# Crushing Analysis and Optimization of Adjacent Variable Thickness Hexagonal Tubes
## 3 Results and Discussion
### Crushing Performance Analysis under Oblique Loading  
Three patterns were applied under each loading angle, and there were nine deformation results for a type of AVTHT. The final deformation modes of all AVTHTs after oblique compression of 120 mm are compared in Figure 10. For a type of AVTHT, when the loading angle is constant, different loading patterns cause different deformation modes. For example, S1525 deforms in inextensional mode with \(\alpha=10^{\circ}\) and \(\beta=10^{\circ}\); however, it exhibits unideal deformation with \(\theta=10^{\circ}\). S1822 shows the bending and sliding mode with \(\alpha=20^{\circ}\) and \(\beta=20^{\circ}\); no bending deformation can be found under \(\theta=20^{\circ}\). Moreover, as the loading angle increases, all the AVTHTs deform with the modes, which range from progressive folding modes to unordered and nonprogressive modes. For a constant loading angle and pattern, different AVTHTs show different deformation modes. Note that under \(\beta=30^{\circ}\), S2020, S1921 and S1822 deform with bending modes at the support end; S1723, S1624, S1525, and S1426 exhibit crease in the plate surface, and the bending position is closer to the midpoint of a tube. Additionally, AVTHTs under the loading patterns of pattern 3 (\(\theta>0^{\circ}\)) are less prone to bending patterns compared with the other loading patterns. The results suggest that the loading angle, loading pattern, and thickness configuration can influence the deformation modes of AVTHTs. The force-displacement curves of all AVTHTs under various loading angles and patterns are compared in Figure 11. When the tubes are under the same loading angles,  
Figure 10: Deformation modes of the AVTHTs under different under oblique loading. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tubes** & **F\({}_{\text{max}}\) (kN)** & **SEA (kJ/kg)** & **CFE (\(\mathbf{l}\))** \\ \hline S2020 & 49.727 & 12.686 & 0.497 \\ S1921 & 47.539 & 11.473 & 0.466 \\ S1822 & 46.998 & 11.227 & 0.466 \\ S1723 & 47.003 & 11.009 & 0.452 \\ S1624 & 47.086 & 10.884 & 0.451 \\ S1525 & 47.363 & 10.649 & 0.434 \\ S1426 & 47.448 & 10.523 & 0.432 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Crashworthiness indices for simulations and experiments. i.e., in every three subgraphs, the force-displacement characteristics are generally similar, whether the curve trends or global force levels. When the loading angle is \(10^{\circ}\), all the corresponding curves exhibit fluctuation trends, and the initial peak force values are also lower compared to the subsequent wave peak values, which is contributed by imperfect progressive folding mode. However, with the loading angle increasing to \(20^{\circ}\) and \(30^{\circ}\), the tubes mainly deform with nonprogressive modes; therefore, the curve platforms drop, and the regular fluctuation is almost non-existent. Another concern is that there are non-ignorable differences between the curves of different AVTHTs under the same loading angles and patterns, especially the when \(\beta=20^{\circ}\) (Figure 11e), \(\theta=20^{\circ}\) (Figure 11f), and \(\theta=30^{\circ}\) (Figure 11i). This is because the plate thickness configuration of the AVTHTs can influence the deformation process under the same loading angle and pattern. The results of F\({}_{\text{max}}\), SEA, and CFE with various loading angles and patterns are obtained by simulations, as shown in Figure 12. For the index of F\({}_{\text{max}}\), the values decrease as the loading angle increases. When the tubes are under certain loading angles, the loading patterns and the plate thickness configuration can have an effect on the F\({}_{\text{max}}\). For example, the F\({}_{\text{max}}\) of S2020 is increased from 31.574 kN with \(\alpha=10^{\circ}\) to 34.096 kN with \(\theta=10^{\circ}\); however, the F\({}_{\text{max}}\) of S1822 is decreased from 31.7314 kN with \(\alpha=10^{\circ}\) to 28.9937 kN with \(\theta=10^{\circ}\). The F\({}_{\text{max}}\) of the tubes under \(\alpha=30^{\circ}\) are considerable close.