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  • STRESS AND FAILURE ANALYSIS OF

    LAMINATED COMPOSITE PINNED JOINTS

    by

    Buket OKUTAN

    December, 2001

    IZMIR

  • STRESS AND FAILURE ANALYSIS OF

    LAMINATED COMPOSITE PINNED JOINTS

    A Thesis Submitted to the

    Graduate School of Natural and Applied Sciences of

    Dokuz Eylul University

    In Partial Fulfillment of the Requirements for

    the Degree of Doctor of Philosophy in Mechanical Engineering, Mechanics

    Program

    by

    Buket OKUTAN

    December, 2001

    IZMIR

  • Ph.D. THESIS EXAMINATION RESULT FORM

    We certify that we have read this thesis, entitled Stress and failure analysis of laminated

    composite pinned joints completed by Buket OKUTAN under supervision of Assoc.

    Prof. Dr. Ramazan KARAKUZU and that in our opinion it is fully adequate, in scope

    and in quality, as a thesis for the degree of Doctor of Philosophy.

    Assoc. Prof. Dr. Ramazan KARAKUZU

    (Supervisor)

    Jury Member Jury Member

    (Thesis Committee Member) (Thesis Committee Member)

    Jury Member Jury Member

    Approved by the

    Graduate School of Natural and Applied Sciences

    Prof. Dr. Cahit HELVACI Director

  • III

    ACKNOWLEDGMENTS

    I would like to express my deep sense of appreciation and gratitude to Assoc. Prof.

    Dr. Ramazan KARAKUZU for his supervision, valuable guidance and continuous

    encouragement throughout this study.

    I would also like to thank Prof. Dr. Onur SAYMAN and Prof. Dr. Tevfik AKSOY for

    their help with valuable suggestions and discussions that they have provided me during

    this research. I also extend my sincere thanks to Prof. Dr. Tevfik AKSOY for his

    permission to use whole facilities of the Department of Metallurgy and Materials

    Engineering during my studies.

    I thank the Dokuz Eylul University Science Organisation and Cumhuriyet University

    Research Foundation for providing financial assistance for this project.

    I thank Mr. Rahmi AKIN and personal of IZOREEL who helped me for the

    manufacture of the glass-epoxy laminates. Thanks goes out to the technician, Ahmet

    YT, in the Construction Laboratory and special thanks to Bahadr UYULGAN and

    Funda AK, for their help during the experimental phase of this study.

    Finally I wish to express sincere thanks to my family for their moral support,

    tolerance and understanding while preparing this thesis.

    Buket OKUTAN

  • IV

    ABSTRACT

    In this study, behavior of pin-loaded laminated composites with different stacking

    sequence and different dimensions has been observed numerically and experimentally.

    The aim is to investigate stresses, failure strength and failure mode of composite

    laminates containing a pin loaded hole when the material exhibits linear and nonlinear

    elastic behavior. For this purpose, PDNLPIN computer code developed by Larry Lessard

    was used. Parametric studies were performed to evaluate the effects of joint geometry,

    ply orientation and material non-linearity on the failure initiation and failure strength.

    The logical methodology for modeling the joint problem uses the three major steps:

    stress analysis, failure analysis, and material degradation rules. A two-dimensional,

    nonlinear, finite element technique was used for the stress analysis. The four-node

    quadrilateral isoparametric element was chosen. Based on the two-dimensional state of

    stress of each element, different failure modes were detected by a set of Hashin-type

    failure criteria. The material property degradation technique was established to degrade

    the material properties of failed elements.

    As the input for the model, the material properties of unidirectional glass/epoxy

    material were characterized under tension, compression, and in-plane shear in static

    loading conditions. An experimental program, by using standard experimental

    techniques was performed for this purpose. The shear characteristics of composite

    material were determined using Iosipescu test apparatus manufactured in the

    Construction Laboratory at Dokuz Eylul University.

  • V

    To investigate and verify to the analytical predictions of mechanical behavior, and to

    observe the failure characteristics of the pin-loaded composites, a series of experiments

    was performed with eight different material configurations, in all, over 160 specimens.

    The edge distance-to-hole diameter ratios and width-to-hole diameter ratios of plate

    were changed from 1 to 5 and 2 to 5, respectively. For this part of study, layered

    composite materials were manufactured at ZOREEL firm.

    The stress distribution around the hole in pin-loaded glass-epoxy laminate was

    performed to compare with the results of PDNLPIN program using ANSYS that is a

    general-purpose finite element computer code. In addition, ANSYS was performed to

    compare effects of different boundary conditions used to simulate the pin load on stress

    distributions around the hole.

  • VI

    ZET

    Bu almada, farkl takviye alar ve farkl boyutlara sahip fiber-glass tabakal

    kompozitlerin, pim yk etkisi altndaki davranlar, nmerik ve deneysel olarak

    incelenmitir. Ama, pim balantl tabakal kompozitlerin, lineer ve nonlineer elastik

    davran gstermesi durumlar iin, pim delii etrafndaki gerilme dalmlarn

    incelemek, hasar tipleri ve hasar mukavemetlerini tespit etmektir. Bu ama iin, Larry

    Lessard tarafndan gelitirilmi PDNLPIN sonlu eleman program kullanlmtr. Tabaka

    oryantasyonunun, balant geometrisinin ve malzeme nonlineerliinin, hasar balangc

    ve hasar mukavemeti zerindeki etkileri incelenmitir.

    Pim balantl problem modeli, temel admdan olumaktadr: Gerilme analizi,

    hasar analizi ve malzeme zelliklerinin indirgenmesi. zmlemede, 4 dml

    isoparametric elemanlar seilip, iki boyutlu nonlinear sonlu eleman tekniklerinden

    faydalanlmtr. Her bir elemandaki gerilme deerleri baz alnarak, farkl hasar tipleri

    iin, farkl tipteki Hashin hasar kriterleri kullanlm ve hasarl elemanlarda malzeme

    zellikleri indirgemesi yaplmtr.

    Modelde kullanlmak zere, standart yntemlerin tercih edildii (statik ykler

    altnda) ekme, basma ve kayma deneyleri yaplp, tek ynl glass-epoxye ait malzeme

    zellikleri tespit edilmitir. Kayma zelliklerinin bulunmasnda ise, Dokuz Eyll

    niversitesi imalat laboratuarnda zel olarak imal edilmi olan Iosipescu test aparat

    kullanlmtr.

  • VII

    Pim balantl tabakal kompozite ait hasar zelliklerini belirleyip, analitik almayla

    karlatrmak iin, toplam 160 numune olmak zere, farkl takviye alarna sahip, sekiz

    eit tabakal kompozit zerinde deneyler yaplmtr. Pim delik merkezi ile plak ucu

    arasndaki uzakln, delik apna oran 1den 5e ve plak geniliinin apa oran 2den

    5e kadar deitirilmitir. Kullanlan tabakal kompozitler ZOREEL firmasnda

    retilmitir.

    Genel amal bir sonlu eleman program olan ANSYSden elde edilen gerilme

    dalm sonular, PDNLPIN sonularyla karlatrlmtr. Ayrca ANSYS program,

    pim ykn ifade etmekte kullanlan plak snr artlarnn, gerilme dalm zerindeki

    etkilerini incelemek iin de kullanlmtr.

  • VIII

    CONTENTS

    Page

    Contents ... VIII

    List of Tables ... XII

    List of Figures .. XIII

    Nomenclature .. XIX

    Chapter One

    INTRODUCTION

    1. Introduction ... 1

    Chapter Two

    JOINTS IN COMPOSITE STRUCTURES

    2.1 Introduction ... 11

    2.2 Comparison of Metals and Composites ..... 13

    2.3 Comparison of Mechanically Fastened and Adhesively Bonded Joints 14

    2.4 Design of Joints ...... 15

    2.5 Mechanically Fastened Joint Design . 16

  • IX

    Chapter Three

    STRESS ANALYSIS

    3.1 Introduction ... 20

    3.2 Stress-Strain Relations for Plane Stress in an Orthotropic Material . 21

    3.3 Material Orientation .. 24

    3.4 Classical Lamination Theory . 26

    3.4.1 Strain and Stress Variations in a Laminate . 27

    3.4.2 Resultant Laminate Forces and Moments ... 31

    3.4.3 Symmetric Laminates ...... 35

    3.5 Finite Element Analysis ........ 37

    3.5.1 Shape Functions .. 37

    3.5.2 Element Stiffness Matrix ..... 39

    3.5.3 Numerical Integration ...... 41

    Chapter Four

    FAILURE ANALYSIS

    4.1 Introduction ... 43

    4.2 Failure Criteria .. 45

    4.2.1 Matrix Tensile Failure ..... 47

    4.2.2 Matrix Compression Failure .... 48

    4.2.3 Fiber / Matrix Shearing Failure ... 48

    4.2.4 Fiber Tensile Failure ....... 49

    4.2.5 Fiber Compression or Bearing Failure .... 49

    4.3 Material Property Degradation .. 50

    4.3.1 Matrix Tension Property Degradation .... 50

    4.3.2 Matrix Compression Property Degradation .... 51

    4.3.3 Fiber / Matrix Shearing Property Degradation .... 52

  • X

    4.3.4 Fiber Tension Property Degradation ... 52

    4.3.5 Fiber Compression (Bearing) Property Degradation ... 53

    4.4 Progressive Damage Model ... 53

    Chapter Five

    EXPERIMENTAL DETERMINATION OF

    BASIC MATERIAL PROPERTIES

    5.1 Introduction ... 56

    5.2 Test Procedures ..... 57

    5.2.1 Determination of the Tensile Properties of Unidirectional Lamina 59

    5.2.2 Determination of the Compressive Properties of Unidirectional Lamina ... 62

    5.2.3 Determination of the Shear Properties of Unidirectional Lamina ... 62

    Chapter Six

    EXPERIMENTAL DETAILS AND RESULTS OF

    A SINGLE PIN-LOADED HOLE IN COMPOSITES

    6.1 Introduction 70

    6.2 Problem Statement . 70

    6.3 Material Selection and Laminate Manufacture . 71

    6.4 Mechanical Properties ... 72

    6.5 Specimen Preparation and Testing Procedures . 73

    6.6 Experimental Results . 75

    6.6.1 Analysis of [0/45]s Laminates .... 75

    6.6.2 Analysis of [90/45]s Laminates .. 79

    6.6.3 Analysis of [0/90/0]s Laminates ... 83

    6.6.4 Analysis of [90/0/90]s Laminates ..... 87

  • XI

    6.6.5 Analysis of [90/0]2s Laminates ..... 91

    6.6.6 Analysis of [45]2s Laminates .. 95

    6.6.7 Analysis of woven[0/90]6 Laminates .... 99

    6.6.8 Analysis of woven[45]6 Laminates .... 103

    6.6.9 Comparison of Strength Values ... 108

    6.7 Discussion of the Experimental Results 111

    Chapter Seven

    NUMERICAL STUDY AND EXPERIMENTAL EVALUATION OF

    A SINGLE PIN-LOADED HOLE IN COMPOSITES

    7.1 Introduction .. 115

    7.2 Modeling of the Problem .. 115

    7.3 Finite Element Model and Boundary Conditions . 116

    7.4 Stress Analysis .. 122

    7.5 Failure Analysis and Experimental Evaluation . 133

    7.6 Comparison Between Predictions and Experimental Results 139

    7.6.1 Comparison of [0/45]s and [90/45]s Laminates 139

    7.6.2 Comparison of [0/90/0]s and [90/0/90]s Laminates .. 149

    7.6.3 Comparison of [90/0]2s and [45]2s Laminates 153

    7.6.4 Comparison of woven[0/90]6 and woven[45]6 Laminates . 153

    Chapter Eight

    CONCLUSIONS

    Conclusions . 160

    References 164

  • XII

    LIST OF TABLES

    Page

    Table 2.1 Comparison of composites and metals on the basis of some important

    mechanical and physical properties relevant to joints .. 13

    Table 2.2 Relative advantages of mechanically fastened joints versus adhesively

    bonded joints .... 14

    Table 5.1 Geometries of the test specimens 58

    Table 5.2 Methods for determination of shear properties of composite

    unidirectional lamina 63

    Table 6.1 Mechanical properties of glass-fiber/epoxy composites . 72

    Table 6.2 The lay-ups and geometries of glass-epoxy samples tested 74

  • XIII

    LIST OF FIGURES Page

    Figure 2.1 Basic joint configurations: (a) bonded joint, (b) singlelap pinned

    joint and (c) double-lap pinned joint . 12

    Figure 2.2 Typical failure mechanisms for the pinned-joint configuration 16

    Figure 2.3 Summary of mechanically fastened joint design methodology . 19

    Figure 3.1 A unidirectional fiber reinforced lamina 21

    Figure 3.2 A lamina in a plane state of stress . 22

    Figure 3.3 A fiber-reinforced lamina with global and material coordinate system 24

    Figure 3.4 A laminate made up of laminae with different fiber orientations . 27

    Figure 3.5 Undeformed and deformed geometries of an edge of a plate under

    the Kirchoff assumptions .. 28

    Figure 3.6 In-plane forces and moments on flat laminate .. 32

    Figure 3.7 Geometry of an N-layered laminate .. 33

    Figure 3.8 Symmetric laminate with identical layers k and k 36 Figure 3.9 The four-node quadrilateral element . 38

    Figure 3.10 The quadrilateral element in , space ... 38 Figure 4.1 Algorithm for progressive damage modeling of the pinned - joint

    problem . 55

    Figure 5.1 Fundamental strengths for a unidirectional reinforced lamina .. 57

    Figure 5.2 Longitudinal tensile test specimen geometry and dimensions .. 59

    Figure 5.3 Typical stress-strain curve of a unidirectional [00]6 glass/epoxy

    specimen under static loading ... 60

    Figure 5.4 Transverse tensile specimen geometry and dimensions 61

    Figure 5.5 Typical stress-strain curve of a unidirectional [900]6 glass/epoxy

    specimen under static loading ... 61

    Figure 5.6 Iosipescu test fixture . 64

    Figure 5.7 Geometry of the Iosipescu shear specimen ... 65

  • XIV

    Figure 5.8 Typical stress-strain curve of notched specimen under static in-plane

    shear loading .. 66

    Figure 5.9 The specimen geometry and dimensions under tensile loading at 450 .. 67

    Figure 5.10 Typical stress-strain curve of [450] specimen under static tension

    loading . 68

    Figure 5.11 Shear test setup 69

    Figure 6.1 Geometry of a laminated composite plate with a circular hole . 71

    Figure 6.2 A schematic description of the testing fixture ... 75

    Figure 6.3 Load/displacement curves for pin-loaded [0/45]s laminates 76

    Figure 6.4 Failure loads for [0/45]s laminates ... 77

    Figure 6.5 The effect of edge distance to diameter ratio on the shearing stress at

    failure for [0/45]s laminates 78

    Figure 6.6 The effect of edge distance to diameter ratio on the bearing strength

    for [0/45]s laminates . 78

    Figure 6.7 The effect of width to diameter ratio on the net-tension stress at

    failure for [0/45]s laminates . 79

    Figure 6.8 The effect of width to diameter ratio on the bearing strength for

    for [0/45]s laminates .... 79

    Figure 6.9 Load/displacement curves for pin-loaded [90/45]s laminates . 80

    Figure 6.10 Failure loads for [90/45]s laminates .. 81

    Figure 6.11 The effect of edge distance to diameter ratio on the shearing stress at

    failure for [90/45]s laminates 82

    Figure 6.12 The effect of edge distance to diameter ratio on the bearing strength

    for [90/45]s laminates 82

    Figure 6.13 The effect of width to diameter ratio on the net-tension stress at

    failure for [90/45]s laminates . 83

    Figure 6.14 The effect of width to diameter ratio on the bearing strength for

    [90/45]s laminates .. 83

    Figure 6.15 Load/displacement curves for pin-loaded [0/90/0]s laminates . 84

    Figure 6.16 Failure loads for [0/90/0]s laminates 85

  • XV

    Figure 6.17 The effect of edge distance to diameter ratio on the shearing stress at

    failure for [0/90/0]s laminates . 86

    Figure 6.18 The effect of edge distance to diameter ratio on the bearing strength

    for [0/90/0]s laminates . 86

    Figure 6.19 The effect of width to diameter ratio on the net-tension stress at

    failure for [0/90/0]s laminates .. 87

    Figure 6.20 The effect of width to diameter ratio on the bearing strength for

    [0/90/0]s laminates .. 87

    Figure 6.21 Load/displacement curves for pin-loaded [90/0/90]s laminates .. 88

    Figure 6.22 Failure loads for [90/0/90]s laminates . 89

    Figure 6.23 The effect of edge distance to diameter ratio on the shearing stress at

    failure for [90/0/90]s laminates 90

    Figure 6.24 The effect of edge distance to diameter ratio on the bearing strength

    for [90/0/90]s laminates .. 90

    Figure 6.25 The effect of width to diameter ratio on the net-tension stress at

    failure for [90/0/90]s laminates .. 91

    Figure 6.26 The effect of width to diameter ratio on the bearing strength for

    for [90/0/90]s laminates .. 91

    Figure 6.27 Load/displacement curves for pin-loaded [90/0]2s laminates .. 92

    Figure 6.28 Failure loads for [90/0]2s laminates .. 93

    Figure 6.29 The effect of edge distance to diameter ratio on the shearing stress at

    failure for [90/0]2s laminates .. 94

    Figure 6.30 The effect of edge distance to diameter ratio on the bearing strength

    for [90/0]2s laminates . 94

    Figure 6.31 The effect of width to diameter ratio on the net-tension stress at

    failure for [90/0]2s laminates . 95

    Figure 6.32 The effect of width to diameter ratio on the bearing strength for

    [90/0]2s laminates 95

    Figure 6.33 Load/displacement curves for pin-loaded [45]2s laminates .. 96 Figure 6.34 Failure loads for [45]2s laminates . 97

  • XVI

    Figure 6.35 The effect of edge distance to diameter ratio on the shearing stress at

    failure for [45]2s laminates ... 98 Figure 6.36 The effect of edge distance to diameter ratio on the bearing strength

    for [45]2s laminates .. 98 Figure 6.37 The effect of width to diameter ratio on the net-tension stress at

    failure for [45]2s laminates . 99 Figure 6.38 The effect of width to diameter ratio on the bearing strength for

    [45]2s laminates 99 Figure 6.39 Load/displacement curves for pin-loaded woven[0/90]6 laminates . 100

    Figure 6.40 Failure loads for woven[0/90]6 laminates .... 101

    Figure 6.41 The effect of edge distance to diameter ratio on the shearing stress at

    failure for woven[0/90]6 laminates .. 102

    Figure 6.42 The effect of edge distance to diameter ratio on the bearing strength

    for woven[0/90]6 laminates .... 102

    Figure 6.43 The effect of width to diameter ratio on the net-tension stress at

    failure for woven[0/90]6 laminates .. 103

    Figure 6.44 The effect of width to diameter ratio on the bearing strength for

    woven[0/90]6 laminates .. 103

    Figure 6.45 Load/displacement curves for pin-loaded woven[45]6 laminates . 104

    Figure 6.46 Failure loads for woven[45]6 laminates . 105

    Figure 6.47 The effect of edge distance to diameter ratio on the shearing stress at

    failure for woven[45]6 laminates .. 106

    Figure 6.48 The effect of edge distance to diameter ratio on the bearing strength

    for woven[45]6 laminates . 106

    Figure 6.49 The effect of width to diameter ratio on the net-tension stress at

    failure for woven[45]6 laminates ... 107

    Figure 6.50 The effect of width to diameter ratio on the bearing strength for

    woven[45]6 laminates .. 107

    Figure 6.51 Effect of fiber orientation on bearing strength . 108

    Figure 6.52 Bearing failure .. 108

  • XVII

    Figure 6.53 Variation of shear stress at failure with edge distance to diameter

    ratio . 109

    Figure 6.54 Shearing failure . 109

    Figure 6.55 Variation of net-tension stress at failure with specimen width to

    diameter ratio .. 110

    Figure 6.56 Net-tension failure 110

    Figure 7.1 Three methods for modeling the pin / hole interface: (a) cosine load

    distribution(b)radial boundary condition and(c)full contact problem 116

    Figure 7.2 The finite element model used cosine load distribution to simulate the

    pin load ... 118

    Figure 7.3 The finite element model used radial boundary condition to simulate

    the pin load . 119

    Figure 7.4 The finite element model used contact elements to simulate the pin

    load . 120

    Figure 7.5 Comparison of the stresses obtained from three different boundary

    conditions for [90/0]2s 121

    Figure 7.6 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 470N ( E/D=1, W/D=2 ) . 123

    Figure 7.7 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 470N ( E/D=1, W/D=5 ) . 124

    Figure 7.8 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 1180N ( E/D=2, W/D=2 ) .. 125

    Figure 7.9 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 1180N ( E/D=2, W/D=5 ) .. 126

    Figure 7.10 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 1850 N (E/D=3, W/D=2) 127

    Figure 7.11 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 1850 N (E/D=3, W/D=5) 128

    Figure 7.12 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at

  • XVIII

    P = 1660 N (E/D=4, W/D=2) . 129

    Figure 7.13 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 1660 N (E/D=4, W/D=5) . 130

    Figure 7.14 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 1180 N (E/D=5, W/D=2) . 131

    Figure 7.15 Variation of stresses of a pin loaded [0/90/0]s case along at ply 1 at P = 1180 N (E/D=5, W/D=5) ..... 132

    Figure 7.16 Illustration of damage propagation of [0/90/0]s at ply 1 at different

    load levels ....... 134

    Figure 7.17 Variation of bearing strength of a [90/0]2s laminate, E/D = 1 to 5

    with W/D = 5 .. 135

    Figure 7.18 Theoretical prediction for failure mechanism (at ply 1) as edge

    distance ratio is varied from 1 to 5, and W/D = 5 is constant . 136

    Figure 7.19 Variation of bearing strength of a [0/45]s laminate, W/D = 2 to 5

    with E/D = 4 137

    Figure 7.20 Theoretical prediction for failure mechanism (at ply 1) as width ratio

    is varied from 2 to 5, and E/D = 4 is constant . 138

    Figure 7.21 Variation of bearing strength of [90/0]2s with W/D = 2 to 5 140

    Figure 7.22 Variation of bearing strength of [90/0]2s with E/D = 1 to 5 . 141

    Figure 7.23 Variation of bearing strength of [0/90/0]s with W/D = 2 to 5 .. 143

    Figure 7.24 Variation of bearing strength of [0/90/0]2s with E/D = 1 to 5 .. 144

    Figure 7.25 Comparison of [0/45]s and [90/45]s laminates .. 146

    Figure 7.26 Comparison of [0/45]s and [90/45]s laminates with E/D = 1 to 5 147

    Figure 7.27 Comparison of [0/90/0]s and [90/0/90]s laminates 150

    Figure 7.28 Comparison of [0/90/0]s and [90/0/90]s laminates with E/D = 1 to 5 .. 151

    Figure 7.29 Comparison of [90/0]2s and [45]2s laminates .. 154

    Figure 7. 30 Comparison of [90/0]2s and [45]2s laminates with E/D = 1 to 5 .. 155

    Figure 7.31 Comparison of woven[0/90]6 and woven[45]6 laminates .. 157

    Figure 7.32 Comparison of woven[0/90]6 and woven[45]6 laminates with

    E/D = 1 to 5.. 158

  • XIX

    NOMENCLATURES

    circumferential coordinate direction non-linearity parameter slope of the laminate middle surface in the x-direction (b)ult bearing strength (s)ult shearing failure strength (t)ult net-tension strength ij poissons ratio ij strains ij stresses ij0 middle surface strains [Bb] strain displacement transformation matrix for bending

    [Db] bending parts of the material matrix

    [Kb] bending stiffness matrix

    [Q ]ij reduced-stiffness matrix

    [Qij] inverse of compliance matrix

    [Sij] compliance matrix

    a, b, h, LT dimensions of the T specimen

    Aij extensional stiffness

    Bij coupling stiffness

    D hole diameter

    Dij bending or flexural stiffness

    E end distance

    Eij elastic moduli in material directions

    Gij shear moduli

    Kij middle surface curvatures

  • XX

    L distance between the hole center and fixed end

    Mij moments

    Ni shape functions

    Nij forces

    P tensile load

    Pult maximum failure load

    S ply shearing strength

    t laminate thickness

    u, v, w displacement components

    Ub strain energy of bending

    V potential energy of external forces

    W laminate width

    Xc ply longitudinal compressive strength

    Xt ply longitudinal tensile strength

    Yc ply transversal compressive strength

    Yt ply transversal tensile strength

  • CHAPTER ONE

    INTRODUCTION

    Composite materials are commonly used in structures that demand a high level of

    mechanical performance. Their high strength to weight and stiffness to weight ratios

    have facilitated the development of lighter structures, which often replace conventional

    metal structures. Due to strength and safety requirements, these applications require

    joining composites either to composites or to metals. Although leading to a weight

    penalty due to stress concentration created by drilling a hole in the laminate, mechanical

    fasteners are widely used in the aerospace industry. In fact mechanically fastened joints

    (such as pinned joints) are unavoidable in complex structures because of their low cost,

    simplicity for assemble and facilitation of disassembly for repair.

    Joint efficiency has been a major concern in using laminated composite materials.

    Relative inefficiency and low joint strength have limited widespread application of

    composites. The need for durable and strong composite joint is even urgent for primary

    structural members made of laminates. Because of the anisotropic and heterogeneous

    nature, the joint problem in composites is more difficult to analyze than the case with

    isotropic materials.

    Mechanical fasteners remain the primary means of load transfer between structural

    components made of composite laminates. As, in pursuit of increasing efficiency of the

    structure, the operational load continues to grow, the load carried by each fastener

    increases accordingly. This increases probability of failure. Therefore, the assessment of

    the stresses around the fasteners holes becomes critical for damage-tolerant design.

    Because of the presence of unknown contact stresses and contact region between the

  • 2

    fastener and the laminate, the analysis of a pin-loaded hole becomes considerably more

    complex than that of a traction-free hole. The accurate prediction of the stress

    distribution along the hole edge is essential for reliable strength evaluation and failure

    prediction. The knowledge of the failure strength would help in selecting the appropriate

    joint size in a given application. An unskillful design of joints in the case of mechanical

    fasteners often causes a reduction of load capability of the composite structure even

    though the composite materials posses high strength. Thus many papers on mechanical

    joints and specifically pin-loaded holes have been conducted in the past.

    Review papers on the strength of mechanically fastened joints in fiber-reinforced

    plastics were written by Godwin & Matthews (1980) and Camanho & Matthews (1997).

    Effects of material properties, fastener parameters and design parameters have been

    summarized and discussed. These parameters are very important for the strength of

    mechanically joints in composite laminate.

    An appreciation of the experimental behavior is necessary before attempting a stress

    analysis or failure analysis prediction. A large of the published information on

    mechanically fastened joints has been related to experimental results.

    Several authors have highlighted the importance of width (W), end distance (E), hole

    diameter (D) and laminate thickness (t) on the joint strength. Kretsis & Matthews (1985)

    showed, using E glass fiber-reinforced plastic and carbon fiber-reinforced plastic, that as

    the width of the specimen decreases, there is a point where the made of failure changes

    from one of bearing to one of tension. A similar behavior between the end distance and

    the shear-out mode of failure was found. They concluded that lay-up had a great effect

    on both joint strength and failure mechanism.

    Hart-Smith (1980) considered that net-tension failure occurs when the bolt diameter

    is a large fraction of the strip width. This fraction depends on the type of material and

    lay-up used. Bearing failure occurs predominantly when the bolt diameter is a small

  • 3

    fraction of the plate width. Shear-out failure can be regarded as a special case of bearing

    failure. This mode of failure can occur at very large end distances for highly orthotropic

    laminates.

    Quinn & Matthews (1977) have studied experimentally the effect of stacking

    sequence on the pin bearing strength in glass fiber reinforced plastic. The results

    suggested that placing the 900 layer (normal to the applied load) at or next to the surface

    increases the bearing strength. Collings (1977) has discussed the effects of variables

    such as ply orientation, laminate thickness and bolt clamping pressure. Collings (1982)

    has also tested CFRP for a range of laminate configurations and hole sizes, and

    investigated the relation between joint strength and W/D, E/D and t/d. Pyner &

    Matthews (1979) have made experimental investigation about comparison of single and

    multi hole bolted joints in glass fiber reinforced plastics. The results suggest that the

    joint strength decreases as the joint geometry becomes increasingly complex. Cohen et

    al. (1989) investigated experimentally for failure loads and failure modes in thick

    composite joints. Thickness effect of pinned joints for composites was also investigated

    by Liu et al.(1999). He has studied the interaction between the pin diameter and

    composite thickness. Results showed that thick composites with small pins and thin

    composites with large pins had lower efficiencies for joint stiffness and joint strength

    than those having similar dimensions between pin diameter and composite thickness.

    Chen et al. (1994) have studied the influence of weave structure on pin-loaded strength

    of orthogonal 3D composites. They evaluated the influences of reinforcement type,

    weave structure, specimen width-to-diameter ratio and edge distance-to-hole diameter

    ratio. Matthews et al. (1982) investigated the bolt bearing strength of glass/carbon

    hybrid composites experimentally. Naby & Hollaway (1993) have investigated behavior

    of bolted joints in pultruded composite materials experimentally. They obtained the

    critical end distance and showed that this distance depends on the width of the joint.

    Khashaba (1996) has conducted an experimental study to determine the notched and pin

    bearing strength of GFRP composites having various values of fiber volume fractions.

    The results show that fiber volume fraction has a significant effect on load-pin bearing

  • 4

    displacement behavior and the value of W/D must be greater than 5 for the development

    of full bearing strength of the composite laminates. Maikuma et al. (1993) found similar

    W/D ratios for PAN-based and pitch-based fiber composites.

    Stockdale & Matthews (1976) investigated the effect of clamping pressure on bolt

    bearing load in glass fiber-reinforced plastics experimentally. Godwin et al. (1982)

    conducted an experimental study of a multi-bolt joint in GRP. For the case of bearing

    mode failures they reported an optimum pitch distance of five or six times the pin

    diameter. Kim et al. (1976) conducted a series of pin bearing tests to examine the effect

    of temperature and moisture on the strength of graphite-epoxy laminates. Zuiker (1995)

    presented an experimental program to measure of metal-matrix composite plates loaded

    through a pinned connection at high temperature. Experimental methods such as photo-

    elasticity (Prabharakaran, 1982 Hyer et al., 1985), moir interferometry (Zimmerman,

    1991 Tsai & Morton, 1990 Serabin & Oplinger, 1987) were applied in search of

    validating the analytical results.

    Although the experimental studies can give both the stiffness and strength of the

    composite joint, it can not give the detail stress information in the structure. Besides, it is

    very costly to perform a great amount of experiments. Hence, analytical and numerical

    methods become very important. Some studies have been done using analytical

    methods. A first approach to analytical determination of stresses around pin-loaded

    holes in orthotropic plates has been given by de Jong (1977), Waszczak & Cruse (1971)

    using the method of complex functions as developed by Muskhelishvili and worked out

    for anisotropic materials by Lekhnitskii (1998). The pin was assumed rigid, the

    uniformly distributed load in the plate was applied at an infinite distance, and a

    cosinusoidal radial stress distribution represented the pin-hole interaction. The solution

    was obtained as the combination of two load cases. Firstly, a pin-loaded hole, where

    loads with the same direction and value were applied at the plate edges. The other case

    was an open hole where the loads with the same value but with opposite directions were

    applied at the plate edges. It was shown that the normal stress distribution at the hole

  • 5

    boundary was highly dependent on the lay-up and width used. In order to account for the

    effect of friction at the pin-hole boundary, Zhang & Ueng (1987) developed a method in

    which normal and shear stresses at the hole boundary were obtained using Lekhnitskiis

    method from displacement expressions.

    The effect of pin elasticity, clearance and friction on the stresses near the pin-loaded

    orthotropic plate has been studied by Hyer et al. (1987). The elasticity problem was

    formulated in terms of complex variable theory. They have found that friction changes

    the sign of the circumferential stress in the bearing stress.

    There has been increased interest in recent years towards mathematical modeling of

    structures. Finite element analysis forms a major part of this process (Zako &

    Tsujimokami, 1994). It has been used for the understanding of how structures behave

    and locations of stress concentration in structures and other related areas. For this

    reason, the stress field around pin-loaded holes in composite plates has also been studied

    numerically (mainly finite element). Several authors considered a plane stress in a pin-

    loaded plate. Usually, two-dimensional finite element models were created and classical

    lamination theory was applied.

    Matthews et al. (1982) applied a finite element analysis and showed that the stress

    distribution around a loaded hole in fiber-reinforced laminates depends on whether the

    load is applied via a pin or bolt. In their model, the hole periphery was loaded via an

    arrangement of pin-joined bars connected between the hole center and element nodes on

    the loaded side of the hole. Load was applied by imposing a longitudinal displacement

    of sufficient magnitude to the center node to give a longitudinal stress resultant at the

    fixed end. Values of stress concentration factors for different geometries of the specimen

    have been given and they have been compared to results from other reference.

    The usual procedure for analyzing of pin joints is to use iterative or inverse methods.

    In the iterative methods, the boundary conditions are continuously modified until the

  • 6

    convergence is observed. In the inverse method a feasible pattern of contact/separation is

    initially specified from physical and symmetry conditions and the magnitude of the

    loading is sought from the elasticity solution.

    Eshwar (1978) used an inverse technique to observe clearance fit pin joints. Crews &

    Naik (1986) have developed a finite element solution by inverse technique to investigate

    the effect of clearance on the stress distribution near the loaded hole. Assuming a

    frictionless contact and rigid pin loading a quasi-isotropic laminate, it was shown that

    the contact angle is a function of clearance. Ramamurty (1988 1990) used an inverse

    technique to study the behaviour of pins fitted with interference. Assuming a frictionless

    contact and rigid pin, it was concluded that the maximum bearing stress varies

    nonlinearly with the load.

    Using an iterative method, the effects of pin elasticity, clearance and friction on the

    stresses in a pin-loaded orthotropic plate were investigated by Hyer & Klang (1985). It

    was concluded that clearance and friction significantly affect load distribution and

    magnitude of the stresses in a way that, in general degrades the load capacity.

    The effect of clearance and interference fits in a pin-loaded cross-ply FGRP laminate

    were investigated by Scalea et al. (1998). A two-dimensional finite element model was

    created using ANSYS software. Speakle Interferometry was used for compression with

    the numerical results. Dano et al. (2000) determined the deformation behavior of the pin-

    loaded joint using a two-dimensional finite element model developed in the commercial

    software ABAQUS. Pierron et al. (2000) used ABAQUS in order to calculate stress

    distribution around the hole of woven composite joint.

    Several approaches have been used to predict the strength of composite laminates

    with fastener holes. Most of the methods developed are based on two-dimensional

    models and only recently have methods considering three-dimensional models have

    been developed. The determination of the joint strength depends on the failure

    criterion. Chang et al. (1982 1986) used a two-dimensional finite element model,

  • 7

    assuming a frictionless contact, a rigid pin and a cosine normal load distribution in the

    pin-hole boundary. The Yamada Sun failure criterion (1978) was applied together with

    a proposed characteristic curve. They used their program to calculate the maximum load

    and the mode of failure of joints involving laminates with different ply orientations,

    different material properties, and different geometries. Results generated by this method

    were compared to data and to existing analytical and numerical solutions. Chang et al.

    (1984, May) performed measuring the characteristic lengths in tension and in

    compression and the rail shear strength of graphite epoxy composites. The characteristic

    length is combined with the Yamada-Sun failure criterion, and the characteristic length

    for compression was determined from the bearing failure test by Hamada et al. (1996).

    Chang & Scott (1984) have extended their analysis for predicting the failure strength and

    failure mode of composite laminates containing two pin-loaded holes placed either or in

    series. In contrary to the previous work (Chang et al., 1982), it was assumed that the

    characteristic distances were functions not only of the material but also of the geometry.

    Changs concept of characteristic curve was used by Lin &Lin (1999). They used a

    two-dimensional direct boundary element method to determine the stresses around a

    loaded hole. It was concluded that the maximum strength decreases when E/D decreases.

    Mahajerin & Sikarskie (1986) also developed boundary element method for loaded hole

    in an orthotropic plate.

    Chang & Chang (1987) also developed a progressive damage model for notched

    laminates subjected to tensile loading. Damage accumulation in laminates was evaluated

    by proposed failure criteria combined with a proposed property degradation model. An

    accumulative damage model based on the damage mechanisms observed from the

    experimental study (Wang, 1996) was developed to simulate the bearing failure in the

    laminated composite joints by Hung & Chang (1996). In other work, Hung & Chang

    (1996) developed an analytical tool to predict the response and estimate the bearing

    strength of mechanically fastened composite joints subjected to multi-axial bypass loads.

  • 8

    Xiong & Poon (1998) conducted the stress analysis of a bi-axially loaded fastener hole

    in a laminate using complex variational approach.

    Lessard and Shokrieh (1995) used two-dimensional linear and non-linear models to

    predict the strengths of pin-loaded holes. In the linear model five types of failure were

    considered. Matrix tensile and compressive failure, fiber/matrix shearing and fiber

    tensile and compressive failure were predicted using Hashin failure criterion. The non-

    linear shear stress-shear strain behavior was also considered Chang et al. (1984).

    Yamada-Sun failure criterion (Sun, 1978) was modified to include non-linear effects.

    The results were compared to data. These comparisons show that for laminates

    exhibiting non-linear behavior their analysis provides the failure strengths and failure

    modes more accurately then the previous method (Chang et al., 1982) employing a linear

    stress-strain.

    Tsujimoto & Wilson (1986) investigated the two-dimensional elasto-plastic finite

    element solution to model the non-linear material response. Using the Hill yield criterion

    in an incremental ply-by-ply failure maps were generated. Agarwal (1980) used the

    NASTRAN software to find the stress distribution around the fastener hole and to

    predict the various modes of laminate failure through the use of average stress criterion.

    Camanho & Matthews (1999) developed a three-dimensional finite element model to

    predict damage progression and strength of mechanically fastened joints in carbon fibre-

    reinforced plastics that fail in the bearing, tension and shear-out modes. Camanho et al

    (1999) created a different three-dimensional model to assess the effects of stacking

    sequence and clamping pressure on the delamination onset loads and surfaces using

    delamination onset criterion. Shoktrieh & Lessard (1996) also established a three-

    dimensional non-linear finite element code to analyze the effect of material non-linearity

    on the initiation load of a pin-loaded laminated composite plate.

  • 9

    The numerical investigations of stress distributions in multi-pin joints were found in

    literature. Both lines (parallel to the load) and rows (perpendicular to the load) of

    fasteners have been considered. Hassan et al. (1996) used a three-dimensional finite

    element model to perform stress analysis of single and multi-bolted double shear lap

    connections of glass-fiber reinforced plastic using ANSYS program. Kim and Kim

    (1995) investigated two bolts in a line and in a row. Using extended interior penalty

    methods, variational formulation was discretized using the finite element method.

    Contact clearances between pins and holes, geometric factors and the load quantity are

    considered as design parameters for three lamination angles. Using the same method,

    Kim et al. (1998) carried out a progressive failure analysis to predict the failure strength

    and modes of pin-loaded composite.

    In the case of two fasteners in tandem, Rowlands et al. (1982) determined the contact

    stresses by using an incremental finite element analysis with iterative solution

    procedure. They discussed the significance of variations in load distribution among

    bolts, friction, material properties, spacing, pin-hole clearance and end distance on the

    contact stresses. Oplinger (1980) also discussed the effects of multiple fasteners in

    parallel or series on the mechanical joint design. Wang et al. (1988) developed a two-

    dimensional finite element model for load distribution of multi-fastener joints. A similar

    model was used by Chutima et al. (1996) to investigate the stress distribution and load

    transfer in multi-fastened composite joints utilizing the I-DEAS software. MeiYing et al.

    (1996) presented a method to compute the load distribution of a joint with three

    fasteners in a row under off-axis tensile loading. Sergeev and Madenci (1998 2000)

    investigated composite laminates with multiple fasteners using the boundary collocation

    technique.

    Today, despite a large number of researches on the behavior of mechanically fastened

    joints of composite materials, not enough advancement has been recorded compared to

    that in homogeneous materials in terms of understanding the fastener behavior.

    Although composite materials exhibit complex behavior, since they employ such looked

  • 10

    after characteristics as high strength, high stiffness and lightness, they attain widely

    increasing areas of application in primary structures. This situation encourages

    researchers in dealing with this subject and puts the failure analysis of composite

    materials pinned-joint on the agenda as current and important. The aim of this study is

    to determine the stresses, strength and life prediction of pinned joints, while capturing

    the effects of geometry, stacking sequence.

    Eight chapters were given in this thesis. Chapter I, introduction, includes the literature

    review, the statement of the problem, the objective of the study and the organization of

    thesis. Chapter II is about joints in composite structures. The comparison of metals and

    composites and the knowledge of mechanically fastened joints are presented. Chapter III

    is about the macro-mechanical behavior of a laminate. The stress-strain relations for

    plane stress in orthotropic materials are explained. Chapter IV, briefly reports failure

    analysis and material property degradation technique involved modeling. Chapter V is

    devoted to experimental characterization of the material properties of a unidirectional

    ply under static loading conditions. In Chapter VI, experimental details and results are

    presented. In the next chapter, the results of the progressive damage model are shown

    for laminated plate compared to experimental results. Finite element method computer

    simulations are shown too.

    Conclusions are drawn and recommendations are made for the future work in Chapter

    VIII.

    In the references section, there are 93 references related to the subject.

    Approximately, half of them are the papers published in various journals. The other half

    of them contains books, thesis and reports.

  • 11

    CHAPTER TWO

    JOINTS IN COMPOSITE STRUCTURES

    2.1 Introduction

    The structures consist of essentially of an assembly of single elements connected to

    form a load transmission path. Joints in components or structures incur a weight penalty,

    are a source of failure and cause manufacturing problems; whenever possible, therefore,

    a designer will avoid using them. Unfortunately, it is rarely possible to produce a

    construction without joints due to limitations on material size, convenience in

    manufacture or transportation and the need for access. All connections or joints are

    potentially the weakest points in the structures so can determine its structural efficiency.

    To make useful structures, consideration must be given to the way structural components

    are joined together.

    The introduction of fiber-reinforced composites has been a major step in the

    evolution of airframe structures. Compared with conventional aluminum alloys,

    optimized use of composites can result in significant weight savings. Additionally,

    composites have many other important advantages, including improvement formability

    and immunity to corrosion and fatigue damage. From the joining viewpoint a very

    important advantage of composite construction is the ability to form large components,

    thus minimizing the number of joints required (Alan Baker, 1997, p.671).

    Basically, two types of composite joints are commonly used: adhesively bonded

    joints, and pinned or bolted joints known as mechanically fastened joints. Figure 2.1

  • 12

    shows that the basic joint configurations for one type of bonded and two types of

    mechanically fastened joints (Larry Lessard, 1995, p.244).

    Figure 2.1 Basic joint configurations: (a) bonded joint, (b) single-lap pinned joint

    and (c) double-lap pinned joint.

    The main methods used for joining metallic parts, mechanical fastening and adhesive

    fastening are also applicable to composites, provided care is taken to allow for the

    characteristics of composites. In mechanical joints loads are transferred between the

    joint elements by compression on the internal faces of the fastener holes with a smaller

    component of shear on the outer faces of the elements due to friction. In bonded joints

    the loads are transferred by mainly shear on the surfaces of the elements. In both cases,

    the load transmission elements, fastener or adhesive, are stressed primarily in shear

    along the joint line; however, the actual stress distribution will be complex (Alan Baker,

    1997, p.672).

  • 13

    2.2 Comparison of Metals and Composites

    Before considering the design of composite-to-composite and composite-to-metal

    joints it is important to appreciate the relevant properties of composites and metals.

    Some of the major points are summarized in Table 2.1 (Alan Baker, 1997, p.673).

    An advantage of composites, compared to metals, is the freedom to tailor mechanical

    properties such as stiffness and strength by judicious selection of fiber type, content and

    orientation. This can be a major benefit, of course, but can cause problems with joints,

    particularly with very non-isotropic lay-ups, resulting in extremely complicated and

    heavy configurations and, also, laminates that are difficult to repair. In addition,

    laminated composites have relatively low through-the-thickness strength and bearing

    strength under concentrated loads (Matthews, 1989, p.119).

    Table 2.1 Comparison of composites and metals on the basis of some important

    mechanical and physical properties relevant to joints

    Laminated composites Metals

    Linear elastic to fracture

    Limited ability to redistribute loads

    Relatively low toughness

    Sensitive to mild stress concentrations

    Easily damaged by mechanical impact

    Yields before failure

    High toughness

    Insensitive to mild stress

    Concentrations, e.g., holes

    Sensitive to hot/wet conditions

    Reduces matrix dominated properties, e.g.,

    compression and shear strength

    Insensitive to hot/wet conditions

    Low through-the-thickness strength/toughness

    Intolerant to out-of-plane loads

    Multiple failure modes possible

    High through - the - thickness

    strength / toughness

    Properties in-plane can be highly directional Properties fairly insensitive to

  • 14

    direction, isotropic

    Low bearing strength, particularly under hot/wet

    conditions

    High bearing strength

    Highly resistant to fatigue

    Critical mode compression

    Insensitive to stress concentrations

    High damage growth threshold

    High damage growth rate

    Prone to fatigue

    Critical mode tension

    Sensitive to stress concentrations

    Low damage growth threshold

    Growth rate fairly slow

    Immune to corrosion Prone to corrosion

    Resistant to fretting Prone to fretting

    Low thermal expansion coefficient High thermal expansion coefficient

    2.3 Comparison of Mechanically Fastened and Adhesively Bonded Joints

    The main advantages of mechanically fastened joints versus adhesively bonded joints

    are summarized in Table 2.2 (Baker, 1997).

    Table 2.2 Relative advantages of mechanically fastened joints versus adhesively

    bonded joints

    Advantages of mechanical joints

    Advantages of bonded joints

    Tolerant to the effects of environment and

    fatigue loading

    Small stress concentration in

    adherends

    Simple inspection procedure Stiff connection

    Simple manufacturing process Relatively lightweight

    Capability for repeated assembly Sealed against corrosion

    High reliability Smooth surface counter

    No thickness limitations

    No major residual stress problem

  • 15

    It is seen that when choosing between the two basic methods their various

    advantages and disadvantages must be kept in mind. Mechanically fastened joints are

    easily disassembled without damage, do not need special surface preparation, are easy

    to inspect but do have high stress concentrations (at the holes) and are heavy. Whilst

    bonded joints have lower stress concentrations and weight penalty, they can not easily

    be disassembled, adequate surface preparation is essential, inspection is difficult and, in

    addition they are sensitive to environmental effects (Matthews, 1989, p.119).

    2.4 Design of Joints

    It is clear that joints must be considered an integral part of the design process. A

    structural joint represents a critical element in virtually all hardware designs. In a

    composite structure, the joint may be made totally or partially of composite materials.

    The method of joining may be adhesive bonding or mechanical fastening; in many

    situations the latter method is preferred because of its nonpermanent nature.

    In all cases a significant weight penalty is incurred by the presence of joints in

    composite structures; furthermore, premature failures have been experienced too

    frequently. The reasons for this situation are fundamentally related to the low strain

    capability of linearly elastic reinforcing fibers; the very low secondary or resin

    dominated properties of composites, and the inherent, localized conditions of bond line

    stresses and bolt hole stress concentrations. Consequently, the design of efficient

    structural attachments represents one of the major challenges in the development of

    composite structures; because of its generic nature, the design is deserving of separate

    treatment as a case study.

    The designer is confronted, in many instances, with a decision as to whether to

    specify a bonded or a bolted joint concept for a given structural attachment. Basic

    considerations that influence this decision usually include the following (Keith T.

    Kedward, 1990, p.22):

  • 16

    The magnitude of the loading, typically expressed as a force per unit joint width, that

    must be transmitted from one end to the other

    The geometrical constraints within which the load transfer must be accomplished

    The desired reliability of the joint

    Environmental factors in joint operation

    A need for repetitive assembly and disassembly

    Joint efficiency desired (the strength-weight factor)

    Cost of manufacture, assembly, and inspection

    2.5 Mechanically Fastened Joint Design

    Although the aim of achieving smooth load transfer from one joint element to

    another is similar in bonding and material fastening, the load transfer mechanisms are

    very different. In mechanical fastening load transfer is by compression (bearing) on the

    faces of holes passing through the joint members by shear (and, less desirably bending)

    of the fasteners (Alan Baker, 1997, p.721).

    It has been observed experimentally that mechanical fastened joints fail under three

    basic mechanisms: net-tension, shear-out and, bearing (in addition, combinations of

    these mechanisms are often given separate names). Typical damage caused by each

    mechanism is shown in Figure 2.2 (Larry Lessard, 1995, p.247).

    Tension failure Shear-out failure Bearing failure

    Figure 2.2 Typical failure mechanisms for the pinned-joint configuration.

  • 17

    To estimate the strength of single pin-loaded specimens, the static strengths are

    defined as;

    Net-tension Strength

    The stress at net-tension section, at failure, is given by

    ( ) ( )tDWPult

    ultt .=

    where Pult is the failing load of the member, W is the joint width at net-section, D is the

    hole diameter and t the joint thickness.

    Bearing Strength

    The bearing strength of a composite material is expressed in the form

    ( )tD

    Pultultb .

    =

    Shearing Strength

    The strength in this case is given as

    ( )tE

    Pultults ..2

    =

    where E is the distance ( parallel to the load) between the hole center and the free edge,

    usually known as the edge distance.

  • 18

    The behavior of joint could be influenced by four groups of parameters (Chen et al.,

    1994).

    Material parameters: fiber types and form, resin type, fiber orientation, laminate

    stacking sequence, etc.

    Geometry parameters: specimen width (W) or ratio of width to hole diameter (W/D),

    edge distance (E) or ratio of the edge distance to hole diameter (E/D), specimen

    thickness (t), hole size (D), and pitch for multiple joints.

    Fastener parameters: fastener type, fastener size, clamping area and pressure,

    washer size and hole size and tolerance.

    Design parameters: loading type (tension, compression, fatigue, etc.), loading

    direction, joint type (single lap, double lap), geometry (pitch, edge distance, hole

    pattern etc.), environment and failure criteria.

    It is clear that, in view of the very large number of variables involved, a complete

    characterization of joint behavior is impossible. Rather, the approach should be to

    determine as thoroughly as possible the behavior of basic joints and to hopefully infer

    the influence of the more important parameters, from which the behavior of joints and

    materials can be predicted.

    An applicable and verifiable approach of the joint design methodology is provided in

    Figure 2.3 (Keith T. Kedward, 1990, p.24).

  • 19

    Figure 2.3 Summary of mechanically fastened joint design methodology.

    ESTABLISH DESIGN GOALS Load Intensity Life Requirements Geometrical Constraints

    DETERMINATION OF DESIGN LOADS

    AND INTERNAL DISTRIBUTION

    Overall Structural Analysis (typical finite

    element model) Sub structural Analysis of Joint Details Fastener flexibility Considerations Identify Critical Loading Conditions

    LAMINATE SELECTION

    EMPIRICAL DATA Stiffness Strength

    DETAILED AT-THE-HOLE

    ANALYSIS

    Detailed Structural Analysis of Joint

    Identify Critical Fastener Locations and Break Down into Simple Loading Conditions

    DESIGN TRIAL JOINT Specify Size, Type and

    Arrangement of Fasteners Adjust Local Thickness as

    necessary Identify Potential Failure

    Modes Tabulate Parameters

    MODIFY DESIGN OR LOCALLY TAILOR LAMINATE

    PREPARE DETAIL JOINT DESIGN DRAWINGS

    DESIGN

    GOALS

    SATISFIED

    Yes

    No

  • 20

    CHAPTER THREE

    STRESS ANALYSIS

    3.1 Introduction

    In this chapter, information is given about the finite element formulation and

    macromechanical behavior of lamina and laminate and the theoretical bases of the model

    were explained. Various authors study these subjects widely. For a detailed description

    one can apply the books, for example, by (Jones, 1975 - Gibson, 1994 - Kaw, 1997 -

    Reddy, 1997 - Hyer, 1998).

    The purpose of a joint is to transfer load between the two parts being joined. As a

    result of this load transfer there will be a stress variation in the components in the joint

    region. In most cases, an accurate understanding of the stress distributions in the joint is

    one of the critical ingredients. Although other methods of analysis may identify the

    origin and mode of failure, stress analysis most often provides a quantitative explanation

    for the cause of failure. Stress analysis procedures for composite materials are complex.

    In order to utilize the full potential of the specific strength of composite materials,

    accurate stress distributions must be known.

    Because composite materials are fabricated by the lamination of highly anisotropic

    plies, a nearly infinite variety of directional moduli and strengths can be achieved.

    Because of their laminated anisotropic construction, significant variations in stress can

    exist within the laminate itself. As a result, consideration must be given to stress and

    failure at both individual ply and the overall laminate levels.

  • 21

    Using the lamination theory and anisotropic theory of elasticity, the state of stress

    depend on the modulus and orientation of each ply is defined and the magnitude of the

    applied load necessary to create failure is determined. It is assumed that the thickness of

    the laminate is small compared with the plate length and width and applied load is in-

    plane (plane stress condition). Finally, only plane stresses are considered (13, 23, 3 = 0). The applied load is symmetric with the respect to the mid-plane and the laminate has

    a symmetric lay-up.

    3.2 Stress-Strain Relations For Plane Stress in an Orthotropic Material

    For an orthotropic material, the compliance matrix components in terms of the

    engineering constants are

    =

    12

    31

    23

    3

    2

    1

    12

    31

    23

    32

    23

    1

    13

    3

    32

    21

    12

    3

    31

    2

    21

    1

    12

    31

    23

    3

    2

    1

    100000

    010000

    001000

    0001

    0001

    0001

    G

    G

    G

    EEE

    EEE

    EEE

    (3.1)

    By assuming two-dimensional orthotropic material properties for each unidirectional

    ply, the number of material properties reduced four. A unidirectional ply is shown in

    Figure 3.1. The 1 and 2 axes are the longitudinal and transverse directions respectively.

    Figure 3.1 A unidirectional fiber reinforced lamina

  • 22

    The most important state of stress in a lamina is plane shown in Figure 3.2, that is:

    3 = 23 = 31 = 0

    In this case, Eq. (3.1) reduces to

    =

    12

    2

    1

    66

    2212

    1211

    12

    2

    1

    0000

    SSSSS

    (3.2)

    where

    =

    12

    21

    12

    2

    21

    1

    100

    01

    01

    G

    EE

    EE

    S

    (3.3)

    Figure 3.2 A lamina in a plane state of stress

  • 23

    which may be written :

    { } [ ]{ }ll S = (3.4)

    where l identifies lamina coordinates, and [S], the compliance matrix, relates the stress

    and strain components in the principal material directions.

    The Eq.(3.2) can be inverted to

    { } [ ] { }ll S 1= (3.5)

    or

    { } [ ]{ }ll Q =

    where the matrix Q is defined as the inverse of the compliance matrix and is known as

    the reduced lamina stiffness matrix.

    =

    12

    2

    1

    66

    2122211

    112

    122211

    12

    2122211

    122

    122211

    22

    12

    2

    1

    100

    0

    0

    S

    SSSS

    SSSS

    SSSS

    SSSS

    (3.6)

    =

    12

    2

    1

    12

    2112

    2

    2112

    212

    2112

    212

    2112

    1

    12

    2

    1

    00

    011

    011

    G

    EE

    EE

    (3.7)

  • 24

    3.3 Material Orientation

    The reduced stiffness and compliance matrices relate stresses and strains in the

    principal material directions of the material. To define the material response in

    directions other than these material coordinates, transformation matrices must be

    developed for the material stiffness. In Figure 3.3, two sets of coordinate systems are

    shown.

    The 1-2 coordinate system corresponds to the principal material directions for a

    lamina, while the x-y coordinates are arbitrary and are related to the 1-2 coordinates

    through a rotation about the axis out of the plane of the Figure 3.3. The angle, , is defined as the rotation from the arbitrary x-y system to the material 1-2 system.

    The transformation of stresses from the 1-2 system to the x-y system follows the rules

    for transformation of tensor components. Thus:

    =

    12

    2

    1

    22

    22

    22

    22

    nmmnmnmnmnmnnm

    xy

    y

    x

    (3.8)

    or

    { } [ ]{ }lx 1=

    Figure 3.3 A fiber reinforced lamina with global and material coordinate system

  • 25

    where m = Cos () and n = Sin (). The subscript x is used to refer to the laminate coordinate system. The same transformation matrix [1] can also be used for the tensor strain components. Thus:

    [ ]

    =

    12

    2

    1

    1

    xy

    y

    x

    or

    =

    12

    2

    1

    22

    22

    22

    2222

    nmmnmnmnmnmnnm

    xy

    y

    x

    (3.9)

    or

    { } [ ]{ }lx 2=

    Given the transformations for stress and strain to arbitrary coordinate systems, the

    relations between stress and strain in the laminate system can be determined.

    Substituting Eq. (3.8) and (3.9) into Eq. (3.5), we obtain:

    { } [ ][ ][ ] { }xx Q 121 = (3.10)

    Thus,

    { } { }xx Q

    = (3.11)

    or

    =

    xy

    y

    x

    xy

    y

    x

    QQQ

    QQQ

    QQQ

    662616

    262212

    161211

    where xy = 2xy

  • 26

    The reduced-stiffness matrix

    Q , relates the stress and strain components in the

    laminate coordinate system. Here:

    [ ][ ][ ] 121 =

    QQ (3.12)

    The terms within

    Q are defined by approximate matrix multiplication to be:

    )()22(

    )2()2(

    )2(2

    )2()2(

    )()4(

    )2(2

    4466

    226612221166

    3662212

    366121126

    226612

    422

    41122

    3662212

    366121116

    )4412

    2266221112

    226612

    422

    41111

    mnQmnQQQQQ

    nmQQQmnQQQQ

    nmQQmQnQQ

    mnQQQnmQQQQ

    nmQnmQQQQ

    nmQQnQmQQ

    +++=++=

    +++=++=

    +++=+++=

    (3.13)

    3.4 Classical Lamination Theory

    A laminate is two or more laminae bonded together to act as an integral structural

    element. A typical laminate is shown in Figure 3.4. Laminated composite materials

    typically have exceptional properties in the direction of the reinforcing fibers, but poor

    to mediocre properties perpendicular (transverse) to the fibers. The problem is how to

    obtain maximum advantage from the exceptional fiber directional properties while

    minimizing the effects of the low transverse properties. The plies or lamina principal

    material directions are oriented in several directions such that the effective properties of

    the laminate match the loading environment. For purposes of structural analysis, it is

    desirable to represent a laminate by set of effective stiffness. The stiffness of such a

    composite material configuration is obtained from the properties of the constituent

    laminae. The procedures enable the analysis of laminates that have individual laminae

  • 27

    with principal material directions oriented at arbitrary angles to the chosen or natural

    axes of the laminate. As a consequence of the arbitrary orientations, the laminate may

    not have definable principal directions.

    It is apparent that overall behavior of a multidirectional laminate is a function of the

    properties and stacking sequence of the individual layers. The so-called classical

    lamination theory predicts the behavior of the laminate within the framework. By use of

    this theory, it can be consistently proceed from the basic building block, the lamina, to

    the end result, a structural laminate.

    Figure 3.4 A laminate made up of lamina with different fiber orientations

    3.4.1 Strain and Stress Variation in a Laminate

    Knowledge of the variation of stress and strain through the laminate thickness is

    essential to the definition of the extensional and bending stiffness of a laminate. In the

    classical lamination theory, the laminate is presumed to consist of perfectly bonded

    lamina. Moreover, the bonds are presumed to be infinitesimally thin as well as non-shear

    deformable. That is, the displacements are continuous across lamina boundaries so that

  • 28

    no lamina can slip relative to another. Thus, the laminate acts as a single with very

    special properties, but nevertheless acts as a single layer of material.

    Accordingly, if the laminate is thin, a line originally straight and perpendicular to the

    middle surface of the laminate is assumed to remain straight and perpendicular to the

    middle surface when the laminate is extended and bent. Requiring the normal to the

    middle surface to remain straight and normal under deformation is equivalent to

    ignoring the shearing strains in planes perpendicular to the middle surface, that is xz = yz = 0 where z is the direction of the normal to the middle surface in Figure 3.5. In addition, the normals are presumed to have constant length so that the strain

    perpendicular to the middle surface is ignored as well, that is, z = 0. The foregoing collection of assumptions of the behavior of the single layer that represents the laminate

    constitutes the familiar Kirchhoff hypothesis for plates and the Kirchhoff-Love

    hypothesis for shells.

    Figure 3.5 Undeformed and deformed geometries of an edge of a plate

    under the Kirchoff assumptions

  • 29

    The implications of the Kirchhoff or the Kirchhoff-Love hypothesis on the laminate

    displacements u, v, and w in the x, y, and z directions are derived by use of the laminate

    cross section in the x-z plane shown in Figure 3.5. The displacement in the x-direction of

    point B from the un-deformed to the deformed middle surface is u0. Since line ABCD

    remains straight under deformation of the laminate, the displacement that

    uc = u0 - zc (3.14)

    But since, under deformation, line ABCD further remains perpendicular to the middle

    surface, is the slope of the laminate middle surface in the x-direction, that is,

    Then, the displacement, u, at any point z through the laminate thickness is

    By similar reasoning, the displacement, v, in the y-direction is

    The laminate strains have been reduced to x, y, xy virtue of the Kirchhoff-Love hypothesis. That is, z = xz = yz = 0. For small strains (linear elasticity), the remaining strains are defined in terms of displacements as

    ) (3.15 0x

    w=

    ) 3.16 ( 00 xwzuu =

    (3.17) 00 ywzvv =

  • 30

    Thus, for the derived displacements u and v Eq. (3.16) and (3.17) and the strains are

    or

    (3.20) 0

    +

    =

    xy

    y

    x

    oxy

    y

    ox

    xy

    y

    x

    KKK

    z

    where the middle surface strains are

    (3.21)

    00

    0

    0

    0

    0

    0

    +

    =

    xv

    yu

    yvxu

    xy

    y

    x

    and the middle surface curvatures are

    xv

    yu

    yvxu

    xy

    y

    x

    +

    ===

    (3.18)

    yxwz

    xv

    yu

    ywz

    yv

    xwz

    xu

    oooxy

    oy

    oox

    +=

    =

    =

    2

    2

    2

    2

    2

    2

    ) (3.19

  • 31

    ) (3.22

    22

    2

    2

    2

    2

    =

    yxw

    ywxw

    KKK

    o

    o

    o

    xy

    y

    x

    The last term in Eq. (3.22) is the twist curvature of the middle surface.

    By substitution of the strain variation through the thickness, Eq. (3.20), in the stress-

    strain relations, Eq. (3.11), the stresses in the kth layer can be expressed in terms of the

    laminate middle surface strains and curvatures as

    Since the ijQ can be different for each layer of the laminate, the stress variation

    through the laminate thickness is not necessarily linear, even though the strain variation

    is linear.

    3.4.2 Resultant Laminate Forces and Moments

    The resultant forces and moments acting on a laminate are obtained by integration of the

    stresses in each layer or lamina through the laminate thickness, for example,

    ) (3.24 2/

    2/

    2/

    2/

    =

    =t

    txx

    t

    txx

    zdzM

    dzN

    (3.23) 0

    0

    0

    662616

    262212

    161211

    +

    =

    xy

    y

    x

    y

    X

    kkxy

    y

    x

    KKK

    zQQQQQQQQQ

    XY

  • 32

    Nx is a force per unit length (width) of the cross section of the laminate. Similarly, Mx is

    a moment per unit length as shown Figure 3.6. The entire collection of force and

    moment resultants for an N-layered laminate is defined as

    Figure 3.6 In-plane forces and moments on flat laminate

    dzdzNNN

    k

    N

    k

    z

    zxy

    y

    xt

    tkxy

    y

    x

    xy

    y

    x k

    k

    =

    =

    =

    1

    2/

    2/ 1

    (3.25)

    zdzdzzMMM

    k

    N

    k

    z

    zxy

    y

    xt

    tkxy

    y

    x

    xy

    y

    x k

    k

    =

    =

    =

    1

    2/

    2/ 1

    (3.26)

    where zk and zk-1 are defined in Figure 3.7. These force and moment resultants do not

    depend on z after integration, but are functions of x and y, the coordinates in the plane of

    the laminate middle surface.

  • 33

    Figure 3.7 Geometry of an N-layered laminate

    The integration indicated in Eq. (3.25) and (3.26) can be rearranged to take advantage

    of the fact that the stiffness matrix for a lamina is constant within the lamina. Thus, the

    stiffness matrix goes outside the integration over each layer. When the Eq. (3.23) are

    substituted, the forces and moments become

    (3.27)

    1

    2

    0

    0

    0

    662616

    262212

    161211

    1 0

    0

    0

    662616

    262212

    161211

    1 1

    1 1

    =

    =

    +

    =

    +

    =

    N

    k

    z

    z

    z

    zxy

    y

    x

    xy

    y

    x

    kxy

    y

    x

    N

    k

    z

    z

    z

    zxy

    y

    x

    xy

    y

    x

    kxy

    y

    x

    k

    k

    k

    k

    k

    k

    k

    k

    dzzKKK

    zdzQQQQQQQQQ

    MMM

    zdzKKK

    dzQQQQQQQQQ

    NNN

    However, x0, y0, xy, Kx, Ky and Kxy are not functions of z but are middle surface values so can be removed from under the summation signs. Thus,

  • 34

    ) (3.28

    662616

    262212

    161211

    0

    0

    0

    662616

    262212

    161211

    662616

    262212

    161211

    0

    0

    0

    662616

    262212

    161211

    +

    =

    +

    =

    xy

    y

    x

    xy

    y

    x

    xy

    y

    x

    xy

    y

    x

    xy

    y

    x

    xy

    y

    x

    KKK

    DDDDDDDDD

    BBBBBBBBB

    MMM

    KKK

    BBBBBBBBB

    AAAAAAAAA

    NNN

    where

    ( ) ( )( ) ( )( ) ( )

    =

    =

    =

    =

    =

    =

    N

    kkkkijij

    N

    kkkkijij

    N

    kkkkijij

    zzQD

    zzQB

    zzQA

    11

    33

    11

    22

    11

    31

    (3.29) 21

    The complete set of the equations can be expressed in matrix form as

    (3.30) 0

    0

    0

    662616662616

    262212262212

    161211161211

    662616662616

    262212262212

    161211161211

    =

    xy

    y

    x

    xy

    y

    x

    xy

    y

    x

    xy

    y

    x

    KKK

    DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA

    MMMNNN

    Or in partitioned form as

    (3.31) 0

    =

    KDBBA

    MN

    Since the applied loads are generally known rather than the deformations, it is often

    necessary to use the inverted form of the laminate force-deformation relationships.

  • 35

    ) (3.32 10

    =

    MN

    DBBA

    K

    The relations above are expressed in terms of three laminate stiffness matrices. [A],

    [B] and [D], which are functions of the geometry, material properties and stacking

    sequence of the individual plies, as described in Eq. (3.29). They are average elastic

    parameters of the multidirectional laminate with the following significance:

    Aij are extensional stiffnesses, or in-plane laminate moduli, relating in-plane loads to

    in-plane strains. Bij are called coupling stiffness, or in-plane flexure coupling laminate

    moduli, relating in-plane loads to curvatures and moments to in-plane strains. Thus, if Bij

    0, in-plane forces produce flexural and twisting deformations; moments produce extension of the middle surface in addition to flexure and twisting.

    Dij are bending or flexural laminate stiffnesses relating moments to curvatures.

    3.4.3 Symmetric Laminates

    A laminate is called symmetric when for each layer on one side of a reference plane

    (middle surface) there is a corresponding layer at an equal distance from the reference

    plane on the other side with identical thickness, orientation, and properties. The laminate

    is symmetric in both geometry and material properties.

    Consider the N-layer laminate in Figure 3.8, where identical layers k and k are symmetrically situated about the reference plane. Then

    ( ) ( )'

    '

    '

    kk

    kijkij

    kk

    hh

    QQtt

    ==

    = (3.33)

    and according to the definition in Eq. (3.29), the coupling stiffness are

  • 36

    Figure 3.8 Symmetric laminate with identical layers k and k

    ( ) ( )( ) ( )( )

    ( ) kkNk

    kijij

    kk

    N

    kkkkijij

    N

    kkkkijij

    thQB

    zzzzQB

    zzQB

    .

    2121

    1

    11

    1

    11

    22

    =

    =

    =

    =

    +=

    =

    (3.34)

    Since

    1

    1 )(21

    =+=kkk

    kkk

    zzt

    zzh (3.35)

    For the conditions of symmetry stated before, the sum above will consist of pairs of

    terms of equal absolute value and opposite signs. Thus, such a symmetry condition when

    substituted in Eq. (3.29) leads to the major simplification that all Bij = 0. This means that

    bending-stretching coupling will not be present in such laminates. Consequently, in-

    plane loads will not generate bending and twisting curvatures that cause out-of-plane

    warping, and bending or twisting moments will not produce on extension of the middle

    surface.

    The load-deformation relations in this case reduce to

  • 37

    662616

    262212

    161211

    0

    0

    0

    662616

    262212

    161211

    =

    =

    xy

    y

    x

    xy

    y

    x

    xy

    y

    x

    xy

    y

    x

    KKK

    DDDDDDDDD

    MMM

    AAAAAAAAA

    NNN

    (3.36)

    3.5 Finite Element Analysis

    A two-dimensional finite element technique is used to investigate pin-loaded

    composite plate problem. By considering the curved boundary around the hole of a

    composite laminate, and in order to apply boundary conditions, mixed Cartesian-

    Cylindrical coordinates are used.

    3.5.1 Shape Functions

    A four-node quadrilateral isoparametric shell element is used. The basic procedure in

    the isoparametric finite element formulation is to express the element coordinates and

    element displacements in the form of interpolations using the natural coordinate system

    of the element. An arbitrary two-dimensional four-node quadrilateral element has been

    drawn in the global x-y plane in Figure 3.9. The quadrilateral element is defined in , coordinates (or natural coordinates), and is square shaped, shown in Figure 3.10. The

    new coordinates , are so arranged that lines of constant and are straight and take up values of 1 at the sides of the quadrilateral. Both values increase along a linear distance scale. Each of the four corner points in the natural coordinates may be

    associated with an element shape function. The shape functions Ni are defined in natural

    coordinate system of the element which has variables , . The shape functions Ni where i = 1,2,3,4, are defined such that Ni is equal to unity at node 1, and is zero at other nodes.

  • 38

    Figure 3.9 The four-node quadrilateral element

    Figure 3.10 The quadrilateral element in , space Thus, the shape functions in terms of the natural coordinates , are

    )1)(1(41

    )1)(1(41

    )1)(1(41

    )1)(1(41

    4

    3

    2

    1

    +=

    ++=

    +=

    =

    N

    N

    N

    N

    (3.37)

    or the compact representation of Eq.(3.37) is

    )1)(1(41

    iiiN ++= (3.38)

  • 39

    The displacement field can be expressed in the following matrix form:

    iy

    x

    n

    i

    i

    i

    i

    i

    y

    x

    wvu

    NN

    NN

    N

    wvu

    =

    0

    0

    1

    0

    0

    00000000000000000000

    (3.39)

    in which n is the total number of nodes and Ni is the shape function at node i. The

    relationship between strains and displacement can be written in the matrix form:

    iy

    x

    xiyi

    yi

    xi

    xiyi

    yi

    xi

    xy

    y

    x

    xy

    y

    x

    wvu

    NNN

    NNNN

    N

    KKK

    =

    0

    0

    ,,

    ,

    ,

    ,,

    ,

    ,

    0

    0

    0

    0000000

    00000000000000