Change search
ReferencesLink to record
Permanent link

Direct link
Numerical analysis of complex instability behaviour using incremental-iterative strategies
KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.ORCID iD: 0000-0002-5819-4544
1999 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 179, no 3-4, 265-305 p.Article in journal (Refereed) Published
Abstract [en]

The paper describes how quasi-static, conservative instability problems can be analysed in a multi-parametric space, using generalised path-following procedures for augmented equilibrium problems. The general formulation of such augmented equilibrium problems is discussed in some detail. The focus is set on two classes of generalised 1D paths: basic equilibrium paths and fold lines, i.e. critical subset paths. The solution methods are seen as extensions to common incremental-iterative strategies, allowing the computation of subsets of equilibrium states which also fulfil some auxiliary conditions, e.g. criticality. In this context, some emphasis is also given to the evaluation of the properties of the problem, at a certain state; the tangential stiffness is here used to evaluate - possibly multidimensional - tangent spaces, and in the isolation of special states, i.e. vanishing variables, turning points and exchanges of stability, being important aspects of instability analyses. A set of carefully chosen numerical examples demonstrate on one hand the ability of the numerical procedures to deal with complex instability phenomena, including coincident or near coincident buckling modes, modal interaction, secondary bifurcations, and, on the other hand, their versatility in performing parameter sensitivity analyses. Finally, comparisons with alternative techniques, based on asymptotic strategies, are also put forth. © 1999 Elsevier Science S.A.

Place, publisher, year, edition, pages
1999. Vol. 179, no 3-4, 265-305 p.
Keyword [en]
Finite elements, Instabilities, Numerical procedures, Shells, Bifurcation (mathematics), Buckling, Finite element method, Iterative methods, Mathematical models, Modal analysis, Problem solving, Sensitivity analysis, Stiffness, Structural analysis, Equilibrium states, Incremental-iterative method, Shells (structures)
National Category
Applied Mechanics
URN: urn:nbn:se:kth:diva-119535OAI: diva2:611577

References: Pacoste, C., Eriksson, A., Element behaviour in post-critical plane frame analysis (1995) Comput. Meth. Appl. Mech. Engrg., 125, pp. 319-343; Eriksson, A., Structural instability analyses based on generalised path-following (1998) Comput. Meth. Appl. Mech. Engrg., 156, pp. 45-74; Pacoste, C., Eriksson, A., Beam elements in instability problems (1997) Comput. Meth. Appl. Mech. Engrg., 144, pp. 163-197; Eriksson, A., (1993) Cutsets and Augmented Equilibrium Equations for Multi-parametric Structural Analysis Models, , Technical Report, Department of Structural Engineering Royal Institute of Technology, Stockholm; Eriksson, A., Fold lines for sensitivity analyses in structural instability (1994) Comput. Meth. Appl. Mech. Engrg., 114, pp. 77-101; Eriksson, A., Equilibrium subsets for multi-parametric structural analysis (1997) Comput. Meth. Appl. Mech. Engrg., 140, pp. 305-327; Rheinboldt, W.C., Riks, E., Solution techniques for nonlinear finite element equations (1983) State-of-the-Art Surveys on Finite Element Technology, pp. 183-224. , in: A.K. Noor, W.D. Pilkey (Eds.), ASME, New York; Rheinboldt, W.C., (1986) Numerical Analysis of Parameterized Nonlinear Equations, , Wiley, New York; Allgower, E.L., Georg, K., (1990) Numerical Continuation Methods. An Introduction, , Springer, Berlin; Allgower, E.L., Georg, K., Continuation and path following (1993) Acta Numerica, pp. 1-64. , in: A. Iserles (Ed.), Cambridge University Press, Cambridge; Crisfield, M.A., New solution procedures for linear and non-linear finite element analysis (1986) The Mathematics of Finite Elements and Applications, 5, pp. 49-81. , in: J. Whitman (Ed.), Academic Press, London; Crisfield, M.A., (1991) Non-linear Finite Element Analysis of Solids and Structures, 1. , Wiley, Chichester; Van Erp, G.M., Menken, C.M., Initial post-buckling analysis with the finite-strip method (1991) Comp. Struct., 40, pp. 1193-1201; Damil, N., Potier-Ferry, M., A new method to compute perturbed bifurcations: Applications to the buckling of imperfect elastic sturctures (1993) Int. J. Engrg. Sci., 28, pp. 943-957; Azrar, L., Cochelin, B., Damil, N., Potier-Ferry, M., An asymptotic-numerical method to compute the postbuckling behaviour of elastic plates and shells (1993) Int. J. Numer. Meth. Eng., 36, pp. 1251-1277; Menken, C.M., Kouhia, R., Groot, W.J., An investigation into non-linear interaction between buckling modes (1994) Thin-Walled Struct., 19, pp. 129-145; Kouhia, R., Menken, C.M., Mikkola, M., Schreppers, G.-J., Computing and understanding interactive buckling (1994) Proceedings of the Fifth Finnish Mechanics Days, , in: R.A.E. Mäkinen, P. Neittaanmäki (Eds.), Jyväskylä; Lanzo, A.D., Garcea, G., Casciaro, R., Asymptotic postbuckling analysis of rectangular plates by hc finite elements (1995) Int. J. Numer. Meth. Engrg., 38, pp. 2325-2345; Lanzo, A.D., Garcea, G., Koiter's analysis of thin-walled structures by a finite element approach (1996) Int. J. Numer. Meth. Engrg., 39, pp. 3007-3031; Salerno, G., Lanzo, A.D., A nonlinear beam finite element for the post-buckling analysis of plane frames by Koiter's perturbation approach (1997) Comput. Meth. Appl. Mech. Engrg., 146, pp. 325-349; Salerno, G., Casciaro, R., Mode jumping and attractive paths in multimode elastic buckling (1997) Int. J. Numer. Meth. Engrg., 40, pp. 833-861; Menken, C.M., Schreppers, G.M.A., Groot, W.J., Petterson, R., Analyzing buckling mode interactions in elastic structures using an asymptotic approach (1997) Comp. Struct., 64, pp. 473-480; Vannucci, P., Cochelin, B., Damil, N., Potier-Ferry, M., An asymptotic-numerical method to compute bifurcating branches (1998) Int. J. Numer. Meth. Engrg., 41, pp. 1365-1389; Ho, D., The influence of imperfections on systems with coincident buckling loads (1972) Int. J. Non-linear Mech., 7, pp. 311-321; Koiter, W.T., (1976) Some Properties of Completely Symmetric Multilinear Forms with An Application to Ho's Theorem for Multi-mode Buckling, , Technical Report, Technological University of Delft, Laboratory Report; Pacoste, C., Co-rotational flat facet triangular elements for shell instability analyses (1998) Comput. Meth. Appl. Mech. Engrg., 156, pp. 75-110; Andersson, B., Falk, U., (1987) Self-adaptive Analysis of Three-dimensional Structures Using a P-version Finite Element Method, , Technical Report FFA TN 1987-31, The Aeronautical Research Institute of Sweden, Bromma; Zdunek, A., (1990) Application of the P-version FE Method to Geometrically Non-linear Problems in Three Dimensions, , Technical Report FFA TN 1990-33, The Aeronautical Research Institute of Sweden, Bromma; Huitfeldt, J., (1991) Nonlinear Eigenvalue Problems - Prediction of Bifurcation Points and Branch Switching, , Technical Report 17, Numerical Analysis Group, Chalmers University, Göteborg; Kouhia, R., Mikkola, M., Strategies for structural stability analysis (1995) Advances in Finite Element Technology, pp. 254-278. , N.E. Wiberg. Barcelona: CIMNE; Griewank, A., Reddien, G.W., Characterization and computation of generalized turning points (1984) SIAM. J. Numer. Anal., 21, pp. 176-185; Rabier, P.J., Reddien, G.W., Characterization and computation of singular points with maximum rank deficiency (1986) SIAM. J. Numer. Anal., 23, pp. 1040-1051; Dai, R.-X., (1988) Characterization and Computation of Foldsets for Parameter-dependent Equations, , Ph.D. Thesis, University of Pittsburgh; Dai, R.-X., Rheinboldt, W.C., On the computation of manifolds of foldpoints for parameter-dependent problems (1990) SIAM. J. Numer. Anal., 27, pp. 437-446; Govaerts, W., Sijnave, B., Rank-deficient matrices as a computational tool (1997) Numer. Linear Algebra Appl., 4, pp. 443-458; Jepson, A., Spence, A., Folds in solutions of two parameter systems and their calculation, Part I (1985) SIAM. J. Numer. Anal., 22, pp. 347-367; Wriggers, P., Wagner, W., Miehe, C., A quadratically convergent procedure for the calculation of stability points in finite element analysis (1988) Comput. Meth. Appl. Mech. Engrg., 70, pp. 329-347; Zdunek, A., (1996) A Sparse Matrix Algorithm for the Determination of Deflated Solutions, , Technical Report FFA TN 1995-34, The Aeronautical Research Institute of Sweden, Bromma; Wriggers, P., Simo, J.C., A general procedure for the direct computation of turning and bifurcation points (1990) Int. J. Numer. Meth. Engrg., 30, pp. 155-176; Keller, H.B., Numerical solution of bifurcation and nonlinear eigenvalue problems (1977) Applications of Bifurcation Theory, , P. Rabinowitz. New York: Academic Press; Decker, D.W., Keller, H.B., Path following near bifurcation (1981) Comm. Pure Appl. Mech., 34, pp. 149-175; Strang, G., (1980) Linear Algebra and Its Applications, 2nd Ed., , Academic Press, New York; Ruhe, A., Algorithms for the non-linear eigenvalue problem (1973) SIAM. J. Numer. Anal., 10, pp. 674-689; Ansell, A., (1996) Frequency Dependent Matrices for Dynamic Analysis of Structures, , Lic. Thesis, Department of Structural Engineering Royal Institute of Technology, Stockholm; Helnwein, P., Mang, H.A., Ab initio estimates of stability limits on nonlinear load-displacement paths: Potential and limitations (1998) In: Fourth World Congress on Computational Mechanics, , Abstracts, Santa Fe, IACM; Potier-Ferry, M., Buckling and post-buckling (1987) Lecture Notes in Physics, 288, pp. 205-223. , Springer, Berlin; Thompson, J.M.T., Hunt, G.W., (1984) Elastic Instability Phenomena, , Chichester: Wiley; Menken, C.M., Groot, W.J., Stallenberg, G.A.J., Interactive buckling of beams in bending (1991) Thin-Walled Struct., 12, pp. 415-434; Poston, T., Stewart, I.N., (1976) Catastrophe Theory and Its Applications, , London: Pitman; El Naschie, M.S., (1990) Stress, Stability and Chaos in Structural Engineering: An Energy Approach, , McGraw-Hill, London; Cochelin, B., A path following technique via an asymptotic-numerical method (1994) Comp. Struct., 53, pp. 1181-1192; Nour-Omid, B., Rankin, C.C., Finite rotation analysis and consistent linearization using projectors (1991) Comput. Meth. Appl. Mech. Engrg., 93, pp. 353-384; Cardona, A., Geradin, M., A beam finite element non-linear theory with finite rotations (1988) Int. J. Numer. Meth. Engrg., 26, pp. 2403-2438; Argyris, J., Tenek, L., Buckling of multilayered composite plates by natural shear deformation matrix theory (1994) Comput. Meth. Appl. Mech. Engrg., 111, pp. 37-59; Argyris, J., Tenek, L., Linear and geometrically nonlinear bending of isotropic and multilayered composite plates by the natural mode method (1994) Comput. Meth. Appl. Mech. Engrg., 13, pp. 207-251; Allman, D.J., Evaluation of the constant strain triangle with drilling rotations (1988) Int. J. Numer. Meth. Engrg., 26, pp. 2645-2655; Babuška, I., Elman, H.C., Performance of the version for the finite element method with various elements h-p (1993) Int. J. Numer. Meth. Engrg., 36, pp. 2503-2523; Surana, K.S., Gan, G.Y., Role of p-version Q(p) and Q′(p) two-dimensional approximation functions in smooth and singular problems (1996) Comp. Struct., 60, pp. 441-470; Zienkiewicz, O.C., (1977) The Finite Element Method, 3rd Ed., , McGraw-Hill, London; Szabo, B., Babuska, I., (1991) Finite Element Analysis, , Chichester: Wiley; Peano, A., Pasini, A., Riccioni, R., Sardella, L., Adaptive approximations in finite element structural analysis (1979) Comp. Struct., 10, pp. 333-342; Stein, M., (1959) The Phenomenon of Change of Buckling Patterns in Elastic Structures, , Technical Report R-39, NASA; Stein, M., (1959) Loads and Deformations of Buckled Rectangular Plates, , Technical Report R-40, NASA; Schaeffer, D., Golubitski, M., Boundary conditions and mode jumping in the buckling of a rectangular plate (1979) Commun. Math. Phys., 69, pp. 209-236; Suchy, H., Troger, H., Weiss, R., A numerical study of mode jumping of rectangular plates (1985) ZAMM, 65, pp. 71-78; Kouhia, R., Mikkola, M., Numerical treatment of multiple bifurcation points (1998) Computational Mechanics, New Trends and Applications, , in: S. Idelsohn, E. Oñate, E. Dvorkin (Eds.), CIMNE, Barcelona; Riks, E., Rankin, C.C., Brogan, F.A., On the solution of mode jumping phenomena in thin-walled shell structures (1996) Comput. Meth. Appl. Mech. Engrg., 136, pp. 59-92; Casciaro, R., Garcea, G., Attanasio, G., Giordano, F., Perturbation approach to elastic post-buckling analysis (1998) Comp. Struct., 66, pp. 585-595; Deml, M., Wunderlich, W., Direct evaluation of the worst imperfection shape in shell buckling (1997) Comput. Meth. Appl. Mech. Engrg., 149, pp. 201-222

NR 20140805Available from: 2013-03-18 Created: 2013-03-18 Last updated: 2013-03-18Bibliographically approved

Open Access in DiVA

No full text

Search in DiVA

By author/editor
Eriksson, Anders
By organisation
Structural Mechanics
In the same journal
Computer Methods in Applied Mechanics and Engineering
Applied Mechanics

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 120 hits
ReferencesLink to record
Permanent link

Direct link