*3.2. Parametric Studies*

In this section, some new results for free vibration of the anti-symmetric cross-ply laminated FG-CNTRC doubly curved shell panels are investigated with respect to FG-CNTRC parameters, curvature, *Rx*/*Ry* ratio, aspect *a*/*b* ratio, and number of layers. The material properties for the matrix and CNT are shown in Table 1.

#### 3.2.1. Effect of FG-CNTRC Parameters

To understand the effect of FG-CNTRC parameters on the free vibration response of different shell panels, non-dimensional frequencies ω of anti-symmetric cross-ply laminated FG-CNTRC doubly curved shell panels with different CNT distribution, CNT volume fraction, and number of CNT layers are examined.

It is observed from Table 3, that the FG-X panels have the highest value of frequency, whereas, the FG-O panels have the lowest one. Therefore, it can be concluded that the type of CNT distribution has a remarkable influence on the stiffness of the FG-CNTRC shell panels. In detail, the CNTs distributed close to the top and bottom surfaces of each FG-CNTRC layer are more efficient than those distributed near the mid-plane of each FG-CNTRC layer in increasing the stiffness of the laminated FG-CNTRC shell panels. This is compatible with conclusions in previous studies in the literature. According to the detailed results, the values of ω can be increased by more than 24% with only 6% increasing CNT volume fraction *V*∗ *CNT* for any other parameters. Thus, by adjusting a small amount of CNT volume, the desired stiffness of the FG-CNTRC panels can be achieved. Table 3 also reveals that the SHP panel has the highest value of ω while the HPR panel has the lowest one. This is because HPR has both sagging and hogging curvature along the two directions, neutralizing the effect of each other, while SHP does not. Table 3, once again confirms the accuracy of the present model by comparing the non-dimensional frequencies of the FG-CNTRC plates with the results of Wang [35].


**Table 3.** Non-dimensional frequencies ω of the simply supported FG-CNTRC doubly curved shell panels (*a*/*b* = 1; *R*/*a* = 5, *a*/*h* = 50).
