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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 12월서울대학교 대학원
에너지시스템공학부 원자핵공학 전공
김 예 지
김예지의 공학박사 학위논문을 인준함
2016년 12월위 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
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
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
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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
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
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
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
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
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
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
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
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
①Until 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.
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.
1.3 Effects of restraint on cracked pipe behavior
②The 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
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.
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.
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)
eXtremelyLow 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, θ
lLBB satisfied
M
NOP(Normal operation)
Crack length θ
M
fault(Faulted condition)
Leak rate = Detectable?
Stable?
2θ
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
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?
Meff J-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 2 2
APP
f
L EJ
T 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
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).
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
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
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 k M
(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
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.
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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.
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
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
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
Figure 2.1 Conceptual diagram of the piping system of IPIRG program experiment 1.3-7 (Schmidt et al., 1992)
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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)
Figure 2.3 Schematic diagram of a restrained pipe containing a circumferential through-wall crack and finite element model (Rahman et al.,
1995a)
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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)
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)
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)
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)
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
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 rest restrained
unrestrained
C M
M
(3.1)
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;
rCOD restrained
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.
Figure 3.1 Diagram of research process
Dynamic load Static load
New Formulation for Meff,app
(Pressure-induced bending case benchmark)
Improvement New Formulation based on Compliance approach
Development of Generalized Formulation Part 1: Development of Effective
Applied Moment Formulation
Benchmark Dynamic Analysis using Cracked Pipe Model Part 2 : Validation of
Developed Formulation
Meff,appComparison under Dynamic Loading Conditions Experimental Result vs Dynamic analysis
vs Formulation
1D piping COD Comparison
with FEA Results
(Pressure-induced bending case benchmark)
Part 3 : Impacts on Pipe Integrity Evaluation
2D piping 3D piping
PIB
Meff,appComparison under Static Loading Conditions
Other types of
loading
Connector Beam
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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
C M
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
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.
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4.1 Effective applied moment formulation for pipe subjected to pressure induced bending
③The 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.
compliance (Gcrack,ψ,M) was defined for the hinge representing a crack., as follows:
, ,
cracked pipe uncracked pipe crack M
G M 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 M
