Skip to main content
SpringerLink
Log in
Book cover

Direct Methods for Limit States in Structures and Materials pp 57–77Cite as

Shakedown Analysis of Kinematically Hardening Structures in n-Dimensional Loading Spaces

  • Chapter

Abstract

Determining the load bearing capacity is essential for the design of engineering structures subjected to varying thermo-mechanical loadings. The according computations can be carried out most conveniently by using shakedown analysis. In order to obtain realistic results, however, limited kinematical hardening needs to be taken into account. Moreover, it is necessary to consider arbitrary numbers of loadings leading to n-dimensional loading spaces. Even so, the numerical tools available for shakedown analysis are—up to now—restricted to either perfectly-plastic material behavior or to a maximum of two independently varying loadings. Thus, the aim of this paper is to present a numerical procedure, which allows the consideration of limited kinematical hardening in n-dimensional loading spaces. The method is based on the lower bound shakedown theorem by Melan, which has been extended to limited kinematical hardening by use of a two-surface model. To solve the resulting nonlinear optimization problem, which is typically characterized by a large number of variables and constraints, an interior-point algorithm is implemented. Finally, the potential of the procedure is shown by application to a flanged pipe subjected to three independently varying thermal and mechanical loadings accounting for different yield stress to ultimate stress ratios.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Akoa FB, Hachemi A, An LTH, Mouhtamid S, Tao PD (2007) Application of lower bound direct method to engineering structures. J Glob Optim 37(4):609–630

    Article  MATH  Google Scholar 

  2. Akrotirianakis I, Rustem B (2000) A primal-dual interior-point algorithm with an exact and differentiable merit function for general nonlinear programming problems. Optim Methods Softw 14(1/2):1–36

    Article  MathSciNet  MATH  Google Scholar 

  3. Andersen ED, Jensen B, Jensen J, Sandvik R, Worsøe U (2009) MOSEK version 6. Technical report TR-2009-3

    Google Scholar 

  4. Andersen ED, Roos C, Terlaky T (2003) On implementing a primal-dual interior-point method for conic quadratic optimization. Math Program 95(2):249–277

    Article  MathSciNet  MATH  Google Scholar 

  5. Ardito R, Cocchetti G, Maier G (2008) On structural safety assessment by load factor maximization in piecewise linear plasticity. Eur J Mech A, Solids 27:859–881

    Article  MathSciNet  MATH  Google Scholar 

  6. Benson HY, Shanno DF, Vanderbei RJ (2002) Interior-point methods for nonconvex nonlinear programming: filter methods and merit functions. Comput Optim Appl 23(2):257–272

    Article  MathSciNet  MATH  Google Scholar 

  7. Benson HY, Shanno DF, Vanderbei RJ (2003) A comparative study of large-scale nonlinear optimization algorithms. In: Di Pillo G, Murli A (eds) High performance algorithms and software for nonlinear optimization. Kluwer Academic, Princeton, pp 95–127

    Chapter  Google Scholar 

  8. Bisbos CD, Makrodimopoulos A, Pardalos PM (2005) Second-order cone programming approaches to static shakedown analysis in steel plasticity. Optim Methods Softw 20(1):25–52

    Article  MathSciNet  MATH  Google Scholar 

  9. Bouby C, De Saxcé G, Tritsch J-B (2009) Shakedown analysis: comparison between models with the linear unlimited, linear limited and non-linear kinematic hardening. Mech Res Commun 36:556–562

    Article  MATH  Google Scholar 

  10. Bousshine L, Chaaba A, De Saxce G (2003) A new approach to shakedown analysis for non-standard elastoplastic material by the bipotential. Int J Plast 19(5):583–598

    Article  MATH  Google Scholar 

  11. Byrd RH, Hribar ME, Nocedal J (2000) An interior point algorithm for large-scale nonlinear programming. SIAM J Optim 9(4):877–900

    Article  MathSciNet  Google Scholar 

  12. Conn AR, Gould NIM, Toint PL (1992) In: LANCELOT: a Fortran package for large-scale nonlinear optimization (release A). Springer series in computational mathematics, vol 17. Springer, Heidelberg

    Google Scholar 

  13. Corigliano A, Maier G, Pycko S (1995) Kinematic criteria of dynamic shakedown extended to nonassociate constitutive laws with saturation nonlinear hardening. Rend Accad Lincei IX 6:55–64

    MathSciNet  MATH  Google Scholar 

  14. El-Bakry AS, Tapia RA, Tsuchiya T, Zhang Y (1996) On the formulation and theory of the Newton interior-point method for nonlinear programming. J Optim Theory Appl 89:507–541

    Article  MathSciNet  MATH  Google Scholar 

  15. Forsgren A, Gill PE, Wright MH (2002) Interior methods for nonlinear optimization. SIAM Rev 44(4):525–597

    Article  MathSciNet  MATH  Google Scholar 

  16. Fuschi P (1999) Structural shakedown for elastic-plastic materials with hardening saturation surface. Int J Solids Struct 36:219–240

    Article  MATH  Google Scholar 

  17. Garcea G, Leonetti L (2011) A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis. Int J Numer Methods Eng 88(11):1085–1111

    Article  MathSciNet  MATH  Google Scholar 

  18. Griva I, Shanno DF, Vanderbei RJ, Benson HY (2008) Global convergence analysis of a primal-dual interior-point method for nonlinear programming. Algorithmic Oper Res 3(1):12–19

    MathSciNet  Google Scholar 

  19. 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):417–433

    Article  MATH  Google Scholar 

  20. Groß-Weege J, Weichert D (1992) Elastic-plastic shells under variable mechanical and thermal loads. Int J Mech Sci 34:863–880

    Article  MATH  Google Scholar 

  21. Hachemi A, An LTH, Mouhtamid S, Tao PD (2004) Large-scale nonlinear programming and lower bound direct method in engineering applications. In: An LTH, Tao PD (eds) Modelling, computation and optimization in information systems and management sciences. Hermes Science, London, pp 299–310

    Google Scholar 

  22. Hachemi A, Mouhtamid S, Nguyen AD, Weichert D (2009) Application of shakedown analysis to large-scale problems with selective algorithm. In: Weichert D, Ponter ARS (eds) Limit states of materials and structures. Springer, Berlin, pp 289–305

    Chapter  Google Scholar 

  23. Hachemi A, Mouhtamid S, Weichert D (2005) Progress in shakedown analysis with applications to composites. Arch Appl Mech 74:762–772

    Article  MATH  Google Scholar 

  24. Heitzer M (1999) Traglast- und Einspielanalyse zur Bewertung der Sicherheit passiver Komponenten. PhD thesis, Forschungszentrum Jülich, RWTH Aachen, Germany

    Google Scholar 

  25. Koiter WT (1960) General theorems for elastic-plastic solids. In: Sneddon IN, Hill R (eds) Progress in solid mechanics. North-Holland, Amsterdam, pp 165–221

    Google Scholar 

  26. König JA (1987) Shakedown of elastic-plastic structures. Elsevier, Amsterdam

    Google Scholar 

  27. König JA, Siemaszko A (1988) Shakedown of elastic-plastic structures. Ing-Arch 58:58–66

    Article  MATH  Google Scholar 

  28. Krabbenhøft K, Damkilde L (2003) A general nonlinear optimization algorithm for lower bound limit analysis. Int J Numer Methods Eng 56:165–184

    Article  MATH  Google Scholar 

  29. Krabbenhøft K, Lyamin AV, Sloan SW (2007) Formulation and solution of some plasticity problems as conic programs. Int J Solids Struct 44:1533–1549

    Article  Google Scholar 

  30. Krabbenhøft K, Lyamin AV, Sloan SW, Wriggers P (2007) An interior-point algorithm for elastoplasticity. Int J Numer Methods Eng 69:592–626

    Article  MATH  Google Scholar 

  31. Magoariec H, Bourgeois S, Débordes O (2004) Elastic plastic shakedown of 3d periodic heterogeneous media: a direct numerical approach. Int J Plast 20(8–9):1655–1675

    Article  MATH  Google Scholar 

  32. Maier G, Pastor J, Ponter ARS, Weichert D (2003) Direct methods of limit and shakedown analysis. In: de Borst R, Mang HA (eds) Comprehensive structural integrity—fracture of materials from nano to macro. Numerical and computational methods, vol 3. Elsevier, Amsterdam, pp 637–684

    Chapter  Google Scholar 

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

    Article  MATH  Google Scholar 

  34. Malena M, Casciaro R (2008) Finite element shakedown analysis of reinforced concrete 3d frames. Comput Struct 86(11–12):1176–1188

    Article  Google Scholar 

  35. Mandel J (1976) Adaptation d’une structure plastique ecrouissable et approximations. Mech Res Commun 3:483–488

    Article  MATH  Google Scholar 

  36. Maratos N (1978) Exact penalty function algorithms for finite dimensional and control opimization problems. PhD thesis, University of London, UK

    Google Scholar 

  37. Melan E (1938) Der Spannungszustand eines ,,Mises-Hencky’schen“ Kontinuums bei veränderlicher Belastung. Sitzungsber Akad Wiss Wien, Math-Nat Kl, Abt IIa 147:73–87

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  39. Mittelmann H (2010) Decision tree for optimization software. http://plato.asu.edu/guide.thml

  40. Morales JL, Nocedal J, Waltz RW, Lie G, Goux J-P (2003) Assessing the potential of interior methods for nonlinear optimization. In: Biegler LT, Ghattas O, Heinkenschloss M, van Bloemen Waander B (eds) Large-scale PDE-constrained optimization, vol 30. Springer, Berlin, pp 167–183

    Chapter  Google Scholar 

  41. Mouhtamid S (2007) Anwendung direkter Methoden zur industriellen Berechnung von Grenzlasten mechanischer Komponenten. PhD thesis, Institute of General Mechanics, RWTH Aachen University, Germany

    Google Scholar 

  42. Ngo N, Tin-Loi F (2007) Shakedown analysis using the p-adaptive finite element method and linear programming. Eng Struct 29(1):46–56

    Article  Google Scholar 

  43. Nguyen Q-S (2003) On shakedown analysis in hardening plasticity. J Mech Phys Solids 51:101–125

    Article  MathSciNet  MATH  Google Scholar 

  44. Pastor F, Loute E (2010) Limit analysis decomposition and finite element mixed method. J Comput Appl Math 234(7):2213–2221

    Article  MathSciNet  MATH  Google Scholar 

  45. Pastor F, Loute E, Pastor J, Trillat M (2009) Mixed method and convex optimization for limit analysis of homogeneous Gurson materials: a kinematic approach. Eur J Mech A, Solids 28:25–35

    Article  MATH  Google Scholar 

  46. Pastor F, Thoré P, Loute E, Pastor J, Trillat M (2008) Convex optimization and limit analysis: application to Gurson and porous Drucker-Prager materials. Eng Fract Mech 75:1367–1383

    Article  Google Scholar 

  47. Pham DC (2007) Shakedown theory for elastic plastic kinematic hardening bodies. Int J Plast 23:1240–1259

    Article  Google Scholar 

  48. Pham DC (2008) On shakedown theory for elastic-plastic materials and extensions. J Mech Phys Solids 56:1905–1915

    Article  MathSciNet  MATH  Google Scholar 

  49. Pham DC, Weichert D (2001) Shakedown analysis for elastic-plastic bodies with limited kinematical hardening. Proc R Soc Lond A 457:1097–1110

    Article  MATH  Google Scholar 

  50. Pham PT, Vu DK, Tran TN, Staat M (2010) An upper bound algorithm for shakedown analysis of elastic-plastic bounded linearly kinematic hardening bodies. In: Proc ECCM 2010

    Google Scholar 

  51. Polizzotto C (1986) A convergent bounding principle for a class of elastoplastic strain-hardening solids. Int J Plast 2(4):359–370

    Article  MATH  Google Scholar 

  52. Polizzotto C (2010) Shakedown analysis for a class of strengthening materials within the framework of gradient plasticity. Int J Plast 26(7):1050–1069

    Article  MATH  Google Scholar 

  53. Ponter ARS (1975) A general shakedown theorem for elastic plastic bodies with work hardening. In: Proc SMIRT-3, p L5/2

    Google Scholar 

  54. Ponter ARS (2002) A linear matching method for shakedown analysis. In: Weichert D, Maier G (eds) Inelastic behaviour of structures under variable repeated loading—direct analysis methods, pp 267–318

    Google Scholar 

  55. Ponter ARS, Chen HF (2005) Direct methods for limits in plasticity. Arch Mech 57:171–188

    MATH  Google Scholar 

  56. Potra FA, Wright SJ (2000) Interior-point methods. J Comput Appl Math 124:281–302

    Article  MathSciNet  MATH  Google Scholar 

  57. Pycko S, Maier G (1995) Shakedown theorems for some classes of nonassociative hardening elastic-plastic material models. Int J Plast 11(4):367–395

    Article  MATH  Google Scholar 

  58. Simon J-W, Chen M, Weichert D (2012) Shakedown analysis combined with the problem of heat conduction. J Press Vessel Technol 134(2):021206

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  60. Simon J-W, Weichert D (2012) Interior-point method for lower bound shakedown analysis of von mises-type materials. In: de Saxcé G, Oueslati A, Charkaluk E, Tritsch J-B (eds) Limit states of materials and structures—direct methods, vol 2. Springer, Berlin, pp 103–128

    Google Scholar 

  61. Simon J-W, Weichert D (2012) Shakedown analysis of engineering structures with limited kinematical hardening. Int J Solids Struct 49(4):2177–2186

    Article  MathSciNet  Google Scholar 

  62. Simon J-W, Weichert D (2012) Shakedown analysis with multidimensional loading spaces. Comput Mech 49(4):477–485

    Article  MathSciNet  MATH  Google Scholar 

  63. Spiliopoulos KV, Panagiotou KD (2012) A direct method to predict cyclic steady states of elastoplastic structures. Comput Methods Appl Mech Eng 223(224):186–198

    Article  MathSciNet  Google Scholar 

  64. Staat M, Heitzer M (2002) The restricted influence of kinematical hardening on shakedown loads. In: Proc WCCM V

    Google Scholar 

  65. Stein E, Zhang G, Huang Y (1993) Modeling and computation of shakedown problems for nonlinear hardening materials. Comput Methods Appl Mech Eng 103(1–2):247–272

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  67. Stein E, Zhang G, Mahnken R, König JA (1990) Micromechanical modelling and computation of shakedown with nonlinear kinematic hardening including examples for 2-D problems. In: Axelard DR, Muschik W (eds) Recent developments of micromechanics. Springer, Berlin

    Google Scholar 

  68. Trillat M, Pastor J (2005) Limit analysis and Gurson’s model. Eur J Mech A, Solids 24:800–819

    Article  MATH  Google Scholar 

  69. Vanderbei RJ (1999) LOQO: an interior point code for quadratic programming. Optim Methods Softw 11–12:451—484

    Article  MathSciNet  Google Scholar 

  70. Vu DK, Staat M (2007) Analysis of pressure equipment by application of the primal-dual theory of shakedown. Commun Numer Methods Eng 23(3):213–225

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  72. Wächter A (2002) An interior point algorithm for large-scale nonlinear optimization with applications in process engineering. PhD thesis, Carnegie Mellon University, Pittsburgh, Pennsylvania

    Google Scholar 

  73. Wächter A, Biegler LT (2005) Line-search filter methods for nonlinear programming: motivation and global convergence. SIAM J Optim 16(1):1–31

    Article  MathSciNet  MATH  Google Scholar 

  74. 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):25–57

    Article  MathSciNet  MATH  Google Scholar 

  75. Waltz RA, Morales JL, Nocedal J, Orban D (2006) An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math Program 107(3):391–408

    Article  MathSciNet  MATH  Google Scholar 

  76. Weichert D, Groß-Weege J (1988) The numerical assessment of elastic-plastic sheets under variable mechanical and thermal loads using a simplified two-surface yield condition. Int J Mech Sci 30(10):757–767

    Article  MATH  Google Scholar 

  77. Weichert D, Hachemi A, Mouhtamid S, Nguyen AD (2008) On recent progress in shakedown analysis and applications to large-scale problems. In: IUTAM symposium on theoretical, computational and modelling aspects of inelastic media, vol 11, pp 349–359

    Chapter  Google Scholar 

  78. Weichert D, Maier G (2000) Inelastic analysis of structures under variable repeated loads. Kluwer Academic, Dordrecht

    Google Scholar 

  79. Weichert D, Ponter ARS (2009) Limit states of materials and structures. Springer, Wien

    Google Scholar 

  80. Wright MH (2004) The interior-point revolution in optimization: history, recent developments and lasting consequences. Bull Am Math Soc 42(1):39–56

    Article  Google Scholar 

  81. Zarka J, Casier J (1981) Elastic-plastic response of a structure to cyclic loading: practical rule. In: Nemat-Nasser S (ed) Mechanics today, vol 6. Pergamon, New York

    Google Scholar 

  82. Zhang T, Raad L (2002) An eigen-mode method in kinematic shakedown analysis. Int J Plast 18:71–90

    Article  MATH  Google Scholar 

Download references

Acknowledgements

I cordially thank Prof. Dieter Weichert for the fruitful discussions and the support, which made this work possible.

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52074, Aachen, Germany

    Jaan-Willem Simon

Authors
  1. Jaan-Willem Simon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jaan-Willem Simon .

Editor information

Editors and Affiliations

  1. Department of Structural Engineering, National Technical University of Athens School of Civil Engineering, Athens, Greece

    Konstantinos Spiliopoulos

  2. Institute of General Mechanics, RWTH-Aachen University, Aachen, Germany

    Dieter Weichert

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Simon, JW. (2014). Shakedown Analysis of Kinematically Hardening Structures in n-Dimensional Loading Spaces. In: Spiliopoulos, K., Weichert, D. (eds) Direct Methods for Limit States in Structures and Materials. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6827-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-6827-7_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-6826-0

  • Online ISBN: 978-94-007-6827-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation