# 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
As can be observed from Figure 6a, with the increase in \(b\), 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 6b, with the increase in \(b\), 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. Comparing Figure 6 with Figure 4, it can be found that the effect of changing \(a\) or \(b\) alone on the deformation characteristics of functionally gradient lining is the same when other parameters remain unchanged. Figure 6: Radial displacement in the single factor test of parameter \(b\). Figure 7: Section moment in the single factor test of parameter \(b\). From Figure 7a, with the increase in \(b\), 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 7b, with the increase in \(b\), the negative section moment at typical points increases linearly, and the positive section moment decreases linearly. The maximum section moment transfers from the section of \(0^{\circ}\) to the section of \(90^{\circ}\), and the maximum section moment decreases. Comparing Figure 7b with Figure 5b, it can be found that with the increase in the section stiffness, the section moment also increases. Under the condition that the other parameters remain unchanged, the influence of \(\lambda\) on the radial displacement and section moment in the range of 0-\(90^{\circ}\) on 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 \(90^{\circ}\) at the axis of functionally graded lining is studied, and the results are shown in Figures 8 and 9. From Figures 8a and 9a, the radial displacement of the lining axis and the section moment are 0 when \(\lambda=1\), which means that when the upper load is equal to the lateral load, the lining is in the safest state. When \(\lambda=0.5\)-\(1.5\), the curves of the radial displacement and the section moment are symmetric with the curve of \(\lambda=1\). Figure 8: Radial displacement in the single factor test of parameter \(\lambda\). Figure 9: Section moment in the single factor test of parameter \(\lambda\). From Figures 8b and 9b, it can be observed that the displacement of the lining axis and section moment changes linearly, and the deformation mode of the lining changes significantly with the increase in \(\lambda\). It turns into the deformation with large lateral pressure to the inside, and the side with small pressure to the outside, which conforms to the actual deformation characteristics.