## 4 Model Illustration of Primary Tunnel Lining Design
* radius of the tunnel arch: \(r_{arch}\) = 3.40 m;
* radius of the tunnel bench: \(r_{bench}\) = 2.65 m;
* radius of the tunnel invert: \(r_{invert}\) = 6.55 m;
* eccentricities of the arches: \(e_{x}\) = 0.55 m; \(e_{y1}\) = 0.51 m; \(e_{y2}\) = 5.00 m.  
The load on primary lining was determined by the deformation and structural strength parameters of the rock mass, which were characterised by the Mohr-Coulomb constitutive model:  
* bulk density of mass \(\gamma_{m}\) = 21 kN/m\({}^{3}\); elastic modulus \(E_{m}\) = 100 MPa; \(v_{m}\) = 0.3; angle of internal friction \(\Phi\) = 28\({}^{\circ}\); cohesion c = 30 kPa; angle of dilation \(\Psi\) = 0\({}^{\circ}\). The tunnel was reinforced with a primary lining of thickness d = 0.2 m. The analysis included ASTA 70 (triangular lattice girder with following transversal geometry: diameter \(D_{1}\) = 32 mm; \(D_{2}\) = 22 mm; H = 124 mm) with an axial distance of 1 m and a reinforcement grid of 6 \(\times\) 100 mm at the face and back sides and a cover of 30 mm in the lining. The components of the primary lining were selected with the following descriptive and deformation parameters:  
* steel (density of steel \(\gamma\) = 78.5 kN/m\({}^{3}\); elastic modulus \(E_{s}\) = 210 GPa; Poisson's ratio \(v_{s}\) = 0.2);
* shotcrete (density of concrete \(\gamma_{c}\) = 25 kN/m\({}^{3}\), elastic modulus as function of time \(E_{c}\) = f(t) (MPa), Poisson's ratio of concrete \(v_{c}\) = 0.2);
* constitutive model: linearly elastic. Typical boundary conditions were selected, i.e., horizontal shifts perpendicular to the relevant planes were limited to the boundary of the model, while the bottom of the model had limited vertical and horizontal shifts in both directions. The numerical model demonstrates cyclic implementation of the primary lining. A reference \"pseudo-time\" of 1 d, during which the driving progress was considered as 1 m/d, i.e., individual increases in the stiffness of the primary lining of the evaluated construction stages observed this assumption. All the construction stages of the lining were adjusted to the selected time periods, particularly the development of the time-dependent parameters of the shotcrete. According to the principles of the NATM, the primary lining was applied shortly after creating a partial burden. It was impossible to capture the influence of the time factor in the lining's defined scale of time-dependent parameters (1 d) or the progress of 1 m/day in the numerical model. Figure 6: Diagram of the arch structure consisting of three circular arches. In total, 20 stages of calculation were performed, representing a distance of 20 m from the tunnel's face to the boundary of the numerical model. The monitored lining cross-section was 10 m from the boundary of the model. The driving of 10 m was therefore reflected in the assessment of the lining and the monitoring of the stress development in the selected lining cross-section (Figure 7). The elastic modulus of the shotcrete \(E_{c}\) was measured via a laboratory test before the deformation parameters of the primary lining were determined for the calculation stages (Table 1). According to a regression analysis, the development of the elastic modulus is described by the function  
\[E_{c}(t)=14.022\cdot t^{0.107}(\mathrm{GPa}) \tag{2}\]  
where \(t\) represents the age of the concrete (d). The coefficient of determination of the function is \(R^{2}\) = 0.98. To increase the objectivity of the regression analysis, the condition of \(E_{c}\) = 0 MPa was considered at \(t\) = 0 d. The function is used only for timeslots ranging from \(t\) = 0 d to \(t\) = 28 d; after 28 d, a constant value of the elastic modulus (\(E_{c}\) = 20.31 GPa) is considered. The development time axis was divided into three intervals A, B and C, which were assigned a representative average value of the elastic modulus of shotcrete in the given solidification stage (Figure 8).