# Investigation of the Dynamic Buckling of Spherical Shell Structures Due to Subsea Collisions
## 1 Introduction  
Deep sea submarines have been used widely in many applications such as investigation, the exploration of oceanographic resources, military action, and so on [1; 2; 3; 4; 5]. In reality, all subsea structures are subjected to significant hydrostatic pressure [6; 7]. Therefore, the pressure hull is the most important component of any deep submersible. It cannot only offer the capability for resisting the external load, it can also provide more workspace for captains or scientists. The shape of the hull is usually designed as a spherical shape [8; 9]. In practice, the exploration vehicle usually works at a sea depth of 1.5-8.5 km under the mean sea level with a low speed [10; 11; 12; 13]. During operations, a collision could occur between the submarine and any obstacle such as a rock wall, iceberg, cliff, or another submarine, as shown in Figure 1. The collisions can induce the damage of the submarine nose structure through the failure of components, and the unstable deformation or buckling of the spherical nose. The buckling behavior and buckling criterion of spherical shell structures subjected to collisions have attracted more research attention over recent years [14; 15; 16]. Many previous studies have been carried out that investigate spherical shells under a uniform pressure [17; 18; 19; 20]. The first and most preeminent researcher who established the buckling theory is [PERSON] [21]. His classical conclusion was derived under the elastic material and small deformationassumption. Afterwards, many studies considering large deformation and more complicated loads have been investigated [22; 23; 24]. [PERSON] [25] evaluated the buckling of a multi-segment underwater hull. In his study, it was revealed that the collapse pressures from numerical analyses agreed well with all five experimental results obtained from the collapse test of a laboratory-scale mild steel structure. [PERSON] and [PERSON] [26] studied the effect of thickness on the buckling strength of egg-shaped pressure hulls. The results showed that the egg-shaped pressure hulls seem to be applicable to deep sea manned/unmanned submersibles, especially to full ocean depth ones (11,000 m). [PERSON] [27] and [PERSON] [28] studied the buckling load using the energy method for subsea collisions. However, they studied the static problem without dynamic effect. It was reported that the deep pressure hull could be buckled without exceeding the limited strength under collision [29; 30], which can be strongly affected by the shell thickness, collision velocity, and properties of materials [31; 32]. Nevertheless, spherical pressure hulls are the most efficient and popular type for the deep-sea pressure hull structures [33; 34; 35], as the spherical hull has the lowest buoyancy factor (weight-to-buoyancy ratio) [36; 37]. [PERSON] and [PERSON] [38] reported the results of a numerical study on the optimal design of a multi-segment spherical hull, which contained details regarding the sensitivity analysis of the design variables, including the thickness and the inner radius of the rib-ring. Furthermore, [PERSON] et al. [39] analyzed the capacity of the ring-stiffened cylindrical shell via the experimental method. The paper exhibited the influence on the stability of the cylindrical shell, but had little analytical discussion based on the derivations of the buckling loads of the cylindrical shell. [PERSON] et al. [40] studied the stability of submarines with and without artificial failure. Several numerical models to evaluate stability were established, but whether or not the results could be extended to the other applications was not discussed. [PERSON] et al. [41] analyzed the nonlinear static and buckling of spherical shells under external pressure, incorporating the effects of temperature. They took von Karman-Donnell geometrical nonlinearity and the external pressure load without a concentrated load into account.