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공학박사 학위논문
Development of Effective Applied Moment
Formulations for Integrity Assessment of
Nuclear Piping Systems under Static and
Dynamic Loading Conditions
정하중 및 동하중 조건에서 원전 배관 건전성
평가를 위한 유효하중 계산식 개발
2017년 2월
서울대학교 대학원
에너지시스템공학부 원자핵공학전공
김 예 지
Development of Effective Applied Moment
Formulations for Integrity Assessment of
Nuclear Piping Systems under Static and
Dynamic Loading Conditions
지도 교수 황 일 순
이 논문을 공학박사 학위논문으로 제출함
2016년 12월
서울대학교 대학원
에너지시스템공학부 원자핵공학 전공
김 예 지
김예지의 공학박사 학위논문을 인준함
2016년 12월
위 원 장 김 재 관 (인)
부위원장 황 일 순 (인)
위 원 최 영 환 (인)
위 원 김 윤 재 (인)
위 원 오 영 진 (인)
i
Abstract
Development of Effective Applied Moment
Formulations for Integrity Assessment of
Nuclear Piping Systems under Static and
Dynamic Loading Conditions
Yeji Kim
School of Energy System Engineering
The Graduate School
Seoul National University
Attentions to the beyond design basis earthquake have been increased
following the Fukushima Daiichi nuclear accidents on March 11, 2011.
Especially, the refinement of the current analysis methodologies has emerged
as one of high priority issues for the piping integrity evaluation. In this
dissertation, a set of generalized formulations has been developed to take into
account the effect of pipe restraint for consistent analysis of the crack opening
displacement and crack stability of nuclear piping containing a postulated
circumferential crack in order to enhance the confidence in the Leak-Before-
Break (LBB) characteristics.
For the current LBB analysis procedure, evaluation models for the
crack opening displacement as well as those for crack stability analysis have
been derived from the assumption that both ends of the pipe under analysis
ii
are free to rotate. In reality, however, the behavior of pipe with a crack can be
restrained by connected components or structures. These aspects of restrained
boundary conditions can make a significant favorable influence on the crack
instability prediction and unfavorable impact on the prediction of leakage size
crack by an underestimation of crack opening displacement (COD).
In this regards, there have been attempts to evaluate the combined
results of the restraint effects from the above two aspects. First, the equations
to determine the onset of a crack extension were developed for various piping
systems and loading conditions case by case. But generalized formulations
which can be employed as the unified practical method has not been derived.
Recently, the analytical expressions to evaluate the restraint effect on COD
were proposed for both linear elastic and elastic-plastic analysis, with its
applicability limited to a straight pipe with fixed ends subjected to pressure
induced bending.
Although significant efforts have been made in earlier studies to deal
with the restraint effect on the calculation of COD and crack stability analysis
separately, these are simultaneous phenomena caused by the decreases in the
applied moment at the cracked section due to the pipe restraint. Therefore, it
is desired to develop a unified formulation to determine the effective applied
moment at a postulated cracked section considering the boundary conditions
that can be utilized to a balanced analysis of both COD and flaw stability.
This dissertation mainly serves to the aims for the development of generalized
solutions that readily enable balanced evaluations of the restraint effect
starting from the following questions:
iii
i) How can we analytically evaluate the effective applied moment at the
cracked section taking into account the pipe restraint effects?
ii) Can the generalized formulations be applicable to various types of the
piping geometries and loading conditions including dynamic
loads including earthquake effect?
iii) Can the developed formulations be verified against both finite
element analysis and experimental results under static and
dynamic loading conditions?
iv) What is the impact of new formulations developed in this dissertation
on pipe integrity analysis and future LBB designs?
The first new formulation has been derived for a one-dimensional pipe
subjected a pressure induced bending that is considered in the earlier studies.
Based on the compliance approach, the formulation was then extended to the
three-dimensional piping system and other types of loading conditions
including the distributed load and relative displacement of supports.
To verify the developed formula, a series of finite element analysis
was conducted for the static and dynamic loading conditions. The static
analyses were performed to evaluate the amount of restraint considering the
anticipated loads of the normal operating conditions. In addition, the crack
stability analysis assumes the faulted dynamic loading condition in which the
seismic load is considered. Furthermore, the dynamic analysis using cracked
pipe model accompanying the comparisons with experimental data also
conducted to demonstrate that restraint coefficient is also available for
transient loading conditions. As results, it is confirmed the generalized
analytical formulations, finite element analysis and experimental data agree
iv
with each other very well in all examined conditions.
Finally, using the developed formula, the effect of restraint on the LBB
evaluation was investigated. All the analysis results of this dissertation
indicated that the restraint effect on the applied moment has more significant
influence on the crack stability evaluation than on COD. Therefore, the
current LBB evaluation procedure, with no attention to the pipe restraint
effect, can predict conservative results compared to the case in which the
restraint effect is considered for the conditions examined herein.
The developed formulation has two implications of the practical
significance. First, if the restraint effect is implemented into the current
practice of deterministic LBB analysis using the developed formulations, the
piping system can be shown to possess greater safety margins. Second, the
time history analysis of the piping system for various crack length can be
replaced with a single uncracked pipe system analysis with the restraint
coefficient without sacrificing accuracy at the significant saving in time and
cost. Therefore, the generalized formulations developed in this dissertation
can greatly help improve the applicability of the probabilistic fracture
mechanics analysis and/or seismic fragility analysis that otherwise require a
significant number of time-consuming calculations.
Keywords: Pipe restraint effect, Leak before break, Crack opening
displacement, Crack stability, Dynamic analysis for cracked pipe, Effective
applied moment formulation
Student Number: 2014-30195
v
Contents
Abstract ...................................................................................... i
Contents ..................................................................................... v
Chapter 1 Introduction ............................................................ 1
1.1 Pipe integrity and the safety of nuclear power plants ......... 1
1.2 Pipe integrity evaluation methods for Leak-Before-Break
design ................................................................................. 3
1.3 Effects of restraint on cracked pipe behavior ...................... 6
1.4 Effective applied moment at cracked section for evaluation of
the restraint effect ................................................................ 9
Chapter 2 Literature Review ................................................ 15
2.1 Effects of pipe restraint on crack stability ......................... 15
2.1.1 Theoretical evaluations .................................................... 15
2.1.2 Experimental observations .............................................. 17
2.2 Effects of restraint of pressure induced bending on COD
evaluation .......................................................................... 18
2.2.1 Investigation of restraint effects on COD using finite
element analysis ............................................................... 18
2.2.2 Numerical expressions of restraint effects on COD ........ 20
2.2.3 Efforts to expand the applicability................................... 21
2.3 Effects of restraint on pipe integrity assessment ............... 23
2.4 Methodology of dynamic analysis for cracked pipe ......... 24
2.4.1 Nonlinear spring model ................................................... 24
2.4.2 Connector element model ................................................ 25
vi
Chapter 3 Rationale and Approach ...................................... 39
3.1 Research rationale from gaps in the literature .................. 39
3.2 Research questions and approaches .................................. 41
Chapter 4 Development of Generalized Formulations on
Effective Applied Moment ..................................................... 45
4.1 Effective applied moment formulation for pipe subjected to
pressure induced bending .................................................. 47
4.2 Compliance approach to improve the formulation ........... 51
4.2.1 Compliance approach ...................................................... 51
4.2.2 Application of compliance approach to 1D pipe subjected to
pressure induced bending ................................................ 52
4.3 Development of generalized formulation.......................... 56
4.3.1 Consideration of the types of applied loading ................. 57
4.3.2 Consideration of the complex piping configurations ...... 60
4.4 Evaluation procedure to determine effective applied moment
68
Chapter 5 Validation of Developed Formulations ............... 87
5.1 Validation under static loading conditions ........................ 88
5.1.1 Evaluation of PIB restraint effects on COD for 1D pipe . 88
5.1.2 Evaluation of effective applied moment for 3D pipe under
static loading conditions .................................................. 97
5.2 Validation under dynamic loading conditions ................. 100
5.2.1 Benchmark dynamic analysis using cracked pipe model100
5.2.2 Validation of developed formulations using experimental
measurements and dynamic analysis results ................. 106
5.2.3 Evaluation of effective applied moment for 3d pipe under
dynamic loading conditions ........................................... 109
vii
Chapter 6 Application of Developed Formulations .......... 143
6.1 Applicability of developed formulations in LBB design 144
6.1.1 Validation methods ........................................................ 144
6.1.2 Validation results of COD and J-integral ...................... 146
6.2 Effects of pipe restraint on LBB evaluation .................... 149
6.2.1 Piping evaluation diagram ............................................. 149
6.2.2 Evaluation methods of the pipe restraint effects on LBB150
6.2.3 Evaluation results .......................................................... 152
Chapter 7 Conclusions and Future Work .......................... 177
7.1 Summary and conclusions ............................................... 177
7.2 Future work ..................................................................... 181
Bibliography ......................................................................... 185
Abbreviation ......................................................................... 195
초 록 ................................................................................. 199
viii
List of Tables
Table 2.1 Differences in leakage crack length and maximum stress due
to restraint of pressure induced bending (Ghadiali et al., 1996)
............................................................................................... 27
Table 5.1 Dimensionless function H4B and H4T in the formula of rotation
due to crack for circumferential through-wall cracked pipe
determined from FEA ........................................................115
Table 5.2 Loading conditions, material property, and pipe geometries
considered for verification of the developed formulation
under the static loading conditions ....................................116
Table 5.3 The material properties applied to the validation analysis of
the experiment 1-1 IPIRG-2 program (ASME, 2010a) .....117
Table 5.4 Comparisons of the natural frequencies of IPIRG-2 piping
system between measured data and FE analysis results ....118
Table 6.1 Matrix of analysis for calculation of COD and J-integral 154
Table 6.2 Matrix of analysis and material properties used for LBB
evaluations ........................................................................ 155
ix
List of Figures
Figure 1.1 Timeline of major events regarding pipe integrity evaluations
............................................................................................... 10
Figure 1.2 Fracture mechanics procedure for leak before break analysis
.............................................................................................11
Figure 1.3 Structure of PRO-LOCA PFM code (Scott et al., 2010)... 12
Figure 1.4 Module structure of xLPR code (Rudland et al., 2015) .... 13
Figure 1.5 Decoupled processes for calculation applied moment and
crack analysis, and restraint effect in the current LBB analysis
............................................................................................ 14
Figure 2.1 Conceptual diagram of the piping system of IPIRG program
experiment 1.3-7 (Schmidt et al., 1992) ............................. 28
Figure 2.2 Net-Section-Collapse analyses predictions, with and without
considering induced bending, as a function of the ratio of the
through-wall crack length to the pipe circumference (Schmidt
et al., 1992) ......................................................................... 29
Figure 2.3 Schematic diagram of a restrained pipe containing a
circumferential through-wall crack and finite element model
(Rahman et al., 1995a) ........................................................ 30
Figure 2.4 Effects of restraint on COD for various restraint lengths and
half angle of circumferential TWC calculated from linear
elastic analysis (Rahman et al., 1995a) ............................... 31
Figure 2.5 Effects of restraint on COD for various restraint lengths and
half angle of circumferential TWC calculated from elastic-
plastic analysis (Kim, 2004) ............................................... 32
Figure 2.6 Statically indeterminate beam model with reduced-thickness
pipe section representing a circumferential crack used to
develop the restrained COD formulation (Miura, 2001) .... 33
Figure 2.7 The schematic diagram of non-linear spring model for
simulation of the crack (Olson et al., 1994)........................ 34
Figure 2.8 Parallel spring-sliders model for simulation of a multi-linear
x
load-displacement curve (Olson et al., 1994) ..................... 35
Figure 2.9 The schematic diagram of connector-beam model for
simulation of the crack and application for piping system
model (Zhang et al., 2010) .................................................. 36
Figure 2.10 Photographs of the 1/3 scale PLR pipe model of JNES’s
experiment (Suzuki and Kawauchi, 2008).......................... 37
Figure 3.1 Diagram of research process ............................................. 44
Figure 4.1 The concept of effective applied moment at the cracked
section ................................................................................. 73
Figure 4.2 Beam model of fixed-ended pipe with a circumferential
crack subjected to a pressure induced bending for
development of the moment restraint coefficient ............... 74
Figure 4.3 Schematic descriptions of the compliance approach ........ 75
Figure 4.4 Beam model and free body diagram of fixed-ended pipe with
a circumferential crack subjected to a pressure induced
bending for development of the moment restraint coefficient
based on the compliance approach ..................................... 76
Figure 4.5 Beam model and free body diagram of fixed-ended pipe with
a circumferential crack subjected to a distributed load for
development of the moment restraint coefficient based on the
compliance approach .......................................................... 77
Figure 4.6 Beam model and free body diagram of fixed-ended pipe with
a circumferential crack subjected to a relative displacement
of the supports for development of the moment restraint
coefficient based on the compliance approach ................... 78
Figure 4.7 Beam model and free body diagram of 2D piping system
containing a circumferential crack for development of the
restraint coefficient based on the compliance approach ..... 79
Figure 4.8 Beam model and free body diagram of the generalized 3D
piping system containing a circumferential crack for
development of the restraint coefficient based on the
compliance approach .......................................................... 80
Figure 4.9 The procedure for calculation of the effective applied
moment and force ............................................................... 81
Figure 4.10 Effect of nonlinear behavior on the applied moment at the
xi
cracked section .................................................................... 84
Figure 5.1 Comparisons of rCOD,LE predicted using the developed
formulations and linear elastic FEA – symmetric model
(Miura, 2001) .....................................................................119
Figure 5.2 Comparisons of rCOD,LE predicted using the developed
formulations and linear elastic FEA – asymmetric model
(Miura, 2001) .................................................................... 120
Figure 5.3 3D FE model of a circumferential through-wall cracked pipe
used for tabulations of new dimensionless functions (H4T, H4B)
.......................................................................................... 122
Figure 5.4 Comparisons of rCOD,EP predicted using the developed
formulations and elastic-plastic FEA – symmetric model
(Kim, 2008) ....................................................................... 123
Figure 5.5 3D FE model of 3D piping system containing a
circumferential through-wall crack used for verification of the
developed formulation ...................................................... 125
Figure 5.6 FE model using beam element of 3D piping system to
calculate the pipe compliance for verification of the
developed formulation ...................................................... 125
Figure 5.7 Comparisons of applied moment and axial force at the
cracked section calculated from finite element analysis .. 126
Figure 5.8 Comparisons of applied nominal stress at the cracked section
due to bending moment and axial force calculated from finite
element analysis ................................................................ 126
Figure 5.9 Comparisons of the restraint coefficient and the ratio of load
reduction calculated from finite element analysis ............ 127
Figure 5.10 FE model used for analysis of simulated seismic pipe
system analysis of IPIRG-2 program ................................ 128
Figure 5.11 Applied moment and rotation due to the crack of
experimental results and input data used for connector
element behavior ............................................................... 129
Figure 5.12 Comparisons of applied moment time history at the cracked
section between experiment and analysis result ............... 130
Figure 5.13 Comparisons of reaction load time history at Node 6
between experiment and analysis results .......................... 131
xii
Figure 5.14 Comparisons of displacement load time history at Elbow 3
between experiment and analysis results .......................... 132
Figure 5.15 Comparisons of displacement load time history at Node 21
between experiment and analysis result ........................... 133
Figure 5.16 3D FE model of pipe containing a surface crack to calculate
the compliance of a crack ................................................. 134
Figure 5.17 Elastic-plastic compliance of the surface crack (Equivalent
crack length of (θ/π) = 0.383) ........................................... 134
Figure 5.18 Applied moment at cracked section calculated from
uncracked pipe analysis for experiment 1-1 of IPIRG-2
program ............................................................................. 135
Figure 5.19 Applied moment and rotation due to the crack applied as
the behavior of connector element .................................... 136
Figure 5.20 Geometries of containment building of OPR-1000 type
plant and FE model (Kim, 2014) ...................................... 137
Figure 5.21 Schematic diagram of procedures and methods to calculate
the effective applied moment to validation of the developed
formulation........................................................................ 138
Figure 5.22 Acceleration response spectrum obtained from containment
building analysis (Kim, 2014) .......................................... 139
Figure 5.23 Displacement time histories of two selected locations
obtained from containment building analysis (Kim, 2014)
.......................................................................................... 140
Figure 5.24 Relative displacement time histories between two selected
locations obtained from containment building analysis (Kim,
2014) ................................................................................. 140
Figure 5.25 Comparisons of the reduction ratios of the applied moment
at the cracked section predicted using the time history
analysis, restraint coefficient compared with the current
practice of LBB ................................................................. 141
Figure 6.1 Summary of analysis case to demonstrate the applicability
of the restraint coefficient in LBB design ......................... 156
Figure 6.2 3D FE model of pipe with a circumferential through-wall
crack used for COD and J-integral calculations ............... 157
Figure 6.3 Tensile property of TP316 stainless steel ........................ 157
xiii
Figure 6.4 Comparisons of COD to validate the restraint coefficient
(Internal pressure was not included) ................................. 158
Figure 6.5 Comparisons of J-integral to validate the restraint coefficient
(Internal pressure was not included) ................................. 161
Figure 6.6 Comparisons of COD to validate the restraint coefficient
(Internal pressure was included) ....................................... 164
Figure 6.7 Comparisons of J-integral to validate the restraint coefficient
(Internal pressure was included) ....................................... 167
Figure 6.8 Schematic diagram of the piping evaluation diagram..... 170
Figure 6.9 Effect of the restrained COD and the effective applied
moment on LBB evaluation .............................................. 171
Figure 7.1 Structure of eXtremely Low Probability of Rupture Code
Version 2.0 (US NRC, 2015) ............................................ 183
xiv
1
Chapter 1 Introduction
1.1 Pipe integrity and the safety of nuclear power plants
From a safety perspective, the nuclear energy should be pursued with the
protection of people and the environment against radiation risks as stated in
the statute of the International Atomic Energy Agency (IAEA). The aging
management of systems, structures and components of nuclear power plant
(NPP) is the key factor for the safe and reliable long-term operation and
economic viability of the plant (IFRAM, 2015). The integrity of the nuclear
piping system, in particular, should be maintained throughout the lifetime of
NPPs because the rupture of pipe can cause the release of radioactivity and
also negatively impact on other safety components.
The design of the NPPs, thus, conservatively assumes the anticipated
loading for normal operating conditions and design basis accidents, and also
the safety and integrity are re-evaluated at regular intervals via the periodic
safety review (PSR). Nevertheless, the unexpected flaws that are exceeding
the criteria in the American Society of Mechanical Engineers (ASME) Boiler
and Pressure Vessel Code Section XI (ASME, 2010c) have been discovered
in the pressure boundary components (Grimmel and Cullen, 2005). As
attentions to the beyond design basis accident have intensified following
Fukushima Daiichi accidents, the refinement and verification of the
methodologies for current analysis of structural integrity have emerged as one
2
of top priority issues in the safety assessment of nuclear power plants,
especially for long-term operation. Particularly, the confidence of predicting
high pressure nuclear piping behavior with postulated cracks became a safety
issue of practical significance. Therefore current practice for the pipe integrity
evaluation is reexamined.
3
1.2 Pipe integrity evaluation methods for Leak-Before-Break
design ①
Until the early 1980s, a major consideration in the design of the NPP was a
double-ended guillotine break (DEBG) of a piping system containing a
circumferential crack. With the anticipated accident propagation upon high
energy piping rupture and the lack of knowledge on the fracture mechanical
behaviors, DEGB assumption has been introduced as conservative design
basis adopted in design rules (Wilkowski et al., 1998).
The pipe evaluation methodology has become more sophisticated with
the improvement of the fracture mechanics and experiences with severe
cracking, as shown by the timeline of major events summarized in Figure 1.1.
LBB approaches were proposed based on the fact that through-wall cracks
with significantly high water leak rate often retain adequate margin to DEGB.
The detailed leak rate analysis considering two-phase flow through a crack
and elastic-plastic fracture analysis demonstrated that the probability of a
DBEG is significantly small given the high confidence of the detectable
leakage from a subcritical crack. (US NRC, 1985). This encouraged the
development of LBB methodology for some of screened piping with the
absence of generic crack growth mechanisms so that excessive conservatism
① This section has been based on the following journal and conference papers:
Kim, Y., Hwang, I.-S., Oh, Y.-J., 2016a. Effective applied moment in circumferential
through-wall cracked pipes for leak-before-break evaluation considering pipe restraint
effects. Nuclear Engineering and Design 301, 175-182.
Kim, Y., Oh, Y.-J., Park, H.-B., 2015. Effect of Pipe Restraint on the Conservatism of Leak-
Before-Break Design of Nuclear Power Plant, ASME 2015 Pressure Vessels and Piping
Conference. American Society of Mechanical Engineers, pp. V06AT06A082-
V006AT006A082.
4
with qualified piping in design can be rationalized by excluding the effect of
DEBG. On this ground, the method of leak-before-break (LBB) is widely
implemented for nuclear piping systems as a means for assuring the piping
integrity when pipe whip restraints are removed, as described in the Standard
Review Plan 3.6.3 (US NRC, 2007b).
The procedure of LBB is schematized in Figure 1.2. At a determined
critical location of the pipe, the length of the leakage size crack (LSC) can be
predicted, considering crack opening displacement (COD) and normal
operating conditions. Then, based on the elastic-plastic fracture analysis, the
bending moment for commencing instability is determined for the given LSC.
The design applied moment under faulted conditions at the postulated critical
location must be lower than the calculated instability moment in order to
satisfy the LBB requirements. In the practical LBB procedure, applied loads
that are calculated from the piping design are used for input in the through-
wall crack analyses. Then, the value of COD and the instability moment are
determined with the assumption that cracked pipe is subjected to the
calculated applied load with its both ends unconstrained (US NRC, 1985,
2007b).
The early methodology of LBB considered the parameters such as the
applied loading, pipe geometry, material properties and cracking mechanism
as a determined value. To deal with the uncertainties, there were extensive
studies to develop the probabilistic fracture evaluation methodology. Rahman
et al. (1995b) proposed the procedure to consider the probability distribution
of parameters while applying the same evaluation method with the
5
deterministic LBB.
Meanwhile, two separate methods to estimate the probability of pipe
rupture were developed in the similar period. First, for the purpose of the re-
evaluation of the emergency core cooling system (ECCS), the PRO-LOCA
probabilistic fracture mechanics (PFM) code was developed (Scott et al.,
2010). Second, the eXtremly Low Probability of Rupture (xLPR) was also
developed to address environmental degradation mechanisms to LBB
approved pipe (Rudland et al., 2015). The basic structures of two codes are
described in Figure 1.3 and Figure 1.4, respectively, in which the crack
detection or COD and crack stability analysis are the essential elements.
Based on these various pipe fracture evaluation methods, the next
milestone is the improvement of safety margin and accuracy by refining
current analytical models and there are several aspects that may influence on
the prediction of crack instability and COD. This dissertation is focused on
one of high priority issues, the pipe restraint effect on the crack behavior.
6
1.3 Effects of restraint on cracked pipe behavior②
The conservatism of the pipe analysis depends on the assumptions and the
methodologies for evaluation of the COD, and the allowable moment of a
pipe containing a circumferential crack. Generally, an applied moment of pipe
at the position of a postulated crack under normal operating or transient
conditions used as the input for the LBB calculations is obtained from the
analysis results on an uncracked piping system as shown in Figure 1.5 (Scott
et al., 2002; Wilkowski et al., 1998). If the piping system contains a crack,
however, the crack driving force is reduced because the stiffness of piping
system lowered due to the crack opening behavior.
Furthermore, in the current procedures, the effect of pipe restraints is
not taken into account in the calculation of the COD and the allowable
moment. But the restraint of pipe can limit the behavior of a crack and
consequently result in the decrease of applied moment. Accordingly, the
current procedure can overestimate the value of the COD and underestimate
the allowable moment. With regards to the conservatism of the pipe integrity
evaluation, these two factors with different influences must be determined in
a consistent manner.
First, an overestimation of COD value can lead to an underestimation
of the leakage size crack under the same operating conditions where net
② This section has been based on the following journal paper:
Kim, Y., Hwang, I.-S., Oh, Y.-J., 2016a. Effective applied moment in circumferential
through-wall cracked pipes for leak-before-break evaluation considering pipe restraint
effects. Nuclear Engineering and Design 301, 175-182.
7
results may result in a non-conservative LBB analysis. In order to evaluate
the effect of pipe restraint on the COD taking into account an internal pressure
effect on bending deformation, a series of finite element analysis (FEA) were
conducted using both linear elastic and elastic-plastic modelings (Kim, 2004;
Rahman et al., 1995a; Rahman et al., 1996; Rahman et al., 1998; Scott et al.,
2005b). Based on the finite element analysis results, the equations for
calculating restrained COD then were developed on the basis of linear elastic
(Miura, 2001; Scott et al., 2005a) and elastic-perfect plastic model (Kim,
2007) for a straight pipe with fixed ends. To improve the applicability, an
effort to use the system stiffness to calculate the unrestrained COD are
followed (Young and Olson, 2015). However, in these studies, the effects of
pipe restraint were only focused on the crack opening displacement not the
load-carrying capacity of a crack.
In case of the crack stability analysis, an overestimation of an applied
moment at cracked section can lead to an underestimation of the allowable
moment for a given LSC, in which net effect may yield conservative results.
E. Smith (1985a, 1985b, 1988a, 1988b, 1990a, 1990b, 1995, 1997, 1999a,
1999b, 2002, 2003) has investigated the instability criterion for growth of a
circumferential through-wall crack, considering various pipe and crack
geometry including boundary conditions. The equations for evaluating the
unstable crack growth were developed for various piping systems and loading
conditions case by case, but the generalized formula which can be employed
practical procedure was not derived.
8
As stated above, the pipe restraint has been the primary factor that
should be considered to enhance the accuracy of the pipe evaluations. While
the restraint effects have been investigated extensively from various
considerations, for practical applications a consistent method is called for to
evaluate both crack stability and COD, warranting further studies.
9
1.4 Effective applied moment at cracked section for
evaluation of the restraint effect
To maintain the consistency of the conservatism of pipe integrity analysis, the
crack and pipe restraint effects should be considered in both COD calculation
and crack stability analysis simultaneously. Earlier studies, however, dealt
with these effects separately just for the limited pipe configurations. These
are simultaneous phenomena caused by the decreases in the applied moment
at the cracked section due to the pipe restraint. Therefore, it is desired to
develop a unified formulation to determine the effective applied moment at a
postulated cracked section considering the boundary conditions that can be
utilized to a balanced analysis of both COD and flaw stability.
In this regards, this dissertation mainly is focused on the development
of generalized solutions that readily enable the evaluations of the restraint
effect on the applied moment at a cracked section for complex configurations
of pipe and/or boundary conditions. It is expected that the generalized
formulation can be applicable to both deterministic and probabilistic pipe
fracture evaluations, and further to the structural analysis for the seismic risk
assessment.
10
Figure 1.1 Timeline of major events regarding pipe integrity evaluations
1970
1980
2000
2002
2002
2012
Deterministic
Fracture Mechanics(FM)
Based Evaluation
Through wall crack detection
and Leakage
(V.C. Summer Plant)
PRObabilistic
-Loss Of Coolant Accident
Through wall crack detection
and Leakage
(Davis Besse, Crystal River)
eXtremely Low Probability
of Rupture
Leak Before Break
(exclusion of active degradation)
Introduction of
Probabilistic FM
11
Figure 1.2 Fracture mechanics procedure for leak before break analysis
Crack Opening
Displacement
Leak rate
Crack Stability
Analysis
Leakage size
crack, θl
LBB satisfied
MNOP(Normal operation)
Crack length
θ
Mfault(Faulted condition)
Leak rate =
Detectable?
Stable?
2θ
12
Figure 1.3 Structure of PRO-LOCA PFM code (Scott et al., 2010)
Realization Module
Parameter Sampling
Time increment
Input
T>Tfinal
Crack Initiation Module
Crack Growth Module
Crack Stability Module
Lear Rate/Inspection
Module
13
Figure 1.4 Module structure of xLPR code (Rudland et al., 2015)
Sampling Structure
Parameter Sampling
Input
Time loop
Crack Initiation
Sampling loop
Crack Propagation
Mitigation
Crack Detection
Probability
14
Figure 1.5 Decoupled processes for calculation applied moment and crack analysis, and restraint effect in the current
LBB analysis
Calculating Applied moment (MNOP, Mfault)
Linear elastic pipe analysis w/o crack
COD
M J-integral
Unrestrained pipe
COD?
MeffJ-integral?
Restrained pipe
Cracked Pipe Analysis for COD/Crack Stability Evaluation
Critical Location
15
Chapter 2 Literature Review
2.1 Effects of pipe restraint on crack stability
2.1.1 Theoretical evaluations
The potential impact of restrained boundary conditions includes the positive
influence on the pipe instability prediction and the underestimation of the
crack opening displacement that is detrimental to the prediction of leakage
size crack. The former aspect was pointed out initially through the efforts of
E. Smith (1984) for extension of the stability analysis of a circumferential
through-wall crack proposed by Tada, Paris and Gamble (Paris et al., 1979;
Tada et al., 1980). Smith tried to apply the tearing modulus approach
developed for a pipe subjected to uniform bending to more practical cases
related with the Boiling Water Reactor piping system. When a straight pipe
of length L and radius R containing a through-wall circumferential crack of
angle 2θ at the center is subjected to displacement controlled uniform bending,
the applied tearing modulus (TAPP) can be represented as
1 22APP
f
L EJT F F
R R
(2.1)
where E, J, and σf are the elastic modulus, J-integral, and flow stress,
respectively(Smith, 1984). F1(θ) and F2(θ) depend upon the crack angle. To
16
extend this approach, the pipe length of Eq. 2.1 was replaced with the
effective pipe length that depends on the crack location, the loading
conditions and the configuration of piping systems. Then the effective lengths
were derived for various situations that might exist in actual piping systems
in power plants (Smith, 1985a, b, 1988a, b, 1990a, b).
Meanwhile, E. Smith (1992) also used the concept of effective pipe
length to express the degree of conservatism of the net-section instability
criterion for estimation of onset of crack extension. He emphasized that the
net-section stress approach can provide conservative results since the applied
stresses at the cracked section is calculated from linear elastic analysis using
a uncracked piping system and the degree of conservatism depends on the
effective pipe length, LEFF. Through a series of research, he has tried to
quantitatively evaluate this effect considering the effect of restraint, crack
position, piping geometrical parameters, and system nonlinearity (Smith,
1995, 1997, 1999a, b, 2002, 2003) case by case. But the methodology to apply
generally the effective pipe length was not suggested.
Among the studies on the development of the effective pipe length, it
is noteworthy that various kinds of applied loading conditions were also
considered. By analyzing the case of a piping system subject to a dead weight,
thermal load, pressure and relative displacement due to seismic events, it was
figured out that the instability criterion using LEFF can be applied irrespective
of loading types (Smith, 1989). More recently, additional investigation
considering time-dependent loading led to the conclusion that inertial loading
arising from earthquake also can be taken into account in the same manner in
17
crack stability analysis(Smith, 1996).
2.1.2 Experimental observations
The effect of restraint on the stability of crack was also confirmed
experimentally through the International Piping Integrity Research Group
(IPIRG) program (Schmidt et al., 1992; Scott et al., 1997). In the IPIRG
experiment 1.3-7, an attempt was made to make a near instantaneous break in
the piping system including surface crack as shown in Figure 2.1. The length
of through-wall crack ligament when double ended guillotine break occurs in
an experiment using the piping system under the pressure loading condition
was found to be significantly longer than the value calculated based on the
net section collapse analyses predictions. It was confirmed that when the fixed
ends are not taken into account, the predicted critical crack length matched
with the experimental value (See Figure 2.2), because the restraint of bending
can increase the load-carrying capacity. This effect can also be verified by the
variations in the accuracy of pipe stability evaluation for pure bending, and
combined bending and pressure-induced tensile loads experiments,
respectively (Wilkowski et al., 1998).
18
2.2 Effects of restraint of pressure induced bending on COD
evaluation
When it comes to the leak before break (LBB) analysis, the crack opening
displacement (COD) is an essential element to calculate the leakage size
crack(US NRC, 1985, 2007b). Because if the pipe ends are not allowed to
rotate freely, the crack opening displacement can be reduced so that the
restraint effect emerged as a significant issue in improving the accuracy of
LBB analysis. This section describes several earlier studies to investigate the
restraint effect on COD and to develop evaluation models for determination
of restrained COD.
2.2.1 Investigation of restraint effects on COD using finite
element analysis
For the purpose of refinement the LBB evaluation methodology, S. Rahman
et al. quantitatively investigated the effects of restraint in case of pressure
induced bending (PIB) which is one of the practical aspects of crack opening
displacement estimation initially (Rahman et al., 1995a; Rahman et al., 1996;
Rahman et al., 1998). Linear-elastic finite element analysis was conducted to
quantify the magnitude of restraint effect using a 3D model of pipe containing
a circumferential crack in the center as illustrated in Figure 2.3. To simulate
the boundary conditions such as piping connected to other components, the
rotations and ovalizations at pipe ends were assumed to be prohibited. The
analysis was also done for the case when the pipe is free to rotate, the amount
19
of restraint effect was then represented as the restrained COD normalized by
unrestrained COD. The results revealed that the restraint effect of PIB
increases as the crack arc length increases and the normalized distance from
the crack to the restraint decreases; i.e., the distance from the crack to fixed-
end normalized by the pipe diameter (see Figure 2.4).
More recently, a round robin analysis was conducted as a part of the
Battelle Integrity of Nuclear Piping (BINP) program (Scott et al., 2005b). The
BINP round robin analysis was designed to check and expand the results of
earlier calculations. All participants were assigned to calculate the crack
opening displacement for restrained and unrestrained cases by linear elastic
FEA. The greater variety of pipe diameters and thicknesses, crack lengths and
restraint lengths were considered as the analysis matrix including the case that
is defined in the study of Rahman et al. (1995a) as well. Major findings were
i) the pipe mean radius to thickness ratio (Rm/t) has more significant
influences on the restraint effect than the pipe diameter, and ii) the restraint
effect for the case when the restraint length on both sides of a crack are
different (asymmetric case) is significant than the symmetric case. The results
of this round robin were used to derive an evaluation model.
Kim (2004) emphasized that elastic-plastic COD evaluation is
required from practical aspects of pipe integrity analysis because a through-
wall crack can deform plastically under the normal and accident conditions.
The results of elastic-plastic FEA showed the same tendency of restrained
effect regarding pipe geometries, crack length and restraint conditions. It
should be noted that when the effects of restraint of PIB considering plastic
20
behavior are considerable compared with the linear elastic analysis, and the
degree of the effects strongly depends on the magnitude of internal pressure
(applied tensile stress) as shown in Figure 2.5.
2.2.2 Numerical expressions of restraint effects on COD
To reflect the restraint effect on the leak before break analysis, several studies
were aimed at developing the analytical solutions to evaluate the restrained
crack opening displacement. N. Miura (2001) developed the evaluation
method for linear elastic COD using the statically indeterminate beam model
including the reduced-thickness section which represents the cracked section
(See Figure 2.6). This model postulated a concentrated vertical load at the
cracked section to represent the pressure induced bending moment. Then he
derived the change of slope at reduced thickness section (rotation due to crack)
which is proportional to the COD. Then COD reduction ratio was derived by
normalizing the rotation due to the crack by that for the case of the
unrestrained pipe. The developed solution was verified with linear elastic
FEA results in which the Paris-Tada formula (Paris and Tada, 1983) was used
for the COD calculations.
In the BINP research program (Scott et al., 2005a), Miura’s model was
improved to be applicable to wide range of the pipe configurations using the
round robin analysis results. In addition to that, Kim (2007) derived the
analytical expression based on the results of elastic-perfectly plastic FEA, and
investigated the impact of restraint effect on COD calculation for the primary
21
piping system of a pressurized water reactor plant (Kim, 2008).
2.2.3 Efforts to expand the applicability
The primary parameter of the evaluation methods introduced in section 2.2.2
is the restraint length normalized by pipe mean diameter (LR/Dm) that is
restricted to the case of fix-ended straight pipe containing a circumferential
TWC. In virtually, the configuration of the piping system are complicated
including elbows, hinge and supports, and therefore, it is difficult to
determine LR/Dm.
In this regards, the BINP program(Scott et al., 2005a) replaced the
restraint length with pipe rotational stiffness (k) which is defined as
applied moment
bending angle
Mk
(2.2)
Then a series of FEA was conducted to derive the relation of k and
LR/Dm. The restrained COD for a complex pipe can be calculated based on the
following procedure.
i) Make a beam model with a hinge representing the crack.
ii) Fix the rotation of the left or right side of the hinge and apply a unit
moment on the opposite side.
iii) Calculate bending angle, and determine k for both sides.
iv) Replace k with LR/Dm and substitute in the COD solutions
22
Above steps were suggested as a method of implementation of a
solution for the 1D pipe to a 3D piping system. Young and Olson (2015)
introduced a similar approach to obtain effective elastic modulus that is the
ratio of the rotational stiffness of target piping system to that of unrestrained
pipe with a crack. These, however, will tend to inaccurately predict because
the deflections or rotations at a specific point in the 3D pipe system are
produced due to a combination of three-directional loads and moments.
Therefore, an improvement from this aspect is required for practical
application of restraint effect on pipe integrity evaluations.
23
2.3 Effects of restraint on pipe integrity assessment
As stated in preceding section, when the pipe is restrained, CODs and crack
driving forces decrease simultaneously compared with the case of the
unrestrained case because the deformation of the crack is limited. Moreover,
the effect on COD decreases the margin of the LBB design, but the effect on
the load carrying capacity increases the margin. Thus, to keep the constancy
of design conservatism, the reduction of the crack driving force needs to be
considered if the restraint effect on COD is accounted for in the LBB design.
These combined results of restraint effects were investigated by
example LBB calculations (Ghadiali et al., 1996) from both deterministic and
probabilistic basis. The results revealed that the effects of restraint on
maximum load are significant then effects on COD calculation, and this is
prominent in small diameter pipe (see Table 2.1). This studies only considered
small and large diameter pipe (4.5, 28 inch), and thus in Chapter 6 of this
dissertation deals with the intermediate pipe sizes that are typically used in
the reactor coolant system of pressurized water reactor.
24
2.4 Methodology of dynamic analysis for cracked pipe
The crack stability analysis is a process that demonstrates that a postulated
crack does not grow unstably even under the accident conditions. The primary
applied loading in faulted conditions arises from an earthquake, and thus in
this dissertation, the seismic analyses for cracked pipe were conducted to
calculate the maximum applied load on cracked section for the purpose of
validation of the applicability of developed solutions. This section describes
the analysis technique for cracked pipe under the dynamic loading conditions
implemented in the earlier researches.
2.4.1 Nonlinear spring model
During the IPIRG program, the nonlinear time history analyses for cracked
pipe were performed to validate the results of surface cracked pipe under the
simulated seismic loading. Since the employment of 3-D solid element can
be extremely time consuming, Olson et al. (1994) developed the simplified
analysis technique using a beam element for pipe and non-linear spring for
crack. Figure 2.7 shows the schematic diagram of the model that consists of
the following elements.
i) Hinge: The element for joining two nodes at the crack point and
allowing the user-defined moment-rotation behavior
ii) Parallel spring-sliders (Figure 2.8): Each spring-slider have different
stiffness and frictional properties, thus the combination of these
25
can behave along the nonlinear moment-rotation curve of a crack.
iii) Break-away element: If the applied moment reaches the maximum
value, this element is removed and does not contribute to the
behavior, so that the transition from surface crack to through-wall
crack can be simulated.
iv) Plastic pin-connected truss: The plastic behavior can be assigned to
this element for simulating the behavior the through-wall crack.
2.4.2 Connector element model
The Engineering Mechanics Corporation of Columbus (EMC2) proposed a
new technique for the simulation of a circumferential crack utilizing a
connector element of ABAQUS (Zhang et al., 2010). One can assign the
moment-rotation curve to a “single” connector element including elastic-
plastic behavior and the simultaneous decrease of load-carrying capacity due
to the crack growth as described in Figure 2.9. The validation analysis
comparing with the pipe experiment under the different loading conditions
led to the conclusions that the use of connector element is very convenient
and accurate to model the crack behavior as a part of beam analysis compared
with existing nonlinear spring model.
This modeling approach using connector element was implemented
the dynamic FEA for analyzing of the experiment conducted by the Japan
Nuclear Energy Safety Organization (JNES). The JNES used a 1/3 scale of
primary loop recirculation system piping containing three surface cracks
26
illustrated in Figure 2.10 and applied uniform excitation using a single
shaking table (Suzuki and Kawauchi, 2008; Suzuki et al., 2006). The results
revealed that the calculated displacements and damage responses have a good
agreement with measured values.
27
Table 2.1 Differences in leakage crack length and maximum stress due to
restraint of pressure induced bending (Ghadiali et al., 1996)
Outside Pipe
Diameter Leakage Crack Length, θ/π Restrained/Unrestrained
Maximum Stresses mm inches Restrained Unrestrained
114.3 4.5 0.7250 0.2360 0.1129
711.2 28.0 0.0219 0.0219 1.007
28
Figure 2.1 Conceptual diagram of the piping system of IPIRG program
experiment 1.3-7 (Schmidt et al., 1992)
29
Figure 2.2 Net-Section-Collapse analyses predictions, with and without
considering induced bending, as a function of the ratio of the through-wall
crack length to the pipe circumference (Schmidt et al., 1992)
30
Figure 2.3 Schematic diagram of a restrained pipe containing a
circumferential through-wall crack and finite element model (Rahman et al.,
1995a)
31
Figure 2.4 Effects of restraint on COD for various restraint lengths and half
angle of circumferential TWC calculated from linear elastic analysis
(Rahman et al., 1995a)
32
Figure 2.5 Effects of restraint on COD for various restraint lengths and half
angle of circumferential TWC calculated from elastic-plastic analysis (Kim,
2004)
33
.
Figure 2.6 Statically indeterminate beam model with reduced-thickness pipe
section representing a circumferential crack used to develop the restrained
COD formulation (Miura, 2001)
34
Figure 2.7 The schematic diagram of non-linear spring model for simulation
of the crack (Olson et al., 1994)
35
Figure 2.8 Parallel spring-sliders model for simulation of a multi-linear
load-displacement curve (Olson et al., 1994)
36
Figure 2.9 The schematic diagram of connector-beam model for simulation
of the crack and application for piping system model (Zhang et al., 2010)
37
Figure 2.10 Photographs of the 1/3 scale PLR pipe model of JNES’s
experiment (Suzuki and Kawauchi, 2008)
38
39
Chapter 3 Rationale and Approach
3.1 Research rationale from gaps in the literature
Starting from the conservative design rules, the pipe evaluation methodology
has become increasingly sophisticated in order to enhance the prediction
accuracy and optimize the safety margins as the fracture mechanics was
advanced. The effects of restraint were considered as an important element
that should be incorporated into the piping fracture evaluation. On this reason
there have been extensive studies to quantify the impact of the restraint effects
on two key steps, including the calculation of crack opening displacement and
crack stability analysis.
However, there are still gaps that were not filled by earlier studies.
First, the restraint effects should be considered in both COD calculation and
crack stability analysis simultaneously, because two aspects have different
influences on the conservatism of pipe integrity evaluations.
Second, the generalized formulation is not available to evaluate the
restraint effects irrespective of the piping configurations. So far, despite that
the variations in the crack stability due to the restraints have been derived for
various boundary conditions case by case, it was not generalized to apply to
complex realistic cases. The formulations for restrained COD were developed
40
so far only for the fixed-ended straight pipe.
Third, the restraint effect should be examined considering various
applied loading conditions: dead weight, thermal load and relative motions of
the supports. Although the crack opening displacement is determined under
the normal operating conditions, the solutions for restrained COD considered
only the pressure induced bending. Moreover, the crack stability analysis is
directly concerned with the accident conditions including earthquakes.
Therefore the dynamic loading conditions should also be included.
41
3.2 Research questions and approaches
This dissertation mainly serves to the aims for the development of a
generalized solution to evaluate the restraint effect that can be utilized to
enhance the practical pipe fracture analysis. Although significant efforts have
been made in earlier studies to deal with the restraint effect on the calculation
of COD and crack stability analysis separately, these are simultaneous
phenomena caused by the decreases in the applied moment at the cracked
section due to the pipe restraint. Therefore, it is desired to develop a unified
formulation to determine the effective applied moment at a postulated
cracked section considering the restrained boundary conditions and a
presence of a crack that can be utilized to a balanced analysis of both COD
and flaw stability. The first question of this dissertation stems from this
perspective as following:
How can we analytically evaluate the effective applied
moment at the cracked section taking into account the pipe
restraint effects?
The restraint effects can be measured as the ratio of the applied
moment at the cracked section for a restrained pipe to unrestrained pipe. In
this dissertation this ratio is defined as the restraint coefficient, as follows;
restraint coefficient restrainedrest
unrestrained
MC
M
(3.1)
42
To underpin the concept of the effective applied moment, the restraint
coefficient was derived first for the fixed-ended pipe subjected the pressure
induced bending for the benchmark against earlier studies. Then the
developed restraint coefficient was valideated by comparing with the rCOD
that represents the ratio of COD for the restrained and unrestrained pipe
defined in the earlier studies (Kim, 2007; Miura, 2001) as follows;
r restrainedCOD
unrestrained
COD
COD
(3.2)
Because this is a very specific case in terms of the pipe geometries and
loading conditions, the applicability of the solutions can be enhanced by
setting the following questions:
Can the generalized formulations be applicable to various
types of the piping geometries and loading conditions
including dynamic loads including earthquake effect?
Can the developed formulations be verified against both finite
element analysis and experimental results under static and
dynamic loading conditions?
In reality, the nuclear pipings can be subjected to not only the pressure
induced bending, but also the various types loading including the dead weight,
thermal load and relative displacements of supports. The restraint of pipe can
occur regardless of the loading type, and the pipe configuration as well. In
43
this regards, the restraint coefficient improved in the form of a general
solution can be practically utilized based on the compliance approach.
To verify the proposed formula, a series of finite element analysis was
conducted for the static and dynamic loading conditions. The static analyses
were performed to evaluate the magnitude of restraint considering the
anticipated loads of the normal operating conditions. In addition, the crack
stability analysis assumes the faulted loading condition in which the seismic
load is considered. Hence, the dynamic analysis using cracked pipe model
accompanying the comparisons with experimental data also conducted to
demonstrate that restraint coefficient is also available for transient loading
conditions.
The effects of restraint have a positive impact for the refinement of
the pipe fracture analysis so that the final question of this dissertations is:
What is the impact of new formulations developed in this
dissertation on pipe integrity analysis and future LBB designs?
An example of leak before break analysis was conducted with
consideration of the restraint effect on both COD calculation and crack
stability analysis, for more practical cases than that was considered in the
literature. On this ground, comprehensive research approach taken in this
dissertation is summarized in Figure 3.1.
44
Figure 3.1 Diagram of research process
Dynamic loadStatic load
New Formulation for Meff,app
(Pressure-induced bendingcase benchmark)
Improvement New Formulation based onCompliance approach
Development of Generalized Formulation
Part 1: Development of Effective Applied Moment Formulation
Benchmark Dynamic Analysis using
Cracked Pipe Model
Part 2 : Validation ofDeveloped Formulation
Meff,app Comparisonunder Dynamic
Loading Conditions
Experimental Resultvs Dynamic analysis
vs Formulation
1D piping COD Comparisonwith FEA Results
(Pressure-induced bendingcase benchmark)
Part 3 : Impacts on Pipe Integrity Evaluation
2D piping
3D piping
PIB
Meff,app Comparisonunder Static
Loading Conditions
Other types of loading
Connector
Beam
45
Chapter 4 Development of Generalized Formulations
on Effective Applied Moment
In this Chapter, generalized analytical formulations have been derived in
order to predict the simultaneous changes in both crack opening displacement
and the allowable moment of a crack due to pipe constraint. The generalized
formulations will be derived based on the concept of effective applied
moment.
Figure 4.1 shows a beam model of the fixed-ended straight pipe
containing a circumferential crack. Initially, a concentrated bending moment
(Mapp) arising from operating conditions was assumed to have been generated
without the restraint effects. The bending moment will force to rotate the pipe
about the cracked section. Then the load can be redistributed due to the
reaction forces and moments from fixed-ends, which results in the decrease
of the initially applied moment at the cracked section. The resultant reduced
moment is defined as the effective applied moment (Meff,app). The ratio of
Meff,app to Mapp represents the fractional moment reduction effects of pipe
restraint which is defined as the moment restraint coefficient, Crest.
, (restrained pipe)restraint coefficient
(unrestrained pipe)
eff app
rest
app
MC
M
(4.1)
The moment restraint coefficient, Crest has been developed first for the
case of the pressure induced bending for the benchmark against earlier studies.
Then the validity of moment restraint coefficient was verified for other types
46
of applied loading. Finally, a set of generalized formulations was derived for
complex piping configurations based on the compliance approach. The
effective applied moment calculated using the moment restraint coefficient
can be utilized to evaluate both the COD and the crack instability moment.
47
4.1 Effective applied moment formulation for pipe subjected
to pressure induced bending③
The earlier studies on development of the solutions for the restrained COD
assumed a straight pipe with built-in ends subjected pressure induced bending
conditions (Kim, 2007; Miura, 2001). This case have been selected for
benchmarking to derive the moment restraint coefficient based on the concept
of the effective applied moment.
First of all, the bending moment that can induce exactly same
rotational displacement under the axial tension load established by the pipe
internal pressures was defined as the pressure equivalent moment (MPress,eq)
for a free-ended pipe. Then, it was assumed that MPress,eq is applied in the
opposite direction to the both sides of the cracked section in the direction of
opening the crack. Figure 4.2 shows a beam model, which represents the
fixed-ended pipe with a circumferential crack subjected to a pressure induced
bending. From the earlier studies, the circumferential crack is represented by
a compliant hinge that can rotate like a rotational behavior of a pure crack.
The effect of the axial displacement of the pure crack and the effect of the
axial force on the crack behavior are ignored. Hence, the rotational
③ This section has been based on the following journal and conference papers:
Kim, Y., Oh, Y.-J., Park, H.-B., 2016b. The Conservatism of Leak Before Break Analysis in
Terms of the Applied Moment at Cracked Section, ASME 2016 Pressure Vessels and Piping
Conference. American Society of Mechanical Engineers, pp. V06AT06A075-
V006AT006A075.
Kim, Y., Oh, Y.-J., Park, H.-B., 2015. Effect of Pipe Restraint on the Conservatism of Leak-
Before-Break Design of Nuclear Power Plant, ASME 2015 Pressure Vessels and Piping
Conference. American Society of Mechanical Engineers, pp. V06AT06A082-
V006AT006A082.
48
compliance (Gcrack,ψ,M) was defined for the hinge representing a crack., as
follows:
, ,
cracked pipe uncracked pipe
crack MGM M
(4.2)
where the bend angle of the pure crack (Δψ) is the difference of rotational
angle between the cracked pipe (ψcracked pipe) and uncracked pipe (ψuncracked pipe)
caused for the same amount of bending moment (M).
The equations that describe the deflections (y) and rotations (ψ) of the
regions 1 and 2 of the pipe as a function of the distance from the left anchor
(x) can be obtained based on the elastic beam theory, as follows;
, ,2
1 1
, ,3 2
1 1 2
, ,2
2 3
, ,3 2
2 3 4
2
6 2
2
6 2
React Rest React Rest
React Rest React Rest
React Rest React Rest
React Rest React Rest
F Mx x C
EI EI
F My x x C x C
EI EI
F Mx x C
EI EI
F My x x C x C
EI EI
(4.3)
In Equation 4.3, FReact,Rest and MReact,Rest, respectively, are the reaction
force and the moment on the pipe with the fixed ends which are induced by
restraint. E and I are the elastic modulus of pipe material and the moment of
inertia of the pipe, respectively. In addition, subscripts 1 and 2 represent the
left (region-1) and right (region-2) side of the crack in Figure 4.2, respectively.
The coefficients C1 to C4 and the reaction force (FReact,Rest) and the moment
(MReact,Rest) can be determined by applying boundary conditions for the pipe
49
with fixed ends, as follows;
1
1
2
2
2 1 1 1 , , , 1 ,
2 1 1 1
0 0 a
0 0 b
2 0 c
2 0 d
e
0 f
crack M Press,Eq React Rest React Rest
x
y x
x L
y x L
x L x L G M F L M
y x L y x L
(4.4)
Boundary conditions (a) to (d) of Eq. 4.4 show that the fixed ends of
the pipe do not experience any displacement and rotation. Both displacement
and rotation at the cracked section must be continuous as stipulated by (e) and
(f) of Eq. 4.4. As shown in (e) of Eq. 4.4, the difference in the rotation between
regions 1 and 2, which means the rotation at the cracked section (x=L1), is
determined from the compliance of the crack and the effective applied
moment at the cracked section.
By inserting Eq. 4.4 into Eq. 4.3, the reaction force and the moment
are obtained as follows:
1, ,
, , 1 2
2 13
4 2 12
NReact Rest Press Eq
crack M N N
LF M
L G EIL L L
(4.5)
1, ,
, , 1 2
2 3
2 6
NReact Rest Press Eq
crack M N N
LM M
G EIL L L
(4.6)
where LN1 and LN2 are normalized value of L1 and L2, respectively, by the total
pipe length (2L). Consequently, the effective applied moment at the cracked
section (MPress,Eq,eff) is determined by using the reaction force and the moment,
50
and the pressure equivalent moment MPress,Eq.
, , 1 ,Press Eq Eff Press,Eq React,Rest React RestM M F L M (4.7)
Substituting Eq. 4.5 and 4.6 into Eq. 4.7 gives an expression for the
moment restraint coefficient, as follows;
, ,
1 2
,1 ,
1
21 1 3
Press,Eq,eff Press,Eq.crack M
N N
Rest D M Press,Eq
M MG EI
L LL
C M
(4.8)
It can be accepted within the linear elastic domain that the applied
moment at cracked section MPress,Eq,eff decreases by the specific ratio from the
pressure equivalent moment MPress,Eq due to the PIB restraint effects. The
reduction ratio of the applied moment was defined as the moment restraint
coefficient (CRest,1D,M). According to Equation 4.8, the restraint coefficient is
a function of the compliance of the crack Gcrack,ψ,M, the restraint length (L1
and L2), E and I. Therefore, it can be inferred that the pipe geometry and
material, the crack length, and the normalized restraint length can affect the
degree of PIB restraint in the same pipe as the earlier studies have concluded
(Miura, 2001; Rahman et al., 1995a; Scott et al., 2005b).
51
4.2 Compliance approach to improve the formulation
4.2.1 Compliance approach
The primary parameter that has an influence on the amount of restraint (Eq.
4.8) is the location of the circumferential crack represented as LN1 and LN2.
However the application of Eq. 4.8 is restricted to the case of fix-ended
straight pipe containing a circumferential crack. In reality, the configuration
of the nuclear piping system are far more complicated by including elbows,
hinge, and supports, and therefore, it is difficult to apply Equation 4.8 to the
practical cases.
For the generalization, the compliance approach is introduced in order
to enable the application of the formulation to complex piping systems, as
described schematically in Figure 4.3. By separating the pipe behavior from
that of the crack, the applied force and the moment at the cut-off section of
pipe are assumed as Fy and Mz, respectively. The deflection y and rotational
displacement ψ at the pipe cut-off end arising from each loads can be written
as follows:
y
y
z
z
yF y
F y
yM z
M z
y G F
G F
y G M
G M
(4.9)
where G is the compliance of pipe for each case that is expressed as a function
of the property of pipe material and geometry of the piping system. In order
52
to determine the compliances, it requires only the responses against the
applied load at the cut-off section of the pipe, irrespective of the
configurations of the connected piping system. Therefore, by replacing the
parameter of the location of a crack (L1N, L2N) of Equation 4.8 with the
compliances of pipe segment, the applicability of the formulation can be
improved.
4.2.2 Application of compliance approach to 1D pipe subjected to
pressure induced bending
This subsection aims at implementing the compliance approach to the 1D pipe
subjected to the pressure induced bending, starting from Eq. 4.8. First, a
concentrated moment (Mapp) due to the pressure induced tension is assumed
to be applied at a crack position as illustrated in Figure 4.4. To introduce the
concept of the pipe compliances, the cracked pipe was cut just before and
after the crack which is represented only by the rotational compliance of
Gcrack,ψ,M. With the lengths of each section given by L1 and L2, respectively. If
it is assumed that the crack behavior is determined by only a bending moment,
the applied moment at cracked section (MC) and pipe ends (Mz) can be
separated, while the sum of two is equal to Mapp. The governing equations in
the preceding section were focused on displacements (y) and rotation (ψ) that
are described as a function of distance from one end of the pipe. In this case,
the displacements and rotations at the cut-off point of the pipe can be
expressed as a function of axial force and bending moment only, as
53
represented in Eq. 4.10.
3 2
1 11
3 2
2 22
2
1 11
2
1 22
( )3 2
( )3 2
( )2
( )2
y z
y z
y z
y z
L Ly F M a
EI EI
L Ly F M b
EI EI
L LF M c
EI EI
L LF M d
EI EI
(4.10)
In Eq. 4.10, the subscript 1 and 2 means the right and left side of the
crack, respectively. At the cut-off section, the boundary conditions are as
follows;
1 2
1 2 , ,
0 ( )
( )
( )
C crack M C
app z C
y y a
G M b
M M M c
(4.11)
Note that the displacement and rotation at the cracked section are
continuous, while MC determines the rotation of ψC at the cracked section.
Elimination of Fy in Eq. 4.10 using the boundary conditions gives the relation
between Mapp and Mz as follows;
2 2
, , 1 1 2 2
3 2 2
1 2 , , 1 1 2 2
4
4
crack M
z app
crack M
G EI L L L LM M
L L G EI L L L L
(4.12)
Eq. 4.12 also leads to a linear relationship between the resultant
moment and the free end moment within the linear elastic domain.
Consequently, the ratio of moment reduction at the cracked section is derived
54
as Eq. 4.13.
3
1 2
3 2 2
1 2 , , 1 1 2 2
,1 ,
14
C z
app app crack M
Rest D M
L LM M
M M L L G EI L L L L
C
(4.13)
It can be shown that the ratio in Eq. 4.13 is equal to the moment
restraint coefficient in Eq. 4.8 (CRest,1D,M). Then the constants regarding the
pipe geometry and the material property in Eq. 4.13 can be replaced with the
compliances as follows;
1 1
2 2
1 1
2 2
1
2
1
2
y z
y z
y z
y z
y F y y M z
y F y y M z
F y M z
F y M z
y G F G M
y G F G M
G F G M
G F G M
(4.14)
Using the same principle, boundary conditions are given as follows;
1 2
1 2 , ,
0
C crack M C
app C z
y y
G M
M M M
(4.15)
Then Eq. 4.14 can be converted as Eq. 4.16.
1 2 1 2
1 2 1 2 , ,
, ,
0
y y z z
y y z z
y F y F y M y M y y
z zF F M M crack M
crack M app
G G G G F FA
M MG G G G G
G M
(4.16)
Consequently, the moment restraint coefficient in terms of the
55
compliance can be derived as:
1 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
,1
, ,
1
, , 1
1
1det
y y
y y z z z y y
y y z z z y y
app zC zRest D
app app app
crack M y F y F
y F y F M M y M y M F F
y F y F M M crack M y M y M F F
M MM MC
M M M
G G G
A
G G G G G G G G
G G G G G G G G G
(4.17)
where the ratio is a function of the compliances of cut pipes and crack. These
compliances of the pipe are not confined to specific pipe geometry. Thus, it
can be inferred that the applicability to complex piping geometry is secured
using the compliance approach.
56
4.3 Development of generalized formulation
It should be noted that the calculation of the crack opening displacement
needs the consideration of the applied loading under the normal operating
conditions in which the leak rate can be measured. In case of the crack
stability analysis, however, the crack stability analyses under the transient
load conditions need to be carried out under both normal operating loads and
safe-shutdown-earthquake loads. Therefore, the loadings applied on the pipe
due to the dead weight, thermal load, inertial load as well as seismic anchor
motion, if any, must be considered.
Because the earlier studies for the formulation of restrained COD only
focused on the pressure induced bending (Kim, 2007; Miura, 2001; Scott et
al., 2005a), there is limitation in applying the formulations to other types of
loading. From the efforts for considering the various loading conditions
conducted by E. Smith (1996), it was confirmed that the inertial load and
displacement controlled load due to a seismic event can be treated in the
common approach, regarding the crack stability analysis. However, the
generalized formulations which can be applied irrespective of the pipe
geometries or loading conditions were still not made available.
From this context, this section aims to enhance the applicability of the
derived formula to generalized load and complexity of pipe geometry thereby
it is expected that the developed method of this dissertation can be generally
used in the calculation of both COD and crack stability analysis.
57
4.3.1 Consideration of the types of applied loading
4.3.1.1 Distributed load
Both inertial loading due to a seismic event and dead weight can be treated as
distributed loads along the piping system. Figure 4.5 shows a diagram of a
cracked beam subjected to a distributed load. Similar with the section 4.2,
displacements and rotations at the cut-off point of pipe can be written as a
superposition caused by each of the individual loads.
1 1 1
2 2 2
1 1 1
2 2 2
1
2
1
2
y z
y z
y z
y z
y w y F y y M z
y w y F y y M z
w F y M z
w F y M z
y G w G F G M
y G w G F G M
G w G F G M
G w G F G M
(4.18)
In Eq. 4.18, w is the load per unit length, Fy and Mz are the applied
force and moment at the cracked section, respectively. To derive the applied
moment at the cracked section for uncracked pipe (MC,UcPipe) and cracked pipe
(MC,CPipe) separately, different boundary conditions regarding the rotations
were used like Eq. 4.19.
1 2
1 2
1 2
0 ( )
0 ( ) ( )
( ) ( )C
y y a
uncracked pipe b
cracked pipe c
(4.19)
By substituting the boundary conditions into Eq. 4.18 and by
eliminating w and Fy, the applied moments at the cracked section for each
58
case can be represented as follows;
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
,
1
y y y y
y y z z z y y
y F y F w w y w y w F F
C UcPipe
y F y F M M y M y M F F
G G G G G G G GM w
G G G G G G G G
(4.20)
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
,
, , 1
y y y y
y y z z z y y
y F y F w w y w y w F F
C CPipe
y F y F M M crack M y M y M F F
G G G G G G G GM w
G G G G G G G G G
(4.21)
By dividing the applied moment of the cracked pipe by that of the
uncracked pipe at the location of the crack (x=L1), the reduction ratio of the
applied moment due to the presence of the crack is derived as follows:
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1,
, , , 1
,1 ,
y y z z z y y
y y z z z y y
y F y F M M y M y M F FC CPipe
C UcPipe y F y F M M crack M y M y M F F
Rest D M
G G G G G G G GM
M G G G G G G G G G
C
(4.22)
where the ratio is equal to the moment restraint coefficient (CRest,1D,M)
obtained from concentrated moment case in the Section 4.2.2.
4.3.1.2 Relative displacements of supports
When a piping system is subjected to a seismic loading, the supports in the
pipe can move differently. Thermal expansion of the structures can also result
in differential displacements. These relative displacements of supports can
have a major influence on the integrity of pipe in particular case (Kim, 2014).
In this regards, the case for displacement controlled load was considered.
Figure 4.6 shows a beam model of which right-hand end moved by an amount
59
of d. The governing equations for displacement and rotations are followings.
1 1
2 2
1 1
2 2
1
2
1
2
y z
y z
y z
y z
y F y y M z
y F y y M z
F y M z
F y M z
y G F G M
y G F G M
G F G M
G F G M
(4.23)
The continuity of the bend angle at the crack position is same with the
previous section while the difference of the vertical displacement should be
equal to d, thereby the boundary conditions are;
1 2
1 2
1 2
( )
0 ( ) ( )
( ) ( )C
y y d a
uncracked pipe b
cracked pipe c
(4.24)
The applied moment at the cracked section for uncracked and cracked
pipe that satisfy the boundary conditions Eq. 4.24 are obtained, as follows:
1 2
1 2 1 2 1 2 1 2
,
y y
y y Z Z Z Z y y
F F
C UcPipe
y F y F M M y M y M F F
d G GM
G G G G G G G G
(4.25)
1 2
1 2 1 2 1 2 1 2
,
, ,
y y
y y Z Z Z Z y y
F F
C CPipe
y F y F M M crack M y M y M F F
d G GM
G G G G G G G G G
(4.26)
whereupon the ratio between two values is derived as Eq. 4.27.
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1,
, , , 1
,1 ,
y y z z z y y
y y z z z y y
y F y F M M y M y M F FC CPipe
C UcPipe y F y F M M crack M y M y M F F
Rest D M
G G G G G G G GM
M G G G G G G G G G
C
(4.27)
In this case as well, the moment reduction ratio is equal to the moment
60
restraint coefficient, CRest,1D,M.
As stated earlier, the earlier studies to express the restraining effects
on COD analytically have only focused on the pressure induced bending. As
Eqs. 4.22 and 4.27 imply, regardless of the types of loadings, the constraint
of pipe has an influence on reducing the applied moment at the cracked
section. This is also the strong evidence to broaden the practical aspects of
the restraint coefficient. By multiplying the restraint coefficient on the
uncracked pipe analysis results, one can calculate the effective applied
moment at the cracked section of a cracked pipe, as follows;
, ,1 , ,C CPipe Rest D M C UcPipeM C M .
(4.28)
This means that, therefore, the restraint coefficient can be used not
only to normalize the pipe restraining effect as derived in Eq. 4.17, but also
to predict the amount of decrease in the driving force caused by the
compliance change of the piping system due to the presence of a crack under
any loading conditions.
4.3.2 Consideration of the complex piping configurations
4.3.2.1 Two-dimensional piping system
In the subsection 4.2.2, the applicability of the derived formulation for the
effective applied moment was improved because the variables that are
61
restricted to the particular geometries were replaced to by the compliance of
pipe. In this subsection, the compliance approach will be extended to a two-
dimensional piping system.
Figure 4.7 shows an arbitrary 2D pipe containing a circumferential
crack. In this case, the piping system may include the elbows bent at various
angles or pipe joints. Unlike the case of 1D pipe, the moment or force applied
to a complex piping system can cause a large axial force on the cracked
section. Therefore, it was assumed that both the bending moment and axial
force at the cracked section are affected by the change of compliance due to
the restraint effect. The rotational compliances of the pure crack then can be
represented as follows;
, ,
, ,
cracked pipe uncracked pipe
crack M
cracked pipe uncracked pipe
crack F
GM M
GF F
.
(4.29)
Suppose that the applied moment and axial force at the postulated
position of a crack calculated from the uncracked pipe analysis are Mapp and
Fapp, respectively. A free body diagram of cracked pipe subjected the Mapp and
Fapp is illustrated based on the compliance approach in Figure 4.7. The pure
crack was separated from the pipe system, and the effective applied moment
and axial force are expressed by MC and FC, respectively. The axial and
vertical forces (Fx, Fy), and bending moment (Mz) at the cut-off section of pipe
then can be described as shown in Figure 4.7. Based on this free body diagram,
vertical displacement (y), axial displacement (x) and rotation (ψ) at the cut-
off point of pipe are expressed as a function of compliances of pipe and
62
loadings as Eq. 4.30.
1 1 1
2 2 2
1 1 1
2 2 2
1 1 1
2 2 2
1
2
1
2
1
2
x y z
x y z
x y z
x y z
x y z
x y z
x F x x F y x M z
x F x x F y x M z
y F x y F y y M z
y F x y F y y M z
F x F y M z
F x F y M z
x G F G F G M
x G F G F G M
y G F G F G M
y G F G F G M
G F G F G M
G F G F G M
(4.30)
In accordance with the continuity of deformation at cracked section,
the boundary conditions are given by the expressions, as follows;
1 2
1 2
1 2 , , , ,
0
0
C crack M C crack F C
x x
y y
G M G F
(4.31)
The equilibrium of force and moment are given as below;
app C z
app C x
M M M
F F F
. (4.32)
By substituting the boundary conditions, Eq. 4.30 can be reduced as
following relation;
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2 , ,
,1 ,2
0
x x y y z z
x x y y z z
x x y y z z
x F x F x F x F x M x Mx
y F y F y F y F y M y M y
zF F F F M M crack M
x x
pipe pipe crack y y
z z
G G G G G G F
G G G G G G F
MG G G G G G G
F F
G G G F A F
M M
, , , ,
0
crack M app crack F appG M G F
(4.33)
63
From the Eq. 4.33, Mz and Fx can be obtained by using Cramer’s rule
(Brunetti, 2014):
3,3
, , , ,
3,1
, , , ,
det
det
z crack M app crack F app
x crack M app crack F app
MM G M G F
A
MF G M G F
A
(4.34)
where Mi,j is the minor of the matrix A that is the determinant of the smaller
matrix formed by eliminating the i-th row and the j-th column from the matrix
A. Solving Eq. 4.34, the reduction ratio of the applied bending moment and
axial force due to the presence of a crack can be derived as follows;
,
3,3
, , , ,
3,3
, , , ,
,
, , , ,
1
det1
1det
1
1
app zC zRest M
app app app
crack M app crack F app
app
app
crack M crack F
app
app xC xRest F
app app app
crack M app crack F ap
M MM MC
M M M
MG M G F
A
M
F MG G
M A
F FF FC
F F F
G M G F
3,1
3,1
, , , ,
det
1det
p
app
app
crack M crack F
app
M
A
F
M MG G
F A
,1 ,2pipe pipe crackwhere A G G G
(4.35)
The restraint coefficient for 2D complex pipe is also expressed as a
function of the compliances of pipe and crack, not the parameters that are
64
limited to the specific pipe configurations. In addition, it should be noted that
the amount of decrease in the bending moment and axial force depends on the
ratio of Mapp and Fapp.
4.3.2.2 Generalized formulation considering three-dimensional piping
system
Finally, the generalized formulation can be derived that can be adopted
regardless of the configurations of the pipe and boundary conditions by
implementing the compliance approach to a piping system in the space.
Figure 4.8 represents a generalized three-dimensional piping system
containing a circumferential crack. In this case, three directions of forces (Fx,
Fy, Fz) and moments (Mx, My, Mz) can produce the deformations in six degrees
of freedom (x, y, z, θ, ϕ, ψ) as depicted in Figure 4.8. By the similar process
employed for the derivation with the 2D pipe, it was assumed that only the
bending moment and axial force can produce the rotational displacement of a
pure crack so that Eq. 4.29 is available in this case.
Based on the free body diagram in Figure 4.8, the relations between
the responses and loadings at the cut-off section of each region of the pipe
can be given by a matrix form, as follows:
65
1
1
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y
x F x F x F x M x M x M
i
y F y F y F y M y M y Mi
z F z F z F z M z M z Mii
i F F F M M M
iF F F M M M
iF F
G G G G G Gx
G G G G G Gy
G G G G G Gz
G G G G G G
G G G G G G
G G G
1
,1
i z i x i y i z
x x
y y
iz z
pipe i
x x
y y
z zF M M M
F F
F F
F FG
M M
M M
M MG G G
(4.36)
where subscript i represents the region index (1 or 2) of the pipe, and Gpipe,i is
the pipe compliance matrix of region i. By the similar process with the
previous section, the assumption that the deformations at the crack position
are continuous except the direction of z-rotation gives boundary conditions,
as follows;
1 2
1 2
1 2
1 2
1 2
1 2 , , , ,
0
0
0
0
0
c crack M C crack F C
x x
y y
z z
G M G F
(4.37)
Next, applying these boundary conditions to 4.36 gives:
,1 ,2
, , , ,
0
0
0
0
0
x
y
z
pipe pipe crack
x
y
crack M C crack F Cz
F
F
FG G G
M
M
G M G FM
(4.38)
Using Cramer’s rule(Brunetti, 2014), one can show that Mz and Fx can
be obtained, as follows:
66
6,6
, , , ,
6,1
, , , ,
,1 ,2
det
det
z crack M app crack F app
x crack M app crack F app
pipe pipe crack
MM G M G F
A
MF G M G F
A
where A G G G
(4.39)
where Mi,j is the minor of the matrix A that is the determinant of the smaller
matrix formed by eliminating the i-th row and the j-th column from the matrix
A. The generalized form of the restraint coefficient is then derived as follows;
6,6
, , ,
6,1
, , ,
1det
1det
z x
z x
app
Rest M crack M crack F
app
app
Rest F crack M crack F
app
F MC G G
M A
M MC G G
F A
.
(4.40)
The efforts in Section 4.3 for the development of the generalized
formulation of the restraint coefficient have resulted in a number of important
findings that may be worthwhile to be summarized before moving into other
subjects. The major findings are summarized as follows;
i) The pipe restraint can reduce the applied moment (crack driving force)
at the cracked section.
ii) The presence of a crack in the piping system may results of decrease
in the applied moment and axial force at the cracked section.
iii) The load reduction due to the pipe restraint and the presence of a crack
67
can be treated as the same phenomenon.
iv) Equation 4.40 is the generalized formulation of the restraint
coefficient that means the load reduction ratio at the cracked
section. This can be utilized to predict the fracture mechanics
parameters of a circumferential crack irrespective of the types of
applied loadings and the configuration of the piping system.
68
4.4 Evaluation procedure to determine effective applied
moment
The restraint coefficient has been defined as the ratio of the effective applied
loading at the cracked section to the anticipated applied loading calculated on
the basis of the pipe analysis without consideration of a crack. Hence, the
restraint coefficient can be used as a unified correction factor for the
calculation of both COD and allowable moment to reflect the effect of
restraint on that. This section describes the step-by-step procedures to
determine the effective applied moment and axial force at the cracked section
for practical cases, as shown in Figure 4.9, as follows;
i) Calculate Mapp and Fapp from the uncracked pipe analysis: First of all,
specify the range of piping of interest including restrained points
such as the intersection with the large components, or rigid
supports. Then, the general piping analysis should be conducted
to calculate the anticipated loading (Mapp and Fapp) without
consideration of a crack. Based on these results, the critical
location of a postulated crack and the orientation are determined
to make the crack subjected to maximum moment MZ’. Set the y’
and z’-axis based on the orientation of the crack.
ii) Define the compliance of the crack, Gcrack: Only the circumferential
crack is considered, and both the surface and through-wall crack
69
are available. The deformation of the crack is allowed only one
direction of rotation due to bending moment and axial force. The
compliance of a crack is a rotation (ψ) due to the crack per unit
moment (M) or axial force (F), and can be easily calculated using
the analytic solutions or the finite element analysis as follows:
, ,
, ,
cracked pipe uncracked pipe
crack M
cracked pipe uncracked pipe
crack F
GM M
GF F
(4.41)
iii) Define the compliances of the pipe in global coordinate (Gpipe,i,global):
The compliance of pipe should be determined for both side of the
crack. To calculate Gpipe,1,global, exclude boundary conditions of
region 2. Then impose the unit forces (Fx, Fy, Fz) and moments
(Mx, My, Mz) on the crack position and calculate the deformation
(x, y, z, θ, ϕ, ψ) to produce the compliance matrix as Eq. 4.42. For
a case of region 2, repeat vice versa and determine Gpipe,2,global.
, ,
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
x F x F x F x M x M x M
y F y F y F y M y M y M
z F z F z F z M z M z M
pipe i global
F F F M M M
F F F M M M
F F F M M M
G G G G G G
G G G G G G
G G G G G GG
G G G G G G
G G G G G G
G G G G G G
(4.42)
70
iv) Transform the compliances of the pipe to the crack coordinate: The
coordinate of the crack decided in step ii) does not always
coincide with the global coordinate. Thus, the compliances of the
pipe in global coordinate should be transformed to the crack
coordinate using the usual tensor transformation rule as Eq. 4.43.
, , , ,
T
pipe i crack pipe i globalG T G T
(4.43)
When the z’-axis (crack coordinate) make the angle α with
respect to the z-axis (global coordinate), then the transformation
matrix T is as follow;
1 0 0 0 0 0
0 cos sin 0 0 0
0 sin cos 0 0 0
0 0 0 1 0 0
0 0 0 0 cos sin
0 0 0 0 sin cos
T
(4.44)
v) Determine the restraint coefficient by substituting the compliance of
the pipes and crack on the Eq. 4.45.
6,6
, , ,
6,1
, , ,
,1, ,2,
6,6 6,1
1det
1det
, minor of matrix A
z x
z x
app
Rest M crack M crack F
app
app
Rest F crack M crack F
app
pipe crack pipe crack crack
F MC G G
M A
M MC G G
F A
A G G G
M M
(4.45)
71
vi) Calculate the effective applied moment and force by multiplying the
restraint coefficient on Mapp or Fapp using Eq. 4.46.
, ,
, ,
eff app Rest M app
eff app Rest F app
M C M
F C F
(4.46)
Although the restraint coefficient has been derived based on the elastic
beam theory, the nonlinearity of crack behavior or pipe material can be
reflected in the developed formulation. When it comes to determine the
compliance, one can consider following combinations:
- Linear elastic pipe (Gpipe,LE) + Elastic-plastic crack (Gcrack,LE)
- Linear elastic pipe (Gpipe,LE) + Elastic-plastic crack (Gcrack,EP)
- Elastic-plastic pipe (Gpipe,EP) + Elastic-plastic crack (Gcrack,EP)
In order to compare the conservatism of above cases, an example
analysis was conducted using the beam model subjected the distributed as
shown in Figure 4.5. 4 cases of the pipe length were analyzed (2L/Do=5, 10,
15, 20) while L1/Do was fixed to 1.
As a result, Figure 4.10 shows the comparisons the effective applied
moment at cracked section (Meff,app) depending upon the Mapp calculated from
the elastic analysis of the uncracked pipe. The difference between graph (a)
and (b) indicates that if only the plastic behavior of a crack is considered, the
applied moment can be reduced more than the case of the elastic model.
However, if the plastic deformation of the pipe occurs, the cracked section
subjected more significant moment than the elastic pipe + plastic crack case
72
due to the load redistribution.
When the elastic-plastic behavior is considered, the compliance
depends on the amount of applied load. Therefore, above steps from i) to vi)
should be repeated to calculate the effective applied moment and axial force.
For convenience, following suggestion regarding the consideration of
nonlinearity can be available: Under the design basis conditions, an
assumption that crack and pipe do not experience the plastic behavior can lead
conservative results while the case that allows the non-linear behavior of
crack only is more accurate. If the applied load is large enough to cause the
plastic deformation of the pipe, the elastic-plastic compliances of crack and
pipe must be considered.
73
Figure 4.1 The concept of effective applied moment at the cracked section
Meff,app
Crack
Mapp
Crack
Reaction
M, F
Initial Applied Moment
Load redistribution
Effective Applied Moment
74
Figure 4.2 Beam model of fixed-ended pipe with a circumferential crack
subjected to a pressure induced bending for development of the moment
restraint coefficient
M
Δψ
Gcrack,ψ,M
Crack :
MPress, Eq.
Region-1 Region-2
MReact,Rest
x
L1 L2
2L=L1+L2
FReact,Rest
75
Figure 4.3 Schematic descriptions of the compliance approach
1. Separate the cracked section from a piping system
2. Define the compliances of pipe (Gpipe) and crack (Gcrack)
3. Derive the restraint coefficient in terms of compliances
Crack
y z
y z
yF y yM z
F y M z
y G F y G M
G F G M
L1 L2
M
Δψ
GCrack
CrackG M
M
Mz
Fy
yψ
Crack
L1 L2
1. Separation of the pipe and crack
2. Definition of the compliance
L1 L2
Crack=
L1 L2
Pure
Crack
+
M
Δψ
Gcrack,ψ,M
, ,crack MG M
76
Figure 4.4 Beam model and free body diagram of fixed-ended pipe with a
circumferential crack subjected to a pressure induced bending for
development of the moment restraint coefficient based on the compliance
approach
Mapp
L1 L2
2L=L1+L2
y
x
z
MC
MZ MZ
Fy Fy
Mapp=MC+MZ
77
Figure 4.5 Beam model and free body diagram of fixed-ended pipe with a
circumferential crack subjected to a distributed load for development of the
moment restraint coefficient based on the compliance approach
wL1 L2
2L=L1+L2
wL1 L2
2L=L1+L2
MC,CPipe
MC,UcPipe
MBMA
FA FB
MBMA
FA FB
78
Figure 4.6 Beam model and free body diagram of fixed-ended pipe with a
circumferential crack subjected to a relative displacement of the supports for
development of the moment restraint coefficient based on the compliance
approach
FA
FB
d
MA
MB
L1 L2
2L=L1+L2
FA
FB
d
MA
MB
L1 L2
2L=L1+L2
MC,CPipe
MC,UcPipe
79
Figure 4.7 Beam model and free body diagram of 2D piping system
containing a circumferential crack for development of the restraint
coefficient based on the compliance approach
MZ
FY
FX
FY
MZ
FX
Mapp=MC+MZ
Fapp = FC + FX
Region-1 Region-2
y
x
z
Mapp, Fapp
MC,FC
80
Figure 4.8 Beam model and free body diagram of the generalized 3D piping
system containing a circumferential crack for development of the restraint
coefficient based on the compliance approach
Mapp, Fapp
MC,FC
Mapp=MC+MZ
Fapp = FC + FX
Region-1 Region-2
y
x
z
Fx,Mx
FY,MY
Fz,Mz
FY,MY
Fz,Mz
Fx,Mx
81
Figure 4.9 The procedure for calculation of the effective applied moment and force
i) Calculate applied moment and axial force at crack position, and determine
orientation of crack to make the crack subjected to maximum MZ
ii) Define the compliances of crack (Gcrack) for M-rotation & F-rotation behavior
Gcrack,ψMz =
y
z
αy'
z'
Global
coordinate
Crack
coordinate
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0x z
crack
F M
G
G G
y
x
z
Crack position
Gcrack,ψFx =
M,
Ψuncrack/2
M,
Ψcrack/2
F,
Ψuncrack/2
F,
Ψcrack/2
82
Figure 4.9 The procedure for calculation of the effective applied moment and force (continued)
iii) Define the compliances of pipe (Gpipe,i,global) for 6 D.O.Fs in global coordinate
<Region 1>
Exclude boundary conditions of region 2
Apply a unit load on crack position and calculate the deformation
Determine Gpipe,1,global
<Region 2>
Exclude boundary conditions of region 1
Apply a unit load on crack position and calculate the deformation
Determine Gpipe,2,global
, ,
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
i x i y i z i x i y i z
x F x F x F x M x M x M
y F y F y F y M y M y M
z F z F z F z M z M z M
pipe i global
F F F M M M
F F F M M M
F F F M M M
G G G G G G
G G G G G G
G G G G G GG
G G G G G G
G G G G G G
G G G G G G
Region-1
y
x
zFZ,MZ
FY,MY
FX,MX
Region-2
FZ,MZ
FY,MY
FX,MX
83
Figure 4.9 The procedure for calculation of the effective applied moment and force (continued)
iv) Transform the compliances of pipe (Gpipe,i,global) to crack coordinate (Gpipe,i,crack)
v) Determine the restraint coefficient (CRest)
vi) Calculate the effective applied moment and axial force (Meff,app, Feff,app)
,1, ,2,
6,6 minor of matrix A
pipe crack pipe crack crackA G G G
M
, , , ,
T
pipe i crack pipe i globalG T G T
Transformation matrixT
y
z
αy'
z'
Global
coordinate
Crack
coordinate
1 0 0 0 0 0
0 cos sin 0 0 0
0 sin cos 0 0 0
0 0 0 1 0 0
0 0 0 0 cos sin
0 0 0 0 sin cos
T
6,6
, , ,
6,1
, , ,
1det
1det
z x
z x
app
Rest M crack M crack F
app
app
Rest F crack M crack F
app
F MC G G
M A
M MC G G
F A
, ,
, ,
eff app Rest M app
eff app Rest F app
M C M
F C F
84
(a) Elastic pipe + Elastic crack
(b) Elastic Pipe + Plastic Crack
Figure 4.10 Effect of nonlinear behavior on the applied moment at the
cracked section
0 100 200 300 400 5000
50
100
150
200
250
Limit of design
moment
Limit moment
of crack
(L1+L
2)/D
o
5
10
15
20
Ap
pli
ed m
om
ent
at c
rack
po
siti
on
of
crac
ked
pip
e [k
N-m
]
Applied moment at crack position
of uncracked pipe [kN-m]
Elastic Pipe + Elastic Crack
0 100 200 300 400 5000
50
100
150
200
250Elastic Pipe + Plastic Crack(L
1+L
2)/D
o
5
10
15
20
Ap
pli
ed m
om
ent
at c
rack
po
siti
on
of
crac
ked
pip
e [k
N-m
]
Applied moment at crack position
of uncracked pipe [kN-m]
Limit moment
of crack
Limit of design
moment
85
(c) Plastic pipe + Plastic crack
Figure 4.10 Effect of nonlinear behavior on the applied moment at the
cracked section (continued)
0 100 200 300 400 5000
50
100
150
200
250
Plastic Pipe + Plastic Crack(L1+L
2)/D
o
5
10
15
20
Ap
pli
ed m
om
ent
at c
rack
po
siti
on
of
crac
ked
pip
e [k
N-m
]
Applied moment at crack position
of uncracked pipe [kN-m]
Limit moment
of crack
Limit of design
moment
86
87
Chapter 5 Validation of Developed Formulations
To verify the generalized analytical formulation and concept of the effective
applied moment derived in this dissertation, benchmark studies are carried
out against both FEA-based numerical analysis as well as well-documented
large-scale experimental data. In the benchmark against numerical analysis,
both static and dynamic loading conditions are included by a series of finite
element analysis. In dynamic loading conditions, a very detailed time history
analysis was included.
First, in the same context as in the development process of the restraint
coefficient, the effects of pressure induced bending restraint on COD were
predicted using FEA from earlier studies and the results from generalized
formulations were compared. Then the static analyses were performed to
evaluate the amount of restraint considering the anticipated loads at the
cracked section under the normal operating conditions. In addition, the crack
stability analysis assumes the faulted loading condition in which the seismic
load is considered. Finally, the dynamic analysis results using cracked pipe
model were compared with experimental data to demonstrate that restraint
coefficient is valid under the dynamic loading conditions with excellent
agreements.
88
5.1 Validation under static loading conditions
5.1.1 Evaluation of PIB restraint effects on COD for 1D pipe④
To represent the PIB restraint effects on COD, rCOD was determined by
dividing the COD of the restrained pipe by that of the unrestrained pipe. For
calculating the COD, the solution from the Electric Power Research Institute
(EPRI) ductile fracture handbook (Zahoor, 1989) can be employed. In this
section, evaluation methods for the PIB restraint effect on COD were
developed for the linear elastic and elastic-plastic models. The reduction
ratios of COD due to the pipe restraint calculated based on the formulation
were then compared with the FEA results from earlier studies.
5.1.1.1 Calculation of the restrained COD for linear elastic model
For the linear elastic model, the COD of the unrestrained pipe (δUnrestraind,LE)
and the COD of the restrained pipe (δRestrained,LE) can be derived by substituting
MPress,Eq and MPress,Eq,eff into the COD formula from the EPRI ductile fracture
④ This section has been based on the following conference paper:
Kim, Y., Hwang, I.-S., Oh, Y.-J., 2016a. Effective applied moment in circumferential
through-wall cracked pipes for leak-before-break evaluation considering pipe restraint
effects. Nuclear Engineering and Design 301, 175-182.
89
handbook (Zahoor, 1989) as follows;
,
, 4 4
44 Press Eq mm bUnrestrained LE
o i
M RR V
E R R
(5.1)
, ,
, 4 4
44 Press Eq eff mm bRestrained LE
o i
M RR V
E R R
(5.2)
Vb is given in the EPRI ductile fracture handbook as a function of the
ratio of the pipe mean radius to pipe thickness (Rm/t) and the crack length
(θ/π). By dividing Eq. 5.1 by Eq. 5.2, the ratio of the COD of a restrained pipe
to the COD of an unrestrained pipe (rCOD,LE) is obtained as Eq. 5.3:
, ,,
, ,1 , ,
, ,
Press Eq effRestrained LE
COD LE Rest D M LE
Unrestrained LE Press Eq
Mr C
M
(5.3)
where the ratio is equal to the moment restraint coefficient, CRest,1D,M,LE. In
case of the 1D pipe, the moment restraint coefficient was derived in Eq.4.8 as
follow;
,1 , ,
, , ,
1 2
1
21 1 3
Rest D M LEcrack M LE
N N
CG EI
L LL
(5.4)
To obtain rCOD,LE, the linear elastic compliance of the crack
(Gcrack,ψ,M,LE) is needed. Gcrack,ψ,M,LE can be determined by dividing rotation
due to the crack (ψC,M,LE) by an applied moment (M).
, ,
, , ,
C M LE
crack M LEGM
(5.5)
The EPRI ductile fracture handbook (Zahoor, 1989) gives the linear
90
elastic rotation due to the crack under a bending moment as follows:
3, , 2'
C M LE
m
B M
E R t
(5.6)
or
, , 3, , , 2'
C M LE
crack M LE
m
BG
M E R t
(5.7)
where E’ denotes E/(1-ν2) for plane strain and E for plan stress (ν is the
Poisson’s ratio.). The dimensionless function B3 is given in the EPRI ductile
fracture handbook as a function of the crack length (θ/π) and pipe mean radius
to thickness ratio (Rm/t). By substituting Eq. 5.7 into Eq.5.4, we can evaluate
the effect of PIB restraint on COD.
5.1.1.2 Comparison with linear elastic finite element analysis results
As described in the preceding section, a series of the linear elastic FEA to
examine the PIB restraint effects on COD was conducted in the BINP
program and in Miura’s research (Miura, 2001; Scott et al., 2005b). Miura has
analyzed three crack length, and two types of restraint models; a symmetric
model (L1=L2) and an asymmetric (L1≠L2). The proposed formula of rCOD,LE
was compared with these results.
Figure 5.1 shows the results of the developed formulation and FEA
for the symmetric model. rCOD,LE predicted by using the moment restraint
coefficient had good agreement with the FEA results regardless of the value
91
of Rm/t and θ/π. The results of the asymmetric model for three restraint lengths
and three crack lengths are represented in Figure 5.2. When the pipe length
of one side of the crack is short (L1/D=1 or L2/D=1), the formula tends to
slightly overestimate rCOD,LE versus FEA. Nevertheless, the comparison
results show that the proposed evaluation method could well predict rCOD,LE
(the effect of PIB restraint) for the linear elastic model.
5.1.1.3 Calculation of the restrained COD for elastic-plastic model
The elastic-plastic behavior was also considered. It is difficult to express the
restraint effect as a closed form formula because of the nonlinearity of the
elastic-plastic model. In the same manner with the linear elastic model, the
COD of the unrestrained pipe and restrained pipe can be derived by
substituting MPress,Eq and MPress,Eq,eff into the elastic-plastic COD formula.
2
2 , 0 2 ,
,
0
n
m Press Eq m Press Eq
Unrestraind EP n
f R M R H M
EI M
(5.8)
2
2 , , 0 2 , ,
,
0
n
m Press Eq eff m Press Eq eff
Restraind EP n
f R M R H M
EI M
(5.9)
The dimensionless functions f2, H2 and M0 are given in the EPRI
ductile fracture handbook (Zahoor, 1989). rCOD,EP can be obtained by dividing
92
Eq. 5.9 by Eq. 5.8 as follow;
2
2 , , 0 0 2 , ,,
, 2
, 2 , 0 0 2 ,
n n
m Press Eq eff m Press Eq effRestraind EP
COD EP n n
Unrestraind EP m Press Eq m Press Eq
f R M M EI R H Mr
f R M M EI R H M
(5.10)
In contrast with the linear elastic case, rCOD,EP depends on the
magnitude of the applied moment due to the nonlinearity. Thus, the additional
calculation procedure is required as follows;
First, the pressure equivalent moment MPress,Eq which means the
bending moment that can induce exactly same rotational displacement with a
given tension load arising from pressure (the bending moment when
ψC,M,EP=ψC,T,EP) should be obtained. However, the EPRI ductile fracture
handbook gives the formula of the rotation due to the crack of the elastic-
plastic pipe caused by a bending moment (ψC,M,EP) but not by an axial tension
load (ψC,T,EP). Thus, a new formula for an axial tension load and coefficient
H4T were developed, and this will be discussed in more detail in the next
subsection. Using two formulae, the equation for calculating the pressure
equivalent moment (MPress,Eq) can be represented as Eq. 5.11.
4 , ,0 4
22
0
2 2
4 0 4
2
0
/ 1 0.5 /
2
/ 1 0.5 /
n
Press Eq Press EqB
m
n
i in T i in
m
f M MH
R tE M
f R P H R P
R tE P
(5.11)
The left-hand side means the rotation due to the crack caused by a
bending moment, and the right-hand side is caused by an internal pressure
(Pin). The pressure equivalent moment MPress,Eq cannot be derived as a closed
93
form from Eq. 5.11; thus, it needs to be determined by an iterative calculation.
Second, the effective applied at the cracked section MPress,Eq,eff needs
to be determined. The relation between MPress,Eq and MPress,Eq,eff is represented
by using the moment restraint coefficient as follows;
, , ,1 , , ,
,1 , ,
, , , 1 2
,2 1 3
Press Eq eff Rest D M EP Press Eq
Rest D M EP
crack M EP N N
M C M
Lwhere C
L G EI L L
(5.12)
To obtain CRest,1D,M,EP, the elastic-plastic compliance of the crack
(Gcrack,ψ,M,EP) is needed. In the same manner as linear elastic case, Gcrack,ψ,M,EP
can be determined by dividing rotation due to the crack (ψC,M,EP) by an applied
moment (M).
, ,
, , ,
C M EP
crack M EPGM
(5.13)
The EPRI ductile fracture handbook (Zahoor, 1989) gives the elastic-
plastic rotation due to the crack under a bending moment as follows:
0 44
, , 22
0/ 1 0.5 /
n
BC M EP n
m
Hf MM
ER t M
(5.14)
or
1, , 0 44
, , , 22
0/ 1 0.5 /
nC b EP B
crack M EP n
m
Hf MG
M ER t M
(5.15)
where the dimensionless functions f4 and H4B are also given in the handbook.
It should be noted that M in Eq. 5.15 is the applied moment at the cracked
94
section MPress,Eq,eff, whereupon Eq. 5.15 can be written as:
1
, ,0 44, , , 22
0/ 1 0.5 /
n
Press Eq effBcrack M EP n
m
MHfG
ER t M
(5.16)
Thus, substituting Eq. 5.16 into Eq. 5.12, we obtain the relation
between MPress,Eq and MPress,Eq,eff .
, ,
,1
, ,0 441 222
0
2 1 3/ 1 0.5 /
Press Eq eff
Press Eqn
Press Eq effBN Nn
m
M
LM
MHfL EI L L
ER t M
(5.17)
To obtain MPress,Eq,eff, Eq. 5.17 needs to be solved iteratively using
MPress,Eq determined from Eq. 5.11. Finally, by substituting MPress,Eq
determined from Eq. 5.11 and MPress,Eq,eff determined from Eq. 5.17 into Eq.
5.10, we can evaluate rCOD,EP.
5.1.1.4 Comparison with elastic-plastic finite element analysis results
As mentioned in the subsection 5.1.1.3, to evaluate rCOD,EP, the rotation due
to the crack caused by an axial tension load (ψC,T,EP) must be calculated. The
new formula was developed referring from the rotation formula of a bending
moment and the axial displacement formula for an axial tension load (P) from
95
the EPRI ductile fracture handbook(Zahoor, 1989) as follow;
0 44
, , 2
0
2
/ 1 0.5 /
n
TC T EP
m
Hf P P
ER t P
(5.18)
The dimensionless function H4T depends on Rm/t, θ/π, and the
Ramberg-Osgood coefficient n. A series of the finite element analyses were
conducted to tabulate the H4T for the case of Rm/t=10, using the finite element
program ABAQUS (Dassault Systémes, 2012). Figure 5.3 shows a half-
symmetry analysis model of a pipe with a circumferential through-wall crack.
The reduced integration 20-noded continuum elements were employed, and a
focused mesh was applied at the crack tip. The multi-point constraint option
in ABAQUS was utilized to make the displacement and rotation at the nodes
on the pipe end plane equal to those of a reference node on the axis of the
pipe. The effects of geometric nonlinearity and ovalization were ignored. For
the material properties, Young’s modulus of 200 GPa and yield stress of 400
MPa were taken. The various crack lengths (θ/π) and Ramberg-Osgood
coefficients (n) were considered, the dimensionless function (H4T) then was
tabulated base on the FEA results.
The development of Eq. 5.18 aims to calculate the pressure equivalent
moment which means the bending moment that can induce exactly same
rotation with given tension load from pressure (the bending moment when
ψC,M,EP= ψC,T,EP) in Eq. 5.11. The dimensionless function such as H4T and H4B
can be strongly controlled by the details of the modeling approach. In this
regards, to maintain consistency, H4B was also newly tabulated using same FE
96
model with the case of H4T as shown in Table 5.1.
The PIB restraint effect on the elastic-plastic COD has been evaluated
by J. Kim through FEA for several cases (Kim, 2008; Kim, 2004). By
comparing with these FEA results, the developed evaluation procedure for the
elastic-plastic model in the subsection 5.1.1.3 was validated. The various
restraint lengths, crack sizes, and materials cases were considered. The results
of the comparisons are shown in Figure 5.4. As the internal pressure (axial
tension load) increases, rCOD,EP decreases because of the effect of the plastic
behavior. These trends are observed in the results of the finite element
analysis and the results predicted from the proposed evaluation procedure as
well. Similar to the case of the linear elastic model, the formula tends to
slightly overestimate rCOD,EP then the FEA results when the pipe length is
short (L1/D=L2/D=1). Generally, the estimated values using Eqs. 5.10 to 5.18
coincide well with the FEA results. Thus, the proposed method can reliably
be applied to the calculation of the restrained COD for the elastic-plastic
model.
In this section, the PIB restraint effects on COD predicted using the
developed formulations were validated with the various FEA results. For both
linear elastic and elastic-plastic model, the restraint coefficient can be used to
adjust the numerical expressions of COD for a free-ended pipe to take into
account the restraint effects.
97
5.1.2 Evaluation of effective applied moment for 3D pipe under
static loading conditions
A crucial aspect of the developed formulation is that the amount of decrease
in the applied load caused by the compliance change of the piping system due
to the presence of a crack can be predicted using the restraint coefficient (see
section 4.3.1). To demonstrate this aspect, the restraint coefficient estimated
based on the procedure in section 4.4 were compared with the ratio of moment
or force reduction predicted from the linear elastic finite element analyses. A
series of FEA was conducted for the uncracked and cracked pipe separately,
the ratio of the applied moment at the crack position between two cases was
calculated.
The particular model shown in Figure 5.5 is analyzed for verification.
The piping system has two horizontal pipe segments of length 10Do, and a
vertical pipe which are linked to two elbows of radius 1.5Do where Do is the
pipe outer diameter. The reduced integration 20-noded continuum elements
were employed, and a focused mesh was applied at the crack tip. The multi-
point constraint option in ABAQUS was utilized to make the axial
displacement and rotation at the nodes on the pipe end plane equal to those of
a reference node on the axis of the pipe while the radial displacement is
allowed. The effects of geometrical nonlinearity were ignored.
The nominal diameter and mean radius to thickness ratio are 12 inch
and 5, respectively. Both pipe ends are fixed rigidly except the radial
displacement, and the horizontal pipe segment connected the anchor 1
contains a circumferential crack at a distance of L1 from the anchor 1 while
98
nine values of L1 (crack position) were considered. For loading condition, not
only three kinds of loading that were assumed for the development of the
formulation, but also the thermal expansion load was considered in this
analysis. Another information about analysis model and material properties
are summarized in Table 5.2.
To determine Gcrack, the typical mesh of a circumferential through-
wall cracked pipe using ABAQUS was used as shown in Figure 5.3. The
rotations and axial displacements of the uncracked pipe and cracked pipe
were obtained respectively, the difference was determined as the rotation or
axial displacement of a pure crack (Zhang et al., 2010). Dividing the rotation
or axial displacement due to the crack by the applied bending moment or axial
force, Gcrack,ψ,Mz and Gcrack,ψ,Fx were obtained as 5.15E-12 rad/N∙mm and
4.03E-10 rad/N for crack length (θ/π) of 0.25, and 4.34E-11 rad/N∙mm and
4.15E-9 rad/N for θ/π=0.5, respectively.
To determine Gpipe, the simple FE model using beam element were
employed as depicted in Figure 5.6. The straight pipe segments and elbows
were created using pipe element and elbow element, respectively. The
kinematic boundary conditions were applied to the connections of an elbow
and straight pipe to account the effects of ovalization and radial expansion at
the elbows while the warping is not allowed.
Figure 5.7 and Figure 5.8 show the applied loads and applied nominal
stress at the cracked section of the uncracked and cracked pipes depending on
the crack positions. Comparing with the results of the uncracked pipe,
moment and force of the cracked pipe are decreased due to the change of
99
system compliance, and the degree of reduction in case of crack length (θ/π)
of 0.5 is larger than 0.25.
The restraint effects on the bending moment and axial force are
summarized in Figure 5.9. The value of 1 means that the behavior of cracked
pipe is same with the uncracked pipe. It can be confirmed again that the crack
and the pipe restraint affect the applied moment at cracked section regardless
of types of loading. The results calculated from proposed evaluation methods
agree well with the FEA results, while some differences are observed. This
could be due to the simplified assumptions considered in the process of
formulations development. Generally, it can be confirmed that using the
restraint coefficient developed in this dissertation, the effective applied
moment at cracked section can be calculated, and consequently, the accurate
COD and allowable moment can be determined.
100
5.2 Validation under dynamic loading conditions⑤
The crack stability analysis is a process that demonstrates that a postulated
crack does not grow unstably even under the accident conditions. The primary
applied loading in faulted conditions arises from earthquake. Therefore, to
utilize the propoased formulation for predicting the allowable moment, the
validation under the dynamic loading conditions should be conducted. In this
regards, the seismic analyses for cracked and uncracked pipe were performed
to calculate the maximum applied load on the cracked section for the purpose
of validation of the applicability of developed solutions.
5.2.1 Benchmark dynamic analysis using cracked pipe model
To predict the behavior of a cracked pipe under the dynamic loading
conditions, the nonlinearity of the materials and crack should be considered.
In this section, the methodology of time history analysis was employed to
conduct detailed analysis of the cracked pipe behavior considering
nonlinearity. Before applying to the validation of the developed formulation,
the time history analysis methods were verified with the experimental results.
⑤ This section has been based on the following conference paper:
Kim, Y., Oh, Y.-J., Park, H.-B., 2016b. The Conservatism of Leak Before Break Analysis in
Terms of the Applied Moment at Cracked Section, ASME 2016 Pressure Vessels and Piping
Conference. American Society of Mechanical Engineers, pp. V06AT06A075-
V006AT006A075.
101
5.2.1.1 Dynamic analysis methods for nonlinear piping system
This subsection describes the procedures of time history analysis using the
finite element (FE) models of the uncracked and cracked piping systems
subjected to a seismic loading. The time history analysis is one of the dynamic
analysis methods, and the process of solving the equation of motion as a
function of time (Chopra, 2007; Kim, 2014). This method can give more
accurate results compared with the response spectrum analysis that is
typically used in the seismic designs of structures. Key procedures and
parameters should be considered in the time history analysis as follows;
i) Finite element (FE) model
The piping system can be simulated by using the beam element to
calculate the applied moment at each point. The number of nodes
should be determined to make the appropriate vibration shapes feed
into the analysis.
For crack modeling, the “connector element” in ABAQUS 6.12
(Dassault Systémes, 2012) can be adopted at the position of a crack
(Zhang et al., 2010). The connector element can join the position of
two nodes and provide a rotational connection. To make the connector
element behave like a crack, the relation between load and
displacement due to the crack is applied as the behavior of the
connector element (Zhang et al., 2010). An additional connector
102
element needs to be used in parallel to limit the crack closure.
ii) Input of applied loading
The seismic inputs of piping system are the excitations of the anchors
or supports that are obtained from reactor building analyses using
ground acceleration time histories. The requirements regarding time
increment and frequency content for the design time histories to
achieve reliable estimations are stated in Standard Review Plan (SRP)
3.7.1 (US NRC, 2007c).
If the supports have different motions, the response time history of
each supports can be applied separately so that the effect of inertial
load and relative anchor motion can be considered simultaneously.
To consider the uncertainties of the natural frequency of a structure,
the analysis can be conducted three or five times for given time history
while changing the time increments (ASME, 2010b).
iii) Modal analysis
Before implementing the time history analysis, the dynamic
characteristics of structures must be evaluated using the modal
analysis. The appropriate boundary conditions and preloading need to
be reflected in the analysis model.
The information obtained from the results of the modal analysis
103
such as the natural frequency, mode shape and effective mass are
utilized in the dynamic analysis in a variety of ways. For example, by
examining the mode shapes and mass participation factors considering
the directions of applied excitations, the minimum and maximum
modes that should be considered in the time history analysis can be
determined. The appropriate time step for the time history analysis
depends on the maximum frequency to be considered.
iv) Consideration of damping
The effects of damping due to the energy dissipation can be considered.
Regulatory guide 1.61 (US NRC, 2007a) gives the acceptable
damping ratio to be used. In the case of the pipe subjected to a safety
shutdown earthquake (SSE), 4% of the damping ratio is recommended.
One of the common damping models for dynamic analysis is
Rayleigh damping model which describes damping ratio as follow:
2 2
nn
n
(5.19)
where ξn and ωn denote the damping ratio and the natural frequency
of the n-th mode, respectively. α and β are the damping coefficients.
The coefficients α and β can be determined using the natural
frequencies of two modes with specific damping ratio (Chopra, 2007).
These modes should be selected so that the dominant vibration mode
104
can be considered in the time history analysis.
v) Calculation of applied load
Using FE model and inputs data described above, one can implement
the time history analysis. The maximum value of the applied moment
at cracked section during loading can be used for calculating crack
opening displacement and allowable moment of a crack.
5.2.1.2 Verification of nonlinear dynamic analysis methods
To validate the applicability of the time history analysis methods using the
cracked pipe model described in the proceeding subsection, detailed analysis
of simulated seismic pipe system experiment (Experiment 1-1) of the second
international piping integrity research group (IPIRG-2) program was
conducted (Hopper et al., 1996; Scott et al., 1996).
The main purpose of Experiment 1-1 was to investigate the behavior
of piping system containing a circumferential surface crack under simulated
seismic loading condition. Figure 5.10 shows the FE model of the piping
system used in the experiment. The straight pipe was fabricated from ASTM
A710, Grade A, Class 3 pipe (sch. 100) and interconnected elbows were of
the type WPHY-65 (sch. 100 and 160). First of all, the straight pipe and elbow
were simulated using PIPE31 which in one of the beam element of the finite
element computer code ABAQUS. The mass of the valve (1950kg) and
105
restraint device (320kg) in the cracked section were represented as the point
mass and distributed mass respectively. The detail information about pipe
configurations are described in the reports (Hopper et al., 1996; Scott et al.,
1996), and Table 5.3 summarizes the material properties applied to the
analysis referred from ASME Code (ASME, 2010a).
For simulation of the crack, two connector elements were adopted that
represent the crack opening and crack closure, respectively, as described in
the preceding subsection. It was assumed that the crack could deform only in
the rotational direction. The measured crack moment versus rotation curve
was used as input for the behavior of the connector element in the crack
opening direction (see Figure 5.11). The decrease of load-carrying capacity
of the crack after reaching the maximum load was not considered.
Prior to the time history analysis, the modal analysis was conducted
first in which only the elastic behaviors of the crack and pipe were considered.
As the preloading, the dead weight, internal pressure (15.5 MPa), and
operating temperature (288 °C) were applied. Table 5.4 shows the
comparisons of the natural frequencies between measured and predicted
results. It can be seen that the analysis predicted the natural frequencies
accurately except the second mode. However, this may not affect the pipe
analysis significantly because the dominant mode shape of the 2nd mode is
in the vertical direction while the simulated seismic loading was applied in E-
W direction, and the piping system dominantly behaves in N-S and E-W
direction during excitation due to the boundary conditions.
The measured value 4.5% was applied as the damping ratio, and the
106
minimum and maximum mode were determined as the 1st (4.60 Hz) and 9th
(43.51 Hz) modes respectively to consider the behavior of the piping system
both in N-S and E-W directions. The Rayleigh damping coefficient in Eq.
5.19 then were calculated as α=2.354257 and β=0.000298.
Based on the results of the modal analysis, the time history analysis
was conducted. Before the dynamic analysis, the initial response (t=0 second)
was calculated using static analysis considering the dead weight, internal
pressure and thermal loading. Figure 5.12 shows the comparison of the
applied moment at the cracked section between the measured and predicted
results. It can be seen that overall waveform of vibration agrees well with the
experiment results. In the experiment, the wall penetration of the crack
occurred at 14.035 second. After that, the difference is observed since the
applied moment is reduced because of the decrease in load-carrying capacity
of crack that is not considered in the analysis. Nevertheless, the maximum
applied moment was accurately predicted (600.01 kN∙m) compared with the
measured value (597.66 kN∙m) within 0.5%. It can be concluded that the time
history method using the elastic-plastic cracked pipe model provides an
accurate estimation under the seismic loading conditions.
5.2.2 Validation of developed formulations using experimental
measurements and dynamic analysis results
In this subsection, an example analysis was conducted to validate the
developed formulations under the dynamic loading condition. The effective
107
applied moment predicted using the restraint coefficient was compared with
the experimental measurements of the experiment 1-1 of IPIRG-2 and
dynamic analysis results of section 5.2.1. According to the evaluation
procedure stated in section 4.4, the effective applied moment was calculated
as follows;
i) Calculate Mapp from the uncracked pipe analysis: An additional
dynamic analysis for an uncracked pipe was conducted as the
same process of section 5.2.1.2. The calculated applied moment
at the crack position is shown in Figure 5.18. The maximum value
of 699.87 kN∙m was determined as Mapp.
ii) Define the compliance of the crack, Gcrack: The test specimen
including the surface crack of IPIRG-2 piping system experienced
the plastic behavior. Thus, the compliance of the crack was
determined based on the elastic-plastic finite element analysis
using the 3D solid model which is depicted in Figure 5.16. The
rotation due to only a bending moment was considered because
the effect of axial force was negligible in the experiment. The
rotational compliance of crack was represented as a function of
applied bending moment as shown in Figure 5.17.
iii) Define the compliances of the pipes: To calculate the compliance
108
matrices, the piping system was separated into region 1 (from
anchor 1 to crack) and region 2 (from crack to anchor 2). Based
on the linear elastic FE analysis using beam model in Figure 5.10,
the compliance matrices were derived as follows;
,1
5.93E-04 6.56E-04 5.82E-06 -6.89E-10 -9.46E-10 1.71E-07
6.56E-04 1.30E-03 2.82E-06 -3.02E-10 -4.72E-10 3.33E-07
5.82E-06 2.82E-06 2.03E-03 -1.14E-07 -4.16E-07 9.77E-10
-6.89E-10 -3.02E-10 -1.14E-07 8.61E-11 1.33E-1pipeG
1 -1.04E-13
-9.46E-10 -4.72E-10 -4.16E-07 1.33E-11 1.04E-10 -1.63E-13
1.71E-07 3.33E-07 9.77E-10 -1.04E-13 -1.63E-13 1.00E-10
,2
3.00E-03 8.99E-04 6.96E-05 0.00E+0 3.51E-08 -4.54E-07
8.99E-04 3.68E-04 1.59E-05 4.63E-10 7.61E-09 -1.89E-07
6.96E-05 1.59E-05 6.15E-04 4.71E-08 2.75E-07 -7.61E-09
0.00E+0 4.63E-10 4.71E-08 5.20E-11 -5.32E-13 0.00E+pipeG
0
3.51E-08 7.61E-09 2.75E-07 -5.32E-13 1.48E-10 -3.84E-12
-4.54E-07 -1.89E-07 -7.61E-09 0.00E+0 -3.84E-12 1.04E-10
iv) Determine the restraint coefficient and calculate the effective applied
moment: When the plastic behavior is considered, the compliance
depends on the magnitude of applied load. Thus an iterative
calculation should be performed to obtain the effective applied
moment using Eq. 5.20.
, , ,
6,6
, , ,1detz
eff app Rest M eff app app
crack M eff app app
M C M M
MG M M
A
(5.20)
6,6
,1 ,2
minor of matrix A
pipe pipe crack
where M
A G G G
Using compliance matrices and Mapp of 699.87 kN∙m which are
109
given above, CRest,M and Meff,app were determined as 0.8511 and 595.69
kN∙m, respectively.
It was found that the prediction based on the developed procedure
(595.69 kN∙m) has good agreement with both the experimental data (597.66
kN∙m) and dynamic analysis (600.01 kN∙m). Therefore, the concept of
effective applied moment can be utilized for the pipe integrity assessment
under transient operating conditions.
5.2.3 Evaluation of effective applied moment for 3d pipe under
dynamic loading conditions
The purpose of this subsection is to verify that the developed solution is
available under the dynamic loading conditions from a practical perspective.
The effective applied moment at the cracked section was calculated using the
current practice of LBB analysis and the time history analysis introduced in
subsection 5.2.1, then compared with that obtained using the restraint
coefficient.
5.2.3.1 Evaluation model and input loading
The same model of the piping system with the experiment 1-1 of IPIRG-2
was employed. In this case, it was assumed that the whole pipe is fabricated
110
from 16 inch nominal diameter scheduled 100 TP 304 stainless steel and has
a circumferential through-wall crack. Three crack lengths (θ/π=0.250, 0.375,
0.500) were considered, and Figure 5.19 describes the relation between the
moment and rotation due to the crack obtained from the finite element
analysis using the model created with the continuum element as shown in
Figure 5.3. In the same manner with the previous section, this was applied to
the behavior of the connector element.
For the input seismic motion, the results of a reactor building analysis
from an earlier study (Kim, 2014) were applied. The reactor building analysis
was conducted using the measured data of El Centro earthquake
(Vibrationdata) to make the response time histories of the anchors and
supports. The geometries of containment building were referred from the
1000 MWe Korean Generation II pressurized water reactor (Optimized Power
Reacter-100, OPR-1000). The simplified FE model was prepared using shell
element without the consideration of the internal walls and other internal
structures as illustrated in Figure 5.20. The building was fabricated from post-
tensioned concrete, and 5% of the damping ratio was considered based on the
regulatory guide 1.61 (US NRC, 2007a).
The piping systems are actually located in a region from the base up
to a height of 27.5 m. Among this region, the bottom (0 m) and top (27.50 m)
positions were selected to extract the response time histories. To make the
piping system be subjected to large loading, it was assumed that the excitation
of 0 m is applied to the anchors and supports of the pipe model (see Figure
111
5.20) and that of 27.5 m is applied to the actuator location.
5.2.3.2 Evaluation methods
To verify that the developed solution is available under the dynamic loading
conditions from a practical perspective, three cases of analysis were prepared
and summarized in Figure 5.21. In this evaluation, only the seismic loading
was considered, and other normal operating loads were not included.
i) The current practice of LBB using response spectrum analysis and
seismic anchor motion (SAM) analysis:
This represents the current procedure of the LBB analysis. The applied
moments due to the inertial load (MRS) and seismic anchor motion
(MSAM) are calculated from the response spectrum analysis and the
static analysis using the relative displacement of anchors, respectively.
The combined applied moment (MCombined) can be determined as Eqs.
5.21 and 5.22 (US NRC, 2007b).
i i iCombined RS SAMM M M
(5.21)
2 2 2
1 2 3Combined Combined Combined CombinedM M M M
(5.22)
where subscript i is the i-th component of moment (i=1, 2, 3). For
response spectrum analysis, the response spectrum obtained from
building analysis in Figure 5.22 was applied. Since it was assumed
that the actuator is connected to different elevation with other supports,
112
the enveloped response spectrum was considered. In addition, to take
into account of the uncertainty of natural frequency of the structure,
the peaks were broadened (US NRC, 1973).
In the case of SAM analysis, the inputs data are the displacement
time histories of each anchor. Figure 5.23 and Figure 5.24 show the
displacement of two locations and relative displacement respectively
(Kim, 2014). The maximum applied moment calculated from SAM
analysis is MSAM.
ii) Time history analysis using linear elastic and elastic-plastic model:
The time history analysis tends to produce more realistic estimations
compared with the above case. In the time history analysis, the effect
of inertial load and relative anchor motion can be considered
simultaneously if the different time histories are applied to the anchors.
The analysis should be performed repeatedly while changing the time
interval of the same time history of input motion to reduce the effects
of natural frequency uncertainty (ASME, 2010b). The maximum
applied moment during the application of seismic load is then
determined as MTH,UcPipe for an uncrakced pipe and MTH,CPipe for a
cracked pipe. Both the linear elastic and elastic-plastic behaviors of
113
crack and pipe material are considered.
iii) Using the restraint coefficient
The developed restraint coefficient is the ratio of the effective applied
moment at the cracked section considering the boundary conditions to
the applied moment at the cracked section calculated from the
uncracked pipe analysis. Therefore, CRest,M can be used as a correction
factor to consider the presence of a crack to the uncracked pipe
analysis results by using following equations;
, ,eff app RS SAM Rest MM M M C (5.23)
, , ,eff app TH UcPipe Rest MM M C (5.24)
5.2.3.4 Evaluation results
Figure 5.25 describes the comparisons of the applied moment at the cracked
section as a function of the crack length for each analysis case. Regarding the
method of dynamic analysis, it is clear that the response spectrum analysis
with SAM analysis overestimates the response than the time history analysis.
If the presence of a crack is considered in the analysis, the applied
moment at the cracked section decreases regardless of analysis methods and
this is coincident with the discussions of the earlier studies (Kim, 2014; Scott
114
et al., 2002). The degree of the reduction increases with the crack length (θ/π).
For example, when the crack length θ/π is 0.25, the applied moment declines
5 % from the case of uncracked pipe. In case of 0.375 and 0.5 of θ/π, the
reduction ratio is about 10 % and 25 %, respectively. Additionally, the elastic-
plastic behaviors also reduce the applied moment. The reduction ratios are 35%
and 60% for the linear elastic and elastic-plastic time history, respectively.
When it comes to the practical perspective regarding the restraint
coefficient, two aspects can be inferred from the evaluation results. First, the
implementing of the restraint coefficient into the current practice LBB
analysis gives more accurate results of cracked pipe behavior without losing
the conservatism due to the overestimation of the response spectrum analysis.
Furthermore, this can be used to secure the margin of the existing pipe
analysis results.
Second, the restraint coefficient can enhance the efficiency of the time
history analysis. The corrected applied moments by using the restraint
coefficient (CRest,M∙MTH,UcPipe) agree with the results predicted from the time
history analysis using the cracked pipe (MTH,CPipe) for all cases (see Figure
5.25). It can be indicated that the time history analysis of the piping system
for various crack length can be replaced with a single uncracked pipe system
analysis using the restraint coefficient for various crack lengths. It may help
to improve the efficiency of the probabilistic fracture mechanics analysis or
seismic fragility analysis that requires a significant number of time-
consuming calculations.
115
Table 5.1 Dimensionless function H4B and H4T in the formula of rotation due
to crack for circumferential through-wall cracked pipe determined from
FEA
θ/π
n
2 3 5 7
R/t=10
H4B
0.125 0.063 0.087 0.126 0.148
0.250 0.357 0.407 0.443 0.445
0.500 1.094 0.854 0.585 0.433
H4T
0.125 0.053 0.073 0.100 0.110
0.250 0.306 0.333 0.302 0.249
0.500 0.817 0.623 0.437 0.331
116
Table 5.2 Loading conditions, material property, and pipe geometries considered for verification of the developed
formulation under the static loading conditions
Loading condition Elastic modulus [psi] Dn [inch] Rm/t Crack Length
[θ/π] (L1/DO, L2/DO)
Internal pressure (2320.6 psi)
Dead weight
Relative displacement
(x axis: 5 inch, y axis: -5 inch)
Thermal load (563 °F)
2.73E+7 12 5 0.25, 0.5
(1,9), (2,8) (3,7),
(4,6), (5,5), (6,4),
(7,3), (9,1)
117
Table 5.3 The material properties applied to the validation analysis of the experiment 1-1 IPIRG-2 program (ASME,
2010a)
(288 °C)
Elastic Modulus
Thermal Expansion
Density
Poisson’s ratio Instantaneous Mean
[MPa] [m/m/C] [m/m/C] [ton/mm3]
A710 (straight pipe) 185469.0 14.76 13.14 7.76E-09 0.3
WPHY 65 (elbow) 185469.0 14.76 13.14 7.76E-09 0.3
TP304 (crack) 176505.8 19.08 17.64 8.03E-09 0.31
118
Table 5.4 Comparisons of the natural frequencies of IPIRG-2 piping system
between measured data and FE analysis results
Natural frequency [Hz]
1st 2nd 3rd 4th
Experiment 4.5 8.5 14.2 19.2
Analysis 4.60 12.99 15.08 18.45
119
(a) Dependence of rCOD,LE on the crack length (Rm/t=5)
(b) Dependence of rCOD,LE on the crack length (Rm/t=10)
Figure 5.1 Comparisons of rCOD,LE predicted using the developed
formulations and linear elastic FEA – symmetric model (Miura, 2001)
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
/= 0.125 : FEA - Miura
/= 0.125 : Eng. Formula
/= 0.250 : FEA - Miura
/= 0.250 : Eng. Formula
/= 0.500 : FEA - Miura
/= 0.500 : Eng. Formula
Symmetric Restraint
Rm/t=5
Res
trai
nt
CO
D r
atio
, r C
OD
,LE
Normalized Restraint Length, L/D
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Res
trai
nt
CO
D r
atio
, r C
OD
,LE
Normalized Restraint Length, L/D
/= 0.125 : FEA - Miura
/= 0.125 : Eng. Formula
/= 0.250 : FEA - Miura
/= 0.250 : Eng. Formula
/= 0.500 : FEA - Miura
/= 0.500 : Eng. Formula
Symmetric Restraint
Rm/t=10
120
(a) Dependence of rCOD,LE on the crack length (L2/D=10)
(b) Dependence of rCOD,LE on the crack length (L2/D=5)
Figure 5.2 Comparisons of rCOD,LE predicted using the developed
formulations and linear elastic FEA – asymmetric model (Miura, 2001)
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Non-symmetric Restraint
Rm/t=10
L2/D=10
/= 0.125 : FEA - Miura
/= 0.125 : Eng. Formula
/= 0.250 : FEA - Miura
/= 0.250 : Eng. Formula
/= 0.500 : FEA - Miura
/= 0.500 : Eng. Formula
Res
trai
nt
CO
D r
atio
, r C
OD
,LE
Normalized Restraint Length, L1/D
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Non-symmetric Restraint
Rm/t=10
L2/D=5
/= 0.125 : FEA - Miura
/= 0.125 : Eng. Formula
/= 0.250 : FEA - Miura
/= 0.250 : Eng. Formula
/= 0.500 : FEA - Miura
/= 0.500 : Eng. Formula
Res
trai
nt
CO
D r
atio
, r C
OD
,LE
Normalized Restraint Length, L1/D
121
(c) Dependence of rCOD,LE on the crack length (L2/D=1)
Figure 5.2 Comparisons of rCOD,LE predicted using the developed
formulations and linear elastic FEA – asymmetric model (Miura, 2001)
(Continued)
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 /= 0.125 : FEA - Miura
/= 0.125 : Eng. Formula
/= 0.250 : FEA - Miura
/= 0.250 : Eng. Formula
/= 0.500 : FEA - Miura
/= 0.500 : Eng. Formula
Res
trai
nt
CO
D r
atio
, r C
OD
,LE
Normalized Restraint Length, L1/D
Non-symmetric Restraint
Rm/t=10
L2/D=1
122
Figure 5.3 3D FE model of a circumferential through-wall cracked pipe used
for tabulations of new dimensionless functions (H4T, H4B)
123
(a) Dependence of rCOD,EP on the crack length (L1/D= L2/D =1)
(b) Dependence of rCOD,EP on the crack length (L1/D= L2/D =10)
Figure 5.4 Comparisons of rCOD,EP predicted using the developed
formulations and elastic-plastic FEA – symmetric model (Kim, 2008)
0 50 100 150 200 2500.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Restraint Length : L1/D=L
2/D=1
Material : Ref. (Kim, 2008)
Rm/t=10
Res
trai
nt
CO
D r
atio
, r C
OD
,EP
Nominal Tensile Stress, t [MPa]
/=0.125 : FEA - Kim
/=0.125 : Eng. Calc.
/=0.250 : FEA - Kim
/=0.250 : Eng. Calc.
/=0.500 : FEA - Kim
/=0.500 : Eng. Calc.
0 50 100 150 200 2500.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Res
trai
nt
CO
D r
atio
, r C
OD
,EP
Nominal Tensile Stress, t [MPa]
/=0.125 : FEA - Kim
/=0.125 : Eng. Calc.
/=0.250 : FEA - Kim
/=0.250 : Eng. Calc.
/=0.500 : FEA - Kim
/=0.500 : Eng. Calc.
Restraint Length : L1/D=L
2/D=10
Material : Ref. (Kim, 2008)
Rm/t=10
124
(c) Dependence of rCOD,EP on material property (L1/D= L2/D =10)
- Mat. Ref. : Reference material property for 304SS (Kim, 2008)
- Mat. 3 : CF8M(288°C) material property (Kim, 2008)
- Mat. 7 : A106(288°C) material property (Kim, 2008)
Figure 5.4 Comparisons of rCOD,EP predicted using the developed
formulations and elastic-plastic FEA – symmetric model
(Kim, 2008) (Continued)
0 50 100 150 200 2500.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Restraint Length : L1/D=L
2/D=10
/=0.25
Rm/t=10
Res
trai
nt
CO
D r
atio
, r C
OD
,EP
Nominal Tensile Stress, t [MPa]
Mat. Ref. : FEA - Kim
Mat. Ref. : Eng. Calc.
Mat. 3 : FEA - Kim
Mat. 3 : Eng. Calc.
Mat. 7 : FEA - Kim
Mat. 7 : Eng. Calc.
125
Figure 5.5 3D FE model of 3D piping system containing a circumferential
through-wall crack used for verification of the developed formulation
Figure 5.6 FE model using beam element of 3D piping system to calculate
the pipe compliance for verification of the developed formulation
L/DO=10
L/DO=10
L1
L2
R/DO=1.5
Anchor 1
Anchor 2
xz
y
Anchor 2
Crack
L/DO=10
L/DO=10
L1
L2
R/DO=1.5
Anchor 1
xz
y
126
Figure 5.7 Comparisons of applied moment and axial force at the cracked
section calculated from finite element analysis
Figure 5.8 Comparisons of applied nominal stress at the cracked section due
to bending moment and axial force calculated from finite element analysis
0 2 4 6 8 10-4.0x10
2
-2.0x102
0.0
2.0x102
4.0x102
6.0x102
8.0x102
Ap
pli
ed m
om
ent
at c
rack
ed s
ecti
on
[k
N-m
]
Crack position from Anchor 1 (L1/D
o)
Applied moment
Uncracked pipe
Cracked pipe (=0.25)
Cracked pipe (=0.50)8.0x10
2
1.0x103
1.2x103
1.4x103
Applied axial force
Uncracked pipe
Cracked pipe (=0.25)
Cracked pipe (=0.50)
Ap
pli
ed a
xia
l fo
rce
at c
rack
ed s
ecti
on
[k
N]
0 2 4 6 8 10-4.0x10
2
-2.0x102
0.0
2.0x102
4.0x102
6.0x102
8.0x102
Nom
inal
str
ess
at c
rack
ed s
ecti
on [
MP
a]
Crack position from Anchor 1 (L1/D
o)
bending
axial force
Uncracked pipe
Cracked pipe (=0.25)
Cracked pipe (=0.50)
127
(a) Bending Moment
(b) Axial force
Figure 5.9 Comparisons of the restraint coefficient and the ratio of load
reduction calculated from finite element analysis
0 2 4 6 8 10-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Res
trai
nt
effe
ct o
n a
pli
ed m
om
ent
Mef
f,ap
p/M
app
Crack position from Anchor 1 (L1/D
o)
Formula FEA
=0.25
=0.50
0 2 4 6 8 100.6
0.7
0.8
0.9
1.0
1.1
1.2
Res
trai
nt
effe
ct o
n a
pli
ed a
xia
l fo
rce,
Fef
f,ap
p/F
app
Crack position from Anchor 1 (L1/D
o)
Formula FEA
=0.25
=0.50
128
Figure 5.10 FE model used for analysis of simulated seismic pipe system
analysis of IPIRG-2 program
129
Figure 5.11 Applied moment and rotation due to the crack of experimental
results and input data used for connector element behavior
130
(a) From 0 to 15 second
(b) Full time history
Figure 5.12 Comparisons of applied moment time history at the cracked section between experiment and analysis result
0 5 10 15-1000
-800
-600
-400
-200
0
200
400
600
800
Ap
pli
ed M
om
ent
at C
rack
ed S
ecti
on
[k
N-m
]
time [sec]
Experiment
Analysis
0 5 10 15 20 25-1000
-800
-600
-400
-200
0
200
400
600
800
Ap
pli
ed M
om
ent
at C
rack
ed S
ecti
on
[k
N-m
]
time [sec]
Experiment
Analysis
131
Figure 5.13 Comparisons of reaction load time history at Node 6 between experiment and analysis results
0 5 10 15 20 25-1000
-750
-500
-250
0
250
500
750
1000
Node
6 R
eact
ion l
oad
[kN
]
time [sec]
Experiment
Analysis
132
(a) N-S
(b) E-W
Figure 5.14 Comparisons of displacement load time history at Elbow 3 between experiment and analysis results
0 5 10 15 20 25-100
-75
-50
-25
0
25
50
75
100
Dis
pla
cem
ent
at E
lbow
3
(N
-S)
[mm
]
time [sec]
Experiment
Analysis
0 5 10 15 20 25-150
-100
-50
0
50
100
150
Dis
pla
cem
ent
at E
lbow
3
(E-W
) [m
m]
time [sec]
Experiment
Analysis
133
(a) N-S
(b) E-W
Figure 5.15 Comparisons of displacement load time history at Node 21 between experiment and analysis result
0 5 10 15 20 25-30
-15
0
15
30
Dis
pla
cem
ent
at N
ode
21
(N-S
) [m
m]
time [sec]
Experiment
Analysis
0 5 10 15 20 25-150
-100
-50
0
50
100
150
Dis
pla
cem
ent
at N
ode
21
(E-W
) [m
m]
time [sec]
Experiment
Analysis
134
Figure 5.16 3D FE model of pipe containing a surface crack to calculate the
compliance of a crack
Figure 5.17 Elastic-plastic compliance of the surface crack (Equivalent
crack length of (θ/π) = 0.383)
0.0 2.0x108
4.0x108
6.0x108
8.0x108
0.00E+000
1.00E-011
2.00E-011
3.00E-011
GC
rack
,PE [
rad
/N-m
]
Applied Moment [N-m]
Equivalent Crack length ()
0.383
135
Figure 5.18 Applied moment at cracked section calculated from uncracked pipe analysis for experiment 1-1 of IPIRG-2
program
0 5 10 15-1000
-800
-600
-400
-200
0
200
400
600
800
Appli
ed M
om
ent
at C
rack
ed S
ecti
on [
kN
-m]
time [sec]
Uncracked pipeMapp = 699.887 kN-m
136
Figure 5.19 Applied moment and rotation due to the crack applied as the
behavior of connector element
0.00 0.02 0.04 0.06 0.08 0.100
200
400
600
800
Ap
pli
ed M
om
ent
[kN
-m]
Rotation due to the crack [rad]
Crack length ()
0.250
0.375
0.500
137
Figure 5.20 Geometries of containment building of OPR-1000 type plant
and FE model (Kim, 2014)
Pipe
region
Anchor &
support
Actuator
138
Figure 5.21 Schematic diagram of procedures and methods to calculate the effective applied moment to validation of the
developed formulation
Displ. Time history
Floor response
spectrum
Containment Analysis
Ground acc.
Acc. Time history
Current practice
(w/o crack, LE)
Response Spectrum Analysis
Using time history analysis
(w/ crack, LE&EP)
MRS
Seismic Anchor Motion Analysis
MTH
MSAM
│MRS│+│MSAM│
Using restraint coefficient
(w/ crack, LE)(│MRS│+│MSAM│)∙CRest
139
Figure 5.22 Acceleration response spectrum obtained from containment
building analysis (Kim, 2014)
10-1
100
101
102
1
2
Acc
eler
atio
n [
g]
Frequency [Hz]
Anchors and supports
Actuator
Enveloped spectrum
140
Figure 5.23 Displacement time histories of two selected locations obtained from containment building analysis (Kim,
2014)
Figure 5.24 Relative displacement time histories between two selected locations obtained from containment building
analysis (Kim, 2014)
0 10 20 30 40 50-120
-100
-80
-60
-40
-20
0
20
40
60
80
100
120
Dis
pla
cem
ent
[mm
]
Time [s]
Anchors and Supports
Actuator
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30-5
-4
-3
-2
-1
0
1
2
3
4
5
Rel
ativ
e D
ispla
cem
ent
[mm
]
Time [s]
141
Figure 5.25 Comparisons of the reduction ratios of the applied moment at
the cracked section predicted using the time history analysis, restraint
coefficient compared with the current practice of LBB
0.0 0.1 0.2 0.3 0.4 0.50.0
0.2
0.4
0.6
0.8
1.0
1.2
Using the restraint coeff. [CRest
(MRS+M
SAM)]
Time history analysis [MTH
] - Linear elastic model
Time history analysis [MTH
] - Elastic plastic model
Using the restraint coeff. [CRest
MTH
] -Linear elastic
Rat
io o
f ap
pli
ed m
om
ent
to c
urr
ent
pra
ctic
e
Crack length []
0
200
400
600
800
Appli
ed m
om
ent
at c
rack
ed s
ecti
on [
kN
-m]
142
143
Chapter 6 Application of Developed Formulations⑥
As stated earlier, the aspects of restrained boundary conditions include the
significant favorable influence on the crack instability prediction and
unfavorable impact on the prediction of leakage size crack by an
underestimation of crack opening displacement (COD). The combined results
of the restraint effects from above two aspects were investigated by example
LBB calculations (Ghadiali et al., 1996) from both deterministic and
probabilistic basis only for the small and large diameter pipe (4.5 and 28 inch).
In this Chapter, the applicability of the restraint coefficient in LBB
design has been investigated first. Then, an example leak before break
analysis is conducted with consideration of the restraint effect on both the
COD calculation and crack stability analysis, for more practical cases than
that was considered in the literature.
⑥ This section has been based on the following conference paper:
Kim, Y., Oh, Y.-J., Park, H.-B., 2015. Effect of Pipe Restraint on the Conservatism of Leak-
Before-Break Design of Nuclear Power Plant, ASME 2015 Pressure Vessels and Piping
Conference. American Society of Mechanical Engineers, pp. V06AT06A082-
V006AT006A082.
144
6.1 Applicability of developed formulations in LBB design
In the previous chapter, it was demonstrated that the restraint coefficient can
predict the reduction ratio of applied moment or axial force at the cracked
section by comparing with the FE analysis results and experimental data. In
this section, the additional FE analyses were conducted to compare the COD
and J integral that are the primary fracture mechanics parameters in LBB
design (see Figure 1.2) to validate the applicability of the developed
formulations from practical aspects.
6.1.1 Validation methods
To validate the applicability of the restraint coefficient to the LBB analysis,
three cases of FE analyses were conducted. The detailed descriptions of each
case are summarized in Figure 6.1. For the loading condition, only the
distributed vertical load along the pipe was considered.
i) Case 1: Not considering pipe restraint effect
This case represents a current LBB evaluation procedure. The applied
moment (Mapp) at the cracked section of an uncracked pipe can be
simply calculated by solving beam equation for pipe subjected a
distributed load. Mapp was then applied to the pipe end of a 3D FE
model containing a circumferential throw-wall crack under free-ended
145
boundary conditions to calculate COD or J-integral.
ii) Case 2: Considering pipe restraint effect – 3D FEA including
restrained boundary conditions
In this case, 3D FEA models of the pipe containing a circumferential
TWC under the fixed-ended boundary conditions were prepared.
Distributed vertical load which can represent various types of loading
in nuclear piping systems was directly applied to the 3D FE models to
calculated COD or J-integral.
iii) Case 3: Considering pipe restraint effect – Using the restraint
coefficient
A 3D FEA model under free-ended boundary conditions same with
that of case 1 was employed. Instead of the applied moment of an
uncracked pipe (Mapp), the effective applied moment (Meff,app) was
used, which is calculated by multiplying the linear elastic restraint
coefficient by Mapp.
A commercial finite element analysis code, ABAQUS (Dassault
Systémes, 2012) was used. A through-wall circumferential cracked pipe was
simulated as a half-model using the 20-noded continuum element with
reduced integration shown in Figure 6.2, and a focused mesh was applied at
the crack tip. The multi-point constraint option in ABAQUS was utilized to
146
make the displacement and rotation at the nodes on the pipe end plane equal
to those of a reference node on the axis of the pipe. The effects of geometric
nonlinearity were ignored.
For the loading condition, the distributed vertical loads and internal
pressures were considered. The distributed load was applied as a type of the
gravity to the lower part of the continuum model. In the case of the internal
pressure, a pressure and corresponding axial force were applied to the pipe
inner surface and pipe ends, respectively. In addition, the half value of the
pressure was applied to the crack face.
TP316 stainless steel 12 inch diameter pipe which is used in a typical
primary side of the nuclear power plant was considered. The high temperature
tensile property (327 ℃) of TP316 is represented in Figure 6.3. Three crack
lengths (θ/π=0.125, 0.25, 0.5) were prepared, and a symmetric model with a
crack in the center of the pipe (L1/Do:L2/Do=5:5, 10:10, 20:20) and an
asymmetric model with a crack in the off-center of the pipe (L1/Do:L2/Do
=1:10, 1:20) were considered. Table 6.2 summarizes details on the analysis
cases.
6.1.2 Validation results of COD and J-integral
Figure 6.4 and Figure 6.5 show the comparison results of COD and J-integral,
in which the internal pressure was not included. X axis of each graph means
the applied moment at the cracked section calculated from the uncracked
147
elastic pipe (Mapp). The difference in the results between for case 1 and 2 is
due to the effects of the restrained boundary conditions. Since the case 2
considered the restraint effect, the COD and J integral are lower than for case
1. The degree of the pipe restraint is increased with the increase of the crack
length (θ/π) and the decrease of the restraint length (L1/Do or L2/Do). Analysis
results in which the internal pressure was included are represented in Figure
6.6 and Figure 6.7. Values of COD and J-integral corresponded with those the
case of without an internal pressure at the pressure equivalent moment.
General trends that were observed in both cases agreed well with each other.
It should be noted that in the case of the symmetric model with a crack
length of 0.125(θ/π), the case 2 tends to overestimate than the case 1 when a
large amount of moment is applied (See graph (a) of Figure 6.5 and graph (a)
of Figure 6.7). If a crack is at the position where the anticipated moment is
relatively low in the piping system, the effective applied moment can be
increased because of the load redistribution due to the plastic deformation of
the pipe. In virtually, this does not likely to occur since the region subjected
to a low value of loading has the low probability of the crack initiation, and
is not considered as a critical location of evaluations. Nevertheless, this
should be carefully discussed under the extremely beyond design basis
conditions.
Results predicted by using the linear elastic restraint coefficient (case
3) agreed well with case 2 while the case 3 overestimated COD and J-integral
when the applied load was large enough for plastic deformations to occur. If
the elastic-plastic restraint coefficient is used, results of case 3 could be close
148
to those of case 2 even in the plastic region.
Generally, it can be seen that the results predicted using the linear
elastic restraint coefficient were closer to the realistic case (case 2) than the
results calculated through the current LBB analysis method (case 1), slightly
overestimating than the case 2. It was confirmed that the restraint coefficient
could enhance the accuracy of the prediction of COD and J integral
considering the pipe restraint effect without losing the conservatism for LBB
evaluation.
149
6.2 Effects of pipe restraint on LBB evaluation
6.2.1 Piping evaluation diagram
As discussed earlier, the overestimations of COD and the applied moment at
the cracked section can affect LBB design differently in terms of the
conservatism. Therefore, to take the pipe restraint effect into account when
conducting the LBB analysis, the decrease of COD and the increase of the
load-carrying capacity must be considered simultaneously. In this section,
combined results of two effects on LBB design were investigated using the
piping evaluation diagram (PED).
The PED was proposed by Fabi et al. (Fabi and Peck, 1994) to conduct
effective LBB design. Comparing the piping analysis results with the PED, it
can be easily checked whether a piping system satisfies the LBB requirements.
LBB analysis processes which were accounted for PED are based on the
NUREG/CR-1061 and Standard Review Plan 3.6.3 (US NRC, 1985, 2007b)
as follows;
i) Determine the leakage size crack corresponding to the detectable
leakage with a margin of 10
ii) Conduct the crack stability analysis for a pipe which has the leakage
size crack with a margin of 2 and is subjected to faulted load.
iii) Conduct the crack stability analysis for a pipe which has the leakage
size crack and is subjected to faulted load with a margin of 1.4.
150
Through this process, the piping evaluation diagram could be derived
as shown in Figure 6.8. When a result of a piping analysis is below the lower
bound of the allowable moment, the pipe is satisfied the LBB design criteria.
6.2.2 Evaluation methods of the pipe restraint effects on LBB
To investigate the effect of the pipe restraint on LBB design, the PEDs
were derived for the case of i) using current LBB procedure (restraint is not
considered), ii) considering only the restrained COD and iii) considering both
the restrained COD and effective applied moment, respectively. The detailed
descriptions of the calculation method are following.
i) Crack opening displacement
The COD of an unrestrained pipe was calculated using the formula in
the ductile fracture handbook (Zahoor, 1989). The restrained COD
was determined based on the calculation process in the subsection
5.1.1.1 using the linear elastic restraint coefficient.
ii) Leakage size crack
The leak rate for a postulated crack length and COD was calculated
using the PICEP code developed by Electric Power Research Institute
(Norris et al., 1984). When the calculated leak rate equals to 10 gpm,
the leakage size crack was determined.
151
iii) Crack stability analysis
Because the material of the model was austenitic stainless steel base
metal, the limit load formula in Eq. 6.1 was used for the crack stability
analysis according to the Standard Review Plan 3.6.3 (US NRC,
2007b).
4 4
,
2
22sin sin
4
0.5 /
2
o i f
instability unrestrained
m
m f
mm in
m
R RM
R
where P
RP P
R t
(6.1)
Minstability,unrestrained, Pm, Pin and σf denotes the instability moment for
unrestrained pipe, the membrane stress, the internal pressure, and the
flow stress of the material, respectively. To consider the restraint effect
on the crack stability analysis, Eq. 6.1 was corrected as:
4 4
,
, ,1 ,
2
, ,1 ,
212sin sin
4
0.5 /
2
o i f
instability restrained
Rest M D LE m
m f
mm Rest M D LE in
m
R RM
C R
where P
RP C P
R t
(6.2)
where the increase of the load-carrying capacity of a crack was
reflected to Pin and Minstability,restrained.
The geometries and material of pipe and the operating conditions were
referred from the typical reactor coolant system of PWR. Other information
about the evaluation matrix and material properties are summarized in Table
152
6.2.
6.2.3 Evaluation results
Figure 6.9 shows the leakage size crack and the piping evaluation diagram of
each case depending upon the normal operating moment (MNOP). The
instability moment and the normal operating moment were normalized by a
limit of the design moment (MASME). The limit of design moment was defined
using Equation 10 of NB-3600, ASME B&PV Code Section III (ASME,
2010b) which is the requirement for piping systems under the normal
operating conditions.
1 1 2
2
23 , 1
2
in OASME m
O
P DIM S C C C
C D t
(6.3)
Sm and Pin are the design stress intensity and the internal pressure,
respectively. C1 and C2 are secondary stress indices. MASME was derived
assuming that the stress term induced by thermal gradient is zero.
Graphs (a) to (f) of Figure 6.9 show the results for the 12 inch pipe. If
the pipe restraint effect on COD is considered, the length of leakage size crack
increases due to the narrowed flow path. When only the restrained COD was
considered (blue lines), instability loads were lower than those predicted
through the current LBB method (black lines, not considering pipe restraint
effect) at the same MNOP. However, if the restrained COD and the effective
applied moment at the cracked section were considered simultaneously (red
lines), instability moment dramatically increased for the same crack length.
153
From this, it was confirmed that the pipe restraint has great influence mainly
by the increase of the load-carrying capacity, rather than the reduction of the
COD.
The degree of the restraint effect increased as the crack length
increased or restraint length decreased. This corresponds to decreases in the
value of the restraint coefficient. Regarding the restraint length, the restraint
effect in (5,5) was insignificant than (1,20) or (1,20). This indicates that the
restraint effect is dominated by the restraint length of shorter side. These
trends are observed for both pipe diameter cases.
The objective of the example LBB analysis is to evaluate the
combined results of effects of restrained COD and increase of the load-
carrying capacity. The analysis results indicated that the restraint effect on the
applied moment has more significant influence on the LBB evaluation than
the restraint effect on COD. Therefore, the current LBB evaluation procedure,
with no attention to the pipe restraint effect, can predict conservative results
compared to the case in which the restraint effect is considered for the
conditions examined herein. In addition, if the restraint effect is implemented
into the current practice of deterministic LBB analysis using the developed
formulations, the piping system can be shown to possess greater safety
margins. Because the value of linear elastic restraint coefficient is greater than
the elastic-plastic restraint coefficient, when plastic deformations occur, the
margin might actually be more significant.
154
Table 6.1 Matrix of analysis for calculation of COD and J-integral
Type Rm [in] Rm/t Internal Pressure [psi] Crack length [θ/π] Restraint length (L1/Do, L2/Do)
Symmetric
5.72 5 0, 2320 0.125, 0.25, 0.5
(5,5), (10,10), (20,20)
Asymmetric (1,10), (1,20)
155
Table 6.2 Matrix of analysis and material properties used for LBB evaluations
Type
Pin
[psi]
Temp.
[°F]
Rm
[in]
Rm/t
Restraint length
(L1/Do , L2/Do)
Material
E
[psi]
α n
σy
[psi]
σu
[psi]
12inch Pipe
2320 563
5.72 5 (1,1), (5,5),
(10,10) (1,5),
(1,10), (1,20)
TP316
(327℃)
2.73∙107 2.361 7.848 23,150 67,380
16inch Pipe 7.2 5
156
Figure 6.1 Summary of analysis case to demonstrate the applicability of the restraint coefficient in LBB design
Not considering
pipe restraint effect Case 1
Considering
pipe restraint effect
Case 2
Case 3
157
Figure 6.2 3D FE model of pipe with a circumferential through-wall crack
used for COD and J-integral calculations
Figure 6.3 Tensile property of TP316 stainless steel
L1
L2
Cracked Section
xz
y
0.00 0.02 0.04 0.06 0.08 0.100
1x104
2x104
3x104
4x104
5x104
Tru
e S
tres
s [p
si]
True Strain
TP 316 SS
327 C
158
(a) Symmetric model – θ/π=0.125
(b) Symmetric model – θ/π=0.25
Figure 6.4 Comparisons of COD to validate the restraint coefficient
(Internal pressure was not included)
0.0 4.0x105
8.0x105
1.2x106
1.6x106
0.000
0.002
0.004
0.006
0.008
0.010
TP 316 base LB
= 0.125
Rm=5.72 inch
Rm/t=5
Pin=0 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 3.0x105
6.0x105
9.0x105
1.2x106
0.000
0.005
0.010
0.015
0.020
TP 316 base LB
= 0.250
Rm=5.72 inch
Rm/t=5
Pin=0 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
159
(c) Symmetric model – θ/π=0.5
(d) Asymmetric model – θ/π=0.125
Figure 6.4 Comparisons of COD to validate the restraint coefficient
(Internal pressure was not included) (Continued)
0.0 2.0x105
4.0x105
6.0x105
8.0x105
0.00
0.01
0.02
0.03
0.04
0.05
0.06
TP 316 base LB
= 0.500
Rm=5.72 inch
Rm/t=5
Pin=0psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
4.0x105
8.0x105
1.2x106
1.6x106
0.000
0.002
0.004
0.006
0.008
0.010
TP 316 base LB
= 0.125
Rm=5.72 inch
Rm/t=5
Pin=0
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
160
(e) Asymmetric model – θ/π=0.25
(f) Asymmetric model – θ/π=0.5
Figure 6.4 Comparisons of COD to validate the restraint coefficient
(Internal pressure was not included) (Continued)
0.0 3.0x105
6.0x105
9.0x105
1.2x106
0.000
0.003
0.006
0.009
0.012
0.015
TP 316 base LB
= 0.250
Rm=5.72 inch
Rm/t=5
Pin= 0 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 2.0x105
4.0x105
6.0x105
8.0x105
0.000
0.005
0.010
0.015
0.020
0.025
0.030
TP 316 base LB
= 0.500
Rm=5.72 inch
Rm/t=5
Pin= 0 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
161
(a) Symmetric model – θ/π=0.125
(b) Symmetric model – θ/π=0.25
Figure 6.5 Comparisons of J-integral to validate the restraint coefficient
(Internal pressure was not included)
0.0 5.0x105
1.0x106
1.5x106
2.0x106
2.5x106
0
200
400
600
800
1000
TP 316 base LB
= 0.125
Rm=5.72 inch
Rm/t=5
Pin=0 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 5.0x105
1.0x106
1.5x106
2.0x106
0
300
600
900
1200
1500
TP 316 base LB
= 0.250
Rm=5.72 inch
Rm/t=5
Pin=0 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
162
(c) Symmetric model – θ/π=0.5
(d) Asymmetric model – θ/π=0.125
Figure 6.5 Comparisons of J-integral to validate the restraint coefficient
(Internal pressure was not included) (Continued)
0.0 3.0x105
6.0x105
9.0x105
1.2x106
1.5x106
0
300
600
900
1200
1500TP 316 base LB
= 0.500
Rm=5.72 inch
Rm/t=5
Pin=0psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 5.0x105
1.0x106
1.5x106
2.0x106
2.5x106
3.0x106
0
200
400
600
800
1000
TP 316 base LB
= 0.125
Rm=5.72 inch
Rm/t=5
Pin=0 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
163
(e) Asymmetric model – θ/π=0.25
(f) Asymmetric model – θ/π=0.5
Figure 6.5 Comparisons of J-integral to validate the restraint coefficient
(Internal pressure was not included) (Continued)
0.0 5.0x105
1.0x106
1.5x106
2.0x106
2.5x106
0
200
400
600
800
1000
1200TP 316 base LB
= 0.250
Rm=5.72 inch
Rm/t=5
Pin= 0 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 3.0x105
6.0x105
9.0x105
1.2x106
1.5x106
0
200
400
600
800
1000
1200
TP 316 base LB
= 0.500
Rm=5.72 inch
Rm/t=5
Pin= 0 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
164
(a) Symmetric model – θ/π=0.125
(b) Symmetric model – θ/π=0.25
Figure 6.6 Comparisons of COD to validate the restraint coefficient
(Internal pressure was included)
0.0 3.0x105
6.0x105
9.0x105
1.2x106
0.000
0.002
0.004
0.006
0.008
0.010
TP 316 base LB
= 0.125
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 2.0x105
4.0x105
6.0x105
8.0x105
1.0x106
0.00
0.01
0.02
0.03
0.04
TP 316 base LB
= 0.250
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
165
(c) Symmetric model – θ/π=0.5
(d) Asymmetric model – θ/π=0.125
Figure 6.6 Comparisons of COD to validate the restraint coefficient
(Internal pressure was included) (Continued)
0 1x105
2x105
3x105
4x105
5x105
6x105
0.00
0.05
0.10
0.15
0.20
TP 316 base LB
= 0.500
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 3.0x105
6.0x105
9.0x105
1.2x106
0.000
0.002
0.004
0.006
0.008
0.010
TP 316 base LB
= 0.125
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
166
(e) Asymmetric model – θ/π=0.25
(f) Asymmetric model – θ/π=0.5
Figure 6.6 Comparisons of COD to validate the restraint coefficient
(Internal pressure was included) (Continued)
0.0 2.0x105
4.0x105
6.0x105
8.0x105
1.0x106
0.000
0.005
0.010
0.015
0.020
0.025
0.030
TP 316 base LB
= 0.250
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0 1x105
2x105
3x105
4x105
5x105
6x105
0.00
0.05
0.10
0.15
0.20
TP 316 base LB
= 0.500
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
Cra
ck o
pen
ing d
ispla
cem
ent
[in]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
167
(a) Symmetric model – θ/π=0.125
(b) Symmetric model – θ/π=0.25
Figure 6.7 Comparisons of J-integral to validate the restraint coefficient
(Internal pressure was included)
0.0 5.0x105
1.0x106
1.5x106
2.0x106
2.5x106
0
300
600
900
1200
1500
1800
TP 316 base LB
= 0.125
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 3.0x105
6.0x105
9.0x105
1.2x106
1.5x106
0
400
800
1200
1600
TP 316 base LB
= 0.250
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
168
(c) Symmetric model – θ/π=0.5
(d) Asymmetric model – θ/π=0.125
Figure 6.7 Comparisons of J-integral to validate the restraint coefficient
(Internal pressure was included) (Continued)
0.0 3.0x105
6.0x105
9.0x105
1.2x106
1.5x106
0
2000
4000
6000
8000
10000
TP 316 base LB
= 0.500
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[5:5] [10:10] [20:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 5.0x105
1.0x106
1.5x106
2.0x106
2.5x106
0
300
600
900
1200
1500
TP 316 base LB
= 0.125
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
169
(e) Asymmetric model – θ/π=0.25
(f) Asymmetric model – θ/π=0.5
Figure 6.7 Comparisons of J-integral to validate the restraint coefficient
(Internal pressure was included) (Continued)
0.0 5.0x105
1.0x106
1.5x106
2.0x106
0
300
600
900
1200
1500TP 316 base LB
= 0.250
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
0.0 3.0x105
6.0x105
9.0x105
1.2x106
1.5x106
0
3000
6000
9000
12000
15000
TP 316 base LB
= 0.500
Rm=5.72 inch
Rm/t=5
Pin=2320 psi
J in
tegra
l [p
si-i
n]
Applied moment at the crack postion of uncracked pipe [lbf-in]
[1:10] [1:20] (=L1/D
o:L
2/D
o)
Case 1
Case 2
Case 3
170
Figure 6.8 Schematic diagram of the piping evaluation diagram
LBB Satisfied
Margin of 1.4 on MFault
Margin of 2.0 on θl
MNOP
MFault,Allow
171
(a) 12 inch, L1/Do=1, L2/Do=5
(b) 12 inch, L1/Do=1, L2/Do=10
Figure 6.9 Effect of the restrained COD and the effective applied moment
on LBB evaluation
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=12 inch, R/t = 5
L1=1D, L2=5D
MASME
=4.01E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=12 inch, R/t = 5
L1=1D, L2=10D
MASME
=4.01E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
172
(c) 12 inch, L1/Do=1, L2/Do=20
(d) 12 inch, L1/Do=1, L2/Do=1
Figure 6.9 Effect of the restrained COD and the effective applied moment
on LBB evaluation (Continued)
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=12 inch, R/t = 5
L1=1D, L2=20D
MASME
=4.01E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=12 inch, R/t = 5
L1=1D, L2=1D
MASME
=4.01E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
173
(e) 12 inch, L1/Do=5, L2/Do=5
(f) 12 inch, L1/Do=10, L2/Do=10
Figure 6.9 Effect of the restrained COD and the effective applied moment
on LBB evaluation (Continued)
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=12 inch, R/t = 5
L1=5D, L2=5D
MASME
=4.01E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=12 inch, R/t = 5
L1=10D, L2=10D
MASME
=4.01E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
174
(g) 16 inch, L1/Do=1, L2/Do=5
(h) 16 inch, L1/Do=1, L2/Do=10
Figure 6.9 Effect of the restrained COD and the effective applied moment
on LBB evaluation (Continued)
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=16 inch, R/t = 5
L1=1D, L2=5D
MASME
=7.99E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=16 inch, R/t = 5
L1=1D, L2=10D
MASME
=7.99E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
175
(i) 16 inch, L1/Do=1, L2/Do=20
(j) 16 inch, L1/Do=1, L2/Do=1
Figure 6.9 Effect of the restrained COD and the effective applied moment
on LBB evaluation (Continued)
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=16 inch, R/t = 5
L1=1D, L2=20D
MASME
=7.99E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=16 inch, R/t = 5
L1=1D, L2=1D
MASME
=7.99E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
176
(k) 16 inch, L1/Do=5, L2/Do=5
(l) 16 inch, L1/Do=10, L2/Do=10
Figure 6.9 Effect of the restrained COD and the effective applied moment
on LBB evaluation (Continued)
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=16 inch, R/t = 5
L1=5D, L2=5D
MASME
=7.99E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
0.0 0.2 0.4 0.6 0.8 1.00.00
0.25
0.50
0.75
1.00
1.25
1.50TP 316
Press=16MPa, Temp.=295ºC
D=16 inch, R/t = 5
L1=10D, L2=10D
MASME
=7.99E+6 [lbf-in]
MIn
stab
ilit
y/M
AS
ME
MNOP
/MASME
Using current LBB method
Considering the restrained COD
Considering the restrained COD
& the effective applied moment
0.0
0.2
0.4
0.6
LSC 2*LSC
Using current LBB method
Considering the restrained COD
Lea
kag
e S
ize
Cra
ck [
]
177
Chapter 7 Conclusions and Future Work
7.1 Summary and conclusions
In this dissertation, a set of generalized formulations has been developed to
take into account the effect of pipe restraint for consistent analysis of the crack
opening displacement and crack stability of nuclear piping containing a
circumferential crack in order to enhance the confidence in the Leak-Before-
Break (LBB) characteristics under both static and dynamic loading conditions.
The impetus of the pipe fracture evaluation methods is the
improvement of safety margin and accuracy by refining current analytical
models, and there remain several issues that may significantly impact. This
dissertation is concerned with one of high priority issues, the piping restraint
effect on the crack behavior, in response to heightened attention to the effect
of beyond design basis earthquake on nuclear safety.
The earlier studies attempted to develop the analytical expression to
evaluate the amount of the restraint effect on the calculation of COD and
crack stability analysis. However, the solutions are difficult to apply in the
practical case because of the limitation of the pipe geometries and applied
loading conditions. Therefore, it is desired to develop a unified formulation
to determine the effective applied moment at a postulated cracked section
considering the boundary conditions that can be utilized to a balanced
analysis of both COD and flaw stability.
178
This dissertation mainly serves to the aims for the development of
generalized solutions that readily enable balanced evaluations of the restraint
effect and have following characteristics:
1. Using the developed formulations, one can evaluate the ratio of the
applied bending moment and axial force at a postulated crack position
of cracked pipe (Meff,app or Feff,app) to those of uncracked pipe (Mapp or
Fapp). In other words, the change in compliance of system due to the
presence of a crack can lead the redistribution of load over the entire
piping, and the extent can be predicted using the restraint coefficient.
Therefore, by substituting the effective applied load to the solutions
for evaluation of crack opening displacement or flaw stability that do
not consider the constraint effects, the reliability of current pipe
evaluation method can be improved.
2. The restraint coefficient consists of the compliance of circumferential
through-wall or surface crack and the compliances of two pipe
segments that can be determined regardless of the piping shapes or
boundary conditions. Therefore, the developed formulations are
generally applicable to irrespective of piping configurations.
3. Based on the validation with experimental data and finite element
analysis results, it was confirmed that the restraint coefficient is
available for both static and dynamic loading conditions: including
the pressure induced bending, distributed load, relative displacement
of the supports and seismic inertial loading. Therefore it can be
introduced to the pipe integrity assessment methods under design
179
basis conditions.
4. Both linear elastic and elastic-plastic behavior can be reflected, and
various combinations also available. For instance, under the design
basis conditions, an assumption that crack and pipe do not experience
the plastic behavior can lead conservative results while the case that
allows the non-linear behavior of crack only is more accurate. If the
applied load is large enough to cause the plastic deformation of the
pipe, the elastic-plastic compliances of crack and pipe must be
considered.
Finally, using the developed formula, the effect of restraint on the LBB
evaluation was analyzed. The analysis results indicated that the restraint effect
on the applied moment has more significant influence on the crack stability
analysis than on COD. Therefore, the current LBB evaluation procedure,
which does not consider the pipe restraint effect, can make more conservative
prediction compared with the case in which the restraint effect is considered.
The application of developed generalized formulations can have two
principal benefits of the practical significance. First, by allowing the restraint
effect implemented into the current practice of deterministic LBB analysis,
the nuclear piping system could possess greater safety margin. Second, the
time history analysis of the piping system for various crack length can be
replaced with a single uncracked pipe system analysis with the restraint
coefficient developed in this dissertation. Therefore, the generalized
formulations can reduce analysis time and cost for piping integrity under both
static and dynamic conditions while improving the accuracy of prediction.
180
This can facilitate the further development of the probabilistic fracture
mechanics analysis or seismic fragility analysis that requires a significant
number of time-consuming calculations.
181
7.2 Future work
Up to the early 1980s, the prevailing principle for the piping design of the
NPP was a double-ended guillotine break (DEBG) of a high energy piping
system containing a circumferential crack. Considering the potential
consequence of high energy piping rupture compounded by the lack of
knowledge on the fracture mechanical understanding, the extremely
conservative DEGB principle was adopted in the NPP design rules
(Wilkowski et al., 1998). In recent years, there have been comprehensive
efforts to employ the probabilistic fracture mechanics (PFM) techniques as a
part of probabilistic risk assessments (Rudland et al., 2016). To this end, the
development of the computer code to calculate the probability of pipe rupture
is being carried out in multiple paths. The generalized formulations on pipe
restraint effect may be well applied to these probabilistic approaches in the
future.
The basic structure of the eXtremely Low Probability of Rupture
(xLPR) code is described in Figure 7.1, as one of two primary PFM computer
codes. It is also concerned with the crack opening displacement and crack
stability, in which the applied bending moment is a significant element. The
current version of xLPR does not consider the effect of pipe restraint, and
results of section 6.2 indicate that the conservatism is inherent in the current
PFM code.
In this regards, future work from this dissertation that may have high
significance can be the application of the restraint effect on the calculation of
182
probability of pipe rupture. The application of analytical generalized
formulations to the probabilistic approach can be done without time-
consuming and costly effort. The expected benefit on improving safety
margin can be very significant, as discussed with deterministic cases herein.
183
Figure 7.1 Structure of eXtremely Low Probability of Rupture Code Version 2.0 (US NRC, 2015)
184
185
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195
Abbreviation
COD Crack opening displacement
DEBG Double-ended guillotine break
EP Elastic-plastic
IPIRG International Piping Integrity Research Group
LBB Leak before break
LE Linear elastic
LSC Leakage size crack
NPP Nuclear Power Plant
PIB Pressure induced bending
PFM Probabilistic fracture mechanics
PSR Periodic safety review
RS Response spectrum
SAM Seismic anchor motion
TH Time history
xLPR eXtremly Low Probability of Rupture
CRest,F,2D,m Force restraint coefficient for the 2D pipe (m=LE,EP)
CRest,F,m Force restraint coefficient for the generalized 3D pipe
(m=LE,EP)
CRest,M,1D,m Moment restraint coefficient for the 1D pipe (m=LE,EP)
CRest,M,2D,m Moment restraint coefficient for the 2D pipe (m=LE,EP)
CRest,M,m Moment restraint coefficient for the generalized 3D pipe
196
(m=LE,EP)
Do Pipe outer diameter
E Elastic modulus of pipe material
E E for plane stress, E/(1-ν2) for plane strain
F Applied axial force
Fapp Applied axial force at a cracked section calculated from
uncracked pipe analysis
FC Applied axial force on pure crack
Feff,app Effective applied axial force at a cracked section considering
the effects of pipe restraint
Fi Applied force (i=x, y, z)
FReact,Rest Reaction force induced by pipe restraint at fixed ends
Gcrack,m Compliance matrix of a crack (m=LE, EP)
Gcrack,i,Fi,m Compliance of a crack for force (i=x, y, z, m=LE,EP)
Gcrack,j,Mi,m Compliance of a crack for moment (i=x, y, z, j=θ, ϕ, ψ, m=LE,
EP)
Gpipe,m Compliance matrix of a pipe (m=LE, EP)
Gpipe,i,Fi,m Compliance of a pipe for force (i=x, y, z, m=LE, EP)
Gpipe,j,Mi,m Compliance of a pipe for moment (i=x, y, z, j=θ, ϕ, ψ, m=LE,
EP)
I Moment of inertia
J J-integral
L Half-length of pipe (2L=L1+L2)
L1 Restraint length-1, pipe length of one side of the crack
L2 Restraint length-2, pipe length of other side of the crack
LN1 Normalized restraint length-1, (L1/2L)
LN2 Normalized restraint length-2, (L2/2L)
197
M Applied bending moment
Mapp Applied moment at a cracked section calculated from
uncracked pipe analysis
MASME Limit of the design moment
Mi Applied moment (i=x, y, z)
Minstability Instability moment of a crack
Mi,j Minor of a matrix (Determinant of smaller matrix formed by
eliminating i-th row and j-th column from a matrix)
MCombined Combined applied moment
MC Applied bending moment on pure crack
MC,CPipe Applied moment at a cracked section for a cracked pipe
MC,UcPipe Applied moment at a cracked section for an uncracked pipe
Meff,app Effective applied moment at a cracked section considering the
effects of pipe restraint
MNOP Normal operating moment
MReact,Rest Reaction moment induced by pipe restraint at fixed ends
MPress,eq Pressure equivalent moment
MPress,eq,eff Effective pressure equivalent moment due to pipe restraint
MRS Applied moment calculated from response spectrum analysis
MSAM Applied moment calculated from seismic motion analysis
MTH Applied moment calculated from time history analysis
n Ramberg-Osgood parameter
P Applied axial tension load
Pin Internal pressure
CODr Ratio of the COD of a restrained pipe to the COD of an
unrestrained pipe
198
,COD LEr CODr for the linear elastic analysis
,COD EPr CODr for the elastic-plastic analysis
TAPP Applied tearing modulus
Rm Pipe mean radius
Ri Pipe inner radius
Ro Pipe outer radius
t Pipe thickness
w Uniformly distributed vertical load per unit length
xi, yi, zi Displacement of pipe of region i (i=1, 2)
α, β Rayleigh damping parameters
δUnrestrained,LE COD of a free-ended pipe for the linear elastic analysis
δUnrestrained,EP COD of a free-ended pipe for the elastic-plastic analysis
δRestrained,LE COD of a fixed-ended pipe for the linear elastic analysis
δRestrained,EP COD of a fixed-ended pipe for the elastic-plastic analysis
εo Reference strain
ψC,M,LE Linear elastic rotation due to crack of a free-ended pipe caused
by a bending moment
ψC,M,EP Elastic-plastic rotation due to crack of a free-ended pipe
caused by a bending moment
ψC,T,EP Elastic-plastic rotation due to crack of a free-ended pipe
caused by an axial tension load
σf Flow stress
θ Half-crack length of a circumferential through wall crack
θi, ϕi, ψi Rotation of pipe of region i (i=1, 2)
ζn Damping ratio of n-th mode
ν Poisson’s ratio
ωn Natural frequency of nth mode
199
초 록
원자력 발전소의 고에너지 배관은 파단될 경우 다른 기기에 큰 영향을
미칠 수 있기 때문에 발전소 수명기간 동안 배관의 건전성을 확보하는
것이 매우 중요하다. 또한 2011년 후쿠시마 제1 원자력발전소 사고 이후
설계기준을 초과하는 사건에 대한 원전 안전성 재확인의 필요성이
높아지고 있는 실정이다. 이에 배관 건전성 평가 방법론의 신뢰성을
높이고 보수성을 정량화하는 것이 주요 현안이 되었다.
최근 경수로 설계에 적용되고 있는 파단전누설 설계를 포함한
배관 파괴역학 평가에서 중요한 요소는 균열열림변위의 계산과 균열
안정성 평가이다. 현재 절차에서는 양 끝단의 회전변위가 구속되어 있지
않은 단순한 배관을 가정하지만, 실제 배관계는 기기 및 지지대 등에
의해 구속되어 있기 때문에 균열의 거동은 제한될 수 있다. 이러한
배관계의 구속효과를 고려할 경우 균열의 한계하중 및 누설균열길이는
기존 절차보다 크게 계산될 수 있고, 이는 배관 건전성 평가의 보수성의
측면에서 서로 다른 영향을 가진다.
구속효과의 영향을 정량적으로 평가하기 위한 선행연구는 크게
두 가지 측면에서 진행되었다. 먼저, 배관계의 형상 및 하중조건이
균열의 한계하중에 미치는 영향을 수식으로 표현하기 위해 다양한
예제에 대한 분석이 수행되었으나 실제 평가 절차에 적용할 수 있는
일반식의 형태로는 도출되지 않았다. 또한 배관의 구속으로 인한
균열열림변위의 감소 비율을 계산할 수 있는 선형탄성 및 선형-완전소성
평가식이 개발되었으나, 관통균열이 포함된 직관에 내압으로 인한
굽힘하중이 인가되는 경우에 대해서만 적용 가능한 형태이다.
200
선행연구에서는 구속효과가 균열의 한계하중 및 균열열림변위에
미치는 영향을 따로 다루었으나, 이는 모두 균열부에 실제로 작용하는
유효 하중이 감소하기 때문에 나타나는 현상이다. 따라서 가상의 균열
위치에 작용하는 유효 하중을 평가할 수 있다면, 위 두 계산을 포함한
다방면의 배관 건전성 평가에 활용할 수 있을 것이다. 이에 본
논문에서는 배관계의 구속효과가 균열부 작용하중에 미치는 영향을
정량적으로 평가할 수 있는 새로운 일반식을 도출하고자 하였다. 따라서
본 논문의 연구질문은 아래와 같다.
1) 배관계의 구속효과가 균열부 유효작용하중에 미치는 영향을
평가식으로 나타낼 수 있는가?
2) 개발된 평가식은 배관계 형상 및 작용하중의 조건에 관계없이
적용 가능한가?
3) 개발된 평가식을 적용하여 계산한 균열부 유효작용하중은 정하중
및 동하중 조건에서 배관계 실험결과 및 유한요소해석 결과와
일치하는가?
4) 평가식을 파단전누설 설계를 포함한 배관 건전성 평가에 활용할
경우 기대효과는 무엇인가?
먼저, 선행연구와 같이 관통균열을 갖는 직관이 내압으로 인한
굽힘하중을 받는 조건을 가정하고 새로운 유효작용하중 평가식을
도출하였다. 이를 복잡한 배관계에 적용할 수 있는 형태로 확장시키기
위해, 길이와 같은 배관 형상 관련 변수를 배관계의 컴플라이언스로
대체하여 일반식을 개발하였다. 또한 평가식은 압력유기굽힘 하중뿐만
아니라 분포하중(자중, 지진으로 인한 관성하중) 및 변위하중(지진으로
인한 변위하중, 열하중) 조건 하에서도 적용될 수 있음을 확인하였다.
개발된 평가식을 검증하기 위해 정하중 및 동하중 조건에서
201
유한요소해석을 수행하였다. 먼저 정적 해석에서는 발전소 정상운전 시
배관이 받게 되는 하중을 고려하였으며 배관계의 구속효과가 균열부
작용하중 및 균열열림변위에 미치는 영향을 계산하여 평가식과
비교하였다. 또한, 균열 안정성 평가에서는 사고상황을 가정하기 때문에
지진을 포함한 동하중 조건에서의 검증이 필요하다. 이에 가상지진하중
하의 균열배관 모사실험 결과 및 동적 유한요소 해석 결과와 검증하여
동적 하중 조건에도 평가식이 유효하게 적용될 수 있음을 확인하였다.
마지막으로 개발된 평가식을 균열열림변위 및 한계하중 계산에
적용하여 배관계 구속효과가 파단전누설 평가에 미치는 영향을
검토하였다. 그 결과 배관계의 구속으로 인해 균열 불안정 하중이
증가하는 효과가 균열림변위 감소하는 정도보다 훨씬 크게 나타나며,
기존 파단전 누설이 보수적인 결과를 도출하고 있음을 확인하였다.
위와 같이 본 논문에서는 배관계 구속효과 및 균열로 인한
배관계의 컴플라이언스 변화를 반영하여 가상의 원주방향 균열의 위치에
유효하게 작용하는 하중을 계산할 수 있는 평가식을 개발하였으며,
배관의 형상 및 작용하중 조건에 관계없이 일반적으로 적용 가능함을
입증하였다. 본 평가식을 기존 결정론적 배관 건전성 평가에 적용할
경우 균열배관 거동 예측에 대한 정확도를 높여 설계 여유도를 확보할
수 있다. 또한 보다 간단한 방법으로 균열부에 작용하는 하중을 평가할
수 있기 때문에 복잡한 배관계에 대한 시간이력해석을 수행하기 전 예비
평가 단계에 적용하거나, 확률론적 배관 건전성 평가 및 지진 기기
취약도 해석에 변수로 사용되는 균열부 작용하중 값을 도출하는 데에도
활용될 수 있을 것이다.
주요어: 배관계 구속효과, 파단전누설 평가, 균열 열림변위, 균열 안정성
평가, 균열배관 동하중 해석, 균열부 유효작용하중 평가식
학 번: 2014-30195