Optimization Method for Stiffened-Plate Layout in Box Structures Based on Load Paths

Abstract

Gantries and beams, as the main load-bearing structures of heavy equipment, usually belong to the box structure consisting of outer walls and inner stiffened plates. The structure of the stiffener layout is bulky due to empirical design, leading to higher material consumption and impacting mechanical performance. There are challenges in effectively identifying load-transferred paths within 3D box structures through direct topological optimization. A method for optimizing the layout of internal stiffened plates of large box structures based on load paths is proposed in this paper. Initially, based on the load conditions acting on the structure, the 3D box structure is decomposed into 2D functional sections. Subsequently, the load paths on the functional cross-section are visualized according to the load path method. Finally, the stiffener layout of the ultimate optimized structure is designed according to the effective load path distribution. Taking the gantry of a heavy-duty aluminum ingot composite processing unit as an example, the optimization results indicate that the maximum stress of the structure decreased by 14.9%, the maximum deformation reduced by 32.95%, and the overall weight decreased by 14.4%. This demonstrates that the approach proposed in this paper is practical and effective for optimizing stiffener layouts in large-box structures.

1. Introduction

Large-box structures are widely used in heavy equipment such as machine tools and cranes, often serving as primary load-bearing structures. Their stiffness, strength, and weight directly influence the mechanical performance and processing quality of the equipment. Currently, the design of box structures primarily relies on empirical analogies, lacking scientific guidance. As a result, excessive design margins are often left, leading to structural bulkiness and impacting the structural mechanical performances. Therefore, large box structures with high stiffness, high strength, and low weight are pursued by the equipment design industry. The box structure is composed of outer skin and internal stiffening plates. The outer skin is sometimes limited by equipment design criteria, therefore, the essence of lightweight design is usually to optimize the layout and dimensional parameters of the internal-stiffening plates [1]. Research on the layout optimization of 2D stiffened panels has garnered widespread attention from numerous scholars [2,3,4], while studies on large 3D box structures are relatively scarce.

As the empirically based stiffened plate layout is more initial, the researchers will also optimize the stiffened plate dimensions based on the static analysis of the box structure. Cong et al. [5] conducted size optimization for the beam of a high-speed vertical machining center that utilizes a gantry structure to obtain better structural performance. Li et al. [6] carried out finite element analysis and optimized the design of the structure of the gantry beam of a guideway grinding machine, which resulted in a significant reduction of the maximum stress and displacement of the gantry beam. Wang et al. [7] addressed the issue of the bulky structure and high material consumption in the gantry-milling machine bed within the current mechanical industry, through the static analysis of the bed along with stress–strain profiles and optimizing its dimensions while meeting stiffness and strength requirements, they achieved a lightweight design of the bed. Lu et al. [8] derived a modal analysis method for variable cross-section beams based on Euler beam theory and the transfer matrix method for gantry machine tool beam structures, which provides a reference for the design of machine tool beam structures. Wu et al. [9] proposed a bilevel accelerated microbial genetic algorithm for lightweight optimization design of sliding pillow block thickness, successfully achieving a weight reduction of 11.51%. However, such designs are limited to empirical approaches and are unable to offer the optimal stiffener layout.

Topology optimization, as a method for optimizing material distribution within a continuous domain towards various objectives, has been widely applied in structural lightweight design. And, some scholars have carried out the optimization of 3D stiffened-plate layouts by topology optimization method. Wu et al. [10] carried out a topology optimization design for medical devices with semi-gantry structures and reduced the weight by 30% with the same accuracy. An et al. [11] proposed a two-level approximation method for multi-objective optimization of composite material stiffened panels. They introduced the concept of base structure in the reinforcement layout and laminate stacking sequence, formulated the design problem using a combination of discrete and continuous variables, and sought to minimize structural weight and maximize the fundamental frequency under given displacement constraints and manufacturing limitations. However, researchers gradually discovered during the research process that classical topology optimization methods face challenges in effectively identifying the stiffener layout within 3D box structures [12]. Moreover, the material distribution patterns provided may be overly complex and difficult to manufacture, making it challenging to effectively guide the optimization and improvement of the stiffener layout [13,14]. Aiming at this kind of problem, Wang et al. [15] proposed a method for the design of stiffener layouts in large box structures under moving load conditions based on multi-case topology optimization. They utilized the principle of equivalent force to partition the model into functional sections for dimension reduction and subsequently conducted multi-case topology optimization to obtain the stiffener layout.

Some scholars have tried to use the structural bionics method to achieve the lightweight design of the box structure. Zhang and Gao et al. [16,17] designed the internal reinforcement plate layout of the structure based on the adaptive method of the natural branching system growth mechanism. They optimized the spindle box, column, and bed of the machine tool by combining them with the dynamic sensitivity analysis, and the simulation and experimental results showed that the overall structure of the machine tool has been greatly improved. Dong et al. [12,18], based on an adaptive growth method inspired by natural branching phenomena, initiated growth from a sowing line and gradually expanded based on design sensitivity. The optimal reinforcement layout is finally obtained with superior performance over the original machine. However, in the selection of biological prototypes and structural design, the design ideas are more subjective and lack a theoretical basis. Additionally, the layout adaptive computation based on the natural branching growth mechanism is relatively complex, leading to very high manufacturing costs, especially for large box structures subjected to moving loads.

In order to solve the troubles caused by the above problems, the load path originating from the civil engineering field provides a new idea. The load path can be seen as the search for the optimal load-transferred skeleton of truss structures to evaluate the force-transferring performance and rationality of the structure. However, this concept cannot be directly applied to the design of solid mechanical structures. Kelly et al. [19] proposed a method of force transmission path extraction for force flow based on the principle of elastic mechanics equilibrium on this basis, which provides a basis for the analysis and evaluation of structures. It has been studied in structural design in recent years. Zhao et al. [20] considered that dynamic factors should not be neglected in analyzing the load-carrying capacity of complex structural components and analyzed the load-carrying characteristics of engine cylinder heads using load path theory. Tamijani, A. et al. [21] used the load function method to obtain the load path of a plate-shell structure, providing crucial insights for the structural design of aerospace components. Wang and Wu et al. [22,23,24,25] optimized and improved automobile rims and control arms based on the load path analysis method. They combined biomimetic design methods to guide the shape optimization of skeletal structures, resulting in significant improvements in both lightweighting and structural strength [26]. These studies effectively demonstrate that load paths can accurately reflect internal load flow within structures, and the arrangement of materials according to the load path can effectively enhance structural stiffness.

In summary, based on the above issues and conclusions, this paper proposes an optimization method for the internal stiffener layout of large box structures guided by load path. Firstly, the box structure is decomposed into functional sections to achieve dimension reduction. Then, on the basis of the working condition analysis, the load path method is used to visualize the path of each functional section. Finally, the layout of the reinforcement plate on each section is determined based on the force flow distribution and path analysis. The remaining parts of this paper are outlined as follows. The optimization methodology proposed in this paper with specific details is given in Section 2. In Section 3, the optimization methodology is validated using an example of key components of an aluminum ingot composite machining unit. Finally, a summary and discussion of the work in this paper are provided.

2. Structural Optimization Method Based on Load Path

The method proposed in this paper consists of three steps. Firstly, an envelope model of a given structure is established based on the working conditions and functional characteristics, and a functional cross-section decomposition of the structural model based on the load distribution is performed to achieve dimensionality reduction. Then, a finite element model (FEM) is established, and the load path theory is used to visualize the load path. Finally, the design guidelines for the external shape and stiffened plate layout are established by analyzing the load transfer law. The design process of each step is described in detail in this section, and the proposed method is validated by the example in the next section.

2.1. Establishment of Envelope Model

Box structures usually contain features such as corners, hollow areas, and thin plates that limit the load path distribution, and it is necessary to exclude such features designed by conventional methods before optimization is carried out. Therefore, the methodology proposed in this paper first requires the construction of an envelope model of a given structure by analyzing its working conditions and functional characteristics. In this process, the following criteria are followed: firstly, the constraints and the load application region need to be kept constant, secondly, the voids in the internal discontinuity region between the two need to be filled in order to provide a basis for the load path analysis. Finally, the outer boundaries of the structure are analyzed, and the complex boundaries are filled with regular boundaries.

2.2. Load Paths Visualization

The theory of load path is specifically discussed in Kelly’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

$$F_{Xa}=F_{Xb}$$

(1)

This requirement must also be met when the point P is subjected to an external load. In order to achieve this, the normal stress $\sigma$_n and the shear stress $\tau$_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 σ and shear stress τ. According to the principle of equilibrium, the stress acting on this point is satisfied the following equation:

$$\{\begin{array}{l}{\sigma }_n\cos \theta -{\tau }_t\sin \theta =0 (x-direction)\\ {\sigma }_n\sin \theta +{\tau }_t\cos \theta =0 (y-direction)\end{array}$$

(2)

Angle α and θ 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 × 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

$$\left[\sigma \right]=\left(\begin{array}{ccc}{\sigma }_{xx}& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{yx}& {\sigma }_{yy}& {\sigma }_{yz}\\ {\sigma }_{zx}& {\sigma }_{zy}& {\sigma }_{zz}\end{array}\right)$$

(3)

where σ_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:

$$\{\begin{array}{l}{\mathbf{V}}_{\mathbf{x}}={\sigma }_x\mathbf{i}+{\tau }_{yx}\mathbf{j}+{\tau }_{zx}\mathbf{k}\\ {\mathbf{V}}_{\mathbf{y}}={\tau }_{xy}\mathbf{i}+{\sigma }_y\mathbf{j}+{\tau }_{zy}\mathbf{k}\\ {\mathbf{V}}_{\mathbf{z}}={\tau }_{xz}\mathbf{i}+{\tau }_{yz}\mathbf{j}+{\sigma }_z\mathbf{k}\end{array}$$

(4)

Using the pointing stress vector, the force acting on the infinitesimal element can be described as the vector dot product:

$$\{\begin{array}{l}{F}_x=\int V_x\cdot \stackrel{\rightharpoonup }{n}dA\\ F_y=\int V_y\cdot \stackrel{\rightharpoonup }{n}dA\\ F_z=\int V_z\cdot \stackrel{\rightharpoonup }{n}dA\end{array}$$

(5)

where $\stackrel{\rightharpoonup }{n}=n_x\mathbf{i}+n_y\mathbf{j}+n_z\mathbf{k}$. 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.

2.3. Optimization Methods for Box Structure and Plates Layout

Differing from traditional finite element stress analysis, regarding material redundancy areas and stress concentration areas, load paths not only visually depict the transmission route of loads but also allow for the determination of the type of internal stresses within the structure based on the posture or direction of the path, offering guidance for design. According to the load path schematic shown in Figure 2, a detailed classification of the force transfer paths was made according to the path departure and termination points, and the paths were roughly classified into six types, corresponding to serial numbers 1 to 6.

In this context, Path 1 fully connects the load point with the constraint point and is reversible in direction. Path 2 forms a U-shaped path around the area where the constraint is applied. Path 3 starts from the load application area and terminates at the structural boundary, also being reversible in direction. Path 4 starts from the structural boundary terminates at the load constraint area and is reversible in direction. Path 5 begins at the boundary, then returns to terminate at the boundary. Regarding Path 6, which represents a vortex-like force flow path, as indicated by Waldman et al. [27], it is understood to result from lower stresses, leading to a recirculating structure in the normalized pointing vectors. It is recommended to eliminate the regions occupied by these vortices during the design process.

Applying the classification principle of force transmission effectiveness, effective load paths are defined as those load paths that establish a complete connection between the load point and the constraint point. While the load paths without a complete connection between load points and constraint points are labeled as invalid load paths. Clearly, Paths 1 and 2 fall into the category of effective load paths, while Paths 3 to 6 belong to ineffective load paths. Effective load paths, serving as the primary load carriers, constitute the efficient region for load transmission, and the layout of stiffeners can be designed based on such path distributions. In contrast, ineffective load paths exhibit low load transmission efficiency or a self-balancing posture, allowing for reduced stiffener distribution in the respective areas and even the removal of original materials to achieve weight reduction. Through this classification method, on one hand, the categorization of the six paths is established, and on the other, this method can provide guidance for optimizing structural design.

3. Application on Gantry of Machine Tool

The main part of the aluminum ingot composite machining unit is composed of gantry, vertical milling shaft parts, crossbeam, and side milling parts, as shown in Figure 3. The machine is equipped with a large 2000 mm cutterhead, which enables rough machining of aluminum ingots with dimensions up to 5000 × 2000 × 850 mm. Accordingly, the large diameter tool with high cutting speed puts high demands on the rigidity and stability of the machine itself in the actual production.

The gantry, as the primary load-bearing part, not only supports the weight of the crossbeam and milling components but also withstands cutting forces. The gantry, with a height of 4020 mm, length of 8010 mm, and width of 3290 mm, is composed of external skin and internal stiffened plates, constituting a typical box structure. Therefore, in order to achieve lightweight construction of the machine tool while ensuring rigidity and stability, it is necessary to optimize the gantry structure and internal stiffened-plate layout.

3.1. 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.

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. Cutting forces under an extreme working condition.

	Main Cutting Force Fc (N)	Feed Resistance Ff (N)	Back Force Fp (N)
Vertical milling shaft	10000	2300	4000
Side milling shaft	1470	485	514

, 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 $F_{\mathrm{f}}$ 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.

$$\{\begin{array}{l}{F}_{xi}=F_r\sin {\theta }_i\\ F_{yi}=F_r\cos {\theta }_i \mathrm{In} \mathrm{XOY}\\ M_{1i}=F_c·R\end{array}$$

(6)

where $F_r$ is the combined force of the cutting forces on the XOY plane, with a magnitude of $\sqrt{F_c^2+F_{\mathrm{f}}^2}$. $M_1$ 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:

$$\{\begin{array}{l}{M}_{2xi}=M_{2i}\sin {\theta }_i\\ M_{2yi}=M_{2i}\cos {\theta }_i \mathrm{In} \mathrm{Z}\text{-}\mathrm{direction}\\ M_{2i}=F_z·R\end{array}.$$

(7)

Based on this, when multiple tools are simultaneously involved in processing, the load at the spindle can be calculated using the following Equation (8).

$$\{\begin{array}{ll}F=\sqrt{{F_x}^2+{F_y}^2+{F_z}^2}& \\ =\sqrt{{\left({\displaystyle \sum _{i=1}^n{F}_{xi}}\right)}^2+{\left({\displaystyle \sum _{i=1}^n{F}_{yi}}\right)}^2+{\left({\displaystyle \sum _{i=1}^n{F}_{zi}}\right)}^2}& \\ M_1={\displaystyle \sum _{i=1}^n{M}_{1i}}& \mathrm{In} \mathrm{XOY}\\ M_2=\sqrt{{\left({\displaystyle \sum _{i=1}^n{M}_{2xi}}\right)}^2+{\left({\displaystyle \sum _{i=1}^n{M}_{2yi}}\right)}^2}& \mathrm{In} \mathrm{Z}\text{-}\mathrm{direction} \end{array}$$

(8)

In the calculation involving the simultaneous use of seven tools, based on the data of angles and cutting forces shown in Table 1 and

Table 2. The angles between cutting tools and the X-axis.

Angle	Θ1	Θ2	Θ3	Θ4	Θ5	Θ6	Θ7
Degree (°)	33	49	72	88	104	131	147

, the resultant force at the spindle connection point is determined to be 61,900 N, with a torque of 8.3265 × 107 N∙mm, and a bending moment of 2.62795 × 107 N∙mm. These forces, compared to the gravity of each component, cannot be negligibly small. Therefore, when optimizing with specificity in mind, the influence of spindle-cutting forces needs to be considered.

The model is imported into the FEA software ANSYS 2024R1, and tetrahedral elements are used for mesh division, as shown in Figure 7a, with an average mesh quality of 0.75. The boundary conditions are set according to the above analysis, as shown in Figure 7b. During the gantry simulation, it is necessary to apply fixed constraints to the bottom. The gravity of the crossbeam and side milling assembly is added to the front wall surface of the gantry (Remote Force). The vertical milling assembly gravity is added to the top connection of the gantry (Remote Force 2), and the simplified cutting forces (Force and Force 2), torque (Moment), and bending moment (Moment 2) are added at the top connection and finally, the standard earth gravity acceleration is added.

Upon solving, the stress and deformation cloud maps as shown in Figure 8 are obtained. It can be observed that the maximum deformation and stress regions are concentrated at the connection of the vertical milling components and the gantry. This implies that the distribution of stiffeners is unreasonable, leading to the concentration of loads at the connection areas. The maximum deformation at the spindle connection is 0.2461 mm. As the main processing area of the aluminum ingot milling, the deformation at the spindle connection will significantly impact the machining quality of the large cutter disk. On the other hand, the maximum stress is 62.43 MPa, significantly lower than the material yield strength (235 MPa). Therefore, the overall strength of the gantry has a sufficient margin, indicating significant potential for lightweight design.

Based on the above boundary conditions, the natural frequencies up to the sixth order of the gantry are obtained as shown in Figure 9. When the natural frequencies of a structure fall within or near the forced frequency, the possibility of the structure resonating is higher. Therefore, during the optimization of the gantry structure, it is necessary to ensure that the natural frequencies of the structure meet stability requirements.

3.2. Load Path Analysis

Using the functional section decomposition method, the functional section decomposition of the gantry is carried out. Since the main load of the gantry comes from the gravity of the spindle parts, beams, side milling parts, and the cutting force, the gravity mainly affects the gantry in the Z-direction, while the torque and bending moment caused by the cutting force act in the X and Y-directions, respectively. At the same time, considering the actual distribution of the stiffened plate, the slices are made from XOY, XOZ, and YOZ planes to realize the decomposition of the functional interface.

According to the load path analysis method given in Section 2, the gantry model is populated to obtain the envelope model as shown in Figure 10. According to the force distribution of the envelope gantry under the same loading conditions, the main functional section of the gantry is to resist the overall bending and torsion deformations. On this basis, the secondary development of APDL is carried out to extract the node stress data for processing, and according to the formula to get the distribution of the P cloud map, and the visualization of load paths in each functional section. Considering the excessive number of slices, the key slices of each functional cross-section shown below have been selected for presentation in the paper.

3.3. Optimization of Gantry Based on Load Path

The gantry is a box structure and the stiffened plates are mainly distributed on the inner side of each wall plate, so the layout optimization of the stiffened plates is also carried out on this basis with reference to the load path distribution. From Figure 11, it can be seen that the load path in the XOZ functional section is obviously biased in the X-positive direction, which is due to the torque effect caused by the cutting force. At the same time, because this machine only exists in the smooth milling of a working condition, so the working state of the machine force flow transfer will only be as shown in Figure 11. However, considering the actual equipment manufacturing requirements, the asymmetric distribution of mass will affect the dynamic performance of the machine tool, so the layout of the stiffened plate is optimized for symmetric design in accordance with the worst working conditions.

From Figure 11, it can be seen that the force transfer path on the XOZ section is divided into three kinds, the first is from the top corner of the gantry with the stress concentration point directly to the bottom surface of the column, the second is from the right side of the gantry with the stress concentration point horizontally through the middle of the gantry and then to the bottom surface of the column on the opposite side of the gantry, and the third is the middle part of the gantry in the form of eddy currents.

If the XOY cross-section (Figure 12) on the slice height is different, the path obtained will also change, in the slice near the top of the existence of an XOZ cross-section (Figure 13) in the stress concentration point of the same X coordinate of a stress concentration line. The stress concentration line to the two sides of the wall plate force flow is close to parallel, meaning that in the top, there is a need to arrange a corresponding stiffened plate, in the middle of the bottom of the gantry near the middle of the wall stress concentration line to the middle of the concentration. The slice at the middle height of the gantry is mainly transferred between two stress concentration points, which can be targeted to arrange the diagonal stiffened plate.

In the slices near the sides of the YOZ section, the load path is shown as flowing from the top of the gantry to the bottom, and in the middle of the gantry, it is shown as extending from the fixing place of the spindle parts to the front and rear walls, where vertical reinforcement plates can be added. Considering the more balanced and weaker force flow close to the two sidewall panels, there is more room for material removal, so adding holes in the sidewall panels can be considered for further weight reduction. Finally, the optimized stiffened plate layout is obtained as shown in Figure 14.

4. Verification of Optimization Result

In order to verify the optimization design effect of the gantry stiffened plate layout guided by the force flow, the new model is taken as the research object, and the same boundary conditions as Figure 7 are used to carry out the finite element analysis here, and the displacement and stress distributions obtained are shown in Figure 15. On the other hand, modal analysis of the optimized gantry structure yields the mode shapes as shown in Figure 16.

A detailed comparison of data before and after optimization is presented in

Table 3. Comparison of gantry mechanical properties before and after optimization.

Properties	Original Gantry	Optimized Gantry	Rate of Change (%)
Max displacement (mm)	0.2461	0.165	−32.95
Max stress (MPa)	62.43	53.115	−14.9
Mass (kg)	61656	52769	−14.4
First order natural frequency (Hz)	46.213	46.641	+0.93
Second order natural frequency (Hz)	54.884	56.525	+2.99
Third order natural frequency (Hz)	74.058	65.394	−11.70
Fourth order natural frequency (Hz)	79.395	67.186	−15.38
Fifth order natural frequency (Hz)	96.24	68.946	−28.36
Sixth order natural frequency (Hz)	101.13	73.394	−27.43

. The maximum stress after optimization is 53.115 MPa, which is reduced by 14.9% compared to the original gantry. The maximum displacement of the gantry after optimization is 0.165 mm, which is reduced by 32.95% compared to the original gantry. Moreover, its distribution has shifted from the original top connection position to the left and right by some distance. In comparison, the deformation at the most critical top connection position has decreased by 40% to 75% compared to the original gantry. This signifies an enhancement in the accuracy of the spindle, which serves as the primary machining component. The overall mass has decreased by 8887 kg, which is a 14.4% reduction compared to the original model. This significant weight reduction demonstrates noticeable lightweighting effects while ensuring the rigidity of the gantry.

Regarding dynamic performance, the optimized gantry shows a slight increase in the first two natural frequencies that are closest to the forced frequency range. The remaining natural frequencies are slightly lower than those of the original model, but still significantly higher than the forced frequency. This indicates that the dynamic performance of the overall gantry is relatively stable after optimization.

5. Conclusions

Aiming at the difficult problem of designing the layout of the internal stiffened plate of large box structures, a method for optimizing the layout of the internal stiffened plate of large box structures based on force-flow guidance is proposed. Initially, an envelope is constructed for analysis, and functional section decomposition is carried out based on load distribution to achieve dimension reduction, transforming the 3D problem into a 2D solution. Then, in conjunction with the load path method, load path visualization operations are conducted to obtain the distribution of load paths across various functional sections. Finally, through the analysis of load paths, ineffective paths are eliminated, and the optimized stiffener layout is obtained by distributing materials along the remaining paths. Taking the gantry of a certain aluminum ingot composite processing unit as an example for validation, the simulation results indicate that with a reduction of 14.9% in maximum stress and 32.95% in displacement, the overall weight of the structure has decreased by 14.4%. While the natural frequency of the structure is improved to reduce the possibility of resonance. This demonstrates that the method proposed in this paper exhibits good practicality and effectiveness for the stiffener layout design of box structures.

The “outer walls + inner stiffened plates” structure is widely present in fields such as aircraft, spacecraft, automotive, marine, and architecture. The method proposed in this paper can significantly reduce structural mass while ensuring structural performance. This effectively achieves cost reduction and revenue enhancement and demonstrates considerable practical value. On the other hand, the research methodology of this paper can be further explored to develop optimization models based on the force path method tailored to different target requirements. This offers a new perspective for optimizing problems related to this type of structure, holding certain research significance.