# Novel Bionic Design Method for Skeleton Structures Based on Load Path Analysis
## 2 Methodology
### Extract the Structural Main-Branch  
In this paper, the main-branch of the new structure is extracted by the load path theory. In general, it can give full play to the load-bearing performance and achieve lightweight at the source if the materials are arranged along the main load path in the design stage. In recent years, more and more researchers are interested in the load path visualization [25; 26; 27; 28]. Among them, [PERSON] et al. [29; 30] proposed a load path method, which holds that the load always transfers along the channel with the largest relative stiffness in the structure according to a given working condition. [PERSON] et al. [31; 32] applied this method to the structural design of automobile body and luggage rack, the main load path is visualized and structural load-bearing performance is improved. Compared with other methods of load path, this method has clear theoretical basis, physical significance and abundant application cases. Therefore, the main load path is extracted by this method as the main-branch of the new structure. The calculation process of the load path visualization based on the relative stiffness [33] is shown in Figure 3. Firstly, the finite element model (FEM) of the envelope structure is established and the strain energy is calculated. Secondly, one node is fixed in turn and the structural strain energy is calculated again. Then, the stiffness between each node and the loading point is calculated according to the strain energy coefficient. Finally, the main load path, i.e., the load transfer skeleton, is fitted according to the strain energy coefficient cloud chart. The detailed calculation process is as follows:  
The elastic structure is simplified to the model shown in Figure 4a, which includes loading point A, supporting point B and an arbitrary point C. It is assumed that the three points are connected by springs which represent the relative stiffness between any two points of the system. Figure 4: Simplified model of an elastic structure: (**a**) original system; (**b**) constrained system. Figure 3: Flow chart of the load path calculation. Figure 2: Establishing the envelope model. (**a**) a cantilever bracket, (**b**) the contrained and loaded areas remain unchanged, (**c**) the lightening holes are removed, (**d**) the regular outline of the bracket. According to the concept of relative stiffness, which is the coupling stiffness between any two points of the structure, the load-displacement relationship among points A, B and C can be expressed as:  
\[\left\{ \begin{array}{c}P_{\rm A}\\ P_{\rm B}\\ P_{\rm C}\end{array} \right\}=\left[ \begin{array}{c}K_{\rm AA}\ K_{\rm AB}\ K_{\rm AC }\\ K_{\rm BA}\ K_{\rm BB}\ K_{\rm BC}\\ K_{\rm CA}\ K_{\rm CB}\ K_{\rm CC}\end{array} \right]\left\{ \begin{array}{c}d_{ \rm A}\\ d_{\rm B}\\ d_{\rm C}\end{array} \right\} \tag{1}\]  
where \(k_{ij(i,j={\rm A,B,C})}\) is the internal stiffness tensor, \(P_{i(i={\rm A,B,C})}\) and \(d_{i(i={\rm A,B,C})}\) represent the load and the displacement vectors, respectively. Applying load \(P_{\rm A}\) at point A produces displacement \(d_{\rm A}\), the total strain energy \(U\) of the model due to the external load is calculated according to the strain energy formula:  
\[U=\frac{1}{2}P_{\rm A}d_{\rm A} \tag{2}\]  
Since point B is fixed (\(d_{\rm B}=0\)), \(P_{\rm A}\) is calculated by Equation (1),  
\[P_{\rm A}= K_{\rm AA}d_{\rm A}+K_{\rm AC}d_{\rm C} \tag{3}\]  
So the total strain energy \(U\) is:  
\[U=\frac{1}{2}P_{\rm A}d_{\rm A}=\frac{1}{2}(K_{\rm AA}d_{\rm A}+K_{\rm AC}d_{ \rm C})d_{\rm A} \tag{4}\]  
In Figure 4b, any point C in the structure is fixed and the same displacement \(d_{\rm A}\) is produced by applying load \(P_{\rm A}^{\prime}\) at point A.