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PVP2015-45575

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Proceedings of the ASME 2015 Pressure Vessels and Piping Conference
PVP2015
July 19-23, 2015, Boston, Massachusetts, USA
PVP2015-45575
VALIDATION OF STRESS INTENSITY FACTOR SOLUTIONS
IN XLPR VERSION 2.0 CODE
Do-Jun Shim†
Engineering Mechanics
Corporation of Columbus
3518 Riverside Dr., Suite 202
Columbus, OH 43221, USA
Jay Wallace‡
US Nuclear Regulatory Commission
Office of Nuclear Regulatory Research
Mail Stop: CSB-5CA24
Washington, DC 20555-0001, USA
Sureshkumar Kalyanam
Engineering Mechanics
Corporation of Columbus
3518 Riverside Dr., Suite 202
Columbus, OH 43221, USA
Michael Benson‡
US Nuclear Regulatory Commission
Office of Nuclear Regulatory Research
Mail Stop: CSB-5CA24
Washington, DC 20555-0001, USA
ABSTRACT
As part of the xLPR (Extremely Low Probability of
Rupture) project, the stress intensity factor (SIF) solutions
required for crack growth calculations were updated. This
involved extension of existing solutions and development of
new solutions as well. For surface cracks, the Universal
Weight Function Method (UWFM) was implemented to handle
complex stress distributions. In addition, a crack transition
model was developed to more accurately capture the surface to
through-wall crack transition. Finally, solutions for idealized
through-wall cracks were extended to various pipe sizes and
longer crack lengths.
In this paper, the SIF (K) solutions in xLPR Version 2.0
were validated against existing experimental data. The SIF
modules in the xLPR code were used to simulate fatigue crack
growth of circumferential cracks in pipes under bending up to
wall penetration and to through-wall crack transition stage
until idealized through-wall cracks were formed.
The
predicted fatigue crack growth results (mainly crack shape
evolution) using the xLPR K-solutions provided good
agreement with experimental data.
†
‡
INTRODUCTION
Stress intensity factor (SIF, K) solutions are used in the
xLPR (Extremely Low Probability of Rupture) Code for crack
growth (fatigue and stress corrosion cracking) calculations [1].
Recently, in xLPR Version 2.0, the SIF solutions required for
crack growth calculations were updated. This involved
extension of existing solutions and development of new
solutions as well. For surface cracks, the Universal Weight
Function Method (UWFM) was implemented to handle
complex stress distributions (e.g., weld residual stress) [2,3,4].
Solutions for idealized through-wall cracks were extended to
various pipe sizes and longer crack lengths and were provided
in closed-form solutions [ 5 , 6 ]. In addition, the crack
transition model (based on K-solutions for non-idealized
through-wall cracks, i.e., through-wall crack with different
crack lengths on the ID and OD surfaces) was developed to
more accurately capture the surface to through-wall crack
transition [7,8].
In this paper, the SIF solutions in the xLPR Version 2.0
code were validated by comparing the crack growth
calculation results with published experimental data from the
literature [9,10]. The SIF solutions were used to simulate
fatigue crack growth of circumferential cracks in pipes under
Corresponding author, djshim@emc-sq.com
The views expressed herein are those of the author and do not represent an official position of the USNRC
Approved for public release; distribution is unlimited
1
Copyright © 2015 by ASME
This work is in part a work of the U.S. Government. ASME disclaims all interest in the U.S. Government’s contributions.
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bending up to wall penetration and to through-wall crack
transition stage until idealized through-wall cracks were
formed. The validation was focused on the crack shape
evolution during the crack growth.
surface crack size, loading condition, and fatigue life (number
of cycles to leak and fracture) are summarized in Table 1.
The initial surface crack was inserted in the pipe using
electric discharge machining or a saw cut. Fatigue tests were
conducted using a four-point bending fixture with outer and
inner spans of 1,000 mm and 245 mm, respectively. Fatigue
load was applied using a sine wave loading with a stress ratio
R=0.1 (frequency 1~5 Hz). All tests were conducted at room
temperature. In addition, crack growth measurements were
made using a stereomicroscope to track beach mark evolution.
Example photographs of fatigue fracture surfaces obtained
from the experiments are provided in Figure 2. As shown in
this figure, the initial surface crack propagated through the
pipe wall and then transitioned into an idealized through-wall
crack. Additional details of the fatigue pipe test procedure
can be found in Ref. [9] and Ref. [10].
EXPERIMENTAL DATA
The experimental data used for the present validation was
developed by Yoo et al. [9,10] where fatigue crack growth in
circumferentially cracked pipes was investigated. In these
experiments, a series of surface cracked pipes was loaded
under cyclic bending until the surface crack penetrated the
pipe wall and transitioned to an idealized through-wall crack
(see Figure 1).
The pipe material tested in Ref. [9,10] was a carbon steel
(STS370) where the yield and ultimate stresses were 227 MPa
and 406 MPa, respectively. The pipe geometry, initial
Figure 1
Table 1
Test ID
(a)
(b)
(c)
Dimensions of (a) semi-elliptical surface crack, (b) non-idealized through-wall crack, and (c) idealized
through-wall crack
Pipe geometry, initial surface crack size, loading condition, and fatigue life from experiments
conducted by Yoo et al. [10]
Pipe geometry
Initial surface crack size
Loading condition
Fatigue life
Ro
t
a
ci
a/t
a/ci
max
min
NL*
NF**
(mm)
(mm)
(mm)
(mm)
-
-
(MPa)
(MPa)
(cycle)
(cycle)
TB-1
51.0
8.1
4.5
22.25
0.556
0.202
200.0
20.0
12,407
21,558
TB-2
51.0
8.1
3.0
6.0
0.370
0.500
210.0
21.0
169,750
197,856
TB-3
51.0
8.1
5.0
5.0
0.618
1.000
325.0
32.5
11,200
17,500
TB-4
51.0
8.1
3.0
18.25
0.370
0.164
200.0
20.0
72,910
862,60
TB-5
51.0
12.7
6.0
6.0
0.236
1.000
220.0
22.2
222,920
245,800
TB-6
51.0
12.7
3.0
6.0
0.236
0.500
261.0
26.1
287,500
301,400
No.
* NL, number of cycles to leak (crack incubation time prior to crack initiation not included)
** NF, number of cycles to fracture (definition of fracture is not clearly defined in Ref. [10])
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(a) TB-4
Figure 2
(b) TB-5
Example of fatigue fracture surfaces from Ref. [10]
(c) TB-6
(a) Cases loaded below yield strength
(b) Cases loaded above yield strength
Figure 3 Fatigue crack growth rate versus SIF range from Ref. [10]
In Ref. [10], the fatigue crack growth rate (FCGR) was
determined from the pipe tests. Figure 3 shows the FCGR
versus the SIF range (K) from Ref. [10] for samples loaded
below the yield strength [Figure 3(a)] and those loaded above
the yield strength [Figure 3(b)]. This figure includes the
crack growth data measured at the deepest and surface points
of the surface crack, and inner and outer surface points of the
through-wall crack. The K-solutions used for these figures
are provided in Ref. [10], where the Newman-Raju solution
(for surface cracked plate) [11] was used for surface cracks
and Zahoor solution (for through-wall cracked pipe) [12] was
used for idealized through-wall cracks. In addition, Yoo et al.
developed a simplified K-solution for non-idealized through-
wall cracks by modifying the Zahoor solution. However, due
to modeling the pipe geometry as a plate, the accuracy of the
K-solutions used for the FCGR determinations in Figure 3 may
have some uncertainty embedded in it. Effect of this
uncertainty on fatigue life predictions is discussed later in this
paper.
VALIDATION OF SIF SOLUTIONS
In order to validate the SIF (K) solutions used in the
xLPR Version 2.0 code, the fatigue crack growth results in
Ref. [10] were predicted using the modules in xLPR Version
2.0. These modules include K-solutions for surface cracks [4],
K-solutions for non-idealized through-wall cracks [7] in the
3
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crack transition module (CTM), and K-solutions for idealized
through-wall cracks [5].
The K-solutions for surface crack assume that the shape of
the surface crack is semi-elliptical, see Figure 1(a). Once the
depth of the surface crack reaches 95% of the wall thickness
(a/t=0.95), the CTM initiates the surface to through-wall crack
transition and forms a through-wall crack. This initial nonidealized through-wall crack has different crack lengths on the
ID and OD surfaces [as illustrated in Figure 1(b)], which are
determined by the CTM.
The crack front shape (or
curvature) of the non-idealized through-wall cracks was
determined from cylindrical transformation of a straight crack
front in a plate (see Ref. [7] for details). The K-solutions in
the CTM are used to grow the non-idealized through-wall
crack until an idealized through-wall crack is approximated
(i.e., 1/2≤1.05) – see Figure 1(c). At this point, the CTM is
turned off and the crack growth is continued using idealized
through-wall crack K-solutions.
In the present study, the FCGR parameters reported in
Ref. [10] were employed. The calculations did not include
the time to crack initiation, i.e., crack incubation time prior to
start of crack growth. After performing a sensitivity study,
the step increment used in the crack growth calculations was
fixed to 100 cycles/increment.
the through-wall crack shape is represented by the normalized
crack angles at the ID and OD surfaces, i.e., 1/ and 2/,
where 1 and 2 are shown in Figure 1(b) and Figure 1(c).
Note that 1=2 for an idealized through-wall crack.
The prediction results in Figure 5 start from the initial
non-idealized through-wall crack determined by the CTM.
As illustrated in this figure, the ID crack angle (1) remains
almost constant during the crack transition, whereas the OD
crack angle (2) increases until it becomes approximately
equal to 1. Once an idealized through-wall crack is formed
(1=2), the through-wall crack maintains that shape as it
continues to grow around the circumference. The predicted
results provided good agreement with the experimental results
throughout the crack transition stage as well as the idealized
through-wall crack growth.
VALIDATION RESULTS
In order to validate the surface crack K-solutions,
predictions from the xLPR code were compared against the
surface crack growth results obtained from the experiments.
Figure 4 provides the change in crack depth (a/t) as well as
crack aspect ratio (a/ci) during surface crack growth. As
demonstrated in Figure 4, overall results show good agreement
between the predicted and experimental results.
The
agreement is excellent for cases where the initial crack aspect
ratio (a/ci) are relatively low, i.e., long surface cracks – TB-1
and TB-4. This indicates that, for these cases, the surface
crack maintains a semi-elliptical shape during the crack growth
[see Figure 2(a)]. Hence, the crack shape evolution is
accurately predicted by the surface crack K-solutions. On the
other hand, for initial surface cracks with a/ci=0.5 (TB-2 and
TB-6), the two results showed good agreement in the earlier
stage of crack growth but then deviated from each other as the
crack depth increased. This is due to the surface crack shape
deviating from the semi-elliptical shape in the experiment –
especially as it gets closer to wall penetration [see Figure
2(c)]. Similar behavior is observed for surface cracks that
have initial value of a/ci=1.0 (TB-3 and TB-5) where the
surface crack shape in the experiment slightly deviated from a
semi-elliptical shape [see Figure 2(b)]. The predicted results
are provided up to a/t=0.95, when the surface crack is
transitioned to a through-wall crack according to the crack
transition module (CTM).
Validations of K-solutions for non-idealized and idealized
through-wall cracks are provided in Figure 5. In this figure,
(a) TB-1, TB-2 and TB-3
Figure 4
4
(b) TB-4, TB-5 and TB-6
Comparison of surface crack growth
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(a) TB-1 and TB-2
Figure 6
Comparison of predicted and measured
crack front shapes during crack growth
(at random cycles)
(b) TB-5 and TB-6
Figure 5
Comparison of crack transition
through-wall crack growth
and
Figure 7 compares the predicted and experimentally
measured number of cycles to wall penetration.
Experimental data is provided in Table 1. Note that the
predictions are based on the FCGR parameters provided in
Ref. [10]. The prediction over-predicted the experimental
results by an average of 37%. This may be due to the less
accurate K-solutions that were used in Ref. [10] to determine
the FCGR (see earlier discussions related to Figure 3 in this
paper). To investigate the effect of FCGR on crack shape
evolution, optimized FCGR parameters were determined from
the experimental data by matching the predicted and measured
number of cycles to wall penetration with some
approximations, as illustrated in Figure 8.
Figure 9
demonstrates that the effect of FCGR constants on crack shape
evolution for TB-1 is negligible.
Figure 6 compares the overall predicted and measured
crack front shapes for two example cases. The comparisons
are made at a random number of cycles since the number of
cycles between beach marks was not presented in Ref. [10].
The crack shape evolutions for surface crack, transitioning
crack, and through-wall crack are well captured by the
predictions. The differences in the crack front shape for
transitioning cracks are due to the assumption that was made in
the development of the non-idealized through-wall crack Ksolutions [7]. However, as demonstrated in Figure 5, the
effect of this assumption on the overall crack growth
predictions are negligible. Furthermore, only crack angles
are used in the actual xLPR code calculations.
5
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Figure 7
Comparison of number of cycles to wall
penetration using FCGR from Ref. [10]
(a) Surface crack
(b) Transition and through-wall crack
Figure 8
Figure 9
Comparison of number of cycles to wall
penetration using optimized FCGR

CONCLUSIONS
In this paper, the SIF (K) solutions (surface crack, nonidealized and idealized through-wall cracks) in xLPR Version
2.0 were validated against limited experimental data on
circumferential fatigue crack growth in pipes. Below are the
conclusions from the present study.
 Predictions from the xLPR Version 2.0 modules
provided good agreement with experimental data in
terms of crack shape evolution (surface crack, nonidealized, and idealized through-wall cracks).

Effect of FCGR on crack growth and crack
shape evolution
Number of cycles to wall penetration was overpredicted compared to the experimental data, which
may be due to the uncertainty in the FCGR
determined from the reference.
By employing an optimized FCGR, it was
demonstrated that the crack shape evolution is not
sensitive to FCGR parameters.
ACKNOWLEDGMENTS
The authors would like to thank the U.S. Nuclear
Regulatory Commission for their support of this work.
6
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REFERENCES
[6] Unpublished work by Working Group on Pipe Flaw
Evaluation, ASME BPVC Section XI.
[7] Shim, D.J., Kurth, R. and Rudland, D., “Development of
Non-Idealized Surface to Through-Wall Crack Transition
Model,” PVP2013-97092, Proceedings of the 2013
ASME Pressure Vessels and Piping Conference, July 1418, 2013, Paris, France.
[8] Shim, D.J., Rudland, D. and Park, J.S., “Surface to
Through-Wall Crack Transition Model for Axial Cracks
in Pipes,” PVP2014-28048, Proceedings of the 2014
ASME Pressure Vessels and Piping Conference, July 2024, 2014, Anaheim, California, USA.
[9] Yoo, Y.S., Ahn, S.H. and Ando, K., “Fatigue Crack
Growth and Penetration Behaviours in Pipes Subjected to
Bending Moment” Proceedings of the 1998 ASME
Pressure Vessels and Piping Conference, PVP-Vol. 371,
High Pressure Technology, pp 63-70, 1998.
[10] Yoo, Y.S. and Ando, K., “Circumferential Inner Fatigue
Crack Growth and Prediction Behaviour in Pipe
Subjected to a Bending Moment,” Fatigue Fract Engng
Mater Struct 23, pp. 1-8, 2000.
[11] Newman, J.C. and Raju, I.S., “An Empirical Stress
Intensity Factor Equation for the Surface Crack,” Engng
Fracture Mech. 15, pp. 185-192, 1981.
[12] Zahoor, A., Ductile Fracture Handbook, Vol. 3, Novetech
Corporation and Electric Power Research Institute, 1989.
[1] Rudland, D., Harrington, C. and Dingreville, R.,
“Development of the Extremely Low Probability of
Rupture (xLPR) Version 2.0 Code,” PVP2015-45134,
Proceedings of the 2015 ASME Pressure Vessels and
Piping Conference, July 19-23, 2015, Boston,
Massachusetts, USA.
[2] Scarth, D. and Xu, S., “Universal Weight Function
Consistent Method to Fit Polynomial Stress Distribution
for Calculation of Stress Intensity Factor,” J. Pressure
Vessel Technol. 134(6), 061204, 2012.
[3] Cipolla, R. and Lee, D., “Stress Intensity Factor
Coefficients for Circumferential ID Surface Flaws in
Cylinders for Appendix A of ASME Section XI,”
PVP2013-97734, Proceedings of the 2013 ASME
Pressure Vessels and Piping Conference, July 14-18,
2013, Paris, France.
[4] Xu, S., Lee, D., Scarth, D. and Cipolla, R., “Closed-Form
Relations for Stress Intensity Factor Influence
Coefficients for Axial ID Surface Flaws in Cylinders for
Appendix A of ASME Section XI,” PVP2014-28222,
Proceedings of the 2014 ASME Pressure Vessels and
Piping Conference, July 20-24, 2014, Anaheim,
California, USA.
[5] Shim, D.J., Xu, S. and Lee, D., “Closed-Form Stress
Intensity Factor Solutions for Circumferential ThroughWall Cracks in Cylinder,” PVP2014-28049, Proceedings
of the 2014 ASME Pressure Vessels and Piping
Conference, July 20-24, 2014, Anaheim, California, USA.
7
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