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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 3568970, 15 pages
http://dx.doi.org/10.1155/2016/3568970
Research Article

Structural Damping Effects on Dynamic Instability of Subtangentially Loaded and Shear Deformable Beck’s Columns

1School of Civil and Architectural Engineering, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon-si 16419, Republic of Korea
2Department of Civil & Environmental Engineering, Dankook University, 126 Jukjeon-dong, Suji-gu, Yongin-si 448-701, Republic of Korea
3Department of Civil Engineering, Semyung University, 579 Sinwoul-dong, Jecheon-si, Chungbuk 390-711, Republic of Korea

Received 14 April 2016; Revised 10 June 2016; Accepted 12 June 2016

Academic Editor: Anders Eriksson

Copyright © 2016 Dong-Ju Min et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Beck, “Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes,” Zeitschrift für angewandte Mathematik und Physik, vol. 3, no. 6, pp. 476–477, 1952. View at Publisher · View at Google Scholar · View at Scopus
  2. H. Ziegler, Principles of Structural Stability, Blaisdell Publishing Company, Waltham, Mass, USA, 1968.
  3. V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press, NewYork, NY, USA, 1963.
  4. H. H. E. Leipholz, Stability of Elastic Systems, SijthotI & Noordhoff, Rijn, The Netherlands, 1980.
  5. B. N. Rao and G. V. Rao, “Stability of a cantilever column under a tip-concentrated subtangential follower force with damping,” Journal of Sound and Vibration, vol. 138, no. 2, pp. 341–344, 1990. View at Publisher · View at Google Scholar · View at Scopus
  6. R. V. Vitaliani, A. M. Gasparini, and A. V. Saetta, “Finite element solution of the stability problem for nonlinear undamped and damped systems under nonconservative loading,” International Journal of Solids and Structures, vol. 34, no. 19, pp. 2497–2516, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. Y. Sugiyama and M. A. Langthjem, “Physical mechanism of the destabilizing effect of damping in continuous non-conservative dissipative systems,” International Journal of Non-Linear Mechanics, vol. 42, no. 1, pp. 132–145, 2007. View at Publisher · View at Google Scholar · View at Scopus
  8. O. N. Kirillov and F. Verhulst, “Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 90, no. 6, pp. 462–488, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. A. Luongo and F. Dannibale, “On the destabilizing effect of damping on discrete and continuous circulatory systems,” Journal of Sound and Vibration, vol. 333, no. 24, pp. 6723–6741, 2014. View at Publisher · View at Google Scholar · View at Scopus
  10. M. A. Langthjem and Y. Sugiyama, “Dynamic stability of columns subjected to follower loads: a survey,” Journal of Sound and Vibration, vol. 238, no. 5, pp. 809–851, 2000. View at Publisher · View at Google Scholar · View at Scopus
  11. I. Elishakoff, “Controversy associated with the so-called ‘follower forces’: critical overview,” Applied Mechanics Reviews, vol. 58, no. 2, pp. 117–142, 2005. View at Publisher · View at Google Scholar · View at Scopus
  12. A. Kounadis and J. T. Katsikadelis, “Shear and rotatory inertia effect on Beck's column,” Journal of Sound and Vibration, vol. 49, no. 2, pp. 171–178, 1976. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Y. Lee, T. Y. Chen, and W. R. Wang, “Non-conservative instability of a Timoshenko beam subjected to a partially tangential follower force,” Journal of Sound and Vibration, vol. 188, no. 1, pp. 25–38, 1995. View at Publisher · View at Google Scholar · View at Scopus
  14. V. Sundararamaiah and G. Venkateswara Rao, “Effect of shear deformation and rotatory inertia on the stability of Beck's and Leipholz's columns,” AIAA journal, vol. 18, no. 1, pp. 124–125, 1980. View at Publisher · View at Google Scholar · View at Scopus
  15. V. Sundararamaiah and G. Venkateswara Rao, “Stability of short Beck and Leipholz columns on elastic foundation,” AIAA Journal, vol. 21, no. 7, pp. 1053–1054, 1983. View at Publisher · View at Google Scholar · View at Scopus
  16. I. Lottati and I. Elishakoff, “Influence of the shear deformation and rotary inertia on the flutter of a cantilever subjected to a follower force-exact and symbolic manipulation solutions,” in Refined Dynamical Theories of Beams, Plates and Shells and Their Applications, pp. 261–273, Springer, Berlin, Germany, 1987. View at Google Scholar
  17. L.-W. Chen and D.-M. Ku, “Stability analysis of a timoshenko beam subjected to distributed follower forces using finite elements,” Computers and Structures, vol. 41, no. 4, pp. 813–819, 1991. View at Publisher · View at Google Scholar · View at Scopus
  18. B. J. Ryu, K. Katayama, and Y. Sugiyama, “Dynamic stability of Timoshenko columns subjected to subtangential forces,” Computers and Structures, vol. 68, no. 5, pp. 499–512, 1998. View at Publisher · View at Google Scholar · View at Scopus
  19. M. M. Attard, J.-S. Lee, and M.-Y. Kim, “Dynamic stability of shear-flexible Beck's columns based on Engesser's and Haringx's buckling theories,” Computers and Structures, vol. 86, no. 21-22, pp. 2042–2055, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. V. V. Bolotin and N. I. Zhinzher, “Effects of damping on stability of elastic systems subjected to nonconservative forces,” International Journal of Solids and Structures, vol. 5, no. 9, pp. 965–989, 1969. View at Publisher · View at Google Scholar · View at Scopus
  21. G. Herrmann and I.-c. Jong, “On the destabilizing effect of damping in nonconservative elastic systems,” Journal of Applied Mechanics, vol. 32, no. 3, pp. 592–597, 1965. View at Google Scholar · View at MathSciNet
  22. O. N. Kirillov and A. P. Seyranian, “The effect of small internal and external damping on the stability of distributed non-conservative systems,” Journal of Applied Mathematics and Mechanics, vol. 69, no. 4, pp. 529–552, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. A. Luongo and F. D'Annibale, “Bifurcation analysis of damped visco-elastic planar beams under simultaneous gravitational and follower forces,” International Journal of Modern Physics B, vol. 26, no. 25, Article ID 1246015, 2012. View at Publisher · View at Google Scholar · View at Scopus
  24. A. Luongo and F. D'Annibale, “Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads,” International Journal of Non-Linear Mechanics, vol. 55, pp. 128–139, 2013. View at Publisher · View at Google Scholar · View at Scopus
  25. S. B. Andersen and J. J. Thomsen, “Post-critical behavior of Beck's column with a tip mass,” International Journal of Non-Linear Mechanics, vol. 37, no. 1, pp. 135–151, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. J.-S. Lee, N.-I. Kim, and M.-Y. Kim, “Sub-tangentially loaded and damped Beck's columns on two-parameter elastic foundation,” Journal of Sound and Vibration, vol. 306, no. 3-5, pp. 766–789, 2007. View at Publisher · View at Google Scholar · View at Scopus
  27. M.-Y. Kim, J.-S. Lee, and M. M. Attard, “Stability of damped columns on a Winkler foundation under sub-tangential follower forces,” International Journal of Structural Stability and Dynamics, vol. 13, no. 2, Article ID 1350020, 27 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. I. Elishakoff, J. Kaplunov, and E. Nolde, “celebrating the centenary of Timoshenko's study of effects of shear deformation and rotary inertia,” Applied Mechanics Reviews, vol. 67, no. 6, Article ID 060802, 2015. View at Publisher · View at Google Scholar · View at Scopus
  29. F. M. Detinko, “Lumped damping and stability of Beck column with a tip mass,” International Journal of Solids and Structures, vol. 40, no. 17, pp. 4479–4486, 2003. View at Publisher · View at Google Scholar · View at Scopus
  30. S. Wolfram, Mathematica, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1991.