## 3 Numerical Simulation Results and Analysis
### Failure Characteristics of the Double U-Shaped Caverns under Vertically Incident Stress Wave  
The direction of incident stress waves is a critical factor for the dynamic response of the cavern. When the dynamic disturbances originate from the vertical direction, the caverns exhibit different failure characteristics compared to those under horizontally incident stress waves. In this section, a vertically incident triangular plane stress wave was applied at the model bottom, with stress wave parameters \(t_{s}/t_{r}\) = 5, \(t_{r}\) = 2 ms. The dynamic response  
Figure 6: Failure patterns of surrounding rock under horizontally incident stress wave under different cavern clearances: (**a**) \(d\) = 3.75 m; (**b**) \(d\) = 2.50 m; (**c**) \(d\) = 1.25 m.  
Figure 7: The evolution process of kinetic energy at B2 under different cavern clearances: (**a**) tangential stress; (**b**) kinetic energy. and failure characteristics of deep double U-shaped caverns under the vertically incident stress waves were investigated by monitoring the stress and energy at key points around the caverns. When a plane stress wave is incident vertically, the identical geometric dimensions and symmetric arrangement of caverns 1 and 2 along the model center result in an approximately symmetrical distribution of the overall stress field of the model. Taking cavern 1 as an example, when the amplitude of the incident stress wave \(P_{d}\) is 15 MPa, the evolution of tangential stress at the sidewalls (B1, C1) and the midpoint of the floor (D1) of the cavern is depicted in Figure 8. The vertically incident stress wave causes the dynamic tensile stress concentration at the cavern floor, and as the cavern clearance decreases, the reduction extent of tangential stress at the midpoint of the floor decreases. Dynamic compressive stress concentration occurs at the sidewalls of the cavern, and as the cavern clearance decreases, the corresponding peak tangential stress increases. Comparing the tangential stress at B1 and C1, it is noted that their evolution processes are similar, but the peak tangential stress at C1 is greater than that at B1, indicating more significant stress concentration in the right sidewall of cavern 1 compared to the left sidewall. When the cavern clearance is 1.25 m, the final tangential stress at point C1 significantly decreases compared to the initial stress, indicating significant damage and failure in the right sidewall of cavern 1. To further investigate the damage characteristics of the surrounding rock under vertically incident stress waves, the stress wave amplitude rises to \(P_{d}=60\) MPa. The damage modes around the cavern with different cavern clearances (\(d=1.25\), 2.50, and 3.75 m) under vertically incident stress waves are depicted in Figure 9. As the stress waves are vertically incident, the stress wave reflects at the cavern floor, generating numerous tensile microcracks parallel to the free surface around the cavern floor, and the spalling phenomena can be observed. As the lateral pressure coefficient is less than 1.0, the in situ stress field  
Figure 8: The evolution process of kinetic energy at different positions of the caverns under vertically incident stress wave: (**a**) midpoint of the left sidewall (B1); (**b**) midpoint of the right sidewall (C1); (**c**) midpoint of the floor (D1). causes the static stress concentrations at the cavern sidewalls. When the vertical stress wave propagates to the sidewalls, dynamic compressive stress concentration occurs at the sidewalls. Combined with the initial static compressive stress, dynamic compressive-shear failure occurs in the sidewalls, generating a large number of rock fragments. Moreover, the damage degree of the adjacent sidewall is more severe than that of the non-adjacent sidewall. As the cavern clearance decreases, stress concentration intensifies in the intermediate rock pillar, and the arch-shaped surfaces generated on the rock pillar boundaries deepen gradually.