Residual Stress

Authors:
Weilin Zang
Jens Gunnars
Jonathan Mullins
Pingsha Dong
Jeong K. Hong
Research
2009:17
Effect of Welding Residual Stresses

on Crack Opening Displacement and
Crack-Tip Parameters
Report number: 2009:17 ISSN: 2000-0456

Available at www.stralsakerhetsmyndigheten.se
Title: Effect of Welding Residual Stresses on Crack Opening Displacement and

Crack-Tip Parameters.
Report number: 2009:17
Authors: Weilin Zang 1), Jens Gunnars 1), Jonathan Mullins 1), Pingsha Dong2) and Jeong K. Hong2)
Inspecta Technology AB, Stockholm, Sweden. 2)Center for Welded Structures Research, Battelle,
Columbus, Ohio
1)
Date: June 2009
This report concerns a study which has been conducted for the Swedish
Radiation Safety Authority, SSM. The conclusions and viewpoints presented in the report are those of the author/authors and do not necessarily coincide with those of the SSM.
Background
Weld residual stresses have a large influence on the behavior of cracks
growing under normal operation loads and on the leakage-flow from a
through-wall crack. Accurate prediction of these events is important in
order to arrive at proper conclusions when assessing detected flaws, for
inspection planning and for assessment of leak-before-break margins.
The fracture mechanical treatment of weld residual stresses in commonly used engineering assessment methods generally use the crack face
pressure method and account neither for the displacement controlled
nature of weld residual stresses, nor for multi-axial residual stresses. In
this report, these effects are studied.
Objectives of the project
The principal objective of the project is to investigate the accuracy of
the commonly used Crack Face Pressure (CFP) method, such as in the
computer code ProSACC, for evaluating the stress intensity factor K for
cracks located in multi-dimensional weld residual stress fields in piping
geometries. Another aim is to investigate the accuracy of existing methods to evaluate the Crack Opening Displacement (COD) for throughwall cracks in pipes.
Results
The following conclusions can be made for cracks located in the vicinity
of welds in piping geometries subjected to a combination of primary
loads (during normal operation) and weld residual stresses:
The CFP method gives a relatively good estimation of K for surface
cracks.
The CFP method cannot in general be demonstrated to give good
accuracy for evaluating K for through-wall cracks even if good
approximations exist for certain cases. The results in the report
should be carefully studied for more guidance.
The CFP method gives a relatively good estimation of the COD
except for very long through-wall cracks.
In the estimation of COD through the CFP method, plasticity effects should be taken into account except for short through-wall
cracks and low load levels.
SSM 2009:17
Effects on SSM
The results of this project will be used by SSM in safety assessments of
welded components with cracks and in assessments of Leak Before Break
(LBB) applications.
Project information
Project leader at SSM: Bjrn Brickstad
Project number: 14.42-200542009, SSM 2008/77
Project Organisation: Inspecta Technology AB has managed the project
with Dr Jens Gunnars as the project manager. Center for Welded Structures Research, Battelle in Columbus, Ohio has been used as subcontractor to Inspecta Technology AB for certain project tasks.
SSM 2009:17
Contents
1
SUMMARY..........................................................................................................3
BACKGROUND ..................................................................................................4
OBJECTIVE ........................................................................................................6
3.1
Basis for crack-tip characterizing parameters .................................................6
3.2
Definition of leak before break parameters .....................................................7
DEFINITION OF EVALUATED CALCULATION METHODS...............................9
4.1
Node relaxation in full 3D residual stress field (Control Method)...................10
4.2
Pr oSACC
Elastic K results using weight functions ( K Elastic
) ........................................10
4.3
Elastic FEM method (EL_b) ..........................................................................10
4.4
COD using weight functions with plastic correction (NURBIT) ......................10
4.5
Elastic-plastic FEM method (EP_b) ..............................................................11
4.6
FEM alternating method using infinite body solution (Alternating) ................11
DEFINITION OF CASE STUDIES.....................................................................12
5.1
Pipe geometry, loading and material properties............................................12
5.2
Crack geometries..........................................................................................13
5.3
Weld geometry assumptions.........................................................................13
CASE ONE - THIN PIPE WITH A THROUGH WALL CRACK ..........................14
6.1
Finite element models...................................................................................14
6.2
2D welding simulation ...................................................................................15
6.3
Crack profiles................................................................................................17
6.4
Maximum crack opening displacements .......................................................21
6.5
Stress intensity factor....................................................................................24
6.6
Crack tip opening displacements ..................................................................32
CASE TWO - THIN PIPE WITH A FULL CIRCUMFERENTIAL INTERNAL

SURFACE CRACK ...........................................................................................35
7.1
Finite element models...................................................................................35
7.2
Crack profiles................................................................................................35
7.3
7.4
8
8.1
CASE THREE - MEDIUM PIPE WITH A THROUGH WALL CRACK................42

Finite element model.....................................................................................42
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8.2
Results from 2D welding simulation ..............................................................43
8.3
Crack profiles................................................................................................45
8.4
Maximum crack opening displacements .......................................................49
8.5
8.6
CASE FOUR - MEDIUM PIPE WITH A FULL CIRCUMFERENTIAL

INTERNAL SURFACE CRACK.........................................................................61
9.1
Finite element model.....................................................................................61
9.2
Crack profiles................................................................................................61
9.3
9.4
10
ALTERNATING FEM METHOD ........................................................................68
10.1
Residual Stress Mapping, 3D Model and Crack Definition............................69
10.2
Results for Surface Cracks ...........................................................................72
10.3
Results for Through-Wall Cracks ..................................................................76
10.4
Comparison to the crack face pressure method............................................80
11
DISCUSSION....................................................................................................82
11.1
Crack opening displacements for leak before break assessments ...............82
11.2
Crack-tip parameters for crack growth assessments ....................................82
11.3
COD for surface cracks and planning of visual inspection ............................83
12
CONCLUSION ..................................................................................................84
13
FUTURE STUDIES ...........................................................................................85
14
REFERENCES..................................................................................................86
SSM 2009:17
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SUMMARY
Weld residual stresses have a large influence on the behavior of cracks growing under normal operation loads
and on the leakage-flow from a through-wall crack. The fracture mechanical treatment of weld residual stresses
in commonly used engineering assessment methods generally use the crack face pressure method and account
neither for the displacement controlled nature of weld residual stresses nor for multi-axial residual stresses. This
work investigates the approximation of the crack face pressure method in the presence of weld residual stresses
under normal operating conditions. Comparisons are made with a control method where the deformation
controlled characteristics of weld residual stresses are considered.
For through wall cracks the crack opening area results are of interest for performing leak before break
assessments. The main conclusion is that good agreement is obtained between the elastic-plastic FE method
(crack face pressure method), NURBIT (crack face pressure method) and the control method for typical
minimum detectable crack sizes and load levels commonly seen in Swedish nuclear reactors (load level 1). The
maximum observed variation between the methods was 26%, where COD was overestimated.
For full circumferential internal surface cracks K estimates are of interest for crack growth predictions. The
main conclusions are:
For the 8 mm thick pipe there is generally good agreement between K predictions for the elastic
ProSACC (crack face pressure) method and the control method for crack depths up to a/t=0.6.
ProSACC method and the control method. The largest variation of 15% was for the higher load level
and deepest crack (a/t=0.8).
For the 32 mm thick pipe, where the axial weld residual stress profile is sinusoidal, the
surface COD due to residual stresses only decreases rapidly, leading to the situation that it is
more difficult to detect a deep crack than a shallow one. This may have implications for VT testing.
For the 32 mm thick pipe, all methods predict crack arrest. For shallow cracks, K was overestimated
by all crack face pressure methods where there was a tensile residual stress near the inner wall.
SSM 2009:17
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BACKGROUND
Weld residual stresses have a large influence on the behavior of cracks growing under normal operation loads
and on the leakage-flow from a through-wall crack. Accurate prediction of these events is important in order to
arrive at proper conclusions when assessing detected flaws, for inspection planning or for assessment of leakbefore-break margin.
A first prerequisite for performing these predictions is reliable prediction of the magnitude and distribution of
welding residual stresses. This issue was addressed in a first stage of this project reported in [1], which had the
purpose to validate and improve the weld residual stress modeling procedure used in Sweden [2].
The fracture mechanical treatment of weld residual stresses in commonly used engineering assessment methods
generally account for neither the displacement controlled nature of weld residual stresses nor for multi-axial
residual stresses. These issues are considered in this stage of the project.
Stress corrosion crack growth under normal operation loads
An important crack growth mechanism that has to be accounted for in performing inspection planning and
structural integrity assessments of stainless steel components is stress corrosion cracking (SCC). For stress
corrosion cracking to take place tensile stresses and an unfavourable environment are required. The mechanism
is generally active under normal operation conditions. Since weld residual stresses usually are very high, they
can have a significant influence on initiation and the subsequent growth of stress corrosion cracks. Crack
growth rates are commonly expressed using the stress intensity factor K which characterizes the mechanical
state at the crack-tip. Thus, the fracture mechanical treatment of weld residual stresses is of a great importance
when predicting cracking of welded components.
Leak-before-break assessments
The leak-before-break (LBB) concept is sometimes applied for nuclear piping, in order to demonstrate that
crack growth will result in a leak that is detectable, long before the crack reaches a critical size resulting in
fracture. For example demonstration of LBB is sometimes applied in order to show that building of pipe break
restraints is not necessary. LBB analyses may show that the margin against guillotine pipe break is very high,
for a given leak detection capability, and hence pipe break restraints are unnecessary for reaching an acceptably
low level of risk for dynamic events. In LBB analyses the weld residual stresses influence the conclusions about
the leak-before-break margin, especially with respect to the estimated crack opening area (or crack opening
displacement COD) and the corresponding leak rate.
Testing by visual inspection technique
One non-destructive inspection method that is increasingly used is the visual inspection technique (VT). In
order to use the method cracks must be detectable for sizes that are smaller than the maximum allowable. In
practice inspection at each location requires that residual stresses are present that open the crack mouth. To
demonstrate that the postulated crack is detectable it is important to have an accurate prediction of the crack
opening displacement (COD) at room temperature.
Determination of K and COD in residual stress fields
The stress intensity factor and crack opening displacement for a crack in a weld residual stress field can be
determined in more than one way. For engineering assessments it is common to utilize a calculation procedure
sometimes called the crack-face pressure method. By this method the stress in an uncracked model is extracted
along the plane were the crack is postulated. Only the stress resolved normal to the crack plane is considered.
This stress is applied to the crack surface in a model where the crack exists. This method is used both for direct
calculation of K by FEM, and for derivation of weight functions by FEM or analytical methods.
SSM 2009:17
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The crack-face pressure method involves an assumption of load-controlled conditions, but is also valid for
situations where remote displacements are applied. As long as the response is essentially linear, for example a
large elastic specimen containing a short crack, and the loading is dominated by the normal stress component,
the crack-face pressure method holds. However, for a crack growing in a weld, the assumptions of remote
loading and linearity do not apply in general:
In a weld, the residual stress field is created by thermal and plastic strains generated in and around the weld
metal. If the crack size is comparable to the radius of this region then the loading conditions should be treated
as local and the remote loading assumption is no longer appropriate.
Significant plastic deformation may occur as the crack grows. The residual stress field must satisfy equilibrium
conditions within the component of concern. For a growing crack, the residual stresses nearby re-distribute as
stress-free surfaces are generated, while for load-controlled conditions the far-field stresses can be assumed to
be stationary.
Weld residual stress fields are triaxial so in these cases it cannot be assumed that loading is dominated by the
normal stress component.
In spite of these limitations the crack-face pressure method is widely used. Thus it is important to investigate
under which conditions the method remains conservative for calculation of the stress intensity factors and crack
openings.
SSM 2009:17
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OBJECTIVE
This work investigates the treatment of weld residual stresses when performing fracture mechanical analysis of
welded components. The objective is to determine if it is conservative to use the crack face pressure method for
crack shapes and loading cases that might be found under normal operating conditions. The magnitude of the
residual stress field and crack characterizing parameters affects crack growth and leak-before-break predictions
in different ways, and accurate and conservative estimates are important.
Detailed simulations are made where a crack is created in full 3D weld residual stress fields by use of the noderelaxation technique. The crack-tip characterizing parameters and the crack opening displacement determined
from this model are compared with results obtained from different implementations of the crack face pressure
method. Of particular interest is comparison and validation of the prediction of the ProSACC stress corrosion
crack growth module and NURBIT. Two pipe geometries with surface and through wall cracks are studied for
normal operation cases where the weld residual stresses constitute a significant part of the loading. Also the
alternating method is considered.
3.1
Basis for crack-tip characterizing parameters
In this investigation crack growth under normal operation is of interest. Relations that describe crack growth
due to fatigue or stress corrosion normally use the elastic stress intensity factor K as the parameter describing
the mechanical state at the crack tip. Values for K can be directly extracted from finite element models through
nodal extrapolation from the near-tip finite element mesh. This method, however, requires a very fine mesh
which makes analysis computationally expensive, especially in 3D models such as those considered in this
study. Further, K is an elastic crack tip parameter and the test cases considered in this study involve ductile
materials for which considerable plastic deformation may occur around the crack tip. For these reasons, the
natural alternative crack tip parameter to consider is the J-integral. It can be shown to characterize the stresses
and strains near the crack-tip in elastic-plastic materials and is also an energy conservation integral.
The standard J-integral is known to be unique for plastic materials provided unloading does not occur.
However, it is path dependent for a growing crack in an elastic-plastic material (due to unloading in the wake)
and also in the presence of a weld residual stress field. This has spawned considerable efforts in recent years to
develop a path-independent form of J [3-6], where the influence of prior plastic deformation is decoupled from
the standard surface integral and included as an additional term in the formulation. These efforts were
successful and the modified J-integral formulation shows path-independence under combinations of residual
stress and mechanical loading. Further, the modified J is also shown to be equivalent to the stress intensity
factor K in cases of small scale yielding. The calculated J in this investigation will be converted to an
equivalent stress intensity factor using the relation,
K J E /(1 2 )
(3.1)
where E is Youngs modulus and is Poissons ratio.

Studies on the modified J-integral [4-7] have also demonstrated that the standard J-integral can be used to
provide an accurate estimate, provided that the evaluation path is suitably near to the crack tip. It has been
found that in an area very close to the crack tip, J evaluated by the standard definition and by the modified
definition both result in an almost identical value [4]. The reason for this is that the contribution from the
additional prior plastic deformation term vanishes to zero when the integral is evaluated very close to the
crack tip [4]. One conclusion from this is that by using the modified
J-integral a coarser mesh and a longer integration path can be used.
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In [4-7] the difference between the standard and the modified J-integral tends to decrease as the primary load
decreases. In this investigation we are focusing on growth and leakage under normal operation loads, and not
on the prediction of fracture due to high primary loads. In order to conclude on the distance at which the two
integrals become identical, it is necessary to calculate both the standard and the modified integral for the
present loading situation. In the present investigation an implementation of the modified J-integral was not
available for 3D cases so, alternatively, near-tip estimates of the standard J-integral are used. It is judged that
the standard J-integral evaluated close enough to the crack-tip is sufficient to demonstrate the key trends for the
present investigation. The standard J-integral evaluated between 0.01 and 0.1 mm from the crack tip is used.
An alternative measure for the near-tip state is CTOD [8]. The definition is illustrated in Figure 3.1. For an
elastic-plastic material when no unloading occurs CTOD has a simple relation with the standard
J-integral [8].
CTOD
Figure 3.1 Definition of CTOD.

While CTOD could provide a valid measure of the mechanical state at the crack tip, the crack growth relations
are usually based upon K, and it is more difficult to convert CTOD to K. It is typically more convenient to use
the J-integral and equivalent K, although CTOD values are reported here.
3.2
Definition of leak before break parameters
The crack opening area COA is a key parameter for estimation of leakage rates in leak-before-break
assessments. In the current investigation we will study the parameters defined below.
Figure 3.2 shows a through wall crack of length 2a approximated by an elliptical function. For such a geometry
the crack opening displacement can be expressed as
2
u e 1 l / a ,
(3.2)
where e is the the crack opening displacement at the mid point of the crack. Integrating the displacement
results in the (elastic) crack opening area
COAe
e a .
2
(3.3)
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Figure 3.2 Elliptical model for crack opening displacement (through wall case).
The COA for each method can be compared directly. As seen above it is also possible to compare an equivalent
maximum crack opening displacement, which will accentuate the difference in the crack opening area
calculated by the different methods. We will present the results as an equivalent crack opening parameter
defined by
2 * COA
,
a *
(3.4)
where COA is evaluated by an integral of the crack opening displacement COD along the crack length.
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DEFINITION OF EVALUATED CALCULATION METHODS
The different analysis methods evaluated for treatment of weld residual stresses are summarized in the table
below.
Table 4.1. Notation and overview of the compared calculation methods.
Method
denotation
Short description of calculation method
Parameters
evaluated
Control
FEM, near tip J-integral calculation

Node relaxation of crack
Full 3D residual stress field
Elastic-plastic model
Considered as the most accurate simulation method.
K, CTOD
COD,
Weight functions, solutions determined for load controlled

boundary conditions using the crack face pressure method
Normal stresses taken from uncracked body
Elastic model
EL_b

Crack face pressure
Elastic model
K, CTOD
COD,
NURBIT
Weight functions, solutions determined for load controlled

boundary conditions using the crack face pressure method
Elastic model using plastic correction
EP_b

Crack face pressure
Elastic plastic model
K, CTOD
COD,
Pr oSACC
K Elastic
Alternating FEM alternating method (coupled, iterative FE/analytical

method)
Infinite body KI solution
Elastic
Full 3D residual stress field applied
Below some more details are given for the different calculation methods.
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K
COD,
4.1
Node relaxation in full 3D residual stress field (Control Method)
This simulation method is designed to best simulate the growth of a crack in a residual stress field. The
procedure is as follows:
A 2D axi-symmetrical welding simulation is performed. The new welding simulation procedure according to [1]
is applied.
Hydro-test pressure is applied and released at 20 C.
The 2D solution is mapped into a 3D model with a crack. At this stage nodes along the crack are constrained.
The operating loads (membrane stress plus bending stress) are applied (load case 1).
Crack growth is activated by instantaneously releasing the constraints on the crack face nodes.
J-integrals, COD and CTOD are calculated.
Finally the primary loads are increased to a slightly higher level, called load case 2.
The J-integral, COD and CTOD are calculated for this load case.
The results generated in this manner are considered as the most accurate results.
4.2
Pr oSACC
Elastic K results using weight functions ( K Elastic
)
This method uses the crack face pressure assumption. Elastic weight functions are used to calculate K values
due to primary and secondary loads. These results are useful for assessment of crack growth calculations.
4.3
Elastic FEM method (EL_b)
This method uses the crack face pressure assumption where the stress in an uncracked model is extracted along
the plane where the crack is postulated. Only the stress resolved normal to the crack plane is considered. This
stress is applied to the crack surface in a model where the crack exists.
This method uses linear elastic finite element calculation and the residual stress obtained from the 2D welding
simulation is applied as the crack face pressure.
The J-integral, COD and CTOD are calculated for two load cases. Results are denoted EL_b. Results for the
Pr oSACC
residual stress only are denoted EL (Resi. only). This model is very similar to the K Elastic
model
4.4
COD using weight functions with plastic correction (NURBIT)
The calculation procedures are similar to those used in ProSACC with the exception that some solutions are
more detailed with regards to through wall crack geometries. In contrast to ProSACC, estimates of COD (and
COA) can be extracted from NURBIT [10]. A plastic correction is also applied and details are available in [10].
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4.5
Elastic-plastic FEM method (EP_b)
The calculation procedure is identical with the elastic FEM method, with the exception that the material
response is elastic-plastic.
The J-integral, COD and CTOD are calculated for two load cases. Results are denoted by EP_b.
Elastic-plastic FEM analyses of cracks using the crack face pressure method are most relevant for COD
assessment of fracture but not for crack growth assessments.
4.6
FEM alternating method using infinite body solution (Alternating)
The method involves an iterative procedure using a coupled finite element and infinite body (analytical)
solution [11, 12]. This method uses an elastic constitutive model. A full 3D residual stress field is applied as an
initial condition in an uncracked FE model. The FEM boundary conditions are successively updated until they
display a (nearly) stress free state along the postulated crack surface. At the same time the crack face pressure
in an infinite body crack model is successively updated until it converges. The alternating method is described
in detail in chapter 10.
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DEFINITION OF CASE STUDIES
The different analysis methods will be evaluated for four cases which represent loading scenarios that occur
under normal operation of a nuclear power plant. The four cases were taken from a LBB study [13]. A thin
and a medium thickness pipe are chosen, containing first a through wall crack and then an internal surface
crack. Four crack sizes are considered for each crack geometry. As discussed in this chapter, two load levels
are specified in an effort to cover the range of conditions experienced under normal operation.
5.1
Pipe geometry, loading and material properties
A thin pipe and a medium pipe are selected for investigation. The geometries and mechanical loadings for the
chosen cases are shown in Table 5.1 and 5.2. Note that in Table 5.2 the axial membrane stress is entirely due to
internal pressure. The load case L2 deviates from [13] and is introduced to represent slightly higher loads than
normal; in this case the bending load specified in load case L1 was doubled. The pipes are made of stainless
steel TP304. The material properties used in the analysis are listed in Table 5.3. Bilinear isotropic hardening
was specified.
Table 5.1 Geometry.
Thin pipe
Medium pipe
Inner pipe radius

(mm)
153.6
146
Thickness, t
(mm)
8.3
31.8
Table 5.2 Applied mechanical loads for load levels 1 and 2.

Internal
pressure
Temperature (MPa)
(C)
Thin pipe
Medium pipe
145
323
4.6
15.5
Load Level 1 (L1)

Axial
Bending
membrane
stress
stress
(MPa)
(MPa)
41
32
32.1
19
Load level 2 (L2)

Axial
Bending
membrane
stress
stress
(MPa)
(MPa)
41
64
32.1
38
Table 5.3 Material properties
E (GPa)
Temperature
(C)
TP304
Weld material
100
195
195
200
185
185
400
170
170
0.3
0.3
100
173
419
200
140
368
400
116
264
y (MPa)
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5.2
Crack geometries
Cracks growing along the weld centerline are studied. Two crack geometries are considered:
A through wall circumferential crack. Four different crack lengths are evaluated, specified as shown in Table
5.4.
A complete circumferential internal crack. Four different crack depths a are considered,
a/t = 0.2, 0.4, 0.6 and 0.8.
In the LBB study [13] the crack size was determined for a leakage rate of 10 gallons per minute. In reality this
crack size is heavily dependent on the crack face morphology and for this reason different crack sizes were
specified in Table 5.4, corresponding to typical observed morphologies.
The detectable through wall crack sizes considered are listed in Table 5.4. The maximum allowable crack size
was also considered, although the shorter crack sizes are of greater interest because weld residual stresses make
a larger contribution to the loading. A further reason why the shorter cracks are of more interest is that a crack
spends most of its lifetime in stable growth, not at the maximum allowable size.
Table 5.4 Detectable through wall crack sizes chosen for evaluation [13].
5.3
Thin pipe
(mm)
137
Medium pipe
(mm)
175
160
218
185
246
350
330
Comments
Crack size resulting in detectable leakage (load
level 1). Assumed crack morphology, fatigue 1.
level 1). Assumed crack morphology, fatigue 2.
level 1). Assumed crack morphology, PWSCC.
Maximum allowable crack size, [13].
Weld geometry assumptions
Four weld passes are used for the thin walled pipe and twelve weld passes for the medium thickness pipe. The
heat input is defined according to the phase 1 report from this project [1]. Welding is simulated using
symmetry conditions at the weld centre line.
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CASE ONE - THIN PIPE WITH A THROUGH WALL CRACK
6.1
Finite element models
The finite element mesh used for 2D welding simulation and the weld pass order is shown in Figure 6.1. The
finite element mesh for the 3D fracture mechanics model with a through wall crack is shown in Figure 6.2. A
small notch with a notch radius of 0.001 mm is introduced at the crack tip this allows an accurate prediction
of near-tip-J and CTOD.
Figure 6.1 Finite element mesh for 2D welding simulation for case 1.
l
Figure 6.2 Case 1 - finite element mesh with a through thickness crack. The coordinate l = 0 corresponds to the
center of the through wall crack, as previously defined in Figure 3.2.
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6.2
2D welding simulation
The welding residual stress distribution at the operating temperature is shown in Figures 6.3 and 6.4. The axial
residual stress along the weld centre is a bending stress, as illustrated in Figure 6.5
Figure 6.3 Welding residual stress in the axial direction at the operating temperature.
Figure 6.4 Welding residual stress in the circumferential direction at the operating temperature.
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500
400
300
200
100
-100
Axial stress
Hoop Stress
-200
-300
0
10
X (mm)
Figure 6.5 Welding residual stresses along the weld centre line. X is measured through the thickness from the
pipes inner surface.
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6.3
Crack profiles
Crack profiles determined by the different evaluation methods are shown in Figures 6.6-6.9 for the four
different through wall crack lengths. Results are shown for the two load levels L1 and L2. In the figures, open
symbols indicate results at load level 1 and the solid symbols for load level 2. The figures are split into two
parts: a) gives crack opening displacement COD results for the pipe inner surface and b) for the pipe outer
surface. Results due to residual stress only calculated by the elastic FEM method are also included.
When compared with the control method, it is observed that the COD calculated by the elastic FEM method
(EL_b) is underestimated, and the COD calculated by the elastic-plastic FEM method (EP_b) is overestimated.
If the results are used for an LBB-analysis or assessment of detection by visual inspection technique, the results
from the elastic FEM method are conservative in the sense that they under-predict the leak rate and the crack
opening. The results from the elastic-plastic FEM method are non-conservative in the same sense.
Note that for one of the analysed cases the crack profile predicted by the elastic FEM method (EL_b) and the
control method coincide, as indicated in Figure 6.6b. This could be explained by an essentially elastic behavior
for this low load level and short crack. As the crack length is increased by 17%, we start to see a discrepancy,
see Figure 6.7b.
0,5
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. Only)
0,4
Control (L2)
EL_b (L2)
EP_b (L2)
0,3
0,2
0,1
0
0
10
20
30
40
50
60
70
l (mm)
Figure 6.6a COD along pipe inner surface for crack length 137 mm. L1 indicates the results at load level 1
and L2 at the higher load level 2. See Table 4.1 for the notation of the methods.
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0,5
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. Only)
0,4
Control (L2)
EL_b (L2)
EP_b (L2)
0,3
0,2
0,1
-0,1
0
10
20
30
40
50
60
70
l (mm)
Figure 6.6b COD along pipe outside surface for crack length 137 mm
0,5
0,4
0,3
0,2
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi only)
0,1
Control (L2)
EL_b (L2)
EP_b (L2)
0
0
10
20
30
40
50
60
70
80
l (mm)
Figure 6.7a COD along pipe inner surface for crack length 160 mm.
18 (86)
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. Only)
Control (L2)
EL_b (L2)
EP_b (L2)
0,5
0,4
0,3
0,2
0,1
-0,1
-0,2
0
10
20
30
40
50
60
70
80
l (mm)
Figure 6.7b COD along pipe outer surface for crack length 160 mm.
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
20
40
60
80
100
l (mm)
19 (86)
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
0,8
0,6
0,4
0,2
-0,2
0
20
40
60
80
100
l (mm)
Figure 6.8b COD along pipe outer surface for crack length 185 mm.
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
0
0
50
100
150
200
l (mm)
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Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
10
0
0
50
100
150
200
l (mm)
Figure 6.9b COD at pipe outer surface for crack length 350 mm.
6.4
Maximum crack opening displacements
As discussed in section 3.2 the integrated crack opening area may be represented by an equivalent crack
opening parameter, Predictions from the FE analyses (from the previous section) are presented in Tables 6.16.2 and in Figure 6.10. Predictions from NURBIT are also shown.
When compared with the control method, it is observed that the calculated by the elastic-plastic FEM method
(EP_b) is slightly overestimated for the shorter cracks (corresponding to the detectable sizes from Table 5.1).
The calculated by NURBIT is similar to the results by EP_b. For the crack lengths up to 185 mm, the crack
opening area is overestimated by 7-26% for the lower load case L1. Comparison to calculated by the elastic
FEM method indicates that plastic effects are substantial even for the moderate crack lengths and low load
level.
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Table 6.1 Equivalent crack opening parameter at pipe inner surface.

Crack
Length
(mm)
137
160
185
350
Load Level
Control
(mm)
Weight fcn
Plastic corrn
NURBIT
(mm)
Elastic FEM
EL_b
(mm)
Plastic FEM
EP_b
(mm)
L1
0.224
0.261
0.205
0.240
L2
0.335
0.407
0.263
0.396
L1
0.280
0.313
0.248
0.310
L2
0.448
0.536
0.324
0.541
L1
0.359
0.394
0.321
0.427
L2
0.610
0.725
0.420
0.766
L1
2.10
1.10
2.74
L2
5.60
1.48
8.07
Table 6.2 Equivalent crack opening parameter at pipe outer surface.

Crack
Length
(mm)
137
160
185
350
Load Level
Control
(mm)
Weight fcn
Plastic corrn
NURBIT
(mm)
Elastic
FEM
EL_b
(mm)
Plastic
FEM
EP_b
(mm)
L1
0.081
0.070
0.083
0.123
L2
0.209
0.268
0.156
0.316
L1
0.140
0.127
0.128
0.199
L2
0.335
0.414
0.221
0.475
L1
0.223
0.218
0.207
0.328
L2
0.511
0.633
0.325
0.720
L1
2.06
1.00
2.74
L2
5.79
1.41
8.40
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0,8
Control (L1)
EL_b (L1)
EP_b (L1)
NURBIT (L1)
Control (L2)
EL_b (L2)
EP_b (L2)
NURBIT (L2)
0,6
0,4
0,2
0
100
120
140
160
180
200
Crack Length (mm)
Figure 6.10a. Equivalent crack opening parameter at inside surface of pipe. In the figure L1 indicates the
results at load level 1 and L2 at load level 2.
1,5
1,2
Control (L1)
EL_b (L1)
EP_b (L1)
NURBIT (L1)
Control (L2)
EL_b (L2)
EP_b (L2)
NURBIT (L2)
0,9
0,6
0,3
0
100
120
140
160
180
200
Crack Length (mm)
Figure 6.10b. Equivalent crack opening parameter at outside surface of pipe.
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6.5
Stress intensity factor
The stress intensity factor calculated in this section is based on the near-tip J-integral. Eq. (3.1) is used to obtain
K. The standard J-integral, when evaluated close enough to the crack-tip will provide an accurate estimate of
the J-integral calculated using the integral modified to account for residual stresses, as discussed in Section 3.1.
The J-integral along different paths is shown in Figure 6.12 for the case of a 160 mm long crack. The crack is
simulated using the control method. Figure 6.12a demonstrates that the standard integral is, in general, path
dependent. However, close enough to the crack tip, path independence is displayed as proposed in [4]. In this
case, within 1 mm from the crack tip the J-integrals are almost path independent. Figure 6.12b shows a detailed
view of the paths very close to the crack-tip. The J-integral is evaluated for three different positions along the
crack front as defined in Figure 6.11.
x
3
2
Figure 6.11. Definition of evaluation points along the crack front of the through wall crack. Crack locations 1, 2
and 3 are also refered to as Loc 1, Loc 2 and Loc 3.
For this case it was chosen to determine K based on J-integrals calculated at 0.1 mm from the crack-tip (path 10
from the tip).
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40
20
-20
Loc 1 (Load 1)
Loc 2 (Load 1)
Loc 3 (Load 1)
Loc 1 (Load 2)
Loc 2 (Load 2)
Loc 3 (Load 2)
-40
r (mm)
Figure 6.12a. J-integrals as function of distance r from the crack-tip for a crack simulated by the control
method (crack length 160 mm).
Loc 1 (Load 1)
Loc 2 (Load 1)
Loc 3 (Load 1)
Loc 1 (Load 2)
Loc 2 (Load 2)
Loc 3 (Load 2)
50
40
30
20
10
-10
0
0,1
0,2
0,3
0,4
0,5
r (mm)
Figure 6.12b. J-integrals as function of distance r from the crack-tip for a crack simulated by the control
method (crack length 160 mm). The 25 near-tip paths are shown.
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The distributions of stress intensity factors along the crack front are shown in Figures 6.13-6.15. Results for
three different methods are presented: control, elastic FEM and plastic FEM.
The most relevant comparison for crack growth assessments is the difference between the control method and
the elastic FE method, because the growth calculations in ProSACC are made using linear elastic solutions. For
the moderate cracks up to 185 mm the results for EL_b and control agree well, except at the pipe inside.
Below is a discussion about the deviation in K that is observed for the control method near the inner surface of
the pipe. As discussed in the background, the different evaluation methods are based on different modeling
assumptions. For example, the high out-of-plane residual stress is only included in the control method. Below
some efforts are made to understand the important assumptions in the different models. Note that for the
control method K actually decreases approaching the pipe inner surface. Such a results has not previously been
observed by the authors and requires further investigation.
We first consider the results for the largest crack (a=350 mm) and the highest loading (L2). In this case the
contribution from the residual stress is minimal as seen in Figure 6.15. This allows a comparison of the control
method and the elastic-plastic FE method for a case dominated by primary loading. It is seen that the K-values
predicted by the control method are consistently lower than those predicted by the elastic-plastic FE method.
Since the contribution from the residual stresses to K is small, the major difference between these methods is
that a high out-of-plane stress is only applied in the case of the control method (see the hoop stress from Figure
6.5). Addition of such an out-of-plane stress will increase the hydrostatic stress component and suppress neartip plasticity which might explain the consistently lower reported K value, where K was converted from J using
equation 3.1.
Considering then the other crack sizes and load levels, a somewhat different trend emerges. Towards the outer
edge of the pipe the results for the elastic-plastic FE and control methods are in reasonably good agreement.
The larger difference in predictions is seen towards the inner pipe surface. In this region the residual stresses
are tensile and constitute a large proportion of the loading. At the outside the residual stress is compressive. It
is hard to differentiate whether the difference towards the inner surface is due to the out-of-plane stress or due
to the displacement controlled nature of the residual stress modelled by the control method.
Comparing the elastic FE and elastic-plastic FE models two observations can be made. For low loads the two
methods are nearly identical. For high loads, and the largest crack size, higher K-predictions are obtained for
the elastic-plastic FE method than for the elastic FE method. These trends may be explained by increased crack
tip plasticity in the elastic-plastic FE model for increases in load and crack size. Increased crack tip plasticity
effectively increases the crack size and therefore also increases the estimated value of K.
A final observation regarding the Figures 6.13 to 6.15 regards application of the different evaluation methods
for crack growth calculations. Compared to the control method, the K-value calculated by the elastic-plastic FE
method is conservative, whereas the elastic FE method is non-conservative.
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150
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
100
50
-50
0
0,2
0,4
0,6
0,8
x/t
Figure 6.13
Stress intensity factors along the crack front for crack length 160 mm. In the figure, x = 0 at the
pipe inner surface, L1 indicates the results at load level 1 and L2 at load level 2.
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
150
100
50
-50
0
0,2
0,4
0,6
0,8
x/t
Figure 6.14
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Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
500
400
300
200
100
-100
0
Figure 6.15
0,2
0,4
0,6
0,8
x/t
The results are also shown in Tables 6.5-6.6. Results from the software package ProSACC are included for
comparison. For crack growth calculations it may be concluded that the elastic ProSACC predictions
overestimate K near the inner surface of the pipe, in comparison to the control results.
Note that the maximum value of K is not always obtained at the inner or outer surfaces for the control method
and EP_b, see Figure 6.15. This could have implications for predictions using ProSACC since only values at
the surfaces are provided from ProSACC.
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Table 6.5 Stress intensity factors estimated using different methods at load level 1.
Crack
Length
(mm)
137
160
185
350
Crack
Location
Control
(MPam)
Pr oSACC
K Elastic
(MPam)
EL_b
(MPam)
EP_b
(MPam)
54.7
85.4
85.2
82.0
46. 6
43.5
46.3
4.1
-3.4
-0.19
-3.0
58.9
94.1
94.4
89.7
56.9
48.5
56.6
3.9
1.9
5.3
3.2
64.3
104.6
109.5
104.3
67.8
61.5
71.1
14.3
7.4
14.4
15.0
121.0
204.0
207. 5
209.2
189.0
136.3
216.7
97.4
50.2
56.3
108.9
Table 6.6 Stress intensity factors estimated using different methods at load level 2.
Crack
Length
(mm)
137
160
185
350
Crack
Location
Control
(MPam)
Pr oSACC
K Elastic
(MPam)
EL_b
(MPam)
EP_b
(MPam)
67.4
105.2
103.8
106.4
72.0
60. 7
76.1
24.5
16.8
18.0
20.3
75.2
117.8
116.5
118.0
88.8
72.0
92.0
37.1
24.1
25.5
30. 5
84.0
132.9
135.9
137.9
106.1
86.7
112.8
48.8
31.8
36.5
47.5
228.3
270.5
308.0
333.7
365.2
183.2
442.5
187.5
88.2
51.7
237.5
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Tables 6.7 and 6.8 are included to show the contribution from the residual stress to K for the load cases chosen
for analysis in this investigation. The contribution at the inner surface is about 50% for the shortest crack and
the low primary loading L1.
Table 6.7 Comparison showing the contribution of the residual stress to K calculated using the elastic-plastic FE
method (load level L1).
Crack
Length
(mm)
137
160
185
350
Crack
Location
Without
residual
stresses,
(MPam)
With
residual
stresses,
(MPam)
44.9
82.0
54.7
65.0
43.7
-3.0
53.0
89.7
64.9
75.6
50.3
3.18
64.1
104.3
77.4
84.4
58.6
15.0
173.7
209.1
222.9
234.7
156.9
108.9
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Table 6.8 Comparison showing the contribution of the residual stress to K calculated using the elastic-plastic FE
method (load level L2).
Crack
Length
(mm)
137
160
185
350
Crack
Location
Without
residual
stresses,
(MPam)
With
residual
stresses,
(MPam)
69.4
106.4
84.8
95.1
67.0
20.3
81.3
118.0
100.6
111.4
77.5
30. 5
97.7
137.9
120.7
127.8
91.1
47.5
298.3
333.7
460.7
472.5
285.5
237.5
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6.6
Crack tip opening displacements
Crack tip opening displacements were calculated at the three locations indicated in Figure 6.11. The results are
listed in Tables 6.9-6.10 and are also plotted in Figure 6.17a and b to allow better visualization. These results
show that the elastic FE method underestimates CTOD. For the three largest crack lengths the elastic-plastic
FE method overestimates CTOD.
Table 6.9 Crack tip opening displacements estimated using different methods load level 1.
Crack
Length
(mm)
137
160
185
350
Elastic FEM
Crack
Location
Control
(mm)
EL_b
(mm)
Elastic-plastic
FEM
EP_b
(mm)
0.0095
0.0018
0.0080
0.0088
0.0009
0.0113
-0.0004
0.0000
0.0053
0.0115
0.0017
0.0112
0.0125
0.0018
0.0164
-0.0001
0.0015
0.0074
0.0142
0.0023
0.0167
0.0175
0.0014
0.0240
0.0004
0.0003
0.0108
0.0924
0.0029
0.1053
0.1332
0.0029
0.1857
0.0361
0.0011
0.3488
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Table 6.10 Crack tip opening displacements estimated using different methods load level 2.
Crack
Length
(mm)
137
160
185
350
Elastic FEM
Crack
Location
Control
(mm)
EL_b
(mm)
Elastic-plastic
FEM
EP_b
(mm)
0.0156
0.0022
0.0201
0.0200
0.0013
0.0296
0.0015
0.0004
0.0149
0.0210
0.0024
0.0287
0.0294
0.0016
0.0420
0.0038
0.0005
0.0213
0.0281
0.0029
0.0456
0.0425
0.0019
0.0600
0.0068
0.0008
0.0326
0.1774
0.0042
0.3289
0.4379
0.0025
0.7011
0.1338
0.0019
0.2936
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Control (Loc 1)
EL_b (Loc 1)
EP_b (Loc 1)
Control (Loc 2)
EL_b (Loc 2)
EP_b (Loc 2)
Control (Loc 3)
EL_b (Loc 3)
EP_B (Loc 3)
0,024
0,016
0,008
-0,008
100
120
140
160
180
200
Crack Length (mm)
Figure 6.17a. Crack tip opening displacements for load level 1.

Control (Loc 1)
EL_b (Loc 1)
EP_b (Loc 1)
Control (Loc 2)
EL_b (Loc 2)
EP_b (Loc 2)
Control (Loc 3)
EL_b (Loc 3)
EP_B (Loc 3)
0,06
0,048
0,036
0,024
0,012
0
100
120
140
160
180
200
Crack Length (mm)
Figure 6.17b. Crack tip opening displacements for load level 2.
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CASE TWO - THIN PIPE WITH A FULL CIRCUMFERENTIAL INTERNAL

SURFACE CRACK
7.1
Finite element models
The pipe geometry considered in case two is the same as for case one with the difference that a full
circumferential, internal surface crack is modeled, as opposed to a through-wall crack. The finite element mesh
used for welding simulation is also the same, as shown in Figure 7.1. The finite element mesh for the fracture
mechanics model is, however, different. A 2D, axisymmetric model was used and is illustrated in Figure 7.1.
The notch radius is 0.001mm, allowing accurate prediction of near-tip-J and CTOD.
Figure 7.1 Case 2 2D axisymmetric finite element mesh for the full circumferential, internal surface crack.
7.2
Crack profiles
Crack profiles through the thickness are shown in Figures 7.2 to 7.5 for the different evaluation methods. Each
figure represents a different crack depth, ranging from a/t=0.2 to 0.8. Results are shown for the two load levels
L1 and L2. In the figures, open symbols indicate results at load level 1 and the solid symbols at load level 2.
Results due to residual stresses only, calculated by the elastic FE method, are also included.
Note that the crack opening displacement due to welding residual stress contributes approximately 50% of the
total COD for both the elastic and elastic-plastic FE methods. Inclusion of weld residual stresses is therefore
very important in computation of COD for these cases.
It is observed that the crack opening displacement predicted by the elastic FE method and the elastic-plastic FE
method are nearly identical when a/t < 0.6. This implies that the effect of plastic deformation in these cases is
very limited. In comparison to the control method, both the elastic and elastic-plastic FE methods
underestimate the crack opening displacement for a/t up to 0.6. For a/t equal to 0.8, the same trend is again
observed for the lower load level L1, but for the higher load level L2, the COD for the elastic-plastic FE
method is overestimated. The difference between the control method and EP_b could be due to the
displacement controlled model and the high out-of-plane stress.
Note that for conclusions on VT inspection results for the control method and residual stresses only would be
needed, since VT is performed in a state without pressure.
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12
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
COD (m)
0
0
0,5
1,5
x (mm)
Figure 7.2 COD for crack depth a/t = 0.2. x = 0 at the pipe inner surface.
25
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
18,75
COD (m)
12,5
6,25
0
0
x (mm)
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70
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
60
Control (L2)
EL_b (L2)
EP_b (L2)
50
40
COD (m)
30
20
10
0
0
x (mm)
160
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
120
COD (m)
80
40
0
0
x (mm)
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7.3
As in Section 6.5 the stress intensity factor calculated in this section is based on the near-tip J-integral. Eq.
(2.1) is used to obtain K. The J-integrals for crack depth a/t = 0.6 are shown in Figure 7.6. It is shown that the
J-integrals are strongly path dependent about 0.05 mm away from crack tip. Within 0.05 mm from the crack tip,
J-integrals are almost path independent.
24
22
20
J (Meth 1 Load 1)
J (Meth 1 Load 2)
18
16
14
12
0
0,1
0,2
0,3
0,4
0,5
x (mm)
Figure 7.6a. J-integrals by Control Method for crack depth a/t = 0.6 (including remote paths).
24
22
20
J (Meth 1 Load 1)
J (Meth 1 Load 2)
18
16
14
12
0
0,01
0,02
0,03
0,04
0,05
x (mm)
Figure 7.6b. J-integrals by Control Method for crack depth a/t = 0.6 (21 paths near the crack tip).
For this case it was chosen to determine K based on J-integrals at 0.015mm from the crack tip (path 10 from the
crack tip).
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Stress intensity factors for the different evaluation methods, including ProSACC, are shown in Table 7.1. The
most relevant comparison for growth assessments is the difference between the control method, the elastic FE
method and the elastic ProSACC method. It is observed that the elastic results from ProSACC provide slight
overestimates of K except for the shallowest crack depth of a/t=0.2 where it is slightly underestimated. The
relative difference is 7% for a/t=0.6 at the lower load level L1.
Table 7.1 Case 2 - stress intensity factors estimated using the different evaluation methods.
Load
Level
Crack
Depth
(a/t)
Control
Method
(MPam)
(MPam)
0.2
27.0
0.4
EL_b
(MPam)
EP_b
(MPam)
26.3
25.0
25.2
39.1
42.4
38.1
38.6
0.6
57.4
61.6
52.7
53.0
0.8
72.4
64.2
67. 6
0.2
29.3
29.3
27.9
28.3
0.4
43.3
48.4
43.3
43.7
0.6
67.2
72.1
61.7
62.8
0.8
89.1
78.0
122.0
Pr oSACC
K Elasitc
Table 7.2 is included to show that the contribution from the residual stress to K is significant. The elasticplastic FE method was used for these calculations.
Table 7.2 Comparison showing the contribution of the residual stress to K, for the elastic-plastic FE method
EP_b.
Load
Level
Crack
Depth
(a/t)
Control
Method
(MPam)
Without
residual
stresses,
(MPam)
With
residual
stresses,
(MPam)
0.2
27.0
7.2
25.2
0.4
39.1
13.3
38.6
0.6
57.4
22.0
53.0
0.8
72.4
36.8
67.6
0.2
29.3
10.3
28.3
0.4
43.3
18.4
43.7
0.6
67.2
31.8
62.8
0.8
89.1
91.2
122.0
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Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
150
125
100
75
50
25
0
0
0,2
0,4
0,6
0,8
x/t
Figure 7.7 Case 2 - stress intensity factors for surface crack. In the figure, x = 0 at the pipe inner surface, L1
indicates the results at load level 1 and L2 at load level 2.
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7.4
The results for CTOD are listed in Tables 7.3-7.4 for load level 1 and load level 2, respectively. In comparison
with the results by the control method, it is clear that the results for both elastic and elastic-plastic methods are
underestimated.
Table 7.3 CTOD at load level 1.
Crack
depth
(a/t)
Control
Method
(mm)
Elastic
FEM
Elasticplastic
FEM
EL_b
(mm)
EP_b
(mm)
0.2
0.00360
0.00058
0.00080
0.4
0.00440
0.00088
0.00188
0.6
0.02000
0.00121
0.00411
0.8
0.03400
0.00151
0.00931

Crack
depth
(a/t)
Control
Method
(mm)
Elastic
FEM
Elasticplastic
FEM
EL_b
(mm)
EP_b
(mm)
0.2
0.00440
0.00070
0.00120
0.4
0.01000
0.00108
0.00288
0.6
0.03400
0.00143
0.00741
0.8
0.05400
0.00181
0.04171
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CASE THREE - MEDIUM PIPE WITH A THROUGH WALL CRACK
8.1
Finite element model
The finite element mesh for 2D axial symmetrical welding simulation is shown in Figure 8.1. In the figure the
weld pass order is also plotted. The 3D finite element mesh for the fracture mechanics model with a through
wall crack is shown in Figure 8.2. Only a quarter of the model is considered due to symmetry. A small notch
with a notch radius 0.001 mm is introduced at the crack tip this allows an accurate estimation of near-tip-J
and CTOD.
Figure 8.1 Finite element mesh for 2D welding simulation for case study 3.
Figure 8.2 Case 3 - finite element mesh with a through thickness crack. The coordinate l = 0 corresponds to the
center of the through wall crack, as previously defined in Figure 3.2.
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8.2
Results from 2D welding simulation
The welding residual stress distribution at the operating temperature (323 C) is shown in Figures 8.3 and 8.4.
The welding residual stress in the hoop and axial directions along the weld centre line are graphed in Figure 8.5.
Figure 8.3 Welding residual stress along axial direction at 323 C.
Figure 8.4 Welding residual stress along circumferential direction at 323 C.
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600
400
200
-200
Axial Stres s
Hoop Stress
-400
-5
10
15
20
25
30
35
X (mm)
Figure 8.5 Case 3 - welding residual stresses along the weld centre line. X is measured through the thickness
from the pipes inner surface.
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8.3
Crack profiles
Crack profiles determined by the different evaluation methods are shown in Figures 8.6-8.9 for the four
different through wall crack lengths. Results are shown for the two load levels L1 and L2. In the figures, open
symbols indicate results at load level 1 and the solid symbols for load level 2. The figures are split into two
parts: a) gives crack opening displacement COD results for the pipe inner surface and b) for the pipe outer
surface. Results due to residual stress only calculated by the elastic FEM method are also included. Note that
for this case the contribution to COD from residual stresses only is small.
When compared with the control method, it is observed that the COD calculated by the elastic FEM method
(EL_b) is underestimated. The COD calculated by the elastic-plastic FEM method (EP_b) is overestimated for
all cases except the 175 mm crack on the outer surface. There the elastic-plastic COD is slightly
underestimated.
If the results are used for an LBB-analysis or assessment of detection by visual inspection technique, the results
from the elastic FEM method are conservative in the sense that they under-predict the leak rate and the crack
opening. The results from the elastic-plastic FEM method are non-conservative in the same sense.
0,3
0,25
0,2
0,15
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
0,1
Control (L2)
EL_b (L2)
EP_b (L2)
0,05
0
0
20
40
60
80
100
l (mm)
Figure 8.6a COD along pipe inner surface for crack length 175 mm. l = 0 at the symmetry plane, L1 indicates
the results at load level 1 and L2 at load level 2. .
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Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
0,4
0,3
0,2
0,1
-0,1
0
20
40
60
80
100
l (mm)
Figure 8.6b COD along pipe outer surface for crack length 175 mm. l = 0 at the symmetry plane, L1 indicates
the results at load level 1 and L2 at load level 2.
0,5
0,4
0,3
0,2
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
0,1
Control (L2)
EL_b (L2)
EP_b (L2)
0
0
20
40
60
80
100
120
l (mm)
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0,6
0,5
0,4
0,3
0,2
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
0,1
Control (L2)
EL_b (L2)
EP_b (L2)
-0,1
0
20
40
60
80
100
120
l (mm)
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
0,6
0,5
0,4
0,3
0,2
0,1
0
0
20
40
60
80
100
120
140
l (mm)
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Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
0,8
0,6
0,4
0,2
-0,2
0
20
40
60
80
100
120
140
l (mm)
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
2,5
1,5
0,5
0
0
50
100
150
200
l (mm)
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Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
3,5
2,5
1,5
0,5
0
0
50
100
150
200
l (mm)
8.4
Maximum crack opening displacements
As discussed in section 2.2 the integrated crack opening area may be represented by an equivalent crack
opening parameter, Predictions from the FE analyses (from the previous section) are presented in Table 8.1
and in Figure 8.10.
When compared with the control method, it is observed that the calculated by the elastic-plastic FEM method
(EP_b) is overestimated. For moderate crack lengths up to 246 mm and the lower load level L1, the crack
opening area is overestimated by up to 12%.
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Table 8.1a Equivalent at pipe inner surface.

Crack
Length
(mm)
175
218
246
330
Load Level
Control
(mm)
EL_b
(mm)
EP_b
(mm)
0.202
0.194
0.207
0.267
0.231
0.276
0.286
0.262
0.304
0.394
0.315
0.446
0.361
0.318
0.403
0.519
0.384
0.625
0.8263
0.566
1.106
1.3933
0.686
2.103
Table 8.1b Equivalent at pipe outer surface.

Crack
Length
(mm)
175
218
246
330
Load Level
Control
(mm)
EL_b
(mm)
EP_b
(mm)
0.230
0.202
0.217
0.321
0.255
0.316
0.337
0.286
0.340
0.480
0.360
0.533
0.428
0.353
0.461
0.630
0.443
0.753
0.965
0.637
1.281
1.646
0.788
2.466
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Figure 8.10a Equivalent crack opening parameter at the inside surface of the pipe. In the figure L1 indicates
Figure 8.10b Equivalent crack opening parameter at the outside surface of the pipe. In the figure L1
indicates the results at load level 1 and L2 at load level 2.
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8.5
The stress intensity factor calculated in this section is based on the near-tip J-integral. Eq. (3.1) is used to obtain
K. The standard J-integral, when evaluated close enough to the crack-tip will provide an accurate estimate of
the J-integral calculated using the integral modified to account for residual stresses, as discussed in Section 3.1.
The J-integral along different paths is shown in Figure 8.11 for the case of a 160 mm long crack. The crack is
simulated using the control method. Figure 8.11a demonstrates that the standard integral is, in general, path
dependent. However, close enough to the crack tip, path independence is displayed as proposed in [4]. In this
case, within 0.5 mm from the crack tip the J-integrals are almost path independent. Figure 8.11b shows a
detailed view of the paths very close to the crack-tip. The J-integral is evaluated for three different positions
along the crack front as defined in Figure 8.11.
40
20
-20
Loc 1_Load 1
Loc 2_Load 1
Loc 3_Load 1
Loc 1_Load 2
Loc 2_Load 2
Loc 3_Load 2
-40
0
r (mm)
Figure 8.11a. J-integrals by control method for crack length 218 mm (including remote paths).
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40
20
0
Loc 1_Load 1
Loc 2_Load 1
Loc 3_Load 1
Loc 1_Load 2
Loc 2_Load 2
Loc 3_Load 2
-20
-40
0
0,125
0,25
0,375
0,5
r (mm)
Figure 8.11b. J-integrals by control method for crack length 218 mm (25 near crack tip paths).
For this case it was chosen to determine K based on J-integrals calculated 0.06 mm from the crack-tip (path 10
from the tip).
The distributions of stress intensity factors along the crack front are shown in Figures 8.12-8.15. Results for
three different methods are presented: control, elastic FEM and plastic FEM.
In this case the predicted stress intensity factors for the different methods typically have a sinusoidal shape
similar in form but different in magnitude to that predicted by the residual stress only case (EL (Resi. only)).
The elastic and elastic-plastic FE methods both appear to be more sensitive to the application of loads than the
control method since the peak to peak amplitude of these methods is always greater than the control method.
For x/t values between 0.25 and 1 the trend in the results is reasonably similar to those reported in case 1. The
elastic FE and elastic-plastic FE methods give reasonably good agreement, especially for the shorter crack
lengths. For x/t values between 0 and 0.25 and for lower K-values (load level L1, short cracks), the elastic FE
and elastic-plastic FE predictions diverge. This deviation at the pipe inside is similar to that for the thin pipe
from section 6.5.
It should also be pointed out that the maximum K-value does not always occur at the pipe inner or outer surface.
In [14], stress intensity factor solutions at five locations along the crack front were provided. Estimates at these
points could be added in ProSACC to provide a more accurate result.
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Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
100
75
50
25
-25
-50
0
0,2
0,4
0,6
0,8
x/t
Figure 8.12
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
125
100
75
50
25
-25
-50
0
0,2
0,4
0,6
0,8
x/t
Figure 8.13
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Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
125
100
75
50
25
-25
-50
0
0,2
0,4
0,6
0,8
x/t
Figure 8.14
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
EP_a (L2)
250
200
150
100
50
-50
0
0,2
0,4
0,6
0,8
x/t
Figure 8.15
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The results are also shown in Tables 8.2 and 8.3. Elastic K results from the ProSACC crack growth module are
included for comparison.
For crack growth calculations it may be concluded that the ProSACC predictions overestimate K in comparison
to the control results near the inner surface of the pipe.
Table 8.2 Stress intensity factors by different methods at load level 1.
Crack
Length
(mm)
175
218
246
330
Crack
Location
Control
(MPam)
Pr oSACC
K Elastic
(MPam)
EL_b
(MPam)
EP_b
(MPam)
46.17
91.0
76.18
63.39
24.56
14.92
14.52
28.62
19.5
29.59
25.01
49.42
106.2
87.67
70.82
40.28
27.39
28.49
35.25
29.5
38.45
34.38
51.76
119.3
96.63
77.77
52.87
36.72
39.67
39.94
36.0
44.34
42.59
61.98
171.6
132.35
112.96
105.04
72.12
100.63
64.12
59.8
63.96
78.54
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Table 8.3 Stress intensity factors by different methods at load level 2.

Crack
Length
(mm)
175
218
246
330
Crack
Location
Control
(MPam)
Pr oSACC
K Elastic
(MPam)
EL_b
(MPam)
EP_b
(MPam)
53.18
102.1
85.35
71.17
42.89
26.46
28.96
38.03
32.6
41.56
37.20
57.26
120.7
99.34
83.52
64.06
41.59
49.71
46.72
44.5
52.16
51.70
60.46
136.3
110.21
94.17
79.82
52.82
68.58
54.11
52.1
59.12
62.55
80.52
198.2
153.06
147.98
154.14
95.02
171.29
90.80
79.3
81.86
125.61
Tables 8.4 and 8.5 show the contribution from the residual stress to K for the load cases chosen for analysis in
this investigation. The elastic-plastic FE method was used for these calculations. The external loads applied are
lower for the medium thickness pipe in case 2 than for the lower thickness pipe geometry in case 1.
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Table 8.4 Comparison showing the contribution of the residual stress to K calculated using the elastic plastic FE
model (load level 1).
Crack
Length
(mm)
175
218
246
330
Crack
Location
Without
residual
stresses,
(MPam)
With
residual
stresses,
(MPam)
24.24
63.39
45.22
14.52
41.25
25.01
31.03
70.82
58.38
28.49
50.09
34.38
37.44
77.77
68.92
39.67
57.90
42.59
70.35
112.96
127.63
100.63
92.62
78.54
Table 8.5 Comparison showing the contribution of the residual stress to K calculated using the elastic plastic FE
model (load level 2).
Crack
Length
(mm)
175
218
246
330
Crack
Location
Without
residual
stresses,
(MPam)
With
residual
stresses,
(MPam)
32.02
71.17
59.66
28.96
53.44
37.20
43.73
83.52
79.60
49.71
67.41
51.70
53.84
94.17
97.83
68.58
77.86
62.55
105.37
147.98
198.30
171.29
139.69
125.61
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8.6
Crack tip opening displacements were calculated at the three locations indicated in Figure 8.11. The results are
listed in Tables 8.6-8.7.
In comparison with the results by the control method, results by the elastic method (EL_b) are underestimated.
Table 8.6 Crack tip opening displacements estimated using different methods (load level 1).
Crack
Length
(mm)
Crack
Location
Control
(mm)
Elastic
FEM
EL_b
(mm)
175
218
246
330
Elasticplastic
FEM
EP_b
(mm)
0.0094
0.0016
0.0033
0.0033
0.0004
0.0093
0.0041
0.0007
0.0063
0.0104
0.0018
0.0050
0.0090
0.0007
0.0166
0.0060
0.0009
0.0096
0.0112
0.0020
0.0070
0.0154
0.0009
0.0247
0.0076
0.0010
0.0129
0.0164
0.0028
0.0328
0.0618
0.0017
0.0882
0.0183
0.0015
0.0394
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Table 8.7 Crack tip opening displacements estimated using different methods (load level 2).
Crack
Length
(mm)
Crack
Location
Control
(mm)
Elastic
FEM
EL_b
(mm)
175
218
246
330
Elasticplastic
FEM
EP_b
(mm)
0.0137
0.0018
0.0054
0.0100
0.0007
0.0172
0.0067
0.0010
0.0110
0.0148
0.0021
0.0101
0.0221
0.0010
0.0335
0.0098
0.0012
0.0183
0.0164
0.0023
0.0162
0.0350
0.0013
0.0518
0.0129
0.0014
0.0260
0.0400
0.0032
0.3837
0.1314
0.0022
0.1902
0.0373
0.0019
0.1060
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9
9.1
CASE FOUR - MEDIUM PIPE WITH A FULL CIRCUMFERENTIAL

INTERNAL SURFACE CRACK
Finite element model
The pipe geometry considered in case four is the same as for case three with the difference that a full
circumferential, internal surface crack is modeled, as opposed to a through-wall crack. The finite element mesh
used for welding simulation is also the same, as shown in Figure 7.1. The finite element mesh for the fracture
mechanics model is, however, different. A 2D, axisymmetric model was used and is illustrated in Figure 9.1.
The notch radius is 0.001mm, allowing accurate prediction of near-tip-J and CTOD.
Figure 9.1 Finite element mesh with an internal surface crack. x = 0 at the inner surface of the pipe.
9.2
Crack profiles
Crack profiles through the thickness are shown in Figures 9.2 to 9.5 for the different evaluation methods. Each
figure represents a different crack depth, ranging from a/t=0.2 to 0.8. Results are shown for the two load levels
L1 and L2. In the figures, open symbols indicate results at load level 1 and the solid symbols at load level 2.
Results due to residual stresses only, calculated by the elastic FE method, are also included.
Note that the crack opening displacement due to welding residual stress is most pronounced at small crack
depths where it contributes more than 50% of the total COD for both the elastic and elastic-plastic FE methods.
For large crack depths the effect is less pronounced. An important observation is that crack closure is predicted
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at a/t=0.4. Inclusion of weld residual stresses is therefore very important in computation of COD for these
cases.
It is observed that the crack opening displacement predicted by the elastic FE method and the elastic-plastic FE
method are nearly identical when a/t < 0.6. This implies that the effect of plastic deformation in these cases is
very limited.
In comparison to the control method, for the higher load level, both the elastic and elastic-plastic FE methods
underestimate the crack opening displacement for a/t up to 0.6. For the lower load level there is no large
difference between the predictions of the control method and the elastic-plastic FE methods.
For a/t equal to 0.8, at the higher load level L2, there is a significant contribution to the COD from plastic
deformation, as predicted by both the control method and the elastic-plastic method for a/t up to 0.6.
Note that the surface COD due to residual stresses only decreases already after a/t=0.2, leading to the situation
that it is more difficult to detect a large crack than a small one, especially for VT inspection methods. The
scope of this study was not broad enough to obtain a full set of data necessary for conclusions regarding the VT
method. Results for the control method for residual stresses only, and at room temperature, not at operating
temperature, would be needed for more complete conclusions.
40
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
30
Control (L2)
EL_b (L2)
EP_b (L2)
COD (m)
20
10
0
0
x (mm)
Figure 9.2 COD for a/t = 0.2. (x = 0 at the pipe inner surface).
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50
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
40
Control (L2)
EL_b (L2)
EP_b (L2)
30
20
COD (m)
10
-10
0
12
15
x (mm)
Figure 9.3 COD for a/t = 0.4.
60
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
45
Control (L2)
EL_b (L2)
EP_b (L2)
30
COD (m)
15
-15
-30
0
12
x (mm)
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16
20
200
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
160
Control (L2)
EL_b (L2)
EP_b (L2)
120
COD (m)
80
40
-40
0
12
18
24
30
x (mm)
9.3
As before, the stress intensity factor calculated in this section is based on the near-tip J-integral. Eq. (2.1) is
used to obtain K. The J-integrals for crack depth a/t = 0.6 are shown in Figure 9.6. The crack is simulated using
the control method. It is shown that the J-integrals are path dependent about 0.1 mm away from crack tip.
Within 0.1 mm from the crack tip, J-integrals are almost path independent. For this case it was chosen to
determine K based on J-integrals at 0.1mm from the crack tip.
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J _Load 1
J_Load 2
0
0
0,2
0,4
0,6
0,8
r (mm)
x (mm)
Figure 9.6a. J-integrals as function of distance r from the crack-tip for a crack simulated by the control method
(crack depth a/t = 0.6).
8
using these J
to calculate K
1
J_Load 1
J_Load 2
0
0
0,1
0,2
0,3
x (mm)
0,4
0,5
r (mm)
Figure 9.6b. J-integrals as function of distance r from the crack-tip for a crack simulated by the control method
(crack depth a/t = 0.6). The 29 near-tip paths are shown.
Stress intensity factors for the different evaluation methods, including ProSACC, are shown in Table 9.1 and
Figure 9.7. The results predicted by the elastic method agree well with the control method, except for the most
shallow cracks and the high load level (L2) where EL_b overestimates K.
An important observation is that for a proposed mechanism of SCC, the crack would arrest at a/t=0.4. This is
predicted by all methods. For the crack depths a/t=0.2 it is observed that the elastic K-values calculated by
ProSACC are always greater than the control method.
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Table 9.1 Case 4 - stress intensity factors estimated using the different evaluation methods.
Load
Level
Crack
Depth
(a/t)
L1
L2
Control
Method
(MPam)
Pr oSACC
K Elastic
(MPam)
19.76
25.0
28.88
29.17
0.4
0.00
-12.6
-8.55
0.00
0.6
18.06
13.3
17.14
17.27
0.8
63.15
62.12
68.06
0.2
23.49
28.2
31.99
32.41
0.4
0.00
-7.2
-3.16
0.00
0.6
28.97
21.3
25.16
25.93
0.8
87.00
74.32
114.28
Control (L1)
EL_b (L1)
EP_b (L1)
EL (Resi. only)
Control (L2)
EL_b (L2)
EP_b (L2)
75
50
25
-25
-50
0
EP_b
(MPam)
0.2
125
100
EL_b
(MPam)
0,2
0,4
0,6
0,8
a/t
Figure 9.7 Stress intensity factors for internal surface crack.
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9.4
The results for CTOD are listed in Tables 9.2-9.3 for load level 1 and load level 2, respectively. In comparison
with the results by the control method, results from the elastic FE method under and over estimate CTOD. Note
that crack arrest would not be predicted at a/t=0.4 if using CTOD and EP_b, though it is predicted for the
control method. Some variation in the results could be attributed to sensitivity to the near-tip mesh design.

Crack
depth
(a/t)
Control
(mm)
Elastic
FEM
EL_b
(mm)
Elasticplastic
FEM
EP_b
(mm)
0.2
0.0040
0.0008
0.0018
0.4
-0.0003
-0.0001
0.0024
0.6
0.0027
0.0004
0.0057
0.8
0.0280
0.0016
0.0172

Crack
depth
(a/t)
Control
(mm)
Elastic
FEM
EL_b
(mm)
Elasticplastic
FEM
EP_b
(mm)
0.2
0.0060
0.0009
0.0025
0.4
-0.0001
0.0000
0.0038
0.6
0.0070
0.0006
0.0091
0.8
0.0670
0.0019
0.0768
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10
ALTERNATING FEM METHOD
In this section an alternating FEM method is evaluated for calculation of K and COD. One of the tasks in the
project is to evaluate stress intensity factors and crack opening displacements for cracks subjected to a given
residual stress field by using the alternating method [11-12]. As discussed in the background, one key aspect in
treating residual stresses in fracture mechanics models is the consideration of displacement-controlled
conditions. This is because a residual stress field must satisfy equilibrium conditions within a component of
concern. For a growing crack, the residual stresses nearby re-distribute as stress-free surfaces are generated,
while in load-controlled conditions, the far-field stresses can be assumed to be stationary.
One convenient method for computing stress intensity factors in a 3D geometry is the finite element alternating
method. As illustrated in Figure 10.1, if stress intensity factors need to be calculated for a cracked body such as
the T-fillet weld shown, a conventional finite element model is all that is needed without the need to model an
actual crack. Therefore, the modeling efforts are significantly simplified.
The stress intensity factor calculation process for the alternating method using FEM and an infinite body
solution can be described as follows:
1. For a given 3D cracked finite body (Figure 10.1b), the associated boundary traction conditions and
crack face pressure are first obtained using a finite element model at the crack location, but without a
crack being modeled.
2. The crack face pressures are then fitted into a polynomial function over the entire crack face so that the
boundary tractions in the infinite body (Figure 10.1c) - but at the actual finite cracked body boundaries can then be calculated using an analytical solution for an embedded elliptical crack in a 3D infinite
body under given crack face pressure.
3. The analytically calculated traction conditions from Figure 10.1c at this stage are different from those
given in Figure 10.1b. By defining the difference in tractions from the FE model and the analytical
model and applying the residual tractions to the FE model, a new estimate of the corresponding crack
face pressure can be obtained for feeding into the 3D analytical solution for calculating a new estimate
of boundary tractions for the finite 3D body.
4. By alternating between the FE based traction solution and that from the analytical solution until the
residual tractions between the two solutions become negligible, a converged solution is obtained.
5. The stress intensity factor can then be calculated by adding up the stress intensity factors calculated
from all iterations.
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FE Model
Without crack
Weld
Crack
,K ,
1
(b)
(a)Problem
Crack
2a
Analytical solution for infinite body

with an elliptical crack
(c)
Figure 10.1
solid model.
Illustration of finite element alternating method for stress intensity factor calculation using 3D
To deal with weld residual stress related stress intensity factor calculations, a mapping procedure is needed to
map a given residual stress field to a 3D welded component so that a full-field residual stress field (e.g. any
potential interaction effects between axial and hoop residual stresses in pipe girth welds) can be properly
considered in stress intensity factor calculations.
10.1
Residual Stress Mapping, 3D Model and Crack Definition
Figures 6.3 and 6.4 summarize the residual stress distribution for a thin-wall pipe girth weld (t = 8.3 mm). The
axial residual stress distribution in Figure 6.3 shows a strong through-wall bending action with tension at the
inner and compression at the outer surfaces.
A representative 3D FE model for the same girth used for K calculations is illustrated in Figure 10.2. A
circumferential surface crack to be considered is also given in the figure. The residual stress distributions in
Figure 6.3 and 6.4 predicted using an axisymmetric model were mapped onto the 3D solid model in Figure
10.2. The resulting distributions in Figure 10.3 are consistent with those in Figure 6.3 and 5.4. Note that the
mesh design in the 3D solid model in Figure 10.2 is intended to provide an adequate resolution regarding
overall stress gradients for calculation of K solutions.
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Weld
Center
Line
Elliptical Surface Crack

Location
Figure 10.2. 3D finite element model for computing stress intensity factors using a finite element alternating
method under displacement-controlled conditions.
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Figure 10.3 Residual stress distribution (axial and hoop direction) after mapping from the axisymmetric model
to the 3D model.
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10.2
Results for Surface Cracks
For the evaluation of K an elliptical crack was defined first for a crack length of 2c = 16 mm and a/t values
between 0.2 and 0.8. The same procedure was then repeated for crack lengths of 24 and 32 mm, and the
results are shown in Figures 10.4 to 10.6. Residual stresses only are considered.
When the surface crack is shallow, K reaches its largest value at the deepest position (or at an elliptical angle
of 90). However, as the crack depth increases, the maximum K values shift to the inner surfaces, as clearly
indicated in Figures 10.4 to 10.6 for a/t = 0.6 and 0.8. The trend becomes less pronounced with the longer
crack sizes. This behavior is directly related to the through wall axial residual stress distribution. The
compressive axial residual stresses become increasingly dominant as the crack depth reaches beyond half
thickness as shown in Figures 10.7a and b, where the results are replotted for fixed a/t ratios and varying crack
lengths.
The above observations suggest that for stress corrosion cracking controlled by K, the crack growth would be
inhibited in the depth direction and promoted in the circumferential direction at the inner surface.
St r ess I nt ensi t y Fact or , MPa- M1/ 2
2. 50E+01
a/ t =0. 2
a/ t =0. 4
a/ t =0. 6
a/ t =0. 8
2. 00E+01
1. 50E+01
1. 00E+01
5. 00E+00
0. 00E+00
- 5. 00E+00
0
45
Inner Surface at Elliptical Angle = 0 ,180

o
Deepest Crack Point at Elliptical Angle = 90
90
135
El l i pt i cal Angl e ( Degr ees)
Elliptical
Surface
Crack
180
Figure 10.4 Stress intensity factor distribution as a function of relative crack depth a/t with 2c = 16 mm
(t = 8.3 mm) . The elliptical angle and crack form parameters are defined in the sub figure.
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2. 50E+01
a/ t =0. 2
a/ t =0. 4
a/ t =0. 6
a/ t =0. 8
2. 00E+01
1. 50E+01
1. 00E+01
5. 00E+00
0. 00E+00
0
45
90
135
180
Figure 10.5 Stress intensity factor distribution as a function of relative crack depth (a/t) with 2c = 24 mm
(t = 8.3 mm).
2. 50E+01
2. 00E+01
1. 50E+01
1. 00E+01
a/ t =0. 2
a/ t =0. 4
a/ t =0. 6
a/ t =0. 8
5. 00E+00
0. 00E+00
0
45
90
135
180
Figure 10.6 Stress intensity factor distribution as a function of relative crack depth (a/t) with 2c = 32 mm
(t = 8.3 mm).
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1. 80E+01
1. 60E+01
1. 40E+01
1. 20E+01
1. 00E+01
8. 00E+00
2c=16mm
2c=24mm
2c=32mm
6. 00E+00
a/ t =0. 2
4. 00E+00
2. 00E+00
0. 00E+00
0
45
90
135
180
2. 50E+01
2. 00E+01
1. 50E+01
2c=16mm
2c=24mm
2c=32mm
1. 00E+01
a/ t =0. 4
5. 00E+00
0. 00E+00
0
45
90
135
180
Figure 10.7a: Stress intensity factor distribution as a function of crack length (2c) with a constant depth (a/t) .
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2. 50E+01
2c=16mm
2c=24mm
2c=32mm
2. 00E+01
1. 50E+01
1. 00E+01
a/ t =0. 6
5. 00E+00
0. 00E+00
0
45
90
135
180
2. 50E+01
2. 00E+01
2c=16mm
2c=24mm
2c=32mm
1. 50E+01
1. 00E+01
a/ t =0. 8
5. 00E+00
0. 00E+00
- 5. 00E+00
0
45
90
135
180
Figure 10.7b: Stress intensity factor distribution as a function of crack length (2c) with a constant depth (a/t) .
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10.3
Results for Through-Wall Cracks
The through wall crack was defined as a full ellipse intersecting the pipe wall and was oriented in the
circumferential direction (illustrated in Figure 10.8). Only residual stresses are considered. The actual crack
front used in simulations is shown in Figure 10.9, and the points at which the stress intensity factor was
estimated are shown in the figure subset. Note that the crack front becomes curved. Figure 10.9 also shows the
variation of stress intensity factor along the crack front for crack sizes of 24 and 36 mm. The stress intensity
factors are plotted from the outer to the inner surface. The two cases (2c = 24 mm and 36 mm) are not
significantly different from each other, except for the case for 2c = 24 mm a slightly higher value of K is
approached at the inner surface.
The corresponding crack opening displacements along both inner and outer surfaces for both cases are
summarized in Figure 10.10. For 2c = 24 mm, the crack opening profiles approximately resemble an elliptical
type of opening, as suggested by classical estimation scheme under remote bending. As the crack length
increases, such as at 2c = 36, the outer crack surface starts to exhibit additional curvature, as shown in Figure
10.10.
To gain insight on the deformation behavior on the crack surface, two through-wall lines were chosen to
examine crack opening displacement distributions. As shown in Figure 10.11, the crack opening displacements
as a function of the through-thickness position is essentially linear, i.e., close to a pure rotation.
x1 Major Axis
x2 Minor Axis
Elliptical Angle
K Calculation Points
X,Z Global Coordinates
x2
o
c
X
x2
x1
a
Outer Surface
Pipe
Inner Surface
Figure 10.8: Through-wall elliptical crack definition in 3D FE alternating model.
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x1
2. 50E+01
2. 00E+01
2c=24mm
2c=36mm
1. 50E+01
1. 00E+01
5. 00E+00
Inner Surface
0. 00E+00
- 5. 00E+00
Outer Surface
- 1. 00E+01
- 1. 50E+01
- 2. 00E+01
- 2. 50E+01
145
155
2c=24 mm & 36
mm
a=8 mm
149o211o
165
175
185
195
205
215
X
x2
Outer
Surface
x1
Inner
Surface
Figure 10.9 Stress intensity factor results for two through-wall elliptical cracks of 2c = 24 mm and 36 mm.
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1. 20E- 02
Inner
Surface
Cr ack Openi ng Di spl acement ( MM)
1. 00E- 02
8. 00E- 03
6. 00E- 03
2c=24mm I nner
2c=24mm Out er
2c=36mm I nner
2c=36mm Out er
4. 00E- 03
2. 00E- 03
0. 00E+00
- 20
- 2. 00E- 03
- 15
- 10
-5
Sur f ace
Sur f ace
Sur f ace
Sur f ace
10
- 4. 00E- 03
15
20
Outer
Surface
- 6. 00E- 03
- 8. 00E- 03
Z Coor di nat e Al ong Pi pe Sur f ace ( MM)
X
x2
Outer
Surface
Displacement Output Points
x1
Inner
Surface
Figure 10.10 Crack opening displacement results for through-wall elliptical cracks of 2c = 24 mm and 36 mm.
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1. 20E- 02
2c=24mm, Cr oss Sect i on
at Z=0 ( Cent er )
at Z=4mm
Cr ack Openi ng Di spl ancement , MM
1. 00E- 02
8. 00E- 03
6. 00E- 03
4. 00E- 03
2. 00E- 03
0. 00E+00
- 2. 00E- 03
- 4. 00E- 03
- 6. 00E- 03
- 8. 00E- 03
0
1
2
3
4
5
6
7
8
Di st ance Al ong Cr oss Sect i on ( Thr ough Thi ckness, MM)
1. 20E- 02
at Z=0 ( Cent er )
at Z=10mm
Cr ack Openi ng Di spl ancement , MM
1. 00E- 02
8. 00E- 03
6. 00E- 03
4. 00E- 03
2. 00E- 03
0. 00E+00
- 2. 00E- 03
- 4. 00E- 03
- 6. 00E- 03
- 8. 00E- 03
0
Di st ance Al ong Cr oss Sect i on ( Thr ough Thi ckness, MM)
X
x2
Outer Surface
Displacement Output Points
x1
Inner Surface
Figure 10.11 Crack opening displacement distributions across thickness through-wall elliptical cracks.
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10.4
Comparison to the crack face pressure method
In this section the results from the analyses by the present alternating method are compared with results from
the crack face pressure method calculated by ProSACC.
Comparison to the results in Section 6.8 from the control method was not possible. With the current alternating
method, the largest through-wall crack length with convergence is the 36 mm crack. It is possible to improve the
convergence in the future by changing the iteration procedure. This is considered as a limitation of the
alternating method.
In Table 10.1 stress intensity factors calculated for through wall cracks by the alternating method are compared
to results from ProSACC (crack face pressure method). The values predicted by ProSACC are up to 95%
higher. In ProSACC straight crack fronts are considered, while by the alternating method the crack front
becomes approximated by a piece of an elliptical curve. This also contributes to the difference.
This deviation could possibly indicate that the CFP-method for through wall cracks is rough and it is possible
that the inclusion of the out-of-plane stress might improve the agreement between the methods.
Table 10.1 Stress intensity factors K for through wall cracks estimated by alternating method and ProSACC.
Crack Length
2c
(mm)
Position
Alternating method
(MPam)
ProSACC, elastic
(MPam)
Inner surface
20.7
30.7
Outer surface
-21.6
-36.4
Inner surface
18.4
34.3
Outer surface
-22.0
-41.8
24
36
Figure 10.12 shows the results for the surface cracks calculated by the alternating method (ALT) compared to
results from the crack face pressure method calculated by ProSACC (CFP). Position A is the result at the
deepest point of the crack front (= 90), and position B is the result at the intersection to the inside surface
(= 0).
The same trends as a function of crack depth are predicted by both methods. This supports the use of the crack
face pressure method for evaluation of K for surface cracks. Slightly higher K-values are predicted by the
alternating method for the deepest point (position A), the values differ up to 15% for the longest crack. For the
surface position B slightly higher values are predicted by ProSACC.
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2c =16mm
25
20
K (MPa m)
CFP Pos A
ALT Pos A
15
CFP Pos B
10
ALT Pos B
5
0
0
0,2
0,4
0,6
0,8
-5
a/t
2c =24mm
30
K (MPa m)
25
20
CFP Pos A
ALT Pos A
15
CFP Pos B
ALT Pos B
10
5
0
0
0,2
0,4
0,6
0,8
a/t
2c =32mm
30
K (MPa m)
25
CFP Pos A
ALT Pos A
20
CFP Pos B
15
ALT Pos B
10
5
0
0
0,2
0,4
0,6
0,8
a/t
Figure 10.12 Comparison of K results for surface cracks. Results calculated by the alternating
method (ALT) and by the crack face pressure method by elastic ProSACC (CFP). Pos
A is the deepest point of the surface crack (= 90), and Pos B is at surface (= 0).
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11
DISCUSSION
11.1
Crack opening displacements for leak before break assessments
The discussion in this section regards the predicted crack opening displacements for through wall cracks. The
crack opening displacement COD is used to predict the crack opening area COA and leakage rates. An
overestimate with respect to COD results in a larger predicted leak rate, which would be non-conservative with
respect to the assessed probability of early detection of leakage.
For a thin walled pipe (t=8.3 mm) containing a through wall crack the COD at the outer-surface is limiting in
terms of the leakage rate. The COD calculated by NURBIT is similar to the results by EP_b. For the crack
lengths up to 185 mm, the crack opening area is overestimated by 7-26% for the lower load case L1 compared
to the control method. Comparison to CODs calculated by the elastic FEM method indicates that plastic
effects are substantial even for the moderate crack lengths and low load level.
For a medium thickness pipe (t=31.8 mm) containing a through wall crack the COD at the inner-surface is
limiting in terms of the leakage rate. This is opposite to the thin-walled case, and can be attributed to the weld
residual stress field. The elastic-plastic method consistently overestimates the COD by up to 15% compared to
the control method. For a low load level and shorter cracks which is more relevant to lead before break
situations, the COD is overestimated to a lesser degree between 2 and 12% for crack lengths between 175 and
246 mm. This implies a small non conservatism with respect to LBB assessment with EP_B.
The elastic FE method underestimates the crack opening area, since plastic deformation is not accounted for.
11.2
Crack-tip parameters for crack growth assessments
The discussion in this section regards the predicted crack-tip parameters for through wall and surface cracks.
The stress intensity factor K is used for prediction of crack growth rates. An underestimation of K results in a
lower growth rate, which is non-conservative with respect to the predicted safe time period, for growth to the
maximum acceptable crack size.
As discussed in section 3.1, the modified J-integral is expected to provide a valid measure of the mechanical
state at the crack tip taking into account for weld residual stresses. Crack growth relations are usually based
upon K and an equivalent stress intensity factor K may be calculated from the J-integral. An implementation of
the modified J-integral accounting for residual stresses in 3D cases was not available, and instead near-tip
estimates of the standard J-integral were used. It was judged that the standard J-integral evaluated close enough
to the crack-tip was sufficient to demonstrate the key trends. The standard J-integral was evaluated between
0.01 and 0.1 mm from the crack tip. This may, however, introduce mesh sensitivity in the results, and use of
the modified J-integral is recommended.
The results for surface cracks are most relevant for drawing conclusions about crack growth.
For a thin walled pipe containing an internal surface crack, K estimates from the elastic ProSACC method, the
elastic FE method and the elastic-plastic FE method are very similar to the control method for crack sizes up to
a/t=0.4. For deeper cracks, the crack face pressure method tends to underestimate K, the exception being the
case EP_b and a/t=0.8.
For a medium thickness pipe containing an internal surface crack, crack arrest was predicted for a/t
approximately equal to 0.4. This was predicted by all methods. For cracks up to a/t=0.2 (shallow cracks), K
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was overestimated by all crack face pressure methods where there was a tensile residual stress near the inner
wall. For the lower load level the elastic (and plastic) FE methods overestimated K by 46% for a/t=0.2.
When calculating inspection intervals, through wall cracks are not allowed for piping, but only for internal
parts. It should be highlighted that the crack growth calculations made for stress corrosion cracking and fatigue
cracking are almost always based upon a purely elastic estimation of K. This includes the growth analysis
modules in ProSACC.
For the thin walled pipe, the shortest through wall crack and the lower load level (L1), the elastic ProSACC
prediction overestimates K by 56% on the pipe inner surface. For the medium thickness pipe, the shortest
through wall crack and the lower load level (L1), the elastic ProSACC prediction overestimates K by 65% on
the pipe inner surface and underestimates it by 32% on the pipe outer surface. It is observed that where K is low
the elastic ProSACC results can underestimate K at some positions along the crack front. However, the
maximum value predicted by ProSACC is always overestimated. Note that the maximum value of K is not
always obtained at the inner or outer surfaces for the control method and EP_b. This could have implications
for predictions using ProSACC since only values at the surfaces are provided from ProSACC.
An investigation was also carried out concerning the alternating method. Only short cracks could be analysed.
Estimates of K were compared with ProSACC results. For the through wall cracks, the ProSACC estimates
were of a higher magnitude they were more positive on the inner pipe surface and more negative on the outer
pipe surface. For surface cracks the same trends are predicted by both methods. Slightly higher K-values are
predicted by the alternating method for the deepest point and the values differ up to 15%. For the surface
position slightly higher values are predicted by ProSACC.
11.3
COD for surface cracks and planning of visual inspection
The results for COD for surface cracks were calculated for normal operating conditions, however, some
conclusions can be made with respect to VT inspection which is conducted at room temperature after reactor
shutdown. For the thin walled pipe, and for crack depths (a/t) up to 0.6, both the elastic and the elastic-plastic
methods underestimate the COD. The results for the elastic and the elastic-plastic FE methods are essentially
the same. It is concluded that the underestimation may be due to either the load controlled nature of these
methods and/or the presence of high out-of-plane stresses that are not captured by these methods.
For the medium thickness pipe there is no large difference between the control method and the elastic/elasticplastic FE methods. This may be because the out-of-plane stress is compressive for this pipe geometry when
the crack is small.
Note also for the medium thickness pipe, where the axial weld residual stress profile is sinusoidal, the surface
COD due to residual stresses only, decreases already after a/t=0.2, leading to the situation that it is more
difficult to detect a deep crack than a shallow one. Further investigation is required to quantify this effect for
more complicated residual stress fields. Results for the control method, for residual stresses only, would be
needed for more complete conclusions.
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12
CONCLUSION
This report investigates the treatment of weld residual stresses when performing fracture mechanical analysis of
welded components. Weld residual stresses have a large influence on the behavior of cracks growing under
normal operation loads and on the leakage-flow from a through-wall crack. Accurate estimates of crack
characterizing parameters such as stress intensity factor K and the crack opening displacement COD are
important for prediction of these events.
Engineering assessment methods that are commonly used account neither for the displacement controlled nature
of weld residual stresses, nor for multi-axial residual stresses. Engineering methods commonly utilize a
calculation procedure called the crack-face pressure method. The stress in an uncracked model is extracted
along the plane were the crack is postulated. The stress normal to the crack plane is applied to the crack surface
in a model where the crack exists, for calculation of K or COD. The crack-face pressure method involves an
assumption of load-controlled conditions but is also used in situations where remote displacements are applied.
These methods generally account neither for the displacement controlled nature of weld residual stresses nor for
multi-axial residual stresses.
The objective of this work is to investigate if the approximation of the crack face pressure method provides
reasonable and yet conservative results for weld residual stresses and normal operating conditions. Detailed
simulations are made where a crack is created in full 3D weld residual stress fields by use of the noderelaxation technique. Crack-tip characterizing parameters and crack opening displacement determined from
this control method model are compared with results obtained from different implementations of the crack face
pressure method. Two pipe geometries with surface and through wall circumferential cracks are studied for
normal operation cases where the weld residual stresses constitute a significant part of the loading.
For through wall cracks the crack opening area results are of interest for performing leak before break
assessments. The main conclusion is that good agreement is obtained between the elastic-plastic FE method
(crack face pressure method), NURBIT (crack face pressure method) and the control method for typical
minimum detectable crack sizes and load levels commonly seen in Swedish nuclear reactors (load level 1). The
maximum observed variation between the methods was 26%, where COD was overestimated.
For full circumferential internal surface cracks K estimates are of interest for crack growth predictions. The
main conclusions are:
ProSACC (crack face pressure) method and the control method for crack depths up to a/t=0.6.
ProSACC method and the control method. The largest variation of 15% was for the higher load level
and deepest crack (a/t=0.8).
For the 32 mm thick pipe, where the axial weld residual stress profile is sinusoidal, the surface COD
due to residual stresses only decreases rapidly, leading to the situation that it is more difficult to detect
a deep crack than a shallow one. This may have implications for VT testing.
For the 32 mm thick pipe, all methods predict crack arrest. For shallow cracks, K was overestimated by
all crack face pressure methods where there was a tensile residual stress near the inner wall.
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13 FUTURE STUDIES
In the present study, a near-tip estimate of the J-integral was used. This was judged to provide an acceptable
indication of the trends but for some cases it was necessary to take results very close to the crack tip and this
may have induced a mesh dependency. Future investigations should use a modified J-integral because it is path
independent in the presence of residual stress fields, prior plastic deformation and material unloading. Further
development work is required for its implementation into 3D finite element models.
The present study has indicated key trends regarding crack face pressure methods for various pipe geometries.
Nevertheless, it is not fully resolved in what situations displacement controlled modeling and when high out-ofplane stress contribute most to the deviation from the crack face pressure methods. A systematic study to
clarify the separate contribution of each of these parameters is required. For example the reason for the large
decrease in K in Figure 6.13 for the control method towards the pipe inner surface is not well understood.
Some observations were made regarding VT inspection. For medium thickness pipes, where the axial weld
residual stress profile is sinusoidal, the surface COD due to residual stresses only, decreases rapidly, leading to
the situation that it is more difficult to detect a large crack than a small one. Further investigations are required
to quantify how this effect applies to more complicated residual stress fields. Results for the control method,
and for residual stresses only at room temperature, would be needed for more complete conclusions.
Note that for through wall cracks in sinusoidal weld residual stress fields the maximum value of K is not always
obtained at the inner or outer surfaces of the pipe when estimates are made using the control method, the elastic
FE method and the elastic-plastic FE method. This could have implications for predictions using ProSACC
since only values at the surfaces are provided from ProSACC. A future version of ProSACC could be updated
to provide estimates of K at several positions through the thickness.
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14
REFERENCES
[1] Zang, W., Gunnars, J., Dong, P. and Hong, J.K., Improvement and validation of weld residual stress
modelling procedure, Inspecta Research Report No. 50002550-1, Rev 0, 2008.
[2] Dillstrm P. et al.,ProSACC Handbook A combined deterministic and probabilistic procedure for
safety assessment of components with cracks, DNV RSE Report No 2004/01 Rev 4-1, Det Norske Veritas AB,
DNV Technology Sweden, 2004.
[3] Moran, B and Shih, C.F., Crack tip and associated domain integrals from momentum and energy
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Effect of Welding Residual Stresses

Report number: 2009:17 ISSN: 2000-0456

Title: Effect of Welding Residual Stresses on Crack Opening Displacement and

Date: June 2009

Basis for crack-tip characterizing parameters .................................................6

Definition of leak before break parameters .....................................................7

DEFINITION OF EVALUATED CALCULATION METHODS...............................9

Node relaxation in full 3D residual stress field (Control Method)...................10

Elastic FEM method (EL_b) ..........................................................................10

COD using weight functions with plastic correction (NURBIT) ......................10

Elastic-plastic FEM method (EP_b) ..............................................................11

FEM alternating method using infinite body solution (Alternating) ................11

DEFINITION OF CASE STUDIES.....................................................................12

Pipe geometry, loading and material properties............................................12

Weld geometry assumptions.........................................................................13

CASE ONE - THIN PIPE WITH A THROUGH WALL CRACK ..........................14

Finite element models...................................................................................14

2D welding simulation ...................................................................................15

Maximum crack opening displacements .......................................................21

Stress intensity factor....................................................................................24

Crack tip opening displacements ..................................................................32

CASE TWO - THIN PIPE WITH A FULL CIRCUMFERENTIAL INTERNAL

Finite element models...................................................................................35

Stress intensity factor....................................................................................38

Crack tip opening displacements ..................................................................41

CASE THREE - MEDIUM PIPE WITH A THROUGH WALL CRACK................42

Results from 2D welding simulation ..............................................................43

Maximum crack opening displacements .......................................................49

Stress intensity factor....................................................................................52

Crack tip opening displacements ..................................................................59

CASE FOUR - MEDIUM PIPE WITH A FULL CIRCUMFERENTIAL

Finite element model.....................................................................................61

Stress intensity factor....................................................................................64

Crack tip opening displacements ..................................................................67

ALTERNATING FEM METHOD ........................................................................68

Residual Stress Mapping, 3D Model and Crack Definition............................69

Results for Surface Cracks ...........................................................................72

Results for Through-Wall Cracks ..................................................................76

Comparison to the crack face pressure method............................................80

Crack opening displacements for leak before break assessments ...............82

Crack-tip parameters for crack growth assessments ....................................82

COD for surface cracks and planning of visual inspection ............................83

FUTURE STUDIES ...........................................................................................85

Basis for crack-tip characterizing parameters

where E is Youngs modulus and is Poissons ratio.

Figure 3.1 Definition of CTOD.

Definition of leak before break parameters

DEFINITION OF EVALUATED CALCULATION METHODS

Short description of calculation method

FEM, near tip J-integral calculation

Weight functions, solutions determined for load controlled

FEM, near tip J-integral calculation

Weight functions, solutions determined for load controlled

FEM, near tip J-integral calculation

Alternating FEM alternating method (coupled, iterative FE/analytical

Node relaxation in full 3D residual stress field (Control Method)

Elastic FEM method (EL_b)

COD using weight functions with plastic correction (NURBIT)