Advertisement

A New Starting Point Strategy for Shakedown Analysis

  • Konstantinos NikolaouEmail author
  • Christos D. Bisbos
  • Dieter Weichert
  • Jaan -W. Simon
Chapter

Abstract

Shakedown analysis is currently implemented by the coupling of finite element methods with techniques of computational optimization. Engineering structures problems contain a large number of variables and constraints, leading to large-scale nonlinear programming problems, since, usually, nonlinear yield criteria are preferred. The respective algorithms use iterative techniques to solve the problem at hand and the selection of a starting point is of crucial importance for their performance. To this goal the elastic limit solution could be applied, which yields a feasible point, since the zero residual stress identically satisfies the null space conditions. The present study proposes a mechanically motivated, simple technique to obtain an initial feasible point with nonzero residual stresses starting from the plastic shakedown analysis. The residual stresses obtained by this problem are generally infeasible and they are projected into the null space of the equilibrium conditions in order to yield a feasible set of self-equilibrating nonzero stresses. Next, this feasible point is completed by a safety factor, obtained from a one-dimensional optimization problem of elastic limit type. The applicability and appropriateness of this approach is studied by numerical comparisons.

Notes

Acknowledgments

The first author would like to thank DAAD for granting a short stay research scholarship at the Institute of General Mechanics (IAM) of RWTH-Aachen University, IAM for the hospitality and especially the head of IAM Prof. B. Markert for financially supporting the trip to Italy. Special thanks to Dr. Hachemi, Dr. Chen and MSc G.Chen from IAM and Dr Skordeli from Aristotle University Thessaloniki, for the fruitful discussions.

References

  1. 1.
    Melan E (1938) Zur Plastizität des räumlichen Kontinuums. Arch Appl Mech 9:116–126zbMATHGoogle Scholar
  2. 2.
    König JA (1987) Shakedown of elastic-plastic structures. Elsevier, AmsterdamGoogle Scholar
  3. 3.
    Weichert D, Maier G (eds) (2000) Inelastic analysis of structures under variable repeated loads. Kluwer Academic, DordrechtGoogle Scholar
  4. 4.
    Weichert D, Ponter ARS (eds) (2009) Limit states of materials and structures: direct methods. Springer, WienGoogle Scholar
  5. 5.
    Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Sturm JF (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11:625–653CrossRefMathSciNetGoogle Scholar
  7. 7.
    Toh KC, Todd MJ, Tütüncü RH (1999) SDPT3—a MATLAB software package for semidefinite programming, version 1.3. Optim Methods Softw 11(1–4):545–581Google Scholar
  8. 8.
    Andersen ED, Roos C, Terlaky T (2003) On implementing a primal-dual interior-point method for conic quadratic optimization. Math Program 95(2):249–277CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Christiansen E, Andersen KD (1999) Computation of collapse states with von Mises type yield condition. Int J Numer Methods Eng 46(8):1185–1202CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Trillat M, Pastor J (2005) Limit analysis and Gurson’s model. Eur J Mech-A/Solids 24(5):800–819CrossRefzbMATHGoogle Scholar
  11. 11.
    Bisbos CD, Makrodimopoulos A, Pardalos PM (2005) Second-order cone programming approaches to static shakedown analysis in steel plasticity. Optim Methods Softw 20(1):25–52CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Krabbenhøft K, Lyamin AV, Sloan SW (2007) Formulation and solution of some plasticity problems as conic programs. Int J Solids Struct 44(5):1533–1549CrossRefGoogle Scholar
  13. 13.
    Skordeli MA, Bisbos CD (2010) Limit and shakedown analysis of 3d steel frames via approximate ellipsoidal yield surfaces. Eng Struct 32(6):1556–1567CrossRefGoogle Scholar
  14. 14.
    Nikolaou K, Skordeli MA-A, Bisbos CD (2013) Limit analysis of aluminium frames via  approximate ellipsoidal yield surfaces. In: 10th HSTAM international congress on mechanics, ChaniaGoogle Scholar
  15. 15.
    Simon J-W, Weichert D (2011) Numerical lower bound shakedown analysis of engineering structures. Comput Methods App Mech Eng 200(41):2828–2839CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Simon J-W, Kreimeier M, Weichert D (2013) A selective strategy for shakedown analysis of engineering structures. Int J Numer Methods Eng 94:985–1014. doi: 10.1002/nme.4476 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Simon J-W, Höwer D, Weichert D (2013) A starting point strategy for interior-point algorithms for shakedown analysis of engineering structures. Eng Optim ISSN:1029–0207. doi: 10.1080/0305215X.2013.791816
  18. 18.
    Akoa F, Hachemi A, Said M, Tao PD (2007) Application of lower bound direct method to engineering structures. J Glob Optim 37(4):609–630CrossRefzbMATHGoogle Scholar
  19. 19.
    Hachemi A, Mouhtamid S, Tao P (2004) Large-scale nonlinear programming and lower bound direct method in engineering applications. In: Modelling, computation and optimization in information systems and management sciences. Hermes Sciences, London, pp 299–310Google Scholar
  20. 20.
    Zouain N, Herskovits J, Borges LA, Feijóo RA (1993) An iterative algorithm for limit analysis with nonlinear yield functions. Int J Solids Struct 30(10):1397–1417CrossRefzbMATHGoogle Scholar
  21. 21.
    Lyamin AV, Sloan SW (2002) Lower bound limit analysis using non-linear programming. Int J Numer Methods Eng 55(5):573–611CrossRefzbMATHGoogle Scholar
  22. 22.
    Krabbenhoft K, Lyamin AV, Sloan SW, Wriggers P (2007) An interior-point algorithm for elastoplasticity. Int J Numer Methods Eng 69(3):592–626CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Pastor F, Loute E (2005) Solving limit analysis problems: an interior-point method. Commun Numer Methods Eng 21(11):631–642CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Vu DK, Yan AM, Nguyen-Dang H (2004) A primal-dual algorithm for shakedown analysis of structures. Comput Methods Appl Mech Eng 193(42):4663–4674CrossRefzbMATHGoogle Scholar
  25. 25.
    Vu DK, Staat M, Tran IT (2007) Analysis of pressure equipment by application of the primal-dual theory of shakedown. Commun Numer Methods Eng 23(3):213–225CrossRefzbMATHGoogle Scholar
  26. 26.
    Bisbos CD, Pardalos PM (2007) Second-order cone and semidefinite representations of material failure criteria. J Optim Theory Appl 134(2):275–301CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Skordeli MA-A (2010) Shakedown analysis of metal structures subjected to ellipsoidal variable repeated loading via robust optimization techniques. PhD Dissertation, Aristotle University of ThessalonikiGoogle Scholar
  28. 28.
    Bisbos CD, Ampatzis AT (2008) Shakedown analysis of spatial frames with parameterized load domain. Eng Struct 30(11):3119–3128CrossRefGoogle Scholar
  29. 29.
    Yildirim EA, Wright SJ (2002) Warm-start strategies in interior-point methods for linear programming. SIAM J Optim 12(3):782–810CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Gertz M, Nocedal J, Sartenar AA (2004) Starting point strategy for nonlinear interior methods. Appl Math Lett 17(8):945–952CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Mehrotra S (1992) On the implementation of a primal-dual interior point method. SIAM J Optim 2(4):575–601CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Stojković NV, Stanimirović PS (2001) Initial point in primal-dual interior point method. Facta Universitatis-Series: Mech Autom Control Robot 3(11):219–222zbMATHGoogle Scholar
  33. 33.
    Castillo E, García-Bertrand R, Mínguez R (2006) Decomposition techniques in mathematical programming: engineering and science applications. Springer, BerlinGoogle Scholar
  34. 34.
    Belytschko T (1972) Plane stress shakedown analysis by finite elements. Int J Mech Sci 14(9):619–625CrossRefGoogle Scholar
  35. 35.
    Hamilton R, Boyle JT, Shi J, Mackenzie D (1996) Shakedown load bounds by elastic finite element analysis. ASME Press Vessel Pip Div Publ PVP 343:205–211Google Scholar
  36. 36.
    Groß-Weege J (1997) On the numerical assessment of the safety factor of elastic-plastic structures under variable loading. Int J Mech Sci 39(4):417–433Google Scholar
  37. 37.
    Muscat M, Mackenzie D, Hamilton R (2003) Evaluating shakedown under proportional loading by nonlinear static analysis. Comput Struct 81(17):1727–1737CrossRefGoogle Scholar
  38. 38.
    Zhang X, Liu Y, Cen Z (2004) Boundary element methods for lower bound limit and shakedown analysis. Eng Anal Bound Elem 28(8):905–917CrossRefzbMATHGoogle Scholar
  39. 39.
    Chen S, Liu Y, Cen Z (2008) Lower bound shakedown analysis by using the element free Galerkin method and nonlinear programming. Comput Methods Appl Mech Eng 197(45):3911–3921CrossRefzbMATHGoogle Scholar
  40. 40.
    Zhang W, Yang LF, Fu CX, Wang J (2012) Evaluating shakedown for structures based on the element bearing-ratio. Appl Mech Mater 137:16–23CrossRefGoogle Scholar
  41. 41.
    Zouain N, Borges L (2002) An algorithm for shakedown analysis with nonlinear yield functions. Comput Methods Appl Mech Eng 191(23):2463–2481CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Konstantinos Nikolaou
    • 1
    Email author
  • Christos D. Bisbos
    • 1
  • Dieter Weichert
    • 2
    • 3
  • Jaan -W. Simon
    • 4
  1. 1.Department of Civil Engineering, Institute of Steel StructuresAristotle University of ThessalonikiThessalonikiGreece
  2. 2.Institute of General MechanicsRWTH Aachen UniversityAachenGermany
  3. 3.Graduate School of Energy ScienceKyoto UniversityKyotoJapan
  4. 4.Institute of Applied MechanicsRWTH Aachen UniversityAachenGermany

Personalised recommendations