## 2 Numerical Models
### Comparison of the Numerical and Analytical Model for the Circular Tunnel Case Study  
For validation purposes of the numerical model subjected to quasi-static loading, the well-known analytical solution proposed by [PERSON] [9], and thereafter improved by [PERSON] [16], was used for comparison with the results obtained from the numerical model. Several authors showed that this solution is efficient and can be used for the seismic design of circular tunnels [12,25]. It should be noted that [PERSON] [16] proposed an expression of the maximum incremental bending moment under the no-slip condition, which was not mentioned by [PERSON] [9]. Parameters shown in Table 1 were adopted in this section as the reference case. The soil and tunnel lining material properties in numerical models were assumed to be linear elastic and massless. These assumptions were also adopted in the analytical solution. An anisotropic stress field was assigned in the numerical model with a lateral earth pressure coefficient at rest, \(K_{0}\), of 0.5.  
Figure 4 illustrates the distribution of the incremental internal forces induced in the tunnel lining under seismic loading. The conditions of lining and soil interaction, when using the [PERSON] solution and FLAC\({}^{3D}\), were considered for both cases of no slip and full slip. The soil and tunnel lining parameters fed into the model are presented in Table 1. The results obtained by numerical and analytical models are in good agreement. Figure 4a,c show that the maximum incremental bending moment in the full-slip case is 14% larger than that obtained in the no-slip case. In contrast, the maximum incremental normal forces in the full-slip case are smaller than those in the no-slip case (Figure 4b,d).  
In the section below, a parametric study is conducted to highlight the behavior of circular tunnel lining subjected to quasi-static loadings considering the effect of Young's modulus \(E\), horizontal seismic acceleration \(a_{H}\), and tunnel lining thickness \(t\) variations. Parameters of the soil and tunnel lining presented in Table 1 are adopted for the reference case study.  
Both maximum and minimum incremental bending moments are presented. They are both labelled \"extreme incremental bending moment\". Similarly, extreme incremental normal forces representing both the maximum and minimum incremental normal forces induced in the tunnel lining are presented.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline _Parameter_ & _Symbol_ & _Unit_ & _Value_ \\ \hline _Soil properties_ & & & \\ Unit weight & \(\gamma\) & \(\mathrm{kN/m^{3}}\) & 18 \\ Young’s modulus & E & MPa & 100 \\ Poisson’s ratio & \(\
u\) & - & 0.34 \\ Internal friction angle & \(\phi\) & degrees & 33 \\ Cohesion & c & kPa & 0 \\ Lateral earth pressure & \(\mathrm{K_{0}}\) & - & 0.5 \\ coefficient & \(\mathrm{H}\) & \(\mathrm{m}\) & 20 \\ Depth of tunnel & \(\mathrm{H}\) & \(\mathrm{m}\) & 20 \\ Peak horizontal & & & \\ acceleration at & \(\mathrm{a_{H}}\) & \(\mathrm{g}\) & 0.5 \\ ground surface & & & \\ _Tumnel lining properties_ & & & \\ Young’s modulus & \(\mathrm{E_{I}}\) & GPa & 35 \\ Poisson’s ratio & \(\
u_{1}\) & - & 0.15 \\ Lining thickness & \(\mathrm{t}\) & \(\mathrm{m}\) & 0.5 \\ External diameter & \(\mathrm{D}\) & \(\mathrm{m}\) & 9.76 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Input parameters for the reference case study.  
Figure 4: _Cont._