This paper presents analytical solutions for bending and buckling of nonlocal functionally graded (FG) Euler–Bernoulli (EB) nanobeams. Material gradation along the thickness direction could be defined by a power function (P-FG), a sigmoidal function (S-FG), and an exponential function (E-FG). Laplace transform is applied to the differential form of the equation of motion of the nonlocal elasticity theory. Closed-form expressions for bending deflection and critical buckling load of FG nanobeams are derived. Effects of material gradations as well as the nonlocal parameter are examined. It is found that bending displacements and critical buckling loads could be controlled by an appropriate choice of material distribution parameter for P-FG nanobeams. The presented results also demonstrate the influences of factors such as the choice of material gradation, power-law index, and nonlocal parameter on bending and buckling behavior.
The nonlocal elasticity theory and the Euler–Bernoulli (EB) beam theory are used to present closed-form analytical expressions for static bending, axial buckling, and free vibration of nanosized beams resting on an elastic foundation. The differential constitutive relations of Eringen are utilized to represent the small-scale effects of the nanobeam’s mechanical response. The governing equation of motion is derived by employing Hamilton’s principle. Utilizing the Laplace transform approach, analytical expressions of the bending displacements, the critical buckling force, and the vibration frequency of nanobeams with simply supported (S-S), clamped, cantilevered, and propped cantilevered boundary conditions are produced. In order to confirm the correctness of the offered closed-form equations, their outputs are compared to those of the available numerical method solutions. The effects of the Winkler parameter, the Pasternak parameter and the nonlocal parameter on bending, buckling, and vibration characteristics of nanobeams have been explained. Presented analytical expressions and graphical representations demonstrate how increasing Winkler and Pasternak parameters reduce bending displacements while raising the critical buckling load and the natural frequency of nonlocal nanobeams. Benchmark numerical results are also presented to investigate and discuss the effects of all parameters on bending deflections, buckling loads, and natural frequencies of nanobeams.
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