# Numerical Studies on Rockbursts in Tunnels with High In Situ Stresses and Geothermally Rich Areas
## 4 Numerical Results
For the cooling area of the rock, the thermal stress is tensile stress, and for the uncooled rock near the cooling area, the thermal stress is compressive stress. However, cooling has an inhibitory effect on the occurrence of rockbursts. Figure 10c,d show the evolution of the two principal stresses in Path 2 after cooling. Comparing Figure 10c,d and Figure 10a,b shows that the trends of the two principal stresses in Path 2 are similar to those in Path 1, so they are not be explained in detail. The above results indicate that to improve the overall stability of the tunnel, the tunnel should be supported before cooling for 3 h (when tensile stress appears) to prevent rock instability and fracture in the later stage. Figure 10: _Cont._  
Figure 9: Magnitude and distribution of the maximum principal stress at \(t=0\)–24 h in the tunnel. (a) \(t=0\) h; (b) \(t=1\) h; (c) \(t=3\) h; (d) \(t=6\) h; (e) \(t=12\) h; (f) \(t=24\) h.  
Figure 10: Variation in the maximum and minimum principal stresses for \(t=0\)–24 h along 4 paths: maximum (**a**) and minimum (**b**) principal stresses along Path 1; maximum (**c**) and minimum (**d**) principal stresses along Path 2; maximum (**e**) and minimum (**f**) principal stresses along Path 3; and maxi-mum (**g**) and minimum (**h**) principal stresses along Path 4. Figure 10e,f show the curves of the maximum and minimum principal stresses of Path 3, respectively. Figure 10e shows that with the cooling of the tunnel, the maximum principal stress (compressive) on the SP#Tunnel face starts decreasing at \(t\) = 0-1 h, and it starts converting from compressive stress to tensile stress at \(t\) = 1-3 h. As cooling continues (\(t\) = 3-24 h), the maximum principal stress (tensile) increases. Figure 10f similarly shows that with the cooling of the tunnel, the maximum principal stress (compressive) on the SP#Tunnel face starts decreasing at \(t\) = 0-6 h, and it starts converting from compressive stress to tensile stress at \(t\) = 6-12 h. As cooling continues (\(t\) = 12-24 h), the maximum principal stress (tensile) increases. The rock is in a three-dimensional tension state on the SP#Tunnel face at \(t\) = 12-24 h, so the rock will fail before cooling for 12 h. The stress evolution trend of Path 4 (Figure 10g,h) is similar to that of Path 3. Compared with that of the tunnel wall, the rock on the SP#Tunnel face is more susceptible to failure and damage. Therefore, the tunnel face should be excavated earlier to prevent subsequent cracks from affecting the construction period. Figure 11 shows the temperature and the distribution of maximum and minimum principal stresses for tunnels with radii of 5 and 10 m. The comparison between Figures 11a-e and 11a'-e' shows that the minimum principal stress, maximum principal stress and temperature distribution behave similarly during cooling for tunnels with radii of 5 and 10 m. Figure 11a,a' show that the stress at locations of concentrated stress in the large tunnel (R = 10 m) can reach 94.8 MPa (see the red box in Figure 11a'), but the stress of the small tunnel (R = 5 m) only reaches 70 MPa at \(t\) = 0 h; thus, it can be inferred that rockbursts more easily occurs in the large tunnel than in the small tunnel. When the tunnel is cooled for 24 h, the maximum stress in the large tunnel (R = 10 m) also converts from compressive stress to tensile stress (see Figure 11d,d'), and the maximum tensile stress can reach 23.1 MPa (see the red box in Figure 11d'), which is also larger than that of the small tunnel (R = 5 m). This indicates that rock failure more easily occurs in large tunnels than in small tunnels. Taken together, the results show that when designing the tunnel size, the diameter should be as small as possible, while still maintaining functionality to effectively reduce the possibility of rockbursts and rock failure. Figure 11: _Cont._