Documentd1

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


3.2 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


4.2 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


5.2 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


failure for [90/45]s laminates 82


for [90/45]s laminates 82


failure for [90/45]s laminates . 83


[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


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


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


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


[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


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


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


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


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


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


for [90/0]2s laminates . 94


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


[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


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


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


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


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


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


failure for woven[45]6 laminates .. 106


for woven[45]6 laminates . 106


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


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

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