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Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: A finite element, linear programming approach

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Sommario

Si adotta la descrizione matriciale del comportamento meccanico fondata sulla discretizzazione dei continui in elementi finiti e sulla linearizzazione a tratti dei domini di plasticizzazione. Con l'impiego di concetti e metodi della programmazione lineare, si delineano in forma compatta i fondamenti di una teoria generale dei fenomeni di assestamento in campo elastico e le basi di metodi risolutivi.

Per leggi sforzi-deformazioni associate il teorema di Koiter viene generalizzato ai casi di distorsioni variabili (per es. escursioni termiche). Per leggi non associate si forniscono due teoremi che consentono di delimitare bilateralmente il coefficiente di sicurezza.

Summary

The matrix description of the mechanical behaviour resting on finite element discretization and piecewise linearization of yield surfaces, is adopted instead of the traditional continuous field description. By the use of linear programming concepts, the essential of a general shakedown theory and the basis of relevant solution procedures are presented in compact form. For systems with associated flow-laws, the second shakedown theorem (Koiter's) is extended in order to allow for variable dislocations (e. g. temperature cycles). For systems with nonassociated flow-laws two theorems are given which supply lower and upper bounds to the safety factor.

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References

  1. F. G. Hodge,Plastic analysis of structures, McGraw-Hill Book Comp., Inc., New York, 1959.

    Google Scholar 

  2. B. G. Neal,The plastic methods of structural analysis, Chapman & Hall Ltd., London, 1963.

    Google Scholar 

  3. W. T. Koiter,General theorems for elastic-plastic solids, “Progr. in Solid Mech.”, North-Holland Pub., Amsterdam, 1960.

    Google Scholar 

  4. V. Franciosi, Scienza delle Costruzioni, Vol. IV,Calcolo a rottura, Liguori, Napoli, 1964.

    Google Scholar 

  5. D. C. Drucker,Coulomb friction, plasticity, and limit loads, “J. Appl. Mech.”, Vol. 21, no. 1, 1954.

  6. D. C. Drucker,Concepts of path indepenence and material stability for soils, “Proc. I.U.T.A.M. Symp. on Rheology and Soil Mechanics, Grenoble, 1964”, Springer, Berlin, 1966.

    Google Scholar 

  7. D. Radenkovic,Theorèmes limites pour un matériau de Coulomb à dilatation non standardisée, Note Ac. Sciences Paris, Meeting of June 12th, 1961.

  8. D. Radenkovic,Theorèmes des charges limites, extension à la mécanique des sols, “Seminaires de Plasticité Ec. Polytech.”, Pub. Sc. et Tech. Minist. Air. n. N. T. 116, 1961.

  9. G. de Josselin de Jone,Lower bound collapse theorem and lack of normality of strain rate to yield surface for soils, “Proc. I.U.T.A.M. Symp. on Rheology and Soil Mechanics, Grenoble, 1964”, Springer, Berlin, 1966.

    Google Scholar 

  10. A. C. Palmer,A limit theorem for materials with non-associated flow laws, “J. de Mecanique”, Vol. V, no. 2, 1966.

  11. G. Sacchi andM. Save,A note on the limit loads of non-standard materials, “Meccanica”, Vol. 3, no. 1, 1968.

  12. W. S. Dorn andH. J. Greenberg,Linear programming and plastic limit analysis of structures, “Techn. Rep. no. 7”, Dept. of Math., Carnegie Inst. of Technology, 1955.

  13. A. Charnes andC. E. Lemke andO. C. Zienkiewicz,Virtual work, linear programming and plastic limit analysis, “Proc. Roy. Soc.”, A, v. 251, 1959.

  14. W. Prager,Lineare Ungleichungen in der Baustatik, “Schweizerische Bauzeitung”, no. 19, May, 1962.

  15. G. Ceradini andC. Gavarini,Calcolo a rottura e programmazione lineare, “Giornale del Genio Civile”, no. 1, 1965.

  16. C. Gavarini,Plastic analysis of structures and duality in linear programming, “Mcccanica”, Vol. I, no. 3–4, 1966.

  17. D. C. A. Koopman andR. H. Lance,On linear programming and plastic limit analysis, “J. Mechs. Phys. Solids”, v. 13, 1965.

  18. R. H. Lance,Duality in the finite-difference method of plastic limit analysis, Dpt. Theor. Appl. Mechs., Cornell Un., Tech. Report NSF GK 687/2, Apr. 1967.

  19. J. L. Tocher andE. P. Popov,Incremental collapse analysis of rigid frames, “Proc. of the Fourth U.S. Nat. Congr. of Appl. Mech.”, 1964.

  20. O. C. Zienkiewicz,The finite element method in structural and continuum mechanics, McGraw-Hill, New York, 1967.

    Google Scholar 

  21. J. S. Przemieniecki,Theory of matrix structural analysis, McGraw-Hill Comp., New York, 1968.

    Google Scholar 

  22. J. F. Besseling,Matrix analysis of creep and plasticity problems, in “Matrix Methods in Structural Mechanics”, Proc. Conference Wright-Patterson AFB, Ohio, 1965.

  23. H. Argyris,Elastoplastic matrix displacement analysis of three dimensional continua, “Jl. R. aeronaut. Soc.”, 69, 1965.

  24. P. V. Marcal andI. P. King,Elastic-plastic analysis of two dimensional stress systems by the finite element method, “Int. J. mech. Sci.”, no. 9, 1967.

  25. O. C. Zienkiewicz andS. Valliappan andI. P. King,Elasto-plastic solutions of engineering problems “initial stress”, finite element approach, “Int. J. Numer. Meth. in Engin.”, Vol. I, no. 1, 1969.

  26. J. F. Besseling,The complete analogy between the matrix equations and the continuous field equations of structural analysis, “Internal Symp. Analogue and Digital Techn. Appl. to Aeron.”, Liege, 1963.

  27. B. Fraeis de Veubeke,Displacement and equilibrium models in the finite element method, in “Stress Analysis”, O. C. Zienkiewicz and G. S. Holister Eds., John Wiley & Sons, Ltd., 1965.

  28. W. T. Koiter,A new general theorem on shake-down of elastic-plastic structures, “Proc. Kon. Nederl. Akad. Wet.”, B 59, 24, 1956.

    Google Scholar 

  29. W. Prager,Shakedown in elastic, plastic media subjected to cycles of load and temperature, “Symp. sulla Plast. nella Scienza delle Costr.”, Varenna, N. Zanichelli Ed., 1956.

  30. G. Hadley,Linear programming, Addison-Wesley, 1962.

  31. H.P. Künzi andW. Krelle,Nonlinear programming, Blaisdell Publ. Comp., Waltham, Mass., 1966.

    Google Scholar 

  32. H. P. Künzi andH. G. Tzschach andC. A. Zehnder,Numerische Methoden der mathematischen Optimierung, B. G. Teubner, Stuttgart, 1967.

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Authors and Affiliations

  1. Istituto di Scienza e Tecnica delle Costruzioni, Politecnico di Milanò, Italy

    Giulio Maier

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  1. Giulio Maier
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Additional information

The results presented in the paper form part of a research supported by the National (Italian) Research Council; they have been partially presented in a lecture at the 1969 annual meeting of the AIMETA, Florence, May 31, 1969.

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Maier, G. Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: A finite element, linear programming approach. Meccanica 4, 250–260 (1969). https://doi.org/10.1007/BF02133439

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