## 7 Numerical Results
### Doubly-Curved Shell of Revolution with Hyperbolic Meridian  
The FC hyperbolic meridian shell, depicted in Figure 4c, is a doubly-curved shell of revolution whose reference surface is defined a branch of a hyperbola. The surface at issue is described by the following position vector  
\[\mathbf{r}(\varphi,\vartheta)=R_{0}(\varphi)\cos\vartheta\,\mathbf{e}_{1}-R_{0 }(\varphi)\sin\vartheta\,\mathbf{e}_{2}+\sqrt{\frac{\left(R_{0}(\varphi) \right)^{2}-a^{2}}{k^{2}-1}}\,\mathbf{e}_{3} \tag{98}\]  
The coordinates \(\varphi,\vartheta\) are defined in the intervals \(\varphi\in[\varphi_{0},\varphi_{1}]\) and \(\vartheta\in[\vartheta_{0},\vartheta_{1}]\), where \(\varphi_{0}=1.741190\), \(\varphi_{1}=1.212026\), \(\vartheta_{0}=0\), \(\vartheta_{1}=2\pi\). On the other hand, the parallel radius \(R_{0}(\varphi)\) assumes the aspect shown below  
\[R_{0}(\varphi)=a\sin\varphi\sqrt{\frac{k^{2}-1}{k^{2}\sin^{2}\varphi-1}} \tag{99}\]  
where \(k=\sqrt{1+a^{2}/b^{2}}\). It should be specified that \(a\) is the parallel radius of the throat section of the structure, and \(b=aD/\sqrt{d^{2}-a^{2}}\) is the characteristic dimension of the shell, in which \(a=1\) m, \(d=2\) m, \(D=4\) m. The meaning of these parameters, as well as the ones needed to sketch the following structure, can be found in Figure 8. For further details concerning the present geometry the reader can refer to the book by [PERSON] et al. [2]. With reference to Figure 8a, the characteristic dimension of the hyperbola can be defined alternatively as \(b=aC/\sqrt{c^{2}-a^{2}}\), where \(c\) is the parallel radius at the top section of the structure and \(C\) represents its coordinate that in the present case is equal to \(C=1\) m. The value of \(c\) can be computed by means of Equation (99). The reference value of thickness is \(h_{0}=0.2\) m and it is measured along the top edge. A cubic variation is applied along the first coordinate \(\varphi\) and it can be defined as follows:  
\[h(\varphi)=h_{0}\Bigg{(}1+2\bigg{(}\frac{\varphi-\varphi_{0}}{\varphi_{1}- \varphi_{0}}\bigg{)}^{3}\Bigg{)} \tag{100}\]  
Figure 7: First three mode shapes for a FC elliptic cylinder with variable thickness assuming \(p=1\) as exponent. It should be noted that the same thickness variation affects each ply, as it can be observed in the cross-section of Figure 4c. The structure is made of five layers with different thickness reference values. The two external laminae are isotropic and fully ceramic (\(h_{0,1}=h_{0,5}=0.02\) m), whereas the inner isotropic core is made of metal and its thickness is equal to \(h_{0,3}=0.06\) m. The remaining two piles have equal thickness \(h_{0,2}=h_{0,4}=0.05\) m and are graded as it can be noted from the stacking sequence depicted in Figure 5b, where the volume fraction distributions of the two constituents are also specified for several values of the exponent \(p^{(2)}=p^{(4)}=p\in[0,\infty]\). In a similar way to the previous example, the boundary values of \(p\) define respectively a layer fully made of Silicon nitride (\(p=0\)) and Stainless steel (\(p=\infty\)). The results of this parametric study are given in Table 4 for the FSDT\({}_{RS}^{\chi=1.2}\), ED2\({}^{\chi=1.2}\), and ED3 models. Due to the presence of various layers with different mechanical properties, the influence of the Murakami's function is also investigated. Thus, the aforementioned structural theories are also embedded with this function. The results related to the FSDTZ\({}_{RS}^{\chi=1.2}\), ED2\({}^{\chi=1.2}\), and EDZ3 models are shown in Table 5. Even in this circumstance, the shear correction factor is used for the first and second order models, whereas the plane stress hypothesis is considered only in the Reissner-Mindlin theory. It is easy to notice that the Murakami's function does not significantly affect the results. Both in Tables 4 and 5, the 3D-FEM solutions are also included for comparison purposes. It should be noted that the finite element model is made of 52,734 brick elements (685,824 degrees of freedom).