# Crasworthiness Analysis and Multi-Objective Optimization for Concave I-Shaped Honeycomb Structure
## 2 Related Theories on the Concave I-Shaped Honeycomb Structure
### Theoretical Calculations  
The cellular structure is centrosymmetric, as shown in Figure 2. The cellular length \(S\), the cellular height \(L\), the vertical short cellular height \(h\), the horizontal short cellular length \(b\)  
Figure 1: Recessed I-shaped honeycomb structure inspired by the combination of the traditional recessed hexagon and I-beam. and the cellular thickness \(t\) all contribute to the cellular structure. The relationship \(L=4\,\times\,h\) must remain constant to guarantee the combined link between cells. A load is given to the cellular structure in the y-direction to investigate the deformation of the cellular structure and the influence of each parameter on the structure's deformation and Poisson's ratio. Since the cellular structure is centrosymmetric, as illustrated in Figure 3, one-fourth of the structure can be used for the investigation. A bending moment of unknown magnitude \(M_{0}\) is applied to the whole structure, while point B is subjected to a force \(F/2\) in the y-direction. At the free end E, the corner deformation coordination condition is satisfied. \[\delta_{11}M_{\rm R1}+\delta_{1F}=0, \tag{1}\]  
where \(\delta_{1F}\) is the angle of rotation of point E when only the load \(F/2\) is applied and \(\delta_{11}\) is the angle of rotation of point E when a unit bending moment is applied at point E.  
The bending moments that are applied to each of the sections \(AB\), \(BC\), \(CD\), and \(DE\) when just \(F/2\) is used are shown in Figure 4. Figure 5 shows that when acting only on the unit bending moment, the bending moment of each section can be found, and using Mohr's theorem [27] to perform Mohr integral on each beam, the \(\delta_{1F}\) and \(\delta_{11}\) can be found. \[M_{AB}=\frac{F}{2}x_{1}\quad\ x_{1}\in\left(0,\frac{S}{2}\right)\quad\ \overline{M_{AB}}=-1, \tag{2}\]  
\[M_{BC}=0\quad\ x_{2}\in\left(0,h\right)\quad\ \overline{M_{BC}}=-1, \tag{3}\]  
\[M_{CD}=\frac{F}{2}x_{3}\quad\ x_{3}\in\left(0,b\right)\quad\ \overline{M_{CD}}=-1, \tag{4}\]  
Figure 3: The internal forces of a \(1/4\) structure. Figure 2: The overall cellular structure under load. \[M_{DE}=\frac{Fb}{2}\ \ \ \ \ x_{4}\in(0,h)\ \ \ \ \ \overline{M_{DE}}=-1, \tag{5}\]  
\[\delta_{1F}=\begin{array}{l}\int_{0}^{\frac{5}{2}}\frac{M_{AB}(x_{1}) \overline{M_{AB}}(x_{1})}{E_{m}l_{m}}dx_{1}+\int_{0}^{h}\frac{M_{DC}(x_{2}) \overline{M_{BC}}(x_{2})}{E_{m}l_{m}}dx_{2}+\\ \int_{0}^{b}\frac{M_{CD}(x_{2})\overline{M_{CD}}(x_{3})}{E_{m}l_{m}}dx_{3}+ \int_{0}^{h}\frac{M_{DE}(x_{4})\overline{M_{DE}}(x_{4})}{E_{m}l_{m}}dx_{4},\\ \delta_{11}= \begin{array}{l}\int_{0}^{\frac{5}{2}}\frac{\overline{M_{AB}}(x_ {1})\overline{M_{AB}}(x_{1})}{E_{m}l_{m}}dx_{1}+\int_{0}^{h}\frac{\overline{M_ {BC}}(x_{2})\overline{M_{BC}}(x_{2})}{E_{m}l_{m}}dx_{2}+\\ \int_{0}^{b}\frac{\overline{M_{CD}}(x_{3})\overline{M_{CD}}(x_{3})}{E_{m}l_{m} }dx_{3}+\int_{0}^{h}\frac{\overline{M_{DE}}(x_{4})\overline{M_{DE}}(x_{4})}{E_ {m}l_{m}}dx_{4},\end{array}  \tag{6}\]  
In Equations (6) and (7), \(E_{m}\) is the modulus of elasticity of the material; \(I_{m}\) is the moment of inertia of the section for the neutral axis. From Equations (6) and (7), \(M_{0}\) and \(\alpha\) are calculated as  
\[M_{0}=-\frac{\delta_{1F}}{\delta_{11}}=\frac{F}{8}\bigg{(}\frac{S^{2}+4b^{2}+8 bh}{S+4h+2b}\bigg{)}=F\alpha, \tag{8}\]  
\[\alpha=\frac{1}{8}\bigg{(}\frac{S^{2}+4b^{2}+8 bh}{S+4h+2b}\bigg{)}, \tag{9}\]  
Using the Castigliano theorem [28], the displacement deformation of each segment in the y-direction under a symmetrical load \(F/2\) in the y-direction can be found:  
\[M_{AB}=\frac{F}{2}x_{1}-M_{0}\ \ \ \ \ x_{1}\in\bigg{(}0,\frac{S}{2}\bigg{)}\ \ \ \ \ \overline{M_{AB}}=x_{1}-2\alpha, \tag{10}\]  
Figure 4: One-fourth structure when only \(F/2\) load is applied. Figure 5: One-fourth structure when acting only on unit bending moment.