The local GDQ method for the natural frequencies of doubly-curved shells with variable thickness: A general formulation

The present paper aims to evaluate the natural frequencies of several doubly-curved shells with variable thickness. The general theoretical formulation allows to take into account various higher-order Equivalent Single Layer (ESL) theories in a unified manner, including the Murakami's function to capture the zig-zag effect. Such approach is able to study very well the dynamic behavior of a laminated composite shell, even in the presence of a soft-core. A general expression, which is able to combine different kinds of variations (such as linear, parabolic, exponential, sine-wave, Gaussian and elliptic shapes), is introduced to define the thickness profiles. In addition, the same formulation can be employed to localize such variations and to define, consequently, ribbed structures. Since the adopted structural model is two-dimensional, the shell reference surface represents the physical domain in which the governing equations are written. Thus, the differential geometry is necessary to define accurately the doubly-curved surfaces at issue. The fundamental system is solved numerically by means of a local approach of the well-known Generalized Differential Quadrature (GDQ) method. The matrices that allow to solve the problem in hand are banded, since only a part of the discrete grid points is considered. As a consequence, the computational effort is lower, if compared to the corresponding global version. The accuracy, reliability and stability of the present approach are proved by the comparison with the results available in the literature and the solutions obtained through three-dimensional FEM models.