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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