# Mechanical Properties of the Functionally Graded Lining for a Deep Buried Subway Tunnel
## 3 Variance analysis of influence factors
### Verification of the Numerical Simulation
From Figure 14, the radial displacement curves of FGM lining almost coincide under the conditions of \(a=1\), \(b=0.9\) and \(a=0.9\), \(b=1\) in the results of both the theoretical analysis and the numerical simulation. There is a certain gap in the results between the theoretical analysis and the numerical simulation, but the theoretical analysis and the numerical simulation results are close at the two places of \(0^{\circ}\) and \(90^{\circ}\) with large displacement, where the gap between the two calculation results is within \(8\%\), so the theoretical calculation results can be used as the basis for the deformation of the lining. From Figure 15, under the condition of the linear distribution of the elastic modulus, the radial deformation of lining is slightly different with different elastic modulus parameters in both the theoretical calculation and the numerical simulation. From Figure 15a, the maximum radial displacement is at \(0^{\circ}\). The maximum radial displacement of homogeneous lining is the smallest and the functional gradient lining with elastic modulus parameter \(a=1\), \(b=0.9\) is the second, and the functional gradient lining with elastic modulus parameter \(a=0.9\), \(b=1\) is the largest. In addition, the difference between the maximum deformation of functionally graded lining with \(a=1\), \(b=0.9\) and that of homogeneous lining is no more than \(5\%\). From Figure 15b, similarly, the maximum radial displacement is at \(0^{\circ}\) in the numerical simulation. The maximum radial displacement of homogeneous lining is the smallest and the functional gradient lining with elastic modulus parameter \(a=1\), \(b=0.9\) is the second, and the functional gradient lining with elastic modulus parameter \(a=0.9\), \(b=1\) is the largest. In addition, the difference between the maximum deformation of functionally graded lining with \(a=1\), \(b=0.9\) and that of homogeneous lining is no more than \(5\%\). In conclusion, although there are some differences in the results between the theoretical analysis and the numerical simulation, the laws shown by them are consistent. When \(\lambda=0.5\), the maximum deformation is at \(0^{\circ}\), and \(a>b\) is conducive to reduce the maximum deformation. From Figure 16, the deformation of the functionally graded lining corresponding to the two elastic modulus functions is similar, but the circumferential stress is quite different. From Figure 16a, the circumferential stress of functional gradient lining corresponding to elastic modulus Function II is larger near \(90^{\circ}\) and is about \(30\%\) larger than that corresponding to Function I. From Figure 16b, the radial displacement of the four functional gradient linings reaches the maximum value at \(0^{\circ}\), and the deformation difference is about \(2\%\). The radial displacement of the functional gradient lining corresponding to Function II with \(a=1\), \(b=0.9\) is the minimum, and that of the functional gradient lining corresponding to Function I with \(a=1\), \(b=0.9\) is the second. It can be observed that Function II has some advantages over Function I in reducing the maximum deformation of the structure, but the advantages are relatively low and the overall stiffness of the functional gradient structure represented by Function II is also larger. Therefore, when Function I can meet the deformation requirements, it is unnecessary to use Function II to reduce the deformation further.