Extensions of the variational theory of complex rays in the general shell theory framework

Nowadays, the interest of aerospace and automotive industries on the prediction of medium-frequency vibrational behavior of shell structures is growing. The development of software capable of predicting the vibrational response in such frequency range is still an open question because classical methods (i.e. FEM, SEA) are not suitable for the medium-frequency bandwidth. In this context the Variational Theory of Complex Rays (VTCR) \cite{Riou2004} is taking place as an \emph{ad-hoc} technique to tackle medium-frequency problems. It is a Trefftz method based on a weak variational formulation. It allows great flexibility because any shape function that satisfies the governing equations can be used. The VTCR applied to the general shell theory has already been proposed in \cite{Riou2004}. This work further develops such theory. The in-plane inertia is addressed and the slight curvature assumption is relaxed extending the theory to shells with constant Lame's parameters. Since shallow shells, cylinders, and cones with a small apex angle satisfy this hypothesis, it is always met in practice. A relevant numerical example is presented to show the strategy.