Bending, Vibration and Vibro-Acoustic Analysis

241 Pages·2014·7 MB·English

Bending, Vibration and Vibro-Acoustic Analysis of Composite Sandwich Plates with
Corrugated Core
by
Rajesh Kumar Boorle

A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Automotive Systems Engineering)
in the University of Michigan-Dearborn
2014

Doctoral Committee:

Professor P.K. Mallick, Chair
Professor Yubao Chen
Professor John Cherng
Associate Professor Hong-Tae Kang
© Rajesh Kumar Boorle

All rights reserved
2014
DEDICATION

to my family members, teachers and friends for their kind affection and support during
my entire research career

ii
ACKNOWLEDGEMENTS

From the bottom of my heart, I have the immense pleasure of expressing my deep
gratitude and sincere thanks to my dissertation committee chair and research supervisor,
Prof. P.K. Mallick, for his invaluable knowledge and outstanding support during this
research work. I greatly admire his ingeniousness, resourcefulness and devotion to the
research. For sure, a word of thanks is not sufficient; given a chance, I would never say
no to work for another research program with him.
I would like to thank Prof. John Cherng, Prof. Yubao Chen and Prof. Hong-Tae
Kang for serving on my dissertation committee. Their suggestions on the dissertation are
deeply acknowledged.
I would also like to express my deep appreciation for the research assistantship
provided by CLAMP (Center for Lightweighting Automotive Material Processing) and
CEEP (Center for Engineering Education and Practice) during my entire PhD program.
I also take this opportunity to express my profound gratitude to my beloved
parents, my family members and relatives for their constant support and having a belief
that I sincerely work to excel in my research carrier. My special thanks are for my dear
wife Dr. Divya, brother Mr. Srikanth (Charted Accountant, Business Finance Analyst),
sister Dr. Sunitha (Henry Ford Hospitals, Detroit, U.S.A) and brother-in-law Dr.
Manohar for their encouraging words during the downhearted time in research.
There are many respectable people who have contributed in many ways to make this
work a success; I wish to thank them all and offer my prayers.

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TABLE OF CONTENTS

DEDICATION .................................................................................................................... ii
ACKNOWLEDGEMENTS ............................................................................................... iii
LIST OF FIGURES ........................................................................................................... ix
LIST OF TABLES ........................................................................................................... xix
ABSTRACT ...................................................................................................................... xx
CHAPTER 1: INTRODUCTION ....................................................................................... 1
1.1 INTRODUCTION .................................................................................................... 1
1.2 APPLICATIONS OF CORRUGATED-CORE SANDWICH CONSTRUCTION . 3
1.3 LITERATURE REVIEW ......................................................................................... 4
1.4 MOTIVATION AND RESEARCH OBJECTIVES ................................................. 7
1.5 ORGANIZATION OF DISSERTATION ................................................................ 8
1.5.1 Chapter 2: Global Bending Response of Composite Sandwich Plates with
Corrugated Core .......................................................................................................... 9
1.5.2 Chapter 3: Global Vibration Response of Composite Sandwich Plates with
Corrugated Core .......................................................................................................... 9
1.5.3 Chapter 4: Local Bending Response of Composite Sandwich Plates with
Corrugated Core .......................................................................................................... 9
1.5.4 Chapter 5: Vibro-Acoustic Response of Composite Sandwich Plates with
Corrugated Core .......................................................................................................... 9
1.5.5 Chapter 6: Conclusions .................................................................................... 10
REFERENCES ............................................................................................................. 11
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CHAPTER 2: GLOBAL BENDING RESPONSE OF COMPOSITE SANDWICH
PLATES WITH CORRUGATED CORE ........................................................................ 13
2.1 INTRODUCTION .................................................................................................. 13
2.2 ANALYTICAL FORMULATIONS ...................................................................... 16
2.2.1 Geometric Parameters ...................................................................................... 17
2.2.2 Stiffness Matrix for the Face and Core members of the Unit Cell .................. 19
2.2.3 Stiffness Matrix of the Equivalent Plate .......................................................... 21
2.2.4 Transverse Shear Stiffness of the Unit Cell ..................................................... 22
2.2.5 Formulation for Global Bending Response using Minimum Potential Energy 27
2.3 GLOBAL DEFLECTIONS .................................................................................... 31
2.3.1 Case Study 1: [0/90] Laminates, t = t = t = t ≤ 1 mm, p = d = 80 mm .. 32
s TF BF c
2.3.2 Case Study 2: [0/90] Laminates, Effects of Web and Face Thickness, p = d =
s
80 mm ....................................................................................................................... 37
2.3.3 Case Study 3: [0/90] Laminates, t = t = 1 mm, t ≤ 1 mm, d = 80 mm, p =
S TF BF c
20, 40, 80, 160 and 320 mm...................................................................................... 48
2.3.4 Case Study 4: [0/90] Laminates, t = t = 1 mm, t ≤ 1 mm, p = 80 mm, d =
S TF BF c
20, 40, 60, 80 mm ..................................................................................................... 53
2.3.5 Case Study 5: Effect of Laminate Construction, (0/α) and (±α) Laminates,
S S
t = t = 1 mm, t ≤ 1 mm, p = d = 80 mm............................................................. 57
TF BF c
2.4 BENDING MOMENTS AND TRANSVERSE SHEAR FORCES ....................... 65
2.4.1 (0/90) Laminate, t = t = 1 mm, t ≤ 1 mm, p = d = 80 mm ...................... 65
s TF BF c
2.4.2 Effect of Pitch, (0/90) Laminate, t = t = 1 mm, t ≤ 1 mm, d = 80 mm p =
s TF BF c
20, 40, 80, 160 and 320 mm...................................................................................... 69
2.4.3 Effect of Laminate Construction: (0/α) and(±α) Laminates ......................... 73
s s
2.5 CONCLUSIONS..................................................................................................... 88
REFERENCES ............................................................................................................. 90
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CHAPTER 3: GLOBAL FREE VIBRATION RESPONSE OF COMPOSITE
SANDWICH PLATES WITH CORRUGATED CORE CONSTRUCTION .................. 91
3.1 INTRODUCTION .................................................................................................. 91
3.2 ANALYTICAL FORMULATION ......................................................................... 92
3.2.1 Kinetic Energy of the Sandwich Plate ............................................................. 93
3.2.2 Calculation of Elastic Strain Energy of the Plate............................................. 94
3.2.3 Calculation of Global Vibration Response using Hamilton's Principle ........... 96
3.3 FLEXURAL MODES ............................................................................................. 97
3.3.1 Flexural Mode Shapes...................................................................................... 98
3.3.2 Case Study 1: t = t = t = t and p = d = 80 mm .......................................... 99
TF BF c
3.3.3 Case Study 2: t = t =1 mm, t ≤ 1 mm and p = d = 80 mm ...................... 108
TF BF c
3.3.4 Case Study 3: t = t =1 mm, t ≤ 1 mm, d =80 mm and p =20, 40, 80, 160
TF BF c
and 320 mm............................................................................................................. 121
3.3.5 Case Study 4: t = t = 1 mm, t ≤ 1 mm, p = 80 mm and d = 20, 40, 60, 80
TF BF c
mm .......................................................................................................................... 122
3.4 EXTENSIONAL MODES .................................................................................... 124
3.4.1 In-plane Extensional Mode Shapes ............................................................... 124
3.4.2 Natural Frequencies ....................................................................................... 125
3.5 CONCLUSIONS................................................................................................... 128
REFERENCES ........................................................................................................... 130
CHAPTER 4: LOCAL BENDING RESPONSE OF COMPOSITE SANDWICH
PLATES WITH CORRUGATED CORE ..................................................................... 132
4.1 INTRODUCTION ................................................................................................ 132
4.2 FINITE ELEMENT FORMULATION ................................................................ 133
4.2.1 Sandwich Plate Geometry .............................................................................. 133
4.2.2 Description of Selected Shell Element in ANSYS ........................................ 134
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4.2.3 Description of Finite Element Model used for Sandwich Plate ..................... 135
4.2.4 Material Properties used in the Finite Element Model .................................. 136
4.2.5 Load Application and Boundary Conditions ................................................. 136
4.2.6 Finite Element Details.................................................................................... 137
4.3 LOCAL DEFLECTIONS ..................................................................................... 143
4.4. BUCKLING ANALYSIS .................................................................................... 145
4.4.1 Buckling Mode Shape .................................................................................... 145
4.4.2 Effect of Fiber Orientation Angle on Critical Buckling Pressure Load ......... 147
4.5 FAILURE ANALYSIS OF COMPOSITE SANDWICH PLATES WITH
CORRUGATED CORE.............................................................................................. 148
4.5.1 Triangular Core (θ = 45º) .............................................................................. 150
4.5.2 Rectangular Core (θ = 90º) ............................................................................ 160
4.6 CONCLUSIONS................................................................................................... 170
REFERENCES ........................................................................................................... 171
CHAPTER 5: VIBRO-ACOUSTIC RESPONSE OF COMPOSITE SANDWICH
PLATES WITH CORRUGATED CORE ..................................................................... 173
5.1 INTRODUCTION ................................................................................................ 173
5.2 DYNAMIC ANALYSIS OF SANDWICH PLATE............................................. 175
5.3 BOUNDARY ELEMENT FORMULATION FOR ACOUSTIC ANALYSIS.... 175
5.3.1 Description of Acoustic Domain used for I-BEM ......................................... 176
5.3.2 Applying Acoustic Velocity as Boundary Condition .................................... 177
5.3.3 Implementing Free Edge and Multiple Connection Boundary Conditions ... 178
5.4 VARIOUS ACOUSTIC MEASURES ................................................................. 179
5.4.1 Sound Power .................................................................................................. 179
5.4.2 Radiation Efficiency ...................................................................................... 180
5.4.3 Decibel Scales ................................................................................................ 180
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5.5 METHODOLOGY FOR VIBRO-ACOUSTIC ANALYSIS ............................... 181
5.6 VIBRO-ACOUSTIC ANALYSIS OF UNDAMAGED SANDWICH PLATES 182
5.6.1 Triangular Core (Web Inclination Angle θ = 45º) ......................................... 182
5.6.2 Rectangular Core (Web Inclination Angle θ = 90º) ....................................... 185
5.7 FUNDAMENTAL VIBRATION FREQUENCY OF PRE-STRESSED
SANDWICH PLATES ............................................................................................... 187
5.8 VIBRO-ACOUSTIC ANALYSIS OF DAMAGED SANDWICH PLATES WITH
CORRUGATED CORE.............................................................................................. 189
5.8.1 Triangular Core (Web Inclination Angle θ = 45º) ......................................... 190
5.8.2 Rectangular Core (Web Inclination Angle θ = 90º) ....................................... 193
5.9 CONCLUSIONS................................................................................................... 196
REFERENCES ........................................................................................................... 198
CHAPTER 6: Conclusions ............................................................................................. 200
6.1 SUMMARY .......................................................................................................... 200
6.1.1 Global Analysis .............................................................................................. 200
6.1.2 Local Analysis ............................................................................................... 203
6.2 CONCLUSIONS................................................................................................... 204
6.2.1 Part 1: Global Response ................................................................................ 204
6.2.1 Part 2: Local Analysis .................................................................................... 205
6.3 RECOMMENDATIONS FOR FUTURE WORK ............................................... 206
APPENDIX A: MATLAB CODES ............................................................................... 207
A1. UNIT CELL STIFFNESS CALCULATION ...................................................... 208
A2. GLOBAL BENDING RESPONSE CALCULATION ........................................ 213
A3. GLOBAL VIBRATION RESPONSE CALCULATION .................................... 216

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LIST OF FIGURES

Figure 1.1: Sandwich structure with corrugated core ......................................................... 2
Figure 1.2: Configuration of corrugated core layer ............................................................ 2
Figure 1.3: Corrugated-core paperboard sandwich of single cells ..................................... 4
Figure 1.4: Summary of the available literature ................................................................. 7
Figure 2.1: Description of the unit cell with unequal face thicknesses ............................ 16
Figure 2.2: Location of the centroidal axis of the unit cell ............................................... 18
Figure 2.3: In-plane normal and shear forces (N , N and N ), transverse shear forces (Q
x y xy x
and Q ), bending and twisting moments (M , M and M ) on a plate .............................. 21
y x y xy
Figure 2.4: Global normal forces, shear forces and bending moments on the unit cell.... 23
Figure 2.5: Unit transverse shear force and horizontal force for moment equilibrium .... 23
Figure 2.6: Half unit cell of corrugated core panel with deflection due ........................... 24
Figure 2.7: Stresses acting in the x-direction on element of the unit cell ......................... 26
Figure 2.8: Thickness of members in unit cells with cross-sectional area of 478 mm2, but
with different web inclination angles ................................................................................ 33
Figure 2.9: Extensional stiffness terms for varying web inclination angle....................... 34
Figure 2.10: Transverse shear stiffness terms for varying web inclination angle ............ 34
Figure 2.11: Flexural stiffness terms for varying web inclination angle .......................... 35
Figure 2.12: Global defection of the sandwich plate corresponding to 45º web inclination
angle and under a uniform pressure load of 1 N/m2 ......................................................... 36
Figure 2.13: Maximum deflection of sandwich plates with various web inclination angles
and with a uniform pressure load of 1 N/m2 ..................................................................... 36
Figure 2.14: Maximum values of rotations α and β of equivalent plate models of
varying web inclination angle ........................................................................................... 37
Figure 2.15: Comparison of member thickness for varying web inclination angle .......... 38
Figure 2.16: Extensional stiffness terms for varying web inclination angle in Case Study
2.1...................................................................................................................................... 38
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