ABSTRACT
Title of dissertation: A STRAIN-BASED EXPERIMENTAL
METHODOLOGY FOR MEASURING
SECTIONAL STIFFNESS PROPERTIES
OF COMPOSITE BLADES
Tyler Sinotte
Doctor of Philosophy, 2020
Dissertation directed by: Professor Olivier Bauchau
Department of Aerospace Engineering
Sectional stiffness properties of composite rotor blades, such as the axial,
torsional, and bending stiffness, play an important role in the design and analysis of
rotorcraft, as they impact the predicted rotor dynamics, structural loads, and stress
and strain fields. Multiple numerical tools exist for predicting the sectional stiffness
properties based on the cross-sectional geometries and materials; however, rotor
blades are often made of composite materials and require complex manufacturing
procedures, making it challenging to provide an accurate prediction of the inputs
needed for these numerical tools. This dissertation therefore focuses on developing
an experimental technique for calculating the sectional stiffness properties, based on
the measurement of a detailed strain field using digital image correlation.
Two primary features distinguish the developed methodology from currently
existing techniques. First, this method is based upon measured strain fields, as
opposed to measurements of displacements or frequencies that are traditionally
used. The strain field provides a description of the local deformation of the blades,
thereby allowing measurements of the sectional stiffness properties to be made at
discrete spanwise locations along the blade and providing the capability to predict
changes in properties due to features commonly associated with helicopter rotors,
such as twist or taper. Second, this method can be used to calculate the full cross-
sectional stiffness matrix based on a combination of experimental measurements and
a numerical warping function, as opposed to only a subset of these properties that
are typically measured in displacement or frequency based approaches.
The developed methodology is first validated using numerical results from
3-D FEA for cross-sections that have received extensive study in literature. A test
setup is then developed for experimental implementation of the proposed method
and applied to five different beams. The material properties and geometries of
these beams were selected to place an emphasis on the unique capabilities of the
method, including the measurement of elastic coupling stiffness components and
spanwise variations in the stiffness properties, with results compared against analytic
solutions and predictions from SectionBuilder. A detailed uncertainty analysis is
also implemented for estimating the impact of measurement errors on the stiffness
properties.
A STRAIN-BASED EXPERIMENTAL METHODOLOGY FOR
MEASURING SECTIONAL STIFFNESS PROPERTIES OF
COMPOSITE BLADES
by
Tyler Mel Sinotte
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2020
Advisory Committee:
Professor Olivier A Bauchau, Chair/Advisor
Professor James Baeder
Professor Inderjit Chopra
Professor Anubhav Datta
Professor Peter W. Chung, Dean?s Representative
? Copyright by
Tyler Mel Sinotte
2020
Dedication
To my family for their love and support throughout my academic journey.
ii
Acknowledgments
My PhD research has not been a sole endeavor and I would have never reached
this point without the help, guidance, and support of many individuals. First and
foremost, I would like to thank my PhD advisor, Dr. Olivier A. Bauchau, for giving
me the opportunity to work on a challenging and interesting project. His wealth of
knowledge in structures and rotorcraft has been invaluable and he has guided me at
many points throughout my research. I would also like to thank the other members of
my committee Dr. James Baeder, Dr. Inderjit Chopra, Dr. Anubhav Datta, and Dr.
Peter Chung for their expertise and guidance throughout my graduate school career.
Their insights have been extremely helpful in the formulation of this dissertation and
their classes have proven instrumental in providing me with a solid understanding of
structural analysis and rotorcraft. I would also like to specifically thank Dr. Chopra
for his effort in recruiting me to come to the University of Maryland, despite the
fact that I had no previous experience in the field of rotorcraft.
This work would not have been possible without the financial support of the
U.S. Army Aviation Development Directorate (ADD) Vertical Lift Research Center
of Excellence (VLRCOE) under cooperative agreement W911W6-17-2-0004 and the
UMD Clark doctoral fellowship program. The VLRCOE yearly reviews served as
great progress checks throughout my studies and I would especially like to thank Dr.
Mahendra Bhagwat (Program Manager and Technical Agent), Dr. Will Staruk, and
Dr. Joon Lim for their contributions and feedback.
My educational experience at UMD would not have been nearly as smooth
without the many staff and senior students who took the time to help me. Dr. VT
Nagaraj is owed many thanks for the countless times he was willing to listen to me
and provide guidance when something wasn?t working in my research. The aerospace
staff members, Otto, Laura, LaVita, Gabby, Tom, Matt, and Leslie, among others,
are also owed thanks for always staying on top of things such as placing orders for
iii
any parts needed in my experiments or making sure I had all the correct paperwork
filled out. Thank you to the senior students in the cube farm, Will, Elizabeth,
and Joe among others, for getting me familiar with UMD when I first arrived and
providing guidance on courses. A special thanks to my fellow labmates within Dr.
Bauchau?s group. Valentin and Alfonso, and more recently Matteo, Giuseppe, and
Minghe, provided much needed guidance and friendship throughout this research.
Nishant, Sheng-wei, Shilei, Ray, and Andrew were always willing to sit through my
presentations and were supportive and helpful whenever I ran into problems. Several
other students, such as Amy, James, Jon, and Lex are also owed many thanks for
their guidance in manufacturing the various test articles, particularly for assistance
in making the composite blades.
Thank you to all my cube farm and aerospace friends from throughout the
years, there are too many of you to list, but you have really made Maryland feel like
a second home and I will greatly miss our interactions. The intramural sports teams
provided a much needed reprieve from research and helped forge many friendships.
A special thanks to Emily, Katie, Laura, and Wanyi ? I have been fortunate to have
you as both friends and roommates at various points over the past several years.
I would also like to specifically thank Bharath, Brandyn, Dan, and Dylan, whose
friendship since I first arrived at Maryland provided many much needed breaks,
whether it was the hiking and camping trips, getting together to play games, or just
hanging out.
Last, but certainty not least, I would like to thank my family. Mom, Dad,
Holly, Shaun, and Breanna, you have always been there for me and I would never
have made it this far without your love and support. My parents are my role models
and have always been there for me throughout more than two decades of school. I
will never forget all the sacrifices you have made to help me get to where I am today
and I hope to have made you proud.
iv
Table of Contents
Dedication ii
Acknowledgements iii
Table of Contents v
List of Tables ix
List of Figures xi
List of Abbreviations xvii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Rotor Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Review of 3-D Beam Theory . . . . . . . . . . . . . . . . . . . 3
1.2.2 Review of Rotorcraft Beam Analysis . . . . . . . . . . . . . . 8
1.2.3 Cross-sectional Analysis . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Advanced Rotor Analysis . . . . . . . . . . . . . . . . . . . . 11
1.2.5 Higher-order Structural Models . . . . . . . . . . . . . . . . . 13
1.3 Sectional Stiffness Measurements . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Deflection Based Measurements . . . . . . . . . . . . . . . . . 15
1.3.2 Rotation Based Measurements . . . . . . . . . . . . . . . . . . 18
1.3.3 Frequency Based Measurements . . . . . . . . . . . . . . . . . 22
1.3.4 Other Notable Measurements . . . . . . . . . . . . . . . . . . 23
1.3.5 Limitations of Existing Measurement Techniques . . . . . . . 24
1.4 Strain Measurement Techniques . . . . . . . . . . . . . . . . . . . . . 25
1.4.1 Contact Based Strain Measurements . . . . . . . . . . . . . . 25
1.4.2 Non-Contact Based Strain Measurements . . . . . . . . . . . . 29
1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.7 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . 35
2 Methodology 36
2.1 Cross-Sectional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.1 Beam Configuration . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.2 Stress and Strain Components . . . . . . . . . . . . . . . . . . 39
v
2.1.3 Semi-Discretization of the Displacement Field . . . . . . . . . 42
2.1.4 The Central Solution . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 Data Reduction Equation . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 Rotation of the Strain Components . . . . . . . . . . . . . . . 46
2.2.2 Local Equilibrium Equations . . . . . . . . . . . . . . . . . . . 49
2.2.3 Compliance Matrix Equations . . . . . . . . . . . . . . . . . . 51
2.2.4 SectionBuilder Data . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.3.1 Matrix Assembly . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.3.2 Compliance Matrix Calculation . . . . . . . . . . . . . . . . . 70
2.3.3 Stiffness Matrix Calculation . . . . . . . . . . . . . . . . . . . 74
2.3.4 Principal Axes of Bending . . . . . . . . . . . . . . . . . . . . 75
2.3.5 Shear Center . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.3.6 Change in Cross-Sectional Frame . . . . . . . . . . . . . . . . 78
3 Verification and Validation 80
3.1 Aluminum Rectangular Beam . . . . . . . . . . . . . . . . . . . . . . 81
3.1.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1.2 3-D Finite Element Model . . . . . . . . . . . . . . . . . . . . 83
3.1.3 Stiffness Results . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2 Tapered Aluminum Rectangular Beam . . . . . . . . . . . . . . . . . 90
3.2.1 3-D Finite Element Model . . . . . . . . . . . . . . . . . . . . 90
3.2.2 Stiffness Results . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3 Composite Box Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.3.1 Classical Laminated Plate Theory . . . . . . . . . . . . . . . . 100
3.3.1.1 Sectional Stiffness Properties . . . . . . . . . . . . . 101
3.3.2 3-D Finite Element Model . . . . . . . . . . . . . . . . . . . . 103
3.3.3 Stiffness Results . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Uncertainty Quantification 118
4.1 Types of Experimental Errors . . . . . . . . . . . . . . . . . . . . . . 118
4.2 Experimental Error Quantification . . . . . . . . . . . . . . . . . . . 120
4.3 Uncertainty Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3.1 Taylor Series Method . . . . . . . . . . . . . . . . . . . . . . . 125
4.3.2 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . 133
4.4 Method Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.4.1 Prismatic Aluminum Rectangular Beam . . . . . . . . . . . . 136
4.4.2 Symmetric Composite Box Beam . . . . . . . . . . . . . . . . 141
4.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 148
5 Experimental Methodology 150
5.1 Test Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.1.1 Test Article Mounting . . . . . . . . . . . . . . . . . . . . . . 152
5.1.2 Load Application . . . . . . . . . . . . . . . . . . . . . . . . . 153
vi
5.1.3 Digital Image Correlation Setup . . . . . . . . . . . . . . . . . 155
5.2 Digital Image Correlation (DIC) . . . . . . . . . . . . . . . . . . . . . 156
5.2.1 DIC Sample Preparation . . . . . . . . . . . . . . . . . . . . . 157
5.2.2 DIC System Setup . . . . . . . . . . . . . . . . . . . . . . . . 158
5.2.3 DIC Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.2.4 DIC Data Extraction . . . . . . . . . . . . . . . . . . . . . . . 166
5.2.5 Strain Measurement Basis . . . . . . . . . . . . . . . . . . . . 170
5.2.6 Rigid Body Motion . . . . . . . . . . . . . . . . . . . . . . . . 174
5.2.7 Uncertainty Identification . . . . . . . . . . . . . . . . . . . . 176
5.3 Load Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.3.1 Load Cell Testing . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.3.2 Load Cell Calibration . . . . . . . . . . . . . . . . . . . . . . . 184
5.3.3 Sectional Load Calculation . . . . . . . . . . . . . . . . . . . . 191
5.4 Test Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.4.1 Prismatic Aluminum Beam . . . . . . . . . . . . . . . . . . . 196
5.4.2 Tailored Composite Beam . . . . . . . . . . . . . . . . . . . . 197
5.4.3 VR-7 Composite Blades . . . . . . . . . . . . . . . . . . . . . 199
5.4.4 Tapered Aluminum Beam . . . . . . . . . . . . . . . . . . . . 202
5.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 203
6 Experimental Results 205
6.1 Prismatic Aluminum Beam . . . . . . . . . . . . . . . . . . . . . . . . 205
6.1.1 DIC measurements . . . . . . . . . . . . . . . . . . . . . . . . 206
6.1.2 Stiffness Matrix Calculation . . . . . . . . . . . . . . . . . . . 208
6.1.3 Comparison to other methods . . . . . . . . . . . . . . . . . . 212
6.2 Tailored Composite Beam . . . . . . . . . . . . . . . . . . . . . . . . 215
6.2.1 DIC measurements . . . . . . . . . . . . . . . . . . . . . . . . 216
6.2.2 Stiffness Matrix Calculation . . . . . . . . . . . . . . . . . . . 217
6.3 VR-7 Composite Blades . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.3.1 DIC measurements . . . . . . . . . . . . . . . . . . . . . . . . 223
6.3.2 Stiffness Matrix Calculation . . . . . . . . . . . . . . . . . . . 226
6.3.3 Centroid and Shear Center . . . . . . . . . . . . . . . . . . . . 232
6.4 Tapered Aluminum Beam . . . . . . . . . . . . . . . . . . . . . . . . 234
6.4.1 DIC measurements . . . . . . . . . . . . . . . . . . . . . . . . 235
6.4.2 Stiffness Matrix Calculation . . . . . . . . . . . . . . . . . . . 238
6.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 242
7 Summary and Conclusions 244
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.2 Key Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 247
7.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
vii
A Singular Value Decomposition 253
A.1 Computation of the SVD . . . . . . . . . . . . . . . . . . . . . . . . . 254
A.2 Reduced SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
B Complex Step Derivative 257
Bibliography 259
viii
List of Tables
2.1 Ratios of warping induced to total strain for the aluminum rectangular
cross-section with various beam curvatures . . . . . . . . . . . . . . . 66
2.2 Ratios of warping induced to total strain for the composite hollow
rectangular cross-section with various beam curvatures . . . . . . . . 67
2.3 Ratios of warping induced to total strain for the composite VR-7
airfoil cross-section with various beam curvatures . . . . . . . . . . . 68
3.1 Applied loads for the 3-D finite element model of the aluminum beam
with a rectangular cross-section . . . . . . . . . . . . . . . . . . . . . 85
3.2 Average stiffness of the aluminum rectangular beam . . . . . . . . . . 88
3.3 Applied loads for the aluminum beam with a taper ratio of 2 . . . . . 92
3.4 Applied loads for the aluminum beam with a taper ratio of 5 . . . . . 92
3.5 Composite Box-Beam Layups . . . . . . . . . . . . . . . . . . . . . . 98
3.6 Applied loads for the composite box-beams . . . . . . . . . . . . . . . 104
3.7 Average stiffness of the uncoupled composite box beam . . . . . . . . 113
3.8 Average stiffness of the symmetric layup composite box beam . . . . 114
3.9 Average stiffness of the antisymmetric layup composite box beam . . 115
3.10 Average compliance of the antisymmetric layup composite box beam 115
4.1 95% confidence intervals for systematic and random uncertainties of a
6-axis load measurement. . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2 Number of Monte Carlo iterations required to converge the uncertainty
bounds for the prismatic aluminum beam. . . . . . . . . . . . . . . . 138
4.3 Comparison of 95% uncertainty levels (in %) for the 6 non-zero stiffness
properties of the prismatic aluminum beam from the Monte Carlo
method (MCM) and Taylor series method (TSM). . . . . . . . . . . . 139
4.4 Number of Monte Carlo iterations required to converge the uncertainty
bounds for the composite box beam. . . . . . . . . . . . . . . . . . . 144
4.5 Comparison of 95% uncertainty levels (in %) from the Monte Carlo
method (MCM) and Taylor series method (TSM) for the stiffness
properties of the composite box beam with symmetric layup. . . . . . 145
5.1 Mean shear strain ?+12 from DIC under multiple torsional loads. . . . 173
5.2 Measured geometric properties of the prismatic aluminum beams. . . 196
ix
6.1 Applied load cases for the aluminum test articles. . . . . . . . . . . . 206
6.2 Average measured stiffness values of the three aluminum test articles. 212
6.3 Applied tip loads for the composite beam. . . . . . . . . . . . . . . . 216
6.4 Comparison of average composite beam stiffness values. . . . . . . . . 222
6.5 Applied tip loads for the composite blade without a spar. . . . . . . . 223
6.6 Applied tip loads for the composite blade with a spar. . . . . . . . . . 223
6.7 Comparison of the diagonal stiffness components for the two composite
blades. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.8 Comparison of the off-diagonal stiffness components for the two com-
posite blades. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.9 Applied load cases for the tapered aluminum beam. . . . . . . . . . . 235
x
List of Figures
1.1 Representative rotor blade. . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Illustrative setups for measurement of bending stiffness based on
deflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Illustrative setup for measurement of torsional stiffness based on
measured rotation angle ?. . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Illustrative setup for measurement of bending stiffness based on the
?mirror method.? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Illustrative setup for measurement of sectional stiffness based on
measured frequencies using a shaker and laser vibrometer. . . . . . . 23
1.6 Illustrative strain gauges. . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.7 Schematic setup of distributed wire sensor. . . . . . . . . . . . . . . . 28
1.8 Example of Fiber Bragg Gratings. . . . . . . . . . . . . . . . . . . . . 29
1.9 Example of results produced using ESPI (from Rastogi 2015 [1]). . . . 30
2.1 Configuration of a naturally curved and twisted beam. . . . . . . . . 37
2.2 Cross-section showing rotation angles between the reference basis B
and material basis B?. . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Discretized cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 External edge bases for example cross-sections. . . . . . . . . . . . . . 47
2.5 ith measurement location in the finite element mesh. . . . . . . . . . . 54
2.6 Basis rotation components along external edge for NACA 0012 and
VR-7 airfoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.7 Example rectangular sections depicting rotation angles on external
edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8 Shear strain components used to predict the compliance matrix com-
ponent S33 for the solid rectangular cross-section. . . . . . . . . . . . 58
2.9 Shear strain components used to predict the compliance matrix com-
ponent S33 for the hollow rectangular cross-section. . . . . . . . . . . 59
2.10 Cross-sections used for determining the r(atio of)warping induced strainto total strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.11 Example beams curved in the plane of b?1, b?2 . . . . . . . . . . . . . . 64
2.12 Run time of SVD implementations for different problem sizes m. . . . 73
2.13 Centroid and principal axes of bending. . . . . . . . . . . . . . . . . . 75
xi
2.14 Shear center and principal axes of shear. . . . . . . . . . . . . . . . . 77
2.15 Translation and rotation into cross-sectional frame FP . . . . . . . . . 79
3.1 Aluminum beam with rectangular cross-section. . . . . . . . . . . . . 81
3.2 Mesh of the aluminum beam with rectangular cross-section. . . . . . 83
3.3 Applied boundary conditions for the three forces. . . . . . . . . . . . 84
3.4 Applied boundary conditions for the three moments. . . . . . . . . . 85
3.5 Strains in the global basis B = (b?1, b?2, b?3) under flap bending and
torsional loads for the aluminum beam with a rectangular cross-section. 86
3.6 Strains in the surface basis E = (e?1, e?2, e?3) under flap bending and
torsional loads for the aluminum beam with a rectangular cross-section. 87
3.7 Spanwise variations of the axial (K11), lag shear (K22), and flap shear
(K33) stiffness using 3-D FEM strains for the aluminum beam with
rectangular cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.8 Spanwise variations of the torsional (K44), flap bending (K55), and
lag bending (K66) stiffness using 3-D FEM strains for the aluminum
beam with rectangular cross-section. . . . . . . . . . . . . . . . . . . 89
3.9 Tapered aluminum beam with rectangular cross-section. . . . . . . . . 91
3.10 Mesh of the tapered aluminum beam with rectangular cross-section. . 91
3.11 Strains in surface basis E = (e?1, e?2, e?3) under axial and torsional loads
for the tapered aluminum beams with a rectangular cross-section. . . 93
3.12 Spanwise variations of the axial stiffness, K11, for the tapered alu-
minum beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.13 Spanwise variations of the lag shear stiffness, K22, for the tapered
aluminum beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.14 Spanwise variations of the flap shear stiffness, K33, for the tapered
aluminum beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.15 Spanwise variations of the torsional stiffness, K44, for the tapered
aluminum beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.16 Spanwise variations of the flap bending stiffness, K55, for the tapered
aluminum beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.17 Spanwise variations of the lag bending stiffness, K66, for the tapered
aluminum beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.18 Composite beam with rectangular box cross-section. . . . . . . . . . . 99
3.19 Cross-sectional mesh of the composite box beams. . . . . . . . . . . . 103
3.20 Shear strain ?+12 in surface basis E = (e?1, e?2, e?3) under an axial load,
f1 for the three composite box beam layups. . . . . . . . . . . . . . . 105
3.21 Axial strain ?+11 in surface basis E = (e?1, e?2, e?3) under a flap bending
load, m2, for the three composite box beam layups. . . . . . . . . . . 106
3.22 Spanwise variations of the axial stiffness, K11, for the composite box
beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.23 Spanwise variations of the lag shear stiffness, K22, for the composite
box beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.24 Spanwise variations of the flap shear stiffness, K33, for the composite
box beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
xii
3.25 Spanwise variations of the torsional stiffness, K44, for the composite
box beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.26 Spanwise variations of the flap bending stiffness, K55, for the composite
box beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.27 Spanwise variations of the lag bending stiffness, K66, for the composite
box beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.28 Spanwise variations of the extension/shear coupling stiffness, K12,
and the bending/torsion coupling stiffness, K45, for the symmetric
composite box beam layup. . . . . . . . . . . . . . . . . . . . . . . . 111
3.29 Spanwise variations of the extension/torsion coupling stiffness, K14,
and the bending/shear coupling stiffnesses, K25 and K36 n, for the
antisymmetric composite box beam layup. . . . . . . . . . . . . . . . 112
4.1 Combined random and systematic error distributions. . . . . . . . . . 119
4.2 Flowchart for uncertainty quantification using Monte Carlo method. . 134
4.3 Convergence of the Monte Carlo method for the uncertainty levels of
the stiffness properties of the prismatic aluminum beam. . . . . . . . 137
4.4 Spanwise variation of 95% uncertainty levels for the axial and shear
stiffnesses of the prismatic aluminum beam. . . . . . . . . . . . . . . 142
4.5 Spanwise variation of 95% uncertainty levels for the torsional and
bending stiffnesses of the prismatic aluminum beam. . . . . . . . . . . 142
4.6 Convergence of the Monte Carlo method for the uncertainty levels of
the stiffness properties of the composite box beam. . . . . . . . . . . 144
4.7 Spanwise variation of 95% uncertainty levels for the axial and shear
stiffnesses of the composite box beam. . . . . . . . . . . . . . . . . . 147
4.8 Spanwise variation of 95% uncertainty levels for the torsional and
bending stiffnesses of the composite box beam. . . . . . . . . . . . . . 147
5.1 Test stand for load application. . . . . . . . . . . . . . . . . . . . . . 151
5.2 Clamp assembly to attach the beams to the test stand and apply loads.153
5.3 Schematics of winch and pulley setups for bending moment application.154
5.4 Full experimental test setup for stiffness property measurement. . . . 156
5.5 Beam at multiple stages in the sample preparation process. . . . . . . 159
5.6 Schematic of the overhead view of the DIC setup. . . . . . . . . . . . 160
5.7 Grid used for DIC calibration. . . . . . . . . . . . . . . . . . . . . . . 161
5.8 Effect of DIC filter size on measured strain. . . . . . . . . . . . . . . 166
5.9 Example 2-D grid of w-displacement generated from DIC. . . . . . . 167
5.10 Updating of the DIC measurement locations into the cross-sectional
basis B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.11 Experimental setup for torsional testing of a circular aluminum tube. 171
5.12 Shear strain ?+12 measured by DIC for a circular tube under a -54.2
N-m torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.13 Shear strain ?+12 distribution along y, measured by DIC under multiple
torques for a circular tube. . . . . . . . . . . . . . . . . . . . . . . . . 173
xiii
5.14 Test description for rigid body motion introduced by a vertical trans-
lation of the cameras. . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.15 Distribution of mean strain values resulting from a vertical translation
of the cameras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.16 Test description for rigid body motion introduced by a lateral transla-
tion of the test article. . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.17 Distribution of mean strain values resulting from a lateral translation
of the test article. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.18 DIC strain fields from averaging images in the undeformed state,
showing the random error in the DIC system. . . . . . . . . . . . . . 177
5.19 Example DIC axial strain distribution in undeformed reference image
at x = 34? 0.05 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.20 Distribution of mean strain offsets in undeformed state. . . . . . . . . 178
5.21 Distribution of standard deviation of y-variations in strain in the
undeformed state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.22 6-axis load cell used to measure reaction loads in the test articles. . . 180
5.23 Test setup for controlled testing of the 6-axis load cell in shear and
bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.24 Calibration beam used for testing the load cell. . . . . . . . . . . . . 181
5.25 Measured force F3 under multiple applied loads and at multiple loca-
tions from the origin of the load cell. . . . . . . . . . . . . . . . . . . 182
5.26 Measured moment M2 under multiple applied loads and at multiple
locations from the origin of the load cell. . . . . . . . . . . . . . . . . 182
5.27 Measured force F2 under multiple applied loads and at multiple loca-
tions from the origin of the load cell. . . . . . . . . . . . . . . . . . . 183
5.28 Measured moment M3 under multiple applied loads and at multiple
locations from the origin of the load cell. . . . . . . . . . . . . . . . . 183
5.29 Results from static testing of the 6-axis load cell with an updated
calibration under a shear force F2. . . . . . . . . . . . . . . . . . . . . 185
5.30 Results from static testing of the 6-axis load cell with an updated
calibration under a shear force F3. . . . . . . . . . . . . . . . . . . . . 186
5.31 Results from static testing of the 6-axis load cell with an updated
calibration under a bending moment M2. . . . . . . . . . . . . . . . . 186
5.32 Results from static testing of the 6-axis load cell with an updated
calibration under a bending moment M3. . . . . . . . . . . . . . . . . 187
5.33 Test setup for controlled testing of the 6-axis load cell under an axial
force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.34 Results from static testing of the 6-axis load cell under an axial force F1.189
5.35 Test setup for controlled testing of the 6-axis load cell in torsion. . . . 190
5.36 Results from static testing of the 6-axis load cell under a torsional
moment M1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.37 Transformation from inertial to local cross-sectional frame. . . . . . . 191
5.38 Translation from inertial frame FI to the cross-sectional frame Fb. . . 193
5.39 Cross-sectional geometry of prismatic aluminum beam for DIC testing.196
xiv
5.40 Three prismatic aluminum beams with speckle pattern applied for
DIC testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.41 Cross-sectional geometry of prismatic aluminum beam for DIC testing.197
5.42 Main components for manufacturing the composite beam with a
rectangular cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.43 Tailored composite beam with speckle pattern applied for DIC testing.199
5.44 Cross-sectional geometries of the composite blades with a VR-7 airfoil.200
5.45 Mold used to manufacturing the composite blades with a VR-7 airfoil. 201
5.46 Two composite blades with speckle patterns applied for DIC testing. . 202
5.47 Tailored composite beam with speckle pattern applied for DIC testing.203
6.1 Axial strain ?+11 under a positive flap bending moment (load case 8).
The upper row of images shows the strain on the z = +h/2 face and
the bottom row shows the strain on the z = ?h/2 face. . . . . . . . . 207
6.2 Axial stiffness K11 measurements for the three prismatic aluminum
beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.3 Lag shear stiffness K22 measurements for the three prismatic aluminum
beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.4 Torsional stiffness K44 measurements for the three prismatic aluminum
beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.5 Flap bending stiffness K55 measurements for the three prismatic
aluminum beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.6 Lag bending stiffness K66 measurements for the three prismatic alu-
minum beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.7 Comparison of the current strain-based method to other common
measurement techniques for the flap bending stiffness. . . . . . . . . . 214
6.8 Comparison of the current strain-based method to other common
measurement techniques for the lag bending stiffness. . . . . . . . . . 214
6.9 Axial strain ?+11 under a positive torsional moment (load case 6). The
upper row of images shows the strain on the z = +t/2 face and the
bottom row shows the strain on the z = ?t/2 face. . . . . . . . . . . 218
6.10 Shear strain ?+12 under a positive torsional moment (load case 6). The
upper row of images shows the strain on the z = +t/2 face and the
bottom row shows the strain on the z = ?t/2 face. . . . . . . . . . . 219
6.11 Axial and shear stiffness components for the tailored composite beam. 220
6.12 Torsional and flap bending stiffness components for the tailored com-
posite beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.13 Lag bending stiffness for the tailored composite beam. . . . . . . . . . 222
6.14 Axial strain ?+11 under an axial force (load case 1) for the blade without
a spar. The upper row of images shows the strain on the upper profile
of the airfoil and the bottom row shows the strain on the lower profile.224
6.15 Axial strain ?+11 under an axial force (load case 1) for the blade with a
spar. The upper row of images shows the strain on the upper profile
of the airfoil and the bottom row shows the strain on the lower profile.225
6.16 Axial stiffness K11 measurements for the composite blades. . . . . . . 227
xv
6.17 Lag shear stiffness K22 measurements for the composite blades. . . . . 227
6.18 Lag shear stiffness K33 measurements for the composite blades. . . . . 228
6.19 Torsional stiffness K44 measurements for the composite blades. . . . . 228
6.20 Flap bending stiffness K55 measurements for the composite blades. . . 229
6.21 Lag bending stiffness K66 measurements for the composite blades. . . 229
6.22 Setup for measuring flap bending stiffness of VR-7 composite blade
with d-shaped spar using tip displacement. . . . . . . . . . . . . . . . 231
6.23 Comparison of flap bending stiffness for a VR-7 blade with a d-spar
using difference measurement methods. . . . . . . . . . . . . . . . . . 232
6.24 Measured and predicted centroid locations for the airfoil without a spar.233
6.25 Measured and predicted centroid locations for the airfoil with a spar . 233
6.26 Measured and predicted shear center locations for the airfoil without
a spar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
6.27 Measured and predicted shear center locations for the airfoil with a spar234
6.28 Axial strain ?+11 under a negative lag bending moment (load case 11)
on the z = +t/2 face. . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
6.29 Axial strain ?+11 under a negative lag bending moment (load case 11)
on the z = ?t/2 face. . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
6.30 Axial stiffness K11 measurements for the tapered aluminum beam. . . 239
6.31 Lag shear stiffness K22 measurements for the tapered aluminum beam.240
6.32 Torsional stiffness K44 measurements for the tapered aluminum beam. 240
6.33 Flap bending stiffness K55 measurements for the tapered aluminum
beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
6.34 Lag bending stiffness K66 measurements for the tapered aluminum
beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
xvi
List of Abbreviations
BECAS Beam Cross Section Analysis Software
CAMRAD Comprehensive Analytical Model of Rotor Aerodynamics
and Dynamics
CCD Charge-Coupled Device
CFD Computational Fluid Dynamics
CMOS Complementary Metal-Oxide-Semiconductor
DIC Digital Image Correlation
ESPI Electronic Speckle Pattern Interferometry
FEA Finite Element Analysis
FEM Finite Element Method
FFT Fast Fourier Transform
HART Higher harmonic control Aeroacoustics Rotor Test
MCM Monte Carlo Method
NACA National Advisory Committee for Aeronautics
PMI Projection Moire? Interferometry
RCAS Rotorcraft Comprehensive Analysis System
SVD Singular Value Decomposition
TRAM Tilt Rotor Aeroacoustic Model
TSM Taylor Series Method
UMARC University of Maryland Advanced Rotorcraft Code
VABS Variational Asymptotic Beam Sectional analysis
VAM Variational Asymptotic Method
X3D Experimental 3-D Dynamic solver
xvii
Chapter 1: Introduction
1.1 Motivation
The design and analysis of rotorcraft is primarily governed by the aeromechanic
behavior of the rotor, including aerodynamics, structural dynamics, and stability and
control. For rotorcraft, these disciplines are strongly coupled and an accurate model
is required for each to accurately predict the response of the rotor. The earliest
aerodynamic models were first developed by Glauert in 1926 [2] which introduced
blade element theory for rotorcraft; since then, a variety of aerodynamics models
have been developed for application to rotorcraft. These aerodynamics models
range from simple steady 2-D linear airfoil theory, to full unsteady models, to 3-D
computational fluid dynamics (CFD), with extensive work done to account for the
unique rotorcraft aerodynamic environment, such as treatment of dynamic stall, in
which flow separation is delayed beyond the expected angle of attack from static
flow conditions due to the dynamic pitching of the aircraft, as well as treatment of
non-uniform inflow, through dynamic inflow models and more accurate free-wake
models.
For the structural dynamics analysis, the blades are conventionally modeled
using beam theory, as rotor blades typically have a much greater radial dimension than
the cross-sectional dimensions. The use of beam theory to model the blades requires
sectional stiffness properties, such as the bending and torsional stiffness, as well
as sectional mass properties, which are both typically predicted through numerical
1
analysis. The aerodynamics and structural dynamics models are then coupled
together to study the aeromechanic behavior of the rotorcraft using a comprehensive
analysis, which also incorporates vehicle trim to determine the controls needed to
maintain a steady flight and to predict the blade deformations, airloads, and rotor
performance in desired flight conditions.
While the capabilities of comprehensive analysis codes have greatly improved
over the past several decades, particularly with the coupling of CFD with comprehen-
sive analysis [3], there is still a large amount of uncertainty in the sectional stiffness
properties for composite blades, which play a pivotal role in the accuracy of compre-
hensive analysis. Since the blades are typically made of anisotropic materials and
present complex cross-sectional geometries that require complicated manufacturing
procedures, the accurate prediction of their stiffness properties using numerical tools
can often be quite challenging. Recent numerical studies have shown the importance
that uncertainties in the sectional stiffness properties can have on the analysis. Using
a Monte-Carlo analysis, Murugan et al. [4] showed the effect of random uncertainty
in the stiffness properties of a BO-105 rotor on the vibratory hub loads, with up to a
400% variations from baseline values. Pfumm et al. [5] performed a similar study for
a UH-60A rotor, which showed up to 12% variations on the elastic blade deflections
in hover and up to 200% variations in vibratory hub forces in forward flight from
their respective baselines.
Because of these uncertainties, it is not uncommon to arbitrarily adjust the
predicted blade stiffness properties to better match experimental data; for example,
Yeo and Johnson [6] found that it was necessary to reduce the predicted flap bending,
chord bending, and torsional stiffness values of the scaled BO-105 blade by 10% to
match the measured nonrotating natural frequencies. While these adjustments can
lead to a satisfactory prediction of the rotor dynamics, inaccuracies in other rotor
component models such as the blade grips or control system components may be the
2
actual cause of the discrepancies between the predicted and measured results, which
can still lead to inaccuracies in structural and vibratory loads and the resulting stress
and strain fields that are critical for the design and analysis of the rotor.
Ultimately, there is a dearth of structural data for actual composite rotor blades,
in spite of its importance for comprehensive analysis. Publicly available stiffness
properties for actual helicopter blades is limited to a few rotors, such as the H-34 [7],
Puma [8], SA 349/2 [9], UH-60A [10?12], and the BO-105 (HART-I and HART-
II) [13,14], with only the BO-105 model containing experimental measurements in
addition to numerical predictions. Even in these cases, the properties are limited to
the axial, torsional, flap bending, and lag bending stiffness at best, which represents
only a subset of the full sectional stiffness characterization. Due to this lack of
available structural data, the overall objective of this research is to develop a new
experimental method that can provide a more complete measurement of the sectional
stiffness properties, so that numerical analysis tools can be used more confidently.
1.2 Rotor Structural Modeling
The response of a helicopter in flight is often computed using comprehensive
analysis, which couples the rotor aerodynamics and structural dynamics with the
full vehicle flight dynamics and controls [15,16]. While the aerodynamics and vehicle
flight dynamics and controls are critical components of a comprehensive analysis,
this dissertation primarily focuses on the structural modeling and properties of the
blade, which are discussed in the subsequent sections.
1.2.1 Review of 3-D Beam Theory
Beam theory is widely used in the analysis of aerospace structures and specifi-
cally in rotorcraft analysis. For a structure to be defined as a beam, it must have one
3
of its characteristic dimensions significantly larger than the other two. A depiction
of a typical helicopter blade is shown in Fig. 1.1; typically the ratio of the radial
dimension R to the chord c, referred to as the aspect ratio, is on the order of 12 to 16
for a rotor blade, thereby making it well suited to be analyzed as a beam. Ultimately,
the aim of beam theory is to solve Saint-Venant?s problem, which refers to the prob-
lem of determining elastic deformations of a three-dimensional beam loaded at its
end sections only. The earliest solutions to this problem were by Saint-Venant, who
derived exact elasticity solutions for these beams under torsion [17] and bending [18].
Figure 1.1: Representative rotor blade.
One of the simplest and most well-known beam theories is the Euler-Bernoulli
beam theory; three underlying assumptions of this theory are that (1) there are no
deformations in the plane of the cross-section, (2) the cross-section remains planar
after deformation, and (3) the cross-section remains normal to the deformed axis
of the beam. Under these assumptions, the displacements can be represented as
rigid-body translations and rotations which lead to a simple kinematic description of
the displacements and strains. These can then be combined with the constitutive
laws and the equilibrium equations to give the following relationships
????? ????? ?? ??? ?? F1(x) ? ? ?? K ? ?11 K15 K16 ????? ?1(x) ?????
?
?? M2(x) ????? = ?? K
?
15 K55 K56 ?????? ?2(x) ???? (1.1)
M3(x) K K K ? ?16 56 66 ?3(x)
where ?1(x) is the sectional axial strain, ?2(x) and ?3(x) are the sectional curvatures
about axis b?2 and b?3 respectively, and F1(x), M2(x), and M3(x) are the axial force
4
and bending moments about axis b?2 and b?3, respectively. The Kij terms are the
sectional stiffness properties, whose measurement are the objective of this thesis.
For an uncoupled beam the off-diagonal stiffness properties become zero leaving the
axial stiffness, K11, the bending stiffness about b?2, K55, and the bending stiffness
about b?3, K66. In rotorcraft applications, K55 and K66 are commonly referred to as
the flap bending and lag bending stiffnesses, respectively.
The beam formulation developed by Timoshenko [19], commonly referred to
as Timoshenko beam theory, relaxed the third assumption from the Euler-Bernoulli
beam theory that the cross-section must remain perpendicular to the beam axis,
meaning that transverse shear deformation was taken into account and no longer
considered negligible. In addition to the three equilibrium equations from Euler-
Bernoulli beam theory, this gave rise to two more equilibrium equations
??? ??? ? ???? ??
?? F2(x) ?? K22 0 ?? ??12(x)
?
= (1.2)
F3(x)
?? 0 K ?? ?? (x) ??33 13
where ??12(x) and ??13(x) are the sectional shear strains and F2(x) and F3(x) are the
shear forces in the b?2 and b?3, respectively. This provides two additional stiffness prop-
erties, K22 and K33, which are the shear stiffness in directions b?2 and b?3, respectively.
Finally, the constitutive law for the torsional behavior of the beam is
M1(x) = K44?1(x) (1.3)
where ?1(x) is the sectional twist rate, M1(x) is the torque, and K44 is the torsional
stiffness.
While Euler-Bernoulli beam theory and Timoshenko beam theory provide a
useful basis for the analysis of beams, it is important to have a beam theory that
unifies all aspects of the deformation ? the axial and bending, shear, and torsional ?
5
into a single system of equations. One of the key steps in having a more complete
structural theory was the development of a beam theory for heterogeneous and
anistropic materials. Ies?an [20, 21] developed a method to obtain the solutions of
Saint-Venant?s problem for straight beams made of heterogeneous materials. Dong et
al. [22?24] then generalized Iesan?s method using a semi-finite element discretization
for the cross-section, while calculating both the warping displacements and sectional
properties of the beam.
The next advancements in a unified beam theory came through the Variational
Asymptotic Method (VAM) pioneered by Berdichevsky [25], in which asymptotic
analysis is applied to the energy functional. In this approach, the three-dimensional
Saint-Venant?s problem is reduced to a two-dimensional analysis over the beam?s
cross-section and a one-dimensional analysis of the nonlinear beam equations. Further
refinements to this method were then made by Atilgan et al. [26,27], Hodges [28], Yu et
al. [29], and Kim et al. [30] to deal with beams with small initial curvatures and made
of anisotropic, composite materials and to account for end effects. These refinements
ultimately lead to a 6 ? 6 sectional stiffness matrix defining the cross-section.
An alternative to the variational asymptotic method is an approach first
developed by Giavotto et al. [31] for beams with arbitrary cross-sectional geometry
and material properties. In this approach, two types of solutions for Saint-Venant?s
problem are identified: the central solution, which defines the behavior through
the entire length of the beam, and extremity solutions, which decay exponentially
away from the boundaries. Taking the central solutions into account leads to a
two-dimensional analysis of the beam?s cross-section using finite elements which
also yields a 6 ? 6 sectional stiffness matrix. Furthermore, this approach allows
the three-dimensional stress and strain fields to be evaluated at any point in the
cross-section once the sectional deformations ? the sectional axial and shear strains
and three curvatures ? are known. This approach was further refined by Borri
6
et al. [32, 33] to introduce the concept of intrinsic warping and also generalized
to initially curved and twisted beams, where the magnitudes of the beam?s initial
curvatures are not required to be small.
Additional refinements have been made over the past several decades to the
method initially proposed by Giavotto. Mielke [34,35] found the center manifold of
Saint-Venant?s problem for a straight beam and showed that it is a finite-dimensional
manifold spanned by twelve generalized eigenvectors associated with the null and
purely imaginary eigenvalues. Six of the modes correspond to the rigid-body modes of
the beam and the other six correspond to the fundamental deformation modes of the
beam (extension, shearing, torsion, and bending). Zhong [36] then developed a novel
analytical technique based on Hamilton?s formalism and showed that the null and
purely imaginary eigenvalues of the Hamiltonian operator give rise to the solution of
Saint-Venant?s problem. He later produced analytical solutions for planar elasticity
problems and three-dimensional straight beams made of anisotropic materials in
which he outlined procedures for determining the twelve generalized eigenvectors [37].
More recently, Morandini et al. [38] developed a similar approach based on
Hamilton?s formalism using numerical techniques to evaluate the generalized eigen-
vectors for straight beams made of both isotropic and anisotropic materials. Druz
et al. [39,40] then accounted for naturally twisted beams and obtained a sectional
stiffness matrix relating the components of the generalized forces and displacement.
Bauchau and Han [41] then developed an approach based on Hamilton?s formalism
that decomposes the solution into its central and extremity components through a
sequence of structure preserving transformations using symplectic matrices. This
leads to a set of linear equations for the nodal warping and sectional compliance
matrix whose solutions are found by projecting the governing equations onto the
subspace associated with the Hamiltonian matrix?s null and purely imaginary eigen-
values. Bauchau and Han [42] then further generalized the approach to initially
7
curved beams undergoing large motion but small strains.
1.2.2 Review of Rotorcraft Beam Analysis
The previous section discussed the development of beam theory in a general
sense; the application of beam theory to rotorcraft analysis, and specifically the
rotor itself, is further discussed in this section. The earliest use of beam theory
for rotorcraft analysis was in the NACA technical report in 1957 by Houbolt and
Brooks [43], which formulated linearized equations of motion for elastic blades while
accounting for the coupling between flap and lag bending and torsional deformation.
This was followed up by a general nonlinear beam formulation (accurate up to
the second-order) by Hodges and Dowell in 1974 for the coupled flap/lag/torsion
dynamics of rotor blades [44]. Subsequent efforts over the next decade primarily
focused on consideration of higher-order terms and exact beam kinematics, including
work by Rosen and Friedmann [45] and Hodges et al. [46, 47].
While these efforts were comprehensive in studying the structural response of
a helicopter rotor, they relied on Euler-Bernoulli formulations for isotropic materials.
However, the use of composite materials in rotor blades was starting to become
common by the 1980s and the development of beam models for anisotropic materials
was required. Some of the earliest developments of an anistropic beam theory for
rotorcraft applications occurred in the late 1980s and early 1990s, including efforts
by Bauchau [48] to develop a beam theory for anisotropic materials; Bauchau and
Hong [49] to develop a nonlinear composite beam theory that included warping effects
and elastic couplings; Hodges [50] to apply variational principles to anisotropic beams
with closed cross-sections and unrestrained warping, and Smith and Chopra [51] to
develop an analysis for blades with elastic couplings. A more complete summary of
the development of nonlinear composite beam theory and its applications to rotorcraft
can be found in the book ?Nonlinear Composite Beam Theor? by Hodges [28].
8
The final key challenge in applying beam theory to rotorcraft was the inclusion
of a multibody dynamics formulation. Helicopter rotors often incorporate kinematic
mechanisms to connect various flexible components and control the blades, with
multibody dynamics playing a particularly important role in advanced concepts
such as tiltrotors, where the rotor and wing have strong interactions, as well as
coaxial rotor systems. One of the earliest efforts was performed by Bauchau and
Kang in 1993 [52], who presented a multibody formulation for the nonlinear dynamic
analysis of a helicopter rotor and considered the ground resonance problem. Since
then, multibody dynamics have become commonplace in rotorcraft analysis and the
combination of these key features, a finite element based beam theory for anisotropic
materials and flexible multibody dynamics analysis, has lead to the current state-of-
the-art in rotorcraft comprehensive analysis tools, including Dymore [53], CAMRAD
II [54], RCAS [55], and UMARC [56].
1.2.3 Cross-sectional Analysis
As previously described in Sec 1.2.1, the 3-D Saint-Venant?s problem can be
reduced into a 2-D cross-sectional analysis and 1-D system of nonlinear equations
using either the variational asymptotic method (VAM) or the approach pioneered
by Giavotto et al. [31]. In both cases, the 2-D cross-sectional analysis is performed
using a finite element model on a cross-section whose geometry and material can
be arbitrary (heterogeneous and anisotropic), to obtain sectional stiffness matrices.
The sectional stiffness matrix K provides a means of relating the sectional stress
resultants F to the sectional deformation measures E according to
F = K E (1.4)
9
The sectional stress resultants is the vector of forces and moments defined by
FT = {F1, F2, F3,M1,M2,M }T3 , where F1 denotes the axial forces and F2 and F3
denote the two shear forces while M1 denotes the torque and M2 and M3 the two
bending moments. The sectional deformation measures is the vector of sectional
strains and curvatures defined by ET = {? T11, ??12, ??13, ?1, ?2, ?3} , where ?11 is the
sectional axial strain and ??12 and ??13 are the sectional shear strains while ?1, ?2 and
?3 are the curvatures about their respective axes. The sectional stiffness matrix is a
symmetric, 6 ? 6 matrix of the form
?? ??? K11 K12 K13 K14 K15 K16? ??? ?? K? 12
K22 K23 K24 K25 K26 ?
??
??
? K13 K23 K33 K34 K35 K36 ?K = ? ?? (1.5)?? K14 K24 K34 K44 K ?45 K46 ??? ?K15 K25 K35 K45 K K ?55 56 ??
K16 K26 K36 K46 K56 K66
where the subscripts 1 through 6 correspond to the extension, shear about axis
b?2, shear about axis b?3, torsion, bending about axis b?2, and bending about axis b?3,
respectively. Any term with mixed subscripts (i.e. Kij where i 6= j) represents a
coupling between the deformation modes of the beam; in the case that no coupling is
present, the stiffness matrix becomes diagonal and reduces to the simple expressions
discussed in Sec 1.2.1.
Once determined, either through measurement or numerically, the sectional
stiffness matrices can then be used as an input to the comprehensive analysis codes
and the resulting sectional forces and moments can be reapplied to the 2-D cross-
sectional models to predict the 3-D stress and strain distributions. For numerical
calculations, the engineering software variational asymptotic beam sectional analysis
(VABS) developed by Hodges et al. [57?59] provides a cross-sectional modeling
10
capability using the variational asymptotic method. For the approach pioneered
by Giavotto, the software SectionBuilder developed by Bauchau et al. [60, 61] is
commonly used, with the beam cross section analysis software (BECAS) [62] also
used in the wind turbine industry.
1.2.4 Advanced Rotor Analysis
Most conventional rotor blades are primarily straight and consist of quasi-
isotropic layups making them well suited for analysis using the beam theories
implemented in comprehensive analysis codes; however, advanced designs that can
also be treated with beam theory have received considerable attention, particularly
blades that are elastically tailored to introduce coupling between deformation modes
as well as curved and twisted blades. Early work by Mansfield and Sobey [63]
and Winkler [64] showed that through a careful selection of the composite layup,
elastically tailored blades can be designed to introduce extension/torsion coupling (i.e.
a non-zero K14 stiffness component) when using an antisymmetric ply sequence or to
introduce bending/torsion coupling (i.e. non-zero K45 and K46 stiffness components)
when using a symmetric ply sequence.
Extension/torsion coupling is of considerable interest in rotorcraft and specifi-
cally for tilt-rotors. Tilt-rotors operate at two different rotational speeds, one for
hover and another for cruise, and have substantially different optimal twist distribu-
tions for each flight mode. Since the rotational environment of the rotor generates
significant extensional loads due to the centrifugal forces, extension-torsion coupling
can be utilized to provide a more optimal twist distribution for the rotor in both
operational conditions. Early application of extension/torsion coupling to a tiltrotor
analysis was performed in 1986 by Bauchau et al. [65] which considered two rotor
designs that matched baseline XV-15 stiffness properties but still provided passive
twist change through extension-torsion coupling. Smith and Chopra [66] in 1991
11
then formulated an analytical model for elastically tailored composite box-beams,
including prediction of a full 6 ? 6 stiffness matrix, which was correlated with
experimental displacement measurements of elastically tailored box beams from
Chandra, Stemple, and Chopra [67]. Significant research since then has focused
on the structural analysis of blades with extension-torsion coupling, including by
Soykasap and Hodges [68], Ozbay et al. [69], and Haynes and Armanios [70], as well
as experimental measurements using an articulated NACA0012 model rotor by Lake
et al. [71].
Bending/torsion coupling, in which bending deformation introduces a change
in twist, has also been studied both for rotorcraft and wind turbine applications.
For rotorcraft, Bao and Chopra [72, 73] performed a study using UMARC and
correlated with experiments for five rotors with composite couplings, showing that
bending/torsion coupling could provide notable reductions to vibratory hub loads.
For wind turbines, Fedorov and Berggreen [74] used an Euler-Bernoulli beam formu-
lation and the cross-sectional analysis BECAS to study bending/torsion coupling
and correlated with experimental results for composite beams. Stablein [75] also
considered bend-twist coupling using a Timoshenko beam element and showed that
the coupling should only be included in certain regions of the blade to ensure tower
clearance is maintained.
While most studies introduce structural coupling by elastic tailoring of the
ply layups, these effects can also be introduced through implementation of twist
or curvature in the blades, known as geometric coupling. An early study in 1987
by Bauchau and Hong [76] presented a nonlinear beam analysis for initially curved
and twisted beams and correlated the tip displacements arising from the geometric
couplings with analytical and experimental solutions. In 1994, Cesnik and Hodges [77]
extended the variational asymptotical method to consider initially curved and twisted
composite beams and showed the importance of accounting for the curvature measures
12
in the 2D cross-sectional analysis; Hodges [78] later correlated results against 3D FEM
for initially curved and twisted beams using VABS, comparing the extension/torsion
and shear/bending couplings with a pre-twisted anistropic beam and the shear/torsion
and extension/bending couplings with an initially curved anistropic beam. Epps
and Chandra [79] in 1996 measured frequencies of rotating composite beams with
tip sweep and compared to predictions using beam theory. Recently, Sinotte and
Bauchau [80] correlated the curved beam models in Dymore and SectionBuilder with
experimental strain results for beams undergoing centrifugal loading in a vacuum,
showing that beam models could accurately capture measured strains in initially
curved beams as long as the curvature is included in the cross-sectional analysis.
1.2.5 Higher-order Structural Models
While beam models are predominately used in computational aeromechanics
analyses, higher order models have also been considered, including shell elements and
fully 3-D finite element models. Bauchau et al. [81?83] investigated the use of shell
elements in a multibody dynamic paradigm and presented energy preserving time
integration schemes. Analysis using shell elements has been applied to wind turbines;
Branner et al. studied the use of shell elements for determining stresses [84] and
compared deformations to experimental data [85]. Bazilevs et al. [86] implemented a
Kirchhoff-Love shell based structural model coupled with 3D computational fluid
dynamics to study the airloads and structural deformations and stresses for a full
scale offshore wind turbine rotor. Application of shell elements to rotorcraft analysis
was performed by Kang et al. [87], in which a shell element was developed within
RCAS. This element was designed to be connected to the existing nonlinear beam
elements to allow for a mixed element model by using shell elements over only a
portion of the blade, which could significantly reduce computational time compared
to that of a full shell model. Comparisons of natural frequencies and deflections
13
for the combined shell and beam models were made to both experimental data and
models consisting of only beam elements.
Analysis using 3-D finite element models for the blades has also been performed
for the aeromechanical response of the rotor. Ortun et al. [88] investigated the
ERATO rotor, a blade which features a double swept profile, using 3D hexahedral
elements for the blade in the finite element code MSC/Marc. The 3-D structural
model was coupled with a 3-D computational fluid dynamics (CFD) model to compare
with a high-speed level flight case (at advance ratio ? = 0.423) from a wind tunnel
experiment. Yeo and Truong et al. [89, 90] also studied the use of 3-D FEA for rotor
analysis using MSC/Marc and comparing the rotating beam frequencies to an RCAS
beam analysis. Most recently, a 3-D finite element method (FEM) solver called X3D
has been developed for the structural analysis of rotor blades based on the work by
Datta and Johnson [91,92]. The code was further refined by Staruk et al. [93] and
coupled with both low-order aerodynamic models as well as a full CFD model to
study the NASA tiltrotor aeroacoustic model (TRAM). Additional studies using the
X3D solver have also been conducted by Ward et al. [94] using a notional UH-60A
like rotor with a free-wake aerodynamics model.
Although these higher-order models have the potential to improve the fidelity
of the analysis, most of these models are still being developed. Moreover, since the
rotor, which dominates the dynamic response of the entire aircraft, can often be well
captured using beam elements in a multibody formulation, at a significant compu-
tation cost savings compared to higher-order structural models, the comprehensive
analysis tools such as Dymore, CAMRAD II, RCAS, and UMARC are still widely
used today. The fact that beam-based comprehensive analyses are still predominately
used and the interest in advanced geometry blades with structural coupling further
underscores the need for more accurate cross-sectional stiffness properties to improve
the accuracy of the aeromechanics calculations.
14
1.3 Sectional Stiffness Measurements
While beam theory is well suited for the structural dynamics analysis of
rotor blades, determining the sectional stiffness properties for use in these codes
is a challenging task. The fact that the blades are typically made of anisotropic
materials and present cross-sectional geometries, often composed of multiple internal
components, can present difficulties in defining an accurate model for use in numerical
cross-sectional analysis tools. Moreover, variations in the material properties that
arise from the manufacturing and curing processes may not be well known. In view
of these difficulties, an experimental measurement of the sectional stiffness properties
seems desirable ? to ensure that the blade cross-section is being modeled correctly
and provide a more accurate analysis.
Traditionally, stiffness properties are calculated based on a measurement of
global parameters, such as deflection or frequency, which are then related to the
stiffness using a beam model, usually from Euler-Bernoulli beam theory. The main
existing measurement techniques can generally be categorized into three groups:
deflection based, rotation based measurements, and frequency based measurements,
with the first two measurement types being static (or quasi-static) and the last one
being dynamic.
1.3.1 Deflection Based Measurements
Deflection based measurements rely on a measurement of the beam?s deflection,
normally at either the tip or the mid-span, to calculate the bending stiffness properties,
with two simple approaches based on this measurement illustrated in Fig. 1.2. The
setup in Fig. 1.2(a) measures the tip displacement wtip of a cantilevered blade of
length L under a static tip force F , while the setup in Fig. 1.2(b), referred to as a
three-point bending test, measures the mid-span displacement wmid of a cantilevered
15
beam simply supported at both ends with a force F applied at the mid-span. From
Euler-Bernoulli theory, the curvature of the beam can be related to the second
derivative of the beam?s deflection (?(x) = d2w(x)/dx2) and the cross-sectional
stiffness can then be found using
d2w
M(x) = EI (1.6)
dx2
where M(x) is the cross-sectional moment due to the applied force, and EI is
the bending stiffness. Equation (1.6) must then be integrated twice to obtain a
relationship for the bending stiffness in terms of the displacement. Normally, the
properties must be assumed to be uniform along the span, so that EI can therefore
be brought outside the integration. For the case of the cantilevered beam setup, the
bending moment is M(x) = F (L? x) and the displacement and slope at the root
are both zero, w(x = 0) = 0 and dw(x = 0)/dx, respectively, and the cross-sectional
bending stiffness can then be determined by
FL3
EI = . (1.7)
3wtip
For the case of the three-point bending setup, the bending moment over the
first half of the beam (0 < x < L/2) is M(x) = ?Fx/2 while the displacement at
the root and the slope at the mid-span of the beam are both zero, w(x = 0) = 0 and
dw(x = L/2)/dx, respectively, and the cross-sectional bending stiffness can then be
determined by
FL3
EI = . (1.8)
48wmid
In the case of the cantilevered setup, the bending stiffness is derived under
the assumption that the slope at the root is zero; however, it can be difficult to
achieve a perfect clamp in an experimental setup, particularly if the blade is very
16
Neutral
Axis
(a) Cantilever setup
Neutral
Axis
(b) Three-point bending setup
Figure 1.2: Illustrative setups for measurement of bending stiffness
based on deflection.
stiff. In this case, the slope boundary condition may no longer be satisfied, resulting
in a measured stiffness that is lower than the expected value. For the three-point
bending setup, it is easier to enforce only a displacement boundary condition at both
ends and errors in the simply supported boundary condition (that M(x) = 0 at the
boundaries) have less of an impact on the measured properties, making it better
suited for measurement of large bending stiffnesses. In the past, these methods
have primarily been applied to measure the bending stiffnesses of the HART I and
HART II rotors in the outboard region of the blade, starting at about 20% of the
radius, where the properties are assumed to be uniform. For the HART II blade [14],
the flap bending stiffness was measured using a cantilever setup with an optical
measurement of the tip displacement. A three point bending setup was also used to
measure the flap and chordwise bending stiffnesses of the Hart I rotor [13], as well
as the chordwise bending stiffness of the HART II rotor [14].
17
1.3.2 Rotation Based Measurements
Rotation based stiffness measurements rely on a measurement of the cross-
sectional rotations, either of the slope to calculate the bending stiffness or of the
twist to calculate the torsional stiffness. One simple method for calculating the
rotation is to measure the displacement at multiple locations on the blade. In the
case of a bending stiffness, the displacement w can be measured at multiple locations
along the span using a cantilever setup similar to Fig. 1.2(a) and the slope can be
calculated between measurement locations using a finite difference approximation
according to
dw ? w(x2)? w(x1)= w (x) = (1.9)
dx x2 ? x1
where x1 and x2 are the spanwise locations of the measurements. The bending
stiffness can then be determined from a single integration of the Euler-Bernoulli
moment-curvature relation (Eq. (1.6)) which gives
Fx(L? x/2)
EI = ? (1.10)w (x)
where x = (x2 + x1)/2. If the slope is measured at the tip, the stiffness can be
approximated by EI = FL2/(2w?(L)).
Figure 1.3: Illustrative setup for measurement of torsional stiffness
based on measured rotation angle ?.
In a similar fashion, the torsional stiffness can also be measured by applying
a torque and measuring the displacement at multiple locations along the chord.
18
While the twist angle could be approximated using a measurement of displacement
at a single location along the chord, the torque is normally not applied as a force
couple but by a single force, offset along the chordwise direction as shown in Fig.
1.3, so a single displacement measurement would not be able to eliminate the flap
displacement arising from the non-zero vertical force. By obtaining displacement
measurements at two different chord locations and making the assumption that the
cross-section deforms rigidly, the flap displacement can be filtered out and the twist
angle ? measured positive nose-up ? can be calculated according to
( )
?1 w(y1)? w(y2)? = tan (1.11)
y1 ? y2
where ? is the rotation angle and y1 and y2 the chordwise measurement locations.
The torsional stiffness GJ is related to the applied torque M1 = Fr and the twist
rate ?1(x) = d?/dx using Eq. (1.3) ? namely GJ = M1/(d?/dx). If the blade is
fixed at its root (?(x = 0) = 0), this relationship can be integrated along the span
to give the torsional stiffness in terms of the measured twist angle as
Frx
GJ = . (1.12)
?
This method, using a slope calculated based on measured displacements, has been
applied to measure both the bending and torsional stiffness of several Mach scale
rotors at the University of Maryland using a digital height gauge [95,96]. Photogram-
metric techniques, which capture the blade motion using cameras, can also be used
to measure the displacement at a finer resolution compared to a height gauge. A
Vicon motion capture system was used to measure the torsional, flap bending, and
chordwise bending stiffness of a Mach scaled NACA 0012 blade at the University of
Maryland [97]. Digital Image Correlation (DIC), another type of photogrammetric
method, was used to measure the flap bending and torsional stiffness of the Hart II
19
rotor in the region near the root (up to 0.22R) where there were significant variations
in properties due to taper and geometry changes [14]. DIC was also used to measure
blade deformations and calculate the torsional as well as flap and lag bending stiffness
in the reinforced root section of Mach scaled VR-12 rotors at the University of Texas
at Austin [98].
An alternative approach to measure the rotation is through the use of the
?mirror method,? in which mirrors are attached to the surface of a beam and laser
light is reflected off the mirror to measure rotation under loading as shown in Fig.
1.4 for a cantilever beam in bending. The deflection of the reflected laser light
? is then measured on a target surface at a distance d from the blade, normally
containing a grid for easy measurement, and used to calculate the reflection angle
? = tan?1 (?/d). In the case of bending, the beam slope is equal to half the reflection
angle (w? = 0.5?), and for an applied torque, the twist is also equal to half of the
reflection angle (? = 0.5?). These measured angles can then be substituted into
Eq. (1.10) and Eq. (1.12) to determine the bending stiffness and torsional stiffness,
respectively. This technique was first applied for the measurement of the torsional
stiffness of composite I-beams by Chandra and Chopra [99]. It has also been applied
for measurements of the torsional, flap bending, and chord bending stiffness values of
Mach scaled VR-12 coaxial rotors at the University of Texas at Austin [98,100]. In
this case, significant differences were observed between the calculated and measured
torsional stiffness values, although a simplified numerical model was used to calculate
the properties. A rigorous uncertainty analysis was also applied to the measurement
of the stiffness properties, which is largely neglected in most other studies.
The mirror method helps reduce some of the errors that can arise from an
imperfect experimental condition compared to a deflection based method; however,
the equations as written still require the assumption that the properties are uniform
along the span if only a single mirror is used in the measurement. If multiple mirrors
20
Neutral
Axis
mirror
laser light
Figure 1.4: Illustrative setup for measurement of bending stiffness
based on the ?mirror method.?
are included along the span and in close proximity to one another, the bending
curvature can instead be approximated using a finite difference method with the slope
measurements of two adjacent mirrors and the bending stiffness can be calculated
using
P (x2 2
EI = 2
? x1) (1.13)
?2 ? ?1
where x1 and x2 indicate the adjacent locations of two spanwise mirrors and ?1 and
?2 the measured laser reflection angle at those locations. The twist rate can also be
approximated in a similar manner between two mirrors to give an updated torsional
stiffness of
2Pr(x2 ? x1)
GJ = . (1.14)
?2 ? ?1
This modified approach has been applied to measure both the torsional and flap
bending stiffness of the Hart I rotor in the region near the root (up to 0.22R)
where there were significant variations in properties due to taper and geometry
changes [13]. This modified ?mirror method? approach relaxes the assumption of
uniform properties along the span, as integration of the Euler-Bernoulli bending-
curvature relation is no longer required, as well as essentially eliminating inaccuracies
arising from imperfect boundary conditions, since the difference of the two measured
slopes will eliminate any rigid body motion present in both measurements. However,
21
the implementation and data collection process using this modified ?mirror method?
for an entire blade can be quite difficult and time consuming due to the large number
of mirrors required and extensive data tracking requirements. In addition, the most
accurate results require a large measurement distance d from the blade, which may
not always be feasible.
1.3.3 Frequency Based Measurements
The previous techniques focused on static deformation measurements for cal-
culating the stiffness properties, but dynamic measurements can also be used to
measure the stiffness properties. The blade frequencies can be measured under
different excitations and boundary conditions, with a shaker commonly used to excite
a cantilever blade as shown in Fig. 1.5. The oscillations of the beam under excitation
must then be recorded, through devices such as laser vibrometers or accelerometers,
and a Fast Fourier Transform (FFT) can be used to extract the natural frequencies.
Typically the blade frequencies are measured since the rotor dynamics play such an
important role in the aeromechanics, regardless of whether the frequencies are used
to calculate the stiffness. Based on the extracted frequencies, the assumption that
properties are uniform along the span, and assumptions on the mass and inertia
distributions, the stiffness can then be calculated from the frequency according to
?2mL4
EI = b
f 4i,b
(1.15)
?2I?L
2
GJ = t
f 2i,t
where ?b and ?t are the non-rotating natural frequencies for bending and torsion,
respectively, m is the mass per unit span, I? the mass moment of inertia per unit
span, and f1,b = 1.87 and f1,t = ?/2 for the first natural frequencies with a cantilever
boundary condition. Because frequency measurements are highly impacted by
22
boundary conditions and require assumptions on the mass distributions, they have
primarily only been used to measure stiffness values in cases when a static test is hard
to perform, such as for the chordwise bending stiffness of the NACA 0012 blade at the
University of Maryland [96]. In addition, there are no simple modifications that can
be performed to account for spanwise variations and frequency based measurements
are only realistically suited for blades with uniform property distributions.
Laser 
Vibrometer
Magnetic 
Shaker
Figure 1.5: Illustrative setup for measurement of sectional stiffness
based on measured frequencies using a shaker and laser vibrometer.
1.3.4 Other Notable Measurements
In addition to the above methods, there have also been a few special purpose
efforts to measure stiffness properties. Jung et al. [101] used an X-ray CT scan
technique to determine the post manufactured geometries and surface boundaries of
the HART II rotor, which were then used in a cross-sectional analysis to evaluate the
cross-sectional stiffness properties and compare with the experimental measurements
from the static discrepancies. While this technique drastically reduced discrepancies
between the actual blade and numerical model in terms of the geometric description
of the cross-section, uncertainties can still exist in the material properties which can
impact the predicted sectional stiffness values. In the wind turbine industry, Fedorov
and Berggreen [74] analyzed the torsional, flap bending, and flap bending/torsion
coupled stiffness values by fitting displacements and rotations measured using DIC
with polynomials up to order 12 and then calculating the curvature and twist rate
23
from the fitted deformations. This is the only known attempt to measure the actual
coupling stiffness for a composite beam, but the equations were specifically derived
under the Euler-Bernoulli assumptions.
1.3.5 Limitations of Existing Measurement Techniques
Previous experiments have focused on measuring the sectional stiffness prop-
erties of beams primarily through static deformations or frequencies. While these
techniques have proven effective, they also have multiple limitations. The first, and
most fundamental, drawback is that none of the existing measurement techniques
can provide the full 6? 6 stiffness matrix. Existing methods have only been used to
calculate torsional and two bending stiffness values and, in the special case of the
wind turbine study by Fedorov and Berggreen [74], also the flap bending/torsion
coupled stiffness. While these stiffness values are generally adequate for an accurate
prediction of the rotor dynamics, they do not provide sufficient information for an
accurate calculation of the stress and strain fields within the blade. Indeed, the
rotational environment of the helicopter rotor generates significant axial strains due
to the centrifugal forces, necessitating an accurate measure of the axial stiffness to
predict.
In addition, most existing measurement techniques rely on the assumption
that the stiffness properties are uniform along the span. While this assumption is
normally true of model scale rotors used for wind tunnel testing, it is not true of most
full-scale helicopter rotors. Finally, when using a single global parameter to measure
stiffness, inaccuracies in the experimental and theoretical boundary conditions can
significantly impact the results; however, several of the methods have been extended
to include multiple measurements along the span to help filter out any rigid body
motion.
24
1.4 Strain Measurement Techniques
Development of a methodology that provides a more complete and accurate
measurement of the beam sectional stiffness requires more than just a measurement of
frequencies or static displacements. Measurement of the strain field within the beam
provides an alternative approach to existing measurement methodologies and can help
address some of their shortcomings. Strain is a measure of the differential deformation
in a structure; while a rigid body motion results in a observable displacement, it
produces no strain and imperfect boundary conditions will have a minimal impact on
the strain and therefore stiffness property measurement. In addition, all six sectional
deformation measures given by E can be related to the strain field, meaning that the
full 6? 6 stiffness matrix can be calculated from the strain field, as opposed to just a
subset of the matrix. Finally, since strain is a local measurement within the beam, it
can be used to predict spanwise variations in properties that most existing methods
aren?t capable of handling. Various techniques exist for measuring the strain field
and can be broadly classified as contact based or non-contact based methods.
1.4.1 Contact Based Strain Measurements
Contact based techniques are those that require direct contact with the sur-
face of a test article in order to obtain the strain measurements. This includes
instrumentation such as strain gauges, extensometers, wire sensors, or fiber optic
sensors. Strain gauges, due to their relatively low cost and versatility, are probably
the most widely used strain measurement technique in aerospace applications. Strain
gauges measure the change in resistance under an applied load to determine the
strain and commonly come in two varieties: metallic foil gauges and semiconductor
gauges [102]. Metallic foil gauges make use of a thin metallic wire looped back and
forth and attached to an insulating film, as shown in Fig. 1.6(a), and are bonded to
25
a structure using an adhesive. When subjected to a tensile strain, the thin metallic
wire stretches, decreasing its cross-sectional area thereby increasing its resistance;
conversely, when subjected to a compressive strain, the metallic wire contracts,
increasing its cross-sectional area and decreasing its resistance. Semiconductor strain
gauges, shown in Fig. 1.6(b), consist of a bar or strip of a semiconductor, normally
silicon or germanium, whose peizoresistive properties result in a change in resistance
under deformation. Semiconductor strain gauges are much more sensitive than metal-
lic foil gauges making them well suited for measurement of small strains, but are
also significantly more expensive, are extremely temperature-sensitive, and typically
more fragile. While both gauge types are designed to measure the strain in only the
longitudinal direction, a strain rosette, with an example layout shown in Fig. 1.6(c),
can be used to measure the shear strain and lateral strain.
Strain gauges have been used extensively within rotorcraft applications. In
1993-1994, full scale testing of the UH-60a airloads included 21 strain gauges for
measuring the flap, lag, and torsional moments [103]. Subsequent testing of the
same rotor system in 2010 included 28 strain gauges for measuring the structural
loads [104]. For model scale rotors, strain gauges have also seen extensive use,
including multiple tests at the University of Maryland to measure the blade loads in
bending/torsion coupled blades [73] and in high advance ratio testing [96,105,106].
Although strain gauges have seen significant applications in rotorcraft, their use
for measurement of the sectional stiffness properties can be problematic. Measurement
of the detailed strain field along the span of the blade would require a large number
of gauges, especially if stiffness measurements are desired for any non-uniform section
of the blade. While an individual gauge typically costs $5 to $10, instrumenting the
large number of required gauges would end up being both costly and very labor-
intensive. Moreover, if the blade is to be tested in a wind tunnel or flown on an
actual helicopter, the inclusion of a large number of gauges could significantly impact
26
(a) Metallic foil strain gauge (b) Semiconductor strain gauge
(c) Strain gauge rosette
Figure 1.6: Illustrative strain gauges.
the aerodynamic performance of the rotor. Finally, measurement of strain in the
rotational environment typically requires a slip-ring to pass electrical signals from the
rotating to non-rotating systems and, because slip-rings introduce additional noise
and have a limited number of signal channels, would therefore limit the practicality
of utilizing all the strain gauges for additional testing.
An alternative method to strain gauges is distributed wire sensors, which are
shape memory wires that are normally embedded into the structure. The changes in
electrical resistance are measured at multiple points along the wire sensor, as shown
in Fig. 1.7, and the displacement and strain field within the beam are interpolated
from these measurements using a finite element formulation. Baz and Poh [107]
devised a wire sensor for measuring distributed bending deformations and strains.
27
Copp [108] extended this to measure the elastic twist of a rotating blade, using
a sawtooth shaped wire sensor. While the embedded nature of distributed wire
sensors is aerodynamically advantageous, these wire sensors require Euler-Bernoulli
assumptions with a finite element formulation to calculate the strains and have seen
limited use in rotorcraft.
Figure 1.7: Schematic setup of distributed wire sensor.
Fiber optic sensors can also be utilized as a distributed strain sensor, usually
through the use of a fiber bragg gratings as shown in Fig. 1.8. As the structure
deforms, the fiber, which is either attached to the external surface or embedded
internally if using composite materials, also deforms. This deformation modifies the
wavelength of the transmitted light which can be directly related to the strain. In
2007, Liu et al. [109] developed a fiber optic strain and pressure sensor for use in
rotors and obtained strain field measurements under centrifugal loading. Fairbanks-
Smith [110] also used fiber-optic strain sensors on a section of a CH-47 rotor blade
to study the impact of ballistic damage, finding good correlation between strain
measurements using a fiber optic sensor and conventional strain gauges. Unlike the
previously mentioned contact based strain sensors, fiber optic strain sensors do not
require an excitation voltage to operate which drastically simplifies their installation.
Fiber-optic sensors are also very lightweight and small and therefore will have a
minimal impact on the blade mass and stiffness properties; however, a network of
fiber optical sensors is still required to get a detailed strain field.
28
Figure 1.8: Example of Fiber Bragg Gratings.
1.4.2 Non-Contact Based Strain Measurements
Non-contact techniques for measuring strains have significantly advanced in
the past several decades due to the advent of the digital camera and increased
computer capabilities. Non-contact techniques have the capability for a much
higher spatial resolution than conventional contact based strain measurements,
which is advantageous for measuring spanwise distributions of stiffness properties.
Many varieties of these optical methods exist, including electronic speckle pattern
interferometry (ESPI), projection Moire? interferometry (PMI), and digital image
correlation (DIC). Note that all optical methods directly measure the object contour
and 3-D displacements; strain is then calculated from this information using an
approach similar to 3-D finite element analysis.
Electronic speckle pattern interferometry (ESPI), uses the fact that when
a rough surface is illuminated with laser light, a high contrast speckle pattern is
observed, as shown in Fig. 1.9(a). An interferogram is then created by superimposing
a reference beam from the same laser source on the speckles. When the structure
is subjected to a load, the surface deforms and the observed speckle interferogram
changes; comparison of the interferograms before and after applying the load generates
a fringe pattern, as shown in Fig. 1.9(b), which provides a measure of the 3-D
displacement that is subsequently used to calculate the strain field on the surface of
the blade. Originally the interferograms were recorded using TV cameras; however,
it is now common to collect digital images using a CCD or CMOS camera and the
method is also referred to as digital speckle pattern interferometry (DSPI) [1]. For
rotorcraft applications, ESPI has primarily been used for nondestructive inspection;
29
Gryzagoridis [111] used ESPI to detect composite de-lamination in a rotor blade.
(a) Example speckle pattern (b) Example fringe pattern
Figure 1.9: Example of results produced using ESPI (from Rastogi
2015 [1]).
Projection Moire? interferometry (PMI) is another optical interferometric mea-
surement method that uses an incoherent light source, normally pulsed laser diodes,
to project a grid of parallel lines onto the test object. Images of the projected
grid lines are then captured in both a baseline undeformed state and a deformed
state under loading, which are compared to a computer generated phase shifted
grid, resulting in interferograms with Moire? fringes. These fringe patterns are then
digitally processed to measure the object contour and displacement field. PMI has
been used in a variety of rotorcraft studies. Fleming and Gorton [112] applied PMI
on a four-bladed, Mach-scaled rotor to measure mean and unsteady blade bending
and twist deformations. Subsequent testing by Fleming et al. [113] on active twist
rotor blades measured the spanwise distribution of flap bending deformation at
multiple azimuthal locations and the tip twist under various actuation amplitudes.
Sekula [114] measured blade deflections using PMI at several different thrust levels
and heights for a rotor in ground effect. While PMI has seen more extensive use
in rotorcraft applications than ESPI, both methods have primarily been used to
30
measure the displacements and not strains.
Alternatively, stereoscopic digital image correlation (DIC) can be used to
measure the 3-D displacements on the surface of an object. DIC uses pairs of cameras
to capture images of an object in an undeformed reference state and a deformed
state; displacements are then calculated by cross correlation of a stochastic speckle
pattern using stereo photogrammmetry. Unlike the interferometric methods, which
generate their patterns optically using lasers, the speckle pattern for DIC is directly
applied to the surface, usually using paint. Of the non-contact, optical measurement
methods, DIC has been used most extensively in rotorcraft applications. Sirohi
and Lawson [115] used DIC to measure the flap and twisting deformation, showing
good correlations with measurements obtained using both a laser displacement
sensor and inclinometer. Sicard and Sirohi [116] extended this study to measure
the extension, lag and flap bending, and torsional deformations of an extremely
flexible spinning rotor, showing that the elastic twist deformation was on the same
order of magnitude as the blade pitch. Uehara and Sirohi [117] then applied DIC
to measure the deformation of a rotating blade, extracting the mode shapes and
frequencies from these measurements using operational modal analysis. For full-scale
rotorcraft, Annett and Littell [118] used DIC to measure the structural response
of two CH-46E airframes during impact testing. The 3-D displacement field was
measured throughout the impact and the residual strains from the impact were also
evaluated.
Optical non-contact measurement techniques provide significant advantages
over traditional contact based strain measurements for evaluating the sectional
stiffness. Minimal preparation is required for measurement of strains using optical
techniques, meaning that the blades can still be used for wind tunnel or flight
testing, without significantly impacting its aerodynamic profile or structural integrity.
In addition, optical measurement techniques provide very high spatial resolution,
31
which is desirable for predicting any spanwise variations in properties. Of the
optical measurement methods, DIC provides a good tradeoff between cost and
strain measurement capabilities and has seen the most widespread use in rotorcraft,
including to measure stiffness properties (based on displacements) and the modal
characteristics of rotating blades, and was thus selected for obtaining the strain
measurements in this work.
1.5 Objectives
The overall objective of the present work is to develop a new measurement
technique for a more complete characterization of the sectional stiffness properties
? to provide better accuracy in the structural loads predicted from comprehensive
analysis codes and the resulting stress and strain fields. The technical approach to
accomplish this can be categorized into the following tasks.
? Develop a new strain-based methodology for measuring the sectional stiffness
properties. This is the main objective of the present dissertation. By using
a strain-based method, as opposed to the displacement or frequency based
methods commonly applied in rotorcraft analysis, the dependency on an
accurate experimental implementation of the expected boundary condition can
be significantly reduced. Moreover, because the strain is a local property within
the beam, this allows the stiffness properties to be measured at discrete radial
locations along the blade which can be used to measure spanwise variations
arising from blade attributes like taper and twist. However, an accurate
measurement of the stiffness properties, particularly of the shear and torsional
terms, using a strain-based approach also requires the definition of a warping
field, which generally needs to be calculated numerically.
? Develop and implement an uncertainty analysis for error estimation. An
32
uncertainty quantification based on a Taylor series method is used to propagate
expected errors in the experimental measurements into the calculated stiffness
properties. The implementation is compared with a Monte Carlo simulation to
verify the accuracy.
? Verify the methodology and implementation. Since a new experimental method
is being explored, the data reduction equations are verified using numerical
strain data from 3-D FEM. This eliminates any unexpected errors that may
arise from experimental measurements and evaluates the methodology under
ideal circumstances.
? Implement an experimental setup for measuring the strains and loads for model
scale rotor blades. A test stand is built to apply six independent load cases of
sufficient magnitude to generate strains with large signal to noise ratios. Since
DIC is used to measure the displacement and strain fields and requires a view of
the surface of the beam for measurement, a system of pulleys is used to generate
loads to keep the view of the beam unobstructed. The mounting adapters
to the test stand and for load application are kept modular to accommodate
beams with a variety of cross-sectional geometries.
? Apply the methodology to experimentally measure the stiffness properties of
various beams. The methodology is evaluated using a variety of test articles,
specifically beams that present elastic coupling terms, beams with spanwise
variations in properties, and model scale composite rotor blades. The ex-
perimental results are then used to provide a comparison with existing 2-D
cross-sectional analysis tools.
33
1.6 Contributions
With these objectives in mind, the following are the key contributions of the
present work
1. Development of a new strain-based method for measuring the cross-sectional
stiffness properties of composite rotor blades. This differs from the conventional
techniques utilizing displacement or frequency measurements.
2. Measurements of the full 6 ? 6 sectional stiffness matrix. Existing measure-
ment techniques focus on only a subset of the matrix and generally include
measurements of only the torsional and flap and lag bending stiffnesses for the
blade.
3. Comparison of results from proposed method to existing displacement based
measurement techniques. This shows some of the advantages that a strain-based
methodology can provide.
4. Detailed uncertainty analysis for predicting expected errors of the stiffness
property measurements. Previous studies have only focused on the measurement
of the stiffness properties itself without focusing on error estimation.
5. Experimental evaluation of 2-D cross-sectional analysis tools, specifically Sec-
tionBuilder. In the past, 2-D cross-sectional analysis codes (both SectionBuilder
and VABS) have primarily been validated with analytic solutions and 3-D
FEM.
6. Detailed strain measurement data sets for model scale composite rotors. These
data sets can be used to validate the stress and strain predictive capabilities of
both 2-D cross-sectional analysis codes as well as 3-D FEM tools, such as X3D.
34
1.7 Organization of the Dissertation
The present chapter provided the motivation for the current work and an
overview of the existing methodologies for predicting and measuring the sectional
stiffness matrices of composite rotor blades, needed for an accurate comprehensive
analysis. Chapter 2 describes the development of a new methodology, combining both
experimental and numerical data, to improve on the measurement of the sectional
stiffness properties. This chapter summarizes the key aspects of this methodology
and highlights the information needed for an accurate measurement. The pro-
posed approach combines experimental data with the warping field predicted by
SectionBuilder to eliminate restrictions on the cross-sectional geometry or materials.
Chapter 3 focuses on validation of the new methodology using strain data
simulated from 3-D finite element models. Models were specifically selected to address
some of the key features of composite rotor blades, including spanwise variations
in properties, elastic coupling, and twist. Chapter 4 then presents a detailed error
analysis for providing uncertainty quantification on the experimental measurements
of the stiffness properties.
Chapter 5 discusses the experimental setup for obtaining the strain and force
measurements needed to calculate the local, sectional stiffness properties. Results of
simple calibration procedures are discussed, to assess the expected errors for each
of the measurements. Chapter 6 then details the stiffness measurements obtained
for various beams using this experimental setup, starting with results for simple
isotropic cross-sections and moving up to more complex composite blades. Finally,
Chap. 7 gives a summary of the present work, provides the key conclusions and
suggestions for future work.
35
Chapter 2: Methodology
This chapter focuses on the mathematical and numerical methodology for the
presented dissertation research. The governing equations for the cross-sectional
analysis are first presented, followed by a discussion of the process to calculate the
stiffness matrix from these equations.
2.1 Cross-Sectional Analysis
The cross-sectional stiffness matrix, K, provides a direct relationship between
the sectional deformation measures F and the sectional stress resultants E , as defined
by Eq. (1.4) (F = K E). While the sectional stress resultants can be measured
experimentally, Eq. (1.4) can not be used to directly calculate the stiffness matrix as
the sectional deformation measures can not be obtained experimentally. Instead, this
relationship must be rewritten in terms of a measurable quantity, the local strains,
through the use of a numerical model. For this thesis, the cross-sectional analysis
tool SectionBuilder was used for the numerical model.
SectionBuilder is a finite element based tool for the analysis of cross-sections
with arbitrary geometry made of anistropic materials, based on a solution approach
of Saint-Venant?s problem using Hamilton?s formalism developed by Bauchau and
Han [41, 60]. It provides an exact solution of the 3D theory of elasticity under
several assumptions: (1) small strains and small warping displacements; (2) uniform
geometry and material properties along the span of the beam; (3) cross-sectional
dimensions are much smaller than the beam?s span. The first two assumptions
36
imply that the beam can be analyzed using Saint Venant?s beam theory, while the
third assumption implies that extremity effects are negligible [119]. A full discussion
of the derivation of the solution process for SectionBuilder can be found in [120];
the following subsections will introduce the key variables and highlight the main
principles needed for providing the relationship between the strains, stiffness matrix,
and sectional stress resultants.
2.1.1 Beam Configuration
In aerospace applications, a beam will generally have a cross-section of arbitrary
geometry and may be initially curved and twisted for aerodynamic purposes, as
depicted in Fig. 2.1. The beam is defined by an arbitrary curve in space, denoted C,
which forms a reference line along the span of the beam. The total length of curve C
is L, with the curvilinear coordinate ?1 defining the intrinsic parameterization of
the curve.
(a) Full beam configuration (b) Cross-sectional configuration
Figure 2.1: Configuration of a naturally curved and twisted beam.
[ ]
The cross-sectional configuration is defined by frame FB = B,B = (b?1, b?2, b?3) ,
where point B lies at the intersection of the cross-sectional plane with the reference
line of the beam. The plane of the cross-section is therefore determined by the two
mutually orthogonal unit vectors, b?2 and b?3. A point within the cross-section can
be defined using the material coordinates ?1, y, and z, where y and z measure the
37
location along b?2 and b?3, respectively.
In general, the orientation of the cross-section will change as it moves along
curve C due to the natural twist and curvature of the beam and basis B will therefore
be a function of ?1. The location of point B with respect to the reference frame
FI = [O, I = (??1, ??2, ??3)] is defined by rB and the rotation tensor that brings basis I
to basis B is R(?1). The motion tensor that brings frame FI to frame FB is then
defined by
??? R r?BR ?
?
C = ? (2.1)
0 R
where r?B is the skew-symmetric matrix defined by
?? ??? 0 ?rB,3 rB,2 ??r?B = ?? rB,3 0 ?r ?B,1 ?? (2.2)
?rB,2 rB,1 0
with rB,i denoting the i
th component of rB. The components of the beam?s curvature
tensor in frame FB are then defined from the motion tensor by
K? ?1 ? ?
? ?
? k? t? ?= C C = ? (2.3)
0 k?
where notation (?)? indicates a derivative with respect to ?1, k is the curvature
vector of the beam, and t? is the unit tangent vector to curve C. For a straight,
untwisted beam, the unit tangent vector is t? = {1, 0, 0}T and the curvature vector is
k = {0, 0, 0}T . For an initially curved or twisted beam, the curvature vector becomes
k = {?1, ?2, ?3}T , where ?1 is the twist rate, ?2 the curvature about axis b2 (i.e. the
anhedral or dihedral), and ?3 the curvature about axis b3 (i.e. sweep).
38
2.1.2 Stress and Strain Components
In the formulation of SectionBuilder, the strains are defined in terms of the
Green-Lagrange strain tensor denoted by
??? ??? ?
? = ?? O ?? (2.4)?
I
where ?T = {?11, 2?12, 2?13} are the out-of-plane strain components and ?T =O I
{?22, ?23, 2?23} are the in-plane strain components. The strain components can be
written in terms of an arbitrary displacement field u(?1, y, z), which describes the
relative displacement of a material point P within the cross-section, according to
1 1
? = ? u?+? D u (2.5a)
O g g O
1
? = ? D u (2.5b)
I g I
where D and D are differential operators defined by
O I
?? ? ?? ?
? ?
? d ?k3 k2 ??? ??? 0 g 0?y ??? ? ? ? ?
D = ???? k3 + g d ?k ?1 , D = ? 0 0 g ? (2.6)O ?y ??? I ??? ?z ???
? ? ?
? ? ? ?
k2 + g k1 d 0 g g
?z ?z ?y
?
with g = t1 ? k3y + k2z and d = ? (t2 ? k1z) ?(?)/?y ? (t3 + k1y) ?(?)/?z. For the
?
case of a straight untwisted beam, the expressions are simplified since g = 1 and
k1, k2, k3, and d are all zero. Using this notation, the Green-Lagrange strain tensor
from Eq. (2.4) can be written as
? = Au? +B u (2.7)
39
where ? ? ? ?
1 ?
A = ? ? I ?? 1 ? D ?, B = ? ? O ? (2.8)
g 0 g D
I
Although the strains must remain small at all times, due to the first assumption for
SectionBuilder, the beam can still undergo large displacements and rotations.
The{stress i}n the beam is represented using the Cauchy stress tensor, denoted
by ?T = ?T T To , ? I , where ? o = {?11, ?12, ?13} are the out-of-plane components and
? I = {?22, ?33, ?23} are the in-plane components. Assuming that the beam is made
of linearly elastic, but generally anisotropic materials, the constitutive relationship
between the Cauchy stress tensor and the Green-Lagrange strain tensor is defined as
? = D? (2.9)
where D is the 6 ? 6 material stiffness matrix resolved in the basis B.
When defining the constitutive relationships, the stress and strain tensors are
commonly partitioned by the normal {(?)11, (?)22, (?)33} and shear {(?)23, (?)13, (?)12}
components; however, because the current partitioning scheme uses out-of-plane and
in-plane components, it is important to make sure the material stiffness matrix is
defined correctly. When resolved in basis B?, whose axis are aligned with the axes of
the material, the material stiffness matrix for an orthotropic material can be written
as ? ?
??? E1 (1? ?23?32) 0 0 E1 (?21 + ?23?31) E1 (?31 + ?21?32) 0?? ?
?
? 0 G12? 0 0 0 0 ????
? 1 ???? 0 0 G13? 0 0 0D = ? ??
?
(2.10)
?
?? E1 (?21 + ?23?31) 0 0 E2 (1? ?13?32) E2 (?32 + ?12?31) 0??
???
E1 (?31 + ?21?32) 0 0 E2 (?32 + ?12?31) E3 (1? ?12?21) 0 ???
0 0 0 0 0 G23?
where ? = (1? ?23?32 ? ?13?31 ? ?12?21 ? 2?32?13?21) and the relationships ?21 =
40
?12E2/E1, ?31 = ?13E3/E1, and ?32 = ?23E3/E2 were used to simplify the matrix.
For a transversely isotropic material, such as a unidirectional layer of composite, the
material stiffness can be simplified using the fact that E3 = E2, G13 = G12, ?13 = ?12,
and G23 = E2/ [2 (1 + ?23)]. For an isotropic material, the material stiffness can
be further simplified using the fact that E1 = E2 = E3 = E, ?12 = ?13 = ?23 = ?,
and G12 = G13 = G23 = E/ [2 (1 + ?)]. Generally, the material basis B? and the
reference basis B are not aligned, and the material stiffness matrix must therefore be
rotated into the reference basis. Figure 2.2(a) shows an example cross-section made
of several layers of composite material. A rotation of ? about b1 brings the reference
basis B to the local basis E . As seen in Fig. 2.2(b), a second rotation of ? about
axis e?3 brings the basis E to the material basis B?, which is aligned with the fibers
of the composite material. The material stiffness matrix in reference basis B is then
D = R? D?R?T (2.11)
? ?
where
?? ?? C2? ? ?S
2
2? 0 S? 0 0 ?
???
??
C
? ?
S2? C?S2? ?
? C2?C? ?C?S? ? 0 S?S? ?? 2 2 ?? ?? S ??S2? S?S2???
C ?2?S? C?C? ? 0 ?C?S?
? 2 2 ??R = ?? ?? (2.12)? ? C2S2? ? ? C
2
?S2? ?S2?S C2? ?C2 S2? ? ?S2?C? ???
????
?
S2S2 S 2 2 2 2
?
? ? ? 2?
S? S2?S? C?S? C? C?S2? ???
S 22?S? S2?S
2
2? C?S2? S
?
2?
S?C2? ? C?C2?
2 2 2 2
with the shorthand notation S(?) = sin (?) and C(?) = cos (?)
41
Layer
(a) Cross-section composed of multiple
(b) Ply orientation in the ith
composite layers
layer
Figure 2.2: Cross-section showing rotation angles between the reference
basis B and material basis B?.
Figure 2.3: Discretized cross-section.
2.1.3 Semi-Discretization of the Displacement Field
In the previous paragraphs, the displacement and warping fields were treated as
general vectors depending on three independent variables, the curvilinear coordinate,
?1, and the cross-sectional coordinates, y and z. However, the equations governing
the displacement field in beam theory are one-dimensional, ordinary differential
equations that are a function of ?1 only. Therefore, to obtain a one-dimensional
formulation, the cross-section of the beam is discretized using two-dimensional
42
elements, as shown in Fig. 2.3. The displacement fields can then be written as
u (?1, y, z) = N (y, z) u? (?1) (2.13)
where N (y, z) are the two-dimensional shape functions used for the discretization
and u? (?1) stores the nodal values of the displacement field. This discretization can
then be introduced in Eq.(2.7) for the Green-Lagrange strain tensor, which gives
? = A u?? +B u? (2.14)
L L
where the strain interpolation matrices, A and B , are defined by
L L
?? ??? Ni 0 0? ????? 0 N 0
1 ?
i ????. . . 0 0 Ni . . .
A = AN = ? ?
L g ???? ?
? (2.15a)
???
0 0 0 ???
0 0 0 ???
0 0 0
?? ??? d ?k3Ni k2Ni??
?
? ?
? k3Ni + gNi,y d ?k1N ?? i ?1 ?? ?
??
? . . . ?k2Ni + gNi,z k1Ni d . . . ?B = BN = ? ? ?? (2.15b)L g ?? 0 gNi,y 0?? ? ?
???
0 0 gN ?i,z ?
? ? ?
0 gNi,z gNi,y
with Ni(?, ?) being the Lagrange shape functions, ? and ? being the local spatial
variables over the element, and Ni,y and Ni,z indicating the derivatives with respect
to y and z, respectively. Since the shape functions and derivatives are only non-zero
43
for the jth element, the strain interpolation matrices A and B are of size 6 ? 3n,
L L
where n is the number of nodes per element.
2.1.4 The Central Solution
Under the assumption that the beam?s span is much greater than the cross-
sectional dimensions, the extremity solutions, which decay exponentially with the
distance from the beam?s ends, can be neglected. The remaining part of the solution
is the central or ?Saint-Venant?s? solution, which propagates along the beam?s span
without decaying. The central solution is an exact solution of the linear theory of
3-D elasticity for beams with uniform geometric and material characteristics along
their span, with the accuracy only limited by the discretization used for the finite
element method.
The detailed derivation of the central solution can be found in Bauchau and
Han [41,60]; an important feature of the central solution is the fact that the nodal
displacements, u? can be expressed as
u? = Z UR +W F c (2.16)
where Z UR represents the rigid-body motion of the nodes an{d W F}c represents
the warping displacement. For the rigid-body motion, UTR = uT , ?TR stores theR
rigid-body displacement and rotation of the entire cross-section, respectively. Matrix
Z is a 3m? 6 matrix, with m being the total number of nodes in the model, storing
the nodal locations for all the nodes in the cross-sectional model, with the rows
corresponding to the ith node defined by
?? ??? 1 0 0 0 zi ?yi? ?Z =i ? ?0 1 0 ?zi 0 0 ??? (2.17)
0 0 1 yi 0 0
44
where yi and zi are t{he y and}z locations of the node, respectively. For the warping
displacement, FTc = F T ,MT denotes the stress resultants at the current apanwise
location, where F contains the three sectional force, consisting of the axial force and
two transverse shear forces, and M contains the sectional moments, consisting of the
twisting and two bending moments, all of which are resolved in the reference basis
B. Matrix W (y, z) is a 3m? 6 matrix, which stores the nodal warping field, with
each column of the matrix representing the warping induced by the respective unit
component of the sectional stress resultant.
For Saint-Venant?s problem, the governing equations reduce to
U ?R + K? UR = S F c (2.18a)
F ? ? K?Tc F c = 0 (2.18b)
where S is the sectional compliance matrix and the inverse of the sectional stiffness
matrix (i.e. K = S?1). Substituting Eqs.(2.16) and (2.18) into the discretized
version of the strain tensor given by Eq.(2.14) and using the fact that a rigid body
motion induces no strain (i.e. the coefficient of UR must vanish), the strain tensor
at any point in the beam is
[ ( ) ]
? (?1, y, z) = A Z S +W K?T +B W F c (?1) (2.19)L L
Equation (2.19) implies that, with knowledge of the nodal warping field, the complete
strain field at any point of the cross-section can be expressed in terms of the six
sectional stress resultants.
45
2.2 Data Reduction Equation
Equation (2.19) provides the basic relationship used to solve for the sectional
compliance matrix. By rearranging the expression on the right-hand side of Eq.
(2.19), the strain tensor can be expressed as
( )
? = A Z S F + A W K?Tc +B W F (2.20)L L L c
The first portion of the equation, ? = A Z S F c, provides a direct relationshipL
between the sectional strains and stress resultants, ? and F c, respectively, which
can both be experiment(ally measured and)the sectional compliance matrix, S. The
remaining expression, A W K?T +B W F c, is a correction factor dependent onL L
the nodal warping field. Since the warping field, which is dependent on both the
geometric and material properties of the cross-section, cannot be measured directly
and an analytical solution cannot be obtained for an arbitrary cross-section, it must
be determined numerically. SectionBuilder is used to calculate the warping field,
whose predominant role is to provide a correction to the shearing and torsional
stiffness properties that would otherwise not be able to be determined experimentally.
Thus, with knowledge of the sectional strains and stress resultants and the nodal
warping field, the sectional compliance matrix, and therefore sectional stiffness
matrix, can be calculated at discrete spanwise locations. However, several additional
steps must be performed to put Eq. (2.20) into a form solvable for the compliance
matrix.
2.2.1 Rotation of the Strain Components
In the current work, DIC is used to measure the deformation and strain on the
external surface of the beam, with the strain measured in the directions normal and
46
tangent to the surface, denoted by basis E = (e?1, e?2, e?3). Consider two representative
cross-sections, as shown in Fig. 2.4. For the triangular cross-section, shown in Fig.
2.4(a), there are three discrete edges, whose tangent bases E are all generally rotated
with respect to the reference basis B. For the airfoil shaped cross-section, shown
in Fig. 2.4(b), the external profile is a continuous curve whose tangent basis E is
constantly changing orientation around the cross-section. Since the strain given
by Eq. (2.20) is in basis B, it is necessary to bring the data into a common basis.
While strain measurements using DIC are the focus of this thesis, other common
measurement techniques, such as strain gauges, also output strains in basis E and
the change in basis will still be necessary, regardless of the measurement technique.
(a) Triangular (b) Airfoil cross-section.
cross-section.
Figure 2.4: External edge bases for example cross-sections.
The transformation between bases B and E consists of single rotation of ?
about axis b?1, as shown in Fig. 2.4. The rotation tensor that brings basis B to E is
therefore defined by
?? ??? 1 0 0? ?? ?R = 0 C? ?S ?? ?? (2.21)?
0 S? C?
and the strain rotation can be expressed as
?? ? ? ?? ?+11 ?+ +12 ?13 ? ? ?11 ?12 ?13 ???? ?? ??+ ?+ ?+ = RT ? ??R (2.22)12 22 23 ?? ? ?? ?12 ?22 ?23 ?? ?
?+ +13 ?23 ?
+
33 ?13 ?23 ?33
47
where notation (?)+ indicates the strain components resolved in basis E . However,
Eq. (2.20) uses the strain tensor in vectoral form and in terms of the engineering
shear strain components instead of the tensor shear strain components (i.e. in terms
of 2?12, 2?13, and 2?23). By expanding the product on the right-hand side and
rearranging into a vector equation, the relationship between the strain vector in basis
B and basis E can be expressed as
?+ = R?1? (2.23)

where ? ?
??? 1 0 0 0 0 0? ???? 0 C? ?S? 0 0 0 ??? ?? ?
?
? 0 S? C? 0 0 0 ?R = ? ?? (2.24) ? 0 0 0 C2 S2? ? ? ?S ?2?? ?? ?0 0 0 S2 C2 S ???? ? 2?
0 0 0 S2?/2 ?S2?/2 C2?
There are two possible ways to bring the measured strains and Eq. (2.20) into a
common basis. The first option is to bring the measured strain components into basis
B using the relationship ? = R ?+. While this relationship is straightforward, only

three of the six strain components can actually be measured using DIC ? ?+ , ?+11 12, and
?+22. The remaining three strain components can be calculated using the constitutive
relationship and the fact that no surface traction is being applied; however, this
requires knowledge of the material stiffness in basis E and the uncertainties in
the assumed material properties and the measured strains will propagate into the
calculation of the strains in basis B. The second option is to rewrite Eq. (2.20) in
basis E , which can be accomplished by multiplying the entire equation by R?1 to

48
give ( )
?+ = R?1A Z S F +R?1 A W K?T +B W F (2.25)
 L  L L
In this case, the rotation is applied only to the matrix of nodal locations, A Z, and
L
the warping correction terms, A W K?T +B W , which minimizes the impact on the
L L
uncertainties in the final stiffness properties.
2.2.2 Local Equilibrium Equations
Although the transformation applied in Eq. (2.20) is correct, the problem still
remains that only three of the six strain components in ?+ can be measured. It
will be shown later that, in most cases, the non-measured components will not be
required; however, special cases may arise where these components are desired. These
three strain components can be calculated using the local equilibrium conditions on
the outer surface of the beam; since there is no applied surface traction, Newton?s
laws imply the vanishing of the stress component normal to the surface (?+33) and
of the two shear stress components acting in the surface plane (?+23 and ?
+
13). These
three conditions provide the additional information needed to evaluate the three
strain components that were not directly measured.
The constitutive relationship can be used to relate the stresses and strains but
needs to be written in basis E , which gives
?+ = D+?+ (2.26)
The material stiffness matrix, D, is naturally defined in the material basis B?, as
given by Eq. (2.10), and needs to be rotated into the local basis E . This consists
only of a rotation of ? about axis e?3, which gives
D+ = R+D?R+T (2.27)
? ?
49
where
?? ?? C2 ?S 2? ? 2? 0 S? 0 0 ????? S2?/2 C2? 0 ?S2?/2 0 0 ?? ?? ??0 0 C+ ?? ? 0 0 ?S? ?R = ? ???? (2.28)? ? S2 S 0 C2? ? 2? ? 0 0?? ?0 0 0 0 1 0 ???
0 0 S? 0 0 C?
In this form, the stresses and strains are partitioned according to the out-of-plane
and in-plane components, respectively. An alternative means of partitioning the
strains is by the measured and non-measured components, denoted by the subscripts
(?)m and (?)n, respectively, leading to??? ??? ? ????+ + + + ??? ? ? ?? m?+ ? ??
D D ?
= mm nm
??
D+T D ?? m (2.29)n ?+ ??nm nn n
{ } { } { }
w{here ?+T =} ?+ , ?+11 12, ?+22 , ?+T = ?+ , ?+13 33, ?+ , ?+T = ?+ + + +Tm n 23 m 11, ?12, ?22 , and ?n =
?+13, ?
+ +
33, ?23 . With this partitioning, D
+ contains the data initially in the first,
mm
second, and fourth rows and columns of Eq. (2.27), D+ contains the data from the
nm
first, second, and fourth rows, and third, fifth, and sixth columns, and D+ contains
nn
the data from the third, fifth, and sixth rows and columns. Using this notation, the
local equilibrium conditions on the surface imply that ?+Tn = 0, and substituting this
into the bottom set of relationships in Eq. 2.29 gives
?+ = D+?1D+T?+ (2.30)
n nn nm m
and the complete strain tensor can thus be determined at any measurement location.
50
2.2.3 Compliance Matrix Equations
As written, Eq. (2.25) contains the sectional compliance in matrix form and
thus can not readily be solved for these properties. Therefore, it is necessary to
restructure the equations such that they represent a linear system of equations for
the compliance matrix entries. The compliance matrix is symmetric, meaning that of
the 36 terms, only 21 of them should be independent. The symmetry can therefore
be imposed either before or after calculating the individual compliance properties
from the measured values; both options will be considered below.
When the symmetry is assumed before calculating the compliance matrix
entries, the product S F c from Eq. (2.25) can be recast as
?? ??? S11 S12 S13 S14 S15 S16? ?
?
???
S ?12 S22 S23 S24 S25 S26 ?
? ?? ?? S13 S23 S33 S34 S35 S36 ?S F = ? ?c ??
?
S14 S24 S ?34 S44 S45 S46
? ??
(2.31)
? S15 S25 S35 S45 S ?55 S56 ??
S16 S26 S36 S46 S56 S66
= G S
s s
where the following terms were defined
?? ? ? ??? F1 F2 F3 M1 M2 M3 ?? ? 0 0 0 0 0? ??? 0 F1 0 0 0 0 ???? ???? ????
??? ?F2 F ?3 M1 M2 M3 ???
0 0 F1 0 0 0 0 F2 0 0 0 ??F = , F =
1 ???? ? 2 ? ?? 0 0 0 F1 0 0? ?
? ?
? ? ?
? 0 0 F2 0 0 ???
0 0 0 0 F1 0 ??? ??? 0 0 0 F ?2 0 ??
0 0 0 0 0 F1 0 0 0 0 F2
51
???? 0 0 0 0? ?
?
? ?
? ?
0 0 0 ?
? ? ?
???
0 0 0 0
? ???
? ??? 0 0 0 ?
? F3 M1 M2 M3 ???? ?
? ??
0 0 0 ??
F = ? ? , F = ?? ?? ,3 4
???
0 F3 0 0 ?? ? M1 M2 M ?3? 0 0 F3 0 ??
? ?
???? ?0 M 0 ??1 ?
? 0 0 ?0 F3 ? ? 0 0 M1??? 0 0 0????
0 0 ???? ???? ???? 0 ??
??
F = ??? 0 0 ?? ?
?
? ? [ ]5 ?? 0 0 ??
? 0 ?
? , F = ? ? , G = F F F F F F ,? 6 ???? ? s 1 2 3 4 5 6? 0 ??? M2 M3 ?
?
? ?? ?0 ???
{ 0 M2 M3 }
STs = S11,S12,S13,S14,S15,S16 S22,S23,S24,S25,S26 S33,S34,S35,S36 S44,S45,S46 S55,S56 S66
If the symmetry is assumed after the compliance matrix entries are calculated, the
product S F c can be rec?ast as
??? S11 S12 S13 S14 S15 S16? ??
??
??????? ?????
?
F1
? ??? S21 S ?? ?? 22
S23 S24 S25 S26 ???????? F2 ?????
???
?? ?S S S ?31 32 33 S34 S35 S36
S F = ? ??
?
c
? S S S S S S ???
F3 ??? 41 42 43 44 45 46? ???? ????
?????
M1 ? (2.32)
S51 S52 S53 S54 S55 S56 ??????
?????M2 ???????S61 S62 S63 S64 S65 S66 M3
= G S
u u
where {
STu = S11,S12,S13,S14,S15,S16 S21,S22,S23,S24,S25,S26 S31,S32,S33,S34,S35,S36}? ? ?
S41,S42,S43,S44,S45,S46 S51,S52,S53,S54,S55,S56 S61,S62,S63,S64,S65,S66 ,
52
? ?
T
??? Fc 0 0 0 0 0?? T? 0 F 0 0 0 0 ?
??
c ?
????
?
0 0 FT ?c 0 0 0 ?
G = ?
u ?? ?0 0 0 FT? c 0 0 ?? ?? 0 0 0 0 FTc 0 ????
0 0 0 0 0 FTc
with 0 being a 1 ? 6 vector of zeros. Using these expressions, Eq. (2.25) can be
recast to a linear system of equations for the compliance properties as
( )
R?1A Z GS = ?+ ?R?1 A W K?T +B W F c (2.33) L  L L
where for the term GS, either the symmetric or unsymmetric form, G Ss or G Ss u u,
respectively, can be used.
2.2.4 SectionBuilder Data
Equation (2.33) was derived using the discretized equations for the finite element
solution for the strain field ? before trying to solve for the compliance properties it
is therefore important to consider what each of the matrices and matrix products
represents. In the experiments, the strains are measured at discrete locations around
the cross-section, with the measurement locations indicated by the circles ( ) in Fig.
2.5(a). The strain measurement at the ith location in the cross-section, indicated
by subscript (?)i , i = 1, 2, . . . , N where N is the number of strain measurements,
is contained within the jth element of the finite element mesh. Because the shape
functions and their derivatives are only non-zero within the current element, the
interpolation matrices A and B are only non-zero for the nodes within the current
L L
element.
Consider first the matrix product A Z; A contains the shape functions as
L L
53
4 7 3
7 3 8 9 6
4 9 6
8 2 1 5 2
1 5
(b) jth
th
element in (c) j element in
(a) Discretized airfoil cross-section. physical space. isoparametric
space.
Figure 2.5: ith measurement location in the finite element mesh.
defined by Eq. (2.15a) and Z contains the nodal locations as defined by Eq. (2.17).
The ith strain measurement is defined at the location (?i, ?i) within the isoparametric
space corresponding to the location (y thi, zi) in basis B for the j element, as seen
in Fig. 2.5(b-c), and the shape functions are evaluated at these isoparametric
coordinates. Expanding the product of theses matrices leads to
??? ? ? ?
?
? Nl ?0 0 ?0 Nlzl ? Nlyl ??
( ) ??? 0 Nl ?0 ?? Nlz
?
? l
0 0
? ??
?
?? 0 0 Nl Nlyl 0 0 ?A Z = ? ?? (2.34)L i ?? 0 0 0 0 0 0 ?? ??? 0 0 0 0 0 0 ???
0 0 0 0 0 0
where all summations go from l = 1 to l = n, with n being the number of nodes
per element. Using the basic pr?operties of the?shape functions, th?ese individual
expressions can be simplified to n n nl=1Nl = 1, l=1 Nlyl = yi, and l=1Nlzl = zi.
Substituting in the above expressions and premultiplying by the inverse of the strain
rotation matrix, given by Eq. (2.24), results in
54
?? ??? 1 0 0 0 zi ?yi??? 0 C ?S ?z C ? y S 0 0 ?
??
( ) ? ? ? i ? i ? ?? ??? ?? 0 S? C? yiC1 ? ? ziS? 0 0 ?U = R A Z = ? ???? (2.35)i  L i ?? 0 0 0 0 0 0?? ?0 0 0 0 0 0 ???
0 0 0 0 0 0
The structure of U leads to some important implications when solving for the
i
compliance matrix. Since the last three rows of U are all null, only three of the initial
i
six equations provided by Eq. (2.33) for the compliance properties are nontrivial.
These equations correspond to the first three strain components in ?+, which are
the out-of-plane strains ?+11, ?
+
12, and ?
+ + +
13; therefore, the in-plane strains ?22, ?33, and
?+23 have no impact on the calculation of the compliance properties. In addition, the
jth column of U corresponds to data used to calculate the jth row of the compliance
i
matrix. Of the three out-of-plane strains, only the first two components, ?+11 and
?+12, are measured while the third component would need to be determined from the
constitutive equations. However, from the structure of U , the first two rows contain
i
non-zero entries in all columns and therefore all the compliance properties can be
predicted using only the measured strains, as long as ? is sufficiently far from integer
multiples of 90? such that both the cosine and sine components are non-zero.
To illustrate the importance of these properties of U , consider two types of
i
representative cross-sectional geometries. The first type consists of external edges
that are curved, such as a circular tube or rotor blade. In this case, the tangent basis
E is constantly changing orientation around the cross-section and the angle ? can
be sufficiently far from the integer multiples of 90? over a significant portion of the
cross-section. Consider, for instance, a NACA 0012 and VR-7 airfoil profile as shown
in Fig 2.6(a) The sine and cosine components of ? are shown in Fig. 2.6(b) for both
55
profiles as a function of the position along the external edge, s. For both airfoils, the
curvature near the leading edge (s ? 0.4 to 0.6) leads to significant sine and cosine
components. For a symmetric airfoil, the sine component is close to zero for only a
small portion of the external edge, which for a NACA 0012 profile occurs near the
quarter chord location. For an unsymmetric airfoil, such as the VR-7 profile, the top
surface tends to exhibit larger curvatures with a greater sine component, while the
bottom surface tends to be flatter with a smaller sine component.
1
0.5
0
-0.5
-1
0 0.2 0.4 0.6 0.8 1
Position along the external edge,
(a) Example airfoil profiles. (b) Sine and cosine of ? for airfoils.
Figure 2.6: Basis rotation components along external edge for NACA
0012 and VR-7 airfoils.
The second type of cross-section contains four perpendicular edges, such as
a beam with a solid rectangular cross-section or a box-beam, which can be rep-
resentative of certain rotor blade spars. While much simpler geometrically than
the airfoil, these types of cross-sections can present more challenges when trying
to measure all the compliance properties. When the edges are aligned parallel to
the reference basis as shown in Fig. 2.7, the angle ? = 0?, 90?, 180?, or 270?. If the
strain were measured on at least two of the perpendicular edges, all 36 components
of the compliance matrix could be measured. However, if one of the cross-sectional
dimensions is sufficiently small, it may not be possible to measure the strain on these
faces. For instance, with the DIC setup used in this thesis, strain measurements can
only be obtained on faces with a minimum edge dimension of about 7 mm to 10 mm.
56
(a) Solid rectangular cross-section. (b) Hollow rectangular cross-section.
Figure 2.7: Example rectangular sections depicting rotation angles on
external edges.
Although the current DIC setup is limited to measurements on faces with edge
dimensions of at least 7 mm to 10 mm, if the beam is sufficiently thinner in one
dimension, there will be minimal shear flow in that direction and therefore the shear
strain in that direction will be small on the perpendicular edges. To illustrate this,
the strains in the cross-sections shown in Fig. 2.7 were simulated for a shear force,
F3, in the direction of b?3. The shear strain under this force is used to measure the
compliance matrix component S33, which is also related to the ?S? component from
the second row and third column of U . For the solid rectangular cross-section, widths
i
of w = 25 mm and w = 100 mm and heights, h, varying from 0.5 mm to 25 mm
were selected, while for the hollow rectangular cross-section, widths of w = 25 mm
and w = 100 mm, a thickness of t = 1.5 mm and heights, h, varying from 4.5 mm to
25 mm were selected, which are representative of some commercially available beams.
A total length of l = 600 mm was defined for the beams, with the magnitude of the
shear force constrained by the resulting bending moment M2 = ?F3l that resulted in
a maximum bending strain of 3000 ?. For both types of cross-sections, an aluminum
(E = 68.9 GPa, ? = 0.33) and a composite material, with unidirectional plies of
carbon-fiber (E1 = 130 GPa, E2 = 8.69 GPa, ?12 = 0.28, ?23 = 0.33, G12 = 4.60
GPa) in a ply sequence of [?45?/45?/0?/90?]s, were considered.
The resulting shear strains under the shear force F3 are shown in Fig. 2.8 for
varying heights for both the aluminum and composite solid rect?angular cross-sections.
Figure 2.8(a) shows the mean shear strain component, ??+ = 1 n13 i=1|?
+
n 13,i
|, on the top
57
and bottom edges, at ? = 0? and ? = 180?, which is not directly measurable using
DIC (or most other strain measurement techniques) and must be calculated using
the cons?titutive relationships. Figure 2.8(b) shows the mean shear strain component,
??+ = 1 n +12 i=1|?12,i|, on the side edges, at ? = 90? and ? = 270?, which can be directlyn
measured using DIC if the height is h ' 10 mm. Similarly, the shear strains for the
hollow rectangular cross-section are shown in Fig. 2.9 for varying cross-sectional
heights. Figure 2.9(a) shows the mean shear strain ??+13 on the top and bottom edges
of the hollow rectangular section, which is not directly measurable, while Fig. 2.9(b)
shows the mean shear strain ??+12 on the side edges, which can be measured if the
height is h ' 10 mm.
20 300
100 
100 
25 250
15
25 
100 
200
100 
25 
10 150 25 
100
5
50
0 0
0 5 10 15 20 25 0 5 10 15 20 25
(a) Shear strain ??+13 on top and (b) Shear strain ??
+
12 on the side edges.
bottom edges.
Figure 2.8: Shear strain components used to predict the compliance
matrix component S33 for the solid rectangular cross-section.
For both the solid and hollow rectangular cross-sections and for both mate-
rial types, the shear strain component used for predicting the compliance matrix
component S33 is much greater on the side edges than on the top and bottom edges.
The shear strain component ?+13 on the top and bottom edges can not be directly
measured and, although it can be calculated from the constitutive equations, since it
is on the order of 10 ?, it will have a very small signal to noise ratio. Because of
this low signal to noise ratio, when the beam height is too small (h / 10 mm) to
58
25
700
100 100 
100 
600 100 
20 25 25 
25 
500 25 
15
400
10 300
200
5
100
0
0
0 5 10 15 20 25
0 5 10 15 20 25
(a) Shear strain ??+13 on top and (b) Shear strain ??+12 on the side edges.
bottom edges.
Figure 2.9: Shear strain components used to predict the compliance
matrix component S33 for the hollow rectangular cross-section.
be able to measure the strains on the side faces, the compliance matrix component
S33 will need to be determined directly from the numerical model. While the airfoil
and rectangular cross-sections considered here are not representative of all possible
cross-sectional geometries, these results show that in most cases, the constitutive
equations provide no additional information for measuring the compliance and stiff-
ness matrices, either because the non-measured strain component is too close to 0
? or because the surface has enough curvature such that both the sine and cosine
components of ? are non-zero.
When solving for the compliance matrix entries, U is also multiplied by the
i
load matrix G. The strain is measured under a discrete loading condition, which
is indicated by superscript (?)(k) , k = 1, 2, ? ? ? , L where L is the number of different
loading conditions. If using the form of the force matrix in which symmetry is not
initially assumed, given by Eq. (2.32), the product H(k) = U G(k) becomes
i i u
59
?? ?(k)T (k)T (k)T?? Fc 0 0 0 ziFc ?yiFc?? ?
?
F (k)
T (k)T (k)T
? 0 C? c ?S?Fc (?ziC ? y S ?? i ?)Fc 0 0
????
??
(k)T T T
0 S?Fc C?F
(k) (k) ?
(y C ? z S )F 0 0 ?
H(k)
c i ? i ? c
= ? ???? (2.36)i 0 0 0 0 0 0
???? ?0 0 0 0 0 0 ???
0 0 0 0 0 0
Several important implications also arise from the form of H(k) when solving for
i
T
the compliance matrix. Each entry in H(k) contains F (k) thc and the j componenti
of F (k)Tc is therefore used to calculate all six components in the jth column of the
compliance matrix. Because each row is dependent on all six components of the stress
resultants, F c, six linearly independent load cases must be applied during testing
in order to measure all components of the compliance matrix uniquely. In addition,
since each row of H(k) has three non-zero entries which are dependent on zi and yi,i
the strain measurements must be made at a minimum of three locations around the
cross-section, with at least two different values of yi and zi for the measurements.
For a cross-section with a curved external edge such as an airfoil, this means that
data only needs to be measured over a portion of the entire external profile such
that a sufficient variation in both yi and zi is obtained, which can be accomplished
by measuring strain along only the upper or lower portion of the profile. However,
for a rectangular cross-section, strains must be measured on at least two of the four
edges to satisfy these requirements.
The remaining ter(ms in Eq. (2.33))are the matrices corresponding to the
warping correction, R?1 A W K?T +B W F c. The first part of the expression in L L
parenthesis, A W , evaluates to a 6 ? 6 matrix containing the components of the
L
warping field at the ith measurement location, with the last three rows all null. The
second part, B W , contains the spatial derivatives of the components of the warping
L
field. The warping correction is generally complicated for a curved beam with an
60
arbitrary cross-section; however, some insight can be provided by first considering
the simple case of a straight beam (i.e. |k| = 0). Because the curvature is zero, the
majority of the terms in both the curvature tensor K? and strain i(nterpolation matri)x
B also evaluate to zero and the warping correction, V = R?1 A W K?T +B W
L i  L L
becomes
?? ??? 0 ?W?1,6,i W?1,5,i 0 0 0? ???? A ?? 1,i
A2,i ?Wy,6,i A3,i +Wy,5,i A4,i A5,i A6,i ?
???
??
B1,i B2,i ?Wz,6,i B3,i +Wz,5,i B4,i B5,i B6,i ?
V = ? ?? (2.37)i ?? C1,i C2,i C3,i C4,i C5,i C6,i?? D D D D D D ??
??
1,i 2,i 3,i 4,i 5,i 6,i ??
E1,i E2,i E3,i E4,i E5,i E6,i
where Wk,l,i corresponds to the component of the warping field in the k
th direction,
induced by the lth unit component of the sectional stress resultant, evaluated at
the ith measurement location on the cross-section and the following notation was
introduced
?W?1,l,i ? ?W?1,l,i ?W( ?1,l,i
?W?1,l,iAl,i = C? S? , Bi i l,i = S? + C??y ?z ?y i ) ?z i
?Wy,l,i 2 ?Wz,l,i 2 ? ?W( z,l,i
?Wy,l,i S) 2?C = C + S + il,i ?y ?i ?z ?i ?y ?z 2
?Wy,l,i 2 ?Wz,l,i 2 ?Wz,l,i ?W( y,l,i
S2?
Dl,i = S? + C? + +
i
?y i ?z i ?y ?z ) 2
?Wy,l,i ? ?Wz,l,i ?Wz,l,i ?Wy,l,iEl,i = S2? S2? + + Ci i 2??y ?z ?y ?z i
When calculating the entries of the compliance matrix, V is multiplied by
i
F (k)c and this quantity then gets subtracted from the strain + to determine the full
right-hand side of the linear system of equations. From the previous discussion of
H(k), rows 1, 5, and 6 of the compliance matrix are calculated from the axial strain
i
?+11 while rows 2, 3, and 4 of the compliance matrix are calculated from the shear
61
strain ?+ . In addition, the jth12 column of the compliance matrix is predicted from
the jth component of the sectional stress resultants. With this information and the
nature of the first two rows of V ? corresponding to ?+ +
i 11,i
and ?12,i, respectively ? it
can be seen that the compliance matrix entries in rows 1, 5, and 6 and columns 1, 4,
5, and 6 have no dependencies on the warping terms and can therefore be calculated
entirely from experimental data. For the simpler case of a cross-section made of an
isotropic material, the warping field components W?1,l,i are 0 for l = 1, 5, and 6, and
therefore only the compliance matrix entries in the smaller subset of rows 2, 3, and
4 and columns 2, 3, and 4 are dependent on the numerical warping correction.
For beams that are initially curved, V will generally be fully populated and all
i
the compliance matrix terms will therefore be impacted by the warping correction.
However, the warping correction will not effect the calculation of each term from
the compliance matrix equally and its importance can be considered by comparing
the ratio of the first and second entries of V F c to the axial and shear strains, ?+i 11,i
and ?+12,i, respectively, for each component of the sectional stress resultant. Since
the strains and V are evaluated at each of the measurement locations around the
i
external profile of the cross-section, the average ratio of these terms can be used,
defined by
????? ? ? ?N1 ?V F ? ?
N ?V F ?
i,(1,j) c,j ??? 1 ? i,(2,j) c,j ?f?+ ,j = + , f?+ ,j = ?? + ?? , (2.38)11 N ? 12 N ?
i=1 11,i,j i=1 12,i,j
where notation V and V is used to indicate the jth column of the first and
i,(1,j) i,(2,j)
second rows of V , respectively, and notation ?+11,i,j and ?
+
12,i,j is used to indicatei
the total axial and shear strains, respectively, at the ith measurement location and
under loading from the jth component of the sectional stress resultants. A ratio of
zero indicates that the warping has no influence on the strain component under the
current loading condition and the compliance matrix entries corresponding to that
62
strain and load are determined entirely from the experimental data. A ratio between
0 and 1 indicates that the warping induced strain is non-zero and of the same sign
as the total strain, with a value of 0.5 indicating that the warping induced strain
accounts for half of the total strain and a value of 1 indicating that the strain is
entirely due to warping effects. A ratio greater than 1 indicates that the strain due
to warping has an opposite effect from the classical strain measurement, given by
U S F c, and the warping correction will significantly impact the calculation of thei
corresponding compliance properties.
(a) Aluminum rectangle. (b) Composite hollow rectangle.
(c) VR-7 airfoil with a D-spar.
Figure 2.10: Cross-sections used for determining the ratio of warping
induced strain to total strain.
These ratios were evaluated for beams with varying curvatures in each of the
three directions, for three representative cross-sections. A non-dimensional form of
the curvature, defined by ??j = ?jd, was used to compare the results for different
cross-sections, where d is the diagonal of the minimal box that fully encompasses
the entire cross-section. An aluminum beam (E =68.9 GPa, nu = 0.33) with a
solid rectangular cross-section, as shown in Fig. 2.10(a), was first considered, with
dimensions of h = 6 mm and w = 75 mm. Next, a beam with a hollow rectangular
cross-section was considered, as shown in Fig. 2.10(b), with dimensions of h = 12 mm,
w = 50 mm, and t = 1.5 mm and made of a composite material, with unidirectional
plies of carbon-fiber (E1 =130 GPa, E2 =8.69 GPa, ?12 = 0.28, ?23 = 0.33, G12 =4.60
63
GPa) in a ply sequence of [?30?/90?/? 45?/0?]s. The ply sequence was selected
to produce extension-torsion and shear-bending coupling, which results in non-zero
expressions for S14/41, S25/52, and S36/63. Finally, a beam with a VR-7 profile was
considered, as shown in Fig. 2.10(c), with a chord of 80 mm, an outer wrap consisting
of a two layers of carbon-fiber at ?45?, a d-shaped spar going from the leading
edge to 0.38c and composed of two layers of carbon-fiber at ?45?, and an internal
foam core (E = 3.6 MPa, ? = 0.38), with the carbon-fiber having the same material
properties as that of the hollow rectangular section. For each cross-section, a beam
with a moderate curvature, ??j = 0.1, and a large curvature, ??j = 1 were evaluated,
wit(h Fig.) 2.11 showing the two levels of curvature for a beam curved in the plane
of b?1, b?2 . The moderate level of curvature is representative of that seen in some
modern helicopter rotors, where, for example, the V-22 blades, which have a 47.5?
twist over a 5.8 m radius, have an average non-dimensional twist of ??1 ? 0.105 [121].
(a) ?? = 0.1 (b() ??3 3 =) 1
Figure 2.11: Example beams curved in the plane of b?1, b?2 .
The ratios of the warping induced strain to the total strain under a load from
the jth component of the sectional stress resultants are shown in Table 2.1 for the
aluminum beam with the rectangular cross-section. As stated earlier, the first,
fifth, and sixth rows of the compliance matrix are calculated using the measured
axial strains, ?+11,i, and the second, third, and fourth rows are calculated using
the measured shear strains, ?+12,i, while the j
th column of the compliance matrix is
calculated from a load in the jth component of the sectional stress resultants. For
64
the straight beam, the axial strain components have no dependency on the warping
correction, so the first, fifth and sixth rows of the compliance matrix, corresponding
to the axial and bending behavior of the beam, can be determined entirely from
the experimental measurements. For moderately curved beams with ??j = 0.1, the
effect of warping remains relatively small on these terms, which means that the axial
and bending behavior is still almost entirely determined by the experimental data,
with the exception of the bending compliance about b?3 when the beam is curved
about axis b?2, where the ratio is f?+ = 0.365. For very highly curved beams, where11,6
??j = 1, these ratios begin to approach and, in some cases, exceed values of 1 and
the warping correction becomes a significant factor for calculating the axial and
bending compliances. For the straight beam, the second components of the second,
third, and fourth rows have a ratio f?+ ? 0.2, which means that the warping effects12,2
will account for about 20% of the total compliance, and therefore shear stiffness,
about axis b?2. Since f?+ = 1, the shear compliance about axis b?3, will be entirely13,2
determined from the warping effects. The ratio f?+ ? 0.5, which means that12,4
the torsional compliance would be calculated at twice its expected value (i.e. the
torsional stiffness would be 50% of its expected value) if the numerical warping
correction were not included. For moderate levels of curvature where ??j = 0.1, the
ratios f?+ , f?+ , and f?+ all remain similar to those of the straight beam and12,2 12,3 12,4
the warping effect on the shear and torsional compliance terms will remain similar.
However, for highly curved beams with ??j = 1, the warping induced strains are
significantly impacted by the curvature, which can be seen in the ratios f?+ and12,2
f?+ .12,4
65
Table 2.1: Ratios of warping induced to total strain for the aluminum
rectangular cross-section with various beam curvatures
Straight Twisted Curved about b?2 Curved about b?3
?? = 0 ??1 = 0.1 ??1 = 1 ??2 = 0.1 ??2 = 1 ??3 = 0.1 ??3 = 1
f?+ 0 0.068 1.273 0.078 1.214 0.006 0.59511,1
f?+ 0 0.497 1.033 0 0 0 011,2
f?+ 0 2.253 1.090 0 0 0 011,3
f?+ 0 1.571 1.749 0 0 0 011,4
f?+ 0 0.002 0.134 0 0 0.016 0.74111,5
f?+ 0 0.012 1.190 0.365 1.101 0.005 0.12111,6
f?+ 0 0.507 0.498 0 0 0 012,1
f?+ 0.199 0.233 8.592 0.228 1.736 0.206 0.67512,2
f?+ 1.000 1.000 1.000 1.000 1.000 0.984 0.97612,3
f?+ 0.500 0.499 0.450 0.520 1.220 0.499 0.75212,4
f?+ 0 23.180 12.233 0 0 0 012,5
f?+ 0 1.000 1.000 0 0 0 012,6
The ratios of the warping induced strains to the total strains for the composite
hollow rectangular and composite VR-7 airfoil cross-sections are shown in Table
2.2 and Table 2.3, respectively. For the straight beam with a composite hollow
rectangular cross-section, the ratios that correspond to the shear and torsional
compliance terms, f?+ = 0.159, f?+ = 1.000, and f?+ = 0.409 are of similar12,2 12,3 12,4
magnitudes to those of the aluminum rectangular cross-section. The ratios f?+ and11,4
f?+ , used when calculating S14 and S41, f?+ and f?+ , used when calculating S2512,1 11,2 12,5
and S52, and f?+ and f?+ , used when calculating S36 and S63, are all important11,3 12,6
due to the elastic coupling introduced by the ply layup. While the compliance
matrix should be symmetric, the warping induced strain does not equally impact
66
the corresponding symmetric terms, which can be seen when comparing the ratios
f?+ = 0 and f?+ = 0.411 used for computing the symmetric terms S14 and S41,11,4 12,1
respectively. Generally, one of these symmetric compliance terms can be calculated
with very little influence from the warping correction, as seen by the fact that
f?+ = 0, f?+ = 0.022, and f?+ = 0.022, while the other compliance component is11,4 12,5 11,3
more heavily influenced by the warping correction. For the straight beam with the
VR-7 airfoil, the selected layup presents no elastic coupling terms and the only ratios
that effect the calculation of the non-zero compliance properties are f?+ , f + ,11,1 ?12,2
f?+ , f?+ , f?+ , and f?+ . For the airfoil, the warping induced strain only accounts12,3 12, 11,5 11,6
for about 20% of the total strain from a torsional load used when calculating the
torsional compliance, which is much smaller than the 40% to 50% in the rectangular
cross-sections. In addition, the ratio f?+ becomes 0.586 compared to 1.000 for both12,3
rectangular sections, and the shear compliance about axis b?3 will no longer be entirely
determined from warping effects. For both the composite hollow rectangular and
Table 2.2: Ratios of warping induced to total strain for the composite
hollow rectangular cross-section with various beam curvatures
Straight Twisted Curved about b?2 Curved about b?3
?? = 0 ??1 = 0.1 ??1 = 1 ??2 = 0.1 ??2 = 1 ??3 = 0.1 ??3 = 1
f?+ 0 0.058 1.824 0.151 0.737 0.011 1.32111,1
f?+ 0.231 0.664 1.046 0.461 1.034 0.446 1.10811,2
f?+ 0.022 0.458 1.309 0.136 0.932 0.050 1.06911,3
f?+ 0 0.233 2.329 0.009 0.851 0.023 1.25011,4
f?+ 0 0.005 1.183 0.000 0.000 0.030 1.17111,5
f?+ 0 0.116 2.288 0.118 1.328 0.009 0.20311,6
f?+ 0.411 0.422 0.418 0.484 1.064 0.428 0.92212,1
f?+ 0.159 0.717 1.105 0.942 1.013 0.218 1.33912,2
f?+ 1.000 1.000 1.000 1.000 1.000 0.992 0.98112,3
f?+ 0.409 0.320 1.533 0.423 0.965 0.412 0.94812,4
f?+ 0.022 0.021 1.106 0.126 1.133 0.046 0.58012,5
f?+ 1.000 1.000 1.000 1.000 1.000 1.016 0.83212,6
67
airfoil cross-sections, moderate levels of curvature only have a minor impact on most
of the ratios important for calculating the non-zero compliance components, with
exception to the ratios f?+ , f + , and f + for the composite hollow rectangular11,2 ?11,3 ?12,2
cross-section and f?+ and f?+ for the composite airfoil.12,2 12,3
Table 2.3: Ratios of warping induced to total strain for the composite
VR-7 airfoil cross-section with various beam curvatures
Straight Twisted Curved about b?2 Curved about b?3
?? = 0 ??1 = 0.1 ??1 = 1 ??2 = 0.1 ??2 = 1 ??3 = 0.1 ??3 = 1
f?+ 0 0.025 1.236 0.066 0.496 0.046 0.81511,1
f?+ 0.398 0.396 0.954 0.562 1.022 0.689 0.91811,2
f?+ 0.910 3.481 1.251 0.840 0.902 0.973 1.00211,3
f?+ 0 0.927 2.816 0.025 0.882 0.159 1.04511,4
f?+ 0 0.007 0.168 0.000 0.000 0.024 0.32811,5
f?+ 0 0.126 1.363 0.022 0.631 0.018 0.21211,6
f?+ 0.364 4.294 5.585 0.938 0.976 0.358 1.07412,1
f?+ 0.506 1.850 1.338 0.978 1.053 0.659 1.02312,2
f?+ 0.586 0.710 0.974 0.902 1.055 0.704 1.05112,3
f?+ 0.198 0.196 0.839 0.203 1.106 0.182 1.04212,4
f?+ 0.790 28.003 0.527 1.388 0.847 0.976 0.72112,5
f?+ 1.358 0.557 0.943 1.261 0.994 1.340 1.56312,6
Overall, the axial and bending compliances can be measured directly in the
experiments with little impact from the numerical warping correction for beams of
arbitrary cross-section that are either straight or have moderate levels of curvature.
However, the shear and torsional compliances require both the experimental mea-
surements and the numerical warping correction, with the impact of the warping
correction dependent on the type of cross-section.
68
2.3 Solution Procedure
The equations discussed up to this point have been for a measurement at a
discrete location within the cross-section under a single loading condition, which
provides at most three linearly independent equations for the 36 unknown terms
in the compliance matrix. In the actual experiment, data needs to be acquired at
multiple locations within the cross-section and under multiple loading conditions to
provide enough information to fully determine the compliance matrix components.
This complete set of data then needs to be collected to provide a fully determined
system of equations that can be used to solve for the compliance properties. Once
the compliance matrix has been calculated, the sectional stiffness matrix can then
be determined.
2.3.1 Matrix Assembly
Using the notation introduced in Sec. 2.2.4, the equations for a single measure-
ment at the ith location and under the kth load can be written in a compact form
as
H(k)S = ?+,(k) ? V F (k)c (2.39)i i i
While this represents a 6 ? 6 system of equations, it was shown that the last three
rows of H(k) are null and the row corresponding to the shear strain component ?+
i 13
often provides no additional information. Therefore, only the first two rows in Eq.
2.39 will be used for calculating the compliance properties. This will be indicated
using the subscript (?) (k)(1:2,1:6) for the matrices, H and V , to indicatei,(1:2,1:6) i,(1:2,1:6)
+,(k)
the first two rows and all 6 columns of these matrices and ?i,(1:2) to indicate the first
+,(k) +,(k)
two entries in the strain vector, ?11,i and ?12,i . The measurements are made at N
locations around the cross-section under L loading conditions and the total data can
69
be collected into a single set of linear equations defined by
?? ? ? ? ? ?? H(1) ?? +,(1)? ?? ?? V F (1) ?? ?1,(1:2,1:6) ??? ??????? 1,(1:2) ??????? ??????? 1,(1:2,1:6) c?? ..? . ??? ???
.
? .
.
?? .
.
?????? ?????? . ?
??????
? ??? ?? H(1) +,(1)? V F (1)
???
? N,(1:2,1:6) ?? ? ?
??? N,(1:2) ???? ???? N,(1:2,1:6) c ????
? H(2) ?? ? ?
?? +,(2)? ???
1,(1:2,1:6) ??? ?????? 1,(1:2) ?? ? ?????? . ? . ?
?? (2) ??
?????? V F ?1,(1:2,1:6) c ????? .. ? .. .
. ??
?? ??S =? H(2) ? ??????
?? +,(2)? ?
.
?????? ??????
? (2.40)
V F (2) ??
? N,(1:2,1:6) ? ? N,(1:2)? N,(1:2,1:6)
c ???
?? .. ?? ??? .. ??????? ??????? ..
??
. . . ?????????
?
H(L) ? ??? +,(L) ??? ??? ? V F (L) ???
? 1,(1:2,1:6) ???? ??????? 1,(1:2)? ?
?
? .. ? ?? .. ?
?
? ??????
? ??????? 1,(1:2,1:6) c ???. ?. ? . ? ?????
. ?????. ??
(L) +,(L)H ? V F (L) ?
N,(1:2,1:6) N,(1:2) N,(1:2,1:6) c
which can be recast into a compact form as
HS = E? V (2.41)
where H will be of size m? n, where m = 2NL and n is either 21 or 36, depending
on whether or not the compliance matrix is initially assumed to be symmetric.
2.3.2 Compliance Matrix Calculation
Since DIC provides data at a significant number of locations around the cross-
section, Eq. (2.41) will generally represent an over-determined set of linear equations
for the components of the sectional compliance matrix. Because there will also be
noise in the experimental measurements, there is no S that will exactly satisfy Eq.
(2.41) and a solution that minimizes the residual
r = HS ? (E? V) (2.42)
70
is therefore desired. This represents a linear least squares problem for the compliance
matrix components, which can be solved using several numerical approaches. Since
the problem is unconstrained but ill-conditioned, with condition numbers generally
between 104 and 106, a method such as the singular value decomposition (SVD) or
QR decomposition is best suited for calculating the solution. The SVD is numerically
stable and less sensitive to perturbations than the QR factorization, but is more
expensive to compute, with a cost of ? 2mn2 + 11n3 floating point operations (flops)
compared to a cost of ? 2mn2? 2n3/3 flops for the QR factorization [122]. However,
because the minimum number of load cases is six and DIC will generally provide at
least 100 strain measurements for a given cross-section, m ' 1200 n and the cost
for both methods will be dominated by the 2mn2 term with the total cost ending up
being similar. Thus, the SVD is used to solve the linear system of equations for S,
since it is numerically stable and has a similar cost to the QR decomposition for the
large problem sizes expected when using DIC.
Applying the singular value decomposition to H gives
H = U ?VT (2.43)
where matrix ? is a m? n matrix with the singular values along its diagonal and U
and V are unitary matrices, of size m?m and n? n, respectively, containing the
left and right singular vectors, as discussed in Appendix A. Substituting Eq. (2.43)
into Eq. (2.41) leads to
U ?VT S = E? V (2.44)
and the solution for the compliance matrix components becomes
S = V ??1 UT (E? V) (2.45)
71
In ? there are only n non-zero singular values, corresponding to the number of
unknowns in the compliance matrix, and only a subset of the left singular vectors
in U are needed for the calculation of S as the majority will multiplied by the null
vectors in ?. Since m n, the computation of these extra left singular vectors is
very costly, both in terms of time and storage, and a reduced form of the SVD can
be calculated that contains only the first n left singular vectors in U , as described in
Appendix A.2. Implementations of the reduced SVD are readily available through
tools like the Fortran package PROPACK and the MATLAB routine svd, with
Fig. 2.12 showing the increased efficiency of the compact SVD over the full SVD
in terms of wall clock time in MATLAB for different values of m in H. For the
experiments in the current work, most beams will be subjected to between 10 and
12 load cases and strain measurements will be obtained at around 250 point per
cross-section, meaning that the total number of rows in H will be around m ? 2500
to 3000. This means that the total time to calculate the SVD would be about 3 s if
using the full SVD compared to about 5 to 10 ms if using the reduced SVD, which
is a reduction in run time by about three orders of magnitude. While additional
time is required for the matrix assembly and multiplications in the full solution
process, the full SVD takes up about 90% of the total solution time and switching
to a compact SVD still reduces the total solution by an order of magnitude. This is
especially important when considering the fact that the DIC data is obtained across
a significant portion of the span of the beam and, if calculating the cross-sectional
properties at multiple span-wise locations, this can reduce the total run time from
several minutes down to several seconds.
The final stage in then to reconstruct the compliance matrix from the solution
given by Eq. (2.45). If the form of the equations in which symmetry was assumed at
the onset was used, corresponding to G and Ss, the compliance matrix can simplys
be reconstructed by substituting the entries Sij into the i
th row and jth column and
72
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
10 100 1000 10000
Rows in
Figure 2.12: Run time of SVD implementations for different problem
sizes m.
jth row and ith column. If the form of the equations in which symmetry was not
initially assumed was used, corresponding to G and Su, some additional steps mustu
be taken to enforce symmetry. In this case, the terms that should be equivalent in
the compliance matrix (Sij and Sji) are calculated from different combinations of the
applied loads and strains, which have different errors associated with them, and are
impacted differently by the numerical warping correction, as discussed in Sec. 2.2.4.
Asymmetries will then be introduced because one of the terms is calculated using
measurements that have relatively low signal to noise ratios, where a small error can
easily produce a significant compliance component that would otherwise be small or
zero. This situation can often arise in terms calculated from applied shear forces, F2
and F3, that have small expected strains, as the bending moments resulting from
these applied shear forces typically generate much larger strains. Therefore, the
73
Wall clock time [s]
following two conditions are applied to eliminate significant asymmetries
if sgn(Su,ij) 6= sgn(Su,ji) then Su,ij = Su,ji = 0 (2.46a)
Su,ij Su,ji
if > 10 or > 10 then Su,ij = Su,ji = 0 (2.46b)
Su,ji Su,ij
where the first condition eliminates terms of opposite signs and the second condition
eliminates terms that differ by more than an order of magnitude. Finally, the
symmetric compliance matrix can be calculated from the unsymmetric form by
S + ST
S = u u (2.47)
2
which averages the non-zero off-diagonal terms. Since the form in which symmetry
is initially assumed has no way of eliminating off-diagonal compliance components
that arise solely from error in a measurement with a very low signal to noise ratio,
the unsymmetric form is used for all the experimental data in this thesis.
2.3.3 Stiffness Matrix Calculation
Once the sectional compliance matrix has been calculated, it becomes straight-
forward to calculate the stiffness matrix as its inverse defined by
K = S?1 (2.48)
While the dimensional form of the stiffness matrix given by Eq. (2.48) is the format
required for most beam codes, it is also useful to consider the stiffness matrix in a
non-dimensional form, where the term in the ith row and jth column is normalized
according to ?
K2ij
Kn,ij = (2.49)
KiiKjj
74
With this definition, the terms along the diagonal, where i = j, evaluate to 1 and
the terms on the off-diagonal, where i 6= j, take on a value between 0 and 1. When
the normalized off-diagonal terms are much less than 1, they are insignificant to the
overall response of the beam, in terms of the static or dynamic behavior and the
resultant stresses and strains. Therefore, this provides a means of determining the
significance of the measured off-diagonal components of the stiffness matrix.
2.3.4 Principal Axes of Bending
The principal axes of bending are defined by frame FC = [C, C = (c?1, c?2, c?3)],
as shown in Fig. 2.13. Their origin C is at the centroid of the cross-section, the
location where an applied axial load, in the direction b?1 = c?1, produces no bending
deformation. The axes are also oriented at an angle ?pb with respect to basis B, with
the angle defined such that the cross-bending stiffness K56 becomes zero and the
bending stiffness components K55 and K66 are maximized [119].
Figure 2.13: Centroid and principal axes of bending.
The location of the centroid can be determined from the stiffness components
relating the axial and bending behavior, K15 and K16, according to
?K16 K15yc = , zc = (2.50)
K11 K11
where yc and zc are the distances along axis b?2 and b?3, respectively. Defining an
intermediate frame F ? = [C, C? = (c??, c??C 1 2, c??3)] whose origin is at the centroid but
whose axes c?? and c??2 3 are parallel to b?2 and b?3, respectively, the axial and bending
75
stiffness components become
Kc?11 = K , K
c? c?
11 15 = 0, K16 = 0,
2
Kc?55 = K55 ?
K15 = K55 ? zcK15,
K11
(2.51)
c? ? K15K16K56 = K56 ,K11
2
Kc?
K
66 = K
16
66 ? = K66 + ycK16
K11
where (?)c? indicates components resolved in basis C?. The orientation of the principal
axes, defined by angle ?pb, is then determined by
( )
?2Kc?
? = 0.5 tan?1 56pb ? (2.52)Kc66 ?Kc?55
and the corresponding axial and bending stiffness components in basis C become
Kc11 = K11, K?c c c1(5 = K16 = K)56 = 0,
Kc? c? c? c?
2
Kc = 55
+K66? K66 ?K5555 ? + (Kc?)
2,
2 ( 2 56 (2.53)
Kc? +Kc? Kc?
)
?Kc? 2
Kc 55 66 66 55 c?66 = + + (K56)
2.
2 2
These definitions imply that the principal axes of bending will always be oriented
such that the bending stiffness Kc55 about axis c?2 will always be less than or equal to
the bending stiffness Kc66 about axis c?3.
2.3.5 Shear Center
The shear center S of the cross-section is the location where an applied shear
force, F2 or F3, produces no torsional deformation [119]. This implies that at
this location, the coupling terms between the shear and torsional behavior, K24
and K34, must vanish. A set of principal axes can then be defined by the frame
76
FS = [S,S = (s?1, s?2, s?3)], as shown in Fig. 2.14, such that the cross-shear stiffness
K23 also vanishes and the shear stiffness components K22 and K33 are maximized.
Figure 2.14: Shear center and principal axes of shear.
The location of the shear center can be determined from the stiffness components
relating the shear and torsional behavior according to
K22K34 ?K23K24 K23K34 ?K33K24
ysc = , zsc = (2.54)
K K ?K222 33 23 K22K33 ?K223
where ysc and zsc are the distances along axis b?2 and b?3, respectively. Defining an
intermediate frame FS? = [S,S? = (s??1, s??2, s??3)] whose origin is at the shear center,
but whose axes s??2 and s?
?
3 are aligned parallel to b?2 and b?3, respectively, the shear
and torsional stiffness components become
Ks? s? s?22 = K22, K33 = K33, K23 = K23,
Ks? = 0, Ks? = 0, (2.55)24 34
Ks?44 = K44 ? yscK34 + zscK24
where (?)s? indicates components resolved in basis S?. The orientation of the principal
axes, defined by angle ?ps, is then determined by
( )
2K23
?ps = 0.5 tan
?1 . (2.56)
K22 ?K33
77
and the corresponding axial and bending stiffness components in basis S become
?( )2
s K22 +K33 K22 ?K33K 222 = +? +K23,2 ( 2 )2
s K22 +K33 (2.57)K33 = ?
K22 ?K33
+K2
2 2 23
,
Ks23 = K
s s s s?
24 = K34 = 0, K44 = K44 = K44 ? yscK34 + zscK24.
These definitions imply that the principal axes of shear will always be oriented such
that the shear stiffness Ks22 in direction s?2 will always be greater than or equal to
the shear stiffness Ks33 in direction s?3.
2.3.6 Change in Cross-Sectional Frame
Depending on the type of analysis, it may be desired to have the stiffness and
compliance properties evaluated at the shear center, the centroid, or another arbitrary
point within the cross-section. In general, this will consist of both a translation and
a rotation into a new cross-sectional frame FP = [P,P = (p?1, p?2, p?3)], where P is the
location within the cross-section about which the stiffness components are desired
and P is the basis in which the stiffness components should be resolved, as shown in
Fig. 2.15. The change in frame consists first of the translation from the origin of
basis B, denoted by B, to point P defined by
rp = 0b?1 + ypb?2 + zpb?3 (2.58)
with yp and zp both resolved in basis B. The change from basis B into basis P also
consists of a rotation by angle ? ab?out axis b?1, with the rotation matrix defined by? ??? 1 0 0 ??R =
? ?? 0 C ?? ?S? ?? . (2.59)
0 S? C?
78
The motion tensor C that brings frame FB into frame FP is then defined byP
?? ?? R r?pRC = ? ? ?? (2.60)
P
0 R
?
with r?p the skew-symmetric representation of rp defined by
?? ??? 0 ?zp yp? ??r?p = ? zp 0 0 ??? . (2.61)
?yp 0 0
Figure 2.15: Translation and rotation into cross-sectional frame FP .
Using the motion tensor C , the stiffness matrix can be brought into frame
P
FP according to
K = CTK C . (2.62)
P P P
Since the compliance matrix is the inverse of the stiffness matrix, it can be brought
into frame FP by inverting Eq. 2.62 according to
( )?1
S = C?1S CT . (2.63)
P p p
79
Chapter 3: Verification and Validation
This chapter presents the verification and validation of the main data reduction
procedure highlighted in Chapter 2. The data reduction procedure is analyzed for
its capability to handle two key features not found in most other measurement
techniques: the ability to calculate the full 6 ? 6 stiffness matrix as well as spanwise
variations in properties associated with features common to rotor blades such as twist
and taper. For the purposes of validation, the experimental results will be simulated
using the 3-D finite element code ANSYS in order to eliminate any uncertainties
and errors that will be introduced through an experimental setup. Three different
beams will be considered, each one with increasing complexity, to address the key
features of the method.
The first validation model considers an aluminum beam with a rectangular
cross-section. For this simple geometry and material, an analytic solution exists for
all the non-zero stiffness components that can be used to verify the data reduction
procedure. For the second model, an aluminum beam with a rectangular cross-section
is also considered, with the dimensions of the cross-section changing linearly along
the length of the beam to verify the prediction of spanwise variations in properties.
The third model considers a composite box-beam, using geometric and material
properties that have been previously studied in literature. Three different ply layups
were considered, to verify the capability to measure various elastic coupling terms.
80
3.1 Aluminum Rectangular Beam
The aluminum beam with a rectangular cross-section provided a useful valida-
tion case, since an analytic solution exists for all the non-zero stiffness components
for a rectangular cross-section made of an isotropic material. The beam, shown
in Fig. 3.1, had a length of l = 60 cm, a cross-sectional width of w = 5 cm, and
a cross-sectional height of h = 2.5 cm, which provided a ratio of the beam length
to the maximum cross-sectional dimension of 12. The beam was made of 6061-T6
aluminum, with a Young?s modulus of E = 68.9 GPa and Poisson?s ratio of ? = 0.33.
(a) 3-D view of beam (b) Cross section geometry
Figure 3.1: Aluminum beam with rectangular cross-section.
3.1.1 Analytical Solution
For a beam with a rectangular cross-section and made of an isotropic material,
there are only six non-zero stiffness terms as long as the cross-sectional axes (b?2, b?3)
are at the center of the cross-section, as shown in Fig. 3.1(b). For the axial and
bending behavior, the stiffness can be readily determined by
Ewh3 Ew3h
K11 = EA = Ewh K55 = EIflap = K66 = EIlag = (3.1)
12 12
The torsional stiffness can then be determined from Saint Venant?s solution for
the torsion of a bar, with the exact solution taking the form of an infinite sum of
81
trigonometric functions given by [123]
? ( ( ))
32Gw3h ? 1 2w hk?
K44 = GJ = 1? tanh (3.2)
?4 k4 hk? 2w
k=odd
The two shear stiffnesses can be determined by equating the sectional strain energy
from 3-D elasticity and beam theory [124], which gives
( ( ))?1
6 36?2Gh2 ?? ??
K22 = ( + (hH4 k + 4 Hkm5Gwh ? E2w6 k=1 m=1 ))? (3.3)? ? 1
6 36?2Gw2 ? ?
K33 = + wHr + 4 Hrs
5Gwh ?4E2h6
r=1 s=1
where the expressions for Hk and Hr are given by
( ( ) )( ( ))?1
h 2k?w k?w
Hk = ( sinh ( ? w k
4 cosh
2k? h ) )( ( h ))?1
w 2k?h k?h
Hr = sinh ? h k4 cosh
2k? w w
and the expressions for Hkm and Hrs are given by
( ( ) )
h ? 2h k?w h sinh (2k?w/h) wHkm = w tanh + + if k = m
2k4 k? h 4k? cosh (2k?w/h) 2 cosh (k?w/h)
(?1)k+m h [(k +m) sin ((k ?m)?) + (k ?m) sin ((k +m)?)]
Hkm = ( ( ) Wkm ) if k =6 mk2m2 (k2 ?m2)?
w 2w k?h w sinh (2k?h/w) h
Hrs = h? tanh + + if r = s
2k4 k? w 4k? cosh (2k?h/w) 2 cosh (k?h/w)
(?1)k+mw [(k +m) sin ((k ?m)?) + (k ?m) sin ((k +m)?)]
Hrs = W if r =6 s
k2m2 (k2 ? 2 rsm )?
with Wkm and Wrs given by ( )
2h (m?w) 2h k?w
Wkm =w ? tanh ? tanh +
m? h k? h
h [(k +m) sinh ((k ?m)?w/h) + (k ?m) sinh ((k +m) ?w/h)]
+
2 (k2 ?m2) ? cosh (k?w/h) cosh (m?w/h)
82
( ) ( )
? 2w m?hWrs =h tanh ?
2w k?h
tanh +
m? w k? w
w [(k +m) sinh ((k ?m) ?h/w) + (k ?m) sinh ((k +m) ?h/w)]
+
2 (k2 ?m2)? cosh (k?h/w) cosh (m?h/w)
3.1.2 3-D Finite Element Model
To numerically simulate the experimental results, ANSYS was used to gen-
erate a 3-D finite element model and calculate the surface strains under a set of 6
linearly independent loads. The model was meshed using 6000 SOLID186 (20-node
hexahedral) elements, with 120 elements along the length, 10 elements along the
width, and 5 elements along the height as shown in Fig. 3.2. The nodes at the left
end (x = 0 cm) were fully constrained in all three degrees of freedom to simulate
a clamped boundary condition, and the loads were then applied at the right end
of the beam (x = 60 cm). The applied loads for the six loading conditions are
summarized in Table 3.1 and shown in Fig. 3.3 and Fig. 3.4 for the forces and
moments, respectively, with the load levels selected to provide a maximum strain of
about 500 ? under an axial load and between 2000 and 2500 ? for the remaining
loads. For the torsional load case, a point load was applied at the four nodes at
the corner of the cross-section in the ?b?3 direction, for a net zero force but a 360
N-m moment about b?1. For the bending moments, the load was applied as a linearly
varying surface traction in the b?1 direction, with the variation in the b?3 direction for
the flap bending moment and in the b?2 direction for the lag bending moment. For
Figure 3.2: Mesh of the aluminum beam with rectangular cross-section.
83
the three forces, the loads were applied as a constant surface traction in the b?1, b?2,
and b?3 direction for the axial, lag shear, and flap shear forcing, respectively.
After the boundary conditions were applied for each of the six loading conditions,
the displacements and strains were then calculated with a few representative results
shown in Figs. 3.5 and 3.6. By default, the strains from the finite element model
are calculated in the global basis B = (b?1, b?2, b?3) as shown in Fig. 3.5; however, to
be consistent with results from an experimental measurement, these strains must
be rotated into the local surface basis E = (e?1, e?2, e?3) using Eq. (2.23) as shown in
Fig. 3.6. It is important to note that the two in-plane shear strains are impacted
(a) Axial force
(b) Lag shear force
(c) Flap shear force
Figure 3.3: Applied boundary conditions for the three forces.
84
Table 3.1: Applied loads for the 3-D finite element model of the
aluminum beam with a rectangular cross-section
Loading Surface Traction (3-D model) Equivalent Beam Load
Axial f1 = 3.445? 107 N/m2 F1 = 4.306? 104 N
Lag shear f2 = 2.392? 106 N/m2 F2 = 2.991? 103 N
Flap shear f3 = 1.196? 106 N/m2 F3 = 1.495? 103 N
Torsional f1,t = 3598 N M1 = 360 N-m
Flap bending m = 1.378? 10102 z N/m2 M2 = 897 N-m
Lag bending m3 = 6.89? 109y N/m2 M3 = 1794 N-m
(a) Torque
(b) Flap bending moment
(c) Lag bending moment
Figure 3.4: Applied boundary conditions for the three moments.
85
by the rotation, while the axial strain remains the same. While the displacements
are not used explicitly in the calculation of the compliance and stiffness matrices
(Eq. (2.41)), they are needed in order to accurately calculate the sectional stress
resultants, F c, from the equivalent beam load applied at the tip of the beam (see
Sec. 5.3.3).
(a) Axial strain, ?11, under flap bending load, m2
(b) Shear strain, ?12, under torsional load, f1,t
(c) Shear strain, ?13, under torsional load, f1,t
Figure 3.5: Strains in the global basis B = (b?1, b?2, b?3) under flap bending
and torsional loads for the aluminum beam with a rectangular
cross-section.
86
(a) Axial strain, ?+11, under flap bending load, m2
(b) Shear strain, ?+12, under torsional load, f1,t
(c) Shear strain, ?+13, under torsional load, f1,t
Figure 3.6: Strains in the surface basis E = (e?1, e?2, e?3) under flap bending
and torsional loads for the aluminum beam with a rectangular
cross-section.
3.1.3 Stiffness Results
Using the sectional stress resultants calculated along the span, the strains
in basis E , and the numerical warping field, the matrices from Eq. (2.41) can be
assembled and used to calculate the compliance and stiffness matrices at discrete
spanwise locations. The strain data was extracted at the nodes along the external
edges of the elements resulting in strain data at every 1 cm along the span. The
87
average stiffness values for the entire beam are shown in Table 3.2 and compared
against the analytic solutions from Sec. 3.1.1. Note that because the beam is isotropic
and the reference axes of the beam are at the geometric center of the cross-section,
which is coincident with both the shear center and centroid for the rectangular
cross-section, all the off-diagonal terms are zero and are omitted in Table 3.2. All six
stiffness values calculated from the 3-D FEM strains showed great agreement with
their respective analytic solutions. The largest difference was in the flap shearing
stiffness K33, which had a discrepancy of 5.3%, while the torsional stiffness had a
discrepancy of 1.9% and the remaining four stiffness components all had differences
of less than 0.5% compared to their respective analytic solutions.
One of the main goals of this thesis was to develop a procedure that could
predict the stiffness coefficients at discrete locations along the beam. Since the strain
data was extracted from the ANSYS model at 1 cm intervals along the span, the
stiffness matrix could also be calculated discretely at each of these spanwise locations
with the results shown in Fig. 3.7 and Fig. 3.8. For each of the six non-zero stiffness
matrix entries, the value calculated using the strains from the 3-D finite element
model are shown as points, with the analytic solution shown as a solid line. For all
six stiffness values, the calculated stiffness has some variations along the first few
centimeters due to the strain concentrations that result from the clamped boundary
Table 3.2: Average stiffness of the aluminum rectangular beam
Stiffness Calculated from Analytic
Term 3-D FEM Strains Solution
K11 8.619? 107 N 8.613? 107 N
K22 2.688? 107 N 2.697? 107 N
K 2.650? 107 N 2.517? 10733 N
K 4716 N-m244 4628 N-m
2
K 2 255 4489 N-m 4486 N-m
K66 1.795? 104 N-m2 1.794? 104 N-m2
88
Figure 3.7: Spanwise variations of the axial (K11), lag shear (K22), and
flap shear (K33) stiffness using 3-D FEM strains for the aluminum beam
with rectangular cross-section.
Figure 3.8: Spanwise variations of the torsional (K44), flap bending
(K55), and lag bending (K66) stiffness using 3-D FEM strains for the
aluminum beam with rectangular cross-section.
89
condition at the root. In addition, the torsional stiffness is affected by the applied
load at the tip, due to the fact that the moment for this loading condition was
generated using loads applied at only four nodes. However, the overall trends for
the local stiffness values are similar to those of the average or global stiffness values
highlighted in Table 3.2 ? with the values calculated from the 3-D FEM strains
showing overall excellent agreement with the analytic solution.
3.2 Tapered Aluminum Rectangular Beam
The tapered aluminum beam with a rectangular cross-section was also consid-
ered in order to verify the capability to predict spanwise variations in properties.
The beam, shown in Fig. 3.9, had a length of l = 60 cm, and a cross-sectional width
and height that decreased linearly along the span, with the greatest dimensions at
the root of the beam. To determine the impact, if any, of the magnitude of the taper,
beams with two different taper ratios, the ratio of the cross-sectional dimensions at
the root to the those at the tip, were considered: the first beam had a moderate
taper ratio of two, while the second beam had a more extreme taper ratio of five.
For both beams, the cross-section at the root had a width of wroot = 6 cm and height
of hroot = 2.5 cm. Thus, for the beam with a taper ratio of two, the dimensions at
the tip were wtip = 3 cm and htip = 1.25 cm, while for the beam with a taper ratio
of five, the dimensions at the tip were wtip = 1.2 cm and htip = 0.5 cm. The two
beams were again made of 6061-T6 aluminum, with a Young?s modulus of E = 68.9
GPa and Poisson?s ratio of ? = 0.33.
3.2.1 3-D Finite Element Model
To numerically simulate the experimental results, ANSYS was again used to
generate a 3-D finite element model and calculate the surface strains under a set of 6
90
(b) Cross section
(a) 3-D view of beam geometry
Figure 3.9: Tapered aluminum beam with rectangular cross-section.
linearly independent loads. Both tapered models were meshed using 3240 SOLID186
elements, with 120 elements along the length, 9 elements along the width, and 3
elements along the height as shown in Fig. 3.10. The nodes at the left end (x = 0
cm) were all fully constrained in all three degrees of freedom to simulate a clamped
boundary condition, and the loads were then applied at the right end of the beam (x
= 60 cm), using the same loading procedure discussed in Sec. 3.1.2 for the prismatic
aluminum beam. However, for the tapered beams, the loading magnitudes were
selected such that the maximum strains at mid-span (x = 30 cm) were about 500?
under an axial load and between 2000 and 2500 ? for the remaining five loading
conditions, with the actual applied loads defined in Table 3.3 and Table 3.4 for the
models with taper ratios of two and five, respectively.
Figure 3.10: Mesh of the tapered aluminum beam with rectangular
cross-section.
Once the boundary conditions were applied for both models under each of the
six loading conditions, the displacements and strains were then calculated with a few
91
representative results shown in Fig. 3.11. The axial strain ?+11 under the axial load
for both taper ratios is shown in Fig. 3.11(a-b), while the shear strain ?+12 under the
torsional load is shown in Fig. 3.11(c-d), with all strain components shown in the
local surface basis E = (e?1, e?2, e?3).
Table 3.3: Applied loads for the aluminum beam with a taper ratio of 2
Loading Surface Traction (3-D model) Equivalent Beam Load
Axial f1 = 7.751? 107 N/m2 F1 = 2.907? 104 N
Lag shear f2 = 4.845? 106 N/m2 F2 = 1.817? 103 N
Flap shear f = 2.019? 106 N/m23 F3 = 757 N
Torsional f1,t = 6466 N M1 = 388 N-m
Flap bending m2 = 9.301? 1010z N/m3 M2 = 454 N-m
Lag bending m3 = 3.876? 1010y N/m3 M3 = 1090 N-m
Table 3.4: Applied loads for the aluminum beam with a taper ratio of 5
Loading Surface Traction (3-D model) Equivalent Beam Load
Axial f1 = 3.101? 108 N/m2 F1 = 1.860? 104 N
Lag shear f2 = 1.550? 107 N/m2 F2 = 930 N
Flap shear f = 6.459? 1063 N/m2 F3 = 388 N
Torsional f1,t = 8277 N M1 = 199 N-m
Flap bending m2 = 1.860? 1012z N/m3 M2 = 233 N-m
Lag bending m3 = 7.751? 1011y N/m3 M3 = 558 N-m
3.2.2 Stiffness Results
After rotating the strains from ANSYS into basis E and combining these
with the sectional stress resultants calculated along the span and the numerical
warping field, the data reduction matrices were assembled and used to calculate
the compliance and stiffness matrices at discrete spanwise locations. The spanwise
variations in the stiffness properties are shown in Figs. 3.12 - 3.17 for the six non-zero
components for both tapered beams, with the results for the beam with a taper ratio
92
(a) Axial strain, ?+11, under axial load, f1, for a taper
ratio of 2
(b) Axial strain, ?+11, under axial load, f1, for a taper
ratio of 5
(c) Shear strain, ?+13, under torsional load, f1,t, for a
taper ratio of 2
(d) Shear strain, ?+13, under torsional load, f1,t, for a
taper ratio of 5
Figure 3.11: Strains in surface basis E = (e?1, e?2, e?3) under axial and
torsional loads for the tapered aluminum beams with a rectangular
cross-section.
93
of 2 shown as hollow circles (?) and for the beam with a taper ratio of 5 shown as
hollow squares (). Although the analytic solutions from Sec. 3.1.1 are derived under
the assumption that the beam is prismatic, meaning it has a constant cross-section
along the span, they still provide a useful reference for comparing the stiffness
properties calculated from the ANSYS strain fields. With the exception of the flap
shear stiffness shown in Fig. 3.14, it can be seen that with there is good agreement
between the reference analytic solution and the stiffness properties calculated from
the ANSYS results. For the axial and flap and lag bending properties, the calculated
stiffnesses are all within 1% of the reference solution, except for near the root where
strain concentrations arise due to the clamped boundary condition. For the lag shear
stiffness K22, the calculated stiffness was on average about 1.5% greater than the
reference stiffness with a maximum difference of about 2% near the root. For the flap
shear stiffness K33, the calculated and reference stiffness had their largest differences
near the root of the beam but showed much better agreement moving closer to the
tip; on average the calculated flap shear stiffness was about 6.5% higher than the
reference stiffness but close to the root was about 13.5% greater. The torsional
stiffness K44 followed a similar trend to the flap shear stiffness with the greatest
differences occurring near the clamped end of the beam but showed smaller overall
discrepancies; the maximum difference was about 9.5% near the root and on average
was about 3.8% greater than the reference stiffness. Overall, these results show
that the data reduction algorithm presented in this thesis is capable of calculating
the cross-sectional stiffness properties locally along the beam, even when significant
spanwise variations in the cross-sectional geometry are present.
3.3 Composite Box Beam
The final beam considered was a thin-walled composite box beam, whose cross-
sectional dimensions and material properties were modeled using the experiments of
94
Figure 3.12: Spanwise variations of the axial stiffness, K11, for the
tapered aluminum beams.
Figure 3.13: Spanwise variations of the lag shear stiffness, K22, for the
tapered aluminum beams.
95
Figure 3.14: Spanwise variations of the flap shear stiffness, K33, for the
tapered aluminum beams.
Figure 3.15: Spanwise variations of the torsional stiffness, K44, for the
tapered aluminum beams.
96
Figure 3.16: Spanwise variations of the flap bending stiffness, K55, for
the tapered aluminum beams.
Figure 3.17: Spanwise variations of the lag bending stiffness, K66, for
the tapered aluminum beams.
97
Chandra and Chopra from 1990 [67]. The composite box beam, shown in Fig. 3.18,
had a length of l = 60 cm, an external width of w = 24.206 mm, an external height
of h = 13.640 mm, and a wall thickness of t = 0.762 mm. This type of box beam
has been widely studied in literature and used to evaluate a variety of numerical and
analytic cross-sectional analyses including those of Smith and Chopra [66], Bauchau
and Hodges [125], Popescu and Hodges [126], Jung et al. [127], and Yu et al. [29].
For this dissertation, three different ply layups, shown in Table 3.5, were considered
to study the ability to measure elastic coupling terms introduced by both symmetric
and antisymmetric ply layups. The lay-up definitions are defined starting with
the innermost ply and ending with the outermost ply with 0? plies oriented with
the axis of the beam (b?1) and a positive ply angle indicating a right-hand rotation
about the local normal to the wall. For all beams, the walls consisted of six plies
of unidirectional AS4/3501-6 graphite/epoxy, with each ply being 0.127 mm thick
and with a longitudinal Young?s modulus of E1? = 141.96 GPa, transverse Young?s
modulus of E2? = 9.79 GPa, shear moduli of G12? = 6 GPa and G23? = 4.80 GPa,
and Poisson?s ratios of ?12? = ?13? = ?23? = 0.42, where notation (?)? is used to
indicate that properties are in the material basis B? with b??1 aligned with the material
fibers.
Table 3.5: Composite Box-Beam Layups
Lay-up Upper wall Right wall Lower wall Left wall
Uncoupled [0?] [0?6 ]6 [0
?]6 [0
?]6
Symmetric [20?] [0?] [?20?] [0?6 6 6 ]6
Antisymmetric [15?] [15?] ? ?6 6 [15 ]6 [15 ]6
The first beam considered had a symmetric, uncoupled layup consisting of six
plies with their fibers all oriented along the axis of the beam ([0?]6), meaning that the
six deformation modes were completely uncoupled and the stiffness and compliance
matrices should be diagonal, as was the case for the previously considered isotropic
98
(a) 3-D view of beam
(b) Cross-section geometry
Figure 3.18: Composite beam with rectangular box cross-section.
beams. The calculated stiffness results for this layup were presented by Popescu and
Hodges [126] using both VABS and NABSA, and can be used to compare with the
results from the 3-D FEM. The second beam consisted of a symmetric, but coupled,
layup with six plies oriented at 0? on the two side walls, six plies oriented at 20?
on the upper wall and six plies oriented at -20? on the lower wall. Generally, a
symmetric layup with not all plies at 0? or 90? will present coupling between the
extension and shear behavior and the torsional and bending behavior. Since the plies
oriented at a non-zero angle are only on the upper and lower walls, it is expected
that a bending/torsion coupling will only be present for the bending about axis
b?2 ? which in rotorcraft is commonly referred to as flap bending ? and not for the
bending about axis b?3 ? commonly referred to as lag bending. The third and final
layup was an antisymmetric layup consisting of all six plies oriented at 15? for the
entire cross-section. Generally, an antisymmetric layup will present coupling between
the extension and torsional behavior and the shearing and bending behavior. The
99
stiffness values for this layup were calculated by Yu et al. using VABS and NABSA
and by Jung et al. using a refined analytical model, which can both be used to
compare with the results obtained here from the 3-D FEM strains. For all three
layups, the analytic expressions for the various stiffness values based on a classical
laminated plate theory presented by Smith and Chopra [66] will also be used for
comparison.
3.3.1 Classical Laminated Plate Theory
A composite laminate is an assembly of plies, each possibly having different
elastic properties, and defined by a stacking sequence describing the orientation
of the composite fibers in each layer. For a ply made of unidirectional composite
material, the material can be assumed to be transversely isotropic meaning that
it has two orthogonal planes of symmetry and is therefore defined by only four
independent material constants, E1? , E2? , G12? , and ?12? . Due to the small thickness
of an individual ply, it can be assumed that each ply is in a state of plane stress with
the stress-strain relationship defined by
????? ? ? ?? ? ? ?? ?11? ????? ? ? ? ? ??? Q ?11 Q12 0 ?????? ?11? ???? ???? ? ?11? ???????? ?22? ????? = ?? Q12 Q22 0 ??????? ?22? ????? = Q???? ?22? ???? (3.4)?12? 0 0 Q ? ? ? ?66 12 ?12?
where Q11, Q12, Q22, and Q66 are related to the material properties by
E1? ?21?E1? E2?
Q11 = , Q12 = , Q =
1? ?12??21? 1?
22
?12??21? 1? ?12??21?
E2??12?
Q66 = G12? , ?21? =
E1?
This stress-strain relationship is derived in the material basis of a single ply; however
a lamina consists of multiply plies that may have different orientations. The stress-
100
strain relationship can be rotated into a consistent frame using the rotation tensor
? ?
?? C2 2? ? S? ?2S?C? ??R = ?? S2 C2? ? ? 2S?C? ??? (3.5)
S C 2 2? ? ?S?C? C? ? S?
where ? is the orientation of the fibers with respect to the global coordinate axes.
The stress-strain relationship for an individual ply in the global coordinate system
can then be?written as???? ?? ? ????? ??
??? ??? ? ??? ??
???
11 ?11 ??? ?? Q?11 Q?12 Q?16 ????? ?11 ???
?????
T
?22 ????? = R QR? ? ????? ?22 ????? = ?
?? Q? ?12 Q?22 Q?26 ??????? ?22 ???? (3.6)?12 ?12 Q?16 Q?26 Q?66 ? ?12
3.3.1.1 Sectional Stiffness Properties
Using the constitutive relationships from the classical laminate plate theory,
defined by Eq. (3.6), some assumptions on the warping of the cross-section and
the transverse in-plane normal stresses, and the equilibrium equations, Smith and
Chopra [66] derived expressions for the cross-sectional stiffness properties of a beam
with a thin-walled composite box cross-section. While the derived expressions do
not provide an exact solution of 3-D elasticity, they do provide a useful check on
both the sign and magnitude of the stiffness coefficients. These expressions for the
stiffness properties are
?? ?? Q?12dA ??h,v
K11 = ?h,?Q?11dA? ???? ?? Q?12dA (3.7a)v Q?h,v 22dA h,v
Q? dA
K12 =? ?? Q?16dA+ ??
h 26 Q?12dA (3.7b)
h ?? Q?22dA??h ??
h
Q?26dA
K13 =? Q?16dA+ v Q?12dA (3.7c)
v Q? dAv 22 v
101
?? ??
K14 =(1 + ?) ??Q?16?dA+ (1? ?) ? Q?16?dA+h v? ?? (3.7d)
(1 + ?) Q?
h 26
?dA? (1? ?) Q?26?dA
+ v?? ?? ?? Q?12dAQ? dAh,v 22 h,v
K22 = ??Q?66dA? ??
??
Q? dA
h 26?? Q?26dA (3.7e)h Q?22dAh h?? ??Q?26?dAK25 =??? Q?16?dA?+?
h 2
?? Q?12? dA (3.7f)Q? ?2h dA?? h
22 h
Q?26dA
K33 = ? Q?66dA?
v Q?26dA (3.7g)
v ? Q? dAv 22 v
K36 =? Q??16??dA+ ?
??? ??Q?26?dAv
2 ??Q? 212? dA (3.7h)v Q?22? dAv v
K44 =(1 + ?)
2 ??Q? ?2dA+ (1? ?)2?? Q? ?266 66 dA+h v [ ??
(1 + ?) Q?26???dA? (1? ?) Q?26?dA+ h v?? ] ?? (1????) Q?26?dA +Q?h,v 22dA v2 (3.7i)Q?26? dA?(1 + ?) ??Q?26?dA ?
2 v
?(?1? ?) ?? Q?26?
2dA+
2
h Q?22? dAh,v v
Q? 2
? 26
? dA
(1 +???)
2?? h Q? 2
2 26
? dA
Q? ? dA
h,v 22 h ?? ??
Q?26?
2dA
K45 =(1 + ?) ?Q?
2
?16? dA? (1 + ?)??
h 2
?? 2 ??Q?12? dA (3.7j)v Q? ? dAh,v 22 h,v
Q? ?2dA
K46 =???(1? ?) Q?16??
2
? dA+ (1?
26
?)?? v Q? ?2?? 12 dA (3.7k)v Q? ?2dAh,v 22 h,v?? Q? ?2dA2 ? h,v 12K55 = 2?? Q?11? dAh,v ???
Q?12? dA (3.7l)
Q? 2
h,v 22
? dA h,v
K = Q? 266 11? dA? ? Q?12?2dA ??h,v Q?12?2dA (3.7m)
h,v Q?22?
2dA
?? h,v
h,v
??
where denotes an integral over the upper and lower walls, denotes an integral
h v
over the side walls, ? is the chordwise coordinate, ? is the heightwise coordinate,
? = (? ? 1)/(1 + ?) and ? = (w/h)(Gv/Gh). Gv and Gh are the effective shear
stiffnesses defined by ( )
2
1 A?16(h,v)
Gh,v = A?66(h,v) ? (3.8)
t A?11(h,v)
102
where
A2
? 12(h,v)
A
? 12(h,v)
A26(h,v)
A?11(h,v) = A11(h,v) , A?16(h,v) = A16(h,v)
A22(h,v) ? A22(h,v)A2 626(h,v) (n) (n)
A?66(h,v) = A66(h,v) ? , Aij(h,v) = Q? t
A ij ply22(h,v) n=1
3.3.2 3-D Finite Element Model
The 3-D finite element models for all three ply layups were generated in ANSYS
using 14400 SOLID186 elements, with 120 elements along the length, 10 elements
along the width, 6 elements along the height, and 2 elements through the thickness,
with the mesh of the cross-section shown in Fig. 3.19. Using the same loading
procedure discussed in Sec. 3.1.2, the loads were applied at the right end (x = 60
cm) of the composite box beams with the left end (x = 0 cm) fully constrain in all
three degrees of freedom. For all three ply layups, the same loading magnitudes
were applied, as summarized in Table 3.6, which were selected to produce maximum
strains of about 500 ? under an axial load and about 2500 ? under the remaining
five loading conditions.
Figure 3.19: Cross-sectional mesh of the composite box beams.
After the boundary conditions were applied for the beams under all six loading
conditions, the displacements and strains were calculated with a few representative
results shown in Fig. 3.20 and Fig. 3.21. The shear strain ?+12 under an axial load
103
for all three ply layups is shown in Fig. 3.20. For the uncoupled beam, shown in
Fig. 3.20(a), the shear strain is nearly zero throughout the entire beam. For the
beam with the symmetric layup under an axial load, shown in Fig. 3.20(b), the shear
strain is non-zero on both the top and bottom walls where the plies are oriented at 20?
and -20?, respectively. This shear strain arises due to the extension/shear coupling
and results in a small shear deformation in the chordwise direction (b?2). For the beam
with an antisymmetric layup under an axial load, shown in Fig. 3.20(e), the shear
strain is again non-zero on both the top and bottom walls and the beam exhibits a
negative torsional rotation about the beam?s axis (b?1) due to the extension/torsion
coupling.
Table 3.6: Applied loads for the composite box-beams
Loading Surface Traction (3-D model) Equivalent Beam Load
Axial f1 = 7.10? 107 N/m2 F1 = 3930 N
Lag shear f 62 = 1.92? 10 N/m2 F2 = 106 N
Flap shear f3 = 1.38? 106 N/m2 F3 = 76.1 N
Torsional f1,t = 39.0 N M1 = 1.89 N-m
Flap bending m2 = 2.60? 107z N/m2 M2 = 45.7 N-m
Lag bending m3 = 1.47? 107y N/m2 M3 = 63.5 N-m
The axial strain ?+11 under a flap bending load for all three ply layups is shown
in Fig. 3.21. For the uncoupled beam shown in Fig. 3.21(a), the axial strain
distribution was nearly constant along the span but varied linearly in the flapwise
direction (b?3) from about -1250 ? to about 1250 ?. For the beams with the
symmetric and antisymmetric layups, shown in Fig. 3.21(b-c), both the displacement
and axial strain were about 2.5 times as much as the for the uncoupled beam. For
the antisymmetric layup, the axial strain was still nearly constant along the span but
varied linearly from about -3300 ? to about 3300 ? in the flapwise direction (b?3);
however for the symmetric layup, strain concentrations arose near both the root and
tip of the beam where the boundary conditions were applied causing larger variations
104
(a) Uncoupled layup
(b) Symmetric layup
(c) Antisymmetric layup
Figure 3.20: Shear strain ?+12 in surface basis E = (e?1, e?2, e?3) under an
axial load, f1 for the three composite box beam layups.
along the span. In addition, for the antisymmetric layup, the beam exhibited a small
amount of lateral deflection in the chordwise direction (b?2) due to the shear/bending
coupling, while for the symmetric layup, the beam exhibited a significant rotation
about the beam?s axis (b?1) due to the torsion/bending coupling present in the beam.
105
(a) Uncoupled layup
(b) Symmetric layup
(c) Antisymmetric layup
Figure 3.21: Axial strain ?+11 in surface basis E = (e?1, e?2, e?3) under a flap
bending load, m2, for the three composite box beam layups.
3.3.3 Stiffness Results
Once the data reduction matrices were assembled, using the sectional stress
resultants along the span, the rotated strains from the ANSYS models, and the
numerical warping field, the compliance and stiffness matrices were calculated at
discrete locations every cm along the span, with the results for the diagonal entries of
the stiffness matrices shown in Figs. 3.22-3.27 for all three composite layups. Since
106
Figure 3.22: Spanwise variations of the axial stiffness, K11, for the
composite box beams.
Figure 3.23: Spanwise variations of the lag shear stiffness, K22, for the
composite box beams.
107
no analytic solution exists for the composite beams, the stiffness results were all
compared to their predicted values from SectionBuilder and predictions available in
literature. For the uncoupled beam, whose values from the 3-D FEM strains are shown
as hollow blue circles (?), the calculated stiffnesses were all nearly constant along
the span and overall showed great agreement with the stiffness values predicted from
SectionBuilder, shown as solid lines. Only in the case of the flap shear stiffness (K33)
and torsional stiffness (K44) are discrepancies observed, with the values calculated
from the 3-D FEM strains about 7% lower than the predicted values. In the case of
the coupled beam with a symmetric layup, whose calculated results from the 3-D
FEM strains are shown as filled orange circles ( ), the stiffness values showed some
variations along the span, particularly near the tip of the beam (x = 60 cm). Overall
the calculated stiffness values and those predicted from SectionBuilder, shown as
dashed lines, showed good agreement except for in the flap shear (K22) and torsional
(K44) stiffnesses, which both had discrepancies of about 10%. For the coupled beam
with the antisymmetric layup, whose results are shown as hollow green squares
(), the stiffness values calculated from the 3-D FEM strains showed the greatest
spanwise variations. Overall, the calculated stiffness values for the antisymmetric
layup show fairly good agreement with the results predicted by SectionBuilder, shown
as dashed-dotted lines; however, the lag bending (K66) and flap shear (K33) were
both about 20% less than the predicted values.
108
Figure 3.24: Spanwise variations of the flap shear stiffness, K33, for the
composite box beams.
Figure 3.25: Spanwise variations of the torsional stiffness, K44, for the
composite box beams.
109
Figure 3.26: Spanwise variations of the flap bending stiffness, K55, for
the composite box beams.
Figure 3.27: Spanwise variations of the lag bending stiffness, K66, for
the composite box beams.
110
While the diagonal entries of the stiffness matrices for the three composite beams
showed generally good agreement with the predicted values from SectionBuilder, the
goal in selecting multiple layups was to also verify the capability to measure the
elastic coupling terms that appear in the off-diagonal entries of the stiffness matrix.
For the beam with the symmetric layup, there should be both a coupling between
the flap bending and torsional deformations, represented by a non-zero stiffness K45,
and between the extension and lag shear deformations, represented by a non-zero
stiffness K12. The spanwise variations of both of these elastic coupling stiffnesses are
shown in Fig. 3.28; the extension/shear coupling stiffness (?) showed overall good
agreement with the value predicted by SectionBuilder, but with some variations near
the beams edges. The bending/torsion coupling () showed a discrepancy of about
10% between the calculated and predicted stiffness value, with significant spanwise
variations near the tip of the beam that mirror the calculated torsional stiffness
Figure 3.28: Spanwise variations of the extension/shear coupling
stiffness, K12, and the bending/torsion coupling stiffness, K45, for the
symmetric composite box beam layup.
111
distribution from Fig. 3.25.
For the beam with the antisymmetric layup, there should be three coupling
terms present; an extension/torsion coupling, represented by a non-zero stiffness
K14, and two elastic couplings between the shear and bending deformations, with a
flap bending/lag shear coupling given by K25 and a lag bending/flap shear coupling
given by K36, with the spanwise distributions shown in Fig. 3.29. Although there
are significant spanwise variations in the extension/torsion coupling stiffness(?), the
mean of the variations was close to the value predicted by SectionBuilder, shown as
a solid line. For the shear bending couplings, the flap bending/lag shear coupling
K25( ) had good agreement with the value predicted by SectionBuilder shown as a
dashed line, but with some variation along the span; however, the lag bending/flap
shear coupling K36() showed a discrepancy of about 30%.
Figure 3.29: Spanwise variations of the extension/torsion coupling
stiffness, K14, and the bending/shear coupling stiffnesses, K25 and K36 n,
for the antisymmetric composite box beam layup.
Although there are some spanwise variations in the stiffness properties cal-
112
culated from the 3-D FEM strains, which arise from the boundary conditions and
the longer diffusion lengths inherent to composite materials, the cross-sectional
geometry and material distribution was constant along the span and it can also
be useful to compare the average stiffness properties along the entire span with
numerical predictions. The uncoupled beam with the [0?]6 layup on all four sides has
been extensively studied in past literature and the calculated results are compared
with those from SectionBuilder, the reported results from VABS [126], and those
calculated using the equations from Smith and Chopra [66] that are summarized
in Sec. 3.3.1 and denoted by CLPT (Composite Laminate Plate Theory). For the
uncoupled beam, the only non-zero stiffnesses were the six entries along the diagonal,
whose average values are compared in Table 3.7. The calculated stiffnesses from 3-D
FEM show good agreement with the numerical predictions, except for in the flap
shear (K33) and torsional (K44) terms which have discrepancies of about 7%. Note
that the stiffness values calculated by treating the walls as composite laminated
plates (CLPT) also show good correlation with the other numerical methods, except
for in the shear stiffness terms which are both overpredicted.
For the symmetric layup, the beam presented non-zero stiffness values for
the extension/shear coupling (K12) and the bending/torsion coupling (K45), whose
average values are compared in Table 3.8 along with the diagonal components. Since
Table 3.7: Average stiffness of the uncoupled composite box beam
Stiffness Calculated from Numerical Predictions
Term 3-D FEM Strains SectionBuilder VABS CLPT
K11 [N] 7.856? 106 7.858? 106 7.858? 106 7.858? 106
K22 [N] 1.968? 105 1.974? 105 1.980? 105 2.144? 105
K 4 433 [N] 7.896? 10 8.452? 10 8.496? 104 11.78? 104
K44 [N-m
2] 21.8 23.4 23.5 23.0
K55 [N-m
2] 249.4 249.4 249.4 249.4
K [N-m266 ] 615.8 616.0 617.0 616.0
113
Table 3.8: Average stiffness of the symmetric layup composite box beam
Stiffness Calculated from Numerical Predictions
Term 3-D FEM Strains SectionBuilder CLPT
K11 [N] 5.648? 106 5.674? 106 6.252? 106
K 5 512 [N] ?7.384? 10 ?7.477? 10 ?10.01? 105
K22 [N] 4.051? 105 4.109? 105 5.437? 105
K33 [N] 9.179? 104 9.974? 104 11.78? 104
K44 [N-m
2] 34.0 38.5 45.0
K 245 [N-m ] 31.0 35.0 52.9
K55 [N-m
2] 130.6 134.0 180.2
K66 [N-m
2] 454.1 454.1 546.5
no results are available in literature for this layup using VABS, the calculated results
are only compared with SectionBuilder and those using the CLPT equations. For
the stiffness values calculated from 3D FEM and the SectionBuilder predictions,
the main discrepancies occurred in the flap shear (K33), torsional (K44), and flap
bending/torsional coupling (K45) stiffnesses with differences of about 10%. In all
cases, the results using the CLPT equations correctly predicted the sign and overall
order of magnitude, but overpredicted the actual stiffness values by 10 to 25%. For
the antisymmetric layup, the beam presented an extension/torsion coupling and two
shear/bending couplings, with the average values of the nine non-zero components
compared to the numerical predictions in Table 3.9, with the VABS results coming
from Yu et al. [29]. The agreement between the calculated values and numerical
predictions from SectionBuilder was good, with the exception of the flap shear
(K33), lag bending (K66), and lag bending/flap shear coupled (K36) stiffnesses, which
showed differences of about 20%,
Since both the data reduction algorithm and the 2-D cross-sectional analysis
codes rely on calculating the sectional compliance matrix and then inverting to
get the sectional stiffness matrix, it can be useful to also compare the compliance
components, specifically for the case of the antisymmetric layup where the largest
114
Table 3.9: Average stiffness of the antisymmetric layup composite box
beam
Stiffness Calculated from Numerical Predictions
Term 3-D FEM Strains SectionBuilder VABS CLPT
K11 [N] 6.211? 106 6.363? 106 6.363? 106 6.363? 106
K22 [N] 3.937? 105 3.940? 105 2.243? 105 4.852? 105
K33 [N] 1.351? 105 1.749? 105 0.915? 105 2.665? 105
K14 [N-m] 1.164? 104 1.203? 104 1.214? 104 1.427? 104
K25 [N-m] ?5697 ?5840 ?3282 ?6012
K36 [N-m] ?4224 ?6307 ?3323 ?6012
K 244 [N-m ] 47.5 48.4 48.2 51.7
K [N-m255 ] 189.0 194.2 152.0 202.0
K66 [N-m
2] 398.3 494.6 384.6 498.8
differences were noticed between the results. For this case, the compliance matrices
calculated from the 3-D FEM strains and from both SectionBuilder and VABS are
compared in Table 3.10. While large differences were observed between the calculated
stiffnesses and those from SectionBuilder for the flap shear, lag bending, and lag
bending/flap shear coupled components, the difference was primarily observed only
Table 3.10: Average compliance of the antisymmetric layup composite
box beam
Stiffness Calculated from Numerical Predictions
Term 3-D FEM Strains SectionBuilder VABS
S11 [1/N] 2.980? 10?7 2.966? 10?7 3.025? 10?7
S22 [1/N] 4.505? 10?6 4.579? 10?6 6.518? 10?6
S33 [1/N] 1.108? 10?5 1.058? 10?5 1.593? 10?5
S14 [1/(N-m)] ?7.300? 10?5 ?7.379? 10?5 ?7.620? 10?5
S25 [1/(N-m)] 1.358? 10?4 1.377? 10?4 1.407? 10?4
S36 [1/(N-m)] 1.175? 10?4 1.350? 10?4 1.376? 10?4
S44 [1/(N-m
2)] 3.896? 10?2 3.904? 10?2 3.999? 10?2
S55 [1/(N-m
2)] 9.384? 10?3 9.289? 10?3 9.618? 10?3
S66 [1/(N-m
2)] 3.757? 10?3 3.743? 10?3 3.789? 10?3
115
in the lag bending/flap shear coupled compliance S36. This shows that, because of
the process of inverting the compliance matrix, a discrepancy in a single compliance
matrix property Sij can propagate into discrepancies in any of the non-zero stiffness
properties with an i or j subscript. Similar observations were observed with the
VABS predicted properties; large discrepancies in the two shear compliance values,
S22 and S33, propagated into discrepancies in the six related stiffness properties, K22,
K25, K33, K36, K55, and K66, despite the fact that the S25, S55, and S66 compliance
properties showed good correlation with one another.
3.4 Summary and Conclusions
The proposed strain-based methodology for calculating the sectional stiffness
properties was validated using strains predicted from 3D finite element models in
ANSYS. The first section considered the case of a prismatic beam with a rectangular
cross-section and made of an isotropic material; the stiffness values calculated using
the data reduction algorithm on the predicted strains from ANSYS were validated
with analytic solutions. The results agreed very well with one another for all six
non-zero stiffness matrix components. Since the beam had a constant cross-section,
the stiffness properties were expected to remain constant along the span with the
only minor spanwise variations in the calculated properties observed near the beam?s
edges where the boundary conditions were applied. This simple beam also verified
that the proposed methodology can measure all six diagonal stiffness terms, as
opposed to only three stiffness values ? K44, K55, and K66 ? that can be measured
with existing techniques.
The second section considered a beam with a rectangular cross-section made of
an isotropic material, with the cross-sectional dimensions decreasing linearly along
the span. Two levels of taper were analyzed to verify the capability of the proposed
data reduction algorithm to handle spanwise variations in properties; with exception
116
to the flap shearing stiffness K33, the calculated stiffness properties showed great
correlation with the analytic solutions that were derived for the case of a prismatic
beam. For the flap shearing stiffness, the boundary condition and taper cause
larger discrepancies of greater than 10% near the clamped end of the beam, but the
results end up showing much better correlation near the tip where the cross-sectional
dimensions were smallest. Overall, this case verified that the current strain-based
method is able to predict the sectional stiffness locally along the beam and allow for
measurements on non-prismatic beams, as opposed to most existing methods which
can only measure a global stiffness value for the entire beam.
Finally, the capability of the implemented data reduction equations to measure
off-diagonal coupling terms is verified using the composite box-beam geometry
initially studied by Chandra and Chopra and subsequently used for validating
various cross-sectional analysis codes. Calculated stiffness values were presented
for three different ply layups, including an uncoupled layup whose only non-zero
stiffness values were the six diagonal components, a symmetric layup with non-zero
K12 and K45 stiffness values in addition to the six non-zero diagonal terms, and an
antisymmetric layup with nonzero K14, K25, and K36 stiffness values in addition
to the six non-zero diagonal terms. While some larger discrepancies were observed
between the calculated and predicted stiffness values, this case verified that the
proposed methodology could measure the full 6 ? 6 stiffness matrix including all
expected non-zero off-diagonal terms, in contrast with past experimental techniques
that have measured the torsion/flap bending off-diagonal stiffness component only.
The discrepancies also highlight a shortcoming of using only the central solution for
a composite beam; since composite materials have a much larger diffusion length,
the end effects propagate significantly farther along the beam?s span than for a beam
made of an isotropic material.
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Chapter 4: Uncertainty Quantification
This chapter focuses on propagation of the uncertainties from the experimental
measurements into the final stiffness matrix calculation. An overview of the types of
experimental uncertainties is first discussed, followed by a discussion of two common
methods for analysis of uncertainty propagation ? a Monte Carlo Method and a
Taylor Series Method. A comparison of the two methods is then made by applying
the uncertainty propagation to the results of the prismatic aluminum beam and the
composite box-beam with extension-torsion coupling from the numerical validation
models of Chapter 3.
4.1 Types of Experimental Errors
When working with experimental data, the total error will consist of contri-
butions from random (or precision) errors and systematic (or bias) errors [128].
The random errors are associated with the noise, or random variations within the
measurement system and, because of this, if the measurement were taken an infinite
number of times and averaged, there would be no impact on the total measurement.
However, it is often impractical to take a large enough number of samples to com-
pletely eliminate the random variations from the measurement and it therefore must
be considered in the uncertainty quantification. The random error for a measure-
ment system can be approximated by taking a sample set of data in a controlled
environment. Typically, this error will take the form of a Gaussian distribution, as
shown in Fig. 4.1(a). If a total of N measurements are obtained in the sample set,
118
the uncertainty can be estimated from the standard deviation ?r of these readings
given by [
1 ? ]N 1/2
?r = (X
(j) ? ?)2 (4.1)
N ? 1
j=1
where X(j) is the value of the jth measurement and ? is the mean value of the
data samples. A common practice when performing an uncertainty analysis is to
provide a 95% confidence interval on the measurement which can be approximated
by Sr = ?2?r [128].
(a) Random error distribution (b) Possible systematic error
from a sample data set distributions
Figure 4.1: Combined random and systematic error distributions.
The second type of error is the systematic error, which is a constant error
within the system resulting from the calibration of the measurement device. This a
fixed error that will be present in the measurement, regardless of how many times
the experiment is performed, which is why it is often regarded as a bias error in the
measurement. Because the true value of any experimental variable, such as mass
or temperature, is not known with 100% accuracy, the systematic error can not
easily be identified or removed from an experiment and must be estimated from
knowledge of the calibration procedure, which is typically provided by the vendor
of the equipment. It is also typically assumed that the systematic error will take
the form of a Gaussian distribution, as seen in Fig. 4.1(b). In this case, the 95%
119
confidence level means that it is 95% certain that the systematic error B is within
the uncertainty range Sb = ?2?b of the true value of the measurement variable,
based on expectations from the calibration procedure.
Oftentimes, the desired results are calculated from multiple measured variables,
each of which has its own uncertainty distributions. The total uncertainty for the ith
measurement variable Xi can then be represented by
u2X = S
2 2
i b,i
+ Sr,i (4.2)
where uX provides the 95% uncertainty interval. Although it is not possible toi
reduce the systematic uncertainty of the experiment, the random uncertainty can
be reduced if the test is repeated M times using the same test setup. In this case,
the results can be averaged across multiple tests allowing for the random noise to
be partially filtered out of the experiment and the associated uncertainty interval
becomes
S2
2 2 r,iuX = Sb,i + (4.3)i M
4.2 Experimental Error Quantification
In the measurement of the sectional compliance and stiffness matrices, there
are multiple variables being measured and used in the data reduction procedure
that have associated errors. These measurements include the loads, the strains, and
the locations of the strain measurements. In addition, while not explicitly used in
the data reduction equations, the material properties are required to predict the
numerical warping field and their associated uncertainties must also be taken into
account. The uncertainty intervals can be represented by either a percent of their
measured values, which can be the case for loads and material properties, or as a
dimensional value, which is more typical for measurement locations and strains.
120
Considering first the applied loads, these are normally measured through the
use of a load cell, either at the root or the tip of the beam. In this work, the test
articles were mounted to a 6-axis load cell at their clamped end to measure the three
reaction forces and three reaction moments under the applied loading conditions.
Extensive calibration was performed for the load cell, as described in Sec. 5.3.2, and
was used to estimate the expected systematic and random uncertainties shown in
Table 4.1. Calibrations were separately calculated at multiple load levels, described
as a low and high loading, so that the 95% confidence intervals could be calculated
as a percentage of the applied load instead of a dimensional loading based on the
maximum rating of the load cell. This was also done so that the uncertainty levels
for low loading conditions would not end up leading to unreasonable uncertainty
levels in the calculated stiffness properties. Note however that the load cell was
still usually most accurate at load levels closest to its maximum rating, which could
generally be observed in comparing the uncertainty bounds between the low and
high loading columns in Table 4.1.
In the measurement locations, there are several uncertainties that can impact
the calculation of the stiffness properties. Since the loads are only measured at
the base of the beam, the cross-sectional loads F c must be calculated from these
measurements based on the spanwise location x and corresponding deflections and
rotations at this location. The coordinates measured by DIC are only relative to
a local coordinate system assumed by the DIC system; markings on the beams
are used to determine the absolute spanwise position which is estimated to have
a systematic uncertainty with a 95% confidence interval of Sb,x = ?1 mm. The
random uncertainty in the spanwise location comes from the variations in defining
this position in the DIC software for each test, with an assumed random uncertainty
of Sr,x = ?0.25 mm. In addition to the uncertainties in the spanwise locations,
there are also uncertainties of the measurement locations within the cross-section.
121
Table 4.1: 95% confidence intervals for systematic and random
uncertainties of a 6-axis load measurement.
Low Loading High Loading
Bias, Sb,F Random,Sr,F Bias, Sb,F Random,Sr,F
Load bounds |F1| < 800 N 800 N < |F1| < 6250 N
F1
Uncertainty (%) 0.75 0.87 0.75 0.87
Load bounds |F2| < 105 N 105 N < |F2| < 2500 N
F2
Uncertainty (%) 1.02 0.64 1.46 0.89
Load bounds |F3| < 105 N 105 N < |F3| < 2500 N
F3
Uncertainty (%) 2.92 1.49 1.29 1.16
Load bounds |M1| < 40 N-m 40 N-m < |M1| < 400 N-m
M1
Uncertainty (%) 1.60 1.26 2.01 2.04
Load bounds |M2| < 60 N-m 60 N-m < |M2| < 400 N-m
M2
Uncertainty (%) 1.00 0.20 1.00 0.10
Load bounds |M3| < 60 N-m 60 N-m < |M3| < 400 N-m
M3
Uncertainty (%) 0.95 0.28 1.10 0.12
From the DIC calibration procedure, the actual measurement locations for the entire
cross-section were calculated to have an uncertainty of about 0.02 pixels, which
corresponds to a 95% confidence interval of Sb,d = ?0.0025 mm. Comparing the
cross-sectional measurement locations across multiple tests provided an estimate for
the 95% confidence interval on the random uncertainty of Sr,d = ?0.013 mm. Finally,
there were also uncertainties in the actual dimensions of the cross-section; because
only two cameras were used in the DIC setup to measure the displacements and
strains on a single external surface, the DIC system could not provide an accurate
measurement of the external dimensions of the cross-section, particularly in the
depth, or thickness, of the cross-section. These dimensions were instead measured
using a digital calipers, with an associated 95% confidence level of Sb,dext = ?0.025
mm. From measurements of the external dimensions at multiple locations along
the span for the various test articles, the 95% confidence interval for the random
uncertainty was estimated at Sr,dext = ?0.8%, resulting from manufacturing and
122
machining tolerances.
The strain is then calculated in the DIC software using derivatives of the
measured displacement fields. The accuracy in the relative displacement between two
points using the DIC software was estimated to be 0.0025 mm, which corresponded
to a 95% confidence interval for the measured strain of 10.9 ?. The cameras were
also mounted on a tripod that was moved vertically during testing and could result
in some rigid body motion between sets of test images; simple tests were performed
by translating the cameras, as discussed in Sec. 5.2.6, to estimate the expected
uncertainty that could result from this rigid body motion, which was found to
introduce about ?16? into the measurement. The total systematic uncertainty in
the strain measurement was then estimated to have a 95% confidence interval of
about Sb,?+ = ?26.9?. The random uncertainty was then estimated based onmean
the variations in the unstrained test images, as discussed in Sec. 5.2.7, with a 95%
confidence interval of Sr,?+ = ?24.0?. These two values represent the uncertaintymean
for the entire strain measurement data set, as indicated by using the subscript ?+mean;
however, there were also random variations observed within the cross-section for
expected states of constant strain. These variations were then measured to provide
a 95% confidence interval on random strain distribution within the cross-section of
Sr,?+ = ?16.1?.std
Finally, there are also uncertainties in the material parameters that need
to be considered. For the case of an isotropic material, any uncertainties in the
Young?s modulus E and poisson?s ratio ? would need to be considered. In this
thesis, aluminum 6061 is the only structural isotropic material and the bounds
for the 95% confidence intervals on the properties are assumed to be Sb,E = ?1
GPa for the Young?s modulus and Sb,? = ?0.01 for the poisson?s ratio. For the
case of a transversely isotropic material, such as the composite materials used in
this dissertation, there may be uncertainties in the material properties, such as the
123
Young?s moduli, E1 and E2, poisson?s ratios, ?12 and ?23, and shear modulus G12, as
well as in the layup orientation angles given by ?ply. From the Composite Materials
Handbook [129], typical coefficients of variance in the measured Young?s moduli and
shear modulus were between 4% and 7.5%. Assuming the upper bound of these
covariances, the 95% confidence intervals for the Young?s moduli and shear modulus
were taken as Sb,E1 = Sb,E2 = Sb,G12 = ?15%. Although the coefficient of variance
for the poisson?s ratio were not described, by comparing materials of similar moduli,
a reasonable estimate on the 95% confidence interval for the poisson?s ratio was
taken as Sb,?12 = Sb,?23 = ?0.025. For the ply orientation angles, it was assumed that
during the manual layup of the plies, the actual layup angle could be produced with a
95% confidence of S ?b,? = ?3 . Since the material parameters are only required forply
the calculation of the warping field, there uncertainties are not expected to impact
all the measured stiffness properties in most cases. Indeed, it was seen in Sec. 2.2.4
that the warping correction had no influence on the predicted axial strains in many
loading conditions.
4.3 Uncertainty Propagation
Determining the uncertainty for the calculated stiffness matrix components
requires propagating the errors from each of the measured variables into the final
calculation. Two popular methods for uncertainty propagation were investigated ? a
Taylor series method and a Monte Carlo method. The nature of the data reduction
equations, namely the fact that the use of DIC to measure the strains leads to a
system of highly overconstrained equations for the compliance matrix components
that is solved using a SVD and that the stiffness matrix is calculated as the inverse of
this compliance matrix, presents additional considerations for applying conventional
techniques to the uncertainty propagation, particularly for the Taylor Series method.
124
4.3.1 Taylor Series Method
The stiffness matrix calculation relies on measurements of the strain, force,
and strain locations, in addition to the material parameters required to calculate
the numerical warping field, all of which have associated uncertainties. A Taylor
series method provides the impact of these uncertainties in the measurements on
the calculated stiffness values through derivatives of the data reduction equation.
To understand the necessary approximations, it is useful to consider the derivation
of the Taylor Series uncertainty method as discussed by Coleman and Steele [128].
If the data reduction equation is a function of measured variables X1 and X2 and
given by
f = f (X1, X2) (4.4)
then the result for the jth set of measurements is given by f (j). The measurement of
(j)
each variable Xi will have an associated systematic and random error, given by Bi
(j)
and Pi , respectively. The value of these variables can be determined from their
parent distributions whose 95% confidence intervals are described by Sb,i = 2?b,i and
Sr,i = 2? . The j
th
r,i measurement of each of these variables is then given by
(j) (true) (j) (j)
Xi = Xi +Bi + Pi (4.5)
(true)
where Xi would be the actual measured value is there were no error present.
Approximating the function f about the true measured values using a Taylor series
expansion gives
( ) ( ) (( ) )2
f (j) = f (true)
?f (j) (j) ?f (j) (j)
+ B1 + P1 + B2 + P2 +O
(j) (j)
Bi + Pi (4.6)?X1 ?X2
(j) (j)
In most cases, the higher-order terms given by O((B 2i +Pi ) ) are assumed to
be negligible, which is valid as long as the higher-order derivatives are of reasonable
125
magnitude and the total errors are small. This small error assumption is true of
the measured loads and strain locations, whose associated uncertainties are all less
than 3%. The small error assumption is also true of the measured strain as long
as it is relatively large ? several hundred or thousand ?; this is normally the case
for the axial strain and the shear strain under a torsional load. However, the shear
strain under a shear force is typically around 100 ?, due to limitations based on
the resulting bending moments induced by the shear forces, and may lead to some
errors in this approximation, which will be investigated in Sec. 4.4 by comparing
results with the Monte Carlo method. Rearranging Eq. (4.6) and neglecting the
higher order terms gives
( ) ( )
(j) (j) ? (true) ?f (j) (j) ?f (j) (j)?f = (f f ) = B1 + P?X 1
+ B2 + P?X 2
(4.7)
1 2
(j)
where ?f is the total error in the j
th calculation of f (j). Using the definition for the
standard deviation given by Eq. (4.1), the variance of the calculated value f can be
computed as
?N1 ( )2 (j) 2?f = ?N f( j=1 ) ?2 ? ?N?f 1 ( ) ? ?N ( ) ?N ( )(j) 2 2 (j) (j) 1 (j) 2= B + B P + P ?+
?X N? 1 N 1 1 N 1( 1 ) j=1 j=1 j=1N N N
?f 2 ? 1 ?( )2 2 ?( ) 1 ?( ) ?(j) (j) (j) (j) 2+ B + B P + P ?+ (4.8)
?X N? 22 N 2 2 N 2( ) j=1? ?N?f ?f 2 ( )
j=1 ?N ( )j=1
(j) (j) 2 (j) (j)
+ B1 B2 + B1 P +?X1 ?X2 N N
2
? j=1 ?j=1N N2 ( ) ?( )(j) (j) 2 (j) (j)
+ P1 B + P P ?N 2 N 1 2
j=1 j=1
Now, assuming that there is no cross correlation between the systematic and
random errors and that there is no covariance between the errors, such that the
126
(j) (j) (j) (j)
terms containing products of Bi Pk and the terms containing products Bi Bk
(j) (j)
and Pi Pk all evaluate to zero, respectively, Eq. (4.8) can be simplified to
( )2 ( )2
2 ?f ( ) ( )? = ?2 + ?2 ?f+ ?2f b,1 r,1 b,2 + ?2?X ?X r,2 (4.9)1 2
These two assumptions, that there is no cross correlation and that the covariance is
zero, are valid for most cases and should apply to the measurements for the stiffness/-
compliance matrix data reduction equation. As an example of a situation where this
is not true would be the case of using a thermometer to measure multiple different
temperatures needed for a data reduction equation or using two thermometers that
are calibrated to the same source.
It is common to represent the uncertainty in the final result f based on a 95%
confidence interval. By multiplying Eq. (4.9) by a factor of 22 and taking the square
root, the 95% confidence interval for the desired result f , denoted by uf , can then
be expressed as [( ) ]2 ( )2 1/2
?f ?f
uf = u
2
X + u
2
?X 1 ?X X
(4.10)
2
1 2
For the general case where f is a function of more than two variables, Eq. (4.10)
can then be easily extended to give
[( ) ]2 ( )2 ( )2 1/2
?f ?f ?f
u 2 2 2f = uX + u + ? ? ?+ u (4.11)?X 11 ?X X2 ?X Xn2 n
For the case of the sectional stiffness matrix, where the individual components
can have various orders of magnitude, it can be desired to have the uncertainty in a
non-dimensional form as a fraction of the total stiffness, which can be accomplished
by dividing both sides of Eq. (4.11) by f and multiplying the individual variable
127
uncertainties by Xi/Xi to give
[( )2( )2 ( )2( )2
X1 ?f 2 X2 ?fU = U + U2f +
f ( ?X
X
) ( 11 ) f ] ?X
X2
2
(4.12)
2 2 1/2
+ ? ? ? Xn ?f+ U2
f ?X Xnn
where UX = uX /Xi is the 95% confidence interval for the measured variables as ai i
fraction of the measured value and Uf = uf/f is the 95% confidence interval of the
result as a fraction of the actual value.
To provide a simple example of an application of the Taylor Series uncertainty
quantification, consider the calculation of the bending stiffness from the measurement
of the tip displacement given by Eq. (1.7) (EI = FL3/(3wtip)). In this calculation of
the bending stiffness, there are three measured variables, the tip force F , the beam
length L, and the tip displacement wtip, each of which has an associated random and
systematic error. Using Eq. (4.12) with f = EI, X1 = F , X2 = L and X3 = wtip,
the 95% uncertainty in the measurement of the bending stiffness can be calculated
by ( ) [ ] (( ) ( ) )u 2 F L3 2 2 2EI Sb,F 1 Sr,F
= ( + +EI E[I 3wtip ]2 (F ) M2 (F ) )2 2L 3L F Sb,L 1 Sr,L
+ + +
[EI 3wtip ] ((L )M L( ) ) (4.13)2 2 2
wtip ?FL3 Sb,w 1 Stip r,w+ + tip
( E)I 3(w( 2tip) ) ( wtip) M wtipu 2 u 2 u 2F L w
= + 9 + tip
F L wtip
In this simple example, because the bending stiffness has a cubic dependency on the
beam length, any uncertainties in the measurement of the length have an impact
nearly an order of magnitude greater then similar uncertainties in the force or the
tip on the final results.
128
Now applying this to the compliance and stiffness matrix calculation discussed
in Chap. 2, the calculation requires measurements of the force, strain, and cross-
sectional location. In addition, the material properties, although not explicitly used
for the calculation of the sectional stiffness properties, are required for the calculation
of the numerical warping field that are ultimately used to calculate these properties.
The uncertainties in the stiffness properties can then be determined with these inputs
using Eq. (4.12), which evaluates to
[?( ) (6 2 ( ) ( ) )1 ?Kij S 2 2F b,Fc,k 1 Sr,FU = + c,kK +ij c,kKij (?Fc,k ) Fc,k n Fc,kk?=12 ?6 2?Kij
+ S2? +??+ ? b,?+mean?k=1 l=1
1k,F
? c,l? ? ??22 6 ( )+ ???Kij ?? 1 2 1S 2r,?+ + S + ++ n mean nN r,?std?k=1 (l=1 ?)?(1k,Fc,l ) (4.14)2 2?Kij 1 1
+ ( ) S
2
b,d ,k+ (Sr,d ,k)
2 (d )2ext,k +S
2 + 2
? ?d
ext
k ?n
ext b,dk
k=1
nE 2 nG ( )2 ?n? ( n)
S
N r,dk
2
?Kij 2 ?Kij 2 ?Kij+ S 2
?E b,E
+ S
k ? ?G b,G
+ S
k ?? b,?( ) kk k kk?=1 k=1 k=12 1/2nply ?Kij
+ + S
2 ?
b,?+?? ply,k
k=1 ply,k
where UK is the non-dimensional 95% confidence interval for the stiffness matrixij
value in the ith row and jth column. Because the data reduction is solved for the
compliance matrix entries using a singular value decomposition and the stiffness
matrix is then calculated as the inverse of the compliance matrix, the partial
derivatives (?Kij/?Xk) can not generally be evaluated analytically and instead must
be approximated numerically. Although a finite difference approximation could be
129
used to estimate these derivatives, a complex step differentiation is used as given by
?Kij 1
= Im [Kij (X1, X2, . . . , Xk + ik, . . . , Xn)] (4.15)
?Xk k
where i is a small imaginary perturbation to the measured variable. The complex
step derivative will typically provide better accuracy in numerical differentiation, as
it does not require a difference of function evaluations, with the derivation for the
complex step derivative discussed in Appendix B.
Considering Eq. (4.14) in detail, the first line describes the contribution to
the total uncertainty from the uncertainty in the loads. From Sec. 2.2.4, it was
found that all six components in the jth column of the compliance matrix were
calculated from the kth component of the sectional loads Fc,k. However, bringing the
compliance matrix into symmetric form by S = (S + ST )/2 and then inverting to
u u
calculate the sectional stiffness matrix leads to the possibility that the uncertainty
in all the sectional loads can impact the uncertainty in the Kij term, requiring the
summation from k = 1 to 6 of the partial derivatives with respect to the kth load
component Fc,k. In fact, this summation can only be simplified for the case when
the only non-zero components are the six diagonal stiffness values, in which case the
derivative is only non-zero for i = j = k. Since the force uncertainties are normally
represented as a percent of the applied load, it is generally easier to calculate the
complex step derivative by perturbing the measurement by iFc,k. In this way, the
complex step differentiation gives
F ?Kij 1c,k = Im [Kij (Fc,k + iFc,k, . . .)] (4.16)
?Fc,k 
and the entire quantity Fc,k?Kij/?Fc,k can be easily calculated, allowing the un-
certainty quantities Sb,F /Fc,k and Sr,F /Fc,k to be input as their percentage (orc,k c,k
fractional) values listed in Table 4.1. This is particularly important because during
130
the actual experiments, multiple load cases with different magnitudes are often
applied to calculate the stiffness matrix, and defining an appropriate value of Fc,k to
use in the calculation would otherwise be difficult.
In addition, the data reduction equation requires the forces and moments within
the current cross-sectional location and not the directly measured loads from the
load cell. Because the measured loads are located at a distance x from the current
cross-sectional location, the cross-sectional bending moments will depend on both
the measured bending moments and measured shear forces according to
M2 = M2,m + xF3,m
(4.17)
M3 = M3,m ? xF2,m
where F2,m and F3,m are the measured forces and M2,m and M3,m are the measured
moments in the load cell, while M2 and M3 are the cross-sectional bending moments.
In this relationship, the additional moments due to the displacement and rotation of
the cross-section (see Sec. 5.3.3) are ignored, as these contributions are generally
small compared to the bending moment generated directly by the shear force (xF3,m
or xF2,m). The moment uncertainties from Table 4.1 then need to be updated to
account for the load cases where a bending moment and shear force are both present,
which can be done by applying a Taylor Series method to Eq. (4.17) to give
???? ( ( ) )2 ( )2S = S2 + S2 Sb,x F3xb,M2 ? b,M2,o b,F
+
3
? ( x ) M( ) ( 2,m?? 2 ) (4.18)2Sb,x F2xS 2 2b,M3 = Sb,M3,o + Sb,F +2 x M3,m
where Sb,M2 and Sb,M3 are the updated moment systematic uncertainties, Sb,M2,o
and Sb,M3,o are the original moment systematic uncertainties from Table 4.1, Sb,F3
and Sb,F2 are the shear force systematic uncertainties, and Sb,x/x is the systematic
131
uncertainty in the cross-sectional location. The same procedure can then be followed
to update the random uncertainties by replacing all the Sb,(?) expressions with Sr,(?).
The second and third lines of Eq. (4.14) represent the contribution of the strain
measurement to the overall uncertainty. Since the axial strain measurements are
used to calculate first, fifth, and sixth rows of the compliance matrix and the shear
strain to calculate the second, third, and fourth rows, it is necessary to consider
their contributions separately as represented by the summation from k = 1 to
2. In addition, the derivatives must be calculated separately for each of the six
sectional loads, as indicated by the second summation from l = 1 to 6. For the mean
systematic strain uncertainty, each partial derivative can be calculated by perturbing
the measurements of the strain component ?+1k by i for each load case separately.
However, for the random uncertainty in the strain measurement, as calculated in
the third line, it is necessary to account for( the) sign of the strain component in
the perturbation (i.e. by taking ?+1k + sgn ?
+
1k i) when calculating the partial
derivative. This is due to the fact that the random uncertainty in the strain will
change each time it is measured, as opposed to the systematic uncertainty which
only changes if the DIC system gets recalibrated. Since the strain uncertainty levels
are represented in dimensional terms (?) and not as a percentage of the measured
strain, the imaginary perturbations used to calculate the partial derivatives should
be of a constant magnitude and therefore not be multiplied by the value of the strain
itself as was done for the case of the load uncertainties.
The fourth line of Eq. (4.14) represents the contribution of the measurement
locations to the overall uncertainty. The derivatives must be calculated separately
for both of the cross-sectional dimensions, as indicated by the summation from
k = 1 to 2. The first two uncertainties, Sb,dext,k and Sr,dext,k, are the systematic
and random uncertainties for the measurements of the external dimensions of the
cross-section ? for a rectangular shaped cross-section this would be the width and
132
height and for an airfoil shaped cross-section the chord and thickness. Since the
random uncertainty was estimated as a percentage based on manufacturing and
machining tolerances, Sr,dext,k is multiplied by its measured dimension dext,k at the
current spanwise location to bring it into a dimensional form. The remaining two
uncertainties, Sb,d and Sr,d , are the systematic and random uncertainties of thek k
location of the strain measurement on the external surface. In order to approximate
the partial derivatives for the measurement locations, the y-coordinates used in
defining H (Eq. (2.36)) are perturbed by i when k = 1 and the z-coordinates are
i
similarly perturbed by i when k = 2.
Finally, the last two lines of Eq. (4.14) represent the uncertainty with respect
to the material parameters. Since the material parameters are only needed for the
calculation of the numerical warping field, they only apply to the strain correction
term V F (k)c in the data reduction equation (Eq. (2.39)). For the case of an isotropici
material, the material is fully defined by its Young?s modulus and poisson?s ratio so
the uncertainties with respect to the shear modulus and ply orientations disappear.
For the case of a composite cross-section, which may consist of layers of isotropic
or transversely isotropic materials, it is necessary to consider the uncertainties for
each material by applying the imaginary perturbation to each layer separately. For
the transversely isotropic materials, the 95% confidence intervals were estimated as
a percentage for the Young?s and shear moduli; therefore, in order to apply these
uncertainties correctly in Eq. (4.14), these percentages must be converted to a
dimensional form by multiplying the terms Sb,E and Sb,G by their correspondingk k
moduli.
4.3.2 Monte Carlo Method
Alternatively, a Monte Carlo simulation can be used to perform the uncer-
tainty analysis on the measured stiffness properties, without requiring the inherent
133
approximations used when propagating the uncertainties through the Taylor Series
method [128]. In the Monte Carlo method, the data reduction procedure is simulated
N times to determine the standard deviations and therefore expected uncertainties
in the measured stiffness values. In performing a Monte Carlo simulation for the
uncertainty analysis, the first step requires defining the systematic and random error
distributions for all the measured variables, which in this dissertation are assumed
to follow a Gaussian distribution. In the jth simulation, a random value for the
uncertainties of all the measured variables is then determined using a Gaussian
random number generator. For each variable, the randomly generated systematic
and random uncertainties are summed together with the initially measured value to
give an updated ?measured? value of the ith variable in the jth simulation as
(j) (j) (j)
Xi = Xi +Bi + Pi (4.19)
(j)
where Xi is the current ?measured? value, Xi is the initially measured value, and
(j) (j)
Bi and Pi are the systematic and random uncertainties, respectively, generated
randomly from their respective parent distributions defined by Sb,i and Sr,i for the i
th
measured variable in the jth simulation. The data reduction procedure is then used
 Simulation, 
repeat    times     
Calculate Standard Deviation 
for each Sti ness Property:
Input Measured 
Variables,     
(Strain, Loads, 
Measurement Location,
 Material Properties)
Input Uncertainty
Distributions for 
 Variables,      and       
Figure 4.2: Flowchart for uncertainty quantification using Monte Carlo
method.
134
to calculate the stiffness matrix for the current simulation, K(j), based on the current
(j)
values of each Xi . After completing the N simulations, the standard deviations of
each component of the stiffness matrix are calculated and multiplied by a factor of
two to provide the 95% confidence intervals based on the measured variables. The
number of required simulations can be determined by considering the convergence of
standard deviations of each stiffness parameter; typically between 1,000 and 7,000
iterations were required for convergence, thus 10,000 simulations were used in the
analysis in this work to ensure sufficient convergence. A summary of this work flow
for calculating the uncertainty using the Monte Carlo method is shown in Fig. 4.2.
4.4 Method Comparison
The previous section described two methods for the uncertainty propagation
from the experimental measurements into the calculated sectional stiffness compo-
nents. With the Taylor series method, the total uncertainty in the stiffness properties
is estimated based on first order partial derivatives of the data reduction equation
with respect to the measured variables ? neglecting higher order derivatives. In the
Monte Carlo method, the total uncertainty is estimated by performing a large num-
ber of simulations with random perturbations of the measured variables, generated
from their respective uncertainty distributions. While the accuracy of the Taylor
series method depends on the reliability of the assumption that the higher-order
derivative terms are negligible, the accuracy of the Monte Carlo method only requires
performing a large enough number of simulations such that the standard deviations
of the uncertainties in the stiffness properties have converged. However, because the
number of simulations for convergence is typically much higher than the number
of measurement variables, and therefore numerical derivatives to calculate in the
Taylor series method, the Monte Carlo method is expected to be more expensive
computationally. To compare the two methods in terms of their accuracy and effi-
135
ciency, measured in terms of computational time, both methods were applied to the
calculation of the stiffness properties for two of the cross-sections from the numerical
models considered in Chap. 3. The first comparison was done for the prismatic
aluminum beam, to assess the methods in the simplest case of a cross-section made
of an isotropic material with the six diagonal entries being the only non-zero stiffness
components. A second comparison was then done for the composite box beam,
specifically the cross-section with the symmetric layup, to assess the two methods
when applied to a cross-section with anisotropic materials and non-zero off-diagonal
stiffness components.
4.4.1 Prismatic Aluminum Rectangular Beam
As presented in Sec. 3.1, the stiffness properties were calculated at every cm
along the span for the prismatic aluminum beam. To compare the two methods for
their accuracy and efficiency, the uncertainty analysis was first applied at a single
spanwise location, which was arbitrarily selected to be at 20 cm from the clamped
boundary, such that it was sufficiently far away from any effect of the boundary
conditions in the 3-D FEM. While the uncertainties are applied based on simulated
data from 3-D FEA, experimental measurements will be done using DIC and it is
important to try and mimic how its use will impact the uncertainties. Since it is
common to only have two cameras for DIC, measuring the strain field on the entire
external surface requires taking four separate sets of images ? one for each face.
Since the calibration doesn?t change between sets of images, the bias uncertainty for
each measurement type should only be generated once per simulation; however, the
random uncertainty would vary for each of the sets of images and therefore needs to
be generated four times during each simulation. In addition, since the applied loads
from the 3-D FEM are mostly outside the bounds listed in Table 4.1, the uncertainty
levels for the high loading condition are used for all the results.
136
For the Monte Carlo method, 10,000 simulations were run with the 3-D FEA
results, applied forces, and material properties perturbed by a randomly generated
value from the normal distributions described by the 2? values from Sec. 4.2. Figure
4.3 shows the variations of the 95% confidence intervals for each of the six non-zero
diagonal stiffness matrix components with the number of simulations, from which it
can be seen that any significant variations disappeared after about 2,000 simulations.
To determine the actual number of iterations required for convergence, the value
after 10,000 simulation runs was taken as the reference value and the number of
required simulations to converge to within 2% and 1% of this value was calculated as
summarized in Table 4.2. With an average run time of about 50 s per 100 simulations,
it would take about 870 s (14.5 min.) to converge all the uncertainty bounds to
within 2% of their reference value or 3,785 s (63.1 min.) to converge to within 1%.
Figure 4.3: Convergence of the Monte Carlo method for the uncertainty
levels of the stiffness properties of the prismatic aluminum beam.
The uncertainty levels were then calculated at the same spanwise location using
the Taylor series method, with a comparison of the two methods shown in Table
137
Table 4.2: Number of Monte Carlo iterations required to converge the
uncertainty bounds for the prismatic aluminum beam.
Uncertainty Converged Iterations for Convergence
Term Value (%) within 2% within 1%
UK11 5.46 977 1278
UK22 14.9 1008 6464
UK33 29.8 810 7567
UK44 2.11 120 2993
UK55 1.44 644 1221
UK66 1.42 1741 4758
4.3. In addition to the total uncertainty level, the contributions from the bias and
random uncertainties for each of the input variables were also compared. For the
Taylor series method, this is easily evaluated since each of the partial derivative terms
is calculated separately; however, to generate these results using the Monte Carlo
method, it was necessary to rerun the simulations by perturbing each variable one at
a time, increasing the computational expense by about an order of magnitude. From
Table 4.3, it can be seen that the uncertainties levels predicted by both methods
showed great agreement in terms of the overall uncertainty as well as most of the
individual measurement uncertainties, with the Taylor series method typically being
slightly more conservative. For the prismatic aluminum beam, the uncertainties in
the strain have the biggest impact on the overall uncertainty for the axial and shear
stiffness terms, due to the relatively low strain levels that can be achieved under their
respective loading; while the uncertainties in the loads and the dimensions have the
biggest impact on the total uncertainties for the torsional and bending stiffnesses.
By considering the uncertainty levels from each of the individual variables,
some special properties of the data reduction equations and solution process can also
be observed. For the case of the systematic strain uncertainty in the DIC system, as
shown in the third row of Table 4.3, the only terms that are significantly impacted
are the axial and torsional stiffness measurements. The shear and bending loads
138
Table 4.3: Comparison of 95% uncertainty levels (in %) for the 6
non-zero stiffness properties of the prismatic aluminum beam from the
Monte Carlo method (MCM) and Taylor series method (TSM).
Uncertainty
Method K11 K22 K33 K44 K55 K66
Term
Force bias MCM 0.749 0.952 0.671 1.30 1.01 1.10
UKii,B (%) TSM 0.750 0.950 0.670 1.29 1.03 1.14F
Force random MCM 0.465 0.370 0.324 0.669 0.063 0.062
UK ,P (%) TSM 0.435 0.290 0.301 0.656 0.101 0.097ii F
Strain bias MCM 4.69 0.005 0.000 0.569 0.000 0.000
UK ,B + (%) TSM 4.70 0.001 0.030 0.621 0.000 0.000ii ?mean
Strain random, mean MCM 2.65 14.81 29.77 0.304 0.632 0.427
UK ,P + (%) TSM 2.48 14.05 32.43 0.478 0.532 0.608ii ?mean
Strain random, std MCM 0.116 0.631 1.22 0.018 0.026 0.032
UK ,P + (%) TSM 0.164 0.927 2.14 0.032 0.035 0.040ii ?
std
Young?s modulus MCM 0.000 0.503 0.693 0.514 0.000 0.000
UK ,B (%) TSM 0.000 0.507 0.698 0.518 0.000 0.000ii E
Poisson?s ratio MCM 0.000 0.200 0.375 0.264 0.000 0.000
UK ,B? (%) TSM 0.000 0.203 0.381 0.262 0.000 0.000ii
Dimension bias MCM 0.000 0.086 0.108 0.294 0.200 0.101
UK ,B (%) TSM 0.000 0.082 0.063 0.272 0.215 0.123ii dext
Dimension random MCM 0.000 0.397 0.538 1.25 0.798 0.790
UK (%) TSM 0.000 0.329 0.433 1.15 0.860 0.980ii,Pdext
Location bias MCM 0.000 0.001 0.019 0.018 0.010 0.007
UK ,B (%) TSM 0.000 0.014 0.015 0.056 0.037 0.021ii d
Location random MCM 0.000 0.001 0.016 0.012 0.006 0.003
UK ,P (%) TSM 0.000 0.002 0.002 0.009 0.006 0.003ii d
Total MCM 5.45 14.87 29.82 2.11 1.44 1.43
UK ,total(%) TSM 5.39 14.13 32.52 2.13 1.47 1.63ii
introduce an antisymmetric strain distribution on the surface of the beam and, when
adding a constant offset to the entire strain field, as is the case for the bias uncertainty
when using DIC, the SVD filters out this offset resulting in a nearly zero uncertainty
level for these stiffnesses. However, since the strain under an axial load and a
139
torsional load are nearly constant around the cross-section, the bias uncertainty is
not filtered out by the SVD for these load cases, resulting in a noticeable uncertainty.
For the strain distribution with a random mean strain uncertainty, each measured
face would have its own associated uncertainty and the SVD no longer filters out
this uncertainty for any of the calculated stiffness components, as can be seen by
significant uncertainty levels for all the stiffness components due to a random strain
perturbation. For the material uncertainties, in both the Young?s modulus and
the poisson?s ratio, it can be seen that they only impact the shear and torsional
stiffnesses, which is a result of the fact that the numerical warping correction is only
non-zero for the shear strain under shear and torsional loads. In addition, because
the numerical warping correction accounts for only a portion of the overall strain
level, the associated uncertainties in the shear and torsional stiffness of about 0.5%
for the Young?s modulus and 0.3% for the poisson?s ratio are significantly less then
the overall estimated uncertainties in the material properties of about 1.5% for the
Young?s modulus and 3% for the poisson?s ratio.
The comparison of the two methods was done in Table 4.3 for a single spanwise
location; however, it is also important to consider any expected variations that may
arise along the span. For the prismatic beam, the bending moments vary linearly
along the span when subjected to the tip shear forces. This spanwise variation in
bending moment also introduces a change in the axial strain field along the span, with
the largest strains near the clamped root of the beam. While the axial strain under
the axial force and the shear strains under the shear forces and torsional moment
are mostly constant along the span, the boundary conditions introduced strain
concentrations at the beam?s edges, which can also impact the expected uncertainty
levels. The spanwise distributions of the uncertainty levels were calculated using only
the Taylor series method, requiring a total of about 16 s to calculate the uncertainty
distributions at all 61 spanwise locations, which is roughly two orders of magnitude
140
faster than the time required to calculate the uncertainty distribution with the Monte
Carlo method at only a single cross-section.
The spanwise variations in the uncertainty levels are shown in Fig. 4.4 for the
axial and shear stiffnesses and in Fig. 4.5 for the torsional and bending stiffnesses,
with all the values normalized by the analytic solutions so that the uncertainty
levels could be plotted as a percentage of the expected stiffness. From Fig. 4.4, it
can be seen that the uncertainty in the axial stiffness is nearly constant along the
span. However, the uncertainty levels for both the lag and flap shear stiffnesses
have larger variations along the span due to the strain concentrations arising from
the boundary conditions. From Fig. 4.5, it can be seen that the uncertainty levels
for the torsional and bending stiffnesses remain nearly constant along the span.
Although the uncertainty in the shear forces provided an additional contribution
to the uncertainty in the local bending moment, as seen in Eq. (4.18), this had
a relatively small impact on the total uncertainty for the bending stiffness, with
the contribution from the bending moment varying between 1.0% and 1.1% at the
root and tip to 1.04% and 1.15% at the mid-span for the flap and lag stiffnesses,
respectively. This small variation across the span is a result of the fact that when
the bending moment due to the shear force is largest near the root, the additional
uncertainty from the shear force is relatively small since the distance x is small, and,
when the additional uncertainty from the shear force is larger due to a large distance
x, its magnitude is smaller and the relative weight when using the SVD is roughly
proportional to the inverse of the magnitude of the bending moment squared.
4.4.2 Symmetric Composite Box Beam
From Sec. 3.3, the sectional stiffness properties of the composite box beam
with a symmetric layup were evaluated and showed a bending/torsion coupling
and extension/shear coupling. Therefore, this composite box beam layup was also
141
Figure 4.4: Spanwise variation of 95% uncertainty levels for the axial
and shear stiffnesses of the prismatic aluminum beam.
Figure 4.5: Spanwise variation of 95% uncertainty levels for the
torsional and bending stiffnesses of the prismatic aluminum beam.
142
considered to compare the two methods on their accuracy and efficiency when off-
diagonal stiffness components are present. The two methods were applied to the
data reduction calculation at the mid-span of the beam (x = 30 cm), such that they
were compared at a location sufficiently far away from any effect of the boundary
conditions in the 3-D FEM.
For the Monte Carlo method, 10,000 simulations were again performed with
the 3-D FEA results perturbed by a randomly generated value from the normal
distributions described by the 2? values from Sec. 4.2. Figure 4.6 shows the variations
of the 95% confidence intervals with the number of simulations for all eight non-zero
stiffness components for the composite box beam, with any significant variations
again dissipating after about 2,000 simulations. As with the prismatic aluminum
beam, the number of required simulations to converge to within 2% and 1% of the
reference uncertainty level were calculated and are summarized in Table 4.4. With
an average run time of about 51 s per 100 simulations, it would take about 1,425 s
(23.7 min.) to converge all the uncertainty bounds to within 2% of their reference
value or 3,500 s (58.3 min.) to converge to within 1%. Note that the inclusion of the
off-diagonal coupling stiffness values had only a minor impact on the convergence
times and number of iterations required for convergence compared to the prismatic
aluminum beam.
The uncertainty levels were then computed at this same spanwise location using
the Taylor series method, with a comparison of the total and individual uncertainty
results shown in Table 4.5. Overall, the two methods still showed good agreement
with one another in terms of the total uncertainty and the contribution of individual
measured variables; however, the Taylor series method was fairly conservative in its
estimation of the uncertainty for the lag shear stiffness K22 and the extension/shear
coupled stiffness K12 due to about a 20% difference in the estimated uncertainty
resulting from the systematic strain error. Although the largest relative differences
143
Figure 4.6: Convergence of the Monte Carlo method for the uncertainty
levels of the stiffness properties of the composite box beam.
Table 4.4: Number of Monte Carlo iterations required to converge the
uncertainty bounds for the composite box beam.
Uncertainty Converged Iterations for Convergence
Term Value (%) within 2% within 1%
UK11 23.21 961 5160
UK12 58.79 913 5160
UK22 24.27 913 5160
UK33 9.10 2790 6861
UK44 5.81 229 1737
UK45 11.20 278 1001
UK55 3.97 271 1001
UK66 1.53 206 2543
between the two methods occurred for the uncertainty in the random location, these
uncertainties were several orders of magnitude smaller than the total uncertainty
bounds and therefore had almost no impact on the overall results.
For the axial stiffness, lag shear stiffness, and extension/shear coupling stiffness,
K11, K12, and K22, respectively, the total uncertainty was almost entirely due
144
Table 4.5: Comparison of 95% uncertainty levels (in %) from the Monte
Carlo method (MCM) and Taylor series method (TSM) for the stiffness
properties of the composite box beam with symmetric layup.
Uncertainty
Method K11 K12 K22 K33 K44 K45 K55 K66
Term
Force bias MCM 0.846 0.602 0.519 0.474 1.37 0.936 1.13 1.10
UK ,B (%) TSM 0.854 0.608 0.517 0.469 1.38 0.969 1.18 1.15ii F
Force random MCM 0.564 0.811 0.420 0.229 0.720 0.586 0.215 0.062
UK ,P (%) TSM 0.487 0.347 0.163 0.211 0.695 0.200 0.133 0.097ii F
Strain bias MCM 20.20 50.17 19.91 0.004 3.09 3.10 0.647 0.000
UKii,B?+ (%) TSM 21.17 60.52 23.76 0.000 3.38 3.61 0.792 0.000mean
Strain random MCM 10.66 27.23 11.75 1.76 3.84 10.28 3.54 0.680
UK (%) TSM 12.35 31.85 12.24 1.43 3.55 9.25 3.17 0.87ii,P?+mean
Strain random MCM 0.458 1.16 0.508 0.075 0.174 0.442 0.152 0.045
UKii,P + (%) TSM 0.455 1.17 0.450 0.053 0.131 0.341 0.117 0.032?
std
Young?s moduli MCM 0.183 1.17 2.17 0.821 1.38 0.565 0.062 0.000
UKii,B (%) TSM 0.184 1.15 2.16 0.807 1.37 0.545 0.059 0.000E
Shear modulus MCM 0.213 3.03 5.17 8.61 0.982 1.46 0.808 0.000
UK ,B (%) TSM 0.194 2.92 5.03 8.40 0.956 1.42 0.784 0.000ii G
Poisson?s ratio MCM 0.042 0.148 0.119 0.024 0.082 0.115 0.031 0.000
UK ,B? (%) TSM 0.043 0.150 0.121 0.025 0.081 0.115 0.031 0.000ii
Ply orientation MCM 0.708 2.733 2.55 0.904 1.88 2.39 0.605 0.004
UKii,B + (%) TSM 0.603 2.50 2.47 0.807 1.88 2.34 0.579 0.000?
ply
Dimension bias MCM 0.027 0.103 0.098 0.180 0.140 0.082 0142 0.102
UKii,B (%) TSM 0.053 0.239 0.254 0.070 0.045 0.195 0.137 0.113dext
Dimension random MCM 0.150 0.685 0.810 0.37 0.804 0.532 0.631 0.796
UK ,P (%) TSM 0.119 0.609 0.815 0.986 1.13 0.693 0.927 1.00ii dext
Location bias MCM 0.018 0.071 0.071 0.032 0.063 0.065 0.011 0.007
UK ,B (%) TSM 0.019 0.083 0.088 0.024 0.016 0.068 0.048 0.039ii d
Location random MCM 0.192 0.504 0.206 0.029 0.019 0.020 0.004 0.002
UK ,P (%) TSM 0.010 0.043 0.046 0.013 0.008 0.035 0.025 0.020ii d
Total MCM 23.21 58.79 24.27 9.10 5.81 11.20 3.97 1.53
UK ,total(%) TSM 24.54 68.52 27.42 8.67 5.83 10.39 3.74 1.76ii
145
to the systematic and random uncertainties in the mean strain values. For the
systematic uncertainty levels in the mean strain, the corresponding uncertainties
in the compliance matrix entries were US11,B + = 2.58%, US12,B + = 39.34%,?mean ?mean
and US11,B + = 0.001%; this shows the significant impact that the inversion of the?mean
compliance matrix can have on the expected stiffness uncertainties when off-diagonal
stiffness components are present. Some additional impacts of the off-diagonal stiffness
components can also be observed from the systematic uncertainties in both the mean
strain and the material property uncertainties. For the uncoupled aluminum beam,
the SVD filtered out the contribution of the systematic uncertainty in the mean strain
for the lag shear, flap shear, flap bending, and lag bending stiffnesses; however, due
to the presence of the extension/shear coupling and the flap bending/torsion coupling
terms in the composite box beam, the systematic uncertainty in the mean strain now
had a noticeable contribution to both the lag shear stiffness K22 and flap bending
stiffness K55. In addition, the warping correction factor was shown in Sec. 2.2.4 to
only impact the shear and torsional compliance terms and it was found that any
uncertainties in the material properties had no impact on the estimated uncertainties
for the axial or bending compliances. However, because of the presence of the off-
diagonal stiffness components and the requirement to invert the compliance matrix
to calculate the stiffness matrix, the material uncertainties ended up propagating
into the uncertainties for both the axial and flap bending stiffnesses.
As with the prismatic aluminum beam, the variations in the uncertainty levels
along the span were also investigated using the Taylor series method, which required
a total of about 28 s for estimating the uncertainty levels at all 61 spanwise locations
shown in Figs. 4.7 and 4.8. This again showed the substantial increase in efficiency
that the Taylor series method provided compared to the Monte Carlo method, which
required nearly two orders of magnitude more time to compute the uncertainty levels
at a single cross-sectional location. From Fig. 4.7, which shows the normalized
146
Figure 4.7: Spanwise variation of 95% uncertainty levels for the axial
and shear stiffnesses of the composite box beam.
Figure 4.8: Spanwise variation of 95% uncertainty levels for the
torsional and bending stiffnesses of the composite box beam.
147
stiffness values and uncertainty levels for the axial, shear, and extension/shear
coupled stiffnesses, it can be seen that while the uncertainty level for the flap
shear stiffness is nearly constant along the entire span, the uncertainty levels have
fairly large variations along the span for the axial, lag shear, and extension/shear
stiffness. These variations are primarily due to the boundary conditions affecting the
shear strain distribution along the span, which end up impacting the lag shear and
extension/shear compliance uncertainties. From Fig. 4.8, which shows the normalized
stiffness values and uncertainties for the torsional, bending, and flap bending/torsional
coupled stiffnesses, it can be observed that the strain concentrations, which arose
from the applied torsional moment at the tip and persisted from about 45 cm to the
tip of the beam, resulted in slightly smaller uncertainty levels in the torsional, flap
bending, and flap bending/torsional stiffness components in this region. However,
the uncertainty levels for these four stiffness components remained nearly constant
over the first two-thirds of the beams length.
4.5 Summary and Conclusions
Uncertainty quantification for the calculation of the stiffness properties from
the data reduction equation was investigated in this chapter. A discussion of the
two types of error, systematic and random, was first presented followed by an
identification of their associated quantities in the experimental measurements. The
systematic uncertainty is a fixed quantity, and because it results from calibration in
which the true value of the calibration variable is never known with 100% accuracy,
is something that can not be eliminated from the test procedure. However, the
impact of the random uncertainty, which results from random fluctuations that occur
in the measurement apparatus, can be reduced by performing a greater number of
tests. The data reduction equation was shown to be dependent on four measured
variables, the loads, strains, and measurement locations, and material parameters.
148
While the first three variables are explicitly measured in the experimental setup,
the material parameters, such as the Young?s moduli, shear moduli, poisson?s ratios,
and ply orientations, are normally taken from the manufacturer, but are required to
calculate the numerical warping field and their expected uncertainties must therefore
also be considered.
For propagating the measurement uncertainties into the final calculated stiffness
matrices, two methods were considered ? a Taylor series method and a Monte Carlo
method. The Taylor series method uses derivatives of the data reduction equation
to propagate the uncertainties from the individual variables, while the Monte Carlo
method randomly perturbs the measured variables over a large number of simulations
to estimate the total uncertainty. Both methods were implemented and compared
for two of the numerical validation models from Chap. 3 ? the prismatic aluminum
beam and the composite box beam with a symmetric layup. While the Taylor
series method ignored higher-order derivatives in its uncertainty propagation and the
first-order derivatives had to be estimated numerically, it was found to be similarly
accurate to the Monte Carlo method for both of the considered models and in most
cases, was slightly more conservative. In addition, the Taylor series method was
found to be significantly cheaper in terms of computation time, by at least two orders
of magnitude and, since the accuracies of the two methods were similar, the Taylor
series method was therefore implemented for analysis of the actual experimental
results. Based on the considered models, the expected strain uncertainties had the
greatest impact on the overall uncertainty level for the stiffness properties; however
the errors in the strain could likely be significantly reduced by using additional
cameras with higher resolution in the DIC measurements. Moreover, the terms with
the largest expected uncertainty levels are generally the shear stiffness components
K22 and K33, which have the smallest overall impact on the dynamic behavior of
composite blades.
149
Chapter 5: Experimental Methodology
This chapter presents the experimental methodology for the presented disserta-
tion research. A general overview of the design for the experimental test stand is first
described. A detailed discussion of the instrumentation used for the strain and force
measurements is then presented, highlighting the calibrations and characterizations
of their associated uncertainties. Finally, the design and fabrication of the test
articles is covered.
5.1 Test Stand
For the experimental test setup, there were two main aspects that needed to
be considered ? the load application and measurement as well as the optical strain
measurement. As discussed in Sec. 2.2.4, the main requirement for the complete
characterization of the sectional stiffness matrix is to measure the out-of-plane strain
components, ?+11 and ?
+
12, under six independent loading conditions. The requirement
for six independent load cases necessitated the ability to apply forces, to produce
axial and shear loads, as well as force couples, to produce torsional and bending
moments. In addition, the use of an optical measurement technique ? digital image
correlation (DIC) ? required an unobstructed view of the test article, in both its
unloaded and loaded states.
With these requirements in mind, a custom test stand was designed and built
for the experimental testing, as shown in Fig. 5.1. The main support structure of the
test stand consisted of an 80/20 frame, which allowed for the necessary structural
150
Figure 5.1: Test stand for load application.
rigidity to support the expected load conditions with significant safety factors, while
allowing for the modularity needed to accommodate a variety of test articles. The
vertical struts of the test stand were spaced with a horizontal separation of 30.5
cm, to ensure that the DIC cameras could capture a fully unobstructed view of the
test article under a wide range of expected deformations. A 6-axis load cell was
mounted at the base of the structure, between the test stand and test article, for
measurement of the full set of reaction loads during testing. A set of four winches,
rated up to 5560 N each, were mounted on the sides of the test setup to generate
the significant loads needed for certain test cases; however, the winches did have
151
a minimum load capability of about 50 N. Therefore, in any load cases where the
desired forces were sufficiently small, such as the flap shear loading F3 and the force
couples for a torsional moment M1, a set of hanging weights were instead used. A
system of adjustable pulleys was incorporated to reorient the generated forces, from
either the winches or weights, into the desired load condition. The entire upper pulley
assembly was also designed to be able to translate along the vertical struts between
tests, thereby allowing testing to be performed for beams with lengths from 25 cm to
78 cm. Based on the overall design, the limitations on the maximum displacements
were about 7 cm in ??2 and 14 cm in ??3. These margins were generally more than
sufficient to generate large enough strains in the test articles, with limitations on the
maximum measurable loads often becoming the main constraint during testing.
5.1.1 Test Article Mounting
One of the main goals in the design of the test setup was to provide the
capability for testing a variety of beams with minimal modifications between test
articles. One of the key aspects of producing this modular design was a clamping
assembly that could be used for a variety of cross-sectional geometries, with a
disassembled schematic shown in Fig. 5.2. The assembly consisted of two outer
clamps, made of aluminum 7075, with external dimensions of 20.3 cm ? 5.08 cm and
a thickness of 19.1 mm. Two adapter clamps, with maximum external dimensions of
12.7 cm ? 1.27 cm, were then manufactured to match the external geometry of each
cross-sectional geometry, with an example of the adapters for an airfoil geometry
shown in Fig. 5.2. The external dimensions of the adapter clamps meant that beams
with external cross-sectional dimensions of up to 12.7 cm ? 2.54 cm could be easily
accommodated in the test setup. The two sets of clamping pieces were then secured
together using a nut and bolt assembly, with the entire clamp structure then secured
directly to the load cell through the circular four hole pattern on the top face.
152
Figure 5.2: Clamp assembly to attach the beams to the test stand and
apply loads.
The same clamping structure was also included at the other end of the test
article for the load application, with the outer hole pattern used to provide a sufficient
moment arm to generate the force couples needed for the torsional and bending
moments. The use of the same clamping assembly at both ends of the beam, which
was used for attaching the beam to the load cell and applying the loads, reduced the
overall number of unique parts that needed to be manufactured for each test article.
It also allowed for the beams to be easily flipped during testing, such that either end
of the beam (root or tip) could be directly mounted to the load cell, allowing for
strain measurements to be collected over a larger portion of the span if necessary.
5.1.2 Load Application
The load application is a crucial step in the experimental procedure. The data
reduction procedure requires six independent loading conditions and, as much as
possible, it is desired to have only a single force or moment applied at a time in
order to maximize the signal to noise ratio for the calculation of each column of the
compliance matrix. The three moments, M1, M2, and M3, can be applied separately
by carefully aligning the pulleys to produce force couples that generate a single
153
moment while minimizing the residual forces in the system, with example pulley
setups shown in Fig. 5.3 for the application of bending moments about ??2 and ??3. A
large axial force F1 could also be applied by using all four winches, while minimizing
(a) Side view under flap bending moment M2.
(b) Front view under lag bending moment M3.
Figure 5.3: Schematics of winch and pulley setups for bending moment
application.
154
the residual in the remaining forces and moments; however the shear forces F2 and
F3 can not be applied without generating a linearly varying bending moment M3 and
M2, respectively, along the span of the beams. An additional constraint on the load
application arises from the use of an optical strain measurement technique, which
necessitates having an unobstructed view of the beam at all points during testing.
Therefore, when using hanging weights, which was typically required for applying a
flap shear force F3 or a torsional force couple M1, it was necessary to make sure that
all the weights were placed behind the test article to avoid having either the weights
or the ropes block the view of the cameras. A positive and negative load case were
also typically applied for all the tests, with the exception of a negative axial force
due to limitations associated with buckling, in order to provide additional data and
reduce the impact of the random uncertainty.
5.1.3 Digital Image Correlation Setup
Digital image correlation (DIC), described in Sec. 5.2, is used in this dissertation
to measure the strain fields under loading by using two cameras pointed at the positive
??3 surface of the beam, with the setup shown in Fig. 5.4. The cameras (5 MP Basler
acA2440-75um with Schneider Xenoplan 28 mm lenses) are mounted on an adjustable
tripod that allows them to be translated by up to 20.3 cm along the b?1 axis of the
beam during testing, such that images can be captured over a significant portion
of the beam?s span. Two high-intensity LED arrays (Visual Instrumentation Corp
Model 901000H) were used to illuminate the test articles during testing to provide
an even light intensity over a significant portion of the beam and reduce the overall
exposure time for the images.
155
Figure 5.4: Full experimental test setup for stiffness property
measurement.
5.2 Digital Image Correlation (DIC)
In the present research, the surface displacements and strains are measured
using the VIC-3D digital image correlation (DIC) system from Correlated Solutions,
Inc, which included software for both image acquisition (Vic-Snap) and image
analysis (Vic-3D v8.0). The two 5 MP Basler cameras are used to capture 2448 ?
2048 pixel images during testing that are then analyzed using the Vic-3D software.
Extensive documentation for obtaining optimal results from DIC was provided with
the software, in the form of both a reference manual and testing guide [130,131]; the
156
key aspects of the various steps taken for the current tests are briefly discussed in
the following subsections for the purpose of repeatability of the experiments and
uncertainty quantification.
5.2.1 DIC Sample Preparation
With DIC, the preparation of the test article is one of the most important
steps, as it can significantly impact the quality of the results. Although DIC is a
non-contact measurement technique, the test articles are required to have a high
contrast, random speckle pattern on the imaging surface to track deformation in the
cross-correlation field between images. To achieve this, the surface of each sample
is first sanded with a fine grit sand paper to remove any minor surface defects. A
base layer of matte white paint is then applied to the surface; it is important to
ensure that matte paint is used in order to limit the reflectivity of light that would
otherwise arise from a glossy finish and cause a glare in the images. The random
speckle pattern is then created using the application kit and black ink provided by
Correlated Solutions, Inc, which consisted of rollers and stamps that could generate
dot sizes from 0.18 mm to 5.1 mm. The selection of a specific dot size is a tradeoff
between spatial resolution and field of view ? ideally an individual speckle should be
? 3 - 5 pixels in the final images. For the current experiments, images were captured
over ? 23 cm of the span and consisted of 2048 pixels in this direction, which meant
that the stamp with a 0.33 mm dot size provided the optimal speckle pattern. Since
both the stamp and roller have finite dimensions, multiple applications needed to
be applied to cover the entire length of the beam. Therefore, it was also important
to sufficiently reorient the stamp or roller between applications to ensure that the
dot pattern remained stochastic by preventing any repeated patterns from occurring
along the span of the beam.
The random speckle pattern itself is the main requirement in preparing the
157
samples for DIC; however, in order to make use of the DIC results in the data
reduction algorithm, two additional steps were necessary. First, the DIC software
only has information about the portion of the beam captured within the images
themselves, but the actual measurement locations are a main requirement in the
data reduction algorithm. Since the full span of the beam normally can not be
captured by a single image, it is therefore important to have references within the
image for knowing the exact locations of the measurements along the beam. To
accomplish this, alphabetic markings were placed every 2.54 cm along the span, such
that multiple markings were included within each set of images that could be used
to define the measurement locations. Second, the DIC software utilizes a coordinate
system aligned with the axes of the images by default, which is not typically the
desired coordinate system. To provide a means of orienting the images, pairs of
vertical lines, roughly 2.5 mm in length and spaced 2.5 cm apart, were created along
the reference axis of the beam, in order to define the orientation of the b?1 axis within
the Vic-3D software. Figure 5.5 shows an example beam at multiple stages within
this sample preparation process.
5.2.2 DIC System Setup
The setup and calibration of the imaging system also has a significant impact
on the quality of the results, with the arrangement used for the current test shown
in Fig. 5.6. When considering the setup of the cameras, selecting an appropriate
stereo angle, ?c, is critical for achieving the best results. For the mid-range lenses
(28 mm) used in this study, the stereo angle should be between 25? and 60? to
maximize the accuracy of the results [131], with an angle of ?c ? 35? used during
testing. Maintaining a large enough depth of field (DOF) is also crucial to ensure
that the image remains in focus even under large deformations. The depth of field is
controlled by the aperture setting (f-number) of the camera F , the lens length f ,
158
Figure 5.5: Beam at multiple stages in the sample preparation process.
the acceptable diameter of the circle of confusion c, and the object distance s, with
the depth of field limits given by [132]
sFc(s? f)
d1 =
f 2 + Fc(s? f)
(5.1)
sFc(s? f)
d2 =
f 2 ? Fc(s? f)
159
where d1 and d2 are the near and far limits, respectively. The maximum acceptable
diameter of the circle of confusion, the circle formed when imaging a point source
that does not come into perfect focus, can be approximated from the lens focal
length as c ? f/1720 [133]. In order to achieve the desired 23 cm image window, the
required distance of the cameras from the beam needed to be s ? 100 cm. When
combined with an expected maximum displacement of ? 9 cm within the imageable
portion of the span, this gave a desired f-number of 5.6. For the beams with larger
bending stiffness, and therefore smaller expected out-of-plane displacements, the
f-number was reduced to 4 to increase the spatial resolution.
Depth of 
Field (DOF)
Focal 
Plane
Tripod Camera
LED Arrays
Figure 5.6: Schematic of the overhead view of the DIC setup.
With the camera locations determined, calibration is then used to establish the
spatial orientations and positions of both cameras within the Vic-3D software, which
are needed to recover the deformations from the cross-correlation procedure. This
calibration is done through the use of a calibration plate, shown in Fig. 5.7. Multiple
sizes of the calibration plates were provided from the vendor; the appropriate size
of the plate is selected based on the requirement that the plate fill up most of the
image during calibration. Hollow dots on the plate provide reference axes with a
160
set of smaller dots providing a reference size that is predefined within the software
and used to determine the approximate size and location of the object being imaged.
The main requirements for the calibration procedure are to capture images of the
plate with significant rotation of the grid about all 3 axes and at multiple distances
within the depth of field. Generally 40-50 images were taken for each test to ensure
accurate results. Based on the calibration images, the software generates a score
describing the average error in pixels between the measured location of a target point
on the calibration grid and the theoretical position of this same point, with a score
of about 0.02 to 0.025 pixels normally achieved during calibration.
Figure 5.7: Grid used for DIC calibration.
5.2.3 DIC Analysis
Once the test article is prepared and the camera calibration is complete, images
of the beams can be collected in the undeformed state as well as in various deformed
states under applied loads. For each load state, 10 images are captured that will be
averaged together to reduce the impact of any dynamic loads or other variabilities
that may arise while capturing the images. In order to provide sufficient data for the
161
data reduction procedure, images are captured of both the front (+z) surface and
back (?z) surface by rotating the entire beam 180? about its span (axis b?1) between
tests. The crux of the DIC technique is then, as its name suggests, correlating the
images between cameras and across loading states to calculate the 3-D shape and
deformation field on the imaged surface. The DIC method has been significantly
improved over the past three decades since its original development in the early
1980?s and a variety of commercial DIC software is now available, with the Vic-3D
v8.0 software used in this dissertation. Because the DIC technique is applicable to a
wide range of possible tests, multiple steps need to be taken during the analysis and
appropriate parameters defined to produce high quality measurements based on the
given setup. The following steps were used in this work to generate the experimental
DIC data for all of the tested beams:
? Apply a mask to the images. A substantial portion of the collected images
contain unnecessary information, such as the 80/20 test frame and background
behind the test article. An area of interest is therefore defined within the
undeformed reference image, encompassing only the speckle pattern, to indicate
which portion of the image should be correlated with images of the deformed
states. In some cases, if the beam experiences significant movement between
images, it may be necessary to define a start point in the image that can be
easily tracked across all images. As the DIC software has no knowledge of the
absolute position of the images with respect to the inertial frame of the load
cell, the alphabetic markings placed on the beams during sample preparation
become important for defining the area of interest. By drawing the edges of
the area of interest tightly around these alphabetic markings, the spanwise
location of the measurements can be easily interpreted on output of the data.
These markings can also serve as a useful reference if the definition of a starting
point is required.
162
? Subdivide the area of interest. Once the area of interest has been defined,
the software needs to subdivide the remaining portion of the image into a
subset of pixels that is used to track the displacement between images. The
subset size must be large enough to provide a distinctive pattern for correlation
purposes, but small enough to maintain an optimal match confidence that
minimizes the uncertainty level and noise in the correlation. A subset size
of 25 ? 25 pixels was the default suggestion by the software based on these
considerations and was used in this work. An additional parameter, the step
size, controls the spacing of the points output by the software ? a step size of
1 indicates that a data point is placed within every pixel while a larger step
size of 5 would correspond to a data point at every 5 pixels. For the DIC
software, the area of interest defines a 2-D grid of pixels within the image and
the analysis time is therefore inversely proportional to the square of the step
size; for the data reduction procedure, the number of cross-sectional points is
proportional to the step size and the analysis time is inversely proportional to
the step size. Therefore, to maintain a high enough spatial resolution while
minimizing the required computational time, a step size of 5 was used for
all the experiments, such that data points were evaluated at about 0.06 cm
intervals in both b?1 and b?2.
? Calculate the 3-D displacements. Once the area of interest, subset size,
and step size have been selected, the software can compute the cross corre-
lations between undeformed and deformed images to recover the 3-D surface
geometry and deformations, with internal algorithms designed to calculate
the displacement at a sub-pixel accuracy. Since the cameras have no absolute
positional reference, an option for an auto plane fit is used to define the initial
orientation of the coordinate system used during the analysis with respect to
the calculated undeformed surface geometry. This origin is placed with its
163
z-coordinate at the ?mass? centroid of all the measured points for a given set
of undeformed and deformed reference images, with the b?3 axis defined normal
to the best-fit plane of the entire data field. This option is selected to ensure
that the average z-coordinate of the measurements is close to 0, making it
easier to define the location of the measurement points within the cross-section
for use in the data reduction algorithm.
? Define the coordinate system. The auto-plane fit option provides a useful
means of closely approximating the cross-sectional coordinates of the beam.
However, due to the orientation of the cameras on the tripod, the software
defaults to the y-coordinate being along the axis of the beam and the cross-
sectional coordinates are thus initially defined by axes b?1 and b?3, in contrast
to the desired axes b?2 and b?3. Once the images have been correlated and the
positions and displacements calculated, a coordinate transformation can be
easily applied within the software by defining the orientation of the b?1 axis
(x-axis), which is done by drawing a line in the software through the axis
markers placed on the beams during sample preparation. Once defined, the
transformation must be applied to the data by selecting this option within the
software. Note that it is important to only apply this transformation once; the
transformation is applied in the software using a rotation matrix calculated
based on the rotation from the initially calculated coordinate system to the
desired coordinate system. If applied more than once, the rotation matrix will
then be multiplied through a second time, resulting in data that is no longer
in the desired coordinate system.
? Calculate the surface strains. After the b?1 axis has been accurately iden-
tified, the surface strains can be calculated in the DIC software to give the
three strain components ?+11, ?
+
12, and ?
+
22 (defined as xx, xy, and yy in the
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software). Because the image correlation generates a grid of x, y, z points with
high resolution and corresponding u, v, w displacements at these points, the
DIC software uses a similar approach to conventional finite element analysis
to numerically calculate the strains based on the displacements. Since data is
calculated only on the surface of the beam (i.e. no through-thickness informa-
tion is available), the set of data points is analyzed internally using 3-noded
triangular elements, similar to conventional plane strain triangular elements.
The strain at nearly all the data points is evaluated within multiple triangles
and therefore must be smoothed by the software, which is done using a local,
center-weighted Gaussian filter. This smoothing is controlled by a filter size
given in terms of neighboring data points ? too large of a filter size will impact
the ability to see local strain concentrations, while too small of a filter will
result in a strain measurement with significant noise levels. To determine an
appropriate filter size, a prismatic aluminum beam was tested under a tip
shear force and the DIC strain measurements were compared to axial strain
measurements from four strain gauges attached to the same beam, as shown
in Fig. 5.8. The measurements from the strain gauges are shown by the solid
lines, with the shaded regions representing ?2.5% of the measured value. The
measurements from the DIC setup were extracted at the same locations for
various filter sizes. The DIC measurement was extracted for various filter sizes
and plotted as points, with an error bar included to represent the two standard
deviation result for all the DIC measurement points within a 3 mm ? 3 mm
box corresponding to the size of the strain gauge. From these results, it can
be observed that above a filter size of about 30 to 40, the noise in the DIC
measurement is significantly reduced and the strains show great agreement
with the corresponding strain gauges. Based on these results, a filter size of 35
was applied for the experimental measurements in this dissertation. As there
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are multiple options for the desired strain tensor, it is also important to make
sure that the strain tensor type is set to engineering to ensure that the shear
strain matches the convention used in the data reduction algorithm.
Figure 5.8: Effect of DIC filter size on measured strain.
? Output the DIC data. Once all the images have been processed and the
displacements and strains calculated, the results of the 10 images taken in
each deformed state are averaged within Vic-3D. These results can then be
exported to file, with options existing for a comma separated variable (.csv)
file, Tecplot ASCII file, plain ASCII file, or MATLAB (.mat) file. Since the
data reduction algorithm was implemented within MATLAB, the DIC data
was exported using the option for the .mat file.
5.2.4 DIC Data Extraction
Once the images have been processed by the DIC software using the above
steps and the data output to file, there are some additional steps that need to be
performed in MATLAB to extract the data into a format for use in the data reduction
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procedure. These steps include extracting data at discrete spanwise locations and
determining the cross-sectional measurement locations, as described below:
? Extracting data at desired spanwise locations. Once the images have
been processed, the data is output in a 2-D grid, with one dimension along the
span and the other along the outer profile of the cross-section. An example
data grid is shown in Fig. 5.9 for the measured out-of-plane displacement wDIC.
Note that for visualization purposes, the grid lines are only shown for every
other point in the DIC data set. Since the data is generated based on the pixels
within the images, the grid will generally never be perfectly aligned with the b?1
and b?2 axes. Therefore, to extract data at a given cross-section, all data points
within ?0.13 cm of the desired x coordinate are selected, which will normally
consist of about five rows of spanwise data from the DIC generated grid. This
selection of data points is shown by the red boundary drawn in Fig. 5.9 for x
= 24 cm. The important values extracted from this 2-D grid of data at each
desired location are then the coordinates (xDIC, yDIC, zDIC), the displacements
(uDIC, vDIC, wDIC) displacements, and the strains ?
+
11 and ?
+
12.
Figure 5.9: Example 2-D grid of w-displacement generated from DIC.
? Determining the measurement locations. Because the DIC software
works from 2-D images, there is no mean of defining the absolute orientation of
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the cross-sectional coordinate system. While the b?1 axis is defined in the DIC
analysis procedure based on its desired orientation, the b?2 and b?3 axes cannot
normally be aligned within the software in their desired orientations, and both
a translation in the b?3 direction and a rotation about b?1 will generally be
required when processing the data. If selected during analysis, the auto-plane
fit option will orient the b?3 axis normal to the best-fit plane of the cross-section.
For the data on the front (+z) surface, the b?3 axis will therefore be close to its
desired orientation; however, for the data on the back (?z) surface, the b?3 axis
is normal to the surface and will be about 180? from its desired orientation. An
example of this is shown in Fig. 5.10 for measurements of a VR-7 airfoil. The
originally measured y and z coordinates are shown in Fig. 5.10(a), which are
not aligned with the coordinates defining the shape of the cross-section. It is
therefore necessary to match the measured coordinates with the cross-sectional
geometry, which can be done by minimizing the function
?N ?
? = (y 2a ? yi) + (za ? zi)2 (5.2)
i=1
where
yi = yDIC,iC? + (zDIC,i + zo)S? (5.3a)
zi = ?yDIC,iS? + (zDIC,i + zo)C? (5.3b)
with i representing a single measurement point, yDIC,i and zDIC,i defining
the measurement locations obtained from DIC, and ya and za defining the
external profile of the cross-section in basis B. The minimization variables
are the offset in the b?3 direction, zo, and the rotation angle ? that brings the
measured coordinates into basis B, such that Eq. (5.3) defines the actual y
and z measurement locations. A standard minimization procedure, using the
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(a) Initial DIC measurement locations
(b) Corrected DIC measurement locations
Figure 5.10: Updating of the DIC measurement locations into the
cross-sectional basis B.
fminsearch function in MATLAB, is implemented to determine these variables,
which for a VR-7 airfoil leads to an average offset of zo = 0.414 cm and rotation
? = ?3.55? for the measurements along the upper edge of the profile and an
average offset of zo = ?0.172 cm and rotation ? = ?181.3? along the lower
edge of the profile. The resulting corrected measurement locations are then
shown in Fig. 5.10(b). Although not explicitly required in the data reduction
equation itself, the displacements are needed to calculate the sectional loads
based on the loads measured by the load cell. Therefore, the displacements
must also be rotated from the DIC coordinate system into the inertial basis I
according to ????? ?? ? ?? ?? u ???? ??? 1 0 0 ??
?
??? ??? ? ?
? u ???DIC ??
???? v ???? = ? 0 C? S? ?????? vDIC ????? (5.4)w 0 ?S? C? wDIC
For the case of a rectangular profile, the minimization procedure can be
skipped if the auto plane fit option is used; for measurements on the +z surface,
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the parameters can be automatically set to zo = h/2 and ? = 0
?, and for
measurements on the ?z surface, the parameters can be automatically set to
z = ?h/2 and ? = 180?o .
5.2.5 Strain Measurement Basis
Based on the stated strain analysis procedure from the DIC vendor, it is
expected that the measured strain components, ?+11, ?
+
12, and ?
+
22, are calculated by
the DIC software in the local surface basis E = (e?1, e?2, e?3). For an airfoil cross-section,
the orientation of the surface basis can have significant variations when moving from
the trailing edge to the leading edge, particularly for the upper surface of the profile.
During DIC testing, images will be taken of both the upper and lower surface of
these airfoils and it is therefore important to ensure that the strains are actually
calculated in the local surface basis. To verify this, a thin-walled circular cylinder,
with an inner radius di = 3.01 cm and outer radius ro = 3.18 cm as shown in Fig.
5.11(a), was tested under torsion, using the setup shown in Fig. 5.11(b). The tube
was made of aluminum 6061 (E = 68.9 GPa and ? = 0.33) with a total length of 61
cm and prepared for DIC testing using the process described in Sec. 5.2.1. For this
simple circular geometry, the shear strain in the local surface basis (in cylindrical
coordinates) should be constant under a torsional load, as given by
+ ? M1r?12 = (5.5)?/32G(d4 ? d4o i )
where M1 is the applied torque, r is the radial measurement location (do/2), and
G = E/(2(1 + ?)) = 25.9 GPa, with the expression in the denominator representing
the torsional stiffness K44. If measured in basis B, the measured strain ?12 would
depend on the measurement location and vary sinusoidal around the cross-section
according to
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? M1r M1y?12 = sin(?) = ? (5.6)
?/32G(d4 ? d4o i ) ?/32G(d4 ? d4o i )
where ? measures the rotation angle from b?2 and y = r sin(?) is the cross-sectional
coordinate of the measurement.
(a) Cross-sectional geometry
(b) Experimental test setup
Figure 5.11: Experimental setup for torsional testing of a circular
aluminum tube.
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DIC testing was performed for multiple torques up to -373.5 N-m, in approx-
imately 50 N-m intervals, with the torque applied as a force couple as shown in
Fig. 5.11(b); the resultant torque was measured by the load cell at the base of the
structure. The cameras were set up to capture images close to the mid-span of the
tube, from about x = 18 cm to x = 41 cm, and from about y = ?2.5 cm to y = 2.5
cm along b?2, with a grid spacing of about 0.075 cm in the DIC measurement points.
An example of the processed strain field component ?+12 under a -54.2 N-m applied
torque is shown in Fig. 5.12, which showed good agreement with the expected strain
from Eq. (5.5) ? ?+12,ana = 218.3? ? but did have some larger variations near the
boundaries of the image in both the b?1 and b?2 directions.
Figure 5.12: Shear strain ?+12 measured by DIC for a circular tube under
a -54.2 N-m torque.
For each load case, the results were then averaged first along b?1 to give the
variations along b?2 as shown in Fig. 5.13. These results showed that the shear strain
remained nearly constant along y, with the only substantial deviations arising near
the outer edges of the DIC measurement window. They were then averaged again
along b?2 to give the mean strain for the entire DIC dataset, with the results for each
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load case summarized in Table 5.1. With exception to a few strain concentrations
that arose near the outer boundaries of the DIC images, the measured shear strain
agreed well with the constant analytic value from Eq. (5.5), which verified that the
DIC strains were indeed measured in the local surface basis E .
Figure 5.13: Shear strain ?+12 distribution along y, measured by DIC
under multiple torques for a circular tube.
Table 5.1: Mean shear strain ?+12 from DIC under multiple torsional
loads.
Torsional Load, Shear Strain ?+12(?)
M1 (N-m) DIC measurement Analytic
-54.7 213.4 218.4
-100.5 390.8 401.2
-150.4 589.8 600.3
-201.1 790.9 803.0
-253.4 1008.7 1011.6
-387.2 1140.4 1146.5
-346.9 1377.7 1385.2
-373.5 1488.8 1491.3
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5.2.6 Rigid Body Motion
An important feature of strain is that a rigid body motion introduces no strain
into a structure. In the experimental testing, it is expected that the boundary
condition in the experimental setup won?t be a perfect clamp condition, due to
a finite stiffness in the load cell and some manufacturing tolerances in the clamp
assembly, resulting in some rigid body motion under loading. Additional rigid body
motion also arises during testing due to the fact that the cameras are translated up
and down on the tripod. It is therefore important to verify that the DIC software is
able to properly filter out various types of rigid body motion. To assess the impact
of a rigid body motion on the strain measurement, two simple tests were carried
out. The first test consisted of translating the cameras vertically on the tripod by
h1 = 2.5 cm and h2 = 5 cm, while defining the same area of interest within the
image, as depicted in Fig. 5.14. These translations were repeated 10 times to assess
the average impact on the measurements. Since the test article is undergoing no
deformation, the measured strain field should remain 0 in all the DIC measurements;
however, some small variations occur during translation of the cameras and the
resulting distribution of the mean strain across the data sets is shown in Fig. 5.15.
From this distribution, the average offset strain was found to be about -16 ?.
Figure 5.14: Test description for rigid body motion introduced by a
vertical translation of the cameras.
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Figure 5.15: Distribution of mean strain values resulting from a vertical
translation of the cameras.
The second test involved rigid-body motion in the other in-plane direction,
which was accomplished by moving the test article laterally by ?2.5 cm while leaving
the cameras stationary, as depicted in Fig. 5.16. This test was again repeated 10
times to assess the average impact on the measurements, with the distribution of the
mean strain across the tests shown in Fig. 5.17. From this lateral motion testing,
the average offset strain was found to be about 15.2 ?, which was similar to the
results from the vertical translation of the cameras and therefore provided a good
estimate for the systematic strain uncertainty resulting from rigid body motion.
Figure 5.16: Test description for rigid body motion introduced by a
lateral translation of the test article.
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Figure 5.17: Distribution of mean strain values resulting from a lateral
translation of the test article.
5.2.7 Uncertainty Identification
As discussed in Chap. 4, a systematic and bias error will be present in every
experimental measurement. To determine the systematic strain error, information
generated during the calibration of the cameras in the DIC software is used. In
particular, a score is displayed that defined the average error in pixels, which generally
was around 0.02 pixels, and could be used to provide an estimate on the strain
systematic error, based on the total image size, of about 10.9 ?. In addition, from
the rigid body motion testing, an average offset of about 16 ? was observed when
the cameras or test articles were subjected to a rigid body motion. Therefore, the
95% confidence interval for the systematic uncertainty can be estimated to be about
26.9 ?.
In order to estimate the random strain error associated with the DIC setup,
information from the DIC analysis of the strain is used. During an individual test, ten
images are captured of the beam in each reference state, including in the undeformed
state. When in the undeformed state, there should be no change between the ten
images and the strain field should remain 0; however, a zero strain state is not
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actually observed, as seen in Fig. 5.18 which shows an example of the calculated
strain from averaging the results of the ten undeformed images. These small, non-
Figure 5.18: DIC strain fields from averaging images in the undeformed
state, showing the random error in the DIC system.
Figure 5.19: Example DIC axial strain distribution in undeformed
reference image at x = 34? 0.05 cm.
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zero strains arise from random variations in the correlation process and result in a
mean offset of the average strain. Since the data reduction procedure utilizes all the
strain measurements at a given x-location to calculate the stiffness matrix, it is also
important to take into account the variations that occur in the b?2 direction (i.e. with
respect to y), as this spread will also impact the uncertainty. Figure 5.19 shows the
extracted axial strain distribution along y at x = 34? 0.05 cm, which had a mean
strain offset of -3.2 ? and a standard deviation of 14.9 ?. These values, the mean
strain offset and standard deviation of the strain along y, were then calculated for
each data set generated from the undeformed reference images across all the tests,
which consisted of about 1000 data sets in total. The distributions of these results
are shown in Figs. 5.20 and 5.21, which can be used to provide an estimate for the
95% confidence levels on the random strain of Sr,?+ = 24? and Sr,?+ = 16? formean std
the mean strain and standard deviation along the cross-section, respectively.
Figure 5.20: Distribution of mean strain offsets in undeformed state.
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Figure 5.21: Distribution of standard deviation of y-variations in strain
in the undeformed state.
5.3 Load Measurement
The other main requirement for the high-quality prediction of the stiffness
properties is an accurate knowledge of the applied load. The use of pulleys, which
introduce friction to the system, and winches when applying forces requires an
independent means of measuring the load during each test. In addition, because
the data reduction procedure needs six independent load cases to calculate the full
stiffness matrix, it is necessary to be able to measure the loads under three forces
(axial and two shears) and three moments (torsion and two bending). To obtain
these load measurements, an ATI Omega 160 6-axis load cell, shown in Fig. 5.22,
was mounted at the base of the test rig such that the reaction loads at the clamped
end of the test articles could be measured. The axes of the load cell were oriented
such that ??1 was along the axis of the beams, ??2 was along the chord or width of
the beams, and ??3 was in the thickness direction of the beams and pointed outwards
towards the DIC camera setup. The load cell was nominally rated up to 6250 N in
axial loading, F1, up to 2500 N in shear, F2 and F3, and up to 400 N-m in torsion
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and bending, M1, M2, and M3. These maximum load ratings were sufficiently large
to generate measureable strain levels for most of the expected test articles.
Figure 5.22: 6-axis load cell used to measure reaction loads in the test
articles.
5.3.1 Load Cell Testing
In order to estimate the random error in the load cell for the uncertainty
analysis, it was necessary to measure applied loads in a controlled environment where
the loading condition could be defined with a high degree of accuracy. To accomplish
this, the load cell was mounted horizontally on the test stand as shown in Fig. 5.23.
A stiff calibration beam was attached to the load cell, which was used to apply
loads through hanging weights under gravity, eliminating any friction that could be
introduced through the pulley setup of the test stand. This calibration beam, shown
in Fig. 5.24, had reference holes placed every 2.54 cm along the span, from 22.9 cm
to 73.7 cm, to accurately define the applied bending moments. Simple testing was
first performed by applying the dead weight loads in the ??3 direction at multiple
locations along the span, to generate various F3 shear force and M2 bending moment
combinations. The entire stand and beam were then rotated by 90? and the same
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Figure 5.23: Test setup for controlled testing of the 6-axis load cell in
shear and bending.
Figure 5.24: Calibration beam used for testing the load cell.
testing was performed with loads now applied in the ??2 direction to consider the shear
force F2 and bending moment M3. The measured forces and moments are shown in
Figs. 5.25 and 5.26, respectively, for applied F3 loads of 24.5 N, 52.5 N, 105.4 N at
various distances x from the origin of the loadcell. From Fig. 5.25, it can be seen
that the measured force F3 (shown as points and dashed lines) had a linear variation
with the point of application x from the load cell, despite a constant magnitude
force being applied. Since the loads are applied at the tip of the beams during the
actual experiments, with the beams expected to have lengths between 0.5 m and
0.75 m, the measured shear force F3 using the initially supplied calibration could be
up to 50% greater than the actually applied load, leading to unreasonable errors in
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Figure 5.25: Measured force F3 under multiple applied loads and at
multiple locations from the origin of the load cell.
Figure 5.26: Measured moment M2 under multiple applied loads and at
multiple locations from the origin of the load cell.
its measurement. From Fig. 5.26, a linear variation in the measured moment with
the point of application of the applied force was also observed; however, this linear
variation is expected for a bending moment and the results showed good overall
agreement with the expected moment. Similar trends were also observed when the
load cell was rotated 90? to apply loads in the ??2 direction as shown in Figs. 5.27 and
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Figure 5.27: Measured force F2 under multiple applied loads and at
multiple locations from the origin of the load cell.
Figure 5.28: Measured moment M3 under multiple applied loads and at
multiple locations from the origin of the load cell.
5.28. The measured bending moment M3 showed good agreement with the applied
moment, while the shear force F2 had the same erroneous linear variation with point
of application, but at a smaller overall maximum error of about 30% of the expected
value.
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5.3.2 Load Cell Calibration
Results from simple static testing of the load cell showed significant errors in
the measurements of the shear force when using the initially supplied calibration
matrix. Therefore, to avoid the impact of these large errors on the shear forces in
the final measurements, it was necessary to recalibrate the load cell before running
the experiments. Because the load cell directly measures and outputs voltages from
six internal strain gauges, the reaction loads are defined according to
Fm = A (V ? V o) (5.7)
where Fm = {F1,m, F2,m, F
T
3,m,M1,m,M2,m,M3,m} are the reaction loads, A is a 6?6
calibration matrix, and V = {V1, V2, V3, V4, V5, V6}T are the 6 voltages output by
the load cell with V o defining the reference voltage measurements under no load.
During the actual experiments, the load cell will be used to measure the reaction
loads based on the measured voltages; however, in a controlled environment where
the forces are generated by hanging weights of known mass, the loads can be directly
calculated based on the mass and distances of application from the reference axes.
Since the voltages are also being measured, Eq. (5.7) can then be used to calculate
the coefficients of the calibration matrix. A single force or moment is related to the
voltages through six calibration coefficients with a total of 36 coefficients required for
all six loads. Therefore, to uniquely define the coefficients, it is necessary to apply
at least six different non-zero loads for each of the forces and moments. Since the
general purpose of this controlled testing is also to quantify the uncertainty in the
force measurement, a larger number of different loads are applied and the calibration
coefficients are then determined using a linear least squares fit of the data.
To calibrate the shear forces and bending moments, the same gravity loading
setup shown in Fig. 5.23 was used. Forces and moments were applied in both the
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positive and negative ??2 and ??3 directions and with small and large magnitudes to
provide data over the full range of expected test loads. For the forces with a small
magnitude, sets of weights ranging from 100g to 2.27 kg were combined together
to generate 22 different loads from 100 g up to 10.7 kg. For larger forces and
moments, sets of weights ranging from 4.54 kg to 20.4 kg were combined together
to generate 12 different loads up to 700 N and bending moments up to 400 N-
m. These loads were then applied at up to 28 different positions along the beam,
varying the distance along ??1 and either ??2 or ??3, which provided more than 3000
data points for use in the calibration. Since the voltage readings from the load
cell tended to have different behavior at the smaller and larger weights and under
positive and negative loads, four sets of calibration coefficients were generated for
each of the shear forces and bending moments, with the recalibrated shear forces
and bending moments shown in Figs. 5.29 - 5.32. In each of these plots, the
measured forces and moments are normalized by their expected values, with squares
used to indicate the data points used in defining the calibration coefficients and
Figure 5.29: Results from static testing of the 6-axis load cell with an
updated calibration under a shear force F2.
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Figure 5.30: Results from static testing of the 6-axis load cell with an
updated calibration under a shear force F3.
Figure 5.31: Results from static testing of the 6-axis load cell with an
updated calibration under a bending moment M2.
the different colors corresponding to the four different calibration sets. Additional
data samples were then collected over subsequent days and the measured loads
were calculated based on the updated calibration coefficents to ensure that these
coefficients were not substantially impacted by the environmental conditions, with
186
Figure 5.32: Results from static testing of the 6-axis load cell with an
updated calibration under a bending moment M3.
these measurements shown as stars. If using the initially provided calibration matrix,
the shear forces exhibited errors of up to 38% and 50% of their expected values for
measurements of F2 and F3, respectively; however, with the updated calibration
matrices, almost all measurements had errors of less than 5% with most having errors
under about 2%. Using this large set of measurements, the random uncertainties
could then be estimated for each of the shear forces and moments, with separate
confidence intervals for the low and high magnitude loads. For the low loading
levels, with |F2,3| < 105 N and |M2,3| < 60 N-m, the 95% confidence intervals for
the random uncertainty were Sr,F2 = 0.0064F2, Sr,F3 = 0.0149F3, Sr,M2 = 0.0020M2,
and Sr,M3 = 0.0028M3. For the high loading levels, the 95% confidence intervals
were Sr,F2 = 0.0089F2, Sr,F3 = 0.0116F3, Sr,M2 = 0.0010M2, and Sr,M3 = 0.0012M3.
To estimate the bias uncertainty, two factors were taken into account. Based on
manufacturing data, the bias uncertainty in the mass of the weights used to apply
the loads was Sb,m = 0.005m and the associated bias uncertainty in the applied
bending moment could be estimated as Sb,M = 0.006M . In addition, at any giveni i
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load level, the average of the resulting measurements calculated from the updated
calibrations generally had a small offset from the expected value. Combining these
two effects, the 95% confidence intervals for the bias uncertainty could be estimated
by Sb,F2 = 0.0102F2, Sb,F3 = 0.0292F3, Sb,M2 = 0.010M2, and Sb,M3 = 0.0095M3 at
the low loading conditions and Sb,F2 = 0.0146F2, Sb,F3 = 0.0129F3, Sb,M2 = 0.010M2,
and Sb,M3 = 0.011M3 at the high loading conditions.
To test the load cell under an axial force F1, the setups shown in Fig. 5.33 were
used. For a negative (compressive) force, the load cell was mounted in its normal
orientation on the test stand with weights stacked directly on top of it as shown in
Fig. 5.33(a), while for a positive (tensile) force, the load cell was mounted upside
down at the top of the structure and weights were hung from a bar directly attached
to the load cell as shown in Fig. 5.33(b). Since total weights of up to only 800 N
could be applied, the full expected axial force range could not be tested using dead
weights. However, based on the results from this testing, as shown in Fig. 5.34,
it was found that the original calibration provided accurate results for the axial
force over the range of applied loads, except for at sufficiently small loads (|F1| / 10
N), and a recalibration was not actually required. From these results, the random
uncertainty was then estimated to be Sr,F1 = 0.0087F1 and assumed to be the same
at both the low and high loading conditions. The bias uncertainty for the axial load
was taken from the manufacturer data, which gave a 95% percent confidence interval
of Sb,F1 = 0.0075F1.
Finally, to test the load cell under a torsional moment M1, the setup from
Fig. 5.23 was modified by attaching a piece of 80/20 to the calibration beam to
produce a large torsional moment arm d, as shown in Fig. 5.35. More than 1000
different torsional moments, M1 = Wd, were then applied by varying the weight W
and moment arm d to generate loading conditions up to 250 N-m, with the results
shown in Fig. 5.36. The measurements using the updated calibrations were generally
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(a) Negative load (b) Positive load
Figure 5.33: Test setup for controlled testing of the 6-axis load cell
under an axial force.
Figure 5.34: Results from static testing of the 6-axis load cell under an
axial force F1.
within 3% of their expected value, which was a significant improvement compared
with the measured torsional moments from the original calibration matrix which
189
showed errors of up to 15%. From these results, the 95% confidence interval for
the random uncertainty in the torsional moment measurement was estimated to be
Sr,M1 = 0.0126M1 for low loading cases (|M1| < 40 N-m) and Sr,M1 = 0.0204M1 for
high loading cases. Similar to the case of the shear forces and bending moments,
the bias uncertainty was estimated by combining the bias uncertainties in the mass
and moment arm location with the mean offset of the calibrated results, to give 95%
confidence intervals of Sb,M1 = 0.016M1 for low loading cases and Sb,M1 = 0.0201M1
for high loading cases.
Figure 5.35: Test setup for controlled testing of the 6-axis load cell in
torsion.
Figure 5.36: Results from static testing of the 6-axis load cell under a
torsional moment M1.
190
5.3.3 Sectional Load Calculation
The 6-axis load cell measures the applied loads Fm at the root of the beam;
however, the calculation of the sectional stiffness properties requires the sectional
load vector F c. The transformation of loads fro[m the inertial fram] e FI = [O, I =
(??1, ??2, ??3)] to the local cross-sectional frame Fb = B,B = (b?1, b?2, b?3) consists of both
a translation, r, and a rotation R, as shown in Fig. 5.37.
Figure 5.37: Transformation from inertial to local cross-sectional frame.
The translation, r, can be written as
r = ru + rd (5.8)
with ru denoting the translation to the cross-sectional origin in the local undeformed
frame Fu = [A,A+ = (a?1, a?2, a?3)] and rd the relative displacement of the cross-section
into frame Fb as the beam undergoes deformation. The translation to the local
undeformed frame can be written as
ru = xu??1 + yu??2 + zu??3 (5.9)
191
where xu is the spanwise location of the cross-section and yu and zu define the offsets
of the cross-sectional origin along ??2 and ??3, respectively, from the origin of the
inertial frame in which the loads are initially measured, as shown in Fig. 5.38(a).
The relative displacement is defined by
rd = u??1 + v??2 + w??3 (5.10)
where u, v, and w are the displacements along inertial axes ??1, ??2, and ??3, respectively,
under the applied load, as seen in Fig. 5.38(b). The 3-D deformation fields, u(x, y, z),
v(x, y, z), and w(x, y, z), can be calculated from the DIC measurements using Eq.
(5.4) and their mean values can then be used to determine the displacements of the
cross-section. Since the tripod is used to manually adjust the vertical position of
the cameras between image collections, there will also generally be some relative
movement of the camera system between subsequent test points which needs to be
taken into account. While not present in the current setup, additional lateral and
out-of-plane movement of the camera system may be desired, specifically in the case
where large displacements bring the beam out of the focal plane of the cameras. The
relative displacements along ??1, ??2, and ??3 can therefore be defined as
u = u?(xu, y, z) + ucam
v = v?(xu, y, z) + vcam (5.11)
w = w?(xu, y, z) + wcam
where u?(xu, y, z), v?(xu, y, z), and w?(xu, y, z) are the average values of the 3-D defor-
mation fields from DIC at the spanwise location xu and ucam, vcam, and wcam are the
relative motion of the camera system from the initial position during image collection
in the undeformed state.
Next, the rotation matrix, R, bringing the forces and moments into the local
192
(b) Translation to local deformed
(a) Translation to local frame
undeformed frame
Figure 5.38: Translation from inertial frame FI to the cross-sectional
frame Fb.
cross-sectional frame Fb must be defined. Conventional means of defining the rotation
matrix, using parameters such as Euler angles or quaternions, are not easily applied
as rotation angles can not be directly measured. However, if the cross-section is
assumed to remain planar after deformation and perpendicular to the elastic axis (i.e.
no warping and no shear deformation), the rotation can be represented in terms of
the beam deflections, v and w, and the elastic twist, ?. The rotation matrix written
in terms of these parameters is
? ?
?? ? v?2 ? w?21 ( ) v? ( w?? 2 2? ) ??? ? ?2 ? ?2R = v C ? w?S (1? v C ? v?w?? ? S 1? w S ? (5.12)2 ) ? ? ( 2 ) ???
v?S ? w?C ? 1? v?2 S ? v? ?2? ? ? w?C 1? w C2 ? 2 ?
with ()? = ?/?x(), C() = cos(), S() = sin(), and ? being the total rotation of the
193
cross section represented by
? xu
? = ?+ ? ? ??twist ? w v dx (5.13)
0
where ?twist is the twist pre-built into the blade [44,134]. Although the displacement
derivatives are not a direct output from DIC, the DIC displacements are measured
at a high spatial resolution of h ? 0.075 cm and they can be used to numerically
approximate the derivatives. Using a second order approximation, the beam slope in
the flap direction is
? dw w?(xu + h, y, z)? w?(xu ? h, y, z)w = = (5.14)
dx 2h
with w? denoting the average deflection of the cross-section. To maintain a second
order approximation, the slopes at the lower and upper boundaries, respectively, are
? 4w?(xmin + h, y, z)? w?(xmin + 2h, y, z)? 3w?(xmin, y, z)w (x = xmin) =
2h
? ? ? (5.15)? w?(xmax 2h, y, z) 4w?(xmax h, y, z) + 3w?(xmax, y, z)w (x = xmax) =
2h
The same expressions are used to approximate the beam slope in the lag direction,
with all expressions for w replaced with v.
The elastic twist, ?, can also be calculated from the beam deflections. Since
the beam deflections are defined in the inertial frame FI , they must first be rotated
into the def?ormed frame using the rotation matrix from Eq. (5.12) and setting
? = ? ? xu w? ??twist v dx, which gives0 ????? ? ?? ( ? ? ?
?
uc xu, y, z) ???? ???? u(xu, y, z)? u?(x , y, z) ?u ??? ????? vc(xu, y, z) ????? = R????? v(xu, y, z)? v?(xu, y, z) ???? (5.16)wc(xu, y, z) w(xu, y, z)? w?(x ?u, y, z)
194
This step is specifically important if the beam exhibits bending/torsion coupling
or undergoes a combination of both bending and torsion loads. For the case of an
uncoupled beam under pure torsion, this preliminary rotation matrix will become an
identity matrix. The elastic twist is then defined as
( )
?1 wc(xu, y, z)? = tan (5.17)
y + vc(xu, y, z)
With definitions for both the translation r and the rotation matrix R, the
motion tensor that brings frame FI into frame Fb is defined by
?? ?? R r?R ?CF = ? (5.18)
0 R
and the sectional loads can be calculated from the measured loads using
( )?1
F Tc = CF Fm (5.19)
5.4 Test Articles
To verify the proposed data reduction procedure for the calculation of the
stiffness matrix, five different test articles were considered. A prismatic aluminum
beam was first manufactured with a solid rectangular cross-section. As with the
numerical validation model, this beam was the simplest case for the data reduction
procedure, as it should have no off-diagonal stiffness matrix components, constant
properties along the span, and can be directly compared with an analytic solution.
The second beam was a prismatic composite beam with a rectangular cross-section,
whose ply layup orientation was tailored to introduce a bending-torsion coupling
in the beam. The third and fourth test articles were composite blades, the first of
which consisted only of a internal foam core and single layer of carbon-fiber on the
195
external surface and the second blade with an additional d-shaped spar along its
leading edge. The fifth and final experimental test article was an aluminum beam
with a solid rectangular cross-section, but whose width was linearly varied along the
span of the beam.
5.4.1 Prismatic Aluminum Beam
The prismatic aluminum beam was the first test article considered and was
made of aluminum 6061-T6, with a Young?s modulus of E = 68.9 GPa and Poisson?s
ratio of ? = 0.33. The beam was nominally designed to have a length of l = 73.6
cm, consisting of a solid rectangular cross-section with a width of w = 10.2 cm
and a thickness of t = 0.49 cm as shown in Fig. 5.39. Three separate beams were
manufactured to ensure that the experimental procedure was repeatable, with the
final prepared samples for DIC testing shown in Fig. 5.40. The geometric parameters
of these three beams were then measured at multiple locations along the span, with
the average properties summarized in Table. 5.2.
Figure 5.39: Cross-sectional geometry of prismatic aluminum beam for
DIC testing.
Table 5.2: Measured geometric properties of the prismatic aluminum
beams.
Parameter Beam 1 Beam 2 Beam 3
Length, l (cm) 73.58 73.58 73.61
Width, w (cm) 10.30 10.27 10.24
Thickness, t (cm) 0.489 0.488 0.489
196
Figure 5.40: Three prismatic aluminum beams with speckle pattern
applied for DIC testing.
5.4.2 Tailored Composite Beam
The next test article considered was a prismatic composite beam with a
rectangular cross-section as shown in Fig. 5.41. The beam was manufactured in-
house using PYROFILTM TR50S 12K uni-directional prepreg, with 16 total plies
in a stacking sequence of [?30? ? ? ?2/902/? 452/02]s which was selected to provide a
significant bending/torsion coupling in the beam. The beam was nominally designed
to have a length of l = 71.1 cm, with a cross-sectional width of w = 7.62 cm and
thickness of t = 0.244 cm.
Figure 5.41: Cross-sectional geometry of prismatic aluminum beam for
DIC testing.
To fabricate the beam, a simple mold consisting of two 2.54 cm thick plates
was used, with the bottom plate containing a 71.8 cm by 8.00 cm cutout to a depth
197
of 0.244 cm. The 16 prepreg layers were cut from the roll of uni-directional prepreg,
using 30 and 45 degree triangular protractors and a steel ruler to precisely cut the plies
at the desired angles. The cut plies were then stacked in the [?30?2/90?2/? 45?2/0?2]s
sequence and wrapped with a release film to facilitate the removal of the cured
beam from the mold. These main components, the mold, an individual ply, and the
release film are shown in Fig. 5.42. Once the plies were assembled and wrapped in
the release film, the beam was placed inside the mold, which was then sealed using
16 bolts along the outer perimeter, and cured in a mechanical convection oven by
heating the oven up to 135?C (275?F), holding for 90 minutes, and then cooling.
Based on this cure cycle, the average material properties provided by the vendor
for the cured prepreg have Young?s moduli of E1? = 130 GPa and E2? = 8.68 GPa,
Poisson?s ratios of ?12 = 0.28 and ?23 = 0.33, and shear modulus of G12? = 4.60
Figure 5.42: Main components for manufacturing the composite beam
with a rectangular cross-section.
198
GPa [135]. The final geometry was then machined from the cured beam, resulting
in dimensions of l = 70.9 cm, w = 7.62 cm, and t = 0.244 cm. Once the composite
beam was machined to its correct size, its surface was prepared for DIC testing with
the final beam shown in Fig. 5.43.
Figure 5.43: Tailored composite beam with speckle pattern applied for
DIC testing.
5.4.3 VR-7 Composite Blades
The next two test articles were composite blades with a VR-7 airfoil cross-
section. The first blade consisted of a ROHACELL?IG-F 31 foam core with a single
ply of HEXPLY?8552 SGP196-PW prepreg wrapped around the exterior at a 45?
orientation, with the cross-sectional geometry shown in Fig. 5.44(a). The second
blade consisted of the same cross-sectional geometry as the first blade, but with
the inclusion of an internal d-shaped spar consisting of two plies of HEXPLY?8552
SGP196-PW prepreg at a 45? orientation and extending from the airfoil leading edge
to 38% of the chord, as shown in Fig. 5.44(b). Although a 45? orientation of the
composite material is used, the plain weave nature of the prepreg meant that the
material properties are nearly identical about b??1 and b?
?
2 and no significant elastic
coupling between the shear/torsion and axial/bending deformations was expected.
However, due to the lack of symmetry in the cross-sectional geometry, the off-diagonal
stiffness components governing the location and orientation of the principle axes of
bending (K15, K16, and K56) and the location and orientation of the principle axes
of shear (K23, K24, and K34) are all expected to be non-zero for both blades. The
blades were nominally designed to have a length of l = 53.3 cm, with a chord of
199
Figure 5.44: Cross-sectional geometries of the composite blades with a
VR-7 airfoil.
c = 8.00 cm and a thickness of t = 0.963 cm.
To fabricate the blades, a two piece mold consisting of two 3.8 cm thick plates
containing the external profile of the VR-7 airfoil was used, as shown in Fig. 5.45.
A piece of ROHACELL?IG-F 31 foam was first cut into a rectangular block with
dimensions of 62.2 cm ? 8.75 cm ? 1.25 cm. This foam block was then placed into
the mold and, after clamping it closed using the 16 bolts along the outer perimeter,
was then placed into the oven and heated at 175?C (350 ?F) for 90 minutes to form
the desired aerodynamic profile. The cooled foam was then removed from the mold
and trimmed to a 7.6 cm chord by removing the portion of the foam at the trailing
edge tab. For the blade with a d-spar, a 1/16 in. end mill was used to cut the foam
at 0.38% of the finished blade chord (3.04 cm from the leading edge). Each piece
of foam was then wrapped with a layer of Cytec 300?film adhesive to prevent any
bleeding of the resin from the prepreg into the foam.
After the foam was formed to the desired shape and wrapped in the adhesive,
200
Figure 5.45: Mold used to manufacturing the composite blades with a
VR-7 airfoil.
a rectangular aluminum 7075 insert of 5.61 cm in length was machined to the shape
of the airfoil near the 1/4 chord to strengthen the blade at the root and tip for
transferring loads during testing. For the first blade, two 14.3 cm ? 5.61 cm sections
of HEXPLY?8552 SGP196-PW prepreg were cut at a 45? orientation to create a
d-shaped spar over these inserts, extending only over the first 5.61 cm at both ends of
the blade. The remainder of the blade, where the measurements were made, was left
as a single piece of foam with no spar. For the second blade a single section of 53.3
? 14.3 cm was cut from the prepreg at a 45? orientation to wrap around the leading
edge foam block to form the d-shaped spar that extended along the entire length of
the blade. For both blades, a 54 cm ? 20.3 cm section of the prepreg was then cut at
a 45? orientation and wrapped around the main internal components, starting from
the trailing edge up over the leading edge and back to the trailing edge to maintain
a smooth aerodynamic profile at the leading edge and avoid potential delamination
issues. Any excess material was then trimmed from the trailing edge and the outer
layer of the prepreg was then smoothed out using a roller to eliminate any wrinkles
or air bubbles, wrapped in a layer of release ply, and placed inside the mold. Once
the mold was clamped shut, it was set in the oven to cure at 175?C (350 ?F) for
150 minutes and left to cool down. Based on this cure cycle, the average material
201
properties provided by the vendor for the cured prepreg have Young?s moduli of
E1 = 84.8 GPa and E2 = 79.9 GPa, Poisson?s ratios of ?12 = 0.30 and ?23 = 0.34,
and shear modulus of G12 = 5.60 GPa [136]. The cured blades were then removed
from the mold and trimmed to a length of l = 53.3 cm and chord of c = 8.00 cm,
with the final prepared blades shown in Fig. 5.46. For the blade with only an internal
foam core, the maximum thickness was measured at t = 0.977 cm, which was about
1.5% larger than the expected thickness. For the blade with the d-shaped spar along
its entire span, the additional internal component resulted in a larger measured
thickness of t = 1.001 cm, which was about 4% larger than the expected thickness.
Figure 5.46: Two composite blades with speckle patterns applied for
DIC testing.
5.4.4 Tapered Aluminum Beam
The final test article was an isotropic beam made of aluminum 6061-T6 (E =
68.9 and ? = 0.33) with a linearly varying width along the span. This beam was
designed to experimentally verify the ability of the data reduction procedure to
measure spanwise variations in properties. The overall length of the beam was
nominally l = 71.1 cm, with the beam having a rectangular cross-section with a
constant thickness of t = 0.79 cm. The beam had a constant cross-sectional width
of w = 7.62 cm from the root up to x = 2.54 cm which was then linearly decreased
202
down to w = 3.81 cm at x = 34.3 cm from the root. The width remained constant
at w = 3.81 cm up to x = 36.8 cm and then was linearly increased back up to a
width of w = 7.62 cm at x = 68.6 cm from the root, to produce a symmetric profile
about the mid-span. This bi-linear taper was machined out of an aluminum 6061-T6
rectangular block using a CNC mill, with the final prepared sample shown in Fig.
5.47.
Figure 5.47: Tailored composite beam with speckle pattern applied for
DIC testing.
5.5 Summary and Conclusions
The design of an experimental test rig for making the strain and load mea-
surements required to calculate the sectional stiffness matrices was described in this
chapter. For the strain measurements, an optical technique, digital image correlation
(DIC), is used, with the sample preparation and system calibration first discussed, as
these are key factors for obtaining accurate measurements. The DIC analysis and the
data extraction process are then highlighted to indicate the required steps for getting
the strain and displacement measurements in a form that can be readily used in the
data reduction algorithm. A simple test case of a cylindrical tube under torsion was
evaluated to ensure that the DIC setup could accurately calculate strains on a highly
curved surface. Additional testing was done by applying rigid body motion to the
system, to verify that the DIC software could accurately filter out the rigid body
motion and produce the expected state of zero strain. The discussion of the DIC
203
setup concluded with the quantification of the expected uncertainty levels for the
strain measurements.
This was followed by a discussion of the load measurement, which was done
using a 6-axis load cell mounted at the base of the test stand to measure the reaction
forces and moments under applied loading. Significant errors in the shear force
measurements based on the original calibration matrix were identified and a rigorous
calibration of the load cell was then performed to reduce these errors. Based on the
calibration testing under a large number of applied load cases, the systematic and
random uncertainties in the loads could be estimated, with most expected errors less
than 2%. Since the data reduction procedure requires the sectional loads while the
load cell measures the reaction loads, the necessary steps to calculate the sectional
loads based on the reaction loads was discussed.
The chapter concluded with a discussion of five different beams fabricated for the
experimental testing. Three prismatic aluminum beams were first made, to provide a
simple validation case and assess the repeatability of the measurements. A composite
beam was then manufactured, with the ply layup tailored to produce significant
bending/torsion coupling. Two composite airfoils were then also manufactured with
the same external profile but differing internal structure to study the impact of
the internal components on the stiffness measurement. An additional aluminum
beam was then made with a linearly varying width along its span to verify the
capability to measure spanwise variations in properties using the proposed strain-
based methodology.
204
Chapter 6: Experimental Results
This chapter presents the experimental results for the stiffness measurements of
the five prepared test articles. For all beams, the spanwise distributions of properties
are measured with the detailed uncertainty analysis applied to the measurements. For
beams with uniform properties along the span, the average properties are calculated
and compared directly with numerical predictions.
6.1 Prismatic Aluminum Beam
For the aluminum beam, the test envelope of anticipated applied loads and
maximum expected strains is summarized in Table 6.1. For all but the axial load
case, both a positive and negative load were applied to provide a redundant set
of data. In the axial load case, the maximum achievable load was limited by the
saturation limit of the load cell. A compressive (negative) load was not applied, as
the buckling limit was estimated at about 330 N (75 lb), which would only produce
a maximum expected strain of 9 ?. Both the lag shear and lag bending loads were
limited by the maximum rating of the load cell. For the flap shear and flap bending
loads, the maximum load was limited by the large deflections that started to bring
the upper portion of the beam out of focus in the camera. For the torsional load, no
specific limitations were observed, and the load was selected to achieve a desirable
strain of about 4000 ?.
205
Table 6.1: Applied load cases for the aluminum test articles.
Expected{ St}rain,Test # Load case Applied Tip Load
max ?
1 Axial F1 = +6250 N 187 ?
2,3 Lag Shear F2 = ?670 N 740 ?
4,5 Flap Shear F3 = ?89 N 2200 ?
6,7 Torsion M1 = ?80 N-m 4000 ?
8,9 Flap Bending M2 = ?49 N-m 1853 ?
10,11 Lag Bending M3 = ?450 N-m 778 ?
6.1.1 DIC measurements
To maximize the amount of DIC data that could be collected along the span,
images were taken at three different locations by adjusting the height of the cameras
on the tripod. These three image sets are denoted by image sets 1, 2, and 3, with
image set 1 collecting data from 5.24 cm to 27.3 cm along the span, image set 2
collecting data from 15.9 cm to 37.5 cm along the span, and image set 3 collecting
data from 26.1 cm to 47.6 cm along the span. All together, this amounted to data
provided over a total length of 39.4 cm with significant portions of the span captured
by multiple image sets. An example set of results are shown in Fig. 6.1 for the
axial strain ?+11 measured by the DIC system for all three image sets on both the
z = +h/2 and z = ?h/2 faces under a positive flap bending load M3 = 49 N-m.
From these results, variations of about 300 ? are observed when moving from a
spanwise location of x = 10 cm to a spanwise location of x = 40 cm. However, while
a pure flap bending moment was desired, the load cell also measured a shear force
F3 ? 20 N, which generated a linear variation in the bending moment along the span
and accounted for most of the observed variation in the axial strain distribution.
This illustrated the importance of being able to directly measure all of the reaction
loads, as the desired load condition may not be able to be applied exactly.
206
Figure 6.1: Axial strain ?+11 under a positive flap bending moment (load
case 8). The upper row of images shows the strain on the z = +h/2 face
and the bottom row shows the strain on the z = ?h/2 face.
207
6.1.2 Stiffness Matrix Calculation
DIC measurements were then obtained for the remaining load cases and the
strain data was extracted at every 0.64 cm along the span using the procedures
discussed in Sec. 5.2. These extracted measurements were then combined with
the calculated sectional loads to evaluate the spanwise distributions of the stiffness
properties, with the results for the axial, lag shear, torsional, flap bending, and lag
bending components of the three beams shown in Figs. 6.2 - 6.6, respectively. For
all plots, the results from the three tested beams are shown as points (?,,?) with
error bars representing the 95% confidence levels based on the expected measurement
uncertainties. The analytic solutions were also calculated using Eqs. (3.1-3.3) and
shown as solid lines for reference. Since measurements were not able to be performed
on the through-thickness faces due to the small thickness of the beam, no data was
obtained for measuring the flap shear stiffness component K33.
For both the axial and lag shear stiffness shown in Figs. 6.2 and 6.3, respectively,
the average uncertainty levels were fairly large due to the small strains under their
corresponding load cases, with about 12% and 23% uncertainty levels for the axial
and lag shear stiffness, respectively. While the uncertainty in the axial stiffness
could be significantly reduced if a much larger axial load were applied, this is not
feasible in the current setup due to the maximum ratings of the load cell. For the lag
shear stiffness, the uncertainty level could be reduced by applying a larger lag shear
force; however, the maximum appliable shear force is primarily constrained by the
bending moment it generates at the root and therefore a substantially larger shear
force would not be possible. Despite the large uncertainty levels in both stiffness
properties, the results for all three beams showed good agreement with one another
and the reference analytic solution. For the lag shear stiffness, the effect of the
clamped boundary condition on the shear strain distribution can be seen up to about
208
15 cm away from the root of the beam, resulting in an increase in the measured shear
stiffness in this region.
For the torsional stiffness shown in Fig. 6.4, nearly perfect agreement was
observed between all three beams and the predicted stiffness. This great agreement
can be largely contributed to the large generated shear strain under the torsional
moment; however, the numerical warping field also accounted for a correction factor
of nearly 50%, due to the relatively small thickness of the beam, and therefore
also significantly contributed to reducing any errors in the measurement. The flap
bending stiffness, shown in Fig. 6.5, also showed good correlation across the tests,
but did exhibit an increase in the calculated properties at spanwise locations above
about 40 cm. At these spanwise locations, the displacement under the flap bending
loads tended to bring the beam near the edges of the depth of field of the cameras,
and likely introduced additional noise into the measurement that could not be easily
accounted for in the uncertainty analysis. For the lag bending stiffness shown in Fig
6.6, the measured results remained fairly constant along most of the span; however,
these measurements were about 10% below the predicted value.
Since these aluminum beams had a constant cross-sectional geometry along
their span, the average values of the stiffness can also be used to provide a direct
comparison across the three test articles, with the calculated average values compared
in Table 6.2. Overall, the average torsional and flap bending stiffness values, K44
and K55, respectively, showed excellent agreement with the analytic solution, with
differences all less than 1.5%. The average measured axial stiffness K11 was between
2.4% and 4.5% greater than the analytic solution and the average lag bending stiffness
K66 was between 6% and 11% lower than the analytic prediction. For the lag shear
stiffness K22, the average measured values were calculated over the spanwise locations
from 15.3 cm up to 45.7 cm to eliminate the initial increase in stiffness resulting
from the strain concentrations near the root. Within this region, the average shear
209
stiffness showed good agreement within 13% of the predicted value.
Figure 6.2: Axial stiffness K11 measurements for the three prismatic
aluminum beams.
Figure 6.3: Lag shear stiffness K22 measurements for the three prismatic
aluminum beams.
210
Figure 6.4: Torsional stiffness K44 measurements for the three prismatic
aluminum beams.
Figure 6.5: Flap bending stiffness K55 measurements for the three
prismatic aluminum beams.
211
Figure 6.6: Lag bending stiffness K66 measurements for the three
prismatic aluminum beams.
Table 6.2: Average measured stiffness values of the three aluminum test
articles.
Stiffness term Analytic Beam 1 Beam 2 Beam 3
K11, N 3.34? 107 3.49? 107 3.44? 107 3.42? 107
K , N 1.05? 107 1.00? 107 9.15? 106 1.12? 10722
K44, N-m
2 92.1 92.3 92.3 92.3
K55, N-m
2 63.1 63.6 62.6 63.0
K66, N-m
2 2.87? 104 2.63? 104 2.55? 104 2.69? 104
6.1.3 Comparison to other methods
In addition to the strain measurements, the DIC setup also directly measures
the detailed 3-D displacement fields. These displacements can be used to directly
212
calculate the bending stiffness under a flap and lag shear load according to
F x23 (3l ? x)
K55 = (6.1a)
6z
F 22x (3l ? x)
K66 = (6.1b)
6y
where the clamped boundary condition was used to generate these expressions.
Because the DIC measurements also generate data at a sufficiently high resolution
along the span, the bending slopes can be approximated numerically and used to
compare with a rotation-based method. For the clamped-free setup used in these
tests, the flap bending and lag bending stiffness values can be calculated along the
span according to
?F3x(2l ? x)K55 = (6.2a)
2w?
F2x(3l ? x)
K66 = (6.2b)
2v?
where w? is the numerical approximation of the flap bending slope and v? is the
numerical approximation of the lag bending slope.
The bending stiffness values from the displacement and rotation based methods
were then calculated from Eqs. (6.1) and (6.2), respectively, with their measured
values shown in Figs. 6.7 and 6.8. The results were also compared with the results
from the current strain-based methodology, with the bending stiffness values from
all three beams averaged for each method and normalized by the analytic solution.
For the flap bending stiffness, shown in Fig. 6.7, the average value calculated from
the displacement method was 54.8 N-m2, which was 86.9% of the predicted value of
63.1 N-m2. When using the slope method, the average calculated stiffness increased
to 58.1 N-m2, which was still only 92.1% of the predicted value. From the observed
spanwise variations for both the displacement and rotation measurements, the flap
213
Expected Value
70 2   63.1 N-m
60
50
40
Current Method
30
"Mirror Method"
Deflection Method
20
10
0
0 10 20 30 40 50
Spanwise Location (cm)
Figure 6.7: Comparison of the current strain-based method to other
common measurement techniques for the flap bending stiffness.
4
10
3
2.5
Expected Value
2
  28,700 N-m
2
Current Method
"Mirror Method"
1.5 Deflection Method
1
0.5
0
0 10 20 30 40 50
Spanwise Location (cm)
Figure 6.8: Comparison of the current strain-based method to other
common measurement techniques for the lag bending stiffness.
bending stiffness was seen to asymptotically approach the predicted stiffness when
moving farther away from the root, which is characteristic of the presence of a small
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2 2
Lag Bending Stiffness, K (N-m ) Flap Bending Stiffness, K (N-m )
66 55
rigid body motion. For the lag bending stiffness, shown in Fig. 6.8, the average
stiffness from the displacement method was 3872 N-m2, which was only 14% of the
expected value of 27800 N-m2. Using the slope method, the average stiffness nearly
doubled to 7757 N-m2, which was still only 27% of the predicted value. These large
discrepancies in the lag bending stiffness, seen in both the displacement and slope
based measurements, were likely a result of the load cell having a finite stiffness close
to the lag bending stiffness of the beam, resulting in a significant rigid body motion
of the beam when subjected to the much larger lag shear force. However, the stiffness
value was predicted with much better accuracy based on the strain measurements,
indicating the expected result that a rigid body motion introduces no strain. This
comparison shows the advantage of using strain-based measurements for calculating
the stiffness properties, particularly if trying to measure large stiffness values.
6.2 Tailored Composite Beam
The ply layup of the composite beam was carefully selected to produce a
significant bending/torsion coupling and thus provided an interesting test case for
the developed methodology. The test envelope of anticipated applied loads for this
beam, and maximum expected strains under these loads, is summarized in Table 6.3.
Tests were again run for 11 different load cases, with the negative load neglected
for the axial case due to buckling concerns. Two main constraints arose during
testing of the composite beam due to its small thickness and the presence of the
flap bending/torsion coupling. First, due to the small thickness, the beam was very
unstable when subjected to either a lag shear or lag bending load, which meant
that in order to apply either of these load cases, a net axial force also needed to be
applied to counteract this instability. This is highlighted in the specified loads for
tests 2 and 3 as well as tests 10 and 11 by the inclusion of a large axial force F1.
Second, due to the bending/torsion coupling the beam experienced both a twist and
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flap deformation when subjected to a flap bending moment or torsional moment.
Therefore, the application of a flap shear load generated a variation in both flap
bending and torsional moment along the span that had to be taken into account
when positioning the pulleys for applying the loads.
Table 6.3: Applied tip loads for the composite beam.
Expected
Test # Load case Applied Tip Load { St}rain,
max ?
1 Axial F1 = +6250 N 635 ?
F1 = 5300 N
2,3 Lag Shear 844 ?
F2 = ?725 N
4,5 Flap Shear F3 = ?3.9 N 1396 ?
6,7 Torsion M1 = ?2.7 N-m 1823 ?
8,9 Flap Bending M2 = ?2.0 N-m 1154 ?
F1 = 4500 N
10,11 Lag Bending 1665 ?
M3 = ?160 N-m
6.2.1 DIC measurements
For the tailored composite beam, three image sets were again taken for both
the front and back surfaces of the beam by adjusting the height of the cameras on
the tripod. Image set 1 collected data from 8.26 cm to 27.3 cm along the span,
image set 2 collected data from 18.4 cm to 37.5 cm along the span, and image set 3
collected data from 28.6 cm to 47.6 cm along the span, with data provided over a
total length of 39.4 cm. An example set of results for the axial and shear strains
measured on both the z = +t/2 and z = ?t/2 faces under a positive torsional
moment M1 = 2.7 N-m are shown in Figs 6.9 and 6.10. From Fig. 6.9, it can be seen
that a significant axial strain ?+11, with a magnitude of about 450 ?, is present, even
though only a torsional moment is being applied and measured. This strain arose
from the bending/torsion coupling in the beam and exhibited the characteristics of
the strain under a positive flap bending moment, with a positive axial strain on the
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z = +t/2 face and a negative axial strain on the z = ?t/2 face that remain nearly
constant along the span. From Fig. 6.10, it can be seen that the applied torsional
moment generated a nearly constant shear strain ?+12 along both x and y as expected,
with the magnitude agreeing well with the predicted 1823 ?.
6.2.2 Stiffness Matrix Calculation
The DIC measurements were then made for the remaining 10 load cases and
the strain measurements were again extracted at every 0.64 cm along the span.
These extracted measurements were combined with the calculated sectional loads
to evaluate the spanwise distributions in the stiffness from x = 8.26 cm to x = 47.6
cm, with the results for the axial, lag shear, torsional, flap bending, and lag bending
components shown in Figs. 6.11 - 6.13, respectively. Experimental measurements
were plotted as points, with error bars representing the 95% confidence intervals for
the measurements. Since no analytic solution was available for this cross-section,
measurements were compared to the results predicted by SectionBuilder, which are
shown as solid lines. Again, because of the small thickness, no data was able to be
measured for calculating the flap shear stiffness component K33.
The results for the axial stiffness K11, the lag shear stiffness K22, and the
axial/lag shear coupled stiffness K12 are shown in Fig. 6.11, with all three components
showing relatively good agreement with their respective predicted values. For the
lag shear stiffness, some impact of the shear strain concentrations arising from the
clamped boundary condition are observed up until about 13 cm from the root. In
addition, the axial/lag shear coupling stiffness was only measured in the region of the
beam from about 28 cm to 45 cm. For the region closer to the root, the conditions
used to imposed symmetry in the compliance matrix, specifically the condition on the
off-diagonal components being measured within an order magnitude of each other,
eliminated this coupling term within this region. With exception to this coupling
217
Figure 6.9: Axial strain ?+11 under a positive torsional moment (load
case 6). The upper row of images shows the strain on the z = +t/2 face
and the bottom row shows the strain on the z = ?t/2 face.
218
Figure 6.10: Shear strain ?+12 under a positive torsional moment (load
case 6). The upper row of images shows the strain on the z = +t/2 face
and the bottom row shows the strain on the z = ?t/2 face.
219
stiffness not being measured at certain locations, the predicted stiffness properties
were almost always within the 95% confidence intervals of the measured stiffness. For
the K12 coupling component, these uncertainty levels were calculated to be about
60% of the actual stiffness, primarily due to the small measured strain, while for
both the axial and lag shear stiffness components, the uncertainty levels were around
20% of the measured values.
Figure 6.11: Axial and shear stiffness components for the tailored
composite beam.
Figure 6.12 shows the measured spanwise variations in the torsional stiffness
K44, flap bending stiffness K55, and flap bending/torsional coupling stiffness K45.
From these results, it can be seen that the coupled nature of these three stiffness
components resulted in similar trends being observed in their variations along the
span. The uncertainty levels were again highest in the coupling stiffness component
at about 11% of its measured value, with the torsional stiffness and flap bending
stiffness having uncertainty bounds of about 5% and 6%, respectively. For the
lag bending stiffness K66, shown in Fig. 6.13, the measured value showed good
220
Figure 6.12: Torsional and flap bending stiffness components for the
tailored composite beam.
agreement with the predicted value, with exception to unexpected jumps in the
measured properties from about 22 cm to 25 cm along the span as well as from
about 45 cm to 47.2 cm along the span. For the lag bending stiffness, the average
uncertainty level was observed to be about 4% of the measured value.
Since the beam is expected to have constant properties along the span, it can
also be useful to compare the average measurements to the predicted properties,
as summarized in Table 6.4. In calculating the average stiffness property for the
axial/lag shear coupling stiffness, only the locations along the span where the property
was measured were used. Great agreement in the average and predicted properties
were observed, with differences of less than 1% for the torsional, flap bending, and
flap bending/torsional coupled stiffnesses. When compared with the predicted values,
the lag bending stiffness and axial stiffness had differences of about 4% and 7%,
respectively, with the axial/lag shear stiffness showing the largest difference at about
15%.
221
Figure 6.13: Lag bending stiffness for the tailored composite beam.
Table 6.4: Comparison of average composite beam stiffness values.
Stiffness term Predicted Measured
K , N 1.00? 107 1.07? 10711
K12, N ?1.96? 106?2.30? 106
K22, N 2.08? 106 2.12? 106
K44, N-m
2 6.02 6.00
K45, N-m
2 3.69 3.72
K55, N-m
2 4.11 4.13
K66, N-m
2 3.98? 103 4.17? 103
6.3 VR-7 Composite Blades
The experimental results for the two blades are considered next, with the test
envelope of anticipated applied loads, and maximum expected strains under these
loads, summarized in Tables 6.5 and 6.6 for the blades without and with the d-spar,
respectively. The loads were selected to provide similar strain levels between the two
blades, to provide a good means of comparing the results. Tests were again run for
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11 different load cases neglecting the negative axial load. The blades were generally
more compliant than the previous beams and also had significant coupling terms,
due to the fact that the loads were not applied at the centroid or shear center, and
strains of several thousand ? could be generated under all the applied load cases.
Table 6.5: Applied tip loads for the composite blade without a spar.
Expected Strain,
Test # Load case Applied Tip Load { }
max ?
1 Axial Fx = +1800 N 3600 ?
2,3 Lag Shear Fy = ?43 N 3428 ?
4,5 Flap Shear Fz = ?7.9 N 2640 ?
6,7 Torsion Mx = ?8.2 N-m 3040 ?
8,9 Flap Bending My = ?3.9 N-m 2825 ?
10,11 Lag Bending Mz = ?23 N-m 2712 ?
Table 6.6: Applied tip loads for the composite blade with a spar.
Expected Strain,
Test # Load case Applied Tip Load { }
max ?
1 Axial Fx = +1550 N 3610 ?
2,3 Lag Shear Fy = ?57 N 3428 ?
4,5 Flap Shear Fz = ?11.5 N 2650 ?
6,7 Torsion Mx = ?13.8 N-m 3059 ?
8,9 Flap Bending My = ?8.2 N-m 2910 ?
10,11 Lag Bending Mz = ?30.2 N-m 2692 ?
6.3.1 DIC measurements
For both composite blades, the inclusion of inserts for the first 5.61 cm at the
root and tip meant that the uniform portion of the blade extended only from 5.61
cm to 47.7 cm along the span. Because of this shorter overall length, only two sets
of images were taken along the span, with image set 1 collected from 6.4 cm to 27.5
cm and image set 2 collected from 16.5 cm to 37.7 cm, measured with respect to
the beginning of the uniform portion of the blade. An example set of the measured
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axial strain ?+11 under the applied axial load F1 is shown in Figs. 6.14 and 6.15 for
the blade without and with the spar, respectively. Because the loads were applied
and measured at 0.6c with respect to the leading edge, a significant variation in the
Figure 6.14: Axial strain ?+11 under an axial force (load case 1) for the
blade without a spar. The upper row of images shows the strain on the
upper profile of the airfoil and the bottom row shows the strain on the
lower profile.
224
Figure 6.15: Axial strain ?+11 under an axial force (load case 1) for the
blade with a spar. The upper row of images shows the strain on the
upper profile of the airfoil and the bottom row shows the strain on the
lower profile.
axial strain along the chord can be observed, particularly on the lower profile of the
blade, due to the off diagonal stiffness components coupling the axial and bending
stiffnesses. While the loads for both blades were selected with the goal of producing
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similar strain levels, the presence of these off-diagonal stiffness components lead to
the blade with the spar having slightly larger (more negative) strain magnitudes near
the leading edge and slightly smaller (less positive) strains near the trailing edge.
6.3.2 Stiffness Matrix Calculation
The DIC measurements were then made for the remaining load cases and the
strain measurements were extracted at every 0.64 cm along the span. These extracted
measurements were combined with the calculated sectional loads to evaluate the
spanwise distributions in the stiffness from x = 7.62 cm to x = 35.6 cm, with
the results for the six diagonal stiffness components shown in Figs. 6.16 - 6.21,
respectively. Experimental measurements were plotted as points, with error bars
representing the 95% confidence intervals for the measurements. Since no analytic
solution was available for this cross-section, measurements were compared to the
results predicted by SectionBuilder, which are shown as solid lines. Comparing the
results for the blades without and with the spars, it can be seen that in all cases,
the inclusion of the spar increases the stiffness, which can be expected due to added
structural material. For the blade without the spar, the stiffness properties had small
variations along the span and showed great agreement with the predicted stiffness
coefficients, with exception to the lag shear stiffness where a difference of about
50% was observed. For the blade with the spar, the measured properties showed
much greater variations along the span and also showed larger discrepancies when
compared to the predicted properties, with the averaged properties along the span
of the blades summarized in Table 6.7. For both blades, the fact that the properties
are being measured about 0.6c and that the airfoil is not symmetric gives rise to
additional off-diagonal stiffness components, whose average properties along the span
are summarized in Table. 6.8.
The discrepancies between the predicted and measured properties likely in-
226
Figure 6.16: Axial stiffness K11 measurements for the composite blades.
Figure 6.17: Lag shear stiffness K22 measurements for the composite
blades.
227
Figure 6.18: Lag shear stiffness K33 measurements for the composite
blades.
Figure 6.19: Torsional stiffness K44 measurements for the composite
blades.
228
Figure 6.20: Flap bending stiffness K55 measurements for the composite
blades.
Figure 6.21: Lag bending stiffness K66 measurements for the composite
blades.
229
Table 6.7: Comparison of the diagonal stiffness components for the two
composite blades.
Stiffness Without Spar With Spar
Entry Numerical Measured Numerical Measured
K11, N 5.65? 105 5.74? 105 9.00? 105 8.06? 105
K22, N 7.21? 105 3.28? 104 1.13? 106 8.15? 105
K33, N 1.66? 104 1.64? 104 1.76? 104 2.00? 104
K44, N-m
2 44.5 43.6 85.3 74.8
K 255, N-m 7.49 7.55 13.2 19.3
K 266, N-m 327 310 678 654
Table 6.8: Comparison of the off-diagonal stiffness components for the
two composite blades.
Stiffness Without Spar With Spar
Entry Numerical Measured Numerical Measured
K15, N-m 780 412 1.40? 103 1.55? 103
K16, N-m ?3.39? 103 ?3.54? 103 ?1.40? 104 ?1.28? 104
K56, N-m
2 ?11.3 ?0.49 ?29.6 ?27.5
K23, N 1.53? 104 7.25? 103 1.15? 104 2.20? 104
K 3 3 324, N-m ?1.58? 10 ?710 ?2.56? 10 ?1.99? 10
K34, N-m 393 414 362 246
dicates that the manufacturing process for the blade with the spar resulted in a
slightly modified geometry than that used for the model. One such difference in
geometry was observed by measuring the external geometry of the blade after curing
in the oven; for the blade without the spar, the thickness was about 1.5% larger than
expected and when including the spar, the difference in the overall airfoil thickness
increased to about 4%. This difference in thickness is particularly important for the
flap bending stiffness, which generally varies with the thickness cubed. Therefore,
a 4% difference would account for about an 12.5% increase in the predicted flap
bending stiffness and reduce the discrepancy between the measured and predicted
stiffness down to about 35%. However, this thickness difference would only have a
marginal impact on the other properties, such as the axial stiffness, that also showed
some large discrepancies between the predicted and measured properties.
230
To ensure that the large difference between the measured and predicted flap
bending stiffness was only a result of a difference between the manufactured and
modeled blade, and not a result of the developed methodology, the flap bending
stiffness was also computed using a displacement based method, with the setup
shown in Fig. 6.22. A tip shear force was applied using various weights from 0.2 kg
to 2 kg under gravity loading, to eliminate any possible effects of friction generated
by the pulleys. The flap bending stiffness was then calculated based on the tip
displacement wtip and applied shear force F3 according to
F l33
K55 = (6.3)
3wtip
where l = 51 cm is the length of the unclamped portion of the beam. The displacement
test was then repeated twice, with the calculated flap bending stiffness results based
on the displacement measurement shown in Fig. 6.23 for each of the applied loads.
The stiffness values based on the tip displacement measurement showed some spread
based on the applied load, but overall showed great agreement with the value
calculated using the proposed methodology. However, even when accounting for
Figure 6.22: Setup for measuring flap bending stiffness of VR-7
composite blade with d-shaped spar using tip displacement.
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the 12.5% change in stiffness due to the manufactured airfoil thickness being about
4% greater than that of the model, there was still a significant difference between
the predicted stiffness value and the measured stiffness values. This indicates that
there are likely some differences between the internal geometry of the modeled and
manufactured blades that are not easily identified, which highlights the importance
of measuring the properties and not solely relying on the designed blade geometry
to predict the stiffness values.
Figure 6.23: Comparison of flap bending stiffness for a VR-7 blade with
a d-spar using difference measurement methods.
6.3.3 Centroid and Shear Center
From Sec. 2.3.4 and 2.3.5, it was shown that the stiffness components coupling
the axial and bending behavior, K15, K16, and K56, can be used to calculate the
centroid and principal axes of bending, while the components coupling the shear and
torsion behavior, K23, K24, and K34, can be used to calculate the shear center and
principal axes of shear. Based on Eq. (2.50), the centroid location calculated from
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the measured properties was found to be yc,m = 0.62 cm (0.52c from the leading edge)
and zc,m = 0.07 cm for the airfoil without a spar and yc,m = 1.53 cm (0.41c from the
leading edge) and zc,m = 0.19 cm for the airfoil with a spar. Based on the predicted
stiffness matrices, the centroid location was yc,p = 0.6 cm (0.53c from the leading
edge) and zc,p = 0.14 cm for the airfoil without a spar and yc,p = 1.71 cm (0.38c
from the leading edge) and zc,p = 0.17 cm for the airfoil with a spar. These values
were then plotted in Figs. 6.24 and 6.25 for the blades without and with a spar,
respectively. For the blade without the spar, the chordwise positions of the measured
and predicted centroid are nearly identical but they do have a small offset in the
thickness direction, while for the blade with a spar, a larger difference is observed in
the chordwise location of the centroid but with the offsets in the thickness direction
showing good agreement.
Origin during Predicted centroid
analysis
Measured centroid
Figure 6.24: Measured and predicted centroid locations for the airfoil
without a spar.
Origin during Predicted centroid
analysis
Measured centroid
Figure 6.25: Measured and predicted centroid locations for the airfoil
with a spar
Based on Eq. (2.54), the shear center calculated from the measured properties
was found to be ysc,m = 2.64 cm (0.27c from the leading edge) and zsc,m = 0.27 cm
for the airfoil without a spar and ysc,m = 1.37 (0.43c from the leading edge) and
zsc,m = 0.26 for the airfoil with a spar. Based on the predicted stiffness matrices,
the shear center location was ysc,p = 2.62 cm (0.27c from the leading edge) and
233
zsc,p = 0.28 cm for the airfoil without a spar and ysc,p = 2.45 cm (0.29c from the
leading edge) and zsc,p = 0.28 cm for the airfoil with a spar. These values were then
plotted in Figs. 6.26 and 6.27 for the blades without and with a spar, respectively.
For both blades, the predicted shear center was fairly close to the 1/4c location,
but the measured shear center location for the blade with the spar was significantly
farther back. However, both the predicted and measured values had similar offsets
in the thickness direction.
Origin during Measured 
analysis shear center
Predicted shear center
Figure 6.26: Measured and predicted shear center locations for the
airfoil without a spar.
Predicted
Origin during 
shear center
analysis
Measured shear center
Figure 6.27: Measured and predicted shear center locations for the
airfoil with a spar
6.4 Tapered Aluminum Beam
The final test article was the aluminum beam with a tapered width along its
span, which provided a means of experimentally verifying the calculation of spanwise
variations in properties. The test envelope of the anticipated applied loads for this
beam, and the maximum expected strains at the mid-span under these loads, is
summarized in Table 6.9. As with all the previous tests, positive and negative load
cases were applied for all but the axial force. With exception to the axial force,
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applied loads could be generated that produced significant strain of several thousand
? at the mid-span of the beam.
Table 6.9: Applied load cases for the tapered aluminum beam.
Expected Strain,
Test # Load case Applied Tip Load { }
max ?
1 Axial F1 = +6000 N 380 ?
2,3 Lag Shear F2 = ?575 N 1510 ?
4,5 Flap Shear F3 = ?143 N 2000 ?
6,7 Torsion M1 = ?81 N-m 4000 ?
8,9 Flap Bending M2 = ?70 N-m 3000 ?
10,11 Lag Bending M3 = ?423 N-m 3000 ?
6.4.1 DIC measurements
For the tapered aluminum beam, a total of six image sets were taken along
the span. The first three image sets were taken with the root of the beam (x = 0
cm) clamped to the load cell with image set 1 collecting data from 5.1 cm to 25.4
cm, image set 2 collecting data from 15.2 cm to 35.6 cm, and image set 3 collecting
data from 25.4 cm to 45.7 cm. The entire beam was then rotated 180? such that
the tip of the blade was now clamped to the load cell and the same test conditions
were repeated, with image set 4 collecting data from 45.7 cm to 66.0 cm, image set
5 collecting data from 35.6 cm to 55.9 cm, and image set 6 collecting data from
25.4 cm to 45.7 cm. This provided data over almost the entire length of the beam,
from 5.1 cm all the way up to 66.0 cm, so that the symmetry of the calculations
about the mid-span could be assessed. An example set of DIC results for the axial
strain ?+11 under a negative lag bending moment M3 = ?423 N-m is shown in Figs
6.9 and 6.10, for the six image sets on the z = +t/2 and z = ?t/2 faces, respectively.
Comparing the images from the upper and lower halves of the span in these two
sets of results, it can be seen that the results exhibit symmetry about the mid-span,
235
Figure 6.28: Axial strain ?+11 under a negative lag bending moment (load
case 11) on the z = +t/2 face.
236
Figure 6.29: Axial strain ?+11 under a negative lag bending moment (load
case 11) on the z = ?t/2 face.
237
which is expected under a constant applied moment.
6.4.2 Stiffness Matrix Calculation
The DIC measurements were then made for the remaining 10 load cases and
the strain measurements were extracted at every 0.64 cm along the span, combined
with the sectional loads, and used to evaluate the spanwise distributions in the
stiffness from x = 5.1 cm to x = 66 cm. The results for the axial, lag shear, torsional,
flap bending, and lag bending stiffness components are shown in Figs. 6.30 - 6.34,
respectively. Experimental measurements were plotted as points, with error bars
representing the 95% confidence intervals for the measurements. The results are
compared with the predictions generated from SectionBuilder, which are plotted
as solid lines. While the thickness of the tapered aluminum beam was starting to
approach the minimum dimension for obtaining DIC measurements in the current
setup, the expected shear strain on these faces under the flap shear load was still
expected to be very small and the measurement was therefore not attempted, meaning
that no data was available for measuring the flap shear stiffness K33.
Figure 6.30 shows the variations in the measured axial stiffness K11 along the
span. The measured and predicted axial stiffness values have the best correlation
closest to the mid-span of the beam, where the axial strain is largest and therefore the
associated uncertainty level is smallest. With exception to a jump in the data around
11 cm, the predicted values were generally within the s95% confidence interval for the
measured axial stiffness. The results for the spanwise variations of the measured lag
shear stiffness K22 along the span are then shown in Fig. 6.31, which have relatively
large uncertainty levels due to having the smallest associated strain of all the applied
load cases. However, with exception to the region from about 20 cm to 25 cm along
the span, the experimental and predicted lag shear stiffness values show reasonable
correlation.
238
Figure 6.32 shows the variations in the measured torsion stiffness K44 along
the span. While the measured and predicted stiffness values showed the same trends,
there was a noticeable offset of about 10% between the two sets of results. The
effect of the clamped boundary condition can also be seen for the results at both
ends of the beam and may contribute to the observed offset in magnitudes. For the
flap bending stiffness K55, shown in Fig. 6.33, a similar effect is observed, with the
measured and predicted stiffness values having nearly identical trends but with a
slight offset in magnitude. The variations in the lag bending stiffness K66 are then
shown in Fig. 6.34, with exceptional agreement observed between the measurements
and predictions. Unlike the other stiffness values that exhibit a linear variation in
properties along the span, the lag bending stiffness exhibited a non-linear variation
due to its cubic dependence on the width.
Figure 6.30: Axial stiffness K11 measurements for the tapered aluminum
beam.
A quantitative comparison between the numerical and measured stiffness
values can be considered by taking the average difference along the span. In this
239
Figure 6.31: Lag shear stiffness K22 measurements for the tapered
aluminum beam.
Figure 6.32: Torsional stiffness K44 measurements for the tapered
aluminum beam.
240
Figure 6.33: Flap bending stiffness K55 measurements for the tapered
aluminum beam.
Figure 6.34: Lag bending stiffness K66 measurements for the tapered
aluminum beam.
241
case, the axial stiffness was the only stiffness value overpredicted, with an average
difference of about 12%. The remaining four stiffness components were all on average
underpredicted compared to their corresponding measurements, with a 12%, 15%,
7%, and 3% average difference in the lag shear, torsional, flap bending, and lag
bending stiffness components, respectively.
6.5 Summary and Conclusions
The experimental measurements of the sectional stiffness properties for five
different beams was presented in this chapter. The first set of tests were presented for
three identical prismatic aluminum beams, with the measurements showing overall
good agreement with one another. With the exception of the lag shear stiffness, which
had a noticeable increase in the measured stiffness close to the clamped boundary,
the measured properties were mostly constant along the span, as expected for a
beam whose cross-sectional geometry remains uniform along its length. Stiffness
measurements were also calculated from these tests based on the displacement and
rotation based methods, which showed generally good agreement in the flap bending
stiffness but showed substantial differences when considering the much larger lag
bending stiffness. In this case, it was expected that rigid body motion resulted in the
substantial differences observed using the displacement and rotation based methods,
which got filtered out in the current method due to the use of strain measurements,
which are invariant to rigid body motion.
Results were next presented for a composite beam whose ply orientation was
tailored to produce significant bending/torsion coupling. The use of composite
materials and a much smaller thickness resulted in some slightly larger spanwise
variations in the measured properties when compare with the aluminum beam;
however, the average properties were still shown to be in good agreement with
the predicted results. The experiments were also able to accurately capture the
242
bending/torsion coupling stiffness along the entire length of the beam as well as the
extension/lag shear coupling stiffness along the middle portion of the beam.
Two composite blades were then tested to evaluate the impact of internal
structural components on the measured stiffness properties. These measurements
showed the expected result that the inclusion of a structural spar resulted in larger
stiffness values when compared to the same blade without a spar. However, it was
also found that the inclusion of the spar resulted in greater spanwise variations in the
stiffness measurements, despite the fact that the cross-sectional geometry should be
uniform along the span for both blades. A significant difference between the measured
and predicted flap bending stiffness of about 50% was observed when including the
d-shaped spar in the cross-sectional geometry. While the additional material resulted
in a slightly larger thickness than in the designed blade, this would only account for
about 12.5% of the observed difference, which would suggest that there are additional
differences internal to the blade that can not easily be assessed. However, these
large discrepancies highlight the importance of experimental measurements as the
manufactured and designed blades can often have unanticipated differences.
Finally, an aluminum beam with a tapered width along its span was tested, with
the measurements showing good agreement in the predicted trends in the spanwise
variations in properties. Since most traditional approaches can not easily handle these
these spanwise variations in properties, these results highlight an important feature
that distinguishes the developed methodology from other existing measurement
techniques.
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Chapter 7: Summary and Conclusions
This chapter presents a summary of this dissertation work, including important
conclusions drawn from the presented work, key contributions of this dissertation,
and suggestions of alterations and additional tests to continue this work into the
future.
7.1 Summary
In this dissertation, a new strain-based methodology for measuring the complete
set of cross-sectional stiffness properties of composite rotor blades is developed
and presented. The developed method provides improvements over most existing
techniques by considering the full 6 ? 6 stiffness matrix, including important off-
diagonal elastic coupling terms, as well as providing the ability to measure the
properties locally within the blade, as opposed to just average properties for the
entire blade.
The technical approach is based on the governing equations for the strain com-
putation from the 2-D cross-sectional analysis code SectionBuilder and is presented
in Chap. 2. It was shown that to calculate the full 6 ? 6 sectional stiffness matrix,
measurements of the applied forces and strains are required, as well as the warping
field. Since the warping field cannot be measured experimentally, it is necessary to
estimate it numerically based on the cross-sectional geometry and material properties.
Investigations on the impact of the numerical warping field were conducted, which
showed that the axial and bending compliances could be calculated entirely from the
244
experimental measurements, but the numerical warping was required for providing a
correction to the experimental measurements of the shear and torsional compliances.
Validation of the method is investigated using strain results from the 3-D FEA
code ANSYS in Chap. 3. The investigated beams had cross-sectional geometries
which have received extensive study in past literature, including an aluminum
rectangular and composite box beam cross-section. Comparisons were made with
analytic solutions, when available, or results from published literature, with the
calculated stiffness properties based on the developed data reduction procedure
generally showing great agreement. An aluminum beam with a spanwise variation in
cross-sectional geometry was also considered, with the calculated results showing
great agreement with numerical predictions of the stiffness properties.
Next, a detailed uncertainty analysis was developed for providing confidence
intervals on the measured properties. A Taylor series method was implemented for
the uncertainty quantification, requiring the calculation of the derivatives of the data
reduction equation with respect to the measured variables. The accuracy of this
method was assessed by comparing the results to a Monte carlo simulation for the
numerical validation models, with the two methods showing good agreement, but
with the Taylor series method being several orders of magnitude faster.
A test stand was then developed to carry out an experimental investigation
of the methodology, with the key features for the strain and load measurements
discussed in Chap. 5. Digital image correlation (DIC) is used to provide an optical
measurement of the strain fields, while also providing a measurement of the 3-D
deformations needed to accurately calculate the sectional loads. The expected errors
in both the DIC system and the 6-axis load cell, used to measure the reaction
loads during testing, are investigated for use in the uncertainty quantification. The
fabrication of five different beams for experimental testing is also discussed, with
the experimental results presented in Chap. 6. The data generated from these tests
245
showed good agreement with numerical predictions generated by the cross-sectional
analysis code SectionBuilder for most of the measured stiffness values. For the
aluminum beam with a rectangular cross-section, results were also compared to
predictions using existing measurement methods, showing the benefits of using a
strain-based method over frequency or displacement based methods.
7.2 Key Conclusions
The following is the list of the key conclusions that were drawn based on this
work.
1. The general formulation of the developed methodology for calculating the
cross-sectional stiffness properties imposed no major restrictions on the types
of cross-sectional geometries or materials that could be evaluated.
2. The measurement of the complete 6 ? 6 stiffness matrix is not always feasible.
For beams with a rectangular cross-section, if one of the cross-sectional dimen-
sions is too small, then the shear compliance about that dimension cannot be
measured. However, the remaining components can all be measured as long as
the strain is large enough.
3. The developed data reduction procedure is able to accurately identify the
presence of off-diagonal components of the sectional compliance matrix. These
off-diagonal components can play a significant role in the analysis of a rotor
and can be used to identify the shear center and centroid of the cross-section.
4. Spanwise variations in properties were demonstrated to be accurately captured
using the developed methodology, based on both experimental and numerically
simulated results.
246
5. A detailed uncertainty analysis showed that using a highly redundant set of
measurements can help reduce the error from the experimental. In addition, the
use of the singular value decomposition can almost entirely eliminate certain
errors from impacting the calculation of the stiffness properties.
6. Strain measurements are invariant to rigid body motion. Therefore, inaccu-
racies in the experimental implementation of a boundary condition did not
significantly impact stiffness properties calculated based on measured strains.
7. Large strains are required for an accurate calculation of the sectional stiffness
properties, although the significant deformations in certain large strain states
can bring the beam near the boundary of the depth of field of the DIC cameras.
For the most accurate results along the entire span of the blade, it is therefore
desired to rotate the beam between tests such that the root and tip portion of
the beams are clamped at the base of the test stand. This may be challenging
for blades that are used in wind tunnel testing since inserts that facilitate
clamping are only included at the root.
8. Prediction of stiffness properties based only on the designed geometry and
material can lead to large errors and experimental results showed the importance
of measuring the stiffness properties. This was particularly indicated by the
measurements of the stiffness properties for the VR-7 airfoil with a spar, where
the manufacturing process resulted in a thickness that was 4% larger than the
initial design, leading to a much greater flap bending stiffness than initially
predicted.
7.3 Original Contributions
The work presented here makes several important original contributions for
the structural analysis of composite rotor blades.
247
1. A new method for calculating the full 6 ?6 sectional stiffness matrix was
developed based on the measured strain under static loading. This provided a
significant improvement over existing measurement techniques, which focus on
only a subset of the matrix and generally include measurements of only the
torsional, flap bending, and lag bending stiffnesses for the blade.
2. A new capability to predict spanwise variations in sectional stiffness properties
that can arise from common rotor features including taper and twist.
3. A detailed uncertainty analysis method for predicting expected errors in the
measured stiffness properties. The effect of each individual parameter can
be identified to help improve the quality of the overall measurement, with
results from the implemented setup showing that the strain errors had the
most significant contributions to the overall uncertainties.
4. A comprehensive experimental database of 3-D displacement and strain fields
for composite airfoils under static loading was generated. These data sets can
be used in the future to validate the stress and strain predictions of numerical
tools for analysis of rotor blades, including 2-D cross-sectional analysis codes
such as SectionBuilder or VABS as well as 3-D FEM codes such as X3D.
7.4 Future Work
This dissertation focused on the development of a new method for measuring
the cross-sectional stiffness properties of composite blades. There are multiple
opportunities to expand on the work carried out in this dissertation, with some
suggestions for further areas of study listed below.
1. Additional testing of model scaled beams and composite blades - this
dissertation focused on the development of a new method for measuring the
248
sectional stiffness properties of beam-like structures, with extensive considera-
tion of how both the numerical and experimental quantities are used in the
data reduction equation. While experimental results were obtained for five
different beams, which showcased the important features of the method, includ-
ing measurement of off-diagonal stiffness properties and spanwise variations
in stiffness properties, additional testing would prove invaluable for further
verification of the method. Possible tests could include:
? Repeatability of measurements - Results from the three prismatic aluminum
beams showed that the experiments had good repeatability for a simple
beam. By manufacturing and testing multiple copies of the remaining
test articles, the variations in the composite manufacturing process could
be assessed. Specifically, this could help identify the reason for the large
discrepancy observed between the measured and predicted flap bending
stiffness for the composite rotor with the D-spar.
? Testing of NACA0012 Airfoils - Model-scaled rotor blades with a NACA0012
airfoil cross-section have been extensively used for various wind-tunnel
tests at the University of Maryland. In most cases, measurements of the
torsional and bending stiffness were performed based on displacement or
frequency-based techniques and the blades could then be used to provide
a comparison with previously measured properties [95,97,137].
? Testing of advanced geometry rotors - Advanced geometry blades have
been fabricated at the University of Maryland for testing on the Maryland
Tiltrotor Rig (MTR) in the Glenn L. Martin wind tunnel [138]. Since
the blades were manufactured in-house, composite molds exist for a blade
with significant twist (37? along its span) and a blade with twist and a
swept tip. These molds could be used to manufacture blades for DIC
testing of advanced geometry, mach-scaled rotors using the developed
249
strain-based methodology.
? Study of the impact of internal structural components - Two blade models
were considered in this work ? a blade consisting of only an foam core
with a single wrap of carbon-fiber around its exterior and the same blade
with the inclusion of an internal d-shaped spar. Most model-scaled rotor
blades also have a leading edge weight for stability purposes and may also
have various instrumentation, such as pressure sensors or strain gauges,
for use in wind tunnel testing. These components are normally neglected
in the calculation of the stiffness properties and an interesting case study
would be to determine the validity of this assumption.
2. Updating the experimental setup - The current experimental setup is
capable of testing a wide range of beams; however, there are multiple stages
in the analysis that could be automated to help achieve more consistent and
possibly accurate results. Since the testing for a single beam requires multiple
days to collect data over the full range of necessary load conditions, this would
be particularly important if applying the method to full rotor systems, where
at least four individual blades need to be fabricated and tested. Camera
calibration, test article loading, and the DIC analysis setup are procedures
that are often repeated in a very similar manner across tests, and automation
of these processes could lead to significant time savings.
In addition, the uncertainty analysis identified the strain measurements as
the key contributor to the overall uncertainty in the stiffness properties. By
incorporating multiple sets of cameras along the span, the cameras would no
longer have to be translated during testing and would eliminate the corre-
sponding errors arising from this rigid-body motion, which could substantially
reduce the overall uncertainty in the measurements. If an array of cameras
250
were also included to capture images on all external surfaces at the same time,
the required time for the experimental tests could be dramatically reduced.
3. Identifying the material properties - Accurately identifying the material
properties for composites is often difficult, requiring specially shaped samples
and specific test procedures. In the developed methodology, a combination of
experimental data and a numerical warping field are required for calculating the
stiffness matrix; however, it was shown that axial and bending properties are
determined entirely from the experimental data. Since the partial derivatives
of the stiffness properties with respect to the material inputs are already
calculated as part of the uncertainty analysis, an optimization procedure could
then be applied based on these derivatives to minimize the residual between
the measured and predicted stiffness matrices. This could then be used to
identify the material properties (Young?s modulus, poisson ratio, and shear
modulus), as well as update other input parameters such as the ply orientation.
4. Measuring the sectional mass properties of the blade - Extensive stud-
ies have been performed at the University of Texas at Austin on the use of
DIC to measure the modal properties of blades through an operational modal
analysis. The cameras used in the current DIC setup only have a frame rate
of 50 fps (frame per second), which is not fast enough to measure the modal
properties; however, with high speed cameras, the two methods could be com-
bined to provide a full characterization of the sectional properties of composite
blades. The methodology from this dissertation could be used to first calculate
the stiffness properties under static loading. The measurement of the mode
shapes and frequencies could then be used to extract the mass properties based
on the already measured stiffness properties.
251
5. Development of a full-scale test rig - Full scale helicopter rotors have
many advanced features, which are well suited for analysis using the developed
methodology and normally aren?t incorporated on model scaled rotors. Since
helicopter companies often already have access to equipment, such as MTS
machines, capable of testing rotor blades under a variety of load conditions,
generating the static load cases necessary to measure the stiffness properties
should be feasible. DIC can be applied at a wide range of scales, but application
to a full-scale rotor blade may lead to unique challenges not present at the
current scale. However, since DIC is a non-contact measurement technique
requiring only a high contrast speckle pattern normally generated using paint
and ink, the testing would be minimally invasive and the blades could still be
used for normal operations.
252
Appendix A: Singular Value Decomposition
The singular value decomposition (SVD) theorem [122] states that given a
matrix A ? Cm?n, with m ? n but not necessarily of full rank, it can be factorized
into the matrix product
A = U ?V ? (A.1)
where U ? Cm?m is unitary, V ? Cn?n is unitary, ? ? Rm?n, and (?)? indicates
the conjugate transpose. Matrix ? contains real, non-negative numbers along only
its main diagonal with the remainder of the components equal to 0. The diagonal
entries ?i of ?, arranged in descending order, are
?1 ? ?2 ? . . . ? ?r > ?r+1 = . . . ?n = 0 (A.2)
where ?i are the singular values of A and r ? n is the rank of A. If A is of full rank,
r = n and the singular values are all positive. Since U and V are unitary matrices,
meaning they are square matrices whose inverse is equivalent to their conjugate
transpose, the inverse of A can be written as
? (A 1 = U ?V ?)?1
= U?1??1V ?1? (A.3)
= U???1V
253
where, because ? is non-square, ??1 is the left inverse defined by ??1 ? = I, whose
only non-zero elements are the reciprocals of the singular values located along its
main diagonal.
A.1 Computation of the SVD
A simple way to compute the SVD is to solve the eigenvalue problem for the
matrix product A?A = V ?2 V ? and then use the eigenvalues and eigenvectors to solve
the system U ? = AV for the unitary matrix U . However, the condition number
of A?A is the square of the condition number of A and this becomes much more
sensitive to perturbations and may become unstable. An alternative, stable method
for reducing the SVD to an eigenvalue problem is to form the (m + n) ? (m + n)
Hermitian matrix defined by ?? ?
H = ? 0 A? ?? (A.4)
A 0
The decomposition given by Eq ( A.1) implies that AV = U ? and A? = V ? which
means that the following equation holds
?? ? ? ?? ?? V V ?H ? ?= ? V V ???? ? 0 ?? (A.5)
U ?U U ?U 0 ??
and describes the eigenvalue decomposition of H. Since the singular values of A
must be non-negative, they are equivalent to the absolute values of the eigenvalues
of H, as seen by the last matrix in Eq. (A.5). This approach will be stable since
the condition number of H is the same as that of A. The eigenvalue problem for
the Hermitian matrix can then be solved numerically using a two-part computation,
first reducing the matrix to a bidiagonal form and then diagonalizing the bidiagonal
matrix; depending on the size of A, this can be efficiently computed using the Golub-
254
Kahan bidiagonalization (m < 5/3n) or the LHC bidiagonalization (m > 5/3n) [122].
In practice, algorithms for computing the SVD are readily available in the scientific
computing packages available for most scripting languages.
A.2 Reduced SVD
While the SVD given in Eq. A.1 provides a full factorization of A, if A is not
square (i.e. m 6= n) then this generally provides additional information that is not
needed in most applications. Consider first the matrix ?; this can be partitioned
according to ? ?
?? ?? ?? = ? (A.6)
0
where ?? ? Rn?n is a square diagonal matrix of the singular values ?i, ?? = diag (?i).
The matrix of left singular vectors U can be partitioned according to
[ ]
U = U? ? (A.7)
where U? ? Cm?n and ? ? Cm?(m?n). The SVD can then be written as
A = U ?V ?
[ ]?? ??? ?
= ?U? ? ? ?V (A.8)
0
= U? ??V ? + ? 0V ?
Since the second term ? 0V ? evaluates to 0, the SVD can be factorized more
compactly as A = U? ??V ? which is known as the reduced SVD. While U? is not
? ?1
unitary, it still forms an orthonormal basis and the equality A?1 = U? ?? V holds.
Moreover this only requires knowledge of the first n left singular vectors, which can
255
result in significant savings over the full SVD, in terms of both computational time
and storage, when m n.
256
Appendix B: Complex Step Derivative
When calculating the derivative of a function, an alternative to finite difference
approximations is a complex step derivative. The derivation of the complex step
derivative can be considered by using a Taylor Series expansion of the function f(x)
about x+ i given by:
df(x) ? 2h
2 d2f(x) 3 3? 3h d f(x)f(x+ i) = f(x) + i  i + ? ? ? (B.1)
dx 2! dx2 3! dx3
Taking the imaginary components of both sides (Im (?)), dividing by , and rearranging
in terms of df(x)/dx gives
df(x) 1
= Im (f(x+ i)) +O(2) (B.2)
dx 
For a small  1, the derivative is then
df(x) ? 1 Im (f(x+ i)) (B.3)
dx 
This is advantageous compared to the conventional central difference approximation
given by
df(x) ? f(x+ )? f(x? ) (B.4)
dx 2
257
as it allows the derivative to be calculated without evaluating a difference of function,
which provides a better accuracy. In addition, taking the real part of Eq. (B.1) gives
f(x) = Re(f(x+ i)) +O(2) (B.5)
which means that f(x) ? Re(f(x+ i)) and therefore both the functional value and
its derivative can be calculated using a single complex function evaluation.
If considering a function of multiple variables, the partial derivatives can be eval-
uated by perturbing each variable separately. For example, if f = f(X1, X2, ? ? ? , Xn)
then the partial derivatives evaluate to
?f ? 1 Im(f(X1 + i1, X2, ? ? ? , Xn))
?X1 1
?f ? 1 Im(f(X1, X2 + i2, ? ? ? , Xn))
?X2 2 (B.6)
...
?f ? 1 Im(f(X1, X2, ? ? ? , Xn + in))
?Xn 2
requiring n complex functional evaluations to compute all the partial derivatives.
258
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