Dynamic analysis of anisotropic doubly-curved shells with general boundary conditions, variable thickness and arbitrary shape

In the present contribution a general formulation is proposed to account for general boundary conditions within the dynamic analysis of anisotropic laminated doubly-curved shell having arbitrary shape and variable thickness. Different analytical expressions are considered for the shell thickness variation along the geometrical principal directions, and the distortion of the physical domain is described by a mapping procedure based on Non-Uniform Rational Basis Spline (NURBS) curves. Mode frequencies and shapes are determined employing higher-order theories within an Equivalent Single Layer (ESL) framework. The related fundamental relations are tackled numerically by means of the Generalized Differential Quadrature (GDQ) method. The dynamic problem is derived from the Hamiltonian Principle, leading to a strong formulation of the governing equations. General external constraints are enforced along the edges of the shell employing a distribution of linear springs distributed on the faces of the three-dimensional solid and accounting for a spatial coordinate-dependent stiffness along both in-plane and out-of-plane directions. Moreover, a Winkler-type foundation with general distribution of linear springs is modelled on the top and bottom surfaces of the shell. A systematic set of numerical examples is carried out for the validation of the proposed theory by comparing mode frequencies with predictions from refined three-dimensional finite element analyses. Finally, we perform a sensitivity analysis of the dynamic response of mapped curved structures for different spring stiffnesses and general external constraints, according to various kinematic assumptions.