Geometrically nonlinear vibrations of beams supported by a nonlinear elastic foundation with variable discontinuity

Abstract Geometrically nonlinear vibrations of a Timoshenko beam resting on a nonlinear Winkler and Pasternak elastic foundation with variable discontinuity are investigated in this paper. A p-version finite element method is developed for geometric nonlinear vibrations of a shear deformable beam resting on a nonlinear foundation with discontinuity. The elastic foundation has cubic nonlinearity with the shearing layer. In the study the p-element which comes from the use of explored special displacement shape functions for damaged beams is used and applied to a model with nonlinear foundation. The novelty of the present study lies in the easy generalisation of the approach of natural frequencies, general mode shapes (transverse and rotations of cross sections), and maximal deflections in nonlinear steady state vibrations of the shear deformable beam for any size and location of discontinuity of the nonlinear elastic support. A new set of nonlinear partial differential equations is developed, and they are solved in the time domain using the Newmark method for obtaining the amplitudes and deformed shapes of a beam in the steady state forced vibration regime. The present work consists of the comparison of the results with various stiffnesses of nonlinear elastic supports of the Winkler and Pasternak type.

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

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

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

[4]  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 .

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

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

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

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

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

[10]  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 .

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

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

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

[14]  D. Griffin,et al.  Finite-Element Analysis , 1975 .

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

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

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

[20]  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 .

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

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

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

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

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

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

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