# Seismic Behavior of Triple Tunnel Complex in Soft Soil Subjected to Transverse Shaking
## 4 Details of the Finite Element Analysis
### Constitutive Model and Boundary Conditions  
The soil tunnel system is modeled using Plaxis 2D, considering the Mohr-Coulomb constitutive model for the soils used by many other researchers [16; 30; 32]. In order to use for the dynamic analysis, Plaxis 2D makes use of the modified Mohr-Coulomb constitutive model. The shear, elastic moduli and other parameters in this model are dependent on the primary input of the shear wave velocity of the soil medium and, thus, are calculated accordingly. Apart from this, shear and elastic moduli variation w.r.t depth and confining stresses are also taken into account. The equations can be given by the equation below. \[G=\gamma V_{s}^{2} \tag{1}\]  
\[E=2G(1+v_{s}) \tag{2}\]  
\[E(y)=E_{ref}+\big{(}y_{ref}-y\big{)}E_{inc} \tag{3}\]  
where \(G\) is the shear modulus, \(V_{s}\) is the shear wave velocity, \(E\) is the elastic modulus, \(v_{s}\) is the soil's Poisson's ratio, \(y\) is the depth in the vertical direction, \(E_{ref}\) is the value of elastic modulus at reference point, \(y_{ref}\) is the depth of the reference point, and \(E_{inc}\) is the increment in value of the elastic modulus [33; 34]. The seismic ground shaking produces cyclic stresses in the soil, which generates a hysteresis loop with the dissipation of energy and damping. The Mohr-Coulomb soil model is an elastic-perfectly plastic soil model and cannot capture this phenomenon. To cater this limitation, the Rayleigh viscous \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Soil Type**} & **Saturated** & **Shear** & \multicolumn{2}{c}{**Permeability (m/s)**} & \multicolumn{2}{c}{**Rayleigh Coefficients**} \\ \cline{3-8}  & & **Unit Weight** & **Strength** & **Horizontal** & **Vertical** & \(\alpha\) & \(\beta\) \\  & & **(kN/m\({}^{3}\))** & **(kPa)** & **Horizontal** & **Vertical** & \(\alpha\) & (\(\times\) 10\({}^{-3}\))** \\ \hline
1 & Silty clay & 18.4 & 29.9 & \(5.5\times 10^{-7}\) & \(2.50\times 10^{-9}\) & 9.660 & 0.776 \\
2 & Very soft silty clay & 17.5 & 27.4 & \(3.5\times 10^{-6}\) & \(1.70\times 10^{-8}\) & 3.893 & 1.926 \\
3 & Very soft clay & 16.9 & 19.8 & \(5.13\times 10^{-8}\) & \(1.91\times 10^{-9}\) & 1.771 & 4.238 \\
4 & Clay & 18 & 26.3 & \(3.40\times 10^{-6}\) & \(3.51\times 10^{-8}\) & 1.744 & 4.301 \\
5 & Silty clay-silty sand & 18.1 & 30 & \(2.13\times 10^{-5}\) & \(2.67\times 10^{-6}\) & 1.706 & 4.397 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Soil properties used in the study. Figure 2: (**a**–**c**) Soil-Tunnel geometry: (**a**) Soil layers with the embedded triple tunnel complex. (**b**) Soil column w.r.t depth and type. (**c**) Enlarged triple tunnel complex section, also showing the truncated parts (units in ‘m’). (**d**) Shear wave velocity profile w.r.t depth. damping coefficients are introduced through a frequency-dependent formulation, which is given by the equation below. \[\alpha=2\omega_{1}\omega_{2}\frac{\omega_{1}\xi_{2}-\omega_{2}\xi_{1}}{\omega_{1} ^{2}-\omega_{2}^{2}},\;\beta=2\frac{\omega_{1}\xi_{1}-\omega_{2}\xi_{2}}{\omega_ {1}^{2}-\omega_{2}^{2}} \tag{4}\]  
While  
\[\omega_{1}(rad/s)=2\pi f_{1},\;\omega_{2}(rad/s)=2\pi f_{2} \tag{5}\]  
\[f_{1}(Hz)=\frac{V_{s}}{4h},\;f_{2}\;(Hz)=\frac{3V_{s}}{4h} \tag{6}\]  
where \(\alpha\) and \(\beta\) are the Rayleigh viscous damping coefficients, \(\omega\) is the angular frequency, \(h\) is the thickness of the soil layer,\(f_{1}\) and\(f_{2}\) are the first and second target frequencies, and \(\xi_{1}\) and \(\xi_{2}\) are the respective damping ratios. The damping ratio is taken as 10% for all of the soil layers that attributes to the soft soil [35]. The damping coefficients are calculated individually for each of the soil layers incorporated into the constitutive model definition. The tunnel lining and inner supporting structure are modeled using linear elastic plate elements as used by many other researchers in the past [8; 9; 16; 17].