## 5 Theoretical Model of Flexural Bearing Capacity of Pre-Cracked Beam
Each cross section subjected to a bending load will rotate at an angle of \(\varphi\), as shown in Figure 17. The curvature of the deflection curve can be obtained according to material mechanics, i.e.,  
\[\varphi=\frac{1}{\rho}=\frac{d^{2}y}{dx^{2}}=\frac{M}{B} \tag{13}\]  
The mid-span deflection of the RC beam can be expressed as  
\[f=\alpha\frac{Ml^{2}}{B} \tag{14}\]  
where \(\alpha\) is the influence coefficient, which is related to support conditions and load forms; \(l\) is the calculated span; \(f\) is the mid-span deflection; and \(M\) and \(B\) are mid-span bending moment and bending stiffness, respectively. Under a three-point bending load, the bending moment at the mid-span of a simply supported RC beam can be expressed as  
\[M=\frac{P_{u}l}{4} \tag{15}\]  
where \(P_{u}\) is the vertical concentrated load and \(l\) is the calculation span of the simply supported beam, \(l=1900\) mm. Figure 17: Schematic diagram of the assumed plane section. For RC flexural members, the flexural stiffness of cracked members can be expressed as  
\[B=\frac{B_{0}}{\left(\frac{M_{\mathrm{cr}}}{M_{\mathrm{s}}}\right)^{2}+\left[1- \left(\frac{M_{\mathrm{cr}}}{M_{\mathrm{s}}}\right)^{2}\right]\frac{B_{0}}{B_{ \mathrm{cr}}}} \tag{16}\]  
The short-term stiffness (\(B_{\mathrm{s}}\)) calculation formula of RC flexural members can be derived as  
\[B_{\mathrm{s}}=\frac{E_{\mathrm{s}}A_{\mathrm{s}}h_{0}^{2}}{1.15\psi+0.2+\frac {6\kappa_{\mathrm{cr}}\rho}{1+3.5\gamma_{f}}} \tag{17}\]  
The meanings and values of related parameters in Equations (16) and (17) can be found in Specification JTG 3362-2018 [41]. Figure 18 shows the schematic diagram of the effective height of the pre-cracked T-beam section. The effective height of the T-beam section considering prefabricated cracks can be expressed as [42]  
\[h_{0}=h-c-\eta\cdot h_{c} \tag{18}\]  
where \(h\) is the height of the cross section; \(c\) is the thickness of the concrete protection layer, which is 41 mm in this study; \(h_{\mathrm{c}}\) is the height of the prefabricated crack; and \(\eta\) is the impact factor of the prefabricated crack. The value of \(\eta\) is taken as 1 when the prefabricated crack is located in the mid-span and the height does not exceed the thickness of the protective layer. The value of \(\eta\) is taken as 1.15 when the height of the prefabricated crack exceeds the thickness of the protective layer. At this time, the thickness of the protective layer is ignored. When the prefabricated crack occurs at the quarter position, \(\eta\) is taken as 0.5 in this study. The measured strains of concrete and steel, and mid-span deflections of RC beams are used to calculate the ultimate load under the bending load of the pre-cracked test beam. Figure 19 shows the results of calculated values and experimental observations, where the \(x\) axis is the experimental values and the \(y\) axis is the theoretical predictions. In addition, 95% confidence intervals (CI) are also provided. It can be seen from Figure 19 that the calculated values all fall within the confidence intervals. The mean of the ratio of the theoretical calculations and measured results is 1.008, and the standard deviation is 0.031. The theoretical calculations are in good agreement with the experimental results, which verifies the validity and accuracy of the proposed calculation model for the flexural capacity of the pre-cracked test beam. Figure 18: Schematic diagram of the cross section of pre-cracked beam.