Classification and Comparison of Crack and Dent Defects in a Metal Pipe Subjected to Variable Amplitude Loading

Abstract

Pipelines are commonly used to transport energy over long distances. If this structure is subjected to an internal pressure of variable amplitude loading, such as water hammer waves, the structural damage caused by the presence of a defect can be exacerbated. Previous research by the authors resulted in the development of finite element models to evaluate crack and dent defects separately. Each model was used to compare and classify defects in their respective categories based on their nocivity in a metal pipe subjected to internal pressure. The primary objective of this paper is to compare the severity of various defect categories on the same scale. A numerical damage assessment model that considers the interaction effect, as well as the loading history, is used to achieve this goal. It takes the output of the two finite element models, as well as the pressure spectrum caused by the water hammer, as inputs. This model is used to analyze the effect of key parameters that influence the severity of the defects, as well as to compare and classify the various types of dent defects with the various types of crack defects found in pipes subjected to variable amplitude loading.

Full text

INTERNATIONAL
JOURNAL OF AUTOMOTIVE AND MECHANICAL ENGINEERING (IJAME)
ISSN: 2229
-8649 e-ISSN: 2180-1606
VOL.
19, ISSUE 1, 9577 – 9592
DOI: https://doi.org/10.15282/ijame.19.1.2022.20.0739
*CORRESPONDING AUTHOR | Zakaria Mighouar |  z.mighouar@gmail.com
9577
© The Authors 2022. Published by Penerbit UMP. This is an open access article under the CC BY license.
ORIGINAL ARTICLE
Classification and Comparison of Crack and Dent Defects in a Metal Pipe Subjected
to Variable Amplitude Loading
Zakaria Mighouar*, Hamza Khatib, Hammadi Chaiti, Laidi Zahiri and Khalifa Mansouri
SSDIA Laboratory, Hassan II University of Casablanca, ENSET of Mohammedia, Post Box No. 159, Mohammedia, Morocco
ARTICLE HISTORY
Received: 28th June 2021
Revised: 20th Dec 2021
Accepted: 18th Mar 2022
KEYWORDS
Damage accumulation;
Defect;
Harmfulness;
Pipeline;
Water hammer
INTRODUCTION
Pipelines are structures used primarily for the pressurized transport of flammable substances, which have higher safety
requirements due to the risk of leakage or explosion [1]. New pipelines are needed to meet the growing demand for
energy, such as gas and oil, among industrial users. Indeed, over the last 50 years, the latter has emerged as the most cost-
effective and safest mode of long-distance transportation for large amounts of energy [2]. The length of pipelines in
Europe was multiplied by four between 1970 and 2007. For the same time period, however, the failure rate was divided
by six [2]. To improve the profitability of this mode of supply, manufacturers have increased both the operating pressure
and the pipe diameter. Between 1910 and 2000, the largest pipeline's diameter increased fourfold, while transport pressure
increased sixtyfold [3]. All of this was made possible by research that improved the mechanical properties of pipelines as
well as tools that allow the severity of defects in these pipes to be assessed.
Undoubtedly, as with every metal structure, flaws in the pipeline may develop over time and cause it to rupture. Pipe
defects can occur during installation, routine maintenance excavations, or new civil engineering work near the pipes [4].
For example, during pipe maintenance operations, mechanical damage may occur as a result of negligence, clumsiness
or a lack of precautions. If site workers are unable to precisely locate the buried pipe, this structure may be subjected to
shock by a tool like bucket teeth or construction machine. Most of the time, the incident goes unnoticed or unreported.
Mechanical interference caused by foreign object contact accounts for approximately 50% of pipeline damage in
Europe and 53.5% in the US [5]. This confirms that external damage causes the vast majority of pipeline ruptures, whether
on land or at sea. These flaws can take the form of dents, cracks, or a combination of the two [6]. The structural damage
induced by the presence of these deficiencies can be exacerbated if the pipeline is subjected to internal pressure of variable
amplitude loading, such as water hammer waves [7]. In fact, transient flows in the pipeline network can be created by
pump failure, pipe rupture, or a sudden change in the state of the valve that controls the flow of fluid through the pipeline.
This can cause a pressure pulse to travel at high speeds along the pipeline in the form of a pressure wave, causing
vibrations that can eventually burst the pipe [8].
The industrialists who specializes in the area of piping networks are concerned about the safety of the population as
well as the environment, given the impact that a major failure can have, especially in the case of flammable gases or
explosives [9]. Besides that, economic and financial aspects must be considered, as financial losses in terms of public
works, pipe replacement, and operating losses are substantial. Thus, breakage prevention is critical, and it is achieved
through inspection and analysis of the harmfulness of discovered defects [10, 11]. This analysis necessitates the use of
specialized tools in order to assess the potential damage caused by a defect in an internally pressurized pipe.
There are several methods in the literature for determining the severity of a crack, dent, notch, or corrosion defect in
a pipeline [9-14]. They are frequently developed using limit analysis, fracture mechanics, and notch fracture mechanics.
Depending on the type of defect, the appropriate tool is selected. The limit analysis is frequently used to assess defects
such as corrosion or dents [9]. In the case of defects such as weld cracks, sharp notches, or a combination of a dent and a
notch, a mechanical fracture approach is preferable.
ABSTRACT – Pipelines are commonly used to transport energy over long distances. If this
structure is subjected to an internal pressure of variable amplitude loading, such as water hammer
waves, the structural damage caused by the presence of a defect can be exacerbated. Previous
research by the authors resulted in the development of finite element models to evaluate crack and
dent defects separately. Each model was used to compare and classify defects in their respective
categories based on their nocivity in a metal pipe subjected to internal pressure. The primary
objective of this paper is to compare the severity of various defect categories on the same scale.
A numerical damage assessment model that considers the interaction effect, as well as the loading
history, is used to achieve this goal. It takes the output of the two finite element models, as well as
the pressure spectrum caused by the water hammer, as inputs. This model is used to analyze the
effect of key parameters that influence the severity of the defects, as well as to compare and
classify the various types of dent defects with the various types of crack defects found in pipes
subjected to variable amplitude loading.

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Dented and cracked pipes in the presence or absence of internal pressure have been studied in recent years in terms of
the risks they pose, using various approaches: theoretical, experimental, or numerical [4-6, 9-15]. In the literature, there
is still a lack of a model that can be adapted to different types of defects. Furthermore, the majority of these studies do
not account for the dynamic effect of loading. However, it has been demonstrated that this effect has an impact on the
defect's harmfulness and is to be considered if precise results are sought [16]. The research presented in this paper is
focused on mechanical damage defects, and more specifically to smooth dents and surface semi-elliptical cracks.
Previous research by the authors allowed for the development of finite element models [17, 18] that were tailored to
each defect category studied (one for cracked pipes and one for dented pipes). The stress intensity factor (SIF) along the
crack front is calculated using a cracked pipe numerical model. The stress concentration factor (SCF) near the dent defect
is calculated using a dented pipe model. These parameters allow the harmfulness of a defect in a metal tube under internal
pressure to be evaluated. Each model was used in a separate study to compare and classify defects of various categories
(e.g., dent and crack) based on their nocivity.
Since the tools used to classify crack defects differ from those used to classify dent defects, it has been impossible to
directly compare defects from these two categories. As a result, the authors [19, 20] developed a numerical damage
assessment model that uses the results of these two finite element models to assess damage accumulation in the defected
pipeline. This dimensionless parameter will make it possible to compare the harmfulness of various defect categories on
the same scale. Analytical models were developed and correlated with results from the finite element models in the
previous studies. The obtained models are then validated in order to replace the numerical calculation, as they produce
nearly identical results in terms of SIF for cracked pipes and SCF for dented pipes. These analytical models are used in a
numerical damage accumulation model developed in Scilab, which takes into account the interaction effect and the
loading history. This model is developed to evaluate the severity of a pipeline defect caused by each loading cycle. This
paper provides a brief overview of the various models. Details on the development and validation of these models can be
found in [19, 20].
The damage assessment model takes as input history of pressure peaks caused by a sudden change in flow velocity.
The characteristic method is used to solve the transient flow equations in the pipe to generate this pressure spectrum.
More information on the calculation process can be found in [16]. Based on an appropriate fatigue life estimation model,
the damage is estimated for each pressure cycle. In the case of dented pipes, fatigue life is derived from SCF and loading
parameters; in the case of cracked pipes, fatigue life is derived from SIF and loading parameters. The accumulation
damage model is validated against published experimental data in the literature for each defect category. This model is
used in this paper to reach conclusions on key parameters that affect the severity of the studied defects, as well as to
compare and classify the different types of dent defects with the different types of crack defects present in a pipe subject
to variable amplitude loading. This will allow for the identification of equivalence between several types and categories
of defects in terms of structural damage.
This paper focuses on the investigation of defective pipes that have been subjected to water hammer waves as a result
of a sudden change in valve status (fast closing or opening) or even a pump failure. The effects of the loading sequence
and the loading history are considered in this study. The considered semi-elliptical cracks are internal and external surface
cracks located in the base material of the pipe, in the welding bead or the heat-affected zone (HAZ). The dents investigated
are of various shapes (spherical and rectangular) and orientations (longitudinal and transverse) in two configurations
(constrained and unconstrained). The following section explains the distinction between constrained and unconstrained
denting.
FINITE ELEMENT MODELS
Dent Defects
In some cases, when a foreign body collides with a pipe, the cross-section of the pipe wall experiences irreversible
plastic deformation known as a dent [21]. Dents can produce stress concentration in the pipe, making the structure more
vulnerable to fatigue damage caused by internal pressurization. The geometric parameters of the dent defect are shown
in Figure 1. A dent can be characterized more precisely in terms of its capacity to move following internal pressure, i.e.,
unconstrained dents can re-round under pressure while constrained dents cannot. A constrained indentation is defined in
general by the presence of a retaining object, usually the indenter [5].
During construction, laying the pipe on a stone or rock might also result in a constrained dent. According to Rosenfeld
[22], it is conceivable for excavating equipment to generate dents at the bottom line of small diameter pipes. Furthermore,
it is likely that local restraint in the vicinity of the dent caused by wall thickness or the nature of the dent prevents it from
responding to pressure fluctuations. These indentations are also known as restrained dents, and this term is used in other
parts of the literature [22]. The broadest definition of the constrained dent was employed in this work to define any dent
that is not able to re-round when the structure is subjected to internal pressure. Cracks are initiated and propagated in the
vicinity of the dent where the stress concentration is highest due to the action of cyclic internal pressure, resulting in
material fatigue. It is consequently critical to be able to forecast the fatigue life of a dented pipe in order to anticipate and
thus avoid structural failure.

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Figure 1. Dent defect parameters.
Several authors [1, 2, 5, 6, 14, 22] have explored the fatigue behavior of pipes with a wide range of diameters and
wall thicknesses indented to varying depths. Furthermore, a substantial amount of study has been conducted to understand
the effect of different forms and sizes of indentations on the behavior of pipelines under static and cyclic pressures [13,
17, 23]. Design fatigue curves have been used to construct a range of semi-empirical and empirical models for estimating
dent fatigue life. The main concept is to use published fatigue curves (S-N) from a relevant design code [5, 21-26] and
account for stress concentration caused by the dent with the stress concentration factor. It should be emphasized that this
method implies that there are no initial cracks in the dent, therefore fatigue life includes the duration required to initiate
and propagate a crack [5].
In addition, there is no standard for calculating the stress concentration factor. Some authors [23, 26] employed
experimental determination, whereas others [1, 6, 13, 17, 22] used finite element modeling. The disadvantage of utilizing
FEM to derive stress concentration factor is that a complete examination of each indentation to be investigated is required,
which is an expensive and time-demanding operation. Some authors, including the paper's authors [20, 27, 28], attempted
to establish an analytical equation based on FEM data to relate the SCF to the geometry of the dent. These formulae can
be used to calculate the stress concentration factor under static or cyclic load for a wide variety of dent and pipe diameter
values.
Cunha et al. [13] used published experimental data to compare the available methodologies for evaluating dent fatigue.
This study advises using the Petrobras model [29] in Eq. (1) because it offers the greatest match to the test data due to the
fact that it accounts for the effect of the mean stress. The S-N curve is derived for given mean stress and is modified by
the SCF caused by a dent in a metal pipe.
1
2
6
1 for N<10 cycles
b
M
AF
NUTS
σ
σσ



= −






(1)
10
1 0.325
log
6
F
UTS
bSCF
σ

×
=
×

(2)
where σA is the alternating stress, σM is the mean stress, σF is the true stress of failure, and UTS is the ultimate tensile
strength.
This model is the one used in our research to estimate the fatigue life of pipes with dents of several dimensions, shapes,
orientations, and configurations. The stress concentration factor in Equation 2 is calculated using a previously developed
FEM of a dented pipe. This numerical model provides the SCF from the Von Mises stress measured around three types
of indentations (spherical, longitudinal rectangular and transverse rectangular) in two configurations (constrained and
unconstrained). The finite element model is developed using the ANSYS code. A validation study revealed that this model
can accurately provide the stress concentration factor in the case of a dented pipe. Using the Petrobras model [29], one
can derive the S-N curves for the indented pipe using this factor. The analyzed pipes are built of API 5L X52 and have
an external diameter of 274 mm, a length of 1100 mm, and a thickness of 13.7 mm. The reason for selecting this material
is that existing pipelines, which are the most prone to defects, are primarily composed of it. The dimensions chosen in
our study represent those of an average pipeline; they were also chosen in order to be able to calibrate and validate our
finite element model of a dented pipe with experimental work conducted by CERP [30].
It should be noted that, as with most FEM software, it is highly preferable that the stress-strain data entered into
ANSYS be true stress-strain data. Rather than the material's initial undeformed state, these curves correlate the current
deformed state with the history of previously performed states. The use of the true stress-strain curve is also due to the
fact that the resulting curve is an increasing function and thus associates a single strain value to stress in the plastic field,
as opposed to stress-strain-engineering curves, which may associate two strain values with one stress value, causing
program confusion. The hardening law adopted for the analysis of nonlinear finite elements is that of isotropic hardening
with as data, the true stress-strain curve of the material (Figure 2). These data are obtained by converting the experimental
data of engineering stress-strain provided by the CERP using Eq. (3) and Eq. (4). Table 1 summarizes the mechanical
parameters of the material required for our study.
( )
1
true engineering engineering
σ=σ
ε
×+
(3)

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( )
1
true engineering
= ln
εε
+
(4)
where σtrue is the true stress, σengineering is the engineering stress, εtrue is the true strain, and εengineering is the engineering
strain.
Figure 2. True stress-strain curve of the API X52 material.
Table 1. Mechanical properties of API X52 material.
Young’s Modulus, E Poisson’s ratio, υ Yield strength, σE
Ultimate tensile
strength, UTS
True stress of
failure, σF
200 GPa
0.3
410 MPa
498 MPa
579 MPa
A spherical indenter with a diameter of 50 mm and a rectangular indenter with a base of 100.34×19.28 mm are used
to simulate the indentations. The rectangular indenter is about the size of an excavating bucket's tooth. The dimensions
of the spherical indenter allow the simulation of a stone being pushed up against the wall of a pipeline.
The longitudinal and transverse indentations are produced by altering the orientation of the rectangular indenter. The
indenters are assumed to be infinitely rigid and to be in frictionless contact with the pipe's outer surface. To reduce
simulation computation time, the numerical model is reduced to one-quarter of the pipe and indenter. Figure 3 shows the
boundary conditions employed in this FEM.
Figure 3. Boundary conditions of the finite element model of a dented pipe.
The mesh of the finite element model is refined to 5 mm at the area to be indented and exhibits a transition on the size
ratio of 10 in the longitudinal direction from the middle of the pipe to its end. The mesh is created using 8 hexahedron
nodes with 6 degrees of freedom. The pipe and indenter mesh is made up of 22176 nodes and 4484 elements.
An internal pressure cycle is applied to the inner surface of the dented pipe model. This allows us to utilize Eq. (5) to
calculate the stress concentration factor by measuring the maximum Von Mises stress surrounding the dent numerically.
The research in [17] covers details on the model's development as well as the validation study.
MAX
NOM
σ
SCF = σ
(5)

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i
NOM
pD
σ=
2t
(6)
where σMAX is the maximum equivalent stress measured by finite element analysis, σNOM is the nominal stress, and p is
the internal pressure of the pipe.
The results of this finite element model are utilized to develop an analytical model for computing the SCF. This
analytical model is then employed in a numerical model for determining the accumulation of damage caused by the
presence of a dent in a pipe subjected to variable amplitude loading.
Crack Defects
Weld and crack flaws may be classified as design, manufacturing, or construction defects according to the
abnormalities discovered during the inspection. The faults seen in welded joints are commonly classified as semi-elliptical
cracks [6], such as those caused by a lack of weld material penetration or a defect in the shape of the weld bead. Having
stated that, mechanical interference is responsible for the vast majority of cracks [5]. Figure 4 depicts the parameters of a
semi-elliptical crack.
Figure 4. Surface crack parameters.
Surface cracks manifest as a local separation of two surfaces known as lips; various investigations have been
undertaken to establish the root causes of this type of defect [31]. According to those studies, cracks caused by the welding
process may appear on the weld metal (WM) or in the heat-affected zone (HAZ). Cracks can arise in either of these two
regions or in the pipe's base material (BM) as a result of external or internal mechanical interference. This type of defect
can also form during solidification or after a prolonged welding operation [31]. In fact, microcracks begin to appear in
the material and propagate at an increasing rate. Failure occurs when the residual thickness of the wall near the crack no
longer allows the structure to withstand the applied stress. This failure occurs only when a sufficient number of load
cycles have been applied to expand the initial crack.
Understanding fatigue failure requires an understanding of fracture mechanics theory. Fracture mechanics predicts
the macroscopic mechanical failure of structures by applying theories of elasticity and plasticity to microscopic flaws
observed in materials [32]. There are two major branches of fracture mechanics [32]; elastic-plastic fracture mechanics
(EPFM) and linear elastic fracture mechanics (LEFM). The LEFM is a method for studying cracked solids that assumes
the material is linear elastic and isotropic. Because it is the foundation upon which the other fracture mechanics theories
are established, the LEFM represents the majority of practical uses of fracture mechanics. The LEFM employs two
approaches; energy calculations and stress intensity calculations. The stress intensity approach is the most often utilized
since it exploits the stress and strain distributions surrounding the crack directly, which is typical in industrial applications.
The range of stress intensity parameters derived as a function of crack geometry and structure loading, according to
LEFM, unequivocally characterizes crack propagation [33].
The literature contains detailed descriptions of structures subjected to constant amplitude loads. However, studies on
fracture damage induced by variable amplitude that have been published in the literature still require additional
development since they are more complex and depend on the load history [32]. Paris et al. [34] assumed that crack
propagation under varying loads would follow the law, commonly referred to as the Paris’ law. This is the first relation
that describes the behavior of crack growth. Several studies [16, 33, 35, 36] later improved on this rule by incorporating
correction coefficients to account for other aspects such as the geometry of the structure, the position and shape of the
surface crack, or the crack closure effect.
Elber [35] discovered the phenomena of crack closing during experimental tests. This phenomenon happens during a
load cycle involving two loading values (i.e., a minimum value of stress cycle, KMIN, and a maximum value of stress
cycle, KMAX), when the crack begins to close prior to the stress intensity factor reaching KMIN as a result of contact between
the crack surfaces. The crack appears to cease closing at a magnitude of the stress intensity known as KOP, which it opens
at. Elber [35] proposes and incorporates an effective stress intensity range, ΔKeff, into the Paris law. This factor is based
on the assumption that the process phase below KOP does not contribute to the growth of fatigue cracks owing that to the
closure effect. This crack closure effect slows the spread of fatigue cracks.
A modified Paris model is utilized in this study to predict the number of cycles to fatigue, N, in the case of a cracked
pipe. This can be stated in the form of Eq. (7), taking into consideration the effect of crack closure and some
rearrangement.
2
0.25 0.5 0.25
m
mMIN
eff MAX MIN
MAX
K
da CK C K K
dN K

=∆= + +



(7)

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where C and m are the material constants of Paris’ law.
In a prior study, the authors [19] developed and validated a finite element model of a cracked pipe. Using the ANSYS
calculation code, this model enables to calculate the effective stress intensity factor induced by semi-elliptical cracks of
various orientations (longitudinal and transverse) and positions (external and internal). These cracks can be found in the
three materials that make up a longitudinally welded pipe (base metal (BM), weld metal (WM), and heat-affected zone
material (HAZ).
The stress intensity factor is determined at the crack front. The acquired results are utilized to estimate the number of
cycles that cause the structure to rupture. The longitudinally welded pipe used in the FEM has the same dimensions as
the one used in the dented pipe analysis. Table 2 presents the mechanical parameters of each of the three zones. Figure 5
depicts the numerical model's boundary conditions. Unlike in the dented pipe analysis, the geometry for the FEM of a
cracked pipe was not limited to a quarter of a pipe. This is primarily due to the fact that the semi-elliptical crack is
embedded with an ANSYS functionality that requires the material of the structures to be visualized.
The pipe mesh is refined using a sphere of influence to improve the mesh around the crack, with an element size of
0.6 mm and a diameter of 1.2×c. To enhance the convergence of the circumferential stress, the mesh is refined on the
circumferential zone encompassing the crack. It should be noted that the results of this FEM are utilized to develop an
analytical model for computing the SIF. This analytical model is then employed in a numerical model to calculate the
accumulation of damage caused by a crack in a pipe subjected to varying amplitude loads.
Table 2. Material properties of three zones of the longitudinally welded pipe.
Material
Paris’ law material constants
Critical fracture toughness
C
m
KIC (MPa.m0.5)
Base metal
3.3 e-09
2.74
53.36
HAZ
1.13 e-09
3.25
53.36
Weld metal
1.04 e-09
3.28
61.02
Figure 5. Boundary conditions of the finite element model of a cracked pipe.
ANALYTICAL MODELS
Dent Defects
According to the literature, the stress concentration factor is directly influenced by the geometry of the component,
with no relevance to material qualities [37]. Two non-dimensional characteristics were discovered to be responsible for
the stress concentration generated by the presence of a smooth dent. These parameters are the outer diameter to pipe
thickness ratio, Do/t, and the width of the dent to pipe outer diameter ratio, d/Do.
A mathematical formulation of the stress concentration factor can be developed based on the data obtained from the
developed finite element model. The SCF calculation formula can be written as a power function in terms of pipe and
dent geometry for numerous types of dent defects [38]. This formula is detailed in Eq. (8).
C
B
o
o
Dd
SCF = 1+ A tD






(8)
The coefficients A, B, and C of the SCF equation are adjusted to ensure the best possible match with the numerical
data achieved from the developed finite element model. The interpolation coefficients were derived through nonlinear
fitting of the appropriate numerical data in [20]. Table 3 lists the dimensionless coefficients for each type of dent.
The mathematical equation's results deviated only slightly from those of the finite element model [20]. The stress
concentration factor of the dented pipes can then be calculated using the previously established equation. The purpose of

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developing this equation is to use it in an algorithm to calculate the damage accumulation induced by a pressure cycle on
a dented pipe.
Table 3. Interpolation coefficients used for the calculation of the SCF by type and configuration of the dent [20].
Dent type
Spherical
Rectangular longitudinal
Rectangular transverse
Coefficient
Unconstrained dents
A
1.483
13.667
1.235
B
0.590
0.800
0.984
C
0.541
1.097
0.850
Constrained dents
A
0.426
2.711
0.284
B
0.638
0.893
1.097
C
0.228
0.818
0.644
Crack Defects
Many researchers have been interested in using various approaches to determine the harmfulness of cracking defects
in pressurized structures [6, 11, 32, 33, 36, 39]. This is due to the fact that the rupture or leakage of these structures poses
a serious risk. The approaches based on the stress intensity factor are the most extensively employed, whether in scientific
research or industry. The first models that have emerged deal with the case of an infinitely large plate subjected to uniaxial
loading [40]. Several researchers [41, 42] have agreed that Newman's model provides the best estimates of the stress
intensity factor for the case of a surface crack. Several studies have been devoted to the development of correction factors
based on this same model and in order to generalize it to other cases of structures. The most important are those that take
plate thickness, structure curvature, and the plastic zone at the crack tip into account [33, 43].
In our case, as with the other researchers, there was a need to improve the convergence of this analytical model for
high pressures and to account for the position of the crack in relation to the wall (internal and external). The reason for
this choice is that industrialists do not follow the recommendations emitted by the organisms operating in the field to the
letter and tend to increase the pressures to the recommended limit. As a result, this study attempt to investigate the
structure's behavior when pressures are very high.
To accomplish this goal, a correction coefficient (f1+af2) has been proposed and calibrated based on a previously
developed and validated finite element model of a cracked tube [16]. Equation (9) depicts the modified Raju and Newman
model with multiple correction coefficients. Equations (10) to (14) show the various terms involved in this model. The f1
and f2 factors are tweaked to produce the best possible concordance with numerical findings from the established finite
element model. Table 4 displays the coefficients of the correction factor used in the SIF equation for each crack position.
( ) ( )
2
11 2
1
fs
I TM
K f a k p aM k a=+⋅ +
(9)
where MTM, k1, k2, k3, and s are correction factors defined as :
( )
12
6.4
111
O
TM
i
aac
Mt DD t
t
−

 

=− ×− +



+×



(10)
13
2
i
D
kk t
π
= × ×
(11)
2
3
1
s
tc
kka
−
=×−
(12)
1.65
3
1.13 0.1 1 1.464
aa
kcc
  
= −× +
  
 




(13)
3
2 8 a
sc

= + 

(14)
The resulting analytical model produces findings that are quite close to those of the FEM one and can thus be used to
determine the stress intensity factor to predict the fatigue life of a cracked pipe [16]. This model is later employed in this
study in a numerical model of damage accumulation.

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Table 4: Interpolation coefficients used for the calculation of the SIF by the position of the crack [16].
Coefficient
Crack position
External
Internal
f1
-0.056
0.0447
f2
0.052
0.00188
DAMAGE ACCUMULATION MODEL
Throughout the operation, the internal loading of pipeline transmission lines varies, causing structural fatigue. Thus,
the material of the pipeline undergoes alterations, resulting in fatigue damage [44-45]. As a result, developing models
that explicitly account for fatigue loading variability is critical for researchers [12]. The authors suggested a nonlinear
model for fatigue damage accumulation [19]. To assess the damage induced for each loading cycle, the fatigue life
prediction models employed in the cracking and denting situations are operated separately based on a modified Miner's
law. The algorithm for this numerical model, depicted in Figure 6, is based on the work of Thun et al. [46]. The effects
of load interaction and load history are considered in this model. It is capable of calculating cumulative damage under
variable amplitude loading levels. The leakage or collapse of the structure may be expected when the cumulative damage
reaches a critical value (equivalent to unity in our study).
Figure 6. Nonlinear damage accumulation model algorithm.
In the instance of cracked or dented pipes, this damage accumulation model can be used to determine the nocivity of
the defect. The dent depth, d/Do, and material endurance limit, σD, are employed as supplementary parameters in the dent
case fatigue life prediction model. If the analyzed defect is a crack, the model for calculating fatigue life includes as added
parameters the crack dimension, a/c, the constants of Paris’ law, C and m, and the critical stress intensity factor, KIC. The
pipe, as shown in Figure 7, is assumed to be subjected to variable amplitude internal pressure. The water hammer
phenomenon causes those pressure peaks, which arise as a result of an abrupt shift in the valve state (opening/closing)
that controls the flow of fluid in the pipeline. This trace of pressure history is generated using the method of characteristics
to the transient flow equations in the pipe [16].

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Figure 7. Internal variable amplitude loading of the pipe caused by the water hammer phenomenon.
Only two pressure blocks are depicted in Figure 7. However, this pressure spectrum will be applied to the structure
until the critical value of the damage is reached in order to estimate the number of blocks (number of cycles) until failure.
To simulate the new trends in industrial pressurization, the two values around which the pressure switches are moderate
pressure and high internal pressure. Figure 8 depicts the stress ratio and amplitude ratio as a function of cycle number to
better visualize the loading conditions experienced by the pipeline. It's worth noting that this is only for a single pressure
block.
Figure 8. Stress and amplitude ratios of pipe loading caused by a single pressure block of the water hammer
phenomenon.
PARAMETRIC ANALYSIS
The analysis performed in this paper is divided into different phases. During the first phase, 144 cases were
investigated to determine the effect of water hammer waves on the severity of a semi-elliptical crack present in a metal
pipe. The dent defect is the focus of the second phase, which includes 48 cases. During these two phases, the same
numerical model will be used to estimate the accumulation of damage in the pipe. The calculation formulas used by this
model are tailored to the type of defect (crack or dent). Throughout the crack and dent studies, as well as during the water
hammer analysis, the pipe's material and dimensions are kept the same. The goal of this last analysis is to estimate the
pressure waves induced by the water hammer phenomenon, which is caused by a sudden change in the state of a valve of
a pipe transmission line carrying water. The obtained pressure spectrum presented in Figure 7 is used as input in the
proposed damage accumulation model (Figure 6). The Do/t ratio of the studied pipe is 20.
The dent depth parameter, d/Do, is considered in the dent defect analysis with the following values: 0.025, 0.05, 0.075,
0.1, 0.125, 0.15, 0.175, and 0.2. The dent types are spherical, longitudinal rectangular, and transverse rectangular. The
impact of the dent's configuration (constrained or unconstrained) is also being investigated. The crack defect under
investigation is a semi-elliptical surface crack. During the crack defect assessment, the initial crack depth is assumed to
be 0.5, 1, 1.5 and 2mm. The crack dimensions, a/c, considered are 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. The crack is assumed
to be in the base metal, weld metal, or heat-affected zone. The location of the crack (whether external or internal) is also
investigated.
RESULTS AND DISCUSSION
The presence of several types of defects in a metal pipe is analyzed using the validated damage accumulation model.
The obtained results are depicted in the form of figures. Figure 9 shows the evolution of the accumulation of damage for
a dented pipe subjected to water hammer pressure waves. These curves will lead to assessing the effect of the dent's depth,
shape, orientation, and configuration on the evolution of the accumulation of damage until the structure collapses. The
rupture of the structure is predicted if the damage accumulation is equal to unity. Figures 10 to 13 show the results of the
evolution of damage accumulation in a longitudinally welded pipe with a crack. This pipe is subjected to variable

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amplitude internal pressure, similar to the dented pipes. The main objective of these figures is to examine the effect of
crack size, location, and initial depth on the evolution of accumulation damage up to structural failure.
A common scale is used on the axis of the number of cycles in Figure 10 to 13. This allows for a visual comparison
of the effect of the crack's location on the evolution of the damage. This was not possible for the figures dealing with
dented pipes due to the large differences in the term of the number of fatigue cycles. To improve the readability of those
results, the axes of the number of cycles in fatigue for the spherical and transverse rectangular indentations are on the
same scale, and on a scale with a factor of 10 for the longitudinal rectangular indentations. A comparative study is carried
out with the goal of classifying the various cases studied in terms of their harmfulness. Figure 14 and 15 depict the study’s
findings.
Figure 9 shows the fatigue life values of 634261, 277750, and 52173 for spherical, rectangular transverse, and
rectangular longitudinal dents, respectively, for an unconstrained dent of 2.5% depth. The fatigue life values of 576243,
364870, and 56562 for spherical, transverse rectangular, and longitudinal rectangular indentations, respectively, can be
read from Figure 9 for a constrained dent of similar depth. The fatigue lifetimes for unconstrained dents in a rectangular
shape are less than those for dents of the same shape in a constrained configuration, as can be seen from these results.
The opposite is true for the 2.5% depth spherical dent, but this is the only case in question. This case confirms the
experimental finding by Alexander [23] that for small dent depths and especially for spherical or quasi-spherical shapes,
constrained dents have shorter lifetimes than unconstrained dents of the same size. Having said that, the authors are more
focused in this research on dents with a consequent depth, which pose a greater risk to the structures of fluid transport
under pressure. However, this could be a very interesting avenue for better understanding the mechanisms underlying the
harmfulness of dented defects, and it should be addressed in future studies to improve the numerical model results.
As a result, except for the dents with depths of 2.5%, all of the other unconstrained cases studied have a shorter fatigue
life than the constrained dents with the same depth and shape. As a result, it's deduced that unconstrained dents are more
harmful than constrained dents. This is explained by the fact that, in the case of constrained smooth indentations, the
indenter rests on the exterior surface of the pipe and serves as an extra boundary condition, reinforcing the structure and
reducing the indentation's potential deformations. It should be noted, however, that in the case of extremely high pressure,
the pipe wall will compress at the level of the indentation.
Figure 9. Effect of the dent configuration, type and depth on fatigue accumulation damage.
For a pipe with a dent having a depth, d/Do, of 10%, there is a deviation of -58.9 and -92.02% for the transverse and
longitudinal dent, respectively, compared to the result obtained in terms of fatigue life for an unconstrained spherical
dent. For a constrained dent of the same depth, a deviation of -40.82 and -90.47% is observed with respect to a spherical
dent for the transverse and longitudinal dent, respectively. These deviations show that the effect of the dent's shape and
orientation is very important and should not be overlooked during the assessment of the harmfulness of dent defect in a
pipe. It is also safe to conclude that the rectangular dent is more harmful than the spherical dent and that this harmfulness
is exacerbated by the effect of the orientation when it shifts from transverse to longitudinal. This observation holds true
regardless of the depth or shape of the dent.
To better understand the effect of dent depth on the fatigue resistance of a dented pipe, and to calculate the deviations
between the values of the number of fatigue cycles (corresponding to a value of accumulation of damage, D, of 1), let us
use the value obtained for an immediately greater dent depth as a reference. This is equivalent to comparing the result
obtained for a dent depth of 10% to that obtained for a dent depth of 12.5%.
For the cases of unconstrained spherical indentations, one can notices a relative deviation of -18.83, -12.31, -9.07,
-7.34, -8.88, -7.09 and -6.56% for the dent depths, d/Do, of 5, 7.5, 10, 12.5, 15, 17.5 and 20% respectively. For the dents
of the same shape in the constrained configuration, a deviation of -39.65, -28.4, -21.99, -17.95, -11.98, -11.28 and -9.78%

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is observed for the dent depths of 5, 7.5, 10, 12.5, 15, 17.5 and 20% respectively. Based on these deviations, it can be
confirmed that the greater the dent depth, the more the fatigue life decreases. It is also observed that this increase in depth
has less and less effect on the fatigue strength of the dented pipe. This observation is valid regardless of the shape and
configuration of the dent.
It's worth noting that the first loading cycle causes very negligible damage, but is not equal to zero. It is of the order
of magnitude 10-4 in all cases of dented pipes. This does not apply to the case of cracked pipes. In fact, one can see from
Figure 10 to 13 that the damage caused by the first pressure cycle of 0.036, 0.073, 0.109, 0.146 and 0.182, which is caused
by the presence of a crack of an initial depth of 0.5, 1, 1.5, 2 and 2.5 mm. These damage values correspond to the initial
depth of the crack divided by the thickness of the pipe. For example, the value of damage of the first cycle for a pipe with
an initial crack of 0.5 is of (0.5 mm /13.7 mm) which is equal to 0.036. This is due to the fact that the damage estimation
model takes the maximum value between the ratios a/t, and KI/KIC for each pressure cycle. This reveals that for all of the
cases of cracked pipes studied in this paper and for the first peak of the pressure spectrum, we obtain a value of aini/t,
greater than KI/KIC.
Consider the case shown in Figure 10 of a pipe with a 0.5 mm initial depth crack located at the HAZ. When the result,
in terms of the number of cycle to fatigue, obtained for each dimension of crack, a/c, is compared to the result obtained
for the immediately lower dimension, a variation of 177, 71.98, 56.6, 42.26, and 38.76% is found for the external cracks
of dimensions 0.5, 0.6, 0.7, 0.8, and 0.9, respectively. Results obtained for pipes with internal cracks with similar locations
and dimensions have a deviation of 176.6, 71.87, 56.51, 42.2 and 38.72%, respectively, for cracks with dimensions 0.5,
0.6, 0.7, 0.8, and 0.9. It is clear from these deviations that the effect of the crack dimension on fatigue life decreases as
a/c increases. It is also noted that the deviations obtained for internal cracks are nearly identical to those obtained for
external cracks. This means that regardless of the position of the crack, the effect of the parameter a/c on the harmfulness
of the crack defect is the same. This observation holds true regardless of the localization of the crack (BM, WM, or HAZ)
or the initial depth of the crack. It can also be concluded that the crack defect is more harmful at low a/c values.
Figure 10. Effect of surface crack with an initial crack depth of 0.5mm on fatigue accumulation damage.
From Figure 11, let us consider the case of a crack with an initial depth of 1 mm and dimensions a/c = 0.6, and use
the values in terms of life in fatigue obtained for pipes with external surface cracks as a reference for calculating the
deviations. There is a difference in the results for the external and internal cracks of -21.33, -23.67, and 28.21% for cracks
located at the base metal, weld metal, and heat-affected-zone. On the one hand, these deviations indicate that internal
cracks are more harmful than external cracks of the same dimensions. On the other hand, the effect of the crack's location
becomes even more pronounced when the crack is located in the HAZ. This is true regardless of the initial depth and a/c
dimensions of the semi-elliptical surface crack.
Figure 11 shows that for cracks a/c = 0. 6 with an initial depth of 1 mm, the fatigue life of -39.56 and -34.29% decreases
when external cracks located in WM and HAZ are compared to the results obtained for a localized crack in the BM. When
comparing the internal cracks in the WM and HAZ to the cracks in the BM, one notices a deviation of -41.35 and -40.04%.
Based on these deviations, it can be concluded that the crack is more harmful when it is located in the WM. It should also
be noted that the consequence of a crack defect located in the WM or HAZ is very similar, especially when compared to
the same crack located in the BM.
The microstructure of the weld at the joint is influenced by the welding technique, bead design, duration of weld
passes, heat input, and other parameters. This, in consequence, influences the extent of the HAZ and residual stresses that
accumulate in the structure’s base metal. These factors impair fatigue strength by increasing the likelihood of crack
nucleation and early development, which will lead to pipeline collapse. Thus, the microstructural alterations have a
significant impact on the fatigue crack development capabilities of each material that makes up the longitudinally welded
pipe. According to the conclusions of this and previous investigations, the HAZ is the most impacted zone.

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Figure 11. Effect of surface crack with an initial crack depth of 1 mm on fatigue accumulation damage.
Figure 12. Effect of surface crack with an initial crack depth of 1.5 mm on fatigue accumulation damage.
Using Figure 10 to 13, the fatigue life obtained for each internal crack with a similar crack of the same dimension
a/c = 0.9 is compared to that with an initial depth immediately below. For cracks in the BM, there is a relative difference
of -24, -20.13, and -16.77% for cracks with initial depths of 1, 1.5, and 2mm, respectively. For cracks in the WM, the
deviation is -26.91, -20.97, and -18.96% for cracks with initial depths of 1, 1.5, and 2 mm, respectively. When it comes
to cracks in the HAZ, there is a relative variation of -26.47, -17.88, and -19.16% for cracks at initial depths of 1, 1.5, and
2 mm, respectively. Based on these deviations, it can be seen that increasing the initial crack depth parameter has less of
an effect on reducing the fatigue life of the defective structure. The shape of the curves also shows that each curve has a
shift to the left when compared to the curve corresponding to a crack of the same dimensions, position, and location but
with an initial depth that is immediately lower.
Let us compare the shape of the curves of evolution of damage accumulation for dented pipes (Figure 9) to those for
cracked pipes (in Figure 10 to 13). The accumulation of damage is clearly slower in the case of dented pipes subjected to
variable amplitude loads. The shape of the evolution of damage accumulation curves increases exponentially with the
number of cycles for all types of defects. When compared to the curves obtained for dented pipes, this exponential effect
is more pronounced in the case of a cracked pipe. For low damage accumulation values, the slope of the curves obtained
for crack defects and that for dent defects is quite close. For high damage accumulation values, however, it is observed
that the rate of increase of the slope in the case of a dented pipe remains quasi-constant, whereas the slope in the case of
a cracked pipe tends towards infinity. Figure 14 and 15 are used to compare and classify the various defects studied based
on their degree of harmfulness in the various stages of damage accumulation evolution.

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Figure 13. Effect of surface crack with an initial crack depth of 2mm on fatigue accumulation damage.
Figure 14. The number of cycles associated with fatigue accumulation damage of 0.25 and 0.5 of the defects
investigated.
Based on the findings depicted in Figure 8 to 13, four points were kept for each curve. These points are the
intersections of horizontal lines representing damage accumulation values of 0.25, 0.5, 0.75, and 1 with the damage
accumulation evolution curve for each type of defect studied. By projecting these points orthogonally on the abscissa
axis, the number of cycles corresponding to damage values of 0.25, 0.5, 0.75, and 1 is obtained. All of these values are
depicted in Figures 14 and 15, with the cases of a defect causing the fewest number of fatigue cycles appearing first. It
should be noted that the y-axis in these figures is on a logarithmic scale to improve readability. For crack defects, the
values of the a/c dimensions are read from the bottom up, from 0.4 to 0.9 with a 0.1 step. For dent defects, the values of
the d/Do dimensions are read from top to bottom, from 2.5% to 20% with a 2.5% step.
These figures not only allow for the classification of the studied defects, but they also allow for the identification of
equivalence between these defects, even for different types, and this for several stages of the structure's life. This is made
possible by using the parameter of damage accumulation as a common adimensional scale. Without this parameter, it is
remarkably difficult to directly compare the harmfulness of various types of defects (cracks and dents).
To demonstrate how these figures can be used, consider the defects framed in Figure 14 and 15. We can conclude
from a simple reader that an internal crack defect with an initial depth of 1.5 mm located in the WM is equivalent in
harmfulness to an external crack defect with an initial depth of 1mm located in the same zone. This observation holds
true for all four stages of life. To find the equivalence in terms of harmfulness for all the studied defects, draw a horizontal
line passing through the defect for which the equivalences are sought. If the value of the a/c or d/Do parameter does not
appear in the figure, linear interpolation can be used to approximate it.

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Figure 15. The number of cycles associated with fatigue accumulation damage of 0.75 and 1 of the defects
investigated.
Let us use Figure 15 as an example to find the defects equivalent to an unconstrained transverse rectangular indentation
of depth d/Do = 5 %. Draw a horizontal line through the defect (see Figure 15) and examine the intersections of this line
with the curves of the various defects. Let us look at some equivalence examples:
i. A constrained transverse rectangular dent defect with a depth of d/Do = 17.5 %;
ii. An unconstrained spherical dent with a depth d/Do of 14.375 % (approximated by linear interpolation);
iii. An internal crack defect with a/c dimension of 0.9 in the HAZ with an initial depth of 1 mm;
iv. An internal crack defect with a/c dimension of 0.9 in the BM with an initial depth of 1.5 mm;
v. An external crack defect with a/c dimension of 0.89 (obtained through linear interpolation) in the WM with an
initial depth of 0.5 mm.
Figure 14 and 15 show that the unconstrained longitudinal dent defect is one of the most harmful defects in the first
phase of the pipe's life. The greater the number of cycles, the less harmful this defect is. In order to better follow the
evolution of the harmfulness of this defect, an arrow has been drawn in these figures. It should be noted that the value
ranges used by all of the parameters studied are those found in the literature and the field. Figure 14 and 15 show that
dent defects are generally less harmful than crack defects, especially near the end of pipe life.
CONCLUSION
In a parametric investigation, a numerical damage accumulation model is utilized to evaluate the nocivity due to the
presence of a crack or dent defect in a pipe. This model takes into account the effects of loading history and interaction
effect. The pressure spectrum induced by the water hammer, which is caused by an abrupt change in the status of a valve
on a water transmission line, is fed into this model. It employs, in its damage calculation process, analytical models
derived from studies on interpolation of FEM model findings.
The main purpose of this study is to compare and classify flaws in diverse categories (surface cracks and dents). In
the parametric analysis concerning dent defects, the effect of the dent's depth, shape (spherical and rectangular),
orientation (longitudinal and transverse), and configuration (constrained and unconstrained) is investigated. The
geometry, initial depth, location (external and internal), and position (in the base metal, weld metal, and heat-affected
zone) effects on surface crack defects are explored in a parametric study involving cracked pipes.
This research not only allows for the classification of the flaws under investigation but also for the identification of
equivalence between them in terms of pipeline damage, even for various categories and at different stages of the structure's
life. This is primarily due to the damage accumulation parameter being used as a common adimensional scale. The
established numerical model can be used to analyze the harmfulness of various flaws in the pipes with minimal alterations.
The same approach can be used to evaluate the impact of other types of structural loading. As a result, equivalences
between flaws throughout a wider spectrum can be found.
ACKNOWLEDGEMENT
The authors would like to acknowledge the facilities, scientific and technical support from Signals, Distributed
Systems and Artificial Intelligence Laboratory in Higher Normal School of Technical Education of Mohammedia.

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