## 3 Numerical Model  
Midas GTS NX is a geotechnical finite element analysis software developed in Korea. It is dedicated to the finite element analysis of geotechnical and tunnel structures and is widely used in geotechnical finite element analysis, such as tunnel and foundation pit calculation. As a finite element software, Midas GTS NX cannot solve some non finite element problems, or the calculated results are not good. However, Midas GTS NX has powerful pre-processing and post-processing functions. The software interface is friendly, the interface interaction is simple and easy to understand, and the learning cycle is short. Figure 1: Reality view of a tunnel with a small curve radius. Figure 2: Schematic diagram of the cross section of the tunnel with a small radius curve. Many researchers used and validated this software [22; 23], so Midas GTS NX was used for analysis. MIDAS GTS NX is used to establish a numerical model of a small radius curved tunnel in a certain section of the Changsha Metro Line 6. The numerical model is 60 m long along the center line of the tunnel double line, 90 m long, and 52 m high in the direction of the tunnel cross section. The schematic diagram is shown in Figure 3.  
In the numerical model, the outer tunnel is firstly excavated, and then the inner tunnel. The soil is simulated by a three-dimensional solid element, considering various engineering properties of the on-site soil measurement, and its constitutive model adopts the modified Mohr Coulomb model [24]. The segment and grouting layer are simulated by solid element. The shield shell is simulated by shell element. Normal and horizontal constraints are applied to the bottom of the model, normal constraints are applied to the sides of the model, and the top of the model is a free surface [25]. In order to simulate the grouting process, an equal thickness and homogeneous equivalent layer elastomer is used to simulate the grouting layer [26]. In addition, the shield machine's own weight is simulated by appropriately increasing the density of the shield machine shell. At the same time, in order to simulate the real situation near the tunnel excavation as much as possible and improve the calculation efficiency, the grid near the tunnel excavation is slightly fine, with a size of 1 m. The grid near the boundary is slightly coarse, and the size is 3 m. In the model, the slurry properties (elastic modulus) are changed to simulate the process of the slurry from soft to hard. The elastic moduli of the grouting layer during the grouting hardening process are 1 MPa and 10 MPa in the softening and hardening stages, respectively [27]. Table 1 shows the physical parameters of each material in the numerical model. The construction load of the shield tunnel is as follows: the grouting pressure is 200 kPa, which is applied to the segment and the surrounding \begin{table}
\begin{tabular}{c c c c c c} \hline
**Different Soil (Rock)** & **Bulk Density** & **Cohesion (kN/m2)** & **Friction Angle (\({}^{\circ}\))** & **Elastic Modulus (kN/m2)** & **Poisson’s Ratio** \\ \hline plain fill & 19 & 16 & 10 & 4000 & 0.286 \\ silty clay & 19 & 15 & 8 & 6000 & 0.31 \\ fully weathered slate & 20 & 35 & 15 & 8500 & 0.231 \\ strongly weathered slate & 23.3 & 50 & 29 & 120,000 & 0.2 \\ segment & 25 & - & - & 3.45 \(\times\) 107 & 0.2 \\ grout & 22 & - & - & 1000 (soft) & 0.2 \\ Shield shell & 78.5 & — & — & 210,000,000 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of each material. Figure 3: Numerical model of small radius curve tunnel. driving pressure is 250 kPa, which is applied to the driving surface, and the jack force is 200 kPa, which acts on the section of the segment. It should be mentioned that the value of each parameter remains unchanged if there is no declaration. Unlike straight tunnel shield excavation, curved tunnel excavation requires the use of an over-excavating knife to over-excavate the inner side of the curved section to help the shield machine turn.