*2.5. Solution Procedure*

The Navier method is employed to formulate the closed-form solution for vibration problems of simply supported anti-symmetric cross-ply laminated FG-CNTRC panels. The simply supported boundary conditions on all four edges can be considered as:

$$w\_0 = w\_b = w\_s = w\_{b,y} = w\_{s,y} = N\_{\text{xx}} = M\_{\text{xx}}^b = M\_{\text{xx}}^s = 0 \text{ at } \mathbf{x} = 0 \text{ and } \mathbf{x} = a \tag{25}$$

$$w\_0 = w\_b = w\_s = w\_{b,x} = w\_{s,x} = N\_{yy} = M\_{yy}^b = M\_{yy}^s = 0 \text{ at } y = 0 \text{ and } y = b \tag{26}$$

These boundary conditions are exactly satisfied by the following double Fourier series forms:

$$\begin{aligned} u(x,y,t) &= \sum\_{\substack{m=1 \\ m \neq 1}}^{\infty} \sum\_{n=1}^{\infty} \mathcal{U}\_{mn} e^{i\omega t} \cos \alpha\_m \mathbf{x} \sin \beta\_n y; \\ v(x,y,t) &= \sum\_{\substack{m=1 \\ m \neq 1}}^{\infty} \sum\_{n=1}^{n} \mathcal{V}\_{mn} e^{i\omega t} \sin \alpha\_m \mathbf{x} \cos \beta\_n y; \\ w\_b(x,y,t) &= \sum\_{\substack{m=1 \\ m \neq 1}}^{\infty} \sum\_{n=1}^{\infty} \mathcal{W}\_{bnm} e^{i\omega t} \sin \alpha\_m \mathbf{x} \sin \beta\_n y; \\ w\_s(x,y,t) &= \sum\_{m=1}^{\infty} \sum\_{n=1}^{\infty} \mathcal{W}\_{snm} e^{i\omega t} \sin \alpha\_m \mathbf{x} \sin \beta\_n y. \end{aligned} \tag{27}$$

where (*Umn*, *Vmn*, *Wbmn*, *Wsmn*) are unknown coefficients to be determined, ω is the circular frequency of vibration, and *<sup>i</sup>* <sup>=</sup> <sup>√</sup> −1, α*<sup>m</sup>* = *m*π/*a*, β*<sup>n</sup>* = *n*π/*b* and *m, n* denote the number of haft-waves in the *x* and *y* directions, respectively.

Substituting the admissible displacement functions of Equation (27) into the equation of motion, Equation (20), one obtains the analytical solution in the following matrix form:

$$
\begin{pmatrix} \begin{bmatrix} s\_{11} & s\_{12} & s\_{13} & s\_{14} \\ s\_{12} & s\_{22} & s\_{23} & s\_{24} \\ s\_{13} & s\_{23} & s\_{33} & s\_{34} \\ s\_{14} & s\_{24} & s\_{34} & s\_{44} \end{bmatrix} - \omega^{2} \begin{bmatrix} m\_{11} & m\_{12} & m\_{13} & m\_{14} \\ m\_{12} & m\_{22} & m\_{23} & m\_{24} \\ m\_{13} & m\_{23} & m\_{33} & m\_{34} \\ m\_{14} & m\_{24} & m\_{34} & m\_{44} \end{bmatrix} \end{pmatrix} \begin{pmatrix} \mathcal{U}\_{mn} \\ \mathcal{V}\_{mn} \\ \mathcal{W}\_{bm} \\ \mathcal{W}\_{bm} \\ \mathcal{W}\_{sm} \end{pmatrix} = \begin{Bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{Bmatrix} \tag{28}
$$

where the matrix elements of Equation (28) are given in the Appendix B.

#### **3. Numerical Results and Discussions**

In this section, several examples are presented and discussed to verify the accuracy and efficiency of the proposed theory in free vibration analysis of simply supported FG-CNTRC doubly-curved panels. Furthermore, the effects of volume fraction of CNTs, distribution type of CNTs, number of layers, CNT fiber orientation and geometrical parameters on the natural frequencies of panels are also investigated in detail. The material properties for the matrix and CNT are given in Table 1 [34,35]. Also, the CNT efficiency parameters η*<sup>j</sup>* (*j* = 1,2,3) associated with a given volume fraction *V*<sup>∗</sup> *CNT* are: η<sup>1</sup> = 0.149 and η<sup>2</sup> = η<sup>3</sup> = 0.934 for the case of *V*<sup>∗</sup> *CNT* = 0.11; η<sup>1</sup> = 0.150 and η<sup>2</sup> = η<sup>3</sup> = 0.941 for the case of *V*∗ *CNT* = 0.14; η<sup>1</sup> = 0.149 and η<sup>2</sup> = η<sup>3</sup> = 1.381 for the case of *V*<sup>∗</sup> *CNT* = 0.17.

**Table 1.** Material properties of carbon nanotube (CNT) and matrix materials.


#### *3.1. Comparison Studies*

To verify the reliability and accuracy of the present model, several comparison studies were carried out with the results of the previous literature [34,35].

Example 1: Free Vibration of the Simply Supported Doubly Curved FG-CNTRC Panels

Free vibration of the simply supported doubly curved single-layered FG-CNTRC panels is further analyzed for the comparison of the results obtained from the present formulation with the existing results developed by Pouresmaeeli and Fazelzadeh [34] based on FSDT formulations. The geometrical dimensions of the panels are taken as *a*/*b* = 1 and *a*/*h* = 20. Values of material parameters are listed in Table 1. From the results presented in Table 2, it is observed that the values of the fundamental frequency for plates, spherical, cylindrical, and hyperbolic paraboloid panels have excellent agreement with the available data.


**Table 2.** Comparison of the non-dimensional frequencies ω = ω *a*2/*h* + ρ*m*/*Em* of the simply supported doubly curved FG-CNTRC panels.
