# Behaviour of Square and Rectangular Tunnels Using an Improved Finite Element Method
## 2 Methods Calculation
### Validation of the HRM Method  
The application of the HRM method in the case of square and rectangular tunnels is verified by comparing the results of internal forces (bending moment M, normal force N, and shear force T) induced in the tunnel lining obtained in this method with those determined by a FEM using Plaxis2D. In this paper, the ground and tunnel lining parameters of the Hanoi metro system (3 rd line Nhon--Kim Ma) are used as input data of the HRM method and FEM. The parameters of the tunnel lining and the soil surrounding the tunnel, i.e., the tunnel lining thickness \(t\) = 0.35 m, [PERSON]'s modulus of soil \(E\) = 10 MPa, and the coefficient of lateral earth pressure \(K_{0}\) = 0.5 were adopted. From the results in Figure 5 and Table 2, in the case of the square tunnel, the following comments can be made: the internal forces in the square tunnel lining calculated by the improved HRM method are in good agreement with results obtained in the FEM. The differences of the maximum bending moment M\({}_{\text{max}}\) and the minimum bending moment M\({}_{\text{min}}\) between the two methods are 2.03% and 0.24%, respectively. Meanwhile, the differences of the maximum normal force N\({}_{\text{max}}\) and minimum normal force N\({}_{\text{min}}\) by the HRM method and FEM are correspondingly 0.55% and 7.89%. Those of the maximum shear force T\({}_{\text{max}}\) and minimum normal shear force T\({}_{\text{min}}\) are 1.91%. Figure 5: Internal forces in the square tunnel lining: (**a**) Bending moments M; (**b**) Normal forces N; (**c**) Shear forces T.  
Figure 6 and Table 2 show the comparative results in the case of the rectangular tunnel. The maximum bending moment M\({}_{\text{max}}\) and the minimum bending moment M\({}_{\text{min}}\) obtained in the HRM method are 0.85% and 1.33% different from the corresponding values of FEM. In the HRM method, the maximum normal force N\({}_{\text{max}}\) and minimum normal force N\({}_{\text{min}}\) are 1.95% and 12.74%, respectively, smaller than those of the FEM model. The differences of maximum shear force T\({}_{\text{max}}\) and minimum normal shear force T\({}_{\text{min}}\) between the HRM method and FEM are both 2.93%. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Tunnel Case/Calculation** & **Value** & **M** & **N** & **T** \\
**Method** & **Value** & **(MN/m)** & **(MN/m)** & **(MN/m)** \\ \hline _Case 1—square tunnel_ & & & & \\ FEM & Max & 0.644 & 1.003 & 1.004 \\  & Min & \(-\)0.702 & 0.437 & \(-\)1.004 \\ HRM & Max & 0.657 & 0.998 & 0.985 \\  & Min & \(-\)0.701 & 0.474 & \(-\)0.985 \\ Difference (\%) & Max & 2.03 & 0.55 & 1.91 \\ _Case 2—rectangular tunnel_ & & & & \\ FEM & Max & 0.781 & 1.073 & 1.071 \\  & Min & \(-\)0.764 & 0.391 & \(-\)1.071 \\ HRM & Max & 0.788 & 1.052 & 1.040 \\  & Min & \(-\)0.754 & 0.441 & \(-\)1.040 \\ Difference (\%) & Max & 0.85 & 1.95 & 2.93 \\  & Min & 1.33 & 12.74 & 2.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the internal forces by the HRM method and FEM. Figure 6: Internal forces in the rectangular tunnel lining: (**a**) Bending moments M; (**b**) Normal forces N; (**c**) Shear forces T.  
The internal forces differences induced in square and rectangular tunnel linings in the HRM method and the FEM could be explained by the fact that the internal forces and deformations of the tunnel lining in the HRM method are mainly affected by the active loads and the soil-lining interaction. These factors are considered through the springs stiffnesses between the ground and the tunnel lining (normal and tangential springs) assigned at the lining beam nodes. In FEM, the ground-structure interaction is taken into account through the interface embedded between the lining and the ground elements. In this study, when using the FEM to simulate square and rectangular tunnels, without an interface between the tunnel lining and the ground elements, the structure and the soil are tied together. It means that no relative displacements (slipping/gapping) are possible between the structure and soil [27].