# On the Design of Bionic Hierarchical H-Type Whip Restraints for Nuclear Power Plants
## 3 Crashworthiness Analysis  
### Crashworthiness Analysis Under Axial Impact  
In this section, a preliminary analysis is conducted on the crashworthiness of five bionic structures under the same mass (14 kg). The parameter settings for the five structures are depicted in Figure 7. Various model parameters and crashworthiness parameters for the five structures are presented in Table 2. The force-displacement curves and energy absorption curves of the structures are shown as Figure 8. From the force-displacement curves, the IPF of the five structures shows minimal variation. There are significant differences in the plateau forces. Compared to NBHMC, the plateau force of the three hierarchical structures exhibits a significant improvement, and the remaining three hierarchical structures demonstrate slightly greater strength than NBIHMC. NBSHMC exhibits the highest plateau force. The results reveal that NBSHMC demonstrates the highest EA, surpassing NBCHMC by 3.87%, and surpassing NBOHMC by 5.92%. The EA is the integral of force over displacement, meaning the plateau force directly determines the EA growth rate. Thus, NBCHMC's elevated plateau force leads to its optimal EA performance. This correlation explains why force fluctuations (e.g., IPF) minimally affect EA, whereas sustained plateau forces dominate cumulative energy absorption. Figure 6: Mesh sensitivity analysis (**a**) Force-displacement curves for different mesh sizes, (**b**) Comparison of EA and IPF for different mesh sizes. Figure 7: Five biomimetic structures and parameter settings: (**a**) NBHMC, (**b**) NBIHMC, **m** = 12, (**c**) NBOHMC, **n** = 48, (**d**) NBSHMC, **TH** = 30 mm, (**e**) NBCHMC. Figure 9 illustrates the comparison of the crashworthiness parameters between bionic structures. Figure 9a is a comparison between IPF and MCF. The maximum IPF occurs at NBIHMC for 743.311 kN, and the minimum IPF is observed at NBSHMC, with a small difference of 1.49% between them. Figure 9b is a comparison between SEA and CFE. SEA exhibits a growth trend similar to MCF under the same mass, and CFE also has the same growth trend as MCF due to the small difference in IPF. Therefore, the crashworthiness of several structures can be judged by solely comparing MCF. The MCF of NBIHMC and NBIHMC was 40.45% and 51.22% higher than that of NBDHMC, respectively. Figure 10 presents the deformed shapes and vertical cross-sections of five structures following compression. This can be used to explain the different MCF of the several structures. For NBHMC, the wrinkles are not entirely flattened, with unfolded sections appearing at various positions, which indicates that NBHMC is not fully compressed. For NBIHMC and NBIHMC, certain sections of the structure demonstrate folding, and others exhibit overall buckling. This indicates that the structures have entered the densification \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Structure** & **t (mm)** & **m (kg)** & **EA (kJ)** & **IFF (kN)** & **SEA (kJ/kg)** & **MCF (kN)** & **CFE (\%)** \\ \hline NBHMC & 2.265 & 14 & 141.279 & 737.934 & 10.091 & 321.461 & 43.562 \\ NBIHMC & 1.76 & 14 & 204.519 & 743.311 & 14.609 & 467.679 & 62.918 \\ NBIHMC & 1 & 14 & 220.503 & 732.929 & 15.750 & 506.620 & 69.123 \\ NBSHMC & 1.052 & 14 & 234.386 & 732.417 & 16.742 & 538.735 & 73.556 \\ NBCHMC & 0.624 & 14 & 225.644 & 734.639 & 16.117 & 518.018 & 70.513 \\ \hline \end{tabular}
\end{table} 
Table 2: Crasworthiness characteristics under the same mass. Figure 8: Force-displacement and energy absorption curves of tubes under the same mass: (**a**) force-displacement curve, (**b**) EA–displacement curve. Figure 9: Crasworthiness performances analysis under the same mass: (**a**) MCF and IPF; (**b**) SEA and CFE. stage. The folding wavelength and number of folds are critical factors influencing the energy absorption efficiency of thin-walled structures. More folds and shorter wavelength lead to greater energy absorption capability. Contrasting NBIHMC and NBOHMC, as depicted in Figure 10, the number of folds appears similar.