## 2 Solution for the Elastoplastic Bending of the Cantilever Beam
### Plastic Regime  
In order to analyze the force-displacement relationship of the cantilever beam in the plastic regime of the material behavior (F \(>\) F\({}_{y}\)), it is necessary to define the value of the bending moment in the cross-section in terms of the size of the plastified part of the cross-section. The stress-strain state of the cantilever beam in the plastic regime of the material behavior is presented in Figure 2. In the stress-strain analysis of the plastified part of the cantilever beam, the hypothesis that plane sections remain a plane is kept, which implies linear distribution of strains along the y axis of the cross-section (Figure 2) [28; 29]. When the intensity of the external force F increases, the plastified zone expands from the most distant fibers toward the neutral axis of the cross-section (along y axis). The plastified part of the cross-section is symmetrical regarding the z axis due to the cross-section symmetry. Besides that, the plastified region propagates along the longitudinal axis of the cantilever beam (x axis). The boundary between the plastic and the elastic region along the cantilever beam axis (x\({}_{p}\)) can be determined from the condition of strain in the most distant fibers of the cross-section, whose position is defined by x\({}_{p}\), is equal to the yield strain of the material (\(\varepsilon_{y}\)):  
\[\varepsilon=\varepsilon_{y},\ \varepsilon=\frac{\sigma}{\mathrm{E}},\ \varepsilon_{y}=\frac{ \mathrm{f}_{y}}{\mathrm{E}}, \tag{23}\]  
\[\frac{\sigma}{\mathrm{E}}=\frac{\mathrm{F}\cdot\left(\mathrm{h}-\mathrm{x}_{p }\right)}{\mathrm{E}\cdot\frac{\mathrm{d}\left(\mathrm{x}_{p}\right)}{64} \cdot\frac{\mathrm{d}\left(\mathrm{x}_{p}\right)}{2}}=\frac{32\cdot\mathrm{F} \cdot\left(\mathrm{h}-\mathrm{x}_{p}\right)}{\mathrm{E}\cdot\mathrm{d}\left( \mathrm{x}_{p}\right)^{3}}, \tag{24}\]  
\[\frac{32\cdot\mathrm{F}\cdot\left(\mathrm{h}-\mathrm{x}_{p}\right)}{\mathrm{E} \cdot\left[\mathrm{d}_{\mathrm{f}}+\left(\mathrm{d}_{\mathrm{c}}-\mathrm{d}_{ \mathrm{f}}\right)\cdot\frac{\mathrm{h}-\mathrm{x}_{p}}{\mathrm{h}}\right]^{3 }}=\frac{\mathrm{f}_{y}}{\mathrm{E}}. \tag{25}\]  
Equation (25) is of the third degree according to the unknown x\({}_{p}\). In cases when the cross-sections diameter ratio is lower than or equal to 1.5, Equation (25) has one real solution used in further analysis (Figure 2). In cases when cross-sections diameter ratio is greater than 1.50, there are two real solutions, where one defines the plastified region along the cantilever beam axis from the critical cross-section toward the fixed support cross-section (x\({}_{p}\)), and the second one defines the plastified region along the cantilever beam axis toward the free end (x\({}_{p}\)) (Figure 2). At a certain force intensity F, the plastified zone expands until it reaches the clamped cross-section, so with a further increasing of the force intensity, Equation (25) has one real solution which defines the plastified zone along the cantilever beam axis, which is analogous to the case when the cross-sections diameter ratio is lower than or equal to 1.50. After plastification of the part of the cross-section, the bending moment can be defined via stress-strain analysis of the cross-section at the arbitrary distance x from the fixed support (Figure 2). The part of the cross-section from the neutral axis (z axis) that is in the elastic regime of the material behavior is defined by the distance \(\eta(\mathrm{x})\), while the shaded part of the cross-section in Figure 2 represents the plastified part. In the elastic part of the cross-section, normal stresses are proportional to the strains, and the coefficient of proportionality is the modulus of elasticity (E). According to the adopted material model, in the plastified part of the cross-section, the normal stresses are proportional to the strains, while the coefficient of proportionality is the tangent modulus (E\({}_{i}\)).