Mechanical Study of an Aircraft’s Structural Condition

Tiago Martinstiago.r.martins@tecnico.ulisboa.pt

Instituto Superior Tecnico, Universidade de Lisboa, Portugal

June 2019

Abstract

The purpose of this paper is the estimation of the Fatigue Lifetime of the Portuguese Air Force’sEpsilon TB-30 aircraft, by strain analysis on critical components of the aircraft. The experimentalstudies and knowledge of aircraft loads was used to reverse engineer a suitable Finite Element Modelof the structure, from which the stress fields in previously identified structural hotspots were retrieved.Cumulative damage on the structure is calculated for reference loading spectra using the Palgrem-Minerrule and suitable fatigue criteria. The Stress Intensity Factors at the critical component’s notchedgeometry were estimated using two different approaches: the Finite Element and Extended FiniteElement Methods from ABAQUS. Crack propagation was then modelled using several laws (Paris,Walker, NASGRO, Forman). A comparison between the two numerical methods and the results ofthe several propagation laws are presented. Parameters for the various equations for AA 2024-T3 areobtained through the fitting of prior experimental data. Lastly, it was suggested new scheduling ofmaintenance activities to adjust the manufacturer’s spectrum results to the loading conditions of theaircraft.Keywords: Fatigue, Crack propagation modelling, XFEM, Structural Health Monitoring

1. Introduction

The Epsilon TB 30 fleet is used by the PortugueseAir Force (PoAF) for pilot training. These are tan-dem, two-seater, low wing aircraft with retractablelanding gear and were acquired by PoAF in 1987.

The aircraft was designed with no specific re-quirements for useful life. For this reason, the man-ufacturer Socata performed real scale tests on theseplanes at the Aeronautics Test Centre of Toulouse(CEAT) in 1996 using load spectra acquired from inservice monitoring of the load factor from verticalacceleration data.

These tests resulted in a recommendation of firstinspection of aircraft components to happen for aoperation time of 12400 Flight hours (FH) with oth-ers to follow at an interval of 4000 FH between in-spections.

The research project SHM TB-30 (methodologyand tools for assessing the structural health andlife extension of aircraft), supported by P2020 withpartners: Critical Materials, PoAF and InstitutoSuperior Tecnico consists on the development of adata analysis system through the implementationof advanced tools for the evaluation of the struc-tural condition of a fleet of aircrafts. The monitor-ing of its operation and structural behaviour willbe used to optimize the plane’s maintenance pro-gramme and may make possible the extension of

the fleet’s useful life in comparison to manufacturerrecommendations.

The present work is inserted in this project withthe objective of providing tools for fatigue lifetimeestimation suited to the aircraft and loads provided.

2. Background and Methodology

CEAT’s testing of the aircraft in question [1] de-termined that the critical component whose fatiguefailure caused the end of the plane’s fatigue lifetime,at 89458 FH, was Frame 2. This component servesas the main attachment of both wings to the air-craft’s body, and its location is illustrated in Fig.1.

Figure 1: Frame 2 location on aircraft structure [1].

1

2.1. Manufacturer and PoAF Load SpectraIn CEAT’s report, a reference load spectrum is ob-tained from in-service data of 1000 FH which wasthen used for the real scale tests. In Table 1, all loadfactor peaks larger than nz = 2 are listed. Peaks oflower value were disregarded due to their reducedimportance on the plane’s structural integrity.

Load factor CEAT PrtAF2 10800 11157

2.5 2700 47293 2700 3467

3.5 1980 17654 520 1071

4.5 205 3155 62 147

5.4 23 05.75 8 646.2 2 42

Table 1: CEAT and PrtAF reference load spectra.Represented in effective level crossings of differentload factors

In a previous work presented by Serrano [2], datafor vertical acceleration on the aircraft was collectedin-service from PoAF operations for a period of70.72 FH to which cycle different counting methodswere applied. An equivalent for 1000 FH was madein order to perform a comparison between manufac-turer and PoAF loading easier. The resulting loadhistory is also presented in Table 1.

In order to adjust the maintenance schedulesto PoAF operation, fatigue life was studied usingstress-life and crack propagation methods, and re-sults obtained from each load spectrum were com-pared.

2.2. Stress-life ApproachFor the stress life approach, the mean stress criteriaof Goodman (eq. 1) and Gerber (eq. 2) were chosendue to their wide use in these applications.

σaσar

+σmSu

= 1 (1)

σaσar

+

(σmSu

)2

= 1 (2)

Where σar is the stress amplitude for stress ratioR = 0 that would cause failure for the same numberof cycles as the desired fatigue life.

A correction to the Goodman criterion proposedby Morrow in the SAE Fatigue Handbook [3] wasalso used due to its accuracy when compared toexperimental data. This criterion uses the fatiguestress coefficient S′f from the S-N curve instead ofthe ultimate stress Su to account for the effect ofmean stress:

σaσar

+σmS′f

= 1 (3)

The S-N curve for the AA 2024-T3 used was ob-tained from [4]. Cumulative damage was calculatedfrom the Miner rule.

2.3. Crack Propagation ApproachThe Paris (eq. 4), Forman (eq. 5), Walker (eq. 6)and NASGRO (eq. 7) equations for crack propaga-tion were used after fitted to the experimental dataavailable [4].

da

dN= CP∆KnP (4)

da

dN=

CF∆KnF

(1 −R)KIC − ∆K(5)

da

dN=

CW∆KnW

(1 −R)(1−γ)(6)

da

dN= CN

[(1 − f)∆K

1 −R

]nN(1 − ∆Kth

∆K

)p(1 − Kmax

Kc

)q (7)

The geometric factor for the specific crackgeometry was obtained using the Finite ele-ment method (FEM) and Extended finite elementmethod (XFEM) incorporated in ABAQUS soft-ware. For the modelling of the crack surface in FEMquarter point crack tip elements (Fig. 2) were usedto model the stress singularity. These special hexa-hedral elements collapse the 8 nodes on one of thefaces into a straight line to make a prismatic ge-ometry element, whose coordinate transformationallows for the modelling of the r1/2 singularity typ-ical of LEFM problems.

Figure 2: Hexahedral Quarter point element.

For XFEM, after defining the semi-circular geom-etry of the crack front, the nodal supports for theenriched functions necessary near the crack tip areautomatically atributed by ABAQUS within a userspecified radius.

The geometric factors are calculated from thestress field for both methods using the J-Integralconcept, requiring the definition of several contourssurrounding the crack tip.

2

Due to the varying amplitude of the load spec-tra, direct integration of the propagation laws isimpossible. A cycle-by-cycle integration scheme isimplemented, where the increase in crack growth isobtained for each cycle while SIF information is up-dated from the numerical results to fit the currentcrack size of the iteration.

The integration of the propagation laws wasperformed using the trapezium rule. Below, themethod is shown for a generic crack growth ratefunction F of the SIF range:

∆a ≈ F (∆K(Ni)) + F (∆K(Ni+1))

2(Ni+1 −Ni)

(8)

3. ImplementationA previous study by the PoAF on fatigue of thecomponents involved in the aircraft was performedby Milharadas [5]. An important part of this studywas the use of a strain gauge in the structuralhotspot identified for Frame 2 in CEAT’s report [1]in combination with accelerometer data to extracta transfer function between vertical load factor andstress at the hotspot, thus making possible furtheranalysis.

This transfer function was identified as the linearrelation:

σ = 31.3nz (9)

This knowledge of the location of a structuralhotspot on the component, along with the value ofstress measured at this site allowed for the reverseengineering of a finite element model representativeof the part’s structural behaviour.

3.1. Construction of a FE model of Frame 2Since CAD geometry for the component in ques-tion was unavailable at the start of this work, amodel had to be constructed using direct measure-ment and observation of the structure. Using a dig-ital pachometer and tape measure the Frame wasmodelled into 3D CAD software

Figure 3: CAD model of Frame 2.

From this geometry a finite element mesh wasconstructed using ABAQUS c©,. Initially, the ob-jective was to use exclusively hexahedral elements,

however certain features of added complexity re-quired the use of tetrahedrals, making the finalmesh predominantly hexahedral with the exceptionof these instances.

The material properties for aluminium alloy2024-T3 were obtained from available bibliographyand are shown in Table 2.

E ν ρ

73100 MPa 0.33 2.8e-9 tonne/mm3

Table 2: Material properties used on all calculationsinvolving Al 2024-T3

A mesh convergence study was performed on thegeometry by constructing several meshes of varyingelement overall size and performing a modal analy-sis of the unconstrained structure. The frequency ofthe first mode of vibration was chosen as the con-vergence criterion. This study found that for anoverall mesh size of 6mm a variation smaller than0.02% existed in regards to solutions from more re-fined meshes. This mesh was therefore used on thefollowing analysis.

3 4 5 6 7 8 9

No of Elements 104

149

149.5

150

150.5

151

151.5

152

Fre

quency o

f F

undam

enta

l M

ode

Figure 4: Mesh convergence plot for Frame C2

The boundary conditions applied to the modelwere varied to obtain the results that most accu-rately depicted real observations. Pin joints areused to carry the aerodynamic loads generated inthe wings to the frame, which then cause the accel-eration of the aircraft. To accurately determine theload on these joints the lift distribution around thewing surface should be determined.

With this information unavailable, this distribu-tion was modelled as elliptical, and formula frombibliography used to determine it. The ellipticaldistribution is the ideal one to minimize drag andincrease efficiency of aircraft, and is used manytimes as a design goal.

From Brederode [6], the lift distribution for min-imum drag was derived for the wings’ geometricalproperties:

d

dyL(y) = Asin

(cos-1

(ys

))(10)

3

Where A is a scaling constant for the distribution,and is obtained from performing a force balance onthe aircraft’s vertical direction.∫

dL

dydy = mTB30nzg (11)

With this, and by constructing a finite elementmodel of the wing spar (since the presence of twopinned joints makes the problem hyperstatic), it ispossible to extract the reactions on the pinned jointsbetween the two components for a given load factornz with which the forces vary linearly

Figure 5: Wing spar free body diagram

RzA[N ] RzB [N ] RyA[N ] RyB [N ]

2475.6nz 3894.4nz 46326.4nz −46326.4nz

Table 3: Reactions on wing spar pinned joints foran elliptical lift distribution as linear functions ofthe load factor

These loads were then applied to the structureusing ABAQUS c© multipoint constraints CBEAMconnecting the centre of the lug to the nodes on theinner surface. These impose equal displacement onthe various nodes connected.

Many iterations for displacement constraintswere performed on the model. In the end, the re-quired structural behaviour was achieved by usingsix springs of equal stiffness to the several boltedjoints which attach the frame to the rest of the air-craft’s body.

Using ABAQUS c© snap to ground option forspring elements provided equal stiffness on all dis-placement components for the nodes positioned onthe centre of these bolted joints. From these, thesame CBEAM MPC were used to connect the cen-tral nodes to the cylindrical surfaces of the joints.The stiffness of these spring elements was varieduntil the value for stress at the hotspot resem-bled that measured by Milharadas [5], convergingat 255000N/mm.

3.2. Fatigue Analysis: Stress LifeThe fatigue analysis presented in this work aims tocompare the manufacturer of the aircraft’s referenceload spectra described in CEAT’s report [1] with theone measured in PoAF operation by Serrano, et al[2].

For this effect, the useful life of the plane wasestimated using stress life methods. The Gerber,

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Spring Stiffness [N/mm]105

-30

-20

-10

0

10

20

30

40

50

60

Sig

ned P

erc

ent E

rror

[%]

Figure 6: Spring Stiffness vs. Stress percent errorrelative to Milharadas [5]

mod-Goodman and Morrow criteria were used toconsider mean stress and calculate the number ofcycles to failure for each cycle in the spectra. TheMiner rule for linear cumulative damage was thenused to obtain an estimate of fatigue life for thecomplete variable amplitude spectra.

Parameters for the SN-curve for Al 2024-T3 inR = 0 (Fig. 3.2) were obtained from previous stud-ies [4].

104 105 106 107 108

Number of cycles, N

0

50

100

150

200

250

300

350

400

Sa [

MP

a]

Regression

Exp. data

Figure 7: SN curve for Al 2024-T3 [4]. Sa =1583.4N−0.1603

The failure of Frame 2 observed in CEAT is es-timated from this approach to happen for a dam-age index D = 0.1329, which is much smaller thanunity. The author considers component geometryto play a very significant role in this deviation.Nonetheless, the percent difference in fatigue lifewas taken as a valuable factor portraying the sever-ity of PoAF loading in relation to the manufac-turer’s. In table 3.2 this differences are shown acrossthe several criteria used.

3.3. Geometric Factor Estimation using FEM andXFEM

For this analysis, two different methods were cho-sen. The Extended Finite Element Method, and thestandard Finite Element Method using collapsedquarter point elements at the crack front will be em-ployed to retrieve the stress intensity factors at thecrack front. To do this, however, a detailed model

4

Criteria FH CFAP FH CEAT % diff.Goodman 85958 120676 28.77

Gerber 29950 47733 37.25Morrow 180744 232016 22.10Average - - 29.373

Table 4: Comparison of fatigue life estimates forthe CFAP and CEAT load spectra, obtained for aDamage index D = 0.1329.

for the crack geometry and its location is required.From CEAT’s report [1], failure was found to

arise from unstable crack growth starting in the rootof the fillet of the structure’s innermost ribs, regionpreviously identified as a stress hotspot. The initialcrack geometry was also found to be semi-circularwith a minimum detectable radius of 0.5mm.

Figure 8: Crack surface (in red) embedded in Frame2

For similitude of results obtained, both methodsused a radius of 0.5mm surrounding the crack frontwhere 5 equally spaced contours for J-integral com-putation are defined in a plane perpendicular to thecrack tip.

In both methods, crack size was varied from 1 to6mm in unit increments. This upper limit was cho-sen so that crack geometry could be always assumedas semi-circular.

Stress intensity factors KI , KII and KIII wereextracted for the five contours at each node alongthe crack front. The average value for the sev-eral contours was taken and plotted along the crackfront. In fig. 3.3 the relevance of the mode I SIF incomparison to the rest is visible, which makes it anacceptable parameter to describe crack growth.

Using XFEM allowed for faster computation ofresults, as well as preparation of the models in-volved, due to the independence between crack ge-ometry and the element mesh. For FEM, not onlythis mesh required updates from one crack lengthto the next, the use of quarter point elements tomodel the crack tip singularity required the use ofquadratic elements which made the solution muchmore time consuming.

0 20 40 60 80 100 120 140 160 180

Angle along crack front [Degrees]

-0.5

0

0.5

1

1.5

2

2.5

3

SIF

[M

Pa m

1/2

]

KI

KII

KIII

Figure 9: Plot of mode I, II and III Stress IntensityFactor results along the crack front for XFEM usinga = 3

The results for KI variation in between contours,however, point to increased accuracy on the FEMside, where the percent variation between the firstand last contour does not exceed 0.026%, whereasfor XFEM the difference was 1000 times larger. InFig. 3.3 the solutions for the 5 contours is plottedfor each method at Θ = 0.

1 1.5 2 2.5 3 3.5 4 4.5 5

Contour

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

KI [

MP

a m

1/2

]

XFEM

FEM

Figure 10: Comparison between KI data obtainedfrom the various contours from XFEM and FEM.The data is for Θ = 0 at a crack length a = 3.

Comparing both solutions for KI for the entirecrack front, XFEM seems to consistently overesti-mate the result in regards to the FEM (Fig. 3.3).The Finite Element solution, in addition to show-ing a smoother variation of the SIF along the crackfront, also displays a sudden decrease when ap-proaching the surface of the component. The for-mer behaviour has been demonstrated to happenfor this sort of semi-circular crack in analytical [7]and numerical [8] calculations.

A function of the geometric factor Y with cracklength was retrieved for both the results of XFEMand FEM using adequate polynomial fittings.

3.4. Propagation Law ParametersIn a previous work [4], Serrano performed testson notched specimens of AA 2024-T3 subjected totwo different constant amplitude loading spectra, ofR = 0.1 named the S1 series, and R = 0.3 named

5

0 20 40 60 80 100 120 140 160 180

Angle along crack front [Degrees]

1

1.5

2

2.5

3

3.5

4

4.5

KI [

MP

a m

1/2

]

FEM: a = 1mm

FEM: a = 3mm

FEM: a = 6mm

XFEM: a = 1mm

XFEM: a = 3mm

XFEM: a = 6mm

Figure 11: Comparison between results of modeI Stress Intensity Factor along the crack front forXFEM and standard FEM

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Crack length (a) [mm]

0.68

0.69

0.7

0.71

0.72

0.73

0.74

0.75

0.76

0.77

0.78

Y

y = 0.00025294*x4 - 0.0051202*x

3 + 0.037846*x

2 - 0.12739*x + 0.86766

Y data XFEM

4th degree

R2 = 0.9992

a)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Crack length (a) [mm]

0.64

0.65

0.66

0.67

0.68

0.69

0.7

0.71

0.72

Y

y = 0.0024114*x2 - 0.030693*x + 0.74432

Y data FEM

quadratic

R2 = 0.9679

b)

Figure 12: Geometric factor Y and polynomial fit-tings for: a) XFEM and b) FEM results.

S2. In this work, the results of these tests were usedto fit parameters for the several propagation laws:Paris, Walker, Forman and NASGRO. Multi-linearregressions were performed for the fittings, after ap-plying logarithms to the exponential equations.

The resulting material parameters are describedin Table 5. For the special case of the Paris law,which does not contemplate the effect of the stressratio R, three fits are provided, one for each dif-ferent ratio, and another for the entire set of data(Fig. ??).

For the Paris and Walker laws, and since thesemodel only the linear part of the log plot of thecrack growth rate vs. SIF range, the data used for

4 6 8 10 12 14 16

dK [MPa m1/2]

10-9

10-8

10-7

10-6

da

/dN

[m

/cycle

]

S1 data

S2 data

NASGRO Eq. - S1+S2 (S1 load data)

NASGRO Eq. - S1+S2 (S2 load data)

Figure 13: Crack growth rate versus SIF range re-gression for NASGRO equation

regression was limited to ∆K ≤ 10.For the Forman and NASGRO (Fig. 3.4) equa-

tions the full range of data was used.

CP nP - - R2

Paris S1 3.4x10−11 3.6 - - 0.99

Paris S2 8.4x10−11 3.5 - - 0.99

Paris S1+2 3.1x10−11 3.8 - - 0.99

CF nF - - R2

Forman 3.3x10−9 2.9 - - 0.82

CW nW γ - R2

Walker 2.6x10−11 3.6 0.2083 - 0.99

CN nN p q R2

NASGRO 2.7x10−9 1.8 1 1 0.82

Table 5: Regression parameters for several laws.Reduced precision was used to be able to list theentire data.

3.5. Fatigue Analysis: Crack PropagationWith material parameters for crack growth now de-scribed, and an estimate of the SIF for the crack’sgeometry, it is possible to obtain estimates for fa-tigue life.

A cycle-by-cycle integration scheme was imple-mented, where the increase in crack growth is ob-tained for each individual SIF range using thetrapezium rule on the crack growth rate computed.

Since geometric factor Y data above 6mm is un-available, and in order to avoid inaccurate extrap-olation of the existing data, the failure criteria as-sumed for this analysis was a maximum crack lengthof a = 6mm which although small in comparison tothat achieved in experimental testing already de-scribes a significant part of the propagation time.

For integration of the Paris Law, two different

6

methods were used. First, only the parametersof the regression for both sets of data S1 and S2were used in the cycle-by-cycle scheme. This setof parameters will be described as R1. The secondapproach was made choosing different parameterswith regards to the stress ratio R specified by eachcycle. In this way cycles with an R ≤ 0.2 would usethe properties extracted from the S1 series of data,and for R > 0.2 the S2 only parameters would beused.

The results of using the XFEM or FEM geomet-ric factor curves were compared. In Fig. 3.5 thiscomparison is presented using the Paris law, wherethe tendency of the higher geometric factors com-puted by XFEM to reduce fatigue life is visible, asexpected.

0 0.5 1 1.5 2 2.5

Flight Hours 104

1

2

3

4

5

6

7

Cra

ck L

en

gth

, a

[m

m]

Paris R1 - FEM

Paris R2 - FEM

Paris R1 - XFEM

Paris R2 - XFEM

Figure 14: Comparison of FH until critical cracklength for Paris law regressions R1 and R2 for geo-metric factors obtained from FEM and XFEM.

This behaviour is observed for all propagationlaws. Due to the increased stability between con-tours obtain from the FEM method, the XFEM re-sults were disregarded when comparing the severityof the two load spectra.

0 1 2 3 4 5 6 7

Flight Hours 104

1

2

3

4

5

6

7

Cra

ck L

en

gth

, a

[m

m]

Paris R1 - CFAP

Paris R2 - CFAP

Walker - CFAP

NASGRO - CFAP

Forman - CFAP

Paris R1 - CEAT

Paris R2 - CEAT

Walker - CEAT

NASGRO - CEAT

Forman - CEAT

Figure 15: Comparison of FH until critical cracklength for all propagation laws subjected to loadspectra CEAT 1000 FH and CFAP 70.72 FH.

In Fig. 3.5 it is seen that PoAF loading reducesfatigue life in comparison to the CEAT load spec-tra by and average of 32.15% through the severalequations used.

The NASGRO equation provides a too large es-timate of fatigue lifetime of 64876 FH from a crack

of 1 to 6mm, which when compared to the resultsfrom the CEAT report makes it a non-conservativeestimate. This may happen due to the effect of thecrack opening function delaying propagation for theoccasional cycles of compressive load. The resultsfrom this equation were disregarded in the forwardanalysis

4. Results

Both the stress life and crack propagation ap-proaches to fatigue analysis concluded similarlythat the operation condition of Epsilon aircraftby the PoAF were more severe than those recom-mended by the manufacturer and tested in CEAT[1] by a difference of close to 30% in total useful life.

Crack -

Forman

Crack -

Paris R

1

Crack -

Paris R

2

Crack -

Walker

Slife -

Gerb

er

Slife -

Goodm

an

Slife -

Morro

w

0

5

10

15

20

25

30

35

40

% D

iffe

rence

Figure 16: Average of the percent differences in fa-tigue lifetime found across all fatigue analysis meth-ods.

To describe the structure’s crack propagation be-haviour along its entire lifetime, the crack size datafrom CEAT [1] was adjusted by a factor determinedfrom the average of the relative differences in fatiguelife obtained by the various methods used, includ-ing stress life and propagation laws. In Fig. 4 it canbe observed that this value varies little around anaverage value of 30.76% across the various methods.The NASGRO eq. for crack propagation has beenexcluded as mentioned in sub-section 3.5

A factor of 0.6924 was thus applied to the CEATcurves, from which it is possible to define a schedulefor maintenance operations.

4.1. First Inspection

The manufacturer defined the time of first inspec-tion from the time expected for a crack of 0.5mmto develop, affected by a safety factor of 3. Fromthe adjusted curve, a operating time of 25896 FH isexpected to lead to a crack of this size. The factorschosen by the manufacturer will be kept in theseproposals, making the suggested time for first in-spection 25896

3 = 8632 FH.

7

0 1 2 3 4 5 6 7 8

FH 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

a [m

m]

Adjusted curve

CEAT curve

Expected critical crack size: 45525/3 FH

1st Inspection: 25896/3 FH

Time between following Inspections: 2729 FH

Figure 17: Crack size a for number of flight hourssimulated FH curve obtained from real scale tests,and resulting adjustment for PoAF loading.

4.2. Following InspectionsThe manufacturer defined a crack of 1.5mm in theframe to be of critical size, which cannot be ex-ceeded in the aircraft’s operating life. For a cracksize of 30mm, the part is in the imminence of fail-ure. The manufacturer suggests the use of a safetyfactor of 6 to the time a crack of 1.5mm would taketo grow to these 30mm. From the adjusted curve,we find that a crack size of 1.5mm is reached for45525 FH and after 61901 FH a crack of 30mm isexpected to be found.

From these, the suggested time between subse-quent inspections is:

61901 − 45525

6= 2729 FH (12)

5. ConclusionsThis study proposes that first inspection for crackdetection in the frame is performed after 8500 FH,with the following inspections executed in intervalsof 2700 FH each, until a crack size of 1.5mm isdetected, signalling the end of the aircraft’s usefullife.

These are a significant reduction from the man-ufacturer’s proposed 12400 FH for first inspectionand 4000 for the period between following inspec-tions, but should prevent premature failure due tothe more severe PoAF load spectra.

From the methods used, it can be concluded that:

• The use of XFEM with a reduced mesh over-estimates the SIF results obtained by FEM. Amore stable solution for the extended methodwould require a finer mesh than the 0.5mm el-ement size used around the crack tip (1 ≤ a ≤6mm)

• For the propagation laws, NASGRO tends toestimate the longest life due to the effect ofthe crack opening function and Paris law esti-mates the shortest lifetime. Test specimens for

negative stress ratios may be necessary for re-sults from the cycle-by-cycle integration closerto those obtained in the CEAT tests [1].

AcknowledgementsThe author would like to thank the PortugueseAir Force, especially the staff from the Departmentof Engineering and Programs, Critical Materials,the IST’s department of Mech. Eng. and Profs.Virgınia Infante, Luıs Sousa and Ricardo Baptista,for the opportunity to work with them and all ofthe support provided.

References[1] CEAT. Epsilon - documents de synthese suite

a la campagne d’essai de fatigue. Technicalreport, SOCATA, Groupe Aeroespatiale, 1996.

[2] B. A. S. Serrano, V. Infante, and B. S. D.Marado. Fatigue life tim prediction of poaf ep-silon tb-30 aircraft - part i: Implementation ofdifferent cycle counting methods to predict theaccumulated damage. In Iberian Conferenceon Fracture and Structural Integrity – CIFIE,2010.

[3] Society of Automotive Engineers. SAE FatigueDesign Handbook, 1st edition, 1968.

[4] B. A. S. Serrano, V. Infante, and B. S. D.Marado. Fatigue life time prediction of poafepsilon tb-30 aircraft – implementation of au-tomatic crack growth based on 3d finite ele-ment method. Engineering Failure Analysis,33:17–28, 2013.

[5] Milharadas. Relatorio de tirocınio. Technicalreport, Forca Aerea Portuguesa, 2004.

[6] Vasco de Brederode. Fundamentos de aerodi-namica incompressivel, chapter 10, pages 563–567. IST Press, 2014.

[7] R. J. Hartranft and G. C. Sih. An approxi-mate three-dimensional theory of plates withapplication to crack problems. InternationalJournal of Engineering Science, 8, 1970.

[8] I. S. Raju and J. C. Newman. Stress-intensity factors for two symmetric cornercracks. In Fracture Mechanics; Proceedings ofthe Eleventh National Symposium, 1979.

[9] A. C. Pickard. The application of 3-dimensional finite element methods to fracturemechanics and fatigue life prediction. Warley,U.K., 1986.

[10] N. E. Dowmling. Mechanical Behaviour of Ma-terials. Pearson Education Limited, 2013.

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[11] David Broek. Elementary Engineering Frac-ture Mechanics. Sijthoff & Noordhoff Interna-tional Publishers B.V., 1978.

[12] S. R. Mettu, J. M. Shivakumar, F. Beek, L. C.Williams, R. G. Forman, McMahon J. J., andJ. C. Newman. Nasgro 3.0: A software foranalyzing aging aircraft. In The Second JointNASA/FAA/DoD Conference on Aging Air-craft.

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