# Experimental and Theoretical Study on Anchorage Loss of Prestressed CFRP-Reinforced Concrete Beams
## 2 FEM Calculation and Theoretical Analysis
### Theoretical Calculation Analysis  
The FEM results indicate that significant deformation primarily occurs at the front end of the steel reaction frame and the connecting bolt. To address the deformation of these components, a TCS is proposed. The deformation at the front end of the steel reaction frame primarily arises from the interaction between this section and the fixed nut, simplifying the analysis to the deformation of the contact area. As depicted in Figure 10, this contact area consists of the vertical plate welded to the side plate of the steel reaction frame. Recognizing that the side plate also experiences deformation, we model the support at both ends as simply supported. Consequently, calculating the deformation of the steel reaction frame simplifies to determining the deflection of a simply supported beam under a concentrated force at its midspan. There are two simplifications to consider in the calculation. The formula for calculating the deformation \(\omega_{1}\) of the front end of the steel reaction frame is as follows:  
\[\omega_{1}=\frac{\text{Fx}^{3}}{48\text{E}_{1}\text{I}}, \tag{1}\]  
where F is the concentrated force, x is the distance between the two simply supported positions, E1 is the modulus of elasticity of Q235, and I is the moment of inertia of the cross-section. The deformation of the connecting screw can be conceptualized as the deformation of a cylinder experiencing axial tensile force, as depicted in Figure 11. Given that the stress primarily accumulates at the three turns of threads in the contact portion of the connecting screw and the resulting deformation is relatively minor, we simplify the analysis by treating the connecting screws at these locations as part of the middle section for uniform deformation calculation. According to the stress-strain relationship, the formula for calculating the deformation \(\omega_{2}\) of the connecting screw under the action of prestress can be obtained as follows:  
\[\omega_{2}=\frac{4\text{F}\big{(}\text{y}_{1}+\text{y}_{2}+\text{y}_{3}\big{)} }{\pi\text{d}_{1}{}^{2}\text{E}_{2}}, \tag{2}\]  
where y1 represents the length of the middle section of the connecting screw, y2 denotes the length of the stressed section of the connecting screw situated at the fixing, y3 denotes the length of the stressed section of the connecting screw positioned at the rear part of the  
Figure 10: Dimension diagram of vertical plate at front end of steel reaction frame. anchor, d\({}_{1}\) represents the effective diameter of the cross-section of the connecting screw, and E\({}_{2}\) represents the modulus of elasticity of steel (Grade 8.8). By combining Equations (1) and (2), we can simplify them to obtain the TCS of the total deformation (a) at the tensioning end. From this, we can derive the formula for calculating the anchorage loss, as shown in the specific equation below:  
\[\text{a}=\omega_{1}+\omega_{2}, \tag{3}\]  
\[\sigma_{1}=\frac{\text{a}\times\text{E}_{\text{effp}}}{\text{L}}, \tag{4}\]  
where \(\omega_{1}\) is the deformation of the steel reaction frame under the action of prestress, \(\omega_{2}\) represents the deformation of the connecting screw under the action of prestress, L denotes the length of the CFRP tendon, and E\({}_{\text{effp}}\) is the modulus of elasticity of the CFRP tendon. By substituting the FEM data into the TCS, with F set to 107.4 kN, x set to 79 mm, E\({}_{1}\) set to 210,000 MPa, I set to 19, 660.8 mm\({}^{4}\), \(y_{1}\) set to 139 mm, \(y_{2}\) set to 7.5 mm, d\({}_{1}\) set to 16.6 mm, and E\({}_{1}\) set to 210,000 MPa, the results are as follows:  
\[\omega_{1}=0.267192\text{ mm }\omega_{2}=0.363915\text{ mm }\text{a}=0.631107\text{ mm }\sigma_{1}=8.70\text{ MPa}.\]  
Figure 12 illustrates a comparison between the results of the FEM and the TCS. It is evident that the deformation at the front end of the steel reaction frame differs by only 1.34%, while the deformation of the connecting screw differs by 0.61%. This comparison highlights the applicability of the theoretical formulas discussed previously.