Differential quadrature application in post-buckling analysis of a hinged-fixed elastica under terminal forces and self-weight

Abstract

Based on geometrically non-linear theory for extensible elastic rods, governing equations of statically post-buckling of a beam with one end hinged and the other fixed, and subjected to a terminal force and a self-weight, are established. The formulation is derived from geometrical compatibility, equilibrium of forces and moments, and constitutive relations, which characterize a complex two-point boundary value problem. By using differential quadrature method (DQM), the non-linear governing equations are solved numerically and the post-buckled configurations of the deformed column are presented. Results are plotted in non-dimensional graphs for a range of density and terminal force, and are in good agreement with available references.

Natural frequencies and mode shapes of statically deformed inclined risers

“…The obtained beam mode shapes are in good agreement with those of Dareing and Huang [3]. Most of the recent research that examines the linear vibration of vertical risers has focused on the mathematical aspects of the methods, such as the segmentation method [5,6], differential transformation methods [7,8], and the varational iteration method [9]. In all of the previous research, the axial stretch was neglected, which motivated Mazzilli [10] to use the method of multiple scales [11] to examine the nonlinear modes of vertical risers.…”

Stability Analysis of Nonlocal Elastic Columns with Initial Imperfection

“…From the effectiveness and accuracy of the proposed methods, we can also conclude that the presented methods can be potentially extended to a broad range of column problems under large deformations, such as the postbuckling problems of shallow arches subjected to lateral loads, problems for columns with initial imperfection having the shape of the second, or higher buckling modes [31], and problems for columns with the inextensibility assumption relaxed to an extensible one. C. Coefficients Appeared in Equations (36) and (38)…”

“…But in general, the first or second approximate is usually sufficient for establishing a large part of the postbuckling path [63]. In view of this and considering that higher approximations require considerable computational efforts, we choose the first approximate solution (36) as the final result, which, by applying the condition 1 (0) = , furnishes us the relation between and 2 in the following form:…”

“…It converts the nonlinear boundary-value problem into an initial-value problem, which is then determined with or without iterations [31,32]. Other newly developed numerical methods include the Lie-group shooting method [33], method of genetic algorithm [34,35], and differential quadrature method [36]. In view that an analytical solution is always convenient for parametric studies and captures clearly the physics of the problem, it appears more appealing than the numerical one.…”