This paper reviews the literature on variational method in limit load analysis and presents both analytical and numerical approaches. One of the most successful applications of variational method in theory of plasticity is limit load analysis. The main objective of the limit load analysis is to estimate the load at the impending plastic limit state of a body. However, for complicated problems it may be very difficult to find the exact limit load. Therefore, based on the extremum principles of limit load analysis, the lower-bound theorem or the upper bound theorem is employed to estimate the limit load directly without considering the entire loading history. In general, limit load analysis plays an important role in design and fitness-for-service assessment of pressurized vessels and piping.
References
1.
Kazinczy
, G.
, 1914
, “Kísérletek Befalazott Tartókkal
,” Betonszemle
, 2
, pp. 101
–104
. (In Hungarian).2.
Kaliszky
, S.
, Sajtos
, I.
, Lógó
, B. A.
, Lógó
, J. M.
, and Szabó
, Z.
, 2015
, “Gábor Kazinczy and His Legacy in Structural Engineering
,” Period. Polytech. Civ. Eng.
, 59
(1
), pp. 3
–7
.3.
Mura
, T.
, and Koya
, T.
, 1992
, Variational Methods in Mechanics
, Oxford University Press
, New York
.4.
Calladine
, C. R.
, 2000
, Plasticity for Engineers
, Horwood Publishing
, Chichester, UK
.5.
Von Trefftz
, E.
, 1926
, “Ein gegenstuck zum ritzschen verfahren
,” Second International Congress of Applied Mechanics
, Zurich, Switzerland, pp. 131
–137
.6.
Von Trefftz
, E.
, 1928
, “Konvergenz und Fehlerschätzung beim Ritzschen Verfahren
,” Mathematische Annalen, pp. 503
–521
.7.
Mura
, T.
, and Lee
, S. L.
, 1963
, “Application of Variational Principles to Limit Analysis
,” Q. Appl. Math.
, 21
(3
), pp. 243
–248
.8.
Mura
, T.
, Kao
, J. S.
, and Lee
, S. L.
, 1964
, “Limit Analysis of Circular Orthotropic Plates
,” ASCE J. Eng. Mech. Div.
, 90
(5), pp. 375
–395
.http://cedb.asce.org/CEDBsearch/record.jsp?dockey=00131139.
Lee
, S. L.
, Mura
, T.
, and Kao
, J. S.
, 1967
, “A Variational Method for the Limit Analysis of Anisotropic Plates
,” Q. Appl. Math.
, 14
(4
), pp. 323
–330
.10.
Sacchi
, G.
, and Save
, M.
, 1968
, “On the Evaluation of the Limit Load for Rigid-Perfectly Plastic Continua
,” Meccanica
, 3
(3
), pp. 199
–206
.11.
Mura
, T.
, Rimawi
, W. H.
, and Lee
, S. L.
, 1965
, “Extended Theorems of Limit Analysis
,” Q. Appl. Math.
, 23
(2
), pp. 171
–179
.12.
Reinhardt
, W. D.
, and Seshadri
, R.
, 2003
, “Limit Load Bounds for the mα Multipliers
,” ASME J. Pressure Vessel Technol.
, 125
(1
), pp. 11
–18
.13.
Pan
, L.
, and Seshadri
, R.
, 2001
, “Limit Load Estimation Using Plastic Flow Parameter in Repeated Elastic Finite Element Analysis
,” ASME J. Pressure Vessel Technol.
, 124
(4
), pp. 433
–439
.14.
Adibi-Asl
, R.
, Fanous
, I. F. Z.
, and Seshadri
, R.
, 2006
, “Elastic Modulus Adjustment Procedures-Improved Convergence Schemes
,” Int. J. Pressure Vessels Piping
, 83
(2
), pp. 154
–160
.15.
Simha
, C. H. M.
, and Adibi-Asl
, R.
, 2012
, “Lower Bound Limit Load Estimation Using a Linear Elastic Analysis
,” ASME J. Pressure Vessel Technol.
, 134
(2
), p. 021207
.16.
Simha
, C. H. M.
, and Adibi-Asl
, R.
, 2015
, “Estimating Lower Bound Limit Loads for Structures Subjected to Multiple Loads
,” ASME J. Pressure Vessel Technol.
, 137
(4
), p. 041205.17.
Seshadri
, R.
, and Mangalaramanan
, S. P.
, 1997
, “Lower Bound Limit Load Using Variational Concepts: The Mα-Method
,” Int. J. Pressure Vessels Piping
, 71
(2
), pp. 93
–106
.18.
Zyczkowski
, M.
, 1981
, Combined Loadings in the Theory of Plasticity
, Polish-Scientific Publishers
, Warszawa, Poland.19.
Martin
, J. B.
, 1975
, Plasticity: Fundamentals and General Results
, MIT Press
, Cambridge, MA
.20.
Davis
, R. O.
, and Selvadurai
, A. P.
, 2002
, Plasticity and Geomechanics
, Cambridge University Press
, Cambridge, UK.21.
Gopinathan
, V.
, 1982
, Plasticity Theory and Its Application in Metal Forming
, Wiley
, NewYork
.22.
Kachanov
, L. M.
, 1971
, Foundations of the Theory of Plasticity
, North-Holland
, Amsterdam, The Netherlands
.23.
Mendelson
, A.
, 1968
, Plasticity: Theory and Application
, Macmillan
, New York
.24.
Rees
, D. W. A.
, 2006
, Basic Engineering Plasticity: An Introduction With Engineering and Manufacturing Applications
, Butterworth-Heinemann
, Oxford, UK
.25.
Chakrabarty
, J.
, 2006
, Theory of Plasticity
, Butterworth-Heinemann
, Oxford, UK
.26.
Hill
, R.
, The Mathematical Theory of Plasticity
, Oxford University Publication
, New York
.27.
Charnes
, A.
, and Greenberg
, H. J.
, 1951
, “Plastic Collapse and Linear Programming
,” Bull. Am. Math. Soc.
, 57
(6), pp. 480–490.28.
Dorn
, W. S.
, and Greenberg
, H. S.
, 1997
, “Linear Programming and Plastic Limit Analysis of Structures
,” Quart. Appl. Math.
, 15
(2
), pp. 155
–167
.29.
Charnes
, A.
, Lemke
, C. E.
, and Zienkiewicz
, O. C.
, 1959
, “Virtual Work, Linear Programming and Plastic Limit Analysis
,” Proc. R. Soc. London. Ser. A, Math. Phys. Sci.
, 251
(1264
), pp. 110
–116
.30.
Koopman
, D. C. A.
, and Lance
, R. H.
, 1965
, “On Linear Programming and Plastic Limit Analysis
,” J. Mech. Phys. Solids
, 13
(2
), pp. 77
–87
.31.
Hodge
, P. G.
, and Belytschko
, T.
, 1968
, “Numerical Methods for the Limit Analysis of Plates
,” ASME J. Appl. Mech.
, 35
(4
), pp. 769
–801
.32.
Biron
, A.
, and Hodge
, P. G.
, 1967
, “Limit Analysis of Rotationally Symmetric Shells Under Central Boss Loadings by a Numerical Method
,” ASME J. Appl. Mech.
, 34
(3
), pp. 644
–650
.33.
Dinno
, K. S.
, and Gill
, S. S.
, 1974
, “A Method for Calculating the Lower Bound Limit Pressure for Thick Shells of Revolution With Specific Reference to Cylindrical Vessels With Torispherical Ends
,” IJMS
, 16
(6
), pp. 415
–427
.34.
Hodge
, P. G.
, 1964
, “Yield-Point Load Determination by Non-Linear Programming
,” 11th ICAM
, pp. 554
–561
.35.
Lyamin
, A. V.
, and Sloan
, S. W.
, 2002
, “Upper Bound Limit Analysis Using Linear Finite Elements and Non-Linear Programming
,” Int. J. Numer. Anal. Meth. Geomech.
, 26
(2
), pp. 181
–216
.36.
Lyamin
, A. V.
, and Sloan
, S. W.
, 2002
, “Lower Bound Limit Analysis Using Non-Linear Programming
,” Int. J. Numer. Meth. Eng.
, 55
(5
), pp. 573
–611
.37.
Zouain
, N.
, Herskovits
, J.
, Borges
, L. A.
, and Feijóo
, R. A.
, 1993
, “An Iterative Algorithm for Limit Analysis With Nonlinear Yield Functions
,” Int. J. Solids Struct.
, 30
(10
), pp. 1397
–1417
.38.
Jones
, G. L.
, and Dhalla
, A. K.
, 1981
, “Classification of Clamp Induced Stresses in Thin Walled Pipe
,” ASME Pressure Vessels and Piping Conference, Denver, CO, June 21–26, pp. 17–23.39.
Marriott
, D. L.
, 1988
, “Evaluation of Deformation or Load Control of Stress Under Inelastic Conditions using Elastic Finite Element Stress Analysis
,” ASME Pressure Vessels and Piping Conference, Pittsburgh, PA, pp. 3–9.40.
Seshadri
, R.
, and Fernando
, C. P. D.
, 1992
, “Limit Loads of Mechanical Components and Structures Using the GLOSS R-Node Method
,” ASME J. Pressure Vessel Technol.
, 114
(2
), pp. 201
–208
.41.
Kraus
, H.
, 1980
, Creep Analysis
, Wiley
, New York.42.
Seshadri
, R.
, and Marriot
, D. L.
, 1993
, “On Relating the Reference Stress, Limit Load, and the ASME Stress Classification Concepts
,” Int. J. Pressure Vessel Piping
, 56
(3
), pp. 382
–408
.43.
Seshadri
, R.
, and Kizhatil
, R. K.
, 1993
, “Notch Root Inelastic Strain Estimates Using GLOSS Analysis
,” Advances Multiaxial Fatigue
, D. L.
Mc-Dowell
, and R.
Ellis
, eds., American Society for Testing and Materials
, Philadelphia, PA
, Standard No. ASTM STP 1191.44.
Seshadri
, R.
, 1994
, “Residual Stress Estimation and Shakedown Evaluation Using GLOSS Analysis
,” ASME J. Pressure Vessel Technol.
, 116
(3
), pp. 290
–294
.45.
Seshadri
, R.
, and Kizhatil
, R. K.
, 1995
, “Robust Approximation Methods for Estimating Inelastic Fracture Parameters
,” ASME J. Pressure Vessel Technol.
, 117
(2
), pp. 115
–123
.46.
Mangalaramanan, S. P., and Seshadri, R., 1997, “
Minimum Weight Design of Pressure Components Using R-Node
,” ASME J. Pressure Vessel Technol.
, 119
(2), pp. 224–230.47.
Seshadri
, R.
, and Wu
, S.
, 2001
, “Robust Estimation of Inelastic Fracture Energy Release Rate (J): A Design Approach
,” ASME J. Pressure Vessel Technol.
, 123
(2
), pp. 214
–219
.48.
Seshadri
, R.
, and Babu
, S.
, 2000
, “Extended GLOSS Method for Determining Inelastic Effects in Mechanical Components and Structures: Isotropic Materials
,” ASME J. Pressure Vessel Technol.
, 122
(4
), pp. 413
–420
.49.
Fanous
, I. F. Z.
, Adibi-Asl
, R.
, and Seshadri
, R.
, 2005
, “Limit Load Analysis of Pipe Bend Using the R-Node Method
,” ASME J. Pressure Vessel Technol.
, 127
(4
), pp. 487
–494
.50.
Seshadri
, R.
, 1998
, “Simplified Methods in Plasticity, Creep and Fracture—Some Recent Developments
,” Trans. Can. Soc. Mech. Eng.
, 22
(4B
), pp. 419
–433
.51.
Mackenzie
, D.
, and Boyle
, J. T.
, 1993
, “A Method of Estimating Limit Loads Using Elastic Analysis—I: Simple Examples
,” Int. J. Pressure Vessels Piping
, 53
(1
), pp. 77
–85
.52.
Seshadri
, R.
, 1991
, “The Generalized Local Stress Strain GLOSS Analysis—Theory and Applications
,” ASME J. Pressure Vessel Technol.
, 113
(2
), pp. 219
–227
.53.
Mackenzie
, D.
, Shi
, J.
, and Boyle
, J. T.
, 1994
, “Finite Element Modeling for Limit Analysis Using the Elastic Compensation Method
,” Comp. Struct.
, 51
(4
), pp. 403
–410
.54.
Boyle
, J. T.
, Hamilton
, R.
, Shi
, J.
, and Mackenzie
, D.
, 1997
, “A Simple Method for Calculating Limit Loads for Axisymmetrivc Thin Shells
,” ASME J. Pressure Vessel Technol.
, 119
(2
), pp. 236
–242
.55.
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
(2), pp. 195
–199
.56.
Nadarajah
, C.
, Mackenzie
, D.
, and Boyle
, J. T.
, 1996
, “Limit and Shakedown Analysis of Nozzle/Cylinder Intersections Under Internal Pressure and In-Plane Moment Loading
,” Int. J. Pressure Vessels Piping
, 68
(3
), pp. 261
–272
.57.
Ponter
, A. R. S.
, and Carter
, K. F.
, 1997
, “Limit State Solution Upon Linear Elastic Solutions With a Spatially Varying Elastic Modulus
,” Comput. Methods Appl. Mech. Eng.
, 140
(3–4
), pp. 237
–258
.58.
Ponter
, A. R. S.
, Fuschi
, P.
, and Engelhardt
, M.
, 2000
, “Limit Analysis for a General Class of Yield Conditions
,” Eur. J. Mech. A/Solids
, 19
(3
), pp. 401
–421
.59.
Ponter
, A. R. S.
, and Chen
, H.
, 2001
, “A Programming Method for Limit Load and Shakedown Analysis of Structures
,” ASME Pressure Vessels and Piping Conference, Atlanta, GA, July 22–26, pp. 155–160.60.
Mackenzie
, D.
, Boyle
, J. T.
, and Hamilton
, R.
, 2000
, “The Elastic Compensation Method for Limit and Shakedown Analysis: A Review
,” J. Strain Anal.
, 35
(3
), pp. 171
–188
.61.
Plancq
, D.
, and Berton
, M. N.
, 1998
, “Limit Analysis Based on Elastic Compensation Method of Branch Pipe Tee Connection Under Internal Pressure and Out-of-Plane Moment Loading
,” Int. J. Pressure Vessels Piping
, 75
(11
), pp. 819
–825
.62.
Mohamed
, A. I.
, Bayoumi
, L. S.
, Megahed
, M. M.
, and Younan
, M. Y. A.
, 1999
, “Applications of Iterative Elastic Techniques for Elastic-Plastic Analysis of Pressure Vessels
,” ASME J. Pressure Vessel Technol.
, 121
(1
), pp. 24
–29
.63.
Hardy
, S. J.
, Gowhari-Anaraki
, A. R.
, and Pipelzadeh
, M. K.
, 2001
, “Upper and Lower Bound Limit and Shakedown Loads for Hollow Tubes With Axisymmetric Internal Projections Under Axial Loading
,” J. Strain Anal.
, 36
(6
), pp. 595
–604
.64.
Yang
, P.
, Liu
, Y.
, Ohtake
, Y.
, Yuan
, H.
, and Cen
, Z.
, 2005
, “Limit Analysis Based on a Modified Elastic Compensation Method for Nozzle-to-Cylinder Junctions
,” Int. J. Pressure Vessels Piping
, 82
(10
), pp. 770
–776
.65.
Calladine
, C. R.
, and Drucker
, D. C.
, 1962
, “Nesting Surfaces for Constant Rate of Energy Dissipation in Creep
,” Q. Appl. Math.
, 20
(1
), pp. 79
–84
.66.
Reinhardt
, W. D.
, and Mangalaramanan
, S. P.
, 2001
, “Efficient Tubesheet Design Using Repeated Elastic Limit Analysis
,” ASME J. Pressure Vessel Technol.
, 123
(2
), pp. 197
–202
.67.
Pan
, L.
, and Seshadri
, R.
, 2002
, “Limit Loads for Layered Structures Using Extended Variational Principles and Repeated Elastic Finite Element Analysis
,” ASME J. Pressure Vessel Technol.
, 124
(4
), pp. 425
–432
.68.
Pan
, L.
, and Seshadri
, R.
, 2002
, “Limit Analysis for Anisotropic Solids Using Variational Principle and Repeated Elastic Finite Element Analyses
,” ASME
Paper No. PVP2002-1321
.69.
Adibi-Asl
, R.
, and Seshadri
, R.
, 2006
, “Modulus Adjustment Procedures (EMAP) in Metal Forming Analysis
,” Trans. Can. Soc. Mech. Eng.
, 30
(2
), pp. 239
–261
.70.
ASME
, 2017
, Boiler and Pressure Vessel Code
, American Society of Mechanical Engineers
, New York
.71.
ASME
, 2017
, B31.1 Power Piping
, American Society of Mechanical Engineers
, New York
.72.
Mangalaramanan
, P.
, 1997
, “Robust Limit Loads Using Elastic Modulus Adjustment Technique
,” Ph.D. thesis
, Memorial University, St. John's, NL, Canada.http://www.collectionscanada.gc.ca/obj/s4/f2/dsk3/ftp04/nq25774.pdf73.
Indermohan
, H. P.
, 2006
, “Variational Principles Based Methods for Integrity Assessments
,” Ph.D. thesis, Memorial University, St. John's, NL, Canada.74.
Adibi-Asl
, R.
, 2008
, “Simplified Limit Load Determination for Integrity Assessment
,” Ph.D. thesis, Memorial University, St. John's, NL, Canada.75.
Reinhardt
, W.
, 2008
, “A Non-Cyclic Method for Plastic Shakedown Analysis
,” ASME J. Pressure Vessel Technol.
, 130
(3
), p. 031209
.76.
Adibi-Asl
, R.
, and Reinhardt
, W.
, 2012
, “Non-Cyclic Shakedown/Ratcheting Boundary Determination—Part 1: Analytical Approach
,” Int. J. Pressure Vessels Piping
, 88
(8–9), pp. 311
–320
.77.
Adibi-Asl
, R.
, and Reinhardt
, W.
, 2012
, “Non-Cyclic Shakedown/Ratcheting Boundary Determination–Part 2: Numerical Implementation
,” Int. J. Pressure Vessels Piping
, 88
(8–9), pp. 321
–329
.78.
API 579
, 2017
, “Recommended Practice for Fitness-For-Service
,” American Petroleum Institute, Washington DC, Standard No. API RP 579-1/ASME FFS-1.79.
R6
, 2015
, “Assessment of Integrity of Structures Containing Defects, Revision 4, With Amendments to Amendment 11
,” EDF Energy, Gloucester, UK.80.
Anon
, 1999
, “SINTAP: Structural Integrity Assessment Procedure, Final Revision, EU-Project
,” Brite Euram Programme, Brussels, Belgium, Standard No. BE 95-1462.81.
AFCEN
, 2005
, “RSE-M: In-Service Inspection Rules for Mechanical Components of PWR Nuclear Islands
,” AFCEN, Paris, France.82.
BSI
, 2015
, Guide on Methods for Assessing the Acceptability of Flaws in Metallic Structures
, British Standard Institute
, London
, Standard No. BS 7910.83.
Seshadri
, R.
, 2005
, “Integrity Assessment of Pressure Components With Local Hot Spots
,” ASME J. Pressure Vessel Technol.
, 127
(2
), pp. 137
–142
.84.
Indermohan
, H.
, and Seshadri
, R.
, 2005
, “Fitness-for-Service Methodology Based on Variational Principles in Plasticity
,” ASME J. Pressure Vessel Technol.
, 127
(1
), pp. 92
–97
.85.
Balakrishnan
, R.
, and Seshadri
, R.
, 2005
, “Fitness for Service Assessment of Corroded Pipelines Based on Variational Principles in Plasticity
,” J. Pipeline Integrity
, 4
(2), pp. 99
–116
.86.
Tantichattanont
, P.
, Adluri
, S. M. R.
, and Seshadri
, R.
, 2007
, “Structural Integrity Evaluation for Corrosion in Spherical Pressure Vessels
,” Int. J. Pressure Vessels Piping
, 84
(12
), pp. 749
–761
.87.
Tantichattanont
, P.
, Adluri
, S. M. R.
, and Seshadri
, R.
, 2007
, “Fitness-for-Service Assessment of Spherical Pressure Vessels With Hot Spots
,” Int. J. Pressure Vessels Piping
, 84
(12
), pp. 762
–772
.88.
Adibi-Asl
, R. R.
, and Seshadri
, R.
, 2011
, “Thermal Hot Spot and Corrosion Damage in Conical Pressure Components
,” ASME J. Pressure Vessel Technol.
, 133
(3
), p. 031203
.89.
Adibi-Asl
, R.
, and Seshadri
, R.
, 2016
, “Thermal Hot Spot Assessment in Pressure Vessels
,” ASME
Paper No. PVP2016-63903
.90.
Szczepinski
, W.
, and Szlagowski
, J.
, 1990
, Plastic Design of Complex Shape Structures
, E.
Horwood
, Warszawa
, PWN
, Chichester, UK
.91.
Drucker
, D. C.
, and Shield
, R. T.
, 1957
, “Design for Minimum Weight
,” Ninth International Congress Application Mechanical, Brussels
, Belgium, pp. 212
–222
.92.
Save
, M.
, and Prager
, W.
, 1985
, Structural Optimization
, Vol. 1
, Plenum Press
, New York
.93.
Foulkes
, J.
, 1955
, “Linear Programming and Structural Design
,” Second Symposium in Linear Programming
, Washington, DC, Jan. 27–29, pp. 177
–184
.94.
Cyras
, A. A.
, 1983
, Mathematical Models for the Analysis and Optimization of Elastoplastic Structures
, Ellis Horwood Lim
, Chichester, UK
.95.
Zavelani
, A.
, 1973
, “A Compact Linear Programming Procedure for Optimal Design in Plane Stress
,” J. Struct. Mech.
, 2
(4
), pp. 301
–324
.96.
NATO Advanced Study Institute, 1977, “
Non-Linear Programming Applications
,” Engineering Plasticity by Mathematical Programming (Editors' Summary): Proceedings of the NATO Advanced Study Institute, University of Waterloo, Waterloo, Canada, 2-12 August 1977
, M. Z. Cohn and G. Maier, eds., Permagon Press, New York, pp. 517–547.97.
Bochenek
, B.
, Kordas
, Z.
, and Zyczkowski
, M.
, 1983
, “Optimal Plastic Design of a Cross-Section Under Torsion With Small Bending
,” J. Struct. Mech.
, 11
(3
), pp. 383
–400
.98.
Egner
, W.
, Kordas
, Z.
, and Zyczkowski
, M.
, 1994
, “Optimal Plastic Shape Design Via the Boundary Perturbation Method
,” Struct. Optim.
, 8
(2–3
), pp. 145
–155
.99.
Egner
, W.
, 2000
, “Optimal Plastic Shape Design of Heads of Plane Tension Members With Skew Bearing Surfaces
,” Eng. Optim.
, 32
(4
), pp. 463
–483
.100.
Dems
, K.
, and Mróz
, Z.
, 1978
, “Multiparameter Structural Shape Optimization by the Finite Element Method
,” Int. J. Numer. Methods Eng.
, 13
(2
), pp. 247
–263
.101.
Capsoni
, A.
, and Corradi
, L.
, 1997
, “A Finite Element Formulation of the Rigid-Plastic Limit Analysis Problem
,” Int. J. Numer. Methods Eng.
, 40
, pp. 2063
–2086
.102.
Adibi-Asl
, R.
, 2011
, “Optimal Shape Design Under Elastic-Plastic Behavior Based on Reference Volume Method
,” ASME
Paper No. PVP2011-57889.
Copyright © 2018 by ASME
You do not currently have access to this content.
