Skip to main content
SpringerLink
Log in

On Bending of Rod Under Strong Longitudinal Compression

  • Conference paper
  • First Online:
Proceedings of the 4th International Conference on Industrial Engineering (ICIE 2018)

Part of the book series: Lecture Notes in Mechanical Engineering ((LNME))

Included in the following conference series:

  • 34 Accesses

Abstract

In this paper, we consider typical processes of rod bending under strong longitudinal compression. The corresponding dynamic equation of bending is considered as a perturbation of the two-dimensional Laplace equation. It is established that, for these processes, expanding of domains of rapid increase of bending begins in small neighborhoods of singularity points of solutions of the limiting Laplace equation. The initial stages of these increases are described using the Hardy integral. Several typical examples of the development of buckling of a rod under the action of strong compression are simulated in the work.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Vol’mir AS (1967) Stability of deformable systems. Nauka, Moscow (in Russian)

    Google Scholar 

  2. Panovko YaG, Gubanov II (1987) Stability and vibrations of elastic systems. Fizmatlit, Moscow (in Russian)

    Google Scholar 

  3. Lavrent’ev MA, Ishlinskii AYu (1949) Dynamic buckling modes of elastic systems. Dokl Akad Nauk 64:779–782 (in Russian)

    Google Scholar 

  4. Il’gamov MA (1969) Vibrations of elastic shells containing a liquid and gas. Nauka, Moscow (in Russian)

    Google Scholar 

  5. Paidouisis MP (1975) Flatter of conservative systems of pipes conveying incompressible fluid. J Mech Eng Sci 17:19–25

    Article  Google Scholar 

  6. Vol’mir AS (1979) Shells in fluid flow. Hydroelastic problems. Nauka, Moscow (in Russian)

    Google Scholar 

  7. Akhtyamov AM, Safina GF (2008) Vibration-proof conduit. J Appl Mech Tech Phys 49:114–121

    Article  MathSciNet  Google Scholar 

  8. Ishlinskii AYu (1976) Applied problems in mechanics. Book 2. Mechanics of elastic and rigid bodies. Nauka, Moscow (in Russian)

    Google Scholar 

  9. Ishlinskii AYu (1985) Mechanics, ideas, problems, applications. Nauka, Moscow (in Russian)

    Google Scholar 

  10. Koning C, Taub J (1933) Impact buckling of thin bars in the elastic range hinged at both ends. Luftfahrtforshung 10:55–64

    Google Scholar 

  11. Il’gamov MA (2010) Reconstruction of harmonics during the dynamic loss of stability in mechanical systems. Dokl Phys 55:297–301

    Article  Google Scholar 

  12. Il’gamov MA (2014) Dependence of dynamic buckling of a rod on the initial conditions. Dokl Phys 59:385–388

    Article  Google Scholar 

  13. Morozov NF, Belyaev AK, Tovstik PE, Tovstik TP (2015) The Ishlinskii-Lavrent’ev problem at the initial stage of motion. Dokl Phys 60:368–371

    Article  Google Scholar 

  14. Morozov NF, Belyaev AK, Tovstik PE, Tovstik TP (2015) The Ishlinskii-Lavrent’ev problem at the initial stage of motion. Dokl Phys 60:519–523

    Article  Google Scholar 

  15. Tarkhanov NN (1995) The Cauchy problem for solutions of elliptic equations. In: Mathematical topics, vol. 7. Akademie Verlag, Berlin

    Google Scholar 

  16. Hardy GH (1917) On the asymptotic value of certain integrals. Mess Math 46:70–73

    Google Scholar 

  17. Riekstynsh EYa (1977) Asymptotic expansions of integrals. Zinatne, Riga

    Google Scholar 

  18. Il’in AM (1992) Matching of asymptotic expansions of solutions of boundary value problems. In: Translations of mathematical monographs. American Mathematical Society, Providence, Rhode Island

    Google Scholar 

  19. Ershov AA, Suleimanov BI (2017) Some features of bending of a rod under a strong longitudinal compression. Russ J Math Phys 24:216–233

    Article  MathSciNet  Google Scholar 

  20. Babich VM, Buldyrev VS (1991) Asymptotic methods in shortwave diffraction theory. Springer, Berlin

    Google Scholar 

  21. Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG (1954) Tables of integral transforms. McGraw-Hill, New York, Toronto, London

    Google Scholar 

  22. Brychkov YuA, Prudnikov AP (1977) Integral transforms of generalized functions. Nauka, Moscow (in Russian)

    MATH  Google Scholar 

Download references

Acknowledgements

For the information, we found very useful when writing this article and we express our gratitude to B. I. Suleimanov, N. F. Valeev, A. R. Danilin, M. A. Il’gamov, and I. V. Mel’nikova. This work was supported by the Russian Foundation for Basic Research, project no. 18-31-00018-mol_a.

Author information

Authors and Affiliations

  1. N.N. Krasovskii Institute of Mathematics and Mechanics, 16, S. Kovalevskaya Str., Yekaterinburg, 620990, Russia

    A. A. Ershov & A. A. Ershova

  2. South Ural State University, 76, Lenin Avenue, Chelyabinsk, 454080, Russia

    A. A. Ershova

Authors
  1. A. A. Ershov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Ershova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Ershov .

Editor information

Editors and Affiliations

  1. South Ural State University, Chelyabinsk, Russia

    Andrey A. Radionov

  2. Platov South-Russian State Polytechnic University, Novocherkassk, Russia

    Oleg A. Kravchenko

  3. South Ural State University, Chelyabinsk, Russia

    Victor I. Guzeev

  4. South Ural State University, Chelyabinsk, Russia

    Yurij V. Rozhdestvenskiy

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ershov, A.A., Ershova, A.A. (2019). On Bending of Rod Under Strong Longitudinal Compression. In: Radionov, A., Kravchenko, O., Guzeev, V., Rozhdestvenskiy, Y. (eds) Proceedings of the 4th International Conference on Industrial Engineering. ICIE 2018. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-95630-5_72

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-95630-5_72

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-95629-9

  • Online ISBN: 978-3-319-95630-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation