# Crushing Analysis and Optimization of Adjacent Variable Thickness Hexagonal Tubes
## 3 Results and Discussion
### Crushing Performance Analysis under Axial Loading  
According to the actual impact conditions and the existing literature [22; 33; 42; 43], we adopted \(0^{\circ}\) (axial loading), \(10^{\circ}\), \(20^{\circ}\), and \(30^{\circ}\) as loading angles and performed simulations to investigate the crushing performance of AVTHTs. As mentioned earlier, three patterns were considered under oblique loading. The thickness configuration of the AVTHTs was as follows: \(t1\) was increased from 1.4 mm to 2.0 mm with an interval of 1 mm; \(t2=4-t1\). In addition, all other settings of simulation were the same as those in Section 2.3.  
The deformation modes of the AVTHTs after 120 mm axial loading are compared in Figure 8. All tubes deform by producing progressive and sequential folds. When the difference of the plate thickness is small (\(t2-t1\leq 0.4\) mm), folds are generated from the support end (fixed plate); however, when \(t2-t1\geq 0.5\) mm, folds are generated from the loading end (moving plate). Moreover, when \(t1<t2\), as the thicker plates bend inward, the thinner plates bend outward and the moving plastic hinge lines are pushed to the thinner plate. If thicker plates bend outward, the thinner plates cannot provide enough force to push the plastic hinge lines to the thicker plates, which causes no moving plastic hinge lines to form in the thicker plate. As shown in Figure 8h, the red dotted lines refer to plastic hinge lines in the thinner plates, and the areas that cannot generate plastic hinge lines in thick plates are circled by blue dotted lines. As the difference between \(t1\) and \(t2\) increases, the distance of every plastic hinge line in the thinner plates moving is increased, and the areas (circled by the black dotted line in Figure 8h) are decreased.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Tubes** & **Fmax (kN)** & **SEA (k/kg)** & **CFE (\(l\))** \\ \hline \multirow{8}{*}{S2020} & Simulation results & 49.727 & 12.686 & 0.497 \\  & Experiment-1 results & 49.70  
For force-displacement characteristics, the force-displacement curves of all tubes are shown in Figure 9. Note that all curves exhibit regular forms and characteristics. In the beginning, the force increases suddenly and reached the initial peak. Then, the force rapidly drops due to the generation of folds. The force-displacement curve of S2020 is obviously different from other curves due to the difference in the deformation mode. Moreover, the maximum initial peak value, 49.727 kN, is contributed by S2020, and its remaining peak values are also basically larger than those of the other curves. When \(t\mathbb{1}\
eq t\mathbb{2}\), the corresponding force-displacement curves of the AVTHTs show high similarity. As the difference in plate thickness increases, the curves of the AVTHTs drop to lower trough values and reach greater values at the last peak force point, which can weaken the EA under axial loading.  
The crashworthiness indices are compared in Table 5. It is found that S2020 reveals the maximal F\({}_{\text{max}}\), SEA and CFE values. When the difference of plate thickness is increased, SEA and CFE are decreased slowly, which indicates that the extensional deformation mode exhibits better energy absorbing characteristics compared with the inextensional deformation mode for AVTHTs, though the F\({}_{\text{max}}\) value of the former is larger. The difference in plate thickness can also attenuate crashworthiness for AVTHTs under axial loading. For  
Figure 8: The simulation results of deformation modes for seven tubes: (**a**) S2020; (**b**) S1921; (**c**) S1822; (**d**) S1723; (**e**) S1624; (**f**) S1525; (**g**) S1424; (**h**) Folds.  
Figure 9: Comparison of the force-displacement curves of the AVTHTs under axial loading.  
the difference of F\({}_{\text{max}}\), the minimum value (S1822) is 5.49% less than the maximum value (S2020). The minimum SEA value (S1426) is 17.05% less than the maximum SEA value (S2020), and the minimum CFE value (S1426) is 13.08% less than the maximum SEA value (S2020).