In the present paper, the vibrational and buckling characteristics of nanotubes with various boundary conditions are investigated considering the coupled e®ects of nonlocal elasticity and surface e®ects, including surface elasticity and surface tension. The nonlocal Eringen theory is adopted to consider the e®ect of small scale size, and the Gurtin-Murdoch model the surface e®ect. Hamilton's principle is employed to formulate the governing equation and di®erential transformation method (DTM) is utilized to obtain the natural frequency and critical buckling load of nanotubes with various boundary conditions. The results obtained match the available ones in the literature. Detailed mathematical derivations are presented and numerical investigations are performed. The emphasis is placed on the e®ects of several parameters, such as the nonlocal parameter, surface e®ect, aspect ratio, mode number and beam size, on the normalized natural frequencies and critical buckling loads of the nanotube. It is explicitly shown that the vibration and buckling of a nanotube is signi¯cantly in°uenced by these e®ects. Numerical results are presented which may serve as benchmarks for future analysis of nanotubes.

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In this paper, the effects of various surface parameters on free vibration behavior of nanobeams are investigated in the presence of nonlocal effect. The GurtinMurdoch model is employed for incorporating the surface parameters including surface density, surface tension and surface elasticity, while the Eringen's nonlocal elasticity theory takes into account the effect of small scales. The governing equations of motion are obtained for different materials such as aluminum and silicon with various boundary conditions. The high-precision semi-analytical differential transformation method is utilized to solve the governing equations. Then, by transforming the governing differential equations and boundary conditions into algebraic equations, the natural frequencies are obtained. The objective of the present study is to explore comprehensively the effect of different combination of various surface effects on nonlocal vibrational behavior of nanobeams for the first time. It is explicitly shown that the vibration characteristics of a nanobeam are significantly influenced by these surface effects. Moreover, it is shown that by increasing the size of beam, the influence of surface effects reduce to zero, and the natural frequency reaches its classical value. Numerical results are also presented to serve as benchmarks for future analyses of nanobeams.