Analysis of nonlinear stability and post-buckling for Euler-type beam-column structure

Abstract

Based on the assumption of finite deformation, the Hamilton variational principle is extended to a nonlinear elastic Euler-type beam-column structure located on a nonlinear elastic foundation. The corresponding three-dimensional (3D) mathematical model for analyzing the nonlinear mechanical behaviors of structures is established, in which the effects of the rotation inertia and the nonlinearity of material and geometry are considered. As an application, the nonlinear stability and the post-buckling for a linear elastic beam with the equal cross-section located on an elastic foundation are analyzed. One end of the beam is fully fixed, and the other end is partially fixed and subjected to an axial force. A new numerical technique is proposed to calculate the trivial solution, bifurcation points, and bifurcation solutions by the shooting method and the NewtonRaphson iterative method. The first and second bifurcation points and the corresponding bifurcation solutions are calculated successfully. The effects of the foundation resistances and the inertia moments on the bifurcation points are considered.

Numerical Methods in Calculating Eigenvalues: Case Studies in Stability of Euler Columns

“…Increase in the number of unknowns means increase in the number of guesses that will further reduce the efficiency of the Shooting Method. Zhu [5] introduce a technique to improve the initial guessing during root finding in shooting method for calculating bifurcation points. A perturbation form of original problem is considered and then bisection method is used to find the perturbation points.…”