stress occurs in the elastic–plastic range.
• The critical value of the width-to-thickness ratio ($b/t$) at which the transition from elastic to plastic buckling occurs depends on the coefficient $n.$ Specifically, as the R-O coefficient ($n$) increases, the corresponding $b/t$ value decreases.
• There is good accuracy in the prediction of ultimate resistance of the aluminium Z-section under uniform compression by means of the DTP procedure.
• The minimum and maximum values of the $N_{u.th}/N_{u.exp }$ ratios are equal to 0.86 and 1.02, respectively.
• The mean value of the $N_{u.th}/N_{u.exp }$ ratio is equal to 0.94, with a standard deviation of $\mathsf{\sigma }=0.05$.
• The European provisions are applied. In particular, the mean value of the $N_{u.\mathrm{E}\mathrm{C}9}/N_{u.exp }$ ratio is equal to 0.79, with a standard deviation of $\mathsf{\sigma }=0.03$.
• The percentage errors were evaluated with respect to the experimental results for both the EC9 rules and the theoretical procedure. The results confirmed that EC9 provides more conservative values. In fact, in the first case, the mean value of the percentage error was equal to 20.59, while it was equal to 6.28 in the second case.

It is important to note that further studies and investigations will be necessary to address the knowledge gap regarding the local instability phenomena of aluminium alloy Z-profiles. As such, potential future developments may include the initiation of new experimental research aimed at evaluating the ultimate capacity of aluminium Z-sections subject to uniform or non-uniform compression, with variations in the mechanical and geometric properties.