## 1 Introduction
Curved panels embedding functionally graded materials were studied by [PERSON] et al. [12]. The free vibrations for panels, with arbitrary boundary conditions on two opposite edges and simply supported boundary conditions on the other sides, were presented. Cylindrical shells were studied by [PERSON] et al. in [13], an analytical method was developed to study the free vibrations of simply supported cylinders. [PERSON] [14] used the power series method to solve the 3D linear equilibrium equations written for functionally graded shells with simply supported edges. [PERSON] [15] evaluated the effects of different boundary conditions on the dynamic response of functionally graded annular plates using the Chebyshev-Ritz method, a convergence analysis was also performed. [PERSON] and [PERSON] [16] studied the advantages that a functionally graded core may give in a sandwich panel in terms of the reduction of stress discontinuities at the interfaces. The closed-form solutions found in literature always consider the problem for particular boundary load conditions. Further interesting exact solutions in the literature are discussed in the following last part. [PERSON] [17] proposed exact solutions for equilateral triangle and elliptic Kirchhoff plates under torsion with no kinematic boundary conditions using the analogy with a beam under torsion. An extension of this solution to flexure was proposed in [18]. The introduction of Functionally Graded Materials in [PERSON] et al. [19] allowed for establishing a relationship between FGM orthotropic Saint-Venant beams and inhomogeneous isotropic Kirchhoff plates. Further details about the employed method can be found in [20; 21]. Even though several solutions were already proposed in the literature, a lack of generality exists. The final goal of the present solution is to give a universal tool capable to include several geometries (plates, cylinders, spherical and cylindrical shells) embedding different material configurations (also including FGMs). This solution was already introduced by the first author for the free vibrationsof FGM structures in [22] and extended to the static analysis of FGM plates and shells subjected to harmonic loads in [23]. The equilibrium equations are written in mixed orthogonal curvilinear coordinates and solved through the thickness via the exponential matrix method. This method may only be used when the coefficients of the differential equations are constant. When FGM layers are introduced, the mechanical properties vary through the thickness, and this feature makes these coefficients a function of the thickness coordinate \(z\). Moreover, when the shell geometries are considered, a further dependence of these coefficients from z exists. This problem may be overcome via the introduction of mathematical layers, where the mechanical properties and the curvature-related terms may be punctually evaluated in their middle. The \(M\) number of mathematical layers and the \(N\) order of expansion for the exponential matrix are two parameters to be determined in order to have exact results. In the convergence analysis of the proposed 3D shell solution, two terms are fundamental, they are the \(N\) order of expansion for the exponential matrix and the \(M\) number of mathematical layers. These two terms depend on several parameters such as: geometry, lamination scheme, material (classical ones or FGMs), FGM law, thickness ratio, performed analysis (static or free vibration analysis), applied loads and so on. For these reasons, a systematic analysis is necessary to understand the better combination of \(M\) and \(N\) values. However, this systematic analysis is possible only if it is separately performed for different cases in order to remark several important features. For these reasons, we have proposed four different works: the present new paper and further three past works [24; 25; 26]. In [24], the convergence study in terms of \(M\) and \(N\) was performed for the free vibration analysis of composite and sandwich plates and shells. Ref.