## 5 Conclusions  
Aiming to quantitatively assess the risk of deeply buried tunnel construction in the case of a hidden karst cave at the bottom of the tunnel, the first deeply buried tunnel construction failure model (PH\(-\)1) is firstly proposed based on the upper bound theorem of limit analysis and Protodyakonov's theory. Thereafter, based on the upper bound theorem of limit analysis and energy dissipation theory, the second deeply buried tunnel construction failure model (PH\(-\)2) is proposed. Furthermore, on the basis of the reliability theory, the dimensionless performance functions of each damage region under these two deeply buried tunnel construction damage modes are constructed. Finally, the instability probability of each damage region under the two damage modes is calculated using the Monte Carlo sampling method. Through comparison and analysis with the safety coefficients obtained by the non-intrusive stochastic finite element method and finite element strength reduction method, the effectiveness and feasibility of the calculation method of the instability probability for deeply buried tunnel construction in the case of a hidden karst cave at the bottom of the tunnel are verified. These two methods of calculating the instability probability in deeply buried tunnel construction were applied to actual projects to analyse and study the consequences of the tunnel diameter, the cohesion of rock and soil masses, the medium pressure inside the karst cave, and the clear distance between the tunnel and the karst cave on the safety and stability of the tunnel structure. The conclusions of the research are the following:  
1. The results calculated on the basis of the PH\(-\)1 failure model could be the smallest, and the results could be more conservative than the safety factors calculated by applying the strength reduction method. And the safety factors based on the PH\(-\)2 failure model decreases significantly with increasing tunnel depth. The curve of the safety factors in the PH\(-\)1 failure model fluctuates only slightly with an increasing tunnel depth. 2. When the tunnel passes through the moderately weathered dolomite strata and slightly weathered dolomite strata, there is not much difference between the results of the PH\(-\)1 failure mode and the PH\(-\)2 failure mode. However, the instability probability calculated from the PH\(-\)1 failure model and the instability probability calculated from the PH\(-\)2 failure model differ considerably when tunnelling in hard plastic red clay strata. Hence, both the PH\(-\)1 failure mode and the PH\(-\)2 failure mode are appropriate for analysing the instability probability in tunnel construction in moderately weathered dolomite and slightly weathered dolomite formations. Moreover, the PH\(-\)1 failure model is not suitable for analysing the probability of instability in tunnel construction in hard plastic red clay strata, while the PH\(-\)2 failure model is appropriate for analysing the probability of instability in tunnel construction in hard plastic red clay strata. 3. The method of calculating the instability probability in tunnel construction based on the two failure modes is applicable when the tunnel is constructed in hard plastic red clay stratum with the cohesion of 100 kPa. And the worse the geological conditions and the larger the diameter of the karst cave, the greater the threat to the safe construction of the tunnel. Furthermore, the smaller the clearance between the karst cave and the tunnel, and the larger the diameter of the karst cave, the more vulnerable the tunnel structure becomes. Conceptualization, B.W. and W.S. ; methodology, B.W. and W.S. ; writing--original draft, WS. ; writing--review and editing, B.W. ; supervision, B.W. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China (Nos. 52278397, 52168055) and the Jiangxi Natural Science Foundation Program (No. 20212 ACB204001). The financial support is gratefully acknowledged.