A large displacement and finite rotation thin-walled beam formulation including cross-section deformation

Abstract This paper presents a new formulation for thin-walled beams that includes cross-section deformation. The kinematic description of the beam emanates from the geometrically exact Reissner–Simo beam theory and is enriched with arbitrary cross-section deformation modes complying with Kirchhoff’s assumption. The inclusion of these deformation modes makes it possible to capture the cross-section in-plane distortion, wall (plate) transverse bending and out-of-plane (warping), which leads to a computationally efficient numerical implementation. Several illustrative numerical examples are presented and discussed, showing that the resulting beam finite element leads to solutions that are in very good agreement with those obtained with standard shell finite elements, albeit involving much less degrees-of-freedom.

[1]  S. Govindjee,et al.  Anisotropic bending-torsion coupling for warping in a non-linear beam , 2003 .

[2]  R. Schardt.  Verallgemeinerte Technische Biegetheorie , 1989 .

[3]  Dinar Camotim,et al.  On the differentiation of the Rodrigues formula and its significance for the vector‐like parameterization of Reissner–Simo beam theory , 2002 .

[4]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[5]  J. M. Davies.  Generalised Beam Theory (GBT) for Coupled Instability Problems , 1998 .

[6]  Jean-Marc Battini,et al.  Plastic instability of beam structures using co-rotational elements , 2002 .

[7]  G. Ruta,et al.  Coupled instabilities in thin-walled beams: a qualitative approach , 2003 .

[8]  J. Z. Zhu,et al.  The finite element method , 1977 .

[9]  J. Argyris.  An excursion into large rotations , 1982 .

[10]  J. C. Simo,et al.  A Geometrically-exact rod model incorporating shear and torsion-warping deformation , 1991 .

[11]  F. Gruttmann,et al.  A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections , 1998 .

[12]  C. Pacoste,et al.  Co-rotational beam elements with warping effects in instability problems , 2002 .

[13]  Dinar Camotim,et al.  GBT-based semi-analytical solutions for the plastic bifurcation of thin-walled members , 2010 .

[14]  Cv Clemens Verhoosel,et al.  Non-Linear Finite Element Analysis of Solids and Structures , 1991 .

[15]  Werner Wagner,et al.  THEORY AND NUMERICS OF THREE-DIMENSIONAL BEAMS WITH ELASTOPLASTIC MATERIAL BEHAVIOUR ∗ , 2000 .

[16]  A. Ibrahimbegovic,et al.  Computational aspects of vector-like parametrization of three-dimensional finite rotations , 1995 .

[17]  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 .

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

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

[20]  Nicola Luigi Rizzi,et al.  Nonstandard Models for Thin-Walled Beams With a View to Applications , 1996 .

[21]  R. Dąbrowski.  Curved thin-walled girders: theory and analysis, , 1972 .

[22]  M. Géradin,et al.  Finite element theory for curved and twisted beams based on exact solutions for three-dimensional solids Part 2: Anisotropic and advanced beam models , 1998 .

[23]  Satya N. Atluri,et al.  Variational approaches for dynamics and time-finite-elements: numerical studies , 1990 .

[24]  Dinar Camotim,et al.  GBT-based Structural Analysis of Thin-walled members: Overview, Recent Progress and Future Developments , 2006 .

[25]  J. Oden.  Mechanics of elastic structures , 1967 .

[26]  Paulo M. Pimenta,et al.  Geometrically Exact Analysis of Spatial Frames , 1993 .

[27]  M. Iura,et al.  A consistent theory of finite stretches and finite rotations, in space-curved beams of arbitrary cross-section , 2001 .

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

[29]  J. Stuelpnagel.  On the Parametrization of the Three-Dimensional Rotation Group , 1964 .