Skip to main content
SpringerLink
Log in

A Non-linear Programming Approach to~Shakedown Analysis for a General Yield~Condition

  • Chapter
Limit States of Materials and Structures

  • 949 Accesses

Abstract

A non-linear programming approach combined with the finite element method is developed to directly calculate the shakedown load of structures. The analysis is based on a general yield condition, which can be used for both isotropic materials (e.g. von Mises’ and Mohr-Coulomb’s criteria) and anisotropic materials (e.g. Hill’s criterion). By means of the associated flow rule, a general, non-linear yield criterion can be directly introduced into the kinematic shakedown theorem without linearization and a non-linear, purely kinematic formulation is obtained. The corresponding finite element formulation is developed as a non-linear mathematical programming problem subject to only a small number of equality constraints. So, the computational effort is very modest. A direct iterative algorithm is proposed to solve the resulting non-linear programming problem, which is validated by numerical simulations.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Bazaraa, M.S., Sherali, H.D., Shetty, C.M. Non-linear Programming: Theory and Algorithms. Wiley, New York (2006)

    Google Scholar 

  2. Boulbibane, M., Weichert, D. Application of shakedown theory to soils with non associated flow rules. Mech. Res. Commun., 24, 513–519 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Carvelli, V., Cen, Z.Z., Liu. Y.H., Maier, G. Shakedown analysis of defective pressure vessels by a kinematic approach. Arch. Appl. Mech., 69, 751–764 (1999)

    MATH  Google Scholar 

  4. Chen, H.F., Ponter, A.R.S. Shakedown and limit analyses for 3-D structures using the linear matching method. Int. J. Pres. Vess. Pip., 78, 443–451 (2001)

    Article  Google Scholar 

  5. Collins, I.F., Cliffe, P.F. Shakedown in frictional materials under moving surface loads. Int. J. Num. Analy. Meth. Geom., 11, 409–420 (1987)

    Article  MATH  Google Scholar 

  6. Collins, I.F., Boulbibane, M. Geomechanical analysis of unbound pavements based on shakedown theory. J. Geot. Geoen. Eng., ASCE, 126, 50–59 (2000)

    Article  Google Scholar 

  7. Feng, X.Q., Liu, X.S. On shakedown of three-dimensional elastoplastic strain-hardening structures. Int. J. Plasticity, 12, 1241–1256 (1997)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  10. Hamadouche, M.A., Weichert, D. Application of shakedown theory to soil dynamics. Mech. Res. Commun., 26, 565–574 (1999)

    Article  MATH  Google Scholar 

  11. Hartley, R. Linear and Non-linear Programming: An Introduction to Linear Methods in Mathematical Programming. Prentice Hall, New York (1985)

    Google Scholar 

  12. Himmelblau, D.M. Applied Non-linear Programming. McGraw-Hill Book Company, New York (1972)

    Google Scholar 

  13. Huh, H., Yang, W.H. A general algorithm for limit solutions of plane stress problems. Int. J. Solid. Struct., 28, 727–738 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Khoi, V.D., Yan, A.M., Huang, N.D. A dual form for discretized kinematic formulation in shakedown analysis. Int. J. Solid. Struct., 41, 267–277 (2004)

    Article  MATH  Google Scholar 

  15. Koiter, W.T. General theorems for elastic-plastic bodies. In: Sneddon IN, Hill R (eds). Progress in Solid Mechanics, North-Holland, Amsterdam, 165–221 (1960)

    Google Scholar 

  16. König, J.A. On upper bounds to shakedown loads. ZAMM, 59, 349–354 (1979)

    Article  MATH  Google Scholar 

  17. König, J.A. Shakedown of Elastic-Plastic Structure. Elsevier, Amsterdam (1987)

    Google Scholar 

  18. Li, H.X., Liu, Y.H., Feng, X.Q., Cen, Z.Z. Limit analysis of ductile composites based on homogenization theory. Proc. Roy. Soc. London A 459, 659–675 (2003)

    MATH  Google Scholar 

  19. Li, H.X., Yu, H.S. Kinematic limit analysis of frictional materials using non-linear programming. Int. J. Solid. Struct. 42, 4058–4076 (2005)

    Article  MATH  Google Scholar 

  20. Liu, Y.H., Cen, Z.Z., Xu, B.Y. A numerical method for plastic limit analysis of 3-D structures. Int. J. Solid. Struct. 32, 1645–1658 (1995)

    Article  MATH  Google Scholar 

  21. Luenberger, D.G., Ye, Y. Linear and Non-linear Programming. Springer, New York (2007)

    Google Scholar 

  22. Maier, G. Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: a finite element linear programming approach. Meccanica, 4, 250–260 (1969)

    Article  MATH  Google Scholar 

  23. Maier, G., Carvelli, G., Cocchetti, G. On direct methods of shakedown and limit analysis. Plenary Lecture, 4th Euromech Solid Mechanics Conference, Metz, June (2000)

    Google Scholar 

  24. Melan, E. Theorie Statisch Unbestimmter Tragwerke aus idealplastischem Baustoff. Sitzungsbericht der Akademie der Wissenschaften (Wien) Abt. IIA 195, 145–195 (1938)

    Google Scholar 

  25. Morelle, P. Numerical shakedown analysis of axisymmetric sandwich shells: an upper bound formulation. Int. J. Num. Meth. Eng., 23, 2071–2088 (1986)

    Article  MATH  Google Scholar 

  26. Ponter, A.R.S., Engelhardt, M. Shakedown limits for a general yield condition: implementation and application for a von Mises yield condition. Eur. J. Mech. A/Solids, 19, 423–445 (2000)

    Article  MATH  Google Scholar 

  27. Raad, L., Weichert, D., Najm, W. Stability of multilayer systems under repeated loads, Transp. Res. Record, 1207, 181–186 (1988)

    Google Scholar 

  28. Radovsky, B.S., Murashina, N.V. Shakedown of subgrade soil under repeated loading. Transp. Res. Record, 1547, 82–88 (1996)

    Article  Google Scholar 

  29. Sharp, R.W., Booker J.R. Shakedown of pavements under moving surface loads. J. Transp. Eng., ASCE, 110, 1–14 (1984)

    Article  Google Scholar 

  30. Shapiro, J.F. Mathematical Programming: Structures and Algorithms. A Wiley-Interscience Publication, New York (1979)

    MATH  Google Scholar 

  31. Shiau, S.H., Yu, H.S. Load and displacement prediction for shakedown analysis of layered pavements. Transp. Res. Record, 1730, 117–124 (2000)

    Article  Google Scholar 

  32. Stein, E., Zhang, G., Konig, J.A. Shakedown with non-linear strain-hardening including structural computation using finite element method. Int. J. Plasticity, 8, 1–31 (1992)

    Article  MATH  Google Scholar 

  33. Stein, E., Zhang, G., Huang, Y. Modeling and computation of shakedown problems for non-linear hardening materials. Comput. Meth. Appl. Mech. Eng., 103, 247–272 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  34. Weichert, D. Shakedown at finite displacement: a note on Melan’s theorem. Mech. Res. Commun., 11, 121–127 (1984)

    Article  MATH  Google Scholar 

  35. Weichert, D., Groß-Weege, J. On the influence of geometrical non-linearities on the shakedown of elastic-plastic structures. Int. J. Plasticity, 2, 135–148 (1986)

    Article  MATH  Google Scholar 

  36. Weichert, D., Hachemi, A., Schwabe, F. Shakedown analysis of composites. Mech. Res. Commun., 26, 309–318 (1999)

    Article  MATH  Google Scholar 

  37. Weichert, D., Hachemi, A., Schwabe, F. Application of shakedown analysis to the plastic design of composites. Arch. Appl. Mech., 69, 623–633 (1999)

    MATH  Google Scholar 

  38. Xue, M.D., Wang, X.F., Williams, F.W., Xu, B.Y. Lower-bound shakedown analysis of axisymmetric structures subjected to variable mechanical and thermal loads. Int. J. Mech. Sci., 39, 965–976 (1997)

    Article  MATH  Google Scholar 

  39. Yu, H.S., Hossain, M.Z. Lower bound shakedown analysis of layered pavements using discontinuous stress field. Comput. Meth. Appl. Mech. Eng., 167, 209–222 (1998)

    Article  MATH  Google Scholar 

  40. Yu, H.S. Three-dimensional analytical solutions for shakedown of cohesive-frictional materials under moving surface loads. Proc. Roy. Soc. London A, 461, 1951–1964 (2005)

    MATH  Google Scholar 

  41. Zhang, P.X., Lu, M.W., Hwang, K. A mathematical programming algorithm for limit analysis. Acta Mechan. Sinica, 7, 267–274 (1991)

    Article  Google Scholar 

  42. Zhang, Y.G. An iteration algorithm for kinematic shakedown analysis. Comput. Meth. Appl. Mech. Eng., 127, 217–226 (1995)

    Article  MATH  Google Scholar 

  43. Zhang, Y.G., Lu, M.W. An algorithm for plastic limit analysis. Comp. Meth. Appl. Mech. Engrg. 126, 333–341 (1995)

    Article  Google Scholar 

  44. Zouain, N., Borges, L., Silveira, J.L. An algorithm for shakedown analysis with non-linear yield functions. Comput. Meth. Appl. Mech. Eng., 191, 2463–2481 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Civil Engineering, University of Nottingham, Nottingham NG7 2RD, UK

    H.S. Yu

  2. Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK

    H.X. Li

Authors
  1. H.S. Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H.X. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Inst. Allgemeine Mechanik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany

    Weichert Dieter

  2. Dept. Engineering, University of Leicester, University Road, Leicester, United Kingdom LE1 7RH

    Ponter Alan

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Yu, H., Li, H. (2009). A Non-linear Programming Approach to~Shakedown Analysis for a General Yield~Condition. In: Dieter, W., Alan, P. (eds) Limit States of Materials and Structures. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9634-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-1-4020-9634-1_14

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-9633-4

  • Online ISBN: 978-1-4020-9634-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation