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Appl. Sci. 2021, 11, 11780. https://doi.org/10.3390/app112411780 www.mdpi.com/journal/applsci
Article
Ultimate Axial Load Prediction Model for X65 Pipeline with
Cracked Welding Joint Based on the Failure Assessment
Diagram Method
Jianping Liu
1,2
, Hong Zhang
1
, Shengsi Wu
1
, Xianbin Zheng
3
, Dong Zhang
1
and Xiaoben Liu
1,
*
1
National Engineering Laboratory for Pipeline Safety, MOE Key Laboratory of Petroleum Engineering,
Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of
Petroleum-Beijing, Beijing 102249, China; liujp@pipechina.com.cn (J.L.); hzhang@cup.edu.cn (H.Z.);
2020210831@student.cup.edu.cn (S.W.); 2021310325@student.cup.edu.cn (D.Z.)
2
PipeChina Engineering Technology Innovation Company, Tianjin 300456, China
3
Natural Gas Marketing Company, PetroChina Company Limited, Beijing 100101, China;
xianbinzheng@petrochina.com.cn
* Correspondence: xiaobenliu@cup.edu.cn
Abstract: Crack defects in the girth welds of pipelines have become an important factor affecting the
safe operation of in-service oil pipelines. Therefore, it is necessary to analyze the factors affecting the
safe operation of pipelines and determine the ultimate load during pipeline operation. Based on the
failure assessment diagram (FAD) method described in the BS 7910 standard, the key factors affecting
the evaluation results of the suitability of X65 pipeline girth welds are analyzed, and the effects of crack
size, pipe geometry, and material properties on the evaluation results are investigated. The results
indicate that the crack depth is more crucial to the safe operation of the pipeline than the crack length.
While the effect of wall thickness is not significant, the misalignment can seriously aggravate the stress
concentration. In general, the higher the yield ratio and tensile strength of the pipe material, the more
dangerous the condition at the weld. The ultimate axial load that a crack-containing girth weld can
withstand under different combinations of the above factors was determined. Furthermore, a data
driven model via the optimized support vector regression method for the ultimate axial load of the
X65 pipe was developed for engineering application, and the comparison results between the FEM
results and the predicted results proved its accuracy and reliability.
Keywords: ultimate axial load; girth weld; failure assessment diagram; fracture assessment; X65
pipeline
1. Introduction
For in-service pipelines, internal inspection can identify possible defects which may
endanger pipeline operation. However, we can control the maximum axial load during
service to ensure that the pipeline continues to operate. Take the Mo-Da line as an exam-
ple, which is the first large-diameter crude oil pipeline in China passing through the al-
pine permafrost. This pipeline is made of APL X65 line pipe steel with a diameter equal
to 813 mm. The pipeline passes through the mountainous island permafrost in the north-
ern part of the Greater Khingan Mountains, where there are numerous swamps and well-
developed river systems. Frost heaving in winter and thaw settlement in summer can eas-
ily cause damage to the pipeline (e.g., pipeline folding, bending, and even rupture), and
can seriously affect its safety [1]. In recent years, several accidents have occurred in the
pipe due to the weld cracks [2].
In general, geometric defects, such as cracks and misalignments, as well as the dis-
continuity of materials, can lead to the concentration of stress at the girth weld joint. This
Citation: Liu, J.; Zhang, H.; Wu, S.;
Zheng, X.; Zhang, D.; Liu, X.
Ultimate Axial Load Prediction
Model for X65 Pipeline with
Cracked Welding Joint Based on the
Failure Assessment Diagram
Method. Appl. Sci. 2021, 11, 11780.
https://doi.org/10.3390/
app112411780
Academic Editor(s): Alberto
Campagnolo and Alberto Sapora
Received: 7 November 2021
Accepted: 8 December 2021
Published: 11 December 2021
Publisher’s Note: MDPI stays neu-
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Copyright: © 2021 by the authors. Li-
censee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and con-
ditions of the Creative Commons At-
tribution (CC BY) license (http://crea-
tivecommons.org/licenses/by/4.0/).
Appl. Sci. 2021, 11, 11780 2 of 20
can result in the reduction of the deformation-bearing capacity of girth welds and even
fracture failure under the axial load. Due to such accidents, girth weld cracking has be-
come an important factor affecting the safe operation of oil and gas pipelines. Therefore,
there is a need to investigate effective fracture control methods for circumferential weld
joints of in-service steel pipelines.
At the early development stages of welded structures, in order to avoid the failure of
oil and gas pipelines and prevent accidents, no measurable crack defects were allowed in
welded joints. However, this failure prevention approach tended to lead to waste of re-
sources and increased costs. To this end, the British Welding Institute proposed the “Fit-
ness-For-Service” principle [3], where the possibility of deviations and defects in welded
structures is acknowledged, and the impact of existing defects on the integrity of the
welded structure is scientifically analyzed based on economic considerations to ensure
that the welded structure will not fail by any known mechanism.
To assess the safety of pipelines with defects, a relatively comprehensive pipeline
integrity management system and related technical standards have been established. In
addition, a variety of safety assessment methods have been developed, including the plas-
tic damage-based evaluation method [4–6], strain-based evaluation method [7,8], and nu-
merical simulation analysis based on the finite element method [9–11]. Among them, the
failure assessment diagram (FAD) method is the most widely used. Bloom et al. [12] con-
ducted experimental research to obtain the load changes in the crack growth stage. By
comparing the results with the FAD, it was verified that the FAD can predict the ductile
tearing behavior of metal samples. Based on the FAD, Milne [13] proposed a failure as-
sessment curve considering strain hardening, which improved the accuracy of applying
the FAD method. The EPRG standard [14] broadens the scope of evaluation of girth weld
defects in long transmission pipelines, which sets tolerance limits for the length and height
of girth defects and enables rapid evaluation of weld defects in the field. Based on full-
scale tests, J. Lukács et al. [15] investigated the role of external reinforcing technics on the
integrity reconstruction of damaged transporting steel pipelines, providing a solution for
the rehabilitation of pipelines containing defects.
First of all, it is necessary to clarify the acceptability for pipeline defects. Several
standards, such as R6 [16], BS 7910-2019 [17], CSA Z662-2015 [18], and API 579-1-2016 [19],
recommend the use of the FAD method to evaluate the applicability of pipeline girth
welds. Among them, after more than 40 years of research, BS 7910 has made significant
developments in both breadth and depth. In particular, BS 7910 is a safe and reliable defect
assessment standard which can provide assurance for new structures in the design phase,
for structural integrity in the manufacturing phase, and for the entire life of the structure
in the operational phase [20–22].
Meanwhile, the calculation of the ultimate axial load of pipelines with crack defects
is challenging and time consuming, since numerous influencing factors affect the failure
behavior of pipes. To this end, the prediction of the ultimate axial load based on data
driven models is necessary to assess the safety of pipelines with cracks. The machine
learning technology has been recently utilized for prediction. Support vector regression
(SVR) is a machine learning algorithm which is highly applicable for nonlinear, and high-
dimensional pattern recognition problems. The algorithm can also achieve a balance be-
tween accuracy and complexity of regression models by simultaneously minimizing the
empirical errors and maximizing the geometric marginal zones.
Behrooz et al. [23] used SVR models to perform prediction analysis on the maximum
pitting corrosion depth in oil and gas pipelines. Their research results demonstrated that
SVR models are capable of providing a more precise prediction. Chang et al. [24] per-
formed prediction research on the crack propagation in aviation aluminum alloy. The re-
sults suggest that their prediction model was effective, since the crack growth was pre-
dicted accurately. Tian et al. [25] performed discrete sizing optimization of a stepped cy-
lindrical silo using the PSO method and implicit dynamic finite element analysis.
Appl. Sci. 2021, 11, 11780 3 of 20
In this paper, a BS 7910 stress-based FAD method is used for the X65 pipeline with a
diameter of 813 mm, i.e., the Mo-da crude oil pipeline in China. Considering crack param-
eters (crack depth ratio; crack length-to-cycle ratio), pipeline geometric parameters (wall
thickness; misalignment), and material parameters (tensile strength; yield ratio), the effect
of various influencing factors on the serviceable evaluation results is systematically ana-
lyzed under axial loads equal to 0.5–0.9 times the minimum yield strength. Moreover, the
effect of various influencing factors on the ultimate axial load is clarified as well. To this
end, an ultimate axial load database of the X65 pipeline with cracks is constructed. With
the help of particle swarm optimization (PSO) and SVR, a prediction model of the ultimate
axial load of the X65 pipeline based on stress failure assessment criteria is developed,
providing reference and guidance for ensuring the intrinsic safety of the X65 pipeline with
cracks.
2. Definition of FAD
When a metal structure with defects is subjected to an external load, it is impossible
to predict brittle cracking or plastic failure in advance. Through stress and material anal-
yses, the FAD method takes the fracture ratio Kr and load ratio Lr as two important deci-
sive factors. As can be seen in Figure 1, for a given circumferential welded structure with
cracks, the corresponding Kr and Lr values can be calculated, and the evaluation point A
(Lr, Kr) in the diagram can be obtained. The final evaluation result can be determined based
on the spatial relationship between the evaluation point and the failure assessment curve.
If the evaluation point A falls within the boundary line, the structural state is considered
as safe; otherwise, the structure is unsafe.
Figure 1. Schematic of the FAD.
As mentioned above, among a series of methods and standards, currently, the com-
monly used standard for girth weld fracture evaluation is the evaluation method specified
in BS 7910. The latest version of the standard is “Guide to methods for assessing the ac-
ceptability of flaws in metallic structures” (BS 7910: 2019). In addition, the API 579 stand-
ard also evaluates the engineering criticality based on fracture mechanics; nevertheless, it
is usually applied in process equipment. Consequently, the safety evaluation of girth weld
defects of the MoDa X65 steel pipeline will be performed according to the failure evalua-
tion method specified in BS 7910.
3. Evaluation Process of the BS 7910 Method
The BS 7910 evaluation method is proposed on the basis of engineering critical as-
sessment (ECA), which provides guidelines and suggestions for assessing the acceptabil-
ity of flaws in metallic structures. The methods described in this standard can be applied
Appl. Sci. 2021, 11, 11780 4 of 20
to all stages of design, manufacturing, and operation over the lifecycle of a structure. The
BS 7910-2019 standard describes three different options, which are summarized as fol-
lows:
(1) Option 1 is a conservative evaluation process, which is relatively simple to apply.
The evaluation level does not require detailed stress-strain data of the analyzed
materials.
(2) Option 2 is an evaluation method based on the application of the stress-strain curve
of materials.
(3) Option 3 uses the numerical simulation method to generate a FAD. The evaluation
method of this level is not limited to materials with ductile tearing.
In this presented study, the true stress-strain data were derived by the Ramberg-Os-
good model and corresponding parameter values. At the same time, considering the con-
venience of calculation and appropriate conservativeness, the assessment of the girth
weld safety of the X65 steel pipeline will be guided by Options 1 and 2 described in BS
7910. Based on the specific design conditions and through stress level analysis of the girth
weld, the reliability of the girth weld defects under specific operation conditions will be
determined through nondestructive testing. In the evaluation process, the load and stress
conditions of the pipeline, fracture toughness, and crack characteristics will be deter-
mined, the material curves and failure assessment curves will be constructed, the load and
fracture ratios will be calculated, and the FAD will be drawn and evaluated (Figure 2).
Are fracture toughness data
(K, J or δ ) available?
Defi ne stres ses
Dete rmine K
mat
Estimate K
mat
from CTOD
Determine materials t ensile
properties
Characterize flaw
Select Option 1 or Option 2
Calculate L
r
and K
r
Plot assessment point
(L
r
,K
r
)on FAD
Structure has been
demonstrated to be safe
Structure cannot be
demonstrated to be safe
Can stres s analysis be
refined?
Recharacterize flaw
YN
YN
Y
N
Y
Flaw tolerable? Can the flaw be
recharacterized?
N
Figure 2. FAD Evaluation Process based on BS 7910-2019.
Appl. Sci. 2021, 11, 11780 5 of 20
4. Introduction of FAD Key Parameters
4.1. Definition for the Stresses in ECA
In the ECA process, the stress of the pipeline can be divided into principal and sec-
ondary stress. The principal stress can reach a certain magnitude, which is sufficient to
cause plastic failure of the pipeline. It includes the membrane stress and bending stress
caused by internal pressure and external loads. The secondary stress is a self-balanced
stress caused by local yield, heat treatment, and other factors. It includes the thermal stress
and participating stress. The secondary stress originates from limited strain or displace-
ment, and although it will not lead to plastic failure, the local conditions at the crack tip
deteriorate. Similar to the principal stress, the secondary stress can also be divided into
membrane and bending stress components.
In this paper, the pipe axial stress σP is the stress component Pm of the principal stress.
Due to misalignment at the pipeline interface, a corresponding bending stress σs is gener-
ated, which is the bending stress component Pb of the principal stress.
4.2. Definition for Fracture Toughness
Fracture toughness is used to characterize the ability of a material to prevent crack
propagation, and can be determined based on the stress intensity factor K, crack tip open-
ing displacement (CTOD), and J integral. In the case of the X65 pipeline in Mo-Da pipeline,
the minimum Charpy impact energy of the weld and heat affected zone is 38 J, and the
average value is 45 J [2]. The apparent fracture toughness of the material can be calculated
according to the method proposed in CSA Z662-2015 [18], and it was determined to be 0.2
mm expressed in CTOD. Based on the equation for calculating the stress intensity factor
K and the CTOD determined through the deep notch test, the fracture toughness was cal-
culated to be 5570.1 MPa·mm0.5.
𝐾 =1.517𝜎/𝜎.𝜎𝛿𝐸
1−𝑣 (1)
where E is the modulus of elasticity (210 GPa); v is the Poisson’ s ratio (0.3); σY and σU are
respectively the yield and tensile strength of the tested material at the fracture toughness
test temperature; and δmat is the fracture toughness expressed by the CTOD (mm).
4.3. Material Properties and Crack Geometry
Taking the data of the Mo-Da pipeline as an example, the methods given as Options
1 and 2 in the BS 7910 standard were used for evaluation. The minimum yield strength
and tensile strength of the X65 material specified in the API 5L were employed in the
Ramberg-Osgood model (Equation (2)) to obtain the engineering stress-strain curve of the
X65 pipe steel to derive conservative results for X65 pipes (Figure 3).
𝜀=𝜎
𝐸+
𝐴
𝜎
𝐸𝜎
𝜎
(2)
𝐴
=𝐸𝜀
𝜎−1
𝑛= 1
0.31−𝜎/𝜎
where Ar and n are the hardening coefficient and power hardening index of steel pipe,
σY/σu is yield ratio (λ).
The engineering stress-strain curve of the pipe was converted into the true stress-
strain curve using Equation (3).
Appl. Sci. 2021, 11, 11780 6 of 20
𝜎 =𝜎1+𝜀
(3)
𝜀 =𝑙𝑛1+𝜀
0% 2% 4% 6% 8% 10% 12%
0
100
200
300
400
500
600
Engineering stress-strain curve
True stress-strain curve
Stress (MPa)
Strain
Figure 3. X65 line-pipe steel stress-strain curve.
In practice, multiple adjacent cracks interact with each other, and the effect of this
interaction is much stronger than that of a single crack. If multiple cracks exist, the inter-
action of each crack with the adjacent cracks should be assessed based on the original size
of each crack, and, if necessary, combined into a single crack. Based on the combined crack
method in BS 7910, a circumferential surface defect was assumed to exist on the outer
surface of the pipe girth weld (Figure 4).
Ri
a
2c
B
Figure 4. Schematic diagram of cracks at the girth weld.
Appl. Sci. 2021, 11, 11780 7 of 20
4.4. Construction of Failure Assessment Curves
Each option of the BS 7910 FAD method consists of an assessment curve, which is
obtained by the curve equation Kr = f(Lr), and a cut-off value Lr,max. If the calculated rating
point (Lr, Kr) is located within the area surrounded by the vertical line of the axis, the rating
line, and the cut-off value Lr,max, then, the crack passes; if it is located outside that area,
then, the crack fails. The cut-off value Lr,max is determined according to Equation (4), in
which the average tensile strength is generally used. In this case, due to a lack of statistical
data regarding the performance of the parent material, the minimum tensile strength of
the X65 steel was used to calculate the cut-off value for conservative consideration.
𝐿,=𝜎+𝜎
2𝜎 (4)
As the stress-strain curve has no yield plateau, the Option 1 failure assessment curve
can be determined according to Equations (5) and (6):
𝑓
𝐿=1+0.5𝐿.0.3+0.7𝑒𝑥𝑝−𝜇𝐿 𝐿≤1
𝑓
𝐿=1𝐿/ 1 <𝐿≤𝐿, (5)
𝜇=min0.001𝐸
𝜎,0.6
(6)
𝑁=1
𝑛=0.31−𝜎
𝜎
where Lr is the load ratio, which is the independent variable in the FAD; E is the modulus
of elasticity; μ and N are the coefficients in Equation (4), for the minimum performance
parameters used in X65 steel, μ = 0.4667 and N = 0.0477.
The Option 2 failure assessment curve can be determined according to Equation (7):
𝑓
𝐿=𝐸𝜀
𝐿𝜎+𝐿𝜎
2𝐸𝜀. 𝐿≤𝐿, (7)
where σref is the reference stress, and εref is the corresponding strain in the true stress-strain
curve of the material, i.e., σref = Lr×σY.
Figure 5 displays the Option 1 and 2 evaluation curves of the X65 base material of the
Ø813 × 17.5 mm pipe used in this paper.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
K
r
L
r
Option 1 FAC
Option 2 FAC
Cut-off Line
Figure 5. Option 1 and 2 evaluation curves of the Ø813 × 17.5 mm X65 pipe base metal.
Appl. Sci. 2021, 11, 11780 8 of 20
4.5. Calculation of Lr and Kr
According to BS 7910, the fracture ratio (Kr) and load ratio (Lr) can be calculated as
follows: 𝐾=𝐾
𝐾
(8)
𝑌𝜎=𝑀
𝑓
𝑘𝑀𝑀𝑃+𝑘𝑀𝑀𝑃+𝑘−1𝑃
√
𝜋𝑎
𝑌𝜎=𝑀𝑄+𝑀𝑄
𝐿=
(9)
where (Yσ)P is the contribution from principle stresses; (Yσ)S is the contribution from sec-
ondary stresses; KI is the tensile stress intensity factor; a is the crack depth; M is the Bulging
correction factor, which is M = 1 for the outer surface circumferential crack; fw is the finite
width correction factor, which in this case is fw = 1.000; Mm and Mb are the stress intensity
magnification factors of membrane and bending stress components, respectively; Mkm and
Mkb are respectively the membrane and bending stress intensity magnification factors at
the weld toe; km is the stress concentration factor due to misalignment; σref is the reference
stress corresponding to the applied load; σY is the base material yield stress, and ktm and
ktb are the membrane and bending stress concentration factors, respectively. Since the spe-
cific equation is not provided in Annex B, the membrane and bending stress concentration
factors are not considered for the time being, and are taken as 1.
4.6. Stress Concentration Factor Due to Misalignment
The presence of misalignment can cause additional bending stress when the girth
weld is subjected to axial tensile load. Therefore, in the calculation of the load and fracture
ratios, it is necessary to consider the stress concentration factor induced by misalignment
and the magnitude of the bending stress. According to Appendix D in BS 7910-2019 [5],
the calculation equation is as follows: 𝑘=1+𝜎
𝑃 (10)
where km is the stress concentration factor due to misalignment; σs is the bending stress
caused by the misalignment; and Pm is the primary stress that the ring weld withstands
axially.
The bending stress for axial misalignment at girth welds in pipes with or without
thickness changes (Figure 6) could be calculated as follows:
𝜎=⎩
⎨
⎧
6𝑒
𝐵1−𝑣1
1+𝐵/𝐵.𝑃 𝜎
𝑃<1
2.6𝑒
𝐵1
1+0.7𝐵/𝐵.𝑃 𝜎
𝑃≥1 (11)
where e is the misalignment level; and B1 and B2 is the wall thickness of the pipe on both
sides of the misalignment of the girth weld. In this paper, it is assumed that the wall thick-
ness on both sides is the same; that is, B1 = B2.
Appl. Sci. 2021, 11, 11780 9 of 20
B
1
R
1
D
1
eR
2
B
2
D
2
Figure 6. Axial misalignment at girth welds in pipes.
5. Parametric Analysis on the Assessment Results
The fracture behavior of the weld joint is mainly affected by three factors: load, flaw,
and material. The effects of crack size, pipeline geometric conditions, and material prop-
erties on the evaluation results are analyzed in the following subsections. In Sections 5.1
and 5.2, the minimum specified values of the X65 steel, i.e., yield strength of 450 MPa and
tensile strength of 535 MPa, were selected for the material properties; in Section 5.3, to
analyze the effect of the material properties on the evaluation results, the maximum and
mean values were used.
5.1. Effect of Crack Size
By analyzing the data of the circumferential weld anomalies detected in each pipe
section obtained through high-definition magnetic flux leakage test results [26], it was
found that most of the circumferential weld anomalies are distributed at a height of 30%
of the wall thickness and the length of 200 mm. Only a small number of defects are found
at heights greater than 40% the wall thickness and length greater than 250 mm. All of the
detected abnormalities in girth welds did not exceed 50% of the wall thickness. To cover
most of the crack size ranges, different crack depth ratios (a/B = 0.1, 0.2, 0.3) and crack
length-to-cycle ratios (2c/πD = 0.01, 0.03, 0.05, 0.07) were selected to determine and analyze
the effect of crack size on the applicability evaluation results of Ø813 × 17.5 mm pipeline
girth welded joints in the Mo-Da pipeline.
By comparing the evaluation results obtained for different crack sizes (Figure 7), it
can be deduced that, the larger the crack size, the closer the pipeline to the critical failure
state, i.e., the higher the risk of pipeline operation. Moreover, based on the effect trends of
changing the size parameters on the evaluation results, it can be seen that the effect of the
crack depth on pipeline safety is far more significant than that of the crack length.
Appl. Sci. 2021, 11, 11780 10 of 20
(a) (b)
Figure 7. Effect of crack size on the evaluation results of Ø813 × 17.5 mm pipeline girth welded joints. (a) a/B = 0.3; (b)
2c/πD = 0.07.
5.2. Effects of Pipe Wall Thickness and Misalignment
In the previous subsection, the effect of the crack size on the evaluation results of the
girth welds on Ø813 × 17.5 mm pipes were compared and analyzed. Based on the current
internal detection accuracy control requirements and crack size control specifications, the
possible maximum crack depth (a) and length (2c) of a pipe girth welding are 2.5 mm and
25 mm, respectively. At the actual position of the crack, the wall thickness and misalign-
ment are not necessarily the same. According to relevant standards [27], the axial stress of
the pipeline under normal service conditions should not exceed 0.9 times the minimum
yield strength of the base metal.
Based on this assumption, in this section, the results of axial pipeline loading with
different misalignment levels and wall thickness values are analyzed under loads equal
to 0.5–0.9 times the minimum yield strength. Figure 8a demonstrates the results of pipe
girth welds with different misalignment and wall thickness values when loaded with an
axial load of 0.9 times the minimum yield strength. In addition, Figure 8b shows the re-
sults of the Ø813 × 11 mm pipe girth welds with different misalignment when loaded with
axial loads of 0.5–0.9 times the minimum yield strength.
According to Figure 8, the smaller the pipe wall thickness, the larger the correspond-
ing load and fracture ratios, and the greater the possibility of failure. Therefore, by in-
creasing the pipeline wall thickness, the safety of the pipeline can be effectively improved.
When there is no misalignment at the girth weld, the difference between the load ratios
corresponding to different wall thickness values is not large. Nevertheless, with the in-
crease of the misalignment level, the distance between the evaluation points correspond-
ing to different wall thickness values increases gradually, indicating that the existence of
misalignment significantly increases the stress concentration, and the greater the misa-
lignment, the more severe the stress concentration.
By comparing the results obtained for different misalignment levels, it can be found
that, when the girth weld does not contain any misalignments, the evaluation results cor-
responding to all wall thickness values meet the safety requirements under loads equal to
0.5–0.9 times the minimum yield strength. When there is a misalignment of 1 mm or more,
not all crack evaluation results under all loading conditions can pass the evaluation. This
is attributed to that a high level of misalignment results in high bending stress, which
makes the bearing state of the circumferential weld extremely dangerous.
0.0 0.4 0.8 1.2 1.6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.80 0.85 0.90 0.95
0.70
0.75
0.80
0.85
K
r
L
r
Option 1
Option 2
Cut-off Line
2c/πD=0.01
2c/πD=0.03
2c/πD=0.05
2c/πD=0.07
K
r
L
r
0.0 0.4 0.8 1.2 1.6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
K
r
L
r
Option 1
Option 2
Cut-off Line
a/B=0.1
a/B=0.2
a/B=0.3
Appl. Sci. 2021, 11, 11780 11 of 20
(a) (b)
Figure 8. Effects of wall thickness and misalignment on the crack evaluation of pipe girth welds. (a) σP = 0.9σY (b) B = 11
mm.
5.3. Effect of Material Properties
When evaluating welded joints, BS 7910 suggested using the tensile properties of
materials with lower strength, which is a more conservative approach. In Option 2, uni-
axial tensile stress-strain curves of materials with lower strength should also be used. The
girth weld joints are generally matched with equal strength or high strength, considering
the effect of strength matching; as such, the bearing capacity of the pipe section depends
mainly on the material properties of the base metal [28]. Even if the same grade of steel
pipe is used for the pipeline, the performance parameters are not completely consistent.
Therefore, in order to verify the effect of the X65 steel performance parameters on the
evaluation results, the maximum and mean values of the tensile strength and yield ratio
were selected to analyze their effect on the evaluation results. The final calculation results
and the corresponding true stress-strain curves are shown in Table 1 and Figure 9, respec-
tively.
From the above section, it can be seen that, the thinner the wall thickness, the higher
the risk of cracks in the pipe; accordingly, the Ø813 × 11 mm pipeline girth weld crack risk
is the highest. Consequently, in this section, the results of the Ø813 × 11 mm pipeline girth
weld with 25 mm × 2.5 mm cracks with different material properties under an axial load
of 0.9 times the minimum yield strength are analyzed (Figure 10).
According to the results shown in Figure 10a,b, under the same tensile strength, the
lower the yield strength, the higher the corresponding cutoff value, and the wider the
envelope range of the failure assessment curve. The increase range is concentrated mainly
in the plastic deformation area. In addition, the higher the yield strength, the higher the
fracture ratio of the evaluation point, which indicates that a too high yield ratio is not
conducive to pipeline safety.
According to Figure 10c,d, under the same yield ratio, the higher the tensile strength
of steel, the wider the envelope range of the corresponding failure evaluation curve. More-
over, the higher the fracture ratio of the evaluation point, the wider the envelope range of
the evaluation curve, which can provide a larger safety margin. However, the increase in
the fracture ratio of the evaluation point increases the risk. Consequently, whether the
increase of the tensile strength of the steel has a positive effect on the evaluation results
requires further investigation.
0.0 0.4 0.8 1.2 1.6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.0 1.1
0.45
0.50
K
r
L
r
Option 1
Option 2
Cut-off Line
B = 11.0 mm
B = 12.5 mm
B = 14.2 mm
B = 16.0 mm
B = 17.5 mm
K
r
L
r
0.00.40.81.21.6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
K
r
L
r
Option 1
Option 2
Cut-off Line
e = 0 mm
e = 1 mm
e = 2 mm
e = 3 mm
Appl. Sci. 2021, 11, 11780 12 of 20
Table 1. Material performance parameters.
Tensile Strength
σu (MPa)
Yield Strength
σY (MPa)
Yield Ratio
λ
Hardening Coefficient
Ar
Power Hardening Index
n
535
450 0.842 1.33 20.98
462.5 0.865 1.27 24.60
475 0.888 1.21 29.72
647.5
545 0.842 0.93 21.06
560 0.865 0.88 24.67
575 0.888 0.83 29.77
760
640 0.842 0.64 21.11
657.5 0.865 0.60 24.72
675 0.888 0.56 29.80
0% 2% 4% 6% 8% 10%
0
100
200
300
400
500
600
700
800
Stress(MPa)
Strain
σ
u
=535MPa,λ=0.842
σ
u
=647.5MPa,λ=0.842
σ
u
=760MPa,λ=0.842
σ
u
=535MPa,λ=0.865
σ
u
=647.5MPa,λ=0.865
σ
u
=760MPa,λ=0.865
σ
u
=535MPa,λ=0.888
σ
u
=647.5MPa,λ=0.888
σ
u
=760MPa,λ=0.888
Figure 9. True stress-strain curves of pipeline steels based on material properties.
(a) (b)
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.00 1.05 1.10
0.45
0.50
0.55
K
r
L
r
λ=0.842,Option 2 FAL
λ=0.865,Option 2 FAL
λ=0.888,Option 2 FAL
λ=0.842,assessment point
λ=0.865,assessment point
λ=0.888,assessment point
K
r
L
r
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.00 1.05
0.55
0.60
K
r
L
r
λ=0.842,Option 2 FAL
λ=0.865,Option 2 FAL
λ=0.888,Option 2 FAL
λ=0.842,assessment point
λ=0.865,assessment point
λ=0.888,assessment point
K
r
L
r
Appl. Sci. 2021, 11, 11780 13 of 20
(c) (d)
Figure 10. Effect of material properties on the evaluation results of a circumferential weld crack on the Ø813 × 11 mm
pipeline. (a) σu = 535 MPa (minimum value); (b) σu = 647.5 MPa (mean value); (c) λ = 0.842; (d) λ = 0.865.
6. Analysis of Ultimate Axial Load under Various Service Conditions
The above analysis indicated that the crack size (depth ratio and length-to-cycle ra-
tio), geometric conditions, and material properties all affect the applicability evaluation
results of circumferential welds with cracks. The fracture behavior of the structure is
mainly affected by three factors: load, flaw, and material. For girth welded joints, if the
fracture toughness and crack size are known, the minimum load can be determined.
Therefore, in this section, the maximum axial load that the ring welded joint can withstand
under different factor combinations (Table 2) is calculated. Moreover, the effects of crack
depth ratio, length-to-cycle ratio, pipe wall thickness, girth welded joint misalignment,
and material properties on the ultimate load are analyzed.
Table 2. Value ranges of the investigated influencing factors.
Influencing Factors Values
Crack depth ratio a/B 0.1, 0.2, 0.3
Crack length-to-cycle ratio 2c/πD 0.01, 0.03, 0.05, 0.07
Wall thickness B (mm) 11.0, 12.5, 14.2, 16.0, 17.5
Misalignment e (mm) 0, 1, 2, 3
Tensile strength σu (MPa) 535, 647.5, 760
Yield ratio λ 0.842, 0.865, 0.888
6.1. Effect of Crack Size
The effect of the crack depth ratio on the ultimate axial load was analyzed based on
the evaluation results without and with 1 mm pipe misalignment under a fixed crack
length-to-cycle ratio of 0.01 (Figure 11).
It can be observed that the larger the crack depth ratio, the lower the ultimate axial
tensile load that the girth welded joint can withstand. The conservative degree of Option
2 evaluation was lower than that of Option 1 evaluation; thus, the ultimate axial load of
the Option 2 evaluation was higher than that of the Option 1 evaluation. According to the
Option 1 evaluation, when the crack depth ratio is 0.3, the maximum axial load should
not exceed 0.93 times the specified minimum yield strength of the pipeline without misa-
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
K
r
L
r
σ
u
=535MPa,Option 2 FAL
σ
u
=647.5MPa,Option 2 FAL
σ
u
=760MPa,Option 2 FAL
σ
u
=535MPa,assessment point
σ
u
=647.5MPa,assessment point
σ
u
=760MPa,assessment point
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
K
r
L
r
σ
u
=535MPa,Option 2 FAL
σ
u
=647.5MPa,Option 2 FAL
σ
u
=760MPa,Option 2 FAL
σ
u
=535MPa,assessment point
σ
u
=647.5MPa,assessment point
σ
u
=760MPa,assessment point
Appl. Sci. 2021, 11, 11780 14 of 20
lignment during operation. Under the same conditions and 1 mm misalignment, the max-
imum axial load should not exceed 0.78 times the specified minimum yield strength dur-
ing operation.
The effect of crack perimeter ratio on the ultimate axial load was analyzed based on
the evaluation results without and with 1 mm pipe misalignment under a fixed crack
depth ratio of a/B = 0.3.
(a) (b)
Figure 11. Ultimate axial load of girth weld cracks in pipes with different wall thickness values (2c/πD = 0.01). (a) e = 0
mm; (b) e = 1 mm.
According to Figure 12, the larger the crack length-to-cycle ratio, the lower the ulti-
mate axial tensile load that the girth welded joint can withstand. In addition, the Option
2 evaluation was less conservative than the Option 1 evaluation; thus, the ultimate axial
load under the Option 2 evaluation was higher than that under the Option 1 evaluation.
According to the Option 1 evaluation, when there is no pipeline misalignment and the
crack length-to-cycle ratio has the maximum value of 0.07, the maximum axial load should
not exceed 0.77 times the specified minimum yield strength during operation. Under the
same conditions and 1 mm misalignment, the maximum axial load should not exceed 0.67
times the specified minimum yield strength during operation.
Without pipeline misalignment, the ultimate axial load value decreases slightly with
increasing wall thickness. This is because the width of the weld toe increases with the
increase of the wall thickness, and the effect of stress concentration becomes more signif-
icant. The load is at the limit of plastic collapse, and its non-conservative degree increases.
However, when there is pipeline misalignment, the ultimate axial load value increases
slightly with the increasing wall thickness, which is attributed to the fact that the stress
concentration of the weld toe is weakened with the occurrence of misalignment. In gen-
eral, the greater the wall thickness, the higher the safety of the pipeline, and the ultimate
axial load value increases as well.
10 11 12 13 14 15 16 17 18
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
Ultimate axial load (*σ
s
MPa)
Wall thickness B (mm)
a/B=0.1,Option 1
a/B=0.1,Option 2
a/B=0.2,Option 1
a/B=0.2,Option 2
a/B=0.3,Option 1
a/B=0.3,Option 2
10 11 12 13 14 15 16 17 18
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
Ultimate axial load (*σ
s
MPa)
Wall thickness B (mm)
a/B=0.1,Option 1
a/B=0.1,Option 2
a/B=0.2,Option 1
a/B=0.2,Option 2
a/B=0.3,Option 1
a/B=0.3,Option 2
Appl. Sci. 2021, 11, 11780 15 of 20
(a) (b)
Figure 12. Ultimate axial load of girth weld cracks in pipes with different wall thickness values (a/B = 0.3). (a) e = 0 mm;
(b) e = 1 mm.
6.2. Effect of Wall Thickness and Misalignment
Due to effects related to the welding environment, labor conditions, and other fac-
tors, girth welded joints often have misalignments. By controlling the maximum and min-
imum crack size, the effects of wall thickness and misalignment on the ultimate axial load
were analyzed and the results are presented in Figure 13.
It can be observed that, the larger the level of misalignment, the lower the ultimate
axial tensile load that the ring welded joint can withstand. According to the Option 1 eval-
uation, when the size of the crack is the smallest and the misalignment is 3 mm, the max-
imum axial load should not exceed 0.62 times the minimum yield strength. Under the
same conditions and when the crack size is the largest, the maximum axial load should
not exceed 0.51 times the minimum yield strength. For different crack sizes with respect
to the depth ratio, the effect of the wall thickness on the axial load does not exhibit a com-
pletely consistent trend, which also indicates that the ultimate axial load that the girth
weld can bear is the result of multi-factor comprehensive effects, and the effect of wall
thickness alone is not significant.
(a) (b)
10 11 12 13 14 15 16 17 18
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
Ultimate axial load (*σ
s
MPa)
Wall thickness B (mm)
2c/πD=0.01,Option 1
2c/πD=0.01,Option 2
2c/πD=0.03,Option 1
2c/πD=0.03,Option 2
2c/πD=0.05,Option 1
2c/πD=0.05,Option 2
2c/πD=0.07,Option 1
2c/πD=0.07,Option 2
10 11 12 13 14 15 16 17 18
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Ultimate axial load (*σ
s
MPa)
Wall thickness B (mm)
2c/πD=0.01,Option 1
2c/πD=0.01,Option 2
2c/πD=0.03,Option 1
2c/πD=0.03,Option 2
2c/πD=0.05,Option 1
2c/πD=0.05,Option 2
2c/πD=0.07,Option 1
2c/πD=0.07,Option 2
10 11 12 13 14 15 16 17 18
0.6
0.8
1.0
1.2
1.4
Ultimate axial load (*σ
s
MPa)
Wall thickness B (mm)
e = 0mm
,
Option 1
e = 0mm
,
Option 2
e = 1mm
,
Option 1
e = 1mm
,
Option 2
e = 2mm
,
Option 1
e = 2mm
,
Option 2
e = 3mm
,
Option 1
e = 3mm
,
Option 2
10 11 12 13 14 15 16 17 18
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Ultimate axial load (*σ
s
MPa)
Wall thickness B (mm)
e = 0mm,Option 1
e = 0mm,Option 2
e = 1mm,Option 1
e = 1mm,Option 2
e = 2mm,Option 1
e = 2mm,Option 2
e = 3mm,Option 1
e = 3mm,Option 2
Appl. Sci. 2021, 11, 11780 16 of 20
Figure 13. Effects of wall thicknesses and misalignment on the ultimate axial load of pipeline girth weld cracks. (a) a/B =
0.1; 2c/πD = 0.01; (b) a/B = 0.3; 2c/πD = 0.07.
6.3. Effect of Material Properties
In this subsection, the effect of material properties on the ultimate axial load of the
girth weld is analyzed considering the maximum and minimum crack size under a wall
thickness of 11 mm and no misalignment.
According to Figure 14, under the same tensile strength, the higher the yield ratio of
the steel, the lower the ratio of ultimate axial load to yield strength that the girth weld can
withstand. This is more apparent when the crack size is small, and the effect of the yield
ratio on the ultimate axial load weakens with increasing crack size. Under the same yield
ratio, the higher the tensile strength, the lower the ratio of ultimate axial load to yield
strength that the girth weld can withstand. This is more apparent when the crack size is
large, and less significant when the crack size is small. The maximum axial load should
not exceed 0.91 times the minimum yield strength specified in the relevant standards for
circumferential welds with maximum size cracks without considering misalignment. This
is consistent with the conclusion in relevant standards that the axial stress of the pipeline
under normal service conditions shall not exceed 0.9 times the specified minimum yield
strength of the base metal.
(a) (b)
Figure 14. Effect of material properties on the ultimate axial load of pipeline girth weld cracks. (a) a/B = 0.1; 2c/πD = 0.01;
(b) a/B = 0.3; 2c/πD = 0.07.
7. Data Driven-Based Prediction of the Ultimate Axial Load
In this paper, based on the BS 7910 FAD method and the stress failure criterion, the
ultimate axial load of a X65 pipeline with cracks under 2160 different working conditions
and different crack parameters, pipeline geometric parameters, and material parameters
has been calculated. The investigated values for each influencing factor are listed in Table
2. In order to accurately assess the effects of the different influencing factors on the pre-
diction results of the ultimate axial load of a X65 pipeline with cracks, this section conducts
a parallel calculation under different working conditions based on different combinations
of material yield ratio, misalignment, wall thickness, and crack size values. In total, 2160
sets of data were produced.
In this paper, the LIBSVM toolkit was used to train the above database and establish
the support vector regression (SVM) model for ultimate axial load prediction. According
to the operation process of the LIBSVM toolkit, the ranges of the penalty function c and
535 647.5 760
1.00
1.04
1.08
1.12
1.16
Ultimate axial load (*σs MPa)
Tensile strength σu (MPa)
λ=0.842,Option 1
λ=0.865,Option 1
λ=0.888,Option 1
λ=0.842,Option 2
λ=0.865,Option 2
λ=0.888,Option 2
535 647.5 760
0.84
0.88
0.92
0.96
Ultimate axial load (*σ
s
MPa)
Tensile strength σ
u
(MPa)
λ=0.842,Option 1
λ=0.865,Option 1
λ=0.888,Option 1
λ=0.842,Option 2
λ=0.865,Option 2
λ=0.888,Option 2
Appl. Sci. 2021, 11, 11780 17 of 20
kernel function g are selected based on experience, and the above parameters are opti-
mized by the PSO algorithm. The PSO algorithm is a population intelligence-based opti-
mization algorithm with good convergence and wide applicability, especially for more
complex engineering problems. The objective function is the mean square error (MSE) of
the prediction results and finite element results. When MSE ≤ 10−4, the optimization of the
parameters is completed. The optimized values are given as cbest and gbest, and the SVR
model is trained by these parameters, in order to accurately predict the evaluation results.
Error analysis is an important method for evaluating the accuracy of prediction mod-
els. The predictive accuracy of the developed SVR model can be determined based on
statistical criteria, such as MSE and R. To assess the accuracy of the evaluation results,
MSE and R are taken as the error analysis indices, which can be obtained using the fol-
lowing equations:
MSE= 1
𝑚𝑦−𝑦
(12)
where m is the total number of samples, i
yrefers to the actual results, and ˆi
ydenotes
the prediction results. 𝑟 =𝑛∑𝑥𝑦−∑𝑥∑𝑦
𝑛∑𝑥−∑𝑥.𝑛∑𝑦−∑𝑦. (13)
where x is the actual result and y the prediction result.
As can be observed in the prediction error diagram of the PSO-SVR model (Figure
15), the error between most prediction results and finite element calculation results was
less than 10%. As can be seen in Table 3, the correlation coefficients of the two models
were higher than 99%, and the predicted results were found to be in good agreement with
the numerical simulation ones, indicating that the proposed model has high prediction
accuracy and reliability. Consequently, the prediction method proposed in this paper can
be utilized for the safety evaluation of X65 pipelines with crack defects.
(a) (b)
Figure 15. Evaluation prediction error diagram. (a) Option 1; (b) Option 2.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2 Option 1 predicting value
Upper 10% error
Lower 10% error
Predicting value
True value
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2 Option 2 predicting value
Upper 10% error
Lower 10% error
Predicting value
True value
Appl. Sci. 2021, 11, 11780 18 of 20
Table 3. Prediction errors.
Parameters Option 1 Prediction Error Option 2 Prediction Error
MSE (mean square error) 1.55 × 10−4 1.39 × 10−4
R (correlation coefficient) 99.75% 99.77%
8. Conclusions
Based on the BS 7910 FAD method, this paper analyzed the important factors affect-
ing the ultimate axial load of the X65 pipeline with cracked girth welds, and developed a
prediction model for the ultimate axial load of the X65 pipeline, which provides a refer-
ence for the applicability evaluation and safe operation of X65 pipeline with girth weld
cracks. The final conclusions are as follows:
(1) The larger the crack size, the higher the risk to the safe operation of the pipeline. For
pipeline girth welds without misalignment, the safety requirements are met if the
axial load does not exceed 0.9 times the yield strength, even when there is a large-
size crack. Moreover, according to the results regarding the effect of crack size pa-
rameters on the evaluation results, the effect of crack depth on pipeline safety is far
greater than that of crack length.
(2) The smaller the pipeline wall thickness, the higher the load and fracture ratios, and
the greater the possibility of damage. The presence of misalignment significantly in-
creases the stress concentration, and the greater the misalignment, the more severe
the stress concentration. Increasing the pipeline wall thickness or reducing misalign-
ment can effectively improve the safety of the pipeline.
(3) Under the same tensile strength, the higher the yield strength, the narrower the en-
velope range of the failure evaluation curve, and the higher the fracture ratio of the
evaluation point. This indicates that a high yield ratio of steel is not conducive to
pipeline safety. On the other hand, whether the improvement of the tensile strength
of steel under the same yield ratio has a positive effect on the evaluation results re-
quires further investigation.
(4) When there is no pipeline misalignment, the ultimate axial load decreases slightly
with increasing wall thickness. When there is pipeline misalignment, the ultimate
axial load increases slightly with increasing wall thickness, which is because the
stress concentration of the weld toe weakens with the occurrence of misalignment. In
general, the larger the wall thickness, the higher the safety of the pipeline, and the
ultimate axial load increases as well.
(5) Under the same tensile strength, the higher the yield ratio of steel, the lower the ulti-
mate axial load that the girth weld can withstand, which is more apparent when the
crack size is small. Therefore, when the crack size increases, the effect of the yield
ratio on the ultimate axial load weakens. Under the same yield ratio, the higher the
tensile strength, the lower the ultimate axial load that the circumferential weld can
bear. This this is more obvious when the crack size is large, and less significant when
the crack size is small.
(6) In this paper, a prediction model for the ultimate axial load of the X65 pipeline based
on stress failure assessment criteria was developed. The correlation coefficients of the
Option 1 and Option 2 evaluation models were above 99%, indicating that the pre-
diction results of the PSO-SVR model are accurate and reliable.
Author Contributions: Conceptualization, H.Z.; data curation, writing and editing, S.W.; Method-
ology and formal analysis, J.L. and D.Z.; Supervision, X.L.; Validation, X.Z. All authors have read
and agreed to the published version of the manuscript.
Funding: This research has been co-financed by Beijing Municipal Natural Science Foundation
(Grant No. 8214053), National Science Foundation of China (Grant No. 52004314), Tianshan Youth
Program (Grant No. 2019Q088), the Open Project Program of Beijing Key Laboratory of Pipeline
Critical Technology and Equipment for Deepwater Oil & Gas Development (Grant No.
Appl. Sci. 2021, 11, 11780 19 of 20
BIPT2020005), PipeChina scientific research and technology development project-Research on fail-
ure mechanism for girth weld of high steel pipeline (No. WZXGL202105), Science Foundation of
China University of Petroleum, Beijing (No. 2462020YXZZ045).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: This work acknowledges the support of the above foundations, as well as the
graduate students as participants from China University of Petroleum, Beijing.
Conflicts of Interest: The authors declare no conflict of interest.
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