## 3 Validation of the Proposed Semi-Analytical Solution
### Validation of the Proposed Semi-Analytical Solution Regarding Numerical Results  
A numerical analysis of the elastoplastic bending of the cantilever beam with linearly variable circular cross-section is conducted via the application of the finite element method and software package Abaqus/Standard. The main aim was to validate the results of the semi-analytical solution presented in the paper using the numerical results. Two cases with different cross-sections diameter ratio were analyzed: \(\mathrm{d_{c}/d_{f}}=30/24\) mm; \(\mathrm{d_{c}/d_{f}}=30/15\) mm. The analysis covers two variants of the span of the cantilever beam: h = 200 mm and h = 500 mm. Besides the material model of steel C45 described in the previous section, the analysis was also conducted in the case when the yield stress of the material was \(\mathrm{f_{y}}=350\) MPa and the tangent modulus was \(\mathrm{E_{t}}=2000\) MPa. The aim was to examine the sensitivity of the proposed semi-analytical solution regarding the physical-mechanical characteristics of the material. All model data are systematized in Table 3.  
In order to define the fixed support in the numerical model, all degrees of freedom in the plane of the cantilever beam cross-section are constrained (Figure 12). Monotonously increasing the load at the free end of the cantilever beam is defined as a concentrated \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & \multicolumn{3}{c}{**VC1**} & \multicolumn{3}{c}{**VC2**} \\ \hline
**Results** & **Semi-Analytical** & **Experiment** & **Difference [\%]** & **Semi-Analytical** & **Experiment** & **Difference [\%]** \\ \hline Force [RN] at displacement of & 15.55 & 15.98 & 2.77 & 2.74 & 2.53 & 8.30 \\ Elastic stiffness \(\mathrm{K_{e}}\) [\(\mathrm{kN/mm}\)] & 2.1817 & 2.36899 & 8.61 & 0.21651 & 0.23122 & 6.79 \\ Post-yield stiffness \(\mathrm{K_{y}}\) [\(\mathrm{kN/mm}\)] & 0.11570 & 0.11616 & 0.40 & 0.01205 & 0.01183 & 1.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Semi-analytical vs. experimental results for vertical components. Figure 11: A material model of the steel material C45 for the semi-analytical solution. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & \(\mathrm{d_{c}}\) **[mm]** & \(\mathrm{d_{f}}\) **[mm]** & **h [mm]** & **E [GPa]** & \(\mathrm{E_{t}}\) **[MPa]** & \(\mathrm{f_{y}}\) **[MPa]** \\ \hline C1 & 30 & 24 & 200 & & & \\ C2 & 30 & 24 & 500 & & & \\ C3 & 30 & 15 & 200 & & 6600 & 430 \\ C4 & 30 & 15 & 500 & 190 & & \\ C5 & 30 & 24 & 200 & & & \\ C6 & 30 & 24 & 500 & & 2000 & 350 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The analyzed models of the cantilever beams. force in the center of gravity of the cross-section in the direction of the x axis of the global coordinate system (Figure 12). Finite element mesh convergence was done for the model C1 in order to obtain the optimal element type and mesh density. The model was meshed with solid hexahedral elements with three translator degrees of freedom per node, which is suitable for stress-strain analysis of the continuum. Tetrahedral elements were not analyzed because they are usually overly stiff and should be avoided whenever possible. When the bending deformation is dominant and solid elements with linear shape functions are applied, large shear stiffness can occur, which is not realistic. This phenomenon is known as \"shear locking\" and it can be prevented by increasing the number of finite elements, using solid elements with quadratic shape functions, or using reduced integration [32; 33]. Therefore, in this study, solid elements with reduced integration for determining elements of the stiffness matrix are applied. In the mesh sensitivity analysis, two variants of mesh density are analyzed. In the first variant, the number of divisions of the circular cross-section of the cantilever is 48, while the size of the finite elements along the longitudinal axis of the cantilever beam is 5 mm.