Archive of Applied Mechanics

, Volume 88, Issue 1–2, pp 317–327 | Cite as

On optimality of column geometry



Critical loads of columns under compressive follower forces have been widely discussed in the literature. By means of shape optimization, improvements by factors of more than eight could be achieved. However, the obtained solutions turn out to be not robust against perturbations of shape or material parameters. The aim of this paper is to explain this sensitivity effect, basing on a study of eigenforms. Further, a robust alternative to the classical optimization approach is proposed.


Columns Shape optimization Stability 


  1. 1.
    Beck, M.: Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes. Z. angew. Math. Mech. 3, 225–228 (1952)MATHGoogle Scholar
  2. 2.
    Bogacz, R., Frischmuth, K., Lisowski, K.: Interface conditions and loss of stability for stepped columns. Appl. Mech. Mater. 9, 41–50 (2008)CrossRefGoogle Scholar
  3. 3.
    Bogacz, R., Irretier, H., Mahrenholtz, O.: Optimal design of structures under nonconservative forces with stability constraints, Bracketing of eigenfrequencies of continuous structures. In: Proceedings of the Euromech, vol. 112, pp. 43–65. Budapest, Hungary (1979)Google Scholar
  4. 4.
    Claudon, J.L.: Characteristic curves and optimum design of two structures subjected to circulatory loads. J. de Méc. 14(3), 531–543 (1975)MATHGoogle Scholar
  5. 5.
    Frischmuth, K., Kosiński, W., Lekszycki, T.: Free vibrations of finite memory material beams. Int. J. Eng. Sci. 31(3), 385–395 (1993)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hanaoka, M., Washizu, K.: Optimum design of Beck’s column. Comput. Struct. 11(6), 473–480 (1980)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Imiełowski, Sz, Bogacz, R.: Fixed point in frequency domain of structures subject to generalized follower force. Mach. Dyn. Probl. 25(3/4), 169–182 (2001)Google Scholar
  8. 8.
    Imiełowski, S., Mahrenholtz, O.: Optimization and sensitivity of stepped columns under circulatory load. Appl. Math. Comput. Sci. 7(1), 155–170 (1997)MathSciNetMATHGoogle Scholar
  9. 9.
    Kerr, A.D.: Stability of a water tower. Ing. Arch. 58, 428–436 (1988)CrossRefGoogle Scholar
  10. 10.
    Katsikadelis, J.T., Tsiatas, C.G.: Regulating the vibratory motion of beams using shape optimization. J. Sound Vib. 292(1–2), 390–401 (2006)CrossRefGoogle Scholar
  11. 11.
    Katsikadelis, J.T., Tsiatas, C.G.: Nonlinear dynamic stability of damped becks column with variable cross-section. Int. J. Nonlinear Mech. 42(1), 164–171 (2007)CrossRefGoogle Scholar
  12. 12.
    Kurnik, W., Bogacz, R.: Ziegler problem revisited–flutter and divergence interactions in a generalized system. Mach. Dyn. Res. 37(4), 71–84 (2013)Google Scholar
  13. 13.
    Preumont, A., Seto, K.: Active Control of Structures. Wiley, New York (2008)CrossRefGoogle Scholar
  14. 14.
    Ringertz, U.T.: On the design of Beck’s column. Struct. Optim. 8, 120–124 (1994)CrossRefGoogle Scholar
  15. 15.
    Tada Y., Matsurnoto, R., Oku, A.: Shape determination of nonconservative structural systems. In: Proceedings of the 1st International Conference on Computer Aided Optimum Design of Structures: Recent Advances, Southampton, pp. 13–21. Springer, Berlin (1989)Google Scholar
  16. 16.
    Tada, Y., Seguchi, Y., Kema, K.: Shape determination of nonconservative structural systems by the inverse variational principle. Mem. Fac. Eng. 32, 45–6l (1985)Google Scholar
  17. 17.
    Tomski, L., Przybylski, J., Gołȩbiowska-Rozanow, M., Szmidla, J.: Vibration and stability of an elastic column subject to a generalized load. Arch. Appl. Mech. 67, 105–116 (1996)CrossRefMATHGoogle Scholar
  18. 18.
    Życzkowski, M.: Stability of bars and bar structures. In: Życzkowski, M. (ed.) Strength of Structural Elements, pp. 242–381. Elsevier, Amsterdam (1991)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.IPPT Polska Akademia NaukWarsawPoland
  2. 2.Politechnika Warszawska, SIMRWarsawPoland
  3. 3.Institut für MathematikUniversität RostockRostockGermany

Personalised recommendations