## 4 Conclusions  
Cantilever beams with linearly variable circular cross-section along the beam axis are often applied in engineering practice. The solution for the elastoplastic deflection of such cantilever beams suitable for practical use has not been developed, to the authors' best knowledge. A semi-analytical solution for determining the deflection of the cantilever beam with linearly variable circular cross-section loaded by shear force at the free end was presented in this paper. A semi-analytical solution was developed for the beams made of homogenous and isotropic materials, which can be described via the bilinear elastoplastic strain hardening model. Based on the results presented in this paper, general conclusions can be summarized as follows:  
* it is confirmed that the Bernoulli-Euler formula grants satisfying results for the analysis of the deflection of the cantilever beam in the elastic regime, while in the plastic regime the differential equation of beam bending has no solution in the closed form; therefore, the problem needs to be solved via incremental analysis;
* the results of the proposed semi-analytical solution are in good correlation with the experimental results of two models of vertical components of a seismic energy dissipation device;
* the results of the proposed semi-analytical solution are in good correlation with the numerical results of several cantilever beam models, where the cross-section diameter, beam span, and mechanical characteristics of material vary;
* taking into account the above-mentioned conclusions, the proposed semi-analytical solution is considered as validated;
* the proposed semi-analytical solution is a simple method for application in practical engineering. Future research can be directed toward developing a solution for different elastoplastic material behavior models and different load types along the axis of the cantilever beam. Furthermore, in the case of significant axial load, an elastoplastic stability analysis of such elements would be very useful. The solution presented in this paper can serve as a good basis for further research. Conceptualization, A.Z. and M.T.-M.; methodology, A.Z. ; software, A.Z. and T.V. ; validation, A.Z., M.T.-M. and T.V. ; formal analysis, A.Z., M.T.-M. and D.Z. ; investigation, A.Z. and D.Z. ; resources, A.Z. and M.T.-M.; data curation, A.Z. ; writing--original draft preparation, A.Z. ; writing--review and editing, A.Z., M.T.-M. and T.V. ; visualization, A.Z. and D.Z. ; supervision, M.T.-M., D.Z. and T.V. ; project administration, A.Z. All authors have read and agreed to the published version of the manuscript. The experimental part of this research was funded by the Innovation Fund of the Republic of Serbia. The APC was funded by the authors. Not applicable. The data presented in this study are available on request from the corresponding author. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**Force at Limit Displacement [kN]**} & \multicolumn{3}{c}{**Elastic Stiffness Ke [kN/mm]**} & \multicolumn{3}{c}{**Post-Vield Stiffness Ky [kN/mm]**} \\ \cline{2-9}  & **Semi** & **FEM** & **Difference** & **Semi** & **Difference** & **Semi** & **FEM** & **Difference** \\  & **Analytical** & **[\%]** & **Analytical** & **[\%]** & **Analytical** & **[\%]** & **Analytical** & **[\%]** \\ \hline C1 & 13.19 & 13.43 & 1.82 & 2.26342 & 2.17084 & 4.26 & 0.11095 & 0.12025 & 8.38 \\ C2 & 4.44 & 4.50 & 1.35 & 0.14486 & 0.14338 & 1.03 & 0.01120 & 0.01221 & 9.02 \\ C3 & 9.83 & 9.93 & 1.02 & 1.40488 & 1.33542 & 5.20 & 0.06873 & 0.07465 & 8.61 \\ C4 & 3.40 & 3.47 & 2.06 & 0.09059 & 0.08884 & 1.97 & 0.00906 & 0.00973 & 7.40 \\ C5 & 9.21 & 9.66 & 4.89 & 2.24202 & 2.15625 & 3.98 & 0.04155 & 0.04522 & 8.83 \\ C6 & 3.41 & 3.50 & 2.64 & 0.14427 & 0.14341 & 0.60 & 0.00532 & 0.00582 & 9.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Semi-analytical vs. numerical results for cantilever beams. **Conflicts of Interest**: The authors declare no conflict of interest.