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Graduate Studies The Vault: Electronic Theses and Dissertations
2019-01-11
Finite Element Modeling of Buried Longitudinally
Welded Large-Diameter Oil Pipelines Subject to
Fatigue
Anisimov, Evgeny
Anisimov, E. (2019). Finite Element Modeling of Buried Longitudinally Welded Large-Diameter Oil
Pipelines Subject to Fatigue (Unpublished master's thesis). University of Calgary, Calgary, AB.
http://hdl.handle.net/1880/109466
master thesis
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UNIVERSITY OF CALGARY
Finite Element Modeling of Buried Longitudinally Welded Large-Diameter Oil Pipelines Subject
to Fatigue
by
Evgeny Anisimov
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
GRADUATE PROGRAM IN MECHANICAL ENGINEERING
CALGARY, ALBERTA
JANUARY, 2019
© Evgeny Anisimov 2019
ii
Abstract
The design and construction of large diameter buried pipelines primarily for crude oil
transportation is governed in Canada by CSA Z662, ASME B31.4, and ASME BPVC Section VIII.
Although these codes provide general guidelines on pipeline design, many aspects of modelling
the pipeline are not given in detail, and the results can vary significantly based on how these details
are modelled. Engineers often adopt a very conservative approach and this results in pipelines that
are overdesigned and therefore unnecessarily costly. Following the design code, this thesis
provides a detailed fatigue analysis (FA) of a large diameter buried liquid pipeline and incorporates
the effects of the stress concentrations associated with manufacturing defects and tolerances. A
stress analysis of the pipe is first performed using the finite element method (FEM), and results
obtained are used in conjunction with both elastic and elastic-plastic FA life assessment models to
predict fatigue damage (FD). The results of a FEM and FA performed on four standard pipeline
OD’s show that a 20% increase in the outside diameter (OD) to wall thickness (WT) ratio can be
achieved when plasticity is considered. This is equivalent to one to two increments of standard
WT or the percent reduction of a pipeline construction cost. In the analyses process, where the
code leaves significant room for interpretation, this thesis provides clarity on appropriate
procedures to follow. Examples include how to accurately model the weld profile, and the
misalignments due to the manufacturing process. Furthermore, a simple calculation tool is
developed that can be used to approximate hot-spot elastic stresses.
Keywords: Large Diameter Pipeline Fatigue, Fatigue of Welded Connections, Elastic-Plastic
Fatigue Analysis, Fatigue Damage.
iii
Preface
The thesis focuses on the design codes related to fatigue analysis of pipelines and discusses
the challenges associated with implementation of the codes, analyzing the procedures of North
American codes in more detail. Special attention is paid to manufacturing misalignments of the
pipes’ weld region and analysis of stresses in that region. The magnification of stresses due to
misalignments is further discussed from the standpoint of fatigue analysis. Addressed the global
aim of the research – development of the simple and easy to use model that can help engineers
with reliable assessment of design stresses in the pipe and its fatigue-safe design.
Chapter 1 provides the background information on the pipe manufacturing processes,
materials used to build the pipelines and their properties related to fatigue degradation. Further
discussion focuses on the elastic and the elastic-plastic models of materials’ behavior, including
the von Mises and the Tresca yielding criteria. Finally, the basis of the three main fatigue
assessment procedures are discussed, including the stress life and the strain life approaches dealing
with non-planar defects (such as pores and others metallurgical defects), as well as the crack
growth approach used for the assessment of structures with planar defects (such as cracks and weld
undercuts).
Chapter 2 is dedicated to manufacturing tolerances used in pipeline design and
manufacturing quality control. The various types of misalignments (manufacturing defects) are
discussed, including weld discontinuity, offset of pipe plate at the weld region, peaking of the weld
region, and ovality of the pipes’ body. As a summary, the research problem is formulated, and the
pipe design parameters selected for the model development, including complete pipe geometry
with manufacturing defects, materials, and pipe loading.
iv
Chapter 3 shows the steps taken toward the development of both the mathematical (Hand
calculations) and the final element (ABAQUS calculations) models used for calculation of stress-
strain states of the pipeline due to various loadings. The computing was focused on obtaining of
the stresses and strains at the most critical location in the pipe – structural hot-spot (at the weld
toe). The models capture the effects of various types of misalignment, internal pressure, soil
pressure, and temperature, on the stress rise at the hot-spot, including bending stress developed.
Chapter 5 provides concluding remarks and discusses major results of the research work
discussed in this thesis. The significant increase of structural stresses due to misalignment was
demonstrated with the help of elastic and elastic-plastic analyses. The stress rise resulted in
dramatic increase of the fatigue damage due to cyclic pressurizing of the pipeline. Another
important observation is the conservatism involved in the elastic fatigue analysis. Elastic-plastic
fatigue analysis suggested the possible reduction of the pipe wall thickness without compromising
the fatigue performance of the modeled pipeline. Newly implemented accounting for the weld
profile, not observed in the standards before, can provide the
Chapter 6 outlines the future work and recommends the areas for improvement. Accounting
for the residual stresses due to manufacturing and during cyclic loading in the model can be very
important for more detailed analysis of stress-strain states at the critical locations and can be
extremely useful when accompanied by the more advanced fatigue assessment methodologies
based on fracture mechanics principles. The crack growth approach in fatigue analysis would
utilize the accurate stress-strain data at the crack tip to yield more accurate predictions of fatigue
damage.
v
Acknowledgements
I would like to acknowledge Dr. Meera Singh and Dr. Les Sudak for their professional
guidance and valuable critical discussions related to the research work discussed in this thesis.
I am grateful for the opportunity to participate in the project that addressed some real-life
challenges that engineers have in the pipeline industry; I was able to research the problems
associated with structural integrity and safety. The research project benefited me professionally
and personally, I was interacting with professionals from the industry, learned many new things
and furthered my knowledge during my studies.
I would also like to thank Darryl Stoyko and Robert Thom from Stress Engineering
Services, Inc. for critical discussions and reviews.
vi
Dedication
I would like to dedicate the thesis to my family, especially to my wife Natalia, to my sons,
Maxim and Denis, and thank them for all the support and understanding provided during my
studies. I also dedicate the thesis to my grandfather Vladislav Anisimov a civil engineer who
sparked my interest to the field of engineering.
vii
Table of Contents
Abstract .............................................................................................................................. ii Preface ............................................................................................................................... iii Acknowledgements ............................................................................................................v Dedication ......................................................................................................................... vi Table of Contents ............................................................................................................ vii
List of Tables .................................................................................................................... ix List of Figures and Illustrations .......................................................................................x List of Symbols, Abbreviations and Nomenclature .................................................... xiii
Epigraph ......................................................................................................................... xvi
Chapter 1 INTRODUCTION ...................................................................................1 1.1. Motivation ...............................................................................................................1 1.2. Background ............................................................................................................2
1.3. Objective .................................................................................................................4 1.4. Thesis Outline .........................................................................................................5
Chapter 2 LITERATURE REVIEW.......................................................................7 2.1 Early FE Analyses ..................................................................................................8 2.2 Recent FE Analyses..............................................................................................12
2.3.1. Hot-Spot Stress ...............................................................................................12
2.3.2. Weld Distortion and Wall Thickness ...........................................................15 2.3 Governing Codes ..................................................................................................19 2.4 Conclusions of Literature Review ......................................................................20
Chapter 3 STANDARD PROCEDURES ..............................................................21 3.1. Pipeline Codes ......................................................................................................21
3.2. Pipeline Geometry ................................................................................................23 3.2.1. Pipe ODs and WTs .........................................................................................24 3.2.2. Weld Misalignments ......................................................................................26
3.2.2.1. Radial Misalignment .............................................................................28
3.2.2.2. Angular Misalignment ..........................................................................30
3.2.2.3. Ovality Misalignment ...........................................................................30 3.2.3. Welding Defects ..............................................................................................31 3.2.3.1. Weld Reinforcement .............................................................................31
3.2.3.2. Welding Cracks .....................................................................................33 3.3. Pipe-Soil Interaction ............................................................................................34 3.4. Pipeline Materials ................................................................................................35 3.5. Loading .................................................................................................................42 3.6. Linearization of Stresses......................................................................................44
3.7. Analytical Model ..................................................................................................45 3.7.1. Stress due to Misalignment ...........................................................................45
3.7.2. Stress due to Soil ............................................................................................47 3.8. Fatigue Assessment ..............................................................................................49 3.8.1. Stress-Life Curves ..........................................................................................50 3.8.2. Elastic Fatigue Analysis .................................................................................52
viii
3.8.3. Modified Elastic Fatigue Analysis ................................................................53 3.8.4. Elastic-Plastic Fatigue Analysis ....................................................................56
3.8.5. Elastic Fatigue Analysis of Welds .................................................................57 3.9. Summary and Problem Definition .....................................................................61
Chapter 4 MODEL DEVELOPMENT .................................................................63 4.1. Static Finite Element Model ................................................................................64 4.1.1. Geometry of Model ........................................................................................64
4.1.2. Material Model ...............................................................................................69 4.1.3. Boundary Conditions .....................................................................................73 4.1.4. Model Meshing and Convergence ................................................................74
4.1.5. Data Extraction ..............................................................................................77
Chapter 5 RESULTS AND DISCUSSION ...........................................................79 5.1. Finite Element Model ..........................................................................................79 5.1.1. Validation of FEM .........................................................................................83
5.2. Fatigue Analysis ...................................................................................................97
Chapter 6 CONCLUSIONS AND FUTURE WORK ........................................105
6.1. Conclusions .........................................................................................................105 6.2. Future Work .......................................................................................................107
APPENDIX A – MATLAB Numerical Solution .........................................................108
APPENDIX B – MATLAB Cycle-Counting ................................................................110
APPENDIX C – ABAQUS Input File ..........................................................................112
APPENDIX D – ABAQUS Report Example ...............................................................118
APPENDIX E – MATLAB Code for the ABAQUS Data ..........................................121
References .......................................................................................................................124
ix
List of Tables
Table 1 Nominal WTs for different ODs ...................................................................................... 24
Table 2 Permissible Specified ODs and WTs ............................................................................... 25
Table 3 Example DFs and WTs used in Keystone pipeline .......................................................... 25
Table 4 Permissible Variation of WT ........................................................................................... 26
Table 5 Permissible radial misalignment for different pipe thicknesses (North America) ........... 29
Table 6 Permissible radial misalignment (BS PD 5500) .............................................................. 29
Table 7 Permissible angular misalignment (BS PD 5500) ........................................................... 30
Table 8 Permissible ovality misalignment (API 5L) .................................................................... 31
Table 9 Permissible weld reinforcement, inch (mm) .................................................................... 32
Table 10 Soil properties ................................................................................................................ 34
Table 11 Constants for a polynomial fit of experimental data in the calculation of number of
cycles to failure ..................................................................................................................... 55
Table 12 Pipeline parameters [in or (mm)] considered in this research ....................................... 61
Table 13 Standard pipeline WTs for selected ODs and steel material at an internal pressure of
10 MPa .................................................................................................................................. 65
Table 14 Pipeline defects .............................................................................................................. 67
Table 15 Weld bead dimensions obtained in this study ................................................................ 68
Table 16 Pipeline steel material .................................................................................................... 69
Table 17 Displacement constraints ............................................................................................... 73
Table 18 Parameters used in calculation of SCFs due to misalignments ..................................... 84
Table 19 Stress magnification at different weld locations ............................................................ 91
Table 20 Results of analysis of the design hoop stresses Sh [MPa] for a pipe of OD 914 mm
and WT 17.5 mm .................................................................................................................. 92
Table 21 Results of fatigue analysis obtained at accumulated fatigue damage of 0.5................ 100
Table 22 Results of fatigue analysis obtained at accumulated fatigue damage of 1.0................ 103
x
Table 23 Construction cost savings associated with WT reduction on a 4700 km pipeline ....... 104
List of Figures and Illustrations
Figure 1 Misalignments of a welded pipe ....................................................................................... 3
Figure 2 Schematic of the DSAW weld profile .............................................................................. 3
Figure 3 Structural stress concept ................................................................................................. 13
Figure 4 Welded connections in: (a) seamless pipeline, (b) pipeline with longitudinal seam,
and (c) pipeline with helical seam; the hatched area shows the plane of connection of
two pipes ............................................................................................................................... 27
Figure 5 Common types of weld misalignment in longitudinally welded pipe: (a) radial, (b)
angular, and (c) ovality ......................................................................................................... 28
Figure 6 Detailed schematic of radial misalignment .................................................................... 29
Figure 7 Schematic of abutting plates before welding .................................................................. 33
Figure 8 Weld widths for different wall thicknesses of plates tapered at 60° .............................. 33
Figure 9 Mechanical properties of pipeline steels ........................................................................ 36
Figure 10 Typical engineering stress-strain tensile curves for some X steels as per API 5L [3] . 36
Figure 11 Typical engineering and true stress-strain tensile curves for X42 steel ....................... 38
Figure 12 Difference between experimental stress-strain tensile curve for X42 steel and
Ramberg-Osgood fit near the yield strength ......................................................................... 39
Figure 13 Tresca and von Mises yield criteria in (a) hydrostatic and (b) plane stresses .............. 40
Figure 14 Stress tensor at the longitudinal weld in pipeline ......................................................... 41
Figure 15 Fatigue loading showing (a) spectrum loading and (b) constant amplitude loading .... 42
Figure 16 In-service pressure history diagram.............................................................................. 43
Figure 17 Cycle-counted in-service pressure history showing Pmin, Pmax, and nk ......................... 44
Figure 18 Through-wall bending stress and ovality of pipe cross-section due to transmitted
pressure ................................................................................................................................. 48
Figure 19 Schematic classification of fatigue life approaches ...................................................... 49
xi
Figure 20 Standardized tensile specimen (a) versus real component (b)...................................... 50
Figure 21 Stress life (S-N) curve .................................................................................................. 51
Figure 22 Neuber’s relationship between linear and non-linear stresses and strains [83] [84]
[85] ........................................................................................................................................ 59
Figure 23 Flow chart showing process of model refinement ........................................................ 63
Figure 24 Geometry of model ....................................................................................................... 65
Figure 25 Schematic of the SAW-processed pipe region ............................................................. 66
Figure 26 Traces of weld profiles used to generate an average weld profile ............................... 67
Figure 27 Geometry of the weld bead profile showing (solid dots) experimental data and
(solid line) 4th-order polynomial approximation ................................................................... 67
Figure 28 Geometry of the weld region including (bold white line) radial and (bold black
line) angular misalignments .................................................................................................. 69
Figure 29 Construction of a tangent to Ramberg-Osgood’s curve from yield point on Hooke’s
curve ...................................................................................................................................... 70
Figure 30 Example of a numerical solution for the tangent point on a Ramberg-Osgood curve . 71
Figure 31 True Stress-Strain curves for pipe steel material .......................................................... 72
Figure 32 Pipe (highlighted by circles) surrounded by a soil box with constraints ...................... 73
Figure 33 Meshing of a pipe showing detailed meshing at the hot-spot (black line indicates
the path used for an SCL) ..................................................................................................... 74
Figure 34 Meshing of soil box around pipe .................................................................................. 75
Figure 35 Refinement of global (away from discontinuity) and local (at the weld toe) meshes
showing von Mises Stress/Strain – Mesh Element Size relationship ................................... 76
Figure 36 Hoop stress distribution maps for a misaligned pipe during elastic loading ................ 80
Figure 37 Hoop stress distribution maps for a misaligned pipe during elastic-plastic loading .... 80
Figure 38 Through-thickness (curved) actual and (linear) linearized stress distributions
obtained for a pipe of 914 mm OD and 14.3 mm WT from an SCL positioned at the hot-
spot (at 0 mm WT coordinate) normal to the pipe wall with no misalignment by using
(a) elastic and (b) elastic-plastic analysis, and with misalignment by using (c) elastic and
(d) elastic-plastic analysis ..................................................................................................... 81
xii
Figure 39 Schematic of (a) the SAW butt joint, represented in the form of (b) fillet in stepped
flat bar, showing (c) equivalent load at the base of reinforcement and (d) real shear
stress diagram with its (dashed line) approximation ............................................................. 85
Figure 40 The (dots) SCFs for different (connected dots) transition radiuses Wr (a)
Wr =0.25WT, (b) Wr =7.145 mm, (c) Wr=3-7/t, and (d) Wr=5 mm ...................................... 89
Figure 41 Secondary bending (curved arrows) due to: (a) axial, (b) angular, and (c) ovality
misalignments, and (d) due to soil; the red dashed line indicates the plane of a
hypothetical crack or SCL, and 1 through 4 are the hot-spot locations ................................ 90
Figure 42 Hoop stress calculated with mathematical model, power-law-fitted, and
extrapolated until solutions of (dashed line) non-misaligned and (solid line) misaligned
conditions intersect (power-law-fitted) ................................................................................. 93
Figure 43 Solutions for Hoop stress (linear fit) in (non)misaligned pipe of OD 914 mm (a)
without and (b) with km.weld accounted .................................................................................. 94
Figure 44 Solutions for Hoop stress (power law fit) in pipe of OD 914 mm (a) without km.weld
and (b) with km.weld ................................................................................................................. 96
Figure 45 Accumulated fatigue damage plots for pipe diameters (a) 610 mm, (b) 864 mm, (c)
914 mm, and (d) 1219 mm, calculated with (solid lines) misalignment and with (contour
lines) no misalignment .......................................................................................................... 99
Figure 46 Relationship between OD and WT at accumulated fatigue damage of 0.5 for (blue)
BS elastic, (red) ASME elastic, and (grey) ASME elastic-plastic analyses ....................... 102
xiii
List of Symbols, Abbreviations and Nomenclature
Symbol Unit Definition
𝐴0, 𝐵0 Geometric constants for a fillet weld
BD [𝑚𝑚] Burial depth
CAE Computer aided engineering
𝐶𝑢𝑠 [0 ÷ 1] Conversion factor
𝐷 [𝑚𝑚] Outside diameter
𝐷𝑓 [0 ÷ 1] Accumulated FD
𝐷𝑙 [1.0 ÷ 1.3] Deflection lag factor
𝑑 Factor for nominal probability of failure
𝐸 [𝑀𝑃𝑎] Young’s modulus
𝐸′ [𝑀𝑃𝑎] Modulus of soil reaction (≈ 0 ÷ 20 for loose to compact soil)
𝐸𝑇,𝑘 [𝑀𝑃𝑎] Young’s modulus at assessed temperature
𝐸𝐹𝐶 [𝑀𝑃𝑎] Young’s modulus of material used to obtain experimental S-N curve
FD Fatigue damage
FEM Finite element method
𝐻 [𝑚𝑚] Pipe BD
𝐾 [𝑀𝑃𝑎] Strength coefficient
𝐾𝑏 [≈ 0.1] Soil bedding constant
𝐾𝑒,𝑘 Fatigue penalty factor
𝑘 Coefficient of deformation of the weld joint
𝑘𝑚 Stress magnification factor
𝑙1,2 [𝑚𝑚] Weld section lengths supporting the shear distributed load
𝑚 Slope of S-N curve
𝑁𝑘 Number of cycles to failure
𝑛𝑘 Number of assessed cycles
𝑛 [0 ÷ 1] Strain hardening exponent
OD [𝑚𝑚] Outside diameter
𝑃𝑖 [𝑀𝑃𝑎] Internal pressure
𝑃𝑠 [𝑀𝑃𝑎] Soil pressure on the pipe above the water table
xiv
𝑅 Stress ratio
SAW Submerged arc-welded
𝑆𝑎 [𝑀𝑃𝑎] Stress amplitude
𝑆𝑎𝑙𝑡,𝑘 [𝑀𝑃𝑎] Effective alternating equivalent stress
𝑆𝑒 [𝑀𝑃𝑎] Fatigue endurance limit
SCF Stress concentration factor
SD Standard deviation
𝑆𝐿 [𝑀𝑃𝑎] Stress in pipe’s longitudinal direction
𝑆𝐻 [𝑀𝑃𝑎] Stress in pipe’s hoop direction
𝑆𝑟 [𝑀𝑃𝑎] Stress range in most critical direction
𝑇 [℃] Assessment temperature
𝑡 [𝑚𝑚] Wall thickness
UOE U-to-O shaped and expanded
𝑈𝑖 [𝑚𝑚] Displacement components
𝑊𝑇 [𝑚𝑚] Wall thickness
𝑊𝑤 [𝑚𝑚] Weld bead width
𝑊ℎ [𝑚𝑚] Weld bead height (reinforcement)
𝑊𝑟 [𝑚𝑚] Weld toe radius
𝑊𝛼 [°] Weld reinforcement angle
𝛼 [℃−1] Coefficient of linear thermal expansion
𝛼𝑓 [°] Friction angle
𝛼𝑑 [°] Dilation angle
𝛿𝑜 [𝑚𝑚] Offset misalignment
𝛿𝑝 [𝑚𝑚] Peaking misalignment
휀 [𝑚𝑚 𝑚𝑚⁄ ] Strain
휀𝑒𝑛𝑔 [𝑚𝑚 𝑚𝑚⁄ ] Engineering strain
휀𝑝 [𝑚𝑚 𝑚𝑚⁄ ] Plastic strain
휀𝑡𝑟𝑢𝑒 [𝑚𝑚 𝑚𝑚⁄ ] True strain
휀𝑦 [𝑚𝑚 𝑚𝑚⁄ ] Yield strain
𝜈 [0 ÷ 0.5] Poisson’s ratio
xv
𝜌 [𝑘𝑔 𝑚3⁄ ] Mass density (unit weight of soil fill)
𝜌𝑚 Material structural characteristic constant
𝜎 [𝑀𝑃𝑎] Normal stress
𝜎𝑏 [𝑀𝑃𝑎] Bending stress (through-wall)
𝜎𝑒𝑛𝑔 [𝑀𝑃𝑎] Engineering stress
𝜎𝑚 [𝑀𝑃𝑎] Membrane stress
𝜎𝑚𝑖𝑗,𝑘 [𝑀𝑃𝑎] Component stresses at the end of cycle
𝜎𝑛𝑖𝑗,𝑘 [𝑀𝑃𝑎] Component stresses at the start of cycle
𝜎𝑢 [𝑀𝑃𝑎] Ultimate tensile stress
𝜎𝑡𝑟𝑢𝑒 [𝑀𝑃𝑎] True stress
𝜎𝑦 [𝑀𝑃𝑎] Yield stress at room temperature
𝜎𝑦𝑇 [𝑀𝑃𝑎] Yield stress at assessed temperature
𝜎𝑦𝐶 [𝑀𝑃𝑎] Cohesion yield stress
𝜏𝑚 [𝑀𝑃𝑎] Shear stress at the fillet weld section
𝜏𝑚′ [𝑀𝑃𝑎] Maximum value of the shear stress at the rectangular reinforcement
∆𝑝𝑖𝑗,𝑘 [𝑚𝑚 𝑚𝑚⁄ ] Change in plastic strain range components for the 𝑘𝑡ℎ loading cycle
∆𝑆𝑃,𝑘 [𝑀𝑃𝑎] Effective equivalent stress range
(∆𝑦
𝐷) Deflection (or ovality) due to soil
∆𝑦 [𝑚𝑚] Vertical deflection of pipe due to soil
∆휀𝑒𝑓𝑓,𝑘 [𝑚𝑚 𝑚𝑚⁄ ] Effective strain range
∆휀𝑒𝑙,𝑘 [𝑚𝑚 𝑚𝑚⁄ ] Elastic strain range
∆휀𝑝𝑒𝑞,𝑘[𝑚𝑚 𝑚𝑚⁄ ] Plastic equivalent stain range
∆𝜎𝑖𝑗,𝑘 [𝑀𝑃𝑎] Range of normal component stresses
∆𝜎 [𝑀𝑃𝑎] Normal stress range
∆𝜎′ [MPa] Additional local stress at upper fillet weld in stepped flat bar
∆𝜎′′ [MPa] Additional local stress at lower fillet weld in stepped flat bar
∆𝜎′′′ [MPa] Additional local stress at both fillet welds in stepped flat bar
∆𝜏 [𝑀𝑃𝑎] Shear stress range
∆𝜃 Ovality misalignment
xvi
Epigraph
In the behavior of structures, truth can usually be found only by testing and observing
genuine structural members built with the materials as they are, with imperfections which cannot
be avoided.
Fritz Leonhardt, 1976
~ Gold Medalist, The Institution of Structural Engineers
1
Chapter 1 INTRODUCTION
1.1. Motivation
According to Natural Resources Canada, Canada is the fourth-largest producer and
exporter of crude oil in the world and owns 10% of the world’s proven oil reserves (as of December
2017) [1]. Approximately 5.1 million barrels of crude oil per day [MMb/d] are transported within
Canada via 840,000 km of pipeline infrastructure, of which 117,000 km is pipeline with a diameter
of up to 48 inches. Currently, 62.4% of large-diameter transmission pipelines in Canada are
federally regulated [1]-[2]. Multiple reports of pipeline rupture events have been accumulated over
40 years by the National Energy Board (NEB) [2], showing that during the last decade alone,
nearly 50% of the reported incidents related to crude oil transportation were caused by cracking
due to fatigue. These fatigue related incidents resulted in more than 54,000 barrels of crude oil
spilled, which is more than 60% of the total amount spilled during the time period.
It is very important to account for all possible factors that may negatively impact the
integrity and reduce the fatigue life of a pipeline. The pipeline fatigue design must incorporate the
manufacturing defects and in-service conditions, both of which introduce stresses/strains, which
for safe operation should not exceed critical values at critical locations. These design
considerations are governed by design codes. In order to avoid failures while performing stress
analyses and fatigue life predictions for large-diameter pipes, engineers generally attempt to follow
the codes and to incorporate a conservative approach in their designs. However, this often results
in a greater wall thickness (WT) than required. This consequently contributes to a considerable
increase in a pipeline’s overall cost. Therefore, optimizing a pipe WT can result in significant
savings when multi-kilometer lines are being designed.
2
1.2. Background
Larger pipes normally have outer diameters (OD’s) ranging between 457.0 and
2,174.0 mm, wall thicknesses (WT’s) ranging between 7.1 and 52.0 mm, and come in standard
lengths between 6.0 and 24.0 m, corresponding to standards API 5L [3] and CSA Z245.1 [4]. In
Canada, such pipes are usually buried 0.6 to 1.2 m below the ground, as per CSA Z662 [5]. They
are normally manufactured by the cold-forming of flat steel plates using the U-ing O-ing and
expanding (UOE) process. During this process, the plate is first formed into a U-shape and then
pressed into an O-shape between two semicircular dies [6]. Subsequently, the longitudinal seam is
welded using the Double Submerged Arc Welding (DSAW) process [7] to connect the abutting
edges of the deformed plate and complete an O-shape. Finally, the pipe is expanded using an
internal mandrel to improve its roundness [8].
Pipelines, particularly liquid-carrying ones, are subjected to repeated thermal and pressure
loads while in service. This cyclic loading can result in fatigue failures at loads much lower than
those observed in static failures. Geometric discontinuities within the loaded pipe act as stress
risers that result in magnification of a local stress [9]. These stress risers promote fatigue failure,
as they act as perfect sites for cracks to initiate. Several such stress risers in large-diameter pipe
can originate from the UOE manufacturing process. Specifically, they can occur due to peaking
and radial misalignment, as shown in Figure 1, in the vicinity of the SAW seam. Furthermore, the
double-SAW welding process used to create the longitudinal seam of a potentially misaligned pipe
can introduce a large stress concentration due to the weld profile, as shown in Figure 2.
3
Figure 1 Misalignments of a welded pipe Figure 2 Schematic of the DSAW weld profile
In order to predict the fatigue life of a large-diameter pipe, all stress risers must be
accounted for. As an example, it was found that cyclic tensile stresses can increase by 50% by
increasing the angle between the weld reinforcement and the base plate, 𝑊𝛼, from 0° to 60° [10]
[11]. A similar strong influence on stress development and fatigue strength has been reported to
be caused by the reinforcement, 𝑊ℎ, the weld radius, 𝑊𝑟, [11] [12], and the axial (offset), 𝛿𝑜, or
angular (peaking), 𝛿𝑝, weld misalignment [12] [13]. Although this topic is being actively
researched, there is a lack of data in the available literature on the analysis of welded pipes with
combined misalignments.
The UOE and SAW processes produce residual stresses in pipe, the most detrimental of
which from a fatigue perspective are the residual tensile stresses observed after welding. Although
the stress distribution after welding [14] can become more uniform after expansion [8], the
uncertainties in the experimentally measured data discussed in [15] [16] and [17] indicate
significant variation in residual stresses that may ambiguously influence fatigue tests on welded
structures [18] or may have a barely noticeable impact on fatigue strength [19]. However, narrower
weld beads were reported to result in slightly lower residual stresses [20]. Another stress
magnification factor is the over-pressurization overloads frequently observed in the engineering
practice, which have been found to accelerate the failure of a component due to fatigue [18].
4
In Canada, pipelines are generally designed with reference to the codes such as API 5L [3],
API 579 [21], ASME BPVC Section VIII Part 2 [22], ASME B31.4 [23], CSA Z245.1 [4],
CSA Z622 [5], BS 7608 [24], and BS 7910 [25]. Although the codes are critical to assuring the
safe operation of pipelines, the application of code requirements is often not straightforward. For
example, the assessment methodology in the British standards BS 7608 [24] and BS 7910 [25],
used to account for the stresses due to weld misalignments, is limited to hand calculations, and its
implementation in FEM analysis is not discussed. A similar situation exists with North American
standards API 579 [21] and ASME BPVC Section VIII Part 2 [22], although FEM is among the
preferred methods for stress-strain analyses. Furthermore, the geometry of weld profiles seems
also not to be well defined in the North American standards. As a result of these uncertainities in
the codes, engineers are often left to provide their best guess on how to move forward in their
analysis. This can lead to underdesign or overdesign of the pipelines analyzed.
1.3. Objective
The overall objective of this project is to provide a systematic assessment methodology for
fatigue life that can be used to design and analyze pipelines so that they are safe from a fatigue
perspective and are also cost-efficient. In this work, some difficulties associated with the
interpretation of codes, modeling misalignments due to the UOE process, and modeling weld seam
profiles are highlighted. The work discusses these challenges in the context of a stress analysis
performed on a standard large-diameter pipe using FEM, and includes the effects of manufacturing
tolerances, misalignments and their combinations, and typical thermo-mechanical loading. The
FEM results are subsequently used to make fatigue life predictions and to guide best practices in
large-diameter pipe design.
5
Therefore, to achieve the objective of the project, the research focuses on the calculation
of acceptable pipe WT, including combinations of commonly observed manufacturing
misalignments from the perspective of fatigue-safe design, including axial and angular
misalignments applied to a range of commonly used pipe diameters. It also uses the Level 2 Fatigue
Assessment Methods from ASME BPVC Section VIII Part 2 [22] and discusses the conservatism
involved in elasticity-based methods when selecting the WT for a given pipe design.
1.4. Thesis Outline
Chapter 2 overviews the published literature associated with the stress and fatigue analyses
of welds used in large-diameter pipelines as well as the pipe manufacturing tolerances governed
by the design codes. Emphasis is placed on allowable pipe weld geometry and weld misalignment
defects.
Chapter 3 provides a discussion on the development of the analytical and FE models used
in the stress analysis of buried oil-carrying pipes, and includes the geometry of the model, the
material model, boundary conditions, and the model meshing. The calculation procedures
described in Chapter 3 closely follow the guidance from pipe design codes, and discussion is
provided regarding the limitations of the codes as they relate to the modeling of weld profiles and
combinations of weld misalignments. Finally, the chapter introduces the elastic and the elastic-
plastic fatigue assessment methodologies used in predictions of fatigue damage, including a
discussion on the rainflow cycle-counting of load history and the linearization of stresses obtained
from FEM.
6
Chapter 4 discusses the results of the analyses described in the previous chapter and
provides a detailed assessment of allowable design wall thicknesses for the studied pipe
geometries. This chapter highlights the significance of combined misalignment with regard to the
fatigue damage of pipes, and discusses the conservatism involved with the elastic fatigue
assessment methods.
Chapter 5 summarizes the results and presents the conclusions of the research work
accomplished in this thesis. It further provides the basis for future work and specific
recommendations to further the understanding of the mechanical behavior of misaligned in-service
large-diameter pipes.
7
Chapter 2 LITERATURE REVIEW
The first oil-carrying pipelines in North America were built in the second half of the 19th
century, shortly after oil was discovered. Pipe manufacturers turned relatively quickly to steel,
which offered higher strength to weight ratios than cast or wrought iron. They also enabled lower
processing temperatures that favored the fabrication of longitudinally welded larger pipes by
means of the lap welding technique [26]. The growing need for oil transportation capacity led to
the development of better-quality steels with fewer defects and to advances in welding processes.
For example, the (double) submerged-arc welding technique was introduced after World War II
and proved to be more reliable than lap joining in the furnace [26]. The design and manufacturing
of pressure vessels and pipes in North America have been governed by ASME VIII code since
1925, by API 5L since 1928, and by ASME B31 since 1935 [27].
Pipeline design has greatly benefited from improvements in quality-assurance methods and
inspection tools. These include hydrostatic testing standards, the introduction of radiographic
inspection in 1948 and in-line inspection in the 1960s [26]. Studies of pipeline failures have
recognized the need for addressing the issues related to fatigue cracking [26]. Miner’s work on
cumulative fatigue damage (1945) and Langer’s design fatigue curves for pressure vessels (1961)
have greatly contributed to the pipeline design capabilities offered by the design codes in the 2000s
[27].
The following sections will discuss in detail the difficulties associated with the FE
modeling of the welded connections when following the governing codes. Specifically, literature
pertaining to FE modeling of weld profiles and weld misalignments in pipelines and their effects
on the stresses and fatigue damage are reviewed.
8
2.1 Early FE Analyses
Pollard and Cover (1972) [10], both American metallurgists, concluded in their literature
review on the fatigue in steel weldments by stating “…fatigue strength is determined primarily by
the geometry of the weldment and the soundness of the weld metal, and so even if the weld
reinforcement is ground off, the fatigue strength may still be influenced by the soundness of the
weld metal”.
Since this thesis focuses on the fatigue performance of UOE pipes influenced by
geometrical tolerances that are acceptable by North American standards, the literature reviewed in
this thesis is concerned with the FE modeling of pipes’ weld profiles and related challenges. FE
modeling has improved greatly over recent years mainly due to advances in computing
capabilities. Therefore, the widely utilized analytical solutions could further benefit from and be
extended by numerical FEM-aided solutions. However, the design codes used for piping do not
provide detailed guidance on FE modeling.
The effects of various geometrical parameters of a full-penetration butt weld on fatigue
strength were studied by Berge and Myhre at Det Norske Veritas in 1977 [28]. Using experimental
and FE analysis the authors showed that misalignment may seriously deteriorate the fatigue
strength of such welds. In Canada, early FE analyses of misaligned line pipes were conducted by
Worswick and Pick (1985) at the University of Waterloo [29]. Their study showed a significant
reduction in limit load (transition from elastic to plastic behaviour) associated with increases of
local bending stress due to weld misalignment, signifying the need to account for these effects in
pipeline design. Later, in 1991, Ferreira and Branco [30] discussed the importance of strength
analysis of misaligned welded joints as opposed to the “good-practical-experience” exercised in
the governing codes at that time when specifying permissible levels of misalignment, which were
9
found to reduce the fatigue strength of welded components markedly. In all three mentioned FE
studies, the mesh size used in the FE models was rather coarse with sharp transitions at the weld
discontinuity, and no convergence study had been performed due to the expenses associated with
computing capabilities at that time [29]. Nonetheless, the study [30] demonstrated a strong
influence of both axial and angular distortions on the local stress at the weld toe in tension (loaded
normal to the plane of a hypothetical crack) and therefore on the fatigue strength of longitudinally
welded tubular structures.
Berge and Myhre [28] used 2D quadrilateral and triangular elements that appear to be first-
order and of an average size of 1.25-2.22 mm for a global mesh, and a size of 0.1 mm was used
for the elements at the stress concentration. The mesh size was refined to a depth of only 1.6 mm
at a total through-thickness size of 20 mm. The authors in [28] concluded that the large scatter in
the published S-N data and derived design curves, which were most probably influenced by
accidental misalignments (𝛿0 𝑡⁄ ≈ 0.125), prohibited close comparison to the FEM results. This
highlighted that the optimization of welds must consider the ratio between the bending and axial
(membrane) stresses 𝜎𝑏 𝜎𝑚⁄ , which depend on the geometry in the vicinity of the weld and
boundary conditions 𝜆, when allowable misalignment is assessed. The study also identified one
main problem with the assessment of 𝜎𝑏 𝜎𝑚⁄ for each joint in question, which was impractical and
led to the selection of conservative values of 𝜆. Nowadays, this may pose much less of a problem
because of the much more advanced computational tools available, which allow for relatively
quick FEM analyses even for a highly detailed 3D model.
Another important observation made by Berge and Myhre [28] was related to the increase
of a toe angle (flank angle), 𝑊𝛼, with increased (axial) misalignment, 𝛿0, that contributes to local
bending stresses and thus impairs the fatigue strength. The bending stresses were found to decrease
10
rapidly with plate thickness. Although, a complete FEM analysis was not done to quantify the
effect of toe angle alone, the combination of 𝛿0 and 𝑊𝛼 indicated an almost 40 % decrease in
fatigue strength at eccentricity 𝛿0 = 𝑡 2⁄ , which resulted in 𝑊𝛼 = 26.6°. It may be obvious to
assume that the combination of axial and angular weld distortions would increase the toe angle,
and therefore the bending stresses at the toe even more.
Worswick and Pick [29] used quadrilateral, i.e., quadratic isoparametric (second-order),
elements that included 8-node quadrilaterals for 2D and 20-node brick elements for a 3D model
for both elastic and the elastic-plastic analyses. Although the authors in [29] used more advanced
mesh elements to address the issue with a singularity at a critical location, no mesh convergence
study was published. The stress-strain curve which was obtained through the uniaxial tensile
testing in the FEM model was approximated by a series of linear segments. The axial misalignment
of 2.286 mm, modeled for a pipe of 11.2 mm WT and 914 mm OD, resulted in approximately a
15% decrease in limit load. The authors in [29] assumed that the FEM approach might be an
alternative to potentially difficult analytical solutions in fracture mechanics techniques.
Ferreira and Branco [30] used isoparametric 8-node elements in 2D model as well. The
authors used strain-life (e-N) approach to calculate crack initiation life and linear fracture
mechanics approach to calculate crack propagation (a-N) life and, therefore, have predicted total
(S–N) life, which was verified experimentally. Aside from the general observation of reduced
fatigue strength with mainly angular (peaking, 𝛿𝑝) misalignment, and that the governing codes
need to shift from good-practical-experience to more detailed strength analysis, the authors showed
that the inner weld toe is a more significant contributor to fatigue strength reduction than the outer
weld toe. Furthermore, the work of Ong and Hoon [31] clearly indicated the need for more strict
regulation of the angular distortion, 𝛿𝑝, and the authors proposed eliminating or moderating it, or
11
otherwise the 1% of diametral deviation (ovality or out-of-roundness) allowable in ASME BPVC
(1989) may yield a non-conservative design. Interestingly, the same allowable ovality can still be
found in recent North American codes, including ASME BPVC (2015).
Andrews [32] modeled distorted welds with cubic isoparametric (third-order) elements and
performed an elastic FE analysis, and showed a reduction of fatigue strength due to axial
misalignment similar to that published by Berg and Myhre [28]. A good correlation between the
experimental results and the results of FEM was found when the stress ranges were measured at a
distance of several millimeters away from the weld toe [32]. Interestingly, stress extrapolation
techniques based on a similar idea are commonly used in modern practice (BS 7608 [24]). It would
be worth adding here the efforts of Berg and Myhre [28] to separately analyze the bending stress,
𝜎𝑏, and membrane stress, 𝜎𝑚, at the critical location (hot-spot). Nowadays, this is implemented
via the stress linearization techniques used in some modern codes (API 579 [21] and ASME BPVC
Section VIII [22]).
Analytical solutions to the problem of weld misalignments were also extensively studied
and supported by FE analysis. The solutions for welded plates developed by Maddox (1985) [33],
whose research contributed greatly to the development of British standards BS 7910 [25] and BS
7608 [24] as well as others, were furthered by Andrews in 1996 [32]. Discrepancies between the
theoretical work of Zeman (1994) [34] [35] and FEM results for longitudinally welded pipes were
critically addressed and refined by Ong and Hoon in 1996 [31].
During the second half of the 20th century, the research on weld misalignments involving
FEM analysis mainly focused on the axial and the angular weld distortions separately. It is worth
noting the observations of Berge and Myhre [28] regarding the effects of weld misalignment on
the weld profile. There are complex synergetic effects due to actual welding processes that
12
influence the geometry of the welded region, and these combinations need to be carefully analyzed
to more accurately predict the fatigue properties of a welded structure.
2.2 Recent FE Analyses
The previous research involving FEM suggested the need for the standardization of some
principles and procedures to bring more agreement in the FEM community.
Since 1998, the ASME BPVC committee, led by Osage, started completely rewriting
Section VIII Division 2, and in 2007 published the modernized version of the code with the critical
changes discussed in [36], and updated it in 2015 [37]. One of the significant changes introduced
in the revised code was the estimation of the minimum WT by using design-by-analysis (DBA)
requirements in Part 5 of the code, which included the (mesh-insensitive) structural stress concept
for computation of the membrane, 𝜎𝑚, and bending, 𝜎𝑏, stresses at the critical location using FEA
as well as the procedures for elastic and elastic-plastic FEM analysis and models for the stress-
strain curves. Recommendations for the linearization of stresses at the critical location were
obtained by using FEA. In the design-by-rule (DBR) requirements, weld efficiency factors were
introduced in order to account for the types of welded connections and the explicit design rules for
combined loadings.
2.3.1. Hot-Spot Stress
The accuracy of FEM predictions of stresses at the critical locations (hot-spots), i.e., the
weld toe in the case of longitudinally welded UOE pipes, subsequently influences the estimation
of fatigue life, and is strongly dependent on the weld details and dimensionality (2D verse 3D) as
well as on the material model used in FEM (elastic verse elastic-plastic) and on the methodology
used to extract these stresses (i.e., extrapolation verse linearization).
13
In 2001 Dong published research [38] on the definition and procedure for obtaining
structural stress, 𝜎𝑠 = 𝜎𝑚 + 𝜎𝑏, which has been claimed to yield FEM mesh-size-insensitive
results even for a high-stress singularity. Assuming equilibrium between the axial, 𝜎, and shear, 𝜏,
stresses, this procedure transforms a complex stress state at a critical location (weld toe) of a
structure into an equivalent simple structural stress state (Figure 3), as opposed to the stress
extrapolation techniques for reference points away from the location of concern (i.e. the weld toe).
Stress extrapolation approximates the local stress at the critical location and cannot correctly
capture the stress concentration effects in locations with apparent weld joint discontinuities, and
thus may not be reliable according to Dong [38]. One of the codes that uses various stress
extrapolation techniques is BS 7608 [24]. The structural stress concept is today used in ASME
BPVC VIII [22].
Figure 3 Structural stress concept
Doerk et al. (2003) [39] systematically examined various methods for calculating the
structural stresses in welded structures by using FEM and discussed the validity of claims made
by Dong [38]. Shell elements (8-node quadratic) and solid elements (20-node isoparametric,
reduced-integration) were used in [39] to model 2D and 3D welded joints, and the use of a finer
mesh of at least 0.4𝑡 was suggested when using higher-order elements. However, one element in
the thickness direction is sufficient. While the weld is frequently omitted from the FEA based on
14
shell elements, or weld geometry is significantly simplified [39], solid elements allow for the weld
to be modeled in greater detail. In contrast to 2D models, the calculation of structural stress was
reported to show mesh-size sensitivity in 3D models, especially when a stress concentration
becomes more localized due to geometry (at the weld toe). Doerk concluded the research in [39]
with the statement “…the structural hot-spot stress approach remains to be relatively coarse,
however, very practical…”. Although the 3D models allow for complex shapes to be analyzed by
FEM and potentially lead to more accurate results, care should be exercised when ensuring their
validity.
Rohart et al. (2015) [40] addressed the conservatism involved in the elastically calculated
design limits of pressure vessels. The modeling of more complex geometries and loading scenarios
leads to more conservative designs compared to results based on a more realistic elastic-plastic
representation of a material’s behaviour and accounts for local effects more accurately. This
observation was supported by Möller et al. (2017) [41], who reported lower scatter in results when
plasticity is considered in the FE model. Therefore, it is not only the geometry problem when
modeling structures with discontinuities, but also in modeling the material’s behaviour.
Goyal, El-Zein, and Glinka (2016) [42] proposed a method for the calculation of mesh-
independent peak stress at the hot-spot via extraction of the through-thickness membrane and
bending components of true non-linear stress from a coarse-mesh FE model by the stress
linearization technique. In [42], cruciform weld joints were investigated, and while the through-
thickness variation of the membrane stress was shown to be mesh-insensitive, the bending
component showed mesh-insensitivity in the middle part of the plate section, 0.75𝑡 ≥ 𝑥 ≥ 0.25𝑡,
and increased at the weld toe with the mesh element size. This was used in [42] as the basis for
that proposed method, which uses linearly distributed bending stresses to calculate the peak stress
15
at the hot-spot. It can be true that a linear distribution of bending stress may exist in the case of a
symmetric geometry of a welded region. However, this may not be true in the case of non-
symmetric welded regions such as the ones found in UOE pipes, especially when they are
misaligned. Moreover, Goyal, El-Zein, and Glinka (2016) [42] reported up to a 10% difference in
calculated peak stress compared to the fine-mesh FE model. The finest mesh elements were
modeled at the weld toe, which had a size of about 0.2 mm, and 6 of these elements comprised the
weld toe radius at 𝑊𝑇 𝑊𝑟⁄ = 2.9. Furthermore, the differences between elastic and elastic-plastic
material models, which are both frequently used in FEM, may further increase the discrepancies.
Therefore, the simplified method for calculation of the peak stress at the hot-spot proposed in [42]
may lead to ambiguous results in the case of welds with combined misalignment.
2.3.2. Weld Distortion and Wall Thickness
Lillemäe et al. (2012) [43] investigated the effect of combined misalignment on the fatigue
strength of thin-wall butt-welded specimens by using both experimental testing and FE modeling,
which included detailed weld profiles. Tensile load was applied to specimens and S–N curves
plotted using notch stress range. The modeled weld was represented by fine mesh elements for
plane strain elements at the weld toes with sizes ranging from 0.025 mm to 0.1 mm, and other
locations were modeled with plane stress elements of 0.1-5.0 mm size. The structural hot-spot
stress was obtained by linear surface stress extrapolation. It was shown that thinner sections
undergo greater straightening of the welded region in tension and thus promote greater bending
compared to thicker sections. The corrected analytical solution for angularly distorted welds
presented by Kuriyama et al. [44] in 1971 accounted for the straightening and yielded a solution
for the stress concentration factor (SCF) that is within the range obtained by the FEM.
Interestingly, a similar formulation is now adopted in BS 7910 [25].
16
Another interesting observation by Lillemäe et al. [43] was that the structural stress could
be under- or over-estimated depending on the weld toe location, and their results showed that the
SCF rapidly increases with angular misalignment, especially at the weld toe located inside the
angled weld profile. This would correspond to the weld toe located inside of an outward peaking
pipe, and structural stresses can increase even more when axial misalignment is also considered,
in which case there will be a need to distinguish between two inside weld toe locations because
one of them would have a greater flank angle, 𝑊𝛼, and thus experience greater bending. The
opposite was reported for weld toes located outside the angled weld profile. Indeed, analytical
solutions available to date seem to be incapable of distinguishing between different weld toe
locations, and the calculated SFC and hot-spot stress is therefore averaged. This may lead to non-
conservative estimates of a design limit. Overall, the study in [43] represents a detailed FE
modeling example that discusses many geometrical parameters of a misaligned weld. However,
the 2D linear-elastic model used in the study may not adequately represent the local behavior of a
real structure, and the 2D model may not adequately capture the stress states of complex geometry.
There are no details on the convergence study of developed FE model. Furthermore, [43] focused
on the behaviour of a weld joint and neglected any surrounding structures.
Nykänen and Björk (2015) [45] analyzed the effects of weld geometry of as-welded butt
joints on stress concentrations at the weld toe using data available in the literature. Constant
amplitude fatigue tensile test results were analysed using the design S–N curve [24] based on
nominal stress. The authors discussed the stress concentrations in these welds due to angular
(0.2-0.3°) and axial (0.1-0.2 mm) misalignments, and mentioned possible variations of the SCF
depending on the weld toe location. The authors attempted to assess the effects of combined
misalignments using conservative analytical solutions when the misalignments were known, or by
17
using the FE-based solutions developed by Anthes et al. [46] in 1993 when only WT, 𝑡 (3-40 mm),
and flank angle, 𝑊𝛼 (14-43°), were known, and the weld toe radius, 𝑊𝑟, was fixed to 1 mm for all
the analyzed cases. The non-linear effects were disregarded because the models used in the
calculations were based on linear material behaviour. This resulted in lower errors for thicker
sections and larger errors for thinner sections. Furthermore, the misalignments assessed in the
study [45] were rather small, i.e., a tenth of a millimeter or a degree, and were not readily available
for many of the sampled data, and may not correctly represent the longitudinal welds in UOE
pipes. Additionally, the detailed analysis of the welded region could benefit from other weld shape-
specific parameters such as weld width and height, and curvature due to pipe diameter.
Lillemäe et al. (2016) [11] observed a noticeable (30%) increase in the fatigue strength of
high-quality welds measures using hot-spot stress in S–N approach. This increase was attributed
to lower stress concentrations at the weld toe due to lower welding distortion such as undercut
(< 0.05 mm), weld height (< 1 mm), flank angle (< 30 °), and higher transition radius (> 0.5 mm).
The linear-elastic 2D FE model was built with detailed weld regions based on the actual geometry
of fatigue test specimens using plane stress elements of 0.2 mm. This model was validated with
the solid 3D model with a 2% difference, and the hot-spot stress was determined using surface
linear extrapolation.
Möller et al. (2017) [41] modeled angularly distorted butt welds with detailed weld profiles
obtained from image analysis, although neglecting the axial misalignment. In contrast to Lillemäe
et al. [43], Möller et al. (2017) [41] considered plasticity in the FE model, which showed lower
scatter in the results. The nominal stresses as well as notch stresses and strains were used to
determine the fatigue resistance from S–N curves. The fatigue life of developed weld profiles in
lower-quality butt joints was reduced, and more significantly in high-cycle fatigue.
18
Pachoud, Manso, and Schleiss (2017) [47] studied the influence of the weld profile
parameters on SCF values, and both the weld height, 𝑊ℎ, and the flank angle, 𝑊𝛼, in axially
misaligned welds were reported to significantly contribute to stress concentrations at the weld toe,
which was calculated using a 2D linear-elastic FE model. The structural stress at the weld toe was
obtained by using both surface extrapolation and through-thickness linearization. For example, in
a 30 mm-thick plate with a butt weld profile of 𝑊ℎ = 0.75 𝑚𝑚, 𝑊𝑤 = 22.3 𝑚𝑚, 𝑊𝛼 = 10 °,
𝑊𝑟 = 1 𝑚𝑚, and misalignment of 𝛿𝑜 = 2.1 𝑚𝑚, the SCF increased to 1.86. The SCF was equal
to 2.02, with an additional change in 𝑊ℎ = 0.75 → 2.1 𝑚𝑚, and the SCF reached 2.52 after an
additional change in 𝑊𝛼 = 10 → 25 °. The authors compared their FE solutions for the SCFs with
analytical solutions from the literature, and aside from the scatter within analytical solutions, they
were observed to generally overestimate the SCFs compared to the FEM solutions, and this trend
increased with WT. Apparently, this difference was due to the fact that the available analytical
solutions did not explicitly account for the weld reinforcement height, 𝑊ℎ, and width, 𝑊𝑤. The
strong influence of 𝑊ℎ and 𝑊𝛼 on the fatigue strength of butt welds has also been reported for
thinner sections [11].
Shiozaki et al. (2018) [19] conducted the S–N fatigue tests and FEM analyses on lap joints
under bending using a 2D elastic model build with plane strain elements of 0.2 mm size, reduced
to approximately 0.04-0.10 mm at the weld toe. The results of this study may be particularly useful
for the analysis of the effects of toe radius in longitudinally welded tubular structures because of
some similarities in the loading and the boundary conditions. Specifically, a simply supported
beam of a lap joint that is allowed to rotate at the supports while under pulsating bending, which
is imposed at the free ends extended outside the supports, would represent the loading condition
of an axially misaligned welded region of a pipe experiencing similar bending during pulsations
19
of the pipe’s internal pressure. Certainly, the pipe’s longitudinal weld would also experience a
tensile component of stress acting in the pipe’s circumferential direction. However, the bending
component of the total stress would contribute to the opening and closing of a hypothetical crack
at the weld toe of an axially misaligned pipe weld, similarly to a lap weld.
Therefore, qualitative analysis of the results published by Shiozaki et al. [19] could
contribute to the selection of weld toe radii for the FEM modeling of misaligned pipes. Although
a lap weld would rather exaggerate the axial weld misalignment in pipes, 𝛿0 = 𝑡, it was also
studied in early research by Berge [28] and Andrews [32]. Reduction in the 𝑊𝑇 𝑊𝑟⁄ ratio from
14.50, 5.8, 2.90, to 1.93 yielded dramatic reduction in the slopes of the S-N curve from 33 × 10−5,
25 × 10−5, 3 × 10−5, to 1 × 10−5 respectively, and showed corresponding increase in the
allowable nominal stress range from 280 MPa, 475 MPa, 620 MPa, to 650 MPa at 106 cycles. The
fatigue strength was improved with increased 𝑊𝑟, and more significantly at 𝑊𝑇 𝑊𝑟⁄ = 5.8 ÷ 2.90.
The authors in [19] also analyzed the residual stresses of welded regions in the as-welded condition
and after heat treatment, and reported insignificant impacts, if any, on fatigue strength.
2.3 Governing Codes
The pipeline governing codes guide the process of selection of WT using well established
rules and analytical solutions (ASME B31.4 [23], ASME BPVC VIII [22], BS 7608 [24], and
BS 7910 [25]). The pipeline design based on these rules (design-by-rule (BDR)) considers the
influence of manufacturing processes and manufacturing tolerances in the form of factors applied
to calculated stresses and WT. Pipeline engineer may utilize additional degree of freedom offered
by the design-by-analysis (DBA) methodology (ASME BPVC VIII [22] or API 579-1/ASME FFS-
1 Fitness-For-Service [21]) when designing a pipeline with specific detailed geometry (i.e.
including weld and misalignments). The DBA analysis is usually performed using FE modeling.
20
However, the governing codes do not provide any guidance on how to approach such modeling.
Therefore, design engineers perform FE modeling based on their experience.
2.4 Conclusions of Literature Review
The discussion of the available literature to date on the problem of misaligned pipe welds
shows a great deal of progress in both physical testing and FEM analysis aimed at assessing
stresses and fatigue strength due to critical details of the welded region. However, the studies
reported in the reviewed literature are mainly limited to the weld part loaded uniaxially and did
not include complete pipe section and effects of typical loading. Therefore, the question remains
open as to how the combination of axial and radial misalignments in large-diameter pipes could
be modeled using FEM based on manufacturing tolerances, and how it would contribute to the
development of local stresses and fatigue behaviour.
21
Chapter 3 STANDARD PROCEDURES
The modern codes used in pipeline design provide the engineer with flexible assessment
methodologies and enable the fatigue design of components with specific geometries by using FE
modeling. However, few details are provided in codes on how to model manufacturing defects and
the combination of misalignments that occur after UOE manufacturing and SAW welding of
pipelines. This Chapter focuses on code procedures and methodologies used in fatigue analysis,
including the definition of permissible pipeline geometry, pipeline materials, and loading. The
stress concentration factor (SCF) was obtained for the SAW weld studied in this research. The
SCF can be equal to one in the case of a smooth transition between the base plate and the weld
bead; however, the real SAW profile of a UOE-processed pipe is rather sharp and magnifies the
applied loads. Moreover, as was discussed earlier, governing standards make use of DFs to account
for a specific welding process and do not take into account the variability of the weld profile
parameters.
3.1. Pipeline Codes
North American codes present the basis for pipeline designs in many countries. The ASME
code for pressure piping, B31, developed in the US, has part B31.4 [23] dedicated to buried
pipeline transportation systems for liquid hydrocarbons and other liquids. The ASME B31 code
was used as a basis for the Canadian code CSA Z662 [5], similarly to CSA 245.1 [4], which is
based on API 5L [3]. Many advanced design rules reference the more complete ASME Boiler &
Pressure Vessel Code (BPVC) Section VIII [22], which is the most comprehensive standard guide
for designing efficient pressurized equipment. The ASME BPVC Section VIII [22] shares similar
design principles with API 579-1/ASME FFS-1 (Fitness-For-Service) [21].
22
A buried piping design is usually based on the minimum specified yield strength (SMYS
or 𝜎𝑦𝑚𝑖𝑛) of selected construction materials and service temperatures below 120°C (ASME B31.4
[23], ASME BPVC VIII [22]). The codes take into account manufacturing defects in the form of
design factors and safety factors. These factors are used to modify 𝜎𝑦𝑚𝑖𝑛 and calculate the allowable
design stress, 𝑆, and obtain the nominal WT, 𝑡 = 𝑃𝐷 2𝑆⁄ , for a pipeline to ensure pressure
containment [48], where 𝑆 = 𝐹 ∙ 𝐸 ∙ 𝜎𝑦𝑚𝑖𝑛, 𝐹 is a safety factor, and 𝐸 is a weld efficiency factor.
The stress acting in the hoop direction, 𝑆𝐻, of a pipe section is usually the largest and, therefore,
should be limited to 𝑆𝐻 ≤ 𝑆. The stress due to thermal gradients and the flexibility stress acting in
the longitudinal direction, 𝑆𝐿, are combined with the hoop stress, 𝑆𝐻, to further adjust the design-
allowable stress. The WT is normally adjusted to a standard value as prescribed in B31.10 [49].
This design method is named as design-by-rule (DBR), which requirements apply to commonly
used pressure vessel shapes and pressure loading.
When the tolerances provided in the code for DBR are exceeded, the engineer may follow
the design-by-analysis (DBA) procedures (from ASME BPVC VIII code [22] or API 579-1/ASME
FFS-1 Fitness-For-Service [21]) to qualify the design. The pipeline design should be evaluated for
each applicable failure mode (plastic collapse, local failure, collapse from buckling, failure from
cyclic loading) to establish the design WT. A fatigue analysis shall be performed on a piping
system to check its suitability for cyclic operating conditions according to B31.3 [50], which
references ASME BPVC VIII code [22].
The DBA methodology utilizes results from a detailed FEM stress analysis to determine
the suitability of a component. However, the DBA methodology does not provide any
recommendations on the stress analysis method. The same issue exists with European codes
23
BS 7608 [24] and BS 7910 [25]. It is expected that the design engineer will provide an accurate
validated analysis, which implies that such analysis is to be performed by a professional with
special expertise. Nevertheless, the code provides the user with all necessary pre-processing
(physical properties, strength parameters, monotonic and cyclic stress-strain curves) and post-
processing (failure modes models) tools to support the FE analysis.
3.2. Pipeline Geometry
Although it has been shown by multiple researchers that the geometrical defects of welded
pipes can greatly influence the service life of pipelines, it is difficult to eliminate the pipeline
defects associated with manufacturing. Therefore, the tolerances prescribed by the pipeline design
governing codes limit the occurrence, size, and location of possible defects. The following parts
of this chapter overview the possible defects associated with the manufacturing processes for larger
pipelines. Extensive knowledge about the existing defects is extremely important in fatigue
assessment of the component.
The tolerances used in the design of a pipeline include those for OD, WT, mechanical
properties of construction materials, operation loading, pipeline location, and weld quality/type.
The following sections focus on the geometrical pipeline tolerances in the following most recent
standards: API 5L [51], ASME BPVC Section VIII [22], CSA Z662 [5], CSA Z245.1 [4],
BS PD 5500 [52], BS 7608 [24], and BS 7910 [25].
24
3.2.1. Pipe ODs and WTs
In [5], the nominal pipe WTs for different pipe ODs are divided into two categories, as
shown in Table 1, depending on how close the pipe section is to the compressor to properly account
for a pressure drop between pumping stations.
Table 1 Nominal WTs for different ODs
OD Close to a compressor Away from a compressor
16-20 0.252 0.189
22-36 0.2-0.3 0.220
38-54 0.311 0.252
It is a common practice to design a pipeline with a larger WT close to a compressor station
and gradually reduce the design WT for pipeline sections farther away from the compressor. This
reduces the cost of pipeline manufacturing. Notably, the values for WTs can be set independently
of distance from a compressor station for ODs in the range of 22-36 in.
The nominal values of ODs and WTs are used in pipeline design as the baseline, which is
usually extended/modified according to the specific design requirements/parameters, including but
not limited to the construction material, manufacturing method, defects, operating conditions,
operating environment, and pipeline location. In general, the design parameters specify the total
critical design stress the pipeline structure must withstand during its service life with no incidents
or hazardous consequences. After all the design factors are accounted for, the pipe dimensions can
be specified as summarized in Table 2 [51]. Similar values can be found in [4], with ODs of 356-
508, 559-914, and 965-1372 mm and WTs of 4.8-7.1, 5.6-7.1, and 6.4-7.1 mm.
25
Table 2 Permissible Specified ODs and WTs
OD, inch (mm) WT, inch (mm)
Light sizes Regular sizes
14-18 (356-457) 0.177-0.281 (4.5-7.1) 0.281-1.771 (7.1-45)
18-22 (457-559) 0.188-0.281 (4.8-7.1) 0.281-1.771 (7.1-45)
22-28 (559-711) 0.219-0.281 (5.6-7.1) 0.281-1.771 (7.1-45)
28-34 (711-864) 0.219-0.281 (5.6-7.1) 0.281-2.050 (7.1-52)
34-38 (864-965) - 0.219-2.050 (5.6-52)
38-56 (965-1422) - 0.250-2.050 (6.4-52)
56-72 (1422-1829) - 0.375-2.050 (9.5-52)
72-84 (1829-2134) - 0.406-2.050 (10.3-52)
An example of utilizing the design factors in the determination of a WT of an oil
transporting pipeline with an OD of 36” made from API 5L X70/X80 steel material, operated at
10 MPa internal pressure, for the TransCanada Keystone XL pipeline design, depending on
transportation, handling, bending, welding, and installation/back-fill, is represented in Table 3
[53]. Notably, the design WT is smaller when higher-grade steels are considered. This can be seen
when comparing the WTs for API 5L X70 and API 5L X80 in Table 3. Furthermore, the total
length of the combined Keystone pipelines is designed to be around 4700 km, which represents a
significant amount of steel to be used in manufacturing and installation processes, and is associated
with great financial expenditure.
Table 3 Example DFs and WTs used in Keystone pipeline
Design Factor DF 0.80 0.72 0.60 0.50
API 5L X70: WT, mm 11.811 13.081 15.697 18.999
API 5L X80: WT, mm 10.312 11.506 13.792 16.510
26
The variations in pipe WT for welded pipes with OD > 457 mm, according to [4], are
shown in Table 4. The variations for WTs of ≤ 5, 5÷15, and ≥ 15 mm, according to [51], should
be limited to ±0.5, ±0.1t, and ±1.5 mm respectively.
Table 4 Permissible Variation of WT
WT, inch (mm) %
< 0.375 (9.5) +17, -8
0.375-0.496 (9.6-12.6) +15, -8
> 0.5 (12.7) +12, -8
Finally, common nominal lengths of a pipe are 20, 30, 40, 50, 60, and 80 ft or 6, 9, 12, 15,
18, and 24 m [51] [4].
3.2.2. Weld Misalignments
The standards governing a pipeline design limit the occurrences and sizes of manufacturing
defects within the tolerances, and prescribe design safety factors for other design parameters. The
manufacturing tolerances and safety factors have been developed through extensive experience in
pipeline operations as well as experimental research.
The plane (location) and direction of the critical stress assessed in the design depend on
the specific pipeline manufacturing process, and are often related to the location of a major
structural defect. In the case of welded structures, the most critical factor would be the plane of a
hypothetical crack at the weld root. The direction of stress depends on the actual loading, and in
most cases is normal to the plane of a hypothetical crack, and referred to as normal stress, 𝜎. Shear
stress, 𝜏, tangential to the critical plane (in-plane stress), can also develop. For instance, in a
seamless straight pipeline, the critical plane is located at the weld root between two abutting
welded pipes parallel to the pipe section, as depicted in Figure 4(a). In a longitudinally welded
27
pipe, such a plane is perpendicular to both the pipe wall and the pipe section (Figure 4(b)), and the
critical plane is located at the weld root almost perpendicular to the pipe wall. Although the
position of a critical plane in a pipe with a helical seam is variable with pipe length, it remains
tangential to the seam and perpendicular to the pipe wall (Figure 4(c)). The critical stress usually
acts normal to the critical plane. In the case of seamless pipe, this stress acts in the pipe’s
longitudinal direction and is referred to as longitudinal stress, 𝜎𝐿. It can be due to pipe-soil
interaction. In longitudinally welded pipe, the critical stress acts in the pipe’s hoop direction
(horizontally and transverse to the pipe) and is referred to as hoop stress, 𝜎ℎ, and can be due to
pipe internal pressure. In a helical seam, the stress is oriented at an angle constant to the pipe’s
longitudinal direction and also tangential to the pipe wall, and can be due to both the pipe-soil
interaction and the internal pressure. In a more conservative estimate, the effects of the welding
process, including the orientation of a weld seam, are accounted for in the form of DFs.
(a) (b) ©
Figure 4 Welded connections in: (a) seamless pipeline, (b) pipeline with longitudinal seam,
and (c) pipeline with helical seam; the hatched area shows the plane of connection of two
pipes
Three major misalignments of a longitudinal weld seam can be observed in SAW-UOE
manufactured pipe, as shown in Figure 5 [24] [52] [25]. It is obvious that the defects associated
Tv L
Th
28
with the weld region would have a tremendous impact on the stress rise and therefore should be
limited, as discussed in the next sections.
(a) (b) (c)
Figure 5 Common types of weld misalignment in longitudinally welded pipe: (a) radial, (b)
angular, and (c) ovality
3.2.2.1. Radial Misalignment
Radial misalignment is also called axial or linear misalignment, or radial offset that arises
due to axial or plate thickness mismatch at the weld region, 𝛿𝑜 (Figure 5(a)). The radial offset of
the abutting edges of the same nominal WT, measured either at the outer or inner surfaces of the
pipe (high-low radial offset) (Figure 6) must not exceed the values presented in Table 5 (CSA and
API), including 10% of WT or 0.8 mm at the pipe ends, and 10% of WT or 1.5 mm away from the
pipe ends, as prescribed in CSA Z245.1 [4] and API 5L [51]. This value shall be limited to 0.063”
(1.6 mm) according to CSA Z662 [5]. Maximum misalignment of the weld beads, 𝑑, measured
between the centers of the weld beads 𝑀1 and 𝑀2 along the middle line of plates, must not exceed
0.1” (3 mm) for WT ≤ 0.8” (20 mm), or 0.16” (4 mm) for greater WT (Figure 6).
Weld
Dmax
Dmin
θ°
δp
2l
δo
29
Figure 6 Detailed schematic of radial misalignment
The offset within the allowable tolerance provided in Table 5 (ASME) must be faired at a
3-to-1 taper transition over the width of the finished weld, and a taper of 3×offset for differences
more than ¼ of the thinnest WT or 0.125” (3 mm) (ASME BPVC Section VIII Part 2 [22]).
Table 5 Permissible radial misalignment for different pipe thicknesses (North America)
CSA Z245.1 and API 5L ASME BPVC Section VIII Part 2
𝑡, inch (mm) ≤ 0.590 (15) 0.590-0.984 (15-25) > 0.984 (25) ≤ 0.5 (13) > 0.5 (13)
𝛿𝑜, inch (mm) 0.060 (1.5) 0.1t 0.098 (2.5) 0.2t 0.094 (2.5)
According to BS PD 5500 [52], the middle line of plates and the surfaces of the plates must
be aligned within the tolerances shown in Table 6. The circumferential tolerance for OD ≤ 25”
(650 mm) is ± 0.197” (5 mm), and 0.25% of circumference for greater ODs.
Table 6 Permissible radial misalignment (BS PD 5500)
Middle line of plates Surfaces of plates
𝑡, inch (mm) 𝛿𝑜𝑚𝑖𝑑𝑑𝑙𝑒, inch (mm) 𝑡, inch (mm) 𝛿𝑜ℎ𝑖𝑔ℎ−𝑙𝑜𝑤
, inch (mm)
≤ 0.394 (10) 0.039 (1) ≤ 0.472 (12) 0.25WT
0.394-1.968 (10-50) 0.1WT or 0.118 (3) 0.472-1.968 (12-50) 0.118 (3)
It can be seen that the different standards prescribe similar tolerances for radial
misalignment. However, the maximum difference can be as large as 1.5 mm. In this research, the
𝛿𝑜high
𝛿𝑜low
d
M1
M2
𝛿𝑜middle
30
value of 1.5 mm for 𝛿𝑜 will be used, corresponding to the portions of pipeline away from the
connection between pipe segments, i.e., pipe ends, according to CSA Z245.1 [4] and API 5L [51].
3.2.2.2. Angular Misalignment
Angular misalignment is also called outward peaking or inward peaking, 𝛿𝑝 (Figure 5(b)).
Peaking shall not exceed 0.125” (3.2 mm) according to API 5L [51], and shall not exceed 3 mm
within 200 mm of each pipe end according to CSA Z245.1 [4]. The maximum permitted values
for peaking (inward or outward), in excess of those shown in Table 7 (BS PD 5500 [52]), are only
permitted when supported by special fatigue analysis but must not exceed 0.197” (5 mm) for
t/D ≤ 0.025, and 0.394” (10 mm) for larger WT/OD ratios (or must not exceed WT).
Table 7 Permissible angular misalignment (BS PD 5500)
𝑡, inch (mm) 𝛿𝑝, inch (mm)
WT < 0.118 (3) 0.059 (1.5)
0.118 (3) ≤ WT < 0.236 (6) 0.094 (2.5)
0.236 (6) ≤ WT < 0.354 (9) 0.118 (3)
0.354 (9) ≤ WT t/3
Although there is insignificant difference between North American standards, European
standards tolerate much larger angular misalignments for vessels of larger WT.
3.2.2.3. Ovality Misalignment
Ovality or ovalization is also called inconsistency of diameter or out-of-roundness
(O-of-R), ∆𝜃 (Figure 5(c)), and shall be limited to the difference between the 𝐷𝑚𝑎𝑥 and 𝐷𝑚𝑖𝑛, not
exceeding 5% [5] or 1% [22]. According to [52], ovality shall not exceed (1
2+
625
𝑂𝐷 𝑚𝑚)% or 1%,
whichever is smaller. The ovality limit due to bending shall be limited to the range ∆𝜃≤ ∆𝜃𝑐𝑟𝑖𝑡=
0.03 ÷ 0.06, where ∆𝜃= 2 (𝐷𝑚𝑎𝑥−𝐷𝑚𝑖𝑛
𝐷𝑚𝑎𝑥+𝐷𝑚𝑖𝑛) is calculated with the OD [5]. Detailed allowance for
ovality misalignment is given in Table 8 [51]. For pipe of 𝐷 𝑡 ≥ 75⁄ the 𝐷𝑚𝑎𝑥 (𝐷𝑚𝑖𝑛) shall be
31
<1% larger (smaller) than the specified 𝐷, and at 𝐷 𝑡 ≤ 75⁄ the maximum differential between the
𝐷𝑚𝑎𝑥 or 𝐷𝑚𝑖𝑛 shall not exceed 12.7 mm for 𝐷 ≤ 1067 mm and 15.9 mm for 𝐷 ≥ 1067 mm [4].
Ovality can be measured using the same method used for measuring peaking [52].
Table 8 Permissible ovality misalignment (API 5L)
𝐷, inch (mm) 𝐷 tolerance, inch (mm) O-of-R tolerance, inch (mm)
Pipe body Pipe end Pipe body Pipe end
6.625-24 (168-610) ± 0.0075 𝐷
max. ± 0.125 (3.2)
± 0.005 𝐷
max. ± 0.063 (1.6)
0.02 𝐷 0.015 𝐷
24-56 (610-1422)
± 0.005 𝐷
max. ± 0.160 (4.0)
max. ± 0.063 (1.6)
0.015 𝐷
max. 0.6 (15)
for 𝐷 𝑡⁄ ≤ 75
0.01 𝐷
max. 0.5 (13)
for 𝐷 𝑡⁄ ≤ 75
Other pipe geometry defects, such as dents, discussed in [51] and [4], are not considered
in this research.
3.2.3. Welding Defects
Welding produces a variety of defects, including metallurgical and geometrical; since the
former is discussed in the codes from the perspective of residual stresses and is mainly considered
in crack propagation studies, this section will be focused on geometrical defects of welds, and
these will be considered in the research.
3.2.3.1. Weld Reinforcement
As-deposited inside and outside weld bead surfaces shall not extend above the adjacent
original parent metal surface by more than the values presented in Table 9, in accordance with
ASME BPVC Section VIII Part 2 [22], API 5L [51], CSA Z662 [5], CSA Z245.1 [4], BS 5500
[52], and shall not be below the prolongation of the applicable adjacent original parent metal
32
surface. Weld beads must be ground down to 0.02” (0.5 mm) from each pipe end to the distance
of 4” (100 mm) [51].
Table 9 Permissible weld reinforcement, inch (mm)
ASME BPVC API 5L CSA Z662 CSA Z245.1 BS 5500
𝑡 𝑊ℎ 𝑡 𝑊ℎ 𝑡 𝑊ℎ 𝑊ℎ 𝑊ℎ
Int. Ext. Ext.
< 0.094 0.031 ≤ 0.512 0.138 0.138 < 0.394 0.098 < 0.157 0.0394 + 0.1𝑊𝑤
< (2.4) (0.8) ≤ (13) (3.5) (3.5) < (10) (2.5) < (4) or (1) + 0.1𝑊𝑤
0.094-0.187 0.062 > 0.512 0.138 0.177 > 0.394 0.138 max. 0.197 (5)
(2.4-4.8) (1.5) > (13) (3.5) (4.5) > (10) (3.5) Int.
0.187-1.0 0.094 Penetration
(4.8-25) (2.5) 0.0394 + 0.3𝑊𝑤
1.0-2.0 0.125 or (1) + 0.3𝑊𝑤
(25-51) (3.0) max. 0.118 (3)
The welding process can negatively impact the geometry of the parent material adjacent to
the weld bead, and this is related to the reduction in WT 𝑡. The reduction of 𝑡 due to welding shall
not exceed 1/32” (1 mm) or 0.1𝑡 (ASME BPVC Section VIII Part 2 [22]). In North American
standards, this defect is referred to as undercut, which shall be not deeper than 0.5 mm (CSA
Z245.1 [4]), or 0.06𝑡𝑛𝑜𝑚 and 0.039” (1 mm) (CSA Z662 [5]). Undercut must be ≤ 0.016” (0.4 mm)
deep, or ≤ 0.031” (0.8 mm) deep when its length is ≤ 0.5𝑡, in any given 12” (300 mm) length of
weld (API 5L [51]). In BS 5500 [52], the reduction of 𝑡 is also referred to undercut, and is limited
to ≤ 0.039” (1 mm) or ≤ 0.025𝑡 in the case of a double-sided full-penetration butt weld between
co-planar plates (type 5.2D in BS 7608 [24]); weld root concavity or shrinkage groove must not
exceed 0.059” (1.5 mm). Since the weld root of the Double SAW method is located at the middle
line of the welded plates, the occurrence of an external shrinkage groove is unlikely.
33
Although the tolerances of weld height 𝑊ℎ can be found in most standards, the weld bead
width calculation, 𝑊𝑤 = 𝑡 2⁄ , as well as the weld root/toe radius definition, 𝑊𝑟 ≥ 0.25𝑡, are found
only in BS 7608 [24]. However, it can be approximately calculated from the geometry of the
abutting plates before welding (Figure 7). For example, the root gap can be set to 0.051-0.075”
(1.3-1.9 mm) for mechanized or automatic welding (CSA Z662 [5]), and the taper angle can be set
to 60° (ASME B31.4 [23]). Therefore, the 𝑊𝑤 for different standard 𝑡 is normally as shown in
Figure 8.
Figure 7 Schematic of abutting plates before welding
Figure 8 Weld widths for different wall thicknesses of plates tapered at 60°
3.2.3.2. Welding Cracks
Planar defects such as cracks, lack of penetration, and incomplete fusion shall be
unacceptable regardless of location [4] [52]. However, surface defects that have a depth ≤ 0.05𝑡
and do not encroach on 𝑡𝑚𝑖𝑛, such as undercuts, can be acceptable [51]. For gas or liquid service,
the maximum imperfection depth may exceed 0.5𝑡 or 0.25𝑊ℎ, provided that an analysis to
determine fatigue crack growth is carried out [5]. The maximum permissible size of any indication
10.00
15.00
20.00
25.00
30.00
35.00
9 14 19 24 29
Wel
d W
idth
, mm
Wall Thickness, mm
60°
Root gap
34
shall be limited to 𝑡 4⁄ or 4 mm, and indications separated by more than 25 mm may have a size
of 𝑡 3⁄ or 6 mm, whichever is less [22]. In any 300 mm-length of weld, elongated slag inclusions
shall not exceed 1.5 × 50 mm, and isolated slag inclusions shall not exceed 2.5 mm or 𝑡 3⁄ ×10
mm [5] [4]. Spherical porosity, circular slag inclusion, or gas pockets shall not exceed 𝑡 4⁄ or
3 mm, and the projected area shall be limited to 3, 4, and 5% at 𝑡 < 14 mm, 𝑡 = 14-18 mm, and
𝑡 > 18 mm respectively in any 150 mm length of weld [5] [4] [52]. Wormholes of length ≤ 6 mm
× width ≤ 1.5 mm and inclusions of length = 𝑡 ≤ 100 mm × width = 𝑡 10⁄ ≤ 4 mm are prescribed
in [52]. Similar tolerances on inclusions are given in [51].
3.3. Pipe-Soil Interaction
Oil pipelines located in non-developed areas in Canada are buried 2 ft (0.6 m) below the
ground (CSA Z622 [5]). The ASME B31.4 [23] code, which is used to design the liquid lines,
references the procedures for the design of restrained underground piping as prescribed in ASME
B31.1 [54] and in Guidelines for the Design of Buried Steel Pipe (American Lifelines Alliance,
2001) [55]. The procedures referenced provide guidance on modeling pipe-soil interaction. For
example, the Winkler model is widely used due to its simplicity in modeling pipe-soil interaction
with soil springs to represent soil forces in three principal directions [55].
The pipe burial conditions and the soil properties specific to the pipeline installation site
should be used in pipeline design to account for the effects of soil on a buried pipe. In this study,
a soil represented by mainly clay with the properties summarized in Table 10 will be considered.
Table 10 Soil properties
Soil 𝐸
[MPa]
𝜈 𝜌
𝑔 𝑚𝑚3⁄
𝛼
[mm/K]
𝜎𝑦𝐶
[MPa]
𝛼𝑓
[°]
𝛼𝑑
[°]
휀𝑝
10 0.3 0.0015 1×10-4 0.03 0 0 0
35
A relatively high coefficient of friction, 𝜇 = 0.3, between pipeline and soil can be selected
in case the pipeline is not provided with the protective coating. The soil friction angle of 𝛼𝑓 = 0°
can be selected (saturated soil showing undrained shear strength) to simplify the analysis [56], in
which case the Mohr-Coulomb criterion reduces to Tresca criterion due to only cohesive behavior
of soil (clay) [57]. Since the study focuses mainly on stresses on the pipe, this would be sufficient
to provide the necessary static pressure of a sustained load on pipeline due to soil and avoid
potential issues with convergence in the FE model. A soil box would provide a more convenient
and more realistic way of modeling the pipe-soil interaction [58].
3.4. Pipeline Materials
Steels manufactured according to the API 5L [51] are widely used in pipeline engineering.
In particular, low-carbon ferrite-perlite grades X42 through X80 are commonly used. Higher-grade
steels (e.g., X90, X100, and X120) are also used, benefiting from the higher strength and toughness
of the bainitic microstructure. While there is a significant difference between maximum yield
strength (𝜎𝑦𝑚𝑎𝑥) and maximum tensile strength (𝜎𝑡
𝑚𝑎𝑥) of lower-grade steels, this difference
reduces dramatically with the steel grade, as shown in Figure 9. This figure, constructed based on
data from [3] per API 5L [51], gives rise to the 𝜎𝑦/𝜎𝑡 ratio from 0.65 to 0.91, resulting in reduction
of elongation at fracture from around 25% to 5%. Materials with a 𝜎𝑦/𝜎𝑡 ratio close to unity may
be brittle because of limited plasticity at 𝜎𝑦 ≈ 𝜎𝑡.
Higher-strength materials contain relatively higher volume fractions of secondary particles
such as carbides 𝐹𝑒3𝐶 on which voids will nucleate and as a consequence, relatively lower fracture
resistances can be observed [59]. This implies that smaller grains and/or no secondary particles is
beneficial for toughness and crack resistance [60] [61]. When a crack is initiated, it can propagate,
36
and both higher-strength and lower-strength steels may be susceptible to sudden changes in the
crack propagation modes. For example, ductile to ductile-brittle mixed-mode types of failure have
been observed on the fracture surfaces [62]. It was shown in [63] that both low-strength and high-
strength steels (Figure 9) can have very similar resistances to fatigue damage.
Figure 9 Mechanical properties of pipeline steels
The increased grades of the X series of steels manufactured as per the API 5L standard [51]
shows an increased strength [64] [65], as shown in Figure 10, reduction of elongation at fracture,
휀, and decreases in both hardening capacity and fracture resistance [59].
Figure 10 Typical engineering stress-strain tensile curves for some X steels as per API 5L
[3]
200
300
400
500
600
700
800
900
1000
1100
1200
40 50 60 70 80 90 100 110 120
Stre
ss, M
Pa
Steel Grade, X
𝜎𝑡𝑚𝑎𝑥
𝜎𝑦𝑚𝑎𝑥
𝜎𝑡𝑚𝑖𝑛
𝜎𝑦𝑚𝑖𝑛
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
σ,
MP
a
1000
800
600
400
200
0
ε, mm/mm
X70 X80
X100
X52
𝑡𝑎𝑛(𝛼) = 𝐸 α
yield strength
37
The linear portion of each stress-strain curve in Figure 10 is referred to as the elastic portion
of the tensile curve. It has the same slope, since the Young’s modulus of elasticity, 𝐸, is the
material property and is almost the same for all X steels, 𝐸 ≈ 207 𝐺𝑃𝑎. This is due to their very
similar chemical compositions regardless of differences in the thermal-mechanical treatment.
Therefore, values of 𝐸 are not prescribed in API 5L [51]. The possible variability in actual 𝐸
reported in the available literature is most likely due to variations in the testing conditions such as
temperature. Hence, all X steels have same Hooke’s relationship for the elastic region,
𝜎 = 휀𝐸, (1)
until the yield strength, 𝜎𝑦, which is unique for each steel grade and signifies the transition to the
elastic-plastic region (the curvilinear portion of each tensile curve), as shown in Figure 10.
When the stress is above the yield strength, the material undergoes plastic deformation,
and the elastic-plastic region of the tensile curve can be described using
𝜎 = 𝐾휀𝑛, (2)
where 𝐾 is the strength coefficient and 𝑛 is the strain hardening exponent. Both can be obtained
experimentally by following the standard-practice ASTM E646 [66].
True stress-strain tensile experimental data can be fitted with the widely used Ramberg-
Osgood material model [67],
휀𝑡𝑟𝑢𝑒 =𝜎𝑡𝑟𝑢𝑒
𝐸+ 𝐾 (
𝜎𝑡𝑟𝑢𝑒
𝜎𝑦)
𝑛
(3)
or
휀𝑡𝑟𝑢𝑒 =𝜎𝑡𝑟𝑢𝑒
𝐸+ 𝛼
𝜎𝑡𝑟𝑢𝑒
𝐸(
𝜎𝑡𝑟𝑢𝑒
𝜎𝑦)
𝑛−1
, (4)
where coefficient 𝛼 can be computed as
38
𝛼 = 𝐾 (𝜎𝑦
𝐸)
𝑛−1
, (5)
and 𝛼𝜎𝑡𝑟𝑢𝑒
𝐸|
𝜎𝑡𝑟𝑢𝑒→𝜎𝑦
= 휀𝑦 is the yield offset (0.2% or 0.002 mm/mm),
true stress,
𝜎𝑡𝑟𝑢𝑒 = (1 + 휀𝑒𝑛𝑔)𝜎𝑒𝑛𝑔, (6)
and true strain,
휀𝑡𝑟𝑢𝑒 = 𝑙𝑛(1 + 휀𝑒𝑛𝑔). (7)
It follows from Eq. (3) that the total strain 휀𝑡𝑟𝑢𝑒 is the sum of the elastic part 𝜎𝑡𝑟𝑢𝑒 𝐸⁄ and
the plastic part 𝐾(𝜎𝑡𝑟𝑢𝑒 𝜎𝑦⁄ )𝑛
, which is present even at lower stresses within the proportionality
region, Eq. (1). For example, this can be seen from the X42 steel tensile data in Figure 11 [68],
cut-off at strain equal to 0.05 for clarity.
Figure 11 Typical engineering and true stress-strain tensile curves for X42 steel
Although the elastic-plastic region seems to have an almost exact fit, the curvature of the
yielding region is clearly omitted from the Ramberg-Osgood fit. A significant deviation from the
linear portion of the elastic region of both the engineering and the true stress-strain curves can be
0 0.01 0.02 0.03 0.04 0.05
σ,
MP
a
500
400
300
200
100
0
ε, mm/mm
true stress-strain engineering stress-strain Ramberg-Osgood fit
39
seen from just above 150 MPa to around 350 MPa, at the intersection of all three curves. This
inaccuracy may be insignificant in the case of softer materials with high strain-hardening capacity
and smooth elastic-plastic transition. However, the inaccuracy needs to be addressed in the case of
higher-strength materials such as X42 and higher-grade steels that show more pronounced yielding
plateaus (see Figure 12) in order to avoid possible errors associated with further use of these data.
Figure 12 Difference between experimental stress-strain tensile curve for X42 steel and
Ramberg-Osgood fit near the yield strength
One of the solutions would be connecting the yield strength point with a line tangent to the
Ramberg-Osgood fit, which can subsequently be utilized in other calculations such as FEM.
The results of uniaxial tensile test of material can be used to predict yielding under multi-
axial loading conditions by calculating an equivalent tensile stress developed by Ludwig Heinrich
Edler von Mises, and, thus, also known as equivalent von Mises stress, 𝜎𝑣, or von Mises yield
criterion,
𝜎𝑣 = √[(𝜎11 − 𝜎22)2 + (𝜎22 − 𝜎33)2 + (𝜎33 − 𝜎11)2 + 6(𝜏122 + 𝜏23
2 + 𝜏312 )] 2⁄ , (8)
which is a circle of a cylinder with axis [1 1 1] and radius √2 3⁄ 𝜎𝑦 in case of hydrostatic stress,
(𝜎11 − 𝜎22)2 + (𝜎22 − 𝜎33)2 + (𝜎33 − 𝜎11)2 = 𝑐𝑜𝑛𝑠𝑡 at 𝜎11 = 𝜎22 = 𝜎33, Figure 13(a), or has
true stress-strain engineering stress-strain Ramberg-Osgood fit
σ,
MP
a
400
300
200
100
0
ε, mm/mm
0 0.0025 0.005 0.0075 0.01
𝜎𝑦𝑡𝑟𝑢𝑒
𝜎𝑦𝑓𝑖𝑡
휀𝑒𝑡𝑟𝑢𝑒
휀𝑒𝑓𝑖𝑡
휀𝑝
∆휀
∆𝜎
40
elliptical shape in plane stress, 𝑐𝑜𝑛𝑠𝑡 = 2𝜎112 |𝜎11→𝜎𝑦
at 𝜎3 = 0, because 𝜎1 − 𝜎2 plane (1 1 0),
would cut the yield surface of a cylinder at 45° angle, resulting in the ellipse projection on the
𝜎1 − 𝜎2 plane, Figure 13(b). The von Mises yield criterion curve would intersect the principal
stress axes, normalized by 𝜎𝑦, at unity when stress reaches yield stress of material.
Figure 13 Tresca and von Mises yield criteria in (a) hydrostatic and (b) plane stresses
The plane stress condition is practical for ductile materials as they are compressible, show
plasticity and shearing at stresses higher than the yield limit, 𝜎𝑦. Although von Mises criterion
provides better agreement with experimental data, the Tresca criterion is still used because of its
simplicity,
𝑚𝑎𝑥 (|𝜎11 − 𝜎22|, |𝜎22 − 𝜎33|, |𝜎33 − 𝜎11|) 2⁄ = 𝜏𝑦 = 𝜎𝑦 2⁄ . (9)
The Tresca yield criterion for plane stress condition is derived from the yield surface of a
hexagonal prism, Figure 13, similarly to von Mises criterion.
The equivalent stress should be limited to some design value, which is usually related to
material’s yield limit, 𝜎𝑦, and is calculated based on actual loading of a component. In
longitudinally welded straight portion of a pipeline, the weld seam would normally experience
𝜎1 𝜎2
𝜎3
𝜎1 = 𝜎2 = 𝜎3
von Mises Tresca
ඨ2
3𝜎𝑦
(a)
-1
𝜎1
𝜎𝑦
𝜎2
𝜎𝑦
-1
1 1
0
Tresca
von Mises
(b)
ඨ2
3𝜎𝑦
√2𝜎𝑦
41
stresses as shown in Figure 14. Both the normal (out-of-plane) and the shear (in-plane) components
of the stress tensor are depicted in Figure 14. The stress tensor shows geometrical relations between
the stress components in a continuous material and directions of acting stresses, wherein 1 = 𝑋,
2 = 𝑌, and 3 = 𝑍.
Figure 14 Stress tensor at the longitudinal weld in pipeline
The stress in hoop direction, 𝜎𝑦𝑦, is usually the largest in longitudinally welded pipes
because it acts normal to the structural discontinuity (weld seam, the red line in Figure 14) and
tend to open the hypothetical crack; longitudinal stress (𝜎𝑥𝑥 in Figure 14) is usually the second
largest stress, and the stress in radial direction (𝜎𝑧𝑧 in Figure 14) is the smallest. For the pipeline
to provide the safe pressure containment, all stresses should be determined, and the equivalent
stress (or stress in most critical direction/plane) should be calculated and limited to the design
value based on mechanical properties of pipeline material used (API 5L [51] and ASME BPVC
Section VIII [22]). The pipeline loading should be properly accounted in the calculation of stresses.
𝜎𝑧𝑧
𝜏𝑧𝑦 𝜏𝑧𝑥
𝜎𝑦𝑦
𝜏𝑦𝑧
𝜏𝑦𝑥
𝜎𝑥𝑥
𝜏𝑥𝑧
𝜏𝑥𝑦
X Y
Z
σLong.= σxx
σRadial= σzz
σHoop= σyy
Z
Y X
σHoop
42
3.5. Loading
A pipeline can be exposed to extreme loading conditions, including aggressive
environments, high temperatures, multiaxial loading states, etc. A pipeline may experience internal
pressure fluctuations and temperature gradients associated with oil and gas transportation and
seasonal weather changes, and location-specific conditions may include winds acting on above-
ground pipes, currents that strain sub-sea pipes, soil pressure in the case of underground pipes,
landslides, dents, etc. All applicable loads and their combinations shall be considered, and a
loading histogram shall be developed. The actual random amplitude loading spectra (Figure 15(a))
can be very complex and are usually analyzed and converted to blocks of constant amplitude
loading (Figure 15(b)) by a standard algorithm, such as rain-flow cycle counting, in order to
simplify further use in fatigue life assessment [21] [22] [24]. The in-service conditions of the
component must be appropriately considered to correctly identify the failure criteria at the design
stage for timely repair/replacement during service.
(a) (b) Figure 15 Fatigue loading showing (a) spectrum loading and (b) constant amplitude
loading
The pressure history of the liquid pipeline, as shown in Figure 16, was obtained from the
literature [69], and was cycle-counted as per ASME BPVC Section VIII Part 2 [22] using the
rainflow counting algorithm from [70], and is shown in Figure 17. This is an example of a typical
time
load
reversal – valley
reversal – peak
mean
range
block 1 block 2 block 3
time
load
max
mean
range
am
pli
tud
e
min
43
loading observed in liquid lines between pumping stations during transportation of oil [69]. Similar
loading spectrum can be found in [71]. Notably, the large variations in pressure are typical. The
pipeline sections located away from pump stations normally experience smaller ranges of pressure
per cycle due to pressure drop, and, therefore, would be considered non-fatigue-prone areas.
However, these sections are usually designed with lower WT and may result in the stress cycles
comparable to those observed in pipeline sections immediately downstream the pump station [71].
Therefore, the actual pressure readings at the location of interest should be used in fatigue design.
Figure 16 In-service pressure history diagram
The selected pressure history contains cycles that exceed the design limit pressure, 𝑃𝑖, of
10 MPa, which can represent a potential worst case loading scenario, as pipelines that transport
liquids, which are generally incompressible, tend to experience transient pressure fluctuations
during service for different reasons [72] [73] [74] [75] [76]. Unfortunately, some of these cycles
can lead to disastrous consequences due to pipeline failure. The overpressure cycles presented in
Figure 17 are expected to accelerate fatigue damage due to pulsating bending of a misaligned weld,
especially for loads with stress ratio 𝑅 ≥ 0, as analyzed in [18].
0
2
4
6
8
10
0 50 100 150 200 250
Pre
ssu
reP
i, M
Pa
Days
44
Figure 17 Cycle-counted in-service pressure history showing Pmin, Pmax, and nk
The cycle-counting process of the data obtained from some real spectra, such as the one
presented in Figure 16, can be controlled as described in APPENDIX B – MATLAB Cycle-
Counting. A complete input file, *.inp, with all programmed features and properties of the elastic-
plastic model is given in APPENDIX C – ABAQUS Input File. The elastic model can be obtained
by deleting the elastic-plastic properties from the elastic-plastic model.
3.6. Linearization of Stresses
Through-thickness stress linearization of the actual non-linear stress distribution is
prescribed by the ASME standard to obtain different components of stress at the hot-spot,
including the membrane stress 𝜎𝑚 and bending stress 𝜎𝑏. After the analysis in ABAQUS is
finished, the calculated stresses for each loading cycle are linearized at the hot-spot along the Stress
Classification Line (SCL). The approximate path of an SCL is used for stress linearization as
required by ASME BPVC Section VIII Part 2 [22].
To calculate the membrane and bending stresses, ABAQUS [77] utilizes the integration of
stresses through the section as:
𝜎𝑚 =1
𝑡∫ 𝜎 ∙ 𝑑𝑥
𝑡
2
−𝑡
2
and 𝜎𝑏 =6
𝑡2 ∫ 𝜎 ∙ 𝑥 ∙ 𝑑𝑥
𝑡
2
−𝑡
2
, (10)
ASME BPVC Section VIII Part 2 [22] utilizes the integration of stresses through the section using
0
20
40
60
0
2
4
6
8
10
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 26 27 28 29 30 Nu
mb
er o
f C
ycle
s n
k
Inte
rnal
Pre
ssu
re
Pm
in, P
ma
x,M
Pa
Cycle
Pmin
Pmax
Cycles
45
𝜎𝑚 =1
𝑡∫ 𝜎 ∙ 𝑑𝑥
𝑡
0 and 𝜎𝑏 =
6
𝑡2 ∫ 𝜎 ∙ (𝑡
2− 𝑥) ∙ 𝑑𝑥
𝑡
0, (11)
where 𝑡 is the WT and 𝜎 is the stress at a coordinate 𝑥 along the SCL path. It should be noted that
both integrations yield the same results.
Assuming that the Level 1 fatigue analysis screening criteria are satisfied (ASME BPVC
Section VIII [22]), and the stresses and strains are obtained using FEA, the fatigue analysis can be
performed by using one or several of the available fatigue assessment methods. These are based
on an elastic stress analysis and equivalent stresses, or based on an elastic-plastic stress analysis
and equivalent strains for smooth bar fatigue curves. They can also be based on the fatigue curves
for welded joints when performing fatigue assessment of welds based on elastic analysis and
structural stress. Each of the fatigue assessment methods available in ASME BPVC Section VIII
[22] follows step-by-step procedures discussed in the next section.
3.7. Analytical Model
The analytical model primarily focuses on the development of stress at the structural hot-
spot of a buried pipe in the hoop direction due to the weld, its misalignment, and soil.
3.7.1. Stress due to Misalignment
Critical stress for a longitudinal weld is the one perpendicular to the longitudinal direction
of a weld seam (⊥ to a defect or discontinuity) acting in the axial direction of the welded steel
plates, i.e., the hoop stress 𝑆ℎ,
𝑆ℎ =𝑃∙𝐷
2∙𝑡, (12)
calculated based on 𝑃 (the internal pressure of a pipe), its 𝐷 (the outside diameter), and 𝑡 (the wall
thickness). For example, 𝑆ℎ = 261 𝑀𝑃𝑎 when 𝑃 = 10 𝑀𝑃𝑎, 𝐷 = 914 𝑚𝑚, and 𝑡 = 17.5 𝑚𝑚.
This stress is a nominal membrane stress that occurs in the pipe wall away from the weld. To
46
obtain the design stress, the nominal stress needs to be multiplied by a design stress magnification
factor, 𝑘𝑚, which should include the effects from all possible stress magnifiers such as the effects
of weld geometry and misalignments, and stress magnification due to soil.
Special attention was paid to the modeling of a weld profile and misalignment to capture
their effects on the hot-spot’s stresses/strains.
The critical spot in the weld is the weld toe, which for a DSAW seam is located on both
the outer and inner surfaces of a pipe at the transition between the plate and weld
crown/reinforcement. The weld toe is a stress concentrator which magnifies the 𝑆ℎ by a factor 𝑘𝑚
(𝑘𝑡) – stress magnification (or stress concentration) factor (SCF), BS 7608 [24],
𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛 = 𝑆ℎ ∙ (1 + ∑ 𝑘𝑚.𝑖𝑛𝑖=1 ). (13)
The contributions to the stress rise due to various types of weld misalignments, discussed
in 3.2.2, will be further addressed. The 𝑛 types of different contributors to the total SCF need to
be considered in design, where each magnification factor comes with a sign depending on the
location of the critical spot considered and represents the contribution to the total stress from the
secondary bending due to a particular effect BS 7910 [25].
Therefore, having abutting plates with equal thicknesses, the SCFs can be calculated for
the axial misalignment (BS 7910 [25]), 𝑘𝑚.𝑎𝑥𝑖𝑎𝑙, as
𝑘𝑚.𝑎𝑥𝑖𝑎𝑙 =𝜎𝑏.𝑎𝑥𝑖𝑎𝑙
𝑃𝑚= 3
𝛿𝑜
𝑡=
6𝛿𝑜
𝑡1(1−𝜈2)(
1
1+(𝑡2 𝑡1⁄ )0.6), (14)
for the angular misalignment (BS 7910 [25]), 𝑘𝑚.𝑎𝑛𝑔𝑢𝑙𝑎𝑟, as
𝑘𝑚.𝑎𝑛𝑔𝑢𝑙𝑎𝑟 =𝜎𝑏.𝑎𝑛𝑔𝑢𝑙𝑎𝑟
𝑃𝑚=
3𝛿𝑝
𝑡(1−𝜈2){
𝑡𝑎𝑛ℎ(𝛽 2⁄ )
𝛽 2⁄}, (15)
and for the ovality misalignment (BS 7910 [25]), 𝑘𝑚.𝑜𝑣𝑎𝑙𝑖𝑡𝑦, as
47
𝑘𝑚.𝑜𝑣𝑎𝑙𝑖𝑡𝑦 =𝜎𝑏.𝑜𝑣𝑎𝑙𝑖𝑡𝑦
𝑃𝑚=
1.5(𝐷𝑚𝑎𝑥−𝐷𝑚𝑖𝑛)𝑐𝑜𝑠2𝜃
𝑡{1+𝑝(1−𝜈2)
2𝐸(
𝐷𝑚𝑡
)3
}, (16)
where parameter 𝛽 is calculated from
𝛽 =2𝑙
𝑡{
3(1−𝜈2)𝜎𝑚𝑚𝑎𝑥
𝐸}
0.5
, (17)
𝛿𝑜 – axial mismatch, 𝑡 – plate (pipe wall) thickness, 𝜎𝑏 – bending stress at the weld toe due
to misalignment, 𝑃𝑚 – applied membrane stress, 𝛿𝑝 – peaking, ν – Poisson’s ratio, 𝑃 – max.
operating pressure, mean diameter 𝐷𝑚 = 𝐷 − 𝑡, E – modulus of elasticity, 𝜎𝑚𝑚𝑎𝑥 – max. membrane
stress (𝑆ℎ), 𝑙 – half of a distance between the points when the peaking departures from the perfect
circle (found geometrically using SolidWorks between tangent points after 𝛿𝑜 and 𝛿𝑝 applied), 𝜃
– angle at which the weld is located w.r.t. the ovalized section (0° and 90° are considered).
3.7.2. Stress due to Soil
It is important to account for the stresses coming from the bending of a pipe wall due to
soil pressure (external pressure). Following the previous methodology, we introduce 𝑘𝑚.𝑠𝑜𝑖𝑙
(Guidelines for the Design of Buried Steel Pipe [55]),
𝑘𝑚.𝑠𝑜𝑖𝑙 =𝜎𝑏.𝑠𝑜𝑖𝑙
𝑃𝑚, (18)
where 𝜎𝑏.𝑠𝑜𝑖𝑙 is a through-wall bending due to soil,
𝜎𝑏.𝑠𝑜𝑖𝑙 = 4𝐸 (∆𝑦
𝐷) (
𝑡
𝐷), (19)
(∆𝑦
𝐷) is a deflection (or ovality) due to soil,
(∆𝑦
𝐷) =
𝐷𝑙𝐾𝑏𝑃𝑠(𝐸𝐼)𝑒𝑞
(𝐷𝑚
2)
3+0.061𝐸′,
(20)
and 𝑃𝑠 is the soil pressure on the pipe above the water table,
48
𝑃𝑠 = 𝜌𝐻, (21)
(𝐸𝐼)𝑒𝑞 – equivalent pipe wall stiffness per unit length of pipe (may include pipe lining and pipe
coating), ∆𝑦 – vertical deflection of pipe, 𝐷𝑚 – pipe mean diameter, 𝐷𝑙 – deflection lag factor
(≈ 1.0-1.5), 𝐾𝑏 – bedding constant (≈ 0.1), 𝐼 is moment of inertia of wall cross section per unit
length of a plane pipe,
𝐼 =𝑡3
12, (22)
𝐸′ – modulus of soil reaction (≈ 0-20 MPa, for loose to compact soil), 𝜌 – unit weight of soil fill
((0 ÷ 125) 𝑙𝑏 𝑓𝑡3⁄ , 120 𝑙𝑏 𝑓𝑡3⁄ = 1.96 × 10−5 𝑁 𝑚𝑚3⁄ , an 𝐻 – pipe burial depth [55] [78], as
shown in Figure 18.
Figure 18 Through-wall bending stress and ovality of pipe cross-section due to transmitted
pressure
H
∆y
P
σb
WT
49
3.8. Fatigue Assessment
The fatigue life assessment of such industrial components as pipelines is critical, as it may
involve substantial financial losses or even human lives in case of a catastrophic failure, as well as
pollution and other environmental impacts [72] [73] [74] [75] [76]. This phenomenon requires
special attention and accompanies developments in different fields of engineering. Different
fatigue assessment methods have been developed and used in engineering practice (Figure 19).
The use of each of the methods summarized in Figure 19 results in the estimation of number of
cycles to failure, 𝑁𝑓, at certain magnitudes of load, stress amplitudes, 𝑆𝑎, or strain amplitudes, 휀𝑎.
With the increasing complexity of components, it is often not sufficient to use the
generalized assessment-specific criteria, which need to be specified more accurately by the user to
reflect the properties of a real component as closely as possible (Figure 20).
Figure 19 Schematic classification of fatigue life approaches
Standardized specimens (Figure 20(a)) and procedures are used to simulate the behavior of
a full-scale component/product (Figure 20(b)). Preferably, the component itself is tested under
service conditions; however, obtaining the experimental data may be very expensive.
Nf (log scale)
Sa (
log
sca
le)
Crack
Initiation Period
Strain Life Approach
(ε-N)
Crack
Growth Period
Fracture
Mechanics (a-N)
Total Life Stress Life Approach
(S-N)
Se
50
(a) (b) Figure 20 Standardized tensile specimen (a) versus real component (b)
This section focuses on design aspects related to fatigue and discusses various S–N
approaches used to determine the fatigue damage/life of a component. These include fatigue
analysis based on elastic stress analysis and equivalent stress ranges (in this work, this is referenced
as Elastic Fatigue Analysis, BS 7608 [24]), fatigue analysis based on elastic stress analysis and
equivalent stresses (in this work, this is referenced as Modified Elastic Fatigue Analysis, ASME
BPVC Section VIII Part 2 [22], Level 2, Method A), and fatigue analysis based on elastic-plastic
stress analysis and equivalent strain (in this work, this is referenced as Elastic-Plastic Fatigue
Analysis, ASME BPVC Section VIII Part 2 [22], Level 2, Method B).
3.8.1. Stress-Life Curves
Methods of fatigue assessment based on stress-life (S-N) fatigue curves (BS 7608 [24],
ASME BPVC VIII [22] or API 579-1/ASME FFS-1 Fitness-For-Service [21]) are also known as
Level 2 methods. The first systematic investigation of S-N curves was carried out in the late 19th
century by the German railway engineer August Wӧhler, whose work was focused on the causes
of fractures in railroad axles. The Wӧhler curves show the number of cycles to failure, 𝑁𝑓, at a
certain stress amplitude, 𝑆𝑎. One of interesting properties of these curves is the stress level called
fatigue strength, also known as fatigue limit or endurance limit, 𝑆𝑒, below which the life of a
component under cyclic loading, is theoretically infinite and where fatigue failure should not
occur.
51
The S-N assessment deals with non-planar flaws, which would apply to the weld profile
and associated geometrical defects. The basic S-N curve is shown in Figure 21 (BS 7608 [24]),
and can be obtained through the constant-amplitude force-controlled fatigue test (ASTM E466
[79]), and it shows the effects of variable amplitude (spectrum loading) and the aggressiveness of
the environment on an SAW type of weld (type 5.2D in BS 7608 [24]).
The basic S-N curves relate predominantly to membrane stress, 𝜎𝑚; however, the fatigue
strength of real structures depends on the degree of through-thickness bending, and thus bending
stress, 𝜎𝑏, BS 7608 [24]. Therefore, the basic S-N curve needs to be modified with all possible
deviations from the idealized scenario to obtain a design S-N curve that will reflect the behavior
of a real component, including in-service loading, surface finishing, notches, welding,
misalignments, environmental effects, etc.
Figure 21 Stress life (S-N) curve
The mentioned effects can be included in the basic S-N curve through modification of the
membrane, 𝜎𝑚, or the nominal, 𝜎𝑛𝑜𝑚, stress in the form of stress amplification factors (SAF),
𝜎𝑡𝑜𝑡𝑎𝑙 = 𝜎𝑛𝑜𝑚(1 + ∑ 𝑘𝑖𝑛𝑖 ). (23)
The next step in S-N fatigue life assessment is calculating the number of cycles to failure
(NCF), 𝑁𝑓. Both European and North American standards use equations derived from
static
limitations
s
Nf (log scale)
Sa (
log
sca
le),
MP
a
Sa.c=53
Sa.v=31
1 m
1 m+2
constant amplitude
5 Nav=
5×
10
7
Nac=
10
7
Sa=2σy
52
experimentally obtained best-fitted data. In BS 7608 [24] this is done based on parameters specific
for different weld connections, and the NCF is computed for a range of cyclic stress, which does
not account for the effects of mean stress. In ASME BPVC [22], the effect of mean stress is
accounted for in the CNF equation, whose parameters are based on a specific group of materials
used to obtain the fatigue curve.
Finally, in the case of constant amplitude loading, blocks of 𝑛 number of cycles can be
used to compute the fatigue damage (FD) of a component using Miner’s rule,
𝐷𝑓 =𝑛
𝑁𝑓, (24)
which can include the FDs caused by each loading block, 𝐷𝑓 = ∑𝑛
𝑁𝑓.
3.8.2. Elastic Fatigue Analysis
The elastic fatigue analysis was adopted from BS 7608 [24] and BS 7910 [25], which are
well established and widely used European standard practices for fatigue analysis of welded
structures. This analysis can be used as a reference for comparison to similar yet more detailed
analyses based on North American standards, ASME BPVC Section VIII Part 2 [22] and API 579-
1 [21]. The type of weld corresponding to a longitudinal SAW seam weld of a large-diameter pipe
is a double-sided full-penetration butt weld between co-planar plates. This corresponds to weld
design class 5.2D according to BS 7608 [24]. The number of cycles to failure, 𝑁𝑘, due to cyclic
loading can be calculated by using
𝑁 = 10𝑙𝑜𝑔(𝐶0)−𝑑∙𝑆𝐷−𝑚∙𝑙𝑜𝑔(𝑆𝑟), (25)
where for a selected weld design class, 𝐶0 = 3.988 × 1012 is a constant, 𝑆𝐷 = 0.2095 is a
standard deviation, 𝑑 = 3 is a factor for a nominal probability of failure of 0.14% equal to three
standard deviations of 𝑙𝑜𝑔(𝑁𝑘) below the mean, 𝑚 = 3 is a slope of the 𝑆 − 𝑁 curve, and 𝑆𝑟 is
53
the applied stress range, i.e., algebraic difference between the two extremes (reversals) of a given
cycle and is equal to ∆𝜎𝑚 + ∆𝜎𝑏 + ∆𝜎𝑝 at the hot-spot, and is obtained through stress linearization.
Although the BS standard states that the use of a stress component, 𝜎𝑖𝑗, acting normal to the weld
toe is sufficient, the use of all stress components is preferred.
The BS code advises that the maximum accumulated FD be limited to 0.5 for critical cases.
Therefore, the calculated values of FD can be normalized to 1.0 to align it with the maximum
allowable FD, as per the ASTM standard.
The elastic fatigue analysis in ASME BPVC [22] and API 579-1/ASME FFS-1 [21] utilizes
similar methodology in Level 2 – Method A, or plasticity corrected Method B.
3.8.3. Modified Elastic Fatigue Analysis
The elastic fatigue analysis can be performed in accordance with API 579-1 [21] and
ASME BPVC Section VIII Part 2 [22], Level 2, Method A. Normally, the effective strain (or the
von Mises equivalent total strain [80]) range, ∆휀𝑒𝑓𝑓,𝑘, one of the main components of the driving
force for fatigue damage, is approximated from an elastic analysis and incorporated in calculation
of fatigue penalty factor (plasticity correction factor), 𝐾𝑒,𝑘. However, a considerable variability in
analysis results can be observed since the stress results are sensitive to the mesh density of the
FEM [37] and the 𝐾𝑒,𝑘 factor can yield inaccurate results for larger stress ranges [81]. Therefore,
the optional modified steps related to the calculation of 𝐾𝑒,𝑘 provided in the ASME standard can
be implemented in order to improve the computational results. Thus, the fatigue analysis in this
section is based on the elastic effective equivalent stress range, ∆𝑆𝑃,𝑘, and the plastic equivalent
strain range, ∆휀𝑝𝑒𝑞,𝑘.
54
The driving force for a fatigue damage – effective alternating equivalent stress (stress
amplitude) for the 𝑘𝑡ℎ cycle, 𝑆𝑎𝑙𝑡,𝑘, which is needed to calculate the number of cycles to failure,
𝑁𝑘, and the total FD, 𝐷𝑓, caused by a fatigue cycle, can be computed using
𝑆𝑎𝑙𝑡,𝑘 =𝐾𝑓∙𝐾𝑒,𝑘∙∆𝑆𝑃,𝑘
2, (26)
where ∆𝑆𝑃,𝑘 is the effective equivalent stress range,
∆𝑆𝑃,𝑘 = √(∆𝜎11,𝑘−∆𝜎22,𝑘)2
+(∆𝜎11,𝑘−∆𝜎33,𝑘)2
+(∆𝜎22,𝑘−∆𝜎33,𝑘)2
+6(∆𝜎12,𝑘2 +∆𝜎13,𝑘
2 +∆𝜎23,𝑘2 )
2
2, (27)
which is obtained using the range of elastic component stresses, ∆𝜎𝑖𝑗,𝑘 = 𝜎𝑖𝑗,𝑘𝑒𝑛𝑑 − 𝜎𝑖𝑗,𝑘
𝑠𝑡𝑎𝑟𝑡, between
the stress tensors of the start 𝜎𝑖𝑗,𝑘𝑠𝑡𝑎𝑟𝑡 and end 𝜎𝑖𝑗,𝑘
𝑒𝑛𝑑 points of a cycle at a hot-spot, including ∆𝜎𝑚,
∆𝜎𝑏, and ∆𝜎𝑝 components, obtained after stress linearization; 𝐾𝑓 is a fatigue strength reduction
factor that accounts for the local weld notch and can be kept equal to unity in the case the FE model
already comprises the weld discontinuity; 𝐾𝑒,𝑘 is a fatigue penalty factor. 𝐾𝑒,𝑘 can be calculated
from the relationship between the plastic and elastic equivalent total strain ranges as
𝐾𝑒,𝑘 =∆ 𝑒𝑓𝑓,𝑘
∆ 𝑒𝑙,𝑘, (28)
where the effective strain range, ∆휀𝑒𝑓𝑓,𝑘, a von Mises-based measure of the strain state [81] for
which a fatigue crack is open during a cycle [82], is
∆휀𝑒𝑓𝑓,𝑘 = ∆휀𝑒𝑙,𝑘 + ∆휀𝑝𝑒𝑞,𝑘 (29)
and the elastic strain range, ∆휀𝑒𝑙,𝑘, is
∆휀𝑒𝑙,𝑘 =∆𝑆𝑃,𝑘
𝐸𝑇,𝑘, (30)
∆휀𝑝𝑒𝑞,𝑘 is a plastic equivalent strain range obtained from ABAQUS for each pressure increment,
55
∆휀𝑝𝑒𝑞,𝑘 =√2[(∆𝑝11,𝑘−∆𝑝22,𝑘)
2+(∆𝑝11,𝑘−∆𝑝33,𝑘)
2+(∆𝑝22,𝑘−∆𝑝33,𝑘)
2+6(∆𝑝12,𝑘
2 +∆𝑝13,𝑘2 +∆𝑝23,𝑘
2 )2
]
3, (31)
where ∆𝑝𝑖𝑗,𝑘 is the change in plastic strain range components at the point under evaluation for the
𝑘𝑡ℎ loading condition or cycle, as per ASME BPVC Section VIII Part 2 [22].
The value of 𝐾𝑒,𝑘 increases with larger plastic strains and becomes larger than unity, and
thus increasing the value of 𝑆𝑎𝑙𝑡,𝑘, used in the calculation of the number of cycles to failure,
𝑁𝑘 = 10𝐶1+𝐶3𝑌+𝐶5𝑌2+𝐶7𝑌3𝐶9𝑌4𝐶11𝑌5
1+𝐶2𝑌+𝐶4𝑌2+𝐶6𝑌3𝐶8𝑌4𝐶10𝑌5 , (32)
based on a smooth bar fatigue curve, where 𝑌 is a coefficient,
𝑌 = (𝑆𝑎𝑙𝑡,𝑘
𝐶𝑢𝑠) (
𝐸𝐹𝐶
𝐸𝑇), (33)
𝐶1 through 𝐶11 are the constants that depend on 𝑆𝑎𝑙𝑡,𝑘 (Table 11), 𝐶𝑢𝑠= 6.894757 is the conversion
factor for stresses in MPa (ASME BPVC Section VIII Part 2 [22]), 𝐸𝐹𝐶=195 GPa for the material
used to obtain the constants 𝐶1 through 𝐶11 (ASME BPVC Section VIII Part 2 [22]), and 𝐸𝑇 is the
Young’s modulus at service temperature.
Table 11 Constants for a polynomial fit of experimental data in the calculation of number
of cycles to failure
Constant 𝑆𝑎𝑙𝑡,𝑘 ≤ 45 𝑀𝑃𝑎 45 𝑀𝑃𝑎 < 𝑆𝑎𝑙𝑡,𝑘 ≤ 214 𝑀𝑃𝑎 214 𝑀𝑃𝑎 < 𝑆𝑎𝑙𝑡,𝑘 ≤ 3999 𝑀𝑃𝑎
𝐶1 0 +2.25451 +7.999502 𝐶2 0 −4.642236 × 10−1 +5.832491 × 10−2 𝐶3 0 −8.312745 × 10−1 +1.500851 × 10−1 𝐶4 0 +8.634660 × 10−2 +1.273659 × 10−4 𝐶5 0 +2.020834 × 10−1 −5.263661 × 10−5 𝐶6 0 −6.940535 × 10−3 0 𝐶7 0 −2.079726 × 10−2 0 𝐶8 0 +2.010235 × 10−4 0 𝐶9 0 +7.137717 × 10−4 0 𝐶10 0 0 0 𝐶11 0 0 0
56
The procedure is repeated for each cycle in the loading history. Miner’s rule is used to
compute the accumulated FD,
𝐷𝑓 = ∑𝑛𝑘
𝑁𝑘≤ 1𝑀
𝑘=1 . (34)
It should be noted that Miner’s rule was derived for elastic analysis and does not account
for local sequencing effects associated with plasticity.
3.8.4. Elastic-Plastic Fatigue Analysis
The elastic-plastic fatigue analysis can also be performed in accordance with API 579-1
[21] and ASME BPVC Section VIII Part 2 [22], Level 2, Method B (the strain-based version of
Method A [37]).
The calculation of FD is based on an approach similar to that described in the Modified
Elastic Fatigue Analysis. However, the range of elastic-plastic component stresses, ∆𝜎𝑖𝑗,𝑘 =
𝜎𝑖𝑗,𝑘𝑒𝑛𝑑 − 𝜎𝑖𝑗,𝑘
𝑠𝑡𝑎𝑟𝑡, shall be used in the calculation of ∆𝑆𝑃,𝑘, Eq. (27). Subsequently, ∆𝑆𝑃,𝑘 from Eq.
(27) and ∆휀𝑝𝑒𝑞,𝑘 from Eq. (31) are substituted in Eq. (29) for the calculation of the effective strain
range, ∆휀𝑒𝑓𝑓,𝑘. Thus, the fatigue analysis in this section is based on the elastic-plastic effective
equivalent stress range, ∆𝑆𝑃,𝑘, and the plastic equivalent stain range, ∆휀𝑝𝑒𝑞,𝑘.
Next, the effective alternating equivalent stress, 𝑆𝑎𝑙𝑡,𝑘, is computed according to
𝑆𝑎𝑙𝑡,𝑘 =𝐸𝑇,𝑘∙∆ 𝑒𝑓𝑓,𝑘
2, (35)
and is used to calculate the number of cycles to failure, as shown in Eq. (32) and Eq. (33).
Miner’s rule, Eq. (34), is used to compute the accumulated FD.
57
3.8.5. Elastic Fatigue Analysis of Welds
Fatigue assessment of welds can also be performed according to ASME BPVC Section
VIII Part 2 [22] and API 579-1 [21] (Level 2 – Method C) and can be used as a reference to the
data obtained by the previously discussed methods. This analysis utilizes the linear elastic stress
analysis of structural stresses and Neuber’s method for local plasticity correction. This method is
recommended for seam weld joints that have not been machined and is based on the structural
stress (comprised of membrane, 𝜎𝑚, and bending, 𝜎𝑏, stresses) normal to the hypothetical crack
plane, i.e., the components of stress in the hoop direction of a pipe.
First, the stress ranges due to pressure cycles can be calculated for structural membrane
stress range, ∆𝜎𝑚,𝑘𝑒 , as:
∆𝜎𝑚,𝑘𝑒 = 𝜎𝑚
𝑚,𝑘𝑒 − 𝜎𝑛
𝑚,𝑘𝑒 , (36)
and for the structural bending stress range, ∆𝜎𝑏,𝑘𝑒 , as:
∆𝜎𝑏,𝑘𝑒 = 𝜎𝑚
𝑏,𝑘𝑒 − 𝜎𝑛
𝑏,𝑘𝑒 , (37)
at the point under evaluation (see structural hot-spot in Figure 33). Subsequently, the elastically
calculated structural stress range, ∆𝜎𝑘𝑒, can be determined as:
∆𝜎𝑘𝑒 = ∆𝜎𝑚,𝑘
𝑒 + ∆𝜎𝑏,𝑘𝑒 , (38)
and the structural strain range, ∆휀𝑘𝑒, as:
∆휀𝑘𝑒 =
∆𝜎𝑘𝑒
𝐸𝑇𝑚𝑒𝑎𝑛
. (39)
These values are further used in the determination of the corresponding ranges of local nonlinear
structural stress, ∆𝜎𝑘, and strain, ∆휀𝑘, by simultaneously solving Neuber’s Rule,
∆𝜎𝑘 ∙ ∆휀𝑘 = ∆𝜎𝑘𝑒 ∙ ∆휀𝑘
𝑒, (40)
58
and modification of Eq. (56) in the form of a model for the material hysteresis loop stress-strain
curve,
∆휀𝑘 =∆𝜎𝑘
𝐸𝑇𝑚𝑒𝑎𝑛
+ 2 (∆𝜎𝑘
2𝐾𝑐𝑠𝑠)
1
𝑛𝑐𝑠𝑠. (41)
The solution for the obtained relationship,
∆𝜎𝑘𝑒∙∆ 𝑘
𝑒
∆𝜎𝑘=
∆𝜎𝑘
𝐸𝑇𝑚𝑒𝑎𝑛
+ 2 (∆𝜎𝑘
2𝐾𝑐𝑠𝑠)
1
𝑛𝑐𝑠𝑠, (42)
i.e., ∆𝜎𝑘 and the corresponding ∆휀𝑘, can be found by the numerical method used in 1.4.1.2 for the
calculation of a tangent point, and modified for low-cycle fatigue by using
∆𝜎𝑘 = (𝐸𝑇𝑚𝑒𝑎𝑛
1−𝜈2 ) ∆휀𝑘, (43)
and implemented similarly to the merged stress-strain curve in APPENDIX A – MATLAB (see
Neuber’s rule).
Level 2 – Method C takes into account local plasticity using Neuber’s rule. A notch, i.e., a
local sharp change (discontinuity) of a component’s geometry, is one of the major contributors to
stress rise. Heinz Neuber developed the model, which relates local linear stress-strain conditions
at the notch root to non-linear stress-strain conditions above the proportionality limit (Figure 22)
[83] [84] [85]. This model has been proven to yield estimates very close to experimental data [83].
Neuber’s rule calculates the local notch-root stress-strain history using the total strain energy
density, 휀 ∙ 𝜎, [84] [85].
59
Figure 22 Neuber’s relationship between linear and non-linear stresses and strains [83] [84]
[85]
Second, the mean stress, 𝜎𝑚𝑒𝑎𝑛,𝑘, and the stress ratio, 𝑅𝑘, for each cycle can be calculated
from
𝜎𝑚𝑒𝑎𝑛,𝑘 =𝜎𝑚𝑎𝑥,𝑘 + 𝜎𝑚𝑖𝑛,𝑘
2 (44)
and
𝑅𝑘 =𝜎𝑚𝑖𝑛,𝑘
𝜎𝑚𝑎𝑥,𝑘 (45)
respectively, where the maximum stress in a cycle was obtained as 𝜎𝑚𝑎𝑥,𝑘 from
𝜎𝑚𝑎𝑥,𝑘 = 𝑚𝑎𝑥[( 𝜎𝑚𝑚,𝑘𝑒 + 𝜎𝑚
𝑏,𝑘𝑒 ), ( 𝜎𝑛
𝑚,𝑘𝑒 + 𝜎𝑛
𝑏,𝑘𝑒 )]. (46)
Similar procedure is used to obtain minimum stress, 𝜎𝑚𝑖𝑛,𝑘 from
𝜎𝑚𝑖𝑛,𝑘 = 𝑚𝑖𝑛[( 𝜎𝑚𝑚,𝑘𝑒 + 𝜎𝑚
𝑏,𝑘𝑒 ), ( 𝜎𝑛
𝑚,𝑘𝑒 + 𝜎𝑛
𝑏,𝑘𝑒 )], (47)
𝜎𝑚𝑒𝑎𝑛,𝑘 and 𝑅𝑘, including the structural stress exponent, 𝑚𝑠𝑠 = 3.6, are subsequently used in the
calculation of the mean stress correction factor, 𝑓𝑀,𝑘,
if 𝜎𝑚𝑒𝑎𝑛,𝑘 ≥ 0.5 ∙ 𝑌𝑆𝑇𝑚𝑎𝑥 && 𝑅𝑘 > 0 && 𝑎𝑏𝑠(∆𝜎𝑚,𝑘
𝑒 + ∆𝜎𝑏,𝑘𝑒 ) ≤ 2 ∙ 𝑌𝑆𝑇𝑚𝑒𝑎𝑛
;
𝑓𝑀,𝑘 = (1 − 𝑅𝑘)^(1/𝑚𝑠𝑠);
else 𝑓𝑀,𝑘 = 1; end
(48)
σ,
MP
a
ε, mm/mm
𝜎𝑛𝑜𝑡𝑐ℎ = 𝑘𝑡𝜎𝑛𝑜𝑚 (𝜎휀)𝑛𝑜𝑡𝑐ℎ = (𝑘𝑡𝜎𝑛𝑜𝑚)2 𝐸⁄
𝜎 = 𝐸휀
𝜎 = 𝐾휀𝑛 𝜎𝑛𝑜𝑡𝑐ℎ = 𝑘𝑡𝜎𝜎𝑛𝑜𝑚
휀 𝑛𝑜
𝑡𝑐ℎ
=𝑘
𝑡휀 𝑛
𝑜𝑚
휀 𝑛𝑜
𝑡𝑐ℎ
=𝑘
𝑡휀 𝑛
𝑜𝑚
휀𝑛𝑜𝑚
𝜎𝑛𝑜𝑚
𝜎𝑦
휀𝑦
Neuber hyperbola Linear stress-strain Non-linear stress-strain Proportionality limit
60
The reader is referred to Eq. (57) used to calculate 𝜎𝑇.
Third, the equivalent structural stress range parameter, ∆𝑆𝑒𝑠𝑠,𝑘, is calculated from
∆𝑆𝑒𝑠𝑠,𝑘 =∆𝜎𝑘
𝑡𝑒𝑠𝑠(
2−𝑚𝑠𝑠2∙𝑚𝑠𝑠
)∙𝐼
1𝑚𝑠𝑠 ∙𝑓𝑀,𝑘
, (49)
where 𝐼 is the correction factor,
𝐼1
𝑚𝑠𝑠 =1.23−0.364𝑅𝑏,𝑘−0.17𝑅𝑏,𝑘
2
1.007−0.306𝑅𝑏,𝑘−0.178𝑅𝑏,𝑘2 , (50)
𝑅𝑏,𝑘 is the ratio,
𝑅𝑏,𝑘 =|∆𝜎𝑏,𝑘
𝑒 |
|∆𝜎𝑏,𝑘𝑒 |+|∆𝜎𝑚,𝑘
𝑒 |, (51)
and the structural stress effective TW, 𝑡𝑒𝑠𝑠, can be defined using the actual WT, 𝑡, using the
statement
if 𝑡 ≥ 16 𝑚𝑚; 𝑡𝑒𝑠𝑠 = 𝑡; else 𝑡𝑒𝑠𝑠 = 16 𝑚𝑚 end. (52)
Fourth, the number of cycles to failure, 𝑁, is calculated from
𝑁 =𝑓𝐼
𝑓𝐸(
𝑓𝑀𝑇
∆𝑆𝑒𝑠𝑠,𝑘)
1
ℎ, (53)
where the fatigue improvement factor, 𝑓𝐼, can be kept as equal to unity in the case no fatigue
improvement method is considered, the environmental modification factor, 𝑓𝐸 , can be selected to
be equal to 4, 𝑓𝑀𝑇 is the temperature adjustment factor defined as
𝑓𝑀𝑇 =𝐸𝑇𝑚𝑒𝑎𝑛
𝐸𝑅𝑇, (54)
and 𝐶 = 11577.9 and ℎ = 0.3195 are the coefficients for the welded joint fatigue curve of ferritic
and stainless steels for lower prediction interval (99%, −3𝜎). Similar to the fatigue assessment
methods previously discussed (see Eq. (32)), no allowance for corrosion is included in Eq. (53).
61
Fifth, the fatigue damage (FD) is calculated similarly to the previous sections, by using
Miner’s rule, Eq. (34).
3.9. Summary and Problem Definition
The research work will focus on elastic (3.8.2 Elastic Fatigue Analysis and 3.8.3 Modified
Elastic Fatigue Analysis) and elastic-plastic (3.8.4 Elastic-Plastic Fatigue Analysis) fatigue
assessment methods (ASME BPVC Section VIII Part 2 [22]), of submerged-arc welded (SAW)
large-diameter oil pipelines with standard outside diameters (ODs) of 610-1219 mm and standard
wall thicknesses (WTs, 𝑡) of 9.53-23.83 mm produced using the UOE process from X56 grade
steel and operated onshore 1000 mm below the ground (BD) at a permanent internal pressure (IP)
of 10 MPa and service temperature of 80 °C. There parameters have been selected with the
reference to the Keystone pipeline design. This will include modeling of longitudinal seam-welded
regions with associated misalignments and defects within the manufacturing tolerances of radial
offset 𝛿𝑜 of 1.5 mm, angular peaking 𝛿𝑝 of 3.2 mm, weld reinforcement 𝑊ℎ of 2 mm, and
reinforcement angle 𝜃 of 27°, with the loading cycles obtained from a common in-service loading
spectrum for a period of 50 years of design fatigue life to provide more accurate service-specific
predictions, as shown in Table 12. An elastic-plastic model (3.8.4) will be included in the analysis
to account for hot-spot plasticity developed during cycling.
Table 12 Pipeline parameters [in or (mm)] considered in this research
Pipeline Geometry Pipeline Defects In-Service Loading
𝐷 𝑡 𝛿𝑜 𝛿𝑝 𝑊ℎ BD 𝑃𝑖 Cycle
16-56 0.188-2.050 ≤ t/10 ≤ 0.125 ≤ 0.157 24 1400* Spectrum
(457-1422) (7.1-52) (1.5) (3.2) (4) (≥ 600) 10** Spectrum
* in [psig]; ** in [MPa]
62
The thesis will review the applicability of the available standards to large-diameter
pipelines and will develop a fatigue assessment methodology that will address the range of
parameters related to pipeline manufacturing and service conditions in order to meet industrial
needs in non-conservative and cost-effective pipeline design, and will provide an optimization
scheme for existing standards. The work will focus on the analysis of mechanical stresses and
strains in a pipeline, with special attention to the effects of misalignment.
The thesis will focus on the calculation of acceptable pipe WTs with combinations of
commonly observed manufacturing misalignments, including axial and angular misalignments,
from the perspective of fatigue-safe designs applied to the range of commonly used pipe diameters
and discusses the conservatism involved in elasticity-based methods when selecting WTs for pipe
design.
63
Chapter 4 MODEL DEVELOPMENT
In Chapter 4 the analytical and FE methods will be used to determine stresses at critical
locations of pipeline and to predict the fatigue life of pipeline based on its geometry tolerable by
design codes and typical loading expected in service. The wall thickness (WT) of a pipeline in the
initially conservative FE model (INPUT) will be reduced gradually until the cumulated fatigue
damage (FD) reaches unity value (GOAL), as shown in Figure 23. This process will be repeated
for elastic and elastic-plastic fatigue assessment methods (ASME BPVC Section VIII Part 2 [22]).
The results of different fatigue methods will be analyzed and compared in the next Chapter.
Figure 23 Flow chart showing process of model refinement
The design of the SAW-UOE manufactured pipeline started first with the models available
in the literature, including the effects of pipe geometry, weld profile, weld misalignments, internal
loading due to the transported medium, and external loading due to soil. Stress concentration
factors due to each effect were calculated and their sum was used as a multiplication factor for the
stress in the hoop direction to calculate the design stresses developed at the structural hot-spot.
This initial step was performed to obtain a mathematical model for a specific case discussed in this
study. Next, the fatigue life predictions were based on the verified FE model (verified hot-spot
stresses) for more detailed study of the component stresses across the WT of a pipe, including
Conservative Model
Reduce WT and Update
the FE Model
Obtain Stresses/Strains
Predict Fatigue Life
YES
NO Cumulated
FD < 1
Non-Conservative Model
GOAL INPUT
64
membrane stress, bending stress, and peak stress components, which are needed for accurate
predictions of FD as prescribed by governing design codes.
The calculations were performed in two steps: first, stresses and strains were obtained at a
structural hot-spot after applying a histogram of service loading onto the FEM modeled pipe; the
second, the FEM results were used in the fatigue analysis, including stress linearization,
calculation of equivalent stresses and strains, numbers of cycles to failure and fatigue damage.
Multiple software packages were used for the calculations, including Computer Aided Engineering
(CAE) software ABAQUS by Dassault Systèmes [77] [86] used in the FEM analysis, and
MATLAB by MathWorks, Inc. [87], which was used in other calculations and analyses.
4.1. Static Finite Element Model
The ABAQUS model comprises a pipe of commonly used standard OD and WT, weld
profile, and soil, as well as thermal loading and pressure history to accurately capture pipeline and
service conditions.
4.1.1. Geometry of Model
Four widely used pipeline ODs were utilized, and the pipeline WT was calculated
according to ASME B31.4 [23] and ASME B36.10 [49] using an internal pressure, 𝑃𝑖, of 10 MPa,
and temperatures of 0-80 °C, normally used in oil transportation, and adjusted to the nearest
standard WT as prescribed in ASME B36.10 [49], Table 13. The WTs were incrementally changed
using standard WTs from ASME B36.10 [49] to study the development of fatigue damage due to
service loading.
65
Table 13 Standard pipeline WTs for selected ODs and steel material at an internal pressure
of 10 MPa
𝐷 [mm] 610 864 914 1219
𝑡 [mm] 11.91 17.48 19.05 23.83
A soil box (Figure 24) was selected over the approximation of interaction between pipe
and soil with springs. The soil box is more convenient due to the weld profile included in the model
and is expected to provide accurate representation of sustained load due to gravity [58].
Figure 24 Geometry of model
The dimensions of the soil box were adjusted to simulate a pipeline trench with the burial
depth (BD) defined by the pipeline construction standard CSA Z662 [5]. In [5], a minimum BD of
0.6 m is required for oil pipelines located in non-developed areas in Canada.
One of the most critical features of pipeline is the welded region. The weld zone is a
complex and heterogeneous region that contains different microstructural features of different
morphologies that have different fatigue properties [88] [89] which have been found to change
with temperature [90]. Furthermore, the welded region of a pipe is highly susceptible to corrosion
and hydrogen embrittlement, especially at the heat-affected zone (HAZ) [91]. A typical double
SAW weld profile from the UOE process is shown in Figure 25. Three distinctly different regions
can be found within the welded region, including base plate, HAZ, and weldment. These regions
normally have different microstructures and mechanical properties.
66
Figure 25 Schematic of the SAW-processed pipe region
It has been shown by other researchers that weld bead geometry has a great impact on
stresses developed at the weld toe, which has been found to be extremely sensitive to weld
reinforcement, its angle to the welded plate, and a weld toe radius, as discussed earlier; bending
due to weld misalignment is also known as a great contributor to stress rise.
Special attention was paid to the modeling of the weld profile to capture its effects on the
hot-spot’s stresses. Although some of the mentioned critical weld parameters are discussed in
standards, some of them are only presented schematically. For instance, the weld bead width
calculation, (𝑊𝑤 = 0.5𝑊𝑇), and the weld root/toe radius definition, (𝑊𝑟 ≥ 0.25𝑊𝑇), are specified
only in BS 7608 [24], and none of the standards explicitly discusses or references a way to combine
radial 𝛿𝑜 and angular 𝛿𝑝 (Figure 1) misalignments. Weld profile curvature is also omitted from the
standards. The standards do however distinguish between weld types by introducing either the
weld joint efficiency factors used as a multiplier of the appropriate allowable stress for a given
material (ASME BPVC Section VIII Part 2 [22]) or weld quality specifications (weld classes) used
to determine the coefficients for fatigue curves (BS 7608 [24]). It seems that there is no unified
approach available in standards to be used as an input for FEM. The authors of this report believe
that there is a need for more accurate and statistically supported definitions of weld geometry and
misalignments that can be used in FEM for elastic and elastic-plastic fatigue analyses. Therefore,
the allowable weld misalignments were included in the model based on tolerances found in the
Base Plate
External
Weldment
Internal
Weldment
HAZ
HAZ
HAZ
HAZ
Base Plate
67
ASME, API, and BS codes, as well as the weld toe radius to more accurately account for the UOE
manufacturing defects (Table 14).
Table 14 Pipeline defects
𝛿𝑜 [mm] 𝛿𝑝 [mm] 𝑊ℎ [mm] 𝑊𝑟 [mm]
≤ 1.5 ≤ 3.2 ≤ 4 5
An average weld profile was obtained from available literature such as [92] [93] by
normalizing the traced SAW weld profiles of various transmission pipelines with ODs ranging
from 864 mm to 1219 mm ODs with WTs between 8 mm and 22 mm. The variations in 30 SAW
weld profiles were studied (Figure 26).
Figure 26 Traces of weld profiles used to generate an average weld profile
The resultant average weld profile is shown in Figure 27.
Figure 27 Geometry of the weld bead profile showing (solid dots) experimental data and
(solid line) 4th-order polynomial approximation
0
1
2
3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19Wel
d H
eigh
tW
h,
mm
Weld Width Ww, mm
68
An average weld bead geometry, obtained through this analysis in the form of a 4th order
polynomial, i.e.,
𝑊ℎ = −9.215 × 10−5𝑊𝑤4 + 3.485 × 10−3𝑊𝑤
3 − 6.291 × 10−3𝑊𝑤2 + 0.5671𝑊𝑤
+ 0.03417,
(55)
was used to obtain 𝑊ℎ [𝑚𝑚] and build the outside weld bead part. The inside weld bead part was
scaled down by a factor of 1.098 to match an average measured weld width 𝑊𝑤 and weld height
𝑊ℎ of a weld reinforcement, as shown in Table 15. The values of 𝑊𝑤 and 𝑊ℎ were found to vary
insignificantly with OD and WT used to obtain Eq. (40). The values for 𝑊ℎ from Table 15 are
well below those prescribed in API 5L [51] and CSA Z662 [4], [5]. The Inside and outside weld
beads were modeled within 0-1 mm of misalignment, which is less than that prescribed in API 5L
[51] for the WTs studied, i.e., 3-4 mm.
Table 15 Weld bead dimensions obtained in this study
Weld Bead Location 𝑊𝑤 [mm] 𝑊ℎ [mm]
Outside 18.98 ±4.71 2.02 ±0.40
Inside 17.29 ±4.58 1.84 ±0.44
The weld profile and misalignments were carefully modelled in order to capture their
effects on the hot-spot’s stresses/strains (Figure 28).
69
Figure 28 Geometry of the weld region including (bold white line) radial and (bold black
line) angular misalignments
The reinforcement angle 𝑊𝛼 was measured to be 27.3° at a hot-spot (circled area in Figure
28) for a non-misaligned weld and 33.5° for a misaligned weld observed due to the combined
effect of axial misalignment (𝛿𝑜 = 1.5 𝑚𝑚) and angular misalignment (𝛿𝑝 = 3.2 𝑚𝑚) specified
according to CSA Z245.1 [4] and API 5 L [51], as shown in Figure 28. The weld toe radius 𝑊𝑟
was set to 5 mm, and assuming there is no weld undercut.
4.1.2. Material Model
The material model for FEA was developed based on the approaches published in the
literature. The calculations were based on the material properties of pipeline steel extracted from
API 5L [51], Table 16. Although the material properties of weld and parent material may differ
and result in different fatigue strengths, the present research focuses on stress concentration due to
geometry of a welded region. Therefore, same properties have been used for all parts of a modelled
pipeline, Table 16.
Table 16 Pipeline steel material
Steel 𝐸
[MPa]
𝜈 𝜌 𝛼
[mm/K]
𝜎𝑦
[MPa]
𝜎𝑢
[MPa]
𝑛 𝐾
[MPa]
X56 207000 0.3 0.0078 1×10-5 358 455 -0.24 684
70
The data from Table 16 were used to generate the stress-strain curves for installation and
service temperatures of 0 °C and 80 °C respectively.
The true stress-strain response is modeled using the Ramberg-Osgood equation,
휀 =𝜎
𝐸+ (
𝜎
𝐾)
1
𝑛, (56)
as dictated by ASME BPVC Section VIII Part 2 [22]. The coordinates of the yield point at the
assessed temperatures 𝑇 were calculated from the yield stress and defined as
𝜎𝑦𝑇= 𝜎𝑦 ∙ 𝑒(𝐶0+𝐶1𝑇+𝐶2𝑇2+𝐶3𝑇3+𝐶4𝑇4+𝐶5𝑇5) (57)
and from the Young’s modulus,
𝐸𝑇 = (7 ∙ 10−17𝑇6 + 10−12𝑇5 − 10−9𝑇4 + 6 ∙ 10−5𝑇2 − 0.0646 ∙ 𝑇 + 203.66) × 103. (58)
In [22], the constants for Eq. (57) were taken as: 𝐶0 = 3.38037095 × 10−2, 𝐶1 =
−1.7355438 × 10−3, 𝐶2 = 8.32638097 × 10−6, 𝐶3 = −2.11471664 × 10−8, 𝐶4 =
3.29874954 × 10−11, and 𝐶5 = −2.69329508 × 10−14 (ASME BPVC Section VIII Part 2 [22]).
One of the connecting points, with coordinates [yield stress, yield strain] on Hooke’s curve,
defined in Eq. (1), lies outside the Ramberg-Osgood curve and can be connected to it by the
tangent line, as shown in Figure 29.
Figure 29 Construction of a tangent to Ramberg-Osgood’s curve from yield point on
Hooke’s curve
71
Therefore, derivation of a function for the tangent line has the initial form of
𝑥1 − 𝑥0 = 𝑚(𝑦1 − 𝑦0), (59)
where [𝑥0, 𝑦0] is the coordinate for the yield point of a material,
[𝑥0, 𝑦0] = [𝜎𝑦
𝐸, 𝜎𝑦],
(60)
[𝑦1, 𝑥1] is the coordinate for point of tangent line point of a tangent line on a Ramberg-Osgood
curve,
[𝑥1, 𝑦1] = [𝜎
𝐸+ (
𝜎
𝐾)
1
𝑛, 𝜎],
(61)
and 𝑚 = 𝑓′(𝜎) is the derivative of Eq. (56) w.r.t. 𝜎,
𝑚 = 𝑓′(𝜎) =𝐾
−1𝑛∙𝜎
1𝑛
−1
𝑛+
1
𝐸.
(62)
This results in
[𝜎
𝐸+ (
𝜎
𝐾)
1
𝑛] −
𝜎𝑦
𝐸= [
𝐾−
1𝑛∙𝜎
1𝑛
−1
𝑛+
1
𝐸] (𝜎 − 𝜎𝑦),
(63)
which can be solved for 𝜎 numerically by finding the lowest difference between the solutions of
the two curves on the left-hand side and right-hand side of the equation, as shown in Figure 30.
Figure 30 Example of a numerical solution for the tangent point on a Ramberg-Osgood
curve
300
350
400
450
500
-0.005 0.005 0.015 0.025 0.035 0.045 0.055 0.065 0.075 0.085 0.095
Stre
ss σ
, MP
a
Strain ε, mm/mm
Solution for left-hand side of Eq. (9) at pipe installation temperatureSolution for right-hand side of Eq. (9) at pipe installation temperatureSolution of Eq. (9) at pipe installation temperatureSolution for left-hand side of Eq. (9) at pipe service temperatureSolution for right-hand side of Eq. (9) at pipe service temperature
72
The equation for the tangent line from the yield point to the Ramberg-Osgood curve can
be found from Eq. (63), which yields as solution for the strain coordinate, 휀 = 𝜎𝑦 𝐸⁄ . The strain
hardening exponent 𝑛 and strength coefficient 𝐾 were adjusted to the installation and service
temperatures using linear interpolation of the experimental data for a low-alloy carbon steel found
in ASME BPVC Section VIII Part 2 [22]. The resultant stress-strain curves are shown in Figure
31. This approach was implemented to incorporate the elastic region into the Ramberg-Osgood
formulation and to avoid errors related to apparent plasticity at stresses lower than the yield stress.
The generated stress-strain curves were implemented in the FEM analysis.
Notably, the obtained cyclic stress-strain curve is very similar to that of the Prager
formulation found in [22] until the tangent point at strain 휀, and after that point the Prager solution
for the stress-strain curve diverges toward lower stresses.
Figure 31 True Stress-Strain curves for pipe steel material
A complete MATLAB implementation of the numerical solution for the merged stress-
strain curve is given in APPENDIX A – MATLAB .
The soil box around the pipe was modelled using Mohr-Coulomb plasticity as clay with
the properties summarized in Table 10 and was added to the model to account for the effects of
soil on buried pipe.
0
100
200
300
400
500
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055
Stre
ss σ
, MP
a
Strain ε, mm/mm
Installation Temperature
Service Temperature
73
4.1.3. Boundary Conditions
The boundary conditions applied to the model are summarized below in Table 17. The pipe
was allowed to move freely in the 𝑋𝑌 plane and to expand radially within the soil box [58] (but
not longitudinally) to simulate the soil-pipe interaction and underground restraint of a pipe [94],
as shown in Figure 32. Gravity was added to the model to account for stresses due to soil pressure
on the pipe. The displacement components 𝑈1, 𝑈2, and 𝑈3 in Table 17 act in the 𝑋, 𝑌, and 𝑍
directions respectively, as shown in Figure 32.
Table 17 Displacement constraints
Entity 𝑈1 𝑈2 𝑈3 Comment
Pipe Ends • Longitudinal displacement restrained
Soil Fronts • Frontal displacement restrained
Soil Sides • • Only vertical displacement allowed
Soil Bottom • • • No displacement allowed
Figure 32 Pipe (highlighted by circles) surrounded by a soil box with constraints
74
A static pressure of 10 MPa was evenly distributed over the inside surface of the pipe in
order to simulate the hydrostatic condition. A loading history from Figure 17 was applied to the
FE model to study the fatigue damage.
Thermal loading due to in-service oil transmission was included to account for stresses due
to differences between the installation and service conditions of 0 °C and 80 °C respectively. While
a temperature of 0 °C was kept constant for the soil, the temperature was increased to 80 °C for
the pipeline before application of the pressure cycles.
4.1.4. Model Meshing and Convergence
The pipe was partitioned to allow for a gradual mesh refinement of the pipe and meshed
with the 20-node quadratic brick elements with reduced integration (C3D20R, hex-dominated
elements [86]). The parts of the pipe away from the weld were meshed with a single through-
thickness 10 mm long elements, reduced in size to 1 mm around the weld region and further
reduced to 0.2 mm at the weld toe (hot-spot), as shown in Figure 33.
Figure 33 Meshing of a pipe showing detailed meshing at the hot-spot (black line indicates
the path used for an SCL)
75
The soil box was meshed with 15-node quadratic triangular prism wedge elements (C3D15,
[86]), which allowed for a sufficiently accurate modeling of surface-to-surface interactions in a
soil-pipe system, as shown in Figure 34. The element size of 50 mm near the external pipe surface
was implemented and refined to 5 mm elements at the pipe weld, and the edges of the soil box
were meshed with 500 mm elements, which was also the size for the elements in the longitudinal
direction throughout the FEM model.
Figure 34 Meshing of soil box around pipe
A convergence test was done on the model to confirm its computational validity. First, the
FEM mesh was refined globally to stabilize the solution for the nominal stresses and strains away
from the structural discontinuity. Second, the FEM mesh in the vicinity of the hot-spot was refined
to accurately capture the stresses and strains at the discontinuity.
The results presented in Figure 35 show a relatively quick convergence of a solution, both
globally and locally.
76
Figure 35 Refinement of global (away from discontinuity) and local (at the weld toe)
meshes showing von Mises Stress/Strain – Mesh Element Size relationship
The convergence of the global solution was found to be strongly related to a surface-to-
surface interaction between pipe and soil. Coarser element sizes in the pipe mesh and an overly
refined mesh in the soil box led to inaccurate solution that is represented by penetration of the
modeled bodies in a final deformed state, specifically in the weld region. Notably, it seems that
there is no similar thorough study in the literature on the modeling of pipe-soil interactions
involving s weld profile. The misaligned weld region is a geometrical irregularity and hence
required special care in the selection of the finite elements. The penetration of the modeled bodies
was dramatically reduced and localized to some pipe-soil contact points when the element size of
280.667
285.241285.129284.844284.969285.009285.040
374.347
355.267354.214352.433352.372352.282
0.001910
0.0019370.0019370.0019350.0019360.0019360.001936
0.003228
0.0049070.004851
0.0047910.0047940.004786
0.0015
0.0025
0.0035
0.0045
0.0055
280
300
320
340
360
380
0.01 0.1 1 10 100
Stra
in ε
, mm
/mm
von
Mis
es S
tres
s σ
, MP
a
Mesh Element Size, mm
Global Mesh - Mises Stress Local Mesh - Mises StressGlobal Mesh - Strain Local Mesh - Strain
77
a pipe mesh reduced to 50 mm. This computational inaccuracy is eliminated at mesh element sizes
below 40 mm, and the solution is stable when elements finer than 20 mm are used.
Another indication of the model’s convergence is that the solution for a nominal hoop
stress, found by subtraction of hoop stresses due to soil and heat from the final state, which also
includes pressure, is in good agreement with a closed form solution for hoop stress,
𝑆𝐻 =𝑃𝐷
2𝑡, (64)
showing only a 0.42% difference. It is important that the solution for the thermal load is also in
reasonable agreement with the procedure in ASME B31.4 [23], where the longitudinal stress due
to heat in an anchored and fully restrained portion of a buried pipe is
𝑆𝐿 = −𝐸𝛼∆𝑇 + 𝑆𝐻𝜈, (65)
showing only a 1.47% difference.
4.1.5. Data Extraction
The results of the simulation in ABAQUS were exported in *.rpt format, and contained
linearized stresses for each cycle, including the start and end points, and converted to *.xlsx format
for convenient use in MATLAB; individual component stresses were extracted by using a
following example command, which helps to separate numerical data from alphabetical.
[v,T,vT]=xlsread(FileName,2,CellName);
a=regexp(vT,'\s+','split');
n=numel(a{1});
m=numel(a); ComponentStress=transpose(reshape(str2double([a{:}]),n,m));
CellName=['A' num2str(Start/End) ':' 'A' num2str(Start/End)];
In (FileName,2,CellName), 2 is the page number in *.xlsx file.
The Start and End coordinates for a cycle within the *.xlsx file were calculated based on
the location of the extracted data. An example of linearized stress at a single cycle point (OD
78
914 mm WT 17.5 mm at a pressure of 2.8 MPa) is presented in APPENDIX D – ABAQUS Report
Example. The number of intervals for the SCL, which is defined in Figure 33, is a variable
parameter selected by the user in ABAQUS during export of the linearized stresses and is set as
constant throughout the analysis, 40, for a convenient subsequent reading of those data in
MATLAB. The “for loop” with the coded data extraction algorithm was used to retrieve the data
points for all pressure cycles sequentially; each sequence calculated the FD for one cycle and added
it to the total value. The number of extracted points is the double of number of pressure cycles, 60
in this research, and the maximum was observed to be more than 170 (106 lines per each cycle in
the report file) depending on the computational iterations during the ABAQUS calculations. The
number of iterations was observed to increase with plasticity due to high-pressure cycles.
ABAQUS produces multiple data points when solving for a cycle due to adjustments to the internal
time increments used in the calculations. The total number of lines can be more than 18,000.
Therefore, a separate subroutine was coded in MATLAB in order to eliminate the intermediate
points between the Start and End cycle points.
An example of conditioning of the ABAQUS report file (*.rpt or *.xlsx) and extraction of
the membrane component of the linearized stress (bending and peak components of stress can be
extracted similarly) is presented in APPENDIX E – MATLAB Code for the ABAQUS Data.
After extraction the data (hot-spot stresses and strains) were used in fatigue analyses
presented in Chapter 3: Elastic Fatigue Analysis (3.8.2), Modified Elastic Fatigue Analysis (3.8.3),
and Elastic-Plastic Fatigue Analysis (3.8.4).
79
Chapter 5 RESULTS AND DISCUSSION
This Chapter summarizes the results of the FEM and fatigue analyses. The effects of the
misalignment on the local stresses and the use of different fatigue assessment methods in the
calculation of allowable WT are discussed. The results of FE model are compared to the results of
an approximate model developed to simply calculate local stresses. Specifically, the hoop stress
and the longitudinal stress obtained from the FE model are verified far from the weld.
Subsequently, a detailed analysis of SCFs due to each loading mode (internal pressure, soil
pressure, temperature), associated with varying degrees of misalignment, is given. Finally,
validation of newly proposed analytical method of obtaining the stresses at four weld toes is
detailed.
5.1. Finite Element Model
Since this study focuses primarily on elastic and elastic-plastic fatigue analyses, it is
important to show the differences in the ABAQUS simulation results between the two models. A
pipe modeled with an OD of 914 mm, WT of 14.3 mm, and a maximum tolerable misalignment
was selected as an example to show the stress distribution plots in the through-thickness direction
of a pipe due to expected in-service loading. Figure 36 and Figure 37 show the hoop stress maps
with the misaligned pipe calculated using the elastic and elastic-plastic models respectively. The
stress in the hoop direction was selected for comparison, as it is of greater importance for a
longitudinally welded pipe, in which the plane of the hypothetical crack is situated perpendicular
to the hoop direction and opens under fluctuations of the hoop stress. The analyses do not show
any difference in the hoop stresses due to soil, as shown in Figure 36(a) and Figure 37(a), or due
to heat, Figure 36(b) and Figure 37(b). However, a significant difference in the stress distributions
can be observed between the models when the design pressure of 10 MPa is applied.
80
(a) Soil/Gravity (b) Soil/Gravity + Heat
(c) Soil/Gravity + Heat + Pressure
Figure 36 Hoop stress distribution maps for a misaligned pipe during elastic loading
(a) Soil/Gravity (b) Soil/Gravity + Heat
(c) Soil/Gravity + Heat + Pressure
Figure 37 Hoop stress distribution maps for a misaligned pipe during elastic-plastic loading
81
Stre
ss σ
, MP
a
(a) (b)
Stre
ss σ
, MP
a
(c) (d)
Wall Thickness t, mm Wall Thickness t, mm Figure 38 Through-thickness (curved) actual and (linear) linearized stress distributions
obtained for a pipe of 914 mm OD and 14.3 mm WT from an SCL positioned at the hot-
spot (at 0 mm WT coordinate) normal to the pipe wall with no misalignment by using (a)
elastic and (b) elastic-plastic analysis, and with misalignment by using (c) elastic and (d)
elastic-plastic analysis
-200
-100
0
100
200
300
400
500
600
700
800
900
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-200
-100
0
100
200
300
400
500
600
700
800
900
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
S11S11mS11mbS22S22mS22mbS33S33mS33mb
-200
-100
0
100
200
300
400
500
600
700
800
900
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-200
-100
0
100
200
300
400
500
600
700
800
900
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Stress along Y axis of model – transverse direction
Linearized stress along Y axis – membrane
Linearized stress along Y axis – membrane + bending
Stress along X axis of model – hoop direction
Linearized stress along X axis – membrane
Linearized stress along X axis – membrane + bending
Stress along Z axis of model – longitudinal direction
Linearized stress along Z axis – membrane
Linearized stress along Z axis – membrane + bending
82
Although the stress distribution patterns for both models seem to be similar, if not the same,
the stress is almost exclusively concentrated at the inside right-hand weld toe when the elastic
model considered, as shown in Figure 36(c); the elastic-plastic behavior of the structure resulted
in dissipation of stress over a much larger volume of material adjacent to the mentioned weld toe
part, Figure 37(c), which experienced larger bending due to misalignment.
This difference between the stress distributions shown in Figure 36 and Figure 37 is due to
local plasticity developed in the elastic-plastic analysis, resulting in a lower through-thickness
stress gradient, which can be much steeper when a sharper weld toe radius or no radius is used.
The stress distributions shown in Figure 36 and Figure 37 are constant in longitudinal direction.
Notably, the through-thickness stress linearization is prescribed by the ASME standard to
obtain different components of stress at the hot-spot, including membrane stress 𝜎𝑚 and bending
stress 𝜎𝑏, as visualized in Figure 38 (see Hoop stress, 𝑆22). Stress linearization clearly eliminates
the discussed difference, e.g., compare Figure 38(a) to Figure 38(b) or Figure 38(c) to Figure 38(d),
with the elastic result showing only 0.18% and 5.44% larger values for 𝜎𝑚 and 𝜎𝑏 respectively.
This fact indirectly signifies the importance of a good convergence required from the FE model.
Comparing Figure 38(a) to Figure 38(c) or Figure 38(b) to Figure 38(d), misalignment was
found to have a significant impact on the stress rise at the hot-spot. While there is no change in the
membrane component of stress, 𝜎𝑚, the dramatic increase in hot-spot stress shown in Figure 38 is
mainly due to the contribution of its bending component, 𝜎𝑏, when weld misalignment is
considered, resulting in a 42.5% increase in total linearized stress, as shown in Figure 38(c-d).
Moreover, the elastic model showed 0.16% and 11.99% larger values for 𝜎𝑚 and 𝜎𝑏 respectively.
Increasing the misalignment results in a larger observed discrepancy between the stresses,
including linearized stresses, obtained through the elastic and elastic-plastic analyses.
83
Therefore, weld misalignment is expected to seriously compromise the fatigue
performance of a UOE-manufactured pipeline and is also expected to cause larger discrepancy
between solutions for FD based on the elastic and elastic-plastic analyses.
Although misalignment was found to significantly increase normal stresses, the shear
stresses of the assessed pipeline were calculated to be below 100 MPa, which is much lower than
normal stresses (> 500 MPa), and the relationship ∆𝜏 ≤ ∆𝜎/3 from ASME BPVC Section VIII
Part 2 [22] was satisfied. Since the structural shear stress range is negligible, multiaxial Elastic
Fatigue Analysis of Welds was not required and was not performed in this study. Furthermore, the
strain-life Level 3 Fatigue Assessment, ASME BPVC Section VIII Part 2 [22], based on the critical
plane approach was not performed either since the FEM results showed that the plane normal to
the hoop stress is the only critical plane.
5.1.1. Validation of FEM
This section is dedicated to the analytical calculation of a SCF that would reflect the actual
weld profile due to the SAW-UOE manufacturing process.
The SCFs due to maximum tolerable misalignments, listed with other parameters in Table
18, are as follows: 𝑘𝑚.𝑎𝑥𝑖𝑎𝑙 = 0.2571 (Eq. (14)), 𝑘𝑚.𝑎𝑛𝑔𝑢𝑙𝑎𝑟 = 0.5981 (Eq. (15)) (𝛽 = 0.3085,
Eq. (17)), 𝑘𝑚.𝑜𝑣𝑎𝑙𝑖𝑡𝑦 = 0.00001 (Eq. (16)). The initial ovality of a pipe does not modify the hoop
stress 𝑆ℎ much (𝑘𝑚.𝑜𝑣𝑎𝑙𝑖𝑡𝑦 = 0.00001 at 𝜃 = 90°), but axial and angular misalignments contribute
the most to the overall stress at the weld toe, resulting in maximum design stress 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛 =
261 × 1.856 = 484 𝑀𝑃𝑎, which is the total stress at the pipe weld toe due to maximum allowable
misalignments and internal pressure inside the pipe.
84
Table 18 Parameters used in calculation of SCFs due to misalignments
𝛿𝑜
[mm]
𝑡
[mm]
2𝑙
[mm]
𝜈 𝜎𝑚𝑚𝑎𝑥
[MPa]
𝐸
[GPa]
𝛿𝑝
[mm]
𝜃
[°]
𝑝
[MPa]
1.5 17.5 92 0.3 261 207 3.2 90 10
In the present work, for the pipe without a coating, the bending stress at the weld toe (hot-
spot) due to soil is 𝜎𝑏.𝑠𝑜𝑖𝑙 = 16 𝑀𝑃𝑎 (Eq. (19)). Therefore, 𝑘𝑚.𝑠𝑜𝑖𝑙 = 0.0620 (Eq. (18)), which
magnifies the hoop stress to the value of 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛 = 261 × (1 + 0.0620) = 277 𝑀𝑃𝑎.
Since the governing codes do not account for the variations in weld profile geometry, the
following part will be dedicated to a method that can be used to calculate SCF due to specific weld
profile. The SCF solution for SAW type of weld connection is not explicitly addressed in the
available literature [95] [96]. The SAW profile was treated as a fillet on stepped flat bar, which
has some features as similar to SAW. However, there are difficulties in obtaining the SCFs due to
weld width, 𝑊𝑤, which range, covered in [95] and [96], is different from that of SAW weld.
Additionally, for a completely automated model, there is a need for the set of equations to obtain
the SCF. Therefore, the following shows the development of the mathematical model for the stress
concentration for a fillet on stepped flat bar at the hot-spot adopted from [84].
The geometry of the SAW profile, shown in Figure 39(a), can be further developed into a
fillet on stepped flat bar, Figure 39(b), which has an approximated weld profile with maintained
weld height (crown), 𝑊ℎ, length of the weld attachment, 𝑊𝑤, weld radius, 𝑊𝑟, and wall thickness,
𝑡 [84]. This approximation has been found to yield an accurate SCF for SAW weld profile [84].
The approximated distributed shear load profiles for the fillet weld are shown schematically in
Figure 39(c) [84].
85
Figure 39 Schematic of (a) the SAW butt joint, represented in the form of (b) fillet in
stepped flat bar, showing (c) equivalent load at the base of reinforcement and (d) real shear
stress diagram with its (dashed line) approximation
The additional local stress on the upper fillet, ∆𝜎′, lower fillet, ∆𝜎′′, and on both fillets,
∆𝜎′′′, due to distributed shear loads on the surface of the reinforcement base, Figure 39(c), can be
obtained from
∆𝜎′ = 𝜏𝑚 (0.212 − 0.25𝑊𝑟
1
(𝑡 2⁄ )1 + 0.093𝑊𝑟
2
(𝑡 2⁄ )2), (66)
∆𝜎′′ = 𝜏𝑚 (−0.215𝑊𝑟
1
(𝑡 2⁄ )1 + 0.123𝑊𝑟
2
(𝑡 2⁄ )2), (67)
and
(a) (b)
(c) (d)
R l1 l
Wr
l l
Ww
t W
h
Ww
l
l1
τ'm
R
τxy
A
l2 x
τ m
a
b
c d
86
∆𝜎′′′ = 𝜏𝑚 (0.212 − 0.125𝑊𝑟
1
(𝑡 2⁄ )1+ 0.03
𝑊𝑟2
(𝑡 2⁄ )2) (68)
respectively, developed in [84], where 𝜏𝑚𝑙1= 𝜏𝑥 and 𝜏𝑚𝑙2
= 𝜏𝑚𝑙1
𝑥3
𝑊𝑟2 are the shear stresses, shown
in Figure 39(d), integrated in a range from 𝑥 = 0 to 𝑥 = 𝑊𝑟, and 𝑙1 and 𝑙2 are the section lengths
supporting the shear distributed load [84].
It follows that the equality of the areas of the rectilinear (𝑎𝑏𝑐) and curvilinear (𝑑𝑏𝑐) shear
stress diagrams in Figure 39(d) results in
𝜏𝑚′ 𝑙1
2= ∫ 𝜏𝑥𝑑𝑥
𝑙
0=
𝑊ℎ∙𝑡∙𝜎
𝑡+2𝑊ℎ, (69)
which yields
𝜏𝑚′ = 𝜎 ∙
2𝑊ℎ∙𝑡
𝑙1(𝑡+2𝑊ℎ). (70)
Similarly, can be found
(𝜏𝑚′ −𝜏𝑚)𝑙1
2=
𝜏𝑚
𝑊𝑟3 ∫ 𝑥3𝑑𝑥
𝑊𝑟
0=
𝜏𝑚𝑊𝑟
4, (71)
which yields
𝜏𝑚 = 𝜏𝑚′ 2𝑙1
2𝑙1+𝑊𝑟. (72)
Therefore, substitution of Eq. (70) into Eq. (72) yields
𝜏𝑚 = 𝜎 ∙2𝑊ℎ∙𝑡
𝑙1(𝑡+2𝑊ℎ)∙
2𝑙1
(2𝑙1+𝑊𝑟), (73)
where 𝜎 is an arbitrary value of the hoop stress in this case.
87
Finally, the SCF can be calculated from
𝑘𝑚.𝑤𝑒𝑙𝑑 =𝜎𝑏.𝑤𝑒𝑙𝑑
𝑃𝑚=
∆𝜎′′′
𝜎=
2𝑊ℎ∙𝑡
𝑙1(𝑡+2𝑊ℎ)×
2𝑙1
(2𝑙1+𝑊𝑟)(0.212 − 0.125
𝑊𝑟1
(𝑡 2⁄ )1 + 0.03𝑊𝑟
2
(𝑡 2⁄ )2), (74)
where 𝑙1 is calculated from
𝑙1 =𝜎
𝜏𝑚′
∙2𝑊ℎ ∙ 𝑡
(𝑡 + 2𝑊ℎ) (75)
and 𝜏𝑚′ , the maximum value of the shear stress in the section under the rectangular reinforcement,
is found from
𝜏𝑚′ = 𝜏𝑥|
𝑥=𝐿
2
𝑚𝑎𝑥 =𝜎
𝐴0𝑠ℎ (𝐵0
𝑊𝑤
2). (76)
The geometrical constant 𝐴0 is obrained as:
𝐴0 =𝑐ℎ(𝐵0
𝑊𝑤2
)−1
𝐵0𝑊ℎ(
𝑡+2𝑊ℎ
𝑡), (77)
and 𝐵0
𝐵0 =2
𝑡√
2𝑡+𝑊ℎ
𝑊ℎ𝑘, (78)
where 𝑘 is the coefficient of deformation of the weld joint
𝑘 = 0.9 (𝑡
𝑡+𝑊ℎ)
2
. (79)
The Fourier hyperbolic functions used in Eq. (76) and Eq. (77) can be calculated from
𝑠ℎ (𝐵0𝑊𝑤
2) =
𝑒𝐵0
𝑊𝑤2 −𝑒
−𝐵0𝑊𝑤
2
2 and 𝑐ℎ (𝐵0
𝑊𝑤
2) =
𝑒𝐵0
𝑊𝑤2 +𝑒
−𝐵0𝑊𝑤
2
2 [84]. (80)
The value of 𝑙1 is independent of the applied stress, as it has a purely geometrical meaning.
In the present research, the values of 𝑊𝑤 and 𝑊ℎ are kept constant, as they have been found
to vary insignificantly with WT during the calculation of an average weld profile. As the focus of
88
the research is on the potential reduction of WT for a pipe with a particular OD, the only variable
is WT for the ODs studied. Different relationships between SCFs, WT, and 𝑊𝑟 are represented in
Figure 40. The SCF decreases with WT when 𝑊𝑟 is computed according to [25] as 𝑊𝑟 = 0.25𝑡, as
shown in Figure 40(a), due to the increase of calculated 𝑊𝑟 with WT. This relationship is used
when the weld undercut is removed and 𝑅 is introduced to improve fatigue properties of the weld
joint when fracture mechanics is considered. The SCF has a parabolic distribution when the 𝑅 is
kept constant, as shown in Figure 40(b), and is lowest for lower WTs.
It would be ideal to have a set of 𝑊𝑟 that allows for the lowest gradient of SCF across the
studied WTs, and the only solution found in this study is shown in Figure 40(c), when 𝑊𝑟 = 3 −7
𝑡.
However, the calculated SCFs have high magnitudes, which would require larger design
parameters for a pipeline, such as OD and/or WT. The value of 𝑊𝑟 = 5 𝑚𝑚 is closer to the
untreated SAW joints and shows a 25 % lower gradient of SCF for WTs of 8.74 mm through
28.58 mm compared to previously computed SCFs. Furthermore, there is no additional
information on the value of 𝑊𝑟 in North American standards [21] [22]. Therefore, for convenience,
the radius 𝑊𝑟 will be kept as equal to 5 mm for all further calculations and simulations in the
present work.
The weld due to SAW processing of the UOE pipe magnifies the hoop stress, resulting in
𝑘𝑚.𝑤𝑒𝑙𝑑 = 0.1524 (Eq. (74)) and the design value 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛 = 261 × (1 + 0.1524) = 301 𝑀𝑃𝑎.
89
Str
ess
Conce
ntr
atio
n F
acto
r k m
.wel
d
Wel
d R
oot
Rad
ius
Wr,
mm
(a) (b)
(c) (d) Wall Thickness t, mm
Figure 40 The (dots) SCFs for different (connected dots) transition radiuses Wr (a)
Wr =0.25WT, (b) Wr =7.145 mm, (c) Wr=3-7/t, and (d) Wr=5 mm
By considering the signs of each modification factor with respect to the location of a critical
spot, as shown in Figure 41, the stresses that may be observed due to different combinations of
discussed effects are presented in Table 19.
2
3
4
5
6
7
8
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.20
1.22
8 10 12 14 16 18 20 22 24 26 28 30
2
3
4
5
6
7
8
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.20
1.22
8 10 12 14 16 18 20 22 24 26 28 30
2
3
4
5
6
7
8
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.20
1.22
8 10 12 14 16 18 20 22 24 26 28 30
2
3
4
5
6
7
8
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.20
1.22
8 10 12 14 16 18 20 22 24 26 28 30
90
(a) (b)
(c) (d)
Figure 41 Secondary bending (curved arrows) due to: (a) axial, (b) angular, and (c) ovality
misalignments, and (d) due to soil; the red dashed line indicates the plane of a hypothetical
crack or SCL, and 1 through 4 are the hot-spot locations
For example, the angle (𝑎𝑏𝑐) tend to increase to (𝑎′𝑏′𝑐′) and to open the hypothetical crack
at hot-spot location 3 when load 𝑃 is applied (due to 𝑆ℎ.𝑛𝑜𝑚𝑖𝑛𝑎𝑙), as shown in Figure 41(a); a similar
situation can be observed at location 1; however, hypothetical crack tend to close at locations 2
and 4, resulting in subtraction of the magnification factor 𝑘𝑚.𝑎𝑥𝑖𝑎𝑙 from the total 𝑘𝑚, while at
locations 1 and 3 the factor 𝑘𝑚.𝑎𝑥𝑖𝑎𝑙 is added to 𝑘𝑚. The same assessment was applied to other
cases, as shown in Figure 41(b) through Figure 41(d).
In Figure 41(b) the load 𝑃 is applied in the direction of the welded plates at the junction
between the true pipe circle (dashed curved line) and the misaligned part (solid line), also
represented by points 𝑎 and 𝑑; with secondary bending being experienced at all points, (𝑎𝑏𝑐𝑑).
Therefore, hypothetical cracks open at hot-spot locations 1 and 4 and close at locations 2 and 3
when internal pressure is applied. There are two locations of the weld assessed in Figure 41(c) and
a b
c d a' b'
c' d'
1
2
3
4
P
P
P P
P
P
91
Figure 41(d), i.e., the upper location and the side location. The analysis of possible scenarios for
calculation of 𝑘𝑚 is summarized in Table 19 and indicates that the part of the weld located at the
inner surface of a pipe at location 1 experiences the largest stresses, while the lowest stress was
obtained at location 2 for the upper weld location (see Figure 41(a) for spot locations/numbers).
Table 19 Stress magnification at different weld locations
Spot 𝑆ℎ.𝑛𝑜𝑚
[MPa]
𝑘𝑚.𝑎𝑥
0.2571
𝑘𝑚.𝑎𝑛
0.5981
𝑘𝑚.𝑜𝑣𝑎𝑙
0.0000
𝑘𝑚.𝑠𝑜𝑖𝑙
0.0620
1+∑𝑘𝑚
1.9172
𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛
[MPa]
𝑘𝑚.𝑤𝑒𝑙𝑑
0.0689
𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛𝑚.𝑤𝑒𝑙𝑑
[MPa]
Upper weld location ( and )
1 261 + + - + 1.9172 501 + 519
2 261 - - + - 0.0828 22 + 40
3 261 + - + - 0.5970 156 + 174
4 261 - + - + 1.4030 366 + 384
Side weld location ( )) and )) )
)) 1 261 + + + - 1.7932 468 + 486
)) 2 261 - - - + 0.2068 54 + 72
)) 3 261 + - - + 0.7210 188 + 206
)) 4 261 - + + - 1.2790 334 + 352
The stress developed at location 1 in a non-misaligned pipe is calculated to be equal to
322 MPa. An upper weld location was selected for development of an ABAQUS-based FEM with
a soil box, including axial and angular misalignments.
The Analytical (Hand) prediction of stresses at the hot-spot based on the analytical model
described in this section shows design hoop stresses, 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛, similar to those obtained by using
92
an FEM approach for both misaligned and the non-misaligned pipeline conditions Table 20; the
result of Hand calculations of stresses seem to be conservative, showing (2 ÷ 20)% larger stresses
than those predicted in ABAQUS. The result of Hand calculations, shown in Table 20, would
represent an actual conservatism that can be involved in design-by-rule governed by the design
codes when compared to the results of a more detail FE modeling. Notably, the stresses calculated
using elastic and elastic-plastic FE analysis, shown in Table 20, are almost the same since the
linearized stresses have been used in both cases.
It can be concluded that the Hand calculations discussed in this work provide pipeline
design engineers with a comprehensive tool for detailed analyses of critical hot-spot stresses
observed at the plate-weld transition due to manufacturing defects as well as in-service conditions,
such as pressure from the transported medium and soil. The advantage of using the FEM approach
is in obtaining the complete set of stress components, including 𝜎𝑚, 𝜎𝑏, and 𝜎𝑝, needed for
advanced fatigue assessment.
Table 20 Results of analysis of the design hoop stresses Sh [MPa] for a pipe of OD 914 mm
and WT 17.5 mm
Misalignment Hot-spot
Location
Method
Hand:
Upper Weld
Hand:
Side Weld
FEM:
Elastic
FEM:
Elastic-Plastic
NO 1 261 261 250 250
YES 1 501 468 431 432
YES 2 22 54 69 69
YES 3 156 188 153 155
YES 4 366 334 345 345
93
The discrepancy between the Hand and ABAQUS calculations is likely due to differences
between the exact geometry of the pipe section modeled with FEM and the geometry used to derive
the closed-form solutions for each individual stress magnification factor, 𝑘𝑚 (Section 3.7). The
Hand calculations are summarized in Figure 42.
Stre
ss S
h.d
esig
n, M
Pa
Wall Thickness t, mm
Figure 42 Hoop stress calculated with mathematical model, power-law-fitted, and
extrapolated until solutions of (dashed line) non-misaligned and (solid line) misaligned
conditions intersect (power-law-fitted)
68, 46
66, 7165, 7763, 109
322
15.20 24.060
100
200
300
400
500
600
700
800
900
1,000
1,100
1,200
1,300
1,400
1,500
1,600
1,700
9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
OD 610 mm
OD 610 mm - Misaligned
OD 864 mm
OD 864 mm - Misaligned
OD 914 mm
OD 914 mm - Misaligned
OD 1219 mm
OD 1219 mm - Misaligned
Intersection
94
The solutions for the 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛 of misaligned pipes of selected ODs (Figure 42) significantly
diverge from the non-misaligned solutions with decreasing WT, 𝑡, because the through-thickness
bending, 𝜎𝑏, becomes the dominant component of total stress 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛. The two conditions for
each pipe have a common solution at certain value of 𝑡 and corresponding 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛, and drift
toward lower 𝑡 and larger 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛 with increasing OD exponentially (Figure 42). A significant
increase in WT of a pipe of OD = 914 mm due to misalignment can be observed (from 15.20 mm
to 24.06 mm) at the cut-off value of allowable stress for the selected material 𝑆 = 0.9𝜎𝑦 = 0.9 ∙
359[𝑀𝑃𝑎] = 322 𝑀𝑃𝑎. The slope of the solution for a misaligned pipe is larger than that for a
pipe without misalignment, as shown in Figure 42. The slope increases with pipe OD. The slopes
were studied in detail for pipe of OD 914 mm and compared to FE results, Figure 43.
Stre
ss S
h.d
esig
n, M
Pa
Wall Thickness t, mm
(a) (b) Figure 43 Solutions for Hoop stress (linear fit) in (non)misaligned pipe of OD 914 mm (a)
without and (b) with km.weld accounted
24.98, 148.00
25.24, 141.00
322
15.07 21.37
14.10
20.470
100
200
300
400
500
600
700
800
14 18 22 26
In-Hand - not misaligned
In-Hand - misaligned
Abaqus - not misaligned
Abaqus - misaligned
24.98, 169.00
25.24, 141.00
322
17.37 21.9714.10
20.47
0
100
200
300
400
500
600
700
800
14 18 22 26
95
Two versions of the calculation of the 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛 are presented in Figure 43; the one with
𝑘𝑚.𝑤𝑒𝑙𝑑 not included in total 𝑘𝑚 (the same as in the standards), Figure 43(a), and the other, with
𝑘𝑚.𝑤𝑒𝑙𝑑 included in total 𝑘𝑚 (as shown in this research), Figure 43(b) (see 𝑆ℎ.𝑑𝑒𝑠𝑖𝑔𝑛𝑚.𝑤𝑒𝑙𝑑 in Table 19).
There is no obvious benefit of accounting for the stress magnification due to weld profile, 𝑘𝑚.𝑤𝑒𝑙𝑑,
in the Hand calculations, as the total stress is simply increases by the 𝑘𝑚.𝑤𝑒𝑙𝑑 factor and the
predictions just become more conservative.
However, when the power law is used to fit the data (i.e., the coefficient of determination
becomes 𝑅2 = 1, resulting in a 100% fit), the use of the 𝑘𝑚.𝑤𝑒𝑙𝑑 factor may be justified, as shown
in Figure 44. The difference in the estimated 𝑡 between the non-misaligned and misaligned
geometries has a factor of 2.52 = (23.72 − 21.68) (15.04 − 14.23)⁄ when 𝑘𝑚.𝑤𝑒𝑙𝑑 is not
considered, as shown in Figure 44(a); however, this factor is more than two times lower, at only
1.18 = (25.22 − 21.68) (17.22 − 14.23)⁄ , when 𝑘𝑚.𝑤𝑒𝑙𝑑 is considered, as shown in Figure
44(b). Therefore, a common factor (or increment of 𝑡) for both the non-misaligned and the
misaligned geometries of a pipe with a particular OD can be used to make a correlation (between
Hand and FEM solutions) based on either method, whether elastic or elastic-plastic stress-strain
analysis. This means that both the non-misaligned and the misaligned geometries can be adjusted
with almost same increment of 𝑡, with discrepancy of only 0.55 𝑚𝑚 = (25.22 − 21.68) −
(17.22 − 14.23), as shown in Figure 44(b), compared to more than double of it, 1.23 =
(23.72 − 21.68) − (15.04 − 14.23), as shown in Figure 44(a), when 𝑘𝑚.𝑤𝑒𝑙𝑑 is not considered.
Moreover, dissonance was found to increase with reduced WT and increased OD, as shown in
Figure 42, which means that the proposed addition to the total 𝑘𝑚 in the form of the 𝑘𝑚.𝑤𝑒𝑙𝑑 factor
may benefit the large-diameter pipelines even more (than small-diameter pipelines).
96
Stre
ss S
h.d
esig
n, M
Pa
(a)
(b)
Wall Thickness t, mm Figure 44 Solutions for Hoop stress (power law fit) in pipe of OD 914 mm (a) without km.weld
and (b) with km.weld
Therefore, the use of the 𝑘𝑚.𝑤𝑒𝑙𝑑 factor can save on computational effort and the associated
costs when a more precise data-to-curve fit is used (Figure 44). It enables the engineer to estimate
the conservatism of the analytical model in WT-equivalent almost independent of misalignment.
62.09, 79.50
60.96, 80.20
322
15.04
23.72
14.23
21.68
25.14, 144.75
0
100
200
300
400
500
600
700
800
14 18 22 26 30 34 38 42 46 50 54 58 62
Hand - not misaligned
Hand - misaligned
Abaqus - not misaligned
Abaqus - misaligned
60.25, 93.50
60.96, 80.20
322
17.22
25.22
14.23 21.68
24.89, 153.98
0
100
200
300
400
500
600
700
800
14 18 22 26 30 34 38 42 46 50 54 58 62
97
Thus, only one series of FE analyses would be needed, for either non-misaligned or misaligned
pipeline model for the range of WTs, to determine the design WT.
5.2. Fatigue Analysis
The results of the fatigue analysis represent the relationships between WTs and FDs
computed for four different ODs as depicted in Figure 45. An example of the FD plots for the pipe
with a non-misaligned weld also included for comparison in Figure 45(c). The obtained data points
collected through the analysis are included in the data-to-curve-fit based on the two constants
power law, which was found to approximate the obtained data accurately, as shown in Table 21(I).
The coefficient of determination, 𝑅2 (Table 21(I)), was calculated to be in the range between 0.97
and 1.00, indicating a 97 to 100 % goodness of data fit.
Although the elastic solutions for the FD obtained in accordance with BS and ASME codes
are very close, as shown in Figure 45(a-d), a significant difference in allowable WT was observed
between the elastic and elastic-plastic solutions, signifying a lower fatigue damage of assessed
pipe design when elastic-plastic analysis considered. The fatigue analysis solutions become
unstable at FDs larger than unity, since decreasing the WT results in an increase of nominal stresses
and through-thickness plasticity can be developed, the stress-strain state known as general
yielding. Therefore, further analysis proceeded with a cut-off value for the maximum acceptable
accumulated FD of 0.5 in order to avoid the influence of excessive plasticity.
Interestingly, the standard design WT obtained from the analysis based on the elastic data,
(Table 21(I)) was found to be exactly the same as the ones predicted using [97] and [94] according
to ASME 31.4 [23]; i.e., WT of 11.9 mm for OD 610 mm, WT of 17.5 mm for OD 864 mm, WT
of 19.1 mm for OD 914 mm, and WT of 23.8 mm for OD 1219 mm. It is worth noting that the
98
actual values may differ depending on actual pressure history and weld misalignment; however,
the trend should not change.
Another important observation noted during analysis of the linearized stresses in Figure 38
was that the fatigue life of a pipeline is significantly reduced with misalignment due to secondary
bending developed at hot-spot. For instance, depending on the assessment method, a pipeline with
an OD of 914 mm can be designed with 23.30% to 44.41% lower actual WT (or with 25.00% to
45.88% lower standard WT) than that of a misaligned pipeline, as shown in the data represented
by full squares in Figure 45(c) and calculations in Table 21(I). Therefore, for the fixed WT and
pressure history, a non-misaligned pipeline can be designed with significantly lower OD
(approximately 610 mm), instead of 914 mm when the pipeline is misaligned. It can be seen from
Figure 45(a) and Figure 45(c). However, an approximately 3-6% reduction in 𝜎𝑦 can be expected
for UOE-manufactured pipes with larger OD/WT ratios, i.e., with lower WTs, based on the
experimental and predicted data discussed in [17]; this would require recalculation of cyclic stress-
strain curves used in ABAQUS simulations for more accurate estimates of FD.
As far as a misaligned pipeline is concerned, a significant WT reduction without
compromising the fatigue performance of a pipeline (fatigue damage value remains 0.5) can be
achieved when the elastic-plastic analysis performed, i.e., between 7.59% and 16.46% when actual
WT considered, or between 7.57% and 19.98% when standard WT considered (Table 21(I)). The
area of envelope between elastic and elastic-plastic solutions in Figure 45(c) increases significantly
with misalignment; this indicates an increase in design conservatism with increased misalignment
when elastic analysis performed. Moreover, conservatism increases for every OD studied at FDs
lower than 0.5, i.e. at larger WTs.
99
Acc
um
ula
ted
Fat
igu
e D
amag
e D
f
(a)
(b)
(c)
(d)
Wall Thickness t, mm
Figure 45 Accumulated fatigue damage plots for pipe diameters (a) 610 mm, (b) 864 mm,
(c) 914 mm, and (d) 1219 mm, calculated with (solid lines) misalignment and with (contour
lines) no misalignment
0.0
0.5
1.0
5 10 15 20 25 30
0.0
0.5
1.0
5 10 15 20 25 30
0.0
0.5
1.0
5 10 15 20 25 30
0.0
0.5
1.0
5 10 15 20 25 30
BS-e (elastic analysis according to [17])
▪ ASME-e/p (elastic-plastic analysis according to [16])
ASME-e (elastic analysis according to [16])
100
Table 21 Results of fatigue analysis obtained at accumulated fatigue damage of 0.5
Method
Data-to-Curve Fit Actual Standard
𝑅2
Adj. 𝑅2
1st const. a
Power b
2nd const. c
WT* [mm]
ΔWT [mm]
ΔWT [%]
WT** [mm]
ΔWT [mm]
ΔWT [%]
I. Power Law Data-to-Curve Fit, 𝑭𝑫 = 𝒂 × 𝑾𝑻𝒃 + 𝒄
OD 610 mm
A (A – B) 0.9903 0.9838 198.0 × 101 -4.601 0.103 10.50 0.41 3.86 11.91 1.60 13.43
B (A – C) 0.9962 0.9937 484.6 × 107 -10.190 0.215 10.09 1.21 11.56 10.31 2.38 19.98
C (B – C) 0.9940 0.9900 264.3 × 1010 -13.270 0.119 9.28 0.81 8.01 9.53 0.78 7.57
OD 864 mm
A (A – B) 0.9999 0.9998 338.3 × 101 -3.066 -0.057 17.14 0.40 2.33 17.48 0.00 0.00
B (A – C) 0.9974 0.9957 325.9 × 106 -7.412 0.223 16.74 2.42 14.11 17.48 1.60 9.15
C (B – C) 0.9987 0.9979 188.3 × 1010 -10.910 0.159 14.72 2.02 12.06 15.88 1.60 9.15
OD 914 mm
A (A – B) 1.0000 1.0000 825.4 × 101 -3.377 -0.019 17.54 0.58 3.30 19.05 1.57 8.24
B (A – C) 0.9985 0.9977 105.9 × 104 -5.216 0.091 16.96 2.89 16.46 17.48 3.17 16.64
C (B – C) 0.9965 0.9947 797.3 × 104 -6.244 0.082 14.65 2.31 13.61 15.88 1.60 9.15
OD 914 mm – No Misalignment
A (A – B) 1.0000 1.0000 361.0 × 10−1 -1.804 -0.093 9.75 -1.74 -15.12 10.31 -1.60 -13.43
B (A – C) 0.9908 0.9814 565.8 × 102 -4.747 -0.026 11.48 -0.85 -7.99 11.91 -1.60 -13.43
C (B – C) 0.9931 0.9862 393.0 × 105 -7.718 0.017 10.59 0.89 7.75 11.91 0.00 0.00
OD 1219 mm
A (A – B) 0.9972 0.9991 104.0 × 1012 -10.780 0.221 22.48 0.13 0.59 23.83 0.00 0.00
B (A – C) 0.9987 0.9996 312.3 × 1012 -11.130 0.199 22.34 1.83 8.13 23.83 3.21 13.47
C (B – C) 1.0000 1.0000 921.6 × 108 -8.651 0.112 20.65 1.70 7.59 20.62 3.21 13.47
101
II. Linear Data-to-Curve Fit, 𝑾𝑻 = 𝒂 × 𝑶𝑫 + 𝒄
OD 610 mm
A (A – B) 0.9726 0.9588 0.01941 -0.5919 11.24 0.48 4.26 11.91 1.60 13.43
B (A – C) 0.9787 0.9681 0.01986 -1.3740 10.76 1.82 16.17 10.31 2.38 19.98
C (B – C) 0.9917 0.9876 0.01850 -1.8600 9.42 1.34 12.44 9.53 0.78 7.57
OD 864 mm
A (A – B) 0.9726 0.9588 0.01941 -0.5919 16.17 0.35 2.18 17.48 0.00 0.00
B (A – C) 0.9787 0.9681 0.01986 -1.3740 15.82 2.05 12.65 17.48 1.60 9.15
C (B – C) 0.9917 0.9876 0.01850 -1.8600 14.12 1.69 10.71 15.88 1.60 9.15
OD 914 mm
A (A – B) 0.9726 0.9588 0.01941 -0.5919 17.14 0.33 1.91 17.48 0.00 0.00
B (A – C) 0.9787 0.9681 0.01986 -1.3740 16.81 2.09 12.20 17.48 1.60 9.15
C (B – C) 0.9917 0.9876 0.01850 -1.8600 15.05 1.76 10.49 15.88 1.60 9.15
OD 1219 mm
A (A – B) 0.9726 0.9588 0.01941 -0.5919 23.06 0.17 0.76 23.83 0.00 0.00
B (A – C) 0.9787 0.9681 0.01986 -1.3740 22.88 2.37 10.26 23.83 3.21 13.47
C (B – C) 0.9917 0.9876 0.01850 -1.8600 20.69 2.19 9.58 20.62 3.21 13.47
A – BS elastic analysis; B – ASME elastic analysis; C – ASME elastic-plastic analysis; A – B – result of B subtracted from A; A – C – result of C subtracted from A; B – C – result of C subtracted from B. * calculated at the accumulated fatigue damage value of 0.5. ** next standard value according to ASME B31.10 larger than that obtained as *.
The experimental data were further refined by linearizing the relationship between the ODs
and WTs at a fatigue damage of 0.5, as shown in Figure 46. Linearization of the OD–WT
relationship yielded the equations for the individual fatigue assessment methods, see Table 21(II),
which closely follow the general solution for WT in the from of Eq. (64), with only 5%
102
discrepancy observed on the studied interval of ODs, as compared to the BS elastic method [24].
One of the contributions of the present study, however, is in discovering the actual difference in
calculated design WT between elastic and elastic-plastic analyses. This difference is at least one
increment of a standard WT. The results of the fatigue analysis suggest that the design of large-
diameter pipelines may benefit from reductions in WT, contributing to significant budget savings
on material when multi-kilometer transmission lines are designed.
Figure 46 Relationship between OD and WT at accumulated fatigue damage of 0.5 for
(blue) BS elastic, (red) ASME elastic, and (grey) ASME elastic-plastic analyses
The WTs were also calculated at FD as equal 1.0, as shown in Table 22, using equations
from Table 21(I). The computed WTs were plotted as OD–WT relationship and linearized in order
to refine the calculations for WT. The results show that savings on WT increased with OD, between
1.06% and 8.57% when the actual WT considered, and between 0.00% and 10.14% when the
standard WT was considered, as presented in Table 22. The trend was similar to that of the data in
Table 21(II) and Table 22 respectively (see bolted data points). In other words, while the ΔWT
increases with OD at FD equal 1.0, at FD equal 0.5 the ΔWT is almost same throughout the ODs.
This demonstrates an increase in the WT savings for larger ODs at FD equal 1.0, while at FD equal
0.5 the WT savings remain relatively unchanged throughout ODs.
9
12
15
18
21
24
600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250
Wal
l Th
ickn
ess
t, m
m
Outside Diameter D, mm
BS-e at 0.5DASME-e at 0.5DASME-e/p at 0.5DLinear (BS-e at 0.5D)Linear (ASME-e at 0.5D)Linear (ASME-e/p at 0.5D)
103
Table 22 Results of fatigue analysis obtained at accumulated fatigue damage of 1.0
Method
Data-to-Curve Fit Actual Standard
𝑅2
Adj. 𝑅2
1st const. a
Power b
2nd const. c
WT* [mm]
ΔWT [mm]
ΔWT [%]
WT** [mm]
ΔWT [mm]
ΔWT [%]
Linear Data-to-Curve Fit, 𝑾𝑻 = 𝒂 × 𝑶𝑫 + 𝒄
OD 610 mm
A (A – B) 0.9982 0.9973 0.01905 -2.802 8.82 -0.48 -5.17 9.53 0.00 0.00
B (A – C) 0.9923 0.9885 0.01844 -1.949 9.30 0.09 1.06 9.53 0.79 8.29
C (B – C) 0.9791 0.9687 0.01633 -1.236 8.73 0.57 6.17 8.74 0.79 8.29
OD 864 mm
A (A – B) 0.9982 0.9973 0.01905 -2.802 13.66 -0.33 -2.33 14.27 0.00 0.00
B (A – C) 0.9923 0.9885 0.01844 -1.949 13.98 0.78 5.74 14.27 0.00 0.00
C (B – C) 0.9791 0.9687 0.01633 -1.236 12.87 1.11 7.94 14.27 0.00 0.00
OD 914 mm
A (A – B) 0.9982 0.9973 0.01905 -2.802 14.61 -0.30 -1.98 15.88 0.00 0.00
B (A – C) 0.9923 0.9885 0.01844 -1.949 14.91 0.92 6.30 15.88 1.61 10.14
C (B – C) 0.9791 0.9687 0.01633 -1.236 13.69 1.22 8.16 14.27 1.61 10.14
OD 1219 mm
A (A – B) 0.9982 0.9973 0.01905 -2.802 20.42 -0.11 -0.53 20.62 0.00 0.00
B (A – C) 0.9923 0.9885 0.01844 -1.949 20.53 1.75 8.57 20.62 1.57 7.61
C (B – C) 0.9791 0.9687 0.01633 -1.236 18.67 1.86 9.06 19.05 1.57 7.61
A – BS elastic analysis; B – ASME elastic analysis; C – ASME elastic-plastic analysis; A – B – result of B subtracted from A; A – C – result of C subtracted from A; B – C – result of C subtracted from B. * calculated at the accumulated fatigue damage value of 0.5. ** next standard value according to ASME B31.10 larger than that obtained as *.
104
It can be seen from Figure 45 that the solutions for WT are almost the same for all three
methods at FD equal to 1.0; the elastic solutions differ more from elastic-plastic solution with
increasing OD. Although this trend is similar in the case of WTs [mm] obtained at an FD of 0.5,
the ΔWT [%] decreases due to changes in the slope ΔWT/OD from
(2.37 − 1.82) (1219 − 610)⁄ = 9.0 × 10−4 to (1.75 − 0.09) (1219 − 610)⁄ = 2.7 × 10−3, as
can be seen from Table 21(II) and Table 22.
Additionally, when the number of cycles to failure in the Elastic-Plastic Fatigue Analysis
method (3.8.4) is calculated according to the equation used in the Elastic Fatigue Analysis of
Welds (3.8.5), Eq. (53), ASME BPVC Section VIII Part 2 [22], the solutions from both methods
were found to be very close, signifying a good agreement with the Neuber approximation.
In summary, larger pipeline material savings can be expected for lines designed with larger
ODs at accumulated FD close to 100%, at accumulated FD close to 50% pipeline designs with
lower ODs can also benefit from the material savings. For example, if the cost of a UOE
manufactured pipeline is in the range of $ (400 ÷ 1000) 𝑡𝑜𝑛⁄ , considering an approximately
4,700 km long pipeline, as in the case of the Keystone project [53], a 10 % reduction in WT
according to Table 22 would result in $(0.07 ÷ 0.17)𝑚𝑙𝑟𝑑 of total savings, as shown in Table 23.
This is a significant amount at the total estimated pipeline cost of $(5 ÷ 8) 𝑚𝑙𝑟𝑑.
Table 23 Construction cost savings associated with WT reduction on a 4700 km pipeline
Method used Standard pipe dimensions Steel
Density
[𝑔 𝑐𝑚3⁄ ]
Pipeline
Weight
[ton]
Pipeline
Cost
[$ 𝑡𝑜𝑛⁄ ]
Budget
Savings
[𝑚𝑙𝑟𝑑$]
OD
[mm]
WT at FD=1
[mm]
Elastic 914 15.88 7.85 1488173 1000
Elastic-Plastic 914 14.27 7.85 1653112 1000 0.17
105
Chapter 6 CONCLUSIONS AND FUTURE WORK
The present work discusses standard fatigue analyses as well as difficulties associated with
the development of the FE model based on standard procedures used in pipeline design. The weld
misalignment and the weld profile both significantly influenced the stresses. Methods of
accounting for those effects analytically and using the FEM are proposed. This included revision
of analytical calculation for the hot-spot stresses and optimization of the FE model due to a
complex interaction between soil, pipe, and profile of the misaligned weld.
6.1. Conclusions
A mathematical representation of an average SAW weld profile was obtained based on 30
SAW welds from the literature and used in the FE model for a more realistic representation of
pipeline geometry.
Analytical models available in standard procedures were reviewed. Proposed the detailed
yet simple methodology for calculating the stresses at all four weld toes of UOE-manufactured
pipe based on analysis of bending stresses at each weld toe. The results of this method were close
to the results of the FE analysis and helped explain the variations in calculated hot-spot stresses.
A discussion on modification of the analytical model for the calculation of SCFs based on
actual weld profile is provided. The use of actual weld profile results in a quick determination of
WT design when a more precise data-to-curve fit is considered; the set of FE models with different
WTs may be built for either non-misaligned or misaligned pipe.
The approximation of the tensile curves used in the elastic-plastic analysis in ABAQUS
was addressed. Specifically, the discrepancy between the real and approximated tensile-curves
near the yield point of higher-strength steels was reduced by modifying the existing standardized
106
model for a true stress-strain curve, which generation via numerical solution was automated in
MATLAB.
A methodology for modeling (combined) weld misalignments in UOE pipes has been
proposed. A detailed pipe model included weld profile, a combination of the axial and angular
weld misalignments, surrounding soil, thermal load and internal pressure. An extensive
convergence study is presented.
It has been shown that the weld misalignment of a longitudinally welded pipeline due to
UOE manufacturing results in significant stress rise at the structural hot-spot. Consequently, it is
detrimental to in-service fatigue performance. For example, at FD equal 0.5, the non-misaligned
pipeline of OD 914 mm witnessed a 23.30% increase of actual WT due to maximum allowable
misalignment when ASME elastic-plastic fatigue analysis considered. A 27.70% increase of WT
was observed when ASME elastic fatigue analysis considered. After the actual WT was adjusted
to the next nearest standard WT, the increase in WT was found to be 25.00% or 31.86% based on
ASME elastic-plastic or ASME elastic fatigue analysis respectively. A similar trend can be
observed at FD equal to 1.0.
The difference was discovered between elastic and elastic-plastic solutions also for FD of
a pipeline due to in-service pressure fluctuations. In the range of studied ODs, the percent savings
on a WT lies in the range between 9.58% and 12.44% for actual WT and between 7.57% and
13.47% for standard WT, at FD equal to 0.5; at FD equal to 1.0, the percent savings on a WT was
between 6.17% and 9.06% for actual WT and between 0.00% and 10.14% for standard WT.
Reported percent values are proportional to the range of 0 to 2 increments of standard WTs and
directly proportional to the weight reduction of a pipe (i.e. associated cost savings).
107
The result of fatigue analysis suggests that the design of large-diameter pipelines may
benefit from reduction of WT and make significant budget savings on material when the multi-
kilometre transmission lines designed for a 100% FD; at lower values of accumulated FD, 50%,
the pipeline designs with lower ODs can also benefit from the material savings.
The results of this study highlighted the WT reduction capabilities of the elastic-plastic
fatigue analysis compared to conservative estimates based on elastic fatigue analysis done on
large-diameter oil pipelines.
6.2. Future Work
Although, the results of this work have been obtained with the use of construction material
free from metallurgical defects, welding is also known to produce residual stresses and
microstructural defects. The weld microstructure is usually heterogeneous and may contain micro-
cracks or voids at the weld toe surface (undercut) or under the surface. Furthermore, welding and
other UOE pipe manufacturing processes produce residual stresses, some of which, specifically
tensile residual stresses, are known as significant contributors to fatigue crack propagation.
Additionally, pipe expanding and hydrotest homogenize the residual stress distribution and
introduce compressive tensile stresses. Therefore, fatigue life of a welded pipeline should be
further analyzed by using a fracture mechanics approach ASME BPVC Section VIII Part 2 [22],
[37], BS 7608 [24], and BS 7910 [25] to support the WT reduction capabilities when the more
advanced methods of fatigue assessment are used.
108
APPENDIX A – MATLAB Numerical Solution
Merged Stress-Strain Curve
% Level of refinement for the solution of a tangent line refine=1000; % Generation of the Stress coordinates varSTmax=zeros(refine+1,1); for i=1:refine+1; S1=(varSTmax(i,1)+(SysTmax-
50)+((Sutst-(SysTmax-50))/refine)*(i-1)); varSTmax(i)=S1; end % varE1=S/Ey+(S/Kcss)^(1/ncss)-Sys/Ey; varE1Tmax=zeros(refine+1,1); for i=1:refine+1;
e1=(varSTmax(i,1)/EyTmax)+(varSTmax(i,1)/KcssBaseTmax)^(1/ncssBaseTmax)-
SysTmax/EyTmax; varE1Tmax(i)=e1; end % varE2=((((Kcss^(-1/ncss))*S^(1/ncss-1))/ncss)+1/Ey)*(S-Sys); varE2Tmax=zeros(refine+1,1); for i=1:refine+1; e2=((((KcssBaseTmax^(-
1/ncssBaseTmax))*varSTmax(i,1)^(1/ncssBaseTmax-
1))/ncssBaseTmax)+1/EyTmax)*(varSTmax(i,1)-SysTmax); varE2Tmax(i)=e2; end % Minimum difference between solutions deltaEzeroTmax=abs(varE2Tmax-varE1Tmax); % Index for the point corresponding to minimum difference MinErrorIndexTmax=find(deltaEzeroTmax == min(deltaEzeroTmax(:))); % Coordinates for the tangent point of Ramberg-Osgood curve from index StressMEITmax=varSTmax(MinErrorIndexTmax,1); StrainMETmax=varE1Tmax(MinErrorIndexTmax,1); % Ramberg-Osgood portion of a merged curve CCStep=20; varScyclicTmax=zeros(CCStep+1,1); for i=1:CCStep+1;
S1cyclic=StressMEITmax+(varScyclicTmax(i,1)+((Sutst-
StressMEITmax)/CCStep)*(i-1)); varScyclicTmax(i)=S1cyclic; end varEcyclicTmax=zeros(CCStep+1,1); for i=1:CCStep+1;
e2cyclic=(varScyclicTmax(i,1)/EyTmax)+(varScyclicTmax(i,1)/KcssBaseTmax)^(1/n
cssBaseTmax); varEcyclicTmax(i)=e2cyclic; end % Elastic portion of a merged curve StressSCurveElasticTmax=transpose([0 SysTmax]); SStrainCurveElasticTmax=transpose([0 SysTmax/EyTmax]); % Tangent portion of a merged curve varStangentTmax=zeros(CCStep+1,1); for i=1:CCStep+1;
S1tangent=SysTmax+(varStangentTmax(i,1)+((Sutst-SysTmax)/CCStep)*(i-1));
varStangentTmax(i)=S1tangent; end varEtangentTmax=zeros(CCStep+1,1); for i=1:CCStep+1;
e2tangent=(StressMEITmax/EyTmax+(StressMEITmax/KcssBaseTmax)^(1/ncssBaseTmax)
)+(((KcssBaseTmax^(-1/ncssBaseTmax))*((StressMEITmax)^(1/ncssBaseTmax-
1)))/ncssBaseTmax+1/EyTmax)*(varStangentTmax(i,1)-StressMEITmax);
varEtangentTmax(i)=e2tangent; end Tangent=[varStangentTmax,varEtangentTmax]; % Merged Curve at Service Temperature StressSCurveBaseTmax=[StressSCurveElasticTmax;varScyclicTmax]; SStrainCurveBaseTmax=[SStrainCurveElasticTmax;varEcyclicTmax]; MergedBaseTmax=[StressSCurveBaseTmax,SStrainCurveBaseTmax]; xlswrite('BaseTangentSolutionTserv.xlsx',varSTmax,1); xlswrite('BaseTangentSolutionTserv.xlsx',varE1Tmax,2); xlswrite('BaseTangentSolutionTserv.xlsx',varE2Tmax,3); xlswrite('BaseTangentSolutionTserv.xlsx',StressMEITmax,4); xlswrite('BaseTangentSolutionTserv.xlsx',StrainMETmax,5);
109
Neuber’s Rule
% Level of refinement for the solution refine=100000; % Generation of the Stress coordinates varSTmax=zeros(refine+1,1); for i=1:refine+1;
S1=(varSTmax(i,1)+((Sutst+100)/refine)*(i-1)); varSTmax(i)=S1; end % varE1=S/Ey+(S/Kcss)^(1/ncss)-Sys/Ey; varEk1Tmax=zeros(refine+1,1); for i=1:refine+1;
e1=deltaSek*deltaEek/(varSTmax(i,1)); varEk1Tmax(i)=e1; end % varE2=((((Kcss^(-1/ncss))*S^(1/ncss-1))/ncss)+1/Ey)*(S-Sys); varEk2Tmax=zeros(refine+1,1); for i=1:refine+1;
e2=(varSTmax(i,1)/EyTmax)+2*(varSTmax(i,1)/(2*KcssBaseTmax))^(1/ncssBaseTmax)
; varEk2Tmax(i)=e2; end % Minimum difference between solutions deltaEzeroTmax=abs(varEk2Tmax-varEk1Tmax); % Index for the point corresponding to minimum difference MinErrorIndexTmax=find(deltaEzeroTmax == min(deltaEzeroTmax(:))); % Coordinates for the tangent point of Ramberg-Osgood curve from index StressMEITmax=varSTmax(MinErrorIndexTmax,1); StrainMETmax=varEk1Tmax(MinErrorIndexTmax,1); figure; plot(varEk1Tmax,varSTmax); hold on; plot(varEk2Tmax,varSTmax); hold on; plot(StrainMETmax,StressMEITmax,'b*');% hold off; suptitle('Solution for the Neubers Rule'); ylabel('Stress S, MPa'); xlabel('Strain e, mm/mm'); legend('Left-hand equation (Neubers Rule)','Right-hand equation (Hysteresis
Curve)','Solution (Intersection)'); set(gcf,'PaperPositionMode','auto','PaperPosition',[0 0 20 10]); deltaSk = StressMEITmax; deltaEk = StrainMETmax;
110
APPENDIX B – MATLAB Cycle-Counting
clear; clc Input = inputdlg({'Agressive Loading History','Average Loading
History','Accuracy - Bin Size'},'Input Data for Rainflow Cycle Counting',[1
40; 1 40; 1
40],{'RainflowAgressiveLoading.xlsx';'RainflowAverageLoading.xlsx';'0.5'}); FileNameAgr = char([Input(1,1)]); FileNameAve = char([Input(2,1)]); AccBinSize=str2double(Input(3,1)); %% Cycles Counted for Min/Max of Agressive Loading History (Larger Bin) s = xlsread(FileNameAgr); tp=sig2ext(s); rf=rainflow(tp); t = transpose(rf); Pamp=t(1:end,1); Pmean=t(1:end,2); Count=t(1:end,3); Min=Pmean-Pamp; Max=Pmean+Pamp; Delta=Max-Min; Cycle=[Min Max]; % Rounding to desired accuracy acc = AccBinSize; Cycle=round(Cycle/acc)*acc; Cycles=[Cycle Count];
% Construction of a plot Cycles=[Min Max Count]; Transp = transpose(Cycles); rfm=rfmatrix(Transp,20,20); % Surface Bar Plot with Color Bar figure suptitle('Rainflow Cycle Counting - Min/Max') subplot(2,2,1); b = bar3(rfm); title('Agressive Loading History (Larger Bin)') zlabel('Number of Cycles') ylabel('Cycle Max') xlabel('Cycle Min') colorbar for k = 1:length(b); zdata = b(k).ZData; b(k).CData = zdata; end % Remove Zeros for i = 1:numel(b) index = logical(kron(rfm(:, i) == 0, ones(6, 1))); zData = get(b(i), 'ZData'); zData(index, :) = nan; set(b(i), 'ZData', zData); end view(45,30)
set(gcf,'PaperPositionMode','auto','PaperPosition',[0 0 20 10]); print(gcf,'Rainflow Cycle Counting Min-Max.png','-dpng','-r300');
111
% Finding the unique combinations of Min&Max U = unique(Cycles(:,[1 2 3]), 'rows'); % Extracting the unique combinations of Min&Max Size=size(U,1); NofT=zeros(Size,1); for i=1:Size; R = [U(i,1) U(i,2) U(i,3)]; % Counting the number of occurences of a unique combination if U(i,3)<=0.5; C = (ismember(Cycles,R,'rows')')/2; else C = ismember(Cycles,R,'rows')'; end S = sum(C); NofT(i)=S; end Table=[U NofT]; Table(:,3) = []; %Delete 3rd column SortedCycles=sortrows(Table,[2 1]); ZeroRow = zeros(1,3); SortedCycles = [SortedCycles ; ZeroRow]; % Delete Cycles with Start=End giving up the number of Cycles to previous
Cycle for i=1:Size; if (SortedCycles(i+1,2)-SortedCycles(i+1,1))<=0; SortedCycles(i,3)=SortedCycles(i,3)+SortedCycles(i+1,3); SortedCycles(i+1,:)=0; end end SortedCycles; SortedCycles = SortedCycles(any(SortedCycles,2),:); Sum = sum(SortedCycles); Size=size(SortedCycles,1) xlswrite('Cycles Counted for Min-Max of Agressive Loading History (Larger
Bin).xlsx',SortedCycles); MinMaxTable = zeros(1,1); for i=1:Size; row1 = SortedCycles(i,1); MinMaxTable = [MinMaxTable ; row1]; row2 = SortedCycles(i,2); MinMaxTable = [MinMaxTable ; row2]; end if MinMaxTable(2,1) - MinMaxTable(3,1) <=0; MinMaxTable(1:3) = []; else MinMaxTable(1) = []; end MinMaxTable xlswrite('Cycles History for Min-Max of Agressive Loading History (Larger
Bin).xlsx',MinMaxTable);
112
APPENDIX C – ABAQUS Input File
**
** PARTS
**
*Part, name=CutSoil
*End Part
**
*Part, name=Pipe
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=Pipe-1, part=Pipe
*Element, type=C3D20R
*Element, type=C3D15
*Nset, nset=Pipeline, generate
*Elset, elset=Pipeline, generate
** Section: Steel
*Solid Section, elset=Pipeline, material=X56
,
*End Instance
**
*Instance, name=CutSoil-1, part=CutSoil
*Element, type=C3D15
*Nset, nset=SoilBox, generate
*Elset, elset=SoilBox, generate
** Section: SoilBox
*Solid Section, elset=SoilBox, material=Soil
,
*End Instance
**
*Nset, nset=PipeEnds, instance=Pipe-1
*Elset, elset=PipeEnds, instance=Pipe-1, generate
*Nset, nset=SoilBottom, instance=CutSoil-1
*Elset, elset=SoilBottom, instance=CutSoil-1
*Nset, nset=SoilFronts, instance=CutSoil-1
*Elset, elset=SoilFronts, instance=CutSoil-1, generate
*Nset, nset=SoilSides, instance=CutSoil-1
*Elset, elset=SoilSides, instance=CutSoil-1
*Nset, nset=Tpipe, instance=Pipe-1, generate
*Elset, elset=Tpipe, instance=Pipe-1, generate
113
*Nset, nset=Tsoil, instance=CutSoil-1, generate
*Elset, elset=Tsoil, instance=CutSoil-1, generate
*Surface, type=ELEMENT, name=PipeExtSurf
*Surface, type=ELEMENT, name=PipeIntSurf
*Surface, type=ELEMENT, name=SoilCavitySurf
*End Assembly
*Amplitude, name=Pcycle
1., 2.75, 2., 2.91, 3., 2.69, 4., 3.17
5., 2.64, 6., 3.43, 7., 2.58, 8., 3.69
9., 2.52, 10., 3.94, 11., 2.47, 12., 4.2
13., 2.41, 14., 4.46, 15., 2.35, 16., 4.72
17., 2.3, 18., 4.98, 19., 2.24, 20., 5.24
21., 2.18, 22., 5.5, 23., 2.13, 24., 5.76
25., 2.07, 26., 6.02, 27., 2.01, 28., 6.28
29., 1.96, 30., 6.54, 31., 1.9, 32., 6.8
33., 1.85, 34., 7.06, 35., 1.79, 36., 7.31
37., 1.73, 38., 7.57, 39., 1.68, 40., 7.83
41., 1.62, 42., 8.09, 43., 1.56, 44., 8.35
45., 1.51, 46., 8.61, 47., 1.45, 48., 8.87
49., 1.39, 50., 9.13, 51., 1.34, 52., 9.39
53., 1.28, 54., 9.65, 55., 1.22, 56., 9.91
57., 1.17, 58., 10.17, 59., 1.11, 60., 10.43
**
** MATERIALS
**
*Material, name=Soil
*Density
0.0015,
*Elastic
10., 0.3
*Expansion
0.0001,
*Mohr Coulomb
0.,0.
*Mohr Coulomb Hardening
0.03,0.
*Material, name=X56
*Density
0.0078,
*Elastic
207000., 0.3
*Expansion
1e-05,
*Plastic
370., 0., 0.
114
423.972, 0.0122871, 0.
428.222, 0.0131397, 0.
432.472, 0.0140509, 0.
436.722, 0.0150241, 0.
440.971, 0.016063, 0.
445.221, 0.0171714, 0.
449.471, 0.0183534, 0.
453.721, 0.0196131, 0.
457.971, 0.020955, 0.
462.22, 0.0223836, 0.
466.47, 0.0239038, 0.
470.72, 0.0255207, 0.
474.97, 0.0272394, 0.
479.22, 0.0290656, 0.
483.469, 0.0310049, 0.
487.719, 0.0330634, 0.
491.969, 0.0352473, 0.
496.219, 0.0375631, 0.
500.468, 0.0400178, 0.
504.718, 0.0426183, 0.
508.968, 0.0453721, 0.
337., 0., 80.
387.223, 0.00825483, 80.
393.31, 0.00909148, 80.
399.398, 0.0100163, 80.
405.485, 0.0110375, 80.
411.572, 0.0121636, 80.
417.659, 0.013404, 80.
423.747, 0.0147686, 80.
429.834, 0.0162683, 80.
435.921, 0.0179144, 80.
442.008, 0.0197192, 80.
448.096, 0.0216957, 80.
454.183, 0.0238579, 80.
460.27, 0.0262207, 80.
466.357, 0.0287999, 80.
472.445, 0.0316124, 80.
478.532, 0.0346761, 80.
484.619, 0.03801, 80.
490.706, 0.0416344, 80.
496.794, 0.0455707, 80.
502.881, 0.0498417, 80.
508.968, 0.0544715, 80.
**
** INTERACTION PROPERTIES
115
**
*Surface Interaction, name=SS
1.,
*Friction, slip tolerance=0.005
0.3,
*Surface Behavior, pressure-overclosure=HARD
**
** PREDEFINED FIELDS
**
** Name: Tpipe Type: Temperature
*Initial Conditions, type=TEMPERATURE
Tpipe, 15.
** Name: Tsoil Type: Temperature
*Initial Conditions, type=TEMPERATURE
Tsoil, 15.
**
** INTERACTIONS
**
** Interaction: SS
*Contact Pair, interaction=SS, type=SURFACE TO SURFACE, adjust=0.0, tied
SoilCavitySurf, PipeExtSurf
** ----------------------------------------------------------------
**
** STEP: Gravity
**
*Step, name=Gravity, nlgeom=NO, inc=1000
*Static
1., 1., 1e-40, 1.
**
** BOUNDARY CONDITIONS
**
** Name: PipeEnds Type: Displacement/Rotation
*Boundary
PipeEnds, 3, 3
** Name: SoilBottom Type: Displacement/Rotation
*Boundary
SoilBottom, 1, 1
SoilBottom, 2, 2
SoilBottom, 3, 3
** Name: SoilFronts Type: Displacement/Rotation
*Boundary
SoilFronts, 3, 3
** Name: SoilSides Type: Displacement/Rotation
*Boundary
SoilSides, 1, 1
116
SoilSides, 3, 3
**
** LOADS
**
** Name: Gravity Type: Gravity
*Dload
, GRAV, 0.0098, 0., -1., 0.
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
** ----------------------------------------------------------------
**
** STEP: Temperature
**
*Step, name=Temperature, nlgeom=NO, inc=1000
*Static
1., 1., 1e-40, 1.
**
** PREDEFINED FIELDS
**
** Name: Tpipe Type: Temperature
*Temperature
Tpipe, 80.
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
117
*End Step
** ----------------------------------------------------------------
**
** STEP: Pressure
**
*Step, name=Pressure, nlgeom=NO, inc=1000
*Static
1., 1., 1e-40, 1.
**
** LOADS
**
** Name: P Type: Pressure
*Dsload
PipeIntSurf, P, 10.
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
118
APPENDIX D – ABAQUS Report Example
********************************************************************************
Statically Equivalent Linear Stress Distribution across a Section,
written on Date Indicated
Source
-------
ODB: C:/*.odb
Step: Pressure
Frame: Increment 1: Step Time = 1.000
Linearized Stresses for stress line 'SCL'
Start point, Point 1 - (9.90777969360352, 438.650207519531, 1000)
End point, Point 2 - (10.3930807113647, 456.264678955078, 1000)
Number of intervals - 40
------------------------------------- COMPONENT RESULTS -------------------------------------
S11 S22 S33 S12 S13 S23
0 1.50064 189.975 -77.1071 -26.9611 0.000914721 2.49023e-05
0.440529 10.7086 157.468 -84.0969 -17.6226 -0.000104373 -3.28035e-06
0.881058 14.3533 136.637 -89.2531 -12.6624 7.09119e-06 -3.40971e-07
1.32159 15.1787 123.072 -93.0747 -9.49782 1.68619e-06 -6.67067e-07
1.76212 14.8283 113.689 -95.9948 -7.21984 -1.1096e-06 -8.46526e-07
2.20264 14.0649 106.76 -98.3025 -5.49221 9.55155e-08 -9.00495e-07
2.64317 13.0834 101.161 -100.277 -4.11358 -9.94643e-08 -9.69991e-07
3.0837 12.0422 96.4699 -101.996 -3.00724 -2.70706e-07 -9.96492e-07
3.52423 11.0827 92.4651 -103.486 -2.1285 -7.29473e-08 -1.14596e-06
3.96476 10.1398 88.9386 -104.827 -1.40159 -1.90878e-08 -9.49357e-07
4.40529 9.28576 85.7652 -106.035 -0.804824 -9.98379e-08 -1.09816e-06
4.84582 8.51961 82.8544 -107.138 -0.316621 -4.35051e-07 -1.00983e-06
5.28635 7.81651 80.1152 -108.17 0.0859991 -6.35731e-07 -1.1078e-06
5.72688 7.11305 77.5458 -109.152 0.429193 -2.72144e-07 -1.74487e-06
6.1674 6.47442 75.1609 -110.059 0.700721 -2.87416e-07 -1.14502e-06
6.60793 5.89226 72.8494 -110.928 0.932453 -3.56561e-07 -1.34228e-06
7.04846 5.36137 70.604 -111.76 1.13076 -3.24729e-07 -1.46821e-06
7.48899 4.87179 68.4115 -112.565 1.30073 4.05415e-08 -1.61171e-06
119
7.92952 4.44361 66.2878 -113.331 1.43976 -1.33458e-06 -1.73046e-06
8.37005 4.03886 64.1871 -114.082 1.56454 -1.87036e-06 -1.85905e-06
8.81058 3.6815 62.1241 -114.808 1.67181 5.07757e-06 -2.30485e-06
9.25111 3.3524 60.0883 -115.518 1.76732 6.20876e-08 -2.30742e-06
9.69164 3.05445 58.0568 -116.217 1.85668 -2.61125e-05 -1.48921e-06
10.1322 2.79171 56.0627 -116.894 1.93788 2.15678e-06 -2.85157e-06
10.5727 2.55446 54.0459 -117.57 2.01811 9.4648e-05 -6.83548e-06
11.0132 2.35023 52.0747 -118.223 2.09557 -8.69029e-06 -3.05306e-06
11.4538 2.172 50.0695 -118.878 2.17593 -0.000362451 1.1048e-05
11.8943 2.02049 48.1103 -119.511 2.25759 -6.92792e-05 -1.22991e-06
12.3348 1.90113 46.1086 -120.147 2.34632 0.00139442 -6.3865e-05
12.7753 1.80143 44.1468 -120.766 2.43878 0.000754958 -3.79915e-05
13.2159 1.73949 42.1653 -121.379 2.54468 -0.00573958 0.00024583
13.6564 1.69255 40.2103 -121.979 2.65446 -0.00544583 0.000235095
14.0969 1.68092 38.2634 -122.567 2.7894 0.0224755 -0.0010039
14.5375 1.68282 36.3355 -123.145 2.92736 0.034734 -0.00155395
14.978 1.72309 34.4469 -123.699 3.09257 -0.0744971 0.00329452
15.4185 1.78724 32.5311 -124.254 3.26926 -0.190682 0.0084657
15.859 1.79555 31.0792 -124.688 3.4821 0.185249 -0.00804487
16.2996 1.88391 29.3371 -125.184 3.81863 0.583377 -0.023698
16.7401 1.66834 28.2721 -125.568 3.91926 -0.0998983 0.00343847
17.1806 1.1641 27.7303 -125.882 3.7096 -1.37179 0.0571848
17.6212 0.552789 27.249 -126.209 3.31325 -2.62428 0.11596
Membrane
(Average) Stress 5.57059 68.5078 -112.326 -0.393342 -0.0558054 0.00241053
Bending
Stress, Point 1 6.8275 54.8041 18.4895 -7.22837 0.166929 -0.00720545
Membrane plus
Bending, Point 1 12.3981 123.312 -93.837 -7.62171 0.111124 -0.00479492
Bending
Stress, Point 2 -6.8275 -54.8041 -18.4895 7.22837 -0.166929 0.00720545
Membrane plus
Bending, Point 2 -1.25691 13.7037 -130.816 6.83502 -0.222735 0.00961598
Peak Stress,
Point 1 -10.8974 66.6635 16.7299 -19.3393 -0.110209 0.00481983
120
Peak Stress,
Point 2 1.8097 13.5453 4.60649 -3.52178 -2.40155 0.106344
------------------------------------- INVARIANT RESULTS -------------------------------------
Bending components in equation for computing
membrane plus bending stress invariants are: S11, S22, S33, S12, S13, S23
Max. Mid. Min. Tresca Mises
Prin. Prin. Prin. Stress Stress
Membrane
(Average) Stress 68.5103 5.56816 -112.327 180.837 159.001
Membrane plus
Bending, Point 1 123.833 11.8769 -93.8371 217.67 188.534
Membrane plus
Bending, Point 2 16.3562 -3.909 -130.816 147.173 138.159
Peak Stress,
Point 1 71.2182 16.7302 -15.4525 86.6707 75.883
Peak Stress,
Point 2 14.5787 5.63313 -0.250339 14.8291 12.9333
121
APPENDIX E – MATLAB Code for the ABAQUS Data
clear; clc %% ABAQUS Input Input = inputdlg({'File Name - Elastic-Plastic Stress','File Name - Elastic-
Plastic Strain','File Name - Elastic Stress','File Name - Rainflow Counted
Cycles','Service Life - Years'},'Input Raw Data',[1 70; 1 70; 1 70; 1 70; 1
70],{'ElasticPlasticStress.xlsx';'ElasticPlasticStrain.xlsx';'ElasticStress.x
lsx';'Cycles Counted for Min-Max of Agressive Loading History (Larger
Bin).xlsx';'50'}); FileNameElasticPlasticStress = char([Input(1,1)]); FileNameElasticPlasticStrain = char([Input(2,1)]); FileNameElastic = char([Input(3,1)]); RainflowCountedCycles = char([Input(4,1)]); ServiceLife = str2double(Input(5,1)); %% Wall Thickness Meaasurement [v,T,vT]=xlsread(FileNameElastic,'A62:A62'); a=regexp(vT,'\s+','split');
n=numel(a{1}); m=numel(a); thickness=transpose(reshape(str2double([a{:}]),n,m)); t=thickness(1,2); xlswrite(['RefinedCell' FileNameElasticPlasticStress],t,1,'D1');
%% Extraction of Elastic Stress Data Points t = readtable(FileNameElastic); [rows, columns] = size(t); Size = round(rows/106); CycleN=Size; % Zeroing the Table of Results Stresses=zeros(1,10); for j=1:CycleN; CycleNumber=j; % Identification of Cycles from the ABAQUS Report file CycleStartCell = CycleNumber; if CycleNumber<=1; LinearStressStart=71; else LinearStressStart=71+106*(CycleStartCell-1); end LinearizedStart=['A' num2str(LinearStressStart) ':' 'A'
num2str(LinearStressStart)]; % Read Loading History from Raw Data [v,T,vT]=xlsread(FileNameElastic,LinearizedStart);
a=regexp(vT,'\s+','split'); n=numel(a{1}); m=numel(a); ElasticStart=transpose(reshape(str2double([a{:}]),n,m)); Stresses = [Stresses ; ElasticStart]; end %xlswrite('ExtractedStresses.xlsx',Stresses); %% Cleaning of Elastic Stress Data (Removing the Intermidiate Points) t = Stresses; [rows, columns] = size(t); % Creation of Min values tmin = zeros(rows+2,columns); for i=1:rows-2; trow1 = t(i+1,1:end); trow2 = t(i+2,1:end); if trow1(1,5) < trow2(1,5); tmin(i+1,1:end) = trow1; end end % Creation of Max values tmin(rows+1,:) = []; tmin(rows+1,:) = [];
122
tmax = t-tmin; Zeros = zeros(2,columns); tmax = [tmax ; Zeros]; for i=1:rows-2; trow1 = t(i+1,1:end); trow2 = t(i+2,1:end); if trow1(1,5) > trow2(1,5); tmax(i+2,1:end) = [0]; end end tmax(rows+1,:) = []; tmax(rows+1,:) = []; tmax(1,1:end) = [0]; % Creation of Min values (Cont.) for i=1:rows-2; trow1 = t(i+1,1:end); trow2 = t(i+2,1:end); if trow1(1,5) < trow2(1,5); tmin(i+2,1:end) = [0]; end end % Combining the Cleaned Min/Max Loading History tMinMax = tmin+tmax; % Deleting Zero Rows tMinMax = tMinMax(any(tMinMax,2),:); [rows, columns] = size(tMinMax); % Restoring Numbering of Rows for i=1:rows; tMinMax(i,1) = i; end tMinMax; xlswrite(['Refined' FileNameElastic],tMinMax);
tMinMax(1:end,2) = [0]; tMinMax(1:end,10) = [0]; [datarow, datacolumn] = size(tMinMax); B = zeros(1,1); for i=1:datarow; A = {[num2str(tMinMax(i,1)) ' ' num2str(tMinMax(i,2)) ' '
num2str(tMinMax(i,3)) ' ' num2str(tMinMax(i,4)) ' ' num2str(tMinMax(i,5)) ' '
num2str(tMinMax(i,6)) ' ' num2str(tMinMax(i,7)) ' ' num2str(tMinMax(i,8)) ' '
num2str(tMinMax(i,9)) ' ' num2str(tMinMax(i,10))]}; B = [B ; A]; end B(1) = []; xlswrite(['RefinedCell' FileNameElastic],B);
%% Extraction of Elastic Stress Data Points (Sm) t = readtable(FileNameElastic); [rows, columns] = size(t); Size = round(rows/106); CycleN=Size; % Zeroing the Table of Results Stresses=zeros(1,9); for j=1:CycleN; CycleNumber=j; % Identification of Cycles from the ABAQUS Report file CycleStartCell = CycleNumber; if CycleNumber<=1; LinearStressStart=65; else LinearStressStart=65+106*(CycleStartCell-1); end LinearizedStart=['A' num2str(LinearStressStart) ':' 'A'
num2str(LinearStressStart)]; % Read Loading History from Raw Data [v,T,vT]=xlsread(FileNameElastic,LinearizedStart);
a=regexp(vT,'\s+','split'); n=numel(a{1}); m=numel(a); ElasticStart=transpose(reshape(str2double([a{:}]),n,m)); Stresses = [Stresses ; ElasticStart]; end
123
%% Cleaning of Elastic Stress Data (Removing the Intermidiate Points) t = Stresses; [rows, columns] = size(t); % Creation of Min values tmin = zeros(rows+2,columns); for i=1:rows-2; trow1 = t(i+1,1:end); trow2 = t(i+2,1:end); if trow1(1,5) < trow2(1,5); tmin(i+1,1:end) = trow1; end end % Creation of Max values tmin(rows+1,:) = []; tmin(rows+1,:) = []; tmax = t-tmin; Zeros = zeros(2,columns); tmax = [tmax ; Zeros]; for i=1:rows-2; trow1 = t(i+1,1:end); trow2 = t(i+2,1:end); if trow1(1,5) > trow2(1,5); tmax(i+2,1:end) = [0]; end end tmax(rows+1,:) = []; tmax(rows+1,:) = []; tmax(1,1:end) = [0]; % Creation of Min values (Cont.) for i=1:rows-2; trow1 = t(i+1,1:end); trow2 = t(i+2,1:end); if trow1(1,5) < trow2(1,5); tmin(i+2,1:end) = [0]; end end % Combining the Cleaned Min/Max Loading History tMinMax = tmin+tmax; % Deleting Zero Rows tMinMax = tMinMax(any(tMinMax,2),:); [rows, columns] = size(tMinMax); % Restoring Numbering of Rows for i=1:rows; tMinMax(i,1) = i; end tMinMax; xlswrite(['Refined' FileNameElastic],tMinMax);
tMinMax(:,2) = []; tMinMax(:,8) = []; [datarow, datacolumn] = size(tMinMax); B = zeros(1,1); for i=1:datarow; A = {[num2str(tMinMax(i,1)) ' ' num2str(tMinMax(i,2)) ' '
num2str(tMinMax(i,3)) ' ' num2str(tMinMax(i,4)) ' ' num2str(tMinMax(i,5)) ' '
num2str(tMinMax(i,6)) ' ' num2str(tMinMax(i,7))]}; B = [B ; A]; end B(1) = []; xlswrite(['RefinedCell' FileNameElastic],B,2);
124
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