# Optimization Method for Stiffened-Plate Layout in Box Structures Based on Load Paths
## 2 Structural Optimization Method Based on Load Path
### Load Paths Visualization  
The theory of load path is specifically discussed in [PERSON]'s literature [19], which defines the load path as a channel for the transfer of an internal force in a specified direction in a structure, starting at the point of action of the force and terminating at the corresponding equilibrium reaction force. In order to clearly explain the physical significance of the load paths, the concept of a hypothetical force stream conduit is introduced, as shown in Figure 1, where the load path is distributed around the conduit. Similar to fluid flow in a conduit, since there is no force (fluid) passing through the boundary, the forces at two ends of the conduit are balanced in the X-direction, that is This requirement must also be met when the point P is subjected to an external load. In order to achieve this, the normal stress \(\sigma_{\text{n}}\) and the shear stress \(\tau_{\text{t}}\) at point P must satisfy the following equations  
In analyzing a point P on the force conduit wall, on a plane tangent to the force conduit wall, there exist normal stress \(\sigma\) and shear stress \(\tau\). According to the principle of equilibrium, the stress acting on this point is satisfied the following equation:  
\[\left\{\begin{array}{l}\sigma_{n}\cos\theta-\tau_{t}\sin\theta=0\ (x-direction)\\ \sigma_{n}\sin\theta+\tau_{t}\cos\theta=0\ (y-direction)\end{array}\right. \tag{2}\]  
Angle \(\alpha\) and \(\theta\) is taken as the direction of load path and the normal line at point P. For a 3D structure, the stress components at a point form a second order tensor; it can be represented in a [3 \(\times\) 3] matrix. If each row gives three stresses acting on a plane that the normal is aligned with one of the coordinate axes [24], then  
\[[\sigma]=\begin{pmatrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\end{pmatrix} \tag{3}\]  
where \(\sigma_{xy}\) is the shear stress acting on the plane that the normal is in the x direction and pointing y direction. Eigenvalues and eigenvectors of this matrix provide the principal stresses and principal stress vectors.  
Thus, the expression for the direction of the load path at each point in the 3D structure can be expressed as:  
\[\left\{\begin{array}{l}\mathbf{V_{x}}=\sigma_{x}\mathbf{i}+\tau_{yx}\mathbf{ j}+\tau_{zx}\mathbf{k}\\ \mathbf{V_{y}}=\tau_{xy}\mathbf{i}+\sigma\mathbf{j}+\tau_{zy}\mathbf{k}\\ \mathbf{V_{z}}=\tau_{xz}\mathbf{i}+\tau_{yz}\mathbf{j}+\sigma_{z}\mathbf{k} \end{array}\right. \tag{4}\]  
Using the pointing stress vector, the force acting on the infinitesimal element can be described as the vector dot product:  
\[\left\{\begin{array}{l}F_{x}=\int V_{x}\cdot\overset{\rightharpoonup}{u}dA \\ F_{y}=\int V_{y}\cdot\overset{\rightharpoonup}{u}dA\\ F_{z}=\int V_{z}\cdot\overset{\rightharpoonup}{u}dA\end{array}\right. \tag{5}\]  
where \(\overset{\rightharpoonup}{n}=n_{x}\mathbf{i}+n_{y}\mathbf{j}+n_{z}\mathbf{k}\). Therefore, the load paths are described by the sequential connection of the pointing stress vectors. Three pointing stress vectors (\(V_{x}\), \(V_{y}\) and \(V_{z}\)) are defined, and they describe the load paths in the corresponding direction  
The stress components can be solved based on the FEM and the direction of load paths in each element can be calculated using Equations (1) and (5). The nodes and elements information, including node number, coordinates, and stress components, are extracted in ANSYS and post-processed for load-path visualization.  
Figure 1: Schematic diagram of the force stream conduit.