Bifurcation and post-buckling analysis of bimodal optimum columns

Abstract

A 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 postbuckling behaviour of several columns with bimodal optimum buckling loads.The stability analysis of continuous and discrete nonlinear elastic systems requires the evaluation of critical (bifurcation) points, post-critical equilibrium states and their stability. A number of important problems of structural stability are characterized by multiple buckling modes associated with the same critical buckling load. For such problems the initial post-buckling behaviour is considerably more complicated compared with problems with simple (unimodal) bifurcation points. An account of the fundamentals of structural stability theory and post-buckling behaviour is given in the books and review articles by Hutchinson and Koiter (1970), Thompson 0020-7683/$ -see front matter Ó

Imperfection sensitivity of thin-walled I-section struts susceptible to cellular buckling

“…A variational model representing the nonlinear mode interaction between weak axis Euler buckling and the elastic local buckling of the flange plates was described. Other structural components such as sandwich struts [9,10,11], stringer-stiffened and corrugated plates [12,13,14,15,16,17,18] and thin-walled beams under uniform bending [19], are also known to suffer from the interaction of global and local modes of instability.…”

Optimum design of thermally loaded beam-columns for maximum vibration frequency or buckling temperature

“…For example, based on the Rayleigh quotient principle, Cheng and Zhong (1983) developed the sensitivity analysis and method of solution for optimal design of clamped-clamped columns under axial compressive forces, and Zhong and Cheng (1986) further proposed a Sequential Quadratic Programming approach for solution of the optimization problems. As recent papers on bimodal optimum design of columns, one may cite Novakovic and Atanackovic (2011) and Olhoff and Seyranian (2008), where the columns in the latter paper rest on a linearly elastic (Winkler) foundation. A notable paper (Bochenek, 1999) on bimodal optimum design of thermal buckling and post buckling behavior of elastic columns is closely related to the present study.…”

Optimal design of clamped columns for stability under combined axial compression and torsion

“…Kiusalaas (1973) was the first who drew attention to a multimodal phenomenon in structural optimization of columns on elastic foundation of the Winkler type under stability constraints. Multimodal optimization of columns on Winkler foundation was considered by Repin (1979), Turner andPlaut (1980), Laritchev (1982), Gajewski (1985), Bochenek (1999), Smaś (2001, 2004), Atanackovic and Novakovic (2006), Smaś (2007) and Olhoff and Seyranian (2008). On the other hand, papers by Kowalski (1967), Życzkowski and Gajewski (1971), Langthjem and Sugiyama (1999) and Wang (2003) deal with optimization of columns under non-conservative loading.…”