# Model Test Study on a Shield Tunnel Adjacent to Pile in the Sub-Clay of a Coastal Area
## 2 Design of Model Test System Crossing Existing Piles by Shielding
### Design of Test Materials
#### 2.2.2 Equivalent Lining  
In order to develop similar material to the lining model test, paraffin wax, clean quartz sand and steel fiber were selected as the suitable main simulation materials. The material coefficients are shown in Table 2.  
Tri-axial tests were conducted. The elastic modulus and fracture angle of the test specimens were detected. The vertical strain gauges were pasted on the surface of the specimen for monitoring. The Poisson's ratio, paraffin, quartz sand content and the elastic modulus relationship was obtained in Figure 8. According to the intersection point of the two curves, the physical and mechanical parameters of different proportions of test pieces are shown in Table 3. The mass ratio of the similarity relationship of the lining material was determined to be 15:90:1. From the physical and mechanical parameters of the prototype and model lining of Table 4, it can be seen that the material bulk density, elastic modulus and Poisson's ratio have a similar relationship. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Grade** & **Bulk Density** & **Deformation Modulus** & **Cohesion** & **Internal Friction Angle** \\  & \(\gamma\)**/(kN/m\({}^{3}\))** & _E_/MPa & _c_/kPa & \(\varphi\)**/\({}^{E}\)** \\ \hline V & 17-20 & 1000–2000 & 50–200 & 40–50 \\ VI & 15-17 & \textless{}1000 & \textless{}100 & 30–40 \\ Similar material & 20.6 & 13.1 & 0.375 & 31.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical-mechanical parameters of prototype and model for surrounding layers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test Group** & **Paraffin Wax (g)** & **Quartz Sand (g)** & **Steel Fiber (g)** & **Scale** \\ \hline
1 & 150 & 0 & 0 & 1:-\textless{} \\
2 & 80 & 160 & 0 & 1:2:- \\
3 & 60 & 240 & 0 & 1:4:- \\
4 & 50 & 300 & 0 & 1:6:- \\
5 & 40 & 360 & 0 & 1:8:- \\
6 (steel-1) & 60 & 240 & 4 & 15:60:1 \\
7 (steel-2) & 45 & 270 & 3 & 15:90:1 \\
8 (steel-3) & 30 & 240 & 2 & 15:120:1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Proportioning table of model material. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Serial Number** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline weight (g) & 105.3 & 145.6 & 168.7 & 189.1 & 206.6 & 232.9 & 242.7 & 254.1 \\ bulk density (kN/m\({}^{3}\)) & 10.4 & 14.5 & 16.8 & 18.8 & 20.6 & 23.2 & 24.2 & 25.3 \\ Poisson ratio & 0.45 & 0.35 & 0.35 & 0.34 & 0.34 & 0.33 & 0.33 & 0.32 \\ \(E\) (MPa) & 79.4 & 105.4 & 190.3 & 250.3 & 275.8 & 262.3 & 303.6 & 330.2 \\ Rupture angle (\({}^{\circ}\)) & — & — & 32 & 36 & 40 & 39 & 42 & 46 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The physical and mechanical parameters of specimens with different proportions. Figure 7: Shear strength-pressure relationship. Bending deformation is an important control parameter of the lining safety. According to the similarity principle of bending deformation [9; 10; 11], the similar lining thickness hm was determined, and the lining structure was equivalent to the thin plate, where the thickness is \(h\), the orthogonal coordinate \(x\)-\(y\) of the cross section, the uniform force \(q\) was applied, and the deflection of the plate satisfies the following equation. \[\frac{\partial^{4}\omega}{\partial x^{4}}+2\frac{\partial^{4}\omega}{\partial x ^{2}\partial y^{2}}+\frac{\partial^{4}\omega}{\partial y^{4}}=\frac{q}{K} \tag{3}\]  
\[K=\frac{Eh^{3}}{12(1-\mu^{2})} \tag{4}\]  
where \(K\) is the bending stiffness, \(E\) is [PERSON]'s modulus, and \(\mu\) is the [PERSON]'s ratio. The geometric similarity scale of the model device was set to \(\alpha_{l}=1/n\), then the model structure and the prototype structure met the Formula (3), to obtain the following similar criteria.