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Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models

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Abstract

The problem of estimating bounds to shakedown loads for problems governed by a class of deviatoric plasticity models including those of Hill, von Mises, and Tresca is addressed. Assuming that an exact elastic solution is available, an upper bound to the elastic shakedown multiplier can be obtained relatively easily using the plastic shakedown theorem. A procedure for computing this upper bound for arbitrary load domains is presented. A number of problems are then examined and it is found that the elastic shakedown factor is given as the minimum of the plastic shakedown factor and the classical limit load factor. Finally, some exact solutions to a number of two dimensional problems are given.

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References

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

    Article  Google Scholar 

  2. Boulbibane M, Collins IF, Ponter ARS, Weichert D (2005) Shakedown of unbound pavements. Int J Road Mater Pavement Des 6:81–96

    Google Scholar 

  3. Boulbibane M, Ponter ARS (2005) Extension of the linear matching method to geotechnical problems. Comput Meth Appl Mech Eng 194:4633–4650

    Article  MATH  Google Scholar 

  4. Carvelli V, Cen ZZ, Liu Y, Maier G (1999) Shakedown analysis of defective pressure vessels by a kinematic approachs. Ach Appl Mech 69:751–764

    MATH  Google Scholar 

  5. Chen HF, Ponter ARS (2001) Shakedown and limit analyses for 3-D structures using the linear matching method. Int J Press Vessels Piping 78:443–451

    Article  Google Scholar 

  6. Corradi L, Zavelani A (1974) A linear programming approach to shakedown analysis of structures. Comp Meth Appl Mech Eng 3:37–53

    Article  MathSciNet  Google Scholar 

  7. Dang van K, Maitournam MH (2002) On some recent trends in modelling of contact fatigue and wear in rail. Wear 253:219–227

    Article  Google Scholar 

  8. EN 13445-3 (2002) Unfired pressure vessels. CEN, Brussels.

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Gross-Weege J (1997) On the numerical assessment of the safetyfactor of elastic-plastic structures under variable loading. Int J Mech Sci 39:417–433

    Article  MATH  Google Scholar 

  12. Hamadouche MA (2002) Kinematic shakedown by the Norton \(\hat{\rm u}\) Hoff \(\hat{\rm u}\) Friaa regularising method and augmented Lagrangian. C.R. Mech 330:305–311

    MATH  Google Scholar 

  13. Johnson KL (1992) The application of shakedown principles in rolling and sliding contact. Eur J Mech/A Solids 11:155–172

    Google Scholar 

  14. Khoi VU, Yan A, Nguyen-Dang H (2004) A dual form of the discretized kinematic formulation in shakedown analysis. Int J Solids Struct 41:267–277

    Article  MATH  MathSciNet  Google Scholar 

  15. Krabbenhoft K, Damkilde L (2003) A general nolinear optimization algorithm for lower bound limit analysis. Int J Numer Meth Eng 56:165–184

    Article  MATH  Google Scholar 

  16. Krabbenhoft K, Lyamin AV, Hjiaj M, Sloan SW (2005) A new discontinuous upper bound limit analysis formulation. Int J Numer Meth Eng 63:1069–1088

    Article  MATH  Google Scholar 

  17. Lang H, Wirtz K, Heitzer M, Staat M, Oettel R (2001) Cyclic plastic deformation tests to verify fem–based shakedown analysis. Nucl Eng Des 206:227–239

    Google Scholar 

  18. Lekarp F, Dawson A (1998) Modelling permanent deformation behaviour of unbound granular materials. Construct Build Mater 12:9–18

    Article  Google Scholar 

  19. Liu Y, Zhang XZ, Cen ZZ (2005) Lower bound shakedown analysis by the symmetric Galerkin boundary element method. Int J Plast 21:21–42

    Article  MATH  Google Scholar 

  20. Lyamin AV, Sloan SW (2002a) Lower bound limit analysis using non-linear programming. Int J Numer Meth Eng 55:573–611

    Article  MATH  Google Scholar 

  21. Lyamin AV, Sloan SW (2002b) Upper bound limit analysis using linear finite elements and non-linear programming. Int J Numer Anal Meth Geomech 26:181–216

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  23. Nguyen-Dang H, Palgen L (1979) Shakedown analysis by displacement method and equilibrium elements. In: Proceedings of SMIRT 5, Berlin. Paper L3/3.

  24. Polizzotto C (1993) On the condition to prevent plastic shakedown of structures: Part II - the plastic shakedown limit load. J Appl Mech 60:20–25

    MATH  MathSciNet  Google Scholar 

  25. Poulos HG, Davis EH (1974) Elastic solutions for soil and rock mechanics. Wiley reprinted by Centre for Geotechnical Research, University of Sydney, 1991.

  26. Sharp RW, Booker JR (1984) Shakedown of pavements under moving surface loads. J Transp Eng 110:1–14

    Article  Google Scholar 

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

    Google Scholar 

  28. Sloan SW (1988) Lower bound limit analysis using finite elements and linear programming. Int J Numer Anal Meth Geomech 12:61–77

    Article  MATH  Google Scholar 

  29. Sloan SW (1989) Upper bound limit analysis using finite elements and linear programming. Int J Numer Anal Meth Geomech 13:263–284

    Article  MATH  Google Scholar 

  30. Sloan SW, Kleeman PW (1995) Upper bound limit analysis using discontinuous velocity fields. Comput Meth Appl Mech Eng 127:293–314

    Article  MATH  Google Scholar 

  31. Stein E, Huang Y (1994) An analytical method for shakedown problems with linear kinematic hardening materials. Int J Solids Struct 31:2433–2444

    Article  MATH  Google Scholar 

  32. Zhang G (1995) Einspielen und dessen numerische Behandlung von Flachentragwerken aus ideal plastischem bzw. kinematisch verfestingendem Material, Berich-nr. F92/i. Institut für Mechanik, University Hannover.

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. Zouain N, Silveira JL (1999) Extremum principles for bounds to shakedown loads. Eur J Mech A/Solids 18:879–901

    Article  MATH  MathSciNet  Google Scholar 

  37. Zouain N, Silveira JL (2001) Bounds to shakedown loads. Int J Solids Struct 38:2249–2266

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Geotechnical Research Group, Discipline of Civil, Surveying and Environmental Engineering, University of Newcastle, Newcastle, NSW, 2308, Australia

    K. Krabbenhøft, A. V. Lyamin & S. W. Sloan

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  1. K. Krabbenhøft
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  2. A. V. Lyamin
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  3. S. W. Sloan
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Correspondence to K. Krabbenhøft.

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Krabbenhøft, K., Lyamin, A.V. & Sloan, S.W. Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models. Comput Mech 39, 879–888 (2007). https://doi.org/10.1007/s00466-006-0076-3

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  • DOI: https://doi.org/10.1007/s00466-006-0076-3

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