Static, Vibration, and Buckling Analysis of Nanobeams

Abstract

Static, vibration, and buckling analysis of nanobeams is studied based on modified couple stress theory (MCST) in this chapter. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new nonclassical beam model reduces the classical beam model when the length scale parameter is set to zero. The finite element formulations are derived for static, free vibration, and buckling problems of nanobeams within MCST and the Euler-Bernoulli beam theory. The effect of the material length scale parameter and geometry parameters on the static, vibration, and buckling responses of the nanobeam is investigated in both the classical beam theory (CBT) and MCST by using finite element method. Also, the difference between the classical beam theory (CBT) and modified couple stress theory is investigated.

A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory

“…Interpolation functions are used to derived the stiffness matrix of the elements that constitute the frame system. These interpolation functions for the axial and transverse displacements are as follows [20,22,57]:…”

“…The above matrix includes both bending and axial effects. Buckling, vibration and bending analysis of nanobeams via matrices that include these both effects were presented by Akbaş [20] using modified couple stress theory. This matrix, which can be used for the solution of a straight nanobeam, is used for the elements of the nanoframe in this study.…”

“…Structures such as rods, beams, plates, and frames have recently been modeled at nano and micro scales and their various analyzes like static, vibration and buckling have been carried out based on the mentioned size dependent theories. Navier's method [5][6][7][8][9][10][11], Fourier sine solution [12][13][14][15][16][17], finite element method [18][19][20][21][22][23][24][25][26][27][28][29], separation of variable procedure [30,31], generalized differential quadrature method [32][33][34][35][36], Ritz method have been frequently used by scholars to present the mechanical responses of various small scale structures such as nanobeam, nanoframe, nanotruss, nanorod, nanoplate, cracked microbeam with functionally graded material, cracked nanobeam, functionally graded nanobeam, porous nanotube etc. The above-mentioned solution methods and others have been also utilized for macro-dimensional porous plate [37][38][39], porous beam [40,41], beam [42], reinforced plate [43,44], functionally graded and laminated beam [45]…”

A Solution Method for Longitudinal Vibrations of Functionally Graded Nanorods

“…The nonlocal elasticity theory has become a frequently performed theory in nanomechanics and micromechanics, as it allows the consideration of small-scale effects. In addition, articles using finite element method to examine the behavior of size-dependent microstructures/nanostructures such as vibration [25][26][27][28][29][30], buckling [29][30][31][32] and bending [29][30][33][34][35] are also found in the literature.…”