A finite rotation shell theory with application to composite structures
Keywords:
nonlinear shell formulation, finite rotations, composite materialAbstract
In this paper we derive a finite element formulation for geometrical nonlinear shell structures. The formulation bases on a direct introduction of the isoparametric finite element formulation into the shell equations. The element allows the occurence of finite rotations which are described by a rotation tensor. A layerwise linear elastic material model for composites is chosen. The consistent linearization of all equations leads to quadratic convergence behaviour within the nonlinear solution procedure. Examples show the applicability and effectivity of the developed element.
Downloads
References
(SIM90) Simo, J. C., Fox, D.D., Rifai, M. S., On a Stress Resultant Geometrically
Exact Shell Model, Camp. Meth. Appl. Mech. Engng. (Part I: 72 267-304 (1989),
Part II: 73 53-92 (1989), Part III: 79 21-70 (1990), Part VII:108 319-339 (1993))
(WAS93] Wagner W., Stein E., A New Finite Element Formulation for Cylindrical
Shells of Composite Material, Composites Engineering, 3, 899-910 {1993)
(RAM77] Ramm, E., A Plate/Shell Element for Large Deflections and Rotations,
US-Germany Symp. on 'Formulations and Computational Algorithms in Finite
Element Analysi8', M.I.T.-Press, Boston, 264-293 (1977)
(RAM76] Ramm, E., Geometrisch nichtlineare Elastostatik und Finite Elemente, Report
No. 76-2, Inst. f. Baustatik Universitat Stuttgart 1976
(TSA88] Tsai, W. S., Composite$ Design, Think Composites, Dayton (1988)
(BUR92] Buchter N., Ramm E., Shell thP.ory versus degeneration - A comparison
in large rotation finite element analysis , Int. J. Num. Meth. Engng. 34 39-59
(1992)
(PIE77] Pietraszkiewicz, W., Introduction to the Non-linear Theory of Shells, Techn.
Rep. lnst. f. Mechanik, Nr. 10, Ruhr-Universitat Bochum (1977)
(BAD85] Bathe, K.J., Dvorkin, E.N., A 4-Node Plate Bending Element based on
Mindlin/Reissner Theory and a Mixed Interpolation, Int. J. Num. Meth. Engng.
367-383 (1985)
(DVB84] Dvorkin, E.N., Bathe,K.J., A Continuum Mechanics based Four-Node Shell
Element for General Nonlinear Analsysis, Engng. Camp. 1 77-88 (1984)
(ZIT89) Zienkiewicz, 0. C. and Taylor R. L., The Finite Element Method, Vol.l-2, 4.
Edn. Me Graw-Hill, London {1989/1991)
(RAM86] Ramm, E., Matzenmiller, A., Large deformation shell analysis based on
the degeneration concept, Finite Element methods for Plate and Shell Structures
(Eds. Hughes,T.J.R., Hinton, E.) Pineridge Press, Swansea 365-393 (1986)
(STM89] Stander,N., Matzenmiller, A., R.amm, E., An assessment of Assumed Strain
Methods in Finite Rotation Shell Analysis, Engng. Comp. 6 57-66 (1989)
[IBR95] lbrahimbegovic A., On Assumed Shear Strain in Finite Rotation Shell Analysis,
Engng. Comp. 11 in press {1995)
[TAB82] Taber, L.A., Large Deflection of a Fluid-Filled Spherical Shell under a Point
Load, J. Appl. Mech., Tran.s. ASME Ser. E, 39, 121-128 (1982)
(WAG90] Wagner, W., A Finite Element Model for Nonlinear Shells of Revolution
with Finite Rotations, Int. J. Num. Jlieth. Engng. 29 1455-1471 {1990)
[MCH85] McNeal, R. H., Harder, R. 1., A proposed standard set of problems to test
finite element accuracy. J. of Finite Elements in Analysis fj Design 1 3-20 (1985)
(HUL81] Hughes, T.J.R., Liu, W.K., Nonlinear Finite Element Analysis of Shells.
Camp. Meth. Appl. Mech. Engng. Part I: Three-dimensional shells, 26 331-362
(1981) Part II: Two-dimensional shells, 27 167-181 (1981)
(SPH86] Stanley, G. M., Park, K.C., Hughes, T.J.R., Continuum-based resultant
shell elements, Finite Element methods for Plate and Shell Structures (Eds.
Hughes, T.J.R., Hinton, E.) Pineridge Press, Swansea, 1-45 (1986)
[GSW89) Gruttmann, F., Stein, E., Wriggers, P., Theory and Numerics of Thin Elastic
Shells with Finite Rotations, lng. Archiv 59 54-67 (1989)
(IBR94] lbrahimbegovic A., Frey F., Stress Resultant Geometrically Nonlinear Shell
Theory With Drilling Rotations, Comp. Meth. Appl. Mech. Engng. (Part I: 118
-284 (1994), Part II: 118 285-308 (1995))
[PAR91) Parisch, H., An Investigation of a Finite Rotation Four Node Shell Element,
Int. Num. Meth. Engng. 31 127-150 (1991)
[GEB90] Gebhardt H., Finite Element K onzepte fur schubelastische Schalen mit
endlichen Drehungen, Report No. 10, lnst. f. Baustatik, Universitii.t Karlsruhe,
[SCH86) Schaap, H., Oberfiii.chenorientierte Schalentheorien endlicher Verschiebungen,
Ing.Archiv 56 427-437 (1986)
[BAD90] B~ar, Y., Ding, Y., Theory and Finite-Element Formulation for Shell
Structures undergoing Finite Rotations, Advances in the Theory of Plates and
Shells (Eds. Voyiadjis, G.Z., Karamanlidis, D.) Elsevier Science Publishers, Amsterdam,
-26 (1990)
[BDM92) B~ar, Y., Ding, Y., Menzel, W., Montag, U., Finite Rotation Shell Elements
via Finite-Rotation Shell Theories, Statik und Dynamik im konstruktiven
lngenieurbau, Festschrift Wilfried B. Krii.tzig, 1992
[DOR90] Dorninger, K., Rammerstorfer, F.G., A Layered Composite Shell Element
for Elastic and Thermoelastic Stress and Stability Analysis at Large Deformations,
Int. Num. Meth. Engng. 30 833-858 (1990)
[RCH85) Reddy, J.N., Chandrashekhara, K., Nonlinear Analysis of Laminated Shells
including Transverse Shear Strains, AIAA Journal23 440-441 (1985)
[WAW88] Wagner, W., Wriggers, P., A Simple Method for the Calculation of Postcritical
Branches, Engineering Computations, 5 103-109 (1988)
[WRG93] Wriggers, P., Gruttmann, F., Thin Shells with Finite Rotations formulated
in Biot-Stresses: Theory and Finite Element Formulation Int. Num. Meth.
Engng., 36, 2049-2071, 1993
[WRG90) Wriggers P., Gruttmann F., Large Deformations of Thin Shells: Theory and
Finite-Element-Discretization, Analytical and Computational Models of Shells
(Eds. Noor, A., Belytschko T., Sima, J. C.) ASME CED-Vol. 3, 135-159 (1990)
[PIJ87) Pinsky, P. M., Jang, J., A C0-elastoplastic shell element based on assumed
covariant strain interpolations, NUMETA 1987, Numerical Methods in Engineering:
Theory and Applications (Eds. Pande, G.N., Middleton, J.) Pineridge Press,
Swansea (1987)
[LAM90) Lammering, R., Structural Analysis and Optimization of a Propfan-Blade
by use of the Finite Element Method, Engineering Computations 7 327-337
(1990)
[HUG87) Hughes, T. J. R., The Finite Element Method, Prentice-Hall Inc., Englewood
Cliffs, New Jersey (1987)