Abstract
In this paper the general non symmetric parametric form of the incremental secant stiffness matrix for non linear analysis of solids using the finite element metod is derived. A convenient symmetric expression for a particular value of the parameters is obtained. The geometrically non linear formulation is based on a Generalized Lagrangian approach. Detailed expressions of all the relevant matrices involved in the analysis of 3D solids are obtained. The possibilities of application of the secant stiffness matrix for non linear structural problems including stability, bifurcation and limit load analysis are also discussed. Examples of application are given for the non linear analysis of pin joined frames.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Badawi, M.; Cusens, A. R. 1992: Symmetry of the stiffness matrices for geometrically non linear analysis, Communications in Appl. Num. Meth., 8: 135–140
Bathe, K. J.; Ramm, E.; Wilson, E. L. 1975: Inestability analysis of free form shells by finite elements. Int. J. Num. Meth. Engng., 9(2): 353–386
Bathe, K. J. 1982: Finite element procedures in non linear analysis. Prentice Hall.
Carrera, E. 1992: Sull'uso dell'operatore secant in analisi non lineare distrutture multistrato con il methodo degli elementi finiti in ATTI, XI Congresso Nazionale AIMETA, Trento, 28 Sept–2 Oct
Duddeck, H.; Kroplin, B.; Dinkler, D.; Hillmann, J.; Wagenhuber, W. 1989: Non linear computations in Civil Engineering Structures (in German), DFG Colloquium, 2–3 March 1989, Springer-Verlag, Berlin
Felippa, C. 1974: Discussions on the paper by Rajasekaran and Murray (1973). J. Struct. Div., ASCE, 100: 2521–2523
Felippa, C.; Crivelli, L. A. 1991: The core congruential formulation of geometrically non linear finite elements. in Non Linear Computational Mechanics. The State of the Art, P.Wriggers and W.Wagner (eds.), Springer-Verlag, Berlin
Felippa, C.; Crivelli, L. A.; Haugen, B. 1994: A survey of the core-congruental formulation for geometrically non linear TL finite element. Archives of Comp. Meth. in Eng. 1(1): 1–48
Frey, F. 1978: L'analyse statique non lineaire des structures per le methode des elements finis et son application à la construction matallique. Ph.D. thesis, Univ. of Liege
Frey, F.; Cescotto, S. 1978: Some new aspects of the incremental total lagrangian description in non linear analysis in “Finite Element in non linear Mehanics”, edited by P. Bergan et al., Tapir Publishers, Univ. of Trøndheim
Geradin, M.; Idelsohn, S.; Hodge, M. 1981: Computational strategies for the solution of large non-linear problems via quasi-Newton methods, Comp. Struct., 13: 73–81
Horrigmoe, G. 1970: Non linear finite element models in solid mechanics. Report 76-2, Norwegian Inst. Tech., Univ Trøndheim
Kroplin, B.; Dinkler, D.; Hillmann, J. 1985: An energy perturbation method applied to non linear structural analysis, Comp. Meth. Appl. Mech. Engng. 52: 885–897
Kroplin, B.; Dinkler, D. 1988: A material law for coupled load yielding and geometric inestability, Engineering Comput. Vol. 5
Kroplin, B.; Dinkler, D. 1990: Some thoughts on consistent tangent operators in plasticity, in Computatinal Plasticity D. R. J. Owen, E. Hinton and E. Oñate (eds.), Pineridge Press/CIMNE
Kroplin, B.; Wilhelm, M.; Herrmann, M. 1991: Unstable phenomena in sheet metal forming processes and their simulation, VDI. Berichte NR. 894: 137–52
Kroplin, B. 1992: Instability prediction by energy perturbation, in Numerical Methods in Applied Sciences and Engineering, H.Alder, J. C.Heinrich, S.Lavanchy, E.Oñate and B.Suarez (eds.) CIMNE, Barcelona
Larsen, P. K. 1971: Large displacement analysis of shells of revolution including creep, plasticity and viscoplasticity, report UC SEEm 71-22, Univ. of California Berkeley
Mallet, R.; Marcal, P. 1968: Finite element analysis of non linear structures, J. Struct. Div., ASCE, 14: 2081–2105
Malvern, L. E. 1969: Introduction to the mechanics of a continuum medium, Prentice Hall
Mondkar, D. P.; Powell, G. H. 1977: Finite element analysis of non linear static and dynamic response, Int. J. Num. Meth. Engng., 11(2): 499–520
Oñate, E.; Oliver, J.; Miquel, J.; Suárez, B. 1986: Finite element formulation for geometrically non linear problems using a secant matrix, in Computational Plasticity' 86, S. Atluri and G. Yagawa (eds.), Springer-Verlag
Oñate, E. 1991: Possibilities of the secant stiffness matrix for non linear finite element analysis, in Non linear Engineering Computation, N. Bicanic et al. (eds), Pineridge Press
Oñate, E. 1994: Derivation of the secant stiffness matrix for non linear finite element analysis of solids and trusses, Research Report No 49, CIMNE, Barcelona
Oñate, E. and Matias, W. T. 1995: A critical displacement approach for structural instability analysis, Research Report No-60, CIMNE, Barcelona
Pignataro, M.; Rizzi, N.; Luongo, A. 1991: Stability, Bifurcation and Postcritical Behaviour of Structures, Elsevier
Rajasekaran, S.; Murray, D. W. 1973: On incremental finite element matrices, J. Struct. Div., ASCE, 99: 7423–7438
Wood, R. D.; Schrefler, B. 1978: Geometrically non linear analysis-A correlation of finite element notations, Int. J. Num. Meth. Engng., 12: 635–642
Yaghmai, S. 1968: Incremental analysis of large deformations in mechanics of solids with applications to axisymmetric shells of revolution, Report No-SESM 68-17, Univ. of California, Berkeley
Zienkiewicz, O. C.; Taylor, R. L. 1989: The finite element method, Vol. I, McGraw-Hill
Zienkiewicz, O. C.; Taylor, R. L. 1991: The finite element method, Vol. II, McGraw-Hill
Author information
Authors and Affiliations
Additional information
Communitated by S. N. Atluri, 20 October 1994
Rights and permissions
About this article
Cite this article
Oñate, E. On the derivation and possibilities of the secant stiffness matrix for non linear finite element analysis. Computational Mechanics 15, 572–593 (1995). https://doi.org/10.1007/BF00350269
Issue Date:
DOI: https://doi.org/10.1007/BF00350269