SpringerLink
Log in

Upper bound shakedown analysis with the nodal natural element method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

In this paper, a novel numerical solution procedure is developed for the upper bound shakedown analysis of elastic-perfectly plastic structures. The nodal natural element method (nodal-NEM) combines the advantages of the NEM and the stabilized conforming nodal integration scheme, and is used to discretize the established mathematical programming formulation of upper bound shakedown analysis based on Koiter’s theorem. In this formulation, the displacement field is approximated by using the Sibson interpolation and the difficulty caused by the time integration is solved by König’s technique. Meanwhile, the nonlinear and non-differentiable characteristic of objective function is overcome by distinguishing non-plastic areas from plastic areas and modifying associated constraint conditions and goal function at each iteration step. Finally, the objective function subjected to several equality constraints is linearized and the upper bound shakedown load multiplier is obtained. This direct iterative process can ensure the shakedown load to monotonically converge to the upper bound of true solution. Several typical numerical examples confirm the efficiency and accuracy of the proposed method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Xu BY, Liu XS (1985) Plastic limit analysis of structures. China Architecture and Building Press, Bei**g

    Google Scholar 

  2. Chen G, Liu YH (2006) Numerical theories and engineering methods for structural limit and Shakedown analyses. Science Press, Bei**g

    Google Scholar 

  3. Melan E (1938) Zur Plastizitä t des räumlichen Kontinuums. Ing-Arch 9:115–126

    Article  Google Scholar 

  4. Koiter WT (1956) A new general theorem on shakedown of elastic-plasitc structures. Proc Konink Ned Akad Wet B 59:24–34

    MathSciNet  MATH  Google Scholar 

  5. Kamenjarzh J, Merzljakov A (1994) On kinematic method in shakedown theory. I: duality of extremum problems. Int J Plast 10(4):363–380

    Article  MATH  Google Scholar 

  6. Kamenjarzh J, Merzljakov A (1994) On kinematic method in shakedown theory. II: modified kinematic method. Int J Plast 10(4):381–392

    Article  Google Scholar 

  7. Khoi VD, Yan AM, Hung ND (2004) A dual form for discretized kinematic formulation in shakedown analysis. Int J Solids Struct 41(1):267–277

    Article  MathSciNet  MATH  Google Scholar 

  8. Khoi VD, Yan AM, Hung ND (2004) A primal-dual algorithm for shakedown analysis of structures. Comput Methods Appl Mech Eng 193(42–44):4663–4674

    Google Scholar 

  9. Tran TN, Liu GR, Nguyen-Xuan H, Nguyen-Thoi T (2010) An edge-based smoothed finite element method for primal-dual shakedown analysis of structures. Int J Numer Meth Eng 82(7):917–938

    MathSciNet  MATH  Google Scholar 

  10. Tran TN (2011) A dual algorithm for shakedown analysis of plate bending. Int J Numer Meth Eng 86(7):862–875

    Article  MATH  Google Scholar 

  11. Stein E, Zhang GB, Konig 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 

  12. Zhang YG (1995) An iteration algorithm for kinematic shakedown analysis. Comput Methods Appl Mech Eng 127(1–4):217–226

    Article  MATH  Google Scholar 

  13. Hachemi A, Weichert D (1998) Numerical shakedown analysis of damaged structures. Comput Methods Appl Mech Eng 160:57–70

    Article  MATH  Google Scholar 

  14. Yan AM, Hung ND (2001) Kinematical shakedown analysis with temperature-dependent yield stress. Int J Numer Meth Eng 50(5):1145–1168

    Article  MATH  Google Scholar 

  15. Maier G, Pan LG, Perego U (1993) Geometric effects on shakedown and ratchetting of axisymmetric cylindrical shells subjected to variable thermal loading. Eng Struct 15:453–465

    Article  Google Scholar 

  16. Borino G, Polizzotto C (1996) Dynamic shakedown of structures with variable appended masses and subjected to repeated excitations. Int J Plast 12(2):215–228

    Article  MATH  Google Scholar 

  17. Tran TN, Kreissig R, Staat M (2009) Probabilistic limit and shakedown analysis of thin plates and shells. Struct Saf 31(1):1–18

    Article  Google Scholar 

  18. Carvelli V (2004) Shakedown analysis of unidirectional fiber reinforced metal matrix composites. Comput Mater Sci 31(1–2):24–32

    Article  Google Scholar 

  19. Li HX, Yu HS (2006) A non-linear programming approach to kinematic shakedown analysis of composite materials. Int J Numer Meth Eng 66(1):117–146

    Article  MATH  Google Scholar 

  20. Li HX, Yu HS (2006) A nonlinear programming approach to kinematic shakedown analysis of frictional materials. Int J Solids Struct 43(21):6594–6614

    Article  MATH  Google Scholar 

  21. Feng XQ, Liu XS (1997) On shakedown of three-dimensional elastoplastic strain-hardening structures. Int J Plast 12(10):1241–1256

    Article  Google Scholar 

  22. Feng XQ, Sun QP (2007) Shakedown analysis of shape memory alloy structures. Int J Plast 23(2):183–206

    Article  MathSciNet  MATH  Google Scholar 

  23. Carvelli V, Cen ZZ, Liu YH, Maier G (1999) Shakedown analysis of defective pressure vessels by a kinematic approach. Arch Appl Mech 69(9–10):751–764

    Article  MATH  Google Scholar 

  24. Chen HF, Ure J, Li TB, Chen WH, Mackenzie D (2011) Shakedown and limit analysis of \(90^{\circ }\) pipe bends under internal pressure, cyclic in-plane bending and cyclic thermal loading. Trans ASME J Press Vessel Technol 88(5–7):213–222

    Google Scholar 

  25. Gross-Wedge 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  Google Scholar 

  26. Shiau SH (2001) Numerical methods for shakedown analysis of pavements under moving surface loads. PhD thesis, University of Newcastle, NSW, Australia

  27. Konig JA, Maier G (1981) Shakedown analysis of elastoplastic structures: a review of recent developments. Nucl Eng Des 66(1):81–95

    Article  Google Scholar 

  28. Panzeca T (1992) Shakedown and limit analysis by the boundary integral equation method. Eur J Mech A-Solids 11(5):685–699

    MATH  Google Scholar 

  29. Zhang XF, Liu YH, Cen ZZ (2004) Boundary element methods for lower bound limit and shakedown analysis. Eng Anal Boundary Elem 28(8):905–917

    Article  MATH  Google Scholar 

  30. Liu YH, Zhang XF, Cen ZZ (2005) Lower bound shakedown analysis by the symmetric Galerkin boundary element method. Int J Plast 21(1):21–42

    Article  MATH  Google Scholar 

  31. Belytschko T, Lu YY, Gu L (1994) Element free Galerkin method. Int J Numer Meth Eng 37(2):229–256

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle method. Int J Numer Meth Fluids 20(8–9):1081–1106

    Article  MathSciNet  MATH  Google Scholar 

  33. Liszka TJ, Duarte CAM, Tworzydlo WW (1996) hp-meshless cloud method. Comput Methods Appl Mech Eng 139(1–4):263–288

    Article  MATH  Google Scholar 

  34. Atluri SN, Zhu T (1998) A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech 22(2):117–127

    Article  MathSciNet  MATH  Google Scholar 

  35. Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Meth Eng 43(5):839–887

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang XG, Liu XH, Song KZ, Lu MW (2001) Least-squares collocation meshless method. Int J Numer Meth Eng 51(9):1089– 1100

  37. Liu GR, Wang JG (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Meth Eng 54(11):1623–1648

    Article  MATH  Google Scholar 

  38. Gu L (2003) Moving Kriging interpolation and element free Galerkin method. Int J Numer Meth Eng 56(1):1–11

    Article  MATH  Google Scholar 

  39. Kim DW, Yoon YC, Liu WK, Belytschko T (2007) Extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative. J Comput Phys 221(1):370–394

    Article  MathSciNet  MATH  Google Scholar 

  40. Bessa MA, Foster JT, Belytschko T, Liu WK (2013) A meshfree unification: reproducing kernel peridynamics. Comput Mech. doi:10.1007/s00466-013-0969-x

  41. Yoon YC, Song JH (2013) Extended particle difference method for weak and strong discontinuity problems: part I. Derivation of the extended particle derivative approximation for the representation of weak and strong discontinuities. Comput Mech. doi:10.1007/s00466-013-0950-8

  42. Yoon YC, Song JH (2013) Extended particle difference method for weak and strong discontinuity problems: part II. Formulations and applications for various interfacial singularity problems. Comput Mech. doi:10.1007/s00466-013-0951-7

  43. Chen SS, Liu YH, Cen ZZ (2008) Lower bound shakedown analysis by using the element free Galerkin method and nonlinear programming. Comput Methods Appl Mech Eng 197:3911–3921

    Article  MATH  Google Scholar 

  44. Chen SS, Liu YH, Li J, Cen ZZ (2010) Performance of the MLPG method for static shakedown analysis for bounded kinematic hardening structures. Eur J Mech A/Solids 30(2):183–194

    Article  MathSciNet  Google Scholar 

  45. Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139(1–4):49–74

    Article  MathSciNet  MATH  Google Scholar 

  46. Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Meth Eng 50(2):435–466

    Article  MATH  Google Scholar 

  47. Chen JS, Yoon S, Wu CT (2002) Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Meth Eng 53(12):2587–2615

    Article  MATH  Google Scholar 

  48. Chen JS, Wu CT, Belytschko T (2000) Regularization of material instabilities by meshfree approximations with intrinsic length scales. Int J Numer Meth Eng 47(7):1303–1322

    Article  MATH  Google Scholar 

  49. Wang D, Chen JS (2004) Locking-free stabilized conforming nodal integration for Mindlin–Reissner plate. Comput Methods Appl Mech Eng 193(12–14):1065–1083

    MATH  Google Scholar 

  50. Wang D, Chen JS (2006) A locking-free meshfree curved beam formulation with the stabilized conforming nodal integration. Comput Mech 39(1):83–90

    Article  MATH  Google Scholar 

  51. Wang D, Chen JS (2008) A Hermite reproducing kernel approximation for thin plate analysis with sub-domain stabilized conforming integration. Int J Numer Meth Eng 74(3):368–390

    Article  MATH  Google Scholar 

  52. Wang D, Lin Z (2011) Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures. Comput Mech 48(1):47–63

    Article  MathSciNet  MATH  Google Scholar 

  53. Yoo JW, Moran B, Chen JS (2004) Stabilized conforming nodal integration in the natural-element method. Int J Numer Meth Eng 60:861–890

    Article  MATH  Google Scholar 

  54. Puso MA, Chen JS, Zywicz E, Elmer W (2008) Meshfree and finite element nodal integration methods. Int J Numer Meth Eng 74:416–446

    Article  MathSciNet  MATH  Google Scholar 

  55. Sibson R (1980) Vector identity for the dirichlet tessellation. Math Proc Camb Philos Soc 87:151–155

    Article  MathSciNet  MATH  Google Scholar 

  56. Braun J, Sambridge M (1995) A numerical method for solving partial differential equations on highly irregular evolving grids. Nature 376:655–660

    Article  Google Scholar 

  57. Sukumar N, Moran B, Semenov AY, Belikov VV (2001) Natural neighbour Galerkin methods. Int J Numer Meth Eng 50(1):1–27

    Article  MathSciNet  MATH  Google Scholar 

  58. Cueto E, Sukumar N, Calvo B, Martinez MA, Cegonino J, Doblare M (2003) Overview and recent advances in natural neighbour Galerkin methods. Arch Comput Methods Eng 10(4):307–384

  59. Gonzalez D, Cueto E, Doblare M (2004) Volumetric locking in natural neighbour Galerkin methods. Int J Numer Meth Eng 61(4):611–632

  60. Thiessen AH (1911) Precipitation averages for large areas. Mon Weather Rep 39:1082–1084

    Google Scholar 

  61. Belikov VV, Ivanov VD, Kontorovich VK, Korytnik SA (1997) The non-Sibsonian interpolation: a new method of interpolation of the values of a function on an arbitrary set of points. Comput Math Math Phys 37(1):9–15

    MathSciNet  Google Scholar 

  62. Hiyoshi H, Sugihara K (1999) Two generalizations of an interpolant based on Voronoi diagrams. Int J Shape Model 5(2):219–231

    Article  Google Scholar 

  63. Alfaro I, Yvonnet J, Chinesta F, Cueto E (2007) A study on the performance of natural neighbour-based Galerkin methods. Int J Numer Meth Eng 71(12):1436–1465

    Article  MATH  Google Scholar 

  64. Cueto E, Doblare M, Gracia L (2000) Imposing essential boundary conditions in the natural element method by means of density-scaled a-shapes. Int J Numer Meth Eng 49:519–546

    Article  MathSciNet  MATH  Google Scholar 

  65. Zhou ST, Liu YH (2012) Upper-bound limit analysis based on the natural element method. Acta Mech Sin 28(5):1398–1415

    Article  MathSciNet  Google Scholar 

  66. Himmelblau DM (1972) Appl Nonlinear Progr. McGraw-Hill Book Company, New York

    Google Scholar 

  67. Belytschko T (1972) Plane stress shakedown analysis by finite elements. Int J Mech Sci 14(9):619–625

    Article  Google Scholar 

  68. Corradi L, Zavelani A (1974) A linear programming approach to shakedown analysis of structures. Comput Methods Appl Mech Eng 3(1):37–53

    Article  MathSciNet  Google Scholar 

  69. Nguyen DH, Palgen L (1980) Shakedown analysis by displacement method and equilibrium finite elements. Trans Can Soc Mech Eng 61:34–40

    Google Scholar 

  70. Genna F (1988) A nonlinear inequality, finite element approach to the direct computation of shakedown load safety factors. Int J Mech Sci 30(10):769–789

    Article  MATH  Google Scholar 

  71. Zouain N, Borges L, Silveira JL (2002) An algorithm for shakedown analysis with nonlinear yield functions. Comput Methods Appl Mech Eng 191(23–24):2463–2481

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  73. Tin-Loi F, Ngo NS (2007) Performance of a p-adaptive finite element method for shakedown analysis. Int J Mech Sci 49(10):1168–1178

    Article  Google Scholar 

  74. Zhang TG, Raad L (2002) An eigen-mode method in kinematic shakedown analysis. Int J Plast 18(1):71–90

    Article  MATH  Google Scholar 

  75. Krabbenhoft K, Lyamin AV, Sloan SW (2007) Bounds to shakedown loads for a class of deviatoric plasticity models. Comput Mech 39(6):879–888

    Article  MathSciNet  Google Scholar 

  76. Chen SS (2009) Lower-bound limit and Shakedown analysis based on meshless methods. PhD thesis, Tsinghua University, Bei**g, People’s Republic of China

  77. Vu DK (2001) Dual limit and shakedown analysis of structures, dissertation. Universite de Liege, Belgium

Download references

Acknowledgments

S. Zhou is supported by the Chinese Postdoctoral Science Foundation (2013M540934). Y. Liu is supported by the National Science Foundation for Distinguished Young Scholars of China (11325211). D. Wang is supported by the National Natural Science Foundation of China (11222221). K. Wang is supported by the Project of Shandong Province Higher Educational Science and Technology Program (J13LG51).

Author information

Authors and Affiliations

  1. Institute of Nuclear and New Energy Technology, The Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education, Tsinghua University, Bei**g, 100084, China

    Shutao Zhou

  2. Department of Engineering Mechanics, AML, Tsinghua University, Bei**g, 100084, China

    Yinghua Liu

  3. Department of Civil Engineering, **amen University, **amen, 361005, Fujian, China

    Dongdong Wang

  4. Department of Mechanical and Electronic Engineering, **an Engineering Vocational Technical College, **an, 250200, Shandong, China

    Kai Wang

  5. Department of Thermal Engineering, Center for Combustion Energy, Tsinghua University, Bei**g, 100084, China

    Suyuan Yu

Authors
  1. Shutao Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yinghua Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dongdong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kai Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Suyuan Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yinghua Liu or Suyuan Yu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, S., Liu, Y., Wang, D. et al. Upper bound shakedown analysis with the nodal natural element method. Comput Mech 54, 1111–1128 (2014). https://doi.org/10.1007/s00466-014-1043-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-014-1043-z

Keywords

Search

Navigation