## 5 Flexural Bearing Capacity  
### Calculation Method of Theoretical Value of Flexural Bearing Capacity of the Normal Section  
When calculating the flexural bearing capacity of BFRP bar concrete beams reinforced with bamboo fiber and steel wire mesh, the following assumptions need to be made:  
1. The section deformation complies with the assumption of a flat section;
2. There will be no relative sliding between the force bar, steel wire mesh, and concrete;
3. The strength of BFRP bars and concrete is determined according to the results of a uniaxial force test, and the elastic modulus of steel wire mesh: \(E_{sm}=1.8\times 10^{5}\) MPa;
4. The influence coefficient of the steel wire mesh on both sides of the beam is obtained according to the literature [19]. The sectional view of the component and related parameters is shown in Figure 16.  
Where \(h_{sm}\) is the distance from the centroid of the steel wire mesh on the bottom of the stirrup of the component to the edge of the concrete under compression, \(A_{s}\) is the cross-sectional area of longitudinal steel bars in the tension zone, \(A^{\prime}_{s}\) is the cross-sectional area of longitudinal steel bars in the compression zone, \(A_{sm}\) is the area of steel wire mesh of the stretched noodles, \(A_{sm1}\) is the area of steel wire mesh of the side face, and \(a_{s}\) is the vertical distance from the resultant point of all lower longitudinal tensile steel bars on the normal section to the tensile edge of the section. The stress-strain relationship of the component section is shown in Figure 17. In the figure, \(\xi h_{0}=0.8x\), \(T_{c}=f_{c}b\xi h_{0}\), \(T^{\prime}_{s}=f^{\prime}_{y}A^{\prime}_{s}\), \(T_{s}=f_{y}A_{s}\), \(T_{sm}=\varepsilon_{sm}E_{sm}A_{sm}\), \(\varepsilon_{cu}=0.0033\), where \(\varepsilon_{cu}\) is the ultimate compressive strain of concrete in the compression zone, \(\varepsilon_{s}\) is the strain of the longitudinal steel bar in the tension zone, \(\varepsilon_{s}^{\prime}\) is the strain of the longitudinal steel bar in the compression zone, \(T_{sm1}\) is the joint force of the steel wire mesh of the side face, \(\xi\) is the relative pressure zone height, \(x\) is the concrete compression zone height, \(T_{c}\) is the joint force of concrete in the compression zone, \(f_{c}\) is the design value of the concrete axial compressive strength, \(T^{\prime}_{s}\) is the joint force of the longitudinal steel bar in the compression zone, \(f^{\prime}_{y}\) is the design value of compressive strength of ordinary steel bars, \(T_{s}\) is the joint force of the longitudinal steel bar in the tension zone, \(f_{y}\) is the design value of the tensile strength of ordinary steel bars, and \(T_{sm1}\) is the joint force of the steel wire mesh of the stretched noodles. Figure 16: Component cross section. The geometric relationship by the section strain:  
\[\epsilon_{sm}=\frac{h_{sm}-x}{x}\epsilon_{cu} \tag{4}\]  
Balanced relationship by force:  
\[T_{c}+T_{s}^{\prime}=T_{sm}+T_{sm1}+T_{s} \tag{5}\]  
To facilitate the calculation of the formula, we can temporarily ignore the influence of the side steel wire mesh, and the force balance equation can be obtained as follows:  
\[T_{c}+T_{s}^{\prime}=T_{sm}+T_{s} \tag{6}\]  
According to the equivalent rectangular stress diagram:  
\[\xi h_{0}=0.8x \tag{7}\]  
\[T_{c}=f_{c}b5h_{0} \tag{8}\]  
\[T_{s}^{\prime}=f_{y}^{\prime}A_{s}^{\prime} \tag{9}\]  
\[T_{s}=f_{y}A_{s} \tag{10}\]  
Figure 17.0  
Substituting Formulas (4) and (7)-(11) into (6), we can get:  
Figure 17.1  
Solving Equation (12), we get:  
Figure 17.2  
where \(A=0.8f_{c}b\), \(B=-\left(f_{y}A_{s}-f_{y}^{\prime}A_{s}^{\prime}-\epsilon_{cu}E_{sm}A_{sm} \right)\), \(C=-\epsilon_{cu}E_{sm}A_{sm}h_{sm}\). Substituting Formula (13) into Formulas (4) and (7) to obtain \(\epsilon_{sm}\) and \(\xi h_{0}\), and from the torque balance relationship, the theoretical value calculation formula is as follows:  
Figure 17.3  
Figure 17.4 Cross-sectional stress and strain distributions of the component.