An analytical investigation on free vibration of FGM doubly curved shallow shells with stiffeners under thermal environment

This paper presents an investigation of free vibration of stiffened doubly curved shallow shells made of functionally graded materials under thermal environment. Two types of temperature rise throughout the shell thickness; namely linear and nonlinear temperature rises are considered in the present investigation. The power law distribution and Mori–Tanaka homogenization scheme are used to describe the material graduation throughout the shell thickness. In order to take into account the significant effects of shear deformation and rotatory inertia of the shell skin and its stiffeners, the first-order shear deformation theory is employed to derive the governing equations used for determining natural frequencies of the stiffened shells. The governing equations can be solved analytically to obtain exact solutions for this problem. The stiffened shells can be specialized into different forms of spherical, cylindrical and hyperbolic shells by setting components of curvature. Several parameters of material volume fraction index, geometrical ratio, temperature change, number of stiffeners, etc. that affect vibration results of the shells are investigated and discussed in detail. Based on the numerical results, it is revealed that increasing number of stiffeners leads to considerable changes in natural frequencies of the stiffened shells.