Free vibration of non-Lévy-type functionally graded doubly curved shallow shells: New analytic solutions

Abstract

• Symplectic superposition framework for free vibration of FG shells is presented. • New analytic solutions of non-Lévy-type FG doubly curved shallow shells are obtained. • Frequencies of FG shells with different boundaries are provided as benchmarks. • Rational and rigorous procedure, fast convergence, and high accuracy are validated. Analytic solutions are of great significance to serve as benchmarks for validating numerical methods and to guide efficient preliminary structural designs. In this paper, we concentrate on seeking new analytic free vibration solutions of non-Lévy-type functionally graded (FG) doubly curved shallow shells, which were not available in the literature to the best of our knowledge. The main challenge in gaining such analytic solutions comes from both the governing system of partial differential equations under complicated boundary conditions. Here, we extend an analytic symplectic superposition method (SSM), which we proposed originally for plate problems, to solve the current tough issue. The solution methodology is implemented via the following three steps: converting an original free vibration problem into two elementary problems and formulating the equations of each elementary problem in the Hamiltonian system; utilizing specific mathematical techniques in the symplectic space for the analytic solutions of the elementary problems; superposing the solutions of the elementary problems to yield the eventual analytic free vibration solutions. The primary advantage of the SSM is that it does not require any assumption of solution forms, but is conducted via rigorous derivation, which is realized by the exceptional mathematical treatments, such as the symplectic space-based separation of variables and the symplectic eigen expansion. Comprehensive new free vibration solutions are presented for non-Lévy-type FG doubly curved shallow shells, through which the solutions of FG cylindrical shell panels are also reported as extreme cases. All tabulated results are well validated by the finite element method via ABAQUS software, guaranteeing the accuracy of the present solutions for benchmark use. Quantitative analyses on the natural frequencies are also conducted to reveal the effects of boundary conditions, FG models, and aspect ratios.