Skip to main content
SpringerLink
Log in

Stability Bounds and Optimal Shape of Elastic Rods

  • Chapter
Modern Problems of Structural Stability

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 436))

  • 263 Accesses

Abstract

In these Lecture notes we treat two problems of determining stability boundary for elastic rods (columns) and two optimization problems in which we determine the shape of the lightest rod, stable against buckling. In the problems of determining stability boundary we shall use generalized rod equations (corresponding to planar and spatial deformations) that take into account both shear deformation and axis extension and examine the influence of these effects on the stability boundary.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alekseev, V. M., Tihomirov, V. M. and Fomin, S. V. (1979). Optimal Control. Moscow: Nauka, (in Russian).

    Google Scholar 

  • Antman, S. S., and Kenny, C. S. (1981). Large Buckled States of nonlinearly Elastic Rods under Torsion, Thrust and Gravity. Archive for Rational Mechanics and Analysis 76: 289–338.

    Article  MATH  MathSciNet  Google Scholar 

  • Antman, S. S., and Nachman, A. (1980). Large Buckled States of Rotating Rods. Nonlinear Analalvsis, Theory Applications 4: 303–327.

    MATH  MathSciNet  Google Scholar 

  • Antman, S. S., and Christoforis, M. L. (1997). Peculiar Instabilities due to the Clamping of Shearable rods. International Journal of Non—Linear Mechanics 32: 31–54.

    Article  MATH  MathSciNet  Google Scholar 

  • Antman, S. S. (1995). Nonlinear Problems of Elasticity. New York: Springer.

    Book  MATH  Google Scholar 

  • Arthurs, A. M. (1980). Complementary Variational Principles. Oxford: Clarendon Press.

    MATH  Google Scholar 

  • Atanackovic, T. M., and Glavardanov, V. B. (1996). Twisted axially loaded rod with shear and compressibility. Acta Mechanica 119: 119–130.

    Article  MATH  Google Scholar 

  • Atanackovic, T. M., and Glavardanov, V. B. (2002) Buckling of a twisted and compressed rod. International Journal of Solids and Structures (in press).

    Google Scholar 

  • Atanackovic, T. M. (2001). On the optimal shape of a rotating rod. Transactions of ASAtE Journal of Applied. Mechanics 68: 860–864.

    Article  MATH  MathSciNet  Google Scholar 

  • Atanackovic, T. M. and Simic, S. S. (1999). On the optimal shape of Pfltiger column. European. Journal of Mechanics, Solids 18: 903–913.

    MATH  MathSciNet  Google Scholar 

  • Atanackovic, T. M. (1997). Stability Theory of Elastic rods. Singapore: World Scientific.

    MATH  Google Scholar 

  • Atanackovic, T. M. (1987). Stability of rotating compressed rod with imperfections. Mathematical. Procedings of Cambridge Philospical. Society 101: 593–607.

    Article  MATH  MathSciNet  Google Scholar 

  • Atanackovic, T. M., Djukic Dj. S., and Jones, S. E. (1991). Effect of shear on the stability and nonlinear behavior of rotating rod. Archive of Applied. Mechanics 61: 285–294.

    Article  MATH  Google Scholar 

  • Atanackovic, T. M. (2001). On a Vujanovic-type variational priniciple with application to rod theory. Ouoterly Journal Mechanics and Applied Mathematics 54: 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  • Bazely, N., and Zwahlen, B. (1968). Remarks on the Bifurcation of Solutions of a Non-linear Eigenvalue problem. Archive of Rational Mechanics and Analysis 28: 51–58.

    Article  Google Scholar 

  • Beck, M., Knickung gerader Stabe durch Druck and conservative Torsion. Ingenieur Archiv 23: 231–255.

    Google Scholar 

  • Béda, B. P., Steindl, A., and Troger, H., Postbuckling of a twisted prismatic rod under terminal thrust. Dynamics and Stability of Systems 7: 219–232.

    Google Scholar 

  • Biezeno, C. B., and Grammel, R. (1953). Technische Dynamik. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Blasius, H. (1914). Träger kleinster Durchbiegung and Stäbe großter Knickfestigkeit bei gegebenem Materialverbrauch. Zeitsch. Math. Physik 62: 182–197.

    Google Scholar 

  • Brenner, J. L., and Alzer, H. (1991). Integral inequalities for concave functions with applications to special functions. Proceedings of Royal Society of Edinburgh 118A: 173–192.

    Article  MATH  MathSciNet  Google Scholar 

  • Chow, S-N., and Hale, J. K. (1982). Methods ofBifurration Theory. New York: Springer.

    Book  Google Scholar 

  • Churchill, R. V. (1969). Fourier Series and Boundary Value Problems. New York: McGraw-Hill.

    Google Scholar 

  • Clausen, T. (1851). Ãœber die Form architektonischer Säulen. Bull. el, phvsico-math. Acad St. Pétersbourg 9: 369–380.

    Google Scholar 

  • Clément, Ph., and Descloux, J. (1991). A Variational Approach to a Problem of Rotating Rods. Archive for Rational Mechanics and Analysis 114: 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  • Cox, S. J. (1992). The Shape of the Ideal Column. The Mathematical Intelligence?, 14: 16–24.

    Article  MATH  Google Scholar 

  • Dacorogna, B., (1989). Direct Methods in the Calculus of Variations. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Golubitsky, M., and Schaeffer, D. (1985). Singularities and Groups in Bifurcation Theory. vol. I, New York: Springer.

    Book  MATH  Google Scholar 

  • Glavardanov, V. G. and Atanackovic, T. M. (2001). Optimal Shape of a twisted and compressed rod. European. Journal ofMechanics, A/Solids 20: 795–809.

    Article  MATH  Google Scholar 

  • Eliseyev, V. V. (1988). The non-linear dynamics of elastic rods. Prikl. Mathem. Mekh., 52: 635–641.

    Google Scholar 

  • Iljohin, A. A. (1979). Spatial problems of Nonlinear theory of Elastic rods. Kiev: Naukova Dumka (in Russian).

    Google Scholar 

  • Karlin, S., and Ziegler, Z. (1975). Some Inequalities for Generalized Concave Functions. Journal ofAppr ximation Theory 13: 267–293.

    MathSciNet  Google Scholar 

  • Keller, J., The shape of the strongest column. Archive for Rational Mechanics and Analysis 5: 275–285.

    Google Scholar 

  • Keyfitz, B. L. (1986). Classification of one-state variable bifurcation problems up to codimension seven. D namics and Stability of Systems 1: 1–41.

    Article  MATH  MathSciNet  Google Scholar 

  • Kovari, K., Raumliche Verzweingungsprobleme des dunnen elastischen Stabes mit endlichen Verformungen. Ing. Archiv 37, 393–416 (1969).

    Article  MATH  Google Scholar 

  • Oeuvres ds Lagrange, Serret, J. A. ed. (1867). Paris: Gauthier-Villars.

    Google Scholar 

  • Libai, A. (1992). Equations for the Nonlinear Planar deformation of Beams. Transactions of ASME Journal of Applied Mechanics 59: 1028–1030.

    Article  Google Scholar 

  • Lime, A. I. (1961). Analytical Mechanics. Moscow: F.M. editors (in Russian).

    Google Scholar 

  • Mitrinovic, D. S. (1970). Analytic Inequalities. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Odeh, F., and Tadjbakhsh, I. (1965). A Nonlinear Eigenvalue Problem for Rotating Rods. Archive of Rational Mechanics and Analysis 20: 81–94.

    Article  MATH  MathSciNet  Google Scholar 

  • Olhoff, N., and Rasmussen, S. (1977). On single and bimodal optimum buckling loads of clamped columns. International Journal of Solids and Structures 13: 605–614.

    Article  MATH  Google Scholar 

  • Parter, S. V. (1970). Nonlinear Eigenvalue Problems for Some Fourth Order Equations: I Maximal solutions, SLIM Journal of Mathematical Analysis 1:437–457, and II: Fixed-point methods 1: 458–478.

    MATH  MathSciNet  Google Scholar 

  • Pflüger, A. (1975). Stabilitätsprobleme der Elastostatik. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Prager, W. and Taylor, J. E. (1968). Problems of optimal structural design. Journal of Applied Mechanics (transactions of ASME) 35: 102–106.

    Article  MATH  Google Scholar 

  • Rabier, P. (1985). Topics in one parameter bifiuration problems. Berlin: Springer.

    Google Scholar 

  • Rabinowitz, P. D. (1977). Bifurcation Theorem for Potential Operators. Journal of Functional Analysis 25: 412–424.

    Article  MATH  MathSciNet  Google Scholar 

  • Ratzersdorfer, J. (1936). Die Knickfestigkeit von Stäben und Stabwerken. Wien: Springer.

    Book  MATH  Google Scholar 

  • Sage, A. P. and White, C. C. (1977). Optimum System Control. New Jersey: Prentice-Hall, 2nd edition.

    Google Scholar 

  • Stodola, A. (1906). Steam Turbines. New York: D. Van Nostrand Company.

    Google Scholar 

  • Struwe, M. (1996). Variational Methods. New York: Springer, 2nd edition.

    Book  MATH  Google Scholar 

  • Seiranian, A. (1984). On the Problem of Lagrange. Inzhenernvi Zh., Mekhanika Tverdoga Tela, 19: 101–111.

    Google Scholar 

  • Seyranian, A. P., Lund, E. and Olhoff, N. (1994). Multiple eigenvalues in structural optimization problems. Structural Optimization 8: 207–227.

    Article  Google Scholar 

  • Seiranyan, A. P. (1995). New Solutions to Lagrange’s Problem. Physics—Doklady 40: 251–253.

    MATH  MathSciNet  Google Scholar 

  • Tadjbakhsh, I., and Keller, J. (1962). Strongest columns and isoperimetric inequalities for eigenvalues. Transactions of ASME Journal of Applied. Mechanics 29: 159–164.

    Article  MATH  MathSciNet  Google Scholar 

  • Timoshenko, S. P., and Gere, J. M. (1961). Theory of Elastic Stability. New York: McGraw-Hill.

    Google Scholar 

  • Troger, H. and Steindl, A. (1991). A Nonlinear Stability and Bifurcation Theory. Wien: Springer.

    Book  Google Scholar 

  • Troesch, B. A. (1967). Integral inequalities for two functions. Archive of Rational Mechanics and Analysis 24: 412–424.

    Article  MathSciNet  Google Scholar 

  • Vujanovic, B., and Jones, S. E. (1989). Variational Methods in Nonconservative Phenomena. Boston Academic Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Yugoslavia

    Teodor M. Atanackovic

Authors
  1. Teodor M. Atanackovic
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Moscow State Lomonosov University, Russia

    Alexander P. Seyranian

  2. Florida Atlantic University, USA

    Isaac Elishakoff

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Wien

About this chapter

Cite this chapter

Atanackovic, T.M. (2002). Stability Bounds and Optimal Shape of Elastic Rods. In: Seyranian, A.P., Elishakoff, I. (eds) Modern Problems of Structural Stability. International Centre for Mechanical Sciences, vol 436. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2560-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-2560-1_1

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83697-2

  • Online ISBN: 978-3-7091-2560-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation