## 3 Finite-Element Modelling
### Lab Model  
A 2D finite-element model was generated on Optum G2 having plan dimensions 1 m \(\times\) 1 m with a circular opening of 30 cm diameter at the center of tunnel box. The main objective of this analysis was to validate our numerical modeling using experimental observations. Tunneling simulation was done through the convergence-confinement method (CCM). CCM has been widely used by various researchers for simulating tunneling problems [46; 47; 48; 49]. CCM is based on the concept of the convergence of the ground, the  
Figure 6: Settlement trough generated through lab experiment and numerical modelling on Optum G2. support pressures, and the resulting confinement losses [46]. The generated numerical model is shown in Figure 7a and the discretized mesh model shown in Figure 7b. In the present study, the elastoplastic soil constitutive Mohr-Coulomb model, were used for modelling of soil. Since the laboratory test was simulated through numerical simulation geotechnical parameters for the soil model used needed to be of the same soil used in the laboratory. Table 1 shows soil properties used for numerical analysis [28,50]. The aluminium lining was simulated using a plate feature in Optum G2. The properties of aluminium (Al 6061-T6) are shown in Table 1[51,52]. The model was provided with a fine mesh having many elements in the range of 1000, and six-node Gaussian type elements were used for both soil and aluminium lining. The mesh adaptivity was adopted to provide recommended fines of mesh in the region. The model was provided with roller support at the two vertical edges having displacement possible in vertical directions and fixed in the horizontal direction. The base of the model was provided with fixed support. The above boundary condition is termed standard fixities \begin{table}
\begin{tabular}{c c} \hline Soil Properties & \\ \hline Unsaturated unit weight & 16 kN/m\({}^{3}\) \\ \hline Saturated unit weight & 19 kN/m\({}^{3}\) \\ \hline Young’s modulus & 1.5 \(\times\) 10\({}^{4}\) kPa \\ \hline Poisson’s ratio & 0.3 \\ \hline Cohesion & 0 kPa \\ \hline Friction angle & 30\({}^{0}\) \\ \hline Dilation angle & 0\({}^{0}\) \\ \hline Aluminum lining (Al 6061-T6) & \\ \hline Density & 26.47 \\ \hline Young’s modulus & 6.9 \(\times\) 10\({}^{4}\) kPa \\ \hline Sectional area & 40 cm\({}^{2}\)/m \\ \hline Plastic-section modulus & 40 cm\({}^{3}\)/m \\ \hline Moment of inertia & 0.534 cm\({}^{4}\)/m \\ \hline Yield strength & 275 MPa \\ \hline Weight & 10.6 kg/m/m \\ \hline \end{tabular}
\end{table} 
Table 1: Properties of soil and aluminium lining. Figure 7: (**a**) Numerical model and (**b**) meshed numerical model. in OptumG2. Analysis was done manually using the convergence-confinement method provided in the Optum G2 examples. Analysis was carried out in three stages for better simulation of the tunnelling problem. Stage 1: This was a greenfield condition having soil modelled on ground conditions. The analysis performed in this stage is known as initial stress analysis. The initial stress state is usually characterized by the earth pressure coefficient Equation (9):  
\[K_{0}=\frac{\sigma_{h}}{\sigma_{v}} \tag{9}\]  
where \(\sigma_{h}\) and \(\sigma_{v}\) are the effective horizontal and vertical stresses, respectively. For a purely frictional Mohr-Coulomb material, the bounds on \(K_{0}\) are the well-known active and passive earth pressure coefficients (Equation (10):  
\[\frac{(1-sin\varphi)}{(1+sin\varphi)}\leq\ K_{0}=\ \frac{\sigma_{h}}{\sigma_{v}} \leq\frac{(1+sin\varphi)}{(1-sin\varphi)} \tag{10}\]  
where, \(\varphi\) is the internal friction angle. Stage 2: An excavation for the tunnel was provided in this stage and an elastoplastic analysis was performed. The tunnel perimeter here was fully supported and a relaxation factor, \(\lambda\), was specified and excavation was provided with full support according to the manual of Optum G2. From an initial state, the tunnel was excavated while keeping the perimeter of the tunnel fully supported.