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Limit State of Materials and Structures pp 19–43Cite as

Decomposition Methods and Strain Driven Algorithms for Limit and Shakedown Analysis

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Abstract

A mathematical programming formulation of strain-driven path-following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in a FEM context, is presented. From the optimization point of view, standard arc–length strain driven elastoplastic analysis, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other nonlinear programming methods, simpler convergence proofs and duality to be exploited. Due to the unified approach in terms of total stresses, the strain driven algorithms become more effective and less nonlinear with respect to a self equilibrated stress formulation and easier to implement in existing codes performing elastoplastic analysis.

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References

  1. Zouain, N.: Shakedown and safety assessment. In: Encyclopedia of Computational Mechanics, vol. 2, pp. 291–334. Wiley, New York (2004)

    Google Scholar 

  2. Armero: Elastoplastic and viscoplastic deformations of solids and structures. In: Encyclopedia of Computational Mechanics, vol. 2. Wiley, New York (2004)

    Google Scholar 

  3. Casciaro, R., Garcea, G.: An iterative method for shakedown analysis. Comput. Methods Appl. Mech. Eng. 191, 5761–5792 (2002)

    Article  MATH  Google Scholar 

  4. Garcea, G., Armentano, G., Petrolo, S., Casciaro, R.: Finite element shakedown analysis of two-dimensional structures. Int. J. Numer. Methods Eng. 63(8), 1174–1202 (2005)

    Article  MATH  Google Scholar 

  5. Boyd, S.: Lecture Slides and Notes. Courses EE364a and EE364b, http://www.stanford.edu/class/ee364b/lectures.html

  6. Nocedal, Wright, S.J.: Numerical Optimization. Springer, Philadelphia (1997)

    Google Scholar 

  7. Bertsekas: Nonlinear Programming. Athena Scientific, Nashua (2003)

    Google Scholar 

  8. Nemirovski, A., Todd, M.: Interior-point methods for optimization. Acta Numer. 17, 191–234 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wright, M.H.: The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull. Am. Math. Soc. 42(1), 39–56 (2005)

    Article  MATH  Google Scholar 

  10. Krabbenhoft, K., Lyamin, A.V., Sloan, S.W., Wriggers, P.: An interior-point algorithm for elastoplasticity. Int. J. Numer. Methods Eng. 69(3), 592–626 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Krabbenhoft, K., Damkilde, L.: A general non-linear optimization algorithm for lower bound limit analysis. Int. J. Numer. Methods Eng. 56(2), 165–184 (2003)

    Article  MATH  Google Scholar 

  12. Pastor, F., Loute, E.: Solving limit analysis problems: an interior-point method. Commun. Numer. Methods Eng. 21(11), 631–642 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Vu, D.K., Yan, A.M., Nguyen-Dang, H.: A primal–dual algorithm for shakedown analysis of structures. Comput. Methods Appl. Mech. Eng. 193, 4663–4674 (2004)

    Article  MATH  Google Scholar 

  14. Nguyen, A.D., Hachemi, A., Weichert, D.: Application of the interior-point method to shakedown analysis of pavements. Int. J. Numer. Methods Eng. 4(75), 414–439 (2008)

    Article  Google Scholar 

  15. Makrodimopoulos, A.: Computational formulation of shakedown analysis as a conic quadratic optimization problem. Mech. Res. Commun. 33, 72–83 (2006)

    Article  MATH  Google Scholar 

  16. Makrodimopoulos, A., Martin, C.M.: Lower: bound limit analysis of cohesive frictional materials using second-order cone programming. Int. J. Numer. Methods Eng. 66(4), 604–634 (2006)

    Article  MATH  Google Scholar 

  17. Makrodimopoulos, A., Martin, C.M.: Upper bound limit analysis using simplex strain elements and second-order cone programming. Int. J. Numer. Anal. Methods Geomech. 31(6), 835–865 (2007)

    Article  MATH  Google Scholar 

  18. Ha, C.D.: A generalization of the proximal point method. SIAM J. Control Optim. 28(3), 503–512 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kaneko, Y., Ha, C.D.: A decomposition procedure for large-scale optimum plastic design problems. Int. J. Numer. Methods Eng. 19, 873–889 (1983)

    Article  MATH  Google Scholar 

  20. Ponter, A.R.S., Martin, J.B.: Some extremal properties and energy theorems for inelastic materials and their relationship to the deformation theory of plasticity. Int. J. Mech. Phys. Solids 20(5), 281–300 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, G.R., Nguyen-Thoi, T., Nguyen-Xuan, H.b., Lam, K.Y.: A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Comput. Struct. 87, 14–26 (2009)

    Article  Google Scholar 

  22. Bilotta, A., Casciaro, R.: An hight–performance element for the analysis of 2D elastoplactic continua. Comput. Methods Appl. Mech. Eng. 196, 818–828 (2007)

    Article  MATH  Google Scholar 

  23. Garcea, G., Trunfio, G.A., Casciaro, R.: Mixed formulation and locking in path following nonlinear analysis. Comput. Methods Appl. Mech. Eng. 165(1–4), 247–272 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. A.S.: The MOSEK Optimization Tools Version 3.2 (Revision 8). User’s Manual and Reference Available from January (2005). http://www.mosek.com

  25. Goldfarb, D., Idnani, A.: A numerically stable dual method for solving strictly convex quadratic programs. Math. Program. 27, 1–33 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bathe, K.J.: Finite Element Procedures. Prentice-Hall, New York (1996)

    Google Scholar 

  27. Groß-Wedge, J.: On the numerical assessment of the safety factor of elastic-plastic structures under variable loading. Int. J. Mech. Sci. 39(4), 417–433 (1997)

    Article  Google Scholar 

  28. Zouanin, N., Borges, L., Silveira, J.L.: An algorithm for shakedown analysis with nonlinear yield function. Comput. Methods Appl. Mech. Eng. 191, 2463–2481 (2002)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Zhang, X., Liu, Y., Cen, Z.: Boundary element methods for lower bound limit and shakedown analysis. Eng. Anal. Bound. Methods 28, 905–917 (2004)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Modellistica per l’Ingegneria, Università della Calabria, Rende, Italy

    Giovanni Garcea & Leonardo Leonetti

Authors
  1. Giovanni Garcea
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  2. Leonardo Leonetti
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Corresponding author

Correspondence to Giovanni Garcea .

Editor information

Editors and Affiliations

  1. Laboratoire de Mécanique de Lille, Bd Paul Langevin, Villeneuve d'Ascq, 59655, France

    Géry de Saxcé

  2. Laboratoire de Mécanique de Lille, Bd Paul Langevin, Villeneuve d'Ascq, 59655, France

    Abdelbacet Oueslati

  3. Laboratoire de Mécanique de Lille, Bd Paul Langevin, Villeneuve d'Ascq, 59655, France

    Eric Charkaluk

  4. Laboratoire de Mécanique de Lille, Bd Paul Langevin, Villeneuve d'Ascq, 59655, France

    Jean-Bernard Tritsch

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© 2013 Springer Science+Business Media Dordrecht

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Garcea, G., Leonetti, L. (2013). Decomposition Methods and Strain Driven Algorithms for Limit and Shakedown Analysis. In: de Saxcé, G., Oueslati, A., Charkaluk, E., Tritsch, JB. (eds) Limit State of Materials and Structures. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5425-6_2

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  • DOI: https://doi.org/10.1007/978-94-007-5425-6_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-5424-9

  • Online ISBN: 978-94-007-5425-6

  • eBook Packages: EngineeringEngineering (R0)

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