Transactions of the Korean Society of Mechanical Engineers A (대한기계학회논문집A)

The Korean Society of Mechanical Engineers (대한기계학회)
권호 표지
DOI QR코드

DOI QR Code

Free Vibration Analysis of Arches Using Higher-Order Mixed Curved Beam Elements

고차 혼합 곡선보 요소에 의한 아치의 자유진동해석

  • 박용국 (대구가톨릭대학교 기계자동차공학부) ;
  • 김진곤 (대구가톨릭대학교 기계자동차공학부)
  • Published : 2006.01.01

Abstract

The purpose of this research work is to demonstrate a successful application of hybrid-mixed formulation and nodeless degrees of freedom in developing a very accurate in-plane curved beam element for free vibration analysis. To resolve the numerical difficulties due to the spurious constraints, the present element, based on the Hellinger-Reissner variational principle and considering the effect of shear deformation, employed consistent stress parameters corresponding to cubic displacement polynomials with additional nodeless degrees. The stress parameters were eliminated by the stationary condition, and the nodeless degrees were condensed by Guyan Reduction. Several numerical examples indicated that the property of the mass matrix as well as that of the stiffness matrix have a great effect on the numerical performance. The element with consistent mass matrix produced best results on convergence and accuracy in the numerical analysis of Eigenvalue problems. Also, the higher-order mixed curved beam element showed a superior numerical behavior for the free vibration analyses.

Keywords

Curved Beam Element;Hybrid-Mixed Formulation;Kinematic Condensation;Free Vibration Analysis

File

Download PDF

References

  1. Dawe, D. J., 1974, 'Numerical Studies Using Circular Arch Finite Elemnets,' Comput. Struct., Vol. 4, pp.729-740 https://doi.org/10.1016/0045-7949(74)90041-8
  2. Stolarski, H. and Belytschko, T., 1983, 'Shear and Membrane Locking in Curved $C^0$ Elements,' Comp. Methods Appl. Mech. Engng., Vol.41, pp. 279-296 https://doi.org/10.1016/0045-7825(83)90010-5
  3. Noor, A. K. and Peters, J. M., 1981, 'Mixed Models and Reduced/Selective Integration Displacement Models for Nonlinear Analysis of Curved Beams,' Int. J. Nnumer. Methods Eng., Vol. 17, pp. 615-631 https://doi.org/10.1002/nme.1620170409
  4. Noor, A. K., Greene, W. H. and Hartely, S. J., 1977, 'Nonlinear Finite Element Analysis of Curved Beams,' Comp. Methods Appl. Mech. Engng., Vol. 12, pp.289-307 https://doi.org/10.1016/0045-7825(77)90018-4
  5. Stolarski, H. and Belyschko, T., 1982, 'Membrane Locking and Reduced Integration for Curved Elements,' J. Appl. Mech., Vol. 49, pp. 172-176 https://doi.org/10.1115/1.3161961
  6. Moon, W. J., Kim, Y. W., Min, O. K. and Lee, K. W., 1996, 'Reduced Minimzation Theory in Skew Beam Element,' Transactions of the KSME, Vol. 20, No. 12, pp. 3792-3803
  7. Babu, C. Ramesh and Prathap, G, 1986, 'A Linear Thick Curved Beam Element,' Int. J. Numer. Methods Eng., Vol. 23, pp. 1313-1328 https://doi.org/10.1002/nme.1620230709
  8. Prathap, G. and Babu, C. Ramesh, 1986, 'An Isoparametric Quadratic Thick Curved Beam Element,' Int. J. Numer. Methods Eng., Vol. 23, pp. 1583-1600 https://doi.org/10.1002/nme.1620230902
  9. Tessler, A. and Spiridigliozzi, L., 1986, 'Curved Beam Elements with Penalty Relaxation,' Int. J. Numer. Methods Eng., Vol. 23, pp. 2245-2262 https://doi.org/10.1002/nme.1620231207
  10. Ryu, H. S. and Sin, H. C., 1996, 'A 2-Node Strain Based Curved Beam Element,' Transactions of the KSME, Vol. 18, No. 8, pp. 2540-2545
  11. Saleeb, A. F. and Chang, T. Y., 1987, 'On the Hybrid-Mixed Formulation $C^0$ Curved Beam Elements,' Comp. Methods Appl. Mech. Engng., Vol. 60, pp. 95-121 https://doi.org/10.1016/0045-7825(87)90131-9
  12. Dorfi, H. R. and Busby, H. R., 1994, 'An Effiective Curved Composite Beam Finite Element Based on the Hybrid-Mixed Formilation,' Comput. Struct., Vol. 53, pp. 43-52 https://doi.org/10.1016/0045-7949(94)90128-7
  13. Kim, J. G. and Kim, Y. Y., 1998, 'A New Higher-Order Hybrid-Mixed Curved Beam Element.' Int. J. Numer. Methods Eng., Vol. 43, pp. 925-940 https://doi.org/10.1002/(SICI)1097-0207(19981115)43:5<925::AID-NME457>3.0.CO;2-M
  14. Kim, J. G., 2000, 'Optimal Interpolation Functions of 2-Node Hybrid-Mixed Curved Beam Elemtent.' Transactions of the KSME, A, Vol. 24, pp. 3003-3009
  15. Kim, J. G. and Kang, S. W., 2003, 'A New and Efficient $C^0$ Laminated Curved Beam Element,' Transactions of the KSME, A, Vol. 27, pp. 559-566 https://doi.org/10.3795/KSME-A.2003.27.4.559
  16. Lee, H. C. and Kim, J. G., 2005, 'A New Hybrid-Mixed Composite Laminated Curved Beam Element,' Journal of Mechanical Science and Technology, Vol. 19, pp. 811-819 https://doi.org/10.1007/BF02916129
  17. Cook, R. D., Malkus, D. S. and Plesha, M. E., 1989, Concepts and Applications of Finite Element Analysis, 3rd Edition, Wiely, New York
  18. Rao, S. S., 2004, Mechanical Vibrations, 4th Edition, Pearson Prentice Hall
  19. Ryu, H. S. and Sin, H. C., 1997, 'The Effiect of the Mass Matrix in the Eigenvalue Analysis of Curved Beam Element,' Transactions of the KSME, A, Vol. 21, pp. 288-296
  20. Leung, A. Y. T. and Zhu, B., 2004, 'Fourier p-Elements for Curved Beam Vibrations,' Thin-Walled Structures, Vol. 42, pp. 39-57 https://doi.org/10.1016/S0263-8231(03)00122-8
  21. Krishinan, A., Dharmaraj, S. and Suresh, Y. J., 1995. 'Free Vibration Studies of Arches,' Journal of Sound and Vibration, Vol. 186, pp. 856-863 https://doi.org/10.1006/jsvi.1995.0493
  22. Raveendranath, P., Singh, G. and Pradhan, B., 1999, 'A Two-Noded Locking-Free Shear Flexible Curved Beam Element,' Int. J. Numer. Methods Eng., Vol. 44, pp. 265-280 https://doi.org/10.1002/(SICI)1097-0207(19990120)44:2<265::AID-NME505>3.0.CO;2-K
  23. Heppler, G. R., 1992, 'An Element for Studying the Vibration of Unrestrained Curved Timoshenko Beams,' Journal of Sound and Vibration, Vol. 158, pp. 387-404 https://doi.org/10.1016/0022-460X(92)90416-U