Abstract
Buckling load estimation of continua modeled by finite element (FE) should be based on non-linear equilibrium. When such equilibrium is obtained by incremental solutions and when sensitivity analysis as well as iterative redesigns are included, the computational demands are large especially due to optimization. Therefore, examples presented in the literature relate to few design variables and/or few degrees of freedom. In the present paper a non-incremental analysis is suggested, and a simple sensitivity analysis as well as recursive redesign is proposed. The implicit geometrical non-linear analysis, based on Green-Lagrange strains, apply the secant stiffness matrix as well as the tangent stiffness matrix, both determined for the equilibrium corresponding to a given reference load, obtained by the Newton-Raphson method. For the formulated eigenvalue problem, which solution gives the estimated buckling load, the tangent stiffness matrix is of major importance. In contrast to formulations based on incremental solutions, the tangent stiffness matrix is here divided into two matrices, the stress stiffness matrix that is linear depending on stresses and the remaining part of the tangent stiffness matrix. Examples verify the effectiveness of the proposed procedure.
Similar content being viewed by others
References
Bathe KJ, Dvorkin EN (1983) On the automatic solution of nonlinear finite element equations. Comput Struct 17(5-6):871–879
Brendel B, Ramm E (1982) Nichtlineare stabilitatsuntersuchungen mit methode der finiten elemente. Ingenieur-Archiv 51:337–362
Cook RD, Malkus DS, Plesha ME, Witt RJ (2002) Concepts and applications of finite element analysis, 4th edn. Wiley, New York. 719 pages
Crisfield MA (1991) Non-linear finite element analysis of solids and structures, vol 1. Wiley, Chichester. 345 pages
Onate E (1995) On the derivation and possibilities of the secant stiffness matrix for non linear finite element analysis. Comput Mech 15:572–593
Onate E, Matias WT (1996) A critical displacement approach for predicting structural instability. Comput Methods Appl Mech Eng 134:135–161
Pedersen NL, Pedersen P (2018) Local analytical sensitivity analysis for design of continua with optimized 3D buckling behaviour. Struct Multidiscip Optim 57(1):293–304
Pedersen P (2005) Analytical stiffness matrices with Green-Lagrange strain measure. Int J Numer Methods Eng 62:334–352
Pedersen P, Pedersen NL (2012) Interpolation/penalization applied for strength designs of 3d thermoelastic structures. Struct Multidiscip Optim 45:773–786
Pedersen P, Pedersen NL (2015) Eigenfrequency optimized 3D continua, with possibility for cavities. J Sound Vib 341:100–115
Roorda J (1965) Stability of structures with small imperfections. ASCE 91(EM1):87–106
Wang W, Clausen PM, Bletzinger KU (2015) Improved semi-analytical sensitivity analysis using a secant stiffness matrix for geometric nonlinear shape optimization. Comput Struct 146:143–151
Author information
Authors and Affiliations
Corresponding author
Additional information
Responsible Editor: Erdem Acar
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Pedersen, N.L., Pedersen, P. Buckling load optimization for 2D continuum models, with alternative formulation for buckling load estimation. Struct Multidisc Optim 58, 2163–2172 (2018). https://doi.org/10.1007/s00158-018-2030-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-018-2030-3