## 3 Comparison and Verification  
### Numerical Analysis  
A numerical simulation method was used to clarify the limited \(c\)-\(\varphi\) soil mechanism between a deep retaining wall and an existing large rigid underground structure. The finite difference numerical analysis model established in FLAC3D (Fast Lagrangian Analysis of three-dimensional Continua) [50; 51; 52; 53] is shown in Figure 8. In which the boundary conditions are: the thickness of the mesh was only 1 m in order to simulate the plane strain state; the depth of the mesh, and the width at the right side of the large rigid underground structure was four times the retaining wall depth [13]. The bottom of the mesh was fixed; the surrounding side can slide vertically; and the top surface was free.  
The burial depth of the large rigid existing underground structure and of the retaining wall were both 12 m. The width between the retaining wall and the existing wall was determined by the limited soil width, which were separately 1, 2, 4, 6, and 8 m. The thickness of the retaining wall was 0.8 m at the top and 3.8 m at the bottom. Its back in contact with soil was vertical. The soil-wall interaction was simulated using an interface element (the rigidness of the interface element was 10 times that of the soil rigidness), and the constitutive model was used the Coulomb shear model. The \(c\)-\(\varphi\) soil was simulated using a Mohr Coulomb model (\(\gamma\) = 24.5 kN/m\({}^{3}\); \(E\) = 15 MPa; \(\mu\) = 0.35; \(c\) = 30 kPa; \(\varphi\) = 25\({}^{\circ}\)). The simulation was carried out for both the retaining structure and the existing large rigid underground  
Figure 8: Numerical calculation model of (**a**) model mesh and (**b**) interface element.  
structure using an elastic element (\(\mu=0.2\); \(\gamma=30\) kN/m\({}^{3}\); \(E=35\) GPa), where \(E\) was the soil elastic modulus, and \(\mu\) was the Poisson's ratio. The min and max limited soil width were separately 0.76 and 6.04 m, according to Equation (19).  
Generally, the retaining wall displacement which forces the soil into the active state is about \(0.2\times 10^{-2}\)\(h\), in which \(h\) is the retaining wall depth [54]. By giving a leftward displacement of about 24 mm to the retaining wall, the calculated cloud map of displacement for different limited \(c\)-\(\varphi\) soil width is shown in Figure 9. The slope of the inclined plane is approximately the same for all the cases. The inclined plane intersects the side wall of the existing large rigid underground structure when the limited \(c\)-\(\varphi\) soil width is less than 6.04 m. The results verify that the theoretical result for the double-fold line slip surface of the limited \(c\)-\(\varphi\) soil obtained in Equation (3) is reasonable.  
The numerical results of the active earth pressure distribution were obtained for \(B/h\) of 0.083, 0.167, 0.333, 0.500, and 0.667, as shown in Figure 10. The results are also compared with the theoretical results obtained in this study and by other scholars. The distribution of the active earth pressure is large in the middle and small at both ends. The active earth pressure decreases as the decreasing of limited soil width in the numerical results and this study. The numerical results are close to the values both from this study and other scholars when the limited soil width is large. However, it is closer to the results by this study when the limited soil width is small. This demonstrates that the theory developed in this study reflects the actual situation well.