## 5 Numerical Simulation of Interaction of Tunnel Lining with Rock Mass under Seismic Loading  
The numerical simulation was performed in the framework of setting a plane strain in the Simulia Abaqus software package. The general design scheme included in the model is shown in Figure 11.  
Calculations on the numerical model were implemented in two stages. Simulation of geostatic loading formation in the tunnel lining from the self-weight of the host rock mass was performed in Stage-1. In Stage-2, the simulation of a seismic impact propagating as a wave disturbance through the calculated area of the rock mass was performed. The model of a linearly deformable body with the parameters presented in Table 3 was used as a mechanical model of a rock mass. Modeling of the behavior of concrete and  
Figure 11: Design scheme of the finite element model. Figure 10: Uniaxial tensile stress versus tensile rupture crack opening according to concrete specimen fracturing simulation. fiber-reinforced concrete of the tunnel lining was performed using the Damaged Concrete Plasticity model. Model parameters were determined based on laboratory tests data and they are summarized in Table 2.  
The boundary conditions on the finite element model were formulated in a specific way to correctly model the seismic impact. The boundary area of the model was divided by a single layer of infinite elements, the main task of which was to extinguish the seismic wave passing through the model boundary. Thus, the effect of wave reflection from the boundary and subsequent related phenomena, such as the possible interference of direct and reflected seismic waves, were excluded. The parameters of the natural stress state of the host rock mass were determined, taking into account the estimated tunnel depth of 120 m.  
The seismic impact was modeled by applying forced accelerations on the inner face of the model, set component-by-component for the directions of the axes of the coordinate system. The accelerogram of a real earthquake that occurred in Friuli in 1976 shows a magnitude of about 6.5. The component-by-component decomposition of this accelerogram is shown in Figure 12.  
The main task of modeling was to state the dependencies of the damage formation in the concrete lining during its dynamic loading under the calculated seismic impact as well as to identify the influence of fiber amount in concrete on its resistance to the accumulation of these damages. To solve this problem, all calculations were performed under the same conditions for the variants when the tunnel lining is made of non-reinforced concrete and of fiber-reinforced concrete with the fiber amount of 3, 5 and 7 kg/m\({}^{3}\). The calculated zones of material damage of the tunnel lining under seismic impact at different time intervals are shown in Figure 13, provided that the tunnel lining is made of non-reinforced monolithic concrete or from fiber-reinforced concrete with a polypropylene macrofiber amount of 3, 5 and 7 kg/m\({}^{3}\). The quantitative damage indicator shows how much of the initial strength and stiffness is lost by concrete in the current time interval due to the crack development. Figure 12: Calculated accelerogram of seismic impact. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Density, kN/m\({}^{3}\)**} & \multicolumn{2}{c}{**Deformation Properties**} & \multicolumn{2}{c}{**The Parameters of Rayleigh Damping**} \\ \cline{3-6}  & & **Young’s Modulus \(E\), GPa** & **Poisson’s Ratio \(\
u\)** & \(\alpha\) & \(\beta\) \\ \hline Mass of Urtites & 0.025 & 30 & 0.2 & 0.037 & 0.0009 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Physical and mechanical properties of rocks [15]. Figure 13: Patterns of formation of damage zones in the tunnel lining material under seismic load. Seismic impact from an earthquake causes the formation of breakaway cracks in the lining of non-reinforced monolithic concrete due to the appearance of significant tensile stresses in certain parts of the lining at particular time intervals.