## 2 Solution for the Elastoplastic Bending of the Cantilever Beam
### Incremental Analysis  
As a practical and sufficiently simplified solution of the above-mentioned nonlinear problem, a novel incremental procedure of the analysis of the elastoplastic strain hardening deflection of the cantilever beam with linearly variable circular cross-section is developed. The starting point of the yield of the material in the critical cross-section corresponds to the angle \(\theta_{\text{e}}=\pi/2\). The full plastification of the cross-section corresponds to the angle \(\theta_{\text{e}}=0\). However, in that case, a strain singularity occurs. Therefore, a sufficiently small value of the angle (\(\theta_{\text{e}}=0.005\pi\) to \(0.010\pi\)) is adopted as the limit state value in the incremental analysis. In the analytical procedure, the plastified zone of the critical cross-section incrementally increases and the value of the angle \(\theta_{\text{e}}\) decreases. With such defined discrete values of the angle \(\theta_{\text{e}}\), the value of the plastic strain of the most distant fibers of the cross-section \(\varepsilon_{\text{pl}}\) can be defined using the value of the yield strain of the material (Figure 2), based on the hypothesis that plane sections remain planes:  
\[ \begin{array}{l}\text{\ }\varepsilon_{\text{pl}}=\left(\frac{1}{\text{sin}\theta_{ \text{e}}(\text{x})}-1\right)\cdot\varepsilon_{\text{y}}=\left(\frac{1}{ \text{sin}\theta_{\text{e}}(\text{x})}-1\right)\cdot\frac{\text{f}_{\text{y}} }{\text{E}}.\end{array}  \tag{40}\]  
Substituting Equation (40) into Equation (39), the bending moment of the plastified cross-section for the incremental analysis of deflection of the cantilever beam can be defined as follows:  
\[ \begin{array}{l}\text{M}=\frac{\text{E}\cdot\varepsilon_{\text{y}}\cdot r^{3 }(\text{x})}{2\cdot\text{sin}(\theta_{\text{e}}(\text{x}))}\cdot\left(\theta_ {\text{e}}-\frac{\text{sin}(4\cdot\theta_{\text{e}}(\text{x}))}{4}\right)+ \frac{4}{3}\cdot\varepsilon_{\text{y}}\cdot\text{E}\cdot r^{3}(\text{x}) \cdot\cos^{3}\theta_{\text{e}}(\text{x})\\ \quad\quad+\left(\frac{1}{\text{sin}\theta_{\text{e}}(\text{x})}-1\right)\frac{ \text{f}_{\text{y}}}{\text{E}}\frac{\text{E}_{\text{t}}r^{3}(\text{x})}{2(1- \text{sin}(\theta_{\text{e}}(\text{x})))}\left[\frac{\pi}{2}-\theta_{\text{e}}( \text{x})+\frac{\text{sin}(4\theta_{\text{e}}(\text{x}))}{4}\right]-\frac{4}{3 }\cdot\left(\frac{1}{\text{sin}\theta_{\text{e}}(\text{x})}-1\right)\frac{\text{ f}_{\text{y}}}{\text{E}}\text{E}_{\text{t}}\frac{\text{sin}(\theta_{\text{e}}(\text{x}))}{1- \text{sin}(\theta_{\text{e}}(\text{x}))}r^{3}(\text{x})\cos^{3}\theta_{\text{e}}( \text{x})\end{array}  \tag{41}\]With discrete values of the bending moment in the critical cross-section defined using Equation (41), the corresponding discrete values of the intensity of the external force F are defined according to Equation (1), where the correct distance between force and the critical cross-section (7) should be taken into account. For each discrete value of the intensity of the external force, the boundary between the plastic and the elastic region along the cantilever beam is determined according to the relation (25). In the plastic region of the cantilever beam, based on the conducted analysis of the bending moment of the plastified cross-section, the function of the curvature cannot be analyzed in the closed form, but the approximation of the curvature function is conducted. The plastified region of the beam is divided into smaller subsegments \(\Delta\mathrm{x_{p}}\), where \(\Delta\mathrm{x_{p}}\) stands for the distance from the fixed support to the subsegment. Based on the distance from the fixed support to the subsegment \(\Delta\mathrm{x_{p}}\), the diameter, and consequently the radius, of the circular cross-section of each subsegment, can be determined using Equation (3). In each subsegment cross-section, the discrete value of the curvature should be determined. The adopted hypothesis that plane sections remain planes implies linear distribution of strains along the y axis of the cross-section in each subsegment cross-section.