A novel meshless particle method for nonlocal analysis of two-directional functionally graded nanobeams

  • M. Rezaiee-PajandEmail author
  • M. Mokhtari
Technical Paper


Based on the Reddy–Bickford beam theory (RBBT), a comprehensive study on high-order bending, buckling and free vibration of two-directional functionally graded (FG) nanobeams is presented. The variation of the material properties is assumed to obey arbitrary power-law form in both axial and thickness directions. The equations of motion are derived using Hamilton’s principle, and small-scale effects are captured by nonlocal elasticity theory of Eringen. The RBBT formulation leads to complicated governing equations, and 2D FGM introduces additional stiffness terms. Accordingly, the governing equations cannot be solved by classical analytical schemes, and thus, symmetric smoothed particle hydrodynamics (SSPH) meshless method is adopted as an efficient numerical solution approach. The revised super-Gauss function is used as the kernel function. To validate the developed SSPH code, benchmark problems are studied and the results are compared to the analytical solutions. In this context, excellent agreement is observed. Several numerical examples are included to illustrate the effects of gradient indexes, boundary conditions, size scale parameters, aspect and elastic modulus ratios on static and dynamic responses of two-directional FG nanobeams.


Symmetric smoothed particle hydrodynamics (SSPH) Two-directional FG nanobeams, meshless method Static bending, buckling Free vibration 

List of symbols


Nonlocal modulus

\( \mu \)

Nonlocal parameter


Elastic modulus


Shear modulus




Volume fraction

\( \sigma_{xx} \)

Axial stress


Beam length


Radius of the CSD


Kernel function


Cross sectional area

\( k_{z} ,k_{x} \)

Gradient indexes


Error norm

\( \varepsilon_{xx} \)

Axial strain

\( \sigma_{xz} \)

Shear stress

\( \gamma_{xz} \)

Shear strain


Potential energy


External work


Kinetic energy

\( I_{i} ,J_{i} \)

Inertia coefficients


Beam height

\( \rho_{0} \)

Scaling factor


Dimensionality of space


Number of nodes (particles)


Mode number

\( \lambda \)

Normalized critical buckling load


Axial displacement


Transverse deflection

\( \phi \)

Bending rotation

\( N_{0}^{cr} \)

Critical buckling load

\( \omega_{n} \)

Natural frequency

\( A_{i} ,B_{i} \)

Stiffness coefficients


Moment of inertia


Beam width

\( h_{0} \)

Smoothing length

\( N_{0} \)

Compressive load


Transverse load

\( \nabla \)

Laplacian operator


Poisson’s ratio



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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringFerdowsi University of MashhadMashhadIran

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