MODES OF VIBRATION OF THE BEAMS UNDER THE INFLUENCE OF DISCONTINUITY IN FOUNDATION

Vibrations of the Timoshenko beams resting on the Winkler and Pasternak elastic foundation with discontinuity are investigated in this paper. A p-version finite element method that accounts for shear deformation is used. This p-element has special displacement shape functions that make it particularly appropriate for dealing with problems with discontinuities such as those introduced in the foundation. A set of ordinary differential equations is derived; geometrical non-linearity is considered in these equations for the sake of generality and for future use. Natural frequencies and mode shapes of vibration (composed by transverse displacements and rotations of cross sections) of the shear deformable beam are presented for diverse sizes and location of the discontinuity in the foundation. Results of the present approach are compared with the ones computed via established finite element software for various stiffness of the elastic support of the Winkler and Pasternak type.

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

[2]  Jen-San Chen,et al.  Steady state and stability of a beam on a damped tensionless foundation under a moving load , 2011 .

[3]  J. E. Stephens,et al.  Natural frequencies of Timoshenko beams on pasternak foundations , 1977 .

[4]  T.-P. Chang,et al.  Dynamic finite element analysis of a nonlinear beam subjected to a moving load , 1996 .

[5]  Pedro Ribeiro,et al.  Free periodic vibrations of beams with large displacements and initial plastic strains , 2010 .

[6]  Howard F. Wolfe,et al.  An Experimental Investigation of Nonlinear Behaviour of Beam an Plates Excited to High Levels of Dynamic Response. , 1995 .

[7]  M. Petyt,et al.  Linear vibration analysis of laminated rectangular plates using the hierarchical finite element method-I. Free vibration analysis , 1996 .

[8]  Seong-Min Kim,et al.  Vibration and dynamic buckling of shear beam-columns on elastic foundation under moving harmonic loads , 2006 .

[9]  A. Kampitsis,et al.  Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads , 2011 .

[10]  J. Béliveau.  Introduction to finite element vibration analysis , 1992 .

[11]  Pedro Ribeiro,et al.  Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods , 2004 .

[12]  Mohammad H. Kargarnovin,et al.  Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition , 2012 .

[13]  A. Houmat.  Free vibration analysis of membranes using the h–p version of the finite element method , 2005 .

[14]  A. R. Vosoughi,et al.  DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges , 2009 .

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

[16]  Stanislav Stoykov,et al.  Nonlinear forced vibrations and static deformations of 3D beams with rectangular cross section: The influence of warping, shear deformation and longitudinal displacements , 2010 .

[17]  Pedro Ribeiro,et al.  A hierarchical finite element for geometrically non‐linear vibration of doubly curved, moderately thick isotropic shallow shells , 2003 .

[18]  Giuseppe Muscolino,et al.  Response of beams resting on viscoelastically damped foundation to moving oscillators , 2007 .

[19]  Pedro Ribeiro,et al.  Non-linear vibration of Timoshenko damaged beams by a new p-version finite element method , 2013 .

[20]  Stefano Lenci,et al.  Nonlinear resonances of a semi-infinite cable on a nonlinear elastic foundation , 2013, Communications in nonlinear science & numerical simulation.

[21]  Pedro Ribeiro,et al.  HIERARCHICAL FINITE ELEMENT ANALYSES OF GEOMETRICALLY NON-LINEAR VIBRATION OF BEAMS AND PLANE FRAMES , 2001 .

[22]  Lu Sun,et al.  A CLOSED-FORM SOLUTION OF A BERNOULLI-EULER BEAM ON A VISCOELASTIC FOUNDATION UNDER HARMONIC LINE LOADS , 2001 .

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

[24]  Guglielmo S. Aglietti,et al.  An h-p finite element vibration analysis of open conical sandwich panels and conical sandwich frusta , 1999 .

[25]  Pedro Ribeiro,et al.  A p‐version, first order shear deformation, finite element for geometrically non‐linear vibration of curved beams , 2004 .

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

[27]  Vladimir Stojanović,et al.  Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory , 2013 .