# Behaviour of Square and Rectangular Tunnels Using an Improved Finite Element Method
## 3 Parametric Study  
In this part, a parametric investigation using both the HRM method and FEM is conducted to highlight the effects of the flexibility ratio (\(F\)) of the lining and the coefficient of lateral earth pressure (\(K_{0}\)) on the behaviour of the tunnel. The square tunnel with dimensions of 5.5 \(\times\) 5.5 m was adopted. The other parameters of the tunnel lining and the soil are shown in Table 1.  
_Influence of the flexibility ratio of the tunnel lining (\(F\))_  
For investigation purposes, the coefficient of lateral earth pressure \(K_{0}\) is kept as constant, \(K_{0}\) = 0.5 and the thickness of the square tunnel lining \(t\) changes from 0.35 m to 0.5 m. It is assumed that Young's modulus of the ground \(E\) is changed in the range from 10 MPa to 100 MPa. The flexibility ratio of the tunnel lining \(F\) is determined by the following equation [6; 7]:  
\[F=\frac{G_{m}}{24}\times\left(\frac{H^{2}W}{E_{l}I_{w}}+\frac{HW^{2}}{E_{l}I_ {R}}\right) \tag{15}\]  
where \(E_{l}\) is Young's modulus of the tunnel lining (MPa); \(G_{m}\) is the shear modulus of ground (MPa); \(I_{w}\) is inertia moment of lining at the sidewall (m4); \(I_{R}\) is inertia moment of lining at the lining roof (m4); \(W\) is the tunnel lining width (m); and \(H\) is tunnel height (m). The lining internal forces calculated by the HRM method and FEM were presented in Figure 7 and Table 3. It is reasonable to conclude that:  
* The internal forces, including, bending moments M, normal forces N, and shear forces T, induced in the tunnel lining decrease when the flexibility ratio of tunnel lining \(F\) increases. This could be explained by the fact that a larger \(F\) value means a more flexible or softer lining and/or a stronger ground. As a consequence, a smaller yielded ground zone that causes active loads acting on the tunnel lining could be predicted. In addition, due to the higher flexibility of the lining, which implies a larger movement/deformation possibility of the lining, a greater relaxation process is induced in the ground surrounding the tunnel and therefore a decrease in active pressure applied in the tunnel. Internal forces induced in the tunnel lining are decreased as mentioned above. * In the range of the small flexibility ratio of tunnel lining (\(F\leq 1\)) (i.e., the thicker tunnel lining and the smaller Young's modulus of soil \(E\)), the deviation of the internal forces obtained using the improved HRM method and FEM is small. For \(F>1\), an increase in the \(F\) value is followed by a larger difference in the results of the two methods. While the absolute bending moments obtained by the HRM method are always greater than the ones of the FEM model, the absolute normal forces and shear forces observed in the HRM results are smaller than those determined in the FEM model. It means that the internal forces estimated in the HRM lead to a more unstable situation of the tunnel lining, which can be acceptable in terms of the tunnel design. The observed larger deviation at the greater flexibility ratio of tunnel lining \(F\) could be concerned  
_Influence of the coefficient of lateral earth pressure \(K_{0}\)_  
The coefficient of lateral earth pressure \(K_{0}\) is varied from 0.5 to 2.0. The ground deformation modulus \(E=10\) MPa, the tunnel lining thickness \(t=0.35\) m, and the flexibility ratio of the tunnel lining \(F=0.41\) are adopted. Figure 8 illustrates the distribution of internal forces along the square tunnel periphery. Figure 9 and Table 4 indicate the dependency of the extreme internal forces in the tunnel lining obtained by the HRM method and FEM on the \(K_{0}\) value. Some following observations can be given:  
* In most investigated cases of \(K_{0}\) values, the HRM gives internal forces results that are in good agreement with those of the FEM model. The deviation is always smaller than 5%;
* An increase in the \(K_{0}\) value is followed by a decrease in the bending moments at the top and tunnel bottom but an increase in the bending moment at the sidewalls (Figure 8a).