Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 940347, 12 pages
http://dx.doi.org/10.1155/2015/940347
Research Article

Higher-Order Hierarchical Models for the Free Vibration Analysis of Thin-Walled Beams

Luxembourg Institute of Science and Technology, 5 Avenue des Hauts-Fourneaux, 4362 Esch-sur-Alzette, Luxembourg

Received 30 July 2015; Accepted 6 September 2015

Academic Editor: Xiao-Qiao He

Copyright © 2015 G. Giunta and S. Belouettar. 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. H. Matsunaga, “Free vibration and stability of thin elastic beams subjected to axial forces,” Journal of Sound and Vibration, vol. 191, no. 5, pp. 917–933, 1996. View at Publisher · View at Google Scholar · View at Scopus
  2. W. Q. Chen, C. F. Lü, and Z. G. Bian, “A semi-analytical method for free vibration of straight orthotropic beams with rectangular cross-sections,” Mechanics Research Communications, vol. 31, no. 6, pp. 725–734, 2004. View at Publisher · View at Google Scholar · View at Scopus
  3. N. F. J. van Rensburg and A. J. van der Merwe, “Natural frequencies and modes of a Timoshenko beam,” Wave Motion, vol. 44, no. 1, pp. 58–69, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. R. Attarnejad, S. J. Semnani, and A. Shahba, “Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams,” Finite Elements in Analysis and Design, vol. 46, no. 10, pp. 916–929, 2010. View at Publisher · View at Google Scholar · View at Scopus
  5. J. B. Gunda, R. K. Gupta, G. R. Janardhan, and G. V. Rao, “Large amplitude free vibration analysis of Timoshenko beams using a relatively simple finite element formulation,” International Journal of Mechanical Sciences, vol. 52, no. 12, pp. 1597–1604, 2010. View at Publisher · View at Google Scholar · View at Scopus
  6. R. Benamar, M. M. K. Bennouna, and R. G. White, “The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part I. Simply supported and clamped-clamped beams,” Journal of Sound and Vibration, vol. 149, no. 2, pp. 179–195, 1991. View at Publisher · View at Google Scholar · View at Scopus
  7. 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
  8. L. Jun, H. Hongxing, S. Rongying, and J. Xianding, “Dynamic response of axially loaded monosymmetrical thin-walled Bernoulli-Euler beams,” Thin-Walled Structures, vol. 42, no. 12, pp. 1689–1707, 2004. View at Publisher · View at Google Scholar · View at Scopus
  9. H. H. Chen and K. M. Hsiao, “Coupled axial-torsional vibration of thin-walled Z-section beam induced by boundary conditions,” Thin-Walled Structures, vol. 45, no. 6, pp. 573–583, 2007. View at Publisher · View at Google Scholar · View at Scopus
  10. H. H. Chen and K. M. Hsiao, “Quadruply coupled linear free vibrations of thin-walled beams with a generic open section,” Engineering Structures, vol. 30, no. 5, pp. 1319–1334, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. H. Duan, “Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open cross section,” Thin-Walled Structures, vol. 46, no. 10, pp. 1107–1112, 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. G. M. Vörös, “On coupled bending-torsional vibrations of beams with initial loads,” Mechanics Research Communications, vol. 36, no. 5, pp. 603–611, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. D. Ambrosini, “On free vibration of nonsymmetrical thin-walled beams,” Thin-Walled Structures, vol. 47, no. 6-7, pp. 629–636, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. D. Ambrosini, “Experimental validation of free vibrations from nonsymmetrical thin walled beams,” Engineering Structures, vol. 32, no. 5, pp. 1324–1332, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. F. de Borbón and D. Ambrosini, “On free vibration analysis of thin-walled beams axially loaded,” Thin-Walled Structures, vol. 48, no. 12, pp. 915–920, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. A. Arpaci and E. Bozdag, “On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections,” Computers and Structures, vol. 80, no. 7-8, pp. 691–695, 2002. View at Publisher · View at Google Scholar · View at Scopus
  17. A. Arpaci, S. E. Bozdag, and E. Sunbuloglu, “Triply coupled vibrations of thin-walled open cross-section beams including rotary inertia effects,” Journal of Sound and Vibration, vol. 260, no. 5, pp. 889–900, 2003. View at Publisher · View at Google Scholar · View at Scopus
  18. 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 Scopus
  19. S.-B. Kim and M.-Y. Kim, “Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frames,” Engineering Structures, vol. 22, no. 5, pp. 446–458, 2000. View at Publisher · View at Google Scholar · View at Scopus
  20. Z. Zhou-Lian, L. Chang-Jiang, H. Xiao-Ting, and C. Shan-Lin, “Free vibration analysis of rectangular orthotropic membranes in large deflection,” Mathematical Problems in Engineering, vol. 2009, Article ID 634362, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. C.-j. Liu, Z.-l. Zheng, X.-t. He et al., “L-P perturbation solution of nonlinear free vibration of prestressed orthotropic membrane in large amplitude,” Mathematical Problems in Engineering, vol. 2010, Article ID 561364, 17 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. E. Carrera, “Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking,” Archives of Computational Methods in Engineering, vol. 10, no. 3, pp. 215–296, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  23. E. Carrera and G. Giunta, “Hierarchical models for failure analysis of plates bent by distributed and localized transverse loadings,” Journal of Zhejiang University Science A, vol. 9, no. 5, pp. 600–613, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. E. Carrera and G. Giunta, “Exact, hierarchical solutions for localized loadings in isotropic, laminated, and sandwich shells,” Journal of Pressure Vessel Technology, vol. 131, no. 4, Article ID 041202, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. E. Carrera and G. Giunta, “Refined beam theories based on a unified formulation,” International Journal of Applied Mechanics, vol. 2, no. 1, pp. 117–143, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. E. Carrera, G. Giunta, and M. Petrolo, Beam Structures: Classical and Advanced Theories, Wiley Series in Computational Mechanics, John Wiley & Sons, 2011.
  27. G. Giunta, F. Biscani, S. Belouettar, and E. Carrera, “Analysis of thin-walled beams via a one-dimensional unified formulation through a navier-type solution,” International Journal of Applied Mechanics, vol. 3, no. 3, pp. 407–434, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. G. R. Cowper, “The shear co-efficient in Timoshenko beam theory,” Journal of Applied Mechanics, vol. 33, no. 10, pp. 335–340, 1966. View at Google Scholar
  29. A. V. K. Murty, “Analysis of short beams,” AIAA Journal, vol. 8, no. 11, pp. 2098–2100, 1970. View at Publisher · View at Google Scholar · View at Scopus
  30. E. Carrera and S. Brischetto, “Analysis of thickness locking in classical, refined and mixed multilayered plate theories,” Composite Structures, vol. 82, no. 4, pp. 549–562, 2008. View at Publisher · View at Google Scholar · View at Scopus
  31. ANSYS, ANSYS v10.0 Theory Manual, ANSYS, Southpointe, Pa, USA, 2006.
  32. E. Madenci and I. Guven, The Finite Element Method and Applications in Engineering Using ANSYS, Springer, New York, NY, USA, 2006.
  33. R. J. Allemang, Investigation of some multiple input/output frequency response function experimental modal analysis techniques [Ph.D. thesis], University of Cincinnati, Cincinnati, Ohio, USA, 1980.
  34. G. Giunta, N. Metla, Y. Koutsawa, and S. Belouettar, “Free vibration and stability analysis of three-dimensional sandwich beams via hierarchical models,” Composites Part B: Engineering, vol. 47, pp. 326–338, 2013. View at Publisher · View at Google Scholar · View at Scopus