Skip to main content
SpringerLink
Log in

Geometry optimization of a slender cantilever beam subjected to lateral buckling

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

The paper deals with the geometry optimization of a slender cantilever beam subjected to a concentrated force acting at the free end. The two-parametric mathematical model of lateral torsional buckling is based on the Bernoulli–Euler beam theory and is given in dimensionless form. The optimization procedure is performed using the optimal control theory and the relation between state and adjoint variables is presented. The boundary value problem derived from the optimization procedure is solved numerically and compared to solutions obtained via an alternative optimization approach called sequential approximate optimization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Atanackovic TM (2007) Optimal design of clamped columns for stability under combined axial compression and torsion. ZAMM 87:399–405

    Article  MathSciNet  MATH  Google Scholar 

  • Atanackovic TM, Simic SS (1999) On the optimal shape of a Pflurger column. Eur J Mech A/Solids 18:903–913

    Article  MathSciNet  MATH  Google Scholar 

  • Atanackovic TM, Jakovljevic BB, Petkovic MR (2010) On the optimal shape of a column with partial elastic foundation. Eur J Mech A/Solids 29:283–289

    Article  MathSciNet  Google Scholar 

  • Braun DJ (2008) On the optimal shape of compressed rotating rod with shear and extensibility. Int J Non-Linear Mech 43:131–139

    Article  MATH  Google Scholar 

  • Choi KK, Kim NH (2005) Structural sensitivity analysis and optimization 1. Linear systems. Springer, New York

    Google Scholar 

  • Cox SJ (1992) The shape of the ideal column. Math Intell 14:16–24

    Article  MATH  Google Scholar 

  • Drazumeric R, Kosel F (2005) Optimization of geometry for lateral buckling process of a cantilever beam in the elastic region. Thin-Walled Struct 43:515–529

    Article  Google Scholar 

  • Drazumeric R, Kosel F (2012) Shape optimization of beam due to lateral buckling problem. Int J Non-Linear Mech 47:65–74

    Article  Google Scholar 

  • Gajewski A, Zyczkowski M (1988) Optimal structural design under stability constraints. Kluwer Academic, Dordrecht

    Book  MATH  Google Scholar 

  • Haftka RT, Gurdal Z (1992) Elements of structural optimization. Kluwer Academic, Dordrecht

    Book  MATH  Google Scholar 

  • Haftka RT, Sobieski JS (2009) Structural optimization: history. In: Faudas CA, Pardalos PM (eds) Encyclopedia of optimization. Springer, New York, pp 3834–3836

    Chapter  Google Scholar 

  • Haug EJ, Choi KK (1982) Systematic occurence of repated eigenvalues in structural optimization. J Opt Theory Appl 38:251–274

    Article  MathSciNet  MATH  Google Scholar 

  • Keller JB (1960) The shape of the strongest column. Arch Ration Mech Anal 5:275–285

    Article  Google Scholar 

  • Kruzelecki J, Ortwein R (2011) Optimal design of clamped columns for stability under combined axial compression and torsion. Struct Multidisc Optim 9:181–196

    Google Scholar 

  • Manickarajah D, Xie YM, Steven GP (2000) Optimization of columns and frames against buckling. Comput Struct 75:45–54

    Article  Google Scholar 

  • Novakovic BN, Atanackovic TM (2009) Optimal shape of a heavy elastic rod loaded with a tip concentrated force against lateral buckling. Int J Struct Stab Dyn 9:83–90

    Article  MathSciNet  Google Scholar 

  • Novakovic BN, Atanackovic TM (2011) On the optimal shape of a compressed column subjected to restrictions on the cross-sectional area. Struct Multidisc Optim 43:683–691

    Article  Google Scholar 

  • Olhoff N, Ramussen SH (1977) On single and bimodal optimum buckling loads of clamped columns. Int J Solids Struct 13:605–614

    Article  MATH  Google Scholar 

  • Olhoff N, Seyranian AP (2008) Bifurcation and post-buckling analysis of bimodal optimum columns. Int J Solids Struct 45:3967–3995

    Article  MATH  Google Scholar 

  • Olhoff N, Taylor JE (1983) On structural optimization. J App Mech 50:1139–1151

    Article  MathSciNet  MATH  Google Scholar 

  • Plaut RH, Virgin LN (2010) Vibrations and large postbuckling deflections of optimal pinned columns with elastic foundations. Struct Multidisc Optim 40:157–164

    Article  MathSciNet  Google Scholar 

  • Popelar CH (1976) Optimal design of beams against buckling: a potential energy approach. J Struct Mech 9:181–196

    Article  Google Scholar 

  • Schmit LA (1960) Structural design by systematic synthesis In: Proceedings of second conference on electronic computation ASCE

  • Seyranian AP (1984) On a problem of Lagrange. Mech Solids 19:100–111

    MathSciNet  Google Scholar 

  • Seyranian AP, Lundt E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8:207–227

    Article  Google Scholar 

  • Tadjbakhsh I, Keller JB (1962) Strongest columns and isoperimetric inequalities for eigenvalues. J Appl Mech 29:159–164

    Article  MathSciNet  MATH  Google Scholar 

  • Timoshenko SP (1961) Theory of elastic stability. McGraw-Hill, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Cimos d.d., Cesta Marezganskega upora 2, 6000, Koper, Slovenia

    Miha Polajnar

  2. Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, 1000, Ljubljana, Slovenia

    Radovan Drazumeric & Franc Kosel

Authors
  1. Miha Polajnar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Radovan Drazumeric
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Franc Kosel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miha Polajnar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Polajnar, M., Drazumeric, R. & Kosel, F. Geometry optimization of a slender cantilever beam subjected to lateral buckling. Struct Multidisc Optim 47, 809–819 (2013). https://doi.org/10.1007/s00158-012-0858-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-012-0858-5

Keywords

Search

Navigation