Comparative Analysis of Composite Material Substitution for a
Structural Metallic Rotor Head Component
by
Kevin A. Laitenberger
An Engineering Research 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-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2011
i
© Copyright 2011
by
Kevin A. Laitenberger
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF EQUATIONS .................................................................................................. viii
LIST OF SYMBOLS ........................................................................................................ ix
GLOSSARY ...................................................................................................................... x
ABSTRACT ..................................................................................................................... xi
1. INTRODUCTION/BACKGROUND .......................................................................... 1
1.1
Yoke Design ....................................................................................................... 2
1.2
Materials ............................................................................................................. 3
2. Methodology ................................................................................................................ 4
2.1
Metallic Materials [1] ......................................................................................... 4
2.2
Composite Materials [1]..................................................................................... 5
2.3
Composite Layup ............................................................................................... 7
2.4
Laminate Material Properties [5] ....................................................................... 8
2.4.1
Compliance and Stiffness Matrices ........................................................ 9
2.4.2
Transformed Reduced Stiffness Matrix ............................................... 10
2.5
Finite Element Analysis ................................................................................... 11
2.6
Flaw Tolerance ................................................................................................. 15
3. Results........................................................................................................................ 17
3.1
Composite Laminate Global Material Properties ............................................. 17
3.2
Finite Element Analysis ................................................................................... 18
3.2.1
Ti 6Al-4V Results ................................................................................ 18
3.2.2
Aluminum 7175-T74 Results ............................................................... 21
3.2.3
HexPly 8552 [0,+45,-45,0] Results ...................................................... 24
3.2.4
HexPly 8552 [02,+45,-45,02] Results ................................................... 27
3.2.5
HexPly 8552 [03,+45,-45,03] Results ................................................... 30
iii
3.3
3.2.6
Modified Yoke Geometry .................................................................... 33
3.2.7
Finite Element Analysis Complete Results .......................................... 37
Flaw Tolerance ................................................................................................. 38
3.3.1
Titanium 6Al-4V Flaw Tolerance ........................................................ 39
3.3.2
Aluminum 7175-T74 Flaw Tolerance.................................................. 41
3.3.3
HexPly 8552 Flaw Tolerance ............................................................... 43
3.3.4
Flaw Tolerance Comparison ................................................................ 46
4. Conclusions................................................................................................................ 47
References........................................................................................................................ 49
Appendix A – Classical Lamination Theory Code .......................................................... 50
iv
LIST OF TABLES
Table 2.1 - Metallic material comparison .......................................................................... 5
Table 2.2 - Prepreg resins available from Hexcel Corp..................................................... 6
Table 2.3 - Prepreg reinforcement fibers available from Hexcel Corp ............................. 6
Table 2.4 - HexPly 8552 physical properties ..................................................................... 6
Table 2.5 - 3-D HexPly 8552 Material Properties ............................................................. 9
Table 3.1 - CLT global material property results ............................................................ 17
Table 3.2 - Layup orientation break down....................................................................... 17
Table 3.3 - TI 6Al-4V Safety Margins ............................................................................ 20
Table 3.4 – Al 7175-T74 Safety Margins ........................................................................ 23
Table 3.5 - [0,+45,-45,0] Laminate Stress Allowables .................................................... 26
Table 3.6 - [0,+45,-45,0] Safety Margins ........................................................................ 27
Table 3.7 - [02,+45,-45,02] Laminate Stress Allowables ................................................. 29
Table 3.8 - [02,+45,-45,02] Safety Margins ..................................................................... 30
Table 3.9 - [03,+45,-45,03] Laminate Stress Allowables ................................................. 32
Table 3.10 - [03,+45,-45,03] Safety Margins ................................................................... 32
Table 3.11 – Modified Geometry Safety Margins ........................................................... 36
Table 3.12 - Safety Margins for all cases ........................................................................ 38
Table 3.13 - Ti 6Al-4V Flawed Stress Increase .............................................................. 40
Table 3.14 – Al 7175-T74 Flawed Stress Increase .......................................................... 42
Table 3.15 - Flawed [02,+45,-45,02] Material Properties ................................................ 43
Table 3.16 – HexPly 8552 Flawed Stress Increase.......................................................... 45
Table 3.17 - Flawed Stress Increase for all Materials ..................................................... 46
v
LIST OF FIGURES
Figure 1.1 - Main rotor hub with blade retention yoke...................................................... 1
Figure 1.2 - Yoke design profile ........................................................................................ 2
Figure 1.3 - Basic yoke with a dummy focal bearing ........................................................ 3
Figure 2.1 - Centrifugal force loading direction ................................................................ 7
Figure 2.2 - Ansys 1/4 symmetry model setup ................................................................ 13
Figure 2.3 - Composite yoke representation split into sections ....................................... 14
Figure 2.4 - Flaw geometry and location ......................................................................... 16
Figure 3.1 - Ti 6Al-4V Normal Stress (X) ...................................................................... 19
Figure 3.2 - Ti 6Al-4V Normal Stress (X) magnified lug area ....................................... 19
Figure 3.3 - Ti 6Al-4V Shear Stress (XY) ....................................................................... 20
Figure 3.4 - Al 7175-T74 Normal Stress (X) .................................................................. 22
Figure 3.5 - Al 7175-T74 Normal Stress (X) magnified lug area ................................... 22
Figure 3.6 - Al 7175-T74 Shear Stress (XY) ................................................................... 23
Figure 3.7 - [0,+45,-45,0] Normal Stress (X) .................................................................. 25
Figure 3.8 - [0,+45,-45,0] Normal Stress (X) magnified lug area ................................... 25
Figure 3.9 - [0,+45,-45,0] Shear Stress (XY) .................................................................. 26
Figure 3.10 - [02,+45,-45,02] Normal Stress (X) ............................................................. 28
Figure 3.11 - [02,+45,-45,02] Normal Stress (X) magnified lug area ............................. 28
Figure 3.12 - [02,+45,-45,02] Shear Stress (XY).............................................................. 29
Figure 3.13 - [03,+45,-45,03] Normal Stress (X) ............................................................. 31
Figure 3.14 - [03,+45,-45,03] Normal Stress (X) magnified lug area ............................. 31
Figure 3.15 - [03,+45,-45,03] Shear Stress (XY).............................................................. 32
Figure 3.16 - Modified yoke geometry ............................................................................ 34
Figure 3.17 - Modified Geometry Normal Stress (X) ..................................................... 35
Figure 3.18 - Modified Geometry Normal Stress (X) magnified lug area ...................... 35
Figure 3.19 - Modified Geometry Shear Stress (XY)...................................................... 36
Figure 3.20 - 6AL-4V Flawed Normal Stress (X) ........................................................... 39
Figure 3.21 - 6AL-4V Flawed Shear Stress (XY) ........................................................... 40
Figure 3.22 - 7175-T74 Flawed Normal Stress (X) ......................................................... 41
Figure 3.23 - 7175-T74 Flawed Shear Stress (XY) ......................................................... 42
vi
Figure 3.24 - [02,+45,-45,02] Flawed Normal Stress (X) ................................................ 44
Figure 3.25 - [02,+45,-45,02] Flawed Shear Stress (XY) ................................................. 44
vii
LIST OF EQUATIONS
Equation 2.1 - Laminate Strain-Stress relationship ........................................................... 9
Equation 2.2 - Compliance Matrix definition .................................................................... 9
Equation 2.3 - Reduced Strain-Stress relationship ............................................................ 9
Equation 2.4 - Laminate Stress-Strain relationship ......................................................... 10
Equation 2.5 - Stiffness Matrix definition ....................................................................... 10
Equation 2.6 - Transformation Matrix definition ............................................................ 10
Equation 2.7 - Transformation symbol definition ........................................................... 10
Equation 2.8 - Force-Stress A value definition ............................................................... 10
Equation 2.9 - Force-Stress B value definition ................................................................ 11
Equation 2.10 - Force-Stress D value definition ............................................................. 11
Equation 2.11 - Force-Strain relationship ........................................................................ 11
Equation 2.12 - Margin of Safety Equation ..................................................................... 14
viii
LIST OF SYMBOLS
[ABD] – Laminate Stiffness Matrix (psi)
[abd] – Laminate Compliance Matrix (1/psi)
CF – Centrifugal Force (lb)
E – Young’s Modulus (psi)
ε – Normal Strain (unitless - in/in)
γ – Shear Strain (unitless - in/in)
Fcu – Ultimate Compressive Strength (psi)
Fcy – Yield Compressive Strength (psi)
Fsu – Ultimate Shear Strength (psi)
Ftu – Ultimate Tensile Strength (psi)
Fty – Yield Tensile Strength (psi)
G – Shear Modulus (psi)
M.S.ult-comp – Compressive Ultimate Margin of Safety (unitless)
M.S.yld-comp – Compressive Yield Margin of Safety (unitless)
M.S.ult-shear – Shear Ultimate Margin of Safety (unitless)
M.S.ult-tensile – Tensile Ultimate Margin of Safety (unitless)
M.S.yld-tensile – Tensile Yield Margin of Safety (unitless)
ν – Poisson’s Ratio (unitless - in/in)
σ – Normal Stress (psi)
σT – Tensile Normal Stress (psi)
σC – Compressive Normal Stress (psi)
[Q] – Layer Stiffness Matrix (psi)
[S] – Layer Compliance Matrix (1/psi)
τ – Shear Stress (psi)
ix
GLOSSARY
ANSYS – a finite element analysis program developed by ANSYS, Inc.
CATIA – a computer aided design program developed by Dassault Systemes.
CLT – Classical Lamination Theory, a method used to determine the material properties
of a complete laminate.
FEA – Finite Element Analysis, the application the principles of the Finite Element
Method.
FEM – Finite Element Method, the technique of splitting a component into several small
elements and nodes to derive the resulting stresses and strains initiating at a
known boundary condition.
Hexcel Corp– composite material supplier and manufacturer based out of Stamford CT.
HexPly – a line of pre-impregnated composite materials offered by Hexcel Corp.
Laminate – a composite material made up of multiple layers or plies.
MATLAB – a numerical computation program developed by The MathWorks, Inc.
Ply – a single thin sheet of a composite material, multiple plies are stacked to create a
laminate.
x
ABSTRACT
The objective of the project described in this report was to develop a quantitative
comparison of a metallic component and its equivalent composite component for a
structural aerospace application. In the field of helicopter design, as with any aerospace
structure, the ultimate goal is to optimize performance by minimizing the weight of the
structure while maintaining strength. Typical helicopter rotor head components are
normally manufactured from high strength alloys. In order to reduce the weight of the
rotor head, replacement of these metallic components with an equivalent composite
component is of increasing interest.
In this project the design of a simple Blade
Retention Yoke using both a metal alloy and a composite layup was considered. Blade
retention yokes experience high Centrifugal forces; therefore, the structural requirements
for the materials are derived from these forces.
The yoke design requires the
determination of the proper materials, based on available strength properties. Using a
basic hoop shaped yoke design, a comparison of the selected materials is performed
using the finite element method. Based on the component design, materials used, and
applied loading, weight reduction and margins of safety were calculated.
Additionally,
the flaw tolerance of each component was evaluated to account for damage that may
occur to the component during operation.
xi
1. INTRODUCTION/BACKGROUND
The function of a helicopter rotor head is to retain the blades during rotation
while allowing articulation to increase thrust and providing control to the aircraft. One
configuration used for the retention of the main rotor blades is a blade retention yoke.
The yoke attaches to the rotor hub via a focal bearing which allows the yoke to pitch,
lag, and flap. The blade is then usually attached to the yoke through a cuff which bolts
to the yoke. This configuration is shown in Figure 1.1. Due to the nature of these
interfaces, a simple hoop style design can be used for the yoke, which leads to the yoke
being a prime candidate for the use of composite materials. The use of composite
materials can significantly reduce weight due to the high strength to weight ratio,
compared to that of common aerospace alloys. However, rotor components can be
complex in nature requiring intricate machining, which limits the number of components
that can incorporate composite materials. On the other hand the composition of the
composite material may also be tailored to optimize structural capabilities is certain
directions where greater strength is necessary; this is not possible with isotropic
materials.
Hub
Cuff
Attachment
Focal
Yoke
Bearing
Figure 1.1 - Main rotor hub with blade retention yoke
1
1.1 Yoke Design
A yoke has two separate functional interfaces, which are the blade/cuff and the focal
bearing. These interfaces must be taken into consideration when sizing the yoke for a
particular aircraft. For the purposes of this project, the yoke will be designed for a heavy
lift helicopter with an estimated gross weight of 35,000 lbs. The basic shape profile of
the yoke design is shown in Figure 1.2 and a view of the yoke with a dummy focal
bearing is shown in Figure 1.3.
Figure 1.2 - Yoke design profile
2
Figure 1.3 - Basic yoke with a dummy focal bearing
1.2 Materials
Blade retention yokes and rotor head components are typically made from metallic
materials.
Previously, designs were limited to mainly aluminum and titanium for
structural aerospace designs due to the high strength and low weight requirements.
Recent advances in technology have brought to light the capabilities and design
possibilities available with composite materials. Composites offer a durable light weight
solution to structural component design. With the current manufacturing processes and
understanding of material properties it is possible to create a cost effective preliminary
design using composite materials. For the purposes of this comparison, it is necessary to
select metallic and composite materials that will fulfill the necessary requirements for
the blade retention yoke. A down selection is required in order to determine the best
materials for this comparison.
3
2. Methodology
2.1 Metallic Materials [1]
A comparison of suitable metallic materials was developed using information and
data from the FAA Metallic Materials Properties Development and Standardization
(MMPDS) report [2]. The initial down selection was based on the size and function of
the yoke, which reduced the materials to alloys available in forging form. Potential
alloys included steels, aluminum, magnesium, titanium, and copper alloys. This list was
further reduced using a combination of strength and weight properties. The intent is to
select an alloy that can sustain the applied loads, while also maintaining a feasible
weight for an aircraft.
While being extremely light, magnesium does not have the necessary strength
requirements.
Copper is too heavy with respect to its strength capabilities, when
compared to the other available alloys. The down selection process further reduced the
applicable alloy forgings to six steel alloys, six aluminum alloys, and six titanium alloys.
Within the material type, the alloys were then ranked with respect to ultimate tensile
strength, density, and strength-to-weight ratio. Since the primary load seen by the yoke
will be centrifugal force, the tensile strengths were used for the comparison opposed to
compressive or shear. The final selection yielded two feasible alloys which are 7175
Aluminum and 6Al-4V Titanium. The aluminum alloy offers a lighter weight, but does
not provide high strength similar to titanium. Therefore, both alloys will be analyzed for
the basic yoke design. The material properties and rankings of all the alloy forgings are
shown in Table 2.1.
4
Table 2.1 - Metallic material comparison
2.2 Composite Materials [1]
Composite materials offer several solutions for material selection depending on the
application of the designed component. Based on the size and loading characteristics of
the yoke, the best material configuration is a resin-fiber solution. This configuration
selection can be further specified to a pre-impregnated product based on availability and
ease of manufacturing. Prepregs are the most widely used type of composite materials
for structural aerospace applications and have the offer widest variety of physical
properties. Also the manufacturing process of the component is made more efficient and
cost effective based on the fact that plies simply need to be stacked together, put under
pressure, and cured.
Unfortunately, very few industry wide composite standards are available due to the
high variation between materials and the private nature of different suppliers’ products;
therefore, the materials used for this comparison are based on the data and availability of
the products offered by Hexcel Corporation [3]. In order to select the proper prepreg
5
composite product, a resin and a fiber type must be selected. The available Hexcel
resins are shown in Table 2.2 and the fibers are shown in Table 2.3. Of the available
resins, Epoxy offers the best strength, easy processing, and is by far the most common
resin found in prepregs on the market. There are several fibers available to reinforce the
epoxy resin; however, carbon is by far the strongest available.
There are generally two types of fiber reinforcement: uni-directional tape and a
woven fabric. The uni-directional tapes offer very high uni-axial strength properties, but
also lacks in shear strength. Woven fabrics, on the other hand, offer a more semiisentropic strength solution. Due to the prominently uni-axial loading characteristics of
the yoke, carbon uni-axial tape was chosen for the reinforcement fiber. Hexcel offers a
few epoxy resin-carbon fiber solutions; however, HexPly 8552 offered the most
favorable physical properties. The characteristics at room temperature (~75-77˚ F) of
HexPly 8552 [4] are shown in Table 2.4.
Table 2.2 - Prepreg resins available from Hexcel Corp
Table 2.3 - Prepreg reinforcement fibers available from Hexcel Corp
Table 2.4 - HexPly 8552 physical properties
6
2.3 Composite Layup
The functional purpose of the yoke is to retain the main rotor blade through flight.
Due to this function, the primary load path seen by the yoke is uni-axial in the outboard
direction; this loading is a result of centrifugal force pulling on the blade and cuff
components. The CF loading direction is shown Figure 2.1. Although the primary
loading is uni-directional, the yoke still experiences shear forces and moments caused by
the blade motions during flight. Due to the nature of the load path, the highest strength
will be required in the 0˚ direction which will align with the outboard CF loading. Yet
additional off-center orientated plies will be required to account for the shear forces and
moments described earlier.
Figure 2.1 - Centrifugal force loading direction
Typically for a thick laminate, it is very important to ensure that the overall layup is
balanced and symmetric. A balanced laminate is defined as a stacking pattern where for
every off axis ply there is a ply with the same properties orientated in the opposite
direction, meaning that the number of +45˚ plies is equal to the number of -45˚ plies.
Symmetric laminates exhibit the same stacking sequence in both directions from the
mid-plane, meaning that every ply has an identical ply at the same distance away from
the mid-plane in the opposite direction. The combination of a balanced and symmetric
laminate greatly reduces the shear coupling in the laminate and simplifies the calculation
7
of overall material properties. With this in mind, three separate layup schemes are
analyzed for the yoke. These schemes are [0,+45,-45,0] repeating, [02, +45,-45, 02]
repeating, and [03,+45,-45,03] repeating. The difference between these layups is that the
02 represents two zero degree plies being applied instead of a single ply, and similarly
the 03 represents three consecutive zero degree plies. These schemes will show the
change in laminate properties with respect to the ratio of 0˚ to +/- 45˚ plies.
2.4 Laminate Material Properties [5]
Composite materials offer great variability and customization of directional
strength; however, unlike isotropic materials, the analysis of a composite component
requires in-depth material property calculations. Due to the layered nature of prepreg
composites, each layer must be taken into consideration when deriving the overall
laminate properties. The most common approach to these derivations is the use of
Classical Lamination Theory (CLT).
To utilize CLT, the thickness, orientation,
directional Young’s moduli, directional shear moduli, and directional Poisson’s ratios
must be known, which in turn will provide the global Young’s moduli, Shear moduli,
and Poisson’s ratios. However, CLT does not provide exact global properties and
requires the incorporation of Kirchhoff’s Hypothesis which assumes:

All laminate layers are bonded perfectly together.

There is no slippage between the laminate layers.

Lines normal to the mid-plane remain normal and straight after deformation.
Meaning that the lines do not deform.

The thickness of each laminate layer remains constant before, during, and
after deformation.
These assumptions result in the finding that there is no through thickness strain (εZ)
and also that there are no shear strains coupled in the Z direction (γZX and γZY). For the
purpose of this analysis, the additional assumption that the HexPly 8552 IM7 Carbon
prepreg material is transversely isotropic is also made. This assumption provides that
the material properties in Z direction are equivalent to those in the Y direction.
8
2.4.1
Compliance and Stiffness Matrices
Before CLT can be applied, the known material properties for each layer must be
used to develop a stress-strain relationship for that layer. This relationship is commonly
known as the compliance matrix [S], and is derived using the Young’s moduli, shear
moduli, and Poisson’s ratio of the material. For the HexPly 8552 [4] material selected
these values are as follows:
Table 2.5 - 3-D HexPly 8552 Material Properties
E1 msi
E2 msi
E3 msi
G12 msi
G13 msi
G23 ksi
ν12
ν13
ν23
23.8
1.7
1.7
2.2
2.2
476.0
.32
.32
.79
These material properties are then used to determine the compliance matrix relationship
defined by:
{ε} = [S]{σ}
[2.1]
Where
𝟏/𝐄𝟏
−𝛎𝟏𝟐 /𝐄𝟐
−𝛎𝟏𝟐 /𝐄𝟐
[𝑺] =
𝟎
𝟎
[
𝟎
−𝛎𝟏𝟐 /𝐄𝟐
𝟏/𝐄𝟐
−𝛎𝟐𝟑 /𝐄𝟐
𝟎
𝟎
𝟎
−𝛎𝟏𝟐 /𝐄𝟐
−𝛎𝟐𝟑 /𝐄𝟐
𝟏/𝐄𝟐
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝟏/𝐆𝟐𝟑
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝟏/𝐆𝟏𝟐
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝟏/𝐆𝟏𝟐 ]
[2.2]
For the evaluation of each layer independently the stress strain relationship can be
reduced:
𝜺𝟏
𝑺𝟏𝟏
𝜺
{ 𝟐 } = [𝑺𝟐𝟏
𝜸𝟏𝟐
𝟎
𝑺𝟏𝟐
𝑺𝟐𝟐
𝟎
𝝈𝟏
𝟎
𝟎 ] { 𝝈𝟐 }
𝑺𝟔𝟔 𝝉𝟏𝟐
[2.3]
The inverse of this relationship is called the stiffness matrix [Q].
relationship is defined as:
9
The stiffness
{ σ } = [Q]{ ε }
[2.4]
Where
[Q] = [S]-1
2.4.2
[2.5]
Transformed Reduced Stiffness Matrix
Composite laminates often have plies that are oriented at an angle from the reenforcement fiber direction. This off axis orientation is incorporated to increase the
strength in the secondary direction, thus to offer better shear stiffness. In order to
account for the varying orientation throughout the laminate thickness, the reduced
stiffness matrix must be transformed for each layer to determine the global properties.
This is done using the transformation matrix [T], which is defined as:
 m2
n2
2mn 
 2
T    n m 2  2mn 
 mn mn m 2  n 2 


[2.6]
[
Where
𝒎 = 𝒄𝒐𝒔𝜽
and
[ ]
𝒏 = 𝒔𝒊𝒏𝜽
[2.7]
Once the transformed reduced stiffness matrix is developed for each individual ply, a
relation between the global applied forces and moments with respect to the resultant
stress can be derived. This force-stress relationship is defined as the ABD matrix, where
the values of the matrix are defined as follows:
N
_
Aij   Q ijk  z k  z k 1 
k 1
10
[2.8]
Bij   Q ijk z k2  z k21 
[2.9]
Dij   Q ijk z k3  z k31 
[2.10]
N
_
k 1
N
_
k 1
Where
_
Q  Transformed Reduced Stiffness Matrix
z  z-directional position of the ply
The derivation of the ABD matrix offers the following force-stress relationship:
 N X   A11
 N  A
 Y   12
 N XY   A16
M    B
 X   11
M Y   B12

 
M XY   B16
A12
A16
B11
B12
A22
A26
B12
A26
A66
B16
B12 B22
B16 B26
D11 D12
B22
B26
D12
D22
B26
B66
D16
D26
0
B16    X 

0 
B26    Y 
 0 
B66   XY 
D16   X0 


D26   0 
 Y 
D66   0 
 XY 
[2.11]
The ABD matrix is considered to be laminate stiffness matrix, as shown previously with
the individual layer stiffness matrix, the inverse provides the laminate compliance matrix
[abd].
Using the equations from section 2.4.1 which derived the individual layer
compliance matrix [S], the abd matrix can be used to derive the overall laminate material
properties. This derivation provides the directional Young’s moduli, shear moduli, and
Poisson’s ratios. These properties were calculated using MATLAB, the code written for
these calculations is provided in Appendix A. The calculated global material properties
provide the necessary information to perform a Finite Element Analysis on the designed
yoke to determine the stress resulting from the applied CF load.
2.5 Finite Element Analysis
A static structural stress analysis, resulting from the applied loads, is suitable to
determine whether composite materials are a viable substitute for metallic materials.
11
This structural analysis is best performed using the Finite Element Method, which splits
the component geometry into several elements and nodes, and analyzes the assembly
beginning at known boundary conditions to evaluate the stresses and strains throughout
the entire component.
This analysis is typically performed using a computational
program; the most common in the Aerospace industry is ANSYS.
ANSYS provides a function which is able to import models created by common
Computer Aided Design (CAD) programs. Via the use of CATIA, a yoke model was
created. To perform a representative analysis of the yoke, it is beneficial to also model
and assemble the components which interface with the yoke.
These additional
components include the focal bearing outer member, the blade cuff, and cuff to yoke
attachment pins. Since the main concern for this analysis is the yoke itself, rough
simplified models were created to represent these components.
Once the models are imported into ANSYS, material properties, contact conditions,
supports, and forces can be assigned to the models. Due to the symmetric nature of the
yoke, it was possible to reduce the assembly model to a quarter model by spitting each
component by the XY and XZ planes. Symmetrically splitting the model provides
quicker computation time; however, it is important to also reduce the loading
accordingly. A standard fine mesh was applied to the yoke geometry for computation;
this mesh remains the same for all material cases. For the sake of providing a model
which represents common configurations, the following constraints were applied in
ANSYS and are shown in Figure 2.2:

Yoke-focal bearing: bonded contact

Yoke-cuff attachment pin: bonded contact

Cuff-yoke attachment pin: bonded contact

Focal bearing: remote displacement support located at the bearing focal
point. Displacement is constrained in the X direction to represent the C F
support provided by the hub.

Cuff: a 25,000 lb force in the positive X direction to represent the C F loading
due to hub rotation.
12
Pin: Ferrium S53
Bonded Contact
(Pin-Cuff)
Bonded Contact (YokeFocal Bearing)
CF
Bonded Contact
(Pin-Yoke)
Focal Bearing:
Al 7175-T74
Cuff: Ferrium S53
Figure 2.2 - Ansys 1/4 symmetry model setup
The constraints applied to the yoke assembly model can be carried over from the
metallic materials to the composite materials. Due to the transversely isotropic nature of
the composite laminate and the curved geometry of the yoke, it is necessary to split the
yoke into sections for this analysis. Each section will have different directional material
properties due to curving of the reinforcement fibers.
For the purposes of this
comparison, the yoke was split into three sections: the lug, the upper radius, and the
lower radius. These sections are shown in Figure 2.3. The material properties for each
are derived by transforming the global X and Z properties by the angle of the tangent
line of each radius at the apex.
13
Yoke Lug Section
Yoke Upper Radius
Section
Yoke Lower Radius
Section
Figure 2.3 - Composite yoke representation split into sections
Using the ANSYS model developed, it is possible to create a representative
comparison between the metallic yoke and the composite yoke. To further evaluate the
available materials, five separate cases were analyzed. These cases include Titanium
6AL-4V, Aluminum 7175-T74, HexPly 8552 [0,+45,-45,0] layup, HexPly 8552
[02,+45,-45, 02] layup, and HexPly 8552 [03,+45,-45, 03] layup. The resultant stresses
can then be compared to the yield and ultimate stresses for each material to predict
failure.
Once the Finite Element Analysis is completed, it is possible to predict failure via
the use of safety margins. Margins of safety effectively compare the calculated stress or
load to the ultimate strength of a material, while also taking into consideration a safety
factor. Based on standard industry practices, a safety factor of 1.15 will be used for
yield margins and an additional safety factor of 1.5 will be used for ultimate margins.
These margins are calculated normal stresses and shear stresses and can be performed as
follows [6]:
𝑴. 𝑺. =
𝑺𝒕𝒓𝒖𝒄𝒕𝒖𝒓𝒂𝒍 𝑺𝒕𝒓𝒆𝒏𝒈𝒕𝒉 (𝒖𝒍𝒕𝒊𝒎𝒂𝒕𝒆 𝒐𝒓 𝒚𝒊𝒆𝒍𝒅)
𝑺𝑭∗𝑨𝒑𝒑𝒍𝒊𝒆𝒅 𝑺𝒕𝒓𝒆𝒔𝒔
14
[2.12 ]
2.6 Flaw Tolerance
In addition to evaluating how a structural component will react to the operational
loading conditions as designed, it is important to also consider how well the component
will tolerate damage during operation. This evaluation of flaw tolerance is of particular
concern when designing a blade retention yoke. If a catastrophic failure were to occur,
the helicopter is likely to lose a blade and crash. Typically, as explained by Adams [7],
flaw tolerance is evaluated via physical testing and reductions in S-N curves. However
since this comparison does not provide the opportunity for testing, a preliminary
evaluation will be performed utilizing flawed Finite Element Models.
For the metallic yoke cases, the flaw tolerance is relatively simple. The yoke
geometry will be modified to include imperfections. Since the outer surfaces of the yoke
are more prone to foreign object damage and the highest stresses occur on the flat lug
section, two imperfections will be added to the upper surface of the yoke. These
imperfections will be conical with a depth of .018” and a diameter of .072”, with one
being located .875” outboard of the cuff attachment bore and the other 1.75” inboard.
The depth of this damage was chosen as a typical condition found in the field on similar
rotor head components. The location of the outboard flaw was chosen to be at the
midpoint between the lug and bore edges. The inboard location was chosen to represent
an unprotected area of the yoke that is relatively close to the yoke bore. A cross-section
of the yoke lug through the imperfections is shown in Figure 2.4. Since Titanium and
Aluminum are isotropic materials, the material properties of the yoke will not change;
however, the flaws will create a stress riser that could lead to crack propagation.
Therefore, the flawed geometry needs to be imported in to ANSYS, and the model can
easily be re-analyzed.
15
.018” Depth
.072” Diameter
Flaws
Cuff Attachment Bore
Figure 2.4 - Flaw geometry and location
Due to the transversely isotropic properties and nature of composite laminates, the
flaw tolerance evaluation is more involved. Since damage to a composite component
would likely lead to fracturing of the affected reinforcement fibers, the laminate loses
structural strength in the damaged area.
To account for this loss, the number of
potentially fractured fibers will be removed from the CLT calculation of the material
properties. Due to the nature of CLT calculating the overall material properties based on
plies and not fibers, this can only be done by removing the entire ply that contains the
damaged fiber. Since the depth of the damage is .018” deep, the three outermost plies
will be removed from the calculations. Once the flawed material properties are derived,
the flawed geometry can be input into the composite yoke ANSYS model for stress
analysis as previously performed on the unflawed model.
To determine the flaw tolerance of each material, the stress experienced after the
flaw will be compared to the original model. This comparison will offer a percent
increase of the stress riser, thus providing a comparable figure of the flaw effects. While
this is a valuable exercise in theory, true flaw tolerance evaluation would need to be
confirmed via physical testing.
However, for the purposes of this preliminary
comparison to determine composite material viability, this comparison will provide a
better understanding of the material reactions to the damage.
16
3. Results
3.1 Composite Laminate Global Material Properties
Through the use of MATLAB a computer program was created based on the
Classical Lamination Theory outlined in section 2.4. This program calculated the global
material properties of the complete laminate based on the layup and the prepreg material
properties provided by Hexcel. Three separate layups were used for the calculations to
provide a better understanding of the affect of ±45˚ plies in the layup. The resulting
global material properties are provided in Table 3.1. The percentage of ±45˚ and 0˚
plies is shown in Table 3.2.
Table 3.1 - CLT global material property results
Layup Ex
νxy
νxz
νyz
2.14
0.48
0.48
0.16
3.57
1.82
0.46
0.46
0.10
3.25
1.62
0.44
0.44
0.08
Ey
Ez
Gxy
Gxz
Gyx
(msi)
(msi)
(msi)
(msi)
(msi)
(msi)
1
15.12
4.95
4.95
4.20
4.20
2
17.87
4.01
4.01
3.57
3
19.25
3.50
3.50
3.25
Table 3.2 - Layup orientation break down
Layup
Total Plies
+45˚ Plies
-45˚ Plies
0˚ Plies
1: [0,+45,-45,0]
150
38 (25.3%)
38 (25.3%)
74 (49.3%)
2: [02,+45,-45,02]
150
26 (17.3%)
26 (17.3%)
98 (65.3%)
3: [03,+45,-45,03]
150
20 (13.3%)
20 (13.3%)
110 (73.3%)
The results from the global calculations clearly show that the percentage of ±45˚
plies has a direct influence on the directional stiffness and rigidity of the component,
based on the Young’s and shear moduli respectively. Based on the loading of the
component in question, the proper layup can be chosen to meet the necessary
requirements. For the purposes of the basic yoke model in this comparison, all three
layup cases will be analyzed due to the limited loading criteria.
17
3.2 Finite Element Analysis
Based on the ANSYS setup defined in section 2.5 for the finite element analysis, the
yoke was analyzed with five material compounds: Ti 6Al-4V, Al 7175-T74, a HexPly
8552 [0,+45,-45,0] layup, a HexPly 8552 [02,+45,-45, 02] layup, and a HexPly 8552
[03,+45,-45, 03] layup.
Using the material properties found in the MMPDS and
calculated using the CLT method, it is possible to determine the resulting stresses
throughout the yoke based on the CF loading conditions. For the purposes of this
comparison, a simple loading condition of 100,000 lbs was used to represent the CF
loading a yoke designed for a heavy lift helicopter would experience. While due to the
nature of a fully articulated rotor head, the yoke would also experience additional shear
loads and bending moments, the initial test to determine viability of composite
substitution would be a simple CF tension case. To facilitate quick analytical results, the
model was reduced via symmetry to a quarter model, thus a 25,000 lb load was applied
in the X direction to the cuff. The results vary with the different materials; however
provide great insight as to how the component would perform.
3.2.1
Ti 6Al-4V Results
Titanium 6Al-4V is known for having high strength and toughness, while also
boasting relatively low density for the magnitude of strength. In addition titanium is
fairly resilient to environmental conditions, generally able to withstand a wide range of
temperatures while maintaining high strength characteristics, and also exhibiting good
corrosion resistance properties.
For these reasons titanium, 6Al-4V, is extremely
common in structural aerospace applications, and is the starting point for this
comparison.
The high strength of Ti 6Al-4V is evident in the tensile and compressive strength
properties, as shown previously in Table 2.1. Based on the results of the finite element
model, the titanium yoke appears to far from the yield strength of the material. The
highest stresses are experienced in the lug region of the yoke where contact is made with
the cuff attachment pin. A second hot spot is found at the end of the upper radius where
the focal bearing support ends. Residual stresses normal to the X axis are shown in
18
Figure 3.1 and a magnified view of the lug, from the bottom, is shown in Figure 3.2.
The resulting shear stresses in the XY plane are shown in Figure 3.3.
Figure 3.1 - Ti 6Al-4V Normal Stress (X)
Figure 3.2 - Ti 6Al-4V Normal Stress (X) magnified lug area
19
Figure 3.3 - Ti 6Al-4V Shear Stress (XY)
The maximum normal stresses experienced are 49.9 ksi tensile and 26.3 ksi in
compression. The yoke also experiences shear stress from the CF loading, based on the
analytical model the maximum shear stress is 19.5 ksi. The normal and shear stress
margins of safety were calculated using the method outlined in section 2.5 and are
shown in Table 3.3.
Table 3.3 - TI 6Al-4V Safety Margins
M.S.yld-tensile
M.S.ult-tensile
M.S.yld-compression
M.S.ult-shear
+1.09
+0.51
+3.07
+1.35
These positive margins show that the titanium yoke is not projected to fail due to
normal or shear stress.
While titanium 6Al-4V offers exceptional strength
characteristics and high positive margins of safety, it is important to also consider the
weight of the titanium component. Based on the material data provided in the MMPDS
[2], the density of Titanium 6Al-4V is 0.160 lb/in3. Inputting this density into the inertia
measurement tool in CATIA provides an estimated yoke weight of 29.0 lbs. From the
material down selection performed in chapter 2, titanium was found to be the densest
20
viable material. Thus since reducing weight is the ultimate goal, the high weight of the
titanium yoke will be considered in the final comparison.
3.2.2
Aluminum 7175-T74 Results
Aluminum is another common material used in the aerospace industry when
designing structural components. When compared to titanium, the density of aluminum
is significantly lower; however there is also a significant reduction in strength. In
addition aluminum is also susceptible to corrosion; however this issue can generally be
resolved via passivation and anodizing. For these reasons aluminum is often used in
place of titanium to reduce the weight of components, therefore the 7175 alloy heat
treated to the T74 condition is also of interest in this comparison and has been analyzed
in ASYS.
The ANSYS results for the aluminum yoke are similar to the titanium; however the
magnitude of the stress is slightly less. The stress hotspots occur in the same places,
with the maximum normal stress being located around the lug bore and a secondary hot
spot at the end of the upper radius, where the focal bearing terminates. Due to the lower
strength of the aluminum, the stresses are significantly closer to the yield point; however
the yoke does not appear to yield in this case. The overall X direction normal stress
distribution is shown in Figure 3.4 while a close up of the lug from the underside is
shown in Figure 3.5. Resultant Shear stresses in the XY plane are shown in Figure 3.6.
21
Figure 3.4 - Al 7175-T74 Normal Stress (X)
Figure 3.5 - Al 7175-T74 Normal Stress (X) magnified lug area
22
Figure 3.6 - Al 7175-T74 Shear Stress (XY)
The CF loading results in a maximum tensile normal stress of 38.2 ksi and
compressive normal stress of 25.4 ksi. The shear stress experienced by the yoke is found
to be a maximum of 17.6 ksi, located in the lug area around the bore. The margins of
safety for the Al 7175-T74 material are shown in Table 3.4.
Table 3.4 – Al 7175-T74 Safety Margins
M.S.yld-tensile
M.S.ult-tensile
M.S.yld-compression
M.S.ult-shear
+0.46
+0.12
+1.23
+0.38
The margins of safety are significantly lower than those derived for the titanium
model; however that was an expected outcome. Regardless, the margins remain positive
and show that an aluminum yoke is not projected to fail due to the simulated loading.
Thus based on the normal and shear stress margins, the use of aluminum is a viable
solution.
To further compare the aluminum option to titanium, the estimate weight of the
aluminum yoke must be considered. Using the inertia calculation tool in CATIA, the
estimated weight of the aluminum yoke is 18.0 lbs. The use of aluminum would result
in an 11 lb reduction in weight per arm, this is a significant reduction. However based
23
on the material research performed in section 2.2, composite materials may still offer a
lighter solution.
3.2.3
HexPly 8552 [0,+45,-45,0] Results
Although composite materials offer a light weight solution for a structural
component, it is important to determine how well the component reacts to the same
loading as the Titanium and Aluminum solution. Based on the material comparison
performed in chapter 2, HexPly 8552 unidirectional carbon tape was found to be the
most promising commercially available product, offering favorable strength and stiffness
characteristics.
As discussed in section 2.5, due to the transverse isotropic nature of the HexPly
8552 composite layup, the yoke model must be split to imitate change in directional
properties. Based on this configuration, it is possible to analyze the stress distribution in
the composite yoke. The finite element analysis of the composite yoke results in similar
stress concentration areas as the metallic examples, with the highest stresses being
located at the cuff attachment bore and the focal bearing termination. Due to the high
stiffness of the HexPly material, the resulting stresses in the composite configuration are
significantly higher.
While splitting the yoke helps to better represent the actual material properties
throughout the component, the results are a function of the number of splits. Since there
are only three sections in this basic model, the change in material properties is abrupt,
thus higher stresses at the section terminations are expected. This is evident at the focal
bearing termination; therefore the stresses calculated in the model are likely higher than
would be seen in actual operation. Therefore to provide a comparison to the metallic
components, the stresses in the bore will also be considered. The normal stress in the X
direction throughout the yoke is shown in Figure 3.7 and a magnified view of the
underside lug area is provided in Figure 3.8. Shear stress distribution is also similar to
that of the metallic yokes; however similar to the normal stress, the magnitude is
significantly greater than that of the metallic examples. The resulting shear stress is
shown in Figure 3.9.
24
Figure 3.7 - [0,+45,-45,0] Normal Stress (X)
Figure 3.8 - [0,+45,-45,0] Normal Stress (X) magnified lug area
25
Figure 3.9 - [0,+45,-45,0] Shear Stress (XY)
Similar to the metallic component analysis, once the maximum stresses are
determined, it is possible to calculate the margins of safety.
However due to the
transversely isotropic nature of the laminate, ultimate stress allowables are not readily
available. To determine the overall allowable for the laminate, the ply orientation and
the orientation ratios must be taken into consideration. These overall allowable normal
and shear stresses are estimated based the design strengths provided by Hexcel [4] and
the laminate layup, and are shown in Table 3.5. Ultimately these strengths would need
to be verified via testing of the proposed laminate layup.
Table 3.5 - [0,+45,-45,0] Laminate Stress Allowables
Ftu ksi
Fcu ksi
Fsu ksi
341.8
208.4
18.6
The maximum stresses determined from the FEA model are found to be 267.7 ksi in
tension and 303.5 ksi in compression. For the sake of comparison, since the maximum
stress was located inside the bore for the metallic yokes, the margins of safety are
calculated for the bore as well. The maximum stresses in the bore are lower and will
provide better safety margins; these maximum stresses are 66.7 ksi in tension and 111.9
26
ksi in compression. The maximum shear stress determined from the FEA model is
found to be 41.8 ksi. All margins of safety are shown in Table 3.6.
Table 3.6 - [0,+45,-45,0] Safety Margins
Maximum
Bore
Maximum
Bore
M.S.ult-tensile
M.S.ult-tensile
M.S.ult-compression
M.S.ult-compression
-0.26
+1.97
-0.60
+0.08
M.S.ult-shear
-0.74
The results of the finite element analysis predict that the composite yoke will fail in
both tension and compression at the focal bearing termination. The normal stresses in
the bore are significantly lower than those at the upper radius where the bearing
terminates. These lower stresses result in positive margins in the bore. Based on the
negative margins of safety at the focal bearing termination, there is some concern that
the [0,+45,-45,0] composite layup will not effectively support the 100,000 lb CF
loading estimated for this comparison. A highly negative shear margin of safety also
predicts failure, thus showing that the composite yoke fabricated with the given material
configuration will fail in multiple modes. To provide a complete comparison of metallic
and composite materials, two additional composite layups are analyzed.
These
additional layups will help to determine the feasibility of a composite replacement.
3.2.4
HexPly 8552 [02,+45,-45,02] Results
To further explore the possibility of replacing a titanium or aluminum yoke with a
composite yoke, a second yoke utilizing a [02,+45,-45,02] repeating ply pattern was
analyzed which increases the number of 0˚ plies in the laminate. The increased 0˚ to
±45˚ ply ratio offers a stronger yoke in the loading direction. Using the material
properties calculated via CLT and the same finite element configuration as the first
composite yoke, the impact of 0˚ plies can be evaluated.
The results of the finite element analysis offer a similar stress distribution to that of
the [0,+45,-45,0] laminate. However the normal stress magnitude is slightly greater,
while the shear stress is slightly lower. This variation in magnitude may prove to be
insignificant due to the parallel variation in the normal and shear strength of the
27
laminate. The normal stress distribution for the entire yoke is provided in Figure 3.10,
while a magnified view of the lug underside is shown in Figure 3.11. The resulting
shear stress throughout the part is shown in Figure 3.12.
Figure 3.10 - [02,+45,-45,02] Normal Stress (X)
Figure 3.11 - [02,+45,-45,02] Normal Stress (X) magnified lug area
28
Figure 3.12 - [02,+45,-45,02] Shear Stress (XY)
Once again, design allowables are estimated for the laminate composition based on
the data provided by Hexcel [4]. The effect of the ply orientation ratio shows an effect
on the ultimate strengths. These allowables are shown in Table 3.7.
Table 3.7 - [02,+45,-45,02] Laminate Stress Allowables
Ftu ksi
Fcu ksi
Fsu ksi
358.5
219.9
19.1
The maximum stresses determined from the FEA model are found to be 260.9 ksi in
tension and 302.7 ksi in compression. As with the first laminate composition, the
maximum stress in the attachment bore is also analyzed for comparison. The maximum
stresses in the bore are 68.6 ksi in tension and 116.2 ksi in compression. The maximum
shear stress determined from the FEA model is found to be 38.0 ksi. The margins of
safety were calculated and are shown in Table 3.8.
29
Table 3.8 - [02,+45,-45,02] Safety Margins
Maximum
Bore
Maximum
Bore
M.S.ult-tensile
M.S.ult-tensile
M.S.ult-compression
M.S.ult-compression
-0.20
+2.03
-0.58
+0.10
M.S.ult-shear
-0.71
The resulting margins of safety are very similar to those found for the [0,+45,-45,0]
laminate, showing only a minor improvement. Again this result predicts that the yoke
will fail in tension and compression at the bearing termination, under the given C F
loading. Also the shear margin of safety remains highly negative, predicting another
mode of failure. These results indicate that the ratio of 0˚ to ±45˚ plies does not have a
considerable effect on the strength of the component. This indication may be a factor of
the increased stiffness caused by the additional 0˚ plies. However, in order to further
validate this observation a third laminate configuration will be analyzed.
3.2.5
HexPly 8552 [03,+45,-45,03] Results
To further explore the possibility of material replacement and to further validate the
effect of the ratio of plies orientations, a third laminate composition is analyzed with 8%
more 0˚ plies. Again the increased 0˚ to ±45˚ ply ratio offers a stronger yoke in the
loading direction. Yet the effect of this increased strength on the margins of safety must
be determined. The normal stress distribution of the yoke is shown in Figure 3.13, and a
magnified view of the lug underside is provided in Figure 3.14.
To provide a
comparison of shear stress, the resulting shear stress is shown in Figure 3.15.
30
Figure 3.13 - [03,+45,-45,03] Normal Stress (X)
Figure 3.14 - [03,+45,-45,03] Normal Stress (X) magnified lug area
31
Figure 3.15 - [03,+45,-45,03] Shear Stress (XY)
As stated previously, the design strengths of the laminate are estimated based on the
layup and material data from Hexcel [4]. These normal and shear strength estimates are
provided in Table 3.9.
Table 3.9 - [03,+45,-45,03] Laminate Stress Allowables
Ftu ksi
Fcu ksi
Fsu ksi
366.9
225.7
19.2
The maximum stresses determined from the FEA model are found to be 256.9 ksi in
tension and 304.4 ksi in compression. Using the maximum stresses in the bore of 69.2
ksi in tension and 117.1 ksi in compression. While the yoke experiences a maximum
shear stress of 36.6 ksi. Based on these maximum stresses and the ultimate strengths
calculated previously, the margins of safety are calculated and shown in Table 3.10.
Table 3.10 - [03,+45,-45,03] Safety Margins
Maximum
Bore
Maximum
Bore
M.S.ult-tensile
M.S.ult-tensile
M.S.ult-compression
M.S.ult-compression
-0.17
+2.07
-0.57
+0.12
32
M.S.ult-shear
-0.70
Once again the resulting margin of safety remains close to that of the previous
laminate compositions. The negative margins of safety for both resulting normal and
shear stresses, lead to some concern.
These results indicate that the effect of the
orientation ratio is insignificant for a yoke of this specific design.
While the margins of safety for this design lead to the conclusion that composite
materials may not be a viable replacement for metallic materials, there are several
variables that may make composites more feasible. The weight advantage of composite
materials is apparent considering that each of the composite yokes analyzed in this
comparison only weigh 10.3 lbs, which is almost an 8 lb and 19 lb reduction when
compared to Aluminum and titanium respectively.
This reduction may not seem
significant, but there are typically between four and seven yokes installed on a
helicopter, which results in a noteworthy reduction in gross weight.
3.2.6
Modified Yoke Geometry
In light of the evident weight advantage and the many variables that can be
modified, composite materials remain particularly interesting for this component. When
reviewing the results of the composite analysis, the overall stress throughout the yoke
does not appear to be excessive and the failure modes are localized in small areas.
Therefore modifying the geometry of the yoke may reduce the stress in these areas to
provide a viable solution.
The highest stress concentrations were located at the cuff attachment bore and at the
focal bearing termination. Therefore a few changes were made to the yoke design to
reduce the stresses in these areas. The modifications include straightening the bow tie
shape of the yoke by removing the scallop and extending the lug. The original yoke
shape along overlaid with the modified design is shown in Figure 3.16. This change in
geometry does render some impacts to the interfacing components, which require
modification as well. Since this is a very preliminary design, intended as a simple trade
study, the changes to interfacing components will not be discussed.
33
Scallop removed to
Lug extended
widen yoke profile
Figure 3.16 - Modified yoke geometry
Based on the results of the three layup configurations, the [02,+45,-45,02] repeating
layup appears to have the best overall margins, although the variation is minimal.
Therefore the modified yoke will be analyzed using the material properties for the
second laminate configuration. The new yoke geometry and existing material properties
can be input into the existing ANSYS model for analysis.
The revised geometry effectively reduces the stress throughout the entire yoke,
resulting in a 50% reduction in both tensile and compressive maximum stress. The
highest stress concentration remains at the focal bearing termination, while the
secondary maximum is still in the bore. The widening of the yoke body does appear to
reduce the normal stress enough to prevent failure and the elongation of the lug appears
to effectively resolve the failure mode at the bore. The normal stresses in the X direction
are shown in Figure 3.17, with a magnified view of the lug in Figure 3.18. The overall
Shear stress actually shows significant increase, unlike the normal stresses.
maximum concentrations in shear remain at the bore and are shown in Figure 3.19.
34
The
Figure 3.17 - Modified Geometry Normal Stress (X)
Figure 3.18 - Modified Geometry Normal Stress (X) magnified lug area
35
Figure 3.19 - Modified Geometry Shear Stress (XY)
To effectively evaluate the effect of the modifications it is necessary to calculate the
margins of safety for the modified yoke. The maximum normal stresses are 129.6 ksi in
tensile and 148.4 ksi in compression.
For a comparison of the effect of the lug
modifications, margins for the bore are also calculated. Based on the FEA, the bore
experiences maximum normal stresses of 119.0 ksi in tension and 132.7 ksi in
compression. Modification of the yoke geometry actually increases the shear stress,
specifically at the bore. The maximum shear stress is found to be 73.2 ksi. The
calculated margins are shown in Table 3.11.
Table 3.11 – Modified Geometry Safety Margins
Maximum
Bore
Maximum
Bore
M.S.ult-tensile
M.S.ult-tensile
M.S.ult-compression
M.S.ult-compression
+0.60
+0.74
-0.14
-0.04
M.S.ult-shear
-0.85
The modifications performed appear to have a considerable effect on the yokes
reaction to the CF loading. The maximum tensile margins and the bore margins have
become positive. While the compressive margins remain negative, there is a significant
reduction. Therefore failure is still evident in compression, but the margins are very
close to showing good. Unlike the normal stress margins the overall shear margin shows
36
a significant negative increase. The geometry modification has resulted in a shear stress
that is nearly double that of the original geometry, and the resulting margin is highly
negative.
Considering the simple modifications performed on the geometry, and the
significant change in results, it appears that some manipulation of the design variables
may lead to an effective composite solution. Other layup or even assembly changes,
such as the installation of a bushing, could be made to result in a sufficient design.
However the most significant advantage of the composite materials, weight, is
drastically reduced as the result composite component weighs 16.4 lbs, which is only a
1.6 lb savings when compared to the aluminum component. Further shape optimization
and analysis would have to be performed to validate a replacement component.
3.2.7
Finite Element Analysis Complete Results
The use of Finite Element Analysis provides a functional comparison of the
different materials used for the yoke. In general the stress concentrations remained in
same location on the component for each case analyzed, thus allowing an evaluation of
the effect of materials on the functionality of the yoke. Based on the results, the metallic
yokes appear to offer a simple solution that will withstand the CF loading applied in
operation. While composite yokes maintaining the exact geometry as the metallic yokes
result in failure at the focal bearing termination and the attachment bore, some slight
modifications offered a significant reduction in stress and resolved the tensile failure
mode. The safety margins for each case is provided in Table 3.12, since the maximum
stress in the metallic yokes occurred at the bore, there is only one margin location for
these cases.
37
Table 3.12 - Safety Margins for all cases
Margin
6AL-
7175-T74
Layup 1 Layup 2
Layup 3 Modified
+0.51
+0.12
-0.26
-0.20
-0.17
+0.60
M.S.yld-tensile +1.09
+0.46
n/a
n/a
n/a
n/a
M.S.ult-tensile
4V
M.S.ult-comp
n/a
n/a
-0.60
-0.58
-0.57
-0.14
M.S.yld-comp
+3.07
+1.23
n/a
n/a
n/a
n/a
M.S.ult-shear
+1.35
+0.38
-0.74
-0.71
-0.70
-0.85
n/a
n/a
+1.97
+2.03
+2.07
+0.74
n/a
n/a
+0.08
+0.10
+0.12
-0.04
Bore
M.S.ult-tensile
Bore
M.S.ult-comp
3.3 Flaw Tolerance
The purpose of this evaluation is to determine the viability of replacing a metallic
structural component such as the yoke with a similar, if not identical, component
comprised of composite materials. While the primary concern for this preliminary
comparison is the ultimate strength of each component as designed, it is also of great
interest to consider the reaction to flaws. The susceptibility to flaws for each material is
evaluated using modified models with flaws incorporated and flawed material properties
for the composite component. These models are input in the same finite element model
as used for the unflawed component, and the stress increase in the flawed areas can be
calculated by comparing the before and after cases.
In light of the results of the analysis of the unflawed components showing that the
composite yoke ply orientation ratio had an insignificant effect on the safety margins,
only the [02,+45,-45,02] laminate will be evaluated for flaw tolerance. The composite
laminate will then be compared to the tolerance of the titanium and aluminum yokes.
38
3.3.1
Titanium 6Al-4V Flaw Tolerance
Based on the flaw geometry and isotropic material properties of Titanium 6Al-4V,
the induced normal stresses and shear stress in the lug area were analyzed. As expected
the flaws created a stress riser in the lug area, however based on the results the increased
stress does not appear to create a failure mode. The resulting normal stress distribution
is shown in Figure 3.20. The shear stress distribution is shown in Figure 3.21. The
stresses at the flaws are flawed with blue labels, which include the magnitude of the
stress.
Figure 3.20 - 6AL-4V Flawed Normal Stress (X)
39
Figure 3.21 - 6AL-4V Flawed Shear Stress (XY)
The stresses in the flawed areas can then be compared to the stresses in the same
area calculated in section 3.2.1. The flawed geometry results in normal stresses of 10.6
ksi at the inboard flaw and -1.9 ksi at the outboard flaw, while the unflawed geometry
had maximum normal stresses of 8.8 ksi tensile at the inboard flaw area and 1.8 ksi
compressive at the outboard flaw area. An increased shear stress in the flawed yoke was
found to be 1.3 ksi at the inboard flaw and -2.7 at the outboard flaw. The initial
evaluation of shear stress in the titanium yoke resulted in stresses of 1.1 ksi at the
inboard flaw and -0.4 ksi at the outboard flaw. To determine the effect of the flaw on
the local stress, the percent difference calculated between the flawed and non-flawed
results. The normal and shear stress increases for both flaws are calculated and shown in
Table 3.13.
Table 3.13 - Ti 6Al-4V Flawed Stress Increase
Inboard
Outboard
Inboard
Outboard
Normal
Normal
Shear
Shear
20.5%
5.5%
18.2%
575%
While the inboard flaw shows a significant increase in normal stress, greater than
20%, the increase at the outboard flaw is minimal and can be considered negligible for a
preliminary analysis such as this. The inboard flaw results in a similar shear stress
40
increase to that of the normal stress, nearly 20%. Yet the outboard flaw experiences
much higher shear stress with the flaw, almost six times the magnitude. However the
original shear stress in this location was minimal and the resulting stress riser remains
well below the ultimate shear strength. Overall the 6Al-4V material maintains structural
integrity with a flaw of this size, but larger flaws may become an issue.
3.3.2
Aluminum 7175-T74 Flaw Tolerance
Based on the isotropic material properties of aluminum the results of the flaw
tolerance can be expected to be fairly similar to those of the titanium case. However the
lower strength and calculated safety margins, there would be some concern that a flaw of
this size may cause failure. Therefore to provide a comprehensive evaluation of all
materials, the 7175-T74 material was evaluated in the same fashion as the titanium yoke.
The normal stress distribution in the X direction is provided in Figure 3.22. In addition
the shear stress riser is also evaluated. The results of the shear case are shown in Figure
3.23, Again the flaws are located at the blue labels.
Figure 3.22 - 7175-T74 Flawed Normal Stress (X)
41
Figure 3.23 - 7175-T74 Flawed Shear Stress (XY)
The flawed aluminum geometry offers a maximum normal stress of 13.0 ksi at the
inboard flaw and -1.6 ksi at the outboard flaw. From the results of section 3.2.2, where
the unflawed yoke was analyzed, the original normal stress magnitudes are 10.0 ksi
tensile at the inboard location and 1.5 ksi compressive at the outboard flaw location.
Shear stress located in the flaws was found to be 1.3 ksi in the inboard flaw and -2.8 in
the outboard flaw. The original aluminum yielded shears stresses of 1.0 ksi at the
inboard location and -0.4 ksi at the outboard location. Both stress risers are evaluated
and percent increase in normal and shear stress is shown in Table 3.14.
Table 3.14 – Al 7175-T74 Flawed Stress Increase
Inboard
Outboard
Inboard
Outboard
Normal
Normal
Shear
Shear
30%
6.6%
30.0%
600%
As expected the stress increase is of similar magnitude to the titanium yoke, yet
slightly greater.
Again the outboard flaw results in an insignificant normal stress
increase for this comparison. While the normal stress values remain well below the
ultimate strengths in these locations, based on the low tensile margin of safety there may
42
be some concern in other areas if stresses were increased by 30%. However this could
likely be resolved via shape optimization, if aluminum were deemed the best solution.
Again, the inboard flaw shows a nearly identical increase in shear stress as it does for
normal stress. The minimal shear stress at the outboard location is once again greatly
increased by the flaw, roughly six fold. Overall the stresses at the flaws do not exceed
the aluminum allowables, however if a significantly larger flaw occurred or if the flaw
were located in a higher stressed region there may be some concern of failure.
3.3.3
HexPly 8552 Flaw Tolerance
Based on the insignificant variation of safety margins with respect to ply orientation
ratio, it was deemed unnecessary to evaluate all three composite layups. As with the
modified geometry analysis the [02,+45,-45,02] layup configuration was chosen for the
flaw tolerance evaluation. The assessment of effect of damage to a component is
slightly more difficult for composite materials. Due to the nature of composites the loss
of structural integrity for the affected reinforcement fibers must be accounted for. The
reduction in structural fibers is simulated by calculating the material properties without
the plies that may be damaged. In the case of this exercise, since the damage is .018”
deep and each ply is .006” thick, the top three plies were removed from the CLT
calculations.
The global material properties were again derived using the same
MATLAB program as for section 3.1, these values are provided in Table 3.15.
Table 3.15 - Flawed [02,+45,-45,02] Material Properties
Ex
Ey
Ez
Gxy
Gxz
Gyx
(msi)
(msi)
(msi)
(msi)
(msi)
(msi)
17.75
4.05
4.05
3.60
3.60
1.83
νxy
νxz
νyz
0.46
0.46
0.10
As shown by the results, damaging or losing a few fibers does not produce a
significant change in the material properties. This observation itself offers a testament
the flaw tolerant nature of composite materials.
However, to fully compare the
composite yoke to the metallic results, these properties were then input into the
composite yoke finite element model along with the flawed geometry for evaluation.
43
The resulting normal stress is shown in Figure 3.24.
The shear stresses are also
evaluated and the results are shown in Figure 3.25.
Figure 3.24 - [02,+45,-45,02] Flawed Normal Stress (X)
Figure 3.25 - [02,+45,-45,02] Flawed Shear Stress (XY)
44
While the overall results of the original finite element model predicted failure of the
yoke, this failure did not occur in the flawed locations.
Therefore an effective
comparison of results is possible. The resulting normal stresses were 49.3 ksi at the
inboard flaw and -22.2 at the outboard flaw, compared to the unflawed yoke which
experienced 31.2 ksi at the inboard flaw location and -15.0 ksi at the outboard location.
The flawed geometry exhibited shear stresses of 10.7 ksi at the inboard flaw and 7.7 ksi
at the outboard flaw. This compares to the initial shear stress analysis which resulted in
7.7 ksi at the inboard location and 4.8 ksi at the outboard location. The calculated stress
increases are shown in Table 3.16.
Table 3.16 – HexPly 8552 Flawed Stress Increase
Inboard
Outboard
Inboard
Outboard
Normal
Normal
Shear
Shear
58.0%
48.0%
39.0%
60.4%
The flawed results show some significantly greater increases than those of both
metallic yokes. The stress risers do not appear to create additional failure modes,
however the magnitude of the increases would lead to some concerns in other locations
with higher initial stress. The normal stress risers are significantly higher than those for
the metallic yokes, especially the outboard flaw which is roughly eight fold higher. The
increase in shear stress at the inboard location is similar in magnitude to the results from
the aluminum yoke. Yet the outboard flaw does not produce the huge shear stress riser as
seen in the titanium and aluminum material. In contrary to the other cases, the yoke is
predicted to fail in shear without the flaws, therefore the lug is a known failure mode that
would require additional attention and likely some shape optimization.
In reality the assumption that the three plies would become ineffective is very
conservative. Impact experienced by the composite material will not necessarily cause
the fibers to split, as the plies underneath will help to absorb the force. Also the loss of
integrity would not occur throughout the part as modeled, instead it would be localized
and the remaining fibers as well as the epoxy resin would remain intact to share the load.
In addition the composite part is likely to have a smoother resulting flaw than the
metallic yokes, which would reduce the possible stress riser due to geometry itself.
45
While the results of this comparison would imply that the titanium and aluminum
materials are more flaw tolerant than the composite material, this is a very conservative
approach and does not fully represent how the composite material would truly react.
This is shown in part by the initial calculation of material properties without the top
three plies, which had a minute effect.
3.3.4
Flaw Tolerance Comparison
While this analytical evaluation of the flaw tolerance for each material case provides
some interesting information and insight, the results are theoretical and do not provide a
clear answer. Ultimately flaw tolerance must be evaluated via physical testing to prove
that the component is capable of withstanding some damage. Analytical review of the
flaw tolerance is rarely performed during preliminary design or in the case of a trade
study. Regardless, the results do provide some insight for this comparison and the
results are shown in Table 3.17.
Table 3.17 - Flawed Stress Increase for all Materials
Material
Inboard Normal
Outboard Normal
Inboard Shear
Outboard Shear
6Al-4V
20.5%
5.5%
18.2%
575%
7175-T74
30.0%
6.6%
30.0%
600%
Layup 2
58.0%
48.0%
39.0%
60.4%
46
4. Conclusions
In the helicopter industry, one of the greatest challenges is optimizing the design by
maximizing strength while simultaneously reducing weight. Particularly in the rotor
head design, structural components are typically manufactured from high strength
metallic alloys. This is due to the high loading that these components experience and the
catastrophic effect of component failure.
However, in recent years the emerging
capabilities of composite materials have triggered the notion that there may be a lighter
alternative to the conventional high strength alloys. To evaluate the feasibility of this
alternative, a comparison between a composite blade retention yoke and a metallic yoke
has been performed.
Ultimately the results of the finite element analysis reveal that composite materials
cannot substitute a metallic component. Titanium 6Al-4V clearly provided the best
strength characteristics and flaw tolerance as analyzed; however, titanium is also the
heaviest material analyzed. The aluminum 7175-T74 material offered a lighter weight
solution than titanium; however, the factors of safety were significantly smaller. Both
metallic materials were able to withstand the centrifugal loading without predicting
failure. The HexPly 8552, on the other hand, while offering exceptional directional
stiffness and light weight, resulted in failure for all three compositions. In general, the
flaw tolerance for all three materials was similar in nature; however, the composite
analysis was very conservative due to the layered nature of a laminate. However,
composite materials offer several variables which can alter the strength of the
component.
In light of the analytical results and opportunity for variation in composite materials,
a substitution remains possible.
Although composites may not be viable direct
substitution for titanium or aluminum using identical geometry, with some modifications
there is potential to design a sufficient yoke. These modifications would likely include a
shape optimization, as shown with the minor geometry changes in section 3.2.6,
significant stress reductions can be achieved by iterating the yoke design. Also, with the
vast array of available composite materials, the laminate could be tailored to reduce
stress.
However these iterations are beyond the purposes of this preliminary
comparison.
47
Conclusively, composite structural components have been proven to be feasible.
Yet detailed analysis and extensive testing of an optimized composite blade retention
yoke is required to validate these results. With increasing performance requirements and
depletion of natural resources, composites are a technology that can be expected to
emerge in the future.
48
References
[1] Sirisalee, P., M. F. Ashby, G. T. Parks, and P. John Clarkson. "Multi-Criteria
Material Selection of Monolithic and Multi-Materials in Engineering Design."
Advanced Engineering Materials 8.1-2 (2006): 48-56. Print.
[2] Metallic Materials Properties Development and Standardization (MMPDS-05).. U.S.
Federal Aviation Administration.
[3] Hexcel Corporation. "Hexcel.com - Prepreg Data Sheets." Hexcel.com - Carbon
Fiber and Composites for Aerospace, Wind Energy and Industrial. 2011. Web.
<http://www.hexcel.com/Resources/prepreg-data-sheets>.
[4] Hexcel Corporation. HexPly® 8552 Epoxy Matrix. Dec. 2007. Raw data. Stamford,
CT.
[5] Hyer, M. W., and S. R. White. Stress Analysis of Fiber-reinforced Composite
Materials. Lancaster, PA: DEStech Publications, 2009. Print.
[6] United States of America. NASA. Goddard Space Flight Center. Structural Stress
Analysis (PD-AP-1318). Apr. 1996. Web.
[7] ADAMS, D. O. Flaw Tolerant Safe-Life Method. Rep. no. ADA389234. Stratford,
CT: Sikorsky Aircraft, 1999. Print.
49
Appendix A – Classical Lamination Theory Code
%Kevin Laitenberger
%Masters Project
%Classic Lamination Theory
clear
clc
r = 150;
r2 = r/2;
%Material Properties
E1 = 23800000000;
E2 = 1700000000;
v12 = .32;
G12 = 2200000000;
%Reduced Compliance Matrix
S=zeros(3,3);
S(1,1) = 1/E1;
S(2,1) = -v12/E1;
S(1,2) = S(2,1);
S(2,2) = 1/E2;
S(3,3) = 1/G12;
S;
C = inv(S);
%Reduced Stiffness Matrix
Q=zeros(3,3);
Q(1,1) = S(2,2)/(S(1,1)*S(2,2)-S(1,2)^2);
50
Q(1,2) = -S(1,2)/(S(1,1)*S(2,2)-S(1,2)^2);
Q(2,2) = S(1,1)/(S(1,1)*S(2,2)-S(1,2)^2);
Q(2,1) = Q(1,2);
Q(3,3) = 1/S(3,3);
Q;
Sbar=zeros(3,3,3);
Qbar=zeros(3,3,3);
%Layup
layup1
=
[0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,-
45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,0,45,45,0,0,45,-45,0,0,45,-45,0,0,45,-45,0,45,-45,-45,45,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0,-45,45,0,0];
layup2
=
[0,0,0,45,-45,0,0,0,0,45,-45,0,0,0,0,45,-45,0,0,0,0,45,-45,0,0,0,0,45,-
45,0,0,0,0,45,-45,0,0,0,0,45,-45,0,0,0,0,45,-45,0,0,0,0,45,-45,0,0,0,0,45,-45,0,0,0,0,45,45,0,0,0,0,45,-45,0,0,45,-45,-45,45,0,0,-45,45,0,0,0,0,-45,45,0,0,0,0,-45,45,0,0,0,0,45,45,0,0,0,0,-45,45,0,0,0,0,-45,45,0,0,0,0,-45,45,0,0,0,0,-45,45,0,0,0,0,-45,45,0,0,0,0,45,45,0,0,0,0,-45,45,0,0,0,0,-45,45,0,0,0];
layup3
=
[0,0,0,0,45,-45,0,0,0,0,0,0,45,-45,0,0,0,0,0,0,45,-45,0,0,0,0,0,0,45,-
45,0,0,0,0,0,0,45,-45,0,0,0,0,0,0,45,-45,0,0,0,0,0,0,45,-45,0,0,0,0,0,0,45,45,0,0,0,0,0,0,45,-45,0,0,0,45,-45,-45,45,0,0,0,-45,45,0,0,0,0,0,0,-45,45,0,0,0,0,0,0,45,45,0,0,0,0,0,0,-45,45,0,0,0,0,0,0,-45,45,0,0,0,0,0,0,-45,45,0,0,0,0,0,0,45,45,0,0,0,0,0,0,-45,45,0,0,0,0,0,0,-45,45,0,0,0,0];
layup = input('Which layup for analysis? (1,2,3) ')
if layup == 1
layup = layup1;
elseif layup == 2
51
layup = layup2;
elseif layup == 3
layup = layup3;
end
%Transformation
for k = 1:r
theta(k)= layup(k);
t(k) = .006;
m(k) = cosd(theta(k));
n(k) = sind(theta(k));
Sbar(1,1,k) = S(1,1)*(m(k)^4)+(2*S(1,2)+S(3,3))*(n(k)^2)*(m(k)^2)+S(2,2)*(n(k)^4);
Sbar(1,2,k) = (S(1,1)+S(2,2)-S(3,3))*(n(k)^2)*(m(k)^2)+S(1,2)*((n(k)^4)+(m(k)^4));
Sbar(2,1,k) = Sbar(1,2,k);
Sbar(1,3,k)
=
(2*S(1,1)-2*S(1,2)-S(3,3))*n(k)*(m(k)^3)-(2*S(2,2)-2*S(1,2)-
S(3,3))*(n(k)^3)*m(k);
Sbar(3,1,k) = Sbar(1,3,k);
Sbar(2,2,k) = S(1,1)*(n(k)^4)+(2*S(1,2)+S(3,3))*(n(k)^2)*(m(k)^2)+S(2,2)*(m(k)^4);
Sbar(2,3,k)
=
(2*S(1,1)-2*S(1,2)-S(3,3))*(n(k)^3)*m(k)-(2*S(2,2)-2*S(1,2)-
S(3,3))*n(k)*(m(k)^3);
Sbar(3,2,k) = Sbar(2,3,k);
Sbar(3,3,k)
=
2*(2*S(1,1)+2*S(2,2)-4*S(1,2)-
S(3,3))*(n(k)^2)*(m(k)^2)+S(3,3)*((n(k)^4)+(m(k)^4));
Qbar(1,1,k)
=
Q(1,1)*(m(k)^4)+(2*Q(1,2)+4*Q(3,3))*(n(k)^2)*(m(k)^2)+Q(2,2)*(n(k)^4);
Qbar(1,2,k)
=
(Q(1,1)+Q(2,2)-
4*Q(3,3))*(n(k)^2)*(m(k)^2)+Q(1,2)*((n(k)^4)+(m(k)^4));
Qbar(2,1,k) = Qbar(1,2,k);
52
Qbar(1,3,k)
=
(Q(1,1)-Q(1,2)-2*Q(3,3))*n(k)*(m(k)^3)+(Q(1,2)-
Q(2,2)+2*Q(3,3))*(n(k)^3)*m(k);
Qbar(3,1,k) = Qbar(1,3,k);
Qbar(2,2,k)
=
Q(1,1)*(n(k)^4)+(2*Q(1,2)+4*Q(3,3))*(n(k)^2)*(m(k)^2)+Q(2,2)*(m(k)^4);
Qbar(2,3,k)
=
(Q(1,1)-Q(1,2)-2*Q(3,3))*(n(k)^3)*m(k)+(Q(1,2)-
Q(2,2)+2*Q(3,3))*n(k)*(m(k)^3);
Qbar(3,2,k) = Qbar(2,3,k);
Qbar(3,3,k)
=
(Q(1,1)+Q(2,2)-2*Q(1,2)-
2*Q(3,3))*(n(k)^2)*(m(k)^2)+Q(3,3)*((n(k)^4)+(m(k)^4));
end
%Full Laminate Stiffness
T = sum(t,2)
T2 = T/2;
z(1) = T2;
for j = 2:r+1
z(j) = (T2)-t(j-1);
T2 = z(j);
zQbar(:,:,j-1) = Qbar(:,:,j-1)*t(j-1);
z2Qbar(:,:,j-1) = Qbar(:,:,j-1)*(z(j)^2-z(j-1)^2);
z3Qbar(:,:,j-1) = Qbar(:,:,j-1)*(z(j-1)^3-z(j)^3);
end
Qbar;
%ABD Matrix
A = sum(zQbar,3)
53
a = inv(A)
B = sum(z2Qbar,3)/2
b = inv(B)
D = sum(z3Qbar,3)/3
d = inv(D)
%Laminate Material Properties
Ex = 1/(a(1,1)*T)
Ey = 1/(a(2,2)*T)
Ez = Ey;
Gxy = 1/(a(3,3)*T)
Gxz = Gxy
vxy = -a(1,2)/a(1,1)
vyz = -a(1,2)/a(2,2)
vxz = vxy
Gyz = Ey/(2*(1+vyz))
%coordinate transformation
theta_u = 20.7;
theta_l = 65.7;
Exglobal_u = Ex*cosd(theta_u)+Ez*sind(theta_u)
Ezglobal_u = Ez*cosd(theta_u)+Ex*sind(theta_u)
Exglobal_l = Ex*cosd(theta_l)+Ez*sind(theta_l)
Ezglobal_l = Ez*cosd(theta_l)+Ex*sind(theta_l)
-*
54