# Damage Characteristics and Dynamic Response of RC Shells Subjected to Underwater Shock Wave
## 4 Numerical Investigation  
### Numerical Model  
The schematic diagram of bottom-sitting shell structures subjected to underwater explosive shock waves is shown in Figure 8. The verified numerical method and material models were employed to construct a fully coupled numerical model of an underwater bottom-sitting shell structure with water explosive through LS-DYNA SMP R11.1.0 finite element software, as shown in Figure 9. Due to the symmetry of the structure, the 1/4 symmetry modeling method was used to reduce the mesh number of the model. The water domain has a 4 m radius at the bottom and a 13 m vertical length of the coupling model of 1/4 semi-spherical RC shell structure with the water explosive. The water domain has a 4 m width at the bottom and a 13 m vertical length of the 1/4 semi-cylindrical RC shell structure-water explosive coupling model. The nodes located on the 1/4 symmetrical surface of the model are set to be symmetrical constraints. The outer surface of the water domain is set to be non-reflecting. The displacements in the \(x\), \(y\), and \(z\) directions at the bottom of the semi-spherical and semi-cylindrical RC bottom-sitting shell are set to zero. The shell structure dimensions of semi-spherical and semi-cylindrical RC shells are shown in Figures 10 and 11, respectively. The RC shell is simulated using the Lagrange algorithm, while the air, water, and TNT are modeled using the Euler algorithm. The interaction between the shell and water is simulated using a fluid-structure coupling method, and the explosive is set as volume fraction form. The steel bars are simulated using beam elements. As shown in Figure 9a, the bottom layer of the semi-spherical RC shell structure is reinforced by steel bars with a protective layer thickness of about 45 mm. As shown in Figure 9b, the bottom layer of the semi-cylindrical shell structure is reinforced by steel bars, and the internal and external layers of side walls at both ends are reinforced by steel bars with a protective layer thickness of about 45 mm. In addition, as shown in Figure 12a, the extension of steel bars is concentrated in the top position of the semi-spherical RC shell. As shown in Figure 12b, the extension of steel bars is concentrated in the bottom center position of the side wall of the semi-cylindrical RC shell. To realize the rational distribution of steel bars, the reinforcement mode that gradually reduces steel bars is adopted in the top center of the semi-spherical RC shell and the bottom center of the side wall of the semi-cylindrical RC shell, as shown in Figure 12. The numerical model whose numerical methods and material models have been verified was used to study the effects of the shock wave transmission medium, explosive equivalent, explosive distance, hydrostatic pressure, and reinforcement on the damage characteristics and dynamic response of bottom-sitting RC shell structures under underwater explosive shock waves. Figure 7: **The comparison of the pressure time history curve between the underwater explosion test [25] and numerical simulation: (a) the pressure time history curve at point 1; (b) the pressure time history curve at point 2. **Figure 8: The schematic diagram of bottom-sitting shell structures subjected to underwater explosive shock waves. Figure 10: The dimensions of semi-spherical RC shells: (**a**) the overall structure diagram; (**b**) the aerial view of the maximum section; (**c**) the front view of the maximum section. Figure 9: The coupling model of the 1/4 RC shell structure with water and underwater explosive shock waves: (**a**) the 1/4 semi-spherical RC shell structure with water and underwater explosive shock waves; (**b**) the 1/4 semi-cylindrical RC shell structure with water and underwater explosive shock waves. Figure 11: The dimensions of semi-cylindrical RC shells: (**a**) the overall structure diagram; (**b**) the aerial view of the maximum section; (**c**) the side view of the maximum section; (**d**) the front view of the maximum section.