A p‐version, first order shear deformation, finite element for geometrically non‐linear vibration of curved beams

A p-version, hierarchical finite element for curved, moderately thick, elastic and isotropic beams is introduced. The convergence properties of the element are analysed and some results are compared with results published elsewhere or calculated using a commercial finite element package. It is verified that, with the proposed element, shear locking does not affect the computation of the natural frequencies and that low dimensional, accurate models are obtainable. Geometrically non-linear vibrations due to finite deformations, which occur for harmonic excitations with frequencies close to the first three natural frequencies of vibration, are investigated using Newmark's method. The influence of the thickness, longitudinal inertia and curvature radius on the dynamic behaviour of curved beams are studied. Copyright © 2004 John Wiley & Sons, Ltd.

[1]  P. Raveendranath,et al.  A three‐noded shear‐flexible curved beam element based on coupled displacement field interpolations , 2001 .

[2]  Maurice Petyt,et al.  Free vibration of a curved beam , 1971 .

[3]  S. H. A. Chen,et al.  Nonlinear Vibration of Plane Structures by Finite Element and Incremental Harmonic Balance Method , 2001 .

[4]  W. J. Austin,et al.  Free vibration of arches flexible in shear. , 1973 .

[5]  Michel Géradin,et al.  Finite element theory for curved and twisted beams based on exact solutions for three-dimensional solids. Part 1: Beam concept and geometrically exact nonlinear formulation , 1998 .

[6]  J. N. Reddy,et al.  Large deflections and large‐amplitude free vibrations of straight and curved beams , 1981 .

[7]  Maurice Petyt,et al.  NON-LINEAR VIBRATION OF BEAMS WITH INTERNAL RESONANCE BY THE HIERARCHICAL FINITE-ELEMENT METHOD , 1999 .

[8]  Leonard Meirovitch,et al.  On the inclusion principle for the hierarchical finite element method , 1983 .

[9]  Moshe Eisenberger,et al.  In‐plane vibrations of shear deformable curved beams , 2001 .

[10]  T. Kaneko.  On Timoshenko's correction for shear in vibrating beams , 1975 .

[11]  Seok-Soon Lee,et al.  Development of a new curved beam element with shear effect , 1996 .

[12]  Q. Bi,et al.  ANALYSIS OF NON-LINEAR DYNAMICS AND BIFURCATIONS OF A SHALLOW ARCH SUBJECTED TO PERIODIC EXCITATION WITH INTERNAL RESONANCE , 2000 .

[13]  J. Łuczko.  Bifurcations and internal resonances in space-curved rods , 2002 .

[14]  Przemysław Litewka,et al.  Free vibrations of shear‐flexible and compressible arches by FEM , 2001 .

[15]  M. Petyt,et al.  Geometrically nonlinear vibration analysis of thin, rectangular plates using the hierarchical finite element method—I: The fundamental mode of isotropic plates , 1997 .

[16]  N. Bardell.  The application of symbolic computing to the hierarchical finite element method , 1989 .

[17]  Stanley B. Dong,et al.  Effect of transverse shear deformation on vibrations of planar structures composed of beam‐type elements , 1973 .

[18]  Maurice Petyt,et al.  Introduction to finite element vibration analysis , 1990 .

[19]  J. Archer.  Consistent matrix formulations for structural analysis using finite-element techniques. , 1965 .

[20]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[21]  Yoshihiro Narita,et al.  Vibrations of Completely Free Shallow Shells of Curvilinear Planform , 1986 .

[22]  S. Timoshenko,et al.  X. On the transverse vibrations of bars of uniform cross-section , 1922 .

[23]  M. Petyt,et al.  Introduction to Finite Element Vibration Analysis , 2016 .

[24]  Pedro Ribeiro,et al.  The second harmonic and the validity of Duffing’s equation for vibration of beams with large displacements , 2001 .

[25]  J. R. Hutchinson.  Shear coefficients for Timoshenko beam theory , 2001 .