Nonlinear isogeometric spatial Bernoulli Beam

A new element formulation of a geometrically nonlinear spatial curved beam assuming Bernoulli theory including torsion without warping is proposed. The element formulation is derived directly from the 3D-continuum by means of nonlinear kinematics, thus accounting for large displacements. The geometric description of the proposed element is adapted from the spatial rod of Greco and Cuomo (2013) and extended to a nonlinear element formulation. The proposed formulation can handle arbitrary orientations of the cross section along the beam. In this publication, NURBS are used as basis functions for discretization, since they can easily provide the required C1-continuity between elements. The presented element formulation has four degrees of freedom, three for displacements and one for the rotation around the center line. In order to prove the accuracy of the developed spatial Bernoulli beam, several numerical examples are presented and compared to analytic solutions and other element formulations.

[1]  Dieter Weichert,et al.  Nonlinear Continuum Mechanics of Solids , 2000 .

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

[3]  Steen Krenk,et al.  Non-linear Modeling and Analysis of Solids and Structures , 2009 .

[4]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[5]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[6]  R. Echter,et al.  A hierarchic family of isogeometric shell finite elements , 2013 .

[7]  Carl T.F. Ross,et al.  Strength of materials and structures , 1992 .

[8]  Alessandro Reali,et al.  Locking-free isogeometric collocation methods for spatial Timoshenko rods , 2013 .

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

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

[11]  Jose Manuel Valverde,et al.  Invariant Hermitian finite elements for thin Kirchhoff rods. II: The linear three-dimensional case , 2012 .

[12]  G. Dupuis,et al.  NONLINEAR MATERIAL AND GEOMETRIC BEHAVIOR OF SHELL STRUCTURES. , 1971 .

[13]  Thomas J. R. Hughes,et al.  A large deformation, rotation-free, isogeometric shell , 2011 .

[14]  Roland Wüchner,et al.  Isogeometric shell analysis with Kirchhoff–Love elements , 2009 .

[15]  Joel Langer,et al.  Lagrangian Aspects of the Kirchhoff Elastic Rod , 1996, SIAM Rev..

[16]  Roland Wüchner,et al.  Integrated design and analysis of structural membranes using the Isogeometric B-Rep Analysis , 2016 .

[17]  Wolfgang A. Wall,et al.  A locking-free finite element formulation and reduced models for geometrically exact Kirchhoff rods , 2015 .

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

[19]  E. Dvorkin,et al.  On a non‐linear formulation for curved Timoshenko beam elements considering large displacement/rotation increments , 1988 .

[20]  I N Bronstein,et al.  Taschenbuch der Mathematik , 1966 .

[21]  Roland Wüchner,et al.  Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures , 2015 .

[22]  Yuri Bazilevs,et al.  Isogeometric rotation-free bending-stabilized cables: Statics, dynamics, bending strips and coupling with shells , 2013 .

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

[24]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[25]  H. Rubin.  Evaluierung der linearen und nichtlinearen Stabstatik in Theorie und Software. Prüfbeispiele, Fehlerursachen, genaue Theorie. Von G. Lumpe, V. Gensichen , 2014 .

[26]  Alain Goriely,et al.  On the Dynamics of Elastic Strips , 2001, J. Nonlinear Sci..

[27]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[28]  C. Gontier,et al.  A large displacement analysis of a beam using a CAD geometric definition , 1995 .

[29]  B. Simeon,et al.  Isogeometric Reissner–Mindlin shell analysis with exactly calculated director vectors , 2013 .

[30]  Wolfgang A. Wall,et al.  An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods , 2014 .

[31]  Leopoldo Greco,et al.  B-Spline interpolation of Kirchhoff-Love space rods , 2013 .

[32]  S. H. Lo,et al.  Geometrically nonlinear formulation of 3D finite strain beam element with large rotations , 1992 .

[33]  Robert A. Heller.  Mechanics of Structures , 2003 .