0
TECHNICAL PAPERS

On Some Issues in Shakedown Analysis

[+] Author and Article Information
G. Maier

Department of Structural Engineering, Technical University (Politecnico), Piazza L. da Vinci, 32, I-20133 Milan, Italy

J. Appl. Mech 68(5), 799-808 (Feb 28, 2001) (10 pages) doi:10.1115/1.1379368 History: Received February 28, 2001
Copyright © 2001 by ASME
Your Session has timed out. Please sign back in to continue.

References

Koiter, W. T., 1960, “General Theorems for Elastic-Plastic Solids,” Progress in Solid Mechanics, Vol. 1, J. N. Sneddon and R. Hill, eds., North-Holland, Amsterdam, pp. 165–221.
König, J. A., 1987, Shakedown of Elastic-Plastic Structures, Elsevier, Amsterdam.
Kamenjarzh, J. A., 1996, Limit Analysis of Solids and Structures, CRC Press, Boca Baton, FL.
Lloyd Smith, D., ed., 1990, Mathematical Programming Methods in Structural Plasticity, Springer-Verlag, New York.
Mróz, Z., Weichert, D., and Dorosz, S., eds., 1995, Inelastic Behavior of Structures Under Variable Loads, Kluwer, Dordrecht, The Netherlands.
Weichert, D., and Maier, G. eds., 2000, Inelastic Analysis of Structures Under Variable Repeated Loads, Kluwer, Dordrecht, The Netherlands.
Maier,  G., Carvelli,  V., and Cocchetti,  G., 2000, “On Direct Methods for Shakedown and Limit Analysis,” Eur. J. Mech. A/Solids (Special Issue), 19, pp. S79–S100.
Koiter,  W. T., 1956, “A New General Theorem on Shakedown of Elastic-Plastic Structures,” Proc. K. Ned. Akad. Wet., B59, pp. 24–34.
Karadeniz,  S., and Ponter,  A. R. S., 1984, “A Linear Programming Upper Bound Approach to the Shakedown Limit of Thin Shells Subjected to Variable Thermal Loading,” J. Strain Anal., 19, pp. 221–230.
Yan,  A., and Nguyen-Dang,  H., 2001, “Kinematical Shakedown Analysis With Temperature-Dependent Yield Stress,” Int. J. Numer. Methods Eng., 50, pp. 1145–1168.
Polizzotto,  C., 1984, “Deformation Bounds for Elastic Plastic Solids Within and Out of the Creep Range,” Nucl. Eng. Des., 83, pp. 293–301.
Ponter,  A. R. S., 1972, “Deformation, Displacement and Work Bounds for Structures in a State of Creep and Subject to Variable Loading,” ASME J. Appl. Mech., 39, pp. 953–959.
Carvelli,  V., Cen,  Z., Liu,  Y., and Maier,  G., 1999, “Shakedown Analysis of Defective Pressure Vessels by a Kinematic Approach,” Arch. Appl. Mech., 69, pp. 751–764.
Carvelli,  V., Maier,  G., and Taliercio,  A., 1999, “Shakedown Analysis of Periodic Heterogeneous Materials by a Kinematic Approach,” Mech. Eng. (Strojnı́cky Časopis), 50, No. 4, pp. 229–240.
Carvelli,  V., Maier,  G., and Taliercio,  A., 2000, “Kinematic Limit Analysis of Periodic Heterogeneous Media,” Comp. Meth. Eng. Sci., 1, pp. 15–26.
Casciaro,  R., and Cascini,  L., 1982, “A Mixed Formulation and Mixed Finite Elements for Limit Analysis,” Int. J. Numer. Methods Eng., 18, pp. 211–243.
König,  J. A., and Kleiber,  M., 1978, “On a New Method of Shakedown Analysis,” Bull. Acad. Pol. Sci., Ser. Sci. Tech., 26, pp. 165–171.
Zhang,  Y. G., 1995, “An Iterative Algorithm for Kinematic Shakedown Analysis,” Comput. Methods Appl. Mech. Eng., 127, pp. 217–226.
Kamenjarzh,  J. A., and Merzljakov,  A., 1994, “On Kinematic Method in Shakedown Theory; I. Duality of Extremum Problems; II. Modified Kinetic Method,” Int. J. Plast., 10, pp. 363–392.
Kamenjarzh,  J. A., and Weichert,  D., 1992, “On Kinematic Upper Bounds for the Safety Factor in Shakedown Theory,” Int. J. Plast., 8, pp. 827–837.
Sloan,  S. W., and Kleeman,  P. W., 1995, “Upper Bound Limit Analysis Using Discontinuous Velocity Fields,” Comput. Methods Appl. Mech. Eng., 127, pp. 293–314.
Teixeira de Freitas,  J. A., 1991, “A Kinematic Model for Plastic Limit Analysis of Solids by the Boundary Element Method,” Comput. Methods Appl. Mech. Eng., 88, pp. 189–205.
Dvorak,  G. J., Lagoudas,  D. C., Huang,  C. M., 1994, “Fatigue Damage and Shakedown in Metal Matrix Composite Laminates,” Mech. Compos. Mat. Struct., 1, pp. 171–202.
Francescato,  P., and Pastor,  J., 1997, “Lower and Upper Numerical Bounds to the Off-Axis Strength of Unidirectional Fiber-Reinforced Composites by Limit Analysis Methods,” Eur. J. Mech. A/Solids, 16, pp. 213–234.
Weichert,  D., Hachemi,  A., and Schwabe,  F., 1999, “Shakedown Analysis of Composites,” Mech. Res. Commun., 26, pp. 309–318.
Liu,  Y. H., Cen,  Z. Z., and Xu,  B. Y., 1995, “A Numerical Method for Plastic Limit Analysis of 3-D Structures,” Int. J. Solids Struct., 32, pp. 1645–1658.
Weichert,  D., Hachemi,  A., and Schwabe,  F., 1999, “Application of Shakedown Analysis to the Plastic Design of Composites,” Arch. Appl. Mech., 69, pp. 623–633.
Hamilton,  R., Boyle,  J. T., Shi,  J., and Mackenzie,  D., 1996, “A Simple Upper-Bound Method for Calculating Approximate Shakedown Loads,” ASME J. Pressure Vessel Technol., 120, pp. 195–199.
Ponter,  A. R. S., and Carter,  K. F., 1997, “Shakedown State Simulation Techniques Based on Linear Elastic Solutions,” Comput. Methods Appl. Mech. Eng., 140, pp. 259–279.
Drucker, D. C., 1963, “On the Macroscopic Theory of Inelastic Stress-Strain-Time-Temperature Behavior,” Advances in Materials Research in the NATO Nations (AGAR Dograph 62), Pergamon Press, New York, pp. 193–221.
Dang Van, K., and Papadopoulos, I. V., 1999, High-Cycle Metal Fatigue From Theory to Applications, CISM, Springer-Verlag, New York.
Ponter,  A. R. S., and Leckie,  F. A., 1998, “Bounding Properties of Metal-Matrix Composites Subjected to Cyclic Loading,” J. Mech. Phys. Solids, 46, pp. 697–717.
Silberschmidt,  V. V., Rammerstorfer,  F. G., Werner,  E. A., Fischer,  F. D., and Uggowitzer,  P. J., 1999, “On Material Immanent Ratchetting of Two-Phase Materials Under Cyclic Purely Thermal Loading,” Arch. Appl. Mech., 69, pp. 727–750.
Lewis R. W., and Schrefler B. A., 1998, The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, John Wiley and Sons, Chichester.
Cocchetti,  G., and Maier,  G., 1998, “Static Shakedown Theorems in Piecewise Linearized Poroplasticity,” Arch. Appl. Mech., 68, pp. 651–661.
Cocchetti,  G., and Maier,  G., 2000, “Shakedown Analysis in Poroplasticity by Linear Programming,” Int. J. Numer. Methods Eng., 47, No. 1–3, pp. 141–168.
Cocchetti, G., and Maier, G., 2000, “Upper Bounds on Post-Shakedown Quantities in Poroplasticity,” Inelastic Analysis of Structures Under Variable Repeated Loads, D. Weichert and G. Maier, eds., Kluwer, Dordrecht, The Netherlands, pp. 289–314.
Maier,  G., 1969, “Shakedown Theory in Perfect Elastoplasticity With Associated and Nonassociated Flow-Laws: A Finite Element, Linear Programming Approach,” Meccanica, 4, pp. 250–260.
Maier,  G., 1970, “A Matrix Structural Theory of Piecewise-Linear Plasticity With Interacting Yield Planes,” Meccanica, 5, pp. 55–66.
Tin-Loi,  F., 1990, “A Yield Surface Linearization Procedure in Limit Analysis,” Mech. Struct. Mach., 18, pp. 135–149.
Comi,  C., and Corigliano,  A., 1991, “Dynamic Shakedown in Elastoplastic Structures With General Internal Variable Constitutive Laws,” Int. J. Plast., 7, pp. 679–692.
Corigliano,  A., Maier,  G., and Pycko,  S., 1995, “Dynamic Shakedown Analysis and Bounds for Elastoplastic Structures With Nonassociative, Internal Variable Constitutive Laws,” Int. J. Solids Struct., 32, pp. 3145–3166.
Du,  S. T., Xu,  B. Y., and Dong,  Y. F., 1993, “Dynamic Shakedown Theory of Elastoplastic Work-Hardening Structures Allowing for Second-Order Geometric Effects,” Acta Mech. Solidica Sinica, 6, pp. 15–26.
Polizzotto, C., 1984, “On Shakedown of Structures Under Dynamic Agencies,” Inelastic Analysis Under Variable Loads, A. Sawczuk and C. Polizzotto, eds., Cogras, Palermo, pp. 5–29.
Polizzotto,  C., Borino,  G., Caddemi,  S., and Fuschi,  P., 1993, “Theorems of Restricted Dynamic Shakedown,” Int. J. Mech. Sci., 35, pp. 787–801.
Ceradini,  G., 1969, “Sull’adattamento dei corpi elastoplastici soggetti ad azioni dinamiche,” Gior. Genio Civile, 415, pp. 239–258.
Corradi,  L., and Maier,  G., 1973, “Inadaptation Theorems in the Dynamics of Elastic-Work Hardening Structures,” Ing. Arch., 43, pp. 44–57.
Corradi,  L., and Maier,  G., 1974, “Dynamic Non-Shakedown Theorem for Elastic Perfectly-Plastic Continua,” J. Mech. Phys. Solids, 22, pp. 401–413.
Corigliano,  A., Maier,  G., and Pycko,  S., 1995, “Kinematic Criteria of Dynamic Shakedown Extended to Nonassociative Constitutive Laws With Saturation Hardening,” Rend. Acc. Naz. Lincei. Sci., Ser. IX, VI, pp. 55–64.
Pham,  D. C., 1996, “Dynamic Shakedown and a Reduced Kinematic Theorem,” Int. J. Plast., 12, pp. 1055–1068.
Maier, G., and Comi, C., 1997, “Variational Finite Element Modelling in Poroplasticity,” Recent Developments in Computational and Applied Mechanics, B. D. Reddy, ed., CIMNE, Barcelona, pp. 180–199.
Maier,  G., and Novati,  G., 1990, “Dynamic Shakedown and Bounding Theory for a Class of Nonlinear Hardening Discrete Structural Models,” Int. J. Plast., 6, pp. 551–572.
Druyanov,  B., and Roman,  I., 1999, “Conditions for Shakedown of Damaged Elastic Plastic bodies,” Eur. J. Mech. A/Solids, 18, pp. 641–651.
Dvorak,  G. J., Lagoudas,  D. C., and Huang,  C. M., 1994, “Fatigue Damage and Shakedown in Metal Matrix Composite Laminates,” Mech. Compos. Mat. Struct., 1, pp. 171–202.
Feng,  X. Q., and Yu,  S. W., 1995, “Damage and Shakedown Analysis of Structures With Strain-Hardening,” Int. J. Plast., 11, pp. 237–249.
Hachemi,  A., and Weichert,  D., 1997, “Application of Shakedown Theory to Damaging Inelastic Material Under Mechanical and Thermal Loads,” Int. J. Mech. Sci., 39, pp. 1067–1076.
Huang,  Y., and Stein,  E., 1996, “Shakedown of a Cracked Body Consisting of Kinematic Hardening Material,” Eng. Fract. Mech., 54, pp. 107–112.
Yan,  A. M., and Nguyen,  D. H., 1999, “Limit Analysis of Cracked Structures by Mathematical Programming and Finite Element Technique,” Comput. Mech., 24, pp. 319–333.
Nayroles,  B., and Weichert,  D., 1993, “La notion de sanctuaire d’elasticite et d’adaptation des structures,” C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers, 316, pp. 1493–1498.
Christiansen,  E., and Andersen,  K. D., 1999, “Computation of Collapse States With von Mises Type Yield Condition,” Int. J. Numer. Methods Eng., 46, pp. 1185–1202.
Tin-Loi,  F., 1989, “A Constraint Selection Technique in Limit Analysis,” Appl. Math. Model., 13, pp. 442–446.
Borges,  L. A., Feijóo,  R. A., and Zouain,  N., 1999, “A Directional Error Estimator for Adaptive Limit Analysis,” Mech. Res. Commun., 26, pp. 555–563.
Franco,  J. R. Q., Oden,  J. T., Ponter,  A. R. S., and Barros,  F. B., 1997, “A Posteriori Error Estimator and Adaptive Procedures for Computation of Shakedown and Limit Loads on Pressure Vessels,” Comput. Methods Appl. Mech. Eng., 150, pp. 155–171.
Cocks,  A. C. F., and Leckie,  F. A., 1988, “Deformation Bounds for Cyclically Loaded Shell Structures Operating Under Creep Condition,” ASME J. Appl. Mech., 55, pp. 509–516.
Polizzotto,  C., 1982, “A Unified Treatment of Shakedown Theory and Related Bounding Techniques,” Solid. Mech. Arch., 7, pp. 19–75.
Genna,  F., 1991, “Bilateral Bounds for Structures Under Dynamic Shakedown Conditions,” Meccanica, 26, pp. 37–46.
Capurso,  M., 1979, “Some Upper Bound Principles for Plastic Strains in Dynamic Shakedown of Elastoplastic Structures,” J. Struct. Mech., 7, pp. 1–20.
Corradi,  L., 1976, “Mathematical Programming Methods for Displacement Bounds in Elastoplastic Dynamics,” Nucl. Eng. Des., 37, pp. 161–177.
Maier,  G., 1973, “Upper Bounds on Deformations of Elastic-Workhardening Structures in the Presence of Dynamic and Second-Order Effects,” J. Struct. Mech., 2, pp. 265–280.
Ponter,  A. R. S., 1975, “General Displacement and Work Bounds for Dynamically Loaded Bodies,” J. Mech. Phys. Solids, 23, pp. 151–163.
Taliercio,  A., 1992, “Lower and Upper Bounds to the Macroscopic Strength Domain of a Fiber-Reinforced Composite Material,” Int. J. Plast., 8, pp. 741–762.

Figures

Grahic Jump Location
Shakedown analysis of a perforated plate: (a) representative volume and finite element mesh; (b) shakedown limit locus (solid line) in the average stress plane for rectangular loading domains like those defined by points A and B; for comparison, the plastic collapse locus in dashed lines (σ0 being the material yield stress)
Grahic Jump Location
Shakedown analysis of a metal-matrix composite subjected to uniaxial average stress Σ: (a) representative volume; (b) incremental collapse mechanism for θ=30 deg; (c) shakedown limit (solid line SDA) versus plastic collapse limit evaluated by the present kinematic method (dashed line LA) and by the static method in 71 (dotted line) (σ0m being the matrix yield stress); (d) convergence on the shakedown limit in the former procedure with penalty factor 106
Grahic Jump Location
Influence of the penalty factor on the plastic collapse limit (dashed line) and on the violation (measured by a norm of the plastic volumetric strain field) of the normality constraint (solid line)
Grahic Jump Location
Incremental collapse mechanism with relevant Mises-equivalent plastic strain rate field (a) for an idealized gravity dam interpreted as a poroplastic system under periodic live load (bα)
Grahic Jump Location
Self-adaptive limit analysis governed by a normalized measure of the plastic strain rate density of the collapse mechanism in piecewise-linearized plasticity
Grahic Jump Location
Upper bounds on the residual displacement at the top of the idealized poroplastic dam model of Fig. 4, as a function of the mesh refinement (point A corresponds to the finite element mesh shown in Fig. 4)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In