1. Introduction
Aluminium cross-sections are present in an increasing number of structural applications in the civil engineering domain. They can be found in lightweight structures, such as trusses and domes, as an appropriate substitute of steel sections, being significantly heavier. The material’s corrosion resistance enables its use in harsh conditioned environments, such as offshore structures that are exposed to humidity and corrosive substances. Aluminium also offers a design flexibility through the extrusion process, which can be useful when a specific aesthetic or architectural design is intended.
Aluminium alloys’ relatively low elastic modulus can be balanced out through the design of complex cross-sections, with geometries able to produce large moments of inertia, resulting in higher cross-sectional stiffness values, EI [
1]. Structural optimisation is a powerful tool towards this scope, since it enables the design of robust components that maintain a low weight through incorporating an optimal geometry and material distribution with regards to particular loading conditions. According to Mei et al. [
2], structural optimisation can fit into four categories: size, shape, topology, and multi-objective optimisation. Topology optimisation, which is addressed herein, refers to the interaction between nodes or joints that comprise a structural member [
2]. The optimal design is therefore produced through the removal of unnecessary members and parts of a cross-section that, given a specific loading, do not contribute to the stress distribution.
Aluminium is an appropriate material for such a use since it can be quite versatile from a geometry standpoint. A variety of shapes can be produced: complex geometries can be manufactured through extrusion, and thus the cross-sectional design is not constrained to conventional shapes and dimensions.
After its initial introduction from Bendsøe and Sigmund [
3], the spatial distribution of material that occurs during topology optimisation was investigated by many researchers [
4,
5]. One of the first applications of optimised cross-sectional design on thin-walled beam sections was conducted by Kim and Kim [
6]. In 2019, Tsavdaridis et al. introduced various novel aluminium section beam and column profiles with unique topologies [
1]. Such optimal design approaches could serve as examples to future similar studies since they combine the innovative spirit of the topology optimisation technique with construction potential and engineering intuition.
While exceling in novelty, geometrically complex cross-sections are hard to incorporate into the standardised design procedure Eurocodes offer. Interaction between the members that form the cross-section shall be accounted for, and thus the need for more inclusive methods of behaviour prediction has emerged. The continuous strength method (CSM) was developed by Gardner et al. [
7,
8,
9,
10,
11,
12,
13] for stainless and carbon steel members, and it caters to this exact need: its scope is to replace the conventional cross-section classification that the European standards (EC3, EC9) [
14] suggest, with a relationship that actively accounts for the cross-section’s slenderness effect in the member’s deformation capacity. According to Gardner et al. [
10,
11,
12,
13], the prediction method has produced increases in the compressive and bending resistances of metals that range from 5% to 20%.
CSM for aluminium structures has been investigated by Ashraf and Young [
15] and Su et al. [
16,
17,
18,
19]. Su et al. [
16,
17,
18,
19] investigated CSM’s application on aluminium alloys through a series of experimental data as well as proposing some additions to better fit aluminium members. Slenderness plays a significant role, since when a cross-section is considered slender, it fails from buckling prior to yielding, and thus it cannot deploy any benefits arising from strain hardening. Additionally, plasticity in aluminium alloy structures is addressed by Ampatzis et al. [
20] and Georgantzia et al. [
21], who demonstrated that the most precise ultimate resistances that employed aluminium’s plasticity were produced by both CSM and the plastic hinge method.
Optimised cold-formed, thin-walled sections do not solely comprise flat-plated elements. Thus, the effective width method is most of the time insufficient to fully capture the geometry’s effects on the member’s resistance. Aiming to simplify the calculation of the elastic buckling stress and the effective properties of optimised steel sections, Schafer and Peköz developed the direct strength method (DSM) in 1998 [
22,
23,
24,
25]. They evaluated its performance in the member resistance estimation through comparing its results with AISI specifications. For some years now, DSM has been officially part of the design specifications featured in
Appendix A of the AISI’s North American specifications [
26,
27]. Additionally, the DSM has also been included in the Australian/New Zealand design specifications [
25] in Section 7 of AS/NZS 4600:2018 [
28]. However, the literature provided above refers to cold-formed steel sections. Zhu and Young slightly altered the DSM equations, aiming to expand the prediction method to aluminium members as well [
29,
30]. They produced a DSM alteration for aluminium members, which was shown to be congruent with the experimental and numerical results.
While topology optimisation is a promising domain in structural applications, there is still some progress to be made in terms of connecting the outputs (products) to the corresponding design guidelines. Eurocode 9 (EC9) covers a variety of standard cross-sections; however, novel forms that could provide increased design flexibility and aesthetic alternatives mentioned above are currently unaccounted for. Consequently, the scope of the herein presented research is to structurally validate novel aluminium sections derived through topology optimisation application, aiming to add reliable alternative profiles that conform to current codification and methodology adopted in this field [
14].
In particular, eight stub columns were examined under concentric compression, and the corresponding limit state of EC9-clause 6 was investigated. Comparison of the FE results and the design guidelines of EC9 was also conducted. In this regard, numerical analyses were performed on aluminium beams of complex cross-sectional shapes. Initially, small modifications were carried out on the beam profiles each time, aiming to understand the effect of each parameter on the overall structural performance of the member. Each modified profile was also evaluated in accordance with the EC9 framework and then compared with the obtained analysis outcome. In this study, CSM was also considered as an approach to address post-yielding material behaviour, specifically in non-slender sections. CSM results are herein compared to both the FEA outcomes and the EC9 design formulae.
This paper attempted to highlight the need for an initial integration of complex geometries to the EC9 design guideline in a standard manner that can be feasible for a designer to follow, without having to rely solely on heavy FEM analyses. The general concept is the evolution of the current design range into a wider spectrum that involves novel, optimal designs that can produce innovative structural applications and reduce the weight without compromising the capacity.
2. Materials and Methods
Figure 1a presents four profiles derived from structural topology optimisation, recently developed by Tsavdaridis et al. [
1]. The proposed dimensions suggest the desired ratios to be kept constant during sizing. Considering that aluminium has a rather low elastic modulus in comparison to steel, an increased cross-sectional moment of inertia is required to mitigate failure risks. Complex cross-section shapes could provide that, without necessarily using large amounts of additional material, thus increasing the structure’s weight.
For the purpose of this study, finite element software Abaqus CAE [
31] was employed to simulate the behaviour of the optimised aluminium alloy stub columns. Before conducting any numerical analysis, two stub column models (a simple rectangular and an optimised one) were validated against experimental work found in the literature [
17], as depicted in
Figure 2. The models used for the purpose of the validation studies were the SV1, SV2, and SV3 (SV—section validation). SV1, which is referenced in [
1,
17] as H70 × 55 × 4.2C-R, is a rectangular cross-section-analysed by Tsavdaridis et al. [
1] and experimentally investigated by Su et al. [
17]. SV2 and SV3, referenced in [
1] as S1 and S2, are optimised cross-sections proposed by Tsavdaridis et al. [
1]. Material properties and dimensions can be seen in
Table 1. The boundary conditions of the columns were constrained by reference points located at the centroid of each section, on the upper and lower end. The lower end was fixed, while the upper end had one degree of freedom, the axial displacement.
A linear eigenvalue buckling analysis was conducted to model the geometric imperfections on the columns. According to the literature [
1,
32,
33,
34], geometrical imperfection was modelled with a value of 0.5 mm, accounting for the 10% of the extruded profile’s thickness. As a general rule, cross-sectional thickness can vary up to 5%, while parts thinner than 5 mm can produce deviations up to 10%. Hollow extruded profiles vary in this manner because of the extrusion process [
35].
In a free mesh consisting of 8 mm sized triangular elements, C3D10 was used, as can be seen in
Figure 2b. The stress distribution plots for the validation sections, SV2 and SV3, are presented in
Figure 2c,d, respectively. The contour plots refer to the entire columns, as well as to the middle cross-section. The validation models were compared with S1 and S2, which were obtained from the literature [
1]. For the static non-linear second-order analysis, Abaqus’ Riks method was employed. The validation models were evaluated in comparison to the experimental specimen in terms of stress–strain curve, as shown in
Figure 3.
For the CSM predictions, the equations that were used, as given by Su et al. [
16], are provided below. Initially, the cross-section slenderness,
λp, was determined. The cross-section slenderness is a non-dimensional form acquired by the square root of the ratio of the yield stress
fy to the elastic buckling stress
σcr of the cross-section, as can be seen in Equation (1). All the cross-sections investigated in this paper were non-slender, since their cross-section slenderness,
λp, was lower than 0.68. In this study, the elastic buckling stress was calculated through a formula acquired from Seif and Schafer [
36], as shown in Equation (2). Thereafter, the cross-section deformation capacity needs to be determined, which is expressed as the ratio of the CSM-predicted level of strain,
εcsm, to the yield strain,
εy. This is given by a prediction provided in Su et al. [
16], which is accurate for aluminium alloy profiles, both stocky and slender, as shown in Equation (3). In addition, the strain hardening slope was calculated by Equation (4), which employs the yield and ultimate strength as well as the yield and ultimate strain,
fy,
fu,
εy, and
εu, respectively. C2 is the proportion of ultimate strain and is assumed to be 0.5, which corresponds to an adequately accurate σ-ε curve. Consequently, the limiting CSM predicting ultimate capacity is given by Equation (5).
Cross-sectional slenderness, CSM:
Elastic buckling stress:
where
= 4 for h/b = 1
Stocky cross-sections: Cross-sectional deformation capacity:
Slender cross-sections: Cross-sectional deformation capacity:
DSM predictions were calculated using the adjusted equations for aluminium cross-sections, provided by Zhu and Young [
37,
38]. In similarity with CSM, the cross-section slenderness
λc is a non-dimensional form determined from the critical buckling load
Pcr and the yield load
Py, as can be seen in Equation (6).
Pcr and
Py are given in Equations (7) and (8), respectively. The DSM resistance for aluminium, non-welded columns, as proposed by Zhu and Young [
37,
38], is given in Equation (9). It is significant to note that the equations’ adjustments covering aluminium members, compared to the original relations that refer to steel sections, are rather minor. For more details on these modifications, the reader can refer to [
37,
38].
Cross-sectional slenderness values according to the DSM:
where
Pcre is the critical elastic buckling load,
and
Pcrl is the critical elastic local buckling load of the cross-section, obtained from Gardner et al. [
39].
DSM prediction resistance:
PDSM = min (
Pne,
Pnl)
EC9 predictions are calculated for the compression limit state of clause 6.2.4, when the cross-sections are not slender, meaning that they fail due to yielding and not buckling. For cross-section S1, which is class 3, the minimum resistance between buckling, Equation (10), and compression, Equation (11), is assumed.
Compression limit state resistance:
Buckling limit state resistance:
where
γM1 = 1.10, and
κ,
χ are provided by EC9 formulae and characteristic values in accordance with the member’s buckling class (either A or B).
The stub column models of the parametric study were square hollow sections with various thicknesses; square hollow sections with a central circular opening, as developed by Kim and Kim [
5]; and optimised cross-sections developed by Tsavdaridis et al. [
1]. The geometries of the cross-sections can be seen in
Figure 4a–d. The alloy was aluminium 6063-T6, with an elastic modulus of 70 GPa; a density of 2.7 gr/mm
3; and a yield and ultimate strength of 160 and 195 MPa, respectively. The Poisson’s ratio was 0.3, and the ultimate strain was 0.106. The column dimensions can be found in
Table 2. The material was class A according to Table 3.2b of EC9, while the columns ranged from Class 1 to 4, EC9-6.1.1.4 [
13]. The column boundary conditions were constrained by reference points located at the centroid of the section, on the upper and lower end. The lower end was fixed, while the upper end had one degree of freedom, the axial displacement. A free mesh consisting of 8 mm sized triangular elements, C3D10, was used. Like in the validation studies, an eigenvalue buckling analysis was conducted first to model the geometric imperfections on the column. As mentioned above, as in [
1,
32,
33,
34], geometrical imperfections were modelled with values that account for the 10% of the profiles’ thicknesses, which was the most unfavourable scenario.