## 3 Validation of the Proposed Semi-Analytical Solution
### Validation of the Proposed Semi-Analytical Solution Regarding Numerical Results
In the fixed support zone, which takes 20% of the length of the cantilever beam, the finite element mesh is finer where the element size is 2 mm. In the second variant, the number of divisions of the circular cross-section of the cantilever is 64 and the size of the finite elements along the longitudinal axis is 3.5 mm, while in the fixed support zone it is 1.5 mm. For each of these variants, the number of nodes per element varies. Namely, Abaqus/Standard element library offers first-order solid elements with 8 nodes and second-order solid elements with 20 nodes [32], and both of these elements are covered in the mesh sensitivity analysis. Taking everything into account, four cases of finite element mesh were analyzed. As a subject of the mesh convergence analysis, the value of the displacement of the free end of the cantilever beam due to the 10 kN intensity of the external load is adopted. Spatial discretization error of the numerical analysis is estimated through a normalized von Mises stress error indicator [32] for the elements in the fixed support zone. All cases for mesh sensitivity analysis, as well as the results, are systematized in Table 4.  
Figure 12: The geometry and boundary conditions of the cantilever beams. Based on the results of the mesh sensitivity analysis, one may conclude that the mesh convergence is achieved by taking into account that the differences of displacement of the free end of the cantilever beam due to 10 kN intensity of the external load are around 3%. Analyzing the value of the normalized von Mises stress error indicator, it can be concluded that the numerical error due to spatial discretization error is negligible and does not have an influence on the results. In further numerical analyses, all cantilever beam models are meshed with 8-node solid finite elements with reduced integration, where the mesh density of case I of the mesh sensitivity analysis was adopted, since it is favorable regarding the other cases from the aspect of required computer resources. Regardless of the same mesh density, there are differences in the number of elements, nodes, and degrees of freedom between analyzed cantilever beam models (Table 5). These differences are consequences of different geometry of the analyzed cantilever beam models. The finite element mesh of the characteristic cantilever beam model is shown in Figure 13. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Number of Divisions** & **Size of the FE * along** & **Number of Nodes** & **Displacement of the** & **Normalized Mises** \\  & **of the Circle** & **the Cantilever Beam** & **per FE * & **Free End** & **Stress Error Indicator** \\ \hline I & 48 & 5 mm (2 mm) & 8 & 6.43 mm & 0.97\% \\ II & 48 & 5 mm (2 mm) & 20 & 6.27 mm & 2.19\% \\ III & 64 & 3.5 mm (1.5 mm) & 8 & 6.38 mm & 0.94\% \\ IV & 64 & 3.5 mm (1.5 mm) & 20 & 6.23 mm & 1.29\% \\ \hline \hline \end{tabular}  
* FE stands for finite element.  
\end{table} 
Table 4: A mesh sensitivity analysis. Figure 13: The finite element mesh of the cantilever beam. The stress-strain analysis is described for the model C1, chosen as a representative sample. Total deformations, von Mises stresses, and maximum principal plastic strains are presented in Figure 14, all for the external load intensity of 12 kN. Under this load von Mises stress is greater than yield stress. Therefore, plastic strains occur in the region of the fixed support and propagate to the free end of the cantilever beam. Taking into account discussion presented in Section 2 and the fact that the cross-sections diameter ratio of this model is lower than 1.50, such a propagation of the plastic strains is expected. It is worth mentioning that stress concentration occurs in a small region of the load application point, but it does not have an influence on the overall stress-strain state of the cantilever beam. The stress concentrations in the numerical analysis are not the subject of this paper, so they will not be further explained.