# A Simulation Study of FRP-PCM Reinforcement for Tunnel Liniings with Void Defects
## 3 Results and Analysis
### The Influence of Void Location and FRP Reinforcement
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**Repair Rate of Safety Factor (\%)**} \\ \hline \multirow{2}{*}{**Position**} & **Void at Crown** & **FRP-PCM** & **FRP-PCM** & **FRP-PCM** \\ \cline{2-5}  & **(45\({}^{\circ}\))** & **(CR4)** & **(CR6)** & **(CR8)** \\ \hline Crown (0\({}^{\circ}\)) & 6.05 & 14.84\% & 27.90\% & 56.89\% \\ \hline Spandrel (45\({}^{\circ}\)) & 3.86 & 12.63\% & 25.59\% & 52.85\% \\ \hline Springline (90\({}^{\circ}\)) & 2.97 & 10.94\% & 23.11\% & 46.07\% \\ \hline Bottom of side wall (135\({}^{\circ}\)) & 4.06 & \(-\)0.95\% & \(-\)1.61\% & \(-\)2.33\% \\ \hline Invert (180\({}^{\circ}\)) & 12.35 & 2.62\% & 5.22\% & 10.30\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Repair rate of safety factor with FRP reinforcement. Figure 9: Bending moment and safety factor. Furthermore, beyond considering internal force changes due to voids at different positions, it is essential to investigate the impact of FRP-PCM reinforcement. Figure 10 compares the reinforcement effects of different types of FRP grids when the top void range is 45 degrees. As shown in Figure 9, the analysis was conducted by comparing the bending moment and safety factor. These two cases exhibit symmetry. Overall, axial forces do not produce isolated tensile or compressive variations due to the presence of a void; the magnitude changes uniformly. After each set of scenarios, reinforcement studies were conducted. Due to the extensive data, the internal force distribution graph includes CR4-grade FRP grids as post-reinforcement cases. Detailed reinforcement comparisons are available in tabular form, including analyses of CR6 and CR8 reinforcement effects. Overall, the pre- and post-reinforcement internal force distributions of the lining exhibit consistency, but specific numerical differences are evident. For evaluation purposes, the safety factor was employed directly. Comparing it with the cross-sectional repair rate revealed that larger FRP grid models enhance lining safety and repair effectiveness. Notably, the repair rate is highest at the centers of the voids, approaching 60%. Furthermore, comparing the safety factor for the void-free condition shows that the efficacy of CR8-type reinforcement surpasses that of the lining in an undamaged state. Additionally, the influence range of reinforcement decreases with increasing distance, aligning with general trends. When voids are of moderate or smaller size, partial reinforcement at specific positions is feasible, reducing economic costs. Finally, the safety factor at the position of the crown is exceptionally high, and due to its distance from voids and reinforcement centers, the variation is minimal. However, the crown's position falls within the load-bearing zone. Through the above two parameters, bending moment and axial force, the change in safety factor could be calculated. Regarding void location, a void located at the spandrel is more dangerous than a void at the crown, especially for the cross range between the crown and the void, and there are severe tension and stress concentration phenomena. According to the simulation results, the safety factor is lower than the standard limit value, and it is very easy for damage to occur. Finally, when the number of voids increases, the safety factor does not linearly  
Figure 10: The safety factor and section repair rate with FRP reinforcement. decrease but reaches a delicate balance. The influence of multiple voids can be further discussed in future studies.