Limit analysis-based calculation of the critical thickness of the waterproof rock mass overlying a karst cavity in TBM tunneling

When a TBM tunnel crosses a karst-developed area, its excavation induces disturbances and compressive stress on water-bearing cavities, potentially triggering water inrush hazards. Based on the upper-bound theorem of limit analysis, a failure model is developed for the waterproof rock mass situated ahead and above the tunnel face, conceptualized as a composite of a truncated cone and a wedge. A stability analysis method is formulated using the virtual work principle. An analytical formula for the critical thickness of the waterproof layer is then derived and validated. Finally, the effects of tunnel design parameters, rock mass properties, and cavity characteristics on the critical failure thickness of the waterproof rock mass are systematically analyzed. The results indicate that the mechanical parameters of the surrounding rock (internal friction angle φ and cohesion c, especially φ) have the greatest impact on the critical thickness of the water-resisting rock mass (H_α). This is followed by the tunnel diameter (D), the karst cavity pressure (p_w), and the face support pressure (p), with the diameter of the karst cavity (D_R) having the least influence. Furthermore, H_α shows a positive linear correlation with D, D_R, and p_w, and a negative linear correlation with p. In contrast, it exhibits a negative exponential relationship with the surrounding rock's φ and c. Additionally, when the c and φ of the surrounding rock increase to a certain level (φ>37.5°, c>3 MPa), the reduction in H_α significantly diminishes. This suggests that strengthening the surrounding rock through methods like grouting is only effective within a certain range for preventing karst water inrush in tunnels, with the effect substantially weakening beyond this range. These findings provide a theoretical basis for predicting the critical thickness of waterproof rock mass above TBM tunnels and provide support for the safe construction of TBM tunneling in karst regions.