Bifurcation and Post-buckling Analysis of Bimodal Optimum Columns

AbstractA mathematical formulation of column optimization problems allowing for bimodal optimum buckling loads is developed in this paper. The columns are continuous and linearly elastic, and assumed to have no geometrical imperfections. It is first shown that bimodal solutions exist for columns that rest on a linearly elastic (Winkler) foundation and have clamped-clamped and clamped-simply supported ends. The equilibrium equation for a non-extensible, geometrically nonlinear elastic column is then derived, and the initial post-buckling behaviour of a bimodal optimum column near the bifurcation point is studied using a perturbation method. It is shown that in the general case the post-buckling behaviour is governed by a fourth order polynomial equation, i.e., near the bifurcation point there may be up to four post-buckling equilibrium states emanating from the trivial equilibrium state. Each of these equilibrium states may be either supercritical or subcritical in the vicinity of the bifurcation point. The conditions for stability of these non-trivial post-buckling states are established based on verification of positive semi-definiteness of a two-by-two matrix whose coefficients are integrals of the buckling modes and their derivatives. In the end of the paper we present and discuss numerical results for the post-buckling behaviour of several columns with bimodal optimum buckling loads

Instability regions for a system with periodically varying moment of inertia

The problem of finding the regions of instability of a system with a periodically varying moment of inertia is considered. An equation is derived which describes small torsional oscillations of a system with periodic coefficients, which depend on four constant parameters, including damping. A method of investigating stability based on an analysis of the behaviour of Floquet multipliers is described. Analytical expressions are obtained for the regions of instability (parametric resonance) in parameter space. Numerical examples are given