Stability of a strut with one end fixed and the other pinned only at the first bifurcation point

Abstract

In the sense of the Lagrange–Dirichlet minimum energy stability criterion, the static stability of a strut with one end fixed and the other pinned only at the first bifurcation point is investigated analytically. The second variation of potential energy expressed by the deflection is semi-positive-definite only at the first bifurcation point and vanishes only in the ‘buckling mode’ in small deflection theory. The fourth variation of potential energy is positive in the ‘buckling mode’. The potential energy of the strut at the first bifurcation point is proved to hold a minimum. Based on the Lagrange–Dirichlet stability criterion, the strut at the first bifurcation point is stable.

Stability to the finite disturbance of the critical equilibrium for three struts

“…Jin and Qi [5] gave the second variation of potential energy in a diagonal quadratic forms for a pinned-pinned strut based on the strain gradient elasticity. However, since the quadratic form is not diagonal in many other cases, for example [6][7][8][9], proofs of the semi-positive-definite of the second variation are tedious. Since terms of higher order than the fourth order in the increment of potential energy are omitted in the stability analysis for a strut in previous studies [1,10], the stability is to the disturbance with infinitesimal value.…”

Initial post-buckling analysis of a fixed–fixed strut

“…For a structure, it should be theoretically proved that the second-order variation of the potential energy is semi-positive definite at the stability limit. For the three types of strut, i.e., a strut with one end fixed and the other clamped in rotation, a fixed-free strut, and a fixed-pinned strut, Jin and Bao [4], and Jin [5], respectively, presented the theoretical proof of this assumption. For the fixed-fixed strut, this paper will present a proof of this assumption and study the initial post-buckling behavior of the strut.…”