Application of Shakedown Theory and Numerical Methods

  • Dieter Weichert
  • Abdelkader Hachemi
Part of the International Centre for Mechanical Sciences book series (CISM, volume 432)


In this lecture, the methodological framework for the numerical application of shakedown analysis is developed. Examples of applications such as the analysis and optimisation of composite materials and the analysis of various plates and shells problems are presented.


Residual Stress Cylindrical Shell Internal Pressure Representative Volume Element Circular Plate 
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  1. Bathe, K.J., Wilson, E.L. and Iding, R.H. (1974). NONSAP: A Structural Analysis Program for Static and Dynamic Response of Nonlinear Systems. Report No. UGSESM 74–3, University of California, Berkeley.Google Scholar
  2. Boulbibane, M. (1995). Application de la Théorie d’Adaptation aux Milieux Elastoplastiques Non-Standards: Cas des Géomatériaux. Ph. D. Dissertation, University of Lille.Google Scholar
  3. Conn, A.R., Gould, N.I.M. and Toint, Ph.L. (1992). LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization (Release A). Berlin: Springer-Verlag.CrossRefMATHGoogle Scholar
  4. Dhatt, G., Touzot, G. and Cantin, G. (1984). The Finite Element Method Displayed. Chichester: John Wiley & Sons.MATHGoogle Scholar
  5. Gallagher, R.H. and Dhalla, A.K. (1975). Direct Flexibility Finite Element Elastoplastic Analysis. New Jersey: Englewood Cliffs.Google Scholar
  6. Giese, H. (1988). On the Application of Shakedown-Theory in Soil Mechanics. IfM-Report, Ruhr-Universität, Bochum.Google Scholar
  7. Gokhfeld, D.A. and Cherniaysky, O.F. (1980). Limit Analysis of Structures at Thermal Cycling. Leyden: Sijthoff and Noordhoff.Google Scholar
  8. Gross-Weege, J. (1988). Zum Einspielverhalten von Flächentragwerken. IfM-Report, No. 58, Ruhr-Universität, Bochum.Google Scholar
  9. Gross-Weege, J. (1990). A unified formulation of statical shakedown criteria for geometrically nonlinear problems. Int. J. Plasticity 6: 433–447.CrossRefGoogle Scholar
  10. Gross-Weege, J. and Weichert, D. (1992). Elastic-plastic shells under variable mechanical and thermal loads. Int. J. Mech. Sci. 34: 863–880.CrossRefMATHGoogle Scholar
  11. Gross-Weege, J. (1997). On the numerical assessment of the safety factor of elastic-plastic structures under variable loading. Int. J. Mech. Sci. 39: 417–433.CrossRefMATHGoogle Scholar
  12. Hachemi, A. (1994). Contribution à l’Analyse de l’Adaptation des Structures Inélastiques avec Prise en Compte de l’Endommagement. Ph. D. Dissertation, University of Lille.Google Scholar
  13. Hachemi, A. and Weichert, D. (1998). Numerical shakedown analysis of damaged structures. Comput. Methods Appl. Mech. Engrg. 160: 57–70.CrossRefMATHGoogle Scholar
  14. Hill, R. (1963). Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11: 357–372.CrossRefMATHGoogle Scholar
  15. Ilyuschin, A. A. (1956). Plasticity. Paris: Eyrolles.Google Scholar
  16. König, J.A. (1969). Shakedown theory of plates. Arch. Mech. Stos. 5: 623–637.Google Scholar
  17. Kreja, I., Schmidt, R., Teyeb, O. and Weichert, D. (1993). Plastic ductile damage finite element analysis of structures. Z. Angew. Math. Mech. 73: T378 - T381.MATHGoogle Scholar
  18. Lemaitre, J. (1985) A continuous damage mechanics model for ductile fracture. J. Engng. Mat. Tech. 107: 83–89.CrossRefGoogle Scholar
  19. Mandel, J. (1976). Adaptation d’une structure plastique écrouissable et approximations, Mech. Res. Comm. 3: 483–488.CrossRefMATHGoogle Scholar
  20. Morelle, P. and Nguyen Dang Hung (1983). Etude numérique de l’adaptation plastique des plaques et coques de révolution par les éléments finis d’équilibre. J. Méc. Théo. Appl. 2: 567–599.MATHGoogle Scholar
  21. Morelle, P. (1989). Analyse Duale de l’Adaptation Plastique des Structures par la Méthode des Eléments Finis et la Programmation Mathématique. Ph. D. Dissertation, University of Liège.Google Scholar
  22. Nguyen Dang Hung and König, J.A. (1976). A finite element formulation for shakedown problems using yield criterion of the mean. Comput. Methods Appl. Mech. Engrg. 8: 179–192.CrossRefMATHGoogle Scholar
  23. Pierre, D.A. and Lowe, M.J. (1975). Mathematical Programming via Augmented Lagrangians. London: Addison-Wesley.MATHGoogle Scholar
  24. Schwabe, F. (2000). Einspieluntersuchungen von Verbundwerkstoffen mit periodischer Mikrostruktur. Ph. D. Dissertation, RWTH Aachen.Google Scholar
  25. Shichun, W. and Hua, L. (1990). A kinetic equation for ductile damage at large plastique strain. J. Mat. Proc. Tech. 21: 295–302.CrossRefGoogle Scholar
  26. Stein, E., Zhang, G. and Huang, Y. (1993). Modeling and computation of shakedown problems for nonlinear hardening materials, Comput. Methods Appl. Mech. Engrg. 103: 247–272.MathSciNetCrossRefMATHGoogle Scholar
  27. Stiefel, E. (1960). Note on Jordan elimination, linear programming and Tchebycheff approximation. Numer. Math. 2: 1–17.MathSciNetCrossRefMATHGoogle Scholar
  28. Suquet, P. (1983). Analyse limite et homogénéisation. C.R. Acad. Sci. 296: 1355–1358.MathSciNetMATHGoogle Scholar
  29. Tritsch, J. B. (1993). Analyse d’Adaptation des Structures Elasto-plastiques avec Prise en Compte des Effets Géométriques. Ph. D. Dissertation, University of Lille.Google Scholar
  30. Tritsch, J. B. and Weichert, D. (1995). Case studies on the influence of geometric effects on the shakedown of structures. In Mroz, Z., Weichert, D. and Dorosz, S., eds. Inelastic Behaviour of Structures under Variable Loads. Amsterdam: Kluwer Academic Publishers. 309–320.CrossRefGoogle Scholar
  31. Weichert, D. and Gross-Weege, J. (1988). The numerical assessment of elastic-plastic sheets under variable mechanical and thermal loads using a simplified two-surface yield condition. Int. J. Mech. Sci. 30: 757–767.CrossRefMATHGoogle Scholar
  32. Weichert, D. and Hachemi, A. (1998). Influence of geometrical nonlinearities on the shakedown of damaged structures. Int. J. Plasticity 14: 891–907.CrossRefMATHGoogle Scholar
  33. Weichert, D., Hachemi, A. and Schwabe, F. (1999a). Shakedown analysis of composites. Mech. Res. Comm. 26: 309–318.CrossRefMATHGoogle Scholar
  34. Weichert, D., Hachemi, A. and Schwabe, F. (1999b). Application of shakedown theory to the plastic design of composites. Arch. Appl. Mech. 69: 623–633.CrossRefMATHGoogle Scholar
  35. Zienkiewicz, O.C. (1984). Methode derfiniten Elemente. München: Carl Hamer Verlag.Google Scholar
  36. Zahl, D.B. and Schmauder, S. (1994). Transverse strength of continuous fiber metal matrix composites. Comput. Mat. Sci. 3: 293–299.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • Dieter Weichert
    • 1
  • Abdelkader Hachemi
    • 1
  1. 1.Aachen University of TechnologyAachenGermany

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