In this paper a simplified technique is presented to determine the shakedown limit load of a 90-degree pipe bend subjected to constant internal pressure and cyclic in-plane closing bending moment using the finite element method. The simplified technique determines the shakedown limit load without performing time consuming full elastic-plastic cyclic loading simulations or conventional iterative elastic techniques. Instead, the shakedown limit load is determined by performing two finite element analyses namely; an elastic analysis and an elastic-plastic analysis. By extracting the results of the two analyses, the shakedown limit load is determined through the calculation of the residual stresses developed in the pipe bend. In order to gain confidence in the simplified technique, the output shakedown limit moments are used to perform full elastic-plastic cyclic loading simulations to check for shakedown behavior of the pipe bend. The shakedown limit moments output by the simplified technique are used to generate the shakedown diagram of the pipe bend for a range of constant internal pressure magnitudes. The maximum moment carrying capacity (limit moment) the pipe bend can withstand and the elastic limit are also determined and imposed on the shakedown diagram of the pipe bend. In order to get acquainted with the simplified technique, it is applied beforehand to a bench mark shakedown problem namely, the Bree cylinder (Bree, J., 1967, J. Strain Anal., 3, pp. 226–238) problem. The Bree cylinder is subjected to constant internal pressure and cyclic high heat fluxes across its wall. The results of the simplified technique showed very good correlation with the analytically determined Bree diagram of the cylinder.

1.
von Kàrmàn
,
Th.
, 1911, “
Über die Formänderung Dünnwandiger Röhre, Insbesondere Federnder Ausgleichröhre
,”
Zeitshrift des Vereines Deutscher Ingenieure
,
55
, pp.
1889
1895
.
2.
Melan
,
E.
, 1938, “
Der Spannungszustand eines Mises-Henckyschen Kontinuums bei veraenderlicher Belastung
,”
Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A
0376-2629,
147
, pp.
73
78
.
3.
Melan
,
E.
, 1938, “
Zur Plastizität des räumlichen Kontinuums
,”
Ing.-Arch.
0020-1154,
8
, pp.
116
126
.
4.
Marriott
,
D. L.
, 1988, “
Evaluation of Deformation or Load Control of Stress Under Inelastic Conditions Using Finite Elements Stress Analysis
,”
ASME-PVP Transactions
,
136
, pp.
3
9
.
5.
Dhalla
,
A. K.
, 1987, “
A Simplified Procedure to Classify Stresses for Elevated Temperature Service
,”
ASME-PVP Transactions
,
120
, pp.
177
188
.
6.
,
R.
, 1991, “
The Generalized Local Stress Strain (GLOSS) Analysis-Theory and Applications
,”
ASME J. Pressure Vessel Technol.
0094-9930,
113
, pp.
219
227
.
7.
Mackenzie
,
D.
, and
Boyle
,
J. T.
, 1993, “
A Simple Method for Estimating Shakedown Load for Complex Structures
,”
ASME J. Pressure Vessel Technol.
0094-9930,
265
, pp.
89
94
.
8.
Leckie
,
F. A.
, and
Penny
,
R. K.
, 1967, “
Shakedown Pressure for Radial Nozzles in Spherical Pressure Vessels
,”
Int. J. Solids Struct.
0020-7683,
3
, pp.
743
755
.
9.
Mohamed
,
A. I.
,
Megahed
,
M. M.
,
Bayoumi
,
L. S.
, and
Younan
,
M. Y. A.
, 1999, “
Applications of Iterative Elastic Techniques for Elastic-Plastic Analysis of Pressure Vessels
,”
ASME J. Pressure Vessel Technol.
0094-9930,
121
, pp.
1
6
.
10.
Muscat
,
M.
, and
Mackenzie
,
D.
, 2003, “
Elastic-Shakedown Analysis of Axisymmetric Nozzles
,”
ASME J. Pressure Vessel Technol.
0094-9930,
125
, pp.
365
370
.
11.
Abdalla
,
H. F.
,
Younan
,
M. Y. A.
, and
Megahed
,
M. M.
, 2005, “
A Simplified Technique for Shakedown Load Determination
,”
Proc. 18th International Conference on Structural Mechanics in Reactor Technology
, Beijing, pp.
1315
1328
.
12.
Megahed
,
M. M.
, 1981, “
Influence of Hardening Rule on the Elasto-Plastic Behaviour of a Simple Structure Under Cyclic Loading
,”
Int. J. Mech. Sci.
0020-7403,
23
, pp.
169
182
.
13.
Bree
,
J.
, 1967, “
Elastic-Plastic Behaviour of Thin Tubes Subjected to Internal Pressure and Intermittent High Heat Fluxes with Application to Fast Nuclear Reactor Fuel Elements
,”
J. Strain Anal.
0022-4758,
2
, pp.
226
238
.
14.
Shalaby
,
M. A.
, and
Younan
,
M. Y. A.
, 1998, “
Limit Loads for Pipe Elbows With Internal Pressure Under In-Plane Closing Bending Moments
,”
ASME J. Pressure Vessel Technol.
0094-9930,
120
, pp.
35
42
.
15.
,
H. M.
, and
Younan
,
M. Y. A.
, 2002, “
,”
ASME J. Pressure Vessel Technol.
0094-9930,
124
, pp.
32
37
.
16.
ASME Boiler and Pressure Vessel Code, 2004, Section III Division I, NG-3000.
17.
Hibbitt, Karlsson and, Sorensen, Inc.
, 2004, ABAQUS/Standard User’s Manual.
18.
Karamanos
,
S. A.
,
Tsouvalas
,
D.
, and
Gresnigt
,
A. M.
, 2004, “
Ultimate Capacity of Pressurized 90 Deg Elbows Under Bending
,”
ASME-PVP Transactions
,
477
, pp.
139
148
.
19.
Sobel
,
L. H.
, and
Newman
,
S. Z.
, 1980, “
Comparison of Experimental and Simplified Analytical Results for the In-Plane Plastic Bending and Buckling of an Elbow
,”
ASME J. Pressure Vessel Technol.
0094-9930,
102
, pp.
400
409
.