# Multi-Objective Optimization of Square Corrugation Multilayer Nested Structures
## 2 The Nested Structure with a Double Octagonal Inner Wall
### The Verification of the FE Model
#### 2.3.2 The Mathematical Model Based on SCOD  
Based on the Simplified Super Folding Element (SSFE) theory, a mathematical model is proposed to predict the mean crushing force (_MCF_) of SCOD [42; 43; 44]. During the collision, the energy balance equation of SCOD can be obtained as shown in Equation (2):  
\[F_{m}2 Hk=E_{b}+E_{m} \tag{2}\]  
where \(F_{m}\) is _MCF_; \(E_{b}\) and \(E_{m}\) are the bending energy and membrane energy, which are primary forms of energy dissipation; \(2H\) represents the wavelengths of the fold; and \(k\) is the effective compression factor, which is 0.7 in this paper [45]. 1. The prediction of \(E_{b}\). The bending energy can be calculated as follows [46]:  
\[E_{b}=2\pi M_{0}L_{c} \tag{3}\]  
where \(L_{c}\) is the perimeter of the cross-section, and \(M_{0}\) is the full plastic bending moment per unit length, which can be obtained as Equation (4):  
\[M_{0}=\frac{1}{4}\sigma_{0}t^{2} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Tube** & _IPCF (kN)_ & _MCF (kN)_ & _EA (kJ)_ & _SEA (kJ/kg)_ \\ \hline \multirow{3}{*}{SC} & Test & 18.34 & 5.70 & 0.34 & 6.14 \\  & FE & 17.95 & 5.60 & 0.34 & 6.08 \\ \cline{1-1}  & Error & 2.17\% & 1.78\% & 1.19\% & 0.98\% \\ \hline \end{tabular}
\end{table} 
Table 2: The comparative result between the experiment test and FE modeling. Figure 5: The force–displacement curves and collapse modes of SC under experiment test and FE simulation. where \(t\) is the thickness of the wall and \(\sigma_{0}\) is the flow stress of the structural material:  
\[\sigma_{0}=0.92\sigma_{u} \tag{5}\]  
B. The prediction of \(E_{m}\). According to SSFE, the SCOD can be divided into several constituent elements in order to analyze theoretically the \(E_{m}\), and then the constituent elements will be simplified [40; 46]. The constituent elements of SCOD are defined as five types, including the shell corner element, T-shape element, Y-shape element, 3-plane corner element, and criss-cross element. The specific situation is shown in Figure 6.  
As shown as Figure 6, the shell corner element can be simplified as a plane corner element (180\({}^{\circ}\)) [47]:  
\[E_{m}^{corner\ element}=\frac{4M_{0}H^{2}}{t}\frac{\tan(\theta/2)}{(\tan( \theta/2)+0.05/\tan(\theta/2))/1.1}=\frac{4.4M_{0}H^{2}}{t} \tag{6}\]  
where \(\theta=180^{\circ}\). The T-shape element is equivalent to the three-plane corner element (30\({}^{\circ}\)) [35], which can be obtained by Equation (7):  
\[E_{m}^{\text{T-shape element}}=\frac{4M_{0}H^{2}}{t}\bigg{(}\frac{tan(\alpha )}{(tan(\alpha)+0.05/tan(\alpha))/1.1}+2 tan\Big{(}\frac{\alpha}{2}\Big{)} \bigg{)}=\frac{5.31M_{0}H^{2}}{t} \tag{7}\]  
The Y-shape element can be regarded as the three-plane corner element with the \(\alpha=90^{\circ}\)[47; 48], which can be obtained by Equation (8):  
\[E_{m}^{\text{Y-shape element}}=\frac{4M_{0}H^{2}}{t}\bigg{(}\frac{tan(\alpha )}{(tan(\alpha)+0.05/tan(\alpha))/1.1}+2 tan\Big{(}\frac{\alpha}{2}\Big{)} \bigg{)}=\frac{12.4M_{0}H^{2}}{t} \tag{8}\]  
The \(E_{m}\) of the criss-cross element is [42]:  
\[E_{m}^{\text{criss-cross}}=\frac{16M_{0}H^{2}}{t} \tag{9}\]  
As mentioned above, the total membrane energy can be summarized as:  
\[E_{m}^{\text{SCOD}}=N_{c,e}.\frac{4.4M_{0}H^{2}}{t}+N_{T-s_{e}}.\frac{5.31M_{ 0}H^{2}}{t}+N_{Y-s_{e}}.\frac{12.4M_{0}H^{2}}{t}+8\frac{12.4M_{0}H^{2}}{t}+8 \frac{16M_{0}H^{2}}{t} \tag{10}\]  
C. The prediction of \(MCF\). Figure 6: The constituent elements of SCOD and simplified elements. By uniting Equations (2), (3) and (10), the predicted _MCF_ of the SCOD can be expressed as:  
Figure 5:0  
In addition, parameter H can be calculated by:  
Figure 5:1  
By calculating Formulas (11) and (12), \(F_{m}^{\text{SCOD}}\) can be solved, and the results predicted by the mathematical model are compared with the FE model  
According to Figure 7, with the increase in thickness \(t_{int}\), \(t_{mid}\), \(t_{out}\), and \(t_{rib}\), the three crashworthiness indicators, namely _IPCF_, _SEA_, and _CFE_, all show an upward trend.