Journal of Mechanical Science and Technology

The Korean Society of Mechanical Engineers (대한기계학회)
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A Finite Thin Circular Beam Element for In-Plane Vibration Analysis of Curved Beams

  • Kim Chang-Boo (Department of Mechanical Engineering, Inha University) ;
  • Park Jung-Woo (Department of Mechanical Engineering, Inha University) ;
  • Kim Sehee (Department of Mechanical Engineering, Inha University) ;
  • Cho Chongdu (Department of Mechanical Engineering, Inha University)
  • Published : 2005.12.01
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Abstract

In this paper, the stiffness and the mass matrices for the in-plane motion of a thin circular beam element are derived respectively from the strain energy and the kinetic energy by using the natural shape functions of the exact in-plane displacements which are obtained from an integration of the differential equations of a thin circular beam element in static equilibrium. The matrices are formulated in the local polar coordinate system and in the global Cartesian coordinate system with the effects of shear deformation and rotary inertia. Some numerical examples are performed to verify the element formulation and its analysis capability. The comparison of the FEM results with the theoretical ones shows that the element can describe quite efficiently and accurately the in-plane motion of thin circular beams. The stiffness and the mass matrices with respect to the coefficient vector of shape functions are presented in appendix to be utilized directly in applications without any numerical integration for their formulation.

Keywords

References

  1. Ashwell, D. G., Sabir, A. B. and Roberts, T. M., 1971, 'Further Studies in the Applications of Curved Finite Elements to Circular Arches,' International Journal of Mechanical Science, Vol. 13, pp. 507-517 https://doi.org/10.1016/0020-7403(71)90038-5
  2. Choi, J. K. and Lim, J. K., 1993, 'Simple Curved Shear Beam Elements,' Communications in Numerical Methods in Engineering, Vol. 9, pp. 659-669 https://doi.org/10.1002/cnm.1640090805
  3. Cowper, G. R., 1966, 'The Shear Coefficient in Timoshenkc's Beam Theory,' Transactions of ASME, Journal of Applied Mechanics, Vol. 33, pp. 335-340 https://doi.org/10.1115/1.3625046
  4. Davis, R., Henshell, R. D. and Warburton, G. B., 1972, 'Constant Curvature Beam Finite Elements for In-Plane Vibration,' Journal of Sound and Vibration, Vol. 25, No. 4, pp. 561-576 https://doi.org/10.1016/0022-460X(72)90478-6
  5. Guimaraes, J. E. F. and Heppler, G. R., 1992, 'On Trigonometric Basis Functions for $C^1$ Curved Beam Finite Elements,' Computers & Structures, Vol. 45, No. 2, pp. 405-413 https://doi.org/10.1016/0045-7949(92)90424-X
  6. Kim, J. G. and Kang, S. W., 2003, 'A New and Efficient $C^0$ Laminated Curved Beam Element,' Transactions of KSME (A), Vol. 27, No. 4, pp. 559-566 https://doi.org/10.3795/KSME-A.2003.27.4.559
  7. Kirkhope, J., 1977, 'In-Plane Vibration of a Thick Circular Ring,' Journal of Sound and Vibration, Vol. 50, No. 2, pp. 219-227 https://doi.org/10.1016/0022-460X(77)90356-X
  8. Krishnan, A. and Suresh, Y. J., 1998, 'A Simple Cubic Linear Element for Static and Free Vibration Analyses of Curved Beams,' Computers & Structures, Vol. 68, pp. 473-489 https://doi.org/10.1016/S0045-7949(98)00091-1
  9. Lee, J. H., 2003, 'In-Plane Free Vibration Analysis of Curved Timoshenko Beams by the Pseudospectral Method,' KSME International Journal, Vol. 17, No. 8, pp. 1156-1163
  10. Lee, P. G. and Sin, H. C., 1994, 'Locking-Free Curved Beam Element Based on Curvature,' International Journal for Numerical Methods in Engineering, Vol. 37, pp. 989-1007 https://doi.org/10.1002/nme.1620370607
  11. Meek, H. R., 1980, 'An Accurate Polynomial Displacement Function for Finite Elements,' Computers & Structures, Vol. 11, pp. 265-269 https://doi.org/10.1016/0045-7949(80)90076-0
  12. Prathap, G. and Babu, C. R., 1986, 'An Isoparametric Quadratic Thick Curved Beam Element,' International Journal for Numerical Methods in Engineering, Vol. 23, pp. 1583-1600 https://doi.org/10.1002/nme.1620230902
  13. Sabir, A. B., Djoudi, M. S. and Sfendji, A., 1994, 'The Effect of Shear deformation on the Vibration of Circular Arches by the Finite Element Method,' Thin-Walled Structures, Vol. 18, pp. 47-66 https://doi.org/10.1016/0263-8231(94)90046-9
  14. Rao, S. S. and Sundararajan, V., 1969, 'In-Plane Flexural Vibrations of Circular Rings,' Transactions of ASME, Journal of Applied Mechanics, Vol. 91, pp. 620-625
  15. Yamada, Y. and Ezawa, Y., 1977, 'On Curved Finite Elements for the Analysis of Circular Arches,' International Journal for Numerical Methods in Engineering, Vol. 11, pp. 1645-1651 https://doi.org/10.1002/nme.1620111102