STRUCTURAL RESPONSE AND DAMAGE
DEVELOPMENT OF CYLINDRICAL COMPOSITE
PANELS
by
Mark A. Tudela
B.S., University of Florida
(1994)
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of
the requirements for the degree of
Master of Science
in Aeronautics and Astronautics
at the.
Massachusetts Institute of Technology
February 1997
© Massachusetts Institute of Technology 1997
Signature of Author _
Department of Aeronautics and Astronautics
September 5, 1996
Certified by
e Professor Paul A. Lagace
MacVicar Faculty Fellow, Professor of Aeronautics and Astronautics
A
Accepted
A
Thesis Supervisor
ON
,Accepted
by
- Professor Jaime Peraire,
Chairman, Departmental Graduate Committee
FEB 101997
'A I
STRUCTURAL RESPONSE AND DAMAGE
DEVELOPMENT OF CYLINDRICAL COMPOSITE
PANELS
by
Mark Tudela
Submitted to the Department of Aeronautics and Astronautics on September 5, 1996 in
partial fulfillment of the requirements for the Degree of Master of Science in Aeronautics and
Astronautics
ABSTRACT
The structural response, including the mechanisms associated with snap-through
buckling, of cylindrical composite shell panels subjected to transverse loading was
investigated via experiments and numerical analysis. Specimens of Hercules AS4/3501-6
graphite/epoxy in [±45n/On]s (n=1,2,3) configurations and with a planar aspect ratio of 1
were tested in static indentation with pinned-free boundary conditions. Structural
parameters (radius, span, and thickness) were varied to encompass values utilized in the
structural configurations of transport aircraft fuselages. Force-deflection response and panel
deformation-shapes were determined during the tests and the damage from the tests was
evaluated using x-ray photography and sectioning techniques. A range of experimental forcedeflection responses was observed including smooth-stable, smooth with an instability
region and nonsmooth responses with an instability region. Deformation-shapes were
generally three-dimensional and exhibited both symmetry and unsymmetry. A switching
between symmetric and unsymmetric deformation-shapes occured in some specimens
corresponding with load-drops or the panel snapping away from the indentor. The geometric
ratio of specimen height to thickness characterizes the structural response as specimens
with larger values of this parameter were more likely to exhibit an instability in the forcedeflection response, unsymmetric deformation-shapes, and panel snap-away. Forcedeflection and deformation-shape behavior for pinned-free and simply-supported-free
boundary conditions were determined using a finite element analysis and the predicted
results for the two boundary conditions either bounded the experimental response or
matched the experimental response well for one of the two boundary conditions The
existence of nonzero in-plane compliance in the test fixture accounts for the variation of the
experimental response with respect to the predicted results as the relative magnitudes of
the in-plane stiffnesses of the shell and of the boundary conditions is a key consideration in
determining the structural response of shell panels. An experimental comparison of different
boundary conditions along the axial edges showed that increased rotational restraint
increases the critical snapping load, decreases the magnitude of the load reduction within
the instability region of the force-deflection response, and prevents the formation of
unsymmetric spanwise deformation-shapes. Damage in the form of matrix cracking and
delaminations in the specimen backside was detected in only the deepest, thickest specimen
geometry. Such damage forms near the critical snapping load and may be similar to that
found in plates due to the localized concave configuration which develops beneath the
loading point resulting in tensile bending stresses. Further work based on these results is
recommended to investigate the effects of unsymmetric deformations, in-plane compliance,
and various boundary conditions on the structural response and damage characteristics of
similar shells. Further experimental work to pinpoint the transition in damage behavior
due to the formation of localized concavity is also suggested.
Thesis Supervisor:
Title:
Paul A. Lagace
MacVicar Faculty Fellow, Professor of Aeronautics and
Astronautics, Massachusetts Institute of Technology
-3-
Acknowledgements
There are many people who contributed, in one way or another, to this
work. First, I've got to thank my wonderful girlfriend Jenna for tolerating the
hectic lifestyle that inevitably accompanies such an endeavor. She sacrificed
many experiences which we would have otherwise shared had I not been
sentenced to two years at the M.I.T correctional facility. I could have never
done it without her. I must also thank my entire family: Mom, Marly, Connie,
Grandma, and Grandpa for enduring the same sacrifices and for listening to my
constant babble about research even though they had no idea what I was
talking about. I would also like to thank my mom for her constant love and
support in everything I've chosen to do throughout my entire life. Whether she
realizes it or not, this has prepared me over the years to undertake and
complete projects such as this.
I'd also like to thank my advisor, Paul Lagace, for giving me tremendous
freedom in my research and for teaching me how to communicate effectively,
both in writing and especially in oral presentations (there's nothing like a good
verbal spanking by Paul during a TELAC presentation). Thanks to Hugh for
introducing me to composites in the 16.222 course (how does that A B D
matrix thingy work again??). Much thanks to Mark for teaching me all about
the failure of composites and for always having an open door. I don't know
what I would have done without my early meetings with Professor Dugundji.
He introduced me to shell theory and more importantly showed incredible
patience when I asked the dumbest of questions. However, the vast majority
of my education from the TELAC faculty came during the weekly TELAC
meetings. Through observation and participation, I was able to learn a great
deal about how quality research is performed and how to professionally
interact with my fellow researchers.
Many other people like Al, Don, Ping, Debbie, and Dick helped to get the
"real" work done. This is where the rubber meets the road here in TELAC.
They all made my life so much easier. Al is the most patient and kind man I
have ever met and, believe me, patience can be a virtue when dealing with me
in my turbo-stress mode. Thanks Ping for all the butterscotch candies and for
explaining to me at least five times how to fill out a requisition. Thanks to
Debbie for adding a much needed spice to the blend of personalities which come
in and out of the main office. Thanks to Don for showing me numerous tricks
of the machining trade and for letting me work late when I needed to. Thanks
to Dick for the invaluable assistance in getting my equipment designed and
built. Dick can find any part you could ever imagine if you give him about a
half an hour. I also can't forget about the numerous talented undergraduate
assistants who helped me out along the way: Marcus, Jason, Rasa, Robby,
Doug, Jimmy, Peter, and Barbara. I think I learned as much from each of
them as any other resource here at M.I.T.
I've also learned quite a bit from my fellow grad students: Bethany,
Chris, Sharath, Steve, Bari, Yuki, Lauren, Hari, David, Brian, Ronan, and
MONGO although these lessons were a bit different. For instance, Ronan and
Brian taught me that alcoholism can actually be a good thing. An ongoing
experiment, headed by Ronan, Brian, myself, and the wonderful people at Red
Dog have made this dream a reality. Hari Budiman always had the answer for
even the toughest questions. His answers were complete with diagrams,
derivations, and a complete bibliography and curriculum vitae of everyone
involved. He even taught me to look out for guys that wear glasses in public
showers. Without question the friendships provided by Hari, Brian, Ronan, and
David were what made TELAC special for me. Thanks to Brian for having
enormous integrity and for always saying it like it is. Thanks to Ronan for
always seeing through my selfish habits and appreciating me for the goodnatured gobshite that I really am. Thanks to Hari for showing the most
genuine concern for me during my entire career at M.I.T. and thanks to David
for being an all around good dude. I've listed a few of my more memorable
experiences with my fellow TELAC'ers here for posterity:
- The VPI trip (thanks to Bethany, Ronan, and Bari for company on the drive)
- The TELAC basketball team: MONGO (sorry, I didn't mean to crush the
bones in your leg dude), David Shia (get that #*!@ outta here), Brian Wardle
(who's reffing next), Hari Budiman (no no David), Paul Lagace (#@!%&**##),
Me (hey MONGO that guys picking on me)
- Sharath's brownness
- Scotch whiskey and cigars at Paul's
- Learning Irish colloquialisms from Ronan such as gobshite
- Ronan's almost scary attraction to the TELAC secretaries..... "would you like
a candy"
- Hugh descending from the Virginia hillside in a trenchcoat and sneakers
Foreword
This work was conducted in the Technology Laboratory for Advanced
Composites (TELAC) in the Department of Aeronautics and Astronautics at
the Massachusetts Institute of Technology. This work was sponsored by the
Federal Aviation Administration under Research Grant 94-G-037 with
additional support from NASA Langley Research Center provided in the form
of computer access for the numerical analysis under NASA Grant NAG-1-991.
In addition, tuition support was provided by the United States Air Force
through the Palace Knight Fellowship Program.
-6-
Table of Contents
List of Figures
8
List of Tables
15
Nomenclature
16
1. INTRODUCTION
18
2. BACKGROUND
22
2.1 Impact of Composite Plates
22
2.1.1
Structural Response
22
2.1.2
Damage Characteristics
25
2.2 Impact of Composite Shells
27
2.2.1
Structural Response
28
2.2.2
Damage Characteristics
32
2.3 Summary
3. APPROACH
35
36
3.1 General Overview
36
3.2 Test Matrix and Specimen Description
39
3.3 Analytical Approach
44
4. EXPERIMENTAL PROCEDURES
49
4.1 Manufacturing Procedures
49
4.2 Curvature and Thickness Mapping
59
4.3 Description of Test Fixture
67
4.3.1
Boundary Conditions
70
4.3.2
Deflection Measurement Assembly
75
4.4 Testing Procedures
81
4.4.1
Specimen Set-up in Fixture
83
4.4.2
Deflection Tests
87
4.4.3
Damage Tests
91
4.5 Damage Evaluation Procedures
92
4.5.1
X-Radiography Technique
92
4.5.2
Sectioning Techniques
93
5. RESULTS
97
5.1 Force-Deflection Behavior
5.1.1
Experimental Results
5.1.2
Numerical Results
5.2 Deformation-Shape Behavior
97
97
113
121
5.2.1
Experimental Results
123
5.2.2
Numerical Results
165
181
5.3 Damage
6. DISCUSSION
189
6.1 Comparison of Experimental and Predicted Results
189
6.2 Deformation-Shape Behavior
196
6.3 Importance of Geometric Parameters
219
6.4 Effects of Boundary Conditions
229
6.5 Damage
243
7. CONCLUSIONS AND RECOMMENDATIONS
247
7.1 Conclusions
247
7.2 Recommendations
250
REFERENCES
252
APPENDIX A
EXPERIMENTAL FORCE-DEFLECTI ON
RESPONSES
263
APPENDIX B
PREDICTED FORCE-DEFLECTION
RESPONSES
282
APPENDIX C
EXPERIMENTAL DEFORMATION-SHAPE
EVOLUTIONS
301
APPENDIX D
PREDICTED DEFORMATION-SHAPE
EVOLUTIONS
356
-8-
List of Figures
Figure 2.1
Illustration of the load-deflection response of a convex
shell under load- and stroke-controlled conditions.
29
Figure 3.1
Illustration of fuselage shell construction showing
stiffening elements.
38
Figure 3.2
Illustration of generic test specimen showing
important parameters.
41
Figure 3.3
Illustration of the grid utilized in the finite element
analysis.
47
Figure 4.1
Illustration of cylindrical mold configuration.
51
Figure 4.2
Schematic of cure assembly
54
Figure 4.3
Nominal temperature, pressure and vacuum profiles
for cure cycle
55
Figure 4.4
Illustration of milling machine cutting apparatus.
57
Figure 4.5
Illustration of mill table channel configuration.
58
Figure 4.6
Locations used for mapping shell thickness.
61
Figure 4.7
Illustration of geometric relation used to calculate
curvature (R) by measuring a and b.
62
Figure 4.8
Illustration of measurements for radii and twist
calculation.
64
Figure 4.9
Side-view illustration of original test fixture with a
convex shell mounted for transverse loading.
68
Figure 4.10
Illustration of the rod-cushion assembly.
69
Figure 4.11
Top view of test fixture top plate showing the slots
and extended cutout.
71
Figure 4.12
Illustration of grooved inserts in the rod-cushion
assembly.
73
Figure 4.13
Schematic of grooved inserts.
74
Figure 4.14
Illustration of knife-edge inserts.
76
Figure 4.15
Illustration of possible locations for measurement of
spanwise deflection.
77
-9Figure 4.16
Illustration of the deflection measurement assembly.
80
Figure 4.17
Illustration of the test fixture as mounted in the
testing machine.
82
Figure 4.18
Illustration of the center finder.
85
Figure 4.19
Schematic of the center deflection intervals used in
the deflection tests.
89
Figure 4.20
Sample planar x-ray picture showing damaged region.
94
Figure 4.21
Sample transcription of the cross-sectional damage.
96
Figure 5.1
Illustration of a smooth stable force-deflection
response (response type I).
99
Figure 5.2
Illustration of a smooth force-deflection response with
an instability (response type II).
100
Figure 5.3
Illustration of a non-smooth force-deflection response
with an instability (response type III).
101
Figure 5.4
Experimental force-deflection response for specimen
R12T3S2.
104
Figure 5.5
Experimental force-deflection response of specimen
R12T3S1.
105
Figure 5.6
Experimental force-deflection response for specimen
R12T3S3.
106
Figure 5.7
Experimental force-deflection response of specimen
R6T2S2.
108
Figure 5.8
Experimental force-deflection response of specimen
R6T2S3.
109
Figure 5.9
Experimental force-deflection response of specimen
R12T1S3.
111
Figure 5.10
Predicted force-deflection responses for geometry
R12T3S1.
118
Figure 5.11
Predicted force-deflection responses for geometry
R6T3S1.
120
Figure 5.12
Predicted force-deflection responses for geometry
R6T2S2.
122
Figure 5.13
Full panel deformation-shape data for specimen
R6T1S2 with a center deflection of 3.4 mm.
124
-10Figure 5.14
Illustration of the central spanwise and axial sections
used in the two-dimensional deformation-shape
presentation.
125
Figure 5.15
Experimental central spanwise deformation-shape
evolution for specimen R6T1S2.
126
Figure 5.16
Experimental central axial deformation-shape
evolution for specimen R6T1S2.
128
Figure 5.17
Illustration of the positive rotations and deflections
defined for the central spanwise and axial sections.
129
Figure 5.18
Experimental central spanwise DFU evolution for
specimen R6T1S2.
130
Figure 5.19
Full panel deformation-shape data for specimen
R12T3S2 (above) in the undeformed state, and
(below) with a center deflection of 0.6 mm.
132
Figure 5.20
Full panel deformation-shape data for specimen
R12T3S2 with a center deflection of (above) 1.1 mm,
and (below) 1.7 mm.
133
Figure 5.21
Full panel deformation-shape data for specimen
R12T3S2 with a center deflection of (above) 2.3 mm,
and (below) 2.8 mm.
134
Figure 5.22
Full panel deformation-shape data for specimen
R12T3S2 with a center deflection of (above) 3.4 mm,
and (below) 4.0 mm.
135
Figure 5.23
Full panel deformation-shape data for specimen
R12T3S2 with a center deflection of (above) 4.5 mm,
and (below) 5.1 mm.
136
Figure 5.24
Full panel deformation-shape data for specimen
R12T3S2 with a center deflection of (above) 5.7 mm,
and (below) 6.2 mm.
137
Figure 5.25
Experimental central spanwise deformation-shape
evolution for specimen R12T3S2.
138
Figure 5.26
Experimental central spanwise DFU evolution for
specimen R12T3S2.
141
Figure 5.27
Experimental central axial deformation-shape
evolution for specimen R12T3S2.
142
Figure 5.28
Full panel deformation-shape data for specimen
R6T2S2 (above) in the undeformed state, and (below)
with a center deflection of 1.1 m.
144
-11-
Figure 5.29
Full panel deformation-shape data for specimen
R6T2S2 with a center deflection of (above) 2.3 mm,
and (below) 3.4 mm.
145
Figure 5.30
Full panel deformation-shape data for specimen
R6T2S2 with a center deflection of (above) 4.5 mm,
and (below) 5.6 mm.
146
Figure 5.31
Full panel deformation-shape data for specimen
R6T2S2 with a center deflection of (above) 6.8 mm
and (below ) 7.9 mm.
147
Figure 5.32
Full panel deformation-shape data for specimen
R6T2S2 with a center deflection of (above) 9.0 mm
and (below) 10.2 mm.
148
Figure 5.33
Full panel deformation-shape data for specimen
R6T2S2 with a center deflection of (above) 11.3 mm
and (below) 12.4 mm.
149
Figure 5.34
Experimental central spanwise deformation-shape
evolution for specimen R6T2S2.
150
Figure 5.35
Experimental central spanwise DFU evolution for
specimen R6T2S2.
151
Figure 5.36
Experimental central axial deformation-shape
evolution for specimen R6T2S2.
153
Figure 5.37
Full panel deformation-shape data for specimen
R6T1S2 (above) in the undeformed state and (below)
with a center deflection of 1.1 mm.
156
Figure 5.38
Full panel deformation-shape data for specimen
R6T1S2 with a center deflection of (above) 2.3 mm
and (below) 3.4 mm.
157
Figure 5.39
Full panel deformation-shape data for specimen
R6T1S2 with a center deflection of (above) 4.5 mm
and (below) 5.7 mm.
158
Figure 5.40
Full panel deformation-shape data for specimen
R6T1S2 with a center deflection of (above) 6.8 mm
and (below) 7.9 mm.
159
Figure 5.41
Full panel deformation-shape for specimen R6T1S2
with a center deflection of 12.7 mm.
160
Figure 5.42
Experimental central spanwise deformation-shape
evolution for specimen R12T1S2.
163
-12-
Figure 5.43
Experimental central axial evolution for specimen
R12T1S2.
164
Figure 5.44
Predicted central spanwise deformation-shape
evolution for specimen R6T3S3 with simplysupported-free boundary conditions.
167
Figure 5.45
Predicted central spanwise DFU evolution for
specimen R6T3S3 with simply-supported-free
boundary conditions.
168
Figure 5.46
Predicted central axial deformation-shape evolution
for specimen R6T3S3 with simply-supported-free
boundary conditions.
170
Figure 5.47
Predicted central spanwise deformation-shape
evolution for specimen R6T3S3 with pinned-free
boundary conditions.
171
Figure 5.48
Predicted central spanwise DFU evolution for
specimen R6T3S3 with pinned-free boundary
conditions.
173
Figure 5.49
Predicted central axial deformation-shape evolution
for specimen R6T3S3 with pinned-free boundary
conditions.
174
Figure 5.50
Predicted central axial deformation-shape evolution
for geometry R12T2S2 with pinned-free boundary
conditions.
176
Figure 5.51
Predicted central spanwise deformation-shape
evolution for specimen R12T2S1 with pinned-free
boundary conditions.
177
Figure 5.52
Predicted central spanwise DFU evolution for
specimen R12T2S1 with pinned-free boundary
conditions.
178
Figure 5.53
Predicted central axial deformation-shape evolution
for specimen R12T2Slwith pinned-free boundary
conditions.
180
Figure 5.54
X-ray photograph for specimen R6T3S3 tested to a
center deflection of 27.7 mm.
182
Figure 5.55
Sectioning transcription of specimen R6T3S3 tested
to a center deflection of 27.7 mm.
183
Figure 5.56
X-ray photograph for specimen R6T3S3 tested to a
center deflection of 18.9 mm.
185
-13-
Figure 5.57
Sectioning transcription of specimen R6T3S3 tested
to a center deflection of 18.9 mm.
186
Figure 5.58
X-ray photograph for specimen R6T3S3 tested to a
center deflection of 23.4 mm.
187
Figure 6.1
Experimental and predicted force-deflection responses
for specimen R6T3S1.
191
Figure 6.2
Experimental and predicted force-deflection responses
for specimen R6T1S2.
193
Figure 6.3
Experimental and predicted force-deflection responses
for specimen R6T2S2.
194
Figure 6.4
Illustration of the important forces in the definition of
197
the "degree-of-pinned" parameter X.
Figure 6.5
Variation of 1with experimental force-deflection
response types I, II, and III.
199
Figure 6.6
Illustration of the important measurements in the
definition of the "degree-of-unsymmetry" parameter 8.
201
Figure 6.7
Force-deflection and 8-deflection responses for
specimen R6T3S1.
203
Figure 6.8
Force-deflection and 8-deflection responses for
specimen R6T2S2.
204
Figure 6.9
Force-deflection and 6-deflection responses of
specimen R6T1S2.
205
Figure 6.10
Geometric illustration of the axial rotation angles used
to characterize the deformation-shapes along the
central axial section.
208
Figure 6.11
Force-deflection and 0-deflection responses of
specimen R12T3S2.
210
Figure 6.12
Force-deflection and 0-deflection responses for
specimen R6T2S2.
211
Figure 6.13
Force-deflection and 6-deflection responses for
specimen R6T1S2.
215
Figure 6.14
Predicted force-deflection and 0-deflection responses
for specimen R6T3S3 with simply-supported-free
boundary conditions.
217
-14Figure 6.15
Predicted force-deflection and 0-deflection responses
for specimen R6T3S3 with pinned-free boundary
conditions.
218
Figure 6.16
Variation of Xwith thickness and radius for a constant
span S2.
220
Figure 6.17
Variation of Xwith span and radius for a constant
thickness T2.
221
Figure 6.18
Illustration of arch with perfectly pinned boundary
conditions.
223
Figure 6.19
Plot of experimental force-deflection response type
with Xand h/T.
226
Figure 6.20
Illustration of the different alignments used with the
double knife-edge fixtures: (top) perfectly aligned and
(bottom) misaligned by 1.6 mm.
230
Figure 6.21
Force-deflection responses for specimen R2T1S1 with
various conditions along the axial edges.
232
Figure 6.22
Central spanwise deformation-shape evolution for
specimen R2S1T1 with misaligned knife-edge
boundary conditions.
235
Figure 6.23
Central spanwise deformation-shape evolutions for
specimen R2T1S1 with grooved boundary conditions.
236
Figure 6.24
Illustration of geometry of arch configuration including
the effective in-plane stiffness of the boundary
conditions.
238
Figure 6.25
Plot of experimental degree-of-pinned paramter Xwith
normalized ratio of thickness to span.
241
-15-
List of Tables
Table 3.1
Test Matrix
45
Table 4.1
Results of Thickness and Curvature Mapping
66
Table 4.2
Locations of axial deflection measurements
in panels of various span.
79
Table 5.1
General Characterization of the Experimental ForceDeflection Responses
102
Table 5.2
Experimental and Predicted (Pinned-Free) Critical
Snapping Loads
112
Table 5.3
Experimental and Predicted (Pinned-Free) Critical
Snapping Displacements
114
Table 5.4
Experimental Peak Force
115
Table 5.5
Experimental Peak Deflection
116
Table 5.6
General Characterization of the Predicted Pinned-Free
Force-Deflection Responses
119
Table 5.7
General Characterization of the Central Spanwise
Deformation-Shapes
139
Table 5.8
General Characterization of the Central Axial
Deformation-Shapes
154
Table 6.1
Values of the parameter X for all specimens
198
Table 6.2
Values of the parameter h/T for all specimens
225
Table 6.3
Characterization of Experimental Force-Deflection and
Deformation-Shape Behavior with h/T
228
-16-
Nomenclature
A
cross-sectional area
E
elastic modulus
h
shell height
H
compressive membrane force
I
moment of inertia
K
in-plane stiffness of the boundary conditions
m
governing parameter for solution to isotropic arch pinned with in-plane
compliance
n
governing parameter for solution to isotropic pinned arch
PD
experimental critical snapping load
Pp
predicted pinned-free critical snapping load
Ps
predicted simply-supported free load at the critical snapping
displacement of the predicted pinned-free response
R
shell radius
Rn
scaled specimen radius
S
shell span
Sn
scaled specimen span
T
shell thickness
Tn
scaled specimen thickness
x
circumferential direction
y
axial direction
z
vertical direction
P
spanwise twist
8
degree-of-unsymmetry parameter
A
center deflection
-17-
7
axial twist
X
degree-of-pinned parameter
OL
axial rotation angle for the left axial portion of the specimen
OR
axial rotation angle for the right axial portion of the specimen
-18-
CHAPTER 1
INTRODUCTION
Composite materials continue to find use in aircraft as they offer a
number of advantages over conventional materials such as aluminum and
titanium. Key advantages of composites are their high specific strength and
stiffness. These attributes allow military aircraft to attain higher levels of
performance and commercial aircraft to be more fuel efficient by simply
decreasing the structural weight. In addition, the properties of laminated
composite structures can be tailored to give greater strength and stiffness in
a preferred direction, further increasing their efficiency.
These
characteristics give aircraft designers more flexibility than they would
otherwise have using conventional materials.
Indeed, many of the exciting
advances on the horizon for the aerospace industry, such as the High Speed
Civil Transport and the Aerospace Plane, will rely heavily on the use of
composite materials. However, the current reality is that composite
structures cannot be utilized to their full potential. Material orthotropy and
the multiplicity of damage modes make composites particularly difficult to
analyze. As a result, large knockdown factors must often be used which
mitigate the previously mentioned advantages over metallic materials [1].
Although they provide a number of advantages, laminated composites
also have several disadvantages. Of particular concern is susceptibility to
damage from transverse loading due to their low through-thickness
strengths. Transverse impact events such as a tool dropped onto a wing
panel, runway debris kicked up during takeoff, and bumping with service
-19vehicles can cause damage which significantly reduces compressive loadcarrying capability in laminated composite structures while leaving little to
no visible damage[2, 3]. A typical mode of damage under these conditions is
delamination.
Such damage could go undetected thereby seriously
compromising the structural
integrity and safety of the aircraft.
Consequently, transverse impact can be the limiting design consideration for
composite structures.
Thus, the advantages of composites cannot be fully
realized until a clear understanding of transverse impact damage
development is established and design methodologies utilizing this
understanding are developed to deal with this issue.
The considerable amount of research involving the impact of
composites has led to the identification of two distinct issues: damage
resistance and damage tolerance[4].
Damage resistance is a measure of the
amount of damage produced in a material/structure due to a particular event
such as impact.
Damage tolerance is a measure of the ability of a
material/structure to perform a certain function, with damage present.
Generally, relationships exist between the two areas.
In particular, an
adequate assessment of a structure's damage tolerance requires a knowledge
of the amount and type of damage present (i.e. damage resistance).
Currently, many aircraft are designed using a damage tolerance philosophy.
Consequently, an understanding of a structure's damage resistance is the
first step in achieving a baseline methodology for a damage tolerant design
with respect to impact. Unfortunately, the current level of understanding
regarding the damage resistance of composites is far from complete.
Limitations due to damage considerations have been an important
consideration in the relegation of composites to mainly secondary structural
applications such as wing flaps and elevators. However, the widespread use
-20of composites in primary load-bearing components such as fuselages remains
an industry goal. This can happen only if the response of these structural
configurations to transverse impact is well understood. Although, a good deal
of research has considered the impact resistance of flat plates, it is
questionable whether this knowledge can be extended to consider realistic
structural configurations such as a fuselage (i.e. shells).
Since most
aerospace components are curved and not flat, a clear need exists to
understand the impact resistance of shells. It is this connection between plate
and shell impact resistance which provides the impetus for the current work.
The impact resistance of realistic fuselage skin panel geometries are
investigated in the present research. This is accomplished by considering
geometries representative of fuselage panel sections in typical commercial
aircraft. A quasi-static approach is utilized to experimentally determine the
forces and deflection shapes which develop under transverse loading. This
approach has recently been validated for the transverse impact of composite
shells[5]. The work will also help establish a better understanding of the
snap-through buckling phenomenon which exists for concave shells under
transverse loading[6-9]
The primary objective is to gain a more detailed
understanding of the mechanisms associated with snap-through buckling and
their relation to the overall structural response and damage development of
convex shells.
The details of this work are described in the following chapters. A
review of the work relating to shell impact is presented in Chapter 2.
The
general approach and objectives of the work are introduced in Chapter 3.
Manufacturing, testing, and damage evaluation procedures are outlined in
Chapter 4. The analytical and experimental results are presented in Chapter
5.
Implications of these results are discussed in Chapter 6.
Finally,
-21conclusions regarding the present work and recommendations for future work
are made in Chapter 7. Appendices containing the load-deflection diagrams,
central, and axial modeshape evolutions are given at the end of the
document.
-22-
CHAPTER 2
BACKGROUND
The significant strength loss caused by the presence of damage has
provided the impetus for predicting and quantifying the amount of damage in
a composite structure. Hence, research pertaining to damage resistance is
reviewed in this chapter. Since a significant knowledge base exists for plate
impact, the major issues with regard to previous work on plate type
configurations are summarized as a prelude to the review of work done for
shells. The damage resistance issues discussed in this review can be divided
into the following two categories: structural response and damage
characteristics.
The sections for both plates and shells are organized
according to these categories for the purpose of providing a clear discussion.
2.1
Impact of Composite Plates
Research into the impact response of plates has been extensive,
producing some fundamental concepts and approaches. The basic insights
gained from plate impact research are presented to establish a framework for
effective discussion of the major issues.
2.1.1 Structural Response
Much has been learned about the structural response of composite
plates subjected to transverse impact. So much so, that numerous review
articles have appeared in the literature which deal mainly with the plate
-23geometry [10-12].
For a complete understanding of the structural impact
response, the time history and spatial distribution of the forces developed at
the point of contact must be determined. Although many important issues
remain unresolved, some general classifications and approaches have been
developed to simplify the treatment of plate impact events.
Impact events have been broken into three rather ambiguous regimes:
low, intermediate, and high velocity [2, 4, 10, 11]. This can be misleading
since knowing the impact velocity is not enough to predict the effect of an
impact event [4].
The impact response of a structure largely depends on
material, geometry, boundary conditions, and the mass and velocity of both
the impactor and a representative part of the structure. There are, therefore,
no rigid boundaries for classifying impact events by velocities alone.
Whether an impact event is termed "low-", "intermediate-" or "high-" velocity
depends on all of the above parameters.
So-called "low velocity" impact events are those with sufficient contact
duration for stress waves to propagate to the boundaries of the structure.
This implies that the response during "low-velocity" impact is global and is
therefore affected by the boundary conditions of the structure. Under such
conditions, a static analysis can be utilized to simulate the structural
response during the impact event. This approach is termed "quasi-static" and
is generally justified for "large mass/low velocity" impacts although boundary
conditions and structural configuration remain important parameters [13-16].
"Low velocity" impacts of aerospace structures can occur in service or even
during routine maintenance. Tools dropped onto the structure and kick-up of
runway debris are common examples. "Intermediate-velocity" impacts refer
to situations where the time required for stress waves to propagate to the
boundaries of the structure are on the same order as the contact duration. As
-24the contact duration becomes much less than the time required for stress
waves to reach the boundaries, the boundary conditions of the structure
become less significant. This situation is characteristic of "high velocity"
impacts such as ballistic encounters.
The interactions that occur between the structure and the impacting
body involve local contact stresses and global structural deformations which
interact. In order to make the problem more tractable, the local and global
responses are typically analyzed seperately and then combined in some
manner [17]. Static contact between two isotropic bodies has been studied
extensively in the classical theory of elasticity [18, 19]. The force-indentation
relationship for elastic isotropic indentation of a half-space has been shown to
follow the Hertzian contact law:
F = Ka1"5
(2.1)
where F is the contact force, K is the constant contact stiffness, and a is the
indentation. Variations of this contact law have been sucessfully used to
model the indentation of composite plates with deflections on the order of the
plate thickness [20, 21].
Oftentimes, the details of the contact force
distribution have little effect on the global plate response, although the
opposite may not be true [22].
However, if the plate undergoes large
deflections, the contact behavior can deviate considerably from Hertzian
behavior as the structure tends to "wrap" around the indentor [17].
Global plate response during impact can be an important component of
the overall structural response [12]. A considerable amount of research has,
therefore, been devoted to predicting the global plate response during impact
[10, 11, 22-26]. These models are primarily concerned with predicting the
force and displacement histories of the impact event. Some models treat the
-25plate as a continuum by using plate theories of varying complexity along with
variational methods to solve for the response [23, 25]. The efficiency and
accuracy of these methods are strongly dependent on the general form of the
deflection shapes which must be assumed a priori. An alternative to this
approach is the finite element method which approximates the solution by
discretizing the plate into small elements and simultaneously solving for the
forces and stresses in each element [13, 15, 27]. Both techniques become
computationally intensive when considering nonlinear behavior such as large
deflections [25].
The aforementioned approaches have proven useful for predicting the
structural response of composite plates.
Particular consideration must
always be given to the pertinent impact event parameters such as plate
geometry, impactor mass and velocity, and boundary conditions to name a
few. In each of the various regimes of interest, techniques exist to predict the
forces and stresses due to the global deformations which develop during
impact. Together with adequate determination of local contact stresses, these
results can be used as input for subsequent damage prediction models.
2.1.2 Damage Characteristics
Predicting the state of impact damage, i.e. damage resistance, in a
composite plate is the ultimate goal of many of the previously mentioned
approaches.
However, a serious shortcoming in the area of damage
prediction exists due to the multiplicity of failure modes and to the lack of
reliable failure criteria [10-12].
Failure can occur in the form of matrix
cracks, delaminations and fiber breakage and the linkage between these
failure modes is not well understood [12]. Nonetheless, a great deal has been
-26learned about plate impact damage and the key findings are briefly reviewed
here.
The impact and structural parameters which affect the structural
response, and hence, the damage resistance of plates, are specimen geometry,
mass, stacking sequence, and boundary conditions, as well as impactor mass,
geometry and velocity [10-12].
A number of studies [28-33] have shown
similar evolutions of damage, in both mode and extent, for composite
laminates. As contact force is applied, damage typically initiates in the form
of matrix cracks. As the force is increased, the matrix cracks grow, coalesce,
and encounter ply interfaces where they form delaminations.
The
delaminations are initally bounded by two matrix cracks, but as the force is
increased further, the delaminations grow in size and become elliptical in
shape. Eventually, fibers begin to break allowing full penetration by the
indentor. This sequence of failure may also be affected by different laminate
thicknesses and boundary conditions [32, 34].
For a given structural configuration, peak impact force has been
identified as an important metric with regard to the damage created [16, 35].
This metric has proven particularly useful for low-velocity/large-mass
impacts. Since a quasi-static analysis is justified in this range, the damage
can be determined by performing static indentation tests up to the same peak
impact force [14, 16, 35].
The different damage modes and sequences of damage development
have been identified for composite plates subjected to transverse loading.
However, the linkages between the stress and damage states, that is the
failure criteria, is a key issue. Consistently reliable failure criteria have not
been established for composites although a large number of failure criteria
can be found in the literature [36, 37]. Thus, this lack of linkage remains a
-27significant shortcoming in the area of damage prediction. As a result, quasistatic testing along with simple metrics such as peak force remain necessary
tools for determining the damage resistance of composite plates.
2.2
Impact of Composite Shells
The impact response of composite shells is more complex than that of
plates due primarily to geometric couplings generated by the presence of
curvature.
However, knowledge gained from plate impact research has
provided direction to the current efforts for shells. The differences from plate
behavior must first be identified to understand what is and is not applicable
to the study of shells.
The issues unique to shells can then be pursued
separately to gain a complete understanding of impact behavior.
As noted, the primary difference between plates and shells is
curvature.
Non-zero curvature causes the bending and membrane
deformations to become coupled. Simple bending loads, such as transverse
loading during an impact event, instantly generate membrane stresses. This
bending-membrane coupling is immediately present in shells, whereas in
plates, it develops gradually as the transverse deflection increases. Thus,
membrane effects generally cannot be ignored during shell impact, regardless
of the magnitude of transverse deflection. This added complexity must be
considered in the damage resistance of shells.
The research pertaining to the damage resistance of composite shells is
reviewed in the following two sections. The first section deals with structural
response and the second covers damage characteristics. As with plates, the
force-deflection relationship for low-velocity/large-mass impact has been
shown to be similar to that of static loading [5]. Thus, a review is given for
-28the static responses of composite shells in the first section. Both static and
dynamic damage studies are reviewed in the second section.
2.2.1 Structural Response
Compressive membrane stresses, which develop during transverse
loading of convex shells, can cause marked differences in the force-deflection
response as compared to plates. Several investigators have shown that the
force-deflection response of convex shells can, in some instances, exhibit a
"snap-through" instability [5-9, 38] as illustrated in Figure 2.1. The forcedeflection response of plates is approximately linear for small deflections and
becomes increasingly stiffer for larger deflections [19]. However, for convex
shells under stroke-controlled conditions, the response is approximately
linear for very small deflections followed by a relaxation of the stiffness as the
deflection increases [9, 39] as illustrated in region O-A, termed the "first
equilibrium path," of Figure 2.1. This large deflection response is opposite to
the plate response.
As the deflection is further increased, the response
changes from relaxation to stiffening. This change may occur at an inflection
point or may occur over a region, known as an "instability region", where,
under deflection-controlled conditions, the slope of the force-deflection curve
becomes negative [9, 39]. This instability region is shown as region A-B in
Figure 2.1. The point A at which the slope changes sign is termed the critical
snapping load. If the test is conducted under load-controlled conditions, the
convex shell can instantaneously "snap through" to a concave configuration,
from point A to point C, upon reaching the critical snapping load. The region
of monotonic stiffening, shown as B-D, is often termed the "second
equilibrium path."
This unique behavior of shells presents additional
challenges to the study of structural impact response.
-29-
Load
Stroke-Controlled Test
- -
-
- Load-Controlled Test
D
First
Equilibrium
Path
0
Figure 2.1
-
m -
m-
-
-
Second
Equilibrium
Path
Deflection
Illustration of the load-deflection response of a convex shell
under load- and stroke-controlled conditions.
-30The force-deflection response of composite shells under static loading
conditions have been examined by several investigators [6-9, 27, 40-44]. A
large displacement analysis based on a shallow orthotropic arch has shown
good correlation with experiment for the load-deflection response of
cylindrical panels subjected to line loads [6]. Membrane forces were assumed
constant throughout the panel, allowing a closed-form solution to be obtained.
The force-deflection response was found to be dependent on a single nondimensional parameter , given by:
4 = A
where R is the radius of curvature,
22 R
2
p 4 /D
22
P is the total arc length
(2.1)
of the panel, A2 2
and D 22 are the circumferential extensional and bending stiffnesses,
respectively. Generally, the parameter X increases with the depth of the
arch. The analysis showed that for very small values of X, all equilibrium
configurations are stable. Panels with larger values of X show a loss of
stability at a limit point and a further increase in X results in a stability loss
at a bifurcation point. The bifurcation point is the intersection of the primary
equilibrium path with a secondary path representing asymmetric equilibrium
configurations. A similar parameter governs the instability response for
shallow isotropic arches [45, 46].
This isotropic parameter is strictly
geometric whereas the orthotropic parameter X depends on the panel
geometry and the ratio of membrane to bending stiffnesses. The analysis in
[6] also captured the general trends of the deformation-shape development.
It was shown that the bifurcation corresponded to the formation of
unsymmetric deformation-shapes.
Analytical methods, which utilize an a priori assumption of the
deflection shapes, have also been used sucessfully to predict the force-
-31deflection response [8, 9, 44]. Linear strain-displacement relations provide
adequate solutions only for transverse deflections on the order of the shell
thickness [44]. Von Karman large displacement kinematics [8, 9] for a plate
with a small initial curvature and Donnell's shallow shell equations [47] have
been sucessfully used to predict the large deflection response for shallow
shells. The accuracy of these simplified kinematics diminish as the depth of
the shell is increased since deeper shells require a more complex formulation
of the kinematics which includes large displacements and large rotations [40,
48]. The resulting solution is often computationally intensive, rivaling the
large computation times of finite element analyses.
As a result, finite
elements are often used to predict the force-deflection response of deep shells
[13, 27, 40, 41, 49].
Experimental data on the large-deflection response of composite shells
is far less abundant than that seen for plates. That which does exist shows
the importance of the snap-through instability. For example, the static
response of shallow cylindrical cross-ply arches to radial line loads showed
the existence of a snap-through instability along with unsymmetric spanwise
deformation-shapes [6]. The loading head was physically bolted to the arches
to prevent them from snapping away from the indentor during a strokecontrolled test. These thin (1.5 mm) laminates developed negative forces in
the force-deflection response, thereby showing the existence of a stable
postbuckled configuration. Such postbuckled configurations were also found
for convex shells of square planform during the stroke-reversal portion of a
stroke-controlled test [5].
Generally, the snap-through response was
concluded to be dependent on the relative contribution of membrane and
bending stiffnesses [5] for the convex shells. Experimental work on thin
unidirectional graphite-reinforced plastic panels showed that panels with
-32-
clamped boundary conditions exhibited snap-through while panels with
simply-supported boundary conditions merely exhibited a mild relaxation in
the force-deflection response [7]. The presence of the snap-through instability
was attributed to the higher compressive membrane stresses for the clamped
case.
The snap-through response of convex shells is clearly different from the
plate response. As a precursor to shell panels, the arch geometry has been
studied analytically to reveal a basic understanding of the snap-through
process. For instance, the snap-through characteristics of orthotropic arches
were shown to be both geometric and material dependent with the possible
formation of unsymmetric deformation-shapes. Although experimental data
regarding the snap-through of general composite shell panels remains sparse,
some basic understanding has been extended from the simple arch geometry.
For instance, compressive membrane stresses, not present in plates under
transverse loading, have been clearly established as the driving force for the
snap-through process for orthotropic arches as well as shell panels [7-9, 39].
Furthermore, analytical tools have been developed to predict the structural
response of general composite shells subjected to static loading. A wide range
of shell geometries can be studied with analyses of varying complexity.
Nonlinear finite element analyses are generally applied to the snap-through
of deep shells while simplified variational approaches are generally utilized
for more shallow geometries. However, it remains difficult to verify and
assess such analyses without sufficient experimental results.
2.2.2 Damage Characteristics
Damage studies involving composite shells have largely utilized the
knowledge base currently available for plates. For instance, during low-
-33velocity/large-mass impact, the use of quasi-static testing and simplified
damage metrics such as peak force have been explored for shells [5, 33, 5052]. This type of approach has identified key similarities and inconsistencies
with plate procedures, all of which are reviewed in this section.
Damage formation in composite shells has been directly compared to
that of plates, with the work concentrating on the effect of the curvature in
the shell configuration as compared to the plates [5, 53, 54]. In the case of
small transverse deflections, the shells showed fiber cracks in the upper
layer, shear cracks in the middle layer and delaminations in the upper and
lower interfaces [53].
A general conclusion was that the stiffer shell
structures had more damage than the plates for these particular conditions.
However, it was unclear whether the differences in damage states were due
to the presence of compressive membrane stresses or simply to the larger
peak force attained by the stiffer shell structure.
It is well established for composite plates that a given contact force
produces a particular state of damage whether it is introduced during a static
or large-mass/low-velocity impact event [14, 29, 31, 55].
Recent evidence
suggests that peak force plays a similar role for composite shells under
similar conditions [5, 33, 50-52]. However, the type and extent of damage for
plates and convex shells subjected to the same impact event can be
significantly different [5]. The typical "peanut-shaped" delamination regions
were found for plates whereas unsymmetric damage states were found for
convex shells that attained a peak force on the first equilibrium path.
Furthermore, average damage extent, defined as the average length of
delaminations, for convex panels was shown to have a linear relationship
with peak force when the peak force occured on the second equilibrium path,
in the same manner as previously shown for plates. However, panels with
-34sufficient stiffness such that the peak force occured on the first equilibrium
path showed significant deviation from the linear trend. This was attributed
to the compressive membrane stresses which exist on the first equilibrium
path [5].
Load and displacement for damage incipience is a function of laminate
layup and thickness [56-58], with matrix cracking and delaminations
occuring before fiber breakage.
These damage characteristics have been
extensively demonstrated for plates indicating similarities between shell and
plate impact damage. Panels with smaller transverse deflections show more
localized damage under the indentor [56, 57] and higher threshold energies
for damage incipience [58]. These results are somewhat contradictory to the
results in [5] which showed that panels with smaller transverse deflections
could experience a larger damage extent depending on whether the peak force
occured on the first or second equilibrium path.
The effects of different boundary or support conditions have been
investigated for a full cylinder configuration by using various forms of
internal support [51]. The damage mode and extent is very sensitive to the
type of boundary or support condition. This is an expected result since it has
been shown extensively for plates. To eliminate the uncertainties associated
with modelling a real structures boundary conditions, impact studies have
been performed on full scale structures such as the XFV-12A composite wing
[59]. Results indicate that the damage found in full scale structures is very
similar to that obtained with laboratory coupons, suggesting that current
techniques may ultimately be applicable to full scale structures.
-352.3
Summary
Although only limited work has been done regarding the damage
resistance of convex shell panels, key differences and similarities with plate
behavior have been identified. The most striking difference is the existence of
snap-through buckling in the response of convex shells. The detailed effects
of this instability must be understood in order to fully elucidate the
differences between plate and shell behavior from a damage resistance
perspective.
For instance, information regarding the global structural
deformations which occur during snap-through may give insight into damage
formation and development. Currently, experimental data regarding these
complex deformations are not available in the literature. Thus, a clear need
exists for the identification of snap-through buckling characteristics.
Key similarities such as the use of peak impact force as a primary
damage metric have also been identified. However, the existence of the snapthrough instability removes the uniqueness of structural state normally
associated with a given force. This calls into question the applicability of any
damage metric associated with force. However, evidence has suggested that
peak impact force may be a good damage metric when one or the other
equilibrium paths is specified [5].
Similarities
to plate damage
characteristics have been identified for shells on the second equilibrium path
[5, 56-58].
Therefore, it becomes important to identify damage incipience
with regard to equilibrium paths. Results also indicate that the behavior is
strongly dependent on the particular shell geometry. Damage studies, to
date, have not considered configurations representative of realistic fuselage
panels. It is, therefore, difficult to draw conclusions regarding the damage
characteristics of a real fuselage structure.
-36-
CHAPTER 3
APPROACH
3.1
General Overview
As pointed out in Chapter 2, a need exists to further understand the
mechanisms involved in snap-through buckling and their relation to the
damage resistance of cylindrical composite panels. Specifically, the deflection
shapes need to be investigated more fully since they represent the most
obvious characterization of the snap-through buckling phenomenon.
Deflection shapes are also important from an analytical point of view.
Variational analyses such as the Rayleigh-Ritz approach require an a priori
selection of the deflection functions, often in series form. Knowledge of the
experimentally-determined deflection shapes allow a prudent choice of
functions to be made, thereby increasing the efficiency of the analysis.
Damage formation during snap-through buckling is a key issue in
characterizing the damage resistance of convex shells. Furthermore, it is
important to determine the damage incipience point with respect to the
primary regions in a typical force-deflection response: the first equilibrium
path, the instability region, and the second equilibrium path, as defined in
Chapter 2. Damage that occurs on the second equilibrium path is likely to be
similar to that seen for plates due to the development of tensile membrane
stresses along this path [5].
However, compressive membrane stresses
dominate on the first equilibrium path and continue to be present in the
instability region [6]. Hence, damage characteristics in these regimes may be
-37-
different from those of plates. In addition, damage which initiates within the
instability region would call into question the use of peak force as a damage
metric since the force steadily decreases in this regime. Knowledge of the
behavior in each regime is, therefore, necessary to better characterize the
damage resistance of composite shells
Due to the increased interest in composites for fuselage construction,
specimen geometries are chosen to represent fuselage sections in typical
commercial aircraft. The current method of construction for aircraft fuselage
panels is to give added support to the cylindrical shell structure through
stringer and ring stiffening elements, as illustrated in Figure 3.1. The small
panels bounded by these stiffening elements are the basis for the specimen
dimensions chosen in this investigation.
The objective of the current work is thus to gain a more detailed
understanding of the mechanisms associated with snap-through buckling and
their relation to the overall structural response and damage development of
realistic fuselage panels. Specifically, effects and mechanisms of the snapthrough instability, under low- velocity/large-mass impact conditions, are
studied. Experimental and analytical studies are conducted to quantify
pertinent variables and explore their relationships. Attention is given to the
portion of the response where compressive membrane loading occurs since
this has been identified as the primary difference from plate behavior [6, 9,
38]. Details of the structural response, such as contact forces and deflection
shapes, are studied experimentally and analytically while damage incipience
and development are investigated experimentally. As discussed in Chapter 2,
quasi-static testing has been shown to produce similar structural responses
and damage states to those seen during low-velocity/large-mass impact
events [5].
Quasi-static testing is, therefore, utilized in the experimental
-38-
Figure 3.1
Illustration of fuselage shell construction showing stiffening
elements.
-39portion of this investigation since it is both easier and more repeatable than
impact testing.
The static force-deflection response of each panel under strokecontrolled conditions is obtained in the first stage of the experimental
program. During each of these tests, the stroke is held at certain values
during which three-dimensional deformation-shapes are recorded for each
panel. The results are compared to those obtained using the commercial
finite element package: Structural Analysis of General Shells (STAGS).
The damage states are investigated in the second stage of the
experimental program. If damage is detected in the first set of experiments,
subsequent panels are tested to reveal the damage incipience and
development.
If no damage is detected in the first set of tests, then the
damage incipience point can be identified as being further along the second
equilibrium path. This implies that the damage development is similar to
that seen in plates and existing evaluation techniques can be utilized.
3.2
Test Matrix and Specimen Description
The three main structural parameters varied herein are radius of
curvature, span, and thickness.
A special nomenclature established in
previous work is used to facilitate discussion [5].
Each parameter is
identified according to the following scaling relation:
(Xn) = n(X1)
(3.1)
with X representing any of the three main structural parameters and n
taking on various values.
As in the previous work [5], the variable X1
represents baseline values of 152 mm (6"), 102 mm (4"), and 0.804 mm
-40-
(0.032") for radius (R), span (S), and thickness (T), respectively.
In the
current investigation, the variable n takes on values of 1, 2 and 3 for both
span and thickness and values of 6 and 12 for the radius. Thus, any given
specimen geometry can be identified by the n values for radius, thickness and
span, e.g. R6T1S1.
A fuselage can be thought of as small cylindrical panels which are
supported by the stiffening elements, as illustrated in Figure 3.1. Thus, all
specimens are of cylindrical curvature with sizes based on actual transport
fuselage configurations. The planform dimensions, or spans, cover typical
stringer spacings in such a transport aircraft fuselage (150 mm to 250 mm)
[60]. A square planform is maintained for consistent comparison of the
structural response as the span is varied [5]. Radii of curvature of 914 mm
(36") and 1829 mm (72") are chosen to represent approximate fuselage
dimensions of general aviation and commercial transport aircraft respectively
[60]. These parameters are depicted in Figure 3.2 for a generic specimen.
The layups chosen are [4 5 n/- 4 5 n/On]s with n varying from 1 to 3 for
comparison with previous impact investigations [5, 31, 34] and to utilize the
"effective ply" concept for damage comparison [61].
During transverse
impact, delaminations, which form at dissimilar ply interfaces, may
constitute a large portion of the resulting damage.
With the current
arrangement, varying n simply changes the effective thickness of each ply,
leaving the number of dissimilar ply interfaces constant and, therefore,
yielding a more controlled damage study.
The material system used in this research is Hercules AS4/3501-6
graphite/epoxy due to its use in related plate impact studies [23, 31, 34] and,
more specifically, to its use in a closely related study of convex shell impact
[5].
-41-
Circumferential
Edge N
Direction
Axial
,Edge
rential
Thickn ess=T
Span=Sn
Direction
II
Radius=Rn
Figure 3.2
Illustration of generic test specimen showing important
parameters.
-42Since the main objectives of this research center around the snapthrough phenomenon, the boundary conditions were chosen to promote its
occurence.
Pinned axial edges resist in-plane motion, resulting in the
compressive membrane stresses necessary to produce an instability. Free
circumferential edges allow full panel rotation which enhances the global
deformations during snap-through. Hence, pinned conditions along the axial
edges and free conditions along the circumferential edges were utilized. It
should be noted that the rotation condition provided by an actual stringer
support falls somewhere between pinned and clamped. Although clamped
axial edges would also provide the essential compressive membrane stresses,
pinned axial edges are more desirable from a damage investigation
standpoint since they are less likely to cause damage at the axial edges which
would complicate the investigation due to multiple damage sites.
The force-deflection response and damage characteristics of convex
shells has recently been shown to be equivalent for quasi-static loading and
low-velocity/large-mass impact conditions [5].
Previous results, for plates
under similar conditions, have proven useful in establishing efficient static
test methods to characterize plate impact damage resistance [14].
Static
tests are desirable since they are easier to conduct and standardize due to the
elimination of impact-related variables. Quasi-static testing is, therefore,
utilized in the present investigation to simulate low-velocity/large-mass
impact conditions.
Each panel is statically loaded on the convex side to simulate exterior
impact of a fuselage panel. From simple geometric considerations, the fully
inverted, or concave, configuration was chosen as a clear point at which
tensile membrane forces exist and, thus, no test was conducted beyond this
point. A 12.7 mm hemispherical indentor is used to apply load to the center
-43of the panel. This indentor size is consistent with previous work performed
for both plates and shells under transverse loading [31, 34, 38]. The first
stroke-controlled test performed on each panel geometry provides the forcedeflection response up to the point where the panel has fully snapped through
to an inverted configuration. During this test, the stroke is held at prechosen increments during which deformation-shape data is taken for the
entire panel. In order to adequately characterize the "deformation-shape
evolutions", stroke increments are chosen to yield roughly ten deflection
scans. In some of the more shallow panels, the evolution is more coarse since
the interval of center displacement is limited by the resolution of the
measurement system.
Panel deformation-shapes are investigated by taking finely spaced
deflection data (approximately 100 data points) in the spanwise direction. A
spanwise deformation-shape is taken at five different axial positions during
each held stroke position.
Coarse axial separations are used since the
variation of deflection in this direction is considered secondary. Adequate
data is obtained to infer the deflection shape of the entire panel at each
stroke interval.
Each panel tested in the first stage of experiments is x-rayed at the
contact point to investigate the state of damage. The x-ray technique gives a
two-dimensional integrated representation of the through-thickness damage
state. After the x-ray is taken, the panel is sectioned along the central span
to investigate the possibility of other spanwise damage locations. Sectioning
also allows the details of the damage state through the thickness to be
investigated.
If damage is detected in these specimens, additional static
indentation tests are conducted up to intermediate stages of snap-through.
These tests are conducted up to key points, such as the critical snapping load
-44and snap-through well, in order to identify damage incipience with respect to
the primary regimes (see Figure 2.1). The damage states of these additional
specimens are also investigated using x-radiography and sectioning. Damage
data from each test helps to reconstruct the incipience and development of
damage in the panels. If no damage is detected in a panel tested to full snapthrough, then the damage incipience point can only be identified as being on
the second equilibrium path. This result indicates that the subsequent
damage development under further application of stroke is similar to that
observed in plates. The knowledge base currently available for plate damage
development can then be applied to these shells.
The test matrix was devised by considering all possible combinations of
the geometric parameters: radius, thickness and span. The complete test
matrix is given in Table 3.1. The matrix is fully populated due to the focused
nature of this work. The full testing program, as outlined above, is carried
out for each specimen geometry. It is desired to provide a large amount of
detailed information about these specific geometries as opposed to providing
more general information for a wider range of geometries.
3.3
Analytical Approach
Analytical tools currently exist for the prediction of the static response
of general composite shells. The force-deflection response, including snapthrough, of convex composite shells under transverse load have been
investigated using finite element analyses [27, 40, 41, 49]. Although these
analyses are also capable of investigating the deformation-shape development
of convex shells, such results are not currently seen in the literature. As
pointed out in Chapter 2, experimental data for the force-deflection response
-45-
Table 3.1
Test Matrix
T2
T1
T3
R6
R12
R6
R12
R6
R12
S1
Xa
X
X
X
X
X
S2
X
X
X
X
X
X
S3
X
X
X
X
X
X
a X indicates one test for deflection shapes and up to three
additional tests for damage evaluation.
-46and deformation-shape evolutions are sparse. A need, therefore, exists to
compare the results of such analyses to the reality of experimental data.
Only then can these analyses be utilized with confidence for the design of
composite structures.
Snap-through buckling of convex shells involves gross changes in the
overall structural configuration which must be taken into account in the
analytical formulation [40, 62]. This can be accomplished either by including
higher order terms in the kinematics of deformation or by incrementally
updating the initial structural configuration (co-rotational procedure) to
include all previous deformations. A commercially-available code, Structural
Analysis of General Shells (STAGS) [49], which is based on the latter
approach, is utilized in the current work.
Previous finite element studies involving snap-through of convex shells
have shown that the force deflection response converges quickly as the grid is
refined [40, 62]. A grid with 8 axial nodes and 12 spanwise nodes was found
to be sufficient for cylindrical shell panels with boundary conditions similar
to those in the current work [62]. A grid refinement beyond that needed for
convergence of the load-deflection response was utilized here to give sufficient
deformation-shape information without creating excessive computation times.
A full panel grid, shown in Figure 3.3 for a typical specimen, with eleven
nodes in the axial direction and 21 nodes in the spanwise direction was found
to give adequate deformation-shapes with total runtimes of approximately
eight minutes. Unsymmetric deformations were possible for the panels in
this research due to the presence of bend-twist coupling. Therefore, the
simplification of a half or quarter shell model could not be utilized. The
STAGS 410 quadrilateral shell element, which has three translational and
three rotational degrees of freedom at each of its four nodes, is used in the
-47-
Figure 3.3
Illustration of the grid utilized in the finite element analysis.
-48analysis. This results in a model with a total of 1386 degrees of freedom.
Each STAGS analysis is carried out either up to the point where the load
reaches zero in the instability region or to where the load exceeds twice the
maximum load found from the experiment. Such a procedure guarantees
that these analytical results provide sufficient output for subsequent
comparison with experiment. The increment in load chosen by the STAGS
code varies throughout the analysis based on an internal convergence
criteria. The increment of center deflection for the predicted results are not,
therefore, uniform as in the experiments. Output from the analysis includes
both force-deflection responses and deformation-shapes which are
subsequently compared to experiments.
-49-
Chapter 4
EXPERIMENTAL PROCEDURES
Procedures related to the manufacture and testing of specimens are
presented in this chapter.
Details of shell manufacturing including
specialized equipment are given.
Existing and newly designed testing
equipment are also described as a prelude to the related testing procedures.
Methods used to investigate the consistency and quality of the specimens are
also included to evaluate the manufacturing procedures.
4.1
Manufacturing Procedures
The procedures used to manufacture the convex shell specimens are
presented in this section. Shells require different manufacturing equipment
and techniques than those normally used for flat specimens. In addition,
unique procedures were developed specifically for the shells used in this
research.
The material used in this research is Hercules AS4-3501-6 net-resin
graphite/epoxy in pre-impregnated tape form. The material is received in 305
mm (12") wide rolls with an uncured areal weight of 150g/m 2 and 35% resin
content. The material has a nominal cured ply thickness of 0.134 mm. The
rolls are kept in airtight packages and stored in a freezer at temperatures
below -180C. To prepare for a layup, the roll is taken out of the freezer and
allowed to warm for 45 minutes while still in the airtight packaging. This
allows the condensation to form on the packaging instead of on the material
-50itself as the material warms. Once at room temperature, the material is cut
into the desired pattern with a utility knife and standard templates. The
templates are designed such that angled plies are formed from two pieces
using only matrix joints. TELAC standard ply sizes (305 mm by 349 mm) are
cut at this stage of the procedure.
The laminate is then assembled by
stacking the individual plies in the proper order. An L-shaped layup jig is
used to align the individual plies in a consistent manner. Once the laminate
is assembled, it is cut with a utility knife into the desired size for curing.
Widths of 102 mm, 203 mm and 305 mm are cut for the S1, S2 and S3
specimens, respectively. The length remains at 349 mm for all laminates.
The backing paper from the surface plies is then removed and peel-ply is
placed on both sides of the laminate with a 50 mm overhang on one of the 349
mm long edges.
Curing of composite shells does not have a standard TELAC procedure.
However, the procedure followed in previous TELAC efforts [5, 63] has
proven successful for manufacture of cylindrical panels.
The primary
difference from the more standardized plate procedures is the use of
cylindrical mold surfaces. Much like their flat counterparts, the molds are
manufactured from 6061-T6 aluminum. A bulkhead/skin construction is
utilized as shown in Figure 4.1. Five 9.53 mm (3/8 in) thick bulkheads are
bolted to a 9.53 mm (3/8 in) thick baseplate of dimensions 737 mm by 838
mm. Aluminum sheets, 0.794 mm (1/32 in) thick, are tightly formed over the
bulkheads and held in place with 9.53 mm (3/8 in) thick clamping bars of
dimensions 711 mm by 102 mm by 9.53 mm and 6.37 mm (1/4 in) bolts. The
sharp bend, at the intersection of the cylindrical and flat surfaces, is created
with a sheet metal bending tool prior to the installation of the clamping bars.
Each bulkhead has a central cutout region to allow equalization of pressure
-51-
'Clamping Bai
Basepl
Bulkhe;
I
Figure 4.1
-
838 mm
-1
Illustration of cylindrical mold configuration.
-52below the thin mold surface.
This prevents collapse of the sheet during
autoclave pressurization.
Standard TELAC cure procedures for the AS4-3501-6 system are
followed with the few exceptions noted in this section.
Further details
regarding the cure procedure can be found in [64]. To begin the preparation
of the cylindrical mold assemblies, the surface is first cleaned with acetone
and nylon scrub cloths. Mold release is applied to the surface in order to
facilitate subsequent cleaning after the next cure. A region of approximately
50 mm in width around the outer circumference is not coated with mold
release in order to provide an adhering surface for subsequent placement of
flash and vacuum tapes. A layer of guaranteed non-porous Teflon (GNPT) is
affixed to the mold surface with flash tape around the outer circumference
leaving roughly 25 mm of uncoated mold surface for placement of the vacuum
tape. The laminates, with peel-ply, are placed onto the molds and 25 mm (1
in) wide and 3.35 mm (1/8 in) thick cork tape is used to build up a snug
enclosure. Two layers of cork tape are used to ensure that the walls of the
enclosure are higher than the laminate and cure materials. There is roughly
430 mm (17") of usable axial length on each mold. Therefore, a mold can cure
any combination of specimen widths, including cork width, which are less
than this dimension. A layer of porous Teflon is placed on top of the laminate
followed by another layer of GNPT. Since a net resin system is used, the
common bleeder paper found for bleed-type systems is not used. Aluminum
top plates, 0.8 mm (1/32 in) thick and with the same planar dimensions as
the laminate, are placed into the cork enclosure. The plates are sufficiently
thin such that their deformations are entirely elastic. The fit is typically
snug which helps to keep them in place. Flash tape is used to further fix the
plates in position. Another layer of GNPT is placed over the entire cure
-53assembly to prevent excess resin from reaching the vacuum ports. Fiberglass
air-breather is placed over the GNPT to allow sufficient airflow into the
vacuum ports and to provide an additional barrier to excess resin flow.
Vacuum bagging material along with vacuum tape are used to seal the entire
contents. A schematic of the entire cure assembly is shown in Figure 4.2.
Vacuum ports protrude from the vacuum bagging through a small hole which
is sealed by rubber washers as the ports are attached to the vacuum bag. As
many as two molds can be placed into the autoclave at once. Vacuum is then
pulled on both molds by linking them to the same vacuum line.
The cure cycle proceeds under a constant pressure of 85 psi and full
vacuum. The cycle begins with a ramp up to 2400 F where it is held for one
hour. This is followed by a ramp in temperature to 3500 F where it is held
for two hours after which temperature and pressure are reduced to
environmental conditions.
The nominal temperature, pressure and vacuum
profiles for the cure are given in Figure 4.3. The cure cycle for this material
system is standard for TELAC and the details can be found in [64]. After
curing, the materials and laminate are carefully removed and separated.
Laminates are post-cured for 8 hours at 1770 C (3500 F).
Due to the snug-fitting cork dams, there is typically some cork
adhering to the laminate edges after cure. This is removed on a table sander
by gently pressing the laminate edge perpendicular to the moving belt.
Uneven resin flow is minimal for this net resin system and areas with excess
resin are typically confined to the outer 6 mm along each edge. The shells
could not be cut along the curved circumferential edges due to interference
with the cutting blade. Hence, the resin rich areas are unavoidable along
these edges. This is assumed to have a negligible effect on the shell response
-54-
Vacuum Bag
Air Breather
I
A
I
.
L
X
X
X
''III
h6
I1111111
IL
' 111111
11111111
1
16
''N
III
L
h. 1h6 16
IIIII
h
''11III,
Nonporous Teflon
(GNPT)
Aluminum Top Plate
Nonporous Teflon
(GNPT)
Porous Teflon
Peel-Ply
Laminate
Peel-Ply
Cork Dam
Nonporous Teflon
(GNPT)
Vacuum
Tape\t
Cylindrical Mold
(Aluminum) Coated with
Mold Release
Figure 4.2
Schematic of cure assembly
-55-
AUTOCLAVE
TEMPERATURE (oC)
177
117
66
25
TIME
0
10
35
95 115
235
(min)
275 280
AUTOCLAVE
PRESSURE (MPa)
0.59
TIME
0
10
(min)
275 280
VACUUM (mm Hg)
760
0
I
- I
280
Figure 4.3
TIME
b
(min)
Nominal temperature, pressure and vacuum profiles for cure
cycle
-56since these are stress free edges located far away from the loading point. The
resin-rich areas along the axial edges are removed by cutting with a watercooled diamond grit saw which is mounted to a milling machine. Specimens
are held in place for cutting on the mill table with a 25 mm by 25 mm by 660
mm aluminum "hold-down" bar. Rubber cushions are mounted to the
underside of the hold-down bar to avoid damage to the laminate. The 220
grit, 1.5 mm thick diamond saw blade extends into a recessed channel on the
mill table. Cutting is performed by running the saw at 1100 rpm and feeding
the table at 4.7 mm per second (11 in per minute). This milling machine
setup is shown in Figure 4.4.
A specific cutting procedure is followed to ensure that each specimen is
cut to size consistently and accurately. First, the resin flash is removed from
one axial edge by cutting away approximately 12 mm. The desired length
(span) is then measured, with a ruler, from this freshly-cut edge and marked
with a white paint marker. The specimen is placed on the table such that the
paint marks are just inside the front wall of the cutting channel. The mill
table is adjusted such that the blade is as near as possible to the rear wall of
the cutting channel. The blade and specimen positioning are illustrated in
Figure 4.5. A cut is made in this position and the panel lengths are measured
at the left and right hand sides of the panel using vernier calipers. If the
lengths measured on the right and left side are different by more than 0.25
mm, the panel is adjusted by hand to compensate for the difference. The
table is moved toward the blade by 1.27 mm (0.050 in) and another cut is
made. The process is repeated until the lengths converge to within 0.25 mm.
Since the width of the cutting channel is 12.7 mm (0.5 in), only seven or eight
cuts are possible before the specimen length becomes too short. Although the
adjustments were entirely by feel, the lengths typically converged to within
-57-
Diamond Saw
Directions of
Manual Adjustment
Figure 4.4
Illustration of milling machine cutting apparatus.
-58-
Blade
Shell
Channel
Front Wall
Channel
Rear Wall
12.7mm
Figure 4.5
Illustration of mill table channel configuration.
-590.25 mm after only three or four iterations. The final cut is made by moving
the mill table in by the difference between the converged length and the
desired specimen length. Total lengths for either side were obtained to
within 0.25 mm (0.010 in) for all specimens.
Specimens used in deflection tests must also be painted white on the
underside for compatibility with the laser displacement transducer. The
laser measures the amount of diffracted laser light as reflected from a target.
Although the laser is reportedly compatible with black targets, it was unable
to recognize the shiny black surface of graphite/epoxy. As a result, the panels
were painted with Krylon flat white according to the following procedures.
Each panel is leaned, on an axial edge, against an appropriate backing
surface. The panel is almost vertical with only enough of an angle to produce
a stable position. The can of spray paint is held approximately 305 mm from
the panel and swept from side to side. Each sweep is continued beyond the
panel itself to ensure that the speed and, hence, the paint coverage, is
uniform across the width. The sweep speed is such that adequate coverage is
obtained in one pass, as determined beforehand on a dummy surface. Each
subsequent pass is overlapped approximately half of the painted width
(approximately 38 mm) until the panel is fully painted. It was discovered
that the paint mist becomes slightly more coarse when the can is less than
one third full. Thus, all cans were discarded at this point.
4.2
Curvature and Thickness Mapping
A convex shell specimen is characterized by the radius of curvature,
twist, thickness, and planform dimensions, the latter having been discussed
in the previous section. To evaluate the manufacturing process, mapping
-60schemes were utilized to determine the radius of curvature, twist and
thickness of each shell. All measurements were taken before the underside of
the panels were painted white.
A nine-point grid is used to determine
various parameters of each shell. Thickness is measured at each grid point
with a deep-throat micrometer with a resolution of 0.001 mm.
The
approximate locations of each grid point are shown in Figure 4.6. The
distances shown in the figure are measured along the surface of the shell so
they are not planar in the circumferential direction.
A simple formula is used to calculate the radius of curvature at three
locations along the shell. The same measurements taken for these radius
calculations are used to determine both axial and spanwise twist. To make
these measurements, a specimen is place in a special jig [5], which supports
three corners of the shell in a plane. The fourth corner of the jig is adjusted
to just make contact with the shell. The jig is mounted onto the table of a
milling machine equipped with a digital position readout. The x-dimension
(circumferential direction) and y-dimension (axial direction) are both
obtained directly from the digital readout with a resolution of 0.012 mm
(0.0005 in). A dial gauge, accurate to 0.025 mm (0.001 in), is mounted into
the vertical head of the milling machine. The gauge measures the vertical
dimension (z) of any point on the shell.
From simple Pythagorean
relationships, the radius of curvature at any axial (y) location can be
calculated by measuring the length (2a) of a straight line connecting two
points on the shell along with the vertical (z) distance from this line at the
midpoint (b) according to equation 4.1:
R=
These
quantities
are
2
a2 + b
(4.1)
2b
illustrated
graphically
in
Figure
4.7.
-61-
Circumferential
Sn
Sn
Figure 4.6
S1:
d = 25.4 mm (1")
S2:
d = 50.8 mm (2")
S3:
d = 76.2 mm (3")
Locations used for mapping shell thickness.
-62-
I-
Figure 4.7
P
Illustration of geometric relation used to calculate curvature
(R) by measuring a and b.
-63-
The measurement scheme followed for each specimen is illustrated in
Figure 4.8. The origin for all measurements is arbitrarily chosen as a point
along the first y-station, as indicated by point 0 in Figure 4.8.
Three
measurements are taken at each axial (y) station. First, the vertical position
zi at the origin is measured with the dial gauge. The mill table is moved in
the x-direction until the dial gauge again reads the same vertical position zi.
This linear distance, xi, is recorded. Finally, the milling machine table is
moved to the midpoint of the xi-dimension and the vertical position zic is
recorded.
The measured quantities are substituted into equation 4.1 to give
a radius of curvature at each axial (y) station:
+ (Zic - Zi)
Ri=
2
;
(4.2)
i = 1,2,3
(Zic - zi)
where the Ri are the radii of curvature at axial (y) location i.
The procedure
is repeated at each y-location as determined from the axial grid locations
shown in Figure 4.6, as the straight-line distance between two points on the
shell, xi, may change at each y-location due to twisting of the shell or changes
in the radius of curvature. At the final y-location, the vertical position at the
xx location, termed Z3x1, is recorded for use in subsequent twist calculations.
Twist about the x- and y-axes can also be calculated from the above
measured quantities. The straight line defined at the yl and y3 stations can
be used to calculate the twist about the x-axis (spanwise twist).
By
measuring the difference in vertical position at the xl location (Z3x1 - z3), the
twist of this straight line can be determined. A similar change in vertical
position, defined as (Z3x1 - zi), is defined to calculate the twist about the y-
-64-
I
II
I
I
I
|0y
X1
x
y
z
II
I
I
I
Z
I
I
II
II
I
z2cI
II
I
X
I
II
I
I
I
II
I
z3xl
z3
Figure 4.8
z2e|
I
I
I
I
II
I
I
II
x I
I
I
,
z3
y3
II
Illustration of measurements for radii and twist calculation.
-65axis (axial twist). Axial and spanwise twist are calculated using the small
angle approximation as given in equations 4.2 and 4.3:
zZII
(4.2)
= tan-' Z3xl - Z3"
(4.3)
=tan-( Z3x
for axial twist, and
for spanwise twist, where the units of y and 13
are radians.
The average radii and thickness are computed for each specimen. In
addition, for each nominal value of radius and thickness, the average and
coefficient of variation over all test specimens is calculated and presented in
Table 4.1.
The specimen evaluation data in Table 4.1 shows that the
manufacturing procedures produce consistent results.
The average
thicknesses are within 8% of the nominal values with acceptable coefficients
of variation (less than 7%). Average radii are within 8% of the nominal
values with coefficients of variation of less than 6%. Average spanwise and
axial twist measurements are 2.00 and 1.10, respectively, and are considered
negligible. All measured radii values are lower than the nominal due to the
thermal deformations associated with cool-down from cure to room
temperature [65].
This is a common phenomenon in curved composite
structures which is termed "spring-in." The R12 panels are more sensitive to
this effect due to their more shallow geometries since a small change in the
depth of an R12 panel leads to a larger change in radius of curvature than in
an R6 panel. Since the sensitivity of the analysis to radius of curvature was
-66-
Table 4.1
Results of Thickness and Curvature Mapping
a
Nominal
Difference
6.9%
0.804 mm
+ 3.2 %
1.703 mm
5.3%
1.608 mm
+ 5.9 %
T3
2.590 mm
5.2 %
2.412 mm
+7.4%
R6
881 mm
3.0%
914 mm
- 3.6 %
R12
1682 mm
5.4%
1829 mm
- 8.0 %
Metric
Average
T1
0.830 mm
T2
C. V.
a Indicates coefficient of variation.
-67-
unknown, these measured radii (average value for each specimen) were used
in the analytical portion of this work.
4.3
Description of Test Fixture
A specially designed test fixture for constraining shells was utilized in
this work. The test fixture, originally designed and used in [5], can be
adjusted to accommodate shells of different curvature and span. The side
view of the original test fixture is shown in Figure 4.9. A fixed brace is used
to give increased resistance to the compressive loads generated at the
boundaries (rods). However, the block cannot be used for the largest panels
(S3) due to space constraints.
Newly-designed equipment was built and
incorporated into the original design, preserving the essential features.
Modifications were also necessary for the present research.
These are
discussed in detail in subsequent sections.
One of the primary difficulties in restraining general shells is
accommodating a range of slopes at the boundaries. Slopes change when the
radius of curvature changes or even when the span of a shell varies. A wide
range of shell geometries are accepted by the use of a special rod-cushion
design. The rod structure houses the details of the boundary conditions along
the axial edges. The rod can be rotated to a variety of angles before being
held in place on the cushion by clamps. The entire rod-cushion assembly is
shown in Figure 4.10.
Two rod-cushion assemblies are mounted on the top plate, one being
fixed and the other having three possible spanwise positions. The top plate
was modified to allow continuous spanwise adjustment. This amounted to
-68-
Steel
Rods
Figure 4.9
Side-view illustration of original test fixture with a convex
shell mounted for transverse loading.
-69-
127 mm
610 mm
Clamps
Cushion
O
O
Rigid stand
NOTE: Not to Scale
Figure 4.10
Illustration of the rod-cushion assembly.
-70-
connecting the adjustable rod/cushion mounting holes into continuous slots.
The slots were also extended slightly to accommodate the newly-designed
boundary conditions. The central cutout in the top plate was also extended in
the spanwise direction to allow greater access for the deflection measurement
assembly. Extending the cutout in the axial direction would also have been
desirable for the same reasons. However, the construction of the fixture did
not allow this due to interference with the supporting legs and crossbeams.
The final dimensions of the top plate and its various features are shown in
Figure 4.11.
The two most significant additions to the original test fixture are the
grooved boundary conditions and the deflection measurement assembly.
Steel grooved inserts, which mount directly into the rods, were designed to
idealize pinned boundary conditions along the axial edges. Frictional effects
are minimized thus allowing full rotations at the boundaries. A deflection
measurement assembly was designed and built to capture the complex
modeshapes which develop during snap-through.
A laser displacement
transducer is mounted to a traverse assembly capable of continuous
movement in the spanwise direction and discrete movement in the axial
direction. All additions to the test fixture were entirely modular so that the
original construction and functionality used in [5] could be easily reproduced.
4.3.1 Boundary Conditions
Free rotation at the axial edges was determined to be an important
feature of the boundary conditions as discussed in Chapter 3. To minimize
the resistance to rotation inevitably created by friction, steel grooved inserts
were designed and built for the test fixture. When a panel undergoes snapthrough, a frictional moment resists the rotation at the boundary.
The
-71-
Holes for
Fixed
Clamp and
Cushion
Continuous Slots
A-
559 mm
I-a
_
Figure 4.11
533
.
mmn
i -_
Top view of test fixture top plate showing the slots and
extended cutout.
-72frictional forces at the boundaries of a panel may be high. However, the
frictional moment may be small if the force acts through a small enough
moment arm. The key feature of the grooved design is that this moment arm
is reduced to be on the order of the panel thickness. The side view of the
grooved inserts, as mounted into the rods, is shown in Figure 4.12.
An
obvious consequence of this design is an inability to resist "pull-out" of the
panel after snap-through when tensile membrane forces develop which, if not
resisted at the boundary, cause the panel to pull away. However, as outlined
in Chapter 3, the snap-through instability occurs under conditions of
compressive membrane loading.
The post snap-through regime, where
tensile membrane forces exist, is not of primary interest in this research.
Thus, the grooved inserts provide the desired boundary conditions.
The detailed drawing of the grooved inserts are given in Figure 4.13.
The inserts were manufactured from 4096 steel flat stock. The groove has a
radius of 1.59 mm (1/16") as cut with a ball end mill. The maximum depth of
the groove is 1.14 mm (0.045") which gives a groove width of 3.05 mm (0.12").
The overall dimensions of the inserts are 38.1 mm by 31.8 mm by 317.5 mm
(1.5" by 1.25" by 12.5").
The thickness dimension of 38.1 mm (1.5") was
chosen to extend the groove location out from within the rod such that the
modeshapes could be captured from below by the laser transducer, as
illustrated in Figure 4.12. Mounting holes are drilled which align with the
existing threaded holes in the rods. Each insert is mounted into the rods
with four 1/4-20 allen-head screws. The holes were also countersunk to avoid
interference with the laser signal during deformation-shape recording.
As a direct extension of the work done in [5], double knife edge inserts
were also constructed.
Preliminary tests showed that these produced
inconsistent rotation conditions due to a large friction moment-arm of 18 mm.
-73-
Adjustable
Grooved Insert
Upper Plate (Rigid Stand)
Laser Signal
From Below
Figure 4.12
Illustration of grooved inserts in the rod-cushion assembly.
-74-
12.5"
-
I
40.12"
S-A
0.675"
0.750"1, 1.875"1,,
7.250"
0.750"
1.875"
, -5/16" Through-..-Holes with 1/2"
countersink
0.40" deep
0.375"
FRONT VIEW
NOTE: Not to Scale
S1.5"
'
0.045"
/16"
1.25"
0.875"
SIDE VIEW
Figure 4.13
Schematic of grooved inserts.
1.25"
-75-
As a result, the grooved inserts were used for the main body of experiments.
The general aspects of the knife edge inserts are discussed here only for
future comparison to the grooved inserts.
The knife edges allow small
rotations while an inner wall resists the compressive membrane loading. The
knife edges as mounted into the rod are shown in Figure 4.14. An attempt
was again made here to bring the panel edge out from within the rod for
access by the laser transducer.
4.3.2 Deflection Measurement Assembly
One of the main objectives of this work is to obtain the complex
deformation-shapes which develop during snap-through buckling.
The
deflection measurement assembly was designed and built for this research
with this objective in mind. A non-contact laser displacement transducer is
utilized to ensure that the measurement process itself does not affect the
panel deflection. The laser measures displacement by reflecting laser light,
in the form of a 1 mm diameter beam, off the target and measuring the
resulting diffraction. A voltage which is proportional to displacement is
ultimately produced. Resolution of 10 gm is obtained for displacements in
the range of 60 mm to 140 mm by the Keyence LB-11/70 transducer.
The laser is mounted on a traverse assembly, which is bolted to the
base plate of the test fixture, allowing the deflection to be "traced out" from
underneath the panel.
The traverse was designed to have continuous
movement in the spanwise direction and discrete axial positions in 12.7 mm
(0.5 in) intervals. Since the panel is approximately point-loaded, spanwise
deflections may be different along the axial direction. Possible paths for the
laser are illustrated in Figure 4.15 for a general panel. Due to interference
with the crossbeams of the test fixture, an axial window of only 203 mm was
-76-
Adjustable Builtout Wall
Rod
Upper Knife
Edge
--~
Shell
Lower Knife
Edge
Note: Not to Scale
Figure 4.14
Illustration of knife-edge inserts.
-77-
.
s
.
.
r
s
•
js
...
s
%%
r
r
t
s
s
z
%
j
'
'
r%
12.7 mm
Spanwise Direction
* Dashed lines indicate
continuous laser movement
Figure 4.15
Illustration of possible locations for measurement of spanwise
deflection.
-78available for deflection measurement. The actual axial positions chosen for
the panels are given in Table 4.2. These positions were chosen to be as
equally spaced as possible considering the available 203 mm wide window of
accessible area and the actual axial dimension of each panel.
The axial position is changed numerous times during a test, so the
manner in which the laser is moved had to be fast, accurate and repeatable.
To achieve this, a special laser holder was designed with portability in mind.
The entire deflection measurement assembly is shown in Figure 4.16. The
laser is rigidly held in the slotted holder with a set screw. The laser holder
has a tongue-in-groove construction ensuring accurate placement in the
spanwise direction while a simple dowel pin was used to locate the discrete
axial positions. The male laser holder sits in a female block which is rigidly
mounted to the movable portion of the traverse. The block contains a groove
with holes spaced in 12.7 mm (0.5 in) intervals to accept the 4.76 mm (3/16
in) dowel pin extending from the lower portion of the laser holder. This
arrangement allows the axial position of the laser holder to be changed easily
by hand. Accuracy and repeatability of placement is obtained by the low
tolerances of less than 0.12 mm (0.010 in) between the mating parts of the
male and female connections. The large mass of the holder ensures that it
seats adequately in the groove under its own weight.
The position of the laser in the spanwise and axial directions must be
measured for each deflection measurement. In the spanwise direction, the
position of the laser is obtained by coupling a rack and pinion assembly to a
potentiometer circuit. The rack is rigidly mounted to the inner wall of the
stationary traverse frame. The pinion, or gear, is mounted onto the shaft of a
precision 10k--potentiometer which is subsequently mounted to the movable
traverse top-plate, as seen in Figure 4.16. As the traverse top-plate moves in
-79Table 4.2
Locations of axial deflection measurementsa in panels of
various span.
S1
S2
S3
-38 mm
-89 mm
-102 mm
-25 mm
-51 mm
-51 mm
0 mm
0 mm
0 mm
25 mm
51 mm
51 mm
38 mm
89 mm
102 mm
a Referenced from the axial centerline.
,
-o
Laser
Laser
Transducer
Holder
--
4
Micro-switches
SC
ircuit
Cabinet
3/16" Dowel Pin
02
#9 (0.196") Holes
o
0.500" apart
Potentiometer/Gear
Assembly
CD
CD
I
I
I
O
0
SCD
Teflon
Bearings
I
Drive
SControls
02l
Flange
I
I
I
I
Rack
Rack
-81the spanwise direction, the pinion passes the stationary rack which turns the
potentiometer, giving a linearly varying voltage with spanwise position.
Calibration of the rack and pinion assembly was performed on a milling
machine table with a digital readout. The maximum error for spanwise
position was determined to be 0.40 mm. The axial position of the laser is
obtained with a standard voltage divider circuit. The circuit contains microswitches located next to each hole which, when tripped, change the output
voltage to indicate the axial position. The switches are tripped by a small leg
built onto the side of the laser holder as shown in Figure 4.16.
The design of the traverse assembly was completed by wiring the
various circuitry to a large utility cabinet which was mounted to one end of
the traverse. The cabinet enclosed the many electrical connections along with
the main direct current drive motor. This was found to help shield the
electromagnetic spikes of the motor from the sensitive electronics of the
testing machine. All connections for external power and data acquisition
output were conveniently mounted to the front face of the cabinet. This
helped keep track of the many data acquisition outputs along with the inputs
from both a fixed and variable voltage source.
4.4
Testing Procedures
All tests were performed on an MTS-810 hydraulic testing machine in
the configuration shown in Figure 4.17. Special procedures were developed
for consistent placement of the specimens into the test fixture. Slots in the
base plate of the test fixture allow mounting directly into the lower grips of
the MTS machine.
A steel 12.7 mm diameter hemispherical indentor is
mounted in series with an 8896 N (2000 lb) load cell which is held rigidly in
-82-
8896 N
(2000 Ib)
Load Cell
Indentor
Specimen
Rod and
Cushion
Test
Fixture
II II
Lower Grip
(moving up)
Figure 4.17
Illustration of the test fixture as mounted in the testing
machine.
-83the upper grips of the MTS machine. During a test, the upper crosshead
remains stationary while the lower crosshead moves up pushing the test
fixture, and, hence the specimen, into the indentor. All tests in this research
are performed under stroke control with the lower crosshead moving the test
fixture toward the indentor at a constant rate.
Particular procedures related to the testing are described herein. The
testing program is divided into two distinct subdivisions: deflection tests and
damage tests. A special procedure is followed to set up the specimens into
the test fixture for all tests in this investigation. These procedures and
others specific to the type of test are described in the following sections.
4.4.1 Specimen Set-up in Fixture
Special procedures are followed to consistently place each specimen
into the test fixture. The purpose of developing such procedures is to obtain
the maximum consistency possible.
Procedures involving both the test
fixture and the MTS machine are necessary to set up each specimen and each
is covered in this section.
The rods, which house the grooved inserts, are rotated to ensure that
the panel impinges perpendicular to the grooved surface as illustrated in
Figure 4.12. To aid in this adjustment, fine gradations of 1 mm were marked
on the outer circumference of each rod. The rotation in arc length can then be
determined by reading off the number of gradations away from the horizontal
position. This rotation is converted to degrees using the known radii of the
rods and matched with the known slope at the panel edge. Once both rods
are rotated to this proper angle, the fixed rod-cushion is locked in place by
tightening the clamps at each end of the rod. The moveable rod-cushion is
brought toward the fixed rod-cushion until the panel is barely supported at
-84the bottom edge of the grooves. At this point, the spanwise position of the
moveable rod-cushion is finely adjusted until the panel sits in the deepest
point, or center, of each groove, noted as point A in Figure 4.12. The threaded
rods which extend from the support block are turned to create fine changes in
the rod-cushion position for the S1 and S2 panels. A light tapping procedure
is used for the S3 specimens since the support block cannot be utilized for
these large panels. The position of the rod-cushion is considered acceptable
when the panel is visibly in the center of each groove and the panel can slide
in the axial direction with light resistance.
The frictional resistance to
sliding was found to be very sensitive to the rod position. A panel which
exhibits almost no detectable resistance to axial sliding can be made almost
immovable by only a small increment of rod-cushion position.
Hence, this
method was determined to yield the best possible accuracy. The moveable
rod is locked into place and the sliding resistance of the panel is again
checked. The spanwise separation of the rods was found to change slightly
when the moveable rod/cushion is locked into place. Sometimes, the change
was large enough to necessitate a repeat of the above procedure, as the
sliding resistance of the panel was also changed. Acceptable alignment was
typically obtained after two or three iterations.
Another important aspect of the specimen set-up is alignment with the
indentor. Since general buckling is sensitive to load eccentricities, random
misalignment of the indentor can lead to significant inconsistencies in the
resulting data.
To minimize such inconsistencies, a special alignment
procedure was followed. Once the rods are locked in place, the indentor is
removed from the steel adapter and replaced with a specially made "center
finder" with a point radius of less than 0.5 mm, as shown in Figure 4.18. The
sharp point of the center finder allows a more accurate visual alignment than
-85-
S,
8896 N
(2000 Ib)
Load Cell
Specimen
Center
Finder
Rod and
Cushion
Test
Fixture
Figure 4.18
Illustration of the center finder.
-86-
possible with the larger radius of the indentor. The upper crosshead is slowly
lowered until the center finder is as close as possible to the panel without
actually making contact. The specimen is adjusted until the cross-hatched
center marking, made with a 0.5 mm wide pencil, is directly below the
pointed tip of the center finder. Adjustment in the axial direction is achieved
by sliding the panel in the grooves. Spanwise adjustment is accomplished by
loosening the mounting bolts in the slotted baseplate of the test fixture. The
entire fixture can then be moved into the proper position before the mounting
bolts are again tightened.
Once alignment is attained, the upper crosshead is moved away from
the specimen to allow the center finder to be unscrewed from the adapter.
The indentor is reinstalled and the upper crosshead is again lowered and
locked when the indentor is within roughly five millimeters of the specimen.
The MTS controller is used to raise the lower grip until the indentor barely
makes contact with the specimen. Contact is determined by observing the
analog load signal, shown on the MTS plotter, as the lower grip is raised.
When a small preload is detected on the plotter, the lower grip is lowered
until the preload is removed. This defines the starting point for each test. By
observing a greatly amplified load signal on the plotter, the preload for each
test is maintained below 0.25 N.
The procedures described above were followed for each quasi-static test
performed in this research. Subsequent procedures differ based on whether
the test is a deflection or damage test. Each type of test is described in the
following sections with the assumption that the specimen set-up procedures
have already been completed.
-874.4.2 Deflection Tests
As mentioned previously, all tests were conducted on the MTS
machine. Both the load and stroke outputs are obtained directly from the
MTS electronics using a standard TELAC data acquisition system. The finest
resolution of the load, as obtained with this system, is 0.659 N.
The
resolution of the stroke varies from 6.2 gm to 31.0 gim depending on the
choice of stroke range. The smaller the range, the better the resolution.
Thus, the more shallow panels have better deflection measurement resolution
than the deeper ones.
The first set of tests performed are for the purpose of obtaining the
deformation-shapes throughout the snap-through process. Tests are carried
out up to the point of full snap-through which is considered to be the limits
for the grooved boundary conditions. Each test is interrupted and held at
certain stroke values during which deflection data is taken. The resulting set
of data for each test captures the three-dimensional deformation-shape
evolutions during snap-through.
The stroke rates are set such that the total uninterrupted application
of forward and reverse stroke takes 12 minutes which is considerably slower
than the rates used in the quasi-static tests of previous TELAC efforts[5, 34].
Since the deflection tests are interrupted and held many times, in order to
take deflection data, the total test time is considerably longer than if the
stroke movement were continuous. In fact, the time spent in the held mode is
often longer than the total time spent applying forward and reverse stroke.
During each held stroke position, the acquisition of deflection data takes
approximately two minutes. Therefore, a test with many different held
stroke positions can spend a considerable portion of the total test time in the
held mode. Since the data acquisition system continues to take data during
-88the stroke holds, unacceptably large data files could be created. The slower
stroke rates used in this work (from 0.004 mm/sec to 0.71 mm/sec) allowed
the use of a slower data sampling rate which ultimately resulted in more
reasonably sized data files.
It was decided that a good deformation-shape evolution would contain
roughly 10 progressive states of deformation. Full snap-through is chosen to
be the point where the panel takes on a fully inverted or concave
configuration.
This is estimated to occur when the loading point has
displaced twice the original panel height. Dividing this into ten intervals
gives a deflection scan to be taken every 0.2h of center deflection, where h is
the height, as illustrated in Figure 4.19. However, a lower limit to the
increment of center displacement was chosen as 0.38 mm in order to make
the laser error small in comparison (roughly 3%). Thus, the interval of center
displacement is chosen to be 0.2h or 0.38 mm, whichever is greater. During
the actual tests, the center deflection, or stroke, is displayed in real time by
the Labview software. This facilitates the manual interruption of the test at
the pre-chosen stroke values.
During each interruption of the stroke-controlled test, the panel
remains held in the same position while deflection scan data is taken. A
Nicolet 206 Digital Oscilloscope with data storage capability is used to
acquire the data. The spanwise position and laser output are connected to
the oscilloscope channels A and B respectively.
As the deflection
measurement assembly moves spanwise across the panel, at a given axial
station, the span and laser output are sampled. The traverse speed is
adjusted to give roughly 100 data points during each sweep by changing the
input voltage to the traverse drive motor. The five spanwise deflection scans
taken at the five different axial positions are sampled one after another and
-89-
led Panel
S,0
A1=0.2h
eigA2 =0.4h
SA3=0.6h
*A4=0.8h
A5 =1 .Oh
A6=1.2h
SA7=1.4h
NA8
=1.6h
I
I A9=1.8h
- --- Alo=2.0h
Inverted Configuration
'
Deflection measurements taken at intervals of center displacement of 0.2h
or 0.38 mm, whichever is greater.
Figure 4.19 Schematic of the center deflection intervals used in the
deflection tests.
-90stored on one Nicolet data record. The actual deflection scans are displayed
on the screen of the Nicolet as the data is sampled, allowing for the fully
inverted configuration to be easily identified. The laser starts at the end
farthest from the control cabinet and moves toward the cabinet for each
recorded spanwise modeshape.
Since the data record is continuous, the
output from the laser is clipped, by physically covering the lens, during the
time it takes to reset the traverse to the farthest end and change axial
positions. The axial position of the laser holder is displayed in real-time,
along with the load and stroke data, by the Labview software. This is used to
properly locate the laser holder in a specific axial position. Once the laser
holder is in the proper axial position and the traverse is reset to the farthest
end, the lens is uncovered and the traverse is again run in the same direction,
tracing out the corresponding deformation-shape. This procedure is repeated
for the five axial stations and the entire data record is saved by the
oscilloscope on a floppy disk.
After all deflection data has been recorded, the stroke-control of the
testing machine is resumed and continued up until the next pre-chosen
stroke value.
The above procedures for deflection data acquisition are
repeated at each held position, while load and stroke data continue to be
recorded continuously throughout the entire test.
Stroke-reversal of each specimen begins when the panel clearly takes
on a concave configuration. This may occur for values of center deflection
greater than 2h, as determined visually from the oscilloscope screen.
Additional deflection scans are taken in this situation, maintaining the
original interval of center deflection for subsequent held stroke positions.
Since the panel would physically pull away from the grooved edges if loaded
beyond the inverted configuration, this center deflection defines the limit to
-91-
the deformations and, hence, the damage for each panel in this work.
Damage results from the deflection tests are then used to define the test
matrix for the damage tests.
4.4.3 Damage Tests
The damage testing program relies directly on the detection of damage
in previous tests.
Damage evaluation consists of x-radiography and
sectioning techniques which are described in the last section of this chapter.
If damage is detected in the deflection test for a given geometry,
subsequent damage tests are performed to determine the damage incipience
and damage development.
The first damage test is carried out up to the
critical snapping load and the damage state is investigated. If damage exists
in this panel, then a test is conducted to the center deflection halfway up to
the critical snapping deflection in the hope of further identifying the region of
damage incipience. If no damage exists in the test to the critical snapping
load, then a test is conducted to the center deflection midway between the
critical snapping deflection and the fully snapped through deflection. By
following this procedure, the damage incipience and development should be
adequately identified.
Damage tests are much more straightforward since the forward
application of stroke proceeds uninterrupted.
Load and stroke (center
deflection) data are taken with the Labview system in the same manner as
for the deflection tests.
The stroke rate is kept the same as for the
corresponding deflection test with the only difference being the absence of
interruptions. Forward application of stroke proceeds directly to the prechosen value of center deflection followed by a reversal of stroke at the same
rate.
-92-
4.5
Damage Evaluation Procedures
Each specimen from both the deflection and damage tests is inspected
for damage. Two types of evaluation procedures are used: x-radiography and
sectioning.
4.5.1 X-Radiography Technique
The dye-enhanced x-radiography procedure provides a throughthickness integrated view of damage including matrix splits and
delaminations. After each test, a 0.79 mm (1/32 in) diameter hole is drilled
through the thickness of the specimen at the point of loading. The hole is
drilled at 2000 rpm on a standard drill press from the convex side through to
the concave side. Flash tape is applied to the concave side of the specimen
and a 1,4 diiodobutane (DiB) dye is injected, with a syringe, into the hole on
the convex side. This dye is opaque to X rays and of low-enough viscosity to
wick into the damaged regions via capillary action. A small bubble of excess
dye is maintained on top of the hole for approximately one hour to ensure
that the dye has fully penetrated into the damaged regions. The excess dye
and flash tape are removed and the specimen is x-rayed using the Scanray®
Torrex 150D X-ray Inspection System. A 50kv potential is used in the
"TIMERAD" mode with the amount of radiation absorbed by the specimen
being adjustable. Most specimens were x-rayed using the Polaroid Type 52
PolaPan film and 260 mR (milliRoentgens) of radiation. If greater detail was
required, the Polaroid Type 55 PolaPan film was used along with 4500 mR of
radiation. The development of this type of film is more complex and is only
used if necessary.
-93DiB-soaked areas show up as dark regions in an x-ray picture. A
sample x-ray picture is shown in Figure 4.20 looking down at the convex side.
The delamination region can be seen as the region bounded by the dark
fringes where the dye has accumulated. The dark lines are matrix splits
where the dye has penetrated between the fibers. The 0o direction in Figure
4.20 is along the vertical axis of the page and positive angles are taken
counterclockwise from that axis. All the x-ray photographs in this work
maintain the same orientation.
The x-ray image is a planar projection of the cylindrically curved
damage area resulting in a slightly smaller representation of the damage
region. The corresponding reduction in damage area is less than 0.02% for
the panel with the most curvature (R6). This effect is thus considered
negligible for subsequent damage measurements. The length of the matrix
splits are measured to the nearest 10 millimeters since the exact beginning
and end of the crack is hard to define. The delamination lengths are also
measured in order to quantify the damage.
4.5.2 Sectioning Techniques
Sectioning allows the details of the damage to be identified through the
thickness of the specimen. After the x-radiography has been completed for a
specimen which revealed some damage, a cut is made along the center span
using the diamond saw of the mill-table cutting apparatus. The cut intersects
with the hole drilled for the x-ray damage evaluation. In order to avoid
interference with the cutting blade, the deeper specimens are first cut into
three shells of smaller span according to the procedures described in section
-94-
fiber direction
(circumferential shell axis)
00
Figure 4.20
Sample planar x-ray picture showing damaged region.
-954.1. This decreases the shell depth for each spanwise cut. The three pieces
are kept together for subsequent damage evaluation. The sectioned edges are
then buffed by a felt bob rotating in a drill press while a slurry mixture of
powder and water is applied. This creates the smooth surface necessary to
identify the location of the damage through the thickness of the laminate
with a microscope.
An Olympus SZ-Tr Zoom Stereo Microscope is used to examine all
specimen cross-sections. The damaged region was magnified 30X to 40X to
identify delaminations, matrix cracks, and fiber damage. Matrix cracks could
be observed as light lines through the dark matrix between the fibers.
Delaminations were seen as lightened areas between plies which had
separated. Manual transcriptions of the damage were made by examining
the specimens under the microscope and a typical example is given in Figure
4.21.
-96-
Load
+450
+450
-450
-450
-450
00
Delamination
I
00
Matrix Crack
o
00
00
00
-450
-450
-450
1 mm
+45
+450
+450
Figure 4.21
Sample transcription of the cross-sectional damage.
-97-
CHAPTER 5
RESULTS
The results presented herein include force-deflection and deflectionshape data taken during the quasi-static tests, and x-ray photographs and
sectioning transcriptions made of the specimens after the tests to determine
the damage state. Results from numerical analyses include force-deflection
responses and deflection-shape development as computed using the STAGS
finite element code.
5.1
Force-Deflection Behavior
The force-deflection results from the stroke-controlled deflection tests
are contained in this section. As explained in Chapter 4, these tests are
carried out up to the point where the convex specimen attains an
approximately inverted, or concave, configuration. These force-deflection
results are presented herein as plots of contact force versus center deflection.
The complete test including the forward and reverse application of stroke is
included in each "force-deflection diagram".
5.1.1 Experimental Results
Three distinct types of experimental force-deflection responses were
observed during the forward application of stroke: smooth and stable, smooth
with an instability region, and non-smooth with an instability region. These
types are characterized as response types I, II, and III, repectively. The
-98stable (type I) force-deflection response initially has a negative concavity
which is termed "softening behavior" since the slope and, hence the stiffness,
decrease with increasing center deflection. The response eventually passes
through an inflection point into a region of positive concavity, thus exhibiting
"stiffening behavior" which continues for the remainder of the test. An
illustration of such behavior is given in Figure 5.1. The type II response
exhibits an instability as shown in Figure 5.2. This response also begins with
softening behavior, however, the key characteristic of this response is the
region of negative slope, termed the instability region, where the contact force
actually decreases with increasing center deflection. This is the same basic
behavior as that shown in previous work [7, 38] and described in Figure 2.1.
Some of the responses which had an instability also displayed non-smooth
behavior, as illustrated in Figure 5.3. These type III responses exhibit a
discontinuity in load, or "load-drop", after passing through the critical
snapping load. The force then decreases to zero in the instability region
which indicates that contact with the indentor is lost as the panels snaps
away into the inverted configuration.
The "type" of force-deflection response, as per the three characteristic
types, exhibited by each specimen in this investigation is listed in Table 5.1.
Specimens with type I responses occupy the upper right portion of the table
while specimens with type III responses reside in the lower left portion.
These two regions are seperated by a relatively diagonal band which contains
specimens with a type II response.
The depth or "height" of the shell is
defined in chapter 4 as the vertical deviation from a flat configuration at the
midspan location. Shell depth increases with increasing span and decreasing
radius. Thus, the deeper thinner shells exhibited type III responses while the
-99-
-J0I
Center Deflection
Figure 5.1
Illustration of a smooth stable force-deflection response
(response type I).
-100-
-a
CO
0
.J
Center Deflection
Figure 5.2
Illustration of a smooth force-deflection response with an
instability (response type II).
-101-
0
.J
Center Deflection
Figure 5.3
Illustration of a non-smooth force-deflection response with an
instability (response type III).
-102-
Table 5.1
General Characterization of the Experimental ForceDeflection Responsesa
T2
T1
T3
Span
R6
R12
R6
R12
R6
R12
S1
II
I
I
I
I
I
S2
III
III
II
II
II
I
S3
III
III
III
II
II
II
a "I"indicates that the response is smooth and stable.
"II"indicates that the response is smooth with an instability
"III" indicates that the response is non-smooth with an instability
-103-
thicker more shallow shells showed type I responses.
Intermediate
combinations of depth and thickness generally produced type II responses.
The force-deflection responses for six out of the eighteen specimens
displayed type I force-deflection responses, as shown for specimen R12T3S2
in Figure 5.4. The force increases monotonically with increasing center
deflection and is, therefore, a stable force-deflection response. An initial
softening region is followed by a region of monotonic stiffening for larger
deflections. The stroke-reversal portion of the response is just below the
forward stroke portion giving a slight hysteresis in the overall response. It
should be noted that the stroke-reversal portion of the response is below the
forward-stroke portion for all of the specimens tested. A much more linear
response is exhibited by specimen R12T3S1 shown in Figure 5.5. This
specimen also shows significantly less hysteresis. The stroke-reversal portion
of the response for both specimens returns to a force of zero at a deflection
very close to zero. This is typical for all specimens that exhibited a stable
force-deflection response.
Generally, force-deflection responses of the deeper shells display an
instability region.
Seven out of the eighteen specimens showed type II
responses, as noted in Table 5.1. The type II force-deflection response for
specimen R12T3S3 is given in Figure 5.6. The response begins with softening
behavior and reaches a peak load, i.e. the critical snapping load. The load
then decreases with increasing center deflection before going through an
inflection point in the instability region and, thereafter, develops stiffening
behavior for the remainder of the instability region and onto the second
equilibrium path. The small deviations from a smooth response on the
underside of the forward-stroke portion of the response represent small
changes in load experienced during the held-stroke positions. This behavior
-104-
500
R12T3S2
400
00
0
.
300
200
100
,
0
0
Figure 5.4
2
4
6
8
Center Deflection (mm)
10
Experimental force-deflection response for specimen R12T3S2.
-105-
500
R12T3S1
400
- 300
-.
200
100
0
Figure 5.5
1
2
Center Deflection (mm)
3
Experimental force-deflection response of specimen R12T3S1.
-106-
500
R12T3S3
400
z
0
-
300.
200
100
0
Figure 5.6
5
10
Center Deflection (mm)
15
Experimental force-deflection response for specimen R12T3S3.
-107was observed for all specimens tested, with the thicker specimens showing
the most pronounced effect. When the stroke was resumed, the response
always returned to the original path giving the curve an overall smooth
shape. These are, therefore, not important points and can be ignored. The
stroke-reversal portion of each test was conducted without interruption and
this portion of the response is without these small deviations. Most of the
type II responses returned to a load of zero at center deflections close to zero.
However, three of the seven specimens with type II responses (R12T2S3,
R6T2S2, and R6T3S3) returned to a load of zero at center deflections greater
than the original shell depth upon stroke-reversal, thus indicating that the
specimen remained in a stable post-buckled configuration upon strokereversal. This phenomenon, also seen in previous work on convex shells [5],
can be seen in the force-deflection response of specimen R6T2S2 in Figure
5.7. These specimens returned to the original convex configuration, i.e. they
"snapped back", when the in-plane fixity of the test fixture was relaxed upon
removal.
The force-deflection responses of the thinnest and deepest specimens
showed an instability along with non-smooth, in fact discontinuous, behavior.
Such type III behavior was observed for five out of the eighteen specimens in
this investigation and is shown for specimen R6T2S3 in Figure 5.8. Two key
features exist in this type III response: a discontinuity in the response, or
"load drop", at a center deflection of 10.2 mm and the attainment of a force of
zero within the instability region.
After the load-drop occurs, the force
decreases in an approximately linear fashion with increasing deflection until,
at a center deflection of 19.9 mm, the force reaches zero. This behavior is
quite different from the previous examples of specimens which reached zero
force only upon stroke-reversal from the second equilibrium path. A zero
-108-
500
R6T2S2
400
z
300
0
.i 200
100
0
Figure 5.7
5
10
Center Deflection (mm)
15
Experimental force-deflection response of specimen R6T2S2.
-109-
1000
R6T2S3
750
a
500
0
250
0
0
Figure 5.8
5
10
15
20
25
Center Deflection (mm)
30
Experimental force-deflection response of specimen R6T2S3.
-110-
force during forward application of stroke implies that the panel is seeking
another stable equilibrium state without the application of further force, i.e.
the panel is dynamically "snapping away" from the indentor into an inverted
configuration. All specimens which exhibited a load drop in this study, also
attained a force of zero within the instability region as they snapped away
from the indentor into the inverted configuration. As explained in Chapter 4,
the application of downward stroke to the indentor was manually stopped
once this inverted configuration was attained and then reversal of the stroke
began. Since the indentor is not in contact with the specimen after "snapaway", the second equilibrium path was not reached in the tested responses
of these specimens. These panels also snapped back into the original convex
configuration when removed from the test fixture.
Specimen R12T1S3 also reached a load of zero in the instability region
but did not exhibit a linear response after the load-drop as seen in Figure 5.9.
The response in the instability region has a positive concavity and closely
approaches a slope of zero until it reaches a load of zero at a center deflection
of 11.5 mm.
All other specimens which reached a force of zero in the
instability region showed an approximately linear response after the loaddrop. Specimen R12T1S3 also has the distinction of being the only specimen
with a load-drop which occurs over a time period longer than the data
sampling interval of two seconds, as the load-drop took a total of 26 seconds
to fully develop. Unlike any other specimen, this load-drop occurred while
the indentor was held stationary for the purpose of deflection-shape data
acquisition.
The experimental force-deflection responses for all specimens can be
found in Appendix A. Key features of each response, such as the critical
snapping loads and critical snapping deflections, can be found in Tables 5.2
-111-
50
R12T1S3
40
S30
.
20 -
0
Figure 5.9
5
10
Center Deflection (mm)
15
Experimental force-deflection response of specimen R12T1S3.
-112Table 5.2
Experimental and Predicted (Pinned-Free) Critical
Snapping Loadsa
T1
T2
T3
Span
R6
R12
R6
R12
R6
R12
S1
9 (51)c
_ b (17)
- (207)
- (51)
- (440)
- (-)
S2
71 (70)
27 (35)
260 (540)
80 (238)
445 (1430)
- (480)
S3
63 (77)
30 (38)
455 (590)
127 (282)
888 (1850)
241 (830)
a All
values in N.
b "-" indicates that an instability was not observed.
c Predicted (Pinned-Free) results are given in parentheses.
-113and 5.3, respectively. Since the peak force reached during any given test may
not coincide with the critical snapping load, the peak forces are also compiled
for all specimens in Table 5.4.
Peak center deflection is an additional
characterization of the force deflection response and is given for each
specimen in Table 5.5. It should be noted that the peak center deflections for
specimens which displayed "panel snap-away" do not correspond with the
maximum center deflections as observed in a typical force-deflection diagram.
The reason is that the center deflection in the force-deflection diagrams
represents the stroke of the indentor as measured by the testing machine,
which is identical to the center deflection of the specimen when the two are in
contact. During "panel snap-away", different values of peak center deflection
exist for the actual specimen, as measured by the laser, and the indentor, as
measured by the stroke output of the testing machine, with the former being
reported in Table 5.5.
5.1.2 Numerical Results
Force-deflection responses were computed using the STAGS finite
element code with both simply-supported-free and pinned-free boundary
conditions. The presentation of the predicted force-deflection responses for
each geometry, given in Appendix B, includes results for both boundary
conditions plotted together for comparison. The predicted responses do not
exhibit the hysteresis exhibited by the experimental data since the analysis
does not include energy absorbing mechanisms such as plastic deformation,
friction, and damage formation. The forward and reverse stroke portions of
these analytical responses, therefore, coincide. Three "types" of predicted
force-deflection responses were observed: stable, unstable, and unstable with
"panel snap-away" and are identified herein as types A, B, and C,
-114-
Table 5.3
Experimental and Predicted (Pinned-Free) Critical
Snapping Displacementsa
T2
T1
T3
Span
R6
R12
R6
R12
R6
R12
Si
1.7 (1.3) c
- b(0 .7 )
- (1.2)
- (0.7)
- (1.1)
- (-)
S2
3.9 (3.3)
2.8 (2.5)
4.5 (4.8)
2.9 (3.0)
4.5 (5.1)
-(2.4)
S3
6.0 (6.0)
3.9 (4.0)
9.0 (7.9)
5.8 (5.1)
9.9 (9.1)
6.3 (5.6)
aAll values in mm.
b "-" indicates that an instability was not observed.
c Predicted (Pinned-Free) results are given in parentheses.
-115-
Table 5.4
Experimental Peak Forcea,b
T2
T1
T3
Span
R6
R12
R6
R12
R6
R12
S1
14
9
170
150
617
375
S2
71
27
260
92
644
368
S3
63
30
455
127
888
321
a All
values in N.
b bold indicates that peak force occured on the second equilibrium path.
116-
Table 5.5
Experimental Peak Deflectiona
T1
T2
T3
Span
R6
R12
R6
R12
R6
R12
S1
2.4
1.7
3.0
2.6
3.4
1.9
S2
9.0
5.3
12.4
6.8
11.3
6.2
S3
19.9
11.5
20.0
14.0
27.7
12.7
a All values in mm.
-117respectively. Types A and B responses are identical to the experimental
types I and II, respectively, while type C is equivalent to the experimental
type III response without the discontinuous "load-drop" behavior observed in
the experimental response. All predicted force-deflections responses are,
therefore, smooth and continuous.
The simply-supported-free responses for all specimen geometries
displayed stable type A force-deflection responses similar to that of specimen
geometry R12T3S1, as shown in Figure 5.10. The concavity of these curves is
typically very small and often difficult to observe in the force-deflection plots,
giving an approximately linear response.
The "types" of force-deflection responses for all pinned-free geometries
are given in Table 5.6. Specimen geometries with types A and B responses
are observed to reside in the upper right corner of the table with the only type
A response (R12T3S1) occupying the extreme upper right position.
The
majority of specimen geometries (15 out of 18) exhibited type C responses and
reside in the lower left and central portion of the table. The upper right
portion of the table corresponds to the thicker, more shallow geometries and
the lower left portion corresponds to the thinner, deeper geometries.
The R12T3S1 geometry was the only pinned-free case which produced
a stable type A force-deflection response, as seen in Figure 5.10. Here the
softening and stiffening regions are more clearly visible with an inflection
point at a center deflection of 1.0 mm and a force of 200 N. It should be noted
that the responses for the two different sets of boundary conditions cross each
other for this and all other specimen geometries.
Type B force-deflection responses exhibit an instability region where
the force decreases with increasing center deflection. The pinned-free forcedeflection response for specimen geometry R6T3S1 given in Figure 5.11
-118-
500
R12T3S1
- - Pinned-Free
- -- Simply-Supported-Free
400
SI
.
300
"
V
0
.j
200
"
-
.,,
-"7
-
~/
100
-
*
/
5l
'A
UI_
Center Deflection (mm)
Figure 5.10
Predicted force-deflection responses for geometry R12T3S1.
-119Table 5.6
General Characterization of the Predicted Pinned-Free
Force-Deflection Responsesa
T2
T1
T3
Span
R6
R12
R6
R12
R6
R12
S1
C
C
C
B
B
A
S2
C
C
C
C
C
C
S3
C
C
C
C
C
C
a "A" indicates that the response is stable.
"B" indicates that the response has an instability.
"C" indicates that the response has an instability with "panel snap-away".
-120-
1000
_
* *
750
A
I R6T3S1
-Pinned-Free
-Simply-Supported- I
Free
,
I
I
I
." I
500
I
U
O
0
.°i
/I
250
-d
Or--re
I
I
.
.
I
I
I
I
I
I I
Center Deflection (mm)
Figure 5.11
Predicted force-deflection responses for geometry R6T3S1.
-121-
shows such behavior.
This region is bounded to the left by the first
equilibrium path, which shows softening behavior, and to the right by the
second equilibrium path, which demonstrates stiffening behavior. As seen in
Table 5.6, two of the eighteen pinned-free geometries showed this type B
behavior.
The predicted type C force-deflection responses show an instability
region along with the attainment of a force of zero within this region. Fifteen
of the eighteen pinned-free cases showed type C force-deflection behavior, as
noted in Table 5.6. An example of such a response is shown in Figure 5.12 for
specimen geometry R6T2S2. The response begins on the first equilibrium
path with softening behavior, enters the instability region where the force
steadily decreases until reaching a value of zero at a positive value of center
deflection. As explained in Chapter 3, the STAGS analysis is terminated
upon reaching a force of zero in the instability region for consistency with the
experimental procedures.
The predicted pinned-free critical snapping loads and critical snapping
deflections for all specimen geometries are given along with the experimental
values in Tables 5.2 and 5.3, respectively.
5.2
Deformation-Shape Behavior
Deformation-shapes for each specimen geometry were obtained
through both experimentation and analysis. Specific schemes were developed
for the presentation of the deformation-shape results and each is explained in
the following sections. Experimental results are presented first, followed by
the analytical results.
-122-
1000
800
R6T2S2
- Pinned-Free
A. nSimply-Supported-Free
0" 600
z
- 400
I
\
!1
200
0
5
10
15
Center Deflection (mm)
Figure 5.12
Predicted force-deflection responses for geometry R6T2S2.
-1235.2.1 Experimental Results
Panel deformations were investigated by obtaining detailed deflection
data at successive stages of the snap-through process. Three-dimensional
plots created from this data set provide a largely qualitative representation of
the panel deformation-shapes, as seen for specimen R6T1S2 in Figure 5.13.
These plots are created by connecting spanwise data points at each axial
station with straight lines and also connecting corresponding spanwise data
points at the five axial stations with straight lines. Small black dots are
included on these plots to indicate the loading point. The deformation shapes
are observed to be truly three-dimensional with variations in both the
spanwise and axial directions.
Such data, taken at roughly ten progressive
states of deformation, is presented in this section to examine the "evolution"
of the full panel deformation-shapes.
In order to provide a more quantitative presentation of the data while
retaining the essential features, deformation-shape evolutions are also
presented for both the central spanwise and axial sections, as defined in
Figure 5.14. These two-dimensional deformation-shape evolutions consist of
panel deformation data, along the central spanwise or central axial station,
for each pre-chosen value of center deflection, plotted on the same set of axes
to show the development of deformation with increasing stroke. Examples of
a central spanwise deformation-shape evolution and a central axial
deformation-shape evolution for specimen R6T1S2 are given in Figures 5.15
and 5.16, respectively. Finely spaced data in the spanwise direction gives a
smooth representation of the deformation-shape at each value of center
deflection, as seen in Figure 5.15. Gridlines are included to mark the vertical
location of the axial edges and the location of the central axial section, that is
the midspan location. Deformation-shapes along the central axial section
-124-
SR6 T2S2
E
6
c
0
5
--
4
CL
3
Ce nter Deflection = 3.4mm
o 2-
20 0
150
Spanwise
100
100
Position (mm)
50
-50
o -100
Figure 5.13
Axial Position (mm)
Full panel deformation-shape data for specimen R6T1S2 with
a center deflection of 3.4 mm.
-125-
Central Spanwise Section
Central Axial Section
Figure 5.14
Illustration of the central spanwise and axial sections used in
the two-dimensional deformation-shape presentation.
-126-
Center Deflections in Millimeters
o
*
0
1.1
E
2.3
*
o
*
3.4
4.5
5.7
6.8
A 7.9
0 12.7
A
10
5
E
E
a-
O
0i)
0
>n
!:
-5
-10L
0
50
100
150
200
Spanwise Position (mm)
Figure 5.15
Experimental central spanwise deformation-shape evolution
for specimen R6T1S2.
-127consist of only five data points, as explained in Chapter 4. As a result, the
central axial deformation-shapes, shown in Figure 5.16, consist of the five
data points along with a cubic spline curve fit to adequately represent the
continuous deformation-shape at each value of center deflection. Gridlines in
these plots mark the vertical position of the axial edges and the location of
the central spanwise section or mid-axis location.
Generally, the
experimental deformation-shapes along these central sections showed two
types of behavior: fully symmetric and partially unsymmetric. These general
deformation-shape characterizations for all specimens are listed in Tables 5.7
and 5.8 for the central spanwise and axial sections, respectively.
The
directions of positive deflections and rotations, as defined for these
presentations, are illustrated graphically in Figure 5.17.
Data representing deflection from undeformed position, along the
central spanwise section, is also presented in a similar "evolution" by
subtracting the original spanwise undeformed shape from each subsequent
spanwise deformation-shape.
This allows the actual deflection behavior
along the central spanwise section, which is not easily discerned from the
deformation-shape evolutions due to the initially curved spanwise shape, to
be directly examined. Such "deflection from undeformed-shape" evolutions,
or DFU evolutions, were not necessary along the central axial section since
the initial configuration is flat in this direction, thereby giving no new
information. An example of a DFU evolution is given in Figure 5.18 for
specimen R6T1S2. The top flat line represents the initial undeformed or
reference configuration and each shape below this line represents successive
states of deformation. Gridlines are again included and mark the midspan
location and the initial undeformed configuration. It should be noted that the
vertical dimension (panel depth) is exaggerated in scale with respect to the
-128-
Center Deflections in Millimeters
-A4
--- - 3.4
-1.1
--
-A- -7.9
- -o -- 12.7
-- 2.3
10
-6.8
R6T1S2
-0--
5
E
E
ill---
-IUa
---.-
-
°rLW
-
-o
0
0
0O
o
=CL
A~
o
A
~
t
--
-5
0----
-10
-100
.
-- .
I.
-50
.
.
.
.
III
I
i
50
-m
-0
i
I
B
100
Axial Position (mm)
Figure 5.16
Experimental central axial deformation-shape evolution for
specimen R6T1S2.
-129-
Positive
Rotation
Positive
Rotation
Positive
Deflection
Central Soanwise Section
itive
ation
Positive
Rotation
Positive
Deflection
Central Axial Section
Figure 5.17
Illustration of the positive rotations and deflections defined for
the central spanwise and axial sections.
-130-
Center Deflections in Millimeters
o
*
0
1.1
[E 2.3
*
*
*
3.4
4.5
5.7
A
A
0
6.8
7.9
12.7
E
E
C,
0
-4
O
-
Cd
-8
a)
-12
-16
50
100
150
200
Spanwise Position (mm)
Figure 5.18
Experimental central spanwise DFU evolution for specimen
R6T1S2.
-131in-plane dimensions in the presentation of deformation-shape data for all
specimens in this investigation since the accurately scaled deformations
would not be distinguishable for such shallow panels.
The specimens which exhibited type I force-deflection behavior
displayed fully symmetric deformation-shape evolutions.
The full panel
deformation-shape evolution for specimen R12T3S2, which has a type I forcedeflection response, is given in Figures 5.19 to 5.24. This specimen can be
seen to deflect in a fully symmetric manner from the original convex
configuration to the inverted concave configuration.
This full panel
deformation behavior is generally characteristic of the specimens with a type
I force-deflection response. Any exceptions are noted in the presentation of
the subseqent two-dimensional representations of the deformation-shapes.
The two-dimensional central spanwise deformation-shape evolution for
specimen R12T3S2, given in Figure 5.25, also shows a fully symmetric
transition to the concave configuration. The central spanwise deformationshapes remain symmetric about the midspan for 13 out of 18 specimens in
this investigation (all but R12T1S2, R12T1S3, R6T1S2, R6T1S3, and
R6T2S3), as noted in Table 5.7. Symmetric central spanwise deformationshapes, therefore, occur for all specimens with types I and II force-deflection
responses, as observed by a comparison of Tables 5.7 and 5.1.
These
deformation-shapes generally have two inflection points, one on each side of
the central loading point. Such inflection points are difficult to observe for
the first couple of deformation-shapes in Figure 5.25 due to the scaling of the
plot, however, they are more clearly detected at a center deflection of 2.8 mm.
Generally, the spanwise location of these inflection points migrate from the
center toward the ends as the panel undergoes larger deflections. It should
be noted that the fully symmetric evolutions never show the truly concave
-132-
R12T3S2
Undeformed
E
E
W4O
0
l
>
2
2
,
1
200
100
SparIise
inn (mm)
Droi
I
lllI
II
I
0
-100
Axial Position (mm)
I I
R12T3S2
Center Deflection = 0.6 mm
E
E
1r00m
4-
0
O
2-
>208.
100
Spar
i,..
.-
Posi W"kIIIII
(
Figure 5.19
\
50m
Axial Position (mm)
Full panel deformation-shape data for specimen R12T3S2
(above) in the undeformed state, and (below) with a center
deflection of 0.6 mm.
I
-133-
R12T3S2
Center Deflection = 1 1 mm
E
E
4-
CD
O
0
>
0
200
Spar iwise
100
100
Pni
inn (mm)
I VVILIVII \IIIIII/
0
-100
-50
Axial Position (mm)
R12T3S2
Center Deflection = 1.7 mm
E
E
4J 4
o
3
0
C
2
>0
>208
Spanwise
Position (mm)
Figure 5.20
50
100
0
-100
100
-50
Axial Position (mm)
Full panel deformation-shape data for specimen R12T3S2 with
a center deflection of (above) 1.1 mm, and (below) 1.7 mm.
-134-
R12T3S2
Center Deflection = 2.3 mm
E
C
3
CD
O
E2
C)
O
>
-1
200
Spanwise
100
Position (mm)
-100
0
-50
100
Axial Position (mm)
R12T3S2
Center Deflection = 2.8 mm
E
E
0
50
3-
0
O
1
20100
Spar
isemmoo
inn (mm)
Poii
I VVLIV., \,,J,,/
Figure 5.21
.....
0
-100
-50
50
Axial Position (mm)
Full panel deformation-shape data for specimen R12T3S2 with
a center deflection of (above) 2.3 mm, and (below) 2.8 mm.
-135-
R12T3S2
Center Deflection = 3.4 mm
E
S3
0O
E2
D,
O
0
o
>200
200
100
Spar nwise
Pni
tion
I V,.,,I,
100
(mm)
\I, II I
.50
0
-100
.0
Axial Position (mm)
R12T3S2
Center Deflection = 4.0 mm
E
E
C3
0O
CD
2
O
0
1
.
0
>
-1
200
100
Spanwise
100
Position (mm)
Figure 5.22
-5050
0
-100
-50
Axial Position (mm)
Full panel deformation-shape data for specimen R12T3S2 with
a center deflection of (above) 3.4 mm, and (below) 4.0 mm.
-136-
R12T3S2
Center Deflection = 4.5 mm
E
E
2-
0
O
1-
0-.
O.
-1-.
>a)
-2
?00
V
100
Spar wise
oo
+inn \(mm)
lllll
I VLI'II
so......
0
-100
R12T3S2
Center Deflection = 5.1
E
Axial Position (mm)
mm
E
0O
C.
050
Spanwise 100
Position (mm)
Figure 5.23
0 -100
100
-50
Axial Position (mm)
Full panel deformation-shape data for specimen R12T3S2 with
a center deflection of (above) 4.5 mm, and (below) 5.1 mm.
-137-
R12T3S2
Center Deflection = 5.7 mm
E
O
C,,
0
O
200
0-1.
-2
200
100
Sparnwise
I
' JILI.I
..
loo
tion (mm)
I
\I
I1I
)
.1o
-50
0
-100
Axial Position (mm)
R12T3S2
Center Deflection = 6.2 mm
0-E
E
O
0
Cn
O
0
-2
208
100
Spanwise
Position (mm)
Figure 5.24
0
-100
Axial Position (mm)
Full panel deformation-shape data for specimen R12T3S2 with
a center deflection of (above) 5.7 mm, and (below) 6.2 mm.
-138-
Center Deflections in Millimeters
E
E
c
O
aa.
o
*
0
0.6
0
1.1
*
*
*
1.7
2.3
2.8
A 3.4
v
A
4.0
0
4.5
[
[
5.1
5.7
6.2
2
1
0
-1
-2
-3
-4
-5
50
100
150
200
Spanwise Position (mm)
Figure 5.25
Experimental central spanwise deformation-shape evolution
for specimen R12T3S2.
-139-
Table 5.7
General Characterization of the Central Spanwise
Deformation-Shapesa
T2
T1
T3
Span
R6
R12
R6
R12
R6
R12
Sl
S
S
S
S
S
S
S2
U
U
S
S
S
S
S3
U
U
U
S
S
S
a "S" indicates that the deformation-shapes were approximately symmetric.
"U" indicates that the deformation-shapes were partially unsymmetric.
-140configuration where the inflection points actually disappear. This is observed
for the final deformation-shape in Figure 5.25 where the inflection points still
visibly exist near the edges. Although deflection data is not obtained for
approximately 15 mm at each end, the panel can also be seen to undergo
negative rotations at the boundaries with increasing center deflection. The
central spanwise DFU evolution for specimen R12T3S2, given in Figure 5.26,
also shows shapes which are symmetric about the midspan point for each
center deflection. This plot demonstrates that the point of maximum vertical
deflection is located at the midspan for all center deflections. Such behavior
was observed for all specimens which exhibited symmetric deformationshapes.
Symmetric behavior is also observed in the central axial deformationshape evolution for specimen R12T3S2, shown in Figure 5.27. Unlike the
central spanwise deformation-shapes, these curves show that each
circumferential edge is free to displace as a result of the free boundary
conditions. Therefore, every point along this axial section displaces during
the deflection process. The uppermost curve, which is approximately flat,
represents the undeformed configuration.
As the center deflection is
increased, symmetric and approximately flat deformation-shapes develop as
seen for a center deflection of 1.7 mm. The deformation-shape has a distinct
local extremum at the loading point with two inflection points located
symmetrically about the mid-axis point. However, when the exaggerated
scaling is taken into consideration, these deformation-shapes could still be
considered approximately flat. For instance, at a center deflection of 5.1 mm,
the vertical position of the central loading point is -2.1 mm while the local
maximums have a vertical position of -1.9 mm. This difference in vertical
position of 0.2 mm between the local extrema is 0.1% of the panel planform
-141-
Center Deflections in Millimeters
o
*
z
0
0.6
1.1
*
1.7
A
3.4
*
2.3
A
4.0
*
2.8
0
4.5
* 5.1
*] 5.7
E 6.2
E
E
E
C)
O
-2
0
(D
0
-4
0)
-6
-8
50
100
150
200
Spanwise Position (mm)
Figure 5.26
Experimental central spanwise DFU evolution for specimen
R12T3S2.
-142-
Center Deflections in Millimeters
-m- - 1.7 -6
-3.4
- - - -5.7
- - -0.6 - - --2.3 - A- -4.0
-- E--1.1 -*---2.8
R-- 6.2
-- 0--4.5 --
- -0
R12T3S2
--~--
E
E
2
C,,
1
O
-*--
- -----
U-
-
-C
-
-
-(
- -0
- -0
-
*--
- -
---
....
-
------
5
-u
O
0
CO
-c)
-0--
-
-1
0--r-
--- C
-- V
-
-2
-
EB ... .I--
-3
--
-
"
"-0
.--
-
-
---
-.
-F
-4
-10(
-10
.
0
m1
$
1.
I*i
.
-50
1.
.
11.
.
I**
. 1 .
0
. 1
.
I*£
.
.
50
.
.
.
.
100
Axial Position (mm)
Figure 5.27
Experimental central axial deformation-shape evolution for
specimen R12T3S2.
-143dimension of 203 mm. Approximately symmetric and flat axial deformationshapes were observed for 9 of the 18 specimens in this study. The type of
behavior exhibited by the central axial deformation-shape evolutions for all
specimens is listed in Table 5.7. Upon comparison with Table 5.1, it is
observed that four out of six specimens with a type I force-deflection response
exhibited symmetric and approximately flat central axial deformationshapes.
The other two specimens, R6T3S1 and R12T1S1, showed
unsymmetric and non-uniform shapes throughout the deformation process.
Specimens which exhibited type II force-deflection responses generally
displayed fully symmetric deformation-shape evolutions.
The full panel
deformation-shape evolution is shown for specimen R6T2S2, which displayed
a type II force-deflection response, in Figures 5.28 to 5.33. Fully symmetric
deformations are observed at each value of center deflection.
This
deformation-shape behavior is characteristic of most specimens with a type II
force-deflection response. Exceptions to this symmetric behavior occured
along the axial direction for specimens R6T1S1 and R12T2S3, as discussed in
the presentation of the central axial deformation-shapes.
The central spanwise deformation-shape evolution for specimen
R6T2S2 is given in Figure 5.34 and is representative of most specimens with
a type II force-deflection response. Inflection points, located symmetrically
about the midspan, migrate toward the edges with increasing center
deflection. These inflection points are clearly observed for this case at a
center deflection of 4.5 mm. Inflection points or inflection regions are still
obvious near the edges in the final deformation-shape at a center deflection of
12.4 mm for this case. The central spanwise DFU evolution for specimen
R6T2S2, given in Figure 5.35, is also seen to have fully symmetric behavior.
The point of maximum deflection migrates slightly from the midspan
-144-
R6T2S2
Undeformed
E
E
8
0
CD
S6
O
M-
4
o
1:
2
>0
200
Spanwise 100
Position (mm)
-100
100
Axial Position (mm)
R6T2S2
Center Deflection = 1.1 mm
E
E
0
0
-50
50
8
0
O
D- 4
-
>0
2
208
-
Spanwise 100
Position (mm)
Figure 5.28
-50
0
-100
100
-50
Axial Position (mm)
Full panel deformation-shape data for specimen R6T2S2
(above) in the undeformed state, and (below) with a center
deflection of 1.1 m.
-145-
R6T2S2
Center Deflection = 2.3 mm
E
E
O
O
1.
Spanwise 100
Position (mm)
0
0
-100
100
50
-50
Axial Position (mm)
>
R6T2S2
Center Deflection = 3.4 mm
0
2 4
2
08o
Spanwise 100
Position (mm)
Figure 5.29
0
S
0
-100
100
-50
Axial Position (mm)
Full panel deformation-shape data for specimen R6T2S2 with
a center deflection of (above) 2.3 mm, and (below) 3.4 mm.
-146-
R6T2S2
Center Deflection = 4.5 mm
E
E
O
O
C
a0)
-2
200
Spanwise 100
Position (mm)
100
so
0
-100
-50
=
Axial Position (mm)
R6T2S2
Center Deflection = 5.6 mm
E
E
0
O
a.
"E
>
-0
200
Spanwise
100
Position (mm)
-50
--
50
100
Axial Position (mm)
Figure 5.30
Full panel deformation-shape data for specimen R6T2S2 with
a center deflection of (above) 4.5 mm, and (below) 5.6 mm.
-147-
R6T2S2
Center I )eflection - 6.8 mm
E
E
t-
O
4-
4-
a.
2-
a0
>
-2
200
Spa nwise 100
Posi tion (mm)
0
0
-100
R6T2S2
Center Deflection -7.9
E
0
O
:
C)
0
0-
100
5o
-50
Axial Position (mm)
mm
6
4
2
> -2
200
Spanwise 100
Position (mm)
Figure 5.31
0
-100
-50
0
50
100
Axial Position (mm)
Full panel deformation-shape data for specimen R6T2S2 with
a center deflection of (above) 6.8 mm and (below) 7.9 mm.
-148-
E
E
rCo
0
4-
R6T2S2
Center Deflection = 9.0 mm
2-
O
0-
0-2-
0)
.;;o -4
200
Spanwise 1oo
Position (mm)
100
so
0
-100
-50
I
Axial Position (mm)
R6T2S2
E
E 4Center Deflection = 10.2 mm
C
C,
0
n
20-
0) S-2-
>
-4
200
Spa nwise 100
Position (mm)
Figure 5.32
0
0 -100
so50
100
-50
Axial Position (mm)
Full panel deformation-shape data for specimen R6T2S2 with
a center deflection of (above) 9.0 mm and (below) 10.2 mm.
-149-
E
R6T2S2
E 2- Center Deflection = 11.3 mm
0
0-
t
C,
0
.
-2-
o
-4-
>
-6.
200
100
Spa nwise 100
Posi tion (mm)
E
2
50
0
-100
-50
\
Axial
Position (mm)
R6T2S2
Center Deflection = 12.4 mm
0o
O
a.
-2
O. -4
>
-6
200
Spanwise 100
Position (mm)
50
0
-100.
-50
.
.
100
..
Axial Position (mm)
Figure 5.33
Full panel deformation-shape data for specimen R6T2S2 with
a center deflection of (above) 11.3 mm and (below) 12.4 mm.
-150-
Center Deflections in Millimeters
o
*
o
0
1.1
2.3
*
*
*
10
E
E
3.4
4.5
5.6
A
A
6.8
7.9
v
0
9.0
E
*
10.2
11.3
12.4
5
E
C
0
O
C'
0
0
a.
0z
-5
-10
50
100
150
200
Spanwise Position (mm)
Figure 5.34
Experimental central spanwise deformation-shape evolution
for specimen R6T2S2.
-151-
Center Deflections in Millimeters
o
*
0
1.1
*
*
3.4
4.5
A
*
2.3
*
5.6
A
6.8
7.9
v
s
10.2
11.3
o
9.0
EB
12.4
0
E
E
-
O
-5
>
-10
-15
50
100
150
200
Spanwise Position (mm)
Figure 5.35
Experimental central spanwise DFU evolution for specimen
R6T2S2.
-152location.
However, it remains relatively close to the midspan and, thus,
basically symmetric throughout the deflection process. The quantification of
such migrations from the midspan location are more fully discussed in
Chapter 6. Fully symmetric central spanwise deformation-shape evolutions
were observed for all specimens with a type II force-deflection response.
Approximately symmetric deformation-shapes are also observed along
the central axial section of specimen R6T2S2, as shown in Figure 5.36. The
central axial deformation-shape behavior exhibited by specimen R6T2S2 is
typical for five out of the seven specimens which exhibited a type II forcedeflection response (all but R6T1S1 and R12T2S3), as denoted in Table 5.8.
Initially, localized deformations near the loading point are observed for the
smaller deflections such as 3.4 mm for this case. At larger deflections a local
minimum is developed at the loading point with two symmetrically located
maxima about the midspan, as seen for specimen R6T2S2 in Figure 5.36 at a
center deflection of 11.3 mm. However, these deformation-shapes could be
considered approximately flat at a center deflection of 11.3 mm, as the
difference in vertical position of 0.4 mm between the local extrema is only
0.2% of the total axial length of the panel. The other specimens, R6T1S1 and
R12T2S3, exhibited asymmetric central axial deformation-shapes even for
small center deflections.
The specimens which exhibited a type III force-deflection response
displayed unsymmetric deformation-shapes at some point during their
evolutions along with "panel snap-away." The deformations are initially
symmetric.
However, an unsymmetric deformation-shape eventually
develops and remains until the panel snaps away from the indentor into the
inverted concave configuration, again giving a fully symmetric deformationshape.
A typical example of such behavior is seen in the full panel
-153-
Center Deflections in Millimeters
- -0- -02.3
-5 2.3
10
-- 3.4
- .- -4.5
-A-
-* -5.6
- -0
-10.2
-6.8
-7.9
-9.0
-A
-
-11.3
-12.4
-N-
R6T2S2
- -
5
E
E
t-
-
- -
-- 0--
-U-
L-
-
°
--
-
- -0
--- 0
-
-n
-
-
--.
O
5+
-u
---
S-
0
CO
--
-
-A-
-
Ir -
.
-5
0---
----------------
[]-
----------------
"
-- -
-
-10(
I
I
I
.
.
.
.
.
.
.
.
.
.
--
.~.~~--
- -- - - - - .
I
I
_-
-- --
-10 0
._ °
I
I
-5 0
I
-
. __- -N--_
I
.. - --
E ...
,
I
0
I
I
I
--
VI
EB-
I
I
50
I
I
I
I
I
100
Axial Position (mm)
Figure 5.36
Experimental central axial deformation-shape evolution for
specimen R6T2S2.
-154-
Table 5.8
General Characterization of the Central Axial
Deformation-Shapesa
T2
T1
T3
Span
R6
R12
R6
R12
R6
R12
S1
U
U
S
S
U
S
S2
U
U
S
S
S
S
S3
U
U
U
U
S
S
a "S" indicates that the deformation-shapes were approximately symmetric.
"U"indicates that the deformation-shapes were partially unsymmetric.
-155deformation-shape evolution of specimen R6T1S2 shown in Figures 5.37 to
5.41.
This loss of symmetry is more clearly illustrated in the central
spanwise deformation-shape evolution of specimen R6T1S2 in Figure 5.15.
Symmetric deformation-shapes exist in the initial stages of the test, followed
by the attainment of an unsymmetric deformation-shape at a center
deflection of 4.5 mm.
This unsymmetric shape remains with further
application of stroke until the panel snaps away from the indentor into a
symmetric concave configuration with a center deflection of 12.7 mm.
Central spanwise DFU evolutions also reveal the existence of unsymmetric
central spanwise deformation-shapes for specimens with a type III forcedeflection response. The point of maximum vertical deflection is initially
located at the midspan. However, this point of maximum deflection migrates
away from the midspan in these cases before returning as the panel snaps
away. The central spanwise DFU evolution for specimen R6T1S2, shown in
Figure 5.18, exemplifies such behavior. The point of maximum deflection is
very near the midspan for center deflections less than 3.4 mm. However, at a
center deflection of 4.5 mm, the shapes become unsymmetric with the point of
maximum deflection clearly migrating from the midspan location.
This
migration continues as the center deflection is increased to 5.7 mm after
which the point of maximum deflection remains somewhat stationary until it
again returns to the midspan location as the panel reaches the inverted
configuration at a center deflection of 12.7 mm.
The central axial deformation-shape evolution for specimens with type
III force-deflection responses also show the development of unsymmetric
deformation-shapes. An approximately symmetric and somewhat localized
deformation is typically observed near the loading point for smaller center
deflections. As the center deflection increases, the specimen generally rotates
-156-
R6T1S2
Undeformed
E
E
8-,
0
c,
O
CD
0
200
>
100
Spanwise
50
100
Position (mm)
0
-100
-50
Axial Position (mm)
R6T1S2
Center deflection = 1.1 mm
E
E
O
O
C.
I
0
200
Spanwise 100
Position (mm)
Figure 5.37
0o
0 -100
100
-50
Axial Position (mm)
Full panel deformation-shape data for specimen R6T1S2
(above) in the undeformed state and (below) with a center
deflection of 1.1 mm.
-157-
R6T1S2
Center deflection = 2.3 mm
E
E
O
O
0_
M)
>
200
Spanwise 100
Position (mm)
100
0
0
-100
-50
Axial Position (mm)
R6T1S2
Center deflection = 3.4 mm
aE
O
E
C.
n
200
Spanwise 100
Position (mm)
Figure 5.38
100
0
-100
-50
Axial Position (mm)
Full panel deformation-shape data for specimen R6T1S2 with
a center deflection of (above) 2.3 mm and (below) 3.4 mm.
-158-
R6T1S2
Center deflection = 4.5 mm
E
E
O
C)
O
o
1)
I
-2
200
50
Spanwise 100
Position (mm)
0
-100
100
-50
Axial Position (mm)
R6T1S2
Center deflection = 5.7 mm
E
E
-
0
64
cD
O
a,
>
-2
200
100
Spa nwise 100
Posi tion (mm)
Figure 5.39
50
0 -100
-0
Axial Position (mm)
Full panel deformation-shape data for specimen R6T1S2 with
a center deflection of (above) 4.5 mm and (below) 5.7 mm.
-159-
R6T1S2
Center deflection = 6.8 mm
E
E
C
cO
O
0-2
a)
-A
200
100
Spanwise 100
Position (mm)
0
0
a-
0
-100
I
Axial Position (mm)
R6T1S2
Center deflection = 7.9 mm
E
E
O
-50
42.
0-
.o -2>,
-40
~00
"'
A
100
Posiltion (mm)
SparIwise
100
50
-100
0 S-100
-50
.
.
.
.
Axial Position (mm)
Figure 5.40
Full panel deformation-shape data for specimen R6T1S2 with
a center deflection of (above) 6.8 mm and (below) 7.9 mm.
-160-
R6T1S2
Center deflection = 12.7 mm
E
E
0-,
C-
-2 -
0O
-4-6,-
>
-8
200
100
Spanwise
10o50
Position (mm)
Figure 5.41
0
-100
-50
.
.
..
h.
Axial P-osition (mm)
Full panel deformation-shape for specimen R6T1S2 with a
center deflection of 12.7 mm.
-161about the spanwise direction to form asymmetric deformation-shapes which
remain for the remainder of the test, including the final snapped-away
configuration.
This general progression is seen in the central axial
deformation-shape evolution for specimen R6T1S2 in Figure 5.16. Symmetric
deformation-shapes are observed up to a center deflection of 4.5 mm followed
by the development of an unsymmetric deformation-shape at a center
deflection of 5.7 mm. At this point, the vertical deflections at positive axial
stations are clearly greater than those at negative axial stations.
The
specimen has effectively rotated about the spanwise direction which comes
out of the page in these plots. This asymmetric behavior can also be detected
in the three-dimensional deformation-shape at a center deflection of 5.7 mm,
shown in Figure 5.39.
The deformation-shape rotates back toward the
approximately flat configuration, but remains asymmetric, as the specimen
snaps away from the indentor to a center deflection of 12.7 mm. This general
deformation-shape behavior is typical for the specimens which exhibited a
type III force-deflection response except for specimen R6T2S3 which showed a
flat configuration after snapping away.
Panel snap-away can be detected on any of these evolution plots by an
inconsistently large interval of center deflection as the inverted configuration
is attained. For instance, the interval of center deflection for specimen
R6T1S2, as controlled by the stroke of testing machine, is very close to 1.1
mm. This well-defined interval of center deflection is obvious in the first
eight deformation-shapes of Figure 5.1, as the specimen remains in contact
with the indentor. However, the interval of center deflection between the
final two deformation-shapes is 4.8 mm. This occurs because the panel is
snapping away from the indentor and, hence, the stroke control of the testing
machine cannot be used to regulate the center deflection of the specimen.
-162Stroke-reversal began once the specimen snapped-away, so no further
deformation-shapes beyond this configuration were recorded.
Specimen R12T1S2 also exhibited unsymmetric deformation-shapes
along with some unique characteristics. The central spanwise deformationshape evolution for specimen R12T1S2 is shown in Figure 5.42.
The
deformation-shapes are initially symmetric followed by a transition to
unsymmetric shapes at a center deflection of 4.5 mm, which is larger than the
undeformed panel height of 3.2 mm. This was the only specimen which
retained symmetric spanwise deformation-shapes for center deflections larger
than the original panel height. The unsymmetric shapes remain until the
panel snaps away to a center deflection of 6.7 mm. Unique behavior was also
observed in the central axial deformation shape evolution shown in Figure
5.43. Slightly unsymmetric deformation-shapes initially develop followed by
the development of a fully asymmetric shape at a center deflection of 4.0 mm.
The specimen then rotates about the spanwise direction as the center
deflection proceeds from 4.0 mm to 4.5 mm, thus creating an asymmetric
deformation-shape with an opposite slope. The rotation is so large that the
deflections for positive axial stations actually decrease even though the
center deflection has increased, causing an overlap of deformation-shapes.
This inversion of the slope of an asymmetric deformation-shape was not
observed for any other specimens. It should be noted that this specimen also
experienced its load-drop at a center deflection of 4.25 mm which is 152% of
the critical snapping displacement of 2.8 mm. All other specimens with type
III responses experienced a load-drop at center deflections within 20% of the
critical snapping displacement, i.e. not nearly as far along the instability
path.
-163-
Center Deflections in Millimeters
0
0.6
1.2
E
E
2
C
O
1
0
0
1.7
2.3
2.8
3.4
4.0
4.5
V
*
5.1
6.7
-1
a>
0I
-2
-3
-4
-5
50
100
150
200
Spanwise Position (mm)
Figure 5.42
Experimental central spanwise deformation-shape evolution
for specimen R12T1S2.
-164-
Center Deflections in Millimeters
-e-
-E - -1.2
E
E
2
C
1
O
0
0
*.-
o
-1
a>
-2
O
--
---- - 1.7
-- e--2.3
-*--- 2.8
-0
-0.6
- - -- 5.1
--&- -6.7
-3.4
- A- -4.0
-3
-4
-5 L
-100
-50
0
50
100
Axial Position (mm)
Figure 5.43
Experimental central axial evolution for specimen R12T1S2.
-165The deformation-shape evolutions along the central sections were
found to adequately characterize the deformation behavior of the specimens
in this investigation. Therefore, the experimental central spanwise and
central axial deformation-shape evolutions for all specimens are given in
Appendix C along with the central spanwise DFU evolutions.
5.2.2 Numerical Results
Deformation-shape evolutions and DFU evolutions were also computed
with the STAGS F.E.M. code for both pinned-free and simply-supported-free
boundary conditions. As explained in Chapter 3, the STAGS code internally
decides the increments of center deflection based on a convergence criteria.
Hence, the deformation-shape evolutions do not show the approximately
uniform interval of center deflection as seen in the experimental results. A
gross change in this experimental interval of center deflection in the
experiment is an indicator of "panel snap-away." The interval of center
deflection cannot, however, reveal whether or not a panel has snapped away
from the indentor in the predicted results since the interval of center
deflection for calculations was not uniform.
The STAGS analysis was
terminated if the force reached zero in the instability region or when the force
reached twice the maximum force observed in the experiments. Thus, in the
former case, the final predicted deformation-shape corresponds to the
configuration just before the panel snaps away. The fully inverted "snappedaway" deformation-shape is not obtained for these cases. Therefore, if the
final deformation-shape in the evolution does not show an approximately
inverted configuration, then the analysis was terminated due to the
attainment of a force of zero, i.e. the panel must have snapped away. If the
panel does not snap away, then the deformation-shape evolutions from the
-166numerical analysis show a complete transition into the approximately
inverted configuration. The presentation of the predicted deformation-shape
evolutions, along the central spanwise and axial sections, is similar to the
experimental results including the central gridlines.
However, all
deformation-shapes from the numerical analysis include the computed results
at each grid point along with a cubic spline curve-fit. This aids in visualizing
the smooth deformation-shape from the relatively coarsely spaced grid points.
The predicted deformation-shapes along the central sections were observed to
be fully symmetric for all specimen geometries and for both boundary
conditions.
The central spanwise deformation-shape evolutions for all simplysupported cases were symmetric with only negative rotations at the
boundaries, as seen for specimen R6T3S3 in Figure 5.44.
Generally,
inflection points for the simply-supported cases become noticeable only for
larger center deflections. It should be noted that snap-away was not observed
in the force-deflection response for any simply-supported case and, therefore,
the deformation-shapes show a complete progression into the approximately
inverted configuration. The central spanwise DFU evolution for specimen
R6T3S3 is given in Figure 5.45. All curves on this plot are symmetric with
the maximum deflections occurring at the midspan location.
Negative
rotations are again observed near the boundaries for all center deflections.
Similar behavior is observed in the central spanwise DFU evolutions for all
simply-supported-free cases in this investigation.
The central axial deformation-shape evolutions for all simplysupported-free cases were symmetric and approximately uniform. As the
center deflection is increased, a local minimum develops at the loading point
with symmetrically located maxima to either side. These deformation-shapes
-167-
Center Deflections in Millimeters
S- -9.0
20- -4.1
20
-- -
- - -- 18.4
13.1
--
--24.6
150
200
250
10
E
aE
O
01
cm
0
C>
a)
-10
-20 L
0
50
100
300
Spanwise Position (mm)
Figure 5.44
Predicted central spanwise deformation-shape evolution for
specimen R6T3S3 with simply-supported-free boundary
conditions.
-168-
Center Deflections in Millimeters
-*-
-4.1
18.4
-0-E 9.0
----
-- m- -13.1
-- *---24.6
R6T3S3
U
E
E
C
O
C-%~~ O
CC
O %
-O ,
CC
,
O
-5
-10
O
0ElY
if
4. .
Cz
0\ W
-20
r4
-25
-30 C-
E0.
0-0
3.Lo.?
ED
-
Qa. -15
0i
C
....I ,
~,
/
s,
.
, ,,
, 1 ,1
50
100
150
I 1 1 1
200
I 1 1 1
250
I
300
Spanwise Position (mm)
Figure 5.45 Predicted central spanwise DFU evolution for specimen
R6T3S3 with simply-supported-free boundary conditions.
-169persist until the panel has reached the approximately inverted configuration.
The difference in the vertical positions of these extrema are typically very
small compared to the total axial length of the panel (less than 1%). These
central axial deformation-shapes may, therefore, be considered approximately
flat. An example of this deformation behavior is given for specimen R6T3S3
in Figure 5.46. A local minimum develops at the central loading point along
with a pair of symmetrically located maxima for all center deflections,
although this is difficult to observe for center deflections. The amplitudes of
these extrema increase with increasing center deflection.
Each local
extremum is clearly visible in the deformation-shape for a center deflection of
18.4 mm. However, the difference in the vertical positions of these extrema is
1.2 mm which is only 0.4% of the total axial length, thereby creating an
approximately uniform shape. As previously noted, no simply-supported-free
case exhibited panel snap-away and, therefore, the final deformation-shape in
these evolutions represents the approximately inverted configuration.
The central spanwise deformation-shapes for the pinned-free cases
which snapped away were fully symmetric about the midspan with visible
inflection points on either side for the larger values of center deflection.
Fifteen out of the eighteen pinned-free geometries exhibited "panel snapaway" (R12T2S1, R12T3S1, and R6T3S1 did not) as illustrated in the central
spanwise deformation-shape evolution for R6T3S3 in Figure 5.47.
As
previously discussed, "panel snap-away" is indicated by the final deformationshape not showing the approximately inverted configuration, which is clearly
seen in Figure 5.47. As with the experimental evolutions, the spanwise
locations of the inflection points migrate toward the boundaries as the center
deflection increases. The deformation-shapes are also observed to initially
deflect upward near the boundaries causing the deformation-shapes to
-170-
Center Deflections in Millimeters
-e-
-0
-- E--9.0
- - -- 18.4
-*-
-4.1
-- N- -13.1
--
-- 24.6
20
R6T3S3
- -e-
E
E
- e-
-- --
--
- e-
- e- -
e- -
10
-
O
C
0
0aCO
.
---
3 -13
E- -
-
-
-
E-
-
-
-- e
- E
-10
0.
-~
~--
~---
----
~--~
-10
-3
-150 -100
-50
0
50
100
150
Axial Position (mm)
Figure 5.46 Predicted central axial deformation-shape evolution for
specimen R6T3S3 with simply-supported-free boundary
conditions.
-171-
Center Deflections in Millimeters
- -
-0
- -E - 3.5 - - -- 9.7
-- A
-19.1
20
E
10
E
C
10
O
>
-10
-2 0 1 . . .,. ,
0
50
1 i I
100
, II
I, , , , I, , , ,i
150 200 250 300
Spanwise Position (mm)
Figure 5.47
Predicted central spanwise deformation-shape evolution for
specimen R6T3S3 with pinned-free boundary conditions.
-172overlap. This behavior near the boundaries can be more clearly observed in
the central spanwise DFU evolution for specimen R6T3S3 in Figure 5.48.
Positive deflections and rotations, as defined in Figure 5.17, are observed
near the boundary as the deflection curves extend above the flat line
representing the undeformed configuration. This behavior was generally
more pronounced for the thinner, deeper geometries with the thicker
shallower geometries exhibiting only negative deflections and rotations. The
curves in these plots are symmetric with the point of maximum deflection
located at the midspan. Such behavior was seen in the central spanwise DFU
evolutions for all pinned-free cases.
The central axial deformation-shape evolutions for the pinned-free
cases which snapped away exhibited initially local deformations at the
loading point followed by the attainment of a more uniform deformationshape for larger center deflections. The uniform deformation-shape develops
as the previously localized deformations propagate toward the edges. Once
the deformations have fully propagated to the free edge of the specimen, they
become approximately uniform or flat. This can be seen in the central axial
deformation-shape evolution for a pinned-free case, R6T3S3, which snapped
away, in Figure 5.49. The deformations are initially localized near the
center, i.e. at the loading point. This is clearly observed at a center deflection
of 6.7 mm, where the deflections at all other axial locations are much less
than at the center. This center deflection corresponds to the maximum
negative rotation observed in the central spanwise deformation-shapes at the
boundaries.
However, as the center deflection is further increased, the
deformations become more uniform along the axial direction. A roughly flat
configuration is attained at a center deflection of 19.1 mm where the panel
begins to snap away from the indentor. This behavior is typical for most of
-173-
Center Deflections in Millimeters
- -E-- 3.5 --o--9.7 -6
- --0
- e- - 1.6 --
E
C
0
-19.1
-6.7 -+- -- 13.1
-5
-10
Cn
o
-
-15
-20
-25 -30
0
I50
50
100
150
200
250
300
Spanwise Position (mm)
Figure 5.48 Predicted central spanwise DFU evolution for specimen
R6T3S3 with pinned-free boundary conditions.
-174-
Center Deflections in Millimeters
- G- -0
- -0- - 3.5 - -
- -9.7
-e-
---m- -6.7 --
-- 13.1
-1.6
20
10
I
-
l--. -m .
-- =-
"
-[
ElL
E-
11
1 ~~C-t
A--
>
R6T3S3
-:
C-.C---C...C..~~
O
-19.1
*-
-4
O
E
E
C
a-
-- A
-I
~C..t--~
A--A
----
--
d
-10
-20 11,
-150 -100
I
I
I
I
I
I
I
,
I
.
-50
II I
.
.
I
..
0
I
I
I
.
I
.
..
50
..
.
.
.
.
100
.
.
I
I
150
Axial Position (mm)
Figure 5.49 Predicted central axial deformation-shape evolution for
specimen R6T3S3 with pinned-free boundary conditions.
-175the pinned-free cases which snapped away from the indentor. Exceptions to
this behavior were noted in eight specimens: R12T1S2, R12T2S2, R12T2S3,
R12T3S2, R12T3S3, R6T1S1, R6T2S2, and R6T3S2.
These specimens
showed significant changes in vertical deflection at points very close to the
extreme axial positions, thereby creating a non-uniform deformation-shape as
the panel begins to snap away from the indentor.
An example of this
behavior is seen in the predicted central axial deformation-shape evolution
for specimen R12T2S2 with pinned-free conditions in Figure 5.50. Initially,
the behavior is similar to the previously described cases, such as that of
specimen R6T3S3 shown in Figure 5.49. However, the deformation-shape at
the point where the panel begins to snap-away (center deflection equal to 4.6
mm in this case) shows an abrupt change in vertical position of 0.3 mm at
each axial extreme. This change in vertical position is large relative to the
maximum change in vertical position of 0.07 mm over all other axial
locations, thereby creating a deformation-shape which is uniform except at
the two axial extremes.
The central spanwise deformation-shape evolutions for pinned-free
cases which did not snap away (R12T3S1, R6T3S1, and R12T2S1) were fully
symmetric with only negative deflections and rotations at the pinned edges.
An example of a central spanwise deformation-shape evolution for a pinnedfree case which did not snap away is given for specimen R12T2S1 in Figure
5.51. This evolution shows symmetric deformation-shapes for all values of
center deflection. Inflection points become obvious only for the larger values
of center deflection. The rotations and deflections at the boundaries are
negative for all center deflections, as further illustrated in the central
spanwise DFU evolution for specimen R12T2S1 in Figure 5.52.
Similar
-176-
Center Deflections in Millimeters
- e- -0
-0-
-
El- -1.2
--M- -2.4
-0.5
-- --4.6
R1 2T2S2
-
- 4
0"-E-
I-m
E
E
-E-
El
--
-Irn]
El.
C
0
C')
0
O
..
I
v
,
4C
aO
Cd
n
Q)
-- +---e-------
.4
-2
-3
-4
-511 I
-100
I
I
I
.
I
-50
I
I
I
I
I
JLJII
50
100
Axial Position (mm)
Figure 5.50 Predicted central axial deformation-shape evolution for
geometry R12T2S2 with pinned-free boundary conditions.
-177-
Center Deflections in Millimeters
- e- -0
-e-
E
E
--- -0.9
-0.2
0.5
O
01
0
O
>
a) -0.5
-1I
0
I
I I
I
20
.
I
40
l
I
60
,
I
.
80
100
Spanwise Position (mm)
Figure 5.51
Predicted central spanwise deformation-shape evolution for
specimen R12T2S1 with pinned-free boundary conditions.
-178-
Center Deflections in Millimeters
-M- - 0.9
- - -0.2
E
E
O
0
aci
-21
0
1
1
1
20
11
1
40
,
60
I
80
100
Spanwise Position (mm)
Figure 5.52 Predicted central spanwise DFU evolution for specimen
R12T2S1 with pinned-free boundary conditions.
-179-
behavior was exhibited by all the pinned-free cases which did not snap away:
R12T2S1, R12T3S1, and R6T3S1.
It should be noted that each of these
geometries exhibited stable type I force-deflection responses in the
experiments.
The central axial deformation-shapes were approximately uniform for
the three pinned-free cases which did not snap away: R12T3S1, R6T3S1, and
R12T2S1, as seen for the R12T2S1 geometry in Figure 5.53.
The
deformation-shapes develop a local minimum at the central loading point
with a local maximum on each side. The amplitude of these local extrema
increase with increasing center deflection. However, they remain a very
small percentage of the total axial length of the panel.
For instance,
specimen R12T2S1 showed a difference in vertical positions of 0.06 mm
between the local extremum at a center deflection of 1.3 mm. Thus, the
difference in vertical position is only 0.06% of the total axial length, thereby
creating an approximately uniform deformation-shape. The slopes at the
axial position of 0 mm were noted to be positive for all values of center
deflection. Due to the symmetry about the mid-axis, the slopes were always
of the opposite sign at an axial position of 102 mm. The same general
behavior is observed for the other pinned-free cases which did not snap away:
R6T3S1 and R12T2S1.
Predicted central spanwise deformation-shape evolutions for all
simply-supported-free and pinned-free cases are included in Appendix D
along with predicted central axial deformation-shape evolutions and
predicted central spanwise DFU evolutions for all cases.
-180-
Center Deflections in Millimeters
-
- -0.2
--- -0.9
R12T2S1
--
-
-- -
--
-
-- e-
-
-
. -E3
- -
-
E
E
C
0,,
O
_W
-
S "
u__
.
.
.-
0
ci
O
>
-1
-40
-20
20
40
Axial Position (mm)
Figure 5.53 Predicted central axial deformation-shape evolution for
specimen R12T2Slwith pinned-free boundary conditions.
-181-
5.3
Damage
The experimental damage results are presented in this section as x-ray
photographs taken of the damage in the plane of the specimen and
transcriptions of the damage of the cross-section as viewed through a
microscope. The x-ray results provide a through-thickness integrated view of
the planar damage shape at the loading point while the transcriptions
identify the through-thickness locations of damage. Since the panels are
sectioned along the central spanwise section, the through-thickness damage
state away from the loading point can also be examined along this section.
As described in Chapter 4, a damage testing program was performed
on similar specimens as those that exhibited damage in the original
deflection test. Only one specimen, R6T3S3, showed damage in the deflection
test, thus providing the only opportunity for an examination of the damage
progression with increasing center deflection. Separate damage tests were
then conducted for this set of specimens up to center deflections of 10.9 mm,
18.9 mm, and 23.4 mm.
The R6T3S3 specimen from the deflection test showed both matrix
cracking and delamination damage when loaded to a center deflection of 27.7
mm, as seen in the x-ray photograph results of Figure 5.54. A 70 mm long
matrix crack in the +450 direction is approximately centered around the
loading point.
A delamination, also centered around the loading point,
extends along this +450 matrix crack for 13 mm. In addition, matrix cracks
along the 00 and -450 direction can be seen leading away relatively
symmetrically from each side of the delaminated region. The transcription of
this same specimen, given in Figure 5.55, shows a matrix crack which
extends through the lower three +450 plies directly under the loading point.
-182-
Figure 5.54
X-ray photograph for specimen R6T3S3 tested to a center
deflection of 27.7 mm.
-183-
Load
+450
+450
-450
-450
-450
Delamination
Matrix Crack
00
00
o
00
00
00
-450
-450
-450
1 mm
+450
+450
+450
Figure 5.55
Sectioning transcription of specimen R6T3S3 tested to a center
deflection of 27.7 mm.
-184This crack intersects a delamination at the interface with the lower -450 ply
group and this delamination extends spanwise 1.0 mm in both directions.
This specimen achieved a peak force of 888 N. It should be noted that this
and all other R6T3S3 specimens attained a peak force at the critical snapping
load.
The test to a center deflection of 10.9 mm (critical snapping
displacement) and a critical snapping load of 870 N did not produce any
damage. However, the tests to 18.9 mm and 23.4 mm did produce detectable
amounts of damage and their damage results are given in Figures 5.56 to
5.58.
The x-ray photograph for specimen R6T3S3 tested to a center
deflection of 18.9 mm is given in Figure 5.56. This specimen had a peak force
of 960 N which occured at the critical snapping load. A matrix crack of 85
mm length is located symmetrically about the loading point in the +450
direction as can be seen in the x-ray photograph. A delamination extends
along the +450 matrix crack for 10 mm with matrix cracks in the 00 and -450
directions leading away from the edges of the delaminated region, all with
similar symmetry of orientation as described for Figure 5.54.
The
transcription for this R6T3S3 specimen tested to a center deflection of 18.9
mm, given in Figure 5.56, shows a matrix crack which extends through the
lower three +450 plies directly under the loading point. This crack intersects
a delamination at the interface with the lower -450 ply group and extends 0.5
mm in both directions. This damage pattern is very similar to that seen for
the test to a center deflection of 27.7 mm, although the length of the
delamination is less in this case.
The x-ray photograph results for specimen R6T3S3 tested to a center
deflection of 23.4 mm is given in Figure 5.58. This specimen had a peak force
-185-
Figure 5.56
X-ray photograph for specimen R6T3S3 tested to a center
deflection of 18.9 mm.
-186-
Load
+450
+450
-450
-450
-450
Delamination
I
Matrix Crack
00
00
00
00
00
00
-450
-450
1 mm
-450
+450
+450
+450
Figure 5.57
Sectioning transcription of specimen R6T3S3 tested to a center
deflection of 18.9 mm.
-187-
Figure 5.58
X-ray photograph for specimen R6T3S3 tested to a center
deflection of 23.4 mm.
-188of 955 N which corresponded to the critical snapping load.
The x-ray
photograph shows a matrix crack 110 mm long which extends symmetrically
about the loading point in the +450 direction. No delamination or further
matrix cracking was observed for this specimen. The transcription did not
show any damage and is, therefore, not included.
-189-
CHAPTER 6
DISCUSSION
The objective of this investigation was to gain an understanding of the
effects and mechanisms associated with snap-through buckling and their
relation to the overall structural response and damage development of
realistic fuselage panels. Specifically, a better understanding of the complex
deformation-shapes which develop are sought along with the identification of
damage incipience and development with respect to the primary regions in
the force-deflection response.
Experimentally determined deformation-
shapes are useful in assessing deflection-functions for use in Rayleigh-Ritz
type analyses. In addition, it is desired to determine the regimes where
damage occurs in order to judge if and how the current understanding of
plate impact damage can be utilized for such shells.
These issues are
addressed in this chapter based on observations of the experimental and
numerical results presented in Chapter 5.
6.1
Comparison of Experimental and Predicted Results
The force-deflection response was investigated through both
experimental and numerical studies as outlined in Chapters 3 and 4.
Grooved inserts were used in the test fixture in the experiment to provide
spanwise in-plane restraint while minimizing the resistance to rotation along
the axial edges. Two different "ideal" boundary conditions were utililzed
along the axial edges in the numerical analysis: perfectly free rotation along
-190with either perfectly rigid (pinned) or perfectly compliant (simply-supported)
in-plane restraint. The conditions for both the experiment and numerical
analysis were perfectly free along the circumferential edges of each shell.
The predicted simply-supported-free responses match the experimental
type I, i.e. smooth and stable, force-deflection responses well. An example of
this is shown in Figure 6.1 where the experimental and predicted forcedeflection responses for specimen geometry R6T3S1 are plotted on the same
set of axes for comparison. A similar correlation between the experimental
and simply-supported-free responses can be made for all type I specimens.
The experimental response follows the simply-supported response very
closely for smaller deflections and eventually dips below the predicted
response for larger center deflections. This behavior is seen for specimen
R6T3S1 in Figure 6.1 where the experimental and simply-supported-free
responses match well up until a center deflection of 2 mm. This deviation
may be due to the specimen pulling away from the grooved boundary
conditions upon developing tensile membrane stresses or simply due to
softening behavior from compressive membrane stresses brought about by
the in-plane restraint. While tensile membrane stresses are known to
produce a stiffening effect in the bending response, the presence of
compressive membrane stresses generally produces a destabilizing or
softening effect [19].
The pinned-free force-deflection responses match the experimental
type III responses very well up to the onset of the instability region. As
described in Chapter 5, the type III responses have an instability along with
a discontinuity or "load-drop" in the response while all the predicted
responses show smooth and continuous behavior. The experimental and
predicted force-deflection responses are given for a representative case,
-191-
1000
750
z
500
0
_J
250
0
1
2
3
4
5
Center Deflection (mm)
Figure 6.1
Experimental and predicted force-deflection responses for
specimen R6T3S1.
-192R6T1S2 in Figure 6.2.
This and all other type III responses match the
pinned-free results up to the critical snapping load. However, at some point
in the instability region, a load-drop in the experimental response causes a
deviation from the continuous pinned-free response.
The pinned-free
response does not, therefore, fully characterize the experimental type III
response throughout the instability region.
However, the pinned-free
response does predict the attainment of a force of zero within the instability
region which is also seen in each experimental type III response. A typical
illustration of this behavior is seen in the force-deflection response of
specimen R6T1S2, shown in Figure 6.2. Here, the experimental and pinnedfree responses match well until a load-drop occurs within the instability
region of the experimental response at a center deflection of 4.5 mm. Both
responses then proceed with different slopes along their respective instability
paths until attaining forces of zero as they snap-away from the indentor.
This deviation of the experimental and pinned-free responses after the loaddrop is likely due to the attainment of an unsymmetric deformation-shape in
the experimental response while the predicted pinned-free configuration
remains symmetric. Differences in the deformation-shapes result in different
effective structural stiffnesses in this region. This relationship between the
force-deflection response and the deformation-shapes is further discussed in a
subsequent section.
The remaining specimens, which exhibited experimental type II forcedeflection responses, i.e. smooth with an instability, were bounded above by
the predicted pinned-free response and below by the predicted simplysupported-free response. The experimental and predicted force-deflection
responses of specimen geometry R6T2S2, given in Figure 6.3, are illustrative
of the general comparisons made for these type II responses. Neither
-193-
250
200
Z
150
-a
0d
_j
100
50
0
10
Center Deflection (mm)
Figure 6.2
Experimental and predicted force-deflection responses for
specimen R6T1S2.
-194-
1000
750
Z
500
0
-j
250
0
Figure 6.3
10
5
Center Deflection (mm)
15
Experimental and predicted force-deflection responses for
specimen R6T2S2.
-195predicted response fully matches the experimental response of these
specimens. The experimental response is similar to the predicted pinned-free
response in the sense that they both possess an instability, although the
experimental critical snapping load is clearly less than that of the prediction.
Furthermore, the experimental response reaches the snap-through well at a
positive force and proceeds onto the second equilibrium path whereas the
predicted pinned-free response attains a force of zero within the instability
region as it begins to snap-away from the indentor. It is interesting to note
that the displacements at the experimental critical snapping load for all
specimens with a type II response are within roughly 25% of the predicted
pinned-free values whereas the experimental critical snapping loads deviate
from the predicted pinned-free values by as much as 80%. This can be seen
for specimen R6T2S2 in Figure 6.3 where the critical snapping displacements
for the experiment and the prediction (pinned-free) are 4.5 mm and 4.8 mm,
respectively, thus giving only a 6.7% difference while the critical snapping
loads from the experiment and the prediction (pinned-free) are 260 N and 540
N, respectively, thereby producing a 52% difference.
In order to characterize and quantify the degree to which any given
experimental response matches the predicted responses, a "degree-of-pinned"
parameter, X, is developed. This parameter is a measure of how close the
experimental behavior is to the pinned-free response relative to the simplysupported-free response. The parameter is computed by taking the ratio of
the difference between the experimental critical snapping load (PD) and the
predicted simply-supported-free load (Ps) at the corresponding critical
snapping displacement of the predicted pinned-free response and the
difference between the critical snapping load of the predicted pinned-free (Pp)
response and Ps:
-196P -P
SPD - PS
PP - PS
(6.1)
The parameter is, therefore, a ratio of the relative differences of the
experimental and predicted pinned-free critical snapping loads as compared
to the corresponding simply-supported load. These loads are illustrated for
the case of the response of specimen R6T2S2 in Figure 6.4. A value of k near
one indicates that the experimental response closely follows the predicted
pinned-free response, whereas values of k near zero indicate that the
experimental response behaves more like the predicted simply-supported-free
response.
The k values for each specimen geometry are listed in Table 6.1 and
the variation of the value of k with experimental response type is shown
graphically in Figure 6.5. The values of k for specimens with a type I
response are close to zero and are generally the smallest whereas the values
of k for specimens with a type III response are the largest and are close to
one. Specimens with a type II response show intermediate values of k. The k
parameter, therefore, characterizes the general nature of the experimental
responses with respect to the predicted responses. In general, the thicker,
more shallow specimens have small values of k whereas the thinner, deeper
specimens show larger values of k . The relationship between k values and
specimen geometries is further explored in a subsequent section of this
chapter.
6.2
Deformation-Shape Behavior
Examination of the deformation-shape behavior gives further insight
into the snap-through process including the load-drop behavior observed in
-197-
1000
V
_0
-J
Cz
0
1
0
Figure 6.4
5
10
Center Deflection (mm)
15
Illustration of the important forces in the definition of the
"degree-of-pinned" parameter X.
-198-
Table 6.1
Values of the parameter , for all specimensa
T2
T1
a
T3
Span
R6
R12
R6
R12
R6
R12
S1
0
0
0.05
0
0
-
S2
1
0.73
0.40
0.17
0.15
0.13
S3
0.80
0.77
0.75
0.37
0.41
0.15
"_"
indicates that an instability was not observed in any of the
experimental or predicted responses.
-199-
1
0.8
I
0.6
0.4
S
0.2
$
0
4
4
I
I
II
III
Experimental Response Type
Figure 6.5
Variation of X with experimental force-deflection response
types I, II, and III.
-200some of the force-deflection responses.
As discussed in Chapter 5, the
deformation-shapes along the central spanwise and axial stations adequately
characterize the deformation-shapes
for the entire panel.
The key
characteristic of the experimental central spanwise deformation-shapes is the
symmetry or lack thereof. The state of symmetry is more readily quantified
by considering the central spanwise deflection-from-undeformed-shapes
(DFU) evolutions. These shapes show a point of maximum deflection at the
midspan for fully symmetric spanwise deformation-shapes whereas
unsymmetric deformation-shapes show a migration of the point of maximum
deflection from the midspan location. The spanwise location of this point of
maximum deflection can therefore be used to quantify the symmetry of the
deformation-shapes along this central spanwise section.
A "degree-of-
unsymmetry" parameter, 8, is defined as the distance the point of maximum
deflection, as detected on a central spanwise DFU, migrates away from the
midspan location, normalized by the panel halfspan:
3=
S/2
(6.2)
This is illustrated in Figure 6.6 where x represents the migration distance of
the point of maximum deflection.
Fully symmetric central spanwise
deformation-shapes would, therefore, have a value of 8 of zero since the point
of maximum deflection remains at the midspan location. Since all predicted
deformation-shapes were fully symmetric for both sets of boundary
conditions, this characterization is not necessary for these cases.
The change in the state of symmetry with increasing center deflection
during a test, i.e. the "8-deflection response", can be easily examined.
In
order to directly compare the behavior of the central spanwise deformationshapes and the force-deflection response, the 8-deflection response and the
-201-
halfspan = S/2
E
0
0
-5
0
-10
I
I
-X
I
-15
c)
00
-20
-
-25
-30
)
50
100
150
200
250
300
Spanwise Position (mm)
Figure 6.6
Illustration of the important measurements in the definition of
the "degree-of-unsymmetry" parameter 5.
-202-
force-deflection response are plotted on the same set of axes. These plots
include a partial gridline to indicate the origin (zero-value) of the 8 axis. An
example of such a plot for a specimen with a type I force-deflection response,
R6T3S1, is given in Figure 6.7. The value of 8 is seen to remain very close to
zero throughout the test, thus indicating that the central spanwise
deformation-shapes remained fully symmetric. Such behavior is typical for
all specimens with a type I or type II force-deflection response. An example
of this for the case of specimen R6T2S2 is shown in Figure 6.8. Again,
symmetric central spanwise deformation-shapes are evident as the value of 8
remains near zero.
However, the lack of symmetry in the central spanwise deformationshapes becomes evident for specimens with type III force-deflection
responses, as seen for specimen R6T1S2 in Figure 6.9. Here, 8 is seen to have
values very near zero until, at the center deflection immediately after the
load-drop, the 6 value increases noticeably. The value of 8 is observed to
increase to 0.10 for specimen R6T1S2 at a center deflection of 4.5 mm which
corresponds to the first set of deformation-shapes recorded after the small
load-drop in the force-deflection response at a center deflection of 4.1 mm. In
this particular case, the value of 8 increases further to a maximum value of
0.22 at a center deflection of 5.7 mm after another larger load-drop in the
force-deflection response. This load-drop occurred as the stroke was resumed
after taking the deformation-shape data at the center deflection of 4.5 mm.
Any relationship between the load-drop and 8 are not, therefore, reflected
until the deformation-shapes are again measured at the next hold in the
stroke position, in this case at a center deflection of 5.7 mm. The increase in
the value of 8 is seen to correspond to the load-drops in the type III forcedeflection response for this and all other such specimens, thus indicating that
-203-
1
2000
1600
0.5
-1200
0 c
-. 800
-0.5
400
-1
0
Center Deflection (mm)
Figure 6.7
Force-deflection and 8-deflection responses for specimen
R6T3S1.
-204-
750
0.5
500
0
..O1
.J
250
-0.5
-1
5
0
Center Deflection (mm)
Figure 6.8
Force-deflection and 8-deflection responses for specimen
R6T2S2.
-205-
1
250
200
V
0.5
150
0')
0
.j
100
-0.5
50
0
-1
5
Center Deflection (mm)
Figure 6.9
Force-deflection and 8-deflection responses of specimen
R6T1S2.
-206the load-drop is caused by the switching from a symmetric to an unsymmetric
state of deformation. The nonzero values of 8 generally remain as the forcedeflection response proceeds along the instability path and the value of 8
returns to zero as the panel snaps away, indicating that this final inverted
configuration is spanwise symmetric. This is observed for specimen R6T1S2
at a center deflection of 12.7 mm. Other specimens showed one distinct loaddrop and the maximum 8 value is, therefore, reached in these cases at the
center deflection immediately after the load-drop.
The formation of unsymmetric central spanwise deformation-shapes is
the primary difference between the experimental and predicted deformationshape results for specimens with a type III force-deflection response. These
deformation-shapes are clearly related to the load-drops in the forcedeflection responses. Although the difference between the predicted (pinnedfree) and experimental type III force-deflection responses beyond the loaddrop involved only a change in slope, the deformation-shapes are
fundamentally different.
It is, therefore, not adequate to completely
characterize the structural response with the force-deflection response, as is
often done for plate geometries.
The possibility of unsymmetric spanwise
deformation-shapes exists for any real specimen and, hence, for any real
structure. Changes in stress distributions and elastic behavior resulting from
unsymmetric deformation-shape behavior can therefore become important
issues in determining the damage resistance of shell structures.
Finite
element analyses which assume a perfect structure are not able to capture
such unsymmetric behavior.
The effects of initial imperfections in the
structural configuration and in the loading should therefore be considered in
further numerical work to investigate the stress states of the entire panel for
unsymmetric deformation-shapes.
Furthermore, this would indicate the
-207locations of maximum stress and, hence, possible damage sites which may
occur away from the loading point during the formation of unsymmetric
deformation-shapes.
The deformation-shapes along the central axial section were also
examined to gain insight into the snap-through process. A slightly different
means of quantification is utilized for the central axial section as compared to
the central spanwise section since the deflection is not constrained to be zero
at the extreme axial locations. The deformation-shapes along the central
axial section were seen, in Chapter 5, to develop symmetric, unsymmetric,
and asymmetric shapes.
Unsymmetric shapes are those which are not
symmetric about the mid-axis line and the asymmetric shapes observed in
this investigation are those which have a constant and nonzero slope. In
order to characterize such deformation behavior, the rotations of the axial
section at each end are computed by taking the inverse tangent of the ratio of
the difference in vertical positions at the central and extreme axial stations
(left and right) and the axial distance between the same two stations:
OL
rZ1
= tan- 1
- Z3
x3 - X1
(6.3)
tan-1
5 - Z3
(6.4)
OR=
(x - x3
where the subsripts on z and x refer to the axial locations, as defined in
Figure 6.10. The development of nonzero positive axial rotation angles OL and
OR indicates that the deformations are greatest at the central axial station, as
seen in the deformed data of Figure 6.10. Small axial rotation angles (less
than 0.10 for specimens in this investigation) indicate that the panel is
deforming along the central axial section in an approximately uniform and
-208-
E
14
-
Undeformed
(
Q
Q
o
E
N
o
12
.
I
SI
SI
10
I
+0
M~
0Z
I
1!
I
I
I
I
II
I
I
~
I
I
I
I
8
I
I
II
I
I
I
I
. I. . . . I . . .
I
SI
I
I
I
. .. I . . . I . . . . I .
I
I
I
Axial Stations
I
-!
--
Defcrmed
II
I
I
I
I
0)
-
Axial Stations
6
T, ,i,
-150
,
I, ,,
-100
,
,,,
.I
-50
,
. .
. .
I . .
50
. .
I .
100
.
. .
150
Axial Position X (mm)
Figure 6.10
Geometric illustration of the axial rotation angles used to
characterize the deformation-shapes along the central axial
section.
-209flat manner. Axial rotation angles of opposite sign indicates that the panel
has taken on an asymmetric deformation-shape.
The axial rotations are
quantified with respect to the center deflection in order to examine the
relationship with the force-deflection response.
This is accomplished by
plotting the axial rotation angles versus center deflection on the same set of
axes as the force-deflection response, thus allowing a direct visual
comparison.
For additional clarification, a partial gridline is included to
indicate the origin of the 0 axis.
The axial rotations show that symmetric and approximately uniform
axial deformation-shapes develop for specimens with a type I force-deflection
response.
The force-deflection and 6-deflection responses for specimen
R6T3S1, which displayed a type I force-deflection response, is shown in
Figure 6.11. The symmetric deformation-shape behavior of specimen R6T3S1
along the central axial section is typical of all specimens with a type I forcedeflection response except R6T3S1 and R12T1S1. The values of OL and OR are
observed to be very close to zero with very little variation as the center
deflection is increased, thus indicating that the axial deformation remained
largely uniform and flat.
Some specimens with a type II force-deflection response show the
development of symmetric localized deformations along the central axial
section. The force-deflection and 6-deflection responses for specimen R6T2S2,
which displayed a type II force-deflection response, are given in Figure 6.12.
The left and right axial angles are seen to both develop nonzero values for
center deflections along the first equilibrium path of the force-deflection
response.
This indicates a more localized deformation at the mid-axis
location. The axial angles again reach values very near zero as the forcedeflection response transitions onto the second equilibrium path.
This
-210-
70
-O
1000
2
750
1
500
0 0(0)
250
-1
0
-2
0
.1
Center Deflection (mm)
Figure 6.11 Force-deflection and 0-deflection responses of specimen
R12T3S2.
-211-
1000
'
Load
o
3
' '
R6T2S2
SOLR
750
V
"0-
2
6L
-*
500
1 O(o)
LI
0
-J
il
ELI
i
250
0 ,-
060)
E l I
i
M- 0
m
u-i
[]-
4
10
0
, -1
15
Center Deflection (mm)
Figure 6.12 Force-deflection and 0-deflection responses for specimen
R6T2S2.
-212indicates that the central spanwise deformation-shape has fully propagated
to the free circumferential edges, thus giving an approximately uniform axial
deformation-shape.
It should be noted that unsymmetric central axial
deformation-shapes were observed for two specimens with a type II forcedeflection response: R6T1S1 and R12T2S3. These specimens showed axial
rotations which changed independently, thereby creating unsymmetric
shapes.
The deformation-shape behavior for responses of types I and II along
the central axial section gives insight into the general distribution of
compressive membrane stresses in these specimens. The specimens with a
type I force-deflection response showed very uniform and flat central axial
deformation-shapes.
Previous observations on the "degree-of-pinned"
parameter X for these specimens suggest that they exhibited a response
which is characteristic of a simple-support condition, i.e. these specimens had
very little restraint in the spanwise direction and did not, therefore, develop
an instability. As a result, these specimens probably developed only small
and relatively uniform compressive membrane stresses in the spanwise
direction. Compressive membrane stresses in the spanwise direction would
have a "softening" effect in the bending response [19], thus creating larger
transverse deflections at axial locations with higher compressive stresses.
The lack of a significant variation of compressive membrane stresses in the
axial direction would, therefore, explain the relatively flat deformationshapes in the axial direction for these specimens. However, the values of X
for specimens with a type II force-deflection response suggest that these
specimens exhibited a more pinned response, i.e. they displayed an
instability. This suggests that sufficient spanwise restraint existed to create
the compressive membrane stresses necessary to produce such an instability.
-213Furthermore, since the loading is applied at the mid-axis location, the
compressive membrane stresses are likely highest at this point. This would
lead to a greater softening of the bending response at the mid-axis location
and, hence, larger transverse deflections at this location.
Such localized
deformations were seen along the central axial section for all specimens with
a type II response at center deflections along the first equilibrium path in the
force-deflection response. Once the force-deflection response passes into the
instability region, the magnitudes of both axial angles are observed to
decrease with increasing center deflection, indicating that the central
spanwise deformation-shape is propagating toward the free edges. This may
suggest that the region of large compressive membrane stresses is also
propagating axially toward the free edges, causing the specimen to soften in
the spanwise direction at points away from the mid-axis. The central axial
deformation-shape generally flattens out upon reaching the second
equilibrium path, suggesting that the compressive membrane stresses are
approaching zero values along the axial direction as the panel attains the
inverted configuration.
It should be noted that these membrane stresses
would eventually become tensile on the second equilibrium path if the
specimens were prevented from pulling away from the grooved boundary
surface.
These relationships between the deformation-shapes and the
distribution of membrane stresses should be explored with experiment and
analysis in future work.
Specimens with a type III force-deflection
response showed
unsymmetric and asymmetric central axial deformation-shapes.
The
deformation-shapes for small center deflections were typically unsymmetric
with the development of localized deformations at the mid-axis.
Typical
force-deflection and 0-deflection responses for a specimen which exhibited a
-214type III force-deflection response, specimen R6T1S2, are given in Figure 6.13.
As can be seen here, the left and right axial angles (OL and OR) initially take
on nonzero values but remain somewhat similar indicating a somewhat
unsymmetric deformation-shape on the first equilibrium path. As discussed
for the type II responses, this localized deformation may be due to the
differential softening in the axial direction produced by a variation of
compressive membrane stresses in this direction.
After passing through the load-drop on the corresponding forcedeflection response, the axial angles then diverge significantly with one
typically taking on a negative value and the other remaining positive, thus
indicating an asymmetric deformation-shape. This also corresponds to the
development of unsymmetric central spanwise deformation-shapes.
The
asymmetric central axial deformation-shapes remain as the force-deflection
response proceeds through the instability region. This can be seen in the left
and right axial angles remaining of opposite sign for the remainder of the
test. However, the two angles both generally decrease in magnitude as the
panel snaps away from the indentor as seen for specimen R6T1S2 in Figure
6.13 where the fully asymmetric shape, indicated by OL and OR having roughly
equal and opposite values (1.530 and -1.200, respectively), is attained at a
center deflection of 7.9 mm.
As mentioned previously, a force of zero was attained in all type III
force-deflection responses in the instability region as the panel snapped
away. When this occurs, the axial deformation-shapes generally show an
asymmetric shape with a smaller slope, i.e. the magnitudes of OL and OR
become smaller but remain of opposite sign. This is seen in Figure 6.13
where the force-deflection response reaches a force of zero in the instability
-215-
o Load (N)
[0.
250
2
200
1
150
-o
0
CI
0
(0)
100
-1
50
S15 - 2
15
0
Center Deflection (mm)
Figure 6.13 Force-deflection and e-deflection responses for specimen
R6T1S2.
-216region and the panel snaps away, with the values of OL and eR converging to
0.630 and -0.280, respectively.
A similar characterization was conducted for the predicted central
axial deformation-shapes. However, since all predicted deformation-shapes
were fully symmetric, only one axial rotation angle 0 was necessary.
The
behavior for all simply-supported-free responses was similar to that seen for
the experimental type I responses, i.e. all axial rotation angles remained very
small throughout the test, as seen for specimen R6T3S3 in Figure 6.14.
However, the axial angle behavior was quite different for the predicted
pinned-free cases, as seen for specimen R6T3S3 in Figure 6.15. The axial
angle is seen to increase for center deflections along the first equilibrium
path. This is followed by a decrease in axial angle as the pinned-free forcedeflection response transitions into the instability region.
The critical
snapping displacement for the force-deflection response corresponded within
15% to the deflection at which the axial angles began to decrease for all
pinned-free specimens.
The center deflection at which a force of zero is
reached in the instability region always corresponded to the recovery of small
axial angles (typically less than 0.10) and, hence, an approximately uniform
deformation-shape.
This overall behavior was similar to that seen
experimentally for specimens with type II force-deflection responses.
However, the final deformation-shapes for some pinned-free cases showed
nonuniform deflections at the extreme axial positions, as mentioned in
Chapter 5. It is unclear whether this is an artifact of the numerical analysis
or if this is a physically realistic phenomenon.
Further work should be
performed to investigate the causes of these nonuniform results.
The relationships established here between the central axial
deformation-shapes and the force-deflection responses indicate that the
-217-
. . . . . . . . . . . . . .
5
-- e--Load
R6T3S3
2500
2000
Z
1500
3m
0
-JO
_1
-
0 e0()
1
U
0_1
U
1000
500
-oa
-G
0
-
rE
0
,
I
,
I
10
,
,
,
I
I
20
I
I
I
I
I
I
I
30
I
I
I
1-5
40
Center Deflection (mm)
Figure 6.14 Predicted force-deflection and 8-deflection responses for
specimen R6T3S3 with simply-supported-free boundary
conditions.
-218-
5000
. .
.
----- Load
4000
3000
F
0 ][
V
0
R6T3S3
f-
-
-j
5
. . . . . . . . . . . . . . . .
0
0 0(0)
2000
1000
S
0
0
5
10
15
20
25
, -5
30
Center Deflection (mm)
Figure 6.15 Predicted force-deflection and 6-deflection responses for
specimen R6T3S3 with pinned-free boundary conditions.
-219support along the circumferential edges may significantly affect the forcedeflection response.
investigation.
Only free conditions were considered in this
However, it is likely that any restraint along these edges
would inhibit deformations at axial locations away from the loading point.
The stresses which resist this deformation could produce a stabilizing or
destabilizing effect depending on if they become tensile or compressive,
respectively [19].
Therefore, the overall response, including snap-through
buckling characteristics, are expected to change as the support along the
circumferential edges change and this should be further explored with both
experiment and numerical analysis.
6.3
Importance of Geometric Parameters
As suggested in the previous section, the characterization of the
experimental force-deflection responses with the parameter X shows a
geometrical dependence. Trends with X and the representative geometrical
parameters of radius, span, and thickness are obvious from Table 6.1. For
example, the value of k increases as the span is increased (radius and
thickness held constant), as the thickness is decreased (span and radius held
constant), and as the radius decreases (span and thickness held constant).
These trends are shown graphically by plotting k versus the intermediate
values of span (S2) and thickness (T2) in Figures 6.16 and 6.17, respectively.
These plots are representative of the trends seen for all values of span and
thickness with the one exception that the trend of increasing values of X with
increasing span is not observed for the R6T1S3 specimen. The trends with
radius can be seen in Figures 6.16 and 6.17 by.considering the values of X for
any constant x-axis location, i.e. holding the span and thickness constant.
-220-
1
U
-
S2
o R12
o R6
0.8
o
0.6
0.4
0.2
0
o
8
-
Thickness (mm)
Figure 6.16
Variation of X with thickness and radius for a constant span
S2.
-221-
1
T2
R12
o R6
0
0.8
0.6
0.4 0.2
0-0
0
100
200
300
400
Span (mm)
Figure 6.17
Variation of X with span and radius for a constant thickness
T2.
-222The values of X are always greater for the smaller radius (R6) under these
conditions. A clear and consistent dependence of X on these geometrical
parameters is evident from the trends shown in Figures 6.16 and 6.17.
These geometrical trends can be more generalized by considering the
relationship between the depth or "height" of a specimen, as described in
Chapter 4, and its span (S) and radius (R).
From simple Pythagorean
relationships, the height h is given by:
h =R
2
S
(6.5)
4
This relationship shows that the height increases with increasing span and
with decreasing radius. Identical trends with span and radius were observed
for k, suggesting that the height may also be an important geometrical
parameter. This observation, along with the previously discussed trend of
increasing X with decreasing thickness, leads to the consideration of a single
geometrical parameter, h/T.
The appropriateness of using the h/T parameter can also be justified by
considering the simpler two-dimensional case of a pinned isotropic arch of
rectangular cross-section subjected to a concentrated transverse load at the
midspan, as shown in Figure 6.18.
This is representative of a two-
dimensional approximation of the geometry in this work.
The full
development of this solution can be found in [39]. The key assumptions and
result are given here. Upon loading, the compressive force H along the arch
is assumed constant. This results in the following relation for the shortening
of the arch:
( HS)
1
AE
2
dy\2
0
ix
dx
dy
-
I(x
Jdx
2 0 dx
(6.6)
-223-
Load
S=Span
Figure 6.18
Illustration of arch with perfectly pinned boundary conditions.
-224-
where y and Y2 are the equations of the centerline for the undeformed and
deformed shape, respectively. The symbols A, E, and S represent the crosssectional area, elastic modulus, and span of the arch. The existence of a limit
point in the force-deflection response, i.e a critical snapping load, then
depends on the quantity:
n= Ah2
41
=3-
T
(6.7)
where h is the height, T is the thickness, and I is the moment of inertia of the
cross-section [39]. Arches with values of n less than one exhibit an entirely
stable response whereas arches with values of n greater than one show an
instability and a critical snapping load which increases with increasing n.
Since n is seen to have a direct dependence on h/T for a rectangular crosssection, it is a logical progression to explore the use of such a parameter for
the three-dimensional case of a convex shell panel. Therefore, further work
should be performed to establish similar parameters which account for the
varying planform of such three-dimensional panels.
Based on the experimental observations and the solutions of simplified
cases such as the isotropic arch, the variation of X with h/T was considered.
Values of h/T for each specimen are shown in Table 6.2. These values are
used to construct a graphical comparison of X with h/T for all specimens as
shown in Figure 6.19. The value of X generally increases with increasing h/T
although there is a slight scatter in the data. Specimen R6T1S3 has an h/T
value of 16 and a X value of 0.8 which places it in the upper right-hand
portion of Figure 6.19.
This indicates that the values of X may actually
asymptote to the value of one as the value of h/T is increased. In general, the
response type is observed to transition from type I to type II at a value of h/T
-225-
Table 6.2
Values of the parameter h/T for all specimens
T1
T2
T3
Span
R6
R12
R6
R12
R6
R12
S1
1.89
1.04
0.92
0.45
0.62
0.31
S2
6.94
3.94
3.77
2.08
2.35
1.24
S3
16
8.32
8.19
4.32
5.4
2.82
-226-
1
0.8 0.6
0.4
[l
o Type I
0.2
o Type II
0% ]
0
Figure 6.19
Type III
-0
0
-..,
,-
I I
5
I
10
h/T
I I I I I I,
15
I
20
Plot of experimental force-deflection response type with X and
h/T.
-227of approximately two and finally to type III at a value of h/T of approximately
six. All specimens with a value of h/T greater than six show type III forcedeflection behavior.
The use of h/T is, therefore, appropriate
for
characterizing the trends in the force-deflection behavior of these composite
shells in addition to isotropic arches. The key conclusion from this plot is
that the thicker, more shallow specimens (large h/T) exhibit a more "pinned"
behavior whereas the deeper, thinner specimens (small h/T) displayed a more
"simply-supported" response, as compared to the numerical results.
The effect of the parameter h/T on the force-deflection
and
deformation-shape behavior "types" is summarized in Table 6.3. This table
shows the trends of each behavior type with h/T. The smallest range of h/T of
zero to 1.3 corresponds to a type I force-deflection response (smooth and
stable) with fully symmetric deformation-shapes along the central spanwise
and axial sections. For a slightly larger range of h/T (1.9 to 5.4) the force
deflection response shows an instability (type II) while the deformationshapes remain fully symmetric along the central spanwise and axial sections.
However, the largest range of h/T (3.9 to 16.0) shows a non-smooth forcedeflection response which attains a force of zero within the instability region
(type III).
In addition, unsymmetric deformation-shapes develop at some
point during the test along both the central spanwise and axial sections.
Although there is a slight overlap of the second and third ranges of h/T, the
trends with this geometrical parameter and the importance of the parameter
are clearly evident.
-228Table 6.3
Range of
h/T
0
-+ 1.3
Characterization of Experimental Force-Deflection and
Deformation-Shape Behavior with h/T
Force-Deflection
Response Typea
Central Spanwise
Deformation-Shape
Typeb
Central Axial
Deformation-Shape
Typeb
I
S
Sc
1.9 -
5.4
II
S
Sd
3.9 -
16.0
III
U
U
a 'ii indicates that the response is smooth and stable.
"II" indicates that the response is smooth with an instability.
"III" indicates that the response is non-smooth with an instability.
b "S" indicates that the deformation-shapes were approximately symmetric.
"U" indicates that the deformation-shapes were partially unsymmetric.
c Specimens R12T1S1 and R6T3S1 showed unsymmetric behavior.
d Specimens R12T2S3 and R6T1S1 showed unsymmetric behavior.
-229-
6.4
Effects of Boundary Conditions
The rotational restraint exhibited along the axial edges by different
mechanical fixtures was investigated in preliminary testing, as indicated in
Chapter 4, to identify the most consistent boundary conditions for the quasistatic tests. Double knife-edge inserts and grooved inserts were both used in
preliminary testing to compare to previous work [5] on the snap-through
response of cylindrical panels. A representative specimen geometry (R2T1S1)
from this previous work was used to compare the structural response of the
different boundary conditions.
Recalling the nomenclature convention
established in Chapter 4, this specimen has a radius of 305 mm (12 in), a
span of 102 mm (4 in), and a thickness of 0.804 mm (0.032 in).
Experimental boundary conditions always produce some degree of nonideal behavior due to dissipative mechanisms such as friction or other
undesired performance. The ideally free rotation which is desired along the
axial edges is, therefore, inevitably restricted by some "resisting moment" to
free rotation caused by frictional forces acting through a nonzero moment
arm or by clamping forces used to restrict the out-of-plane motion.
In order to investigate the effect of rotational restraint, the knife edges
were tested under conditions of perfect alignment and also with a
misalignment of 1.6 mm (1/16 in) to provide a larger resisting moment, as
illustrated in Figure 6.20. No such adjustment was possible with the grooved
fixtures. In addition, the force-deflection response for specimen geometry
R2T1S1 from a previous study which used double knife-edge boundary
conditions [5], is also used for comparison. It was expected that the grooved
boundary conditions would provide the smallest resistive moment due to the
small frictional moment arm (on the order of the specimen thickness), as
-230-
12.7 mm,
Perfect Alignment
Upper Knife
Edge
Toward Loading
Point
Built-out
F,1ullEL
L
Laminate
Lower Knife
I
Edge
Misalignment = 1.6 mm
I
-
i
I
Upper Knife
Edge
I
I
I
I
I
I
I
I
I
I
a
Toward Loading
Point
Built-out
Wall
Laminate
Lower Knife
Edge
Note: Not to Scale
Figure 6.20
Illustration of the different alignments used with the double
knife-edge fixtures: (top) perfectly aligned and (bottom)
misaligned by 1.6 mm.
-231discussed in Chapter 4. The resistive moment produced by the misaligned
knife-edges is coupled with the moment due to a large frictional moment arm
of 13 mm. Therefore, the resistance to rotation is expected to be the least for
the grooved boundary conditions and the most for the misaligned knife-edges
with the others falling somewhere in between.
The force-deflection response differed significantly as the axial edge
conditions and, thus, the rotational conditions along the axial edges, were
changed.
The force-deflection responses under each different axial edge
condition are plotted in Figure 6.21. The initial response is observed to be
similar for all axial edge conditions in the current study, whereas the initial
response from the previous work [5] is approximately 25% stiffer. However,
the critical snapping load is observed to change with the axial edge condition.
The knife-edges produced the highest critical snapping loads of 154 N, 205 N,
and 198 N for the aligned condition, misaligned condition, and the unknown
alignment condition of the previous study, respectively. The case with the
misaligned condition and that of the previous study showed a smooth
response with a small instability region. The case with the aligned knifeedges showed a larger instability region with a somewhat "jagged" response,
i.e. continuous regions seperated by small discontinuities in the response due
to load-drops. As seen in Figure 6.20, the double knife-edge design resists inplane motion through normal forces at a wall which is 12.7 mm from the
point of rotation. As the panel rotates about the knife-edge contact points,
the panel must also slide along the wall. As a result of the finite normal
forces at the wall, frictional forces also exist which produce a resisting
moment by acting through the 12.7 mm moment arm. The jagged response
for the perfectly aligned knife-edges was visibly seen to correspond to the
panel "skidding" along the wall in a stepwise fashion. The slope of the
-232-
300
250
200
V
150
0
j
100
50
0
2
4
6
8
10
12
Center Deflection (mm)
Figure 6.21
Force-deflection responses for specimen R2T1S1 with various
conditions along the axial edges.
-233response is observed to change with each discontinuity, i.e. each time the
specimen skidded to a new point along the wall. This indicates that the
degree of rotational restraint is changing during the test as the panel skids
up the wall. The grooved fixtures produced a significantly smaller critical
snapping load of 125 N and an instability region which attained a force of
zero, thus indicating that the panel snapped away from the indentor during
the test. Small discontinuities in the grooved response were also observed in
the instability region. The significant changes observed in the responses with
these boundary conditions suggests that the effect of different boundary
conditions along all edges of the panel should be further explored.
The knife-edges were physically clamped onto the specimen in order to
ensure that the out-of-plane motion was prevented.
With any small
misalignments of these knife-edges, a clamping force or "pre-moment" is
introduced into the panel which may have changed the initial configuration
and, hence, the force-deflection response. The general behavior is, however,
still a useful comparison of the different fixtures. The results of Figure 6.21
show that an increased rotational restraint generally increases the critical
snapping load and decreases the magnitude of load reduction in the
instability region. In addition, the rotational restraint can inhibit the panel
snap-away phenomenon, as seen for all responses except for that with the
grooved edge conditions. As explained in Chapter 5, the attainment of a force
of zero within the instability region is an indicator that the panel lost contact
with the indentor and, hence, dynamically snapped-away, as seen for the
response with grooved edge conditions in Figure 6.21.
The central spanwise deformation-shapes were investigated for the two
extreme cases: misaligned knife-edges and grooves. The central spanwise
deformation-shape evolution for the misaligned knife-edges is shown in
-234Figure 6.22.
The deformation-shapes are observed to remain roughly
symmetric although a detectable eccentricity in the initial configuration is
carried throughout the subsequent deformation-shapes.
As previously
discussed, a misalignment of the knife-edges causes a "pre-moment" to be
applied to the specimen. If this "pre-moment" is different along each axial
edge, an eccentricity would be introduced into the initial configuration of the
specimen and this is suspected to be the cause of the eccentricity observed in
the initial configuration of Figure 6.22. In addition, the rotation at the edges
is clearly limited even for the larger center deflections. The central spanwise
deformation-shape
evolution for the grooved fixtures shows initially
symmetric deformation-shapes followed by the development of unsymmetric
deformation-shapes for larger center deflections. Furthermore, the specimen
undergoes a full rotation to the concave configuration at the edges, as seen in
Figure 6.23.
The rotational restraint exhibited by the misaligned knife-
edges, therefore, promotes the formation of fully symmetric deformationshapes in this case. This result has been previously shown for isotropic
arches with clamped boundary conditions, i.e. the resistance to rotation
provided by the clamped edges causes the deformation-shapes to remain fully
symmetric [66].
The effect of in-plane restraint was also considered.
A seperate
experimental study was not carried out to control the in-plane fixity.
However, a simple analysis of an isotropic arch can help explain the forcedeflection behavior observed in the main body of experiments. In this section,
the possibility of in-plane compliance of the test fixture is considered in the
arch analysis.
If the effective in-plane stiffness of the test fixture is lumped into a
spring of stiffness K, the conditions of the analysis can be represented as
-235-
E
E
4
Oc
0
0
Q.
C-
>i
-4
-8
20
40
60
80
100
Spanwise Position (mm)
Figure 6.22
Central spanwise deformation-shape evolution for specimen
R2S1T1 with misaligned knife-edge boundary conditions.
-236-
R2T1S1
E
E
4
_oo
O
0O
-0
-
00
OOCO
O0
OO
0 0
c
O,--
?ooo ooooooooooo
00
000000
0
0
oo
0o
0o
00000000000
(o
0000000
00
0
9
00
00
0
"-
t'
00000
-4
-8
20
40
0000
60
80
100
Spanwise Position (mm)
Figure 6.23
Central spanwise deformation-shape evolutions for specimen
R2T1S1 with grooved boundary conditions.
-237shown in Figure 6.24.
The full solution for such a situation is in [39] with
the key points of the solution method noted herein. The compressive force H
along the arch is again assumed constant but now the displacement at the
boundary due to the in-plane compliance is included in the consideration of
the shortening of the arch:
CHS)
-
AE
H
K
_
-
1 (dy
2
2
dx
2
-
2
dx
dx
(6.8)
The corresponding solution isgoverned by a controlling parameter m which
reflects the inclusion of the in-plane stiffness and is no longer purely
geometric:
m =3
T
I
AE
KS
(6.9)
where h is the specimen height, T is the thickness, S is the span, A is the
cross-sectional area, E is the elastic modulus, and K is the effective in-plane
stiffness of the boundary conditions [391.
As in the previous analytical
solution, a value of m less than one gives an entirely stable response for an
arch while geometries with values of m greater than one have an instability
with an increasing critical snapping load with increasing m.
The expression for m is composed of two parts: the parameter n from
the case of a pinned arch and a modifier term. Using equations (6.7) and (6.9)
gives the expression:
m=nLE
(6.10)
The bracketed term which multiplies n, or the "modifier term", is in a form
similar to that seen for the effective stiffness of two springs in series. Thus,
-238-
Load
Figure 6.24
Illustration of geometry of arch configuration including the
effective in-plane stiffness of the boundary conditions.
-239this configuration can be considered as two springs in series. One spring
constant is the effective stiffness of the arch and the other spring constant is
the effective stiffness of the boundary condition.
Specific trends can be
discerned as the stiffnesses of the two "springs", i.e. that of the arch and the
boundary conditions, are varied relative to each other.
If the in-plane
stiffness of the boundary conditions K is much greater than the extensional
stiffness of the arch (AE/S), then the modifier term approaches one and the
parameter m reduces to the one found in the previous section, i.e. a perfectly
pinned condition. If, however, the in-plane stiffness is vanishingly small
compared to the extensional stiffness of the arch, then the modifier term and
the parameter m both reduce to zero, thus guaranteeing an entirely stable,
i.e. simply-supported, force-deflection response. These trends are consistent
with the physical model of two springs in series. With an infinitely stiff
boundary condition spring, the arch spring is effectively pinned.
If the
boundary condition spring is infinitely compliant, then the arch spring is
supported as on a roller (simply-supported).
The behavior exhibited by the shells in this investigation are believed
to follow a similar trend. The change in the force-deflection behavior, i.e. the
degree-of-pinned behavior, with differing specimen geometry is due to both
the geometry and the relative stiffness of the specimen in relation to the inplane stiffness of the test fixture. The changes in the in-plane stiffness of the
arch, AE/S, with changing geometry can be examined by first considering
that the effective modulus in the spanwise direction (A 11/T) of the T1, T2, and
T3 configurations is identical since the same basic layup is considered in each
case. Therefore, modulus (E) does not vary from specimen to specimen and is
arbitrarily set equal to one for comparative purposes. The cross-sectional
area A is the thickness T multiplied by a unit width since this is a two-
-240dimensional approximation.
The AE/S term, therefore, reduces to the
specimen thickness divided by the span: T/S. The AE/KS component of the
modifier term thus varies as follows:
AE C- T
KS KS
(6.11)
The variation of the modifier term with specimen geometry clearly depends
on the relative magnitudes of TIS and K.
The variation of the experimental degree-of-pinned parameter k with
the geometric portion of the modifier term T/S, normalized by T1/S
1,
is
shown graphically in Figure 6.25 for all specimens. This seperates the effects
of the in-plane stiffness of the boundary conditions, K, from the geometrical
contribution of the specimen and thus allows the trends with panel geometry
to be examined seperately.
The k parameter generally decreases with
increasing TIS and is observed to asymptote to zero as T/S approaches
infinity, i.e. the response becomes more simply-supported as T/S becomes
larger.
This indicates that there is an interaction between the in-plane
stiffness of the panel with the in-plane stiffness of the test fixture which
affects the structural response in a manner similar to that seen for isotropic
arches with in-plane compliance in the boundary conditions. The values of k
for the specimens with the largest radius (R12 equal to 1.828 meters) are
generally the smallest for a given value of T/S. This trend is likely due to
the dependence of k on height, as previously seen in Figure 6.19, since
specimens with a larger radius of curvature and similar span have a smaller
height.
An examination of the specimens with extreme values of T/S illustrates
the interaction between the in-plane stiffnesses of the shell (T/S) and of the
boundary conditions (K). For example, a thin specimen with a large span,
-241-
1.2
R12
. R6
0
1.0
0.8
O1
0.6
0.4
0.2
0.0
I
I
I
I
I
II
I
I1
*
I I I
I
I
2
1
Normalized T/S
Figure 6.25
3
Plot of experimental degree-of-pinned parameter X versus
normalized ratio of thickness to span.
-242-
such as R12T1S3, would have the smallest value of T/S.
This specimen
showed a value of X of 0.77 which is somewhat close to one indicating that the
response largely exhibited pinned behavior. This suggests that the effective
in-plane stiffness of the boundary conditions K is much greater than this
panel stiffness thereby resulting in a mostly pinned response.
Thicker
specimens with the same span, such as specimen R12T3S3, have larger
values of T/S. Specimen R12T3S3 showed a value of X of 0.15 which indicates
that the response largely exhibited simply-supported behavior although this
particular response still displayed an instability region. This could reflect the
interaction between the two in-plane stiffnesses, K and T/S, which may have
similar magnitudes.
Specimens with the largest values of T/S, such as
R6T3S1, may have stiffnesses which are comparable or even greater than the
effective stiffness of the test fixture, thereby creating a fully simply-supported
response which is evidenced by the value of X of zero for specimen R6T3S1.
The trends in the experimental force-deflection responses indicate a clear
interaction between the in-plane stiffnesses of the shell and that of the test
fixture.
Since all real structures possess some degree of compliance, it is
expected that the in-plane support provided for such convex shell panels can
play a key role in determining the response of the overall structure.
For
example, in an aircraft fuselage structure a grid of structural elements,
commonly called frames and stringers, are used to provide support to the
cylindrical fuselage panels. This grid of stiffening elements allows the overall
structure to be considered as a grouping of small cylindrical shell panels with
boundary support provided by these stiffening elements.
The in-plane
compliance of such structures would need to be considered relative to the
stiffness of the panels themselves in order to adequately assess the structural
-243performance. Furthermore, if peak force is an applicable damage metric, the
effect of in-plane compliance on the damage resistance of such structural
configurations could be explored.
It is likely that the larger in-plane
compliances in the structural boundary conditions would increase the damage
resistance of a composite shell structure due to the attainment of smaller
contact forces and larger structural deformations. Further experimental and
analytical work regarding the effect of the in-plane stiffness of the structural
boundary conditions on the damage resistance of convex shells should,
therefore, be performed.
6.5
Damage
The damage progression observed for specimen R6T3S3 indicates that
damage incipience is in the form of a matrix crack along the +450 direction.
The x-ray photograph for the test to a center deflection of 23.4 mm shows a
110 mm long matrix crack along the +450 direction.
No other forms of
damage are seen in this x-ray photo suggesting that this matrix crack is the
incipient form of damage for this specimen. Other damage results showed
this matrix crack along with other forms of damage such as delamination and
matrix cracking in the 00 and -450 plies. However, no damage results showed
the existence of these other forms of damage without the existence of the +450
matrix crack which would support the conclusion that the +450 matrix crack
is the incipient form of damage for this specimen geometry.
The damage progression for specimen R6T3S3 shows the development
of matrix cracking and delaminations as the center deflection is increased.
For instance, the test with the smallest center deflection which produced a
detectable amount of damage (18.9 mm), showed a delamination area along
-244with matrix cracking in the 00, +450, and -450 directions. The 85 mm long
matrix crack in the +450 direction and the delamination (10 mm long and 2
mm wide) at the interface between the lower +450 and -450 ply groups are
also detected in the sectioning transcriptions. However, the other matrix
cracks are not. This may be due to the fact that the cracks are extending
from the edges of the delamination region where high stress concentrations
exist. It is also possible that these matrix cracks in the -450 direction interact
with the 00 plies to produce the matrix cracks in this direction as well. When
the center deflection is increased to 23.4 mm, only one large matrix crack of
110 mm in length is visible in the x-ray photograph.
The case of the
specimen tested to a center deflection of 27.7 mm shows damage, as viewed in
the x-ray photographs, in the form of delaminations and matrix cracks in the
00, +450, and -450 directions.
The matrix cracking in the -450 direction is
similar to that seen in the test to a center deflection of 18.9 mm. However,
the size of the delamination region is larger (14 mm long and 4 mm wide) and
the +450 matrix crack is smaller (70 mm long) than for the test to a center
deflection of 18.9 mm. The region of matrix cracking in the 00 plies (30 mm
long) is also smaller than that seen in the test to a center deflection of 18.9
mm.
Although the damage results are limited to that found in specimen
R6T3S3, several issues regarding damage incipience and development for
these shells can be discussed. For instance, the sectioning results showed a
matrix crack in the lower +450 ply group. This suggests that the failure is
due to tensile stresses caused by membrane and bending action at the loading
point.
The deformation-shape results show that the point of maximum
curvature and, hence, maximum bending stress is at the midspan location for
these specimens, which results in the tensile failure. As seen for plates with
-245-
tensile stresses due to transverse loading, transverse cracks branch at the
first ply mismatch interface to form a delamination [10]. This behavior is
also observed for the R6T3S3 specimen which further suggests that the
failure was due to tensile stresses caused by the transverse loading. The lack
of a clear and consistent evolution of the different subsequent damage modes,
i.e matrix cracking and delaminations, with increasing center deflection can
be explained by the large variability in damage results typically observed for
composite laminates [12].
Furthermore, the results tend to confirm that peak load is a critical
metric for these configurations [5]. The damage incipience point may have
been very close to the critical snapping load and a slight material variability
could account for the presence or absence of damage at this point. Continued
increase in the center deflection beyond that corresponding to the critical
snapping load results in a decrease in the force.
Thus, the peak force
experienced during the test remains the critical snapping load. Relatively
consistent damage resulted from the large range of center deflections (18.9
mm to 27.7 mm).
This suggests that center deflection is not likely a
controlling parameter. Therefore, the evidence suggests that peak force may
be the appropriate damage metric since the peak force remained similar for
each of these center deflections.
The fact that damage was not detected in these tests for the large
majority of specimen geometries suggests that the knowledge base currently
available for plate impact damage may be applicable for many of these
shallow geometries. As discussed in Chapter 4, these tests are conducted
under conditions of predominantly compressive membrane stresses. The
damage which may result upon further application of stroke would occur
under conditions of predominantly tensile membrane stresses, i.e. further
-246along the second equilibrium path. Previous work has shown that the
damage development for shells under these conditions is very similar to that
seen for plates [5] since shells in this state may be considered as plates with
large deformations.
Therefore, shells with geometries representative of
realistic fuselage panels may exhibit damage behavior very similar to that
seen for plates if the damage occurs near or beyond the critical snapping
displacement since these regions are likely to have tensile stresses at the
underside of the loading point caused by bending or membrane action.
Previous work has, however, shown that the damage development in shells
may be significantly different from plates if the damage incipience occurs at
center deflections which are small compared to the critical snapping
displacements [5]. Specifically, in that work damage occured on the top side
of the convex shell underneath the loading point for small center deflections.
From the deformation-shape data gathered in the current work, it is clear
that the panel configurations remain convex at smaller center deflections
suggesting that the primary stresses are compressive in this regime.
However, as the displacement increases, the deformations become somewhat
localized at the loading point, thereby creating a locally concave configuration
due to bending deformations as can be seen in Figure 5.15. This locally
concave configuration produces tensile bending stresses on the backside of
the laminate which likely account for the tensile failures observed in this
investigation.
The combined effects of bending and membrane action determine the
overall stress state and are therefore key in determining the damage
behavior of convex shells as the localized configuration beneath the loading
point changes from convex to concave. Further work should be conducted to
better pinpoint and understand this transition in damage behavior.
-247-
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
This work was conducted to investigate the mechanisms of snapthrough buckling and their relation to the overall structural response and
damage resistance of realistic composite fuselage panels, i.e. convex
cylindrical shells, under transverse loading.
To do this, structural
parameters such as radius of curvature, span, and thickness were varied
while contact forces and deformation-shapes
were
obtained via
experimentation and numerical analysis.
7.1
Conclusions
The following conclusions are drawn based on the results presented
and discussed in the previous chapters:
1.
The deformation-shapes which form can be fully three-dimensional and
can have both unsymmetric and symmetric components.
2.
Different experimental force-deflection responses occur depending on
the geometric characteristics of the shell, and can be classified into
three types: smooth-stable, smooth with an instability region, and
nonsmooth with a sharp instability.
-2483.
The predicted force-deflection responses with simply-supported
boundary conditions match the smooth-stable responses; the predicted
responses with pinned-free boundary conditions match the nonsmooth
responses with an instability; while the experimental responses of
smooth with an instability are bounded by the predicted results for the
two different boundary conditions.
4.
Load-drops in the experimental force-deflection response and
subsequent "panel snap-away" are associated with a switching between
symmetric and unsymmetric deformation-shapes.
5.
Numerical analyses which assume a perfect structural and loading
configuration cannot capture the load-drops in the force-deflection
response, the corresponding formation of unsymmetric deformationshapes, and the subsequent panel response prior to "snap-away".
6.
The experimental force-deflection responses generally approach the
predicted pinned-free results with increasing span, decreasing radius,
and decreasing thickness.
7.
The ratio of the panel height to thickness is a geometric parameter
which provides a clear characterization of the trends seen in the
experimental structural responses as the occurence of an instability in
the force-deflection
response, the formation of unsymmetric
deformation-shapes, and the panel snap-away phenomenon become
more likely as the value of this paramter increases.
-2498.
Increased rotational restraint in the boundary conditions along the
axial edges increases the critical snapping load, decreases the
magnitude of load reduction within the instability region of the forcedeflection response, and inhibits the formation of unsymmetric
spanwise deformation-shapes.
9.
The in-plane stiffness/compliance of the boundary conditions is a key
consideration in determining the structural response of convex
composite shells as the existence of finite in-plane compliance in the
boundary conditions reduces the overall stiffness of the structural
configuration, changes the overall response, and can inhibit the
formation of an instability region.
10.
The relative magnitudes of the in-plane stiffnesses of the shell and of
the boundary conditions can be combined in a manner similar to two
springs in series to determine the overall effective stiffness of the
structural configuration.
11.
The ratio of the panel thickness to span is a geometrical parameter
which shows the relative importance of the shell effective in-plane
stiffness to the in-plane stiffness of the boundary condition.
12.
Damage incipience for many convex shells with structural
configurations similar to actual fuselages may be similar to that
observed for plates if there is no damage prior to snapping through to
the inverted, i.e. concave, configuration.
-25013.
Damage development for convex shells with damage incipience at
center deflections near the critical snapping displacement may also be
similar to that seen for plates due to the localized concave
configuration which develops beneath the loading point.
14.
The utility of peak force as a damage metric is reinforced for convex
composite shells with the configurations considered in this work.
7.2
Recommendations
The following recommendations are made based on the results
presented and discussed in the previous chapters:
1.
The effect which eccentricities in the initial structural configuration
and in the loading have on the structural response characteristics
including the possible formation of unsymmetric deformation-shapes
should be investigated via numerical analysis and associated
experiments.
2.
The relationship between the distribution of compressive membrane
stresses and panel deformation-shapes should be further explored via
experimentation and analysis.
3.
The prediction of nonuniform deformations near the extreme axial
locations for some panels in this work should be further investigated
via numerical analysis and experimentation.
-2514.
Experiments and analyses should be performed to investigate the
effect of varying planform ratio on the structural response
characteristics as the planform ratio may modify the key geometric
parameters
5.
Tests should be performed and numerical analyses conducted to
investigate the effects of various boundary conditions along all edges
on the structural response and damage characteristics.
6.
The in-plane stiffness/compliance of test fixtures and actual
configurations should be quantified via experimental and numerical
techniques in order to further understand the role of the in-plane
stiffness of the boundary conditions in the overall structural response
and damage resistance.
7.
The stress state of the entire panel should be investigated in order to
determine the magnitudes and locations of maximum stresses and,
hence, possible damage sites which may occur away from the loading
point during the formation of unsymmetric deformation-shapes.
8.
An experimental damage study and associated numerical work should
be performed on convex shells which will sustain damage at the
loading point in both the locally convex and locally concave
configurations in order to better investigate the transition in damage
behavior for shells in this regime.
-252-
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27.
Reddy, J. N., "Bending of Laminated Anisotropic Shells by a Shear
Deformable Finite Element", Fiber Science and Technology, Vol. 17,
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28.
Lee, S. and Zahuta, P., "Instrumented Impact and Static Indentation of
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29.
Kwon, Y. S. and Sankar, B. V.,
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Sjoblom, P. O., Hartness, J. T., and Cordell, T. M., "On Low-Velocity
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31.
Wolf, E., "Impact Damage Mechanisms in Several Laminated Material
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32.
Cantwell, W. J. and Morton, J., "Geometrical Effects in the Low
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33.
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34.
Thomas, S., "Effects of Structural Parameters on the Static
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35.
Lagace, P. A., Williamson, J. E., Tsang, P. H. W., Wolf, E., and
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36.
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37.
Nahas, M. N., "Survey of Failure and Post-Failure Theories of
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38.
Wardle, B. L. and Lagace, P. A., "Importance of Instability in the
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37th Structures, Structural
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1363-1373.
39.
Timoshenko, S., Theory of Elastic Stability, Engineering Societies
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40.
Palazotto, A. N. and Dennis, S. T., Nonlinear Analysis of Shell
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41.
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42.
Mustafa, B., Li, S., Soden, P. D., Reid, S. R., Leech, C. M., and Hinton,
M. J., "Lateral Indentation of Filament Wound GRP Tubes",
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443-457.
43.
Li, S., Rosen, P. D., Reid, S. R., and Hinton, M. J., "Indentation of
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No. 5, 1993, pp. 407-421.
44.
Razi, H. and Lindsay, V. C., "Analysis of Simply Supported Orthotropic
Cylinders
Subjected
to
Low-velocity
Impact",
AIAAIASMEIASCEIAHS 28th Structures, StructuralDynamics, and
Materials Conference, Monterey, CA, 1987, pp. 438-442.
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45.
Schreyer, H. L. and Masur, E. F., "Buckling of Shallow Arches",
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46.
Fung, Y. C. and Kaplan, A., "Buckling of Low Arches or Curved Beams
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Technology, November, 1952.
47.
Ramkumar, R. L. and Thakar, Y. R., "Dynamic Response of Curved
Laminated Plates Subjected to Low-velocity Impact", Journal of
EngineeringMaterialsand Technology, Vol. 109, No. 1, 1987, pp. 67-71.
48.
Simitses, G. J., Shaw, D., and Sheinman, I., "Stability of Cylindrical
Shells, by Various Shell Theories", ZAMM-Z. Angew. Math. u. Mech,
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50.
Manders, P. W., Bader, M. G., Hinton, M. J., and Flower, P. Q.,
"Mechanisms of Impact Damage in Filament Wound Glass-fibre/Epoxyresin Tubes", Third InternationalConference on Mechanical Behaviour
of Materials,Cambridge, UK, 1979, pp. 275-284.
51.
Christoforou, A. P., Swanson, S. R., Ventrello, S. C., and Beckwith, S.
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pp. 964-973.
52.
Alderson, K. L. and Evans, K. E., "Failure Mechanisms During the
Transverse Loading of Filament-wound Pipes Under Static and Low
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53.
Lin, H. J. and Lee, Y. J., "Impact-Induced Fracture in Laminated
Plates and Shells", Journalof Composite Materials,Vol. 24, November,
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54.
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Structures", TELAC Report 88-9, S. M. Thesis, Massachusetts
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55.
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Laminates", Composite Materials: Testing and Design (Sixth
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56.
Palazotto, A., Perry, R., and Sandhu, R., "Impact Response of
Graphite/Epoxy Cylindrical Panels", AIAA Journal, Vol. 30, No. 7,
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57.
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Clamped Edges", Air Force Institute of Technology, M. S. Thesis, 1990.
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59.
Gause, L. W., Rosenfeld, M. S., and Vining, R. E., "Effect of Impact
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62.
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Wong, M., "The Effects of Fabrication and Moisture on the Curvatures
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Course Notes", TELAC Report 88-4B, Massachusetts Institute of
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-263-
APPENDIX A
EXPERIMENTAL FORCE-DEFLECTION
RESPONSES
The experimental force-deflection data for all specimens is presented in
this appendix as plots of contact force versus center-deflection. Five different
force scales and five different center-deflection scales were used to provide a
means of comparison while retaining the specifics of each response.
-264-
50
R12T1S1
40
30
0
-j
20
100
0
O o000o
1
2
3
Center Deflection (mm)
Figure A.1
Experimental force-deflection response for specimen R12T1S1.
-265-
50
40
30
0
I.
20
10 0
0
0
2
4
6
8
10
Center Deflection (mm)
Figure A.2
Experimental force-deflection response for specimen R12T1S2.
-266-
50
40
30
-j
20-
10
0
5
10
15
Center Deflection (mm)
Figure A.3
Experimental force-deflection response for specimen R12T1S3.
-267-
250
R12T2S1
200
z 1500
-
100-
50
0
0
1
2
3
Center Deflection (mm)
Figure A.4
Experimental force-deflection response for specimen R12T2S1.
-268-
250
R12T2S2
200
N 150
"0
0
- 100-
50
0
2
4
6
8
10
Center Deflection (mm)
Figure A.5
Experimental force-deflection response for specimen R12T2S2.
-269-
250
R12T2S3
200
0f
150
0
100-
Z
"
50-
0
5
10
15
Center Deflection (mm)
Figure A.6
Experimental force-deflection response for specimen R12T2S3.
-270-
500
R1 2T3S1
400
3000
J
200-
100
0
1
2
3
Center Deflection (mm)
Figure A. 7
Experimental force-deflection response for specimen R12S3T1.
-271-
500
R12T3S2
400-
3000
J 200-
100
0
2
4
6
8
10
Center Deflection (mm)
Figure A.8
Experimental force-deflection response for specimen R12T3S2.
-272-
500
R12T3S3
400-
z
0
300-
J 200
100
0
5
10
15
Center Deflection (mm)
Figure A.9
Experimental force-deflection response for specimen R12T3S3.
-273-
50
40
30
0
-- 20
10.
0
1
2
3
Center Deflection (mm)
Figure A. 10
Experimental force-deflection response for specimen R6T1S1.
-274-
250
200
0150
0
-i
100
50-
0
2
4
6
8
10
Center Deflection (mm)
Figure A. 11
Experimental force-deflection response for specimen R6T1S2.
-275-
250
R6T1S3
250
200
- 150
0
-J
100
50
0
5
10
15
20
25
30
Center Deflection (mm)
Figure A.12
Experimental force-deflection response for specimen R6T1S3.
-276-
250
200
0"k 150-
z
0
-
100
50-
0
1
2
3
Center Deflection (mm)
Figure A. 13
Experimental force-deflection response for specimen R6T2S1.
-277-
500
R6T2S2
400
3000
-
200
100
0
5
10
15
Center Deflection (mm)
Figure A.14
Experimental force-deflection response for specimen R6T2S2.
-278-
1000
R6T2S3
800-
6000
J
400-
200
0
5
10
15
20
25
30
Center Deflection (mm)
Figure A.15
Experimental force-deflection response for specimen R6T2S3.
-279-
1000
R6T3S1
800
z
600
J 400-
200
0
1
2
3
4
5
Center Deflection (mm)
Figure A.16
Experimental force-deflection response for specimen R6T3S1.
-280-
1000
R6T3S2
800-
6000
-
400-
200
0
5
10
15
Center Deflection (mm)
Figure A.17
Experimental force-deflection response for specimen R6T3S2.
-281-
2000
R6T3S3
1500
- 1000
0
.j
500
OC
0
5
10
15
20
25
30
Center Deflection (mm)
Figure A. 18
Experimental force-deflection response for specimen R6T3S3.
-282-
APPENDIX B
PREDICTED FORCE-DEFLECTION
RESPONSES
The predicted force-deflection responses for all specimens are
presented in this appendix. As discussed in Chapter 5, both pinned-free and
simply-supported-free boundary conditions were utilized in the analysis. The
presentation in this appendix includes the predicted force-deflection
responses for both sets of boundary conditions plotted on the same set of axes
for direct comparison.
-283-
50
R12T1 S1
-- --Pinned-Free
-- A-- Simply-Supported-Free
40
30
0
--
20 -
10
/
0
\ ,
1
2
3
Center Deflection (mm)
Figure B.1
Predicted force-deflection responses for geometry R12T1S1.
-284-
50
-
40
0Z
-
R12T1S2
-Pinned-Free
Simply-Supported-Free
-A--
30 -I
\
I
o
00
20
I
10 A
i
0
2
4
6
8
10
Center Deflection (mm)
Figure B.2
Predicted force-deflection responses for geometry R12T1S2.
-285-
50
-
40
-
---A-
R12T1 S3
Pinned-Free
Simply-Supported-Free
I'
30
I
zCz
I\
o
20-
A-
I
O-
0
5
10
15
20
25
30
Center Deflection (mm)
Figure B.3
Predicted force-deflection responses for geometry R12T1S3.
-286-
250
7 R12T2S1
I
200-
150-
0-1
z
-0
I
I
I
0
150
0 -A
II
,100
,
50
0
-Pinned-
Free
-- A--SimplySupported-
a'/
- Ao
--
-
Free
1
2
3
4
5
Center Deflection (mm)
Figure B.4
Predicted force-deflection responses for geometry R12T2S1.
-287-
500
-
400
z
-
-
----
R12T2S2
- Pinned-Free
Simply-Supported-Free
300
200
/
100L
0
2
4
6
8
10
Center Deflection (mm)
Figure B.5
Predicted force-deflection responses for geometry R12T2S2.
-288-
500
R12T2S3
- - - Pinned-Free
400
~----
Simply-Supported-Free
300
z
?]
o
-0
200I
\
100
0
-
5
10
15
20
25
30
Center Deflection (mm)
Figure B.6
Predicted force-deflection responses for geometry R12T2S3.
-289-
1000
R12T3S1
-
800
I
-PinnedFree
-- A-- Simply-
I
I
SupportedFree
I
600
A
O/
0
-J
400
A4
200
V,
i
I
I.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Center Deflection (mm)
Figure B. 7
Predicted force-deflection responses for geometry R12T3S1.
-290-
1000
R12T3S2
- - Pinned-Free
800
-
---- Simply-Supported-Free
600
- 400 200CIp
0
2
4
6
8
10
Center Deflection (mm)
Figure B.8
Predicted force-deflection responses for geometry R12T3S2.
-291-
2000
R12T3S3
- E- Pinned-Free
-
-A-- Simply-Supported-Free
1500
0
.J
1000
500
./1
-'
I I
v-
I
I
O
0
-
w-
-
10
I
I
I
15
Center Deflection (mm)
Figure B.9
Predicted force-deflection responses for geometry R12T3S3.
-292-
250
200
z
0
-
R6T1 S1
- Pinned-Free
--
---- Simply-Supported-Free
150
J 100
50
-
0
0
1
2
3
4
Center Deflection (mm)
Figure B.10
Predicted force-deflection responses for geometry R6T1S1.
5
-293-
250
200
z
0
-- -- , l.,Jy-JUp
I LVu-, IV
150
J 100
50 -
0
5
10
15
Center Deflection (mm)
Figure B.11
Predicted force-deflection responses for geometry R6T1S2.
-294-
250
R6T1S3
- - -Pinned-Free
-- -Simply-Supported-Free
200
150
0
-J
100
JRL
50
I
I
4i1.-
nrI ~l~8
R
.
.
lgl~I;
.
.
I
rn .
.
10
~
il-I-
15
IL
i
i
20
25
30
Center Deflection (mm)
Figure B.12
Predicted force-deflection responses for geometry R6T1S3.
-295-
500
-
R6T2S1
- Pinned-Free
400 - -A- Simply-Supported-Free
z
0
300
- 200
O,,
100
-
00
0
,,
1
2
3
4
Center Deflection (mm)
Figure B.13
Predicted force-deflection responses for geometry R6T2S1.
5
-296-
1000
R6T2S2
- - - Pinned-Free
800
z
-
--- - Simply-Supported-Free
600
- 400-
/
200
\
0
5
-"
10
15
Center Deflection (mm)
Figure B.14
Predicted force-deflection responses for geometry R6T2S2.
-297-
2000
--
-
R6T2S3
Pinned-Free
-Simply-Supported-Free
1500
-o 10000
-1j
500 -,
0
7
-
0
5
10
15
20
25
30
Center Deflection (mm)
Figure B.15
Predicted force-deflection responses for geometry R6T2S3.
-298-
2000
.
R6T3S1
- - Pinned-Free
--- -Simply-Supported-Free
1500
Z
-0 1000O
0I
/
/
500 0
0
-a
1
2
3
4
Center Deflection (mm)
Figure B.16
Predicted force-deflection responses for geometry R6T3S1.
5
-299-
2000
- - Pinned-Free
-- A-- Simply-Supported-Free
R6T3S2
1500
-
10 0 0
P
I
500
o'
I
,ib
.*
CCI
0 0,
0
5
10
15
Center Deflection (mm)
Figure B.17
Predicted force-deflection responses for geometry R6T3S2.
-300-
2000
R6T3S3
/
----
/
1500
-Pinned-Free
Simply-
Supported-Free
I
I
/
/
1000
0
a
.
-
I
0
500
.0
-I
0)
I
,
I
10
I
I
I
I
i
i
i\1
I
i
i
15
I
25
I
I
30
Center Deflection (mm)
Figure B.18
Predicted force-deflection responses for geometry R6T3S3.
-301-
APPENDIX C
EXPERIMENTAL DEFORMATION-SHAPE
EVOLUTIONS
The experimental deformation-shape evolutions for each specimen are
presented in this appendix. The deformation-shape evolutions along the
central spanwise section are presented in Figures C.1 to C.18; the central
axial deformation-shape evolutions are presented in Figures C.19 to C.36;
and the central spanwise DFU evolutions are presented in Figures C.37 to
C.54.
-302-
Center Deflections in Millimeters
o
0
*
0.4
l
0.8
c
1.5
U
1.2
*
1.9
E
E
C
O
4-
0
,0
0
a)
0
-1
-2
20
40
60
80
100
Spanwise Position (mm)
Figure C.1
Experimental central spanwise deformation-shape evolution
for specimen R12T1S1.
-303-
Center Deflections in Millimeters
E
E
o
*
0
0.6
o
1.2
*
o
*
1.7
2.3
2.8
A
A
0
3.4
4.0
4.5
*
*
5.1
6.7
2
c
0
1
0
0
0
-1
C>
-2
-3
-4
-5
50
100
150
200
Spanwise Position (mm)
Figure C.2
Experimental central spanwise deformation-shape evolution
for specimen R12TIS2.
-304-
Center Deflections in Millimeters
o
*
0
1.3
o
2.6
10
*
0
*
A
A
0
3.9
5.1
6.4
7.7
V
13.4
9.0
10.2
5
t-
E
E
C
O
0
0
-5
-101 ....
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.3
Experimental central spanwise deformation-shape evolution
for specimen R12T1S3.
-305-
Center Deflections in Millimeters
o
*
E
0
0.3
0.7
*
o
*
1.1
1.5
1.8
A
2.2
0%
E
E
C
0
-
0
Q.
CO
>
-2
20
40
60
80
100
Spanwise Position (mm)
Figure C.4
Experimental central spanwise deformation-shape evolution
for specimen R12T2S1.
-306-
Center Deflections in Millimeters
o 0
* 0.5
m 1.1
n
1.7
E
E
a-
2
o
0
A
2.3
2.8
3.4
o
*
rq
A
3.9
fi
o
*
4.5
5.0
5.6
6.2
v
6.8
1
-1
-2
-3
-4
-5
50
100
150
200
Spanwise Position (mm)
Figure C.5
Experimental central spanwise deformation-shape evolution
for specimen R12T2S2.
-307-
Center Deflections in Millimeters
o
0
*
3.8
A
7.6
*
1.2
2.5
*
5.1
A
*
6.3
0
8.9
10.1
D
v
r
i
11.4
12.7
14.0
10
5
E
E
4O
0
0
-1)
-5
-10L
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.6
Experimental central spanwise deformation-shape evolution
for specimen R12T2S3.
-308-
Center Deflections in Millimeters
o
0O
Ei
0.8
*
1.5
e
0.4
m
1.2
*
1.9
E
E
O
Co
0
0
>
CO)
CD)
-2 L
0
20
40
60
80
100
Spanwise Position (mm)
Figure C.7
Experimental central spanwise deformation-shape evolution
for specimen R12T3S1.
-309-
Center Deflections in Millimeters
E
2
C:
1
E
0
0
0
a-
-1
o
0
*
0.6
l
1.1
*
o
*
1.7
2.3
2.8
A
3.4
4.0
0
4.5
A
v
*
a
5.1
5.7
6.2
-2
-3
-4
-5
50
100
150
200
Spanwise Position (mm)
Figure C.8
Experimental central spanwise deformation-shape evolution
for specimen R12T3S2.
-310-
Center Deflections in Millimeters
o
0
*
3.8
a
7.6
v
11.4
*
1.3
2.6
*
*
5.1
6.4
A
8.9
10.2
*
12.7
]
o
10
5
E
E
a-
0
0
O
>
-5
-101
0
.,,
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.9
Experimental central spanwise deformation-shape evolution
for specimen R12T3S3.
-311-
Center Deflections in Millimeters
o
*
o
0
0.2
0.5
*
o
*
0.9
1.3
1.7
A
A
2.1
2.4
E
E
C
0
0
c
-1
-2 L
0
20
40
60
80
100
Spanwise Position (mm)
Figure C.10
Experimental central spanwise deformation-shape evoution for
specimen R6T1S1.
-312-
Center Deflections in Millimeters
o
*
0
1.1
D
2.3
10
E
E
E
*
*
*
3.4
4.5
5.7
*
6.8
A
7.9
0
12.7
5
0
0
c>
-5
-10L
0
50
100
150
200
Spanwise Position (mm)
Figure C.11
Experimental central spanwise deformation-shape evolution
for specimen R6T1S2.
-313-
Center Deflections in Millimeters
o
*
*
20
E
E
t-
0
2.5
5.1
* 7.7
*> 10.2
* 12.8
15.4
17.9
28.8
A
A
0
10
C
O
0
0
Ci
-10
-201 _I_
__
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.12
Experimental central spanwise deformation-shape evolution
for specimen R6T1S3.
-314-
Center Deflections in Millimeters
o
*
0
0.4
o
0.8
*
o
*
1.1
1.5
1.9
A
A
0
2.3
2.7
3.0
E
E
C
O
0
0
CL
>
-1
-2
20
40
60
80
100
Spanwise Position (mm)
Figure C.13
Experimental central spanwise deformation-shape evolution
for specimen R6T2S1.
-315-
Center Deflections in Millimeters
o
0
*
1.1
o
2.3
*
o
*
3.4
4.5
5.6
A
A
0
6.8
7.9
v
*
*
9.0
10
10.2
11.3
12.4
5
E
E
aO
0
nO
o
CL
-5
-10
50
100
150
200
Spanwise Position (mm)
Figure C.14
Experimental central spanwise deformation-shape evolution
for specimen R6T2S2.
-316-
Center Deflections in Millimeters
o
*
0
*
*
*
0
2.5
5.2
7.7
10.2
12.8
*
15.4
A
17.9
0
24.8
20
E
E
10
O
0
0_
n
-10
-20
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.15
Experimental central spanwise deformation-shape evolution
for specimen R6T2S3.
-317-
Center Deflections in Millimeters
E
E
o
0
*
1.1
A
*
0.3
o
1.5
A
2.3
2.6
E
0.7
*
1.9
0
3.0
3.4
v
1
O
0
C
0
a13L)
-1
-2
-3
20
40
60
80
100
Spanwise Position (mm)
Figure C.16
Experimental central spanwise deformation-shape evolution
for specimen R6T3S1.
-318-
Center Deflections in Millimeters
o
*
0
1.1
*
<
[E 2.2
*
10
3.3
4.5
5.6
A
A
6.7
0
9.0
7.9
v
[
10.2
11.3
5
E
E
C
O
0
0
>
0)
-5
-10
50
100
150
200
Spanwise Position (mm)
Figure C.17
Experimental central spanwise deformation-shape evolution
for specimen R6T3S2.
-319-
Center Deflections in Millimeters
o
*
0
2.6
o
5.3
20
E
E
*
o
*
7.7
10.3
12.9
A
15.4
A
18.0
0
20.6
*
*
23.2
25.6
10
E
C
O
0
0
>
-10
-20
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.18
Experimental central spanwise deformation-shape evolution
for specimen R6T3S3.
-320-
Center Deflections in Millimeters
-
- E- - 0.8
- -0
-- - - 1.2
- *- -0.4
R12T1S1
1
E
E
t-
-
-
-
O
-0-
El .
-
0
o
01
-l-.
A--
--
-
- -
- -
- ---
0
-1
I
-2 S40
-40
I
I
-20
I
I
I
I
2I
20
40
Axial Position (mm)
Figure C.19
Experimental central axial deformation-shape evolution for
specimen R12T1S1.
-321-
Center Deflections in Millimeters
-0-
---
-0
-1.7
-*- -0.6
-- -- 1.2
-A
-3.4
--
-
-4.0
---- 6.7
--5.1
-- 0--4.5
R12T1S2
8z0--
_
E
E
- q>- -
-.
--
2
-4
A-..
1
-
-- I
A
-
0
cn
-1
2-
A--
--
-~ -
-2
-3
°
°T
-
-
-
--
-
-
-41
-5 -
-100
-5I ,
-50
50
100
Axial Position (mm)
Figure C.20
Experimental central axial deformation-shape evolution for
specimen R12T1S2.
-322-
Center Deflections in Millimeters
-
G-
-0
-7.7
-- m- -3.9
- - 1.3
-
S- 2.6
-*-6.4
-- 5.1
- -- 13.4
-A- -8.9
-- 0--10.2
10
R12T1S3
5
-
--- --
E
- -t
-
-.-. . ....
0 ---
0>--
0-
..
K..---
0
o
>
-0.
_
-O
-5
v-.P
-10
-150
-100
-50
50
100
150
Axial Position (mm)
Figure C.21
Experimental central axial deformation-shape evolution for
specimen R12T1S3.
-323-
Center Deflections in Millimeters
- e- -0
- - -0.3
- -0.7
-2.2
--- -1.1
-- 0--1.5
-* --- 1.8
R12T2S1
E
E
0
C)
0
- -
a-
.
I '-
-
-
"- -
0
O
CL
..
-1
in
>
-~
I
I
1
"
1
I
l
1
I
lI
-2
-3
-40
-20
20
40
Axial Position (mm)
Figure C.22
Experimental central axial deformation-shape evolution for
specimen R12T2S1.
-324-
Center Deflections in Millimeters
-
-5.0
--- -5.6
-- - -1.7
-0
S-3.4
--0.5 - -- 2.3 - A- -3.9
E- - 1.1 -+ - 2.8
-0- -4.5
0-
R12T2S2
-E-
El -
-4 I- -
-
-
0..
-
E
E
-- -I
4
1
C
O
>d
0f
-U----
F---
-e
A-
-
0
4_0
0
)
-0
]---
.....
2
--
-
Al.
-- 0--
-1
-
•
- -
- °-
T
V-V-
-2
-3
-4
-5
-1 50
I
I
!
I
I
SI
I
I
I
I
-100
I
I II
I
-50
I
I
I (II
0
iL
I
-
I I
i
50
I~ I
I
I I
I
100
I
150
Axial Position (mm)
Figure C.23
Experimental central axial deformation-shape evolution for
specimen R12T2S2.
-325-
Center Deflections in Millimeters
e- -0
--- -3.8
*- -1.2 -- -5.1
---0--2.5
6.3
- -0
10
I
W-
5
E
E
- -'--11.4
- -12.7
-- E -- 14.0
-7.6
-8.9
-10.1
-- A
cll
1-
.--
lo
0--
O
R12T2S3
I°'
..
cn
0
0
0o
.
A-
C-
_.-4
_-
.
--- A
-__
-i
-
A,
L-
-
A-
-
0z
-
-
=-0-0
-5
-ED-
E
--
-
-
--
-10
1,,
-150
I,
-100
|
•
.
.
..
-50
.
.
.
0
.
.
.
.
_]
B---
.
.
I.
.
50
.
.
.
I.
.
100
I
.
I
.
I
.
I.
150
Axial Position (mm)
Figure C.24
Experimental central axial deformation-shape evolution for
specimen R12T2S3.
-326-
Center Deflections in Millimeters
S-2.3 - - 3.4
-- 1.2
-*- -0.4 -- --1.5
- - -2.7 --i- -3.8
--+-- 1.9 --0--3.1 -E--4.2
n -0.8
-e-
-0.
R12T3S1
1
E
E
0
c
0O
0
0.
>d
0
---
-
-
-
--
--
-1
-A
-2
-3
•-
- O---.-.---
-
_-... -
---------
--O -
-4
-5
l I ,
-40
,
i
I
-20
i
i
I
I
I
20
,
I
I
40
Axial Position (mm)
Figure C.25
Experimental central axial deformation-shape evolution for
specimen R12T3S1.
-327-
Center Deflections in Millimeters
--- -1.7 -- -3.4
-o- -0.6 --O --2.3 -k -4.0
----1.1 -,-- 2.8 --0o--4.5
- -
-0
-
-
-
V"
-E
-5.1
-5.7
-6.2
R12T3S2
0-
E
E
0
a)
0-[--_
---00
_ [
UI-
- -U--
-
-
-
----
O---
--
I
---- 4
0
0
--
-
2
1
0
4
--------
SA---
A
-1
-2
-v-
-
-
-.
-
°-
-Ois]
r
---- [
-3
-4-5 1
-100
I
I
I
I
i
-5 0
I
0
.
i
I
t
50
100
Axial Position (mm)
Figure C.26
Experimental central axial deformation-shape evolution for
specimen R12T3S2.
-328-
Center Deflections in Millimeters
--i- -3.8
- -1.3 --O--5.1
E--2.6 ----- 6.4
-0
---A
-A- -0 -
-v -11.4
- -12.7
-7.6
-8.9
-10.2
10
R12T3S3
5
E
E
t-
E-
- -0
0LI-
-B
-D
° .
-U
U--.
_ -
--
O
--
0
0
0-
.°
-
,/I-
A.-
IN
-- A
~--0--
S------S--
-5
-
-
I
III
I
I
--- -- V
i:
-1011
-150
I1
-100
I
I
I
I
I
-50
I
I I
I
0
I
I
I
50
I
I
100
I
I
150
Axial Position (mm)
Figure C.27
Experimental central axial deformation-shape evolution for
specimen R12T3S3.
-329-
Center Deflections in Millimeters
-
-2.1
-
-0.9
- - -0.2
A- -2.4
I
E
E
R6T1S1
-Z
E11 -
1
~
-0-
El--
.- - -.-
.El
0
0
O
_
a,
.m
0
n13.0
.
. ...-
.
.-A--..- 4
...
-
.
-1
-2
nI
I
-40
,
, ,
I
-20
I
,
I
20
,
I
40
Axial Position (mm)
Figure C.28
Experimental central axial deformation-shape evolution for
specimen R6T1S1.
-330-
Center Deflections in Millimeters
- -*-
--- -3.4
--0--4.5
-*---5.7
-0
-1.1
-E--2.3
10
---
-6.8
- A- -7.9
-- 0--12.7
R6T1S2
E
E
cO
5
G-
--fb.
--
0
..-
-.
c'
.
I-I-._----
AkA_
~
-U
o
,7
C)
1
O
-5
o
-101
-10 0I
o---I
I
=- .
I
I
-50
II
I
.
0
50
I
100
Axial Position (mm)
Figure C.29 Experimental central axial deformation-shape evolution for
specimen R6T1S2.
-331-
Center Deflections in Millimeters
-0
-2.5
-5.1
20
-U- -7.7
-- o.-- 10.2
-A
-*- -- 12.8
-- 0--28.8
-15.4
-17.9
-A-
R6T1S3
15
- --
-e
E
E
C
10
I
>-
-~
-e
-
-U
0--
-3-
5
0
O
Zk--.....
"
.... ,.
0
-
. _
_.
.
A-,
0
>C
0n
-5
-10
-
-15
O-o
i--
-20
-150
-100
-50
0
50
100
150
Axial Position (mm)
Figure C.30 Experimental central axial deformation-shape evolution for
specimen R6T1S3.
-332-
Center Deflections in Millimeters
- E
- 1.1
-A
-2.3
---
1.5
-A
-2.7
--
-- 1.9
-0
---
- -0.4
- -E- -0.8
-
R6T2S1
-
-4
E
E
1-
-
-
o~
I
-
- --n-
- 1 E...
O
A I. -
0
.-.......
o
C
00
0L
0)
-.
O
~-----A---
_ -O ...
- -
-1
-0
-2
-40
A
_ ,
0..----
-20
20
40
Axial Position (mm)
Figure C.31 Experimental central axial deformation-shape evolution for
specimen R6T2S1.
-333-
Center Deflections in Millimeters
-
10
-
--1.1
EL
2.3
--- - 3.4 -A -6.8
-- --4.5 -A- -7.9
----5.6 --0--9.0
V- -10.2
-11.3
-12.4
R6T2S2
0-
5
E
E
-
-
-[--
-
G-
-(
--
-i-
R-------A
A
1-~-
-
-
h-1
----
0
0
0
-
A
A----
0ii
O
O
it
13
0-------------
i,
---
-5
I ..
-
-1011
-100
I
-..
I
-
I
-5 0
--
-
--O.
-----.-
--
I
-
-
E- - -.--
I
-0-
I
- --
I
I
I
0
E- ....
I, I
I
,
50O
[]-
i
I
100
Axial Position (mm)
Figure C.32
Experimental central axial deformation-shape evolution for
specimen R6T2S2.
-334-
Center Deflections in Millimeters
---
-e- -2.5
S-Ei--5.2
- 7.7
- 10.2
----
20
-15.4
-17.9
-A-
-- 0--24.8
12.8
R6T2S3
e
e-
-
E
E
'-
10
I...-
-o
e
ElI .
-
-
--
El-O
O
0
40
n0
----
- ----
Cz
-10
I
.
.
I
.
.
-0n
-150
-150
-100
-50
50
100
150
Axial Position (mm)
Figure C.33
Experimental central axial deformation-shape evolution for
specimen R6T2S3.
-335-
Center Deflections in Millimeters
- &- -0
-2.3
-2.6
-- 0--3.0
--M- -1.1 --A
--O-- 1.5 -k-
- - -0.3
-- -- 0.7 ----
1.9
R6T3S1
E
E
-oQ
*I
F- -
- - -
"El
O
+\
0
o
a-
"
VI-
0-
n
-3
0
* -V.
-2
V
,
.
-40
I.
.I
-20
I
I
I
I
I
I
I
I
20
•
S
,
II
I
I
40
Axial Position (mm)
Figure C.34
Experimental central axial deformation-shape evolution for
specimen R6T3S1.
-336-
Center Deflections in Millimeters
-e-
-0
A -6.7 - v-- 10.2
3.3 --- - -1.1 -- --4.5 - A- -7.9 --n- -11.3
- -2.2 -- --5.6 --0--9.0
10
---
R6T3S2
S-0--
5
E
E
-
-_.-
-
G
-o
.
_
A-
c
0
0
O..--. ----
a)
V.-
-10
-150
- - -
-V.
-
" " "
.
-T
-5
-
- -
-100
-50
.
50
.
100
150
Axial Position (mm)
Figure C.35
Experimental central axial deformation-shape evolution for
specimen R6T3S2.
-337-
Center Deflections in Millimeters
e--0
--- -7.7
.- -2.6 --- 10.3
-- 5.3 -- +- -12.9
---A a- -0
10
I
0-
E
E
-
-15.4 - - v--23.2
-18.0 --- -25.6
-20.6
5
0---.
R6T3S3
_ -E>
--
-e--B
-
S-
-
-
- ----
-
-
. -0
-I-
U--
----
C
0
0
CL
O
ci
0
-
o
-j
------...
-5
S.
-1
I
-100
-50
-0
-. -
o
----------
H
,
-150
-
S-0----
--.
a,
.~--
0
I
I
50
I
I
I
i
.
100
.
a
150
Axial Position (mm)
Figure C.36 Experimental central axial deformation-shape evolution for
specimen R6T3S3.
-338-
Center Deflections in Millimeters
o
0
o
0.8
*
1.5
*
0.4
*
1.2
*
1.9
0
E
E
C
0
aO
0
-1
0
>
-2
-3
20
40
60
80
100
Spanwise Position (mm)
Figure C.37
Experimental central spanwise DFU evolution for specimen
R12T1S1.
-339-
Center Deflections in Millimeters
o
*
o
0
0.6
1.2
n
o
*
1.7
2.3
2.8
A 3.4
A
4.0
0
4.5
V
5.1
N
6.7
E
E
c
-2
O
0
a.
-4
-6
-8
50
100
150
200
Spanwise Position (mm)
Figure C.38
Experimental central spanwise DFU evolution for specimen
R12T1S2.
-340-
Center Deflections in Millimeters
o
*
w
0
1.3
2.6
*
*
*
3.9
5.1
6.4
a
A
0
7.7
v
13.4
8.9
10.2
E
E
C)
0
-4
-0
O
O
0z
-8
>
-12
-16
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.39
Experimental central spanwise DFU evolution for specimen
R12T1S3.
-341-
Center Deflections in Millimeters
o
0
*
*
0.3
2> 1.5
D
0.7
*
A
1.1
2.2
1.8
0
E
E
O
-1
CL
>
-2
-31 -20
0
20
40
60
80
100
Spanwise Position (mm)
Figure C.40
Experimental central spanwise DFU evolution for specimen
R12T2S1.
-342-
Center Deflections in Millimeters
o
*
*
0
0.5
1.1
n
o
*
1.7
2.3
2.8
A
A
o
3.4
3.9
4.5
v
[
a
5.0
5.6
6.2
[E 6.8
E
E
O
-2
0
-4
-6
-8
50
100
150
200
Spanwise Position (mm)
Figure C.41
Experimental central spanwise DFU evolution for specimen
R12T2S2.
-343-
Center Deflections in Millimeters
o
*
*
4
0
1.2
2.5
m 3.8
*
*
5.1
6.3
A
A
0
7.6
8.9
10.1
11.4
12.7
14.0
v
*
Ea
0
E
E
C)
0
-4
O
0O
-8
-12
-16
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.42
Experimental central spanwise DFU evolution for specimen
R12T2S3.
-344-
Center Deflections in Millimeters
o
0
o
0.8
*
0.4
*
1.2
1.5
.
1.9
0
E
E
C
0O
>
-1
-2
-3
20
40
60
80
100
Spanwise Position (mm)
Figure C.43
Experimental central spanwise DFU evolution for specimen
R12T3S1.
-345-
Center Deflections in Millimeters
o
0
*
0.6
E
1.1
1.7
> 2.3
* 2.8
*
a
3.4
A
4.0
0
4.5
*
*
*
5.1
5.7
6.2
E
E
C)
0
-2
O
O
n
-4
-CL
-6
-8
50
100
150
200
Spanwise Position (mm)
Figure C.44
Experimental central spanwise DFU evolution for specimen
R12T3S2.
-346-
Center Deflections in Millimeters
o
0
*
1.3
LI
2.6
*
*
*
3.8
5.1
6.4
A
7.6
v
A
8.9
10.2
*
o
11.4
12.7
E
E
C,
-4
4-0
0n
-8
0
O
Cz
-12
-16
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.45
Experimental central spanwise DFU evolution for specimen
R12T3S3.
-347-
Center Deflections in Millimeters
o
0
*
0.2
0.5
o
*
o
*
0.9
1.3
1.7
a
2.1
A
2.4
0
E
E
C
O
0
-1
n
-2
-3 L
0
20
40
60
80
100
Spanwise Position (mm)
Figure C.46
Experimental central spanwise DFU evolution for specimen
R6T1S1.
-348-
Center Deflections in Millimeters
o
0
*
1.1
LI
2.3
m 3.4
< 4.5
* 5.7
A
A
0
100
150
6.8
7.9
12.7
E
E
O
0
-4
O
-8
-12
-16
50
200
Spanwise Position (mm)
Figure C.47
Experimental central spanwise DFU evolution for specimen
R6T1S2.
-349-
Center Deflections in Millimeters
o
*
0
2.5
o
5.1
*
*
*
7.7
10.2
12.8
A
15.4
A
0
17.9
28.8
10
5
0
E
E
C
O
.0
0
-5
-10
-15
-20
a>
-25
-30
-35
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.48
Experimental central spanwise DFU evolution for specimen
R6T1S3.
-350-
Center Deflections in Millimeters
o
*
E
0
0.4
0.8
*
o
*
1.1
1.5
1.9
A
2.3
2.7
0
3.0
A
E
E
-1
C)
0
O
aC.)
:2
-2
-3
-4
20
40
60
80
100
Spanwise Position (mm)
Figure C.49
Experimental central spanwise DFU evolution for specimen
R6T2S1.
-351-
Center Deflections in Millimeters
0
1.1
2.3
3.4
4.5
5.6
A
A
0
v 10.2
rg 11.3
m 12.4
6.8
7.9
9.0
R6T2S2
0
E
E
O
C
0
O
-5
aC.)
-10
-15
I
I
I
I
I
50
.
.
.
100
I
II
I
150
.
,
I
200
Spanwise Position (mm)
Figure C.50
Experimental central spanwise DFU evolution for specimen
R6T2S2.
-352-
Center Deflections in Millimeters
E
E
C)
o
0
o
*
0
2.5
*
5.2
*
o
*
a
7.7
10.2
12.8
A
0
15.4
17.9
24.8
-5
-10
0
-15
0
>z
-20
-25 -30 1....
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.51
Experimental central spanwise DFU evolution for specimen
R6T2S3.
-353-
Center Deflections in Millimeters
o
*
0
0.3
o
0.7
N
1.1
A
2.3
o
*
1.5
1.9
A
0
2.6
3.0
v
3.4
E
E
c-
O
0
n
-1
-2
aO
-
-3
-4
20
40
60
80
100
Spanwise Position (mm)
Figure C.52
Experimental central spanwise DFU evolution for specimen
R6T3S1.
-354-
Center Deflections in Millimeters
o
0
m 3.3
*
o
1.1
2.2
o
*
4.5
5.6
A
6.7
A 7.9
0 9.0
'
[
10.2
11.3
0
E
E
0
0
,0
-5
O
>)
-10
-15L
0
50
100
150
200
Spanwise Position (mm)
Figure C.53
Experimental central spanwise DFU evolution for specimen
R6T3S2.
-355-
Center Deflections in Millimeters
E
E
c
O
o
0
*
7.7
A
15.4
v
23.1
*
2.6
o
10.3
A
18.0
r
25.6
*
5.3
*
12.9
0
20.6
-5
-10
CD
o0
a- -15
-20
-25-30
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure C.54
Experimental central spanwise DFU evolution for specimen
R6T3S3.
-356-
APPENDIX D
PREDICTED DEFORMATION-SHAPE
EVOLUTIONS
The predicted deformation-shape evolutions are presented for all
specimens with both sets of boundary conditions in this appendix.
As
explained in Chapter 5, a cubic spline curve fit was used for all predicted
deformation-shapes for purposes of clarity. The predicted central spanwise
deformation-shape evolutions with simply-supported-free and pinned-free
boundary conditions are presented in Figures D.1 to D.18 and Figures D.19 to
D.36, respectively. The predicted central axial deformation-shape evolutions
with simply-supported-free
and pinned-free boundary conditions are
presented in Figures D.37 to D.54 and Figures D.55 to D.72, respectively.
The predicted central spanwise DFU evolutions with simply-supported-free
and pinned-free boundary conditions are presented in Figures D.73 to D.90
and Figures D.91 to D.108, respectively.
-357-
Center Deflections in Millimeters
-e-
-0
-.-
-0.1
--- -0.5
-1.8
El --0.3
E
E
C
O
0
id
tj
-1
,
-2
0
_
A A
20
I
I
40
,
II
60
80
100
Spanwise Position (mm)
Figure D.1
Predicted central spanwise deformation-shape evolution for
geometry R12T1S1 with simply-supported-free boundary
conditions.
-358-
Center Deflections in Millimeters
5
-
-0
-5
-1.0 --- -3.1
- -1--2.1
--- -7.7
E
E
C
O
0
0
nz
-1
-2
-3
-4
-5
50
100
150
200
Spanwise Position (mm)
Figure D.2
Predicted central spanwise deformation-shape evolution for
geometry R12T1S2 with simply-supported-free boundary
conditions.
-359-
Center Deflections in Millimeters
-0
-e-
10
-0
-2.1
- El- -4.5
--- -6.8
--
-14.9
- -- -12.2
E
E
C
O
-5
1
-10 . I I I I I I I I
0
50
100
I 1. I I ,,
1,I,
,,,,,1,
150 200 250 300
Spanwise Position (mm)
Figure D.3
Predicted central spanwise deformation-shape evolution for
geometry R12T1S3 with simply-supported-free boundary
conditions.
-360-
Center Deflections in Millimeters
- G- -0
2
-
-
- -0.2
-- 0.5
--m- -0.8
---- 1.9
E
E
O
0
CD
-1
-2
20
40
60
80
100
Spanwise Position (mm)
Figure D.4
Predicted central spanwise deformation-shape evolution for
geometry R12T2S1 with simply-supported-free boundary
conditions.
-361-
Center Deflections in Millimeters
5
- 0- -0
- - -2.0
- e
---
-0.8
-3.1
-- --- 6.8
E
E
C
0
4O
CL
Cz
0E
0
50
100
150
200
Spanwise Position (mm)
Figure D.5
Predicted central spanwise deformation-shape evolution for
geometry R12T2S2 with simply-supported-free boundary
conditions.
-362-
Center Deflections in Millimeters
-o0
-Ge-
----
-@- -1.4
-E -- 3.2
10
-4.9
-14.0
-A
1--7.2
--- 10.3
E
E
O
O
0
0.
(U
ci
-5
-10 1,,
0
.
.
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.6
Predicted central spanwise deformation-shape evolution for
geometry R12T2S3 with simply-supported-free boundary
conditions.
-363-
Center Deflections in Millimeters
-
- E- -0.7
-0
--- -1.3
- --0.3
R1 2T3S1
1
E
E
C,,
O
0
0
-. W-
.. --
EL-1Or-.--
- -.
..
80
100
a.
o
>
S
-
0)
a
,,
6\0
I I
.
20
40
60
Spanwise Position (mm)
Figure D.7
Predicted central spanwise deformation-shape evolution for
geometry R12T3S1 with simply-supported-free boundary
conditions.
-364-
Center Deflections in Millimeters
--
-0.5
-- E -- 1.6
-e-
-0.5
-
- -2.5
-*- - - 5.9
E
E
C
0
CD,
0
aO
c)
>
mm
0
50
100
150
200
Spanwise Position (mm)
Figure D.8
Predicted central spanwise deformation-shape evolution for
geometry R12T3S2 with simply-supported-free boundary
conditions.
-365-
Center Deflections in Millimeters
-
- -0
-*-
--14.9
--- - 4.8
-1.1
-
10
3.0
E
E
C
O
0O
>
-5
-10 11
0
. . . 1. ,
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.9
Predicted central spanwise deformation-shape evolution for
geometry R12T3S3 with simply-supported-free boundary
conditions.
-366-
Center Deflections in Millimeters
-
-
-o0
-9-
E--- 0.6
--- - 1.0
-0.2
E
E
CJ
0
0
O
-2
1
0
.1 1
I
20
I
I
1
40
1
,
I
1
60
.
I
80
100
Spanwise Position (mm)
Figure D.10
Predicted central spanwise deformation-shape evolution for
geometry R6T1S1 with simply-supported-free boundary
conditions.
-367-
Center Deflections in Millimeters
0-
- 1.9 ---
-
10
-0- - 3.0 - ---
5.3 --- -9.0
7.0 - A- - 10.7
-4.0 ----
R6T1S2
G-
e- - &
E
E
0
-
Q.
C
0'
-
- O-
r-o-
-
0-O
-0
-
El
"
E.- W
~
--
Xr1 ].-B-!72]
-
" -.
0..O
.
-.
I
I
e~nW
*A,-*-.-
~
-
0
.t
~2 119
ed
a,
A~A
-5
-10
0
&
A-A
I.
SI
I
I
I
50
I
~I
I
100
I
I
I
I
I
I
150
I
200
Spanwise Position (mm)
Figure D.11 Predicted central spanwise deformation-shape evolution for
geometry R6T1S2 with simply-supported-free boundary
conditions.
-368-
Center Deflections in Millimeters
- &- -0
- e- - 4.2 ----
20
E
E
- -n- - 9.3
----
- 12.9 --
16.5--
-23.7
-- 19.3 - A- - 28.0
10
C
aO
c)
:E
(1)
-10
-201 I I I I I
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.12
Predicted central spanwise deformation-shape evolution for
geometry R6T1S3 with simply-supported-free boundary
conditions.
-369-
Center Deflections in Millimeters
- - 0.8
-e-
-0
-*-
-0.3
-U- -1.3
R6T2S1
v o~
/
-
K
-
.-
E
E
E-El:-
2-1-
0O
/1
l~
_
- " - "--- -
\o- 0/-El
0--N-
- l."
El
0)
Cz
40
0
>n
-s.
'.7
-2
I
I
I
SI
20
I
I
I
40
I
II
I
60
I
i
80
I
I
I
100
Spanwise Position (mm)
Figure D.13 Predicted central spanwise deformation-shape evolution for
geometry R6T2S1 with simply-supported-free boundary
conditions.
-370-
Center Deflections in Millimeters
-
e- - 0
- -E- -2.5 - ---
- 0- - 1.0 --
10
-11.8
5.8 -
-3.8 -*- --8.4 - A- - 13.6
E
E
O
a.
O
>
-5
-10
50
100
150
200
Spanwise Position (mm)
Figure D.14 Predicted central spanwise deformation-shape evolution for
geometry R6T2S2 with simply-supported-free boundary
conditions.
-371-
Center Deflections in Millimeters
-e-
-0
-2.7
E -- 5.9
--- - 8.6
S-24.0
---
20
I-
E
--
- 16.5
10
E
C
O
,0
0
a)
-20111111111111
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.15
Predicted central spanwise deformation-shape evolution for
geometry R6T2S3 with simply-supported-free boundary
conditions.
-372-
Center Deflections in Millimeters
-
-0.3
--
-e-
-0.3
--- -1.1
--0.7
--+---2.9
E
E
C
O
-2 L
0
20
40
60
80
100
Spanwise Position (mm)
Figure D.16
Predicted central spanwise deformation-shape evolution for
geometry R6T3S1 with simply-supported-free boundary
conditions.
-373-
Center Deflections in Millimeters
- e- -o0
-e- -1.8
10
---- -6.5
-*--
14.7
E
E
O
a.
0
CL
0i
-5
-10 1
0
.
I
1 I . .'
50
I. . . . I I I
100
150
200
Spanwise Position (mm)
Figure D.17
Predicted central spanwise deformation-shape evolution for
geometry R6T3S2 with simply-supported-free boundary
conditions.
-374-
Center Deflections in Millimeters
S
- e- -4.1
20
- -9.0
-u--13.1
- - - 18.4
--
-- 24.6
E
E
C,
0
0,
aC)
Ci
-20 1
0
I 1 li
250 300
1. ,....,... . ,,,, I 1 I
50
100
150
200
Spanwise Position (mm)
Figure D.18
Predicted central spanwise deformation-shape evolution for
geometry R6T3S3 with simply-supported-free boundary
conditions.
-375-
Center Deflections in Millimeters
- e- -0
-- m- -0.5
-e- -0.1
- E- 0.3
-- 0.7
--- --0.9
A
-1.1
R12T1S1
1
E
a-
01
-I
I--
-
,
&__6
,
,
(
,
,
0-
,
,
,
,
,
4-f-~e
z
r
18
-
L
20
40
60
80
100
Spanwise Position (mm)
Figure D.19
Predicted central spanwise deformation-shape evolution for
geometry R12T1S1 with pinned-free boundary conditions.
-376-
Center Deflections in Millimeters
5
- e- -0
- -E - -1.2
- - -0.5
---
-A
-4.6
-2.1
E
E
C
0O
cl
0
a0U
0
50
100
150
200
Spanwise Position (mm)
Figure D.20
Predicted central spanwise deformation-shape evolution for
geometry R12T1S2 with pinned-free boundary conditions.
-377-
Center Deflections in Millimeters
-
10
- -0
- e- -2.1
-- -- 3.5
---- -4.6
--- -9.8
-*- - 8.0
10%
E
E
C,
0
4C)
0i
-5
-10
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.21
Predicted central spanwise deformation-shape evolution for
geometry R12T1S3 with pinned-free boundary conditions.
-378-
Center Deflections in Millimeters
-
El-
- -0.2
-@- -0.2
-0.6
--m- -0.9
R12T2S1
c~-o-c)
E
E
/O
0.5
,o
0
,,
v
1r
I
G\
---00
0
OL
\o\
o
-
o\
o
n
El-
- El-
I F
n. I - - ~I-
0
0
-W
/0
or .
El
/o
E].- El- E- El-
El-El-
E - E]-
EILEL~
rD- E -.
El=
-- - m-N6
--A'
'\1
01;
E~u
F 0-
0
0
>
-0.5
-1
"0-
C)
I
I
,
I
I
I
.
20
,
I'
,
I'
'
40
-
'
'
"
,
I
60
,
n
,
.
,
•
I
.
80
,
.
.
,
I
100
Spanwise Position (mm)
Figure D.22
Predicted central spanwise deformation-shape evolution for
geometry R12T2S1 with pinned-free boundary conditions.
-379-
Center Deflections in Millimeters
E
-@-
-0.5
--- -2.4
-- -- 4.6
01
C
0
Ci)
- -E- 1.2
1
c
0
-0.5
2
E
oa
--
-1
-2
-3
-4-5 L
0
50
100
150
200
Spanwise Position (mm)
Figure D.23
Predicted central spanwise deformation-shape evolution for
geometry R12T2S2 with pinned-free boundary conditions.
-380-
Center Deflections in Millimeters
-
-
-e-
10
-0
- -E - 2.4 -
-1.0
-----4.2 -*---
8.0
100
200
-5.9 -
-10.1
E
E
0
I,
-5
-101 1
0
I
I1,
50
150
250
300
Spanwise Position (mm)
Figure D.24
Predicted central spanwise deformation-shape evolution for
geometry R12T2S3 with pinned-free boundary conditions.
-381-
Center Deflections in Millimeters
--
-0.3
-- 0--0.7
-*-
-0.3
---- -1.1
-- -- 2.4
R12T3S1
E
E
C:
0
0n
~-"~-~-
ero"
0-0OE -EI-S-Vr-r
%&
G G-
- . 1-4U-
-"
t-"l
-.
.*'(3~.
B-*-
CL
()
-
.
rn.
"
RnR_-sU ~-U-
C
.a
0.4
.
20
Er'
ea
0)
>~
-2
8:
- n- El- El- 0-
. .
.
I
.
.
40
.
.
.
O
.
.
.
I
60
"
II
I
I
80
,
m
I
I
100
Spanwise Position (mm)
Figure D.25 Predicted central spanwise deformation-shape evolution for
geometry R12T3S1 with pinned-free boundary conditions.
-382-
Center Deflections in Millimeters
-e- -0
-
-
---
- -0.3
- - - 0.7
-A
2.9
-+---
- 1.6
-3.8
E
E
C
0
,-'
0
a.
C>
-2
)
-3
-4
-5
1
0
I
I
I
50
I
I
I
I
II
100
,
I
I
I
I
150
I
200
Spanwise Position (mm)
Figure D.26 Predicted central spanwise deformation-shape evolution for
geometry R12T3S2 with pinned-free boundary conditions.
-383-
Center Deflections in Millimeters
-
10
-
- -1
- -1.0
- --- 2.1
-A
-9.7
---A- -4.0
5
E
E
01
C:
0>
-5
-10 1 ....
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.27
Predicted central spanwise deformation-shape evolution for
geometry R12T3S3 with pinned-free boundary conditions.
-384-
Center Deflections in Millimeters
--
-0.2
-0-
-0.2
-1.6
-
--- -0.7
- A- -2.0
E
E
C,
0
O
O
-1
-2
1
0
.II
I
20
I
I I
I
40
I
II I
I
60
I I
I
I
80
100
Spanwise Position (mm)
Figure D.28
Predicted central spanwise deformation-shape evolution for
geometry R6T1S1 with pinned-free boundary conditions.
-385-
Center Deflections in Millimeters
- a- -0
10
10
-
- -1.1
- -0- - 2.2
-- 0--4.4
-
-8.2
--- - 3.4
O
O
-5
-10L
0
50
100
150
200
Spanwise Position (mm)
Figure D.29
Predicted central spanwise deformation-shape evolution for
geometry R6T1S2 with pinned-free boundary conditions.
-386-
Center Deflections in Millimeters
- &- -0
-
20
0
- - E - 3.3 -
- -1.5 -
- -5.3 -
- - 8.0
-- A
-13.3
-- 10.2 - A- -19.0
10
C
01
O
>
-10
-20
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.30
Predicted central spanwise deformation-shape evolution for
geometry R6T1S3 with pinned-free boundary conditions.
-387-
Center Deflections in Millimeters
-e- -0
-*-0.2
- 0--0.4
--- -0.6
-A
-A-
-1.5
-2.0
E
E
C,
O
0
o
-2 L
0
20
40
60
80
100
Spanwise Position (mm)
Figure D.31
Predicted central spanwise deformation-shape evolution for
geometry R6T2S1 with pinned-free boundary conditions.
-388-
Center Deflections in Millimeters
10
-
-0
- - - 1.8
-0-
-0.8
---- -3.4
-A
-8.8
E
E
O
Cd
0
0>
0L
-5
-1
0
50
100
150
200
Spanwise Position (mm)
Figure D.32
Predicted central spanwise deformation-shape evolution for
geometry R6T2S2 with pinned-free boundary conditions.
-389-
Center Deflections in Millimeters
- &- -0
- 0- -2.0 ---
20
-
-- E - 4.1
-6.7 -*---
-15.9
12.0 - A- - 19.4
0
E
a.0
O
Cz
0
ed
>
-201 _ 1
0
I 11,
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.33
Predicted central spanwise deformation-shape evolution for
geometry R6T2S3 with pinned-free boundary conditions.
-390-
Center Deflections in Millimeters
-e-*-
-M- - 1.3
-0
-0.4
-3.6
-
E - - 0.9
E
E
C
0
40
Ci
>z
-30
0
I1
20
20
,40
40
I
60
I
I
80
100
Spanwise Position (mm)
Figure D.34
Predicted central spanwise deformation-shape evolution for
geometry R6T3S1 with pinned-free boundary conditions.
-391-
Center Deflections in Millimeters
-
-
-@
10
El--
-0
-1.2
2.6
--m- - 4.7
R6T3S2
98
E
E
U-
3- .
U-.-_
i...
u
Li-
{- -[
0
o
-U----1a
"
a.
/
C,
.
%OI
W
.
.
-4
o
00
0
-5
-10
0
I
I
I
I
I
50
I
a
I
I
II
100
I
,
,
,
I
150
I.
I.
.I
I
I
200
Spanwise Position (mm)
Figure D.35
Predicted central spanwise deformation-shape evolution for
geometry R6T3S2 with pinned-free boundary conditions.
-392-
Center Deflections in Millimeters
-
-
20
E
E
El- -3.5 - -
-0
- - 1.6 ---
-6.7 ---
--9.7 -A-
-19.1
- 13.1
10
4-
O
C
01
-10
-20 1 .. i ,
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.36 Predicted central spanwise deformation-shape evolution for
geometry R6T3S3 with pinned-free boundary conditions.
-393-
Center Deflections in Millimeters
--
-- 0-
--M- -0.5
-- - 0.8
-- -1.2
-0
-0.1
-0.3
-- A
-1.8
-A-
-2.4
R12T1S1
E
E
)-
C
t-
-l-E
- ---Q--G
----- -F- -- Iq - --
---
El- ]
.
-
]
-
_
El-W--
- i--
-•--
0- -
0-
-El -
0-
0_--
El-
El--
-
-
U---
-
a'O
O
(D
>,
A-
-12
A--
A-,A-
a
I
-40
AA
A-,
I- ,~t--~--~
-A-
-2
A~-
S I
I
-20
I
20
,
I
40
Axial Position (mm)
Figure D.37 Predicted central axial deformation-shape evolution for
geometry R12T1S1 with simply-supported-free boundary
conditions.
---
-394-
Center Deflections in Millimeters
-
-0
- -El--2.1 -
--4.3 --
--
-1.0
---
-- 5.9
-3.1
--
-7.7
R12T1S2
31
E
E
- -
-
-G-
-0t--O
-
-
G-
----
-
-
-0-
-@-
E--E
El " - ---- E
El-
El- - -
- - El-
E
-.
O
-I-
-U-
----
-U-
--
-*~-
---
_I
0
00
>
-1
-,:--)-
0,
o--.0-
A--
A
-
----
0-
--
-0--4
-2
-4-51
-100
A--
I
I
""""""'~''~''
I
I
I
-50
A-I
I
I
I
I
I
z
I
50
I
I
I
I
100
Axial Position (mm)
Figure D.38 Predicted central axial deformation-shape evolution for
geometry R12T1S2 with simply-supported-free boundary
conditions.
-395-
Center Deflections in Millimeters
- - -
- t
-- El - 4.5 -- 0--9.5
-2.1 ---
-6.8 --
-14.9
A--
--- 12.2
10
R12T1S3
--
E
E
- -
G--
- -
GG--G-EG-9
E - - - El
-
E-
E
ED-
E -
E
-
-
El
C:
(0
-"
O
O
--
-i-----
----
--- A
-----
-
A----
in
o
CL
-5
-'
--
"--i
--
I 'I I t' t' '
-10
-150
-100
-50
--
i .
I.
-
I-e--
. . I . . .
50
--
I
100
I-
. . . II
150
Axial Position (mm)
Figure D.39 Predicted central axial deformation-shape evolution for
geometry R12T1S3 with simply-supported-free boundary
conditions.
-396-
Center Deflections in Millimeters
-e-
S-- 0.5
-0
-e- -0.2
---- 1.9
-0.8
---
R12T2S1
t - ~- - ~
0E
E
-0
---
00
-
-
-
- ~ - ~- - ~- - ~ - ~ - ~ - ~ -
-
R --- - - - -
-UI--U
----
-
t
El--
----------- Ei--i--
El-
- - - W--E-I--
E
- -
-- I
0
1I_
0
i
>
-2
-3
I
-40
-20
40I
I
20
40
Axial Position (mm)
Figure D.40 Predicted central axial deformation-shape evolution for
geometry R12T2S1 with simply-supported-free boundary
conditions.
-397-
Center Deflections in Millimeters
---
- -E--2.0
0
-@- -0.8
---
-3.1
R12T2S2
S--G--
-
-
---
-
-
e-
---
-
E
E
C
O
.-
-~----U--rn--UI-.-E-----------. . 0-I
.
.
-E
- _E- -
E. .
0
0
Nk-
-2
-3
-4
-5
-100
-50
50
100
Axial Position (mm)
Figure D.41 Predicted central axial deformation-shape evolution for
geometry R12T2S2 with simply-supported-free boundary
conditions.
-398-
Center Deflections in Millimeters
-0- -0
-@- -1.4
-A
--- -4.9
-- --7.2
-*- --10.3
--- 3.2
-14.0
10
R12T2S3
-
E
E
5
I-
- - - &-
-@-
--
--
-
-@--
--
E
C'
V-
0
0
-I--
- I--
-m--
- -
0--
e-
-
-
---
E - - -
E-
----
-
- -- El-
-
W_
-I-
-i-
*
-*U-
-U-
-i
r
0O
n
O
:e
>
~
-5
A7-
-10 l,,
-150
i, i,
A
I,
I,
-100
--
I,
I
,
I
,
I
.
A-
I.
-50
I
.
I.
I.
.
0
A-
I
.
I
.
I.
AB--
I
.
I.
.
~--
I. I. .I
50
~-
I
.
.
100
.
I
.
I
I
150
Axial Position (mm)
Figure D.42 Predicted central axial deformation-shape evolution for
geometry R12T2S3 with simply-supported-free boundary
conditions.
-399-
Center Deflections in Millimeters
- e- -0
- - 0.7
-- - -1.3
-o- -0.3
R12T3S1
E
E
----......
----
-----
-
--
-m------------
-
I
0
-
I-
e -
-
'--
-
E
-
aCz
-2
-3
El
-40
n
-
-20
I
,
20
,I
,
40
Axial Position (mm)
Figure D.43 Predicted central axial deformation-shape evolution for
geometry R12T3S1 with simply-supported-free boundary
conditions.
-400-
Center Deflections in Millimeters
--
-El- 1.6
- -0
-*-
-0.5
--- -2.5
R12T3S2
E
E
-- G- -- G-
--
-09-
H--0- H
-E-l-
O
-
V_
--L---
- El-
-
II--- - 0-
- - - -
-
-(
O- -- - -
-I---
- --
I]- - - D -
--
-
U --
-M-
F - .
E
- a--
II
0
Cz
-1
fl)
-2
-~--~~~~-~-.~
-4
-5100
-100
I
I
I
I
,
I
-50
50
, 100I
100
Axial Position (mm)
Figure D.44 Predicted central axial deformation-shape evolution for
geometry R12T3S2 with simply-supported-free boundary
conditions.
-401-
Center Deflections in Millimeters
--
-0
-*-
-1.1
---A -14.9
--- -4.8
--------
-rEI--3.0
10
710.9
10.9
R12T3S3
- ----
-
-e--
---
E
E
]- - -r1l-
-
- -
--
-
-
D - - - E-----
- --
----
-
-@
---
-
- -0-
-
-
-
-
_
e- -
- -0--i
El- - - E- -
--
- - u-
e- -E
-u-
E
--
cn
C
O
- - "-
--._.
._- - - -
-5
A-
&-
A-.A-.A-
A'
A
I..
-101L,
-150 -100
I1
I
1
1
1*I**** 11
-50
1
1
0
S
1
50
I
100
150
Axial Position (mm)
Figure D.45 Predicted central axial deformation-shape evolution for
geometry R12T3S3 with simply-supported-free boundary
conditions.
-402-
Center Deflections in Millimeters
- a- -0
-- E- -0.6
-e-
--
-0.2
-1.0
R6T1S1
E
E
O
C
0
0
a.
- o
---
S3-
--
-
--
----
-----
-
--
--
-
----
.
-- e--
-- e- -
E
----
E-----------E
-
-
0
1I.
Od
>
-2
-40
-20
20
40
Axial Position (mm)
Figure D.46 Predicted central axial deformation-shape evolution for
geometry R6T1S1 with simply-supported-free boundary
conditions.
-403-
Center Deflections in Millimeters
--- -4.0
-e-
-0
-.-
-1.9
S- - 3.0
10
-
-9.0
- A- -10.7
-A_
-7.0
R6T1S2
-
E
E
---
-0-E-
O
--
- --
- -
-
-
- -E-- -
-
- e- - e- -
-
-----r-------o----.._
- -E
E
--- 0- --
-
--
-- -
-
-
0
---- -
-- ---
a-
-
-- --
<
CL
O
A--
>
--
-
AA- --
-
A-
AA-
-10
-100
-50
50
100
Axial Position (mm)
Figure D.47 Predicted central axial deformation-shape evolution for
geometry R6T1S2 with simply-supported-free boundary
conditions.
-404-
Center Deflections in Millimeters
- 9-0-
-- -
-o0
-4.2
-- A
-12.9
- 16.5
-23.7
- A- -28.0
0E- -9.3
20
R6T1S3
-- (-
S-
E
--
--
--
--
-
-
--
O
--
O--
-e-
10
-
-
4-E
-
[
S----
-
---
0
R------
F-
O
--
-----:> ~-
--
- . _
-------------
>
-10
I
-20
-150
I
I
-100
A--
l
i
----
- __
--
---
l
-50
-
I
l
0
I
I
--
e----
A-A
A--
l
-
I
,._~~---~---~---$-~_~
.G
-1---- ---
----
A-
-
-
I
l
50
A---
l
l
--
-
A--
l
100
150
Axial Position (mm)
Figure D.48 Predicted central axial deformation-shape evolution for
geometry R6T1S3 with simply-supported-free boundary
conditions.
-405-
Center Deflections in Millimeters
- 0-
-0
--
--0.8
-0-
-0.3
---
-1.3
R6T2S1
-
--
F
-
O-
E
E
C:
-L]
O
-0
-0(-
- i-
--
-
-
-
-
--
- -- 0-
-
S--E
_
U--
-6ED
-
-
E-].-
-
-E - -- 0--
-U--
-U--
E
-U--
- w-
- 6--
El- -
-E-
0
n
Cz
o
>
----t----~
~~---4+--- -4
I-
-2 ''""""""'"''"
-20
-40
S
I
, I
,
*
I
I
I
20
S
I
I
I
,
40
Axial Position (mm)
Figure D.49 Predicted central axial deformation-shape evolution for
geometry R6T2S1 with simply-supported-free boundary
conditions.
-406-
Center Deflections in Millimeters
-G-
-0
-0-
-1.0
--
-11.8
-3.8
-A-
- 13.6
10
R6T2S2
-
/
E
E
5
- - t -- -e- --i-
- -j--
El -o
-0-- -0--
i-
-
---
-E- - -- El--
U-
-
-
-E-
" - E - S -ED
- I
-
--
- I
O
C
0
0
--+-- ~~ ~-----~--.~
0
>
-5
-10
-100
A-
A-A
A--
IA-- -A-
-
I
I
I
I
, S7-
A,
A-
I
I
-50
-
l
l
l
A-,
&-
k-
,
50
.
I
I
I
I
100
Axial Position (mm)
Figure D.50 Predicted central axial deformation-shape evolution for
geometry R6T2S2 with simply-supported-free boundary
conditions.
-407-
Center Deflections in Millimeters
-0-
--- - 8.6
-- -- 12.1
---16.5
-0
-@- -2.7
E- --5.9
S-24.0
20
R6T2S3
G-
-
E
E
-O-
-- 0-
1-----
0-
-0-
--
--
-i
W
=I
_
I---
I-
---
F
--
UI- -
- - -- - - F-
"]M.
0
-
W-
- - -0
-
-
- W-
- I-
El"
- I
-
0
a_
-10
A-
A-
-
A
-
A-
-A
-20
-150
-100
-50
0
50
100
150
Axial Position (mm)
Figure D.51 Predicted central axial deformation-shape evolution for
geometry R6T2S3 with simply-supported-free boundary
conditions.
-408-
Center Deflections in Millimeters
S0
--E -
- - -0.3
-0.7
-----1.1
R6T3S1
E
E
-
t -
-
t-
e-
-
-
0-
-0-
-
-(
-
-
0-
---
- - El
8-
e-
--.
- --- - ---- 1---- -
.
4-
O
-i-
-U--
-- w
---
-
---
-- -
C
n
i ~ B - ~a
-- B-~--B-~
.e -- e-
Bt
-ce~c..cC.--*---4--~
-3
I
I
-40
I
I
,
,,
I
i
I
-20
|
I
i
I.
.
.
.
.
.
.
20
.
.
.
40
Axial Position (mm)
Figure D.52 Predicted central axial deformation-shape evolution for
geometry R6T3S1 with simply-supported-free boundary
conditions.
-409-
Center Deflections in Millimeters
10
- 0-
-0
-0-
-1.8
El-- 4.4
--M- -6.5
R6T3S2
-
------
eG-
---
G--
-
E
E
C:
3-
- -E]
E- - -El-
- -E -
E -
El-
0
0
I-tl--------
-
---
--''"--
_I--
-
E-
.E
E--
--------
-
-
-- '--
.-
IC.)
U)
-5
S.-
-10
-1 00
.-
---
-50
,---
.
.
. - --
50
i-'--
"
100
Axial Position (mm)
Figure D.53 Predicted central axial deformation-shape evolution for
geometry R6T3S2 with simply-supported-free boundary
conditions.
-410-
Center Deflections in Millimeters
-e-
-0.
-e-
-4.1
-- E--9.0
-- 0-- 18.4
-- m- -13.1
-- -- 24.6
20
R6T3S3
-- G- -G-
10
- 0- -
-
- --
--
I;
-0-
-
C
E
-E-------
n
0
--
-
- -)- e-
-
-
-
E
- 0
El- - - I-- - - -
-
- - ,.
0._.
_ .
0- . _ . |
-10
-20
-0-
:-
E
aO
e-
-Q.-- -0- -- -_
..
-+-----+-----+--
=
|
-150
II
11111111111(1111111
-100
-50
11111111
0
50
100
150
Axial Position (mm)
Figure D.54 Predicted central axial deformation-shape evolution for
geometry R6T3S3 with simply-supported-free boundary
conditions.
-411-
Center Deflections in Millimeters
- -
-o0
--A -1.1
--- -0.5
- -0.1
l- - 0.3
R12T1S1
E
E
I
VI2I....
a
O
.--
--
*--w
U-E]
i --
0
0
A-
i~-
-
--
A-
--
io
>
-1
-2 ''''''"""'''''''''
,
-40
,
,
I
-20
20
40
Axial Position (mm)
Figure D.55 Predicted central axial deformation-shape evolution for
geometry R12T1S1 with pinned-free boundary conditions.
-412-
Center Deflections in Millimeters
-
- -0
-0-
- El--1.2
---
-0.5
-A
-4.6
-2.1
R12T1S2
4I
3E
E-
-0-
-4
E
E
C
1-
R1-
U
-
O
0
.~~----~--.~.~
-~
A-
-AA
A-- r--
A-
A-
&_---
A--
6
-2
-3
-4
-511 1
-100
I
I
-50
I
I
I
I1
I
I
I
I
I
50
I
100
Axial Position (mm)
Figure D.56 Predicted central axial deformation-shape evolution for
geometry R12T1S2 with pinned-free boundary conditions.
-413-
Center Deflections in Millimeters
- e- -o
-
-0-
--- -4.6
-2.1
- -3.5
-*----8.0
10
I
~- -G--
-G--
}-- -- 0I-o
-
a- --
-e-
--
_S_
--I
---
~~e
.0-l
-_4
El.
.
1-.- o
01,
C
R12T1S3
--..
-- -I-.
E
-
S-9.8
-- 0--6.1
-
-H1--
I
- - -0 -- -E _.
0
U)
&-
>
--+
...
4
V... --- -""-- """+
O
-
6 .
- - --- -------
~ A-
A-A-A-A-
"--
A-
A-
-5
•
-150
-100
III
•
•
-
-50
I I
-
-
i
I.
I
.
I.
I
.
I
.
.
50
.
.
I
I
100
I
I
150
Axial Position (mm)
Figure D.57 Predicted central axial deformation-shape evolution for
geometry R12T1S3 with pinned-free boundary conditions.
-414-
Center Deflections in Millimeters
-
- -0.2
---- -0.9
R12T2S1
---
E
E
----
e-
- E-
- --
_.
e--e-E-e-R_
_ _
_ __
-
-
0
-
C)
0
40
--
_I-.--'
-~
--
-
C.)
CO
O
>
-40
-20
20
40
Axial Position (mm)
Figure D.58 Predicted central axial deformation-shape evolution for
geometry R12T2S1 with pinned-free boundary conditions.
-415-
Center Deflections in Millimeters
-
-0
0-
-0-
--- -2.4
-0.5
R12T2S2
-0
3
RT
-
E
E
-0
0
a,
.
-
. E]
E"
2
1
-
0
- - ,0.
.._
0
-
,q
-1
-2
-3
-4
-5100
-100
I
I
I
I
I
-50
I
I
I
I
I
50
I
I
100
Axial Position (mm)
Figure D.59 Predicted central axial deformation-shape evolution for
geometry R12T2S2 with pinned-free boundary conditions.
-416-
Center Deflections in Millimeters
-
- -0
-
- -1.0
- -E--2.4
- --
- 4.2 --
--
-5.9
- 10.1
--8.0
10
R12T2S3
E..
~-0G
EL.
E
E- -4
F-l.
El-
FJ
IW
.-e-,
-U--
-
E
O
C
0
-~----~-- -.
0
C)
AA- A- A-A-
>
A-
A-~
-5
-10
,
-150
i
i
.
.
I
.
-100
.
.
.
I.
-50
-
-
-
0
i
,i
.
.
.
I. .
I
50
.
100
150
Axial Position (mm)
Figure D.60 Predicted central axial deformation-shape evolution for
geometry R12T2S3 with pinned-free boundary conditions.
-417-
Center Deflections in Millimeters
- -E- 0.7
-G
-0
-
- -0.3
-+-
--m- -1.1
-- 2.4
R12T3S1
E
E
C
O
- e- -
- -
-
0- -
I- -.-
A,-
U l-L
4-
-Fm.--n
0- -in -
-- -U-
i_ -
- a-- -
-
0-
-0-
-
-
-0-
-0-
.I'
--
IF-- - M..
ii-
-
-
a- -
E
-
4
Fl--S-- - F1"
-"
-0-
-0-
" -
--
e- -
e- -
-I-
- --
- I--- - a-
-
.-- ~-----~--
.4---.--..
(D
-2
-3
'
I
-40
,
•
,
I
-20
,- "
',
,
,
I
20
,
.
-
40
Axial Position (mm)
Figure D.61 Predicted central axial deformation-shape evolution for
geometry R12T3S1 with pinned-free boundary conditions.
-418-
Center Deflections in Millimeters
- -
-0
--- -1.6 --- --2.9
-@- -0.3
I
3
E
E
O
0
E-
-L
R12T3S2
-0-
2
-E--0
- W.
1
0
-1
Cz
-3.8
--A
- - - -0.7
A-
A-
A--
A--
A-
A---
A---
-2
-3
-4
-5 1 - """"""'"''''
-100
-50
.
I
m
m
i
m
50
,
,
I
100
Axial Position (mm)
Figure D.62 Predicted central axial deformation-shape evolution for
geometry R12T3S2 with pinned-free boundary conditions.
-419-
Center Deflections in Millimeters
- E- -2.1
- e- -0
-
-- 1.0
----
-*---8.1
4.0
10
I
R12T3S3
(
--
~0-
E.
;g
-9.7
-
5!
.~=9
EP
-I
.
aE
O
-I
O
--.
I
-
OC
B---- A---
>
--
A--- -
A
18 _
A-1--
-5
-101 I I I I I
-150 -100
I
I I I I I I
-50
I
0
I
I
50
100
150
Axial Position (mm)
Figure D.63 Predicted central axial deformation-shape evolution for
geometry R12T3S3 with pinned-free boundary conditions.
-420-
Center Deflections in Millimeters
-
-.-
-
-A -1.6
--m- -0.7
-0
-A-
-0.2
-2.0
- --0.5
R6T1S1
E
C~
E
F
-0- .
O
0
LI~=~-
'l - ..
I-
O
I
-u-i-
EL
r-
-
C'
0O
a)
o
0
>
-1
-2
-40
-20
20
40
Axial Position (mm)
Figure D.64 Predicted central axial deformation-shape evoluti(on for
geometry R6T1S1 with pinned-free boundary conditions.
-421-
Center Deflections in Millimeters
-
- --
10
E- -2.2
- -0
S-8.2
1.1 --m- -3.4
R6T1S2
-
E
E
I-
-
(
-0-
El
C
-
El"
-U-
-~ --- s~,
0O
O
s-
>
A-
A-
A-
A-
A--
A-
A-
A-
-5
-1011
-100
,
I
I
I
I
''''~~~~~~~~~~~~~~~
I
-50
I
I
I
I
I
I
I
I
50
I
I
I
I
100
Axial Position (mm)
Figure D.65 Predicted central axial deformation-shape evolution for
geometry R6T1S2 with pinned-free boundary conditions.
-422-
Center Deflections in Millimeters
- --0
-
3.3
-
-13.3
--m- -5.3
-
A- -19.0
-
-0- -1.5
20
R6T1S3
15
--
--
E
E
-
El
-- - 0--
i=--
10'
L7
-W_"
-
O
O
C,
i
-5
- A- A ---
t
-10
-150
l
l
l
-100
l
l
-A A
--
A-
i
l
-50
i
t
-A
iI
i
0
--
-
A-
llli
50
100
150
Axial Position (mm)
Figure D.66 Predicted central axial deformation-shape evolution for
geometry R6T1S3 with pinned-free boundary conditions.
-423-
Center Deflections in Millimeters
- G-e-
-0
-0.2
-
-1.5
- A- -2.0
-- m- -0.6
-EL--0.4
R6T2S1
- - &
-
0-
-e-
-
-e-0-
-EL
]-
--
° o]
--
-E
W
-
-El
.0--
E
E
CI
.0
A--
A-
o
d)
A--
-2 ''"""-'-----'----'
-40
-20
,
I
,
I
I
I
A--
I
I
20
I
I
I
40
Axial Position (mm)
Figure D.67 Predicted central axial deformation-shape evolution for
geometry R6T2S1 with pinned-free boundary conditions.
-424-
Center Deflections in Millimeters
- e- -0
--- -3.4
-0.8
-0-
10
-- A -8.8
-1.8
E
R6T2S2
E
E
5' " ~e _g
--
-Ei
1-"
E.
U--
-
mo
C:
O
04 ----
-
.
0>
0.
0
-
_*
- - .&
--
---- A
1
4
+
---
A-
A -A --A
A--
_ -
""_
"
A--,
A---
-5
-1011 , ''''"~~'~~~~~~~~~'
I
-100
I
I
-50
I
I
I
I
I
I
I
I
50
I
I
I
I
I
100
Axial Position (mm)
Figure D.68 Predicted central axial deformation-shape evolution for
geometry R6T2S2 with pinned-free boundary conditions.
-425-
Center Deflections in Millimeters
- a- -0
- --
-0-
--- -6.7
-2.0
- 4.1
S-15.9
- A- -19.4
20
R6T2S3
- -- - e- - -
10
A
E
-
- --
A
-
A-
-- --
C
C-8
a
-10 1-
_-1 I
-150
a
-
i-
A
i
i
A-
-
-
-
---
--
--
-
-
-
-
-
--
-
i
-100
-50
0
50
100
150
Axial Position (mm)
Figure D.69 Predicted central axial deformation-shape evolution for
geometry R6T2S3 with pinned-free boundary conditions.
-426-
Center Deflections in Millimeters
- G- -o0
-*- -0.4
E - - 0.9
--A
--- -1.3
-3.6
R6T3S1
E
E
C_
- -
-
-
a-
e-
0
I- -
0O
-
-
-
U-
Cz
E-
0cn
o
V'
>d
+-
- -,+-
-
_-
0
-
-0-
-
-I- -
E
-MF_
W_-
S
"El-
"" El
-I -
- WU-
OF~
0_~-.8--~..8--.-<
--
--
. ,
---- - - ,--
- - - --
-€
A--
-3
-
e-
-
a-
-4.
..
-*~
1~_--~----
It=
CD
-
- 0- -
1-- - -*
S- ElFI-
a-
II
-40
A---
.I
.I
. I.
.
A-
e!
I.
-20
.
..
.I.
I
20
,
,
I
,
40
Axial Position (mm)
Figure D.70 Predicted central axial deformation-shape evolution for
geometry R6T3S1 with pinned-free boundary conditions.
-427-
Center Deflections in Millimeters
10
-r0--2.6
---
-0
-*-
-1.2
--
-4.7
R6T3S2
E
E
5
0w
" -
El
. E
E-IF
O
0
-- U-----
0
.-+- -41
C-
>
-5
_-In
1 0
-100
II
.
.
---41--- -~
--I
.
..
-
-- --- -- -- -- --- --
.
I
.
-.
..
''"""''"'''''''
-50
50
100
Axial Position (mm)
Figure D.71 Predicted central axial deformation-shape evolution for
geometry R6T3S2 with pinned-free boundary conditions.
-428-
Center Deflections in Millimeters
- - - -3.5
- -0
-
-
-19.1
- 6.7 -*----13.1
- *- - 1.6 --
20
R6T3S3
E
E
10, -
-E-
-" E -
-
-
E
-
-
--4-i-
0O
O
0
0
i
--....
o
a-
A
A
A -.....
-"
- "
--
Cz
-10
-20 III
-150
i
l
n
-100
n
I
I
.
-50
I
..
0
I
I
I
50
I l .
100
.
I
150
Axial Position (mm)
Figure D.72 Predicted central axial deformation-shape evolution for
geometry R6T3S3 with pinned-free boundary conditions.
-429-
Center Deflections in Millimeters
-A
---- -0.5
-- --0.8 -.-0.1
- -E --0.3 -- -- 1.2
-1.8
- -2.4
E
O
c,
O
>
-1
-2
-3 L
0
20
40
60
80
100
Spanwise Position (mm)
Figure D.73 Predicted central spanwise DFU evolution for geometry
R12T1S1 with simply-supported-free boundary conditions.
-430-
Center Deflections in Millimeters
- & -0
- -o- -2.1 - - -- 4.3 --
- *- - 1.0 ---m- - 3.1 --
-7.7
- -5.9
R12T1S2
L4
j
a
C
0-
"
-E.
-
-2
0
r
0C
t
g0
_
-
C2
O an
O
-
%
;
-
_0
ir :--
. El. E]-.1-
-r
Ip
-iE' r0"
~YEY
El,
ci
00
'
C
o -
V*
0
0
Or
0
-4
.*
4K
a)
-6
/P
A
-8
0
I
I
I
I
50
"A
*1A
I
~II
100
I
150
200
Spanwise Position (mm)
Figure D.74 Predicted central spanwise DFU evolution for geometry
R12T1S2 with simply-supported-free boundary conditions.
-431-
Center Deflections in Millimeters
-
-0
-*-
-2.1
-4.5
-1- -6.8
-- - 9.5
-* - - 12.2
A -14.9
R12T1S3
E
E
O
--E
-4
0_0.
E" F,
ni
/El
"0 -E-
0
0O
El,
,
'
-8
/0
0z
) , , I I I I
-16
. .
. .
,
I , I ,
w
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.75 Predicted central spanwise DFU evolution for geometry
R12T1S3 with simply-supported-free boundary conditions.
-432-
Center Deflections in Millimeters
-- --0.5
--- - 0.8
- - -0.2
-- -- 1.9
E
E
C
O
4-0
O
-1
0
aC,
0
-2
-31
0
I
I
I
20
I
I
II I
I
40
I
60
I
I
I
I
80
100
Spanwise Position (mm)
Figure D.76 Predicted central spanwise DFU evolution for geometry
R12T2S1 with simply-supported-free boundary conditions.
-433-
Center Deflections in Millimeters
E-- 2.0
- *- -0.8
-
-3.1
R12T2S2
E
E
O
C:
\L
"
W
El E-El E
-2
*'
E El'.
or 0
-- i E EE
n.zO
0
F--
-4
I
>
-6
-8
I
I
I
\-- I
.P
I
es s e
I lll l i
50
100
I
,
150
l i s a
200
Spanwise Position (mm)
Figure D.77 Predicted central spanwise DFU evolution for geometry
R12T2S2 with simply-supported-free boundary conditions.
-434-
Center Deflections in Millimeters
-
-
0
- -u - - 3.2 --.--
- - 1.4 -
7.2 --
- 14.0
-4.9 -*- -- 10.3
E
E
C)
O
-4
o
O
-8
-
-12
-161
0
.1 I I
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.78 Predicted central spanwise DFU evolution for geometry
R12T2S3 with simply-supported-free boundary conditions.
-435-
Center Deflections in Millimeters
- -0.3
-
---
- 1.3
R12T3S1
E
E
'' 01 ~ ~ ' ' ' ' ~' ' ' ' ' ' ' ~" ' ' ' ' ' ' ' ' ' ' 's-' ' ' ' ' )m/
F
ES
c, IE
\ELJ
e-
-
* *
i
-
.
,(
/~
E'
E
-l
O
0O
ETU'
EU"
- 0- 0_ 0_
-/
-1
/
0_
o
:E
a)
-2
10.
-3 L
0
I
I
I
I
20
I
I
40
I
-0
Ii
I
I
60
I , 80
80
100
Spanwise Position (mm)
Figure D.79 Predicted central spanwise DFU evolution for geometry
R12T3S1 with simply-supported-free boundary conditions.
-436-
Center Deflections in Millimeters
-El-- 1.6
- *- -0.5
--
- 2.5
E
E
C
O
-2
0
0a
13_
-4
-6
-8
50
100
150
200
Spanwise Position (mm)
Figure D.80 Predicted central spanwise DFU evolution for geometry
R12T3S2 with simply-supported-free boundary conditions.
-437-
Center Deflections in Millimeters
-.-
-1.1
-E--3.0
---- 4.8
--.--7.5
----- 10.9
--
-14.9
E
E
O
-4
C
O
a.
-8
-12
-16L
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.81 Predicted central spanwise DFU evolution for geometry
R12T3S3 with simply-supported-free boundary conditions.
-438-
Center Deflections in Millimeters
-e -0
--
1
-0.2
-o--0.6
----
--0-- 1.5
-+- -- 2.4
-1.0
E
E
C
0
C,
0'
-31
0
1 1
I 1 I I I I I1
20
40
I 1 1 1 I
60
80
100
Spanwise Position (mm)
Figure D.82 Predicted central spanwise DFU evolution for geometry
R6T1S1 with simply-supported-free boundary conditions.
-439-
Center Deflections in Millimeters
-.--
-*- -1.9
- - 3.0
-- -4.0
--0-- 5.3
----- 7.0
S-9.0
- A- -10.7
E
E
c
-4
0
0
-)
Cd
-8
-12
-16L
0
50
100
150
200
Spanwise Position (mm)
Figure D.83 Predicted central spanwise DFU evolution for geometry
R6T1S2 with simply-supported-free boundary conditions.
-440-
Center Deflections in Millimeters
-- - 12.9
---- 16.5
- - -4.2
- - --9.3
-- 6 -23.7
- 28.0
R6T1S3
_
E
E
:
0
_
1K\\n\
0"- -9_ O_
-5
.
-
.o..;
..
m OrOrID
L-:J
"11, E "E]
E E
I I -. E
.10
-
.o- =
o,
ll"
1\
\O
0
/ -/
I
SIt
/J
C- -15
A3/
C)
)
.20
P
/7
~.
1< /
/
-25 -30
I I
O0
I
I I
50
I .I
II
100
I I I.
II
I
150
I
I
200
I
I
250
. I
300
Spanwise Position (mm)
Figure D.84 Predicted central spanwise DFU evolution for geometry
R6T1S3 with simply-supported-free boundary conditions.
-441-
Center Deflections in Millimeters
-
- -0.3
--m- -1.3
E
E
C
a-
0-
-2
-3
-4 L
0
20
40
60
80
100
Spanwise Position (mm)
Figure D.85 Predicted central spanwise DFU evolution for geometry
R6T2S1 with simply-supported-free boundary conditions.
-442-
Center Deflections in Millimeters
- o--0
- e- - 1.0
-a- -3.8
---- 5.8
--E- -2.5
--
-A
-11.8
- A- -13.6
--8.4
01
0E
E
O
-5
CL
Cz
-10
-15
50
100
150
200
Spanwise Position (mm)
Figure D.86 Predicted central spanwise DFU evolution for geometry
R6T2S2 with simply-supported-free boundary conditions.
-443-
Center Deflections in Millimeters
- - -0
- - -2.7
- -- --- 5.9
E
E
Co
0
aO
-----8.6
---- 12.1
-- -- 16.5
S-24.0
-5
-1
C.
O
-1
-20
-25
-30
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.87 Predicted central spanwise DFU evolution for geometry
R6T2S3 with simply-supported-free boundary conditions.
-444-
Center Deflections in Millimeters
--E--0.7
- *--0.3
--m--1.1 -- -- 2.9
R6T3S1
El.
E
E
L4)
C:
0
Lis~
El 0- O..
EI.E
13
-1
El
. El
El- - -
El
FI 0 . FI.
o-o-
l
j"
0
'
0
U-u.
0
>
CL
Cz
- , -
."
08
-2
0/
,*-4.*---
-3
-4
I
I
I
I
I
20
40
I
60
,
,
I
80
---
100
Spanwise Position (mm)
Figure D.88 Predicted central spanwise DFU evolution for geometry
R6T3S1 with simply-supported-free boundary conditions.
-445-
Center Deflections in Millimeters
-o- -1.8
--a- - 6.5
--
--- 14.7
E
E
C:
aO
-5
Cm
>
-10
-15L
0
50
100
150
200
Spanwise Position (mm)
Figure D.89 Predicted central spanwise DFU evolution for geometry
R6T3S2 with simply-supported-free boundary conditions.
-446-
Center Deflections in Millimeters
-- o--18.4
-E-0--9.0
-e-
---
-4.1
- 13.1
R6T3S3
co o- o;
S-
i
O
or
E
E
-5
l E
E
C
0
-
-10
E
E
@
/
/
.
-
It
-15
U-
P
p1
0
C
(D
E]
rlE
F-l"
_I 1 7
ElY
\]
-20
-
-25
-30
, , ,
C)
I ,
50
,
I ,
100
, ,
Ar,
-
150
200
250
300
Spanwise Position (mm)
Figure D.90 Predicted central spanwise DFU evolution for geometry
R6T3S3 with simply-supported-free boundary conditions.
-447-
Center Deflections in Millimeters
-0-
-0.1
-----0.5
--O--0.7
-A4 -1.1
- --0.3
E
E
C
4O
0
-1
0
0)
-2
-3
0
I
II
I
20
1
I
40
6I
60
80
100
Spanwise Position (mm)
Figure D.91 Predicted central spanwise DFU evolution for geometry
R12T1S1 with pinned-free boundary conditions.
-448-
Center Deflections in Millimeters
- - - - 1.2 - - - - 2.8
--
-0
-o-
-0.5 ---
-
-4.6
- 2.1
E
E
C'
O
-2
0
a.
-4
-6
-81 1I I I I I I I I I 1 1 1
0
50
100
I
150
200
Spanwise Position (mm)
Figure D.92 Predicted central spanwise DFU evolution for geometry
R12T1S2 with pinned-free boundary conditions.
-449-
Center Deflections in Millimeters
-
-e-
E- - 3.5
-0
-2.1
---
-9.8
--
-4.6
E
E
O
-4
tO
_
-8
"-E
-12
-161 1
0
11
50
11I 1.1 . I..1 111 1I. I
100
150
200
1.. I.
.I . ..
250
300
Spanwise Position (mm)
Figure D.93 Predicted central spanwise DFU evolution for geometry
R12T1S3 with pinned-free boundary conditions.
-450-
Center Deflections in Millimeters
-
-0
- - 0.2
--
-0.6
--
- 0.9
E
E
C
O
nO
a)
-21
0
I I I I I I I I II I I I I I I
20
40
60
80
100
Spanwise Position (mm)
Figure D.94 Predicted central spanwise DFU evolution for geometry
R12T2S1 with pinned-free boundary conditions.
-451-
Center Deflections in Millimeters
- -E--1.2
-e-
-0.5
--a- -2.4
E
E
0
-2
0
0-
-4
-6
-8
50
100
150
200
Spanwise Position (mm)
Figure D.95 Predicted central spanwise DFU evolution for geometry
R12T2S2 with pinned-free boundary conditions.
-452-
Center Deflections in Millimeters
--
- -E- - 2.4 -
-0
- 9- - 1.0 -- m- -4.2 --
- -5.9
-10.1
- - 8.0
E
O
-4
0
_0
-8
-
-161 ,,,,,,
0
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.96 Predicted central spanwise DFU evolution for geometry
R12T2S3 with pinned-free boundary conditions.
-453-
Center Deflections in Millimeters
E
- - -0
--- -0.7
- e- -0.3
--m- -1.1
---- 2.4
01
E
C
O
a0z
-1
>C
-2
-3
20
40
60
80
100
Spanwise Position (mm)
Figure D.97 Predicted central spanwise DFU evolution for geometry
R12T3S1 with pinned-free boundary conditions.
-454-
Center Deflections in Millimeters
- - -0
- --0.7 -- -- 2.2 -----3.8
- *- -0.3 ---m- -1.6 --
-- 2.9
E
E
C
-2
40
Ci
-4
-6
-8 L
0
50
100
150
200
Spanwise Position (mm)
Figure D.98 Predicted central spanwise DFU evolution for geometry
R12T3S2 with pinned-free boundary conditions.
-455-
Center Deflections in Millimeters
- &- -0
-- n --2.1
-0- -1.0
--- -4.0
-9.7
--A
E
E
C,
0
-4
O
0
-8
0
-12
-16
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.99 Predicted central spanwise DFU evolution for geometry
R12T3S3 with pinned-free boundary conditions.
-456-
Center Deflections in Millimeters
-- e.
.
[]
-
-0
-0.2
-0.5
-U- .
-+,..
-0.7
.
-0.9
-1.3
-1.6
-2.0
- A- A-
0
E
C,
O
0
O
>
-2
-3 1
0
, .
.
.
.
20
.
.
.i ,
40
,
. I
60
.
.
.
80
100
Spanwise Position (mm)
Figure D.100
Predicted central spanwise DFU evolution for geometry
R6T1S1 with pinned-free boundary conditions.
-457-
Center Deflections in Millimeters
-
- -o
- -E- -2.2 -- --4.4 --
-8.2
- e- -1.1 --- -3.4 --, --6.0
R6T1S2
E
E
O
I
'E\
I
0
.
.
E3
0.)
. .,
E]
E3
I
i
El
-4
-81
a)
Oz
-12~
-16
50
100
150
200
Spanwise Position (mm)
Figure D.101
Predicted central spanwise DFU evolution for geometry
R6T1S2 with pinned-free boundary conditions.
-458-
Center Deflections in Millimeters
- - -0
--E--3.3
- e- - 1.5
-- m- -5.3
-A
-
---10.2
150
200
-13.3
- A- -19.0
-5
0E
E
I-
-10
O
-15
a> -20
-25
-30
50
100
250
300
Spanwise Position (mm)
Figure D.102
Predicted central spanwise DFU evolution for geometry
R6T1S3 with pinned-free boundary conditions.
-459-
Center Deflections in Millimeters
--m -0.6
- - -0.2 --- 0.9 -A-l - --0.4 -- -- 1.2
-.-
0
-1.5
-2.0
0
0
-1
O
CL
-2
o
a)
-3
-4 L
0
20
40
60
80
100
Spanwise Position (mm)
Figure D.103
Predicted central spanwise DFU evolution for geometry
R6T2S1 with pinned-free boundary conditions.
-460-
Center Deflections in Millimeters
- &- -0
- -E- - 1.8 --
- *- -0.8 --
o--5.0
-- a
-8.8
-3.4
0
0
E
-5
01
0
50
100
150
200
Spanwise Position (mm)
Figure D.104
Predicted central spanwise DFU evolution for geometry
R6T2S2 with pinned-free boundary conditions.
-461-
Center Deflections in Millimeters
- - -0
- E- -4.1 -- o--8.9 -
- o- -2.0 ----
E
E
C,,
0-
a)
-6.7 --
- 15.9
-- 12.0 - A- - 19.4
-5
-10
-15
-20
-25
-30
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.105
Predicted central spanwise DFU evolution for geometry
R6T2S3 with pinned-free boundary conditions.
-462-
Center Deflections in Millimeters
-9-
-0.4
--- - 1.3
-- --1.7
--A
-3.6
----- 2.4
E
E
E
C-
0
O
n
0
0
a_
0
20
40
60
80
100
Spanwise Position (mm)
Figure D.106
Predicted central spanwise DFU evolution for geometry
R6T3S1 with pinned-free boundary conditions.
-463-
Center Deflections in Millimeters
-o- -1.2
-m- - 4.7
-- --7.9
E
E
c
C
a) -1
-1
0
50
100
150
200
Spanwise Position (mm)
Figure D.107 Predicted central spanwise DFU evolution for geometry
R6T3S2 with pinned-free boundary conditions.
-464-
Center Deflections in Millimeters
-e- -0
- -in-3.5 --s--9.7
- 9- - 1.6 -m- -6.7 --
E
E
-5
O
-10
0
0n
-19.1
-- 13.1
-15
C> -20
-25
-30
50
100
150
200
250
300
Spanwise Position (mm)
Figure D.108
Predicted central spanwise DFU evolution for geometry
R6T3S3 with pinned-free boundary conditions.