Keywords: toroids, circular and elliptical cross-sections, external pressure, buckling.

Progress in understanding of the role of the oceans in human development is dependent on the provision of a variety of manned and unmanned underwater vehicles being available. Manned submersibles, currently in operation, are capable of reaching the depth of 6 km. Pressure hulls of manned submersibles have traditionally been metallic, e.g., from high strength steel HY80 (or higher), aluminium or titanium alloys. Ring reinforced cylinders/cones closed by domed ends are the most frequent shape layouts for shallow seawaters. Thick spheres have primarily been used for deep seawaters [1,2,3].

Recent decades have seen attempts at utilising new materials for the construction of the main pressure hull in both manned and unmanned submersibles. This includes acrylic plastic, Glass Reinforced Plastics (GRP), Carbon Fibre Reinforced Plastics (CFRP), Metal Matrix Composites (MMC), transparent glass ceramics and chemically tempered glass. Acrylic plastic has been extensively researched for underwater applications. Due to acrylic transparency its primary use is for windows but various acrylic pressure hulls were also built. This includes spherical hulls. Plain and ring reinforced cylinders closed by hemispheres were also manufactured from acrylic. An operational diving depth of up to 934 m was reported for a pressure hull made entirely from acrylic. GRP pressure hulls for manned vehicles with operational depth of 260 m and 450 m were built. The use of CFRP for submersibles has also been pursued over the last decades [3,4].

Studies into the static and dynamic stability of toroidal shells made from steel have shown that this kind of pressure vessel for underwater applications has several advantageous features. Firstly, a change of cross-section from circular to elliptical can increase the buckling pressure by a factor of three or more. Secondly, toroids appear to have relatively small sensitivity to initial geometric imperfections. This is directly opposite to sensitivities in hemispheres, in domed ends of various shapes and in cylinders, all of which are currently used as standard geometries for pressure hulls [5]. The shape of the pressure vessel and the aimed application dictate the manufacturing route. Some of the above mentioned components cannot be filament wound due to shape constraints, for example. Alternative techniques would normally include hand lay-up, resin transfer moulding, vacuum assisted resin infusion, or fibre placement. Toroidal shells belong to the class of doubly curved components and it appears that literature on doubly curved, filament wound shells, aimed at underwater applications, is scarce. It is worth noting here that it has only recently been possible to filament wind closed toroidal shells for internal pressure applications.

This paper presents results of a numerical study into static stability of filament wound CFRP toroids. The following topics are covered: (i) the effect of boundary conditions on the load carrying capacity, (ii) the effect of constant versus variable wall thickness, (iii) the effect of lamination stacking versus geometrical parameters of toroidal shells, and (iv) the effect of moving from circular to elliptical cross- sections.

Results of this numerical study confirm the observation made for metallic shells that their static buckling pressures can be significantly increased by departure from a circular cross-section to an elliptical one with the semi-axes ratio . In CFRP toroidal shells the transition from the eigenmode, , to the eigenmode, , carries a different sensitivity to the initial geometric imperfections taken in the form of the buckling shape. Also, the transition from to occurs at much smaller values of -ratio than in metallic shells. The buckling strength of toroids with prolate cross-sections which fail into mode are not much stronger than similar shells with circular cross-section. The opposite situation exists for metallic shells where large increases of buckling strength have been obtained for buckling mode.

The above conclusions are based on a single, symmetric, lamination. It would therefore be needed to include other stacking of ply before wider generalisation can be made about the buckling performance of composite toroids. Ideally, this should include modelling of winding path with the resulting wall thickness variation. The current study offers timely results for possible experimental investigations. As such, the paper contributes to the on-going search for a more efficient underwater pressure hull.

1

C.S. Smith, "Design of marine structures in composite materials", Elsevier Appl. Sci., London NY, 1990, 389p.

2

W.A. Nash, "Hydrostatically loaded structures", Pergamon, Kidlington, UK, 1995, 183p.

3

P. Davies, P. Cauchot, "Composites for marine applications: underwater structures", in `Mechanics of Composite Materials and Structures', (eds) C.A. Mota Soares et al, NATO ASI, Troia, 1998, vol. I, pp. 267-277.

4

D. Graham, "Composite pressure hulls for deep ocean submersibles", Composite Structures, vol. 32, 1995, 335-343. doi:10.1016/0263-8223(95)00028-3

5

J. Blachut, O.R. Jaiswal, "On buckling of toroidal shells under external pressure", Computers Structures, vol. 77, 2000, 233-251. doi:10.1016/S0045-7949(99)00226-6

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