The Complex Non-Linear Dynamics of Imperfection Sensitive Shells

Abstract

Shell structures have been used for centuries in all engineering fields. This is largely due to the important role played by these shells as efficient loading carrying members leading to very light and slender structures. However, when subjected to a compressive stress state in one or more directions they often exhibit a highly non-linear response and may loose their stability under quasi-static or dynamic conditions at load levels well below those predicted by linearized eigenvalue-problems or by non-linear analyses of a perfect system [1–3]. This is mainly due to the deleterious effects of imperfections (e. g., load, geometrical and boundary imperfections) and, in certain circumstances, to the small degree of safety of these systems on account of the fast decrease or erosion of the basin of attraction associated with the stable solution whose stability must be preserved. This can be explained by the fact that many shells under quasi-static loads display a symmetric unstable or an asymmetric bifurcation along the fundamental equilibrium path [2,3]. In these circumstances there are at least two equilibrium points for load levels lower than the critical load and, induced by the topology of the associated potential function, these shells may display subharmonic and superharmonic bifurcations, period-multiplying bifurcations, multiple solutions, chaotic motions and dangerous jumps caused by the presence of competing potential wells and non-linear resonance curves within each well [4]. Shells under compressive loads display not only cubic geometric non-linearities, like any other structural element, but also strong quadratic non-linearities due to the simultaneous effects of the initial curvature, prestresses and geometric imperfections. This strong non-linearity explains the complex dynamics of these systems and controls the various modes of elastic instability.