Institute of Structural Univ.Prof. DI Dr. Martin Schagerl

Submitted byPratik Patel

Submitted atInstitute of StructuralLightweight Design

SupervisorUniv.Prof. DI Dr.Martin Schagerl

Co-SupervisorDipl.-Ing.Markus Winklberger

July 2020

JOHANNES KEPLERUNIVERSITY LINZAltenbergerstraße 694040 Linz, Österreichwww.jku.atDVR 0093696

Analysis of static stressdistributions and throughcracks in straight attachmentlugs

Master Thesisto obtain the academic degree of

Master of Sciencein the Master’s Program

Polymer Technologies and Science

AcknowledgementThis thesis is the final work that concluded my master degree in Polymer Technologiesand Science. This thesis work is carried out at Johannes Kepler University Linz. Thesubject of the thesis was proposed in collaboration with my co-supervisor Dipl.-Ing.Markus Winklberger.

I wish to express my deep gratitude for the support and help offered by the followingindividuals in completing this thesis. First, I would like to thank supervisor Univ.Prof.DI Dr. Martin Schagerl for allowing me to write the thesis at the Institute of StructuralLightweight Design. Especially, I would like to express my gratitude to co-supervisorDipl.-Ing. Markus Winklberger for the guidance and constructive discussions relatedto my thesis work. I greatly benefited from his comprehensive knowledge and goodcounselling. Additionally, I would like to thank my academic friends and my family forsupporting me through the whole tenure of higher education.

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AbstractAttachment lugs are connective elements with a hole carrying static and fatigue loads,and they are dominantly used in are in aircraft industries, railways, agriculture machin-ery etc. The hole edges of attachment lugs are subjected to stress concentration underthe application of load. An acceptable design of the straight attachment lugs againststatic and fatigue failure are based on detailed knowledge of the value of the peak stressat the hole edge of the lug, the resulting stress concentration factor and stress intensityfactor at developing cracks. For such case, there are numerous guidelines which coveringthese topics and suited as a guideline to engineers in aircraft industries, i.e. HandbookStructure Analysis (HSB). In this work, a comparative analysis of the SCFs in an un-flawed straight attachment lug by empirical and numerical methods under application ofstatic loading is accomplished. The identical analysis is conducted in an unflawed finiteplate containing a centrally located hole for acquiring a comprehensive understandingof the behaviour of the stress concentration factor. Additionally, comparative analysisof empirical and numerical methods for the stress intensity factor is conducted for thefinite plate with a centrally located through-crack. For a straight attachment lug, thecomparative analysis of the stress intensity factor of a through-crack is based on theliterature ESDU 81029 and numerical calculations. The stress concentration factor inthe straight attachment lug with a hole is, in general, depends on its geometry whichmeans its width and hole diameter. The effect of the thickness parameter was not con-sidered for the analysis. While the SIF depends on crack configuration, the geometry ofthe material and applied load. In this work, the SCF for both above-defined structures(straight attachment lug and finite plate) are analysed for W/D ratio ranges from 1.5to 4. The stress intensity factor is analysed for relative crack length ranges 0.04 to 0.8for a finite plate and range 0.025 to 0.8 for a straight attachment lug. Furthermore, amesh convergence study is carried out on an unflawed straight attachment lug having awidth-to-diameter ratio of two. The finite element software tool Abaqus CAE is utilizedfor modelling and numerical analysis. The behaviour of tangential stress across the lughole and away from the lug axis under application of a static axial load in a straightattachment lug is analysed using finite element analysis. Also, the von Mises stressesalong the circumference of the lug hole are analysed. These analyses are carried outon lug geometry having width-to-diameter ratios 1.5, 2, 3 and 4. It is concluded thatthe results of the SCF and SIF calculated with formulas from literature are in closeagreement with numerical calculations.

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List of Notations and Symbols

Notation DefinitionSCF Stress concentration factorSIF Stress intensity factorKt Stress concentration factor in finite and infinite platesKtb Stress concentration factor in a straight attachment lugK Stress intensity factor at the crack tipKI Stress intensity factor at the crack tip in a crack opening mode of

failureKII Stress intensity factor at the crack tip in an in-plane shear mode of

failureKIII Stress intensity factor at the crack tip in an out-of-plane shear mode

of failureKIC Fracture toughness of the material for an opening mode of failureKIIC Fracture toughness of the material for an in-plane shear mode of failureKIIIC Fracture toughness of the material for an out-of-plane shear mode of

failureσpeak Peak stress at the root of a notchσnominal Nominal stressS∞ Uniaxial remote stress in an infinite plateS Uniaxial applied stress in a finite plateD Diameter of a holeW Lug width and plate widthL Length of the plate with a holea Length of a crackρ Radius of a root of the notchθ θ = 0° — aligns with remote loading directionφ Angle made by a small element of stress from a crack tip to the crack

planeβ Dimensionless geometrical factorσxx Tensile or compressive stress in the x-directionσyy Tensile or compressive stress in the y-directionτxy Shear stress in the xy-directionσrr Radial stress in an infinite plate containing a holeσθθ Hoop stress in an infinite plate containing a holeτrθ Shear stress in an infinite plate containing a holer Distance of a stress element from a crack tipR0 Radius of the outside edge of an attachment lugδ Oblique loading angle made with longitudinal axisP Failure load of a lugKBR Bearing efficiency factorFtu Ultimate strength of a lug material

iv

v

t Thickness of a lugk1 and k2 Dimensionless factorsA Material parameter, MPaB Dimensionless material parameterN Number of loading cycle to failureR Stress ratioRm Ultimate tensile strength, MPaφ Dimensionless factorσa Net-sectional stress amplitude of the analysing lugσa,sl Net-sectional stress amplitude of the standard lugσn Gross stress acting on the lug or nominal stressα Dimensionless geometrical factor in a straight attachment lug contain-

ing a crack

Contents

1 Introduction 1

2 Theoretical background 3

2.1 Phases of fatigue life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Crack initiation period . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Crack growth period . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Stress concentration factor . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Infinite plate with a centrally located hole . . . . . . . . . . . . . 9

2.2.2 Finite Plate with a centrally located hole . . . . . . . . . . . . . . 11

2.3 Stress intensity factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Infinite plate with a centrally located crack . . . . . . . . . . . . . 15

2.3.2 Finite plate with a centrally located crack . . . . . . . . . . . . . 17

2.3.3 Crack in arbitrary geometry using FE method . . . . . . . . . . . 18

2.4 Lug shapes and parameters . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Lug materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Lug loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Static analysis of lugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7.1 Ekvall’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7.2 HSB 26101 calculation guideline . . . . . . . . . . . . . . . . . . . 23

2.8 Fatigue analysis of lugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8.1 ESDU 81029 calculation guideline . . . . . . . . . . . . . . . . . . 24

2.8.2 HSB 63511 calculation guideline . . . . . . . . . . . . . . . . . . . 26

2.9 Fatigue crack growth in lugs . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Finite element calculations 30

3.1 Static analysis of the finite plates and the straight attachment lugs . . . 30

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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CONTENTS vii

3.1.2 Module Part - FE modelling of a finite plate and a straight at-tachment lug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.3 Module Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.4 Module Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.5 Module Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.5.1 Modelling a crack using conventional FE method . . . . 35

3.1.5.2 Modelling a crack using XFEM method . . . . . . . . . 35

3.1.6 Module Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.6.1 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.6.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 39

3.1.7 Module Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.7.1 Mesh Density . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.7.2 Mesh Elements . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.7.3 Mesh Control . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.7.4 Element Shape . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.7.5 Meshing techniques . . . . . . . . . . . . . . . . . . . . . 40

3.1.7.6 Meshing Algorithm . . . . . . . . . . . . . . . . . . . . . 41

3.1.8 Mesh Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.9 Verification of the FE model and analysis procedure . . . . . . . . 41

3.1.10 Stress concentration factor . . . . . . . . . . . . . . . . . . . . . . 42

3.1.11 Stress intensity factor . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Mesh convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Stress distribution in a straight attachment lug . . . . . . . . . . . . . . 44

4 Results and discussion 47

4.1 Comparison of stress concentration factors . . . . . . . . . . . . . . . . . 47

4.2 Comparison of stress intensity factors . . . . . . . . . . . . . . . . . . . . 49

5 Conclusions 51

6 Future work 52

List of Figures

1.1 Necked lugs within tie rods attachment lugs during operation [1]. . . . . 1

1.2 Airbus A300 rudder hinge lug [1]. . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Lug joint under tension [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Different phases of fatigue life and relevant factors [3]. . . . . . . . . . . . 4

2.2 Cycle slip leads to crack nucleation [3]. . . . . . . . . . . . . . . . . . . . 5

2.3 Cross-section of microcrack [3]. . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Top view of a crack with the crack front passing through many grains [3]. 7

2.5 Stress distribution gradient in a finite strip with a centrally-located hole. 8

2.6 Infinite plate containing a hole with representation of remote uniaxialload and infinitesimal stress element [4]. . . . . . . . . . . . . . . . . . . 9

2.7 Tangential stress around a circular hole in an infinite plate loaded by atensile stress S∞ [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.8 Effect of shape of the hole on Kt in an infinite plate loaded under tension[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.9 A graph depicts the SCF decreases towards two with increase in diameter-to-width ratio in a finite plate according to 2.11, 2.12 and 2.13. . . . . . . 12

2.10 The comparison of ratios σpeak/σnominal and σpeak/S in a finite plate. . . . 13

2.11 Three different opening modes [3]. . . . . . . . . . . . . . . . . . . . . . . 15

2.12 Different types of cracks initiating in the material from the edge of a hole[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.13 Infinite plate containing a crack with representation of remote uniaxialload. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.14 Relationships between dimensionless geometrical factor β and relativecrack length a/b according to 2.28, 2.29 and 2.30. . . . . . . . . . . . . . 17

2.15 The sketch depicts the layers of mesh element around the crack tip withtheir contour integrals order. . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.17 Sketch of tie-rod containing straight tube, sleeve and two necked lugs atthe ends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.18 Necked lug [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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LIST OF FIGURES ix

2.19 Failure modes in straight attachment lug [6]. . . . . . . . . . . . . . . . . 22

2.20 Relation between the bearing efficiency factor KBR and the SCF Ktb forthe straight attachment lugs loaded at 0° and 45° and tapered lugs loadedat 0°, 45° and 90° to the lug axis [7]. . . . . . . . . . . . . . . . . . . . . 23

2.21 Sketch depicts the straight attachment lug configuration utilized for itsstatic analysis by HSB 26101 calculation guideline [8]. . . . . . . . . . . . 24

2.22 List of α values of a symmetrical through-cracks for the straight attach-ment lug of various configuration under axial loading state [9]. . . . . . . 25

2.23 Sketch depicts the symmetrical through-cracks at two opposite sides of ahole in straight attachment lug [9]. . . . . . . . . . . . . . . . . . . . . . 25

2.24 Straight attachment lug configuration utilized for fatigue analysis by HSB63511 guideline [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.25 Crack growth curves of four specimens with artificial cracks [11]. . . . . . 28

2.26 Comparison between the crack growth curves of natural cracks and arti-ficial cracks [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 The FE model of a finite plate having centrally located hole with rep-resentation of boundary conditions and an applied load utilized for theanalysis of SCFs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 The half-symmetrical FE models of a) a straight attachment lug withrepresentation of boundary conditions and b) a pin with representationof an applied load, utilized for the analysis of SCFs. . . . . . . . . . . . . 32

3.3 The half-symmetrical FE model of a finite plate with representation of acentrally located crack, boundary conditions and an applied load utilizedfor the analysis of SIFs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 a) a straight attachment lug with representation of symmetrical edgecracks, boundary conditions and b) a pin with representation of the ap-plied load, utilized for the crack analysis by conventional FE method. . . 32

3.5 Abaqus CAE window of the step module. . . . . . . . . . . . . . . . . . . 33

3.6 Stress distribution in a two-dimensional FE model of a finite plate witha centrally located hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Stress distribution in a two-dimensional FE model of a straight attach-ment lug. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.8 Two dimensional half-symmetrical FE model of a finite plate with a crackand spider-shaped mesh elements distribution around a crack tip. . . . . 36

3.9 Three dimensional half-symmetrical FE model of a finite plate with aXFEM crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

LIST OF FIGURES x

3.10 Two dimensional FE model of a straight attachment lug with symmetricalcracks at two opposite ends of the lug hole utilized for crack analysis usingthe conventional FE method. . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.11 Three dimensional FE model of straight attachment lug with XFEM cracks. 37

3.12 Abaqus CAE window of XFEM crack . . . . . . . . . . . . . . . . . . . . 37

3.13 XFEM model: a) A straight attachment lug with representation of sym-metrical XFEM edge cracks and boundary conditions. b) A pin withrepresentation of the applied load. . . . . . . . . . . . . . . . . . . . . . . 38

3.16 Convergence of a maximum von Mises stress and a maximum tangentialstress arising at the hole edge in a straight attachment lug with increasein mesh density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.17 Stress distribution along the y-axis in the unflawed straight attachmentlugs for four distinct values of width-to-diameter ratios. . . . . . . . . . . 45

3.18 Von Mises stress along the edge of the lug hole for a pin loaded in 0°direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Comparison of analytically and numerically calculated SCFs of finiteplates with a centrally located hole. . . . . . . . . . . . . . . . . . . . . . 48

4.2 Comparison of analytically and numerically obtained SCFs of a straightattachment lugs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Comparison of empirically and numerically calculated results of finiteplates with a centrally located crack. . . . . . . . . . . . . . . . . . . . . 50

4.4 Comparison of data given by ESDU 81029 and numerical results of straightattachment lugs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

List of Tables

3.1 Modified guidance on FE modelling with respect to derivation of the SCFs. 42

3.2 Modified guidance on FE modelling with respect to derivation of the SIFs. 43

4.1 Comparison of SCFs according to analytical and numerical methods forthe finite plates with a centrally located hole. . . . . . . . . . . . . . . . 47

4.2 Comparison of SCFs according to analytical and numerical methods forthe straight attachment lugs. . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Calculation of SIFs according to analytical and numerical methods forfinite plates with a centrally located crack. . . . . . . . . . . . . . . . . . 49

4.4 Calculation of SIFs according to ESDU 81029 and numerical results forthe straight attachment lugs. . . . . . . . . . . . . . . . . . . . . . . . . . 49

xi

1 Introduction

Figure 1.1: Necked lugs within tie rods attachment lugs during operation [1].

Figure 1.2: Airbus A300 rudder hinge lug [1].

Lug attachments are connecting elements dominantly used for structural supports inairframe structures, railways, agricultural machinery and other engineering applications[12]. Lug attachments consist of a lug (male lug) and a fork (female lug) held togetherby a single bolt (pin), as shown in Figure 1.3. All these components of a lug attachmentoffer a high amount of flexibility in assembly and disassembly during installation andrepair [2].

The main purpose of most of the lug attachment is to transmit loads from one elementto another during the service operation. They are mostly used to connect differentstructural elements [13]. For example, wing-fuselage attachment (lugs transfer shearand bending loads from the wing elements to the fuselage), landing gear links, etc [13].

1

1. Introduction 2

Figure 1.3: Lug joint under tension [2].

Attachment lugs are simple in geometry and help efficiently transmit high loads. At-tachment lugs are quick and simple in installation. As lug attachments are used forconnecting major components, it is necessary to analyse them under static and fatigueloading cases.

The unavoidable hole in these attachment lugs leads to stress concentrations at theedge of the lug hole during loading. Surveys conducted on aircraft structural failures bythe United States Air Force revealed that bolt and rivet holes comprise over one-thirdof observed failure origins [2]. In general, aircraft attachment lugs are exposed to cyclicloading. Hence, the combination of high-stress concentrations, fretting and corrosionleads to premature failure of the attachment lug owing to crack initiation and its prop-agation [14]. The main aim of this thesis is the assessment of stress concentrations atthe edge of a notch and the intensity of stress distributions at the tip of a crack inthe straight attachment lugs and finite plates through common parameters: The stressconcentration factor (SCF) and the stress intensity factor (SIF).

2 Theoretical backgroundLug attachments are carrying static as well as fatigue loads during the service opera-tion. Hence, it is crucial to understand the fundamental base of the origin of failures.Nowadays, complete failures under static loading are rare, in contrast to fatigue inducedfailures, which are complex to assess due to numerous influence factors.

Many serious fatigue failures were reported in the 19th century. August Wöehler hadconducted noteworthy research on fatigue failure. He observed that the static load act-ing on material and which is far below its static strength did not do any damage whilethe equivalent load when applied repeatedly causes complete failure to the material [15].Most scientists working with materials say that such fatigue failures are insidious, asthey occur with little warning or none at all [16]. The frequency of unexpected failureson mechanical elements is due to propagating cracks that remain undetected. 60 % offailures occur owing to fatigue in aircraft concerning a study carried out by NationalBureau of Standard (now known as NIST) on 230 failed aircraft components [17]. Anaircraft is rarely failing due to a static overload during its service life [18]. As previouslymentioned, one basic application of lug attachments in aircraft is to connect the wing tothe fuselage [19]. The lug attachments in aircraft are usually utilized for the transfer ofload from stationary and moving mechanical elements to other elements of a structure[14]. In the 19th century, the fatigue phenomenon was considered a mysterious problemin the material as damage could not be foreseen and failures occurred without warning.In the 20th century, owing to the research of Woehler and other researchers, it wasdetermined that repeated load application leads to nucleation of microcracks followedby crack growth, and ultimately complete failure [3]. Therefore, phases of fatigue lifewhich help to understand the behaviour of stress concentration at notches and stressseverity at the crack tip were studied in detail.

2.1 Phases of fatigue life

The phases of fatigue life comprise two main phenomena: Firstly, a crack initiation pe-riod and secondly a crack growth period [3]. In a specimen subjected to fatigue loading,nucleation of a crack is originated at a microscopic level which grows to macroscopicsize, and finally into a specimen failure as depicted in Figure 2.1. The time requiredfor the whole process to take place is the fatigue life of that specimen. The crack ini-tiation period includes nucleation of a crack by cyclic slip, together with microscopiccrack growth that is invisible to naked eyes for a considerable part of the fatigue life [3].During the crack growth period, the microcrack originated in the crack initiation periodgrows further until complete failure [3]. Several environmental conditions influence thecrack initiation period and barely affect the crack growth period [3]. For example, sev-eral surface conditions such as surface roughness affect the initiation period and have a

3

2. Theoretical background 4

Figure 2.1: Different phases of fatigue life and relevant factors [3].

no effect on the crack growth period [3].

Furthermore, predicting the fatigue phenomenon or processes is different for both pe-riods. For example, the SCF and SIF are crucial parameters for examining the crackinitiation period and the crack growth period (e.g. Paris law) respectively [3]. Addition-ally, fracture toughness is a material property which is calculated with the laboratoryspecimen test and is prerequisite for the analysis of crack growth period. All threemodes of fatigue failures have their respective fracture toughness value and are denotedby KIC , KIIC and KIIIC for opening under loading in tension, in in-plane shear and intransverse or out-of-plane shear respectively.

2.1.1 Crack initiation period

The fatigue crack nucleation begins in slip bands. A slip band simply is a part of ma-terial where cyclic deformation occurs during the application of fatigue load, which isdepicted in Figure 2.2. This crack nucleation originates when cyclic stress exceeds thefatigue limit of that material. After the nucleation of a crack, this microcrack formationprocess can still be slow or erratic.

In the crack initiation period, when the cyclic load exceeds the fatigue limit, the cor-responding cyclic plastic deformation (cyclic slip or dislocation activities in the crystallattice) leads to the origin of crack nucleation [3]. In general, the process of crack ini-tiation occurs at stress amplitudes below yield stress as mentioned previously. In theelastic limit region, plastic deformation is limited to only a few crystal lattices or grainsof material and this kind of microplastic deformation occurs normally at the surface ofthe structure, owing to less constraint on the slip at the surface [3]. In contrast, thegrains available in the inside of the material are surrounded by other neighbouring grainswhich prohibit the cyclic deformation (a high degree of slip constraint in the material’sinside region compared to the surface region).


Figure 2.2: Cycle slip leads to crack nucleation [3].

Materials which suffer from cyclic slip require an application of cyclic shear stress on it[3]. A material experiencing cyclic load is prone to cyclic shear stress. At a microscopiclevel, this cyclic shear stress varies from one grain to another in the material (genera-tion of shear stress gradient). There are two important phenomena occur during cyclicslip. Firstly, when slip occurs at the material surface, a slip step will be created at thematerial surface, as shown in Figure 2.2. And, owing to slip step, part of new mate-rial is exposed to an outside gaseous or liquid environment. This part of the exposedmaterial immediately starts reacting with the oxygenated environment and leads to anoxide layer formation. Such an oxide layer strongly adheres to the material surface andare not easy to remove [3]. Therefore, deformed material will not come to an originalundeformed state after complete unloading, even if the load is within the elastic limit.Secondly, slip during the increase of loading leads to some strain hardening in the slipband [3]. Therefore, after unloading larger shear stress in the opposite direction remainsand the material will not come to its original undeformed state. Both these phenomenaoccur within the material’s elastic region [3]. The reversed slip occurs in the same slipband but on an adjacent parallel slip plane and produces an intrusion or extrusion, asschematically illustrated in Figure 2.2 [3]. Intrusion or extrusion depends on the direc-tion of an applied cyclic load. The same sequence of events occurs in further loadingcycles, as indicated in Figure 2.2.

This is the theory of lower constraint on the cyclic slip at the material surface formicrocrack initiation. Besides this theory, additional arguments for the cause of crackinitiation at the material surface are the inhomogeneous stress distribution in structurescontaining a notch, owing to stress concentration at the surface edge of a notch [3]. Fur-thermore, surface roughness, corrosion pits and fretting fatigue damage have negativeeffect on crack initiation at the surface [3]. Therefore, the crack initiation period offatigue life is a surface phenomenon [3]. Whenever a mechanical component containingnotches undergoes tension or compression loading, stress concentration always occursat the notch edges [3]. The raised level of stress at notches can be described with the


stress concentration factor (SCF), see Section 2.2. However, stress concentration occursat the tip of the generated microcrack. The parallel outward lines at the tip of a mi-crocrack show stress concentration in Figure 2.3. Unfortunately, these microcracks arenot visible, as previously discussed. Therefore, the SCF is utilized to evaluate the stressconcentration intensity at the edge of notches neglecting microcrack effect [3].

2.1.2 Crack growth period

As described in Section 2.1.1, in the crack initiation period microcracks are generatedin elastically anisotropic material with several slip systems at the micro-level [3]. Inho-mogeneous stress distributed at microlevel with stress concentration at the tip of themicrocracks owing to the presence of these microcracks in the material [3]. In this pe-riod, a microcrack grows through a grain with no restrictions from the surface of thematerial to the inside boundary of the same grain [3]. Later, when this growing crackreaches the boundary of an adjacent grain, its movement is restricted. Therefore, thisgrowing crack deviates its direction at the boundary to the other less restricted direc-tion. It means a crack grows through multiple slip bands [3]. The microcrack growingdirection also depends on the direction of the load application [3].

When the crack-front of these microcracks pass through a substantial number of grainsinside the material, as schematically illustrated in Figure 2.4, and as the crack front iscontinuous, the crack growth rate becomes more independent on the adjacent restrictionof grains boundary [3]. Furthermore, due to the restriction of a coherent crack front, acrack cannot grow in any arbitrary direction. This continuity prevents a large gradientof the crack’s growth rate. The growth rate depends on the crack growth resistanceof the material. Therefore, two important parameters (previously discussed) which areresponsible for crack nucleation are no longer effective in the crack growth period. Addi-tionally, the parameters surface roughness and other surface conditions are not affectingthe crack growth period [3]. Therefore, material’s resistance to crack growth depends onthe bulk property, and crack growth is not anymore a surface phenomenon [3]. There-fore, the SCF is useless in the crack growth period, owing to singularity of stress at thecrack tip [3]. In this period, the SIF is utilized to predict the stress (singular stress)distribution at the tip of a crack which is illustrated in Section 2.3.

2.2 Stress concentration factor

In a mechanical structure, geometrical notches (e.g. holes) are unavoidable. Becausein a structure, geometrical notches are always needed for assembly, disassembly of sub-structures and load transfer application. Notches lead to inhomogeneous stress distri-bution in the adjacent structure. In the following, the stress concentration is explainedusing a simple plate with a hole. In Figure 2.5, the maximum amount of tangentialstress is concentrated at the notch root and then it decreases across the width [3]. Thestress concentration occurs because of the reduced section available near the hole for


Figure 2.3: Cross-section of microcrack [3].

Figure 2.4: Top view of a crack with the crack front passing through many grains [3].


Figure 2.5: Stress distribution gradient in a finite strip with a centrally-located hole.

load transfer. The intensity of stress distribution at the edge of notches in the materialis represented by the SCF. The SCF theory is established on the assumption of elasticmaterial behaviour (materials deformation within elastic limit during load application)[3]. In general, the SCF is defined as the ratio of peak stress at the notch root σpeakand nominal stress across the remaining net section across the hole σnominal, see (2.1)[3]. It is a dimensionless parameter and only a function of geometry. Therefore, theintensity of stress concentration depends only on the geometrical configuration of thenotch (shape of the notch) together with the specimen’s geometry [3]. Additionally,the nominal stress increases rapidly with increase in hole diameter or decrease in widthcomparably to the respective peak stress in a structure.

Kt = σpeakσnominal

(2.1)

The evaluation of the SCF in an attachment lug is crucial for further strength analysisto calculate a failure load [7]. Therefore, the behaviour of the SCF is studied in detailinitially with two basic examples which are explained in following sections.


2.2.1 Infinite plate with a centrally located hole

The stresses at the hole edge in an infinite plate is evaluated with Kirsch’s solution [4].Figure 2.6 depicts an infinite plate is loaded under application of uniaxial remote stressS∞, where a is the hole radius, r is the radial coordinates and θ = 0° aligns with remoteloading direction.

Figure 2.6: Infinite plate containing a hole with representation of remote uniaxial loadand infinitesimal stress element [4].

According to Bob McGinty [4], the stress state solutions are illustrated as follows:

σrr = S∞2

[1−

(a

r

)2]+ S∞

2

[1− 4

(a

r

)2

3(a

r

)4]cos 2θ (2.2)

σθθ = S∞2

[1 +

(a

r

)2]− S∞

2

[1 + 3

(a

r

)4]cos 2θ (2.3)

τrθ = −S∞2

[1 + 2

(a

r

)2

− 3(a

r

)4]sin 2θ (2.4)

At the hole edge at a = r and θ = 90°, the above three equations modifies to [4]:

σrr = τrθ = 0 (2.5)

σθθ = 3S∞ (2.6)


Because σpeak = σθθ and σnominal = S∞ the ratio σθθ/S∞ represent the SCF. Relationship(2.6) represent the largest stress at a hole in the plate. Therefore, the exact analyticalsolution for the SCF in an infinite plate containing a circular hole equals to three [4].The SCF in an infinite plate is independent on hole size [4]. Furthermore, all three stresscomponents at the hole surface are independent of hole size [4].

Additionally, the tangential or hoop stress at the end of the vertical axis (y = b, x = 0)is a compressive stress which is same as the uniaxial remote stress S∞ as schematicallyshown in Figure 2.7 [3]. The tangential stress changes along the hole edge from pointP, +3S to Q, -S and tangential stress is zero at ϕ = 60°which follows from (2.7) [3].

σϕ = S∞(1 + 2 cos 2ϕ) (2.7)

Figure 2.7: Tangential stress around a circular hole in an infinite plate loaded by atensile stress S∞ [3]

An exact analytical solution for the SCF in an infinite plate containing an elliptical holeis illustrated as follows [3]:

σpeak = S∞

(1 + 2a

b

)= S∞

(1 + 2

√a

ρ

)(2.8)

Kt = 1 + 2ab

= 1 + 2√a

ρ(2.9)

ρ = b2

a(2.10)


where, a is the semi-major length and b is the semi-minor length of an elliptical hole.

According to relations (2.8), (2.9) and (2.10), the classical value of SCF variation in aninfinite plate regarding a/b ratios is schematically illustrated in Figure 2.8.

(a) (b) (c)

Figure 2.8: Effect of shape of the hole on Kt in an infinite plate loaded under tension[3].

2.2.2 Finite Plate with a centrally located hole

Finite plate problems are regularly encountered in engineering practice, such as can-tilever beams and aircraft airfoils [20]. Therefore, stability and strength determinationis an essential subject for the finite plates containing notches. The most simple case ofsuch a finite plate problem is a single strip with a centrally-located hole, see Figure 2.5.The effect of a hole size significantly affects the value of SCF in a finite plate, unlike aninfinite plate, where the SCF is unaffected by the hole size. There are three empiricalsolutions available for the SCF calculation and which are proposed by multiple authors.


Figure 2.9: A graph depicts the SCF decreases towards two with increase in diameter-to-width ratio in a finite plate according to 2.11, 2.12 and 2.13.

The SCF according to Pilkey and Peterson [21] is given as

Kt = 3− 3.14(D

W

)+ 3.667

(D

W

)2

− 1.527(D

W

)3

(2.11)

,

Howland [22] proposed the formula

Kt = 2 + 0.284(

1− D

W

)− 0.600

(1− D

W

)2

+ 1.32(

1− D

W

)3

(2.12)

,

and Heywood’s formula [23] is

Kt = 2 +(

1− D

W

)(2.13)

.

The behaviour of the SCFs versus diameter-to-width ratios according to (2.11), (2.12)and (2.13) are plotted in Figure 2.9 and most part of their curves fit to each others[21, 22, 23]. It is noticeable that the SCFs decreases towards two as D/W approachesone. When D/W increases, the load-carrying cross-section across the hole in the platedecreases too. This corresponds to the increase in the SCF. While this is not the case,as the SCF is the ratio of σpeak and σnominal in a finite plate. Both the numerator anddenominator increase with raise in D/W , but σnominal value increases more rapidly thanσpeak which leads to a decrease in the SCF value.


Figure 2.10: The comparison of ratios σpeak/σnominal and σpeak/S in a finite plate.

The general relation of the SCF in a finite plate is represented in the form with σnominalas shown in (2.14),

Kt = σpeakσnominal

(2.14)

, where σnominal = P/Dt, P is an applied load, D is a diameter and t is a thickness ofa plate.

Acording to Figure 2.5 force balance relation in a finite plate, at sections with andwithout the hole is written as [4]

SW = σnominal(W −D) (2.15)

.

The relationship (2.15) can be re-written as:

σnominal = W

W −DS (2.16)

σpeakS

=( σpeakσnominal

)(σnominalS

)= Kt

W

W −D(2.17)

A relation (2.17) is formulated with rearrangement of terms and using (2.16). Thebehaviour of σpeak to S is plotted in Figure 2.10 using (2.17) together with a curvegenerated from empirical solution 2.11. It is noticeable that the SCF is initially flat andequal to three for a small D/W ratio. After which, it rises with an increase in D/W .The initial flatness at the small D/W is due to a small change in σpeak, as the width W


decreases. It’s a matter of representation of the SCFs with either considering σnominal oruniaxial applied stress S. The most important interpretation for a hole in a finite plateis σpeak always rises with either an increase in hole size or with a decrease in width ofthe finite plate [4].

2.3 Stress intensity factor

Cracks originate at the region of stress concentration [24]. Subsequently, the loadingis concentrated at the crack tip during application of load on a structure containing acrack, causing a stress singularity with infinitely large stress [25]. In reality, looking atsingular stress result at the crack tip to predict whether a crack will propagate does notwork well [25]. As previously discussed, the SCF is utilized to describe the concentra-tion of stress at the notch edge of the structure [3]. In the crack growth period, theSCF is no longer meaningful to indicate the severity of stress distribution around thecrack tip [3]. This is due to the radius of a crack tip which is more or less equal tozero. With a zero tip radius, the SCF goes to infinity, see (2.9), and independent onthe crack length [3]. However, a new parameter is introduced to predict the intensity ofdistributed stress at the tip of a crack which is called the stress intensity factor (SIF)[3]. The SIF is denoted as K and its unit isMPa

√mm. The SIF elucidate, how quickly

the stress increases towards the crack tip. The SIF theory is based on linear elasticfracture mechanics (LEFM) assumptions similar to the SCF,e.g. linear elastic materialbehaviour and small scale plasticity [3].

The fracture mechanics theory describes three linearly independent crack opening modes,which are schematically illustrated in Figure 2.11. These three modes describe the crackopening under loading in tension, in in-plane shear and transverse or out-of-plane shear.Their respective SIFs are represented by KI , KII and KIII . Most failures occur owingto mode I which is crack opening under tension. Therefore, an analysis of mode I ismost essential in engineering design [3]. The SIFs for KI in a finite plate and a straightattachment lug are analysed using Abaqus CAE and their respective empirical solutionsin the following sections.

The standard equation for the SIF reads [3],

K = βS∞√πa (2.18)

. Where β is a dimensionless geometrical factor, S∞ is applied stress and a is a cracklength. The magnitude of SIF depends on the geometry, applied load, substantially onthe crack size and its location [11].

Stress distribution around crack tip ∝ f(K) (2.19)


Figure 2.11: Three different opening modes [3].

Figure 2.12: Different types of cracks initiating in the material from the edge of a hole[3].

There are mainly three types of cracks, which are originating at the edge of a notch,as schematically illustrated in Figure 2.12: Through-cracks, corner cracks and semi-elliptical cracks [3]. In many practical cases, the SIF solution of a crack for a complexgeometry can be approximated with the help of available solutions of simple geometries,or it can be acquired with finite element calculations. The evaluation of SIFs in straightattachment lugs can help to predict their fatigue life. The SIF is studied in more detailwith two basic examples which are as follows:

2.3.1 Infinite plate with a centrally located crack

A crack in an infinite plate can be formulated by considering the semi-minor axis lengthb of an elliptical hole equal to zero, see Section 2.2.1. For an infinite plate, the dimen-sionless geometrical factor β = 1. Therefore, (2.18) is rewritten as

K = S∞√πa. (2.20)


The approximate solution for the stress distribution near the crack tip in an infiniteplate are illustrated in terms of the polar coordinate system [26]:

σxx = S∞√πa√

2πrcos φ2

(1− sin φ2 sin 3φ

2)

(2.21)

σyy = S∞√πa√

2πrcos φ2

(1 + sin φ2 sin 3φ

2)

(2.22)

and

τxy = S∞√πa√

2πrcos φ2 sin φ2 sin 3φ

2 (2.23).

Figure 2.13: Infinite plate containing a crack with representation of remote uniaxialload.

The above three stress state relationships are further simplified with introducing theSIF:

σxx = K√2πr

f(φ) (2.24)

σyy = K√2πr

g(φ) (2.25)

τxy = K√2πr

h(φ) (2.26)


2.3.2 Finite plate with a centrally located crack

Finite plate problems containing a crack or microcracks are regularly encountered inengineering practice [20]. As the strength of such structures is extremely dependenton the geometrical configuration, loading condition, and the crack behaviour. In orderto predict the stability of such structures, it is crucial to implement theoretical andnumerical studies to produce the parameters (e. g. SIF) that control the stability of thestructure.

K = Sβ√πa (2.27)

The empirical dimensionless geometrical factor β in a finite plate containing a centrallylocated crack is proposed by different authors.Feddersen [3]:

β =√

sec πa2b (2.28)

David Percy Rooke, David John Cartwright [27]:

β =[1− a

2b + 0.326(ab

)2√1− a

b

](2.29)

and Dietmar Gross and Thomas Seelig [28]:

β =[

1− 0.025(ab)2 + 0.06(a

b)4√

cos πa2b

](2.30)

The dimensionless geometrical factor β versus relative crack length a/b for the abovethree relations are plotted in Figure 2.14. The plotted curves are mostly overlappingwith each other. Therefore, any of these three relations can be used for the calculationof the SIF using 2.27 [3, 27, 28].

Figure 2.14: Relationships between dimensionless geometrical factor β and relative cracklength a/b according to 2.28, 2.29 and 2.30.


2.3.3 Crack in arbitrary geometry using FE method

The SIFs at the crack tip in a mechanical structure containing cracks can be obtainedusing Abaqus CAE [29]. The SIFs are obtained by two methods in Abaqus CAE [29]:

1. Conventional finite element method:In this method, it is pre-requisite to mesh the crack geometry, assign a crack frontand a virtual crack extension direction. Additionally, the proper assignment ofthe mesh around a crack tip or crack front is needed for an efficient calculation ofcontour integrals (layers around a crack front for which the integral is calculated),see Figure 2.15. Each ring of elements along the crack is taken for the evaluation ofa contour integral and afterwards, Abaqus CAE processes these results to producethe respective SIF [29]. The results of the first two contour integrals are commonlyneglected and contour integral three to five are utilized for analysis of a crack[29]. The organized circular mesh around a crack tip is called as spider-shapedmesh. Furthermore, the evaluation of a contour integral in the two-dimensionalmodel is faster compared to a three-dimensional model. Whereas, in the three-dimensional model the meshing of the crack is cumbersome and time-consuming.The spider-shaped mesh can not be regenerated during the propagation of a crack.Therefore, the method applies to the analysis of stationary cracks only. All theseshortcomings which are proper mesh to the crack geometry, spider-shaped mesharound the crack tip, assigning a crack front and crack extension direction, can beavoided using a new method, developed in 1999 by Belytschko and Black, whichis called the XFEM method (eXtended finite element method) [25].

Figure 2.15: The sketch depicts the layers of mesh element around the crack tip withtheir contour integrals order.

2. XFEM method:The eXtended finite element method allows defining a crack on any arbitrary plane


even through the mesh elements in an FE model. There are no prerequisites ofassigning a mesh to the geometry of a crack, spider-shaped mesh around the cracktip and a definition of the crack front in the model [29]. The analysis of non-stationary cracks is also feasible. In this method also the evaluation of contourintegrals around crack tips is carried out.

Contour integral evaluation method:A contour integral evaluates the rate of energy release during the fracture [29]. Conven-tional finite element and XFEM methods require an estimation of the contour integralaround the tip of a crack in order to estimate the SIF [29]. The SIF depends on thecontour integral in FE numerical calculation, which is illustrated as follows [29]. TheJ-Integral is a specific type of contour integral and is calculated with

J = 18πK

TB−1K (2.31)

where K = [KI , KII , KIII ]T is the combined matrix of SIFs in all three modes, B isthe pre-logarithmic energy factor matrix. For a homogeneous isotropic material B is adiagonal matrix and therefore the J-Integral relation (2.31) is re-written as,

J = 1E

(K2I +K2

II) + 12GK

2III (2.32)

. Where, E = E for a plane stress problems, G is the shear modulus and E = E/(1− ν2)for a plane strain problems.

Afterwards, Abaqus CAE evaluates the SIFs for each mode using (2.32). In this thesis,the conventional FE method and the XFEM are utilized for the analysis of through-cracks in a finite plate and a straight attachment lug. The complete modelling techniquesare explained in Section 3.1.

2.4 Lug shapes and parameters

In general, attachment lugs are classified into three types according to their shapes:

1. Straight attachment lug:The geometrical parameters influencing the strength of straight attachment lugsare width W , hole diameter D, outside radius R0, load and loading direction [7],see Figure 2.16a. In this thesis, a straight attachment lug with W = 2R0 isutilized for the analysis and this type of lug is called as the symmetrical straightattachment lug.

2. Tapered attachment lug:The geometrical parameters influencing the strength of the tapered attachment lugare hole diameter D, outside radius R0, taper angle α, load and loading directionsee Figure 2.16b.


Figure 2.17: Sketch of tie-rod containing straight tube, sleeve and two necked lugs atthe ends.

(a) Straight attachment lug.

(b) Tapered attachment lug.

3. Necked or Waisted lug:Necked lugs are utilized in combination with tie-rods. Necked lugs are connectedto the two ends of a tie-rod. Tie-rods are connecting elements used in an aircraft toattach interior components (e.g., galleys, lavatories or storage bins) to the fuselagestructure. They consist of a straight tube and two screw adapters at the ends toattach necked lugs. The complete length of an assembly is adjusted with these twoindividual elements [30]. The geometrical parameter influencing necked lugs areouter diameter D, inside diameter d, neck radius Rt and diameter W , see Figure2.18


Figure 2.18: Necked lug [5].

2.5 Lug materials

The material selection for designing an attachment lug can be categorized into capitalcosts and structural performance. The capital cost comprises material and maintenancecost. In general, the most common isotropic materials used for designing lightweightattachment lugs are aluminium alloys, steels and titanium [31]. Common non-metallicmaterials are thermoplastic polymers and polymer composite [12]. The key materialproperties which affect the maintenance cost and structural performances are density,stiffness, strength, durability, damage tolerance and corrosion resistance [12].

2.6 Lug loading

Lugs are loaded in an axial, transverse and oblique direction from the lug axis. In-planeaxial loading is the most practical case at the industrial level [14]. In this thesis, straightattachment lugs under the application of an in-plane static axial loading are analysed.

Failure modes: The analysis of attachment lug is complex as there are several si-multaneous, interacting failure modes [6]. These failure modes related to different areasof lug as schematically illustrated in Figure 2.19. In general, four possible kinds of fail-ure modes occur in the lug [6].

1. Tensile failure mode across net section: It occurs over the cross-section highlightedwith the numerical digit 1 in Figure 2.19.

2. Shear failure along two planes: It occurs over the two shear planes highlightedwith the numerical digit 2.

3. Bearing failure mode: It occurs between the surface of the pin and inner surfaceof the lug hole highlighted with the numerical digit 3 [6].


4. Hoop tension failure: The hoop tension failure plane is highlighted with the nu-merical digit 4.

In this thesis, a tensile failure mode across the net section is studied by analysing theSCF and SIFs at the tip of a crack.

Figure 2.19: Failure modes in straight attachment lug [6].

2.7 Static analysis of lugs

Two analyses are crucial in a straight attachment lug under the constant application ofan axial load. Firstly, the analysis of the tension failure across the hole’s net sectionbecause of stress concentration at the edge and secondly, shear tear-out or bearingfailure analysis between the pin surface and hole surface of the lug [7]. The SCF for thecorresponding lug geometry is needed for the evaluation of the bearing efficiency factor[7]. The failure load of straight and tapered attachment lugs can be estimated with thebearing efficiency factor.

2.7.1 Ekvall’s method

A method is developed by J. C. Ekvall for the prediction of the static strength of apin-loaded symmetrical straight and 45° symmetrical tapered attachment lugs whichare loaded between 0° to 90° direction from the lug axis [7]. The SCFrs of lug geome-tries were evaluated with the help of finite element analysis. subsequently, the empiricalrelationship (2.33) for evaluating the SCFs was compared to the FE results and wasfitted with an acceptable tolerance of 0.5% [7].

In the course of analysis, the bearing efficiency factor KBR is calculated using rela-tions (2.33) and (2.20) [7]. Eventually, the relation 2.34 is utilized for the calculation ofthe failure load.


Ktb = 2.75[W

D− 1

]−0.675

(2.33)

Figure 2.20: Relation between the bearing efficiency factor KBR and the SCF Ktb forthe straight attachment lugs loaded at 0° and 45° and tapered lugs loaded at 0°, 45°and 90° to the lug axis [7].

Additionally, the bearing efficiency factor KBR is a function of lug geometry materialand is uniquely related to the SCF, see Figure 2.20 [7]. However, the bearing efficiencyfactor is crucial for estimating the ultimate strength of the lug.

P = KBRDtFtu (2.34)

2.7.2 HSB 26101 calculation guideline

This calculation guideline is valid when the combination of a male and a female straightand tapered attachment lug is connected by using a pin and bushings, see Figure 2.21[8]. The method is applicable under two conditions. Firstly, no surface gaps betweenmale and female lugs and secondly, the applied load should be concentrated on the cen-treline of the lugs (the horizontal continuous straight line in Figure 2.21 represents thecentreline) [8]. According to the method, a lug is analysed for various loading directionfrom the lug axis.


The method of analysis is based on experiments carried out on the lug specimens madeup of aluminium, titanium alloys and steel materials including test data utilized byEkvall [8]. In all the tests, a solid bolt (higher stiffness of the bolt than the lug) isused, which did not fail during the tests. In the course of analysis, the bearing efficiencyfactor is calculated using the empirical relation (2.35) [8]. Herein, the SCF is evaluatedusing Ekvall’s expression (2.33) [8]. Eventually, the evaluated bearing efficiency factoris utilized in the strength analysis of attachment lugs.

Figure 2.21: Sketch depicts the straight attachment lug configuration utilized for itsstatic analysis by HSB 26101 calculation guideline [8].

KBR = 7.502[Ktb + 1.125]−1.637 (2.35)

2.8 Fatigue analysis of lugs

The behaviour of a straight attachment lug under the application of fatigue loading isa major topic in aircraft industries. Cyclic fatigue loading is the most likely cause ofa lug failure [5]. A lug has low fatigue strength due to stress concentrations at thelug hole [32]. The complete fatigue phenomenon and its causes have been illustratedin Section 2.1. A stress concentration in combination with fretting lead to low fatiguestrength. The attachment lug can fail catastrophically under fatigue load [32]. Onemust adopt low-stress levels in the lugs to achieve a slow growth of fatigue cracks. Aslugs are important components in aircraft, it is imperative to predict their fatigue lifeas accurate as possible. Fatigue life predictions depend on accurate evaluation of theallowable stress amplitude and SIFs. There are two calculation guidelines stated for theevaluation of these factors in straight attachment lugs:

2.8.1 ESDU 81029 calculation guideline

The calculation guideline is utilized to calculate the SIF of stationary through-cracksoriginating at the edge of a hole in straight attachment lug [9]. The method is establishedon finite element analysis [9], and α values for various crack lengths and lug geometries


Figure 2.22: List of α values of a symmetrical through-cracks for the straight attachmentlug of various configuration under axial loading state [9].

are tabulated in Figure 2.22. The SIF obtained using this method is utilized further forthe damage tolerance analysis [9]. The SIF is calculated in a straight attachment lugcontaining two symmetrical cracks using (2.36).

K = Sα√πa (2.36)

where α is a dimensionless geometrical factor, a is a crack length and S is an appliedstress.

Figure 2.23: Sketch depicts the symmetrical through-cracks at two opposite sides of ahole in straight attachment lug [9].


2.8.2 HSB 63511 calculation guideline

In this calculation guideline, allowable stress amplitudes are evaluated for the fatiguestrength analysis of straight attachment lugs. It is applicable if there is a constantamplitude loading in the axial direction applied and the criteria described below aresatisfied [10].

1. The guideline is applicable to a lug which is more or less geometrically same as thestandard lug, see Figure 2.24. A lug with a configuration asl = csl = dsl = 10mm iscalled a “standard lug”. The standard lug is utilized as a reference for the analysisof similarly configured lug.

2. The pre-stress of a bushing in a lug should be similar to that of the standard lug.

3. Bolt fit criteria (no clearance between hole and pin).

4. The ratio of lug thickness to hole diameter tdhas to meet the criteria: t

d≤ 0.8 for

titanium/steel and td≤ 1.0 for light metals.

5. A condition −1 ≤ R < +1 must be satisfied.

Figure 2.24: Straight attachment lug configuration utilized for fatigue analysis by HSB63511 guideline [10].

There are two methods stated in the following sections for the evaluation of allowablestress amplitude, if and only if the above criteria are satisfied:

1. Method 1:The method is applicable when material properties, stress ratio R, cycle numberN and HAIGH-diagram of the reference lug are available. (The HAIGH diagram


represents the fatigue life of a material with the help of an equation used tomeasure the interaction of the mean and the alternative stresses [10]. The equationis typically presented as a linear curve of mean stress versus alternating stressthat provides the maximum number of alternating stress cycles a material willwithstand before failing due to fatigue).The allowable stress amplitude for a lug loaded at a stress ratio equal to zero iscalculated using 2.37.

σa(R = 0) = σa,sl(R = 0)[1 + φ(k1k2 − 1)] (2.37)

,where σa is an allowable stress amplitude, R is a stress ratio, σa,sl is an allowablestress amplitude of a standard lug, φ is a dimensionless factors and their valuesdepends on cycle numbers, k1 and k2 are dimensionless factors and are evaluatedwith geometrical equations.

2. Method 2:The method is utilized when HAIGH diagram of a reference lug is not available,but the S-N curve, as well as material parameters A and B are available. In thismethod, the allowable stress amplitude of a lug is calculated using one of the twoapplicable cases:

Case 1: When applied stress ratio R ≥ 0For Al/Mg alloy,

σa(R ≥ 0) = σa,sl(R = 0)1

1+φ(k1k2−1) + σa,sl(R=0)A

2R1−R

(2.38)

.For Ti alloy/steel,

σa(R ≥ 0) = σa,sl(R = 0)1

1+φ(k1k2−1) + σa,sl(R=0)Rm

2R1−R

(2.39)

, where Rm is a ultimate tesnile strength of a lug.Case 2: When applied stress ratio R<0For Al/Mg alloy, Ti alloy and steel,

σa(R < 0) = σa,sl(R = 0)B(1−R)1−R + (B − 1)(1 +R) [1 + φ(k1k2 − 1)] (2.40)

.

2.9 Fatigue crack growth in lugs

The theory of fatigue cracks nucleation and its propagation during application of fatigueload as previously illustrated in Section 2.1. There are three types of cracks originating


Figure 2.25: Crack growth curves of four specimens with artificial cracks [11].

in the stress concentrated region of lugs and which are propagating through the mate-rial [11]: Through-cracks, corner cracks and semi-elliptical cracks as illustrated in Figure2.12 [11]. The crack front of through-cracks is a straight line and normal to the surfaceof the material [11]. Therefore, SIFs are more or less constant along the entire crackfront [11]. In the case of corner and surface cracks, the crack front is quarter-ellipticaland semi-elliptical respectively. Therefore, the SIF varies along the entire crack frontand due to that effective value of SIF is calculated [11].

Natural cracks initiating at the stress concentrated region along the circumference of alug hole, owing to fretting (relative movement between a pin and a lug) will be most ofthe time corner (quarter-elliptical shape) or surface cracks (semi-elliptical shape) [11].These natural cracks grow further under application of cyclic load and later form athrough-crack. This behaviour is depicted in the experiment on straight attachmentlugs with an artificial crack at the hole edge, see Figure 2.25 [11]. In the beginning, anartificially introduced corner crack begins to grow with a quarter-elliptical shape andafter reaching a length equal to the thickness of the lug, it takes the shape of a through-crack. Furthermore, life of crack growth is significantly different for all crack types [11]and which is depicted in Figure 2.25.

Sometimes, more than one natural crack nucleates simultaneously at the adjacent loca-tion along the circumference of a lug hole. Later, during crack growth, these nucleatedcracks converge into one another and form a single ridge. The single converged ridgecrack propagates further until failure of the lug [11]. The crack front of a natural crackis irregular compared to an artificial crack. The growth length of natural cracks dependson the time and location of their nucleation. Therefore, crack growth curves of naturalcracks are more scattered than artificial cracks as shown in Figure 2.26 [11]. After thesubstantial growth of a crack, all the crack curves (artificial as well as natural crackcurves) are more or less parallel to each other, as shown in Figure 2.26 [11]. The nat-ural cracks show parallel behaviour after crack growth reaching a length equal to the


Figure 2.26: Comparison between the crack growth curves of natural cracks and artificialcracks [11].

thickness of the lug. Furthermore, the more complex geometry and growth behaviourof natural cracks lead to slow crack growth, which is favourable with regard to damagetolerance [11]. Additionally, the artificial fatigue through-crack grows faster than natu-ral crack, as shown in Figure 2.26. Therefore, the artificial through-crack is taken to bea “worst-case” in the engineering design [11]. Advantageously, a through-crack is easierto analyse as compared to surface or corner crack [11].

3 Finite element calculationsAbaqus CAE is utilized in this thesis for the analysis of SCFs and SIFs in a finiteplate and a straight attachment lug. Attention was given to the mesh element sizeespecially in the stress concentrated region of the model to avoid the artificial increaseof stiffness. Hence, more or less square and linear mesh elements are used in regions ofstress concentrations. Mesh convergence studies were carried out on all SCFs analysedlug geometries. Furthermore, the tangential stress distribution was analysed in fourdifferent lug geometries. The finite element models utilized in the thesis are explainedin the following sections.

3.1 Static analysis of the finite plates and the straightattachment lugs

This section covers three main procedures carried out with the finite element softwareAbaqus CAE [29]:

1. Determination of SCFs at the edge of a hole of the defined two-dimensional modelof finite plates and straight attachment lugs [29].

2. Determination of SIFs for a centrally located crack in two-dimensional models offinite plates and straight attachment lugs containing symmetrical cracks at theedge of a hole using conventional finite element method [29].

3. Determination of SIFs for symmetrical edge cracks at the hole in three dimensionalmodels of straight attachment lugs and a centrally located crack in two dimensionalmodels of finite plates using XFEM [29].

3.1.1 Introduction

The evaluation of SCFs and SIFs are executed by finite element analysis (FEA) soft-ware named Abaqus/CAE [29]. Abaqus environment allows pre-processing (modelling),processing (evaluation and simulation) and post-processing (visualization and results)by analysis products Abaqus/Standard or Abaqus/Explicit which can solve complexfinite element models. All three activities in Abaqus CAE are divided into modules andeach module explain a logical aspect of the modelling activity: sketch, part, property,assembly, step, interaction, load, mesh, optimization, job and visualization. When finiteelement model is carried through module to module, Abaqus CAE generates an inputfile that is submitted later to Abaqus/Standard product. Eventually, Abaqus executesthe analysis of submitted jobs within a monitored progress that creates results saved

30

3. Finite element calculations 31

Figure 3.1: The FE model of a finite plate having centrally located hole with represen-tation of boundary conditions and an applied load utilized for the analysis of SCFs.

in a output database which is viewed in Abaqus CAE Visualisation module. Each FEmodel was designed in the same manner with stepwise consideration of all Abaqus CAEmodules which are illustrated as follows.

3.1.2 Module Part - FE modelling of a finite plate and a straightattachment lug

1. SCFs: The FE models of a finite plate and a half symmetrical straight attachmentlug with a clearance-fit pin are schematically illustrated in Figure 3.1 and 3.2respectively. They comprise of circular hole, which are modelled with the help ofgeometrical parameters from Tables 4.1 and 4.2.

2. SIFs: The FE models of a half symmetrical finite plates and a straight attachmentlug with a clearance-fit pin are illustrated in Figure 3.3 and 3.4 respectively. Theyare comprise a stationary crack, which are modelled with the help of geometricalparameters from Tables 4.3 and 4.3.

Modelling space of 2D planer is utilized for designing each part whose base features con-sidered to be shell deformable. After designing all parts, material- and section propertiesare allocated in module window called “Property”. The aluminium material propertiesare defined to the models of a finite plate and a straight attachment lug: Young’s mod-ulus E(Lug, plate) = 70000 MPa, E(Pin)= 210000 MPa, Poisson’s ratio ν(Lug, plate,pin) = 0.3. The pin’s stiffness is assumed to be three times the lug and which is imple-mented in the models. However, stiffness difference between the pin and lug has a smalleffect on the peak tension stresses [7].

3.1.3 Module Assembly

In the assembly module multiple functions (e.g. instance part, translate instance, rotateinstance and merge instance) have been utilized to assemble the pin and the lug into one


Figure 3.2: The half-symmetrical FE models of a) a straight attachment lug with rep-resentation of boundary conditions and b) a pin with representation of an applied load,utilized for the analysis of SCFs.

Figure 3.3: The half-symmetrical FE model of a finite plate with representation of acentrally located crack, boundary conditions and an applied load utilized for the analysisof SIFs.

Figure 3.4: a) a straight attachment lug with representation of symmetrical edge cracks,boundary conditions and b) a pin with representation of the applied load, utilized forthe crack analysis by conventional FE method.


model. The function Create instance was utilized for locally importing each modelledpart from Part module to Assembly module. Meanwhile, translate instance and rotateinstance were utilized to acquire distance and angle between assemble parts respectively.

3.1.4 Module Step

The Step module is used to define one or more analysis steps within a simulation. In thecourse of the analysis in the model, the step order is an appropriate way to differentiateloads and boundary conditions of the model. Within this module the field and historyoutput are defined according to the desired output variables [29].

In each analysis of finite plates and straight attachment lugs, only one step has beengenerated. For all perfomed simulations the procedure step is selected to be Static, Gen-eral and the incrementation is set to default settings as schematically shown in Figure3.5.

Figure 3.5: Abaqus CAE window of the step module.

3.1.5 Module Interaction

Interaction module allows to define the interaction type, e.g. the friction coefficientbetween two or more parts and cracks in the model. In the course of analysis, thedefinition of part interaction is substantial to investigate local behaviour of assembledparts in the model. In the analysis process of SCFs in a finite plate, no interactionsare defined. Whereas the interaction module is utilized for the SCFs analysis process ina straight attachment lug. It is also utilized for the SIF analysis in a finite plate anda straight attachment lug. In the interaction module the Constraint toolset in general


defines the degrees of freedom of the analysis. The type of constraint applied in theanalysis of lug is “Rigid Body” with region type “Tie”. The rigid body constraint en-ables to constrain the movement of region of the assembled parts to the movement ofreference point where load is added. This means that the relative position of the rigidbody area remains constant during the numerical analysis [29].

Figure 3.2 and 3.4 illustrated the disassembled model of a pin and straight attach-ment lug. The contact surfaces of a pin and a lug were considered as “Master Surface”and “Slave Surface”, respectively. In the thesis, the contact between a pin and a lug wasmodelled in Abaqus CAE using, frictionless tangential behaviour and a “hard” contactin normal direction.

Figure 3.6: Stress distribution in a two-dimensional FE model of a finite plate with acentrally located hole.

Figure 3.7: Stress distribution in a two-dimensional FE model of a straight attachmentlug.


3.1.5.1 Modelling a crack using conventional FE method

The conventional finite element method procedure is utilized in this thesis for definingthe through-cracks in two dimensional models of finite plate and straight attachment lug.Firstly, a partition is created representing the crack. Then the Seam tool of Interactionmodule is utilized for defining the seam to generate separate nodes on elements on eachside of a crack. Subsequently, the Create Crack tool is utilized for defining the cracksurface, crack tip and crack extension direction to obtain crack related output. In orderto capture the stress singularity around the crack tip special mesh elements are usedwhere the midside node is shifted by a factor of 0.25 to the crack tip, see Figure 3.14a[29]. This method needed a spider-shaped mesh layers around the crack tip, as shownin Figure 3.10. For the creation of such a mesh, additional partitions are created in themodel. Subsequently, defining a crack and creating a mesh in the model, the HistoryOutput is requested appropriately, see Figure 3.14b.

3.1.5.2 Modelling a crack using XFEM method

The XFEM method is an extension of the conventional finite element method and isbased on concept of partition of unity which permits local enrichment functions to beeasily integrated into finite element approximation [29]. It does not needed the spider-shaped mesh at the crack tip geometry to evaluate the contour integral [29]. It is notrequisite to make partition around the crack tip, as assigning a mesh is simpler aroundthe crack tip region in XFEM method [29]. It permit the presence of discontinuities inthe interior mesh elements and which is possible through addition of enrichment func-tions in conventional finite element method [29].

The three dimensional models of straight attachment lug and finite plate are modelledbased on Tables 4.4 and 4.3 respectively, thickness of 5 mm is considered in both thecases. Modelling space of 3D is utilized for modelling a lug, a pin and a plate whose basefeatures considered to be solid extrusion deformable. Where as the crack is model using2D planer whose base feature is considered to be shell deformable. Then, two crackssurfaces, a lug and a pin from the Part module are imported in Assembly module incase of straight attachment lug. Subsequently, multiple functions from assembly modulehave been used to assemble the pin, lug and two cracks into one model, see Figure 3.11.The position of a stationary cracks are selected at the edge of a hole in straight attach-ment lug and are directed towards the outer region of the lug. The enrichment featureof XFEM cracks are defined in the Interaction module of Abaqus CAE with introducingcrack domain. The similar procedure is applied to a finite plate, see Figure 3.8. Thefive layers of elements around the crack are selected to captured the

ularity in straight attachment lug and finite plate, depicted in Figure 3.11. The appli-cation of corresponding loads and boundary conditions for a straight attachment lugand a finite plate are illustrated in Figures 3.13 and 3.3 respectively. The post process-ing XFEM results of centred node along the thickness direction are compared to theconventional FE and empirical results in Figures 4.3 and 4.3.


Figure 3.8: Two dimensional half-symmetrical FE model of a finite plate with a crackand spider-shaped mesh elements distribution around a crack tip.

Figure 3.9: Three dimensional half-symmetrical FE model of a finite plate with a XFEMcrack.

Figure 3.10: Two dimensional FE model of a straight attachment lug with symmet-rical cracks at two opposite ends of the lug hole utilized for crack analysis using theconventional FE method.


Figure 3.11: Three dimensional FE model of straight attachment lug with XFEM cracks.

Figure 3.12: Abaqus CAE window of XFEM crack

3.1.6 Module Load

The load modules allows to defines various types of loads and boundary conditions fora model. For the analysis of finite plates, a pressure load and two boundary conditionsare used, whereas for the straight attachment lug a point load at the defined referencepoint and two boundary conditions have been generated in one step.

3.1.6.1 Load

Tables 4.1 and 4.2 summarises the applied loads for the evaluation of SCFs in a finiteplate and a straight attachment lug respectively. Whereas Tables 4.3 and 4.4 coversthe corresponding loads applied for the evaluation of SIFs. To compare with literature


Figure 3.13: XFEM model: a) A straight attachment lug with representation of sym-metrical XFEM edge cracks and boundary conditions. b) A pin with representation ofthe applied load.

(a) Abaqus CAE window of Create Cracktool.

(b) Abaqus CAE window of HistoryOutput Request for the defined crack.


results, only axial loads and no in-plane and out-of-plane moments has been considered[7, 9]. The investigated load cases are: Balanced axial load and point load applied in thefinite plate and the straight attachment lug models respectively as schematically shownin Figures 3.1 and 3.2.

3.1.6.2 Boundary Conditions

Boundary condition (BC) for the respective models of a finite plate and a straight at-tachment lug are depicted in Figures 3.1, 3.2, 3.3 and 3.4 by red triangles. The boundarycondition type fixed was utilized which produces zero translational and rotational move-ment in all directions. Additionally, symmetry boundary condition is utilized for thehalf symmetrical models. In the case of XFEM crack analysis in straight attachmentlug boundary condition fixed is utilized to the pin and loading is applied on the lug, seeFigure 3.13 [29].

3.1.7 Module Mesh

Mesh module allows to creates meshes on whole assembled models or meshes can becreated separately on each part of the assembly model. The creation of mesh maydiffer and is depended on the model geometry. The Mesh module has various tools andspecification to control the elements size from coarse to fine, which depends on the areaof interest.

3.1.7.1 Mesh Density

The mesh density is one of vital parameter, which allows to adjust the level of meshgeneration in the model. In this thesis, the scale of mesh density is varied from coarseto very fine density. The value of mesh density is measured in millimetre. Additionally,a finite plate and a straight attachment lug are partitioned into several faces to applylocal mesh density and also generating a spider-shaped mesh for the crack analysis, seeFigures 3.7 and 3.10.

3.1.7.2 Mesh Elements

Mesh elements in Abaqus CAE include predefined elements in two- and three- dimen-sional shapes. Each predefined element in these two categories becomes available de-pending on base feature of the created part in the Part module [29]. The finite plateand the straight attachment lug model are shells, and therefore only two-dimensionalelements shape become accessible. For determination of SCFs and SIFs, quadrilateraland triangular mesh elements are defined as four-node bilinear plane stress quadrilat-eral elements with reduced integration and hourglass control (CPS4R) aw well as threenode linear plain stress triangle elements (CPS3) respectively. Whereas, hexagonal ele-


(a) Abaqus CAE window of mesh control for twodimensional elements.

(b) Abaqus CAE window of mesh controlfor three dimensional elements.

ments are utilized as 8-node linear bricks with reduced integration and hourglass control(C3D8R) especially in the XFEM crack domain of the three dimensional models.

3.1.7.3 Mesh Control

With the Mesh Controls in Abaqus CAE one can define the element shape, the meshingtechnique and meshing algorithm of specified regions of the finite element model. Themesh control option which are needed for the mesh generation in a finite plate anda straight attachment model, are depicted in Figure 3.15a. There are two meshingmethodologies available in Abaqus CAE: top down and bottom up. Top down meshingproduces a mesh by working down from the geometrical region to individual mesh nodesand elements [29]. This kind of mesh creation process is default and autonomous inAbaqus CAE. This process may create difficulties to generate a high quality mesh onareas with complex shapes. Bottom up meshing produces a mesh by working up fromtwo-dimensional bodies (geometrical faces, element faces, or two dimensional elements)to generate three dimensional mesh [29]. This type of mesh generation methodology isonly available for solid three-dimensional models created by manual process. The topdown and bottom up mesh methodology are utilized in this thesis.

3.1.7.4 Element Shape

Abaqus CAE allows three different elements shape as illustrated in Figure 3.15a. In thisthesis Quadratic, traingular, wedge and bricks elements shapes are utilized for differentregions of the models..

3.1.7.5 Meshing techniques

Meshing techniques comprise three major options of mesh generation which are: Freemeshing, structured meshing and sweep meshing, see Figure 3.15a and 3.15b. Free


meshing is the flexible top down mesh technique which do not use pre-created controlledmesh pattern and applies to any shape of the model. As free meshing doesn’t predictthe mesh pattern, which in turn provided minimum control over the mesh generation.The second top down mesh techniques is structured mesh technique, which providesmore control over the mesh compared to free meshing techniques, as it applies pre-created mesh pattern to the model topologies. Sweep meshing is the third top downmesh techniques, which produce mesh in sweeped paths. Sweep meshing also appliespre-created mesh patterns but is limited to models with certain topologies. AbaqusCAE auto generates the adequate mesh technique for the created model and indicatesit by colour coding. In this thesis, all three types of mesh techniques are utilized invarious region of the models according to requirement.

3.1.7.6 Meshing Algorithm

There are two options for mesh algorithms in Abaqus CAE as illustrated in Figure 3.15aand they are: Medial axis and Advancing front. The medial axis mesh algorithm pro-vides mesh generation with decomposing the area to be meshed into simpler regions andlater uses structured meshing techniques to fill up each simple area with elements [29].The mesh generation occurs much faster and mesh quality remain poor compared to ad-vancing front algorithm. “Minimize the mesh transition” option is useful for improvingmesh quality, which improve quality of the mesh to some extent but not effective enoughfor the whole model. Whereas the advancing front algorithm allows the generation ofmesh with quadrilateral elements at the boundary of the region it continues to generatequadrilateral elements as it moves orderly to the interior of the region [29]. The ad-vancing front algorithm generates mesh slower than medial axis algorithm. Instead itprovides a good quality mesh especially during selection of “mapped meshing” option.In this thesis, “Medial axis” algorithm with minimize mesh transition option is utilizedfor the mesh generation in the models.

3.1.8 Mesh Verification

Abaqus CAE allows examining the quality of mesh generated in mesh module withthe help of Mesh verification option. In the analysis procedure, “Verify Mesh” tool isutilized for the mesh verification and in turn, the optimal quality mesh is utilized foreach analysis. Figures 3.6, 3.7, 3.8, 3.9, 3.10 and 3.11 depicts the verified mesh elementsand their distribution for the respective models utilized in the analysis.

3.1.9 Verification of the FE model and analysis procedure

The FE model and its analysis procedure in Sections 3.1.2-3.1.8 have been consideredin the thesis. The accuracy of SCF and SIF results from FE analysis for a finite plateand a straight attachment lug are verified against their empirical solution and literatureresults. In Abaqus CAE, there are four different geometries modelled to carry out theSCF and the SIF analysis:


1. A finite plate with a centrally located hole was modelled and acquired results ofthe SCFs were verified against relation (2.11).

2. A half symmetrical model of a straight attachment lug was modelled and acquiredresults of the SCFs were verified against relation (2.33).

3. A half symmetrical model of a finite plate with a centrally located crack wasmodelled and acquired results of the SIFs were verified against relation (2.30).

4. A straight attachment lug containing a symmetrical crack at a hole edge wasmodelled and acquired results of the SIFs were verified against literature [9].

Area of Interests: To compare empirical approach the SCFs in Abaqus CAE areinvestigated at the hole edge in a finite plate and a straight attachment lug. The SIFsare investigated at the tip of a crack in their corresponding models.

3.1.10 Stress concentration factor

The SCF in finite plate and straight attachment lug models are evaluated based ondescribed procedure in Sections 3.1.2-3.1.8. The values of peak von Mises stresses areutilized for the SCFs calculation, see relations (2.14) and (2.33) - (2.34) respectively.Table 3.1 shows a modified guidance on FE modelling with respect to derivation of theSCFs [29].

Material behaviour Linear elasticModel body ShellElement type 4-noded quadrilateral elements at the stress concentrated region

Mesh refinement/quality Fine mesh at region of stress concentrationHole edge Maximum von Mises stress at the hole edge

Table 3.1: Modified guidance on FE modelling with respect to derivation of the SCFs.

3.1.11 Stress intensity factor

The SIF in finite plate and straight attachment lug models are evaluated based ondescribed procedure in Section 3.1.2-3.1.8 [29]. The Abaqus CAE evaluates the SIFaccording to the brief procedure illustrated in Section 2.3.3. The values of SIFs aredirectly acquired in the post-processing [29]. Table 3.2 shows a modified guidance onFE modelling with respect to derivation of the SIFs [29].


Material behaviour Linear elasticModel body Shell

Element type (Conventional FEmehod)

4-noded quadrilateral and 3-nodedtriangular elements in the

spider-shaped region

Element type (XFEM mehod) 8-noded hexagonal elements utilizedaround the crack tip region

Mesh refinement and quality Fine mesh around a crack tipContour integral considered for the

evaluation of SIFsResults of 3rd to 5th contour integrals

are averaged

Table 3.2: Modified guidance on FE modelling with respect to derivation of the SIFs.

3.2 Mesh convergence study

In finite element modelling, a finer mesh typically results in a more accurate solution.On the other side, as a mesh is made finer, the computation time increases. Therefore,a mesh that balances both accuracy and computing resources is obtained by performinga mesh convergence study [33]. According to St. Venant’s principle, the local stresses inone region of a structure are considered to do not influence the stresses elsewhere [33].Therefore, mesh convergence test of a model are carried out by refining the mesh onlyin the region of interest which is around the edge of lug hole.

The following key points discusses the mesh convergence study using static stress analy-sis on a two dimensional model of straight attachment lug with width-to-diameter ratio(W/D) of two:

1. Created a mesh using a reasonably small number of elements (e.g. ten) from theedge of the lug hole, along the y-axis and analysed the model, see Figure 3.16.

2. Re-meshed the model with a denser element distribution, re-analysed it and com-pared the peak stress results to those of the previous experiments.

3. The mesh density is continued to be increased and the model is re-analysed untilthe peak stress results began to converge satisfactorily.

For example, it begun to converged around elements along the y-axis equal to twohundred as shown in Figure 3.16. For 200 elements (element size ∼ 0.06 mm) the cor-responding stress equal to 610 MPa. For 800 elements the calculation time increasedsignificantly, however the stress increased only little by 23 MPa.

The mesh convergence study is essential to obtain an accurate solution with suffi-ciently fine mesh and not overly demanding computational costs [33]. The convergencestudy is used for the accurate calculation of the SCFs and SIFs. The four-node bilinearplane stress quadrilateral, mesh elements with reduced integration and hourglass control


(CPS4R) are assigned around the edge of the lug hole. For each simulation run of themesh convergence study, the number of mesh elements and their sizes are varied alongthe y-axis to get finer and square mesh elements.

Figure 3.16: Convergence of a maximum von Mises stress and a maximum tangentialstress arising at the hole edge in a straight attachment lug with increase in mesh density.

3.3 Stress distribution in a straight attachment lug

The primary interest in the analysis of attachment lugs is the stress distribution analysiswhen no crack is nucleated in the lug (unflawed attachment lug). This is vital for threeimportant reasons [34]:

1. The location of the most critical crack is dependent on the peak tangential stress[35].

2. The SIF for a small crack is proportional to the SCF [34].

3. The weight function method (the method is not discussed in the thesis) for esti-mating the SIF, requires the stress distribution on the crack surface of the unflawedlug [34].

The conventional FE method is utilized to obtain the tangential stress along the y-direction from the edge of a lug hole, see Figure 3.17. The geometrical parameters usedfor modelling the straight attachment lugs are depicted in Table 4.2. Figure 3.7 depictsthe two-dimensional half-symmetrical model of a straight attachment lug with a typicaldistribution of mesh elements, utilized in the analysis. Similar modelling techniques,as in Sections 3.1.2 - 3.1.8, are utilized for analysing the tangential stress distribution.The lug and pin were modelled by a set of constant-strain triangular and quadrilateral


linear elements. A point load at reference point and two boundary conditions have beenapplied according to (3.1.5). Along the lug-pin contact, an equal amount of the meshelements are used in the attachment lug and the pin. A neat-fit pin with frictionlesssurfaces of the pin and the lug are assumed [29].

The calculated tangential stresses along the y-direction from edge of the lug hole arenormalized by the average bearing stress 314.73 MPa and the distances along the y-axisare normalized by the average distance 14.05 mm, see Figure 3.17. The pin is loadedat 0° direction for all four FE experiments whose W/D ratios are 1.5, 2, 3 and 4. Thestress distribution gradient near to the edge of the lug hole is very steep especially fora lug having smaller W/D ratio, see Figure 3.17.

Figure 3.17: Stress distribution along the y-axis in the unflawed straight attachmentlugs for four distinct values of width-to-diameter ratios.

Figure 3.18: Von Mises stress along the edge of the lug hole for a pin loaded in 0°direction.


The evaluated von Mises stresses along the edge of the lug hole for a pin loaded in the0° direction are depicted in Figure 3.18 for width-to-diameter ratios of 1.5, 2, 3 and 4.As anticipated the maximum von mises stress is located more or less close to 90° awayfrom the lug axis and at the edge of the lug hole, and minimum stresses are located atthe 30° location, see Figure 3.18. Therefore, 90° is the critical location for the cracknucleation.

4 Results and discussionA comprehensive investigation of SCFs and SIFs is accomplished. This chapter coverstwo following comparisons:

1. The SCFs in finite plates and straight attachment lugs according to Sections 2.14,2.7.1 and 3.1 [29].

2. The SIFs at the crack tip in finite plates and straight attachment lugs accordingto Sections 2.3.2, 2.8.1 and 3.1 [29].

4.1 Comparison of stress concentration factors

Comparison of SCFs for investigated finite plates and straight attachment lugs are sum-marized in Tables 4.1 and 4.2 respectively. These tables presents a summary betweenSCFs obtained by parametric equations and finite element analysis. The correspondingcomparisons are also plotted in Figures 4.2 and 4.3.

W D W/D t P σnominal Kt (Analytical method) Kt (FE method)mm mm mm N MPa

50 33.40 1.5 5 5000 301.20 2.0836 2.092350 25 2 5 5000 200 2.1559 2.174050 16.70 3 5 5000 150.15 2.3034 2.315750 12.50 4 5 5000 133.33 2.4203 2.4300

Table 4.1: Comparison of SCFs according to analytical and numerical methods for thefinite plates with a centrally located hole.

W D W/D t P σBR Ktb (Ekvall method) Ktb (FE method)mm mm mm N MPa

50 33.40 1.5 5 30000 179.64 4.391 4.16450 25 2 5 30000 240 2.750 2.61950 16.70 3 5 30000 359.28 1.722 1.76750 12.50 4 5 30000 480 1.310 1.453

Table 4.2: Comparison of SCFs according to analytical and numerical methods for thestraight attachment lugs.

47

4. Results and discussion 48

Figure 4.1: Comparison of analytically and numerically calculated SCFs of finite plateswith a centrally located hole.

Figure 4.2: Comparison of analytically and numerically obtained SCFs of a straightattachment lugs.

Figures 4.1 and 4.2 lead to the conclusion that the empirical solutions of SCFs for afinite plate and a straight attachment lug are in close agreements with their numericalresults respectively.


4.2 Comparison of stress intensity factors

Comparison of SIFs for investigated finite plates and straight attachment lugs are sum-marized in Tables 4.3 and 4.4 respectively. These tables present a summary betweenSIFs obtained by parametric equations and finite element analysis. The correspondingcomparison are also plotted in Figures 4.1 and 4.4.

2b 2a a/b σKI(Analytical

method)KI(ConventionalFE method) KI(XFEM)

mm mm MPa MPa√mm MPa

√mm MPa

√mm

50 2 0.04 200 354.83 353.40 392.650 10 0.2 200 812.07 811.00 82550 20 0.4 200 1243.24 1242.00 126750 25 0.49 200 1460.38 1456.00 148850 30 0.6 200 1788.58 1784.00 175550 40 0.8 200 2876.32 2858.00 2957

Table 4.3: Calculation of SIFs according to analytical and numerical methods for finiteplates with a centrally located crack.

W d a b a/b α σKI(ESDUmethod)

KI(Conventional FEmethod) KI(XFEM)

mm mm mm mm MPa MPa√mm MPa

√mm MPa

√mm

60 30 12 15 0.8 4.24 50 1302 1398 130060 30 7.5 15 0.5 3.94 50 956 856.2 808.260 30 3.75 15 0.25 4.68 50 803 631.7 607.960 30 1.5 15 0.1 5.68 50 617 461.8 501.860 30 0.375 15 0.025 6.57 50 357 268 255.7

Table 4.4: Calculation of SIFs according to ESDU 81029 and numerical results for thestraight attachment lugs.


Figure 4.3: Comparison of empirically and numerically calculated results of finite plateswith a centrally located crack.

Figure 4.4: Comparison of data given by ESDU 81029 and numerical results of straightattachment lugs.

Figure 4.3 lead to the conclusion that the empirical solutions of the SIF for a finite plateare in close agreements with its numerical results. The numerically obtained SIF resultsof a straight attachment lugs are compared with ESDU 81029 calculation guideline, seeFigure 4.4. It is noticeable that the numerical results are in close agreement with ESDUresults.

5 ConclusionsThe parameters that have influenced the calculation of the SCFs and SIFs other than thegeometrical parameters in finite element analysis are: boundary conditions, 3-noded, 4-noded and 8-noded mesh elements and mesh refinement around the stress concentrationregion. These parameters resulted in a deviation of results of finite element simulationsfrom empirical equations to some extent.

The maximum von Mises stress is located more or less at 90° location away from the lugaxis and at the edge of the lug hole. The location of maximum von Mises stress is highlysusceptible to crack nucleation and later its propagation and finally failure through thenet-section of a lug. The tangential stress distribution gradient near to the edge of a lughole is very steep, especially for a lug having a smaller width-to-diameter ratio. In otherwords, peak tangential stress is higher for a lug having a smaller width-to-diameter ratiowhich corresponds to a higher stress concentration. Therefore, a straight attachmentlug with the smaller width-to-diameter ratio is more vulnerable to failure.

Optimum results of the SCF in a finite element model of a straight attachment lugwere obtained by using a mesh convergence study. Increase in mesh density in a modelleads to the magnification of computational time. Therefore, it is necessary to balanceboth accuracy and computational time of the analysis. In the thesis, a straight attach-ment lug of width-to-diameter ratio equal to 2 has converged at mesh elements equal to200, respective square mesh element size equal to 0.06 mm and von Mises stress equalto 610 MPa

The SCF in an infinite plate with a centrally located circular hole is always three.The hole size does not influence the SCF. In a finite plate, when the diameter-to-widthratio is increased, the nominal stress increases rapidly in comparison to the peak stress,which leads to a decrease in SCF. The peak stress at a hole edge always increases withincrease in hole diameter or decrease in width in a finite plate. In straight attachmentlugs, the SCF is decreasing with increase in width-to-diameter ratio. A straight attach-ment lug with a high SCF value is highly vulnerable to failure.

In a finite element model, the spider-shaped mesh around a crack tip is importantto obtain optimum results of SIFs. In the case of conventional finite element and XFEMmethod for the SIF calculation, the SIF results begin to converge for contour integralshigher than 2nd order. In a finite plate, the SIF results which are acquired by numericalanalysis are in close agreement with the empirical solution. The SIFs result acquiredby numerical analysis are in close agreement with ESDU 81029 results in a straightattachment lug. Therefore, the SIF for through-cracks can be directly calculated usingthe dimensionless geometrical factor results which are accessible in ESDU 81029.

51

6 Future workThe areas of peak tangential stress at the edges of a lug hole, von Mises stress along theedge circumference of a lug hole, SCFs and SIFs evaluation in a straight attachment lugare the major topics in this thesis. Future work has to be done in:

1. Evaluation of empirical solution of the SCF at the hole edge in a necked lug.

2. Prediction of the precise location of crack nucleation and the rate of crack growthto complete failure in a necked lug.

3. Influence of an introduced bushing element in lug joints on the stress concentrationin the lug.

4. Development of lug joints using composite materials and prediction of the locationof crack nucleation and the crack growth rate to complete failure. Evaluation ofempirical solutions of the SCFs and the SIFs in lug joints made of compositematerials for further strength analysis.

52

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Institute of Structural Univ.Prof. DI Dr. Martin Schagerl