Beam Buckling Analysis by Nonlocal Integral Elasticity Finite Element Method

Abstract

In this study, a finite element method (FEM) based on the size dependent nonlocal integral elasticity theory is implemented for buckling analysis of nanoscaled beams with various boundary conditions. The method is based on the principle of total potential energy. The variations of buckling load with respect to the scaling effect parameter and to the length-to-thickness ratio are investigated. Furthermore, the effect of attenuation function type on the buckling load is examined. The results are compared with the corresponding solutions of governing stability equations which are derived in the context of nonlocal differential elasticity theory. It is found that the small scale coefficient has a noticeable effect on the buckling load of nanobeams.

Free vibration characteristics of nanoscaled beams based on nonlocal integral elasticity theory

“…However, it should be noted that only for some special kernels, the integral form of nonlocal theory could be converted to the differential form. Moreover, the latter equality is only valid when the limits of integration are infinity or x is far enough from the boundaries (Taghizadeh et al., 2016), otherwise this method would lead to some inaccurate results. In addition, handling the simply-supported and free boundary conditions would be ambiguous for nanoplates considering the differential nonlocal equations.…”

“…Taghizadeh et al. (2015, 2016) have investigated the buckling and bending problems of a nanobeam considering various types of loadings and boundary conditions using a FEM based on nonlocal integral elasticity theory. By comparing the results with those of the nonlocal differential elasticity, it has been shown that the nonlocal differential elasticity may lead to noticeable errors in some cases.…”

Nonlocal integral approach to the dynamical response of nanobeams

“…Furthermore, Taghizadeh et al [58,59] presented finite element formulation in the context of nonlocal integral elasticity for the bending and the buckling problem of a nanobeam, respectively, with different types of loading and boundary conditions. By comparing the results of nonlocal integral elasticity with those of nonlocal differential elasticity for the bending problem, they concluded that the nonlocal integral form successfully handled all the problems investigated [58].…”

“…By comparing the results of nonlocal integral elasticity with those of nonlocal differential elasticity for the bending problem, they concluded that the nonlocal integral form successfully handled all the problems investigated [58]. In addition, the work [59] focused on the buckling behavior of beams with various boundary conditions by employing the NL-FEM. They solved the corresponding eigenvalue problems, then made a comparison between the deriving results and the corresponding solutions of the nonlocal differential stability equations and drew to conclusion that the nonlocal integral buckling loads exceed those of the differential model.…”

Buckling of elastically restrained nonlocal carbon nanotubes under concentrated and uniformly distributed axial loads

“…Previous studies on the buckling of nanotubes mostly involved rigid boundary conditions such as clamped, hinged and free boundaries. Recent studies on the buckling of nanotubes under a tip load and subject to classical boundary conditions include Ansari et al (2011), Sahmani and Ansari (2011), Kumar (2016), Kumar and Deol (2016) and Taghizadeh and Ovesy (2016). In these studies the buckling loads for carbon nanotubes were obtained subject to classical boundary conditions.…”