## 4 Model Illustration of Primary Tunnel Lining Design
The representative value of shotcrete \(E_{c\_A,B,C}\) corresponding to construction stages A, B and C was determined as the arithmetic average of discrete \(E_{c}\) values divided by the time axis by 0.1 d in the set time interval of the relevant stage, as follows:  
\[E_{c\_A,B,C}=\frac{T\cdot\Sigma E_{c\_A,t}}{\Delta t}, \tag{3}\]  
where \(E_{c\_A,B,C}\) represents the average elastic modulus of the shotcrete in stages A, B and C (GPa), and \(E_{c\_A}\)_t_ represents the elastic modulus of the shotcrete (GPa) at time \(t\) according to Formula (1). \(\Delta t\) represents the duration (d) of the stage (A, B or C), and \(T\) represents the fineness of the division of the time axis (d), which was 0.1 d in this study. \begin{table}
\begin{tabular}{c c} \hline \hline \(t\) (d) & \(E_{c}\) (GPa) \\ \hline
1 & 14.550 \\
10 & 18.320 \\
28 & 20.310 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measured time-dependent elastic modulus of concrete \(E_{c}\) (according to VSB). Figure 7: 3D model of the lining. When this relationship is applied, the elastic modulus of construction stage A, for example, can be determined as follows:  
\[E_{c\_A}=\frac{0.1\cdot\Sigma\left(0+11.554+12.384+...+14.546\right)}{1}=\frac{0. 1\cdot 115.58}{1}=11.558\;\text{GPa} \tag{4}\]  
To capture the development of the elastic modulus of shotcrete in further detail, more construction stages can be used in the numerical model, but at the expense of a more articulate numerical model and the need to use more calculation stages. The method for setting the elastic modulus of shotcrete \(E_{c}\) is shown graphically in Figure 8.  
Table 2 presents the elastic modulus of the shotcrete and resulting modules of the homogenised cross-section for the calculation stages. The heterogeneous structure of the lining was subsequently homogenised using the software Homogenisation. The input parameters and results are presented in Table 2, including the inputs and outputs of the homogenisation process for individual construction stages can be found. Figure 9 presents the construction stages in the numerical model. The figure clearly shows the gradual increase in the elastic modulus of the homogenised lining Ehm, which was due to shotcrete solidification. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Stage**} & \multirow{2}{*}{
\begin{tabular}{c} **Time** \\ **(d)** \\ \end{tabular} } & \multirow{2}{*}{**Alternative**} & \(\begin{array}{c}\mathbf{E_{c}}\\ **(GPa)** \\ \end{array}\) & \multirow{2}{*}{\(\mathbf{E_{lm}}\) **(GPa)**} \\ \hline \multirow{2}{*}{A} & \multirow{2}{*}{0–1} & I. & \multirow{2}{*}{11.558} & 12.130 \\ \cline{3-3}  & & II. & & 13.807 \\ \hline \multirow{2}{*}{B} & \multirow{2}{*}{1–5} & I. & \multirow{2}{*}{15.883} & 16.442 \\ \cline{3-3}  & & II. & & 18.029 \\ \hline \multirow{2}{*}{C} & \multirow{2}{*}{5–28} & I. & \multirow{2}{*}{18.671} & 19.224 \\ \cline{3-3}  & & II. & & 20.786 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Deformation parameters of the lining in the considered construction stages. Figure 8: Deformation parameters of the lining in the considered construction stages. Alternative II using steel arches and reinforcement grids in cross-section homogenisation exhibited identical (variation of \(<\)1%) internal forces to Alternative I, in which the reinforcement elements were modelled with \"beam\" elements and reinforcement grids were only considered in the cross-section homogenisation. Thus, the procedure allowing the numerical model to be simplified via homogenisation of the lining provides sufficiently reliable results for application in practice. Potential differences between the results of the model and monitoring are caused by an individual modelling approach (the selection of the constitutive relationship, etc.) and often also by errors in the monitoring. Further evaluation only focuses on model Alternative II, which uses arch frames and grids in the cross-section homogenisation.