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T. J. Griesbach
Senior Nuclear Engineer,
Combustion Engineering, Inc.,
Windsor, Conn. 06095;
presently, Project Manager,
Nuclear Safety Analysis Center,
Palo Alto, Calif.
Mem.ASME
Dynamic Elastic-Plastic Behavior
of Circumferential Cracks in a Pipe
Subject to Seismic Loading
Conditions
An analytical study is performed to investigate the structural stability, and inherent
integrity, of a Pressurized Water Reactor primary coolant loop cold leg pipe
containing a through-wall circumferential crack. The purpose of this study is to
gain a better understanding of the mechanisms and extent of crack opening
behavior in a real piping system, and, thus, establish the basis for improved pipe
break criteria. Cracks extending one-half the circumference or greater are considered in the pipe. Combined operating pressure plus external seismic loads are
applied simultaneously to cause a maximum crack opening effect. A dynamic
elastic-plastic analysis is performed to calculate stress intensity factors at the crack
tip. The results are compared with experimental fracture toughness data to assess
the material's capability to resist crack extension. A fatigue crack growth study is
also performed to determine the range of initial flaw sizes which could grow and
extend during normal operation to threaten the integrity of the cold leg pipe.
1
Introduction
In the design of nuclear power plants to withstand a loss of
coolant accident (LOCA), regulatory guidelines [1] and industry standards [2] require that pipe whip restraint devices be
installed to mitigate the effects of an instantaneous guillotine
type of pipe failure. The present pipe break criteria [3]
presume that an instantaneous gillotine pipe break can occur
without determining the likelihood of such an event from a
consideration of the physical mechanisms of crack growth
and crack opening behavior in a real piping system. By
choosing the guillotine failure as the worst case accident, it is
assumed that this will "envelope" any instance of a circumferentially propagating crack in a pipe. However, in the
design of piping systems to prevent pipe whip, the required
pipe whip restraints could cause interference with the system
during normal operation. In addition, the size of such
restraint mechanisms hinder access for in-service inspection of
the piping and maintenance of adjacent components. This
may reduce plant safety while increasing cost and complexity.
Determination of the margin of safety against a large break
in the cold leg piping system of a Pressurized Water Reactor
(PWR) power plant has been the subject of a study at Battelle
[4]. Their results conclude that initial defects in the piping
system could possibly grow through the wall and cause a leak
in less than one plant life (40 yr). The calculated leakage rate
from such a crack in the pipe was predicted to be large enough
to be detectable prior to the formation of a large pipe break.
It was determined that the time between leak and large pipe
break would be sufficient for plant shutdown even if a Safe
Shutdown Earthquake (SSE) occurs immediately after formation of leak.
Probabilistic fracture mechanics analysis was performed by
Lawrence Livermore Laboratory as part of the USNRC Load
Combination Program [5]. The purpose of the study was to
estimate the probability of a seismic induced loss-of-coolant
accident (LOCA) in the primary system of a commercial
PWR. The results indicate that the probability of a doubleended guillotine pipe break directly caused by a seismic event
during the plant life is very small, on the order of 10 ~12.
The purpose of the present analysis is to further investigate
the integrity of the discharge leg pipe in a PWR under severe
loading conditions using deterministic methods. In particular,
the behavior of a partially cracked pipe subjected to the
combined effects of normal operating loads plus external
seismic loads is examined to determine the largest stable
through-wall circumferential crack size. Furthermore, a
fatigue crack growth study is performed to evaluate the
mechanism and extent to which initially present inside surface
flaws could grow to be critical in size during the operating
plant life. The results of this analysis emphasize the need, and
provide a basis, for more realistic pipe break criteria in the
design of nuclear piping systems.
Contributed by the Pressure Vessels and Piping Division for publication in
'he JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received by the
Pressure Vessel and Piping Division, October 28, 1980; revised manuscript
received December 6,1982.
Journal of Pressure Vessel Technology
2 Method of Analysis
A previous study by Griesbach and Ayres [6] considered the
Copyright © 1983 by ASME
FEBRUARY 1983, Vol. 105/63
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crack opening effects of a circumferential crack at the reactor
vessel inlet nozzle end of the pipe. Such a crack was presumed
to initiate at a time during the Safe Shutdown Earthquake
loading transient which would produce the most severe
loading condition at the crack tip. The results of the previous
analysis demonstrated several significant effects:
A/—NV-PUMP IB
©
(a) The end constraint afforded by the reactor vessel, the
reactor coolant pump and the support structures severely
limit the displacements and rotations in the pipe which
contribute to circumferential crack opening.
(b) Under such constraint conditions, seismic loading has
only a small influence on the amount of crack opening as
well as the crack-tip stress intensity factor resulting from a
circumferential crack in the pipe.
(c) A circumferential crack larger than two-thirds of the way
around the pipe would be necessary before the effects of
operating pressure plus SSE loading could cause crack
instability and, thus, produce a full circumference pipe
break at the inlet nozzle.
The present study is a continuation of the effort described
in the foregoing to demonstrate that seismic loading cannot
cause a guillotine break in the middle of the pipe where the
end constraint effects are at a minimum. The method of
analysis used here is the same as that which was established
and utilized in the previously described work [6], The details
of this method will be discussed further in Section 3. Briefly, a
three-dimensional finite element model of one leg of the
primary system is constructed, including the reactor vessel,
the reactor coolant pump, and the vertical and horizontal
structure supports. A circumferential through-thickness crack
is postulated to occur in the discharge leg pipe half-way
between the reactor vessel inlet nozzle and the reactor coolant
pump. Static operating pressure loading is applied, as well as
a dynamic time history loading function representing a Safe
Shutdown Earthquake. The crack is presumed to initiate at
the most severe time during the SSE loading transient. A
detailed elastic-plastic dynamic analysis is performed to
evaluate the behavior of the piping system containing a crack.
The value of the crack-tip stress intensity factor, K,, is
calculated as a function of time using the /-integral method.
For values of K, which are below the measured fracture
toughness, KIC, the crack is assumed to remain stable. By
applying this technique for various different crack sizes, the
critical crack size is determined as the minimum crack size for
which the calculated stress intensity factor exceeds the
fracture toughness of the material. All finite element computations are performed using the MARC general purpose
nonlinear finite element program.
The second phase of this study evaluates the fatigue crack
growth mechanisms by which initial flaws could enlarge and
grow to threaten the integrity of the piping system. Various
initial flaw sizes are hypothesized and the stress intensities for
design basis operating transients are calculated. Crack growth
rates are determined by performing an iterative process
considering the frequency and severity of operating cycles,
PUMP i A / ^ S j T
o
o
-PUMP IA
PUMP 2B-\V^>
STEAM GENERATOR No. 1
Fig. 1
STEAM GENERATOR No. 2
Nuclear steam supply system arrangement
and the flaw size is updated and monitored to the point of
first leak when the flaw depth exceeds the wall thickness. By
continuing the analysis for circumferential crack growth, the
required time for a through-thickness crack to grow to be
critical in size is determined. Critical crack size is defined as
the minimum circumferential through-thickness crack which
could propagate unstably to become a full circumference pipe
break during a Safe Shutdown Earthquake. The margin
between the point of leak and possible break during SSE is
calculated for a wide range of initial (pre-service) flaw depths
and lengths.
3
Model Studies
The primary system for a typical Combustion Engineering
two-loop PWR contains the reactor vessel and two heat
transfer loops, each loop containing a steam generator and
two recirculating reactor coolant pumps. Figure 1 illustrates
the arrangement of the components, the primary system
piping, and the horizontal and vertical structural supports.
The reactor vessel is supported vertically by four rectangular
columns, each column is bolted to a pad under a vessel inlet
nozzle. Each reactor coolant pump is supported by a set of
Nomenclature
K, --= crack-tip stress intensity
factor
J --= /-integral value
G --= strain energy release rate
E == elastic modulus
v == Poisson's ratio
g == acceleration due to gravity
t == time
P == period of vibration
64 / Vol. 105, FEBRUARY 1983
w = natural frequency of vibration
Kic = plane strain fracture
toughness
Jre = critical /-integral value for
crack growth
a0 = initial surface flaw depth
C„ = initial circumferential halflength of flaw
da
dN
da/dN
C
n
j3,7
=
=
=
=
=
=
change in crack depth
number of fatigue cycles
fatigue crack growth rate
scaling constant
fatigue growth rate exponent
parameters of Newmark
dynamic operator
A = increment
Transactions of the ASME
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Fig. 3 System arrangement showing finite element model
Fig. 2 Finite element model
four vertical columns, four horizontal columns, and a
horizontal snubber. The steam generator is supported vertically on a base plate which slides on low-friction bearings. A
system of hydraulic snubbers and keys with mating keyways
are attached to the upper portion of the steam generator to
provide horizontal support. The arrangement of the supports
allows for essentially free thermal expansion of the components and piping (excluding friction), while providing
support for static weight, seismic, and pipe break effects.
The three different types of piping runs which are shown in
Fig. 1 are the "hot leg piping," "suction leg piping," and
"discharge leg piping." Only one leg of the discharge leg
piping is considered in this analysis since, by symmetry
conditions, each leg is identical. The piping is made from SA516 Grade 70 Carbon Steel. The discharge leg piping is 30 in.
(762 mm) inside diameters with a nominal wall thickness of 3
in. (76.2 mm). The pipes are of relatively short length and
large diameter, resulting in a stiff piping system.
In analyzing the dynamic behavior of the discharge leg
pipe, the mass and stiffness associated with the major
components and the structural supports have a significant
effect on the overall response of the piping system. This
phenomenon if often ignored when considering the integrity
of a pipe containing a cracklike defect; the discharge leg is a
single component in a more complex system, and it is not
valid to analyze the dynamic response of this component
without including the interacting effects with the adjacent
structural members. Taking these effects into account the
finite element representation used in this analysis is presented
in Fig. 2. The model is made up of a combination of 169 beam
and shell-type elements with approximately 1800 degrees-offreedom. Three-dimensional beam elements, with the appropriate section properties, were used to model the reactor
vessel, the reactor coolant pump, the horizontal and vertical
supports, and the snubber. Lumped masses were used for
these main components in accordance with the mass
distribution from a system flexibility anlaysis [7]. The
discharge leg was modeled using a combination of beam
elements and shell elements with uniform distributed mass
being used for the elements representing the pipe. The threedimensional rectangular curved shell element was chosen to
model the section of the pipe containing a crack for several
Journal of Pressure Vessel Technology
Fig. 4 Schematic showing finite element representation
reasons: It is a rapidly convergent element, it can handle
rigid-body motions exactly, distributed mass terms are accurately represented, through-thickness cracks can be
modeled with relative ease, localized plasticity effects can be
included, and the /-integral technique can be utilized to
calculate the stress intensity at the crack tip. The choice of
element mesh size represents a reasonable compromise between using a very coarse mesh near the crack tip which would
be too crude to model the localized effects in this region, and
the prohibitive constraint (in terms of computer time) of using
a more refined crack-tip mesh.
The crack-tip stress intensity factor, K,, was calculated
directly using the /-integral technique. J is an energy term
which is used to express the change in potential energy per
unit change in crack extension. For the case of linear elastic,
nonlinear elastic, and small-scale yield fracture mechanics
analyses, the parameter J is identical to the strain energy
release rate, G, defined for a linear elastic body.
(1-v2)
J=G=
K,2 (for plane strain)
(1)
E
FEBRUARY 1983, Vol. 105/65
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10.0
MAGNIFIED CRACK OPENING DISPLACEMENT
Fig. 5 Displaced view showing crack opening displacement for onehalf circumference crack in pipe
0.1
1.0
Fig. 6
10.0
FREQUENCY, CPS
SSE in-structure response spectrum
100.0
where K, is the stress intensity factor, E is Young's modulus,
and v is Poisson's ratio. Writing K, in terms of / gives the
relation:
the finite element model containing a one-half circumference
crack under static pressure loading. A "bowing-out" of the
J>E (assuming plane strain)
(2) pipe, and the corresponding crack opening effect, is apparent
K,=
in this figure for a crack occurring in the middle of the
The validity of the /-integral approach has been demon- discharge leg pipe. The maximum deflection for the static
strated in its ability to produce accurate results in the load case with a one-half circumference crack was determined
determination of crack-tip stress intensity factors without the to be less than 0.1 in. (2.54 mm) as compared to the total
need for special crack-tip elements using the stiffness length of the pipe which is 218 in. (8.58 m). These results
derivative technique [8]. Furthermore, in the absence of gros,s confirm the kinematic basis of the analysis in that the
plastic deformation, the value of the /-integral is only slightly deflections and rotations of the pipe are justifiably small so
affected by mesh size variation provided the /-integral that large displacement effects need not be considered.
The static analysis for a one-half circumference crack in the
contour is taken far enough away from the crack tip [9]. Since
the pipe is treated as a thin shell, the calculated value for / , or middle of the discharge leg pipe produced a crack tip stress
Vm). This correKj, represents the mean value across the thickness of the pipe intensity value, K,, of 106 ksi Vin (117 MPa
2
2
wall. This is consistent with the formulation of the /-integral sponds to a crack opening area of 1.98 in (1278 mm ) due to
which is established for two-dimensional geometries, the pipe pressure loading only. For comparison purposes, the results
in this case being considered as a two-dimensional reference from [6] for a one-half circumference crack at the inlet nozzle
ksi
surface in the Gaussian (surface) coordinate system. Although end of the discharge leg pipe gave a value for K, of 92
MPa Vm) and a crack opening area of 1.36 in2 (877
transverse shear effects at the crack tip are not accounted for Jin (101
2
in this formulation, the thin shell approximation is fairly mm ) due to static pressure loading.
good for long cracks predominantly under pressure (i.e.,
Considering the static behavior of the structure containing
membrane) loading [10] as is the case in this analysis.
a two-thirds circumference crack in the middle of the pipe, the
Figure 3 is a plan view and Fig. 4 is a perspective view calculated value for the stress intensity at the crack tip was
superimposing the finite element representation and the determined to be 183 ksi Vin (202 MPa Vm). A crack opening
corresponding system components. Compatibility between area of 7.142 (4607 mm2) resulted from this geometry and
the intersection of the beams and shells was accomplished by a loading condition. In comparing the results for a two-thirds
set of imposed constraint equations (i.e., nodal point circumference crack at the inlet nozzle end of the pipe [6], the
"trying") to maintain the Euler-Bernoulli condition that calculated value for K, of 132 ksi Vin (145 MPa Vm) and the
plane sections remain plane before and after deformation. crack opening area of 2.36 in2 (1520 mm2) show that the
Also, a similar application of nodal point trying between tendency for crack opening is considerably less for a crack at
nodes on the shell surface permitted the introduction of a the end of the pipe than if such a crack existed in the middle of
crack in the pipe by the releasing of tide nodes.
the pipe. This demonstrates the constraint effect produced by
The kinematic behavior of the finite element model was the system components and the structural supports to limit the
evaluated under static conditions to verify the overall range of motion and restrict any large opening effects due to a
response of the system to operating pressure loading. Internal circumferential crack in the pipe.
pressure of 2250 psi (15.5 MPa) was applied as a distributed
The upper shelf, dynamic fracture toughness for SA-516
load to the shell elements forming the pipe section which Gr. 70 material is expected to exceed the static, initiation
produced the state of axial and circumferential stresses toughness value of 250 ksi Vin (275 MPa Vm) assumed in the
present in the pipe during normal operating conditions. It was analysis. This is based upon two experimental observations:
assumed that the presence of a through-thickness leaking initiation toughness measurements underestimate the real
crack would not depressurize the system and reduce the stress fracture toughness of ductile materials; and, the upper shelf
level in the pipe.
dynamic fracture toughness for ductile materials exceeds the
For both the static and dynamic analyses, circumferential static toughness value. Due to specimen size limitations, there
cracks were chosen to occur around the outer side of the pipe are no valid Klc measurements for this material on the upper
where the resulting deflections and crack opening effects shelf. However, a conservative lower-bound estimate for Krc
would ba at a maximum. Figure 5 shows a displaced plot of can be made from JIC measurements [11] of SA-516 Gr. 70
66/Vol. 105, FEBRUARY 1983
Transactions of the ASME
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2.0
1
i
i
i
300
i
ACCELERATION
270 -
1.0
i ,IM Jr\M.ltak
0
r
-1.0
1
-2.0
100
i
1
'C 240
wry
1
1
i
i
1
i
VELOCITY
50
0
-50
-100
i
0
1
1
2
1
i
1
3
-
1
5
TIME, SECONDS
Fig. 7
20
Time histories generated from reference spectrum
40
60
80
100
TIME, MILLISECONDS
Fig. 9 K; versus time (following appearance of crack) for a one-half
circumference crack in the middle of the pipe
4
0.2
0.3
STRAIN, IN/IN
Fig. 8
Stress-strain curve for SA 516-Grade 70 carbon steel at 550°F
and similar steels which indicate a static, initiation toughness
in excess of 250 ksi Vin (275 MPa Vm) [12, 13].
The tests demonstrate that these steels have steep Jrcsistance curves and a resulting true fracture resistance
significantly higher than the assumed value. A second conservative assumption is the use of the static toughness for the
calculation in lieu of the dynamic fracture toughness SA-516
Gr. 70. The fracture toughness of ferritic materials shows a
significant increase with loading rate on the material upper
shelf when the cleavage fracture mode is precluded, as is
demonstrated in reference [14]. For these reasons, the
assumed fracture toughness for SA-516 Gr. 70 is appropriate
for this analysis.
By comparing the static results of the analysis with the
value for the critical fracture toughness of the piping material,
both a one-half and a two-thirds circumferential crack in the
middle of the pipe are determined to remain stable during
normal operation. This is a significant result when compared
to the finding for the critical crack size in a pipe under
operating pressures and having "free-end" conditions [8],
which showed that the largest stable crack size is only one-half
circumference with a corresponding crack opening area of 5.3
in2 (3420 mm2). Thus, when considering the pipe as an integral part of the reactor coolant system, the "in-system"
behavior of the pipe results in a larger tolerable crack size
Prior to instability for a circumferential crack in the pipe.
Journal of Pressure Vessel Technology
Seismic Loading Conditions
In evaluating the dynamic response of the system of seismic
loading, a seismic excitation was applied to the model as
support motion time histories at the structural attachments to
the foundation. The loading function for a seismic event is
defined by "ground response spectra" corresponding to the
peak ground accelerations. The model evaluated here is
considered to be a subsystem of a much more complex
configuration of nuclear power plant basemat, containment
building, internal structure, and pressurized water reactor
coolant system. It has been demonstrated that an uncoupled
subsystem analysis produces accurate results provided that the
excitations applied to the uncoupled model are derived from a
coupled system model to account for interaction effects [15].
For this analysis, "in-structure" response spectra were
determined at support locations from a complex threedimensional structural model of the building and reactor
coolant system [7], and a conservative envelope of the
response spectra at the various support locations was
determined for SSE conditions, as shown in Fig. 6.
From the envelope which defines the spectral input of the
seismic event, and using the Fourier transform method [16,
17], it is possible to generate an artifical time history which
maintains the identical spectrum over the frequency range of
interest. The resulting acceleration and velocity time histories
for horizontal support motion are given in Fig. 7. From this
figure it is noted that peak accelerations greater than 1.0 g are
present in the SSE loading function. Typically, maximum
ground accelerations on the order of 0.2-0.4 g are used for
design basis earthquakes [15]. By comparison, the seismic
loading used in this analysis represents a "very severe"
earthquake. A more detailed description of the seismic
loading procedure is given in [6].
5
Dynamic Elastic-Plastic Analysis
The combined effects of pressure plus seismic loading on
the behavior of the discharge leg pipe are evaluated by an
incremental dynamic analysis. The purpose of the dynamic
analysis is to examine the overall behavior of the system
following initiation of a through-wall circumferential crack.
An earlier study by Ayres considered the dynamic response of
a free-end pipe following the instantaneous appearance of a
stable circumferential crack [18]. Similar studies have
FEBRUARY 1983, Vol. 105/67
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Table 1 Loading transients analyzed and life occurrences
Life occurrences
500
500
15,000
15,000
2000
2000
10s
400
200
10
40
40
5
200
Loading transients
Plant heatup, 100°F/hr
Plant cooldown, 100°F/hr
Plant loading, 5 percent/min
Plant unloading, 5 percent/min
10 percent step load increase
10 percent step load decrease
Normal plant variation (± 100 psi, ± 10°F)
Reactor trip
Leak test, 2250 psia, 100-400°F
Hydrostatic test, 3125 psia, 100-400°F
Loss of reactor coolant flow <a) (o)
Loss of turbine generator load
Loss of secondary pressure (a)
Operating basis earthquake (a>
PRESSURE
(<0Abnormal
transient conditions
SSE LOADING
60
80
100
TIME, MILLISECONDS
120
140
Fig. 10 K/ versus time (following appearance of crack) for a two-thirds
circumference crack in the middle of the pipe
T
1—I I M i l !
1
1—I 1 I 1 I I J
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1—I I I I 11!
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DATA
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o
AS ME SECTION XI
"WATER" LINE -
1—1
t—
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10' 10
\
i i i i nl
|
i
l i i i ill
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'
i i i i
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10
100
1000
STRESS INTENSITY FACTOR RANGE, A K , ksi-in 1/2
TIME, HOURS
PLANT HEATUP
Fig. 11
TIME, HOURS
Fig. 12
Reference fatigue crack growth curve
P U N T COOLDOWN
Plant heatup and cooldown loading transients
examined the propagation phenomena of unstable cracks
produced by an instantaneous initiation [19], as well as the
fluid-structure interaction effects due to fluid depressurization [20]. While fluid depressurization effects may
be significant for some piping systems in consideration of
stability, particularly in pipes with large length-to-diameter
ratios, these effects are judged to be small in this case since
L/D = l, and including them is beyond the scope of the
present study.
In the present analysis, the response of the system due to an
impulse load will be understandably different than the freeend pipe. The large stiffness contribution of the end constraints, and the associated inertia of the system components,
will have a dominant effect over such very short duration
events. Thus, the crack-opening behavior evaluated in this
study is the results of the fundamental frequency
displacement and rotations of the pipe following the sudden
appearance of a crack. It was determined that the natural
frequency of the pipe without a crack is 16-17 Hz, which is
within the range of frequencies which would be active during
seismic loading.
The pipe is initially considered to be pressurized and un68/V01.105, FEBRUARY 1983
cracked (at time / = 0.0), and the motion time history is applied as a loading function to all supports. Dynamic time steps
of 0.1 s were used during the "buildup" phase of the seismic
event, and direct integration of the dynamic equations was
performed using the Newmark /3 method, with 7 = 1/2, (3 =
1/4 (trapezoid rule).
From the time history plots in Fig. 7, it is apparent that the
maximum positive seismic excitation occurs at a time of
t = 1.0 s as noted by the coincident peaks in acceleration and
velocity. Because the discharge leg is a high-energy pipeline,
and the seismic oscillations contribute additional kinetic
energy to the system, the pipe is presumed to remain uncracked until the time of peak seismic loading for which the
stored energy is at a maximum. The initiation of the crack at
this point in time causes the largest amount of pipe whip
behavior. Since the ends of the pipe are not free to whip, the
greatest degree of loading is directed toward the middle of the
pipe and, ultimately, to the crack tip.
The integration time step was reduced to .01 s just prior to
the time of crack initiation to trace the crack opening
behavior of the pipe. The choice of time step for this part of
the analysis was based on the criterion that:
Af=l/6p=l/6«l/w
(3)
Transactions of the ASME
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•
*
•
SSE
f
!
—~"~1
III
1M
Through Wall Crack
•
1'
,
I
U-2c —1
Surface Crack
/
/
a0-.50IN
Z CQ - 39.0 IN
/
'
'
'
20
YEARS
Fig. 13 Fatigue crack growth versus years of operation - a 0 = 0.50
in.,2C 0 = 39.0 in.
-
11
-
7
1
SSE a
M •
<
cc
<->
x
u
40
-
Fig. 15 Fatigue crack growth versus year of operation-a 0 = 0.35 in.,
2Cn
45.5 in.
h—2c
c
— |
/-
T]
"I
Through Wall Crack
o
•
0
i
t
• / 1 U^ M
-J . |
y
a 0 = 1.0 IN
2C f l -3d.0IN
2c
S u rface C rack
25 £
<
•
20
YEARS
Fig. 14 Fatigue crack growth versus years of operation - a 0
2C„ = 34.0 in.
a.
0
1.0 in,
where p is the period and w is the natural frequency of the
pipe. Frequencies above 20-25 Hz would be filtered out by the
coarseness of the time step. However, because of the large
inertia of the system, the higher frequencies do not have a
significant effect on crack opening behavior.
A small amount of stiffness damping was included with a
damping factor of 1.0 x 10"5 which imposes less than 0.05
percent damping over the frequency range of interest. No
artificial damping is induced in using the Newmark /3
operator; however, small periodicity errors may be introduced
in the dynamic solution as a function of the time step size.
Localized plasticity effects are considered when stresses in
the pipe exceed the yield strength of the material. An initial
yield point value of 35 ksi was determined experimentally for
SA-156 Gr. 70 carbon steel at 550°F (288°C), and the
corresponding work hardening behavior used in the analysis is
as shown in the stress-strain curve given in Fig. 8. The crack
tip stress intensity factor, Kj, is determined as a function of
time for both a one-half and a two-thirds circumference
crack, and the results are compared with the material fracture
Journal of Pressure Vessel Technology
Fig. 16 Fatigue crack growth results for a range of initial flaw sizes - 2
C 0 versus A0.
toughness, KIC. Using this criterion, a critical crack size is
determined for the onset of crack instability due to the
combination of operating pressure plus external seismic
loading.
6
Results of Dynamic Analysis
Stress levels exceeding the yield strength were calculated
near the crack-tip region following the appearance of the
crack. The development of a plastic zone near the tip of the
crack was very localized, and the calculated value for plastic
FEBRUARY 1983, Vol. 105/69
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1
1
-
>S.
\
1
TIMETO CAUSE LEAK
^ T I M E T O CAUSE
^
BREAK DURING
^CvMAXSSE
-*
a „ - .50 IN
N^
-
i
1
20
YEARS
Fig. 17 Margin between time to cause leak and large pipe break
during SSE for various initial flaw lengths, 2C0(a0
= 0.50 in)
strain in the locally yielded zone ahead of the crack tip was
less than 1 percent. No other portion of the pipe experienced
stresses large enough to cause plastic deformation.
The dynamic response of the pipe showed a noticeable
decrease in the natural frequency following initiation of a
crack, which indicates a corresponding change in the stiffness
of the system as a result of the crack. This is consistent with
previous findings for the effects of a circumferential crack at
the inlet nozzle end of pipe [6].
The dynamic crack opening effects are described in terms of
the crack-tip stress intensity factor, KIt and the calculated
values for K, versus time are shown in Fig. 9 for a one-half
circumference crack in the middle of the pipe. The influence
of seismic loading produces a peak value for K, of 150 ksi Vin
(166 MPa Vm), which is a 57 percent increase in the stress
intensity factor K, above the static value due to pressure
loading only. By comparison, the results of the previous study
[6] for a one-half circumference crack at the reactor vessel
inlet nozzle end of the pipe showed an increase due to seismic
loading of only 16 percent above the static value due to
pressure. The fact that K, is less than KIC indicates that the
crack would remain stable under these conditions. The
calculated values for K, versus time for a two-thirds circumference crack are shown in Fig. 10. It is noted that the
dynamic behavior of the pipe containing a two-thirds circumference crack produces a peak value for K, of 272 ksi
Vin (300 MPa Vm). This is significantly above the static value
due to pressure of 183 ksi Vin (202 MPa Vm), and, in fact,
exceeds the fracture toughness of the material which indicates
that such a crack would not remain stable.
The results of the dynamic analysis are significant in that
the critical crack size is identified to be a very large crack
greater than one-half the circumference. Furthermore, the
existence of such a crack would have the greatest effect on the
integrity of the discharge leg if the crack occurred in the
middle of the pipe rather than at the ends and was subjected to
the combination of pressure plus seismic loading conditions.
7
Fatigue Crack Growth Study
The results of the dynamic analysis are significant in the
determination of the critical defect size. Extensive pre-service
and in-service inspections are conducted to ensure the quality
of the piping against such flaws. However, the fact that
70/Vol. 105, FEBRUARY 1983
uncertainties exist in the performance of such inspections, and
the possibility that initially small flaws can enlarge during
normal operation by fatigue crack growth, poses an additional concern. While it has been demonstrated that a
significantly large (i.e., greater than one-half circumference),
through-wall crack would be required before crack instability
could occur during an SSE, a fatigue crack growth study was
performed to determine the range of initial defect sizes that
could grow to be critical in size during the 40-yr design life of
the plant.
The primary piping system and related components must be
designed to withstand both the normal and abnormal loading
transients as outlined in Table 1. The stress levels due to
pressure and thermal loads in the pipe are determined by
ASME code accepted procedures for pressure vessels and
piping from the pressure and temperature design. Typical
design curves for plant heatup and cooldown as a function of
time are shown in Fig. 11. The effects of stress concentration
due to a flaw in the pipe are evaluated using the methods for
K, determination in the ASME Code [21].
Section XI of the ASME Code [22] defines a fatigue crack
growth rate law of the form:
da
= C'(AKI)n
~dN
(4)
where n is the slope of the log {da/dN) versus log (AK,) curve,
and C" is a scaling constant. This material property curve has
been determined experimentally, and the material constants
for fatigue crack growth in a water environment are: C" =
3.795 x 10" 10 and n = 3.726. The rate of crack growth
(da/dN) is measured in inches per cycle from this relationship.
This crack growth law is intended to be a conservative upper
bound to the experimental data, however, recent fatigue crack
growth studies have produced data which lie above this curve
[23]. The solid line in Fig. 12 shows the da/dN versus AK,
curve which has been proposed as a revision to Section XI of
the ASME Code and is seen to envelope all of the fatigue
crack growth data [24]. A threshold stress intensity of 2.4 ksi
Vin (2.6 MPa Vm) was used to bound the lower range of the
fatigue crack growth rate curve based on data for A-106-B
piping steel [25]. The results of this study include the upper
bound to the crack growth rate curve as given in Fig. 12.
Semi-elliptical shaped inner surface flaws were hypothesized for various initial crack depth, a0, and lengths,
2C0. A computer methodology was used to evaluate the stress
intensity factor for a given flaw size and loading function and
then compute the growth rate of the flaw under cyclic loading
conditions. The method of analysis is based on the Section XI,
Appendix A flaw evaluation procedure which calculates the
stress intensity factor, K,, as a function of flaw geometry and
stress state [26]. Using this method, the AK/ level for a
loading cycle is calculated based on the crack size and loading
conditions. From the number of cycles of loading in a given
time period, crack growth rates are calculated and the
corresponding changes in crack size. The time to produce first
leak is determined when an existing flaw enlarges and, subsequently, "pops-through" the thickness of the pipe. This
"pop-through" phenomenon is what is meant by a suddenlyappearing through-thickness crack. In relating these results to
the dynamic analysis, the circumferential length of the
through-wall crack is important to the determination of crack
stability.
For cracks which penetrate the wall thickness, the subsequent calculation of the stress intensity factor, K,, was
evaluated using the finite element method since the Section XI
flaw evaluation procedure does not extend to through-wall
cracks. The finite element technique enabled the determination of Kj as a function of the circumferential crack
length, 2C, and the applied load, and this information was
Transactions of the ASME
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incorporated into the crack growth procedure. In this manner,
the fatigue crack growth study was continued for circumferential growth of through-wall cracks. Using the
criterion for crack instability during SSE loading, the margin
between leak and possible large pipe break is determined for a
given initial defect size.
A wide range of initial flaw sizes and shapes were considered, and the resulting crack growth rates were calculated
for the design basis loading transients and corresponding
frequencies of occurrence given in Table 1. An example of the
predicted growth behavior for a defect with an initial depth of
0.50 in. (12.7 mm) and an initial length of 39.0 in. (990 mm) is
shown in Fig. 13. For the prescribed cyclic loading history the
flaw was calculated to become a leaking crack in 21 yr of
operation, which is much less than the normal (40 yr) plant
life. It should be emphasized that the loading histories used in
this analysis are a conservative representation of the design
transients and are intended to describe the upper limit of
possible reactor operating experience. If, after formation of a
through-wall crack, the defect remains undetected and
operation of the reactor continues, the calculated fatigue
crack growth rate is such that it would require another 18 yr
for the crack to extend circumferentially to become a large
pipe break in the event of a Safe Shutdown Earthquake occurring near the end of the reactor plant life. This illustrates
that a significant leak-before-large-break margin exists for a
initial defect of this size.
Figure 14 shows a similar plot of crack size versus years of
operation for an initial flaw with dimensions a0 = 1.0 in.
(25.4 mm) and 2C0 = 34.0 in. (863 mm). This initial flaw size
also results in the formation of a large circumferential
through-wall crack which could become a large pipe break
during SSE loading within one plant life of 40 yr. However, it
was determined that an initial flaw 1.0 in. deep and 34.0 in.
long would grow to become a leaking crack in only 4 yr of
reactor operation. This produces a leak-before-large-break
margin of almost 36 yr in which to detect and repair the
leaking crack.
The fatigue crack growth results for a 0.35-in. (8.9-mm)
deep and 45.5-in. (1156-mm) long initial flaw size are
presented in Fig. 15. The time required to cause a leaking
crack for this case is 38 yr, and the time to produce a critical
length circumferential crack during SSE loading would be 40
yr of reactor life. The calculated leak-before-large-break
margin for this initial defect size is only 2 yr.
It is possible to construct a curve describing the range of
initial flaw sizes (2C0 versus a0) which could produce a
leaking crack in 40 yr of reactor operation, as well as the locus
of initial flaw lengths and depths which could grow to become
a large pipe break during a Safe Shutdown Earthquake in the
40th year of plant life as shown in Fig. 16. It is seen from this
plot that for initial flaws less than 0.33 in. (7.6 mm) in depth
there could be no through-wall penetration by fatigue crack
growth to become a leaking crack. Similarly, for initial flaws
greater than 0.33 in. (7.6 mm) in depth, but less than 30.0 in.
(762 mm) in length, the results indicate that a leak could
develop but no large pipe break during the plant operating
life. For flaws which are initially deeper than 0.33 in. (7.6
mm) but less than 60 in. (1524 mm) in length, a sufficient
leak-before-large-break margin would exist to insure
detectability by leakage rate prior to the development of an
unstable through-thickness crack. For example, considering
an initial flaw depth of 0.50 in. (12.77 mm) and a range of
initial flaw lengths, the time required to produce a leaking
crack and the time for which a circumferential crack could
grow to become a possible large break during SSE loading are
shown in Fig. 17. It is noted that initial flaws must exceed 60
in. (1524 mm) in length before little or no leak-before-largebreak margin exists. However, for initial defects less than 60
Journal of Pressure Vessel Technology
in. (1524 mm) in length the margin is substantial. Considering
an initial flaw depth of 0.50 in. (12.7 mm) and an initial
length of 40 in. (1016 mm), the margin between leak and
possible break during SSE is determined to be 14 yr.
The major concern is for those flaws which could grow to
leak and, within a short period of time, cause a large pipe
break during SSE loading. The results shown here demonstrate that such a flaw have to initially exceed 60 in. (1524
mm) in length and be greater than 0.33 in. (7.6 mm) in depth.
The Code allowable limit for an inside surface indication of
very long length (a0/2C0 « 0.0) is less than 0.15 in. (3.8 mm)
depth for flaws detected during pre-service inspection [27].
Because the magnitude of a potentially dangerous flaw size is
greater than twice the Code allowable limit, the ability to
demonstrate that no such flaw could exist in the piping system
prior to operation rules out the practical possibility of the
guillotine pipe break failure.
8
Conclusions
From this study it can be concluded that a throughthickness crack would have to extend more than halfway
around the circumference in the middle of the discharge leg
pipe before the effects of pressure plus SSE loading could
cause unstable crack propagation.
A range of initial defect sizes were identified which could
cause a primary system pipe to leak, or cause a pipe to break
in the event of a Safe Shutdown Earthquake, as based on
conservative design basis plant operating conditions and crack
growth rate data. The results of this study indicate that no
leak or pipe break could occur for initial (pre-service) inside
surface defects in the piping system less than 0.33 in (7.6 mm)
deep.
A significant margin exists between the time to cause a
leaking crack and the time for a crack to become critical in
size to produce a possible pipe break (caused by the combination of pressure plus SSE) for initial flaws greater than
0.33 in. (7.6 mm) deep and less than 60 in. (1524 mm) in
circumferential length.
The range of initial flaw sizes which could grow to threaten
the integrity of the system represent large defects which
should be easily detected during in-process inspection.
On the basis of the large critical crack sizes and, similarly,
the large initial defect sizes required before significant fatigue
crack growth could occur, the results of this study provide
evidence that the occurrence of a guillotine type of pipe break
in the discharge leg is virtually impossible.
Acknowledgments
The author would like to acknowledge the contributions of
N. A. Lebedinsky for helpful discussions in dynamic
modeling, R. J. Fabi for providing programming assistance,
and T. U. Marston for assistance with fracture toughness
determination.
References
1 "Protection Against Pipe Whip Inside Containment," Regulatory Guide
1.46, U.S. Atomic Energy Commission, May, 1973.
2 "Design Basis for Protection Against Pipe Whip," ANSI N176,
American Nulcear Society, June, 1973.
3 "Design Basis Pipe Breaks for the Combustion Engineering Two Loop
Reactor Coolant System," CENPD-168-A, Combustion Engineering, Windsor,
Conn., June 1977.
4 Mayfield, M. E., Forte, T. P., Rodabaugh, E. C , Leis, B. M., andEiber,
R. J., "Cold Leg Integrity Evaluation," Battelle Columbus Laboratories
Report to USNRC, NUREG/CR-1319, Feb. 1980.
5 Lu, S., Streit, R. D., and Chou, C. K., "Probability of Pipe Fracture in
the Primary Coolant Loop of a PWR Plant, Vol. 1, Summary Load Combination Program," Project I Final Report, Lawrence Livermore Laboratory,
NUREG/CR-2189, Vol. 1, Sept. 1981.
6 Griesbach, T. J., and Ayres, D. J., "Opening and Extension of Circumferential Cracks in a Pipe Subject to Dynamic Loads," Nuclear
Engineering and Design, Vol. 57, No. 1,1980, pp. 141-152.
FEBRUARY 1983, Vol. 105/71
Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 10/26/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use
7 Kassawara, R. P., Austin, S. C , and Izor, R. C , "The Effects of
Reactor Coolant System Pipe Rupture Motion on Tributary Piping and Attached Equipment," presented at 5th International Conference on Structural
Mechanics in Reactor Technology, Berlin (West), Germany, Aug. 1979.
8 Parks, D. M., " A Stiffness Derivative Finite Element Technique for
Determination of Elastic Crack Tip Stress Intensity Factors," International
Journal of Fracture, Vol. 10, No. 4, Dec. 1974.
9 Shih, C. F., et al., "Crack Initiation and Growth Under Fully Plastic
Conditions: A Methodology for Plastic Fractor," EPRI Ductile Fracture
Research Review Document, EPRI NP-701-SR, Feb. 1978.
10 Barsoum, R. S., Loomis, R. W., and Stewart, fi. D., "Analysis of
Through Cracks in Cylindrical Shells by the Quarter-Point Elements," International Journal of Fracture, Vol. 15, No. 3, June 1979, pp. 259-280.
11 "Standard Test for JIc, A Measure of Fracture Toughness," ASTM
StandSLvdESn-Sl, Annual Book of ASTM Standards, Part 10, ASTM, 1981.
12 Menke, B. H., Hiser, A. L., Hawthore, J. R., and Loss, F. J., "R Curve
Characterization of Low Strength Structural Steels," Materials Engineering
Associates, Inc., MEA-EPRI Research Program RP2055-7, EPRI NP-2715,
Nov. 1982.
13 Oldfield, F. M., Strickler, T., and Oldfield, W., "Description and
Catalogue of the Nuclear Reactor Material Databases," Materials Research
and Computer Simulation Corp., MRCS-EPRI Research Program RP2055-2,
Topical Report, Apr. 1982.
14 Server, W. L., Oldfield, W., and Wullaert, R A., "Experimental and
Statistical Requirements for Developing a Well-Defined Kjg Curve," Fracture
Control Corp., FCC-EPRI Research Program RP696-1, EPRI NP-372, May
1977.
15 Gerdes, L. D., "Dynamic Structural Analysis of Uncoupled Subsystems," Paper K6/18, 4th International Conference on Structural Mechanics
in Reactor Technology, San Francisco, Aug. 1977.
16 Scanlan, R. H., and Sachs, K., "Earthquake Time Histories and
Response Spectra," Journal of Engineering Mechanics Division, ASCE,
Volume 100, Aug. 1974, pp. 635-655.
17 Scanlan, R. H., and Sachs, K., "Floor Response Spectra for MultiDegree-of-Freedom Systems by Fourier Transform," Paper K5/5, 3rd International Conference on Structural Mechanics in Reactor Technology,
London, U.K., Sept. 1975.
18 Ayres, D. J., "Determination of the Largest Stable Suddenly Appearing
Axial and Circumferential Through Cracks in Ductile Pressurized Pipe," Paper
F7/1, 4th International Confernce on Structural Mechanics in Reactor
Technology, San Francisco., Aug. 1977.
19 Emery, A. F., Kobayashi, A. S., and Love, W. J., "An Analysis of the
Propagation of a Brittle Circumferential Crack in a Pipe Subjected to Axial
Stresses," Paper No. 78-PVP-101, Joint ASME/CSME Pressure Vessel and
Piping Conference, Montreal, Canada, June 1978.
20 Emery, A. F., Kobayashi, A. S., Love, W. J., and Jain, A., "Dynamic
Propagation of Circumferential Cracks of Two Pipes with Large-Scale
Yielding,'' ASME JOURNAL OF PRESSURE VESSEL TECHNOLOGY, Vol. 102, No. 1,
Feb. 1980, pp. 28-32.
21 ASME Boiler Pressure Vessel Code, Section XI, Article A-3000,
"Method forKj Determination," 1977.
22 ASME Boiler and Pressure Vessel Code, Section XI, Article A-4000,
"Definition of Material Properties," 1977.
23 Bamford, W. H., "Application of Corrosion Fatigue Crack Growth Rate
Data to Integrity Analyses of Nuclear Reactor Vessels," ASME Journal of
Engineering Materials and Technology, Vol. 101, July, 1979, pp. 182-190.
24 Minutes of ASME Section XI Evaluation Working Group Meeting,
Bethesda, Maryland, Nov. 1979.
25 Mukherjee, B., and Vanderglas, M. L., "Fatigue Threshold Stress Intensity and Life Estimation of ASTM-A106B Piping Steel," ASME JOURNAL OF
PRESSURE VESSEL TECHNOLOGY, Vol. 102, Aug. 1980, pp. 294 -302.
26 Cipolla, R. C , "Computational Method to Perform the Flaw Evaluation
Procedure as Specified in the ASME Code, Section XI, Appendix A , " EPRINP-1181, Part 1, Electric Power Research Inst., Palo Alto, Calif., Sept., 1979.
27 ASME Boiler Pressure Vessel Code, Section XI, Article IWB-3000, Table
IWB-3514-2, "Acceptance Standards for Flaw Indications," 1977.
plications, and components (reliability, qualification and
testing, seismic qualification, valves, pump piping, piping
supports, flow-induced vibration, fluid structure interaction,
inspection, maintenance, repair, and behavior of cracks);
1984 PVP Conference and Exhibit
lifeline earthquake engineering (seismic design of oil and gas
pipelines and storage facilities and refineries, telecomThe 1984 Pressure Vessel and Piping Conference and munications, experiments and field observations in lifelines,
Exhibit, sponsored by ASME's Pressure Vessels and Piping modes of failure in fault movements, slope instability and
Division, will be held June 17-21, 1984, in San Antonio, liquidation, post-earthquake recovery in water supply and
natural gas systems, offshore oil and gas platforms); and high
Texas.
Papers are solicited in the following area: design and pressure technology (modes of failure, fabrication practices,
analysis (design methods for pressure vessels and piping, testing, high temperature effects, system safety, and in-service
process equipment design, elevated temperature analysis, inspection). Ideas for and developers of poster sessions are
limit analysis, stress analysis, seismic analysis, dynamic also sought.
Abstracts of the proposed papers are due August 30, 1983.
analysis, code requirements, dynamic stress criteria, testing
technology, field services, fatigue, creep, and fracture damage Manuscripts will be required by December 15, 1983, for
analysis); materials and fabrication (modes of failure, review. Address inquiries and send abstracts to:
nondestructive examination, environmental effects, bimetallic
G.E.O. Widera,
welds, fabrication methods, quality assurance, specifications,
Technical Program Chairman,
technology transfer, fatigue, and elevated temperature efMechanical Engineering Department,
fects); computer technology (structural analysis software,
University of Illinois at Chicago,
optimum design methods, graphics, vertification, water
Box 4348,
hammer analyses, data base usage, analysis of welds, and
Chicago, IL 60680;
parameter identification techniques); operations, apTel: (312)996-5317
Call for Papers
72 / Vol. 105, FEBRUARY 1983
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