SpringerLink
Log in

Periodic Greenhill’s Problem for Twisted Elastic Rod

  • Chapter
  • First Online:
Fundamentals of Structural Optimization

Part of the book series: Mathematical Engineering ((MATHENGIN))

  • 109 Accesses

Abstract

In this chapter the authors demonstrate the isoperimetric inequality arising in exactly solvable structural optimization problem of stability under torque load. The periodic Greenhill problem describes the forming of a loop in an elastic bar under torsion. The inequality for infinite rod with periodical cross-section with two types of supports is rigorously verified. The optimal shape of the twisted rod is constant along its length and the optimal shape of cross-section is the equilateral triangle. The technique to demonstrate of isoperimetric inequalities exploits the variational method and the Hölder inequality.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 139.09
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
EUR 181.89
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Greenhill, A. G. (1883). On the strength of shafting when exposed both to torsion and to end thrust. Institution of Mechanical Engineers, Proceedings, 182–225.

    Google Scholar 

  2. Kruzelecki, J., & Ortwein, R. (2012). Optimal design of clamped columns for stability under combined axial compression and torsion. Structural and Multidisciplinary Optimization, 45, 729–737.

    Article  MathSciNet  MATH  Google Scholar 

  3. Ziegler, H. (1977). Principles of structural stability, 2. ed. Birkhäuser.

    Google Scholar 

  4. Pachpatte, B. G. (2005). Mathematical inequalities. North-Holland Mathematical Library, 67. Elsevier.

    Google Scholar 

  5. Bullen, P. S. (2003). Handbook of means and their inequalities mathematics and its applications. Kluwer Academic Publishers.

    Google Scholar 

  6. Ting T. W. (1963). Isoperimetric inequality for moments of inertia of plane convex sets. Transactions of the American Mathematical Society, 107(3), 421–431.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Natural Sciences, University of Siegen, Paul-Bonatz-Str. 9-11, 57068, Siegen, Germany

    Vladimir Kobelev

Authors
  1. Vladimir Kobelev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Kobelev .

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kobelev, V. (2023). Periodic Greenhill’s Problem for Twisted Elastic Rod. In: Fundamentals of Structural Optimization. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-031-34632-3_8

Download citation

Publish with us

Policies and ethics

Search

Navigation