# Mechanical Properties of the Functionally Graded Lining for a Deep Buried Subway Tunnel
## 2 Analytical Solutions for the Mechanical Properties of the Functionally Graded Lining
### Single Factor Test on the Lining Model of the Functionally Graded Lining  
The dimensionless parameter \(K\) is employed to reflect the displacement at the lining axis, and the dimensionless parameter \(P\) is employed to reflect the section moment. The definitions of \(K\) and \(P\) are given by  
\[K=\omega/\frac{p_{0}R_{\mathrm{H}}^{4}}{E_{0}I}. \tag{13}\]  
\[P=\frac{M_{\alpha}}{p_{0}R_{\mathrm{H}}^{2}} \tag{14}\]  
The single factor tests are carried out to study the influence of the lateral pressure coefficient \(\lambda\), the elastic modulus parameters \(a\) and \(b\) on \(K\), and the typical parameters are given by \(a=1\), \(b=1\) and \(\lambda=0.5\). In the single factor test, one of the factors is changed in turn, and the other factors are fixed according to the typical parameters. The variation range of each factor is summarized in Table 1.  
Under the condition that the other parameters remain unchanged, the influence of \(a\) on the radial displacement and section moment in the range of \(0\sim 90^{\circ}\) at the axis of functionally graded lining is studied. In addition, the influence of \(a\) on the typical points of \(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), \(75^{\circ}\) and \(90^{\circ}\) at the axis of functionally graded lining is studied, and the results are shown in Figures 4 and 5. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Factors** & **Variation Range** & **Step** \\ \hline \(a\) & 0.7–1.3 & 0.1 \\ \(b\) & 0.7–1.3 & 0.1 \\ \(\lambda\) & 0.5–1.5 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factor levels of the single factor test. As can be observed from Figure 4a, with the increase in \(a\), the radial displacement at the lining axis decreases, but the deformation trend of the lining is consistent with the maximum displacement at \(0^{\circ}\) and \(90^{\circ}\) and the minimum displacement near \(45^{\circ}\). In addition, from Figure 4b, with the increase in \(a\), the slope of the radial displacement curve of each typical point decreases gradually. It shows that the increase in \(a\) can enhance the stiffness of the lining, but the reduction in deformation becomes smaller and smaller. From Figure 5a, with the increase in \(a\), the distribution trend of section moment is consistent with the maximum moment at \(0^{\circ}\) and \(90^{\circ}\) and the minimum moment near \(45^{\circ}\). In addition, from Figure 5b, with the increase in \(a\), the negative section moment at typical points decreases linearly, and the positive section moment increases linearly. The maximum section moment transfers from the section of \(90^{\circ}\) to the section of \(0^{\circ}\), and the maximum section moment decreases. It can be observed from the distribution of the displacement and internal force that the optimal mode of the elastic modulus is to configure the larger elastic modulus at \(0^{\circ}\) and \(90^{\circ}\) to improve the rigidity, and the smaller elastic modulus at \(45^{\circ}\) to achieve the full utilization  
Figure 4: Radial displacement in the single factor test of parameter \(a\). Figure 5: Section moment in the single factor test of parameter \(a\). of the material. The optimized elastic modulus function is given by Equation (15), which is defined as Function II. \[E(\alpha)=E_{0}\left[\frac{a-b}{\left(\pi/4\right)^{\mathrm{n}}}\left|\alpha- \frac{\pi}{4}\right|^{\mathrm{n}}+b\right] \tag{15}\]  
Under the condition that the other parameters remain unchanged, the influence of \(b\) on the radial displacement and section moment in the range of 0-90\({}^{\circ}\) at the axis of functionally graded lining is studied. In addition, the influence of the typical points of 0\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 75\({}^{\circ}\) and 90\({}^{\circ}\) on the axis of functionally graded lining is studied, and the results are shown in Figures 6 and 7.