13th Annual (International) Mechanical Engineering Conference – May 2005
Isfahan University of Technology, Isfahan, Iran
Influence of initial stresses on cleavage fracture of pressure vessel
containing a part circumferential surface crack
Saeid HadidiMoud 1, David Smith 2
Mechanical Engineering, University of Bristol, BS8 1TR, UK 1,2
Mechanical Engineering, Ferdowsi University of Mashhad, Iran 1
Saeid.HadidiMoud@Bristol.ac.uk , David.Smith@Bristol.ac.uk
Abstract
One critical issue in the integrity assessment of pressurised vessels to be addressed is characterising the
influence of initial residual stresses (arising from fabrication processes especially around welds, where the
appearance of cracks is also a likely event) on cleavage fracture. Using a specific crack configuration, this
paper explores the stress state at crack tip area under similar loading conditions with and without the presence
of initial stresses as well as their interaction and compares the stress fields in the context of stress based
statistical modelling of cleavage fracture toughness. A recently developed local stress based approach to
failure probability prediction for cleavage fracture of ferritic steels is used to predict toughness distributions.
It is found that the presence of initial residual stresses dramatically change the final fracture conditions.
Keywords: Interaction, residual stress, cleavage fracture, pressure vessel, finite element method
Introduction
Proof loading or autofrettage, where no defects or
initial residual stresses are assumed, improves the
load carrying capacity of pressure vessels. This is
principally because of local compressive residual
stresses generated at the crack tip. In presence of
cracks the benefit from proof loading depends on the
combination of crack geometry and loading
conditions. If there exist residual stresses in a vessel
containing defects, the stress field due to the applied
mechanical loading will interact with the residual
stresses. Determination of fracture parameters in
residual stress fields has received great interest in
recent years [1]. Local statistical approaches to
predicting cleavage fracture toughness, mostly based
on the Weibull distributions, have been developed
[2-4]. Recently the authors developed a local stress
based model that suggested using parameters
calibrated based on material response at the specified
test temperature regardless of the crack /loading
configuration [5]. This model was used to predict
the influence of warm pre-stressing on cleavage
fracture in standard fracture mechanics specimens
(e.g. C(T) and SENB) [6]. Recently the model was
1
2
examined to explore its applicability to cases where
initial residual stresses interact with the mechanical
loading [7,8]. Initial RS field were generated by
either local compression, proof loading, or in plane
compression and the test specimens were
subsequently subjected to proof loading prior to final
loading to fracture [9] at a temperature lower than
the proof load. This paper further explores the role
of the type and the nature of initial residual stress
fields on the subsequent resistance to fracture for a
ferritic steel pressure vessel structure. This study is a
continuation of the work reported by authors on use
of the local approach to predict failure probability
for a cracked vessel with prescribed initial stress
[10].
The Finite Element Simulations
The model used in this study was a 432 mm diameter
pipe with a wall thickness of 19.6 mm. The
dimensions were adopted from a test configuration
reported by Bouchard and Bradford [11]. Depending
on the type of loading, whole or half pipe FE models
were produced in ABAQUS/CAE and used in the
corresponding analyses. In case of symmetry, only
- Assistant Professor, Ferdowsi University, Iran / visiting fellow, University of Bristol, UK; Dr.
- Faculty Director of Research and Head of Structural Integrity; University of Bristol, UK; Prof.
13th Annual (International) Mechanical Engineering Conference – May 2005
Isfahan University of Technology, Isfahan, Iran
one half of the cylinder was modelled with the plane
of symmetry in the crack plane normal to the pipe
axis. The whole pipe was modelled for use with
general loading schemes where symmetry rules did
not apply. In the whole model the partial
circumferential crack was introduced artificially by
the addition of new nodes at the coordinates
coincident with the nodes representing the crack face
and assigning those nodes to elements on one side of
the crack. For the half model the crack was identified
by assigning boundary conditions to the remaining
ligament at the crack plane, the plane of symmetry.
The partial circumferential crack was introduced on
the outer surface of the pipe as shown in Figure 1.
The crack front was 50 mm wide and 5mm through
the wall at the deepest point. The material used in
the analyses was A533 steel for which appropriate
models were available from previous research [12].
In all cases the introduction or generation of residual
stresses as well as proof loading took place at room
temperature, whereas the material was cooled to –
170C before final loading to fracture.
Based on the results of an initial displacement
controlled analysis, the load level was adjusted for
load controlled analyses so that the level of prior
loading resulted in significant level of plastic
deformation at the crack tip region thus producing a
residual stress field of noticeable extent. The initial
analysis was also used to identify the appropriate
node/element sets at the crack front area in order to
obtain the stress data files containing the crack front
plastic zone for use in the local approach to cleavage
fracture [5]. Thus stress and plastic strain data from
ABAQUS results were only extracted over the
region required and were stored in a data file for use
with the failure prediction model. Throughout the
load-controlled analyses an appropriate amplitude
function was used to control the load application
scheme.
The FE simulations were conducted in two stages.
Firstly the as-received (AR) or cool-fracture (CF)
and load-unload-cool-fracture (LUCF) analyses were
performed. In the next stage, initial residual stress
fields were introduced and the local approach was to
predict fracture probability in presence of initial
stresses. A series of analyses were undertaken as
summarised below.


Trial analysis: an un-cracked pipe with and
without the initial residual stresses (IS)
subjected to internal pressure until plasticity
spread throughout the wall.
As received (AR) or cool-fracture (CF):
Cracked pipe with no residual stress subjected
to axial loading to fracture at low temperature

Load-unload-cool-fracture (LUCF): Cracked
pipe with no residual stress subjected to axial
loading to fracture at low temperature following
WPS
 Initial stress-cool-fracture (ISCF): Cracked pipe
with IS subjected to axial loading to fracture at
low temperature
 Initial residual stress-load-unload-cool-fracture
(ISLUCF): Cracked pipe with IS subjected to
axial loading to provide proof loading at room
temperature, unloaded and then reloaded to
fracture at low temperature (CF)
Finally the results for a whole range of analyses were
used in conjunction with a stress based local
approach to fracture [5], to explore the predictions
for probability distributions and the consistency of
the predictions with the stress distributions.
Description of analyses and results
The residual stresses introduced to the cracked pipe
were produced at two levels. “Short range” or
“local” IS fields were generated by carrying out
separate finite element simulations of load-unload
cycles and editing the resulting stress distributions
for use as initial conditions in the main analyses.
Prescribed IS fields were also artificially generated
using an arbitrary IS file and re-distributing over an
initial analysis step. The latter procedure was used to
introduce IS fields of significant level and extent.
Analytical stress intensity factors for crack
configurations in cylindrical vessels are available
from Anderson [13] and others [14-16]. An
approximate solution taking into account the finite
size effect was used to estimate K values. Fracture
toughness was calculated from incremental fracture
loading at low temperature during the last step of
each analysis for the specific crack configuration.
The local approach predictions for fracture in the
cylinder used the same Weibull parameters
calibrated for the C(T) specimen in as-received
conditions [5]. The failure probability distribution
was calculated using the routine “local”, and plotted
against stress intensity (or fracture toughness in the
context of probability). The predictions were
promising especially with regard to the validation of
calibrated Weibull parameters for various
configurations of geometry and crack for the same
material. The distribution obtained for cylinder was
found very much consistent with those of previous
research for standard fracture mechanics specimens
(C(T) and SENB) [5]. This is illustrated by
comparing the as received distributions for the two
problems for A533B steel loaded to fracture at –
170°C in figure 2.
13th Annual (International) Mechanical Engineering Conference – May 2005
Isfahan University of Technology, Isfahan, Iran
Various analyses as described were performed and
required data was extracted for the examination of
the local approach. The routine “local” used the
material yield point in all cases to evaluate the
Weibull stress hence the failure probability
distributions. Consistent with C(T) test results,
significant increase in fracture toughness distribution
resulting from LUCF was observed (figure 2).
A series of analyses were performed for the ISCF
and ISLUCF cycles. For low levels of initial residual
stresses at the crack tip area, the shift in failure
probability distributions were only marginal.
However, the influence of initial residual stresses
became more obvious where sufficiently high level
of tensile stress normal to the crack plane and around
the crack front was introduced. The crack front
initial stress field remained significant when
rebalanced through an analysis step. The results of
these analyses were also used to predict fracture
distribution using the local approach. Significant
shift in predictions was observed when compared
with the predictions corresponding to the effect of
low-level initial stresses. This is shown in Figure 3.
The required data files for the routine local were also
prepared for all the analyses and the probability
distributions were predicted. As figure 3 indicates
the predictions are apparently different from the
predictions found earlier in presence of secondary
residual stresses in C(T) specimens [7].
Interaction of RS distribution with
subsequent fracture loading
Interaction of initial residual stresses (distribution /
redistribution) with the crack development process
was also studied. Autofrettage loading was applied
to the cylinder using the boundary conditions that
represent the un-cracked cylinder. Using a loadunload (LU) cycle, a global RS field was generated.
The RS field was then used to create a file
containing stresses at significant level, both in
magnitude and the extent. This stress field was then
used as initial conditions. Subsequently, the crack
was introduced and the residual stresses were
allowed to rebalance. Cool-fracture (CF) and proof
load –cool-fracture (LUCF) loading cycles were then
applied in presence of the incorporated initial RS
fields.
For all the cases described above axial and hoop
stresses throughout the analyses were obtained to
explore the differences in fracture response based on
changes in the stress-state. Results are discussed in
the next section in relation to the local approach
predictions for the corresponding analyses.
Finally a separate analysis was performed similar to
ISCF as described above, but using an initial RS
field defined intentionally with significant level of
axial stress and no other stress component.
Introducing such stress field into a cracked cylinder
and redistributing the RS through one analysis step
using the elastic material response, resulted in
appearance of hoop stresses of significant level and
also dramatic reduction in the initially introduced
axial RS component when the RS field was
redistributed due to the crack introduction.
Comparison of the stress level at similar loading with
the analyses that used low level of residual stresses,
suggests that much further influence of the initial
global RS field on the proof load effect is expected.
This was the case both when initial RS was followed
by LUCF and when it was followed by CF.
probability distribution predictions presented in
figure 3, suggested that unlike the secondary RS
field, a global initial RS field with sufficient level
and extent does influence the final fracture
conditions.
The completed analyses include fracture loading of
the cylinder with or without initial residual stress
field and in both cases with or without proof loading.
The residual stresses as well as the stress
distributions on final loading step for various
analyses were monitored and compared. Normal to
the crack face residual stresses, axial for cylinder,
and the hoop residual stress distributions through the
thickness of the pipe wall are plotted in figures 4a
and 4b respectively. Similarly stress distributions on
reloading for identical load levels for various
analyses are shown in figures 5a and 5b.
Discussion of results and conclusions
The performed analyses indicate that the nature of IS
introduced has significant influence on subsequent
fracture. This is believed to be due to the nature of
the RS field that was introduced as initial condition.
Where the RS field was generated by means of
autofrettage of un-cracked cylinder, the level of axial
RS was forced to remain significantly lower than
hoop RS to maintain equilibrium conditions.
The analyses results were used to explore the stress
distributions throughout various load cycles and to
extract the required input data for the routine ”local”
to investigate the effect of global RS field on the
proof loading effect and to compare the effect with
previous analyses. In earlier study it was shown that
secondary residual stresses used as initial RS field
had effectively no influence on the final fracture
conditions following proof loading as the RS field
was completely wiped out as a result of pre-loadunload stage of the loading cycle [7]. The analyses
performed based on the introduction of initial low
13th Annual (International) Mechanical Engineering Conference – May 2005
Isfahan University of Technology, Isfahan, Iran
level residual stress field (created from load-unload
of cracked cylinder as described) are essentially
considered as a verification of previous findings
obtained from the C(T) model Analyses [7].
However, the analyses performed based on the
introduction of initial residual stress field of
significant extent, i.e. arbitrary introduced high axial
RS field to the un-cracked cylinder followed by
crack introduction and rebalancing, suggested
dramatic decrease in the probability distribution as
shown in figure 3. From figure 4 it can be concluded
that there exists consistent residual stress
distributions, especially in the normal to crack
direction, and their corresponding failure probability
distributions in all analyses as shown in figure 3.
From figure 5 it can be concluded that this
consistency is also the case for stress distributions
for similar load levels for various analyses and their
corresponding probability distributions.
The present study only looks at a specific crack
configuration whereas experimental observations
suggest that cracks with different geometry and
direction are likely to appear in pressure vessels. A
comprehensive study is underway that considers
further simulations for various crack configurations
to further explore the stress and failure issues. These
investigations when put together would provide a
framework for possible suggestions regarding the
improvements in currently established integrity
assessment procedures. Currently available integrity
assessment codes include design curve approaches,
FAD approach according to R6/BS 7910, fully
plastic solutions of EPRI type, CEGB reference
stress method, the engineering treatment model and
the European SINTAP procedure [17]. The results of
these studies may be compared to those of the
established methods in order to assess the credibility
of the local approach in fracture prediction.
References
Lei, Y., “J-integral evaluation for case involving
non-proportional stressing”, Engng. Fract. Mech.,
2004, in press.
2. Margolin B Z, Kostylev V I, Keim E,
“Prediction of brittle fracture of RPV steels under
complex loading on the basis of a local probabilistic
approach”, Int. J Pressure Vessels and Piping, 81,
2004, pp 949-959
3. Gao X, Zhang G and Srivatsan T S, “Prediction
of cleavage fracture in ferritic steels: a modified
Weibull stress model”, Materials Science
Engineering, 2005, in press
4. Bordet S R, Karstensen A D, Knowels D M and
Wiesner CS, “A new statistical local criterion for
cleavage fracture in steel. Part I: model
presentation”, Engineering Fracture Mechanics, 72,
2005, pp 435-452
1.
Hadidi-Moud, S., Mirzaee-Sisan, A., Truman,
C.E., and Smith, D.J.,“A local approach to cleavage
fracture in ferritic steels following warm prestressing,” Fatigue Fract. Engng. Mater. Struct., 27,
2004, pp 931-942
6. S Hadidi-Moud, A Mirzaee-Sisan, C E Truman
and D J Smith, “Comparison of Global and Local
Approaches to Predicting Warm Pre-stress effect on
Cleavage Fracture toughness of ferritic steels”, Key
Engineering Materials, Volume 261, 2004, pp 69-74.
Advances in Fracture and Failure Prevention, ISBN:
0-87849-938-5
7. S Hadidi-Moud, A H Mahmoudi, C E Truman
and D J Smith: “Combined Effect of Residual Stress
and Loading History on Brittle Fracture”, ICM9: 9th
Int. Conference on Mechanical Behaviour of
Materials, Geneva, Switzerland, 25-29 May, 2003, in
proceedings
8. A Mirzaee-Sisan, S. Hadidi-Moud, CE Truman
and D.J Smith, “Application of the Local Approach
to predict brittle fracture following Local
Compression”, in proceedings, 15th European
Conference on Fracture (ECF15), August 2004,
Sweden, in proceedings
9. Mirzaee-Sisan A, Mahmoudi A H, Truman C E
and Smith D J, “Application of the local approach to
predict load history effects in ferritic steels”,
Proceedings of PVP2005: ASME Pressure Vessels
and Piping Division Conference, July 17-21, 2005,
Denver, Colorado, PVP2005-71606
10. S. Hadidi-Moud, CE Truman and D.J Smith:
“Interaction of mechanical loading with residual
stresses in pressure vessels”, APCFS04, Oct 2004,
Jeju, S Korea, to appear in Key Engineering
Materials, 2005
11. Bouchard, J. and Bradford, C., “Validated Axial
Residual Stress Profiles for Fracture Assessment of
Austenitic Stainless Steel Pipe Girth Welds,”
PVP2001, Vol. 422: Fracture and Fitness, Atlanta,
USA, 2001
12. Smith DJ, Hadidi-Moud S and Fowler H, “The
Effects of Warm Pre-Stressing on Cleavage Fracture,
Part 1: Evaluation of Experiments”, Engineering
Fracture Mechanics, Volume 71, Issues 13-14, 2004,
pp 2015-2032
13. Anderson T L, Fracture Mechanics (2nd
edition), CRC Press, 1994
14. Lin X B and Smith R A, “ Stress intensity
factors for semi elliptical internal surface cracks in
autofrettaged thick-walled cylinders”, Journal of
Strain Analysis, vol. 32, No 5, 1997, pp 351-363
15. Fett T, “Estimation of stress intensity factors for
semi-elliptical surface cracks”, Engineering Fracture
Mechanics, 66, 2000, pp 349-356
16. Brighenti R, “Surface cracks in shells under
different hoop stress distributions” Int. J Pressure
Vessels and Piping, 77, 2000, pp 503-509
5.
13th Annual (International) Mechanical Engineering Conference – May 2005
Isfahan University of Technology, Isfahan, Iran
100
AR; C(T)
AR
80
Failure Probability; %
17. Zerbst U, Schwalbe K H and Ainsworth R A,
“Comprehensive structural Integrity,” vol. 7, pp.289347, Edited by Ainsworth, R.A., Schwalbe, K. –H.,
Elsevier Ltd., 2003
ISCF
LUCF
ISLUCF
60
40
20
0
0
20
40
60
80
100
120
Fracture Toughness; MPa-m0.5
Figure 3: Failure probability predictions for various analyses
for A533B
steel cracked pipe at
-170C
Figure 3: Failure
probability
predictions
for
various analyses for A533B steel cracked
pipe at -170C
600
Initial RS-no crack
RS-Cracked-no LU
RS-Cracked-LU
no RS-Cracked-LU
Axial RS, initial; MPa
400
200
0
-200
-400
-600
-800
Figure 1: Geometry and crack details for
the finite element model
0
1
2
3
4
5
6
7
8
9
10
distance from outer surface; mm
Fig 4a. Axial residual stresses
for various analyses
100
AR; C(T)
AR or CF ; Pipe
LUCF; Pipe
300
Initial RS-no crack
60
RS-Cracked-no LU
RS-Cracked-LU
no RS-Cracked-LU
200
40
Hoop RS; MPa
Failure Probability; %
80
20
0
0
20
40
60
80
100
0
120
Fracture Toughness; MPa-m0.5
Figure 2: Comparison of AR predictions for CT and pipe
100
Figure 2: Comparison
ARsteel
predictions
for
and the effect of WPSof
for A533B
at -170C
CT and pipe and the effect of local RS and
WPS for A533B steel at -170Cf
-100
-200
0
1
2
3
4
5
6
7
8
distance from outer surface; mm
Fig 4b. Hoop residual stresses
for various analyses
9
10
13th Annual (International) Mechanical Engineering Conference – May 2005
Isfahan University of Technology, Isfahan, Iran
1000
AR or CF
AR or CF
LUCF
800
LUCF
400
ISCF
ISCF
Hoop stress; MPa
Axial stress; MPa
ISLUCF
ISLUCF
600
400
200
0
200
0
-200
-200
0
1
2
3
4
5
6
7
8
9
10
Distance from outer surface; mm
Fig 5a: Comparison of Axial stress field on reloading
Fig 5a: Comparison
of Axial
for various
analysesstress field on
reloading for various analyses
0
1
2
3
4
5
6
7
8
9
10
Distance from outer surface; mm
Fig 5b. Comparison of Hoop stress on reloading
Fig 5b. Comparison
of Hoop stress field on
for various analyses
reloading for various analyses