# 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
According to significant variations in the bending moment values at different locations, the maximum-to-minimum ratio can reach up to fivefold. To facilitate an intuitive statistical comparison, the natural logarithm of the bending moment values was uniformly summarized. By comparing the effects of voids in different positions in tunnel lining structures, it becomes evident that the safety factor is lower at the center of a void. Notably, the bending moment distribution graph we created reveals tensile forces acting at this location. For concrete structures, the tensile capacity is typically one-tenth of the compressive capacity. Consequently, when voids occur behind the lining, the safety factor significantly decreases for this structural position. The most pronounced reduction occurs when the void is located at the arch crown, as the stress concentration combines with gravitational effects. This position lies at the tunnel's highest point and along its central axis, preventing the transfer of pressure to other void-free areas. Figure 6 shows the displacement (U2) contours of a tunnel lining under deformation, both without FRP reinforcement and with three different types of FRP reinforcement: CR4, CR6, and CR8. The color scale represents the magnitude of displacement. In contrast to bending moments, the axial force variations do not exhibit specific characteristics related to void positions. Instead, they uniformly increase after the development of a void. Since safety factors are calculated based on both bending moments and axial forces, only axial force changes for void-free and arch crown void scenarios are presented. The most significant improvement was observed with the CR8 reinforcement. The displacement across the tunnel lining was more uniform, with only a small amount of orange and yellow at the bottom. The maximum displacement was reduced to +3.323 \(\times\) 10\({}^{-3}\), representing the strongest performance among the reinforcement types. The bottom of the tunnel lining experienced the greatest displacement in all cases, but this was significantly mitigated with FRP reinforcement. CR8 provided the most effective reduction in displacement. Figure 7 shows the distribution of internal forces in the tunnel lining under different void ranges, where the void range gradually expands from 30 degrees to 60 degrees. Based  
Figure 6: The deformation analysis of the lining structure. on actual tunnel conditions, voids larger than 60 degrees are rarely encountered. During the expansion of a void, the change in bending moment is particularly noticeable. The bending moment at the top of the tunnel transitions from compressive to tensile, and the rate of increase in the tensile bending moment is higher than the rate of increase in the void range. Since the bending moment values have different directions, and there is a large difference between the maximum and minimum values, the bending moment values shown in this diagram are the natural logarithms of the values. The range of \(\ln(-190\) to \(190)\) is from \(-6\) to \(6\). Additionally, as the void range increases, the area affected by the tensile bending moment also increases. Given that the tensile strength of concrete is much lower than its compressive strength, assessing the safety factor becomes particularly important. The safety factor analysis could be achieved through an analysis of Figure 7c, showing that the safety factor at the center of the void decreased most significantly, and the rate of decrease could be calculated. The rates of decrease or increase in the safety factor for every one-degree expansion of the void are as follows:  
* In the range of 0-30 degrees, the rate of decrease in the tensile bending moment is 16.67%;
* In the range of 30-45 degrees, the rate of increase is 20%;
* In the range of 45-60 degrees, the rate of decrease is 56.67%. The axial force change presents a linear trend.