Refined quasi-3D shear deformation theory for buckling analysis of functionally graded curved nanobeam rested on Winkler/Pasternak/Kerr foundation

In the present work, a novel refined three-variable quasi-3D shear deformation theory incorporates a correction factor is developed to analyze the buckling behavior of multi-directional functionally graded (FG) curved beams. The proposed displacement field is formulated in accordance with the Euler-Bernoulli beam theory. The research investigated two types of coated Functionally Graded (FG) nanobeams: Hardcore (HC) FG curved nanobeams and Softcore (SC) FG curved nanobeams. Three different material distributions are taken into consideration: a bidirectional material distribution referred to as "2D-FG," a unidirectional transverse material distribution known as "T-FG," and a unidirectional axial material distribution called "A-FG." Eringen's nonlocal elasticity theory is employed to capture small-scale effects. The total potential energy principle is utilized to derive the equilibrium equations of curved nanobeams. A novel solution, utilizing Galerkin's method, has been developed to effectively address a range of boundary conditions. The curved FG beam is supported by an elastic foundation following the Winkler/Pasternak/Kerr model. A comprehensive analysis has been conducted to examine the impacts of various FG schemes, curved beam geometry, nonlocal parameter, elastic foundations, and various boundary conditions on the dimensionless critical buckling loads. This analysis aims to provide a comprehensive understanding of how each of these factors influences the critical buckling loads of the nanobeams.