# Comparison of Stress Concentration Factors Obtained by Different Methods
## 6 Stress FEM Analysis of Shape-Discontinuity-Dimensions-Load Configuration (a)
For all combinations of \(D/d\) and \(r/d\) ratios, the model was loaded by a single force F = 2 kN applied axially in the direction of the longitudinal axis of the element (shaft). The model is loaded by tensile forces. Since the results of numerical simulations are mainly influenced by the quality and size of the finite elements, it was advisable to define the location of the transition curve with the smallest possible elements. In this case, an element size of 0.25 mm was chosen for the investigated region. For the needs of local refinement of the mesh, a non-uniform distribution of elements along the edges of the model was chosen. Along the curve and its transition to a small shaft radius, a fixed size of 0.25 mm was defined, and towards both ends of the shaft, a bias function was used to determine the element size, which prescribed an increase in element size from 0.25 mm to 4.0 mm with an increasing tendency towards the edge. The critical region investigated was 8 layers of elements interleaved with a regular arrangement towards the volume of the component. Although the choice of finite element type may not have a significant effect on the accuracy of the results, a more regular arrangement of the grid elements was achieved solely by the choice of four-node quadratic elements. Quadratic quadrilateral elements of type CAX8R were used for discretization and the FEM analysis was performed in the ABAQUS programming environment. In total, several dozen models were created for different combinations of \(D/d\) and \(r/d\) ratios. In order to increase the accuracy of the analysis, it was necessary to make adequate adjustments to the FE mesh for each numerical model, while ensuring the regularity of the arrangement of the elements in the investigated region. The final choice of the element size while gradually reducing the element size was completed at the stage when a negligible change in stress had already been achieved by the previous refinement of the mesh. The main monitored output of each simulation was the maximum normal stresses identified at the point of intersection of adjacent diameters. The maximum values of the equivalent stresses \(\sigma_{\text{VM}}\) according to the von Mises theory and the principal normal stresses \(\sigma_{1}\) in the longitudinal axis direction, i.e., in the direction of the applied loading force \(F\), were monitored. The many results obtained from FEM analyses are illustrated by several examples in Figures 8-11. Figure 8 shows the equivalent stress field according to the [PERSON] theory on a quarter 3D model with a close-up of the 2D view in Figure 9 at the location of the transition arc. Figure 10 shows the field of principal normal stresses in the longitudinal axis direction on the quarter 3D model with detail in Figure 11 in 2D view at the location of the transition arc. Figure 8: Example of a equivalent stress field according to [PERSON] theory on a quarter 3D model. Regression and Correlation Analysis of Stress Concentration Factor Values Obtained from FEM Analysis  
Based on the values of the normal stresses \(\sigma_{\text{VM}}\) and \(\sigma_{1}\) obtained by FEM analysis, the stress concentration factors \(\alpha_{\text{VM}}\) and \(\alpha_{\text{S1}}\) were then expressed by dividing the corresponding stresses and \(\sigma_{\text{VM}}\), \(\sigma_{1}\), respectively, by the corresponding nominal value \(\sigma_{\text{nom}}\) according to the relations  
\[\alpha_{\text{VM}}=\frac{\sigma_{\text{VM}}}{\sigma_{\text{nom}}},\;\alpha_{ \text{S1}}=\frac{\sigma_{1}}{\sigma_{\text{nom}}}. \tag{13}\]  
For the purpose of comparison of the data thus obtained, the values of the stress concentration factors of \(\alpha_{\text{Cl}}\) and \(\alpha_{\text{T1}}\) from the imported sources were expressed in parallel.