1. Introduction
Aluminum alloy thin-walled structures, due to their high energy absorption efficiency and lightweight structures, are an essential requirement in many industries. Aerospace, elevator, automotive, offshore structures, and liquid storage tanks inevitably use energy absorbers [
1,
2,
3,
4,
5,
6]. Different section shapes of aluminum alloy thin-walled structures such as square and hexagonal have been well documented [
7,
8], and the majority of research focuses on the even sides. However, odd sides’ tubes such as triangular are also widely used in bridges and buildings et al. Therefore, more attention has been paid to study these structures in recent years. Alexander [
9] presented the first theoretical model of the thin-walled structures’ plastic collapse, which modeled the procedure as one stationary hinge and folding wave. Abramowicz [
10] expanded the theory by proposing the effective crushing distance concept. Wierzbicki [
11,
12] proposed the Super Folding Element (SFE) theory, which combined the concepts from plasticity with moving hinge line collapse theories. Chen [
13] proposed a simplified SFE based on energy mechanisms. Zhang [
14] studied the collapse performance of X-shaped elements with various angles and modified the inextensional mode of corner element. Later, he established the theoretical collapse model of a three-plate element with an arbitrary central angle [
15]. With the advances of the thin-walled structure theories and the maturity of the computer simulation, the thin-walled structures of varies cross-section were extensively studied. Wang [
16] inferred the theoretical prediction of the mean crushing force of the thin-walled multi-cell structures based on the simplified super folding element theory. Then he investigated a combined five-cell thin-walled structure assembled in high speed. Tran [
17] proposed a nested tubular thin-walled structure, and studied the crush behavior and energy absorption performance under dynamic axial loading. Zhang [
18] proposed a novel type of self-locking multi-cell structure. On the other hand, the dynamic and static experimental tests were conducted to look into the energy absorption characteristics. Gui [
19] presented a comprehensive crashworthiness design method of the automotive structure plastic frame model, which produced a thin-walled beam with an arbitrary cross-sectional shape. Fang [
20] proposed a topology optimization method based on a modified artificial bee colony algorithm in order to make more efficient use of the material of the multi-cell tube under out-of-plane crushing. Zhang [
21] designed a quadric-arc multi-cell honeycomb and investigated the in-plane energy absorption property and dynamic crushing behavior under various impact loads.
Recently, there has been a trend to adopt bioinspired design principles in the structure design. Nature employs fractal designs to serve a mechanical function in many cases [
22,
23,
24]. Fractal geometry is one of the most suitable choices to improve the collapse performance of the thin-walled structure. Inspired by the ’pomelo peel’s unique microstructure, Zhang [
25] constructed a new hierarchical honeycomb. Xu [
26] proposed a new self-similar hierarchical column and the crashworthiness was studied to improve the performance. Inspired by Koch topology, Wang [
27] presented a specific multi-corner fractal structure with the highest specific energy absorption performance. Zhang [
28] proposed a bio-inspired hierarchical circular tube to enhance the structural crashworthiness performance. Later, Xu [
29] analyzed and optimized the hybrid multi-cell structures. Dadrasi [
30] presented steel thin-walled square columns, which were reinforced with three types of reinforcers. Zhang [
31] presented a fractal-appearing self-similar regular hexagonal hierarchical honeycombs, and achieved further improvement by design of the fractal geometries. Ajdari [
32] investigated the mechanical behavior of two-dimensional hierarchical honeycomb structures by using analytical, numerical and experimental methods.
As mentioned above, the existing studies mainly focused on the application scope and summarizing it into a design method remains unexplored. In the present work, a bioinspired design strategy of the cross-section of the thin-walled column is proposed. A novel type of triangle thin-walled columns is presented based on the strategy. The out-of-plane crashworthiness theoretical model is established. The finite element analysis is carried on to investigate the mechanical characteristics of various structures. In addition, parameter studied is done to find the effect of different design variables, such as columns height, side length, and panel thickness. There are three main contributions of this paper. Firstly, a bioinspired design strategy is proposed, which provides new insight into the designing of thin-walled columns with high energy absorption performance. Secondly, a novel fractal thin-walled triangle column up to 2nd-order is presented. Thirdly, the energy absorption properties in theoretical and numerical manners are investigated, and a positive relevant relationship is identified between the thickness and the crashworthiness.
The rest of the paper is organized as follows.
Section 2 gives the numerical model of the structure and conduct an experimental validation. The theoretical models of the columns are established in
Section 3. The comparison of theoretical and simulation results of different orders is discussed in
Section 4.
Section 5 conducts the parametric studies of crashworthiness to identify the effects of side length, height and wall thickness.
3. Theoretical Model Establishment
Based on the Super Folding Element theory [
12], the external work done during the deformation is equal to the energy dissipation. The dissipation of energy consists of bending energy and membrane energy. It can be expressed as:
where
Pm denotes the average crush force,
H is the half fold length,
η represents the effective crushing distance coefficient.
Ebending and
Emembrane are the bending dissipation energy and the membrane dissipation energy respectively.
In the experiment, the mean crush force is calculated by:
where
Pe is the experimental mean crush force,
Sd is the effective crushing length,
P(x) is the crushing force at the distance of
x.
The bending energy is evaluated by the energy dissipation in the plastic hinge lines [
13]. Due to theoretical deduction, the panel should be perfect flatten during the collapse. The bending energy for a multi-cell column is obtained as:
where
M0 =
σ0t2/
4 represents the wholly plastic bending moment per unit width,
b is the panel width,
p is the number of the panel in the multi-cell column.
σ0 refers to the flow stress of the material, and
t is the thickness of the column.
where
σy,
σu, are the yield stress of the material and the ultimate stress of the material respectively.
n is power law exponent.
For the 0th FTTC, it has three panels width
D0 and the bending energy is calculated as:
For the 1st FTTC, it has nine panels width
D1 and the bending energy is calculated as:
For the 2nd FTTC, it has 27 panels width
D2 and the bending energy is calculated as:
According to Ref [
12,
39], the irregularity section is divided into a small element and the fractal columns’ cross-section consists of two-panel and four-panel angle element, as shown in
Figure 7. The membrane energy can be calculated as:
where
α and
β are the angles in the element.
There are three two-panel angle elements in the 0th profile of the column. Therefore, for the 0th-order FTTC, the membrane energy is calculated as:
For the 1st-order FTTC, there are three two-panel angle elements and three four-panel angle elements. As a result, the membrane energy is calculated as:
The profile of the 2nd-order FTTC constitutes of three two-panel elements and twelve four-panel elements. The membrane energy is calculated as:
The calculation of theoretical solution of the average crushing force is introduced in this section, taking the 0th fractal column as an example. Substituting Equation (15) and Equation (20) into Equation (11). The mean crush force can be obtained as:
The half folding wavelength is determined by the stationary condition of the mean crushing force:
Substituting Equation (25) into Equation (23), the average crush force in quasi-static loading can be calculated as:
Using the same procedure, the mean crush force of the other fractal columns can be expressed as:
4. Crashworthiness Comparison
For the sake of maintaining the same material usage, all the specimens have the same mass (same section area). According to
Table 1, the dimensions of the FTTC series are listed in
Table 4.
The comparison of the FEA and theoretical prediction of the average crush force is illustrated in
Table 5. It can be seen the maximum relative error of theoretical to the simulation is 14.54%. For the progressive folding 0th-order one, the error is relatively low. Therefore, the theoretical model is sufficient accuracy to predict PCF.
The deformation process of the three columns is illustrated in
Figure 8. As well as the corresponding force-displacement curves of the three columns are illustrated in
Figure 9. It can be observed that columns have an acute peak at the beginning stage, then followed by a fluctuating load, which is the most energy dissipated place. The 0th-order column exhibits a typical progressive folding. The column shows a bending tendency with the increase of the fractal order. Thus, the force-displacement behave more irregular. The comparison of the indicators of the crashworthiness is shown in
Table 6. Energy absorption is a key indicator that reflects energy absorption performance and SEA is employed to evaluate the energy absorption performance. As shown in
Table 6, the 0th-order FTTC has the smallest SEA of 3.76 kJ/kg, and the 2nd-order FTTC has the biggest SEA of 7.13 kJ/kg, which is 89.6% higher than the 0th-order. There is an average increase of 30%-40% SEA via increasing the fractal order. CFE represents the load uniformity of a structure. The 2nd-order FTTC shows the most uniformity load due to its high rigidity. It can be observed that the CFE becomes larger with the increase of the fractal order.
6. Conclusions
A bioinspired design strategy has been proposed in the present work. Based on that, a novel fractal thin-walled triangle column (FTTC) has been designed. It is investigated up to the second-order and shows a great improvement in the crash indicators. A parameter study has been conducted to study the influence of geometric details. The conclusions are summed up as follows:
(1) A bioinspired design strategy is proposed and shows great potential to advance the crashworthiness of the thin-walled column. The fractal order has a major influence on performance. In particular, SEA of the 2nd-order FTTC is 89.6% higher than that of the 0th-order.
(2) The theoretical models have been deduced to predict the average crush force of FTTC, and it shows a good agreement with the FEA model.
(3) The different collapse mode of the FTTC has been studied. It finds that the fractal order has a significant influence on the collapse mode. With the increase of t and D0, it shows a tendency to transfer from the unstable to stable, whereas the change of parameter He shows the opposite trend.
(4) Changing the fractal thickness of different orders has an effect on crashworthiness by rearranging the material distribution. The wall thickness change between 0th-order and 1st-order has a bigger effect on the SEA and PCF.