# Comparison of Stress Concentration Factors Obtained by Different Methods
## 6 Stress FEM Analysis of Shape-Discontinuity-Dimensions-Load Configuration (a)  
The subject of numerical stress analyses for subsequent comparison and statistical evaluation was the shape-discontinuity-dimensions-load configuration (a) according to Figure 1. The overall dimensions were defined by a trio of parameters: the larger diameter \(D\), the smaller diameter \(d\), and the transition radius or the radius of curvature of the transition curve \(r\). The aim of the numerical analyses was to determine the nominal and maximum equivalent and principal normal stresses in the critical, i.e., transition region of the individual diameters. These should then be the main characteristic defining the peak stress of the notching effect. The data obtained were then used to express (quantify) the stress concentration factors for specific values of diameters \(D\), \(d\), radius \(r\), and ratios \(D/d\) and \(r/d\), respectively, in the axial tension loading regime. Since the FEM analysis of configuration (a) was the case of an axially loaded rotationally (axially)-symmetric element, the numerical modelling could be made more efficient by using a two-dimensional axisymmetric shell model with a longitudinal axis of rotation and with appropriate consideration of the configuration in the definition of the boundary conditions (Figure 7). The definition of the transition curve for the case \(t>r\) was implemented in the form of a circular arc with radius \(r\) passing through the edge of the larger shaft cross-section and tangent to diameter \(d\). In the other cases, the circular arc took the form of a tangent to the respective edges. The appropriate working values for the \(D/d\) and \(r/d\) ratios and their intercombinations were chosen to ensure that the analyses included the full applicable range of dependencies for the stress concentration factors imported from the other sources being compared. Another important criterion was to maintain a certain minimum statistical significance of the obtained results but without a huge increase in the number of individual simulations required. Based on these criteria, 4 individual \(r/d\) ratios with values of 0.01, 0.05, 0.08 and 1.10 were selected for the 4 individual selected \(D/d\) ratios with values of 1.01, 1.02, 1.05 and 1.10, respectively. Similarly, for 4 individual selected \(D/d\) ratios with values of 1.2, 1.5, 2.0 and 3.0, 4 individual \(r/d\) ratios were selected with values of 0.01, 0.05, 0.08 and 1.10. This provided the basis for 2 \(\times\) 4 \(\times\) 4 = 32 combinations of input data. For the selected \(D/d\) and \(r/d\) ratios and the selected diameter size value \(D\) = 50 mm, the \(d\) and \(r\) values were \begin{table}
\begin{tabular}{c c c} \hline  & \(0.1\leq t/r\leq 2.0\) & \(2.0\leq t/r\leq 20.0\) \\ \hline \(c_{3}\) & \(0.365-2.098\sqrt{t/r}+0.878\,t/r\) & \(-0.593-0.028\sqrt{t/r}-0.106t/r\) \\ \hline \(c_{2}\) & \(-0.302+3.977\sqrt{t/r}-1.744\,t/r\) & \(2.198-0.486\sqrt{t/r}+0.165t/r\) \\ \hline \(c_{1}\) & \(0.012-3.036\sqrt{t/r}+0.961\,t/r\) & \(-1.805-0.346\sqrt{t/r}-0.038t/r\) \\ \hline \(c_{0}\) & \(0.926+1.157\sqrt{t/r}-0.099\,t/r\) & \(1.200+0.860\sqrt{t/r}-0.022t/r\) \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensionless coefficients \(c_{i}\) as functions of dimensionless \(t/r\) ratios to determine stress concentration factors according to relation (10). Figure 7: Boundary conditions and loading used for finite element analysis. expressed to 3 decimal places in the next step. For each such case, a separate geometric model of the shaft was then generated, made as an axisymmetric shell element with a longitudinal axis of rotation. In the range of elastic deformations, the stress concentration factors were assumed to be independent of the material characteristics (properties) and the magnitude of the load, assuming isotropy of the material properties. The material considered was a standard steel, with the following characteristics: Young's modulus \(E=210\) GPa and Poisson's ratio \(\mu=0.3\).