ArticlePDF Available

Plastic Limit Pressure and Stress Intensity Factor for Cracked Elbow Containing Axial Semi-Elliptical Part-Through Crack

Authors:
  • Independent Researcher
  • University of Slavonski Brod
  • University of Slavonski Brod

Abstract and Figures

The aim of this study is to provide a solution for the plastic limit pressure and stress intensity factor of the elbows containing a part-through axial semi-elliptical crack by considering various crack sizes. The supporting system and loading conditions of the pipeline are described. The critical part of the observed pipeline was isolated for analysis and subjected to various sizes of semi-elliptical cracks. By performing numerical analysis, results were obtained for crack dimension ratios of c/a, and depth/thickness ratios of a/t. The obtained results include plastic limit pressure and stress intensity factor. The results were analyzed with a symbolic regression algorithm, and closed-form solutions for the limit pressure and stress intensity factor were proposed. To validate pipeline integrity, the Structural Integrity Assessment Procedure (SINTAP) was applied, and the FAD (Failure Assessment Diagram) was generated for cracks below the FAD function. The failure pressure was calculated by determining the points where the loading paths intersect the FAD function.
Content may be subject to copyright.
Citation: Damjanovi´c, B.; Konjati´c, P.;
Katini´c, M. Plastic Limit Pressure and
Stress Intensity Factor for Cracked
Elbow Containing Axial Semi-
Elliptical Part-Through Crack. Appl.
Sci. 2024,14, 8390. https://doi.org/
10.3390/app14188390
Academic Editor: JoséAntónio
Correia
Received: 29 August 2024
Revised: 10 September 2024
Accepted: 15 September 2024
Published: 18 September 2024
Copyright: © 2024 by the authors.
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/).
applied
sciences
Article
Plastic Limit Pressure and Stress Intensity Factor for Cracked
Elbow Containing Axial Semi-Elliptical Part-Through Crack
Božo Damjanovi´c 1, *, Pejo Konjati´c 2and Marko Katini´c 2
1Numikon d.o.o., Radniˇcka cesta 177, 10000 Zagreb, Croatia
2Mechanical Engineering Faculty, University of Slavonski Brod, Trg Ivane Brli´c-Mažurani´c 2,
35000 Slavonski Brod, Croatia; pkonjatic@unisb.hr (P.K.); mkatinic@unisb.hr (M.K.)
*Correspondence: bdamjanovic@unisb.hr
Abstract: The aim of this study is to provide a solution for the plastic limit pressure and stress
intensity factor of the elbows containing a part-through axial semi-elliptical crack by considering
various crack sizes. The supporting system and loading conditions of the pipeline are described.
The critical part of the observed pipeline was isolated for analysis and subjected to various sizes of
semi-elliptical cracks. By performing numerical analysis, results were obtained for crack dimension
ratios of c/a, and depth/thickness ratios of a/t. The obtained results include plastic limit pressure
and stress intensity factor. The results were analyzed with a symbolic regression algorithm, and
closed-form solutions for the limit pressure and stress intensity factor were proposed. To validate
pipeline integrity, the Structural Integrity Assessment Procedure (SINTAP) was applied, and the FAD
(Failure Assessment Diagram) was generated for cracks below the FAD function. The failure pressure
was calculated by determining the points where the loading paths intersect the FAD function.
Keywords: plastic limit pressure; stress intensity factor; cracked elbow; numerical analysis; FAD
1. Introduction
To ensure the safety and reliability of conventional and nuclear power plants, it is
essential to adopt a proactive approach to preventing component failures. The failure of a
single component can result in the shutdown of the entire plant. Securing the structural
integrity of a piping system leans on two key elements: ensuring a sufficient pipe wall
thickness and installing appropriate supports. A pipe elbow is one of the most common
pipeline fittings. It contains higher flexibility compared to a straight pipe and can be used to
reduce reaction forces and moments within the piping system. Pipe elbows are subjected to
various damage mechanisms, manifested on internal or external surfaces, intrados, crowns,
or extrados, and exposed to fluids and the external environment. In a long-term operation,
piping as pressure equipment may contain crack-like defects [17].
Applying fracture mechanics principles is crucial when determining a structural
integrity assessment of flawed pipelines. In this context, the limit load (limit pressure) and
the stress intensity factor are essential parameters while evaluating structural integrity.
To prevent pipe failure, the internal pressure (P) must remain below the plastic limit
pressure (P
L
), indicated as (P<P
L
). In scenarios involving flawed pipes, the plastic
limit load corresponds to the local limit load, representing a loading level at which gross
plasticity occurs in the un-cracked ligament. This may have implications for ligament
failure. Numerous studies have investigated the plastic limit pressure on straight pipes
featuring circumferential, axial, and part-through or through-wall cracks that appear on
thin- and thick-walled geometries [
8
18
]. Various solutions rely on experimental data,
while others are examined using finite element analysis. However, these solutions cannot
be implemented on pipe elbows due to the impact of the elbow curvature, leading to an
increase in hoop stress [19].
Appl. Sci. 2024,14, 8390. https://doi.org/10.3390/app14188390 https://www.mdpi.com/journal/applsci
Appl. Sci. 2024,14, 8390 2 of 15
In the late 1970s, Griffiths [
20
] published experimental data on pipe elbows under
in-plane bending, which were relatively limited. Miller and Zahoor’s publications from
the late 1980s rely on Griffith’s experimental data [
21
,
22
]. Yahiaoui et al. [
23
] verified
that in-plane opening bending, contributing to crack circumferential opening, poses a
higher risk to the elbow than closing bending. In later studies, numerous publications
focused research on pipe elbows with cracks, subjecting them to in-plane opening or closing
bending loads [
24
26
]. Some publications simultaneously combine bending moments with
internal pressure to estimate the limit load of cracked elbows [
27
30
]. Additionally, several
publications investigated exclusively circumferential part-through or through-wall cracks
on elbows [
30
34
]. The latest research has concentrated on developing plastic limit pressure
solutions for elbows with through-wall cracks, considering idealized and non-idealized
crack shapes [
35
,
36
]. Yield load solutions play a significant role in failure assessment
procedures in the widely used European Structural Integrity Procedure SINTAP and the R6
procedure [3739].
This paper aims to provide a solution for the plastic limit pressure and stress intensity
factor of the elbows containing a part-through axial semi-elliptical crack by considering
various crack sizes. Contrary to the mentioned studies, this study will not consider the
elbow as an isolated case but will consider the influence of the entire pipeline along with
pipe supports. In 2011, Lei proposed the ‘Equivalent straight pipe method’ that relies on
the similarity in geometry and loading between an elbow and its corresponding straight
pipe [
40
,
41
]. While numerous studies have focused on isolated cases of pipe elbows,
our investigation takes a different perspective by considering not just the behavior of
cracked elbows but the interactions of various components within the pipeline system. This
approach ensures a more complete analysis of structural integrity under realistic operating
conditions offering a safer and more reliable way for evaluating pipeline integrity.
2. Problem Description
The observed analyzed geometry is part of the feed-water system in the combined
cycle power plants. A calculated feed-water pipeline is used for water transportation in
combined cyclic gas turbines (CCGT). Such power plants work so that the exhaust gases of
the gas turbine transfer their heat to the boiler and thus convert water into steam. The steam
further circulates in a circular process and operates as the usual power plant. Combined
power plants usually run on natural gas, fuel oil, and synthesis gases. Coal or biofuels
can also be used as an additional energy source [
42
]. The feed water system prepares the
water for re-introduction into the boiler, this includes venting, collecting water into the
tank, and pumping it to the boiler. The feed water is sucked in through the suction side of
the feed water pump and redistributed through the discharge side. The observed pipeline
is connected to a feed water pump suction nozzle, which allows the distribution of water to
the pump [
43
,
44
]. The geometry was analyzed, and at the critical section, where the highest
principal stress is concentrated, an axial semi-elliptical part-through crack is present (on the
intrados of the elbow). The pipeline is subjected to internal pressure of 3.5 MPa. The study
aims to investigate the influence of hypothetically persistent cracks at the critical location
on the observed pipeline. Since hoop stress dominates the critical area, crack initiation will
occur at the critical location, perpendicular to the highest principal stress. According to [
37
],
to simplify calculations, the actual crack geometry will be replaced with a semi-elliptical
one. Pipeline dimensions are shown in Figure 1.
Appl. Sci. 2024,14, 8390 3 of 15
Appl.Sci.2024,14,xFORPEERREVIEW3of15
Figure1.Pipelinedimensions(alldimensionsareinmillimeters)[43].
3.Methodology
Todeterminetheplasticlimitpressure,theniteelementmethodwasapplied.The
nominaldiameterofthepipelineisDN125(outerdiameter139.7mm)withawallthick-
nessof5.6mm.Theelbowradiusisequalto190mm.Figure2displaystheboundary
conditionsofthepipelinealongwiththeequivalentstressofthecalculatedcriticalelbow.

(a)(b)
Figure2.Boundaryconditions:(a)Supports;(b)LoadsandvonMissesstressdistributiononcritical
location.
Boundaryconditionsandgeometryarecharacteristicofpipelinesinpowerplants,
andbesidesinternalpressure,themassesofvalvesandangesareconsidered.
Inadditiontoniteelementanalysis,thestructuralintegrityofthepipelinewasas-
sessedusingtheSINTAPprocedure.Thisprocedureintegratestheyieldloadsolution
withinthefailureassessmentdiagram(FAD)toassessthestructuralintegrityandensure
thesafeoperationoftheevaluatedstructure.Observedresultscanbeusedforthestruc-
turalintegrityassessmentprocedure(FADphilosophy).TheFADfunctionisgivenasK
r
=f(L
r
),whereoption1Aisusedasthematerialisexpectedtodisplayayieldplateau(σ
Y
=
Figure 1. Pipeline dimensions (all dimensions are in millimeters) [43].
3. Methodology
To determine the plastic limit pressure, the finite element method was applied. The
nominal diameter of the pipeline is DN125 (outer diameter 139.7 mm) with a wall thickness
of 5.6 mm. The elbow radius is equal to 190 mm. Figure 2displays the boundary conditions
of the pipeline along with the equivalent stress of the calculated critical elbow.
Appl.Sci.2024,14,xFORPEERREVIEW3of15
Figure1.Pipelinedimensions(alldimensionsareinmillimeters)[43].
3.Methodology
Todeterminetheplasticlimitpressure,theniteelementmethodwasapplied.The
nominaldiameterofthepipelineisDN125(outerdiameter139.7mm)withawallthick-
nessof5.6mm.Theelbowradiusisequalto190mm.Figure2displaystheboundary
conditionsofthepipelinealongwiththeequivalentstressofthecalculatedcriticalelbow.
(a)(b)
Figure2.Boundaryconditions:(a)Supports;(b)LoadsandvonMissesstressdistributiononcritical
location.
Boundaryconditionsandgeometryarecharacteristicofpipelinesinpowerplants,
andbesidesinternalpressure,themassesofvalvesandangesareconsidered.
Inadditiontoniteelementanalysis,thestructuralintegrityofthepipelinewasas-
sessedusingtheSINTAPprocedure.Thisprocedureintegratestheyieldloadsolution
withinthefailureassessmentdiagram(FAD)toassessthestructuralintegrityandensure
thesafeoperationoftheevaluatedstructure.Observedresultscanbeusedforthestruc-
turalintegrityassessmentprocedure(FADphilosophy).TheFADfunctionisgivenasK
r
=f(L
r
),whereoption1Aisusedasthematerialisexpectedtodisplayayieldplateau(σ
Y
=
Figure 2. Boundary conditions: (a) Supports; (b) Loads and von Misses stress distribution on
critical location.
Boundary conditions and geometry are characteristic of pipelines in power plants, and
besides internal pressure, the masses of valves and flanges are considered.
In addition to finite element analysis, the structural integrity of the pipeline was
assessed using the SINTAP procedure. This procedure integrates the yield load solution
within the failure assessment diagram (FAD) to assess the structural integrity and ensure
the safe operation of the evaluated structure. Observed results can be used for the structural
integrity assessment procedure (FAD philosophy). The FAD function is given as K
r
=f(L
r
),
where option 1A is used as the material is expected to display a yield plateau (
σY
=R
p0.2
).
Appl. Sci. 2024,14, 8390 4 of 15
K
r
represents the ratio between the stress intensity factor and fracture toughness of the
material, while the ligament-yielding parameter L
r
is defined as the ratio between internal
pressure and plastic limit pressure [37].
Figure 3illustrates the dimensions of defects, where arepresents the crack depth, cis
half the crack length, and Windicates the remaining ligament.
Appl.Sci.2024,14,xFORPEERREVIEW4of15
R
p0.2
).K
r
representstheratiobetweenthestressintensityfactorandfracturetoughnessof
thematerial,whiletheligament-yieldingparameterL
r
isdenedastheratiobetweenin-
ternalpressureandplasticlimitpressure[37].
Figure3illustratesthedimensionsofdefects,wherearepresentsthecrackdepth,cis
halfthecracklength,andWindicatestheremainingligament.
Figure3.Pipeelbowwithcrack.
Theanalyzeddefectsaredescribedbyusingdimensionratiosofc/a=0.5,1,1.5,2,2.5,
3,4,and5,anddepth/thicknessratiosofa/t=0.15,0.3,0.45,0.6,and0.75.Dierentc/a
ratiosareobservedtoexplorebehaviorsthatmayoccurwithvarioussemi-ellipticalcrack
shapes.Additionally,dierenta/tratiosareexaminedtoinvestigatebothdeeperandshal-
lowercracks.Furthermore,suchpreparedinputdataandresultscanbestatisticallypro-
cessed.
Asmentionedearlier,similarlyloadedcrackedpipeswillexhibitlocalplasticcol-
lapseoftheligamentasafailuremechanism.Thereareseveralapproachesthatcanbe
usedtodeterminelimitload.Inthisstudy,thelimitpressureasloadwillbeobtained.To
calculatethelimitpressureusingtheniteelementmethod,nonlinearmaterialproperties
mustbeconsidered.Abilinearelastic-plasticmaterialmodelwithalowlevelofisotropic
hardeningwasusedtoachievefasterconvergenceoftheresults.Materialdesignproper-
tiesforP235aregiveninTab l e1.
Tab l e1.PropertiesofP235steel[45].
MaterialPropertyVal ue
Young’smodulus(E):200
,
000MPa
Density(ρ):7700kg/m
3
Yieldstrength(R
p0.2
):235MPa
Ultimatestrength(R
m
):360MPa
Poisson’sratioinelasticrange(ν):0.3
Asimpliedmodelwascreatedtocalculatethelimitpressureusingthesubmodeling
technique.Thisapproachenablestheapplicationofanermesharoundthecrackarea.
CalculationswereperformedinAnsysWorkbench2022[46].Thesubmodelmeshingwas
performedusingnearly700.000tetrahedralniteelements,whilethecrackwasmeshed
withdominantlyhexahedralelements.Themeshdensityincreasesfromtheedgesofthe
modeltowardsthecrack,withtheelementsizeof10mmattheedgesand0.5mmaround
thecracktiplocation.Additionally,themeshwasstructuredtocoveratleastthreeele-
mentsspanningthethicknessofthepipe.Meshissimilarforallconsideredcrack
Figure 3. Pipe elbow with crack.
The analyzed defects are described by using dimension ratios of c/a= 0.5, 1, 1.5,
2, 2.5, 3, 4, and 5, and depth/thickness ratios of a/t= 0.15, 0.3, 0.45, 0.6, and 0.75. Dif-
ferent c/aratios are observed to explore behaviors that may occur with various semi-
elliptical crack shapes. Additionally, different a/tratios are examined to investigate both
deeper and shallower cracks. Furthermore, such prepared input data and results can be
statistically processed.
As mentioned earlier, similarly loaded cracked pipes will exhibit local plastic collapse
of the ligament as a failure mechanism. There are several approaches that can be used to
determine limit load. In this study, the limit pressure as load will be obtained. To calculate
the limit pressure using the finite element method, nonlinear material properties must be
considered. A bilinear elastic-plastic material model with a low level of isotropic hardening
was used to achieve faster convergence of the results. Material design properties for P235
are given in Table 1.
Table 1. Properties of P235 steel [45].
Material Property Value
Young’s modulus (E): 200,000 MPa
Density (ρ): 7700 kg/m3
Yield strength (Rp0.2): 235 MPa
Ultimate strength (Rm): 360 MPa
Poisson’s ratio in elastic range (ν): 0.3
A simplified model was created to calculate the limit pressure using the submodeling
technique. This approach enables the application of a finer mesh around the crack area.
Calculations were performed in Ansys Workbench 2022 [46]. The submodel meshing was
performed using nearly 700.000 tetrahedral finite elements, while the crack was meshed
with dominantly hexahedral elements. The mesh density increases from the edges of the
model towards the crack, with the element size of 10 mm at the edges and 0.5 mm around
the crack tip location. Additionally, the mesh was structured to cover at least three elements
spanning the thickness of the pipe. Mesh is similar for all considered crack dimensions.
Figure 4shows the meshed submodeled region of interest containing a semi-elliptical crack.
Appl. Sci. 2024,14, 8390 5 of 15
Appl.Sci.2024,14,xFORPEERREVIEW5of15
dimensions.Figure4showsthemeshedsubmodeledregionofinterestcontainingasemi-
ellipticalcrack.
Figure4.Submodeledelbowwithsemi-ellipticalcrack.
4.Results
Afterrunningtheanalysisforallcrackshapes,theresultsshowedthatthetypical
failurepaernforallcrackdimensionsinvolveslocalizedmaterialyieldingonbothsides
ofthecracksurfacepoints,inaplaneparalleltothecrack.Localizedmaterialyielding
nearthecracksurfaceisawell-knownphenomenoninfracturemechanics,primarilybe-
causeofthestressconcentrationsdevelopedatthecracktip.Thisphenomenonisillus-
tratedinFigure5,whichshowsthestressdistributionfortwospeciccracksizeratios:c/a
=1(a/t=0.15)andc/a=2.5(a/t=0.6).Theredzoneindicatesstresslevelsexceedingthe
yieldstrength,leadingtoplasticcollapse.
(a)(b)
Figure5.Plasticcollapseincrackzone:(a)c/a=1;(b)c/a=2.5.
Figure 4. Submodeled elbow with semi-elliptical crack.
4. Results
After running the analysis for all crack shapes, the results showed that the typical
failure pattern for all crack dimensions involves localized material yielding on both sides of
the crack surface points, in a plane parallel to the crack. Localized material yielding near the
crack surface is a well-known phenomenon in fracture mechanics, primarily because of the
stress concentrations developed at the crack tip. This phenomenon is illustrated in Figure 5,
which shows the stress distribution for two specific crack size ratios:
c/a=1(a/t= 0.15
)
and c/a= 2.5 (a/t= 0.6). The red zone indicates stress levels exceeding the yield strength,
leading to plastic collapse.
Appl.Sci.2024,14,xFORPEERREVIEW5of15
dimensions.Figure4showsthemeshedsubmodeledregionofinterestcontainingasemi-
ellipticalcrack.
Figure4.Submodeledelbowwithsemi-ellipticalcrack.
4.Results
Afterrunningtheanalysisforallcrackshapes,theresultsshowedthatthetypical
failurepaernforallcrackdimensionsinvolveslocalizedmaterialyieldingonbothsides
ofthecracksurfacepoints,inaplaneparalleltothecrack.Localizedmaterialyielding
nearthecracksurfaceisawell-knownphenomenoninfracturemechanics,primarilybe-
causeofthestressconcentrationsdevelopedatthecracktip.Thisphenomenonisillus-
tratedinFigure5,whichshowsthestressdistributionfortwospeciccracksizeratios:c/a
=1(a/t=0.15)andc/a=2.5(a/t=0.6).Theredzoneindicatesstresslevelsexceedingthe
yieldstrength,leadingtoplasticcollapse.
(a)(b)
Figure5.Plasticcollapseincrackzone:(a)c/a=1;(b)c/a=2.5.
Figure 5. Plastic collapse in crack zone: (a)c/a= 1; (b)c/a= 2.5.
There is also present a yielding of the crack tip, particularly noticeable in Figure 6,
where the crack size ratio is c/a= 2.5 (a/t= 0.75). As the crack size increases (higher c/a and
a/t ratios), the stress concentration intensifies, which explains why the yielding becomes
more evident in these regions, particularly for deeper cracks.
Appl. Sci. 2024,14, 8390 6 of 15
Appl.Sci.2024,14,xFORPEERREVIEW6of15
Thereisalsopresentayieldingofthecracktip,particularlynoticeableinFigure6,
wherethecracksizeratioisc/a=2.5(a/t=0.75).Asthecracksizeincreases(higherc/aand
a/tratios),thestressconcentrationintensies,whichexplainswhytheyieldingbecomes
moreevidentintheseregions,particularlyfordeepercracks.
Figure6.EquivalentvonMisesstressdistributionfortimestepwhenlimit(yield)loadisreached
(c/a=2.5anda/t=0.75).
Thestressintensityfactor(K
I
)wascalculatedusingastraightforwardapproachbased
onfracturemechanicsprinciples.Thecrackgeometryisdened,andtheworkingpres-
sureappliedissetasaloadintheniteelementmodel.Afterrunningthesimulationsfor
eachcrackconguration,theresultsatthecracktip,consideredthecriticallocation,were
obtained.Thestressintensityfactorwasdeterminedbasedonthecracktipstressdistri-
bution,whichniteelementanalysis(FEA)calculatesbyevaluatingthelocalstresselds
aroundthecrackfront.Thestressintensityfactorvaluesatthecracktipwereobtained
directlyfromtheFEApost-processingcorrelatingwiththeboundaryconditions,applied
pressure,andcrackgeometry.
Unlikethestressintensityfactor,theplasticlimitpressure(P
L
)wasobtainedbyusing
aniterativeprocess.IntheFEAsimulation,theinternalpressurewassetasaload,and
loadincreasesineachiterativecalculationuntilthematerialaroundthecracktipyields
throughtheentirethicknessofthepipe.Thepressureatwhichfullyieldingofthecrack
ligamentsectionoccursisconsideredtheplasticlimitpressure,asitrepresentsthepoint
wherefailureduetoplasticcollapseisreached.
TovalidateFEAsolutionsforthecrackedelbow,ananalyticalsolutionfortheplastic
limitpressureofstraightpipeP
LS
withthesamediameterandthicknesswascalculated,
basedonthemethodologydemonstratedinKiefneretal.[8].Thisanalyticalsolutionfor
thestraightpipeservesasabenchmarkforcomparisontoevaluatetheaccuracyofthe
FEAresultsforthemorecomplexgeometryofthecrackedelbow.
Thecomparisonshowsthattrendsinbothsetsofresultsareconsistent(Table2).As
thecrackdepthorgeometrychanges,boththeFEAresultsfortheelbowandtheanalytical
solutionsforthestraightpipeexhibitsimilartrendsinpressurereduction.Elbowcomplex
geometrycreatessomevariationintheexactvalues.Theprimaryreasonforthesevaria-
tionsisthecurvatureoftheelbow.Elbowsexperienceanon-uniformdistributionofstress
duetobendingandtorsionaleects.Whenanelbowisloadedwithinternalpressure,the
curvatureoftheelbowcreatesregionsofincreasedstress,particularlyontheintrados
wherethestressconcentrationstendtobehigher.Thegeometryofanelbowintroduces
additionalcomplexitiesthataectboththeplasticlimitpressureandthestressintensity
Figure 6. Equivalent von Mises stress distribution for time step when limit (yield) load is reached
(c/a= 2.5 and a/t= 0.75).
The stress intensity factor (K
I
) was calculated using a straightforward approach based
on fracture mechanics principles. The crack geometry is defined, and the working pressure
applied is set as a load in the finite element model. After running the simulations for each
crack configuration, the results at the crack tip, considered the critical location, were ob-
tained. The stress intensity factor was determined based on the crack tip stress distribution,
which finite element analysis (FEA) calculates by evaluating the local stress fields around
the crack front. The stress intensity factor values at the crack tip were obtained directly
from the FEA post-processing correlating with the boundary conditions, applied pressure,
and crack geometry.
Unlike the stress intensity factor, the plastic limit pressure (P
L
) was obtained by using
an iterative process. In the FEA simulation, the internal pressure was set as a load, and
load increases in each iterative calculation until the material around the crack tip yields
through the entire thickness of the pipe. The pressure at which full yielding of the crack
ligament section occurs is considered the plastic limit pressure, as it represents the point
where failure due to plastic collapse is reached.
To validate FEA solutions for the cracked elbow, an analytical solution for the plastic
limit pressure of straight pipe P
LS
with the same diameter and thickness was calculated,
based on the methodology demonstrated in Kiefner et al. [
8
]. This analytical solution for
the straight pipe serves as a benchmark for comparison to evaluate the accuracy of the FEA
results for the more complex geometry of the cracked elbow.
The comparison shows that trends in both sets of results are consistent (Table 2).
As the crack depth or geometry changes, both the FEA results for the elbow and the
analytical solutions for the straight pipe exhibit similar trends in pressure reduction. Elbow
complex geometry creates some variation in the exact values. The primary reason for these
variations is the curvature of the elbow. Elbows experience a non-uniform distribution
of stress due to bending and torsional effects. When an elbow is loaded with internal
pressure, the curvature of the elbow creates regions of increased stress, particularly on
the intrados where the stress concentrations tend to be higher. The geometry of an elbow
introduces additional complexities that affect both the plastic limit pressure and the stress
intensity factor in comparison to a straight pipe. The FEA approach effectively captures the
mechanical behavior under similar loading conditions.
Appl. Sci. 2024,14, 8390 7 of 15
Table 2. Solutions for stress intensity factor (KI) and plastic limit pressure (PL).
c/a a (mm) c(mm) a/t
K
I(MPamm) PL(MPa) PLS (MPa)
0.5
0.84 0.42 0.15 49.811 17.7 19.62
1.68 0.84 0.30 71.584 17.6 19.62
2.52 1.26 0.45 89.724 17.35 19.60
3.36 1.68 0.60 106.6 17.1 19.54
4.2 2.1 0.75 123.044 16.9 19.36
1
0.84 0.84 0.15 56.137 17.6 19.61
1.68 1.68 0.30 82.088 17.5 19.60
2.52 2.52 0.45 106.439 17.35 19.52
3.36 3.36 0.60 131.995 17.15 19.29
4.2 4.2 0.75 159.593 16.9 18.63
1.5
0.84 1.26 0.15 61.258 17.75 19.62
1.68 2.52 0.30 91.864 17.3 19.57
2.52 3.78 0.45 120.197 17.15 19.39
3.36 5.04 0.60 148.292 16.95 18.91
4.2 6.3 0.75 183 16.55 17.59
2
0.84 1.68 0.15 68.386 17.7 19.61
1.68 3.36 0.30 103.613 17.6 19.53
2.52 5.04 0.45 138.654 17.4 19.23
3.36 6.72 0.60 175.518 17.2 18.42
4.2 8.4 0.75 211.721 16.5 16.4
2.5
0.84 2.1 0.15 73.142 17.4 19.61
1.68 4.2 0.30 112.172 17.1 19.47
2.52 6.3 0.45 153.451 16.55 19.02
3.36 8.4 0.60 199 16 17.87
4.2 10.5 0.75 245.813 14.5 15.19
3
0.84 2.52 0.15 75.898 17.7 19.60
1.68 5.04 0.30 117.541 16.9 19.41
2.52 7.56 0.45 164.599 16.3 18.79
3.36 10.08 0.60 218.923 15.5 17.27
4.2 12.6 0.75 277.304 14.4 14.06
4
0.84 3.36 0.15 80.051 17.2 19.58
1.68 6.72 0.30 127.446 16.7 19.26
2.52 10.08 0.45 185.401 16.2 18.27
3.36 13.44 0.60 258.365 14.8 16.10
4.2 16.8 0.75 344.46 13.1 12.17
5
0.84 4.2 0.15 82.771 18 19.56
1.68 8.4 0.30 135.069 17 19.09
2.52 12.6 0.45 202.795 15.9 17.71
3.36 16.8 0.60 295.03 14.5 15.02
4.2 21 0.75 407.75 11 10.75
The results presented in Table 2are used to plot the variation in stress intensity factor
and plastic limit pressure against crack length and pipe thickness ratio (a/t), with respect
to the crack dimension ratio (c/a). These plots are illustrated in Figures 7and 8.
As expected, the observed plots demonstrate two trends: as the a/tratio increases, the
stress intensity factor increases, while the plastic limit pressure decreases. There is also a
trend of an increased slope for the stress intensity factor with a higher c/aratio, while for
plastic limit pressure, the slope decreases with a higher c/aratio.
Appl. Sci. 2024,14, 8390 8 of 15
Figure 7. Stress intensity factor plot regarding a/tand c/aratio.
Appl.Sci.2024,14,xFORPEERREVIEW8of15
Figure7.Stressintensityfactorplotregardinga/tandc/aratio.
Figure8.Plasticlimitpressureplotregardinga/tandc/aratio.
Asexpected,theobservedplotsdemonstratetwotrends:asthea/tratioincreases,the
stressintensityfactorincreases,whiletheplasticlimitpressuredecreases.Thereisalsoa
trendofanincreasedslopeforthestressintensityfactorwithahigherc/aratio,whilefor
plasticlimitpressure,theslopedecreaseswithahigherc/aratio.
Toobtainclosed-formexpressionsfromniteelementanalysis(FEA)results,asta-
tisticalsymbolicregressionanalysiswasperformedbyusingTuri ngBotsoftwarev2.19
[47].Turin gBotisanadvancedtooldesignedforsymbolicregression.Unliketraditional
linearregression,symbolicregressionaemptstondthebest-ingequationbytesting
dierentmathematicalexpressions,whichcanbenonlinearorinvolvemultipleinterac-
tionsamongthevariablesThismakesitwellsuitedforcomplexengineeringapplications
wheretherelationshipsbetweenvariables,suchascrackdimensionsandstressresponses,
arenotpurelylinear.Crackpropagationandplasticlimitpressureinmaterialsoftenin-
volvenonlinearity,especiallyincasesofcomplexgeometrieslikesemi-ellipticalcracks.
Thisapproachenabledthegenerationofclosed-formexpressionsforcalculatingthestress
0
50
100
150
200
250
300
350
400
450
500
0.15 0.3 0.45 0.6 0.75
𝐾I[MPamm]
a/t
c/a=0.5
c/a=1
c/a=1.5
c/a=2
c/a=2.5
c/a=3
c/a=4
c/a=5
11
12
13
14
15
16
17
18
19
0.15 0.3 0.45 0.6 0.75
PL[MPa]
a/t
c/a=0.5
c/a=1
c/a=1.5
c/a=2
c/a=2.5
c/a=3
c/a=4
c/a=5
Figure 8. Plastic limit pressure plot regarding a/tand c/aratio.
To obtain closed-form expressions from finite element analysis (FEA) results, a statis-
tical symbolic regression analysis was performed by using TuringBot software v2.19 [
47
].
TuringBot is an advanced tool designed for symbolic regression. Unlike traditional linear
regression, symbolic regression attempts to find the best-fitting equation by testing different
mathematical expressions, which can be nonlinear or involve multiple interactions among
the variables This makes it well suited for complex engineering applications where the
relationships between variables, such as crack dimensions and stress responses, are not
purely linear. Crack propagation and plastic limit pressure in materials often involve
nonlinearity, especially in cases of complex geometries like semi-elliptical cracks. This ap-
proach enabled the generation of closed-form expressions for calculating the stress intensity
factor K
I
(
MPamm
and plastic limit pressure P
L
(MPa), where aand care measured in
millimeters (mm):
KI= (7.8071 + 1.49956 ×a)×(6.94365 (3.49515/(a×(c+ 1.76901))) + c) (1)
Appl. Sci. 2024,14, 8390 9 of 15
PL= 17.4989 (0.00745158 ×(a×a+ (c((2.24832)/(2.21352 + (c/a))))) ×c) (2)
The quality of the regression models was evaluated using two key parameters, the
R-squared (R
2
) and the root mean square (RMS) error. For stress intensity factor K
I
the R
2
is 0.998518, showing that the regression model explains 99.85% of the variance in the FEA
results. The RMS error for K
I
is 1.90855, indicating that there is an average deviation of the
predicted values from the observed FEA results. For the limit pressure P
L
, the regression
model observed an R-squared value of 0.981582, meaning that the model explains 98.16%
of the variance in the FEA results. The RMS error for P
L
is 0.1885232, showing a high level
of accuracy in the predictions. Lower RMS error values generally indicate a better model
fit, with errors between 0.2 and 0.5 being considered highly accurate in many statistical
engineering applications. The high R
2
value confirms that the models capture the key
trends in the data, while the low RMS error indicates that the models make highly accurate
predictions within the specified range of crack dimensions.
Figures 9and 10 illustrate the regression models for the stress intensity factor and
plastic limit pressure comparing finite element analysis (FEA) results with statistically
obtained expressions (1 and 2). The scatter plots represent the observed FEA values, while
the regression lines demonstrate the relationship modeled by the statistical analysis. The
high degree of alignment between the FEA data and the regression models suggests a
strong correlation between them. This indicates that the proposed expressions accurately
represent the relationship between the crack dimensions within the tested range. The tight
grouping of data points around the regression lines confirms the validity of the statistical
models for predicting stress intensity factor and plastic limit pressure. This demonstrates
that generated expressions are reliable for practical engineering calculations.
Although the regression models have shown a strong fit within the tested range,
future work could explore the models’ behavior beyond the tested crack sizes and include
parameters such as pipe thickness and radius, which may further increase their accuracy
and applicability. It is also essential to validate these models against experimental data or
other analytical solutions to ensure their robustness in different scenarios.
Appl.Sci.2024,14,xFORPEERREVIEW9of15
intensityfactorKI(MPamm󰇜andplasticlimitpressurePL(MPa),whereaandcaremeas-
uredinmillimeters(mm):
KI=(7.8071+1.49956×a)×(6.94365−(3.49515/(a×(c+1.76901)))+c)(1)
PL=17.4989−(0.00745158×(a×a+(c−((2.24832)/(2.21352+(c/a)))))×c)(2)
Thequalityoftheregressionmodelswasevaluatedusingtwokeyparameters,theR-
squared(R2)andtherootmeansquare(RMS)error.ForstressintensityfactorKItheR2is
0.998518,showingthattheregressionmodelexplains99.85%ofthevarianceintheFEA
results.TheRMSerrorforKIis1.90855,indicatingthatthereisanaveragedeviationofthe
predictedvaluesfromtheobservedFEAresults.ForthelimitpressurePL,theregression
modelobservedanR-squaredvalueof0.981582,meaningthatthemodelexplains98.16%
ofthevarianceintheFEAresults.TheRMSerrorforPLis0.1885232,showingahighlevel
ofaccuracyinthepredictions.LowerRMSerrorvaluesgenerallyindicateabeermodel
t,witherrorsbetween0.2and0.5beingconsideredhighlyaccurateinmanystatistical
engineeringapplications.ThehighR2valueconrmsthatthemodelscapturethekey
trendsinthedata,whilethelowRMSerrorindicatesthatthemodelsmakehighlyaccurate
predictionswithinthespeciedrangeofcrackdimensions.
Figures9and10illustratetheregressionmodelsforthestressintensityfactorand
plasticlimitpressurecomparingniteelementanalysis(FEA)resultswithstatisticallyob-
tainedexpressions(1and2).ThescaerplotsrepresenttheobservedFEAvalues,while
theregressionlinesdemonstratetherelationshipmodeledbythestatisticalanalysis.The
highdegreeofalignmentbetweentheFEAdataandtheregressionmodelssuggestsa
strongcorrelationbetweenthem.Thisindicatesthattheproposedexpressionsaccurately
representtherelationshipbetweenthecrackdimensionswithinthetestedrange.Thetight
groupingofdatapointsaroundtheregressionlinesconrmsthevalidityofthestatistical
modelsforpredictingstressintensityfactorandplasticlimitpressure.Thisdemonstrates
thatgeneratedexpressionsarereliableforpracticalengineeringcalculations.
Figure9.Stressintensityfactorscaerplot.
0
50
100
150
200
250
300
350
400
450
500
0 100 200 300 400 500
Expected KI [MPamm]
Obtained KI[MPamm]
Figure 9. Stress intensity factor scatter plot.
Appl. Sci. 2024,14, 8390 10 of 15
Appl.Sci.2024,14,xFORPEERREVIEW10of15
Figure10.Plasticlimitpressurescaerplot.
Althoughtheregressionmodelshaveshownastrongtwithinthetestedrange,fu-
tureworkcouldexplorethemodels’behaviorbeyondthetestedcracksizesandinclude
parameterssuchaspipethicknessandradius,whichmayfurtherincreasetheiraccuracy
andapplicability.Itisalsoessentialtovalidatethesemodelsagainstexperimentaldataor
otheranalyticalsolutionstoensuretheirrobustnessindierentscenarios.
5.FADProcedure
Thefailureassessmentdiagram(FAD)providesagraphicalmethodofassessingthe
structuralintegrityofthepipelinebyploingtherelationshipbetweencrackdimensions
andsafeoperatingconditions.Figure11illustratestheFADdiagramthroughwhichan
evaluationofthedimensionsofthecrackslocatedinthesafedomainisshown.Theesti-
matedassessmentpointsaredeterminedwithcoordinatesKrandLrbyusingtheFEA
results.ThepointsillustratedliebelowtheFADfunction(doedline),indicatingthatthey
fallwithinthesafedomainofthechart.WhenapointonaFADdiagramfallswithinthe
safedomain,itsuggeststhatthepipelinecancontinuetooperatesafelyunderthecurrent
loadingconditions.Thepipelineisnotlikelytofailduetocrackpropagationoryielding
untilloadsremainwithinthematerialcapacity.Thekeypointtonotehereisthatthere-
lationshipbetweenKrandLrappearsapproximatelylinearforthesecrackcongurations
withinthesafedomain(belowthedashedline).Thislinearbehavioriscommoninfracture
mechanicswhendealingwithsmall-to-moderatecracksizesandloadlevels.Thelinearity
indicatesthat,astheloadonthepipelineincreases(representedbyincreasingLr),the
stressintensityfactoralsoincreasesproportionally(leadingtoariseinKr).Thispropor-
tionalityholdstrueuntilthematerialbeginstoyield,ortheloadapproachesthelimitload.
Oncetheplasticlimitpressureisreached,thematerialgoesthroughplasticcollapse,and
therelationshipbetweenKrandLrbecomesnonlinear[48].Additionally,theplasticlimit
pressureisalmostvetimeshigherthantheworkingpressureofthepipe.Thepointnear
theFADfunction(a=1.68mm,c=3.36mm)stillfallswithinthesafedomainbutisclose
totheFADthresholdwherethestructuremayfail.Forengineers,thisproximitytothe
boundaryraisesconcernsaboutthereliabilityofthesystemunderadditionalloadsorun-
expectedconditions,suchaspressureoscillationsorfurthercrackgrowth.Incaseslike
this,thereistheoptiontoconsiderpreventativeactions,suchasreducingtheoperating
pressure,arrangingimmediaterepairs,oreventakingthecomponentoutofservicefora
moredetailedinspection.
8
10
12
14
16
18
20
8 101214161820
Expected PL [MPa]
Obtained PL [MPa]
Figure 10. Plastic limit pressure scatter plot.
5. FAD Procedure
The failure assessment diagram (FAD) provides a graphical method of assessing the
structural integrity of the pipeline by plotting the relationship between crack dimensions
and safe operating conditions. Figure 11 illustrates the FAD diagram through which an
evaluation of the dimensions of the cracks located in the safe domain is shown. The
estimated assessment points are determined with coordinates Kr and Lr by using the FEA
results. The points illustrated lie below the FAD function (dotted line), indicating that they
fall within the safe domain of the chart. When a point on a FAD diagram falls within the
safe domain, it suggests that the pipeline can continue to operate safely under the current
loading conditions. The pipeline is not likely to fail due to crack propagation or yielding
until loads remain within the material capacity. The key point to note here is that the
relationship between K
r
and L
r
appears approximately linear for these crack configurations
within the safe domain (below the dashed line). This linear behavior is common in fracture
mechanics when dealing with small-to-moderate crack sizes and load levels. The linearity
indicates that, as the load on the pipeline increases (represented by increasing L
r
), the stress
intensity factor also increases proportionally (leading to a rise in K
r
). This proportionality
holds true until the material begins to yield, or the load approaches the limit load. Once
the plastic limit pressure is reached, the material goes through plastic collapse, and the
relationship between K
r
and L
r
becomes nonlinear [
48
]. Additionally, the plastic limit
pressure is almost five times higher than the working pressure of the pipe. The point near
the FAD function (a= 1.68 mm, c= 3.36 mm) still falls within the safe domain but is close
to the FAD threshold where the structure may fail. For engineers, this proximity to the
boundary raises concerns about the reliability of the system under additional loads or
unexpected conditions, such as pressure oscillations or further crack growth. In cases like
this, there is the option to consider preventative actions, such as reducing the operating
pressure, arranging immediate repairs, or even taking the component out of service for a
more detailed inspection.
Appl. Sci. 2024,14, 8390 11 of 15
Appl.Sci.2024,14,xFORPEERREVIEW11of15

Figure11.Failureassessmentdiagram.
Tabl e3presentsthefailurepressurevalues,whichwerecalculatedbydetermining
thepointswheretheloadingpathsintersecttheFADfunction.Thesevaluesarecriticalfor
assessingtheintegrityofthepipelineandpredictingtheconditionsunderwhichfailure
mightoccur.
Tab l e3.FailurepressurevaluesdeterminedfromFADintersections.
a(mm)c(mm)
P
F(MPa)
0.840.427.1
1.680.845.05
2.521.264.08
0.840.846.35
1.681.684.45
0.841.265.85
1.682.524
0.841.685.3
1.683.363.55
0.842.14.95
0.842.524.75
0.843.364.5
0.844.24.38
Safetydependsonthedierencebetweentheworkingandfailurepressures.Forex-
ample,iftheworkingpressureissignicantlylowerthanthefailurepressure,itmeans
thatthepipelinecanoperatesafelyundernormalworkingconditions.However,ifthe
workingpressureapproachesorexceedsthefailurepressure,theriskoffailureincreases
signicantly,requiringimmediate