Within the framework of Timoshenko beam theory, the buckling of nano sandwich beams is developed. The material properties are assumed to vary arbitrarily in both axial and thickness directions. These types of beams are referred to as bi-directional functionally graded -BDFG- beams. Two types of nano sandwich beams with different material distribution patterns and immovable supports are considered. Since the size effects play a significant role in mechanical behavior of nanostructures, the small-scale effects are captured by Eringen’s nonlocal theory of elasticity. The governing equations are derived using the variational formulation. Symmetric smoothed particle hydrodynamics -SSPH- and the Galerkin method are adopted as numerical solution approaches. As a truly meshless method, the convergence of the SSPH technique mainly depends on the smoothing length value and distribution of particles in the compact support domain of the kernel function. The Revised Super Gauss Function is used as the kernel function and an optimum value for the smoothing length that bears the fastest convergence rate is obtained. The solution methods are verified through benchmark problems found in the literature. Numerical and illustrative results show that various parameters, including the aspect ratio, nonlocal parameter, gradient indexes, and cross-sectional types have significant effects on the buckling responses of BDFG nano sandwich beams.

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