## 4 Numerical Simulation for CFRP-Strengthened Corroded RC Beams
### Fatigue Simulation Results and Correctness Verification  
Using the method described above, the test beams in Table 1 were modeled and numerically analyzed. Due to the limited experimental data provided in the literature [23], a comparison of fatigue loading number to median deflection curves of the strengthened beams was chosen for verification, and the comparison results are shown in Figure 7. The comparison of the simulated values of the mid-span deflection of the strengthened beam with the specific values of the experimental results is given in Table 3.  
Figure 4: Concentration force loading setting.  
Figure 5: Finite element model diagram.  
Figure 6: Mesh of the FE model.  
Figure 7: Comparison of fatigue loading times and mid-span deflection curves of strengthened beams.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Beam ID** & \begin{tabular}{c} **Fatigue Cycles** \\ **\(N\)(10\({}^{\text{th}}\))** \\ \end{tabular} & \begin{tabular}{c} **Experimental** \\ **Result\(f_{e}\) (mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Simulation** \\ **Result\(f_{s}\) (mm)** \\ \end{tabular} &
\begin{tabular}{c} **[\(f_{e}-f_{s}\)/\(f_{e}\)] (%)** \\ \end{tabular} \\ \hline \multirow{4}{*}{Beam No. 8} & 0 & 0.93 & 1.02 & \(-\)9.68 \\  & 5 & 1.11 & 1.18 & \(-\)6.31 \\  & 25 & 1.19 & 1.29 & \(-\)8.40 \\  & 50 & 1.21 & 1.33 & \(-\)9.92 \\  & 110 & 1.34 & 1.44 & \(-\)7.46 \\  & 134 & 6.66 & 6.17 & 7.36 \\ \hline \multirow{4}{*}{Beam No. 10} & 0 & 1.00 & 1.15 & \(-\)15.00 \\  & 1 & 1.16 & 1.26 & \(-\)8.62 \\ \cline{1-1}  & 20 & 1.42 & 1.54 & \(-\)8.45 \\ \cline{1-1}  & 50 & 1.50 & 1.67 & \(-\)11.33 \\ \cline{1-1}  & 60 & 4.22 & 3.59 & 14.93 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison between mid-span deflection simulation results and experimental results.  
As shown in Figure 7, the mid-span deflection of the strengthened beam increases rapidly at the beginning of fatigue loading. With the increase in the number of fatigue loading cycles, the mid-span deflection enters a steady development stage and the increase in mid-span deflection slows down. When the strengthened beam approaches fatigue damage, the mid-span deflection rises steeply and quickly reaches fatigue brittle damage. The development trends of both sets of comparison curves are basically consistent, indicating that the simulation results of both beams are in good agreement with the experiment results.  
As shown in Table 3, there are certain errors between the simulated and tested values of the mid-span deflection. Possible sources of error include the fact that the fatigue constitutive models for each material are based on certain assumptions and may differ from the actual stress states and fatigue performance of each material in the experiment, and that there may be differences in the boundary conditions of the strengthened beam between the experiment and the simulation.  
On the whole, the simulation results of the strengthened beams are basically consistent with the variation trend of the experimental results, and the error range is not more than 15%, which confirms the feasibility and accuracy of the finite element model establishment method used in the simulation, and further follow-up research can be carried out.