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Beam elements in instability problems
1997 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 144, no 1-2, 163-197 p.Article in journal (Refereed) Published
Abstract [en]

This paper deals with the formulation of beam elements for the numerical analysis of instability phenomena in frame-type structures. Total versus co-rotational approaches are discussed comparatively, for both two-dimensional and three-dimensional problems, and the similarities between the two types are outlined. In the context of 3D beam elements, special attention is given to the parameterisation of the orthogonal transformation used to define the rotational field of the beam. The technique advocated in the paper is based on the so-called rotational vector. This leads to symmetric stiffness matrices and avoids the need for special updating procedures for the rotational variables. A set of test problems, for which the critical behaviour is governed by fold, cusp and butterfly catastrophes, is used to assess the performances of the considered element types. It is shown that analytically verified identities in element formulation, also hold in numerical application. The examples also show how complex instability behaviour can be reproduced by all elements, where sufficient accuracy is introduced into the kinematic expressions. The analytical derivation of element expressions, with symbolic manipulations from stated basic assumptions, is consistently used in the paper.

Place, publisher, year, edition, pages
1997. Vol. 144, no 1-2, 163-197 p.
Keyword [en]
Finite element method, Mathematical models, Mathematical transformations, Stability, Stiffness matrix, Vectors, Beam elements, Structural analysis
National Category
Applied Mechanics
Identifiers
URN: urn:nbn:se:kth:diva-117945OAI: oai:DiVA.org:kth-117945DiVA: diva2:603974
Note

References: Pacoste, C., Eriksson, A., Element behaviour in post-critical plane frames analysis (1995) Comput. Methods Appl. Mech. Engrg., 125, pp. 319-343; Reissner, E., On one-dimensional, large displacement, finite-strain beam theory (1973) Stud. Appl. Math., 52, pp. 87-95; Antman, S.S., Kirchhoff problem for nonlinear elastic rods (1974) Quart. J. Appl. Math., 32 (3), pp. 221-240; Antman, S.S., Jordan, K.B., Qualitative aspects of the spatial deformation of non-linearly elastic rods (1975) Proc. Roy. Soc. Edinburgh. Sect. A, 73 (5), pp. 85-105; Reissner, E., On finite deformations of space curved beams (1981) J. Appl. Math. Phys., 32, pp. 734-774; Parker, D.F., The role of Saint Venant's solutions in rods and beam theories (1979) J. Appl. Mech., 46, pp. 861-866; Pleuss, P., Sayir, M., A second order theory for large deflection of slender beams (1983) J. Appl. Math. Phys., 34, pp. 192-217; Eriksson, A., Pacoste, C., Beam elements in instability problems (1995) Proc. of the Eighth Nordic Seminar on Computational Mechanics, , Göteborg; Iura, M., Effects of coordinate system on the accuracy of corotational formulation for Bernoulli-Euler's beam (1994) Int. J. Solids Struct., 31, pp. 2793-2806; Goto, Y., Kasugai, T., Nishino, F., On the choice of local moving coordinates in the finite displacement analysis of planar frames (1990) Proc. JSCE, 386 (8), pp. 311-320; Argyris, J.H., An excursion into large rotations (1982) Comput. Methods Appl. Mech. Engrg., 32, pp. 85-155; Simo, J.C., Vu-Quoc, L., A three dimensional finite-strain rod model. Part II: Computational aspects (1986) Comput. Methods Appl. Mech. Engrg., 58, pp. 79-116; Buechter, N., Ramm, E., Shell theory versus degeneration - A comparison in large rotation finite element analysis (1992) Int. J. Numer. Methods Engrg., 34, pp. 39-59; Crisfield, M.A., A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements (1990) Comput. Methods Appl. Mech. Engrg., 81, pp. 131-150; Cardona, A., Geradin, M., A beam finite element non-linear theory with finite rotations (1988) Int. J. Numer. Methods Engrg., 26, pp. 2403-2438; Ibrahimbegović, A., Frey, F., Kožar, I., Computational aspects of vector like parametrization of three-dimensional finite rotations (1995) Int. J. Numer. Methods Engrg., 38, pp. 3653-3673; Nour-Omid, B., Rankin, C.C., Finite rotation analysis and consistent linearization using projectors (1991) Comput. Methods Appl. Mech. Engrg., 93, pp. 353-384; Eriksson, A., Fold lines for sensitivity analyses in structural instability (1994) Comput. Methods Appl. Mech. Engrg., 114, pp. 77-101; Char, B.W., Geddes, K.O., Gonnet, G.H., Leong, B.L., Monagan, M.B., Watt, S.M., (1991) Maple V Language Reference Manual, , Springer-Verlag; Argyris, J.H., Scharpf, D.W., Some general considerations on the natural mode technique. Part I. Small displacements (1969) Aeron. J. Roy. Aeron. Soc., 73, pp. 219-226; Argyris, J.H., Scharpf, D.W., Some general considerations on the natural mode technique. Part II. Large displacements (1969) Aeron. J. Roy. Aeron. Soc., 73, pp. 361-362; Crisfield, M.A., (1991) Non-linear Finite Element Analysis of Solids and Structures, , John Wiley & Sons, Chichester; Simo, J.C., A finite strain beam formulation. the three-dimensional dynamic problem. Part I (1985) Comput. Methods Appl. Mech. Engrg., 49, pp. 55-70; Eriksson, A., Equilibrium subsets for multi-parametric structural analysis (1996) Comput. Methods Appl. Mech. Engrg., , accepted for publication; Eriksson, A., (1993) Cutsets and Augmented Equilibrium Equations for Multi-parametric Structural Analysis Models, , Dept. Struct. Engng., Royal Inst. Techn., Stockholm; Fertis, D.G., (1993) Nonlinear Mechanics, , CRC Press, Boca Raton, FL; Goto, Y., Yoshimitsu, T., Obata, M., Elliptic integral solutions of plane elastica with axial and shear deformations (1990) Int. J. Solids Struct., 26, pp. 375-390; Williams, F.W., An approach to the nonlinear behavior of members of a rigid jointed plane framework with finite deflections (1964) Q. J. Mech. Appl. Math., 17, pp. 451-469

NR 20140805Available from: 2013-02-07 Created: 2013-02-07 Last updated: 2017-12-06Bibliographically approved

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