Interpolation non lineaire pour un element fini de poutre en grandes rotations tri-dimensionnelles
Keywords:
Reissner's three-dimensional beam, finite rotations, geometric nonlinearityAbstract
We propose in this paper a finite element formulation for three-dimensional beams undergoing large displacement and large rotations but small strains, many structures of mechanical and civil engineering belong to this class of problems. The main feature of this formulation concerns the treatment of finite rotations : by using the orthogonal matrix parametrization of 3D finite rotations and a nonlinear finite element interpolation of linearized rotations, we are able to provide a symetric tangent stiffness matrix leading to quadratic convergence of the incremental solution procedure, and the consistent linearization being affected by this approach. Several numerical examples demonstrate the efficiency of this development.
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(ARG 79] ARGYRIS J.H., H. BALMER, J.St. DOLTSINIS, P.C. DUNNE. M.
HAASE, M. KLEIBER, G.A. MALEJANNAKIS, H.P. MLEJNEK and D.W.
SCHARPF, Finite Element Method- The Natural Approach, Comput. Methods
Appl. Mech. Eng., 17/18, [1979]1-106
(ARG 82] ARGYRIS J.H., An Excursion Into Large Rotations, Comput. Methods
Appl. Mech. Eng., 32, [1982] 85-155
[BAT 79] BATHE K.J. and S. BOLOURCHI, Large Displacement Analysis of ThreeDimensional
Beam Structures, Int. J. Numer. Methods Eng., 14, [1979]961-986
[BAT 90] BATOZ J.L. and G. DHATT, Modelisation des structures par elements
finis, I. Solides elastiques, II. Poutres et plaques, III. Coques, Hermes, Paris,
[CAR 88] CARDONA A. and M. GERADIN, A Beam Finite Element Non-linear
Theory With Finite Rotations, Int. J. Numer. Methods Eng., 26, [1988] 2403-
[CIA 88] CIARLET Ph. G., Mathematical Elasticity. Volume I: Three-Dimensional
Elasticity, North-Holland, Amsterdam, [1988]
(CRI 90] CRISFIELD M.A., A Consistent Co-Rotational Formulation for Non-Linear,
Three-Dimensional Beam Elements, Comput. Methods Appl. Mech. Eng., 81,
131-150
(DAD 75] DADEPPO D.A. and R. SCHMIDT, Instability of Clamped-Hinged Circular
Arches Subjected to a Point Load, J. Appl. Mech., 97, (1975] 894-896
(FRE 78] FREY F. and S. CESCOTTO, Some New Aspects of the Incremental Total
Lagrangian Description in Nonlinear Analysis, in Finite Elements in Nonlinear
Mechanics, (eds. P.G. Bergan et al.), Tapir Publisher, Trondheim, (1978] 323-
(GOL 80] GOLDSTEIN H., Classical Mechanics, Addison-Wesley, Reading, MA,
(IBR 93) IBRAHIMBEGOVIC A., F. FREY, Finite Element Analysis of Linear
and Nonlinear Planar Deformation of Elastic Initially Curved Beams, Int. J.
Numer. Methods Eng., 36, (1993] 3239-3258
(IBR 95a] IBRAHIMBEGOVIC A., On FE Implementation of Geometrically Nonlinear
Reissner's Beam Theory: Three-Dimensional Curved Beam Elements,
Comput. Methods Appl. Mech. Eng., 122, (1995] 11-26
(IBR 95b] IBRAHIMBEGOVIC A., F. FREY and I. KOZAR, Computational Aspects
of Vector-Like Parameterization of Three-Dimensional Finite Rotations,
Int. J. N umer. Methods Eng., (1995) in press
(IBR 95c] IBRAHIMBEGOVIC A., H. SHAKOURZADEH, J-L. BATOZ, M. AL
MIKDAD and Y-Q. GUO, On the Role of Geometrically Exact and Second
Order Theories in Buckling and Post-buckling Analysis of Three-dimensional
Beam Structures, Comp. and Structures, (1995] in press
[LAN 85] LANG S., Differential Manifolds, Springer-Verlag, Berlin, (1985)
(LEE 94) LEE H., D-W. JUNG, J-H. JEONG and I. SEYOUNG, Finite Element
Analysis of Lateral Buckling for Beam Structures, Comp. and Structures, 53,
1357-1371
(MAR 83] MARSDEN J.E. and T.J.R. HUGHES, Mathematical Foundations of
Elasticity, Prentice-Hall, Englewoods Cliffs, N.J., (1983)
(REI 81) REISSNER E., On Finite Deformations of Space Curved Beams, J. Appl.
Math. Phys., 32, [1981] 734-744
(SIM 84) SIMO J.C. and L. VU-QUOC, A Finite Strain Beam Formulation. The
Three-Dimensional Dynamic Problem. Part I, Comp. Methods Appl. Mech.
Eng., 49, [1984] 55-70
(SIM 86) SIMO J.C. and L. VU-QUOC, A Three-Dimensional Finite-Strain Rod
Model. Part II: Computational Aspects, Comp. Methods Appl. Mech. Eng.,
, [1986) 79-116
[SIM 92] SIMO J.C., The (Symmetric) Hessian for Geometrically Nonlinear Models
in Solid Mechanics: Intrinsic Definition and Geometric Interpretation, Comp.
Methods Appl. Mech. Eng., 96, [1992] 189-200
[SPR 86] SPRING I<.W., Euler Parameters and the use of Quaternion Algebra in the
Manipulation of Finite Rotations: A Review, Mechanism and Machine Theory,
, [1986] 365-373
[TIM 61] TIMOSHENKO S.P. and J.M. GERE, Theory of Elastic Stability, McGrawHill,
London, [1961]
[ZIE 89] ZIENKIEWICZ O.C. and R.L. TAYLOR, The Finite Element Method:
Basic Formulation and Linear Problems, vol I, McGraw-Hill, London, [1989
