# Investigation of the Dynamic Buckling of Spherical Shell Structures Due to Subsea Collisions
## 4 Discussion  
According to Equation (5), when the pressure tends to be zero, which means that there is no external pressure under the shell, the buckling forces can increase without boundary. Physically speaking, if the pressure ratio is less than 0.05, the force ratio would be larger than 2.8. In such a case, it can be considered that the structural buckling of the shell cannot occur.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Case** & \(q^{\prime}\) & \(\mathbf{p^{\prime}/p^{\prime}}\) & **Ratio** & **Case** & \(q^{\prime}\) & \(\mathbf{p^{\prime}/p^{\prime}}\) & **Ratio** & **Case** & \(q^{\prime}\) & \(\mathbf{p^{\prime}/p^{\prime}}\) & **Ratio** \\ \hline \multirow{3}{*}{\(t=0.04\)} & 0.10 & 3.0/1.40 & 2.12 & \multirow{3}{*}{\(t=0.04\)} & 0.18 & 1.5/0.73 & 2.12 & \multirow{3}{*}{\(t=0.04\)} & 0.16 & 1.9/0.84 & 2.26 \\  & 0.15 & 1.7/0.90 & 1.97 & & 0.26 & 1.0/0.47 & 2.22 & \multirow{3}{*}{\(t=0.03\)} & 0.23 & 1.3/0.55 & 2.36 \\  & 0.20 & 1.3/0.65 & 1.99 & & \(R=1.5\) & 0.12 & 2.0/1.15 & 1.74 & \multirow{3}{*}{\(R=2.5\)} & 0.31 & 0.9/0.37 & 2.43 \\  & 0.25 & 1.0/0.50 & 2.03 & & \(E=210\) & 0.14 & 1.7/0.97 & 2.12 & \multirow{3}{*}{\(E=210\)} & 0.39 & 0.6/0.27 & 2.22 \\ \hline \hline \end{tabular}
\end{table}
Table 13: The ratio of criteria in the random parameters (unit: meters, GPa).  
Figure 21: The comparison of prediction and numerical simulation.  
During the collision impacts, the maximum force can be considered to be 1.70 times the average force derived from Equation (7). The impulse of collision is linear to the velocity and added mass, but has a negligible change due to the thickness and elastic modulus. In addition, the quasi-static buckling force deduced by Equation (5) is about only 0.48 times that of the maximum force during the collision. It should be noted that Equation (5) is based on the elastic material, as the buckling usually occurs before the plastic stage. The effect of strain rate is neglected for the low velocity analysis in this study. It is noted that the spherical shell could reach up to the plastic stage or over the limited stress under the collision before buckling. In that case, the structure will be damaged or failed for the lack of strength, so it is suggested that the static analysis should be analyzed before the buckling analysis.  
For applications from the outcome of this study, the critical force under a single point load and a single uniform pressure load, which are named \(p_{c}\) and \(q_{c}\), respectively, can be calculated firstly by Equation (1) and Equation (2) relevant to the property of the material and the parameters of the geometry. Then, the external pressure \(q\) can be determined by the operation depth under the sea; therefore, Equation (5) can be used by the designer to determine the quasi-static buckling force. After obtaining the normalized force ratio of \(p^{\prime}\), it can be used to identify the critical velocity, which may induce the dynamic buckling by Equation (13). By the mechanism described above, the dynamic buckling of the spherical shell under the collision impacts and external pressure can be evaluated.