**Figure 1.**
The diagram of a cylindrical coordinate.

**Figure 1.**
The diagram of a cylindrical coordinate.

**Figure 2.**
Frequency comparison between this study and Donnell’s theory.

**Figure 2.**
Frequency comparison between this study and Donnell’s theory.

**Figure 3.**
The calculation sketch of energy when detonation is in the middle along the width.

**Figure 3.**
The calculation sketch of energy when detonation is in the middle along the width.

**Figure 4.**
The plot of pressure vs. time in theory and numerical simulation. (**a**) The pattern of blast load, (**b**) The time history of blast load in simulation

**Figure 4.**
The plot of pressure vs. time in theory and numerical simulation. (**a**) The pattern of blast load, (**b**) The time history of blast load in simulation

**Figure 5.**
The experiment model and the numerical model. (**a**) Experiment geometry model (mm). (**b**) Numerical model with hiding air element.

**Figure 5.**
The experiment model and the numerical model. (**a**) Experiment geometry model (mm). (**b**) Numerical model with hiding air element.

**Figure 6.**
The comparison of the displacement time history of C1 and C2.

**Figure 6.**
The comparison of the displacement time history of C1 and C2.

**Figure 7.**
The sketch of the finite element model. (**a**) The mesh in the Z=0 plane, (**b**) The mesh of 3-D

**Figure 7.**
The sketch of the finite element model. (**a**) The mesh in the Z=0 plane, (**b**) The mesh of 3-D

**Figure 8.**
Model of the vertical view, the plan view and the detonation arrangement. (**a**) Vertical view. (**b**) Plan view.

**Figure 8.**
Model of the vertical view, the plan view and the detonation arrangement. (**a**) Vertical view. (**b**) Plan view.

**Figure 9.**
The displacement mode when t = 0.26 s, t = 0.76 s, t = 1.26 s, P11. (**a**) t = 0.26 s, (**b**) t = 0.76 s, and (**c**) t = 1.26 s.

**Figure 9.**
The displacement mode when t = 0.26 s, t = 0.76 s, t = 1.26 s, P11. (**a**) t = 0.26 s, (**b**) t = 0.76 s, and (**c**) t = 1.26 s.

**Figure 10.**
The displacement mode when t = 0.26 s, t = 0.76 s, t = 1.26 s, P13. (**a**) t = 0.26 s, (**b**) t = 0.76 s, and (**c**) t = 1.26 s.

**Figure 10.**
The displacement mode when t = 0.26 s, t = 0.76 s, t = 1.26 s, P13. (**a**) t = 0.26 s, (**b**) t = 0.76 s, and (**c**) t = 1.26 s.

**Figure 11.**
The displacement vs. time and correspondent result of FFT, P11, and P13. (**a**) The displacements of points at the condition P11. (**b**) The corresponding result of FFT at the condition P11. (**c**) The displacements of points at the condition P13. (**d**) The corresponding result of FFT at the condition P13.

**Figure 11.**
The displacement vs. time and correspondent result of FFT, P11, and P13. (**a**) The displacements of points at the condition P11. (**b**) The corresponding result of FFT at the condition P11. (**c**) The displacements of points at the condition P13. (**d**) The corresponding result of FFT at the condition P13.

**Figure 12.**
The displacement mode when t = 0.24 s, t = 0.60, t = 1.26 s, P21. (**a**) t = 0.24 s, (**b**) t = 0.60 s, and (**c**) t = 1.26 s.

**Figure 12.**
The displacement mode when t = 0.24 s, t = 0.60, t = 1.26 s, P21. (**a**) t = 0.24 s, (**b**) t = 0.60 s, and (**c**) t = 1.26 s.

**Figure 13.**
The displacement mode when t = 0.24 s, t = 0.60 s, t = 1.26 s, P23. (**a**) t = 0.24 s, (**b**) t = 0.60 s, and (**c**) t = 1.26 s.

**Figure 13.**
The displacement mode when t = 0.24 s, t = 0.60 s, t = 1.26 s, P23. (**a**) t = 0.24 s, (**b**) t = 0.60 s, and (**c**) t = 1.26 s.

**Figure 14.**
The displacement vs. time and correspondent result of FFT, P13, and P23. (**a**) The displacements of points at the condition P21. (**b**) The corresponding result of FFT at the condition P21. (**c**) The displacements of points at the condition P23. (**d**) The corresponding result of FFT at the condition P23.

**Figure 14.**
The displacement vs. time and correspondent result of FFT, P13, and P23. (**a**) The displacements of points at the condition P21. (**b**) The corresponding result of FFT at the condition P21. (**c**) The displacements of points at the condition P23. (**d**) The corresponding result of FFT at the condition P23.

**Figure 15.**
The plot of blast load in the condition of P1 and P2.

**Figure 15.**
The plot of blast load in the condition of P1 and P2.

**Figure 16.**
The plot of global and shell part energy vs. time. (**a**) The energy plot of all parts in the simulation, (**b**) The energy plot of shell part

**Figure 16.**
The plot of global and shell part energy vs. time. (**a**) The energy plot of all parts in the simulation, (**b**) The energy plot of shell part

**Figure 17.**
The plot of energy vs. time and the integration of shell part energy.

**Figure 17.**
The plot of energy vs. time and the integration of shell part energy.

**Figure 18.**
The contour of the radius-displacement when the thickness is 0.01 m, P12.

**Figure 18.**
The contour of the radius-displacement when the thickness is 0.01 m, P12.

**Figure 19.**
The counter of the radius-displacement when the thickness is 0.01 m, P22.

**Figure 19.**
The counter of the radius-displacement when the thickness is 0.01 m, P22.

**Figure 20.**
The velocity counter of the air part with a thickness of 0.01m, P12, when t = 1.20 s. (**a**) Air part outside. (**b**) Air part inside.

**Figure 20.**
The velocity counter of the air part with a thickness of 0.01m, P12, when t = 1.20 s. (**a**) Air part outside. (**b**) Air part inside.

**Figure 21.**
The velocity counter of air part with thickness 0.01 m, P22, when t = 1.20 s. (**a**) Air part outside. (**b**) Air part inside.

**Figure 21.**
The velocity counter of air part with thickness 0.01 m, P22, when t = 1.20 s. (**a**) Air part outside. (**b**) Air part inside.

**Figure 22.**
The plot of displacement time history when the thickness is 0.03 m, P12.

**Figure 22.**
The plot of displacement time history when the thickness is 0.03 m, P12.

**Figure 23.**
The plot of x-displacement vs. time when the thickness is 0.03 m, P22.

**Figure 23.**
The plot of x-displacement vs. time when the thickness is 0.03 m, P22.

**Figure 24.**
The configuration of the shell with displacement multiplied 50 times, t = 1.30 s.

**Figure 24.**
The configuration of the shell with displacement multiplied 50 times, t = 1.30 s.

**Figure 25.**
The energy time history with thickness t = 0.01 m, x = 0.0 m and z = 0, z = L/2 respectively. (**a**) z = 0.0 m. (**b**) z = L/2 m.

**Figure 25.**
The energy time history with thickness t = 0.01 m, x = 0.0 m and z = 0, z = L/2 respectively. (**a**) z = 0.0 m. (**b**) z = L/2 m.

**Figure 26.**
The plot of the ratio vs. thickness when x = 0.

**Figure 26.**
The plot of the ratio vs. thickness when x = 0.

**Figure 27.**
The comparison between Equations (14) and (18).

**Figure 27.**
The comparison between Equations (14) and (18).

**Figure 28.**
The comparison between Equations (14) and (18).

**Figure 28.**
The comparison between Equations (14) and (18).

**Figure 29.**
Plot of ratio vs. thickness when x = 0.

**Figure 29.**
Plot of ratio vs. thickness when x = 0.

**Figure 30.**
The comparison of decreasing ratio in the conditions of Z = 0, Z = L/4 and theory.

**Figure 30.**
The comparison of decreasing ratio in the conditions of Z = 0, Z = L/4 and theory.

**Figure 31.**
The plot and fitting curve of energy vs. time with E = 140 and 470 in case 1, respectively.

**Figure 31.**
The plot and fitting curve of energy vs. time with E = 140 and 470 in case 1, respectively.

**Figure 32.**
The comparison of energy and decrease ration vs. elastic modulus.

**Figure 32.**
The comparison of energy and decrease ration vs. elastic modulus.

**Figure 33.**
The plot of energy vs. time with E = 70, 470 GPa in case 1, respectively.

**Figure 33.**
The plot of energy vs. time with E = 70, 470 GPa in case 1, respectively.

**Figure 34.**
The plot of energy vs. elastic modulus with t = 0.01 m, detonation position, x = 0, z = 0.

**Figure 34.**
The plot of energy vs. elastic modulus with t = 0.01 m, detonation position, x = 0, z = 0.

**Figure 35.**
The plot of energy vs. time at density $\rho $ = 10 ton/m^{3}.

**Figure 35.**
The plot of energy vs. time at density $\rho $ = 10 ton/m^{3}.

**Figure 36.**
The plot of decrease ratio vs. density (t/m^{3}).

**Figure 36.**
The plot of decrease ratio vs. density (t/m^{3}).

**Figure 37.**
The plot of energy vs. time at the density = 7.8, 2.0, respectively.

**Figure 37.**
The plot of energy vs. time at the density = 7.8, 2.0, respectively.

**Figure 38.**
The comparison of internal energy and kinetic energy at density = 7.8, 4.0 ton/m^{3}.

**Figure 38.**
The comparison of internal energy and kinetic energy at density = 7.8, 4.0 ton/m^{3}.

**Table 1.**
The total energy with different density of mesh.

**Table 1.**
The total energy with different density of mesh.

**Density of Mesh (m)** | 2.0 | 1.0 | 0.8 | 0.5 | 0.3 | 0.2 | 0.1 |

**Total Energy (10**^{4} J) | 4.52 | 4.21 | 3.93 | 3.90 | 3.89 | 3.89 | 3.88 |

**Max Displacement (mm)** | 82 | 73 | 70 | 69 | 68 | 67 | 67 |

**Table 2.**
The initial total energy (×10^{4} J) and the ratio $\beta $ at different $T$ /m.

**Table 2.**
The initial total energy (×10^{4} J) and the ratio $\beta $ at different $T$ /m.

$\mathit{T}$ | $\mathit{z}/\mathit{L}$ | ${\mathit{E}}_{0}$ | $\mathit{\beta}$ | $\mathit{z}/\mathit{L}$ | ${\mathit{E}}_{0}$ | $\mathit{\beta}$ | $\mathit{z}/\mathit{L}$ | ${\mathit{E}}_{0}$ | $\mathit{\beta}$ | $\mathit{z}/\mathit{L}$ | ${\mathit{E}}_{0}$ | $\mathit{\beta}$ |
---|

0.010 | 0/4 with x = 0 R = 5 L = 30 | 6.2 | 3.87 | 1/4 with x = 0 R = 5 L = 30 | 12 | 3.74 | 2/4 with x = 0 R = 5 L = 30 | 12.5 | 3.93 | 2/4 with x = 0 R = 7.5 L = 30 | 9.9 | 3.89 |

0.015 | 4.1 | 2.53 | 8.1 | 2.46 | 8.25 | 2.59 | 6.6 | 2.59 |

0.020 | 3.2 | 2.03 | 6.2 | 2.04 | 6.31 | 2.16 | 4.9 | 1.94 |

0.025 | 2.5 | 1.44 | 4.8 | 1.72 | 4.94 | 1.89 | 4.0 | 1.55 |

0.030 | 2.0 | 0.87 | 3.9 | 0.77 | 4.03 | 0.83 | 3.3 | 1.26 |

0.040 | 1.5 | 0.69 | 2.9 | 0.65 | 2.94 | 0.69 | 2.3 | 0.86 |

0.060 | 1.0 | 0.38 | 2.0 | 0.39 | 1.96 | 0.38 | 1.5 | 0.33 |

0.100 | 0.6 | 0.24 | 1.2 | 0.24 | 1.25 | 0.24 | 1.0 | 0.23 |

0.200 | 0.3 | 0.12 | 0.6 | 0.13 | 0.59 | 0.12 | | |

0.500 | 0.1 | 0.02 | 0.2 | 0.03 | 0.19 | 0.03 | | |

**Table 3.**
The initial total energy (×10^{4} J) and the ratio $\beta $ at different $T$ /m.

**Table 3.**
The initial total energy (×10^{4} J) and the ratio $\beta $ at different $T$ /m.

$\mathit{T}$ | $\mathit{z}/\mathit{L}$ | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | $\mathit{\beta}$ | $\mathit{\alpha}$ | $\mathit{z}/\mathit{L}$ | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | $\mathit{\beta}$ | $\mathit{\alpha}$ | $\mathit{z}/\mathit{L}$ | ${\mathit{E}}_{0}$ | $\mathit{\beta}$ |
---|

0.010 | 0/4 with ×2.5 R = 5 | 5.2 | 1.6 | 5.12 | 0.38 | 1/4 with ×2.5 R = 5 | 9.8 | 3.6 | 4.46 | 0.32 | 2/4 with ×0.0 R = 7.5 | 9.9 | 3.89 |

0.015 | 3.2 | 1.4 | 3.35 | 0.26 | 5.9 | 3.4 | 3.57 | 0.28 | 6.6 | 2.59 |

0.020 | 2.0 | 1.5 | 2.95 | 0.32 | 4.1 | 3.0 | 3.01 | 0.25 | 4.9 | 1.94 |

0.025 | 1.6 | 1.2 | 1.74 | 0.24 | 3.1 | 2.5 | 1.81 | 0.24 | 4.0 | 1.55 |

0.030 | 0.8 | 1.5 | 1.30 | 0.37 | 2.4 | 2.4 | 1.86 | 0.27 | 3.3 | 1.26 |

0.040 | 0.8 | 0.9 | 1.24 | 0.15 | 1.6 | 1.9 | 1.41 | 0.17 | 2.3 | 0.86 |

0.060 | 0.6 | 0.6 | 1.07 | 0.13 | 1.3 | 1.1 | 1.18 | 0.13 | 1.5 | 0.33 |

0.100 | 0.3 | 0.5 | 1.11 | 0.17 | 0.6 | 0.9 | 1.07 | 0.16 | 1.0 | 0.23 |

**Table 4.**
The initial total energy (×10^{4} J) and the ratio $\beta $ at different E and $\upsilon $, with R = 5, L = 30.

**Table 4.**
The initial total energy (×10^{4} J) and the ratio $\beta $ at different E and $\upsilon $, with R = 5, L = 30.

$\mathit{E}$ | $\mathit{\upsilon}$ | ${\mathit{E}}_{0}$ | $\mathit{\beta}$ | $\mathit{\upsilon}$ | ${\mathit{E}}_{0}$ | $\mathit{\beta}$ | $\mathit{\upsilon}$ | ${\mathit{E}}_{1}$ | $\mathit{\beta}$ | ${\mathit{E}}_{2}$ | $\mathit{\alpha}$ |
---|

910 | Case 1 0.23 Z = 0/4 × 0.0 | 1.89 | 3.29 | Case 2 0.13 Z = 0/4 × 0.0 | 1.91 | 3.28 | Case 3 0.07 Z = 2/4 × 2.5 | 3.37 | 3.25 | 5.28 | 0.30 |

470 | 3.46 | 3.45 | 3.52 | 3.45 | 2.11 | 5.09 | 3.53 | 0.61 |

210 | 6.16 | 3.82 | 5.91 | 3.86 | 10.8 | 6.41 | 3.73 | 0.55 |

170 | 6.76 | 4.01 | 7.26 | 3.95 | 12.7 | 7.19 | 3.10 | 0.46 |

140 | 7.81 | 4.29 | 8.25 | 4.53 | 12.2 | 7.34 | 4.40 | 0.52 |

70 | 10.2 | 4.55 | 11.7 | 4.62 | 13.0 | 7.82 | 5.65 | 0.37 |

40 | 11.8 | 4.29 | 11.7 | 4.42 | 11.5 | 9.13 | 6.78 | 0.31 |

10 | 14.3 | 4.31 | 13.8 | 4.37 | 8.64 | 14.6 | 19.7 | 0.05 |

**Table 5.**
Simulation results at plastic kinematic model.

**Table 5.**
Simulation results at plastic kinematic model.

$\mathit{\rho}$**(ton/m**^{3}) | 10 | 7.8 | 4.0 | 2.0 | 1.0 | 0.6 | 0.3 |

${\mathit{E}}_{\mathbf{0}}$**(10**^{4} J) | 1.94 | 2.13 | 2.31 | 2.31 | 2.18 | 1.95 | 1.80 |

$\mathit{\beta}$ | 4.62 | 3.84 | 2.18 | 1.16 | 1.73 | 2.33 | 3.53 |

**Table 6.**
Plastic kinematic model, $E$ = 210 GPa, $\upsilon $ = 0.23, ${f}_{y}$ = 210 MPa, ${E}_{t}$ = 4.0 GPa.

**Table 6.**
Plastic kinematic model, $E$ = 210 GPa, $\upsilon $ = 0.23, ${f}_{y}$ = 210 MPa, ${E}_{t}$ = 4.0 GPa.

$\mathit{\rho}$**(ton/m**^{3}) | 10 | 7.8 | 4.0 | 2.0 | 1.0 |

${\mathit{E}}_{\mathbf{0}}$**(10**^{4} J) | 5.62 | 6.08 | 7.52 | 7.86 | 8.21 |

$\mathit{\beta}$ | 4.63 | 3.76 | 3.12 | 1.48 | 2.75 |

$\mathit{\alpha}$ | / | / | 0.12 | 0.03 | 0.06 |