## 3 Results and Discussion
### Numerical Modelling Analyses  
To justify the use of the rheological behaviours of rocks theory in the present study, a comparison of two case studies was performed. The first one concerned comparing the experimental results of rock beams to those gained from the analytical method, and modelling using a FEM approach reflecting the potential insitu conditions simulating the scenario considered for empirical analyses. The second one was to analyse the total time-dependent vertical displacements of circular tunnel walls subjected to hydrostatic insitu stress, using Equation (25) for both analytical and FEM approaches. The first case considered rock beams with a rectangular cross section of 20 \(\times\) 20 mm and a length of 160 mm, subjected to various values of concentrated load applied to the centre points of the beams so that the values of generated tensile stress were equal to 1.8; 2.25; and 2.7 MPa, respectively. The modulus of elasticity of rock was 14,700 MPa. Tests of simply supported rock beams were simulated using the finite element analysis using MIDAS GTS NX V.1.1 software. Based on the obtained results it was noted that the values of the instantaneous deflection y0 were different, corresponding to the various values of subjected tensile stress. However, thanks to the homogeneity of the materials for all tests, the creep function remained constant, as shown in (19). The comparison of the deflection of rock beams achieved from experiments [3], to those attained using Equation (19) and finite element analysis, is presented in Figure 6. The figure shows good agreement between all the approaches used in the study; the best match is observed for the analytical and modelling method. The biggest discrepancies could be observed for samples tested at 1.8 MPa tensile stress. This is observed especially for an experimental approach that could be referred to as extended boundary conditions applied in the computational mode, due to the nature of expanded boundaries of FEM modelling. Selected results of the FEM analyses of rock beam deflections for different testing times are shown in Figure 7.  
The second case considered a circular tunnel with a radius of 3.5 m at a depth of 200 m. The parameters fed into the modeling were as follows:  
* The hydrostatic in situ stress field took the value of 5.2 MPa;
* T = The modulus of elasticity Et of the rock mass obeyed the law shown in Equation (23) with the E of 15.2 GPa;  
Figure 6: Comparison of results on rectangular beam deflection in time, subjected to different ranges of tensile stress, obtained via three investigation approaches (experimental, analytical and FEM). The vertical displacement formula of a circular tunnel, in this case, was calculated according to [3], taking into account the GSI, based on [PERSON]'s empirical constants and Mohr-Coulomb failure criterion, as expressed in (26). The adopted formula for vertical displacements was as follows:  
\[u=\frac{\gamma HR(1+v)}{E\Big{[}0.02+\frac{1-D/2}{1+e^{(\left(\theta+1\right) \mathrm{D}-GSI)/11}}\Big{]}}\left[1+\frac{3}{2(1+v)}\frac{\delta}{(1-\alpha)}t ^{1-\alpha}\right] \tag{27}\]  
Comparative results obtained from Equation (27) and the FEM approach are represented in Figure 8. The results reveal that the best agreement between the approaches was observed for the highest values of GSI (60) which was due to the increased stress resistance of tested material. Selected results of the vertical time-dependent displacement of the tunnel wall over different periods at 40, 50, and 60 GSI levels are shown in Figure 9. Analyses of the modelling results revealed that tunnel wall vertical displacement obtained using the FEM approach gave close values to those calculated using Formula (27). The comparative results are in good agreement with those obtained by [1; 3; 5; 6]. Thereby, the effect of the GSI on stand-up time calculation is convincing. This allows for the optimisation of supporting structure design in tunnelling and provides an opportunity to avoid potential risks and hazards during tunnel excavation works.