University of Calgary PRISM: University of Calgary's Digital Repository 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 Finite Element Modeling of Buried Longitudinally Welded Large-Diameter Oil Pipelines Subject to Fatigue (Unpublished master's thesis).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 University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca
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University of Calgary
PRISM: University of Calgary's Digital Repository
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Pipelines Subject to Fatigue (Unpublished master's thesis).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
University of Calgary graduate students retain copyright ownership and moral rights for their
thesis. You may use this material in any way that is permitted by the Copyright Act or through
licensing that has been assigned to the document. For uses that are not allowable under
copyright legislation or licensing, you are required to seek permission.
Downloaded from PRISM: https://prism.ucalgary.ca
UNIVERSITY OF CALGARY
Finite Element Modeling of Buried Longitudinally Welded Large-Diameter Oil Pipelines Subject
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 2 LITERATURE REVIEW.......................................................................7 2.1 Early FE Analyses ..................................................................................................8 2.2 Recent FE Analyses..............................................................................................12
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
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
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
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
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 *.
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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
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
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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
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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).
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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.
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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;
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;
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;
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);
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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;
; 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
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)
% 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
%% 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,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
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%% 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)) ' '