The Design and Analysis of a Double Transtibial Composite Prosthesis and the
Effect of Lateral Movement Loads
by
Max A. Willer
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-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2014
i
CONTENTS
The Design and Analysis of a Double Transtibial Composite Prosthesis
and the Effect of Lateral Movement Loads
i
LIST OF TABLES
iii
LIST OF FIGURES
iv
TERMINOLOGY / LIST OF SYMBOLS / ACRONYMS
v
ACKNOWLEDGEMENTS
vii
ABSTRACT
viii
1. Introduction
1
1.1. Background
1
1.2. Challenges of this Study
3
2. Theory and Methodology
7
2.1. Reverse Engineering the Flex Foot Design
7
2.2. Lateral Forces, Friction, and Angle of Attack
11
2.3. Analytic Method Summary
13
3. Results and Discussion
17
3.1. Model Setup
17
3.2. Lateral Movement
21
4. Remarks and Conclusion
25
5. References
30
6. Appendices
32
6.1. Graphical Data
ii
LIST OF TABLES
Table 2-1: Properties of the Flex Foot
8
Table 2-2: Consistent Units for Analysis
8
Table 2-3: Coefficients of Friction Equations
10
Table 3-1: 0/90 Ply Orientation, Stress and Stiffness vs Load
18
Table 3-2: 0/-45/90/45 Stiffness at 1500 N
20
Table 3-3: Summary of Baseline Stresses and TSAIW
24
Table 4-1: Deflection and Modulus Response of Selected Materials
26
Table 4-2: Model Dimension Lengths
27
Table 4-3: Weight of Prosthetic Using Selected Materials
27
iii
LIST OF FIGURES
Figure 1-1: History of Leg Prostheses
1
Figure 1-2: Official Rendering of the Flex Foot Prosthetic
2
Figure 1-3: Ankle Power Response Curve
3
Figure 1-4: Diagram of Tennis Sole
4
Figure 1-5: Definitions of ankle eversion and inversion
5
Figure 1-6: Connection bolts of Flex Foot bolts
5
Figure 1-7: Ankle eversion in tennis lateral motion
6
Figure 2-1: Geometric Analysis of Flex Foot Prosthetic
7
Figure 2-2: Dimensions of Baseline Prosthetic
9
Figure 2-3: Comparison of Baseline Model to Original
9
Figure 2-4: Plot of Friction Equations
11
Figure 2-5: Maximum Force and Angle of Attack
12
Figure 2-6: Loads and Constraints
13
Figure 2-7: Assembly with Fixed Reference Point Shown
14
Figure 2-8: Assembly with Contact Interaction Method Shown
14
Figure 2-9: Tangential Behavior
15
Figure 2-10: Profile of Assembly
15
Figure 3-1: Mesh Experiment
17
Figure 3-2: Visualization of Linear Stiffness vs Load
19
Figure 3-3: Various Ply Stiffness at 1500 N
19
Figure 3-4: 0/-45/90/45 Stiffness at 1500 N, Various Plies
20
Figure 3-5: [0/90] Vertical Stress and TSAIW
21
Figure 3-6: [0/90] 45 Degree Stress and TSAIW
22
Figure 3-7: [0/45/90/-45] Vertical Stress and TSAIW
23
Figure 3-8: [0/45/90/-45] 45 Degree Stress and TSAIW
23
Figure 4-1: Modulus Response of Selected Materials
26
Figure 4-2: Dimensions of Baseline Prosthetic
27
Figure 4-3: Weight of Prosthetic Using Selected Materials
28
iv
TERMINOLOGY / LIST OF SYMBOLS / ACRONYMS
Transtibial – occurring across or involving the tibia
Abduction/Adduction – Ankle rotation around the shin axis
Plantar Flexion / Dorsiflexion – Ankle rotation about the ankle joint axis
Inversion / Eversion = Ankle rotation about the foot axis
FEA – Finite Element Analysis
FBD – Free Body Diagram
2D – 2 Dimensions
E – Modulus of Elasticity (Msi)
G – Modulus of Rigidity (Msi)
ν – Poisson’s Ratio
ρ – Density (lbf/in3)
tp – Ply Thickness (in)
YS – Yield Strength (ksi)
UTS – Ultimate Tensile Strength (ksi)
σ1t – Tensile strength in the 1 (longitudinal) direction (ksi)
σ1c – Compressive strength in the 1 (longitudinal) direction (ksi)
σ2t – Tensile strength in the 2 (transverse) direction (ksi)
σ2c – Tensile strength in the 2 (transverse) direction (ksi)
τ12f – Shear Strength (ksi)
[Orientation number of plies]S – Laminate Layup which is characterized by ply orientation,
number of plies and symmetry about the mid-plane (S, if applicable).
v
Abaqus – Computer Software used to perform modeling and FEA
Isotropic – Same properties in all directions
Orthotropic – Different properties in different directions
TSAIW – An abbreviation for Tsai-Wu Abaqus uses
vi
ACKNOWLEDGMENTS
I would first like to thank my Project Adviser, Professor Ernesto, for all of the help he
has given me and for his patience. I would like to thank Professor Hufner for his help
with composites analysis. I would also like to thank all those who are working in the
field of prosthetics. I would also like to thank the faculty of Rensselaer Polytechnic
Institute at Groton for their support throughout the Masters of Engineering program.
vii
ABSTRACT
The purpose of this project is to evaluate the performance of a prosthetic device similar
to the Flex Foot Cheetah design used by Olympian sprinters. This transtibial prosthetic
has been evaluated before to determine if it offers more mechanical advantage in a
forward sprint to a runner than does an anatomical leg, ankle and foot. This paper
determines the properties of such a prosthetic and then analyzes what changes, if any,
would benefit the prosthetic wearer in sports that require lateral movement in addition to
forward sprinting.
First a model was created by gathering all non-proprietary information about the Flex
Foot Cheetah model. Dimensions were ascertained by image analysis and from given
information by previous papers. The dimensioned model was then analyzed with two ply
orientation types to determine orientation, number, and thickness of composite plies to
have a similar baseline mechanical performance as the Flex Foot.
These models were then analyzed for stresses in the lateral direction at the maximum
amount of force seen by a tennis player changing direction. Modifications are proposed
to the Flex Foot design to accommodate these forces, such as resizing of the equivalent
ankle area and widening the contact foot for additional stability and traction.
viii
1. Introduction
Background:
The use of prosthetic limbs as a medical solution for lost limb functionality has been in
practice for most of human history, from wooden crutches to wooden legs all the way up
to the articulating hands of the present and future. Those who use prosthetic limbs have
long been perceived as at a disadvantage in physical fitness and sport to those with full
use of their limbs. This perception however is on the precipice of change as advanced
material technology and fabrication techniques are applied towards the design and manufacturing of prosthetic devices. In fact, some prosthetic limbs have become so advanced,
that they have caused controversy in the sporting world for giving an unfair advantage
over natural human limbs.
Figure 1-1: History of Leg Prostheses, Reference 1
Oscar Pistorius is a South African sprint runner and a double transtibial amputee, which
means the prosthetics he wears replace legs and feet that have been amputated below the
knee. In 2007 he gained attention by participating in able-bodied international sprinting
competitions which resulted in the International Association of Athletics Federations
(IAAF) banning "any technical device that incorporates springs, wheels or any other element that provides a user with an advantage over another athlete not using such a device."
(Reference 2). This led to several studies as to whether there was indeed an advantage in
1
using the Flex Foot prostheses that Oscar Pistorius has become famous for using. One of
the most famous of theses is the Bruggemann study, a series of tests published in 2008
that reported that Pistorius uses twenty-five percent less energy expenditure than ablebodied runners with use of their lower legs, ankles, and feet. (Reference 3). A series of
other studies followed showing that Pistorius used less metabolic energy and foot to
ground force than able-bodied runners, further pointing toward the mechanical advantage
of the Flex Foot prosthesis over conventional running (Reference 4). Regardless of the
controversy in the sport, the fact is that prosthetic devices have become so advanced as to
give their wearers equal or higher functionality than those with natural limbs.
Given that prosthetics are becoming so advanced that they give their wearers an ad-
Figure 1-2: Official Rendering of the Flex Foot Prosthetic, Reference 5
vantage over those with human limbs, it is natural to desire to expand the practical uses
to other uses, including other sports. The purpose of the Flex Foot design, as well as
many other leg prosthetic designs historically, is to emulate the spring action of the
lower leg as a runner is moving forward. While the design works exceedingly well for
that one function, it lacks the specialization to be useful for sports that include movements outside of the forward direction. The Flex Foot’s narrow design gives an effective
2
running spring stiffness and an aerodynamic profile for speed, but in sports where the
athlete needs traction and ankle mobility for lateral movements, the Flex Foot design
may come up short. Thus it is this paper’s purpose to analyze the Flex Foot design, and
modify it to properly suit leg movements in sports that include lateral motion.
Challenges of this Study:
The Flex Foot Cheetah by description is a laid up composite laminate. Most challengingly it is a proprietary design, which means to analyze it will require reverse
engineering it. This will mean estimating the dimensions, materials, and composite layup
structure to arrive at appropriate baseline model from which to design a lateral force
transtibial prosthesis. To accomplish this, this paper uses the data collected from the
Bruggemann study, which illustrates the static and dynamic responses of the Flex Foot
Cheetah. Getting within a reasonable degree of accuracy to this model will help ensure a
functional lateral movement prosthesis.
Figure 1-3: Ankle Power Response Curve, Reference 3
The next challenge of this study will be to accurately model the ground reaction forces
seen in lateral movement of the human body. Side to side movement sees greater forces
in the human leg than running forward, since the foot and leg has to support the weight
of the body as well as stop its horizontal momentum in an appropriate amount of time in
3
the direction of gravity and in the opposite direction of the body’s momentum. Depending on the playing surface, different sports solve withstanding lateral moving force with
various shoe surfaces. Field sports like soccer and American football give their athletes
traction with cleats while ice sports give their athletes steel blades. However, court
sports like basketball, racquetball, and tennis require their athletes to simply use the traction of a flat sole for movement. Luckily, studies have been done on shoe materials and
patterns best suited for the lateral movements of tennis. These studies have detailed data
on the force required of the athletes interface with the ground as well as materials used.
Thus the next challenge will be to use these studies to determine the contact surface area
of the new lateral movement prosthetic as well as the lateral force against the ground it
can resist.
After modelling the ground forces, the dynamic response of the baseline Flex Foot de-
Figure 1-4: Diagram of Tennis Sole, Refer-
ence 6
sign will be studied to examine the critical points to redesign to allow the maximum
effectiveness for lateral movement. The design will be iterated keeping in mind the following main requirement:

A large majority of the weight of the prosthetic design is in the connecting assembly to the remaining lower leg. Thus the weight of the mechanical spring part
of the prosthetic should be kept at a minimum.
4
It is for this requirement that the Flex Foot original design utilizes a composite layup
with strengthening carbon fiber laminates, and this study will use this line of reasoning
during material selection.
With the main requirement in mind, the first iterations will likely expand the under-foot
contact area in order to meet the amount of friction force required by the lateral movement of tennis or other court sports. Just like the standard tennis shoe is wider at the base
than the standard running shoe to accommodate lateral forces, so will the new prosthetic
device need to be wider to grant it the traction to resist lateral movement.
Figure 1-6: Definitions of an-
Figure 1-5: Connection
kle eversion and inversion,
bolts of Flex Foot pros-
Reference 7
thetic, Reference 8
The next iterations will lead to fine tuning the ankle support axis that contributes towards inversion and eversion of the foot as related to the shin axis. This twisting action
is important to get tuned correctly. Too loose and the under-foot contact area will overturn on its leading edge causing the leading edge to dig in and possibly lead to injury or
send the player off balance. If the twisting action support is too stiff, the design will not
allow the full under-foot contact area to make contact with the ground and provide full
traction.
Additional iterations beyond the scope of this project could include fine tuning the shin
axis of the prosthetic device. The current Flex Foot design only utilizes two bolts to
5
sandwich the mechanical spring to the leg connecting section of the prosthetic. These iterations will likely beef up the lateral moment of inertia of the mechanical spring section
and perhaps iterate on the bolt connecting pattern.
There are many factors that go into a prosthetic limb design, just as there are many factors as to way our limbs evolved the way they did. This project attempts to be an open
study into the factors that would allow this type of prosthetic to be used in sports that require more movement than just in a straight line.
Figure 1-7: Ankle eversion in tennis lateral motion
6
2. Theory and Methodology
2.1 Reverse Engineering the Flex Foot Design
The first step in redesigning the Flex Foot design is understanding the original design.
Unfortunately, the original design of the Flex Foot is proprietary to Ossur, the Icelandic
prosthetic design company that created the Flex Foot prosthetic. However, by using the
resources of the Bruggeman study, Reference 3, we can work backwards using the dimensional and material properties reported to come to an educated guess at the original design.
Figure 5 of Reference 3 displays the power through time stance phase via percentage of
the time elapsed as its argument the Flex Foot design allows a runner to be more energy
efficient than a runner with natural limbs. Table 2-1 of this paper uses the same basic
mechanical properties Reference 3 uses to make its analysis.
2.1.1
Dimensions
Using Figure 2-1, we can determine the neutral position geometric dimensions of the Flex
Foot prosthetics. We take the figure and determine the pixel coordinates of significant
points in the design. Then using Table 3 of Reference 3, we see the neutral height of the
Flex Foot is 46 cm, we can determine the relationship between the pixels of the image and
length dimensions.
Figure 2-1: Geometric Analysis of Flex Foot Prosthetic
7
Table 2-1: Properties of the Flex Foot, Reference 3
𝐻𝑒𝑖𝑔ℎ𝑡,
𝑝𝑖𝑥𝑒𝑙 ℎ𝑒𝑖𝑔ℎ𝑡,
𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜,
ℎ = 0.46𝑚
ℎ𝑝 = 584𝑝𝑥 − 316𝑝𝑥 = 268𝑝𝑥
𝑐=
0.46𝑚
= 0.0017164 𝑚⁄𝑝𝑥
268𝑝𝑥
(2.1)
(2.2)
(2.3)
Knowing the conversion rate, we can discover the rest of the significant dimensions
of the Flex Foot, and input them into the model. This study uses Abaqus analysis software,
for which there are no internal units analysis. Thus we need to pick a consistent units
system, indicated below.
Table 2-2: Consistent Units for Analysis
8
Figure 2-2: Dimensions of Baseline Prosthetic
In this way we guess at the dimensions of the Flex Foot. The below figure demonstrates the baseline model’s approximation to the original design. While this appears
satisfactory for the geometry, we must next iterate on the possible composite material
properties to accurately represent the spring response of the prosthetic.
Figure 2-3: Comparison of Baseline Model to Original
𝐸𝑥𝑡𝑟𝑢𝑠𝑖𝑜𝑛 𝑤𝑖𝑑𝑡ℎ,
9
𝑤 = 0.07𝑚
2.1.2
Material
Having gotten reasonably close to the Flex Foot geometric design, we now attempt to get
reasonably close to the linear stiffness measured by Table 2-1.
𝐿𝑖𝑛𝑒𝑎𝑟 𝑆𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠,
𝐸 = 38.7 𝑘𝑁⁄𝑚
As mentioned in numerous sources, as well as the product manufacturer’s description,
the prosthetic is made out of a carbon-fiber composite lay-up. Within the industry this
could refer to any number for material properties, laminate orientations, and intermediary
materials. In this study, we use Reference 9 as an initial guide to the material properties
and common orientations of performance composites.
Composites can be used in a variety of applications with various requirements for
stiffness and strength. Some applications require the high stiffness and strength that a carbon-fiber Nomex honeycomb sandwich provides, while other applications require the
lower stiffness spring-type profile of many thin laminate plies.
In this case the closest mechanical analogue is that of a leaf-spring, which in most
light applications such as this one, uses the thin laminate ply style of composite design.
Since the thicknesses of the plies can be of various thickness and. The above attempt at
ascertaining the geometry of the Flex Foot worked well for the large geometries like the
length and width but not as well for the thickness which is significantly smaller and more
difficult to ascertain. In addition, the thickness of the plies used must be guessed at.
Thus the initial material section will iterate on the ply thickness, orientations, and
number of plies. A load will be placed at the top of the Flex Foot model, along with constraints to ensure the load and displacement only occurs in the vertical direction as seen in
Figure 2-4. The displacement of the top of the model with then be calculated, resulting in
a linear stiffness to compare to the Bruggeman study.
10
2.2 Lateral Forces, Friction, and Angle of Attack
Table 2-3: Coefficients of Friction Equations, Reference 6
Figure 2-4: Plot of Friction Equations, Reference 6
It is important at this point to research the loads that the average tennis player impacts
on the ground during side-to-side lateral movement. Luckily the Clarke paper that concentrated on shoe patterns and traction effects summarized it nicely. According to the
Clarke Paper:
“In order to mechanically test under conditions that best represent real-life
play, ground reaction forces from a study conducted by Damm et al. were
examined to understand the forces exerted by a tennis player during shoesurface interactions. Damm et al. measured three-dimensional ground reaction forces of tennis players performing a side jump followed by a pushoff movement on an acrylic hard court surface. The mean peak normal
force found during the initial impact phase of the movement was found to
be approximately 1,150 N, and during the phase of forefoot push-off the
normal force reduced to relatively constant value of approximately 650 N.”
11
The Clarke paper summarizes its material results into a trend line graph of the shoe
traction force to the normal force of the foot on the ground of a relatively rough tennis
court surface:
𝐹𝑡 = 0.84𝐹𝑁 + 206.62
(2.4)
𝑅 2 = 0.98
(2.5)
𝐹𝑡 = 0.84(1150 𝑁) + 206.62
(2.6)
𝐹𝑡 = 1172.62 𝑁
(2.7)
Figure 2-5: Maximum Force and Angle of Attack
Maximum Angle of Attack:
1150 𝑁
𝜃 = arctan (
) = 44.44 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
1172.62
(2.8)
Maximum Axial Force
𝐹 = √11502 + 1172.622 = 1642.4 𝑁
(2.9)
This equation is put into analysis software as the governing friction equation in the
dynamic analysis of the prosthetic coming into contact with the ground. The peak normal
force of 1642.4 N applied downward axially along the shin at a 44.44 degree angle to the
ground as described in the Clarke paper will be used as the analysis goal for what the
prosthetic design can handle laterally.
12
2.3 Analytic Method Summary
2.3.1
Mesh
We first take the baseline design and attempt at optimizing the mesh of the part. In this
first part we don’t worry about the accuracy of the linear stiffness result, we just adjust the
mesh fineness up until an acceptable point.
2.3.2
Boundary Conditions
We simulate the loading on the prosthetic by applying a vertical load on the top set of
nodes, this can be seen in Figure 2-6. These nodes are constrained in all directions but the
vertical. This simulates the bolting pattern on the prosthetic that constrains it to the axis
of the shin.
Figure 2-6: Loads and
Constraints
The ground is constrained via tying it to a Reference Point in the Abaqus Interaction constructor. This ensures the ground is fixed in all degrees of freedom as seen in Figure 2-7.
13
Figure 2-7: Assembly with Fixed Reference Point Shown
The interaction between the prosthetic model and the ground model is governed by a
Kinematic Contact method between the lower face of the horizontal portion of the prosthetic and the upper face of the ground model. This sets up the contact rules for the
modelling and can be seen in Figure 2-8.
Figure 2-8: Assembly with Contact Interaction Method Shown
The Interaction is also governed by an interaction property that defines the tangential
behavior, in this case the sliding friction. We input the friction coefficient from Equation
2.4 as the governing interaction property as can be seen in Figure 2-9.
Finally, we place the prosthetic model slightly above the ground model. This ensures
that in the initial steps of the analysis, nodes on the prosthetic model cannot be construed
14
as being below the ground model, which would skew the results. An image of this is shown
in Figure 2-10.
Figure 2-9: Tangential Behavior
Figure 2-10: Profile of Assembly
15
2.3.3
Number of Plies
We then attempt to come to an accurate representation of the linear stiffness result of Reference 3. We do this simply be increasing the number of plies of the composite. We begin
with a standard [0/90] orientation. This lets half of the fiber laminate assist with the linear
stiffness in the longitudinal tension mode where composite laminate is strongest.
2.3.4
Ply Orientation
After we arrive at a reasonable approximation of the linear stiffness, we determine the
effect of the ply orientation. Various ply orientations will have different effects when we
analyze lateral forces on the prosthetic. Before we test these effects, we need to ensure
that the ply orientation types we have also match up with the measured linear stiffness of
the baseline design.
2.3.5
Lateral Forces
We then apply the lateral forces on the prosthetic and observe the stresses and deformations of the laminate. This will be achieved by applying a force downward on the
prosthetic design and tilting the ground at increasing angle. This will reveal the stresses
and deformations on the prosthetic design that will form the basis of the design iterations.
2.3.6
Design Iterations
It is the hypothesis of this paper that the prosthetic design will not be adequately resistant
to the maximum lateral forces applied at maximum tennis shoes traction. For a robust and
accurate ankle simulation, this paper hypothesizes that the design will have to be thickened
in the abduction/adduction axis to prevent excess twist in addition to being tweaked in the
inversion/eversion axis for proper ground contact and stability.
16
3. Results and Discussion
3.1 Model Setup
3.1.1
Mesh
Before attempting to discern the material properties of the Flex Foot, we first make sure
that the baseline design’s mesh is fine enough to make accurate calculations. We begin
with a coarse mesh, a 200 N load downward axially on the shin, a 20 ply [0 / 90] composite
and a level ground.
Maximum von Mises Stress (Pa e8)
Maximum von Mises Stress
3
2.5
2
1.5
Maximum von Mises
Stress
1
0.5
0
0
0.02
0.04
0.06
0.08
Mesh Size (m)
Figure 3-1: Mesh Experiment
As shown in Figure 3-1, adjusting the mesh size leads to slightly erratic maximum stresses.
The most stable of the meshes was around 0.03 to 0.02. Thus we will be using a mesh of
0.03 to enable quicker compiling of the stress data when the plies of the laminate increase.
3.1.2
Number of Plies
As stated in the methodology section, the thickness of the composite prosthetic design is
difficult to ascertain via visual inspection. In addition the thickness directly correlates with
the mechanical performance of the prosthetic, so estimating the number of plies is of the
utmost importance.
17
The first step of the material methodology was to take the assumed geometry definition of
the baseline model and see what material properties let us arrive at the Bruggemann study
linear elasticity measurement of the Flex Foot device stated in Figure 2-1.
𝐿𝑖𝑛𝑒𝑎𝑟 𝑆𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠,
𝐸 = 38.7 𝑘𝑁⁄𝑚
We begin with our baseline model as stated in section 3.1.1. We then add successive increasing loads in the shin axial direction and measure the total deflection. As we can see,
the original 20 ply model has nowhere near the linear stiffness required by the Bruggeman
study. We also see with this 20 ply model that the linear stiffness is not a constant as the
load increases.
0/90 Vertical Load (N)
20 Ply Displacement (m)
20 Ply Linear Stiffness (Pa)
24 Ply Displacement (m)
24 Ply Linear Stiffness (Pa)
30 Ply Displacement (m)
30 Ply Linear Stiffness (Pa)
40 Ply Displacement (m)
40 Ply Linear Stiffness (Pa)
50 Ply Displacement (m)
50 Ply Linear Stiffness (Pa)
60 Ply Displacement (m)
60 Ply Linear Stiffness (Pa)
100
0.0280
3571
0.0240
4167
0.0175
5714
0.0125
8000
0.0095
10526
0.0080
12500
200
0.0400
5000
0.0310
6452
0.0210
9524
0.0190
10526
0.0110
18182
0.0081
24691
300
0.0540
5556
0.0375
8000
0.0250
12000
0.0200
15000
0.0120
25000
0.0093
32258
400
0.0600
6667
0.0440
9091
0.0400
10000
0.0240
16667
0.0130
30769
0.0100
40000
500
0.0720
6944
0.0525
9524
0.0450
11111
0.0275
18182
0.0150
33333
0.0108
46296
600
0.0770
7792
0.0560
10714
0.0550
10909
0.0325
18462
0.0170
35294
0.0115
52174
700
0.0800
8750
0.0600
11667
0.0400
17500
0.0240
29167
0.0175
40000
0.0120
58333
800
0.0910
8791
0.0660
12121
0.0440
18182
0.0260
30769
0.0180
44444
0.0125
64000
Bruggeman Linear Stiffness (Pa)
38700
38700
38700
38700
38700
38700
38700
38700
Table 3-1: 0/90 Ply Orientation, Stress and Stiffness vs Load
We can visualize this data in a chart and compare it to the ideal stiffness value from the
Bruggeman study. The plots of Figure 3-2 show how the variation in linear stiffness at
different load increases as the number of plies increases. While in a simple beam this
behavior would seem odd, due to the complex shape of the prosthetic, it can be expected
that the prosthetic would not have a simple linear stiffness as the load is supported by the
different sections.
We now repeat the Bruggeman test in paragraph 2.2 of reference 3 which tests the prosthetic device on a “material testing machine T1-FR020TN.A50 (Zwick GmbH & Co, Ulm,
Germany)” [3] at a controlled deflection up to a load of 1500 N. In this way they come to
the estimation of linear stiffness. The results of this test simulation are in Figure 3-3.
18
Figure 3-2: Visualization of Linear Stiffness vs Load
As we can see from the data in Figure 3-3, at 1500 N, a prosthetic design incorporating 40
plies comes closest to the Bruggeman test results of 38.7 kN/m linear stiffness at a loading
of 1500 N. At a laminate thickness of 0.00025 as estimated by reference 9, this leaves us
with a prosthetic thickness of 1 cm. From the limited specifications and images given, this
thickness and linear stiffness best estimates the physical and mechanical properties of the
Flex Foot baseline design.
Figure 3-3: Various Ply Stiffness at 1500 N
19
3.1.3
Orientation of Plies
The 0/90 orientation of plies is certainly not the only orientation option available for carbon fiber composites, though it is one of the more common. The 0/90 orientation is simple
to manufacture, and well suited for an environment where the laminates are under a simple
tensile load from one direction. However, since this study aims to test a composite prosthetic in more than one direction and brings twist of the composite into the equation, other
laminate orientations that can accommodate twist were considered, the most common of
which being a simple 0/-45/90/-45 degree pattern. This pattern, according to reference 9,
has less linear stiffness in the two main longitudinal and transverse directions, but has a
higher stiffness for in-plane shear.
Thus we also test a material with the 0/-45/90/-45 degree pattern and see how many plies
it must have to meet the Bruggeman study linear stiffness. As can be seen in Table 3-2
and Figure 3-4, the lower longitudinal and transverse moduli correlate to a higher number
of plies to maintain the same linear stiffness at 1500 N.
Load
1500 N Ply
Test
20
28
36
40
48
1500 N
Vertical Displacement
(m)
0.1250
0.0830
0.0620
0.0450
0.0335
Linear Stiffness
(Pa)
12000
18072
24194
33333
44776
Bruggeman Stiffness
(Pa)
38700
38700
38700
38700
38700
Table 3-2: 0/-45/90/45 Stiffness at 1500 N
Figure 3-4: 0/-45/90/45 Stiffness at 1500 N, Various Plies
20
As you can see, the [0/-45/90/45] ply pattern would need more plies to fully meet the
Bruggeman Linear Stiffness value. This would lead to a thicker prosthetic, which would
likely be heavier and worse off in the forward sprinting mode. In the next section we’ll
see how it performs in a lateral movement mode.
3.2 Lateral Movement
3.2.1
Stresses and Tsai Wu
Now that a general ply structure has been analyzed, we now look to see what the effects
of lateral movement are on the stresses and failure criteria. In the vertical position, the
baseline prosthetic has a maximum stress of 2.815e8 and a maximum Tsai Wu failure
criterion of 0.47 as seen in Figure 3-5. The stress concentration is mainly in the equivalent
of the ankle region.
Figure 3-5: [0/90]Vertical Stress and TSAIW
21
In the 45 degree model, stresses are still concentrated around the ankle area, but asymmetric stresses on the sole of the prosthetic become apparent. The maximum von Mises stress
decreases while the Tsai Wu failure maximum increases. This is likely due to the increase
amount of torsion in the inversion/eversion axis. In this model, stresses and Tsai Wu criteria still don’t approach any failure point as designed. See Figure 3-6. This is important
to note as we iterate the design.
3.2.2
Orientation of Plies
Next we look at how the ply orientation affects the maximum stresses and Tsai Wu criterion. The additional number of plies will reduce the maximum stresses in the prosthetic
and the additional diagonally oriented plies in the [0/45/90/-45] ply structure will likely
reduce the maximum Tsai Wu failure criterion as well. stresses seen in the prosthetic. As
Figure 3-6: [0/90] 45 Degree Stress and TSAIW
can be seen in the 90 degree orientation in Figure 3-7, these intuitions hold true. Note that
boundary conditions have slightly skewed the Tsai Wu criterion scale. The largest Tsai
Wu values are seen in the ankle area with a value of 0.3016. Figure 3-8 shows the results
22
of the [0/45/90/-45] ply structure in the 45 degree orientation. Again, this orientation generates higher Tsai Wu failure numbers since a large amount of torsional shear is present
in the ankle area in this location.
Figure 3-7: [0/45/90/-45] Vertical Stress and TSAIW
Figure 3-8: [0/45/90/-45] 45 Degree Stress and TSAIW
23
In addition to being located in the ankle region, the Tsai Wu values are particularly high
for this laminate configuration. This high failure value means the laminate is quite close
to delaminating due to the high shear stresses present in the ankle area. So while the
[0/45/90/-45] laminate type was significantly better at stress distribution in the vertical
direction, the [0/90] laminate behaves better in the 45 degree orientation. This is the behavior we want to see from a lateral movement prosthetic.
Laminate
[0/90]
[0/45/90/-45]
Vertical
45 Degree
Stress (Pa)
TSAIW
Stress (Pa)
TSAIW
2.82E+08
0.471
2.71E+08
0.64
1.09E+08
0.3016
1.71E+08
0.925
Table 3-3: Summary of Baseline Stresses and TSAIW
24
4. Remarks and Conclusion
4.1 Prosthetic Design
The scope of this project began as taking a simple prosthetic limb design and redesigning it to fully model the mechanics of an ankle for sports with lateral movement. As the
project progressed, it became apparent that even emulating the original design of the
Flex Foot was a challenge. Composite design is many faceted, with millions of permutations of structures, materials, fibers, epoxies, and laminate patterns. A sports composite
such as a golf club, racing fairing, bicycle helmet, or prosthetic already have years of
design work behind them, which is why composites are even now just coming out to
revolutionize the sporting industry.
Thus taking a design that has already been tested and created and attempting to
make even a close replica is a difficult task. Fortunately studies existed for this particular
composite, particularly the Bruggemann study which gave some dimensions, spring
constants, and energy curves with which to get a baseline design.
4.2 Composite Choices
Only two basic composite choices were considered for this study. Given more scope,
more composite patterns and materials could be considered, not to mention use of metals
and other plastics. The [0/90] composite configuration was a bit of an all-purpose
composite pattern. While this paper had the hypothesis early on that the [0/45/90/-45]
composite pattern would have a better response to the lateral force considerations, it
turned out that the [0/90] was further from failure in the weakest portion of the prosthetic, the ankle. The [0/90] configuration also required less layers to reach the adequate
spring constant proposed by the Bruggemann paper. This spring constant was a target to
hit to get close to the Flex Foot’s original design, but with more scope changing this
number would have led to an interesting dynamic analysis of a runner’s stride as well as
that of a lateral sports player.
4.2.1 Metallic Side Study
As a quick side study, we briefly take a look at other cutting edge materials that
could be used in place of laminate composite for a prosthetic design similar to the Flex
25
Foot. Keeping in mind our model from Figure 2-2 and Figure 2-3, we are seeking a
design that matches the response of the Flex Foot design while maintaining as little
weight as possible.
We first select a few high strength metals, Aluminum 7078 T6, TiAl4v, and SS 174, and determine the modulus of elasticity of the prosthetic part with various thicknesses:
Prosthetic Response at:
1500 N
Thickness:
0.004
Al 7075 T6
TiAl4v
SS 17-4
Max Deflection (m)
0.110
0.060
0.080
0.047
0.070
0.035
0.035
0.028
0.023
Al 7075 T6
TiAl4v
SS 17-4
Modulus of Elasticity (Pa)
13636
25000
18750
31915
21429
42857
42857
54545
66667
0.006
0.008
Table 4-1: Deflection and Modulus Response of Selected Materials
Prosthetic Modulus (Pa)
70000
60000
50000
40000
Al 7075 T6
30000
TiAl4v
20000
SS 17-4
Bruggeman
10000
0
0
0.002
0.004
0.006
0.008
0.01
Thickness (m)
Figure 4-1: Modulus Response of Selected Materials
We can see that the different materials require different thicknesses to emulate at
least the same spring constant of the prosthetic design. We can then compare the densities, thicknesses and the volumes to see which design is advantageous in terms of
weight.
26
First we analyze our model to determine its total length and width:
Section
Shin
Ankle Arc 1
Ankle Arc 2
Heel Arc
Toe Arc
Radius (m)
NA
0.15
0.05
0.25
0.11
Arc Angle (degrees)
NA
38
90
58
39
Length (m)
0.190
0.099
0.079
0.253
0.075
Total Length
Width of Prosthetic
0.696 m
0.070 m
Table 4-2: Model Dimension Lengths
Figure 4-2: Dimensions of Baseline Prosthetic
Then we look at how much each material would weigh at the thickness it has to be
to support the prosthetic’s mechanical response.
Material
Composite
Al 7075 T6
TiAl4v
SS 17-4
Density
(kg/m^3)
1600
2810
4430
7800
Thickness (m)
0.008
0.007
0.0065
0.0055
Volume
(m^3)
0.000389744
0.000341026
0.000316667
0.000267949
Table 4-3: Weight of Prosthetic Using Selected Materials
27
Weight
(kg)
0.624
0.958
1.403
2.090
Weight (kg)
2.500
Weight
2.000
1.500
1.000
0.500
0.000
Composite
Al 7075 T6
TiAl4v
SS 17-4
Material
Figure 4-3: Weight of Prosthetic Using Selected Materials
As you can see, even the strongest of the materials, the SS 17-4, could not be thin
enough to overcome its much higher density than the composite. Even the lightest of the
metals was almost a pound heavier. While this might not be a deal breaker for many, in a
competitive sport, every pound counts.
4.3 Design Iterations
The immediate design iterations to consider after the results found in this study would be
adjusting the ankle section dimensions of the prosthetic and adjusting the width of the
prosthetic footprint. Both of these ideas have parallels in sporting goods design. A
running shoe is vastly different than a tennis shoe in these two areas. While a running
shoe favors lightness and minimalism, a tennis shoe requires a large amount of ankle
support due to the lateral movements. This support, in the form of a thicker and higher
shoe wall helps the tennis player resist the higher forces of moving from side to side. On
the same vein, the footprint of a tennis shoe is wider than that of a running shoe. Again
in this case the running shoe prefers thinness and lightness while the tennis shoe requires
a wider sole for ankle support and for lateral traction.
These are the two obvious iterative areas to explore, tough other design changes exist
that may help the Flex Foot design in a lateral sport. These include the connection point
28
between the leg connection of the prosthetic and the foot itself. In the original design, the
foot is simply bolted on. But when a large amount of torsion is introduced by the lateral
forces, a more integral connection could be considered to prevent high local stresses at
the bolts.
4.4 Conclusion
Based on the initial results of this analysis, the current Flex Foot design is structurally
adequate enough to survive the larger forces of lateral movement in sports. Simple
composite patterns can be used to build this prosthetics baseline design. However like in
other sporting goods design, tweaks can be made to the foot to allow for better lateral
dynamics.
29
5. References
1. “Perfecting the Prosthetic Leg: How incremental innovation works for patients.”
Eucomed
Medical
Technology
www.eucomed.org,
http://www.eucomed.org/uploads/_key_themes/mdd/eucomed_incremental_innovati
on.jpg
2. Knight, Tom. “IAAF call time on Oscar Pistorius’ dream”. The Telegraph. 10 Jan
2008.
http://www.telegraph.co.uk/sport/othersports/athletics/2288489/IAAF-call-
time-on-Oscar-Pistorius-dream.html
3. Bruggemann, Arampatzis, Emrich, and Potthast. “Biomechanics of double transtibial
amputee sprinting using dedicated sprinting prostheses.” Institute of Biomechanics
and Orthopaedics, German Sport University, Cologne, Germany. Sports Technology
2008, 1, No. 4-5, 220-227. DOI: 10.1002/jst.63
4. Weyand, Peter G.; Bundle, Matthew W.; McGowan, Craig P.; Grabowski, Alena;
Brown, Mary Beth; Kram, Rodger; Herr, Hugh (17 February 2009), The fastest runner on artificial legs: different limbs, similar function?, Journal of Applied
Physiology, retrieved 19 September 2012
5. “Flex Foot Cheetah Product Description.” Ossur Americas. Web. 2014.
http://www.ossur.com/prosthetic-solutions/products/feet/feet/cheetah
6. Clarke, Dixon, Damm, and Carré. The effect of normal lad force and roughness on
the dynamic traction developed at the shoe-surface interface in tennis.” Department
of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S1
3JD, UK. Sports Eng (2013) 16:165-171. DOI 10.1007/s12283-013-0121-3. Tennis
Friction Paper
7. “Unit VIII – The Foot and Ankle.” Virginia Commonwealth University Course
Lectures. Web. 2014. http://www.courses.vcu.edu/DANC291-003/unit_8.htm
8. http://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Oscar_Pistoris_at_Inter
national_Paralympic_Day%2C_Trafalgar_Square%2C_London__20110908.jpg/640px-Os-car_Pistorius_at_International_Paralympic_Day%2C_Trafalgar_Square%2C_London_-_20110908.jpg
30
9. Mechanical Properties of Carbon Fibre Composite Materials, Fibre / Epoxy resin
(120
degree
Cure).
Performance
Composites
Ltd.
Web.
2009.
http://www.performance-composites.com/carbonfibre/mechanicalproperties_2.asp
10. Metals Handbook, Vol.2 - Properties and Selection: Nonferrous Alloys and SpecialPurpose Materials, ASM International 10th Ed. 1990
11. SS 17-4 Data Bulletin. AK Steel Corporation. 9227 Centre Pointe Drive, West
Chester, OH 45069. 2007.
12. Materials Properties Handbook: Titanium Alloys, R. Boyer, G. Welsch, and E. W.
Collings, eds. ASM International, Materials Park, OH, 1994.
13. Metals Handbook, Vol. 3, Properties and Selection: Stainless Steels, Tool Materials
and Special-Purpose Metals, Ninth Edition, ASM Handbook Committee., American
Society for Metals, Materials Park, OH, 1980.
14. Structural Alloys Handbook, 1996 edition, John M. (Tim) Holt, Technical Ed; C. Y.
Ho, Ed., CINDAS/Purdue University, West Lafayette, IN, 1996.
31
6. Appendix A
6.1 Mesh Experiment
Mesh Size Results
Mesh Size (m)
Maximum von Mises
Stress (108 Pa)
1.991
1.991
1.354
1.189
1.409
1.484
2.392
0.07
0.06
0.05
0.04
0.03
0.02
0.01
Maximum von Mises Stress (10e8 Pa)
Maximum von Mises Stress (Pa e8)
6.1.1
3
2.5
2
1.5
Maximum von Mises
Stress (108 Pa)
1
0.5
0
0
0.02
0.04
0.06
Mesh Size (m)
32
0.08
6.1.2
Mesh Raw Data
6.1.2.1 Mesh Size 0.07
6.1.2.2 Mesh Size 0.06
33
6.1.2.3 Mesh Size 0.05
6.1.2.4 Mesh Size 0.04
34
6.1.2.5 Mesh Size 0.03
6.1.2.6 Mesh Size 0.02
35
6.1.2.7 Mesh Size 0.01
6.1.2.8 Mesh Size 0.005
Mesh Size 0.005 turned out to be too fine for efficient processing.
36
6.2 Number of Plies
6.2.1
Summary Table
0/90 Vertical Load (N)
20 Ply Displacement (m)
20 Ply Linear Stiffness (Pa)
24 Ply Displacement (m)
24 Ply Linear Stiffness (Pa)
30 Ply Displacement (m)
30 Ply Linear Stiffness (Pa)
40 Ply Displacement (m)
40 Ply Linear Stiffness (Pa)
50 Ply Displacement (m)
50 Ply Linear Stiffness (Pa)
60 Ply Displacement (m)
60 Ply Linear Stiffness (Pa)
100
0.0280
3571
0.0240
4167
0.0175
5714
0.0125
8000
0.0095
10526
0.0080
12500
200
0.0400
5000
0.0310
6452
0.0210
9524
0.0190
10526
0.0110
18182
0.0081
24691
300
0.0540
5556
0.0375
8000
0.0250
12000
0.0200
15000
0.0120
25000
0.0093
32258
400
0.0600
6667
0.0440
9091
0.0400
10000
0.0240
16667
0.0130
30769
0.0100
40000
500
0.0720
6944
0.0525
9524
0.0450
11111
0.0275
18182
0.0150
33333
0.0108
46296
600
700
800
0.0770 NA
NA
7792 #VALUE! #VALUE!
0.0560 NA
NA
10714 #VALUE! #VALUE!
0.0550 NA
NA
10909 #VALUE! #VALUE!
0.0325
0.0240
0.0260
18462
29167
30769
0.0170
0.0175
0.0180
35294
40000
44444
0.0115
0.0120
0.0125
52174
58333
64000
Bruggeman Linear Stiffness (Pa)
38700
38700
38700
38700
38700
38700
6.2.2
38700
38700
Summary Figure
70000
20 Ply Linear Stiffness (Pa)
60000
24 Ply Linear Stiffness (Pa)
50000
30 Ply Linear Stiffness (Pa)
40000
40 Ply Linear Stiffness (Pa)
30000
50 Ply Linear Stiffness (Pa)
20000
60 Ply Linear Stiffness (Pa)
10000
Bruggeman Linear Stiffness
(Pa)
0
100
200
300
400
500
600
700
37
800
6.2.3
Experiment Setup
6.2.4
20 Ply, Displacement Response to Specific Load Values
6.2.4.1 20 Ply, 100 N
38
6.2.4.2 20 Ply, 200 N
6.2.4.3 20 Ply, 300 N
6.2.4.4 20 Ply, 400 N
39
6.2.4.5 20 Ply, 500 N
6.2.4.6 20 Ply, 600 N
6.2.4.7 20 Ply, 700 N
40
6.2.4.8 20 Ply, 800 N
6.2.5
24 Ply, Displacement Response to Specific Load Values
6.2.5.1 24 Ply, 100 N
6.2.5.2 24 Ply, 200 N
41
6.2.5.3 24 Ply, 300 N
6.2.5.4 24 Ply, 400 N
6.2.5.5 24 Ply, 500 N
42
6.2.5.6 24 Ply, 600 N
6.2.5.7 24 Ply, 700 N
6.2.5.8 24 Ply, 800 N
43
6.2.6
30 Ply, Displacement Response to Specific Load Values
6.2.6.1 30 Ply, 100 N
6.2.6.2 30 Ply, 200 N
6.2.6.3 30 Ply, 300 N
44
6.2.6.4 30 Ply, 400 N
6.2.6.5 30 Ply, 500 N
6.2.6.6 30 Ply, 600 N
45
6.2.6.7 30 Ply, 700 N
6.2.6.8 30 Ply, 800 N
6.2.7
40 Ply, Displacement Response to Specific Load Values
6.2.7.1 40 Ply, 100 N
46
6.2.7.2 40 Ply, 200 N
6.2.7.3 40 Ply, 300 N
6.2.7.4 40 Ply, 400 N
47
6.2.7.5 40 Ply, 500 N
6.2.7.6 40 Ply, 600 N
6.2.7.7 40 Ply, 700 N
48
6.2.7.8 40 Ply, 800 N
6.2.8
50 Ply, Displacement Response to Specific Load Values
6.2.8.1 50 Ply, 100 N
6.2.8.2 50 Ply, 200 N
49
6.2.8.3 50 Ply, 300 N
6.2.8.4 50 Ply, 400 N
6.2.8.5 50 Ply, 500 N
50
6.2.8.6 50 Ply, 600 N
6.2.8.7 50 Ply, 700 N
6.2.8.8 50 Ply, 800 N
51
6.2.9
60 Ply, Displacement Response to Specific Load Values
6.2.9.1 60 Ply, 100 N
6.2.9.2 60 Ply, 200 N
6.2.9.3 60 Ply, 300 N
52
6.2.9.4 60 Ply, 400 N
6.2.9.5 60 Ply, 500 N
6.2.9.6 60 Ply, 600 N
53
6.2.9.7 60 Ply, 700 N
6.2.9.8 60 Ply, 800 N
54