GRI-04/0229
Guidelines for Reliability Based Design and
Assessment of Onshore Natural Gas Pipelines
Final Report
Prepared by:
Maher Nessim, PhD, PEng
Wenxing Zhou, PhD, PEng
C-FER Technologies
200 Karl Clark Road
Edmonton, AB T6N 1H2 Canada
Prepared for:
GAS RESEARCH INSTITUTE
GRI Contract No. 8565
GRI Project Manager
Charles E. French
July 2005
LEGAL NOTICE
This Report was prepared by C-FER Technologies (1999) Inc., as an account of work sponsored
by Gas Research Institute ("GRI"). Neither GRI, members of GRI, nor any person acting on
behalf of any of these parties:
a.
MAKES ANY WARRANTY OR REPRESENTATION, EXPRESS OR
IMPLIED, WITH RESPECT TO THE ACCURACY, COMPLETENESS, OR
USEFULNESS OF THE INFORMATION CONTAINED IN THIS REPORT, OR
THAT THE USE OF ANY INFORMATION, APPARATUS, METHOD, OR
PROCESS DISCLOSED IN THIS REPORT MAY NOT INFRINGE PRIVATELY
OWNED RIGHTS, OR
b.
ASSUMES ANY LIABILITY, INCLUDING WITHOUT LIMITATION,
SPECIAL OR CONSEQUENTIAL DAMAGES, WITH RESPECT TO THE USE
OF, OR FOR ANY AND ALL DAMAGES RESULTING FROM THE USE OF,
ANY INFORMATION, APPARATUS, METHOD OR PROCESS DISCLOSED IN
THIS REPORT.
ii
ACKNOWLEDGMENTS
This report is based on a project funded by the Gas Research Institute and carried out under
supervision of the Design, Construction and Operations Committee of the Pipeline Research
Council International (PRCI). The work was an extension of a previous project on the same
topic that was carried out by C-FER Technologies and funded by TransCanada PipeLines and
BP Exploration Operating Company. Members of the PRCI ad hoc group for this project are
gratefully acknowledged for their guidance throughout the project. Special thanks are due to
Joe Zhou, Brian Rothwell, Martin McLamb, Louis Fenyvesi, Rick Gailing, Keith Leewis, and
Alan Glover for their ongoing advice and contributions to key decisions throughout the work.
iii
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July 2005
Final Report
4. TITLE AND SUBTITLE
5. FUNDING NUMBERS
Guidelines for Reliability Based Design and Assessment of Onshore Natural
Gas Pipelines
GRI Contract No. 8565
6. AUTHOR(S)
Maher Nessim, PhD, PEng and Wenxing Zhou, PhD, PEng
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
8. PERFORMING ORGANIZATION
C-FER Technologies (1999) Inc.
200 Karl Clark Road
Edmonton, Alberta T6N 1H2
Canada
L080-1
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
10. SPONSORING/MONITORING
REPORT NUMBER
GRI
1700 S. Mt. Prospect Rd.
Des Plaines, IL 60018
AGENCY REPORT NUMBER
GRI-04/0229
11. SUPPLEMENTARY NOTES
12a. DISTRIBUTION/AVAILABILITY STATEMENT
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 words)
A set of guidelines for the application of Reliability Based Design and Assessment (RBDA) of onshore natural gas
pipelines has been developed. These guidelines contain a general overview of RBDA methods and a discussion of
the key issues associated with applying them to pipelines. Requirements for the application of RBDA are also given,
specifying the design conditions that must be considered and the reliability targets that are to be met. In addition, the
guidelines provide the technical information required to apply RBDA including methodologies to identify relevant
limit states, construct limit states functions, develop probabilistic models for uncertain input parameters and estimate
the lifetime reliability. Two example applications are given: one for the design of a new pipeline segment, and the
other for a class upgrade assessment of an existing segment.
15. NUMBER OF PAGES
14. SUBJECT TERMS
235
16. PRICE CODE
17. SECURITY CLASSIFICATION
OF REPORT
18. SECURITY CLASSIFICATION
OF THIS PAGE
19. SECURITY CLASSIFICATION
OF ABSTRACT
Unclassified
Unclassified
Unclassified
NSN 7540-01-280-5500
20. LIMITATION OF ABSTRACT
Standard Form 298 (Rev.2-89)
Prescribed by ANSI Std 239-1B
298-102
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TABLE OF CONTENTS
Project Team and Revision History
Legal Notice
Acknowledgements
Report Documentation Page
List of Figures and Tables
Research Summary
Executive Summary
1.
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INTRODUCTION..................................................................................................................1
1.1 Purpose
1.2 Scope and Focus
1.3 Organization
1
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1
2.
DEFINITIONS.......................................................................................................................3
3.
OVERVIEW OF RELIABILITY BASED DESIGN AND ASSESSMENT ..............................7
3.1
3.2
3.3
3.4
3.5
Introduction
Historical Perspective
Sources of Uncertainty
Limit States
Reliability and Probability of Failure
3.5.1 Basic Concepts
3.5.2 Limit State Function
3.5.3 Calculation Methodology
3.6 Reliability Based Design and Assessment
3.7 Benefits
4.
RBDA METHODOLOGY FOR PIPELINES .......................................................................16
4.1 Introduction
4.2 Key Issues for Pipeline Reliability
4.2.1 Time Variability
4.2.2 Impact of Maintenance
4.3 Implementation Methodology
4.4 Applicability
5.
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DESIGN AND ASSESSMENT REQUIREMENTS .............................................................22
5.1
5.2
General Requirements
Limit States
5.2.1 Categories of Limit States
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5.3
6.
5.2.2 Loads and Limit States
Reliability Targets
5.3.1 Introduction
5.3.2 Ultimate Limit States
5.3.2.1 Approach
5.3.2.2 Format
5.3.2.3 Safety Criteria
5.3.2.4 Reliability Targets
5.3.2.5 Meeting the Targets
5.3.3 Leakage Limit States
5.3.4 Serviceability Limit States
IDENTIFICATION OF RELEVANT LIMIT STATES...........................................................37
6.1 Introduction
6.2 Deterministic Screening
6.3 Probabilistic Screening
6.3.1 Introduction
6.3.2 Continuously Applied Loads
6.3.3 Discrete Loads
7.
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DEVELOPING A LIMIT STATE FUNCTION .....................................................................44
7.1 Introduction
7.2 Generalized Definition of a Limit State Function
7.3 Overview of Development Procedure
7.4 Defining the Limiting Condition
7.5 Developing the Limit State Model
7.5.1 Introduction
7.5.2 Example 1 – Using a Simple Analytical Model
7.5.3 Example 2 – Using a Numerical Finite Element Model
7.5.4 Sources of Relevant Information
7.6 Model Uncertainty
7.6.1 Introduction
7.6.2 Characterizing Model Error
7.6.2.1 General
7.6.2.2 Proportional Error
7.6.2.3 Independent Error
7.6.2.4 Model Selection
7.6.3 Example
8.
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PROBABILISTIC CHARACTERIZATION OF INPUT VARIABLES..................................63
8.1 Introduction
8.2 Frequency of Random Events
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8.2.1 Introduction
8.2.2 The Poisson Process
8.2.3 Estimation of the Rate of Occurrence
8.2.4 Example
8.3 Probability Distributions of Time-independent Variables
8.3.1 Introduction
8.3.2 Data Analysis
8.3.3 Distribution Selection Based on Data
8.3.3.1 Introduction
8.3.3.2 Selection of Candidate Distribution Types
8.3.3.3 Distribution Parameter Estimation
8.3.3.4 Best Fit Distribution Selection
8.3.4 Other Distribution Selection Methods
8.4 Time-dependent Variables
8.4.1 Introduction
8.4.2 Discrete Random Process
8.4.2.1 Process Characterization
8.4.2.2 Maximum Load Distribution
8.4.2.3 Asymptotic Extremal Distributions
8.4.2.4 Estimation of Return Periods
8.4.3 Continuous Random Processes
8.5 Effect of Sample Size
8.5.1 Introduction
8.5.2 Example 1 – Occurrence Rate of a Poisson Process
8.5.3 Example 2 – Mean of a Distribution with Known Standard Deviation
8.5.4 General Procedure
8.5.5 Comments
9.
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RELIABILITY ESTIMATION ..............................................................................................92
9.1 Introduction
9.2 Single Time-independent Limit State
9.2.1 Introduction
9.2.2 General Methodology
9.2.2.1 Failure Rate
9.2.2.2 Conditional Failure Probability
9.2.2.3 Example
9.2.3 Special Methodology for Seismic Limit States
9.2.3.1 Failure Rate
9.2.3.2 Conditional Failure Probability
9.3 Single Time-dependent Limit State
9.3.1 Introduction
9.3.2 Failure Rate
9.3.3 Conditional Failure Probability
9.3.4 Impact of Rehabilitation
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9.3.4.1 Approach
9.3.4.2 Detection Capability
9.3.4.3 Sizing Accuracy
9.3.4.4 Defect Excavation and Repair
9.3.4.5 Failure Rate Calculation
9.3.5 Example
9.4 Multiple Limit States
9.4.1 Introduction
9.4.2 Example 1: Yielding and Burst of Defect-free Pipe
9.4.3 Example 2: Equipment Impact
9.4.4 Example 3: Corrosion
9.5 Reliability Calculation Tools
10.
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EXAMPLE APPLICATIONS ............................................................................................117
10.1 Example 1 – New Pipeline Design
10.1.1 Introduction
10.1.2 Pipeline Information
10.1.3 Applicable Limit States
10.1.4 Reliability Targets
10.1.5 Limit State Functions
10.1.6 Probabilistic Characterizations of Input Parameters
10.1.7 Reliability Calculation
10.1.8 Design Process
10.1.9 Results
10.1.10
Sensitivity to Pressure
10.2 Example 2 – Class Upgrade Deferral
10.2.1 Introduction
10.2.2 Limit States
10.2.3 Reliability Targets
10.2.4 Reliability Analysis
10.2.5 Results for Enhanced Maintenance
10.2.6 Comparison to Conventional Class Upgrade Approaches
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11.
CONCLUDING REMARKS..............................................................................................137
12.
REFERENCES.................................................................................................................138
APPENDICES
Appendix A
Appendix B
Appendix C
Appendix D
Appendix E
Limit State Functions for Key Failure Causes
Probabilistic Models for Basic Variables
Methodology to Characterize Combined Proportional and Independent
Model Error
Basic Probability Concepts
Failure Probability Calculation for Seismic Loading
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LIST OF FIGURES AND TABLES
Figures
Figure 3.1 Illustration of Load Effect and Resistance Distributions
Figure 3.2 Illustration of the Limit State Surface
Figure 3.3 Illustration of Reliability Estimation for Internal Pressure
Figure 4.1 Types of Loading Processes Applicable to Onshore Pipeline
Figure 4.2 Types of Resistance Processes
Figure 4.3 Steps Involved in Implementing Reliability Based Design and Assessment
Figure 5.1 Reliability Targets from All Three Criteria Considered
Figure 5.2 Reliability Targets by Class
Figure 5.3 Relative Expected Number of Fatalities for Large Leaks and Ruptures
Figure 7.1 Procedure for Developing a Limit State Function
Figure 7.2 Illustration of an Excavator Impacting a Pipeline
Figure 7.3 Puncture Model Results Versus Test Data
Figure 7.4 Excavator Mass Versus Digging Force
Figure 7.5 Illustration of Frost Heave Loading Scenario
Figure 7.6 Applied Curvature from Finite Element Versus Regression Model
Figure 7.7 Illustration of Proportional Model Error
Figure 7.8 Illustration of Independent Model Error
Figure 7.9 Actual Burst Pressure Versus Model Results
Figure 7.10 Proportional and Independent Error Plots for the Corrosion Data in Figure 7.9c
Figure 7.11 Plot of Final Model with an Error Band of One Standard Deviation on Each Side
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List of Figures and Tables
Figure 8.1 Histogram Plot for the Yield Strength Data in Table 8.1
Figure 8.2 Cumulative Probability Plot for the Yield Strength Data in Table 8.1
Figure 8.3 Steps Involved in Fitting a Distribution to Statistical Data
Figure 8.4 Illustration of Goodness-of-Fit Test Statistics – a) Chi-square Test b) K-S Test
Figure 8.5 Probability Paper Plots for the Yield Strength Data
Figure 8.6 Illustration of Time-dependent Random Variables (or Random Processes)
Figure 8.7 Distributions of Extremes for a Normal Parent Distribution
Figure 8.8 Extremal Load Distributions for Fixed and n (expected value of n = 10)
Figure 8.9 Exact and Gumbel Approximation of the Maximum Annual Impact Load
Figure 8.10 Illustration of Methods to Discretize a Continuous Random Process
Figure 8.11 Probability Distribution of Impact Rate for Different Sample Sizes
Figure 8.12 The 90% Probability Interval as a Function of Observation Period
Figure 8.13 Probability Distributions of the Mean Toughness for Various Sample Sizes
Figure 8.14 Toughness Probability Distributions for Various Sample Sizes
Figure 8.15 Cumulative Toughness Distributions for Various Sample Sizes
Figure 9.1 Probability Density Function of the Safety Margin Showing the Probability of Failure
Figure 9.2
Idealization of a Time-dependent Load as a Time-independent for Reliability
Calculations
Figure 9.3 Illustration of the Rehabilitation Process
Figure 9.4 Probability of Detection as a Function of Defect Depth
Figure 9.5 Illustration of Measurement Error Band and Corresponding Probability
Figure 9.6 Failure Rate by Burst as a Function of Time
Figure 9.7 Impact of Rehabilitation on the Average Defect Depth Distribution
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List of Figures and Tables
Figure 9.8 Impact of Rehabilitation on the Failure Rate for Burst
Figure 9.9 Impact of a Specific Rehabilitation Plan on the Future Failure Rate for Burst
Figure 9.10 Limit States for Yielding and Burst Under Internal Pressure
Figure 9.11 Limit States for Different Failure Modes Associated with Equipment Impact
Figure 9.12 Limit States for Different Failure Modes Associated with Corrosion
Figure 10.1 Variation of the Population Density along the Right-of-Way
Figure 10.2 Calculated Versus Target Reliability for Section A
Figure 10.3 Calculated versus Target Reliability for Section B
Figure 10.4 Calculated Versus Target Reliability for Section C
Figure 10.5 Reliability Compared to Target for Status Quo and Enhanced Maintenance
Figure 10.6 Reliability Comparisons of Replacement, Pressure Reduction and Enhanced
Maintenance
Tables
Table 4.1 Classification of Limit States with Respect to Time Dependence
Table 5.1 Load Cases and Limit States Relevant to Onshore Pipelines
Table 5.2 Tolerable Societal Risk Levels Calibrated to ASME B31.8
Table 5.3 Population Density by Class Based on Structure Data for Actual Pipelines (Nessim
and Zhou 2005)
Table 6.1 Probability Estimates for Soil Displacement
Table 7.1 Actual and Calculated Burst Pressure for Corroded Pipe Specimens
Table 8.1 Yield Strength Data for X60 Steel
Table 8.2 Range and General Shape of Some Commonly Used Probability Distributions
Table 8.3 Results of Goodness-of-Fit Tests for Yield Strength Data
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List of Figures and Tables
Table 8.4 Gumbel Distribution Parameters for a Number of Parent Distribution Types (Maes
1985)
Table 9.1 Deterministic Pipeline Parameters for Example
Table 9.2 Probability Distributions of Basic Random Variables for Example
Table 9.3 Basic Variable Distributions Used in the Example
Table 10.1 Preliminary Applicable Limit States for Segment 1
Table 10.2 Final List of Applicable Limit States
Table 10.3 Pipeline Segments and Reliability Targets
Table 10.4 Probability Distributions for Uncertain Limit State Input Parameters
Table 10.5 Parameters used in Calculating Equipment Impact Frequency
Table 10.6 Metal Loss Inspection Tool Accuracy Specifications
Table 10.7 Wall Thickness and Equivalent Design Factors
Table 10.8 Calculated Reliability for SLS Under Hydrostatic Test Pressure
Table 10.9 Wall Thickness and Equivalent Design Factors
Table 10.10 Inspection Interval and Defect Repair Criteria
Table 10.11 Limit States Analyzed in Example 2
Table 10.12 Target Reliability Levels
Table 10.13 Probability Distributions for Uncertain Limit State Input Parameters
Table 10.14 Basic and Enhanced Failure Prevention Measures for Equipment Impact
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RESEARCH SUMMARY
TITLE:
Guidelines for Reliability Based Design and Assessment of
Onshore Natural Gas Transmission Pipelines
CONTRACTOR:
C-FER Technologies
PRINCIPAL
INVESTIGATORS:
Maher A. Nessim, PhD, PEng
Wenxing Zhou, PhD
REPORT PERIOD:
January 2003 – July 2005
OBJECTIVES:
The objective of this project was to develop a set of guidelines
for the application of Reliability Based Design and
Assessment (RBDA) to onshore natural gas pipelines.
TECHNICAL
PERSPECTIVE:
The guidelines provided a set of reliability targets and a
methodology to demonstrate that the targets are met. They are
intended to facilitate the application of RBDA in practical
situations and ensure that the resulting pipelines are safe and
serviceable.
TECHNICAL
APPROACH:
The design and assessment requirements developed for this
project were based on risk analysis principles. They have
been developed to ensure that the average safety of pipelines
designed and operated using the RBDA approach equals or
exceeds the average safety associated with new pipelines that
conform to current codes and best practice. The steps
involved in the reliability methodology include identification
of the relevant limit states, development of limit state models,
characterizing the uncertainties associated with the limit state
parameters, calculating reliability, and comparing the
calculated reliability to specified minimum targets.
RESULTS:
The guidelines consist of a main document describing the
methodology and containing two example applications, as well
as a number of appendices containing supporting information
on limit state functions, statistical parameter definitions, and
failure probability calculation for seismic limit states.
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Research Summary
PROJECT
IMPLICATIONS:
RBDA is a tool to support engineering decision-making based
on rigorous analysis. Its benefits include consistent safety
levels, best use of resources and an ability to deal with nonstandard problems. The guidelines in this document will help
make these benefits accessible for onshore natural gas
pipelines. To facilitate use by pipeline engineers, these
guidelines provide explicit procedures and examples for the
steps involved in applying RBDA techniques.
The reliability targets used in this document were developed
as minimum requirements to ensure that adequate human
safety is maintained throughout the life of a pipeline.
Economic considerations were not taken into account because
they vary widely for different pipelines. In some situations,
lifetime costs could be minimized by exceeding the targets
used in this document. As with any tool, RBDA methodology
should be applied with good engineering judgment.
PROJECT
MANAGER:
Charles E. French, P.E.
Program Manager
Compression and Measurement
Gas Operations
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C-FER Technologies
EXECUTIVE SUMMARY
General
This document contains a set of guidelines for the application of Reliability Based Design and
Assessment (RBDA) to onshore natural gas transmission pipelines. The guidelines describe the
reliability analysis framework and give detailed guidance on how to develop the deterministic
and probabilistic models required to apply it to specific pipelines. They also contain state-of-theart models for some key design conditions and failure causes including yielding and burst,
equipment impact, and corrosion, making analysis of these failure causes possible without any
further development. To facilitate use by pipeline practitioners, the guidelines provide explicit
procedures and illustrative examples for the various steps involved in applying reliability-based
design and assessment methods.
The guidelines are applicable to decisions that influence the structural integrity of a pipeline.
These include design decisions for new pipelines, fitness-for-service evaluation for existing lines,
assessment of changes in operational parameters (e.g. location class changes, fluid changes,
damage) and evaluation of maintenance alternatives.
Overview of Probabilistic Limit States Design
A limit state is formally defined as a state beyond which the structure no longer satisfies a
particular design requirement. Depending on the design requirement that is violated, pipeline
limit states can be grouped into: 1) ultimate limit states, such as large leaks and ruptures, which
are concerned with loss of containment events that could lead to significant safety consequences;
2) leakage limit states, which are defined as small leaks that do not lead to significant safety
consequences; and 3) serviceability limit states, such as ovalization and denting, which affect
functionality without jeopardizing pressure containment.
The essence of the limit states concept is to identify the true failure modes of the pipeline and to
make decisions that ensure appropriate conservatism, considering the severity of the failure
consequences. Instead of designing a pipeline primarily against hoop yield as required by
current elastic limit design codes, the limit states approach suggests that the pipeline should be
designed for the above-mentioned limit state categories.
Since the consequences of
serviceability limit states are much less serious than those of ultimate limit states, more
conservatism is required for the latter. This ensures proper consideration of the true failure
mechanisms, such as corrosion and equipment impact, resulting in more consistent safety levels
than the stress limit design approach.
Acknowledging the uncertainties associated with structural performance, RBDA uses reliability
as a measure of structural safety. Reliability with respect to a particular limit state category is
defined as the probability that a given length of the pipeline will not reach any limit states within
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Executive Summary
that category for a specified period of time. As such it is equal to the probability of reaching a
limit state (i.e. probability of failure per km-year) subtracted from one.
The probability of failure for a given limit state is calculated as the probability that the load
effect will exceed the corresponding resistance (i.e. combined probability of overload and underresistance). The load effect and resistance distributions are usually estimated from other (more
basic) variables using analytical models. Examples are the estimation of earthquake load effects
from peak ground accelerations or the calculation of pipeline pressure resistance from yield
strength, diameter and wall thickness. Ultimately, a limit state function is formulated, which
defines combinations of the basic parameters that lead to failure. Standard methods are available
to calculate the failure probability from the limit state function and the probability distributions
of the basic parameters.
RBDA is a design and assessment method, in which the pipeline is designed and operated to
meet a pre-defined set of target reliability levels. The targets must be met along the entire
pipeline throughout its operational life. Failure consequences are accounted for by requiring
more stringent target reliability levels for limit states with more severe consequences.
Benefits of the Reliability Based Approach
•
Design for the true structural behaviour. By identifying the true modes of pipeline failure
and making decisions that mitigate the actual consequences of these failures, unrealistic
design criteria and excessive conservatism are avoided.
•
Consistent safety levels. By requiring lower failure rates (or higher reliability levels) for
pipelines with more severe failure consequences, a consistent safety level can be achieved.
This is an improvement over the use of fixed safety factors, which result in unknown and
highly variable risk levels for different pipelines.
•
Cost savings. Conservatism is placed where it is most needed (e.g. higher reliability for
ultimate limit states than for serviceability limit states), leading to minimum cost solutions
for a given level of overall safety.
•
Adaptability to new problems. Reliability levels are calculated from basic principles, making
the approach well suited to new problems (e.g. stress corrosion cracking), unconventional
environmental conditions (e.g. northern pipelines) and the application of new technology
(e.g. the use of high strength steels).
•
Integration of design and operational practices. Since reliability is a function of both design
and operational parameters, reliability gains due to in-service maintenance activities can be
incorporated at the design stage, resulting in potential reductions in capital expenditures.
This could have significant economic benefits in view of the recent and on-going
improvements in inspection and maintenance technologies and practices.
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Executive Summary
Summary of the Guidelines
Applying reliability-based methods to onshore pipelines requires two key departures from
current design and assessment practice:
•
Design and assessment checks. Since the reliability-based limit states approach requires
consideration of the actual failure causes, dominant failure mechanisms such as corrosion,
mechanical damage and, for some pipelines, ground movement must be considered more
explicitly in the design and assessment process.
•
Life cycle reliability. Since reliability varies with time for some of the major failure
mechanisms such as corrosion, design should be based on lifetime reliability, taking the
impact of maintenance into consideration.
The steps involved in implementing reliability-based design and assessment for a specific
segment of a given pipeline are summarized in the figure shown below, along with the main
inputs required for each step. The process is an iterative one in which various alternatives are
identified, analyzed and evaluated against the appropriate safety and economic criteria, until the
most economic alternative that satisfies the safety requirements is identified. The figure assumes
that the pipeline route and main operating parameters (e.g. throughput, diameter and pressure)
are defined.
The details of each step shown in the figure are addressed in a separate section of the guidelines,
with example applications given in the final section. A brief description of these steps is given in
the following.
1. Identify relevant limit states. The relevant limit states are identified based on the fluid being
transported, internal pressure and external loads. Chapter 5 provides a list of possible limit
states and Chapter 6 describes a procedure that can be followed to determine their
applicability to a given pipeline. The procedure involves sequential application of a number
of simple, conservative checks (both deterministic and probabilistic) to the limit state under
consideration. Passing any one of these checks implies meeting the target reliability
requirements, thus eliminating the need for a detailed reliability calculation.
2. Develop limit state functions. Guidelines for developing limit state functions are given in
Chapter 7, along with some information on availability of the required deterministic pipe
behaviour models. A simple procedure is given for limit state functions utilizing semiempirical models, which are available for most pipeline limit states including yielding, burst,
corrosion and equipment impact. A second procedure is provided for limit states involving
more complex structural analyses, such as frost heave and thaw settlement. Appendix A
gives limit state functions for some of the key limit states associated with onshore pipelines.
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Executive Summary
Route Data and
Loading Conditions
Identify Relevant
Limit States
Deterministic
Behaviour Models
Develop Limit State
Functions
Statistical Data
Develop Probabilistic
Models of Basic
Variables
Operational
Parameters and
Regulations
Select Design
Parameters and
Maintenance Plan
Probability
Calculation Method
Calculate Reliability
Target Reliability
Levels
Reliability
Target met ?
No
Yes
Economic
Criteria met ?
No
Yes
Acceptable Design
Steps Involved in Implementing Reliability Based Design and Assessment
3. Develop probabilistic models for basic variables. The uncertain parameters (referred to as
basic random variables) used in each limit state function must be characterized by
appropriate probabilistic models. Definition of these models is usually based on statistical
data, theoretical considerations and judgment. Chapter 8 provides guidelines for the selection
of appropriate probabilistic models. A procedure is given for selecting probability
distributions to model time-independent parameters, such as yield strength and wall
thickness. Also described is the development of stochastic process models for parameters
that vary randomly with time, such as wind speed and equipment impact loads. The use of
stochastic process models to estimate the extreme parameter values required for reliability
calculation (e.g. maximum load and minimum resistance) is also covered. Procedures to deal
with the additional uncertainties associated with small data sets are included. Appendix B
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Executive Summary
gives a review of publicly available data and models for the basic random variables used in
key pipeline limit states.
4. Select design parameters and maintenance plan. An initial set of design and maintenance
parameters (including, for example, inspection frequencies, repair criteria, equipment impact
prevention measures) should be proposed, taking into account any regulatory or policy
constraints. In addition to defining the steel grade and wall thickness, it is necessary to
define the set of supplementary measures that will be used to ensure reliable operation of the
pipeline throughout its design life. These measures include quality assurance plans such as
material testing and weld inspection procedures; corrosion mitigation strategies such as
coating type and cathodic protection system characteristics; damage prevention activities
such as burial depth, right-of-way patrols and first call system; and in-line inspection plans
including tools to be used, inspection frequency and repair criteria. This information is
required to evaluate life cycle reliability as discussed earlier. For existing pipelines, some of
these parameters will be already defined and can be treated as constraints.
5. Calculate reliability. A reliability calculation methodology suitable for the main limit state
functions affecting onshore pipelines is described in Chapter 9. The methodology addresses
both time-independent and time-dependent limit states. A time-independent limit state is
based on load and resistance processes that do not change systematically with time, and
therefore the corresponding failure rate does not change with time. Examples include
yielding and rupture of new pipe, and failure due to accidental equipment impacts. Timedependent limit states involve systematic changes in the failure rate due to changes in the
underlying load or resistance processes. Examples include gradual deterioration mechanisms
such as corrosion or fatigue crack growth and slow developing loads such as deformations
induced by frost heave. In addition, methodologies are described for simultaneous
consideration of multiple limit states, which is required in cases involving multiple failure
mechanisms (e.g. puncture of gouged dent failure due to equipment impact) and/or multiple
failure modes (e.g. leaks and ruptures).
6. Compare to target reliability. The calculated reliability levels for various limit states are
compared to the target values. The target values must be pre-defined based on an overall
safety philosophy, which takes into account the severity of the consequences associated with
each class of limit states. Chapter 5 summarizes the target reliability levels selected for
natural gas pipelines. These targets define minimum requirements to ensure adequate safety
levels throughout the life of a pipeline. Economic considerations were not taken into account
because they vary widely for different pipelines. According to the limit state categories
defined earlier, only ultimate limit states have significant safety-related consequences. The
reliability targets for ULS were therefore developed using a risk-based approach that ensures
consistent and adequate safety levels for all pipelines. Reliability targets for leakage (i.e.
small leaks) and serviceability limit states were defined on the basis of historical information
and published precedent. Details of the methodology used in developing the reliability
targets are described in a separate document (Nessim and Zhou 2005).
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Executive Summary
7. Assess economic implications. This step involves a check to ensure that the safety criteria
are met at a reasonable cost and without undue conservatism. It may involve comparing
various design alternatives that meet the target reliability levels for the purpose of selecting
the minimum cost alternative. An example would be comparing a thick walled design
coupled with a standard maintenance plan to a thinner walled pipeline combined with an
enhanced maintenance plan. This analysis involves calculating the life cycle costs associated
with each design/maintenance alternative. Although this is a key step in developing an
optimal design, it is highly project-specific and is therefore not addressed in detail in this
document.
Example applications involving the design of a new pipeline and the assessment of an existing
pipeline are given in Chapter 10. The examples demonstrate use of the approach and provide
some comments on how the results compare to conventional approaches.
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1. INTRODUCTION
1.1 Purpose
These guidelines have been prepared in partial fulfillment of the requirements of a project carried
out by C-FER Technologies for the Gas Research Institute (GRI) under the supervision of the
Design, Construction, and Operations Committee of PRCI. The objectives of the project were to
develop a set of guidelines for applying Reliability Based Design and Assessment (RBDA)
methods to onshore natural pipelines.
The document is intended as a tool to guide pipeline engineers through the process of applying
RBDA to the planning, design and operation of natural gas pipelines. It can also be used as a
guide for carrying out the analysis required to develop deterministic reliability-based design and
assessment checks such as those based on Load and Resistance Factor Design (LRFD) methods.
1.2 Scope and Focus
These guidelines describe the reliability analysis framework and give detailed guidance on how
to apply it to the design and assessment of natural gas transmission pipelines. They list the key
failure modes that threaten pipelines and specify a set of reliability targets that must be met to
ensure adequate safety and serviceability. They also describe the procedure used to calculate
reliability for a particular pipeline and evaluate the results in relation to the targets. This
procedure involves the use of deterministic structural behaviour models and statistical data on the
input parameters to these models. The guidelines contain a state-of-the-art compilation of
available models and data for some key conditions including yielding and burst, equipment
impact, corrosion, seismic loading. Limit states for upheaval buckling and ground deformations
are outlined in separate reports that are being prepared in conjunction with this project (Xie et.al.
2004 and Zhou 2005). They also contain a detailed description of how the required information
can be developed for new and unique design conditions.
The guidelines focus on application and implementation rather than theoretical background.
Specific procedures and illustrative examples are provided for different steps to ensure that the
process can be followed without ambiguity. Reference is made to other documents that contain
the required theoretical background.
Finally, the term Reliability Based Design and Assessment (and the acronym RBDA) is used
throughout the document to indicate that the methodology is applicable to decision making in a
general sense, including design decisions for new pipelines, fitness-for-service evaluation for
existing lines, assessment of changes in operational parameters (e.g. location class or pressure
changes), and evaluation of inspection and maintenance alternatives.
1.3 Organization
Chapter 2 contains definitions of the technical terms used in the document – first appearance of
each term defined in Chapter 2 is italicized. Chapter 3 contains a general overview of Reliability
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Based Design and Assessment (RBDA), including some background information, essential
definitions, basic calculations and benefits. Chapter 4 describes implementation of the
methodology for pipelines, discussing some of the key issues that are specific to pipeline
reliability estimation.
Requirements for the application of RBDA to onshore natural gas transmission pipelines are
given in Chapter 5. These requirements specify the design conditions (limit states) that must be
considered and the reliability targets that need to be satisfied. Chapters 6 through 9 provide the
technical information required to apply RBDA including methodologies to identify relevant limit
states, construct limit states functions, develop probabilistic models for uncertain input
parameters and estimate the lifetime reliability.
Chapter 10 gives two example applications, one for the design of a new pipeline segment and the
other for a class upgrade assessment of an existing segment.
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2. DEFINITIONS
Accidental Load: Load resulting from an accidental event such as equipment impact due to
excavation activity. Accidental events are typically discrete rare.
Allowable Stress Design: Design method in which the elastic stresses in the pipeline are limited
to a specified fraction of the minimum resistance.
Assessment Area: Area within which the occupants of buildings and facilities are counted for the
purpose of calculating the population density.
Basic Variable: Random variable (x) used in a limit state function. The basic variables can
include loads, pipe geometry, pipe mechanical properties, and defect properties.
Characteristic Value: The parameter value used in a deterministic (e.g. LRFD) design check. It
is typically defined as the value corresponding to a specified probability level (on the upper tail
for load parameters and lower tail for resistance parameters).
Continuous Load: A load resulting from a continuous random process.
Continuous Random Process: A random process, whose parameter changes continuously with
time (e.g. wind load). Although the parameter may assume an instantaneous value of zero, its
value is generally non-zero.
Design Factor: Design load effect divided by design resistance. It is the inverse of the factor of
safety.
Discrete Load: A load resulting from a discrete random process (e.g. seismic or equipment
impact loads).
Discrete Random Process: A random process, whose parameter assumes non-zero values only
at discrete points in time.
Distributed Limit State: A limit state that is equally likely to occur anywhere along the pipeline
segment. This includes continuously applicable limit states, such as yielding of defect free pipe
under internal pressure, and limit states with unknown locations, such as equipment impact.
Environmental Loads: Loads caused by environmental processes, which are generally variable
with respect to time. They include loads due to thermal variations, ground movement,
earthquakes and wind.
Evaluation Length. Maximum pipeline length over which the reliability targets must be met.
Extreme Distribution: The probability distribution of the maximum or minimum value occurring
in a number of realizations of a random variable.
FORM: First Order Reliability Method
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Definitions
Factor of Safety: Design resistance divided by design load effect. It is the inverse of the design
factor.
Failure: A condition in which the pipeline violates one of its limit states.
Fatigue Limit States: Limit state resulting from fatigue under cyclic loading.
Independent Model Error: A random model error component whose magnitude is independent
of the model output.
Individual Risk: Annual probability of fatality due to a pipeline incident for an individual
situated at a particular location.
Intermittent Random Process: A continuous random process that is interrupted by periods
during which the process parameter is zero.
Leakage Limit State: A limit state characterized by a small leak (less than 10 mm in diameter),
leading to limited loss of containment that does not normally result in a safety hazard.
Limit State: A state beyond which the pipeline no longer satisfies a design requirement.
Limit State Function: Function of the basic variables that assumes negative values when the
limit state is exceeded (i.e. the pipeline fails) and positive values when the limit state is not
exceeded (i.e. the pipeline is safe).
Limit State Surface: A surface in the basic variable space that is defined by setting the value of
the limit state function to zero. It defines the boundary between random variable combinations
leading to failure and random variable combinations leading to safe performance.
Load and Resistance Factor Design (LRFD): Design method in which reliability-calibrated load
and resistance factors are used. The design procedure is deterministic, but the design method is
considered probabilistic, as the load and resistance factors are calibrated to meet specified
reliability targets.
Load Effect: Effect of a single load or combination of loads on the pipeline. The load effect can
be defined in terms of such parameters as force, stress, strain, deformation, or displacement.
Location-specific limit state. A limit state that occurs at a known location, such as failure of a
known corrosion defect or at a known moving slope. The probability of failure for a locationspecific limit state is defined on a per location basis.
Margin of Safety: Load effect subtracted from resistance.
Model Bias: The average value of model error.
Model Scatter: The random scatter associated with model error.
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Definitions
Operational Loads: Loads associated with normal activities during construction or operation.
They are generally variable with respect to time and include internal pressure, weight of
contained fluids, thermal forces due to construction-operation temperature differential and
variable surcharge (e.g. crossing traffic).
Parent Distribution: The probability distribution of a single realization of a random variable.
The term parent is used in the context of external analysis to distinguish the probability
distribution of the random variable from the probability distributions of its extreme values
(maxima or minima).
Partial Safety Factors: Factors by which the characteristic value of a design variable is
multiplied to give the design value. Partial safety factors can be divided into load factors and
resistance factors.
Permanent Loads: Constantly applied loads whose value does not change with time. They
include pipe weight, weight of permanent equipment and coatings, and permanent overburden.
Probabilistic Design: Design method that uses reliability as a measure of structural safety.
Probability of Failure: The probability that a component or a system will fail during a specified
time interval (usually taken as one year). It is equal to the reliability subtracted from one.
Proportional Model Error: A random model error component whose magnitude is proportional
to the model output.
Reliability: The probability that a component or system will perform its required function
without failure during a specified time interval (usually taken as one year). It is equal to the
probability of failure subtracted from one.
Reliability Based Design and Assessment: Design and assessment method in which the pipeline
is designed and operated to meet specified target reliability levels.
Resistance: The maximum load effect that can be borne by a pipeline without leading to a limit
state being exceeded (i.e. without leading to failure).
Return Period: Average time period between occurrences of an uncertain event such as
exceedance of a particular value of a given random variable.
Risk: Probability of failure multiplied by a measure of the adverse failure consequences.
Risk Based Design and Assessment: Design and assessment method in which the pipeline is
designed to meet specified tolerable risk levels.
SORM: Second Order Reliability Method
Safety Class: A classification of the criticality of the pipeline system or part thereof.
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Definitions
Serviceability Limit State: A limit state that limits ability of the structure to meet its functional
requirements, without jeopardizing the primary structural function or leading to safety or
environmental risks. For natural gas pipelines, it is defined as a limit state that violates a design
or service requirement without leading to loss of containment.
Societal Risk: A measure of the overall expected number of fatalities occurring due to pipeline
failures. It can be defined as the expected number of fatalities, in which case it reflects a
constant level of risk for incidents. Alternatively, it can be defined as the expected value of the
number of fatalities raised to a power greater than one, in which case it reflects societal aversion
to incidents causing a large number of fatalities.
Target Reliability Levels: Minimum reliability levels that are considered acceptable for a
specific limit state or class of limit states.
Time-Dependent Reliability: Reliability with respect to a limit state for which the annual
probability of failure changes as a function of time.
Time-Independent Reliability: Reliability with respect to a limit state for which the annual
probability of failure does not change as a function of time.
Ultimate Limit State: A limit state relating to loss of the primary structural function and is likely
to have adverse safety and environmental consequences. For natural gas pipelines, it is defined
as a limit state that leads to loss of containment and results in a safety hazard.
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3. OVERVIEW OF RELIABILITY BASED DESIGN AND ASSESSMENT
3.1 Introduction
The basic objectives of structural design and operation are to ensure that: 1) a structure can
sustain all anticipated loads and deformations during its design life with an adequate margin of
safety against failure; and 2) the performance of the structure does not conflict with its functional
and operational requirements. For many types of structures, including pressure vessels and
pipelines, these objectives were historically achieved by using the allowable stress design
approach, in which the elastic stresses in the structure are limited to some fraction of the
anticipated minimum material strength. Safety against failure in this approach is based on a
factor of safety, defined as the material strength divided by the operating stress (or a design
factor, defined as the inverse of the factor of safety). Minimum factors of safety were
established by code writing bodies on the basis of past experience and professional judgment.
Codes generally started by using conservative safety factors, but as more experience was gained
and material quality improved, less conservative factors were adopted. In the case of the ASME
Boiler and Pressure Vessel Code (BPVC) for instance, the minimum factor of safety on the
tensile strength of steel was originally given a value of 5.0 in 1931, modified to 4.0 in 1950 and
finally set to its current value of 3.0 (Farr 1982). This is a reflection of the fact that safety factors
are meant to compensate for the uncertainties associated with structural systems and, therefore,
these factors can be made less conservative as uncertainty decreases.
The essence of the Reliability Based Design and Assessment (RBDA) approach is to quantify
design uncertainties and use them to calculate a probabilistic safety measure that forms the basis
for evaluating specific designs. This measure, which is referred to as the reliability, is defined as
the probability that failure will not occur during a specified period of time. This section contains
an overview of RBDA methods as they apply to pipelines.
3.2 Historical Perspective
The classical theory of structural reliability, which is the basis for all probabilistic design
methods, was developed after the Second World War as a tool to model the uncertainties
associated with the performance of structures (Pugsley 1951 and Freudenthal et al. 1966). Key
publications, such as Freudenthal (1947), Pugsley (1966) and Ferry-Borges and
Castenheta (1971) describe the foundations of the theory.
Initially, there was little application of the theory in practical design situations because
deterministic design methods had been well established and probabilistic design was viewed as
complex and computationally demanding. This began to change, however, when it was shown
that reliability could be related to a set of deterministic safety factors applied to the load effect
and resistance (Lind 1973). This development made it possible to define deterministic design
checks that are calibrated to meet certain reliability targets, which meant that the benefits of the
approach could be realized through practical design methods based on the Load and Resistance
Factor Design (LRFD) approach.
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In the last three decades, probabilistic design methods have been used as a basis for many
structural design codes. Building codes pioneered this effort on the basis of a number of key
research studies (CIRIA 1977, Ellingwood et al. 1980, MacGregor 1976, Kennedy and Gad
Aly 1980, and Ravindra and Galambos 1978). Major codes that adopted this methodology in the
early 1970’s include ACI (1971), CSA (1973) and CEB (1975). LRFD codes are now used
almost exclusively in North America for designing steel and reinforced concrete buildings
(AISC 1986, CSA S16.1 1994, ACI 318 1983, and CSA A23.3 1994).
Significant advances in reliability theory, software tools and computer hardware over the last two
decades made reliability calculations much more efficient. Most significant in the 1970’s and
1980’s was the development of first and second order reliability concepts (Hasofer and Lind
1974, Rackwitz and Fiessler 1978, and Madsen et al. 1986), which have resulted in several orderof-magnitude reductions in computational requirements. Although these methods made it much
more practical to apply the theory with the computers of that era, they are based on specific
assumptions (see Section 9.2.2.2) and do not necessarily work for all cases. Recent increases in
the processing speed of computers have provided a solution to this problem by allowing a return
to simulation methods for problems that cannot be solved using first and second order reliability
methods. Due to these advances, the calculations required to implement probabilistic design
have become much more efficient, and this resulted in the quick evolution of probabilistic design
methods for many types of structures. Examples are offshore structures (e.g. API RP 2A-LRFD
1993, DNV 1989, FIP 1985 and CSA 1992), bridges (CSA S6 1988), and nuclear containment
structures (Nessim and Hong 1993, and Hwang et al. 1986).
Although reliability-based methods have not yet been widely adopted in the pipeline industry,
interest in these approaches has been growing in recent years, as their potential to achieve
consistent safety at a lower cost is better recognized. This is evidenced by the adoption of these
methods as a basis for the DNV Rules for Submarine Pipeline Systems (DNV 1996), and the
ongoing development of a new ISO standard on Reliability Based Limit State Methods for
pipelines (ISO DIS 16708 2004). Probabilistic methods are also mentioned as a possible design
philosophy in the Canadian Standards Association’s pipeline design standard Z662 (CSA 2003)
and the draft International Standards Organization’s pipeline design code (ISO DIS 13623 2004).
In addition to these design applications, reliability-based methods have been used in the industry
as a basis for increasing the pressure in operating lines (Francis et al. 1998) and making
inspection and maintenance decisions (Nessim and Pandey 1997, and Nessim and Stephens
1998).
3.3 Sources of Uncertainty
Different classification schemes for the sources of uncertainty have been proposed in the
literature (e.g. Ditlevsen 1981b, Der Kiureghian 1989, Melchers 1999 and ISO 2001). The
classification proposed in this document is based on the premise that all uncertainties arise from
lack of precise knowledge of the value of a quantity that is required to forecast the behaviour of a
given pipeline. Uncertainties are classified with respect to the source of lack of knowledge into
the following categories:
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a) Random variations - defined as uncertainty regarding the future value of a parameter that
changes randomly with time. Examples are internal pressure, environmental loads (such as
wind loads) and the forces resulting from equipment impact. This uncertainty stems from the
fact that pipelines are designed and operated to perform adequately under future conditions.
Some of the parameters defining these conditions cannot be determined with certainty at the
time of making the required design or operational decisions, even if perfect models and data
were available.
b) Measurement uncertainty - defined as uncertainty regarding the value of a fixed parameter
due to limitations on the ability to measure its value. Examples are material properties,
defect sizes and defect growth rates. Although it is possible, in principle, to measure these
parameters with great precision, it is usually not practical to do so. For example, the yield
strength at a particular location of a pipeline can only be determined from a destructive
coupon test. Similarly, in-line inspection tools, which are the most practical means of
measuring defect sizes, have some accuracy limitations.
c) Model uncertainty - defined as uncertainty regarding the value of a calculated physical
parameter due to the assumptions and idealizations associated with the model used in the
calculation. An example is uncertainty about the remaining strength of corroded pipe as
calculated from ASME Standard B31G (ASME 1991). This uncertainty can be reduced by
developing better (physical) models.
d) Statistical uncertainty - defined as uncertainty regarding a hypothesized probabilistic model
(or distribution) used to characterize an uncertain parameter. For example, the probability
distribution of the fracture toughness is required to determine the probability of crack failure.
The type and parameters of the fracture toughness distribution may themselves be uncertain
if they are estimated from a limited amount of data. This uncertainty can be reduced by
obtaining more data. It may be interpreted as a subset of model uncertainty (see item c
above) relating to a probabilistic rather than a physical model. To simplify the terminology,
model uncertainty will be used to refer to uncertainties arising from physical models and
statistical uncertainty to denote uncertainty arising from probabilistic models.
3.4 Limit States
A limit state is defined as a state beyond which the structure no longer satisfies a particular
design requirement. It can be regarded as a failure mode, where “failure” is understood in the
broad sense of failing to meet a design requirement. To maintain consistent risk for all failures,
limit states are typically classified into categories with similar failure consequences and higher
reliability targets assigned to limit states with more severe failure consequences.
There are two basic limit state categories that are used in all structural codes:
1. Ultimate limit states (ULS) are concerned with loss of the primary structural function. They
usually refer to loss of strength or stability and are likely to have adverse safety and
environmental consequences. Examples of ultimate limit states for pipelines are burst and
rupture.
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2. Serviceability limit states (SLS) are concerned with the ability of the system to meet its
functional requirements. They often refer to excessive deformations that affect functionality
without jeopardizing the structural integrity or leading to safety or environmental risks.
Examples of serviceability limit states for pipelines include ovalization and denting.
Some references define other limit state categories that overlap the ULS category. For example,
DNV (2000) and ISO (2004) define Fatigue Limit States (FLS) and Accidental Limit States
(ALS) as separate categories. In these codes, fatigue limit states relate to failure resulting from
cyclic loading (e.g. weld cracks), and accidental limit states address severe, rare accidental
loading conditions such as fires or dropped objects. The above codes assign the same reliability
targets to ULS, FLS and ALS.
The essence of the limit states concept is to identify the true failure modes of the pipeline and to
make design decisions that ensure appropriate conservatism, considering the severity of the
failure consequences. For example, the commonly used approach of designing a pipeline
primarily against hoop yield (elastic limit design) could be challenged on the basis that a small
amount of yielding does not necessarily have adverse effects on the pipeline. The real concern is
the possibility of burst leading to loss of containment. Given that the ratio of burst pressure to
yield pressure varies significantly with the post-yield stiffness of the steel, designing against
yield will result in variable safety against burst. In this example, a limit states approach would
lead one to consider burst as an ultimate limit state and ensure that the design is appropriately
conservative considering the corresponding consequences.
3.5 Reliability and Probability of Failure
3.5.1 Basic Concepts
Reliability, R, is defined as the probability that the pipeline will meet all of its design
requirements for a specified period of time. Selecting the time period to be used as a basis for
the definition of reliability is a question of units that does not have a significant impact on the
results. The time period is usually taken as one year. Reliability is related to the probability of
failure, pf, during the same time period by:
R = 1− p f
[3.1]
If the probability of failure due to corrosion is 10-4 per km-year, for example, then the reliability
with respect to corrosion is 1-10-4 or 0.9999 per km-year. This simple one-to-one relationship
between R and pf means that knowledge of one implies knowledge of the other. In practice, pf is
calculated from the probability distributions of the load and resistance and R is calculated from pf
using Equation [3.1]. Since reliability is typically very close to 1.0, it is difficult to read its value
in a simple fractional form (e.g. 0.9999 or 0.99999). It has therefore been customary to calculate
and use the probability of failure as an indication of reliability (e.g. if the probability of failure is
10-5 per km-year, then reliability is expressed as 1-10-5 rather than 0.99999 per km-year).
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Figure 3.1 shows two probability distributions representing the load effect and resistance
corresponding to a specific limit state for a given structural member. It shows that the resistance
is generally higher than the load effect but that the two distributions have a small overlap. This
overlap represents situations in which the load effect exceeds the resistance, leading to the limit
state being exceeded (i.e. failure).
Probability Distribution of the
Resistance (r)
Probability Distribution of the
Load (l)
Mean Load
Mean
Resistance
Load or Resistance
Figure 3.1 Illustration of Load Effect and Resistance Distributions
The probability of failure depends on the degree of overlap between the two distributions, which
is a function of:
•
Separation between the two distributions as determined, for example, by the ratio between
the mean resistance and the mean load effect. Higher values of this ratio imply that the two
distributions will be further apart, leading to smaller overlap area and lower probabilities of
failure.
•
Uncertainty associated with the distributions as measured by its standard deviation or
Coefficient of Variation (COV). For a given ratio between the mean load and mean
resistance, a higher COV results in a distribution that is more “spread out”, resulting in a
larger overlap area and a higher probability of failure.
The basic idea of RBDA is to make decisions that maintain a minimum required level of
reliability (referred to as a reliability target) or, synonymously, a maximum permissible failure
rate. Reliability targets are usually selected to maintain uniform risk, where risk is defined as the
failure probability multiplied by the failure consequences. To achieve this, higher reliability
targets (i.e. lower permissible failure probability levels) are usually specified for limit states with
more severe consequences.
3.5.2 Limit State Function
The probability of failure, pf, can be expressed mathematically as:
p f = p (r ≤ l ) = p (m = r − l ≤ 0)
[3.2]
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Overview of Reliability Based Design and Assessment
where r is the resistance, l is the load effect and m is the margin of safety defined as the
difference between the resistance and the load effect. The load effect and resistance distributions
are usually estimated from other (more basic) variables using analytical models. Examples are
the estimation of earthquake load effects from peak ground accelerations or the calculation of
pipeline pressure resistance from yield strength, diameter and wall thickness. Given this, the
margin of safety, m, can be expressed as a function of a set of n basic variables (denoted by
vector x = x1, x2,….,xn) that determine the load effect and resistance. This function is denoted
g(x), and is called the limit state function. Equation [3.2] becomes:
p f = p[m = g ( x ) ≤ 0]
[3.3]
A simple example can be constructed for burst of a pipeline under internal pressure. In this case,
the load is calculated as the product of the internal pressure, P, and diameter, D (i.e. l = P D).
The resistance equals twice the product of the wall thickness, t, and the flow stress, σf, multiplied
by a factor, a, representing model uncertainty (i.e. r = 2 a t σf). Using this information in
Equation [3.1], leads to the following limit state function:
g = 2 a t σf – P D
[3.4]
Since g equals the safety margin as indicated in Equation [3.3], g(x) ≤ 0 indicates a negative
safety margin, which implies failure, while g(x) > 0 indicates a positive safety margin, which
implies that failure will not occur (safe). This means that the g(x) = 0 separates combinations of
x that lead to failure from those that lead to a safe pipeline. This is illustrated in Figure 3.2 for a
special case with two basic random variables. Because g(x) = 0 defines the boundary between
the failure region and the safe region (see Figure 3.2), it represents the failure condition and is
usually referred to as the limit state surface.
X2
Safe region
g (X1 , Y2 ) > 0
Limit State Surface
g (X1 ,X2 ) = 0
g (X1 , X2 ) < 0
Failure region
X1
Figure 3.2 Illustration of the Limit State Surface
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3.5.3 Calculation Methodology
Estimation of the probability of failure involves solving Equation [3.3] for a given function g and
a given set of probability distributions of the basic variables x. This is illustrated in Figure 3.3
for the limit state function in Equation [3.3]. Assuming that the uncertainties associated with D
are relatively small, the probability of failure can be calculated from Equation [3.4] and the
probability distributions of t, P, σf and a as illustrated in the figure.
Wall thickness
data and
tolerances
Frequency
Several approaches are available to carry out this calculation, including First and Second Order
Reliability Methods (Madsen et al. 1986, Thoft-Christensen and Baker 1982, and Gollwitzer
et al. 1988) and various simulation techniques including Monte Carlo (Rubinstein 1981),
importance sampling (Engelund and Rackwitz 1992) and Latin Hypercube (L’Ecuyer 1994, and
Avramidis and Wilson 1996). A discussion of the basic approach and advantages/limitations of
each of these methods is given in Section 9.2.2.2. The important point for the purpose of this
section is to recognize that, with the variety of available methods and the power of recent
computers, estimating the probability of failure does not present a practical obstacle to the
application of RBDA, provided that the limit state function and input variable distributions are
available.
Operating pressure
profile
Yield strength
data
Maximum Pressure
Reliability
Estimates
Frequency
Random pressure
fluctuations
Probability Density
Wall Thickness
Failure model
uncertainties
Test Results
Yield Stress
x
x
x
x x
x
xx
x
x x
Model Results
Figure 3.3 Illustration of Reliability Estimation for Internal Pressure
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3.6 Reliability Based Design and Assessment
The essence of RBDA is to use reliability (as defined in Section 3.5) as a measure of the safety of
a given structure or facility. This is a rational measure of the degree of success in achieving the
main goal of structural design and operation, namely to ensure safety by minimizing any
chanceof failure. Because reliability is a direct indication of the ultimate safety objective, it
provides a meaningful and consistent measure of the effectiveness of various design and
operational options in achieving the objective. In addition, it allows all structures of a given type
to be evaluated and compared on an equal basis.
The implementation of RBDA involves designing and operating the structure or facility to meet
specified target reliability levels for all applicable limit states. Different target reliability levels
are usually defined for different limit state categories, with higher targets being required for limit
states with more severe consequences. For example, higher reliability targets are usually
specified for ultimate limit states than for serviceability limit states. This helps achieve overall
risk consistency by ensuring that failures with more severe consequences are less likely to occur.
3.7 Benefits
The benefits of RBDA include the following:
1. Design for the true structural behaviour. Reliability-based limit states methods identify the
true modes of pipeline failure and result in solutions that mitigate the actual consequences of
these failures. This avoids unrealistic criteria that lead to undue conservatism. For example,
it may be unrealistic to design pipelines to remain elastic in areas subject to large ground
movements (e.g. areas subject to frost heave and thaw settlement). A reliability-based
approach allows the designer to recognize that a certain amount of plastic deformations could
be acceptable as long as it does not lead to loss of containment or impaired operations.
2. Consistent safety levels. The objective of industry is to achieve adequate safety levels.
Recognizing the uncertainties associated with pipeline design and operation, adequate safety
can most readily be achieved by limiting the probability of a failure to a tolerable level. It is
also reasonable to maintain consistent risk levels (and consequently consistent safety levels)
by requiring lower failure probabilities (or higher reliability levels) for pipelines with more
severe failure consequences. This approach is more consistent than the approach used in
current codes, which use fixed safety factors that result in unknown and highly variable
reliability levels for pipelines with similar consequences.
3. Optimal use of resources. Although higher safety levels are always desirable, the resources
available to improve safety are finite. The best design approach is one that achieves the
highest possible overall level of safety for a given cost. The reliability-based approach
achieves this by ensuring that resources are not wasted on unnecessary conservatism.
4. Adaptability to new problems. The safety measures used in the reliability-based approach
(i.e. risk or reliability) can be calculated from basic principles. Because of this, they are less
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Overview of Reliability Based Design and Assessment
dependent than traditional design methods on a successful track record of application.
Reliability-based methods are therefore suitable for unique projects involving newly
recognized problems (e.g. stress corrosion cracking), unconventional environmental
conditions (e.g. frost heave and thaw settlement) and the application of new technology
(e.g. the use of high strength steels).
5. Integration of design and operational decisions. Reliability-based methods are capable of
evaluating the lifetime reliability of a pipeline considering the impact of operational practices
and integrity maintenance activities. This allows design and operational decisions to be
considered simultaneously, leading to cheaper overall solutions. For example, the reliability
gains due to in-service maintenance activities to be incorporated at the design stage, resulting
in potential reductions in capital expenditures. This could have significant economic benefits
in view of the recent and on-going improvements in inspection and maintenance technologies
and practices.
6. Unified safety measure. The reliability targets used in RBDA provide an objective and direct
measure of safety that is consistent with risk assessment principles. Industry-accepted
reliability targets combined with a standardized approach to reliability estimation provide the
necessary tools for both industry and regulators to measure and evaluate safety performance
in the context of design, operations and risk assessment.
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4. RBDA METHODOLOGY FOR PIPELINES
4.1 Introduction
There are a number of key issues that arise in applying of RBDA to pipeline systems. These
issues include reliability variation as a function of time (due to deterioration mechanisms such as
corrosion and SCC) and periodic reliability improvements due to maintenance and rehabilitation.
The purpose of this chapter is to describe these issues and present an overall methodology that
takes them into account in applying RBDA to pipelines.
Pipeline-specific issues that need to be considered in estimating and evaluating are described in
Section 4.2. This description focuses on introducing the issues and explaining them to the extent
required to follow the overall RBDA methodology described in Section 4.3. A more detailed
description of how these issues are taken into consideration in estimating reliability is given later
in Chapter 9.
A step-by-step process for applying the RBDA methodology to pipelines is described in
Section 4.3. An overview of each step in the process is provided, describing its purpose,
outlining the basic information required for its execution, and showing how it fits in with other
steps. The major analytical steps in this process are addressed in detail in separate subsequent
chapters of these guidelines.
Section 4.4 provides a discussion of the applicability of RBDA to decision-making in the context
of pipeline design and operations.
4.2 Key Issues for Pipeline Reliability
4.2.1 Time Variability
Reliability varies with time for some of the major pipeline failure mechanisms such as corrosion
and ground movements. In the case of corrosion, for example, defects grow continuously with
time and this causes resistance to internal pressure to drop. This means that, without
intervention, the resistance distribution in Figure 3.1 will continue to move closer to the load
distribution, resulting in an ongoing increase in failure probability. Because of this, reliability
must be estimated as a function of time, and this requires information on the rate of change of the
parameters governing deterioration (e.g. corrosion growth or ground movement rates).
In general, pipeline limit states can be classified as either time-dependent if the reliability
changes with time or time-independent if the reliability does not change with time. This
classification depends on the time characteristics of the load and resistance processes involved.
Figure 4.1 shows the types of load processes relevant to onshore pipelines. They are:
a) Time-independent. The load is fixed with respect to time, although its value may be
uncertain (i.e. a random variable). This category includes permanent loads such as dead
load, which does not change with time but could be uncertain because of variability in wall
thickness.
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Load
Load
Time
b) Time-dependent stationary - continuous
Time
a) Time-independent
Load
Load
Time
c) Time-dependent stationary - discrete
Time
d) Time-dependent increasing
Figure 4.1 Types of Loading Processes Applicable to Onshore Pipeline
b) Time-dependent stationary – continuous. The load is continuously applied to the pipeline,
but its value changes randomly as a function of time. Stationary means that, although the
load value changes randomly as a function of time, the statistical properties of the load
process do not change due to a shift of the time scale. The figure shows that the load could
be changing either continuously or at specific points in time. Intermittent processes are also
included in this category as they can be treated as continuous processes applied for a certain
proportion of time. Examples of loading processes in this category include wind and internal
pressure loads (continuous and continuously changing), operational loads (continuous and
changing at specific points in time) and snow and ice loads (intermittent).
c) Time-dependent stationary - discrete. The load occurs at specific (discrete) points in time
and has a very short duration when it occurs. Its value changes randomly between different
occurrences, but its statistical properties do not change due to a shift of the time scale
(stationary). Examples are equipment impact, earthquakes and severe storms.
d) Time-dependent increasing. The load is applied continuously and has an increasing value as
a function of time. The increase is not subject to random fluctuations, although the
parameters governing the change may be uncertain. An example is ground movement due to
frost heave, which will increase continuously with time. The change is governed by
uncertain parameters such as the soil properties and moisture content but is not subject to
significant random time fluctuations.
Resistance processes fall into one of two major categories (Figure 4.2):
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Resistance
Resistance
Time
Time
a) Time-independent
b) Time-dependent decreasing
Figure 4.2 Types of Resistance Processes
•
Time-independent. The resistance is fixed with respect to time, although its value may be
uncertain (i.e. a random variable). Examples are the yield and burst resistance for defect-free
pipe and pipe resistance to equipment impact loads.
•
Time-dependent decreasing. Resistance decreases with time without being subject to random
fluctuations. Examples include resistance at growing defects or deterioration mechanisms
such as corrosion, SCC or weld cracks. The parameters governing the change (such as defect
growth rates) may be uncertain.
Table 4.1 shows the type of limit state arising from each combination of load and resistance
processes described earlier. Combinations that are unlikely to apply to onshore natural gas
pipelines are denoted as N/A. The table shows that a time-dependent stationary load or
resistance results in a time-independent limit state. This is the case because reliability with
respect to a given limit state is a function of the statistical properties of the load or resistance
process, which do not change with time in the case of a stationary process. Only systematically
increasing or decreasing load or resistance processes lead to time-dependent limit states.
Loading Process
Resistance Process
Time-independent
Time-dependent
decreasing
Time-independent
Time-dependent
Stationary continuous
Time-dependent
Stationary discrete
Time-dependent
increasing
Time-independent
Time-independent
Time-independent
Time-dependent
N/A
Time-dependent
N/A
N/A
Table 4.1 Classification of Limit States with Respect to Time Dependence
The classification given here is not entirely comprehensive or strictly representative of all
possible limit states. It is, however, adequate to represent the great majority of onshore gas
pipeline problems and is therefore used as a basis for the models included in these guidelines.
Special cases in which these idealizations are not deemed appropriate must be addressed from
first principles.
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4.2.2 Impact of Maintenance
As mentioned in Section 4.2.1, reliability with respect to time-dependent limit states such as
corrosion will decrease with time as defects grow. A maintenance event such as an inline
inspection followed by appropriate repairs will eliminate the most critical defects resulting in an
immediate increase in reliability. Maintenance can also influence reliability for timeindependent limit states. For example, the probability of failures due to equipment impact can be
reduced by improving damage prevention measures, such as public awareness programs, one-call
systems and pipe location and excavation procedures.
The foregoing indicates that a correct forecast of reliability as a function of time must take
account of all maintenance and prevention activities affecting the limit states being considered.
This implies that maintenance activities must be planned at the evaluation stage and incorporated
in the reliability calculations.
4.3 Implementation Methodology
The steps involved in implementing RBDA for a specific pipeline segment are shown in
Figure 4.3, along with the main inputs required for each step. The process is applicable to design
decisions involving the selection of wall thickness and material properties, as well as operational
decisions involving replacements, inspections, rehabilitation, pressure testing and damage
prevention planning.
The following is a discussion of steps involved in Figure 4.3 with reference to other sections of
the guidelines where these steps are described in more detail.
1. Identification of relevant limit states. The limit states relevant to a given pipeline are
identified based on the loading conditions associated with the proposed route and operational
conditions. Section 5.2.2 provides a list of possible limit states and Chapter 6 describes a
procedure that can be followed to determine their applicability in a given situation.
2. Development of limit state functions. For each limit state, a limit state function is required
(see Section 3.5.2). Limit state functions are deterministic models that can be developed
based on the structural behaviour of the pipe. Guidelines for developing limit state functions
and comments on the availability of relevant structural models are described in Chapter 7.
Appendix A gives limit state functions for some of the key limit states associated with
onshore pipelines.
3. Development of probabilistic models for basic variables. The uncertain parameters (referred
to as basic random variables) used in each limit state function must be characterized by
appropriate probabilistic models. Definition of these models may be based on statistical data,
theoretical models or engineering judgment. Chapter 8 provides guidelines for the selection
of appropriate probabilistic models, and Appendix B gives a review of publicly available data
and models for the basic random variables used in key pipeline limit states.
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Route Data and
Loading Conditions
Identify Relevant
Limit States
Deterministic
Behaviour Models
Develop Limit State
Functions
Statistical Data
Develop Probabilistic
Models of Basic
Variables
Operational
Parameters and
Regulations
Select Design
Parameters and
Maintenance Plan
Probability
Calculation Method
Calculate Reliability
Target Reliability
Levels
Reliability
Target met ?
No
Yes
Economic
Criteria met ?
No
Yes
Acceptable Design
Figure 4.3 Steps Involved in Implementing Reliability Based Design and Assessment
4. Selection of design parameters and maintenance plans. All the parameters required to
evaluate lifetime reliability are required for this step. These include material properties;
design parameters; corrosion mitigation strategies such as coating type and cathodic
protection system characteristics; damage prevention activities such as burial depth, right-ofway patrols and first call system; and in-line inspection plans including tools to be used,
inspection frequency and repair criteria. Depending on the application, some of these
parameters will be treated as decision variables, while others will be treated as constraints.
For a design application, there is a high degree of flexibility, and most of required parameters
can be treated as decision variables. For existing pipelines, the pipeline physical attributes
are fixed, and decisions are typically limited to operational aspects such as defining a safe
operating pressure or a suitable inspection interval. The purpose of this step is to obtain the
values of the parameter that will be treated as constraints and select a set of viable initial
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values for the parameters that will be treated as decision variables. Selection of a reasonable
set of initial values must take any regulatory or policy constraints into consideration.
5. Reliability calculation. For a given limit state, standard probabilistic analysis techniques can
be used to calculate the reliability from the probabilistic models of the basic random variables
and the deterministic limit state surface. A reliability calculation methodology suitable for
the main limit states affecting pipelines is described in Chapter 9.
6. Reliability evaluation. The calculated reliability levels for various limit states are compared
to the target values. The target values must be pre-defined based on an overall safety
philosophy, which takes into account the severity of the consequences associated with each
class of limit states. Section 5.3 states the target reliability levels developed for natural gas
pipelines. The approach used in developing these targets is described in detail by Nessim and
Zhou (2005). If the reliability targets are not met, then steps four, five and six must be
repeated with a new set of decision variables.
7. Economic assessment. This step determines that the safety criteria are met at a reasonable
cost and without undue conservatism. It may involve comparing various design alternatives
that meet the target reliability levels for the purpose of selecting the minimum cost
alternative. An example would be comparing a thick walled design coupled with a standard
maintenance plan to a thinner walled pipeline combined with enhanced maintenance. This
analysis involves calculating the life cycle costs associated with each alternative. Although
this is a key step in developing an optimal design, it is highly project-specific and is therefore
not discussed further in this guideline.
4.4 Applicability
The methodology described in Section 4.3 is equally applicable to new pipeline design and
assessment of existing pipelines for the purpose of making operational and maintenance
decisions. For new pipeline design, the probability distribution of the load effect (refer to
Figure 3.1) is typically pre-determined based on the relevant operational and environmental
parameters. In this case, reliability targets are met by influencing the probability distribution of
the resistance through selection of such parameters as wall thickness, grade and burial depth.
For the assessment of existing pipelines, the basic parameters influencing the resistance
distribution (such as wall thickness and grade) are pre-determined. However, deterioration
mechanisms, such as corrosion and SCC, will cause the resistance distribution to change with
time. The changes could include a reduction in average resistance, which would be reflected as a
shift to the left of the resistance distribution in Figure 3.1 and/or an increase in variability, which
means an increase in uncertainty or distribution “spread”. As discussed in Section 3.5.1, both of
these changes would result in an increase in failure probability, which can be prevented by either
reducing the load through a pressure de-rating or controlling the deterioration of resistance
through inspection and maintenance.
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5. DESIGN AND ASSESSMENT REQUIREMENTS
5.1 General Requirements
This chapter describes the requirements that must be met to ensure adequate overall reliability for
a natural gas pipeline. It specifies the limit states that need to be considered and categorizes
them into three groups: ultimate, leakage and serviceability (Section 5.2). It also specifies the
minimum reliability targets that must be met for each limit state category to ensure adequate
safety (Section 5.3). In addition to the specific requirements in Sections 5.2 and 5.3, the
following general requirements apply:
•
Reliability targets must be met along the entire pipeline or pipeline segment being
considered. Given the variations in conditions that occur along a pipeline route, it will
typically be required to divide the pipeline into segments over which the parameters affecting
reliability (e.g. pressure, population density, land use and soil conditions) are reasonably
uniform and establish reliability on a segment-by-segment basis. The number of segments is
a compromise between accuracy and level of effort.
•
For a given pipeline segment, the reliability targets for a particular limit state category must
be met considering the combined contributions to the failure probability from all limit states
in that category. This means that the failure probability with respect to ultimate limit states
for a pipeline segment that is subject to a number of independent failure causes, such as
corrosion and equipment impact, should be calculated as the sum of the individual failure
probabilities for all applicable causes. The reliability can then be calculated as the total
failure probability subtracted from one, and must be higher than or equal to the appropriate
reliability target.
•
If the reliability changes along a given pipeline segment, it is necessary to specify a
maximum length over which the reliability targets are to be met (this will be referred to as the
evaluation length). This length is defined as the lesser of the segment length and 1600 m
(1 mile). The evaluation length defines the size of the “unit” over which the reliability
targets must be consistently met. If the evaluation length is large (e.g. hundreds of
kilometers), the reliability targets can be met over the total length, despite the presence of
short low-reliability portions within the evaluation length. This implies that a single
reliability check is required for segments that are shorter than or equal to 1600 m. For
segments longer than 1600 m, the number of reliability checks required equals the number of
positions of the 1600 m evaluation length producing unique values of the average failure
probability. This number depends on the manner in which the failure probability varies along
the segment length.
•
If the probability of failure changes, or if location-specific limit states (such as known
corrosion defects or moving slopes) exist within the evaluation length, the probability of
failure used in the reliability check should be an average value per km-year, calculated as the
total probability of failure divided by the evaluation length.
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•
Load combinations resulting from all applicable operating and environmental conditions
must be appropriately considered in calculating reliability. For example, in calculating the
probability of buckling due to ground movement, conditions involving pressurized and
unpressurized pipe must be appropriately considered.
•
Reliability targets must be met throughout the operating life of the pipeline. Since reliability
generally decreases gradually with time due to time-dependent deterioration mechanisms
such as corrosion or SCC and increases suddenly after specific maintenance events such as
inline inspection and repair, the critical time points for meeting the reliability targets will
occur immediately prior to maintenance events. To meet the reliability targets, reliability can
be increased throughout the operational life by changing the basic design parameters such as
wall thickness. Alternatively, maintenance intervals can be reduced to ensure that reliability
minima at critical time points do not drop below the target.
5.2 Limit States
5.2.1 Categories of Limit States
Based on the general limit state classification described in Section 3.4 and the pipeline-specific
considerations discussed in Nessim and Zhou (2005), the following limit state categories have
been adopted for natural gas pipelines:
•
Ultimate Limit State (ULS). A limit state that leads to loss of containment and results in a
safety hazard. This category expressly includes large leaks and ruptures (as defined in
Section 2.2), which may result from defect failures, equipment impact or tensile longitudinal
bending. It may also include other structural conditions such as local or global buckling or
section collapse, if these conditions progress into large leaks and ruptures.
•
Leakage Limit State (LLS). A leak of less than 10 mm in diameter leading to limited loss of
containment that does not result in a safety hazard.
•
Serviceability Limit State (SLS). A limit state that violates a design or service requirement
without leading to loss of containment. This category includes yielding, ovalization, denting
and excessive plastic deformation. It may also include other structural conditions, such as
local or global buckling, if it is demonstrated through detailed analysis or implementation of
an appropriate monitoring and maintenance program that these conditions will not progress to
cause loss of containment.
5.2.2 Loads and Limit States
Table 5.1 provides a list of load/limit state combinations that are applicable to onshore pipelines.
The list was compiled partly on the basis of the design conditions mentioned in various codes.
The first column in the table specifies the life cycle phase, which is defined as a phase of the
pipeline life with distinct loading conditions (and consequently with distinct limit states). The
three life cycle phases considered are transportation, construction and operation. Column 2
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identifies load cases that occur during each phase and Column 3 lists companion load cases that
can occur in combination with each case in Column 2.
Load Case
Life Cycle Phase
Transportation
Companion
Load Cases
Limit State Type
Stress limit
Strain limit
Time
Dependent
1 Accidental impact
Denting / gouging
SLS
9
No
2 Cyclic bending
Fatigue crack growth
SLS
9
Yes
3 Stacking weight
Ovalization
SLS
9
4 Cold field bending
Local Buckling
SLS
5 Bending during installation
Construction
Limit State
6 Directional drilling tension and bending
7 Hydrostatic test
9 Overburden and surface loads
8
SLS
9
9
No
Local Buckling
SLS
9
9
No
Girth weld tensile fracture
SLS
9
No
Local buckling
SLS
9
No
Excessive plastic deformations
SLS
9
No
Burst of defect-free pipe
SLS
9
No
Burst at dent-gouge defect
SLS
9
Burst at seam weld defect
SLS
9
SLS or ULS1
9
No
Burst at corrosion defect
ULS
9
Yes
Small leak at corrosion defect
LLS
9
Yes
Burst at environmental crack (SCC)
ULS
9
Yes
Small Leak at environmental crack (SCC)
LLS
9
Yes
Burst of a manufacturing defect
ULS
9
Yes
Small leak of a manufacturing defect
LLS
9
Yes
Ductile fracture propagation
ULS
9
No
Burst of a weld defect
ULS
9
Yes
11 Above ground span support settlement
12 Wind on above-ground spans
13 Slope instability, ground movement
Operation
14 Seismic loads
15 Restrained thermal expansion
8
8,10
8,10
8,14,15
16 Frost heave
8,14,15
17 Thaw settlement
8,14,15
18 Loss of soil support (e.g., subsidence)
8,15
9
Yes
9
No
SLS
9
No
Formation of mechanism by yielding
SLS or ULS1
9
No
Local buckling
SLS or ULS1
9
No
Girth weld tensile fracture
SLS or ULS1
9
No
Local buckling
SLS or ULS1
9
Yes
Yes
Girth weld tensile fracture
Dynamic instability
Burst of crack by fatigue
8,15
20 Buoyancy
8,15
23 Sabotage
8
8
9
9
No
ULS
9
Yes
9
Yes
ULS
9
Yes
SLS or ULS1
9
No
ULS
9
No
Local buckling
SLS or ULS1
9
No
Upheaval buckling
SLS or ULS1
Local buckling
SLS or ULS1
9
Yes
Girth weld tensile fracture
Local buckling
Girth weld tensile fracture
No
9
ULS
9
Yes
SLS or ULS1
9
Yes
9
ULS
Excessive plastic deformation
SLS or ULS1
Local buckling
SLS or ULS1
9
Yes
Yes
9
Yes
9
Yes
ULS
9
Dynamic instability
SLS or ULS1
9
Formation of mechanism by yielding
SLS or ULS1
9
Local buckling
SLS or ULS1
9
No
ULS
9
No
Girth weld tensile fracture
21 Outside force
ULS
SLS or ULS1
SLS or ULS1
Local buckling
Girth weld tensile fracture
Girth weld tensile fracture
19 River bottom erosion
No
LLS
Plastic collapse
8,15, 13 or 16 Local buckling
or 17
Girth weld tensile fracture
8
No
9
SLS or ULS1
Small leak of a weld defect
Ovalization
10 Gravity loads on above-ground spans
No
Plastic collapse
Excessive plastic deformations
8 Internal pressure
No
9
No
No
Floatation
SLS or ULS1
Denting
SLS
9
No
Puncture
ULS
9
No
Burst of a gouged dent
ULS
9
No
Small leak of a gouged dent
LLS
9
No
Rupture
ULS
9
No
No
1 Starts as a serviceability limit state, but could progress to an ultimate limit state
Table 5.1 Load Cases and Limit States Relevant to Onshore Pipelines
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The limit states corresponding to each life cycle phase and loading process combination are
given in Column 4 and classified as ultimate (ULS), serviceability (SLS) or leakage (LLS) in
Column 5. Limit states that are classified as “SLS or ULS” start as serviceability limit states, but
could progress into ultimate limit states. In these cases, classification of the limit state requires
judgment regarding the potential for an SLS to progress into a ULS. A limit state should be
classified as an SLS only if it can be demonstrated that it will not progress into a ULS. For
example, local buckling due to frost heave can be treated as a serviceability limit state if there is
an active monitoring program to identify and repair locations experiencing buckling.
Columns 6 and 7 classify limit states as stress-based or strain-based. This is intended to
highlight the formulation that would typically be used rather than exclude the other (unchecked)
option. A strain-based limit state is one that is deformation-controlled, so that strains cannot
increase unless further deformations are imposed (e.g. frost heave and thaw settlement). A
stress-based limit state is one that is load-controlled, causing strains to increase dramatically once
the applied load reaches the load carrying capacity. (See CSA Z662 Clause C.4.4.2.2 for more
detailed definitions). This classification is useful in identifying situations in which an SLS is
likely to progress into a ULS (see above paragraph). For example, local buckling due to frost
heave is strain-controlled and can therefore be treated as an SLS provided that it can be detected
and repaired before the slowly increasing strains lead to a ULS. By contrast, local buckling
under gravity loads is load-controlled and unless adequate reserve load capacity can be
demonstrated, it should be treated as a ULS.
Although Table 5.1 includes most key limit states, it is not intended as a comprehensive listing,
and should not be used as evidence that other possible design conditions can be excluded.
5.3 Reliability Targets
5.3.1 Introduction
The reliability targets given in this document were developed as minimum requirements to
ensure that adequate human and environmental safety levels are maintained throughout the life of
a pipeline. Since the environmental risks associated with natural gas pipelines are negligible in
comparison to human safety risks, the reliability targets were developed to ensure adequate levels
of human safety. Economic considerations were not taken into account because they vary widely
for different pipelines. Since economic impact is borne by the pipeline operator, it is considered
prudent for operators to carry out economic assessments in conjunction with RBDA applications
related to high-cost pipelines. An economic assessment may show that cost is minimized by
exceeding the reliability targets given in this section.
According to the limit state categories defined in Section 5.2.1, only ultimate limit states have
significant safety-related consequences. The reliability targets for ULS were therefore developed
using a risk-based approach that ensures consistent and adequate safety levels for all pipelines.
On the other hand, the consequences of leakage (i.e. small leaks) and serviceability limit states
are primarily economic, including the costs of repair and possible service interruption. Minimum
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LLS and SLS reliability targets were defined on the basis of historical information and published
precedent.
Details of the methodology used in developing the reliability targets are described in a separate
document (Nessim and Zhou 2005). This section summarizes the targets and provides an
overview of the approach and criteria used to develop them.
5.3.2 Ultimate Limit States
5.3.2.1 Approach
A risk-based approach was used to define the maximum permissible failure probability for given
pipeline segment. The approach follows from the basic definition of risk, r, as:
r = p×c
[5.1]
where p is the probability of failure per km-year of pipeline and c is a measure of the failure
consequences. Based on Equation [5.1], the maximum permissible failure probability, pmax, can
be calculated from the maximum permissible risk level, rmax, as:
p max = rmax / c
[5.2]
Since reliability, RT, is defined as the annual probability that the pipeline will not fail (see
Equation [3.1]), the tolerable reliability can be calculated as:
R T = 1 − pmax = 1 − rmax / c
[5.3]
Equations [5.2] and [5.3] show that the maximum permissible failure rate (and consequently the
target reliability) is a function of the maximum tolerable level of risk, rmax, and the failure
consequences, c. By using a given tolerable risk level for all pipelines and substituting the
appropriate pipeline-specific consequences, the reliability targets resulting from Equation [5.3]
ensure that the risk level for all pipelines is fixed and equal to the tolerable value.
5.3.2.2 Format
The consequences of failure, c, can be measured by the number of fatalities, N, resulting from
exposure to heat emitted from a gas fire. The expected number of fatalities in a given incident
can be calculated from:
c = N = pi a h ρ τ
[5.4]
where pi is the probability of ignition given a failure, ah is the size of the hazard area (defined as
the area within which people would be exposed to a lethal heat dosage), ρ is the population
density (occupants per unit area) and τ is the occupancy probability (defined as the probability of
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an occupant being present in the hazard zone at the time of the incident). It can be shown (see
Nessim and Zhou 2005) that the hazard area is proportional to PD2 and that the probability of
ignition is approximately proportional to D, where P and D are the pipeline pressure and
diameter. Using this in Equation [5.4], and assuming that τ is constant for a given location, leads
to:
c ∝ ρ PD 3
[5.5]
Substituting Equation [5.5] in [5.2] demonstrates that, for a given value of tolerable risk, rmax, the
maximum permissible failure rate, pmax, is inversely proportional to ρPD3. Equivalently,
substituting in Equation [5.3] demonstrates that a reliability target that meets a particular
tolerable risk level is an increasing function of ρPD3. This approach, which is used in defining
the format of the ULS reliability targets in these guidelines, implies the reasonable expectation
that reliability targets should become more stringent for larger pipelines operating at higher
pressures in more heavily populated areas.
5.3.2.3 Safety Criteria
Safety criteria are expressed in terms of the maximum tolerable risk level, which is used in
Equation [5.3] to define the ULS reliability targets. Because of the complex issues associated
with quantifying risk, a number of measures that focus on different aspects of risk have been
used in the industry. To ensure comprehensive consideration of all safety-related aspects,
appropriate criteria corresponding to each of these measures were considered in developing the
reliability targets. These criteria can be classified into two major categories, namely societal risk
and individual risk. The remainder of this section explains these risk measures and describes the
tolerable risk levels associated with each.
5.3.2.3.1 Societal risk
Societal risk is a measure of the overall risk of fatality due to pipeline incidents. It can be
quantified using one of two approaches. The first approach, referred to here as societal risk with
fixed expectation, is to use the expected number of fatalities as a direct measure of the risk.
This measure implies that the risk associated with a low probability incident causing a large
number of fatalities is equivalent to the risk associated with a higher probability incident causing
a proportionately lower number of fatalities. The second approach, referred to here as societal
risk with aversion function, is to measure the risk by the expected value of the number of
fatalities raised to a power greater than one. This measure implies that the risk increases
exponentially with the number of fatalities, which means that a low probability incident causing
a large number of fatalities represents a higher risk than a higher probability incident causing a
proportionately lower number of fatalities (see Nessim and Zhou 2005 for a detailed discussion).
This trend represents society’s aversion to incidents causing large numbers of fatalities – the
power to which the number of fatalities is raised represents the degree of aversion.
Tolerable societal risk levels were generated by calibration to existing codes including ASME
B31.8 (ASME 1999), ASME B31.8S (ASME 2002) and 94CFR192.327 (US Federal Regulations
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1971). Since new pipelines designed and maintained to the requirements of these codes are
widely accepted as safe, the average level of societal risk implied by current codes for the
existing pipeline network can be considered tolerable. Based on this, the maximum tolerable
societal risk levels were specified as equal to the calculated average societal risk for a network of
new pipelines that are designed, operated and maintained according to the above-mentioned
codes and regulations. The risk levels implied by B31.8 were estimated using the calibration
process described in Section 3.2 of Nessim and Zhou (2005). The societal risk measures and
corresponding tolerable risk levels resulting from this process are given in Table 5.2, in which p
is the probability of failure and N is the number of fatalities.
Criterion
Fixed expectation
Societal risk with aversion function
Risk Measure
Tolerable Risk (per km-year)
pxN
p x N1.6
1.6 x 10-5
3.6 x 10-5
Table 5.2 Tolerable Societal Risk Levels Calibrated to ASME B31.8
5.3.2.3.2 Individual risk
Individual risk is a measure of risk to specific individuals who are exposed by virtue of their
regular presence at a particular location (e.g. home or workplace). It is usually measured by the
annual probability of fatality due to a pipeline incident for a person located at a specific point
within the pipeline hazard zone. An individual risk criterion is required because societal risk
criteria could lead to high permissible failure probabilities in sparsely populated areas where the
number of expected fatalities is low. This would imply that the societal risk, although tolerable,
is concentrated in a small number of individuals who may not be adequately protected.
Maximum tolerable individual risk criteria used in this work were selected based on information
published by HSE (2001) and, MIACC (1995). Although these sources do not explicitly specify
target individual risk levels by location class, the information they provide was used to select
annual tolerable risk levels of 10-4 in Class 1, 10-5 in Class 2, and 10-6 in Classes 3 and 4, where
the various classes are defined according to ASME B31.8. The decrease in tolerable individual
risk as a function of class reflects a requirement to decrease risk as the number of people exposed
increases. This is an established method to include aversion to large incidents (see discussion
under societal risk criteria) in an approach based on individual risk.
5.3.2.4 Reliability Targets
5.3.2.4.1 Based on Population Density
Figure 5.1 shows the reliability targets as a function of ρPD3 for all societal and individual risk
criteria discussed in Section 5.3.2.3. The individual risk targets are class-dependent because the
underlying risk criteria vary by location class. Each individual risk target curve is applicable in a
limited range of ρPD3 that corresponds to the population density for the underlying location
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class. Selected pipelines with specific combinations of pressure, diameter and class are marked
on the figure to create reference points for interpreting the quantity on the horizontal axis.
1 - 1E-09
1000 psi,14-inch,Class
Target Reliability (per km-yr)
1 - 1E-08
1000 psi,14-inch,Class
2
1 - 1E-07
1 - 1E-06
Proposed Targets
1 - 1E-05
1 - 1E-04
3000 psi,26-inch,Class 3
1400 psi,20-inch,Class 4
1 - 1E-03
Risk aver
1 - 1E-02
Fixed expectation
1 - 1E-01
Individual risk
1 - 1E+00
1.E+04
1.E+05
1.E+06
1.E+07
3
1.E+08
1.E+09
1.E+10
1.E+11
3
ρ PD (people/hec-psi-in )
Figure 5.1 Reliability Targets from All Three Criteria Considered
The proposed targets are defined as the upper envelope for all criteria, which means that the most
conservative criterion is always selected. The figure suggests that the individual risk criterion
produces the highest target reliability level for small value of ρPD3. It also shows that societal
risk with fixed expectation produces the highest target level for medium values of ρPD3, and
societal risk with aversion function produces the highest target for large values of ρPD3. For a
population density of zero, the societal risk targets are not applicable and an individual risk level
of 10-4 per year is used as a minimum requirement. The target reliability levels, RT, in Figure 5.1
can be calculated from the following equations:
72

ρ =0
1 − ( PD3 )0.66

9

1
ρPD3 ≤ 1.0 ×105
−
 (ρPD3 )0.66

RT = 
1 − 450
1.0 ×105 < ρPD3 ≤ 6.0 ×107
3
 ρPD

7
1 − 2.1×10
ρPD3 > 6.0 ×107
3 1.6
 (ρPD )
[5.6]
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The reliability targets given in Figure 5.1 and Equation [5.6] are defined as a direct function of
population density, ρ. This approach makes the application of RBDA independent of the
location class concept. When this approach is used, the population density at any point along the
pipeline is calculated as the number of occupants of all buildings and facilities within an
assessment area centred on that point, divided by the size of the assessment area. The number of
occupants in this calculation should be the average number of people in the building or facility
during its normal use. No reduction should be made to this number based on the fraction of time
during which the building or facility is occupied because this reduction is already built into the
targets (Nessim and Zhou 2005).
For the purpose of pipeline segmentation, the population density at any point along the pipeline
is the lesser of the two values calculated using the following two definitions of the assessment
area (Zhou and Nessim 2005):
•
A rectangle with a length of 1600 m and width of 0.33 PD 2 m, where P is the pressure in
psi and D is the diameter in inches, with the length is parallel to the pipeline axis.
•
A square with sides equaling 0.33
diameter in inches.
PD 2 m, where P is the pressure in psi and D is the
The value of 0.33 PD 2 represents the diameter of the equivalent hazard area around the
pipeline. Using a 1600 m long rectangle provides a realistic value of the population density, but
does not give a correct indication of the appropriate boundaries between segments. Using a
length of 0.33 PD 2 gives an accurate characterization of segment boundaries, but gives
unrealistic sharp increases in population density around isolated structures. Using the minimum
of the density values calculated from the two methods is equivalent to using the first approach to
calculate the population density and the second approach to define the segment boundaries.
To limit risk variations within a given segment, it is suggested that limits should be placed on
population density variations within a segment. Based on the population density categories
described in Nessim and Zhou (2005), it is suggested that these limits should be as follows:
•
The maximum population density along a segment that has unpopulated portions shall not
exceed 0.4 people per hectare.
•
The ratio between the maximum and minimum population density along any given segment
shall not exceed 10.
Once the pipeline has been segmented, the average population density for a given segment
should be used to determine the target reliability level for the segment. The average population
density for the segment can be calculated as the number of occupants of all buildings and
facilities within half an assessment width on either side of the pipeline along the segment,
divided by the product of the assessment width and the segment length. If the segment length is
less than 1600 m, this calculation should be based on a 1600 m segment created by extending the
original segment equally on either end.
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5.3.2.4.2 Based on Location Class
Target Reliability (per km-yr)
Reliability targets for the ASME B31.8 and CSA Z662 Classes 1 through 4 are given in
Figure 5.2 and Equation [5.6]. These targets were derived by using the average population
density for each location class (see Table 5.3) in Figure 5.1 and Equations [5.6]. They represent
the highest of the individual risk, fixed expectation societal and risk averse societal targets. The
slope change in the Class 1 target line corresponds to a transition between targets governed by
individual risk and fixed expectation societal risk criteria. The change in slope for the Classes 2,
3 and 4 target lines corresponds to a transition between targets governed by fixed expectation and
risk averse societal risk criteria.
1 - 1E-09
1000 psi, 10 in
1 - 1E-08
Class 1
Class 2
Class 3
Class 4
1 - 1E-07
1200 psi, 20 in
1400 psi, 42 in
1.E+07
1.E+08
1 - 1E-06
1 - 1E-05
1 - 1E-04
1 - 1E-03
1 - 1E-02
1 - 1E-01
1.E+05
1.E+06
3
3
1.E+09
PD (psi-in )
Figure 5.2 Reliability Targets by Class
Class
Pipeline Length
(km)
Average Population Density
(people per hectare)
1
18845
0.04
2
315
3.3
3
37
18
4
0
100 (assumed value)
2
Note: 1 hectare = 10,000 m = 2.47 acre
Table 5.3 Population Density by Class Based on Structure Data for Actual Pipelines (Nessim and Zhou 2005)
The class-specific societal risk targets, RTsc, and individual risk targets, RTi, are given by the
following equations:
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75

1 − ( PD3 )0.66
RT = 
1 − 11250

PD3
 135
1 − PD3
RT = 
3.1 ×106
1 −
 ( PD3 )1.6
25

1 − PD3
RT = 
2.0 ×105
1 −
 ( PD3 )1.6
4.5

1 − PD3
RT = 
13250
1 −
3 1.6
 ( PD )
PD3 ≤ 2.5 ×106
Class 1
[5.7a]
Class 2
[5.7b]
Class 3
[5.7c]
Class 4
[5.7d]
PD > 2.5 ×10
3
6
PD3 ≤ 1.8 ×107
PD > 1.8 ×10
3
7
PD3 ≤ 3.3 ×106
PD3 > 3.3 ×106
PD3 ≤ 6.0 ×105
PD3 > 6.0 ×105
5.3.2.5 Meeting the Targets
5.3.2.5.1 Large Leaks versus Ruptures
Although the ULS reliability targets given here are intended to apply to large leaks and ruptures
combined, they were derived based on the conservative assumption that the consequences of
rupture apply to both large leaks and ruptures. A simple and conservative approach for applying
these targets is to ensure that the reliability level calculated from the total probability of large
leaks, pLL, and ruptures, pRU exceeds the targets, RT, i.e.
1 − ( p LL + p RU ) > RT
[5.8]
This approach does not require distinction between large leaks and ruptures in the reliability
calculations. The failure probability calculations for corrosion and equipment impact for
example, would only require consideration of a single limit state for burst; a second limit state
for unstable growth of the resulting hole would not be required. It is conservative, however, as it
does not take advantage of the relatively small magnitude of leak consequences.
If the reliability calculation model used in the assessment provides separate probability estimates
for large leaks and ruptures, a less conservative approach can be used in which leaks are
converted to “equivalent ruptures”. To produce an equivalent risk level, the probability of the
equivalent ruptures is defined as the probability of a large leak multiplied by the ratio between
the consequences of a large leak, cLL, and the consequences of a rupture, cRU. The ratio, cr,
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between the consequences of large leaks and ruptures can be calculated from Equation [5.9] or
Figure 5.3 (Nessim and Zhou 2005).
cr =
45.8
D3
[5.9]
1.0E+00
cr
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E+02
1.0E+03
1.0E+04
3
1.0E+05
3
D (inch )
Figure 5.3 Relative Expected Number of Fatalities for Large Leaks and Ruptures
In this case, the target reliability can be achieved by ensuring that the reliability level
corresponding to the sum of the probabilities for “equivalent ruptures” and ruptures exceeds the
reliability target, i.e.:
45.8


1 − [ p LL c r + p RU ] = 1 −  p LL 3 + p RU  > RT
D


[5.10]
Since reliability decreases with time due to time-dependent failure causes (e.g., corrosion), and
increases suddenly after maintenance events that are aimed at managing these causes (e.g., a
corrosion inspection and repair), the critical points in time for meeting the targets will occur
immediately prior to a maintenance event. Meeting the targets at those points in time implies
that reliability will be higher than the targets for most of the pipeline life, and, therefore, the
average reliability will be higher than the target. This means that the actual average reliability
associated with using the proposed approach will be higher than the average reliability implied
by current codes for new well-maintained pipelines.
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5.3.2.5.2 Location-Specific Limit States
The reliability targets in Section 5.3.2.4 are defined on a per km-year basis. This format is
directly applicable to distributed limit states, defined as limit states that have the same
probability of occurrence at all locations along the pipeline. Distributed limit states include limit
states that apply continuously at all cross sections of the pipeline, such as excessive deformation
under hoop stress, and limit states that are equally likely to occur anywhere along the length,
such as equipment impact or corrosion (if the locations of corrosion defects are unknown). The
reliability check in this case is given by:
1 − p f > RT
[5.11]
A special case arises for limit states that apply at known locations. These limit states, which will
be referred to here as location-specific limit states, include excessive strains due to movement of
a particular slope or burst of known corrosion defects (e.g., based on the results of an inline
inspection). The probability of failure for these limit states is a finite value defined at the
location in question, which cannot be directly compared to a target that is defined on a per km
basis.
To address this case, special design checks have been developed to ensure that the risk criteria
underlying the targets are met for pipelines involving location-specific limit states. These checks
account for the fact that location-specific limit states contribute differently to societal risk and
individual risk. In the case of societal risk, the targets are intended to limit total risk over the
evaluation length, and therefore, the contributions of distributed and location-specific limit states
are aggregated over the evaluation length (1600 m in most cases). For individual risk, the targets
are intended to limit the individual risk at specific points along the pipeline alignment, and
therefore, the probabilities are summed up over the interaction length for a given point.
For societal risk, this leads to the following check:

∑ p fi 

i =1, 2 ,..,n
1 −  p fd +
 ≥ RTS
l




[5.12]
where pfd is the probability of failure (per km-year) for distributed limit states, pfi is the
probability of failure (per year) due to the ith location-specific limit state on the evaluation length,
n is the number of location-specific limit states on the evaluation length, and RTS is the societalrisk-based reliability target given by (see Section 5.3.2.4):
450

1 − ρPD3

RTS = 
7
1 − 2.1×10
 (ρPD3 )1.6
ρPD3 ≤ 6.0 ×107
[5.13]
ρPD > 6.0 ×10
3
7
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Equation [5.12] can be interpreted as a version of the basic reliability check in Equation [5.11],
with the failure probability being calculated as an average over the evaluation length. The check
must be met for all possible positions of the evaluation length within a given segment.
For individual risk, the reliability check is given by:

∑ p fi 

i =1, 2 ,..,m
1 −  p fd +
 ≥ RTI
IL 



[5.14]
where m is the number of location-specific limit states within 0.5 IL on either side of the location
of interest, IL is the equivalent interaction length given by 0.33 PD 2 (see Nessim and Zhou
2005), and RTI is the individual-risk-based target reliability given by (see Section 5.3.2.4):

9
RTI = 1 −
3 0.66
 (ρPD )
ρ >0
[5.15]
Equation [5.14] can be interpreted as a version of the basic reliability check in Equation [5.11],
with the failure probability being calculated as an average over an evaluation length that equals
the equivalent interaction length. The check must be satisfied for all possible positions of the
interaction length on the pipeline alignment.
Because the reliability checks for individual risk and societal risk targets are different, it is not
possible to pre-select a single reliability target (i.e., based on either individual or societal risk) for
a given pipeline section as suggested in Section 5.3.2.4. This is why it is necessary to carry out
independent checks on the societal and individual risk targets for pipeline lengths involving
location-specific limit states. Since societal risk targets are not relevant if the population density
is zero (see Section 5.3.2.4.1), a single population-density-independent individual-risk-based
target is applicable in that case. This is given by (see Section 5.3.2.4):
RT = 1 −
72
( PD3 ) 0.66
ρ =0
[5.16]
and must be met for all possible positions of the interaction length using the individual risk check
in Equation [5.14].
5.3.3 Leakage Limit States
The specified reliability target for LLS (i.e. small leaks) is 1-10-2 per km-year. This target was
selected based on a combination of leak impact analysis, historical leak rates and calibration to
ASME B31.8 (see Nessim and Zhou 2005 for details of the analysis and justification of the
target). Because the human and environmental safety consequences of a small leak are
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insignificant and fairly uniform for all pipelines, the target is a fixed value that does not depend
on pipeline characteristics.
The LLS targets are only likely to govern small diameter and low pressure pipelines. This is
because large diameter and high-pressure pipelines require thick walls for pressure containment
(i.e. will be governed by ULS targets) and will therefore have small leak rates that are much
smaller than the targets. In cases where LLS targets govern, it is likely that the targets will be
met through enhancements to inspection and maintenance rather than through increases in wall
thickness. Further, operators may choose to exceed this target for economic or political reasons.
Factors that should be considered in selecting an appropriate actual reliability level may include:
•
Ease of access and repair costs.
•
Security of supply and availability of alternate product sources.
•
Regulatory response and public reaction.
5.3.4 Serviceability Limit States
The specified reliability target for SLS is 1-10-1 per km-year. This target is based on values
suggested in other standards (CSA 1992a and ISO 2004). It is based on the SLS definition used
in these guidelines, which explicitly excludes any conditions leading to loss of containment (see
Section 5.2.1), thus ensuring that an SLS does not have any significant safety or environmental
consequences.
Two comments are made regarding the application of this value:
•
To use this target, it must be established through detailed analyses or maintenance programs
that the deformation level associated with the SLS will not progress to loss of containment
and a ULS. If this cannot be established, then the limit should be conservatively treated as a
ULS.
•
Since the consequences of exceeding an SLS are mainly economic (repair costs and possible
operational delays), operators may choose to exceed this target in cases where the probability
of exceeding an SLS and/or the cost of repair is high.
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6. IDENTIFICATION OF RELEVANT LIMIT STATES
6.1 Introduction
Inclusion of a limit state in a reliability-based design and assessment procedure requires a certain
level of effort to select and implement an appropriate limit state function and model the basic
variables involved. It is therefore important to avoid including limit states that are known
beforehand not to influence the decisions being made. This section provides an approach for
screening limit states to determine their relevance to a particular pipeline segment.
Identification of applicable limit states is primarily a process of exercising sound engineering
judgment to determine conditions that require an explicit check. The screening process presented
in this section is intended as a tool to facilitate (rather than substitute for) engineering judgment
in deciding on the applicability of borderline limit states.
The screening process consists of a number of sequential deterministic and probabilistic
screening checks (see Sections 6.2 and 6.3). Preliminary values of the pipeline parameters must
be developed beforehand to provide the required context for implementing these checks. For
example, the wall thickness is required to assess the applicability of the limit state representing
plastic deformation under internal pressure. The probabilistic checks also require that target
reliability values be defined for each limit state (see Section 6.3).
Since the purpose is to avoid expending the effort necessary to carry out a full reliability-based
check for cases that do not govern, the procedure does not contemplate detailed probability
calculations. Whenever probability estimates are required, they are assumed to be order-ofmagnitude values defined on the basis of subjective assignments, previous knowledge of similar
situations or analysis of available data on individual parameters characterizing load and
resistance. Since the consequences of a non-conservative probability estimate could be the
elimination of an applicable limit state, it is important to ensure that probability estimates are
conservative.
The deterministic checks are listed before the probabilistic ones, and within each category
simpler checks are listed before more detailed ones. To minimize the amount of work involved,
the various checks should be implemented in the order given. The initial check in each category
is simplest to implement but will exclude only limit states that are inapplicable by a large margin.
If a limit state cannot be eliminated on the basis of the initial check, subsequent checks may be
used. Checks that are lower on the list require more information to implement but have a
progressively higher chance of eliminating an inapplicable limit state.
This entire screening process is assumed to be carried out by a team of professionals with
expertise in all aspects related to the pipeline and design or assessment decisions being made. As
mentioned earlier, the process should be applied conservatively, eliminating only limit states that
are confidently determined to be inconsequential. If the information required for a given check is
not available or the required judgments (e.g. regarding probability assignments) cannot be made,
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the check can be omitted and the limit state would have to be considered applicable, unless
eliminated by another check.
6.2 Deterministic Screening
The first step in the selection process is to review the list of limit states and, if necessary,
supplement it with additional ones. Once a final list is available, two deterministic methods can
be used to assess each limit state:
1. Subjective assessment. The first level of screening is based on a subjective examination that
relies on previous experience and knowledge. The purpose of this is to identify limit states
that are clearly either applicable or inapplicable. Limit states that are clearly applicable or
clearly inapplicable need not be considered further. The screening process should be
continued only for limit states that are potentially applicable.
Example: Plastic deformations or rupture under internal pressure is clearly applicable to
any high pressure pipeline, whereas buckling due to thaw settlement induced ground
movement is not applicable to a pipeline outside of the geographic regions affected by
permafrost. Gravity loads on an above ground span are potentially applicable depending on
such factors as the span length and pipeline diameter.
2. Worst-case analysis. A worst-case analysis involves estimating the highest credible load
effect and the lowest credible resistance. In this context, the highest credible load and lowest
credible resistance are conservative estimates of the maximum possible load and minimum
possible resistance. For example, the highest credible thermal expansion stresses may be
calculated based on conservative estimates of the lowest possible installation temperature and
the highest possible operating temperature. If a check on these values indicates that the
worst-case resistance exceeds the worst-case load effect, the limit state can be ignored,
otherwise, the probabilistic checks in Section 6.3 can be used.
Example: Consider a 16-in diameter gas pipeline operating at 1000 psi. The pipeline is of
grade X65 material. The product density at the operating pressure is 47 kg/m3. The pipeline
is designated as a Class 1 pipeline, which leads to a design factor of 0.72 according to
ASME B31.8. The required pipe wall thickness based on the design factor is 4.34mm. Since
the industry-accepted minimum wall thickness for a 16-in pipeline, which equals 5.56mm, is
greater than 4.34mm, the pipe wall thickness is selected to be 5.56mm.
The pipeline has a self-supporting crossing over a 10 m wide stream. A limit state
representing yielding of the pipe due to combined gravity, thermal stresses and internal
pressure is considered. A worst-case analysis is carried out to determine whether this limit
state requires further consideration.
The following (worst case) assumptions were made regarding the applied stress:
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Identification of Relevant Limit States
•
The crossing is completely restrained from expanding to allow for thermal expansion
differential. The maximum differential between installation and operation temperatures
was assumed to be 30°C.
•
The crossing was modeled as a simply supported beam spanning the stream. Thus
applied moment at mid-section is equal to wl2/8 where w is the distributed load along the
pipeline length and l is the stream width. The distributed load consists of self-weights of
the pipeline and contents, and snow loads. The self-weight of the coating is ignored.
•
The snow load was calculated by assuming a ground snow load of 4kN/m2 and the
pipeline having a flat surface with the width equal to the pipe diameter.
The maximum longitudinal stress was calculated by combining the axial stress due to
internal pressure (hoop stress multiplied by Poisson’s ratio), bending stress due to gravity
loads and expansion stress due to temperature variations. The maximum longitudinal stress
was 49.2MPa. The effective stress was also calculated by combining the axial and hoop
stresses using σ e = σ a + σ h − σ aσ h obtaining a maximum value of 231MPa.
2
2
The specified minimum yield strength for X65 steel is 448MPa. Conservatively assuming that
the actual minimum (worst case) is 10% below specified, gives a minimum value of 403MPa,
which is larger than the worst case stress of 231MPa by a large margin. This indicates that
this limit state need not be considered further.
6.3 Probabilistic Screening
6.3.1 Introduction
The probabilistic screening methods are based on estimating a lower bound value of the annual
reliability and comparing it to the target annual reliability for the limit state. The limit state can
be eliminated if the reliability lower bound is greater than or equal to the target value or, in
mathematical terms, if
RLB ≥ RT
[6.1]
where R is the annual reliability and subscripts LB and T denote lower bound and target values,
respectively. Given that R is related to the annual probability of failure, pf, by R = 1 – pf, and
using subscripts UB and M to denote upper bound and maximum allowable, respectively,
Equation [6.1] can be re-written as (1 − pfUB ) ≥ (1 − pf M ) , or
pfUB < pf M
[6.2]
which states that the limit state can be eliminated if an upper bound of the annual probability of
failure is less than the maximum allowable annual value for the limit state. It is emphasized that
(1 - pfM) represents the target reliability for the individual limit state being considered. As
discussed in Section 5.1, target reliability levels must be met considering the aggregate for all
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limit states contributing to the category (e.g. 1 – 10-5 per km-year for ULS due to all causes). To
apply Equation [6.2] in such cases, a certain proportion, α, of the target must be “assigned” to the
limit state. If the total target is denoted 1 – pfMT, then pfM in Equation [6.2] can be replaced by
pfMT / α, where the value of α is at the discretion of the user.
The relevant probabilistic checks are dependent on whether the load is the result of a continuous
or discrete loading process (see Section 4.2.1). The checks for these two cases are discussed in
Sections 6.3.2 and 6.3.3.
6.3.2 Continuously Applied Loads
The annual probability of failure, pf, for a continuously applied load can be calculated from
pf = p(r ≤ l )
[6.3]
where r is the minimum resistance and l is the maximum annual load effect (see Section 8.4).
Two upper bounds are presented for limit states that correspond to continuously applied loads.
Check No. 1
The check involves defining or estimating the following quantities:
•
x
an arbitrary value of the load effect;
•
P(l > x)
the probability that the maximum annual load exceeds x; and
•
P(r < x)
the probability that the minimum resistance is less than x.
Basic probabilistic logic can be used to show that an upper bound to the probability of failure can
be calculated from
pfUB1 = p(l > x) + p(r < x)
[6.4a]
The limit state can be disregarded if
pfUB1 < pf M
[6.4b]
Example: Consider a limit state representing rupture of a girth weld due to bending created by
lateral wind load applied to an above ground pipeline. Selecting a wind speed of 80 km / hour (x
= 80), it is assumed that meteorological information shows that the annual probability of the
wind speed exceeding 80 km / hour is no higher than 10-5 per year [i.e. p(l > x) = 10-5]. It is also
estimated that the probability that a typical span cannot resist the load resulting from a wind
speed of 80 km / hour is 10-5 [i.e. p(r < x) = 10-6]. The upper bound of the annual failure
probability for a randomly selected span is given by Equation [6.4a]
pfUB1 = 10-5 + 10-5 = 2 x 10-5
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It is assumed that the maximum permissible annual probability, pfMT, of a ULS (i.e. rupture) at a
girth weld is 10-4 per weld. This means that pfUB1 < pfM and the limit state can be eliminated.
It is noted that the effectiveness of this check is dependent on the selected value of x. Since the
upper bound is the sum of two individual terms representing p(l > x) and p(r < x) , it is always
greater than the larger of these two terms. Selecting a value of x that leads to a high value of
either term could render the check ineffective. In the above wind load example, if a frequently
occurring value of x is selected (say 30 km per hour), p(l > x) will be high, leading to a high
pfUB1 that would not satisfy the condition for eliminating the limit state. This can be avoided by
trying a number of x values and using the lowest upper bound.
Check No. 2
The check involves defining or estimating the following quantities:
•
x1 and x2
two arbitrary values of the load effect with x1 < x2
•
P(l > x1)
the probability that the maximum annual load exceeds x1
•
P(l > x2)
the probability that the maximum annual load exceeds x2
•
P(r < x1) the probability that the minimum resistance is less than x1
•
P(r < x2) the probability that the minimum resistance is less than x2
Basic probabilistic logic can be used to show that an upper bound to the probability of failure can
be calculated from
pfUB 2 = p(r < x1 ) + p(l > x1 ) × p(r < x 2 ) + p(l > x2 )
[6.5a]
The limit state can be disregarded if
pfUB 2 < pf M
[6.5b]
Note that PFUB2 will generally be significantly less than PFUB1.
Example: Consider a limit state representing buckling on the compression side of a pipeline
subject to imposed bending deformations due to thaw settlement. Values of soil displacement, x,
corresponding to various lifetime exceedance probabilities have been determined as shown in
Columns 1 and 2 of Table 6.1. The corresponding estimates of p(r < x) are shown in Columns 3,
where r is the resistance expressed in terms of allowable soil displacement. Values of upper
bound No. 1, PFUB1, were calculated for each value of x and are shown in Column 4 for
comparison.
The value of pfUB2 obtained using x1 = 1.5 and x2 = 3.5 is 10-6 + 10-3x10-2 + 10-5 = 2.1x10-5.
Assuming that the maximum allowable lifetime probability of failure for the individual limit state
(pfM) is 10-4, compressive buckling due to thaw settlement can be eliminated. The example
illustrates that PFUB2 is significantly lower than the smallest value of pfUB1 (6x10-4).
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x (m)
Probability of
Exceedance, p(l>x)
p(r<x)
UB1(x)
1.5
1e-3
1e-6
1e-3
2.5
1e-4
5e-4
6e-4
3.5
1e-5
1e-2
1e-2
Table 6.1 Probability Estimates for Soil Displacement
6.3.3 Discrete Loads
The annual probability of failure, pf, for a discrete load process can be calculated from
pf = p E × p F |E
[6.6]
where pE is the probability of the loading event (e.g. slope failure) and pF|E is the probability of
failure given the event. This equation states that the probability of failure is equal to the
probability of occurrence of the loading event multiplied by the probability of failure if the event
occurs. If the event itself is unlikely to occur, then the limit state can be ignored without
considering the probability of failure given the event. In mathematical terms, since pf, pE and
pF|E are all smaller than one, pE is itself an upper bound for pf, and this provides the basis for the
first check for discrete loads.
Check No. 1
To apply this check, estimate the probability of occurrence of the loading event, pE, and classify
the limit state as inapplicable if
p E < pf M
[6.7]
Example: Consider a limit state representing bending failure due to loads imposed by a
potential flood on a pipeline river crossing. If the probability of occurrence of a flood that raises
the water level to the pipeline location is 10-5 per year and the target reliability for the individual
limit state is 10-4 per year, this limit state can be ignored without considering the probability of
failure if the flood occurs.
Check No. 2
In this check the probability of failure is calculated from Equation 6.6, in which the probability
of failure given the loading event, pF|E, is calculated from the upper bound calculation in check
No.1 in Section 6.3.2 (Equation 6.4a).
Example: In the above river-crossing example, assume that the annual probability of a flood
reaching the pipe location (pE) is 10-3 (instead of 10-5). An estimate of the probability of pipe
failure given that flood level (pF|E) is 10-2. The upper bound of the annual failure probability can
then be calculated from Equation 2.6 as
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pfUB2 = 10-3 x 10-2 = 10-5
Since this upper bound is lower than the maximum allowable failure probability of 10-4 per year,
the limit state can be eliminated.
Check No. 3
In this check the probability of failure is calculated from Equation 6.6, in which the probability
of failure given the loading event (PF|E) is calculated from the upper bound given in check No.2
in Section 6.3.2 (Equation 6.5a).
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7. DEVELOPING A LIMIT STATE FUNCTION
7.1 Introduction
Limit state functions are readily available for many of the key limit states for natural gas
pipelines. Appendix A includes a set of limit state functions for yielding, bursting, corrosion and
equipment impact. These limit state functions are based on state-of-the-art structural behaviour
models and can be used directly in RBDA. This chapter describes the method used to develop a
limit state function, which is included to provide an understanding of how the limit state
functions in Appendix A have been constructed, how they can be modified, and how new limit
state functions can be developed for conditions that are not included in the appendix.
7.2 Generalized Definition of a Limit State Function
In Section 3.5.2, a limit state function, g(x), was defined as a mathematical function of a set of
basic random variables, x = x1, x2, …., xn,. The limit sate function is defined such that g(x) ≤ 0 if
failure occurs and g(x) > 0 if failure does not occur. For pipelines, the basic random variables (x)
include loads, pipe geometry, pipe mechanical properties and defect properties.
A limit state function, however, is a general concept that can be used to separate any two distinct
states of structural behaviour. For example, a limit state may be defined as the formation of a
through-wall crack from a surface crack. Considering that a surface crack has no adverse
consequences in itself, the corresponding limit state surface separates safe performance from
failure by leak. Once the through-wall crack has formed, another limit state function can be
defined to determine whether rupture will occur due to fracture initiation from the crack. The
limit state surface in this case defines the boundary between two failure modes (leak or rupture)
rather than between safety and failure. For this case, the convention used in this document is to
formulate the limit state function such that g(x) ≤ 0 for the subsequent “less safe” state
(i.e. fracture initiation) and g(x) > 0 for the initial “safer” state (i.e. no initiation).
Given the above, a limit state function, g(x), is defined as a function of a set of basic random
variables x, which separates two distinct states of structural behaviour. The function will be
defined such that g(x) ≤ 0 for the state with more serious consequences and g(x) > 0 for the state
with less serious consequences.
The main steps involved in developing a limit state function are shown in Figure 7.1. They are:
1. Define the limiting condition. The limiting condition defines the boundary between the
structural responses separated by the limit state surface. It may be defined in terms of such
parameters as stresses, strains, deformations, defect sizes or pipe geometry.
2. Model the limiting condition in terms of the basic parameters. A model must be developed
to express the failure condition as a function of the basic uncertain parameters including
material properties, geometry and loading conditions. The model used may be analytical,
empirical or numerical.
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3. Characterize model error. Every deterministic model involves idealizations and limitations
that lead to some error in the calculated quantities. Model error can be included by adding
appropriate model error factors that are quantified by comparing model results to
experimental data. Since model error dominates other sources of uncertainty in some cases,
it is essential to quantify and include it in any reliability analysis.
These steps are discussed in more detail and illustrated by examples in Sections 7.3 through 7.5.
7.3 Overview of Development Procedure
Define limiting
condition
Develop limit
state model
Characterize
model error
Figure 7.1 Procedure for Developing a Limit State Function
7.4 Defining the Limiting Condition
The limiting condition can be defined in terms of any parameter or set of parameters that are
required to satisfy specific criteria of strength or serviceability. In general terms, this can be
expressed as the demand reaching the capacity. For structural design, this will often correspond
to a load effect, l, reaching the corresponding resistance, r, i.e. r = l or
g = r −l = 0
[7.1]
The value of g in Equation [7.1] will be less than 0 if the load exceeds the resistance (failure) and
greater than 0 if the resistance exceeds the load (safe). The parameters forming the basis for
these criteria (i.e. the definition of r and l) may include stresses, strains, deformations or
geometric properties. Some examples are given in the following.
•
Force-based criteria. For equipment impact the limiting condition may be defined as the
impact load, applied by the excavation equipment, exceeding the load required to create a
critical gouged dent in the pipe wall.
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•
Stress-based criteria. In the case of a defect failure under internal pressure the limiting
condition may be defined as the applied hoop stress reaching the hoop stress required to fail
the defect.
•
Strain-based criteria. In the case of local buckling due to ground movement the limiting
condition can be expressed as the axial compressive strain reaching a value that leads to a
significant buckle.
•
Deformation-based criteria. A deformation-based criterion is relevant for mill pressure
testing for example, where the limiting condition could be defined as the total plastic hoop
deformation of the pipe reaching the upper bound of diameter tolerance. A similar criterion
may also be applicable to the field hydrostatic test, where a limiting condition may be defined
as the plastic hoop deformation reaching a value that results in coating damage.
•
Geometry-based criteria. A pinhole corrosion leak can be represented by a criterion limiting
the maximum corrosion depth to the pipe wall thickness.
•
Defect size-based criteria. For fracture initiation, the limiting condition may be expressed by
limiting the crack size (depth and length combination) to the critical size for initiation of a
part through wall flaw.
Although this list includes most of the criteria commonly used for onshore pipelines, it is not
intended to be comprehensive. It is also noted that the parameter used as a basis for the criterion
associated with a given limit state is not necessarily unique. For example, an equivalent limit
state can be defined for defect failure based on internal pressure instead of hoop stress.
7.5 Developing the Limit State Model
7.5.1 Introduction
The purpose of this step is to express the limit state condition as a function of a number of basic
parameters for which statistical data can be obtained and appropriate probability distributions (or
other probabilistic models) defined. The limit state function is typically developed by expressing
the load, l, and resistance, r, in Equation [7.1] in terms of the appropriate influencing parameters
representing material properties, geometry, defect characteristics and loading conditions.
In principle, any analytical, empirical or numerical model may be used in developing a limit state
function, however, there are practical limitations related to subsequent use of the model in
probabilistic calculations. Although these limitations depend on the method used for
probabilistic calculations (see Section 9.2.2.2), there are some generally desirable attributes that
apply for all methods. These include:
•
Simplicity. The function should be easy to program and link to a probabilistic calculation
algorithm or software program. Functions based on complex algorithms involving numerical
(finite element) analyses are generally less desirable.
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•
Efficiency. Probabilistic calculations involve repetitive use of the limit state function.
Depending on the problem and method used, hundreds, thousands and possibly millions of
calls to the limit state function may be required. To ensure efficiency of the probability
calculations, the function should be computationally efficient.
The following sub-sections illustrate the process of developing a limit state function by simple
examples and provide suggestions for the sources of information required to develop limit state
functions for other limit states.
7.5.2 Example 1 – Using a Simple Analytical Model
This example illustrates a case for which a simple analytical model that meets the criteria of
simplicity and efficiency is available. The specific example considered is a limit state
representing pipe puncture due to impact by an excavator tooth (Figure 7.2). The steps involved
in developing a limit state function are described in the following.
Figure 7.2 Illustration of an Excavator Impacting a Pipeline
1. Basic limit state function. As mentioned earlier, a limit state function can be expressed as
g = r −l
[7.2]
where r is the puncture resistance of the pipeline and l is the impact force.
2. Resistance model. The resistance, r, can be calculated from a semi-empirical model based on
punching shear and membrane resistance of the pipe wall (Driver and Zimmerman 1998
based on Corbin and Vogt 1997):
r = (1.17 − 0.0029 ⋅
D
) ⋅ (lt + wt ) ⋅ t ⋅ σ u
t
[7.3]
where D is the pipe diameter, lt and wt are the excavation tooth length and width, t is the wall
thickness and σu is the tensile strength of the pipe material. The coefficients 1.17 and 0.0029
determine the relative importance of punching shear versus membrane resistance. These
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coefficients were evaluated empirically using data from tests conducted by EPRG and
Battelle. The model results are shown versus the experimental data in Figure 7.3.
600
Experimental Puncture Resistance (kN)
500
400
300
200
100
0
0
100
200
300
400
Predicted Puncture Resistance (kN)
500
600
Figure 7.3 Puncture Model Results Versus Test Data
3. Load model. The impact load, l, is represented by the maximum quasi-static force applied by
the excavator to the pipeline. This force is expressed as a function of excavator weight using
l = 16.5 we 0.6919
[7.4]
where we is the excavator weight. This model is based on regression analysis of various
excavator manufacturer specifications regarding the maximum quasi-static force that can be
applied by an excavator of given weight (see Figure 7.4).
4. Final limit state function. The limit state function can be obtained by substituting r and l
from Equations [7.3] and [7.4] into Equation [7.2]. This gives
g = (1.17 − 0.0029 ⋅
D
0.6919
) ⋅ (lt + wt ) ⋅ t ⋅ σ u − 16.5 we
t
[7.5]
The basic variables used in this limit state function include pipe geometry (diameter and wall
thickness), material properties (ultimate strength) and excavator characteristics (weight, tooth
length and tooth width). It is noted that, to simplify the example, model error was not considered
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in developing this function. In most cases, model error is a key contributor to the uncertainty
that should quantified and included in the analysis. This topic is discussed in detail in
Section 7.6.
450
400
Digging Force (kN)
350
300
250
200
150
100
50
0
0
20
40
60
80
100
Excavator Mass (tonnes)
Figure 7.4 Excavator Mass Versus Digging Force
7.5.3 Example 2 – Using a Numerical Finite Element Model
In some cases, simplified models such as the one described in Section 7.5.2 do not exist, and the
structural response must be determined using more sophisticated approaches. An example of this
is the limit state corresponding to bending deformations due to frost heave of Northern pipelines
(see Figure 7.5). In this case, the pipe deformations are dependent on soil stiffness and soil
deformation, both of which are variable with time and dependent on the pipe properties and
operating temperature. The most realistic approach to model this problem is a detailed finite
element analysis, which would be impractical to incorporate in a reliability calculation.
The recommended approach to developing a limit state function in this case is to use the finite
element model to produce a reasonable number of data points representing the relevant range of
input parameters. Regression analysis can then be used to develop a simple model of the data,
which can be utilized in the reliability analysis. Such a simple model is referred to as a
“response surface”. In most cases, the response surface is likely to be a non-linear model of
multiple parameters.
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Uplift Resistance
Pipeline
Transition
Frost Stable Zone
Frost Susceptible Zone
Figure 7.5 Illustration of Frost Heave Loading Scenario
The steps involved in applying this approach for frost heave are described in the following:
1. Basic limit state function. Two limiting conditions are applicable in this case: one based on
compressive strain and the other on tensile strain in the pipe section. The limit state function
can be expressed as
g t = rt − lt
[7.6a]
g c = rc − l c
[7.6b]
where r is the critical strain (capacity or resistance) of the pipeline and l is the applied strain
(demand). The subscripts t and c denote tension and compression.
2. Resistance model. The critical tensile strain is typically defined as a fixed value, which is
dependent on the quality of girth welds. The critical compressive strain is a function of a
number of parameters including wall thickness, diameter, hoop stress and post-yield stiffness.
Deriving critical compressive strains is quite complex, requiring experimental data, finite
element modeling and application of the regression analysis approach discussed here (see 3
below). A model for compressive strain was developed by Zimmerman et al. (1995) and is
used here. Based on the above, the tensile and compressive strain limits are defined as
follows:
rt = ε tc
[7.7a]
  t  2 120 − D / t 2

rc = 8.5  +
σ h + 0.0021 (1.53 − 0.018 n)
5000
  D 

[7.7b]
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where εtc is the critical tensile strain, t is the wall thickness, D is the diameter, σh is the hoop
stress and n is the Ramberg-Osgood exponent (characterizes the post-yield behaviour of the
stress-strain relationship).
3. Load model. The load model for predicting pipeline strains due to frost heave can be
developed by de-coupling the geothermal aspects of the problem from the
structural/geotechnical aspects. The steps involved in developing this model are as follows
(Chen 1994):
•
Define soil heave versus time relationships at different contact pressures for various
combinations of pipeline diameter and soil type, based on the operating temperature of
the pipeline. These relationships take into account the geothermal aspects of the problem
including the development of the frost bulb and its dependence on contact pressure.
•
Develop a soil spring finite element model of the pipe and surrounding soil. The input
soil displacement is defined using the heave-time relationships defined in the previous
bullet. This model can be used to predict the applied tensile and compressive strains as a
function of time. If a large deformation model based on moment-curvature is used,
curvature and axial strains can be calculated.
•
Using the above model, calculate curvature and axial strain corresponding to the
maximum tensile and compressive strains for all relevant combinations of diameter, wall
thickness and soil type (as reflected by the heave relationship and soil response
relationships). Use the resulting data to develop a multi-parameter regression model that
correlates applied curvature and/or axial strain with diameter, D, wall thickness, t, and
free heave, h. For instance, the model for applied curvature, φ, has the form φ = qts Du hv,
where q, s, u and v are the regression coefficients. The values calculated from this
relationship are plotted against the finite element results in Figure 7.6. The scatter in the
figure is an indication of the accuracy lost by using the regression model instead of the
finite element results. Further, the model has a conservative bias at high strain values.
Although these aspects are important in realistic applications, they are not considered
here for simplicity. Based on this model, and assuming that the axial strain (which can be
modeled using a similar regression analysis) is εa, the tensile and compressive strains can
be written as
lt = ε a + q t s D u h v D / 2
[7.8a]
lc = ε a − q t s D u h v D / 2
[7.8b]
4. Final limit state function. The limit state function can be obtained by substituting r and l
from Equations [7.7] and [7.8] into Equation [7.6]. This gives
(
g t = ε tc − ε a + q t s D u h v D / 2
)
[7.9a]
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  t  2 120 − D / t 2

g c = 8.5  +
σ h + 0.0021 (1.53 − 0.018n ) − (ε a − q t s D u h v D / 2)
5000
  D 

[7.9b]
The basic variables used in the loading side of this limit state function include only pipe
diameter, wall thickness, free heave and axial strain. This implies that all other relevant
parameters, including pressure, temperature and material properties, have been fixed and that the
model is only applicable for these fixed values. The list of variable parameters can be expanded,
but this requires more finite element runs and tends to increase the scatter associated with the
regression model. This demonstrates that a trade-off must be made between the accuracy and
generality of models developed using this approach. It is noted that model error was not
considered in developing this function. Analyzing and including model error is discussed in
Section 7.6.
Curvature from regression (1/mm)
3.50E-04
3.00E-04
2.50E-04
2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00
0.00E+00
5.00E-05
1.00E-04
1.50E-04
2.00E-04
2.50E-04
Curvature from finite element analysis (1/mm)
Figure 7.6 Applied Curvature from Finite Element Versus Regression Model
7.5.4 Sources of Relevant Information
As demonstrated in Sections 7.5.2 and 7.5.3, the main requirement for developing a limit state
function is to find appropriate deterministic models to express the load effect and resistance in
terms of material properties, pipe geometry and loading conditions. Many of the required models
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have been developed in connection with current deterministic design and integrity assessment
requirements. Some of the major sources of such models are as follows:
1. The Pipeline Defect Assessment Manual (APA 2002). This manual was developed by
Andrew Palmer and Associates (APA) under a joint industry project. The manual specifies
pipe resistance models for defect free pipe and pipe with defects under various loading
conditions. For defect free pipe, burst under internal pressures as well as response to axial
load, bending load, thermal load and combined loads are included. Models are also given for
pipes with corrosion defects, dents, gouges, and combined dent-gouges. Future work for the
manual will add models for girth weld defects, seam weld defects, environmental cracking,
leak and rupture effects, and fracture propagation.
2. NEN 3650 – Requirements for Steel Pipeline Transport Systems (NEN 1992). This Dutch
code of practice provides general requirements for the design of oil and gas transmission
systems. NEN is a limit state design code with a strong structural emphasis for both onshore
and offshore pipelines. This code provides pipe resistance models for internal pressure (both
hydrostatic test and operation), local buckling under gravity loads on above-ground
supported spans, restrained thermal expansion, and loss of soil support.
3. DNV Submarine Pipeline Systems 2000 OS-F101 (DNV 2000). This code deals mostly with
structural aspects such as external pressure, internal pressure, bending, thermal stresses, axial
loads and combined loading. The code is specifically designed for offshore pipelines but has
many relevant models for onshore lines. The principles presented in DNV OS-F101 were
used for the development of the APA Pipeline Defect Assessment Manual (see 1 above).
4. Guidelines for Seismic Design of Oil and Gas Pipelines (ASCE 1984). This reference,
prepared by the ASCE committee on Gas and Liquid Fuel Lifelines, details analytical
methods for the determination of loads due to ground movement, surface faulting, soil
liquefaction and loss of soil support. Analytical methodologies are presented for the
determination of a maximum axial stress or a maximum change in length that the pipeline
can withstand due to the various loading conditions.
5. Plastic Design of Buried Steel Pipelines in Settlement Areas (Gresnigt 1986). This reference
was developed in the Netherlands jointly by Delft University of Technology and TNOInstitute for Building Materials and Structures. Its emphasis is on buckling behaviour of
pipelines experiencing settlement. It addresses the effect of loads such as earth pressure,
axial force, shear force, and torsional moment in conjunction with internal pressure.
6. Other documents that are not specific to pipelines, but contain relevant information are API
Recommended Practice 579 for Fitness For Service (API 2000) and British Standard BS
7910 for assessing the acceptability of flaws in structures (BSI 1999). The API document
addresses a variety of conditions including brittle fracture, general and local metal loss,
pitting corrosion, blisters, lamination, and crack-like flaws. It also addresses such issues as
misalignment, creep and fire damage. BS 7910 covers fracture, fatigue, and plastic collapse.
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7.6 Model Uncertainty
7.6.1 Introduction
A limit state function is essentially a combination of physical models that calculate a load effect
and corresponding resistance. As discussed in Section 3.3, there is some uncertainty associated
with the results of such models because of the assumptions and idealizations made. For example,
some simplified models for the burst pressure of a corrosion defect idealize the typically irregular
shape of the defect cross section as a rectangle or parabola.
Model error may have a systematic component (model bias) and a random component (model
scatter). Model bias represents an error component with a fixed value, which causes the model
results to be different, on average, from the actual values. Model scatter represents an error
component that changes randomly from one application of the model to another, causing the
results to fluctuate randomly around the average prediction. Model error may be characterized
by a number of random model error factors. The mean value of a model error factor represents
model bias, whereas the standard deviation represents model scatter.
7.6.2 Characterizing Model Error
7.6.2.1 General
Characterization of model error requires comparison between model results and actual data
obtained from experimental measurements. For example, the error associated with a model to
estimate the remaining pressure resistance of pipe with a corrosion feature can be defined using
burst test results for pipe sections with actual or simulated corrosion features. To compare this
data to the values calculated from the model, the model input parameters (yield strength, wall
thickness, diameter, defect depth and defect length in this case) must be known for each test
specimen.
It is recognized the measured values of model results (e.g. burst pressure) and model input
parameters (e.g. yield strength and defect sizes) are subject to measurement errors. Due to this,
the observed discrepancy between measured and calculated values includes the combined effects
of model error and measurement error. It is possible, under some simplifying assumptions, to
develop approaches for separating measurement errors from model errors (Ellingwood et al.
1980); however, application of these approaches is contingent upon quantifying measurement
errors. Since the information required to do so is not available, measurement errors are typically
not considered – the total discrepancy between experimental and theoretical results is typically
attributed to model error. This approach is conservative because it exaggerates the value of
model error. Since measurement errors are likely to be small in comparison to model error in
most cases, the degree of conservatism introduced by this approach is not likely to be excessive.
The appropriate format for characterizing model error depends on the relationship between the
error magnitude and the quantity being estimated by the model. Although any relationship is
possible, some special assumptions are typically used to simplify the analysis. The two most
common assumptions are those of proportional error and independent error. These two cases
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are discussed in Sections 7.6.2.2 and 7.6.2.3. A more general model that considers a
combination of proportional and independent errors is discussed in Appendix C.
7.6.2.2 Proportional Error
The simplest and most common approach to characterize model error is to define a factor that
equals the ratio between the actual value and model estimate of the quantity being estimated.
This implies the following relationship
y a / y m = e1
[7.10]
where e1 is a random variable representing model error factor, ya is the actual value (i.e. the test
result) and ym is the value calculated from the model. The parameter y is used generically to
represent a load effect, l, or a resistance parameter, r, that is used in a given limit state function.
Equation [7.10] implies that the actual value, ya, can be estimated from the model outcome, ym,
using
y a = e1 y m
[7.11]
Equation [7.11] can be re-written as y a − y m = (e1 − 1) y m , which implies that the difference
between the actual value and model result is proportional to the model result. In this model, e1
will be referred to as a proportional model error factor. The model is illustrated in Figure 7.7a,
in which the width of the error band around the best-fit regression line for ya versus ym increases
in proportion to ym. A perfect model is plotted in Figure 7.7 for comparison. The perfect model
goes through the origin, has a slope of 1.0 and no associated scatter.
Figure 7.7b shows a plot of ya/ym versus ym. Since e1 is independent of ym, ya/ym is also
independent of ym. Therefore, a regression line of ya/ym versus ym is horizontal and the error band
has a constant average width. The mean value of e1 (denoted µe1) represents model bias and the
standard deviation of e1 (denoted σe1) represents model scatter. Given a set of data points
representing yai and ymi, i = 1,…,n, the values of µe1 and σe2 can be estimated by the mean and
standard deviation of yai/ymi.
7.6.2.3 Independent Error
Although, a proportional model error is representative in some cases, other relationships between
ya and ym may be applicable in other cases. This is demonstrated in Figure 7.8a, which shows a
model error band with a fixed width. A perfect model is plotted in Figure 7.8 for comparison. As
mentioned earlier, the perfect model goes through the origin, has a slope of 1.0 and no scatter.
In this case the model error can be represented as the difference between the actual value and
model estimate, with the latter multiplied by the slope of the regression line in Figure 7.8a
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Regression Line
Error Band
Perfect Model
ya/ym
ya
σe1 (scatter)
1.0
µe1 (bias)
ym
ym
(a)
(b)
Figure 7.7 Illustration of Proportional Model Error
y a − e1 y m = e2
[7.12]
where e1 is a deterministic constant representing the slope of the regression line and e2 is a
random variable. The actual value can be estimated from the model outcome using this equation:
y a = e1 y m + e2
[7.13]
Equation [7.12] implies that the difference between ya and e1 ym is independent of the model
result. In this model, e2 is referred to as an independent model error factor.
Figure 7.8b shows a plot of (ya – e1 ym ) versus ym. In this case (ya - e1 ym) is independent of ym,
and therefore a regression line of (ya - e1 ym) versus ym is horizontal and the error band has a
constant width. The mean value of e2 (denoted µe2) represents model bias and the standard
deviation of e2 (denoted σe2) represents model scatter. Similar to the proportional error case, the
values of µe2 and σe2 can be estimated by the mean and standard deviation of (yai - e1 ymi), where
yai and ymi, i = 1,…,n represent a set of corresponding actual values and calculated model results.
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Regression Line
Error Band
Perfect Model
ya
ya - e1 ym
σe2 (scatter)
0.0
Slope = e1
µe2 (bias)
ym
ym
(b)
(a)
Figure 7.8 Illustration of Independent Model Error
7.6.2.4 Model Selection
The first step in selecting the most representative model for a given data set is visual inspection
on plots similar to Figures 7.7 and 7.8. Definition of the axes in such plots may have a
significant impact on the data scatter and appropriate model error format. For example,
corrosion defect resistance data may be plotted as the actual versus calculated burst pressure or
the actual versus calculated reduction in burst pressure from the burst pressure of perfect pipe.
The data may also be normalized by dividing the burst pressure by the yield strength. Such
normalization may reduce the overall scatter by eliminating the effect of variations in parameters
that have a neutral impact on the model result. If the model error analysis is performed on a
transformation of the main parameter (burst pressure in the corrosion example), the
transformation must be later reversed to retrieve the value of the required parameter.
The selection of a proportional, independent or combined model error format can in many cases
be made by visual inspection of the data. If the scatter appears similar to Figure 7.7, then
proportional error dominates. If, on the other hand, it is similar to Figure 7.8, independent error
will be dominant. It is also possible to use the combined model error format descried in
Appendix C to test a given data set to evaluate the significance of each of the proportional and
independent error components and determine whether one of them could be eliminated.
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7.6.3 Example
This section illustrates the steps involved in selecting a model error format and defining the
distributions of the corresponding model error parameters. The example used as a basis for the
discussion is a model that estimates the burst pressure at a corrosion defect. This model is as
follows:
Pm = 2.3
t σy
D
(
1− d / t
)
1− d / mt
m = 1 + 0.6275
m = 0.032
l2
l4
− 0.003375
dt
d 2t 2
l2
+ 3.3
dt
[7.14a]
for
l2
≤ 50
dt
[7.14b]
for
l2
> 50
dt
[7.14c]
where,
= burst pressure calculated using model
Pm
t
= wall thickness
σy
= yield strength
D
= pipe diameter
d
= average defect depth
m
= Folias factor
l
= axial length of the corrosion feature.
The model error is evaluated using a set of test data representing the measured burst pressure of
pipe sections with corrosion defects (Kiefner and Vieth 1989). It is recognized that other data
are available that could be included in Table, 7.1, however, the current data set is considered
sufficient for the illustrative nature of this example. Table 7.1 gives the parameters required to
calculate the burst pressure from Equations [7.14] for each test specimen (Columns 1 to 5), the
burst pressure calculated from the model, Pm, (Column 6) and the actual burst pressure, Pa,
(Column 7). It important to recognize that the model error analysis should be based on the actual
values rather than nominal values of the parameters used in the calculation. For example, the
yield strength data in Table 6.1 is obtained from coupon tests of the pipe samples used and not
based on SMYS. The importance of this is that using a nominal value introduces an additional
error in the model calculation and results in an unrealistic characterization of model error.
The steps involved in quantifying model error are as follows:
1. Plot the data and linear regression line using various possible parameter transformations
(Figure 7.9), and select a parameter definition for the model error analysis. Visual inspection
of Figure 7.9 gives no clear reason to select one format over another. The format in
Figure 7.9c was selected because the model is intended to characterize the reduction in burst
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pressure due to the corrosion defect and because this format gives a dimensionless
characterization of that reduction. This means that the parameter used in the analysis is
defined as
y = ( P0 − P) / σ y
[7.15a]
P0 = 2.3 t σ y / D
[7.15b]
where P0 is the burst pressure of the perfect pipe assuming a flow stress of 1.15σy, P is the
burst pressure of the corroded pipe, t is the wall thickness, σy is the yield strength and D is
the pipe diameter. The measured and actual values of y (ym and ya) are given in Columns 8
and 9 in Table 7.1.
1
2
3
4
5
6
7
8
9
10
D
t
σy
d
l
Pm
Pa
ym
ya
e i2
0.0188
0.0176
0.0185
0.0067
0.0115
0.0144
0.0091
0.0073
0.0061
0.0050
0.0052
0.0042
0.0035
0.0051
0.0040
0.0047
0.0040
0.0054
0.0018
0.0011
0.0040
0.0026
0.0019
0.0030
0.0020
0.0028
0.0010
0.0008
0.0032
0.0109
0.0082
0.0168
0.0198
0.0162
0.0071
0.0111
0.0143
-0.0003
0.0077
-0.0014
-0.0014
-0.0018
0.0028
0.0007
0.0000
-0.0020
-0.0019
0.0009
0.0019
-0.0006
0.0008
-0.0006
0.0006
-0.0013
0.0026
-0.0018
0.0019
-0.0013
-0.0016
-0.0018
0.0114
0.0092
-0.004877
-0.000533
-0.005109
-0.000667
-0.002075
-0.002333
-0.010842
-0.000799
-0.008413
-0.007146
-0.007804
-0.002032
-0.003378
-0.005896
-0.006619
-0.007284
-0.003682
-0.004305
-0.002658
-0.000557
-0.005214
-0.002369
-0.003495
-0.000842
-0.00407
-0.001239
-0.002504
-0.002473
-0.005579
-0.001187
-0.000296
(in)
(in)
(psi)
(in)
(in)
(psi)
(psi)
24
24
20
22
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
36
0.375
0.375
0.260
0.198
0.365
0.375
0.376
0.381
0.375
0.376
0.372
0.375
0.370
0.377
0.373
0.378
0.378
0.379
0.370
0.370
0.382
0.375
0.375
0.375
0.375
0.375
0.375
0.375
0.375
0.298
0.330
42000
48800
61000
60967
58600
68770
52000
52000
60264
59030
60750
58800
58700
61794
60173
62082
61221
65419
58700
58700
63542
62000
61300
63800
62000
66200
66500
70600
64400
71000
65000
0.210
0.230
0.174
0.089
0.178
0.208
0.156
0.129
0.109
0.080
0.076
0.127
0.093
0.092
0.083
0.070
0.088
0.094
0.069
0.097
0.065
0.142
0.059
0.086
0.071
0.105
0.104
0.114
0.076
0.146
0.155
33.50
9.00
16.00
6.00
16.00
27.00
12.00
12.00
12.00
20.00
36.00
4.75
2.50
12.00
9.00
33.00
8.00
14.00
4.25
2.25
20.00
2.75
5.50
5.50
4.50
4.00
2.00
1.60
7.50
63.00
16.00
721
894
697
851
969
984
1024
1137
1365
1408
1418
1444
1457
1473
1479
1509
1532
1548
1560
1598
1609
1623
1644
1645
1661
1721
1844
1974
1643
850
838
804
788
835
828
987
992
1515
1120
1815
1785
1844
1525
1623
1789
1840
1916
1720
1775
1700
1620
1902
1745
1840
1670
1895
1775
2000
2140
1970
815
775
Table 7.1 Actual and Calculated Burst Pressure for Corroded Pipe Specimens
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2500
1200
1000
2000
800
1500
Pa
Po-Pa
600
1000
400
200
500
0
-200
0
0
500
1000
1500
2000
0
2500
200
400
600
Pm
P 0 -P m
a)
b)
800
1000
1200
0.020
(Po-Pm) / σ y
0.015
0.010
0.005
0.000
-0.005
0
0.005
0.01
0.015
0.02
(P o-P m ) / σ y
c)
Figure 7.9 Actual Burst Pressure Versus Model Results
2. Find the best model error format by plotting the chosen parameter using various formats.
Figure 7.10 shows the data in Figure 7.9c plotted in proportional and independent formats
analogous to Figures 7.7b and 7.8b. In Figure 7.10a the best-fit line has a significant upward
trend and the error band has a significant narrowing trend. Figure 7.10b, on the other hand,
has a horizontal best-fit line and a constant error band. This indicates that the independent
model error format is representative and Equation [7.13] can be used. Based on the slope of
the best linear fit to the data in Figure 7.9c, e1 is 1.154. The model error format is given by
y a = 1.154 y m + e2
[7.16]
3. Analyze the data to produce a set of values corresponding to the model error factor e2. In this
case, data can be produced using e2i = yai – 1.154 ymi. This gives the data in Column 10 of
Table 7.1. The probability distribution of e2 can be derived from this data using the methods
described in Section 7.3. The data shows that the mean (bias) and standard deviation
(scatter) of e2 are -0.00375 and 0.00275. The model is plotted in Figure 7.11 along with an
error band that has a width of one standard deviation on either side.
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4. Reverse the parameter transformation to obtain the final model. By using [7.15] for ya and ym
in [7.16] and re-arranging, the final model is obtained as
Pa = 1.154 Pm − 0.15 Po + e2 σ y
[7.17]
2.0
y a /y m
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
0.000
0.005
0.010
0.015
0.020
0.015
0.02
ym
a) Proportional
0
y a- e 1 y m
-0.002
-0.004
-0.006
-0.008
-0.01
-0.012
0
0.005
0.01
ym
b) Independent
Figure 7.10 Proportional and Independent Error Plots for the Corrosion Data in Figure 7.9c
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0.025
(P 0 -Pa )/ σ y
0.020
0.015
0.010
0.005
0.000
-0.005
-0.010
0
0.005
0.01
0.015
0.02
(P 0 -P m )/ σ y
Figure 7.11 Plot of Final Model with an Error Band of One Standard Deviation on Each Side
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8. PROBABILISTIC CHARACTERIZATION OF INPUT VARIABLES
8.1 Introduction
This section describes methods of assigning probabilistic models to the uncertain (random)
variables and events used in calculating reliability. The possible sources of uncertainty for these
variables were described in detail in Section 3.3. They include random variations, measurement
uncertainty, model uncertainty and statistical uncertainty. To understand and apply the methods
described in this section, the reader must be familiar with the basics of probability theory
including the mathematical definition of probability, the basic rules of probabilistic algebra,
random events, random variables and probability distributions. These concepts are explained in
many standard probability texts and are summarized for convenience in Appendix D.
Throughout this document, probability is interpreted as a degree of belief that a certain outcome
will occur. This is called the subjective interpretation because it implies that probability is
assigned by a certain person and that it is a function of that person’s knowledge and judgment
about the processes or phenomena involved. There is a competing interpretation of probability,
in which it is viewed as a relative frequency of occurrence of a certain outcome in an infinite
number of trials. The frequency definition makes the assignment of probability conditional on
having a large (infinite) amount of data, thus excluding application of probability to problems for
which data are limited or unavailable. This philosophy is likely the reason that shortage of data
is commonly cited as a reason for avoiding the use of probabilistic methods in engineering.
The subjective interpretation allows the engineer to assign a probability (or probability
distribution) based on available evidence (including data, experience and simple logic) and to
change the probability assignment when new information is obtained. Recognizing limitations
on the quality and quantity of available information cannot be compensated for by sophisticated
analyses, data-related uncertainties can and should be incorporated into the probability
assignment and propagated through the analysis. This results in final solutions that incorporate
an appropriate amount of additional conservatism to compensate for data limitations. The
underlying rationale is that making design and operational decisions with imperfect information
is a normal part of engineering. The RBDA approach is viewed as a tool to facilitate the
decision-making process by providing a rational basis to quantify uncertainty and reflect it in the
decisions made.
The discussion in this section is divided into three main parts. Section 8.2 describes probabilistic
models for randomly occurring events such as equipment impact on pipelines. Section 8.3
provides an overview of distribution fitting methods for time-independent random variables such
as pipe diameter and yield strength. Probability models for time-dependent random variables
such as internal pressure and environmental loads are discussed in Section 8.4. Section 8.5
describes approaches that can be used to deal with statistical uncertainties associated with small
data sets.
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8.2 Frequency of Random Events
8.2.1 Introduction
A random event is one for which individual occurrences are independent in time. The main
condition defining independence is that the time to occurrence of the next event is independent of
the time since occurrence of the last event. For onshore pipeline reliability, random event
processes may be used to model:
•
•
•
•
•
•
•
Failure incidents
Equipment impact events
Earthquakes
Initiation of new corrosion defects
Slope failures
Overpressure due to operator’s error
Severe storms
8.2.2 The Poisson Process
The Poisson process models events that occur randomly in time. It is fully defined by the rate of
occurrence of the event per unit time, λ. The Poisson process implies the following relationships
(Benjamin and Cornell 1970):
•
The average time between events µT is given by:
µT = 1/ λ
•
The probability that the next event will occur before time, t, has elapsed is given by the
exponential distribution:
F (t ) = 1 − exp(−λt )
•
[8.1]
[8.2]
The probability that n events will occur during a time period, t, is given by the Poisson
distribution:
(λt ) n exp(−λt )
P ( n) =
n!
[8.3]
8.2.3 Estimation of the Rate of Occurrence
The rate of occurrence is estimated as the number of events observed (no) divided by the time
interval of observation (to). This can be written as:
λ = no / t o
[8.4]
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8.2.4 Example
A pipeline company has recorded eight incidents of excavation equipment impacts with a certain
pipeline during a period of four years. Assuming that equipment impact events are independent
and therefore represented by a Poisson process, the following quantities can be calculated:
•
The annual rate of impact is two per year. This is calculated from Equation [8.4] with no = 8
impacts and to = 4 years.
•
Assuming the same level of construction activity, the probability of one or more impacts
occurring some time within the next year can be calculated from Equation [8.2] with λ = 2
impacts per year and t = 1 year, giving a probability of 0.865.
•
The probability that exactly five impacts will occur next year can be calculated from
Equation [8.3] with λ = 2 impacts per year, t = 1 year, and n = 5 impacts. This gives a
probability of 0.036.
8.3 Probability Distributions of Time-independent Variables
8.3.1 Introduction
A time-independent random variable has a fixed value that cannot be determined with certainty
due to measurement limitations. A good example of this is the group of parameters representing
the mechanical properties of pipe steel including yield strength and fracture toughness. Due to
uncontrollable variability in the manufacturing process, these parameters vary from location to
location, even within the same pipe joint. Since destructive testing is normally required to
determine mechanical properties, it is not possible to know the exact values of these parameters
at any specific pipe location. Because of this, mechanical properties must be treated as uncertain
for the purpose of analyses relating to a specific pipe location (such as fitness for service
assessment of a given defect).
Time-independent random variables of relevance to reliability calculations for onshore pipelines
can be grouped into five categories (see Appendix B for a summary of published data and
distributions for these parameters):
1. Loads. Time-independent loads include pipeline weight, weight of permanent equipment and
overburden. The analysis described here is also required (although not sufficient) for timedependent loads such as internal pressure, equipment impact and environmental loads. As
discussed in Section 8.4, time-dependent loads are characterized by a frequency of
occurrence and a severity when they occur. The analysis described in this section is required
to characterize the load severity (e.g. the load in a given impact event).
2. Mechanical properties. These include yield strength, flow stress, tensile strength, yield-totensile ratio, fracture toughness, modulus of elasticity and strain hardening coefficient.
3. Pipe geometry. This group includes wall thickness, diameter, ovality and straightness.
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4. Damage characteristics. The geometric attributes required for various types of defects such
as corrosion, SCC, weld cracks, dents, gouges and environmental cracks are determined by
the models used to assess pipe resistance. They typically include depth, length, aspect ratio,
width and in some cases a complete geometric profile of the defect. This group also includes
defect growth parameters such as growth rates or various growth model parameters.
Locations subject to progressive ground movements may also be treated as damage locations
and characterized by appropriate parameters such as pipe curvature.
5. Model error factors (see Section 7.6).
The purpose of this section is to explain how a probability distribution can be assigned to a timeindependent random variable for subsequent use in a reliability analysis. Distributions are
typically based on statistical data and this is discussed in Sections 8.3.2 and 8.3.3. The
distribution model is defined by a simple formula and two (or three) parameters and is therefore
much more convenient to store and manipulate than the complete data set. Distribution models
also allow extrapolation of the parameter beyond the range of observed data, although this must
be done with caution. Other methods that may be used in selecting probability distributions
include theoretical models, logical arguments and judgment. These methods, which may be used
if data are not available, are discussed in Section 8.3.3.
Previously published data and distributions for the parameters required for pipeline reliability
analysis are given in Appendix B.
8.3.2 Data Analysis
Table 8.1 gives a random sample consisting of 50 yield strength values for X60 pipe steel. Data
analysis can be used to generate useful information about the statistical characteristics of the
random variable, which can be used to facilitate the selection of an appropriate probability
distribution. Assuming that a random data sample x1, x2, x3, ….xn is available, the following
information can be generated:
1. Basic Statistics. The sample mean, µx, and standard deviation, σx, can be estimated from the
well known relationships:
µx =
σx =
1 n
∑ xi
n i =1
1 n
2
∑ ( xi − µ x )
n − 1 i =1
[8.5]
[8.6]
For the yield strength data set shown in Table 8.1, the sample mean and standard deviation
are 453.2 MPa and 18.6 MPa respectively. The mean value is a measure of the location of
the distribution middle and can be seen as a best estimate of the variable. The standard
deviation is a measure of the degree of variability or randomness of x. The ratio between the
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standard deviation and the mean is referred to as the Coefficient Of Variation (COV) and is a
measure of relative variability. For the yield strength data in Table 8.1, the COV is 0.04 or
4.0%. The mean and standard deviation are the most commonly used statistics, but there are
other statistics that can also be helpful in understanding the random variable. These include
the coefficient of skewness, which measures lack of symmetry around the mean value, and
the coefficient of kurtosis, which measures the degree of peakedness around the distribution
middle. Expressions to calculate these coefficients can be found in Blank (1980).
Rank -i
Yield
Strength
(Mpa)
Cumulative
Frequency = i
/ (n +1)
Rank -i
Yield
Strength
(Mpa)
Cumulative
Frequency = i
/ (n +1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
408.1
420.4
423.9
424.9
428.8
430.4
433.5
436.7
436.8
437.1
438.2
442.3
442.4
442.8
442.8
443.4
444.6
445.6
446.6
447.2
447.7
447.9
447.9
448.8
450.4
0.020
0.039
0.059
0.078
0.098
0.118
0.137
0.157
0.176
0.196
0.216
0.235
0.255
0.275
0.294
0.314
0.333
0.353
0.373
0.392
0.412
0.431
0.451
0.471
0.490
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
451.4
455.2
456.7
457.6
457.7
459.6
460.6
460.7
461.1
462.5
462.6
464.3
465.8
467.5
469.3
469.4
470.4
471.6
472.4
473.4
473.5
485.0
487.7
491.0
495.2
0.510
0.529
0.549
0.569
0.588
0.608
0.627
0.647
0.667
0.686
0.706
0.725
0.745
0.765
0.784
0.804
0.824
0.843
0.863
0.882
0.902
0.922
0.941
0.961
0.980
Table 8.1 Yield Strength Data for X60 Steel
2. Histogram Plot. A histogram is developed by dividing the range of the random variable into
a number of intervals, called bins, and counting the number of data points in each bin. The
relative frequency associated with each bin is equal to the proportion of data points that fall
within that bin. The histogram plots the relative frequency in each bin over the associated
interval. A histogram for the yield strength data is presented in Figure 8.1.
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0.30
Relative Frequency
0.25
0.20
0.15
0.10
0.05
0.00
405 415 425 435 445 455 465 475 485 495
Yield Strength (MPa)
Figure 8.1 Histogram Plot for the Yield Strength Data in Table 8.1
3. Cumulative Frequency Plot. The vertical axis of a cumulative frequency plot represents the
relative frequency of the random variable taking a value less than or equal to the
corresponding value on the horizontal axis. One common method to plot the cumulative
frequencies is to arrange the data in an increasing order and use the relative rank, i, of each
data point to estimate the associated cumulative frequency, F(x) using:
F ( xi ) = i /(n + 1)
[8.7]
where n is the total number of data points. Because the data are arranged in increasing order,
there are i points with a value equal to or less than xi. Therefore, dividing i by the total
number of points n provides the relative frequency of occurrence of a value less than xi. One
reason for using n+1 in Equation [8.7] is to prevent the probability estimate from reaching
one at the last data point, thereby acknowledging that higher values than the maximum
observed are possible. A further discussion of this issue is given in Ang and Tang (1975).
The difference in the cumulative frequency value resulting from the use of n+1 instead of n is
small for large values of n. The cumulative frequency plot for the yield strength data in
Table 8.1 is given in Figure 8.2. Note that a cumulative frequency plot can also be generated
from the histogram by plotting the upper bound value of each bin (on the horizontal axis)
against the sum of the relative frequencies for all bins up to and including the current bin (on
the vertical axis).
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Cumulative Frequency
1.00
0.80
0.60
0.40
0.20
0.00
400
420
440
460
480
500
Yield Strength (MPa)
Figure 8.2 Cumulative Probability Plot for the Yield Strength Data in Table 8.1
8.3.3 Distribution Selection Based on Data
8.3.3.1 Introduction
There are different distributions that can be used to model a random variable (see Table D.1 in
Appendix D for the most commonly used distributions, and Christensen (1989) for a more
comprehensive listing). A good distribution choice will model the data as closely as possible,
especially in the region of relevance to reliability analyses. Since failure occurs due to a
combination of high load and low resistance, the high end of the distribution of any parameter
that increases load or reduces resistance and the low end of the distribution of any parameter that
reduces load or increases resistance are more important for reliability calculations. The high and
low ends of a distribution are often referred to as the upper and lower distribution tails. The
issue of obtaining a good fit in the appropriate tail region will be discussed further in
Section 8.3.3.4.
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Select candidate
distribution types
Estimate
distribution
parameters
Select best fit
distribution
Figure 8.3 Steps Involved in Fitting a Distribution to Statistical Data
The steps involved in selecting a distribution to model a set of data are shown in Figure 8.3.
These are discussed in detail in Sections 8.3.3.2 through 8.3.3.4. Section 8.3.3.5 gives a brief
overview of software tools that can be used to facilitate the fitting process.
8.3.3.2 Selection of Candidate Distribution Types
The selection of candidate distributions to model a set of data should be based on the statistical
characteristics of the data (see Section 8.3.2). Table 8.2 gives the range and general shape of
some commonly used distributions. Consideration should be given to matching the data range to
the distribution range and the histogram shape to the distribution shape. Based on these criteria,
and considering the distributions in Table 8.2, the yield strength data may be modeled by a
Lognormal, Gamma or Weibull distribution. All of these distributions match the data range (0 to
∞) and shape (values in the middle of the range occur most frequently). It is also possible to
include the Normal and Gumbel distributions because, although their theoretical range is -∞ to ∞,
the high mean and a low standard deviation of the data would result in all negative values having
negligible probabilities. Note that Table 8.2 shows a limited number of distributions that are
presented for illustration. Other references such as Christensen (1989) include many other
distributions that could be considered.
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Distribution Name
Range of Definition
Shape
f
Rectangular
a≤x≤b
f
Normal
− ∞ < x < +∞
x
x
f
Lognormal
0 ≤ x < +∞
x
f
Exponential
0 ≤ x < +∞
x
f
Gamma
0 ≤ x < +∞
x
f
Gumbel
(Extr. value type I)
− ∞ < x < +∞
x
f
Weibull
(Extr. value type III)
0 ≤ x < +∞
x
Table 8.2 Range and General Shape of Some Commonly Used Probability Distributions
8.3.3.3 Distribution Parameter Estimation
Each distribution type (e.g. Normal) represents a family of distributions with general
characteristics such as range and shape. A specific distribution from a given family is defined by
a number of parameters (called distribution parameters) that are directly related to the mean and
standard deviation of the random variable being modeled (see Table A.1 and Christensen 1989).
There are three methods that may be used to estimate the distribution parameters from the data.
1. Method of moments. The distribution parameters are selected such that the mean and
standard deviation of the distribution match those of the data.
2. Method of maximum likelihood. The distribution parameters are selected such that the
likelihood of obtaining the data sample from the resulting distribution is maximized. This
can be interpreted as selecting the distribution (as defined by the parameters) from which the
existing data would have been the most likely outcome of random sampling.
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3. Least square fit. The distribution parameters are selected such that the resulting distribution
represents a least square fit to the data on the appropriate probability paper plot (see below
for a description of probability paper plots).
The calculations involved in estimating distribution parameters from data require familiarity with
the mathematical forms of probability distribution functions. These standard calculations are
typically performed using a fitting software package and are therefore not described further in
this document.
8.3.3.4 Best Fit Distribution Selection
8.3.3.4.1 Goodness-of-fit Tests
These tests are based on a calculated statistic representing the deviation of each distribution from
the data. The deviation statistic is then used as a basis for evaluating goodness-of-fit. There are
two commonly used tests. The chi-square test is based on the sum of the squared deviations
between the bin frequencies from the data histogram and the corresponding probabilities
estimated from the distribution (Figure 8.4a). The Kolmogorov-Smirnov test is based on the
maximum deviation between the cumulative frequency values calculated from the data and the
corresponding cumulative probabilities estimated from the distribution (see Figure 8.4b).
The deviation statistic from either test can be used to calculate a quantity called the level of
significance, which represents the likelihood that the distribution selection is valid given the data.
A better distribution fit will produce a lower deviation statistic and a higher level of significance.
Table 8.3 gives the deviation statistic and level of significance for each of the four distributions
considered for the yield strength data given in Table 8.1. It is noted that this is an informal
interpretation of goodness-of-fit tests - the classical interpretation is to use the test statistic to
reject (but not necessarily confirm) a potential distribution based on a pre-defined level of
significance. A detailed description of these tests can be found in Ang and Tang (1975) or
Benjamin and Cornell (1970).
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Distribution
Sum of deviations used
in chi-square test
Parameter
Cumulative Frequency
a)
Distribution
Maximum deviation
used in K-S test
Parameter
b)
Figure 8.4 Illustration of Goodness-of-Fit Test Statistics – a) Chi-square Test b) K-S Test
Chi-square
Normal
Gumbel
Gamma
LogNormal
Kolmogorov - Smirnov
Test statistic
Level of
significance
Test statistic
Level of
significance
7.13
0.4150
0.073
0.955
12.65
0.0810
0.081
0.899
13.21
0.0400
0.073
0.952
23.54
0.0006
0.185
0.064
Table 8.3 Results of Goodness-of-Fit Tests for Yield Strength Data
Despite their objective and straightforward appearance, goodness-of fit tests have some serious
limitations. These include:
1. Sensitivity to bin width selection. The results of the chi-square test are sensitive to the bin
width selected to group the data. A larger number of bins (i.e. smaller bin width) will result
in an increased deviation statistic and a reduced level of significance. There are no accepted
criteria for bin width selection.
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2. Sensitivity to the number of data points. As the number of data points increase the level of
significance decreases and the chi-square or K-S tests are more likely to result in distribution
rejection.
3. Lack of sensitivity to tail deviations. The tests are more influenced by the mid-range of the
random variable. This is because a cumulative probability plot on a natural scale is very flat
in the tail regions, and therefore, the deviations between data and distribution are much
smaller in the tail region than in the mid-range. The deviation statistic is therefore dominated
by the mid-range characteristics making the results insensitive to a poor fit in the tail region.
8.3.3.4.2 Visual Comparison on Probability Paper Plots
A probability paper plot is a cumulative distribution function, which is rescaled to plot as a
straight line. The required scaling depends on the mathematical distribution formula and
therefore probability paper plots are unique to each distribution type. (The mathematical details
of producing probability paper plots can be found in Ang and Tang 1975, Chapter 6.).
Figure 8.5 Probability Paper Plots for the Yield Strength Data
Figure 8.5 shows probability paper plots of the yield strength data for four different distribution
types. The distribution selection is made subjectively based on inspection of these plots. A good
fit has two attributes:
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1. The data plots as a straight line.
2. The deviations from the line are small.
The main advantage of this method is that it allows meaningful comparison between the data and
distribution over the full range of the parameter, including the tail area. If the tail area is of
special importance, the above-mentioned criteria can be applied to (or emphasized in) the tail
region. This is possible because probability paper plots do not suffer from the flat tail limitations
of cumulative probability plots on a natural scale. The best-fit distribution is selected based on
visual comparison of the probability paper plots for all candidate distributions. For the yield
strength data, Figure 8.5 shows that Normal distribution gives the best fit.
8.3.3.4.3 Recommended approach
A rigid procedure for distribution selection cannot be developed because of the unique aspects
associated with each data set and its intended application. The final selection will therefore
involve a certain degree of judgment based on the information generated from the methods
discussed earlier.
The final distribution selection should be based on both goodness-of-fit tests and probability
paper plots. Because of the limitations associated with goodness-of-fit tests, their use within the
usual classical framework of acceptance or rejection at a given significance level is not
recommended. These tests are better used to compare different distributions based on the
deviation statistic (lower is better) and significance level (higher is better). Probability paper
plots add another tool for comparison, which should be considered along with goodness-of-fit
tests.
In the yield strength example used throughout this section the chi-square and K-S tests
(Table 8.3), as well as visual inspection of probability paper plots (Figure 8.4) point to the
Normal distribution as the best fit, making the selection straightforward. In cases where
goodness-of-fit tests and probability paper plots point to different distributions it is recommended
that probability paper plots should weigh more heavily in making the distribution selection. It is
also recommended that in examining the probability paper plots, attention should be given to the
fit in the appropriate tail area (upper tail for loads and lower tail for resistance) as this area has a
significant influence on the accuracy of the calculated failure probabilities.
A number of commercial software packages are available to facilitate probability distribution
fitting and selection. Examples are BestFit by Palisade Corporation (www.palisade.com), NY,
Statrel by RCP Gmbh, Munich, Germany (www.strurel.com).
8.3.4 Other Distribution Selection Methods
Sufficient statistical data are not always available or required to assign probability distributions.
Other methods that can be used are:
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1. Previous knowledge of the statistical characteristics of the parameter. Analyses of many data
sets on the yield strength of recent pipe steel show some consistent trends. The data tends to
fit a Normal or Lognormal distribution. The mean value is approximately 10% higher than
SMYS and the COV is around 3.5%. In the absence of project-specific data, this information
can be used to derive a probability distribution of yield strength from the steel grade. In
some cases, the distribution may be only partially known. For example, experience shows
that corrosion defect depth data tend to fit a Weibull distribution with a COV of 40 to 60%.
The mean value of the defect depth, however, is dependent on the pipeline location, age and
cathodic protection condition. If the amount of data available for a given pipeline is
sufficient to estimate the mean but not to fit a complete distribution, the COV and
distribution type may be assigned based on previous knowledge.
2. Theoretical basis. The distribution type can be selected on a theoretical basis in some cases.
These relevant theoretical solutions are usually correct when some parameter becomes very
large (i.e. asymptotically valid), however, they often provide reasonable approximations even
if the parameter is not very large. These cases include:
•
The Normal distribution is a good model for the sum of a large number of independent
identically distributed random variables. This is based on the central limit theorem,
which is detailed in Feller (1971).
•
The Lognormal distribution is a good model for the product of a large number of
identically distributed independent random variables. Again, this is based on the central
limit theorem, which is detailed in Feller (1971).
•
The distributions of extremes (Gumbel, Frechet and Weibull) are good models for the
maximum or minimum value out of a large number of independent samples of a given
random variable (details of the conditions under which each distribution is valid can be
found in Ang and Tang 1990).
In these cases, the distribution type is assigned on a theoretical basis; however, the
distribution mean and standard deviation must be assigned using other methods (e.g. data
analysis).
3. Logical argument. A logical argument is often used to assign a uniform probability
distribution to random variables that are equally likely to take any value within a bounded
range. An example is the location of a corrosion defect along a pipeline segment with length
l. If all the parameters affecting corrosion (e.g. soil type, coating condition and CP
condition) are uniform along the segment, there would be no reason to assume that the
corrosion defect is more likely to occur at any given location over another. In this case, the
random variable representing distance from one end of the segment to the corrosion defect
can be modeled by a uniform probability distribution between 0 and l.
4. Parameter tolerances. Variability in parameters that are subject to quality checks may be
estimated based on the control criteria applied. API 5L, for example, specifies that the
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tolerance on the diameter of seamless pipe with a diameter greater than 20” is ±1.00%.
Assuming that the distribution type is Normal, the mean value can be assumed to fall half
way between the upper and lower bounds (i.e. equal to the specified value). Also, if it is
conservatively assumed that 95% of all pipes will fall within the specified range, standard
Normal distribution calculations can be used to determine that the corresponding COV is
0.5%. This method has also been used to “back calculate” weld defect size distributions
based on knowledge of the capability of weld inspection techniques (Jiao et al. 1995). In
using this method to select a probability distribution to a given parameter, it may be
necessary to consider tolerances associated with another related parameter; for example,
weight tolerances may influence the probability distribution of wall thickness.
5. Subjective judgment. Judgment may be used when data are not available and none of the
above methods are applicable. The assignment should be made by a person familiar with the
physical aspects involved, and the process may be facilitated by using a formal approach to
soliciting subjective probabilities (e.g. Spetzler and Stael von Holstein 1972). Probability
solicitation approaches offer strategies to ensure accurate understanding of input
requirements and consistency of the final distribution. Strategies are also available to
develop distributions that represent a reasonable compromise between the opinions of a
number of experts.
8.4 Time-dependent Variables
8.4.1 Introduction
Time-dependent random variable is one that changes randomly as a function of time. Timedependent random variable models are referred to as random processes. They are typically used
to model variable loads, including operational, environmental and accidental loads. There are
two basic types of random processes (see Figure 8.6), namely a discrete random process
(Figure 8.6a) and a continuous random process (Figure 8.6 b). Specific definitions and
examples of these two types are given in Section 4.2.1.
For the pipeline to survive a certain time period under a variable load, it must withstand the
maximum load applied during that period (see Figure 8.6). Reliability (or probability of failure)
must therefore be calculated from the probability distribution of the maximum (or extreme) load
for the specified time period. If reliability is calculated on an annual basis for example, a
variable load must be characterized by the probability distribution of its annual maximum. The
analysis required to define this distribution is referred to as extremal analysis. Sections 8.4.2 and
8.4.3 describe the methods used to characterize discrete and continuous random processes and to
define the probability distributions of their extremes.
If the resistance is represented by a time-dependent random variable, the concepts in the previous
paragraph hold, except that the distribution of annual minimum resistance would be required.
This is not a common requirement in pipeline reliability analyses, because time-dependent
parameters influencing resistance (such as the size of corrosion defects) are typically modeled in
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terms of time-independent growth variables (such as the constants of a defect growth function).
Because of this, the discussion in Sections 8.4.2 and 8.4.3 is focused on loading processes.
Largest Load
Load
Hits
Time
a) Discrete
Global peak
Pressure
Local peaks
Time
b) Continuous
Figure 8.6 Illustration of Time-dependent Random Variables (or Random Processes)
8.4.2 Discrete Random Process
8.4.2.1 Process Characterization
A discrete random process is characterized by:
1. Frequency. Occurrences of the event are modeled by a Poisson Process, which is
characterized by the rate of occurrence, λ (see Section 8.2). An example is the number of
equipment impact events per year for a given pipeline.
2. Severity. The severity of the process is characterized by the probability distribution of the
maximum process parameter given an event. An example is the maximum equipment impact
load on the pipeline in a given incident. This distribution is often referred to as the parent
distribution to distinguish it from the extremal distribution for the same process.
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8.4.2.2 Maximum Load Distribution
The maximum load, y, occurring in a given year, can be expressed as
y = max( x1 , x 2 ,......., x n )
[8.8]
where x is the load in a given event and n is the number of events during the year. The
cumulative probability distribution, Fy, of y can be obtained by recognizing that y will be less
than a given value z if x1, x2,…., xn are all less than z. This gives:
Fy ( z ) = p( y ≤ z ) = p( x1 ≤ z I x 2 ≤ z I .......x n ≤ z )
[8.9]
where I denotes co-occurrence (intersection) of events. Since the loads resulting from different
events are independent variables belonging to the same distribution, Fx, Equation [8.9] gives:
n
Fy ( z ) = p( x1 ≤ z ) × p( x 2 ≤ z ) × ....... p( x n ≤ z ) = Fx ( z )
[8.10]
This means that the cumulative distribution of the maximum of n values of x is obtained by
raising the parent cumulative distribution to the power n. Figure 8.7 shows the probability
distributions of the maximum (extreme) of 10, 100 and 1000 values out of a Normal load
distribution. It shows that the extremal distribution moves further to the right (higher values) as
n increases, reflecting the intuitive expectation that the likelihood of encountering a high load
increases with the number of loading events.
0.30
"Parent"
Extreme of 10
Probability Density
0.25
Extreme of 100
Extreme of 1000
0.20
0.15
0.10
0.05
0.00
0
5
10
15
20
25
30
35
40
45
Load (kN)
Figure 8.7 Distributions of Extremes for a Normal Parent Distribution
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In reality, the frequency of occurrence of the event in a given period of time is not known with
certainty but can be modeled by a Poisson Process as discussed in Section (8.2). To take this into
account, the extremal distribution can be derived as a sum of the extremals for different possible
values of n, each weighted by the probability of occurrence of the corresponding value of n. This
leads to:
Fy ( z ) = exp[−λt{1 − Fx ( z )}]
[8.11]
where λ is the rate of occurrence in events per unit time and t is the time period being considered.
Figure 8.8 shows a comparison between the extremal distributions calculated from
Equation [8.10] for a fixed value of ten events (n = 10), and Equation [8.11] for a Poisson
distributed random number of events with an expected number of ten events (λt = 10). The
difference between the two results is small and can be shown to become much smaller as n
increases. It is therefore reasonable to ignore the randomness of n and use Equation [8.10] for
most applications.
0.16
"Parent"
Probability Density
0.14
Equation [7.10]
Equation [7.11]
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
5
10
15
20
25
30
35
40
45
Load (kN)
Figure 8.8 Extremal Load Distributions for Fixed and n (expected value of n = 10)
8.4.2.3 Asymptotic Extremal Distributions
There are a number of standard probability distributions that can be used to model the extremes
(maxima and minima) of random variables. In general, these distributions are derived
theoretically based on two assumptions:
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1. A large value of n. The distributions are valid asymptotically as n tends to infinity.
2. A certain format of the parent distribution tail. Figure 8.7 shows that extremal distributions
have significant probability levels only in the tail region of the parent distribution. They are
therefore determined by the shape of the parent distribution tail. Since the tails of standard
distributions can be classified into a limited number of mathematical forms, their extremes
tend to converge to certain common distributions.
There are three extremal distributions referred to as Extremal Type I, II and III or alternatively as
Gumbel, Frechet and Weibull. Details of the derivations and applicability of these distributions
can be found in (Ochi 1990, and Ang and Tang 1990). The Gumbel distribution is the one most
commonly used in engineering applications and is therefore given here as an illustration of the
general methodology. The cumulative distribution function of the Gumbel distribution is given
by:
F ( y ) = exp[− exp{−(
y−a
)}]
b
[8.12]
where a and b are two distribution parameters. The mean µy and standard deviation σy can be
calculated from a and b using µy ≈ a + 0.577b and σy ≈ 1.282b.
Name
Cdf FX(x)
Range
Characteristic
Extreme an
Dispersion
Factor bn
Exponential
 x −α 

1 − exp −
β


αK+ ∞
α + β lnn
β
Gamma
u α −1e − u / β du
∫0 Γ(α ) β α
0K + ∞
Normal
(de Finetti)
N ( x | µ ,σ 2 )
− ∞K + ∞
µ + σ 2 l n ( 0 .4 n )
σ (2lnn) −1 / 2
0K + ∞
exp(a n ,normal )
a n bn ,normal
Lognormal
(Standardized)
x
1
∫
2π
ln x
−∞
e
−t 2 / 2
dt
 n 

 Γ(α ) 
β ln

n 

β  ln
 Γ(α ) 
(n>10)
Rayleigh
 x2 

1 − exp −
2 
 2β 
0K + ∞
β 2lnn
β 2lnn
Weibull
  x α 
1 − exp −   
 β 


0K + ∞
β (lnn)1 / α
β
(lnn) α
α
1−α
1−α
Table 8.4 Gumbel Distribution Parameters for a Number of Parent Distribution Types (Maes 1985)
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The parameters of the Gumbel distribution for the extreme of a given process can be calculated
from the parameters of the parent distribution and the frequency of the process. The appropriate
relationships for this calculation depend on the parent distribution type. Table 8.4 gives the
required relationships for some of the most commonly used distributions. The steps involved in
calculating the distribution of extreme are:
1. Fit a distribution to the parent using the methods described in Section 8.3.3.
2. Find the expected number of events n based on the process occurrence rate and the time
interval considered.
3. Use n and the parent distribution parameters in the appropriate formulas in Table 8.4 to
calculate a and b.
4. Use a and b in Equation [8.11] to get the required extremal distribution.
Example: Assume that the (parent) probability distribution of the load in a given equipment
impact event is Normal with a mean of 117 kN and a standard deviation of 36 kN and that the
expected number of impacts is 0.01 per km-year. The probability distribution of maximum
annual impact load on a 5000 km pipeline was estimated using the procedure described above.
The expected number of impacts is 0.01 impact per km-year x 5000 km = 50 impacts per year.
Using the Normal distribution formulas from Table 8.4, the Gumbel distribution parameters of
the annual maximum load are a = 205.1 kN and b = 12.87 kN. The resulting distribution is
plotted in Figure 8.9 along with the parent Normal distribution and the (exact) maximum annual
distribution from Equation [8.10]. The Figure shows that the Gumbel gives a reasonably close
conservative approximation for this case. The approximation would improve as n increases.
0.03
"Parent"
0.03
Probability Density
Exact Annual Maximum
0.02
Gumbel Annual Maximum
0.02
0.01
0.01
0.00
0
50
100
150
200
250
300
Load (kN)
Figure 8.9 Exact and Gumbel Approximation of the Maximum Annual Impact Load
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8.4.2.4 Estimation of Return Periods
The return period is defined as the average time between occurrences of a certain event. For
example, if the impact rate for a given pipeline is 0.1 per year, then the return period for an
impact event is 1/0.1 = 10 years. If the event is defined as a random variable exceeding a certain
value, y, then the return period, RP(y) can be estimated from the cumulative probability
distribution of the annual maximum, F(y), using the following relationship:
RP( y ) = 1 /[1 − F ( y )]
[8.13]
Since F(y) represents the probability that y will not be exceeded in a given year, the denominator
of the right hand side of Equation [8.13] represents the probability of exceeding y in a year. The
average time between exceedances of a given value is then calculated as the inverse of the
probability of exceeding that value in a year.
Example: The Gumbel probability distribution for the impact load generated in Section 8.4.2.3
can be used to calculate the return period associated with a load value of 265 kN. The first step
is to calculate the probability that a load of 265 kN will not be exceeded in one year. This is
done by substituting a = 205.1 kN, b = 12.87 kN (as calculated in the example in Section 8.4.2.3)
and y=265 kN in Equation [8.12], leading to F(265) = 0.99 per year. Substituting this value for
F(y) in Equation [8.13] gives a return period R(265) = 100 years.
8.4.3 Continuous Random Processes
A continuous random process is typically characterized by the parameter value measured at a
regular time interval. The reported value of the process is often averaged over a small period of
time (for example, the mean hourly wind speed or temperature). The time period over which the
parameter measurements may be averaged is application dependent. It is important to recognize
that a longer averaging period will result in missing parameter variations within the averaging
period.
Depending on the time variation patterns of the process, values of the parameter that are
separated by a certain time interval can be correlated. For example, wind speeds measured ten
minutes apart are likely to be highly correlated. Typically the correlation is high for values
separated by a small time interval, converging to zero as the time interval increases. The
correlation is often described by an autocorrelation function, which gives the correlation between
two values of the process as a function of time period separating them. Random processes may
be characterized by decomposing them into the sum of sine waves with different frequencies
using spectral analysis. The theory of stochastic processes is described in detail in Ochi (1990).
A simple method to calculate the extremes of a continuous process is to select discrete points
from the process and analyze them using the techniques described in Section 8.4.2. The discrete
points can be selected in such a way as to minimize the effects of autocorrelation, so that the
process can be treated as independent. Two approaches are presented here:
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1. Equal interval sampling (Figure 8.10a). The time axis is broken into equal intervals, t*, and
the maximum value in each interval is recorded. The interval maxima are then treated as a
discrete random process with a rate of occurrence of 1/t* and a parent distribution that is
derived by fitting a distribution to the interval maxima. In this approach, the interval must be
sufficiently long to ensure that the interval maxima are independent.
2. Peak-over-threshold sampling (Figure 8.10b). A threshold of the parameter is defined. Each
time the parameter exceeds the threshold is treated as an occurrence of the process. The
corresponding parameter value is then taken as the maximum value reached before the
parameter drops below the threshold. The process rate equals the rate of threshold
upcrossings and the parent distribution is developed by fitting the peaks. Independence
between the individual peaks is achieved by selecting a sufficiently high threshold.
Pressure
t*
t*
Time
a) Equal Interval Sampling
Pressure
Time
b) Peak-over-threshold
Figure 8.10 Illustration of Methods to Discretize a Continuous Random Process
The methods discussed in this section are also applicable to intermittent processes, which are
treated as continuous processes that are interrupted by periods during which the process is not
applied. An example is the internal pressure in a pipeline, which may be interrupted by
shutdowns. The only difference in these cases is that the frequency of the process should be
adjusted to account for the interruptions. For example, if the process is applied only 75% of the
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time, then the frequency calculated from a continuous portion of the process should be multiplied
by 0.75.
8.5 Effect of Sample Size
8.5.1 Introduction
As discussed in Section 8.3.2, the parameters of a given distribution are estimated from the mean
and standard deviation of a given data set. The use of a limited sample in estimating these
parameters implies that there is some uncertainty regarding the estimate. The random sample
can be seen as n actual values drawn from the probability distribution of the variable. Two
samples of size n will have different data points leading to different parameter estimates (e.g.
means and standard deviations). This means that there is some statistical error (referred to as
statistical uncertainty in Section 3.3) associated with using sample statistics to estimate
distribution parameters. Statistical uncertainty is large for small samples, reducing to zero for
very large (infinite) samples.
Statistical uncertainty has some important implications with respect to the overall uncertainty
associated with the reliability calculation. These include:
1. Sample size. What is an adequate sample size to estimate a certain distribution parameter?
2. Total uncertainty. How can statistical uncertainty be incorporated with other sources of
uncertainty (such as random variations or measurement uncertainty) to produce a
characterization of the total uncertainty?
3. Conservatism. How can appropriate conservatism be incorporated in the reliability analysis
to ensure that statistical uncertainty is compensated for?
The basic method used to account for statistical uncertainty is to recognize that distribution
parameters estimated from data are themselves random variables and can therefore be modeled
by probability distributions. The probability distributions of various statistics depend on the
distribution from which the sample is drawn and the sample size. These distributions and their
parameters can be defined analytically for several practical cases and can be found numerically
in other cases.
Sections 8.5.2 and 8.5.3 demonstrate how statistical uncertainty analysis can be used to answer
the three questions raised earlier for a number of special cases that can be solved analytically.
Section 8.5.4 gives a general numerical procedure that can be used to achieve the same
objectives.
8.5.2 Example 1 – Occurrence Rate of a Poisson Process
In the equipment impact example presented in Section 8.2.3 consider the following three data
sets:
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•
•
•
8 impacts in 4 years (no = 8 and to = 4)
40 impacts in 20 years (no = 40 and to = 20)
160 impacts in 80 years (no = 160 and to = 80)
The rate of occurrence calculated from Equation [8.4] is two impacts per year for all three data
sets, but the sample size, and consequently the magnitude of statistical uncertainty, is different in
each case. As mentioned in Section 8.5.1, statistical uncertainty can be accounted for by treating
the impact rate as a random variable. It is convenient in this case (see Benjamin and Cornell
1970, page 633) to assume that the rate of the Poisson process is Gamma distributed.
f (λ ) =
exp(−λt o ) × (λt o ) no −1
to
Γ ( no )
[8.14]
where Γ is the incomplete Gamma function. The mean and standard deviation of this distribution
are given by:
µ λ = no / t o
[8.15]
σ λ = no / t o 2 = µ λ / t o
[8.16]
Equation [8.16] shows that for a given mean value the standard deviation of λ is inversely
proportional to the square root of the observation interval, which means that the standard
deviation of the rate estimate is reduced as the observation period increases. This is illustrated in
Figure 8.11, which shows the probability distribution of the impact rate for each of the assumed
data samples mentioned earlier. It shows that the distribution becomes narrower, reflecting
lower uncertainty, as the observation period increases.
3
160 hits in 80 years
Probability Density
2.5
2
40 hits in 20 years
1.5
1
8 hits in 4 years
0.5
0
0
1
2
3
4
5
Impact Rate (per year)
Figure 8.11 Probability Distribution of Impact Rate for Different Sample Sizes
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Figure 8.12 shows the 90% probability interval for the impact rate as a function of the
observation period. The 90% probability interval is the interval within which there is a 90%
chance that the impact rate will fall, given the data sample. For a given observation period, the
upper and lower bounds of this interval can be calculated as the inverse cumulative probability of
the corresponding Gamma distribution at 5% and 95% probability. Again, the figure shows that
the interval becomes narrower as the observation period increases.
90% Probability Interval
3.5
3
2.5
2
1.5
1
0.5
0
0
20
40
60
80
Observation Period (years)
Figure 8.12 The 90% Probability Interval as a Function of Observation Period
This information can be used to determine the observation period required to achieve a certain
level of confidence on the process rate estimate (answer to question 1 stated in Section 8.5.1).
Given a certain data sample, the mean value, standard deviation and 90% probability interval of
the process can be determined as explained above, and evaluated against pre-defined criteria.
Two criteria may be used
1. Based on the COV. Assume for example that the data sample gives 8 impacts in 4 years (no
= 8 and to = 4). Using Equations [8.15] and [8.16] the mean and standard deviation of the
impact rate are 2 and 0.70, leading to a COV of 35%. If the maximum allowable COV is
20%, for example, Equation [8.15] can be used to calculate that the maximum standard
deviation is 0.40 and Equation [8.16] to calculate that the required minimum observation
period is 12.5 years. Note that this analysis is based on the current estimate of the mean,
which is based on four years of observation.
2. Based on probability intervals. For the same values as in criteria one above, assume that the
rate estimate is required to be within ± 25% (i.e. within the range of 1.5 and 2.5 impacts per
year) with a probability of 90%. From Figure 8.12 (which can be generated using the mean
estimate as calculated from the 4-year sample), this shows a required observation period of
40 years.
It is noted that this analysis is contingent on having an estimate of the mean value of the process
rate. In the above example, this estimate was available from a data sample based on four years of
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process observation. If no data were available, a subjective estimate would have to be made (e.g.
based on similar pipelines). In either case, an incremental sampling process can be developed in
which an initial estimate of the mean is used to define the required sample size to meet the
specified criteria. Once this data set is collected, a better estimate of the mean and standard
deviation can be made and further sampling carried out if necessary.
8.5.3 Example 2 – Mean of a Distribution with Known Standard Deviation
Consider a fitness for service assessment of an old natural gas pipeline with unknown notch
toughness distribution. Assume that information from similar pipelines indicates that notch
toughness is normally distributed with a standard deviation of 5 J. To determine the notch
toughness distribution for the pipeline, an estimate of the mean value is required. The required
notch toughness samples can only be obtained from pipeline material cutouts, which is a costly
and disruptive process. It is therefore important to understand the implications of sample size in
order to avoid excessive sampling.
Assuming that a sample x1, x2,…., xn of notch toughness values is available, the mean toughness
can be estimated by the mean of the sample, which is given by
µ x = ( x1 + x 2 + ........ + x n ) / n
[8.17]
Because x1 to xn are independent samples from f(x), their probability distributions are identical to
that of x. Given that x is normally distributed, it can be shown that µx is also normally distributed
with a mean value given by [8.17] and a standard deviation, σµx, given by
σ µx = σ x / n
[8.18]
This result is valid for any value of n if f(x) is normal and for large values of n regardless of the
distribution type of x. Equation [8.17] shows that the standard deviation of the estimate of the
mean toughness decreases in proportion to the square root of the sample size. Figure 8.13 shows
the probability distributions of the mean toughness for three assumed sample sizes that have the
same mean of 40 J. The figure shows that the distribution is wider (more uncertain) for smaller
sample sizes. Similar to Section 8.5.2, this information can be used to select a sample size that
limits the standard deviation or probability interval of the mean toughness to specified (small)
values.
Since the mean notch toughness can take a number of values, there are a number of possible
toughness distributions, each corresponding to a possible value of the mean. These are called the
conditional fracture toughness distributions. The unconditional fracture toughness distribution
can be derived using the total probability rule (see Appendix D) as the sum of all possible
conditional distributions, each weighted by the likelihood of the corresponding mean value. This
can be written as:
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∞
f ( x | n) = ∫ f ( x | µ x , n) f ( µ x | n) dx
[8.19]
0
The final distribution of x is conditional on the sample size, n, because the distribution of the
mean is conditional on n. This means that there is a different distribution of x for each sample
size. Three of these distributions corresponding to n = 2, 5 and 20 are shown in Figure 8.14. The
distributions show more spread (uncertainty) for smaller values of n, but in this case the
difference is not very significant. The distributions shown in Figure 8.14 reflect the total
uncertainty about the value of x, including statistical uncertainty about the mean value of x. This
answers the second questions stated in Section 8.5.1.
0.40
Probability Density
0.35
n=2
n=5
0.30
n=20
0.25
0.20
0.15
0.10
0.05
0.00
30
35
40
45
50
Mean Charpy V-Notch Toughness (J)
Figure 8.13 Probability Distributions of the Mean Toughness for Various Sample Sizes
0.09
Probabiltiy Density
0.08
n=2
0.07
n=5
0.06
n=20
0.05
0.04
0.03
0.02
0.01
0
20
25
30
35
40
45
50
55
60
Charpy V-Notch Toughness (J)
Figure 8.14 Toughness Probability Distributions for Various Sample Sizes
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The implications of statistical uncertainty are illustrated in Figure 8.15. This figure shows the
cumulative probability distributions corresponding to the density functions in Figure 8.14. When
defining the value of a resistance parameter to be used in a structural assessment, a value with a
small cumulative probability (i.e. a small probability of not being exceeded) is selected to ensure
a small probability of the resistance being lower than the design value. For a cumulative
probability of 0.01, the design toughness values are approximately 27, 28 and 29 (J) for sample
sizes of 2, 5 and 20. This demonstrates that a lower (more conservative) resistance is
automatically used if the sample size is small. The level of conservatism can be reduced if more
data are available. This answers the third question stated in Section 8.5.1.
1
Cumulative Probabiltiy
n=2
n=5
0.1
n=20
0.01
0.001
0.0001
15
20
25
30
35
40
Charpy V-Notch Toughness (J)
Figure 8.15 Cumulative Toughness Distributions for Various Sample Sizes
8.5.4 General Procedure
Analytical formulas (such as the ones discussed in Sections 8.5.2 and 8.5.3) for the statistical
uncertainty associated with distribution parameters are available only for a limited number of
cases. For other cases a general numerical procedure called bootstrapping can be used. The
procedure, which is described in detail in Efron and Tibshirani (1993) is as follows:
1. Given a data sample of size n, generate m random samples of size n from the cumulative data
plot or a distribution fitted to it.
2. Calculate the required distribution parameters (e.g. mean and standard deviation of the
random variable) from each sample.
3. Use the m statistical estimates to find the mean, standard deviation and (if required)
probability distribution of each of the distribution parameters calculated in 2).
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4. Use the information from 3) to find the required sample size or the unconditional distribution
of the random variable as demonstrated in Section 8.5.3.
The advantage of this procedure is that it is conceptually simple and is valid for any distribution
parameter (or combination of parameters).
8.5.5 Comments
1. The stage at which the statistical uncertainty is incorporated in the analysis (i.e. the
integration in Equation [8.19]) is an important aspect of modeling the effect of sample size.
In the example in Section 8.5.3, the statistical uncertainty was incorporated into the notch
toughness distribution. Assume that the toughness distribution was later used to calculate the
distribution of pressure resistance of a gouged pipe. In that case, there are two possible
choices to deal with statistical uncertainty. The first is to incorporate it in the toughness
distribution (as was done in Section 8.5.3) and then use the final toughness distribution in
subsequent analyses. The second choice is to derive the probability distribution of gouge
resistance for each conditional toughness distribution (i.e. for each possible value of the mean
toughness), then integrate the gouge resistance distributions over the distribution of the mean
toughness. The results of the two approaches are not identical. In general, the correct
approach to model the effect of statistical uncertainty on a calculated parameter is to calculate
the conditional value of that parameter, then integrate statistical uncertainty at the end. In
practice, the problem can be simplified by incorporating statistical uncertainty at intermediate
stages of the analysis.
2. The probability distributions of distribution parameters may be generated using the Bayesian
approach. In this approach a prior distribution of the parameter in question is assumed based
on previous evidence. Bayes theorem is then used to update the prior distribution with the
information contained in the data sample. This theory is described in more detail in
Benjamin and Cornell (1970).
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9. RELIABILITY ESTIMATION
9.1 Introduction
As mentioned in Section 3.5.1, reliability with respect to a given limit state category, Ri, is
defined as the probability that none of the limit states within the category are exceeded during a
given period of time. It is related to the probability of failure, pf, by
Ri = 1 − p fi = 1 − ∑ p fij
[9.1]
all j
where i refers to the limit state category and j to a specific limit state within the category. As a
convention, the time period will be taken as 1 year and the pipe length as 1 km. Reliability is
therefore defined on a per km-year basis.
As discussed in Section 8.2, failures can be modeled by a Poisson process, which is characterized
by a failure rate (in failures per km-year). For the small failure rates that are typical of
transmission pipelines (smaller than 0.1 failures per km-year), Equation [8.3] shows that the
probability of multiple failures in a given km-year is negligible and that the probability of one
failure per km-year is approximately equal to the expected failure rate, λf. The reliability can
therefore also be expressed as
Ri = 1 − λ fi = 1 − ∑ λ fij
[9.2]
all j
As mentioned in Section 4.3.1, pipeline limit states are classified into two major categories: those
with time-dependent reliability and those with time-independent reliability. Sections 9.2 and 9.3
describe the failure probability calculation approach for a single time-independent limit state, and
a single time-dependent limit state, respectively. For time-dependent reliability, a general model
to quantify the impact of prevention and maintenance activities on the failure probability is
presented.
In some cases, a number of limit states are linked to a single loading scenario and must be
considered simultaneously. An example of this is failure due to equipment impact, which may
occur by puncture under the load imposed by the excavator tooth or leakage of a gouged dent
after removal of the excavator tooth. The limit states for these conditions are correlated because
they depend on the same parameters (e.g. excavator load, tooth geometry, wall thickness and
grade). Simultaneous consideration of correlated limit states is discussed in Section 9.4.
9.2 Single Time-independent Limit State
9.2.1 Introduction
As discussed in Section 4.2.1 (Table 4.1), time-independent reliability problems arise when a
time-independent or time-dependent stationary load is coupled with a time-independent
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resistance. The random stationary loading process may be continuous (e.g. bending due to wind
load on a free span, stresses resulting from internal pressure) or discrete (e.g. equipment impact
and permanent ground deformations due to seismic events). A general probability calculation
methodology that applies to all time-independent limit states, except those relating to seismic
loading, is presented in Section 9.2.2. A special methodology for seismic limit states is
presented in Section 9.2.3.
9.2.2 General Methodology
9.2.2.1 Failure Rate
With the exception of seismic-related limit states, the failure rate, λf (failures per km-year), for
time-independent limit states can be calculated from
λf =ω× pf
[9.3]
where pf is the conditional probability of failure and ω is the frequency associated with the
conditioning event for which pf is calculated. Equation [9.3] assumes that individual
conditioning events are independent. The definitions of ω and pf are situation-dependent as
illustrated by the following examples.
Example 1: Equipment impact
The conditioning event can be defined as a randomly selected equipment impact incident. Based
on this, ω is defined as the frequency of impact events (impacts per km-year) and pf is the
probability of failure for a given impact (per impact). The conditional failure probability pf is
calculated from the probability distributions of the load in a randomly selected impact and the
resistance at a randomly selected pipeline location. This assumes that failures during individual
impacts are independent events. The frequency of impact is determined by such line attributes as
land use and burial depth, as well as damage prevention measures such as frequency of right-ofway patrols, one-call system, public awareness programs and excavation procedures.
Improvements to these mitigation measures can be used to reduce the frequency of impact and
the overall failure rate (Chen and Nessim 1999).
Example 2: Burst of defect-free pipe
In this case, pf can be defined as the annual probability of failure for a randomly selected joint
within a 1 km length of the line. This can be calculated by using the maximum annual load (see
Sections 8.4.2 and 8.4.3) and the resistance of a randomly selected joint in a limit state
comparing the internal pressure to the burst resistance. By using the annual maximum load, the
effect of time is already included in pf, which will be defined per joint-year. The frequency, ω, is
defined as the number of joints per km, so that the product of ω (joints per km) and pf (per jointyear) gives the failure rate per km-year.
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An equivalent characterization of this example is to define ω as the frequency of pressure peaks
per year and pf as the failure probability per peak-km. In this case pf can be calculated by using
the probability distributions of a randomly selected pressure peak and the minimum resistance
for a single pipe joint in a 1 km length. Alternatively, pf can be calculated as the probability of
failure per joint per pressure peak and ω as the number of joints per km multiplied by the
number of pressure peaks per year. In this case, pf must be calculated using the probability
distributions of the pressure at a randomly selected peak and the resistance of a randomly
selected joint. Since all of these formulations are equivalent, the choice should be based on ease
of defining the required inputs.
9.2.2.2 Conditional Failure Probability
The simple formulation given in this section applies to most time-independent reliability
problems such as equipment impact, gravity loads, wind loads and slope failures. The common
characteristic of these cases is that the conditional probability is a simple one-step calculation of
the probability that the load (e.g. the equipment impact load) will exceed the resistance
(e.g. puncture resistance to equipment impact load).
As discussed in Section 3.5.3, the conditional probability of failure, pf, can be calculated as the
probability that the resistance, r, is less than the maximum load effect, l:
p f = p(r ≤ l ) = p (m = r − l ≤ 0)
[9.4]
in which m is the margin of safety defined as the difference between the resistance and load
effect. If the load effect and/or resistance are estimated from other basic variables x = x1,
x2,….,xn (see Section 6.4 for examples), Equation [9.4] becomes:
p f = p[m = g ( x ) ≤ 0]
[9.5]
Figure 9.1 shows that the probability of failure is equal to the area under the probability density
function of m for all values of m ≤ 0. This means that Equation [9.5] can be written as
p f = p[m ≤ 0] =
∫ f (m) dm
[9.6]
m≤ 0
Substituting g(x) for m in Equation [9.6] gives:
p f = p[g ( x ) ≤ 0] =
∫ f( x ) d x
[9.7]
g ( x) ≤ 0
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Safe
Failure
pf = p(m ≤ 0)
0
Safety Margin = r - l
Figure 9.1 Probability Density Function of the Safety Margin Showing the Probability of Failure
Because x is a vector of basic random variables, Equation [9.7] is a multi-dimensional integral of
the joint probability density function of x over the failure domain (characterized by g(x) ≤ 0). A
mathematical solution to this integral does not exist for the majority of practical cases.
Numerical and approximate analytical solutions that have been developed to solve this problem
are described in detail in standard texts such as Thoft-Christensen and Baker (1982), Madsen et
al., (1986), and Melchers (1999). The most commonly used methods are the following:
•
First and Second Order Reliability Methods (FORM and SORM). FORM and SORM are
approximate analytical methods that utilize a simple closed form solution, which exists for
Equation [9.7] if the random variables x are all normal and independent and the limit state
function g(x) is linear of the form g(x) = a1 x1 + a2 x2 +…..+ an xn. The essence of FORM is
to define an equivalent problem with these special characteristics that has approximately the
same solution as the original problem. SORM improves on FORM by relaxing the
assumption of a linear limit state function and using a second order approximation to better
match the original g function. These methods are computationally efficient because they
require a relatively small number of calls to the limit state function, g. In addition, the
number of calls is not sensitive to the probability level being calculated. These aspects give
FORM and SORM an advantage over other methods for computationally intensive limit state
functions and small probability levels. On the other hand, FORM and SORM require the g
function to be once (FORM) or twice (SORM) differentiable, which limits the generality of
the limit state functions to which they can be used.
•
Monte Carlo simulation. The basic idea of simulation methods is to generate a large number
of samples from the distribution of the basic variables f(x). Generating random samples from
specific distributions is addressed in detail in Rubinstein (1981). Each sample of x values is
substituted in the limit state function to determine whether failure occurs. The number of
simulated failures divided by the total number of simulations is then used as an estimate of
the failure probability. The Monte Carlo simulation method uses only point values of g(x)
and does not require g(x) to have any analytical properties. This makes it simple, robust and
capable of handling any type of limit state functions. Its main limitation is that it can be
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computationally intensive for small probabilities, since the number of calls required to reach
a certain level of confidence in the result is inversely proportional to the failure probability
being estimated.
•
Fast simulation methods. Other simulation techniques exist that have better efficiency than
the Monte Carlo method. These include importance sampling, Latin Hypercube sampling,
adaptive sampling, conditional simulation methods and directional simulation. These
methods are effective for some problems, but generally suffer similar limitations to those of
FORM and SORM.
Detailed descriptions of these methods can be found in
Rubinstein (1981) for general problems and Melchers (1999) for structural reliability
applications.
Equation [9.7] represents the most basic calculation required for pipeline reliability analyses.
The methods described in this section are discussed and evaluated only with respect to their
capability to solve this basic problem. The following sections show that more complex
calculations are required for problems involving correlations, time-dependent limit states and
multiple limit state functions. The same basic solution methods discussed above are used to
carry out these calculations; however, additional limitations apply for more complex problems.
These issues are discussed further in Section 9.5.
9.2.2.3 Example
The annual probability of rupture of a joint of defect-free pipe under internal pressure is
calculated using the characterization given in Example 2 of Section 9.2.2.1. The limit state
function, g, is formulated by subtracting the maximum annual load from the resistance. In this
case, the load is calculated as the product of the internal pressure, P, and diameter, D
(i.e. l = PD). The resistance is equal to, twice the product of the wall thickness, t, and flow
stress, σf, multiplied by a model error factor c (i.e. r = 2 c t σf). If the flow stress is defined as
95% of the ultimate tensile strength σu, the following limit state function results:
g = 1.9 c t σu – P D
[9.11]
The deterministic pipeline design parameters are given in Table 9.1, and the probability
distributions of the basic random variables are given in Table 9.2. The annual probability of
yield in this case is 3.0 x 10-8 per joint. This probability was calculated using Monte Carlo
simulation. It is defined on an annual basis because the load was defined as an annual maximum.
The probability distributions of the yield strength and wall thickness are defined for a randomly
selected joint of pipe to obtain the probability of failure per joint.
Given that there are approximately 80 pipe joints per km, and assuming independence between
failures of these joints, the failure rate equals 80 x 3.0 x 10-8 = 2.4 x 10-6 per km-year. Since the
pressure distribution is based on the maximum allowable operating pressure, this solution is valid
immediately downstream of a compressor station for a pipeline operating at capacity.
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Parameter
Specified pipe diameter (16”)
Maximum allowable operating pressure (1000 psi)
Specified minimum tensile strength (X70)
Nominal wall thickness
Symbol
Unit
Value
D
MAOP
SMTS
tN
mm
MPa
MPa
mm
406
6.895
566
4.03
Table 9.1 Deterministic Pipeline Parameters for Example
Parameter
Symbol
Unit
Distribution Type
Mean
COV
Tensile strength
Wall thickness
Maximum annual pressure
Model error factor
σu
MPa
mm
MPa
-
Normal
Normal
Gumbel
Normal
1.1 SMTS
tN
1.03 MAOP
1.0
0.04
0.25 / tN
.013
.04
t
P
c
Table 9.2 Probability Distributions of Basic Random Variables for Example
9.2.3 Special Methodology for Seismic Limit States
9.2.3.1 Failure Rate
Although a seismic event typically originates from a specific seismic source, the resulting ground
movement hazard could extend to a large area around the source. Since many seismic sources
could exist in an active area, events from several sources may contribute to the seismic hazard at
a particular location. Based on this, the failure probability due to seismic events can be
calculated from:
λ f = ∑ ωk × p fk
[9.8]
all k
where the summation is for all seismic sources (k= 1, 2, 3,….), pfk is the conditional probability
of failure given a seismic event at source k and ωk is the frequency of a seismic event from
source k. Equation [9.8] assumes that failures in individual seismic events are independent.
9.2.3.2 Conditional Failure Probability
The two types of seismic hazard considered for pipelines are fault displacement and ground
liquefaction (O’Rourke and Liu 1999). The structural impact of a seismic event on a pipeline is
modeled as a sequence of events representing:
1. Occurrence of the seismic hazard at the pipeline location. Depending on its magnitude and
source, any particular seismic event may or may not result in ground liquefaction or fault
displacement across the pipeline route. Applicability of the limit state is conditional on
occurrence of one of these hazards.
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2. Severity of the seismic hazard. Provided that the hazard occurs at a particular location,
failure will take place if the magnitude of the hazard is sufficiently large. For example, if
ground liquefaction occurs, failure will take place if the resulting Permanent Ground
Deformation (PGD) leads to excessive strains in the pipeline.
Both the probability of hazard occurrence given a seismic event, p0|w, and the probability of
failure given hazard occurrence, pf|w, are conditional on the characteristics of the seismic event
(e.g. magnitude, epicentral distance and peak ground acceleration). Assuming that W is a vector
denoting the characteristics of the seismic event, the failure probability, pf, given a seismic event
from a given source can be calculated as:
p f = ∫ Pf w P0 w fW ( w ) d w
[9.9]
where fW(w) is the joint probability density function of W. Although this equation is specific to a
given seismic source k (see Equation [9.8]), the subscript k is dropped for simplicity.
The probability of failure given the hazard, pf|w can be calculated using the same approach
discussed in Section 9.2.2.2 for non-seismic events. This gives:
p f |w = p[m = g ( x ) ≤ 0] =
∫ f( x|w ) d x
[9.10]
g ( x )≤ 0
where x is a vector of basic random variables in the limit state function (e.g. permanent ground
deformation, pipe diameter, pipe wall thickness, steel grade, internal pressure and soil strength)
and g is a limit state function representing the occurrence of excessive strains in the pipe due to
ground movement. An approach that can be used to develop such a function is described in
detail in (Xie et.al., 2004). Equation [9.10] is essentially the same as Equation [9.7], with the
exception that some of the basic variables in x (e.g. permanent ground deformation) are
conditional on W.
The interpretation of Equations [9.9] and [9.10] is as follows:
•
Calculate the probability of failure for a given seismic event (from a given source) with a
specific set of characteristics, W. This probability equals the product of the probability that
the seismic hazard (e.g. liquefaction) will occur and the probability of failure given
occurrence of the hazard. This calculation must take into account the fact that Pf|w and P0|w
are correlated as they are both conditional on W.
•
Calculate the total probability of failure as the sum of the failure probabilities for all possible
values of W, each weighted by the corresponding probability of W. The summation is
performed for all seismic event characteristics from the seismic source under consideration.
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The different terms in Equation [9.9] can be derived from seismic and geotechnical data using
empirical seismic assessment techniques. Details of these calculations are described in
Appendix E.
The solution approach of Equations [9.9] and [9.10] is an expanded version of the FORM/SORM
or simulation methods described in Section 9.2.2.2 that takes into account conditionality on the
seismic event characteristics, W, and the resulting correlation between the probability of hazard
occurrence, P0|w, and the probability of failure given the hazard, Pf|w,. Possible solution
approaches include:
•
Monte Carlo simulation. The Monte Carlo approach is well suited to this problem because of
its flexibility in addressing parameter dependencies and correlations. The simulation would
generate specific seismic events and track their consequences through hazard occurrence,
hazard magnitude and impact on the pipeline. Similar to the discussion in Section 9.2.2.2,
the probability of failure can be estimated by the relative frequency of simulations leading to
a failure.
•
Nested first or second order reliability method. This method can address conditional
reliability problems such as the one in Equation [9.9]. The essence of the method is to map
the conditional limit state function onto an equivalent unconditional one that uses an
auxiliary random variable (Wen and Chen 1987). This converts the conditional reliability
problem into an unconditional one that can be efficiently solved using FORM or SORM.
•
Hybrid methods. It is possible to utilize a combination of methods, in which conditionality
on W is addressed using simulation methods and FORM or SORM are used to carry out the
basic unconditional probability calculation. In this approach, seismic events can be randomly
simulated from the appropriate probability distributions of seismic characteristics. Once a
seismic event is simulated, W becomes deterministic and the calculation reduces to a simple
reliability problem that can be solved using any of the methods discussed in Section 9.2.2.2.
The final probability of failure is then estimated by the average of the calculated values for
all simulated seismic events.
Advantages and limitations of these methods are the same as those of the underlying basic
methods discussed in Section 9.2.2.2.
9.3 Single Time-dependent Limit State
9.3.1 Introduction
As discussed in Section 4.2.1 (Table 4.1), time-dependent reliability arises in two cases:
•
Time-dependent increasing load and time-independent resistance. An example of this
case is a limit state representing excessive deformations under gradually increasing
imposed soil deformations.
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•
Time-dependent stationary load and decreasing resistance. Limit states that fall in this
category include failures caused by growing defects or deterioration mechanisms under
internal pressure or environmental loads. Examples include failure of corrosion and SCC
defects under internal pressure. To simplify the problem in this case the random load
process is conservatively replaced by a maximum credible sustained load value, which is
assumed to be applicable on a continuous basis (Figure 9.2). The maximum load is
assumed to be time-independent but can be modeled as a random variable. This
simplification avoids the need to solve a problem involving the crossing of a decreasing
resistance and a load represented as a random process.
Resistance
Maximum Load
Failure
Load
Time
Figure 9.2 Idealization of a Time-dependent Load as a Time-independent for Reliability Calculations
9.3.2 Failure Rate
The failure rate, λf(τ), in failures per km-year, is calculated from the following:
λ f (τ ) = ω × p f (τ )
[9.12]
where ω is the frequency of potential failure locations per km, pf(τ) is the conditional probability
of failure for a randomly selected failure location in failures per location per year and τ is time.
For corrosion and other types of defects, for instance, ω represents the average number of defects
per km and pf(τ) is the conditional annual probability of failure at a randomly selected defect.
Both the failure rate and conditional failure probability are defined as functions of time.
Equation [9.12] assumes that failures at individual defects or loading locations are independent
events. This implies that all the random variables affecting failure at a given defect or location
(e.g. yield strength, defect dimensions and growth rates) are independent. It is recognized that
this assumption is not strictly valid in all cases; for example, correlation would exist between the
yield strength values at defects located within the same pipe joint. System reliability techniques
(e.g., Thoft-Christensen and Murotsu 1986) show that the assumption of independence gives an
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upper bound of the failure probability. Therefore, in the absence of sufficient information to
characterize possible correlations, the conservative assumption of independence may be adopted.
9.3.3 Conditional Failure Probability
Although the discussion in this section is based on the case of a time-independent load and
decreasing resistance (see Figure 9.2), the results are equally applicable to the case of increasing
load and fixed resistance. Figure 9.2 shows that failure occurs at a given location when sufficient
time has elapsed for the resistance to drop below the load. The probability that failure will occur
before time, τ, has elapsed is equal to the probability that the time to failure is less than τ, which
(by definition) is equal to the cumulative probability distribution of the time to failure. The
cumulative probability of τ can be expressed as:
F (τ ) = p[r (τ ) ≤ l ] = p[m(τ ) = r (τ ) − l ≤ 0]
[9.13]
where l is the load, r(τ) is the resistance at time τ, and m(τ) is the safety margin. Substituting the
basic random variables for r and l in Equation [9.13] leads to (see the derivation of Equation
[9.7] for details):
F (τ ) = p[ g ( x ,τ ) ≤ 0]
[9.14]
The cumulative probability distribution of the time to failure F(τ), can be used to calculate the
conditional probability of failure during a specific time interval (τ1 to τ2) using the following
relationship (Madsen et al. 1986, page 287):
p fc (τ 1 , τ 2 ) = p (τ 1 < τ < τ 2 | τ > τ 1 ) =
F (τ 2 ) − F (τ 1 )
1 − F (τ 1 )
[9.15]
Equation [9.15] represents the probability of failure between τ1 and τ2, conditional on failure not
occurring before τ1. It calculates this probability as the probability of failure before the end of
the interval less the probability of failure before the beginning of the interval, all divided by the
probability that failure will not occur before the interval begins.
Equation [9.15] can be used to calculate the required probabilities as follows:
•
Cumulative probability of failure before time τ’ (conditional on the fact that the pipeline is
safe at time τ = 0) can be obtained by setting τ1 = 0 and τ2 = τ’, in Equation [9.15] leading to:
p f (0, τ ′) = p(0 < τ < τ ' | τ > 0) =
•
F (τ ' ) − F (0)
1 − F (0)
[9.16]
Annual probability of failure in year τ’ (conditional on the fact that the pipeline is safe at
time τ = 0) can be calculated from:
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p f (τ ′) = p (τ ′ − 1 < τ < τ ′ | τ > 0)=
F (τ ′) − F (τ ′ − 1)
1 − F (0)
[9.17]
Equation [9.17] shows that the final annual probability of failure per defect or loading location is
a direct function of the cumulative probability distribution of the time to failure, F(τ). Since
failure occurs when g ( x ,τ ) = 0 (see Equation 9.14]), the time to failure can be expressed as:
τ = g ′( x )
[9.18]
Using Equation [9.7] and recalling that F(τ) is defined as the probability that time to failure will
have a value less than or equal to τ, F(τ) can be calculated as:
F (τ ) = p[ g ′( x ) ≤ τ ] =
∫ f( x )d x
[9.19]
g ′( x ) ≤τ
Equation [9.19] can be solved for any value of τ using FORM, SORM or simulation techniques
as discussed in Section 9.2.2.2. To obtain the complete cumulative distribution of τ, repeated
solutions at different τ values are required.
9.3.4 Impact of Rehabilitation
9.3.4.1 Approach
Rehabilitation improves reliability by reducing the number of growing damage sites (or defects).
Targeted rehabilitation methods such as hydrostatic testing or in-line inspection coupled with
appropriate repairs, also improve reliability by reducing the percentage of large damage features
in the population. For time-dependent failure causes, rehabilitation may involve a hydrostatic
test that eliminates defects with a pressure capacity lower than the test pressure. It may also
involve excavation and repair of selected defects based on the results of an above ground survey
or in-line inspection. The repair may be limited to methods that prevent further growth of the
damage feature such as a coating repair for corrosion defect or slope stabilization for a pipe
segment on a moving slope. In addition, pipe repairs, such as sleeving a corrosion defect or
stress relieving pipe segments on a moving slope, may also be necessary.
When an imperfect inspection method is used to detect the damage sites, the degree of
improvement depends on the accuracy of the inspection and the degree of conservatism built into
the criteria used to excavate and repair damage sites (or defects). In this section, the term
“defect” is used in a general sense to represent any detectable damage feature. This includes
actual defects such as corrosion and SCC, as well as other damage features such as excessive
curvature or ovality due to ground movement.
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Size of Existing
Defects
Inspection
Size of Detected
Defects
Size of Undetected
Defects
Inspection
Measurement
Error
Measured Size of
Detected Defects
Excavation
Criterion
Size of
Excavated
Defects
Size of
Unexcavated
Defects
Repair Criterion
Size of
Remaining
Defects
Size of
Unrepaired
Defects
Size of Repaired
Defects
Figure 9.3 Illustration of the Rehabilitation Process
Figure 9.3 gives a conceptual model of the inspection process. Before inspection, the pipeline
will have a certain defect population. The severity of damage is characterized by the probability
distribution of a set of attributes, a, defining the geometry of a randomly selected defect. For a
corrosion defect the set of attributes may include defect depth, length and width, whereas for
lateral ground movement it may consist of pipe curvature. The probability distribution of a at the
time of inspection can be estimated based on inspection results and/or defect growth rates.
As shown in Figure 9.3, the inspection and repair process acts as a filter that removes defects
above a certain size. For each detected defect an indication of severity will typically be available
from the inspection. Defects that exceed certain severity criteria will be excavated and repaired,
thereby being eliminated from the defect population. The defect population after rehabilitation
will consist of undetected defects and detected defects that are not repaired. Details of the
various steps involved in this process are given in the following sections.
Once the probability distribution of the defect size after the rehabilitation is calculated, it can be
used in the model described in Section 9.3.3 to calculate the updated conditional probability of
failure.
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9.3.4.2 Detection Capability
The detection capability of any inspection tool can be represented by the probability of detection,
pd, which may be modelled as:
p d = h( a , q )
[9.20]
where q is a set of constants defining tool accuracy. Equation [9.20] acknowledges that the
probability of detection is a function of defect attributes, a, (e.g. deeper corrosion defects are
more likely to be detected by a magnetic flux tool).
The form of Equation [9.20] can be defined for a specific inspection tool based on vendor
specifications. The inspection divides the population of original defects into two separate
populations: one including defects that are detected and the other including defects that are not
detected. The size distributions of detected and undetected defects may be different because
there is usually a higher chance of detecting a larger defect, and therefore defects that are
detected are more likely to be large than defects that are not detected.
Example: For corrosion defects Rodriguez and Provan (1989) suggested an exponential
probability-of-detection relationship of the form:
pd = 1 − e − qa
[9.21]
where pd is the probability of detection for a defect of depth a, and q is a constant that
determines the overall detection power of the tool. The constant q can be determined from tool
specifications, which will typically give the defect depth for a given probability of detection,
POD (Shell International 1998). This information defines corresponding values of a and pd,
which can be substituted in Equation [9.21] to calculate q. Figure 9.4 gives a number of
detection probability curves with different values of POD for a defect 1 mm deep.
Probability Of Detection (POD)
1
0.8
0.6
POD of 1 mm defect
0.4
95%
90%
0.2
80%
0
0
0.5
1
1.5
2
Defect Depth (mm)
Figure 9.4 Probability of Detection as a Function of Defect Depth
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9.3.4.3 Sizing Accuracy
Defect attributes estimated from inspection data are subject to random measurement errors
because of the accuracy limitations of inspection tools. Because of this, the measured size of a
defect will generally differ from its actual size. Given the measurement, the actual defect size
can be estimated by subtracting the measurement error from the measured size. The probability
distribution of measurement error may be determined from tool specifications as illustrated in the
following example.
Example: In-line inspection tool accuracy is characterized by the probability, pe, that the
measurement error will fall within certain bounds, emin and emax (Specifications and
Requirements for Intelligent Pig Inspection of Pipelines, 1998). Consider a tool that estimates
corrosion defect depth within an error band of -10% to +10% of the pipe wall thickness, with a
probability of 0.90. This information can be used to calculate the mean value, µe, and standard
deviation, σe, of the measurement error, e, for any distribution type. For a Normal distribution,
which is commonly used for measurement errors (Dally et al. 1983), the mean and standard
deviation can be calculated from (see Figure 9.5):
µ e = (emin + emax ) / 2
[9.22]
σ e = (emax − µ e ) /[Φ −1 (
1 + pe
)]
2
[9.23]
where Φ −1 is the inverse standard Normal distribution function (obtained from Normal
distribution tables). In this example, Equations [9.22] and [9.23] give µe = 0 and σe = 6% of the
wall thickness. It is noted that measurement error need not be symmetric around the measured
value (e.g. emin = -10% and emax = +15%).
Mean error, µe
Probability Density
Probability that error
is within band
emin
emax
Error band
-20
-15
-10
-5
0
5
10
15
20
Measurement Error, e (% wt)
Figure 9.5 Illustration of Measurement Error Band and Corresponding Probability
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9.3.4.4 Defect Excavation and Repair
The decision to excavate a defect is based on the measured values of the defect attributes
estimated by the inspection tool. It can be based on a criterion of the form
he ( a m ) < he *
[9.24]
where am is a vector of the measured value of the set of defect attributes and he* is the threshold
value for excavation.
Example: A low-resolution in-line inspection tool for metal loss corrosion provides an estimate
of the defect depth. Therefore, the excavation threshold, he* is defined as a minimum allowable
remaining wall. The excavation criterion is:
he (d m ) = (t − d m ) / t ≤ he *
[9.25]
where t is the nominal wall thickness and dm is the measured maximum defect depth. Typical
values of he* are in the order of 50%.
In some cases excavation of a defect implies that it will be eliminated as a potential threat
(regardless of whether or not a pipe repair by a sleeve or cut-out replacement is carried out). For
SCC, for example, operators will grind defects to a safe depth and re-coat the pipe to ensure that
further growth does not occur.
Once a defect is excavated, there are two possible scenarios:
1. Excavation provides access for exact defect measurement. In this case the decision on
further repairs of the pipe (i.e. by sleeving or cut-out-replacement), will be based on the
actual defect attribute values as obtained from the in situ measurement. For external
corrosion, for example, an excavation provides an opportunity to measure defect attributes in
situ, and this means that even if the excavation decision is based on a low resolution tool
(i.e. depth-based), the repair decision can be based on depth and length measurements
(i.e. pressure-based). In this case, the measurement error is negligible and therefore the
actual defect attributes, a, are assumed to be known. The repair criterion takes the form of:
hr ( a ) < hr *
[9.26]
Example: The repair criterion for a corrosion defect may be defined as the ratio between the
pressure resistance at the defect and the Maximum Allowable Operating Pressure (MAOP).
The repair criterion is, therefore, defined on the basis of a model that calculates the pressure
resistance at a corrosion defect (see Section 7.6.3). It is given by:
 2.3 t σ y 


 1− d / t 
−0.15+ µ e 2 σ y  / MAOP < hr *
hr (d , l )= 
1.154
 D 

 1 − d / mt 

[9.27]
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where MAOP is the maximum allowable operating pressure and all other parameters are the
nominal values of the parameters with the same notation in Section 7.6.3.
2. Excavation provides no additional access to the defect. This is applicable to internal
corrosion defects for example. In this case all excavated defects will be repaired and a
separate repair criterion is not required.
9.3.4.5 Failure Rate Calculation
To calculate the failure rate taking into account the impact of rehabilitation, the adjusted defect
population (i.e. including only defects that are not repaired) can be used in the approach
described in Section 9.3.3 to calculate the annual conditional failure probability after
rehabilitation. The result can then be used to calculate the failure rate as described in Section
9.3.2.
The characteristics of the population of remaining defects can be determined using the Monte
Carlo simulation process. For rehabilitation based on inspection and repair, this involves
generating individual defects, checking whether they are found and eliminated at the time of a
given inspection, and using the remaining defects to determine the probability distribution of
defect characteristics. For hydrostatic testing, a similar process can be used with the exception
that a defect is eliminated if its failure pressure is lower than the test pressure.
Bayesian updating can also be used to define the impact of rehabilitation on reliability. For
example, knowledge that the pipeline has withstood a hydrostatic test without failure
demonstrates that the pipeline is free from critical defects and increases confidence in its
reliability. Similarly, an inspection provides information regarding defect populations that can
be used to upgrade or downgrade prior reliability estimates.
Reliability updating using Bayesian techniques is a standard problem that is described in many
references (Madsen 1986, Section 9.3.3, and Madsen 1987). Bayesian techniques assume that a
prior probability assignment is modified based on new information. For example, a prior
distribution of defect sizes and numbers may be assumed, but once an inspection is carried out,
this distribution can be updated based on the inspection results. Similarly, a certain probability
of failure can be initially calculated as p[ g ( x ) < 0] based on a set of prior probability
distributions of x, but if the pipeline is hydrostatically tested without failures, the probability can
be updated by the knowledge that parameter combinations leading to failure in the test are not
possible.
Bayesian updating can be very useful in accounting for the impact of maintenance. However, the
key issue in applying it relates to the definition of prior distributions and the weighting that
should be assigned to new information versus prior assumptions. For example, estimating
reliability with respect to corrosion requires an estimate of the defect population in the pipeline.
Prior to an inspection, the distribution of defect sizes would be based on experience with similar
pipelines that have been inspected. Once an inspection of the specific line under consideration is
carried out, and given the accuracy of current metal loss inspection tools, reliable information on
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the actual defects in the pipeline would be available. It is likely in this situation that little weight
would be assigned to the prior defect distributions, and therefore, instead of applying Bayesian
updating, the prior distribution would be discarded and the inspection data used directly in
calculating an updated reliability.
9.3.5 Example
The failure probability for the limit state function representing burst at a corrosion defect is used
in this section to illustrate the results of the models described in Sections 9.3.1 through 9.3.4. A
Class 1, 36” (914 mm), X70 (SMYS = 483 MPa) pipeline, with a maximum operating pressure of
7000 kPa is considered. The nominal wall thickness calculated using a design factor of 0.72 is
9.16 mm. The limit state function used is defined in Appendix A. It is assumed that the pipeline
has one defect per km and that the probability distributions of the basic random variables are as
given in Table 9.3. The maximum pressure distribution (which represents the maximum credible
sustained load as defined in Section 9.3.1) is assumed to correspond to the maximum operating
pressure, implying that the calculated failure rates are applicable immediately downstream from
a compressor station. Figure 9.6 shows the failure rate by burst as a function of time. This was
calculated using the models described in Section 9.3.2 and 9.3.3.
Basic Variable
Average defect depth ( mm )
Distribution Type
Mean Value
Standard
Deviation
Weibull
2
1
Lognormal
60
60
Shifted Lognormal
2.08
1.06
Defect depth gowth rate ( mm/yr )
Weibull
0.1
0.05
Defect length growth rate ( mm/yr )
Lognormal
1
0.5
Maximum pressure ( kPa )
Gumbel
7350
118
Wall thickness ( mm )
Normal
9.16
0.25
Yield strength ( MPa )
Normal
531
18.6
Model error factor 1
Deterministic
1.15
0
Model error factor 2
Normal
-0.003
0.0027
Defect length ( mm )
Max. to avg. defect depth ratio
Table 9.3 Basic Variable Distributions Used in the Example
To assess the impact of immediate rehabilitation on the failure rate, a high-resolution in-line
inspection tool is assumed. The tool is estimated to have a 95% probability of detecting a defect
with an average depth of 1 mm. The probability distributions of the measurement error are
assumed to be Normal with a mean of 0 and standard deviation of 0.75 mm for the average
defect depth, and a mean of 0 and standard deviation of 7.5 mm for the defect length (see
Section 9.3.4.3). These are based on an 80% probability error band of approximately ±10% t for
the defect depth and ±10 mm for defect length. The excavation criterion is defined as a pressure
safety factor in the form of Equation [9.27]. All excavated defects are assumed to be eliminated
using an appropriate repair method.
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Failure Rate (per km.year)
1.00E-02
1.00E-03
1.00E-04
1.00E-05
1
5
9
13
17
21
25
29
Year
Figure 9.6 Failure Rate by Burst as a Function of Time
Figure 9.7 shows the impact of inspection on the probability distribution of the average defect
depth for two repair thresholds of 1.75 and 1.85. These distributions were calculated using the
approach described in Section 9.3.4. The repair thresholds of 1.75 and 1.85 are higher than usual
because the criterion used here (Equation [9.27]) is calibrated to eliminate the conservatism that
is typically built into repair criteria in current use. For example, an overall safety factor (which
is equivalent to the repair threshold defined here) of up to 1.39 is used in DNV recommended
practice RP-F101 (DNV 1999). In addition to this overall factor, DNV specifies that the failure
pressure should be calculated using defect depth and length values that are adjusted upward to
account for measurement error. This means that the actual safety factor built into the DNV
criterion is higher than the apparent value.
Probability density
1
Before inspection
0.8
Repair threshold = 1.75
0.6
Repair threshold = 1.85
0.4
0.2
0
0
1
2
3
4
5
6
7
Average corrosion defect depth (mm)
Figure 9.7 Impact of Rehabilitation on the Average Defect Depth Distribution
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Figure 9.7 shows that the defect depth distribution is shifted toward smaller defects with the
more conservative repair criterion (higher required safety factor) resulting in a larger shift.
Because the inspection and repair process finds and eliminates larger defects, it might be
expected that the results would be a sudden truncation of the original defect depth distribution.
This is not the case because the truncation is based on pressure resistance, which is a function of
both defect depth and defect length. Therefore, depending on the defect length, the truncation
may occur at a range of defect depths, resulting in a smooth transition rather than a sudden
truncation. The transition is made even smoother because the process is affected by a relatively
large measurement error.
Figure 9.8 shows the failure rates after rehabilitation as calculated using the updated defect depth
(Figure 9.7) and length (not shown) distributions.
Failure Rate (per km.year)
1.00E-03
1.00E-04
1.00E-05
1.00E-06
1.00E-07
Original Pipeline
Repair threshold = 1.75
1.00E-08
Repair threshold = 1.85
1.00E-09
1
3
5
7
9
11
Year
Figure 9.8 Impact of Rehabilitation on the Failure Rate for Burst
Finally, Figure 9.9 shows that the methodology can be used to determine the impact of future
planned maintenance on reliability. This figure corresponds to one rehabilitation event carried
out after ten years. The inspection is assumed to use the same inspection tool and repair criteria
described in the previous paragraph. The figure shows that this rehabilitation plan maintains the
failure rate below 5x10-4 per km-year, which means that it maintains a minimum reliability level
of 1-5x10-4 per km-year for a period of at least 24 years. This type of analysis can be used to
develop maintenance plans that meet a specified target reliability level.
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Failure Rate (per km.year)
1.00E-02
1.00E-03
1.00E-04
1.00E-05
1.00E-06
Rehabilitation in year 10
1.00E-07
Original Pipeline
1.00E-08
1
5
9
13
17
21
Year
Figure 9.9 Impact of a Specific Rehabilitation Plan on the Future Failure Rate for Burst
9.4 Multiple Limit States
9.4.1 Introduction
Multiple limit states need to be considered in the following cases:
•
Multiple failure mechanisms. An example of this is an immediate failure due to equipment
impact incidents. In this example, there are at least two credible failure mechanisms with
distinct limit state functions. The first is puncture under the excavator tooth and the second is
failure of a resulting gouged dent after removal of the excavator load (see Appendix A for
details). The mechanism that will cause failure in a given impact is the one with the lower
resistance. In this case, the probability of failure is the joint probability of violating either of
the individual limit states. For n limit state functions, g1, g2,……, gn, this can be written as
(where the symbol U means AND/OR):
p f = p[( g1 ≤ 0) U ( g 2 ≤ 0) U .......... U ( g n ≤ 0)]
•
[9.28]
Distinguishing between different failure modes. Pipelines typically fail in one of three
distinct modes, namely small leaks, large leaks and ruptures. Distinguishing between these
failure modes will involve analyzing multiple limit states. In the case of corrosion, for
example, failure may occur by either one of two distinct limit states representing a pinhole
leak at the deepest part of the corrosion defect, or by burst of the corrosion feature under
internal pressure. Further, in the case of burst an additional limit state function will be
required to determine whether the initial large leak will lead to a rupture (see Appendix A for
the specific limit state functions related to failure by corrosion). The probability of failure by
a given mode, pfm, can be written as:
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p fm = p( fm | f ) × p f
[9.29]
where p(fm | f) is the probability of failure mode, fm given failure and pf is the probability of
failure. By definition of conditional probabilities, p( fm | f ) × p f is equal to p( f | fm) × p fm
(see Appendix D), which means that pfm is also given by:
p fm = p ( f | fm) × p fm
[9.30]
The individual limit states and the model according to which they interact depend on the failure
cause being considered. Three examples are presented in Sections 9.4.2 through 9.4.4 to
demonstrate the modeling approach.
In principle, Equations [9.28] and [9.29] can be evaluated using any of the methods discussed in
Section 9.2.1 (i.e. simulation, fast simulation, or first and second order reliability methods) in
conjunction with system reliability techniques (see Madsen et al. 1986 and Thoft-Christensen
and Murotsu 1986 for detailed information). However, FORM, SORM and fast simulation
solutions have not been developed for pipeline problems. Because of the constraints applicable
to these methods (see Section 9.2.2.2), such solutions are likely to require significant effort and
involve many simplifications. Because of this, the discussion in Sections 9.4.2 to 9.4.4 focuses
on Monte Carlo simulation.
9.4.2 Example 1: Yielding and Burst of Defect-free Pipe
The limit states representing yielding and burst of defect-free pipe are shown in Figure 9.10 in
which g1 and g2 represent the limit state functions for yielding and burst, respectively (see
Appendix A for details of these limit state functions). The failure probabilities can be calculated
from:
p y = p ( g1 ≤ 0)
[9.31]
p fb = pb = p( g 2 ≤ 0)
[9.32]
p fy = p( g1 ≤ 0) − p( g 2 ≤ 0)
[9.33]
where py is the probability of yielding, pb is the probability of burst, pfb is the probability of
failure by burst (the final outcome is burst) and pfy is the probability that the pipe will yield but
not burst (i.e. the final outcome is failure by yielding). In the case of burst, pfb = pb because if the
burst limit state is exceeded, the final outcome will be burst. Equation [9.33] acknowledges that
yielding will always occur before burst and therefore for yielding to be the final outcome, the
yield limit state must be exceeded but not the burst limit state. Equations [9.32] and [9.33] can
be used to calculate the required probabilities directly from the individual limit state exceedance
probabilities, which can be calculated using the methods described in Section 9.2.2. This simple
solution is possible because the burst domain is a subset of the yielding domain.
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Yield
PD
Burst
Safe
g1 = 0
g2 = 0
Yield
No Yield
Burst
No Burst
2 σy t
Figure 9.10 Limit States for Yielding and Burst Under Internal Pressure
9.4.3 Example 2: Equipment Impact
The limit states required to calculate the probabilities of failure by leak or rupture due to
equipment impact are illustrated in Figure 9.11. For illustration purposes the figure is based on
two random variables representing load and resistance, but in reality, there will be a number of
random variables representing impact load, internal pressure, wall thickness, yield strength,
fracture toughness, gouge geometry and model error.
Leak
Load
g3 = 0
Rupture
Gouged Dent
Failure
Safe
Large leak
g2 = 0
Rupture
No Gouged
Dent Failure
Puncture
g1 = 0
No puncture
Resistance
Figure 9.11 Limit States for Different Failure Modes Associated with Equipment Impact
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The following three limit states are assumed (see Appendix A for detailed descriptions of the
individual limit state functions):
•
Puncture (g1). This failure mechanism occurs if the load imposed by the excavator tooth
exceeds the combined shear and membrane resistance of the pipe wall.
•
Gouged Dent (g2). This failure mechanism occurs if the load is not sufficient to cause
puncture, but is large enough to cause a gouged dent that fails under pressure after removal of
the load.
•
Fracture initiation (g3). A puncture or gouged dent failure will result initially in a leak. If the
length of the resulting breach is large enough, unstable axial growth could occur leading to a
rupture.
Figure 9.11 illustrates combinations of load and resistance leading to a safe pipeline (no failure)
and to failure by a leak or rupture. The safe area is the area on the safe side of both g1 and g2.
The failure domain is the area on the failure side of either or both of g1 and g2. The failure
domain is split into two areas representing conditions that lead to rupture or leak.
Calculating the failure rate for this case is a time-independent reliability problem that requires
solution of Equation [9.28]. The problem can be solved using a modified version of the model
and simulation scheme described in Section 9.2.2.2, in which the margin of safety is expressed
as:
m = min[ g1 ( x1 ) , g 2 ( x 2 ) ]
[9.34]
where x1 and x2 are the two sets of basic variables for g1 and g2 (which will include some
common parameters). Equation [9.34] combines the two failure mechanisms of puncture and
gouged dent as a weakest link model. The proportion of simulations resulting in a failure
according to Equation [9.34] is an estimate of the total probability of failure by leak or rupture,
pf. The value of g3(x3) can be used to distinguish between leaks and ruptures for each simulated
failure. The probabilities of leak and rupture given failure, p(fm|f), can be estimated by the
corresponding proportion of simulated failures and used in Equation [9.29] to calculate the
probabilities of failure by leak or rupture.
9.4.4 Example 3: Corrosion
The limit states required to calculate the probabilities of failure by small leak, large leak or
rupture due to corrosion are illustrated in Figure 9.12. For illustration purposes the figure is
based on two random variables representing defect depth and length, but in reality, there will be a
number of random variables representing internal pressure, wall thickness, yield strength, defect
dimensions and model error.
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Fail
Rupture
Defect length
g2 = 0
g3 = 0
Large Leak
No Fail
Small leak
Rupture
Large leak
Large leak
Rupture
Small leak
g1 = 0
Leak
No leak
Wall
thickness
Defect depth
Figure 9.12 Limit States for Different Failure Modes Associated with Corrosion
The following three limit states are applicable (see Appendix A for detailed descriptions of the
individual limit state functions):
•
Small Leak (g1). This failure mode occurs if the maximum defect depth exceeds the wall
thickness. As indicated in the figure, small leaks only occur for defects that are short enough
to corrode through the wall without violating the burst criterion.
•
Burst (g2). This failure mode occurs if the internal pressure exceeds the burst resistance at
the corrosion defect. It is a function of both defect depth and length. As indicated in the
figure, burst occurs only for defects that are long enough to violate the burst criterion before
corroding through the wall.
•
Rupture (g3). Burst of a corrosion defect will result initially in a leak. If the length of the
resulting breach is large enough, the axial unstable growth could occur leading to a rupture.
Calculating the probabilities of failure for corrosion is a time-dependent reliability problem. An
algorithm to solve this problem using the probability distribution of the time to failure, τ, was
described in Section 9.3.3. Assuming that all defects begin with zero depth and length, the time
to failure is defined as the time required for a defect to cross the limit state surface from the safe
domain into the failure domain. Figure 9.12 shows that the boundary between the safe and
failure domains is divided into three zones defining small leak, large leak and rupture. A defect
that crosses this boundary along the small leak limit state surface (g1 = 0) will fail as a small leak.
A defect that crosses along the burst limit state surface (g2 = 0) will be a large leak if it crosses
on the large leak side of g3 and a rupture if it crosses on the rupture side of g3.
The probability of each failure mode can be calculated by a modified version of the model and
algorithm described in Section 9.3.3 for a single time-dependent limit state function. When
Equation [9.18] is solved to calculate the time to failure, τ, the failure mode should also be
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identified. The simulation data can then be analyzed separately for each failure mode resulting in
the failure rate by mode. This gives the probability of failure for each mode, p(f | fm). The
probability of failure for each failure mode, pfm, can be estimated by the proportion of
simulations leading to that failure mode. Equation [9.30] can then be used to calculate the
conditional failure probability. In this approach, the impact of maintenance can be included by
applying the model described in Section 9.3.4 for each individual failure mode.
9.5 Reliability Calculation Tools
There are a number of commercial software packages that can be used to calculate reliability for
a user-defined limit state function and user-selected probability distributions of the basic
variables. Some of the leading packages are as follows:
•
General-purpose structural reliability tools. STRUREL (www.strurel.com) is one of the most
widely used reliability calculation packages, with a focus on FORM, SORM and fast
simulation methods. The package is capable of solving time-dependent and multiple limit
(system reliability) state problems. Other software packages for FORM and SORM
calculations are summarized in Appendix F of Melchers (1999). The use of these packages
requires an understanding of the underlying techniques and experience in interpreting and
validating the results.
•
General-purpose Monte Carlo simulation packages. The most well known of these packages
are @RISK (www.palisade.com) and Crystal Ball (www.decisioneering.com). Both of these
are spreadsheet add-ons that are easy to use but tend to be slow. Because of this, their use in
reliability calculations may be limited to simple problems and exploratory analyses.
Simulation analyses can also be implemented in standard analysis packages such as
MATLAB, Mathcad or Excel. Each of these packages has its own limitations with respect to
speed, ease of implementation and the number of available distribution functions.
•
Pipeline-specific packages. PRISM is a specialized package for pipeline reliability
calculation that has been developed by C-FER (www.cfertech.com). This package uses the
Monte Carlo simulation approach, implemented as an interactive algorithm that allows the
user to determine the required number of simulations as the solution proceeds. PRISM
addresses both time-independent and time-dependent reliability problems, allowing the user
to define and link limit state functions. The program supports calculation of the impact of
maintenance and rehabilitation events applied at user-specified times, based on a
characterization of the accuracy of the inspection method and excavation/repair criteria used.
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10. EXAMPLE APPLICATIONS
10.1 Example 1 – New Pipeline Design
10.1.1 Introduction
The purpose of this example is to demonstrate the steps involved in applying RBDA to the
design of a new pipeline segment. The example is based on route information for an existing
pipeline. The design pressure, pipeline diameter and grade were specified, and the goal of the
design was to select a wall thickness profile and an appropriate maintenance plan for the
segment.
To simplify the example, the selected route does not involve major crossings, unstable slopes,
significant seismic activity, or frost/heave problems. Based on this, the design focused on a
limited number of limit states corresponding to corrosion, equipment impact, hydrostatic testing,
and restrained thermal expansion. The design considered all three types of limit states, namely,
ULS, LLS, and SLS.
10.1.2 Pipeline Information
A 4.5 km length (between Station 3420.35 km and Station 3424.85 km) of an NPS20 natural gas
pipeline is considered. The design pressure for the pipeline is 1400 psi and the selected steel
grade is API 5L X70 (SMYS = 482MPa). The pipeline will be Fusion Bonded Epoxy (FBE)
coated and cathodically protected. A 50-year design life is considered.
A database containing the structure information along the pipeline right of way is available.
Based on the database, the approach described in Section 5.3.2.4.1 was used to calculate the
population density at 25m intervals along the pipeline (see Figure 10.1). The pipeline was then
divided into three segments (denoted A, B and C in Figure 10.1) following the segmentation
criteria described in Section 5.3.2.4.1. The average population density was found to be 4.0, 1.54,
and 0.18 people per hectare for Segments A, B and C, respectively.
10.1.3 Applicable Limit States
The list of potentially applicable limit states is provided in Section 5.2.2 (Table 5.1). These limit
states were evaluated using the approach described in Chapter 6 in order to eliminate ones that
are not applicable or critical for the current pipeline segment. Table 10.1 summarizes the results
of this analysis. Limit states that are considered applicable are highlighted in the table. The
rationale for excluding the remaining limit states is explained in the last column of the table. The
majority of the exclusions were based on two considerations: either the load case corresponding
to the limit state is not applicable (e.g., ground movements or above ground span loads), or the
limit state can be addressed through other construction or operational practices (e.g. weld
inspection or proper cold bending procedures).
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Example Applications
5
4
Segment B
3
Segment C
Segment
A
2
1
0
3420
3421
3422
3423
3424
3425
Station (km)
Figure 10.1 Variation of the Population Density along the Right-of-Way
The limit state corresponding to local buckling under restrained thermal expansion is applicable;
however, it was thought at the outset that the thermal strains might be sufficiently low to justify
exclusion from the probabilistic analysis. To check this, a worst-case analysis was carried out as
described in Section 6.2. The longitudinal strain, ε L , due to combined internal pressure and
restrained thermal expansion can be calculated from:
ε L = νσ h / E s − α∆T
[10.1]
where ε L is tensile strain if positive and compressive strain if negative, v = 0.3 is Poisson’s ratio,
σ h =P(D-2t)/2t is the hoop stress due to internal pressure, t is the pipe wall thickness, α =
12×10-6/°C is the thermal expansion coefficient, Es =2×10-5MPa is the elastic modulus, and ∆T is
the difference between the maximum operating temperature and the tie-in temperature. A
conservative estimate of the maximum possible value of ∆T for this pipeline was determined to
be 30°C. Since the axial tension due to internal pressure is proportional to the hoop stress, the
worst-case scenario for local buckling corresponds to the minimum hoop stress, which results
from a large wall thickness and/or low internal pressure. The lower bound of the operating
pressure is assumed to be 60% of the design pressure, and Segment A would have the largest
wall thickness of 10.2 mm according to ASME B31.8. The corresponding longitudinal
compressive strain calculated from Equation [10.1] is 0.015%, which is only 1.5% of the
anticipated buckling strain limit of approximately 1%. This confirms that it is appropriate to
exclude buckling under thermal expansion from the list of limit states to be subjected to a
detailed probabilistic analysis.
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Load Case
Life Cycle Phase
Transportation
Limit State
Time
Dependent
Included in the
Analysis
SLS
No
No
1 Accidental impact
Denting / gouging
2 Cyclic bending
Fatigue crack growth
SLS
Yes
No
3 Stacking weight
Ovalization
SLS
No
No
4 Cold field bending
5 Bending during installation
Construction
Limit State
Type
6
Local Buckling
SLS
No
No
Plastic collapse
SLS
No
No
Local Buckling
SLS
No
No
No
Directional drilling tension and Girth weld tensile fracture
bending
Local buckling
7 Hydrostatic test
8 Internal pressure
9 Overburden and surface loads
No
No
No
Excessive plastic deformations
SLS
No
Yes
Burst of defect-free pipe
SLS
No
No
Burst at dent-gouge defect
SLS
No
No
Burst at seam weld defect
SLS
No
No
Excessive plastic deformations
SLS
No
No
Practically impossible
Burst at corrosion defect
ULS
Yes
Yes
Small leak at corrosion defect
LLS
Yes
Yes
External corrosion considered as a major failure cause for dry gas
pipelines
Burst at environmental crack (SCC)
ULS
Yes
No
Small Leak at environmental crack (SCC)
LLS
Yes
No
Burst of a manufacturing defect
ULS
Yes
No
Small leak of a manufacturing defect
LLS
Yes
No
Ductile fracture propagation
ULS
No
No
Burst of a weld defect
ULS
Yes
No
Small leak of a weld defect
LLS
Yes
No
Plastic collapse
SLS
No
No
SLS
No
No
SLS or ULS1
No
No
SLS or ULS1
No
No
SLS or ULS1
No
No
Local buckling
ULS
Yes
No
Girth weld tensile fracture
ULS
Yes
No
Gravity loads on above-ground
Local buckling
spans
Girth weld tensile fracture
Above ground span support
settlement
12 Wind on above-ground spans
Dynamic instability
Burst of crack by fatigue
13
Slope instability, ground
movement
Operation
Local buckling
Girth weld tensile fracture
Local buckling
14 Seismic loads
Girth weld tensile fracture
15 Restrained thermal expansion
16 Frost heave
Local buckling
21 Outside force
23 Sabotage
SLS or ULS1
Yes
No
ULS
Yes
No
SLS or ULS1
No
No
ULS
No
No
Only critical for large diameter and high pressure pipelines given
the level of notch toughness resulting from the current
manufacturing process
Not applicable to the pipeline - no major road or highway
crossings
Not applicable to buried pipelines
Not applicable to buried pipelines
Not applicable to buried pipelines
No unstable slopes on the pipeline route
Low seismicity area
No
Excluded by a deterministic check
No
No
Route does not cross areas prone to upheaval buckling
Local buckling
SLS or ULS1
Yes
No
ULS
Yes
No
SLS or ULS1
Yes
No
ULS
Yes
No
Excessive plastic deformation
SLS or ULS1
Yes
No
Local buckling
SLS or ULS1
Yes
No
ULS
Yes
No
Dynamic instability
SLS or ULS1
No
No
Formation of mechanism by yielding
SLS or ULS1
No
No
Local buckling
SLS or ULS1
No
No
Girth weld tensile fracture
20 Buoyancy
No
Manufacturing defects can be eliminated by proper welding and
inspection processes
No
Girth weld tensile fracture
19 River bottom erosion
No
Yes
FBE coating is not susceptible to SCC.
ULS
Local buckling
Loss of soil support (e.g.,
subsidence)
No
ULS
Practically impossible based on previous calculations
Defect failure during test will be repaired
SLS or ULS1
Girth weld tensile fracture
18
SLS or ULS1
Not applicable to the pipeline
Ensures coating integrity
Upheaval buckling
Girth weld tensile fracture
17 Thaw settlement
Addressed by proper bending method
SLS
Formation of mechanism by yielding
11
Addressed by proper handling during transportation
SLS
Ovalization
10
Rationale for Including or Excluding the Limit State
Route does not contain ice rich soils
The pipeline route does not cross discontinuous permafrost
Route not prone to subsidence
Pipeline does not cross any waterways
ULS
No
No
Floatation
SLS or ULS1
No
No
Pipeline does not cross any waterways
Denting
SLS
No
No
Gouged dents due to equipment impact are dominant
Puncture
ULS
No
Yes
Burst of a gouged dent
ULS
No
Yes
Small leak of a gouged dent
LLS
No
Yes
Rupture
ULS
No
No
Equipment impact on buried pipelines is a major failure cause
Assume that sabotage is not applicable to the pipeline
Table 10.1 Preliminary Applicable Limit States for Segment 1
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The final list of applicable limit states are summarized in Table 10.2.
No
Load Case
Limit State
Limit State Category
1
Hydrostatic test
Excessive plastic deformation
SLS
2
Equipment impact
Small leak at a gouged dent
LLS
3
Internal pressure
Small leaks at a corrosion defect
LLS
4
Internal pressure
Burst at a corrosion defect
ULS
5
Equipment impact
Puncture
ULS
6
Equipment impact
Burst of a gouged dent
ULS
Table 10.2 Final List of Applicable Limit States
10.1.4 Reliability Targets
As explained in Section 5.3.2, reliability targets for ULS are defined as a function of ρPD3,
where ρ is the population density, P is the maximum operating pressure and D is the diameter.
Since P and D are constant along the segment, the reliability targets depend only on ρ. To select
appropriate reliability targets, the pipeline is divided into a number of segments as indicated in
Section 5.1. Although the specific segmentation scheme used is a subjective choice, the intent is
to select the segments such that the variation in population density within each segment is as
small as possible. For this example, three segments were defined to capture high, intermediate,
and low population density values on the segment (see Figure 10.1). Segment boundaries,
lengths, and average population densities are given in Table 10.3. The ULS reliability targets
were selected for each section by using the corresponding average population density in
Equations [5.6] (Section 5.3.2.4). These targets are also given in Table 10.3.
Segment
Start
Station
End
Station
Length
(km)
Average
Population
Density
(people/hectare)
Target
Reliability
(per km-yr)
Allowable
Failure
Probability
(per km-yr)
A
3420+350
3420+850
0.5
4.0
1-1.00E-05
1.00E-05
B
3420+850
3422+700
1.85
1.54
1-2.61E-05
2.61E-05
C
3422+700
3424+850
2.15
0.18
1-2.23E-04
2.23E-04
Table 10.3 Pipeline Segments and Reliability Targets
Assuming 3 people per dwelling, the average population densities in Table 10.3 translate to the
equivalent of 85, 33 and 4 dwellings within the B31.8 location class assessment area of
1600×400 m. This means that according to B31.8, Segments A, B and C would be designated as
Classes 3, 2 and 1, respectively.
The reliability target is 1-10-2 for LLS (see Section 5.3.3) and 1-10-1 for SLS (see Section 5.3.4).
These targets are independent of the pipeline segment characteristics.
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10.1.5 Limit State Functions
The limit state functions required for the limit states in Table 10.2 were selected from those
described in Appendix A. These are as follows:
•
Excessive plastic deformation under hydrostatic test pressure. The purpose of this limit state
is to ensure that plastic hoop deformations under the test pressure do not exceed the strain
capacity of the coating in order to prevent coating damage and subsequent corrosion. Since
NACE RP0394-98 specifies a minimum tensile strain capacity of 1.3% for epoxy coating, a
limit of 1% plastic hoop strain represents a reasonable limit. A limit state function based on
initial yielding would be conservative, and this approach was adopted here for simplicity.
The limit state function used is described in Section A.2.1.
•
Small leak, large leak or rupture under equipment impact load. The limit state functions used
for this load case are described in Section A.3. They include individual limit state functions
for puncture, dent gouge and unstable axial growth, which interact according to the approach
described in Section 9.4.3. These limit state functions identify leaks and ruptures, but they
do not distinguish between small and large leaks, neither do they account for delayed failures
– these issues are addressed in Section 10.1.7.
•
Small leak, large leak or rupture due to corrosion. The limit state functions used for
corrosion limit states are described in Section A.4. They include individual limit state
functions for through-wall perforation, burst and unstable axial growth, which interact
according to the approach described in Section 9.4.4.
Model error factors and their probability distributions for these limit states are also defined in the
corresponding sections of Appendix A.
10.1.6 Probabilistic Characterizations of Input Parameters
The list of basic random variables required for this example can be derived from the limit state
function descriptions in Appendix A. Table 10.4 provides a summary of the probabilistic models
used, which were based on the information in Appendix B. Although corrosion is a timedependent limit state, the approach described in Section 9.3 was used to express the probability
of failure in terms of a number of time-independent corrosion growth parameters. Therefore, all
the parameters in Table 10.4 are defined by simple probability distributions. Pipe diameter was
not treated as a random variable based on previous experience indicating that randomness of this
parameter has a negligible influence on reliability.
10.1.7 Reliability Calculation
The approaches described in Chapter 9 were used to calculate the failure probability for the limit
states in Table 10.2. These calculations were carried out using the PRISM pipeline reliability
software (see Section 9.5).
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Parameter
Units
Wall thickness
mm
Yield strength
Initial corrosion defect size:
Average depth
Maximum length
Max. to avg. defect depth
ratio
Growth rate of significant
corrosion defects:
Average depth
Maximum length
Annual maximum operating
pressure
Notch toughness
Operating pressure
Excavator tooth length
Excavator weight
Gouge depth
Gouge length
MPa
1.0 x
Nominal
530
mm
mm
ratio
mm/year
mm/year
%
MAOP
Joules
%MAOP
mm
tonne
mm
mm
Mean
Standard
Deviation
Distribution
Type
0.25
Normal
18.6
Normal
0.005
30
2.08
0.0025
15
1.08
Weibull
Lognormal
Exponential
0.06
1.0
0.03
0.5
Weibull
Lognormal
99.3
3.4
Beta(1)
130
86.5
90
15.2
1.2
201
52
8.4
28.4
10.8
1.1
372
Lognormal
Beta(2)
Rectangular
Gamma
Weibull
Lognormal
(1) Lower bound = 80% MAOP, upper bound = 110% MAOP
(2) Lower bound = 60% MAOP, upper bound = 110% MAOP.
Table 10.4 Probability Distributions for Uncertain Limit State Input Parameters
As indicated in Chapter 9, failure probability calculations require an indication of the frequency
of occurrence of each limit state and a characterization of the associated maintenance approach.
In addition, historical information was in the case of equipment impact limit states to adjust the
results. Details of these aspects are discussed in the following:
Hydrostatic testing
Hydrostatic test pressures of 1.4 MAOP, 1.25 MAOP and 1.1 MAOP were applied for Segments
A, B and C (according to ASME B31.8 for Location Class 3, Class 2, and Class 1, Division 2
respectively). Assuming that yielding of each joint in a given segment is an independent event,
the frequency of application of the test pressure per km is equal to the number of joints per km.
This number was taken as 80 joints per km.
Equipment impact
The frequency of occurrence of equipment impact events was calculated from a fault tree model
developed by Chen and Nessim (1999). This model calculates the frequency of hits (per kmyear) from the frequency of excavation activities and the effectiveness of the damage prevention
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measures that are assumed to be in place. Activity rates and typical damage prevention practices
were used for the three segments (Table 10.5).
Parameter
Segment A
Segment B
Segment C
Pipe burial depth (m)
0.91
0.91
0.91
Mechanical protection
No
No
No
Activity rate (/km-yr)
0.1
0.06
0.02
Above ground alignment
markers
No
No
No
Buried alignment markers
Yes
Yes
Yes
Explicit signage
At selected
strategic
locations
At selected
strategic
locations
Closely
spaced and
highly visible
Dig notification requirement
Required but
not enforced
Required but
not enforced
Required and
enforced
Dig notification response
Locate and
mark with no
site
supervision
Locate and
mark with no
site
supervision
Locate and
mark with site
supervision
One-call system
Unified system
to min.
standard
Unified system
to min.
standard
Unified system
to min.
standard
Right-of-way indication
Intermittent or
variable
Intermittent or
variable
Intermittent or
variable
Public awareness level
Average
Average
Average
Surveillance method
Aerial
Aerial
Aerial
Surveillance interval
Monthly
Monthly
Monthly
Table 10.5 Parameters used in Calculating Equipment Impact Frequency
The methodology used for calculating the failure probability due to equipment impact only
considers immediate failures. To account for delayed failures, the calculated failure rate was
increased by the ratio of total failures to immediate failures. This ratio was determined to be
1.29 based on historical information (Kiefner et al. 2001). Further, the equipment impact failure
probability calculation model does not distinguish between small and large leaks. Based on
historical information (EGIG 2001), this split was assumed to be 1/3 small leaks and 2/3 large
leaks.
Corrosion
The frequency of significant corrosion defects was assumed to be 4 defects per km. Corrosion
maintenance was assumed to involve in-line inspection using a high-resolution metal loss tool
and will repair defects based on a required combination of minimum wall thickness and burst
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pressure safety factor. Accuracy characteristics of the assumed tool are given in Table 10.6 (see
Section 9.3.4).
Defect depth at 90% probability of detection
10% of wall thickness
80% confidence interval on maximum defect depth measurement
±10% of wall thickness
80% confidence interval on defect length measurement
±20mm
Table 10.6 Metal Loss Inspection Tool Accuracy Specifications
10.1.8 Design Process
As mentioned in Section 10.2.1, parameters that could be varied in this example to achieve the
reliability targets include the wall thickness, corrosion inspection intervals and repair criteria, and
equipment impact prevention measures. Since the number of parameter combinations that could
be attempted is very large, judgment was used to reduce the number of options. For this
example, mechanical damage prevention measures were not treated as decision variables because
the most effective measures are deeper burial and mechanical protection, both of which are
relatively expensive in comparison to minor increases in wall thickness. Further, variations in
corrosion maintenance were limited to changes in the inspection interval; repair criteria were not
treated as a decision variable because operators are unlikely to deviate from industry-accepted
criteria.
The design process was as follows:
1. An initial maintenance plan was defined based on the inspection intervals and repair criteria
in ASME B31.8S. The plan was uniform along the entire segment.
2. A wall thickness that satisfies the ULS reliability targets was selected for each of the three
segments. This was based on an iterative process of selecting a wall thickness and
calculating the lifetime reliability with respect to all ultimate limit states. Based on Equation
[5.11], the reliability was checked against the target using
 cor
ei 60,000
} PD 3 + {p RUcor (τ ) + p RUei } > RTU ,
1 − {p LL
(τ ) + p LL


0 < τ ≤ 50
[10.2]
where LL and RU denote large leak and rupture, cor and ei refer to corrosion and equipment
impact, τ is time in years, and RTU is the ULS reliability target. The process was repeated
until an acceptable wall thickness was found.
3. The reliability with respect to small leaks was calculated and compared to the LLS target for
each of the three segments. This check was based on the following equation
cor
ei
}> RTL ,
1 − {p SL
(τ ) + p SL
0 < τ ≤ 50
[10.3]
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where SL refers to small leaks and RTL is the LLS reliability target. If required, the inspection
intervals were varied until the LLS target was met.
4. The serviceability limit state relating to yielding under hydrostatic test pressure was checked
for all segments using the following equation
1 − pY > RTS
[10.4]
where pY is the probability of yielding under the hydrostatic test pressure and RTS is the SLS
reliability target. The wall thickness was changed as necessary to meet the SLS target.
This process ensures that the final design meets the reliability target requirements for all limit
state types, throughout the design life.
10.1.9 Results
Table 10.7 shows the wall thickness values required for the three pipeline segments. The table
also shows the equivalent design factor calculated using PD/(2⋅t⋅SMYS), where t is the wall
thickness calculated from the RBDA approach. The ASME B31.8 design factor is also given for
each segment for comparison. In-line inspections were required in years 10, 20, 30, 36, 41, and
46, and the criterion calls for a minimum remaining wall thickness of 50% and a minimum safety
factor of 1.39 on the failure pressure calculated using B31G (ASME B31G 1991).
Segment
Wall Thickness
(mm)
Equivalent
Design Factor
ASME B31.8
Design Factor
Governing Condition
A
9.0
0.565
0.5
ULS target
B
7.9
0.64
0.6
ULS target
C
6.1
0.83
0.72
SLS target
Table 10.7 Wall Thickness and Equivalent Design Factors
Figures 10.2 through 10.4 show the final calculated and target reliability levels associated with
ULS and LLS for segments A, B and C. Although the calculations typically produce the
probability of failure, the results are given in terms of reliability, which is calculated by
subtracting the failure probability from 1. The reliability levels with respect to the serviceability
limit state representing yielding under hydrostatic test pressure are shown in Table 10.8.
Segment
Reliability for Yielding under
Hydrostatic Test Pressure
Target Reliability Level
A
≈1
1-1E-1
B
≈1
1-1E-1
C
1-7.3E-2
1-1E-1
Table 10.8 Calculated Reliability for SLS Under Hydrostatic Test Pressure
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Segment A - ULS
Reliability (/km-yr)
1-1.0E-06
Section A Target
WT = 9.0mm
1-1.0E-05
1
6
11
16
21
26
31
36
41
46
51
46
51
Year
Segment A - LLS
1-1.0E-06
Reliability (/km-yr)
1-1.0E-05
1-1.0E-04
LLS-Target
1-1.0E-03
WT = 9.0mm
1-1.0E-02
1-1.0E-01
1
6
11
16
21
26
31
36
41
Year
Figure 10.2 Calculated Versus Target Reliability for Section A
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Segment B - ULS
Reliability (/km-yr)
1-1.0E-05
Section B Target
WT = 7.9mm
1-1.0E-04
1
6
11
16
21
26
31
36
41
46
51
Year
Segment B - LLS
1-1.0E-06
Reliability (/km-yr)
1-1.0E-05
1-1.0E-04
LLS-Target
1-1.0E-03
WT = 7.9mm
1-1.0E-02
1-1.0E-01
1
6
11
16
21
26
31
36
41
46
51
Year
Figure 10.3 Calculated versus Target Reliability for Section B
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Segment C - ULS
Reliability (/km-yr)
1-1.0E-05
1-1.0E-04
Section C Target
WT = 6.1mm
1-1.0E-03
1
6
11
16
21
26
31
36
41
46
51
46
51
Year
Segment C - LLS
1-1.0E-06
Reliability (/km-yr)
1-1.0E-05
1-1.0E-04
LLS-Target
WT = 6.1mm
1-1.0E-03
1-1.0E-02
1-1.0E-01
1
6
11
16
21
26
Year
31
36
41
Figure 10.4 Calculated Versus Target Reliability for Section C
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The following observations are made:
•
Reliability-based design results in reduced wall thickness for all segments of this pipeline.
The maximum wall thickness reduction is 15% for Segment C (lowest population density).
•
For Segments A and B, the wall thickness is governed by ULS targets. Equipment impact
dominates the failure probability for ULS in the early stages of the design life, whereas
corrosion makes a significant contribution later in the design life. The wall thickness
selected is slightly higher than would be required to meet the targets for equipment impact
loading, with the difference creating an allowance for decrease of reliability between
inspections later in the design life. The inspection interval during the later stages of the
pipeline life were governed by ULS related to corrosion.
•
For Segment C, the wall thickness is governed by the SLS target corresponding to excessive
yielding under hydrostatic test pressure. This condition resulted in capping the equivalent
design factor at a value of 0.83. As mentioned earlier, the limit state function used is
conservative because it corresponds to initial yielding rather than an acceptable amount of
plastic deformation. A more accurate limit state function would result in a further reduction
in wall thickness. This condition is expected to govern for thinner walled pipelines in areas
with a low population density, where the hoop stress due to test pressure approaches or
exceeds SMYS.
•
The wall thicknesses selected, combined with the B31.8S maintenance plans satisfied the
LLS targets, although small leak rates close to the targets occurred just before the planned
inspection in some cases.
It must be emphasized that all analyses for corrosion-related limit states are based on a
hypothetical defect population, which was defined based on experience from similar pipelines.
Once the first inspection is carried out, a more accurate definition of the defect population will be
available, and the maintenance plan will be adjusted accordingly using the same type of analysis
discussed here. The actual maintenance plan may be more aggressive if actual corrosion is worse
than hypothesized, or less aggressive if it is better than hypothesized. The purpose of
considering corrosion at the design stage is to demonstrate that sufficient reliability allowance is
made to manage corrosion based on operational information collected during operations.
10.1.10 Sensitivity to Pressure
The example discussed in Sections 10.1.2 through 10.1.9 was re-analyzed for a pressure of 900
psi instead of 1400 psi, and a steel grade of X60 instead of X70. The resulting wall thickness
values are summarized in Table 10.9 and the inspection/repair plan is given in Table 10.10.
In this case, Segments A and B were governed by ULS targets, while Segment C was governed
by minimum wall thickness requirements (from Table 3.1 of Nessim and Zhou 2005). RBDA
results in roughly the same wall thickness as ASME for Segments A and C, and in 10% thicker
walls for Segment B. The reason for this is that pressure containment requirements are met at a
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thinner wall that does not provide sufficient reliability with respect to equipment impact and
corrosion. This is consistent with the general finding that RBDA leads to cost reductions for
larger diameter, higher pressure pipelines in low population areas, and safety improvements for
smaller diameter lower pressure pipelines in more heavily populated areas. Table 10.10 indicates
that the first inspection is scheduled at Year 9 with slightly more stringent defect repair criteria
than the rest of the inspections. This is driven by the need to meet the LLS target for Segments B
and C within the first 20 years of the design life.
Segment
Wall Thickness
(mm)
Equivalent
Design Factor
ASME B31.8
Design Factor
Governing Condition
A
7.8
0.49
0.5
ULS target
B
7.0
0.54
0.6
ULS target
C
5.6
0.68
0.72
Minimum wall
thickness requirement
Table 10.9 Wall Thickness and Equivalent Design Factors
Inspection
Time
(year)
Defect Repair Criteria
Remaining wall thickness/nominal
wall thickness (%)
Calculated failure
pressure/MAOP
9
60
1.39
20
50
1.39
30
50
1.39
40
50
1.39
45
50
1.39
Table 10.10 Inspection Interval and Defect Repair Criteria
10.2 Example 2 – Class Upgrade Deferral
10.2.1 Introduction
The purpose of this example is to demonstrate the application of RBDA to the assessment of an
existing pipeline. The example considers a 1.57 km segment of an NPS30, X52 pipeline in
Canada. The pipeline has a wall thickness of 0.36 inches (9.1 mm) and an MAOP of 1000 psi.
This implies a utilization factor of 0.80, which is currently allowed by CSA Z662 (CSA 2003)
for Class 1 pipelines. The pipeline was inspected with a high-resolution inline metal loss tool in
2004 (see Table 10.6 for tool accuracy characteristics), but required repairs have not yet been
carried out.
The assessment is required because a single-family housing development is being built within the
location class assessment area for the segment. This development changes the location class for
the segment from Class 1 to Class 2, which changes the maximum utilization factor from 0.8 to
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0.72 (CSA 2003). This requirement can be met through a reduction in the operating pressure or
replacement with a thicker walled pipe.
Both the pressure reduction and replacement options are costly. The purpose of the assessment
carried out in this case is to determine whether adequate reliability can be demonstrated through
enhanced maintenance. If successful, the results can be used as a basis to apply to the regulator
for a deferral of the Class upgrade. Depending on the duration of the deferral, the analysis may
need to be updated in the future to obtain further deferrals. Since corrosion and equipment
impact were determined to be the dominant failure causes for this segment, maintenance options
considered include enhancing equipment impact prevention measures and more frequent in-line
inspections.
10.2.2 Limit States
Since the maintenance measures considered are targeted at reducing the failure probabilities due
to external corrosion and equipment impact, this example focuses on the limit states associated
with these two failure causes. Given that the pipeline is relatively old, other failure causes such
as manufacturing defects and SCC may be applicable. To account for these failure causes, the
failure rate due to corrosion and equipment impact combined was increased by a fixed factor (see
Section 10.2.4 for details).
The limit states that were explicitly considered in the failure probability analysis are summarized
in Table 10.11. The most relevant SLS relates to excessive plastic deformation under internal
pressure (see Example 1 for a detailed explanation). This limit state was not considered because
it is easily satisfied at the utilization factor used (0.80).
No
Failure Cause
Limit States
Limit State
Category
1
Equipment Impact
Small Leak of a Gouged
Dent
LLS
2
Internal Pressure
Small Leaks of Corrosion
Defects
LLS
3
Internal Pressure
Burst of Corrosion Defects
ULS
4
Equipment Impact
Puncture
ULS
5
Equipment Impact
Burst of a Gouged Dent
ULS
Table 10.11 Limit States Analyzed in Example 2
10.2.3 Reliability Targets
It is assumed that detailed building count and population density information was not available at
the time of the assessment. Therefore, the class-based ULS reliability targets given in
Equations [5.7] (Section 5.3.2.4) were used. Table 10.12 summarizes these targets. It shows that
upgrading the existing pipeline from Class 1 to Class 2 requires a two order-of-magnitude
reduction in the allowable failure probability for ULS.
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Location
Class
ULS
LLS
Target
Reliability
(per km-yr)
Allowable Failure
Probability
(per km-yr)
Target
Reliability
(per km-yr)
Allowable Failure
Probability
(per km-yr)
1
1-4.17×10-4
4.17×10-4
1-10-2
10-2
2
1-3.99×10-6
3.99×10-6
1-10-2
10-2
Table 10.12 Target Reliability Levels
10.2.4 Reliability Analysis
The failure probabilities due to external corrosion and equipment impact were calculated using
the same limit state functions and reliability calculation methods used for Example 1. To
account for other failure causes that are relevant to older pipelines (e.g. SCC and manufacturing
defects), the calculated total failure probability was increased by a factor of 50%. This factor is
based on historical failure data, which indicate that corrosion and equipment impact account for
approximately 2/3 of all failures (Nessim and Zhou 2005). The factor is considered reasonable
for ULS failures, but conservative for LLS failures, which are dominated by corrosion.
The basic random variables required for this example are summarized in Table 10.13. The inline
inspection data indicate that the frequency of significant defects is 10 per km. Probabilistic
characteristics of the defect size and growth rate were derived from the detected significant
defects using the distribution selection techniques described in Section 8.3.3. The rate of
construction activity on the right-of-way was assumed to be 0.02 per km for Class 1 and 0.06 per
km for Class 2. Probabilistic characteristics of notch toughness, excavator weight and tooth
length, and gouge depth and length were based on available information in the literature (see
Appendix B).
10.2.5 Results for Enhanced Maintenance
The reliability was calculated over 30 years for the status quo and for a combination of proposed
maintenance enhancements. For the status quo, it was assumed that subsequent inspections
would be carried out every 10 years, using the same high-resolution inspection tool and that
defect repairs would be carried out based on a required remaining wall thickness of 50% and a
pressure safety factor of 1.25 applied to the burst pressure as calculated based on ASME B31G
(1991). Typical mechanical damage prevention measures (see Column 2 of Table 10.14) were
assumed to be in place.
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Parameter
Units
Mean
Wall thickness
mm
Yield strength
Corrosion defect size:
Average depth
Maximum length
Max. to avg. defect depth ratio
Growth rate of significant
corrosion defects:
Average depth
Maximum length
Annual maximum operating
pressure
Notch toughness
Operating pressure
Excavator tooth length
Excavator weight
Gouge depth
Gouge length
MPa
1.0 x
Nominal
395
mm
mm
ratio
mm/year
mm/year
%
MAOP
Joule
%MAOP
mm
tonne
mm
mm
Standard
Deviation
Distribution
Type
0.25
Normal
13.8
Normal
2.5
33
2.08
1.25
33
1.08
Weibull
Lognormal
Exponential
0.044
1.12
0.022
1.12
Weibull
Lognormal
99.3
3.4
Beta(1)
54
86.5
90
15.2
1.2
201
7.24
8.4
28.4
10.8
1.1
372
Normal
Beta(2)
Rectangular
Gamma
Weibull
Lognormal
(3) Lower bound = 80% MAOP, upper bound = 110% MAOP
(4) Lower bound = 60% MAOP, upper bound = 110% MAOP.
Table 10.13 Probability Distributions for Uncertain Limit State Input Parameters
Parameter
Basic Measures
Enhanced Measures
Burial Depth (m)
Mechanical protection
0.76
No
Above ground alignment
markers
Buried alignment markers
Explicit signage
No
0.76
Painted concrete slab
or steel plate
No
No
At selected strategic
locations
Required but not
enforced
Locate and mark with no
site supervision
Unified system to min.
standard
Intermittent or variable
Average
Aerial
Monthly
Yes
At selected strategic
locations
Required but not
enforced
Locate and mark with
site supervision
Unified system to min.
standard
Intermittent or variable
Above Average
Aerial
Two times a week
Dig notification requirement
Dig notification response
One-call system
Right-of-way indication
Public awareness level
Surveillance method
Surveillance interval
Table 10.14 Basic and Enhanced Failure Prevention Measures for Equipment Impact
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The enhanced maintenance case involved the following actions:
•
Corrosion. The same tool and criteria for defect maximum depth were used as for the status
quo. The pressure safety factor was increased from 1.25 to 1.30. The inspection frequency
was determined by scheduling inspections at times when the reliability is forecast to drop
below the target. This resulted in inspections being required in years 2008, 2013, 2019, and
2026.
•
Equipment impact. The prevention enhancements included are highlighted in column 3 of
Table 10.14. The selected enhancements were based on experience regarding the
effectiveness and practicality of various options.
The reliability analysis results are plotted in Figure 10.5, which shows the target reliability levels
and the calculated reliability for both the status quo and the enhanced maintenance option. Note
that the status quo and enhanced maintenance have the same reliability levels in 2004, i.e. the
reliability improvements due to enhanced maintenance appear in 2005 after the maintenance is
implemented. The figure shows that the pipeline in its current state does not meet the ULS target
for Class 1. If the repairs indicated by the recent inspection are executed and subsequent
inspections are implemented every 10 years, the reliability of the status quo will meet the Class 1
ULS target but not the Class 2 ULS target. Figure 10.5 also shows that the enhanced
maintenance measures increase the reliability and maintain it above the Class 2 target for the
next 29 years. The LLS target is not met at present, but it is expected to be met immediately
after carrying out the repairs indicated by the recent inspection. Future inspections result in
maintaining the reliability above the LLS target over a period of 29 years for both the status quo
and enhanced maintenance option.
10.2.6 Comparison to Conventional Class Upgrade Approaches
The reliability was also calculated for the two options available to reduce the utilization factor
from 0.80 to 0.72, namely pressure reduction and replacement with a thicker walled pipe. The
pressure reduction option requires a pressure drop from 1000 psi to 900 psi. It was assumed that
the pressure reduction was applied after the in-line inspection in 2004, but before carrying out the
required repairs. Since no changes are made to the pipeline segment, all input parameters are the
same as the status quo.
The replacement option was based on using the same steel grade (X52) and resulted in a wall
thickness of 10.2 mm. All input parameters and distributions used were the same as the status
quo, except for the wall thickness, fracture toughness and initial corrosion defect sizes, which
were modified to reflect the new pipe. The wall thickness was assigned a normal distribution
with a mean of 10.2 mm (nominal wall thickness) and standard deviation of 0.25 mm. The
probability distributions of initial defect sizes and fracture toughness given in Table 10.4 were
used for the replacement pipe. The frequency of significant defects was assumed to be the same
as the status quo, i.e. 10 defects per km.
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Figure 10.6 shows the reliability associated with the pressure reduction and replacement options
compared to the enhanced maintenance option discussed in section 10.2.5. All cases have the
same reliability levels in 2004 because the positive impact on reliability of pressure reduction or
replacement options takes effect in 2005 after these actions are implemented. The figure shows
that enhanced maintenance is the only option that meets the ULS and LLS targets for the design
period. The potentially more costly options of replacement and pressure reduction, although
permitted under the current standard, do not meet the ULS target for Class 2. This is because the
mechanical damage prevention methods proposed for the enhanced maintenance option are much
more effective in reducing the probability of failure than either the pressure reduction or
replacement options.
ULS
1-1.0E-06
Reliability (/km-yr)
Class 1-Target
Class 2-Target
1-1.0E-05
Status Quo
Enhanced Maintenance
1-1.0E-04
1-1.0E-03
2004
2009
2014
2019
Year
2024
2029
2034
LLS
1-1.0E-07
Reliability (/km-yr)
1-1.0E-06
1-1.0E-05
1-1.0E-04
LLS-Target
1-1.0E-03
Status Quo
Enhanced Maintenance
1-1.0E-02
1-1.0E-01
2004
2009
2014
2019
2024
2029
2034
Year
Figure 10.5 Reliability Compared to Target for Status Quo and Enhanced Maintenance
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ULS
Reliability (/km-yr)
1-1.0E-06
1-1.0E-05
1-1.0E-04
Class 1-Target
Class 2-Target
Enhanced Maintenance
Replacement
Pressure Reduction
1-1.0E-03
2004
2009
2014
2019
Year
2024
2029
2034
2024
2029
2034
LLS
1-1.0E-07
Reliability (/km-yr)
1-1.0E-06
1-1.0E-05
1-1.0E-04
1-1.0E-03
1-1.0E-02
1-1.0E-01
2004
LLS-Target
Enhanced Maintenance
Replacement
Pressure Reduction
2009
2014
2019
Year
Figure 10.6 Reliability Comparisons of Replacement, Pressure Reduction and Enhanced Maintenance
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11. CONCLUDING REMARKS
Reliability Based Design and Assessment (RBDA) is a design and assessment method in which
the pipeline is designed and operated to meet a pre-defined set of target reliability levels for all
applicable limit states (or failure modes). Reliability, defined as the probability of failure
subtracted from 1, is used as a measure of structural safety because it reflects the uncertainties
involved in pipeline design and operation. Failure consequences are accounted for by requiring
more stringent target reliability levels for limit states with more severe consequences.
RBDA methods have been used successfully for many structural systems including buildings,
bridges, nuclear containments, and offshore structures. They have also been used in the pipeline
industry as a basis for designing offshore pipelines, carrying our corrosion assessments, and
justifying pressure increases and class upgrade exceptions. They offer an integrated approach to
design and maintenance decisions that considers the true structural behaviour and actual failure
modes. Their benefits include achievement of demonstrated and consistent safety levels at the
lowest possible cost and adaptability to new problems involving unique loads or new
technologies.
This document contains a set of guidelines for the application of RBDA to onshore natural gas
transmission pipelines. The guidelines give detailed requirements (including a set of reliability
target) that must be met to demonstrate that a given pipeline is designed and operated safely.
They also describe the overall process of calculating reliability and comparing it to the targets in
order to demonstrate that the requirements are met. The reliability calculation process is
described in some detail, including guidance on how to develop the requisite deterministic and
probabilistic models from first principles if necessary. State-of-the-art models are given for
some key design conditions and failure causes including yielding and burst, equipment impact,
and corrosion. This makes analysis of these failure causes possible without any further
development. To facilitate use by pipeline practitioners, the guidelines provide explicit
procedures and illustrative examples for the various steps involved in the methodology.
The guidelines are applicable to all decisions that influence the structural integrity of a pipeline.
These include design decisions for new pipelines, fitness-for-service evaluation for existing lines,
assessment of changes in operational parameters (e.g. location class changes, fluid changes,
damage), and evaluation of maintenance alternatives.
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12. REFERENCES
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American Concrete Institute (ACI) 1971. Building Code Requirements for Reinforced Concrete
(ACI-318-71). ACI., Detroit, Michigan.
American Concrete Institute (ACI) 1983. Building Code Requirements for Reinforced Concrete
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American Institute of Steel Construction (AISC) 1986. Load and Resistance Factor Design
Specification for Structural Steel Buildings. September.
American Petroleum Institute (API) RP 2A-LRFD. 1993. Recommended Practice for Planning,
Designing and Constructing Fixed Offshore Platforms - Load and Resistance Factor
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American Society of Civil Engineers (ASCE) 1984. Guidelines for the Seismic Design of Oil
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Andrew Palmer and Associates (APA) 2002. Pipeline Defect Assessment Manual (PDAM) –
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Ang, A.S. and Tang, W.H. 1975. Probability Concepts in Engineering Planning and Design
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Ang, A.S. and Tang, W.H. 1990. Probability Concepts in Engineering Planning and Design.
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ASME 1991. ASME B31G, Manual for determining the Remaining Strength of Corroded
Pipelines. A Supplement to ASME B31 Code for Pressure Piping.
ASME 1999. ASME B31.8, Gas Transmission and Distribution Piping Systems. The American
Society of Mechanical Engineers, New York.
ASME 2002. ASME B31.8S-2001, Managing System Integrity of Gas Pipelines.
American Society of Mechanical Engineers, New York, NY.
The
Avramidis, A.N. and Wilson, J.R. 1996. Integrated Variance Reduction Strategies for
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References
Benjamin, J.R. and Cornell, C.A., 1970. Probability, Statistics, and Decision for Civil
Engineers. McGraw-Hill Publishing Company, USA.
Blank, L., 1980. Statistical Procedures for Engineering, Management, and Science. McGrawHill, Inc., USA.
British Standards Institute (BSI) 1999. BS 7910 Guide on Methods for Assessing the
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Canadian Standards Association (CSA) 1988. CAN/CSA-S6, Canadian Highway Bridge Design
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Chen, Q. 1994. Risk-based Calibration of Reliability Levels for Limit States Design of
Pipelines. Submitted to National Energy Board, C-FER Report 94004, March.
Chen, Q. and Nessim, M.A. 1999. Reliability-Based Prevention of Mechanical Damage to
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APPENDIX A – LIMIT STATE FUNCTIONS FOR KEY FAILURE CAUSES
A.1 NOTATION ..........................................................................................................................2
A.2 YIELDING AND BURST OF DEFECT-FREE PIPE.............................................................3
A.2.1
A.2.2
Yielding
Burst
3
3
A.3 EQUIPMENT IMPACT .........................................................................................................4
A.3.1
A.3.2
A.3.3
A.3.4
Introduction
Puncture
A.3.2.1 Background
A.3.2.2 Limit State Function
Dent-Gouge Failure
A.3.3.1 Background
A.3.3.2 Limit State Function
Differentiating Leaks and Ruptures
A.3.4.1 Background
A.3.4.2 Limit State Function
4
4
4
4
5
5
5
6
6
7
A.4 CORROSION .......................................................................................................................8
A.4.1
A.4.2
A.4.3
Introduction
Small Leaks
Large Leaks and Ruptures
A.4.3.1 Background
A.4.3.2 Limit State Function for Through Wall Failure
A.4.3.3 Differentiating Large Leaks and Ruptures
8
8
8
8
9
9
A.5 REFERENCES...................................................................................................................11
A.1
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Appendix A
A.1 NOTATION
The following notation for common parameters is used in this appendix:
t
D
σy
σu
P
wall thickness
pipe diameter
yield strength
tensile strength
internal pressure
Unless otherwise stated, the units used are MPa for pressure and stress, mm for dimensions, and
kN for force.
A.2
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Appendix A
A.2 YIELDING AND BURST OF DEFECT-FREE PIPE
A.2.1 Yielding
The limit state function for yielding of defect free pipe is given by:
g 1 =2σ y t − PD
[A.1]
A.2.2 Burst
The limit state function for burst of defect free pipe is given by:
g 2 =2 c σ f t − PD
[A.2]
where σf is the flow stress and c is a model error factor. If the flow stress is defined as 0.953 σu
(see SUPERB (1995)) Equation [A.2] becomes:
g 2 = 1.906 c σ u t − PD
[A.3]
where c is a model error factor that accounts for uncertainty regarding the definition of the flow
stress. Based on information in SUPERB (1995), c has a Normal distribution with a mean of 1.0
and a COV of 4%.
A.3
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Appendix A
A.3 EQUIPMENT IMPACT
A.3.1 Introduction
The limit state models presented here are appropriate for impacts by the tooth of an excavator
bucket. Previous work (Chen and Nessim 2000) has shown that the majority of impacts (75%)
result from backhoes. For a modern large diameter pipeline, only excavators, which impart a
much larger force than backhoes, present a significant failure hazard. Limit state functions for
two failure mechanisms are presented. Puncture occurs if the force on the excavator tooth is
large enough to penetrate the pipe wall. Dent-gouge failure occurs if the force is insufficient to
puncture the wall impact but creates a gouged dent that fails under pressure upon removal of the
excavator tooth.
A.3.2 Puncture
A.3.2.1 Background
The limit state function described here is appropriate for a load generated by an indentor having a
shape corresponding to that of an excavator bucket tooth. The model was developed by C-FER
(Driver and Playdon 1997, Driver and Zimmerman 1998) following a review of existing models
(Spiekhout et al. 1987, Spiekhout 1995 and Corbin and Vogt 1997), theoretical considerations
and available test data. The C-FER model has a term of the form a + b D/t where a and b are
empirical coefficients. The coefficients a and b were determined based on regression analysis of
a set of test data produced by the EPRG (Muntiga 1992, Hopkins et al. 1992, and Chatain 1993)
and Battelle (Maxey 1986). This model was calibrated for values of t between 4 and 12.5 mm, D
between 168 and 914 mm, and steel grades up to X70.
A.3.2.2 Limit State Function
The limit state function g1 for puncture is as follows:
g1 = ra – q
where:
ra
is the estimated resistance including model error, given by
ra = [1.17 – 0.0029 (D t)] (lt + wt) t σu + e;
σu
is the tensile strength;
lt
is the cross-sectional length of the indentor;
wt
is the cross-sectional width of the indentor;
q
is the normal impact force (kN), given by
q = 16.5w0.6919 RD RN
w
RD
[A.4]
[A.5a]
[A.5b]
is the excavator mass (tonne);
is the dynamic impact factor and equals 2/3;
A.4
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Appendix A
RN
e
is the normal load factor, which is the ratio between the component force normal
to the pipe wall and the total force, characterized by a random quantity uniformly
distributed between 0 and 1; and
is a model error term, characterized by a normal distribution with a mean of
0.833 kN and a standard deviation of 26.7 kN.
A.3.3 Dent-Gouge Failure
A.3.3.1 Background
The limit state function for dent-gouge failure is a version of the EPRG semi-empirical model.
The dent depth is calculated from the impact force using a model published by
Linkens et al. (1998). A model error term has been developed for this relationship by C-FER
based on confidence intervals specified in Corder and Chatain (1995). The resulting gouged dent
is checked for failure under hoop stress using a model developed for EPRG (Hopkins et al. 1992)
and later modified by Francis et al. (1997). The method is based on an evaluation of the fracture
ratio, Kr, and the load ratio, Sr, according to the British Standards PD6493 procedure for defect
assessment. Model uncertainty is not considered in this limit state function. The reason for this
is that the scatter of the experimental data on which the model was used is small. Further the
model makes the conservative assumption that all gouges have an axial orientation, which
overestimates the stresses acting on the gouge.
A.3.3.2 Limit State Function
The limit state function, g2, for dent-gouge type failures is as follows, given an impact with a
force q normal to the pipe wall (as defined by Equation [A.5b]) and a gouge of length lg and
depth dg.
[A.6]
g2 = σc - σh
where:
σc
is the calculated critical hoop stress resistance, determined as a solution of [A.7]
σh
is the hoop stress resulting from internal pressure = PD /2t
Note that model error is not considered in this. The critical hoop stress resistance is calculated
from
  π 2  b  2 K 2 
2
IC
 2 
σc =
Arc cos exp−

π b2
  8  b1  π d g 


[A.7]
where
KIC
is the critical stress intensity, defined as
 E ⋅ cv 0 

KIC = 
a
 C 
0.5

c
⋅  v 2 / 3 
 cv 0 
0.95
;
[A.8]
A.5
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Appendix A
b1 and b2
are given by
b1 = ( S m Ym + 5.1Yb d do / t ) ;
[A.9]
m
dg 


S m 1 −
m t 

;
b2 =
 dg 

1.15 σ y 1 −
t 

is the Folias factor defined as
dd0
 0.52 ⋅ l g 2 
 ;
m = 1 +


D
⋅
t


is the dent depth at zero pressure, approximated by
[A.10]
0.5


q
d d 0 = 1.43

0.25
 0.49 (lt ⋅ σ y ⋅ t ) ⋅ (t + 0.7 ⋅ P ⋅ D / σ u ) 
Sm, Ym and Yb are given by
S m = (1 − 1.8 d do / D) ,
2
Ym
=
2.381
;
[A.12]
[A.13]
3
4
 dg 
d 
d 
d 
 + 10.6 g  − 21.7 g  + 30.4 g  ,
1.12 − 0.23
 t 
 t 
 t 
 t 
2
and
[A.11]
3
[A.14]
4
 dg 
d 
d 
d 
[A.15]
 + 7.32 g  − 13.1 g  + 14.0 g  ;
=
1.12 − 1.39
Yb
 t 
 t 
 t 
 t 
the basic input parameters are defined as follows
E
is Young’s modulus;
cv2/3 is the Charpy energy of 2/3 size specimens (2/3 of full size specimen energy);
cv0
is an empirical coefficient equal to 110.3 Joule;
aC
is the cross-sectional area of 2/3 Charpy V-Notch specimens, equal to 53.55 mm2;
and
is the cross-sectional length of the excavator tooth.
lt
A.3.4 Differentiating Leaks and Ruptures
A.3.4.1 Background
Given puncture or failure of a dent-gouge, the mode of failure is determined based on whether or
not unstable axial growth of the resulting through-wall defect occurs. The defect is assumed to
fail as a through-wall crack-like (i.e. sharp) defect. The initial defect length is assumed to equal
the indentor length in the case of puncture and the gouge length in the case of gouged dent
failure. If the length of the resulting through-wall crack exceeds the critical defect length for
A.6
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Appendix A
unstable growth, as determined from the criterion developed by Kiefner et al. (1973), the failure
is classified as a rupture. Otherwise the failure is classified as a leak.
A.3.4.2 Limit State Function
The limit state function g for rupture is:
g3 = Scr - σh
where
S cr =


2(σ y + 68.95) −1 
125π EC v
cos exp− 

2
π MT

 c(σ y + 68.95) Ac 

c2
c4 
M T = 1 + 1.255 − 0.0135 2 2 
Rt
Rt 

c2
+ 3.3
Rt
= pipe radius = D/2 (mm);
M T = 0.064
R
[A.16]
1/ 2
[A.17a]
for
c2
≤ 25
Rt
[A.17b]
for
c2
> 25
Rt
[A.17c]
c
is ½ the defect length (mm) where 2c equals the indentor cross-sectional length
for the case of puncture and equals the gouge length in the case of dent-gouge
failure;
Cv
= full-size Charpy V-notch plateau energy (J);
Ac
= full-size Charpy shear area (mm2); and
E
= elastic modulus (MPa).
A.7
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Appendix A
A.4 CORROSION
A.4.1 Introduction
In the limit state functions used for corrosion, a corrosion defect is characterized at any point in
time by its maximum axial length lc, average depth da, and maximum depth dmax. A small leak
occurs if the maximum depth reaches the pipe wall thickness. Burst occurs if the defect reaches
a critical overall size defined in terms of maximum axial length and average depth. Given burst,
the defect ruptures if the length of the resulting hole exceeds the critical value for unstable
growth of a through-wall defect. Otherwise, the failure is classified as a large leak.
A.4.2 Small Leaks
The limit state function for small leak due to corrosion is:
g1 = t - dmax
[A.18]
A.4.3 Large Leaks and Ruptures
A.4.3.1 Background
The limit state function used for burst at a corrosion defect is a variant of the semi-empirical
model for failure of a ductile pipe with a longitudinally oriented metal loss defect, which was
developed by Battelle (Kiefner 1969) and later modified by Kiefner and Vieth (1989) and
Bubenik et al. (1992). A model developed by C-FER (Brown et al. 1995) followed the same
basic format of the semi-empirical model, however some input parameters were redefined in
order to achieve better accuracy.
Recent work on predicting the failure stress of corroded pipelines revealed that the abovementioned models are not accurate for modern, high toughness pipeline steels (Stephens 2000).
To address this, Advantica (Fu 1999) and Battelle (Stephens 1999) proposed improved prediction
models with the flow stress defined based on the ultimate tensile strength (σf = 0.9 σu) rather than
the yield strength (σf = 1.15 σy). Models based on ultimate tensile strength are applicable to for
all X-grade steels, while the models based on yield strength are more applicable to grades A and
B as well as the lower X-grades (up to X60).
To ensure applicability in the entire range of steel grades, two separate models are used for X
grade steels (SMYS > 35 ksi or 241 MPa), and A & B grade steels (SMYS ≤ 35 ksi or 241 MPa).
Both models were calibrated with burst tests carried out on corroded segments of pipe removed
from service (Kiefner and Vieth 1989). Although the flow stress definition is different for the
two models, they both employ the same defect depth measure (average depth) and Folias factor.
The condition used for unstable defect growth is taken from Kiefner et al. (1973) and Shannon
(1974). The model error factors were based on 25 burst test data points for X grade steels and 38
points for A and B grade steels.
A.8
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Appendix A
A.4.3.2 Limit State Function for Through Wall Failure
The limit state function g2 for plastic collapse at a surface corrosion defect with total axial length
l and average depth da is defined as follows
g 2 =ra − P
[A.19]
where
ra is the estimated pressure resistance including model error, which is given by.
r0
rc
ra = e1 rc + (1 − e1 ) r0 − e2σ u ,
SMYS > 35 ksi ( 241 MPa )
[A.20a]
ra = e3 rc + (1 − e3 ) r0 − e4σ y ,
SMYS ≤ 35 ksi (241 MPa )
[A.20b]
is the pressure resistance for perfect pipe, given as
tσ
r0 = 1.8 u ,
SMYS > 35 ksi ( 241 MPa )
D
tσ
r0 = 2.3 y ,
SMYS ≤ 35 ksi ( 241 MPa )
D
is the calculated pressure resistance, defined as
d 

 1− a 
t ,
rc = r0 ⋅ 
da 

1−

 m⋅t 
m
[A.21a]
[A.21b]
[A.22]
is the Folias factor, defined as
l2
l4
m = 1 + 0.6275
− 0.003375 2 2
D⋅t
D ⋅t
l2
for
≤ 50
D ⋅t
[A.23a]
l2
l2
[A.23b]
+ 3.293
for
> 50
D ⋅t
D ⋅t
is a deterministic multiplicative model error term that equals 1.04,
is an additive model error term, defined by a normally distributed random variable
with a mean of -0.00056 and a standard deviation of 0.001469,
is a deterministic multiplicative model error term that equals 1.17 for A and B
grade pipe and 1.15 for X grade pipe, and
is an additive model error term, defined by a normally distributed random variable
with a mean of -0.007655 and standard deviation of 0.006506.
m = 0.032
e1
e2
e3
e4
A.4.3.3 Differentiating Large Leaks and Ruptures
Failure of a corrosion defect is assumed to result in a through wall defect with axial length l.
Rupture occurs if unstable growth of the through wall defect takes place. The limit state function
for rupture is
A.9
C-FER Technologies
Appendix A
1.8 ⋅ t ⋅ σ u
− P,
SMYS > 35 ksi ( 241 MPa )
m⋅D
2 .3 ⋅ t ⋅ σ y
g3 =
− P,
SMYS ≤ 35 ksi ( 241 MPa )
m⋅D
where m is the Folias factor defined by [A.25a] and [A.25b].
g3 =
[A.24a]
[A.24b]
A.10
C-FER Technologies
Appendix A
A.5 REFERENCES
Brown, M., Nessim, M. and Greaves, H. 1995. Pipeline Defect Assessment: Deterministic and
Probabilistic Considerations. Second International Conference on Pipeline Technology,
Ostend, Belgium, September.
Bubenik, T.A., Olson, R. J., Stephens, D.R. and Francini, R.B. 1992. Analyzing the Pressure
Strength of Corroded Line Pipe. Proceedings of the Eleventh International Conference
on Offshore Mechanics and Arctic Engineering, Volume V-A, Pipeline Technology
ASME.
Chatain, P. 1993. An Experimental Evaluation of Punctures and Resulting Dents in
Transmission Pipelines. Proc., 8th Symposium on Line Pipe Research, American Gas
Association, Sep. 26-29, Houston, Texas, pp. 11-1 to 11-12.
Chen, Q. and Nessim, M.A. 2000. Reliability-Based Prevention of Mechanical Damage to
Pipelines. Submitted to the Pipeline Research Committee International, American Gas
Association, Project PR-244-9729, C-FER Report 97034, August.
Corbin, P. and Vogt, G. 1997. Future Trends in Pipelines. Proc., Banff/97 Pipeline Workshop:
Managing Pipeline Integrity - Planning for the Future, Banff, Alberta.
Corder, I. and Chatain, P. 1995, EPRG Recommendations for the Assessment of the Resistance
of Pipelines to External Damage, EPRG/PRC 10th Biennial Joint Technical Meeting On
Line Pipe Research.
Driver, R. and Playdon, D. 1997. Limit States Design of Pipelines for Accidental Outside
Force, Report to National Energy Board of Canada.
Driver, R.G. and Zimmerman, T.J.E. 1998. A Limit States Approach to the Design of Pipelines
for Mechanical Damage. Proceedings of the Seventeenth International Offshore &
Arctic Engineering Conference, OMAE98-1017, Lisbon, Portugal, July.
Francis A., Espiner R., Edwards A., Cosham A., and Lamb M. 1997. Uprating an In-Service
Pipeline Using Reliability-Based Limit State Methods, Risk-Based and Limit State
Design and Operation of Pipelines, Aberdeen, UK, 21st-22nd, May, 1997.
Fu, B. and Batte, A.D. 1999. New Methods for Assessing the Remaining Strength of Corroded
Pipelines. Proceedings EPRG/PRCI 12th Biennial Joint Technical Meeting on Pipeline
Research, May.
Hopkins, P., Corder, I. and Corbin, P. 1992. The Resistance of Gas Transmission Pipelines to
Mechanical Damage. Proc., CANMET International Conference on Pipeline Reliability,
vol. 2, June 2-5, Calgary, AB, pp. VIII-3-1 to VIII-3-18.
A.11
C-FER Technologies
Appendix A
Kiefner, J.F. 1969. Fracture Initiation. 4th Symposium on Line Pipe Research, Paper G,
American Gas Association Catalogue No. L30075, November.
Kiefner, J F. and Vieth, P.H. 1989. Project PR 3-805: A modified Criterion for Evaluating the
Remaining Strength of Corroded Pipe. A Report for the Pipeline Corrosion Supervisory
Committee of the Pipeline Research Committee of the American Gas Association.
Kiefner, J.F., Maxey, W.A., Eiber, R.J. and Duffy, A.R. 1973. Failure Stress Levels of Flaws in
Pressurized Cylinders. Progress in Flaw Growth and Fracture Toughness Testing,
ASTM STP 536, American Society for Testing and Materials, pp. 461 - 481.
Linkens D., Shetty N., and Bilo M., 1998. A Probabilistic Approach to Fracture Assessment of
Onshore Gas-Transmission Pipelines, Pipes & Pipeline International.
Maxey, W.A. 1986. Outside Force Defect Behavior. Proc., 7th Symposium on Line Pipe
Research, American Gas Association, Oct, Houston, Texas.
Muntiga, T.G. 1992. Wall Thickness in Relation to Puncture - Summary of GU Results Report
to EPRG, N.V. Nederlandse Gasunie, Oct. 2.
Shannon, R.W.E. 1974. The Failure Behaviour Line Pipe Defects. International Journal of
Pressure Vessel and Piping, Vol. 2, pp. 243 - 255.
Spiekhout, J. 1995. A New Design Philosophy for Gas Transmission Pipelines - Designing for
Gouge-Resistance and Puncture-Resistance. Proc., 2nd International Conference on
Pipeline Technology, vol. I, Sept. 11-14, Ostend, pp. 315-328.
Spiekhout, J., Gresnigt, A.M. and Kusters, G.M.A. 1987. The Behaviour of a Steel Cylinder
Under the Influence of a Local Load in the Elastic and Elasto-Plastic Area. Proc.,
International Symposium on Shell and Spatial Structures: Computational Aspects (July,
1986), Leuven, Belgium. Springer-Verlag, Berlin, Germany, pp. 329-336.
Stephens, D.R., Leis, B.N., Kurre, M.D. and Rudland, D.L. 1999. Development of an
Alternative Criterion for Residual Strength of Corrosion Defects in Moderate-to-High
Toughness Pipe. PRCI Report PR-3-9509, January.
SUPERB, 1995. Jiao, G., Sotberg, T. and Ingland, R. SUPERB 2M – Statistical Data: Basic
Uncertainty Measures for Reliability Analysis of Offshore Pipelines. Report Number
STF70 F95212 submitted to SUPERB JIP Members, June 8.
A.12
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APPENDIX B – PROBABILISTIC MODELS FOR BASIC VARIABLES
B.1 LOADING PARAMETERS .....................................................................................................2
B.1.1
B.1.2
B.1.3
Internal Pressure
Equipment Impact
B.1.2.1 Impact Frequency
B.1.2.2 Impact Severity
Ground Movement
B.1.3.1 Soil Strength
B.1.3.2 Soil Stiffness
2
2
2
3
4
4
5
B.2 MECHANICAL PROPERTIES ...............................................................................................7
B.2.1
B.2.2
Summary of Available Data
Discussion
B.2.2.1 Yield and Ultimate Tensile Strength
B.2.2.2 Fracture Toughness
7
8
8
9
B.3 PIPE GEOMETRY ................................................................................................................11
B.4 DEFECT CHARACTERISTICS............................................................................................12
B.4.1
B.4.2
B.4.3
External Corrosion
Dents and Gouges
Seam Weld Cracks
12
13
14
B.5 REFERENCES......................................................................................................................15
B.1
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Appendix B
B.1 LOADING PARAMETERS
B.1.1 Internal Pressure
As discussed in Section 7.4.3, internal pressure is a time-dependent load that is characterised by
the probability distribution of its maximum value during a specified time period as well as its
value at an arbitrary point in time. Although pipeline operators routinely maintain pressure
histories at key pipeline locations, little public data is available on this parameter. Available
information on these parameters include:
•
Based on proprietary pressure records from one pipeline operator, C-FER found that the
ratio between the annual maximum pressure and design pressure can be assigned a Beta
distribution with a mean of 0.993, COV of 0.034, lower bound of 80%, and upper bound
of 110%. The ratio between the arbitrary-point- in-time operating pressure and design
pressure was also assigned a Beta distribution with a mean of 0.865, COV of 0.084, lower
bound of 60%, and upper bound of 110%. Note these probabilistic characteristics are
based on the assumption that the pipeline is operating at its capacity, i.e. the maximum
operating pressure equals the design pressure.
•
Based on assumptions regarding the probability of over-pressure as a function of normal
pressure control settings and the reliability of pressure control systems, Jiao et al. (1995)
have proposed a Gumbel distribution with a mean of between 1.03 and 1.07 and a COV
between 1 and 2% to model the ratio between the maximum annual pressure and the
design pressure. This relationship is applicable at locations operating at the design
pressure (i.e. immediately downstream of a compressor or pumping station). Since the
pressure drops along the line between compressor stations, this distribution is a
conservative estimate of the maximum annual normal operating pressure at locations that
are further down stream of a compressor or pumping station.
It should be mentioned that the operating pressure could rise above the value corresponding to
the normal operating profile at any location along the pipeline during special operating
conditions such as hydrostatic testing, line pack or reversal of flow direction. These conditions
should be taken into account in selecting a probability distribution of the maximum annual
operating pressure.
B.1.2 Equipment Impact
B.1.2.1 Impact Frequency
Chen and Nessim (1999) describe a fault tree model to derive impact frequency from the
frequency of construction activity and the damage mitigation measures implemented for a given
pipeline. Damage mitigation measures considered in this model include right-of-way patrols,
marking and signs, one-call system, excavation procedures and public awareness programs.
B.2
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Appendix B
Activity rates and estimates of the effectiveness of various mitigation measures were quantified
based on survey responses from 15 pipeline companies. Excavation rates of between 0.076 per
km.year in agricultural areas and 0.52 per km- year in commercial and industrial areas were
reported. With typical prevention measures, these lead to hit rates of between 0.004 per km.year
for undeveloped areas and 0.05 for developed areas. These values are consistent with the hit
rates reported in a GRI study (Doctor et al. 1995).
It is noted that approximately 75% of equipment impacts are by backhoes, which are too small to
cause significant damage to the range of diameters typical of transmission pipelines (greater than
about 8”). The rates of significant hits (i.e. by excavators) are therefore approximately 25% of
the numbers quoted in the previous paragraph.
B.1.2.2 Impact Severity
Table B.1 summarizes the probability distributions of the parameters required to characterize the
maximum load resulting from excavator impact. Driver and Zimmerman (1998) developed the
probability distribution of excavator mass based on a survey of excavators sold by major
manufacturers in North America. This distribution was combined with the relationship between
excavator mass and maximum force to generate a distribution of the maximum quasi-static force
imparted to a pipeline by the bucket tooth of an excavator. The excavator tooth width and length
distributions given by Chen and Nessim (2000) were based on the assumption that all values are
equally likely within a range that was obtained from a survey of excavator manufacturers.
B.3
C-FER Technologies
Appendix B
Variable
Units
Distribution Type
Mean
COV
(%)
Source
Impact rate
per
km.year
Deterministic
Up to
0.02
0
Chen and Nessim
(1999)
(1)
Beta
(2)
Excavator weight
(1)
Wolvert et al.
(2004)
(2)
Wolvert et al.
(2004)
(3)
Chen and Nessim
(2000)
(4)
(1)
8.0
(2)
10.8
(3)
38
(4)
5.7
Gamma
15.2
-
29
-
20.7
18
Chen and Nessim
(2000)
tonne
Excavator force
kN
Shifted Gamma
164
45
Driver and
Zimmerman (1998)
Excavator bucket
tooth length
mm
Uniform
90
32
Chen and Nessim
(2000)
Excavator bucket
tooth width
mm
Uniform
3.5
25
Chen and Nessim
(2000)
(1)
(2)
(3)
(4)
(5)
(5)
For urban and semi-urban areas from usage survey data, lower bound = 0.5 tonnes, upper bound = 34 tonnes
For rural area based excavator sales data
For excavators in class 1 and 2 areas only
For excavators in class 3 and 4 areas only
Since the smallest data point was approximately 92 kN
Table B.1 Parameter Distributions for Load Parameters
B.1.3 Ground Movement
B.1.3.1 Soil Strength
Soil strength is characterized by the ultimate bearing capacity, which depends on the type and
strength of the soil, direction of loading, depth of soil cover and pipe diameter. For undrained
cohesive soils, the ultimate horizontal resistance per unit area of pipe σz is given by:
σ z = Nch Su
[B.1]
B.4
C-FER Technologies
Appendix B
where Su is the undrained shear strength and Nch is the horizontal bearing capacity factor. The
horizontal bearing capacity factor is a function of the pipe embedment ratio (i.e. the distance
from ground surface to mid- height of the pipe, divided by pipe diameter) and the degree to which
soil separation occurs on the back side of the pipe.
For granula r soils the ultimate horizontal resistance per unit pipe area is given by:
σ z = N qh γ s H
[B.2]
where γs is the effective unit weight of soil, Nqh is the horizontal bearing capacity factor for
frictional soils, and H is equal to the cover depth plus one-half of the pipe diameter. The
horizontal bearing capacity factor is a function of the embedment ratio and the soil friction angle.
According to the Committee on Gas and Liquid Fuel Lines (1984), the results of numerical
analysis by Rowe and Davis (1982) support the use of a model developed by Hansen (1961) for
transversely loaded rigid piles, as a basis for estimating Nch and Nqh . This model was used in
conjunction with typical pipe embedment ratios, and the properties of generally accepted
cohesive soil categories, to derive typical values of σz (see Table B.2).
Soil Type
Sand or Gravel
Clay
Soil Strength (kPa)
Loose
100
Medium
150
Dense
200
Soft
90
Medium
180
Stiff
400
Very Stiff
800
Hard
1300
Table B.2 Typical Values of Soil Strength During Horizontal Transverse Ground Movement
No statistical information is available to characterize the probability distribution of soil strength.
The values in Table B.2 may be seen as reasonable estimates of distribution means. Evidence
indicates that a COV of 30% is considered reasonable for soil strength parameters (e.g. Orr
1994).
B.1.3.2 Soil Stiffness
Soil stiffness is characterised by the modulus of subgrade reaction, which depends on soil type,
strength and loading direction. It is defined as the slope of the soil pressure versus soil
displacement curve, which generally exhibits non- linear behaviour. Where large displacements
are involved, equivalent stiffness values that fall somewhere between the secant modulus and the
B.5
C-FER Technologies
Appendix B
tangent modulus are typically used in pipe-soil interaction calculations (NEN 3650 1992,
Rajani et al. 1995, Trigg and Rizkalla 1994).
For transverse horizontal movement in sandy soils the tangent modulus is most commonly used
to characterize the soil stiffness. Typical values of the tangent modulus were obtained using the
pressure-displacement curves provided by the Committee on Gas and Liquid Fuel Lines (1984)
and assuming typical pipe embedment ratios and soil parameters representative of generally
accepted granular soil type categories (see Table B.3).
For transverse horizo ntal movement in clayey soils the Dutch standard for steel pipeline systems
(NEN 3650 1992) recommends using 0.7 times the downward stiffness, where the downward
stiffness is estimated to be twice the secant modulus for a pressure-displacement curve. The
representative stiffness values given in Table B.3 for clayey soils were obtained using this
calculation method with the horizontal secant modulus being used as a lower bound. The secant
modulus estimates were obtained from the equations for ultimate soil strength and corresponding
displacement as given by the Committee on Gas and Liquid Fuel Lines (1984) assuming typical
pipe embedment ratios and soil parameters representative of generally accepted cohesive soil
type categories.
The values in Table B.3 provide reasonable estimates of the mean soil stiffness. As in the case of
soil strength a COV of 30% may be assumed (Orr 1994).
Soil Type
Sand or Gravel
Clay
Soil Stiffness (MPa/m)
Loose
5
Medium
10
Dense
30
Soft
2
Medium
5
Stiff
10
Very Stiff
20
Hard
35
Table B.3 Typical Values of Soil Subgrade Reaction During Horizontal Transverse Ground Movement
B.6
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Appendix B
B.2 MECHANICAL PROPERTIES
B.2.1 Summary of Available Data
Table B.4 summarizes available statistical data and distributions for yield strength, tensile
strength, flow stress, ultimate tensile strain, Charpy V-Notch upper plateau impact energy,
CTOD fracture toughness, and Young’s modulus.
Variable
Yield strength/SMYS
Units
-
Distribution
Type
Pipe body Charpy
V-notch impact
energy
Source
(1)
1.11
3.4
Jiao et al. (1997a)
X60
Normal
(1)
1.08
3.3
Jiao et al. (1997a)
X65
1.07–1.10
2.6-3.6
Jiao et al. (1995)
X80
1.08
4
Sotberg and Leira
(1994)
1.1
3.5
C-FER
N/A
(2)
(3)
Normal or
(4)
Lognormal
Ultimate tensile strain
COV (%)
Normal
Lognormal
Tensile strength/
SMTS
Mean
Normal
(5)
1.12
3.0
Jiao et al. (1995)
(X60 hoop)
Normal
(5)
1.12
3.5
Jiao et al. (1995)
X65 hoop
Normal
(5)
1.07
2.6
Jiao et al. (1995)
X60 axial
Normal
(5)
1.08
2.9
Jiao et al. (1995)
X65 axial
-
N/A
(6)
1.12
-
GRI – X60
N/A
(6)
1.14
-
GRI – X65
N/A
(6)
1.13
-
GRI – X70
Lognormal
36
6
Jiao et al. (1995)
X60
Lognormal
46
7
Jiao et al. (1995)
X65
-
30 – 70
Formula
Lognormal
149 - 259
18 – 24
110
13
%
J
-
(9)
(7)
Leewis (1997)
(8)
(9)
Jiao et al. (1995)
(10)
X60 & X65
Hillenbrand (1999)
(10)
X80
B.7
C-FER Technologies
Appendix B
Variable
Units
Seam weld Charpy
V-notch impact
energy
J
Pipe body critical
crack tip opening
displacement (CTOD)
mm
Seam weld critical
crack tip opening
displacement (CTOD)
mm
Young’s Modulus
GPa
Distribution
Type
Mean
COV (%)
-
176 – 183
16 - 18
Gartner et al. (1992)
(10)
X80
131 - 291
4 – 47
Jiao et al. (1995)
(11)
X60 & X65
Lognormal
-
40
Sotberg and Leira
(10)
(1994)
Beta
0.67
41
Jiao et al.
(11,12)
(1995)
Beta
-
30 - 60
ISO 2001
Normal
210
4%
Sotberg and Leira
(1994)
Lognormal
(9)
Source
(11)
(1) Based on a mixture of various samples from different mills (760 tests for X60 and 2,753 tests for X65).
(2) Based on 3 sets of tests.
(3) No information on original source provided.
(4) Based on mill data for X60 to X70 pipe
(5) Based on similar numbers of samples as (1).
(6) Based on an analysis of GRI proprietary data by C-FER
(7) Equation [B.3] in text.
(8) Pre 1971 pipes of all grades.
(9) See also Equation [B.4] in text.
(10) Modern steels.
(11) Modern welding techniques.
(12) Based on a range of welding techniques, pipe suppliers and wall thicknesses.
Table B.4 Parameter Distributions for Pipe Mechanical Properties
B.2.2 Discussion
B.2.2.1 Yield and Ultimate Tensile Strength
Published distributions of yield and tensile strength are believed to be reliable as they are based
on data from routine mill tests. The reliability of this information is also evidenced by the
consistency of the distributions cited in different references. The mean value and COV of the
ratio between actual and specified minimum yield strength are quite consistent for different
grades. Jiao et al. (1997a) show that the COV can vary by a factor of 2 depending on the quality
of the mill and therefore mill-specific data should be used where possible. It should be noted that
the yield strength distribution obtained from mill coupon tests is likely to be a conservative
estimate of the actual distribution because of the positive impact of low strength joint rejection
and field pressure testing.
The yield and tensile strength could be different in the hoop and axial directions. This is
demonstrated in Table B.4, which shows that both the mean and COV of the tensile strength are
B.8
C-FER Technologies
Appendix B
lower in the axial than in the hoop direction. The average ratio between actual and minimum
specified tensile strength is reasonably consistent (around 1.12 to 1.14) for X60 to X70 pipe.
Jiao et al. (1997a) provide data indicating that tensile strength is higher for welds than for the
pipe body.
B.2.2.2 Fracture Toughness
The most commonly used parameters to define fracture toughness are the Charpy V-notch impact
energy and the critical crack tip opening displacement (CTOD). Charpy impact energy is the
only toughness measure typically available for older line pipe whereas CTOD data is a better
toughness measure for modern high strength steels. While fracture toughness has been shown to
be largely independent of steel grade it is highly dependent on construction vintage and specific
purchaser’s requirements. In addition, pipe body toughness can be significantly different from
that of the weld metal and heat affected zone.
The Charpy energy for older line pipe, manufactured prior to the 1970’s (when no specific
toughness requirements existed), exhibits considerable variability. Analysis of data from North
American mills provided by Leewis (1997) indicates that the mean value of the Charpy upper
plateau energy for the body of older pipe can range between 30 and 70 Joules. There is also
concern that some older pipe may exhibit brittle behaviour as it may be operating below the
transition temperature. However, if a credible estimate of the mean Charpy energy is available
for a given order of pipe, regression analysis by C-FER of statistical data on line pipe reported by
AGA (1977) and Eiber (1977) suggests that the COV can be estimated from the mean value
using the following equation:
cov = 0.0223µ 0. 46
[B.3]
This formula is considered applicable to the pipe body of lower strength steels (up to grade X70)
with mean Charpy energies not significantly exceeding 100 Joules.
More modern steels, and in particular the higher strength grades, are typically associated with
much higher Charpy plateau energies. Expanding the data set that formed the basis for
Equation [B.3], to include modern X60, X65, and X80 material (from Gartner et al. 1992, Graf et
al. 1993, Jiao et al. 1995, and Hillenbrand 1999), yie lds the following expression for the COV of
the Charpy energy:
cov = 0.0421µ 0.29
[B.4]
This formula gives similar results to Equation [B.3] for steels with mean Charpy values below
100 Joules and is applicable for higher values as well.
With regard to the seam weld, the Charpy plateau energies associated with the weld zone have
been shown to be dependent on weld vintage, weld process and wall thickness. For older line
B.9
C-FER Technologies
Appendix B
pipe there is insufficient data in the public domain to facilitate the development of a
representative probabilistic characterisation of Charpy toughness. However, it is commonly
assumed that the weld zone toughness in older steels is significantly less than that of the pipe
body, with older problematic welds (e.g., low-frequency ERW pipe) exhibiting weld zone
toughness values as low as 3 J (Kiefner 1992). For modern line pipe employing modern seam
welding techniques it is possible to achieve weld zone toughness comparable to that of the pipe
body, however, weld zone toughness typically exhibits a higher COV (e.g. Jiao et al. 1995).
Both normal and lognormal distribution types have been shown to provide a good fit to Charpy
test data, however, the lognormal distribution type is generally considered to be the most
appropriate (ISO 2001).
CTOD data is not generally available for older line pipe. For modern line pipe the mean CTOD
value is highly dependent on the line pipe specifications. A summary of published CTOD
distributions is given in Table B.4.
B.10
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Appendix B
B.3 PIPE GEOMETRY
Table B.5 gives published distributions for pipe diameter and wall thickness. There is a
significant amount of information regarding these parameters, although there are indications that
different mills may produce different levels of variability. Defining these distributions is also
aided by geometric parameter and mass tolerances specified in various pipeline codes (see
Section 7.4.3). Jiao et al. (1995) for example suggest that the standard deviation may be
calculated by assuming that the width of the tolerance interval equals three standard deviations
on either side of the mean value. Given the small variability in pipe diameter, it is often treated
as deterministic in reliability calculations.
Variable
Units
Diameter/nominal
diameter
-
Diameter/nominal
diameter
-
Wall thickness /
nominal wall thickness
Distribution Type
Deterministic
-
(1)
Normal
COV (%)
Source
1.0
0
Jiao et al. (1995)
1.0
0.06%
Zimmerman et al.
(1998)
Normal
(2)
1.0
0.25
/nominal (3)
Jiao et al. (1997a)
Normal
(4)
1.1
3.3%
Jiao et al. (1997a)
1.01
1.0%
Zimmerman et al.
(1998)
Normal
(1)
(2)
(3)
(4)
Mean
Based on 16 – 56” pipe, COV < 0.1%.
For welded pipe with thickness values between 15 and 37 mm.
Equivalent to a fixed standard deviation of 0.25 mm.
For seamless pipe based on permissible tolerances.
Table B.5 Parameter Distributions for Pipe Geometry
B.11
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Appendix B
B.4 DEFECT CHARACTERISTICS
B.4.1 External Corrosion
Corrosion dimensions and growth rates are highly dependent on pipeline-specific attributes such
as coating type and condition, level of cathodic protection and soil corrosivity. Significant
amounts of relevant data are being collected by high-resolution in- line inspections, but this
information is usually treated as proprietary. Although a comprehensive database that allows
developing the required correlations between corrosion defect characteristics and related line
attributes is not available, organisations with access to data from several pipelines have published
representative values and ranges (Table B.6).
Variable
Units
Distribution
Type
Mean
COV (%)
Source
Defect length
mm
Lognormal
27 - 105
35 - 130
C-FER
Weibull
0.01 – 0.20
30 - 70
C-FER
Non-standard
0.05 – 0.15
45 - 140
ISO (2001)
Shifted
(3)
Lognormal
2.08
50
C-FER
Growth rate of average
defect depth
mm/year
Ratio of maximum to
average defect depth
-
(1)
(1)
(2)
(4)
(1) Based on in-house data (9 pipelines of 700 km total length), information in literature, and judgement. Majority of pipe
was tape coated.
(2) From Appendix D of the standard, which presents an example based on a project carried out by Advantica to justify
pressure uprating of an offshore pipeline.
(3) Minimum value = 1.0.
(4) Based on geometric defect data from corroded pipe section taken out of service (Keifner and Veith 1989)
Table B.6 Parameter Distributions for External Corrosion Defect Characteristics
Information obtained from ISO (2001) is based on in- line inspection data obtained by Advantica
for pipelines with different ages. The defect depth growth rates quoted here were back calculated
from a series of age-specific depth distributions that were given in the original reference. The
specific characteristics of the pipelines represented in this database are not known. The
information provided by C-FER is based on nine pipelines with a combined length of 700 km.
Eight of these pipelines had tape coating and one had polyethylene coating with tape at the joints.
Most of these lines were between 20 and 25 years old.
Depth growth rates were estimated assuming linear growth from a depth of zero over the pipeline
life. This approach may lead to underestimating the growth rate because little corrosion activity
occurs in the first few (up to 10) years of the pipeline life. This is offset by the fact that
corrosion growth is likely to be an exponentially decaying, rather than a linear, function of time.
The values in the table are further supported by the results of field tests for unprotected steel
(e.g. Crews 1976 and Matsushima 2000). Test results published by Matsushima (2000), for
example, give pitting corrosion rates between 0.033 and 0.33 mm per year. Given that the ratio
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Appendix B
between maximum and average defect depths is approximately 2, the growth rate for average
defect depth is between 0.016 to 0.16 mm per year. The COV of growth rate is also corroborated
by Sheikh et al. (1990) who suggested a Weibull distribution with a COV of 60% for corrosion
growth rates in water injection lines.
Defect length growth rate could not be estimated because defects are expected to have a finite
length at initiation (equal to the length of damaged coating). Although it is believed that the
length of a coating damage feature (and consequently of the resulting corrosion defect) grows
over time, available data does not allow distinction between original length and accumulated
length.
The range of observed defect density is very wide (between 0.1 and several thousand per km).
The median of the data available to C-FER was 1.75 defects per km. High corrosion density
values typically correspond to pipelines with problematic coating. If specific coating problems
are not indicated, the defect density is likely to be near the low end of the range.
B.4.2 Dents and Gouges
Available data on the geometry of dents and gouges are summarized in Table B.7, which shows
considerable variations in data obtained from different sources. Proprietary information obtained
by C-FER supports the gouge depth distribution given in ISO (Advantica’s data) and the mean
gouge length given by Wattis and Noble (1998). However, the COV of gouge length given in
that database is significantly higher tha n the value given by Wattis and Noble (1998).
Variable
Units
Dent depth
mm
Gouge depth
mm
Distribution Type
Mean
COV
(%)
Source
Weibull
13
95
ISO 2001
Deterministic
50
0
Jiao et al. (1992)
Weibull
1.2
92
Fuglem (2003)
Exponential
0.53
100
Jiao et al. (1992)
0.5
100
Fuglem et al. (2001)
-
-
ISO 2001
Weibull
249
125
Wattis and Noble (1998)
Weibull
153
125
Fuglem et al. (2001)
Uniform
π/4
58
Fuglem et al. (2001)
Weibull
Offset Logistic
Gouge length
Gouge Orientation
mm
rad
(3)
(1,2)
(1)
(1) From Appendix D of the ISO standard, which presents an example based on a project carried out by Advantica to
justify pressure uprating of an onshore pipeline.
(2) For a number of pipelines with different characteristics.
(3) Defined by distribution parameters (see ISO 2001) – mean and COV of data not given.
Table B.7 Parameter distributions for defect characteristics
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Appendix B
B.4.3 Seam Weld Cracks
Table B.8 summarizes available information related to seam weld crack size and growth rate
constants. Information on the size of seam weld defects is very sparse in the literature. The
distribution shown in Table B.7 has been derived by Jiao et al. (1995). It represents the
distribution of crack sizes that are likely to escape detection by QA procedures of typical
accuracy.
Variable
Units
Distribution Type
Mean
COV
(%)
Source
Seam weld defect
depth
mm
Exponential
0.18 mm
100
Jiao et al. (1995)
Lognormal
2.5 x 10
-13
54
Stacey et al. (1996)
Lognormal
1.1 x 10
-13
55
Jiao et al. (1997a)
Deterministic
3.0
0.0
Stacey et al. (1996)
Deterministic
3.1
-
Jiao et al (1997a)
Flaw growth parameter
gh1
-2/3
N mm
Flaw growth parameter
gh2
(1,2)
(3)
(3)
-
(1) Based on cracks that are likely to escape QA process.
(2) Author’s assume constant crack aspect ratio a/c = 0.2 where a is the depth (mm) and c is the half length (mm).
(3) For base metal – evidence presented by Mayfield and Maxey suggests that these values are still conservative for the
weld zone.
Table B.8 Parameter Distributions for Defect Characteristics
Flaw depth growth by fatigue is assumed to take place according to the “Paris law” which gives
the change in crack depth h per stress cycle N as:
dh
g
= g h1 (∆K ) h 2
dN
for
∆K ≥ ∆K 0
[B.5]
where ∆K is the stress range and gh1 and gh2 are growth rate parameters that are estimated from
regression of experimental data representing dh/dN versus ∆K. In characterizing the uncertainty
on fatigue crack growth rates it is common practice to treat the growth model exponent gh2
(i.e. the slope of the best-fit line) as a constant and associate all of the growth model uncertainty
with the growth model constant gh1 . Typical values of these parameters are given in Table B.8.
Although these values were developed for the base metal, evidence presented by Mayfield and
Maxey (1982) suggests that their use in the weld zone is conservative.
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Appendix B
B.5 REFERENCES
American Gas Association (AGA) 1977. Statistical Properties of the Arrest Criterion as
Related to the Pipe Mill Data - Appendix B. AGA Sixth Symposium on Line Pipe
Research.
Chen, Q. and Nessim, M.A. 1999. Reliability- Based Preve ntion of Mechanical Damage to
Pipelines. Proceedings from the EPRG/PRCI 12th Biennial Joint Technical Meeting
on Pipeline Research, Groningen, The Netherlands, May 17-21, 1999, pp. 25-1 - 2512.
Chen, Q. and Nessim, M.A. 2000. Reliability- Based Prevention of Mechanical Damage to
Pipelines. Submitted to the Pipeline Research Committee International, American
Gas Association, Project PR-244-9729, C-FER Report 97034, August.
Committee on Gas and Liquid Fuel Lines 1984. Guidelines for the Seismic Design of Oil
and Gas Pipeline Systems. American Society of Civil Engineers, New York.
Crews, D.L. 1976. Interpretation of Pitting Corrosion Data from Statistical Prediction
Interval Calculations. Galvanic and Pitting Corrosion - Field and Laboratory Studies.
ASTM STP 576, American Society for Testing and Materials, pp. 217 - 230.
Doctor, R.H., Dunker, N.A. and Santee, N.M. 1995. Third-Party Damage Prevention
Systems. Prepared by NICOR Technologies Inc., GRI-95/0316, Gas Research
Institute, Chicago.
Driver, R.G. and Zimmerman, T.J.E. 1998. A Limit States Approach to the Design of
Pipelines for Mechanical Damage. Proceedings of the Seventeenth International
Offshore & Arctic Engineering Conference, OMAE98-1017, Lisbon, Portugal, July.
Eiber, R.J. 1977. Investigation of Charpy Plateau Energy Variation in Production Pipe
Orders. Proceedings of the Nineteenth Mechanical Working and Steel Processing
Conference, pp. 55-73.
Fuglem, M. 2003. Software for Estimating the Lifetime Cost of High Strength, High Design
Factor Pipelines. Report Prepared for Gas Research Institute, GRI-8505, April.
Fuglem, M., Chen, Q., and Stephens, M. 2001. Pipeline Design for Mechanical Damage.
Submitted to the Pipeline Research Committee International, Project PR-244-9910,
C-FER Report 99024.
Gartner A.W., Graf M.K., and Hillenbrand, H.G. 1992. A Producer’s View of Large
Diameter Linepipe in the Next Decade. International Conference on Pipeline
Reliability, 1992.
Gräf, M.K., Hillenbrand, H.G. and Neiderhoff, K.A. 1993. Production of Large Diameter
Linepipe and Bends for the World’s First Long-Range Pipeline in Grade X80
(GRS550). 8th Symposium on Line Pipe Research, Houston (Texas), September 2629.
B.15
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Appendix B
Hansen, J.B. 1961. The Ultimate Resistance of Rigid Piles Against Transverse Forces.
Bulletin 12, Danish Geotechnical Institute, Copenhagen, Denmark.
Hillenbrand, H., Koppe, T., and Niederhoff, K. 1999. Manufacture of Longitudinally
Welded Large-Diameter Pipe from Fully Martensitic Low-Carbon 13% Chromium
Steels, EPRG/PRCI 12 Biennial Joint Technical Meeting on Pipeline Research,
Gronigen, The Netherlands.
ISO 2001. “Petroleum And Natural Gas Industries – Pipeline Transportation Systems –
Reliability Based Limit State Methods”. ISO Standard - ISO CD 16708, Revision
No. 02, October 2000.
Jiao G., Sotberg T., Bruschi R., Igland, R. 1997a. The Superb Project: Linepipe Statistical
Properties and Implications in Design of Offshore Pipelines, 1997 OMAE – Vol. V,
Pipeline Technology, ASME 1997, pg 45-56.
Jiao, G., Sotberg, T. and Bruschi, R. 1992. Probabilistic Assessment of the Wall Thickness
Requirement for Pressure Containment of Offshore Pipelines. Proceedings of the 11th
International Conference on Offshore Mechanics and Arctic Engineering, Volume VA, Pipeline Technology, ASME.
Jiao, G., Sotberg, T. and Igland, R. 1995. Report No. STF70 F95212 – SUPERB 2M
Statistical Data – Basic Uncertainty Measures for Reliability Analysis of Offshore
Pipelines. Submarine Pipelines – Superb Project No. 700411, June.
Kiefner, J.F. 1992. Installed Pipe, Especially Pre-1970, Plagued by Problems (Part 1).
Pressure Management Key to Problematic ERW Pipe (Part 2). Oil & Gas Journal,
Part 1 – August 10th , and Part 2 - August 17th .
Kiefner, J.F. and Vieth, P.H. 1989. Project PR 3-805: A modified Criterion for Evaluating
the Remaining Strength of Corroded Pipe. A Report for the Pipeline Corrosion
Supervisory Committee of the Pipeline Research Committee of the American Gas
Association.
Leewis, K. 1997.
Gas Research Institute (GRI), Arlington, Virginia, personal
communication.
Matsushima 2000. Carbon Steel - Corrosion by Soils, Uhlig’s Corrosion Handbook, John
Wiley & Sons, Inc.
Mayfield, M.E. and Maxey, W.A. 1982. Final Report on ERW Weld Zone Characteristics.
NG-18 Report No. 130 submitted to the American Gas Association, June 18.
NEN 3650 1992. Requirements for Steel Pipeline Transportation Systems.
(Netherlands). September.
NNI
Orr, T.L.L. 1994. Probabilistic Characterization of Irish Till Properties. Risk and Reliability
in Ground Engineering, Edited by B.O. Skipp, Published on behalf of the Institution
of Civil Engineers by Thomas Telford Services Ltd.
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Appendix B
Rajani, B., Robertson, P.K. and Morgenstern, N. 1995. Simplified Design Methods for
Pipelines Subject to Transverse and Longitudinal Soil Movement. Canadian
Geotechnical Journal, Vol. 32.
Rowe, R.K. and Davis, E.H. 1982. The Behaviour of Anchor Plates in Clay. Geotechnique,
Vol. 32, No. 1.
Sheikh, A.K., Boah, J.K. and Hansen, D.A. 1990. Statistical Modelling of Pitting Corrosion
and Pipeline Reliability. Corrosion, Vol. 46, No. 3, March.
Sotberg, T. and Leira, B.J. 1994. Reliability-based Pipeline Design and Code Calibration.
Proceedings of the Thirteenth International Conference on Offshore Mechanics and
Arctic Engineering, Vol. V, pp. 351 - 363.
Stacey, A., Burdekin, F.M., and Maddox, S.J. 1996. The Revised BS PD 6493 Assessment
Procedure – Application to Offshore Structures. Proceedings of the 15th International
Conference on Offshore Mechanics and Arctic Engineering, Volume III, Materials
Engineering, ASME.
Stephens, M. and Nessim, M. 2001. PIRAMID Technical Reference Manual, 2001.
Trigg, A. and Rizkalla, M. 1994. Development and Application of a Closed Form Technique
for the Preliminary Assessment of Pipeline Integrity in Unstable Slopes. OMAE
Vol V, Pipeline Technology. pp. 127-139.
Wattis, Z.E. and Noble, J.P. 1998. Development of a Mathematical Model for the
Minimization of Total Cost of Building, Operating, and Maintaining a Buried,
Onshore, Gas Transmission Pipeline. Conference Documentation - Risk Based and
Limit State Design and Operation of Pipelines, IBC UK Conferences Ltd.
Wolvert, G., Mures Z., Rousseau D. and Andrieux, C. 2004. Probabilistic Assessment of
Pipeline Resistance to Third Party Damage: Use of Surveys to Generate Necessary
Input Data. Proceedings of the International Pipeline Conference, IPC04-0656.
October.
Zimmerman, T.J.E., Cosham, A., Hopkins, P. and Sanderson, N. 1998. Can Limit States
Design be Used to Design a Pipeline Above 80% SMYS? Proceedings of the
Seventeenth International Conference on Offshore Mechanics and Arctic
Engineering, OMAE98-902, Lisbon, Portugal, July.
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APPENDIX C – METHODOLOGY TO CHARACTERIZE COMBINED PROPORTIONAL AND
INDEPENDENT MODEL ERROR
C.1 INTRODUCTION..................................................................................................................2
C.2 METHODOLOGY.................................................................................................................3
C.3 MODEL SELECTION...........................................................................................................6
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Appendix C
C.1 INTRODUCTION
In Section 7.6 of the main body of this document, two types of model error were defined. The
first, which is referred to as proportional error, defines the random model error component as
proportional to the quantity being calculated by the model. The second defines the model error
as independent of the quantity being calculated and is therefore referred to as independent model
error. Models to characterize proportional and independent error were described in Section 7.6.
In general, the model error band may take any shape, which means that the model error will not
necessarily conform to either of these two formats. In this Appendix, a model that incorporates a
combination of proportional and independent model errors is described.
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Appendix C
C.2 METHODOLOGY
Figure C.1 shows a plot of the model result, ym, versus the actual value (or test result), ya. The
plot reflects the assumption that the relationship between the actual value and model result is
linear and that the error band around the regression line is also linear. It shows that the error
band has a finite width at the origin and changes (increases) in width as ym increases. This
indicates that model error is represented by a combination of proportional and independent
components. To take this into account the origin is shifted to a location that eliminates the
independent error component (see Figure C.2). If the new origin has coordinates a and b, the
model is given by:
( y a − b) /( y m − a) = e1
[C.1]
where e1 is a random variable representing model error factor, and a and b are deterministic
constants.
ya
Regression Line
Error Band
Perfect Model
ym
Figure C.1 Illustration of Model Error Format Adopted in These Guidelines
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Appendix C
Regression line
Error band
Perfect Model
ya
ya’ = [ya - b)] / (ym - x)
σe1 (scatter)
1.0
µe1 (bias)
O’
(a, b)
O
(0,0)
ym’ = ym- x
ym
(a)
(b)
Figure C.2 Illustration of Approach to Define Independent and Proportional Error Components
This model can also be written as
y a = e1 y m + e2
[C.2]
where e2 = (b − a e1 ) , implying that e1 and e2 are dependent random variables with the same
distribution type and the following relationships between the distribution parameters
µ e 2 = b − a µ e1
[C.3a]
σ e 2 = a 2 σ e1 2
[C.3b]
The values of a and b, which are initially not known, are determined by the relative magnitudes
of proportional and independent errors as implied by the data points (yai, ymi). Since a and b are
defined to eliminate the independent error component in the transformed coordinate set, a
regression line of ( y ai − b) / ( y mi − a ) against ( y mi − a ) , will have only a proportional error
component. Similar to Equation [7.10] in the main report, this regression line must have a slope
of zero and an error band with constant width. These two conditions can be used to set up a
constrained optimization problem that can be solved to determine a and b, which define the
location of the new origin. The solution procedure is as follows:
1. Define y a ' = ( y ai − b) / ( y mi − a ) and x a ' = ( y mi − a) .
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Appendix C
2. Define the slope, sy(a, b), of the regression line for ya’ against xa’.
3. Divide the xa’ range into n intervals with an equal number of data points.
4. For each interval, define the standard deviation, σya, of ya’ and the mid point, xm, of the xa’.
5. Define the slope, sσy(a, b), of the regression line for σya against xm.
6. Find a and b that minimise sσy(a, b) subject to sy(a, b) = 0.
The probability distribution of e1 (and consequently e2) can be derived from a set of data points
representing yai and ymi, i = 1,…,n. This is done by substituting ymi and yai along with the
calculated values of a and b in Equation [6.14] to generate corresponding data points e1i, i =
1,…,n, which can be used to find a best fit distribution.
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Appendix C
C.3 MODEL SELECTION
The model described in C.2 can be used to test a given data set to evaluate the significance of
each of the proportional and independent error components and determine whether one of them
could be eliminated. This could be achieved by applying the following criteria to the calculated
values of the constants a and b and the mean and standard deviation of e1:
•
Dominant proportional error. Small values for both a and b indicate that the origin in
Figure C.2 does not need to be moved significantly to eliminate the independent error
component, which implies that proportional error dominates. This can be verified using
Equation [C.1], which reduces to the proportional error format (Equation [7.10] in the main
body of the guidelines) if zero is substituted for both a and b.
•
Dominant independent error. If the independent error dominates, the origin would
theoretically have to be moved to negative infinity to find the intersection point between the
regression line and error bounds. Equation [C.2] shows that this would be satisfied if a and b
are large negative numbers and e1 has a standard deviation close to 0.0, such that the mean
and standard deviation of e2 as given in Equations [C.3] are finite. This means that the
proportional error component, e1, will be deterministic and the independent error component
e2 will be random, and the model reduces to the independent format in Equation [7.12] in the
main body of thee guidelines.
For illustration, the example described in Section 7.6.3 in the main body of the guidelines was
solved using the model described in Section C.2. The resulting parameter values were found to
be: a = -92852.476, b = -107125.004, µe1 = 1.153711833, σe1 = 3.078 x 10-8. The large negative
values of a and b and the small value of σe1 confirm that the independent error dominates. The
mean and standard deviation of independent error component e2 can be calculated by using these
values in Equation [C.3]. This gives µe2 = -0.00375 and σe2 = 0.00286, which are essentially the
same values obtained from the independent model error calculation used in Section 7.6.3 of the
main guidelines.
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APPENDIX D - BASIC PROBABILITY CONCEPTS
D.1 INTRODUCTION..................................................................................................................2
D.2 DEFINITION OF PROBABILITY..........................................................................................3
D.3 BASIC PROBABILITY RULES............................................................................................4
D.3.1
D.3.2
D.3.3
D.3.4
Venn Diagrams and Probability Axioms
Intersection and Union of Events - The Addition Rule
Conditional Probabilities and Independence - The Multiplication Rule
The Rule of Total Probability
4
4
7
9
D.4 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS.......................................10
D.4.1
D.4.2
D.4.3
Discrete Random Variables and Distributions
Continuous Random Variables and Distributions
Distribution Types and Parameters
10
12
14
D.5 REFERENCES...................................................................................................................19
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Appendix D
D.1
INTRODUCTION
This appendix outlines the basic probability concepts required to follow the subjects covered in
the main body of these guidelines. It covers the following topics:
•
Definition of probability (Section D.2).
•
Basic probability rules and axioms including event intersections and unions, conditional
probabilities and independence, and the rule of total probability (Section D.3).
•
Random variables and the use of probability distributions to model them (Section D.4).
The presentation in this appendix is informal, focusing on conceptual aspects and avoiding
mathematical details. Detailed information on the basics of probability theory and its
engineering applications can be found in Benjamin and Cornell (1970), Chapter 2, and Ang and
Tang (1975), Chapters 2 and 3.
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Appendix D
D.2 DEFINITION OF PROBABILITY
Probability of an event is defined as “A degree of belief” that the event will occur. This
definition implies that probability is defined by a certain person(s) referred to as the probability
assignor. Probability is not a characteristic of the event itself, but a reflection of the state of
knowledge (or ignorance) of the assignor regarding the event. The ability to assign probabilities
is not dependent on observations or data (although this type of information is very helpful), but
can be based on other concepts such as logic or judgment.
Example D.1: Assume that a ball is drawn from a bag containing 100 balls some of which are
red and the rest are black. What is the probability of this ball being red? Without any
information regarding the number of black or red balls, the most reasonable probability
assignment would be 0.5. This reflects complete ignorance: there is as much chance, in the
assignor’s mind, of the bag containing n black balls and (100-n) red balls, as of it containing n
red balls and (100-n) black balls. If 10 balls are drawn randomly (with replacement) and 8 of
them found to be red, it would be reasonable for the assignor to change the probability of
drawing a red ball to approximately 0.8.
It must be noted that there is no consensus among statisticians regarding the above definition of
probability (called the subjective definition). Some prefer to use another definition called the
relative frequency definition. The subjective probability definition has been adopted because it
is the most generally applicable and most consistent with the requirements of reliability-based
design. The intention is not to minimize the importance of good data and information in
reaching appropriate conclusions, but to ensure flexibility in applying the methodology. The
approach allows sensitivity analyses to weigh the costs and benefits associated with acquiring
new information and make informed decisions on data collection.
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Appendix D
D.3 BASIC PROBABILITY RULES
D.3.1
Venn Diagrams and Probability Axioms
Venn diagrams represent an uncertain event as a closed region within a rectangle (e.g. event A
in Figure D.1). The rectangle represents all possible events that can occur and is referred to as
the sample space. For example, event A may represent “pipeline segment failure”, in which case
the remainder of the sample space represents “no failure”. For the purposes of this document it
will be assumed that the area corresponding to a given event as a ratio of the total area of the
rectangle represents the probability of the event. Probability of the event A is denoted p ( A) .
This illustration can be used to verify that the following rules or axioms apply:
•
Probability is a number between zero and 1.0.
•
The sum of the probabilities of all possible events (i.e. the probability of the sample space
itself) is 1.0. This implies that p(not A) = 1.0 − p ( A) as illustrated in Figure D.1.
Not A
Sample Space
A
Figure D.1 Venn Diagram Illustration of a Random Event
D.3.2
Intersection and Union of Events - The Addition Rule
Figure D.2a illustrates a Venn diagram with two events A and B. In this representation the
intersection area between A and B (see Figure D.2b) represents the probability of occurrence of
both events. It is referred to as the intersection of events A and B and is denoted p( A I B). The
area encompassed by both events (see Figure D.2c) represents the probability that either or both
of the events occur. This is called the union of events A and B and is denoted p( A U B).
Example D.2: Consider a cable with two strands and define the events A=“failure of the first
strand” and B=“failure of the second strand”. In this case p ( A I B ) is the probability of failure
of both strands (intersection), which is equivalent to the probability of failure of the cable. If a
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Appendix D
chain with two links is considered with A=“failure of the first link” and B=“failure of the
second link”, then failure of the chain takes place if either or both of the links fail. This
corresponds to the definition of the union of A and B and the probability of failure of the chain is
given by p( A U B).
The union and intersection probabilities of two events are related by:
p ( A U B) = p( A) + p( B) − p ( A I B)
[D.1]
This equation results from adding the areas corresponding to the two events and subtracting the
intersection area that was included twice in the sum. Events that cannot occur together are called
mutually exclusive events. In this case p ( A I B ) is zero and Equation [D.1] becomes:
p( A U B) = p( A) + p( B)
[D.2]
Example D.3: The events A=“making a right turn” and B=“making a left turn” at a given
intersection are mutually exclusive.
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Appendix D
Sample
Space
B
A
a) Two Events A and B
Sample
Space
AI B
b) Intersection of A and B
Sample
Space
AU B
c) Union of A and B
Figure D.2 Venn Diagram Illustrations of Event Intersections and Unions
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Appendix D
D.3.3
Conditional Probabilities and Independence - The Multiplication Rule
The probability of event A given that event B has occurred is called the conditional probability
of A given B and is denoted p ( A | B) . Figure D.3 shows that this probability corresponds to the
chance of a certain point being in A given that we know that it is in B. Because we know that B
has occurred, the part of the sample space outside of event B is no longer a possible outcome and
can be eliminated. This means that B itself becomes the sample space. The probability p( A | B)
can then be calculated as the ratio of the portion of B that includes A as well (i.e. p ( A I B ) ) to
the total area of B. This means that:
p(A | B) = p(A I B)/p(B)
[D.3]
p(A I B) = p(A | B) × p(B)
[D.4]
or
and similarly
p(A I B) = p(B | A) × p(A)
Original
Sample
Space
[D.5]
Eliminated
Because B
Occurred
A|B
B
Figure D.3 Conditional Probability of A given B
Example D.4 (from Ang and Tang 1975): Consider a 100 km pipeline and define the events
A=“a failure in km 0 to 30” and B=“ a failure in km 20 to 60” (see Figure D.4). Assuming that
failures are equally likely to occur anywhere on the pipeline, then if a failure occurs we can
assign p( A) = (30 − 0) / 100 = 0.3 and p ( B) = (60 − 20) / 100 = 0.4 . Now if we know that B is true
(i.e. a failure has occurred in the interval 20 to 60 km), what is the probability that A is true (i.e.
that the failure is also in the interval 0 to 30 km)? This is the conditional probability p( A | B) ,
which can be estimated by the proportion of B that is overlapped by A.
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Appendix D
Figure D.4 shows that this is given by p( A | B) = 10 / 40 = 0.25 , which corresponds to the
application of Equation [D.3].
km 20 to 60
km 0 to 30
100 km Pipeline
Figure D.4 Illustration of Pipeline Example of Conditional Failure
If the occurrence (or nonoccurrence) of event A has no influence on one’s judgment regarding
the probability of B, then A and B are independent, which implies that p( A | B) = p( A) and
Equation [D.4] becomes:
p(A I B) = p(A) × p(B)
[D.6]
which states that the probability of occurrence of both A and B is simply the product of their
individual probabilities. Also for independent events Equation [D.1] becomes:
p(A U B) = p(A) + p(B) - p(A) × p(B)
[D.7]
Example D.5: Consider the cable mentioned in Example D.2. Assume that the probability of
failure of each strand is p( A) = p( B) = 0.01 and that failures of the strands are independent
events. In this case the probability of failure of the cable is given by Equation [D.6]
as p( A I B) = 0.01 × 0.01 = 10 −4 . Under the same assumptions the probability of failure of the
chain
in
example
D.2
is
given
by
Equation
[D.7]
as
p( A U B) = 0.01 + 0.01 − (0.01) 2 = 0.0199 ≅ 0.02 . Note that the influence of the product term
(0.01)2 is negligible because p(A) and p(B) are small. This is often omitted to simplify the
analysis for small probabilities that are typical of the failure probabilities encountered in
engineering applications.
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Appendix D
D.3.4
The Rule of Total Probability
The rule of total probability is used when the probability of an event A cannot be assigned
directly but can be assigned conditionally for a number of mutually exclusive events B1, B2,....,
Bn. This is shown in Figure D.5, which demonstrates that the probability of A can be calculated
as the sum of the probabilities of its occurrence with each of the events B1, B2,...., Bn. This
corresponds to:
p(A) = p(A I B2 ) + ..... + p( A I Bn )
[D.8]
Note that Equation [D.8] applies only if B1, B2,...., Bn occupy the whole sample space. If not, the
right hand side must be divided by p( B1 U B2 U K U Bn ) . Using Equation [D.5], [D.8] becomes:
p(A) = p(A | B1 ) × p(B1 ) + p(A | B2 ) × p(B2 ) + ..... + p(A | Bn ) × p(Bn )
[D.9]
which can be interpreted as the sum of the conditional probabilities of A given B1, B2,..., Bn, each
weighted by the probability of occurrence of the corresponding B event.
Example D.6: An engineer submitted two alternative plans for a pumping system and is awaiting
the final management decision on which system will be installed. The two plans involve two
different types of pumps. In the mean time the engineer needs to do some analysis on down time
of the system which requires defining the probability that a pump will be out of service at least
once during any given month (event A). The following probabilities are assumed:
•
event B1 = “management selects first type of pumps” with p(B1) = 0.7
•
event B2 = “management selects second type of pumps” with p(B2) = 0.3
•
probability of first pump being out of service is p(A|B1) = 0.08
•
probability of second pump being out of service is (A|B2) = 0.06
Using Equation [D.9] the probability of A is given by p( A) = 0.7 × 0.08 + 0.3 × 0.06 = 0.074 .
B1
B2
B3
Bn
Sample
Space
A
Figure D.5 Illustration of the Rule of Total Probability
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Appendix D
D.4 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
D.4.1
Discrete Random Variables and Distributions
A discrete random variable is one that can assume only a discrete number of values. For
example the number of cracks (N) in a given weld can only be an integer number, N = 0, 1, 2, .....
Figure D.6a shows an example of a discrete probability distribution. The probability distribution
is denoted pN(n), where N is the random variable itself and n is a specific value of n. According
to one of the probability axioms discussed in Section D.3.1, the sum of pN(n) for all values of n is
equal to 1.0.
The Cumulative Distribution Function CDF is defined as a function of n whose value is the
probability that N is less than or equal to n. This means that it is the probability that N = 0, 1,
2,…, or n. This is written as:
FN ( n ) = p( N = 0 U N = 1 U K U N = n )
[D.10]
Considering that the events in Equation [D.10] are mutually exclusive (because N can only take
one value), Equation [D.2] can be used leading to:
FN ( n ) = p N (0) + p N (1) + ...... p N ( n )
[D.11]
Figure D.6b shows the CDF corresponding to the probability distribution in Figure D.6a. It can
be seen from Equation [D.11] that the CDF is an increasing function of n in the range of 0.0
to 1.0.
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Appendix D
0.3
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
5
6
7
n
a) Probability Distribution
1
0.9
0.8
Cumulative Probability
Probability
0.25
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
7
n
b) Cumulative Distribution
Figure D.6 Probability Models for Discrete Random Variable
D.11
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Appendix D
D.4.2
Continuous Random Variables and Distributions
A continuous random variable is one that can assume any value within a given range. For
example, the load, X, resulting from an excavator hit on a pipeline can take any positive value 0
< X < ∞. Figure D.7a shows an example of a continuous probability density function (PDF).
The PDF is denoted fX(x). In this notation X represents the random variable itself and x
represents a specific value of X. The PDF does not give probability values for different values of
x. The reason for this is that there are an infinite number of values that a continuous random
variable X can take. This implies that the probability of any specific value must be zero because,
if a finite probability value is attached to each of the (infinite) possible values of X, the sum of
these probabilities would be infinite as well. This would violate the probability axiom stating
that the sum of probabilities must be 1.0. Instead a PDF, is defined as a function of x such that
the area under the PDF between any two values of x is equal to the probability of X having a
value between these two values (see Figure D.7a). This is why the abscissa of the curve in
Figure D.7a is referred to as a “probability density” rather than a probability. The above
definition of a PDF can be written mathematically as:
p( x 1 < X < x2 ) = ∫
x2
x1
f X ( x )dx
[D.12]
Where the integration calculates the area under the PDF in the specified range of x. The total
area under the PDF represents the probability of all possible values of x and is therefore equal
to 1.0.
The Cumulative Distribution Function (CDF) is defined as a function of x whose value is
equal to the probability that X is less than or equal to x. This is denoted FX(x), and according to
the above definition of the PDF is equal to the area under the PDF for all values less than and up
to x. This is given by:
x
FX ( x ) = ∫ f X ( x )dx
0
[D.13]
Figure D.7b shows the CDF corresponding to the probability distribution in Figure D.7a. As for
discrete random variables, the CDF is an increasing function of x in the range of 0.0 to 1.0.
Example D.7: The probability density function in Figure D.7b can be used to calculate the
probability that a certain excavator load value will be exceeded in an impact. For an excavator
load of x = 400 kN for example the CDF value is 0.98. This means that p( x ≤ 400) = 0.98 . The
probability p ( x > 400) = 1.0 − p ( x ≤ 400) = 1 − FX (400) = 0.02 . This information is useful for
the selection of a design load value. It can also be verified that the probability of the load being
between 350 and 400 kN is given by p(350) < x < 400) = F (400) − F (350) = 0.03 .
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Appendix D
0.008
Probability Density
0.007
0.006
0.005
0.004
0.003
0.002
p (x 1 < X < x2 )
0.001
0
0
100
x1 x2
200
300
400
500
300
400
500
Excavator X (kN)
a) Probability Density
Cumulative Probability
1
0.8
0.6
0.4
0.2
0
0
100
200
Excavator X (kN)
b) Cumulative Distribution
Figure D.7 Probability Models for Continuous Random Variables
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Appendix D
D.4.3
Distribution Types and Parameters
In addition to being a function of the random variable, a probability distribution is also a function
of a number of other distribution parameters (most distributions are defined by two parameters).
Changing these parameters produces a family of curves that can fit different random variables.
The normal distribution for example is given by:
 1  x − µ 2 
exp − 
f X ( x) =
 
σ 2π
 2  σ  
1
[D.14]
where µ and σ are the mean and standard deviation of the random variable x. Figure D.8 shows
how changes in the mean and standard deviation of a random variable with a normal distribution
affect the distribution. Increasing the mean value at a constant standard deviation results in
shifting the distribution to the right while maintaining the same shape (Figure D.8a). The mean
value is therefore referred to as the position or central tendency measure. On the other hand,
increasing the standard deviation without changing the mean results in flatter or more spread
distributions that have the same central value (Figure D.8b). This implies that the random
variable has a wider range of variability. The standard deviation is therefore referred to as a
measure of spread or variability.
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Probability Density
Appendix D
0.1
0.08
0.06
0.04
0.02
0
0
20
40
60
80
100
Random Variable
Probability Density
a) fixed standard deviation - different mean values
0.1
0.08
0.06
0.04
0.02
0
0
20
40
60
80
100
Random Variable
b) fixed mean - different standard deviations
Figure D.8 Effect of Distribution Parameters on Shape of Distribution
The normal distribution is a special case in that its parameters are the mean and standard
deviation themselves. Parameters for other distributions are generally not equal to the mean and
standard deviation, but are related to the mean and standard deviation by simple formulae.
Table D.1 gives a summary of some common distributions and defines their basic characteristics,
their parameters and how these parameters relate to the mean and standard deviation.
The selection of an appropriate distribution for a specific random variable depends on the
characteristics of the random variable and how they compare to the characteristics of the
distribution. These characteristics include for example the type of the random variable (discrete
or continuous) and its range (e.g. -∞ to ∞, or 0.0 to ∞). There are also statistical tests and
procedures that can assist in selecting the best distribution type among those that satisfy the basic
characteristics of the random variables. These methods use data describing the value of the
random variable under consideration. This topic is discussed in more detail in Section 7 of the
main guideline.
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Appendix D
Distribution
Name
Rectangular
Normal
Lognormal
Exponential
Gamma
Range of
Definition
Density
Function
Mean
a≤ x≤b
1
b−a
a+b
2
− ∞ < x < +∞
0 ≤ x < +∞
1
1
xδ 2π
− ∞ < x < +∞
αe
General Shape
of PDF
(b − a )
f
12
f
1  ln ( x / m ) 
− 

2
δ

µ
σ
f
me
δ2 /2
me
δ2 /2
(Extr. value
0 ≤ x < +∞
type III)
Rayleigh
0 ≤ x ≤ +∞
−1
1
1
λ
λ
k
k
λ
λ
0.577
π
α
α 6
 1
uΓ1 + 
 k
 2
 1
u Γ1 +  − Γ 2 1 + 
 k
 k
− λx
k −1
e
 x
− 
u
k
 x2 
exp
−
2 
α2
 2α 
x
e
δ2
x
−α ( x −u )−e − α ( x −u )
k  x
 
uu
x
2
u+
type I)
Weibull
x
2
1
λ (λx )k −1 e −λx
Γ(k )
Gumbel
(Extr. value
e
λe
0 ≤ x < +∞
0 ≤ x < +∞
e
σ 2π
1  x−µ 
− 

2 σ 
Standard
Deviation
α
π
2
f
x
f
x
f
x
f
x
f
α ( 2 − π / 2)
x
Table D.1 Common Probability Distributions
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Appendix D
D.5 REFERENCES
Ang, A.S. and Tang, W.H., 1975. Probability Concepts in Engineering Planning and Design
Volume 1 - Basic Principles. John Wiley and Sons, N.Y.
Benjamin, J.R. and Cornell, C.A., 1970. Probability, Statistics, and Decision for Civil
Engineers. McGraw-Hill Publishing Company, USA.
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APPENDIX E – FAILURE PROBABILITY CALCULTION FOR SEISMIC LOADING
E.1 OVERVIEW OF SEISMIC HAZARD FOR BURIED PIPELINES .........................................2
E.2 SURFACE FAULTING.........................................................................................................3
E.3 LIQUEFACTION-INDUCED LATERAL SPREADING.........................................................6
E.3.1
E.3.2
Introduction
PRCI Approach
E.3.2.1 Liquefaction Potential
E.3.2.2 Liquefaction-Induced Ground Movement
E.3.2.3 Pipeline Failure Probability Estimate
E.3.3 HAZUS Approach
E.3.3.1 Liquefaction Potential
E.3.3.2 Liquefaction-Induced Ground Movement
E.3.3.3 Guidance on Characterizing Soil Liquefaction Susceptibility
E.3.3.4 Pipeline Failure Probability Estimate
6
6
6
9
10
14
14
15
16
18
E.4 REFERENCES...................................................................................................................19
E.1
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Appendix E
E.1 OVERVIEW OF SEISMIC HAZARD FOR BURIED PIPELINES
This appendix presents a methodology to evaluate the failure probability for a buried pipeline
due to seismic hazards. There are generally two types of seismic hazards for buried pipelines,
namely seismic wave propagation hazard and Permanent Ground Deformations (PGD)
(O’Rourke and Liu 1999). Wave propagation hazard is characterized by the transient strain and
curvature that occur due to travelling wave effects. The corresponding potential damage occurs
over large areas and the associated pipeline failure rates are low (O’Rourke and Liu 1999).
Because of this, wave propagation is not considered further in this appendix.
PGD damage typically happens over small areas and can result in significant pipeline failure
rates. PGD can result from surface faulting, lateral spreading due to soil liquefaction, soil
settlement and landslides. This study will focus on PGD hazard due to surface faulting and
liquefaction-induced lateral spreading, as they are the predominant causes of pipeline failures
during seismic events (O’Rourke and Liu 1999).
Surface faulting is the relative movement between two portions of earth crust along an active
fault. Surface fault movement is typically localized and abrupt. Lateral spreading due to
liquefaction occurs when a loose saturated sandy soil deposit loses its shear strength due to
seismic shaking, resulting in lateral movement of liquefied soil. PGD hazards are characterized
by the amount of the PGD as well as the geometry and spatial extent of the PGD zone.
The annual failure rate of the pipeline due to a certain type of PGD hazard can be evaluated using
Equations [9.8], [9.9], and [9.10] given in Chapter 9. For convenience of reference, Equations
[9.9] and [9.10] are repeated in this appendix as follows:
p f = ∫ p f w p0 w fW ( w ) d w
p f |w = p[m = g ( x ) ≤ 0] =
[9.9]
∫ f( x|w ) d x
[9.10]
g ( x )≤ 0
The term g(x) in Equation [9.10] represents a limit state function representing failure due to a
specified ground movement event. Let U denote the magnitude of permanent ground
deformation during an earthquake and V the vector of random variables (other than U) that
determine the performance of pipelines subjected to permanent ground deformation. Thus, g(x)
can be rewritten as g(v, u). Note that U is generally independent of V. Models that can be used
to define g(v, u) are described in (C-FER 2004). Equation [9.10] can be rewritten as:
pf w =
∫ f ( v) f (u w)dvdu
V
g ( v ,u )<0
U
[E.1]
where fV (v ) is the joint probability density function of V and fU(u|w) is the probability density
function of U conditional on the earthquake characteristics (represented by a vector w). The
following sections will discuss the evaluation of fU(u|w), p0|w, and fW(w) for PGD hazards due to
surface faulting and liquefaction-induced soil lateral spreading.
E.2
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Appendix E
E.2 SURFACE FAULTING
Pipelines that intersect an active fault are likely to be subject to fault movement if faulting
reaches the surface during an earthquake. There are mainly three types of fault movement,
namely strike-slip, normal-slip and reverse-slip movements (O’Rourke and Liu 1999). A detailed
site-specific seismologic analysis is preferred to estimate the probability of surface faulting and
the magnitude of surface fault movement given a seismic event. This type of analysis requires
such information as moment magnitude on the fault, fault surface rupture length, fault slip type
and slip rate, and rigidity of the earth’s crust at fault rupture location (Honegger and Nyman
2001). In lieu of a detailed analysis, the amount of surface fault movement can be estimated
from the earthquake moment magnitude on the fault. Many empirical equations have been
proposed in the literature relating surface fault movement to earthquake moment magnitude. The
models proposed by Wells and Coppersmith (1994) are adopted here because they were
developed based on a large database of historical earthquakes worldwide. Wells and
Coppersmith (1994) suggested that the fault movement can be estimated as follows:
log u fa = −6.32 + 0.90me ,σ fa = 0.28
log u fm = −7.03 + 1.03me ,σ fm = 0.34
log u fa = −4.45 + 0.63me ,σ fa = 0.33
log u fm = −5.90 + 0.89me ,σ fm = 0.38
log u fa = −0.74 + 0.08me ,σ fa = 0.38
log u fm = −1.84 + 0.29me ,σ fm = 0.42
for Strike-slip fault
[E.2]
for Normal-slip fault
[E.3]
for Reverse-slip fault
[E.4]
where u fa is the mean (for different earthquakes) of the average fault displacement (m) in a
given earthquake, conditional on a given moment magnitude me, u fm is the conditional mean of
the maximum fault displacement (m), σfa and σfm are the conditional standard deviations of the
average and maximum fault displacements (m), respectively. The conditional fault displacement
can be modelled by a lognormal distribution (Wells and Coppersmith 1994). Note that σfa and
σfm do not depend on the moment magnitude. If a fault type is unknown or unclear, Wells and
Coppersmith (1994) suggested using an all-type-slip equation shown below to estimate u fa and
u fm .
log u fa = −4.80 + 0.69me ,σ fa = 0.36
log u fm = −5.46 + 0.82me ,σ fm = 0.42
for all fault types
[E.5]
They further suggested that Equation [E.5] could also be used to estimate fault displacements for
normal and reverse-slip faults.
E.3
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Appendix E
Honegger and Nyman (2001) suggest that the use of maximum or average fault displacement for
pipeline design should depend on the product being transported in the pipeline and the location
class of the pipeline.
The probability of hazard occurrence given a seismic event, p0 w , consists of two components:
the probability of faulting reaching surface for a given seismic event, p1, and the probability of
the pipeline being subjected to the surface fault movement, p2. Models to evaluate p1, are not
available; however, historical data suggest that surface faulting rarely occurs for earthquakes
with magnitudes less than 5.0 (PRCI Seismic Design Guidelines). Therefore, p1 can be
approximated as follows:
0
p1 = 
1
me ≤ 5.0
me > 5.0
[E.6]
Note that Equations [E.2] to [E.5] are developed based on data from historical earthquakes with
magnitudes between about five to eight.
Since surface faulting is associated with a certain surface rupture length, p2 can be estimated by
evaluating the probability of the pipeline being inside the surface rupture length given an
earthquake. Let Sr denote the surface rupture length for a given earthquake. It is assumed that
earthquakes are equally likely to occur at any point along an active fault of length Lf, that is, the
location of a given earthquake on the fault is uniformly distributed with a probability density
function equal to 1/ Lf (Cornell 1968). For a pipeline that crosses such a fault, p2 given surfact
faulting equals Sr / Lf (Sr / Lf ≤ 1.0) (see Figure E.1). If the magnitude of the earthquake is
assumed to be independent of its location along the fault, p0 w is then simply the product of p1
and p2:
 0
p0 w = p1 ⋅ p 2 = 
S r / L f
me ≤ 5.0
[E.7]
me > 5.0
pipeline
active fault
Sr
Lf
Figure E.1 Illustration of Surface Rupture Length
E.4
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Appendix E
Wells and Coppersmith (1994) proposed empirical models that relate the surface rupture length
to the earthquake moment magnitude. They suggested that an all-slip-type model be used to
assess the surface rupture length regardless of the fault slip type. This model is as follows:
log S r = −3.22 + 0.69me
[E.8]
where Sr is in km.
Based on Equations [9.9] and Equations [E.2] to [E.8], the failure probability of pipelines due to
surface faulting, p f , can be evaluated as follows:
m
1 u
pf =
p S r ( me ) f M e ( me )dme
L f 5∫.0 f me
pf m =
e
∫ f (v) f
V
g ( v ,u fm )<0
U fm
(u fm me )dvdu fm
[E.9]
[E.10]
Equations [E.9] and [E.10] are obtained by substituting Equation [E.7] into Equation [9.9] and by
replacing W and U in Equations [9.9] and [E.1] with moment magnitude, Me, and maximum
surface fault movement Ufm, respectively. Note that f M e (me ) is the probability density function
of the earthquake magnitude on the fault and that mu is the upper bound of the moment
magnitude. The notation S r (me ) is used to emphasize that the surface rupture length depends on
the moment magnitude ( S r (me ) ≤ Lf).
The probability distribution of the earthquake magnitude can be derived from the well-known
Gutenberg-Richter relationship, which takes the form:
log N = a − bme
[E.11]
where N is the annual number of earthquakes with magnitudes greater than me, a and b are
parameters determined from existing seismic data using, for example, the least squares method or
the maximum likelihood method (Reiter 1990). Cornell and Vanmarcke (1969) proposed a
truncated exponential distribution for the earthquake magnitude based on Equation [E.11]. The
truncation in the distribution results from the introduction of lower and upper bound earthquakes.
The cumulative distribution function of the earthquake magnitude, FMe(me), can be expressed as:
1 − FM e (me ) =
exp(− β (me − m0 )) − exp(− β (mu − m0 ))
1 − exp(− β (mu − m0 ))
[E.12]
where m0 and mu are lower and upper bound magnitudes, respectively. β is related to b in
Equation [E.11] by β = b ln10. The lower bound magnitude is typically about 5.0, which is
believed to be a conservative estimate of the minimum earthquake magnitude that could have a
deleterious effect on well-engineered structures (Reiter 1990). The determination of the upper
E.5
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Appendix E
bound magnitude is highly site-specific and often needs extrapolation from the existing seismic
data (Reiter 1990). Weichert and Milne (1979) suggested that the upper bound can be
determined from regional geology and/or tectonics using a suitable physical model.
E.3 LIQUEFACTION-INDUCED LATERAL SPREADING
E.3.1
Introduction
Pipelines that pass through saturated sandy soils may be subject to PGD hazards due to soil
liquefaction-induced lateral spreading. The ground movement due to lateral spreading includes
both horizontal and vertical components. However, the vertical component is typically small
(O’Rourke and Liu 1999). Therefore, only horizontal ground movement due to soil lateral
spreading is discussed in this document.
Two approaches for quantifying the PGD hazard due to soil liquefaction are presented. One
approach is suggested in the PRCI Seismic Design Guidelines (1998). The other approach is
based on HAZUS, the natural hazard loss estimation methodology developed by the Federal
Emergency Management Agency (FEMA 1999), and extended by C-FER. Both approaches
employ site-specific seismic and geotechnical information as well as empirical models developed
based on data collected in past earthquakes. Each of the two approaches can be used in
conjunction with Equations [9.9] and [E.1] to estimate the annual failure probability of pipelines
subject to liquefaction-induced ground movement.
E.3.2
PRCI Approach
E.3.2.1
Liquefaction Potential
The approach recommended in the PRCI Seismic Design Guide uses the peak ground
acceleration (PGA) and subsoil properties to quantify the probability of soil liquefaction. Soil
properties are obtained through subsurface investigations using standard penetration and cone
penetration tests (SPT and CPT). The liquefaction potential is assessed by comparing the
liquefaction resistance, measured by the cyclic resistance ratio (CRR), to the cyclic stress ratio
(CSR), which is primarily a function of PGA and in-situ soil stress. Liquefaction is likely to
occur if CSR exceeds CRR. CSR can be evaluated as follows:
σ 
CSR = 0.65 ⋅ a PG ⋅  v' 0  ⋅η
 σ v0 
[E.13]
where aPG is the peak ground acceleration (m/s2), σvo = γz, is the total stress at the test location,
which is z below the ground surface; σ'vo = γz – γw(z-Dw), is the effective stress at the test
location; γ is the unit weight of soil; γw is the unit weight of water; Dw is the depth of water table
at the time of the test, and η is the stress-reduction coefficient and can be evaluated as:
η=
1 − 0.4113 z + 0.04052 z + 0.001753z1.5
1 − 0.4177 z + 0.05729 z − 0.006205 z1.5 + 0.001210 z 2
[E.14]
E.6
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Appendix E
Note that CSR does not depend on SPT or CPT data.
The evaluation of CRR depends on whether soil properties are obtained using SPT or CPT. If
SPT data are used, CRR can be calculated as follows:


4.8(10) −2 − 4.721(10) −3 x + 6.136(10) −4 x 2 − 1.673(10) −5 x 3
 MSF ⋅ Kσ
 
1
−
3
2
−
4
3
−
6
4
CRR =   1 - 1.248(10) x + 9.578(10) x − 3.285(10) x + 3.714(10) x 
0.05 ⋅ MSF ⋅ K
σ

x≥3
x<3
[E.15]
x = ( N1 ) 60 FC
[E.16]
103.74
me4.33
[E.17]
MSF =
where


 1
0.897
0.411

−
Kσ = 0.514 +
'
 σ v 0   σ v' 0  2


 


 Pa   Pa 

 0.6
σ vo' ≤ Pa
Pa < σ vo' < 10 Pa
[E.18]
σ vo' ≥ 10 Pa
and x = ( N1 ) 60 FC is the normalized SPT data corrected for fines content; MSF is the magnitude
scaling factor for a 32% chance of liquefaction, K σ is the overburden pressure correction factor
for σ'vo > Pa, and Pa is the atmospheric pressure (=100kPa = 14.5psi ≈ 1tsf).
Two steps are needed to correct the measured SPT value, N, to obtain ( N1 ) 60 FC . The first
correction is to account for specific test procedures and normalize N to an effective overburden
pressure equal to Pa. The second correction is to account for the greater liquefaction resistance
of soils with larger quantities of fine-grained material. The SPT value after the first correction,
(N1 ) 60 , can be calculated as:
( N1 ) 60 = N ⋅ C N ⋅ C E ⋅ C B ⋅ C R ⋅ C S
[E.19]
where C N is the overburden correction, C N = Pa / σ v' 0 ≤ 2.0 ; C E is the hammer energy
correction, C E = Eeff / 60 or 1.0 if Eeff unknown, and Eeff is the percentage of hammer energy
delivered to the sampling rod in SPT test; C B is the borehole diameter correction and can be
assumed as 1.0 if unknown; C R is the rod length correction, C R = 0.75 + 0.25( Lrod − 4) (0.75 ≤
C R ≤ 1.0), and Lrod is the length of rod connected to SPT sampler, C S is the sample liner
correction, C S equals 1.2 for no liner and can be assumed as 1.0 if unknown.
E.7
C-FER Technologies
Appendix E
( N1 ) 60 FC is obtained by further correcting (N1 ) 60 for the effects of fines content (FC) exceeding
5% in silty sands. The fines content is determined as the percentage of material passing through
a #200 sieve (0.074mm diameter). ( N1 ) 60 FC can be calculated from the following equations:
( N1 ) 60 = α + β ( N1 ) 60
[E.20]
 0
190
 1.76− ( FC
)2
α = e
 5

[E.21]
FC ≤ 5%
5% < FC < 35%
FC ≥ 35%
1.0

( FC )1.5

β = 0.99 +
1000

 1.20
FC ≤ 5%
5% < FC < 35%
FC ≥ 35%
[E.22]
If CPT data are used, CRR can be calculated as follows
 0.833 ⋅ (qc1N ) FC

+ 0.05  MSF

1000


CRR =  
3

  93 (qc1N ) FC  + 0.08  MSF

   1000 


(qc1N ) FC < 50
[E.23]
50 ≤ (qc1N ) FC < 160
where (qc1N ) FC is the normalized tip resistance (kPa) corrected for fines content exceeding 5%.
(qc1N ) FC can be obtained from the following equations:
(qc1N ) FC = K c ⋅ qc1N
[E.24]
qc1N = CQ ⋅ (qc / 100)
[E.25]
CQ = ( Pa / σ v' 0 ) n
[E.26]
1.0

Kc = 
2
3
4
- 17.88 + 33.75 I n − 21.63I n + 5.581I n − 0.403I n
I n = (3.47 − log Qn ) 2 + (1.22 + log F ) 2
(q − σ v 0 )  100 


Qn = c
100  σ v' 0 
I n ≤ 1.64
I n > 1.64
[E.27]
[E.28]
n
[E.29]
E.8
C-FER Technologies
Appendix E
F = [ f s /(qc − σ v 0 )] ⋅100%
[E.30]
where fs and qc are the tip bearing and friction forces, respectively, measured from the CPT. The
value of n in Equation [E.26] is determined based on the soil type, n =1.0 for clayey soil, n = 0.5
for granular soil, and n = 0.7 for silty soil.
E.3.2.2
Liquefaction-Induced Ground Movement
Honegger and Nyman (2001) suggested that a finite element analysis is the most appropriate
method for evaluating the lateral spread movement of liquefied soil. In lieu of such an analysis,
empirical models proposed by Youd et al (Honegger and Nyman 2001) can be used to estimate
the expected ground movement due to liquefaction. These models are developed from lateral
spread and soil data collected in past earthquakes in the United States and Japan. For gently
sloping ground conditions (see Figure E.2(a)), the expected ground movement (m), uls, is
estimated from:
loguls = −17.614 + 1.581me − 1.518 log r * − 0.011r + 0.343 log S +
0.547 log T15 + 3.976 log(100 − F15 ) − 0.923 log( D5015 + 0.1)
[E.31]
For free face conditions (see Figure E.2(b)), uls, is estimated as:
loguls = −18.084 + 1.581me − 1.518 log r * − 0.011r + 0.551 log S w +
0.547 log T15 + 3.976 log(100 − F15 ) − 0.923 log( D5015 + 0.1)
[E.32]
In Equations [E.31] and [E.32], me is the earthquake moment magnitude (6.0 < me <8.0), r is the
epicentral distance (km), r* = r + 100.89m-5.64 (km), S is the ground slop (%) (0.1 < S < 6.0) (see
Figure E.2(a)), Sw is the free face ratio (%) (1 < W < 20)(see Figure E.2(b)), T15 is the thickness
(m) of saturated cohensionless soils with (N1 ) 60 < 15 (1 < T15 < 15), F15 is the average fines
content (%) in the soil layer corresponding to T15 (0 < F15 < 50), and D5015 is the average median
particle size (mm) the soil layer corresponding to T15 (0 < D5015 < 50).
B
A
(a) Gently Sloped Ground Condition, S = 100 A/B
E.9
C-FER Technologies
Appendix E
B
A
Slip Surface
(b) Free Face Condition, Sw = 100 A/B
Figure E.2 Sloping Ground and Free Face Conditions (Elevation View)
E.3.2.3
Pipeline Failure Probability Estimate
The equations used to evaluate CSR and CRR indicate that the liquefaction potential depends on
the PGA and moment magnitude. Equations [E.31] and [E.32] indicate that the expected amount
of ground displacement depends on the moment magnitude and the epicentral distance between
the site and the earthquake energy source. Although the PGA, moment magnitude, and
epicentral distance are all uncertain quantities, the PGA is related to the moment magnitude and
epicentral distance through the so-called attenuation relationship, which represents the decrease
in ground motion with increasing distance from the fault rupture plane. Different attenuation
relationships have been proposed in the literature for different geologic and tectonic conditions.
For near-source earthquakes (prevalent in the western United States), the attenuation model
given in Equation [E.33] is recommended (Honegger and Nyman 2001).
[
ln (a PG ) = −3.512 + 0.904me − 1.328 ln rs2 + 0.149e 0.647 me
]+
2
[1.125 − 0.112 ln(rs ) − 0.0957me ]F + [0.440 − 0.171 ln(rs )]S SR +
[0.405 − 0.222 ln(rs )]S HR + f A ( D)
[E.33]
σ ln( A ) = 0.889 − 0.0691me
PG
where a PG is the mean of the PGA (m/s2) conditional on given values of moment magnitude me
and distance from the site to the seismogenic rupture rs (km); F equals 0 for normal and strikeslip faulting, 1 for reverse or thrust faulting, and 0.5 for unknown faulting type; SSR equals 0 for
hard rock, alluvium, and firm soil, and 1 for soft rock; SHR equals 1 for hard rock and 0 for soft
rock, alluvium, and firm soil; fA(D) equals 0 if D ≥ 1km, and {[0.405-0.22ln(rs)]-[0.440.171ln(rs)]SSR}(1-D)(1-SSR) otherwise, and σ ln( APG ) is the conditional standard deviation of the
natural logarithm of the PGA. rs can be calculated from:
rs = r 2 + d s2
[E.34]
E.10
C-FER Technologies
Appendix E
where ds = (HB + HT + Wd sin(α))/2 ≥ HS (km) is the average depth to the top of the seismogenic
rupture zone, HB is the depth to the bottom of the fault (km), HS is the depth to the top of the
seismogenic part of the crust (km), HT is the depth to the top of the fault (km), Wd = 10-1.01+0.32me
(km) is the expected down-dip fault width, and α is the fault dip angle.
For Pacific Northwest (Northern California to Washington and beyond), also known as Cascadia
earthquake zones, the attenuation models shown in Equations [E.35a] and [E.35b] are
recommended (Honegger and Nyman 2001).
(
)
ln (a PG ) = −0.6687 + 1.438me − 2.329 ln r + 1.097e 0.617 me +
0.00648H + 0.3643Z T
for soil
[E.35a]
for rock
[E.35b]
σ ln( A ) = 1.45 − 0.1me
PG
(
)
ln (a PG ) = −0.2418 + 1.414me − 2.552 ln r + 1.7818e 0.554 me +
0.00607 H + 0.3846Z T
σ ln( A ) = 1.45 − 0.1me
PG
where H is the rupture depth (km), and ZT equals 0 for interface event (i.e. along boundaries
between tectonic plates) and 1 for intraslab event (i.e. within a given tectonic plate).
Note that the conditional standard deviations in Equations [E.33] and [E.35] depend on the
moment magnitude but not the hypocentral distance. Also, note that the peak ground
acceleration conditional on a given moment magnitude and epicentral distance is commonly
assumed to be lognormally distributed in the literature.
To estimate the conditional failure probability due to liquefaction-induced soil spreading, the
vector of earthquake characteristics denoted W in Equation [9.9] is represented by the moment
magnitude and the epicentral distance. Assuming that the magnitude is independent of the
epicentral distance, f W (w ) can be written as:
f W (w ) = f M e (me ) ⋅ f R (r )
[E.36]
where f R (r ) is the probability density function of the epicentral distance. The evaluation of
f R (r ) depends on whether the potential source of earthquake is a point, line or area source. The
rationale for treating an earthquake energy source as a point, line or area source can be found in
Cornell (1968). For point sources, there is no uncertainty in the epicentral distance, and
f R (r ) = 1 (see Figure E.3). For line sources (see Figure E.4), f R (r ) can be evaluated using
Equation [E.37] by assuming that the earthquake is equally likely to occur anywhere along the
source:
E.11
C-FER Technologies
Appendix E
2r


2
2
L r −∆
f R (r ) =  f
r

 L f r 2 − ∆2

∆ ≤ r ≤ r1
[E.37]
r1 ≤ r ≤ r2
where Lf is the fault length, ∆ is the perpendicular distance from the site to a line on the surface
vertically above the fault at the focal depth d, r1 and r2 are defined in Figure E.4. Similarly,
f R (r ) for area sources could be evaluated as:
f R (r ) =
2r
r − ∆20
2
0
∆ 0 ≤ r ≤ r0
[E.38]
where r0 and ∆0 are defined in Figure E.5. Note that Equation [E.38] is derived by assuming that
the earthquake is equally likely to occur anywhere within the source area.
site
r
Epicenter
rh
d
Earthquake focus
Figure E.3 Point Source of Earthquake (Perspective View)
Lf
Epicenter
r1
∆
R
r2
Line directly above the fault
site
Figure E.4 Line Source of Earthquake (Plan View)
E.12
C-FER Technologies
Appendix E
Area directly above the source
Epicenter
∆0
r0
R
site
Figure E.5 Area Source of Earthquake (Plan View)
The liquefaction potential for a given earthquake magnitude and epicentral distance, p0|w, can be
calculated as follows:
p0 w = p0 me ,r =
∫ f (b) f
B
CRR −CSR ≤0
APG
(a PG me , r )dbda PG
[E.39]
where f B (b) is the joint probability density function of random variables other than the PGA in
Equations [E.13] to [E.30], and f APG ( a PG me , r ) is the conditional probability density function of
the PGA with the mean and standard deviation given in Equation [E.33] or [E.35]. The
integration in Equation [E.39] is evaluated over the domain where CRR ≤ CSR.
The failure probability of a pipeline due to liquefaction-induced ground movement for an
earthquake with given magnitude and location, i.e. p f w , can be evaluated as follows:
p f w = p f me ,r =
∫ f ( v) f (u m , r )dvdu
V
g ( v ,uls )<0
U ls
ls
e
ls
[E.40]
where fU ls (uls me , r ) is the conditional probability density function of the liquefaction-induced
ground movement. Since only expected values of uls are given in Equations [E.31] and [E.32],
Equation [E.40] is simplified by treating uls as a deterministic parameter:
p f w = p f me ,r =
∫ f ( v ) dv
V
g ( v ,uls me , r )<0
[E.41]
The pipeline failure probability due to liquefaction-induced PGD hazard given the occurrence of
one earthquake, p lsf , can be calculated by substituting Equations [E.36], [E.39] and [E.41] into
Equation [9.9].
E.13
C-FER Technologies
Appendix E
It should be noted that Equation [9.9] together with Equations [E.36], [E.39] and [E.41] estimates
p lsf for a single earthquake source, which could be a point, line or area source. If there exist
several independent earthquake sources that could cause soil liquefaction, p lsf should be
calculated for each individual source. The annual failure rate of the pipeline, λ f , can be
obtained using Equation [9.8] in Chapter 9.
E.3.3
E.3.3.1
HAZUS Approach
Liquefaction Potential
The HAZUS approach developed by FEMA (1999) and extended by C-FER assumes that the
liquefaction potential for a given earthquake, pLIQ, is a function of the PGA, the earthquake
moment magnitude, the groundwater depth, and the liquefaction susceptibility of the soil deposit.
pLIQ can be estimated from:
pLIQ =
where a1, a2
g
Pml
KW
KM
(a1a PG / g − a2 )Pml
K M KW
0 ≤ (a1aPG/g – a2) ≤ 1.0
[E.42]
= acceleration coefficients (see Table E.1);
= acceleration of gravity (m/s2);
= a proportion factor (see Table E.2);
= a groundwater depth correction factor (see Table E.3);
= a moment magnitude correction factor; and
= 0.0027 me3 − 0.0267 me2 − 0.2055 me + 2.9188.
Soil Liquefaction
Susceptibility
Acceleration Coefficients
a1 (g-1)
a2
Very high
9.09
0.82
High
7.67
0.92
Moderate
6.67
1.00
Low
5.57
1.18
Very low
4.16
1.08
None
N/A
N/A
Table E.1 HAZUS Acceleration Coefficients
E.14
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Appendix E
Soil Liquefaction
Susceptibility
Pml
Very high
0.25
High
0.20
Moderate
0.10
Low
0.05
Very low
0.02
None
0.00
Table E.2 HAZUS Proportion Factor
Groundwater Depth
Kw
Shallow (< 5 m)
1.1
Intermediate (5 to 10 m)
1.4
Deep (> 10 m)
1.7
Table E.3 Groundwater Correction Factor
The proportion factor, Pml, reflects the effects of soil variation within a geologic map unit, which
tends to reduce the likelihood of liquefaction. The groundwater correction factor, KW, reflects
the effect of groundwater depth on the likelihood of soil liquefaction and constitutes a three-step
approximation of the continuous function given in HAZUS.
E.3.3.2
Liquefaction-Induced Ground Movement
The amount of ground movement associated with lateral spreading of liquefied soil is assumed to
be a function of the liquefaction susceptibility of the soil deposit, the surface topography
(i.e. ground slope), the PGA, and the earthquake magnitude. The ground movement (m), uls, is
estimated by:
uls = (b1aPG /(gaPGT) – b2)K∆ KS
0 ≤ ( b1 aPG /(gaPGT) – b2)K∆ ≤ 2.55 m
[E.43]
where aPGT = the threshold acceleration necessary to induce liquefaction (g) (see Table E.4);
b1, b2 = displacement coefficients (see Table E.5);
K∆
= a displacement correction factor;
= 0.0086 me3 – 0.0914 me2 + 0.4698 me – 0.9835; and
= a ground slope correction factor (see Table E.6).
KS
E.15
C-FER Technologies
Appendix E
Soil Liquefaction
Susceptibility
aPGT
Very high
0.09
High
0.12
Moderate
0.15
Low
0.21
Very low
0.26
None
N/A
Table E.4 HAZUS Acceleration Thresholds for Liquefaction
Normalized aPG Range
b1
b2
aPG/(gaPGT) ≤ 1
0
0
1 < aPG/(gaPGT) ≤ 2
0.31
0.31
2 < aPG/(gaPGT) ≤ 3
0.46
0.61
aPG/(gaPGT) > 3
1.78
4.57
Table E.5 HAZUS Displacement Coefficients
Surface Topography
KS
Flat (slope < 5%)
1.0
Rolling (slope 5% to 50%)
2.0
Steep (slope > 50%)
3.5
Table E.6 Slope Factor
Note that the empirical relationship given in HAZUS for estimating the permanent ground
displacement was derived using historical data from relatively flat areas where ground slopes
were typically in the range of 0.5 to 5%. To acknowledge the potential for increased soil
displacement in areas with steeper slopes, a slope factor, KS, was introduced by C-FER. The
factors adopted for the ‘rolling’ and ‘steep’ topography categories are consistent with the
empirical ground displacement models developed by Barlett and Youd (1992), assuming that flat,
rolling and steep terrain are associated with representative slopes of 3%, 20% and 80%,
respectively.
E.3.3.3
Guidance on Characterizing Soil Liquefaction Susceptibility
The soil liquefaction susceptibility required by Equations [E.42] and [E.43] (see Tables E.1, E.2
and E.4) is characterized in terms of six relative susceptibility categories. These susceptibility
categories are consistent with the conventional notation used in seismic hazard analysis. For
selected geographical regions (e.g. California and south western British Columbia) soil
liquefaction susceptibility maps employing these categories have been developed and are
E.16
C-FER Technologies
Appendix E
publicly available. In areas where liquefaction susceptibility is not known in terms of the
adopted categories, the classification system shown in Table E.7, which is based on formation
age, depositional environment and material type, can be used to characterize liquefaction
susceptibility. This categorization approach was developed by Youd and Perkins (1978) and has
been adopted by HAZUS. Note that areas characterized as rock or rock-like, and other areas that
are assumed to present no liquefaction hazard, are assigned a liquefaction susceptibility of
‘none’.
Type of Deposit
Distribution
of
Cohesionless
Sediments
in Deposits
Liquefaction Susceptibility of Saturated Cohesionless
Sediments (by age of deposit)
Modern
(less than
500 yrs)
Holocene Pleistocene Pre-Pleistocene
(11,000 to
(over 2 million
(500 to 11,000
2 million yrs)
yrs)
yrs)
Continental Deposits
River channel
Flood plain
Alluvial fan and plain
Marine terraces and plains
Delta and fan-delta
Lacustrine and playa
Colluvium
Talus
Dunes
Loess
Glacial till
Tuff
Tephra
Residual soils
Sebkha
Coastal Zone
Delta
Estuarine
Beach (high wave energy)
Beach (low wave energy)
Lagoonal
Fore shore
Locally variable
Locally variable
Widespread
Widespread
Widespread
Variable
Variable
Widespread
Widespread
Variable
Variable
Rare
Widespread
Rare
Locally variable
Very high
High
Moderate
N/A
High
High
High
Low
High
High
Low
Low
High
Low
High
High
Moderate
Low
Low
Moderate
Moderate
Moderate
Low
Moderate
High
Low
Low
High
Low
Moderate
Low
Low
Low
Very low
Low
Low
Low
Very low
Low
High
Very low
Very low
N/A
Very low
Low
Very low
Very low
Very low
Very low
Very low
Very low
Very low
Very low
Very low
N/A
Very low
Very low
N/A
Very low
Very low
Widespread
Locally variable
Widespread
Widespread
Locally variable
Locally variable
Very high
High
Moderate
High
High
High
High
Moderate
Low
Moderate
Moderate
Moderate
Low
Low
Very low
Low
Low
Low
Very low
Very low
Very low
Very low
Very low
Very low
Artificial
Uncompacted fill
Compacted fill
Variable very
Variable
High
Low
N/A
N/A
N/A
N/A
N/A
N/A
Table E.7 Liquefaction Susceptibility of Sedimentary Deposits
E.17
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Appendix E
E.3.3.4
Pipeline Failure Probability Estimate
Evaluation of the pipeline failure probability based on the HAZUS liquefaction hazard model is
similar to the approach presented in Section E.3.2.3. The vector of random variables W in
Equation [9.9] is represented by the earthquake magnitude and epicentral distance whereas the
PGA is evaluated using Equation [E.33] or [E.35] for given values of magnitude and epicentral
distance. However, p0 w and p f w in Equation [9.9] cannot be evaluated separately in this case
since Equations [E.42] and [E.43] indicate that both the probability of liquefaction and the
ground displacement depend on the PGA. Thus,
(
)
p0 w ⋅ p f w = ∫ p LIQ p f aPG f APG (a PG me , r )da PG
p f aPG =
∫ f ( v ) dv
V
g ( v ,uls aPG )≤0
[E.44]
[E.45]
where p f aPG is the pipeline failure probability for a given peak ground acceleration. The PGA is
assumed to be lognormally distributed conditional on the moment magnitude and epicentral
distance with the conditional mean and standard deviation obtained from Equation [E.33] or
[E.35]. The integration in Equation [E.44] accounts for the uncertainty in the PGA for a given
earthquake magnitude and epicentral distance. Note that the ground movement, uls, is uniquely
determined by Equation [E.43] for a given PGA.
The failure probability of the pipeline due to liquefaction-induced PGD hazard for a single
earthquake source, p lsf , can be calculated by substituting Equations [E.36], [E.44], and [E.45]
into Equation [9.9]. If there are several independent earthquake sources, p lsf should be evaluated
for each of them. The annual failure rate of the pipeline is thus the sum of the failure rates due to
all sources and can be calculated using Equation [9.8].
E.18
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Appendix E
E.4 REFERENCES
C-FER 2004. Development of Limit State Functions for Ground Movement. Report to GRI. In
preparation.
Cornell, C. A. 1968. Engineering Seismic Risk Analysis. Bull. Seism. Soc. Am., Vol. 58, No.
5, October, pp. 1583-1606.
FEMA 1999. HAZUS 99 Technical Manual. developed by the Federal Emergency Management
Agency, Washington, D.C., through a cooperative agreement with the National Institute
of Building Sciences, Washington, D.C.
Honegger, D. G. and Nyman, D. J. 2001. Guidelines for the Seismic Design and Assessment of
Natural Gas and Liquid Hydrocarbon Pipelines. Prepared for the Pipeline Research
Council International, Project PR-268-9823, May.
O’Rourke, M. J. and Liu, X. 1999. Response of Buried Pipelines Subject to Earthquake Effects.
Multidisciplinary Center for Earthquake Engineering Research, University of Buffalo,
Buffalo, NY 14261.
Reiter, L. 1990. Earthquake Hazard Analysis: Issues and Insights. Columbia University Press,
New York.
Weichert, D. H. and Milne, W. G. 1979. On Canadian Methodologies of Probabilistic Seismic
Risk Estimation. Bull. Seism. Soc. Am., Vol. 69, No. 5, October, pp. 1549-1566.
Wells, D. L. and Coppersmith, K. J. 1994. New Empirical Relationships among Magnitude,
Rupture Length, Rupture Width, Rupture Area, and Surface Displacement. Bull. Seism.
Soc. Am., Vol. 84, No. 4, August, pp. 974-1002.
E.19