Optimal shape of an elastic column on elastic foundation

Atanacković, Teodor M.; Novakovic, Branislava N.

doi:10.1016/j.euromechsol.2005.06.008

Cited by 23 publications

(16 citation statements)

References 19 publications

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“…3. For the compressed rod described by the Bernoulli-Euler theory it was shown that the optimal shape of the rod obtained by bimodal optimization agrees with the ones presented in Atanackovic and Novakovic (2006), Olhoff and Rasmussen (1977), Seyranian (1984), Cox and Overton (1992) and Egorov (2004). As a generalization of this case a compressible and shearable compressed rod is examined.…”

Section: Closuresupporting

confidence: 58%

Does generalized elastica lead to bimodal optimal solutions?

Spasic

Glavardanov

2009

International Journal of Solids and Structures

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Section: Closuresupporting

confidence: 58%

Does generalized elastica lead to bimodal optimal solutions?

Spasic

Glavardanov

2009

International Journal of Solids and Structures

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“…the books by Gajewski and Zyczkowski (1988), Zyczkowski (1989), the survey articles by Olhoff (1980) and Olhoff and Taylor (1983), and the papers by Olhoff and Niordson (1979), Olhoff and Plaut (1983), Plaut and Olhoff (1983), Seyranian (1984), Masur (1984), Seyranian et al (1994), Seyranian and Privalova (2003), and Atanackovic and Novakovic (2006). In the paper by Seyranian and Mailybaev (2001) it is shown that typical conservative multi-parameter systems exhibit multiple r-fold critical eigenvalues (and eigenmodes) if the number of parameters of the system exceeds r(r + 1)/2.…”

Section: Introductionmentioning

confidence: 98%

Bifurcation and post-buckling analysis of bimodal optimum columns

Olhoff

Seyranian

2008

International Journal of Solids and Structures

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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

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“…Examples of this work involving clamped-clamped columns include Olhoff and Rasmussen (1977), Blachut and Zyczkowski (1984), Masur (1984), Plaut et al (1986), Tada and Wang (1995), and more recently the papers by Manickarajah et al (2000), Maalawi (2002), Olhoff and Seyranian (2002), Egorov (2002Egorov ( , 2003Egorov ( , 2004, and Atanackovic and Novakovic (2006), where several issues concerning the optimization of columns have been studied with the bimodal buckling behaviour receiving special attention. The finite-element method was used for the optimization of columns by Simitses et al (1973), and of frames by Szyszkowski and Watson (1988) and Szyszkowski et al (1989).…”

Section: Introductionmentioning

confidence: 98%

Optimization of clamped columns under distributed axial load and subject to stress constraints

Cagdas

Adali

2007

Engineering Optimization

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The optimum designs are given for clamped-clamped columns under concentrated and distributed axial loads. The design objective is the maximization of the buckling load subject to volume and maximum stress constraints. The results for a minimum area constraint are also obtained for comparison. In the case of a stress constraint, the minimum thickness of an optimal column is not known a priori, since it depends on the maximum buckling load, which in turn depends on the minimum thickness necessitating an iterative solution. An iterative solution method is developed based on finite elements, and the results are obtained for n = 1, 2, 3 defined as I = α n A n , with I being the moment of inertia, and A the crosssectional area. The iterations start using the unimodal optimality condition and continue with the bimodal optimality condition if the second buckling load becomes less than or equal to the first one. Numerical results show that the optimal columns become larger in the direction of the distributed load due to the increase in the stress in this direction. Even though the optimal columns are symmetrical with respect to their mid-points when the compressive load is concentrated at the end-points, in the case of the columns subject to distributed axial loads the optimal shapes are unsymmetrical.

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Optimal shape of an elastic column on elastic foundation

Cited by 23 publications

References 19 publications

Does generalized elastica lead to bimodal optimal solutions?

Does generalized elastica lead to bimodal optimal solutions?

Bifurcation and post-buckling analysis of bimodal optimum columns

Optimization of clamped columns under distributed axial load and subject to stress constraints

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