Composite Structures

Volume 256, 15 January 2021, 113090
Composite Structures

Nonlinear forced vibrations analysis of imperfect stiffened FG doubly curved shallow shell in thermal environment using multiple scales method


The nonlinear forced vibrations of imperfect stiffened functionally graded doubly curved shallow shells.

The multiple scales method and Galerkin’s procedure are employed.

The shell is resting on the nonlinear elastic foundations and exposed to thermal and external harmonic loads.

Using the classical shell theory and non-linear von Kármán relationships.

Effects of the stiffeners and other parameters on dynamic behaviers of the shells are considered.


This study investigates the non-linear vibrations of stiffened imperfect functionally graded double-curved shallow shells, as rested on nonlinear elastic foundations. The shells are exposed to external harmonic excitation and are placed in the thermal situations. The modeling of shells is derived according to the classical shell theory and the non-linear geometric von Kármán relationships. It is considered that the distribution of material properties changes along the thickness direction based on a power law index. The smeared stiffener technique is considered to model the stiffened shells. An approximation, according to Galerkin’s approach, is utilized to reduction of the shell governing equations into the non-linear coupled ordinary differential relations. The ODE equations are analytically solved and analyzed through the perturbation methodology for investigating the resonance behavior of shells. Simulation results are reported to examine the influences of stiffeners, initial imperfection, foundation coefficients, thermal environment, and geometrical characteristics on the non-linear primary resonance response of doubly curved shallow shells. Also, the nonlinear dynamic behaviors are analyzed by numerical methods through the bifurcation diagrams, and the nonlinear dynamical behaviors of the shell for different value of parameters are examined.


Stiffened double curved shallow shell
Non-linear forced vibration
Imperfect FGM shells
Multiple scales method
Resonance analysis
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