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A Historical View on Shakedown Theory

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The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering

Part of the book series: Lecture Notes in Applied Mathematics and Mechanics ((LAMM,volume 1))

Abstract

Plastic design started in the early 20th century with the arrival of steel constructions in civil engineering. The objective was to determine the load carrying capacity in particular of steel bridges and steel skeleton buildings beyond the elastic limit. The related studies were first focused on monotonically increasing, “dead” loading. From this point of view they were directly related to the ancient question of determining the load carrying capacity of masonry construction like domes of churches.

It was in the extension of these studies that the problem of plastic design under variable loads came into the picture. Martin Grüning was the first to be attracted by the beneficial effect of limited plastic deformation in redundant elements in hyperstatic structures and opened the door to the fascinating theory of shakedown.

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References

  1. Benvenuto, E.: An Introduction to the History of Structural Mechanics, Part I, Statics and Resistance of Solids. Springer (1991)

    Google Scholar 

  2. Poleni, G.: Memorie istoriche della gran cupola del Tempio Vaticano, Padova (1748)

    Google Scholar 

  3. Heyman, J.: Poleni’s problem. Proc. Instn Civ. Engrs, Part 1 84, 737–759 (1988)

    Article  Google Scholar 

  4. Maier-Leibnitz, H.: Beitrag zur Frage der tatsächlichen Tragfähigkeit einfacher, durchlaufender Balkenträger aus Baustahl St. 37 und Holz, Bautechnik, 6. Jg. Heft 1, 11–14 und Heft 2, 27–31 (1928)

    Google Scholar 

  5. Schaim, J.H.: Der durchlaufende Träger unter Berücksichtigung der Plastizität, Stahlbau (1930)

    Google Scholar 

  6. Fritzsche, J.: Die Tragfähigkeit von Balken aus Baustahl bei beliebig oft wiederholter Belastung. Bauingenieur 12, 827 (1931)

    Google Scholar 

  7. Grüning, M.: Die Statik des ebenen Tragwerks. Springer, Berlin (1925)

    Book  MATH  Google Scholar 

  8. Grüning, M.: Die Tragfähigkeit statisch unbestimmter Tragwerke aus Stahl bei beliebig häufig wiederholter Belastung. Springer, Berlin (1926)

    Google Scholar 

  9. Bleich, H.: Über die Bemessung statisch unbestimmter Stahltragwerke unter Berücksichtigung des elastisch-plastischen Verhaltens des Baustoffes. Der Bauingenieur, Heft 19/20, 261–267 (1932)

    Google Scholar 

  10. Bleich, F., Melan, E.: Die gewöhnlichen und partiellen Differentialgleichungen der Baustatik. Springer (1927)

    Google Scholar 

  11. Melan, E.: Der Spannungszustand eines Mises-Hencky’schen Kontinuums bei veränderlicher Belastung, Sitzungsberichte, vol. 147, pp. 73–87. Akademie der Wissenschaften, Wien (1938)

    Google Scholar 

  12. Prager, W.: Problem Types in the Theory of Perfectly Plastic Materials. Journal of the Aeronautical Sciences, 337–341 (June 1948)

    Google Scholar 

  13. Melan, E.: Theorie unbestimmter Systeme aus idealplastischem Baustoff. Sitzungsberichte, Akademie der Wissenschaften 145, 195–218 (1936)

    Google Scholar 

  14. Symonds, P.S., Prager, W.: Elastic-plastic analysis of structures subjected to loads varying arbitrarily between prescribed limits. J. Appl. Mech. 17, 315–324 (1950)

    MathSciNet  Google Scholar 

  15. Symonds, P.S.: Shakedown in Continuous Media. J. Appl. Mech. 17, 85–89 (1951)

    MathSciNet  Google Scholar 

  16. Neal, B.G.: Plastic-Collapse and Shake-Down Theorems for Structures of Strain Hardening Material. J. Aero. Sci. 17, 297–307 (1950)

    Article  MathSciNet  Google Scholar 

  17. Symonds, P.S., Neal, B.G.: Recent progress in the plastic methods of structural analysis. J. Franklin Inst. 252(6), 469–492 (1951)

    Article  Google Scholar 

  18. Symonds, P.S., Neal, B.G.: The Calculation of Failure Loads on Plane Frames unbbder Arbitrary Loading Programs. J. Inst. Civil Engrs. 35, 41–61 (1951)

    Article  Google Scholar 

  19. Neal, B.G., Symonds, P.S.: A Method for Calculating the Failure Load for a Framed Structure Subjected to Fluctuating Loads. J. Inst. Civil Engrs. 35, 186 (1951)

    Article  Google Scholar 

  20. Symonds, P.S.: An early upper bound method for shakedown. In: Weichert, D., Maier, G. (eds.) Inelastic Analysis of Structures under Variable Loads – Theory and Engineering Applications. Series, Solid Mechanics and its Applications, vol. 83, pp. 1–9. Kluwer Academic Publishers, Dordrecht (2000)

    Chapter  Google Scholar 

  21. Baker, J., Horne, M.R., Heyman, J.: The steel Skeleton, vol. II, ch. 9. Cambridge University Press (1956)

    Google Scholar 

  22. Horne, M.R.: The Effect of Variable Repeated Loads in the Plastic Theory of Structures. Research, Engineering Structures Supplement, Colston Papers II, 141 (1949)

    Google Scholar 

  23. Koiter, W.T.: A new general theorem on shakedown of elastic-plastic structures. Proc. Kon. Ne. Ak. Wet. B59, 24–34 (1956)

    MathSciNet  Google Scholar 

  24. Koiter, W.T.: General theorems for elastic-plastic structures. In: Sneddon, I.N., Hill, R. (eds.) Progress in Solid Mechanics, ch. IV, pp. 165–221. North-Holland Publ. Co., Amsterdam (1960)

    Google Scholar 

  25. Symonds, P.S.: Basic Theorems in the Plastic Theory of Structures. J. Aero. Sci. 17, 669–670 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maier, G.: Shakedown theory in perfect elastoplasticity with associated and non-associated flow-laws, a finite element, linear programming approach. Meccanica 4(3), 250–260 (1969)

    Article  MATH  Google Scholar 

  27. Maier, G.: A shakedown matrix theory allowing for workhardening and second-order geometric effects. In: Sawczuk, A. (ed.) International Symposium, Warsaw, August 30-September 2. Foundations of Plasticity, vol. 1, pp. 417–433. Noorhoff, Leyden (1972, 1973)

    Google Scholar 

  28. Corradi, L., Maier, G.: Dynamic non-shakedown theorem for elastic perfectly-plastic continua. Journal of the Mechanics and Physics of Solids 22, 401–413 (1974)

    Article  MATH  Google Scholar 

  29. König, A., Maier, G.: Shakedown analysis of elastoplastic structures, a review of recent developments. Nuclear Engineering and Design 66, 81–95 (1981)

    Article  Google Scholar 

  30. Maier, G.: A generalization to nonlinear hardening of the first shakedown theorem for discrete elastic-plastic structures. Rendic. Acc. Naz. dei Lincei, Serie Ottava, 161–174 (1988)

    Google Scholar 

  31. Maier, G., Novati, G.: Dynamic shakedown and bounding theory for a class of nonlinear hardening discrete structural models. Int. J. of Plasticity 6(5), 551–572 (1990)

    Article  MATH  Google Scholar 

  32. Maier, G., Pan, L., Perego, U.: Geometric effects on shakedown and ratchetting of axisymmetric cylindrical shells subjected to variable thermal loading. Engineering Structures 15(6), 453–466 (1993)

    Article  Google Scholar 

  33. Corigliano, A., Maier, G., Pycko, S.: Dynamic shakedown analysis and bounds for elastoplastic structures with nonassociative, internal variable constitutive laws. Int. J. Solids and Structures 32(21), 3145–3166 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Cocchetti, G., Maier, G.: Static shakedown theorems in piecewise linearized poroplasticity. Archive of Applied Mechanics 68, 651–661 (1998)

    Article  MATH  Google Scholar 

  35. Carvelli, V., Maier, G., Taliercio, A.: Shakedown analysis of periodic heterogeneous materials by a kinematic approach. Journal of Mechanical Engineering 50(4), 229–240 (1999)

    Google Scholar 

  36. Carvelli, V., Cen, Z.Z., Liu, Y., Maier, G.: Shakedown analysis of defective pressure vessels by a kinematic approach. Archive of Applied Mechanics 69, 751–764 (1999)

    Article  MATH  Google Scholar 

  37. Cocchetti, G., Maier, G.: A shakedown theorem in poroplastic dynamics. Rend. Mat., Accademia Nazionale dei Lincei, s. 9 13, 43–53 (2002)

    Google Scholar 

  38. Maier, G., Cocchetti, G.: Fundamentals of direct methods in poroplasticity. In: Weichert, D., Maier, G. (eds.) Inelastic Behaviour of Structures Under Variable Repeated Loads, Direct Analysis Methods, CISM, pp. 91–113. Springer, Wien (2002)

    Google Scholar 

  39. Maier, G., Carvelli, V.: A kinematic method for shakedown and limit analysis of periodic composites. In: Weichert, D., Maier, G. (eds.) Inelastic Behaviour of Structures Under Variable Repeated Loads, Direct Analysis Methods, CISM, pp. 115–132. Springer, Wien (2002)

    Google Scholar 

  40. Maier, G., Pastor, J., Ponter, A.R.S.: Direct Methods in Limit and Shakedown Analysis. In: De Borst, R., Mang, H.A. (eds.) Numerical and Computational Methods, vol. 3. Elsevier-Pergamon, Amsterdam (2003); Milne, I., Ritchie, R.O., Karihaloo, B. (eds.) Comprehensive Structural Integrity

    Google Scholar 

  41. Cocchetti, G., Maier, G.: Elastic-plastic and limit-state analyses of frames with softening plastic-hinge models by mathematical programming. Int. J. Solids and Structures 40, 7219–7244 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Martin, J.: Plasticity, Fundamentals and General Results. MIT Press (1975)

    Google Scholar 

  43. Ponter, A.R.S., Williams, J.J.: Work bounds and the associated deformation of cyclically loaded cree** structures. ASME, J. Appl. Mech. 40, 921–927 (1973)

    Article  Google Scholar 

  44. Ponter, A.R.S.: On the relationship between plastic shakedown and the repeated loading of cree** structures. Trans ASME, J. Appl. Mech., Series E 38, 437–440 (1971)

    Google Scholar 

  45. Ponter, A.R.S.: Deformation, displacement and work bounds for structures in a state of creep and subject to variable loading. Trans. ASME, J. Appl. Mech, 39, Series E, 953–959 (1972)

    Google Scholar 

  46. Ponter, A.R.S.: On the creep modified shakedown limit. In: Ponter, A.R.S., Hayhurst, D.R. (eds.) 3rd IUTAM Symposium on “Creep in Structures”, Leicester University (September 1980), pp. 264–278. Springer (1981)

    Google Scholar 

  47. Ponter, A.R.S.: An upper bound on the small displacement of elastic-plastic structures. Trans. ASME, J. Appl. Mech. 39, Series E, 959–964 (1972)

    Google Scholar 

  48. Ponter, A.R.S.: General displacement and work bounds for dynamically loaded bodies. J. Mech. Phys. Solids 23, 151–163 (1975)

    Article  MathSciNet  Google Scholar 

  49. Ponter, A.R.S.: A general shakedown theorem for inelastic materials. In: Proc. SMiRT-3, London, paper L5/2 (1975)

    Google Scholar 

  50. Bree, J.: Elasto-plastic behavior of thin tubes subjected to internal pressure and in-termittent high-heat fluxes with applications to Fast Reactor Fuel Elements. J. Strain Analysis, 2(3), 226–238 (1967)

    Google Scholar 

  51. Ponter, A.R.S., Karadeniz, S.: An extended shakedown theory for structures that suffer cyclic thermal loading, Part I: Theory. Trans. ASME, J. Appl. Mech. 52, 877–882 (1985)

    Google Scholar 

  52. Ponter, A.R.S., Karadeniz, S.: An extended shakedown theory for structures that suffer cyclic thermal loading, Part II: Applications. Trans. ASME, J. of Appl. Mech. 52, 883–889 (1985)

    Google Scholar 

  53. Ponter, A.R.S., Carter, K.F.: The ratchetting of shells subjected to severe thermal loading. In: Tooth, A.S., Spence, J. (eds.) Applied Solid Mechanics - 2, pp. 303–320. Elsevier Applied Science (1987)

    Google Scholar 

  54. Ponter, A.R.S., Cocks, A.C.F.: The incremental strain growth of an elastic-plastic body loaded in excess of the shakedown limit. Trans. ASME, J. Appl. Mech. 51(3), 465–469 (1984)

    Article  MATH  Google Scholar 

  55. Ponter, A.R.S., Cocks, A.C.F.: The incremental strain growth of elastic-plastic bodies subjected to high levels of cyclic thermal loading. Trans. ASME J. Appl. Mech. 51(3), 470–474 (1984)

    Article  MATH  Google Scholar 

  56. Ponter, A.R.S., Karadeniz, S., Carter, K.F.: The computation of shakedown limits for structural components subjected to variable thermal loading - Brussels Diagrams. Directorate General for Science, Research and Development, Office for Official Publications of the E.C., Report EUR12686EN, Brussels, 170 pages (1990)

    Google Scholar 

  57. Ponter, A.R.S., Chen, H.F., Chiavarella, M., Specchia, G.: Shakedown analysis for rolling and sliding contact problems. Int. J. Sol. Struct. 43, 4201–4219 (2001)

    Article  Google Scholar 

  58. Ponter, A.R.S., Leckie, F.A.: Bounding Properties of Metal Matrix composites subjected to Cyclic loading. Jn. Mech. Phys. Solids 46, 697–717 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  59. Ponter, A.R.S., Leckie, F.A.: On the behaviour of metal matrix composites subjected to cyclic thermal loading. Jn. Mech. Phys. Solids 46, 2183–2199 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  60. Ponter, A.R.S., Engelhardt, M.: Shakedown limits for a general yield condition. European Journal of Mechanics, A/Solids 19, 423–445 (2000)

    Article  MATH  Google Scholar 

  61. Ponter, A.R.S., Chen, H.F.: A Minimum theorem for cyclic load in excess of shakedown, with applications to the evaluation of a ratchet limit. Euro. Jn. Mech., A/Solids 20, 539–553 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  62. Chen, H.F., Ponter, A.R.S., Ainsworth, R.A.: The Linear Matching Method ap-plied to the high temperature life assessment of structures, Part 1. Assessments involving constant residual stress fields. Int. J. Pressure Vessels and Pi** 83, 123–135 (2006)

    Article  Google Scholar 

  63. Chen, H.F., Ponter, A.R.S., Ainsworth, R.A.: The Linear Matching Method applied to the high temperature life assessment of structures, Part 2. Assessments beyond shakedown involving changing residual stress fields. Int. J. Pressure Vessels and Pi** 83, 136–147 (2006)

    Google Scholar 

  64. Gokhfeld, D.A.: On the possibility of increase of plastic deformation due to thermal cycling effects. Calculations of Strength (7) (1961) (in Russian)

    Google Scholar 

  65. Gokhfeld, D.A.: Bearing capacity of structures under thermal cycles. Mashinostroenie publ., Moscow, p. 259 (1970) (in Russian)

    Google Scholar 

  66. Gokhfeld, D.A., Cherniavsky, O.F.: Theory of shakedown and strain accumulation under thermal cyclinges. In: Proceedings of the All-USSR. Symp. on Low-cycle Fatigue at Elevated Temperatures, Chelyabinsk, vol. (3), pp. C3–C31 (1974)

    Google Scholar 

  67. Gokhfeld, D.A., FCherniavsky, O.: Limit analysis of structures at thermal cycling. Sijthoff and Noordhoff. Int. Publ. Alphen aan den Rijn, The Netherlands – Rockville, USA (1980)

    Google Scholar 

  68. G. Ceradini, Sull’ adattamento dei corpi elasto-plastic isoggetti ad azioni dinamiche. Giornaledel Genio Civile 415, 239–258 (1969)

    Google Scholar 

  69. Polizzotto, C.: Workhardening adaptation of rigid-plastic structures. Meccanica 10, 280–288 (1975)

    Google Scholar 

  70. Polizzotto, C.: A unified approach to quasi-static shakedown problems for elasto-plastic solids with piecewise linear yield surfaces. Meccanica 13, 109–120 (1978)

    Google Scholar 

  71. Polizzotto, C.: On workhardening adaptation of discrete structures under dynamic loadings. Arch. Mech. Stos. 32, 81–99 (1980)

    Google Scholar 

  72. Polizzotto, C.: A unified treatment of shakedown theory and related bounding techniques. S.M. Arch. 7, 19–75 (1982)

    MATH  Google Scholar 

  73. Polizzotto, C.: A convergent bounding principle for a class of elastoplastic strain-hardening solids. Int. J. Plasticity 2, 357–370 (1986)

    Article  Google Scholar 

  74. Polizzotto, C.: On the condition to prevent plastic shakedown of structures ASME. J. Appl. Mech. I, II(60), 15–25, 318–330 (1993)

    Google Scholar 

  75. Polizzotto, C., Borino, G.: Shakedown and steady state responses of elastic-plastic solids in large displacements. Int. J. Sol. Struct. 33, 3415–3437 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  76. Polizzotto, C., Borino, G., Caddemi, S., Fuschi, P.: Shakedown problems for material models with internal variables. Int. J. Mech. Sci. 35, 787–801 (1993)

    Article  MATH  Google Scholar 

  77. Polizzotto, C., Borino, G., Fuschi, P.: An extended shakedown theory for elastic-damaged models. Eur. J. Mech. - A/Solids 15, 825–858 (1996)

    MathSciNet  MATH  Google Scholar 

  78. Polizzotto, C., Borino, G., Fuschi, P.: Weak forms of shakedown for elastic-plastic structures exhibiting ductile damage. Meccanica 36, 49–66 (2001)

    Google Scholar 

  79. Życzkowski, M.: Combined Loadings in the Theory of Plasticity. Polish-Scientific Publ. (1981)

    Google Scholar 

  80. König, J.A.: Shakedown of strainhardening structures. 1st Canad. Cong. Appl. Mech., Quebec. (1967)

    Google Scholar 

  81. König, J.A.: A shakedown theorem for temperature dependent elastic moduli. Bull. Acad. Polon. Sci. Sér. Sci. Tech. 17, 161–165 (1969)

    Google Scholar 

  82. König, J.A.: Deflection bounding at shakedown under thermal and mechanical loadings. In: Second SMiRT Conf., Berlin, paper L7/3 (1973)

    Google Scholar 

  83. König, J.A.: A method of shakedown analysis of frames and arches. Int. J. Sol. Struct., 327–344 (1971)

    Google Scholar 

  84. König, J.A.: Shakedown deflections, a finite element approach. Teoret. I Priloż. Meh. 3, 65–69 (1972)

    Google Scholar 

  85. König, J.A.: Shakedown of Elastic-Plastic Structures. Elsevier, Amsterdam (1987)

    Google Scholar 

  86. Borkowski, A.: Analysis of Skeletal Structural Systems in the Elastic, and Plastic Range. PWN and Elsevier, Warsaw (1988)

    Google Scholar 

  87. Borkowski, A., Kleiber, M.: On a numerical approach to shakedown analysis of structures. Comput. Methods Appl. Mech. Engng. 22, 101 (1980)

    Google Scholar 

  88. Dorosz, S.: An improved upper bound to maximum deflections of elasticplastic structures at shakedown. J. Struct. Mech. 6, 267–287 (1978)

    Article  Google Scholar 

  89. Dorosz, S., König, J.A., Sawczuk, A., Biegus, A., Kowal, Z., Seidel, W.: Deflections of elastic-plastic hyperstatic beams under cyclic loading. Arch. Mech. 33, 611–624 (1981)

    Google Scholar 

  90. Dorosz, S., König, J.A.: Iterative method of evaluation of elastic-plastic deflections of hyperelastic structures. Ing. Archiv. 55, 202–222 (1985)

    Google Scholar 

  91. Janas, M., Pycko, S., Zwoliński, J.: A min-max procedure for the shakedown analysis of skeletal structures. Int. Journal of Mechanical Sciences 37, 629–649 (1995)

    Article  MATH  Google Scholar 

  92. Janas, M., König, J.A.: A cylindrical tank response as an example of shakedown of non-Clapeyronian systems. Arch. Mech. 43, 49–56 (1991)

    Google Scholar 

  93. Kleiber, M., König, J.A.: Incremental shakedown analysis in the case of thermal effects. J. Num. Methods Engng. 20, 15–67 (1984)

    Article  Google Scholar 

  94. König, A., Kleiber, M.: On a new method of shakedown analysis. Bull. Acad. Pol.- Sci., Ser. Sci. Technol. 26, 165 (1978)

    Google Scholar 

  95. König, J.A., Siemaszko, A.: Strainhardening effects in shakedown process. Ing.-Archiv 58, 58–66 (1988)

    Google Scholar 

  96. König, J.A., Maier, G.: Shakedown of elastoplastic structures, a review of recent developments. Nuccl. Eng. Design 66, 81–95 (1981)

    Google Scholar 

  97. König, J.A., Pycko, S.: Shakedown analysis in the case of imposed displacements. Mech., Teor. Stos. 28, 101–108 (1990)

    Google Scholar 

  98. Mróz, Z.: On the theory of steady plastic cycles in structures. In: Proc. 1st SMIRT, L6, 489501 (1971)

    Google Scholar 

  99. Orkisz, J., Orringer, O., Holowinski, M., Pazdanowski, M., Cecot, W.: Discrete analysis of actual residual stress resulting from cyclic loadings. Comput. Struct. 35, 397 (1990)

    Google Scholar 

  100. Pycko, S., Mróz, Z.: Alternative approach to shakedown as a solution of min-max problem. Acta Mechanica 93, 205 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  101. Pycko, S., König, J.A.: Elastic-plastic structures subjected to variable repeated imposed displacements and mechanical loads. Int. J. Plasticity 8, 603–618 (1992)

    Article  MATH  Google Scholar 

  102. Sawczuk, A.: Evaluation of upper bounds to shakedown loads of shells. J. Mech. Phys. Solids 17, 291–301 (1969)

    Article  Google Scholar 

  103. Sawczuk, A.: Shakedown analysis of elastic-plastic structures. Nucl. Eng. Design 28, 121–136 (1974)

    Article  Google Scholar 

  104. Sawczuk, A.: Mechanics and Plasticity of Structures. Ellis-Horwood/PWN, Chichester/Warsaw (1989)

    Google Scholar 

  105. Siemaszko, A., Mróz, Z.: Sensitivity of plastic optimal structures to imperfections and non-linear geometrical effects. Structural Opimization 3, 99–105 (1991)

    Google Scholar 

  106. Siemaszko, A., König, J.A.: Analysis of stability of incremental collapse of skeletal structures. J. Struct. Mech. 13, 301–321 (1985)

    Article  Google Scholar 

  107. Siemaszko, A., König, J.A.: Shakedown optimisation accounting for non-linear geometrical effects. ZAMM 71, 294–296 (1991)

    Google Scholar 

  108. Zwoliński, J., Bielawski, G.: An optimal selection of residual stress for determination limit and shakedown multiplier. In: Proc. of Conf. on Comp’. Meth. struct. Mecł., Jadwisin, p. a59 (1987) (in Polish)

    Google Scholar 

  109. Skoczeń, B., Skrzypek, J., Bielski, J.: Shakedown and inadaptation mechanisms of bellows subject to constant pressure and cyclic axial forces. Mech. Struct. Mach. 20, 119 (1992)

    Article  Google Scholar 

  110. Skoczeń, B., Skrzypek, J.: Inadaptation mechanisms in bellows subject to sustained pressure and cyclic axial loadings in terms of finite deformations. In: Mróz, Z., Weichert, D., Dorosz, S. (eds.) Inelastic Behaviour of Structures under Variable Loads, pp. 341–361. Kluwer Academic Publishers (1995)

    Google Scholar 

  111. Mahrenholtz, O., Leers, K., König, J.A.: Shakedown of tubes: a theoretical analysis and experimental investigations. In: Reid, S.R. (ed.) Metal Forming and Impact Mechanics, W. Johnson Commemorative Volume, pp. 155–172. Pergamon Press (1984)

    Google Scholar 

  112. Leers, K.: Experimentelle und theoretische Shakedownuntersuchung an Rohren. VDI-Verlag, Düsseldorf (1985)

    Google Scholar 

  113. Leers, K., Klie, W., König, J.A., Mahrenholtz, O.: Experimental investigations of shakedown of tubes. In: Sawczuk, A., Bianchi, G. (eds.) Plasticity Today, pp. 259–275. Elevier Appl. Sci. Publ., London (1985)

    Google Scholar 

  114. Stein, E., Zhang, G., Mahnken, R., König, J.A.: Micromechanical Modeling and Computation of Shakedown with Nonlinear Kinematic Hardening including examples for 2-D problems. In: Proc. CSME Mechanical Engineering Forum, Toronto, pp. 425–430 (1990)

    Google Scholar 

  115. Stein, E., Zhang, G., König, J.A.: Shakedown with Nonlinear Hardening including Structural Computation using Finite Element Method. Int. J. Plasticity 8, 1–31 (1992)

    Article  MATH  Google Scholar 

  116. Stein, E., Zhang, G., Mahnken, R.: Shakedown Analysis for Perfectly Plastic and Kinematic Hardening Materials. In: Stein, E. (ed.) Progress in Computational Analysis of Inelastic Structures, pp. 175–244. Springer (1993)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  118. Huang, Y., Stein, E.: Shakedown of a cracked body consisting of kinematic hardening material. Engineering Fracture Mechanics 54, 107–112 (1996)

    Article  Google Scholar 

  119. Huang, Y., Stein, E.: Prediction of the fatigue threshold for a cracked body by using shakedown theory. Fatigue & Fracture of Engineering Materials & Structures 18(3), 363–370 (1995)

    Article  Google Scholar 

  120. Huang, Y., Stein, E.: Shakedown of a CT-specimen with St52-steel: Experimental, analytical and numerical investigations. Journal of Strain Analysis 30(4), 283–289 (1995)

    Article  Google Scholar 

  121. Wiechmann, K., Barthold, F.-J., Stein, E.: Optimization of elasto-plastic structures using the finite element method. In: 2nd World Congress of Structural and Multidisciplinary Optimisation, pp. 1013–1018 (1997)

    Google Scholar 

  122. Wiechmann, K., Barthold, F.-J., Stein, E.: Shape Optmization under Shakedown Constraints. In: Weichert, D., Maier, G. (eds.) Inelastic Analysis of Structures under Variable Loads, pp. 49–68. Kluwer Academic Publishers (2000)

    Google Scholar 

  123. Stein, E.: Private communication

    Google Scholar 

  124. Rafalski, P.: Minimun Principles in Plasticity, Mitteilungen aus dem Institut für Mechanik Bochum. Band 13 (1978)

    Google Scholar 

  125. Weichert, D.: On the influence of geometrical nonlinearities on the shakedown of elastic-plastic structures. Int. J. Plast. 2(2), 135–148 (1986)

    Article  MATH  Google Scholar 

  126. Weichert, D., Groß-Weege, J.: 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 (1989)

    Article  Google Scholar 

  127. Raad, L., Weichert, D., Haidar, A.: Analysis of full-depth asphalt concrete pavements using shakedown-theory. Transactions of the Transportation Research Board (NRC), Transportation Research Record 1227, 53–65 (1989)

    Google Scholar 

  128. Raad, L., Weichert, D., Najim, W.: Stability of multilayer systems under repeated loads. Transactions of Transportation Research Board (NRC), Transportation Research Record 1207, Pavement Design, 181–186 (1988)

    Google Scholar 

  129. Weichert, D., Raad, L.: Extension of the static shakedown-theorem to a certain class of materials with variable elastic coefficients. Mech. Res. Comm. 19(6), 511–517 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  130. Boulbibane, M., Weichert, D., Raad, L.: Numerical application of shakedown theory to pavements with anisotropic layer properties, Paper No. 99-0342. Journal of the Transportation Research Board (NRC), Transportation Research Record 1687, 75–81 (1999)

    Google Scholar 

  131. Boulbibane, M., Collins, I.F., Weichert, D., Raad, L.: Shakedown analysis of anisotropic asphalt concrete pavements with clay subgrade. Geotech. J. 37, 882–889 (2000)

    Article  Google Scholar 

  132. Hachemi, A., Weichert, D.: An extension of the static shakedown-theorem to a certain class of damaging inelastic material. Arch. of Mech. 44(5-6), 491–498 (1992)

    MATH  Google Scholar 

  133. Hachemi, A., Weichert, D.: Application of shakedown theorems to damaging inelastic material under mechanical and thermal loads. Int. J. Mech. Sci. 39(9), 1067–1076 (1997)

    Article  MATH  Google Scholar 

  134. Belouchrani, M., Weichert, D.: An extension of the static shakedown theorem to inelastic cracked structures. Int. J. Mech. Sci. 41, 163–177 (1999)

    Article  MATH  Google Scholar 

  135. Belouchrani, M.A., Weichert, D., Hachemi, A.: Fatigue threshold computation by shakedown theory. Mech. Res. Comm. 27(3), 287–293 (2000)

    Article  MATH  Google Scholar 

  136. Hachemi, A., Weichert, D.: Numerical shakedown analysis of damaged structures. Comp. Meth. Appl. Mech. Engng. 160, 57–70 (1998)

    Article  MATH  Google Scholar 

  137. Hamadouche, M.A., Weichert, D.: Application of shakedown theory to soil dynamics. Mech. Res. Comm. 26(5), 565–574 (1999)

    Article  MATH  Google Scholar 

  138. Weichert, D., Schwabe, F., Hachemi, A.: Composite design by shakedown analysis for low cycle fatigue service conditions. In: Khan, A.S., Zhang, H., Yuan, Y. (eds.) Proceedings of the 8th Int. Symp. on Plasticity and its Current Applications (PLASTICITY 2000), held at Whistler, Canada, July 16-20, pp. 505–507. Neat Press (2000)

    Google Scholar 

  139. Weichert, D., Hachemi, A.: Shakedown- and Limit Analysis of periodic composites. Journal of Theoretical and Applied Mechanics 40(1), 273–289 (2002)

    MATH  Google Scholar 

  140. Hachemi, A., Weichert, D.: On the problem of interfacial damage in fibre-reinforced composites under variable loads. Mech. Res. Comm. 32, 15–23 (2005)

    Article  MATH  Google Scholar 

  141. Hachemi, A., Mouhtamid, S.D.: Weichert, Progress in shakedown analysis with applications to composites. Archives of Applied Mechanics, 1–11 (2005)

    Google Scholar 

  142. Weichert, D., Hachemi, A.: A shakedown approach to the problem of damage of fiber-reinforced composites. In: Sadowski, T., (ed.) Proc. IUTAM-Symposium on “Multiscale Modelling of Damage and Fracture Processes in Composite Materials”, KazimierzDolny, Poland, May 23-27, Solid Mechanics and its Applications, vol. 135, pp. 41–48. Springer (2006)

    Google Scholar 

  143. Nguyen, A.D., Hachemi, A., Weichert, D.: Application of the interior-point method to shakedown analysis of pavements. Int. J. Numer. Meth. Engng. 75, 414–439 (2008)

    Google Scholar 

  144. Weichert, D., Hachemi, A.: Recent advances in lower bound shakedown analysis. In: Proceedings of Pressure Vessel and Pi** Conference 2009, Prague, Czech Republic, July 26-30. ASME (2009)

    Google Scholar 

  145. Simon, J.-W., Chen, M., Weichert, D.: Shakedown analysis combined with the problem of heat conduction. In: Proceedings of the ASME 2010 Pressure Vessels & Pi** Division/K-PVP Conference, PVP 2010, Bellevue, Washington, USA, July 18-22 (2010)

    Google Scholar 

  146. Simon, J.-W., Weichert, D.: An improved Interior-Point Algorithm for Large-Scale Shakedown Analysis PAMM. Proc. Appl. Math. Mech. 10, 223–224 (2010)

    Article  Google Scholar 

  147. Morelle, P.: Numerical shakedown analysis of axisymmetric sandwich shells: An upper bound formulation. Int. J. Num. Meth. 23(11), 2071–2088 (1986)

    Article  MATH  Google Scholar 

  148. Hung, N.-D., König, J.A.: Finite element formulation for shakedown problems using a yield criterion of the mean. Computer in Applied Mechanics and Engineering 8(2), 179–192 (1976)

    Google Scholar 

  149. Hung, N.-D., Palgen, L.: Shakedown analysis by displacement method and equilibrium finite element. Transactions of the CSME 6(1), 32–39 (1980)

    Google Scholar 

  150. Hung, N.-D.: Shakedown analysis by finite element method and linear programming techniques. Journal de Mécanique Appliquée 2(4), 587–599 (1983)

    Google Scholar 

  151. Hung, N.-D., Yan, A.-M.: Direct finite element kinematical approaches in limit and shakedown analysis of shells and elbows. In: Inelastic Analysis of Structures under Variable Loads, Theory and Engineering Applications, pp. 233–254. Kluwer Academic Publishers (2000)

    Google Scholar 

  152. Hung, N.-D., Vu, D.K.: Primal-dual algorithm for shakedown analysis of structures. Computer Methods in Applied Mechanics and Engineering 193(42-44), 4663–4674 (2004)

    Article  Google Scholar 

  153. Staat, M., Heitzer, M., Yan, A.M., Vu, D., Hung, N.-D., Voldoire, F., Lahousse, A.: Limit Analysis of Defects (In collaboration with, Berichte des Forschungszentrums Jülich 3746, Jül-3746) (2000)

    Google Scholar 

  154. Vu, D., Staat, M., Tran, I.T.: Analysis of pressure equipment by application of the primal-dual theory of shakedown. Commun. Numer. Meth. Engng. 23(3), 213–225 (2007)

    Article  MATH  Google Scholar 

  155. Vu, D., Staat, M.: Shakedown analysis of structures made of materials with temperature-dependent yield stress. International Journal of Solids and Structures 44(13), 4524–4540 (2007)

    Article  MATH  Google Scholar 

  156. Heitzer, M., Staat, M.: Reliability analysis of elasto-plastic structures under variable loads. In: Weichert, D., Maier, G. (eds.) Inelastic Analysis of Structures under Variable Loads: Theory and Engineering Applications, pp. 269–288. Kluwer, Academic Press, Dordrecht (2000)

    Chapter  Google Scholar 

  157. Tran, T.N., Phạm, P.T., Vu, D.K., Staat, M.: Reliability analysis of ine-lastic shell structures under variable loads. In: Weichert, D., Ponter, A.R.S. (eds.) Limit States of Materials and Structures: Direct Methods, pp. 135–156. Springer, Netherlands (2009)

    Chapter  Google Scholar 

  158. Spiliopoulos, K.V.: On the automation of the force method in the optimal plastic design of frames. Comp. Meth. Appl. Mech. Eng. 141, 141–156 (1997)

    Article  MATH  Google Scholar 

  159. Spiliopoulos, K.V.: A fully automatic force method for the optimal shakedown design of frames. Comp. Mech. 23, 299–307 (1999)

    Article  MATH  Google Scholar 

  160. Spiliopoulos, K.V.: Force Method-based procedures in the limit equilibrium analysis of framed structures. In: Weichert, D., Ponter, A.R.S. (eds.) Limit States of Materials and Structures: Direct Methods, pp. 233–252. Springer, Netherlands (2009)

    Google Scholar 

  161. de Saxcé, G., Massonet, C., Morelle, P.: Discussion of paper Plastic collapse, shakedown and hysteresis. Int. J. Struct. Eng. ASCE 112(9), 2177–2183 (1987); S.A. Grualnik, S. Singh, T. Erber (eds.)

    Google Scholar 

  162. de Saxcé, G.: Une generalization de l’inégalité de Fenchel et ses applications aux lois constitutives. C.R. Acad. Sci. t.314. série II, 125–129 (1992)

    Google Scholar 

  163. Bodoville, G., de Saxcé, G.: Plasticity with non linear kinematic hardening: modeling and shakedown analysis b the bipotential approach. Int. J. Plast. 17(1), 21–46 (2001)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  165. Moreau, J.J.: On unilateral constraints, friction and plasticity, in: new variational techniques in Mathematical Physics. CIME Course, pp. 173–322. Springer (1974)

    Google Scholar 

  166. Debordes, O., Nayroles, B.: Sur la théorie et le calcul à l’adaptation des structures élasto-plastiques. J. Mécanique 20, 1–54 (1976)

    MathSciNet  Google Scholar 

  167. Nayroles, B.: Tendences récentes et perspectives à moyen terme en élastoplastici-té asymptotique des constructions. In: Congrès Français de Mécanique, Grenoble, France (1977)

    Google Scholar 

  168. Nayroles, B., Weichert, La, D.: notion de sanctuaire d’élasticité et l’adaptation des structures. C.R. Acad. Sci. Paris 316, Série II, 1493–1498 (1993)

    Google Scholar 

  169. Nguyen, Q.-S.: Extension des théorèmes d’adaptation et d’unicité en écrouissage non linéaire. C.R. Acad. Sc. 282, 755–758 (1976)

    Google Scholar 

  170. Nguyen, Q.-S.: Min-Max Duality and shakedown theorems in plasticity. In: Alart, P., Maisonneuve, O., Rockafellar, R.T. (eds.) Nonsmooth Mechanics and Analysis, Theoretical and Numerical Advances, ch. 8. Springer (2006)

    Google Scholar 

  171. Nguyen, Q.-S., Pham, D.: On shakedown theorems in hardening plasticity. C.R. Acad. Sci. 329, 307–314 (2001)

    MATH  Google Scholar 

  172. Halphen, B.: Q-S Nguyen, Sur les matériaux standard généralisés. J. Mécanique 14, 1–37 (1975)

    Google Scholar 

  173. Mandel, J.: Adaptation d’une structure plastique écrouissable. Mech. Res. Comm. 3, 251–256 (1976)

    Google Scholar 

  174. Radenkovic, D.: Théorèmes limites pour un materiau de Coulomb à dilatation nonstandardisée. Comptes Rendus de l’Académie des Sciences Paris 252, 4103–4104 (1961)

    MathSciNet  MATH  Google Scholar 

  175. Zarka, J., Frelat, J., Inglebert, G., Kasmaï-Navidi, P.: A new approach to inelastic analysis of structures, CADLM edition, France (1989)

    Google Scholar 

  176. Inglebert, G., Zarka, J.: On a simplified inelastic analysis of structures. Nucl. Eng. 57, 333–368 (1980)

    Article  Google Scholar 

  177. Hassine, T., Inglebert, G., Pons, M.: shakedown and damage analysis applied to rocket machines. In: Weichert, D., Maier, G. (eds.) Inelastic Analysis of Structures under Variable Loads: Theory and Engineering Applications, pp. 255–2267. Kluwer, Academic Press, Dordrecht (2000)

    Chapter  Google Scholar 

  178. Sharp, R.W., Booker, J.R.: Shakedown of pavements under moving surface loads. Journal of Transportation Engng. ASCE 110(1), 1–14 (1984)

    Article  Google Scholar 

  179. Krabbenhøft, K., Lyamin, A.V., Sloan, S.W.: Bounds to shakedown loads for a class of deviatoric plasticity models. Comp. Mech. 39(6), 879–888 (2007)

    Article  Google Scholar 

  180. Krabbenhøft, K., Lyamin, A.V., Sloan, S.W.: Shakedown of a cohesive-frictional half-space subjected to rolling and sliding contact. Int. J. Solids and Structures 44(11-12), 3998–4008 (2007)

    Article  Google Scholar 

  181. Zhao, J.D., Sloan, S.W., Lyamin, A.V., Krabbenhøft, K.: Bounds for shakedown of cohesive-frictional materials under moving surface loads. Int. J. Solids and Structures 45(11-12), 3290–3312 (2008)

    Article  MATH  Google Scholar 

  182. Collins, I.F., Wang, A., Saunders, L.: Shakedown Theory and the design of un-bound pavements. Road and Transport Research 2, 28–39 (1993)

    Google Scholar 

  183. Collins, I.F., Wang, A., Saunders, L.: Shakedown in layered pavements under moving surface loads. Int. J. Numerical and Analytical Methods in Geomechanics 17, 165–174 (1993)

    Article  Google Scholar 

  184. Grundy, P.: Shakedown of Bars in Bending and Tension. J. Eng. Mech. Div. ASCE 95(EM3), 519–529 (1969)

    Google Scholar 

  185. Alwis, W.A.M., Grundy, P.: Shakedown Analysis of Plates. Int. J. Mech. Sci. 27(1/2), 71–82 (1985)

    Article  MATH  Google Scholar 

  186. Shiau, S.H., Yu, H.S.: Finite element method for shakedown analysis of pavements. In: 16th Australasian Conference on the Mechanics of Structures and Materials, pp. 17–22 (1999)

    Google Scholar 

  187. Shiau, S.H., Yu, H.S.: Shakedown analysis of flexible pavements. In: The John Booker Memorial Symposium, pp. 643–653 (2000)

    Google Scholar 

  188. Yu, H.S.: Foreward: shakedown theory for pavement analysis. International Journal of Road Materials and Pavement Design 6(1), 7–9 (2005)

    Article  Google Scholar 

  189. Li, H.X., Yu, H.S.: A non-linear programming approach to kinematic shakedown analysis of composite materials. International Journal of Numerical Methods in Engineering 66(1), 117–146 (2006)

    Article  MATH  Google Scholar 

  190. Yu, H.S., Wang, J.: Three-dimensional shakedown solutions for cohesive-frictional materials under moving surface loads. Int. J. Sol. Struct. 49(26), 3797–3807 (2013)

    Article  Google Scholar 

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  1. Institute of General Mechanics, RWTH-Aachen University, Templergraben 64, 52056, Aachen, Germany

    Dieter Weichert

  2. Department of Engineering, University of Leicester, Leicester, LE 7RH, U.K.

    Alan Ponter

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  1. Institute of Mechanics and Computational Mechanics (IBNM), Gottfried Wilhelm Leibniz Universität Hannover, Hannover, Germany

    Erwin Stein

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Weichert, D., Ponter, A. (2014). A Historical View on Shakedown Theory. In: Stein, E. (eds) The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering. Lecture Notes in Applied Mathematics and Mechanics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39905-3_11

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