## 1 Introduction
In the first paper, the differential quadrature method was used to solve numerically the governing equations based on the classical plate theory, whereas the free vibration problem was solved by means of the polynomial-Ritz method in the second paper. The main aim of the present paper is to investigate the natural frequencies of shell structures with variable thickness made of FGMs. A unified formulation is employed to derive the governing equations for several Higher-order Shear Deformation Theories (HSDTs) [93, 94, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114]. As highlighted by [PERSON] and [PERSON] [115], the introduction of innovative materials in the stacking sequence of the structure could generate some effects that classical and first-order theories that are commonly neglected. For the sake of completeness, it should be noted that the same observation was illustrated also in the works [116, 117, 118, 119, 120, 121]. In the present paper, the field equations, as well as the corresponding boundary conditions are solved by means of the Generalized Differential Quadrature (GDQ) method developed by [PERSON] [122]. A local approach which does not take into account all the sampling points of the domain is employed to approximate the derivatives of the governing equations, following the adaptive procedure illustrated by [PERSON] [123]. Further details concerning the numerical method can be found in [124, 125, 126, 127, 128]. It should be underlined that the authors have considered the topic of variable thickness shells in the works [104, 109], where laminated composite structures are investigated. In particular, a global version of the GDQ method is employed in the paper by [PERSON] et al. [109], in which several comparisons with the results available literature allowed to validate both the structural model and the numerical technique. On the other hand, the convergence and accuracy features of the local approach of the GDQ are studied in the work by [PERSON] et al. [104], where a general formulation to define arbitrary smooth thickness variations was presented. Nevertheless, in these papers the mechanical properties of the materials were assumed constant in each layer. The novelty introduced in this paper is to employ FGM plies. Thus, all the structures are characterized by a continuous gradual variation of the mechanical properties along the shell thickness. In addition, several lamination schemes and different through-the-thickness distributions of the constituents are considered to define sandwich configurations, without discontinuities at the interfaces between two contiguous layers. In the aforementioned papers, in fact, all the laminae are orthotropic and their mechanical parameters do not depend on the thickness coordinate. As far as the thickness profiles are concerned, the general approach shown in the work by [PERSON] et al. [104] represents the mathematical tool employed even in this paper to describe the thickness variation by means of several functions, such as sinusoidal or power-law. The present work is structured as follows. The mechanics of FGMs is described preliminarily. Then, some details concerning the definition of the shell geometry and the higher-order structural models are presented. The main aspects of the local GDQ method are briefly illustrated, followed by some numerical results. Finally, some concluding remarks are presented.