## 7 Numerical Results
### Free vibrations of Laminated Composite Shells
[13] & ELD1\({}_{RS}^{\chi=1.2}\) & ELD2\({}^{\chi=1.2}\) & ELD3 & ELD4 \\  
\hlineThe first 10 natural frequencies of this structure are shown in Table 7 for different higher-order ELW models. As in the previous tests, each order of kinematic expansion is taken with and without the Murakami's function to investigate its contribution. It is easy to notice that in the present case all the theories give the same results, independently from this function. The accuracy of the numerical technique is proven also by the comparison with the 3D-FEM solution (8000 brick elements \"Hexa20\"), which is nearly equal to the proposed ones. It should be recalled that the same considerations illustrated above concerning the use of both the shear correction factor and the reduced elastic coefficients are still valid. Finally, the first nine mode shapes for the shell structure at issue are depicted in Figure 5.  
The following structure is a CCCC doubly-curved revolution shell, whose reference surface is generated by the rotation about the revolution axis of a branch of a catenary (Figure 4d). \[ \begin{split}\mathbf{r}(\varphi,\theta)&=a\cosh \Bigl{(}\operatorname{arcsinh}\Bigl{(}\tfrac{1}{\tan\varphi}\Bigr{)}\Bigr{)} \cos\vartheta\,\mathbf{e}_{1}-a\cosh\Bigl{(}\operatorname{arcsinh}\Bigl{(} \tfrac{1}{\tan\varphi}\Bigr{)}\Bigr{)}\sin\vartheta\,\mathbf{e}_{2}+\\ &+a\operatorname{arcsinh}\Bigl{(}\tfrac{1}{\tan\varphi}\Bigr{)} \mathbf{e}_{3}\end{split}  \tag{56}\]  
where \(\varphi,\vartheta\) are the curvilinear coordinates of the surface defined in the intervals \(\varphi\in[2.626124,0.515469]\) and \(\vartheta\in[-\pi/3,\pi/3]\), with \(a=0.75\,\mathrm{m}\). The lamination scheme is chosen to define a so-called sandwich structure in which the core exhibits mechanical properties that are considerably different from the ones that characterize the external face-sheets, especially along the transverse direction. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\boldsymbol{f}\) [Hz] & \(\mathbf{ELD_{RS}^{\boldsymbol{\chi=1.2}}}\) & \(\mathbf{ELD2^{\boldsymbol{\chi=1.2}}}\) & \(\mathbf{ELD3}\) & \(\mathbf{ELD4}\) & \(\mathbf{3D-FEM}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 7: First 10 natural frequencies ([Hz]) for a CCFC laminated cylindrical surface with (90/30/45/60/0) as lamination scheme for several higher-order ELW approaches, with and without the Murakami’s function. The Cheb-Gau-Lob grid distribution is used for the numerical solution assuming \(I_{N}=31\) and \(I_{M}=21\). In other words, if the stiffness of the constituent materials along the transverse direction is really different the zig-zag effect is generated. As a consequence, the Murakami's function is embedded in the kinematic model to represent the effective behavior of the structure. The lamination scheme here is given by \((45/\text{Core}/-45)\), assuming that the external orthotropic layers have equal thickness \(h_{1}=h_{3}=0.02\,\text{m}\) and are made of Graphite-Epoxy. On the other hand, the isotropic core is made of Ceramic Foam and it is defined by a thickness equal to \(h_{2}=0.06\,\text{m}\). With reference to the mechanical properties of Table 2, it is easy to notice that \(E_{3}^{(1)}=E_{3}^{(3)}\gg E_{3}^{(2)}\). Thus, the shell falls into the class of sandwich structures. The first 10 natural frequencies are shown in Table 8 and are evaluated by using the Cheb-Gau-Lob grid distribution with \(I_{N}=I_{M}=31\). In the same table, the 3D-FEM solution is also reported for comparison purposes. It should be noticed that the first two orders of kinematic expansion (\(N=1,2\)) are taken into account with and without the shear correction factor. From Table 8 it can be noted that the use of the shear correction factor \(\chi=1.2\) leads to a solution that are slightly different from the others. On the other hand, if the correction is not applied (which means \(\chi=1\)) the natural frequencies are similar to the ones obtained for higher-order theories, as well as the ones given by the 3D-FEM (11560 brick elements \"Hexa20\").