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Casey, Ryan
A Design and Optimization Study of a Composite
Rotor System Tension-Torsion Strap
by
Ryan T. Casey
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravette, Engineering Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
May, 2010
© Copyright 2010
by
Ryan T. Casey
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ....................................................................................................... vii
ABSTRACT .................................................................................................................... viii
1. Introduction .................................................................................................................. 1
1.1
Preamble ............................................................................................................. 1
1.2
Previous Works .................................................................................................. 2
1.3
Composites ......................................................................................................... 3
1.4
1.3.1
Fiber Materials ....................................................................................... 4
1.3.2
Matrix Materials ..................................................................................... 5
Tension-Torsion Strap Characteristics ............................................................... 6
2. Methodology ................................................................................................................ 9
2.1
Classical Laminate Theory ................................................................................. 9
2.1.1 Assumptions of Laminate Theory ............................................................. 9
2.1.2
Fundamental Equation of Laminate Theory ......................................... 10
2.1.3
Equivalent Laminate Properties for Quasi-isotropic Laminates .......... 11
2.2
Preliminary Tension-Torsion Strap Sizing ....................................................... 13
2.3
FEA Model Development ................................................................................ 16
3. Results ........................................................................................................................ 19
3.1
Determination of Laminate Properties ............................................................. 20
3.2
Determination of Stress Concentrations ........................................................... 21
3.3
Lug Analysis ..................................................................................................... 23
3.3.1
Net Tension Failure .............................................................................. 23
3.3.2
Bearing Failure ..................................................................................... 24
3.4
Effect of Flatwise Bending ............................................................................... 25
3.5
Shear Stress Response to Varying Loads ......................................................... 26
iii
3.6
Strain due to CF Loading ................................................................................. 27
3.7
Optimization of Geometry................................................................................ 28
3.7.1
Optimization: Pitch Section Width....................................................... 28
3.7.2
Optimization: Pitch Section Length ..................................................... 29
3.7.3
Optimization: Effect of Additional Plies .............................................. 30
4. Conclusions ................................................................................................................ 32
4.1
Final Configuration and Conditions ................................................................. 32
4.2
Future Fatigue Substantiation & Testing.......................................................... 32
4.3
Closing Remarks .............................................................................................. 34
5. REFRERENCES ........................................................................................................ 35
Appendix A: MATLAB Code for Laminate Properties................................................... 36
Appendix B: Preliminary Sizing for Max Shear, Axial Strain/Stress .............................. 41
Appendix C: Analysis of Laminate Cases ....................................................................... 47
iv
LIST OF TABLES
Table 2-1: Normal Stress Due to CF vs Cross-Sectional Area ........................................ 14
Table 2-2: Max Shear, Varying Pitch Value & Cross-Section ........................................ 15
Table 3-1: Equivalent Properties of Quasi-Isotropic Laminates ...................................... 20
Table 3-2: Ply Orientations .............................................................................................. 20
Table 3-3: Lug Stress Allowables .................................................................................... 23
Table 3-4: Interlaminar Shear Stress Due to Flatwise Bending Load.............................. 25
Table 4-1: Final Configuration Data ................................................................................ 32
Table 4-2: Shear Stresses & Axial Strain due to Static Loading ..................................... 32
v
LIST OF FIGURES
Figure 1-1: Common Main Rotor Configuration with Tension-Torsion Strap .................. 1
Figure 1-2: Rotorhead of MD 500E Helicopter [12] ......................................................... 2
Figure 1-3: Tensile Modulus of IM7/8552 ....................................................................... 7
Figure 1-4: Tensile Strength of IM7/8552 ......................................................................... 7
Figure 1-5: Shear Strength of IM7/8552 ............................................................................ 8
Figure 2-1: Coordinates for Laminate Theory ................................................................. 12
Figure 2-2: ANSYS Model Creation .............................................................................. 17
Figure 2-3: Sphere of Influence Application ................................................................... 18
Figure 2-4: Mesh Resulting from Sphere of Influence .................................................... 18
Figure 3-1: Geometrical Features of the Tension-Torsion Strap ..................................... 19
Figure 3-2: Thin, Flat Plate Loading for KT Determination............................................ 21
Figure 3-3: Kt vs e/d Ratio ............................................................................................... 22
Figure 3-4: Stress Concentration Study ........................................................................... 23
Figure 3-5 Shear Stress Locations of Concern................................................................. 25
Figure 3-6: Interlaminar Shear VS Angular Displacement .............................................. 26
Figure 3-7: XY Shear Stress in Transition Radius ........................................................... 27
Figure 3-8: Stretching Due to CF Load ........................................................................... 27
Figure 3-9: Effect of Pitch Section Width on Shear Stress .............................................. 28
Figure 3-10: Effect of Pitch Section Width on Normal Stresses ..................................... 29
Figure 3-11: Effect of Pitch Section Width on Weight .................................................... 29
Figure 3-12: Effect of Pitch Section Length on Shear Stresses ....................................... 30
Figure 3-13: Effect of Adding Plies on Shear Stresses .................................................... 31
vi
LIST OF SYMBOLS
A
a
a*
In-Plane Stiffness Matrix
Length of Side of Square Cross-section
Laminate Compliance Matrix
Abearing
B
CF
D
d
E
e
F
in
2
in
Projected Contact Area (Bearing)
Bending-Stretching Coupling Matrix
Centrifugal Force
Bending Stiffness Matrix
Hole diameter
Young's Modulus
Edge Distance
Force
in
psi
in
lb
Fnt
Laminate net tension stress allowable
psi
Fso
G
H
J
Kt
L
l
Laminate bearing stress allowable
Shear Modulus of Elasticity
1/2 Total Laminate Thickness
Polar Moment of Inertia
Stress Concentration Factor
Length of Beam / Total Length of Strap
Pitch Section Length
psi
psi
in
4
in
in
in
Bearing Contact Length
in
lbearing
MSbearing
MSnt
Papplied
lbf
Margin of Safety (Bearing Stress Failure)
Margin of Safety (Net Tension Failure)
Applied Tensile Load
lbf
Pnt
Qij
Net Tension Load to Failure
lbf
qij
t
T
W
z
γ
ε
ν
σ
Reduced, Transformed Stiffness Coefficients
Total laminate thickness
Applied Torque
Lug width
Midplane Offset
Shear Strain
Tensile Strain
Poisson's ratio
Normal Stress
Reduced Stiffness Coefficients
σbearing
τ
φ
Bearing Stress
Shear Stress
Torsional Displacement (Pitch Angle)
in
lb-in
in
in
in/in
in/in
psi
psi
psi
deg
-Variables are in units listed, unless otherwise noted-
vii
ABSTRACT
The objective of this project was the design, stress analysis and optimization of a
helicopter rotor system composite tension-torsion strap. A helicopter rotor system may
be designed with the use of a tension-torsion strap for blade retention outboard of the
effective flap and lead/lag hinge.
This strap needs to endure tensile load due to
centrifugal force, twisting due to blade pitch inputs, and flatwise bending loads due to
pitch housing deflection. Classical laminate theory is utilized for the creation of a
MATLAB calculator which determines laminate properties for different ply
configurations.
Finite element analysis models created in ANSYS aid in the
determination of stress concentrations in the inboard and outboard lug attachment holes
and are used for static structural analysis of preliminary designs. A resulting composite
tension-torsion strap design along with a number of educated recommendations for
fatigue life prediction and testing are found and discussed.
viii
1. Introduction
1.1 Preamble
A helicopter rotor system may be designed with a tension-torsion strap assembly such
that the inboard pin attachment of the strap lies outboard of both the flap and lead/lag
hinge for the system, but is fixed in terms of pitch. The outboard pin of the tensiontorsion strap would generally be attached to some form of pitch housing. The strap is
designed to carry centrifugal force (CF) and to endure twist about the pitching axis. A
design engineer must also account for a small amount of flatwise bending, depending on
whether the designed pitch housing will be expected to deflect. Figure 1-1 displays an
example rotor configuration from hub attachment (flap hinge) to the blade attachment
with the use of a tension-torsion strap.
Pitch Housing
Blade Attachment
& Lead/Lag Hinge
Flap Hinge
Tension-Torsion Strap
Figure 1-1: Common Main Rotor Configuration with Tension-Torsion Strap
Within the pitch housing, there is a bearing element to restrict all degrees of freedom
other than that of pitchwise rotation.
The small flatwise bending term, which the
tension-torsion strap carries, is due to deflection within these bearings or within the pitch
housing itself. The flap hinge is located inboard of the strap in this configuration, and
1
the lead/lag hinge is located outboard at the blade attachment. Typically, a damper is
used at the lead/lag hinge to restrict the blade’s lead-lag motions.
Figure 1-2 shows the five bladed rotor head of the McDonnell Douglas 500E helicopter.
This rotor system utilizes steel tension-torsion straps as described above. The straps are
located within the pitch housings, which, in this case, attach directly to the blade root via
two large blade pins. The large cylindrical canisters are the lead-lag dampers and attach
to the inboard end of the pitch housings. Pitch is actuated by translating (collective) and
“tipping” (cyclic) the swashplate located below the rotor plane. The pitch motion is
transferred from the rotating swashplate to the blades by way of control rods (attached to
the leading edge of the pitch housing).
Figure 1-2: Rotorhead of MD 500E Helicopter [12]
1.2 Previous Works
Tension-torsion straps have been designed for production use as buildups of long, thin
steel shims separated by thin metal or Teflon spacers.
2
As with many aerospace
components, the objective here is to find the next strong yet lightweight material capable
of performing the job of a heavier (in this case steel) component. Due to the potential of
reducing weight, as well as documented display of good performance under high tensile
loads, composite materials appear to be a good candidate for this application.
In 2009, Moon and Fermen-Coker produced an analysis of a “tie-bar strap” used on the
Boeing H-46 and discussed methodologies that were created to develop stress spectra
and to predict fatigue failures likely to be induced during testing [1]. Transfer functions
of load to stress/strain were developed to capture combined loading scenarios of CF
tensile loading, torsional twisting, and flatwise bending. These functions were used to
develop a nonlinear FEA model. The model provided notch factors and multi-axial
fatigue failure theories that were utilized to produce an expected fatigue life range.
Moon and Fermen-Coker’s report provides important insight into certain characteristics
that would be desired for the optimal performance of a tension-torsion strap.
As Moon and Fermen-Coker discuss the determination of stress concentrations in holes
for metallic parts, Henshaw, Sorem Jr. and Glaessgen provide an in-depth look at stress
concentrations due to holes in composite structures [2]. They investigated the effect of
utilizing multiple adjacent holes of different sizes and locations and provided stress
concentration studies for both tensile and shear loading cases. Cheung et al investigated
stress concentrations due to holes in composites and developed a finite element failure
model based on Tsai-Wu theory and progressive damage evolution [3]. Norton has
outlined the necessary failure modes of attachment lugs [4] and Hibbeler has provided
direction on determining shear stresses due to torsion in beams of various cross-sections
[5]. These works provide direction for preliminary sizing of this composite tensiontorsion strap.
1.3 Composites
Composite materials, as used in the field of engineering, are those that are formed from
at least two dissimilar materials. The product is a finished structure made of combined
materials which, while maintaining their individual physical and chemical properties,
3
produce a part with improved characteristics.
The use of composite materials has
become the clear direction in the future development of aerospace structural technology
mostly due to the impressive strength to weight ratios of these materials. A major
tradeoff of this important benefit is not only the difficulty of fabrication, but difficulty in
aligning the properties (strength, stiffness, etc) of the resulting fabrication with those
predicted analytically. Analytical methods have been devised to predict conservative
behavior of composites, but frequently the results do not reflect initial calculations,
especially if there is a lack of control in manufacturing methods.
Composite structures for many aerospace applications are formed with the use of two
phases - fiber and matrix. In most cases, the fiber dictates the material properties of the
composite and the matrix is used for reinforcement. A short discussion of the many
types of fiber and matrix used in composites engineering follows.
1.3.1
Fiber Materials
Composite fibers have been made from a number of materials since the initial
development of glass fibers in the 1930’s. Twenty years later, fibers with high specific
stiffness and specific strength were being refined for use in engineering structures.
Today, the four major fiber classes are: glass, organic, carbon, and ceramics. The most
common glass fibers used are E-Glass and S-2. E-glass has a relatively high strength
and electrical resistivity. S-2 glass is ideal for components which may experience a wide
operating temperature range and corrosive environments. Both E-glass and S-2 have a
fairly low modulus of elasticity when compared to the other fiber classes, which can
limit their applicability.
The organic fiber class is made up exclusively of aramid, which is also widely known as
Kevlar. Aramid forms tend to be fairly flexible and strong in tension, but have only
moderate modulus of elasticity and poor compressive strength.
The carbon class of fibers is made up more specifically of carbon and graphite type
fibers. The most notable difference between the two is that carbon fibers are composed
4
of up to 85% carbon, while graphite fibers are composed of as much as 95% carbon.
Graphite is achieved through a process called graphitization where the material is heated
to 5400 degrees Fahrenheit in an inert atmosphere, whereas carbon fibers are heated only
as high as 3600 degrees. The temperature difference during fabrication results in carbon
fibers that have a high tensile strength, while graphite fibers have a high modulus of
elasticity.
The use of ceramics as fibers for composites is fairly widespread. One type of ceramic
fiber is boron, which is technically a standalone composite itself. It is composed of
boron filaments which are created using chemical vapor deposition on a tungsten core.
Alumina fibers are a ceramic which exhibit exemplary strength retention when exposed
to high temperatures. Lastly, Silicon carbide (SiC) is a ceramic fiber which is well
known as two separate forms, SCS-6 and nicalon. SCS-6 is also created using chemical
vapor deposition and is generally very stiff. Nicalon is much more flexible due to its
filament size being as much as one-tenth smaller than that of SCS-6.
1.3.2
Matrix Materials
The materials most commonly used for composites are polymers, metals, and ceramics.
The two main types of polymers are themoplastics and thermosets.
A thermoplastic is a polymer that can be shaped and reshaped using a controlled heat
and pressure environment. They have an operating temperature range of up to 437°F.
Some common thermoplastics used in composite matrix are nylon, polyurethane,
polypropylene, polyvinyl chloride, polyphenylene sulfide (PPS), and poly-ether-etherketone (PEEK). These materials are generally low cost.
Thermosets can not be reshaped upon heating because they become “cross-linked”
during fabrication [6]. Epoxies, polyimides, and polyesters are some thermosets which
have been used extensively for aerospace applications.
The temperature range for
thermosets is fairly restrictive; with polyimides having the highest usage temperature of
5
about 570°F. Unfortunately, polyimides are also the most difficult thermoset polymer
to fabricate.
The matrix can also be composed using metals, including copper, titanium, and
aluminum. The biggest drawback to using metals is the increase in weight of the
composite, even if the volume ratio of the matrix is relatively low. However, utilizing a
metal for a matrix can have its benefits. Typically, usage temperatures, toughness,
transverse strength, and thermal conductivity will all be higher when using metallic
matrices.
Lastly, ceramics may be used for matrix as well, but their application must take into
account their very brittle behavior. Still, carbon, silicon nitride, and silicon carbide
continue to be utilized as ceramic matrix materials.
1.4 Tension-Torsion Strap Characteristics
Design considerations when using composite laminates can be quite numerous,
depending on the application. A few immediate constraints are imposed on the design of
this tension-torsion strap to place bounds on the project. First, the designer will be
dealing with symmetric laminates only. The benefit of this will become quite evident in
the section discussing laminate theory.
Composite material will be restricted to
IM7/8552 unidirectional tape. This material has a graphite fiber with an epoxy matrix.
The flexibility of carbon fibers is important for this application so that the strap is able to
withstand torsional twisting motions.
Epoxy is used due to its relative ease of
fabrication and resistance to shrinking on curing when compared to polyimides or
polyesters.
Tables 1-3 through 1-5 contain material properties for individual plies of IM7/8552
carbon/epoxy matrix, as supplied on a material data sheet by Hexcel Corporation [7].
An expanded discussion on the chosen designs is provided in the methodology section of
the report.
6
Tensile Modulus vs Temperature,
IM7/8552
25
Tensile Modulus (Msi)
20
15
10
5
0
-100
0
100
200
300
400
500
Temperature (deg F)
0 Deg, Dry
90 Deg, Dry
Figure 1-3: Tensile Modulus of IM7/8552
Tensile Strength vs Temperature,
IM7/8552
Tensile Strength (ksi)
500
400
300
200
100
0
-100
-50
0
50
100
150
Temperature (deg F)
0 deg, dry
90 deg, dry
Figure 1-4: Tensile Strength of IM7/8552
7
200
250
Shear Strength vs Temperature,
IM7/8552
Shear Strength (ksi)
25
20
15
10
5
0
0
50
100
150
200
250
Temperature (deg F)
0 Deg Short Beam, Dry
0 Deg Short Beam, Wet
+/- 45 Deg In-plane, Dry
Figure 1-5: Shear Strength of IM7/8552
8
2. Methodology
2.1 Classical Laminate Theory
Laminate theory has been developed to analyze a laminated composite which undergoes
in-plane loads and bending moments. This methodology results in the determination of
effective material properties (tensile and shear modulus of elasticity and Poisson’s ratio).
A full discussion on laminate theory has been provided by Herakovich [6].
A
mathematical model was developed using MATLAB which uses this theory to arrive at
material properties for each specified case.
2.1.1 Assumptions of Laminate Theory
The following assumptions are used to develop equivalent material properties for
composite laminates [6]:
1. The laminate consists of layers which are perfectly bonded.
2. Properties of each individual layer may be isotropic, orthotropic, or
transversely isotropic.
3. Individual layers are homogeneous and have known material properties.
4. Plane stress states are considered.
5. Kirchoff assumptions are followed for analysis of thin plate deformation
(bending and extension). This requires that an element initially normal to the
midplane will remain straight and normal to the midplane once it has been
deformed, which means that γzx and γzy shear strains will be zero. Kirchoff
also states that normals to the midplane will maintain constant length under
plate bending and stretching.
In other words, the z-displacement of the
midplane will only be a function of x and y coordinates.
The nature of layers being perfectly bonded is important as it means that there will be
zero displacement between plies. Isotropic materials have equivalent material properties
in all directions. Orthotropic materials may have varying material properties in all of
their three principal directions. A transversely isotropic material has effective properties
9
which are isotropic in directions normal to its primary axis.
The IM7/8552
unidirectional tape will be assumed to be transversely isotropic. Therefore,
E11 ≠ E 22
E11 ≠ E33
E 22 = E33
G12 ≠ G23
G13 ≠ G23
G12 = G13 .
Additionally, the following assumptions are upheld for Poisson’s ratios
υ12 = υ13
υ 23 = υ epoxy = 0.35
An in-depth look at transverse Poisson’s ratio ( υ 23 ) generated by testing was performed
by Philippidis and Theocaris [8]. This study was discovered towards the conclusion of
the project, so the designer will make a recommendation to consider its usage for similar
work in the future.
2.1.2
Fundamental Equation of Laminate Theory
Developed by Herakovich [6], the following equation is the fundamental equation of
lamination theory:
 N   A B  ε o 
 =
 
M   B D   κ 
Also, in expanded form it may be shown as:
 N x   A11
N  
 y   A12
 N xy   A16
=

 M x   B11
 M y   B12
 

 M xy   B16
A12
A16
B11
B12
A22
A26
B12
B22
A26
A66
B16
B26
B12
B16
D11
D12
B22
B26
D12
D22
B26
B66
D16
D26
B16   ε x o 


B26   ε y o 
B66  γ xy o 


D16   κ x 
D26   κ y 


D66   κ xy 
The N-terms and the M-terms on the left side of the equation represent the in-plane
A B
forces per unit length and moments per unit length, respectively. The 
 term is
B D
symmetric, and it should be noted that if [B] = 0, in-plane and bending responses will be
decoupled. Further development of this fundamental equation for symmetric cross-ply,
angle-ply, and other orthotropic laminates has been performed [6].
10
2.1.3
Equivalent Laminate Properties for Quasi-isotropic Laminates
A MATLAB model was developed which calculates equivalent in-plane properties for
laminates based on their orientation, thickness, and position. First, use the known
properties of IM7/8552 Carbon/Epoxy unidirectional tape (dry):
E11 = 23.8 Msi
E22 = 1.7 Msi
G12 = 2.2 Msi
v12 = .32
in the following explicit expressions to acquire reduced stiffness coefficients Qij
Q11 = E11/(1-v12E22/E11v12)
Q22 = E22/(1-v12E22/E11v12)
Q12 = (v12E22/(1-v12v12E22)/E11)
Q66 = G12.
These values are then used to determine the explicit equations for the reduced,
transformed stiffness coefficients, which are
q11 = (Q11m4)+(2(Q12+2Q66)m2n2)+(Q22n4)
q12 = (Q11+Q22-4Q66)(m2n2)+Q12(n4+m4)
q22 = Q11n4+(2(Q12+2Q66)(m2n2))+(Q22m4)
q16 = [(Q11-Q12-2Q66)m2+(Q12-Q22+2Q66)n2]mn
q26 = [(Q11-Q12-2Q66)n2+(Q12-Q22+2Q66)m2]mn
q66 = (Q11+Q22-2Q12-2Q66)m2n2+Q66(n4+m4)
where m = cos(θ) and n = sin(θ). These values are used directly to populate the
following matrices which are required components of the aforementioned Fundamental
Equation of Laminate Theory [6]:
N
[A] = ∑ [q ]k ( zk − zk −1 )
in-plane stiffness matrix
k =1
N
[B ] = ∑ [q ]k ( zk2 − zk2−1 )
bending-stretching coupling matrix
k =1
11
N
[D] = ∑ [q ]k ( zk3 − zk3−1 )
bending stiffness matrix
k =1
where z is the location of the layer of interest with respect to the midplane. Figure 2-1
defines the orientation of the kth and Nth layers.
1
-H
z0
z1
y
zk-1
H
zk
k
zN
N
x
z
Figure 2-1: Coordinates for Laminate Theory
For all symmetric matrices, the bending-stretching coupling matrix [B] reduces to zero,
which greatly simplifies the system. The laminate compliance equation
a * = 2 H [ A]−1
is used to solve, in its expanded form
 ε x0   a11*
 0  *
 ε y  = a12
γ 0  a *
 xy   16
a12*
*
a 22
a 26*
a16*  σ x  
  
* 
a 26
 = σ y  
* 
  
a 66
 τ xy  
As a final step, the following material properties are determined by
Ex =
1
a11*
Ey = −
1
*
a22
ν xy = −
12
a12*
a11*
Gxy =
1
.
*
a66
2.2 Preliminary Tension-Torsion Strap Sizing
The generation of effective material properties leads to the ability for the designer to
apply some preliminary shape and size constraints to the strap design.
For an initial
consideration of wall thickness adjacent to a bolt hole, a minimum e/d of 2 should be
used for composite structures, where E is the edge distance from the bolt axis and d is
the diameter of the hole. This design in this project utilized a bolt diameter of 0.75
inches, therefore maintaining an edge distance of 1.5 inches. Total strap length will be
limited to 18 inches maximum from centerline to centerline of the attachment lugs.
As discussed, the loading regime for the tension-torsion strap will consist of blade CF,
torsional twist due to blade pitch motion, and a small magnitude of flatwise bending.
These three components should be analyzed statically. As a first pass of the response of
the part due to CF loads, determine the stress due to pure tension loading by
σ=
F
,
A
where A is the known cross-sectional area. Preliminary normal strain due to CF can also
be found from
ε=
σ
E
where E is the elastic modulus in the axial direction. The normal stress above can easily
be calculated for a range of cross-sections and CF values. Table 2-1 displays the
calculated stresses due to CF loading for a square cross-section with both sides equal to
length. The known yield strength of IM7/8552 carbon/epoxy unidirectional tape is
398,000 psi. The expectation for this design is that the interlaminar shear stresses will
be the limiting factor. For optimization of the tension-torsion strap, we will limit normal
stress due to static tensile loading to 100,000 psi.
Once the effect of CF loading has been investigated, the designer can attempt to bound
the design further by analyzing the max shear strain due to torsional twist for a tensiontorsion strap of a specified cross-section. This is a two step process which requires the
knowledge of Gyz (the shear modulus normal to the pitching axis of the part), the length
13
Normal Stress (psi) due to CF
CF (lbf)
a (in)
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
10000
20000
30000
40000
50000
160000
111111
81633
62500
49383
40000
33058
27778
23669
20408
17778
15625
13841
12346
11080
10000
9070
8264
7561
6944
320000
222222
163265
125000
98765
80000
66116
55556
47337
40816
35556
31250
27682
24691
22161
20000
18141
16529
15123
13889
480000
333333
244898
187500
148148
120000
99174
83333
71006
61224
53333
46875
41522
37037
33241
30000
27211
24793
22684
20833
640000
444444
326531
250000
197531
160000
132231
111111
94675
81633
71111
62500
55363
49383
44321
40000
36281
33058
30246
27778
800000
555556
408163
312500
246914
200000
165289
138889
118343
102041
88889
78125
69204
61728
55402
50000
45351
41322
37807
34722
Table 2-1: Normal Stress Due to CF vs Cross-Sectional Area
of the pitching section L, the value φ of total torsional displacement throughout the
length of the part, and the polar moment of inertia J. These values can be found by a
general torsional displacement equation:
ϕ=
TL
, where T is applied torque
GJ
Solving for T, we have
T=
ϕGJ
L
.
The torque can then be used to solve for the maximum shear stress by
τ max =
Tr
,
J
where r is the distance to the position of interest. Simplified equations specific to the
cross-section type of interest can be used. As an example, for a shaft of square crosssection the following simplified equations can be used to find T and τ max
14
T=
ϕGa 4
and
7.10 L
τ max =
4.81T
,
a3
where a, again, is the length of one side of the square [5]. For a square cross section
with torque applied about its center, the max shear stress will be located at the midpoint
between adjacent corners along each edge of the square. A similar study of maximum
shear stresses for varying cross-sections and torsional displacements is shown in Table
2-2. Results provided will help set limits to the design. In this case, the yield shear of
IM7/8552 is given as 19.9 ksi. S-N curves for the initiation of shear and tensile
delaminations in unidirectional IM7/8552 tape show that loading at 60 percent of shear
strength, will initiate delamination between 10E6 and 10E7 cycles. 10E6 cycles
translates to 65 flight hours for a part which sees 4 load cycles per second, which is
typical of dynamically loaded helicopter rotors parts. Obviously, one would rather
10E7 cycles (650 flight hours), which could be achieved by further reducing loads.
Recommendations will be made for further fatigue analysis and most importantly, for
testing at the conclusion of this paper. In the interim, the designer will make 60 percent
of given shear strength (12 ksi) the max shear for the entirety of the analysis to follow.
Max Shear
(psi)
phi
a
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
10
L
15
18
20
25
30
4598
5518
6438
7357
8277
9196
10116
11036
11955
12875
13795
14714
15634
16554
17473
18393
6897
8277
9656
11036
12415
13795
15174
16554
17933
19313
20692
22071
23451
24830
26210
27589
9196
11036
12875
14714
16554
18393
20232
22071
23911
25750
27589
29429
31268
33107
34946
36786
11496
13795
16094
18393
20692
22991
25290
27589
29888
32188
34487
36786
39085
41384
43683
45982
13795
16554
19313
22071
24830
27589
30348
33107
35866
38625
41384
44143
46902
49661
52420
55179
Table 2-2: Max Shear, Varying Pitch Value & Cross-Section
15
Preliminary shear and tensile stress prediction charts for square cross-sections can be
found in Appendix B.
2.3 FEA Model Development
A number of finite element models are created in ANSYS to aid in the substantiation of
the tension-torsion strap design. A model is developed with the goal of determining
stress concentrations in the inboard and outboard attachment lugs of the strap. Another
linear model is created to first load the part in CF and then to apply the twist and
bending. The method for determining the stress concentrations is discussed in detail in
the Results section.
The ANSYS model created to analyze actual tension-torsion straps (as shown in Figure
2-2) is fairly simple to set up. The strap for consideration is created in Catia V5 as a part
body along with two separate bodies which represent bolts.
Once imported into
ANSYS, the cylindrical faces of the bolt and strap holes are constrained to one another
using bonded contact. A “Static Structural” analysis is created, and all surfaces of the
inboard bolt are given a fixed constraint (A), where all degrees of freedom are removed.
The CF load (30000 lbs) is applied (B) to the outboard bolt in the X-direction at time
step 1. Being a linear analysis, the force applied is increased linearly from 0 pounds at 0
seconds and 30000 pounds at 1 second. From time step 1 to time step 2, a remote
displacement (C) is applied to the outboard bolt which consists of 10 degrees of twist
about the x-axis and 0.5 degrees about the y-axis of the coordinate system local to the
inboard bolt.
16
Figure 2-2: ANSYS Model Creation
This basic model will be used to analyze conditions after time step 1 (CF only) and after
time step 2 (all loading conditions).
It will be used to investigate the following
conditions and effects:
•
Effect of adding plies to the 80 ply baseline configuration
•
Effect of modifying the pitch section length and width
•
Effect of flatwise bending on overall stresses
•
Relationship between CF and Normal Stresses
•
Relationship between xy and zx interlaminar shear and applied twist
•
Effect of variation of applied twist and CF on shear in the transition from strap
lugs to pitch section
The meshing of the parts in ANSYS is fairly straightforward. A fair portion of the part
receives moderate mesh sizing, while sections of high interest receive what ANSYS
refers to as “Spheres of Influence.” These user defined spheres create a spherical
boundary which has independently assigned mesh sizing within the boundary. Figures
2-3 and 2-4 show the application of spheres of influence in ANSYS for meshes.
17
Figure 2-3: Sphere of Influence Application
Figure 2-4: Mesh Resulting from Sphere of Influence
Once the results have been properly examined, conclusions can be drawn, along with
recommendations for further study and improvement.
18
3. Results
At the onset of the design of almost any load bearing component, the designer must first
understand the how the part will be expected to function. Bounds must be placed on
features such as material types, weight allocations, manufacturing and recurring costs,
and fatigue life requirements. Due to the open-ended nature of many design projects,
these bounds will help to align achievable goals with demanding schedule constraints.
Once material types have been selected, one needs to begin to lay out the geometrical
constraints for the component so that the feasibility of meeting those goals can be
understood.
For this tension-torsion strap design, bounds have been placed on material types and
properties, loading conditions, and a general idea of how the part is expected to function.
Models were created in ANSYS which determine stress concentrations in lug holes,
identify the effect of flatwise bending in this application, and assist the designer in
optimizing the dimensions of the part. The stress concentrations generated through the
use of finite element modeling result in the ability to properly size the inboard and
outboard attachment lugs.
In the sections that follow, a few geometrical features of the tension-torsion strap will be
highlighted. Reference Figure 3-1 for identification of the “pitch section” and “”lug to
pitch section transition.”
L
l
Pitch Section Length
Lug – Pitch Section Transition
Figure 3-1: Geometrical Features of the Tension-Torsion Strap
19
3.1 Determination of Laminate Properties
Through the course of this study, laminate properties for three separate cases were
configured. All of the cases utilized IM7/8552 carbon/epoxy unidirectional tape with
specific symmetric layup orientations. Table 3-1 contains laminate properties of the
three major cases. Ply orientations for each case are listed in Table 3-2. The first case
includes 0 degree, positive and negative 45 degree, and 90 degree plies. After a number
of early analyses, it became evident that the using the 90 degree plies was not an
efficient way to build stiffness in the part, mainly due to the absence of transverse
loading. The positive and negative 45 degree plies are required to prevent shear out of
the attachment lugs. This ply configuration resulted in shear stresses that were well over
the self-imposed limits discussed in the methodology.
Equivalent Properties via Classic Laminate Theory
(E & G modulus in Msi)
Case
Ex
Ey
Ez
Gxy
Gyz
Gzx
ν12
ν23
ν31
1
9.38
6.68
6.68
3.70
2.48
3.70
0.45
0.35
0.45
2
13.0
3.61
3.61
3.32
1.34
3.32
0.74
0.35
0.74
3
18.3
2.68
2.68
1.97
0.991
1.97
0.63
0.35
0.63
Table 3-1: Equivalent Properties of Quasi-Isotropic Laminates
Case
Orientation
Number of Plies
1
(02 / +/-452 / 902 / +/-452 / 02)
80
2
(0 / +/-45 / 0)
80
3
(03 / +/-45 / 03)
80
Table 3-2: Ply Orientations
The second and third ply orientations were limited to the use of 0 degree and positive
and negative 45 degree orientations. Case two was a significant improvement over the
first attempt for this application. Case three utilized less 45 degree plies, and ended up
returning the final configuration which was the only case that which met the stress
20
targets. Appendix A contains script from the MATLAB code responsible for generating
these ply configuration properties.
3.2 Determination of Stress Concentrations
The preliminary design of a tension-torsion strap involves material selection and
determination of smeared laminate properties, along with basic hand calculations of
tensile and shear stresses to be endured by critical cross-sections. However, to properly
size the lug geometry, a study of the effect of the inboard and outboard attachment holes
under part loading is required. When loaded, localized high stress regions will generally
be found on or adjacent to the hole surfaces. The effect that the hole will have on local
stresses must be quantified to appropriately size the lugs. By calculating the stress
concentration factor (Kt) due to these holes, the lugs can be effectively sized.
The method used to determine the stress concentration of the holes involved the analysis
of two thin flat plates. The two plates are identical, with the exception that the second
plate will have a hole centered on the part equivalent in size to the one to be used for the
attachment holes of the tension-torsion strap. Figure 3-1 shows the second of the two
plates. The thickness of the plates (t) should be much less than either the width (W)
W
e = W/2
Figure 3-2: Thin, Flat Plate Loading for KT Determination
21
or the length (L). The theory is that the first plate is long enough that it may be fixed at
one end and loaded in tension at the other such that the normal stress in the center of the
part lengthwise is unaffected by any edge constraint effects. The normal stress is taken
at a location halfway along the length (it should be constant across the height) and is
considered the reference stress. This stress is compared to localized normal stress in the
hole surfaces when the second plate is loaded. The stress concentration factor (Kt) is
calculated
Kt =
σ hole
σ ref
Geometrically, the stress concentration factor is dependent on a few features. The first is
the edge to hole diameter ratio (e/d). In general, a decrease in e/d will result in an
increased stress concentration factor. Increasing edge distance while keeping the hole
diameter constant is beneficial for stresses, but its impact is minimal for high ratios up
around 3 to 5. At this point, it becomes a space and weight tradeoff when attempting to
drive down stresses. Deviation of the thickness of the plates will have no affect on stress
concentration of the hole, as the reference and hole normal stresses will increase or
decrease together in a linear fashion. Stress concentration will also vary depending on
material selection and ply configuration. In this case, the stress concentration was
determined for IM7/8552 for the designs final layup configuration of the design. Figure
3-3 shows how stress concentration varies with a changing e/d ratio. These values were
generated using an ANSYS model of the aforementioned plate configurations. In the
case of Figure 3-4, the normal stresses were analyzed for the plate without holes (left)
and the plate containing the hole (right) with an e/d ratio of 2. The stress concentration
calculated is also shown.
Stress Concentration (Kt) vs E/d
e/d Ratio
(for (03 / +/-45 / 03, IM7/8552)
Stress
Concentration (Kt)
5
4
3
2
1
0
0
0.5
1
1.5
2
E/d Ratio
e/d
Stress
Concentration(Kt)
(Kt)
Ratio
Stress Concentration
vs vs
e/dE/d
Ratio
Figure 3-3: Kt vs e/d Ratio
22
2.5
3
Kt =
σ hole 146460 psi
=
= 3 .4
σ ref
43400 psi
Figure 3-4: Stress Concentration Study
3.3 Lug Analysis
As previously mentioned, the prediction of required cross-sectional areas for tensile
loading and torsional displacement of the pitch section alone is not sufficient for
preliminary design of a composite tension-torsion strap.
The stress concentrations
determined above allow the designer to effectively size the inboard and outboard lug
geometries. For this type of design, it is important to analyze two major failure modes,
net tension failure and bearing failure. Table 3-3 contains the stress allowables to be
used in this analysis. However, due to the lack of publicly available data on IM7/8552
graphite/epoxy composite tape, it must be noted that these are assumed values. These
values should not be used for future design, they are simply values assumed to complete
the exercise of lug sizing. Allowables for other similar composite materials have been
generated through testing and are readily available.
Material Lug Failure Allowables
Net Tension Allowable
Bearing Stress Allowable
Fnt
120,000 psi
σbearingallow
120,000 psi
Table 3-3: Lug Stress Allowables
3.3.1
Net Tension Failure
The net tension load to failure (Pnt) also known as tearout failure [7] can be calculated
with the following equation
23
Pnt =
(W − d )t
Fnt ,
Kt
where the dimensional features represented in Figure 3-1 along with calculated Kt for
the holes gives
Pnt =
(3in − .75in).48in
(120,000 psi ) = 38118lb
3 .4
Taking this net tension load and comparing it to the planned max applied tension load
due to CF gives a calculation of margin of safety for this failure mode
MS nt =
Pnt
Papplied
−1 =
38118lb
−1
30,000lb
or
MS nt = 0.27
This margin of safety for net tension load, while close to going negative, is acceptable.
The stress concentration for the hole which was 3.4 was fairly high with what was
expected going into this exercise. This may have built some level of conservatism into
the margin of safety calculation.
Next, a similar method will be applied for the
calculation of the bearing failure mode.
3.3.2
Bearing Failure
Direct bearing failure is a possibility whenever two surfaces are pressed against one
another. Bearing stress is an attempt to crush a hole rather than shear it, as it is
compressive in nature. [4] Assuming zero clearance between the bolt and hole surface,
the area is defined as the length of the bearing contact times the diameter of the hole.
This is known as a projected contacted area. The bearing stress is defined as
σ bearing =
P
Abearing
where
Abearing = lbearing d
For this case
σ bearing =
30000lb
= 83333.33 psi
(.48in)(.75in)
With the assumed bearing stress allowable, the margin of safety calculation is
performed:
24
MS bearing =
120,000 psi
− 1 = 0.44
83333.33 psi
3.4 Effect of Flatwise Bending
The three major loads that the tension-torsion strap will experience are tensile load due
to CF, interlaminar stresses due to torsional displacement (pitching) of the blade, and up
to 0.5 degrees of flatwise bending. The prediction would be that the flatwise bending
case should only have a small impact on part design. This expectation was validated
early on in the project. Using the 02/+/-452/902/+/-452/02 material properties, along with
e/d of 2, the stresses found in Table 3-5 were calculated using an ANSYS finite element
model. The locations of those stresses are shown in Figure 3-5. From left to right, the
figure displays the locations of interlaminar shear along the transition (XY), interlaminar
shear (XY) and interlaminar shear (ZX). The bending term contributes the most to shear
stress in the transition, but it still creates less than 3 percent of change in stress compared
to the case which ignores it. This difference is not necessarily going to change the
design it simply aides in understanding the cause of the stress conditions which are seen
along the design process.
Shear XY
Transition
Shear XY
Shear ZX
CF&10degTwist
14900 psi
9024 psi
7300 psi
CF, 10 deg Twist, &
Bending
15247 psi
8975 psi
7305 psi
Percent Variation (%)
2.33
0.54
0.08
Table 3-4: Interlaminar Shear Stress Due to Flatwise Bending Load
Figure 3-5 Shear Stress Locations of Concern
25
3.5 Shear Stress Response to Varying Loads
The assumption can be made that for a tension-torsion strap with a long pitch section,
the CF loads will have no affect on interlaminar shear stresses along the pitch section far
enough away from the transition. Using ANSYS to measure shear stress while varying
CF loads from 0 to 30,000 lbs and pitch angle from 0 to 10 degrees, this assumption was
verified. The ply layup was that of Case 1, with identical dimensions of that from the
flatwise bending effect study. Interlaminar shear stresses are plotted with varying pitch
values in Figure 3-6.
Interlaminar Shear vs Angular Displacement
IM7/8552, Case 1
10000
9000
Shear Stress (psi)
8000
7000
6000
5000
4000
3000
2000
1000
0
0
2.5
5
7.5
10
Ang Displacement (deg)
Interlaminar Shear (XY)
Interlaminar Shear (ZX)
Figure 3-6: Interlaminar Shear VS Angular Displacement
Next, a study was performed on the xy shear stress in the lug to pitch section transition.
CF loads and pitch angles were varied similarly to investigate the effects of each on
shear stresses. From the data in Figure 3-7, it is noted that a majority of the xy shear
stress in this region was present due to CF load.
26
Shear XY in Transition Radius for CF vs
Angular Displacement
IM7/8552, Case 1
16000
Shear Stress (psi)
14000
12000
10000
8000
6000
4000
2000
0
0
10000
20000
30000
CF (lb)
0 Deg Ang Displacement
7.5 deg Ang Displacement
2.5 deg Ang Displacement
10 deg Ang Displacement
5 deg Ang Displacement
Figure 3-7: XY Shear Stress in Transition Radius
3.6 Strain due to CF Loading
One of the material properties of composites which render them useful is their
elongation to failure. IM7/8552 graphite/epoxy has an elongation limit of 1.9%. In
looking at the strains due to CF, this tension-torsion strap would stretch well below that
limit. Figure 3-8 shows the strains experienced due to CF for the final configuration.
From the strains measured, the elongation for the portion of the strap which sees the
highest amount of stretching (green) was 0.4%. This is acceptable and will be a value to
take forward for an in-depth fatigue analysis.
Figure 3-8: Stretching Due to CF Load
27
3.7 Optimization of Geometry
As a designer moves toward solidifying a design, it is important early on to understand
how the geometric constraints he or she places on the design affects the behavior of the
overall product. General practice guidelines may be followed for creation of initial
geometric shapes, but seemingly small optimization studies can result in dramatic
improvements.
3.7.1
Optimization: Pitch Section Width
Here, the feature of consideration is the width of the section which will be expected to
experience a majority of the pitch motion. ANSYS models were run for multiple cases
of identical ply configuration, varying the width of the pitch section for each one.
Figure 3-9 displays the resulting shear stress states due to CF, torsional displacement,
and flatwise bending for each of the cases. Figure 3-10 contains the variation of normal
stress present with change in pitch section width. Lastly, the weight impact that this
change in pitch section width would create is displayed in Figure 3-11. The designer
must make close observations on the impact of decisions, which will affect weight. If
composite materials are being considered for a design, chances are that weight impact is
of great concern for overall performance. In this case, going from a pitch section width
of 0.80 inches to 1.5 inches, shear stress XY is almost unchanged while a substantial
reduction is experienced in the other interlaminar shear stresses and normal stress. This
may be a significant modification, but it is not achieved without a cost, as the total
component weight is increased by 30%.
Effect of Pitch Section Width on Shear Stresses
(For 80 Ply Laminate Thickness)
14000
Shear Stress (psi)
12000
10000
8000
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Width (in)
Shear XY
Shear XY Transition
Shear ZX
Figure 3-9: Effect of Pitch Section Width on Shear Stress
28
1.6
Effect of Pitch Section Width on Normal Stresses (X)
(For 80 Ply Laminate Thickness)
Normal Stress (psi)
120000
80000
40000
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Pitch Section Width (in)
Normal Stress (X) vs Pitch Section Width
Figure 3-10: Effect of Pitch Section Width on Normal Stresses
Effect of Pitch Section Width on Weight
(For 80 Ply Laminate Thickness)
1.20
1.00
Weight (lb)
0.80
0.60
0.40
0.20
0.00
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Pitch Section Width (in)
Weight vs Pitch Section Width
Figure 3-11: Effect of Pitch Section Width on Weight
Moving forward in the analysis, the 0.72 inch pitch section width was selected and
would represent the weight-conscious thinking of the designer.
3.7.2
Optimization: Pitch Section Length
The next geometrical feature identified and explored was the effect of the change in total
constant area pitch section length. The longer the pitch section length, the shorter
distance that the inboard and outboard lug section transitions into the pitch section.
Therefore, the shorter the pitch section length, the more gradual the transition from the
lugs. As mentioned in the previous optimization exercise, 0.72 inch pitch section width
was maintained throughout. Figure 3-12 contains data which describes the effect of
29
pitch section length on interlaminar shear stresses. The normal stresses will remain
constant, as the cross-sectional area is unchanged for the pitch section, and the changes
in weight are so small that they can be neglected for this discussion. As the optimization
continues, the 8 inch pitch section length will be used. This data point keeps all three
shear stresses of interest under 12,000 psi, which is the target for this project.
Effect of Pitch Section Length on Shear Stresses
80 ply, IM7/8852 Graphite Epoxy
20000
Shear Stress (psi)
15000
Shear XY
Shear XY Transition
10000
Shear ZX
5000
0
4
6
8
10
12
14
Pitch Section Length (in)
Figure 3-12: Effect of Pitch Section Length on Shear Stresses
3.7.3
Optimization: Effect of Additional Plies
The motivation behind attempting to add plies to the strap design comes from the
potential improvement of many features. First, adding plies would help to increase
margins of safety for the lugs. As it currently stands, the 80 ply configuration provides a
positive, yet low margin for net tension lug failure. Adding plies would also decrease
the normal stresses due to CF loading.
These two improvements are fairly well
understood. To be able to quantify the effect of adding plies to the current 80 ply, 8 inch
long and 0.72 inch wide pitch section, an ANSYS model was developed and analyzed.
As is almost immediately evident by Figure 3-13, that adding plies will certainly have
adverse effects on the interlaminar shears in the tension-torsion strap. Shear stress in
both the XY and ZX orientation see significant increases with additional plies, and the
XY shear stress in the transition only decreases slightly.
Besides the effect on
interlaminar shear, adding plies also means added weight, another drawback. Therefore,
30
the setbacks of increased shear stresses and weight in an attempt to improve already
relatively low normal stresses and positive margins in the lugs are not worthwhile
tradeoffs.
Effect of Adding Plys on Shear Stresses
16000
Shear Stress (psi)
14000
12000
10000
8000
6000
4000
2000
0
80
96
112
Number of Plies (.006" each)
Shear XY
Shear XY Corner
Shear ZX
Figure 3-13: Effect of Adding Plies on Shear Stresses
31
4. Conclusions
4.1 Final Configuration and Conditions
At the completion of the numerous studies of tension-torsion strap features, one clear
configuration emerged as the one which would be recommended for further study. It
was clear at the onset that to reduce interlaminar shear stresses, the designer would
utilize the high end of the limited total length range. Therefore, the final strap is 18
inches long and is made from the third configuration of ply layup. Other dimensions
associated with the strap design can be seen in Table 4-1. This design, with all of its
dimensions, was arrived upon through the use of classic laminate theory and the
implementation of FEA models using ANSYS to both validate assumptions and to fully
understand the effects of the load cases.
Ply Orientation
Number of Plies (n)
Ply Thickness (in)
Total Thickness (t, in)
(03 / +/-45 / 03)
80
0.006
0.48
Lug Edge Distance (e, in)
Hole Diameter (d, in)
Pitch Length (l, in)
Pitch Section Width (w)
1.50
0.75
8.00
0.72
Table 4-1: Final Configuration Data
This tension-torsion strap was sized for preliminary loads and was then optimized based
on those loads and stress and strain limits. Table 4-2 contains results for this final
design. Results of each ply layup configuration case analyzed for this final geometry are
found in Appendix C.
Normal Stress (ksi)
XY Shear Stress (ksi)
XY Shear Stress (ksi)
ZX Shear Stress (ksi)
Strain (X, in/in)
Location
Pitch Section
Pitch Section
Transition Radius
Pitch Section
Pitch Section
Value
86.8
11.0
11.2
9.2
0.004
Table 4-2: Shear Stresses & Axial Strain due to Static Loading
4.2 Future Fatigue Substantiation & Testing
As this project has progressed, the need for further study prior to fabrication of such a
tension-torsion strap became increasingly evident. A number of topics which should be
addressed include:
•
Interlaminar stress distributions and near-edge conditions
32
•
Investigation of transverse Poisson’s ratio by studying works of Philippidis
and Theocaris [8]
•
Behavior of fiber endings at free edges
•
Fabrication techniques for working with the lug to pitch section transition
and fibers running out into free edges
•
Thermodynamic Effects
•
Damage Theory
•
In-depth look into potential failure mechanisms
o Fiber fracture, kinking, fiber/matrix disbond, matrix cracking, etc
•
Applying bushings and clamping composite tension-torsion straps
At the beginning of this project, the intent was to develop multi-axial fatigue failure
methods for this application which would correlate with testing results. It quickly
became evident that, without prior composites design knowledge or experience, it would
be more beneficial to focus on designing this part for static conditions and to then make
recommendations for working in the fatigue realm in the future.
A number of useful publications on fatigue analysis of composites which would be
integral to the direction of future work were investigated.
Nyman developed a
simplified approach for composite design of components with combined loading [9]. He
established fatigue failure functions on the laminate level and proved that they reflect the
overall behavior of the laminate. Fatigue tests were performed by Reis et al [10], which
studied the behavior due to different stress ratios and for variable block loadings.
Estimated and experimental fatigue lives were compared for carbon/epoxy laminates
with good agreement.
An important study to take into consideration for future work was performed by Ferry et
al [11]. This involved the design of an experimental device to apply bending and torsion
to unidirectional glass-fiber/epoxy composite bars. Fiber failure, delamination and
cracking were studied and showed that the bars exhibited damage which varied
33
depending on the stress ratio between bending and torsion and on the ratio between
minimum and maximum stress.
4.3 Closing Remarks
As has been expressed, with the preliminary design of composites more than almost any
other structural material which can be used, initial analysis should be taken and verified
extensively through component fabrication and testing.
The great unknown for
composites design is the tightness of control on fabrication of these kinds of parts. All
geometrical, structural, and behavioral assumptions should be validated through
vigorous testing of multiple manufactured components.
34
5. REFRERENCES
[1] Moon, S. and Fermen-Coker, M. June 2009. H-46 Tie Bar Fatigue Failure
Predictions by Multiaxial Fatigue Theory. Journal of the American Helicopter
Society 54 (042006): 1-17.
[2] Henshaw, J.M., Sorem, J.R., Jr. and Glaessgen, E.H., “Finite Element Analysis of
Ply-by-Ply and Equivalent Stress Concentrations in Composite Plates with
Multiple Holes under Tensile and Shear Loading,” Journal of Composite
Structures, Vol. 36, pp. 45-58, 1996.
[3] Cheung, C.K., Liaw, B.M., Delale, F. and Raju, B.B. June 2004. Composite Strips
with a Circular Stress Concentration Under Tension. Proceedings of the SEM X
International Congress & Exposition on Experimental and Applied Mechanics.
Section 89: Composites, Paper No. 387.
[4] Norton, R. L. 2009. Machine Design: An Integrated Approach, 3rd edition, Upper
Saddle River, NJ: Prentice Hall.
[5] Hibbeler, R.C. 1997. Mechanics of Materials. Upper Saddle River, NJ: Prentice Hall.
[6] Herakovich, C. T. 1998. Mechanics of Fibrous Composites. USA: John Wiley &
Sons, Inc.
[7] “HexPly® 8552 Epoxy Matrix: Mid-Toughened, High Strength, Damage-Resistant,
Structural Epoxy Matrix Product Data” Web. December 2007. Found
electronically at: http://www.hexcel.com/NR/rdonlyres/9229D78D-51BC-44609248-CC256BC6B6A4/0/HexPly_8552_2_22_US.pdf
[8] Philippidis, T.P. and Theocaris, P. S. 1994. The Transverse Poisson’s Ratio in Fiber
Reinforced Laminae by Means of a Hybrid Experimental Approach. Journal of
Composite Materials Vol. 28, No. 3 (1994).
[9] Nyman, T. Composite Fatigue Design Methodology: A Simplified Approach.
Composite Structures 35 (1996) 183-194.
[10] Reis, P.N.B., Ferreira, J.A.M., Costa, J.D.M., and Richardson, M.O.W. Fatigue
Life Evaluation for Carbon/Epoxy Laminate Composites Under Constant and
Variable Block Loading. Composites Science and Technology 69 (2009): 154160.
[11] Ferry, L., Perreux, D., Varchon, D. and Sicot, N. 1999. Fatigue Behaviour of
Composite bars Subjected to Bending and Torsion. Composites Science and
Technology 59 (1999): 575-582.
[12]
"MD500E Rotorhead." Web. 17 Apr 2010. Found electronically
http://upload.wikimedia.org/wikipedia/commons/2/29/Md500e-rotorheadaradecki-070316-01.jpg
35
at:
Appendix A: MATLAB Code for Laminate Properties
%MATLAB Code for Equivalent Laminate Properties
%This code calculates Equivalent properties for laminate with number of
plies N = 56
%and (0 / +/-45 / 0) ply orientation.
%Similar code can be created for a laminate with any number of plies
for symmetric, quasi-isotropic configurations.
%As long as care is taken to ensure that the ply orientation is
symmetric
%about the mid-plane, the material properties are valid for
clc;
clear all;
E11=23400000; %selected from material properties for IM7/8552
E22=1420000;
G12=620000;
v12=.32;
%Find the Reduced Stiffness Coefficients Qij...
Q11=E11/(1-v12*E22/E11*v12);
Q22=E22/(1-v12*E22/E11*v12);
Q12=(v12*E22/(1-v12*v12*E22/E11));
Q66=G12;
%These Steps Calculate the Transformed, Reduced Stiffness Coefficients
qijx
%for varying values of theta (0deg, 30deg, 45deg, 60deg, 90deg, -45deg
theta=0;
m=cos(theta*pi/180);
n=sin(theta*pi/180);
q11a=(Q11*m^4)+(2*(Q12+2*Q66)*m^2*n^2)+(Q22*n^4);
q12a=(Q11+Q22-4*Q66)*(m^2*n^2)+Q12*(n^4+m^4);
q22a=Q11*n^4+(2*(Q12+2*Q66)*(m^2*n^2))+(Q22*m^4);
q16a=((Q11-Q12-2*Q66)*m^2+(Q12-Q22+2*Q66)*n^2)*m*n;
q26a=((Q11-Q12-2*Q66)*n^2+(Q12-Q22+2*Q66)*m^2)*m*n;
q66a=(Q11+Q22-2*Q12-2*Q66)*m^2*n^2+Q66*(n^4+m^4);
theta=30;
m=cos(theta*pi/180);
n=sin(theta*pi/180);
q11b=(Q11*m^4)+(2*(Q12+2*Q66)*m^2*n^2)+(Q22*n^4);
q12b=(Q11+Q22-4*Q66)*(m^2*n^2)+Q12*(n^4+m^4);
q22b=Q11*n^4+(2*(Q12+2*Q66)*(m^2*n^2))+(Q22*m^4);
q16b=((Q11-Q12-2*Q66)*m^2+(Q12-Q22+2*Q66)*n^2)*m*n;
q26b=((Q11-Q12-2*Q66)*n^2+(Q12-Q22+2*Q66)*m^2)*m*n;
q66b=(Q11+Q22-2*Q12-2*Q66)*m^2*n^2+Q66*(n^4+m^4);
theta=45;
m=cos(theta*pi/180);
36
n=sin(theta*pi/180);
q11c=(Q11*m^4)+(2*(Q12+2*Q66)*m^2*n^2)+(Q22*n^4);
q12c=(Q11+Q22-4*Q66)*(m^2*n^2)+Q12*(n^4+m^4);
q22c=Q11*n^4+(2*(Q12+2*Q66)*(m^2*n^2))+(Q22*m^4);
q16c=((Q11-Q12-2*Q66)*m^2+(Q12-Q22+2*Q66)*n^2)*m*n;
q26c=((Q11-Q12-2*Q66)*n^2+(Q12-Q22+2*Q66)*m^2)*m*n;
q66c=(Q11+Q22-2*Q12-2*Q66)*m^2*n^2+Q66*(n^4+m^4);
theta=60;
m=cos(theta*pi/180);
n=sin(theta*pi/180);
q11d=(Q11*m^4)+(2*(Q12+2*Q66)*m^2*n^2)+(Q22*n^4);
q12d=(Q11+Q22-4*Q66)*(m^2*n^2)+Q12*(n^4+m^4);
q22d=Q11*n^4+(2*(Q12+2*Q66)*(m^2*n^2))+(Q22*m^4);
q16d=((Q11-Q12-2*Q66)*m^2+(Q12-Q22+2*Q66)*n^2)*m*n;
q26d=((Q11-Q12-2*Q66)*n^2+(Q12-Q22+2*Q66)*m^2)*m*n;
q66d=(Q11+Q22-2*Q12-2*Q66)*m^2*n^2+Q66*(n^4+m^4);
theta=90;
m=cos(theta*pi/180);
n=sin(theta*pi/180);
q11e=(Q11*m^4)+(2*(Q12+2*Q66)*m^2*n^2)+(Q22*n^4);
q12e=(Q11+Q22-4*Q66)*(m^2*n^2)+Q12*(n^4+m^4);
q22e=Q11*n^4+(2*(Q12+2*Q66)*(m^2*n^2))+(Q22*m^4);
q16e=((Q11-Q12-2*Q66)*m^2+(Q12-Q22+2*Q66)*n^2)*m*n;
q26e=((Q11-Q12-2*Q66)*n^2+(Q12-Q22+2*Q66)*m^2)*m*n;
q66e=(Q11+Q22-2*Q12-2*Q66)*m^2*n^2+Q66*(n^4+m^4);
theta=-45;
m=cos(theta*pi/180);
n=sin(theta*pi/180);
q11f=(Q11*m^4)+(2*(Q12+2*Q66)*m^2*n^2)+(Q22*n^4);
q12f=(Q11+Q22-4*Q66)*(m^2*n^2)+Q12*(n^4+m^4);
q22f=Q11*n^4+(2*(Q12+2*Q66)*(m^2*n^2))+(Q22*m^4);
q16f=((Q11-Q12-2*Q66)*m^2+(Q12-Q22+2*Q66)*n^2)*m*n;
q26f=((Q11-Q12-2*Q66)*n^2+(Q12-Q22+2*Q66)*m^2)*m*n;
q66f=(Q11+Q22-2*Q12-2*Q66)*m^2*n^2+Q66*(n^4+m^4);
%Laminate Configuration 0deg thickness (.006per)
%Next step calculate distances from the mid-plane for each ply
%These values will not contribute to the material properties, as the
[B]
%and [D] matrices are dependent on ply distance from mid-plane.
Because
%[B] = 0 for symmetric matrices, bending stiffness matrix [D] is
decoupled
%from the in-plane response [A]. Material Properties are found
directly
%from [A] for transversely isotropic material.
n=56;
t=.06;
z0=-28*t;
37
z56=-z0;
z1=-27*t;
z55=-z1;
z2=-26*t;
z54=-z2;
z3=-25*t;
z53=-z3;
z4=-24*t;
z52=-z4;
z5=-23*t;
z51=-z5;
z6=-22*t;
z50=-z6;
z7=-21*t;
z49=-z7;
z8=-20*t;
z48=-z8;
z9=-19*t;
z47=-z9;
z10=-18*t;
z46=-z10;
z11=-17*t;
z45=-z11;
z12=-16*t;
z44=-z12;
z13=-15*t;
z43=-z13;
z14=-14*t;
z42=-z14;
z15=-13*t;
z41=-z15;
z16=-12*t;
z40=-z16;
z17=-11*t;
z39=-z17;
z18=-10*t;
z38=-z18;
z19=-9*t;
z37=-z19;
z20=-8*t;
z36=-z20;
z21=-7*t;
z35=-z21;
z22=-6*t;
z34=-z22;
z23=-5*t;
z33=-z23;
z24=-4*t;
z32=-z24;
z25=-3*t;
z31=-z25;
z26=-2*t;
z30=-z26;
z27=-1*t;
z29=-z27;
z28=0;
38
%Combining the Transformed, Reduced Stiffness Matrices qijx which are
used
%for this ply configuration into the In-Plane Stiffness Matrix [A]
[A]=[2*q11a+q11f+q11c,2*q12a+q12f+q12c,2*q16a+q16f+q16c;2*q12a+q12f+q12
c,2*q22a+q22f+q22c,2*q26a+q26f+q26c;2*q16a+q16f+q16c,2*q26a+q26f+q26c,2
*q66a+q66f+q66c]*(14*t);
%[B]=0 for symmetric configurations
%[B]=[q11a,q12a,q16a;q12a,q22a,q26a;q16a,q26a,q66a]*((z1^2-z0^2)+(z2^2z1^2)+(z3^2-z2^2)+(z4^2-z3^2)+(z5^2-z4^2)+(z6^2-z5^2)+(z7^2z6^2)+(z8^2-z7^2))
%Calculating Bending Stiffness Matrix [D] as an exercise (Material
Props
%are dependent only on the In-Plane Stiffness Matrix)
[D0]=(1/3)*([q11a,q12a,q16a;q12a,q22a,q26a;q16a,q26a,q66a]*((z1^3z0^3)+(z4^3-z3^3)+(z5^3-z4^3)+(z8^3-z7^3)+(z9^3-z8^3)+(z12^3z11^3)+(z13^3-z12^3)+(z16^3-z15^3)+(z20^3-z19^3)+(z21^3-z20^3)+(z24^3z23^3)+(z25^3-z24^3)+(z28^3-z27^3)+(z29^3-z28^3)+(z32^3-z31^3)+(z33^3z32^3)+(z36^3-z35^3)+(z37^3-z36^3)+(z40^3-z39^3)+(z41^3-z40^3)+(z44^3z43^3)+(z45^3-z44^3)+(z48^3-z47^3)+(z49^3-z48^3)+(z52^3-z51^3)+(z53^3z52^3)+(z56^3-z55^3)));
%[D90]=(1/3)*([q11e,q12e,q16e;q12e,q22e,q26e;q16e,q26e,q66e]*((z7^3z6^3)+(
%z8^3-z7^3)+(z17^3-z16^3)+(z18^3-z17^3)+(z29^3-z28^3)+(z30^3z29^3)+(z31^3-z30^3)+(z32^3-z31^3)+(z43^3-z42^3)+(z44^3-z43^3)+(z53^3z52^3)+(z54^3-z53^3)));
[D45]=(1/3)*([q11c,q12c,q16c;q12c,q22c,q26c;q16c,q26c,q66c]*((z3^3z2^3)+(z7^3-z6^3)+(z11^3-z10^3)+(z15^3-z14^3)+(z19^3-z18^3)+(z23^3z22^3)+(z27^3-z26^3)+(z30^3-z29^3)+(z34^3-z33^3)+(z38^3-z37^3)+(z42^3z41^3)+(z46^3-z45^3)+(z50^3-z49^3)+(z54^3-z53^3)));
[Dn45]=(1/3)*([q11f,q12f,q16f;q12f,q22f,q26f;q16f,q26f,q66f]*((z2^3z1^3)+(z6^3-z5^3)+(z10^3-z9^3)+(z14^3-z13^3)+(z18^3-z17^3)+(z22^3z21^3)+(z26^3-z25^3)+(z31^3-z30^3)+(z35^3-z34^3)+(z39^3-z38^3)+(z43^3z42^3)+(z47^3-z46^3)+(z51^3-z50^3)+(z55^3-z54^3)));
%Combining Components of the Bending Stiff Matrix into [D]
[D]=[D0]+[D45]+[Dn45];
%H is the height from the mid-plane to the free edge
H=n*t/2;
%Laminate Compliance Equation
[a]=([A]^-1)*2*H
[d]=[D]^-1
%NOTE: Run At this point SAVE AND RUN the model to acquire the [a]
matrix
%values. Input those values into the respective equations for a11,
a22,
%a12, and a66 below and then RERUN to solve for Ex, Ey, vxy, and Gxy
a11=.077*10^-6
a22=.2769*10^-6
39
a12=-.0570*10^-6
a66=.3014*10^-6
%d11=.1104*10^-4
%d12=-.0361*10^-4
%d22=.2295*10^-4
%d66=.6213*10^-4
%Equivalent Laminate Properties
Ex=1/a11
Ey=1/a22
vxy=-a12/a11
Gxy=1/a66
40
Appendix B: Preliminary Sizing for Max Shear, Axial
Strain/Stress
Required Torque and Max Shear for 12 inch Beam Length and Varying Cross-Sectional
Areas (a) and Pitch Angles (phi)
REQUIRED TORQUE (in*lb)
L
phi
12
5
10
15
20
0.25
11
22
34
45
0.3
23
46
70
93
Max Shear
in
L
25
12
20
in
30
5
10
15
25
30
56
67
3449
6897
10346
13795
17243
20692
116
139
4138
8277
12415
16554
20692
24830
a
0.35
43
86
129
172
215
258
4828
9656
14484
19313
24141
28969
0.4
73
147
220
294
367
441
5518
11036
16554
22071
27589
33107
0.45
118
235
353
470
588
706
6208
12415
18623
24830
31038
37246
0.5
179
358
538
717
896
1075
6897
13795
20692
27589
34487
41384
0.55
262
525
787
1050
1312
1575
7587
15174
22761
30348
37935
45522
0.6
372
743
1115
1487
1858
2230
8277
16554
24830
33107
41384
49661
0.65
512
1024
1536
2048
2560
3072
8967
17933
26900
35866
44833
53799
0.7
689
1377
2066
2754
3443
4132
9656
19313
28969
38625
48281
57938
0.75
907
1815
2722
3630
4537
5445
10346
20692
31038
41384
51730
62076
0.8
1175
2349
3524
4699
5873
7048
11036
22071
33107
44143
55179
66214
0.85
1497
2994
4491
5988
7485
8982
11725
23451
35176
46902
58627
70353
0.9
1882
3763
5645
7527
9408
11290
12415
24830
37246
49661
62076
74491
0.95
2336
4672
7008
9344
11680
14016
13105
26210
39315
52420
65525
78630
1
2868
5736
8604
11472
14340
17207
13795
27589
41384
55179
68973
82768
1.05
3486
6972
10458
13944
17430
20916
14484
28969
43453
57938
72422
86906
1.1
4199
8398
12597
16796
20995
25193
15174
30348
45522
60696
75871
91045
1.15
5016
10032
15048
20064
25080
30096
15864
31728
47592
63455
79319
95183
1.2
5947
11894
17841
23788
29735
35681
16554
33107
49661
66214
82768
99322
1.25
7002
14003
21005
28007
35009
42010
17243
34487
51730
68973
86217
103460
1.3
8191
16382
24573
32764
40955
49146
17933
35866
53799
71732
89665
107598
1.35
9526
19052
28577
38103
47629
57155
18623
37246
55868
74491
93114
111737
1.4
11017
22035
33052
44069
55087
66104
19313
38625
57938
77250
96563
115875
1.45
12678
25355
38033
50710
63388
76066
20002
40005
60007
80009
100011
120014
1.5
14519
29038
43556
58075
72594
87113
20692
41384
62076
82768
103460
124152
1.55
16554
33107
49661
66214
82768
99322
21382
42763
64145
85527
106909
128290
1.6
18795
37590
56385
75181
93976
112771
22071
44143
66214
88286
110357
132429
1.65
21257
42514
63771
85028
106285
127542
22761
45522
68284
91045
113806
136567
1.7
23953
47906
71859
95812
119765
143719
23451
46902
70353
93804
117255
140705
1.75
26898
53796
80694
107592
134489
161387
24141
48281
72422
96563
120703
144844
1.8
30106
60212
90319
120425
150531
180637
24830
49661
74491
99322
124152
148982
1.85
33593
67187
100780
134373
167967
201560
25520
51040
76560
102080
127601
153121
41
Required Torque and Max Shear for 15 inch Beam Length and Varying Cross-Sectional
Areas (a) and Pitch Angles (phi)
REQUIRED TORQUE (in*lb)
L
phi
15
Max Shear
in
L
5
10
15
20
25
0.25
9
18
27
36
0.3
19
37
56
74
0.35
34
69
103
0.4
59
117
176
0.45
94
188
0.5
143
287
15
in
30
5
10
15
20
25
30
45
54
2759
5518
8277
11036
13795
16554
93
112
3311
6621
9932
13243
16554
19864
138
172
207
3863
7725
11588
15450
19313
23175
235
294
352
4414
8829
13243
17657
22071
26486
282
376
470
564
4966
9932
14898
19864
24830
29796
430
574
717
860
5518
11036
16554
22071
27589
33107
a
0.55
210
420
630
840
1050
1260
6070
12139
18209
24279
30348
36418
0.6
297
595
892
1189
1487
1784
6621
13243
19864
26486
33107
39729
0.65
410
819
1229
1638
2048
2457
7173
14346
21520
28693
35866
43039
0.7
551
1102
1653
2203
2754
3305
7725
15450
23175
30900
38625
46350
0.75
726
1452
2178
2904
3630
4356
8277
16554
24830
33107
41384
49661
0.8
940
1880
2819
3759
4699
5639
8829
17657
26486
35314
44143
52971
0.85
1198
2395
3593
4791
5988
7186
9380
18761
28141
37521
46902
56282
0.9
1505
3011
4516
6021
7527
9032
9932
19864
29796
39729
49661
59593
0.95
1869
3737
5606
7475
9344
11212
10484
20968
31452
41936
52420
62904
1
2294
4589
6883
9177
11472
13766
11036
22071
33107
44143
55179
66214
1.05
2789
5578
8366
11155
13944
16733
11588
23175
34763
46350
57938
69525
1.1
3359
6718
10077
13437
16796
20155
12139
24279
36418
48557
60696
72836
1.15
4013
8026
12038
16051
20064
24077
12691
25382
38073
50764
63455
76147
1.2
4758
9515
14273
19030
23788
28545
13243
26486
39729
52971
66214
79457
1.25
5601
11203
16804
22406
28007
33608
13795
27589
41384
55179
68973
82768
1.3
6553
13106
19659
26211
32764
39317
14346
28693
43039
57386
71732
86079
1.35
7621
15241
22862
30483
38103
45724
14898
29796
44695
59593
74491
89389
1.4
8814
17628
26442
35256
44069
52883
15450
30900
46350
61800
77250
92700
1.45
10142
20284
30426
40568
50710
60853
16002
32004
48005
64007
80009
96011
1.5
11615
23230
34845
46460
58075
69690
16554
33107
49661
66214
82768
99322
1.55
13243
26486
39729
52972
66214
79457
17105
34211
51316
68421
85527
102632
1.6
15036
30072
45108
60144
75181
90217
17657
35314
52971
70629
88286
105943
1.65
17006
34011
51017
68022
85028
102034
18209
36418
54627
72836
91045
109254
1.7
19162
38325
57487
76650
95812
114975
18761
37521
56282
75043
93804
112564
1.75
21518
43037
64555
86073
107592
129110
19313
38625
57938
77250
96563
115875
1.8
24085
48170
72255
96340
120425
144510
19864
39729
59593
79457
99322
119186
1.85
26875
53749
80624
107499
134373
161248
20416
40832
61248
81664
102080
122497
42
Required Torque and Max Shear for 18 inch Beam Length and Varying-Cross Sectional
Areas (a) and Pitch Angles (phi)
REQUIRED TORQUE (in*lb)
L
phi
18
Max Shear
in
L
5
10
15
20
25
30
0.25
7
15
22
30
37
45
0.3
15
31
46
62
77
93
0.35
29
57
86
115
143
0.4
49
98
147
196
245
0.45
78
157
235
314
0.5
119
239
358
478
0.55
175
350
525
0.6
248
496
743
0.65
341
683
phi
18
10
15
20
25
30
2299
4598
6897
2759
5518
8277
9196
11496
13795
11036
13795
16554
172
3219
6438
9656
294
3679
7357
11036
12875
16094
19313
14714
18393
392
470
4138
8277
22071
12415
16554
20692
597
717
4598
9196
24830
13795
18393
22991
27589
700
875
1050
5058
10116
15174
20232
25290
30348
991
1239
1487
5518
11036
16554
22071
27589
33107
1024
1365
1706
2048
5978
11955
17933
23911
29888
35866
38625
a
0.7
459
918
1377
1836
2295
2754
6438
12875
19313
25750
32188
0.75
605
1210
1815
2420
3025
3630
6897
13795
20692
27589
34487
41384
0.8
783
1566
2349
3133
3916
4699
7357
14714
22071
29429
36786
44143
0.85
998
1996
2994
3992
4990
5988
7817
15634
23451
31268
39085
46902
0.9
1254
2509
3763
5018
6272
7527
8277
16554
24830
33107
41384
49661
0.95
1557
3115
4672
6229
7786
9344
8737
17473
26210
34946
43683
52420
1
1912
3824
5736
7648
9560
11472
9196
18393
27589
36786
45982
55179
1.05
2324
4648
6972
9296
11620
13944
9656
19313
28969
38625
48281
57938
1.1
2799
5599
8398
11197
13996
16796
10116
20232
30348
40464
50580
60696
1.15
3344
6688
10032
13376
16720
20064
10576
21152
31728
42304
52880
63455
1.2
3965
7929
11894
15858
19823
23788
11036
22071
33107
44143
55179
66214
1.25
4668
9336
14003
18671
23339
28007
11496
22991
34487
45982
57478
68973
1.3
5461
10921
16382
21843
27303
32764
11955
23911
35866
47821
59777
71732
1.35
6351
12701
19052
25402
31753
38103
12415
24830
37246
49661
62076
74491
1.4
7345
14690
22035
29380
36725
44069
12875
25750
38625
51500
64375
77250
1.45
8452
16903
25355
33807
42259
50710
13335
26670
40005
53339
66674
80009
1.5
9679
19358
29038
38717
48396
58075
13795
27589
41384
55179
68973
82768
1.55
11036
22071
33107
44143
55179
66214
14254
28509
42763
57018
71272
85527
1.6
12530
25060
37590
50120
62650
75181
14714
29429
44143
58857
73572
88286
1.65
14171
28343
42514
56685
70857
85028
15174
30348
45522
60696
75871
91045
1.7
15969
31937
47906
63875
79844
95812
15634
31268
46902
62536
78170
93804
1.75
17932
35864
53796
71728
89660
107592
16094
32188
48281
64375
80469
96563
1.8
20071
40142
60212
80283
100354
120425
16554
33107
49661
66214
82768
99322
1.85
22396
44791
67187
89582
111978
134373
17013
34027
51040
68054
85067
102080
43
Required Torque and Max Shear for 24 inch Beam Length and Varying Cross-Sectional
Areas (a) and Pitch Angles (phi)
REQUIRED TORQUE (in*lb)
L
phi
24
Max Shear
in
L
24
in
5
10
15
20
25
30
5
10
15
20
25
30
0.25
6
11
17
22
28
34
1724
3449
5173
6897
8622
10346
0.3
12
23
35
46
58
70
2069
4138
6208
8277
10346
12415
0.35
22
43
65
86
108
129
2414
4828
7242
9656
12070
14484
0.4
37
73
110
147
184
220
2759
5518
8277
11036
13795
16554
0.45
59
118
176
235
294
353
3104
6208
9311
12415
15519
18623
0.5
90
179
269
358
448
538
3449
6897
10346
13795
17243
20692
22761
a
0.55
131
262
394
525
656
787
3794
7587
11381
15174
18968
0.6
186
372
558
743
929
1115
4138
8277
12415
16554
20692
24830
0.65
256
512
768
1024
1280
1536
4483
8967
13450
17933
22416
26900
0.7
344
689
1033
1377
1721
2066
4828
9656
14484
19313
24141
28969
0.75
454
907
1361
1815
2269
2722
5173
10346
15519
20692
25865
31038
0.8
587
1175
1762
2349
2937
3524
5518
11036
16554
22071
27589
33107
0.85
749
1497
2246
2994
3743
4491
5863
11725
17588
23451
29314
35176
0.9
941
1882
2822
3763
4704
5645
6208
12415
18623
24830
31038
37246
0.95
1168
2336
3504
4672
5840
7008
6552
13105
19657
26210
32762
39315
1
1434
2868
4302
5736
7170
8604
6897
13795
20692
27589
34487
41384
1.05
1743
3486
5229
6972
8715
10458
7242
14484
21727
28969
36211
43453
1.1
2099
4199
6298
8398
10497
12597
7587
15174
22761
30348
37935
45522
1.15
2508
5016
7524
10032
12540
15048
7932
15864
23796
31728
39660
47592
1.2
2973
5947
8920
11894
14867
17841
8277
16554
24830
33107
41384
49661
1.25
3501
7002
10503
14003
17504
21005
8622
17243
25865
34487
43108
51730
1.3
4096
8191
12287
16382
20478
24573
8967
17933
26900
35866
44833
53799
1.35
4763
9526
14289
19052
23814
28577
9311
18623
27934
37246
46557
55868
1.4
5509
11017
16526
22035
27543
33052
9656
19313
28969
38625
48281
57938
1.45
6339
12678
19016
25355
31694
38033
10001
20002
30003
40005
50006
60007
1.5
7259
14519
21778
29038
36297
43556
10346
20692
31038
41384
51730
62076
1.55
8277
16554
24830
33107
41384
49661
10691
21382
32073
42763
53454
64145
1.6
9398
18795
28193
37590
46988
56385
11036
22071
33107
44143
55179
66214
1.65
10628
21257
31885
42514
53142
63771
11381
22761
34142
45522
56903
68284
1.7
11977
23953
35930
47906
59883
71859
11725
23451
35176
46902
58627
70353
1.75
13449
26898
40347
53796
67245
80694
12070
24141
36211
48281
60352
72422
1.8
15053
30106
45159
60212
75265
90319
12415
24830
37246
49661
62076
74491
1.85
16797
33593
50390
67187
83983
100780
12760
25520
38280
51040
63800
76560
44
Normal Stresses due to in square cross-section with varying area (a) and CF Load
Stress
(Psi)
10000
20000
30000
40000
50000
0.25
160000
320000
480000
640000
800000
0.3
111111.1
222222.2
333333.3
444444.4
555555.6
0.35
81632.65
163265.3
244898
326530.6
408163.3
0.4
62500
125000
187500
250000
312500
0.45
49382.72
98765.43
148148.1
197530.9
246913.6
0.5
40000
80000
120000
160000
200000
0.55
33057.85
66115.7
99173.55
132231.4
165289.3
0.6
27777.78
55555.56
83333.33
111111.1
138888.9
0.65
23668.64
47337.28
71005.92
94674.56
118343.2
0.7
20408.16
40816.33
61224.49
81632.65
102040.8
0.75
17777.78
35555.56
53333.33
71111.11
88888.89
CF
a
0.8
15625
31250
46875
62500
78125
0.85
13840.83
27681.66
41522.49
55363.32
69204.15
0.9
12345.68
24691.36
37037.04
49382.72
61728.4
0.95
11080.33
22160.66
33241
44321.33
55401.66
1
10000
20000
30000
40000
50000
1.05
9070.295
18140.59
27210.88
36281.18
45351.47
1.1
8264.463
16528.93
24793.39
33057.85
41322.31
1.15
7561.437
15122.87
22684.31
30245.75
37807.18
1.2
6944.444
13888.89
20833.33
27777.78
34722.22
1.25
6400
12800
19200
25600
32000
1.3
5917.16
11834.32
17751.48
23668.64
29585.8
1.35
5486.968
10973.94
16460.91
21947.87
27434.84
1.4
5102.041
10204.08
15306.12
20408.16
25510.2
1.45
4756.243
9512.485
14268.73
19024.97
23781.21
1.5
4444.444
8888.889
13333.33
17777.78
22222.22
1.55
4162.331
8324.662
12486.99
16649.32
20811.65
1.6
3906.25
7812.5
11718.75
15625
19531.25
1.65
3673.095
7346.189
11019.28
14692.38
18365.47
1.7
3460.208
6920.415
10380.62
13840.83
17301.04
1.75
3265.306
6530.612
9795.918
13061.22
16326.53
1.8
3086.42
6172.84
9259.259
12345.68
15432.1
1.85
2921.841
5843.682
8765.522
11687.36
14609.2
45
Strain values for square cross-sectional areas (a) and varying CF Load,
with E1=18.3Msi.
Strain (in/in)
CF
10000
20000
30000
40000
50000
0.25
0.009
0.017
0.026
0.035
0.044
0.30
0.006
0.012
0.018
0.024
0.030
0.35
0.004
0.009
0.013
0.018
0.022
0.40
0.003
0.007
0.010
0.014
0.017
0.45
0.003
0.005
0.008
0.011
0.013
0.50
0.002
0.004
0.007
0.009
0.011
0.55
0.002
0.004
0.005
0.007
0.009
0.60
0.002
0.003
0.005
0.006
0.008
0.65
0.001
0.003
0.004
0.005
0.006
0.70
0.001
0.002
0.003
0.004
0.006
0.75
0.001
0.002
0.003
0.004
0.005
0.80
0.001
0.002
0.003
0.003
0.004
0.85
0.001
0.002
0.002
0.003
0.004
0.90
0.001
0.001
0.002
0.003
0.003
0.95
0.001
0.001
0.002
0.002
0.003
1.00
0.001
0.001
0.002
0.002
0.003
1.05
0.000
0.001
0.001
0.002
0.002
1.10
0.000
0.001
0.001
0.002
0.002
1.15
0.000
0.001
0.001
0.002
0.002
1.20
0.000
0.001
0.001
0.002
0.002
1.25
0.000
0.001
0.001
0.001
0.002
1.30
0.000
0.001
0.001
0.001
0.002
1.35
0.000
0.001
0.001
0.001
0.001
1.40
0.000
0.001
0.001
0.001
0.001
1.45
0.000
0.001
0.001
0.001
0.001
1.50
0.000
0.000
0.001
0.001
0.001
1.55
0.000
0.000
0.001
0.001
0.001
1.60
0.000
0.000
0.001
0.001
0.001
1.65
0.000
0.000
0.001
0.001
0.001
1.70
0.000
0.000
0.001
0.001
0.001
1.75
0.000
0.000
0.001
0.001
0.001
1.80
0.000
0.000
0.001
0.001
0.001
1.85
0.000
0.000
0.000
0.001
0.001
a
46
Appendix C: Analysis of Laminate Cases
Comparison of Three Laminate Cases
For this Study, each case analyzed with identical geometries & loading conditions
Loads: CF, 10 deg Pitch Angle, 0.5 deg Flatwise Bending
Equivalent Properties via Classic Laminate Theory
(E & G modulus in Msi)
Case
Ex
Ey
Ez
Gxy
Gyz
Gzx
ν12
ν23
ν31
1
9.38
6.68
6.68
3.70
2.48
3.70
0.45
0.35
0.45
2
13.0
3.61
3.61
3.32
1.34
3.32
0.74
0.35
0.74
3
18.3
2.68
2.68
1.97
0.991
1.97
0.63
0.35
0.63
Case
Orientation
Number of Plies
1
(02 / +/-452 / 902 / +/-452 / 02)
80
2
(0 / +/-45 / 0)
80
3
(03 / +/-45 / 03)
80
Stresses in Tension-Torsion Strap
Case 1
Case 2
Normal
XY Shear
XY Shear Stress
ZX Shear
Stress (psi)
Stress (psi)
Transition (psi)
Stress (psi)
12639
13147
10630
86802
15428
11946
12978
86802
Case 3
10970
11241
86806
* Values in Red are above self-imposed Limits for this design
9185
See the following pages for FEA models of each of these cases with measured
stress locations.
47
Case 1
Normal Stress (x)
Interlaminar Shear Stresses (XY & XY Transition)
Interlaminar Shear Stress XZ
48
Case 2
Normal Stress (x)
Interlaminar Shear Stresses (XY & XY Transition)
Interlaminar Shear Stress XZ
49
Case 3
Normal Stress (x)
Interlaminar Shear Stresses (XY & XY Transition)
Interlaminar Shear Stress XZ
50
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