Buckling of Cassini Oval Pressure Hulls Subjected to External Pressure

The paper focuses on Cassini oval pressure hulls under uniform external pressure. The Cassini oval pressure hull is proposed based on the shape index of Cassini oval. The buckling of a series of Cassini oval pressure hulls with the shape index of 0.09–0.30 and one spherical pressure hull with the diameter of 2 m is devoted. Such hulls are numerically studied in the case of constant volume, material properties, and wall thickness. The results show that Cassini oval pressure hulls with the shape index of 0.10–0.11 can resist about 4% more external pressure than the spherical one. This deviates from the classical mechanics conclusion that spherical shell is the optimal shape for underwater pressure resistant structures.


Introduction
Pressure hulls have attracted much attention for hundred years to meet the deep submergence requirement (Zoelly, 1915). Hulls can act as critical pressure vessels and buoyancy units for submarines, submersibles, and underwater facilities. Generally, they are shells of revolution subjected to uniform external pressure, which can provide safe living and working spaces for the carried crews and devices Cui, 2010a, 2011;Du et al., 2015;Ross, 2006;Ross et al., 2013). However, these pressure hulls are prone to instability in the case of hydrostatic pressure, which are significantly influenced by geometry, material, and imperfections (Zingoni, 2015;Arbocz and Starnes, 2002;Błachut, 2013;Błachut and Magnucki, 2008a;Thompson, 2015).
It has been long evidenced that spherical configuration is the most optimal shape for underwater pressure hulls. Spherical pressure hulls have an ideal geometry and the best load carrying capacity because of the equally distributed stress and deflection in the material. Consequently, such hulls are widely used in underwater vehicles especially in deep sea cases Cui, 2010a, 2011), and many studies have been devoted to their buckling analyses (Pan et al., 2010b(Pan et al., , 2012Cui, 2013;Cui et al., 2014;Zhang et al., 2017d). However, spherical pressure hulls are highly sensit-ive to the initial geometrical imperfections. This disadvantage not only costs a lot to fabricate such hulls, but also yields much difficulty to reliably assess their buckling pressures.
To alleviate this sensitivity, various non-spherical shells of revolution with positive Gaussian curvature have been proposed as well as the evaluations of their buckling behaviour. For example, the buckling of Cassini oval (Jasion and Magnucki, 2015a), clothoidal-spherical (Jasion and Magnucki, 2015b), circular arc (Magnucki and Jasion, 2013;Błachut and Smith, 2008b), generalized ellipsoidal shells (Błachut, 2003) are studied analytically, numerically or experimentally. More recently, the buckling of a family of egg-shaped pressure hulls have been systematically studied using bionic, analytical, numerical, and experimental approaches (Zhang et al., 2017a(Zhang et al., , 2017b(Zhang et al., , 2017c(Zhang et al., , 2017e, 2018a. However, the critical buckling loads of above non-spherical shells seem to be less than those of spherical ones especially in the thin-walled cases. Until now, the non-spherical pressure hulls have been rarely used in deep sea manned submersibles. There are two main reasons, on the one hand, compared with spherical pressure hulls, non-spherical pressure hulls have more complex contours, thus, the manufacture of non-spherical pressure hulls is difficult, and the cost is very high. On the other hand, compared with spherical pressure hulls, the ultimate bearing capacity of non-spherical pressure hulls is low. For example, the critical buckling loads of egg-shaped pressure shells are lower than that of spherical ones (Zhang et al., 2018a). So, it is necessary to explore a new type of hull with high load carrying capacity.
Recently, based on the results of Jasion and Magnucki (2015a), Zhang and his co-investigators have evaluated the elastic buckling of resin Cassini oval shells with various shape indices under external pressure (Zhang et al., 2018b). Interestingly, the load carrying capacity of Cassini oval shell with the shape index of 0.1 is even better than that of the spherical shell. However, this work only examines the elastic buckling of resin Cassini oval shells in laboratory scale, and whether the Cassini oval configuration can be used to steel pressure hull or not requires further investigation.
Therefore, the current work presents a family of Cassini oval pressure hulls with various shape indices. Both linear elastic and nonlinear elastic-plastic buckling of such pressure hulls are provided numerically using finite element method, which are also compared with the equivalent spherical one. The proposed Cassini oval pressure hulls with the shape indices ranging from 0.11 to 0.14 are found to carry much more external pressure than the spherical one, which can provide a theoretical guide to design and fabricate high buckling resistance pressure hulls for underwater vehicles. This discovery deviates from the classical mechanics conclusion that spherical shell is the optimal shape for underwater pressure resistant structures.

Materials and methods
The study involves the Cassini oval pressure hull and the spherical pressure hull. These hulls are designed with the same volume, material and yield strength for the purpose of comparison.

Design of Cassini oval pressure hulls
According to Cassini oval, an innovative thin-walled shell of revolution named as Cassini oval pressure hull, with a major axis, l, and a minor axis, w. Then, the shape index, k c , varies from 0.09 to 0.3, which controls the shape of Cassini oval pressure shell, and can be defined by Eq. (1). The geometry of such a pressure hull is determined using Eqs.
(1)-(4), as shown in Fig. 1. (1) The radius of the pressure shell of revolution is as follows: and where a and c are the equation parameters.
In addition, the size of shell is determined by assuming that the hull has the same volume as the widely used spherical pressure hull with median radius of 1 m. The volume of Cassini oval pressure hull is given as follows (Jasion and Magnucki, 2015a): Combined with the volume equation of spherical pressure hull: The major axis, l, and the minor axis, w of a series of Cassini oval pressure hulls under various shape indices have been designed and are given in Table 1.
Let all pressure hulls be made of titanium alloy Ti-6Al-4V ELI (TC4 ELI), and the material properties are as follows: Young modulus , yield strength , Poisson ratio (Zhang et al., 2018c). This kind of material is extensively used to make manned cabins of deep manned submersibles . Additionally, the wall thicknesses of all pressure hulls are assumed to be uniform with the magnitude of 10 mm.

Numerical models
For thin-walled shells of revolution, buckling is the main design constraint because such shells often tend to lose stability either in an elastic range or in an elastic-plastic range (Błachut, 2013;Błachut and Magnucki, 2008a). Consequently, this section is devoted to the buckling of several volume equivalent Cassini oval pressure hulls under various shape indices listed in Table 1. Also, the classical spherical pressure hull is evaluated for a like-for-like comparison with Cassini oval pressure hulls. For this purpose, the finite element method (FEM) available in ABAQUS code is implemented to perform linear eigenvalue analysis and nonlinear arc length analysis. Such analyses are in accordance with the existing rules (CCS, 2013; ENV, 2007), which have been widely adopted in previous studies (Zhang et al., 2017d(Zhang et al., , 2018c.
As can be seen from Fig. 2, the finite element mesh adopts the fully integrated S4 shell element to avoid hourglassing. The element number is determined using the convergence analysis of the mesh density, resulting in around 20000 elements for each model. A unit uniform external pressure, , is imposed on the surface of pressure hulls. In this way, the obtained eigenvalue in the linear analysis corresponds to the linear buckling load, whilst the obtained maximum load proportionality factor in the nonlinear analysis corresponds to the critical buckling load. To prevent rigid body motion, three nodes of each model along the axis of revolution are constrained according to CCS2013 (CCS, 2013), as follows: U y =U z =0, U x =U y =0, U y =U z =0. This kind of boundary condition does not introduce an overconstraint into the problem owing to the equally applied pressure and has been extensively adopted in the numerical buckling analysis of pressure hulls (Zhang et al., 2017c(Zhang et al., , 2017b(Zhang et al., , 2018a(Zhang et al., , 2018c. The material is assumed to be elastic in the linear eigenvalue analysis and to be perfectly elastic-plastic in nonlinear arc length analysis. Moreover, the eigenmode of each pressure hull obtained from the linear eigenvalue analysis is introduced as the initial geometrical imperfections to perform nonlinear arc length analysis, which is the common practice in the imperfect shell buckling analysis (ENV, 2007;Schmidt, 2000). The small imperfection amplitude is assumed to be 1, 2, 3 and 4 mm, respectively. The results obtained using FEM are shown in Figs. 3-6 and Table 1.

Linear buckling analysis
As can be seen from Table 1, the linear buckling pressure of Cassini oval pressure hull decreases monotonously with the increase in its shape index. Further, the linear buckling pressures of all Cassini oval pressure hulls are smaller than those of spherical pressure hull. Additionally, As can be seen from Fig. 3, the linear buckling modes of all pres-  TANG Wen-xian et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 503-508 505 sure hulls are similar and take the form of several circumferential waves (n), along with one longitudinal half-wave (m=1), which is typical for shells of revolution with positive Gaussian curvature Magnucki, 2015a, 2015b;Magnucki and Jasion, 2013). The reason for these phenomena may be that under uniform external pressure, such shells either tend to lose stability with the form of several circumferential waves and one-half meridional wave in the linear case or lose stability with the form of a local dent in the nonlinear case. Consequently, the obtained buckling mode is not caused by support conditions but by the shape and load of shells. Such linear buckling performances are consistent with previous studies about resin Cassini oval shells under uniform external pressure (Zhang et al., 2018b).

Nonlinear buckling analysis
On the other hand, as can be seen from the nonlinear results in Fig. 4, the critical buckling pressure decreases significantly with an increase in the imperfection amplitude. At the same amplitude, the critical buckling pressure of Cassini oval pressure hull first increases with an increase in the shape index, k c . This continues for up to k c =0.10, beyond which the critical buckling pressure decreases steadily with an increase in the shape index, k c . This phenomenon is more obvious in the case of small imperfection. Also, the critical buckling pressures of k c =0.09-0.15 Cassini oval pressure hulls are higher than those of spherical pressure hull. These observations extend our previous work (Zhang et al., 2018b), confirming that thin-walled Cassini oval shells with a rational shape index can support more pressure than the equivalent spherical shell.
Interestingly, the results show that Cassini oval pressure hulls with the shape indices of 0.10-0.11 can resist the external pressure about 4% higher than that of the spherical one. This result, if testified by experiments, can certainly be used to design the actual pressure resistant structures. More importantly, the fabricating tolerances should fall into this range to ensure a good load carrying capacity of pressure hull. On the other hand, the Cassini oval-shaped initial geometrical imperfection refers to the machining error in the manufacturing process, resulting in a deviation in the size of the spherical shell. However, the deviation is very small, so Cassini oval pressure shells themselves can be considered as spherical shells with Cassini oval-shaped initial geometrical imperfections. It can be seen that the importance of the shape, influencing the performance of the pressure shell. And the load carrying capacity can be improved in a suitable shape. Generally, for nonlinear buckling analysis on spherical hull, the first mode shape of Eigen value analysis is used as the initial geometrical defect, which will lead to decrease of the strength. The comparisons between Cassini oval pressure shell with different k c and pressure shells also reflect the importance of choosing correct initial geometrical defect shape, and such imperfection is beneficial and may increase its load carrying capacity. Similar phenomena can be found for cylindrical shell with circumferential and longitudinal corrugations. Owing to small corrugated imperfections, the buckling load of cylindrical shell is significantly improved (Ghanbari Ghazijahani et al., 2015a. The reason for these phenomena may be that either Cassini ovalshaped or corrugated shells are less sensitive to initial geometric imperfections than spherical or cylindrical ones.
The equilibrium paths of Cassini oval-shaped and spherical pressure hulls are similar, which demonstrates an un-

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TANG Wen-xian et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 503-508 u max /t σ y = 830 MPa stable character. Thus, the equilibrium paths of Cassini oval pressure hulls with different shape indices k c at the imperfection amplitude of 2 mm in Fig. 5, it plots the applied pressure, p, versus the maximum deflection-to-wall thickness ratio, . As can be seen from Fig. 5, the applied load first increased monotonically with an increase in the deflection up to the critical buckling point, beyond which the applied load decreased considerably. This trend is typical of shells of revolution subjected to uniform external pressure (Bazant and Cedolin, 2003). Also, as shown in Fig. 6, the critical buckling mode is consistent with the corresponding linear buckling mode, whilst the post buckling mode at the equator is of the form of a local dent. Furthermore, the maximum Huber-Mises stress of such pressure hull at the critical buckling point is 842.4 MPa, which is more than the yield strength of material ( ). It is indicated that the yield phenomenon occurs before the buckling due to the initial geometrical imperfections, and the Cassini oval pressure hull buckles in the elastic-plastic regime.

Conclusion
The present work provides a family of Cassini oval pressure shells, along with the results of a numerical study into their buckling performances. A like-for-like comparison is conducted between Cassini oval-shaped and spherical pressure hulls, as well. Owing to initial geometrical imperfections, all pressure hulls buckle in an elastic-plastic regime even though they are thin-walled structures. The critical buckling pressures 0.09-0.15 of Cassini oval pressure hulls are more than those of spherical pressure hull. Interestingly, Cassini oval pressure hulls with the shape indices of 0.10-0.11 can resist about 4% higher external pressure than the spherical one, which can provide a theoretical guide to design and fabricate high buckling resistance pressure hulls for underwater vehicles.