## 5 Results and Discussion
### Full-Scale Numerical Study of Tunnels  
A Two-dimensional elastoplastic study for full-scale field simulation of the tunnelling problem was carried out using finite element-based software Optum G2. Analysis was performed for five different lining shapes used in common tunnelling constructions, namely (i) circular, (ii) horseshoe, (iii) arch, (iv) elliptical, and (v) square shapes. The CCM method was adopted for numerical simulation of the problem. The schematic diagram of the model with a circular tunnel is shown in Figure 8. The geometric dimensions for tunnel linings are shown in Figure 9.  
The study was to analyze ground settlement and plot the ground-settlement trough for different shapes of the lining. The tunnel constructed with elliptical lining had a minimum value of maximum settlement followed by horseshoe, circular, arch, and square-shaped linings. Figure 12 shows the plotted ground settlement for various shapes of tunnel linings. The square-shaped tunnel had the highest value for maximum settlement with a magnitude 4 cm and the elliptical-shaped tunnel had the least maximum settlement value among all shapes with magnitude of 1.4 cm. Displacement in the soil can be characterized as horizontal displacement or vertical displacement. Horizontal displacement does not pose any vulnerability to tunnel stability and was therefore not computed. Vertical displacements,  
Figure 11: Principal stresses in soil. (**a**) horizontal plane and (**b**) vertical plane. Figure 10: Vertical displacement contours for (**a**) soil mass and (**b**) aluminium lining. on the contrary, can cause significant damage to the tunnel, so the determination of vertical displacement is thought important and necessary. Vertical displacement contours were plotted for all cases in Figure 13. The maximum settlement was recorded for the square lining and least for the horseshoe lining. The stresses developed in soil are an important parameter to analyze. The model was analyzed to determine the principal stresses in the horizontal plane, \(\sigma_{xx}\) and the  
Figure 12: Ground-settlement trough for different shapes of tunnel linings. Figure 13: Vertical displacement in soil having (**a**) circular-, (**b**) arch-, (**c**) horseshoe-, (**d**) elliptical-, and (**e**) square-shaped openings. principal stress in the vertical plane, \(\sigma_{yy}\), developed in soil media. Tensile stresses are developed throughout the soil mass with the only exception being the area in close vicinity to the ground surface for both the principal plane directions. Results showed that for the horizontal plane, maximum stresses were at invert locations for all shapes within the range 209-252 \(\mathrm{kN}/\mathrm{m}^{2}\) with a maximum for the elliptical-shaped lining. Vertical-plane stresses were maximum around the springer location within the range 336-422 \(\mathrm{kN}/\mathrm{m}^{2}\) with a maximum for the circular-shaped lining and stresses around the crown and invert showing large variations. Figure 14 shows a contour plot for horizontal-plane stresses and Figure 15 shows vertical-plane stresses for all shapes of the lining. Stresses at the springer location for the horizontal plane were localized for all shapes except the square opening due to the absence of arch action. There was no effect of the shape of lining at the bottom of soil as the value was approximately the same for all cases in both the principal planes. The results showed that lining shapes such as horseshoe and square experience lower stresses at the springer location than other shapes due to the vertical plate boundaries. the springer locations in the horseshoe- and square-shaped linings due to vertical plate boundaries whereas for other shapes there was a variation along the vertical lining due to arching effect. The stability of an elliptical shape can be understood through the concept of arches and the distribution of forces in them. Maximum displacements were observed at three crucial points on tunnel lining--crown, springer, and invert.