# Optimization Method for Stiffened-Plate Layout in Box Structures Based on Load Paths
## 3 Application on Gantry of Machine Tool
### Finite Element Analysis of Gantry Model  
As shown in Figure 4a, the internal stiffened plate distribution within the gantry is complex and characterized by numerous subtle features. This complexity can significantly impact the efficiency and accuracy of finite element analysis, leading to issues such as stress concentration in the final analysis results. Therefore, in order to enhance computational efficiency and ensure accuracy, it is necessary to simplify the gantry model before conducting finite element analysis. This simplification process involves filling unnecessary voids, removing irrelevant auxiliary components from the gantry model such as bolt holes, small fillets, and reserved weld seams, etc. The simplified model is obtained as shown in Figure 4b. Figure 4: Simplified solid model of the gantry. (**a**) Internal stiffened plate display. (**b**) Simplified gantry model. Figure 3: Main body of the aluminum ingot composite machining unit. (**a**) Overall view. (**b**) Profile view. The machine tool structure is primarily welded from Q235 steel plates. Due to their similar properties, in the finite element analysis, structural steel embedded in the software is used as a substitute for simulation. From the machine structure shown in Figure 4, it is easy to see that the deformation of the gantry is mainly caused by the weight of itself and the auxiliary components, and the cutting forces. The gantry weighs 61,656 kg, the spindle part weighs 21,745 kg, the crossbeam weighs 9510 kg, and each set of side milling components weighs 6160 kg. The cutting force of the machine under typical machining conditions is shown in Table 1, in which the cutting force of the side milling is much smaller than its own gravity, so, only gravity is taken as a factor contributing to the deformation of the gantry for the side milling module. Regarding the cutting forces on the vertical milling shaft, due to the large size of the cutterhead, multiple cutting tools will be involved in the typical smooth milling process as shown in Figure 5. Taking any tool as an example, it is subjected to a three-way force as shown in Figure 6, where the main cutting force \(F_{c}\) and the feed resistance \(F_{\text{f}}\) act in the XOY plane and the back force \(F_{p}\) acts in the Z-direction. For ease of subsequent analysis and computation, it is necessary to simplify the loads on the spindle to the gantry. Initially, focusing on the forces acting on the XOY plane, the forces are simplified to the spindle at the center of the cutterhead based on the principle of force translation. \[\left\{ \begin{array}{l}F_{xi}=F_{r}\sin\theta_{i}\\ F_{yi}=F_{r}\cos\theta_{i}\end{array} \right.\text{ In XOY} \tag{6}\]  
where \(F_{r}\) is the combined force of the cutting forces on the XOY plane, with a magnitude of \(\sqrt{F_{c}^{2}+F_{\text{f}}^{2}}\). \(M_{1}\) denotes the moment of this resultant force when translated to the spindle. Subsequently, simplifying the back force in the Z-axis direction acting on any tool to the spindle will result in a moment \(M_{2}\) on the plane formed by the tool and the spindle. For ease of computation, this moment is orthogonally resolved to obtain: \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Main Cutting Force \(F_{c}\) (N)** & **Feed Resistance \(F_{f}\) (N)** & **Back Force \(F_{p}\) (N)** \\ \hline Vertical milling shaft & 10000 & 2300 & 4000 \\ Side milling shaft & 1470 & 485 & 514 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cutting forces under an extreme working condition. Figure 5: Schematic diagram of spindle cutting forces. Based on this, when multiple tools are simultaneously involved in processing, the load at the spindle can be calculated using the following Equation (8). \[\left\{