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Shock and Vibration
Volume 2014 (2014), Article ID 153532, 11 pages
http://dx.doi.org/10.1155/2014/153532
Research Article

On the Flexural-Torsional Vibration and Stability of Beams Subjected to Axial Load and End Moment

Department of Aerospace Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3

Received 21 January 2014; Revised 19 June 2014; Accepted 19 June 2014; Published 13 October 2014

Academic Editor: Reza Jazar

Copyright © 2014 M. Tahmaseb Towliat Kashani 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. S. Timoshenko, Vibration Problems in Engineering, Van Nostrand Reinhold, New York, NY, USA, 1964.
  2. S. M. Hashemi and M. J. Richard, “A Dynamic Finite Element (DFE) method for free vibrations of bending-torsion coupled beams,” Aerospace Science and Technology, vol. 4, no. 1, pp. 41–55, 2000. View at Publisher · View at Google Scholar · View at Scopus
  3. S. M. Hashemi, Free vibrational analysis of rotating beam-like structures: a dynamic finite element approach [Ph.D. thesis], Department of Mechanical Engineering, Laval University, Quebec, Canada, 1998.
  4. E. Dokumaci, “An exact solution for coupled bending and torsion vibrations of uniform beams having single cross-sectional symmetry,” Journal of Sound and Vibration, vol. 119, no. 3, pp. 443–449, 1987. View at Publisher · View at Google Scholar · View at Scopus
  5. C. Mei, “Coupled vibrations of thin-walled beams of open section using the finite element method,” International Journal of Mechanical Sciences, vol. 12, no. 10, pp. 883–891, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. M. Tanaka and A. N. Bercin, “Free vibration solution for uniform beams of nonsymmetrical cross section using Mathematica,” Computers and Structures, vol. 71, no. 1, pp. 1–8, 1999. View at Publisher · View at Google Scholar · View at Scopus
  7. J. R. Banerjee, “Coupled bending-torsional dynamic stiffness matrix for beam elements,” International Journal for Numerical Methods in Engineering, vol. 28, no. 6, pp. 1283–1298, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. J. R. Banerjee, S. Guo, and W. P. Howson, “Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping,” Computers and Structures, vol. 59, no. 4, pp. 613–621, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. W. H. Wittrick and F. W. Williams, “A general algorithm for computing natural frequencies of elastic structures,” The Quarterly Journal of Mechanics and Applied Mathematics, vol. 24, pp. 263–284, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. J. R. Banerjee and H. Su, “Free transverse and lateral vibration of beams with torsional coupling,” Journal of Aerospace Engineering, vol. 19, no. 1, pp. 13–20, 2006. View at Publisher · View at Google Scholar · View at Scopus
  11. S. M. Hashemi and A. Roach, “A dynamic finite element for the free vibration analysis of extension-torsion coupled composite beams,” Mathematics in Engineering, Science and Aerospace, vol. 1, no. 3, pp. 221–239, 2010. View at Google Scholar
  12. S. M. Hashemi and E. J. Adique, “A quasi-exact dynamic finite element for free vibration analysis of sandwich beams,” Applied Composite Materials, vol. 17, no. 2, pp. 259–269, 2010. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Borneman and S. M. Hashemi, “A dynamic finite element for the vibration analysis of tapered composite beams,” in Proceeding of the 5th Canadian-International Composite Conference, Vancouver, Canada, 2005.
  14. J. R. Banerjee and F. W. Williams, “Free vibration of composite beams—an exact method using symbolic computation,” Journal of Aircraft, vol. 32, no. 3, pp. 636–642, 1995. View at Google Scholar
  15. J. R. Banerjee, “Free vibration of sandwich beams using the dynamic stiffness method,” Computers & Structures, vol. 81, no. 18-19, pp. 1915–1922, 2003. View at Publisher · View at Google Scholar · View at Scopus
  16. J. R. Banerjee and A. J. Sobey, “Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam,” International Journal of Solids and Structures, vol. 42, no. 8, pp. 2181–2197, 2005. View at Publisher · View at Google Scholar · View at Scopus
  17. J. R. Banerjee, C. W. Cheung, R. Morishima, M. Perera, and J. Njuguna, “Free vibration of a three-layered sandwich beam using the dynamic stiffness method and experiment,” International Journal of Solids and Structures, vol. 44, no. 22-23, pp. 7543–7563, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. J. R. Banerjee, H. Su, and C. Jayatunga, “A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings,” Computers & Structures, vol. 86, no. 6, pp. 573–579, 2008. View at Publisher · View at Google Scholar · View at Scopus
  19. S. R. Borneman, S. M. Hashemiand, and H. Alighanbari, “Vibration analysis of doubly coupled cracked composite beams: an exact dynamic stiffness matrix,” International Review of Aerospace Engineering, vol. 1, no. 3, pp. 298–309, 2008. View at Google Scholar
  20. W. L. Hallauer and R. Y. L. Liu, “Beam bending-torsion dynamic stiffness method for calculation of exact vibration modes,” Journal of Sound and Vibration, vol. 85, no. 1, pp. 105–113, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. S. M. Hashemi, M. J. Richard, and G. Dhatt, “A new Dynamic Finite Element (DFE) formulation for lateral free vibrations of Euler-Bernoulli spinning beams using trigonometric shape functions,” Journal of Sound and Vibration, vol. 220, no. 4, pp. 601–624, 1999. View at Publisher · View at Google Scholar · View at Scopus
  22. S. M. Hashemi and M. J. Richard, “Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic finite element,” Computers and Structures, vol. 77, no. 6, pp. 711–724, 2000. View at Publisher · View at Google Scholar · View at Scopus
  23. J. R. Banerjee and F. W. Williams, “Coupled bending-torsional dynamic stiffness matrix of an axially loaded timoshenko beam element,” International Journal of Solids and Structures, vol. 31, no. 6, pp. 749–762, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. A. Y. T. Leung, “Natural shape functions of a compressed Vlasov element,” Thin-Walled Structures, vol. 11, no. 5, pp. 431–438, 1991. View at Publisher · View at Google Scholar · View at Scopus
  25. J. R. Banerjee and S. A. Fisher, “Coupled bending-torsional dynamic stiffness matrix for axially loaded beam elements,” International Journal for Numerical Methods in Engineering, vol. 33, no. 4, pp. 739–751, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. L. Jun, S. Rongying, H. Hongxing, and J. Xianding, “Coupled bending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effects,” Applied Acoustics, vol. 65, no. 2, pp. 153–170, 2004. View at Publisher · View at Google Scholar · View at Scopus
  27. M. Murthy and J. Neogy, “Determination of fundamental natural frequencies of axially loaded columns and frames,” Journal of the Institution of Engineers (India) Civil Engineering Division, vol. 49, pp. 203–212, 1969. View at Google Scholar
  28. M. Gellert and J. Gluck, “The influence of axial load on eigen-frequencies of a vibrating lateral restraint cantilever,” International Journal of Mechanical Sciences, vol. 14, no. 11, pp. 723–728, 1972. View at Publisher · View at Google Scholar · View at Scopus
  29. A. Bokaian, “Natural frequencies of beams under compressive axial loads,” Journal of Sound and Vibration, vol. 126, no. 1, pp. 49–65, 1988. View at Publisher · View at Google Scholar · View at Scopus
  30. A. Bokaian, “Natural frequencies of beams under tensile axial loads,” Journal of Sound and Vibration, vol. 142, no. 3, pp. 481–498, 1990. View at Publisher · View at Google Scholar · View at Scopus
  31. F. J. Shaker, “Effect of axial load on mode shapes and frequencies of beams,” NASA Technical Note TN D-8109, 1975. View at Google Scholar
  32. A. Joshi and S. Suryanarayan, “Coupled flexural-torsional vibration of beams in the presence of static axial loads and end moments,” Journal of Sound and Vibration, vol. 92, no. 4, pp. 583–589, 1984. View at Publisher · View at Google Scholar · View at Scopus
  33. A. Joshi and S. Suryanarayan, “Unified analytical solution for various boundary conditions for the coupled flexural-torsional vibration of beams subjected to axial loads and end moments,” Journal of Sound and Vibration, vol. 129, no. 2, pp. 313–326, 1989. View at Publisher · View at Google Scholar
  34. A. Joshi and S. Suryanarayan, “A unified solution for various boundary conditions for the coupled flexural-torsional instability of closed thin-walled beam-columns,” International Journal of Solids and Structures, vol. 20, no. 2, pp. 167–178, 1984. View at Publisher · View at Google Scholar · View at Scopus
  35. A. Joshi and S. Suryanarayan, “Iterative method for coupled flexural-torsional vibration of initially stressed beams,” Journal of Sound and Vibration, vol. 146, no. 1, pp. 81–92, 1991. View at Publisher · View at Google Scholar · View at Scopus
  36. R. Pavlović and P. Kozić, “Almost sure stability of the thin-walled beam subjected to end moments,” Theoretical and Applied Mechanics, vol. 30, no. 3, pp. 193–207, 2003. View at Google Scholar · View at MathSciNet
  37. R. Pavlović, P. Kozić, P. Rajković, and I. Pavlović, “Dynamic stability of a thin-walled beam subjected to axial loads and end moments,” Journal of Sound and Vibration, vol. 301, no. 3–5, pp. 690–700, 2007. View at Publisher · View at Google Scholar · View at Scopus
  38. J. S. Przemieniecki, Theory of Matrix Structural Analysis, McGraw-Hill, New York, NY, USA, 1968.
  39. W. F. Chen and T. Atsuta, Theory of Beam Columns Space Behaviour and Design Volume, McGraw-Hill, New York, NY, USA, 1977.
  40. K. J. Bathe, Finite Element Procedures, Prentice Hall, New York, NY, USA, 1996.
  41. L. Meirovitch, Analytical Methods in Vibrations, The MacMillan, 1967.
  42. L. Meirovitch, Computational Methods in Structural Dynamics, Sijthoff & Noordhoff, 1980. View at MathSciNet
  43. F. Y. Chen, Matrix Analysis of Structural Dynamics: Applications and Earthquake Engineering, Marcel Decker, New York, NY, USA, 2001.
  44. J. S. Wu, Analytical and Numerical Methods for Vibration Analysis, John Wiley & Sons, Singapore, 2013.
  45. S. Laux, Estimation of axial load in timber beams using resonance frequency analysis [M.S. thesis], Chalmers University of Technology, Gothenburg, Sweden, 2012.