## 8 Conclusions  
A new higher-order approach is presented to analyze the natural frequencies of thin and thick laminated composite shells. The method, named Equivalent Layer-Wise (ELW) method, is used to evaluate the overall mechanical and geometric properties directly on the shell middle surface, as commonly done in the Equivalent Single Layer (ESL) approaches, and the displacement field has been described by using the same kinematic expansions employed in the most typical Layer-Wise (LW) models, but without considering each layer separately. The governing equation are solved numerically by the GDQ method. This new approach has been validated by comparison with the results available in the literature and those obtained using a commercial finite element code. It has been illustrated that the solutions are in excellent agreement with the reference ones for several boundary conditions and various mechanical configurations. When compared to the corresponding three-dimensional finite element model, the present methodology requires a lower number of degrees of freedom to reach the same level of accuracy. In addition, since the differential geometry is employed to describe the reference doubly-curved surface, several geometric shapes have been easily obtained and analyzed.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(f\)**[Hz]**} & \multicolumn{2}{c}{**ED1**} & \multirow{2}{*}{\(\text{{ELD}}_{\text{RS}}^{\text{X=1.2}}\)**} & \multicolumn{2}{c}{**EDZ1**} & \multirow{2}{*}{\(\text{{ELD}}_{\text{RS}}^{\text{X=1.2}}\)**} & \multirow{2}{*}{**LD1**} & \multirow{2}{*}{**3D – FEM**} \\  & **Ref. [61]** & & & & & & \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 11: Comparison of the natural frequencies for a laminated composite conical shell [61; 83]. The solutions obtained through the present approach are compared to the results given by the corresponding ESL and Layer-Wise (LW) models.  
In fact, the domain has been defined using only one element, whereas the FEM models required many brick or plate elements to describe the same geometry accurately. It should be noted that the present methodology provides a physical meaning to the degrees of freedom of the problem for each order of kinematic expansion, different from the ESL approach, in which the generalized displacements related to higher-order expansions represent a mathematical parametrization of these quantities. Furthermore, the structural analysis of a laminated composite structure has not required the same computational resources needed by the LW methods. Thus, the computational time has been noticeably reduced when compared to the LW models. It should be noted also that the use of the Murakami's function has been also investigated by taking into account a sandwich structure. The linear static analysis of laminated composite shell structures can be seen as a future development of the present approach. Finally, it should be highlighted that the presented methodology could be used to deal with some typical aerospace applications which require an efficient tool to analyze composite structures characterized by complex geometries, with different material and thickness distributions.  
The research topic is one of the subjects of the Centre of Study and Research for the Identification of Materials and Structures (CIMEST)-\"[PERSON]\" of the University of Bologna (Italy).  
All the authors contributed equally to this work.  
The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.