The presented work is focused on the development of a simplified analytical method to study the struc- tural response of a deeply immersed cylinder submitted to the primary shock wave of an underwater explosion. It relies on a methodology developed by hoo Fatt and Wierzbicki where the two dimensional boundary value problem for a cylindrical shell is converted to an equivalent one-dimensional problem of a plastic string on a non-linear plastic foundation. unstiffened cylinders immersed in shallow water have already been investigated by the authors, taking into account fluid structure interaction effects. The aim of the proposed work is to adapt the formulations to a deep immersed cylinder. The analytical developments will be presented for unstiffened cylinders. The resulting plastic dents are compared to experimental and numerical results. Although some limitations are pointed out, it is shown that this method is promising and may be advantageously used to assess rapidly the damage of a deep immersed cylinder submitted to an underwater explosions.
fluid structure interaction, immersed cylinder, rigid-plastic analysis, Underwater explosion.
 Brochard, K., Le Sourne, H. & Barras, G., A simplified method to assess the damage of an immersed cylinder subjected to underwater explosion. Proceedings of the 6th International Conference on Marine Structures (MARSTRUCT 2017), pp. 405–413, 2017.
 Brochard, K., Le Sourne, H. & Barras, G., Extension of the string-on-foundation method to study the shock wave response of an immersed cylinder. International Journal Of Impact Engineering, 117, pp. 138–152, 2018. https://doi.org/10.1016/j.ijimpeng.2018.03.007 Figure 7: C omparison of analytical and numerical results : (a) Shell final deflection _ f – (b) Increase in shell final deflection ._ f . 108 K. Brochard, et al., Int. J. of Safety and Security Eng., Vol. 9, No. 2 (2019)
 Wierzbicki, T. & Hoo Fatt, M.S., Damage assessment of cylinders due to impact and explosive loading. International Journal of Impact Engineering, 13(2), pp. 215–241, 1993. https://doi.org/10.1016/0734-743x(93)90094-n
 Gupta, S., Matis H., LeBlanc, J.M. & Shukla, A., Shock initiated instabilities in underwater cylindrical structures. Journal of the Mechanics and Physics of Solids, 95, pp. 188–212, 2016. https://doi.org/10.1016/j.jmps.2016.05.034
 Biglarkhani, M. & Sadeghi, K., Incremental explosive analysis and its application to performance-based assessment of stiffened and unstiffened cylindrical shells subjected to underwater explosion. Shock and Vibration, 2017, pp. 1–17, 2017. https://doi.org/10.1155/2017/3754510
 Wierzbicki, T. & Suh, M.S., Indentation of tubes under combined loading. International Journal of Mechanical Sciences, 30(3–4), pp. 229–248, 1988. https://doi.org/10.1016/0020-7403(88)90057-4
 Cole, R.H., Underwater Explosions, Dover Publications, New York, NY, 1965.
 Geers, T.L., Doubly asymptotic approximations for transient motions of submerged structures. Journal of Acoustical Society of America, 64(5), pp. 1500–1508, 1978. https://doi.org/10.1121/1.382093
 Taylor, G.I., The pressure and impulse of submarine explosion waves on plates. Underwater Explosion Research (Office of Naval Research, Washington D. C., 1950), 1, pp. 1155–1174, 1941.
 Pinna, R. & Ronalds, B., Hydrostatic buckling of shells with various boundary conditions. Journal of Constructional Steel Research, 56(1), pp. 1–16, 2000. https://doi.org/10.1016/s0143-974x(99)00104-2