A new doubly-curved shell element for the free vibrations of arbitrarily shaped laminated structures based on Weak Formulation IsoGeometric Analysis

The main aim of the paper is to present a new numerical method to solve the weak formulation of the governing equations for the free vibrations of laminated composite shell structures with variable radii of curvature. For this purpose, the integral form of the stiffness matrix is computed numerically by means of the Generalized Integral Quadrature (GIQ) method. A two-dimensional structural model is introduced to analyze the mechanical behavior of doubly-curved shells. The displacement field is described according to the basic aspects of the general Higher-order Shear Deformation Theories (HSDTs), which allow to define several kinematic models as a function of the free parameter that stands for the order of expansion. Since an Equivalent Single Layer (ESL) approach is considered, the generalized displacements evaluated on the shell middle surface represent the unknown variables of the problem, which are approximated by using the Lagrange interpolating polynomials. The mechanical behavior of the structures is modeled through only one element that includes the double curvature in its formulation, which is transformed into a distorted domain by means of a mapping procedure based on the use of NURBS (Non-Uniform Rational B-Splines) curves, following the fundamentals of the well-known Isogeometric Analysis (IGA). For these reasons, the presented methodology is named Weak Formulation Isogeometric Analysis (WFIGA) in order to distinguish it from the corresponding approach based on the strong form of the governing equations (Strong Formulation Isogeometric Analysis or SFIGA), previously introduced by the authors. Several numerical applications are performed to test the current method. The results are validated for different boundary conditions and various lamination schemes through the comparison with the solutions available in the literature or obtained by a finite element commercial software.