## 4 Calculation of the Bending Bearing Capacity
### Calculation Formula of Bearing Capacity
The distance \(y_{\rm c}\) between the application point of the compressive stress force \(F_{\rm c}\) and the edge of the compressive zone of concrete is as follows:  
\[y_{c}=x_{c}\cdot\frac{1-0.25\beta_{c}}{3-\beta_{c}} \tag{6}\]  
The deformation coordination condition is as follows:  
\[\frac{\varepsilon_{c}}{\varepsilon_{EM}^{u}}=\frac{x_{c}}{h+t_{EM}-x_{c}} \tag{7}\]  
\[\varepsilon_{c}=\frac{x_{c}}{h+t_{EM}-x_{c}}\cdot\varepsilon_{EM}^{u} \tag{8}\]  
\[\beta_{c}=\frac{\varepsilon_{c}}{\varepsilon_{\rm c0}}=\frac{\varepsilon_{c }\varepsilon_{EM}^{u}}{\varepsilon_{EM}^{u}\varepsilon_{\rm c0}}=\frac{x_{c }}{h+t_{EM}-x_{c}}\cdot\frac{\varepsilon_{EM}^{u}}{\varepsilon_{\rm c0}} \tag{9}\]  
\[\frac{\varepsilon_{s}}{\varepsilon_{EM}^{u}}=\frac{h_{0}-x_{c}}{h+t_{EM}-x_{ c}} \tag{10}\]  
\[\varepsilon_{s}=\frac{h_{0}-x_{c}}{h+t_{EM}-x_{c}}\cdot\varepsilon_{EM}^{u} \tag{11}\]  
The equilibrium equation in the direction of the horizontal force is as follows:  
\[f_{c}bx_{c}\beta_{c}(1-\frac{\beta_{c}}{3})=\left[f_{y}+E_{2}(\varepsilon_{\rm s }-\varepsilon_{y})\right]\cdot A_{\rm s}+E_{EM}\varepsilon_{EM}^{u}A_{EM} \tag{12}\]  
where \(x_{\rm c}\) is found by Formulae (7), (10), and (12), and then we solve for \(F_{\rm c}\), \(y_{\rm c}\). The moment at the center of the reinforcement layer is as follows:  
\[ \begin{array}{rl}M_{du}&=f_{c}bx_{c}\beta_{c}(1-\frac{\beta_{c}}{3})(h+ \frac{t_{EM}}{2}-y_{c})\\ &-\left[f_{y}+E_{2}(\varepsilon_{\rm s}-\varepsilon_{y})\right]\cdot A_{\rm s }(a_{\rm s}+\frac{t_{EM}}{2})\end{array}  \tag{13}\]  
The distance of the concrete force in the compression zone is as follows:  
\[ \begin{array}{rl}M_{du}&=\left[f_{y}+E_{2}(\varepsilon_{\rm s}-\varepsilon_ {y})\right]\cdot A_{\rm s}(h_{0}-y_{c})\\ &+E_{EM}\varepsilon_{EM}^{u}A_{EM}(h+\frac{t_{EM}}{2}-y_{c})\end{array}  \tag{14}\]  
(2) When \(\varepsilon_{\rm c0}<\varepsilon_{\rm c}<\varepsilon_{\rm cu}\), the resultant force of the concrete compressive stress is as follows:  
\[ \begin{array}{rl}F_{c}=\int_{0}^{y_{0}}f_{\rm c}[1-(1-\frac{\varepsilon_{c}} {\varepsilon_{\rm c0}}\cdot\frac{y}{x})^{2}]bdy\\ &+\int_{y_{0}}^{x_{c}}f_{\rm c}bdy=f_{\rm c}bx_{\rm c}(1-\frac{\varepsilon_{ \rm c0}}{3\varepsilon_{\rm c}})\end{array}  \tag{15}\]  
Let \(\beta_{c}=\frac{\varepsilon_{c}}{\varepsilon_{\rm c0}}\), then  
\[F_{c}=f_{c}bx_{\rm c}(1-\frac{1}{3\beta_{\rm c}}) \tag{16}\]  
The distance \(y_{\rm c}\) between the application point of the compressive stress force \(F_{\rm c}\) and the edge of the compressive zone of the concrete is as follows:  
\[y_{\rm c}=x_{c}\cdot\left(1-\frac{1.5\beta_{c}^{2}-0.25}{3\beta_{c}^{2}-\beta _{c}}\right) \tag{17}\]The equilibrium equation in the direction of the horizontal force is as follows:  
\[f_{c}bx_{\text{c}}(1-\frac{1}{3\beta_{\text{c}}})=\left[f_{\text{y}}+E_{2}( \varepsilon_{\text{s}}-\varepsilon_{\text{y}})\right]\cdot A_{\text{s}}+E_{EM} e_{EM}^{u}A_{EM} \tag{18}\]  
where \(x_{\text{c}}\) is found by Formulae (7), (10), and (18), and then we solve for \(F_{\text{c}}\), \(y_{\text{c}}\). The center of the reinforcement layer is as follows:  
\[ \begin{array}{ll}M_{du}&=f_{c}bx_{\text{c}}(1-\frac{1}{3\beta_{\text{c}}})( h+\frac{t_{EM}}{2}-y_{c})\\ &-\left[f_{\text{y}}+E_{2}(\varepsilon_{\text{s}}-\varepsilon_{\text{y}}) \right]\cdot A_{\text{s}}(a_{s}+\frac{t_{EM}}{2})\end{array}  \tag{19}\]  
Or the distance of the concrete force in the compression zone is as follows:  
\[ \begin{array}{ll}M_{du}&=\left[f_{\text{y}}+E_{2}(\varepsilon_{\text{s}}- \varepsilon_{\text{y}})\right]\cdot A_{\text{s}}(h_{0}-y_{c})\\ &+E_{EM}e_{EM}^{u}A_{EM}(h+\frac{t_{EM}}{2}-y_{c})\end{array}  \tag{20}\]  
When the above formula is applied, it can be calculated as 0 \(<\varepsilon_{\text{c}}<\varepsilon_{\text{c0}}\) first. If the obtained \(\beta c<1\) in the middle calculation, it is assumed that the calculation continues; otherwise, it is calculated as \(\varepsilon_{\text{c0}}<\varepsilon_{\text{c}}<\varepsilon_{\text{cu}}\).