Upper Bound Theorem

  • Sergey AlexandrovEmail author
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)


Plastic limit analysis is a convenient tool to find approximate solutions of boundary value problems. In general, this analysis is based on two principles associated with the lower bound and upper bound theorems. The latter is used in the present monograph to estimate the limit load for welded structures with and with no crack. A proof of the upper bound theorem for a wide class of material models has been given by Hill (1956). The only reliable output of upper bound solutions is the load required to initiate the process of plastic deformation. Any upper bound limit load is higher than or equal to the actual load. This statement becomes more complicated in the case of multiple load parameters. Upper bound solutions are not unique and their accuracy significantly depends on the kinematically admissible velocity field chosen.


Velocity Field Plastic Zone Limit Load Shear Yield Stress Velocity Jump 
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  1. S. Alexandrov, Plastic limit load solutions for highly undermatched welded joints, in Welding: Processes, Quality, and Applications, ed. by R.J. Klein (Nova Science Publisher, Hauppauge, 2011)Google Scholar
  2. S. Alexandrov, M. Kocak, Effect of three-dimensional deformation on the limit load of highly weld strength undermatched specimens under tension. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 222, 107–115 (2008)CrossRefGoogle Scholar
  3. S. Alexandrov, O. Richmond, On estimating the tensile strength of an adhesive plastic layer of arbitrary simply connected contour. Int. J. Solids Struct. 37, 669–686 (2000)zbMATHCrossRefGoogle Scholar
  4. S. Alexandrov, O. Richmond, Singular plastic flow fields near surfaces of maximum friction stress. Int. J. Non-Linear Mech. 36, 1–11 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. S. Alexandrov, G. Mishuris, W. Mishuris et al., On the dead zone formation and limit analysis in axially symmetric extrusion. Int. J. Mech. Sci. 43, 367–379 (2001)zbMATHCrossRefGoogle Scholar
  6. A.N. Bramley, UBET and TEUBA: fast methods for forging simulation and perform design. J. Mater. Process. Technol. 116, 62–66 (2001)CrossRefGoogle Scholar
  7. C.C. Chang, A.N. Bramley, Forging perform design using a reverse simulation approach with the upper bound finite element procedure. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 214, 127–136 (2000)CrossRefGoogle Scholar
  8. D.C. Drucker, W. Prager, H.J. Greenberg, Extended limit design theorems for continuous media. Q. J. Appl. Mech. 9, 381–389 (1952)MathSciNetzbMATHGoogle Scholar
  9. S. Hao, A. Cornec, K.-H. Schwalbe, Plastic stress-strain fields and limit loads of a plane strain cracked tensile panel with a mismatched welded joint. Int. J. Solids Struct. 34, 297–326 (1997)zbMATHCrossRefGoogle Scholar
  10. R. Hill, The Mathematical Theory of Plasticity (Clarendon Press, Oxford, 1950)zbMATHGoogle Scholar
  11. R. Hill, New horizons in the mechanics of solids. J. Mech. Phys. Solids 5, 66–74 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  12. P.G. Hodge Jr, C.-K. Sun, General properties of yield-point load surfaces. Trans. ASME. J. Appl. Mech. 35, 107–110 (1968)CrossRefGoogle Scholar
  13. A.G. Miller, Review of limit loads of structures containing defects. Int. J. Press. Vessels Pip. 32, 197–327 (1988)CrossRefGoogle Scholar
  14. J. Rychlewski, Plain plastic strain for jump of non-homogeneity. Int. J. Non-Linear Mech. 1, 57–78 (1966)CrossRefGoogle Scholar
  15. A. Sawczuk, P.G. Hodge Jr, Limit analysis and yield-line theory. Trans. ASME J. Appl. Mech. 35, 357–362 (1968)CrossRefGoogle Scholar
  16. K.-H. Schwalbe, On the beauty of analytical models for fatigue crack propagation and fracture—a personal historical review. J. ASTM Int. 7(8), Paper ID 102713 (2010)CrossRefGoogle Scholar
  17. W.R.D. Wilson, A simple upper-bound method for axisymmetric metal forming problems. Int. J. Mech. Sci. 19, 103–112 (1977)CrossRefGoogle Scholar
  18. W.-C. Yeh, Y.-S. Yang, A variational upper-bound method for plane strain problems. Trans. ASME J. Manuf. Sci. Eng. 118, 301–309 (1996)CrossRefGoogle Scholar
  19. U. Zerbst, R.A. Ainsworth, K.-H. Schwalbe, Basic principles of analytic flaw assessment methods. Int. J. Press. Vessels Pip. 77, 855–867 (2000)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.A.Yu. Ishlinskii Institute for Problems in MechanicsRussian Academy of SciencesMoscowRussia

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