On the derivation and possibilities of the secant stiffness matrix for nonlinear finite element analysis. (English) Zbl 0826.73059

Summary: The general non symmetric parametric form of the incremental secant stiffness matrix for nonlinear analysis of solids using the finite element method is derived. A convenient symmetric expression for a particular value of the parameters is obtained. The geometrically nonlinear formulation is based on a generalized Lagrangian approach. Detailed expressions of all the relevant matrices involved in the analysis of three-dimensional solids are obtained. The possibilities of application of the secant stiffness matrix for nonlinear structural problems including stability, bifurcation and limit load analysis are also discussed. Examples of application are given for the nonlinear analysis of pin joined frames.


74S05 Finite element methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G60 Bifurcation and buckling
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[1] Badawi, M.; Cusens, A. R. 1992: Symmetry of the stiffness matrices for geometrically non linear analysis, Communications in Appl. Num. Meth., 8: 135-140 · Zbl 0754.73083
[2] 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 · Zbl 0304.73060
[3] Bathe, K. J. 1982: Finite element procedures in non linear analysis. Prentice Hall. · Zbl 0528.65053
[4] 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
[5] 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
[6] Felippa, C. 1974: Discussions on the paper by Rajasekaran and Murray (1973). J. Struct. Div., ASCE, 100: 2521-2523
[7] 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
[8] 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
[9] 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
[10] 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 · Zbl 0411.73065
[11] 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 · Zbl 0456.73072
[12] Horrigmoe, G. 1970: Non linear finite element models in solid mechanics. Report 76-2, Norwegian Inst. Tech., Univ Trøndheim
[13] 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 · Zbl 0552.73072
[14] Kroplin, B.; Dinkler, D. 1988: A material law for coupled load yielding and geometric inestability, Engineering Comput. Vol. 5
[15] 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
[16] Kroplin, B.; Wilhelm, M.; Herrmann, M. 1991: Unstable phenomena in sheet metal forming processes and their simulation, VDI. Berichte NR. 894: 137-52
[17] 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
[18] 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
[19] Mallet, R.; Marcal, P. 1968: Finite element analysis of non linear structures, J. Struct. Div., ASCE, 14: 2081-2105
[20] Malvern, L. E. 1969: Introduction to the mechanics of a continuum medium, Prentice Hall · Zbl 0181.53303
[21] 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 · Zbl 0353.73065
[22] 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
[23] 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
[24] 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
[25] Oñate, E. and Matias, W. T. 1995: A critical displacement approach for structural instability analysis, Research Report No-60, CIMNE, Barcelona · Zbl 0883.73033
[26] Pignataro, M.; Rizzi, N.; Luongo, A. 1991: Stability, Bifurcation and Postcritical Behaviour of Structures, Elsevier
[27] Rajasekaran, S.; Murray, D. W. 1973: On incremental finite element matrices, J. Struct. Div., ASCE, 99: 7423-7438
[28] Wood, R. D.; Schrefler, B. 1978: Geometrically non linear analysis-A correlation of finite element notations, Int. J. Num. Meth. Engng., 12: 635-642 · Zbl 0372.73002
[29] 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
[30] Zienkiewicz, O. C.; Taylor, R. L. 1989: The finite element method, Vol. I, McGraw-Hill · Zbl 0991.74002
[31] Zienkiewicz, O. C.; Taylor, R. L. 1991: The finite element method, Vol. II, McGraw-Hill · Zbl 0991.74002
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