Mechanical Study of an Aircraft's Structural Condition Mechanical

Mechanical Study of an Aircraft’s Structural Condition

Tiago Alexandre Rosado Martins

Thesis to obtain the Master of Science Degree in Mechanical Engineering

Supervisors: Prof. Virgínia Isabel Monteiro Nabais Infante Prof. Luís Alberto Gonçalves de Sousa

Examination Committee Chairperson: Prof. Luis Filipe Galrão dos Reis Supervisor: Prof. Virgínia Isabel Monteiro Nabais Infante Member of the Committee: Prof. Ricardo Miguel Gomes Simões Baptista

June 2019 ii Acknowledgments

I would like to express my gratitude to professors Lu´ıs Sousa and Virg´ınia Infante for the invitation to write this thesis and for their continuous support and attention whenever necessary. I also wish to thank the kind people of the Portuguese Air Force’s Department of Programs and Engineering, Capts. Serrano, Prieto, Cardoso and Bastos and Lts. Diniz, ... for their support and guidance. My thanks to the staff of Esquadra 101 at Base Aerea´ no 1 for having received me and allowing me to spend many hours around their stunning aircraft. I would also like to thank the staff at Critical Materials, especially Doc. Paulo Antunes for the numerous advices regarding ﬁnite element analysis and software. Also, many thanks to Prof. Ricardo Baptista for the advice regarding crack propagation and his willingness to give them. To my classmates which provided me with hours of much welcome company in the writting of this thesis, Bogdan Sandu, Lu´ıs Abreu, Ricardo Ferreira and Beatriz Lopes. Lastly, I would like to thank my family and friends, near and far, whose love and support has kept me going through all of this life’s trials. Without you the journey so far would been much less joyful.

iii Resumo

O objectivo do presente trabalho e´ a estimaçao˜ do tempo de vida a` fadiga da frota Epsilon TB-30 da Força Aerea´ Portuguesa, e a instalaçao˜ de um sistema de aquisiçao˜ de dados de aceleraçao˜ e extensometria nos componentes cr´ıticos em voo, que servira´ no futuro como input para a interface de Structural Health Monitoring PRODDIA c . Informaçao˜ de estudos experimentais anteriores e dos carregamentos na aeronave foi usada num processo de engenharia inversa na construçao˜ de um modelo de elementos finitos do componente cr´ıtico do aviao.˜ Deste modelo, estimaram-se os campos de tensoes˜ para hotspots estruturais previamente identificados. O dano acumulado na estrutura foi calculado com base em carregamentos de referencia,ˆ usando a regra de Miner e criterios´ de fadiga apropriados. Os factores de intensidade de tensao˜ para as fendas previstas por analises´ anteriores foram estimados utilizando os metodos´ dos elementos finitos, e o mais recente XFEM integrados no software ABAQUS c . A propagaçao˜ de fenda foi depois modelada fazendo uso das leis de Paris, Walker, NASGRO e Forman. Os parametrosˆ utilizados nas varias´ equaçoes˜ foram obtidos atraves´ de dados experimentais de ensaios anteriores para a liga AA 2024-T3. Por ultimo,´ o autor sugere uma revisao˜ ao programa de manutençao˜ das aeronaves atraves´ do ajuste dos per´ıodos de inspeçao˜ definidos pelo fabricante.

Palavras-chave: Fadiga, Propagac¸ao˜ de Fenda, Structural Health Monitoring, XFEM

iv Abstract

The purpose of this work is the estimation of the Fatigue Lifetime of the Portuguese Air Force’s Epsilon TB-30 fleet, and the installation of a data acquisition system to measure vertical acceleration and strain on critical components of the aircraft which will later serve as input to the Structural Health Monitoring interface PRODDIA c . Experimental studies and prior knowledge of aircraft loads was used to reverse engineer a suitable Finite Element Model of 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-Miner rule and adequate fatigue criteria. The Stress Intensity Factors at the critical component’s notched geometry were estimated using the Finite Element and Extended Finite Element Methods from ABAQUS c . Crack propagation was then modelled using various laws (Paris, Walker, NASGRO, Forman). A comparison between the two numerical methods and the results of the several propagation laws is presented. Parameters for the various equations for AA 2024-T3 are obtained through the fitting of experimental data. Lastly, the author suggests a revision of the aircraft’s maintenance program through the adjustment of the inspection periods defined by the manufacturer.

Keywords: Fatigue, Crack propagation modelling, Structural Health Monitoring, XFEM

v Contents

Acknowledgments ...... iii Resumo...... iv Abstract...... v List of Tables...... ix List of Figures ...... x List of Abbreviations...... xiii List of Symbols...... xv

1 Introduction 1 1.1 Background...... 1 1.2 Motivation and Objectives...... 3 1.3 Thesis Outline ...... 4

2 Literature Review 5 2.1 An overview on the study of Fatigue ...... 5 2.2 Physical Mechanisms of Fatigue ...... 7 2.3 Stress-Life Approach to Fatigue in Metals...... 8 2.3.1 Stress-Life (S-N) curves...... 8 2.3.2 Mean Stress Inﬂuence and Criteria...... 9 2.3.3 Cumulative Damage...... 11 2.3.4 Other Factors Affecting Fatigue Life ...... 12 2.4 Linear Elastic Fracture Mechanics (LEFM)...... 12 2.4.1 The Grifﬁth Criterion...... 16 2.4.2 The J-Integral Concept ...... 17 2.5 Crack Propagation...... 18 2.6 Finite Element Method applied to Fracture Mechanics ...... 21 2.6.1 Displacement Extrapolation Method ...... 22 2.6.2 Crack tip elements...... 22 2.6.3 The Energy Method ...... 23 2.6.4 The J-Integral Method...... 24 2.7 Extended Finite Element Method (XFEM) ...... 25 2.8 Structural Health Monitoring...... 26 2.8.1 The Forward Problem ...... 27

vi 2.8.2 The Inverse Problem...... 27 2.9 Manufacturer’s Results for Aircraft Useful Life...... 27

3 Experimental Activities 28 3.1 Strain Gauges ...... 29 3.1.1 Sensor Principle and Description...... 29 3.1.2 Testing and Calibration ...... 29 3.1.3 Aircraft installation...... 31 3.2 Accelerometers...... 31 3.2.1 Sensor Principle and Description...... 31 3.2.2 Testing and Calibration ...... 32 3.2.3 Aircraft Installation...... 32

4 Finite Element Modelling of Aircraft Structures 33 4.1 Objective of Finite Element Modelling ...... 33 4.2 Real Structure ...... 33 4.3 CAD Modelling...... 35 4.4 Finite Element Mesh...... 36 4.5 Boundary Conditions...... 38 4.5.1 Loading Conditions ...... 38 4.5.2 Constraints...... 44 4.5.3 Structural Hotspots...... 47 4.6 Dynamic Analysis ...... 48

5 Fatigue Analysis 51 5.1 Load Spectra for Fatigue Analysis ...... 51 5.2 Stress-Life Approach...... 52 5.3 PRODDIA c stress-life calculation ...... 53 5.4 Crack Propagation...... 54 5.4.1 Crack Location and Geometry ...... 55 5.4.2 Geometric Factor Estimation: XFEM...... 55 5.4.3 Geometric Factor Estimation: FEM...... 58 5.4.4 Parameter Fitting for Propagation Laws ...... 60 5.4.5 Fatigue Lifetime Estimation using Crack Propagation Models ...... 65

6 Final Remarks 69 6.1 Proposal of Maintenance Scheduling...... 69 6.1.1 First Inspection...... 70 6.1.2 Following Inspections ...... 71 6.2 Future Work...... 71 6.2.1 Relating Sensor Data to Frame damage...... 71

vii 6.2.2 Fatigue Analysis of Wing Spar ...... 72 6.2.3 Application of Crack Arresting Solutions...... 72

References 75

Appendix A Technical Datasheets 79

viii List of Tables

Table 1.1 Aircraft Parameters ...... 2

Table 3.1 Young’s modulus computed from linear regression of sensor data, and respective percent error to the theoretical value...... 31

Table 4.1 Material properties used on all calculations involving AA 2024-T3 ...... 36 Table 4.2 Reactions on wing spar pinned joints for a rectangular lift distribution, calculated through ﬁnite element model ...... 42 Table 4.3 Reactions on wing spar pinned joints for an elliptical lift distribution for a load factor

nz = 1, calculated using a ﬁnite-element model...... 43 Table 4.4 Maximum Principal Stress and Biaxiality ratio at relevant hotspots...... 48

Table 5.1 CEAT 1000FH load spectrum used in real scale testing [2]. Represented in cumulated and effective level crossings of different load factors ...... 52 Table 5.2 CFAP 70.72FH load spectrum equivalent for 1000FH[4]. Represented in cumulated and effective level crossings of different load factors ...... 52 Table 5.3 Comparison of fatigue life estimates for the CFAP and CEAT load spectra, obtained for a Damage index d = 0.1329...... 53 Table 5.4 Regression parameters for Paris Law...... 63 Table 5.5 Comparison of fatigue life estimates for the CFAP and CEAT load spectra, from crack propagation methods assuming an initial crack of 1mm...... 68

ix List of Figures

Figure 1.1 PrtAF Epsilon TB-30 aircraft...... 1 Figure 1.2 Crack propagation on the part as reported by the manufacturer [2]...... 2 Figure 1.3 Accelerometer installed by Milharadas [3] and location on aircraft...... 2 Figure 1.4 Strain gauge installed by Milharadas [3] and location on aircraft...... 3

Figure 2.1 Stress versus life (S-N) curve fitted to data from unnotched specimens of AA 7075- T6 in rotating bending [8]. The plot on the left is presented in a linear scale, while on the right, the same values are shown with cycles to failure in log scale...... 6 Figure 2.2 Cyclic constant amplitude loading and descriptive parameters [8]...... 8 Figure 2.3 Typical S-N curve for steels, adapted [5]...... 9 Figure 2.4 Effect of mean stress on fatigue life for AA 7075-T6 [8]...... 10 Figure 2.5 Normalized stress amplitude vs mean stress plot for AA 7075-T6 [8]...... 10 Figure 2.6 In flight load factor data from Epsilon TB-30 aircraft...... 11 Figure 2.7 Crack (light area) growing from a large non-metallic inclusion (dark area) in a steel artillery tube [8]...... 13 Figure 2.8 The three modes of crack loading [15]...... 13 Figure 2.9 Crack in infinite plate in Mode I, adapted [15]...... 14

Figure 2.10 Elastic stress σy distribution close to the crack tip [15]...... 14

Figure 2.11 Approximate σy stress distribution at crack tip including plastic zone [15]...... 15 Figure 2.12 The Griffith criterion for fixed grips [15]...... 16 Figure 2.13 Experimental data for fatigue crack growth on an A533B-1 steel alloy, adapted from Dowling [8]...... 19 Figure 2.14 Influence of the stress ratio of crack growth rate data [17]...... 19 Figure 2.15 Typical mesh refinement around a crack tip [26]...... 23 Figure 2.16 Collapsed quarter point Elements...... 23 Figure 2.17 Integral contours selection in Abaqus [27]...... 24

Figure 3.1 Schematic of the implementation of the SHM system. FDR: Flight Data Recorder, SDR: Strain Data Recorder, OLM: Operational Load Monitoring, RUL: Remaining Useful Life, MTBF: Mean Time Between Failure...... 28 Figure 3.2 DT 3757-5 full bridge strain gauge, taken from the model’s datasheet ...... 29 Figure 3.3 Aluminium 2024-T3 specimen for tensile test...... 30

x Figure 3.4 Strain vs. Force data from tensile test and respective error compared to theoretical values...... 30 Figure 3.5 Strain vs. Force data from tensile test and respective error compared to theoretical values...... 31 Figure 3.6 SA-102MFTB accelerometer...... 32

Figure 4.1 Airplane general structure [37] ...... 34 Figure 4.2 Detail of the pinned connection between Frame C2 and the Main wing spar (CAD model)...... 34 Figure 4.3 Components’ visible geometry during measurement ...... 35 Figure 4.4 CAD models of aircraft components ...... 36 Figure 4.5 Finite element mesh of frame C2 using 8mm overall element size...... 37 Figure 4.6 Detail of the finite element mesh of the frame using 8mm overall element size. Note the 3 elements across the thickness of the small rib and the use of tetrahedral elements on the filleted region...... 37 Figure 4.7 Finite element mesh of the Main wing spar using 16mm overall element size . . . 37 Figure 4.8 Mesh convergence plot for Frame C2...... 38 Figure 4.9 Mesh convergence plot for wing spar...... 38 Figure 4.10 Airplane Free Body Diagram in Steady Level Flight...... 39 Figure 4.11 Airplane Free Body Diagram using the Load Factor for vertical force...... 40 Figure 4.12 Wing spar Free Body Diagram ...... 41 Figure 4.13 Max. Principal Stress Plot for constrained structure...... 44 Figure 4.14 Max. Principal Stress Plot for unconstrained structure...... 45 Figure 4.15 Spring Stiffness vs. Stress percent error relative to Milharadas [3]...... 46 Figure 4.16 Maximum Stress plot for Spring Model...... 46 Figure 4.17 Details of maximum stress plot (scale in a) )...... 47 Figure 4.18 Maximum Principal Stress plot with identified hotspots...... 48 Figure 4.19 Load factor vs. time plot of the data provided ...... 49 Figure 4.20 Load factor vs. time plot of the relevant data sample...... 49 Figure 4.21 Hotspot Max. Princ. Stress vs. vertical acceleration. Plots provided by Critical Materials ...... 50

Figure 5.1 S-N curve regression for experimental data obtained by Serrano [4] ...... 52 Figure 5.2 Rainﬂow counting method applied to max. principal stress results from FE analysis. 54 Figure 5.3 Hotspot damage computed from post-processing of FEM results by Critical Materials. 54 Figure 5.4 Detail of the Hotspot region showing gradient of max. principal stress...... 55 Figure 5.5 Crack surface (in red) embedded in Frame 2 ...... 56 Figure 5.6 Crack surface (in red) embedded in XFEM mesh...... 56 Figure 5.7 Plot of mode I, II and III Stress Intensity Factor results along the crack front for XFEM using a = 3 ...... 57

xi Figure 5.8 Plot of mode I Stress Intensity Factor results along the crack front with varying crack length for XFEM...... 57 Figure 5.9 Geometric factor Y and polynomial ﬁtting for XFEM analysis...... 58 Figure 5.10 Details of maximum stress plot (scale in a) )...... 59

Figure 5.11 Comparison between KI data obtained from the various contours from XFEM and FEM. The data is for Θ = 0 at a crack length a = 3...... 59 Figure 5.12 Plot of mode I, II and III Stress Intensity Factor results along the crack front for standard FEM using a = 3. Re-sampled to display 1 point every 2.5o ...... 60 Figure 5.13 Comparison between results of mode I Stress Intensity Factor along the crack front for XFEM and standard FEM ...... 60 Figure 5.14 Geometric factor Y and polynomial ﬁtting for FEM analysis...... 61 Figure 5.15 Crack propagation specimen used by Serrano [4]...... 61 Figure 5.16 Crack Length vs number of cycles data from experimental testing, translated from [4]...... 62 Figure 5.17 Crack Growth rate versus SIF range regressions for Paris Law...... 64 Figure 5.18 Crack growth rate versus SIF range regressions for Walker and Forman equations. 64 Figure 5.19 Crack growth rate versus SIF range regressions for Walker and Forman equations. 65 Figure 5.20 Comparison ofFH until critical crack length for Paris law regressions R1 and R2 for geometric factors obtained from FEM and XFEM...... 66 Figure 5.21 Comparison ofFH until critical crack length for various propagation laws for geometric factors obtained from FEM and XFEM...... 67 Figure 5.22 Comparison ofFH until critical crack length for all propagation laws subjected to load spectra CEAT 1000FH and CFAP 70.72FH...... 67

Figure 6.1 Average of the percent differences in fatigue lifetime found across all fatigue analysis methods...... 69 Figure 6.2 Crack size a for number of ﬂight hours simulatedFH curve obtained from real scale tests, and resulting adjustment for PrtAF loading...... 70 Figure 6.3 Inspection times for PrtAF loading...... 70 Figure 6.4 Cracks found on the wing spars on real scale tests [2]...... 72 Figure 6.5 Improved fatigue life from varying diameter stop holes [47]...... 73 Figure 6.6 Improved fatigue life from CFRP reinforcements on both sides of a steel specimen [50]...... 73

xii List of Abbreviations

ASTM American Society for Testing and Materials.7

CAD Computer Aided Design. vii, xi, 34–36

CEAT Centre D’Essais Aeronautiques´ de Toulouse (Toulouse Centre of Aeronautic Testing). ix, xii,1, 2, 27, 33, 44, 45, 51–53, 55, 60, 67, 68

CFRP Carbon Fiber Reinforced Polymer. xii, 73

CTOD Crack Tip Opening Displacement.7

DAU Data Acquisition Unit. 29, 32, 79

EFGM Element Free Galerkin Methods. 25

EPFM Elastic Plastic Fracture Mechanics.7

FDR Flight Data Recorder.x, 28

FEM Finite Element Method. xii,4, 22, 25, 45, 55, 57–62, 66, 67

FE Finite Element. 22, 24, 33, 35

FH Flight Hours. ix, xii, 27, 52, 53, 66–68, 70, 71

GNP Gross National Product.5

IST Instituto Superior Tecnico.´ 3

LEFM Linear Elastic Fracture Mechanics. vi,6, 12, 15, 22, 23, 25

MPC Multi Point Constraint. 41, 44, 45

MTBF Mean Time Between Failure.x, 28

NDT Non Destructive Testing. 70

OLM Operational Load Monitoring.x, 28

PUM Partition of Unity Method. 25

PU Partition of Unity. 25

xiii PrtAF Portuguese Air Force.x, xii,1–4, 29, 30, 51–53, 69–72

RUL Remaining Useful Life.x, 27, 28

SAE Society of Automotive Engineers. 11

SDR Strain Data Recorder.x, 28

SHM Structural Health Monitoring.x,3,4, 26–28, 33, 71

SIF Stress Instensity Factor. xii,3,4,6, 16–18, 20, 22–24, 56–59, 63–67

XFEM Extended Finite Element Method. vi, vii, xi, xii,4, 25, 55, 57–60, 66, 67

xiv List of Symbols

Greek symbols

δ Finite element displacement Vector.

∆σ Range of stress in a load cycle, deﬁned as the difference from maximum to minimum stress.

General designation for strain in a material.

ηj Parameters to be ﬁtted by multi-linear regression.

Γ Aerodynamic circulation.

γ Walker law load ratio control parameter.

λ Biaxiality ratio.

ν Material poisson’s ratio.

φ Finite element shape functions.

Π Potential energy from body deformation.

Ψ Vector of asymptotic functions ψ used in xFEM.

ψ Asymptotic function used in xFEM.

ρ Air density.

σ General designation for stress in a material.

σx, σy, σz Normal components of stress.

σar SN curve alternating stress for the speciﬁc case of R = 0.

∆σ σa Alternating stress in a load cycle, deﬁned as 2 .

σij Stress tensor.

σmax Maximum stress in a load cycle.

σmin Minimum stress in a load cycle.

σmax+σmin σm Mean stress in a load cycle, deﬁned as 2 .

xv σo Plastic ﬂow stress.

τxy Shear stress in plane xy.

Θ Angle describing the contour of thee semi-circular crack front.

θ Angular coordinate in polar coordinate system.

ζ Arbitrary contour for J-integral computation.

ζc Contour described by crack boundary.

Su Material ultimate stress from tensile testing.

Sy Material yield stress from tensile testing.

Roman symbols

∆A Variation in crack surface area between iterations

∆K Range of stress intensity factors over a load cycle.

∆Keff Effective stress intensity factor range for which the crack tip is open.

∆Kth Threshold stress intensity factor range.

da dN Crack growth rate. a Length of a crack.

A0,A1,A2,A3 Coefﬁcients for Newman crack tip oppening function az Vertical acceleration.

AR Wing aspect ratio b Exponent for the S-N curve.

Bn Coefﬁcients for elliptical lift distribution c Constants in linear regression equation. c0, c2 Constants of ﬁnite element coordinate transformation

CP ,CW ,CN ,CF Coefﬁcients for propagation laws. Paris, Walker, NASGRO and Forman, respectively.

D Damage index from cummulative damage laws.

E Material Young’s Modulus. ei Regression error of the the ith point in the data set.

F Finite element force vector. f Newman’s crack tip opening function.

xvi Fx,Fy,Fz Loads acting on a structure in the x,y and z directions. fz Vertical force on a wing per unit length. fij Geometric function describing the stress ﬁeld near a crack.

G Elastic energy release rate. g Constant of gravitational acceleration 9.81 m/s2

H Heaviside function. i, j Vector or matrix indices.

J J-Integral. k Finite element stiffness matrix.

K,KI ,KII ,KIII stress intensity factor, and its respective mode I, II and III contributions.

KIc Critical stress intensity factor for mode I loading.

Kmax Maximum stress intensity factor in a load cycle.

Kmin Minimum stress intensity factor in a load cycle.

L, D Aerodynamic forces, lift and drag respectively.

M Sets of nodes affected by the different shape functions in XFEM m Total mass of the aircraft.

M c,M f Sets of nodes whose shape function supports are intersected by the crack surface and the crack front, respectively.

N General designation for number of cycles in fatigue loading.

Nf Fatigue life in number of cycles to failure.

Ni Number of cycles for technical crack initiation.

Ni Number of load cycles of a speciﬁc type existing in a spectrum. ni Number of load cycles of a speciﬁc type that a component can endure without failure.

Np Number of cycles for propagation of existing crack to failure. nP , nW , nN , nF Exponents for propagation laws. Paris, Walker, NASGRO and Forman, respectively. nz Aircraft vertical load factor p, q NASGRO eq. curvature control parameters.

R Load ratio, deﬁned as the ratio between minimum and maximum stress in a load cycle.

xvii r Radial coordinate in polar coordinate system.

Reff Effective load ratio.

RyA,RyB,RzA,RzB Horizontal (y) and vertical z reactions on the frame’s pin joints A and B. s Wing span.

0 Sf Coefﬁcient for the S-N curve.

Sa Obtained from the S-N curve, it is the value of alternating stress that will cause part failure for a number of cycles N.

6 Se Fatigue endurance limit of a material, deﬁned as the stress causing component failure for 10 load cycles.

So Crack closure stress. t Traction vector for J-Integral calculation

U Crack surface energy. u Displacement function over a domain. uf Finite element displacement function. ui Displacement components.

U∞ Undisturbed air velocity, equal to aircraft velocity. v, w Nodal values for XFEM shape functions.

W Strain energy. x, y, z Spatial coordinates in catesian coordinate system. x∗ Spatial coordinate of the point on the crack boundary closest to the point x in study, from XFEM. xi, yi Abcissa of the points to be ﬁtted in a multi-linear regression and corresponding images of the ﬁtting function.

Xij Matrix of regression equations.

Y Geometric factor from linear elastic frature mechanics.

xviii Chapter 1

Introduction

1.1 Background

The Epsilon TB-30 fleet, shown in Fig.1.1 is used by the Portuguese Air Force (PrtAF). These are tandem, two-seater, low wing aircraft with retractable landing gear. A six horizontal cylinder engine powers the single constant speed propeller located at the nose of the airplane with a fuelling device which allows fuel injection to the engine even in inverted flight. Its ability to withstand vertical accelerations from -3.35G to 6.7G, and its design of the dashboard commands and instrument displays make it resemble a small commercial light fighter aircraft [1]. Additional specifications are provided in table 1.1.

Figure 1.1: PrtAF Epsilon TB-30 aircraft

To determine fatigue life of the aircraft and establish critical locations of its structure, real scale tests were performed by the manufacturer on the Centre D’Essais Aeronautique de Toulouse (CEAT), where its design loads were recreated in laboratory condition [2]. The structure was afterwards checked for structural failure, and it was concluded that the component of the structure where failure should ﬁrst occur is the airplane frame 2 which attaches the wings to the aircraft’s body through pinned joints. Failure occurred due to fatigue crack propagation (Fig. 1.2).

1 Wingspan Wing Area Aspect Ratio 7.92m 9m2 7

Never Exceed Speed Cruise Speed Range 520km/h 358km/h 1300km

Max. Takeoff Weight Empty Weight Engine Power 1300kg 930kg 300HP

Table 1.1: Aircraft Parameters

Figure 1.2: Crack propagation on the part as reported by the manufacturer [2]

These real scale tests estabilished the fatigue lifetime and maintenance schedules of the aircraft based on the manufacturer’s design loads. The PrtAF however, to be able to compare these values and verify this fatigue lifetime when subjected to their missions’ load spectra, instrumented one aircraft to extract the fleet’s specific load spectrum, using both experimental and numerical methods to compare the expected fatigue lifetime with that obtained by the CEAT experiment in order to adapt their own maintenance schedule. This was the objective of Milharadas [3], whose work included the placement of a strain gauge on the hotspot of Frame 2 (Fig. 1.4) identified by the CEAT report and monitor stresses along with their variation with aircraft load factor measured by an accelerometer (Fig. 1.3). A transfer function which related component stress with vertical load was obtained.

Figure 1.3: Accelerometer installed by Milharadas [3] and location on aircraft

2 Figure 1.4: Strain gauge installed by Milharadas [3] and location on aircraft

Later, the development of representative notched specimens of the hotspot region were constructed and tried to failure using both an equivalent of the measured load spectrum for the PrtAF operations and the manufacturer’s spectrum to compare the fatigue lifetime expected for each. With the acquisition of a new load spectrum representative of the aircraft’s operation, Serrano [4] would repeat this testing using the same design of test specimens. Serrano coupled this analysis with a computational crack propagation approach which calculated the Stress Intensity Factors (SIF) on the notched specimen geometry to estimate crack growth for the applied spectra. For this thesis, the author proposes the calculation of SIF on the modelled component geometry to provide a more accurate description of crack growth with applied loading. The measurement of strain data will also be extended to other identiﬁed critical components allowing future reﬁnement of the aircraft useful life calculations.

1.2 Motivation and Objectives

The research project SHM TB-30 (methodology and tools for assessing the structural health and life extension of aircraft), supported by P2020 with partners: Critical Materials, PrtAF and Instituto Superior Tecnico´ (IST) consists on the development of a data analysis system through the implementation of advanced tools for the evaluation of the structural condition of a fleet of aircraft. The monitoring of its operation and structural behaviour will be used to optimize the plane’s maintenance programme and in the future this information can be used to better evaluate the fatigue lifetime. The present thesis is inserted in this project and its objective is to help build the system responsible for flight data acquisition, and also to develop knowledge and analysis tools on the mechanical behaviour of the structure and for the estimation of fatigue lifetime of its critical components subjected to PrtAF operation. The work advanced in this thesis will help to define better schedules for preventive maintenance of the Epsilon TB-30 as well as develop computational tools and define new methods for future work of this kind for similar parts of the TB-30 and other PrtAF aircraft.The development of representative finite element models of the aircraft’s critical structures serves as input to refine the algorithms used on the structural

3 health monitoring platform PRODDIA c , which is to be implemented for this speciﬁc aircraft.

1.3 Thesis Outline

This thesis contains six chapters, including this introduction. In chapter 2, a literature review is presented, explaining the various concepts used throughout the thesis. Its introduced with a brief explanation on the concept and mechanisms of fatigue of materials and the importance for modern engineering of modelling this process. The stress-life and crack propagation approaches for fatigue analysis are presented. The Finite Element Method (FEM) and the more recent Extended Finite Element Method (XFEM) are explained as tools for the determination of crack Stress Intensity Factor. A small introduction to Structural Health Monitoring (SHM) systems is also presented. Chapter 3 shows the installation of the Data Acquisition unit which will monitor the aircraft’s structural hotspots and acquire mission data as input to the PRODDIA c SHM platform. The calibration procedures for the instruments used are also described. In chapter 4 a reverse engineering approach is followed to model the stresses on Frame 2, using knowledge of aircraft loading conditions and prior experimental data. A ﬁnite element model is produced and boundary conditions are varied to obtain the expected structural behaviour, which culminates on the suggestion of additional stress hotspots to monitor. A dynamic analysis of the structure is used to validate the linearity of prior observed stress transfer functions. Chapter 5 deals with the fatigue analysis of the critical component, starting with the stress-life, cumulative damage approach and comparison between both the manufacturer and the PrtAF’s load spectra. The Finite Element and Extended Finite Element Methods are employed to estimate the SIFs for various crack sizes. Existing experimental data is ﬁtted to a variety of propagation laws which are later used to compare the component lifetime results between the manufacturer and PrtAF’s loading conditions. Conclusions are taken in chapter 6, where a new scheduling of maintenance activities is suggested for the aircraft based on the results from fatigue analysis, along with the implementation of reinforcements and other solutions to delay fatigue crack growth.

4 Chapter 2

Literature Review

2.1 An overview on the study of Fatigue

As described by Budynas [5], fatigue failure is observed in components subjected to time-varying stresses, many times well below the static limits of the material. Unlike static failure, where considerable deformation may occur prior to component failure, fatigue causes most materials to break in a brittle manner making it much more dangerous as it is harder to predict when the part will fail. The overall cost of fracture in advanced nations is estimated as taking up 4% of the GNP for Europe and America [6]. Within these costs, 80% are considered to be due to fatigue failure exclusively, making it the most frequent cause of component failure with costs comparable to some of the largest budget items of the economy of a country such as the U.K. The increase in knowledge in this sector has a signiﬁcant contribution to reducing expenditures on component maintenance as well as to the increased safety of consumers. The ﬁrst systematic studies on fracture, according to Krupp [7] may have started in the 15th century, when the genius Leonardo da Vinci tested the strength of metallic wires, and concluded from his observations, that the strength of the wires decreased with their increased length due to the increasing presence of defects in the material. It wasn’t however until the 18th century, with the invention of the steam engine by James Watt that the use of rotating machinery started to attract attention to the problem of complex cyclic loadings. Frequent accidents caused by mechanical failure of metal axles made engineers realize there is a limited service life to such components. One particular accident that gave mobility to the study of this topic occurred in May 1842 when a train making its way from king Louis Phillipe’s birthday celebration in Versailles to Paris crashed due to the mechanical failure of an axle causing the deaths of a presumed 50 people. Upon inspection of the fractured axle, several theories arose to explain the event. Some attributed the accident to changes in time to the material’s microstructure due to vibrations, heat and magnetic induction, which decreased its properties and caused sudden failure. Others, such as William J. Rankine however, discarded the time-dependent changes to the material, and upon inspection of the fracture

5 surface, suggested a reduction of the load-carrying surface to a point where it cannot withstand the load, causing failure. Later, experimental studies regarding fatigue were conducted by August Wohler¨ who measured strain during train journeys on a 5000km reference route. It was his successor Spangenberg however who summarized his results on a table and plotted the stress amplitudes vs number of cycles, as exempliﬁed in Fig. 2.1, resulting in the S-N curve still known today as the Wohler¨ curve.

Figure 2.1: Stress versus life (S-N) curve ﬁtted to data from unnotched specimens of AA 7075-T6 in rotating bending [8]. The plot on the left is presented in a linear scale, while on the right, the same values are shown with cycles to failure in log scale

Despite the efforts of Wohler¨ and his successors however, there was still a lack of knowledge about the phenomenon of fatigue by the start of the 20th century. It wasn’t until the rise of high resolution microscopy in the 20th century that it was possible to observe the microstructural mechanisms leading to crack formation. Ewing and Humfrey’s studies in 1903 on the growth and formation of slip bands on the polished surface of rotating bending steel specimens, eventually developing into fatigue cracks was among the first direct observations of fatigue’s microstrucural mechanisms. By the end of the first half of the twentieth century, most of the stress-life methods were already defined. A new concept however, revisiting the ideas of Da Vinci of present defects in materials had become a subject of study, named fracture mechanics. This concept focuses on the local study of these inhomogeneities, especially cracks, by the definition of stress parameters as tools for the estimation of crack propagation. From World War II onwards, and due to the increasing number of plane crashes caused by fatigue failures, aircraft construction became a driving force for fatigue related research. The rise of electronic microscopy made possible for newer developments concerning the mechanisms of fatigue to be attained. During the second half of the 20th century, many contributions were made to the fracture mechanical treatment of crack propagation. Despite the known occurrence of plasticity at the crack tip region Linear Elastic Fracture Mechanics (LEFM) became a well established tool to study propagation, using the Stress Intensity Factor (SIF) as a universal parameter independent of part dimensions. By the 1970s, due to the importance of crack tip plasticity in many applications, Elastic Plastic Fracture

6 Mechanics (EPFM) methods using the J-Integral and Crack Tip Opening Displacement (CTOD) parameters were developed to study crack propagation. By 1969, over 21000 publications regarding fatigue had already been published [9]. This number is estimated to have grown exponentially to common days. However, despite the many advances in fatigue, this is still a very active ﬁeld of research and failure of components still drives researchers to ﬁnd new methods for the design of more durable structures and more accurate estimates of fatigue life.

2.2 Physical Mechanisms of Fatigue

The ASTM[10] defines fatigue as a ”the process of progressive localized permanent structural change occurring in a material subjected to conditions that produce fluctuating stresses and strains at some point or points and that may culminate in cracks or complete fracture after a sufficient number of fluctuations”. The processes of initiation and propagation of these cracks are caused by different mechanisms. The total fatigue life Nf of an unnotched specimen can be separated into two phases [11]:

Nf = Ni + Np (2.1)

Where Ni is the number of cycles leading to crack initiation, and Np the cycles causing crack propagation until the eventual failure of the component. The ratio between these two can vary considerable, whether one is speaking of high stress/strain loadings, where the propagation phase dominates, or low stress/strain where the opposite is the case. The initiation phase generally takes place at smaller scales, as stated by Krupp [7], one can go as small as considering initiation to occur due to the overcoming of inter-atomic forces. It is however more relevant to consider the events leading to initiation in the same scale as the material’s signiﬁcant microstructural features, such as grain size or the size of the inherently present defects. At the scale of a metal’s crystal grains, the material cannot be described as either homogeneous or isotropic [7], making its local behaviour dependent on grain shape and orientation. The presence of small voids or inclusions of different particles provide additional inhomogeneity to the material’s microstructure and contribute to the already complex distribution of local stresses. Fatigue damage starts preferentially at points of concentration of stress. In more ductile materials, cracks are usually initiated from slip bands forming in grains with an orientation which makes them less resistant. Various slip bands may form and as they grow more and more severe, develop into cracks. These may eventually grow out of grain boundaries and join with others to become long cracks and enter the propagation phase. In more brittle materials, such as ceramics or high alloy metals, crack initiation is likely to occur in pre- existing voids or other ﬂaws promoting stress concentration.

7 2.3 Stress-Life Approach to Fatigue in Metals

It is the existence of a ﬂuctuating load which makes failure due to fatigue possible. These loads may be cyclic, with their most simpliﬁed form being constant amplitude loading (Fig. 2.2). This exhibits alternating stress between a minimum (σmin) and a maximum value (σmax), and is used in performing most fatigue tests on materials [8].

Figure 2.2: Cyclic constant amplitude loading and descriptive parameters [8].

Some important values to describe this loading are the stress range (∆σ) (eq. 2.2), average stress (σm)

(eq. 2.3), and stress amplitude (σa) (eq. 2.4).

∆σ = σmax − σmin (2.2) σ + σ σ = max min (2.3) m 2 ∆σ σ = (2.4) a 2

Another parameter of the loading is the stress ratio, deﬁned as:

σ R = min (2.5) σmax For values of R > 0, and assuming the maximum stress is positive, the loading is exclusively tensile, with the minimum stress falling to zero for the common case of R = 0. For negative values of the ratio, minimum stresses become increasingly more compressive, reaching R = −1 for the fully reversed situation where σm = 0.

2.3.1 Stress-Life (S-N) curves

The plotting of S-N curves from series of experimental data from specimens subjected to different variable loading started from the work of Wohler¨ and is a powerful method in determining fatigue life. The process of plotting these curves involves extensive experimental data. Several specimens of a given material are built and tested to failure while varying the stress range [5]. While the range is decreased, the number of cycles to failure is increased in a close to exponential trend. A ﬁtting expression is often employed to describe this behaviour:

8 0 b Sa = Sf (2N) (2.6)

0 Also known as the Basquin equation [12], where Sf and b are coefficients depending on the material, and 2N being twice the number of cycles signifies the number of load reversals. For a component to last a given amount of cycles N, an alternating stress Sa cannot be exceeded. One must note however that the data used to fit these curves is subjected to some scatter, making the use of adequate safety factors mandatory for a good design.

Figure 2.3: Typical S-N curve for steels, adapted [5].

For steels, these tests usually result in a threshold stress, below which specimens do not fail (or at least not for a feasible amount of applied load cycles), as can be seen in Fig. 2.3. Parts in this situation are said to have inﬁnite life, and the threshold stress for which this occurs is called the endurance limit Se and is assumed to happen for 106 cycles, although recent studies have disproved the existence of this limit, particularly for the gigacycle ( > 109 cycles) range [13]. For aluminium and other metals and alloys a threshold is not clearly identiﬁed, (as shown in Fig.2.1) and the decreasing stress with number of cycles trend is maintained. It is common for aluminium alloys to designate an equivalent endurance limit stress for 108 cycles.

2.3.2 Mean Stress Inﬂuence and Criteria

Stress amplitude is not the only driving factor affecting fatigue failure. Increasing mean stress has been shown to reduce fatigue life for the same stress range. As the maximum stresses approach static yield, and the entire stress range becomes more tensile, crack propagation becomes faster. Dowling [8] shows in Fig. 2.4 the effect of mean stress in fatigue life.

As σm decreases, the number of cycles to failure for the same stress amplitude is increased. The values of the S-N curve for σm = 0 are called σar. Plotting the normalized stress-amplitude σa/σar versus the

9 Figure 2.4: Effect of mean stress on fatigue life for AA 7075-T6 [8].

mean stress σm, we can obtain a useful graph where fatigue failure data is comprised for various loading situations.

Figure 2.5: Normalized stress amplitude vs mean stress plot for AA 7075-T6 [8].

In Fig.2.5 these normalized stress amplitude curves are plotted with the addition of various criteria for stress-life analysis. These criteria estimate fatigue failure from a combination of static material properties and results from S-N testing, and are useful due to their availability for any case of load cycle. Among them, the Gerber criterion:

σ σ 2 a + m = 1 (2.7) σar Su And the modiﬁed Goodman criterion:

10 σ σ a + m = 1 (2.8) σar Su These are some of the most used criteria in fatigue design, with special attention given to the Goodman being the most conservative [5]. Another criterion is specified in Fig. 2.5 with increased accuracy in failure estimation when compared to the previous. This is the result of a correction of the Goodman criteria suggested by Morrow [14] in the SAE Fatigue Design Handbook, where ultimate stress is replaced by the stress coefficient from the S-N curve achieving a better fitting to experimental data:

σa σm + 0 = 1 (2.9) σar Sf

2.3.3 Cumulative Damage

For most engineering applications, loading history is far from the constant amplitude case. In many cases, variations in load are irregular and of variable amplitude between different cycles. In Fig. 2.6a vertical load history over time for the aircraft analysed in this thesis is presented as example.

4.5

3.5

3 z

2.5

2 Load Factor n 1.5

0.5

0 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 Time [s]

Figure 2.6: In ﬂight load factor data from Epsilon TB-30 aircraft

In these cases, the number of cycles to failure cannot be adequately estimated from S-N data directly. One of the most widely used methods to approach this, is the Palgrem-Miner rule:

X ni = d (2.10) Ni Where d is called the damage index, and failure is expected to occur once d = 1 is reached, although in many cases, it may happen for different values. Variable ni is the number of times a cycle of a certain mean and amplitude of stress occurs in the spectrum used, and Ni the number of cycles for which failure is estimated to occur for these stress parameters. This estimate is made either from the S-N curve directly, or using stress criteria. The number of cycles ni is counted from a load spectrum such as

11 the one in Fig.2.6 using cycle counting methods, of which the rainﬂow method is the most widely used. Some authors argue that the Miner rule estimate, by not contemplating the sequence in which loads are applied can sometimes provide estimates of lower accuracy than similar methods. However its simplicity of use makes it one of the most used in fatigue analysis [5].

2.3.4 Other Factors Affecting Fatigue Life

Experimental tests performed to obtain a material’s S-N curves are done in controlled laboratory conditions. In design for fatigue life, appropriate safety factors must be used to account for miscellaneous factors that may contribute to a difference in the expected lifetime. The following are some of the important factors to consider in this analysis [5]:

• Environment: Temperature and corrosion can cause considerable differences between laboratory results and in-service components. Corrosion is known to cause small defects in the material which may severely reduce crack initiation time, resulting in a shorter life. Low temperature is known to cause materials to become more brittle, accelerating crack propagation. Conversely, high temperatures tend to lower the yield stress, and for very high temperature creep may be induced on the component with very adverse effects on fatigue life. Some experimental tests may use corrosion environments and/or different temperatures to account for these differences, however, this kind of data is scarce for most materials.

• Manufacturing: The presence of initial defects will cause initiation of fatigue cracks. The process of extracting and shaping the material used is generally the cause for the presence of these flaws. The processes used can therefore hinder or accelerate the formation of these cracks. In general, those processes which provide smooth surface finishes, such as precision machining and the use of finishing operations are preferred for fatigue situations to other methods such as casting, since crack initiation is usually started on the component’s surface, and irregularities can act as local stress raisers accelerating the process.

• Design: Specimens for fatigue testing such as the rotating bending specimen are ideally unnotched and of speciﬁc shapes and sizes. The presence of sharp variations in geometry on a part can cause stress concentration and consequently different fatigue behaviour than the expected from testing. Propagation of cracks in a part of given shape will very likely be different than the one occurring in these specimens. Appropriate geometries for fatigue should at all costs avoid notches which could lead to a reduction in useful life.

2.4 Linear Elastic Fracture Mechanics (LEFM)

Fracture mechanics concerns the study of the propagation of cracks along a component, leading to its eventual brittle failure once the load carrying section is too small. This propagation is assumed to start

12 from small defects present in the material, such as inclusions, gaps or other inhomogeneities from which the progressive cycling of loads produces the development of cracks (Fig. 2.7).

Figure 2.7: Crack (light area) growing from a large non-metallic inclusion (dark area) in a steel artillery tube [8].

The presence of cracks in a component is known to cause local stress concentration allowing the regions surrounding the crack to surpass yield stress leading to its increase in size. As stated by Broek [15], a crack in a solid can be stressed in three different modes. These are Mode I, the ”opening mode” caused by stresses which are normal to the crack plane, mode II or ”sliding mode” caused by in-plane shear, and mode III or ”tearing mode” where out-of-plane shear causes the crack surface to move in parallel with the edge of the crack. These three modes, depicted in Fig 2.8 are superposed in the case of 3-dimensional cracks, but in general, the contribution of mode I to crack propagation is the most critical.

Figure 2.8: The three modes of crack loading [15].

Using methods from the theory of linear elasticity, it is possible to obtain solutions for the stress ﬁeld in the vicinity of a crack for a variety of simple geometries. Such is the case for a through the thickness crack of length 2a, under mode I loading in an inﬁnite plate as illustrated in Fig. 2.9. The solution, described by Broek [15], can be shown to be:

13 Figure 2.9: Crack in inﬁnite plate in Mode I, adapted [15].

r a θ θ 3θ σ ' σ cos 1 − sin sin (2.11a) x 2r 2 2 2 r a θ θ 3θ σ ' σ cos 1 + sin sin (2.11b) y 2r 2 2 2 r a θ θ 3θ τ ' σ sin cos cos (2.11c) xy 2r 2 2 2

σz = 0 for plane stress (2.11d)

σz = ν(σx + σy) for plane strain (2.11e)

Where σ is the nominal tensile stress away from the crack region, σx, σy and σz are the normal components of the stress tensor, and τxy is the shear stress in the xy plane. These equations represent the first term of a series which gives a sufficiently accurate description for the components of stress in the vicinity of the crack. These show a variation with the square root of the crack size a and approach infinity at the crack tip for values of the tip radius r approaching zero (ideally sharp crack). In Fig. 2.10 the distribution of σy is represented for θ = 0.

Figure 2.10: Elastic stress σy distribution close to the crack tip [15].

14 These functions in polar coordinates r and θ with origin on the crack tip can be written in a more generalized form as:

KI √ σij = √ fij(θ) ,with KI = σ πa (2.12) 2πr

Where KI is the Stress Intensity Factor and is the commanding variable for the stress field around the crack. Additionally, for the infinite plate situation, it is observed that, for the same value of KI , cracks of different size a show the same stress field making this generalized form an expression of the similarity between stress fields. A concentration of infinite stress however is only a theoretical concept since for real materials, not only can a crack of infinite tip radius not exist, but also stresses that high would either cause the brittle failure of the component, or be accommodated by localized plastic deformation around the crack tip, leaving the maximum stress to be a finite value close to the yield stress of the material. This region of localized plastic deformation is called the plastic zone and its effects on the crack tip stress distribution are represented in Fig. 2.11.

Figure 2.11: Approximate σy stress distribution at crack tip including plastic zone [15].

It is found however, that for a plastic zone sufficiently small, the elastic stress field still holds for a region outside of it, called the K-field. The size of the plastic zone is therefore the criteria for choosing between the methods of Linear Elastic Fracture Mechanics or Elastic Plastic Fracture Mechanics. Dowling [8] states that, for the applicability of LEFM, the crack size parameter must exceed the size of the plastic zone by at least four times, leading to the condition:

4 K 2 a ≥ (2.13) π Sy

Where Sy is the yield stress of the material. The infinite plate solution for the stress field presented above is a helpful approximation, but is not an accurate model for a wide variety of crack geometries observed in engineering applications. For more complex cases it is typical to add to the infinite plate expression a term which accounts for the differences

15 in the stress ﬁeld.

√ KI = σ πa Y (2.14)

Where Y becomes a shape factor which varies for different crack geometries and is usually a function of the crack’s size and other shape parameters as it propagates. Several textbook solutions for this factor are available for common crack geometries, determined either from analytical elasticity theory, numerical solutions or even empirical data. There is a critical value for the stresses at the crack tip for which brittle fracture ensues. This critical value can be described by means of the SIF and is known in fracture mechanics as the fracture toughness

KIc. This parameter is expected to be a property of the material, however variation is observed for measurements of KIc with different specimen thickness, showing an increasing tendency with the reduction of the component’s thickness which is due to the transition from plane strain to plane stress conditions.

Tabled values for KIc are therefore obtained from plane strain situations for consistency of results, with the advantage of providing a conservative approximation of the parameter for real situations.

2.4.1 The Grifﬁth Criterion

In 1921, even before the main developments in Fracture Mechanics, Grifﬁth [16] proposed an equation to model crack growth using a different approach. Broek [15] explains this concept by considering an inﬁnite cracked plate of unit thickness with a central crack of length 2a, where a stress σ is applied.

Figure 2.12: The Grifﬁth criterion for ﬁxed grips [15].

In the present case, the load application is made by displacement of the ends of the plate. As the plate is extended, elastic energy is increased, as shown by line OA in Fig. 2.12, however, the increase in length of the crack will decrease the plate’s stiffness, and since the displacement is imposed on the plate, the load applied must decrease, leading to a drop in elastic energy by a magnitude corresponding to area OCB. Grifﬁth assumed this drop in stored elastic energy would be used to increase the surface energy in the

16 amount required by the crack’s growth. Under this assumption, an energy criterion for crack propagation was proposed:

d (W − U) = 0 (2.15) da

Where W is the strain energy and U the crack’s surface energy. The equation above states that an increase of crack length a will only happen in the event of both variations caused on these quantities to dU be balanced. For an elliptical ﬂaw, da then becomes:

dU 2πσ2a = (2.16) da E

This is usually replaced by a quantity known as the elastic energy release rate G:

πσ2a G = (2.17) E

The assumption that surface energy is the driving force of the crack propagation is a good approximation for the situation of glass and other brittle materials, such as the ones used by Griffith in the experiments leading to this theory. For materials exhibiting higher plasticity however, this is no longer the case. This energy approach does not consider the stress field, and it may be the case that crack growth is energetically favourable, but the stresses at the tip are not sufficient to allow it. For these materials, a similar condition is employed, making use of the SIF:

K2 G = (2.18) E

For plane stress, and:

K2 G = (1 − ν2) (2.19) E

For plane strain.

2.4.2 The J-Integral Concept

The methods above are applicable only for the case where plasticity around the crack tip is negligible and linear elasticity can be assumed for the entire stress-ﬁeld around the crack. If there is appreciable plasticity, G cannot be determined from the elastic stress ﬁeld [15]. An energy release rate estimate in these cases can still be made however, through the use of the J-Integral. For two-dimensional problems, this quantity can be written as:

Z ∂u J = W dy − t ds (2.20) ζ ∂x Where t is the traction vector perpendicular to ζ in the outside direction, and W is the energy density, computed as:

17 Z W = σd (2.21) 0 The contour ζ is an arbitrary one which must only enclose the crack tip. This quantity provides an adequate energy estimate for both the linear-elastic case and cases of considerable plasticity, which means that, in an elastic situation J = G, making this a very attractive variable to use in crack propagation applications.

2.5 Crack Propagation

The influence of regular service loading on a component may cause a pre-existing defect to develop into a small crack. This crack may not be prone to cause failure at the applied stresses, but may eventually grow due to corrosion, or fatigue until it is of such a size that brittle failure ensues [15]. The stress intensity factor, being the leading quantity for the stress field around a crack in an assumable elastic situation, becomes an important parameter in determining propagation. A fatigue load cycle is defined as containing a peak and valley of maximum and minimum applied loads. in a given cycle, the stress intensity factor will vary accordingly with the load over a range ∆K =

Kmax − Kmin. The growth of a crack in a given cycle must therefore be correlated with the range of stress intensity factor. Experimental data from crack propagation in several materials shows a large region (II) of steady growth, where the length of the crack is increased exponentially, preceded by an initial region (I) of fast variation of the growth rate for very small crack sizes, and reaching a region of fast unsteady propagation (III) as ∆K reaches critical values. Plotting the data on a log-log scale reveals a straight line in region II, where the crack growth rate can be modelled by the Paris Law:

da = C ∆KnP (2.22) dN P

Parameters CP and nP are properties of the material. Although being a great ﬁt for region II data, it is not at all appropriate for any of the two other regions. Tensile stresses, which promote the opening of the crack surface provide the largest contribution to propagation [15]

Values of R approaching unity are descriptive of increasingly tensile stresses, and increase Kmax = √ Y σmax πa making it approach KIc. This inﬂuence makes the crack growth rate also a function of R. From Fig. 2.14 it is observed that the crack growth rate is severely increased for higher values of R for the same ∆K, causing a horizontal shift in the logarithmic representation. In light of these observations, one can argue that the Paris Law parameters can only be descriptive for a precise value of R, with several ﬁts being needed for this data to be usable for variable amplitude loads.

A shift is also noticeable for ∆Kth. Higher stress ratios make for a lower value of the threshold SIF range causing earlier initiation of the propagation process, and consequently reaching failure sooner.

18 Figure 2.13: Experimental data for fatigue crack growth on an A533B-1 steel alloy, adapted from Dowling [8].

Figure 2.14: Inﬂuence of the stress ratio of crack growth rate data [17].

Due to this effect, several researchers tried to include the inﬂuence of R in propagation laws, with the hope of obtaining a single curve which would model crack growth for any given load. One such equation which gained wide acceptance was the Walker eq.:

19 da ∆K nW = C (2.23) dN W (1 − R)1−γ

Where CW , nW and γ are material parameters, Cw theoretically being the same as the Paris Law coefﬁcient for the special case of R = 0. Another equation proposed to deal with the inﬂuence of R was the Forman equation ( 2.24 ). Not only ∆K does this equation deal with the effect of R, it also introduces an asymptote as Kmax = 1−R approaches

KIc, approximating both regions II and III from the crack growth rate curve.

da ∆KnF = CF (2.24) dN (1 − R)(KIc − Kmax) In 1971, Elber [18] stated that cracks in a material under compressive stress, or in some cases even under low tensile stress, could sometimes be closed. This closure caused part of the load cycle to be ineffective in propagating the crack. To account for this effect, the use of a corrected Stress Intensity

Factor is suggested. The ∆Keff , describes the range of SIF for which the crack tip is open, and is deﬁned as:

! √ 1 − S0 ∆K = Y (σ − S ) πa = σmax ∆K (2.25) eff max 0 1 − R

Where S0 is the stress at crack closure. This ∆Keff will always be smaller or equal than ∆K due to the approximation of the closure process. The problem now resides in estimating S0, for which Newman [19] suggested the use of a crack tip opening function, ﬁtted to experimental data:

S0 2 3 = A0 + A1R + A2R + A3R (2.26) σmax

Where coefficients A0, A1, A2 and A3 are functions of the stress field characteristics, and are defined as:

1/a 2 σmax A0 = (0.825 − 0.34α + 0.05α )cos π (2.27a) 2σ0 σ A1 = (0.415 − 0.071α) max (2.27b) σ0

A2 = 1 − A0 − A1 − A3 (2.27c)

A3 = 2A0 + A1 − 1 (2.27d)

Where σ0 is the material’s ﬂow stress, which can be approximated by the average between yield stress and the ultimate stress in a static test:

S + S σ = y u (2.28) 0 2

And α = 3 for plane strain and α = 1 for plane stress. One equation which contemplates these effects is the NASGRO eq. (eq.2.29), originally published by

20 Forman and Mettu [20] as the Forman-Newman-de Koning crack growth law. This equation provides the added modelling of all regions of the crack growth rate curve due to the use of the asymptotic term present in the Forman eq. as well as a new term which makes the curve fall to zero at ∆K = ∆Kth

p nN ∆Kth da 1 − f 1 − ∆K = CN q (2.29) dN 1 − R 1 − ∆Kmax ∆KIc

Where f = S0/σmax is the Newman crack tip opening function, and CN , nN , p and q are material parameters. The models above, although widely used, do not take into account the interaction between different loads. The sequence of load application becomes signiﬁcant when dealing with such phenomena as overloads and underloads. Overloads are a mechanism that leads to retardation of the crack growth [8]. The occurrence of occasional higher loads in the loading spectrum can lead to a considerable increase of the plastic zone around the crack tip, introducing compressive residual stresses, that will cause retardation of the propagation process for the subsequent loadings until the crack reaches outside of this overload plastic zone. Underloads on the other hand cause the acceleration of crack propagation, or more commonly, a reduction in the retardation caused by the counterpart overloads. In this mechanism, a compressive load leads to the appearance of tensile residual stresses in the regions surrounding the crack tip, accelerating its growth for the following cycles.

Willenborg [21] proposed a model which accounted for the effect of overloads using corrected ∆Keff and Reff factors varying with the size of the plastic zone. Later, Gallagher and Hughes [22] suggested a generalized Willenborg model, later adapted into the modiﬁed generalized Willenborg model for use in Nasgro 3.0 software [23], which accounts for both over and underload interactions. This model however, requires the use of empirical data not available in the scope of this thesis. As such, the effect of sequential overloads will be disregarded, arguably resulting in a conservative approximation of the component’s useful life, due to the overall dominance of tensile stresses observed in the application in this thesis and studied in section 5.1.

2.6 Finite Element Method applied to Fracture Mechanics

The problem of most crack propagation applications is many times, the calculation of the geometric factor Y for complicated geometries, where analytical solutions are unavailable or too time costly to make use of. The Finite Element Method is a numerical tool ﬁrst presented by Turner et al. [24], which consists of the division of a continuous domain into a discrete number of subdomains, or elements, where the governing equations are approximated by the use of variational methods. These elements intersect at vertices called nodes, where numerical values for the various solution ﬁelds are approximated by their degrees of freedom.

21 One of the most typical applications of the FEM is elasticity theory, where the deformation of a component of given material properties is estimated when subjected to a set of loads. Stresses and strains can be estimated from the displacement ﬁeld which are useful quantities in the computation of Fracture Mechanics variables such as the SIF. Pickard [25] explains three methods by which these LEFM quantities can be estimated in post-processing of an available FEM solution for a cracked body.

2.6.1 Displacement Extrapolation Method

The first approach derives directly from the definition of the SIF and its relation to the stress concentration around cracks, and consists on the extrapolation of the displacement field near the crack tip. Perhaps a more obvious way of performing this would be the direct extrapolation of the stress field, however, the accuracy of the displacement solution (being the primary variable in theFE problem) is higher and therefore advised. The displacement extrapolation method takes nodal values of the solution along radial lines from the crack tip which are then correlated to the theoretical tip solution obtained from (2.11) to derive the SIF. For a crack in a two-dimensional body, the displacement components in the vicinity of the notch can be described as:

1 2(1 + v) r 2 u = (K f (θ, ν) + K g (θ, ν)) (2.30) i E 2π I i II i

Where index i represents each of the three displacement components x, y and z. Functions fi and gi are constant for radial lines and so KI and KII can be estimated after proper ﬁtting of the near-tip solution.

2.6.2 Crack tip elements

For this or any of the other methods to provide an appropriate estimation of the SIF, there should be sufficient nodes in the proximity of the crack tip, ideally distributed in radial lines [25]. A typical mesh refinement around the crack tip for the use of FEM is represented in Fig. 2.15. These elements surrounding the crack tip are usually distributed into a circular envelope as the one represented, or a square one. At the crack tip however, there is a singularity in the stress (and consequently displacement) field. As the radial distance to the crack tip r is decreased, the stress tends to infinity as previously described. To model such a distribution with linear or even normal quadratic elements would require a significant refinement to obtain approximate results. Special elements are used to deal with this region. Starting from a three-dimensional 20-node quadratic element and collapsing the 6 nodes on one side into a straight line, one can model a singularity where the Jacobian of the transformation will make it so that in the problem’s reference frame:

22 Figure 2.15: Typical mesh reﬁnement around a crack tip [26].

du c = 0 + c (2.31) dx r 2

Where c0 and c2 are constants depending on the elements’ shape. The 1/r singularity can be useful in Elatic Plastic Fracture Mechanics problems where plasticity at the crack tip cannot be disregarded, √ whereas for LEFM the typical 1/ r is obtained from moving the midside nodes of the resulting elements to 1/4 of the distance along the element side [25], achieving the quarter point element depicted in Fig. 2.16.

Figure 2.16: Collapsed quarter point Elements

du c0 = 1 + c2 (2.32) dx r 2 This element is normally used directly adjacent to the crack front followed by regular hexahedral elements.

2.6.3 The Energy Method

A second approach, called the energy method, is based on the deﬁnition of the energy release rate G, and attempts to numerically approximate this quantity by the difference in elastic potential energy between two slightly different crack size geometries subjected to the same set of boundary conditions. This approach, although more computationally expensive has been shown to give a better estimation than the displacement extrapolation method except for mixed mode situations, where SIFs for different

23 modes cannot be easily separated using the energy method. In a ﬁnite element analysis, the potential energy Π of a cracked body can be retrieved as:

1 Π = δT kδ − δT F (2.33) 2

Where k, δ and F are the stiffness matrix, the displacement and force vectors of theFE problem respectively. Making two different meshes separated by a small difference in crack size ∆a, G is approximated as:

∆Π G ' − (2.34) ∆A

Where ∆A is the difference in crack areas.

2.6.4 The J-Integral Method

The third method is based on the computation of the J-integral and is the one used by ABAQUS c software [27] due to being an adequate approximation for many different kinds of applications including crack tip plasticity as well as mixed mode behaviour. The J-integral is calculated from the ﬁnite element solution for a number of contours to ensure stability of the solution. Since any contour enclosing the crack tip is adequate, the estimates from the various contours should show negligible variation for a good solution to be obtained. The contours are chosen from the layers of elements around the crack tip as shown in Fig. 2.17. The expression in Eq. 2.20 is computed using numerical integration methods.

Figure 2.17: Integral contours selection in Abaqus [27].

From the J-integral, SIFs for all three modes of crack propagation can be estimated by:

K2 + K2 ν + 1 J = I II + K2 (2.35) E E III

24 2.7 Extended Finite Element Method (XFEM)

Typical numerical models for the propagation of a crack in a component depend on an adapting mesh which accompanies the development of the crack refining the elements around it for the calculation of the stresses surrounding it, and the LEFM quantities that derive from them, such as ∆K or the J-Integral which are necessary for further calculations. This process of re-meshing for every advancement of the crack front is a cumbersome one, and in complex geometries it can become very computationally demanding. Regarding this problem, during the last several decades, some solutions have been presented for crack propagation analysis without re-meshing. One such solution was proposed in 1997 by Babuska˘ et al. [28] and was described as the Partition of Unity Method (PUM), and is one example of a meshless finite element method. The concept behind this method is to provide a better approximation of the exact solution using shape functions chosen a priori which are known to adequately model the reality of the problem, instead of relying on the polynomial shape functions used in FEM. The authors refer this possibility of inputting knowledge of the local behaviour of the solution as a way of handling geometrical singularities, such as the case of fatigue cracks, and also highly oscillatory solutions or even boundary layers in a more cost-effective way, since the FEM would require a very refined mesh to approximate this behaviour, whereas the PUM uses higher order functions. Another class of meshless finite element methods are the Element Free Galerkin Methods (EFGM) proposed by Belitschko et al. [29], where the authors justified the appeal of these methods realizing that the process of mesh generation for linear analysis was frequently more time consuming than the assembly and solution of the finite element equations. This is a method that, constructed upon the concept of the diffuse element method proposed by Nayroles et al. [30], produces a mesh of nodes where interpolating polynomials are fit to the nodal values using least-squares approximations. In this way, the method has no need for an element mesh. Later on Strouboulis et al. [31] developed the Generalized Finite Element Method, combining the classical FEM with the PUM. This employs the standard finite element space and enriches it with functions from known analytical solutions. The authors’ approach took advantage of the partition of unity concept to model singularities where there was a need to, and kept the standard form of the finite element method to circumvent one problem ofPU based methods which is the application of boundary conditions, that were typically made using Lagrange multipliers, which are less stable than the direct application made in the FEM. In 1999 Moes¨ et al. [32] proposed a method for crack modelling based on the standard FEM with the added use of enriched shape functions on the elements surrounding the crack, which allowed an improved description of the stresses in this region without the need to refine the mesh around it. This method later on developed into the extended finite element method proposed by Sukumar et al. [33]. These enriched shape functions, described in Eq. 2.36 added a discontinuous generalized Heaviside function H(x) (Eq. 2.37) term to the standard element shape function which allowed for a discontinuity

25 in displacement across crack faces to be modelled, along with an asymptotic term Ψ(x) (Eq. 2.38) which treated the stress singularity at the crack front.

4 ! X X X J u(x) = uf (x) + φI (x)H(x)vI + φJ ψl(x)wl (2.36) I J l=1 c f nI ∈M nJ ∈M

Where the set M f consists of the nodes where the nodal shape function support intersects the crack front and the set M c consists of those nodes whose nodal shape function support is intersected by the crack and which do not belong to M f . The generalized Heaviside function for the discontinuity in displacement is shown in Eq. 2.37 where x∗ is deﬁned as the closest point to x on the inner boundary which describes the crack ζc along the normal to the crack plane.

 1 if x − x∗ ≥ 0 H(x) = (2.37) −1 otherwise

The asymptotic function Ψ(x) is based on the radial and angular behaviour of the two dimensional crack-tip displacement ﬁeld. The variables θ and r are the polar coordinates centred on the crack-tip and deﬁned on Fig. 2.11. along a plane perpendicular to the crack front.

√ θ √ θ √ θ √ θ Ψ(x) = [ψ (x), ψ (x), ψ (x), ψ (x)] = r cos , r sin , r sin(θ)sin , r sin(θ)cos 1 2 3 4 2 2 2 2 (2.38) The use of these enriched functions means that the crack can be modelled without the need to position element boundaries along the crack faces, or reﬁning the mesh near the crack tip, which not only solves the problem of mesh actualization on complex geometries, but also makes the solution for the crack-tip displacement (and the related stress) ﬁeld independent of the mesh size.

2.8 Structural Health Monitoring

According to Sohn, et al. [34] Structural Health Monitoring (SHM) is deﬁned as: ”the process of implementing a damage detection strategy for aerospace, civil and mechanical engineering infrastructures”. This process involves the instrumentation and data analysis from these instruments on structures to estimate their remaining useful life. Doebling, et al. [35] describe four levels of damage identiﬁcation using SHM Systems:

• Level 1 : Determination that damage is present in the structure

• Level 2 : Determination of the geometric location of the damage

• Level 3 : Quantiﬁcation of the severity of the damage

• Level 4 : Prediction of the remaining service life of the structure

26 A large part of these methods measure changes in the dynamic behaviour of the structure over time, namely the frequency of the structure’s fundamental modes of vibration, to estimate the location and size of ﬂaws.

2.8.1 The Forward Problem

Level 1 of damage identification includes the use of mathematical models to acquire theoretical data of the dynamic behaviour of the unnotched structure and in the present of defects. There are numerous examples of such studies, one of them being performed by Vandiver [36] simulated the change in frequencies for various modes of vibration of a fixed offshore when reduction in cross area due to rust and even failure of some of the members occurred. By comparison with the changes in frequency due to the sloshing of stored fluids, Vandiver concluded that the former changes in structural integrity could be indentified by measurement of the bending and torsional modes of the platform, making them suitable for SHM applications.

2.8.2 The Inverse Problem

Levels 2 and 3 concern the treatment of sensor data and its comparison to the results of the mathematical models in order to estimate the presence and extent of damage. A wide variety of methods are available in literature for this kind of analysis, many of them described in Doebling, et al. [35], which usually require information from several of the structure’s modes of vibration to isolate the different causes of damage from the data available. Level 4 prediction generally makes use of methods of fatigue life analysis and fracture mechanics to estimate the Remaining Useful Life (RUL) of the component.

2.9 Manufacturer’s Results for Aircraft Useful Life

During the design phase, SOCATA did not define critical structural locations along the aircraft, nor a programme for tracking the plane’s remaining useful life. As a result, the manufacturer proceeded to perform real scale tests on finished aircraft, resulting in a report from CEAT. The reference load spectra used was obtained by the manufacturer as an equivalent to the loads felt in 1000 Flight Hours (FH). In the report, total failure of the aircraft is described as occurring after 89458FH simulated. This failure was due to the fracture of Frame 2, illustrated in Fig. 1.2, which was identified as the critical component controlling fatigue life [2]. Applying a safety factor of 3, the estimated useful life for Epsilon aircraft was then stated as 29800FH. The plane’s first inspection was therefore scheduled to happen at 12400FH followed by an interval of 4000FH between successive inspections until a critical crack size of 1.5mm is reached.

27 Chapter 3

Experimental Activities

The activities executed during this thesis are part of the SHM TB-30 project which aims to describe the structural condition of the Epsilon ﬂeet, and to implement systems designed to monitor it, in order adapt maintenance scheduling to the aircraft’s operation.

Figure 3.1: Schematic of the implementation of the SHM system. FDR: Flight Data Recorder, SDR: Strain Data Recorder, OLM: Operational Load Monitoring, RUL: Remaining Useful Life, MTBF: Mean Time Between Failure

An essential part of this project which is described in Fig. 3.1 is the full instrumentation of two aircraft, used to perfect the Structural Health Monitoring algorithms, which make use of neural networks to estimate the plane’s condition from measured data and will be later applied to the entire ﬂeet using reduced sensorization. The fully instrumented aircraft will make use of one accelerometer and several strain gauges on key structural components in order to relate aircraft loads directly to the structural behaviour in these previously designated points, which will be referred to as hotspots. Other planes will use acceleration sensors already built-in to deﬁne aircraft operation, from which the

28 remaining useful life will be estimated During the making of the present thesis, the author worked directly with the PrtAF in the calibration of these sensors and setting up the Data Acquisition Unit (DAU) which would power and receive data from these instruments. In this chapter, a brief description of the setting up of the DAU and of the laboratory testing performed for calibration of the various sensors will be provided.

3.1 Strain Gauges

3.1.1 Sensor Principle and Description

The strain sensors chosen by PrtAF to be used on the system were acquired from Columbia Research Laboratories, with Part Number: DT 3757-5 (Fig. 3.2) and its datasheet is presented in Appendix A. These are full bridge strain gauges which are flight qualified and fulfil the requirements of FAA DO-160 standard for airborne electrical equipment and instruments.

Figure 3.2: DT 3757-5 full bridge strain gauge, taken from the model’s datasheet

Full bridge sensors are formed by four electrical resistances arranged in parallel pairs of two. The value of these resistances vary linearly with length, but for the specific case of the DT 3757-5, only two of these resistances are made to have variable value, and are arranged in a cross like manner. For a strain gauge bonded to a component, as the part is stretched by the presence of material strains, the length of the variable resistances is also increased, changing the difference of potential at the measured nodes by a quantity which should be linearly proportional to strain in the component, with a constant of proportionality called the gauge factor in the sensor’s range. From this change of voltage between the two nodes, and knowing the sensor’s gauge factor, one can extract the value of strain at the region measured. The full-bridge configuration has the added advantage of being less sensitive to temperature changes than other half or quarter bridge type sensors. This specific sensor also incorporates an integrated power electronics circuit to amplify the output signal, therefore increasing its signal to noise ratio.

3.1.2 Testing and Calibration

To ensure that these sensors were correctly used in the aircraft, a spare strain gauge was used to test the ﬁxing procedure, and afterwards the signal acquisition and conditioning in order to interpret strain

29 data. An aluminium 2024-T3 specimen was prepared with three different strain sensors: the DTD 3757- 5 described above, an older DTD 2684-1 sensor spare from previous studies by the PrtAF, and a conventional strain gauge used in the materials laboratory at IST.

Figure 3.3: Aluminium 2024-T3 specimen for tensile test.

Tensile testing was performed at the Air Force Academy in Sintra using an MTS 810 machine (Fig. 3.3). The specimen was loaded to 10kN and then discharged, with values recorded every 2kN. The data acquired from the three sensors was compared with the theoretical strain calculated for the part. The zero offset for all sensors was taken as the voltage measured when the specimen was held at full strength by the grips and with 0 tensile load. The gauge factor of each sensor was obtained from the respective data sheets.

Figure 3.4: Strain vs. Force data from tensile test and respective error compared to theoretical values.

From Fig.3.4 the increase in accuracy of the recently acquired DTD 3757-5 is visible in comparison to

30 the other two, with a maximum error of 10% in regards to the theoretical solution. Extracting the Young’s Modulus for each set of sensor data, the DTD 3757 is once again the one with the least error, not exceeding 2.14% (Table 3.1).

DTD 3757-5 DTD 2864-1 IST sensor Theoretical Young’s Modulus [MPa] 71432 86207 82989 73100 Error [%] 2.14 −18.1 −13.69 -

Table 3.1: Young’s modulus computed from linear regression of sensor data, and respective percent error to the theoretical value.

3.1.3 Aircraft installation

A total of ﬁve strain gauges were installed in the aircraft. One was placed on the structural hotspot identiﬁed in Frame 2, while the other four were installed on different locations on the wing spar, as per Fig. 3.5.

Figure 3.5: Strain vs. Force data from tensile test and respective error compared to theoretical values.

A bonding procedure was designed based on Columbia c recommendations and tested on the sensors installed on the tensile specimen. The strain gauges were bonded on the aircraft accordingly. Prior to data acquisition, the null offset of these sensors will have to be determined in a ready to ﬂy situation. The procedure for this calibration is still in discussion, with options being: taking the offset with the aircraft in ground with no fuel on the wings, and taking it with no fuel and also supporting the tips of the wings to remove load caused by the wing’s own weight.

3.2 Accelerometers

3.2.1 Sensor Principle and Description

The accelerometers used were also acquired from Columbia Research Laboratories with Part Number: SA-102MFTB (Fig. 3.6), and with its datasheet in Appendix A. These are force balance accelerometers, which are the preferred sensors for low frequency accelerations, such as those felt by the aircraft (below 25Hz) and especially in DC current applications, which is also the case. Furthermore, the model in question uses a pendulous mass to measure the accelerations.

31 Figure 3.6: SA-102MFTB accelerometer.

The working principle behind this sensor is the use of the suspended pendulous mass made out of a high magnetic permeability material whose accelerating force is counteracted by a magnetic coil. As the mass is displaced further from its null position, the electric current ﬂowing through the coil, with the help of an ampliﬁer circuit is increased to balance the accelerating force. The resulting displacement of the mass is very small (typically smaller than one thousandth of a mm) making hysteresis a rare problem for this type of accelerometer.

3.2.2 Testing and Calibration

The sensor is previously calibrated by Columbia c , so that the DAU is able to automatically read the acceleration data in G’s with no necessary interaction from the user. Nonetheless, the signal was tested for error using an inclinometer. The accelerometer was positioned on top of a straight surface and rotated from completely vertical orientation of the direction of measurement, to horizontal, and upside down. A Lucas AngleStar c digital inclinometer was used, and for each angle registered, the sensor signal was compared to the theoretical value obtained from the cosine of the sensor’s inclination in relation to vertical. Errors in sensor acquisition were found to be negligible (less than 0.02G), as expected.

3.2.3 Aircraft Installation

The accelerometer will be positioned very close to the aircraft’s centre of gravity, to neglect errors coming from rotation components of acceleration, with the axis of measurement oriented vertically. The same positioning has been used in the past by the manufacturer. A support will be constructed which makes use of four bolted joints, already existing in the aircraft’s structure, to hold the sensor in place.

32 Chapter 4

Finite Element Modelling of Aircraft Structures

When there is a necessity of modelling the structural behaviour of 3D parts of complex geometry, analytical models are often inaccurate or require an oversimpliﬁcation of the problem at hand. Thus, in the absence of analytical solutions, one must turn to numerical methods. Among the existing numerical methods, the Finite Element method is the most widely used in the structural analysis of components with complex geometry [25]. In the last decades theFE method has developed considerably which led to an availability of a variety of commercial codes to choose from. During the writing of this thesis, the author had access to such codes as SIEMENS NX c , ANSYS c and ABAQUS c . The latter, ABAQUS c was chosen due to Critical Materials’ knowledge of the software and to make it easier for the models and results fabricated to be imported into their own computational tools.

4.1 Objective of Finite Element Modelling

As part of the SHM TB30 project, there is interest in the accurate estimation of accumulated damage and remaining useful life of the Epsilon aircraft. A numerical tool for stress analysis will serve at ﬁrst to give conﬁdence in the structural hotspots found in the real scale tests performed in CEAT, and to direct attention at other, previously unknown critical regions of the structure, as well as determining relations between aircraft load factor and stresses to aid in experimental testing of these components. Later on, these models can be used to estimate fatigue damage in the entire structure and to simulate crack propagation.

4.2 Real Structure

The CEAT real scale tests concluded that the critical component which determined the useful life of the aircraft was Frame C2. This frame acts as the main connection between the wing structure and the

33 plane’s fuselage, aided by the smaller Frame C3. In Fig. 4.1 a detailed depiction of the aircraft’s structure is presented, where it can be seen that the wing is composed by an outer metal shell supported by a series of ribs and spars, the ribs traversing the wing from front to back, and the spars from side to side. There are a total of 12 ribs and two spars. Of these, the forward most is named the main spar, and the rear most the rear small spar, both of these connect through pinned joints to Frames C2 and C3 respectively.

Figure 4.1: Airplane general structure [37]

The connection between Frame C2 and the Main spar is made with 4 instances of 3 thick sections on the frame’s side intertwined with two on the spar’s side and a tightly ﬁtted 25mm diameter metal pin passing between these, allowed by a series of through holes (Fig. 4.2).

Figure 4.2: Detail of the pinned connection between Frame C2 and the Main wing spar (CAD model)

The use of multiple load carrying sections such as this is a typical fail safe design used in aerospace

34 structures, where the failure of a member does not imply the immediate failure of the aircraft, since there is another member capable of carrying the load. Frame C2 is connected to the plane’s fuselage through 6 M12 bolts that attach to metal shells, which in turn connect to the 4 spars that give shape to the aircraft’s fuselage. The Main wing spar is divided into 2 parts bolted together by several bolts along its span, one facing the front of the plane, and another facing the rear. Both Frame C2 and the Main wing spar are precision machined out of aluminium alloy 2024-T3.

4.3 CAD Modelling

The ﬁnite element method is built upon the discretization of the problem’s originally continuous domain into a mesh of small elements. This means that in order to achieve accurate solutions using theFE method, there must be an accurate model of the component’s geometry. The ideal situation would be to have access to the CAD models produced by the manufacturer at the time of the aircraft’s design. Since this was not possible, these models had to be built by direct inspection and measurement of the parts. Furthermore, since the parts could not be disassembled from the aircraft, the use of a 3D scanner was not possible, so these measurements needed to be made using simple measurement equipment. For this purpose, the author was allowed to directly inspect the aircraft stationed at the Air Force Academy, from which several panels were previously disassembled, allowing visibility of Frame C2 and the front half of the Main wing spar (Fig. 4.3). The following analysis was therefore restricted to these two parts.

a) Frame C2 visible geometry b) Main wing spar visible geometry

Figure 4.3: Components’ visible geometry during measurement

The instruments used to measure the parts were a 10m Tape measure with a precision of 0.5mm for the larger outside dimensions, and a digital pachometer with 0.01mm precision for the smaller thickness measurements. The outline of the parts was ﬁrst constructed using the tape measure, by taking dimensions like the vertical and horizontal distances between holes, the largest width and height of the part, and distances between distant features.

35 Then the detailed thicknesses of all relevant features was added measuring with the pachometer. The complete CAD models are depicted in Fig.4.4.

a) Frame C2 complete CAD model b) Main wing spar complete CAD model

Figure 4.4: CAD models of aircraft components

4.4 Finite Element Mesh

The finite element method relies on the discretization of the geometry in question to a number of small elements. The size of these elements must be such that it allows an accurate modelling of the geometry’s various features. This discretization implies that the finer these elements are, the more detailed the mesh will be, and the precision of the solutions obtained is increased. There are however limitations on computational requirements which make the use of very fine meshes infeasible, primarily due to the increasing time consumption of the analysis. Performing a mesh convergence study is a convenient way of determining an adequate degree of mesh refinement, for which the solutions obtained are of sufficient precision. In the making of the finite element meshes of the components studied, certain quality specifications were defined beforehand. These were to have mostly hexahedral elements through the entire frame, reducing as much as possible the use of tetrahedral elements, and to have at least three elements through the thickness of all structurally relevant sections and to impose the constrained nodes to be absolutely centred in their positions on the frame. Through adequate partitioning and seeding the meshes were constructed using the ABAQUS c interface (Figs. 4.5 and 4.7). The elements used were ABAQUS c 3D stress first order elements, prioritizing 8 node hexahedral elements and using tetrahedals only for regions of transition of the geometry where hexahedrals would be too heavily distorted (Fig. 4.6). As for material properties, both components are made from Aluminium 2024-T3 alloy. The material’s elastic properties used were extracted from available bibliography [38] and are listed in Table 4.1.

Young’s Modulus Poisson’s ratio Density 73100 MP a 0.33 2.8e-9 tonne/mm3

Table 4.1: Material properties used on all calculations involving AA 2024-T3

36 Figure 4.5: Finite element mesh of frame C2 using 8mm overall element size

Figure 4.6: Detail of the ﬁnite element mesh of the frame using 8mm overall element size. Note the 3 elements across the thickness of the small rib and the use of tetrahedral elements on the ﬁlleted region.

Figure 4.7: Finite element mesh of the Main wing spar using 16mm overall element size

A mesh convergence study was performed on both components by constructing several meshes of varying element overall size while monitoring the frequency of the ﬁrst mode of vibration of the unconstrained structures. The frequency of this ﬁrst mode was chosen as the convergence criterion, since later on Modal dynamic analysis, which relies on information from the modes of vibration, was intended to be performed to simulate the dynamic behaviour of Frame C2. This modal dynamic analysis however, was disregarded due to the loss of accuracy in comparison to direct integration. For Frame C2, starting with a 12mm overall element size, and decreasing 2mm per iteration, the solution was found to be reasonably converged at a overall mesh size of 6mm, as seen in Fig. 4.8 with a variation of 0.02% in frequency between this and the 4mm solution. For the Main wing spar, starting with a 16mm element size, convergence was observed at a overall

37 152

151.5

151

150.5

150

149.5 Frequency of Fundamental Mode

149 3 4 5 6 7 8 9 No of Elements 104

Figure 4.8: Mesh convergence plot for Frame C2 element size of 10mm, as shown in Fig 4.9 with a variation of 0.17% in frequency between this and the 8mm solution.

21.1

20.9

20.8

20.7

20.6 Frequency of fundamental mode

20.5 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 No of Elements 104

Figure 4.9: Mesh convergence plot for wing spar

4.5 Boundary Conditions

With the domain now described, boundary conditions that try to describe the reality of the loading and constraints to which the parts are subjected is necessary. The problem becomes one of taking the inverse approach and infer the boundary conditions from the measurable results.

4.5.1 Loading Conditions

To accurately simulate the behaviour of aircraft components, it is important to understand how the forces acting on the structure are developed. For this, a general understanding of ﬂight mechanics is needed. In ﬂight, the plane is directed by aerodynamic forces, thrust forces generated by the plane itself, and gravity. Balancing these three components is the key to directing the aircraft. The aerodynamic forces are generated by all surfaces in contact with outside air and its components are called lift (L), for the component of the force in the transverse direction of the plane’s trajectory, and

38 drag (D), for the component in the contrary direction of the aircraft’s motion. Thrust forces (T ) are generated by the aircraft’s own system of propulsion and, along with aerodynamic drag, will be of little interest to this work, since only vertical acceleration data will be provided, and so its contribution to damage must be considered negligible. The aircraft’s weight is always a vertical downwards force, which is the product of the plane’s mass which, although it varies during a ﬂight situation, will be considered constant and equal to the plane’s maximum load to yield conservative results, multiplied by the gravitational acceleration g. Steady Level ﬂight is a situation in which the airplane is generating lift in a direction parallel to gravity (vertical) and no acceleration is produced, meaning all forces are in static equilibrium (Fig. 4.10).

Figure 4.10: Airplane Free Body Diagram in Steady Level Flight.

Solving the equilibrium in this situation readily yields:

X Fz = 0 ⇐⇒ L = mg (4.1)

X Fx = 0 ⇐⇒ T = D (4.2)

Meaning Lift force must match the aircraft’s weight, and thrust must match drag to maintain constant velocity. Any disturbance to this equilibrium will cause acceleration in a given direction. The load factor is used to quantify the forces involved in situations outside static equilibrium. In aeronautics, the load factor is a non-dimentional quantity which represents a global measure of the stress to which the structure is subjected. It is calculated as the ratio between lift and the aircraft’s weight. Using the load factor, lift is calculated at a given moment of the flight by comparing it to the steady level flight situation (Fig. 4.11). Using data of the aircraft’s vertical acceleration, the load factor for any given moment of a flight can be calculated through the equilibrium equations using its definition.

X Fz = 0 ⇐⇒ nzL − mg − maz = 0 (4.3)

Since (eq.4.1) holds:

39 Figure 4.11: Airplane Free Body Diagram using the Load Factor for vertical force.

a n = 1 + z (4.4) z g

Acceleration data is available for the instrumented aircraft and therefore, one must only find transfer functions that relate the loads and stresses on the structure to the load factor to be able to infer damage to the aircraft during flight. For analysis purposes the vertical force acting on the airplane will therefore equal the lift generated in steady level flight times the load factor nz which can be treated like a dynamic lift force in itself.

L = nzmTB30 g (4.5)

For the Epsilon TB-30, the maximum range of load factors in operation specified by the manufacturer is between −2.75 and +6.2. The maximum service load of the aircraft is 1300kg and this value shall be the assumed aircraft weight in all flight situations for the calculations performed, in order to achieve conservative results. For this analysis the effects of the horizontal forces acting on the airplane and moments generated by the aerodynamic forces are neglected, since values for these forces cannot be easily calculated with the data available and their influence on the structural integrity of the aircraft is reduced in comparison to the vertical forces. As a first approach, the pressure distribution on the wing will be modelled as constant along the entire surface. This is known to be incorrect since tip effects in finite wings make lift distribution fall to zero as it approaches the tip, but will serve as an initial approximation. We will also assume that all lift is generated in the wings and no other lifting surfaces. This is also an acceptable approximation, since most of the total lift is known to be generated on the wing surfaces. The wing spar is attached to the frame of the plane through two pinned joints which fix the wing’s motion in a hyperstatic situation. Some information although can be retrieved through a simple free body diagram (Fig. 4.12).

X Z Fz = 0 ⇐⇒ RzA + RzB = fz dy (4.6)

40 Figure 4.12: Wing spar Free Body Diagram

X Fy = 0 ⇐⇒ RyA = −RyB (4.7)

X Z M xA = 0 ⇐⇒ RyB h = fzy dy (4.8)

The equilibrium equation in the vertical direction shows that the sum of the vertical forces on the supports must match the total vertical force acting on the wing. No more information can be achieved regarding these forces using only a simple free-body diagram. Later, a static ﬁnite-element analysis will be used to obtain a ratio between the loads in each of the joints. The lift distribution is modelled by a funtion of the spanwise coordinate y. The origin of this axis system will be conveniently placed, for ease of calculation, at the root of the wing. Assuming that only the wing surfaces generate any lift, this funtion becomes:

n m g f(y) = z TB30 (4.9) 2s

Where s is the span of each wing. The equilibrium in the horizontal direction shows that the horizontal forces on the supports must be opposite and equal. Additionally, equilibrium of moments around support A allows for the calculation of the relation between total lift and the horizontal reaction (for a constant pressure distribution) in B as:

n m gs R = z TB30 (4.10) yA 4h

To retrieve the ratio between the vertical forces on each support, and because this structure is hyperstatic, a model that takes the spar’s deformation into account must be used. Unfortunately, there is much information lacking from the wing spar model, since the rear half of the part was not visible for measurement as well as the many ribs belonging to the assembly. We will use the model of the Main wing spar to extract an approximation of the vertical loads on both supports, with the knowledge however that this is an oversimpliﬁcation of the assembly. To simulate the lug joint at the root of the wing, ABAQUS c CBEAM Multi Point Constraints (MPC) were used, connecting the inner surfaces of the lug joint to a node positioned on the centre of these cylindrical surfaces. These are kinematic constraints which impose that all the displacement degrees of freedom of the nodes involved are equal on the solution obtained. As for boundary conditions, the displacement of these centred nodes was imposed as 0 in the three

41 translation components, and the loads were varied as will be described below. It is important that the rotation components are not constrained, since the parts are connected using metal pins. Since the conditions by which loads are carried from the wing surface to the spar are unknown, the load carrying surfaces were iterated between the top surface of the spar, the bottom surface, and both surfaces simultaneously. The results for reaction force on the supports showed little variation, granting conﬁdence in these results.

The loads obtained on the joints, for a load factor nz = 1 are shown in table 4.2

Model Figure RzA[N] RzB[N] RyA[N]

Loads through top surface 2360.4 4016.1 53368

Loads through bottom surface 2351.2 4018.8 53040

Loads through both surfaces 2354.7 4015.3 53182

Table 4.2: Reactions on wing spar pinned joints for a rectangular lift distribution, calculated through ﬁnite element model

The use of a constant distribution along the span is a conservative approximation, since it is known that tip effects on the ﬂow around the wing would make lift approach zero near the tip pf the wing, which in turn would decrease the horizontal loads on the lug joints. An elliptical distribution of lift was used, although it still may be very different from the real wing lift distribution as it still does not account for several geometric factors like chord distribution, twist and dihedral, for which a detailed description, and at the moment an infeasible one, of the wing geometry would be necessary. Nonetheless, including tip effects in the estimation of these loads will undoubtedly increase accuracy. Once again the axis system for the lift function of the span will have its origin at the root of the wing and the y coordinate aligned with the span. The expression from lifting line theory [39] acceptable for aspect ratio AR > 3 as is the case (for the Epsilon aircraft AR = 7) and which correlates the circulation distribution to the spanwise location y is:

X y Γ(y) = 4sU B sin n cos−1 (4.11) ∞ n s

Where Bn are the coefﬁcients of the elliptical distribution. Lift is obtained from circulation through the Kutta-Joukowski theorem:

dL(y) = ρU∞Γ(y) dy (4.12)

d h X y i L(y) = ρU 4sU B sin n cos-1 (4.13) dy ∞ ∞ n s

For this approach, only the ﬁrst coefﬁcient for the distribution will be different than 0, since there is no information available for a better approximation. This corresponds to a wing with minimum induced drag,

42 and will yield an elliptical distribution of lift.

The remaining coefﬁcient B1 will therefore be chosen so that the integral of the distribution matches the lift generated:

y Γ(y) = 4sU B sin cos−1 (4.14) ∞ 1 s

Z s 2 −1 y L = 4ρsU∞B1 sin cos dy (4.15) −s s

2 L = 2ρsU∞B1π (4.16)

2 P = 4sU∞B1 (4.17)

π n m g = P ⇐⇒ P = 8110.5 (4.18) z TB30 2

d y L(y) = P n sin cos-1 (4.19) dy z s

Using the same method for the calculation of the loads on the lugs, now with the elliptical distribution the forces described in table 4.3 are obtained.

RzA[N] RzB[N] RyA[N] RyB[N] 2475.6 3894.4 46326 -46326

Table 4.3: Reactions on wing spar pinned joints for an elliptical lift distribution for a load factor nz = 1, calculated using a ﬁnite-element model

The elliptical distribution approximation was found to be a more refined one, and therefore providing greater confidence in its results. The loads applied in the following analysis were the ones derived from this approximation. These loads were calculated for a steady level flight situation, where the load factor equals unity. If a linear variation between the load factor and the loads on the lug joint is assumed, the values on table 4.3 can be treated as the coefficients of linear functions that directly relate the loads felt with the load factor, i.e. for the horizontal loads on lug joint A, these would be of the form:

RyA = 46326nz (4.20)

And conversely for the remaining loads. To assume this linear variation one must disregard the variation in load application conditions, and in the shape of the deformation of the structure. Since the aircraft will not be subjected to accelerations larger than those required for the structure to yield, changes in the deformation due to plasticity can be

43 disregarded. As for the conditions of load application, since the lift distribution is dependent on the wing span, the deformation felt by the wing which will shorten its horizontal length can be source of non-linearity.

However, the displacement measured in the ﬁnite element analysis for nz = 1 along the y direction is less than 2mm (1.914) which is 0.065% of the span of one half of the wings and can be neglected.

4.5.2 Constraints

With the conditions for load application defined, there is need only for constraints on the displacement to be able to produce a static analysis for Frame C2. The first suggestion for displacement constraints comes from the use of the six bolted connections dispersed along the frame as no displacement constraints, by assuming that not only the bolted connections but also the structures they connect to are sufficiently rigid to yield negligible displacement in these places.

Fully Constrained Model

The same CBEAM MPC constraints used for load application on the lug joints were used to connect the hole surfaces to a central node that is constrained on all 6 displacement degrees of freedom, and applying the loads for the elliptic lift distribution, a static analysis of the frame was performed. The results however did not match at all to the results from the CEAT tests, or those performed by Milharadas, where the critical region of the structure is shown to be on the root of the ﬁllets on the bottom of the part.

Figure 4.13: Max. Principal Stress Plot for constrained structure.

The solution (Fig. 4.13) showed low stresses developing in the central areas of the frame, since the constraints imposed restricted deformation to the area ranging from the lug joints where there is a load input to the displacement constraints.

Unconstrained (Free) Model

A different approach was to remove the vertical loads from the analysis, so that the Frame is in static equilibrium, and use only sufﬁcient constraints so that the solver can yield a solution. This neglecting of the vertical component is an approximation which although it is stating that there is no vertical component of force to dynamically support the aircraft, structurally, the change in magnitude of the force applied compared to the total force determined earlier is 8.4%, and its change in direction less than 5o (4.805o).

44 Applying displacement constraints to all DOF of the node in centred in the bottom of the frame, and to every DOF except z-translation on the centred node on top of the frame, a model is obtained that is only sufﬁciently constrained to yield a solution, but not so much that these constraints would alter the deformed shape of said solution.

Figure 4.14: Max. Principal Stress Plot for unconstrained structure.

The results (Fig. 4.14) provided from this unconstrained structure yield the expected results from the CEAT tests, which place fatigue hotspots at the root of the ﬁllets of the inner ribs. The magnitude of the stresses obtained however is much larger than the 31.3 MP a obtained by Milharadas [3]. This overestimation of stress can be attributed to numerous factors, some regarding approximations made for load estimation, where it was considered that the full aerodynamic lift force is generated solely by the wings, and the portion of the load carried by frame C3 was neglected, due to its small stiffness in relation to C2. These approximations however are not far from the aircraft’s reality. Another source of error is the obvious underestimation of the rest of the aircraft’s stiffness when using the unconstrained structure. The bolted connections from the frame to the adjacent sheet structures as well as the various rivet connections along the frame which were not modelled due to limited access provide additional rigidity which will inevitably change the deformation of the component.

Added Spring Stiffness Model

It is infeasible to model the added stiffness of every riveted joint, but it is possible to model the bolted connections using 6 spring elements with high rigidity. ABAQUS c allows the use of spring elements that are snapped to ground, meaning the force produced is the stiffness of the element multiplied by the total displacement in any direction of the node where the spring is applied. These were applied to the central nodes in the same MPC beam structures used to enforced the no displacement constraints, and the same stiffness was applied to all 6. This conﬁguration of constraints will allow the full loads (horizontal and vertical) calculated for the lug joints to be used, further increasing the reality of the model. The solution provided will inevitably be an intermediate one between the unconstrained, and zero displacement solutions, which means that the value of stress on the critical hotspots can be tuned to match the values measured by Milharadas [3], hopefully providing a realistic approximation to the entire stress ﬁeld on the component. The values for this stiffness were varied iteratively until the maximum principal stress in the structural hotspots matched the 31.3 MP a found by Milharadas. In Fig. 4.15 a plot of the signed percent error of the FEM stress relative to the measured value is presented.

45 60

10 % Error (Signed) 0

-10

-20

-30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Stiffness [N/mm] 105

Figure 4.15: Spring Stiffness vs. Stress percent error relative to Milharadas [3].

For a stiffness of 255000 N/mm a sufficiently small error of 0.04% was found, and while the stress field (Fig. 4.16) on the component varied significantly, the expected structural hotspots remain regions of very significant stress, with new regions of highest stress appearing on the elements close to the lug joint (Fig 4.17). It should be noted, however that this region in the real structure will not be carrying loads in the same way the model is applying them, since the pin joint can only transmit load by pushing the pin against the cylindrical walls around it, and never by pulling (the load in this joint is pulling away from the centre of the frame). The stresses here were therefore overlooked in the analysis. An accurate representation of the stress in this region would require the modelling of the contact between the metal pin and the lug joint which was deemed too computationally expensive for the gain in accuracy.

Figure 4.16: Maximum Stress plot for Spring Model

It is also important to bear in mind that the loads used may be overestimated, due to other members in the aircraft’s structure absorbing them partially (Frame C3, wing load overestimation), which would result in using a larger spring stiffness and removing stress from the expected hotspots as was the case for the fully constrained solution. Despite this, the present model was deemed an acceptable approximation of the static behaviour of the real structure.

46 a) Detail of max. principal stress at lug b) Detail of max. principal stress at the joints Hotspot

Figure 4.17: Details of maximum stress plot (scale in a) )

4.5.3 Structural Hotspots

From these static results, and trusting the accuracy of the model, one can extract other structural hotspots to monitor in the component. Not only that, but it is possible to infer transfer functions that relate stresses in one monitored location on the aircraft, to other unmonitored spots of structural importance. As a preliminary analysis, using maximum principal stress as a criterion for regions more highly subjected to fatigue damage, one can ﬁnd the areas where this parameter is more intense to determine potential structural hotspots. Furthermore, one can relate the intensity of these stresses to get a somewhat detailed picture of fatigue damage across the component. It is important however, to note that this criterion is less accurate the more stresses in a certain point differ from uniaxial. For the expected hotspot, the maximum principal stress calculated from the present model equals the 31.3 MP a found by Milharadas. The components of the stress tensor in this point (Eq. 4.21) were found to be very close to uniaxial and oriented closely perpendicular to plane xz, since components other than

σy are all found to be many orders of magnitude below this larger stress.

  1.70 −1.03 0.0187     σij =  −1.03 31.1 0.482  (4.21)   0.0187 0.482 0.169

The biaxiality ratio is the quotient between the first and second principal stresses and is a relevant parameter in choosing between multiaxial fatigue criteria [40]. For a biaxiality ratio (λ) close to 0, the maximum principal stress is acceptable as a fatigue criterion. For the presented hotspot, the biaxiality ratio is λ = 0.04. Additionally, other hotspots with considerable values of maximum principal stress were identified and are represented in Fig. 4.18. Table 4.4 compiles their values, along with the biaxiality ratio found in each. These identified hotspots will be useful later on during aircraft instrumentation, since they signal additional regions where stress should be monitored. Special attention should be given to hotspot B, where stress is only slightly lower than the critical hotspot A’s stress.

47 Figure 4.18: Maximum Principal Stress plot with identiﬁed hotspots

Hotspot Max. Principal Stress (nz = 1) λ A 31.31 MP a 0.0387 B 30.37 MP a 0.0437 C 29.26 MP a 0.0193 D 29.06 MP a -0.0003 E 28.37 MP a 0.0792 F 27.47 MP a 0.0083 G 26.8 MP a 0.001 H 26.23 MP a 0.0082

Table 4.4: Maximum Principal Stress and Biaxiality ratio at relevant hotspots

4.6 Dynamic Analysis

As stated by Bruhn [41], when elastic structures are subjected to dynamic loads with high variation in time, there is a risk that the structures’ own inertia will produce stresses larger than those predicted by a linear static analysis. To accurately predict stresses on a component, one must therefore assess its behaviour when subjected to dynamic loadings. The objective of the following analysis will be the verification of the linearity of the transfer functions previously found when subjected to loads of varying frequency. It is important however to note, that even before this analysis is performed, there is no indication that the frequency of load application will have any influence on the stresses of the structure at hand. As can be seen from Fig. 4.8, the first natural frequency of Frame C2 is close to 150Hz, while the frequency of applied loads in a typical flight situation does not exceed 15Hz [41] and is therefore too low to excite the modes of vibration of the structure. This analysis will therefore serve to increase confidence in the use of a transfer function from load to stresses that is linear and independent of frequency. Firstly, a load spectrum that adequately represents the flight dynamics of the structure must be specified. To do this, the author was given data from the accelerometers previously installed on the Epsilon aircraft by Milharadas [3] for a real flight situation. The data is plotted in Fig. 4.19 and refers to a training flight,

48 starting acquisition when the system is turned on, with the included taxiing stages from before and after the ﬂight when the aircraft is being carried on the runway, and with the data referring to the ﬂight itself spanning through almost two hours.

5 Flight

3 Taxi Taxi 2 Load Factor

Sample Spectrum -1 0 1000 2000 3000 4000 5000 6000 Time

Figure 4.19: Load factor vs. time plot of the data provided

The taxi phases are easily identiﬁed by their much lower amplitude loads, and are of no relevance to the analysis. A dynamic simulation of the whole spectrum is possible, but too computationally expensive. Instead, the author decided to use a small sample of the entire loading, representative of the full range of frequencies to which the aircraft is subjected. A sample of ninety seconds of ﬂight data (represented in red in Fig. 4.20) was chosen around the highest loading peak present in the spectrum.

2 Load Factor

-1 3930 3940 3950 3960 3970 3980 3990 4000 4010 4020 Time

Figure 4.20: Load factor vs. time plot of the relevant data sample

The sample spectrum contains high amplitude and low frequency peaks and valleys, as well as low amplitude and high frequency variations which may be a combination of signal noise and actual aircraft vibration, which was left in the spectrum in order to have more high frequency information more likely to excite the structure. An added advantage of this spectrum is the highest peak which reaches a load

49 factor of nz = 5.2 which is very close to the aircraft’s maximum allowed, putting the component close to its critical conditions, and also providing high gain for the results making numerical errors less relevant. The previous model with added springs was used, and load application was made using the loading values for nz = 1 (Table 4.3) and multiplying by the load factor found in each step. This was made by using ABAQUS c amplitude feature applied to the load, which means that for any given step the vertical load on joint A, for example, will equal F zA = 2475.6nz and conversely for all other lug joint loads, where nz is extracted from the load spectrum curve for each step. An initial step of 0.04 seconds, equal to the sampling period of the data, was used, while later step intervals were automatically produced by ABAQUS c to increase stability. Outputs were extracted for every second of data in the sample. In Fig. 4.21, the stress at Hotspot A is plotted against the load factor nz, where the linearity of the stress with load factor can be easily seen, which, as expected, points towards the independence of the stress from the frequency of load application.

Figure 4.21: Hotspot Max. Princ. Stress vs. vertical acceleration. Plots provided by Critical Materials

The linear approximation ﬁts the dynamic model with a coefﬁcient of correlation of R2 = 0.999 making evident that for the service loads the aircraft is subjected to, one can approximate the critical stresses on the frame as a function σ = 31.4nz which is a very slight increase from the static analysis’ constant of 31.3, indicating a faint underestimation of the stresses in the purely static analysis.

Lastly, the maximum stress felt by the component is estimated to be nz,max × 31.3 = 194.06MP a which leaves the part at 56% of the yield stress. This part is therefore at a safety factor close to 2 for static failure, and rupture can be estimated to occur only for high cycle fatigue.

50 Chapter 5

Fatigue Analysis

Using the manufacturer’s reference load spectrum for fatigue analysis and the one constructed by the PrtAF from in flight acceleration data, several methods will be used to estimate fatigue lifetime. A first approach using cumulative damage from stress-life material data available is performed, followed by a fracture mechanics approach using various propagation laws fitted to experimental data. A comparison between the use of the Extended finite element method and the standard Finite Element approach using collapsed quarter point elements to obtain Stress Intensity Factors along a crack boundary will also be presented.

5.1 Load Spectra for Fatigue Analysis

In the reports of the real scale testing performed by the manufacturer in CEAT, the load spectrum used is presented, which is meant to reﬂect an equivalent for fatigue testing of the loads obtained during 1000 hours of aircraft operation. This information is presented in table form, compiling the number of times a certain load factor threshold is crossed. This process is called level cross counting. For every load peak occurring in ﬂight, every level below that peak is crossed in the process. Due to this, to transform the cumulated number of each level crossings into the number of peaks of the same load factor value, one must only subtract, for every value of measured g’s, the number of crossings for the value directly above. This results in an effective number of crossings of each level, corresponding to all the times a load reversal is observed in the designated range. In Table 5.1 the cumulated level crossings presented in the CEAT report [2] are shown along with the effective crossings calculated for all load factors where nz ≥ 2. To assemble this data into a representative load spectrum, the PrtAF suggests the use of only the positive load entries into an algorithm consisting of [3]:

• Disregarding all load factor counts for nz < 2 since their structural importance is reduced

• Randomizing the sequence of consecutive peaks

• After each load factor peak lower than 4 in the new sequence, insert a valley of 0.5

51 Load factor N Cumulated N Effective Load factor N Cumulated N Effective 2 19000 10800 2 22757 11157 2.5 8200 2700 2.5 11600 4729 3 5500 2700 3 6871 3467 3.5 2800 1980 3.5 3404 1765 4 820 520 4 1639 1071 4.5 300 205 4.5 568 315 5 95 62 5 253 147 5.4 33 23 5.4 106 0 5.75 10 8 5.75 106 64 6.2 2 2 6.2 42 42

Table 5.1: CEAT 1000FH load spectrum Table 5.2: CFAP 70.72FH load spectrum used in real scale testing [2]. Represented equivalent for 1000FH[4]. Represented in cumulated and effective level crossings of in cumulated and effective level crossings of different load factors different load factors

• After each peak higher or equal to 4 in the new sequence, insert a valley of -0.5

The output is a sequence of variable amplitude cycles of random occurrence. In Serrano [4], the acceleration measurements of 70.72 hours of Epsilon ﬂight operations is documented. The author of the thesis in question proceeded to count the cumulated number of crossing for each level, and multiplying by a factor of 14.44 extrapolated them to match the 1000FH simulated by the CEAT spectrum. These results are presented in Table 5.2 along with the effective number of crossings calculated. Through the comparison of these two load spectra in fatigue analysis, the author intends to provide a proposal for the scheduling of maintenance activities more suited to the necessities of the PrtAF ﬂeet.

5.2 Stress-Life Approach

Using stress based methods it is possible to provide an estimate of fatigue life. In his thesis, Serrano [4], through experimental testing of unnotched specimens of AA 2024-t3 provided parameters for the material’s S-N curve.

400 Regression Exp. data 350

300

250

200 [MPa] a S 150

100

0 104 105 106 107 108 Number of cycles, N

Figure 5.1: S-N curve regression for experimental data obtained by Serrano [4]

52 The results are plotted in Fig. 5.1, and the ﬁtting expression follows as:

−0.1603 Sa = 1583.4N (5.1)

The modified-Goodman criterion, along with Gerber and Morrow’s correction to the Goodman were used to provide an estimate of the number of cycles the part could withstand Ni for the various load cycles involved. Since PrtAF procedures indicate the use of a valley of load factor 0.5 and −0.5 after peaks lower and higher or equal then 4 in the applied spectrum respectively, a reduced number of terms was used to make the linear sum of cumulative damage using the Palgrem-Miner rule. The estimates for fatigue lifetime for both the PrtAF and CEAT load spectra were obtained using each of the three different criteria were compared (Table 5.3), and propose an average difference of 29.37% between them. One should note that these estimates indicate that the failure witnessed in CEAT at 89458FH happened for an average damage index of 0.1329 from the Miner rule, which is much smaller than unity. For this reason, the estimates for fatigue life using the PrtAF load spectrum will not be used as absolute values, but only the difference in regards to the CEAT one will be considered. The author suggests that this discrepancy will likely be due to the presence of a notch in the component in the form of the filleted feature where crack initiation was found and the significant difference between part geometry and that of the specimens used.

Criteria FH CFAP (d = 0.1329) FH CEAT( d = 0.1329) % difference Mod-Goodman 85958 120676 28.77 Gerber 29950 47733 37.25 Morrow 180744 232016 22.10 Average - - 29.373

Table 5.3: Comparison of fatigue life estimates for the CFAP and CEAT load spectra, obtained for a Damage index d = 0.1329.

5.3 PRODDIA c stress-life calculation

Critical Materials’ own software PRODDIA c was also used as a post processing tool to the dynamic analysis performed in section 4.6, allowing the application of the criteria mentioned above directly to the results of the FE analysis. From the results of maximum principal stress over time for the relevant structural hotspots, a rainﬂow counting method was applied (Fig. 5.2). The resulting spectrum of load cycles was used with the Goodman criterion and Miner rule to obtain an elementwise damage estimate directly on the FE mesh (Fig. 5.3). Since the dynamic analysis requires some degree in continuity for the loads applied, the load spectra described in section 5.1 could not be applied, and the data from section 4.6 was chosen. Although the use of such a short load spectrum disregards these results for comparison with the more descriptive CEAT and PrtAF spectra, this is a powerful tool that disregards some of the approximations

53 Figure 5.2: Rainﬂow counting method applied to max. principal stress results from FE analysis.

Figure 5.3: Hotspot damage computed from post-processing of FEM results by Critical Materials. made for aircraft operational data and structural behaviour to provide fatigue life estimates. In the future this approach may be used with more extensive operational data to obtain results of greater conﬁdence.

5.4 Crack Propagation

For this analysis, two different methods were chosen. The Extended Finite Element Method, and the standard Finite Element Method using collapsed quarter point elements at the crack front will be

54 employed to retrieve the stress intensity factors for the crack’s geometry. To do this, however, a detailed model for said geometry is required.

5.4.1 Crack Location and Geometry

From CEAT[2], failure was found to arise from unstable crack growth starting in the root of the fillet of the structure’s central ribs, region previously identified as a stress hotspot. The initial crack geometry was also found to be semi-circular and somewhere on the top surface of the lower part of the rib. To further determine the approximate location of the initial crack, the mesh for the static model of the frame was locally refined to find the exact spot of maximum principal stress.

Figure 5.4: Detail of the Hotspot region showing gradient of max. principal stress.

As seen in Fig. 5.4 the highest value for maximum principal stress is found exactly at the root of the fillet and centred in between the central reinforcement and the boundary of the structure. This spot can be assumed as the approximate location of the initial crack. As modelled by Milharadas [3] and in accordance to the findings of CEAT[2], the crack is assumed to have semi-circular geometry. Ideally, the centre of the half-circle would be positioned in the exact location of maximum stress, however, since the filleted region is of complex geometry and is modelled using tetrahedral elements, the author found it best to offset the crack by a small distance of 1mm along the y-axis (Fig. 5.5), making the local refinements around the crack surface easier, and allowing it to be embedded completely in hexahedral elements.

5.4.2 Geometric Factor Estimation: XFEM

Since XFEM has a lower dependence on the mesh surrounding the crack, due to the enriched shape functions used, local reﬁnements were less intense than those applied using standard FEM. However, to ensure that the crack surface was modelled correctly, with precise size and shape, the mesh was made of a uniform size of 0.5mm in a small area surrounding it (Fig. 5.6). A radius of 0.5mm surrounding the crack front was deﬁned to include 5 contours where the Stress Intensity Factors for all three modes of loading are calculated. These 5 contours were chosen for similarity to the FEM solution described below. These contours are calculated along a plane perpendicular to the crack front for each node present along this line.

55 Figure 5.5: Crack surface (in red) embedded in Frame 2

Figure 5.6: Crack surface (in red) embedded in XFEM mesh

Since the crack is of semi-circular geometry, it is convenient to describe the position of a given point on its front in a polar reference frame using radial (r) and angular (Θ) coordinates. In such a coordinate system, the entire semi-circle described by the crack front can be obtained by ﬁxing r = a and 0o < Θ < 180o. By averaging K-values from the different contours, it is possible to obtain a representative value of the SIF for each node along the crack front. The resulting plot in Fig. 5.7 provides a picture of the stress state along the crack.

The dominance of mode I is apparent, with KII and KIII being close to zero for the entire crack front when compared to KI . Furthermore, one can see an almost symmetric curve with a decrease of SIF at the regions of the crack which are deeper within the thickness. In truth the curve is not symmetric and reveals a lower peak of SIF for the side of the crack that is nearest to the central rib of the part, this is apparent from Fig. 5.8 for higher crack lengths, and is likely caused from the apparent decrease in stress close to the rib shown in Fig. 5.4. The crack size was varied from 1 to 6mm, which was deﬁned as the maximum allowed crack size, since it occurs on an area of 7mm and any crack exceeding this size would no longer be semicircular and would further complicate adequate meshing. The variation was done in unit increments which seemed

56 3

2.5

2 ] 1/2 1.5 K I K 1 II K

SIF [MPa m III

0.5

-0.5 0 20 40 60 80 100 120 140 160 180 Angle along crack front [Degrees]

Figure 5.7: Plot of mode I, II and III Stress Intensity Factor results along the crack front for XFEM using a = 3

to be sufﬁcient to capture the variation of SIF. In Fig. 5.8 the KI SIF along the crack front for three different crack lengths is plotted, where one can observe that the shape of the curve of SIF along the crack remains similar with increasing scale as a increases.

4.5

4 a = 1mm a = 3mm a = 6mm 3.5 ]

1/2 3

2.5 [MPa m I K

1.5

1 0 20 40 60 80 100 120 140 160 180 Angle along crack front [Degrees]

Figure 5.8: Plot of mode I Stress Intensity Factor results along the crack front with varying crack length for XFEM

For the propagation analysis, it was assumed a priori that the crack would maintain its original semi- circular geometry throughout. This is likely an approximation, as the SIF distribution would imply faster propagation of the superﬁcial regions of the crack front distorting the semi-circular shape. However, taking this into account would require very computationally expensive analysis with standard FEM.

Thus, the SIF was averaged along the crack boundary to obtain an average KI which would control the increase in radius of the crack semi-circle.

From the resulting KI values, the geometric factor was then calculated using the nominal stress as the

57 previously mentioned value of 31.3MP a since all analysis used a loading situation of nz = 1. Y is retrieved from the SIF for the various crack lengths a, from the expression:

K Y = √I (5.2) σ πa

The resulting curve was then plotted with the varying crack length and a suitable polynomial ﬁtting was performed to obtain a function Y (a) for all crack lengths within the data obtained (Fig. 5.9).

0.78

0.77 Y data XFEM 4th degree 0.76 y = 0.00025294*x4 - 0.0051202*x3 + 0.037846*x2 - 0.12739*x + 0.86766 0.75 R2 = 0.9992 0.74

Y 0.73

0.72

0.71

0.7

0.69

0.68 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Crack length (a) [mm]

Figure 5.9: Geometric factor Y and polynomial ﬁtting for XFEM analysis.

For these values, a 5th order polynomial was chosen, and the expression obtained was:

Y (a) = 0.00025294a4 − 0.0051202a3 + 0.037846a2 − 0.12739a + 0.86766 (5.3)

5.4.3 Geometric Factor Estimation: FEM

The use of standard FEM to model the crack surface requires very fine meshing of the surrounding regions. A square geometry swept along the crack boundary was constructed, bearing 5 elements from the outer limit of the region to the crack boundary defining 5 contours for the calculation of the stress intensity factor. At the centre, the collapsed quarter point elements were employed, (Fig. 5.10) requiring a more costly quadratic mesh to be used on this model. This change from linear to quadratic elements increased computational time but the results are of much greater confidence than its XFEM counterpart, where a higher variation in SIF is visible from one contour to the next (Fig. 5.11), suggesting a less stable solution and a need for a more refined mesh which contradicts the theoretically mesh-independent nature of XFEM. The percent variation between the first and last contours obtained in FEM is 0.026% which is over 1000 times smaller than the XFEM variation. The XFEM data also appears to estimate a higher average value for the SIF than the standard FEM.

58 Figure 5.10: Details of maximum stress plot (scale in a) )

3.1

2.9

2.8 XFEM FEM 2.7 1 K 2.6

2.5

2.4

2.3

2.2 1 1.5 2 2.5 3 3.5 4 4.5 5 Contour

Figure 5.11: Comparison between KI data obtained from the various contours from XFEM and FEM. The data is for Θ = 0 at a crack length a = 3.

Again the data obtained by FEM shows that Mode I dominates the propagation, since KI is at least one order of magnitude above either KII and KIII . The need for a ﬁner mesh also allows for greater resolution of the angular variation. The plot in Fig. 5.12 features only half the available points and displays a much smoother variation of SIF than the one obtained by XFEM.

A sharp decrease of KI can be observed in a small region close to the intersection between the crack front and the surface of the component. Frangi [42] calls this region a boundary layer where classical stress intensity factors obtained analytically, such as those of Hartranft et al. [43] either approach zero, or a lower non-zero value depending on the part’s geometry. Raju and Newman [44] provide examples of numerical solutions containing the same behaviour. In Fig. 5.13 it becomes apparent that the XFEM results provide a higher estimate of SIF than their FEM counterpart. A smoother curve from the FEM data is also seen , with the added difference of the boundary layer effect which is only present in FEM data. The resulting geometric factor was calculated along with an adequate polynomial ﬁtting (Fig. 5.14). Note however, that for these results, it is XFEM which presents a smoother tendency in geometric factor data,

59 3

2.5

2 ] 1/2 1.5 K1 K2 1 K3 SIF [MPa m

0.5

-0.5 0 20 40 60 80 100 120 140 160 180 Angle along crack front [Degrees]

Figure 5.12: Plot of mode I, II and III Stress Intensity Factor results along the crack front for standard FEM using a = 3. Re-sampled to display 1 point every 2.5o

4.5

FEM: a = 1mm 4 FEM: a = 3mm FEM: a = 6mm XFEM: a = 1mm XFEM: a = 3mm 3.5 XFEM: a = 6mm ]

1/2 3

2.5 [MPa m I K

1.5

1 0 20 40 60 80 100 120 140 160 180 Angle along crack front [Degrees]

Figure 5.13: Comparison between results of mode I Stress Intensity Factor along the crack front for XFEM and standard FEM although both indicate a decreasing tendency with the increase of crack length. The resulting expression:

Y (a) = 0.0024114a2 − 0.030693a + 0.74432 (5.4)

5.4.4 Parameter Fitting for Propagation Laws

In his thesis Serrano [4] monitored the crack size on aluminium 2024-T3 specimens with a crack geometry simulating that of the reports of CEAT for Frame 2 (see Fig. 5.15). These specimens were subjected to two different constant amplitude loadings. The ﬁrst specimen, designated by S1, was tested at a stress ratio R = 0.1, with a varying load between 65.25 and 6.525kN which corresponds to a nominal stress

60 0.72

Y data FEM 2 0.71 y = 0.0024114*x - 0.030693*x + 0.74432 quadratic R2 = 0.9679 0.7

0.69

Y 0.68

0.67

0.66

0.65

0.64 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Crack length (a) [mm]

Figure 5.14: Geometric factor Y and polynomial ﬁtting for FEM analysis. range at the crack site of 80.4065MP a.The second specimen, S2 was subjected to loads between 90 and 27kN which translates to R = 0.3 and a stress range of 86.2598MP a .

a) Specimen geometry b) Detail of specimen notch

Figure 5.15: Crack propagation specimen used by Serrano [4].

The author took the results from these experiments (Fig. 5.16) and used them to fit to crack propagation models which will later allow the estimation of the fatigue lifetime of the structure subjected to varying loadings. One must note the importance of having at least two different stress ratios R tested, since this is the minimum recommended number of tests [8] to fit those laws which have an explicit dependence of R for reasonable accuracy. A more accurate fitting for these equations would also include tests performed at negative load ratios. To retrieve the crack growth rate from crack length versus number of cycles data, numerical derivation procedures must be used. The central differences method was applied for all points except the first and last, where forward and rearward differences, respectively, were the only available option.

61 Number of cycles from 2a > 3.1mm

Figure 5.16: Crack Length vs number of cycles data from experimental testing, translated from [4].

da a − a = i+1 i−1 (5.5) dN i Ni+1 − Ni−1 To obtain ∆K one must know the geometric factor Y for the crack geometry used. The values calculated above for Y correspond to an estimation for cracks on the frame, which for the case of the specimen may not be as accurate as the ones obtained by FEM simulations of the cracked specimen. The geometric factor functions obtained by Serrano [4] were chosen for this analysis since these deal directly with the geometry of the specimen:

Y (Θ, a) = 0.7578 − 0.392Θ − 44.52a + 0.2806Θ2 + 0.2806Θa + 2.098x104a2 − 0.101Θ3

− 13.1Θ2a − 3.506x104Θa2 − 1.719x106a3 + 0.01647Θ4 − 12.32Θ3a

+ 1.063x104Θ2a2 + 3.109x106Θa3 − 7.38x107a4 + 1.132Θ4a + 487.9Θ3a2

− 1.132x106Θ2a2 + 3.22x107Θa2 + 9.089x109a5

π Where Θ is the angle along the crack front and should be used fixed as 2 . The typically exponential nature of crack propagation laws makes direct application of conventional multiple linear regression impossible. Fortunately, by taking logarithms of the various terms, it is possible to express these equations as linear dependencies of the various parameters. Following the methods of Draper et al. [45], to perform a linear regression, one must start from a function f(x) to be fitted to a set of points. This function depends not only of the values of x, but also of the parameters to be fitted, denoted as ηj.

yi = f(xi; ηj), (5.6)

Where yi are the images of the ﬁtting function at points xi. Taking as an example the NASGRO equation

62 arranged into logarithmic form, the absolute error ei between experimental values and the regression at point xi can be expressed as:

da 1 − fi ∆Kth Kmax,i ei = log − logCW − nW log ∆K − p log 1 − + q log 1 − (5.7) dNi 1 − Ri ∆Ki KIc

da To write this in the aforementioned notation, one must take log dN = yi, and logCN = η1, nN = η2, p = η3 q = η4. For each point in the set to be ﬁtted there is an equation of the form above. The system of equations can be written by deﬁning a matrix Xij, such as:

yi = Xijηj + ei + ci (5.8)

Where constant c was added to allow use for equations such as the Forman law, where the logarithmic form contains terms not multiplied by the regression parameters. X then becomes a matrix of coefﬁcients, which for the speciﬁc case of the NASGRO equation is:

  1−f1 ∆Kth Kmax,i 1 log 1−R ∆K log 1 − ∆K log 1 − KIc  1 i    X =  ::::  (5.9)  K  1 log 1−fm ∆K log 1 − ∆Kth log 1 − max,m 1−Rm ∆Km KIc

Following the methods in Draper [45], deriving eT e (the sum of square errors) in respect to the parameters vector η and setting the resulting expression equal to zero yields:

T −1 T T ηfit = (X X) (X y − X c) (5.10)

Which for the case of linear equations corresponds to the minimum of the sum of square errors. In the case of the Paris Law, which does not account for the effects of R, various ﬁttings were performed, including S1 series data only, S2 data and lastly, both S1 and S2 series data. Furthermore, the full range of data was not used, since this equation deals only with region II of crack growth, where the logarithmic plot of the crack growth rate curve is linear with the SIF range. Data which featured seeming non-linearity were thus excluded from these ﬁts. The results are presented in Fig. 5.17.

Paris Law - S1 + S2 Paris Law - S1 Paris Law - S2

−11 −11 −11 CP 3.0603x10 3.4381x10 8.3559x10

nP 3.8123 3.6113 3.5131 R2 0.9943 0.9992 0.9993

Table 5.4: Regression parameters for Paris Law.

For those laws which contemplate the effects of the stress ratio, only one ﬁtting was performed, containing data from both series of experimental data. For the Walker equation only data in the linear section was used, but for the Forman and NASGRO equations, which feature asymptotes for SIF approaching critical, the full spectrum of data was used. For the Walker and Forman equations, the regression and

63 10-6

10-7

10-8

da/dN [m/cycle] S1 data S2 data Paris Law - S1+S2 Paris Law - S1 Paris Law - S2 10-9

3 4 5 6 7 8 9 10 11 12 dK [MPa m1/2]

Figure 5.17: Crack Growth rate versus SIF range regressions for Paris Law. corresponding parameters are shown in Fig. 5.18.

10-6 10-6

S1 data S2 data

10-7 Walker - S1 10-7 Walker - S2

10-8 10-8 da/dN [m/cycle] da/dN [m/cycle]

S1 data S2 data Forman Eq. - S1+S2 (S1 load data) 10-9 10-9 Forman Eq. - S1+S2 (S2 load data)

3 4 5 6 7 8 9 10 4 6 8 10 12 14 16 dK [MPa/m1/2] dK [MPa m1/2]

Walker Equation - S1 + S2 Forman Equation - S1 + S2 −11 −9 CW 2.6380x10 CF 3.3469x10 nW 3.5898 nF 2.8724 γ 0.2083 R2 0.8168 R2 0.9983

a) Walker equation regression b) Forman equation regression

Figure 5.18: Crack growth rate versus SIF range regressions for Walker and Forman equations.

In the case of the NASGRO equation a ﬁrst attempt at minimizing square error was made by freely changing all four parameters of the equation, however, negative values for the asymptote curve control parameters p and q were obtained, which contradict fundamental mechanisms of crack propagation, e.g. while approaching KIc a negative value of q would cause the growth rate to decrease, which should clearly be dismissed. This was solved by ﬁxing these parameters between zero and one, and optimizing only C and n while changing p and q manually between iterations. The results for both regressions are

64 presented below in Fig. 5.19.

10-6 10-6

10-7 10-7

10-8 10-8 da/dN [m/cycle] da/dN [m/cycle]

S1 data S1 data S2 data S2 data NASGRO Eq. - S1+S2 (S1 load data) NASGRO Eq. - S1+S2 (S1 load data) 10-9 10-9 NASGRO Eq. - S1+S2 (S2 load data) NASGRO Eq. - S1+S2 (S2 load data)

4 6 8 10 12 14 16 4 6 8 10 12 14 16 dK [MPa m1/2] dK [MPa m1/2]

NASGRO Equation - S1 + S2 NASGRO Equation - S1 + S2 −10 −9 CN 3.3548x10 CN 2.6868x10 nN 3.3183 nN 1.8462 p 0.2388 p 1 q −0.4450 q 1 R2 0.7331 R2 0.8162

a) NASGRO equation regression with all b) NASGRO equation regression with ﬁxed parameters free. parameters p and q.

Figure 5.19: Crack growth rate versus SIF range regressions for Walker and Forman equations.

The best ﬁt resulted from leaving both p and q equal to unity, and coincidentally resulted in a better higher R2 than the solution for all four free parameters.

5.4.5 Fatigue Lifetime Estimation using Crack Propagation Models

With material parameters for crack growth now described, and an estimate of the SIF for the crack’s geometry, it is possible to obtain estimates for the number of cycles required for an existing crack in the hotspot identiﬁed to grow and cause fatigue failure of the component. Due to the varying amplitude of the load spectra, direct integration of the propagation laws is impossible. A cycle-by-cycle integration scheme is implemented, where the increase in crack length is obtained for each individual cycle of a speciﬁc SIF range. The integration of the propagation laws was performed using the trapezium rule. Below an example of the method is shown for the case of the Paris Law:

Z Ni+1 da da n n dN (∆K(Ni)) + dN (∆K(Ni+1)) C∆Ki + C∆Ki+1 ∆a = f(N)dN ≈ (Ni+1 − Ni) = x1 (5.11) Ni 2 2

Since geometric factor Y data above 6mm is unavailable, and in order to avoid inaccurate extrapolation of the existing data, the failure criteria assumed for this analysis was a maximum crack length of a =

65 6mm which although small in comparison to that achieved in experimental testing already describes a significant part of the propagation time, since crack growth rate increases exponentially making propagation very fast once this threshold is passed. For integration of the Paris Law, two different methods were used. First, only the parameters of the regression for both sets of data S1 and S2 were used in the cycle-by-cycle scheme. This set of parameters will be described as R1. The second approach was made choosing different parameters with regards to the stress ratio R specified by each cycle. In this way cycles with an R ≤ 0.2 would use the properties extracted from the S1 series of data, and for R > 0.2 the S2 only parameters would be used in hopes of more precisely describing each step of the crack growth, while the first approach would provide an average propagation for each cycle. This set of parameters will be described as R2.

3 Crack Length, a [mm]

Paris R1 - FEM

2 Paris R2 - FEM Paris R1 - XFEM Paris R2 - XFEM

1 0 0.5 1 1.5 2 2.5 Flight Hours 104

Figure 5.20: Comparison ofFH until critical crack length for Paris law regressions R1 and R2 for geometric factors obtained from FEM and XFEM.

The results using the geometric factor obtained by XFEM and FEM were compared (Fig. 5.20). As expected, the higher values of SIF estimated by the XFEM cause earlier expected failure of the component. Also comparing the approaches using Paris Law parameters, regression R1 is shown to estimate earlier component failure than R2. For the remaining propagation laws which contemplate R there was only one set of parameters which was used for each load spectrum simulated. The results for cycle-by-cycle integration of the growth rate for these equations are in Fig. 5.21. The introduction of the crack opening functions contemplated by the NASGRO equation seems to delay the expected propagation when compared to the other methods, presenting the latest estimate for component failure. Due to the presence of occasional negative values of R which the Paris law does not account for, the higher values ofFH to failure exhibited by the Forman and Walker laws in relation to the former seem expected. Between the Forman and Walker equations, the ﬁrst shows the earliest estimate for failure, possibly due to its modelling of the unstable growth region (III) of crack propagation. Again, the XFEM ﬁtting for geometric factor provides a more conservative result than the FEM one.

66 7

3 Walker - FEM

Crack Length, a [mm] NASGRO - FEM Forman - FEM Walker - XFEM 2 NASGRO - XFEM Forman - XFEM

1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Flight Hours 104

Figure 5.21: Comparison ofFH until critical crack length for various propagation laws for geometric factors obtained from FEM and XFEM.

The difference in ﬂight hours between the two methods reaches an average decrease of 23.89% for the XFEM propagation. Nonetheless, due to the improved stability observed in the SIF solutions, the author chose to use the FEM obtained ﬁttings for the following comparison between different load spectra.

4 Paris R1 - CFAP Paris R2 - CFAP Walker - CFAP 3 NASGRO - CFAP Crack Length, a [mm] Forman - CFAP Paris R1 - CEAT Paris R2 - CEAT 2 Walker - CEAT NASGRO - CEAT Forman - CEAT 1 0 1 2 3 4 5 6 7 Flight Hours 104

Figure 5.22: Comparison ofFH until critical crack length for all propagation laws subjected to load spectra CEAT 1000FH and CFAP 70.72FH.

From the curves in Fig. 5.22 it is observable that load spectrum CFAP 70.72FH is more severe on the structure than the one suggested by the manufacturer, making critical crack size be reached sooner. Averaging the difference in number of ﬂight hours to critical crack length in both spectra for the various propagation laws, an increase of 32.15% is computed for the PrtAF spectrum, which is a very signiﬁcant amount. In Table 5.5 the total amount of FH estimated for each equation and spectrum is presented, with the

67 Prop. Law FH CFAP FH CEAT % difference Paris R1 18774 29402 36.15 Paris R2 24643 38083 35.29 Walker 25861 36418 28.99 Forman 26095 35634 26.77 Average - - 31.8

Table 5.5: Comparison of fatigue life estimates for the CFAP and CEAT load spectra, from crack propagation methods assuming an initial crack of 1mm. exclusion of NASGRO. Due to severely overestimating propagation time, the NASGRO equation was removed from further analysis since a propagation time of 64876 hours from 1 to 6mm for the CEAT spectra would be inconsistent with the results from the real scale tests which register 89458FH for failure of the unnotched component.

68 Chapter 6

Final Remarks

Both the stress life and crack propagation approaches to fatigue analysis concluded similarly that the operating conditions of Epsilon aircraft by the PrtAF were more severe than those recommended by the manufacturer and tested in CEAT [2] by a difference of close to 30% in total useful life. In the following sections, these estimates will be used to suggest new scheduling of maintenance activities.

6.1 Proposal of Maintenance Scheduling

To describe the structure’s crack propagation behaviour along its entire lifetime, the crack size data from CEAT [2] was adjusted (Fig. 6.2) by a factor determined from the average of the relative differences in fatigue life obtained by the various methods used, including stress life and propagation laws. In Fig. 6.1 it can be observed that this value varies slightly around an average value of 30.76% across the various methods. The NASGRO eq. for crack propagation has been excluded as mentioned in section 5.4.5.

20 % Difference 15

Slife - Gerber Slife - Morrow Crack - Forman Crack - Paris R1 Crack - Paris R2 Crack - Walker Slife - Goodman

Figure 6.1: Average of the percent differences in fatigue lifetime found across all fatigue analysis methods.

69 A scaling factor of 0.6924 was thus applied to the CEAT curves, from which it is possible to deﬁne a schedule for maintenance operations.

30 Adjusted curve CEAT curve 25

20 a [mm]

0 0 1 2 3 4 5 6 7 8 9 FH 104

Figure 6.2: Crack size a for number of ﬂight hours simulatedFH curve obtained from real scale tests, and resulting adjustment for PrtAF loading.

6.1.1 First Inspection

The manufacturer defined the time of first inspection from the time expected for a crack of 0.5mm to develop, affected by a safety factor of 3. This crack length of 0.5mm is defined as the minimum crack size for which the Non-Destructive Testing (NDT) methods suggested can identify a crack. The PrtAF should make use of Eddy Current testing to identify cracks of such a small size.

Adjusted curve 4.5 CEAT curve

3.5 Time between following Inspections: 2729 FH 3

2.5 a [mm] 2

Expected critical crack size: 45525/3 FH 1.5

1st Inspection: 25896/3 FH 0.5

0 0 1 2 3 4 5 6 7 8 FH 104

Figure 6.3: Inspection times for PrtAF loading.

From the adjusted curve (Fig. 6.3), a operating time of 25896 FH is expected to lead to a crack of this

70 size. The factors chosen by the manufacturer will be kept in these proposals, making the suggested time 25896 for ﬁrst inspection 3 = 8632 FH.

6.1.2 Following Inspections

The manufacturer deﬁned a crack length of 1.5mm in the frame to be the critical size. When a crack of this size or larger is found on the component the end of aircraft operations is due. For a crack size of 30mm, the part is in the imminence of failure. In light of this information, CEAT agreed with the French Air Force to make the inspection times following the ﬁrst spaced by the amount of time between these two crack sizes divided by a safety factor of 6, making it very unlikely that an un-inspected crack could cause failure. From the adjusted curve, it is seen that a crack size of 1.5mm is reached for 45525 FH and after 61901 FH a crack of 30mm is expected to be found. From these, the suggested time between subsequent inspections is:

61901 − 45525 = 2729 FH (6.1) 6

The author proposes that first inspection for crack detection in the frame is performed after 8500 FH, with the following inspections executed in intervals of 2700 FH each, until a crack size of 1.5mm or more is detected, signalling the end of the aircraft’s useful life. These are a significant reduction from the manufacturer’s proposed 12400 FH for first inspection and 4000 for the period between following inspections, but should prevent premature failure due to the more severe PrtAF load spectra.

6.2 Future Work

6.2.1 Relating Sensor Data to Frame damage

One important study to be made for implementation in the SHM system of the ﬂeet would be estimating the changes in measured data from the increase in fatigue damage to Frame 2 over time. Doing this depends on knowing the accurate position of the strain gauge placed in the part, to be able to get theoretical stresses measured from the numerical model. By taking this data for different crack sizes, ideally in static and dynamic conditions, one could afterwards take the inverse approach, and estimate crack length from available sensor data, consequently leading to remaining useful life computations from the methods used in this thesis. A similar analysis for the wing spar would also be of use, but would require the modelling of stress on this part during operation.

71 6.2.2 Fatigue Analysis of Wing Spar

Another of the aircraft’s major structures which exhibited fatigue damage during the real scale tests is the wing spar. On this component, cracks were identiﬁed as starting on the riveted connections to the outer skins and propagating to the web of the spar, as is shown in Fig. 6.4.

Figure 6.4: Cracks found on the wing spars on real scale tests [2].

Although the critical component commanding fatigue damage in the aircraft is frame 2, whose failure caused the conclusion of the real scale tests, these cracks are also of great concern since some are already being monitored in Epsilon planes belonging to the PrtAF. The situation of stress analysis on the wing spar is a complex one, since many elements which are connected to the component, such as wing ribs and the outer skins, provide increased local stiffness to the part in some areas. Also, since crack initiation is found at riveted connections, a detailed modelling of these joints would also be necessary. In total, to be able to estimate stresses in the part from a numerical model would require modelling of a signiﬁcant part of the assembly surrounding it. The analysis performed in this thesis for frame 2 was only possible due to the simpliﬁed connections to the surrounding elements, which could be neglected with reasonable approximations. This is not possible for the spar. Efforts are already being made to instrument critical locations on the spar with strain gauges to estimate these stresses directly from the real part, as is described in chapter3. From the data acquired, a transfer function from vertical load to the stresses at these locations will hopefully be obtained, making it possible to estimate crack propagation and fatigue life in general for the component.

6.2.3 Application of Crack Arresting Solutions

In an attempt to further stretch the useful life of these aircraft, the author suggests the study of some crack arresting solutions.

Stop Hole

Perhaps one of the most widely known methods of prolonging the propagation life of a component is the drilling of Stop Holes at the crack front, with examples of its use including aircraft bodies, bridges,

72 among others [46]. Doing this removes the stress singularity at the crack tip, making stress concentration reduce to a manageable amount and improving fatigue life as a consequence. In 2004, Song et al. [47] performed tests on aluminium specimens with stop holes placed at the crack front and of varying diameter, obtaining fatigue lives up to 443% higher than for the un-repaired specimens (Fig. 6.5).

a) Al 6061-T651 specimen with stop hole. b) Crack length vs Number of cycles results

Figure 6.5: Improved fatigue life from varying diameter stop holes [47].

Positive results on fatigue life have also been found from introducing residual compressive stress around the stop hole by placing pins into them [48].

Crack Arrestor

A crack arrestor is a reinforcement that is applied to the component with the objective of reducing stresses in the crack region, making crack propagation considerably slower [49]. Some common designs include the welding or riveting of metal plates bridging the crack blocking its opening. These are very common solutions in pipe and ship hull repairs.

a) Specimen conﬁguration b) Crack length ratio vs Number of cycles results

Figure 6.6: Improved fatigue life from CFRP reinforcements on both sides of a steel specimen [50].

73 More recent studies have used bonded composite plates as reinforcements, achieving very good results on fatigue life. Ellyin [50] reports such an experiment with 3 layers of prepreg carbon ﬁber on both sides of notched steel specimens with considerable increase in propagation life (Fig. 6.6). A similar study by Daghyani et al. [51] reports crack arresting results for glass ﬁber reinforcements applied to a single side, and to both sides of an aluminium notched specimen. This solution has the added advantage that reduction of the load carrying section by drilling holes in the component is not required. The bonding of composite plates can be easily performed by wet layups directly on the part.

74 References

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78 Appendix A

Technical Datasheets

Datasheets for the sensors used in the DAU

79 Strain Sensors Model DT3757 Series • High Signal to Noise Ratio Columbia's strain sensors take the work out of strain sensing. They are complete and thermally • Ease of Installation compensated strain sensing solutions. Columbia's strain sensors are much easier to install compared to foil • Signal Gain of 10 strain gages by themselves and provide additional accuracy given the thermal compensation. Columbia's strain sensors have been flight qualified, including requirements to FAA DO-160, and have been used on • Thermally Compensated military and commercial aircraft for more than 30 years. Columbia's strain sensors have been used on • High Reliability various aircraft including large cargo airplanes, fighter aircraft, and helicopters. The DT3757 series takes Columbia's strain sensors to the next level by integrating an amplified within the sensor body providing a higher signal to noise ratio. Columbia's DT3757 has a signal 10 times higher than standard strain gages. In addition, the DT3757 series is interchangeable with Columbia's DT2684 series. The simplicity of use and reliability make them ideal for other applications including structural monitoring, geophysical measurements, and laboratory use. Note: Exports from the UnitedStates are subject to the licensing requirementsof the Export AdministrationRegul ations (EAR) and/orthe International Traffic in Arms Regulations (/TAR).

SPECIFICATIONS Operation1 Series DT3757 Input Resistance 10000, ±2% Sensitivity 10.25mVN/1000µ€ nominal Rated Excitation Voltage 7 to 10.0VDC Internal Amplifier Current 120µA@10VDC maximum 0.28 Linearity ±1.0% maximum Zero Offset ±0.5mVN typical Operating Range ±3000µ€ 100 cycles / ±2000µ€ 10,000 cycles Output Resistance 1000Ω, ±2% Sensitivity Shift ±0.013%/° F maximum Hysteresis, Repeatability ±0.5% maximum Zero Shift ±0.00025mVN/° F typical 2 Environmental Temperature Range -40° to 200°F Vibration 30g, 10Hz to 2KHz 3 Humidity MIL-STD-202 Method 1036 3 Salt Spray MIL-STD-202 Method 101D (168 Hours) 3 Insulation Resistance 100 Meg. min @ 500VDC Dielectric Strength 1050VRMS, 60Hz, 1 Min. Altitude Sea Level to 70,000 Ft. 3 Flammability MIL-STD-202 Method 111A 3 Shock 100g, 11mSec 3

Fluids Resistance to short term exposure to fuel, lubricating oils and hydrolic fluids 3

Physical Size 0.562" Square x 0.28" Thick Encapsulation Silicone Rubber per MIL-S-23586A Type I, Class 2, Grade A Weight Approx. 13gms (Depending on length of leads) Matrix 0.001" Polyimide Leads #26AWG, Teflon Ins, SPC, 12" Min. 1 @25°C 21nstalled Gage 3Preliminary/Qualified by Similarity/Pending Testing Fig. 2 Installation of Columbia StrainSensor Fig. 1 • TypicalIns tallation of Old Style Strain Gages 1. Bolt or rivet removed fromassembly 1. Strain Sensor bonded to surfaceunder test. 2. Dummy gage(s) bonded to "Z Tab" of same material as 2. Leads connected to wire harness. structure. 3. Coat sensor and wires with waterproofing material. 3. Activegage bonded to structure under test. ADVANTAGES 4. "Z Tab" mounted to structure with bond or rivet. 5. Strain gage leads interwiredand soldered to junction • Higher level accuracy • 10X the output block. • Less Installation time • No loss of structural Integrity • Optimum temperaturecompensation t 6. Entire unit covered with protectivematerial. ColumbtUResearchLaboratories,Inc. 1925 MacDade Blvd. Woodlyn, PA 19094 USA Phone: 1.800.813.8471 I Fax: 610.872.3882 I email: [email protected] I Web: www.crlsensors.com Rev. A Subject to change without notice Linear Accelerometers

SA-102MFTB, SA-302MFTA * Miniature Airborne Sensors * +28 VDC Operation The Columbia Models SA-102MFTB and SA-302MFTA are precision miniature force balance accelerometers designed to operate from +24 to +32 VDC aircraft * High Reliability power. The high level output signal and inherent self-regulation qualities of the 1 & 3 Axis Configurations servo loop design make these accelerometers excellent candidates for airborne * telemetry applications. The output signal format is compatible with most airborne VCO’s and signal processing systems. Both configurations incorporate the Columbia HP torquer system, which provides exceptional performance and reliability in the severe vibration and shock environments associated with aircraft and missile flights. The Model SA- 302MFTA is a compact triaxial version of the single axis Model SA-102MFTB. Consult the factory for customized versions of these sensors. Note: Exports of accelerometers from the United States are subject to the licensing requirements of the Export Administration Regulations (EAR) and/or the International Traffic in Arms Regulations (ITAR).

Specifications SA-102MFTB SA-302MFTA SA-102MFTB SA-302MFTA Operational Ranges Available +1 G To +50 G +1 G To +35 G Output Voltage 0.2 To 4.8 VDC into 100K Load Excitation +24 To +32 VDC, <30 mA Output Impedance <5000 Ohms Sensitive Axis Alignment Better Than 0.5 Deg Scale Factor Tolerance +1% Scale Factor Temp Coefficient 0.02% / Deg C Max. Zero Bias 2.5 +0.025 VDC Null Temp Sensitivity 0.005% F.R./Deg C Natural Frequency 50 To 350 Hz Dependent Upon Range Damping 0.7 +0.2 Cross Axis Sensitivity 0.002 G/G Exclusive of Case Alignment Output Noise <0.02% F.R. Non-Linearity 0.1% F.R. Hysteresis & Non-Repeatability 0.1% F.R. I / O Terminal Pin Functions: Threshold & Resolution 0.001% F.R. SA-102MFTB SA-302MFTA Pin Function Pin Function Environmental 1 +28 VDC A +28 VDC Temperature, Operating -50 To +90 Deg C 2 Pwr / Sig Grd B Ground Temperature, Storage -55 To +100 Deg C 3 Signal Out C “X” Signal Vibration Survival 4 Test (+) D “X” Ground 10 G RMS, 0.75” Disp D.A. (2 To 2,000 Hz) 5 Test (-) E “Y” Signal F “Y” Ground Shock Survival 1000 G, 0.5 mSec 500 G, 0.5 mSec G “Z” Signal Ambient Pressure 0 To 5 Atmospheres H “Z” Ground Humidity 90% R.H. I “X” Test J “Y” Test

Physical K “Z” Test Weight 2 Oz (57 Gm) Max. 14 Oz (397 Gm) Max. 1.60 In L x 1.56 In W x 1.36 In H 2.50 In Sq x 1.00 In H Ordering Information: Size (4.1 cm L x 4.0 cm W x 3.5 cm H) (6.35 cm Sq x 2.5 cm H) SA-102MFTB(+/- X G) Case Material Nickel-Plated Aluminum SA-302MFTA(+/- X G) Sealing Environmental Standard Accelerometer Electrical Interface 5 Terminal Pins 11 Terminal Pins Range +/- X G (Required)

R112805

Columbia Research Laboratories, Inc. 1925 Mac Dade Blvd. Woodlyn, PA 19094 USA Phone: 1.800.813.8471 / Fax: 610.872.3882 / email: [email protected]