Large amplitude free vibration analysis of shear deformable FGM shallow arches on nonlinear elastic foundation

Abstract

Current investigation deals with the small and large amplitude free vibrations of a curved beam (arch) resting on a nonlinear elastic foundation. The arch is made of a functionally graded material, where the properties are graded across the thickness. Uniform temperature elevation in the arch is also considered and material properties are assumed to be temperature dependent. The governing motion equations of the arch are established using a higher order arch theory which satisfies the traction free boundary conditions and the von Kármán type of non-linearity. The governing equations of the arch are solved for the case of an immovable pinned arch using the two step perturbation technique. Closed form expressions are given to estimate the nonlinear frequencies of the arch as a function of the mid-span deflection. Numerical results of this study are validated for the case of flat FGM beams on elastic foundation. Afterwards, novel numerical results are given to explore the influences of power law index, elastic foundation coefficients, length to thickness ratio, length to radius ratio, and the temperature effects.