Shakedown of Structures Subjected to Dynamic External Actions and Related Bounding Techniques

  • Castrenze Polizzotto
  • Guido Borino
  • Paolo Fuschi
Part of the International Centre for Mechanical Sciences book series (CISM, volume 432)


The shakedown theory for dynamic external actions is expounded considering elastic-plastic internal-variable material models endowed with hardening saturation surface and assuming small displacements and strains as long with negligible effects of temperature variations on material data. Two sorts of dynamic shakedown theories are presented, i.e.: i) Unrestricted dynamic shakedown, in which the structure is subjected to (unknown) sequences of short-duration excitations belonging to a known excitation domain, with no-load no-motion time periods in between and for which a unified framework with quasi-static shakedown is presented; and ii) Restricted dynamic shakedown, in which the structure is subjected to a specified infinite-duration load history. Two general bounding principles are also presented, one is non evolutive in nature and holds for repeated loads below the shakedown limit, the other is evolutive and holds for a specified load history either below and above the shakedown limit. Both principles are applicable in either statics and dynamics to construct bounds to the actual plastic deformation parameters. A continuum solid mechanics approach is used throughout, but a class of discrete models (finite elements with piecewise linear yield and saturation surfaces and plastic deformability lumped at Gauss points) are also considered. Extensions of shakedown theorems to materials with temperature dependent yield and saturation functions are also presented.


Plastic Strain Gauss Point Saturation Function Load History Repeated Load 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bathe, K. -J., and Wilson, E. L. (1976). Numerical Methods in Finite Element Analysis. Englewood Cliffs, N.J.: Prentice Hall.MATHGoogle Scholar
  2. Borino, G. (2000). Consistent shakedown theorems for materials with temperature dependent yield functions. International Journal of Solids and Structures. 37: 3121–3147.MathSciNetCrossRefMATHGoogle Scholar
  3. Borino, G., Caddemi, S., and Polizzotto, C. (1990). Mathematical programming methods for the evaluation of dynamic plastic deformations. In Lloyd Smith, D., ed., Mathematical Programming Methods in Structural Plasticity. Wien: Springer-Verlag. 349–372.Google Scholar
  4. Borino, G., and Polizzotto, C. (1995). Dynamic shakedown of structures under repeated seismic loads. Journal of Engineering Mechanics 121: 1306–1314.CrossRefGoogle Scholar
  5. Borino, G., and Polizzotto, C. (1996). Dynamic shakedown of structures with variable appended masses and subjected to repeated excitations. Int. Journal of Plasticity 12: 215–228.CrossRefMATHGoogle Scholar
  6. Borino, G, and Polizzotto, C. (1997). Shakedown theorems for a class of materials with temperature-dependent yield stress. In Owen, D. R. J., Oiiate, E. and Hinton, E., eds., Computational Plasticity. Barcelona, Spain, CIMNE. Part 1, 475–480.Google Scholar
  7. Capurso, M., Corradi, L., and Maier, G. (1978). Bounds on deformations and displacements in shakedown theory. In Ecole Politechnique Palaiseau, ed., Materials and Structures under Cyclic Loads. Paris: Association Amicale des Ingénieurs Anciens Elèves, Ec61e National Ponts et Chaussees. 231–244.Google Scholar
  8. Capurso, M. (1979). Some upper bounds principles for plastic strains in dynamic shakedown of elastoplastic structures. Journal of Structural Mechanics 7: 1–20.CrossRefGoogle Scholar
  9. Ceradini, G. (1969). On shakedown of elastic-plastic solids under dynamic actions. Giornale del Genio Civile 4–5: 239–258 (in Italian).Google Scholar
  10. Comi, C., and Corigliano, A. (1991). Dynamic shakedown in elastoplastic structures with general internal variable constitutive laws. Int. Journal of Plasticity 7: 679–692.CrossRefMATHGoogle Scholar
  11. Corradi, L., and Maier, G. (1973). Inadaptation theorems in the dynamics of elastic-workhardening structures. Ingenieur Archiv 43: 44–57.MathSciNetCrossRefMATHGoogle Scholar
  12. Corradi, L., and Maier, G. (1974). On non-shakedown theorems for elastic perfectly plastic continua. Journal of Mechanics and Physics of Solids 22: 401–413.CrossRefMATHGoogle Scholar
  13. Corradi, L., and De Donato, O. (1975). Dynamic shakedown theory allowing for second-order geometric effects. Meccanica 10: 93–98.CrossRefMATHGoogle Scholar
  14. Drucker, D. C. (1960). Plasticity. In Goodier, J. N., and Hoff, J. H., eds., Structural Mechanics. London: Pergamon Press. 407–455.Google Scholar
  15. Fuschi, P., and Polizzotto, C. (1998). Internal-variable constitutive model for rate-independent plasticity with hardening saturation surface. Acta Mechanica 129: 73–95.MathSciNetCrossRefMATHGoogle Scholar
  16. Fuschi, P. (1999). Structural shakedown for elastic-plastic materials with hardening saturation surface. Int. Journal of Solids and Structures 36: 219–240.CrossRefMATHGoogle Scholar
  17. Halphen, B. and Nguyen, Q. S. (1975). Sur les matériaux standard généralisées. Journal de Mécanique 14: 39–63.MATHGoogle Scholar
  18. Halphen, B. (1979). Accomodation et adaptation des structures élasto-visco-plastiques et plastiques. In Ecole Politechnique Palaiseau, ed., Matériaux et Structures Sous Chargement Cyclique. Paris: Association Amicale des Ingénieurs Anciens Elèves, Ec61e National Ponts et Chaussées. 203–229.Google Scholar
  19. Koiter, W. J. (1960). General theorems for elastic-plastic solids. In Hill, R., and Sneddon, I., eds., Progress in Solid Mechanics. North-Holland: vol. I. 167–221.Google Scholar
  20. König, A, and Maier, G. (1981). Shakedown analysis of elastoplastic structures: a review of recent developments. Nuclear Engineering and Design 66: 81–95.CrossRefGoogle Scholar
  21. König, A. (1982). Shakedown criteria in the case of loading and temperature variations. Journal de Mécanique Appliquée Special Issue, 99–108.Google Scholar
  22. Leckie, F. A. (1974). Review of bounding techniques in shakedown and ratchetting at elevated temperature. Welding Research Council Bullettin, 195: 1–11.Google Scholar
  23. Maier, G. (1973). A shakedown matrix theory allowing for workhardening and second-order geometric effects. In Sawczuk, A., eds., Foundations of Plasticity. Leyden, Noordhoff. 417–433.Google Scholar
  24. Maier, G. (1987). A generalization to nonlinear hardening of the first shakedown theorem for discrete elastic-plastic models. Atti Accademia dei Lincei, Rendiconti di Fisica 8, LXXXI: 161174.Google Scholar
  25. Maier, G., and Novati, G. (1990). Dynamic shakedown and bounding theory for a class of nonlinear hardening discrete structural models. Int. Journal of Plasticity 7: 679–692.Google Scholar
  26. Maier, G., Carvelli, V., and Cocchetti, G. (2000). On direct methods for shakedown and limit analysis. European Journal of Mechanics A/Solids (to appear).Google Scholar
  27. Owen, D. R. J., and Hinton, E. (1980). Finite Element in Plasticity. Swansea, U.K.: Pineridge Press Ltd.Google Scholar
  28. Panzeca, T., Polizzotto, C., and Rizzo, S. (1990). Bounding techniques and their application to simplified plastic analysis of structures. In Lloyd Smith, D., ed., Mathematical Programming Methods in Structural Plasticity. Wien: Springer-Verlag. 315–348.Google Scholar
  29. Pham, D. C. (1996). Dynamic shakedown and reduced kinematic theorems. Int. Journal of Plasticity 12: 1055–1068.MathSciNetCrossRefMATHGoogle Scholar
  30. Polizzotto, C. (1982a). A unified treatment of shakedown theory and related bounding techniques. Solid Mechanics Archives 7: 19–75.MATHGoogle Scholar
  31. Polizzotto, C. (1982b). Bounding principles for elastic-plastic-creeping solids loaded below and above the shakedown limit. Meccanica 17: 143–148.CrossRefMATHGoogle Scholar
  32. Polizzotto, C. (1984a). On shakedown of structures under dynamic agencies. In Sawczuk, A., and Polizzotto, C., eds., Inelastic Analysis Under Variable Loads. Palermo: Cogras. 5–29.Google Scholar
  33. Polizzotto, C. (1984b). Dynamic shakedown by modal analysis. Meccanica 19: 133–144.MathSciNetCrossRefGoogle Scholar
  34. Polizzotto, C. (1984c). Deformation bounds for elastic-plastic solids within and out of the creep range. Nuclear Engineering and Design 83: 293–301.CrossRefGoogle Scholar
  35. Polizzotto, C. (1985a). New developments in the theory of dynamic shakedown. In Sawczuk, A., and Bianchi, G., eds., Plasticity Today. Amsterdam: Elsevier. 343–360.Google Scholar
  36. Polizzotto, C. (1985b). Bounding techniques and their application to the analysis of elastic-plastic structures. In Zarka, J., ed., Simplified Analysis of Inelastic Structures Subjected to Statical and Dynamical loadings. Palaiseau, France: Ecole Polytechnique.Google Scholar
  37. Polizzotto, C. (1986). A convergent bounding principle for a class of elastoplastic strain-hardening solids. Int. Journal of Plasticity 2: 359–370.CrossRefMATHGoogle Scholar
  38. Polizzotto, C., Borino, G., Caddemi, S., and Fuschi, P. (1991). Shakedown problems for material models with internal variables. European Journal of Mechanics A/Solids 10: 621–639.MathSciNetMATHGoogle Scholar
  39. Polizzotto, C., Borino, G., Caddemi, S., and Fuschi, P. (1993). Theorems of restricted dynamic shakedown. Int. Journal of Mechanical Sciences 35: 787–801.CrossRefMATHGoogle Scholar
  40. Ponter, A. R. S. (1972). An upper bound on the small displacements of elastic perfectly plastic structures. Journal of Applied Mechanics 39: 959–963.CrossRefGoogle Scholar
  41. Ponter, A.R. S. (1975). General displacement and work bounds for dynamically loaded bodies. Journal of Mechanics and Physics of Solids 23: 157–163.MathSciNetGoogle Scholar
  42. Prager, W. (1956). Shakedown in elasticplastic media subjected to cycles of load and temperature In Proceedings of A. Danusso Symposium on: La Plasticità nella Scienza delle Costruzioni, Bologna, Nicola Zanichelli Editore, 239–244.Google Scholar
  43. Tajimi, H. (1960). A statistical method to determining the maximum response of a building structure during an earthquake. In Proceedings of the 2nd World Conference on Earthquake Engineering, Tokyo, 2: 781–797.Google Scholar

Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • Castrenze Polizzotto
    • 1
  • Guido Borino
    • 1
  • Paolo Fuschi
    • 2
  1. 1.University of PalermoPalermoItaly
  2. 2.University of Reggio CalabriaReggio CalabriaItaly

Personalised recommendations