**Vu Van Tham, Tran Huu Quoc \* and Tran Minh Tu**

Faculty of Industrial and Civil Engineering, National University of Civil Engineering, Hanoi 100000, Vietnam; thamvv@nuce.edu.vn (V.V.T.); tutm@nuce.edu.vn (T.M.T.)

**\*** Correspondence: thquoc@gmail.com; Tel.: +84-916-600-081

Received: 31 October 2019; Accepted: 28 November 2019; Published: 1 December 2019

**Abstract:** In this paper, a new four-variable refined shell theory is developed for free vibration analysis of multi-layered functionally graded carbon nanotube-reinforced composite (FG-CNTRC) doubly curved shallow shell panels. The theory has only four unknowns and satisfies zero stress conditions at the free surfaces without correction factor. Five different types of carbon nanotube (CNTs) distribution through the thickness of each FG-CNT layer are considered. Governing equations of simply supported doubly curved FG-CNTRC panels are derived from Hamilton's principle. The resultant eigenvalue system is solved to obtain the frequencies and mode shapes of the anti-symmetric cross-ply laminated panels by using the Navier solution. The numerical results in the comparison examples have proved the accuracy and efficiency of the developed model. Detailed parametric studies have been carried out to reveal the influences of CNTs volume fraction, CNTs distribution, CNTs orientation, dimension ratios and curvature on the free vibration responses of the doubly curved laminated FG-CNTRC panels.

**Keywords:** free vibration analysis; doubly-curved shell and panel; nano-composites; functionally graded carbon nanotube-reinforced composite (FG-CNTRC); four-variable refined shell theory
