## 5 Theoretical Model of Flexural Bearing Capacity of Pre-Cracked Beam  
The experimental results showed that the failure mode in the limit state was cracking in the tensile zone of the concrete. Then, the tensile steel bar reached the yield strength. Finally, the concrete in the compression zone was crushed. At this stage, the concrete strain (\(\varepsilon_{\rm c}\)) in the compression zone was greater than the peak strain (\(\varepsilon_{0}\)) in the elastic stage, which was the plastic stage. According to the design code GB 50010-2010 [40], the stresses of concrete in the compression zone and the steel bar in the tension zone reached the design strength at the ultimate stage. The concrete compression zone (\(\varepsilon>\varepsilon_{0}\)) was equivalent to a rectangular distribution, and the stress of the concrete compression zone (\(\varepsilon\leq\varepsilon_{0}\)) was a curved distribution. The geometric relationship can be expressed as  
\[\frac{\varepsilon_{c}}{\xi_{n}h_{0}}=\frac{\varepsilon_{tu}}{h_{tu}}=\frac{ \varepsilon}{y}=\frac{\varepsilon_{s}}{h_{0}(1-\xi_{n})} \tag{5}\]  
where \(\varepsilon_{\rm c}\) is concrete strain; \(\xi_{n}\) is the height of the relative boundary compression zone; \(h_{0}\) is the height of the effective section; \(\varepsilon_{\rm tu}\) is the section strain in the tension zone; \(h_{\rm tu}\) is the height of the current section from the neutral axis; \(\varepsilon_{\rm s}\) is rebar strain. The meaning of the parameters is shown in Figure 16.  
The equilibrium equation can be expressed as  
\[\left\{ \begin{array}{l}\sigma_{s}A_{s}=\int_{0}^{\xi_{n}h_{0}}\sigma_{c}b(y )dy\\ \sigma_{s}A_{s}=\int_{0}^{y_{0}}\sigma_{0}(\frac{2\phi y}{\varepsilon_{0}}- \frac{\sigma^{2}y^{2}}{\varepsilon_{0}^{2}})bdy+\int_{y_{0}}^{\xi_{n}h_{0}} \sigma_{0}b(y)dy\end{array} \right. \tag{6}\]  
where \(\sigma_{0}\) is the peak stress in the elastic stage of concrete; \(\phi\) is geometric parameters; \(b\) is the width of the cross section. The height of the neutral axis can be obtained according to the limit condition of concrete compressive strain (\(\varepsilon_{\rm c}=0.0033\)), i.e.,  
\[\xi_{n}h_{0}=\frac{f_{y}\rho h}{\sigma_{0}(1-\frac{\varepsilon_{0}}{ \varepsilon_{\rm c}})} \tag{7}\]  
where \(f_{y}\) is the tensile design strength of reinforcement; \(\rho\) is the reinforcement ratio; \(h\) is the height of the beam. Assuming \(\varepsilon_{\rm u}=0.0001\), the crack growth height can be expressed as  
\[h_{cr}=h-0.95_{n}^{x}h_{0} \tag{8}\]  
Figure 16: Stress of concrete in plastic stage. Generally, the tensile steel bar also reached the ultimate tensile strength when the concrete in the compression zone entered plastic failure, i.e., \(\varepsilon_{c}=0.0033\) and \(\varepsilon_{s}=0.01\). The geometric relationship can be written as  
\[\frac{\varepsilon_{c}}{\xi_{n}h_{0}}=\frac{\varepsilon_{s}}{h_{0}(1-\xi_{n})} \tag{9}\]  
Then, \(\xi_{n}=0.248\) can be obtained. The maximum height of the crack can be expressed as  
\[h_{cr}=h-0.255h_{0} \tag{10}\]  
At this time, the load carried by the RC beam was the ultimate bending moment. To simplify the calculation, the stress curve of the concrete -plastic zone was simplified as a triangle shape. The geometric relationship can be derived as  
\[\left\{ \begin{array}{l}y_{0}=\frac{\varepsilon_{0}}{\varepsilon_{c}}\xi_{n }h_{0}=\frac{0.002}{0.0035}\xi_{n}h_{0}=0.57\xi_{n}h_{0}\\ M_{u}=(h_{0}-\xi_{n}h_{0}+\frac{2}{3}y_{0})\sigma_{s}A_{s}+(\frac{y_{0}h_{0}-y _{0}}{2}+\frac{y_{0}}{3})\sigma_{c}b(\xi_{n}h_{0}-y_{0})\end{array} \right. \tag{11}\]  
where \(M_{u}\) is the ultimate bending moment. Combining Equations (5)-(11), the ultimate bending moment can be written as  
\[M_{u}=(h_{0}-0.62\xi_{n}h_{0})\sigma_{s}A_{s}+0.18\sigma_{c}b\xi_{n}^{2}x_{0}^ {2} \tag{12}\]  
Generally, RC bending members work with cracks, in practical service.