RSDM-S: A Method for the Evaluation of the Shakedown Load of Elastoplastic Structures

  • Konstantinos V. SpiliopoulosEmail author
  • Konstantinos D. Panagiotou


To estimate the life of a structure, or a component, which are subjected to a cyclic loading history, the structural engineer must be able to provide safety margins. This is only possible by performing a shakedown analysis that belongs to the class of direct methods. Most of the existing numerical procedures addressing a shakedown analysis are based on the two theorems of plasticity and are formulated within the framework of mathematical programming. A different approach is presented herein. It is an iterative procedure and starts by converting the problem of loading margins to an equivalent loading of a prescribed time history. Inside an iteration, the recently published RSDM direct method is used, which assumes the decomposition of the residual stresses into Fourier series and evaluates its coefficients by iterations. It is proved that a descending sequence of loading factors is generated which converges, from above, to the shakedown load factor when only the constant term of the series remains. An elastic-perfectly plastic with a von Mises yield surface is currently assumed. The method may be implemented in any existing FE code and its efficiency is demonstrated by a couple of applications.


Direct methods Shakedown analysis RSDM 


  1. 1.
    Drucker DC (1959) A definition of stable inelastic material. ASME J Appl Mech 26:101–106zbMATHMathSciNetGoogle Scholar
  2. 2.
    Frederick CO, Armstrong PJ (1966) Convergent internal stresses and steady cyclic states of stress. J Strain Anal 1:154–169CrossRefGoogle Scholar
  3. 3.
    Melan E (1938) Zur Plastizität des räumlichen Kontinuums. Ing Arch 9:116–126CrossRefzbMATHGoogle Scholar
  4. 4.
    Koiter W (1960) General theorems for elastic-plastic solids. In: Sneddon IN, Hill R (eds) North-Holland, AmsterdamGoogle Scholar
  5. 5.
    Weichert D (1986) On the influence of geometrical nonlinearities on the shakedown of elastic-plastic structures. Int J Plast 2:135–148CrossRefzbMATHGoogle Scholar
  6. 6.
    Belouchrani MA, Weichert D (1999) An extension of the static shakedown theorem to inelastic cracked structures. Int J Mech Sci 41:163–177CrossRefzbMATHGoogle Scholar
  7. 7.
    Pham DC (2005) Shakedown static and kinematic theorems for elastic–plastic limited linear kinematic hardening solids. Eur J Mech A/Solids 24:35–45CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Simon J-W (2013) Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening. Int J Plast 42:141–167CrossRefGoogle Scholar
  9. 9.
    Bousshine L, Chaaba A, de Saxcé G (2003) A new approach to shakedown analysis for non-standard elastoplastic material by the bipotential. Int J Plast 19:583–598CrossRefzbMATHGoogle Scholar
  10. 10.
    Polizzotto C (2008) Shakedown theorems for elastic–plastic solids in the framework of gradient plasticity. Int J Plast 24:218–241CrossRefzbMATHGoogle Scholar
  11. 11.
    Maier G (1969) Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: a finite element, linear programming approach. Meccanica 4:1–11Google Scholar
  12. 12.
    Stein E, Zhang G, König JA (1992) Shakedown with nonlinear strain-hardening including structural computation using finite element method. Int J Plast 8:1–31CrossRefzbMATHGoogle Scholar
  13. 13.
    Heitzer M, Staat M (2003) Basis reduction technique for limit and shakedown problems. In: Staat M, Heitzer M (eds) Numerical methods for limit and shakedown analysis. NIC Series, vol 15, pp 1–55Google Scholar
  14. 14.
    Andersen KD, Christiansen E, Overton ML (1998) Computing limit loads by minimizing a sum of norms. SIAM J Sci Comput 19:1046–1062CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Vu DK, Yan AM, Nguyen-Dang H (2004) A primal-dual algorithm for shakedown analysis of structures. Comput Methods Appl Mech Eng 193:4663–4674CrossRefzbMATHGoogle Scholar
  16. 16.
    Simon J-W, Höer D, Weichert D (2014) A starting-point strategy for interior-point algorithms for shakedown analysis of engineering structures. Eng Optim 46(5):648–668CrossRefMathSciNetGoogle Scholar
  17. 17.
    Spiliopoulos K, Weichert D (2014) Direct methods for limit states in structures and materials. Springer, BerlinCrossRefGoogle Scholar
  18. 18.
    Ponter ARS, Carter KF (1997) Shakedown state simulation techniques based on linear elastic solutions. Comput Methods Appl Mech Eng 140:259–279CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Mackenzie D, Boyle T (1993) A method of estimating limit loads by iterative elastic analysis. I—Simple examples. Int J Press Vessel Pip 53:77–95CrossRefGoogle Scholar
  20. 20.
    Pisano AA, Fuschi P, de Domenico D (2013) A kinematic approach for peak load evaluation of concrete elements. Comput Struct 119:125–139CrossRefGoogle Scholar
  21. 21.
    Chen H, Ponter ARS (2010) A direct method on the evaluation of ratchet limit. ASME J Press Vessel Technol 132(4):1–8CrossRefGoogle Scholar
  22. 22.
    Spiliopoulos KV, Panagiotou KD (2012) A direct method to predict cyclic steady states of elastoplastic structures. Comput Methods Appl Mech Eng 223–224:186–198CrossRefMathSciNetGoogle Scholar
  23. 23.
    Spiliopoulos KV, Panagiotou KD (2014) A residual stress decomposition based method for the shakedown analysis of structures. Comput Methods Appl Mech Eng 276:410–430CrossRefGoogle Scholar
  24. 24.
    Gokhfeld DA, Cherniavsky OF (1980) Limit analysis of structures at thermal cycling. Sijthoff & Noordhoff, Alphen aan den RijnGoogle Scholar
  25. 25.
    König JA, Kleiber M (1978) On a new method of shakedown analysis. Bull Acad Polon Sci Ser Sci Tech 26:165–171zbMATHGoogle Scholar
  26. 26.
    Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New YorkzbMATHGoogle Scholar
  27. 27.
    Ponter ARS, Engelhardt M (2000) Shakedown limits for a general yield condition: implementation and application for a von Mises yield condition. Eur J Mech A/Solids 19:423–445CrossRefzbMATHGoogle Scholar
  28. 28.
    Carvelli V, Cen ZZ, Liu Y, Maier G (1999) Shakedown analysis of defective pressure vessels by a kinematic approach. Arch Appl Mech 69:751–764CrossRefzbMATHGoogle Scholar
  29. 29.
    Tran TN, Liu GR, Nguyen XH, Nguyen TT (2010) An edge-based smoothed finite element method for primal-dual shakedown analysis of structures. Int J Numer Eng 82:917–938zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Konstantinos V. Spiliopoulos
    • 1
    Email author
  • Konstantinos D. Panagiotou
    • 1
  1. 1.Department of Civil Engineering, Institute of Structural Analysis and Antiseismic ResearchNational Technical University of Athens, Zografou CampusAthensGreece

Personalised recommendations