## 2 Materials and Methods
### Description of Test Machines, Fatigue Tests Setup, and Specim Geometry
#### 2.2.2 Geometry of the Specimens for Rim Section Fatigue Tests  
The customized geometry of the specimen for the rim section tests was designed to obtain the fatigue crack development exactly in the area of interest--the radius shown in Figure 4.  
The final geometry is represented in Figure 7. It is the result of the optimization conducted by FEA, using the software SolidWorks Simulation 2022. Figure 5: Section test machine (**a**) and detail of the rim section specimen test setup (**b**). Figure 6: Example of the load function used for the rim section fatigue testing. The main aim of the FEA was to obtain a proper shape to guarantee the crack development in the radius under investigation, avoiding comparable stress values in the remaining regions of the specimen. Therefore, a simplified symmetrical model was adopted, simulating only the clamping system and the specimen. The symmetry constraint was used to save computational time. A static linear elastic analysis was conducted, using a general structural steel as a material model, with mechanical properties according to Table 2 and [PERSON]'s ratio \(\
u=0.3\). The load conditions of the test machine were applied--a fixed boundary condition to the fixed clamp and a unidirectional pulling force to the clamp directly joint to the machine hydraulic actuator. The finite element model is shown in Figure 8.  
The contact condition between the clamps and the specimen surface was simulated using a frictional coefficient of 0.3 and a proper preload was applied to the connection bolts. Parabolic solid elements, with 16 integration points, were used for the mesh generation. A 2 mm mesh size was used, resulting in a total number of 455,986 elements and 670,942 nodes. The mesh size was chosen after a preliminary sensitivity analysis, the results of which are shown in Figure 9. The comparison between three different mesh sizes showed a maximum difference below 3% in the stress distribution along the specimen axial direction. The choice of the final mesh size allowed us to optimize the overall computational time without influencing the accuracy of the result. Figure 8: Finite element model of the optimized specimen. Figure 7: Geometry of the specimen used for rim section fatigue testing. Dimensions in unit [mm]. The finite element model of the specimen was developed in a simplified way because the basic target of the analysis was the design of a proper shape to obtain the fatigue crack development where needed. For this reason, no further effort was made to exactly recreate the test machine behavior in the model and the FEA result must be considered from a qualitative point of view. The shape optimization resulted in the introduction of a notch at both sides of the specimen, near the radius under investigation, as shown in Figure 7. It is important to notice that, given the load conditions and the curved geometry of the specimen, the notch acts as a stress raiser only for the central part of the remaining section and not at the root of the notch itself. This is an essential difference, if compared to standard fatigue tests of notched specimens. The optimization study focused on two parameters of the notch geometry: the notch length and the notch radius. The notch length was optimized to reduce the specimen cross-section, thus increasing the stress in the area of interest without affecting the test machine setup. In fact, the stiffness of the specimen had to be sufficient to allow proper test loads in the sensitivity range of the machine. The notch radius was optimized to not create the maximum stress peak at the root of the notch itself, thus avoiding an undesired failure mode. As a final result, considering the maximum principal stress along the axial direction of the specimen (x, in Figure 8), the stress ratio between the radius under investigation and the region next to the fixed clamp, where the bending moment is maximum, was 1.53.