FINITE ELEMENT AND ANALYTICAL MODELS FOR LOAD TRANSFER
CALCULATIONS FOR STRUCTURES UTILIZING METAL AND
COMPOSITES WITH LARGE CTE DIFFERENCES
A Thesis by
Uday Sankar Meka
Bachelor of Engineering, Madras University, India, 2004
Submitted to the Department of Aerospace Engineering
and the faculty of the Graduate School of
Wichita State University
in partial fulfillment of
the requirements for the degree of
Master of Science
May 2007
FINITE ELEMENT AND ANALYTICAL MODELS FOR LOAD TRANSFER
CALCULATIONS FOR STRUCTURES UTILIZING METAL AND
COMPOSITES WITH LARGE CTE DIFFERENCES
I have examined the final copy of this thesis for form and content and recommend that it be
accepted in partial fulfillment of the requirements for the degree of Master of Science, with a
major in Aerospace Engineering.
__________________________________
Charles Yang, Committee Chair
We have read this thesis and recommend its acceptance.
___________________________________
K. Suresh Raju, Committee Member
___________________________________
Hamid Lankarani, Committee Member
ii
DEDICATION
To my parents and friends
iii
ACKNOWLEDGEMENTS
I express my sincere gratitude to my advisor Dr. Charles Yang for his continuous support
and guidance in helping me to complete my thesis. I acquired much knowledge from him during
the entire course of my master’s degree. He always helped me in understanding the technical
details and overcoming difficulties. For his valuable help, patience, and encouragement, I owe
him the deepest gratitude. It’s hard to express in words my thanks towards him.
I thank Dr. K. Suresh Raju and Dr. Hamid Lankarani for reviewing my thesis and making
valuable suggestions. I also thank the many people who have supported me: engineers from the
National Institute for Aviation Research, especially Waruna Senivaratne and Dimuthu
Tilekaratne, for their kind assistance in carrying out experiments and helping with various
applications. I am indebted to my colleagues Ananthram, Wenjun Sun, Santhosh Kumar, Peter
Chou, Kannan, Alireza Chadegani, and Bava and, I thank all the direct and indirect support that
helped me complete this thesis.
Lastly and most importantly, I want to thank my parents who have been a source of
encouragement and inspiration throughout my life.
iv
ABSTRACT
Large composite structures have been increasingly used in the aviation industry. In order to
achieve higher fuel efficiency, the use of light-weight, high-strength composite materials, such as
carbon/epoxy, needs to be fully explored. New applications of composite materials include
primary structures such as aircraft fuselages. This study dealt with thermal stresses induced in a
composite aircraft fuselage, in which the fuselage skin was made of carbon/epoxy composite and
was fastened to aluminum beams. These stresses resulted from the large coefficient of thermal
expansion (CTE) difference and also the large temperature difference between the time of
assembly, which was 75ºF and the actual flight condition, which was -65ºF). This temperature
difference of around 140ºF induced high thermal stresses, not only in the fasteners but also in the
aluminum beams and composite panels.
The two main objectives of the study are as follows:
• To investigate the thermally induced stresses in the aluminum beams.
• To investigate the feasibility of thermally isolating the aluminum beams from the
composite fuselage skins.
An experimental program was conducted to measure the strains on the top surface of an
aluminum beam, which was fastened to the composite panel from thermal loads due to
temperature difference and CTE mismatch. An approach was also designed to study the effects
of the length of the aluminum beam on stresses. An analytical model was developed to evaluate
the fastener load transfer and the thermally induced stress within the fastened
aluminum/composite assemblies. Five parameters were used to develop an analytical model to
calculate the load transfer between the aluminum/composite hybrid structures: equivalent area of
the aluminum beam and composite panel, equivalent temperatures of the aluminum beam and
v
composite panel, and equivalent fastener stiffness were determined using three-dimensional
finite element analysis.
An attempt has been made to study the effect of fastener diameter, fastener spacing,
material of the metallic beam, size of the metallic beam, thickness of the composite panel on the
five parameters required to find the load transfer so that a relation could be established for a
working engineer to determine these parameters without doing any finite element work.
Equations correlating the five parameters with geometric and material properties were provided.
vi
TABLE OF CONTENTS
Chapter
1.
Page
INTRODUCTION ...............................................................................................................1
1.1
1.2
1.3
2.
EXPERIMENTAL METHODS.........................................................................................18
2.1
2.2
2.3
2.4
2.5
3.
Objective ................................................................................................................18
Experimental Setup................................................................................................18
Specimen Configuration and Fabrication ..............................................................27
2.3.1 Aluminum/Composite Assembly Tests without Insulation ........................27
2.3.2 Aluminum/Composite Assembly Tests with Insulation .............................28
Testing for Mechanical Properties .........................................................................29
Testing for Coefficient of Thermal Expansion (CTE)...........................................30
ANALYTICAL APPROACH ...........................................................................................32
3.1
3.2
3.3
3.4
4.
Background ..............................................................................................................1
1.1.1 Composites in Aircraft Industry....................................................................2
1.1.2 Mechanical Joints in Aircraft Structures.......................................................3
1.1.3 Design Methods ............................................................................................5
1.1.4 Importance of Bolt Preload...........................................................................6
1.1.5 Thermo-Mechanical Loading........................................................................8
1.1.6 Defining Pretension in ABAQUS.................................................................8
1.1.7 Defining Contact in ABAQUS ...................................................................10
Literature Review...................................................................................................11
Objective ................................................................................................................15
1.3.1 Solution Approach ......................................................................................17
Objective ................................................................................................................32
Governing Equations of Aluminum Beam ............................................................33
Governing Equations of Composite Panel.............................................................35
Governing Equations of Fasteners .........................................................................38
FINITE ELEMENT MODELS..........................................................................................40
4.1
4.2
Objective ................................................................................................................40
Finite Element Model Development......................................................................41
4.2.1 Finite Element Mechanical Model to Determine Aa ...................................41
4.2.2 Finite Element Mechanical Model to Determine Ac ...................................46
4.2.3 Finite Element Thermal Model...................................................................49
4.2.4 Sequentially Coupled Finite Element Analysis to Determine Ta ................50
vii
TABLE OF CONTENTS (continued)
Chapter
Page
4.2.5 Sequentially Coupled Finite Element Analysis to Determine Tc ................52
4.2.6 Sequentially Coupled Finite Element Analysis to Determine Kf ................54
5.
RESULTS AND DISCUSSION OF PANEL TESTS .......................................................58
5.1
5.2
5.3
5.4
5.5
6.
RESULTS AND DISCUSSION OF PARAMETRIC STUDY.......................................108
6.1
6.2
6.3
6.4
6.5
7.
Results from Material Property Tests ....................................................................58
Results from Chamber Tests of Fastened Aluminum/Composite Assembly.........58
5.2.1 Group 1 Test Results...................................................................................60
5.2.2 Group 2 Test Results...................................................................................65
5.2.3 Group 3 Test Results...................................................................................69
5.2.4 Group 4 Test Results...................................................................................73
Analytical Model Validation..................................................................................77
5.3.1 Comparison of Tests of Thick Panel without Insulation ............................77
5.3.2 Comparison of Tests of Thick Panel with Insulation (Cirlex®)..................83
5.3.3 Comparison of Tests of Thin Panel without Insulation ..............................88
5.3.4 Comparison of Tests of Thin Panel with Insulation (Cirlex®)....................93
Analytical Model Validation with Finite Element Model .....................................98
Load Transfer Prediction .....................................................................................100
Equivalent Area of the Metallic Beam (Aa) ........................................................110
6.1.1 Comparison of Aa Obtained from FEM and Equation ..............................111
Equivalent Area of the Composite Panel (Ac) .....................................................125
6.2.1 Comparison of Ac Obtained from FEM and Equation ..............................127
Equivalent Temperature of Metallic Z-Beam (Ta) ..............................................127
6.3.1 Comparison of Ta Obtained from FEM and Equation ..............................128
Equivalent Temperature of Composite Panel (Tc) ..............................................131
6.4.1 Comparison of Tc Obtained from FEM and Equation ..............................131
Equivalent Stiffness of Fastener (Kf)..……………………………………… 144
6.5.1 Comparison of Kf Obtained from FEM and Equation…………………...144
CONCLUSIONS AND RECOMMENDATIONS ..........................................................147
7.1 Conclusions………………………….……………………………………………147
7.2 Recommendations……..…………….……………………………………………148
REFERENCES ............................................................................................................................149
viii
LIST OF TABLES
Table
Page
5.1
Material Properties.............................................................................................................58
5.2
List of Aluminum/Composite Assembly Tests..................................................................59
5.3
Temperature Distributions Recorded from Group 1 Tests ...............................................61
5.4
Temperature Distributions Recorded from Group 2 Tests ................................................65
5.5
Temperature Distributions Recorded from Group 3 Tests ................................................69
5.6
Temperature Distributions Recorded from Group 4 Tests .................................................74
5.7
Parameters for Fastened Aluminum/Composites Assembly with Thick
Composite Panel without Insulation ...................................................................................78
5.8
Equivalent Area for Aluminum Beam for Aluminum/Composites Assembly
Composite Panel without Insulation ...................................................................................78
5.9
Equivalent Temperature for Aluminum with Thick Composite Panel
without Insulation ...............................................................................................................78
5.10 Equivalent Area for Thick Composite Panel without Insulation ........................................79
5.11
Equivalent Temperature for Thick Composite Panel without Insulation ...........................79
5.12
Parameters for Fastened Aluminum/Composites Assembly with Thick
Composite Panel and Co-Cured Cirlex® .............................................................................84
5.13 Equivalent Area for Aluminum Beam for Aluminum/Composites Assembly
Composite Panel and Co-Cured Cirlex® .............................................................................84
5.14 Equivalent Temperature for Aluminum with Thick Composite Panel
and Co-Cured Cirlex® .........................................................................................................84
5.15 Equivalent Area for Thick Composite Panel and Co-Cured Cirlex®..................................84
5.16
Equivalent Temperature for Thick Composite Panel and Co-Cured Cirlex® .....................85
5.17
Parameters for Fastened Aluminum/Composites Assembly with Thin
Composite Panel without Insulation ...................................................................................89
ix
LIST OF TABLES (continued)
Table
Page
5.18 Equivalent Area for Aluminum Beam for Aluminum/Composites Assembly
with Thin Composite Panel without Insulation ..................................................................89
5.19
Equivalent Temperature for Aluminum with Thin Composite Panel
without Insulation ...............................................................................................................89
5.20 Equivalent Area for Thin Composite Panel without Insulation..........................................89
5.21
Equivalent Temperature for Thin Composite Panel without Insulation .............................90
5.22
Parameters for Fastened Aluminum/Composites Assembly with Thin
Composite Panel and Co-Cured Cirlex® .............................................................................94
5.23 Equivalent Area for Aluminum Beam for Aluminum/Composites Assembly
with Thin Composite Panel and Co-Cured Cirlex® ............................................................94
5.24
Equivalent Temperature for Aluminum with Thin Composite Panel
and Co-Cured Cirlex® .........................................................................................................94
5.25 Equivalent Area for Thin Composite Panel and Co-Cured Cirlex® ...................................94
5.26
Equivalent Temperature for Thick Composite Panel and Co-Cured Cirlex® .....................95
6.1
Variables used to Calculate Aa, Ac, Ta, Tc, and Kf .............................................................108
6.2
Z-Beam Dimensions According to Beam Thickness tb ....................................................109
6.3
¼ Fractional Factorial Design............................................................................................110
x
LIST OF FIGURES
Figure
Page
1.1
Deformation of Bolt and Joint Members when Tightened ..................................................2
1.2
Examples of Bolted Composite Joints .................................................................................4
1.3
Overall Design Procedure ....................................................................................................6
1.4
Basic Joint Diagrams ...........................................................................................................7
1.5
Pre-Tension Section .............................................................................................................9
1.6
Prescribed assembly load is given at the pre-tension node and applied in direction n......10
1.7
Contact Surfaces. ...............................................................................................................11
1.8
Schematic of Aluminum Beam/Composite Skin Assembly ..............................................16
2.1
Side View of the New Environmental Chamber used for Testing.....................................19
2.2
Environmental Chamber used for Testing .........................................................................20
2.3
Thermocouple and Strain Gage Locations for 66-Fastener Configuration........................21
2.4
Thermocouple and Strain Gage Locations for 48-Fastener Configuration ........................22
2.5
Thermocouple and Strain Gage Locations for 32-Fastener Configuration ........................23
2.6
Thermocouple and Strain Gage Locations for 24-Fastener Configuration........................24
2.7
Thermocouple and Strain Gage Locations for 16-Fastener Configuration.........................25
2.8
Thermocouple and Strain Gage Locations for 8-Fastener Configuration...........................26
2.9
Dimensions of Cross Section of Z-Shape Aluminum Beam ..............................................27
2.10
Configuration of Chamber Tests with Cirlex® Co-Cured inside Composite panel ............29
2.11
CTE of Aluminum Sample .................................................................................................31
3.1
Aluminum/Composite Assembly........................................................................................32
3.2
Free Body Diagram of Z-Shape Aluminum Beam .............................................................33
xi
LIST OF FIGURES (continued)
Figure
Page
3.3
Free Body Diagram of Flat Composite Panel .....................................................................36
3.4
Area influenced by the external load in the wide composite plate .....................................38
4.1
Mechanical FEM of Z-Shape Aluminum Beam Fastened to Composite Panel..................42
4.2
Point of application of tensile load .....................................................................................43
4.3
Displacement Contour ........................................................................................................43
4.4
Displacement measurements for the aluminum beam ........................................................44
4.5
Mechanical FEM of Flat Composite Panel with 20 Unit....................................................46
4.6
Location at which the concentrated load is applied ...........................................................47
4.7
Displacement Contour of Flat Composite Panel with 20 Units.........................................47
4.8
Displacement measurements from the composite panel....................................................48
4.9
Thermal Finite Element Model of Aluminum/Composite Assembly with Eight Units ....49
4.10
Temperature Distribution in Steady State..........................................................................50
4.11
Sequentially Coupled Finite Element Model of Z-shape Aluminum Beam ......................51
4.12
Displacement Contour of Eight Unit Z-Shape Aluminum
Beam due to Thermal Load………………………………………………………………52
4.13
Sequentially Coupled Finite Element Model of Composite Panel ....................................53
4.14
Displacement Contour of Eight Unit Flat Composite Panel due to Thermal Load ...........54
4.15
Sequentially Coupled Finite Element Model of Eight Units of Assembly........................55
4.16
Deformation of Fastened Aluminum/Composites Assembly due to Thermal Load...........56
5.1
Thermocouple Locations of Chamber Tests without Insulation........................................61
5.2
Mechanical Strains of 66-Fastener Setup of Thick Panel without Insulation....................62
5.3
Mechanical Strains of 48-Fastener Setup of Thick Panel without Insulation....................62
xii
LIST OF FIGURES (continued)
Figure
Page
5.4
Mechanical Strains of 32-Fastener Setup of Thick Panel without Insulation....................63
5.5
Mechanical Strains of 24-Fastener Setup of Thick Panel without Insulation....................63
5.6
Mechanical Strains of 16-Fastener Setup of Thick Panel without Insulation....................64
5.7
Mechanical Strains of 8-Fastener Setup of Thick Panel without Insulation......................64
5.8
Thermocouple Locations of Chamber Tests with Insulation.............................................65
5.9
Mechanical Strains of 66-Fastener Setup of Thick Panel with Cirlex® .............................66
5.10
Mechanical Strains of 48-Fastener Setup of Thick Panel with Cirlex® .............................67
5.11
Mechanical Strains of 32-Fastener Setup of Thick Panel with Cirlex® .............................67
5.12
Mechanical Strains of 24-Fastener Setup of Thick Panel with Cirlex® .............................68
5.13
Mechanical Strains of 16-Fastener Setup of Thick Panel with Cirlex® .............................68
5.14
Mechanical Strains of 8-Fastener Setup of Thick Panel with Cirlex® ...............................69
5.15
Mechanical Strains of 66-Fastener Setup of Thin Panel without Insulation .....................70
5.16
Mechanical Strains of 48-Fastener Setup of Thin Panel without Insulation .....................71
5.17
Mechanical Strains of 32-Fastener Setup of Thin Panel without Insulation .....................71
5.18
Mechanical Strains of 24-Fastener Setup of Thin Panel without Insulation .....................72
5.19
Mechanical Strains of 16-Fastener Setup of Thin Panel without Insulation .....................72
5.20
Mechanical Strains of 8-Fastener Setup of Thick Panel without Insulation......................73
5.21
Mechanical Strains of 66-Fastener Setup of Thin Panel with Cirlex®...............................74
5.22
Mechanical Strains of 48-Fastener Setup of Thin Panel with Cirlex®...............................75
5.23
Mechanical Strains of 32-Fastener Setup of Thin Panel with Cirlex®...............................75
5.24
Mechanical Strains of 24-Fastener Setup of Thin Panel with Cirlex®...............................76
xiii
LIST OF FIGURES (continued)
Figure
Page
5.25
Mechanical Strains of 16-Fastener Setup of Thin Panel with Cirlex®...............................76
5.26
Mechanical Strains of 8-Fastener Setup of Thin Panel with Cirlex® ................................77
5.27
Configuration of Chamber Tests without Insulation .........................................................79
5.28
Mechanical Strains Comparison between Tests and Analytical Model for
66-Fastener Setup without Insulation (Thick Panel) .........................................................80
5.29
Mechanical Strains Comparison between Tests and Analytical Model for
48-Fastener Setup without Insulation (Thick Panel) .........................................................81
5.30
Mechanical Strains Comparison between Tests and Analytical Model for
32-Fastener Setup without Insulation (Thick Panel) .........................................................81
5.31
Mechanical Strains Comparison between Tests and Analytical Model for
24-Fastener Setup without Insulation (Thick Panel) .........................................................82
5.32
Mechanical Strains Comparison between Tests and Analytical Model for
16-Fastener Setup without Insulation (Thick Panel) .........................................................82
5.33
Mechanical Strains Comparison between Tests and Analytical Model for
8-Fastener Setup without Insulation (Thick Panel) ...........................................................83
5.34
Mechanical Strains Comparison between Tests and Analytical Model for
66-Fastener Setup with Co-Cured Cirlex® (Thick Panel) ..................................................85
5.35
Mechanical Strains Comparison between Tests and Analytical Model for
48-Fastener Setup with Co-Cured Cirlex® (Thick Panel) ..................................................86
5.36
Mechanical Strains Comparison between Tests and Analytical Model for
32-Fastener Setup with Co-Cured Cirlex® (Thick Panel) ..................................................86
5.37
Mechanical Strains Comparison between Tests and Analytical Model for
24-Fastener Setup with Co-Cured Cirlex® (Thick Panel) ..................................................87
5.38
Mechanical Strains Comparison between Tests and Analytical Model for
16-Fastener Setup with Co-Cured Cirlex® (Thick Panel) ..................................................87
5.39
Mechanical Strains Comparison between Tests and Analytical Model for
8-Fastener Setup with Co-Cured Cirlex® (Thick Panel) ....................................................88
xiv
LIST OF FIGURES (continued)
Figure
Page
5.40
Mechanical Strains Comparison between Tests and Analytical Model for
66-Fastener Setup without Insulation (Thin Panel) ...........................................................90
5.41
Mechanical Strains Comparison between Tests and Analytical Model for
48-Fastener Setup without Insulation (Thin Panel) ...........................................................91
5.42
Mechanical Strains Comparison between Tests and Analytical Model for
32-Fastener Setup without Insulation (Thin Panel) ...........................................................91
5.43
Mechanical Strains Comparison between Tests and Analytical Model for
24-Fastener Setup without Insulation (Thin Panel) ...........................................................92
5.44
Mechanical Strains Comparison between Tests and Analytical Model for
16-Fastener Setup without Insulation (Thin Panel) ...........................................................92
5.45
Mechanical Strains Comparison between Tests and Analytical Model for
8-Fastener Setup without Insulation (Thin Panel) .............................................................93
5.46
Mechanical Strains Comparison between Tests and Analytical Model for
66-Fastener Setup with Co-Cured Cirlex® (Thin Panel)....................................................95
5.47
Mechanical Strains Comparison between Tests and Analytical Model for
48-Fastener Setup with Co-Cured Cirlex® (Thin Panel)....................................................96
5.48
Mechanical Strains Comparison between Tests and Analytical Model for
32-Fastener Setup with Co-Cured Cirlex® (Thin Panel)....................................................96
5.49
Mechanical Strains Comparison between Tests and Analytical Model for
24-Fastener Setup with Co-Cured Cirlex® (Thin Panel)....................................................97
5.50
Mechanical Strains Comparison between Tests and Analytical Model for
16-Fastener Setup with Co-Cured Cirlex® (Thin Panel)....................................................97
5.51
Mechanical Strains Comparison between Tests and Analytical Model for
8-Fastener Setup with Co-Cured Cirlex® (Thin Panel)......................................................98
5.52
Temperature Distribution in Steady State for Aluminum/Composite
Assembly without Insulation (Thin Panel) ........................................................................99
5.53
Deformation of Fastened Aluminum/Composites Assembly due to Thermal Load..........99
xv
LIST OF FIGURES (continued)
Figure
Page
5.54.
Load Transfer Comparison for 24 units (half of 48 F case) for
Aluminum/Composite Assembly (Thin Panel)................................................................100
5.55
Load Transfer Comparison between with and without Insulation
Material for 66-Fastener Setup (Thick Panel) .................................................................101
5.56
Load Transfer Comparison between with and without Insulation
Material for 48-Fastener Setup (Thick Panel) .................................................................102
5.57
Load Transfer Comparison between with and without Insulation
Material for 32-Fastener Setup (Thick Panel) .................................................................102
5.58
Load Transfer Comparison between with and without Insulation
Material for 24-Fastener Setup (Thick Panel) .................................................................103
5.59
Load Transfer Comparison between with and without Insulation
Material for 16-Fastener Setup (Thick Panel) .................................................................103
5.60
Load Transfer Comparison between with and without Insulation
Material for 8-Fastener Setup (Thick Panel) ...................................................................104
5.61
Load Transfer Comparison between with and without Insulation
Material for 66-Fastener Setup (Thin Panel) ...................................................................104
5.62
Load Transfer Comparison between with and without Insulation
Material for 48-Fastener Setup (Thin Panel) ...................................................................105
5.63
Load Transfer Comparison between with and without Insulation
Material for 32-Fastener Setup (Thin Panel) ...................................................................105
5.64
Load Transfer Comparison between with and without Insulation
Material for 24-Fastener Setup (Thin Panel) ...................................................................106
5.65
Load Transfer Comparison between with and without Insulation
Material for 16-Fastener Setup (Thin Panel) ...................................................................106
5.66
Load Transfer Comparison between with and without Insulation
Material for 8-Fastener Setup (Thin Panel) .....................................................................107
6.1
Dimensions of Metallic Z-Beams ....................................................................................109
6.2
Equivalent Area Aa (aluminum) for D=0.1875",tb=0.13",tp=0.125"................................112
xvi
LIST OF FIGURES (continued)
Figure
Page
6.3
Equivalent Area Aa (Titanium) for D=0.1875",tb=0.13",tp=0.125" .................................112
6.4
Equivalent Area Aa (Steel) for D=0.1875",tb=0.13",tp=0.125"........................................112
6.5
Equivalent Area Aa (Aluminum) for D=0.1875",tb=0.25",tp=0.08".................................113
6.6
Equivalent Area Aa (Titanium) for D=0.1875",tb=0.25",tp=0.08" ...................................113
6.7
Equivalent Area Aa (Steel) for D=0.1875",tb=0.25",tp=0.08"..........................................113
6.8
Equivalent Area Aa (Aluminum) for D=0.1875",tb=0.13",tp=0.08".................................114
6.9
Equivalent Area Aa (Titanium) for D=0.1875",tb=0.13",tp=0.08" ...................................114
6.10
Equivalent Area Aa (Steel) for D=0.1875",tb=0.13",tp=0.08"..........................................114
6.11
Equivalent Area Aa (Aluminum) for D=0.1875",tb=0.5",tp=0.125".................................115
6.12
Equivalent Area Aa (Titanium) for D=0.1875",tb=0.5",tp=0.125" ...................................115
6.13
Equivalent Area Aa (Steel) for D=0.1875",tb=0.5",tp=0.125"..........................................115
6.14
Equivalent Area Aa (Aluminum) for D=0.1875",tb=0.25",tp=0.125"...............................116
6.15
Equivalent Area Aa (Titanium) for D=0.1875",tb=0.25",tp=0.125" .................................116
6.16
Equivalent Area Aa (Steel) for D=0.1875",tb=0.25",tp=0.125"........................................116
6.17
Equivalent Area Aa (Aluminum) for D=0.25",tb=0.13",tp=0.08".....................................117
6.18
Equivalent Area Aa (Titanium) for D=0.25",tb=0.13",tp=0.08" .......................................117
6.19
Equivalent Area Aa (Steel) for D=0.25",tb=0.13",tp=0.08"..............................................117
6.20
Equivalent Area Aa (Aluminum) for D=0.25",tb=0.5",tp=0.125".....................................118
6.21
Equivalent Area Aa (Titanium) for D=0.25",tb=0.5",tp=0.125" .......................................118
6.22
Equivalent Area Aa (Steel) for D=0.25",tb=0.5",tp=0.125"..............................................118
6.23
Equivalent Area Aa (Aluminum) for D=0.25",tb=0.25",tp=0.125"...................................119
xvii
LIST OF FIGURES (continued)
Figure
Page
6.24
Equivalent Area Aa (Titanium) for D=0.25",tb=0.25",tp=0.125" .....................................119
6.25
Equivalent Area Aa (Steel) for D=0.25",tb=0.25",tp=0.125"............................................119
6.26
Equivalent Area Aa (Aluminum) for D=0.25",tb=0.25",tp=0.08".....................................120
6.27
Equivalent Area Aa (Titanium) for D=0.25",tb=0.25",tp=0.08" .......................................120
6.28
Equivalent Area Aa (Steel) for D=0.25",tb=0.25",tp=0.08"..............................................120
6.29
Equivalent Area Aa (Aluminum) for D=0.375",tb=0.25",tp=0.125".................................121
6.30
Equivalent Area Aa (Titanium) for D=0.375",tb=0.25",tp=0.125" ...................................121
6.31
Equivalent Area Aa (Steel) for D=0.375",tb=0.25",tp=0.08"............................................121
6.32
Equivalent Area Aa (Aluminum) for D=0.375",tb=0.5",tp=0.08".....................................122
6.33
Equivalent Area Aa (Titanium) for D=0.375",tb=0.5",tp=0.08" .......................................122
6.34
Equivalent Area Aa (Steel) for D=0.375",tb=0.5",tp=0.08"..............................................122
6.35
Equivalent Area Aa (Aluminum) for D=0.375",tb=0.13",tp=0.125".................................123
6.36
Equivalent Area Aa (Titanium) for D=0.375",tb=0.13",tp=0.125" ...................................123
6.37
Equivalent Area Aa (Steel) for D=0.375",tb=0.13",tp=0.125"..........................................123
6.38
Equivalent Area Aa (Aluminum) for D=0.375",tb=0.5",tp=0.125"...................................124
6.39
Equivalent Area Aa (Titanium) for D=0.375",tb=0.5",tp=0.125" .....................................124
6.40
Equivalent Area Aa (Steel) for D=0.375",tb=0.5",tp=0.125"............................................124
6.41
Equivalent Area Ac of 0.25" Thick Composite Panel for
Different Fastener Diameters...........................................................................................125
6.42
Equivalent Area Ac of the Composite Panel as a Function of x .......................................126
6.43
Equivalent Area Ac as a Function of x for Different Panel Thicknesses..........................126
xviii
LIST OF FIGURES (continued)
Figure
Page
6.44
Comparison of Equivalent Area Ac as a Function of x
for Different Panel Thicknesses.......................................................................................127
6.45
Comparison of Ta Obtained from FEM and Equations as a function
of Fastener Diameter D for tp=0.13", tb=0.125"...............................................................128
6.46
Comparison of Ta Obtained from FEM and Equations as a function
of Fastener Diameter D for tp=0.25", tb=0.08".................................................................129
6.47
Comparison of Ta Obtained from FEM and Equations as a function
of Fastener Diameter D for tp=0.5", tb=0.125".................................................................129
6.48
Comparison of Ta Obtained from FEM and Equations as a function
of Fastener Diameter D for tp=0.25", tb=0.125"...............................................................130
6.49
Comparison of Ta Obtained from FEM and Equations as a function
of Fastener Diameter D for tp=0.5", tb=0.08"...................................................................130
6.50
Comparison of Tc (Aluminum) for D=0.1875", tp=0.13", tb=0.125" ...............................132
6.51
Comparison of Tc (Titanium) for D=0.1875", tp=0.13", tb=0.125"..................................132
6.52
Comparison of Tc (Steel) for D=0.1875", tp=0.13", tb=0.125" ........................................132
6.53
Comparison of Tc (Aluminum) for D=0.1875", tp=0.25", tb=0.08" .................................133
6.54
Comparison of Tc (Titanium) for D=0.1875", tp=0.25", tb=0.08"....................................133
6.55
Comparison of Tc (Steel) for D=0.1875", tp=0.25", tb=0.08" ..........................................133
6.56
Comparison of Tc (Aluminum) for D=0.1875", tp=0.13", tb=0.08" .................................134
6.57
Comparison of Tc (Titanium) for D=0.1875", tp=0.13", tb=0.08"....................................134
6.58
Comparison of Tc (Steel) for D=0.1875", tp=0.13", tb=0.08" ..........................................134
6.59
Comparison of Tc (Aluminum) for D=0.1875", tp=0.25", tb=0.04" .................................135
6.60
Comparison of Tc (Titanium) for D=0.1875", tp=0.25", tb=0.04"....................................135
6.61
Comparison of Tc (Steel) for D=0.1875", tp=0.25", tb=0.04" ..........................................135
xix
LIST OF FIGURES (continued)
Figure
Page
6.62
Comparison of Tc (Aluminum) for D=0.1875", tp=0.25", tb=0.125" ...............................136
6.63
Comparison of Tc (Titanium) for D=0.1875", tp=0.25", tb=0.125"..................................136
6.64
Comparison of Tc (Steel) for D=0.1875", tp=0.25", tb=0.125" ........................................136
6.65
Comparison of Tc (Aluminum) for D=0.25", tp=0.5", tb=0.125" .....................................137
6.66
Comparison of Tc (Titanium) for D=0.25", tp=0.5", tb=0.125"........................................137
6.67
Comparison of Tc (Steel) for D=0.25", tp=0.13", tb=0.125" ............................................137
6.68
Comparison of Tc (Aluminum) for D=0.25", tp=0.13", tb=0.08" .....................................138
6.69
Comparison of Tc (Titanium) for D=0.25", tp=0.13", tb=0.08"........................................138
6.70
Comparison of Tc (Steel) for D=0.25", tp=0.13", tb=0.08" ..............................................138
6.71
Comparison of Tc (Aluminum) for D=0.25", tp=0.25", tb=0.125" ...................................139
6.72
Comparison of Tc (Titanium) for D=0.25", tp=0.25", tb=0.125"......................................139
6.73
Comparison of Tc (Steel) for D=0.25", tp=0.25", tb=0.125" ............................................139
6.74
Comparison of Tc (Aluminum) for D=0.25", tp=0.25", tb=0.08" .....................................140
6.75
Comparison of Tc (Titanium) for D=0.25", tp=0.25", tb=0.08"........................................140
6.76
Comparison of Tc (Steel) for D=0.25", tp=0.25", tb=0.08" ..............................................140
6.77
Comparison of Tc (Aluminum) for D=0.375", tp=0.25", tb=0.125" .................................141
6.78
Comparison of Tc (Titanium) for D=0.375", tp=0.25", tb=0.125"....................................141
6.79
Comparison of Tc (Steel) for D=0.375", tp=0.25", tb=0.125" ..........................................141
6.80
Comparison of Tc (Aluminum) for D=0.375", tp=0.13", tb=0.125" .................................142
6.81
Comparison of Tc (Titanium) for D=0.375", tp=0.13", tb=0.125"....................................142
6.82
Comparison of Tc (Steel) for D=0.375", tp=0.13", tb=0.125" ..........................................142
xx
LIST OF FIGURES (continued)
Figure
Page
6.83
Comparison of Tc (Aluminum) for D=0.375", tp=0.5", tb=0.125" ...................................143
6.84
Comparison of Tc (Titanium) for D=0.375", tp=0.5", tb=0.125"......................................143
6.85
Comparison of Tc (Steel) for D=0.375", tp=0.5", tb=0.125" ............................................143
6.86
Comparison of Kf Obtained from FEM and Equations as a Function
of Fastener Diameter D for tp=0.25", tb=0.125"...............................................................145
6.87
Comparison of Kf Obtained from FEM and Equations as a Function
of Fastener Diameter D for tp=0.13", tb=0.125"...............................................................145
6.88
Comparison of Kf Obtained from FEM and Equations as a Function
of Fastener Diameter D for tp=0.5", tb=0.125".................................................................146
6.89
Comparison of Kf Obtained from FEM and Equations as a Function
of Fastener Diameter D for tp=0.5", tb=0.08"...................................................................146
xxi
CHAPTER 1
INTRODUCTION
1.1 Background
In structural applications such as aircraft and spacecraft, composite components are often
fastened to other structural components (composites or metal) by mechanical means or by
adhesive bonding. While bonded joints have advantages such as low weight, distributed load
transfer, and perfect sealing of the structure, the low resistance of a composite joint to interlaminar stresses limits the loads that can be transferred. Bolted joints are thus preferred for
transferring high loads and have particular relevance for future primary structures. They are also
preferred in situations where disassembly is required for inspection or repair. Combining these
two techniques has been considered unnecessary in terms of structural performance, since the
adhesive provides a stiffer load path and hence transfers the majority of the load. Mechanically
fastened joints are used frequently and are one of the most important elements in composite
structures such as aircraft. Regardless of the combination of material in the parts joined, the joint
is a critical element whose design is vital for overall structural performance. Improper design
may lead to overweight or defective structures.
Bolted joints are classified by the service loads placed on them. If those loads are applied
parallel to the axes of the bolts, the joint is called a tensile joint. If the line of action of loads is
essentially perpendicular to the axes of the bolts, the joint is called a shear joint. The purpose of
bolts in all tensile and most shear joints is to create a clamping force between the joint members
to prevent them from separating. If the joint is exposed to shear loads, the bolts must also prevent
the joint members from slipping.
1
In the case of a tension joint, both joint members are bolts, which behave like stiff springs,
one being compressed and the other being stretched, as shown in Figure 1.1.
Figure 1.1. Deformation of Bolt and Joint Members Elastically when Tightened [1].
Like springs, they acquire potential energy when tightened. When released, they suddenly
snap back to their original dimension [1]. This stored energy allows bolts to maintain the
clamping force between the joint members after they are tightened.
In shear joints, bolts resist slip by acting as shear pins, and joint integrity is determined by
shear strength of the bolts and joint members. In many shear-loaded joints, slip is prevented by
friction restraint between joint members. These friction forces are created by the clamping load.
1.1.1 Composites in the Aircraft Industry
A composite material may be defined as the combination of two or more materials on a
macroscopic scale to form another useful material. Examples of composite materials range from
common reinforced concrete to glass fiber reinforced plastics (GFRP) and carbon fiber
composites, to name only a few. The term “advanced composites” is usually used for the
composite materials consisting of polymer matrix and glass or carbon/graphite fibers. Composite
materials, if properly used, offer many advantages over metals. Such advantages include: high
2
strength and high stiffness-to-weight ratio, good fatigue strength, corrosion resistance, and low
thermal expansion. In order to achieve higher fuel efficiency, the use of light-weight, highstrength composite materials, such as carbon/epoxy, needs to be fully explored. Large composite
structures have been increasingly used in the aviation industry. New applications of composite
materials include primary structures such as aircraft fuselages. These large composite parts are
usually attached, either by fasteners or adhesive bonding, to metallic structures. Computer
software and test data are well established for calculating mechanical loads of aircraft structures
during flight, take off, and landing. However, thermally induced loads due to large coefficient of
thermal expansion (CTE) mismatch between the metallic and composite structures, such as those
between aluminum frames and composite fuselage skins, have not been thoroughly investigated.
Due to the large CTE mismatch between composite materials and metallic materials, the
temperature change from the aircraft assembly line to the actual flight condition induces high
thermal stresses, which might result in premature failure.
1.1.2 Mechanical Joints in Aircraft Structures
Joints which are fastened mechanically are critical elements in aircraft structures.
Therefore, design of the joint is of great importance, because improper design may lead to
overweight or defective structures and high life cycle costs of the aircraft. For joints in composite
structures, this is even more pronounced because of their inability to yield, the low transverse
strength, anisotropy, and sensitivity to temperature and moisture. Mechanically fastened joints,
such as those that are bolted and riveted, are preferable where disassembly for inspection or
maintenance is important.
3
Typical applications of mechanically fastened joints in composite aircraft structures are the
skin to spar/rib connections in, for example, a wing structure, the wing to fuselage connection,
and the attachment of fittings. Examples of such joints are shown in Figure 1.2.
Figure 1.2. Examples of Bolted Composite Joints.
Mechanically fastened joints can be classified into two general types by the amount of load
being transferred as lightly loaded and highly loaded joints. An Example of lightly loaded joints
is the connection between substructure and skin. Root joint of a wing is an example of a highly
loaded joint. Dissimilar materials can be fastened by means of mechanical joints. This feature is
used intensively in the aircraft industry to join aluminum components with composite structures.
Most of the mechanical joints encountered in aircraft structures have multiple fasteners. The
number and type of fasteners needed to transfer the given loads are usually established by
airframe designers relative to available space, productivity, and assembly.
4
1.1.3 Design Methods
Improper design of bolted joints may lead to overweight or structural problems and high life
cycle costs of the aircraft. Mechanically fastened joints are difficult to design due to many
parameters and complex phenomena such as contact and friction involved in the behavior of the
joints. The tensile strength of bolts are of primary interest since it determines the amount of
preload that can be applied to the bolt and the amount of working load it can see thereafter. It is
important to recognize, therefore, that the load carrying ability of a given fastener is reduced if
the fastener also experiences torsion or shear loads as well as tensile loads. Several analysis
programs for composite bolted joints are available for composite structure designers. Snyder et
al, [3] examined six different analytical programs and discussed their merits and disadvantages.
In Figure 1.3 the overall design procedure for the case when finite element analysis (FEA)
is used in each step is presented. First, the internal load distribution in the joint is determined.
The more important the joint is considered to be, the more detailed the modeling that is
performed. Then, the local stress distribution around the fastener is determined by detailed FEA
or by analytically based methods, and the strength is predicted using the appropriate failure
criteria. The determination of the local stress distribution in a bolted laminate is, in general, a
three-dimensional problem. The three-dimensional stress state is due to the effects of bending
and clamping of the fastener.
Load distribution analysis is performed to calculate the load distribution between the
fasteners in a multi-fastener joint and a cut-out part of the laminate around the fastener. Load
5
distribution analyses are often performed on large structures, which result in rather coarse
meshes and simplified modeling of the joints.
Global Structural Analysis
Load Distribution Analysis
Local Stress Analysis
Figure 1.3. Overall Design Procedures.
1.1.4 Importance of Bolt Preload
A bolt is tightened by applying torque to the head and/or nut. As the bolt is tightened, it is
stretched (preloaded). Preload tension is necessary to keep the bolt tight, increase joint strength,
6
create friction between parts, and improve fatigue resistance. As a nut is rotated on a bolt's
screw thread against a joint, the bolt is extended. Because internal forces within the bolt resist
this extension, a tension force or bolt preload is generated. The reaction to this force is a clamp
force that is the cause of the joint being compressed. Bolt extension and joint compression are
shown in Figure 1.4. The slope of the lines represents the stiffness of each part. The clamped
joint is usually stiffer than the bolt.
Figure 1.4. Basic Joint Diagrams.
Preload has been shown to be the critical factor in ensuring the static and dynamic
reliability of a bolted joint [4]. In the control of preload using the torque control method, the bulk
relationship between the applied torque and the clamping force can be expressed by the
following equation, which is widely used in industry:
T = kDP
(1.1)
where D is the nominal diameter of the bolt, K is the torque coefficient, and P is the clamping
force created in the fastener.
The amount of clamping force provided by installed fasteners is quite important in the
mechanical behavior of bolted joints. As has been observed, mechanically fastened joints with
high clamp-ups yield better strength to applied static loading than joints with low clamp-ups [5].
7
The main reason for this is that, in bolted joints with highly tightened fasteners, a larger part of
the applied load is transferred by friction forces than in joints with low-tightened fasteners.
1.1.5 Thermo-Mechanical Loading
The analysis of the thermal phenomena occurring in a bolted joint is complex since heat
flow depends on many independent parameters such as surface roughness, surface waviness,
thermal conductivity, Poisson’s ratio, yield stress, hardness, applied load, and geometry of the
system. When a structure is exposed to a change in operating temperature, thermally induced
stresses arise from constrained thermal expansion or contraction. In addition, both mechanical
and thermal properties may change over the range of temperature. The combined effect of altered
thermal environment and additionally applied mechanical loading results in a complex thermomechanical loading of the structure, which might result in premature failure. Thermal effects in
addition to mechanically applied loads create a state of thermo-mechanical loading for the bolted
joint. Elevated temperatures may degrade the load-carrying capabilities of the joint and
consequently impose additional challenges on the joint design. Unfortunately, design tools such
as those developed by Ram Kumar et al, [6] do not include temperature effects.
1.1.6 Defining Pretension in ABAQUS
ABAQUS, a pre-processing and post-processing software is used to define the pretension section as a surface inside the fastener that cuts it into two parts, as shown in Figure 1.5.
The surface, which contains the element and face information, is defined with the *SURFACE,
TYPE=ELEMENT option [7]. The *PRE-TENSION SECTION option is used along with the
required SURFACE parameter to convert the surface into a pre-tension section across which pretension loads are applied. The required NODE parameter is used to assign a controlling node to
the pre-tension section. The assembly load is transmitted across the pre-tension section by
8
means of the pre-tension node. The pre-tension node should not be attached to any element in the
model. It has only one degree of freedom (degree of freedom 1), which represents the relative
displacement at the two sides of the cut in the direction, of the normal as shown in Figure 1.6.
The coordinates of this node are not important. A concentrated load of 1167 pounds is applied to
the pre-tension node by using the *CLOAD option. This load is the self-equilibrating force
carried across the pre-tension section, acting in the direction of the normal on the part of the
fastener underlying the pre-tension section (the part that contains the elements that are used in
the definition of the pre-tension section).
Figure 1.5. Pre-Tension Section.
Controlling the Pre-Tension Node during the Analysis
9
The initial adjustment of the pre-tension section is maintained by using the *BOUNDARY,
FIXED option once an initial pre-tension is applied in the fastener; this option enables the load
across the pre-tension section to change according to the externally applied loads to maintain
equilibrium. If the initial adjustment of a section is not maintained, the force in the fastener will
remain constant.
Figure 1.6. Prescribed Assembly Load Given at Pre-Tension Node Applied in Direction n.
1.1.7 Defining Contact in ABAQUS
When two plates are mechanically fastened, it is necessary to define interactions between
the surfaces in contact. The contact between the upper plate and lower plate, fastener and
cylindrical hole of the upper plate, and fastener and cylindrical portion of the lower plate can be
defined using the contact pair approach in ABAQUS. This approach is based on the master-slave
concept, and the contact problem is solved using the Lagrange multiplier method [8]. The contact
pairs are defined from surfaces, which in turn are defined from free-element faces. Since sliding
between different parts is assumed to be small, the ‘small sliding’ option can be used. The small
sliding option implies that a possible contact between master and slave nodes are defined at the
10
beginning of the analysis and are not redefined during the analysis. The different surfaces used to
define contact pairs when tow plates are joined with a fastener are shown in Figure 1.7.
Figure 1.7. Contact Surfaces.
1.2 Literature Review
Computer software and test data are well established for calculating mechanical loads of
aircraft structures during flight, take-off, and landing. However, the thermally induced loads due
to large CTE mismatch between the metallic and composite structures, such as those between
aluminum frames and composite fuselage skins, have not been thoroughly investigated.
In a single-lap composite joint, the stress field in the vicinity of the bolt hole in a composite
is three dimensional because inter-laminar stresses are present at the free edges, and also,
bending of the bolt due to stresses and secondary bending in the composite lap joint create a nonuniform stress distribution between the bolt and hole edge. Despite this, most of the studies on
bolted joints in composite structures are two dimensional. Ireman [8] conducted a threedimensional stress analysis of bolted single-lap composite joints to determine the non-uniform
stress distributions through the thickness of composite laminates in the vicinity of the bolt hole.
11
A number of different joint configurations including laminate lay-up, bolt diameter, bolt type,
bolt pretension, and lateral support condition were studied. Ireman set up an experimental
program to measure deformations, strains, and bolt loads on test specimens for validation of the
numerical model developed. The total load transferred was measured using three strain gauges
positioned between two fasteners, and the load transferred by the bearing was measured using an
instrumented steel bolt [8].
Bolted joints are widely used in composite structures. However, most studies of composite
structural joints are concerned with single-bolt joint. The problems of a multi fastener are more
complex than a single bolted joint [9]. Determining the loading magnitude and direction is the
main concern. Xiao Peng, et al, studied the load distribution of multi-fastener composite joints
under applied shearing load or off-axis tensile load. In this study, finite element analysis was
used, and the bolts were assumed to be rigid. The effects of material and geometric properties on
bolt-load distribution were considered. It was concluded that for multi-bolt joints under off-axis
tensile load distribution changes with various angles of off-axis loading and also the degree of
non-uniformity of a bolt-loading distribution under a shearing load is larger than under tensile
load, and the degree of non-uniformity between the two rows will be improved with an increase
in the number of bolts in a row.
Snyder et al, [3] examined six different analytical programs, but none of these programs
focused on analyzing multiple-row joints under generalized in-plane loading. Eriksson [10]
carried out the analysis of multiple-row bolted composite joints under in-plane loading
conditions. The stress analysis is typically done in two steps.
12
• Source analysis (load distribution analysis), the objective of which is to determine the
distribution of applied load between fasteners and the load distribution far away from the
fastener holes.
•
Target analysis (detailed stress analysis), the objective of which is to determine the stress
distributions around the bolt holes, which are required for the subsequent failure analysis.
The failure analysis was done according to simple point stress criteria. Experiments were
performed using single-bolt specimens to validate the target and failure analyses for a
graphite/epoxy material.
The stresses and strengths of a bolted joint depend on many factors such as end distance,
plate width, bolt clearance, number of bolts, bolt spacing, and load distribution between bolts.
Most analyses assume the contact stresses, ignore friction, consider only perfectly fitting bolts,
ignore the way in which the individual bolt holes load up and neglect bolt spacing. Considering
the above factors, Rahman[11] developed a geometrically nonlinear finite element analysis to
determine the stresses in a double-bolted mechanical joint of orthotropic plates. The bolts were
in series, and the loaded plates were considered to be under plane stress and to behave
orthotropically linear elastic. This problem is nonlinear due to the change in contact area with the
change in load. In an externally loaded multiple bolted joint, the amount of load taken by each
bolt (hole) and the boundary of each loaded hole that contacts the bolt were determined using
this approach. The results could be extended readily to an additional number of bolts.
Polymer matrix composites are generally joined by mechanical fastening or adhesive
bonding. Since adhesive provides a stiffer load path and transfers the majority of load,
combining these two methods was considered unnecessary in terms of structural performance.
Hybrid joining techniques (combining mechanical fastening and hybrid bonding) could be
13
motivated in non-aerospace applications like joining polymer matrix composite composites.
Kelly [12] investigated the distribution of load in hybrid (bolted/bonded) composite single-lap
joints to identify joint configurations and conditions where the method offers improved structural
performance. Load transferred by the adhesive and the bolt in a hybrid joint was calculated using
both the finite element method and experimentally using a specially designed instrumented bolt.
Finite element analysis included a parametric study to investigate the effect of selected geometric
and material parameters on the load distribution through the joint. The parametric study indicated
an increase in load transferred by the bolt with an increase in adherend thickness, increase in
adhesive thickness, and increase in overlap length, pitch, and adhesive modulus.
Lin[13] investigated the response of bolted composite single-lapped joints loaded in
tension. The failure behavior of single-lapped mixed composite joints with variations in bolting
arrangement and clamping pressure was studied. The C-scan NDT method was used to observe
the failure mechanism. This Author also conducted a three-dimensional failure analysis to
predict the site of damage initiation in the joint.
Johan Ekh [14] investigated the load transfer in multi-fastener single-shear joints. Plates
with different thicknesses, and stiffnesses, and CTEs (coefficient of thermal expansion) were
used. A finite element model was validated using an instrumented fastener in the experimental
program. Any variation in clearance between different holes implies that the load is shifted to the
fastener where the smallest clearance occurs. Temperature influenced the load distribution
because of different bolt clearances at the holes caused by the eccentricity of the holes. In
composite laminates used in aircraft structures, the CTE can be smaller than that of aluminum,
which implies that hole eccentricity changes when temperature changes. It was observed that the
effective clearance was more at 20ºC than at -50ºC which approximately corresponds to a
14
situation where the joint was manufactured and assembled at room temperature and the aircraft
was operating at a temperature of -50ºC. This resulted in more load transfer at 20ºC.
In 1979, Perry and Hyer established testing procedures [15]. Testing of 16-ply double-lap
quasi-isotropic specimens was conducted at temperatures of -250ºF, 75ºF, and 600ºF. Wichorek
observed a 30% reduction in bearing strength at the 600ºF testing temperature, in comparison to
room temperature, and an 18% increase in bearing strength at the low temperature of -250ºF, in
comparison to room temperature. Other experimental investigations also revealed strength
reductions at elevated temperatures for bolted composite material joints [16-17].
Another area where effects due to mismatch in coefficient of thermal expansion is studied
is in bonded repairs. Experiments were conducted to measure the thermal residual strains in
bonded repairs [18]. The load distribution among fasteners in a bolted joint were experimentally
measured using strain gages, which generally involve the use of many gages that are bonded to
either surface of the bolted plate [18]. Two analytical models are available to determine the
thermal residual stresses due to CTE mismatch between the skin material and the repair material.
Experimental results from an investigation that examines the combined effects of temperature,
joint geometry, and out-of-plane constraint upon the response of mechanically fastened
composite joints are presented by Wilson and Pipes [19].
1.3 Objective
Composites are currently being used in primary structures such as aircraft fuselages. These
composite structures are usually attached to metallic structures either by fasteners or bonding.
Thermal stresses that arise due to CTE mismatch between composite and metallic structures have
not been thoroughly investigated. Due to large CTE differences and temperature change from an
aircraft assembly line, where the temperature is around 75ºF, to actual flight conditions, where
15
the temperature is around -65ºF large thermal stresses are induced, which may result in
premature failure. When a composite aircraft fuselage is fastened to aluminum beams, the
temperature difference of 140ºF (75ºF at assembly and -65ºF during flight) induces stresses not
only in the fasteners but also in the aluminum beams and composite skins. These high-thermal
stresses can be reduced by decreasing the temperature change from the aircraft assembly line,
where the aluminum/composite structures are assembled, to the actual flight. Therefore, an
approach to maintain aluminum beams at higher temperature by adding a layer of insulating
material between the beam and the composite panel was investigated since it is difficult to
maintain the outer fuselage skin at a higher temperature, as shown in Figure 1.8.
Figure 1.8 Schematic of Aluminum Beam/Composite Skin Assembly.
Hence, the two maintain objectives of this study were as follows:
•
To investigate the thermally induced stresses in the aluminum beams.
•
To investigate the feasibility of thermally isolating the aluminum beams from the
composite fuselage skins.
16
Another important task of this study was to predict the load transfer due to CTE mismatch
without performing any finite element modeling for different geometries and materials of the
assembly.
1.3.1 Solution Approach
In the effort to investigate the thermally induced stresses on the aluminum beam, strain
gages were attached at the locations of interest to record strains during testing. Strain readings
were analyzed to obtain separate thermal and mechanical strains so that the mechanical loads due
to temperature change could be determined. Analytical models were developed to evaluate the
strains and load transfer for each fastener. Parameters included in the analytical model, such as
equivalent areas of the aluminum beam and composite panel, equivalent temperatures of
aluminum beam and composite panel, and fastener stiffness, were obtained using finite element
models.
To investigate the feasibility of thermally isolating the aluminum beams from the
composite fuselage skins, experiments were designed to test aluminum/composites assemblies
with the composite side exposed to a temperature that changed from 75ºF to -60ºF, which
simulates the ambient temperature from the manufacturer assembly line to the actual flight
condition. The Kapton®-based insulating material, Cirlex®, was identified and used in the tests
to thermally isolate the aluminum beams to reduce heat loss to the outside ambient air so that the
temperature change of the aluminum beams could be minimized. Temperatures were recorded at
representative locations throughout the test specimens. Finite element analysis approaches were
developed to simulate the temperature distribution. The load transfer for different geometries and
materials of the assembly due to CTE mismatch, equations correlating the five parameters
included in the analytical model with geometric and material properties were provided.
17
CHAPTER 2
EXPERIMENTAL METHODS
2.1 Objective
The two main objectives of carrying out tests of the Z-shape aluminum beam/flat
composite panel were as follows: (1) to investigate the feasibility of thermally isolating the
aluminum beams from the flat composite panel so that stress due to temperature changes could
be reduced, and (2) to validate the load transferred across the fasteners obtained from the
analytical model.
2.2 Experimental Setup
To investigate the effects of size (length) on thermally induced stresses, an environmental
chamber was built to accommodate the aluminum/composite assembly (68 inches by 36 inches).
Each of the three bays was 12 inches wide so that the central bay could be isolated from the two
side bays in order to minimize any edge effects from the assembly. The environmental chamber
consisted of a warm chamber at the bottom and a cold chamber at the top, as shown in Figures
2.1 and 2.2. A heat blanket was used to supply a constant rate of heat (400 W) to the warm
chamber. A baffle was added between the heat blanket and the test assembly to ensure a uniform
temperature distribution. The outer wall of the chamber was made of plywood, and the inner
walls were insulated by a 1.5-inch-thick rigid sheet of PVC foam. Thus, the warm chamber
controlled the amount of heat supplied to the test assembly. The top portion of the environmental
chamber was fixed with a cold chamber in which liquid nitrogen was injected at a controlled rate
to cool the air inside the chamber down to a stable temperature of -65°F. Two fans were
installed on the walls to generate air circulation. Weather strips were used between the brackets
and the composite panel to prevent cold air from leaking.
18
Figure 2.1. Side View of the Environmental Chamber used for Testing.
For studying the effects of length, six test setups were defined, based on finite element and
analytical model simulations. For the longest setup, 66 fasteners were used for each of the three
beams with a fastened length of 65 inches. After the tests were performed with 66 fasteners, the
same number of fasteners were taken out from each of the two ends (lengthwise) to shorten the
fastened length to 48-fastener, 32-fastener, 24-fastened, 16-fastener, and 8-fastener lengths. For
each setup, different numbers of thermocouples and strain gages were applied at different
locations to record the temperature and strain data of the assembly. Figure 2.3 to Figure 2.8
show the numbers and locations of thermocouples and strain gages used in different
configurations.
19
Figure 2.2. Environmental Chamber used for Testing.
20
Figure 2.3. Thermocouple and Strain Gage Locations for 66-Fastener Configuration
(10 TCs and 30 SGs).
21
Figure 2.4. Thermocouple and Strain Gage Locations for 48-Fastener Configuration
(15 TCs and 25 SGs).
22
Figure 2.5. Thermocouple and Strain Gage Locations for 32-Fastener Configuration
(15 TCs and 24 SGs).
23
Figure 2.6. Thermocouple and Strain Gage Locations for 24-Fastener Configuration
(15 TCs and 25 SGs).
24
Figure 2.7. Thermocouple and Strain Gage Locations for 16-Fastener Configuration
(15TCs and 19 SGs).
25
Figure 2.8. Thermocouple and Strain Gage Locations for 8-Fastener Configuration
(14 TCs and 14 SGs).
26
2.3 Specimen Configuration and Fabrication
A total of four test assemblies, consisting of a Z-shape aluminum beam fastened to a
composite panel were used. Two test specimens consisted of 0.25-inch composite panel fastened
to an aluminum beam (one of which had Cirlex® embedded) and the other two test specimens
consisted of 0.13" composite panel fastened to the aluminum beam (one of which had Cirlex®
embedded). A detailed description of the test specimens follow.
2.3.1 Aluminum/Composite Assembly Tests without Insulation
The test specimen consisted of three 68 inch-long Z-shape aluminum beam, fastened to the
composite panel using titanium fasteners 0.25 inch with a pitch of one inch. The cross section of
the Z-shape aluminum beam is shown in Figure 2.9.
Figure 2.9. Dimensions of Cross Section of Z-Shape Aluminum Beam.
A torque of 87.5 lb-in was applied to tighten the fasteners. The composite panel was
fabricated with uni-directional and plain-weave graphite/epoxy prepeg. The laminate stacking
sequence was balanced, symmetric, and quasi-isotropic, consisting of thirty four plies [0f/[45/0/45/90]4]s obtaining the cured thicknesses of 0.25 inch and [0f/[45/0/-45/90]2]s consisting of 18
plies obtaining a thickness of 0.13 inch. The edges of the cured panel were trimmed to obtain the
27
dimensions of 68 inch by 36 inch. Dimensions of the substrates and fasteners were taken prior to
assembly to accommodate in the finite element model.
2.3.2 Aluminum/Composite Assembly Tests with Insulation
When the temperature was reduced, the aluminum beams contracted more than the
composite panel, and due to interaction, through the fasteners, between the aluminum beams and
the composite panel, resulting in tensile stresses on the aluminum beams and compressive
stresses on the composite panel. Proper insulation material helped to reduce the thermally
induced stresses by maintaining the aluminum beams at a higher temperature. Cirlex® (solid
Kapton® sheet from Fralock of Canoga Park, CA) was used for this purpose. Considering the
structural integrity of the assembly, three Cirlex® strips each 68 inch by 2.75 inch by 0.06 inch in
size for the 0.25-inch thick composite panel and 68 inch by 2.75 inch by 0.12 inch in size for the
0.13-inch thick composite panel were embedded at the mid-plane of the composite panel located
at the point where the three aluminum beams were attached, as shown in Figure 2.10. For
maintaining proper adhesion between Cirlex® and the composite prepeg, a layer of 0.0025-inchthick T-576 adhesive was added at each Cirlex®/prepreg interface, making the final lay-up
0f[45/0/-45/90]4/T-576/Cirlex®/T-576/[90/-45/0/45]4/0f for the 0.25-inch-thick composite panel
and 0f[45/0/-45/90]2/T-576/Cirlex®/T-576/[90/-45/0/45]2/0f for the 0.13-inch-thick composite
panel. The entire panels were co-cured in an autoclave.
28
Figure 2.10. Configuration of Chamber Tests with Cirlex® Co-Cured Inside Composite Panel.
2.4 Testing for Mechanical Properties
The tensile moduli of both the aluminum and composite laminates were obtained according
to ASTM D3039, Standard Test Method for Tensile Properties of Polymer Matrix Composites
(D3039) procedure. Rectangular test specimens machined to have a width and length of 1 inch
and 9 inches, respectively. Specimens were untabbed, because the strength was not evaluated.
However, each end was wrapped with two layers of emery cloth to minimize damages to the
specimen from grip pressure. Tensile testing was conducted in a servo-hydraulic material test
system (MTS) at a displacement controlled rate of 0.05 inch/min. In order to obtain tensile
modulus and Poisson’s ratio, longitudinal and transverse strains were measured using a biaxial
extensometer. Tensile stress, σt, was calculated by
σt =
P
A
29
(2.1)
where P and A correspond to tensile load and the cross-sectional area of the gage section,
respectively. Tensile strain was calculated as the ratio between extensometer displacement and
the extensometer gage length for each direction. Then, the stress-strain was plotted, and the
tensile modulus was calculated from the slope of the linear fit of the strain range given in the
ASTM D3039 test procedure. Typically, it is between 1,000 and 3,000 micro strains.
2.5 Testing for Coefficient of Thermal Expansion (CTE)
The coefficient of thermal expansion (CTE) is defined as the fractional change in length or
volume of a material for a unit change in temperature. When the temperature of an object
increases, the object usually expands. Solid bodies expand in all directions when heated, but
composite tend to expand and contract at different rate on all direction depend on the fiber
orientation of the composite part.
The CTE test was conducted in accordance with the ASTM E831, Standard Test Method
for Linear Thermal Expansion of Solid Materials by Thermo-mechanical Analysis on the Perkin
Elmer thermo-mechanical analyzer (TMA). The temperature ramp method was used, where the
force is constants, and displacement is monitored under a linear temperature ramp to provide
intrinsic property measurements.
The temperature signal of the TMA was calibrated using indium and zinc reference
materials with known CTE values.
Test specimens were machined to sizes of no more than 10 mm (0.394 inch) by 10 mm in
length and width, and no more than 10 mm in height. They were placed inside the TMA using
the quartz holder of the TMA instrument. Then, a quartz probe was lowered onto the specimen to
apply a pre-load to hold the specimen in place. This fixture was made of quartz because of
negligible CTE so that the total displacement during temperature application could be assumed
30
was due to displacement (expansion) of the specimen. Each test specimen was then heated to the
temperature of interest at a constant temperature ramp rate of 5.0°C/min. The coefficient of
linear thermal expansion was calculated between the desired temperature ranges.
Figure 2.11 shows the use of the expansion probe to accurately measure small CTE
changes in an aluminum sample between -80°C and 40°C at a temperature ramp rate of 5°C/min.
Figure 2.11. CTE of Aluminum Sample.
31
CHAPTER 3
ANALYTICAL APPROACH
3.1 Objective
This chapter focuses on the analytical models developed to calculate the load transferred
across the fasteners in an aluminum/composite joint subjected to thermal load. Thermal loads or
stresses arise due to change in temperature between the assembly level and the actual flight
conditions, and are also due to mismatch of coefficients of thermal expansions between the
aluminum beam and the flat composite panel. The analytical models developed by Ananthram
[21] can be used for any fastened assembly with dissimilar metals and different substrate
dimensions. While deriving the analytical model, one bay of Z-shape aluminum beam fastened to
a composite panel was considered instead of three bays, shown in Figure 3.1.
Figure 3.1. Aluminum/Composite Assembly.
32
3.2 Governing Equations of Aluminum Beam
The free body diagram of each unit of aluminum beam is shown in Figure 3.2. The length
of each unit is one inch. Due to symmetry in length, only the left half of the entire assembly was
considered, and the right ends represent the center of the assembly. In the model derivation only
in-plane load transfer in the x-direction is considered since out of plane displacements in the zdirection are very small. As shown in the figure, the right end, which represents the center of the
assembly, is fixed in the x-direction, and the left end of the assembly is free to contract.
Figure 3.2. Free Body Diagram of Z-Shape Aluminum Beam.
Based on the free body diagram of Figure 3.2, the force equilibrium conditions of unit 1
can be written as
P1a − F1 − P2a = 0
(3.1)
Similarly, for the ith unit of the aluminum, the force equilibrium conditions become
Pi a − Fi − Pi a+1 = 0
33
(3.2)
where P is the bypass force, F is the load transferred by the fastener, i = 1… N specifies the unit
number, and the superscript a denotes the aluminum beam. The force boundary conditions are
PNa+1 = 0
(3.3)
The different coefficients of thermal expansion of the aluminum beam and composite panel and
changes in temperature would result in different expansion and contraction. Therefore, tensile
and compressive forces would be acting on the aluminum beam and composite panel.
Each unit of the aluminum beam reduces its length because of the change in temperature.
The reduction in length of the ith unit of aluminum beam ∆uiTa due to temperature change from To
to Ta is
∆u iTa = α a (T o − T a ) La
(3.4)
where αa is the CTE of aluminum, To represents the temperature at which the aluminum
beam/composite panel is assembled, and Ta represents the “equivalent” temperature of the
aluminum beam at which the thermally induced stresses are to be determined. The length of
each unit of the aluminum beam La, is one inch.
If N units of the aluminum beam are considered with unit N at the left end and unit 1 at the
a
, due to mechanical
right end, then the change in length of the ith unit of aluminum beam ∆u iM
loads as a result of CTE mismatch, is given by
∆u
a
iM
Fi L fa
=− a a −
E A0.625
N
F j La
∑E
j = i +1
a
A aj −i
(3.5)
Here, the mechanical loads are the summation of deformations from the fastener forces
from the ith unit to the very left edge of the assembly.
If an assembly of eight units is considered, the change in length of unit 1, which is located
close to the center of the assembly, is given by
34
∆u
a
1M
F3 La
F1 L fa
F2 La
F4 La
=− a a − a a − a a − a a
E A0.625 E A1 E A2 E A3
(3.6)
It should be noted in equation (3.5) that Aia which represents the “equivalent” cross
sectional area of the aluminum beam unit, is a function of distance and will be determined by
finite element analysis discussed in the next chapter. Lfa, which is 0.625 inch, represents the
distance from the left edge of the fastener hole to the right edge of the unit, and Ea is the Young’s
modulus of aluminum. Therefore, reduction in length of the ith unit of aluminum beam is
a
∆uia = uia − uia−1 = ∆uiTa + ∆uiM
(3.7)
From equations (3.4) and (3.5), the kinematics equation of the ith unit of aluminum beam
becomes
u −u
a
i
a
i −1
F L fa
= α (T − T ) L − a i a
−
E A@ 0.625
a
o
a
a
N
∑E
j = i +1
F j La
a
A@a j −i
(3.8)
where i = 1… N . The displacement boundary condition is
u 0a = 0
(3.9)
where u 0a is the displacement at the center of the assembly.
3.3 Governing Equations of Composite Panel
Similarly, for the composite panel based on the free body diagram shown in Figure 3.3, the
force equilibrium condition on the composite panel is
− Pi c + Fi + Pi +c 1 = 0
35
(3.10)
Figure 3.3. Free Body Diagram of Flat Composite Panel.
where P is the bypass force, F is the load transferred by the fastener, i = 1… N specifies the unit
number, and the superscript c denotes the composite panel. The force boundary condition is
PNc +1 = 0
(3.11)
The reduction in length of the ith unit of composite panel due to temperature change ∆uiTc is
∆u iTc = α c (T o − Ti c ) Lc
(3.12)
where αc is the CTE of composite panel, To represents the temperature at which the aluminum
beam/composite panel is assembled, and Tc represents the “equivalent” temperature of the
composite panel at which the thermally induced stresses are to be determined. Tc is a function of
distance and is determined from finite element analysis.
The length of each unit of the
composite panel, La, is one inch.
The load transferred by fastener Fi and the bypass force Pi c+1 compress the ith unit of the
composite panel. Similar to the aluminum beam, the equivalent area Aic of the ith unit of the
36
composite panel determined from finite element analysis varies from unit to unit because the
composite panel is wide, and the area affected by the fastener and bypass loads is a function of
the distance from where the load is applied as shown in Figure 3.4.
Therefore, the reduction in the length of ith unit of the composite panel due to mechanical
loads ∆uiMc is the summation of the deformations due to all fastener forces from the ith unit to the
very left edge of the assembly.
∆u
c
iM
Fi L fc
= c c +
E A0.375
N
F j Lc
∑E
j = i +1
c
(3.13)
A cj −i
When the assembly of eight units is considered, the change in length of unit 1 of the composite,
panel which is close to the center of the assembly, is given by
∆u1cM =
F3 Lc
F1 L fc
F2 Lc
F4 Lc
+
+
+
E c A0c.375 E c A1c E c A2c E c A3c
(3.14)
where Ec is the Young’s modulus of the composite plate, Lfc is the actual length in of each unit
that is deformed by the fastener load, which is the distance from the right edge of fastener hole to
the right edge of the unit, and Lc is the length of each unit.
Therefore, the reduction in length of the ith unit of the composite panel is
∆uic = uic − uic−1 = ∆uTc + ∆u Mc
(3.15)
From equations (3.12) and (3.13), the kinematics equation of the ith unit of the composite panel
becomes
u −u
c
i
c
i −1
Fi L fc
= α (T − Ti ) L + c c
+
E A@ 0.375
c
o
c
c
N
∑E
j = i +1
F j Lc
c
A@c j −i
(3.16)
where i =1… N , and the displacement boundary condition is
u 0c = 0
37
(3.17)
where u 0c is the displacement at the center of the assembly.
Figure 3.4 Area influenced by the external load in the wide composite plate.
3.4 Governing Equations of the Fasteners
The fastener force is related to its deformation as
Fi = K f (u ia − u ic )
(3.18)
where Kf represents the equivalent fastener stiffness, which is calculated from the finite element
model. The description of how to determine Kf is given in Section 4.2.6.
Based on equations (3.1) to (3.18), there are a total of 5N+4 unknowns
Pi a , Pi c ,
(i = 1...N + 1)
u , u , (i = 0...N )
a
i
c
i
Fi , (i = 1...N )
38
which are N + 1 bypass forces on the aluminum beam, N + 1 bypass forces on the composite
panel, N + 1 displacements of the aluminum units, and N + 1 displacements of the composite
units and N fastener loads.
A total of 5N governing equations in terms of the unknowns and all the parameters can be
obtained from equations (3.1) to (3.18). With the inclusion of four boundary conditions based on
equations (3.3), (3.9), (3.11), and (3.17) there are a total of 5N+4 equations. The mathematical
solver Maple 10 was used to solve these 5N+4 equations.
The mechanical strains obtained on the bottom flange of the aluminum beam between the
fasteners were compared with strains obtained from the tests. Once the strains were validated, the
load transferred by the fasteners in the assembly could be predicted.
39
CHAPTER 4
FINITE ELEMENT MODELS
4.1 Objective
The objective of finite element modeling is to determine the five parameters needed in the
analytical model used to calculate load transfer across the fasteners. Based on the analytical
model, six finite element models were developed and executed to determine equivalent area of
the Z-shape aluminum beam Aa, equivalent area of the composite panel Ac, equivalent
temperatures of the Z-shape aluminum beam Ta and composite panel Tc, and stiffness of the
fasteners Kf. An additional model to determine β, the bending factor to compare the strains
obtained from test results and analytical models, was also developed. Finite element models were
developed and executed by using the general pre-processing and post-processing finite element
software ABAQUS/Standard.
Mechanical Finite Element Models
In order to determine the equivalent area of the Z-shape aluminum beam Aa, threedimensional finite element models were constructed consisting of eight units of Z-shape
aluminum beam fastened to the composite panel using titanium fasteners. A flat composite panel
of 20 units was built to determine the equivalent area Ac.
Thermo-Mechanical Finite Element Models
To compare the temperature distribution within the Z-shape aluminum/composite panel
assembly during the environmental chamber test, a steady-state heat transfer finite element
analysis of eight units of Z-shape aluminum/composite panel was carried out, which gave the
temperature distribution at each node. Three sequentially coupled finite element analysis models
40
were developed and executed to determine equivalent temperatures of the aluminum beam Ta
and composite panel Tc, and stiffness of the fastener Kf.
4.2 Finite Element Model Development
A three-dimensional bolted joint assembly for both the mechanical and thermo-mechanical
analysis is described in this section. ABAQUS/Standard 6.5 was utilized to develop all the
mechanical models, thermal models, and mechanical/thermal coupled models. Non-linear
analyses using linear eight-node three-dimensional brick elements were conducted. Contact and
friction were included in the finite element models.
Model Description
As shown in Figure 4.1, the model for finite element analysis consisted of a Z-shape
aluminum beam fastened to a wide composite panel using countersunk titanium fasteners. It
should be noted that a total of eight units, each one inch in length, were constructed while
performing finite element analysis, although the actual testing length varied up to 66 inches.
4.2.1 Finite Element Mechanical Model to Determine Aa
When the Z-shape aluminum beam was fastened to the composite panel, most of the load
was taken by the bottom flange, so it was required to find the equivalent area of the aluminum
beam. For this purpose, an aluminum/composite assembly of eight units was constructed using
ABAQUS 6.5. The meshed model with boundary conditions is shown in the Figure 4.1. “Surface
contact” was defined between the aluminum/composite contact surfaces, the fastener and the
aluminum hole, and the fastener and the composite hole. The finite element model was
developed using three-dimensional brick elements. Eight-node linear brick elements enhanced
with incompatible modes (C3D8I) were used for developing the aluminum beam, composite
panel, and titanium fasteners. The behavior between perpendicular surfaces in contact was
41
established as “hard contact,” which meant that no penetration was allowed. The tangential
surface behavior was governed by the penalty friction formulation. The coefficients of friction
used were 0.2 for aluminum/composites surfaces and 0.07 for the fastener/aluminum holes and
fastener/composite holes. A pre-load or pretension load of 1,750 pounds was applied in the
pretension step. Right end of the assembly was fixed and a load of 1,000 pounds was applied at
the free end, as shown in Figure 4.2. The tensile load was applied at the center of the flat
composite panel.
Figure 4.1. Mechanical FEM of Z-Shape Aluminum Beam Fastened to Composite Panel.
Contact was defined between the fastener and the aluminum hole, the fastener and the
composite panel hole, and between the aluminum beam and the composite panel. The contact
pair approach based on a master-slave algorithm was used with finite sliding allowed between
surfaces in contact. Since this process involves several surfaces in contact with each other,
nonlinear contact analysis was included. A friction coefficient of 0.07 was assumed between the
fastener and the holes, and 0.2 between the aluminum beam and the composite panel. The
42
displacement contour shown in Figure 4.3 reveals that the displacement is not uniform along the
cross section of aluminum beam, and most of the load transferred across the fastener is taken by
the bottom flange. Therefore, it is not reasonable to use the entire cross-sectional area of the Zshape beam.
Figure 4.2. Point of Application of Tensile Load.
Figure 4.3. Displacement Contour.
43
4.2.1.1 Extracting Aa from Finite Element Model
Analyses were performed in two steps. First, a bolt-clamping load was applied through
application of A pre-tension load of 1,750 pounds to the mid-surface of the bolt. Application of
the pre-tension load is discussed in Section 1.1.6. Second, a tensile load of 1,000 pounds was
applied at the free end, as described earlier.
After the analysis was completed, the displacement of each unit at the center node of the
aluminum beam’s bottom flange following the end of the load step was recorded, as shown in
Figure 4.4. In addition, contact forces, including those between the aluminum beam and the
composite panel, and between the fastener and the aluminum hole were also recorded.
Figure 4.4. Displacement Measurements for the Aluminum Beam.
44
As discussed in the previous chapter, the equivalent area of the aluminum beam is a
function of distance x. From Equation (3.5) of Chapter 3, the change in length of unit 8 of the
aluminum beam ∆u 8a , close to the applied load, is given by
F8 L fa
∆u = − a a
E A0.625
a
8
(4.1)
Again, the change in length of unit 7 of the aluminum beam is given by
∆u 7a = −
F7 L fa
F8 La
−
E a A0a.625 E a A1a
(4.2)
.
.
.
.
Similarly, for unit 1 the change in length is given by
∆u1a = −
8
F j La
F1 L fa
−
∑
E a A0a.625 j = 2 E a A aj−1
(4.3)
Therefore, for eight units of the aluminum/composite assembly, there are eight equations, where
Fi is the load transferred by the ith fastener, which was obtained from the finite element analysis;
La is the length of each unit of the aluminum beam (one inch), Lfa; which is 0.625 inch,
represents the distance from the left edge of the fastener hole to the right edge of the unit; Ea is
the Young’s modulus of aluminum; and ∆ uia , obtained from the results of finite element
analysis, represents the deformation of the ith unit, i=1....8. Therefore, the above eight equations
were solved simultaneously to obtain eight values of Aa. These values of Aa signify that the
equivalent area Aa increases while the distance of the load increases up to a certain value and
then remains constant.
45
4.2.2 Finite Element Mechanical Model to Determine Ac
To find the area of the 12-inch-wide composite panel as a function of distance, 20 units of
the composite panel, each one inch in length were constructed. Figure 4.5 shows the meshed
model of 20 units of the composite panel with one end fixed and a compressive load of 1,132
pounds applied at the center of the edge of the hole of the last unit, which is at a distance of
0.625 inch from the left end of unit 20, as shown in Figure 4.6. The load was applied at this
location with the assumption that the load from the fastener was transferred onto the composite
panel at this point. Three-dimensional linear brick elements (C3D8I) were used for developing
20 units of the composite panel.
Figure 4.5. Mechanical FEM of Flat Composite Panel with 20 Units.
46
Figure 4.6 Location at which Concentrated Load is Applied.
As shown in Figure 4.7, displacement of the units near the point where load was applied is
very non-uniform, and as the location of interest is moved away from that point of load
application, displacement becomes more uniform hence, the finite element model with 20 units
was required to describe this trend.
Figure 4.7. Displacement Contour of Flat Composite Panel with 20 Units.
47
Displacements of the center nodes at the left and the right edges of each unit i (or i-1) were
recorded from the output database file and are represented as ui and ui-1 respectively. The relative
displacement ∆uic of each unit was calculated as ui - ui-1. The equivalent area of unit 20 A1c ,
which is closest to the load, was then calculated as
PLc
A = c c
E ∆u 20
c
1
(4.4)
Similarly, the equivalent area of unit 19 A2c was calculated as
A2c =
PLc
E c ∆u19c
(4.5)
where P is the compressive force applied, Ec is the Young’s modulus of the composite panel, and
Lc is the length of the unit under consideration. Hence, equivalent areas of the subsequent units
were also obtained in a similar manner.
Figure 4.8 Displacement Measurements from the Composite Panel.
48
4.2.3 Finite Element Thermal Model to Describe Temperature Distribution of
Aluminum/Composite assembly
A steady-state heat transfer analysis was done to obtain proper temperature distribution of
the Aluminum/Composite assembly. The finite element model of eight units of the assembly is
shown in Figure 4.9. Linear three-dimensional eight-node hex diffusion (DC3D8) elements were
used in this analysis. The aluminum beam was exposed to a room temperature of 77°F, the
bottom surface of the composite panel was exposed to -65°F air, and all other surfaces were
insulated in the heat transfer analysis.
Figure 4.9. Thermal Finite Element Model of Aluminum/Composite Assembly with Eight Units.
49
The temperature distribution at the end of analysis is shown in Figure 4.10. It can be seen
that temperature of the aluminum beam is higher at the top flange and lower at the bottom flange,
and that the temperature of the composite panel underneath the aluminum beam is much lower
than the composite panel away from the beam.
Figure 4.10. Temperature Distribution in Steady State.
4.2.4 Finite Element Analysis to Determine Ta
Due to the non-uniform temperature distribution shown in Figure 4.10, a sequentially
coupled thermal-stress analysis was required to determine the equivalent temperature of the
aluminum beam Ta. This is the most common approach thermal-stress analysis, where the
thermal field is the driving force for stress analysis, i.e., nodal temperatures from the thermal
model described in Section 4.2.3 were fed as the thermal load. The eight units of aluminum beam
50
were modeled with one end fixed, as shown in Figure 4.11, and the rest of the beam was free to
deform under the thermal load. In this analysis, three-dimensional eight-node stress elements
(C3D8I) were used. The displacement contour at the end of thermal/mechanical analysis is
shown in Figure 4.12.
Figure 4.11. Coupled Finite Element Model of Z-shape Aluminum Beam.
After execution of the finite element model, the reduction in length of each aluminum unit
∆ uia , which is calculated as ui - ui-1, was recorded at the bottom flange, and the equivalent
temperature of the ith aluminum beam unit Tia was back-calculated as
Ti a = T o −
51
∆uia
α a La
(4.6)
where the initial room temperature To is 77°F, αa is the CTE of aluminum, and the unit length of
each aluminum beam unit La is one inch.
Figure 4.12. Displacement Contour of an Eight-Unit Z-Shape Aluminum Beam due to Thermal
Load.
4.2.5 Finite Element Analysis to Determine Tc
Similarly, eight units of composite panel were used in performing the thermal/mechanical
analysis to determine the equivalent temperature of the composite panel Tc, as shown in Figure
52
4.13, with one end of the panel fixed and the rest of the panel free to expand under the thermal
load. Eight-node, three-dimensional stress elements (C3D8I), as described in Section 4.2.4, were
used.
Figure 4.13. Coupled Finite Element Model of Composite Panel.
The displacement of an eight-unit model of a composite panel due to thermal loads is
shown in Figure 4.14. As discussed in Section 4.2.3, the temperature of the portion of the
composite panel underneath the aluminum beam was much higher than the rest of the composite
panel because aluminum beam acted as a heat source for the composite panel. Due to the higher
temperatures at the center of the beam, concentrated tensile forces were developed at the center
of the panel. Hence, as described in Section 3.3, the same concept, which was used to determine
the equivalent area of the composite panel Ac, was used to determine Tic. The equivalent
temperature of unit i was calculated as
53
∆u ic
Ti = T − c c
α L
c
o
(4.7)
where ∆ u ic is the reduction of lengths of the ith unit of the composite at the central location, αc is
the CTE of composite, and the length, Lc, of each unit of the composite panel is one inch.
Figure 4.14. Displacement Contour of Eight Unit Flat Composite Panel due to Thermal Load.
4.2.6
Finite Element Analysis to Determine Kf
To determine the stiffness of the fastener, a thermal/mechanical analysis of eight units of
aluminum/composite assembly was constructed and executed, as shown in Figure 4.15. Nodal
temperatures obtained from Section 4.2.3 were used as thermal input. Three-dimensional eightnode elements (C3D8I) were used the right end of the assembly was fixed, and the rest of the
aluminum/composite assembly were free to expand under the thermal load. It must be noted that
this analysis included friction between the joints, and hence it is assumed that part of the load
was transferred through the contact surfaces of the aluminum beam and composite panel and the
54
rest through the fasteners. Similar to what was explained in Section 4.2.1, the analyses were done
in two steps. In the first step, a pre-tension load of 1,750 pounds was applied at the mid-plane of
the fastener, and in the second step, the nodal temperatures obtained from the thermal model
were fed as input to the coupled model in the form of thermal load.
Figure 4.15. Coupled Finite Element Model of Eight Units of Assembly.
From the displacement contour shown in Figure 4.16, it can be seen that contraction of the
bottom flange of the aluminum beam due to temperature changes was restricted by the composite
panel through loads from the fasteners. The composite panel contracted more than the amount
due to thermal load because of the compressive load from aluminum beam through the fasteners.
55
Displacements uia and uic of each unit of the aluminum beam and composite panel were
recorded from the FEA model at the center node of each unit at the end of load step. The load
transferred by the ith unit of the fastener, which is the sum of contact or friction force between the
aluminum surface and composite surface Fj, and the contact force between the fastener and hole
Fb was obtained as output from the ABAQUS model.
Finally, the fastener stiffness was
calculated as
Kf =
Fb + F j
u ia − u ic
(4.8)
Figure 4.16. Deformation of Fastened Aluminum/Composite Assembly due to Thermal Load.
Mechanical strains obtained from the physical tests were compared with strains obtained
from the analytical model. In the physical tests, strains were recorded on the top surface of the
56
bottom flange of the aluminum beam, but the strains obtained from the analytical models were
those at the mid-plane of the bottom flange of aluminum beam. Therefore, in order to be able to
compare the strains, a factor that converts the mid-plane strain to the top surface was necessary.
The bending conversion factor is defined as βi for the ith unit as
βi =
ε iTop
ε icenter
(4.9)
where the strain at the top surface of the bottom flange ε iTop and at the center of the bottom flange
ε icenter are extracted from the finite element model. Because only mechanical strains were
considered, the Z-shape aluminum beam is under tension loads from the fasteners at the bottom
flange. The bottom flange bends up, so the strain at the top surface is smaller than the strain at
center. Therefore, the value of β is less than one. Because the bending effect is small, the value
of β is very close to one. This also validates the assumption that only the in-plane force needs to
be considered.
57
CHAPTER 5
RESULTS AND DISCUSSIONS OF PANEL TESTS
5.1 Results from Material Property Tests
Results from tests conducted to obtain material properties are shown in Table 5.1 along
with the values found in literature. The numbers obtained from tests were used in analytical
models and finite element modeling.
Table 5.1. Material Properties.
Material
Aluminum
Beam
Composite
Panel
Titanium
Fastener
Cirlex®
Young’s Modulus
(Msi)
Poisson’s
Ratio
Thermal Conductivity
@73° F
(×10-6 Btu/in·sec·°F)
Literature
Test
Literature
Test
Test
10.6
10.878
0.328
1,750
-
7.975
0.323
16.9
-
0.264
0.556
CTE
(×10-6/°F)
Literature
Test
-
10.90
10.90
-
4.75
-
1.446
0.31
100
-
4.8
-
0.327
2.27
1.025
11.1
46.95
5.2 Results from Chamber Tests of Fastened Aluminum/Composite Assembly
A total of four group tests were conducted for using the fabricated environmental chamber.
Details of these tests are shown in Table 5.2.
In each of the specimens, three Z-shape aluminum beams were fastened on the composite
panel using 0.25 inch Hi-Lock fasteners. Three Cirlex® strips each of 68 inches in length and
2.75 inches in width, were embedded in the mid-plane of composite panel under the three
aluminum beams for Groups (2) and (4) specimens.
58
Table 5.2. List of Aluminum/Composite Assembly Tests.
# of Fasteners
Composite Type
Group 1
Group 2
Group 3
Group 4
Thick Panel
Without Insulation
Thick Panel With
Cirlex®
Thin Panel
Without Insulation
Thin Panel With
Cirlex®
Dimensions of Flat
Composite Panel
68"×36"×0.25"
66,48,32,24,16,8
2
68"×36"×0.25" with
Cirlex® 68"×2.75"×0.06"
embedded
66,48,32,24,16,8
68"×36"×0.13"
66,48,32,24,16,8
68"×36"×0.13" with
Cirlex® 68"×2.75"×0.12"
embedded
# of Tests
for Each
Case
2
2
2
66,48,32,24,16,8
The temperature distribution of the assembly and the strains on the top surface of the
bottom flange of the aluminum beam were recorded. Strain readings ε taken during the test
actually included two components:
• Temperature output εT0 – the output of the strain gage due to the change of electric
resistance of the gage material from the temperature change, which was truncated by
applying the polynomial supplied by the strain gage manufacturer (Vishay MicroMeasurements), calculated as
ε T 0 = −3.09 × 10 2 + 6.52 × 10 0 T − 3.49 × 10 −2 T 2 + 4.28 × 10 −5 T 3
(5.1)
where T represents the temperature at the instance at which strains were recorded
• Actual deformation of the strain gage, which includes the following: (1) thermal strain εT –
due to thermal contraction, and (2) mechanical strain εM – due to tensile stress. The thermal
strain can be calculated by
ε T = α (T − T o )
59
(5.2)
where α is the CTE of the aluminum beam, and To represents the initial temperature (77°F).
Therefore, the mechanical strain was obtained by truncating the original strain reading ε by the
thermal output εT0 and the thermal strain εT as
ε M = ε − εT 0 − εT
5.2.1
(5.3)
Group 1 Test Results
Six different configurations with different fastened lengths were tested to study length
effects: 66-fastener (fastened length was 65 inches with one inch pitch) setup, 48-fastener
(fastened length was 47 inches) setup, 32-fastener (fastened length was 31 inches) setup, 24fastener (fastened length was 23 inches) setup, 16-fastener (fastened length was 15 inches) setup
and, 8-fastener (fastened length was 7 inches) setup. The 66-fastener setup was tested first and
fasteners were taken out evenly from the two ends to reduce the fastened length to 47 inches, 31
inches, 23 inches, 15 inches, and finally 7 inches. Each setup was tested at least twice to ensure
repeatability.
The strain gage locations along the cross-section of the aluminum/composite assembly are
shown in Figure 5.1. The temperature distributions in the steady state were recorded and are
shown in Table 5.3. The thermocouple locations were shown previously in Figures 2.3 to 2.8.
With an inside chamber temperature of -65°F, the temperature difference between the bottom
surface of the composite panel and the top surface of the aluminum bottom flange varied from
34ºF to 50ºF.
60
Figure 5.1. Thermocouple Locations of Chamber Tests without Insulation.
Table 5.3. Temperature Distributions Recorded from Group 1 Tests.
No. of Fasteners
TC1
TC2
TC3
TC4
TC5
TC7
66F
48F
32F
24F
16F
8F
16
19
22
25
26
30
15
18
21
24
24
28
14
18
20
22
23
27
11
12
15
19
18
20
9
11
12
18
17
18
-25
-28
-31
-32
-27
-27
For each setup, the mechanical strains between fasteners at the top surface of the aluminum
bottom flange are shown in Figures 5.2 to 5.7, respectively. The strain gage locations are shown
previously in Section 2.2. For all cases with different fastened lengths, the peak strain of
aluminum always occurred around the center of the fastened assembly because the strain
accumulated from the free ends and reached the maximum value at the center. Moreover, the
peak strain increased when the fastened length increased until the fastened length reached a
certain value, and the peak strain remained constant.
61
66 Fasteners
500
Mechanical Micro Strain
400
300
200
100
Test1
0
Test2
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.2. Mechanical Strains of 66-Fastener Setup of Thick Panel without Insulation.
48 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test1
-200
Test2
-300
0
10
20
30
40
50
60
x (in)
Figure 5.3. Mechanical Strains of 48-Fastener Setup of Thick Panel without Insulation.
62
32 Fasteners
Mechanical Micro Strain
400
300
200
Test1
Test2
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.4. Mechanical Strains of 32-Fastener Setup of Thick Panel without Insulation.
24 Fasteners
Mechanical Micro Strains
400
300
200
100
0
Test1
Test2
-100
-200
0
10
20
30
40
50
60
x (in)
Figure 5.5. Mechanical Strains of 24-Fastener Setup of Thick Panel without Insulation.
63
16 Fasteners
Mechanical Micro Strain
400
300
200
100
0
Test1
Test2
-100
-200
0
10
20
30
x (in)
40
50
60
Figure 5.6. Mechanical Strains of 16-Fastener Setup of Thick Panel without Insulation.
8 Fasteners
Mechanical Micro Strain
200
100
0
Test1
Test2
-100
-200
0
10
20
30
40
50
60
x (in)
Figure 5.7. Mechanical Strains of 8-Fastener Setup of Thick Panel without Insulation.
64
5.2.2
Group 2 Test Results
Six setups for different fastened lengths were tested with Cirlex® co-cured inside the 0.25
inch thick composite panel. These six setups were the same as those used for the assembly
without Cirlex®. At least two repeatable tests were completed for each setup. The average
temperature distribution in the steady-state condition is shown in Table 5.4, and the
thermocouple locations are shown previously in Figures 2.3 to 2.8. For the tests with Cirlex®,
under the same temperature boundary conditions as the tests without insulation, the temperature
difference between the bottom surface of the composite panel and the top surface of the
aluminum bottom flange varied from 29°F to 38°F for the tests with.
Figure 5.8. Thermocouple Locations of Chamber Tests with Insulation.
Table 5.4. Temperature Distributions Recorded from Group 2 Tests.
No. of Fasteners
TC1
(deg F)
TC2
(deg F)
TC3
(deg F)
TC4
(deg F)
TC5
(deg F)
TC7
(deg F)
66F
48F
32F
24F
16F
8F
22
27
22
25
25
25
20
26
20
23
22
22
19
24
19
21
20
21
17
22
16
17
16
16
11
15
8
11
11
10
-26
-22
-26
-18
-23
-22
65
The mechanical strains for configurations with different fastened lengths are shown in
Figures 5.9 to 5.14. Strain gage locations were the same as in the tests without insulation. The
trend of strain distributions was also similar. The peak values of mechanical strain were less
than the peak strains of tests without insulation, as expected because the aluminum beam was
relatively warmer when a Cirlex® layer was added in the composite panel.
66 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test1
-100
Test2
-200
-300
0
10
20
30
x (in)
40
50
60
Figure 5.9. Mechanical Strains of 66-Fastener Setup of Thick Panel with Cirlex®.
66
48 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test1
-200
Test2
-300
0
10
20
30
40
50
60
x (in)
Figure 5.10. Mechanical Strains of 48-Fastener Setup of Thick Panel with Cirlex®.
32 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test1
-100
Test2
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.11. Mechanical Strains of 32-Fastener Setup of Thick Panel with Cirlex®.
67
24 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test1
-200
Test2
-300
-400
0
10
20
30
40
50
60
x (in)
Figure 5.12. Mechanical Strains of 24-Fastener Setup of Thick Panel with Cirlex®.
16 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test1
-100
Test2
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.13. Mechanical Strains of 16-Fastener Setup of Thick Panel with Cirlex®.
68
8 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test1
-100
Test2
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.14. Mechanical Strains of 8-Fastener Setup of Thick Panel with Cirlex®.
5.2.3 Group 3 Test Results
Again, six setups for different fastened lengths were tested without insulation material.
These six setups were the same as the previous two groups. At least two repeatable tests were
completed for each setup. The average temperature distribution in the steady-state condition is
shown in Table 5.5, and the thermocouple locations are shown previously in Figures 2.3 to 2.8.
The temperature difference between the bottom surface of composite panel and the top surface of
aluminum bottom flange varied from 26°F to 38°F.
Table 5.5. Temperature Distributions Recorded from Group 3 Tests.
No. of Fasteners
TC1
TC2
TC3
TC4
TC5
TC7
66F
48F
32F
24F
16F
8F
14
19
24
25
29
30
13
16
21
22
25
27
9
14
19
19
23
24
6
10
15
15
19
21
1
6
8
8
10
15
-25
-24
-25
-24
-22
-23
69
The mechanical strains for configurations with different fastened lengths are shown in
Figures 5.15 to 5.20. Strain gage locations were the same as in the tests without insulation. The
trend of strain distributions was also similar. The peak values of mechanical strain were less than
the peak strains of tests without insulation, as expected because the aluminum beam was
relatively warmer when a Cirlex® layer was added in the composite panel.
66 Fasteners
500
Mechanical Micro Strain
400
300
200
100
Test1
0
Test2
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.15. Mechanical Strains of 66-Fastener Setup of Thin Panel without Insulation.
70
48 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test1
-200
Test2
-300
0
10
20
30
40
50
60
x (in)
Figure 5.16. Mechanical Strains of 48-Fastener Setup of Thin Panel without Insulation.
32 Fasteners
Mechanical Micro Strain
400
300
200
100
0
Test1
Test2
-100
-200
0
10
20
30
40
50
60
x (in)
Figure 5.17. Mechanical Strains of 32-Fastener Setup of Thin Panel without Insulation.
71
24 Fasteners
Mechanical Micro Strains
400
300
200
100
0
Test1
Test2
-100
-200
0
10
20
30
40
50
60
x (in)
Figure 5.18. Mechanical Strains of 24-Fastener Setup of Thin Panel without Insulation.
16 Fasteners
Mechanical Micro Strain
400
300
200
100
0
Test1
Test2
-100
-200
0
10
20
30
x (in)
40
50
60
Figure 5.19. Mechanical Strains of 16-Fastener Setup of Thin Panel without Insulation.
72
8 Fasteners
Mechanical Micro Strain
200
100
0
Test1
Test2
-100
-200
0
10
20
30
40
50
60
x (in)
Figure 5.20. Mechanical Strains of 8-Fastener Setup of Thin Panel without Insulation.
5.2.4
Group 4 Test Results
Similar to group 3, six setups for different fastened lengths-66F, 48F, 32F, 24F, 16F, 8F
were tested with Cirlex® co-cured inside the 0.13 inch thick composite panel. Two tests were
done for each case. The average temperature distribution in the steady-state condition is shown
in Table 5.6, and the thermocouple locations are shown previously in Figures 2.3 to 2.8. Under
the same temperature boundary conditions as the tests without insulation, the temperature
difference between the bottom surface of the composite panel and the top surface of the
aluminum bottom flange varied from 32°F to 41°F for the tests with Cirlex®.
73
Table 5.6. Temperature Distributions Recorded from Group 4 Tests.
No. of Fasteners
TC1
TC2
TC3
TC4
TC5
TC7
66F
48F
32F
24F
16F
8F
23
27
25
27
29
30
22
25
23
25
25
28
20
24
22
24
23
27
17
20
18
20
19
23
13
16
14
16
10
19
-23
-22
-23
-21
-22
-22
The mechanical micro strains for both tests for different fastened lengths are shown in
Figures 5.21 to 5.26. Strain gage locations were the same as in the tests without insulation. The
trend of strain distributions was also similar. The peak values of mechanical strain were less
than the peak strains of tests without insulation, as expected because the aluminum beam was
relatively warmer when a Cirlex® layer was added in the composite panel.
66 Fasteners
Mechanical Micro Strain
400
300
200
100
Test1
0
Test2
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.21. Mechanical Strains of 66-Fastener Setup of Thin Panel with Cirlex®.
74
48 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test1
-200
Test2
-300
0
10
20
30
40
50
60
x (in)
Figure 5.22. Mechanical Strains of 48-Fastener Setup of Thin Panel with Cirlex®.
32 Fasteners
M echanical M icro Strain
400
300
200
Test1
Test2
100
0
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.23. Mechanical Strains of 32-Fastener Setup of Thin Panel with Cirlex®.
75
24 Fasteners
Mechanical Micro Strain
400
300
200
Test1
Test2
100
0
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.24. Mechanical Strains of 24-Fastener Setup of Thin Panel with Cirlex®.
16 Fasteners
Mechanical Micro Strain
300
200
Test1
100
Test2
0
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.25. Mechanical Strains of 16-Fastener Setup of Thin Panel with Cirlex®.
76
8 Fasteners
Mechanical Micro Strain
200
100
Test1
0
Test2
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.26. Mechanical Strains of 8-Fastener Setup of Thin Panel with Cirlex®.
5.3 Analytical Model Validation
An analytical model was developed using six parameters, as described in Chapter 3. These
six parameters required for the analytical model were obtained from finite element analysis, as
described in Chapter 4. The mechanical micro strains obtained from the tests are compared with
those obtained from the analytical model. Once the mechanical strains were validated the load
transferred by each fastener could be obtained from the analytical model.
5.3.1
Comparison of Tests of Thick Panel without Insulation
As described in Chapter 4, six parameters, (equivalent area for aluminum beam and
composite panel, equivalent temperature for aluminum beam and composite panel, equivalent
fastener stiffness, and bending factor) were determined by finite element analyses. These results
are shown in Tables 5.7 to 5.11. The x-direction is consistent with the x-direction shown in
Figure 5.27. The x-coordinate used in Tables 5.8 and Table 5.10 is the distance between the
77
point where the equivalent area is calculated and where the load is applied. However, the xcoordinate used in Table 5.11 is the global x-coordinate, as specified in Figure 5.27. It can be
seen that the equivalent area Ac increases when x increases until Ac reaches the geometric limit
(12 in. × 0.25 in. = 3 in.2).
The calculated bending factor β is 0.9, which means that the strain on the top surface of the
aluminum bottom flange is very close to the strain at the mid-plane of the aluminum bottom
flange. In another words, the bending effects in the assembly are very small.
Table 5.7. Parameters for Fastened Aluminum/Composites Assembly with Thick
Composite Panel without Insulation.
Equivalent
Equivalent Temperature
Material
Equivalent Area A (in.2)
Stiffness K
T (°F)
(lb/in)
Varied
Varied
Aluminum Beam
(shown in Table 5.8)
(shown in Table 5.9)
Varied
Varied
Composite Panel
(shown in Table 5.10)
(shown in Table 5.11)
Titanium Fastener
4.5×105
Table 5.8. Equivalent Area for Aluminum Beam for Aluminum/Composites
Assembly with Thick Composite Panel without Insulation.
Distance x
0
1
2
3
4
5
6
(in)
Aa (in2)
0.146
0.214
0.246
0.261
0.267
0.271
0.272
7
0.272
Table 5.9. Equivalent Temperature for Aluminum with Thick Composite Panel without
Insulation.
Fasteners Ta (°F)
66
10
48
12
32
13
24
19
16
18
8
19
78
Table 5.10. Equivalent Area for Thick Composite Panel without Insulation.
Distance x
(in)
0.375
1
2
3
4
5
6
7
8
9
10
Ac (in2)
Distance x
(in)
0.35
0.47
0.80
1.11
1.40
1.67
1.92
2.14
2.33
2.50
2.64
11
12
13
14
15
16
17
18
19
20
Ac (in2)
2.74
2.83
2.89
2.93
2.95
2.96
2.97
2.98
2.99
3
c
2
Note: The equivalent area of composite panel A remains 3 in. when distance x equals or exceeds
20 inches.
Table 5.11. Equivalent Temperature for Thick Composite Panel without Insulation.
X
T (°F) 66F and 48F
Tc (°F) 32F and 24F
Tc (°F) 16F and 8F
c
Note:
0
-21.73
-19.49
-16.03
1
-33.49
-25.15
-22.18
2
-41.82
-35.78
-33.66
3
-45.84
-41.87
-40.20
4
-48.18
-45.44
-44.03
5
-49.63
-47.63
-46.38
6
-50.49
-48.92
-47.77
7
-50.89
-49.53
-48.42
The equivalent temperature of the composite panel Tc remains constant when x equals or exceeds
8 inches.
Figure 5.27. Configuration of Chamber Tests without Insulation.
79
The temperature distributions from finite element model of each case with different
fastened length closed matched the test results.
After the six parameters for the fastened aluminum/composite assembly without insulation
were substituted into the governing equations, the mechanical strains were solved and compared
with the test results as for the six test configurations of thick composite panel without Cirlex®,
shown in Figures 5.28 to 5.33.
66 Fasteners
Mechanical Micro Strain
500
400
300
200
100
Test
0
Analytical Model
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.28. Mechanical Strains Comparison between Tests and Analytical Model for
66-Fastener Setup without Insulation (Thick Panel).
80
48 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test
-200
Analytical Model
-300
0
10
20
30
40
50
60
x (in)
Figure 5.29. Mechanical Strains Comparison between Tests and Analytical Model for
48-Fastener Setup without Insulation (Thick Panel).
32 Fasteners
Mechanical Micro Strain
400
300
Test
200
Analytical Model
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.30. Mechanical Strains Comparison between Tests and Analytical Model for
32-Fastener Setup without Insulation (Thick Panel).
81
24 Fasteners
Mechanical Micro Strains
400
300
200
100
0
Test
Analytical Model
-100
-200
0
10
20
30
40
50
60
x (in)
Figure 5.31. Mechanical Strains Comparison between Tests and Analytical Model for
24-Fastener Setup without Insulation (Thick Panel).
16 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test
Analytical Model
-100
-200
0
10
20
30
x (in)
40
50
60
Figure 5.32. Mechanical Strains Comparison between Tests and Analytical Model for
16-Fastener Setup without Insulation (Thick Panel).
82
8 Fasteners
Mechanical Micro Strain
200
100
0
Test
Analytical Model
-100
-200
0
10
20
30
40
50
60
x (in)
Figure 5.33. Mechanical Strains Comparison between Tests and Analytical Model for
8-Fastener Setup without Insulation (Thick Panel).
5.3.2
Comparison of Tests of Thick Panel with Insulation (Cirlex®)
Similar to the analytical model for assemblies without insulation, six parameters for the
assembly with the thick composite panel and co-cured Cirlex® were obtained from finite element
analyses. The results are shown in Tables 5.12 to 5.16. Again, the x-coordinate used in Tables
5.13 and 5.15 is the distance between the point where the equivalent area is calculated and where
the load is applied, while the x-coordinate used in Table 5.16 is the global x-coordinate as
specified in Figure 5.27. The bending factor β for the assembly with co-cured Cirlex® is also
0.9.
83
Table 5.12. Parameters for Fastened Aluminum/Composites Assembly with
Thick Composite Panel and Co-Cured Cirlex®.
Equivalent
Equivalent Temperature
Material
Equivalent Area A (in.2)
Stiffness K
T (°F)
(lb/in)
Varied
Varied
Aluminum Beam
(shown in Table 5.13)
(shown in Table 5.14)
Varied
Varied
Composite Panel
(shown in Table 5.15)
(shown in Table 5.16)
Titanium Fastener
4.8×105
Table 5.13. Equivalent Area for Aluminum Beam for Aluminum/Composites Assembly with
Thick Composite Panel and Co-Cured Cirlex®.
Distance x
(in)
0
Aa (in2)
0.146
1
0.214
2
3
4
5
6
7
0.246
0.261
0.267
0.271
0.272
0.272
Table 5.14. Equivalent Temperature for Aluminum with Thick Composite Panel and Co-Cured
Cirlex®.
Fasteners Ta (°F)
66
18
48
21
32
10
24
16
16
12
8
12
Table 5.15. Equivalent Area for Thick Composite Panel with Co-Cured Cirlex®.
Distance x
(in)
0.375
1
2
3
4
5
6
7
8
9
10
Ac (in2)
Distance x
(in)
0.35
0.79
1.11
1.38
1.61
1.82
2.01
2.18
2.32
2.44
2.53
11
12
13
14
15
16
17
18
19
20
Ac (in2)
2.59
2.64
2.67
2.69
2.70
2.71
2.71
2.71
2.71
2.71
c
2
Note: The equivalent area of composite panel A remains 2.71 in. when distance x equals or exceeds 20
inches.
84
Table 5.16. Equivalent Temperature for Thick Composite Panel with Co-Cured Cirlex®.
x (in)
Tc (°F) 66F & 32F
Tc (°F) 48F
Tc (°F) 24F,16F & 8F
Note:
0
-37.81
-36.72
-37.28
1
-50.26
-49.81
-50.11
2
-56.64
-55.74
-56.53
3
-59.54
-58.13
-59.31
4
-60.63
-59.32
-60.42
5
-61.23
-60.04
-61.11
6
-61.74
-60.51
-61.62
7
-61.83
-60.74
-61.71
The equivalent temperature of composite panel Tc remains constant when x equals or exceeds 8
inches.
Substituting the six parameters into the governing equations, the mechanical strains were
solved and compared with the test results for the assembly with thick composite panel and cocured Cirlex®, as shown in Figures 5.34 to 5.39. Again, the analytical results correlate fairly well
with the test results.
66 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test
-100
Analytical Model
-200
-300
0
10
20
30
x (in)
40
50
60
Figure 5.34. Mechanical Strains Comparison between Tests and Analytical Model for
66-Fastener Setup with Co-Cured Cirlex® (Thick Panel).
85
48 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test
-200
Analytical Model
-300
0
10
20
30
40
50
60
x (in)
Figure 5.35. Mechanical Strains Comparison between Tests and Analytical Model for
48-Fastener Setup with Co-Cured Cirlex® (Thick Panel).
32 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test
-100
Analytical Model
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.36. Mechanical Strains Comparison between Tests and Analytical Model for
32-Fastener Setup with Co-Cured Cirlex® (Thick Panel).
86
24 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test
-200
Analytical Model
-300
-400
0
10
20
30
40
50
60
x (in)
Figure 5.37. Mechanical Strains Comparison between Tests and Analytical Model for
24-Fastener Setup with Co-Cured Cirlex® (Thick Panel).
16 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test
-100
Analytical Model
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.38. Mechanical Strains Comparison between Tests and Analytical Model for
16-Fastener Setup with Co-Cured Cirlex® (Thick Panel).
87
8 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test
-100
Analytical Model
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.39. Mechanical Strains Comparison between Tests and Analytical Model for
8-Fastener Setup with Co-Cured Cirlex® (Thick Panel).
5.3.3
Comparison of Tests of Thin Panel without Insulation
As described in the two previous sections, six parameters, (equivalent area for aluminum
beam and composite panel, equivalent temperature for aluminum beam and composite panel,
equivalent fastener stiffness, and bending factor) were determined by finite element analyses.
The results are shown in Tables 5.17 to 5.21. The x-coordinate used in Tables 5.18 and 5.20 is
the distance between the point where the equivalent area is calculated and where the load is
applied. The x-coordinate used in Table 5.21 is the global x-coordinate, as specified in Figure
5.27.
The bending factor β calculated is still 0.9, which means the strain on the top surface of
aluminum bottom flange is very close to the strain at the mid-plane of the aluminum bottom
flange. In another words, the bending effects in the assembly are very small.
88
Table 5.17. Parameters for Fastened Aluminum/Composites Assembly with Thin Composite
Panel without Insulation.
Material
Aluminum Beam
Composite Panel
Titanium Fastener
Equivalent Area A (in.2)
Equivalent Temperature
T (°F)
Varied
(shown in Table 5.18)
Varied
(shown in Table 5.20)
-
Varied
(shown in Table 5.19)
Varied
(shown in Table 5.21)
-
Equivalent
Stiffness K
(lb/in)
4.0 ×105
Table 5.18. Equivalent Area for Aluminum Beam for Aluminum/Composites Assembly with
Thin Composite Panel without Insulation.
Distance x
(in)
0
Aa (in2)
0.165
1
0.217
2
3
4
5
6
7
0.268
0.289
0.302
0.311
0.316
0.318
Table 5.19. Equivalent Temperature for Aluminum without Insulation for
Thin Composite Panel
Fasteners Ta (°F)
66
9
48
10
32
11
24
10
16
13
8
16
Table 5.20. Equivalent Area for Thin Composite Panel without Insulation.
Distance
x (in)
0.375
1
2
3
4
5
6
7
8
9
10
Ac (in2)
Distance
x (in)
0.27
0.46
0.633
0.784
0.92
1.04
1.15
1.253
1.336
1.4
1.45
11
12
13
14
15
16
17
18
19
20
Ac (in2)
1.49
1.51
1.53
1.538
1.54
1.542
1.545
1.55
1.56
1.56
Note:
The equivalent area of composite panel Ac remains 1.56 in.2 when distance x equals or exceeds 20
inches.
89
Table 5.21. Equivalent Temperature for Thin Composite Panel without Insulation.
0
-20.49
-19.64
-17.74
X
Tc (°F) 66F & 48F
Tc (°F) 32F & 24F
Tc (°F) 16F & 8F
1
-25.92
-25.23
-23.55
2
-37.16
-36.71
-35.51
3
-43.24
-42.88
-41.94
4
-46.70
-46.40
-45.61
5
-48.78
-48.52
-47.82
6
-50.01
-49.77
-49.12
7
-50.59
-50.36
-49.73
After the six parameters for the fastened aluminum/composite assembly with the thin panel
without insulation were substituted into the governing equations, the mechanical strains were
solved and compared with the test results, as shown in Figures 5.40 to 5.45 for the six test
configurations without Cirlex®. As can be seen from the figures, the analytical results correlate
reasonably well with the test results.
66 Fasteners
Mechanical Micro Strain
500
400
300
200
100
Test
0
Analytical Model
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.40. Mechanical Strains Comparison between Tests and Analytical Model for
66-Fastener Setup without Insulation (Thin Panel).
90
48 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test
-200
Analytical Model
-300
0
10
20
30
40
50
60
x (in)
Figure 5.41. Mechanical Strains Comparison between Tests and Analytical Model for
48-Fastener Setup without Insulation (Thin Panel).
32 Fasteners
Mechanical Micro Strain
400
300
200
100
0
Test
-100
Analytical Model
-200
0
10
20
30
40
50
60
x (in)
Figure 5.42. Mechanical Strains Comparison between Tests and Analytical Model for
32-Fastener Setup without Insulation (Thin Panel).
91
24 Fasteners
Mechanical Micro Strains
400
300
200
100
0
Test
-100
Analytical Model
-200
0
10
20
30
40
50
60
x (in)
Figure 5.43. Mechanical Strains Comparison between Tests and Analytical Model for
24-Fastener Setup without Insulation (Thin Panel).
16 Fasteners
400
Mechanical Micro Strain
300
200
100
0
Test
-100
Analytical Model
-200
0
10
20
30
x (in)
40
50
60
Figure 5.44. Mechanical Strains Comparison between Tests and Analytical Model for
16-Fastener Setup without Insulation (Thin Panel).
92
8 Fasteners
Mechanical Micro Strain
200
100
0
Test
Analytical Model
-100
-200
0
10
20
30
40
50
60
x (in)
Figure 5.45. Mechanical Strains Comparison between Tests and Analytical Model for
8-Fastener Setup without Insulation (Thin Panel).
5.3.4
Comparison of Tests of Thin Panel with Insulation (Cirlex®)
Similar to the analytical model for assemblies without insulation, six parameters for the
assembly with thin composite panel and co-cured Cirlex® were obtained from finite element
analyses. The results are shown in Tables 5.22 to 5.26. Again, the x-coordinate used in Tables
5.23 and 5.25 is the distance between the point where the equivalent area is calculated and where
the load is applied, while the x-coordinate used in Table 5.26 is the global x-coordinate, as
specified in Figure 5.27. The bending factor β for the assembly with co-cured Cirlex® is also
0.9.
93
Table 5.22. Parameters for Fastened Aluminum/Composites Assembly with Thin Composite
Panel and Co-Cured Cirlex®.
Material
Aluminum Beam
Composite panel
Titanium Fastener
Equivalent Area A (in.2)
Equivalent Temperature
T (°F)
Varied
(shown in Table 5.23)
Varied
(shown in Table 5.25)
-
Varied
(shown in Table 5.24)
Varied
(shown in Table 5.26)
-
Equivalent
Stiffness K
(lb/in)
4.4×105
Table 5.23. Equivalent Area for Aluminum Beam for Aluminum/Composites Assembly with
Thin Composite Panel and Co-Cured Cirlex®.
Distance x
0
1
2
3
4
5
6
7
(in)
Aa (in2)
0.165
0.217
0.268
0.289
0.302
0.311
0.316
0.318
Table 5.24. Equivalent Temperature for Aluminum with Thin Composite Panel and Co-Cured C
Cirlex®.
Fasteners Ta (°F)
66
13
48
15
32
14
24
15
16
19
8
19
Table 5.25. Equivalent Area for Thin Composite Panel with Co-Cured Cirlex®.
Distance x
(in)
0.375
1
2
3
4
5
6
7
8
9
10
Ac (in2)
Distance x
(in)
0.32
0.47
0.61
0.72
0.8
0.86
0.92
0.96
0.99
1.02
1.04
11
12
13
14
15
16
17
18
19
20
Ac (in2)
1.05
1.06
1.065
1.07
1.07
1.08
1.08
1.09
1.1
1.1
Note:
The equivalent area of composite panel Ac remains 1.1 in.2 when distance x equals or exceeds 20
inches.
94
Table 5.26. Equivalent Temperature for Thick Composite Panel with Co-Cured Cirlex®.
X (in)
Tc (°F) 66F & 32F
Tc (°F) 48F
Tc (°F) 24F,16F & 8F
Note:
0
-27.61
-26.72
-27.24
1
-34.54
-34.81
-34.16
2
-44.22
-43.74
-43.58
3
-51.46
-50.13
-51.54
4
-56.66
-55.32
-56.27
5
-58.42
-58.04
-58.15
6
-58.79
-58.51
-57.62
7
-60.83
-60.74
-59.65
The equivalent temperature of composite panel Tc remains constant when x equals or exceeds 8
inches.
Substituting the six parameters into the governing equations, the mechanical strains were
solved and compared with the test results for the assembly with the thick composite panel and
co-cured Cirlex®, as shown in Figures 5.46 to 5.51. Again, the analytical results correlate fairly
well with the test results.
66 Fasteners
Mechanical Micro Strain
400
300
200
100
Test
0
Analytical Model
-100
-200
-300
0
10
20
30
40
50
60
x (in)
Figure 5.46. Mechanical Strains Comparison between Tests and Analytical Model for
66-Fastener Setup with Co-Cured Cirlex® (Thin Panel).
95
48 Fasteners
400
Mechanical Micro Strain
300
200
100
0
-100
Test
-200
Analytical Model
-300
0
10
20
30
40
50
60
x (in)
Figure 5.47. Mechanical Strains Comparison between Tests and Analytical Model for
48-Fastener Setup with Co-Cured Cirlex® (Thin Panel).
32 Fasteners
Mechanical Micro Strain
400
300
Test
Analytical Model
200
100
0
-100
0
10
20
30
40
50
60
x (in)
Figure 5.48. Mechanical Strains Comparison between Tests and Analytical Model for
32-Fastener Setup with Co-Cured Cirlex® (Thin Panel).
96
24 Fasteners
Mechanical Micro Strain
400
300
Test
Analytical Model
200
100
0
-100
0
10
20
30
40
50
60
x (in)
Figure 5.49. Mechanical Strains Comparison between Tests and Analytical Model for
24-Fastener Setup with Co-Cured Cirlex® (Thin Panel).
16 Fasteners
Mechanical Micro Strain
300
Test
200
Analytical Model
100
0
-100
0
10
20
30
40
50
60
x (in)
Figure 5.50. Mechanical Strains Comparison between Tests and Analytical Model for
16-Fastener Setup with Co-Cured Cirlex® (Thin Panel).
97
8 Fasteners
Mechanical Micro Strain
200
Test
Analytical Model
100
0
-100
0
10
20
30
40
50
60
x (in)
Figure 5.51. Mechanical Strains Comparison between Tests and Analytical Model for
8-Fastener Setup with Co-Cured Cirlex® (Thin Panel).
5.4 Analytical Model Validation with Finite Element Model
The analytical model developed using the equations in Chapter 3 were validated using a
finite element model. For this purpose, 24 units (48F case) of Z-shape aluminum fastened to a
thin composite panel without insulation was built and was subjected to temperature boundary
conditions obtained from the 48 fastener case in the group 3 tests. The load transfer obtained
from the analytical model was compared with the load transfer obtained from the finite element
sequentially coupled analysis.
The temperature distribution under the steady-state condition at the end of thermal analysis
is shown in Figure 5.52. The temperature distribution obtained at the end of thermal analysis was
fed as input to the sequentially coupled analysis of 24 units of the aluminum/composite assembly
with the right end fixed. The same procedure as described in Section 4.2.6 was followed to find
the load transfer. The final deformation of the entire assembly is bent up, as shown in Figure
5.53.
98
Figure 5.52. Temperature Distribution in Steady State for Aluminum/Composite Assembly
Without Insulation (Thin panel).
Figure 5.53. Deformation of Fastened Aluminum/Composites Assembly due to Thermal Load.
99
In each unit, the load transferred by the ith fastener Fi is defined as the summation of
contact force between the fastener and hole, and the friction between the aluminum and
composite surfaces. Both the contact force and friction were obtained from finite element
analysis. Figure 5.54 shows the comparison of load transfers obtained from finite element
analysis using ABAQUS and the analytical model. It can be seen from this graph that load
transfers for each unit obtained from the analytical model are close to those obtained from finite
element analysis. Hence, the analytical model derived is a valid one and can be used to obtain the
load transferred by each fastener for different lengths of the aluminum/composite assembly, once
the strains from the test data are validated.
450
400
load transfer
350
300
250
200
150
abaqus
100
maple program
50
0
0
4
8
12
16
20
24
28
unit #
Figure 5.54. Load Transfer Comparison for 24 Units (half of 48F case) for Aluminum/Composite
Assembly (Thin Panel).
5.5 Load Transfer Prediction
The load transfer of each fastener in the aluminum beam/composite panel assemblies can
be predicted after the validation of the analytical model. Figures 5.55 to 5.66 show the load
transfer prediction for different fastened lengths. Load transfers are compared for assemblies
100
without insulation and for assemblies with the co-cured Cirlex® layer for both thin and thick
panel.
It is noted that the peak stress of the aluminum beam occurs at the center of the assembly.
This is because stress “accumulates” from the free ends and finally peaks at the center.
However, the load transfer through the fasteners has the opposite trend. The end fasteners take
the highest load while the fasteners at the center carry very little load. It is also noted that the
peak load transfer is approximately reduced by 21 percent when the 0.06-inch Cirlex® layer is
added in the thick composite panel and by 25 to 32 percent when the 0.12-inch Cirlex® layer is
added in the thin composite panel.
66-Fastener
400
Load Transfer (lb)
Without Insulation
300
With Cirlex
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.55. Load Transfer Comparison with and without Insulation Material for 66-Fastener
Setup (Thick Panel).
101
48-Fastener
400
Load Transfer (lb)
Without Insulation
With Cirlex
300
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.56. Load Transfer Comparison with and without Insulation Material for
48-Fastener Setup (Thick Panel).
32-Fastener
400
Without Insulation
Load Transfer (lb)
With Cirlex
300
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.57. Load Transfer Comparison with and without Insulation Material for
32-Fastener Setup (Thick Panel).
102
24-Fastener
400
Without Insulation
Load Transfer (lb)
With Cirlex
300
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.58. Load Transfer Comparison with and without Insulation Material for
24-Fastener Setup (Thick Panel).
16-Fastener
400
Without Insulation
With Cirlex
Load Transfer (lb)
300
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.59. Load Transfer Comparison with and without Insulation Material for
16-Fastener Setup (Thick Panel).
103
8-Fastener
Load Transfer (lb)
300
Without Insulation
200
With Cirlex
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.60. Load Transfer Comparison with and without Insulation Material for
8-Fastener Setup (Thick Panel).
66-Fastener
400
Load Transfer (lb)
Without Insulation
300
With Cirlex
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.61. Load Transfer Comparison with and without Insulation Material for
66-Fastener Setup (Thin Panel).
104
48-Fastener
400
Load Transfer (lb)
Without Insulation
With Cirlex
300
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.62. Load Transfer Comparison with and without Insulation Material for
48-Fastener Setup (Thin Panel).
32-Fastener
400
Load Transfer (lb)
Without Insulation
With Cirlex
300
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.63. Load Transfer Comparison with and without Insulation Material for
32-Fastener Setup (Thin Panel).
105
24-Fastener
400
Load Transfer (lb)
300
200
100
Without Insulation
With Cirlex
0
0
10
20
30
40
50
60
x (in)
Figure 5.64. Load Transfer Comparison with and without Insulation Material for
24-Fastener Setup (Thin Panel).
16-Fastener
Load Transfer (lb)
400
300
200
100
Without Insulation
With Cirlex
0
0
10
20
30
40
50
60
x (in)
Figure 5.65. Load Transfer Comparison with and without Insulation Material for
16-Fastener Setup (Thin Panel).
106
8-Fastener
400
Without Insulation
Load Transfer (lb)
With Cirlex
300
200
100
0
0
10
20
30
40
50
60
x (in)
Figure 5.66. Load Transfer Comparison with and without Insulation Material for
8-Fastener Setup (Thin Panel).
107
CHAPTER 6
RESULTS AND DISCUSSION FOR PARAMETRIC STUDY
A method was developed for the working engineer to predict the load transfer due to CTE
mismatch without doing any finite element modeling for different geometries and materials of
the assembly. The five parameters required to determine the load transfer, as discussed in
Chapter 3, are the equivalent areas of metallic beam and composite panel Aa and Ac, equivalent
temperatures of metallic beam and composite panel Ta and Tc, and equivalent stiffness of the
fastener Kf.
The variables included in this study are the metal beam cross-section size and length,
composite panel thickness, fastener diameter and spacing, and material of metal beam, as listed
in Table 6.1. It should be noted that as the thickness of the beam changes the cross-section
geometry also changes, i.e., the height of the metallic beam and lengths of the top and bottom
flanges change with thickness of the beam.
Therefore, the Z-beam geometries used for
calculation are defined by Figure 6.1, with the values listed in Table 6.2.
Table 6.1. Variables used to Calculate Aa, Ac, Ta, Tc, and Kf.
Fastener Diameter (D)
0.1875"
0.25"
0.375"
4D
5D
6D
Composite Panel Thickness (tp)
0.13"
0.25"
0.5"
Metal Beam Thickness (tb)
0.04"
0.08"
0.125"
Aluminum
Titanium
Steel
Fastener Spacing (p)
Metal Material
108
Figure 6.1. Dimensions of Metallic Z-Beams.
Table 6.2. Z-Beam Dimensions According to Beam Thickness tb.
Beam Thickness
(tb) (inch)
0.125
α (inch)
1
β
(inch)
1.75
γ
(inch)
4
0.08
0.64
1.13
2.58
0.04
0.32
0.56
1.29
Using a design of experiments approach, the final matrix of finite element models to be
conducted for five parameters (Aa, Ac, Ta, Tc and Kf) was developed. Fractional factorial designs
were set up for all of them. The idea behind fractional factorial designs is to deliberately
introduce aliasing in a controlled way. One-quarter fractional factorial design consisting of 27
models for each metal material was created for Aa, Ta, and Kf. For Tc, three models for different
thicknesses of 0.13-inch, 0.25-inch, and 0.5-inch were developed for Ac . The 27 models in Table
6.3 were modeled and analyzed as described in Chapter 5.
109
Table 6.3. One-Quarter Fractional Factorial Design.
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Fastener
Diameter
(D) (inch)
0.1875
0.1875
0.1875
0.1875
0.1875
0.1875
0.1875
0.1875
0.1875
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.375
0.375
0.375
0.375
0.375
0.375
0.375
0.375
0.375
Fastener
Spacing
4D
4D
4D
5D
5D
5D
6D
6D
6D
4D
4D
4D
5D
5D
5D
6D
6D
6D
4D
4D
4D
5D
5D
5D
6D
6D
6D
Thickness of
Composite Panel
(inch)
0.13
0.25
0.5
0.13
0.25
0.5
0.13
0.25
0.5
0.13
0.25
0.5
0.13
0.25
0.5
0.13
0.25
0.5
0.13
0.25
0.5
0.13
0.25
0.5
0.13
0.25
0.5
Thickness of
Metal Beam
(inch)
0.125
0.08
0.04
0.08
0.04
0.125
0.04
0.125
0.08
0.08
0.04
0.125
0.04
0.125
0.08
0.125
0.08
0.04
0.04
0.125
0.08
0.125
0.08
0.04
0.08
0.04
0.125
6.1 Equivalent Area of Metallic Beam (Aa)
The equivalent area of the metallic beam is influenced by its thickness (tb), thickness of the
composite panel (tp), fastener diameter (D), fastener spacing (p) and material of the beam. Three
different metallic beam materials (aluminum, titanium and steel) were used, as shown in Table
6.1.
110
As discussed in Chapter 3, most of the load in the Z-beam was taken by the bottom flange
of the beam. Therefore, more displacement was observed here. It can be noted from Section 3.1,
that the equivalent area of the Z-beam is a function of distance. It can also be seen that the Zbeam equivalent area Aa, increased up to certain distance and then remained constant. After
obtaining the equivalent area as a function of distance for the above 27 cases, the data was
analyzed by a statistician, and equations in terms of thickness of the metallic beam tb, thickness
of the composite panel tp, fastener diameter D, and fastener spacing were developed for the three
metallic beam materials.
A a (aluminum) = −3.325 − 9.718(t b t p ) 2 +
0.78t b
0.02401D 0.05306t b
−
+ 3.1715e tb −
tp
tp
De x p
A a ( titanium) = −4.418 + 2.0808( Dt p ) 2 − 7.057t b t p D + 4.107e tb −
A a ( steel ) = −4.272 − 0.02857 pD − 1.593t b t p D + 4.107e tb −
1.34t b
De x p
1.34012t b
De x p
p: Spacing Parameter=4 for Fastener Spacing 4D, Spacing Parameter=5 for Fastener Spacing
5D, Spacing Parameter=6 for Fastener Spacing 6D
6.1.1 Comparison of Aa Obtained from FEM and Equation
In this section, the equivalent area of the metallic beam obtained from finite element
analysis is compared with that obtained from the developed equation for various cases mentioned
in Table 6.3. From the comparisons shown in Figures 6.2 to 6.40, it can be noted that the
developed equation predicts the equivalent area of the metallic beam very closely.
111
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (AL)
EQUATION (AL)
0.05
0
0
1
2
3
4
5
6
x (in)
Figure 6.2. Equivalent Area Aa (aluminum) for D=0.1875",tb=0.13",tp=0.125".
0.4
0.35
Aa (in-in)
0.3
0.25
0.2
0.15
FEM (TI)
0.1
EQUATION (TI)
0.05
0
0
1
2
3
4
5
6
x (in)
a
Figure 6.3. Equivalent Area A (titanium) for D=0.1875",tb=0.13",tp=0.125".
0.4
0.35
Aa (in-in)
0.3
0.25
0.2
0.15
FEM (ST)
0.1
EQUATION (ST)
0.05
0
0
1
2
3
4
5
6
x (in)
a
Figure 6.4. Equivalent Area A (steel) for D=0.1875",tb=0.13",tp=0.125".
112
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (AL)
EQUATION (AL)
0.025
0
0
1
2
3
4
5
6
x (in)
Figure 6.5. Equivalent Area Aa (aluminum) for D=0.1875",tb=0.25",tp=0.08".
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (TI)
EQUATION (TI)
0.025
0
0
1
2
3
4
5
6
x (in)
a
Figure 6.6. Equivalent Area A (titanium) for D=0.1875",tb=0.25",tp=0.08".
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (ST)
EQUATION (ST)
0.025
0
0
1
2
3
4
5
6
x (in)
a
Figure 6.7. Equivalent Area A (steel) for D=0.1875",tb=0.25",tp=0.08".
113
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (AL)
EQUATION (AL)
0.025
0
0
1
2
3
4
5
6
7
x (in)
Figure 6.8. Equivalent Area Aa (aluminum) for D=0.1875",tb=0.13",tp=0.08".
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (TI)
EQUATION (TI)
0.025
0
0
1
2
3
4
5
6
7
x (in)
Figure 6.9. Equivalent Area Aa (titanium) for D=0.1875",tb=0.13",tp=0.08".
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (ST)
EQUATION (ST)
0.025
0
0
1
2
3
4
5
6
7
x (in)
Figure 6.10. Equivalent Area Aa (steel) for D=0.1875",tb=0.13",tp=0.08".
114
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (AL)
EQUATION (AL)
0.05
0
0
1
2
3
4
5
6
7
x (in)
Figure 6.11. Equivalent Area Aa (aluminum) for D=0.1875",tb=0.5",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (TI)
EQUATION (TI)
0.05
0
0
1
2
3
4
5
6
7
x (in)
Figure 6.12. Equivalent Area Aa (titanium) for D=0.1875",tb=0.5",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (ST)
EQUATION (ST)
0.05
0
0
1
2
3
4
5
6
7
x (in)
a
Figure 6.13. Equivalent Area A (steel) for D=0.1875",tb=0.5",tp=0.125".
115
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (AL)
EQUATION (AL)
0.05
0
0
1
2
3
4
5
6
7
8
9
x (in)
Figure 6.14. Equivalent Area Aa (aluminum) for D=0.1875",tb=0.25",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (TI)
EQUATION (TI)
0.05
0
0
1
2
3
4
5
6
7
8
9
x (in)
Figure 6.15. Equivalent Area Aa (titanium) for D=0.1875",tb=0.25",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (ST)
EQUATION (ST)
0.05
0
0
1
2
3
4
5
6
7
8
9
x (in)
Figure 6.16. Equivalent Area Aa (steel) for D=0.1875",tb=0.25",tp=0.125".
116
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (AL)
EQUATION (AL)
0.025
0
0
1
2
3
4
5
6
7
8
x (in)
Figure 6.17. Equivalent Area Aa (aluminum) for D=0.25",tb=0.13",tp=0.08".
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (TI)
EQUATION (TI)
0.025
0
0
1
2
3
4
5
6
7
8
x (in)
a
Figure 6.18. Equivalent Area A (titanium) for D=0.25",tb=0.13",tp=0.08".
0.15
Aa (in-in)
0.125
0.1
0.075
0.05
FEM (ST)
EQUATION (ST)
0.025
0
0
1
2
3
4
5
6
7
8
x (in)
a
Figure 6.19. Equivalent Area A (steel) for D=0.25",tb=0.13",tp=0.08".
117
0.35
FEM (AL)
0.3
EQUATION (AL)
Aa (in-in)
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
x (in)
Figure 6.20. Equivalent Area Aa (aluminum) for D=0.25",tb=0.5",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (TI)
EQUATION (TI)
0.05
0
0
1
2
3
4
5
6
7
8
x (in)
Figure 6.21. Equivalent Area Aa (titanium) for D=0.25",tb=0.5",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (ST)
EQUATION (ST)
0.05
0
0
1
2
3
4
5
6
7
8
x (in)
Figure 6.22. Equivalent Area Aa (steel) for D=0.25",tb=0.5",tp=0.125".
118
0.35
0.3
Aa (in-in)
0.25
FEM (AL)
0.2
EQUATION (AL)
0.15
0.1
0.05
0
0
1
2
3
4
5
6
x (in)
7
8
9
10
Figure 6.23. Equivalent Area Aa (aluminum) for D=0.25",tb=0.25",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (TI)
EQUATION (TI)
0.05
0
0
1
2
3
4
5
6
7
8
9
10
x (in)
a
Figure 6.24. Equivalent Area A (titanium) for D=0.25",tb=0.25",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (ST)
EQUATION (ST)
0.05
0
0
1
2
3
4
5
6
7
8
9
10
x (in)
Figure 6.25. Equivalent Area Aa (steel) for D=0.25",tb=0.25",tp=0.125".
119
0.125
Aa (in-in)
0.1
FEM (AL)
0.075
EQUATION (AL)
0.05
0.025
0
0
2
4
6
x (in)
8
10
12
Figure 6.26. Equivalent Area Aa (aluminum) for D=0.25",tb=0.25",tp=0.08".
0.125
Aa (in-in)
0.1
0.075
0.05
FEM (TI)
0.025
EQUATION (TI)
0
0
2
4
6
8
10
12
x (in)
a
Figure 6.27. Equivalent Area A (titanium) for D=0.25",tb=0.25",tp=0.08".
0.125
Aa (in-in)
0.1
0.075
0.05
FEM (ST)
0.025
EQUATION (ST)
0
0
2
4
6
8
10
12
x (in)
a
Figure 6.28. Equivalent Area A (steel) for D=0.25",tb=0.25",tp=0.08".
120
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (AL)
0.05
EQUATION (AL)
0
0
2
4
6
x (in)
8
10
12
Figure 6.29. Equivalent Area Aa (aluminum) for D=0.375",tb=0.25",tp=0.125".
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (TI)
0.05
EQUATION (TI)
0
0
2
4
6
8
10
12
x (in)
Figure 6.30. Equivalent Area Aa (titanium) for D=0.375",tb=0.25",tp=0.125".
0.35
0.3
a
A (in-in)
0.25
0.2
0.15
0.1
FEM (ST)
EQUATION (ST)
0.05
0
0
2
4
6
8
10
12
x (in)
a
Figure 6.31. Equivalent Area A (steel) for D=0.375",tb=0.25",tp=0.125".
121
Aa (in-in)
0.1
0.075
0.05
FEM (AL)
0.025
EQUATION (AL)
0
0
2
4
6
x (in)
8
10
12
Figure 6.32. Equivalent Area Aa (aluminum) for D=0.375",tb=0.5",tp=0.08".
0.125
a
A (in-in)
0.1
0.075
0.05
FEM (TI)
0.025
EQUATION (TI)
0
0
2
4
6
8
10
12
x (in)
a
Figure 6.33. Equivalent Area A (titanium) for D=0.375",tb=0.5",tp=0.08".
0.125
a
A (in-in)
0.1
0.075
0.05
FEM (ST)
0.025
EQUATION (ST)
0
0
2
4
6
8
10
12
x (in)
a
Figure 6.34. Equivalent Area A (steel) for D=0.375",tb=0.5",tp=0.08".
122
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (AL)
0.05
EQUATION (AL)
0
0
2
4
6
8
10
12
14
x (in)
a
Figure 6.35. Equivalent Area A (aluminum) for D=0.375",tb=0.13",tp=0.125".
0.35
0.3
a
A (in-in)
0.25
0.2
0.15
0.1
FEM (TI)
0.05
EQUATION (TI)
0
0
2
4
6
8
10
12
14
x (in)
a
Figure 6.36. Equivalent Area A (titanium) for D=0.375",tb=0.13",tp=0.125".
0.35
0.3
a
A (in-in)
0.25
0.2
0.15
0.1
FEM (ST)
EQUATION (ST)
0.05
0
0
2
4
6
8
10
12
14
x (in)
Figure 6.37. Equivalent Area Aa (steel) for D=0.375",tb=0.13",tp=0.125".
123
0.35
0.3
Aa (in-in)
0.25
0.2
0.15
0.1
FEM (AL)
0.05
EQUATION (AL)
0
0
2
4
6
8
10
12
14
16
18
x (in)
Figure 6.38. Equivalent Area Aa (aluminum) for D=0.375",tb=0.5",tp=0.125".
0.35
0.3
a
A (in-in)
0.25
0.2
0.15
0.1
FEM (TI)
0.05
EQUATION (TI)
0
0
2
4
6
8
10
12
14
16
18
x (in)
a
Figure 6.39. Equivalent Area A (titanium) for D=0.375",tb=0.5",tp=0.125".
0.35
0.3
a
A (in-in)
0.25
0.2
0.15
0.1
FEM (ST)
EQUATION (ST)
0.05
0
0
2
4
6
8
10
12
14
16
18
x (in)
a
Figure 6.40. Equivalent Area A (steel) for D=0.375",tb=0.5",tp=0.125".
124
6.2 Equivalent Area of Composite Panel (Ac)
The equivalent area Ac of the composite panel is a function of distance from where the load
is applied. The area of the panel increases up to a certain distance from the free end and then
remains constant. The equivalent area Ac of the panel does not depend on fastener size (D) and
fastener spacing (p), which is explained in Figures 6.41 and 6.42 for a composite panel of
thickness 0.25 inch thick and D=0.25 inch and three different fastener spacings of 4D, 5D, and
6D, respectively. However, as the thickness of the panel tp increases, the amount of area that
takes load also increases, which can be seen in Figure 6.43.
Ac for different diameters
4
3.5
Ac (in-in)
3
2.5
D=0.1875"
2
D=0.25"
D=0.375"
1.5
1
0.5
0
0
5
10
15
x (in)
20
25
30
Figure 6.41. Equivalent Area Ac of 0.25" thick Composite Panel for
Different Fastener Diameters.
125
4
4D
2
c
A (in-in)
3
3D
6D
1
0
0
5
10
15
20
25
30
x (in)
Figure 6.42. Equivalent Area Ac of Composite Panel as a Function of x
(Steel Z-Beam with for tb=0.125", D=0.25", and Different p).
7
tp=0.13"
tp=0.25"
tp=0.5"
6
c
A (in-in)
5
4
3
2
1
0
0
4
8
12
16
20
x (in)
Figure 6.43. Equivalent Area Ac as a Function of x for Different Panel Thicknesses.
From Chapter 3, it can be seen that the composite panel equivalent area of composite panel
Ac, increases up to certain distance and then remains constant. Therefore, an equation for this
equivalent area Ac as a function of distance x was developed by analyzing data obtained from
finite element analysis of three models of thicknesses 0.13 inch, 0.25 inch, and 0.5 inch.
A c = −0.257 + 12.73t p e − x − 0.01025t p x 2 + 6.07t p ln x − 3.63t p
126
2
6.2.1 Comparison of Ac Obtained from FEM and Equation
7
6
c
A (in-in)
tp=0.13" (fem)
5
tp=0.25" (fem)
4
tp=0.5" (fem)
tp=0.13" (equation)
3
tp=0.25" (equation)
tp=0.5" (equation)
2
1
0
0
4
8
12
16
20
x (in)
Pigure 6.44. Comparison of Equivalent Area Ac as a Function of x for Different panel
Thicknesses.
6.3
Equivalent Temperature of Metallic Z-beam (Ta)
The equivalent temperature of the Z-shape metallic beam Ta is a specific number, and it
depends on parameters like fastener size D and spacing, material of the metallic beam, thickness
of the metallic beam tb, and thickness of the composite panel tb. As discussed in Section 4.2.4,
equivalent temperatures of the metallic beam were found for 27 cases as listed in Table 6.3.
Using the data obtained from finite element analysis of these models, three equations for
different beam materials were developed by a statistician.
T a (aluminum) = −4.42 + 702.06t p t b − 4905.4(t p t b ) 2
T a ( titanium) = −11.05 + 784t p t b − 6212(t p t b ) 2 − 124t b D + p
T a ( steel) = −8.23 + 810t p t b − 6437(t p t b ) 2 − 118.9t b D + 1.04 p
p: Spacing Parameter=-1 for Fastener Spacing 4D, Spacing Parameter=0 for Fastener Spacing
5D, Spacing Parameter=1 for Fastener Spacing 6D
127
6.3.1 Comparison of Ta Obtained from FEM and Equation
The equivalent temperature of the aluminum beam, Ta as a function of diameter of the
fastener D was compared for various cases, keeping constant both thickness of the metallic beam
tb and thickness of the composite panel tp. It can be seen that the equivalent temperatures of the
metallic beam are lower when the titanium beam is used and higher when the aluminum beam is
used, which is evident from the fact that the CTE of aluminum is higher than steel which is
higher than that of titanium. Figures 6.45 to 6.49 show that the equation predicts the results very
closely.
t b and t p are constant
10
FEM(AL)
6
FEM(ST)
2
EQUATION(AL)
0
Ta ( F)
FEM(TI)
EQUATION(TI)
-2
EQUATION(ST)
-6
-10
0
0.1
0.2
0.3
0.4
D (in)
Figure 6.45. Comparison of Ta Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.13", tb=0.125".
128
t b and t p are constant
10
FEM(AL)
6
FEM(ST)
2
EQUATION(AL)
0
Ta ( F)
FEM(TI)
EQUATION(TI)
-2
EQUATION(ST)
-6
-10
0
0.1
0.2
0.3
0.4
D (in)
Figure 6.46. Comparison of Ta Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.25", tb=0.08".
t b and t p are constant
20
FEM(AL)
FEM(TI)
0
Ta ( F)
16
FEM(ST)
12
EQUATION(AL)
EQUATION(TI)
8
EQUATION(ST)
4
0
0
0.1
0.2
0.3
0.4
D (in)
Figure 6.47. Comparison of Ta Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.5", tb=0.125".
129
t b and t p are constant
20
FEM(AL)
FEM(TI)
0
Ta ( F)
16
FEM(ST)
12
EQUATION(AL)
EQUATION(TI)
8
EQUATION(ST)
4
0
0
0.1
0.2
0.3
0.4
D (in)
Figure 6.48. Comparison of Ta Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.25", tb=0.125".
t b and t p are constant
20
FEM(AL)
FEM(TI)
0
Ta ( F)
16
FEM(ST)
12
EQUATION(AL)
EQUATION(TI)
8
EQUATION(ST)
4
0
0
0.1
0.2
0.3
0.4
D (in)
Figure 6.49. Comparison of Ta Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.5", tb=0.08".
130
6.4
Equivalent Temperature of Composite Panel (Tc)
The equivalent temperature of the composite panel Tc is a function of distance from the free
end and depends on different factors such as fastener size and spacing, material of the metallic
beam, thickness of the metallic beam, and thickness of the composite panel. Similarly, three
equations for different materials of the metallic beam were developed using the data obtained
from the 27 finite element models as
T c (aluminum) = −18.1 + 0.1134 x 2 − 14.325 x + 45.55t b FS1 + 28.76t b FS 2 + 151.5t p t b − 18.66 D
FS1=1 for Fastener Spacing 4D otherwise 0, FS2=1 for Fastener Spacing 5D otherwise 0.
T c ( titanium ) = −12.2 + 0.894 x 2 − 11.97 x + 13.24t b FS1 − 5.4 ln D + 9.89 ln t b
FS1=1 for Fastener Spacing 4D otherwise 0
T c ( steel ) = −7.71 + 0.0948 x 2 − 12.824 x + 13.9t b FS1 − 5.71 ln D + 10.79 ln t b
FS1=1 for Fastener Spacing 4D otherwise 0
In all three equations, FS1 and FS2 are Spacing Parameters
6.4.1 Comparison of Tc Obtained from FEM and Equations
Figures 6.50 to 6.85 show the comparison of equivalent temperatures of the composite
panel as a function of distance from the free end. It can be seen that the equivalent temperatures
of the panel are lower when a titanium beam was used and higher when an aluminum beam was
used. These figures reveal a close prediction of Tc.
131
-10
FEM (AL)
-20
EQUATION
-30
c
0
T ( F)
0
-40
-50
-60
0
1
2
3
4
5
6
x (in)
c
Figure 6.50. Comparison of T (aluminum) for D=0.1875", tp=0.13", tb=0.125".
-10
FEM (TI)
-20
EQUATION
( )
-30
c
0
T ( F)
0
-40
-50
-60
0
1
2
3
4
5
6
x (in)
Figure 6.51. Comparison of Tc (titanium) for D=0.1875", tp=0.13", tb=0.125".
Tc ( 0F)
0
-10
FEM (ST)
-20
EQUATION
(S )
-30
-40
-50
0
1
2
3
x (in)
4
5
6
Figure 6.52. Comparison of Tc (steel) for D=0.1875", tp=0.13", tb=0.125".
132
0
-10
FEM (AL)
Tc ( 0F)
-20
EQUATION
-30
-40
-50
-60
0
1
2
3
4
5
6
x (in)
Figure 6.53. Comparison of Tc (aluminum) for D=0.1875", tp=0.25", tb=0.08".
Tc ( 0F)
0
-10
FEM (TI)
-20
EQUATION
( )
-30
-40
-50
-60
0
1
2
3
x (in)
4
5
6
Figure 6.54. Comparison of Tc (titanium) for D=0.1875", tp=0.25", tb=0.08".
Tc ( 0F)
0
-10
FEM (ST)
-20
EQUATION
-30
-40
-50
-60
0
1
2
3
4
5
6
x (in)
Figure 6.55. Comparison of Tc (steel) for D=0.1875", tp=0.25", tb=0.08".
133
-10
FEM (AL)
-20
EQUATION
-30
c
0
T ( F)
0
-40
-50
-60
0
2
4
6
8
x (in)
Figure 6.56. Comparison of Tc (aluminum) for D=0.1875", tp=0.13", tb=0.08".
Tc ( 0F)
0
-10
FEM (TI)
-20
EQUATION
( )
-30
-40
-50
-60
0
2
4
x (in)
6
8
Figure 6.57. Comparison of Tc (titanium) for D=0.1875", tp=0.13", tb=0.08".
Tc ( 0F)
0
-10
FEM (ST)
-20
EQUATION
-30
-40
-50
-60
0
2
4
x (in)
6
8
Figure 6.58. Comparison of Tc (steel) for D=0.1875", tp=0.13", tb=0.08".
134
0
-10
FEM (AL)
Tc ( 0 F)
-20
EQUATION
-30
-40
-50
-60
-70
0
1
2
3
4
x (in)
5
6
7
Figure 6.59. Comparison of Tc (aluminum) for D=0.1875", tp=0.25", tb=0.04"
0
-10
FEM (TI)
Tc ( 0 F)
-20
EQUATION
-30
-40
-50
-60
-70
0
1
2
3
4
x (in)
5
6
7
Figure 6.60. Comparison of Tc (titanium) for D=0.1875", tp=0.25", tb=0.04"
0
-10
FEM (ST)
Tc ( 0 F)
-20
EQUATION
(S )
-30
-40
-50
-60
-70
0
1
2
3
4
5
6
7
x (in)
Figure 6.61. Comparison of Tc (steel) for D=0.1875", tp=0.25", tb=0.04".
135
-10
FEM (AL)
-20
EQUATION
( )
-30
c
0
T ( F)
0
-40
-50
-60
0
1
2
3
4
5
x (in)
6
7
8
9
Figure 6.62. Comparison of Tc (aluminum) for D=0.1875", tp=0.25", tb=0.125".
Tc ( 0 F)
0
-10
FEM (TI)
-20
EQUATION
-30
-40
-50
-60
0
1
2
3
4
5
x (in)
6
7
8
9
Figure 6.63. Comparison of Tc (titanium) for D=0.1875", tp=0.25", tb=0.125".
-10
FEM (ST)
-20
EQUATION
-30
c
0
T ( F)
0
-40
-50
-60
0
1
2
3
4
5
x (in)
6
7
8
9
Figure 6.64. Comparison of Tc (steel) for D=0.1875", tp=0.25", tb=0.125".
136
-10
FEM (AL)
-20
EQUATION
( )
c
0
T ( F)
0
-30
-40
-50
0
1
2
3
4
5
6
7
8
x (in)
Figure 6.65. Comparison of Tc (aluminum) for D=0.25", tp=0.5", tb=0.125".
Tc ( 0 F)
0
-10
FEM (TI)
-20
EQUATION
-30
-40
-50
-60
0
1
2
3
4
x (in)
5
6
7
8
Figure 6.66. Comparison of Tc (titanium) for D=0.25", tp=0.5", tb=0.125".
Tc ( 0 F)
0
-10
FEM (ST)
-20
EQUATION
(S )
-30
-40
-50
-60
0
1
2
3
4
x (in)
5
6
7
8
Figure 6.67. Comparison of Tc (steel) for D=0.25", tp=0.5", tb=0.125".
137
Tc ( 0 F)
0
-10
FEM (AL)
-20
EQUATION
-30
-40
-50
-60
0
1
2
3
4
x (in)
5
6
7
8
Figure 6.68. Comparison of Tc (aluminum) for D=0.25", tp=0.13", tb=0.08".
Tc ( 0F)
0
-10
FEM (TI)
-20
EQUATION
( )
-30
-40
-50
-60
0
1
2
3
4
5
6
7
8
x (in)
Figure 6.69. Comparison of Tc (titanium) for D=0.25", tp=0.13", tb=0.08".
0
-10
FEM (ST)
Tc ( 0F)
-20
EQUATION
-30
-40
-50
-60
0
1
2
3
4
x (in)
5
6
7
8
Figure 6.70. Comparison of Tc (steel) for D=0.25", tp=0.13", tb=0.08".
138
Tc ( 0 F)
0
-10
FEM (AL)
-20
EQUATION
-30
-40
-50
-60
0
1
2
3
4
5
6
x (in)
7
8
9
10
Figure 6.71. Comparison of Tc (aluminum) for D=0.25", tp=0.25", tb=0.125".
Tc ( 0 F)
0
-10
FEM (TI)
-20
EQUATION
( )
-30
-40
-50
-60
0
1
2
3
4
5
6
x (in)
7
8
9
10
Figure 6.72. Comparison of Tc (titanium) for D=0.25", tp=0.25", tb=0.125".
0
-10
FEM (ST)
Tc ( 0F)
-20
EQUATION
(S )
-30
-40
-50
-60
0
1
2
3
4
5
6
x (in)
7
8
9
10
Figure 6.73. Comparison of Tc (steel) for D=0.25", tp=0.25", tb=0.125".
139
0
-10
FEM (AL)
Tc ( 0 F)
-20
EQUATION
( )
-30
-40
-50
-60
-70
0
2
4
6
x (in)
8
10
12
Figure 6.74. Comparison of Tc (aluminum) for D=0.25", tp=0.25", tb=0.08".
0
-10
FEM (TI)
Tc ( 0F)
-20
EQUATION
-30
-40
-50
-60
-70
0
2
4
6
x (in)
8
10
12
Figure 6.75. Comparison of Tc (titanium) for D=0.25", tp=0.25", tb=0.08".
0
-10
FEM (ST)
Tc ( 0 F)
-20
EQUATION
-30
-40
-50
-60
-70
0
2
4
6
x (in)
8
10
12
Figure 6.76. Comparison of Tc (steel) for D=0.25", tp=0.25", tb=0.08".
140
0
-10
FEM (AL)
Tc ( 0F)
-20
EQUATION
( )
-30
-40
-50
-60
0
2
4
6
x (in)
8
10
12
Figure 6.77. Comparison of Tc (aluminum) for D=0.375", tp=0.25", tb=0.125".
-10
FEM (TI)
-20
EQUATION
-30
c
0
T ( F)
0
-40
-50
-60
0
2
4
6
x (in)
8
10
12
Figure 6.78. Comparison of Tc (titanium) for D=0.375", tp=0.25", tb=0.125".
0
-10
FEM (ST)
Tc ( 0 F)
-20
EQUATION
(S )
-30
-40
-50
-60
0
2
4
6
x (in)
8
10
12
Figure 6.79. Comparison of Tc (steel) for D=0.375", tp=0.25", tb=0.125".
141
0
-10
FEM (AL)
Tc ( 0F)
-20
EQUATION
( )
-30
-40
-50
-60
-70
0
2
4
6
8
x (in)
10
12
14
Figure 6.80. Comparison of Tc (aluminum) for D=0.375", tp=0.13", tb=0.125".
0
-10
FEM (TI)
Tc ( 0F)
-20
EQUATION
( )
-30
-40
-50
-60
-70
0
2
4
6
8
10
12
14
x (in)
Figure 6.81. Comparison of Tc (titanium) for D=0.375", tp=0.13", tb=0.125".
Tc ( 0 F)
0
-10
FEM (ST)
-20
EQUATION
(S )
-30
-40
-50
-60
0
2
4
6
8
10
12
14
x (in)
Figure 6.82. Comparison of Tc (steel) for D=0.375", tp=0.13", tb=0.125".
142
Tc ( 0 F)
0
-10
FEM (AL)
-20
EQUATION
( )
-30
-40
-50
-60
0
2
4
6
8
10
x (in)
12
14
16
18
Figure 6.83. Comparison of Tc (aluminum) for D=0.375", tp=0.5", tb=0.125".
0
-10
FEM (TI)
Tc ( 0F)
-20
EQUATION
-30
-40
-50
-60
0
2
4
6
8
10
x (in)
12
14
16
18
Figure 6.84. Comparison of Tc (titanium) for D=0.375", tp=0.5", tb=0.125".
0
-10
FEM (ST)
Tc ( 0F)
-20
EQUATION
(S )
-30
-40
-50
-60
0
2
4
6
8
10
x (in)
12
14
16
18
Figure 6.85. Comparison of Tc (steel) for D=0.375", tp=0.5", tb=0.125".
143
6.5
Equivalent Stiffness of the Fastener (Kf)
Similarly, equivalent stiffness of the fastener Kf also depends on fastener size, spacing,
thickness of the metallic beam and composite panel, and also material of the metallic beam. As
discussed in Chapter 3, the equivalent stiffness of the fastener was calculated from the
displacements of each unit of the Z-beam and composite panel, and the summation of contact
forces. The equations obtained from SAS analysis of 27 models for calculating equivalent
stiffness of the fastener for three different metallic beams are
2
K f (aluminum) = 2754421 −
22299256t b
90727
− 7216639043( t b t p ) 4 +
− 2522476e tb D
2
Dp
( pD)
K f ( titanium ) = 8677548 +
148242t b
4
− 9155528e −tb − 1954562t p
D
2
K f ( steel ) = −2193016 + 8133495t b p + 1506070e D +
26486
D2
p: Spacing Parameter=4 for Fastener Spacing 4D, Parameter=5 for Fastener Spacing 5D,
Parameter=6 for Fastener Spacing 6D
6.5.1 Comparison of Kf Obtained from FEA and Equations
This section compares the values of stiffness of the fastener obtained using FEA and those
obtained using the developed equation in Figures 6.86 to 6.89. Since the stiffness of the fastener
is a single number, the comparison is done by varying the diameter of the fasteners, keeping the
thickness of the metallic beam and thickness of the composite panel constant.
144
16
14
FEM (AL)
12
Kf (lb/in)10
5
FEM (TI)
10
FEM (ST)
8
EQUATION (AL)
6
EQUATION (TI)
EQUATION (ST)
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
D (in)
Figure 6.86. Comparison of Kf Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.25", tb=0.125".
16
14
FEM (AL)
12
Kf (lb/in)10
5
FEM (TI)
10
FEM (ST)
8
EQUATION (AL)
6
EQUATION (TI)
EQUATION (ST)
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
D (in)
Figure 6.87. Comparison of Kf Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.13", tb=0.125".
145
16
14
FEM (AL)
12
Kf (lb/in)10
5
FEM (TI)
10
FEM (ST)
8
EQUATION (AL)
6
EQUATION (TI)
EQUATION (ST)
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
D (in)
Figure 6.88. Comparison of Kf Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.5", tb=0.125".
6
Kf (lb/in)10
5
FEM (AL)
FEM (TI)
4
FEM (ST)
EQUATION (AL)
EQUATION (TI)
2
EQUATION (ST)
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
D (in)
Figure 6.89. Comparison of Kf Obtained from FEM and Equations as a Function of Fastener
Diameter D for tp=0.5", tb=0.08".
146
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
An analytical model was developed to investigate the behavior of a fastened
aluminum/composite assembly under thermal loads. Necessary parameters, such as equivalent
area, equivalent temperature, and equivalent fastener stiffness, were determined by finite element
analyses. Two 68 inches by 36 inches by 0.13 inches (thin panel) composite panel/aluminum
beam assemblies and two 68 inches by 36 inches by 0.25 inches (thick panel) fabricated were
tested using an environmental chamber. Two of the four composite panels have three Cirlex®
strips (68"×2.75"×0.06" for the 0.25-inch thick panel and 68"×2.75"×0.12" for the 0.13-inch thin
panel) embedded in the mid-plane of the composite panel and co-cured with the prepreg. The
Cirlex® strips, located where the aluminum beams were fastened, acted as a thermal barrier to
reduce heat loss and raise the temperature of the aluminum beams. The effect of beam length
was also studied.
7.1 Conclusions
The following conclusions related to load transfer behavior in a bolted joint are drawn from
this investigation:
• The developed analytical model for a thermal-loading condition predicts accurate load
transfer across the fasteners. Hence, the model can be used to predict and reduce thermal
stresses in hybrids of aluminum and composite structures.
• The analytical model can be used to calculate load transfer across the fasteners in joints of
different shapes and material properties, which would assist designers.
• The peak load transfer can be reduced when the Cirlex® layer is added to the composite
panel. It is also noted that the peak load transfer is approximately reduced by 21 percent
147
when the 0.06-inch Cirlex® layer is added in the thick composite panel and by 25 to 32
percent when the 0.12-inch Cirlex® layer is added in the thin composite panel.
• Parameters included in the analytical model, such as equivalent areas, Young’s modulus,
CTEs of the substrates, and temperature change of the aluminum/composite assemblies,
can be changed to analyze assemblies with different materials, geometries, and dimensions
in other applications. In other words, the developed models can be applied not only to the
aluminum/composite assemblies but also to any fastened assemblies with dissimilar
materials. Effects from assembly geometry, such as substrate length, on stress levels can
be predicted. Therefore, the models can assist designers in reducing structure loads due to
CTE mismatch.
• The concept of using a β factor is efficient to compare the strain results obtained from the
analytical model with the experimental surface strains, and this method of determining
joint flexibility can be used to design the joints effectively.
• To investigate the bolted joint behavior, it is not necessary to build the FE model of the
complete joint, thereby reducing the computational time.
• Equations correlating the five parameters with geometric and material properties can be
provided.
7.2
Recommendations
• The analytical model developed can be further modified to calculate the out-of-plane load
transfer due to the warping effect of the joint subjected to temperature change.
• The analytical model developed can be verified when an I-beam or any other crosssections are used instead of a Z-beam.
148
REFERENCES
149
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151