A geometrically non-linear formulation of a three-dimensional beam element for solving large deflection multibody system problems

A beam finite element formulation for large deflection problems in the analysis of flexible multibody systems has been proposed. In this formulation, a set of independent discrete deformation modes are defined for each element which are related to conventional small deflection beam theory in a co-rotational frame. The paper examines the applicability of this formulation for a shear-deformable three-dimensional Timoshenko beam model, in which geometric non-linearities due to large deflections, buckling loads and post-buckling are included. The geometric non-linearities are accounted for by additional second-order terms in the expressions for the deformation modes. Some numerical examples including large deflections are presented and discussed in order to illustrate the influence of these terms on the accuracy and rate of convergence. The influence of these terms on the displacements is small, except for bifurcation points where the load–deflection characteristics change drastically. It is demonstrated, by comparison with available results in the literature, that highly accurate solutions can be obtained with the present beam finite element formulation

[1]  J. Meijaard.  Direct determination of periodic solutions of the dynamical equations of flexible mechanisms and manipulators , 1991 .

[2]  Jan B. Jonker,et al.  Dynamics of Flexible Mechanisms , 1984 .

[3]  J. G. Jalón,et al.  Dynamics of flexible multibody systems using cartesian co‐ordinates and large displacement theory , 1991 .

[4]  A. Mikkola,et al.  Description of Elastic Forces in Absolute Nodal Coordinate Formulation , 2003 .

[5]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[6]  Gordan Jelenić,et al.  Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics , 1999 .

[7]  R. Y. Yakoub,et al.  Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory , 2001 .

[8]  M. Géradin,et al.  A beam finite element non‐linear theory with finite rotations , 1988 .

[9]  Ignacio Romero,et al.  The interpolation of rotations and its application to finite element models of geometrically exact rods , 2004 .

[10]  E. Reissner,et al.  On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory , 1973 .

[11]  K. Hsiao,et al.  A consistent co-rotational finite element formulation for geometrically nonlinear dynamic analysis of 3-D beams , 1999 .

[12]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[13]  J. B. Jonker.  A finite element dynamic analysis of spatial mechanisms with flexible links , 1989 .

[14]  K. Bathe,et al.  Large displacement analysis of three‐dimensional beam structures , 1979 .

[15]  R. Y. Yakoub,et al.  Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications , 2001 .

[16]  Anders Eriksson,et al.  Beam elements in instability problems , 1997 .

[17]  Johannes Gerstmayr,et al.  On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach , 2008 .

[18]  Jan B. Jonker,et al.  SPACAR — Computer Program for Dynamic Analysis of Flexible Spatial Mechanisms and Manipulators , 1990 .

[19]  T. Belytschko,et al.  Applications of higher order corotational stretch theories to nonlinear finite element analysis , 1979 .

[20]  Alan Jennings,et al.  Frame Analysis including Change of Geometry , 1968 .

[21]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

[22]  J. F. Besseling.  Non-linear analysis of structures by the finite element method as a supplement to a linear analysis , 1974 .

[23]  Jack B. Kuipers,et al.  Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality , 2002 .

[24]  Ignacio Romero,et al.  A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations , 2008 .

[25]  Carlos A. Felippa,et al.  A unified formulation of small-strain corotational finite elements: I. Theory , 2005 .

[26]  J. F. Besseling.  Derivatives of deformation parameters for bar elements and their use in buckling and postbuckling analysis , 1977 .

[27]  G. Cowper.  The Shear Coefficient in Timoshenko’s Beam Theory , 1966 .

[28]  J. F. Besseling.  Non-linear theory for elastic beams and rods and its finite element representation , 1982 .

[29]  S. Atluri,et al.  Dynamic analysis of planar flexible beams with finite rotations by using inertial and rotating frames , 1995 .

[30]  J. Mäkinen.  Total Lagrangian Reissner's geometrically exact beam element without singularities , 2007 .

[31]  A. Ibrahimbegovic.  On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements , 1995 .

[32]  M. Crisfield.  A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements , 1990 .

[33]  M. Crisfield,et al.  Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[34]  J. P. Meijaard,et al.  Validation of flexible beam elements in dynamics programs , 1996 .

[35]  D. W. Scharpf,et al.  Finite element method — the natural approach , 1979 .

[36]  Dannis Michel Brouwer,et al.  Analytical and experimental investigation of a parallel leaf spring guidance , 2009 .

[37]  Arend L. Schwab,et al.  Comparison of Three-Dimensional Flexible Beam Elements for Dynamic Analysis: Classical Finite Element Formulation and Absolute Nodal Coordinate Formulation , 2010 .

[38]  E. Reissner.  On one-dimensional finite-strain beam theory: The plane problem , 1972 .

[39]  Johannes Gerstmayr,et al.  Deformation modes in the finite element absolute nodal coordinate formulation , 2006 .

[40]  M. Géradin,et al.  Flexible Multibody Dynamics: A Finite Element Approach , 2001 .

[41]  Jan B. Jonker,et al.  Definition of deformation parameters for beam elements and their use in flexible multibody system analysis , 2009 .