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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





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





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


a circumferential crack subjected to a distributed load for

development of the moment restraint coefficient based on the

compliance approach .......................................................... 77


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


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


(Internal pressure was included) ....................................... 164


(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

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:




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)

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


V006AT006A075.




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




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


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


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


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


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;

, ,

, ,


crack M


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

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:

, ,

, ,


crack M


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.

, ,








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


app

app


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


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


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

, ,








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


app

app


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


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

]



Limit moment

of crack

Limit of design

moment

85

(c) Plastic pipe + Plastic crack


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

]



Limit moment

of crack

Limit of design

moment

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:




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


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)


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


/= 0.500 : FEA - Miura


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


/= 0.250 : FEA - Miura


/= 0.500 : FEA - Miura


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)


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


/= 0.250 : FEA - Miura


/= 0.500 : FEA - Miura


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


Rm/t=10

L2/D=5

/= 0.125 : FEA - Miura


/= 0.250 : FEA - Miura


/= 0.500 : FEA - Miura


Res

trai

nt

CO

D r

atio

, r C

OD

,LE


121

(c) Dependence of rCOD,LE on the crack length (L2/D=1)


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


/= 0.250 : FEA - Miura


/= 0.500 : FEA - Miura


Res

trai

nt

CO

D r

atio

, r C

OD

,LE



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)


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


/=0.125 : FEA - Kim

/=0.125 : Eng. Calc.

/=0.250 : FEA - Kim

/=0.250 : Eng. Calc.

/=0.500 : FEA - Kim

/=0.500 : Eng. Calc.


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)


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


2/D=10

/=0.25

Rm/t=10

Res

trai

nt

CO

D r

atio

, r C

OD

,EP


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



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]


o)

bending

axial force

Uncracked pipe



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


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


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]

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:




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


(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]


[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


(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]


[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]


[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



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]


[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]


[1:10] [1:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

161




(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]


[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]


[5:5] [10:10] [20:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

162





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]


[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]


[1:10] [1:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

163





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]


[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]


[1:10] [1:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

164




(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]


[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]


[5:5] [10:10] [20:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

165




(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]


[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]


[1:10] [1:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

166





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]


[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]


[1:10] [1:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

167




(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]


[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]


[5:5] [10:10] [20:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

168





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]


[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]


[1:10] [1:20] (=L1/D

o:L

2/D

o)

Case 1

Case 2

Case 3

169





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]


[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]


[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


& the effective applied moment

0.0

0.2

0.4

0.6

LSC 2*LSC



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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


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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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


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





0.0

0.2

0.4

0.6

LSC 2*LSC



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)

185

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2015 Regulatory Information Conference.

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(United States). Div. of Engineering Technology; Battelle, Columbus,

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Circumferential Crack Opening Displacements: A Rotational Stiffness

Approach, ASME 2015 Pressure Vessels and Piping Conference.

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V006AT006A079.

Zahoor, A., 1989. Ductile fracture handbook. Electric Power Research

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Zhang, T., Brust, F., Shim, D., Wilkowski, G., Nie, J., Hofmayer, C., 2010.

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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)

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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

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,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

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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

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