Axiomatic/asymptotic PVD/RMVT-based shell theories for free vibrations of anisotropic shells using an advanced Ritz formulation and accurate curvature descriptions

Abstract

The Hierarchical Trigonometric Ritz Formulation (HTRF) has earlier been successfully developed for plates [1–4] and shells [5] using the Principle of Virtual Displacements (PVD). In this paper the HTRF is significantly extended with the help of Reissner’s Mixed Variational Theorem (RMVT) so as to deal with the free vibrations of doubly-curved anisotropic laminated composite shells. The interlaminar equilibrium of the transverse normal and shear stresses is fulfilled a priori by exploiting the use of Lagrange multipliers. The transverse normal and shear stresses thus become primary variables within the formulation and are always modeled with a Layer-Wise kinematics description. Equivalent Single Layer, Zig-Zag and Layer-Wise approaches are instead efficiently used for the displacement primary variables. Appropriate expansion orders for each displacement or stress unknown are selected depending on the required accuracy and the computational cost. Axiomatic/asymptotic shell theories are then developed by virtue of a deep study on the effectiveness of each term both in the displacements and in the transverse stresses fields. Next exact and/or accurately approximated curvature descriptions are taken into account. Cylindrical, spherical and hyperbolic paraboloidal shells are investigated. The proposed advanced quasi-3D shell models are assessed by comparison with 3D elasticity solutions.