# Investigation of the Dynamic Buckling of Spherical Shell Structures Due to Subsea Collisions
## 5 Conclusions  
This paper has studied the buckling of a spherical shell structure subjected to both external pressure and point load in order to establish criteria to predict the dynamic buckling of a spherical shell structure under collision impacts. In the present study, the standard for static buckling, including external pressure and concentrated load, was established as Equation (5). Furthermore, the relationship between static force and dynamic force under subsea collisions has been studied based on the theory that the static buckling criteria can be used to reflect the dynamic collision problem. By extensive parametric analyses, a precise formulation has been formed and validated using multiple regression analyses. During the collisions, the maximum and mean contact forces are proportional to the velocity and thickness of the shell, and are also linear to the total mass and elastic modulus. For the collision impulses, they are proportional to the velocity and added mass, but the elastic modulus and shell thickness do not affect their momentum change. In addition, the same skewness ratio of the contact forces during different parameters can be considered the same value: 1.70. In that case, the contact force during collision can be described as a sine function with respect to time.  
Overall, this study has established a fundamental framework and novel theory to simulate, design, and validate the dynamic buckling for a spherical shell structure under collision impacts and external pressure. The precise formulations shown in the present paper enable the prediction of the dynamic buckling of spherical shell structures subjected to subsea collisions. Such insights and outcomes will benefit the potential applications involving submarines and deep sea exploration, etc.  
L.P., S.K. and Z.D.C. developed and initialized the concept and methods; L.P. developed model and carried out analyses; S.K. provided analysis tools and validated the model; Z.D.C. and S.S.W. provided industry guidance and verified results with national standards; L.P., S.K., Z.D.C. and S.S.W. wrote and reviewed the paper; L.P. and S.S.W sought funding. This work was supported by the national Natural Science Foundation of China (grant numbers 51508238) and by the Jiangsu Postdoctoral Research Plan (grant numbers 1601014B). The APC is sponsored by the University of Birmingham Library, U.K.  
The second author wishes to gratefully acknowledge the Japan Society for Promotion of Science (JSPS) for his JSPS Invitation Research Fellowship (Long-term), Grant No L15701, at Track Dynamics Laboratory, Railway Technical Research Institute and at Concrete Laboratory, the University of Tokyo, Tokyo, Japan. The JSPS financially supports this work as part of the re-search project, entitled \"Smart and reliable railway infrastructure\". Special thanks to European Commission for H2020-MSCA-RISE Project No. 691135 \"RISEN: Rail Infrastructure Systems Engineering Network\" (www.risen2 rzil.eu) (2010). In addition, the collaboration and assistance from EU Cost Action TU1409 (Mathematics for Industry Network) are highly appreciated.  
The authors declare that there are no conflicts of interest regarding the publication of this paper.