Free vibration analysis of axisymmetric functionally graded doubly-curved shells with un-uniform thickness distribution based on Ritz method

This paper analyzed the free vibration of functionally graded (FG) doubly-curved shells with un-uniform thickness distribution based on Ritz method. The energy method and first-order shear deformation theory are adopted to derive the formulas. In this paper, the stepped FG doubly-curved shells are divided into their segments in axial direction according to the steps of the structures, and the displacement functions of shell segments are consisted with Jacobi polynomials along axial direction and standard Fourier series along circumferential direction. In addition, the boundary conditions at ends of the stepped FG doubly-curved shells and the continuity conditions at two adjacent segments were enforced by penalty method. Then the final solutions can be obtained based on Ritz method. Finally, to confirm the validity of proposed method, the results with the same conditions are compared by Finite Element Method (FEM) and experiment. The results show that the proposed method has the advantage of fast convergence, high accuracy and simple boundary simulation etc.