Summary
The incremental finite element equations for geometrically non linear problems are obtained via the full incremental form of the principle of virtual displacements using a Generalized Lagrangian approach. This leads to the expression of the tangent matrix in a straight forward manner and an example of application for 2D elasticity is presented. For large displacements/large rotations beam/shell problems the incremental equations are derived using a quadratic approximation for the increment of the reference vectors in terms of the nodal rotation increments. It is shown how this approach leads to a complete tangent matrix which in the examples analyzed seems to be competitive with respect to the simplified form obtained by linearizing the changes in the reference vectors.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Zienkiewicz, O.C.: The finite element method. McGraw-Hill, 1975.
Bathe, K.J.: Finite Element Procedures in Non Linear Analysis. Prentice Hall, 1982.
Yaghmai, S.: Incremental analysis of large deformations in mechanics of solids with applications to axisymmetric shells of revolution. Report No. SESM 68–17, Univ. California, Berkeley, 1968.
Larsen, P.K.: Large displacement analysis of shells of revolution including creep, plasticity and viscoelasticity. Report SESM 71–22, Univ. California, Berkeley, 1971.
Bathe, K.J.; Ramm, E. and Wilson, E.L.: Instability analysis of free form shells by finite elements. Int. J. Num. Meth. Engng., 9, 2, pp. 353–386, 1975.
Horrigmoe, G.: Non linear finite element models in solid mechanics. Report 76–2, Norwegian Inst. Tech., Univ. Trondheim, 1970.
Mondkar, D.P. and Powell, G.H.: Finite element analysis of non linear static and dynamic response. Int. J. Num. Meth. Engng., 11, 3, pp. 499–520, 1977.
Frey, F. and Cescotto, S.: Some new aspects of the incremental total lagrangian description in non linear analysis in “Finite Element in Non Linear Mechanics ”, edited by P. Bergan et al. Tapir Publishers, Univ. of Trondheim, 1978.
Oiiate, E.; Oliver, J.; Miquel-Canet, J. and Suârez, B.: A finite element formulation for geometrically non linear problems using a secant matrix: Applications to 3-D Trusses in Computational Mechanics., edited by G. Yagawa y S.N. Atluri. Springer-Verlag, Tokyo, 1986.
Ovate, E.: On the obtention of secant and tangent matrices for the analysis of geometrically non linear problems using finite elements. Internal Report, E.T.S. Ingenieros de Caminos, Technical University of Catalonia, Spain, 1988.
Argyris, J.: An excursion into large rotations. Comp. Meth. Appl. Mech. Engng., 32, pp. 85–155, 1982.
Dvorkin, E.; Ovate, E. and Oliver, J.: On a non linear formulation for curved Timoshenko beam elements considering large displacements/rotation increments. Int, J. Num. Meth. Engng., 1988.
Mattiason, K.: Numerical results from large deflection beam and frame problems analyzed by means of elliptic integrals. Int. J. Num. Meth. Engng., 17, 145–153, (1981).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Oñate, E., Dvorkin, E., Canga, M.E., Oliver, J. (1990). On the Optention of the Tangent Matrix for Geometrically Nonlinear Analysis Using Continuum Based Beam/Shell Finite Elements. In: Krätzig, W.B., Oñate, E. (eds) Computational Mechanics of Nonlinear Response of Shells. Springer Series in Computational Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84045-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-84045-6_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-84047-0
Online ISBN: 978-3-642-84045-6
eBook Packages: Springer Book Archive