On the differentiation of the Rodrigues formula and its significance for the vector‐like parameterization of Reissner–Simo beam theory

In this paper we present a systematic way of differentiating, up to the second directional derivative, (i) the Rodrigues formula and (ii) the spin-rotation vector variation relationship. To achieve this goal, several trigonometric functions are grouped into a family of scalar quantities, which can be expressed in terms of a single power series. These results are then applied to the vector-like parameterization of Reissner–Simo beam theory, enabling a straightforward derivation and leading to a clearer formulation. In particular, and in contrast with previous formulations, a relatively compact and obviously symmetric form of the tangent operator is obtained. The paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples. Copyright © 2002 John Wiley & Sons, Ltd.

[1]  K. C. Gupta,et al.  An historical note on finite rotations , 1989 .

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

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

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

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

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

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

[8]  D. Camotim,et al.  Integration algorithm forJ2 elastoplasticity under arbitrary mixed stress-strain control , 2001 .

[9]  Filip C. Filippou,et al.  Non-linear spatial Timoshenko beam element with curvature interpolation , 2001 .

[10]  A. Ibrahimbegovic,et al.  Finite rotations in dynamics of beams and implicit time-stepping schemes , 1998 .

[11]  Dinar Camotim,et al.  Work-conjugacy between rotation-dependent moments and finite rotations , 2003 .

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

[13]  John Argyris,et al.  Nonlinear finite element analysis of elastic systems under nonconservative loading—natural formulation part II. Dynamic problems , 1981 .

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

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

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

[17]  J. C. Simo,et al.  Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms , 1995 .

[18]  J. C. Simo,et al.  The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: intrinsic definition and geometric interpretation , 1992 .

[19]  Stuart S. Antman,et al.  Kirchhoff’s problem for nonlinearly elastic rods , 1974 .

[20]  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.

[21]  E. Reissner.  On finite deformations of space-curved beams , 1981 .

[22]  Byung Chai Lee,et al.  GEOMETRICALLY NON-LINEAR AND ELASTOPLASTIC THREE-DIMENSIONAL SHEAR FLEXIBLE BEAM ELEMENT OF VON-MISES-TYPE HARDENING MATERIAL , 1996 .

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

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

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

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

[27]  E. Stein,et al.  On the parametrization of finite rotations in computational mechanics: A classification of concepts with application to smooth shells , 1998 .

[28]  E. Ramm,et al.  Shell theory versus degeneration—a comparison in large rotation finite element analysis , 1992 .

[29]  M. Géradin,et al.  Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra , 1988 .

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

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