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Stress analysis of a double-doubler joint in composite structure

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Title:
Stress analysis of a double-doubler joint in composite structure
Creator:
Park, Myung Kyun, 1956-
Publication Date:
Language:
English
Physical Description:
xvi, 183 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Adhesives ( jstor )
Aluminum ( jstor )
Axial stress ( jstor )
Bending ( jstor )
Bonded joints ( jstor )
Coordinate systems ( jstor )
Lap joints ( jstor )
Shear stress ( jstor )
Stress analysis ( jstor )
Stress distribution ( jstor )
Aerospace Engineering, Mechanics, and Engineering Science thesis Ph. D ( lcsh )
Composite construction ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics, and Engineering Science -- UF ( lcsh )
Joints (Engineering) ( lcsh )
Strains and stresses ( lcsh )
Structural analysis (Engineering) ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 179-182).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Myung Kyun Park.

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STRESS ANALYSIS OF A DOUBLE-DOUBLER JOINT
IN COMPOSITE STRUCTURE

















By

MYUNG KYUN PARK


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1990














ACKNOWLEDGEMENTS


The author deeply appreciates the efforts of his committee chairman, Professor L. E. Malvern. For his advice, guidance, and encouragement, the author is greatly indebted.

The author is grateful to Professor B. V. Sankar

especially for suggesting study of the stress singularity treatment and for helpful suggestions in carrying out all the analyses.

Special appreciation is extended to Professors C. T. Sun, S. Y. Lu, and Professor David C. Wilson for their assistance and serving on the supervisory committee.

The author thanks the Department of Aerospace

Engineering, Mechanics and Engineering Science and the Center for Studies of Advanced Structural Composites of the University of Florida for their assistance and financial support on this research.

The author thanks all his family members for their encouragement on his pursuing an advanced degree in the United States.

Finally, true thanks are due to the author's wife, Boeun, who with her prayers, love and forbearance supported the author to finish his study at the University of Florida.


ii












TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS ....................................... ii

LIST OF TABLES ... ...................................... vi

LIST OF FIGURES ........................................ vii

ABSTRACT ............................................... xv

CHAPTERS

1 INTRODUCTION ................................... 1

1.1 Introduction ............................. 1
1.2 Objective and Scope .......................... 2

2 REVIEW OF PREVIOUS WORK ........................ 5

2.1 Introduction ............................. 5
2.2 Analytical Study ......................... 7
2.2.1 Volkersen's Theory ................... 7
2.2.2 Goland and Reissner Theory ........ 9 2.2.3 Plantema's Analysis .................. 13
2.2.4 Kelsey and Benson Theory ............ 14
2.2.5 Pahoja's Analysis .................... 15
2.2.6 Renton and Vinson's Analysis ...... 19 2.2.7 Hart-Smith's Analysis ............... 21

2.3 Finite Element Studies ................... 23
2.3.1 Introduction ...................... 23
2.3.2 Ahluwalia's Analysis ................ 24
2.3.3 Wooley and Carver's Analysis ...... 25

2.4 Experimental Studies ..................... 26
2.4.1 Mclaren and MacInnes's
Investigation ..................... 26
2.4.2 Tuzi and Shimada's
Investigation ..................... 27

3 LINEAR ELASTIC STRESS ANALYSIS .................... 29

3.1 Introduction ....... ...................... 29
3.2 Joint Description --...................... 30
3.3 Determining Material Properties ............ 33
3.4 Finite Element Model Representation ...... 36


iii








3.5 Finite Element Results ...................


3.5.1 Case 1. Aluminum Adherends and
Epoxy adhesive in
Simple Tension ...................... 43
3.5.2 Case 2. Aluminum Central
Adherend, Epoxy Adhesive, and
Composite Outer Adherend
in Simple Tension ................... 52
3.5.3 Case 3. Aluminum Adherends and
Epoxy Adhesive in Pure Bending..... 53
3.5.4 Case 4. Aluminum Central
Adherends, Epoxy Adhesive, and
Composite Outer Adherend
in Pure Bending ..................... 68

4 PARAMETRIC STUDY OF THE
DOUBLE-DOUBLER JOINT ............................. 78

4.1 Introduction ................................ 78
4.2 The Material Parameter .................... 79
4.2.1 Influence of E/Ea. ................. 80
4.3 The Geometric Parameters ................... 85
4.3.1 Influence of Overlap Length ....... 87

5 TREATMENT OF STRESS SINGULARITY
AT BONDING CORNERS IN THE DOUBLEDOUBLER JOINT .................................. 89

5.1 Introduction ............................. 89
5.2 Stress Singularity at Interface
Corners in Bonded Dissimilar Materials ... 90
5.3 Calculation of Order and Intensities
of Stress Singularities at the Bonding
Corners of a Double-doubler Joint ........ 98
5.3.1 Case 1. Half Plane Bonded
to a Quarter Plane ................ 98
5.3.2 Case 2. Three Quarter Plane
Bonded to a Quarter Plane ......... 103
5.4 Specified Boundary Displacement Method ... 106
5.4.1 Introduction ....................... 106
5.4.2 Procedure .......................... 107
5.4.3 Benefits of Submodelling .......... 110
5.4.4 Coarse Model and Submodel ......... 110

6 RESULTS AND DISCUSSION .......................... 117

6.1 Introduction ............................... 117
6.2 Summary and Conclusions ................... 118
6.3 Recommendations for Future Work .......... 121


iv


40









APPENDICES


A CASE 1. THE AXIAL STRESS DISTRIBUTIONS
OF DOUBLE-DOUBLER REINFORCEMENT OF A
CONTINOUS MEMBER AT DIFFERENT
LEVELS OF Y .................................... 124

B CASE 1. THE AXIAL STRESS DISTRIBUTIONS
OF A DOUBLE-DOUBLER JOINT AT
DIFFERENT LEVELS OF Y .......................... 129

C CASE 2. THE AXIAL STRESS DISTRIBUTIONS
OF DOUBLE-DOUBLER REINFORCEMENT OF A
CONTINOUS MEMBER AT DIFFERENT
LEVELS OF Y .................................... 134

D CASE 2. THE AXIAL STRESS DISTRIBUTIONS
OF A DOUBLE-DOUBLER JOINT AT
DIFFERENT LEVELS OF Y .......................... 139

E CASE 3. THE AXIAL STRESS DISTRIBUTIONS
OF DOUBLE-DOUBLER REINFORCEMENT OF A
CONTINOUS MEMBER AT DIFFERENT
LEVELS OF Y .................................... 144

F CASE 3. THE AXIAL STRESS DISTRIBUTIONS
OF A DOUBLE-DOUBLER JOINT AT
DIFFERENT LEVELS OF Y .......................... 153

G CASE 4. THE AXIAL STRESS DISTRIBUTIONS
OF DOUBLE-DOUBLER REINFORCEMENT OF A
CONTINOUS MEMBER AT DIFFERENT
LEVELS OF Y .................................... 162

H CASE 4. THE AXIAL STRESS DISTRIBUTIONS
OF A DOUBLE-DOUBLER JOINT AT
DIFFERENT LEVELS OF Y .......................... 171

LIST OF REFERENCES ..................................... 179

BIOGRAPHICAL SKETCH ..................................... 183


V














LIST OF TABLES
Table Page

3.1. Elastic material properties used in
analysis ...................................... 33

4.1. Shear stress concentration factor
and the order of stress singularity A
for case 1 in Chapter 5.3 .............. 82

5.1. Selected material properties for
singular behavior analysis ............. 100

5.2. Different values Cy, C xy for Case 1
in r, direction in simple tension
loading case ................................. 114

5.3. Different values C,, Cy, C for Case 2
in the r2 direction in simple tension
loading case ................................. 114

5.4. Different values C for Case 1 in the
r3 direction in simple tension
loading case ... ........................... 115

5.5. Different values C., CY for Case 2
in the r4 direction in simple tension
loading case ........................... 115


vi











LIST OF FIGURES


Figure Page

1.1 Various types of joints ...................... 4

2.1 The single lap adhesive bonded joint .... 6

2.2 Identification of the various parts of
a single lap joint ........................... 6

2.3 Systems considered by Goland and
Reissner ....... ............................. 10

2.4 The single lap joint analyzed by
Pahoja ....... ............................... 17

3.1 Double-doubler joint configuration ...... 31

3.2 Double-doubler joint geometry ........... 32

3.3 Material principal coordinate system .... 34

3.4 Displacement boundary conditions for
simple tension and pure bending ......... 38

3.5 The finite element mesh coordinates
for the high stress concentration
regions ....... .............................. 39

3.6 The layout of finite element mesh in
simple tension ................................ 41

3.7 The layout of finite element mesh in
pure bending .................................. 42

3.8 Case 1. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the central adherend and adhesive ....... 44

3.9 The different levels of y in the adhesive
strip ........ ...---........................ 46

3.10 Case 1. The stress distributions of
double-doubler reinforcement of a continous
member along the boundary between
the outer adherend and adhesive ......... 48


vii








3.11 Case 1. The stress distributions of
a double-doubler joint along the boundary
between the central adherend and
adhesive .................................... 50

3.12 Case 1. The stress distributions of
a double-doubler joint along the boundary
between the outer adherend and
adhesive ..... .................. .......................... 51

3.13 Case 2. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the central adherend and adhesive ....... 54

3.14 Case 2. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the outer adherend and adhesive ......... 55

3.15 Case 2. The stress distributions of
a double-doubler joint along the boundary
between the central adherend and
adhesive .................................... 56

3.16 Case 2. The stress distributions of
a double-doubler joint along the boundary
between the outer adherend and
adhesive .................................... 57

3.17 Case 3. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the central adherend and upper adhesive . 60

3.18 Case 3. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the upper adherend and adhesive.......... 61

3.19 Case 3. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the central adherend and
lower adhesive .......................... 62

3.20 Case 3. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the lower outer adherend and adhesive ... 63


viii








3.21 Case 3. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the central adherend and
upper adhesive ................................ 64

3.22 Case 3. The stress distributions of
a double-doubler joint along the boundary
between the upper outer adherend and
adhesive .................................... 65

3.23 Case 3. The stress distributions of
a double-doubler joint along the boundary
between the central adherend and
lower adhesive ................................ 66

3.24 Case 3. The stress distributions of
a double-doubler joint along the boundary
between the lower outer adherend and
adhesive ....... ............................. 67

3.25 Case 4. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the central adherend and
upper adhesive ................................ 70

3.26 Case 4. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the upper outer adherend and adhesive ... 71

3.27 Case 4. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the central adherend and
lower adhesive ................................ 72

3.28 Case 4. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the lower outer adherend and adhesive ... 73

3.29 Case 4. The stress distributions of
double-doubler reinforcement of a
continous member along the boundary between
the central adherend and
upper adhesive ................................ 74


ix








3.30 Case 4. The stress distributions of
a double-doubler joint along the boundary
between the upper outer adherend and
adhesive ...................................... 75

3.31 Case 4. The stress distributions of
a double-doubler joint along the boundary
between the central adherend and
lower adhesive ................................ 76

3.32 Case 4. The stress distributions of
a double-doubler joint along the boundary
between the lower outer adherend and
adhesive ...................................... 77

4.1 Axial lateral normal stress distribution
on the boundary between the central
adherend and adhesive for four values
of E/E.................................. - --83

4.2 Axial shear stress distribution
on the boundary between the central
adherend and adhesive for four values
of E/E ....................................... 84

4.3 Axial lateral normal stress distribution
on the boundary between the outer
adherend and adhesive for four values
of E/E. ............................--... 85

4.4 Axial shear stress distribution
on the boundary between the outer
adherend and adhesive for four values
of E/Ea .-------- ------........................ 86

5.1 Two edge-bonde elastic wedges of
different materials under normal and
shear loading ........................... 91

5.2 Case 1. Half plane bonded to a quarter
plane .................................... 99

5.3 Case 2. Three quarter plane bonded to
a quarter plane ---..................... 104

5.4 Specified boundary displacement method .. 108

5.5 The coarse model and sub-model
in case 1 ...... ................ .......112

5.6 The coarse model and sub-model
in case 2 .........----................... . .113


x








A.1 Case 1. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.06375 ...... 124

A.2 Case 1. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.0675 ....... 125

A.3 Case 1. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.08 ......... 126

A.4 Case 1. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.0925 ....... 127

B.1 Case 1. Stress distributions of
a double-doubler joint at the
level of y = 0.06375 ....................... 129

B.2 Case 1. Stress distributions of
a double-doubler joint at the
level of y = 0.0675 ........................ 130

B.3 Case 1. Stress distributions of
a double-doubler joint at the
level of y = 0.08 ............................ 131

B.4 Case 1. Stress distributions of
a double-doubler joint at the
level of y = 0.0925 ........................ 132

C.1 Case 2. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.06375 ...... 134

C.2 Case 2. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.0675 ....... 135

C.3 Case 2. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.08 ......... 136

C.4 Case 2. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.08 ......... 137


xi








D.1 Case 2. Stress distributions of
a double-doubler joint at the
level of y = 0.06375 ............... ..... ... 139

D.2 Case 2. Stress distributions of
a double-doubler joint at the
level of y = 0.0675 ........................ 140

D.3 Case 2. Stress distributions of
a double-doubler joint at the
level of y = 0.08 ............................ 141

D.4 Case 2. Stress distributions of
a double-doubler joint at the
level of y = 0.0925 ........................ 142

E.1 Case 3. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = -0.095 ....... 144

E.2 Case 3. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = -0.08 ........ 145

E.3 Case 3. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = -0.07 ........ 146

E.4 Case 3. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = -0.065 ....... 147

E.5 Case 3. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.065 ........ 148

E.6 Case 3. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.07 ......... 149

E.7 Case 3. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.08 ......... 150

E.8 Case 3. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.095 ........ 151

F.1 Case 3. Stress distributions of
a double-doubler joint at the
level of y = -0.095 ..................... 153


xii








F.2 Case 3. Stress distributions of
a double-doubler joint at the
level of y = -0.08 ........................... 154

F.3 Case 3. Stress distributions of
a double-doubler joint at the
level of y = -0.07 ........................... 155

F.4 Case 3. Stress distributions of
a double-doubler joint at the
level of y = -0.065 ........................ 156

F.5 Case 3. Stress distributions of
a double-doubler joint at the
level of y = 0.065 ......................... 157

F.6 Case 3. Stress distributions of
a double-doubler joint at the
level of y = 0.07 ............................ 158

F.7 Case 3. Stress distributions of
a double-doubler joint at the
level of y = 0.08 ............................ 159

F.8 Case 3. Stress distributions of
a double-doubler joint at the
level of y = 0.095 ......................... 160

G.1 Case 4. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = -0.095 ....... 162

G.2 Case 4. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = -0.08 ........ 163

G.3 Case 4. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = -0.07 ........ 164

G.4 Case 4. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = -0.065 ....... 165

G.5 Case 4. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.065 ........ 166

G.6 Case 4. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.07 ......... 167


xiii








G.7 Case 4. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.08 ......... 168

G.8 Case 4. Stress distributions of doubledoubler reinforcement of a continous
member at the level of y = 0.095 ........ 169

H.1 Case 3. Stress distributions of
a double-doubler joint at the
level of y = -0.095 ........................ 171

H.2 Case 3. Stress distributions of
a double-doubler joint at the
level of y = -0.08 ........................... 172

H.3 Case 3. Stress distributions of
a double-doubler joint at the
level of y = -0.07 ........................... 173

H.4 Case 3. Stress distributions of
a double-doubler joint at the
level of y = -0.065 ........................ 174

H.5 Case 3. Stress distributions of
a double-doubler joint at the
level of y = 0.065 ........................... 175

H.6 Case 3. Stress distributions of
a double-doubler joint at the
level of y = 0.07 ............................ 176

H.7 Case 3. Stress distributions of
a double-doubler joint at the
level of y = 0.08 ............................ 177

H.8 Case 3. Stress distributions of
a double-doubler joint at the
level of y = 0.095 ..-----.................. 178


xiv














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfilment of
the Requirements for the Degree of Doctor of Philosophy


STRESS ANALYSIS OF A DOUBLE-DOUBLER JOINT IN COMPOSITE STRUCTURE


BY

MYUNG KYUN PARK

May, 1990

Chairman: Lawrence E. Malvern Major Department: Aerospace Engineering, Mechanics and Engineering Science This study first provides a linear elastic stress analysis using the finite element method for a double-doubler

adhesive joint. The joint is modelled such that the variation of stresses and strains through the adhesive thickness can be determined. The linear elastic stress analysis is focused on three areas: first, the interfacial stress distribution

between the adherend and adhesive; second, cohesive stress distribution in the adhesive; and third, stress distribution

across the thickness of the adhesive layer at the ends of layer.

The second step is to carry out a parametric study on the double-doubler joint. Since the stress distribution in

the adhesive layer depends on several parameters (geometrical


xv














and material), an understanding of the influence of these various parameters will lead to better joint design for specific requirements.

The third step is to devise parameters that describe the singular behavior near the bonding edges between the adherend and the adhesive layer by applying the existing elastic solution for stress singularities to the double-doubler joint problem. The specified boundary displacement method is then

used to determine the values of parameters. These parameters can be used as a means of failure prediction by comparing the calculated parameters with known critical values of the parameters for the material and comparison of stresses very near the bonding corners where stress singularities occur.


xvi













CHAPTER 1
INTRODUCTION


1.1 Introduction

In the design of material structures components must be

joined in such a manner that overall the structure retains its structural integrity while performing its intended function subjected to loads and environment. Joining metallic structures is a well-developed technology involving riveting, bolting, welding, glueing, brazing, soldering, and combinations thereof. However, for polymer matrix fiber reinforced composites only adhesive bonding and mechanical fasteners (bolts and rivets) can be utilized.

Recent advances in composite structures and the new

adhesive bonding techniques based upon very strong epoxy type adhesives have made feasible the adhesive joining, stiffening and repairing of structural elements subjected to extreme enviromental and loading conditions. This type of joining and stiffening has been used in the aerospace, automobiles, electronics, sporting goods and appliance industries.

Consequently the importance of adhesive bonding in technology has been recognized. Various types of adhesive joints are shown in Figure 1.1. Among the various types of joints, the double-doubler joint is selected for this study.


1








2


Adhesive bonding provides structures having lighter weight, more fatigue resistance, improved aerodynamic smoothness,

better distribution of imposed stress, better thermal and electrical insulation and greater crack retardance over

mechanical fasteners. Therefore, adhesive bonding is very desirable for use in composite material structures.

The use of adhesives is growing rapidly but designers lack the definitive approach that gives confidence in routinely incorporating adhesive bonding into design.



1.2 Obiective and Scope

The analysis of the behavior and the failure of an adhesive joint must include both the understanding of the mechanism of adhesion, which is closely related to the

microstructure between the adhesive polymer and the adherends, and the analysis of stresses in the joint. This study will be focused on the latter part. Failure in an adhesive joint

can be either "adhesive," which occurs at the interfaces between the adhesive and the adherend, or "cohesive, " which occurs either in the adhesive or in the adherends.

Therefore the first objective of this thesis is to do

a linear elastic stress analysis for the double-doubler joint by means of finite element method (using ANSYS). The linear elastic stress analysis is focused on three areas: first, the interfacial stress distribution between the adherend and the adhesive; second, cohesive stress distribution in the








3
adhesive; and finally, the stress distributions at various levels of y in the adhesive layer near the ends of layer.

The second objective is to carry out a parametric study

on the double-doubler joint. Since the stress in the adhesive layer depends on several parameters (geometrical and material), an understanding of the influence of these various

parameters will lead to better joint design for specific requirements.

The third objective is to devise parameters that describe the singular behavior near the bonding edges between the adherend and the adhesive layer by applying an existing elastic solution for stress singularities to the doubledoubler adhesive joint problem. The specified boundary

displacement method (SBD) is used to devise the parameters. These parameters can be used as a means of failure prediction

by comparing the calculated parameters with known critical values of these parameters for the material and comparison of stresses very near the bonding corners where stress singularities occur.









4


Single Lap Joint


I


Double Lap Joint


Single Doubler Joint


Double Doubler Joint


Stepped Lap Joint


Scarf Joint


Figure 1.1 Various types of joints


I














CHAPTER 2
REVIEW OF PREVIOUS WORK


2.1 Introduction

A considerable amount of experimental and theoretical research has been carried out on adhesive bonded joints. Theoretical and experimental stress analyses of various types of adhesive-bonded joints have been made by many authors. Due to the relatively simple geometry and wide application, many

theoretical and experimental analyses have been devoted to the single lap joint. (See Figure 2.1)

In this review the important investigations of the single lap joint and known investigation of the double lap joints are included. The following review provides a history and

background of the present state of knowledge in adhesively bonded joints. First the investigations of the single-lap joint are reviewed here because the earlist analyses and the

analyses which are used as the basis for most subsequent analyses were made on the single-lap joint.

The terminology used in identifying various parts of the joint is shown for a single lap joint in Figure 2.2. The

"overlap" is the length over which the adhesive extends, and the "overhang" is the part of the adherend outside the overlap. The adhesive layer has two interfaces and two free


5







6


adherend 1


t 1 [


adhes i ve


ta


t2


-4--


adherend 2


Figure 2.1 The single lap adhesive bonded joint

Iead i ng end


interface


adherend I


adhes i ve
free edge trailing end


trailing end


leading end


~1


:adherend 2


Cover I ap V-over hang -0



Figure 2.2 Identification of the various parts
of a single lap joint


p


7


F-00-


-4 i








7
edges. The "leading end" of an adherend is where the overhang begins after the overlap. The "trailing end" of the same

adherend is at the opposite end of the overlap. Corresponding parts of the double-lap joints are identified with the same terminology. A typical single-lap joint is under tensile loading.



2.2 Analytical Studies

2.2.1 Volkersen's Theory

Classical analysis of an adhesive bonded joint was made

by Volkersen [1]. He derived the load distribution of a multi-row riveted lap joint in tension, where he idealized the rivets by replacing them by a continous medium of given

shear flexibilty, and considered the case of an adhesive bond. Bending of the adherends and the associated lateral normal and axial normal stresses were not considered. Therefore the shear stress was forced to be constant through the thickness of adhesive. By setting up the differential equations of linear elasticity, he showed that shear stress at any point

along the length of the overlap was given with respect to dimensionless quantites:

a) the overlap ratio

D = Gc2/E2tat2 2.1

b) the adherend stiffness

Eit1 + E~t,
W Eit+ ) 2.2








8
The shear stress distribution in the adhesive layer was

nonuniform in the x-direction, but assumed uniform constant in the y-direction. Peaks of the shear stress occurred at the ends of the overlap. He compared the maximum shear stress, r. at the ends of the overlap with the mean stress, rm and found the stress concentration factor, n , for joints with nonidentical adherend.



rax 1/2 (W-1)cosh[(DW/2]
1= = (D/W) 1 2.3
rm sinh[ (DW)/2

where D = Gc2/E2t2t.

W = (E1t1+E2t2)/E1t1

rm = averaged applied shear stress



This is for the case Elt1 > E2t2- He showed that the stress concentration factor increased with increasing shear modulus of the adhesive and with decreasing moduli of elasticity and thickness of the adherends.

The theory developed by Volkersen has two important shortcomings.

a) The bending deformation was neglected. (There should

be a bending moment applied to the joint in addition

to the in-plane tension, since the two applied forces

are not collinear.)








9

b) The assumption was made that the shear stress does not

vary across the thickness.



2.2.2 Goland and Reissner Theory

Goland and Reissner [2] studied the stress distribution for adhesive joints with identical adherends under the action of tensile forces at the ends of the adherends. The forces act in the direction parallel to the bonding line before any

deformation takes place, but the line of action of the tensile force R is assumed to pass through the midpoint of the

adhesive layer for the deformed equilibrium configuration. They recognized that the bending of the adherends beyond the bonded region has a significant effect on the stress distributions in the joint itself. They approached the

problem in two steps. First they determined the loads at the edges of the joint which were transmitted from the loads at the ends of the adherends. In the first step of the problem, they treated the adherends as cylindrically bent plates, and

found that forces transmitted to the edges of the joint consisted of a tensile force P, a small shear force V, and a bending moment M, as shown in Figure 2.3 in which



P R

V KP[3(l - v,2 )P/Elt 112 2.4

M = KPt1/2








10


y


t I El



L loo


-a - -a


El


V


k


adherend adhes i ve adherend


L


T

V


Figure 2.3 Systems considered by Goland and Reissner


tI

-I-


x


M p


p


M


-


E a


E a


L


C








11
where K is the bending moment factor which is the ratio of the magnitude of the edge moment M and the value of this moment in a nondeformable system:



1
K =2.5

1+(2,2)tanh( (c/4t,) !6(1-v1) P/Eiti ]



The value K=1 indicates that the adherends exhibit no deformation due to bending, and K < 1 indicates that a certain amount of bending deformation takes place.

In the second step of the problem, they assumed that the

normal stress in the adhesive parallel to the layer can be neglected, and that the lateral normal stress and shear stress do not vary across the thickness of the adhesive layer. They used a plane strain analysis and solved the problem for two limiting cases:

1) The case where the adhesive layer is so thin and stiff that its deformation may be neglected. This case requires that the condition ta/Ea << t1/El. The results show that, while the lateral normal stress is high at the edge of the joint, the shear stress is zero there. The shear stress, however, rises rapidly to a maximum value in close proximity to the edge.

2) The case in which the adhesive layer is soft and flexible and the joint flexibility is mainly due to the








12
deformation of the adhesive layer. This case requires the condition tdE >> t/El. The shear stress r and lateral normal stress a are expressed as follows.




r =Bc cosh(Bx/t) +
4c 2t, (1+3K) cosh(Bc/2t,) + 3(1-K) 2.6



a =(l/A) (P/tl) (2t./c)2 [ (R2A2 K/2 +

AK'coshA cosA)cosh(2Ax/c) cos(2Ax/c) 2.7

+ (RA2K/2 + AK'sinhA sinA)sinh(2Ax/c) sin(2Ax/c)]

where

B = (8Gt/ Elt)12

A = (c/2t,) (6Etl/Elt) 4

R, = coshA sinA + sinhA cosA

R2 = sinhAcosA - coshAsinlA

A = (1/2) (sinh2A + sin2A)

K'= Vc/2Pt,


The maximum value of r and a are found at the edges of the joint.

(t1( ) 4= [ = L - (1+3K) coth + 3(1-K) ] 2.8




-,,- c 2 _ 2K sinh2A - sin2A cosh2A + cos2A
(P/ti) 2tA 2 sinh2A + sin2A + AK' sinh2A + sin2A


2.9








13
Compared with the factor D used by Volkersen, it can be seen that Bc/2t, = (2D)12. For large values of D, coth(2D)"2 approaches the value of 1, and therefore the shear stress concentration factor, q1, can be written, for large D, as



0ii = r.,/ (p/c) ~ (1/4) (1+3K) (2D) "2 + 3 (1-K)/4 2.10


where D = Gc2/E2t2ta



The limitations of Goland and Reissner theory include a) It is valid only for identical adherends.

b) It is valid only for two special cases; the first case

is for tY/Ea << t1/El, while the second case requires

that t./Ea >> tE1.

c) The stress across the adhesive layer thickness

is assumed to be constant, which results in inaccurate predictions in the region adjacent to the edges of the

overlap.



2.2.3 Plantema's Analysis

Plantema [3] combined the Goland-Reissner theory with Volkersen's theory in the analysis of a single-lap adhesive bonded joint. The factor K from the Goland-Reissner theory

was employed to calculate the differential strain of the members due to bending, while Volkersen's theory was used to








14
calculate the differential strain of the members due to shear, and the stress at the edges of the overlap.

The shear stress concentration factor was given as



i = [D(1+3K)/2]12coth[D(1+3K)/2] 12 2.11



He did not make an attempt to calculate the lateral normal stress.



2.2.4 Kelsey and Benson Theory

Kelsey and Benson [4] employed the complementary energy method to determine the shear and lateral normal stresses in a single lap joint. A linear variation of the lateral normal stress was allowed across the the adhesive layer, while the shear stress was assumed constant across the adhesive thickness. The result was a differential equation of higher

order than that obtained by Volkersen, which enabled the boundary condition of zero shear stress at the ends of the overlap to be satisfied. The shear and lateral normal stress obtained are as follows:

Let A = pm1m2c

D = (micoshmlc/2) (sinhm2c/2) - (m2sinhmic/2) (coshm2c/2)





= (A/4D) [ (coshmlx) (coshm2c/2) - (coshmIc/2) (coshm2X) ]


2.12








15
a = (At/8D) [ (mlsinhmlx) (coshm2c/2) - (m2coshmlc/2) (sinhm2c/2) 3 2.13


The shear stress concentration factor y = T/(p/c) is given by


_ mmc2 (coshm, (xc/2) ) (coshmic/2) (coshmc/2) (coshm, (xc/2)
' ~ 4 m,(coshm1c/2) (sinhm2c/2) -m2(sinhM2c/2) (coshm2c/2)


2.14
where

M, = 2 ( 1 + m3 )
Gta


M2 = 6E. M3
-Gt a~ 1r



M3 = ( : - 2G2t. 1/2
3EaEiti


Kelsey and Benson imposed the zero shear stress boundary condition which causes a sharp drop of shear stress from its peak value at positions very near the ends to zero at ends. They reasoned that this is necessary for equilibrium because the adhesive/air boundary can not sustain a shear stress.



2.2.5 Pahoia's Analysis

Pahoja [5] made the analysis of the joint under tensile, shear, and bending loads as shown in Figure 2.4. The joint

was subjected to a general loading condition. He employed the theory of minimum potential energy to study the axial normal,








16
lateral normal and shear stresses in the adhesive layer. He defined the displacements in the adhesive in terms of

polynominals with m terms for the displacements in the x direction and n terms for the displacement in the y direction. The total energy was expressed in terms of the displacement components in series form. By minimizing the total potential energy of the system, a system of governing linear ordinary

differential equations and natural boundary conditions were obtained. The equations were reduced to algebraic and first

order differential equations by substituting new variables for the displacements and their derivatives. By solving these equations numerically, the stress distribution and stress concentrations were found. The stress concentrations for identical adherends have the follwing forms:



Shear stress

(Tmax/ av) = 1 + (6/7) [ (19/18) - Vf] (Sf - 1)

for Vf < 1 and Sf > 4

2.15
(rTx/Tav) = -1 + (18/19) (Vf - 17/18) (Sf - 1)

for Vf > 1 and Sf > 5



Lateral normal stress

(ay)./rav = (t2/c) (1.4 - 1.675 Vf) Nf2 2.16









17


y


1

1t2


ta--)


I x


adrer adhes i ve


M2

V

end I I E2
P
E a Ii


adherend I El


V


P




M1'


M2'

VP






x


Figure 2.4 The single lap joint
analyzed by Pahoja


I!


ti


Li


P





M1


I


I
If


Y


t








18


Axial normal stress

(L7)max/ra,, = (t1/c) (0.85 - 0.95Vf) Nf2 2.17



where

Vf V c
Vf = T [ta + Q, t, (t + (4/3)tl


Sf = (c/tl) (Eti/Eta)1/2 Nf = (c/tl) (Et/Eita) 1/4


Q= EJ/(l-v12)


S= Qta2 /6 + Qitita + Q1t12


Q = E/(1-v2


Pahoja concluded that the single lap joint with identical adherends was the best design. This joint was most efficient when loaded such that the shear stress distribution was uniform, i.e., Vf = 1. He compared his theoretical results with the results of photoelastic analysis, and these are in good agreement. The limitation of this analysis is that the free boundary condition at the ends of the adhesive were violated just as in Volkersen's and Goland and Reissner's analyses.








19


2.2.6 Renton and Vinson's Analysis

Renton and Vinson [6,7] made the most complete analysis of a single lap joint. They clearly identify the extent of the influence of the various parameters on the stress distribution in the adhesive and adherends. The analysis was very general and applicable to isotropic and anisotropic

adherends as well as to dissimilar and similar adherends. The method of analysis that they developed involve the following assumptions and/or limitations:

1) The laminated adherends are symmetric about their own

midsurfaces (i.e., no bending stretching coupling).

2) Each ply or lamina in each adherend is orthotropic 3) The effective elastic mechanical properties of the

adhesive are accounted for.

4) Both shear and lateral normal stresses are accounted

for in the adhesive; they vary in the load direction,

but do not vary in the thickness direction.

5) The adhesive thickness is much smaller than the

adherend thickness, i.e., ta << t, and ta << t2

6) Shear deformation and lateral normal strains are

accounted for in each adherend.



Composite material laminated plate theory was used as the building block or starting point for developing the method of analysis. The lap joint structure was divided into five portions for analysis: the adhesive, the adherends on either








20
side of the adhesive, and the adherend portions from the end of lap joint to the load application. Twenty-six boundary conditions are specified and satisfied.

The analytical solutions obtained for stresses and

deformations throughout the structure were programmed on computer for easy calculation. After developing a valid closed form solution for the single lap joint problem, they ascertained the influence of certain important parameters on the stress distributions in the adhesive.

The influence of the joint parameters on the stress distributions in the adhesive are summarized below:



1) The greater the difference in the moduli of the

adherends, the higher is the magnitude of the maximum

shear stress.

2) For dissimilar adherends, the maxima of both the

shear and the lateral normal stresses in the adhesive occured at the leading end of the less rigid adherend.

3) The greater the ratio of the moduli of the adherend

and the adhesive, the more uniform the shear stress

distribution.

4) The larger the ratio of the moduli of the adherends

and the adhesive, the smaller is the magnitude of the

maximum lateral normal stress.








21
In the design of single lap adhesive bonded joints, they recommended that

1) The adherends of joints should be identical.

2) The modulus of the adherends should be high compared

to that of adhesive.

3) The length of overlap should be about ten times the

minimum thickness of the adherend.

4) The joint's intended function should influence

adhesive selection: in static bodies, adhesive

ultimate strength should be high; in fatigue, adhesive

fracture toughness should be considered.



2.2.7 Hart-Smith's Analysis

The analysis of a double lap joint by Hart-Smith [8] was

designed to calculate the joint efficiency of a double lap joint in terms of the strength of the joint.

The adhesive was characterized as elastic-plastic in shear. In the analysis, the shear stress distribution was constant across the adhesive thickness.

In his analysis, the deformation of the adhesive was greatest at the ends of the overlap, and consequently the behavior of the adhesive was characterized as plastic at the ends of the lap.

In the elastic analysis, the shear stress in the adhesive was expressed in terms of the displacements in the outer and center adherends. The differential equation governing the








22
shear stress distribution was determined from the stress strain relationship and the force-equilibrium equation.

In the elastic-plastic analysis, the plastic shear stress and the distance over which the adhesive was plastic were determined from the boundary condition for the plastic region, the equation ensuring the continuity of the elastic and

plastic shear strains, and the gross horizontal equilibrium equation of the joint. The expression for the plastic shear

stress was then rearranged to express the applied load in terms of the shear strain energy per unit area of the adhesive layer.

The lateral normal stress in the adhesive was considered to be independent of the shear stresses. The analysis for the lateral normal stress was applicable only when the adhesive

was deformed beyond the elastic range in joints with long overlaps. The lateral displacement of the outer adherends was related to the moment at the ends of the overlap by the theory of bending of plates.

The coefficients of the normal displacements, which were expressed in terms of trigonometric and hyperbolic functions, were determined from the boundary conditions in terms of the

maximum shear stress. The lateral normal stress was then given in terms of an "effective" tension modulus of the adhesive, whose value is difficult to establish theoretically. The conclusions drawn by Hart-Smith on the stress distributions are








23
1) The highest shear stress in the adhesive was at the

end of overlap near the leading end of the less stiff

adherend.

2) For long overlaps, the magnitude of the maximum shear

stress did not vary with length; only the stress in

the central portion of the joint was lowered by

increasing length.

3) Lateral normal stresses were reduced by using a thin

outer adherend, or an adherend with a low extensional modulus, or a thick adhesive layer. Thus, the lateral normal stresses can be reduced at the expense of the

shear stresses.

4) The characteristic extent of the end zones was

independent of the length of overlap.



2.3 Finite Element Studies

2.3.1 Introduction

The use of finite element methods of analysis means, in principle, that the simplifying assumptions made in the classical analytical methods need not be made. The technique does allow the solution of problems that are totally intractable by classical methods. Basically, one can choose between a large number of primitive elements, i.e., elements

in which strain is assumed to be constant, or a smaller number of more sophisticated elements, ones with linear or quadratic strain variation. Whatever idealization is adopted, a higher








24
density (finer mesh) of elements is necessary in regions of rapidly varying stress.

Finite element analysis of an adhesive-bonded joint was

first performed in 1969. Considerable work has been done with finite element analysis of adhesive-bonded joints of fiber reinforced composite materials.



2.3.2 Ahluwalia's Analysis

The first finite element analysis of an adhesive bonded

joint was made by Ahluwalia [9]. He analyzed a double-lap joint. The finite element solution for the double lap joint was for plane strain with 120 nodal points and 188 triangular elements. The adhesive was divided into ten equal elements

along its length and was made of one element across the thickness. Consequently, the high magnitude of the stresses at and near the ends of the overlap were not correctly

determined, and the stresses did not vary across the thickness of the adhesive. Smaller elements were not defined at the

ends of the overlap where shear and lateral normal stress concentrations exist. The shear and lateral normal stress distributions were nonuniform with peaks at the ends of the overlap. Large sizes of elements in areas of very sharply rising peak stresses result in lower magnitude of the calculated stresses.








25


2.3.3 Wooley and Carver's Analysis

Wooley and Carver [10] analyzed a single lap adhesivebonded joint with the finite element method. The finite

element solution for the single lap adhesive-bonded joint was for plane stress problem with 497 nodal points and 394 elements.

The solution was based on the computer program "ARBAS" by E.L. Wilson of the University of California at Berkeley.

The locating of nodal points was accomplished by dividing the configuration into three parts, the section of the lap joints, and the sections to each side of the lap. In the program,

quadrilateral elements, composed of four triangular elements, were used to describe the joint. The displacements of the triangular elements were described by linear functions which resulted in a constant strain and, therefore, constant stress over each element.

The adhesive layer was divided into two equal elements

through its thickness and twenty equal elements along its length. The maximum aspect ratio of a quadrilateral element in the adhesive was 100:1. The single-lap adhesive-bonded joint was made of identical adherends with an elastic adhesive. The Poisson's ratios of the adhesive and adherends

were 0.3 for all the cases. The range of the ratio of the elastic moduli of the adherend and adhesive, E/Ea was from 0.1 to 1000. The range of the ratio of the thickness of the adhesive and adherends, t8/t was from 0.02 to 0.1. The length








26
of overlap was taken as 5, 10 and 20 times the adherend thickness.

The stress distributions at the adhesive-adherend

interface were not given. The stress distributions at the mid plane of the adhesive layer were nonuniform with peaks near

the ends of the overlap for both shear and lateral normal stresses. They compared the stress concentrations obtained by each method at three values of the geometric and material

parameters. With two elements across the adhesive thickness, the variation of the stresses across the adhesive thickness can not be adequately represented.



2.4 Experimental Studies

2.4.1 Mclaren and MacInnes's Investigation

Mclaren and MacInnes [11] conducted photoelasticity experiments on single lap joints. The object of their

investigation was to verify the mathematical analysis of Goland and Reissner. Two series of tests were performed. In

the first series, the complete joint was made of Araldite. In the second series of tests, the adherends were made of aluminum and the adhesive of Araldite. The ratio of the

modulus of the adhesive to that of the adherend was 1/20 in the second series of tests. The results were presented in terms of the fringe order, and no attempt was made to determine the magnitude of the stresses. The difference

between these two model systems was in the transfer of load








27
from one adherend to the other. The adhesive in the second series was in longitudinal shear across its entire thickness; thus, the transfer of load was primarily by shear. Their

investigation supports the theory of Goland and Reissner in that the qualitative aspects of the theory are well corroborated.



2.4.2 Tuzi and Shimada's Investigation

Tuzi and Shimada [12] conducted a photoelastic

investigation of a single lap joint. Their investigation concentrated mainly on stress concentration in the joint. Single lap joint models having different adhesive shapes at the ends of overlap; rectangular type, with a fillet and with convex and concave semi-circular ends were used in the photoelastic investigation.

The adhesive layer of the joint was made of epoxy rubber and the adherends were represented by epoxy resin and duralumin. In the first experiment, the adherends were made of epoxy resin KT-102 and the adhesive layer was made of an epoxy rubber. The adhesive was bonded to the adherends by Araldite D. In the second experiment, the adherends were made of duralumin, the adhesive layer was made of epoxy resin KT102, and the adhesive was bonded to the adherends by Araldite D. Their experimental results indicated that the joint with concave semicircular ends was the best, the one with rectangular ends was second best, and the one with convex








28
semicircular ends was the worst as regards the stress

concentration in the joint. They also studied the stress distributions in scarf and butt joints photoelastically. Their photoelastic models were made of epoxy resin and epoxy rubber and also epoxy resin and metal. Tension and bending tests were carried out on bars bonded at various angles. For the joints with a soft adhesive layer bonded to hard adherends, the stress concentrations were maximum at the ends of the bonded boundary with an obtuse angle. For the joints

with a hard adhesive layer bonded to soft adherends, the stress concentrations were found at the ends of the bonded boundary with an acute angle.













CHAPTER 3
LINEAR ELASTIC STRESS ANALYSIS


3.1 Introduction

The joint under consideration is a double-doubler joint (Figure 3.1). This joint is subjected to two different loading cases (constant tensile loading and pure bending). In this chapter the three materials (central adherend,

adhesive, outer adherend) are assumed to behave in a linear elastic manner under the applied loads. The elastic

properties, Young's modulus and the Poisson's ratio are E, and VC for central adherend; E, and v. for the adhesive layer; E0 and v, for the outer adherend. The material properties are extended to orthotropic material properties for the application to composite adherends.

The overall problem is to find the stress distributions in the double-doubler adhesive bonded joint under the applied load. The distributions along the axial direction of the stresses on the interface between the central adherend and adhesive, and of the cohesive stresses at various levels in the adhesive are the particular interest in this study. The

analysis can be simplified to either a plane stress or a plane strain problem. For most practical purposes the width of an

adhesive joint is large compared to the thickness of the


29








30
adherend and the adhesive. Therefore the plane strain

analysis option is used throughout. First of all linear

elastic stress analysis for double-doubler reinforcement of a continuous member (Figure 3.la) was carried out. The

results, are compared with the previous experimental results and they show good agreement. After that the linear elastic stress analysis for a double-doubler joint (Figure 3.lb) was investigated. The finite element method of stress analysis is applied to this plane problem involving different

materials. The reason why the finite element method is chosen over any other method is that geometrically complex bodies of

different materials can be easily represented and arbitrary distributions of displacement and stress boundary conditions can be specified.

For this linear elastic stress analysis the finite element program ANSYS is employed in the finite element studies of the double-doubler joint.



3.2 Joint Description

The geometry and dimension of the joint are shown in Figure 3.2. The basic material properties used in this joint are given in Table 3.1.














outer adherend




central adherend


y


adhesive


~~II~J


-A +


a) the double-doubler reinforcement
of a cont inous member


outer adherend central adherend


\___ I


y


I! /


/


I


I1


adhesive


x


b') the double-doubler joint


Figure 3.1 Double-doubler joint configuration


31


x


11


I









32


Y


ta = 0. 04", 2t 0. 12"


_____________________I _______________________


-J


0 . Oe,"


I_____


0. 02"

c= 2.56"


4.0"


Figure 3.2 Double-doubler joint geometry








33


Table 3.1. Elastic Material Properties Used in Analysis Mateial Modulus,E Poisson's ratio Shear Modulus,G

aluminum 10.6E6 psi 0.33

epoxy 3.6E6 psi 0.39

graphite 22.8E6a psi 0.3c 0.81E6d psi
epoxy 1.58E6b psi

a: Longitudinal modulus of unidirectional composite i.e.
fiber direction
b: Transverse modulus of composite
c: Major Poisson's ratio of unidirectional composite
d: Inplane shear modulus


3.3 Determining Material Properties

The strain-stress relations for anisotropic materials can be expressed as

C= si aJ ij = 1,2,-----,6 3.1

where ei are the strain components, sij are the elements of the compliance matrix, and aj are the stress components. Equation 3.2 gives the form of the relations for a unidirectional fiber reinforced lamina with axes x,, x2,x3 aligned with the principal material directions as shown in Figure 3.3. (xl corresponds to fiber direction, x2 is the inplane coordinate perpendicular to x1, and x3 is the tranverse coordinate which is perpendicular to the x1-x2 plane.)


C1 Sl S12 S13 0 0 0 i1
C2 S12 S22 S23 0 0 0 02
C3 S13 S23 S33 0 0 0 03 3.2
E4 0 0 0 S4 0 0 04
E 0 0 0 0 S55 0 a5
C6 0 0 0 0 0 S66 06








34


Figure 3.3 Material principal coordinate system


2


X X X I








35
The compliances can be expressed in terms of engineering constants as follows:

S11 1/Ejj S12 = -i2/E S13 = -va/El

S22 = 1/E22 S23 =-23/E22 S33 = l/E33 3 . 3

S4 = 5/G12 S55 1/G23 S66 = 1/G13

where

Ell, E22, E33 are Young's moduli in 1,2 and 3 directions

respectively.

Vi is Poisson's ratio determined from contraction

in the xi direction during a tensile test in the xi

direction

Gij is shear moduli in the xi-xj planes



We know that the following relationships exist between the Young's moduli and the Poisson's ratios due to symmetry of the compliance

Ellv21 = E22"12, E22/32 = E331/23, E33"13 = E L 3.4 Thus for an orthotropic material, constitutive relations are


1/E11 -vL1/El -iV13/E11 0 0 0 a1

C2 -/12/El l/E22 -1/23/E22 0 0 0 U2

f3 = -13/Ell -v23/E22 1/E33 0 0 0 a3

C4 0 0 0 1/G12 0 0 a4

f 0 0 0 0 1/G23 0 r5

C 0 0 0 0 0 1/G13 a6


3.5








36
For a transversely isotropic material in which the x2-x3 plane is the plane of isotropy, we find E22 = E33, G13 = G01, V12 = V13 and G23 = E22(+L23). If we invert Equation 3.5, for a material having tranverse isotropy relative to the x2 - X3 plane, we obtain the elastic stiffness, C%, in terms of engineering constants with the following results: C11 = (1 -V223) Ej/V

C13 = C12 =V12(1 + V23) E22/V C23 = (V23 + L 2E22/El) E22/V

C33 = C22 = (1 - v 12E22/Ell) E22/V 3.6

C44 = G23 = E22/2 (1 + L23)

C55 = C66 = G12

where V = [ (1 + V23) (1 - "23 - 2v 212E22/E11)]

For the plane strain case,E1, E22, E33, L2, '13, L23, G12, G13, G23 would be used as input data for ANSYS. If G12,G13,G23 are not specified, they are computed as: G12 = E1E2/(E1 + E2 + 2V21El)

G23 = E2E3/(E2 + E3 + 2v32E2) 3.7

G13 = E1E3/(E1 + E3 + 21/13E,)



3.4 Finite Element Model Representation

Only one quarter of the model is selected for analysis

of the simple tension loading case, because it has double symmetry in geometry, materials, boundary, and loading conditions. One half of the model is chosen for the pure bending loading case.








37
The ANSYS two-dimensional isoparametric element which accomodates orthotropic material properties is used. The element is defined by four nodal points, each having two degree of freedom. The plane strain option is used. The displacement boundary conditions imposed on simple tension and pure bending case are shown in Figure 3.4. In simple tension

cases analyzed in this study, the double-doubler joint is subjected to 50 lb at the outer end of the central adherend. In pure bending case, a linear varying force boundary condition equivalent to a couple is used at the outer end of the central adherend. For the input of the orthotropic material properties the classical thin plate lamination theory is used to calculate the effective material properties for the laminate. For unidirectional composite lamina, properties in

the thickness direction are calculated assuming that E22 = E33, M12 = 1/13 and &23 = matrix Poisson's ratio. The material principal coordinate system (1,2,3) refers to the fiber and two transverse matrix directions, respectively. (See Figure 3.3) The computer program can be used to calculate effective material properties for composite laminates. The mesh is so generated that around the bonding

corner and on the boundary between two different materials the grids are much finer than those of other locations for better

results in these regions of high stress gradients. The finite element dimensions which are used in the analysis for the high stress concentration regions are shown in Figure 3.5.








38


y


pe b dryn rrrn m rr t on ldrrrinx
aD imposed boundary condition on simple tension loading


Irr


IC

A/-,


y


roI Ing support


u X= 0


x


b0 imposed boundary condition on pure bending






Figure 3.4 Displacement boundary conditions for
simple tension and pure bending


C0
10 C0











0.0925 0.08 0. 0675 0.06375
0.06
0.05625 0.0525

0.045


y


A


2.54 2.56 2.58


outer adaerend


adhesive


2.55 2.57


central aanerena


0.0925
-9. 08


.- .


x


unit = inch


0.0675


-0


.06375 J.06 05625 .0525

.045


_-__9 015
ILIE -.03
0.015

o.193 0.2 0.207
0.1965 0.2035





Figure 3.5 The finite element mesh coordinates for the high
stress concentration regions


39


ad res i e


-4- -4- - - - - i -


u.
n

0








40
A total of 1591 nodes and 1456 elements for the simple tension case, and 2819 nodes and 2672 elements for the pure bending case are used. (See Figure 3.6 and Figure 3.7) The numbers in circles relate to the nodal-point numbers. Those within the mesh relate to element numbers. The model is constructed using ANSYS STIF42.



3.5. Finite Element Results

In all the stress distribution results, the stresses are normalized with respect to the average applied axial stress,

Pa,' acting on the central adherend. Accordingly a = ay = ay/pav, gy = ay/pav, while x = x/c.

where ax: axial normal stress

ay: lateral normal stress

ao : shear stress

Pav: applied average stress

c: 2.56 inches



For nodal stress, the stresses at each node are computed by averaging the nodal stresses of the node being processed for

all elements that are connected to that node. In other words, the component nodal stresses that are automatically stored per element are averaged at a node whenever two or more elements connect to the same node. For example average nodal stress ax at the node N can be calculated as below.





















1365 1456




1181


241 121



1


-f


Figure 3.6 The layout of finite element mesh in simple tension


41


159


B1~


RB47


92



93


a










42


y


1_193-.


I-,


2075





.x


372)


2


Figure 3.7 The layout of finite element mesh in pure bending


2448


745


251 2672


-r 4


737
545


736


EtE


92








43


elements From element 1 ax at node N
N From element 2 ax at node N
element2 element3 From element 3 ax at node N


average nodal stress = (U1X + 02x + 03)/3 Four cases are considered in the following four subsections. Case 1 and 2 are for two different choices of adherends with simple tension loading. Case 3 and 4 are for the same two choices of adherends as case 1 and 2, but with pure bending loading. Each of the four cases is treated first for doubledoubler reinforcement of a continuous member, and then for a double-doubler lap joint.



3.5.1 Case 1. Aluminum Adherends and Epoxv Adhesive in
Simple Tension

The model for the first case is composed of aluminum adherends and an epoxy adhesive. For double-doubler reinforcement of a continuous member (Figure 3.la), the axial distribution of the lateral normal stress (cy), and the shear stress (axy) along the boundary between central adherend and adhesive are plotted from the averaged nodal stress values. The reason why the averaged nodal stress values are used is

that the lateral normal (ay) and shear stress (axy) are continous across the interface. These stress distributions

are shown in Figure 3.8. From these graphs and the known symmetry about the mid-length, we can see that the stress distributions (ayay) along the axial direction are very










44

0.28


V)
V) 0.2
U]


0.2
-j


Cr 0.16
0









< -0.12
a

- J

- 0.01
U]
C -
uLJ
N o.o
-j


0
Z -0.02
0 0.2 0.4 0.6 0.8
NOMALIZED AXIAL COODINATE x/c
0


-0.01


V) -0.02
U) UIJ



S-0.04


V) -0.06
M
LU
N -0.06


-0.07

0
Z -0.00


-0.090 0.2 0.4 0.6 0.8
NORMAL IZED AXIAL COORIDINDATE x/c


Figure 3.8 Case 1. The stress distributions of double-doubler
reinforcement of a continous member along the
boundary between the central adherend and adhesive








45
uniform and at a very low level for -0.75< x/c <0.75. But near the free edges (for 0.75 < x/c < 1 and -1 < x/c <-0.75) we can see sharp variation of stresses and see the peak stresses at the free end of adhesive. Different peak stress

values are obtained depending upon the size of the mesh element. This means that calculated stress fields near this area are not quite reliable.

In linear elastic analysis, stress fields near the two bonding edges show singular behavior. In order to describe the stress fields near these corners, singular stress parameters are used in Chapter 5.

The cohesive stress distributions along the axial

direction at different levels of y of the adhesive were obtained. These different levels of y in the adhesive strip is shown in Figure 3.9. According to the results which are shown in Appendix A, the axial distribution of shear stress

(ay) at the level of y=0.06375 shows a similar trend to that in the stress distribution along the boundary between central

adherend and adhesive. The axial distribution of axial normal stress (a,) at the same level of y=0.06375 shows a similar trend as that in the stress distribution along the boundary between central adherend and adhesive, except that the axial

normal stress distribution drops toward zero near the free end of the adhesive layer. The axial distribution of lateral normal stress (ay) near the free end shows a similar trend to that in the stress distribution along the boundary between









46


y


outer adherend A 0. OW5


>.O_ adhesive
0.0I75


. 375 1


0.06


central adherend


x


0.07


0.06






-0.0.
-0.07


outer adherend


central adherend


0 .09


7171


SI I17


Figure 3.9 The different levels of y in the adhesive strip


Ac


y


-0.1
outer adherend


=


i


i








47
central adherend and adhesive, except that the peak stress value becomes lower than previous peak stress on the boundary.

The axial distribution of axial normal stress (a,) and lateral normal stress (ay) at the higher level (y=0.0675) than the previous case (y=0.06375) shows a similar trend, except that the peak stress values are lower than ones of the previous case. The axial distribution of shear stress (an)

near the free end zone, which is taken from the level of y=0.0675, rises steeply, far beyond its average level, and drops steeply near the free end to near zero. These stress

distribution trends are similar to the case which is taken from the upper level of the adhesive at y=0.08. The stress distribution at the level of y=0.0925 shows same trend as the

previous case except that the lateral normal stress (ay) changes its sign and drops to a negative value which means compressive stress, immediately after the peak positive value in the end zone.

The axial distributions of stresses (ay, anY) along the boundary between adhesive and outer adherend are plotted in the same way as for the boundary between the central adherend and adhesive. The results are shown in Figure 3.10. They show that the shear stress (an) drops to near zero near the free end. The lateral normal stress (ay) shows that after the peak positive stress value near the end zone, it suddenly drops to the negative value which means compressive stress.

We find that along the boundary between the central or

















U) U)
w


_j





0
z

-j











C




0
z


0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


Figure 3.10


Case 1. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the outer adherend and adhesive


48


0.04



0.02



0



-0.02



-0.04



-0.06



-0.08



-0. 1



-0.12




0




-0.01




-0.02




-0.03




-0.04




-0.05




-0.06




-0.07


V) U) U)








0




z


0 0.2 0.4 0.6 0.81
NORMALIZED AXIAL COORDINATE xlc
-


0


i








49
outer adherend and adhesive the stress is not well defined at

the ends of adhesive. Therefore one would not expect the free end condition (a,=0, ax,=0) to be satisfied at the very near end zone. However, the end condition (aXY=a1=0) at the various levels inside the adhesive, which means no external traction

force in the x or y direction, is almost satisfied even by using the displacement finite element method .

To check the stress variation through the thickness

direction of the adhesive, the interlaminar adhesive layer is divided into four longitudinal strips as shown in Figure 3.9. The results from the data plotted for various levels of y in

Appendix A show that through the adhesive thickness the shear, axial and lateral normal stresses are pretty uniform through the adhesive thickness within the zone -0.975 < x/c < 0.975, but near the free end zones (0.98 < x/c < 1 and -1 < x/c < 0.98) the stress changes rapidly through the adhesive

thickness. If we assume that the stresses are uniform through the adhesive thickness, we can not predict the compressive lateral normal stress of the adhesive layer.

For a double-doubler lap joint (Figure 3.lb), the stress distributions on the boundaries between central adherend and

adhesive and between outer adherend and adhesive are shown in Figure 3.11 and Figure 3.12, respectively. The axial

distributions of stress at different levels of y are shown in Appendix B. From the results we find that the axial

distribution of the lateral normal stress (ay) , axial normal























m










z
















0
z


0 0.2


AXIAL


0. i


COORDINATE


Figure 3.11 Case 1. The stress distributions of a doubledoubler joint along the boundary between

the central adherend and adhesive


0.3


50


I)


0.25 0.2


0.15 0.1.


0.05


0


-0.05



-0.1


-0.15


-0.2


0


-0.01


-0.02


-0.03


-0.04


-0.05


-0. 06


-0.07

-0.06


-0.09




-0.11


-0.12


-0.13


I I I I I I I
0 0.2 0.4 0.6 0.8 1


AXIAL COORDINATE x/c





















-


E













N



0--


0.6


0.8


x/c


I

















0.04


0.2


AXIAL


0. 4


0.6


COORDINATE


0.x

x/c


Figure 3.12


Case 1. The stress distributions doubler joint along the boundary the outer adherend and adhesive


of a doublebetween


-r

















o 0.2 0.4 0.6 0.81


AXIAL COORDINATE x/c


51


r34 E)





0
z


0.02



0



-0.02



-0.04



-0.06



-0.08



-0.1.



-0.A2


0


-0.01




-0.02




-0.03




-0.04




-0.05




-0.06 -


U) E




















zi


-0.07


0


1


1








52
stress (a.) and the shear stress (ay) are very uniform and at a very low level between 0.3 < x/c <0.8 and -0.3 < x/c <-0.8. But near the corner edge of adhesive line and bonded corner at the butted region between two central adherends, there is a very high stress concentration and very complicated stress

distribution. The peak stress values obtained from the finite element method are not quite reliable, since the peak values vary depending upon the mesh size. The stress distributions,

except near the bonded corner in the butted region, show similiar trends to the ones of double-doubler reinforcement of a continous member. Near the bonded corner of the butted region, the shear stress (aY) is very large. The lateral normal stress (ay) has its highest peak value at the edge of the interface between central adherend and adhesive. This

lateral normal stress (ay) becomes compressive at the level of y=0.0925 which is very near the interface between adhesive and outer adherend, and at that interface it shows very high compressive lateral normal stress (ay).



3.5.2 Case 2. Aluminum Central Adherend, Epoxy Adhesive,
and Composite Outer Adherend in Simple Tension

The model for the second case is composed of aluminum central adherend, epoxy adhesive and graphite/epoxy composite outer adherend. A unidirectional graphite/epoxy laminate is used for outer adherend.








53
For double-doubler reinforcement of a continous member, the axial distribution of the cohesive lateral normal stress (a.), shear stress (ay) and the axial normal stress (a,) are plotted from the averaged nodal stress values and are shown in Appendix C for various levels of y. The axial

distribution of the lateral normal stress and shear stress along the boundary between the central adherend and adhesive are plotted from the averaged nodal stress values stress and

are shown in Figure 3.13. The stress distributions on the boundary between the adhesive and outer adherend are plotted in the same way and are shown in Figure 3.14.

For a double-doubler joint, the axial distributions of the cohesive stress at different levels of y are plotted and

shown in Appendix D. The stress distributions on the boundary between the central adherend and adhesive, and the outer adherend and adhesive, are shown in Figure 3.15 and Figure 3.16, respectively. The stress distribution trends are similar to Case 1, but higher peak stress values are obtained from Case 2.



3.5.3 Case 3. Aluminum Adherends and Epoxy Adhesive in
Pure Bending

The model for Case 3 is composed of aluminum adherends and epoxy adhesive and is under pure bending in the positive

sense (concave upward). A linear varying force boundary
















()
L3



U)

_j



0
z

-j

LU
H

-j

0

N
-j



z


0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


Figure 3.13


Case 2. The stress distributions of double-doubler reinforcement of a continous member along the
boundary between the central adherend and adhesive


0.16


54


0.14


0.12 0.1 0.08 0.06 0.04 0.02


0


-0.02


-T














0 0.2 0.4IA 0.6 0.9 iO"X
NORMALIZED AXA CORDNT x/


-


0


-0.01


V) V)
LU




H
V)




U)

0

N
-



z


-0.02


-0.03


-0.04


-0.06


-0.06


-0. 07


-0. 08


-0.09


1


- -















U) U)
LU


_j


U)

-J

_j



-J
N


-j

0



z


0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


1


Figure 3.14


Case 2. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the outer adherend and adhesive


55


0.04




0.02




0




-0.02




-0.04




-0.06


U)
U)

I
V)


r

U)


N




0
z


0 0.2 0.4 0.6 0.8 1
NORMALIZED AXIAL COORDINATE x/c
-


0



-0.01



-0.02



-0.03



-0.04



-0. 05



-0.06



-0.07



-0.08
















U)

U)








-J





C
z


0 0 .2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


i


Figure 3.15


Case 2. The stress distributions of a doubledoubler joint along the boundary between the central adherend and adhesive


56


I I ~ I I I I I I I I I
O 0.2 0.4 0.6 0.8 1
NORMALIZED AXIAL COORDINATE x/c


















-


0.3






0.2






0.1






0-


-0.1


~1


0




-0.02
U) U)

F- -0.04
(I)




0.06

0
N
_J -0.08

0

z
-0.1i










57


cn
(I)
w
cr
I
U)
-J


0
z
-J
cr






z uJ
H


N




0
z


0.04 0.02






-0.02


-0.04



-0.06


-0.0D8


-0.1


0









-0.02






-0.04-0.06






-0.00 -


i


Figure 3.16 Case 2. The stress distributions of a doubledoubler joint along the boundary between
the outer adherend and adhesive


0 0.2 0.4 0.6 0.6
NORMALIZED AX IAL COORD INATE x/c


U) U)
w



Cr
U)



0

N
-



z


I I I I I I I 1 - 1
0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


4r---


i








58
condition equivalent to a couple is used at the outer end of the central adherend.

For double-doubler reinforcement of a continous member,

the axial distributions of the cohesive stress are plotted from the averaged nodal stress values and shown in Appendix E. We have four overlap interfaces in the pure bending case. The axial stress (a,, aY) distributions on the boundaries between the central adherend and upper adhesive and between the upper outer adherend and adhesive are plotted from

averaged nodal stress values and shown in Figure 3.17 and Figure 3.18, respectively. The axial stress distributions (ay, aY) on the boundaries between the central adherend and lower adhesive and between the lower outer adherend and adhesive are shown in Figure 3.19 and Figure 3.20,

respectively. As expected, the shear stress is symmetric with respect to the mid plane, while the lateral normal stress is approximately antisymmetric.

For a double-doubler lap joint, the axial distributions of the cohesive stress for various levels of y in the upper and lower adhesive are shown in Appendix F. The axial stress distributions (ay, a1,y) on the boundary between the central adherend and upper adhesive and between the upper outer

adherend and adhesive are shown in Figure 3.21 and Figure 3.22, respectively. The axial stress distributions (aY, a,) on the boundary between the central adherend and lower








59
adhesive and between the lower outer adherend and adhesive are shown in Figure 3.23 and Figure 3.24, respectively.












U) U)







0
z
-j




r-)


N




0
z


0.12


0.1 0.08 0.06 0.04 0.02


0


0 0.2
NORMALIZED


Figure 3.17


0.4 0.6 0.8 AXIAL COORDINATE


x/c


I


Case 3. The stress distributions of double-doubler reinforcement of a continous member along the
boundary between the central adherend and upper adhesive


60


0.05


0


-0.05


-0.1


-0. 15


-0.2


-0.25


-0.3


-0.35


0 0.2 0.4 0.6 0.8 L.
NORMALIZED AXIAL COORDINATE x/c


U]
Ln U)


I

U)


N




0
z















()
V) V)









-LJ LU



N





0
z





V)
U)


V) U)

u-)
N





z


0. 2
NORMAL IZED


0.4
AX I AL


0.6
COORD I


0. 8
NATE x/c


I


Figure 3.18


Case 3. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the upper outer adherend and adhesive


61


0.08


0.06 0.04 0.02


0


-0.02


-0.04


-0.06


-0.08


-0.1




0.12




0.1




0.08




0. 06




0.04




0.02




0


0 0.*2 0.1 0.6 0.81
NORMALIZED AXIAL COORDINATE x/c


0












62


o.35



0.3 -


En U)
w


_j


U)




-J





0
w
N


0


0.25



0.2 0.1s



0. 1 0.05



0



-0.05


0.12 0 1 0.00 0.06





0.04





0.02


0


0.2


0.4 0.6
AXIAL COORDINATE x/c


0.8


I


Figure 3.19 Case 3. The stress distributions of double-doubler

reinforcement of a continous member along the
boundary between the central adherend and lower
adhesive


0 0.2 0.4 0.6 0.8

AX IAL COORD INATF x/ c


(I) U/)

I


U)



0
w
N
-j


/















-
























U3











LU



0
Lu
N


0.2


0.4


0.6


0.8


AXIAL COORDINATE x/c


Figure 3.20 Case 3. The stress distributions of double-doubler

of a continous member along the boundary
between the lower outer adherend and adhesive


63


0.1 0.08 0.06



0.04 0.02



0


-0.02


-0. 04


-0.06


-0.08





0.12





0.1





0.00





0.06





0.04





0.02


LuJ NI
Hj




0



z


0 0.2 0.4 0.6 0.8


AXIAL COORDINATE x/c


0


0


i










64

0.05

V)
U) 0


V)
-J
< -0.1




-J
< -0.2
uJ
H F

uJ

N -J

-0.4
0
z

0 0.2 0.4 0.6 0.e
NORMALIZED AXIAL COORDINATE x/c
0.14



0.13

U)

0.
U)


I

< 0.08
U)

0 0.06
N
-J
< 0.04

m 0
z
0.02




0 0.2 0.4 0.6 0.8 1
NORMALIZED AXIAL COORDINATE x/c


Figure 3.21 Case 3. The stress distributions of a doubledoubler joint along the boundary between
the central adherend and upper adhesive











65


U) U)



U)





0
z

-J










N





0
z









V)
V)

I
U)
CE uJ

U)

0
U] IN
-J



0
z


0.12 0.1


0.08 0.06 0.04 0.02



-0.01


-0.03


-0.0s


-0.07


-0.09



0.12




0.1










0. 060. 04




0.02




0-


i


Figure 3.22 Case 3. The stress distributions of a doubledoubler joint along the boundary between
the upper outer adherend and adhesive


I I I I I I
0 0.2 0.4 0.6 0.8 1
NORMALIZED AXIAL COORDINATE x/c


I I I I r
0.2 0.4 0.6 0.6
NORMAL IZED AXIAL COORDINATE x/c


0


-











































It I I I I I T I I
0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


0.2 0.4
NORMALIZED AXIAL


0.6 0.0
COORDINATE x/c


Figure 3.23


Case 3. The stress distributions of a doubledoubler joint along the boundary between the central adherend and lower adhesive


66


LO U)

I
U)

-J





-J



I

0
w
N
-J

'I
0
z


0.4 -J


0.3


0.2





0.1





0-


-0.05


1


0.14



0.12
U) U)

0.1







< 0.08 7
U)




one o 0.0or



W
N < 0.04



0.02



0


a


I














































i I I I I I I I I I
0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


Figure 3.24


Case 3. The stress distributions of a doubledoubler joint along the boundary between the lower outer adherend and adhesive


67


V) U)







-J

2


-J



I

Fr
0
w


0.08 0.06 0.04 0.02


0


-0.02-0.04


-0.06


-0.06


-0.1


-0.12-


0.12




0, L U/) U)


I-0.08 U)



1: 0.06


C N 0.04

-J



o 0.02
z



0


1.


J








68


3.5.4 Case 4. Aluminum Central Adherend, Epoxy Adhesive.
and Composite Outer Adherend in Pure Bending

The model for Case 4 consists of aluminum central

adherend, epoxy adhesive, and graphite/epoxy outer adherend under pure bending in the positive sense.

For double-doubler reinforcement of a continous member,

the axial distributions of the cohesive stress at various levels of y in the upper adhesive are shown in Appendix G. The axial stress distributions (ay, ay) on the boundary between the central adherend and upper adhesive and between

the upper outer adherend and adhesive are shown in Figure 3.25 and Figure 3.26, respectively. The axial stress distributions (Uy, aY) on the boundary between the central adherend and lower adhesive and between the lower outer adherend and adhesive are shown in Figure 3.27 and Figure 3.28, respectively.

For a double-doubler lap joint, the axial distributions of the cohesive stresses for various levels of y in the upper adhesive are shown in Appendix H. The stress distributions (ay, a,,y) on the boundary between the central adherend and upper adhesive and between the upper outer adherend and adhesive are shown in Figure 3.29 and Figure 3.30, respectively. The stress distributions (ay, a,) on the boundary between the central adherend and lower adhesive and

between the lower outer adherend and adhesive are shown in Figure 3.31 and Figure 3.32, respectively. The stress








69
distribution trends are similar to Case 3, but higher peak stress values are obtained from Case 4.











70

0.05

En
U) 0
Er
Ln _)


11 -0.1
0 z
-j
Er



N
-0.2
-j
0 w
N -j
-0.3
Er
0 z

0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c
0.12




0.1
V) V)


0.08
LU




T_ 0.06
V..



- 0.06
U)

0
z 0.04



0



0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


Figure 3.25 Case 4. The stress distributions of double-doubler
reinforcement of a continous member along the
boundary between the central adherend and upper
adhesive










71


0.04
V) U)
Lu
0.02
Er


< 00
-0.02







CD -0.06N
-0.0



Z -0.1
0 0.2 0.4 0.6 0.8 1
0.12 NORMALIZED AXIAL COORDINATE x/c




0.1






0. 1+
U) U)










W



T_ 0602040. .
U)


N .0
uj

0 0.06






o0


0 0.2 0.4 0.6 0.0
NORMALIZED AXIAL COORDINATE x/c


Figure 3.26 Case 4. The stress distributions of double-doubler
reinforcement of a continous member along the boundary between the upper outer adherend and
adhesive











72

0.36


LI)
v) 0.3
LU

U) 0.2Z5
-j

0.2
0
z









N
0.L
_j

- 0.





-0.05
0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c
0.1




0.1
U)
V)





I-
-- 0.08
U)
uJ

:. 0.06
U)

uJ N
-j
4:

0
Z 0.02





0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


Figure 3.27 Case 4. The stress distributions of double-doubler

reinforcement of a continous member along the
boundary between the central adherend and lower
adhesive
















































v I I I I I I I I
0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


0.2 0.4 0.6 0.
NORMALIZED AXIAL COORDINATE x/c


Figure 3.28


Case 4. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the lower outer adherend and adhesive


0.1


73


o.o



0.06 0.04 0.02



0



-0.02



-0.04 -


U)
(n

r F
U)

-j



z


r

I




N

-j



z


U)
w

U)


U)

I

U)

N
-


-9


0.12 0.1 0.08




0.06




0.04 0.02


1 .


i


0


-I.


0


1










74

0.05-V)
U) 0




-
U)


-0.12
0 z







-0.3



W N


-0.4



0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


0.14



0.12



U)
Ir


0.08



U) 0.06
0

N 0.04
-J


0.02
0
z

0
0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c

Figure 3.29 Case 4. The stress distributions of a doubledoubler joint along the boundary between
the central adherend and upper adhesive










75


V) 0.08
V)
UJ

- 0.06
V)

0.04


0
z 0.02
-J



-0.02
-J

WU -0.04

-J


0
Z -0.08


0 0.2 0.4 0.6 0.8 1
NORMALIZED AXIAL COORDINATE x/c
0.12




U) 0.1
U)
W


U) 0.08


cr

U) o.oC6


N
_J 004


0
z 0.02




0
0 0.2 0.4 0.6 0.8 1
NORMALIZED AXIAL COORDINATE x/c


Figure 3.30 Case 4. The stress distributions of a doubledoubler joint along the boundary between
the upper outer adherend and adhesive









76


U)
U)9 0.4
W

U)
-J





0.3
0 z
-J
0.2
Lu H -J
0 0.1
N -J

S 0
0

-0.05
0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


0.14



U)

U) 0.1


2: 0.08
U)

W
N0.06
-J

0.04
0 z
0.02


0
0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c

Figure 3.31 Case 4. The stress distributions of a doubledoubler joint along the boundary between
the central adherend and lower adhesive











77


(r) 0.08
U)

F- 0.06
U) -J
0.04


o 0.02
z
-J
0
ix w

< -0.02
-J

U -0.0
N
-J
< -0.06


0
- 0.08


0 0.2 0.4 0.6 0.8 I
NORMALIZED AXIAL COORDINATE x/c
0.12




0. 1

U)

U)




WI 0.06
U)
M
uLJ
N o o4




O 0.02
z



0
0 0.2 0.4 0.6 0.8
NORMALIZED AXIAL COORDINATE x/c


Figure 3.32 Case 4. The stress distributions of a doubledoubler joint along the boundary between
the lower outer adherend and adhesive














CHAPTER 4
PARAMETRIC STUDY OF THE DOUBLE-DOUBLER JOINT


4.1 Introduction

The objective of this chapter is to study the parameters which influence the stress distributions, so that designers can design the adhesive bonded joints more efficiently. The

ANSYS finite element program is used as the computational tool to determine the influence of various joint parameters on stress distributions in the double-doubler joint under tesile loading.

The stress distributions presented in this parametric study are normalized with respect to the applied average stress pa, acting on the central adherend. The influence of the parameters is studied by varying one parameter at a time

from a double-doubler joint which is described in Figure 3.1.

The geometric and material parameters which are examined are

a) bonding half length c

b) the ratio of the adherend modulus to the

modulus of the adhesive E/Ea


For the material parameter study, the central and outer adherends are made of the same isotropic material.


78








79
The thickness of the central adherend, 2t,, is constant through the study. All geometric parameters are

nondimensionalized by dividing by t,. Note that x/c =

(x/t)/(c/t.) . Material parameters are nondimensionalized by dividing the adherend modulus E by the modulus of the adhesive Ea

The influence of E/Ea on the stresses for simple tension is dicussed in Section 4.2. The influence of the length of

overlap on the stresses for simple tension is examined in Section 4.3.



4.2 The Material Parameter

The material parameters consist of the mechanical

properties of the adherend and adhesive. The mechanical

properties of the any isotropic elastic material are the elastic modulus and the Poisson's ratio. Stresses are

affected by variation in Poisson's ratio if the changes in Poission's ratio are very large. Since the range of the variation of Poisson's ratio for common adhesive systems is very limited, only a single material parameter is considered

in this analysis. This single material parameter is the ratio of the adherend modulus to the adhesive modulus. To study the influence of the material parameters four joints are analyzed. The adherends in the four joints have four different moduli of elasticity but the same adhesive. The modulus of

elasticity is the same for central adherend and outer








80
adherend. The Poisson ratio of the adherend is constant at

0.33.

The variation in the shear and lateral normal stress distributions in the adhesive are examined for E/Ea = 1, E/Ea = 30, E/E. = 100, and E/Ea = 200. In this study the modulus of the adhesive Ea is 0.356x106 psi and the constant Poisson's ratio of the adhesive is 0.39. The stress

distribution and singularity parameters in this double-doubler joint are studied for four ratios of E/Ea.



4.2.1 Influence of E/E,

The influence of the ratio of the modulus of adherends, E, to the modulus of the adhesive, Ea, on the stresses in a double-doubler joint is examined here. The influence of E/E. on the axial distributions of shear stress at the interface with the center and outer adherend is shown in Figure 4.2 and 4.4, respectively. Each figure consists of two parts. One

frame is the neighborhood of the free end surface of the adhesive; the other frame is the neighborhood of the butted

adhesive corner. The magnitude of the shear stress decreases rapidly from the neighborhood of the end of adhesive corner and is almost uniform, and equal to the nominal shear stress, over the middle three-fifths of the overlap (0.2 < x/c < 0.8 and -0.2 < x/c <-0.8). The shear stress then drops to zero at the free end surface of adhesive and outer adherend. As E/E, increases, the shear stress concentration factor (rmra)








81
decreases and the order of stress singularity ,A, decreases. (See chapter 5.3 for singularity order A) The magnitude of the highest shear stress is greatest when E = EaThe location of maximum shear stress is not affected when E/Ea is varied. The variation of stress concentration factor and

singularity order A with respect to E/E. is given in Table 4.1. The ideal double-doubler joint would have a uniform shear stress distribution and would also have no lateral normal stress. The distributions of lateral normal stress on

the boundary between the central adherend and adhesive and between the adhesive and outer adherend are shown in Figure

4.1 and Figure 4.3, respectively. Each figure consists of two parts. One frame is the neighborhood of the free end surface of the adhesive; the other frame is the neighborhood of the butted adhesive corner. The lateral normal stress is tensile on the end of boundary between central adherend and adhesive and is compressive on the end of boundary between adhesive and outer adherend. The lateral normal stresses are almost uniform and almost zero in magnitude over the middle threefifths of the overlap (-0.2 < x/c <-0.9 and 0.2 < x/c < 0.9) but rise rapidly to tension and compression peaks at the respective ends of the overlap. The axial distribution of lateral normal stress near the free end of the adhesive layer

also have tension and compression peaks depending upon the level of y. It is quite obvious that compression and tension lateral normal stress must simultaneously exist in order to








82
satisfy the vertical equilibrium condition of the adhesive layer. If we assume that the stresses through the adhesive

thickness are uniform, we can not predict any compressive lateral normal stress in the adhesive layer. The tensile

lateral normal stress concentration factor at the interface between central adherend and adhesive is always greater than

that at the interface between adhesive and outer adherend. This concentration factor decreases as E/Ea increases. The greatest concentration factor is obtained at both interfaces

for E = Ea- It is seen from Figures 4.1 to 4.4 that the magnitude of the maximum lateral normal stress is always greater than the magnitude of the maximum shear stress in a given joint. Therefore the magnitude of the lateral normal stress will be the governing criterion for joint failure in this double-doubler joint.



Table 4.1 Shear stress concentration factor and the order of stress singularity A for case 1 in Chapter 5.3


shear stress order of
concentration factor stress singularity A

E/Ea = 1 23.2 0.465

E/Ea = 30 3.71 0.355

E/Ea = 100 1.95 0.348

E/Ea = 200 1.45 0.346

The shear stress concentration factor is calculated with respect to average shear stress














-7


(/n u
U]




U
r
0








u
-J
r



0
z


0


0.04


0.08


0.12


0.16


0.2 0.24


NORMALIZED AXIAL COORDINATE x/c


Figure 4


.1 Axial lateral normal stress distribution on the
boundary between the central adherend and adhesive
for four values of E/E,


0.4


83


0.85 0.87 0.89 0.91 0.93 0.95 0.97 0.99

NORMALIZED AXIAL COORDINATE x/C




----------- --



7Z-------- - - - E/ E-=





E/ Ea=30

E/Ea=100 .. E/Ea=200






-


-----E/Ea=1

E/ Ea=30

__---E/Ea=1DO

------- E / Ea=20 0


3~~ D II (


0.3





02 -


-U. 1


0.02


U)
If)

U





-J








0
z:


0

-0.02

-0.04

-0.06


-0.0 D


-0. 14

-0.16

-0. ia

-0.2
















- - -- - - - - - - -












E/ Ea=1

E/ Ea=30

E/Ea=100 E/ Ea=200


i i I I I I I I
0 89 0 8 0 89 0 91 0 93 0 95 0 97 0

NOPMALIZED AXIAL COOPIDINATE c







---------- - ---E/ Ea=1

E/Ea=30

------- E/ Ea=100

- _ -----E/ Ea=200


0 0 04 0 08 0 12 0.16 0 2 0 24

'RMDMAL(CED AxIAL COORD1NATZ


0 zq


Figure 4.2 Axial shear

between the four values


stress distribution on the boundary central adherend and adhesive for of E/Ea


0



-0.04


84


V)
LU I


LI


LU

N


Uf)
Uf) Lii




Uf)



f-)

Lu
N


-1

0
z


-o 08



-0 12



-0 16



-0.2z


-0 24



-0 28 -


-0.02


-0.04


-0 06


-0.08e


-0 1


-0.12


-0 14


-0.16


-0.18


-0.2




Full Text

PAGE 1

STRESS ANALYSIS OF A DOUBLE-DOUBLER JOINT IN COMPOSITE STRUCTURE By MYUNG KYUN PARK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1990

PAGE 2

ACKNOWLEDGEMENTS The author deeply appreciates the efforts of his committee chairman, Professor L. E. Malvern. For his advice, guidance, and encouragement, the author is greatly indebted. The author is grateful to Professor B. V. Sankar especially for suggesting study of the stress singularity treatment and for helpful suggestions in carrying out all the analyses. Special appreciation is extended to Professors C. T. Sun, S. Y. Lu, and Professor David C. Wilson for their assistance and serving on the supervisory committee. The author thanks the Department of Aerospace Engineering, Mechanics and Engineering Science and the Center for Studies of Advanced Structural Composites of the University of Florida for their assistance and financial support on this research. The author thanks all his family members for their encouragement on his pursuing an advanced degree in the United States. Finally, true thanks are due to the author's wife, Boeun, who with her prayers, love and forbearance supported the author to finish his study at the University of Florida. ii

PAGE 3

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii LIST OF TABLES v i LIST OF FIGURES vii ABSTRACT XV CHAPTERS 1 INTRODUCTION 1 1 . 1 Introduction 1 1.2 Objective and Scope 2 2 REVIEW OF PREVIOUS WORK 5 2 . 1 Introduction 5 2 . 2 Analytical Study 7 2.2.1 Volkersen ' s Theory " ] 7 2.2.2 Goland and Reissner Theory 9 2.2.3 Plantema's Analysis 13 2.2.4 Kelsey and Benson Theory 14 2.2.5 Pahoja's Analysis 15 2.2.6 Renton and Vinson's Analysis 19 2.2.7 Hart-Smith's Analysis 21 2.3 Finite Element Studies 23 2.3.1 Introduction \ 23 2.3.2 Ahluwalia's Analysis .] 24 2.3.3 Wooley and Carver's Analysis 25 2.4 Experimental Studies 26 2.4.1 Mclaren and Maclnnes's Investigation 26 2.4.2 Tuzi and Shimada's Investigation 27 3 LINEAR ELASTIC STRESS ANALYSIS 29 3.1 Introduction 29 3 . 2 Joint Description 30 3 . 3 Determining Material Properties ." .' ." ." 33 3 . 4 Finite Element Model Representation ' ." .' ." .* .' 36 iii

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3 . 5 Finite Element Results 40 3.5.1 Case 1 . Aluminum Adherends and Epoxy adhesive in Simple Tension 43 3.5.2 Case 2. Aluminum Central Adherend, Epoxy Adhesive, and Composite Outer Adherend in Simple Tension 52 3.5.3 Case 3 . Aluminum Adherends and Epoxy Adhesive in Pure Bending 53 3.5.4 Case 4 . Aluminum Central Adherends, Epoxy Adhesive, and Composite Outer Adherend in Pure Bending 68 4 PARAMETRIC STUDY OF THE DOUBLE-DOUBLER JOINT 78 4.1 Introduction 78 4.2 The Material Parameter 79 4.2.1 Influence of E/E a 80 4.3 The Geometric Parameters 85 4.3.1 Influence of Overlap Length 87 5 TREATMENT OF STRESS SINGULARITY AT BONDING CORNERS IN THE DOUBLEDOUBLER JOINT 89 5.1 Introduction 89 5.2 Stress Singularity at Interface Corners in Bonded Dissimilar Materials ... 90 5.3 Calculation of Order and Intensities of Stress Singularities at the Bonding Corners of a Double-doubler Joint 98 5.3.1 Case 1. Half Plane Bonded to a Quarter Plane 98 5.3.2 Case 2. Three Quarter Plane Bonded to a Quarter Plane 103 5.4 Specified Boundary Displacement Method ... 106 5.4.1 Introduction ^06 5.4.2 Procedure [\ 107 5.4.3 Benefits of Submodelling no 5.4.4 Coarse Model and Submodel no 6 RESULTS AND DISCUSSION 117 6.1 Introduction 117 6 . 2 Summary and Conclusions I '. ns 6.3 Recommendations for Future Work 121 iv

PAGE 5

APPENDICES A CASE 1. THE AXIAL STRESS DISTRIBUTIONS OF DOUBLE-DOUBLER REINFORCEMENT OF A CONTINOUS MEMBER AT DIFFERENT LEVELS OF Y 124 B CASE 1. THE AXIAL STRESS DISTRIBUTIONS OF A DOUBLE-DOUBLER JOINT AT DIFFERENT LEVELS OF Y 129 C CASE 2. THE AXIAL STRESS DISTRIBUTIONS OF DOUBLE-DOUBLER REINFORCEMENT OF A CONTINOUS MEMBER AT DIFFERENT LEVELS OF Y 1 34 D CASE 2. THE AXIAL STRESS DISTRIBUTIONS OF A DOUBLE-DOUBLER JOINT AT DIFFERENT LEVELS OF Y 13 9 E CASE 3. THE AXIAL STRESS DISTRIBUTIONS OF DOUBLE-DOUBLER REINFORCEMENT OF A CONTINOUS MEMBER AT DIFFERENT LEVELS OF Y 144 F CASE 3. THE AXIAL STRESS DISTRIBUTIONS OF A DOUBLE-DOUBLER JOINT AT DIFFERENT LEVELS OF Y 15 3 G CASE 4. THE AXIAL STRESS DISTRIBUTIONS OF DOUBLE-DOUBLER REINFORCEMENT OF A CONTINOUS MEMBER AT DIFFERENT LEVELS OF Y 162 H CASE 4. THE AXIAL STRESS DISTRIBUTIONS OF A DOUBLE-DOUBLER JOINT AT DIFFERENT LEVELS OF Y 171 LIST OF REFERENCES 179 BIOGRAPHICAL SKETCH 183 v

PAGE 6

LIST OF TABLES Table Page 3.1. Elastic material properties used in analysis 33 4.1. Shear stress concentration factor and the order of stress singularity x for case 1 in Chapter 5.3 82 5.1. Selected material properties for singular behavior analysis 100 5.2. Different values Cy, for Case 1 in r x direction in simple tension loading case 114 5.3. Different values C x , Cy, C^ for Case 2 in the r 2 direction in simple tension loading case 114 5.4. Different values C^ for Case 1 in the r 3 direction in simple tension loading case 115 5.5. Different values C x , C^ for Case 2 in the r A direction in simple tension loading case 115 vi

PAGE 7

LIST OF FIGURES Figure Page 1.1 Various types of joints 4 2.1 The single lap adhesive bonded joint 6 2.2 Identification of the various parts of a single lap joint 6 2.3 Systems considered by Goland and Reissner 10 2.4 The single lap joint analyzed by Pahoja 17 3.1 Double-doubler joint configuration 31 3.2 Double-doubler joint geometry 32 3 « 3 Material principal coordinate system 34 3.4 Displacement boundary conditions for simple tension and pure bending 38 3 « 5 The finite element mesh coordinates for the high stress concentration regions 3g 3.6 The layout of finite element mesh in simple tension 41 3.7 The layout of finite element mesh in pure bending 42 3 8 Case 1. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and adhesive 44 J. 9 The different levels of y in the adhesive Stri P 46 MO Case 1. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the outer adherend and adhesive 48 vii

PAGE 8

Case 1. The stress distributions of a double-doubler joint along the boundary between the central adherend and adhesive Case l. The stress distributions of a double-doubler joint along the boundary between the outer adherend and adhesive Case 2 . The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and adhesive Case 2. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the outer adherend and adhesive Case 2. The stress distributions of a double-doubler joint along the boundary between the central adherend and adhesive Case 2. The stress distributions of a double-doubler joint along the boundary between the outer adherend and adhesive Case 3. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and upper adhesive . Case 3. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the upper adherend and adhesive Case 3. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and lower adhesive Case 3. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the lower outer adherend and adhesive viii

PAGE 9

Case 3. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and upper adhesive Case 3 . The stress distributions of a double-doubler joint along the boundary between the upper outer adherend and adhesive Case 3 . The stress distributions of a double-doubler joint along the boundary between the central adherend and lower adhesive Case 3. The stress distributions of a double-doubler joint along the boundary between the lower outer adherend and adhesive Case 4. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and upper adhesive Case 4. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the upper outer adherend and adhesive . . . Case 4. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and lower adhesive Case 4. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the lower outer adherend and adhesive . . . Case 4. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and upper adhesive ix

PAGE 10

3.3 0 Case 4. The stress distributions of a double-doubler joint along the boundary between the upper outer adherend and adhesive 75 3.31 Case 4. The stress distributions of a double-doubler joint along the boundary between the central adherend and lower adhesive 76 3.32 Case 4. The stress distributions of a double-doubler joint along the boundary between the lower outer adherend and adhesive 77 4 • 1 Axial lateral normal stress distribution on the boundary between the central adherend and adhesive for four values of E/E a 83 4.2 Axial shear stress distribution on the boundary between the central adherend and adhesive for four values of E/E a 84 4 « 3 Axial lateral normal stress distribution on the boundary between the outer adherend and adhesive for four values of E/E a 85 4.4 Axial shear stress distribution on the boundary between the outer adherend and adhesive for four values of */* 86 5.1 Two edge-bonde elastic wedges of different materials under normal and shear loading 5.2 Case 1. Half plane bonded to a quarter plane 5.5 The coarse model and sub-model in case l 5.6 The coarse model and sub-model in case 2 91 99 5.3 case 2. Three quarter plane bonded to a quarter plane 104 5.4 Specified boundary displacement method .. 108 112 113 x

PAGE 11

A.l Case 1. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.06375 124 A « 2 Case l. Stress distributions of doubledoubler reinforcement of a continous member at the level of y 0.0675 125 A. 3 Case 1. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.08 126 A. 4 Case 1. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.0925 127 B. l Case 1. Stress distributions of a double-doubler joint at the level of y = 0.06375 12 g B.2 Case 1. Stress distributions of a double-doubler joint at the level of y = 0.0675 B « 3 Case 1. Stress distributions of a double-doubler joint at the level of y = 0.08 B 4 Case 1. Stress distributions of a double-doubler joint at the level of y = 0.0925 130 131 132 C.l Case 2. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.06375 134 C.2 Case 2. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.0675 135 C.3 case 2. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.08 13 6 C.4 case 2. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.08 137 xi

PAGE 12

Case 2. Stress distributions of a double-doubler joint at the level of y = 0.06375 Case 2. Stress distributions of a double-doubler joint at the level of y = 0.0675 Case 2. Stress distributions of a double-doubler joint at the level of y = 0.08 Case 2. Stress distributions of a double-doubler joint at the level of y = 0.0925 Case 3 . Stress distributions of doubledoubler reinforcement of a continous member at the level of y = -0.095 Case 3. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = -0.08 Case 3. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = -0.07 Case 3. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = -0.065 Case 3. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.065 Case 3. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.07 Case 3. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.08 Case 3. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.095 Case 3. Stress distributions of a double-doubler joint at the level of y = -0.095 xii

PAGE 13

Case 3. Stress distributions of a double-doubler joint at the level of y = -0.08 , Case 3. Stress distributions of a double-doubler joint at the level of y = -0.07 Case 3. Stress distributions of a double-doubler joint at the level of y = -0.065 Case 3. Stress distributions of a double-doubler joint at the level of y = 0.065 Case 3. Stress distributions of a double-doubler joint at the level of y = 0.07 Case 3. Stress distributions of a double-doubler joint at the level of y = 0.08 Case 3. Stress distributions of a double-doubler joint at the level of y = 0.095 Case 4. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = -0.095 Case 4. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = -0.08 Case 4. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = -0.07 Case 4. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = -0.065 Case 4. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.065 Case 4. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.07 xiii

PAGE 14

Case 4. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.08 Case 4. Stress distributions of doubledoubler reinforcement of a continous member at the level of y = 0.095 Case 3. Stress distributions of a double-doubler joint at the level of y = -0.095 Case 3. Stress distributions of a double-doubler joint at the level of y = -0.08 Case 3. Stress distributions of a double-doubler joint at the level of y = -0.07 Case 3. Stress distributions of a double-doubler joint at the level of y = -0.065 Case 3. Stress distributions of a double-doubler joint at the level of y = 0.065 Case 3. Stress distributions of a double-doubler joint at the level of y = 0.07 Case 3. Stress distributions of a double-doubler joint at the level of y = 0.08 Case 3. Stress distributions of a double-doubler joint at the level of y = 0.095 xiv

PAGE 15

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy STRESS ANALYSIS OF A DOUBLE-DOUBLER JOINT IN COMPOSITE STRUCTURE BY MYUNG KYUN PARK May, 1990 Chairman: Lawrence E. Malvern Major Department: Aerospace Engineering, Mechanics and Engineering Science This study first provides a linear elastic stress analysis using the finite element method for a double-doubler adhesive joint. The joint is modelled such that the variation of stresses and strains through the adhesive thickness can be determined. The linear elastic stress analysis is focused on three areas: first, the interfacial stress distribution between the adherend and adhesive; second, cohesive stress distribution in the adhesive; and third, stress distribution across the thickness of the adhesive layer at the ends of layer. The second step is to carry out a parametric study on the double-doubler joint. since the stress distribution in the adhesive layer depends on several parameters (geometrical xv

PAGE 16

and material) , an understanding of the influence of these various parameters will lead to better joint design for specific requirements. The third step is to devise parameters that describe the singular behavior near the bonding edges between the adherend and the adhesive layer by applying the existing elastic solution for stress singularities to the double-doubler joint problem. The specified boundary displacement method is then used to determine the values of parameters. These parameters can be used as a means of failure prediction by comparing the calculated parameters with known critical values of the parameters for the material and comparison of stresses very near the bonding corners where stress singularities occur. xvi

PAGE 17

CHAPTER 1 INTRODUCTION 1 . 1 Introduction In the design of material structures components must be joined in such a manner that overall the structure retains its structural integrity while performing its intended function subjected to loads and environment. Joining metallic structures is a well-developed technology involving riveting, bolting, welding, glueing, brazing, soldering, and combinations thereof. However, for polymer matrix fiber reinforced composites only adhesive bonding and mechanical fasteners (bolts and rivets) can be utilized. Recent advances in composite structures and the new adhesive bonding techniques based upon very strong epoxy type adhesives have made feasible the adhesive joining, stiffening and repairing of structural elements subjected to extreme enviromental and loading conditions. This type of joining and stiffening has been used in the aerospace, automobiles, electronics, sporting goods and appliance industries. Consequently the importance of adhesive bonding in technology has been recognized. Various types of adhesive joints are shown in Figure 1.1. Among the various types of joints, the double-doubler joint is selected for this study.

PAGE 18

2 Adhesive bonding provides structures having lighter weight, more fatigue resistance, improved aerodynamic smoothness, better distribution of imposed stress, better thermal and electrical insulation and greater crack retardance over mechanical fasteners. Therefore, adhesive bonding is very desirable for use in composite material structures. The use of adhesives is growing rapidly but designers lack the definitive approach that gives confidence in routinely incorporating adhesive bonding into design. 1.2 Objective and Scope The analysis of the behavior and the failure of an adhesive joint must include both the understanding of the mechanism of adhesion, which is closely related to the microstructure between the adhesive polymer and the adherends, and the analysis of stresses in the joint. This study will be focused on the latter part. Failure in an adhesive joint can be either "adhesive," which occurs at the interfaces between the adhesive and the adherend, or "cohesive," which occurs either in the adhesive or in the adherends. Therefore the first objective of this thesis is to do a linear elastic stress analysis for the double-doubler joint by means of finite element method (using ANSYS) . The linear elastic stress analysis is focused on three areas: first, the interfacial stress distribution between the adherend and the adhesive; second, cohesive stress distribution in the

PAGE 19

adhesive; and finally, the stress distributions at various levels of y in the adhesive layer near the ends of layer. The second objective is to carry out a parametric study on the double-doubler joint. Since the stress in the adhesive layer depends on several parameters (geometrical and material) , an understanding of the influence of these various parameters will lead to better joint design for specific requirements . The third objective is to devise parameters that describe the singular behavior near the bonding edges between the adherend and the adhesive layer by applying an existing elastic solution for stress singularities to the doubledoubler adhesive joint problem. The specified boundary displacement method (SBD) is used to devise the parameters. These parameters can be used as a means of failure prediction by comparing the calculated parameters with known critical values of these parameters for the material and comparison of stresses very near the bonding corners where stress singularities occur.

PAGE 20

S i ng I e Lap Jo i nt Double Lap joint Single Doubler Joint Double Doubler Joint Stepped Lap Joint Scarf Joint Figure 1.1 Various types of joints

PAGE 21

CHAPTER 2 REVIEW OF PREVIOUS WORK 2 . 1 Introduction A considerable amount of experimental and theoretical research has been carried out on adhesive bonded joints. Theoretical and experimental stress analyses of various types of adhesive-bonded joints have been made by many authors. Due to the relatively simple geometry and wide application, many theoretical and experimental analyses have been devoted to the single lap joint. (See Figure 2.1) In this review the important investigations of the single lap joint and known investigation of the double lap joints are included. The following review provides a history and background of the present state of knowledge in adhesively bonded joints. First the investigations of the single-lap joint are reviewed here because the earlist analyses and the analyses which are used as the basis for most subsequent analyses were made on the single-lap joint. The terminology used in identifying various parts of the joint is shown for a single lap joint in Figure 2.2. The "overlap" is the length over which the adhesive extends, and the "overhang" is the part of the adherend outside the overlap. The adhesive layer has two interfaces and two free

PAGE 22

adherend 1 adhesive Figure 2.1 The single lap adhesive bonded joint l ead i ng end ^-overlap -^_ overhang _^ Figure 2.2 Identification of the various parts of a single lap joint

PAGE 23

7 edges. The "leading end" of an adherend is where the overhang begins after the overlap. The "trailing end" of the same adherend is at the opposite end of the overlap. Corresponding parts of the double-lap joints are identified with the same terminology. A typical single-lap joint is under tensile loading. 2.2 Analytical Studies 2.2.1 Volkersen ' s Theory Classical analysis of an adhesive bonded joint was made by Volkersen [1]. He derived the load distribution of a multi-row riveted lap joint in tension, where he idealized the rivets by replacing them by a continous medium of given shear flexibilty, and considered the case of an adhesive bond. Bending of the adherends and the associated lateral normal and axial normal stresses were not considered. Therefore the shear stress was forced to be constant through the thickness of adhesive. By setting up the differential equations of linear elasticity, he showed that shear stress at any point along the length of the overlap was given with respect to dimensionless quantites: a) the overlap ratio D = Gc 2 /E 2 t a t 2 21 b) the adherend stiffness

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The shear stress distribution in the adhesive layer was nonuniform in the x-direction, but assumed uniform constant in the y-direction. Peaks of the shear stress occurred at the ends of the overlap. He compared the maximum shear stress, r max at the ends of the overlap with the mean stress, r m and found the stress concentration factor, 7 , , for joints with nonidentical adherend. ii = 1/2 (W-l)cosh[ (Dwj 1 / 2 ] = (d/w) 1/2 . ; sinh[ (DW) 1/2 ] 2.3 where D = Gc 2 /E 2 t 2 t a W = (Eiti+Eat^/Eiti r m = averaged applied shear stress This is for the case E x t, > E 2 t 2 . He showed that the stress concentration factor increased with increasing shear modulus of the adhesive and with decreasing moduli of elasticity and thickness of the adherends. The theory developed by Volkersen has two important shortcomings . a) The bending deformation was neglected. (There should be a bending moment applied to the joint in addition to the in-plane tension, since the two applied forces are not collinear.)

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9 b) The assumption was made that the shear stress does not vary across the thickness. 2.2.2 Goland and Reissner Theory Goland and Reissner [2] studied the stress distribution for adhesive joints with identical adherends under the action of tensile forces at the ends of the adherends. The forces act in the direction parallel to the bonding line before any deformation takes place, but the line of action of the tensile force R is assumed to pass through the midpoint of the adhesive layer for the deformed equilibrium configuration. They recognized that the bending of the adherends beyond the bonded region has a significant effect on the stress distributions in the joint itself. They approached the problem in two steps. First they determined the loads at the edges of the joint which were transmitted from the loads at the ends of the adherends. In the first step of the problem, they treated the adherends as cylindrically bent plates, and found that forces transmitted to the edges of the joint consisted of a tensile force P, a small shear force V, and bending moment M, as shown in Figure 2.3 in which a P = R >n n ,, 2 \ t>/i? 4. -i 1/2 2.4 V = KP[3(1 i^JP/EitJ 1 ' 2 M = KPt x /2

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Figure 2.3 Systems considered by Goland and Reissner

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11 where K is the bending moment factor which is the ratio of the magnitude of the edge moment M and the value of this moment in a nondeformable system: K 1 2.5 1+ (2/2) tanh[ (c/itj J&{l-v\) P/Eiti ] The value K=l indicates that the adherends exhibit no deformation due to bending, and K < 1 indicates that a certain amount of bending deformation takes place. In the second step of the problem, they assumed that the normal stress in the adhesive parallel to the layer can be neglected, and that the lateral normal stress and shear stress do not vary across the thickness of the adhesive layer. They used a plane strain analysis and solved the problem for two limiting cases: 1) The case where the adhesive layer is so thin and stiff that its deformation may be neglected. This case requires that the condition ta /E a « t l/El . The results show that, while the lateral normal stress is high at the edge of the joint, the shear stress is zero there. The shear stress, however, rises rapidly to a maximum value in close proximity to the edge. 2) The case in which the adhesive layer is soft and flexible and the joint flexibility is mainly due to the

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12 deformation of the adhesive layer. This case requires the condition t./E a » t^E^ The shear stress r and lateral normal stress a are expressed as follows. T = P r Be cosh(fix/t ,l 4c I TET (1+3K) coBhcflcAti) + 3(1_K) ] 2 6 a =(1/A) (P/tJ (2^0)2 [ (R 2 A 2 K/2 + AK'coshA cosA) cosh(2Ax/c) cos(2Ax/c) 2.7 + (RiA 2 K/2 + AK'sinhA sinA ) sinh (2Ax/c) sin(2Ax/c)] where B = (SGtJ E x t) 1/2 A = fe/2t a ) (SEti/Eit) 1 '* R : = coshA sinA + sinhA cosa R 2 = sinhAcosA coshAsinlA a = (1/2) (sinh2A + sin2A) K«= Vc/2Pt! The maximum value of r and a are found at the edges of the joint. fP/tJ ( t, ) T t -2t[ (1+3K) coth -ff+ 3 (1-K) ] 2.8 To/Ti < It > 2 Si " h ? A ~ S . in2A + \jri CQSh2A + soggA (P/tJ 2t, 2 smh2A + sin2A + sinh2A + sin2A 2.9

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13 Compared with the factor D used by Volkersen, it can be seen that &c/2t x = (2D) 1/2 . For large values of D, coth(2D) 1/2 approaches the value of 1, and therefore the shear stress concentration factor, r ll , can be written, for large D, as 1l = r max /(p/c) « (1/4) (1+3K) (2D) 1/2 + 3(l-K)/4 2.10 where D = Gc 2 /E 2 t 2 t a The limitations of Goland and Reissner theory include a) It is valid only for identical adherends. b) it is valid only for two special cases; the first case is for VE, « ti/Ei, while the second case reguires that tJE a » t a Ex. c) The stress across the adhesive layer thickness is assumed to be constant, which results in inaccurate predictions in the region adjacent to the edges of the overlap. 2.2.3 Plantema's Analysis Plantema [3] combined the Goland-Reissner theory with Volkersen -s theory in the analysis of a single-lap adhesive bonded joint. The factor K from the Goland-Reissner theory was employed to calculate the differential strain of the members due to bending, while Volkersen -s theory was used to

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14 calculate the differential strain of the members due to shear, and the stress at the edges of the overlap. The shear stress concentration factor was given as 91 = [D(l+3K)/2] 1/2 coth[D(l+3K)/2] 1/2 2.11 He did not make an attempt to calculate the lateral normal stress. 2.2.4 Kelsey and Benson Theory Kelsey and Benson [4] employed the complementary energy method to determine the shear and lateral normal stresses in a single lap joint. A linear variation of the lateral normal stress was allowed across the the adhesive layer, while the shear stress was assumed constant across the adhesive thickness. The result was a differential equation of higher order than that obtained by Volkersen, which enabled the boundary condition of zero shear stress at the ends of the overlap to be satisfied. The shear and lateral normal stress obtained are as follows: Let A = pm^c D = (%coshm lC /2) (sinhm 2 c/2)-(m 2 sinhm 1 c/2) (coshm 2 c/2) r (A/4D) [(cosh^x) (coshm 2 c/2)-(coshm lC /2) (coshm 2 x) ] 2.12

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(At/8D) [ (niiSinhmiX) (coshm 2 c/2) 15 -(m 2 coshm 1 c/2) (sinhm 2 c/2) ] 2.13 The shear stress concentration factor r) = r/(p/c) is given by v = m^c^coshro, (xc/2) ) (coshm-c/2) (coshm,c/2) fcoshm, (xc/2) 4 m 1 (coshm 1 c/2) (sinhm 2 c/2) -m 2 (sinhm 2 c/2) (coshm 2 c/2) 2.14 where m 2 = Gt '» 'a m 3 ( i 2G2fc B ) V2 Kelsey and Benson imposed the zero shear stress boundary condition which causes a sharp drop of shear stress from its peak value at positions very near the ends to zero at ends. They reasoned that this is necessary for equilibrium because the adhesive/air boundary can not sustain a shear stress. 2.2.5 Pahoia's Analysis Pahoja [5] made the analysis of the joint under tensile, shear, and bending loads as shown in Figure 2.4. The joint was subjected to a general loading condition. He employed the theory of minimum potential energy to study the axial normal,

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16 lateral normal and shear stresses in the adhesive layer. He defined the displacements in the adhesive in terms of polynominals with m terms for the displacements in the x direction and n terms for the displacement in the y direction. The total energy was expressed in terms of the displacement components in series form. By minimizing the total potential energy of the system, a system of governing linear ordinary differential equations and natural boundary conditions were obtained. The equations were reduced to algebraic and first order differential equations by substituting new variables for the displacements and their derivatives. By solving these equations numerically, the stress distribution and stress concentrations were found. The stress concentrations for identical adherends have the follwing forms: Shear stress ('maxAav) = 1 + ( 6/7 )[( 19/18 ) V f ] (S f 1) for V f < l and S f > 4 2.15 ('maxA.v) = "1 + (18/19) (V f 17/18) (S f 1) for V f > 1 and S f > 5 Lateral normal stress K^axAav = (t2/c)(1.4 1.675 V f ) N f 2 2.16

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17 Figure 2.4 The single lap joint analyzed by Pahoja

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18 Axial normal stress (OnmxAav = (ti/c)(0.85 0.95V £ ) N f 2 2.17 where V c V f = T [t. + Q, t/ ft + M/fTy j S f = (c/tj (Et^Et.) 1 ' 2 N f = (C/tJ (Et^Eit,) 1 '* Qi Ei/fl-i/^) * = Qt. 2 /6 + Qjtjt, + Qiti 2 Q = E/(l-„ 2 ) Pahoja concluded that the single lap joint with identical adherends was the best design. This joint was most efficient when loaded such that the shear stress distribution was uniform, i.e., v f = l. He compared his theoretical results with the results of photoelastic analysis, and these are in good agreement. The limitation of this analysis is that the free boundary condition at the ends of the adhesive were violated just as in Volkersen's and Goland and Reissner's analyses.

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19 2.2.6 Renton and Vinson's Analysis Renton and Vinson [6,7] made the most complete analysis of a single lap joint. They clearly identify the extent of the influence of the various parameters on the stress distribution in the adhesive and adherends. The analysis was very general and applicable to isotropic and anisotropic adherends as well as to dissimilar and similar adherends. The method of analysis that they developed involve the following assumptions and/or limitations: 1) The laminated adherends are symmetric about their own midsurfaces (i.e., no bending stretching coupling). 2) Each ply or lamina in each adherend is orthotropic 3) The effective elastic mechanical properties of the adhesive are accounted for. 4) Both shear and lateral normal stresses are accounted for in the adhesive; they vary in the load direction, but do not vary in the thickness direction. 5) The adhesive thickness is much smaller than the adherend thickness, i.e., t a « t, and t a « t 2 6) Shear deformation and lateral normal strains are accounted for in each adherend. Composite material laminated plate theory was used as the building block or starting point for developing the method of analysis. The lap joint structure was divided into five portions for analysis: the adhesive, the adherends on either

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20 side of the adhesive, and the adherend portions from the end of lap joint to the load application. Twenty-six boundary conditions are specified and satisfied. The analytical solutions obtained for stresses and deformations throughout the structure were programmed on computer for easy calculation. After developing a valid closed form solution for the single lap joint problem, they ascertained the influence of certain important parameters on the stress distributions in the adhesive. The influence of the joint parameters on the stress distributions in the adhesive are summarized below: 1) The greater the difference in the moduli of the adherends, the higher is the magnitude of the maximum shear stress. 2) For dissimilar adherends, the maxima of both the shear and the lateral normal stresses in the adhesive occured at the leading end of the less rigid adherend. 3) The greater the ratio of the moduli of the adherend and the adhesive, the more uniform the shear stress distribution. 4) The larger the ratio of the moduli of the adherends and the adhesive, the smaller is the magnitude of the maximum lateral normal stress.

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21 In the design of single lap adhesive bonded joints, they recommended that 1) The adherends of joints should be identical. 2) The modulus of the adherends should be high compared to that of adhesive. 3) The length of overlap should be about ten times the minimum thickness of the adherend. 4) The joint's intended function should influence adhesive selection: in static bodies, adhesive ultimate strength should be high; in fatigue, adhesive fracture toughness should be considered. 2.2.7 Hart-Smith's Analysis The analysis of a double lap joint by Hart-Smith [8] was designed to calculate the joint efficiency of a double lap joint in terms of the strength of the joint. The adhesive was characterized as elastic-plastic in shear. In the analysis, the shear stress distribution was constant across the adhesive thickness. In his analysis, the deformation of the adhesive was greatest at the ends of the overlap, and consequently the behavior of the adhesive was characterized as plastic at the ends of the lap. In the elastic analysis, the shear stress in the adhesive was expressed in terms of the displacements in the outer and center adherends. The differential equation governing the

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22 shear stress distribution was determined from the stress strain relationship and the force-equilibrium equation. In the elastic-plastic analysis, the plastic shear stress and the distance over which the adhesive was plastic were determined from the boundary condition for the plastic region, the equation ensuring the continuity of the elastic and plastic shear strains, and the gross horizontal equilibrium equation of the joint. The expression for the plastic shear stress was then rearranged to express the applied load in terms of the shear strain energy per unit area of the adhesive layer. The lateral normal stress in the adhesive was considered to be independent of the shear stresses. The analysis for the lateral normal stress was applicable only when the adhesive was deformed beyond the elastic range in joints with long overlaps. The lateral displacement of the outer adherends was related to the moment at the ends of the overlap by the theory of bending of plates. The coefficients of the normal displacements, which were expressed in terms of trigonometric and hyperbolic functions, were determined from the boundary conditions in terms of the maximum shear stress. The lateral normal stress was then given in terms of an "effective" tension modulus of the adhesive, whose value is difficult to establish theoretically. The conclusions drawn by Hart-Smith on the stress distributions are

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23 1) The highest shear stress in the adhesive was at the end of overlap near the leading end of the less stiff adherend . 2) For long overlaps, the magnitude of the maximum shear stress did not vary with length; only the stress in the central portion of the joint was lowered by increasing length. 3) Lateral normal stresses were reduced by using a thin outer adherend, or an adherend with a low extensional modulus, or a thick adhesive layer. Thus, the lateral normal stresses can be reduced at the expense of the shear stresses. 4) The characteristic extent of the end zones was independent of the length of overlap. 2.3 Finite Fi gment Sfcyfllfeg 2.3.1 Introduction The use of finite element methods of analysis means, in principle, that the simplifying assumptions made in the classical analytical methods need not be made. The technique does allow the solution of problems that are totally intractable by classical methods. Basically, one can choose between a large number of primitive elements, i.e., elements in which strain is assumed to be constant, or a smaller number of more sophisticated elements, ones with linear or quadratic strain variation. Whatever idealization is adopted, a higher

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24 density (finer mesh) of elements is necessary in regions of rapidly varying stress. Finite element analysis of an adhesive-bonded joint was first performed in 1969. Considerable work has been done with finite element analysis of adhesive-bonded joints of fiber reinforced composite materials. 2.3.2 Ahluwalia's Analysis The first finite element analysis of an adhesive bonded joint was made by Ahluwalia [9]. He analyzed a double-lap joint. The finite element solution for the double lap joint was for plane strain with 120 nodal points and 188 triangular elements. The adhesive was divided into ten equal elements along its length and was made of one element across the thickness. Consequently, the high magnitude of the stresses at and near the ends of the overlap were not correctly determined, and the stresses did not vary across the thickness of the adhesive. Smaller elements were not defined at the ends of the overlap where shear and lateral normal stress concentrations exist. The shear and lateral normal stress distributions were nonuniform with peaks at the ends of the overlap. Large sizes of elements in areas of very sharply rising peak stresses result in lower magnitude of the calculated stresses.

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25 2.3.3 Wooley and Carver's Analysis Wooley and Carver [10] analyzed a single lap adhesivebonded joint with the finite element method. The finite element solution for the single lap adhesive-bonded joint was for plane stress problem with 497 nodal points and 394 elements. The solution was based on the computer program "ARBAS" by E.L. Wilson of the University of California at Berkeley. The locating of nodal points was accomplished by dividing the configuration into three parts, the section of the lap joints, and the sections to each side of the lap. in the program, quadrilateral elements, composed of four triangular elements, were used to describe the joint. The displacements of the triangular elements were described by linear functions which resulted in a constant strain and, therefore, constant stress over each element. The adhesive layer was divided into two equal elements through its thickness and twenty equal elements along its length. The maximum aspect ratio of a quadrilateral element in the adhesive was 100:1. The single-lap adhesive-bonded joint was made of identical adherends with an elastic adhesive. The Poisson's ratios of the adhesive and adherends were 0.3 for all the cases. The range of the ratio of the elastic moduli of the adherend and adhesive, E/E a was from 0.1 to 1000. The range of the ratio of the thickness of the adhesive and adherends, tjt was from 0.02 to 0.1. The length

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26 of overlap was taken as 5, 10 and 20 times the adherend thickness. The stress distributions at the adhesive-adherend interface were not given. The stress distributions at the mid plane of the adhesive layer were nonuniform with peaks near the ends of the overlap for both shear and lateral normal stresses. They compared the stress concentrations obtained by each method at three values of the geometric and material parameters, with two elements across the adhesive thickness, the variation of the stresses across the adhesive thickness can not be adequately represented. 2 . 4 Experimental Studies 2.4.1 Mclaren and Maclnnes's Investigation Mclaren and Maclnnes [11] conducted photoelasticity experiments on single lap joints. The object of their investigation was to verify the mathematical analysis of Goland and Reissner. Two series of tests were performed, in the first series, the complete joint was made of Araldite. In the second series of tests, the adherends were made of aluminum and the adhesive of Araldite. The ratio of the modulus of the adhesive to that of the adherend was 1/20 in the second series of tests. The results were presented in terms of the fringe order, and no attempt was made to determine the magnitude of the stresses. The difference between these two model systems was in the transfer of load

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27 from one adherend to the other. The adhesive in the second series was in longitudinal shear across its entire thickness; thus, the transfer of load was primarily by shear. Their investigation supports the theory of Goland and Reissner in that the qualitative aspects of the theory are well corroborated . 2.4.2 Tuzi and Shimada's Investigation Tuzi and Shimada [12] conducted a photoelastic investigation of a single lap joint. Their investigation concentrated mainly on stress concentration in the joint. Single lap joint models having different adhesive shapes at the ends of overlap; rectangular type, with a fillet and with convex and concave semi-circular ends were used in the photoelastic investigation. The adhesive layer of the joint was made of epoxy rubber and the adherends were represented by epoxy resin and duralumin. In the first experiment, the adherends were made of epoxy resin KT-102 and the adhesive layer was made of an epoxy rubber. The adhesive was bonded to the adherends by Araldite D. in the second experiment, the adherends were made of duralumin, the adhesive layer was made of epoxy resin KT102, and the adhesive was bonded to the adherends by Araldite D. Their experimental results indicated that the joint with concave semicircular ends was the best, the one with rectangular ends was second best, and the one with convex

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28 semicircular ends was the worst as regards the stress concentration in the joint. They also studied the stress distributions in scarf and butt joints photoelastically . Their photoelastic models were made of epoxy resin and epoxy rubber and also epoxy resin and metal. Tension and bending tests were carried out on bars bonded at various angles. For the joints with a soft adhesive layer bonded to hard adherends, the stress concentrations were maximum at the ends of the bonded boundary with an obtuse angle. For the joints with a hard adhesive layer bonded to soft adherends, the stress concentrations were found at the ends of the bonded boundary with an acute angle.

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CHAPTER 3 LINEAR ELASTIC STRESS ANALYSIS 3.1 Introduction The joint under consideration is a double-doubler joint (Figure 3.1). This joint is subjected to two different loading cases (constant tensile loading and pure bending) . In this chapter the three materials (central adherend, adhesive, outer adherend) are assumed to behave in a linear elastic manner under the applied loads. The elastic properties, Young's modulus and the Poisson's ratio are E c and j/ c for central adherend; E a and »/ a for the adhesive layer; E 0 and i/ Q for the outer adherend. The material properties are extended to orthotropic material properties for the application to composite adherends. The overall problem is to find the stress distributions in the double-doubler adhesive bonded joint under the applied load. The distributions along the axial direction of the stresses on the interface between the central adherend and adhesive, and of the cohesive stresses at various levels in the adhesive are the particular interest in this study. The analysis can be simplified to either a plane stress or a plane strain problem. For most practical purposes the width of an adhesive joint is large compared to the thickness of the 29

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30 adherend and the adhesive. Therefore the plane strain analysis option is used throughout. First of all linear elastic stress analysis for double-doubler reinforcement of a continuous member (Figure 3.1a) was carried out. The results, are compared with the previous experimental results and they show good agreement. After that the linear elastic stress analysis for a double-doubler joint (Figure 3.1b) was investigated. The finite element method of stress analysis is applied to this plane problem involving different materials. The reason why the finite element method is chosen over any other method is that geometrically complex bodies of different materials can be easily represented and arbitrary distributions of displacement and stress boundary conditions can be specified. For this linear elastic stress analysis the finite element program ANSYS is employed in the finite element studies of the double-doubler joint. 3.2 Joint Description The geometry and dimension of the joint are shown in Figure 3.2. The basic material properties used in this joint are given in Table 3.1.

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31 a} the doub I e-doub I er reinforcement of a continous member b} the doub I e-doub ler joint Figure 3.1 Double-doubler joint configuration

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32 y i t a = 0 . 04' T i 2t = D. 12" c 0.08" 0. 02" C= 2.56" 4.0' Figure 3.2 Double-doubler joint geometry

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33 Table 3.1. Elastic Material Properties Used in Analysis Mateial Modulus , E Poisson's ratio Shear Modulus, G aluminum 10.6E6 psi 0.33 epoxy 3.6E6 psi 0.39 graphite epoxy 22.8E6" psi 1.58E6 b psi 0.3 C 0.81E6 d psi a: Longitudinal modulus of unidirectional composite i.e, fiber direction b: Transverse modulus of composite c: Major Poisson's ratio of unidirectional composite d: Inplane shear modulus 3 . 3 Determining Material Properties The strain-stress relations for anisotropic materials can be expressed as € i = s u °i ifj = 1,2, ,6 3.1 where £i are the strain components, Sij are the elements of the compliance matrix, and o 3 are the stress components. Equation 3.2 gives the form of the relations for a unidirectional fiber reinforced lamina with axes x lf x 2 ,x 3 aligned with the principal material directions as shown in Figure 3.3. (x x corresponds to fiber direction, x 2 is the inplane coordinate perpendicular to x lt and x 3 is the tranverse coordinate which is perpendicular to the Xj-x 2 plane.) S n ^12 s 13 0 0 0 °\ ^22 S 2 3 0 0 0 °2 e 3 ^13 ^23 s 33 0 0 0 °3 0 0 0 0 0 °k 0 0 0 0 s 55 0 *6 0 0 0 0 0 ^66 3.2

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Figure 3.3 Material principal coordinate system

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35 The compliances can be expressed in terms of engineering constants as follows: S n =1/E n 3.3 S i2 = -^12/Eu S 13 = -*/ 13 /E u S 22 = 1/E 22 S 23 = -u 23 /E 22 S 33 = I/E33 S A4 = 1/G 12 S 55 = 1/G 23 S 66 = 1/G 13 where E iif E 22 , E 33 are Young's moduli in 1,2 and 3 directions respectively. i/ij is Poisson's ratio determined from contraction in the Xj direction during a tensile test in the x A direction G l3 is shear moduli in the x^Xj planes We know that the following relationships exist between the Young's moduli and the Poisson's ratios due to symmetry of the compliance &U*21 = E 22 »/ 12/ E 22i / 32 = E 33 i, 2 3, E 33l / 13 = E ul / 31 3.4 Thus for an orthotropic material, constitutive relations are «2 « 5 «6 1/En -^12/Eu -'is/En 0 0 0 1/E 22 _1/ 2 3/E 22 0 0 0 -"la/Eu _,/ 23/E 22 VE33 0 0 0 0 0 0 0 0 0 0 0 0 1/G 12 0 0 0 1/G 23 0 0 0 1/G 13 3.5

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36 For a transversely isotropic material in which the x 2 -x 3 plane is the plane of isotropy, we find E 22 = E 33 , G 13 = G 12 , v n = i/ 13 and G 23 = E 22 /2 (1+j/ 23 ) . If we invert Equation 3.5, for a material having tranverse isotropy relative to the x, x 2 3 plane, we obtain the elastic stiffness, C^, in terms of engineering constants with the following results: C u = (1 -» 2 23 ) Ejj/V c i3 C 12 =»/ 12 (l + i/ 23 ) E 22 /V C 23 = (f 23 + " 2 i 2 E 22 /E u ) E 22 /V C33 = C 22 = (1 v\ 2 E 22 /E n ) E 22 /V 3>6 c *a = G 23 = E 22 /2(l + »/ 23 ) C 55 = C 66 = G 12 where V [(1 + „„) (1 „ 23 2„ 2 12 E 22 /E n ) ] For the plane strain case,E u , E 22 , E 33 , , 12 , „„, Cuf ^ G 23 would be used as input data for ANSYS . If G 12 ,G 13/ G 23 are not specified, they are computed as: G 12 = E 1 E 2 /(E 1 + E 2 + 2«/ 21 Bi) G 23 = E 2 E 3 /(E 2 + E 3 + 2i/ 32 E 2 ) 3
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37 The ANSYS two-dimensional isoparametric element which accomodates orthotropic material properties is used. The element is defined by four nodal points, each having two degree of freedom. The plane strain option is used. The displacement boundary conditions imposed on simple tension and pure bending case are shown in Figure 3.4. In simple tension cases analyzed in this study, the double-doubler joint is subjected to 50 lb at the outer end of the central adherend. In pure bending case, a linear varying force boundary condition equivalent to a couple is used at the outer end of the central adherend. For the input of the orthotropic material properties the classical thin plate lamination theory is used to calculate the effective material properties for the laminate. For unidirectional composite lamina, properties in the thickness direction are calculated assuming that E 22 = e 33 , "12 = "is and j/ 23 = matrix Poisson's ratio. The material principal coordinate system (1,2,3) refers to the fiber and two transverse matrix directions, respectively. (See Figure 3.3) The computer program can be used to calculate effective material properties for composite laminates. The mesh is so generated that around the bonding corner and on the boundary between two different materials the grids are much finer than those of other locations for better results in these regions of high stress gradients. The finite element dimensions which are used in the analysis for the high stress concentration regions are shown in Figure 3.5.

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o o o — CJ ^ »W 7T7-F 71 a} imposed boundary condition on simple tension loading rol I ing support 0 imposed boundary condition on pure bendi ng Figure 3.4 Displacement boundary conditions for simple tension and pure bending

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39 y i 0.0925 0.08 0. 0675 0.06375 0.06 0.05625 0.0525 0.045 outer aanerena \ — i adhes i ve V254 2.56 2.58 2.55 2.57 centra I aanerena 1 — 0 e e g 0 B-r( 8 8 n -9.06375 -0.06 43.0525 -9.045 0.193 0.2 0.207 0.1965 0.2035 unit = inch Figure 3.5 The finite element mesh coordinates for the high stress concentration regions

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40 A total of 1591 nodes and 1456 elements for the simple tension case, and 2819 nodes and 2672 elements for the pure bending case are used. (See Figure 3.6 and Figure 3.7) The numbers in circles relate to the nodal-point numbers. Those within the mesh relate to element numbers. The model is constructed using ANSYS STIF42. 3.5. Finite Element Results In all the stress distribution results, the stresses are normalized with respect to the average applied axial stress, p av , acting on the central adherend. Accordingly a, = a~/p , £y " ^y/Pav/ £xy = <*xy/P.v/ While X = X/C. where a x : axial normal stress a y : lateral normal stress o^: shear stress p av : applied average stress c: 2.56 inches For nodal stress, the stresses at each node are computed by averaging the nodal stresses of the node being processed for all elements that are connected to that node, m other words, the component nodal stresses that are automatically stored per element are averaged at a node whenever two or more elements connect to the same node. For example average nodal stress a x at the node N can be calculated as below.

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Figure 3 . 6 The layout of finite element mesh in simple tension

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42 Figure 3.7 The layout of finite element mesh in

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43 elementl element 2 -N 1 element 3 From element 1 a x at node N From element 2 a x at node N From element 3 cr_ at node N average nodal stress = (a lx + a 2x + a 3x )/3 Four cases are considered in the following four subsections. Case 1 and 2 are for two different choices of adherends with simple tension loading. Case 3 and 4 are for the same two choices of adherends as case 1 and 2, but with pure bending loading. Each of the four cases is treated first for doubledoubler reinforcement of a continuous member, and then for a double-doubler lap joint. 3.5.1 Case 1. Aluminum A dherends and Eooxy Adhesivg in Simple Tension The model for the first case is composed of aluminum adherends and an epoxy adhesive. For double-doubler reinforcement of a continuous member (Figure 3.1a), the axial distribution of the lateral normal stress (a y ) , and the shear stress (a^) along the boundary between central adherend and adhesive are plotted from the averaged nodal stress values. The reason why the averaged nodal stress values are used is that the lateral normal (a y ) and shear stress { 0xy ) are continous across the interface. These stress distributions are shown in Figure 3.8. From these graphs and the known symmetry about the mid-length, we can see that the stress distributions {o y , axy ) along the axial direction are very

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44 o. ze CO CO 0.24 UJ cr hcn 0.2 cr O < cr LU Q LU N cr o 0.12 o. oe 0.04 o -0. 02 -0.01 CO CO LU 00 -0.03 cr -o.o4 < LU x CO -o.os Q LU |SJ -0.06 cr o -0.08 -0.09 0.2 I 0.4 NORMALIZED AXIAL COORDINATE' x/c —> 1 i 1 1 1 1 1 02 01 0.6 0 8 NORMALIZED AXIAL COORD I NDATE x/c Figure 3.8 Case 1. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend an/adhesive

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45 uniform and at a very low level for -0.75< x/c <0.75. But near the free edges (for 0.75 < x/c < 1 and -1 < x/c <-0.75) we can see sharp variation of stresses and see the peak stresses at the free end of adhesive. Different peak stress values are obtained depending upon the size of the mesh element. This means that calculated stress fields near this area are not quite reliable. In linear elastic analysis, stress fields near the two bonding edges show singular behavior. In order to describe the stress fields near these corners, singular stress parameters are used in Chapter 5. The cohesive stress distributions along the axial direction at different levels of y of the adhesive were obtained. These different levels of y in the adhesive strip is shown in Figure 3.9. According to the results which are shown in Appendix A, the axial distribution of shear stress ( 0xy ) at the level of y=o. 06375 shows a similar trend to that in the stress distribution along the boundary between central adherend and adhesive. The axial distribution of axial normal stress (c x ) at the same level of y=o. 06375 shows a similar trend as that in the stress distribution along the boundary between central adherend and adhesive, except that the axial normal stress distribution drops toward zero near the free end of the adhesive layer. The axial distribution of lateral normal stress K ) near the free end shows a similar trend to that in the stress distribution along the boundary between

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46 y k outer adherend y=0.1 j ,0.0925 4 0.08 adhes i ve 0.06 l I central adherend 0.07 -0.07 outer adherend centra l adherend outer adherend 3.095 Figure 3.9 The different levels of y in the adhesi strip

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47 central adherend and adhesive, except that the peak stress value becomes lower than previous peak stress on the boundary. The axial distribution of axial normal stress (a x ) and lateral normal stress (a y ) at the higher level (y=0.0675) than the previous case (y=0. 06375) shows a similar trend, except that the peak stress values are lower than ones of the previous case. The axial distribution of shear stress ( CTxy ) near the free end zone, which is taken from the level of y=0.0675, rises steeply, far beyond its average level, and drops steeply near the free end to near zero. These stress distribution trends are similar to the case which is taken from the upper level of the adhesive at y=0.08. The stress distribution at the level of y=0.0925 shows same trend as the previous case except that the lateral normal stress ( CTy ) changes its sign and drops to a negative value which means compressive stress, immediately after the peak positive value in the end zone. The axial distributions of stresses (a y , 0xy ) along the boundary between adhesive and outer adherend are plotted in the same way as for the boundary between the central adherend and adhesive. The results are shown in Figure 3.10. They show that the shear stress ( 0xy ) drops to near zero near the free end. The lateral normal stress (a y ) shows that after the peak positive stress value near the end zone, it suddenly drops to the negative value which means compressive stress. We find that along the boundary between the central or

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4 0. 04 CO CO ID 0. 02 CO _l < cr Q -0.02 < cr LU < _J Q LU N cr O -0. 04 -0. 06 -0.08 -0.1 -0. 12 -0.01 V) in LU -0.02 LT h00 (X -0.03 < LU I L0 -0. 04 Q LU N _J -0. 05 o -0.06 -0. 07 0 2 0 « 0.6 o.B NORMALIZED AXIAL COORDINATE x/c NORMALIZED AXIAL COORD I NATE "^x/c Figure 3.10 Case 1. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the outer adherend and adhesive

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49 outer adherend and adhesive the stress is not well defined at the ends of adhesive. Therefore one would not expect the free end condition (a x =0, ^=0) to be satisfied at the very near end zone. However, the end condition (o^o^O) at the various levels inside the adhesive, which means no external traction force in the x or y direction, is almost satisfied even by using the displacement finite element method . To check the stress variation through the thickness direction of the adhesive, the interlaminar adhesive layer is divided into four longitudinal strips as shown in Figure 3.9. The results from the data plotted for various levels of y in Appendix A show that through the adhesive thickness the shear, axial and lateral normal stresses are pretty uniform through the adhesive thickness within the zone -0.975 < x/c < 0.975, but near the free end zones (0.98 < x/c < l and -1 < x/c < 0.98) the stress changes rapidly through the adhesive thickness. If we assume that the stresses are uniform through the adhesive thickness, we can not predict the compressive lateral normal stress of the adhesive layer. For a double-doubler lap joint (Figure 3.1b), the stress distributions on the boundaries between central adherend and adhesive and between outer adherend and adhesive are shown in Figure 3.11 and Figure 3.12, respectively. The axial distributions of stress at different levels of y are shown in Appendix B. From the results we find that the axial distribution of the lateral normal stress { o y ) , axial normal

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in w £ Eh in < « Pi O s; < OS W E< w IS) M « S o 0.25 0. 15 0.05 -0.1 -0.2 AXIAL COORDINATE x/c ' 1 1 ' 1 1 1 — — i 1 — 02 °* 0.6 0.8 , AXIAL COORDINATE x/c Figure 3.11 Case 1. The stress distributions of a doubledoubler joint along the boundary between the central adherend and adhesive

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CO w OS o to u E< w CO I — ( < OS o to 0.04 0.02 -0.02 -0.04 -0.06 -0.1 -o.oi co CO w -0.02 CO
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52 stress (a x ) and the shear stress ( CTxy ) are very uniform and at a very low level between 0.3 < x/c <0.8 and -0.3 < x/c <-0.8. But near the corner edge of adhesive line and bonded corner at the butted region between two central adherends, there is a very high stress concentration and very complicated stress distribution. The peak stress values obtained from the finite element method are not guite reliable, since the peak values vary depending upon the mesh size. The stress distributions, except near the bonded corner in the butted region, show similiar trends to the ones of double-doubler reinforcement of a continous member. Near the bonded corner of the butted region, the shear stress ( 0xy ) is very large. The lateral normal stress (a y ) has its highest peak value at the edge of the interface between central adherend and adhesive. This lateral normal stress (o y ) becomes compressive at the level of y=0.0925 which is very near the interface between adhesive and outer adherend, and at that interface it shows very high compressive lateral normal stress (a ) . * y / 3.5.2 Case 2. Aluminum Central . Adhere , E poxv Ari h^-i^ and Composi te ggg Adherent g g&g g < The model for the second case is composed of aluminum central adherend, epoxy adhesive and graph ite/epoxy composite outer adherend. A unidirectional graphite/epoxy laminate is used for outer adherend.

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53 For double-doubler reinforcement of a continous member, the axial distribution of the cohesive lateral normal stress (CT y ), shear stress (a^) and the axial normal stress (o x ) are plotted from the averaged nodal stress values and are shown in Appendix C for various levels of y. The axial distribution of the lateral normal stress and shear stress along the boundary between the central adherend and adhesive are plotted from the averaged nodal stress values stress and are shown in Figure 3.13. The stress distributions on the boundary between the adhesive and outer adherend are plotted in the same way and are shown in Figure 3.14. For a double-doubler joint, the axial distributions of the cohesive stress at different levels of y are plotted and shown in Appendix D. The stress distributions on the boundary between the central adherend and adhesive, and the outer adherend and adhesive, are shown in Figure 3.15 and Figure 3.16, respectively. The stress distribution trends are similar to Case 1, but higher peak stress values are obtained from Case 2. 3 ' 5 ' 3 Bendin minUI " Adher P nds and R P QXV ArihP
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54 0. 16 cr l/l 0. 12 < o 2 < or LU 0. 1 0. 08 0. 06 0. 04 Q LU N 0.02 cr o z -0. 02 -0.09 I 1 — i r 1 r 02 °.i o.6 0.8 NORMALIZED AXIAL COORDINATE x/c NORMALIZED AXIAL COORD I NATE °" 8 x/c * Figure 3.13 Case 2 . The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and adhesive

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55 -o.oi in -0.02 m LU cr \-0.03 in DC < LU -0.04 X Q -0.05 LU N _l -0.06 < cr -0.07 -0.08 " r ' i 1 1 1 r 02 0* 0.6 0.8 NORMALIZED AXIAL COORDINATE X/C 0. 2 T NORMALIZED AXIAL COORD I NATE °" "x/ c Figure 3.14 Case 2. The stress distributions of double-doubler reinforcement of a continous member alonq the boundary between the outer adherend and adhesive

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CO if) 0.3 DC \— o 0.2 < cc 01 LU Q LU N o o -0.1 -0.02 -0.04 -0.06 _J -0.08 -0.1 1 1 I 1 02 «>•« 0.6 0.8 NORMALIZED AXIAL COORDINATE x/C NORMALIZED AXIAL COORD I NATE ' *x/C Figure 3 . 15 Case 2 . The stress distributions of a double doubler ]omt along the boundary between the central adherend and adhesive

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Figure 3.16 Case 2. The stress distributions of a double doubler joint along the boundary between the outer adherend and adhesive

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58 condition equivalent to a couple is used at the outer end of the central adherend. For double-doubler reinforcement of a continous member, the axial distributions of the cohesive stress are plotted from the averaged nodal stress values and shown in Appendix E. We have four overlap interfaces in the pure bending case. The axial stress (a y , 0xy ) distributions on the boundaries between the central adherend and upper adhesive and between the upper outer adherend and adhesive are plotted from averaged nodal stress values and shown in Figure 3.17 and Figure 3.18, respectively. The axial stress distributions (a y , 0xy ) on the boundaries between the central adherend and lower adhesive and between the lower outer adherend and adhesive are shown in Figure 3.19 and Figure 3.20, respectively. As expected, the shear stress is symmetric with respect to the mid plane, while the lateral normal stress is approximately antisymmetric. For a double-doubler lap joint, the axial distributions of the cohesive stress for various levels of y in the upper and lower adhesive are shown in Appendix F. The axial stress distributions (o y , »,) on the boundary between the central adherend and upper adhesive and between the upper outer adherend and adhesive are shown in Figure 3.21 and Figure 3.22, respectively. The axial stress distributions ( 0y , Qxj ) on the boundary between the central adherend and lower

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59 adhesive and between the lower outer adherend and adhesive are shown in Figure 3.23 and Figure 3.24, respectively.

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60 0.05 CD in LU o cr ILO o < cr _ 0 , LU 0 h< I -0.Z6 o LU N -° 3 < -0.36 o z 0. 12 -i CO 0.1 t/1 LU cr f0.08 cr < LU X 0. 06 CO Q LU 0. 04 N AL 0.02 cr O z 0 -1 1 1 1— 0.2 0.4 i r 0. 6 1 r o. e NORMALIZED AXIAL COORDINATE X/C NORMALIZED AX"lAL COORDINATE" x/c Figure 3.17 Case 3. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the central adherend and upper adhesive ^

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61 0. 06 cn UJ cr i— CO _l < cr o < cr LU < Q LU N cr o -0.04 -o.i 0. 12 NORMALIZED AXIAL COORDINATE x/c ' CO 0.1 CO LU cr 00 0. 08 cr < LU X CO 0 06 Q LU N 0. 04 < o 2 0.02 02 °« 0.6 o.e NORMALIZED AXIAL COORDINATE X/C Figure 3 . 18 Case 3 . The stress distributions of double-doubler reinforcement of a continous member along the boundary between the upper outer adherend and adhesive

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62 0. 35 CO Ul LU a. I< tr 2 < cr Q LU N < LT 2 Q LU N d O 2 -0. 05 0. 12 0.08 0. 06 0.04 0. 02 -i 1 r 0* 0.6 AXIAL COORDINATE x/c Figure 3.19 Case 3. The stress distributions of double-doubler reinforcement of a oontinous member along the adhesi^ betWeen Central adherend ™* later

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63 V) in LU cr < cr LU t— < D LU N GO CO 111 cr hcr < LU X CO Q UJ N 0.1 0.06 0.06 0.04 0.02 -0.04 -0.06 0.1 0. 08 0.06 0.04 . 02 AXIAL COORDINATE x/C AXIAL COORDINATE x/c Figure 3.20 Case 3 . The stress distributions of double-doubler of a continous member along the boundary between the lower outer adherend and adhesive

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. 05 10 LU cr to < -o. t a. LU Q -0. 3 LU N LT -0.4 o C/D LU cr o. it cr < LU X in Q LU N or o 0. 1 0. 08 0. 06 0. 02 0 2 0.4 0.6 0.8 NORMALIZED AXIAL COORDINATE x/c °2 °< 0.6 0.8 NORMALIZED AXIAL COORDINATE X/C Figure 3.21 Case 3. The stress distributions of a double doubler joint along the boundary between the central adherend and upper adhesive

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0.12 CO to LU 0-1 cr hm o. ob < 0. 06 cr O 0. 04 0. 12 to 10 UJ cr ito cr < LU X lO LU N 0. 04 cr o ~Z. 002 NORMALIZED AXIAL COORD I NATE "^x/c 0.2 0.4 NORMALIZED AXIAL COORD I NATe" " x/ c Figure 3.22 Case 3. The stress distributions of a doubledoubler joint along the boundary between the upper outer adherend and adhesive

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66 CO CO o« LLI tr co _j < 0.3 DC < LT LLI 0.2 0.14 0.12 CO CO LU cr h0.1 co cr < 0.08 LU X CO Q 0.06 LU N _l < 0.04 NORMALIZED AXIAL COORDINATE x/c 02 0.4 0.6 o.o NORMALIZED AXIAL COORDINATE x/c Figure 3.23 Case 3. The stress distributions of a doubledoubler joint along the boundary between the central adherend and lower adhesive

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0.12 0.1 If) If) LU cc h0. 06 m cr < LU X 0. 06 \f) Q LU N 0.04 _J < LT NO 0. 02 1 i 1 1 1 r 1 1 0-2 0.4 0.6 O.B NORMALIZED AXIAL COORDINATE x/C 02 0* 0.6 o B NORMALIZED AXIAL COORDINATE x/C Figure 3.24 Case 3. The stress distributions of a doubledoubler nomt along the boundary between the lower outer adherend and adhesive

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68 3.5.4 Case 4. Aluminum C entral Adherend . Epoxv Adhesive , and Composite Outer Ad herend in Pure Rending The model for Case 4 consists of aluminum central adherend, epoxy adhesive, and graphite/ epoxy outer adherend under pure bending in the positive sense. For double-doubler reinforcement of a continous member, the axial distributions of the cohesive stress at various levels of y in the upper adhesive are shown in Appendix G. The axial stress distributions (a y , a v ) on the boundary between the central adherend and upper adhesive and between the upper outer adherend and adhesive are shown in Figure 3.25 and Figure 3.26, respectively. The axial stress distributions K, a v ) on the boundary between the central adherend and lower adhesive and between the lower outer adherend and adhesive are shown in Figure 3.27 and Figure 3.28, respectively. For a double-doubler lap joint, the axial distributions of the cohesive stresses for various levels of y in the upper adhesive are shown in Appendix H. The stress distributions K, a v ) on the boundary between the central adherend and upper adhesive and between the upper outer adherend and adhesive are shown in Figure 3.29 and Figure 3.30, respectively. The stress distributions K , 0xy) on tne boundary between the central adherend and lower adhesive and between the lower outer adherend and adhesive are shown in Figure 3.31 and Figure 3.32, respectively. The stress

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69 distribution trends are similar to Case 3, but higher peak stress values are obtained from Case 4.

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70 0. 05 m LU cr cr -o.i —i LU < -0.2 D HI N < . 0.3 0.12 -, 0.1 to RE IS o.oe cr < LU I 0.06 lO D LU N _l 0.04 or o 0.02 z 0 2 0.4 0.6 o.B NORMALIZED AXIAL COORDINATE x/c 02 0.4 0.6 0 B NORMAL I ZED AX I AL COORD I NATE ' x/c Figure 3.25 Case 4. The stress distributions of double-doubler reinforcement of a continous member along the adhesive C6ntral adherend and upper

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71 CO CO LU CC 0. 08 co cr < LU 0. 06 CO D LU M 0. 04 1 : ON 0.02 02 0.* 0.6 0.8 NORMALIZED AXIAL COORDINATE x/C Figure 3.26 Case 4 The s ^ distributions of double-doubler reinforcement of a continous member along the adhere *" ^ UPPGr ° Uter ad ^end and

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7 Figure 3.27 Case 4. The stress distributions of double-doubler ^n^T^ ° f 3 Vinous member along the SSESlS StWeen CGntral adherend ™* 1-er

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73 02 0.4 D.6 0.8 NORMALIZED AXIAL COORDINATE X/c O.i w LU cr 0.08 cr < LU I 0.06 LO O 111 N _J 0.04 cr o 0.02 z ° 02 0.4 0.6 08 NORMALIZED AXIAL COORDINATE x/c Figure 3.28 Case 4. The stress distributions of double-doubler reinforcement of a continous member along the boundary between the lower outer adherend and adhesive

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0.05 in in LU cr H cr o < LU LU N -0. 3 cr O -0.4 02 0.* 0.6 0.8 NORMAL I ZED AXIAL COORDINATE x/c NORMALIZED AXIAL COORDINATE x/c Figure 3.29 Case 4. The stress distributions of a double doubler joint along the boundary between the central adherend and upper adhesive

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0. 12 02 0.4 0.6 oe NORMALIZED AXIAL COORDINATE X/C °2 0 * 0.6 0.8 NORMALIZED AXIAL COORDINATE x/c Figure 3.30 Case 4 . The stress distributions of a doubl doubler : omt along the boundary between the upper outer adherend and adhesive

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NORMALIZED AXIAL COORDINATE x/c Figure 3.31 Case 4. The stress distributions of a doubledoubler joint along the boundary between the central adherend and lower adhesive

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0 2 0.4 0.6 0.8 NORMALIZED AXIAL COORDINATE x/C 0. 12 0.1 0.08 CO ID cr i— CO CC < LU X CO Q LU N 0. 04 O 0. 02 02 °« 0.6 0.8 NORMALIZED AXIAL COORDINATE x/c Figure 3.32 Case 4. The stress distributions of a doubledoubler joint along the boundary between the lower outer adherend and adhesive

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CHAPTER 4 PARAMETRIC STUDY OF THE DOUBLE-DOUBLER JOINT 4 . 1 Introduction The objective of this chapter is to study the parameters which influence the stress distributions, so that designers can design the adhesive bonded joints more efficiently. The ANSYS finite element program is used as the computational tool to determine the influence of various joint parameters on stress distributions in the double-doubler joint under tesile loading. The stress distributions presented in this parametric study are normalized with respect to the applied average stress p av acting on the central adherend. The influence of the parameters is studied by varying one parameter at a time from a double-doubler joint which is described in Figure 3.1. The geometric and material parameters which are examined are a) bonding half length c b) the ratio of the adherend modulus to the modulus of the adhesive E/E a For the material parameter study, the central and outer adherends are made of the same isotropic material. 78

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79 The thickness of the central adherend, 2t c , is constant through the study. All geometric parameters are nondimensionalized by dividing by t e . Note that x/c = (x/t c )/(c/t c ) . Material parameters are nondimensionalized by dividing the adherend modulus E by the modulus of the adhesive E.. The influence of E/E a on the stresses for simple tension is dicussed in Section 4.2. The influence of the length of overlap on the stresses for simple tension is examined in Section 4.3. 4.2 The Material ParaTne>1-gTThe material parameters consist of the mechanical properties of the adherend and adhesive. The mechanical properties of the any isotropic elastic material are the elastic modulus and the Poisson's ratio. Stresses are affected by variation in Poisson's ratio if the changes in Poission's ratio are very large. since the range of the variation of Poisson's ratio for common adhesive systems is very limited, only a single material parameter is considered in this analysis. This single material parameter is the ratio of the adherend modulus to the adhesive modulus. To study the influence of the material parameters four joints are analyzed. The adherends in the four joints have four different moduli of elasticity but the same adhesive. The modulus of elasticity is the same for central adherend and outer

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80 adherend. The Poisson ratio of the adherend is constant at 0.33. The variation in the shear and lateral normal stress distributions in the adhesive are examined for E/E a = l, E/E a = 30, E/E a = loo, and E/E a = 200. in this study the modulus of the adhesive E a is 0.356xl0 6 psi and the constant Poisson 1 s ratio of the adhesive is 0.39. The stress distribution and singularity parameters in this double-doubler joint are studied for four ratios of E/E a . 4.2.1 Influence of E/E a The influence of the ratio of the modulus of adherends, E, to the modulus of the adhesive, E a , on the stresses in a double-doubler joint is examined here. The influence of E/E a on the axial distributions of shear stress at the interface with the center and outer adherend is shown in Figure 4.2 and 4.4, respectively. Each figure consists of two parts. One frame is the neighborhood of the free end surface of the adhesive; the other frame is the neighborhood of the butted adhesive corner. The magnitude of the shear stress decreases rapidly from the neighborhood of the end of adhesive corner and is almost uniform, and equal to the nominal shear stress, over the middle three-fifths of the overlap (0.2 < x/c < 0.8 and -0.2 < x/c <-o.8). The shear stress then drops to zero at the free end surface of adhesive and outer adherend. As E/E a increases, the shear stress concentration factor (r max /r av )

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81 decreases and the order of stress singularity ,A, decreases. (See chapter 5.3 for singularity order A) The magnitude of the highest shear stress is greatest when E = E a . The location of maximum shear stress is not affected when E/E a is varied. The variation of stress concentration factor and singularity order a with respect to E/E a is given in Table 4.1. The ideal double-doubler joint would have a uniform shear stress distribution and would also have no lateral normal stress. The distributions of lateral normal stress on the boundary between the central adherend and adhesive and between the adhesive and outer adherend are shown in Figure 4.1 and Figure 4.3, respectively. Each figure consists of two parts, one frame is the neighborhood of the free end surface of the adhesive; the other frame is the neighborhood of the butted adhesive corner. The lateral normal stress is tensile on the end of boundary between central adherend and adhesive and is compressive on the end of boundary between adhesive and outer adherend. The lateral normal stresses are almost uniform and almost zero in magnitude over the middle threefifths of the overlap (-0.2 < x/c <-o.9 and 0.2 < x/c < 0.9) but rise rapidly to tension and compression peaks at the respective ends of the overlap. The axial distribution of lateral normal stress near the free end of the adhesive layer also have tension and compression peaks depending upon the level of y. it is quite obvious that compression and tension lateral normal stress must simultaneously exist in order to

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82 satisfy the vertical equilibrium condition of the adhesive layer. If we assume that the stresses through the adhesive thickness are uniform, we can not predict any compressive lateral normal stress in the adhesive layer. The tensile lateral normal stress concentration factor at the interface between central adherend and adhesive is always greater than that at the interface between adhesive and outer adherend. This concentration factor decreases as E/E a increases. The greatest concentration factor is obtained at both interfaces for E = E a . It is seen from Figures 4.1 to 4.4 that the magnitude of the maximum lateral normal stress is always greater than the magnitude of the maximum shear stress in a given joint. Therefore the magnitude of the lateral normal stress will be the governing criterion for joint failure in this double-doubler joint. Table 4.1 Shear stress concentration factor and the order of stress singularity a for case l in Chapter 5.3 shear stress concentration factor order of stress singularity a E/Ea = 1 23.2 0.465 E/Ea =30 3.71 0.355 E/Ea = 100 1.95 0.348 E/Ea =200 1.45 0.346 respect to average shear stress

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0. 4 CD W 1 1 1 LLf cr h0. 3 w i < cr 0 2 Z < a: LU 1— 0.1 LA' Q LU N D AL cr o -0.1 0 . 0. 02 -] in in LU EX 1-0.O2 10 _l -0.04 < -0 . 06 — cr O 0 . 08 — AL -0. 1 cr LU -0 . 12 — < _J -0. 14 Q LU N -0.16 IAL -0.18 ;> cr ON -0.2 -1 1 1 1 ) ( E/Ea=1 E/ Ea=30 / ( E/ Ea=lO0 r 1 r I E/Ea=2Q0 i 1 i 1 / J r 1 ' 1 < J 1 ' I /is LJ /i r ' 1 1 1 i 1 r 1 1 1 1 r / / s 0 . 87 NORMALIZED AXIAL COORDINATE x/c 0 99 NORMALIZED AXIAL COORDINATE x/c Figure 4.1 Axial lateral normal stress distribution on the

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111 cr \in a < LU X 111 N < in in LU cr hm cr < IT) O LU N < cr -0.12 -0. 16 -0. 2 -0 24 E/Ea=l E/Ea=:30 E/Ea=100 E/Ea=200 ~ i 1 1 1 1 1 1 — 0. 85 0 87 0 89 0 91 0. 93 0 95 NORMALIZED AXIAL COORDINATE x/C -0.02 -0. 04 -o. oe -o : -0.12 -0.16 -0.18 -C ! " 1 1 0 97 0 99 NORMALIZED AXIAL COORDINATE X/C Figure 4.2 Axial shear stress distribution on the boundary between the central adherend and adhesive for four values of E/E a or

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0. 05 in CO LU cr h< cr < LU Q LU N Q CO 00 LU LT f00 < EC LU h< o LU N CL o 0. 04 0. 03 0. 02 0. 01 0 --0.01 -0. 02 -0. 03 -0. 04 -0. 05 E/ Ea= 1 E/ Ea=30 E/ Ea=100 E/ Ea=200 0.1 0. 0B 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 NORMALIZED AXIAI COORD I NATE x/r E/Ea=1 •E/ Ea=30 — E/ Ea=100 E/Ea=200 i i i i 1 r 0.87 0.89 0.91 0. 93 1 I I 1 1 0.95 0.97 0.99 Figure 4 NORMALIZED AXIAL COORDINATE x/c 3 Axial lateral normal stress distribution on the boundary between the outer adherend and adhesive four values of E/E

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86 to CO UJ cr hcr < LU X 10 D LU N 01 o -0.04 -0. 14 — E/ Ea= 1 E/ Ea=30 E/Ea=100 E/ Ea=200 D.2 , 4 NORMALIZED AXIAL COORDINATE x/c on t/D UJ cr hCO LT < LU I u r ' D LU N < o -0.12 -0.16 -0.2 -0.24 E/Ea=1 E/Ea=30 -E/ Ea=100 -E/ Ea=200 -0.28 t — r 0. 8 I I I 1 1 0. 84 0. 88 i i i — i 1 — r 0.92 0.96 NORMALIZED AXIAL COORDINATE x/c Figure 4.4 Axial shear stress distribution on the boundary between the outer adherend and adhesive for four different values of E/E

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87 4.3 The Geometric Parameters The influence of the geometric parameters is investigated by varying one dimension while all others are held the same as that of original joint. The dimensions of the joint are given in the notation shown in Figure 3.2. half length of overlap: c = 2.56 thickness of central adherend: 2t c = 0.12" thickness of outer adherend: t G = 0.04" thickness of adhesive: t a = 0.04" The thickness of the central adherend is held constant throughout the analysis. The material parameters of the adherends and adhesive are the same for all the joints analyzed for the study of the influence of the geometric parameters : adherend modulus: E = 10.6E6 psi adherend Poisson's ratio: v = 0.33 adhesive modulus: E = 3.6E6 psi adhesive Poisson's ratio v = 0.39 4.3.1 Influenc e of Overlap T,Pngth The length of overlap is usually considered to be the most important parameter to the designer. As the applied load is increased, the overlap length must be increased in order to reduce the magnitudes of the average stresses. The stress distributions in three double-doubler joints are analyzed here. All geometries of the three joints are the same as in

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88 original joint (c/t c = 43) except the overlap lengths which range between c/t c = 30 and c/t e = 50. The shear stress distributions on the boundary between the central adherend and adhesive and between the outer adherend and adhesive for the three double-doubler joints (c/t c = 30, c/t c = 43, c/t c = 50) show that the shear stress concentration factors for the three cases are almost the same. There is no significant difference in stress distribution shape and normalized by p av peak stress values on the range between c/t c =30 and c/t c =50. Although the concentration factors are about the same, the actual peak stresses are significantly higher for the shorter overlap, because the average stresses are higher. The lateral normal stress distribution on the boundary between the central adherend and adhesive, and between the outer adherend and adhesive also shows that there is no significant change on the range between c/t e = 30 and c/t c = 50. For overlap length c/t c = l (not shown) the highest shear stress occurs at the butt corner of the overlap between two central adherends. As the length of overlap increases significantly, the position of the highest shear stress approaches the ends of the overlap. The maximum lateral normal stress occurs at the ends of the overlap for all three joints.

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CHAPTER 5 TREATMENT OF STRESS SINGULARITY AT BONDING CORNERS IN THE DOUBLE-DOUBLER JOINT 5.1 Introduction In the linear elastic analysis, stress fields near the two bonding edges between the adherend and the adhesive layer show singular behavior. However, the finite element method solutions provide finite valued stresses at the bonding corners and in these regions the maximum stresses, calculated by the finite element stress analysis, depend on the size of element meshes and are therefore not guite reliable, m order to describe the stress fields around these corners, singular stress parameters are used to evaluate the magnitude of the stress fields near the singularity point of the double-doubler adhesive joint. The assumption is that at some small but finite distance from the singular point where the material is actually elastic, the singularity solution gives a closer approximiation to the stress variation with distance from the corner than the finite element solution does. This singularity solution method uses two parameters, ClJ and a = iPl . Th e Se parameters express the stress distribution near a bonding edge along the interface, as follows. 89

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90 "H = C i3 /r x 5.1 where a i6 is stress, r is distance from singularity point, 0^ is intensity of stress singularity, A is order of stress singualrity. In this chapter the ways to find the and A for doubledoubler joints are presented. For most practical purposes the width of an adhesive joint is large compared to the thickness of the adherend and the adhesive. Therefore plane strain analysis is used throughout. 5.2 Stress Singularities at Interface Cnrnprs in Bonded Dis similar Materials The typical study of the stress singularity at the interface corner in bonded dissimilar elastic wedges has been done extensively by Bogy [13, 14, 15]. Let D2, Dl denote the open two-dimensional regions occupied by the cross section of the two wedges of angles, a, b, respectively (a+b < 2*) , and suppose that they have one straight segment B of their boundaries in common (Figure 5.1) . Denote their remaining straight boundary segments by Bl, B2 . Let (r,e) be plane polar coordinates of a point with rectangular cartesian coordinates (x,y) . The material properties are represented by their respective shear moduli

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91 y A Figure 5.1 Two edge-bonded elastic wedges of different materials under normal and shear loading

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92 and Poission's ratios (G^i/i) and (G 2 ,i/ 2 ). Shear and normal tractions are applied to their free boundaries: t x (r) and n x (r) on Bl, and t 2 (r) and n 2 (r) on B2. The traction is independent of the coordinate perpendicular to the cross section. Therefore we consider this boundary value problem in the plane theory of linear elastostatics. The problem now is to find the stress and displacement fields with plane polar components related to the Airy stress function $ X/ $ 2 , which are suitably defined and satisfy the bi-harmonic equation V* * = 0 in Dl, D2 5>2 The stresses at any point (r,e) can be derived by l a i a 2 + r ar r 2 ae 2 a 2 ^ 2 , dr 1 d 2 <(> l d
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93 aue ug i au r i i d 2 i a 3r r r 39 g r arae r 2 ae which satisfy following boundary conditions on Bl, B2 (*i)ee(r,-b) = n x (r) (^ 2 )ee(r,a) = n 2 (r) (^i)re(r,-b) = t x (r) (a 2 ) re (r,a) = t 2 (r) 5.4 as well as the conditions of bonding on B (continuity of traction and displacement components) (*i)ee(r,0) = (a 2 ) 6e (r f 0) (aJ^f^O) = (a 2 ) r6 (r,0) 5.5 (uj r (r,0) = (u 2 ) r (r,0) (uJ.fr^O) = (u 2 ) e (r,0) In these equations the subscripts 1 and 2 indicate quantities in wedges I and II, respectively. m addition we shall require the stress components of the two stress fields to specify the regularity conditions °rr, a ze 0(r~ ) as r t » for every s > 0 55

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94 The Mellin transform is the most convenient tool for solving the boundary-value problem. We denote the Mellin transform of a function f defined and suitably regular on (0, «) by g(f '* S) = L f (r) r * ldr 5 ' 7 where s is the complex transform parameter. Let t(a,B), S„(s,e), a M (a f e) f a re (s,e), u r (s,e), u.(s,e) ( n(s), €(s) in this order denote the Mellin transforms with respect to r of *(r,e), r 2 a rr (r,e), r^^e), r 2 a r9 (r,e), ru r (r,e), rue(r,e), r 2 n(r) , and r 2 t(r). A formal application of the Mellin transform to Equation 5.2 yields an ordiary differential equation d 2 d 2 .2 + s 3 C —T + ( s + 2) 2 ] $ (s ,e) = 0 5.8 The general solution for this differential equation is given by Bogy [13, 14, 15] by $(s,e) = a(s)sin(se) + b(s)cos(se) + c(s)sin(s + 2 e) + d(s)cos(s +26) 5 9

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95 The functions a(s) , b(s) , etc., (al, bl, etc., for £ and a2, b2, etc., for £ 2 ) are to be determined through the transforms of Equation 5.3 from the transforms of the boundary condititons Equation 5.4 and Equation 5.5. After use of Equation 5.7 these transformed equations appear as a a 2 *rr(S,e) = ( -f Q 2 ~ S) £( S ,0) £ee(s,6) = s(s + 1) $(s,e) $«e(s,e) = (s+1) -| e $(s r e) 5.10 u r (s,e) = 1/2G [s £(s,e) +mc(s)sin(se+2e)+md(s)cos(se+2e) ] u e (s,e) = 1/2G [_d_ $(s,e) + mc(s)cos(se + 29) md(s) sin(se+26) ] ( *i)ee(s,-b) = njs) ( Sj^s^b) = t l(s) 5.11 ( S 2 )ee(s, a) = n 2 (s) ( S 2 ) re ( S ,a) = € 2 ( S ) ( ^i)ee(s,0) ( ^,(8,0) " ( ^2)ee(s,0) = ( u 2 ) r (s,0) ( *l)r9(S,0) ( U 2 ) e (s,0) = ( *2)re(S,0) 5.12 = ( Ui)8(s,0)

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96 By substitution of Equation 5.9 and Equation 5.10 into Equation 5.11 and Equation 5.12, a system of eight equations for the eight unknown functions al(s), bl(s), cl(s), dl(s), a2(s), b2(s), c2(s), d2(s) is obtained. Then Bogy is concerned with determining the dependence of the order of the singularity in the stress fields as r -» 0 on the angle, a, b and the material properties G lr u x , G 2 , u 2 . It is shown in reference [13, 14, 15] that the stress fields have a singularity of order r'^log r, or 1 as r -» 0 according as the determinant of the coefficient in the system of eight equations has a zero s in -2 < Re(s)< -l, has no zero in this open strip but s = -2 is a zero of order two, has no zero in the strip and s = -2 is only a simple zero. Therefore attention is focused on the determinant of the coefficients in the system of equations with a view toward locating its zeros in -2 < Re(s) < -i. By letting Q denote the value of the determinant of the coefficients of these eight equations the following equation is obtained. Q(a,b,a,fl;p) = A(a,b;p)B 2 + 2B(a,b;p)aB + C(a,b;p)a 2 +2D(a,b;p)B + 2E(a,b;p) Q + F(a,b;p) 5.13 where A(a,b;p) = 4M(p,a)M(p,b) B(a,b;p) = 2p 2 sin 2 (a)M(p,b) + 2p 2 sin 2 (b)M(p,a) C(a,b;p) = 4p 2 (p 2 -i)sin 2 (a)sin 2 (b) + M[p, (a-b) ] 5. 14 D(a,b;p) = 2p 2 [sin 2 (a)sin 2 (pb) sin 2 (b) sin 2 (pa) ]

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97 E(a,b;p) = -D(a,b;p) + M(p,b) M(p,a) F(a,b;p) = M[p, (a+b) ] in which the auxiliary function M(p,x) is defined by M(p,x) = sin 2 (px) p 2 sin 2 (x) 5 . 15 In Equation 5.13, Equation 5.14 and Equation 5.15 p = p + iri is a complex varible. a = G i m ? " G ? m i a _ G,(m, 2) G,fm, 2) + ' ~ Gl m 2 + G 2 m, 5 ' 16 where in 4(1 v) for plane strain = 4/(1 + u) for generalized plane stress If we let Q=0 (Equation 5.13) and solve for the smallest value of p, which is called singularity parameter, Pl , then the orders of singularity in the stress field of wedges I and II are found to be °i, o 2 = 0(r p l ) if p= Pl is real Otr^cosf^log r) or r^sin^log r) ] 5.17 lf Pi = £1 + »?i is complex The value of the singularity parameter Pl in Equation 5.17 is dependent upon the two composite material constants * and 6

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98 from Equation 5.16. Stress near the corner point can thus be expressed for real p x =p as °ti = c" ij f ij (e)r p 1 5 . 18 where c" u is a constant and f^fe) is a function of e. Along the interface, e = 0, stresses do not depend on e, and we can write " c ij rP 1 5.19 where c'y = C ^(0) is a constant. In component form for e = 0 Equation 5.19 can be written as follows for real Pl =p o x = a rr = c» x r p 1 5.20 o„ a re = C' r p_1 5.3 Calculation of Order and intensifies of Stress Singularities at the Bonding Corners nf Doubl e-noubler Jninr In a double-doubler joint model we can specialize the foregoing discussions for the two different cases. 5.3.1 Case 1. Half Plane R p nded to a Quarts Piano We have four different locations for this case (Figure 5.2). However, the singularity parameter Pl has the same value at each of the four locations for the simple tension case because of the symmetric loading and geometric conditions. The expressions in Equation 5.14 can be written as follows:

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100 A = 4sin 2 (pw) [sin 2 (pw/2) p 2 ] B = 2p 2 sin 2 (pjr) C = sin 2 (p*/2) 5>21 D = -2p 2 sin 2 (p*) E = (2p 2 l)sin 2 (p*) + sin 2 (p,r/2) p 2 F = sin 2 (3p w /2) p 2 Substitute this Equation 5.21 into the Equation 5.13. Then Equation 5.13 becomes Q(*,>r/2,a,B;p) = 4sin 2 (p w ) [sin 2 (p7r/2 p 2 ]B 2 + 2x2p 2 sin 2 (p7r)a6 + sin 2 (p^/2) Q 2 5.22 + 2[-2p 2 sin 2 ( P} r)]B + 2[(2p 2 l)sin 2 (p,r) + sin 2 (p w /2) p 2 ] + sin 2 (3 pJr /2) p 2 The solution of Q=0 in Equation 5.13 for the smallest p yields the singularity parameter Pl . This parameter Pl depends on only two factors; the material constants a and B. From the given material properties (Table 5.1) , we find the value « and B using the Equation 5.16. Table. 5.1. Selected material properties for material Young's Modulus Poission's ratio Aluminum 10.6E+6 psi 0.33 Epoxy 3.56E+5 psi 0.39

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101 A computer program was written to solve the characteristic equation (Equation 5.22). The Muller method [42] was used for this computer program. This method is used to find any prescribed number of zeros, real or complex, of an arbitrary function. The values found for a, B are -0.93 and -0.165, respectively. Using these a and 6 values we find that p 1 = 0.645. Then Equation 5.20 can be rewritten in the following form by dividing by the average applied axial normal stress (Pav = P/2t c ) acting on the central adherend. In the case that r goes in the negative x-direction along the interfacial plane between central adherend and adhesive (r 1 direction in Figure 5.2), the Equation 5.20 can be written in the follwing form. Here the axial normal stress a x is not considered since it is not well defined on the interface boundary between central adherend and adhesive. , °y . r Pi-l r -0.355 ( — ) = Cy ( ) =C y ( ) Pav C c 5.23 , Qxy . r Pi-l r -0.355 ( — ) " ( ) = Cxy ( ) Pav C c For the case that r goes in the positive y direction at point B (r 2 direction in Figure 5.2), the Equation 5.20 can be written in the following form. Here we assume that joint boundary BBis parallel to the y axis at x=2.56. It is found

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102 that a x , ff^y are almost zero near the free end of adhesive layer since there is no traction force in the x or y direction on this free end of the adesive layer. Therefore the intensity of stress singularities C x and are very small. Only the a y field is considered. °y r p^l r -0.355 ( — )=C y ( ) =C y ( ) 5.24 P.v t a t a c x/ Cy/ Cxy are called the "intensities of stress singularity", which are constant for a specific joint under a specific loading conditions. If we know these factors for a joint, the stresses near the bonding corner along a bond can easily be found by using Equation 5.23 and Equation 5.24. These explicit equations give infinite stress right at the corner. The stress intensity parameters can be used as a means of failure prediction by comparing the calculated parameters with known critical values of these parameters for the material and comparison of stresses very near the bonding corners where stress singularity occur. In order to find the desired intensities of stress singulaity for the doubledoubler joint, the following procedure is used. 1) Using the specified boundary displacement (SBD) method described in the following section , calculate stress values for a x , a^, a y from the end of the adhesive interface corner (r/c = 0) to several nodal points in the r lt r 2 directions,

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103 which are very near the end of the adhesive interface corner. (See Figure 5.2) 2) Substitute these values into equations Equation 5.23 and Equation 5.24, and find several values for C x , Cy, C^, respectively. 3) Choose the maximum values from the computed values for C x/ The maximum values are taken as the desired intensities of the stress singularity. Substitution of these values into Equation 5.23 and Equation 5.24 gives a set of stress distributions near the singular point. 5.3.2 Case 2. Three Quarter Plan e Bonded to a Quarter Plang (composite full plane a=3w/2, b=jr/2) In order to apply the same general solution, Equation 5.13, to the neighbourhood of the butted end, it is assumed that the end of the central adherend is de-bonded from the adhesive between the two central adherends. This is equivalent to assuming a straight crack that meets the plane of bond at a=3,r/2 and b=*/2. We then have four different singularity locations as with Case 1. (See Figure 5.3) The singularity parameter Pl has the same value at each of the four locations for the simple tension case because of the symmetric loading and geometric conditions as Case l. The expressions in Equation 5.14 can be written as follows.

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104

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105 A = 4 [sin 2 (3p*/2) p 2 ] [sin 2 (p,r/2)-p 2 ] B = 2p 2 [sin 2 (p*/2) + sin 2 (3p w /2) -2p 2 ] C = 4p 2 (p 2 -l) + [sin 2 ( P} r)] D = 2p 2 [sin 2 (p*/2) sin 2 (3p w /2)] 5.25 E = (2p 2 -l) [sin 2 (3p*/2) sin 2 (p*/2)] F = sin 2 (2pw) Substitute this Equation 5.24 into the Equation 5.13. Then Equation 5.13 becomes Q(3*/2,,r/2,ar,B;p) = 4 [sin 2 (3p,r/2) -p 2 ] [sin 2 (p w /2) -p 2 ]B 2 + 4p 2 [sin 2 (p J r/2)+sin 2 (3p*/2)+2p 2 ]aB 5.26 + [4p 2 (p 2 +l) + sin 2 (p ff )] Q 2 + 4p 2 [sin 2 (p ff /2) sin 2 (3p*/2)]B + (4p 2 +2) [sin 2 (3p*/2) sin 2 (p w /2)] Q + sin 2 (2p7r) The solution of Q = 0 for the smallest p yields the singularity parameter Pl . The values found for a, B are a = -0.93 B = -0.165. Using these a and 6 values, we find from Equation 5.26 that Pl = 0.543. The same SBD procedure is used to find the desired intensities of stress singularity as was used for Case 1. Then Equation 5.20 can be rewritten in the following form in same way as Case l. For the case that r goes along the positive x direction of the interface between central adherend

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106 and adhesive from B" (r 3 direction in Figure 5.3), Equation 5.20 can be written in the following form. Here only the axial shear stress is considered since it is the dominant stress field. °xy r p x -i r -0.457 ) = C xy ( ) = Cxy ( ) 5.27 For the case that r goes along the positive y direction from B" (r« direction in Figure 5.3), Equation 5.20 is rewritten in the following form. Axial normal stress (a x) , shear stress (ct^) are considered here since these are the major stress fields here. , a x r p x -i r -0.457 ( — ) = C x ( — ) C, ( — ) Pav t. t . 5.28 ff *y , r Pi -1 r -0.457 — ) = C *y ( — ) = ( ) Pav t. t . 5.4 Specified Boundar y pjspl cement Mat-hod rSRm 5.4.1 Introduct i on In many applications of finite element structural analysis, it becomes necessary to determine concentrated stress in a small portion of a structure. The stress field surrounding a sharp fillet is characterized by rapidly

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107 changing stress gradients, which call for a comparatively fine mesh. It can be difficult to include sufficient mesh refinement in an entire model for accurate stress computation at the concentration. One of approaches to this problem is the specified boundary displacement method (SBD) . The specified boundary displacement method is a technique whereby the boundaries of a specialized model of a sub-region within a structure assume the displacement field from a previous solution of the entire structure. The analyses of the subregion and of the entire structure are separate. What is important is that the initial analysis of the entire structure yields an accurate displacement field where the boundaries of the subregion occur and that the analysis of the subregion accurately recovers stress and strain at the points of interest. Analyses of structures containing stress concentration are excellent applications for the specified boundary displacement method. In reality, it can be often observed that the more localized is the effect of a stress concentration, the greater is the need for a specialized model to recover accurate stress and strain. See, for example, [25, 26, 33]. 5.4.2 Procedure Analysis of a stress concentration by the SBD method proceeds as described below and shown in Figure 5.4. This procedure follows that in [33].

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spec i f i ed boundar ies Figure 5.4 Specified boundary displacement method

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109 1) A finite element model of a large portion of the structure is prepared. This model must have sufficient refinement to obtain the overall displcement field, but need not have sufficient refinement for accurate computation of stress at the concentration. A stress solution is performed for the desired loading with this coarse model. 2) A detailed finite element model of the stress concentration is prepared. This submodel extends some arbitrary distance away from the concentration. 3) Displacements along the specified boundaries of the submodel are obtained from the coarse model solution. For each node along the specified boundary, the appropriate element of the coarse model is determined. The corner displacements of that element are interpolated using its shape function to produce the displacement constraints to be applied to the specified boundary nodes. 4) The submodel stress solution is performed using these interpolated displacements along with any other appropriate boundary conditions. The SBD method permit independent definition of the coarse and sub models. it allows review of coarse model results before decisions are made about local models.

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110 5.4,3 Benefits of Submodel ling Acoording to [25,26], the benefits of submodelling are that 1) It reduces the need for complicated transition regions in solid finite element models. 2) It enables the analyst to decide which areas of a structure will receive detailed analysis after the initial analysis is complete. 3) It enables the analyst to study the effect of local geometric changes around the stress discontinuity of alternate designs. 4) It enables the analyst to demonstrate the adequacy of mesh refinement. 5) The refined mesh boundary doesn't have to follow coarse mesh element boundaries. 5.4.4 Coarse M odel and Submodel Accurate results can be obtained through the proper selection of coarse model and submodel. Two different coarse models and three different sub-models were prepared for the double-doubler joint. From these models the combination of coarse model and submodel that gave the highest peak stresses was selected. Two cases were considered. The submodel for Case 1 is in the corner at the free end of the adhesive as shown in Figure 5.5. For Case 2 the submodel is in the adhesive near the butted corner of the central adherend, as

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Ill shown in Figure 5.6. In both cases the joint is loaded in simple tension. The selected coarse and submodel for Case 1 are shown in Figure 5.5. For Case 2 the same coarse model is used as for Case 1 and another submodel is generated as shown in Figure 5.6. The analysis was carried out using the finite element computer program ANSYS. For the model which is under simple tension loading, only one guarter of the double-doubler joint is selected for this finite element model because of symmetry. For the model which is under pure bending, half of the double-doubler joint will be selected for this finite element model. Both coarse model and sub-model are constructed using ANSYS STIF42, a four-noded isoparametric plane strain element. Using the stress values obtained from the sub-models and substituting these values into Eguation 5.23, Eguation 5.24, Eguation 5.27 and Eguation 5.28, we obtain several different values for C x , Cxy/ C y for Case 1 and Case 2 for two different loading cases, respectively. Here only the simple tension loading case is presented to obtain the values for C x , C^, C, because the procedure to obtain these values for pure bending case is exactly the same as for simple tension. The obtained different values are shown in Table 5.2 and Table 5.3 for Case 1, and in Table 5.4, and Table 5.5 for Case 2 for the simple tension loading.

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112 331 C2-5^0.0B3 3-41 236 141 1 1 1 1 1 1 1 1 1 1 1 1 1 I 211 / y / C2. 56,0. OB) / i 1 ; / /325 / / / 1 170 ca n 175 submode I 231 coarse mode I Figure 5.5 The Coarse model and sub-model

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113 296 211 CO. 18.0.073 y 106 ' CO. 18,0. 06: 325 231 CO. 02,0. 073 170 CO. 18,0.05^ 175 126 TO. 22,0. 06} 11 " C0'2, 0.053 submode I 21 CO. 22,0.053 coarse mode 35 Figure 5.6 The coarse model and submodel in case 2

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114 Table 5.2. Different values Cy, for Case 1 in r x direction in simple tension loading case *1
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115 Table 5.4. Different values for Case 1 in the r 3 direction in simple tension loading case r 3 r 3 /c ''xy/Pav .004 0.00156 0.273 0.014 .008 0.00312 0.22 0.0157 .012 0.00469 0.178 0.0153 .016 0.00625 0.156 0.0153 .02 0.0078 0.141 0.0153 where r 3 is the distance originating from the butted corner point to the positive x direction of interface plane (See Figure 5.3) Table 5.5. Different values C x , for Case 2 in the r 4 direction in simple tension loading case r. r,/t. ^x/Pav ^xy/Pav c x .0025 0.0625 0.3612 0.1512 0.102 0.04 .005 0.125 0.2712 0.07 0.1048 0.027 .01 0.25 0. 1888 0.046 0.1 0.024 .015 0.375 0. 1332 0.038 0. 08 0. 024 a *£ tne ^stance originating from the butt, point to the positive y direction (See Figure 5.3)

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116 From these values, we choose the maximum values in each direction. The maximum values for Case 1 in the r 1 direction are C y = 0.014, C^ = 0.0077 and in the r 2 direction Cy = 0.067. For Case 2 the maximum values in the r 3 direction are C = 0.016 and C x = 0.105, and in the r 4 direction C^ = 0.04. Substituting these values into Equation 5.23, Equation 5.24, Equation 5.27 and Equation 5.28, we obtain the final equations, which give a set of stress distributions near the singular point as follows. Case 1 for the simple tension case in the direction -0.355 (
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CHAPTER 6 RESULTS AND DISCUSSION 6 . 1 Introdntrti nn The results of finite element stress analysis of a double-doubler adhesive bonded joint are presented in this thesis. Two different loading cases (a simple tension and a pure bending applied at the ends of the adherends) are considered to evaluate the traction distributions on the interfaces and the cohesive stresses in the interior of the adhesive layer. Results of a parametric study of the joint properties is presented to show the influences of the various joint parameters on the stress distributions in the joint. A refined treatment for the stress field near the singular points is presented. A set of "stress singularity parametersare devised to represent the stresses in the regions close to and including the singular points. The summary of the results of this analysis is presented in Section 6.2. Suggestions for more efficient joint design are discussed. Recommendations for future work are in the concluding section, Section 6.3. 117

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118 6.2 Summary and Conclusions The double-doubler joint with identical adherends and non-identical adherends, shown in Figure 3.1, was analyzed. The computational tool for the analysis is the ANSYS finite element program. All three materials involved are assumed in linear elastic under the applied load. The stress analysis is treated as a plane strain problem. A linear elastic stress analysis of the double-doubler joint is given in Chapter 3. A parametric study of the double-doubler joint is given in Chapter 4, where the influences of two important joint parameters are analyzed. In Chapter 5, "stress singularity parameters" are devised to describe stress fields near the singularity zone. The important stresses in a double-doubler joint are the shear and lateral normal stresses in the adhesive and axial normal stresses in the adherends. The maximum magnitudes of these stresses and their gradients occur at the end zone of adhesive overlap and at the bonded corner zone at butted region between two central adherends in the simple tension case. In the pure bending case the maximum magnitude of these stresses and their gradients occur at the end zone of adhesive over lap. The critical areas of the adhesive are, therefore, at the ends of the overlap and at the bonded corner of butted region. These critical areas extend over approximately 10% of the overlap length at the ends of the overlap and 10% of the

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119 overlap length at the bonded corner of the butted region. The axial distributions of stress (a xf CTy , a^) in the adhesive are almost uniform at a low level over the middle 75% of overlap. The axial normal stress distribution in the adhesive layer varies linearly through the adhesive layer thickness near the end. Along the boundary between the central adherend or outer adherend and adhesive, axial normal stress (aj is not well defined. Therefore one would not expect the free end condition (a x = 0xy = 0) to be satisfied at the very near end zone. However, the end condition (a x = = 0) at the various levels inside the adhesive, which means no external traction force in the x or y direction on the end, is almost satisfied by using the displacement finite element method. The lateral normal stress has its highest peak value at the edge of the interface between the central adherend and adhesive. The lateral normal stress (a y ) becomes compressive in part of the upper level of adhesive near the interface between the adhesive and outer adherend, and at that interface it shows an interval of very high compressive stress, if we assume that stresses through the adhesive thickness are uniform, we can not predict the compressive lateral normal stress regions of the adhesive layer. From the parametric study of the joint with adherends of the same isotropic material, it is concluded that the single most important parameter influencing the stresses is the ratio of the adherend modulus to the modulus of the adhesive, E/E .

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120 The stresses in the adherends and adhesive are influenced by change in E/E a . The stress peaks are greatest when E/E a = l and decrease with increase in E/E a . The stresses in the adhesive and adherends do not change appreciably for values of E/E, > 100. The influence on the stress peaks due to change in overlap length, 2c, is similar to the influence of E/E a . As c/t c increases, the peak stresses and stress gradients decrease in magnitude. The stress singularity parameters together with the order of singularity fully describe the stress fields near the ends of adhesive overlap and at the bonded corner of the butted region between two central adherends: »ij/P«v = C„ (r/c)*1 where c 4J are the stress singularity parameters and (p-i) is the singularity order (the strength of the singularity) . The singularity at the bonding corners of two different materals depends on the materal constants a and 6 (Eguation 5.16), which are functions of Young's moduli and Poisson's ratios of the adhesive and adherends. The strength of the singularity ( singularity order) a = p-i decreases as the ratio, E/E., increases. The singularity order, a, increment is very small over the range E/E a > 50 and is noticeable below the range E/E a < 30. Efficient double-doubler joints can be designed based on a study of these influences. The parameters of the joint actually designed will depend on the particular

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121 design requirements and the functions the joint has to accomplish. In the model of adhesive bonded joints for finite element analysis, it is important that the critical regions of the adhesive be modeled with very small elements. It has been shown that the stress gradients are high over the 10% of the overlap length at each end of the overlap. Additionally, 5% of the overlap length at each end of the overlap and 15% of the adhesive thickness from each interface must be modeled with several elements in order to get more accurate peak stress near the boundaries. The specified boundary displacement method for analyzing fine-mesh submodels in the critical region is an efficient method for solving such problem. 6.3 R ecommendations for Future Work The double-doubler bonded joint with linear elastic adhesives was analyzed in this study. Future study should be extended to joints with viscoelastic adhesives subjected to constant thermal stresses, transient thermal stresses and to dynamic loads. The stress singularity parameters may be considered as a means of failure prediction of an adhesive joint by comparing the calculated parameters with known critical values of these parameters for the material. it should be pointed out, however, that this kind of comparison can be made only for machined joints where contact angles

PAGE 138

122 between the adherends and the adhesive layer do not vary (e.g., 90" for the model of this study). In actual manufacturing processes, however, the uniform contact angles do not occur naturally. Further study should be done to evaluate singularity for different shapes of edge corner with identical anisotropic adherends. Some shape optimization technigue could be used to reduce the singularity. Furthermore, the stress singularity problem is encountered only in the linear elastic stress analysis. If yielding occurs during the loading process, the maximum stress concentrations will be significantly reduced due to plastic deformation. This should be further studied.

PAGE 139

APPENDIX A CASE 1. THE AXIAL STRESS DISTRIBUTIONS OF DOUBLE-DOUBLER REINFORCEMENT OF A CONTINOUS MEMBER AT DIFFERENT LEVELS OF Y (ALUMINUM ADHERENDS AND EPOXY ADHESIVE IN SIMPLE TENSION)

PAGE 140

124

PAGE 141

125 in in ijj a. 5 ••" o LU 0. U4 -o 01 in -0.09 in UJ STR -O.03 1 -o.a* Q LU -O.M N —1 -o .u -O 07 -C.OD 1 f 1 1 1 — — i 1 1 r NORMALIZED AXIAL COORD I NATE "x/c NORMALIZED AXIAL COORDINATE X/C -i 1 r NORMALIZED AXIAL COORDINATE x/c" Figure A. 2 Case 1. Stress distributions of double-doubler of 3 0 0675* ° f 9 COntinous men iber at the level

PAGE 142

126 Figure A. 3

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127 a.» r« r NORMALIZED AXIAL COORDINATE X/C NORMAL IZED AX I A°L COORD I NATE X/C Figure A. 4 Case 1. Stress distributions of double-doubler of 1 yiofS92r t ° f 9 COntinous member a t the level

PAGE 144

APPENDIX B THE AXIAL STRESS DISTRIBUTIONS OF A DOUBLE DOUBLER JOINT AT DIFFERENT LEVELS OF Y (ALUMINUM ADHERENDS AND EPOXY ADHESIVE IN SIMPLE TENSION)

PAGE 145

CD cn cj en a o < ri Q W N H J 1 I 1 1 r 1 1 1— 02 ».t •.• o.i AXIAL COORD I If ATE x/c co » « I -1 1 r— 0 1.3 tn [0 u 3 Em a « « H N S p 95 -0.02 1 — 3 — r— 0 6 AXIAL COORDINATE x/c Figure B.l Case 1. stress distributions of a double-doubler joint at the level of y=o. 06375 129

PAGE 146

130 10 m a a • » 2 < — 0.M Q n N 0.03 1 in in U 10 < « w tsi M J < PS o o • « ' 1 1 1 1 1 1 1— 0.1 it 0 1 AXIAL COORDINATE x/c 1 i 1 1 1 1 r•" « 1.4 |.| AXIAL COORDINATE x/c AXIAL COORDINATE x/c Figure B.2 Case 1; stress ditributions of a double-doubler joint at the level of y=0.0675

PAGE 147

131 0 w ' 1 1 I 1 1 1 — 02 0.4 „. t AXIAL COORDINATE x/c Figure B.3 Case 1; Stress distributions of a double-doubler joint at the level of y=0.08

PAGE 148

132 < w o < i — i <* p w N M H a o BS M s < m u P < B S o « « OB AXIAL COORDINATE x/c cn w w « en < « n M r: g AXIAL COORDINATE x/c Figure B.4 Case 1. stress distributions of a double-doubler joint at the level of y=0.0925

PAGE 149

APPENDIX C CASE 2. THE AXIAL STRESS DISTRIBUTIONS OF DOUBLE-DOUBLER REINFORCEMENT OF A CONTINOUS MEMBER AT DIFFERENT LEVELS OF Y (ALUMINUM CENTRAL ADHEREND, EPOXY ADHESIVE AND COMPOSITE OUTER ADHEREND IN SIMPLE TENSION)

PAGE 150

D.OM I i 1 1 i i i 7 1 1 0 0.4 0.6 0.1 1 NORMALIZED AXIAL COORDINATE X/C V9 I T 1 1 1 — 1 1 , NORMALIZED AXIAL COORD I NATE "x/r in -o. w a la cr jfi *' i g N T" NORMAL I ZED AXIAL COORD I NATE ° "x/ c Figure C.l Case 2. Stress distributions of double-doubler reinforcement of a continous member at the level of y=0. 06375 134

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135 in w LU k a. iS I (/I Q LU N NORMALIZED AXIAL COORDINATE x/c NORMALIZED AXIAL COORDINATE ' X/ Figure c.2 Case 2. Stress distributions of double-doubler reinforcement: of a continous member at the level

PAGE 152

136 in m UJ cr $ cr J < x < Q LU N 0. DOS 0 DM m 0.09 10 LU cr ST o.oi _i D. 06 NOP i. M _i RA 0.03 UJ LA 0.02 o LU M 0.01 _l NOR* -0.01 -m in EC 5 to o LU N cr NORMALIZED AXIAL COORDINATE X/c NORMALIZED AXIAL COORDINATE x/c NORMAL IZD AXIAL COORDINATE x/c Figure C.3 Case 2. Stress distributions of double-doubler o? yio^s ° f 9 COntinous ^er at the le^el

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137 Figure C.4

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APPENDIX D CASE 2. THE AXIAL STRESS DISTRIBUTIONS OF A DOUBLE -DOUBLER JOINT AT DIFFERENT LEVELS OF Y (ALUMINUM CENTRAL ADHEREND, EPOXY ADHESIVE, AND COMPOSITE OUTER ADHEREND IN SIMPLE TENSION)

PAGE 155

o.a T -I1 1 NORMALIZED AXIAL COORD I NATE ° 'x/C NORMAL I ZED AXIAL COORDINATE' X/C Figure D.l Case 2. Stress distributions of a double-doubler joint at the level of y=o. 06375 139

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140 NORMALIZED AXIAL COORD I NATE ° 'x/c NORMAL I ZED AX I AL COORD I NATE ' X/ c Figure D.2 Case 2. stress distributions of a double-doubler joint at the level of y=0.0675

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141 0. OS NORMALIZED AXIAL COORDINATE x/C Figure D.3 Case 2. Stress distributions of a double-doubler joint at the level of y=0.08

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Figure D.4 Case 2. Stress distributions of a double-doubler joint at the level of y=o.0925

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APPENDIX E CASE 3. THE AXIAL STRESS DISTRIBUTIONS OF DOUBLE-DOUBLER REINFORCEMENT OF A CONTINOUS MEMBER AT DIFFERENT LEVELS OF Y (ALUMINUM ADHERENDS AND EPOXY ADHESIVE IN PURE BENDING)

PAGE 160

to 1/5 at IX 0.1 ILO cr o z 5 CM D LU N o < o 2 "I 1 r a D 12 in (fl 1X1 1/3 _J < < cr LU o.l NORMALIZED AXIAL COORD I NATE X/C N cr O — i 1 1— ••• 0.0 I 1 0.0 NORMALIZED AXIAL COORDINATE X/C 10 0.1 in LU a 110 0.00 a < LU I 0.00 CO a LU N O.M _J NORMALIZED Ax'lAL COORDINATE X/C ' Figure E.l Case 3. Stress distributions of double-doubler reinforcement of a continous member at the level of y=-0. 095 J-evei 144

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145 NORMAL I ZD AXIAL COORDINATE X/C NORMALIZED AXIAL COORD I NATE X/c ' NORMALIZED AXIAL COORDINATE X/C ' Figure E.2 Case 3. Stress distributions of double-doubler reinforcement of a continous member at the level

PAGE 162

146 in LU CC (w X < Q LU N cr 2 1/3 1/3 LU cr P 1/3 o 2 < cr LU Q LU N -i 1 rNORMAL I ZED AXIAL COORDINATE X/C ) 1 -1 1 1 — — 1 1 — — NORMALIZED AXIAL COORDINATE x/C 10 1/3 LU i1/3 cr l a X Q 111 1 1 1 1 1 T •« D.t g.t NORMALIZED AXIAL COORDINATE x/c Figure E.3 Case 3. Stress distributions of double-doubler reinforcement of a continous member at the level

PAGE 163

147 NORMALIZED AXIAL COORDINATE x/c a *L _ in NORMALIZED AXi'aL COORDINATE X/C Figure E.4 Case 3. Stress distributions of double-doubler reinforcement of a continous member at the level of y=-0 . 065

PAGE 164

148 m m ID a tm _i < X < Q UJ N NORMALIZED Ax'lAL COORD I NATE ' X/C in in LU cr Im < cr -0 » LU h< _J Q -a m LU M -0.4 -J -O.40 cr ON O.li -. LO 0.t4 m LU TR D.1S m or 0.1 < LU I in D.OB Q LU Q.Oi N _l 0.0* cf s 0 01 0 NORMALIZED Ax'lAL COORDINATE' X/C NORMALIZED AXIAL COORDINATE X/C Figure E.5 Case 3. Stress distributions of double-doubler of OoS" 1 * ° f 3 COntinous membe r at the level

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149 in in UJ tr t— (/) _i < 2 < IT O UJ 1 1 1 1 1 NORMAL i ZED AXIAL i 1 1 1 1 COORDINATE X/C 1.04 A -0.0* -a ct -t.ii • -0 11 -o-i -on 1 1 1 1 -r NORMALIZED AXIAL COORDINATE' x/c NORMALIZED AXIAL COORDINATE x/c Figure E.6 Case 3. Stress distributions of double-doubler reinforcement of a continous member at the level of y=0.07 J-eve±

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150 0-0* tn in LU a. 01 cr cr 2 -0.0 UJ < -o.oi _J O -o.t UJ N ! -o.ii NORMALIZED AXIAL COORD I NATE X/C —1 1 1 r 0.1 0.4 | im NORMALIZED AXIAL COORDINATE X/C ' in LU O.l cr ttr o.« < UJ I 10 Q LU N 0.06 cr O 0.03 2 -i 1 r 0.1 o.t NORMALIZED AXIAL COORDINATE x/c' Figure E.7 Case 3. Stress distributions of double-doubler reinforcement of a continous member at the level

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I T NORMAL TzED AX°IAL COORDINATE'' X/C LU D.03 £ K | ...» . S LU < Q ft * 1 1 1 r 0 * 0. ft NORMALIZED AXIAL COORDINATE x/c ' 1/3 o.i LU LT o.oe or < LU in a ft o 2 -r— t t.l i l M 151 NORMALIZED AXIAL COORDINATE X/C ' Figure E.8 Case 3. Stress distributions of double-doubler reinforcement of a continous member at the level of y=0.095 61

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APPENDIX F THE AXIAL STRESS DISTRIBUTIONS OF A DOUBLE-DOUBLER JOINT AT DIFFERENT LEVELS OF Y (ALUMINUM ADHERENDS AND EPOXY ADHESIVE IN PURE BENDING)

PAGE 169

0 « 10 to LU a —J i 2 < i s in '' H w LU a t— 0. w In o 2 < LU on a LU ISI g -O.I ;ss ».i cr LO EAR 0.01 3HS IZED 0 06 MAL 0 0* NOR "I I I 1 -1 1 1 1 I a.« o.s t.i NORMALIZED AXIAL COORDINATE X/C 0 1 * t ' ^ r NORMAL I ZED AX I AL COORD I NATE *' x/C — i r 1 r NORMALIZED AXIAL COORD I NATE ° 'x/ c Figure F.l Case 3; Stress distributions of a double-doubler ]oint at the level of y=-0.095 153

PAGE 170

154 Figure F.2 Case 3. Stress distributions of a double-doubler joint at the level of y=-0.08

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155 • « o« t. t ,, NORMALIZED AXIAL COORDINATE X/C °-« »« ».» o.« NORMALIZED AXIAL COORD I NATE X/C Figure F.3 Case 3. Stress distributes of a double-doubler joint at the level of y=-o.07

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156 Figure F.4 Case 3. Stress distributions of a double-doubler joint at the level of y=-0.065

PAGE 173

157 to in w a: in tr I 0.1 0.6 D 6 NORMALIZED AXIAL COORDINATE X/C 1 1 T ' r " NORMAL I ZED AXIAL COORD I NATE "x/C Figure F.5 Case 3. Stress distributions of a double-doubler joint at the level of y=0.065

PAGE 174

158 Figure F.6 Case 3. Stress distributions of a double-doubl joint at the level of y=0.07

PAGE 175

159 NORMALIZED AXIAL COORDINATE X/C Figure F.7 Case 3. Stress distributions of a double-doubler joint at the level of y=0.08

PAGE 176

160 in in LU cr < D LU N 02 0.4 o.i o.t NORMAL I ZED AXIAL COORDINATE X/c HI i . .. uj -«•«» < _l -0.04 Q m N 5 10 w 111 (X 0M cr u5 X 0.04 w Q LU N — a at ' ^ ' 7, 1 1 ' 1 1 NORMALIZED AXIAL COORD I NATE ° 'x/ c a § 0.0 IM ' ' '" ^ ' ' NORMALIZED AXIAL CCORD I NATE° 'x/c * Figure F.8 Case 3. Stress distributions of a double-doubler joint at the level of y=0.0925

PAGE 177

APPENDIX G CASE 4. THE AXIAL STRESS DISTRIBUTIONS OF DOUBLE DOUBLER REINFORCEMENT OF A CONTINOUS MEMBER AT DIFFERENT LEVELS OF Y (ALUMINUM CENTRAL ADHEREND, EPOXY ADHESIVE, AND COMPOSITE OUTER ADHEREND IN PURE BENDING)

PAGE 178

Figure G.l Case 4. Stress distributions of double-doubler reinforcement of a continous member at the level 162

PAGE 179

163 ••» 0.1 OH II NORMAL I ZED AXIAL COORDINATE X/C -I 1 r NORMALIZED AXIAL COORDINATE °'x/c " *•• •.» •• NORMALIZED AXIAL COORDINATE X/C Figure G.2 Case 4. Stress distributions of double-doubler reinforcement of a continous member at the level of y=-0.08

PAGE 180

164 NORMALIZED AXIAL COORDINATE X/C in 1/5 LU a. O o 11 z < 0.13 cr HI i< 1 oi Q UJ o.o< M NORMALIZED AXIAL COORD I NAT E° *x/C "P NORMALIZED AXIAL COORDINATE x/C Figure G.3 Case 4. stress distributions of double-doubler reinforcement of a continous member at the level of y=-0 . 07

PAGE 181

165 in •« in Bj 5 ttn < D W N ~v t — 1 — r ~ — ' — ' — 1 — 1 — r NORMAL I ZED AXIAL COORDINATE *X/C ~r 1 1 i rNOHMAL I ZED AXIAL COORD I NATE* 'x/c NORMAL I ZED AXIAL COORDINATE " "x/c Figure G.4 Case 4. Stress distributions of double-doubler reinforcement of a continous member at the level of y=-0. 065

PAGE 182

166 T 0.4 D t 0. • NORMALIZED AXIAL COORDINATE x/c < -ri 1 1 1 i «« •.« 0» 1.1 NORMALIZED AXIAL COORDINATE X/c Figure G.5 Case 4. Stress distributions of double-doubler reinforcement of a continous member at the level of y=0.065

PAGE 183

167 •< 0 1 o.i o » NORMALIZED AXIAL COORDINATE x/c NORMALIZED AXIAL COORD I NATE "'x/C Figure G.6 Case 4. Stress distributions of double-doubler reinforcement of a continous member at the level of y=0.07

PAGE 184

168 D CM co o.o3 id £ s s .... -o.o» HI IQ 111 -0 1 IM NORMALIZED AXIAL COORDINATE X/C I Tj 1 1 1 1 , 1 NORMALIZED AXIAL COORD I NATE ' 'x/C T NORMALIZED AXIAL COORD I NATE "'x/C Figure G.7 Case 4. Stress distributions of double-doubler reinforcement of a continous member at the level of y=0.08

PAGE 185

169 ~i 1 1NORMAL I ZED AXIAL COORD I NATE° 'x/C NORMALIZED AXIAL COORDINATE x/c Figure G.8 Case 4. Stress distributions of double-doubler reinforcement of a continous member at the level of y=0.095 1

PAGE 186

APPENDIX H CASE 4; THE AXIAL STRESS DISTRIBUTIONS OF A DOUBLE-DOUBLER JOINT AT DIFFERENT LEVELS OF Y (ALUMINUM CENTRAL ADHEREND, EPOXY ADHESIVE, AND COMPOSITE OUTER ADHEREND IN PURE BENDING)

PAGE 187

Figure H.l Case 4. Stress distributions of a double-doubler joint at the level of y=-o.095 171

PAGE 188

172 oot o ci o ' ' ' NORMAL I ZED AXIAL COORD I NATE "'x/ c °» °< o.t oe NORMALIZED AXIAL COORDINATE X/C Figure H.2 Case 4; Stress distributions of a double-doubler joint at the level of y=-0.08

PAGE 189

173 Figure H.3 Case 4. Stress distributions of a double-doubler joint at the level of y=-o.07

PAGE 190

174 Figure H.4 Case 4. Stress distributions of a double-doubler joint at the level of y=-0.065

PAGE 191

175 Figure H.5 Case 4. Stress distributions of a double-doubler joint at the level of y=0.065

PAGE 192

176 Figure H.6 Case 4. Stress distributions of a double-doubler joint at the level of y=0.07

PAGE 193

177 < °> at a( NORMALIZED AXIAL COORDINATE x/C Figure H.7 Case 4. Stress distributions of a double joint at the level of y=0.08 -doubler

PAGE 194

178 Figure H.8 Case 4. Stress distributions of a double-doubler joint at the level of y=0.095

PAGE 195

LIST OF REFERENCES [1] O. Volkersen, "Die Nietkraftverteilung in Zugbeanpruchten Nietverbindungen mit Konstanten Laschenquerschnitten, " Luf tf ahrtf orschuna . 1938, pp. 41-47. [2] M. Goland and E. Reissner, "The Stresses in Cemented Joints," Journal of Applied Mechanics, 1944, pp. A17-A27 [3] F. J. Plantema, De Schuif spanning in een Liimnaad . National Luchtvaartlaboratorium, Amsterdam, Report M1181, 1949. [4] S. Kelsey, and N. K. Benson, Institut fur Statik und Dynamik, Technische Hochschule, Stuttgart, ISD Report No. 10, 1966. [5] M. K. Pahoja, Stress Analysis of as Adhesive Lap Joint Subjected to Tension. Shear Force and Bending Moments . Ph.D. Thesis and T.& A.M. Report No. 361, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, 1972. [6] W. J. Renton and J. R. Vinson, The Analysis and Design of Anisotropic Bonded Joints , Report No. 2, University of Delaware, Newark, Delaware; Air Force Office of Scientific Research, Scientific Report AFOSR TR 75-0125, August 1974. [7] W. J. Renton and J. R. Vinson, "The Efficient Design of Adhesive Bonded Joints," Journal of Adhesion , March 1975 pp. 175-193. [8] L. J. Hart-Smith, Adhesive-Bonded Double-Lap Joints , McDonnell Douglas Corporation, Long Beach, California, NASA CR-112235, January 1973. [9] K. S. Ahluwalia, Stress Analysis of Double Shear Bonded Joints bv the Finite Element Method r Master's Thesis, Engineering Science Department, Louisiana State University, Baton Rouge, Louisiana, January, 1969. [10] G. R. Wooley and D. R. Carver, "Stress Concentration Factors for Bonded Lap Joints," Journal of AircraftOctober 1971, pp. 817-820. " '' 179

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180 [11] A. S. McLaren and Z. Maclnnes, "The Influence on the Stress Distribution in an Adhesive Lap Joint of Bending of the Adherend Sheets," British Journal of Applied Mechanics . Vol. 9, February 1958, pp. 72-77. [12] I. Tuzi and H. Shimada, "Photoelastic Investigation of the Stress Distribution in Cemented Joints Bulletin of Japanese Society of Mechanical Engineers . Vol.7, No. 26, 1964, pp. 263-267. [13] D. B. Bogy, "Edge-Bonded Dissimilar Orthogonal Elastic Wedges Under Normal and Shear Loading," Journal of Applied Mechanics . September 1968, pp. 460-466. [14] D. B. Bogy and K. C. Wang, "Stress Singularities at Interface Corners in Bonded Dissimilar Isotropic Elasic Materials," International Journal of Solids and Structures . Vol. 7, 1971, pp. 993-1005. [15] D. B. Bogy, "Two Edge-Bonded Elastic Wedges of Different Materials and Wedge Angles under Surface Traction," Journal of Applied Mechanics . Vol.93, 1971, pp. 377-386. [16] F. L. Mattews, P. F. Kitty, and E. W. Godwin, "A Review of the Strength of Joints in Fiber Reinforced Plastics, Part 2-Adhesively Bonded Joints," Composites . Vol. 13 No. 1, 1982, pp. 29-31. [17] J. R. Vinson and R. L. Sierakowski, The Behavior of Structures Com posed of Composite Materials (BostonNijhoff, 1986) pp. 239-262. [18] F. Erdogan and M. Ratwani, "Stress Distribution in Bonded Joints," Journal of Composite Materials . Vol. 5, 1971 pp. 378-393. [19] R. M. Barker and F. Hatt, "Analysis of Bonded Joints in Vehicular Structures," AIAA Journal . Vol. 11, 1973. pp 1650-1654. ^ [20] K. L. Devries and M. L. Williams, "Adhesive Fracture of a Lap Shear Joint," Experimental Mechanics . March 1974 pp. 89-97. [21] R. s. Alwar and Y. R. Nagaraja, "Elastic Analysis of Adhesive Butt Joints," Journal of Adhesion , Vol. 6. 1976 pp. 279-287. [22] I. u. Ojalvo, "Optimization of Bonded Joints," AIAA Journal, Vol. 23, October 1985, pp. 1578-1582.

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181 [23] D. W. Schmueser, N. L. Johnson and R. T. Foister, "Stress Analysis of Adhesively Bonded Electroprimed Steel Lap Shear Joints," Journal of Adhesion r Vol. 24, 1987, pp. 47-64. [24] M. D. Wright, "The Stress Analysis of a Butt Strap Joint in Carbon Fiber Reinforced Plastic," Composites . Vol. 4, October 1978, pp. 259-262. [25] D. J. Schwartz, "Practical Analysis of Stress Raisers in Solid Structures," 4th International Conference on Vehicle Structural Mechanics. Society of Automotive Engineers . Warrendale, Pa., 15096, November 1981. [26] F. S. Kelly, "Mesh Requirement for the Analysis of a Stress Concentration by the Specified Boundary Displacement Method," Proceedings of the Sennnd Interna tional Computer Engineering Conference . Computer Engineering Division, ASME, August 1982. [27] M. D. Wright, "Stress Distribution in Carbon Fiber Reinforced Plastic Joints," Composites . January 1980, pp. 46-50. [28] H. L. Groth, "Calculation of Stresses in Bonded Joints Using the Substructuring Technique," International Journal of Adhesion and Adhesives r Vol. 6, January 1986, pp. 31-35. [29] P. A. Gradin and H. L. Groth, "A Fracture Criterion for Adhesive Joints in terms of Material Induced Singularities," Proceedi ngs of the Third International Conference on Num erical Methods in Fracture Mechanins Pineridge Press, Swansea, 1984. ' [30] U. Yuceoglu and D. P. Updike, "Comparison of Continuum and Mechanical Spring Model of Adhesive Layers in Bonded Joints," 1981 Advances in Aerospace Structures arH Materials, ASME Publication, November 1981. [31] U. Yuceoglu and D. P. Updike, "Stress Analysis of Bonded Plates and Joints," ASCE Journal of The Engineering Mechanics Division, February 1980, pp. 37-56. [32] W. s. Johnson and S. Mall, "A Fracture Mechanics Approach for Designing Adhesively Bonded Joints, " Delamination Debonding of Materials , STP 876, American Society for Testing and Materials, Philadelphia, 1985, pp. 189-199

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182 [33] G. J. DeSalvo and J. A. Swanson, ANSYS Engineering Analysis System User's Manual , Swanson Analysis Systems, Inc., Houston, PA., February 1982. [35] R. D. Cook, Concepts and Application of Finite Element Analysis . John Wiley & Sons, New York, 1974. [36] Yona Kaplevatsky and Vitaly Raevsky, "On the Theory of the Stress-Strain State in Adhesive Joint," Journal of Adhesion . 1976, Vol. 7, pp. 65-77. [37] F. Delale, F. Erdogan, and M. N. Aydinoglu, "Stresses in Adhesively Bonded Joints: A Closed-Form Solution," Journal of Composite Materials . 1981, Vol. 15, pp. 249-267. [38] R. L. Spilker and S. C. Chou, "Edge Effects in Symmetric Composite Laminates: Importance of Satisfying the Traction-Free-Edge Condition," Journal of Composite Materials . 1980, Vol. 14, pp. 2-20. [39] R. L. Spilker, S. C. Chou, and O. Orringer, "Alternate Hybrid-Stress Elements for Analysis of Multilayer Composite Plates," Journal of Composite Materials . 1977, Vol. 11, pp. 57. [40] S. Aivazzadeh and G. Verchery, "Stress Analysis at the Interface in Adhesive Joints by Special Finite Elements," International J ournal of Adhesion and Adhesives . 1986 Vol. 6, pp. 185-188. [41] Y. Yamaguchi and S. Sato, "Stress Distribution and Mechanical Strength of Tand L-type Adhesive-Bonded Joints Subjected to Applied Bending Moments," International J ournal of Adhesion and Adhesives . 1986 Vol. 6, pp. 81-87. ~ ' [42] S. D. Conte and C. de Boor, Elementary Numer i r.a 1 Analysis: Algorithmic Apprnanh (McGraw-Hill Book Co 1980) pp. 202-204. '

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BIOGRAPHICAL SKETCH Myung Kyun Park was born December 3, 1956, in Pusan, Korea. In March 1979, he received his Bachelor of Science degree with major in mechanical engineering from the Yonsei University, Seoul, Korea. He worked for the Agency for Defense Development for five years in Korea as a mechanical engineer. In August 1984, he came to the University of Florida for graduate study in the area of solid mechanics. He received his Master of Science degree from the department of Aerospace Engineering, Mechanics and Engineering Science in December 1985. He continued his Ph.D. study in the same department. He was awarded the degree of Doctor of Philosophy in May 1990. He married Boeun in 1983 and they have a son named James. 183

PAGE 200

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Lawrence E. Malvern, Chairman Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. C . T . Sun Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. B. V. Sankar Assistant Professor of Aerospace Engineering, Mechanics and Engineering Science

PAGE 201

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. S. Y. Lu Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David C. Wilson Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1990 rWinfrejd M. Phillips a Ltean, College of Engineering Madelyn M. Lockhart Dean, Graduate School


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INGEST IEID ES9RF9R1R_XEOTGO INGEST_TIME 2015-04-22T21:43:47Z PACKAGE AA00029883_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES