Mathematical Programming Methods for the Evaluation of Dynamic Plastic Deformations

  • G. Borino
  • S. Caddemi
  • C. Polizzotto
Part of the International Centre for Mechanical Sciences book series (CISM, volume 299)


Dynamic plastic deformation can be evaluated with two accuracy levels, nemely either by a full analysis making use of a step-by-step procedure, or by a simplified analysis making use of a bounding technique. Both procedures can be achieved by means a unified mathematical programming approach here presented. It is shown that for a full analysis both the direct and indirect methods of linear dynamics coupled with mathematical programming methods can be successfully applied, whereas for a simplified analysis a convergent bounding principle, holding both below and above the shakedown limit, can be utilized to produce an efficient linear programming-based algorithm.


Plastic Strain Mathematical Programming Problem Perturbation Vector Plastic Strain Increment Mathematical Programming Method 


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  1. 1.
    Zienkiewicz, O.C.: The finite element method in engineering science, McGraw-Hill, 1971.Google Scholar
  2. 2.
    Owen, D.R.J., Hinton, E.: Finite elements in plasticity, Pineridge Press Ltd., Swansea, U.K., 1980.MATHGoogle Scholar
  3. 3.
    Owen, D.R.J.: Implicit finite element methods for the dynamic transient analysis of solids with particular reference to nonlinear situations, Advanced Structural Analysis, ed. by J. Donea, Applied Science Publs., London, 1980, pp. 123–152.Google Scholar
  4. 4.
    Hughes, T.J.R.: Analysis of transient algorithms with particular reference to stability behaviour, Computational Methods for Transient Analysis, ed. by T. Belytschko and T.J.R. Hughes, North-Holland, Amsterdam, 1983, chap. 2.Google Scholar
  5. 5.
    Polizzotto, C.: Elastoplastic analysis method for dynamic agencies, Jour. Eng. Mech., Div. ASCE, March 1986, pp. 293–310.Google Scholar
  6. 6.
    Polizzotto, C.: A solution method for elastoplastic structures under dynamic agencies, Numerical Methods in Engineering: Theory and applications, ed. by J. Middleton and G.N. Pande, A.A. Balkema, Rotterdam, 1985, pp. 243–251.Google Scholar
  7. 7.
    Polizzotto, C., Borino, G.: Mathematical programming formulation of dynamic elastoplasticity analysis problems, Proc. Int. Conf. on Computational Mechanics, Tokyo, May 25–29, 1986, pp. VI/105–110.Google Scholar
  8. 8.
    Borino, G., Polizzotto, C.: Time integration algorithms and quadratic programming in the dynamic analysis of elastoplastic structures, Proc. 8-th Congress of the Associazione Italiana di Meccanica Teorica ed Applicata, AIMETA, Torino, Sept. 29 -Oct. 3, 1986, Vol. 2, pp. 731–736.Google Scholar
  9. 9.
    Borino, G., Polizzotto, C.:’ A mathematical programming approach to dynamic elastoplastic structural analysis, Meccanica (to appear).Google Scholar
  10. 10.
    Polizzotto, C.: A bounding technique for dynamic plastic deformations of damped structures, Nuclear Engineering and Design, June 1984, Vol. 79, No. 3, pp. 363–376.CrossRefGoogle Scholar
  11. 11.
    Polizzotto, C.: A quadratic programming approach to dynamic elastoplasticity, Trans. Int. Conf. on Structural Mechanics in Reactor Technology, SMIRT-8, Brussels, August 19–23, 1985, paper 84/5.Google Scholar
  12. 12.
    Muscolino, G., Polizzotto, C.: Un approccio alla dinamica elastoplastica basato sulla programmazione matematica, (in italian), VII Nat. Congress of the Associazione Italiana di Meccanica Teorica ed Applicata, AIMETA, Trieste, Oct. 1984, Vol. 5, pp. 181–192.Google Scholar
  13. 13.
    Di Paola, M., Polizzotto, C.: Metodo per la determinazione delle deformazioni plastiche in una struttura soggetta ad azioni sismiche, (in Italian), Proc. II Nat. Congress “L’Ingegneria Sismica in Italia”, Rapallo, June 6–9, 1984, pp. 7/143–161.Google Scholar
  14. 14.
    Capurso, M., Maier, G.: Incremental elastoplastic analysis and quadratic optimization, Meccanica, Vol. 5, No. 2, June 1970, pp. 107–116.CrossRefMATHGoogle Scholar
  15. 15.
    Maier, G.: Mathematical programming methods in structural analysis, Proc. Int. Conf. on Variational Methods in Engineering, Vol. II, 8, Southampton Univ. press, August 1978.Google Scholar
  16. 16.
    Cohn, M.Z., Maier, G., (eds.): Engineering plasticity and mathematical programming, Pergamon Press, New York, 1979.Google Scholar
  17. 17.
    De Donato, O., Maier, G.: Historical deformations analysis of elastoplastic structures as a parametric linear complementarity, Meccanica, Vol. 11, No. 3, 1976, pp. 166–171.CrossRefMATHGoogle Scholar
  18. 18.
    Hodge, P.G., Belytschko, T., Merakovich, C.T.: Quadratic programming and plasticity, ASME, Computational Approaches in Applied Mechanics, ed. by E. Sevin, New York, 1969, p. 73.Google Scholar
  19. 19.
    Sayegh, A.F., Rubistein, N.F.: Elastic-plastic analysis by quadratic programming, Journal of the Engineering Mechanics Division, ASCE, No. EM6, Dec. 1972, pp. 1547–1572.Google Scholar
  20. 20.
    Colonnetti, G.: L’equilibre des corps deformables, Dunod, Paris, 1955.Google Scholar
  21. 21.
    Polizzotto, C.: Dynamic shakedown by modal analysis, Meccanica, Vol. 17, No. 2, 1984, pp. 133–144.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Capurso, M., Corradi, L., Maier, G.: Bounds on deformations and displacements in shakedown theory, in Proc. of the Seminary on Materials and Structures Under Cyclic Loads, Laboratoire de Mécanique des Solides, Ecole Polytechnique, Palaiseau, France, Sept. 28–29, 1978, pp. 231–144.Google Scholar
  23. 23.
    König, J.A., Maier, G.: Shakedown analysis of elastoplastic structures: A review of recent developments, Nucl. Engng. Des., 66, 1981, pp. 81–95.Google Scholar
  24. 24.
    Ponter, A.R.S.: General displacements and work bounds for dynamically loaded bodies, J. Mech. Phys. Solids, 23, 1975, pp. 157–163.MathSciNetGoogle Scholar
  25. 25.
    Polizzotto, C.: A unified treatment of shakedown theory and related bounding techniques, S.M. Archives, 7, 1982, pp. 19–75.MATHGoogle Scholar
  26. 26.
    Polizzotto, C.: Bounding principles for elastic-plastic-creeping solids loaded below and above the shakedown limits, Meccanica, 17, 1982, pp. 143–148.CrossRefMATHGoogle Scholar
  27. 27.
    Polizzotto, C.: Deformation bounds for elastic-plastic solids within and out of the creep range, Nuclear Engng. Design, Vol. 83, 1984, pp. 293–301.Google Scholar
  28. 28.
    Polizzotto, C.: On shakedown of structures under dynamic agencies, in Polizzotto C. and Sawczuk A., (eds.), Inelastic Structures under Variable Loads, Cogras, Palermo, 1984, Proc. Euromech Colloquium, Palermo, 1983.Google Scholar
  29. 29.
    Polizzotto, C.: A convergent bounding principle for a class of elastoplastic strain-hardening solids, J. of Plasticity, Vol. 2, 1986, pp. 359–370.CrossRefMATHGoogle Scholar
  30. 30.
    Borino, G., Caddemi, S., Polizzotto, C.: A linear programming method for bounding plastic deformations, International Conference on Computational Engineering Science, ICES ‘88, Atlanta, Georgia, U.S.A., April 10–14, 1988.Google Scholar
  31. 31.
    Corradi, L.: On compatible finite element methods for elastic-plastic analysis, Meccanica, Vol. 13, 1978, p. 133.CrossRefMATHGoogle Scholar
  32. 32.
    Corradi, L.: A•displacement formulation for finite element elastoplastic problems, Meccanica, Vol. 18, 1983, pp. 77–91.CrossRefMATHGoogle Scholar
  33. 33.
    Corradi, L., Maier, G.: Finite element elasto-plastic and limit analysis, in Nonlinear F.E. Analysis in Structural Mechanics, ed. by Winderlich, E. Stein, K.J. Bathe, Springer-Verlag, 1981, p. 290.CrossRefGoogle Scholar
  34. 34.
    Panzeca, T., Polizzotto, C.: A finite element model for dynamic elastoplastic structural analysis, in Computational Plasticity, ed. by D.R.J. Owen, E. Hinton, E. Onate, Swansea, Pineridge Press, 1987, pp. 1247–1262.Google Scholar
  35. 35.
    Bathe, K.J.: Computational methods in structural dynamics, in R.F Hartung (ed.), Computing in Applied Mechanics, The Am. Society of Mechanical Engineers, AMD-18, 1976, pp. 163–176.Google Scholar
  36. 36.
    Bathe, K.J., Wilson, E.L.: Numerical methods in finite element analysis, Englewood Cliffs, N.J., Prentice-Hall, 1976.MATHGoogle Scholar
  37. 37.
    Belytschko, T.: Explicit time integration of structure–Mechanical Systems, in J. Donea (ed.), Advanced Structural Dynamics, Appl. Science Pubs., London, 1980, pp. 97–122.Google Scholar
  38. 38.
    Zienkiewicz, O.C., Wood, W.L., Hine, N.W., Taylor, R.L.: A unified set of single step algorithms, Part 1, Int. J. Num. Meth. Engng., Vol. 20, 1984, pp. 1529–1552.MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Wood, W.L.: A unified set of single step algorithms, Part 2, Int. J. Num. Meth. Engng., Vol. 20, 1984, pp. 2303–2309.CrossRefMATHGoogle Scholar
  40. 40.
    Kuhn, H.W., Tucker, A.W.: Linear inequalities and related systems, Ann. Math. Stat., 1956.Google Scholar
  41. 41.
    De Donato, 0., Franchi, A.: Elastic-plastic analysis by finite elements, Engineering Plasticity and Mathematical Programming, ed. by Cohn M.Z. and Maier G., Pergamon Press, New York, 1979, pp. 413–432.Google Scholar
  42. 42.
    Feijòo, R.A.: Variational methods in the theory of plasticity, Seminar held at the Dept. of Structural and Geotechnical Engineering, University of Palermo, Oct. 1986.Google Scholar
  43. 43.
    Feijòo, R.A., Zouain, N.: Variational formulation for rates and increments in plasticity, Proc. Int. Conf. on Computational Plasticity, Barcellona, 1987, pp. 33–57.Google Scholar
  44. 44.
    Borino, G., Caddemi, S., Polizzotto, C.: Mathematical programming methods for evaluating dynamic deformations of elasto-plastic structures, Proc. Int. Conf. on Computational Plasticity, Barcellona, 1987, pp. 1231–1245.Google Scholar

Copyright information

© Springer-Verlag Wien 1990

Authors and Affiliations

  • G. Borino
    • 1
  • S. Caddemi
    • 1
  • C. Polizzotto
    • 1
  1. 1.University of PalermoPalermoItaly

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