Skip to main content
SpringerLink
Log in

A numerical procedure for the shakedown analysis of structures under cyclic thermomechanical loading

  • Special
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

Determining safety margins for a structure or a component against excessive inelastic deformations is an important issue for engineering design. Direct methods and particularly shakedown analysis constitute a convenient tool towards this direction. Most of the developed approaches in shakedown analysis are based on optimization algorithms. In this paper, a procedure for the shakedown analysis of structures under thermo-mechanical loads is presented. The approach makes use of the recently published Residual Stress Decomposition Method (RSDM) which assumes the decomposition of the residual stress field into Fourier series with respect to time. Starting from a high loading factor, the shakedown limit is estimated through an iterative procedure that updates the Fourier coefficients, reducing at the same time this loading factor until the only remaining term of the Fourier series is the constant term. The method is formulated within the finite element method and is applied to two-dimensional structures under thermal and mechanical cyclic loading.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Drucker D.C.: A definition of stable inelastic material. ASME J. Appl. Mech. 26, 101–106 (1959)

    MATH  MathSciNet  Google Scholar 

  2. Melan E.: Zur Plastizität des räumlichen Kontinuums. Ing. Arch. 9, 116–126 (1938)

    Article  MATH  Google Scholar 

  3. Koiter, W.: In: Sneddon, I.N., Hill, R. (eds.) General Theorems for Elastic–Plastic Solids. North-Holland, Amsterdam (1960)

  4. Prager, W.: Shakedown in Elastic–Plastic Media Subjected to Cycles of Load and Temperature, Symp. su la Plasticita nella Scienza delle Construzioni, Bologna, pp. 239–244 (1957)

  5. Donato O.: Second shakedown theorem allowing for cycles of both loads and temperature. Rend. Ist. Lombardo Scienza Lettere (A) 104, 265–277 (1970)

    MATH  Google Scholar 

  6. Weichert D.: On the influence of geometrical nonlinearities on the shakedown of elastic–plastic structures. Int. J. Plast. 2, 135–148 (1986)

    Article  MATH  Google Scholar 

  7. Pham D.C.: Shakedown static and kinematic theorems for elastic–plastic limited linear kinematic hardening solids. Eur. J. Mech. A/Solids 24, 35–45 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Stein E., Zhang G., König J.A.: Shakedown with nonlinear strain-hardening including structural computation using finite element method. Int. J. Plast. 8, 1–31 (1992)

    Article  MATH  Google Scholar 

  9. Bousshine L., Chaaba A., de Saxcé G.: A new approach to shakedown analysis for non-standard elastoplastic material by the bipotential. Int. J. Plast. 19, 583–598 (2003)

    Article  MATH  Google Scholar 

  10. Polizzotto C.: Shakedown theorems for elastic–plastic solids in the framework of gradient plasticity. Int. J. Plast. 24, 218–241 (2008)

    Article  MATH  Google Scholar 

  11. Borino G., Polizzotto C.: Dynamic shakedown of structures with variable appended masses and subjected to repeated excitations. Int. J. Plast. 12, 215–228 (1996)

    Article  MATH  Google Scholar 

  12. Maier G.: Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: a finite element, linear programming approach. Meccanica 4, 1–11 (1969)

    Google Scholar 

  13. Heitzer, M., Staat, M.: 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. 1–55 (2003)

  14. Zouain N., Borges L., Silveira J.L.: An algorithm for shakedown analysis with nonlinear yield function. Comp. Methods Appl. Mech. Eng. 191, 263–2481 (2002)

    Article  MathSciNet  Google Scholar 

  15. Andersen K.D., Christiansen E., Overton M.L.: Computing limit loads by minimizing a sum of norms. SIAM J. Sci. Comput. 19, 1046–1062 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bisbos C.D., Makrodimopoulos A., Pardalos P.M.: Second-order cone programming approaches to static shakedown analysis in steel plasticity. Optim. Methods Softw. 20, 25–52 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pastor F., Loute E.: Solving limit analysis problems: an interior-point method. Commun. Numer. Meth. Eng. 21, 631–642 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Tran T.N., Liu G.R., Nguyen-Xuan H., Nguyen-Thoi T.: An edge-based smoothed finite element method for primal-dual shakedown analysis of structures. Int. J. Numer. Meth. Eng. 82, 917–938 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Simon J-W., Weichert D.: Numerical lower bound shakedown analysis of engineering structures. Comp. Methods Appl. Mech. Eng. 200, 2828–2839 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Garcea G., Leonetti L.: A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis. Int. J. Numer. Meth. Eng. 88, 1085–1111 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Simon J.-W., Kreimeier M., Weichert D.: A selective strategy for shakedown analysis of engineering structures. Int. J. Numer. Meth. Eng. 94, 985–1014 (2013)

    Article  MathSciNet  Google Scholar 

  22. Spiliopoulos K., Weichert D.: Direct Methods for Limit States in Structures and Materials. Springer, Berlin (2014)

    Book  Google Scholar 

  23. Zarka, J., Engel, J.J., Inglebert, G.: On a simplified inelastic analysis of structures, Nucl. Eng. Des. 333–368 (1980)

  24. Ponter A.R.S., Carter K.F.: Shakedown state simulation techniques based on linear elastic solutions. Comput. Methods Appl. Mech Eng. 140, 259–279 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  25. Mackenzie D., Boyle J.T., Hamilton R.: The elastic compensation method for limit and shakedown analysis: a review. J. Strain Anal. 35, 171–188 (2000)

    Article  Google Scholar 

  26. Ponter A.R.S., Engelhardt M.: Shakedown limits for a general yield condition: implementation and application for a von Mises yield condition. Eur. J. Mech. A/Solids 19, 423–445 (2000)

    Article  MATH  Google Scholar 

  27. Chen H.F., Ponter A.R.S.: Shakedown and limit analyses for 3-D structures using the linear matching method. Int. J. Press. Vess. Piping 78, 443–451 (2001)

    Article  Google Scholar 

  28. Boulbibane M., Collins I.F., Ponter A.R.S., Weichert D.: Shakedown of unbound pavements. Road Mat. Pavem. Des. 6, 81–96 (2005)

    Article  Google Scholar 

  29. Boulbibane M., Ponter A.R.S.: The linear matching method for the shakedown analysis of geotechnical problems. Int. J. Numer. Anal. Meth. Geomech. 30, 157–179 (2006)

    Article  MATH  Google Scholar 

  30. Ponter A.R.S., Chen H.: A minimum theorem for cyclic load in excess of shakedown, with application to the evaluation of a ratchet limit. Eur. J. Mech. A/Solids 20, 539–553 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Chen H.F., Ponter A.R.S.: A method for the evaluation of a ratchet limit and the amplitude of plastic strain for bodies subjected to cyclic loading. Eur. J. Mech. A/Solids 20, 555–571 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  32. Spiliopoulos K.V., Panagiotou K.D.: A direct method to predict cyclic steady states of elastoplastic structures. Comp. Methods Appl. Mech. Eng. 223–224, 186–198 (2012)

    Article  MathSciNet  Google Scholar 

  33. Frederick C.O., Armstrong P.J.: Convergent internal stresses and steady cyclic states of stress. J. Strain Anal. 1, 154–169 (1966)

    Article  Google Scholar 

  34. Gokhfeld D.A., Cherniavsky O.F.: Limit Analysis of Structures at Thermal Cycling. Sijthoff & Noordhoff, Alphena (1980)

    Google Scholar 

  35. Polizzotto C.: Variational methods for the steady state response of elastic–plastic solids subjected to cyclic loads. Int. J. Solids Struct. 40, 2673–2697 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  36. König J.A.: Shakedown of Elastic–Plastic Structures. Elsevier, Amsterdam (1987)

    Google Scholar 

  37. Tolstov G.P.: Fourier Series. Dover, New York (1962)

    Google Scholar 

  38. König J.A., Kleiber M.: On a new method of shakedown analysis. Bull. Acad. Polon. Sci. Ser. Sci. Tech. 26, 165–171 (1978)

    MATH  Google Scholar 

  39. Luenberger D.G., Ye Y.: Linear and Nonlinear Programming. Springer, New York (2008)

    MATH  Google Scholar 

  40. Bree J.: Elastic–plastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fluxes with application to fast-nuclear-reactor fuel elements. J. Strain Anal. 2, 226–238 (1967)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Institute of Structural Analysis and Antiseismic Research, National Technical University of Athens, Zografou Campus, 157-80, Athens, Greece

    K. V. Spiliopoulos & K. D. Panagiotou

Authors
  1. K. V. Spiliopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. D. Panagiotou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. V. Spiliopoulos.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Spiliopoulos, K.V., Panagiotou, K.D. A numerical procedure for the shakedown analysis of structures under cyclic thermomechanical loading. Arch Appl Mech 85, 1499–1511 (2015). https://doi.org/10.1007/s00419-014-0947-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-014-0947-6

Keywords

Search

Navigation