Reliability Analysis Of Pipelines Containing Cracks And ......rupture of the pipeline. Based on fatigue reliability analysis, statistical methods are applied to assess the probability

Reliability Analysis Of Pipelines Containing Cracks And

Corrosion Defects

Main authors

Lie Zhang, Robert Adey C M BEASY Ltd

United Kingdom

[email protected]

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© Copyright 2008 IGRC2008

1. ABSTRACT

This paper is concerned with predicting the probability of failure (POF) of pipelines containing

cracks and corrosion defects. The current work was conducted within the framework of the

European Union NATURALHY project.

The aim of the NATURALHY project is to investigate the possibility of using the existing natural

gas transmission pipelines to deliver hydrogen or natural gas/hydrogen mixtures. Hydrogen has

been demonstrated to accelerate defect growth which may affect the safety of pipeline and

make it more expensive to operate and maintain. Failure of a structural member such as a

pipeline may occur when a crack becomes unstable or metal loss corrosion leads to leakage or

rupture of the pipeline. Based on fatigue reliability analysis, statistical methods are applied to

assess the probability of failure of pipelines containing cracks like defects and corrosion defects.

A software package has been developed based on the stochastic approach to assess the POF of

the gas pipeline due to the existence of cracks and corrosion defects. The inspection and repair

procedures are also included in the tool. With various parameters such as defect sizes, material

properties, internal pressure modelled as uncertainties, a reliability analysis based on BS 7910

level-2 failure assessment diagram is conducted through stratified Monte Carlo simulation.

Inspection and repair programmes and realistic pipeline maintenance scenarios can be

simulated. In the data preparation process, the accuracy of the probabilistic definition of the

uncertainties is crucial as the results show that the POF is very sensitive to certain variables

such as the defect sizes and the defect growth rate. The POF for a single defect with known

dimensions and the POF for a system containing multiple defects can be computed separately.

Various inspection and repair criteria are available in the Monte Carlo simulation whereby an

optimal maintenance strategy can be achieved by comparing different combinations of repair

procedures. In addition, the hazard function can be obtained based on the POF results, so that

the POF calculation tool can be used to satisfy certain target reliability requirement. Several

examples are presented in the paper comparing the POF for a natural gas pipeline with the POF

for a pipeline containing natural gas/hydrogen mixtures.

Nomenclature ()f probability distribution function c half crack length

a crack depth B pipe wall thickness

( )g x ( ) 0g x ≤ represents the failure domain Φ normal distribution function

ix stochastically independent variable PK primary stress intensity factor

SK secondary stress intensity factor thK∆ threshold stress intensity factor

ρ plastic correction factor maxrL permitted limit of rL

Yσ yield strength of material Uσ ultimate tensile strength

ICK Toughness of material refσ reference stress

n number of loading cycles *

N total number of simulations

( )f

p n cumulative POF of a single defect q total number of cracks

AN average number of cracks repaired

pfC probable cost of have a failure

|f totalP probability of failure for multiple defects

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C O N T E N T S

1. Abstract ................................................................................................. 2

2. Introduction ........................................................................................... 4

3. Theory.................................................................................................... 4

3.1. Fundamentals of structural reliability theory ...........................................................4

3.2. Failure assessment for corrosion defects and cracks.................................................5

3.3. Modelling of inspection and repair program.............................................................7

4. Monte-Carlo simulation as POF evaluation method................................. 9

4.1. Direct Monte-Carlo simulation ...............................................................................9

4.2. Stratified Monte-Carlo simulation method .............................................................10

4.2.1. Random number generation..........................................................................10 4.2.2. Stratified Monte-Carlo simulation ..................................................................11

5. Sample problems.................................................................................. 12

5.1. Probability of failure calculation for corrosion defects .............................................12

5.2. Probability of failure calculation for cracks ............................................................14

6. Conclusions .......................................................................................... 15

7. Acknowledgements .............................................................................. 15

8. References ........................................................................................... 16

9. List of Tables ........................................................................................ 17

10. List Of Figures ...................................................................................... 17

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2. INTRODUCTION

Currently major research effort is focused on the use of hydrogen as an energy vector both in

Europe and the US. An obvious pragmatic solution to transporting hydrogen, which will be

necessary to make the European hydrogen economy feasible, is to transport a mixture of

natural gas and hydrogen in the existing natural gas pipeline network. Within the European

project NATURALHY†, 39 European partners have combined their efforts to assess the effects of

the presence of hydrogen on the existing gas network. Key issues are durability of pipeline

material, integrity management, safety aspects, life-cycle and socio-economic assessment and

end-use. The work described in this paper was performed within the NATURALHY work package

on ’Integrity Management’.

Failure of a structural member such as a pipeline may occur when a crack or a corrosion defect

propagates in an unstable manner to cause leakage or rupture of the pipeline. Failure analysis

combined with a probabilistic approach has been utilized in many fields involving important

structural components such as pressure vessels and nuclear piping. Based on the failure

analysis for cracks and corrosion defects, statistical methods can be applied to assess the

reliability of pipeline containing crack-like defects [1], in other words, to provide a figure which

represents the probability that a pipeline will fail. One aim of the Naturalhy project is to assess

the durability and integrity of natural gas infrastructures for transporting and distributing

hydrogen/natural gas mixtures. In particular, the mechanical integrity of the gas piping is a

matter of great importance for both economical and safety reasons. Among all the parameters

the probability of failure is one of the most important factors for the integrity management of

pipeline infrastructures. Although evidence suggests that crack-like defects are most likely to be

affected by the presence of hydrogen due to hydrogen embrittlement the probability of failure

calculation for corrosion defects is also included in this paper for the sake of completeness.

3. THEORY

3.1. Fundamentals of structural reliability theory

According to the reliability theory if one and only one defect exists in a pipeline, the failure

probability is given by:

( ) ( ) ( ) ( )1

11 1

( ) 0 NNf X X N N

g x xP f x f x d x d x

⋅⋅⋅ ≤= ⋅⋅⋅ ⋅ ⋅⋅∫ (3.1)

where 1 Nx x⋅ ⋅ ⋅ are random variables such as defect sizes, yield strength, defect growth

parameters and applied stresses. ( )Nx N

f x denotes the probability density function of the input

variable N

x . The integration will be performed over the failure domain where 1( ) 0Ng x x⋅ ⋅ ⋅ ≤ .

1( )Ng x x⋅ ⋅ ⋅ is also called a limit state in some cases. For the case of crack-like defects, semi-

elliptical surface cracks and the through-thickness cracks will be considered. The initial surface

crack has the crack depth a and the crack length 2c as shown in Figure 1. The above

integration is rarely solved directly once the number of unknowns rises above 3. Therefore, in

order to solve practical engineering problems alternative approaches are required.

††NATURALHY is an Integrated Project, co-financed by the European Commission’s Sixth Framework Programme

(2002-2006) for research, technological development and demonstration (RTD). For more details please visit

www.naturalhy.net

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Figure 1 Geometry of surface/through thickness crack model used in the analysis

One such approach is called First Order Reliability Method (FORM). The concept behind the

approach is that instead of integrating the density functions directly a reliability index β is

computed to define the reliability of the structure thereby the probability of failure being

obtained. β is computed through the following equation [2].

( )f

P β= Φ − (3.2)

where Φ is the standard cumulative normal distribution function. However, there are some restrictions and disadvantages associated with FORM. First, the limit state equation has to be

continuous and smooth. Secondly, all variables must be continuous. Thirdly, all non-normally

distributed variables must be transformed to normal distribution. These conditions are often

infeasible in solving a practical engineering problem and some times multiple solutions/design

points exist and thus a more reliable method is needed to tackle the problem.

One promising approach is called Monte-Carlo simulation (MCS), which is essentially an

approximation of the exact integration but it is very flexible and reliable as long as there are

enough samples. The details of MCS and its enhanced version will be discussed in section 4.

3.2. Failure assessment for corrosion defects and cracks

For pipeline industry leakage normally refers to the situation where a defect penetrates through

the wall of a pipeline so the gas inside start leaking from the pipe. If a leak becomes unstable,

although not always the case, it could cause rupture, meaning a section of a pipe is torn apart

as a result of the fast propagation of the through-thickness defect. Different levels of failure

severity such as leakage or rupture etc. can be incorporated into the POF calculation. For

example the transition between leakage and rupture is modelled in the simulation. For each

failure mode a limit state function is defined.

For corrosion defects two levels of severity are considered: i.e. leakage and rupture. The limit

state function for a leakage failure takes the form:

( )g x r P= − (3.3)

where r is the estimated pressure resistance and P is the internal pressure of the pipeline.

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2

burstW S

rD

×= (3.4)

1

1burst flow

d

WS Sd

m W

−

= −

×

(3.5)

20.8

1l

mD W

= +×

(3.6)

where W is the wall thickness, D for pipe diameter, burstS for burst stress, flowS for flow

stress, d for depth of a defect, m for Folias factor and l is the total axial length of the

corrosion defect.

For cracks the basic failure criterion is based on the BS 7910 level two failure assessment

diagram (FAD). Equation (3.7) is used to describe the FAD curve, which has been shown in

Figure 2, where the region under the curve is considered safe. As with corrosion defects two

levels of severity are taken into consideration i.e. leakage and rupture, therefore a surface crack

model and a through thickness model are used in the analysis to calculate the probability of

failure for these two situations

Kr=f(Lr)

Lr

Kr

0.20

0.40

0.60

0.80

1.00

0.20 0.40 0.60 0.80 1.00 1.20 1.400

Safe region

Plastic collapse

cut-off

Brittle fracture dominated region

Figure 2 The BS7910 FAD curve used to define the limit state function

{ }2 6max

max

for (1 0.14 ) 0.3 0.7exp( 0.65 )

for 0

r r r r r

r r r

L L K L L

L L K

≤ = − + −

> = (3.7)

where rK measures the proximity to brittle fracture and rL represents the likelihood of

plastic collapse. The failure criterion includes both brittle fracture and plastic collapse. For

BS7910 level 2A FAD, they are given by:

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P Sr

IC

ref

r

Y

K KK

K

L

ρ

σ

σ

+= +

=

(3.8)

ρ is a parameter that takes plastic interaction between primary and secondary stress into

consideration. For materials that exhibit a yield discontinuity (referred to as Lüders plateau) rL

is restricted to 1.0 if no additional data is available [3]. Otherwise it is calculated through:

max2

Y ur

Y

Lσ σ

σ

+= (3.9)

Since the cumulative probability of failure over a given timeframe is required, crack propagation

due to cyclic loading must be included. The PARIS law with a threshold th

K∆ is selected to

calculate the crack length and depth with regard to the corresponding number of load cycles:

0 for <

for

th

m

th

K Kda

dn C K K K

∆ ∆=

∆ ∆ ≥ ∆ (3.10)

where C and m are fatigue growth parameters. An equivalent equation applies for the

second axis of the semi-ellipse. However it has been observed during studies that the crack

length hardly changes during fatigue analysis for a surface crack until it penetrates through the

pipe wall and becomes a through-thickness crack. The initial crack depth and length are

modelled as variables and the fatigue property parameters can also be modelled as random

variables based on certain distributions. As crack propagation could lead to fracture or leakage

of the pipeline after a certain period of time, f

P is a function of loading cycles n .

( )f

P f n= (3.11)

fP denotes the cumulative probability which monotonically increases with time or load cycles.

The inclusion of inspection and repair program can slow down this process so as to meet certain

target reliability targets.

3.3. Modelling of inspection and repair program

As part of the maintenance program of pipelines, inspections are performed using intelligent

‘pigs’ to detect defects in the pipeline. However, due to the sensitivity of the pigs not all defects

can be detected. Also it is not practical and economical to repair all the cracks detected

therefore according to Bayes' theory the process of inspection and repair at a given time

interval will change the distribution of defect depths and lengths in the pipeline. The exact

distribution will depend upon the repair strategy adopted, the frequency of inspection and the

sensitivity of the inspection tool. Not all the remaining defects will lead to failure but only those

missed by the inspection tool could cause either leakage or rupture of the pipeline. In order to

illustrate the nature of the inspection and repair program an event tree for crack-like defects

has been plotted to help understand the continuous updating process. From Figure 3, we can

clearly see that only the defects not repaired or missed by the inspection tool will contribute to

the cumulative probability of failure.

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Load cyclesn=0 n1 n2 n3

D

ND

R

NR

1st Inspection 2nd Inspection 3rd Inspection

D

ND

D

ND

R

NR

R

NR

Maintenance event tree:n,n1,n2..: load cyclesD: cracks detected; ND: cracks not detected;R: cracks repaired; NR: cracks not repaired;

Figure 3 Maintenance event tree for crack like defects

The Probability of Detection (POD) is mainly dependant on the defect depth and is usually

treated as an increasing function of defect depth and defined as an exponential function [4].

/ 1a

D aP e

λ−= − (3.12)

where a is the defect depth and λ is a parameter defining the POD curve. /D aP can be

expressed as the cumulative distribution function for the detectable depth of the inspection tool.

Hence, the detectable depth of the inspection tool follows the exponential distribution function

and both the average detectable depth and the standard deviation of the crack depth

equal1/ λ . Therefore the overall defect population can be divided into detected defects and undetected defects as shown in Figure 3. According to Bayes' theory [5] the posterior

probability density function for the un-detected defect with a depth of a is equal to:

( )0

( ) ( )

( ) ( )

NDUD

ND

P a f af a

P a f a da∞

=

∫ (3.13)

where ( )f a is the overall distribution of the defect depth and ( )NDP a can be interpreted as

the non-detection probability, which is equal to /1 D aP− . ( )UDf a can be used to calculate the

hazard function *

fP i.e. the probability of failure per year given that the failure has not yet

occurred. *

/

1

f

f

f

dP dnP

P=

−, where

fP is the cumulative probability of failure and n is the number

of load cycles.

The measurement error of the inspection tool is also taken into account during the POF

calculation, which is modelled by adding a measurement error to the actual defect depth such

that:

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m a

a a e= + (3.14)

where m

a denotes the measurement error; a for the actual defect depth and a

e the random

measurement error associated with the measurement of a . The measurement error can follow

normal, Weibull or other common types of distribution.

4. MONTE-CARLO SIMULATION AS POF EVALUATION METHOD

4.1. Direct Monte-Carlo simulation

The solution to equation (3.1) with MCS is obtained by interpreting the integral as the mean

value of a stochastic experiment after a large number of independent tests. A flowchart of the

whole simulation process is shown in Figure 4. There are two separate models used in the

analysis in order to compute leakage and rupture rate, which have been shown in Figure 4. As

the leakage always happens before the pipeline ruptures, rupture failures are considered as part

of leakage failure following the relation:

(leakage) (leakage only/no rupture) (rupture)f f f

P P P= + (4.1)

If Nf(n) failures i.e. leakages or ruptures are observed after N* repetitions, the POF is obtained

via:

*

( )( )

f

f f

N nP P n

N= = (4.2)

Equation (4.2) is only valid for pipelines which contain one defect. If there is more than one

defect in the pipeline, the cumulative probability of failure of a pipeline containing q defects

after n load cycles can be derived according to the definition:

| 1 (1 )q

f total fP P= − − (4.3)

where q is the number of cracks per km length of the pipeline. In fact, a kilometre long pipe

can be regarded as a series system [2] with q elements such that the failure of any individual

element contributes to the overall probability of failure.

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Start

Generate random

variables?

Surface crack growth for time period t

Leakage?

Nf(leakage)=Nf(leakage)+1

Yes

Yes

Yes

NoSimulation

ends

Through crack growth for time period t

No

Rupture?

Nf(rupture)=Nf(rupture)+1

No

Leakage happened in the previous year?

Nf(leakage)=Nf(leakage)-1

No

Yes

Figure 4 MC simulation including a transition from leakage to rupture failure modes

4.2. Stratified Monte-Carlo simulation method

4.2.1. Random number generation

Once a random number between 0 and 1 has been generated by the multiplicative congruent

random generator, it can be used to generate required pseudo-random variables with a given

probability distribution function. The inverse transform method is a common method to achieve

this, which is based on the observation that continuous cumulative distribution functions (cdfs)

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range uniformly over the interval (0, 1). If u is a uniform random number on (0, 1), then a

random number x from a continuous distribution with selected cdf F can be obtained using:

1( )x

x F u−= (4.4)

However, for normal distribution the inverse of the cumulative distribution function cannot be

found analytically. Hence, other methods such as the Box-Müller method and the envelop-

rejection method [6] are used in the simulation .

4.2.2. Stratified Monte-Carlo simulation

The variance of f

P is *

(1 )f f

P P

N

−. There are some variance reduction techniques which can

help to 'speed up' the simulation process. One of them is called Stratified Monte-Carlo

Simulation method (SMCS) [6, 7] and it has been employed in the current version of

PipeSafety© developed for the Naturalhy project. The basic idea is that the sampling of the

variables is carried out in the integration domain { } { }[0, ] 2 [0, ]R a B c= ∈ × ∈ ∞ which is

further partitioned into mutually disjunctive strata ( )1iD i m= K as shown in Figure 5.

Defect depth

De

fect

len

gth

Figure 5 The stratified Monte-Carlo simulation

Therefore, the original formula for calculating f

P can be written as:

( ) ( ) ( ) ( )1

11 1

( ) 0,

( ) ( )

( )

NN i

m

f X X N Ng x x D

i i

m

f i i

i i

m

f i

i i

P f x f x d x d x

Q D B D

P D

⋅⋅⋅ ≤=

=

=

= ⋅⋅⋅ ⋅ ⋅ ⋅

=

=

∑∫

∑

∑

(4.5)

For each stratum i

D , ( )f i

Q D is calculated through MCS with i

n number of samples for each

stratum and ij

q is used to count the number of failures in that stratum. Furthermore, a

normalization factor ( )i

B D is multiplied by the ( )f i

P D . ( )i

B D is calculated through:

( ) ( ) ( )i

iD

B D f a f c dcda= ∫ (4.6)

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Figure 6 Snapshot of PipeSafery software

5. SAMPLE PROBLEMS

5.1. Probability of failure calculation for corrosion defects

The fist example is used to demonstrate how the proposed framework can be used for the

integrity management of the pipeline. As the probability of failure (either leakage or rupture)

will increase as time goes by, therefore a proper inspection and repair program is required to

ensure that the POF will not rise above the target value. The following table summarises the

input data used in the analysis. In this analysis, the distribution of the corrosion defects is

assumed to be known beforehand and there is an average of 1 corrosion defect per km. In

addition it is assumed that no new corrosion defect will be considered for this pipeline during its

lifetime. If new corrosion defects are to be included, a series system needs to be constructed

therefore the total probability of failure can be evaluated. The target cumulative failure

probability value set for leakage is 1e-5 per km and 1e-7 per km for rupture.

Corrosion

defect

Internal

pressure

(Mpa)

Wall

thickness

(mm)

Pipe

diameter

(mm)

Flow

stress

(MPa)

defect

depth

(mm)

Defect

length

(mm)

defect

depth

growth rate

(mm/year)

defect

length

growth rate

(mm/year)

Distribution

type constant normal constant normal normal normal normal normal

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Mean 6.5 13 900 416 3.7 97 0.2 1.2

Standard

deviation 0 0.3 0 15 1 12 0.02 0.3

Table 1 Summary of the input data for POF evaluation of corrosion defects

The simulation results for cumulative POF evaluation without introducing inspection and repair

program are shown in Figure 7. As can be seen from the results the probability of leakage

reaches 1e-5 per km at the beginning of year 8, therefore we should conduct inspection and

repair work to prevent the probability of leakage from reaching this target value.

POF v.s year

1.00E-14

1.00E-12

1.00E-10

1.00E-08

1.00E-06

1.00E-04

1.00E-02

1.00E+00

0 5 10 15 20 25 30 35 40 45

year

pro

bab

ilit

y o

f fa

ilu

re

Leakage probability Rupture probability

Target leakage probability Target rupture probability

Figure 7 Leakage and rupture probabilities with regard to service life

Assume we have an inspection tool with the POD of 90% at 30% wall thickness and the

minimum detectable corrosion depth being 0.2mm. Also due to the limited repair budget the

detected defects with depths smaller than 1mm will not be repaired. It can be expected that the

probability of leakage and rupture will fall sharply following the inspection and repair program.

Therefore we should arrange an inspection and repair program by the end of year 7. It is worth

mentioning that the cumulative POF after inspection is conditional, assuming no actual failure

has happened before inspection. The results are shown in Figure 8.

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POF v.s year

1.00E-14

1.00E-13

1.00E-12

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 5 10 15 20 25 30 35 40 45

year

pro

bab

ilit

y o

f fa

ilu

re

Leakage probability Rupture probability

Target leakage probability Target rupture probability

Figure 8 Leakage and rupture probabilities following the inspection

It is evident that the probability of having a leakage and a rupture is now within the target

range for the first 15 years. A second inspection and repair program should therefore be

arranged by the end of year 15 in order to meet the same goal as before.

5.2. Probability of failure calculation for cracks

In this section we will investigate the impact of hydrogen on the probability of leakage and

rupture. We assume that the cracks are longitudinally oriented as the tensile stresses applied to

the crack are the maximal in this orientation. The input data is summarised in the list below

(data is based on the experiments carried out within the NATURALHY project):

No. Parameter Mean Standard

deviation

Distribution

type

1 Pipe diameter (mm) 600 0 Fixed

2 Wall thickness (mm) 10 0 Fixed

3 Initial crack depth (mm) 2.85 0.3 Log-normal

4 Initial crack length (mm) 130 70 Log-normal

5 Pressure (MPa) 6.06 0.2 Normal

6 Residual stress (MPa) Base metal- 0 0 Fixed

7 Fracture toughness

(MPa*mm1/2)

4743.4 for 100% NG

2383.9 for 100% H2

2383.9 for 50% H2/NG

0 Fixed

8 Yield strength (MPa) 358 20 Normal

9 Tensile strength (MPa) 455 25 Normal

10 Threshold toughness

(MPa*mm1/2)

338.36 for 100% NG

275.12 for 100% H2

316.23 for 50% H2/NG

10 Normal

11 Fatigue parameter c 2.32e-14 for 100% NG

2.6e-19 for 100% H2

5.2e-14 for 50% H2/NG

0 fixed

12 Fatigue parameter m 3.4 for 100% NG

5.57 for 100% H2

3.32 for 50% H2/NG

0 fixed

13 Pressure drop ratio 0.35 0 fixed

14 Service life (years) 50 0 fixed

15 No. of load cycles per

year

365 0 fixed

Table 2 Input data for crack model

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As with the previous example it is assumed that there is only one crack per km pipeline. Figure

9 shows how the cumulative POFs for pipelines carrying natural gas-hydrogen mixtures change

over time. It is clear from the results that hydrogen does have a significant impact on the life of

the pipeline and therefore risk mitigation measures must be taken to ensure that the pipelines

carrying hydrogen are checked and repaired in time.

POF v.s year

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 10 20 30 40 50 60year

pro

bab

ilit

y o

f fa

ilu

re

100% natural gas 100% hydrogen 50% natural gas/hydrogen mixtures

Figure 9 Probability of leakage for pipelines carrying natural gas-hydrogen mixtures

6. CONCLUSIONS

A methodology has been developed to assess the integrity of pipelines carrying natural gas and

hydrogen mixtures. The simulation based method has demonstrated its flexibility and reliability

in solving complex problems involving large number of variables and complex procedures. The

stratified Monte-Carlo sampling scheme has been successfully employed to accelerate the

simulation process without compromising the accuracy of results. The proposed method for POF

evaluation is of practical importance as the integrity and risk mitigation procedures can be

planned based on the information provided by PipeSafety©. The sample results demonstrate

that POFs of pipelines increase in the presence of hydrogen but the extent of such increase can

be reduced by mixing hydrogen and natural gas.

In addition to the material properties defect size distributions also have a huge influence on the

final POF results. Therefore the POF is very sensitive to the means and standard deviations

defined for the defect depth and length. In this sense, data collection and interpretation is the

most important step for a successful simulation. There are many effective ways to approximate

the distribution of variables such as the maximum likelihood method and the bootstrap

sampling method (when data is scarce). These methods should be first implemented to obtain

as accurate data as possible in advance of the use of PipeSafety©.

7. ACKNOWLEDGEMENTS

The authors would like to thank the NATURALHY project partners for providing data and

information and the European Union for kindly providing the financial support for this study.

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8. REFERENCES

1. A.Bruckner and D.Munz, Determination of Crack Size Distributions from Incomplete Data

Sets for the Calculation of Failure Probabilities. Reliability Engineering, 1985. 11: p.

191-202.

2. Thoft-Christensen, P. and M. J.Baker, Structural Reliability Theory and Its Applications.

1982: Springer-Verlag.

3. BS7910: Guide to methods for assessing the acceptability of flaws in metallic structures.

2005: British Standards. 23-60.

4. E.S.Rodrigues and J.W.Provan, Development of a general failure control system for

estimating the reliability of deteriorating structures. Corrosion, 1989. 45: p. 3.

5. R.Benjamin, J. and C.A. Cornell, Probability, Statistics, and Decision for Civil Engineers.

1970: McGraw-Hill Book Company.

6. J.S.Dagpunar, Simulation and Monte Carlo with Applications in Finance and MCMC.

2007: John Wiley & Sons, Ltd.

7. A.Bruckner-Foit, Th.Schmidt, and J.Theodoropoulos, A comparison of the PRAISE code

and the PARIS code for the evaluation of the failure probability of crack-containing

components. Nuclear Engineering and Design, 1989. 110: p. 395-411.

9. LIST OF TABLES

Table 1 Summary of the input data for POF evaluation of corrosion defects ...........................13 Table 2 Input data for crack model...................................................................................14

10. LIST OF FIGURES

Figure 1 Geometry of surface/through thickness crack model used in the analysis................... 5 Figure 2 The BS7910 FAD curve used to define the limit state function .................................. 6 Figure 3 Maintenance event tree for crack like defects ......................................................... 8 Figure 4 MC simulation including a transition from leakage to rupture failure modes...............10 Figure 5 The stratified Monte-Carlo simulation...................................................................11 Figure 6 Snapshot of PipeSafery software .........................................................................12 Figure 7 Leakage and rupture probabilities with regard to service life ...................................13 Figure 8 Leakage and rupture probabilities following the inspection......................................14 Figure 9 Probability of leakage for pipelines carrying natural gas-hydrogen mixtures ..............15

Documents

Reliability Analysis Of Pipelines Containing Cracks And ......rupture of the pipeline. Based on fatigue reliability analysis, statistical methods are applied to assess the probability