# A Simulation Study of FRP-PCM Reinforcement for Tunnel Liniings with Void Defects
## 2 Research Methods and Evaluation Criteria
### Safety Factors for Evaluating the Stable State of Lining  
The basic model based on the actual situation of the void behind the lining is presented in Figure 3. In this study, the influence of the typical patterns of a void behind a tunnel's lining on tunnel deformation was investigated in detail. According to the JTG D70-2004 Design Specification for Highway Tunnels [31], the ultimate strength of the lining material was used to determine the ultimate bearing capacity \(N_{u}\) of the tunnel structure under eccentric compression. The safety factor, \(K\), of the calculated section was obtained by comparing it with the axial force, \(N\). One can determine whether the safety requirements are met by using Formula (1). \[K=\frac{N_{u}}{N}\geq\ K_{s} \tag{1}\]  
In the formula above, \(K\) is the safety factor, \(N_{u}\) is the ultimate bearing capacity of the tunnel structure under eccentric compression, \(N\) is the axial force of the tunnel lining under compression, and \(K_{s}\) is the limit value of the safety factor specified in the specifications. When the axial eccentricity \(e_{0}\leq 0.20\) h, the ultimate bearing capacity of the compressive strength of the lining structure under eccentric compression of a concrete rectangular section should be calculated according to Equation (2):  
\[N_{u}=\varphi\alpha R_{a}bh \tag{2}\]  
In the formula above, \(R_{a}\) is the compressive ultimate strength of concrete (MPa); \(b\) is the width of the lining section (m), taken as 1 m; \(h\) is the thickness of the lining section (m), taken according to the actual thickness; \(\varphi\) is the longitudinal bending coefficient of the component, taken as 1 for the tunnel lining; and \(\alpha\) is the eccentricity influence coefficient of axial force. The compressive strength safety factor of the lining structure section was calculated according to Formula (3):  
\[K=\frac{N_{u}}{N}=\frac{\varphi\alpha R_{a}bh}{N} \tag{3}\]  
In this formula, N represents the axial force (kN). \begin{table}
\begin{tabular}{c c c c c} \hline  & & **Young’s Modulus** & **Compressive Strength** & **Tensile Strength** & **Design Thickness** \\  & **(MPa)** & **(MPa)** & **(MPa)** & **(mm)** \\ \hline \multirow{3}{*}{FRP Grid} & CR4 & & & 2.0 \\  & CR6 & 100,000 & - & 1400 & 4.0 \\  & CR8 & & & 5.0 \\ \hline PCM & & 26,000 & 59.3 & 4.6 & - \\ \hline \end{tabular}
\end{table} 
Table 4: Mechanical parameters of the FRP grid reinforcement material. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Volume Weight of Soil** & **Elastic Modulus** & **Poisson’s Ratio** & **Cohesive Force** & **Internal Friction Angle** \\ \(\gamma\)**/(kN/m\({}^{3}\))** & **E/GPa** & \(\mu\) & **c/kPa** & \(\varphi\)/(c\({}^{*}\)) \\ \hline
20 & 2.4 & 0.32 & 200 & 30 \\ \hline \end{tabular}
\end{table} 
Table 2: Mechanical parameters of the surrounding rock. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Volume Weight** & **Elastic Modulus** & **Poisson’s Ratio** & **Cohesive Force** & **Internal Friction Angle** \\ \(\gamma\)**/(kN/m\({}^{3}\))** & **E/GPa** & \(\mu\) & **c/kPa** & \(\varphi\)/(c\({}^{*}\)) \\ \hline
20 & 2.4 & 0.32 & 200 & 30 \\ \hline \end{tabular}
\end{table} 
Table 3: Mechanical parameters of the tunnel lining. When \(e_{0}>0.20\) h, the lining section must be determined based on tensile strength verification. According to the crack resistance requirements, the ultimate bearing capacity of the tensile strength of the lining structure under eccentric compression of a rectangular concrete section should be calculated according to Equation (4). \[N_{u}=\frac{1.75R_{t}bh\varphi}{\frac{6e_{0}}{h-1}} \tag{4}\]  
In the formula above, \(R_{t}\) is the ultimate tensile strength of concrete (MPa). The safety factor for the tensile strength of the lining structure section was calculated according to Formula (5):  
\[K=\frac{N_{u}}{N}=\frac{1.75R_{t}bh\varphi}{N\left(\frac{6e_{0}}{h-1}\right)} \tag{5}\]  
The reinforcement mechanism and internal structure of the FRP grid are shown in Figure 5[22].