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Citation: Baji´c, D. Toughness
Properties of a 50-Year-Old Pipeline
Material. Sustainability 2023,15, 5143.
https://doi.org/10.3390/su15065143
Academic Editor: Tao Zhou
Received: 9 February 2023
Revised: 28 February 2023
Accepted: 6 March 2023
Published: 14 March 2023
Copyright: © 2023 by the author.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
sustainability
Article
Toughness Properties of a 50-Year-Old Pipeline Material
Darko Baji´c
Faculty of Mechanical Engineering, University of Montenegro, Cetinjski put bb, 81000 Podgorica, Montenegro;
darko@ucg.ac.me
Abstract:
To ensure the reliable functioning of the main pipeline for water transport in hydropower,
which has been operational for 50 years, a detailed analysis of the pipeline’s material condition is
necessary. The pipeline material undergoes changing exploitation and working conditions, including
extreme load phases such as periodic intensive emptying and filling with water for inspection. In
this paper, we analyse the material resistance of NIOVAL47 steel using the method for determining
R-curves for three states: Normalised state (2 samples), aged state (1 sample) which underwent
10% cold deformation and was heated for 30 min at +250
◦
C, and deformed state (1 sample) which
underwent 10% cold deformation. The test results indicate that analysing the material’s aged and
deformed states is crucial for obtaining a reliable picture of the structural integrity of the NO2500
mm pipeline, which has been in operation for five decades. The study should demonstrate the
current state and level of reliability of the pipeline in order to ensure the sustainability of the energy
facility. Additionally, these tests provide a realistic picture of the necessity of introducing an online
monitoring system for the pipeline.
Keywords:
pipes; material state; mechanical properties; R-curve; fracture toughness; semi-elliptical
crack; stress intensity factor
1. Introduction
The reliability of pipe systems is of vital importance for the functioning of electricity-
producing power plants [
1
–
5
]. Periodic inspections which use NDT (Non-Destructive
Testing) methods control the state of critical places [
6
–
8
]. These places are welded joints,
arcs, reductions, and places where pipe fittings are installed.
The analysed structure is a pipeline ND2500
×
32 mm with an internal working
pressure of 50.9 bar. It is an energy facility within the HPP Peru´cica, Montenegro. The
pipeline has been in operation and has been exploited for 50 years.
The parameter that defines the reliability of the structure is the critical value of frac-
ture toughness [9–12]. Determination of one of the parameters (KI—stress intensity factor,
J—integral or critical crack opening, and CTOD (Crack Tip Opening Displacement)) is pre-
ceded by defining the mechanical tensile characteristics of NIOVAL47 steel. The experiment
must be performed in accordance with the ASTM 1820 standard [13].
2. Materials and Methods
The material used for the construction of the pipe is NIOVAL47 steel, manufactured
by SIJ—Slovenian Steel Group (ex-Steelworks Jesenice, Slovenia).
Three states of NIOVAL47 steel were analysed:
-
State A: The normalised state of the material before its deformation and pipe making
(undeformed sheets).
-
State B: The aged and deformed state, which implies 10% cold deformation and heated
for 30 min at +250 ◦C.
-
State C: The deformed state of the material which is in the range of 10% cold deformation.
Sustainability 2023,15, 5143. https://doi.org/10.3390/su15065143 https://www.mdpi.com/journal/sustainability
Sustainability 2023,15, 5143 2 of 16
The mechanical properties of this material are given in Table 1.
Table 1. Mean values of the tensile characteristics of the material NIOVAL47 at room temperature.
Material E, GPa Rp0.2, MPa Rm, MPa Em, % Ef, % σ0, MPa N
State A 196 442 610 13.7 27 27 27
State B 193 647 726 4.7 12 12 12
State C 192 627 684 4.5 11 11 11
3. Results Analysis and Discussion
3.1. Tensile Testing
Tensile test specimens were obtained from rolled sheets of sheet metal to determine the
mechanical properties of the material perpendicular to the rolling direction. This direction
is typically less favourable in terms of mechanical properties. The test was performed
on a sample that represents the least favourable direction. Additionally, the sampling
method was chosen to characterise the properties of the material when the pipeline opens
(breaks) longitudinally.
The test specimens (Figure 1) were taken from 20 mm thick sheet metal plates (the
basic delivered state of the material) and 18 mm thick sheet metal plates (in the deformed
and aged state of the material). The tensile test was conducted according to the standard
that specifies a test specimen needs to have a 5 mm diameter.
Sustainability 2023, 15, x FOR PEER REVIEW 2 of 16
The mechanical properties of this material are given in Table 1.
Table 1. Mean values of the tensile characteristics of the material NIOVAL47 at room temperature.
Material E, GPa Rp0,2, MPa Rm, MPa Em, % Ef, % σ0, MPa N
State A 196 442 610 13.7 27 27 27
State B 193 647 726 4.7 12 12 12
State C 192 627 684 4.5 11 11 11
3. Results Analysis and Discussion
3.1. Tensile Testing
Tensile test specimens were obtained from rolled sheets of sheet metal to determine
the mechanical properties of the material perpendicular to the rolling direction. This
direction is typically less favourable in terms of mechanical properties. The test was
performed on a sample that represents the least favourable direction. Additionally, the
sampling method was chosen to characterise the properties of the material when the
pipeline opens (breaks) longitudinally.
The test specimens (Figure 1) were taken from 20 mm thick sheet metal plates (the
basic delivered state of the material) and 18 mm thick sheet metal plates (in the deformed
and aged state of the material). The tensile test was conducted according to the standard
that specifies a test specimen needs to have a 5 mm diameter.
Figure 1. A tension test specimen with a 5 mm diameter.
The tests were performed at the temperature of +22 °C, using the servo-hydraulic
testing machine INSTRON 1255 with the 250 kN capacity, with a constant speed of the
cylinder piston at 1 mm/min. Based on the data for the limit of proportionality σ0 and the
deformation strengthening exponent N (Table 1), we obtained curves of mean values that
characterise the correlation between true stress and true strain σ‒ε. The tension curves are
shown in Figure 2. For the numerical calculation, it is necessary to take into account the
material input data corresponding to the state of material B (aged and deformed state).
Since the material NIOVAL47 was already plastically deformed during the construction
of this pipeline in 1974, due to the technical operation of rolling, the mechanical properties
of the material in the certificate of the supplier SŽ Jesenice no longer correspond to the
actual state of the material of the structure in question. As can be seen (Table 1 and Figure
1), the value of the yield strength of the material in the deformed and aged state (state B)
is significantly higher than the value of the yield point of the material in the delivered
normalised state (state A), which has traditionally been taken as the initial state for
numerical calculation.
Figure 1. A tension test specimen with a 5 mm diameter.
The tests were performed at the temperature of +22
◦
C, using the servo-hydraulic
testing machine INSTRON 1255 with the 250 kN capacity, with a constant speed of the
cylinder piston at 1 mm/min. Based on the data for the limit of proportionality
σ0
and
the deformation strengthening exponent N(Table 1), we obtained curves of mean values
that characterise the correlation between true stress and true strain
σ
–
ε
. The tension curves
are shown in Figure 2. For the numerical calculation, it is necessary to take into account
the material input data corresponding to the state of material B (aged and deformed state).
Since the material NIOVAL47 was already plastically deformed during the construction of
this pipeline in 1974, due to the technical operation of rolling, the mechanical properties
of the material in the certificate of the supplier SŽ Jesenice no longer correspond to the
actual state of the material of the structure in question. As can be seen (Table 1and
Figure 1), the value of the yield strength of the material in the deformed and aged state
(state B) is significantly higher than the value of the yield point of the material in the
delivered normalised state (state A), which has traditionally been taken as the initial state
for numerical calculation.
Sustainability 2023,15, 5143 3 of 16
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Figure 2. Tensile curves for the three material states.
3.2. Examination of Fracture Mechanics
Experimental determination of fracture toughness was performed using compact
tension specimens (CT) in accordance with the ASTM 1820 standard. Two test specimens
were tested against the delivered state (state A), one for the aged and deformed state (state
B) and one for the deformed state, respectively (state C). The test specimens were sampled
from sheet metal plates with a thickness of 20 mm (as the new delivered state of the
material) and from metal sheet plates with a thickness of 18 mm (in the deformed and
aged state of the material). The required 10% deformation was achieved by cold rolling
from a thickness of 20 mm to 18 mm. As all plates were uneven, and most of them were
deformed (bending) after the thickness reduction from 20 mm to 18 mm, it was necessary
to straighten them. With this technological procedure, their thickness was reduced by
another 2 mm, i.e., by another 10%. The test specimens are made of 200 × 500 mm plates
(length × width) with a mechanical notch in the direction of rolling. In this way, states are
created for determining the fracture toughness of the material, which is relevant to the
opening of longitudinal cracks. Furthermore, the fracture toughness values for the
sampled test specimens with a mechanical notch perpendicular to the rolling direction are
higher than the equivalent ones measured in the sheet rolling direction [14]. In this way,
the state of conservatism is met, because the measured fracture toughness values with a
notch perpendicular to the rolling direction will be higher than the values with a notch in
the rolling direction. The nominal dimensions of the test specimen are shown in Figure 3.
Before starting the fracture toughness test, the test specimen was fatigued in order to
achieve the required crack length with the smallest possible plastic zone at the
microstructural level [13,15,16]. Material fatiguing was performed on the servo-hydraulic
testing machine INSTRON 8500+, as shown in Figure 4. In accordance with the standard,
the length of the fatigue crack must be greater than 5% of the entire length of the crack a0,
i.e., greater than 1.5 mm.
Figure 3. Shape and dimensions of compact tension specimens (CT specimen) for determining
fracture toughness (CT specimen thickness B = 16 mm (states B and C) and B = 18 mm (state A), CT
specimen width W = 62.44 mm).
Figure 2. Tensile curves for the three material states.
3.2. Examination of Fracture Mechanics
Experimental determination of fracture toughness was performed using compact
tension specimens (CT) in accordance with the ASTM 1820 standard. Two test specimens
were tested against the delivered state (state A), one for the aged and deformed state
(state B) and one for the deformed state, respectively (state C). The test specimens were
sampled from sheet metal plates with a thickness of 20 mm (as the new delivered state of
the material) and from metal sheet plates with a thickness of 18 mm (in the deformed and
aged state of the material). The required 10% deformation was achieved by cold rolling
from a thickness of 20 mm to 18 mm. As all plates were uneven, and most of them were
deformed (bending) after the thickness reduction from 20 mm to 18 mm, it was necessary
to straighten them. With this technological procedure, their thickness was reduced by
another 2 mm, i.e., by another 10%. The test specimens are made of 200
×
500 mm plates
(length
×
width) with a mechanical notch in the direction of rolling. In this way, states
are created for determining the fracture toughness of the material, which is relevant to
the opening of longitudinal cracks. Furthermore, the fracture toughness values for the
sampled test specimens with a mechanical notch perpendicular to the rolling direction
are higher than the equivalent ones measured in the sheet rolling direction [
14
]. In this
way, the state of conservatism is met, because the measured fracture toughness values
with a notch perpendicular to the rolling direction will be higher than the values with a
notch in the rolling direction. The nominal dimensions of the test specimen are shown
in Figure 3. Before starting the fracture toughness test, the test specimen was fatigued in
order to achieve the required crack length with the smallest possible plastic zone at the
microstructural level [13,15,16]. Material fatiguing was performed on the servo-hydraulic
testing machine INSTRON 8500+, as shown in Figure 4. In accordance with the standard,
the length of the fatigue crack must be greater than 5% of the entire length of the crack a
0
,
i.e., greater than 1.5 mm.
Sustainability 2023, 15, x FOR PEER REVIEW 3 of 16
Figure 2. Tensile curves for the three material states.
3.2. Examination of Fracture Mechanics
Experimental determination of fracture toughness was performed using compact
tension specimens (CT) in accordance with the ASTM 1820 standard. Two test specimens
were tested against the delivered state (state A), one for the aged and deformed state (state
B) and one for the deformed state, respectively (state C). The test specimens were sampled
from sheet metal plates with a thickness of 20 mm (as the new delivered state of the
material) and from metal sheet plates with a thickness of 18 mm (in the deformed and
aged state of the material). The required 10% deformation was achieved by cold rolling
from a thickness of 20 mm to 18 mm. As all plates were uneven, and most of them were
deformed (bending) after the thickness reduction from 20 mm to 18 mm, it was necessary
to straighten them. With this technological procedure, their thickness was reduced by
another 2 mm, i.e., by another 10%. The test specimens are made of 200 × 500 mm plates
(length × width) with a mechanical notch in the direction of rolling. In this way, states are
created for determining the fracture toughness of the material, which is relevant to the
opening of longitudinal cracks. Furthermore, the fracture toughness values for the
sampled test specimens with a mechanical notch perpendicular to the rolling direction are
higher than the equivalent ones measured in the sheet rolling direction [14]. In this way,
the state of conservatism is met, because the measured fracture toughness values with a
notch perpendicular to the rolling direction will be higher than the values with a notch in
the rolling direction. The nominal dimensions of the test specimen are shown in Figure 3.
Before starting the fracture toughness test, the test specimen was fatigued in order to
achieve the required crack length with the smallest possible plastic zone at the
microstructural level [13,15,16]. Material fatiguing was performed on the servo-hydraulic
testing machine INSTRON 8500+, as shown in Figure 4. In accordance with the standard,
the length of the fatigue crack must be greater than 5% of the entire length of the crack a0,
i.e., greater than 1.5 mm.
Figure 3. Shape and dimensions of compact tension specimens (CT specimen) for determining
fracture toughness (CT specimen thickness B = 16 mm (states B and C) and B = 18 mm (state A), CT
specimen width W = 62.44 mm).
Figure 3.
Shape and dimensions of compact tension specimens (CT specimen) for determining
fracture toughness (CT specimen thickness B= 16 mm (states B and C) and B= 18 mm (state A), CT
specimen width W= 62.44 mm).
Sustainability 2023,15, 5143 4 of 16
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(a) (b)
Figure 4. Fatigue (a) and testing (b) of the compact tension specimen on the testing machine
INSTRON 8500+.
An additional requirement of the standard is that the length of the crack meets the
state, i.e., that the ratio of the length of the crack a and the width of the compact tension
specimen W is within 0.45 ≤ a/W ≤ 0.6. Determination of fracture toughness was carried
out at a constant displacement speed of v = 1 mm/min. The test was controlled using the
testing machine INSTRON 8500+ management console and the “INSTRON View-maker”
software. During the experiment, we monitored the dependence of force—displacement
(F—Δ) on the point of impact of the force as well as force—Crack Mouth Opening
Displacement (F—CMOD). Figure 5 shows the F—CMOD curves for four compact tension
specimens (3 material states). The diagram shows that, with the compact tension
specimens in state A, there is a blunting of the crack and plastic deformation with the force
increasing to its maximum value only for CMOD values between 3 and 4 mm. In the case
of compact tension specimens for the material in state B or state C, the maximum force
values were reached for a CMOD value of up to 1 mm, followed by a stable crack growth
with a drop in the force value. They quickly reached maximum force and the decreasing
curve shows that the materials in states B and C have a lower resistance to stable crack
growth, even though these materials do not experience a brittle fracture [17].
Figure 5. Curves force—Crack Mouth Opening Displacement (F—CMOD).
After the completion of the first phase of the fatigue test, the specimens were again
subjected to dynamic loading to determine the length of the stable crack growth. In the
case of broken specimens, the initial and final average lengths of the crack were measured.
These data are necessary to determine the fracture mechanics parameters pertaining to
the test specimens. The normalisation technique [18–20] was used to determine the
Figure 4.
Fatigue (
a
) and testing (
b
) of the compact tension specimen on the testing machine
INSTRON 8500+.
An additional requirement of the standard is that the length of the crack meets the state,
i.e., that the ratio of the length of the crack aand the width of the compact tension specimen
Wis within 0.45
≤
a/W
≤
0.6. Determination of fracture toughness was carried out at a
constant displacement speed of v= 1 mm/min. The test was controlled using the testing
machine INSTRON 8500+ management console and the “INSTRON View-maker” software.
During the experiment, we monitored the dependence of force—displacement (F—
∆
) on
the point of impact of the force as well as force—Crack Mouth Opening Displacement
(F—CMOD). Figure 5shows the F—CMOD curves for four compact tension specimens
(3 material states). The diagram shows that, with the compact tension specimens in state
A, there is a blunting of the crack and plastic deformation with the force increasing to
its maximum value only for CMOD values between 3 and 4 mm. In the case of compact
tension specimens for the material in state B or state C, the maximum force values were
reached for a CMOD value of up to 1 mm, followed by a stable crack growth with a drop in
the force value. They quickly reached maximum force and the decreasing curve shows that
the materials in states B and C have a lower resistance to stable crack growth, even though
these materials do not experience a brittle fracture [17].
Sustainability 2023, 15, x FOR PEER REVIEW 4 of 16
(a) (b)
Figure 4. Fatigue (a) and testing (b) of the compact tension specimen on the testing machine
INSTRON 8500+.
An additional requirement of the standard is that the length of the crack meets the
state, i.e., that the ratio of the length of the crack a and the width of the compact tension
specimen W is within 0.45 ≤ a/W ≤ 0.6. Determination of fracture toughness was carried
out at a constant displacement speed of v = 1 mm/min. The test was controlled using the
testing machine INSTRON 8500+ management console and the “INSTRON View-maker”
software. During the experiment, we monitored the dependence of force—displacement
(F—Δ) on the point of impact of the force as well as force—Crack Mouth Opening
Displacement (F—CMOD). Figure 5 shows the F—CMOD curves for four compact tension
specimens (3 material states). The diagram shows that, with the compact tension
specimens in state A, there is a blunting of the crack and plastic deformation with the force
increasing to its maximum value only for CMOD values between 3 and 4 mm. In the case
of compact tension specimens for the material in state B or state C, the maximum force
values were reached for a CMOD value of up to 1 mm, followed by a stable crack growth
with a drop in the force value. They quickly reached maximum force and the decreasing
curve shows that the materials in states B and C have a lower resistance to stable crack
growth, even though these materials do not experience a brittle fracture [17].
Figure 5. Curves force—Crack Mouth Opening Displacement (F—CMOD).
After the completion of the first phase of the fatigue test, the specimens were again
subjected to dynamic loading to determine the length of the stable crack growth. In the
case of broken specimens, the initial and final average lengths of the crack were measured.
These data are necessary to determine the fracture mechanics parameters pertaining to
the test specimens. The normalisation technique [18–20] was used to determine the
Figure 5. Curves force—Crack Mouth Opening Displacement (F—CMOD).
After the completion of the first phase of the fatigue test, the specimens were again
subjected to dynamic loading to determine the length of the stable crack growth. In the
case of broken specimens, the initial and final average lengths of the crack were measured.
These data are necessary to determine the fracture mechanics parameters pertaining to the
test specimens. The normalisation technique [
18
–
20
] was used to determine the fracture
Sustainability 2023,15, 5143 5 of 16
toughness. For each tested specimen, J-R curves of material resistance with the correspond-
ing final length of the stable crack growth were formed. The test results were evaluated in
accordance with the ASTM E-1820-05a standard [
21
] and are shown graphically in Figure 6
for all four tested specimens. The relevant values are given in Table 2.
Sustainability 2023, 15, x FOR PEER REVIEW 5 of 16
fracture toughness. For each tested specimen, J-R curves of material resistance with the
corresponding final length of the stable crack growth were formed. The test results were
evaluated in accordance with the ASTM E-1820-05a standard [21] and are shown
graphically in Figure 6 for all four tested specimens. The relevant values are given in Table
2.
Figure 6. J-R resistance curves for four test specimens.
Table 2. Test results for determining fracture mechanics parameters.
Parameter Label A1 A2 B C
Initial crack length a0
,
mm 25.052 28.089 27.797 29.766
The ratio of the length of the
crack to the width of the com-
pact tension specimen
a0/W 0.561 0.562 0.556 0.595
Stable increase in crack length Δastab, mm 3.495 4.020 4.412 4.525
Crack opening 0.2BL * J0.2,BL, N/mm 298 305 250 219
Crack length increment at
0.2BL
Δa0.2BL, mm 0.570 0.540 0.307 0.336
Maximum force Fmax, kN 39.1 41.9 42.9 41.2
Value at Fmax Jm, N/mm 635 740 340 315
Fracture toughness of material
PSS **
KI,mat, MPa∙m0.5 242 244 220 206
Fracture toughness of material
PSN **
KI,mat, MPa∙m0.5 253 256 230 216
*—BL—Blunting Line. **—fracture toughness of the material at J0.2,BL, for PSS—plane state of stress,
PSN—plane state of strain.
3.3. Calculation of Critical Crack Length
In accordance with the Structural INtegrity Assessment Procedure (SINTAP
procedure) [22,23], a calculation was made for the longitudinal and meridian crack on the
flat part of the pipeline, in several places in front of branch junction 6A (Figure 7), behind
the fork and on the drainage pipeline towards fork 6B. As the numerical calculations using
the software for the finite element method KOMIPS [24] showed that the stresses only
locally reach the value of 288 MPa, at the junction of the elliptical connecting plate and the
distribution pipes of the fork (the main part of the pipeline and drain), then it is
appropriate to estimate the critical crack size at the point of the pipeline where the wall
thickness decreases. The stresses in the pipe walls at those places correspond to the
theoretical stresses due to the internal pressure of the p = 50.9 bar. This approach is
conservative, but with it, we can achieve greater reliability when calculating the integrity
Figure 6. J-Rresistance curves for four test specimens.
Table 2. Test results for determining fracture mechanics parameters.
Parameter Label A1 A2 B C
Initial crack length a0, mm 25.052 28.089 27.797 29.766
The ratio of the length of the crack to
the width of the compact tension
specimen
a0/W0.561 0.562 0.556 0.595
Stable increase in crack length ∆astab, mm 3.495 4.020 4.412 4.525
Crack opening 0.2BL * J0.2,BL, N/mm 298 305 250 219
Crack length increment at 0.2BL ∆a0.2BL, mm 0.570 0.540 0.307 0.336
Maximum force Fmax, kN 39.1 41.9 42.9 41.2
Value at Fmax Jm, N/mm 635 740 340 315
Fracture toughness of material PSS ** KI,mat, MPa·m0.5 242 244 220 206
Fracture toughness of material PSN ** KI,mat , MPa·m0.5 253 256 230 216
*—BL—Blunting Line. **—fracture toughness of the material at J
0.2,BL
, for PSS—plane state of stress, PSN—plane
state of strain.
3.3. Calculation of Critical Crack Length
In accordance with the Structural INtegrity Assessment Procedure (SINTAP proce-
dure) [
22
,
23
], a calculation was made for the longitudinal and meridian crack on the flat
part of the pipeline, in several places in front of branch junction 6A (Figure 7), behind the
fork and on the drainage pipeline towards fork 6B. As the numerical calculations using the
software for the finite element method KOMIPS [
24
] showed that the stresses only locally
reach the value of 288 MPa, at the junction of the elliptical connecting plate and the distri-
bution pipes of the fork (the main part of the pipeline and drain), then it is appropriate to
estimate the critical crack size at the point of the pipeline where the wall thickness decreases.
The stresses in the pipe walls at those places correspond to the theoretical stresses due to
the internal pressure of the p= 50.9 bar. This approach is conservative, but with it, we can
achieve greater reliability when calculating the integrity of the structure. This practically
means that the critical size (length) of the crack will be at least equal to the calculated value,
and may be higher, which is in accordance with the SINTAP procedure approach.
Sustainability 2023,15, 5143 6 of 16
Sustainability 2023, 15, x FOR PEER REVIEW 6 of 16
of the structure. This practically means that the critical size (length) of the crack will be at
least equal to the calculated value, and may be higher, which is in accordance with the
SINTAP procedure approach.
Figure 7. The branch junction 6A for which the calculation of the critical crack size is performed.
Individual places on the pipeline for which the hypothetical critical crack length was
calculated are shown in Figure 8. These are places of reduction to a smaller pipe thickness,
where longitudinal or meridian cracks could occur due to higher stress concentration. The
shape of the internal longitudinal semi-elliptical crack is shown schematically in Figure
9a. In addition to the internal longitudinal semi-elliptical crack, it is necessary to calculate
the critical crack size for the crack through the pipeline wall (semi-elliptical through a
crack). This is an example of the presence of a crack where water could leak rather than
destroy the pipeline. Two cases were analysed: the existence of a longitudinal passing
crack (Figure 9b) and a meridian internal crack (Figure 9c). In the case of an internal semi-
elliptical crack, the change in the length of the crack (2c) and the depth (a) of the crack in
the wall are monitored. In the case of a crack passing through the thickness of the wall,
only its length on the surface (2a) is monitored.
Figure 8. Places on the pipeline at branch junction 6A for which the calculation of the critical crack
size is performed.
Figure 7. The branch junction 6A for which the calculation of the critical crack size is performed.
Individual places on the pipeline for which the hypothetical critical crack length was
calculated are shown in Figure 8. These are places of reduction to a smaller pipe thickness,
where longitudinal or meridian cracks could occur due to higher stress concentration. The
shape of the internal longitudinal semi-elliptical crack is shown schematically in Figure 9a.
In addition to the internal longitudinal semi-elliptical crack, it is necessary to calculate the
critical crack size for the crack through the pipeline wall (semi-elliptical through a crack).
This is an example of the presence of a crack where water could leak rather than destroy the
pipeline. Two cases were analysed: The existence of a longitudinal passing crack (Figure 9b)
and a meridian internal crack (Figure 9c). In the case of an internal semi-elliptical crack,
the change in the length of the crack (2c) and the depth (a) of the crack in the wall are
monitored. In the case of a crack passing through the thickness of the wall, only its length
on the surface (2a) is monitored.
Sustainability 2023, 15, x FOR PEER REVIEW 6 of 16
of the structure. This practically means that the critical size (length) of the crack will be at
least equal to the calculated value, and may be higher, which is in accordance with the
SINTAP procedure approach.
Figure 7. The branch junction 6A for which the calculation of the critical crack size is performed.
Individual places on the pipeline for which the hypothetical critical crack length was
calculated are shown in Figure 8. These are places of reduction to a smaller pipe thickness,
where longitudinal or meridian cracks could occur due to higher stress concentration. The
shape of the internal longitudinal semi-elliptical crack is shown schematically in Figure
9a. In addition to the internal longitudinal semi-elliptical crack, it is necessary to calculate
the critical crack size for the crack through the pipeline wall (semi-elliptical through a
crack). This is an example of the presence of a crack where water could leak rather than
destroy the pipeline. Two cases were analysed: the existence of a longitudinal passing
crack (Figure 9b) and a meridian internal crack (Figure 9c). In the case of an internal semi-
elliptical crack, the change in the length of the crack (2c) and the depth (a) of the crack in
the wall are monitored. In the case of a crack passing through the thickness of the wall,
only its length on the surface (2a) is monitored.
Figure 8. Places on the pipeline at branch junction 6A for which the calculation of the critical crack
size is performed.
Figure 8.
Places on the pipeline at branch junction 6A for which the calculation of the critical crack
size is performed.
Based on the obtained experimental results, and in accordance with the procedures
for assessing the integrity of the structure (SINTAP, EPRI, R6, WES 2805), a calculation can
be made in order to determine the critical length of the crack for the adopted load.
Due to the height of the water column, the defined load of the branch junction pipeline
is a working internal pressure of 50.9 bar (p= 5.09 MPa). For different pipeline diameters
and wall thicknesses, different sizes of crack opening stress are obtained in both the
longitudinal and meridian directions.
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(a)
(b)
(c)
Figure 9. Location of the crack: (a) Internal longitudinal semi-elliptical crack in the pipeline wall
under the effect of internal pressure p; (b) Longitudinal passing crack through the pipeline wall
under the effect of internal pressure p; (c) Meridian passing crack through the pipeline wall under
the influence of internal pressure p.
Based on the obtained experimental results, and in accordance with the procedures
for assessing the integrity of the structure (SINTAP, EPRI, R6, WES 2805), a calculation
can be made in order to determine the critical length of the crack for the adopted load.
Due to the height of the water column, the defined load of the branch junction
pipeline is a working internal pressure of 50.9 bar (p = 5.09 MPa). For different pipeline
diameters and wall thicknesses, different sizes of crack opening stress are obtained in both
the longitudinal and meridian directions.
For the longitudinal crack opening, the crack opening stress value ( lon
σ
) is
determined from the following expression:
T
Dp u
lon ⋅
⋅
=2
σ
, (1)
For the meridian crack opening, the stress value ( mer
σ
) of the crack opening is determined
from the following expression:
T
Dp u
mer ⋅
⋅
=4
σ
, (2)
Figure 9.
Location of the crack: (
a
) Internal longitudinal semi-elliptical crack in the pipeline wall
under the effect of internal pressure p; (
b
) Longitudinal passing crack through the pipeline wall
under the effect of internal pressure p; (
c
) Meridian passing crack through the pipeline wall under the
influence of internal pressure p.
For the longitudinal crack opening, the crack opening stress value (
σlon
) is determined
from the following expression:
σlon =p·Du
2·T(1)
For the meridian crack opening, the stress value (
σmer
) of the crack opening is determined
from the following expression:
σmer =p·Du
4·T(2)
where pbar is internal pressure, D
u
mm is the internal pipe diameter, and Tmm is the pipe
wall thickness.
The required mechanical properties for the deformed and aged state of the material
(state B) were obtained by testing and are given in Table 1. The relevant value of the fracture
toughness at which crack growth occurs for the material in state B is J
0.2,BL
= 250 N/mm,
that is, for the plane state of stress (PSS)—K
I,mat
= 220 MPa
·
m
0.5
or for the plane state of
strain (PSN)—KI,mat = 230 MPa·m0.5.
Following the original documentation [
25
–
29
] and the assumed position and location
of the cracks (Figures 8and 9) on the branch junction pipeline, the values of the pipeline
diameter and wall thickness were taken, as indicated in the following tables for each crack.
Sustainability 2023,15, 5143 8 of 16
SINTAP Procedure, Level “1”
The SINTAP procedure is based on two mutually equivalent procedures for the
calculation of structural integrity:
- R6, which was developed by British Energy [30,31] and
- ETM, which was developed by HZG (Helmholtz-Zentrum Hereon) [32].
The R6 procedure was used in the calculation, which is based on the FAD (Failure
Assessment Diagram) diagram for assessing the critical size of the crack. The FAD concept
is based on the FAC (Failure Assessment Curve) curve for assessing the acceptability of a
crack-type defect in materials. The diagram is formed based on the normalised load L
r
and
the error tolerance function f(Lr), (Figure 10). The error tolerance evaluation curve f(Lr) is
limited by the Lrmax value in the area of plastic collapse.
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where p bar is internal pressure, Du mm is the internal pipe diameter, and T mm is the
pipe wall thickness.
The required mechanical properties for the deformed and aged state of the material
(state B) were obtained by testing and are given in Table 1. The relevant value of the
fracture toughness at which crack growth occurs for the material in state B is J0.2,BL= 250
N/mm, that is, for the plane state of stress (PSS)—KI,mat = 220 MPa∙m0.5 or for the plane state
of strain (PSN)—KI,mat = 230 MPa∙m0.5.
Following the original documentation [25–29] and the assumed position and location
of the cracks (Figures 8 and 9) on the branch junction pipeline, the values of the pipeline
diameter and wall thickness were taken, as indicated in the following tables for each crack.
SINTAP Procedure, Level “1”
The SINTAP procedure is based on two mutually equivalent procedures for the
calculation of structural integrity:
- R6, which was developed by British Energy [30,31] and
- ETM, which was developed by HZG (Helmholtz-Zentrum Hereon) [32].
The R6 procedure was used in the calculation, which is based on the FAD (Failure
Assessment Diagram) diagram for assessing the critical size of the crack. The FAD
concept is based on the FAC (Failure Assessment Curve) curve for assessing the
acceptability of a crack-type defect in materials. The diagram is formed based on the
normalised load Lr and the error tolerance function f(Lr), (Figure 10). The error tolerance
evaluation curve f(Lr) is limited by the Lrmax value in the area of plastic collapse.
Figure 10. FAD diagram for determining the critical crack size.
The process of assessing the integrity of the structure is carried out with constant
exposure of the pipe to internal pressure, and with the simulation of crack growth. The
procedure is repeated as many times as necessary so that the points in Figure 10 from the
area of reliable operation cross the FAC error tolerance curve. Therefore, at the critical
dimension and load, the following state [33] will be fulfilled:
()
rr LfK =. (3)
The dimension of the crack at which we found the intersection of the curve of change
in crack size and acceptance of the error is the critical dimension of the crack (ac). For each
crack length, at the internal pressure of p = 5.09 MPa, it is necessary to determine the
normalised value of the stress intensity factor (Kr) and the normalised value of the load
(Lr).
The normalised value of the stress intensity factor Kr is the ratio of the stress intensity
factor KI(a, σ) and the measured fracture toughness of the material Kmat [33]:
mat
I
rK
aK
K),(
σ
=. (4)
Figure 10. FAD diagram for determining the critical crack size.
The process of assessing the integrity of the structure is carried out with constant
exposure of the pipe to internal pressure, and with the simulation of crack growth. The
procedure is repeated as many times as necessary so that the points in Figure 10 from the
area of reliable operation cross the FAC error tolerance curve. Therefore, at the critical
dimension and load, the following state [33] will be fulfilled:
Kr=f(Lr)(3)
The dimension of the crack at which we found the intersection of the curve of change
in crack size and acceptance of the error is the critical dimension of the crack (a
c
). For
each crack length, at the internal pressure of p= 5.09 MPa, it is necessary to determine
the normalised value of the stress intensity factor (K
r
) and the normalised value of the
load (Lr).
The normalised value of the stress intensity factor K
r
is the ratio of the stress intensity
factor KI(a,σ) and the measured fracture toughness of the material Kmat [33]:
Kr=KI(a,σ)
Kmat (4)
Using K
I
(a,
σ
) or J
I
, the load level due to pressure (p), the position and length of the
crack (a), as well as the geometric shape of the structural component—the pipeline—are
taken into account.
K
mat
is the fracture toughness parameter of the material, and in the case of brittle
behaviour, Kmat =KIc.
The calculation of the permissible crack size was made with the measured value of the
J—integral J0.2,BL =250 N/mm.
The standard load (Lr) is the ratio of the working pressure (p) and the pressure at the
yield point of the material (
pY
), at which the stress will be equal to the yield stress of the
material (σY):
Lr=σ
σY
=p
pY
(5)
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The stress
σ
is equal to the circumferential stress in the pipeline wall and depends on
its diameter and wall thickness at a given location [33].
For materials without Lüders plateau (Lüders bands) [
34
,
35
] in tensile behaviour,
the curve for assessing the acceptability of the FAC error is determined by the following
equation [33]:
f(Lr)=h1+1
2L2
ri−1/2·h0.3 +0.7e−0.6L6
ri
0≤Lr≤Lrmax
(6)
Plastic collapse occurs at a value of L
r
> 1. As component failure will not occur
immediately after reaching L
r
= 1, it is adopted that the limit engineering value L
rmax
corresponds to the arithmetic mean of the yield strength (R
p0.2
) and the tensile strength
(Rm) of the material:
Lr,max =(Rp0.2 +Rm)
2·Rm(7)
Equations from (3) to (7) are valid regardless of the direction of crack growth or the
way and direction the structural component is loaded. This is explained by the fact that
both parameters, K
I
(a,
σ
) and L
r
, are determined by the geometric characteristics of the
structure and the way it is loaded.
In the SINTAP procedure, already confirmed mathematical and empirical solutions are
used for K
I
(a,
σ
) and L
r
, so that for each individual crack configuration, it is stated according
to which solutions for KI(a,σ) and Lrwe performed the calculation and whether the states
are reached.
Longitudinal crack
Figures 11–13 show the diagrams of determining the critical size of the crack (semi-
elliptical) at locations #1, #2, and #3 (Figure 8), and for the following ratio: a/c= 0.1. Table 3
shows the calculated opening stress and crack geometry on the inner surface for this ratio.
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Figure 11. Determination of the critical crack size (semi-elliptical) at location #1 for the ratio a/c = 0.1.
Figure 12. Determination of the critical crack size (semi-elliptical) at location #2 for the ratio a/c = 0.1.
Figure 13. Determination of the critical crack size (semi-elliptical) at location #3 for the ratio a/c = 0.1.
Table 3. Calculation for the ratio a/c = 0.1 between crack depth a and half of its length on surface 2c
(Figure 9a).
Location T, mm Du, mm a/c σ, MPa ac, mm 2cc, mm Comment
#1 28.0 2500 0.1 227.2 25.05 500.1 Fracture before
highlighting
#2 18.0 1200 0.1 169.7 17.0 * 340.0 * The condition for
LBB
#3 28.0 2350 0.1 213.2 26.08 521.6 Fracture before high-
lighting
Du—pipe diameter; T—wall thickness; σ—opening stress; ac—critical crack depth; 2cc—critical crack
length on the inner surface; *—the last value taken for the crack size that is not yet critical.
Figure 11.
Determination of the critical crack size (semi-elliptical) at location #1 for the ratio a/c= 0.1.
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Figure 11. Determination of the critical crack size (semi-elliptical) at location #1 for the ratio a/c = 0.1.
Figure 12. Determination of the critical crack size (semi-elliptical) at location #2 for the ratio a/c = 0.1.
Figure 13. Determination of the critical crack size (semi-elliptical) at location #3 for the ratio a/c = 0.1.
Table 3. Calculation for the ratio a/c = 0.1 between crack depth a and half of its length on surface 2c
(Figure 9a).
Location T, mm Du, mm a/c σ, MPa ac, mm 2cc, mm Comment
#1 28.0 2500 0.1 227.2 25.05 500.1 Fracture before
highlighting
#2 18.0 1200 0.1 169.7 17.0 * 340.0 * The condition for
LBB
#3 28.0 2350 0.1 213.2 26.08 521.6 Fracture before high-
lighting
Du—pipe diameter; T—wall thickness; σ—opening stress; ac—critical crack depth; 2cc—critical crack
length on the inner surface; *—the last value taken for the crack size that is not yet critical.
Figure 12.
Determination of the critical crack size (semi-elliptical) at location #2 for the ratio a/c= 0.1.
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Figure 11. Determination of the critical crack size (semi-elliptical) at location #1 for the ratio a/c = 0.1.
Figure 12. Determination of the critical crack size (semi-elliptical) at location #2 for the ratio a/c = 0.1.
Figure 13. Determination of the critical crack size (semi-elliptical) at location #3 for the ratio a/c = 0.1.
Table 3. Calculation for the ratio a/c = 0.1 between crack depth a and half of its length on surface 2c
(Figure 9a).
Location T, mm Du, mm a/c σ, MPa ac, mm 2cc, mm Comment
#1 28.0 2500 0.1 227.2 25.05 500.1 Fracture before
highlighting
#2 18.0 1200 0.1 169.7 17.0 * 340.0 * The condition for
LBB
#3 28.0 2350 0.1 213.2 26.08 521.6 Fracture before high-
lighting
Du—pipe diameter; T—wall thickness; σ—opening stress; ac—critical crack depth; 2cc—critical crack
length on the inner surface; *—the last value taken for the crack size that is not yet critical.
Figure 13.
Determination of the critical crack size (semi-elliptical) at location #3 for the ratio a/c= 0.1.
Table 3.
Calculation for the ratio a/c= 0.1 between crack depth aand half of its length on surface 2c
(Figure 9a).
Location T, mm Du,
mm a/cσ, MPa ac, mm 2cc, mm Comment
#1 28.0 2500 0.1 227.2 25.05 500.1 Fracture before
highlighting
#2 18.0 1200 0.1 169.7 17.0 * 340.0 * The condition
for LBB
#3 28.0 2350 0.1 213.2 26.08 521.6 Fracture before
highlighting
D
u
—pipe diameter; T—wall thickness;
σ
—opening stress; a
c
—critical crack depth; 2c
c
—critical crack length on
the inner surface; *—the last value taken for the crack size that is not yet critical.
The stress intensity factor is calculated according to reference [
36
] with the note that
the range of validity is as follows: 0 < T/D<2,0<a/c< 1.
From Figures 11–13, it is visible that the crack will reach the critical depth (a
c
) rather
than the critical length (2c
c
) on the inner surface. The basis for this conclusion is the fact
that higher K
r
values were calculated for the angle of
φ
= 90
◦
than for 0
◦
. Additionally, the
critical depth is reached before the crack passes through the thickness of the wall, which
shows that the conditions for leakage of water before breaking (LBB—Leak Before Break)
are not met. This indicates it is necessary to do online monitoring for cracks at places #1
and #3.
In Figure 12, it can be seen that all the points for the hypothetical crack depths remain
in the diagram within the region of safe operation (below the f(L
r
) curve), which indicates
the possibility of leakage before failure (LBB). As we continued our work, the calculation
was repeated for the example of a passing crack, to determine the critical size of the crack
when water leaks from the pipeline.
The limit load is calculated according to reference [
37
] with the note that the range of
validity is 0 < a/T≤0.8, 0.2 ≤a/c≤1.
If the crack depth exceeds 0.8T, the results for the ultimate load are outside the valid
range for the ratio of a/c = 0.1. Therefore, a calculation was made for the ratio of a/c = 0.2 to
obtain valid values for the calculation of the critical crack and limit loads (Table 4).
At all three locations, the diagrams in Figures 14–16 show that none of the cracks are
critical for failure to occur even though there is only 1 mm left to break through the pipe
wall. Since at the last value, at which the calculation is stopped, the character of the crack
changes, instead of an internal semi-elliptical (surface) crack, a crack that passes through
the pipe wall (through the crack) appears. In the case of a crack that has broken through
the pipe wall, leakage occurs and only one geometric parameter of the crack is relevant: Its
surface length 2c. The calculation results are shown in Figures 17–19 and in Table 5.
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Table 4.
Calculation for the ratio a/c = 0.2 between the depth of the crack (a) and half of its length on
the surface (2c) (Figure 8a).
Location T, mm Du, mm a/cσ, MPa a, mm 2c, mm Comment
#1 28.0 2500 0.2 227.2 27.0 * 270.0 * The condition
for LBB
#2 18.0 1200 0.2 169.7 17.0 * 170.0 *
#3 28.0 2350 0.2 213.2 27.0 * 270.0 *
D
u
—pipe diameter; T—wall thickness;
σ
—opening stress; a—crack depth; 2c—crack length on the inner surface;
*—the last value taken for the crack size that is not yet critical.
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The stress intensity factor is calculated according to reference [36] with the note that
the range of validity is as follows: 0 < T/D < 2, 0 < a/c < 1.
From Figures 11 and 13, it is visible that the crack will reach the critical depth (ac)
rather than the critical length (2cc) on the inner surface. The basis for this conclusion is the
fact that higher Kr values were calculated for the angle of ϕ = 90° than for 0°. Additionally,
the critical depth is reached before the crack passes through the thickness of the wall,
which shows that the conditions for leakage of water before breaking (LBB—Leak Before
Break) are not met. This indicates it is necessary to do online monitoring for cracks at
places #1 and #3.
In Figure 12, it can be seen that all the points for the hypothetical crack depths remain
in the diagram within the region of safe operation (below the f(Lr) curve), which indicates
the possibility of leakage before failure (LBB). As we continued our work, the calculation
was repeated for the example of a passing crack, to determine the critical size of the crack
when water leaks from the pipeline.
The limit load is calculated according to reference [37] with the note that the range of
validity is 0 < a/T ≤ 0.8, 0.2 ≤ a/c ≤ 1.
If the crack depth exceeds 0.8T, the results for the ultimate load are outside the valid
range for the ratio of a/c = 0.1. Therefore, a calculation was made for the ratio of a/c = 0.2
to obtain valid values for the calculation of the critical crack and limit loads (Table 4).
Table 4. Calculation for the ratio a/c = 0.2 between the depth of the crack (a) and half of its length on
the surface (2c) (Figure 8a).
Location T, mm Du, mm a/c σ, MPa a, mm 2c, mm Comment
#1 28.0 2500 0.2 227.2 27.0 * 270.0 * The condition
for LBB
#2 18.0 1200 0.2 169.7 17.0 * 170.0 *
#3 28.0 2350 0.2 213.2 27.0 * 270.0 *
Du—pipe diameter; T—wall thickness; σ—opening stress; a—crack depth; 2c—crack length on the
inner surface; *—the last value taken for the crack size that is not yet critical
At all three locations, the diagrams in Figures 14–16 show that none of the cracks are
critical for failure to occur even though there is only 1mm left to break through the pipe
wall. Since at the last value, at which the calculation is stopped, the character of the crack
changes, instead of an internal semi-elliptical (surface) crack, a crack that passes through
the pipe wall (through the crack) appears. In the case of a crack that has broken through
the pipe wall, leakage occurs and only one geometric parameter of the crack is relevant:
its surface length 2c. The calculation results are shown in Figures 17–19 and in Table 5.
Figure 14. Determination of critical crack size (semi-elliptical) at location #1 for ratio a/c = 0.2.
Figure 14. Determination of critical crack size (semi-elliptical) at location #1 for ratio a/c= 0.2.
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Figure 15. Determination of critical crack size (semi-elliptical) at location #2 for ratio a/c = 0.2.
Figure 16. Determination of the critical size of the crack (semi-elliptical) at location #3 for ratio a/c =
0.2.
Figure 17. Determination of the critical size of the longitudinal passing crack at location #1.
Figure 18. Determination of the critical size of the longitudinal passing crack at location #2.
Figure 15. Determination of critical crack size (semi-elliptical) at location #2 for ratio a/c= 0.2.
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Figure 15. Determination of critical crack size (semi-elliptical) at location #2 for ratio a/c = 0.2.
Figure 16. Determination of the critical size of the crack (semi-elliptical) at location #3 for ratio a/c =
0.2.
Figure 17. Determination of the critical size of the longitudinal passing crack at location #1.
Figure 18. Determination of the critical size of the longitudinal passing crack at location #2.
Figure 16.
Determination of the critical size of the crack (semi-elliptical) at location #3 for ratio
a/c= 0.2.
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Figure 15. Determination of critical crack size (semi-elliptical) at location #2 for ratio a/c = 0.2.
Figure 16. Determination of the critical size of the crack (semi-elliptical) at location #3 for ratio a/c =
0.2.
Figure 17. Determination of the critical size of the longitudinal passing crack at location #1.
Figure 18. Determination of the critical size of the longitudinal passing crack at location #2.
Figure 17. Determination of the critical size of the longitudinal passing crack at location #1.
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Figure 15. Determination of critical crack size (semi-elliptical) at location #2 for ratio a/c = 0.2.
Figure 16. Determination of the critical size of the crack (semi-elliptical) at location #3 for ratio a/c =
0.2.
Figure 17. Determination of the critical size of the longitudinal passing crack at location #1.
Figure 18. Determination of the critical size of the longitudinal passing crack at location #2.
Figure 18. Determination of the critical size of the longitudinal passing crack at location #2.
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Figure 19. Determination of the critical size of the longitudinal passing crack at location #3.
Table 5. Calculation for a passing crack that is parallel—longitudinal to the flow of water (Figure
9b).
Location T, mm Du, mm σ, MPa 2cc, mm Comment
#1 28.0 2500 227.2 257.0 *
The condition
for LBB
#2 18.0 1200 169.7 251.2 *
#3 28.0 2350 213.2 270.2 *
Du—pipe diameter; T—wall thickness; σ—opening stress; 2cc—critical crack length on the inner
surface; *—the last value taken for the crack size that is not yet critical.
Figures 17–19 show the determination of the critical size of a longitudinal passing
crack through the pipe wall. Asll determined critical sizes of cracks are of such dimensions
that, in the event of their possible formation on a real object, they can be monitored
(controlled) by a system of linear measuring tapes, which would measure the considerable
relaxation of the material. In addition, the calculation shows that, in the case of subcritical
cracks, water will flow out of the pipeline.
The stress intensity factor KI was calculated according to the reference [38], and the
limit load calculation [39]. “Limit Load Solution” references were used and they give valid
results for all shown hypothetical crack dimensions, with a note that the range of validity
is 0.01 < T/D < 0.4, 0.5 < c/T < 25.
Meridian crack
Since the opening stresses in the case of a meridian crack are two times smaller than
the corresponding stresses in the case of a longitudinal crack, we determined the critical
size of the crack that passes through the wall (passing crack) and leads to the discharge of
water from the pipeline.
Figures 20–22 show the diagrams for determining the critical size of a transient
meridian crack at locations #1, #2, and #3 (Figure 8).
Figure 20. Determining the critical size of a transient meridian crack at location #1.
Figure 19. Determination of the critical size of the longitudinal passing crack at location #3.
Table 5.
Calculation for a passing crack that is parallel—longitudinal to the flow of water (Figure 9b).
Location T, mm Du, mm σ, MPa 2cc, mm Comment
#1 28.0 2500 227.2 257.0 * The
condition for
LBB
#2 18.0 1200 169.7 251.2 *
#3 28.0 2350 213.2 270.2 *
D
u
—pipe diameter; T—wall thickness;
σ
—opening stress; 2c
c
—critical crack length on the inner surface; *—the
last value taken for the crack size that is not yet critical.
Figures 17–19 show the determination of the critical size of a longitudinal passing crack
through the pipe wall. Asll determined critical sizes of cracks are of such dimensions that,
in the event of their possible formation on a real object, they can be monitored (controlled)
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by a system of linear measuring tapes, which would measure the considerable relaxation of
the material. In addition, the calculation shows that, in the case of subcritical cracks, water
will flow out of the pipeline.
The stress intensity factor K
I
was calculated according to the reference [
38
], and the
limit load calculation [
39
]. “Limit Load Solution” references were used and they give valid
results for all shown hypothetical crack dimensions, with a note that the range of validity
is 0.01 < T/D< 0.4, 0.5 < c/T< 25.
Meridian crack
Since the opening stresses in the case of a meridian crack are two times smaller than
the corresponding stresses in the case of a longitudinal crack, we determined the critical
size of the crack that passes through the wall (passing crack) and leads to the discharge of
water from the pipeline.
Figures 20–22 show the diagrams for determining the critical size of a transient merid-
ian crack at locations #1, #2, and #3 (Figure 8).
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Figure 19. Determination of the critical size of the longitudinal passing crack at location #3.
Table 5. Calculation for a passing crack that is parallel—longitudinal to the flow of water (Figure
9b).
Location T, mm Du, mm σ, MPa 2cc, mm Comment
#1 28.0 2500 227.2 257.0 *
The condition
for LBB
#2 18.0 1200 169.7 251.2 *
#3 28.0 2350 213.2 270.2 *
Du—pipe diameter; T—wall thickness; σ—opening stress; 2cc—critical crack length on the inner
surface; *—the last value taken for the crack size that is not yet critical.
Figures 17–19 show the determination of the critical size of a longitudinal passing
crack through the pipe wall. Asll determined critical sizes of cracks are of such dimensions
that, in the event of their possible formation on a real object, they can be monitored
(controlled) by a system of linear measuring tapes, which would measure the considerable
relaxation of the material. In addition, the calculation shows that, in the case of subcritical
cracks, water will flow out of the pipeline.
The stress intensity factor KI was calculated according to the reference [38], and the
limit load calculation [39]. “Limit Load Solution” references were used and they give valid
results for all shown hypothetical crack dimensions, with a note that the range of validity
is 0.01 < T/D < 0.4, 0.5 < c/T < 25.
Meridian crack
Since the opening stresses in the case of a meridian crack are two times smaller than
the corresponding stresses in the case of a longitudinal crack, we determined the critical
size of the crack that passes through the wall (passing crack) and leads to the discharge of
water from the pipeline.
Figures 20–22 show the diagrams for determining the critical size of a transient
meridian crack at locations #1, #2, and #3 (Figure 8).
Figure 20. Determining the critical size of a transient meridian crack at location #1.
Figure 20. Determining the critical size of a transient meridian crack at location #1.
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Figure 21. Determining the critical size of a transient meridian crack at location #2.
Figure 22. Determining the critical size of a transient meridian crack at location #3.
Figures 20–22 show the critical size of a meridian crack that passes through the
thickness of the pipe wall. All critical sizes are of significantly high dimensions (Table 6),
so any subcritical crack that may have appeared can be registered in a timely manner. At
subcritical sizes, water will flow out of the pipeline, but not rupture the pipeline material.
The stress intensity factor KI was calculated according to reference [38] with the
validity range of 0.01 < T/D < 0.4, 0.5 < c/T < 25. The limit load calculation [40] “Limit Load
Solution” is within the scope of validity (thin-walled cylinders). References provide valid
results for all hypothetical crack dimensions shown.
Table 6. The calculation for a passing crack that is meridian to the water direction (Figure 9c).
Location T, mm Du, mm σ, MPa 2c, mm Comment
1 28.0 2350 106.8 248.0
They are fulfilled
conditions for LBB
2 28.0 2500 113.6 251.4
3 18.0 1200 84.8 209.2
Du—pipe diameter; T—wall thickness; σ—opening stress; 2c—the length of the crack along the inner
surface.
4. Conclusions
Applying the SINTAP procedure provides answers to the following questions
important for the safe operation of pipelines:
1. What is the size of the fatigue crack at which an unstable fracture would occur?
2. Is it possible to observe such a crack with the naked eye, that is, by using the
available measuring equipment?
3. Will there be leakage of the medium (water) from the pipeline at a subcritical crack
size, which may indicate the need for a timely shutdown of the pipeline system?
Figure 21. Determining the critical size of a transient meridian crack at location #2.
Sustainability 2023, 15, x FOR PEER REVIEW 14 of 16
Figure 21. Determining the critical size of a transient meridian crack at location #2.
Figure 22. Determining the critical size of a transient meridian crack at location #3.
Figures 20–22 show the critical size of a meridian crack that passes through the
thickness of the pipe wall. All critical sizes are of significantly high dimensions (Table 6),
so any subcritical crack that may have appeared can be registered in a timely manner. At
subcritical sizes, water will flow out of the pipeline, but not rupture the pipelin