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

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=0013113
9.
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.pdf
73.
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.
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