Postbuckling Analysis of a Nonlocal Nanorod Under Self-Weight

Abstract

<jats:p> This paper presents the postbuckled configurations of simply supported and clamped-pinned nanorods under self-weight based on elastica theory. Numerical solution is considered in this work since closed-form solution of postbuckling analysis under self-weight cannot be obtained. The set of nonlinear differential equations of a nanorod including the effect of nonlocal elasticity are investigated. The constraint equation at boundary condition technique is introduced for the solution of postbuckling analysis. In order to solve the set of nonlinear differential equations, the shooting method is utilized, where the set of these equations along with boundary conditions are integrated by the fourth-order Runge-Kutta algorithm. Numerical results are obtained and the highlighting influences of the nonlocal elasticity on postbuckling behavior of nanorods are discussed. The obtained results indicate that the rotation angle and the postbuckled configurations of nanorods are varied by changing the nonlocal elasticity parameter. The effect of nonlocal elasticity shows the softening behavior in comparison with the Euler beam. The present formulation together with constraint boundary condition technique is an effective solution for postbuckling analysis of a nanorod under self-weight including the effect of nonlocal elasticity. </jats:p>

Simple Approximate Formulas for Postbuckling Deflection of Heavy Elastic Columns

“…This deflection, called columnar buckling, is a crucial consideration in structural design because the column becomes unstable under compressive stress considerably lesser than the failure strength of the constituent materials [1]. In addition to the field of structural design, columnar buckling is relevant in a wide range of fields in which slender column structures are involved; plant science, nanomaterial engineering [2], and robotics [3] are only a few to mention. In plant science, for example, columnar buckling has been presumed to impose an upper limit on the growth height of upright stems and trunks [4][5][6].…”

“…Due to the nonlinearity, this differential equation cannot be solved in an elementary way, unless it is approximated that the column deflection is exceedingly small. Therefore, various methods have been proposed for solving it to realize high-accuracy analysis of columnar buckling behaviors under different mechanical conditions, including the geometry, loading conditions, and boundary conditions [2,. Most methods have been based on either the Bessel function, elliptic integral, or other nonelementary functions/integrals, ensuring high accuracy in the analysis of column buckling behavior even with large deflections; those based on generalized hypergeometric functions [29] and the singular perturbation method [31] are only a few to mention.…”

Simple approximate formulas for postbuckling deflection of heavy elastic columns

“…Due to the nonlinearity, this differential equation cannot be solved in an elementary way, unless it is approximated that the column deflection is exceedingly small. Therefore, various methods have been proposed for solving it to realize high-accuracy analysis of columnar buckling behaviors under different mechanical conditions, including the geometry, loading conditions, and boundary conditions 2, . Most meth-ods have been based on either the Bessel function, elliptic integral, or other nonelementary functions/integrals, ensuring high accuracy in the analysis of column buckling behavior even with large deflections; those based on generalized hypergeometric functions 29 and the singular perturbation method 31 are only a few to mention.…”