Using the large-scale general-purpose finite element analysis software ANSYS, based on the finite element analysis of the cross shaft of a 1700mm rolling mill universal joint in a hot rolling mill, a three-dimensional solid optimization analysis of the cross shaft was performed to meet its strength and stiffness requirements. The ANSYS system contains the Parametric Design Language (APDL), which has high-level language elements such as parameters, mathematical functions, macros, decision branches, and loops. It is an ideal program flow control language, well-suited for finite element calculations and optimization analyses. The finite element method and optimization methods are two of the most important mathematical tools in engineering analysis. Organically combining the two, fully utilizing the accuracy of numerical calculations in the finite element method and the efficiency of finding extrema in optimization methods, will exert tremendous power in engineering analysis. 1. Finite Element Analysis Calculation of the Cross Shaft The driving and driven shafts of the universal joint apply two pairs of forces to the cross shaft through bearings via their forks. These forces form a pair of equal-magnitude, opposite-direction couples (Figure 1). These two pairs of couple vectors lie in the plane determined by the active and passive axes. If the inclination angles of the two axes are neglected (which are very small and can be ignored), then the forces constituting the two couples are all in the plane of the cross axis. 1.1 Model Establishment Since the structure and load of the cross axis are symmetrical about the two sections II and II-II (Figure 1), it can be cut from the two sections II and II-II, and 1/4 of the cross axis can be taken as the research object (Figure 2). As shown in Figure 1, the dimensions of the cross axis are as follows: L=865mm, A=327mm, B=325mm, D=242mm, H=174mm, R=90mm, d=50mm, r=10mm. The cross axis is meshed using three-dimensional solid units, and a total of 41,904 elements are divided. The finite element model is shown in Figure 2. 1.2 Boundary Conditions In Figure 2, the two 45° sections A and B of the computational model, as well as the Y=0 plane, are symmetry planes of the cross-shaft structure and the load. The constraint conditions of the computational model are as follows: nodes at Y=0 on both planes A and B are constrained in three directions (X, Y, Z); nodes at Y≠0 on both planes A and B are constrained in two directions (X, Z), with free direction in the Y direction. 1.3 Load Application As shown in Figure 3, the load is distributed in a trapezoidal shape along the axial direction of the cross-shaft. In the XY plane, the load is distributed on the surface of the outer cylindrical surface of the cross-shaft, following a cosine law on the arc, with arc AB being 120°. 1.4 Finite Element Calculation Results Analysis Figure 4 shows the maximum principal stress distribution of the cross-shaft bearing under a torque of 240 kN·m. The stress is greatest on the loaded side of the transition section from the cylinder to the cone of the cross-shaft (i.e., the R90 transition section on the loaded side), and there is severe stress concentration. The maximum principal stress value is 498.15 MPa. 2. Structural Optimization Design of the Cross Shaft 2.1 Selection of Design Variables The selection of design variables is considered from two aspects: First, given that the main drive system of the rolling mill already exists and the outer frame dimensions of the universal coupling are already determined, the structural optimization design of the cross shaft is carried out. These dimensions can only be used as given design parameters (as shown by L in Figure 1). Second, based on the three-dimensional finite element analysis of the cross shaft above, the stress at the R90 arc transition point of the cross shaft is the greatest and is in an alternating stress state, making it a dangerous area. Changing certain dimensions cannot reduce the stress concentration at this point or has only a slight effect (as shown by A, B, d, and r in Figure 1). Therefore, the design variable can be x = [x1, x2, x3] = [R, D, H] where: R is the radius of the transition arc; D is the diameter of the cross shaft; and H is the width of the rolling bearing. 2.2 Determination of Objective Function The purpose of structural design for the cross shaft is to reduce the bending fatigue stress (manifested as the maximum principal stress) at that location, making it as close as possible to or less than the bending fatigue strength. Therefore, when transmitting a torque of 240 kN·m, the objective function is to minimize the maximum principal stress borne by the cross shaft, i.e.: f(x) = S1max. Where: S1 is the maximum principal stress borne by the cross shaft when transmitting a torque of 2400 kN·m. 2.3 Determination of Constraint Function Since the design of the cross shaft only considers reducing the stress at the fatigue failure point (i.e., the transition arc on the loaded side), without other limiting factors, there are no constraint functions in this optimization process. For each design variable, the boundary constraints are as follows: 222mm≤D≤264mm; 164mm≤H≤184mm. 2.4 Selection of Optimization Method and Optimization Process The ANSYS program provides two optimization methods: the zero-order method and the first-order method. Both use the penalty function (SUMT) method to transform the constrained optimization problem into an unconstrained problem for solution. Due to the complex force and deformation of the cross shaft, the zero-order method is used here to ensure the smooth progress of optimization. The optimization process is a series of analysis processes, namely a series of preprocessing-solving-postprocessing-optimization cycles. 3. Comparison between Before and After Optimization Through a series of iterations, the optimal design result value is obtained. Figure 5 shows the change of the objective function with the number of iterations and the change of variables with the number of iterations. The optimization results show that when the transition radius R increases from 90mm to 94.20mm, the cross shaft diameter D increases from 242mm to 259.80mm, and the rolling bearing width H decreases from 174mm to 166.43mm, the maximum principal stress that the cross shaft bears when transmitting a torque of 2400kN·m decreases from 498.15 MPa to 416.07MPa. The optimized result is 19.71% lower than the result before optimization.