Advertisement

KSCE Journal of Civil Engineering

, Volume 8, Issue 1, pp 43–48 | Cite as

Free vibrations of arches with inclusion of axial extension, shear deformation and rotatory inertia in Cartesian coordinates

  • Byoung Koo Lee
  • Tae Eun Lee
  • Dae Soon Ahn
Structural Engineering

Abstract

The differential equations governing free vibrations of the elastic, parabolic arches with unsymmetric axes are derived in Cartesian coordinates rather than in polar coordinates. The formulation includes the effects of axial extension, shear deformation, and rotatory inertia. Frequencies and mode shapes are computed numerically for arches with clamped-clamped and hinged-hinged ends. The convergent efficiency is highly improved under the newly derived governing equations in Cartesian coordinates. The four lowest natural frequency parameters are reported as functions of four non-dimensional system parameters: the rise to chord length ratio, the span length to chord length ratio, the slenderness ratio and the shear parameter. Typical mode shapes of vibrating arches are also presented.

Keywords

cartesian coordinates natural frequency axial extension shear deformation rotatory inertia 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al-Khafaji, A.W. and Tooley, J.R. (1986).Numerical Method in Engineering Practice, Holt, Reinhardt and Winston, Inc.Google Scholar
  2. Borg, S.F. and Gennaro J.J. (1959).Advanced Structural Analysis, New Jersey: Van Nostrand.Google Scholar
  3. Chidamparam, P. and Leissa, A.W. (1993). “Vibrations of planar curved beams, rings and arches.”Applied Mechanics Reviews, Vol. 46. No. 9, pp. 467–483.CrossRefGoogle Scholar
  4. Henrych, J. (1981).The Dynamics of Arches and Frames. Amsterdam: Elsevier.MATHGoogle Scholar
  5. Ine, T., Yamada, G., and Tanaka, K. (1981). “Natural frequencies of in-plane vibration of arcs.”Journal of Applied Mechanics, ASME, Vol. 50, pp. 511–525.Google Scholar
  6. Laura, P.A.A. and Maurizi, M.J. (1987). “Recent research on vibration of arch-type structure.”The Shock and Vibration Digest, Vol. 19, pp. 6–9.CrossRefGoogle Scholar
  7. Lee, B.K., Oh, S.J., and Park, K.K. (2002). “Free vibrations of shear deformable circular curved beams resting on an elastic foundation.”International Journal of Structural Stability and Dynamics, Vol. 2, No. 1, pp. 77–97.CrossRefMathSciNetGoogle Scholar
  8. Lee, B.K. and Wilson, J.F. (1989). “Free vibrations of arches with variable curvature.”Journal of Sound and Vibration, Vol. 136, No. 1, pp. 75–89.Google Scholar
  9. Oh, S.J. (1996).Free vibrations of arches with variable cross-section, Ph.D. Dissertation, Wonkwang University, Iksan, Korea.Google Scholar
  10. Oh, S.J., Lee, B.K. and Lee, I.W. (1999). “Natural frequencies of non-circular arches with rotatory inertia and shear deformation.”Journal of Sound and Vibration, Vol. 219, No. 1, pp. 23–33.CrossRefGoogle Scholar
  11. Romanelli, E. and Laura, P.A.A. (1972). “Fundamental frequency of non-circular, elastic, hinged arcs.”Journal of Sound and Vibration, Vol. 24, No. 1, pp. 17–22.MATHCrossRefGoogle Scholar
  12. Tufekci, E. (2001). “Exact solution of free in-plane vibration of shallow circular arches.”International Journal of Structural Stability and Dynamics, Vol. 1, No. 3, pp. 409–428.CrossRefGoogle Scholar
  13. Veletsos, A.S., Austin, A.J., Pereira, C.A.L., and Wung, S.J. (1972). “Free in-plane vibrations of circular arches.”Journal of Engineering Mechanics Division, ASCE, Vol. 93, pp. 311–329.Google Scholar
  14. Wang, T.M. and Moore, J.A. (1973). “Lowest natural extensional frequency of clamped elliptic arcs.”Journal of Sound and Vibration, Vol. 30, No. 1, pp. 1–7.CrossRefGoogle Scholar
  15. Wilson, J.F. and Lee, B.K. (1995). “In-plan free vibrations of catenary arches with unsymmetric axes.”Structural Engineering and Mechanics, Vol. 3, No. 5, pp. 511–525.Google Scholar

Copyright information

© KSCE and Springer jointly 2004

Authors and Affiliations

  1. 1.School of Civil, Environmental and Urban EngineeringWonkwang UniversityKorea
  2. 2.Department of Civil and Environmental EngineeringWonkwang UniversityKorea

Personalised recommendations