## 5 Calculation Method of Shear Capacity Based on Truss-Arch Model
The overall compression exerted on the compression bar is  
\[C=\sigma_{c}bz\cos\varphi \tag{3}\]  
The vertical projection of the truss compression \(C\) constitutes the shear force borne by the truss compression bar. \[V_{c}=C\sin\varphi=\sigma_{c}bz\cos\varphi\sin\varphi \tag{4}\]  
(3) Shear capacity provided by arch  
In an ideal scenario, when \(C=T\), as depicted in Figure 18, the truss compression bar and the truss tie rod concurrently attain their maximum strength and fail simultaneously. Nevertheless, in actual circumstances, the stirrups are often provided in excessive amounts. In this situation, the stirrups fail to yield, and the concrete is crushed before attaining its compressive strength. In another case, when the stirrups yield, the concrete has not yet reached its compressive strength, and the failure is ductile with a certain degree of post-yielding strain and a shear force acting in the form of an arch. At this point, the stress in the RC beam is as follows:  
Figure 14:0  
\(\sigma_{c}\) can be expressed as follows:  
Figure 14:1  
Figure 14:2 Truss model for RC beams under stress. Figure 14:3 The arch model for RC beam. The excess strength of an RC beam, \(\sigma_{a}\), can be represented as follows:  
Figure 14:4  
where \(\
u\) is the softening coefficient of concrete, which will be discussed later; \(f_{c,hy}\) is the compressive strength of epoxy mortar. Substituting Equation (6) into Equation (7) provides the following:  
Figure 14:5  
At this moment, the shear force by the arch is as follows:  
Figure 14:6  
Based on the geometric relations depicted in Figure 18, the following can be derived:  
Figure 14:7  
where L is the span of the specimen, providing the following:  
Figure 14:8  
Figure 14:9  
As can be observed from the figure, the horizontal projected length of the cracking zone is \(zcot\varphi\), and the height of the compression zone of the RC beam within the height of the compression zone is \(x_{n}=h\)sin\(\varphi\)cot\(\varphi\), which provides the following:  
Figure 15.0  
(4) Shear capacity calculation  
Substituting Equation (6) into Equation (3) and incorporating the concrete softening coefficient \(\
u\), the concrete of the RC beam and the epoxy mortar are calculated separately, providing the following:  
Figure 15.1  
This truss-arch model comprehensively considers the compressive effect of concrete, the tensile effect of reinforcement, and the tensile and compressive effects of epoxy mortar in the components. Therefore, the shear capacity of the specimens is the sum of the shear capacities of the truss model and the arch model, which is expressed by the formula as follows: \(V_{u}=V_{tu}+V_{c}+V_{a}\). By substituting Equations (2), (13), and (14), the calculation formula for the shear capacity of an RC beam reinforced with high-strength and high-toughness epoxy mortar based on the truss-arch model can be obtained:  
\[ \begin{array}{l}V_{u}=\left(\rho_{s}f_{yv}b+f_{t}b_{m}\right)z\cot\varphi+ \left(\
u\rho_{s}f_{yv}b+\frac{f_{t}b_{m}^{2}}{b}\right)z\cos\varphi+\\ \sin\theta\sin\varphi\cos\varphi h\Big{[}\left(\
u f_{c}b+f_{c,hy}b_{m}\right) -\cos(\varphi-\theta)\left(\frac{\rho_{s}f_{yv}b}{\sin\varphi}+\frac{f_{t}b_{ m}^{2}}{\sin\varphi b}\right)\Big{]}\end{array}  \tag{15}\]The experiments indicate that, in the failure process of the strengthened beam, the original concrete inside the strengthened beam is damaged prior to the reinforcement layer, and this will cause the epoxy mortar of the reinforcement layer to peel from the original beam and be unable to bear the force in coordination with the original beam. Therefore, the shear capacity of the reinforced beam cannot increase linearly with the increase in the thickness of the reinforcement layer.