Fatigue Analysis of a Bone Implant Construct

Fatigue Analysis of a Bone Implant Construct Mert GÖNÜL
Department of Mechanical Engineering, Yeditepe University
1
A Graduation Project Report
FATIGUE ANALYSIS OF A BONE IMPLANT CONSTRUCT
By
MERT GÖNÜL
Dep artment o f Mec hanic al Engineering
F ac ulty o f Engineering and Arc hitec ture
Yed itep e Univers ity
S ep temb er 2010, Is tanb ul, T urkey

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FATIGUE ANALYSIS OF A BONE IMPLANT CONSTRUCT
By
MERT GÖNÜL
DATE OF APPROVAL:14 September 2010
APPROVED BY:
Asst. Prof. A. FETHİ OKYAR
Thesis Supervisor
Dep artment o f Mec hanic al Engineering
F ac ulty o f Engineering and Arc hitec ture
Yed itep e Univers ity
S ep temb er 2010, Is tanb ul, T urkey

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ACKNOWLEDGEMENT
I am grateful and thankful especially to my supervisor, Asst. Prof. A. Fethi Okyar,
whose encouragement, guidance and support from the initial to the final level enabled me to
develop an understanding of the subject.
I would also like to thank to all mechanical engineering department lecturers and
professors for my background information, and to ones who work for this implant complex
for their helps and case studies.
Finally ı owe my deepest gratitude to my parents who never give up supporting me.
Mert GÖNÜL

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ABSTRACT
In today’s world; medicine itself is not enough to overcome some certain patient-
related circumstances. A new branch of science “biomedicine” comes out to help the patients
physically by producing health care medical products mainly. Lengthening the limbs and
treating the bones by using intramedullary devices is a common approach of using medical
devices in biomedical engineering. For kinds of applications like that, reliability is the most
common issue especially for the consolidation phase in this subject, which comes after the
lengthening period and occurs during walking (gait cycle). Many studies are investigated
upon it and it is seen that the distal locking screw at the top end of the device (nail) carries the
weight of the body on its shoulders; acting like a shear or load pin. The overloading on the
screw causes failure due to that. The aim of this project is to apply a sample calculation based
on fatigue analysis of this distal interlocking screw attached in an intramedullary nail that is
placed in the femur bone of a patient of height 1.90 m, weighing 80 kg.

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ÖZET
Günümüzde tıp, hastayla ilgili bazı konuların üstesinden tek başına gelememektedir.
Yeni bir bilim dalı olan biyotıp, hastalara sağlık hizmeti sunabilecek tıbbi ürünler üretip,
fiziksel olarak yardım etmek amacıyla ortaya çıkmıştır. İntramedüller aygıtlar kullanarak uzuv
uzatma ve kemik tedavisi biyomedikal mühendislikte sık kullanılmaktadır. Bu tür
uygulamalarda güvenilirlik, özellikle uzuv uzatma safhasından sonra gelen konsolidasyon
(yürüyüş hali) periyodunda en öne çıkan konudur. Bu konuda birçok araştırma yapılmıştır ve
bu intramedüller aygıtların üst kısmına yerleştirilen distal kilitli vidanın, emniyet pimi gibi
tüm yükü omuzlarında taşıdığı gözlemlenmiştir. Vidaya yapılan fazla yükleme, bozulma ya da
kırılmaya yol açmaktadır. Projedeki amaç; 1.90 m lik ve 80 kg ağırlığa sahip bir bireyin femur
kemiğine yerleştirilen aygıtın distal vidasına bir örnek üzerinden yorulma analizi
yapılmasıdır.

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TABLE OF CONTENT
ACKNOWLEDGEMENT ……………………………………………………………...… 6
ABSTRACT …………………………………………………………………………..…. 6
ÖZET ………………………………………………………………………………….... 6
LIST OF FIGURES …………………………………………………………………….... 6
1. INTRODUCTION …………………………………………………..........................1
2. LOAD ANALYSIS .………………………………………………………………….8
2.1. The Gait Cycle …………………………………………………..….....8
3. STRESS ANALYSIS …………..……………………………………………………13
3.1. Sample geometry and material …………………………………………13
3.2. Bending stress ……………………………………………………….….15
4. FATIGUE ANALYSIS …..………………………………………………………....20
4.1. Rain-flow cycle …………………………………………………………20
5. DISCUSSION and CONCLUSION ……….………………………………………..26
REFERENCES……………………………………………………………………………….28

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LIST OF FIGURES
Figure 1: The post-operative phase of the entire process [1] ….…….…………….….……1
Figure 2: Ilizarov surgery applied on a patient ……………….….…………………...……3
Figure 3: The solid model of the implant construct [3] ………..…………………..………4
Figure 4: IM nail schematics[2] ……………………….……….…………………………..4
Figure 5: A model of the implant [2] ……………….……….….…………………..………5
Figure 6: An example of double shear in flat plate ….…….…………….…………………6
Figure 7: Gait cycle phases ……………………….……….…………..……………………8
Figure 8: Reaction forces that are occurring in upper femur bone and footing [6] …….…10
Figure 9: Reaction forces on the upper leg ………………….……….……..………..........11
Figure 10: Resultant forces and moments occur in femur ………………………….….….12
Figure 11: Solid model of the screw, nail and femur bone of 60 mm cut view …..………13
Figure 12: Assembly of the screw, nail and femur bone …….……………………….….…13
Figure 13: Technical drawing of the assembly …………..………………………….….…..14
Figure 14: Resultant axial force and moment of femur……………………..………………14
Figure 15: Load distribution on screw……………………………………….……………..15
Figure 16: Free body diagram of the middle part ………….………………..………..….…16
Figure 17: Free body diagram of the left part ……………...……………….…….………...16
Figure 18: Shear and moment diagrams of the screw ………..…………………..………...17

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Figure 19: Theoretical stress concentration factor chart for a notched round specimen …...19
Table 20: List of peak and valleys in Figure 14………………..……….…………...……….21
Figure 21: Stress versus time graph …………………….……………………………..….....21
Figure 22: a) Rain-flow cycle counting – STEP 1 ….….…………………………………....22
Figure 22: b) Rain-flow cycle counting – STEP 2 …….……………….…………………....22
Figure 22: c) Rain-flow cycle counting – STEP 3 …………………….………………....….23
Table 23: Rain-flow table …………..………………….…………………………………….24

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1. INTRODUCTION
In today’s world, a new concept comes into account called biomedical engineering
which is a bridge between engineering and medicine. It is an engineering discipline that uses
most of its capabilities for improving human life physically. It is also a highly
interdisciplinary branch of science aiming to release long lasting, reliable, tough products to
the public. These are the major characteristics of the products that are created or designed by
biomedical engineers for use in various applications.
Prominent biomedical engineering applications include development of biocompatible
prostheses, various diagnostic and therapeutic medical devices ranging from clinical
equipment to micro-implants, common imaging equipment such as MRIs and EEGs,
biotechnologies such as regenerative tissue growth, and pharmaceutical drugs and
biopharmaceuticals.
The subject of this project is a device used to lengthen human limbs; “intramedullary
distractor nail device”. In the development of such a device, mechanical engineering
knowledge in failure analyses play an important role in the life-cycle assessment of the final
product (see Figure 1). In the consolidation phase, the strained tissue relaxes decaying the
axial compressive load. However, to do that fatigue analysis has to be performed to prevent
undesired failures of the product.
Figure 1: The post-operative phase of the entire process [1]

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In this study such an analysis is going to be applied to an intramedullary distractor nail
and its locking screws, which is used in limb lengthening by placing it inside the femur bone.
But before presenting the details about this procedure, one must know about operational
techniques, the model and characteristics of the implant, materials and devices, add to that
what the limb lengthening is and what it is used for.
Limb lengthening is a surgical process used to reconstruct skeletal deformities and
lengthening the long bones of the body. It is composed of two phases. During the first phase
which is called “the distraction phase”, a corticotomy is used to fracture the bone into two
segments, and the two bone ends of the bone are gradually moved apart, allowing new bone to
form in the gap.
Such a method of distraction osteogenesis was first developed by Gavriel Ilizarov, a
Russian orthopedic surgeon, in 1951. This method is called “Ilizarov surgery” and is applied
by exclusively an external fixator, which has become the common method for limb
lengthening for decades.
Ilizarov discovered that by carefully severing a bone without severing the periosteum
around it, one could separate two halves of a bone slightly and fix them in place, and the bone
would grow to fill the gap. He also discovered that bone regrows at a fairly uniform rate
across people and circumstances. These experiments led to the design of what is known as an
Ilizarov apparatus, which makes the extention of a bone possible by a desired amount. Add to
that, by using this technique some complications like limb deformities, limb length
inequalities, malunion and deformation of new bone, joint contracture or stiffness, treatment
prolongation and nerve palsy were observed, also including some minor complications like
paresthesia and pin-track infections.

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Figure 2: Ilizarov surgery applied on a patient
However, because of the Ilizaroy method’s complications, a new method came under
the name of “intramedullary nail” during the last decade. The intramedullary nail is placed
within the femur, allowing lengthening to take place internally, thereby drastically reducing
the risk of infections and scarring. The device is energized four times, daily, each time for
several seconds such that a relative motion of about 0.25 mm is obtained across the
longitudinal direction. This way, a distraction of about 1 mm is achieved every day. Finally
the desired length is achieved in a few weeks.
After the desired or possible length is reached, a consolidation phase follows in which
the bone is allowed to heal (see Figure 1). However at this phase, the loading spectrum is
more complex although loads have a smaller order of magnitude. These loads are primarily
due to the bodyweight that occur during the gait (walking) cycle. A fatigue and fracture
analysis must be performed to account for this phase.
An intramedullary rod (shown in Figure 3), also known as an intramedullary nail (IM
nail) or inter-locking nail is used to align and stabilize fractures. IM rods are inserted into the
bone marrow canal (see Figure 4) in the center of the long bones of the extremities (e.g. femur
or tibia). The DC motor shown in Figure 3 delivers torque to the lead screw to carry out the
lengthening.

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Figure 3: Solid model of the implant construct [1]
Figure 4: IM nail schematics [2]
One of the significant advantages of IM rods over other methods of fixation is that the
IM rods share the load with the bone, rather that entirely support the bone. Because of this,
healing process takes a shorter time and patients are able to use the extremity more quickly.
The IM nail can usually stay inside the patients’ bone forever, if designed properly
considering all kinds of failures.
The nail must be attached to the bone by some mechanical elements, like screws. It is
intended to fix the dynamic nail, into the femur bone by two proximal (lag) and one distal
locking screw as seen in Figure 5.

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Figure 5: A model of the implant [2]
It is seen that the failures usually occur at the distal interlocking screw location.
Because of it the most crucial and critical element is the distal interlocking screw. So the life
estimation is done considering mostly this element. Broaching the subject, the distal
interlocking screw basically performs the same task as a radially inserted shear pin under
double shear in mechanical systems. This brings us the necessity of broad information about
shear pins.
An interlocking pin is a device that is used for fastening flat or cylindrical surfaces
together. The pin itself may be a plain metal rod inserted through a tube and a rod; the pin
diameter being carefully chosen to allow for reliable operation during the entire lifecycle of
the product.
In this project the distal interlocking screw functions as a shear pin under double shear
which means the shear is symmetrical (see Figure 6). It is an advantageous kind of shear that
the balanced shear relationship which eliminates bending, prying and tension loads on shear
attachments while distributing an applied shear load over two shear planes in each attachment.
However, in our case there may well be a local bending effect due to the clearance between
the nail and the intramedullary canal, which accounts for fatigue failure of these elements.
Proximal interlockingscrew
Distal interlockingscrew

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Figure 6: An example of double shear in flat plate
Because of the interlocking screw’s critical importance, a broad literature search is done
especially to have more knowledge about it, its usage and load carrying characteristics. The
papers found are mostly finite elements studies on the mechanics of biomedical nails, the
fatigue characteristics of interlocking screws and associated load distributions on them.
The type of interlocking screw and usage is the key point determining the device’s life.
In Cheung’s paper [3] for example, certain regions of high stress concentrations are shown
and it is pointed out that stress shielding (the reduction in bone density as a result of removal
of the normal stress from the bone by an implant) and torsional deformity both in bone and in
the implant would probably cause failure or bending, loosening of the interlocking screw,
failure of the nail through the screw hole and bone refracture at the end of the nail or through
the screw holes. They also noted that a majority of failures are mostly due to fatigue of the
distal interlocking screw. So they suggest using more screws or to change the type of screw
material.
A somewhat similar failure mechanism to that analyzed here is studied in a paper
written by O.S. Es-Said [4], about the load analysis on a shear pin suggests that in a municipal
water filtration plant; the flocculator drive shear pin fails prematurely after only one week of
operation. The cause of failure is attributed to fatigue bending stresses in the shear pin caused
by misalignment and wobble in the flocculator drive coupling.

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Another paper published by Erich Schneider [5] mentions the benefits of
intramedullary interlocking nail in limb rotation ability and in maintaining the desired limb
length while under load. It is argued that it is necessary to convert the interlocking screw from
static locking into dynamic locking (loading by dynamization which causes the screw to carry
higher loads). Also the changing load due to fracture consolidation and the implant loading in
vivo, plus the general skeleton loading are investigated.
In our study, load and fatigue analysis of the interlocking screw is performed to
understand failure of the pin and to support material selection and redesign of a new pin. After
the analysis it is seen that the main problem is that the shear pin was initially designed for
direct shear stress and was not able to withstand the additional stress induced by the bending
fatigue.
In the next chapter, the load distribution on the device at the consolidation regime, the
caused stresses with related calculations and the materials and methods of the fatigue and
fracture analysis on a case study is going to be introduced. In the following chapter, the result
of the analysis is going to be discussed based upon the fatigue and fracture criteria.

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2. LOAD ANALYSIS
In this section the load analysis is going to be performed on the distal interlocking
screw and its hole. In a previous case study “Kinetic analysis during limb lengthening of
human walking gait” [6], the forces that are created in a gait cycle are obtained.
2.1.The Gait Cycle
The gait cycle is used to describe the complex activity of walking or our gait pattern.
This cycle describes the motions from initial placement of the supporting heel on the ground
to when the same heel contacts the ground for a second time. It is usually considered to be
composed of 8 stances (see Figure 7) and the load distribution (values of tension,
compression, bending and torsion) is different in all these stances.
Figure 7: Gait cycle phases

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These are; initial contact where the knee is extended and the ankle is neutral (or
slightly plantarflexed). Normally, the heel contacts the ground first. However the patients with
pathological gait patterns, the contact of the entire foot or the toes to the ground initially can
be seen.
In the loading response stance, the phase corresponds to the gait cycle's first period of
double limb support and ends with contralateral toe off, when the opposite extremity leaves
the ground. During loading, knee flexes 15 deg while ankle plantarflexes 15 degrees, which is
an energy-conserving mechanism. Throughout first phase of stance, hamstrings and ankle
dorsiflexors remain active. Quadriceps and gluteal muscles act during loading and throughout
early midstance to maintain hip and knee stability
Midstance begins with contralateral toe off and ends when the center of gravity is
directly over the reference foot. At this stance, the knee is extended & ankle is neutral again.
The triceps surae acts to control tibial advancement preventing the tendency for the ankle to
dorsiflex due to body weight and inertia.
Terminal stance begins when the center of gravity is over the supporting foot and ends
when the contralateral foot contacts the ground. As a note; terminal stance and midstance are
the only phases when the centre of gravity truly lies over the base of support.
Pre-swing begins at contralateral initial contact and ends at toe off. It corresponds to
the gait cycle's second period of double limb support. At this stance, knee flexes 35 degrees
and ankle plantarflexes 20 degrees. In these last phases of stance, the toes, which have been
neutral, dorsiflex at the metatarsophalangeal joints.
Initial swing begins at toe off and continues until maximum knee flexion (60 degrees)
occurs. The contraction of the quadriceps, initiated before toe off and serves two purposes
which are the prevention of heel from rising too high in a posterior direction and help to
initiate the forward swing of the leg
Mid-swing starts from maximum knee flexion until the tibia is vertical or
perpendicular to the ground.

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Terminal swing begins where the tibia is vertical and ends at initial contact. At this
stance, the hamstrings muscles become active to decelerate forward swing of the leg and
thereby control the position of the foot at heel strike
Due to the gait cycle, the reaction forces that exist at the foot affect the whole leg and
create forces and moments at the edge of the upper leg where the femur meets the hip. X2, Y2,
M2 (shown in Figure 9) are the reaction forces and moments that are going to be used to find
the resultant forces M, P, V at the interlocking screw interface in order to analyze the fatigue
characteristics of the interlocking screw itself. The forces that are created on the remaining
part are not any of our concern.
Position of the distal interlocking screw
V M
P
Figure 8: Reaction forces that are occurring in upper femur bone and footing [6]
To understand where the reaction forces are acting, some illustrations are presented in
the following page by transforming the solid model into a 2D form (see Figure 9). The lower
side of the bone is the knee side, and the upper side of the bone is the hip side, where the
distal interlocking screw occupies in the middle part.

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M2
Y2
X2
V
P
M
Figure 9: Reaction forces on the upper leg [2]
The resultant forces due to reaction forces were calculated in Matlab in a case study [6]
to acquire the minimum and maximum force values which are important for the fatigue
analysis. In the same study, the 8-stage cycle is taken to be 1 second and the resultant forces
and moments are shown due to it in Figure 10.

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However these are the forces that are applied on the leg (nail, bone and muscles)
during the gait cycle. For the purpose of the simplicity, we consider only the axial force
fluctuation within a single gait cycle and then use the rain-flow cycle count method [7] in
order to extract the appropriate cyclic loading pattern. After that, it would be possible to apply
a fatigue analysis on it.
Figure 10: Resultant forces and moments occur in femur [6]

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3. STRESS ANALYSIS
3.1 Sample geometry
To apply a stress analysis on the distal interlocking screw, the dimensions of the screw
and device must be known. For understanding the subject deeper, solid modeling software [8]
is used to show the screw, nail, bone interactions and dimensions (see Figure 11-13).
Figure 11: Solid model of the screw, nail and femur bone of 60 mm cut view
Figure 12: Assembly of the screw, nail and femur bone

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Note: It is important to mention that the screw is assumed to be M5x0.8; means it has a 5
mm diameter and a threading pitch of 0.8 mm. Bottom view of the screw head has a 2.3x
magnification. A 1 mm clearance is taken between the nail and the intramedullary canal.
M5x0.8 thread
Figure 13: Technical drawing of the assembly
Figure 14: Resultant axial force and moment of femur

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3.2. Bending stress
The axial load “P” is read off from the peaks of Figure 14. It causes bending, bearing
and shear stresses on the distal locking screw. Because the screw is designed for double shear
and not for direct shear, bending effect becomes much more dominant; creating a necessity to
the effect of bending stress into the fatigue analysis. However, for practical purposes, the
effects of bearing and shear stresses on the distal interlocking screw have been neglected in
this study. Now the calculation of the stresses are illustrated by a sample study conducted on a
person that weigh 80 kg including a pretension of 500 N within his muscle.
Sample calculation
The loads are acting as distributed loads and are shown in the Figure 15. The
maximum bending moment is calculated; also the shear and moment diagrams are drawn to
present the mechanical changes on the screw. To reduce complexity the length of the middle
part (diameter of the nail) is taken to be 10 mm instead of 8 mm. The distributed loads are
transformed to non-distributed loads while calculating the values of shear force and bending
moment to apply equations 1 and 2. The sample calculation is done considering P= -430 N
which is the first peak in the axial load diagram.
Figure 15: Load distribution on screw

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Shear diagram is drawn due to the below illustration of the middle part. The shear
forces are found applying the equality ∑Fy= 0. (eqn. 1)
10 mm
y
V= 215 N V= 215 N x
P= -430 N
Figure 16: Free body diagram of the middle part
Moment diagram is drawn due to the below illustration of the left part (same for the
right part).
20
3
mm
10
3
mm
215 N
V M
Figure 17: Free body diagram of the left part
∑ M= 0 (eqn. 2) => -M + 215N * 10mm – 215N *
20
3
= 0
=> M= 731 Nmm

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V
215 N
x
-215 N
M
x
-731
N mm
Figure 18: Shear and moment diagrams of the screw

26
The moments occur at the red regions in the previous sketch in Figure 18 and are
going to be used to find the maximum bending stresses at those regions.
σo =
M∗c
I
: Bending stress (eqn. 3)
c: The perpendicular distance from the neutral axis to a point farthest away from the neutral
axis, where σ max acts
I: Moment of inertia
I=
π
4
* r4 for circular cross-sections (eqn.4)
I=
π
4
* (2.5)4 = 30.7 mm4
I=
π
4
* (2.3)4 = 22 mm4 => Moment of inertia of the notched section
σo =
731Nmm∗2.5mm
30.7
= 59.5 MPa
σo =
731N∗2.3mm
22
= 76.4 MPa => Bending stress at the notched section
A handbook [11] is used for threading characteristics. According to the handbook the
minimum thread root radius is taken as the notch radius which is given as r=0.125p where p
denotes the pitch p. However we take twice the minimum notch radius as the average notch
radius for the worst case scenario.
The notched section is under a higher amount of stress as expected; which means the
failure occurs at the notched section of the screw. It is because the cross-sectional area of the
notch is smaller, causing a stress raiser effect and producing a higher fracture risk on the
region. This is simply a consequence of the locally higher stresses causing fatigue cracks to
start at such locations. So only the maximum bending stress of the notched section is going to
be taken into account. To do that the stress concentration factor must be found using the graph
of Figure 19.

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Figure 19: Theoretical stress concentration factor chart for a notched round specimen [9]
To find the stress concentration factor, two proportions have to be known.
D
d
=
5 mm
4.6 mm
= 1.09
r
d
=
0.2 mm
4.6 mm
= 0.04
Using the two constants above, the stress concentration factor is found to be Kt = 2.25.
To find the maximum bending stress, below equality is going to be used.
σb,max = Kt * σo = 76.4 MPa * 2.25 = 172 MPa (eqn. 5)
This means that the screw would be fractured from the notched part, if a moment of
172 MPa occurs at the red regions (see Figure 18).

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4. FATIGUE ANALYSIS
4.1 Rain-flow Cycle
Because the load distribution is different in all 8 stages, the loading is not uniform, it is
a spectrum loading. To overcome this complexity in the graph, a method called “Rain-flow
Method [7]” is used in such fatigue analysis. This method is used in order to reduce a
spectrum of varying stress into a set of simple stress reversals by allowing the application of
“Miner’s Rule” for assessing the fatigue life of a structure subject to complex loading.
The rain-flow counting algorithm is one of the most popular methods of cycle-counting
algorithms and generally used for the fatigue analysis of wind turbine components because of
turbulence [7]. It is necessary in non-uniform or non-periodic loadings. The algorithm
consists of 8 steps, which are:
1. Reduce the time history to a sequence of (tensile) peaks and (compressive) valleys
which are points where the direction of loading changes.
2. The irregular time history also consists of the stress differences measured between
peaks and valleys or valleys and peaks.
3. Count the number of half-cycles by looking for both valleys and peaks.
4. Assign a magnitude to each half-cycle equal to the range between its start and
termination.
5. Pair up half-cycles of identical magnitude (but opposite sense) to count the number of
complete cycles. Typically, there are some residual half-cycles.
Using rain-flow counting method and miners rule on the axial load graph (in Figure
14) would let us find the mechanical life of the interlocking screw. The number of cycles to
failure is going to be determined to find the screw’s mechanical life.
To apply rain-flow cycling method, the force values must be transformed into stresses.
To find these stresses and to create a S-t graph; a procedure which has been adopted to gather
the maximum bending moment from the maximum applied load -430 N, would be also
adopted to all the possible edge points of the curves in the Figure 14. That means all the
corresponding moments for each point in the load-time graph have to be found.

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The maximum bending stresses are shown in Table 20. With the calculated stresses the
stress versus time graph can be created and counting can be applied.
Table 20: List of peak and valleys in Figure 14
Figure 21: Stress versus time graph [10]
The stress-time graph is drawn as above and is now suitable to apply the counting. To
do the counting, a cycle must be chosen. Points B and H are very close to each other so they
can be assumed as they have the same stress value. So we start the counting with the degraded
graph shown in Figure 22.
Point Time(s) Load(N) Moment (Nmm) Stress(MPa)
A 0 -265 -442 -104
B 0,110 -430 -731 -172
C 0,368 0 0 0
D 0,384 -25 -42 -10
E 0,436 100 167 39.5
F 0,560 -215 -358 -84.5
G 0,670 -190 -317 - 75
H 0,890 -410 -683 -161
J 0,976 -225 -375 -88

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E
C
D G
F
B H
Figure 22: a) Rain-flow cycle counting – STEP 1
Counting a cycle with rain-flow method depends on the below condition:
In a peak-valley-peak or valley-peak-valley combination (an irregular stress
history), the second range must be equal to the first range or larger than it in order to be
counted. After counting each cycle, the graph is degraded; means the time history is
rearranged.
In this case C-D can be counted as a cycle because the D-E is a larger range. The counted
cycles are erased from the graph.
E
G
F
B H
Figure 22: b) Rain-flow cycle counting – STEP 2

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As it is seen F-G is the second cycle because the G-H range is larger.
E
B H
Figure 22: c) Rain-flow cycle counting – STEP 3
The ranges of B-E and E-H are equal, so the third cycle is B-E cycle. All of the
history is exhausted and the counting is completed. After finding the mean and amplitude of
the maximum and minimum stresses of the found 3 cycles, a table of necessary parameters
can be created to be able to use Miner rule.
Stress amplitude: σa=
∆σ
2
=
σ max −σmin
2
(eqn. 6)
Mean stress: σm =
σ max + σmin
2
(eqn. 7)
The values that are obtained from counting are for the compression side of the screw
and need to be replaced by their counterpart in the tension side as it is the tensile stress that
plays a major role in the propogation of a fatigue crack. The maximum bending stress values
are multiplied by “minus” according to that and the below table is created. The negative part
in cycle B-E has been replaced by zero in order to neglect the effect of compression in fatigue
(crack closure under compression).

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Cycle j Nj σmax σmin σa σm
C-D 1 1 10 0 5 5
F-G 2 1 84.5 75 4.8 80
B-E 3 1 172 -39.5(0) 105.8 66.2
Table 23: Rain-flow table
Now the number of cycles to the failure (Nfj) must be calculated for each 3 cycles to
be used in Miner’s rule. To do that, eqn. 8 and eqn. 9 must be used. However we need two
constant values (which are σ'f and b) to use the equalities. So a screw material assumption is
done as steel. Considering it as a Man-Ten (hot rolled) steel; the σ'f value is found as 1089
and b is found to be -0.115 from table 9.1 in [7].
Miner’s Rule
Miner’s rule [7] states that where there are k different stress magnitudes in a spectrum,
Si (1 ≤ i ≤ k), each contributing Nj (Si) cycles, then if Nfj (Si) is the number of cycles to failure
of a constant stress reversal Si, failure occurs when the below formulae is satisfied with C=1).
∑
Nj
Nfj
k
j=1
= C
(σmax*σa)0.5 = σ'f (2Nf)b (σmax>0) (eqn. 8)
Nf = ∞ (σmax≤0) (eqn.9)

33
The σmax value is greater than 0 in all 3 cycles, so eqn.8 would be applied.
Nf (C-D) = 0.5 * √
(σmax∗σa)^0.5
σ′f
b
= 0.5 * √
(10 MPa∗5 MPa)^0.5
1089
−0.115
= 1.3*108 cycles
The number cycles to the failure of cycles F-G and B-E are found is same way as above.
Nf (F-G) = 2.6*106 cycles
Nf,(B-E) = 1920 cycles
The number of cycles (Nj) is 1 for each 3 cycles. So by knowing all the necessary parameters,
the estimated number of repetitions to failure is:
Bf = 1/∑
Nj
Nfj
k
j = 1/(
1
1.3∗10^8
+
1
2.6∗10^6
+
1
1920
)
= 5.23*104
repetitions are needed for failure

34
5. DISCUSSION and CONCLUSION
The bone treatment and limb lengthening are significant medical terms that also have
to include mechanical concepts to be reliable and produce healthy and durable products.
Using the technique of intramedullary device placement instead of old fashioned, traditional
methods like Ilizarov’s overcomes many complications like limb deformities or inequalities,
malunion and deformation of new bone joint contracture or stiffness, and so on. Because of
these advantages, it is used widely. The device is composed of one proximal, one distal
interlocking screw and a nail that is usually placed inside one of the leg bones of a person.
Some studies have been done on the intramedullary devices and it is observed that the
distal interlocking screw is the key point that determines the life of these devices. So to
undertake mechanical design of these devices, a fatigue analysis has to be done on the distal
interlocking screw to determine the lifespan of the product.
The aim of this study is to make a sample calculation of a Man-Ten(hot rolled) steel by
applying fatigue analysis to a distal interlocking screw of an intramedullary nail that is placed
into the femur bone of a patient of height 1.90 m, weighing 80 kg.
The sample calculation begins with load analysis on a gait cycle to determine the axial,
torsional and bending loads that are applied on the device while walking (at the consolidation
regime). However, only the axial loads on the device are taken into account because of the
design criteria of the distal interlocking screw. It is produced to endure direct shear instead of
double shear, causing it to be effected from bending in critical amount. The maximum axial
force applied on the screw is found to be -430 N.
By using the maximum axial load, the maximum bending moment applied on screw is
determined as -731 Nmm. The maximum bending stress on the most critical part, “the
notched region” is obtained as 172 MPa, using shear and moment diagrams and considering
the notch effect.
Finally a method called rain-flow cycle counting is applied to the maximum bending
moments applied on the screw by dividing the gait cycle period into sequences. Miner rule is
applied to find the total life cycle of the screw. It is found that the device’s life is 5.23*104
repetitions.

35
In this project it is shown that the failure occurs due to the notched region of the distal
interlocking screw because of the stress raiser effect of the notch. The screw’s life is 5.23*104
cycles, so the number of cycles to the failure is nearly at the limit between low and high cycle
fatigue which means the fatigue is accompanied by both plastic and elastic deformation. It is
the B-E cycle that causes the failure to occur earlier.
The screw is desired to have the maximum life as far as possible because, placing and
removing the nail regularly is so hard both for the patient and the authorized person add to
that it is dangerous and unhealthy. To increase the mechanical life of the product, the stresses
applied on the screw must be reduced by preferring an increased diameter screw or changing
the type of the material. Increasing the number of distal interlocking screw can also be a
solution to this. Also the clearance between the screw and nail may be rebalanced to produce
more reliable devices.

36
REFERENCES
[1] A. Fethi Okyar, Koray K. Safak and Nilufer Egrican. Mechanical Design and
Prototyping Considerations for an Intramedullary nail for Extending Bone Sections.
Department of Mechanical Engineering. ASME 2010 10th Biennical Conference on
Engineering Systems Design and Analysis, June 12-14, 2010, Istanbul, Turkey.
[2] Isa Demir and Rıza Bayoglu. A Methodology for the Performance Assestment of
Intramedullary Nails Based on Finite Element Analysis. A Graduation Project, June 2010,
Yeditepe University, Istanbul, Turkey.
[3] G. Cheung, P. Zalzal, M. Bhandari, JK. Spelt, M. Papini. Finite Element Analysis of a
Femoral Retrograde Intramedullary Nail Subject to Gait Loading. Med Eng Phys26
(2004)
[4] M. Smith, F. Fisher, M. Romios, O.S. Es-Said. On the Redesign of a Shear Pin Under
Cyclic Bending Loads. Department of Mechanical Engineering, Loyola Marymount
University, Los Angeles. 7 November 2005.
[5] Erich Schneider, Markus C. Michel, Martin Genge, Kurt Zuber, Reinhold Ganz,
Stephan M. Perren. Loads Acting in an Intramedullary Nail during Fracture Healing in the
Human Femur. University of Bern, Switzerland. 22 February 2001.
[6] Mehmet Baser. Kinetic Analysis During Limb Lengthening of Human Walking Gait.
A Graduation Project. Department of Mechanical Engineering, Yeditepe University.
[7] Norman E. Dowling. Mechanical Behavior of Materials - Engineering Methods for
Deformation, Fracture and Fatigue – third edition – Pearson International Edition P/391-
470.
[8] Solidworks 2010, Solid Modeling Software.
[9] Richard G. Budyans, J. Keith Nisbett. Shigley’s Mechanical Engineering Design. 8th
Edition in SI Units.p/1006.
[10] Matlab Software
[11] Machinery’s Handbook, 26th edition. Industrial press, 2000. Newyork.

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