Abstract
For the prebuckling range an extensive literature of effective solution techniques exists for the numerical solution of structural problems but only a few algorithms have been proposed to trace nonlinear response from the pre-limit into the post-limit range. Among these are the simple method of suppressing equilibrium iterations, the introduction of artificial springs, the displacement control method and the “constant-arc-length method” of Riks/Wempner. It is the purpose of this paper to review these methods and to discuss the modifications to a program that are necessary for their implementation. Selected numerical examples show that a modified Riks/Wempner method can be especially recommended.
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References
Riks, E.: The Application of Newton’ s Method to the Problem of Elastic Stability. J. Appl. Mech. 39 (1972) 1060–1066.
Riks, E.: Anlncremental Approach to the Solution of Snapping and Buckling Problems. Int. J. Solids Struct. 15 (1979) 529–551.
Wempner, G. A.: Discrete Approximations Related to Nonlinear Theories of Solids. Int. J. Solids Struct. 7 (1971) 1581–1599.
Bathe, K. -J., Ramm, E., Wilson, E. L.: Finite Element Formulations for Large Deformation Dynamic Analysis. Int. J. Num. Meth. Engng. 9 (1975) 353–386.
Bergan, P. G.: Solution Algorithms for Nonlinear Structural Problems. Int. Conf. on “Engng. Appl. of the F. E. Method”, HOvik, Norway 1979, published by A. S. Computas.
Wright, E. W., Gaylord, E. H.: Analysis of Unbraced Multi-Story SteelRigid Frames. Proc. ASCE, J. Struct. Div. 94 (1968) 1143–1163.
Sharifi, P., Popov, E. P.: Nonlinear Buckling Analysis of Sandwich Arches. Proc. ASCE, J. Engng. Div. 97 (1971) 1397–1412.
Ramm, E.: Geometrisch nichtlineare Elastostatik und finite Elemente. Habilitationsschrift, Universität Stuttgart, 1975.
Argyris, J. H.: Continua and Discontinua. Proc. 1st Conf. “Matrix Meth. Struct. Mech. ”, Wright-Patterson A. F. B., Ohio 1965, 11–189.
Pian, T. H. H., Tong, P.: Variational Formulation of Finite Displacement Analysis. IUTAMSymp. on “High Speed Computing of Elastic Structures”, Liège 1970, 43–63.
Zienkiewicz, O.C.: Incremental Displacement in Non- Linear Analysis. Int. J. Num. Meth. Engng. 3 (1971) 587–588.
Lock, A. C., Sabir, A. B.: Algorithm for Large Deflection Geometrically Nonlinear Plane and Curved Structures. In “Mathematics of Finite Elements and Applications” (ed. J. R. Whiteman ), Academic Press, N. Y. 1973, 483–494.
Haisler, W., Stricklin, J., Key, J.: Displacement IncrementationinNonlinear Structural Analysis by the Self-Correcting Methods. Int. J. Num. Meth. Engng. 11 (1977) 3–10.
Nemat-Nasser, S., Shatoff, H. D.: Numerical Analysis of Pre-and Postcritical Response of Elastic Continua at Finite Strains. Comp. Struct. 3 (1973) 983–999.
Batoz, J. -L., Dhatt, G.: Incremental Displacement Algorithms for Nonlinear Problems. Int. J. Num. Meth. Engng. 14 (1979) 1262–1267.
Wessels, M.: Das statische und dynamische Durchschlags-problem der imperfekten flachen Kugelschale bei elastischer rotationssymmetrischer Verformung. Dissertation, TU Hannover, 1977, Mitteil. Nr. 23 des Instituts für Statik.
Crisfield, M. A.: A Fast Incremental/Iterative Solution Procedure that Handles “Snap-Through”. Proc. Symp. on “Computational Methods in Nonlinear Structural and Solid Mech. ”, Washington, Oct. 1980.
Brendel, B., Häfner, L., Ramm, E., Sättele, J. M.: Programmdokumentation - Programmsystem NISA. Bericht, Institut für Baustatik, Universität Stuttgart, 1977.
Ramm, E.: A Plate/Shell Element for Large Deflections and Rotations. Symp. “Formulations and Computational Algorithms in F. E. Analysis”, Cambridge 1976, MIT Press 1977.
Sabir, A. B., Lock, A. C.: The Application of Finite Elements to the Large Deflection Geometrically Nonlinear Behaviour of Cylindrical Shells. In “Variational Methods in Engng;’ (ed. C. A. Brebbia and H. Tottenham), Southampton, University Press (1972) 7/66–7/75.
Brendel, B., Fischer, D., Ramm, E., Rammerstorfer, F.: Linear and Nonlinear Stability Analysis of Thin Cylindrical Shells under Windloads. To be published, J. Struct. Mech. 1981.
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Ramm, E. (1981). Strategies for Tracing the Nonlinear Response Near Limit Points. In: Wunderlich, W., Stein, E., Bathe, KJ. (eds) Nonlinear Finite Element Analysis in Structural Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81589-8_5
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DOI: https://doi.org/10.1007/978-3-642-81589-8_5
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