Citation
A biomechanics approach to the interaction of host implant systems

Material Information

Title:
A biomechanics approach to the interaction of host implant systems
Creator:
Miller, Gary Joel, 1947-
Publisher:
University of Florida
Publication Date:
Language:
English
Physical Description:
xiv, 190 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Bending ( jstor )
Bone fractures ( jstor )
Bones ( jstor )
Femur ( jstor )
Geometric properties ( jstor )
Material properties ( jstor )
Moduli of elasticity ( jstor )
Moments of inertia ( jstor )
Stiffness ( jstor )
Structural deflection ( jstor )
Bones -- Surgery ( lcsh )
Dissertations, Academic -- Mechanical Engineering -- UF
Mechanical Engineering thesis, Ph.D.
City of Jacksonville ( local )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 184-188.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Gary Joel Miller.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
025864853 ( ALEPH )
03418761 ( OCLC )
AAV4586 ( NOTIS )

Aggregation Information

UFIR:
Institutional Repository at the University of Florida (IR@UF)
UFETD:
University of Florida Theses & Dissertations
IUF:
University of Florida

Downloads

This item has the following downloads:

UF00084167_00001 ( .pdf )

00006.txt

00199.txt

00206.txt

00026.txt

00047.txt

00080.txt

00058.txt

00105.txt

00060.txt

00054.txt

00092.txt

00051.txt

00177.txt

00055.txt

00061.txt

00153.txt

00162.txt

00137.txt

00205.txt

00183.txt

00067.txt

00142.txt

00181.txt

00037.txt

00033.txt

00100.txt

00096.txt

00145.txt

00108.txt

00174.txt

00062.txt

00002.txt

00112.txt

00146.txt

00076.txt

00057.txt

00148.txt

00182.txt

00158.txt

00087.txt

00066.txt

00186.txt

00073.txt

00075.txt

00194.txt

00007.txt

00127.txt

00027.txt

00063.txt

00114.txt

00091.txt

00071.txt

00120.txt

00059.txt

00136.txt

00150.txt

00042.txt

00012.txt

00201.txt

00156.txt

00125.txt

00023.txt

00167.txt

00039.txt

00122.txt

00163.txt

00133.txt

00072.txt

00081.txt

00020.txt

00038.txt

00188.txt

00179.txt

00193.txt

00151.txt

00101.txt

00011.txt

00190.txt

00160.txt

UF00084167_00001_pdf.txt

00034.txt

00010.txt

00083.txt

00157.txt

00143.txt

00024.txt

00110.txt

00093.txt

00117.txt

00152.txt

00184.txt

00022.txt

00204.txt

00119.txt

00189.txt

00168.txt

00111.txt

00154.txt

00019.txt

00203.txt

00126.txt

00135.txt

00172.txt

00191.txt

00170.txt

00169.txt

00070.txt

00032.txt

00138.txt

00068.txt

00107.txt

00128.txt

00140.txt

00064.txt

00008.txt

00035.txt

00095.txt

00200.txt

00090.txt

00196.txt

00016.txt

00116.txt

00118.txt

00005.txt

00103.txt

00166.txt

00197.txt

00017.txt

00139.txt

00178.txt

00097.txt

00050.txt

00121.txt

00085.txt

00195.txt

00018.txt

00098.txt

copyright.txt

00113.txt

00052.txt

00144.txt

00084.txt

00069.txt

00134.txt

00004.txt

00088.txt

00187.txt

00029.txt

00175.txt

00074.txt

00132.txt

00077.txt

00041.txt

00053.txt

00164.txt

00198.txt

00104.txt

00185.txt

00115.txt

00078.txt

00149.txt

00141.txt

00131.txt

00021.txt

00028.txt

00031.txt

00009.txt

00046.txt

00147.txt

00044.txt

00013.txt

00001.txt

00109.txt

00099.txt

00102.txt

00180.txt

00040.txt

00129.txt

00094.txt

00159.txt

00014.txt

00086.txt

00130.txt

00049.txt

00079.txt

00048.txt

00165.txt

00123.txt

00065.txt

00106.txt

00015.txt

00056.txt

00192.txt

00045.txt

00161.txt

00171.txt

00176.txt

00173.txt

00202.txt

00030.txt

00089.txt

00082.txt

00155.txt

00036.txt

00124.txt

00043.txt

00025.txt

00003.txt


Full Text















A BIOMECHANICS APPROACH TO THE INTERACTION OF HOST
IMPLANT SYSTEMS













By

GARY JOEL MILLER


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA


1977
















ACKNOWLEDGEMENTS

I would like to thank Dr. George Piotrowski for suggesting

the area of intramedullary fixation for study, and for his guidance

and encouragement throughout this work. I am also indebted to

Dr. Larry L. Hench and Dr. Robert B. Gaither for their continual

support and valuable comments and criticisms.

My special thanks go to Dr. William C. Allen, now at the

University of Missouri Medical School, for his comments and

assistance with the many clinical aspects of this research endeavor;

and to my wife, Suzy, without whose encouragement and understanding

this work would have been impossible.

I would like to acknowledge the technical assistance of

Ms. Karla Smith, Mr. Thomas Carr, and Mr. Mike Hardee. I would

also like to thank Mrs. Loretta Thigpin, Ms. Susan deFalco,

Ms. Dee Schneider, and Ms. Pam Blalock for their assistance in

preparing this manuscript. Thanks are also due Mr. Robert Hagen

for his assistance in preparing the illustrations.

This research was aided by a grant from the Orthopaedic

Research and Education Foundation.

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS .. ii

LIST OF TABLES . v

LIST OF FIGURES . .. ... vi

ABSTRACT . .. xii

Chapter

I. INTRODUCTION . .... 1

Brief History of Intramedullary Fixation 1
Previous Work . 4
The Present Research . 7

II. PROPERTIES OF THE SYSTEM COMPONENTS: A REVIEW. 8

The Intact Femur. . 8
The Intramedullary Nail ... 16
Summary . 21

III. DETERMINATION OF THE GEOMETRIC PROPERTIES
OF THE SYSTEM COMPONENTS. ... 23

The Femur . 23
The Intramedullary Nail .. 69

IV. MODELING OF THE OSTEOSYNTHESIS. ... 75

The Preliminary Idealized Model .. 75
Intramedullary Interfacial Behavior .. 83
A General Intramedullary Osteosynthesis Model 90

V. EXPERIMENTAL VERIFICATION OF THE MODELING
OF THE OSTEOSYNTHESIS . 121

Well Characterized Systems. .. 121
In vitro Osteosynthesis .. 134
Conclusions . 145










VI. BONAIL PREDICTION OF THE OSTEOSYNTHESIS
DEFLECTION. .... .148

APPENDIX A. DEVELOPMENT OF THE DIFFERENTIAL
BEAM DEFLECTION EQUATIONS. ... 178

LIST OF REFERENCES ...................... .184

BIOGRAPHICAL SKETCH ................... 189
















LIST OF TABLES
Table Page

3.1 Values for the mean side-to-side difference,
Tp, and standard deviations (in parentheses)
for the paired femurs .. 34

3.2 Geometric properties of the 14 mm Fluted type
nail, with and without flutes .... .72

4.1 Finite difference computational molecules ..... 107

4.2 System properties for optimization evaluation ..... 119

















LIST OF FIGURES


Figure


Page

2


15


17


1.1 Bones of the lower extremity .

2.1 The effect of strain rate on the material
properties of bone ..

2.2 Cross-sectional shapes of commonly used
intramedullary nails .

2.3 Structural properties of commonly used
intramedullary nails .

2.4 Elastic modulus for several implant materials. .

3.1 Schematic representation of femoral
length measurements. .

3.2 Serial cross-sections of a representative femur
with size reference marks in place .

3.3 Definition of axes and distances for geometrical
property determination .

3.4 Mohr's circle construction for locating principal
axes and directions of the cross-sections. .

3.5 Gross cortical bone area as a function of
position for the left femurs .

3.6 Gross cortical bone area as a function of
position for the right femurs. .

3.7 Area moment of inertia, Ixx, as a function
of position for the left femurs. .

3.8 Area moment of inertia, Ixx, as a function
of position for the right femurs .

3.9 Area moment of inertia, lyy, as a function
of position for the left femurs. .


. 19

. 20


. 24


. 26


. 28


. 30


. 35


. 36


. 37


. 38


. 39









Figure


3.10 Area moment of inertia, I as a function
of position for the right lemurs .. 40

3.11 Principal moment of inertia, I as a function
of position for the left femurs. ... 41

3.12 Principal moment of inertia, I T, as a function
of position for the right femurs .. 42

3.13 Principal moment of inertia, I C, as a function
of position for the left femurs. .... 43

3.14 Principal moment of inertia, IC, as a function
of position for the right femurs .. 44

3.15 Polar moment of inertia, J, as a function of
position for the left femurs ... 45

3.16 Polar moment of inertia, J, as a function of
position for the right femurs. ... 46

3.17 Effective polar moment of inertia, Jeff, as a
function of position for the left femurs ...... 47

3.18 Effective polar moment of inertia, Jeff, as a
function of position for the right femurs. ... 48

3.19 Principal angle orientation and relative
magnitude of the principal moments for
the femurs in group I. ... 51

3.20 Principal angle orientation and relative
magnitude of the principal moments for
the femurs in group II .... 53

3.21 Principal angle orientation and relative
magnitude of the principal moments for
the femurs in group III. ... 55

3.22 Section modulus, Z, as a function of length
along the femur for two sample femurs compared
to data adapted from Koch (1917) .. 59

3.23 Graphical representation of overall femoral
length versus average cortical bone area ...... 66

3.24 Graphical representation of lesser trochanter
to femoral condyle length (Llesser) versus
average cortical bone area .... 67


Page










Figure


3.25 Graphical representation of the outer midshaft
diameter versus average cortical bone area .. 68

3.26 Calculated geometric properties for several
intramedullary nail shapes .... 71

4.1 Schematic representation of preliminary idealized
osteosynthesis with applied four-point loading
and point contacts . 76

4.2 Free body diagram of the outer tube with external
and point contact reaction loads .. 78

4.3 Free body diagram of the inner rod with point
contact reactions. . .. 78

4.4 Idealized angular mismatch representation. ... 80

4.5 Photograph showing the intramedullary nail in
place in the cast bone section for interfacial
stiffness determination. ... 84

4.6 Schematic drawing of the self-aligning compression
fixture for interfacial stiffness determination. 86

4.7 Load-deflection curves for interfacial behavior
of Kuntscher, Fluted and Schneider nails
in cancellus bone. . ... 87

4.8 Load-deflection curves for interfacial behavior
of Kuntscher, Fluted and Schneider nails
in cortical bone . 88

4.9 Schematic diagram of a segment of the general
osteosynthesis model under distributed load q(x) 92

4.10 Free body diagram of an element of the bone
with external distributed load and interfacial
reaction load. .. .. 92

4.11 Free body diagram of the nail with interfacial
reaction load. .... 92

4.12 Schematic diagram of an osteosynthesis under
equally spaced four-point loading with a
midshaft transverse fracture .. 102


viii


Page











4.13 Schematic diagram of a healed osteosynthesis
under equally spaced four-point loading with
the nail remaining in situ .. 105

4.14 Finite difference representation of the
osteosynthesis under four-point loading. ... 110

4.15 Modified computational molecules for the
describing nondimensional differential
equations of the coupled system. ... 114

5.1 Schematic representation of the four-point
loading apparatus used for in vitro verification
studies. . ... ... 122

5.2 Intact human femur with small dimples at 2 cm
intervals to aid in deflection measurement .. .124

5.3 Intact human femur mounted in four-point loading
fixture for in vitro testing .... 125

5.4 Intact femur, with dial indicator in position,
in the four-point loading fixture. ... 126

5.5 Graphical comparative deflection results obtained
from the testing and modeling of a uniform steel
beam . ... 129

5.6 Graphical comparative deflection results obtained
from the testing and modeling of a tapered
steel beam . ... 130

5.7 Measured deflections and BONAIL predictions for
an unnailed intact human femur under four-point
load . ... ... .. 131

5.8 Comparative results obtained by measurement and
BONAIL prediction of a four-point loaded intact
femur with 12 mm Fluted nail in place. ... 132

5.9 Close-up photograph of the fracture site of a
femur with Fluted nail in place. ... 136

5.10 Osteosynthesis in the test fixture with the
fracture surfaces distracted to show the
nail and fracture site location. ... 138

5.11 Osteosynthesis in the testing fixture for
in vitro verification of BONAIL predicted
deflection response. ... 138


Figure


Page









Figure


5.12 Graphical presentation of actual deflection
measurement and BONAIL prediction for the
12 mm Fluted intramedullary osteosynthesis .. .140

5.13 Presentation of results of deflection
measurement and BONAIL prediction for a 14 mm
Fluted intramedullary osteosynthesis ... 141

6.1 Nondimensional midspan bone deflection for the
healed osteosynthesis with intramedullary
nail in situ .... .. '. 151

6.2 Nondimensional interfacial loading profiles for a
healed osteosynthesis with interfacial stiffness
of 5 x 106 N/m/m at various rigidity ratios. ... .152

6.3 Nondimensional interfacial loading profiles for a
healed osteosynthesis with interfacial stiffness
of 5 x 107 N/m/m at various rigidity ratios. ... .153

6.4 Nondimensional interfacial loading profiles for a
healed osteosynthesis with interfacial stiffness
of 2 x 108 N/m/m at various rigidity ratios. ... .154

6.5 Nondimensional interfacial loading profiles for a
healed osteosynthesis at a modulus ratio of 13.0
for various interfacial stiffnesses. ... 155

6.6 Nondimensional interfacial loading profiles
using different properties and stiffnesses
to produce equal midspan bone deflections. ... 156

6.7 BONAIL predicted midspan deflections for a fractured
osteosynthesis as a function of modulus ratio for
various interfacial stiffnesses. ... 158

6.8 BONAIL predicted interfacial reaction profiles for
fractured osteosyntheses at a modulus ratio of
4.34 for several interfacial stiffnesses ... 160

6.9 BONAIL predicted interfacial reaction profiles for
fractured osteosyntheses at a modulus ratio of
8.7 for several interfacial stiffnesses. .. .161

6.10 BONAIL predicted interfacial reaction profiles for
fractured osteosyntheses at a modulus ratio of
13.0 for several interfacial stiffnesses 162


Page









Figure


6.11 Nondimensional fracture surface mismatch
as a function of modulus ratio for various
interfacial stiffness values ... .163

6.12 Fracture surface angulation as a function of
modulus ratio for various interfacial
stiffness values . .. 164

A.1 A segment of a beam having been curved by
an applied moment Mb .. .179

A.2 Stress distribution profiles developed in
the curved beam. . .. 179

A.3 A segment of a beam under distributed
load, q(x) . ... 182

A.4 A differential element of the beam with
distributed load, q(x) and reaction
moments and shears .... 182


Page
















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy


A BIOMECHANICS APPROACH TO THE INTERACTION OF HOST
IMPLANT SYSTEMS

By

Gary Joel Miller

June 1977

Chairman: Robert B. Gaither
Cochairman: George Piotrowski
Major Department: Mechanical Engineering

A general model of an intramedullary osteosynthesis of the

femur is postulated. The model allows for variable geometric

properties along the length of the bone, and allows homogeneous

and isotropic material properties to be chosen. The model includes

an intramedullary nail of arbitrary material and geometric proper-

ties running the full length of the bone and unvarying with position.

The nail and bone are assumed to be separated by a linearly elastic

layer which transmits the nail-bone interfacial loading.

Nondimensional multicomponent beam equations which relate

external loading to deflection are presented. Boundary conditions

which describe four-point loaded osteosyntheses with transverse

midshaft fractures and osteosyntheses consisting of healed bones

with intramedullary nails remaining in situ are included. The

resulting equations are cast into finite difference form and are









solved with the aid of a digital computer program (BONAIL) using

numerical relaxation techniques.

Results of in vitro verification studies involving steel

beams, intact femurs, and femurs with intramedullary fixation are

presented and serve to substantiate the accuracy of the BONAIL

model. Both the effect of relative flexural rigidity of the com-

ponents and the character of the interfacial layer are found to

play a major role in the solutions obtained.

Graphical results are presented for the nondimensional

midspan bone deflection, the transverse deflection mismatch and

angulation of the fracture surfaces, and the nail-bone interfacial

loading profiles. These results are plotted as a function of the

relative nail-bone elastic modulus ratios for various interfacial

layer stiffnesses.

These data indicate that the more flexurally rigid devices

reduce the overall midspan deflection, deflection mismatch, and

angulation. However, the use of different combinations of modulus

ratios and interfacial stiffnesses are found to result in comparable

deflections. This is caused by the different interfacial reaction

distributions and intensities which develop. Nonetheless, the

overall results appear to support using nails of high flexural

rigidity to effect lower deflections and therefore enhance primary

healing of the fracture. However, healed bones, with more rigid

nails remaining in situ, are subjected to significant load sharing

between the components, and the bone is partially protected from

stress. By being so protected, the functional remodeling at the


xiii









fracture may be hampered and result in a weakened bone which is

liable to refracture upon removal of the intramedullary nail.

The lack of information in the literature concerning the

geometric properties of the femur and the character of the nail-bone

interface for use in modeling the system led to a parallel experi-

mental study. Ten pairs of human femurs were processed to determine

the gross cortical area, area moments of inertia, principal moments

of inertia and principal angles, and polar moments of inertia at

serial points from the level of the lesser trochanter to the distal

metaphysis.

The results of this experimental study are presented graphi-

cally as a function of position along the femoral length. While

considerable variation in properties is found from one pair to the

next ( 25% or more), no statistically significant right-left bias

was observed.

In vitro experimental results of the interfacial stiffness

behavior of several nail types were also obtained in different bone

material. These results ranged over several orders of magnitude

and were dependent on both the nail cross-sectional shape and the

nature of the medullary bone, i.e. cancellus or cortical.















CHAPTER I

INTRODUCTION


The femur is one of the major load carrying members in the

human body. It is the single bone which joins the knee to the hip

(Figure 1.1) and is therefore called upon to support essentially

the entire weight of the body during standing. During walking or

strenuous activity, this bone is subjected to loads several times

larger than the weight of the body. Fractures of the femur are

not uncommon, and present a management problem to the orthopaedic

surgeon. One common mode of treatment for transverse femoral

fractures involves the insertion of an intramedullary nail through

the marrow cavity of the bone, using this metal rod as a strut to

hold the two pieces of bone in alignment while they heal together.


Brief History of Intramedullary Fixation

Lambotte of Belgium (1913) has been credited with using the

first intramedullary nail in a clavicle in 1907. It was not until

1918, however, that Hey Groves of England used intramedullary nailing

for fixation of a fracture of the femur. Rush and Rush (1937, 1939)

presented the first cases of intramedullary fixation in the United

States using nails of round cross-section. Kuntscher's (1940)

classic treatise, "The Intramedullary Nailing of Fractures," served

to lend strong support to intramedullary nailing as a viable method














'~~~r~r ': *T .. I


Figure 1.1. Bones of the lower extremity









of fixation of fractured long bones. At that time, Kuntscher intro-

duced his V-shaped cross-section nail which was said to be more

easily driven into the intramedullary canal than a solid nail because

its open cross-section allowed it to compress on insertion. This

shape was subsequently modified to a cloverleaf pattern which fol-

lowed a guide pin more easily and provided greater nail strength.

Street et al. (1947) reported the use of the diamond

cross-section nail which had been developed by Dr. Hansen in

1945 to provide greater flexural and torsional rigidity. Schneider

designed a four flanged nail in 1950, which could be driven into

the intramedullary canal and engaged the endosteal surfaces with

its flanges to maintain torsional stability. Its cross-section was

also said to provide higher flexural rigidity than previous designs.

(Schneider, 1968). In 1967, the Fluted nail was used for the first

time (Allen, 1977). This nail was designed to provide increased

torsional and bending rigidity and better torsional gripping

of the intramedullary surfaces.

The reader gets the impression from a cursory review of these

clinical reports on the evolution of intramedullary devices that the

author's primary concern in redesign is to increase the torsional

and bending rigidity of the device in order to affect a better

osteosynthesis. There is, however, no experimental verification

that continued increases in rigidity will, in fact, produce this

desired result.









Previous Work

McKeever (1953) has summarized the general criteria for

design and application of an effective intramedullary device.

His conclusions, and those of the above investigators, can be

summarized as follows:

1. The nail should maintain alignment and apposition
of the fracture surfaces, i.e. close fit at the
fracture site.

2. It should permit compression of the apposing fracture
surfaces by allowing axial sliding of the bone seg-
ments along the nail.

3. It should prevent torsion of the fractured bone
segments with respect to each other.

4. The nail must be constructed from materials which
meet biocompatibility requirements.

5. It should be able to withstand.the synamic bending
stresses applied by the bone segments and their
related muscle groups.

6. The nail must be rigid enough to provide suitable
bending stabilization so that new bone can form.

Many authors have directed their attention to meeting these

criteria which are generally accepted as necessary for the attainment

of good results using intramedullary nails in fracture fixation.

Lindahl (1962, 1967) stated that rigidity (6) and immobilization (1 and

3) are of the highest priority. Braden and Brinker (1973), Andersen

(1965), Andersen et al. (1962), and Hutzschenreuter et al. (1969) have

supported Lindahl and reiterate the need for "rigid fixation" to

affect proper healing using intramedullary nailing. Taylor (1963) has

also noted that the use of intramedullary nails which do not produce

complete rotational fixation (3) will lead to a higher incidence of

complications such as malunions and nonunions.









In clinical use, variable success has been reported in the

literature regarding intramedullary fixation. Wickstrom et al. (1968),

in a review of 324 total cases, cited many of the complications that

occur in the use of intramedullary fracture fixation. While many of

these complications were biological in nature (i.e. infections, etc.),

others were the result of improper implant choice or application.

These biological and biomechanical complications led to 3 1/2% of

the patients having delayed or nonunion of their fractures. These

data support the work of Nichols (1963), who also found a 3% delayed

union rate in intramedullary osteosyntheses.

Success or failure of this surgical procedure depends on the

mechanical interaction between the nail and the bone. Peters (1972)

has put the problem into perspective.

Studies of the most appropriate material for
immobilization of fractures must be correlated with
determination of the biomechanical factors which act
to increase and decrease fracture healing. The
material and method of application can then be de-
signed to minimize the disruptive stress at the
fracture site and enhance those that increase
healing... If the healing process can be materially
speeded and non-union prevented, months and years of
patient disability could be eliminated...(p.18)

Attempts to improve intramedullary fracture fixation have

focused on increasing bending rigidity, torsional rigidity, ultimate

strength, and biocompatibility of the nail, by experimenting with

variations in cross-sectional geometry, length, and material com-

position of the nails. Results reported to date are, however,

qualitative and have not been correlated with mechanical factors.

Both good and bad results have been reported using similar surgical

techniques and intramedullary nails.









The above review of intramedullary fixation leads to two

major questions. Do the relative flexural rigidities of the nail

and bone have a significant effect on the displacements of the

fracture surfaces of the loaded osteosynthesis? Are there other

parameters which affect the performance of the system?

Allen et al. (1968) underscored the need for understanding

this interrelationship of the geometric, material, and structural

properties of the nail and bone in an intramedullary osteosynthesis.

However, quantification of the deflection at the fracture interface

of an osteosynthesis has not been published to date. Bourgois and

Burny (1972) took a first step in describing this complex problem.

In analyzing various loading configurations, they developed relation-

ships between fracture callus rigidity and percentage healing. The

model used consisted of a cantilever beam (the intramedullary nail)

surrounded by bone, with a fracture callus present at the midsection.

Deflections were calculated for this type of system under a trans-

verse end load. To carry out these calculations, an assumption of

complete bonding of the nail to the bone was implicitly made (i.e.

the interface between bone and nail transmitted shear stresses).

This transmission of shear stresses at the interface affects the

deflection results significantly and is not seen clinically with

intramedullary nails. However, first approximations on the effects

of variations in geometry, material, and fracture callus properties

on deflections were obtained. No other quantitative studies which

deal with intramedullary fixation have been found, and the following

investigation was therefore undertaken.









The Present Research

The present work is aimed at quantification of the nail-bone

interactions and the relative displacements of the fracture surfaces

present during the bending of an osteosynthesis. Utilizing engine-

ering techniques and analyses, supported by in vitro experimental

verification, the ability of different nail shapes and sizes to

maintain alignment and fracture surface apposition are presented.

A model of the femur is used which involves varying geometric

properties and constant material properties along its length. An

intramedullary nail of constant material and cross-sectional

properties is within the bone. It is assumed that a four-point

bending load is applied; consequently, interfacial loading is

assumed to develop at the nail-bone interface due to relative

differences in displacements of the nail and bone. Osteosyntheses

with and without fractures present are considered. Emphasis is

placed on the effects of the observed differences of structural

properties of currently used intramedullary nails on the inter-

facial deflections at the fracture site and the expected contact

forces between the nail and the bone in the medullary canal.
















CHAPTER II

PROPERTIES OF THE SYSTEM COMPONENTS: A REVIEW


The engineer is often faced with the need to analyze the

deflection behavior of load carrying machine elements. In order

to proceed, it is important that the engineer consider the elements

in a two-fold manner. The first involves understanding of the

components as geometrical entities, i.e. their shape and/or dis-

tribution of material. The second involves the appreciation of

the properties of the materials from which the elements or devices

are fabricated. By combining these two views, a quantitative

evaluation of the performance and interaction of the devices can

be performed. It is believed that understanding and modeling the

intramedullary osteosynthesis requires the same approach. There-

fore, this thesis begins with an examination of the existing

information on the structural and material properties of the

components of the intramedullary osteosynthesis.


The Intact Femur

Geometrical Properties

Koch (1917) was one of the first to apply engineering

principles to the determination of the stress response of the

loaded femur. Using time consuming graphical methods, he analyzed

the shape and distribution of bone in a single human femur. He









showed that the external form of the bone, and its internal

structure, were proportioned in such a way as to resist axial,

bending and shear loads applied during normal activity with a

minimum use of material.

Photoelastic techniques (Pauwels, 1951) and stress lacquers

(Evans and Lissner, 1948, and Frankel, 1960) have also been used

to determine stress levels in bones under axial and bending loading.

More recently, Toridis (1969) and Rybicki et al. (1972) have pre-

sented analyses using engineering beam theory to predict stress

levels throughout the femur. Their results indicate that engine-

ering beam theory is applicable to analysis of the long bones in

the midshaft region. (The bone ends, with their flared structures,

are not adequately modeled using these methods.) Piziali et al.

(1976) analytically examined several assumptions inherent in the use

of classical engineering beam theory to long bones (specifically

tibias), and found that deflections due to shear deformations are

negligible, and that the effects of initial curvature are not

significant. The mismatch between the neutral axis of the

cross-section and the applied moment, however, introduces de-

flection components perpendicular to applied loads. Their assumption

of linearly elastic properties led to predicted deflections which

were inversely proportional to the modulus of elasticity of the

bone with shear deformations causing negligible deviation in this

behavior. Yamada and Evans (1970) presented empirical load-deflection

curves for long bones but did not relate these data to the actual

geometry of the bones involved.









Long bones have an irregular cross-sectional geometry

which varies considerably along their length. The determination

of geometric properties, such as the area moments of inertia, is

necessary to apply the above cited strength results to other bones.

As Evans (1957) pointed out, much of the work on structural per-

formance of long bones requires approximation of the geometric

properties of bones. Allen et al. (1968), realizing this problem,

predicted the structural rigidity of a femur in bending and torsion.

They used tedious numerical integration techniques which yielded

the moments of inertia necessary for calculation of the bending and

torsional rigidity of the femur. This work was, however, limited

to only three cross-sections. Piotrowski and Wilcox (1971)

developed a computer program which performed the numerical

determination of the geometric properties and stresses in solid

(singly connected) cross-sections and allowed the analysis of

over 200 cross-sections of canine fibulae (Piotrowski, 1975).

The program was updated (Piotrowski and Kellman, 1973) to take

into account the presence of internal holes (the intramedullary

canal). Minns et al. (1975) used finite element techniques to

analyze four left tibias and reported variations in geometrical

properties as a function of length. They also found that geometrical

properties between bones can differ by 30% or more. Piziali et al.

(1976) determined the geometric properties of two femurs, two

fibulae and four tibiae. However, they presented only the tibial

information, which corresponded to that reported by Minns et al.

(1975).









It is important to note that adequate understanding of the

femur as a structure required the knowledge of the geometric

properties and response to load of the system under consideration.

Knowledge of only the load response is not sufficient to allow

for application of the results to the modeling of other loading

situations or bone configurations.


Material Properties

The material properties of bone have received more attention

than the geometric properties. Many hypotheses have been offered

to explain the microscopic behavior of bone. Currey (1964), Katz

(1970), and Kraus (1968) discuss the many theoretical lumped

parameter models used to describe the mechanical behavior of bone

substance on a microscopic scale. The present work, however, is

concerned with the macroscopic behavior of bone, and the interested

reader is referred to the above review articles for further dis-

cussion of microscopic behavior and modeling.

As Kummer (1972) has pointed out in his review of the bio-

mechanics of bone, bone is neither morphologically homogeneous nor

isotropic. In fact, Reilly and Burstein (1974) indicate that a

transversely isotropic model would be more accurate than the

homogeneous and isotropic representations which are often used in

bone modeling. Whether or not bone can be regarded as homogeneous

with respect to its macroscopic behavior, however, remains to be

seen.










Reilly (1974) and Reilly and Burstein (1974) have presented

a comprehensive, critical review of the literature on the mechanical

properties of human cortical bone. Their review gives a detailed

discussion of the many variations in mechanical properties of bone

presented throughout the literature. These include the effects

of specimen handling, and testing procedures for determination of

these elastic properties. Several articles referred to are dis-

cussed below.

Reilly (1974) measured the following average properties for

freshly frozen human bone tested at high strain rates:


E' = 17.1 x 109 N/m2


E = 11.5 x 109 N/m2

v = 0.46

v' = 0.58


G' = 3.28 x 109 N/m2

where E is the modulus of elasticity, v is the Poisson's ratio, and

G is the shear modulus.- The primed values refer to properties

found parallel to the long axis of the bone, and the unprimed values

refer to the radial and transverse properties. There are large dif-

ferences in properties in these two orthogonal directions. The

values reported for E and E' are associated with standard deviations

of approximately 15%-20% of the modulus if determined in tension,

and 7%-10% if determined in compression. The Poisson's ratios had a

standard deviation which represented 30% of their mean values.

Standard deviations for the shear modulus were typically 10%.









These results for the variations to be expected in determin-

ation of the material properties correlate well with the results

cited by Evans and Lebow (1951), who made additional observations

concerning the local variations in properties along the length of

a given bone. They found 2%-8% variations in the modulus of

elasticity and similar variations in the ultimate strength and

hardness as well. These results indicate that, while there is

some variation to be expected in the properties measured at dif-

ferent locations along a single bone, these variations are not

statistically significant when considering the overall scatter in

results obtained at several locations on several different femurs

as indicated by Reilly.

As Reilly (1974) noted, the average results he obtained in

measuring material properties were for freshly frozen, nonembalmed

bone. Many authors have dealt in considerable detail with the

effects of different treatment on the mechanical properties of

human cortical bone (Evans and Lebow, 1951; Evans, 1957; McElhaney

et al. 1964; Sedlin and Hirsch, 1966; Kraus, 1968; Kummer, 1972; and

Burstein et al. 1972). They made several salient observations:

1. There was no significant temperature effect on the
maximum failure stress (210 370). The modulus
of elasticity was found to increase slightly along
with maximum deflection at 370.

2. Air drying alters the physical properties in only
a few seconds, and a marked increase, as much as
25% or more, in the modulus of elasticity is observed.
A decrease in the observed plastic deformation occurs
as well.

3. While freezing increases the strength of the specimen
slightly, it does not alter any other properties.










4. Fixation in formalin does not appear to alter the
modulus of elasticity.

5. Significant differences were found in all the prop-
erties of bone from individual to individual and
these are not correlated with age.

The results on the effects of embalming on the properties of

bone are equivocal. It has been suggested (Allen, 1977) that

embalming may have a highly variable effect on the mechanical

properties of the bone due to the differences in vascularity and,

therefore, disposition of the embalming medium.

McElhaney and Byars (1965), testing bovine bone in compression,

Panjabi et aZ. (1973), testing rabbit femora in torsion, and Burstein

and Frankel (1968), testing human bone in torsion, also point out

that the physical properties of bone may vary with changes in load

or strain application rates. Figure 2.1 indicates the large

variations in properties which may be expected.

Thus, as Reilly and Burstein (1974) have pointed out, data

can be extracted from the literature to support or contradict many

hypotheses. This is not due to any inherent ambiguities in the

data, but is rather due to the absence of standardization or unifi-

cation of experimental approaches and goals. These large variations

in the reported data indicate the need for more work before the

intrinsic and extrinsic variations of bone properties are clearly

understood.
























HUMAN

'max -Suc


BOVINE


A Elastic Modulus

* Ultimate Strain


i I I I L L `


10-4 10-3 10-2 10-1 100


102 10j 10*


STRAIN RATE


Figure 2.1.


The effect of strain rate on the material
properties of bone (Bovine data from McElhaney
and Byars, 1965; Human data from Burstein
and Frankel, 1968)


Co
(0
0-

C\J

CC
)



C

Cu X
E

t-










The Intramedullary Nail


Geometrical Properties

Some of the more commonly used nail shapes are shown in

Figure 2.2. The American Society for Testing and Materials (1976)

has developed standard specifications for the geometry of Kuntscher,

or cloverleaf nails (F339), and for solid cross-section intramedullary

nails (F455) which include the Schneider and Hansen-Street configura-

tions. There has been no specification adopted to date for the
**
newly developed Fluted nail. It is important to note that these

are voluntary standards and that the manufacturers are not bound

by them in any way. In actuality, the dimensions obtained from

the implants themselves appear to vary considerably from nail to

nail and manufacturer to manufacturer for a given cross-sectional

configuration. It is, therefore, important to consider these

devices on an individual basis for determining their geometric

properties. There has been little done in this area in the way of

characterizing the important geometric properties of intramedullary

devices.

Allen et aZ. (1968) examined the overall structural per-

formance of various nail types in torsion and bending. These

included the Kuntscher, Hansen-Street, and Schneider nails. They

neglected to note, however, the nail size being considered. The



F339, Specification for Cloverleaf Intramedullary Pins;
F455 Specification for Solid Cross-Section Intramedullary Nails.

Sampson Corporation, Pittsburgh, PA.
Sampson Corporation, Pittsburgh, PA.































































Figure 2.2. Cross-sectional shapes of commonly
used intramedullary nails









Sampson Corporation (1976) used proposed ASTM standard test methods,

which appeared to be F383, to evaluate the bending rigidity, bending

strength, and torsional rigidity of these devices. Their comparative

results appear in Figure 2.3 a-c. In this test method, however, the

geometrical properties were not explicitly considered. The data

needed to further model nail use in an osteosynthesis is, therefore,

only partially known.


Material Properties

The ASTM has also devised several recommended standard

specifications for the materials commonly used in implant manufacture

(F55, F56, F67, F75, F90, F136, F138, F139). These include the

chemical requirements, and the minimum acceptable mechanical

properties, including ultimate tensile strength, yield strength,

elongation, and reduction of area. Because of the effects of

machining, cold working, and processing techniques in fabrication

of implants from the annealed metals, these values for the unworked


*
F55, Specification for stainless steel bars and wire for
surgical implants; F56, Specification for stainless steel sheet
and strip for surgical implants; F67, Specification for unalloyed
titanium for surgical implants; F75, Specification for cast
cobalt-chromium-molybdenum base alloy for surgical implants;
F90, Specification for wrought cobalt-chromium alloy for surgical
implants; F136, Specification for titanium 6A1-4V ELI alloy for
use in clinical evaluation of surgical implant materials; F138,
Specification for stainless steel bars and wire for surgical
implants (Special quality); F139, Specification for stainless
steel sheet and strip for surgical implants (Special quality).


















BONE 316SS INTRAMEDULLARY
{inside by outside DEVICES (diamelers in mm)
diameters in mm) 0 ( >

17X 29.,
16x28 (0I) BONE
S} SAMPSON
iSX27T. I KUNTSCHER

14X26 SCHNEIDER

13X25 *ANSESTREET
APPROPRIATE
12X24 \ROD SIZE
X2 \ due to FLUTES
II 23
OX 22
I16
14 7



13 .*1
:12 .11
11-111\0 .8.9
<) ^"7y ^"~9


Figure 2.3.


Structural properties of commonly used intramedullary nails
(Data taken from Technical Memorandum 1-76, Sampson Corp.
Pittsburgh, Pa.)


600
soo


s
P 500
it



(3
Z 400



W
I--
S300

I.-
U)
200
z
a
a 100


0


BONE 316SS INTRAMEDULLARY
(inside by outside DEVICES diameters in mm
diameters in mm) 8 ( '


() BONE
a SAMPSON
400
o () KUNTSCHER
SCHNEIDER
S(0 UNSEN-STR EET
APPROPRIATE
SIE \ ROD SIZE
S6 due to FLUTES


Q
.- /

20 200



14X26
16X 25 2
S 4 -X-26 12
100 12X 24
11X23
tOX 22
1 2 -10 .13

I1\ V2 12. I .^ 1*II
9 I7 5 o .O
0 4~ 1




























a
0
C)
25


u 20


S15



10
"o
0
2


Cr Co Mo Alloy

cold
worked
annealed
Stainless n a
Steel


Ti6A14V


Bone


Figure 2.4. Elastic modulus for several implant materials










material have little use in evaluating the performance of the

devices themselves. Figure 2.4 gives the average values for the

moduli of elasticity of several alloys used in implants. These

values have been shown as ranges due to the effects of fabrication

techniques on the mechanical properties of the nails.


Summary

The review of the literature on the mechanical and geometric

properties of the components of an intramedullary osteosynthesis

of the femur, as presented above, has led to several observations.



1. While many investigations have been carried out in an
attempt to characterize the mechanical properties of
the femur, the large variations in results indicate
that careful consideration must be given to the use
of these results in the modeling of an osteosynthesis.
Indiscriminate use can lead to significant differences
in the results obtained.

2. Determination of the geometric properties of the femur
has received little attention. The information reviewed
above concerns itself with the rigidity and ultimate
properties of the nail and the bone. While the flexural
rigidity is related to the modulus of elasticity and
the area moment of inertia, the nonuniform geometry and
difficulty in specifying the modulus of elasticity of
the bone does not allow for the direct calculation of
the moment of inertia or other geometric properties.

3. The material properties of intramedullary nails have
been found to be adequate. Here again, however, care
must be exercised in the use of these properties so
that the effects of machining and other processing are
taken into account (Utah Biomedical Test Laboratory,
1975).

4. The load-deflection behavior of several intramedullary
nails has been carried-out, and the geometric properties
necessary for modeling their use in an osteosynthesis
can be derived from this information.







22

Before a general modeling of the osteosynthesis can be carried

out, quantitative characterization of the geometric properties of the

femur must be performed. The first portion of the experimental work

presented here concerns itself with this characterization.
















CHAPTER III

DETERMINATION OF THE GEOMETRIC PROPERTIES
OF THE SYSTEM COMPONENTS


The Femur


Materials and Methods

Ten pairs of embalmed human femurs were obtained and cleaned

of all soft tissue and periosteum. The femurs were then allowed

to air dry for one day. Information regarding age, sex, height,

and weight of the subjects was not available. Two reference

lengths were measured as shown in Figure 3.1. The first, L r
maj or'
was defined as the overall length of the femur as measured from

the most proximal portion of the femoral head to the most distal

portion of the femoral condyles. The second length, Llesser was

defined as the distance from the midpoint of the lesser trochanter

to the most distal portion of the femoral condyles. With the femur

resting on the posterior aspect of the femoral condyles and greater

trochanter, a small hole, approximately 3 mm diameter, running

approximately anterior to posterior, was drilled in the femur at

the level of the midpoint of the lesser trochanter. A 3 mm dowel

was inserted to act as a reference for position measurements along

the femur. The bone was placed in a plywood mold, with the dowel

marker in a vertical orientation, and plaster poured over it to

make a rectangular slab. After the plaster had set, 20 mm thick



























































Figure 3.1.


Schematic representation of the femoral
length measurements: Llesser, the lesser
trochanter to femoral condyle length, and
Major, the femoral head to condyle length









sections were marked off using the lesser trochanter marker as the

starting point. Approximately 16 serial sections were obtained

from each bone. These represented positions (measured proximal to

distal) from approximately 20% of the total femoral length, the

level of the lesser trochanter, to 80%, the level of the distal

metaphysis.

After sectioning and numbering, each section was carefully

cleaned of bone marrow and a size reference marked directly on the

surface of the plaster blocks (Figure 3.2). The size reference

marks were made so that the magnification and the location,

relative to a fixed axis, of each cross-sectional image could

be maintained. Each block was projected on a large sheet of paper

using an opaque projector. A tracing of the cortical bone of each

cross-section and its reference line was produced. These tracings

were used to generate a set of coordinate points on computer punch

cards, representing the outside and inside boundaries of each

section, using a digitizing machine. The punched data deck of the

boundary coordinate points became the input information for the

SCADS computer program (Piotrowski and Kellman, 1973; Piotrowski

and Wilcox, 1971). The SCADS program was used to calculate the

geometrical properties of the cross-section as defined below.

Cortical bone area. The computer program determines the

area of cortical bone in each cross-section by integrating all



Department of Radiation Therapy, J. Hillis Miller Health
Center, Gainesville, Florida





















2 6 10 14



3 7 11 15
99 '

4 12 16


I _
S5 9 13 17







Figure 3.2. Serial cross-sections of a representative
femur with size reference marks in place









the differential area elements such that


AORT = dA = AA (3.1)
(CORT3
fA i


Area moments of inertia. The area moments of inertia, or

second moments of the area, are generally used in the calculation

of bending stresses and deflections and are a measure of the

distribution of the load-carrying material about a set of orthogonal

axes. For this work the orthogonal reference axes were chosen to

be the medio-lateral (x-axis), and the antero-posterior (y-axis)

planes. The moments of inertia are defined, referring to Figure

3.3, as


I = x2 dA = x dA. (3.2)
xx A i I


I = y2 dA = E y? dA. (3.3)
yy i 1 1


I = xy dA = xiy. dA. (3.4)
xy J 1 1
i


Principal angle and moments of inertia. Another set of

orthogonal axes, n and C, whose origin is at the centroid of the

cross-section can be determined such that


I = n dA = 0 (3.5)
JA

These axes are known as the principal axes, and correspond with

the axes of symmetry in cross-sections which are symmetrical. For

nonsymmetrical cross-sections these axes represent the least and















ANTERIOR


LATERAL


POSTERIOR


Figure 3.3.


Definition of axes and distances for
geometrical property determination


MEDIAL









most rigid directions of the cross-section. The principal moments

are then defined with reference to the principal axes or directions

as


n = n2 dA (3.6)


and


I = Cf 2 dA (3.7)

These quantities can be obtained from a Mohr's circle construction

(Beer and Johnston, 1962) as shown in Figure 3.4. The principal

angle, 0, between the x-axis and the n-axis is calculated as


0 = arctan ( xy ) (3.8)
yy xx

The principal area moments of inertia are also readily computed

from I I and I as
xx yy xy

I +1 I -I
I xx yy + 12 + xx yy (39)
nn 2 xy 2 J

and
I +1 / I I 2
I x yy 2 xx y (3.10)
2 2 xy 2


Polar moment of inertia. Used in determination of torsional

shear stresses and deflections of round cross-sections, the polar

moment of inertia, or polar second moment of area, is defined as


J = fr2 dA (3.11)
-/A




























xx,-Ixy


lIlnn l ii


Iyy,l


Figure 3.4.


Mohr's circle construction for locating
principal moment magnitudes and directions
of the cross-sections










where r is.measured from the centroid of the cross-section. The

polar moment may be calculated using the previously determined

values of the moments of inertia, I and I Using trigonometry
xx yy
and referring to Figure 3.3


r2 = x2 + y2 (3.12)


Therefore, equation 3.11 can be rewritten as


J = f (x2 + y2)dA = x2 dA + f y2 dA (3.13)


and substituting equations 3.2 and 3.3 yields


J = I + I (3.14)
xx yy

The effective polar moment of inertia. The last property to

be considered is the "effective" polar moment of inertia, Jeff. As
eff
den Hartog (1952) has shown in his discussion of the torsion of

machine elements, a torsional stiffness, C, may be defined as the

torque, T, per unit twist, 0. That is


C T (3.15)


Further analysis of elements of circular cross-section leads to

the result that


C = G Jeff (3.16)


where G is the shear modulus of the material and Jeff is the
eff
effective polar moment of inertia. For circular cross-sections

Jeff reduces to J. The noncircular cross-section requires a more










complex derivation to obtain the geometrical determination of the

torsional stiffness. Prandtl's "membrane analogy," as discussed

by den Hartog (1952) and Crandall (1956) allows for the determination

of an effective polar moment of inertia; thus the torsional stiffness

of noncircular cross-sections may still be expressed as a product of

a material and geometric property. Thus, for noncircular sections,


T
S= C = G J (3.17)
0 eff

The effective polar moment of inertia is still a measure of the

area's distribution. However, in this case its value cannot be

obtained by the simple integration of differential area elements

as shown for the polar moment of inertia since it also depends on

the convolutions of the sections periphery. The development of the

equations for determination of Jeff are beyond the scope of this

work,however, and the interested reader is referred to the above

references for further discussion. The SCADS computer program

contains the algorithm for mathematically modeling the "membrane

analogy," and thus calculating Jeff and these results have been

included for comparison with the polar moment of inertia, J, as

defined in the previous section.


Results of the Geometric Properties of the Femur

Raw data for the gross cortical area (A), area moments of

inertia (Ixx and I ), principal moments of inertia (I and I ),

principal angle (0), polar moments of inertia (J), and effective

moment of inertia (Jeff), were extracted from the SCADS computer









runs for the ten femur pairs. The left-to-right variation of these

data, except for the principal angles to be discussed later, were

first evaluated at sequential points along the bone length (from

20% to 80% of the total bone length in 10% increments). As dis-

cussed by Miller and Piotrowski (1974), the side-to-side difference,

d of any property is defined as

P P
d = (3.18)
p PL + PR


where PL and PR are the values of the property for the left and

right femurs respectively. The differences computed at each

incremental position were averaged to obtain the mean value of the

side-to-side difference, dp, and its standard deviation, S.D.,

for each pair of femurs. Table 3.1 summarizes these results. The

accumulated values, obtained by pooling the results from all ten

femur pairs, are also shown. The standard deviations have been

corrected for small sample sizes using the Bessel's Correction

(Moroney, 1951).

Figures 3.5-3.18 have been constructed to indicate the

overall values obtained for the geometrical properties of the right

femurs and the left femurs. The shaded band indicates the total

range over which the raw data falls. A line has been drawn to

indicate the calculated mean value of the property from 20% to

80% of the total bone length. Also included are vertical bars

which represent a range of plus and minus one standard deviation

from the mean.









Table 3.1. Values for the mean side-to-side difference, d ,
and standard deviations (in parentheses) for
the paired femurs

Bone dA d dl d d d
Bone # dA xx Vy Tn dJ Jeff

1 .005 -.079 .006 .014 -.005 .014 .003
(.022) (.058) (.033) (.034) (.034) (.032) (.041)

2 -.018 .035 -.059 -.007 -.034 -.016 -.0134
(.048) (.129) (.046) (.059) (.057) (.053) (.055)

3 -.006 -.036 -.038 -.004 -.002 -.018 -.009
(.019) (.155) (.035) (.033) (.015) (.023) (.0191)

4' .005 .054 -.004 -.010 .061 .018 .032
(.020) (.037) (.036) (.030) (.026) (.021) (.021)

5 .003 .043 -.024 .032 .010 .025 .026
(.030) (.042) (.043) (.037) (.060) (.032) (.041)

6 -.001 -.064 -.030 -.055 -.045 -.046 -.032
(.037) (.032) (.034) (.031). (.036) (.026) (.049)

7 .056 .024 .041 .020 .024 .034 .058
(.024) (.081) (.067) (.052) (.099) (.048) (.034)

8 -.002 .019 -.020 .012 -.024 .006 .002
(.016) (.043) (.055) (.040) (.057) (.032) (.043)

9 -.007 .002 .025 -.005 .018 .005 .006
(.020) (.080) (.049) (.074) (.063) (.064) (.052)

10 -.022 .012 .006 .010 .011 .014 .011
(.037) (.054) (.028) (.043) (.028) (.045) (.053)

Pooled
mean .001 .001 -.010 .001 .002 .004 .0035
Data
S.D. (.030) (.085) (.046) (.024) (.057) (.040) (.047)

t .11 .04 .69 .13 .11 .32 .24

P >.9 >.9 >.5 >.9 >.9 >.7 >.8


n = 10 for all cases











































20 30 40 50 60 70 80
PERCENTAGE OF TOTAL LENGTH


Figure 3.5.


Gross cortical bone area as a function
of position for the left femurs


0









6 -


o


PERCENTAGE OF TOTAL LENGTH


Figure 3.6.


Gross cortical bone area as a function
of position for the right femurs


t I I IL








































oT-
0


PERCENTAGE OF TOTAL LENGTH


Figure 3.7.


Area moment of inertia, Ixx, as a function
of position for the left femurs


I I r IL














51


oI

o0


PERCENTAGE OF TOTAL LENGTH


Figure 3.8.


Area moment of inertia, Ixx, as a function
of position for the right femurs










6r


5


. E







LU
U-
U-
LU
-







2







01-O I I I i I ,
0 20 30 40 50 60 70 80

PERCENTAGE OF TOTAL LENGTH



Figure 3.9. Area moment of inertia, Iyy, as a function of
position for the left femurs











































I-- I


40 50

PERCENTAGE OF TOTAL LENGTH


Figure 3.10.


Area moment of inertia, lyy, as a function of
position for the right femurs


Io-
0


ir










































.-4
0


PERCENTAGE OF TOTAL LENGTH


Figure 3.11.


Principal moment of inertia, ln, as a function
of position for the left femurs


__ ^1 __ __











































0 30 n
0 20 30 an n~ a


PERCENTAGE OF TOTAL LENGTH


Figure 3.12.


Principal moment of inertia, I as a function
of position for the right femurs



















4







V) 3


LLu

Ur-
-J 2







1-







0 II I
0 20 30 40 50 60 70 80

PERCENTAGE OF TOTAL LENGTH

Figure 3.13. Principal moment of inertia, IE as a function
of position for the left femurs
























































0 20 30 40 50 60 70

PERCENTAGE OF TOTAL LENGTH


Figure 3.14.


Principal moment of inertia, IEE, as a function
of position for the right femurs


.



CA
cr,

w
LJ

I-

0
CU



wu
*JJ


































E C I I I I T I
o 20 30 40 50 60 70 8
PERCENTAGE OF TOTAL LENGTH


Figure 3.15.


Polar moment of inertia, J, as a function
of position for the left femurs


111111














`K>


Ij


0 20 30 40 50 60 70 8
PERCENTAGE OF TOTAL LENGTH


Figure 3.16. Polar moment of inertia, J, as a function
of position for the right femurs























.1. 1111111 jIl


0 ,I I I 6 7
0 20 30 40 50 60 70
PERCENTAGE OF TOTAL LENGTH


Figure 3.17.


Effective polar moment of inertia, Jeff, as a
function of position for the left femurs















8








LJ -6



4-





2





0 I I
0 20 30 40 50 60 70 80
PERCENTAGE OF TOTAL LENGTH


Figure 3.18. Effective polar moment of inertia, Jeff, as a co
function of position for the right femurs


10r










The values of the principal angles are more difficult to

present. These values appear to have fallen into three main

groupings. Representatives of these three groups are shown

graphically in Figures 3.19-3.21. Here, the values of principal

moments along the length of a pair of femurs are shown as scaled

orthogonal lengths in their actual principal angle orientation.

In the first two cases, there is a gradual angular displacement

to the principal axes, going proximal to distal. The principal

axes were not well defined in case III.


Discussion of the Geometric Properties

Right-left variability. Using the pooled means which were

calculated from the data presented in Table 3.1, the Student's

t Test (Moroney, 1951) was applied to compare the resulting dis-

tributions with the hypothesis that there is no right-left bias

in the geometric properties, i.e. d = 0. In this particular case,
p
the value of t is indicative of the ratio of the mean to the

standard deviation. The nonzero mean values seen in Table 3.1

are quite likely to occur by chance since they are smaller than

the data's inherent scatter, represented by the standard deviations.

The actual statistical results have been included in the table and

indicate that P, the probability of t being this large or larger

by chance, is greater than 0.5 in all cases. One can thus conclude,

with considerable certainty, that there was no right-left bias in

the geometric properties of the femurs investigated.





































Figure 3.19.


Principal angle orientation and relative magnitude
of the principal moments as a function of position
for a representative pair of femurs from group I
(the darker line indicates the larger moment
of inertia, I )








DISTAL


Left Right


7LF9


PROXIMAL




































Figure 3.20. Principal angle orientation and relative magnitude
of the principal moments as a function of position
for a representative pair of femurs from group II
(the darker line indicates the larger moment of
inertia, I )








DISTAL


Left


Right


3LF9


PROXIMAL




































Figure 3.21. Principal angle orientation and relative magnitude
of the principal moments as a function of position
for a representative pair of femurs from group III
(the darker line indicates the larger moment of
inertia, I )
rnn






DISTAL


+e


Left


Right


PROXIMAL


8LF9










Miller and Piotrowski (1974) showed that one can expect the

standard deviation of the side-to-side difference in the torsional

strength of paired animal bones to be about 9%. The results obtained

above for the standard deviation of the side-to-side difference in

the polar moment of inertia was of the order of 4%. This indicates

that, while the geometric variability has some effect on the observed

strength variations of whole bones, it does not account for all of

it. The strength of whole bones, however, depends on both geometric

and material properties. Reilly (1974) reported that one can expect

variability in ultimate torsional strength of bony material to be

approximately 4%. Thus, it would appear that the combination of

geometric and material variability adequately describes the scatter

found in the strength of whole bones.

Gross Cortical Area. The flared outward shape of the femur

might imply a similar cortical bone area distribution. This is not

the case, however, due to the thinning of the cortex which occurs

in the flared metaphysial regions. The cross-sectional area of

bone as shown graphically in Figures 3.5 and 3.6, tends to increase

only slightly from the level of the lesser trochanter to approxi-

mately 40% of the bone length. This is where the isthmus, or

narrowest section of the intramedullary canal, is found. The cortical

area then decreases monotonically to the knee. At the most distal

data point taken, 80% from proximal, the area is approximately

2.5 cm2. This resulting area, just proximal to the knee, is slightly

larger than the proximal tibia area, just distal to the knee, pre-

sented by Minns et al. (1975) and Piziali et al. (1976). Thus,










ignoring the discontinuity at the knee, the lower limb appears to

be a tapered column of bone mass. This shape reduces the amount

of bone appearing distally, and results in a minimization of the

bones' contribution to the mass moment of inertia of the limb about

the hip. Thus, compared to a limb with bone mass evenly distributed

along its length, the torques necessary to accelerate the extremity

during locomotion are reduced, with the result that the energy

necessary to accelerate the limb is also decreased and higher

efficiency can be maintained.

The slight increase in area seen in the proximal portion of

the femur is due to the presence of the gluteal tuberosity and the

beginning of the linea aspera. These areas allow for the attachment

of the muscle groups which, while providing for rotations about the

hip, also apply significant axial loading in the region due to their

poor mechanical advantage. These axial loads tend to reduce the

tensile stresses set up by bending of the femur due to the loading

of the femoral head, but also requires the need for additional bone

area to maintain suitable compressive stress levels.

Area moments of inertia. It is important to note at the

outset of this discussion of the area moments of inertia (Figures

3.7-3.10) and polar moment of inertia (Figures 3.15 and 3.16) that

there is a relationship between the two, as shown in equation (3.14).

These properties influence the stress levels caused by different

forms of loading, namely bending and torsion. These properties

also influence the rigidity of the system under these different

loading configurations.










The bending stresses, o, developed in the outer surface of

a beam with an applied bending moment, M, may be represented by


Mc
a = M (3.19)


where c is the distance from the centroid of the section to the

outer surface, and I is the area moment of inertia. The quantity

I/c is often referred to as the section modulus, Z, of the beam.

Figure 3.22 shows the results obtained from the calculation of the

section modulus for two representative femurs. Also included is

the result obtained by Koch (1917) in his detailed analysis of a

single femur. While the magnitudes are quite similar, the results

obtained here indicate that the section modulus decreases slightly

from proximal to distal. Koch (1917) and Rybicki et al. (1972)

also note that when a downward load is applied to the femoral head,

the bending moment in the x-plane increases rapidly from the femoral

head to the level of the lesser trochanter and then decreases

monotonically to the knee. Thus, the bending stresses, equal to

M/Z, decrease from the lesser trochanter to the knee, but more

gradually than the bending moment itself. This corresponds well

with the bending stress distribution reported by Rybicki in his

analytical modeling of the femur.

The increase in the area moment of inertia I in the
xx'
proximal femur, also results in an increase in the flexural rigidity

of the bone in this region. This is due to the direct proportionality

between the rigidity and the moment of inertia. The femoral condyles,

which are the bearing surfaces of the knee joint, allow muscular


































2 *



**.. data from Koch (1917)


1- Femur 3L


-- Femur 3R


Ia


SPROXIMAL


DISTAL


Figure 3.22. Section modulus, Z, as a function of length
along the femur for two sample femurs compared
to data derived from Koch (1917)


E



I I
NI










insertions, and provide areas for attachment of the ligaments which

stabilize the knee joint, are responsible for the increase in I
xx
in the last 15%-20% of length. This results in an increase in the

flexural rigidity of the femur in this region as well.

The antero-posterior moment of inertia, I (Figures 3.9

and 3.10) is smaller in magnitude than I in the proximal femur.
xx
It also decreases in magnitude over the proximal 40% of'length,

and then remains almost constant until the femoral condyles are

reached. One reason for the initially larger relative values of

I is the large bending loads which appear in this plane due to
Yy
the femoral head loading, as well as forces generated by the large

gluteus maximus muscle used in extending the hip.

Principal moments and directions. The principal moments of

inertia, In and I were calculated and their principal angle,

relative to the medio-lateral direction, was determined. The

magnitudes of these principal moments show a curved profile

(Figures 3.11-3.14) similar to that for the values of I
xx
However, the orientation of the principal axes was quite remarkable.

Study of the principal axes of the ten femur pairs led to the obser-

vation that the femoral cross-sectional shape at the midshaft

appears to fall into three basic categories. A midshaft section

from the left femur of representatives of each of these groups has

been included in Figures 3.19-3.21 along with scaled orthogonal

lines representing the relative magnitudes of the principal moments.

(The heavier of the two lines indicates the larger value, I .)

Of the ten femur pairs examined, four pairs were found to be










in the first group, four pairs in the second group and two pairs

in the third group.

The first group, shown in Figure 3.19,has an elliptical

cross-sectional shape whose major axis, I runs antero-posterior

at the midshaft. There is definite right-left symmetry as seen in

this figure. At the proximal end of the femur, the major axis is

inclined at an angle which provides greatest rigidity in the

approximate direction of the femoral neck and head. As the midshaft

is approached, the linea aspera dominates the cross-section, result-

ing in a principal angle of approximately 90 (the antero-posterior

direction). In the distal portion, the major axis rotates and

comes to an almost medio-lateral orientation corresponding well

with the femoral condyles.

The second group, Figure 3.20, has an elliptical cross-section

at the midshaft as well, but is perhaps less pronounced than in the

previous group. However, the I axis begins in a more medio-lateral

position and remains as such throughout the length of the femur.

Thus the more rigid direction, corresponding to the I direction,

is now 90 from that of the first group. Here again, the right-left

symmetry is maintained.

The third group, Figure 3.21, has a profile in'the proximal

segment which is similar in magnitude and direction to the second

type described above. However, as the midshaft is approached, the

magnitudes of I and I are almost equal, in contrast to groups

I and II, and determination of the principal direction is uncertain

due to the relatively small size of the Mohr's circle in comparison










to the magnitudes of I and I Thus, the bone has no preferen-
xx yy
tially rigid direction until the femoral condyles are reached. The

only generalization which can be made here is that in all groups

the principal direction is medio-lateral at the femoral condyles.

An attempt to explain the above observations led to the

consultation of the anthropological literature. Comas (1960) makes

several observations relative to the point of femoral shape. The

linea aspera, in the femur of people of the more modern civilizations,

is more pronounced in males than females. The more robust skeleton

of the male has broad areas for muscular attachment and the linea

aspera may develop in such a way as to produce a large, rough, crest

shaped protrusion not unlike that seen in group I (Figure 3.19).

Antero-posterior flattening, as represented by the second group

presented above, is observed in the female skeleton.

This observation was reiterated by Maples (1977) who indicated

that the anthropological and forensic medicine literature suggests

that a pronounced linea aspera is indicative of the male and a

flattened diaphyseal shape of the femur may be indicative of the

female. Maples cautioned, however, that generalization is very

tenuous in this regard.

From a biomechanics point of view, the large linea aspera of

the male increases the rigidity of the system in the plane needed

to resist the large bending loads of the flexors and extensors

of the thigh and knee so that moderate stress levels are maintained.

One might also conjecture that the wider pelvis of the female

results in larger bending loads in the medio-lateral plane and a









principal axis oriented in that plane. Unfortunately, the sex of

the femoral samples studied here was unknown, and substantiation of

this hypothesis must remain for further study.

Polar moment of inertia. The polar moment of inertia, J, is

distributed in a manner similar to I (Figures 3.15 and 3.16), due
XX
to its dependence on I and I as discussed in the previous
xx yy
section. In the proximal femur, the relative increase in this

property appears to be due to the several internal and external

rotators of the hip which attach in this area and apply torsional

loading to the bone.

In the midshaft region, the values for J do not vary con-

siderably due to the uniform torsional loading which is distributed

along the length of the bone. As was discussed in the previous

section on area moments of inertia, the distal femur must also

maintain a fairly large area for support of the cartilagenous

bearing surface of the knee. It accomplishes this by using a

preponderance of cancellus bone which aids in distribution of the

joint loading force. In order to maintain the torsional strength

of the system, however, a thin layer of cortical bone is distributed

at the periphery of this area and the polar moment of inertia is

maintained with efficient use of material. This larger polar

moment of inertia also aids in reducing the loads in the collateral

ligaments necessary to maintain joint stability.

Effective polar moment of inertia. The effective polar

moment of inertia calculated for the noncircular bone cross-sections,

using the SCADS computer program, are shown in Figures 3.17 and 3.18.









While the profile of the distribution is similar to that found for

the polar moment of inertia given in the previous section, the

magnitudes of Jeff are approximately 10%-15% less. The greatest

differences appear in the distal portion of the bone. Thus, the

nonsymmetry of the bone cross-sections result in a decrease in the

actual torsional rigidity of the bone.


Discussion of the Observed Scatter in Properties

The magnitude of the observed scatter in the geometric cal-

culations, characterized by the standard deviations, ranged up to

20%-25% of the mean values. The somewhat larger standard deviations

present at the ends of the bones are due to the difficulty in

discerning the cortico-cancellus bone interface of the intra-

medullary canal. In light of the discussion of the inherent

differences in individuals presented above, this large range does

not seem unreasonable. The maximum error of the SCADS computer

program, for calculations on well defined geometries, has been

shown to be 5% or less (Piotrowski and Kellman, 1973).

In an attempt to quantify the remaining 20% of variation,

nondimensionalization of the observed properties using other

dimensions of the bones was carried out. It was hoped that a

general characteristic shape of each of the property distributions

along the bone length could be generated in this way. Using the

overall length, Lmajor, and the lesser trochanter to femoral condyle

length, Llesser, (Figure 3.1) as nondimensionalization parameters,

did not, however, reduce the magnitudes of the scatter. In fact,










as can be seen in Figures 3.23 and 3.24, there is no correlation

between these lengths and the average calculated bone area. There

is also no correlation between the average area and the midshaft

outer.diameter (Figure 3.25). One must conclude, therefore, that

there are significant differences in the bone properties of dif-

ferent individuals and that these differences are dependent on many

variables. This is consistent with the thought (Comas, 1960) that

it is build and/or body weight, sex, age and general physical con-

dition and not height alone, which determines the size and shape of

an individuals bones. Unfortunately, this information was not

available for consideration in this study.


Conclusion

Even with the substantial scatter present in the geometric

properties discussed in this chapter, the general property dis-

tribution profiles are valid. Several conclusions can be made

from the results presented.

1. No statistically significant right-left bias is evident
in the geometric properties presented.

2. The geometric properties calculated using the SCADS
computer program adjoins the tibial information pre-
sented by Minns et al. (1975) and Piziali et al. (1976).

3. The area distribution profiles do not mirror the out-
ward appearance of the femur. In fact, the entire
lower limb has a "tapered column" distribution of
bone area, with the greater area appearing in the
proximal femur. This distribution of bone aids in
efficient ambulation.

4. The flared shape of the femur is more pronounced in the
medio-lateral plane than in the antero-posterior plane,
and I and I the respective area moments of inertia,
mirroxthis ard appearance.
mirror this onward appearance.













0 AR vs. Lmajor r=0.21


* AL vs. Lmajor r=0.41



0 0


52.0


50.0


48.0


46.0


44.0


42.0


40.0


0


* 0
0o


L-^----- I ----- 1 --- I ------
2.5 3.0 3.5 4.0 4.5 5.

AVERAGE AREAS (cm2)


Figure 3.23.


Graphical representation of the overall femoral length versus
average cortical bone area with the respective correlation
coefficients, r, for the right and left femurs


0
0

















o AR vs. Lesser r=0.31


* AL vs. Llesser- r=0.50

0 O


44.0


42.0


40.0


38.0


36.0


AVERAGE


AREAS (cm2)


Figure 3.24.


Graphical representation of the lesser trochanter to femoral
condyle length, Llesser, versus average cortical bone area
with respective correlation coefficients, r, for the right
and left femurs


00
0
*


34


I I 1
r


-Ot
















0 AR vs. dR r=0.38


* AL vs. dL- r=0.42


S00


0
0


S0


AVERAGE AREAS (cm2)


Figure 3.25.


Graphical representation of the outer midshaft diameter
versus average cortical bone area with the respective
correlation coefficients, r, for the right and left femurs


3.2


3.0


2.8


2.6


I I 1


4










5. The principal directions were found to fall into three
major groups. There are those femurs which are most
rigid in the antero-posterior plane; those which are
most rigid in the medio-lateral plane; and those which
appear to have no preferential orientation. Anthro-
pological information available suggests that these
differences are sex-linked.

6. The calculated effective polar moment of inertia
using the "membrane analogy" indicates that the
noncircularity of the bone cross-section results
in torsional rigidities which are 10%-15% less than
is predicted by the symmetrical polar moment 'of
inertia calculations.

7. The geometric property profiles are distributed along
the bone length in a highly efficient use of the material
so that "strength to weight" optimization appears to be
maintained under complex loading configurations.

The results which have been obtained in this geometric study

of the femur are consistent with the small amount of information

available in the literature and the data presented in Figures

3.5-3.22 can be used in the subsequent structural modeling of this

bone.


The Intramedullary Nail


Materials and Methods

Determination of the geometric properties of five intra-

medullary nails was carried out. These included four nails, of

differing types (Figure 2.2), intended for implantation in a 13 mm

reamed intramedullary canal (13 mm Schneider, 13 mm Kuntscher,

13 mm x 11 mm Hansen-Street, and 12 mm Fluted). A 14 mm Fluted

nail, intended for subsequent use in an in vitro osteosynthesis


model, was also processed.










A transverse section, approximately 2 mm thick, was cut

from the midportion of commercially supplied intramedullary nails

of the above nominal sizes. The sections were then ground and

carefully deburred. In a manner similar to that described in the

previous section for femoral geometry determinations, these sections

were projected and an enlarged tracing of their cross-sectional

outline made. Coordinate points of the boundaries were produced

using the digitizing unit and were input to the SCADS computer

program for geometric property determination. The 14 mm Fluted

nail was processed with and without its flute-like protrusions to

appraise the significance of these gripping points on the overall

geometric properties of the device.


Results

Figure 3.26 shows the results obtained for the geometric

property determination of the four nail types considered. These

results include the gross area, area moments of inertia, polar

moment of inertia, and principal moments of inertia as defined at

the beginning of this chapter. Also included in the figure are the

axes orientations used for geometric property determination. The

principal angles were.zero in all cases due to the orientations

used, i.e. the nails were oriented in their principal directions

during processing. Table 3.2 shows the results obtained from

the geometric processing of the 14 mm Fluted intramedullary nail.










xx lyy Ill 122


L -e- -e-


0.251-


0.20)


0.15-


0.10-


4-e


0.031-


LEGEND

-- FLUTED (12mm)

- HANSEN-STREET (13mm)


-- SCHNEIDER

-9- KUNTSCHER
ORIENTATION
y

S ,X Fluted


-4


0.051


5~~x


(13mm)

(13 mm)


Hansen-Street


Schneider


Kuntscher


Figure 3.26. Calculated geometrical properties for several intramedullary nail shapes
(all nails are intended for use in a nominal 13 mm reamed medullary canal)


1.11


0.09F


0.95


0.7F


0.071


0.5-


0.05


0.3-


I


- .. _










Table 3.2. Geometric properties of
with and without flutes


the 14 mm Fluted type nail,


14 mm Nail


With Without
Flutes Flutes

Area (.cm2) 1.12 1.00

Ix (cm4) 0.19 0.16
xx
I (cm4) 0.19 0.16
yy
J (cm4) 0.38 0.32

Ill (cm4) 0.19 0.16

122 (cm4) 0.19 0.16

0 0.0 0.0









Discussion

The results obtained using the SCADS computer program are

approximate due to the difficulties inherent in processing the

intricate geometry of these cross-sections. Independent cal-

culations indicate that the results obtained are within 5% of the

exact values. The Fluted nail has the largest values for all the

geometric properties determined. This has a pronounced effect on

its relative bending and torsional rigidity. Its very high

cross-sectional symmetry also makes it insensitive to the orienta-

tion of the applied loading. The Schneider nail also has a high

degree of symmetry, while the Hansen-Street and Kuntscher nails

are sensitive to orientation due to large differences observed in

their orthogonal geometric properties.

The results obtained from the processing of the 14 mm

Fluted nail were quite interesting. The flute-like projections

of this nail, primarily intended to maintain torsional stability

of the bone fragments, have a significant effect on the area and

polar moment of inertia. These properties increase almost 20%

with the addition of the flutes to the cross-section while

increasing the area of the section by only 12%.

It is important to note that only one sample of each nail

type was processed. Measurement of the outer dimensions of a few

other samples of similar nails from different manufacturers indicate

that there is considerable difference in the cross-sections of nails

made by different companies. Changes in the wall thickness of a







74.

Kuntscher nail, for example, without altering the outside dimensions,

causes significant changes in the flexural rigidity of the device.

The preliminary results reported here are sufficient to allow for

verification of the osteosynthesis modeling to be carried out at

this time. However, a more detailed study is indicated so that

the effect of implant variability on geometric properties of the

devices can be accurately determined.
















CHAPTER IV

MODELING OF THE OSTEOSYNTHESIS


The Preliminary Idealized Model

To gain insight into the parameters affecting the deflections

occurring in an osteosynthesis under bending loading, an elementary

model was analyzed using engineering beam theory. This model, as

shown in Figure 4.1, consisted of an outer tube with a transverse

cut, representing the fractured bone, and a central rod, representing

the intramedullary nail, passing the full length of the system.

Four-point bending loading was assumed to be applied to the outer

tube with equal spacing between the loading points. Clearances

were assumed to be such that the rod and tube interacted only at

idealized point contacts, as shown in Figure 4.1, and that no

deformations occurred at these points. The space between the

fracture surfaces was assumed to be just wide enough so that these

surfaces did not contact during deflection. The moduli of elasticity

of the outer tube and inner rod, Eb and En, were assumed to be con-

stant throughout, and the components were restricted to linearly

elastic behavior. The cross-sectional geometry of these components

was also assumed to be constant along the system length. Thus, the

moments of inertia of the tube and rod, Ib and In, were unvarying

along the length of the model.







































Figure 4.1. Schematic representation of the preliminary idealized
osteosynthesis model with equally spaced four-point
external loading and point contact reactions between
the nail and bone fragments









Free Body Diagram of the Outer Tube

Due to the symmetry of this system, a free body diagram for

half of the outer tube is constructed as shown in Figure 4.2. With

an applied load P at each of the external loading points, and rod

reactions R1 and R2 as shown, vertical force equilibrium yields


P + R1 + R2 = 0 (4.1)


and


R1 = R2 = R (4.2)


Static moment equilibrium about the left end of the tube yields


PL RL
3 =- 0 (4.3)
3 2

so that

2
R = R =R P (4.4)


Free Body Diagram of the Inner Rod

As shown in Figure 4.3, a free body diagram of the inner

rod can be constructed with reaction loads as shown. Note the

idealized point contacts of the nail-bone interaction.


Deflection Equations

The flexural rigidities of the inner rod (N) and the outer

tube (B) are defined as


N = E I (4.5)
n n


B = Eb Ib


(4.6)












P
L/3 L/6





R1 R





P


Figure 4.2.


Figure 4.3.


Free body diagram of half of the
outer tube with external and point
contact reaction loads


Free body diagram of the inner rod
with idealized point contact reactions


IT6P


'12 % PIIY '12


/3









where En and Eb are the modulus of elasticity of the inner rod and

outer tube respectively, and In and Ib are the area moments of

inertia for the inner rod and outer tube respectively.

The reaction loads on the rod are applied to each end and the

midspan of the component (Figure 4.3). The midspan deflection, z,

for the three point loaded span in this configuration can be written

(Roark, 1954) as
pL3
z= (4.7)
36N

where P is the load applied to the outer tube, of length L, at its

four loading points.

Similarly, from the outer tube free body diagram (Figure 4.2),

an angular deflection of the fracture surface, relative to its

original centerline, can be defined as


PL2
0 PL2 (4.8)
b 64.8B

The outer tube has been constrained to contact the inner rod

only at its idealized contact points, and the centerline of the

inner rod is now inclined at an angle 0 which may be approximated,

for small deflections, as


S z (4.9)
n (L/2)

The fracture interfacial deflection angle, Ot, relative to the

vertical axis, can be obtained, as shown in Figure 4.4, by super-

imposing equations (4.8) and (4.9) as defined above. Due to the

symmetry of the system, the total angular mismatch of the system,













































Figure 4.4. Idealized angular mismatch representation
with the overall angulation, Ot, equal to
the difference in nail angulation, 0 ,
and bone angulation, 0b








T0 is twice 0t and if


pL2 pL2
t n b 18N 64.8B10)

then

PL2 B
T 2 x 2 (3.6 1) (4.11)
TP@L/3 64.8B N


The values obtained for the angular deflection of the fracture

interface are dependent on the applied load configuration. For

example, if the downward applied loads are assumed to be located

at positions L/4 in from each end of the system, instead of at L/3

as discussed above, the resulting angular deflection becomes

PL2 B
2 x 128B (2.66 1) (4.12)
P@L/4


While the form of the equations remain the same, their

coefficients take on new values. Nonetheless, several observations

can be made. The application of bending loads to an idealized

system such as the one shown indicates that an optimum flexural

rigidity ratio exists (N/B = 3.6 or 2.66 depending on the configura-

tion) which results in no angular mismatch of the fracture surfaces.

This implies that in an actual osteosynthesis, where the outer tube

represents the bone, and the rod represents the intramedullary nail,

some optimum property ratio could result in a minimum fracture

surface angulation.

While the many assumptions used to derive the results obtained

above were necessary to allow for the application of elementary








engineering beam equations, these assumptions are not necessarily

consistent with the configuration of an actual intramedullary

osteosynthesis as discussed in Chapter I. The geometry of the

bone, as discussed in Chapter III, is not uniform along its length.

The intimate contact between the nail and the bone, needed to

maintain torsional rigidity of the fragments, does not allow for

large enough clearances to assure only point contacts. 'Large

clearances would also allow for unacceptable relative transverse

motion at the fracture site under shearing loads. Also, as loads

are applied to the system, the reactions cause simultaneous, but

unequal, deflections to take place in the components. Interactions

at other than the original contact points will take place once the

mismatch exceeds the clearance, and, as the loads increase, local

deformations at the contact points will occur. These deformations

result in redistribution of the reaction forces along the nail-bone

interface, changing the loading configurations and the relative

deflections.

While the simplified model indicates that an optimum relation-

ship between the component properties may minimize deflections at

the fracture site, a more general model, which more closely

duplicates the actual osteosynthesis, is necessary before a

quantitative evaluation of this hypothesis can be obtained.

In light of the previous discussions, the material properties

of the nail and the bone, and the geometric properties of the nail,

may be considered to be arbitrary constants whose values depend on

the nail and bone under consideration. The model should, however,









account for the nonuniformity of the bone's geometric properties so

that the bone's flexural rigidity is properly represented as it

varies along its length. It should also provide intimate contact

between the nail and the bone, but should allow for longitudinal

sliding of the components. The nail-bone interface should be

assumed to develop an arbitrary interfacial loading profile which

is dependent on the relative deflections occurring between the nail

and the bone. Contact between the fracture surfaces will be assumed

to be nonexistent.

The previous work cited in proceeding chapters allows for

quantification of all the parameters discussed in the previous

paragraph except for the nail-bone interfacial reactions. In order

to obtain quantitative information on this interaction, an experi-

mental investigation was performed.


Intramedullary Interfacial Behavior


Methods and Materials

A matched pair of embalmed femurs was cleaned of soft tissue

and periosteum. At all times the bones were kept moist using a

saline solution drip. The bones were cast in plaster, and as soon

as the plaster had set, several 20 mm thick sections were cut from

the diaphysial and metaphysial regions of the bone. These sections

were then placed in saline solution until testing.

A Kuntschner nail (13 mm), Schneider nail (13 mm), or Fluted

nail (12 mm) was placed in the intramedullary canal of these

sections as shown in Figure 4.5. Each of these samples was placed






















.4


Figure 4.5.


Photograph showing the intramedullary nail in
place in the cast bone section for interfacial
stiffness determination










on a self-aligning compression fixture (Figure 4.6) for loading

of the nail-bone interface. Using an Instron universal testing

machine at a cross-head speed of 0.02 mm/min, the nails were pressed

against the endosteal surface, which contained either cancellus or

cortical bone. Load-deflection curves were obtained, along with a

measurement of the section thickness, for further calculations.

The results obtained from the load-deflection curves were

used to obtain the interfaciall stiffness" of the cancellus and

cortical bone of the femur in contact with different nail shapes.

This stiffness was defined as the slope of the load-deflection

curve divided by the specimen thickness. Thus, a parameter relating

force per deflection per unit length could be obtained for subsequent

use in the general modeling of the osteosynthesis.


Results

Representative curves for the load-deflection response of the

different nail types tested in cancellus and cortical bone are shown

in Figures 4.7 and 4.8 along with the testing configurations used.

Even with the small number of samples tested, the similarity

in the load-deflection response of the different nail types during

testing in cancellus bone led to the definition of three overall

interfacial stiffness regimes. The first interfacial stiffness range

was low in magnitude and appeared to be due to the initial crushing

of the cancellus bone. A transition region was observed as the

deflection increased. This was followed by a higher stiffness

level after significant deflection and packing had taken place.

These different regimes are shown as different line types in the






























INTRAMEDULLARY
NAIL


BONE
IN


SECTION
PLASTER


SELF-ALIGNING
SUPPORT


Figure 4.6.


Schematic drawing of the self-aligning
compression fixture for interfacial
stiffness determination




Full Text

PAGE 1

$ %,20(&+$1,&6 $3352$&+ 72 7+( ,17(5$&7,21 2) +267 ,03/$17 6<67(06 %\ *$5< -2(/ 0,//(5 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( &281&,/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

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

PAGE 3

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

PAGE 4

9, %21$,/ 35(',&7,21 2) 7+( 267(26<17+(6,6 '()/(&7,21 $33(1',; $ '(9(/230(17 2) 7+( ',))(5(17,$/ %($0 '()/(&7,21 (48$7,216 /,67 2) 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ LY

PAGE 5

/,67 2) 7$%/(6 7DEOH 3DJH 9DOXHV IRU WKH PHDQ VLGHWRVLGH GLIIHUHQFH FIS DQG VWDQGDUG GHYLDWLRQV LQ SDUHQWKHVHVf IRU WKH SDLUHG IHPXUV *HRPHWULF SURSHUWLHV RI WKH PP )OXWHG W\SH QDLO ZLWK DQG ZLWKRXW IOXWHV )LQLWH GLIIHUHQFH FRPSXWDWLRQDO PROHFXOHV 6\VWHP SURSHUWLHV IRU RSWLPL]DWLRQ HYDOXDWLRQ Y

PAGE 6

/,67 2) ),*85(6 )LJXUH 3DJH %RQHV RI WKH ORZHU H[WUHPLW\ 7KH HIIHFW RI VWUDLQ UDWH RQ WKH PDWHULDO SURSHUWLHV RI ERQH &URVVVHFWLRQDO VKDSHV RI FRPPRQO\ XVHG LQWUDPHGXOODU\ QDLOV 6WUXFWXUDO SURSHUWLHV RI FRPPRQO\ XVHG LQWUDPHGXOODU\ QDLOV (ODVWLF PRGXOXV IRU VHYHUDO LPSODQW PDWHULDOV 6FKHPDWLF UHSUHVHQWDWLRQ RI IHPRUDO OHQJWK PHDVXUHPHQWV 6HULDO FURVVVHFWLRQV RI D UHSUHVHQWDWLYH IHPXU ZLWK VL]H UHIHUHQFH PDUNV LQ SODFH 'HILQLWLRQ RI D[HV DQG GLVWDQFHV IRU JHRPHWULFDO SURSHUW\ GHWHUPLQDWLRQ 0RKUnV FLUFOH FRQVWUXFWLRQ IRU ORFDWLQJ SULQFLSDO D[HV DQG GLUHFWLRQV RI WKH FURVVVHFWLRQV *URVV FRUWLFDO ERQH DUHD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV *URVV FRUWLFDO ERQH DUHD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV $UHD PRPHQW RI LQHUWLD ,[[ DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV $UHD PRPHQW RI LQHUWLD ,[[ DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV $UHD PRPHQW RI LQHUWLD ,\\! DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV

PAGE 7

)LJXUH 3DJH $UHD PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV 3ULQFLSDO PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV 3ULQFLSDO PRPHQW RI LQHUWLD A DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV 3ULQFLSDO PRPHQW RI LQHUWLD ,AMU DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV f 3ULQFLSDO PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV 3RODU PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV 3RODU PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV (IIHFWLYH SRODU PRPHQW RI LQHUWLD -HII DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV (IIHFWLYH SRODU PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV 3ULQFLSDO DQJOH RULHQWDWLRQ DQG UHODWLYH PDJQLWXGH RI WKH SULQFLSDO PRPHQWV IRU WKH IHPXUV LQ JURXS 3ULQFLSDO DQJOH RULHQWDWLRQ DQG UHODWLYH PDJQLWXGH RI WKH SULQFLSDO PRPHQWV IRU WKH IHPXUV LQ JURXS ,, 3ULQFLSDO DQJOH RULHQWDWLRQ DQG UHODWLYH PDJQLWXGH RI WKH SULQFLSDO PRPHQWV IRU WKH IHPXUV LQ JURXS ,,, 6HFWLRQ PRGXOXV = DV D IXQFWLRQ RI OHQJWK DORQJ WKH IHPXU IRU WZR VDPSOH IHPXUV FRPSDUHG WR GDWD DGDSWHG IURP .RFK f *UDSKLFDO UHSUHVHQWDWLRQ RI RYHUDOO IHPRUDO OHQJWK YHUVXV DYHUDJH FRUWLFDO ERQH DUHD *UDSKLFDO UHSUHVHQWDWLRQ RI OHVVHU WURFKDQWHU WR IHPRUDO FRQG\OH OHQJWK //HVVHUf YHUVXV DYHUDJH FRUWLFDO ERQH DUHD YLL

PAGE 8

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f )UHH ERG\ GLDJUDP RI DQ HOHPHQW RI WKH ERQH ZLWK H[WHUQDO GLVWULEXWHG ORDG DQG LQWHUIDFLDO UHDFWLRQ ORDG )UHH ERG\ GLDJUDP RI WKH QDLO ZLWK LQWHUIDFLDO UHDFWLRQ ORDG 6FKHPDWLF GLDJUDP RI DQ RVWHRV\QWKHVLV XQGHU HTXDOO\ VSDFHG IRXUSRLQW ORDGLQJ ZLWK D PLGVKDIW WUDQVYHUVH IUDFWXUH YLLL

PAGE 9

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

PAGE 10

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

PAGE 11

)LJXUH 3DJH 1RQGLPHQVLRQDO IUDFWXUH VXUIDFH PLVPDWFK DV D IXQFWLRQ RI PRGXOXV UDWLR IRU YDULRXV LQWHUIDFLDO VWLIIQHVV YDOXHV )UDFWXUH VXUIDFH DQJXODWLRQ DV D IXQFWLRQ RI PRGXOXV UDWLR IRU YDULRXV LQWHUIDFLDO VWLIIQHVV YDOXHV $O $ VHJPHQW RI D EHDP KDYLQJ EHHQ FXUYHG E\ DQ DSSOLHG PRPHQW $ 6WUHVV GLVWULEXWLRQ SURILOHV GHYHORSHG LQ WKH FXUYHG EHDP $ $ VHJPHQW RI D EHDP XQGHU GLVWULEXWHG ORDG T[f $ $ GLIIHUHQWLDO HOHPHQW RI WKH EHDP ZLWK GLVWULEXWHG ORDG T[f DQG UHDFWLRQ PRPHQWV DQG VKHDUV [L

PAGE 12

$EVWUDFW RI 'LVVHUWDWLRQ 3UHVHQWHG WR WKH *UDGXDWH &RXQFLO RI WKH 8QLYHUVLW\ RI )ORULGD LQ 3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ $ %,20(&+$1,&6 $3352$&+ 72 7+( ,17(5$&7,21 2) +267 ,03/$17 6<67(06 %\ *DU\ -RHO 0LOOHU -XQH &KDLUPDQ 5REHUW % *DLWKHU &RFKDLUPDQ *HRUJH 3LRWURZVNL 0DMRU 'HSDUWPHQW 0HFKDQLFDO (QJLQHHULQJ $ JHQHUDO PRGHO RI DQ LQWUDPHGXOODU\ RVWHRV\QWKHVLV RI WKH IHPXU LV SRVWXODWHG 7KH PRGHO DOORZV IRU YDULDEOH JHRPHWULF SURSHUWLHV DORQJ WKH OHQJWK RI WKH ERQH DQG DOORZV KRPRJHQHRXV DQG LVRWURSLF PDWHULDO SURSHUWLHV WR EH FKRVHQ 7KH PRGHO LQFOXGHV DQ LQWUDPHGXOODU\ QDLO RI DUELWUDU\ PDWHULDO DQG JHRPHWULF SURSHUn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

PAGE 13

VROYHG ZLWK WKH DLG RI D GLJLWDO FRPSXWHU SURJUDP %21$,/f XVLQJ QXPHULFDO UHOD[DWLRQ WHFKQLTXHV 5HVXOWV RI LQ YLWUR YHULILFDWLRQ VWXGLHV LQYROYLQJ VWHHO EHDPV LQWDFW IHPXUV DQG IHPXUV ZLWK LQWUDPHGXOODU\ IL[DWLRQ DUH SUHVHQWHG DQG VHUYH WR VXEVWDQWLDWH WKH DFFXUDF\ RI WKH %21$,/ PRGHO %RWK WKH HIIHFW RI UHODWLYH IOH[XUDO ULJLGLW\ RI WKH FRPn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

PAGE 14

IUDFWXUH PD\ EH KDPSHUHG DQG UHVXOW LQ D ZHDNHQHG ERQH ZKLFK LV OLDEOH WR UHIUDFWXUH XSRQ UHPRYDO RI WKH LQWUDPHGXOODU\ QDLO 7KH ODFN RI LQIRUPDWLRQ LQ WKH OLWHUDWXUH FRQFHUQLQJ WKH JHRPHWULF SURSHUWLHV RI WKH IHPXU DQG WKH FKDUDFWHU RI WKH QDLOERQH LQWHUIDFH IRU XVH LQ PRGHOLQJ WKH V\VWHP OHG WR D SDUDOOHO H[SHULn PHQWDO VWXG\ 7HQ SDLUV RI KXPDQ IHPXUV ZHUH SURFHVVHG WR GHWHUPLQH WKH JURVV FRUWLFDO DUHD DUHD PRPHQWV RI LQHUWLD SULQFLSDO PRPHQWV RI LQHUWLD DQG SULQFLSDO DQJOHV DQG SRODU PRPHQWV RI LQHUWLD DW VHULDO SRLQWV IURP WKH OHYHO RI WKH OHVVHU WURFKDQWHU WR WKH GLVWDO PHWDSK\VLV 7KH UHVXOWV RI WKLV H[SHULPHQWDO VWXG\ DUH SUHVHQWHG JUDSKLn FDOO\ DV D IXQFWLRQ RI SRVLWLRQ DORQJ WKH IHPRUDO OHQJWK :KLOH FRQVLGHUDEOH YDULDWLRQ LQ SURSHUWLHV LV IRXQG IURP RQH SDLU WR WKH QH[W s b RU PRUHf QR VWDWLVWLFDOO\ VLJQLILFDQW ULJKWOHIW ELDV ZDV REVHUYHG ,Q YLWUR H[SHULPHQWDO UHVXOWV RI WKH LQWHUIDFLDO VWLIIQHVV EHKDYLRU RI VHYHUDO QDLO W\SHV ZHUH DOVR REWDLQHG LQ GLIIHUHQW ERQH PDWHULDO 7KHVH UHVXOWV UDQJHG RYHU VHYHUDO RUGHUV RI PDJQLWXGH DQG ZHUH GHSHQGHQW RQ ERWK WKH QDLO FURVVVHFWLRQDO VKDSH DQG WKH QDWXUH RI WKH PHGXOODU\ ERQH LH FDQFHOOXV RU FRUWLFDO [LY

PAGE 15

&+$37(5 ,1752'8&7,21 7KH IHPXU LV RQH RI WKH PDMRU ORDG FDUU\LQJ PHPEHUV LQ WKH KXPDQ ERG\ ,W LV WKH VLQJOH ERQH ZKLFK MRLQV WKH NQHH WR WKH KLS )LJXUH f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f KDV EHHQ FUHGLWHG ZLWK XVLQJ WKH ILUVW LQWUDPHGXOODU\ QDLO LQ D FODYLFOH LQ ,W ZDV QRW XQWLO KRZHYHU WKDW +H\ *URYHV RI (QJODQG XVHG LQWUDPHGXOODU\ QDLOLQJ IRU IL[DWLRQ RI D IUDFWXUH RI WKH IHPXU 5XVK DQG 5XVK f SUHVHQWHG WKH ILUVW FDVHV RI LQWUDPHGXOODU\ IL[DWLRQ LQ WKH 8QLWHG 6WDWHV XVLQJ QDLOV RI URXQG FURVVVHFWLRQ .XQWVFKHUnV f FODVVLF WUHDWLVH 7KH ,QWUDPHGXOODU\ 1DLOLQJ RI )UDFWXUHV VHUYHG WR OHQG VWURQJ VXSSRUW WR LQWUDPHGXOODU\ QDLOLQJ DV D YLDEOH PHWKRG

PAGE 16

+LS -RLQW .QHH -RLQW f§f§f§ 7LELD )LEXODr 6WMEX $QNOH -RLQW )LJXUH %RQHV RI WKH ORZHU H[WUHPLW\

PAGE 17

RI IL[DWLRQ RI IUDFWXUHG ORQJ ERQHV $W WKDW WLPH .XQWVFKHU LQWURn GXFHG KLV 9VKDSHG FURVVVHFWLRQ QDLO ZKLFK ZDV VDLG WR EH PRUH HDVLO\ GULYHQ LQWR WKH LQWUDPHGXOODU\ FDQDO WKDQ D VROLG QDLO EHFDXVH LWV RSHQ FURVVVHFWLRQ DOORZHG LW WR FRPSUHVV RQ LQVHUWLRQ 7KLV VKDSH ZDV VXEVHTXHQWO\ PRGLILHG WR D FORYHUOHDI SDWWHUQ ZKLFK IROn ORZHG D JXLGH SLQ PRUH HDVLO\ DQG SURYLGHG JUHDWHU QDLO VWUHQJWK 6WUHHW HW DW f UHSRUWHG WKH XVH RI WKH GLDPRQG FURVVVHFWLRQ QDLO ZKLFK KDG EHHQ GHYHORSHG E\ 'U +DQVHQ LQ WR SURYLGH JUHDWHU IOH[XUDO DQG WRUVLRQDO ULJLGLW\ 6FKQHLGHU GHVLJQHG D IRXU IODQJHG QDLO LQ ZKLFK FRXOG EH GULYHQ LQWR WKH LQWUDPHGXOODU\ FDQDO DQG HQJDJHG WKH HQGRVWHDO VXUIDFHV ZLWK LWV IODQJHV WR PDLQWDLQ WRUVLRQDO VWDELOLW\ ,WV FURVVVHFWLRQ ZDV DOVR VDLG WR SURYLGH KLJKHU IOH[XUDO ULJLGLW\ WKDQ SUHYLRXV GHVLJQV 6FKQHLGHU f ,Q WKH )OXWHG QDLO ZDV XVHG IRU WKH ILUVW WLPH $OOHQ f 7KLV QDLO ZDV GHVLJQHG WR SURYLGH LQFUHDVHG WRUVLRQDO DQG EHQGLQJ ULJLGLW\ DQG EHWWHU WRUVLRQDO JULSSLQJ RI WKH LQWUDPHGXOODU\ VXUIDFHV 7KH UHDGHU JHWV WKH LPSUHVVLRQ IURP D FXUVRU\ UHYLHZ RI WKHVH FOLQLFDO UHSRUWV RQ WKH HYROXWLRQ RI LQWUDPHGXOODU\ GHYLFHV WKDW WKH DXWKRUnV SULPDU\ FRQFHUQ LQ UHGHVLJQ LV WR LQFUHDVH WKH WRUVLRQDO DQG EHQGLQJ ULJLGLW\ RI WKH GHYLFH LQ RUGHU WR DIIHFW D EHWWHU RVWHRV\QWKHVLV 7KHUH LV KRZHYHU QR H[SHULPHQWDO YHULILFDWLRQ WKDW FRQWLQXHG LQFUHDVHV LQ ULJLGLW\ ZLOO LQ IDFW SURGXFH WKLV GHVLUHG UHVXOW

PAGE 18

3UHYLRXV :RUN 0F.HHYHU &Of KDV VXPPDUL]HG WKH JHQHUDO FULWHULD IRU GHVLJQ DQG DSSOLFDWLRQ RI DQ HIIHFWLYH LQWUDPHGXOODU\ GHYLFH +LV FRQFOXVLRQV DQG WKRVH RI WKH DERYH LQYHVWLJDWRUV FDQ EH VXPPDUL]HG DV IROORZV 7KH QDLO VKRXOG PDLQWDLQ DOLJQPHQW DQG DSSRVLWLRQ RI WKH IUDFWXUH VXUIDFHV LH FORVH ILW DW WKH IUDFWXUH VLWH ,W VKRXOG SHUPLW FRPSUHVVLRQ RI WKH DSSRVLQJ IUDFWXUH VXUIDFHV E\ DOORZLQJ D[LDO VOLGLQJ RI WKH ERQH VHJn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f VWDWHG WKDW ULJLGLW\ f DQG LPPRELOL]DWLRQ DQG f DUH RI WKH KLJKHVW SULRULW\ %UDGHQ DQG %ULQNHU f $QGHUVHQ f $QGHUVHQ HW DW f DQG +XW]VFKHQUHXWHU HW DW f KDYH VXSSRUWHG /LQGDKO DQG UHLWHUDWH WKH QHHG IRU ULJLG IL[DWLRQ WR DIIHFW SURSHU KHDOLQJ XVLQJ LQWUDPHGXOODU\ QDLOLQJ 7D\ORU f KDV DOVR QRWHG WKDW WKH XVH RI LQWUDPHGXOODU\ QDLOV ZKLFK GR QRW SURGXFH FRPSOHWH URWDWLRQDO IL[DWLRQ f ZLOO OHDG WR D KLJKHU LQFLGHQFH RI FRPSOLFDWLRQV VXFK DV PDOXQLRQV DQG QRQXQLRQV

PAGE 19

,Q FOLQLFDO XVH YDULDEOH VXFFHVV KDV EHHQ UHSRUWHG LQ WKH OLWHUDWXUH UHJDUGLQJ LQWUDPHGXOODU\ IL[DWLRQ :LFNVWURP HW DO f LQ D UHYLHZ RI WRWDO FDVHV FLWHG PDQ\ RI WKH FRPSOLFDWLRQV WKDW RFFXU LQ WKH XVH RI LQWUDPHGXOODU\ IUDFWXUH IL[DWLRQ :KLOH PDQ\ RI WKHVH FRPSOLFDWLRQV ZHUH ELRORJLFDO LQ QDWXUH LH LQIHFWLRQV HWFf RWKHUV ZHUH WKH UHVXOW RI LPSURSHU LPSODQW FKRLFH RU DSSOLFDWLRQ 7KHVH ELRORJLFDO DQG ELRPHFKDQLFDO FRPSOLFDWLRQV OHG WR b RI WKH SDWLHQWV KDYLQJ GHOD\HG RU QRQXQLRQ RI WKHLU IUDFWXUHV 7KHVH GDWD VXSSRUW WKH ZRUN RI 1LFKROV f ZKR DOVR IRXQG D b GHOD\HG XQLRQ UDWH LQ LQWUDPHGXOODU\ RVWHRV\QWKHVHV 6XFFHVV RU IDLOXUH RI WKLV VXUJLFDO SURFHGXUH GHSHQGV RQ WKH PHFKDQLFDO LQWHUDFWLRQ EHWZHHQ WKH QDLO DQG WKH ERQH 3HWHUV f KDV SXW WKH SUREOHP LQWR SHUVSHFWLYH 6WXGLHV RI WKH PRVW DSSURSULDWH PDWHULDO IRU LPPRELOL]DWLRQ RI IUDFWXUHV PXVW EH FRUUHODWHG ZLWK GHWHUPLQDWLRQ RI WKH ELRPHFKDQLFDO IDFWRUV ZKLFK DFW WR LQFUHDVH DQG GHFUHDVH IUDFWXUH KHDOLQJ 7KH PDWHULDO DQG PHWKRG RI DSSOLFDWLRQ FDQ WKHQ EH GHn VLJQHG WR PLQLPL]H WKH GLVUXSWLYH VWUHVV DW WKH IUDFWXUH VLWH DQG HQKDQFH WKRVH WKDW LQFUHDVH KHDOLQJ ,I WKH KHDOLQJ SURFHVV FDQ EH PDWHULDOO\ VSHHGHG DQG QRQXQLRQ SUHYHQWHG PRQWKV DQG \HDUV RI SDWLHQW GLVDELOLW\ FRXOG EH HOLPLQDWHGSf $WWHPSWV WR LPSURYH LQWUDPHGXOODU\ IUDFWXUH IL[DWLRQ KDYH IRFXVHG RQ LQFUHDVLQJ EHQGLQJ ULJLGLW\ WRUVLRQDO ULJLGLW\ XOWLPDWH VWUHQJWK DQG ELRFRPSDWLELOLW\ RI WKH QDLO E\ H[SHULPHQWLQJ ZLWK YDULDWLRQV LQ FURVVVHFWLRQDO JHRPHWU\ OHQJWK DQG PDWHULDO FRPn SRVLWLRQ RI WKH QDLOV 5HVXOWV UHSRUWHG WR GDWH DUH KRZHYHU TXDOLWDWLYH DQG KDYH QRW EHHQ FRUUHODWHG ZLWK PHFKDQLFDO IDFWRUV %RWK JRRG DQG EDG UHVXOWV KDYH EHHQ UHSRUWHG XVLQJ VLPLODU VXUJLFDO WHFKQLTXHV DQG LQWUDPHGXOODU\ QDLOV

PAGE 20

7KH DERYH UHYLHZ RI LQWUDPHGXOODU\ IL[DWLRQ OHDGV WR WZR PDMRU TXHVWLRQV 'R WKH UHODWLYH IOH[XUDO ULJLGLWLHV RI WKH QDLO DQG ERQH KDYH D VLJQLILFDQW HIIHFW RQ WKH GLVSODFHPHQWV RI WKH IUDFWXUH VXUIDFHV RI WKH ORDGHG RVWHRV\QWKHVLV" $UH WKHUH RWKHU SDUDPHWHUV ZKLFK DIIHFW WKH SHUIRUPDQFH RI WKH V\VWHP" $OOHQ HW DW f XQGHUVFRUHG WKH QHHG IRU XQGHUVWDQGLQJ WKLV LQWHUUHODWLRQVKLS RI WKH JHRPHWULF PDWHULDO DQG VWUXFWXUDO SURSHUWLHV RI WKH QDLO DQG ERQH LQ DQ LQWUDPHGXOODU\ RVWHRV\QWKHVLV +RZHYHU TXDQWLILFDWLRQ RI WKH GHIOHFWLRQ DW WKH IUDFWXUH LQWHUIDFH RI DQ RVWHRV\QWKHVLV KDV QRW EHHQ SXEOLVKHG WR GDWH %RXUJRLV DQG %XUQ\ f WRRN D ILUVW VWHS LQ GHVFULELQJ WKLV FRPSOH[ SUREOHP ,Q DQDO\]LQJ YDULRXV ORDGLQJ FRQILJXUDWLRQV WKH\ GHYHORSHG UHODWLRQn VKLSV EHWZHHQ IUDFWXUH FDOOXV ULJLGLW\ DQG SHUFHQWDJH KHDOLQJ 7KH PRGHO XVHG FRQVLVWHG RI D FDQWLOHYHU EHDP WKH LQWUDPHGXOODU\ QDLOf VXUURXQGHG E\ ERQH ZLWK D IUDFWXUH FDOOXV SUHVHQW DW WKH PLGVHFWLRQ 'HIOHFWLRQV ZHUH FDOFXODWHG IRU WKLV W\SH RI V\VWHP XQGHU D WUDQVn YHUVH HQG ORDG 7R FDUU\ RXW WKHVH FDOFXODWLRQV DQ DVVXPSWLRQ RI FRPSOHWH ERQGLQJ RI WKH QDLO WR WKH ERQH ZDV LPSOLFLWO\ PDGH LH WKH LQWHUIDFH EHWZHHQ ERQH DQG QDLO WUDQVPLWWHG VKHDU VWUHVVHVf 7KLV WUDQVPLVVLRQ RI VKHDU VWUHVVHV DW WKH LQWHUIDFH DIIHFWV WKH GHIOHFWLRQ UHVXOWV VLJQLILFDQWO\ DQG LV QRW VHHQ FOLQLFDOO\ ZLWK LQWUDPHGXOODU\ QDLOV +RZHYHU ILUVW DSSUR[LPDWLRQV RQ WKH HIIHFWV RI YDULDWLRQV LQ JHRPHWU\ PDWHULDO DQG IUDFWXUH FDOOXV SURSHUWLHV RQ GHIOHFWLRQV ZHUH REWDLQHG 1R RWKHU TXDQWLWDWLYH VWXGLHV ZKLFK GHDO ZLWK LQWUDPHGXOODU\ IL[DWLRQ KDYH EHHQ IRXQG DQG WKH IROORZLQJ LQYHVWLJDWLRQ ZDV WKHUHIRUH XQGHUWDNHQ

PAGE 21

7KH 3UHVHQW 5HVHDUFK 7KH SUHVHQW ZRUN LV DLPHG DW TXDQWLILFDWLRQ RI WKH QDLOERQH LQWHUDFWLRQV DQG WKH UHODWLYH GLVSODFHPHQWV RI WKH IUDFWXUH VXUIDFHV SUHVHQW GXULQJ WKH EHQGLQJ RI DQ RVWHRV\QWKHVLV 8WLOL]LQJ HQJLQHn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n IDFLDO GHIOHFWLRQV DW WKH IUDFWXUH VLWH DQG WKH H[SHFWHG FRQWDFW IRUFHV EHWZHHQ WKH QDLO DQG WKH ERQH LQ WKH PHGXOODU\ FDQDO

PAGE 22

&+$37(5 ,, 3523(57,(6 2) 7+( 6<67(0 &20321(176 $ 5(9,(: 7KH HQJLQHHU LV RIWHQ IDFHG ZLWK WKH QHHG WR DQDO\]H WKH GHIOHFWLRQ EHKDYLRU RI ORDG FDUU\LQJ PDFKLQH HOHPHQWV ,Q RUGHU WR SURFHHG LW LV LPSRUWDQW WKDW WKH HQJLQHHU FRQVLGHU WKH HOHPHQWV LQ D WZRIROG PDQQHU 7KH ILUVW LQYROYHV XQGHUVWDQGLQJ RI WKH FRPSRQHQWV DV JHRPHWULFDO HQWLWLHV LH WKHLU VKDSH DQGRU GLVn WULEXWLRQ RI PDWHULDO 7KH VHFRQG LQYROYHV WKH DSSUHFLDWLRQ RI WKH SURSHUWLHV RI WKH PDWHULDOV IURP ZKLFK WKH HOHPHQWV RU GHYLFHV DUH IDEULFDWHG %\ FRPELQLQJ WKHVH WZR YLHZV D TXDQWLWDWLYH HYDOXDWLRQ RI WKH SHUIRUPDQFH DQG LQWHUDFWLRQ RI WKH GHYLFHV FDQ EH SHUIRUPHG ,W LV EHOLHYHG WKDW XQGHUVWDQGLQJ DQG PRGHOLQJ WKH LQWUDPHGXOODU\ RVWHRV\QWKHVLV UHTXLUHV WKH VDPH DSSURDFK 7KHUHn IRUH WKLV WKHVLV EHJLQV ZLWK DQ H[DPLQDWLRQ RI WKH H[LVWLQJ LQIRUPDWLRQ RQ WKH VWUXFWXUDO DQG PDWHULDO SURSHUWLHV RI WKH FRPSRQHQWV RI WKH LQWUDPHGXOODU\ RVWHRV\QWKHVLV 7KH ,QWDFW )HPXU *HRPHWULFDO 3URSHUWLHV .RFK f ZDV RQH RI WKH ILUVW WR DSSO\ HQJLQHHULQJ SULQFLSOHV WR WKH GHWHUPLQDWLRQ RI WKH VWUHVV UHVSRQVH RI WKH ORDGHG IHPXU 8VLQJ WLPH FRQVXPLQJ JUDSKLFDO PHWKRGV KH DQDO\]HG WKH VKDSH DQG GLVWULEXWLRQ RI ERQH LQ D VLQJOH KXPDQ IHPXU +H

PAGE 23

VKRZHG WKDW WKH H[WHUQDO IRUP RI WKH ERQH DQG LWV LQWHUQDO VWUXFWXUH ZHUH SURSRUWLRQHG LQ VXFK D ZD\ DV WR UHVLVW D[LDO EHQGLQJ DQG VKHDU ORDGV DSSOLHG GXULQJ QRUPDO DFWLYLW\ ZLWK D PLQLPXP XVH RI PDWHULDO 3KRWRHODVWLF WHFKQLTXHV 3DXZHOV f DQG VWUHVV ODFTXHUV (YDQV DQG /LVVQHU DQG )UDQNHO f KDYH DOVR EHHQ XVHG WR GHWHUPLQH VWUHVV OHYHOV LQ ERQHV XQGHU D[LDO DQG EHQGLQJ ORDGLQJ 0RUH UHFHQWO\ 7RULGLV f DQG 5\ELFNL HW DO f KDYH SUHn VHQWHG DQDO\VHV XVLQJ HQJLQHHULQJ EHDP WKHRU\ WR SUHGLFW VWUHVV OHYHOV WKURXJKRXW WKH IHPXU 7KHLU UHVXOWV LQGLFDWH WKDW HQJLQHn HULQJ EHDP WKHRU\ LV DSSOLFDEOH WR DQDO\VLV RI WKH ORQJ ERQHV LQ WKH PLGVKDIW UHJLRQ 7KH ERQH HQGV ZLWK WKHLU IODUHG VWUXFWXUHV DUH QRW DGHTXDWHO\ PRGHOHG XVLQJ WKHVH PHWKRGVf 3L]LDOL HW DO f DQDO\WLFDOO\ H[DPLQHG VHYHUDO DVVXPSWLRQV LQKHUHQW LQ WKH XVH RI FODVVLFDO HQJLQHHULQJ EHDP WKHRU\ WR ORQJ ERQHV VSHFLILFDOO\ WLELDVf DQG IRXQG WKDW GHIOHFWLRQV GXH WR VKHDU GHIRUPDWLRQV DUH QHJOLJLEOH DQG WKDW WKH HIIHFWV RI LQLWLDO FXUYDWXUH DUH QRW VLJQLILFDQW 7KH PLVPDWFK EHWZHHQ WKH QHXWUDO D[LV RI WKH FURVVVHFWLRQ DQG WKH DSSOLHG PRPHQW KRZHYHU LQWURGXFHV GHn IOHFWLRQ FRPSRQHQWV SHUSHQGLFXODU WR DSSOLHG ORDGV 7KHLU DVVXPSWLRQ RI OLQHDUO\ HODVWLF SURSHUWLHV OHG WR SUHGLFWHG GHIOHFWLRQV ZKLFK ZHUH LQYHUVHO\ SURSRUWLRQDO WR WKH PRGXOXV RI HODVWLFLW\ RI WKH ERQH ZLWK VKHDU GHIRUPDWLRQV FDXVLQJ QHJOLJLEOH GHYLDWLRQ LQ WKLV EHKDYLRU
PAGE 24

/RQJ ERQHV KDYH DQ LUUHJXODU FURVVVHFWLRQDO JHRPHWU\ ZKLFK YDULHV FRQVLGHUDEO\ DORQJ WKHLU OHQJWK 7KH GHWHUPLQDWLRQ RI JHRPHWULF SURSHUWLHV VXFK DV WKH DUHD PRPHQWV RI LQHUWLD LV QHFHVVDU\ WR DSSO\ WKH DERYH FLWHG VWUHQJWK UHVXOWV WR RWKHU ERQHV $V (YDQV f SRLQWHG RXW PXFK RI WKH ZRUN RQ VWUXFWXUDO SHUn IRUPDQFH RI ORQJ ERQHV UHTXLUHV DSSUR[LPDWLRQ RI WKH JHRPHWULF SURSHUWLHV RI ERQHV $OOHQ HW DO f UHDOL]LQJ WKLV SUREOHP SUHGLFWHG WKH VWUXFWXUDO ULJLGLW\ RI D IHPXU LQ EHQGLQJ DQG WRUVLRQ 7KH\ XVHG WHGLRXV QXPHULFDO LQWHJUDWLRQ WHFKQLTXHV ZKLFK \LHOGHG WKH PRPHQWV RI LQHUWLD QHFHVVDU\ IRU FDOFXODWLRQ RI WKH EHQGLQJ DQG WRUVLRQDO ULJLGLW\ RI WKH IHPXU 7KLV ZRUN ZDV KRZHYHU OLPLWHG WR RQO\ WKUHH FURVVVHFWLRQV 3LRWURZVNL DQG :LOFR[ f GHYHORSHG D FRPSXWHU SURJUDP ZKLFK SHUIRUPHG WKH QXPHULFDO GHWHUPLQDWLRQ RI WKH JHRPHWULF SURSHUWLHV DQG VWUHVVHV LQ VROLG VLQJO\ FRQQHFWHGf FURVVVHFWLRQV DQG DOORZHG WKH DQDO\VLV RI RYHU FURVVVHFWLRQV RI FDQLQH ILEXODH 3LRWURZVNL f 7KH SURJUDP ZDV XSGDWHG 3LRWURZVNL DQG .HOOPDQ f WR WDNH LQWR DFFRXQW WKH SUHVHQFH RI LQWHUQDO KROHV WKH LQWUDPHGXOODU\ FDQDOf 0LQQV HW DO f XVHG ILQLWH HOHPHQW WHFKQLTXHV WR DQDO\]H IRXU OHIW WLELDV DQG UHSRUWHG YDULDWLRQV LQ JHRPHWULFDO SURSHUWLHV DV D IXQFWLRQ RI OHQJWK 7KH\ DOVR IRXQG WKDW JHRPHWULFDO SURSHUWLHV EHWZHHQ ERQHV FDQ GLIIHU E\ b RU PRUH 3L]LDOL HW DO f GHWHUPLQHG WKH JHRPHWULF SURSHUWLHV RI WZR IHPXUV WZR ILEXODH DQG IRXU WLELDH +RZHYHU WKH\ SUHVHQWHG RQO\ WKH WLELDO LQIRUPDWLRQ ZKLFK FRUUHVSRQGHG WR WKDW UHSRUWHG E\ 0LQQV HW DO f

PAGE 25

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f .DW] f DQG .UDXV f GLVFXVV WKH PDQ\ WKHRUHWLFDO OXPSHG SDUDPHWHU PRGHOV XVHG WR GHVFULEH WKH PHFKDQLFDO EHKDYLRU RI ERQH VXEVWDQFH RQ D PLFURVFRSLF VFDOH 7KH SUHVHQW ZRUN KRZHYHU LV FRQFHUQHG ZLWK WKH PDFURVFRSLF EHKDYLRU RI ERQH DQG WKH LQWHUHVWHG UHDGHU LV UHIHUUHG WR WKH DERYH UHYLHZ DUWLFOHV IRU IXUWKHU GLVn FXVVLRQ RI PLFURVFRSLF EHKDYLRU DQG PRGHOLQJ $V .XPPHU f KDV SRLQWHG RXW LQ KLV UHYLHZ RI WKH ELRn PHFKDQLFV RI ERQH ERQH LV QHLWKHU PRUSKRORJLFDOO\ KRPRJHQHRXV QRU LVRWURSLF ,Q IDFW 5HLOO\ DQG %XUVWHLQ f LQGLFDWH WKDW D WUDQVYHUVHO\ LVRWURSLF PRGHO ZRXOG EH PRUH DFFXUDWH WKDQ WKH KRPRJHQHRXV DQG LVRWURSLF UHSUHVHQWDWLRQV ZKLFK DUH RIWHQ XVHG LQ ERQH PRGHOLQJ :KHWKHU RU QRW ERQH FDQ EH UHJDUGHG DV KRPRJHQHRXV ZLWK UHVSHFW WR LWV PDFURVFRSLF EHKDYLRU KRZHYHU UHPDLQV WR EH VHHQ

PAGE 26

5HLOO\ f DQG 5HLOO\ DQG %XUVWHLQ f KDYH SUHVHQWHG D FRPSUHKHQVLYH FULWLFDO UHYLHZ RI WKH OLWHUDWXUH RQ WKH PHFKDQLFDO SURSHUWLHV RI KXPDQ FRUWLFDO ERQH 7KHLU UHYLHZ JLYHV D GHWDLOHG GLVFXVVLRQ RI WKH PDQ\ YDULDWLRQV LQ PHFKDQLFDO SURSHUWLHV RI ERQH SUHVHQWHG WKURXJKRXW WKH OLWHUDWXUH 7KHVH LQFOXGH WKH HIIHFWV RI VSHFLPHQ KDQGOLQJ DQG WHVWLQJ SURFHGXUHV IRU GHWHUPLQDWLRQ RI WKHVH HODVWLF SURSHUWLHV 6HYHUDO DUWLFOHV UHIHUUHG WR DUH GLVn FXVVHG EHORZ 5HLOO\ f PHDVXUHG WKH IROORZLQJ DYHUDJH SURSHUWLHV IRU IUHVKO\ IUR]HQ KXPDQ ERQH WHVWHG DW KLJK VWUDLQ UDWHV (n [ 1P ( [ 1P Y Yn *n [ 1P ZKHUH ( LV WKH PRGXOXV RI HODVWLFLW\ Y LV WKH 3RLVVRQnV UDWLR DQG LV WKH VKHDU PRGXOXV 7KH SULPHG YDOXHV UHIHU WR SURSHUWLHV IRXQG SDUDOOHO WR WKH ORQJ D[LV RI WKH ERQH DQG WKH XQSULPHG YDOXHV UHIHU WR WKH UDGLDO DQG WUDQVYHUVH SURSHUWLHV 7KHUH DUH ODUJH GLIn IHUHQFHV LQ SURSHUWLHV LQ WKHVH WZR RUWKRJRQDO GLUHFWLRQV 7KH YDOXHV UHSRUWHG IRU ( DQG (n DUH DVVRFLDWHG ZLWK VWDQGDUG GHYLDWLRQV RI DSSUR[LPDWHO\ bb RI WKH PRGXOXV LI GHWHUPLQHG LQ WHQVLRQ DQG bb LI GHWHUPLQHG LQ FRPSUHVVLRQ 7KH 3RLVVRQnV UDWLRV KDG D VWDQGDUG GHYLDWLRQ ZKLFK UHSUHVHQWHG b RI WKHLU PHDQ YDOXHV 6WDQGDUG GHYLDWLRQV IRU WKH VKHDU PRGXOXV ZHUH W\SLFDOO\ b

PAGE 27

7KHVH UHVXOWV IRU WKH YDULDWLRQV WR EH H[SHFWHG LQ GHWHUPLQn DWLRQ RI WKH PDWHULDO SURSHUWLHV FRUUHODWH ZHOO ZLWK WKH UHVXOWV FLWHG E\ (YDQV DQG /HERZ f ZKR PDGH DGGLWLRQDO REVHUYDWLRQV FRQFHUQLQJ WKH ORFDO YDULDWLRQV LQ SURSHUWLHV DORQJ WKH OHQJWK RI D JLYHQ ERQH 7KH\ IRXQG bb YDULDWLRQV LQ WKH PRGXOXV RI HODVWLFLW\ DQG VLPLODU YDULDWLRQV LQ WKH XOWLPDWH VWUHQJWK DQG KDUGQHVV DV ZHOO 7KHVH UHVXOWV LQGLFDWH WKDW ZKLOH WKHUH LV VRPH YDULDWLRQ WR EH H[SHFWHG LQ WKH SURSHUWLHV PHDVXUHG DW GLIn IHUHQW ORFDWLRQV DORQJ D VLQJOH ERQH WKHVH YDULDWLRQV DUH QRW VWDWLVWLFDOO\ VLJQLILFDQW ZKHQ FRQVLGHULQJ WKH RYHUDOO VFDWWHU LQ UHVXOWV REWDLQHG DW VHYHUDO ORFDWLRQV RQ VHYHUDO GLIIHUHQW IHPXUV DV LQGLFDWHG E\ 5HLOO\ $V 5HLOO\ f QRWHG WKH DYHUDJH UHVXOWV KH REWDLQHG LQ PHDVXULQJ PDWHULDO SURSHUWLHV ZHUH IRU IUHVKO\ IUR]HQ QRQHPEDOPHG ERQH 0DQ\ DXWKRUV KDYH GHDOW LQ FRQVLGHUDEOH GHWDLO ZLWK WKH HIIHFWV RI GLIIHUHQW WUHDWPHQW RQ WKH PHFKDQLFDO SURSHUWLHV RI KXPDQ FRUWLFDO ERQH (YDQV DQG /HERZ (YDQV 0F(OKDQH\ HW DO 6HGOLQ DQG +LUVFK .UDXV .XPPHU DQG %XUVWHLQ HW DO f 7KH\ PDGH VHYHUDO VDOLHQW REVHUYDWLRQV 7KHUH ZDV QR VLJQLILFDQW WHPSHUDWXUH HIIHFW RQ WKH PD[LPXP IDLOXUH VWUHVV r rf 7KH PRGXOXV RI HODVWLFLW\ ZDV IRXQG WR LQFUHDVH VOLJKWO\ DORQJ ZLWK PD[LPXP GHIOHFWLRQ DW r $LU GU\LQJ DOWHUV WKH SK\VLFDO SURSHUWLHV LQ RQO\ D IHZ VHFRQGV DQG D PDUNHG LQFUHDVH DV PXFK DV b RU PRUH LQ WKH PRGXOXV RI HODVWLFLW\ LV REVHUYHG $ GHFUHDVH LQ WKH REVHUYHG SODVWLF GHIRUPDWLRQ RFFXUV DV ZHOO :KLOH IUHH]LQJ LQFUHDVHV WKH VWUHQJWK RI WKH VSHFLPHQ VOLJKWO\ LW GRHV QRW DOWHU DQ\ RWKHU SURSHUWLHV

PAGE 28

)L[DWLRQ LQ IRUPDOLQ GRHV QRW DSSHDU WR DOWHU WKH PRGXOXV RI HODVWLFLW\ 6LJQLILFDQW GLIIHUHQFHV ZHUH IRXQG LQ DOO WKH SURSn HUWLHV RI ERQH IURP LQGLYLGXDO WR LQGLYLGXDO DQG WKHVH DUH QRW FRUUHODWHG ZLWK DJH 7KH UHVXOWV RQ WKH HIIHFWV RI HPEDOPLQJ RQ WKH SURSHUWLHV RI ERQH DUH HTXLYRFDO ,W KDV EHHQ VXJJHVWHG $OOHQ f WKDW HPEDOPLQJ PD\ KDYH D KLJKO\ YDULDEOH HIIHFW RQ WKH PHFKDQLFDO SURSHUWLHV RI WKH ERQH GXH WR WKH GLIIHUHQFHV LQ YDVFXODULW\ DQG WKHUHIRUH GLVSRVLWLRQ RI WKH HPEDOPLQJ PHGLXP 0F(OKDQH\ DQG %\DUV f WHVWLQJ ERYLQH ERQH LQ FRPSUHVVLRQ 3DQMDEL HW DW f WHVWLQJ UDEELW IHPRUD LQ WRUVLRQ DQG %XUVWHLQ DQG )UDQNHO f WHVWLQJ KXPDQ ERQH LQ WRUVLRQ DOVR SRLQW RXW WKDW WKH SK\VLFDO SURSHUWLHV RI ERQH PD\ YDU\ ZLWK FKDQJHV LQ ORDG RU VWUDLQ DSSOLFDWLRQ UDWHV )LJXUH LQGLFDWHV WKH ODUJH YDULDWLRQV LQ SURSHUWLHV ZKLFK PD\ EH H[SHFWHG 7KXV DV 5HLOO\ DQG %XUVWHLQ f KDYH SRLQWHG RXW GDWD FDQ EH H[WUDFWHG IURP WKH OLWHUDWXUH WR VXSSRUW RU FRQWUDGLFW PDQ\ K\SRWKHVHV 7KLV LV QRW GXH WR DQ\ LQKHUHQW DPELJXLWLHV LQ WKH GDWD EXW LV UDWKHU GXH WR WKH DEVHQFH RI VWDQGDUGL]DWLRQ RU XQLILn FDWLRQ RI H[SHULPHQWDO DSSURDFKHV DQG JRDOV 7KHVH ODUJH YDULDWLRQV LQ WKH UHSRUWHG GDWD LQGLFDWH WKH QHHG IRU PRUH ZRUN EHIRUH WKH LQWULQVLF DQG H[WULQVLF YDULDWLRQV RI ERQH SURSHUWLHV DUH FOHDUO\ XQGHUVWRRG

PAGE 29

)LJXUH 7KH HIIHFW RI VWUDLQ UDWH RQ WKH PDWHULDO SURSHUWLHV RI ERQH %RYLQH GDWD IURP 0F(OKDQH\ DQG %\DUV +XPDQ GDWD IURP %XUVWHLQ DQG )UDQNHO f

PAGE 30

7KH ,QWUDPHGXOODU\ 1DLO *HRPHWULFDO 3URSHUWLHV 6RPH RI WKH PRUH FRPPRQO\ XVHG QDLO VKDSHV DUH VKRZQ LQ )LJXUH 7KH $PHULFDQ 6RFLHW\ IRU 7HVWLQJ DQG 0DWHULDOV f KDV GHYHORSHG VWDQGDUG VSHFLILFDWLRQV IRU WKH JHRPHWU\ RI .XQWVFKHU RU FORYHUOHDI QDLOV )f DQG IRU VROLG FURVVVHFWLRQ LQWUDPHGXOODU\ QDLOV )f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f H[DPLQHG WKH RYHUDOO VWUXFWXUDO SHUn IRUPDQFH RI YDULRXV QDLO W\SHV LQ WRUVLRQ DQG EHQGLQJ 7KHVH LQFOXGHG WKH .XQWVFKHU +DQVHQ6WUHHW DQG 6FKQHLGHU QDLOV 7KH\ QHJOHFWHG WR QRWH KRZHYHU WKH QDLO VL]H EHLQJ FRQVLGHUHG 7KH N ) 6SHFLILFDWLRQ IRU &ORYHUOHDI ,QWUDPHGXOODU\ 3LQV ) 6SHFLILFDWLRQ IRU 6ROLG &URVV6HFWLRQ ,QWUDPHGXOODU\ 1DLOV N N 6DPSVRQ &RUSRUDWLRQ 3LWWVEXUJK 3$

PAGE 31

)LJXUH &URVVVHFWLRQDO VKDSHV RI FRPPRQO\ XVHG LQWUDPHGXOODU\ QDLOV

PAGE 32

6DPSVRQ &RUSRUDWLRQ f XVHG SURSRVHG $670 VWDQGDUG WHVW PHWKRGV ZKLFK DSSHDUHG WR EH ) WR HYDOXDWH WKH EHQGLQJ ULJLGLW\ EHQGLQJ VWUHQJWK DQG WRUVLRQDO ULJLGLW\ RI WKHVH GHYLFHV 7KHLU FRPSDUDWLYH UHVXOWV DSSHDU LQ )LJXUH DF ,Q WKLV WHVW PHWKRG KRZHYHU WKH JHRPHWULFDO SURSHUWLHV ZHUH QRW H[SOLFLWO\ FRQVLGHUHG 7KH GDWD QHHGHG WR IXUWKHU PRGHO QDLO XVH LQ DQ RVWHRV\QWKHVLV LV WKHUHIRUH RQO\ SDUWLDOO\ NQRZQ 0DWHULDO 3URSHUWLHV 7KH $670 KDV DOVR GHYLVHG VHYHUDO UHFRPPHQGHG VWDQGDUG VSHFLILFDWLRQV IRU WKH PDWHULDOV FRPPRQO\ XVHG LQ LPSODQW PDQXIDFWXUH ) ) ) ) ) ) ) )fr 7KHVH LQFOXGH WKH FKHPLFDO UHTXLUHPHQWV DQG WKH PLQLPXP DFFHSWDEOH PHFKDQLFDO SURSHUWLHV LQFOXGLQJ XOWLPDWH WHQVLOH VWUHQJWK \LHOG VWUHQJWK HORQJDWLRQ DQG UHGXFWLRQ RI DUHD %HFDXVH RI WKH HIIHFWV RI PDFKLQLQJ FROG ZRUNLQJ DQG SURFHVVLQJ WHFKQLTXHV LQ IDEULFDWLRQ RI LPSODQWV IURP WKH DQQHDOHG PHWDOV WKHVH YDOXHV IRU WKH XQZRUNHG r ) 6SHFLILFDWLRQ IRU VWDLQOHVV VWHHO EDUV DQG ZLUH IRU VXUJLFDO LPSODQWV ) 6SHFLILFDWLRQ IRU VWDLQOHVV VWHHO VKHHW DQG VWULS IRU VXUJLFDO LPSODQWV ) 6SHFLILFDWLRQ IRU XQDOOR\HG WLWDQLXP IRU VXUJLFDO LPSODQWV ) 6SHFLILFDWLRQ IRU FDVW FREDOWFKURPLXPPRO\EGHQXP EDVH DOOR\ IRU VXUJLFDO LPSODQWV ) 6SHFLILFDWLRQ IRU ZURXJKW FREDOWFKURPLXP DOOR\ IRU VXUJLFDO LPSODQWV ) 6SHFLILFDWLRQ IRU WLWDQLXP $9 (/, DOOR\ IRU XVH LQ FOLQLFDO HYDOXDWLRQ RI VXUJLFDO LPSODQW PDWHULDOV ) 6SHFLILFDWLRQ IRU VWDLQOHVV VWHHO EDUV DQG ZLUH IRU VXUJLFDO LPSODQWV 6SHFLDO TXDOLW\f ) 6SHFLILFDWLRQ IRU VWDLQOHVV VWHHO VKHHW DQG VWULS IRU VXUJLFDO LPSODQWV 6SHFLDO TXDOLW\f

PAGE 33

Ef Ff LJXUH 6WUXFWXUDO SURSHUWLHV RI FRPPRQO\ XVHG LQWUDPHGXOODU\ QDLOV 'DWD WDNHQ IURP 7HFKQLFDO 0HPRUDQGXP 6DPSVRQ &RUS 3LWWVEXUJK 3Df

PAGE 34

)LJXUH U] /7 &/ ;f 2 R ,6f UR ( ; 2 &U &R 0R $OOR\ FROG ZRUNHG DQQHDOHG 6WDLQOHVV UD 6WHHO + 7L $,9 %RQH + (ODVWLF PRGXOXV IRU VHYHUDO LPSODQW PDWHULDOV

PAGE 35

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f 7KH ORDGGHIOHFWLRQ EHKDYLRU RI VHYHUDO LQWUDPHGXOODU\ QDLOV KDV EHHQ FDUULHGRXW DQG WKH JHRPHWULF SURSHUWLHV f§ QHFHVVDU\ IRU PRGHOLQJ WKHLU XVH LQ DQ RVWHRV\QWKHVLV FDQ EH GHULYHG IURP WKLV LQIRUPDWLRQ

PAGE 36

%HIRUH D JHQHUDO PRGHOLQJ RI WKH RVWHRV\QWKHVLV FDQ EH FDUULHG RXW TXDQWLWDWLYH FKDUDFWHUL]DWLRQ RI WKH JHRPHWULF SURSHUWLHV RI WKH IHPXU PXVW EH SHUIRUPHG 7KH ILUVW SRUWLRQ RI WKH H[SHULPHQWDO ZRUN SUHVHQWHG KHUH FRQFHUQV LWVHOI ZLWK WKLV FKDUDFWHUL]DWLRQ

PAGE 37

&+$37(5 ,,, '(7(50,1$7,21 2) 7+( *(20(75,& 3523(57,(6 2) 7+( 6<67(0 &20321(176 7KH )HPXU 0DWHULDOV DQG 0HWKRGV 7HQ SDLUV RI HPEDOPHG KXPDQ IHPXUV ZHUH REWDLQHG DQG FOHDQHG RI DOO VRIW WLVVXH DQG SHULRVWHXP 7KH IHPXUV ZHUH WKHQ DOORZHG WR DLU GU\ IRU RQH GD\ ,QIRUPDWLRQ UHJDUGLQJ DJH VH[ KHLJKW DQG ZHLJKW RI WKH VXEMHFWV ZDV QRW DYDLODEOH 7ZR UHIHUHQFH OHQJWKV ZHUH PHDVXUHG DV VKRZQ LQ )LJXUH 7KH ILUVW / PDMRUr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

PAGE 38

)LJXUH 6FKHPDWLF UHSUHVHQWDWLRQ RI WKH IHPRUDO OHQJWK PHDVXUHPHQWV /OHVVHU} WKH OHVVHU WURFKDQWHU WR IHPRUDO FRQG\OH OHQJWK DQG /PDMRU! WKH IHPRUDO KHDG WR FRQG\OH OHQJWK

PAGE 39

VHFWLRQV ZHUH PDUNHG RII XVLQJ WKH OHVVHU WURFKDQWHU PDUNHU DV WKH VWDUWLQJ SRLQW $SSUR[LPDWHO\ VHULDO VHFWLRQV ZHUH REWDLQHG IURP HDFK ERQH 7KHVH UHSUHVHQWHG SRVLWLRQV PHDVXUHG SUR[LPDO WR GLVWDOf IURP DSSUR[LPDWHO\ b RI WKH WRWDO IHPRUDO OHQJWK WKH OHYHO RI WKH OHVVHU WURFKDQWHU WR b WKH OHYHO RI WKH GLVWDO PHWDSK\VLV $IWHU VHFWLRQLQJ DQG QXPEHULQJ HDFK VHFWLRQ ZDV FDUHIXOO\ FOHDQHG RI ERQH PDUURZ DQG D VL]H UHIHUHQFH PDUNHG GLUHFWO\ RQ WKH VXUIDFH RI WKH SODVWHU EORFNV )LJXUH f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f 7KH 6&$'6 SURJUDP ZDV XVHG WR FDOFXODWH WKH JHRPHWULFDO SURSHUWLHV RI WKH FURVVVHFWLRQ DV GHILQHG EHORZ &RUWLFDO ERQH DUHD 7KH FRPSXWHU SURJUDP GHWHUPLQHV WKH DUHD RI FRUWLFDO ERQH LQ HDFK FURVVVHFWLRQ E\ LQWHJUDWLQJ DOO N 'HSDUWPHQW RI 5DGLDWLRQ 7KHUDS\ +LOOLV 0LOOHU +HDOWK &HQWHU *DLQHVYLOOH )ORULGD

PAGE 40

I 7 W n0Ln f Wn W _M n D i I I F !‘ fO/ 9 I )LJXUH 6HULDO FURVVVHFWLRQV RI D UHSUHVHQWDWLYH IHPXU ZLWK VL]H UHIHUHQFH PDUNV LQ SODFH

PAGE 41

WKH GLIIHUHQWLDO DUHD HOHPHQWV VXFK WKDW $&257 f $UHD PRPHQWV RI LQHUWLD 7KH DUHD PRPHQWV RI LQHUWLD RU VHFRQG PRPHQWV RI WKH DUHD DUH JHQHUDOO\ XVHG LQ WKH FDOFXODWLRQ RI EHQGLQJ VWUHVVHV DQG GHIOHFWLRQV DQG DUH D PHDVXUH RI WKH GLVWULEXWLRQ RI WKH ORDGFDUU\LQJ PDWHULDO DERXW D VHW RI RUWKRJRQDO D[HV )RU WKLV ZRUN WKH RUWKRJRQDO UHIHUHQFH D[HV ZHUH FKRVHQ WR EH WKH PHGLRODWHUDO [D[LVf DQG WKH DQWHURSRVWHULRU \D[LVf SODQHV 7KH PRPHQWV RI LQHUWLD DUH GHILQHG UHIHUULQJ WR )LJXUH DV [ G$ e [ G$ [[ -D L \ G$ \" G$ \\ -$ L [\ G$ [\ G$ [\ -D f f f 3ULQFLSDO DQJOH DQG PRPHQWV RI LQHUWLD $QRWKHU VHW RI RUWKRJRQDO D[HV S DQG e ZKRVH RULJLQ LV DW WKH FHQWURLG RI WKH FURVVVHFWLRQ FDQ EH GHWHUPLQHG VXFK WKDW I Se G$ Q -D f 7KHVH D[HV DUH NQRZQ DV WKH SULQFLSDO D[HV DQG FRUUHVSRQG ZLWK WKH D[HV RI V\PPHWU\ LQ FURVVVHFWLRQV ZKLFK DUH V\PPHWULFDO )RU QRQV\PPHWULFDO FURVVVHFWLRQV WKHVH D[HV UHSUHVHQW WKH OHDVW DQG

PAGE 42

)LJXUH 'HILQLWLRQ RI D[HV DQG GLVWDQFHV IRU JHRPHWULFDO SURSHUW\ GHWHUPLQDWLRQ

PAGE 43

PRVW ULJLG GLUHFWLRQV RI WKH FURVVVHFWLRQ 7KH SULQFLSDO PRPHQWV DUH WKHQ GHILQHG ZLWK UHIHUHQFH WR WKH SULQFLSDO D[HV RU GLUHFWLRQV DV DQG f f 7KHVH TXDQWLWLHV FDQ EH REWDLQHG IURP D 0RKUnV FLUFOH FRQVWUXFWLRQ %HHU DQG -RKQVWRQ f DV VKRZQ LQ )LJXUH 7KH SULQFLSDO DQJOH EHWZHHQ WKH [D[LV DQG WKH QD[LV LV FDOFXODWHG DV \ DUF WDQ f§f f \\ [[ 7KH SULQFLSDO DUHD PRPHQWV RI LQHUWLD DUH DOVR UHDGLO\ FRPSXWHG IURP , DQG DV [[ \\ [\ DQG f f 3RODU PRPHQW RI LQHUWLD 8VHG LQ GHWHUPLQDWLRQ RI WRUVLRQDO VKHDU VWUHVVHV DQG GHIOHFWLRQV RI URXQG FURVVVHFWLRQV WKH SRODU PRPHQW RI LQHUWLD RU SRODU VHFRQG PRPHQW RI DUHD LV GHILQHG DV U G$ f

PAGE 44

)LJXUH 0RKUnV FLUFOH FRQVWUXFWLRQ IRU ORFDWLQJ SULQFLSDO PRPHQW PDJQLWXGHV DQG GLUHFWLRQV RI WKH FURVVVHFWLRQV

PAGE 45

ZKHUH U LV PHDVXUHG IURP WKH FHQWURLG RI WKH FURVVVHFWLRQ 7KH SRODU PRPHQW PD\ EH FDOFXODWHG XVLQJ WKH SUHYLRXVO\ GHWHUPLQHG YDOXHV RI WKH PRPHQWV RI LQHUWLD A DQG 8VLQJ WULJRQRPHWU\ DQG UHIHUULQJ WR )LJXUH U [ \ f 7KHUHIRUH HTXDWLRQ FDQ EH UHZULWWHQ DV [ \fG$ [ G$ \ G$ -D -D -D DQG VXEVWLWXWLQJ HTXDWLRQV DQG \LHOGV f , [[ \\ f 7KH HIIHFWLYH SRODU PRPHQW RI LQHUWLD 7KH ODVW SURSHUW\ WR EH FRQVLGHUHG LV WKH HIIHFWLYH SRODU PRPHQW RI LQHUWLD $V HI I GHQ +DUWRJ f KDV VKRZQ LQ KLV GLVFXVVLRQ RI WKH WRUVLRQ RI PDFKLQH HOHPHQWV D WRUVLRQDO VWLIIQHVV & PD\ EH GHILQHG DV WKH WRUTXH 7 SHU XQLW WZLVW 7KDW LV FI f )XUWKHU DQDO\VLV RI HOHPHQWV RI FLUFXODU FURVVVHFWLRQ OHDGV WR WKH UHVXOW WKDW & HI I f ZKHUH LV WKH VKHDU PRGXOXV RI WKH PDWHULDO DQG LV WKH HII HIIHFWLYH SRODU PRPHQW RI LQHUWLD )RU FLUFXODU FURVVVHFWLRQV A UHGXFHV WR 7KH QRQFLUFXODU FURVVVHFWLRQ UHTXLUHV D PRUH

PAGE 46

FRPSOH[ GHULYDWLRQ WR REWDLQ WKH JHRPHWULFDO GHWHUPLQDWLRQ RI WKH WRUVLRQDO VWLIIQHVV 3UDQGWOnV PHPEUDQH DQDORJ\ DV GLVFXVVHG E\ GHQ +DUWRJ f DQG &UDQGDOO f DOORZV IRU WKH GHWHUPLQDWLRQ RI DQ HIIHFWLYH SRODU PRPHQW RI LQHUWLD WKXV WKH WRUVLRQDO VWLIIQHVV RI QRQFLUFXODU &URVVVHFWLRQV PD\ VWLOO EH H[SUHVVHG DV D SURGXFW RI D PDWHULDO DQG JHRPHWULF SURSHUW\ 7KXV IRU QRQFLUFXODU VHFWLRQV HI I f 7KH HIIHFWLYH SRODU PRPHQW RI LQHUWLD LV VWLOO D PHDVXUH RI WKH DUHDn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f DUHD PRPHQWV RI LQHUWLD DQG f SULQFLSDO PRPHQWV RI LQHUWLD DQG Bf [[ \\n A A QQ .. SULQFLSDO DQJOH f SRODU PRPHQWV RI LQHUWLD -f DQG HIIHFWLYH PRPHQW RI LQHUWLD -AAf ZHUH H[WUDFWHG IURP WKH 6&$'6 FRPSXWHU

PAGE 47

UXQV IRU WKH WHQ IHPXU SDLUV 7KH OHIWWRULJKW YDULDWLRQ RI WKHVH GDWD H[FHSW IRU WKH SULQFLSDO DQJOHV WR EH GLVFXVVHG ODWHU ZHUH ILUVW HYDOXDWHG DW VHTXHQWLDO SRLQWV DORQJ WKH ERQH OHQJWK IURP b WR b RI WKH WRWDO ERQH OHQJWK LQ b LQFUHPHQWVf $V GLVn FXVVHG E\ 0LOOHU DQG 3LRWURZVNL f WKH VLGHWRVLGH GLIIHUHQFH G RI DQ\ SURSHUW\ LV GHILQHG DV G 3 3/ 35 3/ 35 f ZKHUH 3 DQG 3 DUH WKH YDOXHV RI WKH SURSHUW\ IRU WKH OHIW DQG ULJKW IHPXUV UHVSHFWLYHO\ 7KH GLIIHUHQFHV FRPSXWHG DW HDFK LQFUHPHQWDO SRVLWLRQ ZHUH DYHUDJHG WR REWDLQ WKH PHDQ YDOXH RI WKH VLGHWRVLGH GLIIHUHQFH G DQG LWV VWDQGDUG GHYLDWLRQ 6' 3 IRU HDFK SDLU RI IHPXUV 7DEOH VXPPDUL]HV WKHVH UHVXOWV 7KH DFFXPXODWHG YDOXHV REWDLQHG E\ SRROLQJ WKH UHVXOWV IURP DOO WHQ IHPXU SDLUV DUH DOVR VKRZQ 7KH VWDQGDUG GHYLDWLRQV KDYH EHHQ FRUUHFWHG IRU VPDOO VDPSOH VL]HV XVLQJ WKH %HVVHOnV &RUUHFWLRQ 0RURQH\ f )LJXUHV KDYH EHHQ FRQVWUXFWHG WR LQGLFDWH WKH RYHUDOO YDOXHV REWDLQHG IRU WKH JHRPHWULFDO SURSHUWLHV RI WKH ULJKW IHPXUV DQG WKH OHIW IHPXUV 7KH VKDGHG EDQG LQGLFDWHV WKH WRWDO UDQJH RYHU ZKLFK WKH UDZ GDWD IDOOV $ OLQH KDV EHHQ GUDZQ WR LQGLFDWH WKH FDOFXODWHG PHDQ YDOXH RI WKH SURSHUW\ IURP b WR b RI WKH WRWDO ERQH OHQJWK $OVR LQFOXGHG DUH YHUWLFDO EDUV ZKLFK UHSUHVHQW D UDQJH RI SOXV DQG PLQXV RQH VWDQGDUG GHYLDWLRQ IURP WKH PHDQ

PAGE 48

7DEOH 9DOXHV IRU WKH PHDQ VLGHWRVLGH GLIIHUHQFH G DQG VWDQGDUG GHYLDWLRQV LQ SDUHQWKHVHVf IRU 3 WKH SDLUHG IHPXUV f§ G7 G G7 G G %RQH G$ ;; Z F QQ K" GII HUI f f f f f f f &f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f 3RROHG PHDQ 'DWD 6' f f f f f f f W 3 ! ! ! Q IRU DOO FDVHV

PAGE 49

BM ‘ ‘ 3(5&(17$*( 2) 727$/ /(1*7+ L )LJXUH *URVV FRUWLFDO ERQH DUHD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV /2 /Q

PAGE 50

)LJXUH *URVV FRUWLFDO ERQH DUHD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV

PAGE 51

)LJXUH $UHD PRPHQW RI LQHUWLD ,[[ DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV /R

PAGE 52

r9rY 3(5&(17$*( 2) 727$/ /(1*7+ )LJXUH $UHD PRPHQW RI LQHUWLD ,[[ DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV 8f

PAGE 53

nH R *2 &8 )LJXUH $UHD PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV

PAGE 54

)LJXUH $UHD PRPHQW RI LQHUWLD ,\\ SRVLWLRQ IRU WKH ULJKW IHPXUV DV D IXQFWLRQ RI IV R

PAGE 55

f§L L LB 3(5&(17$*( 2) 727$/ /(1*7+ )LJXUH 3ULQFLSDO PRPHQW RI LQHUWLD A RI SRVLWLRQ IRU WKH OHIW IHPXUV DV D IXQFWLRQ

PAGE 56

f§W / 3(5&(17$*( 2) 727$/ /(1*7+ )LJXUH 3ULQFLSDO PRPHQW RI LQHUWLD A DV D IXQFW RI SRVLWLRQ IRU WKH ULJKW IHPXUV

PAGE 57

f§LW L 3(5&(17$*( 2) 727$/ /(1*7+ )LJXUH 3ULQFLSDO PRPHQW RI LQHUWLD RI SRVLWLRQ IRU WKH OHIW IHPXUV DV D IXQFWLRQ X!

PAGE 58

)LJXUH f§O 3(5&(17$*( 2) 727$/ /(1*7+ 3ULQFLSDO PRPHQW RI LQHUWLD ,m DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV 2r

PAGE 59

)LJXUH 3RODU PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV

PAGE 60

f§ , 3(5&(17$*( 2) 727$/ /(1*7+ )LJXUH 3RODU PRPHQW RI LQHUWLD DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV 2n

PAGE 61

S R HQ FX 3(5&(17$*( 2) 727$/ /(1*7+ )LJXUH (IIHFWLYH SRODU PRPHQW RI LQHUWLD -HII IXQFWLRQ RI SRVLWLRQ IRU WKH OHIW IHPXUV DV D W

PAGE 62

m BL } L O L L 3(5&(17$*( 2) 727$/ /(1*7+ )LJXUH (IIHFWLYH SRODU PRPHQW RI LQHUWLD -HII DV D IXQFWLRQ RI SRVLWLRQ IRU WKH ULJKW IHPXUV 3n RR

PAGE 63

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nV W 7HVW 0RURQH\ f ZDV DSSOLHG WR FRPSDUH WKH UHVXOWLQJ GLVn WULEXWLRQV ZLWK WKH K\SRWKHVLV WKDW WKHUH LV QR ULJKWOHIW ELDV LQ WKH JHRPHWULF SURSHUWLHV LH GA ,Q WKLV SDUWLFXODU FDVH WKH YDOXH RI W LV LQGLFDWLYH RI WKH UDWLR RI WKH PHDQ WR WKH VWDQGDUG GHYLDWLRQ 7KH QRQ]HUR PHDQ YDOXHV VHHQ LQ 7DEOH DUH TXLWH OLNHO\ WR RFFXU E\ FKDQFH VLQFH WKH\ DUH VPDOOHU WKDQ WKH GDWDnV LQKHUHQW VFDWWHU UHSUHVHQWHG E\ WKH VWDQGDUG GHYLDWLRQV 7KH DFWXDO VWDWLVWLFDO UHVXOWV KDYH EHHQ LQFOXGHG LQ WKH WDEOH DQG LQGLFDWH WKDW 3 WKH SUREDELOLW\ RI W EHLQJ WKLV ODUJH RU ODUJHU E\ FKDQFH LV JUHDWHU WKDQ LQ DOO FDVHV 2QH FDQ WKXV FRQFOXGH ZLWK FRQVLGHUDEOH FHUWDLQW\ WKDW WKHUH ZDV QR ULJKWOHIW ELDV LQ WKH JHRPHWULF SURSHUWLHV RI WKH IHPXUV LQYHVWLJDWHG

PAGE 64

)LJXUH 3ULQFLSDO DQJOH RULHQWDWLRQ DQG UHODWLYH PDJQLWXGH RI WKH SULQFLSDO PRPHQWV DV D IXQFWLRQ RI SRVLWLRQ IRU D UHSUHVHQWDWLYH SDLU RI IHPXUV IURP JURXS WKH GDUNHU OLQH LQGLFDWHV WKH ODUJHU PRPHQW RI LQHUWLD f P9

PAGE 66

)LJXUH 3ULQFLSDO DQJOH RULHQWDWLRQ DQG UHODWLYH PDJQLWXGH RI WKH SULQFLSDO PRPHQWV DV D IXQFWLRQ RI SRVLWLRQ IRU D UHSUHVHQWDWLYH SDLU RI IHPXUV IURP JURXS ,, WKH GDUNHU OLQH LQGLFDWHV WKH ODUJHU PRPHQW RI LQHUWLD f f QQ

PAGE 67

',67$/ 352;,0$/

PAGE 68

)LJXUH 3ULQFLSDO DQJOH RULHQWDWLRQ DQG UHODWLYH PDJQLWXGH RI WKH SULQFLSDO PRPHQWV DV D IXQFWLRQ RI SRVLWLRQ IRU D UHSUHVHQWDWLYH SDLU RI IHPXUV IURP JURXS ,,, WKH GDUNHU OLQH LQGLFDWHV WKH ODUJHU PRPHQW RI LQHUWLD f UP

PAGE 69

',67$/ 352;,0$/

PAGE 70

0LOOHU DQG 3LRWURZVNL f VKRZHG WKDW RQH FDQ H[SHFW WKH VWDQGDUG GHYLDWLRQ RI WKH VLGHWRVLGH GLIIHUHQFH LQ WKH WRUVLRQDO VWUHQJWK RI SDLUHG DQLPDO ERQHV WR EH DERXW b 7KH UHVXOWV REWDLQHG DERYH IRU WKH VWDQGDUG GHYLDWLRQ RI WKH VLGHWRVLGH GLIIHUHQFH LQ WKH SRODU PRPHQW RI LQHUWLD ZDV RI WKH RUGHU RI b 7KLV LQGLFDWHV WKDW ZKLOH WKH JHRPHWULF YDULDELOLW\ KDV VRPH HIIHFW RQ WKH REVHUYHG VWUHQJWK YDULDWLRQV RI ZKROH ERQHV LW GRHV QRW DFFRXQW IRU DOO RI LW 7KH VWUHQJWK RI ZKROH ERQHV KRZHYHU GHSHQGV RQ ERWK JHRPHWULF DQG PDWHULDO SURSHUWLHV 5HLOO\ f UHSRUWHG WKDW RQH FDQ H[SHFW YDULDELOLW\ LQ XOWLPDWH WRUVLRQDO VWUHQJWK RI ERQ\ PDWHULDO WR EH DSSUR[LPDWHO\ b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n PDWHO\ b RI WKH ERQH OHQJWK 7KLV LV ZKHUH WKH LVWKPXV RU QDUURZHVW VHFWLRQ RI WKH LQWUDPHGXOODU\ FDQDO LV IRXQG 7KH FRUWLFDO DUHD WKHQ GHFUHDVHV PRQRWRQLFDOO\ WR WKH NQHH $W WKH PRVW GLVWDO GDWD SRLQW WDNHQ b IURP SUR[LPDO WKH DUHD LV DSSUR[LPDWHO\ FP 7KLV UHVXOWLQJ DUHD MXVW SUR[LPDO WR WKH NQHH LV VOLJKWO\ ODUJHU WKDQ WKH SUR[LPDO WLELD DUHD MXVW GLVWDO WR WKH NQHH SUHn VHQWHG E\ 0LQQV HW D= f DQG 3L]LDOL HW D= f 7KXV

PAGE 71

LJQRULQJ WKH GLVFRQWLQXLW\ DW WKH NQHH WKH ORZHU OLPE DSSHDUV WR EH D WDSHUHG FROXPQ RI ERQH PDVV 7KLV VKDSH UHGXFHV WKH DPRXQW RI ERQH DSSHDULQJ GLVWDOO\ DQG UHVXOWV LQ D PLQLPL]DWLRQ RI WKH ERQHVn FRQWULEXWLRQ WR WKH PDVV PRPHQW RI LQHUWLD RI WKH OLPE DERXW WKH KLS 7KXV FRPSDUHG WR D OLPE ZLWK ERQH PDVV HYHQO\ GLVWULEXWHG DORQJ LWV OHQJWK WKH WRUTXHV QHFHVVDU\ WR DFFHOHUDWH WKH H[WUHPLW\ GXULQJ ORFRPRWLRQ DUH UHGXFHG ZLWK WKH UHVXOW WKDW WKH HQHUJ\ QHFHVVDU\ WR DFFHOHUDWH WKH OLPE LV DOVR GHFUHDVHG DQG KLJKHU HIILFLHQF\ FDQ EH PDLQWDLQHG 7KH VOLJKW LQFUHDVH LQ DUHD VHHQ LQ WKH SUR[LPDO SRUWLRQ RI WKH IHPXU LV GXH WR WKH SUHVHQFH RI WKH JOXWHDO WXEHURVLW\ DQG WKH EHJLQQLQJ RI WKH OLQHD DVSHUD 7KHVH DUHDV DOORZ IRU WKH DWWDFKPHQW RI WKH PXVFOH JURXSV ZKLFK ZKLOH SURYLGLQJ IRU URWDWLRQV DERXW WKH KLS DOVR DSSO\ VLJQLILFDQW D[LDO ORDGLQJ LQ WKH UHJLRQ GXH WR WKHLU SRRU PHFKDQLFDO DGYDQWDJH 7KHVH D[LDO ORDGV WHQG WR UHGXFH WKH WHQVLOH VWUHVVHV VHW XS E\ EHQGLQJ RI WKH IHPXU GXH WR WKH ORDGLQJ RI WKH IHPRUDO KHDG EXW DOVR UHTXLUHV WKH QHHG IRU DGGLWLRQDO ERQH DUHD WR PDLQWDLQ VXLWDEOH FRPSUHVVLYH VWUHVV OHYHOV $UHD PRPHQWV RI LQHUWLD ,W LV LPSRUWDQW WR QRWH DW WKH RXWVHW RI WKLV GLVFXVVLRQ RI WKH DUHD PRPHQWV RI LQHUWLD )LJXUHV f DQG SRODU PRPHQW RI LQHUWLD )LJXUHV DQG f WKDW WKHUH LV D UHODWLRQVKLS EHWZHHQ WKH WZR DV VKRZQ LQ HTXDWLRQ f 7KHVH SURSHUWLHV LQIOXHQFH WKH VWUHVV OHYHOV FDXVHG E\ GLIIHUHQW IRUPV RI ORDGLQJ QDPHO\ EHQGLQJ DQG WRUVLRQ 7KHVH SURSHUWLHV DOVR LQIOXHQFH WKH ULJLGLW\ RI WKH V\VWHP XQGHU WKHVH GLIIHUHQW ORDGLQJ FRQILJXUDWLRQV

PAGE 72

7KH EHQGLQJ VWUHVVHV R GHYHORSHG LQ WKH RXWHU VXUIDFH RI D EHDP ZLWK DQ DSSOLHG EHQGLQJ PRPHQW 0 PD\ EH UHSUHVHQWHG E\ f ZKHUH F LV WKH GLVWDQFH IURP WKH FHQWURLG RI WKH VHFWLRQ WR WKH RXWHU VXUIDFH DQG LV WKH DUHD PRPHQW RI LQHUWLD 7KH TXDQWLW\ F LV RIWHQ UHIHUUHG WR DV WKH VHFWLRQ PRGXOXV = RI WKH EHDP )LJXUH VKRZV WKH UHVXOWV REWDLQHG IURP WKH FDOFXODWLRQ RI WKH VHFWLRQ PRGXOXV IRU WZR UHSUHVHQWDWLYH IHPXUV $OVR LQFOXGHG LV WKH UHVXOW REWDLQHG E\ .RFK f LQ KLV GHWDLOHG DQDO\VLV RI D VLQJOH IHPXU :KLOH WKH PDJQLWXGHV DUH TXLWH VLPLODU WKH UHVXOWV REWDLQHG KHUH LQGLFDWH WKDW WKH VHFWLRQ PRGXOXV GHFUHDVHV VOLJKWO\ IURP SUR[LPDO WR GLVWDO .RFK f DQG 5\ELFNL HW DO f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

PAGE 73

)LJXUH 6HFWLRQ PRGXOXV = DV D IXQFWLRQ RI OHQJWK DORQJ WKH IHPXU IRU WZR VDPSOH IHPXUV FRPSDUHG WR GDWD GHULYHG IURP .RFK f

PAGE 74

LQVHUWLRQV DQG SURYLGH DUHDV IRU DWWDFKPHQW RI WKH OLJDPHQWV ZKLFK VWDELOL]H WKH NQHH MRLQW DUH UHVSRQVLEOH IRU WKH LQFUHDVH LQ [[ LQ WKH ODVW b4b RI OHQJWK 7KLV UHVXOWV LQ DQ LQFUHDVH LQ WKH IOH[XUDO ULJLGLW\ RI WKH IHPXU LQ WKLV UHJLRQ DV ZHOO 7KH DQWHURSRVWHULRU PRPHQW RI LQHUWLD )LJXUHV DQG f LV VPDOOHU LQ PDJQLWXGH WKDQ LQ WKH SUR[LPDO IHPXU ,W DOVR GHFUHDVHV LQ PDJQLWXGH RYHU WKH SUR[LPDO b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f VLPLODU WR WKDW IRU WKH YDOXHV RI ;; +RZHYHU WKH RULHQWDWLRQ RI WKH SULQFLSDO D[HV ZDV TXLWH UHPDUNDEOH 6WXG\ RI WKH SULQFLSDO D[HV RI WKH WHQ IHPXU SDLUV OHG WR WKH REVHUn YDWLRQ WKDW WKH IHPRUDO FURVVVHFWLRQDO VKDSH DW WKH PLGVKDIW DSSHDUV WR IDOO LQWR WKUHH EDVLF FDWHJRULHV $ PLGVKDIW VHFWLRQ IURP WKH OHIW IHPXU RI UHSUHVHQWDWLYHV RI HDFK RI WKHVH JURXSV KDV EHHQ LQFOXGHG LQ )LJXUHV DORQJ ZLWK VFDOHG RUWKRJRQDO OLQHV UHSUHVHQWLQJ WKH UHODWLYH PDJQLWXGHV RI WKH SULQFLSDO PRPHQWV 7KH KHDYLHU RI WKH WZR OLQHV LQGLFDWHV WKH ODUJHU YDOXH f QQ 2I WKH WHQ IHPXU SDLUV H[DPLQHG IRXU SDLUV ZHUH IRXQG WR EH

PAGE 75

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n LQJ LQ D SULQFLSDO DQJOH RI DSSUR[LPDWHO\ r WKH DQWHURSRVWHULRU GLUHFWLRQf ,Q WKH GLVWDO SRUWLRQ WKH PDMRU D[LV URWDWHV DQG FRPHV WR DQ DOPRVW PHGLRODWHUDO RULHQWDWLRQ FRUUHVSRQGLQJ ZHOO ZLWK WKH IHPRUDO FRQG\OHV 7KH VHFRQG JURXS )LJXUH KDV DQ HOOLSWLFDO FURVVVHFWLRQ DW WKH PLGVKDIW DV ZHOO EXW LV SHUKDSV OHVV SURQRXQFHG WKDQ LQ WKH SUHYLRXV JURXS +RZHYHU WKH D[LV EHJLQV LQ D PRUH PHGLRODWHUDO SRVLWLRQ DQG UHPDLQV DV VXFK WKURXJKRXW WKH OHQJWK RI WKH IHPXU 7KXV WKH PRUH ULJLG GLUHFWLRQ FRUUHVSRQGLQJ WR WKH GLUHFWLRQ LV QRZ r IURP WKDW RI WKH ILUVW JURXS +HUH DJDLQ WKH ULJKWOHIW V\PPHWU\ LV PDLQWDLQHG 7KH WKLUG JURXS )LJXUH KDV D SURILOH LQnWKH SUR[LPDO VHJPHQW ZKLFK LV VLPLODU LQ PDJQLWXGH DQG GLUHFWLRQ WR WKH VHFRQG W\SH GHVFULEHG DERYH +RZHYHU DV WKH PLGVKDIW LV DSSURDFKHG WKH PDJQLWXGHV RI A DQG DUH DOPRVW HTXDO LQ FRQWUDVW WR JURXSV DQG ,, DQG GHWHUPLQDWLRQ RI WKH SULQFLSDO GLUHFWLRQ LV XQFHUWDLQ GXH WR WKH UHODWLYHO\ VPDOO VL]H RI WKH 0RKUnV FLUFOH LQ FRPSDULVRQ

PAGE 76

WR WKH PDJQLWXGHV RI DQG 7KXV WKH ERQH KDV QR SUHIHUHQ [[ \\ U WLDOO\ ULJLG GLUHFWLRQ XQWLO WKH IHPRUDO FRQG\OHV DUH UHDFKHG 7KH RQO\ JHQHUDOL]DWLRQ ZKLFK FDQ EH PDGH KHUH LV WKDW LQ DOO JURXSV WKH SULQFLSDO GLUHFWLRQ LV PHGLRODWHUDO DW WKH IHPRUDO FRQG\OHV $Q DWWHPSW WR H[SODLQ WKH DERYH REVHUYDWLRQV 7HG WR WKH FRQVXOWDWLRQ RI WKH DQWKURSRORJLFDO OLWHUDWXUH &RPDV f PDNHV VHYHUDO REVHUYDWLRQV UHODWLYH WR WKH SRLQW RI IHPRUDO VKDSH 7KH OLQHD DVSHUD LQ WKH IHPXU RI SHRSOH RI WKH PRUH PRGHUQ FLYLOL]DWLRQV LV PRUH SURQRXQFHG LQ PDOHV WKDQ IHPDOHV 7KH PRUH UREXVW VNHOHWRQ RI WKH PDOH KDV EURDG DUHDV IRU PXVFXODU DWWDFKPHQW DQG WKH OLQHD DVSHUD PD\ GHYHORS LQ VXFK D ZD\ DV WR SURGXFH D ODUJH URXJK FUHVW VKDSHG SURWUXVLRQ QRW XQOLNH WKDW VHHQ LQ JURXS )LJXUH f $QWHURSRVWHULRU IODWWHQLQJ DV UHSUHVHQWHG E\ WKH VHFRQG JURXS SUHVHQWHG DERYH LV REVHUYHG LQ WKH IHPDOH VNHOHWRQ 7KLV REVHUYDWLRQ ZDV UHLWHUDWHG E\ 0DSOHV f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

PAGE 77

SULQFLSDO D[LV RULHQWHG LQ WKDW SODQH 8QIRUWXQDWHO\ WKH VH[ RI WKH IHPRUDO VDPSOHV VWXGLHG KHUH ZDV XQNQRZQ DQG VXEVWDQWLDWLRQ RI WKLV K\SRWKHVLV PXVW UHPDLQ IRU IXUWKHU VWXG\ 3RODU PRPHQW RI LQHUWLD 7KH SRODU PRPHQW RI LQHUWLD LV GLVWULEXWHG LQ D PDQQHU VLPLODU WR )LJXUHV DQG f GXH WR LWV GHSHQGHQFH RQ DQG DV GLVFXVVHG LQ WKH SUHYLRXV [[ \\ U VHFWLRQ ,Q WKH SUR[LPDO IHPXU WKH UHODWLYH LQFUHDVH LQ WKLV SURSHUW\ DSSHDUV WR EH GXH WR WKH VHYHUDO LQWHUQDO DQG H[WHUQDO URWDWRUV RI WKH KLS ZKLFK DWWDFK LQ WKLV DUHD DQG DSSO\ WRUVLRQDO ORDGLQJ WR WKH ERQH ,Q WKH PLGVKDIW UHJLRQ WKH YDOXHV IRU GR QRW YDU\ FRQn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

PAGE 78

:KLOH WKH SURILOH RI WKH GLVWULEXWLRQ LV VLPLODU WR WKDW IRXQG IRU WKH SRODU PRPHQW RI LQHUWLD JLYHQ LQ WKH SUHYLRXV VHFWLRQ WKH PDJQLWXGHV RI DUH DSSUR[LPDWHO\ bb OHVV 7KH JUHDWHVW GLIIHUHQFHV DSSHDU LQ WKH GLVWDO SRUWLRQ RI WKH ERQH 7KXV WKH QRQV\PPHWU\ RI WKH ERQH FURVVVHFWLRQV UHVXOW LQ D GHFUHDVH LQ WKH DFWXDO WRUVLRQDO ULJLGLW\ RI WKH ERQH 'LVFXVVLRQ RI WKH 2EVHUYHG 6FDWWHU LQ 3URSHUWLHV 7KH PDJQLWXGH RI WKH REVHUYHG VFDWWHU LQ WKH JHRPHWULF FDOn FXODWLRQV FKDUDFWHUL]HG E\ WKH VWDQGDUG GHYLDWLRQV UDQJHG XS WR bb RI WKH PHDQ YDOXHV 7KH VRPHZKDW ODUJHU VWDQGDUG GHYLDWLRQV SUHVHQW DW WKH HQGV RI WKH ERQHV DUH GXH WR WKH GLIILFXOW\ LQ GLVFHUQLQJ WKH FRUWLFRFDQFHOOXV ERQH LQWHUIDFH RI WKH LQWUDn PHGXOODU\ FDQDO ,Q OLJKW RI WKH GLVFXVVLRQ RI WKH LQKHUHQW GLIIHUHQFHV LQ LQGLYLGXDOV SUHVHQWHG DERYH WKLV ODUJH UDQJH GRHV QRW VHHP XQUHDVRQDEOH 7KH PD[LPXP HUURU RI WKH 6&$'6 FRPSXWHU SURJUDP IRU FDOFXODWLRQV RQ ZHOO GHILQHG JHRPHWULHV KDV EHHQ VKRZQ WR EH b RU OHVV 3LRWURZVNL DQG .HOOPDQ f ,Q DQ DWWHPSW WR TXDQWLI\ WKH UHPDLQLQJ b RI YDULDWLRQ QRQGLPHQVLRQDOL]DWLRQ RI WKH REVHUYHG SURSHUWLHV XVLQJ RWKHU GLPHQVLRQV RI WKH ERQHV ZDV FDUULHG RXW ,W ZDV KRSHG WKDW D JHQHUDO FKDUDFWHULVWLF VKDSH RI HDFK RI WKH SURSHUW\ GLVWULEXWLRQV DORQJ WKH ERQH OHQJWK FRXOG EH JHQHUDWHG LQ WKLV ZD\ 8VLQJ WKH RYHUDOO OHQJWK / DQG WKH OHVVHU WURFKDQWHU WR IHPRUDO FRQG\OH PDMRU OHQJWK OHVVHUf )LJXUH f DV QRQGLPHQVLRQDOL]DWLRQ SDUDPHWHUV GLG QRW KRZHYHU UHGXFH WKH PDJQLWXGHV RI WKH VFDWWHU ,Q IDFW

PAGE 79

DV FDQ EH VHHQ LQ )LJXUHV DQG WKHUH LV QR FRUUHODWLRQ EHWZHHQ WKHVH OHQJWKV DQG WKH DYHUDJH FDOFXODWHG ERQH DUHD 7KHUH LV DOVR QR FRUUHODWLRQ EHWZHHQ WKH DYHUDJH DUHD DQG WKH PLGVKDIW RXWHU GLDPHWHU )LJXUH f 2QH PXVW FRQFOXGH WKHUHIRUH WKDW WKHUH DUH VLJQLILFDQW GLIIHUHQFHV LQ WKH ERQH SURSHUWLHV RI GLIn IHUHQW LQGLYLGXDOV DQG WKDW WKHVH GLIIHUHQFHV DUH GHSHQGHQW RQ PDQ\ YDULDEOHV 7KLV LV FRQVLVWHQW ZLWK WKH WKRXJKW &RPDV f WKDW LW LV EXLOG DQGRU ERG\ ZHLJKW VH[ DJH DQG JHQHUDO SK\VLFDO FRQn GLWLRQ DQG QRW KHLJKW DORQH ZKLFK GHWHUPLQHV WKH VL]H DQG VKDSH RI DQ LQGLYLGXDOV ERQHV 8QIRUWXQDWHO\ WKLV LQIRUPDWLRQ ZDV QRW DYDLODEOH IRU FRQVLGHUDWLRQ LQ WKLV VWXG\ &RQFOXVLRQ (YHQ ZLWK WKH VXEVWDQWLDO VFDWWHU SUHVHQW LQ WKH JHRPHWULF SURSHUWLHV GLVFXVVHG LQ WKLV FKDSWHU WKH JHQHUDO SURSHUW\ GLVn WULEXWLRQ SURILOHV DUH YDOLG 6HYHUDO FRQFOXVLRQV FDQ EH PDGH IURP WKH UHVXOWV SUHVHQWHG 1R VWDWLVWLFDOO\ VLJQLILFDQW ULJKWOHIW ELDV LV HYLGHQW LQ WKH JHRPHWULF SURSHUWLHV SUHVHQWHG 7KH JHRPHWULF SURSHUWLHV FDOFXODWHG XVLQJ WKH 6&$'6 FRPSXWHU SURJUDP DGMRLQV WKH WLELDO LQIRUPDWLRQ SUHn VHQWHG E\ 0LQQV HW DO f DQG 3L]LDOL HW DL f 7KH DUHD GLVWULEXWLRQ SURILOHV GR QRW PLUURU WKH RXWn ZDUG DSSHDUDQFH RI WKH IHPXU ,Q IDFW WKH HQWLUH ORZHU OLPE KDV D WDSHUHG FROXPQ GLVWULEXWLRQ RI ERQH DUHD ZLWK WKH JUHDWHU DUHD DSSHDULQJ LQ WKH SUR[LPDO IHPXU 7KLV GLVWULEXWLRQ RI ERQH DLGV LQ HIILFLHQW DPEXODWLRQ 7KH IODUHG VKDSH RI WKH IHPXU LV PRUH SURQRXQFHG LQ WKH PHGLRODWHUDO SODQH WKDQ LQ WKH DQWHURSRVWHULRU SODQH DQG DQG WKH UHVSHFWLYH DUHD PRPHQWV RI LQHUWLD PLUURU WKLV R6]ZDUG DSSHDUDQFH

PAGE 80

29(5$// )(025$/ /(1*7+ FPf 2 $S YV /PDMU U f $MB YV /PDMRU f§ U 2 f 2 f f 2 2 f R R ‘ ‘ f§ $9(5$*( $5($6 FPf M )LJXUH *UDSKLFDO UHSUHVHQWDWLRQ RI WKH RYHUDOO IHPRUDO OHQJWK YHUVXV DYHUDJH FRUWLFDO ERQH DUHD ZLWK WKH UHVSHFWLYH FRUUHODWLRQ FRHIILFLHQWV U IRU WKH ULJKW DQG OHIW IHPXUV 21 21

PAGE 81

( R 2f 6! ;f 2U 2 $S YV /_HVVHU U f§ f $O 96 /OHVVHU U 2 2 2 2 2 2 f R R II $9(5$*( $5($6 FPf )LJXUH *UDSKLFDO UHSUHVHQWDWLRQ RI WKH OHVVHU WURFKDQWHU WR IHPRUDO FRQG\OH OHQJWK /LHVVHU YHUVXV DYHUDJH FRUWLFDO ERQH DUHD ZLWK UHVSHFWLYH FRUUHODWLRQ FRHIILFLHQWV U IRU WKH ULJKW DQG OHIW IHPXUV 2n

PAGE 82

0,'6+$)7 ',$0(7(56 FPf U 2 $S YV GS f§ U f $>B YV G O f§ U 2 R R f f f R f R R + $9(5$*( $5($6 FPf f RR BO )LJXUH *UDSKLFDO UHSUHVHQWDWLRQ RI WKH RXWHU PLGVKDIW GLDPHWHU YHUVXV DYHUDJH FRUWLFDO ERQH DUHD ZLWK WKH UHVSHFWLYH FRUUHODWLRQ FRHIILFLHQWV U IRU WKH ULJKW DQG OHIW IHPXUV RQ &2

PAGE 83

7KH SULQFLSDO GLUHFWLRQV ZHUH IRXQG WR IDOO LQWR WKUHH PDMRU JURXSV 7KHUH DUH WKRVH IHPXUV ZKLFK DUH PRVW ULJLG LQ WKH DQWHURSRVWHULRU SODQH WKRVH ZKLFK DUH PRVW ULJLG LQ WKH PHGLRODWHUDO SODQH DQG WKRVH ZKLFK DSSHDU WR KDYH QR SUHIHUHQWLDO RULHQWDWLRQ $QWKURn SRORJLFDO LQIRUPDWLRQ DYDLODEOH VXJJHVWV WKDW WKHVH GLIIHUHQFHV DUH VH[OLQNHG 7KH FDOFXODWHG HIIHFWLYH SRODU PRPHQW RI LQHUWLD XVLQJ WKH PHPEUDQH DQDORJ\ LQGLFDWHV WKDW WKH QRQFLUFXODULW\ RI WKH ERQH FURVVVHFWLRQ UHVXOWV LQ WRUVLRQDO ULJLGLWLHV ZKLFK DUH bb OHVV WKDQ LV SUHGLFWHG E\ WKH V\PPHWULFDO SRODU PRPHQW RI LQHUWLD FDOFXODWLRQV 7KH JHRPHWULF SURSHUW\ SURILOHV DUH GLVWULEXWHG DORQJ WKH ERQH OHQJWK LQ D KLJKO\ HIILFLHQW XVH RI WKH PDWHULDO VR WKDW VWUHQJWK WR ZHLJKW RSWLPL]DWLRQ DSSHDUV WR EH PDLQWDLQHG XQGHU FRPSOH[ ORDGLQJ FRQILJXUDWLRQV 7KH UHVXOWV ZKLFK KDYH EHHQ REWDLQHG LQ WKLV JHRPHWULF VWXG\ RI WKH IHPXU DUH FRQVLVWHQW ZLWK WKH VPDOO DPRXQW RI LQIRUPDWLRQ DYDLODEOH LQ WKH OLWHUDWXUH DQG WKH GDWD SUHVHQWHG LQ )LJXUHV FDQ EH XVHG LQ WKH VXEVHTXHQW VWUXFWXUDO PRGHOLQJ RI WKLV ERQH 7KH ,QWUDPHGXOODU\ 1DLO 0DWHULDOV DQG 0HWKRGV 'HWHUPLQDWLRQ RI WKH JHRPHWULF SURSHUWLHV RI ILYH LQWUDn PHGXOODU\ QDLOV ZDV FDUULHG RXW 7KHVH LQFOXGHG IRXU QDLOV RI GLIIHULQJ W\SHV )LJXUH f LQWHQGHG IRU LPSODQWDWLRQ LQ D PP UHDPHG LQWUDPHGXOODU\ FDQDO PP 6FKQHLGHU PP .XQWVFKHU PP [ PP +DQVHQ6WUHHW DQG PP )OXWHGf $ PP )OXWHG QDLO LQWHQGHG IRU VXEVHTXHQW XVH LQ DQ LQ YLWUR RVWHRV\QWKHVLV PRGHO ZDV DOVR SURFHVVHG

PAGE 84

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

PAGE 85

K &0 FH F + R &+ 8L 2 FH ( R Of§ FF /O( XB R FR f§ ] /8 4& Bf 2 4 \ \ nOO n /(*(1' k k k )/87(' PPf +$16(1675((7 PPf 6&+1(,'(5 PPf .8176&+(5 PPf )LJXUH &DOFXODWHG JHRPHWULFDO SURSHUWLHV IRU VHYHUDO LQWUDPHGXOODU\ QDLO VKDSHV DOO QDLOV DUH LQWHQGHG IRU XVH LQ D QRPLQDO PP UHDPHG PHGXOODU\ FDQDOf

PAGE 86

7DEOH *HRPHWULF SURSHUWLHV RI WKH PP )OXWHG W\SH QDLO ZLWK DQG ZLWKRXW IOXWHV PP 1DLO :LWK )OXWHV :LWKRXW )OXWHV $UHD FPf FPf ;; FPf \\ M FPf [Q FPf ; FPf

PAGE 87

'LVFXVVLRQ 7KH UHVXOWV REWDLQHG XVLQJ WKH 6&$'6 FRPSXWHU SURJUDP DUH DSSUR[LPDWH GXH WR WKH GLIILFXOWLHV LQKHUHQW LQ SURFHVVLQJ WKH LQWULFDWH JHRPHWU\ RI WKHVH FURVVVHFWLRQV ,QGHSHQGHQW FDOn FXODWLRQV LQGLFDWH WKDW WKH UHVXOWV REWDLQHG DUH ZLWKLQ b RI WKH H[DFW YDOXHV 7KH )OXWHG QDLO KDV WKH ODUJHVW YDOXHV IRU DOO WKH JHRPHWULF SURSHUWLHV GHWHUPLQHG 7KLV KDV D SURQRXQFHG HIIHFW RQ LWV UHODWLYH EHQGLQJ DQG WRUVLRQDO ULJLGLW\ ,WV YHU\ KLJK FURVVVHFWLRQDO V\PPHWU\ DOVR PDNHV LW LQVHQVLWLYH WR WKH RULHQWDn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b ZLWK WKH DGGLWLRQ RI WKH IOXWHV WR WKH FURVVVHFWLRQ ZKLOH LQFUHDVLQJ WKH DUHD RI WKH VHFWLRQ E\ RQO\ b ,W LV LPSRUWDQW WR QRWH WKDW RQO\ RQH VDPSOH RI HDFK QDLO W\SH ZDV SURFHVVHG 0HDVXUHPHQW RI WKH RXWHU GLPHQVLRQV RI D IHZ RWKHU VDPSOHV RI VLPLODU QDLOV IURP GLIIHUHQW PDQXIDFWXUHUV LQGLFDWH WKDW WKHUH LV FRQVLGHUDEOH GLIIHUHQFH LQ WKH FURVVVHFWLRQV RI QDLOV PDGH E\ GLIIHUHQW FRPSDQLHV &KDQJHV LQ WKH ZDOO WKLFNQHVV RI D

PAGE 88

.XQWVFKHU QDLO IRU H[DPSOH ZLWKRXW DOWHULQJ WKH RXWVLGH GLPHQVLRQV FDXVHV VLJQLILFDQW FKDQJHV LQ WKH IOH[XUDO ULJLGLW\ RI WKH GHYLFH 7KH SUHOLPLQDU\ UHVXOWV UHSRUWHG KHUH DUH VXIILFLHQW WR DOORZ IRU YHULILFDWLRQ RI WKH RVWHRV\QWKHVLV PRGHOLQJ WR EH FDUULHG RXW DW WKLV WLPH +RZHYHU D PRUH GHWDLOHG VWXG\ LV LQGLFDWHG VR WKDW WKH HIIHFW RI LPSODQW YDULDELOLW\ RQ JHRPHWULF SURSHUWLHV RI WKH GHYLFHV FDQ EH DFFXUDWHO\ GHWHUPLQHG

PAGE 89

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

PAGE 90

)LJXUH 6FKHPDWLF UHSUHVHQWDWLRQ RI WKH SUHOLPLQDU\ LGHDOL]HG RVWHRV\QWKHVLV PRGHO ZLWK HTXDOO\ VSDFHG IRXUSRLQW H[WHUQDO ORDGLQJ DQG SRLQW FRQWDFW UHDFWLRQV EHWZHHQ WKH QDLO DQG ERQH IUDJPHQWV

PAGE 91

)UHH %RG\ 'LDJUDP RI WKH 2XWHU 7XEH 'XH WR WKH V\PPHWU\ RI WKLV V\VWHP D IUHH ERG\ GLDJUDP IRU KDOI RI WKH RXWHU WXEH LV FRQVWUXFWHG DV VKRZQ LQ )LJXUH :LWK DQ DSSOLHG ORDG 3 DW HDFK RI WKH H[WHUQDO ORDGLQJ SRLQWV DQG URG UHDFWLRQV DQG DV VKRZQ YHUWLFDO IRUFH HTXLOLEULXP \LHOGV 3 5 5 f DQG f 6WDWLF PRPHQW HTXLOLEULXP DERXW WKH OHIW HQG RI WKH WXEH \LHOGV VR WKDW 5 f§ 5A 5 f f )UHH %RG\ 'LDJUDP RI WKH ,QQHU 5RG $V VKRZQ LQ )LJXUH D IUHH ERG\ GLDJUDP RI WKH LQQHU URG FDQ EH FRQVWUXFWHG ZLWK UHDFWLRQ ORDGV DV VKRZQ 1RWH WKH LGHDOL]HG SRLQW FRQWDFWV RI WKH QDLOERQH LQWHUDFWLRQ 'HIOHFWLRQ (TXDWLRQV 7KH IOH[XUDO ULJLGLWLHV RI WKH LQQHU URG 1f DQG WKH RXWHU WXEH %f DUH GHILQHG DV 1 ( Q Q f % ? K f

PAGE 92

3 )LJXUH )UHH ERG\ GLDJUDP RI KDOI RI WKH RXWHU WXEH ZLWK H[WHUQDO DQG SRLQW FRQWDFW UHDFWLRQ ORDGV )LJXUH )UHH ERG\ GLDJUDP RI WKH LQQHU URG ZLWK LGHDOL]HG SRLQW FRQWDFW UHDFWLRQV

PAGE 93

ZKHUH ( DQG ( DUH WKH PRGXOXV RI HODVWLFLW\ RI WKH LQQHU URG DQG Q E RXWHU WXEH UHVSHFWLYHO\ DQG DQG DUH WKH DUHD PRPHQWV RI LQHUWLD IRU WKH LQQHU URG DQG RXWHU WXEH UHVSHFWLYHO\ 7KH UHDFWLRQ ORDGV RQ WKH URG DUH DSSOLHG WR HDFK HQG DQG WKH PLGVSDQ RI WKH FRPSRQHQW )LJXUH f 7KH PLGVSDQ GHIOHFWLRQ ] IRU WKH WKUHH SRLQW ORDGHG VSDQ LQ WKLV FRQILJXUDWLRQ FDQ EH ZULWWHQ 5RDUN f DV 3/ 1 f ZKHUH 3 LV WKH ORDG DSSOLHG WR WKH RXWHU WXEH RI OHQJWK / DW LWV IRXU ORDGLQJ SRLQWV 6LPLODUO\ IURP WKH RXWHU WXEH IUHH ERG\ GLDJUDP )LJXUH f DQ DQJXODU GHIOHFWLRQ RI WKH IUDFWXUH VXUIDFH UHODWLYH WR LWV RULJLQDO FHQWHUOLQH FDQ EH GHILQHG DV E 3/ % f 7KH RXWHU WXEH KDV EHHQ FRQVWUDLQHG WR FRQWDFW WKH LQQHU URG RQO\ DW LWV LGHDOL]HG FRQWDFW SRLQWV DQG WKH FHQWHUOLQH RI WKH LQQHU URG LV QRZ LQFOLQHG DW DQ DQJOH A ZKLFK PD\ EH DSSUR[LPDWHG IRU VPDOO GHIOHFWLRQV DV pQ f /f f 7KH IUDFWXUH LQWHUIDFLDO GHIOHFWLRQ DQJOH UHODWLYH WR WKH YHUWLFDO D[LV FDQ EH REWDLQHG DV VKRZQ LQ )LJXUH E\ VXSHUn LPSRVLQJ HTXDWLRQV f DQG f DV GHILQHG DERYH 'XH WR WKH V\PPHWU\ RI WKH V\VWHP WKH WRWDO DQJXODU PLVPDWFK RI WKH V\VWHP

PAGE 94

)LJXUH ,GHDOL]HG DQJXODU PLVPDWFK UHSUHVHQWDWLRQ ZLWK WKH RYHUDOO DQJXODWLRQ kW HTXDO WR WKH GLIIHUHQFH LQ QDLO DQJXODWLRQ DQG ERQH DQJXODWLRQ A

PAGE 95

A LV WZLFH DQG LI SM SM k c B W Q E 1 % f WKHQ 3#/ [ 3/ % % 1 f f 7KH YDOXHV REWDLQHG IRU WKH DQJXODU GHIOHFWLRQ RI WKH IUDFWXUH LQWHUIDFH DUH GHSHQGHQW RQ WKH DSSOLHG ORDG FRQILJXUDWLRQ )RU H[DPSOH LI WKH GRZQZDUG DSSOLHG ORDGV DUH DVVXPHG WR EH ORFDWHG DW SRVLWLRQV / LQ IURP HDFK HQG RI WKH V\VWHP LQVWHDG RI DW / DV GLVFXVVHG DERYH WKH UHVXOWLQJ DQJXODU GHIOHFWLRQ EHFRPHV [ 3, % A f 3#/ f :KLOH WKH IRUP RI WKH HTXDWLRQV UHPDLQ WKH VDPH WKHLU FRHIILFLHQWV WDNH RQ QHZ YDOXHV 1RQHWKHOHVV VHYHUDO REVHUYDWLRQV FDQ EH PDGH 7KH DSSOLFDWLRQ RI EHQGLQJ ORDGV WR DQ LGHDOL]HG V\VWHP VXFK DV WKH RQH VKRZQ LQGLFDWHV WKDW DQ RSWLPXP IOH[XUDO ULJLGLW\ UDWLR H[LVWV 1% RU GHSHQGLQJ RQ WKH FRQILJXUDn WLRQf ZKLFK UHVXOWV LQ QR DQJXODU PLVPDWFK RI WKH IUDFWXUH VXUIDFHV 7KLV LPSOLHV WKDW LQ DQ DFWXDO RVWHRV\QWKHVLV ZKHUH WKH RXWHU WXEH UHSUHVHQWV WKH ERQH DQG WKH URG UHSUHVHQWV WKH LQWUDPHGXOODU\ QDLO VRPH RSWLPXP SURSHUW\ UDWLR FRXOG UHVXOW LQ D PLQLPXP IUDFWXUH VXUIDFH DQJXODWLRQ :KLOH WKH PDQ\ DVVXPSWLRQV XVHG WR GHULYH WKH UHVXOWV REWDLQHG DERYH ZHUH QHFHVVDU\ WR DOORZ IRU WKH DSSOLFDWLRQ RI HOHPHQWDU\

PAGE 96

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n VKLS EHWZHHQ WKH FRPSRQHQW SURSHUWLHV PD\ PLQLPL]H GHIOHFWLRQV DW WKH IUDFWXUH VLWH D PRUH JHQHUDO PRGHO ZKLFK PRUH FORVHO\ GXSOLFDWHV WKH DFWXDO RVWHRV\QWKHVLV LV QHFHVVDU\ EHIRUH D TXDQWLWDWLYH HYDOXDWLRQ RI WKLV K\SRWKHVLV FDQ EH REWDLQHG ,Q OLJKW RI WKH SUHYLRXV GLVFXVVLRQV WKH PDWHULDO SURSHUWLHV RI WKH QDLO DQG WKH ERQH DQG WKH JHRPHWULF SURSHUWLHV RI WKH QDLO PD\ EH FRQVLGHUHG WR EH DUELWUDU\ FRQVWDQWV ZKRVH YDOXHV GHSHQG RQ WKH QDLO DQG ERQH XQGHU FRQVLGHUDWLRQ 7KH PRGHO VKRXOG KRZHYHU

PAGE 97

DFFRXQW IRU WKH QRQXQLIRUPLW\ RI WKH ERQHnV JHRPHWULF SURSHUWLHV VR WKDW WKH ERQHf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n PHQWDO LQYHVWLJDWLRQ ZDV SHUIRUPHG ,QWUDPHGXOODU\ ,QWHUIDFLDO %HKDYLRU 0HWKRGV DQG 0DWHULDOV $ PDWFKHG SDLU RI HPEDOPHG IHPXUV ZDV FOHDQHG RI VRIW WLVVXH DQG SHULRVWHXP $W DOO WLPHV WKH ERQHV ZHUH NHSW PRLVW XVLQJ D VDOLQH VROXWLRQ GULS 7KH ERQHV ZHUH FDVW LQ SODVWHU DQG DV VRRQ DV WKH SODVWHU KDG VHW VHYHUDO PP WKLFN VHFWLRQV ZHUH FXW IURP WKH GLDSK\VLDO DQG PHWDSK\VLDO UHJLRQV RI WKH ERQH 7KHVH VHFWLRQV ZHUH WKHQ SODFHG LQ VDOLQH VROXWLRQ XQWLO WHVWLQJ $ .XQWVFKQHU QDLO PPf 6FKQHLGHU QDLO PPf RU )OXWHG QDLO PPf ZDV SODFHG LQ WKH LQWUDPHGXOODU\ FDQDO RI WKHVH VHFWLRQV DV VKRZQ LQ )LJXUH (DFK RI WKHVH VDPSOHV ZDV SODFHG

PAGE 98

)LJXUH 3KRWRJUDSK VKRZLQJ WKH LQWUDPHGXOODU\ QDLO LQ SODFH LQ WKH FDVW ERQH VHFWLRQ IRU LQWHUIDFLDO VWLIIQHVV GHWHUPLQDWLRQ

PAGE 99

RQ D VHOIDOLJQLQJ FRPSUHVVLRQ IL[WXUH )LJXUH f IRU ORDGLQJ RI WKH QDLOERQH LQWHUIDFH 8VLQJ DQ ,QVWURQ XQLYHUVDO WHVWLQJ PDFKLQH DW D FURVVKHDG VSHHG RI PPPLQ WKH QDLOV [HUH SUHVVHG DJDLQVW WKH HQGRVWHDO VXUIDFH ZKLFK FRQWDLQHG HLWKHU FDQFHOOXV RU FRUWLFDO ERQH /RDGGHIOHFWLRQ FXUYHV ZHUH REWDLQHG DORQJ ZLWK D PHDVXUHPHQW RI WKH VHFWLRQ WKLFNQHVV IRU IXUWKHU FDOFXODWLRQV 7KH UHVXOWV REWDLQHG IURP WKH ORDGGHIOHFWLRQ FXUYHV ZHUH XVHG WR REWDLQ WKH LQWHUIDFLDO VWLIIQHVV RI WKH FDQFHOOXV DQG FRUWLFDO ERQH RI WKH IHPXU LQ FRQWDFW ZLWK GLIIHUHQW QDLO VKDSHV 7KLV VWLIIQHVV ZDV GHILQHG DV WKH VORSH RI WKH ORDGGHIOHFWLRQ FXUYH GLYLGHG E\ WKH VSHFLPHQ WKLFNQHVV 7KXV D SDUDPHWHU UHODWLQJ IRUFH SHU GHIOHFWLRQ SHU XQLW OHQJWK FRXOG EH REWDLQHG IRU VXEVHTXHQW XVH LQ WKH JHQHUDO PRGHOLQJ RI WKH RVWHRV\QWKHVLV 5HVXOWV 5HSUHVHQWDWLYH FXUYHV IRU WKH ORDGGHIOHFWLRQ UHVSRQVH RI WKH GLIIHUHQW QDLO W\SHV WHVWHG LQ FDQFHOOXV DQG FRUWLFDO ERQH DUH VKRZQ LQ )LJXUHV DQG DORQJ ZLWK WKH WHVWLQJ FRQILJXUDWLRQV XVHG (YHQ ZLWK WKH VPDOO QXPEHU RI VDPSOHV WHVWHG WKH VLPLODULW\ LQ WKH ORDGGHIOHFWLRQ UHVSRQVH RI WKH GLIIHUHQW QDLO W\SHV GXULQJ WHVWLQJ LQ FDQFHOOXV ERQH OHG WR WKH GHILQLWLRQ RI WKUHH RYHUDOO LQWHUIDFLDO VWLIIQHVV UHJLPHV 7KH ILUVW LQWHUIDFLDO VWLIIQHVV UDQJH ZDV ORZ LQ PDJQLWXGH DQG DSSHDUHG WR EH GXH WR WKH LQLWLDO FUXVKLQJ RI WKH FDQFHOOXV ERQH $ WUDQVLWLRQ UHJLRQ ZDV REVHUYHG DV WKH GHIOHFWLRQ LQFUHDVHG 7KLV ZDV IROORZHG E\ D KLJKHU VWLIIQHVV OHYHO DIWHU VLJQLILFDQW GHIOHFWLRQ DQG SDFNLQJ KDG WDNHQ SODFH 7KHVH GLIIHUHQW UHJLPHV DUH VKRZQ DV GLIIHUHQW OLQH W\SHV LQ WKH

PAGE 100

,175$0('8//$5< 1$,/ %21( 6(&7,21 ,1 3/$67(5 )LJXUH 6FKHPDWLF GUDZLQJ RI WKH VHOIDOLJQLQJ FRPSUHVVLRQ IL[WXUH IRU LQWHUIDFLDO VWLIIQHVV GHWHUPLQDWLRQ

PAGE 101

)LJXUH /RDGGHIOHFWLRQ FXUYHV IRU LQWHUIDFLDO EHKDYLRU RI .XQWVFKHU )OXWHG DQG 6FKQHLGHU QDLOV WHVWHG LQ FDQFHOOXV ERQH WHVWLQJ VSHHGf§ PPPLQf

PAGE 102

'()/(&7,21 PPf )LJXUH /RDGGHIOHFWLRQ FXUYHV IRU WKH LQWHUn IDFLDO EHKDYLRU RI .XQWVFKHU )OXWHG DQG 6FKQHLGHU QDLOV LQ FRUWLFDO ERQH WHVWLQJ VSHHGf§ PPPLQf

PAGE 103

ILJXUHV DQG RFFXU DW GLIIHUHQW ORDGV DQG GHIOHFWLRQV GHSHQGLQJ RQ WKH QDLO SURILOH DQG ERQH VXEVWDQFH WHVWHG 7KH ILUVW UHJLPH H[n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n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

PAGE 104

:KHQ WHVWHG LQ FRUWLFDO ERQH )LJXUH f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n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

PAGE 105

)LJXUH 6FKHPDWLF GLDJUDP RI D VHJPHQW RI WKH JHQHUDO RVWHRV\QWKHVLV PRGHO ZLWK ERQH RI YDULDEOH FURVVVHFWLRQ XQGHU DUELWUDU\ GLVWULEXWHG ORDG T[f DQG QDLO UXQQLQJ WKH OHQJWK RI WKH V\VWHP )LJXUH )UHH ERG\ GLDJUDP RI DQ HOHPHQW RI WKH ERQH ZLWK H[WHUQDO GLVWULEXWHG ORDG T[f DQG LQWHUIDFLDO UHDFWLRQ ORDG Z[f )LJXUH )UHH ERG\ GLDJUDP RI DQ HOHPHQW RI WKH QDLO ZLWK LQWHUIDFLDO UHDFWLRQ ORDG Z[f

PAGE 107

IUDFWXUH KDV JHRPHWULF DQG PDWHULDO SURSHUWLHV ZKLFK DUH DVVXPHG WR EH XQYDU\LQJ DORQJ WKH OHQJWK RI WKH V\VWHP &ORVH QDLOERQH LQWHUIDFLDO FRQWDFW DOORZV IRU ORQJLWXGLQDO VOLGLQJ DORQJ WKH LQWHUIDFH EXW DV GHIOHFWLRQV WDNH SODFH XQGHU WKH [WHUQDO ORDG T[f ORFDO GHIRUPDWLRQV UHVXOW LQ GLVWULEXWHG QRUPDO LQWHUIDFLDO ORDGV 7KHVH ORDGV Z[f DUH DVVXPHG WR EH GLUHFWO\ SURSRUWLRQDO WR WKH UHODWLYH GHIOHFWLRQV RI WKH QDLO DQG ERQH 'HVFULELQJ (TXDWLRQV 7KH GHYHORSPHQW RI WKH GHVFULELQJ HTXDWLRQV IRU WKH W\SH RI RVWHRV\QWKHVLV GHVFULEHG DERYH LV H[HFXWHG LQ WZR EDVLF VWDJHV 7KH ILUVW LV WKH HYROXWLRQ RI D JHQHUDO PXOWLFRPSRQHQW EHDP HTXDWLRQ WR GHVFULEH WKH ORDGGHIOHFWLRQ UHVSRQVH RI WKH V\VWHP 7KH VHFRQG LV WKH DSSOLFDWLRQ RI VXLWDEOH ERXQGDU\ FRQGLWLRQV ZKLFK FRPSOHWH WKH GHVFULSWLRQ RI WKH IRXUSRLQW ORDGLQJ RI WKH V\VWHP ZLWK DQG ZLWKRXW D WUDQVYHUVH PLGVKDIW IUDFWXUH RI WKH ERQH IRU FRQVLGHUDWLRQ LQ WKLV ZRUN $ VHJPHQW RI WKH V\VWHP RI OHQJWK G[ LV FRQVLGHUHG $SSURSULDWH IUHH ERG\ GLDJUDPV RI WKH QDLO DQG ERQH DUH GUDZQ ZLWK WKHLU DVVXPHG UHDFWLRQ PRPHQWV VKHDUV DQG QRUPDO LQWHUIDFLDO ORDGV DV VKRZQ LQ )LJXUHV DQG 7KH IUHH ERG\ GLDJUDP RI WKH ERQH 7KH ERQH GHIOHFWLRQ \ LV GHILQHG DV SRVLWLYH GRZQZDUG 9HUWLFDO IRUFH HTXLOLEULXP FRQn GLWLRQV DUH WKHQ DSSOLHG WR WKH ERQH VHFWLRQ )LJXUH f UHVXOWLQJ LQ WKH HTXDWLRQ 4 T[fG[ Z[fG[ 4 G4f f

PAGE 108

ZKHUH 4 LV WKH VKHDU ORDG DSSOLHG WR WKH OHIW IDFH RI WKH VHFWLRQ 7KLV HTXDWLRQ UHGXFHV WR WKH GLIIHUHQWLDO UHODWLRQVKLS 0 G[ Z[f T[f f 6LPLODUO\ PRPHQW HTXLOLEULXP \LHOGV WKH UHVXOWLQJ HTXDWLRQ G[ >T[fG[ Z[fG[@ 4 G4fG[ 0A G0Af f ZLWK 0 UHSUHVHQWLQJ WKH UHDFWLRQ PRPHQW DSSOLHG WR WKH OHIW HGJH RI WKH ERQH VHJPHQW (TXDWLRQV f DQG f DUH WKHQ FRPELQHG WR JLYH 4G[ G0A ZKLFK DV G[ DSSURDFKHV ]HUR \LHOGV f G[ 4 f 'LIIHUHQWLDWLQJ HTXDWLRQ f DQG VXEVWLWXWLQJ LQWR HTXDWLRQ f JLYHV G0 E GA r V f ;7 Z[f T[f G[ G[ f $V VKRZQ LQ WKH GHYHORSPHQW RI WKH HQJLQHHULQJ EHDP HTXDWLRQV GHVFULEHG LQ $SSHQGL[ $ WKH EHQGLQJ PRPHQW 0 RQ D EHDP VHFWLRQ LV UHODWHG WR WKH FXUYDWXUH RI WKH EHDP DSSUR[LPDWHG E\ G\G[ E\ WKH HTXDWLRQ 0 B G\ (O G[ f

PAGE 109

ZKHUH (O LV WKH EHDPnV IOH[XUDO ULJLGLW\ DQG \ LV WKH EHDP GHIOHFWLRQ 7KH ERQH IOH[XUDO ULJLGLW\ LV GHILQHG DV %[f (E ,E[f f ZKHUH (E LV WKH ERQH PRGXOXV RI HODVWLFLW\ DQG ,E[f LV WKH DUHD PRPHQW RI LQHUWLD RI WKH ERQH 5HPHPEHULQJ WKDW ,E[f DQG WKHUHIRUH %[f LV D IXQFWLRQ RI SRVLWLRQ DORQJ WKH OHQJWK RI WKH EHDP HTXDWLRQ f LV UHFDVW WR JLYH % [f G\ G[ ZKLFK LV GLIIHUHQWLDWHG WZLFH WR SURGXFH f A AE B J G\ B A G% G \ G% G\ G[ G[r G[ G[ G[ f 6XEVWLWXWLQJ HTXDWLRQ f LQWR WKH DERYH HTXDWLRQ UHVXOWV LQ WKH ORDGf§GHIOHFWLRQ UHODWLRQVKLS IRU WKH ERQH f G\ f G% G\ G% G\ [ 1 % f§ M ‘r T [f Z[f G[ ; G[ G[ G[ f 7KH IUHH ERG\ GLDJUDP RI WKH QDLO $SSO\LQJ VLPLODU DUJXPHQWV WR WKH IUHH ERG\ GLDJUDP RI WKH QDLO )LJXUH f OHDGV WR WKH YHUWLFDO IRUFH HTXLOLEULXP HTXDWLRQ 9 Z[fG[ 9 G9f f ZKHUH 9 LV WKH VKHDU IRUFH DFWLQJ RQ WKH OHIW IDFH RI WKH VHJPHQW DQG ] LV WKH GRZQZDUG GHIOHFWLRQ RI WKH QDLO (TXDWLRQ f FDQ

PAGE 110

WKHQ EH VLPSOLILHG WR \LHOG e Z[f f $OVR VXPPLQJ WKH PRPHQWV JLYHV 0 >Z[fG[@ f§ 9 G9fG[ 0 G0 f f Q Q Q ZKHUH LV WKH PRPHQW ZKLFK LV DSSOLHG WR WKH OHIW IDFH RI WKH QDLO VHJPHQW 7KXV DV G[ EHFRPHV VPDOO HTXDWLRQ f UHGXFHV WR G0 GA Y 'LIIHUHQWLDWLQJ HTXDWLRQ f DQG VXEVWLWXWLQJ LQWR HTXDWLRQ f \LHOGV G0 f§ Z[f f G[ 7KH QDLO IOH[XUDO ULJLGLW\ LV GHILQHG DV 1 ( Q Q f ZKHUH (A WKH QDLO PRGXOXV RI HODVWLFLW\ DQG WKH QDLO DUHD PRPHQW RI LQHUWLD DUH DVVXPHG FRQVWDQW DORQJ WKH OHQJWK RI WKH V\VWHP 7KH EHDP HTXDWLRQ f FDQ EH DSSOLHG WR WKH QDLO WR \LHOG 0 Q 1 G] G[ f

PAGE 111

(TXDWLRQ f LV LQFRUSRUDWHG DIWHU GLIIHUHQWLDWLQJ LW WZLFH DQG FRPELQLQJ LW ZLWK HTXDWLRQ f WR \LHOG WKH ORDGGHIOHFWLRQ GLIIHUHQWLDO HTXDWLRQ 1 f§ Z[f f G[ (TXDWLRQV f DQG f DUH WKH GLIIHUHQWLDO HTXDWLRQV IRU WKH GHIOHFWLRQV RI WKH ERQH DQG QDLO UHVSHFWLYHO\ ZLWK WKH IRUPHU FRQWDLQLQJ VHYHUDO DGGLWLRQDO WHUPV GXH WR WKH QRQFRQVWDQW JHRPHWULF SURSHUWLHV RI WKH ERQH 7KHVH WZR HTXDWLRQV DUH UHODWHG E\ WKH LQWHUIDFLDO ORDGLQJ Z[f DV GLVFXVVHG EHORZ 7KH QDLOERQH LQWHUIDFLDO UHDFWLRQ $ FRXSOLQJ RI WKH WZR GHIOHFWLRQ HTXDWLRQV f DQG f UHTXLUHV WKDW DQ DVVXPSWLRQ EH PDGH DV WR WKH LQWHUDFWLRQ EHWZHHQ WKH ERQH DQG WKH QDLO 7KLV LV DFFRPSOLVKHG LQ WKLV PRGHO E\ DVVXPLQJ WKDW VRPH OLQHDU UHODWLRQVKLS EHWZHHQ WKH LQWHUIDFLDO ORDG DQG WKH UHODWLYH GLVn SODFHPHQW RI WKH QDLO DQG WKH ERQH H[LVWV VXFK WKDW Z[f N \ ]f f ZKHUH N LV D PHDVXUH RI WKH HIIHFWLYH QDLOERQH LQWHUIDFLDO VWLIIQHVV RU VSULQJ FRQVWDQW DV GLVFXVVHG LQ WKH SUHYLRXV VHFWLRQV 7KH VLPXOWDQHRXV HTXLOLEULXP ORDGGHIOHFWLRQ GLIIHUHQWLDO HTXDWLRQV &RPELQLQJ WKH DERYH HTXDWLRQ ZLWK WKH GLIIHUHQWLDO HTXDWLRQV f DQG f UHVXOWV LQ WZR VLPXOWDQHRXV HTXLOLEULXP

PAGE 112

GLIIHUHQWLDO HTXDWLRQV ZKLFK UHODWH WKH V\VWHP ORDGLQJ WR WKH GHIOHFWLRQV E f§ A A G\ G[ G[ [ [ [ T[f N\ ]f Df DQG 1 f§ N\]f Ef G[ 1RUPDOL]DWLRQ RI WKH 'HVFULELQJ (TXDWLRQV ,Q RUGHU WR PDLQWDLQ WKH JHQHUDOLW\ RI WKH UHVXOWV DQG VLPSOLI\ IXUWKHU GHULYDWLRQ RI WKH VROXWLRQ RI WKH VLPXOWDQHRXV HTXDWLRQV Df DQG Ef QRQGLPHQVLRQDOL]DWLRQ RI VHYHUDO YDULDEOHV LV FDUULHG RXW 7KH IROORZLQJ UHODWLRQVKLSV DUH GHILQHG IRU XVH EHORZ ; [/ < \\R = ]\R 4 TT4 Df Ef Ff Gf ZKHUH TA LV VRPH FKDUDFWHULVWLF ORDGLQJ LQWHQVLW\ \A LV VRPH FKDUDFWHULVWLF GHIOHFWLRQ / LV WKH OHQJWK RI WKH ERQH DQG WKH QDLO DQG ; < = DQG 4 DUH QRZ QRQGLPHQVLRQDO YDULDEOHV (TXDWLRQV Df DQG Ef DUH ILUVW UHZULWWHQ DV G% GMA % G[ G[ B G% G\ A G[ G[ \ R % Df DQG G ] G[ ] Ef

PAGE 113

6XEVWLWXWLRQ RI HTXDWLRQV DGf LQWR HTXDWLRQV Df DQG Ef \LHOGV WKH QRQGLPHQVLRQDO GLIIHUHQWLDO HTXDWLRQV IRU WKH V\VWHP < L n L L % / \ L L L %n n/ NO A < ]@ 9 4 Df DQG ] 77 & f>= a <@ Ef ZKHUH WKH SULPHV UHIHU WR GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR ; (TXDWLRQV Df DQG Ef DUH WKH FRXSOHG GLIIHUHQWLDO HTXDWLRQV ZKLFK ZKHQ VROYHG ZLWK VXLWDEOH ERXQGDU\ FRQGLWLRQV UHODWH WKH QRQGLPHQVLRQDO ORDGLQJ 4;f RI DSDUWLFXODU RVWHRV\QWKHVLV WR WKH QRQGLPHQVLRQDO GHIOHFWLRQV < DQG = RI WKH ERQH DQG QDLO UHVSHFWLYHO\ $ FRPPHQW RQ WKH UHODWLYH PDJQLWXGHV RI WKH FRHIILFLHQWV LQ HTXDWLRQV Df DQG Ef LV KHOSIXO LQ XQGHUVWDQGLQJ WKH LPSRUWDQFH RI WKH FRQWULEXWLRQ RI HDFK RI WKH WHUPV RI WKH HTXDWLRQV ,I RQH DVVXPHV XQLIRUP PDWHULDO SURSHUWLHV RI WKH ERQH DORQJ LWV OHQJWK WKHQ WKH FRHIILFLHQWV RI
PAGE 114

PRUH WKDQ D IHZ SHUFHQW ,Q WKH OLPLWLQJ FDVH RI XQLIRUP ERQH JHRPHWU\ DQG PDWHULDO SURSHUWLHV WKHVH FRHIILFLHQWV ZLOO EH ]HUR DQG WKH WHUPV ZLOO YDQLVK N/r 7KH FRXSOLQJ FRHIILFLHQW f§ ‘ f§ KDV D PRUH SURQRXQFHG HIIHFW 7KLV FRHIILFLHQW LV D VFDOHG UDWLR RI WKH LQWHUIDFLDO VWLIIQHVV DQG ERQH IOH[XUDO ULJLGLW\ DQG LV D PHDVXUH RI WKH LQWHUDFWLRQ RU FRXSOLQJ RI WKH WZR FRPSRQHQWV XQGHU ORDG ,Q FOLQLFDOn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f WKH FRXSOLQJ FRHIILFLHQW LV PRGLILHG E\ WKH UHODWLYH IOH[XUDO ULJLGLWLHV RI WKH ERQH DQG QDLO ZKLFK LV W\SLFDOO\ RI RUGHU 7KXV WKH DERYH GLVFXVVLRQ LQGLFDWHV WKDW LW LV WKH UDWLR RI LQWHUIDFLDO SURSHUWLHV DQG IOH[XUDO ULJLGLWLHV RI WKH FRPSRQHQWV ZKLFK GRPLQDWH WKH VROXWLRQ

PAGE 115

$SSOLFDWLRQ RI %RXQGDU\ &RQGLWLRQV 7KH DSSOLFDWLRQ RI ERXQGDU\ FRQGLWLRQV ZKLFK GHVFULEH WKH LQGLYLGXDO RVWHRV\QWKHVHV XQGHU FRQVLGHUDWLRQ LV WKH QH[W VWHS LQ WKH VROXWLRQ RI WKH ORDGGHIOHFWLRQ UHODWLRQVKLSV 7ZR VLWXDWLRQV VLPXODWLQJ D IUDFWXUHG DQG KHDOHG ERQH ZHUH VWXGLHG $Q RVWHRV\QWKHVLV ZLWK D IUDFWXUH SUHVHQW )RU WKH RVWHRn V\QWKHVLV VKRZQ LQ )LJXUH ZLWK D IUDFWXUH DQG LQWUDPHGXOODU\ IL[DWLRQ XQGHU IRXUSRLQW ORDGLQJ DQG HTXDO VSDFLQJ EHWZHHQ WKH ORDGLQJ SRLQWV WKH IROORZLQJ DVVXPSWLRQV FDQ EH PDGH ZLWK UHIHUHQFH WR WKH ERQH LQWHUDFWLRQV $W SRLQWV $ DQG WKH GHIOHFWLRQV RI WKH ERQH DUH GHILQHG DV ]HUR 7KHUH DUH QR H[WHUQDO PRPHQWV DSSOLHG WR WKH ERQH DW SRLQWV $ % & DQG 7KHUH LV QR VKHDU SUHVHQW LQ WKH ERQH DW SRLQWV % DQG & 5HIHUULQJ WR $SSHQGL[ $ RQH FDQ ZULWH WKH IROORZLQJ HTXDOLWLHV (O\nnf 9 f (O\ 0 f ZKLFK FDQ EH FDVW LQWR QRQGLPHQVLRQDO IRUP (O\ r n 9 f 0 f

PAGE 116

)LJXUH 6FKHPDWLF GLDJUDP RI DQ RVWHRV\QWKHVLV XQGHU HTXDOO\ VSDFHG IRXUSRLQW ORDGLQJ ZLWK D PLGVKDIW WUDQVYHUVH IUDFWXUH

PAGE 117

7KXV WKH DERYH ERXQGDU\ FRQGLWLRQV EHFRPH L \D \G <$ r <% <& r \Gn <%nn < $ODf Ef Df Ef Ff Gf Df Ef 6LPLODUO\ DVVXPSWLRQV DUH PDGH DV WR WKH ERXQGDU\ FRQGLWLRQV RI WKH QDLO 6LQFH WKH QDLO FDQ PRYH UHODWLYH WR WKH ERQH QR FRQGLWLRQV UHJDUGLQJ GHIOHFWLRQ FDQ EH VSHFLILHG 7KHUH DUH QR PRPHQWV DSSOLHG DW SRLQWV $ DQG 7KHUH DUH QR VKHDUV DW SRLQWV $ DQG $JDLQ DSSO\LQJ WKH EHDP HTXDWLRQV RI $SSHQGL[ $ WKHVH DVVXPSWLRQV EHFRPH Df Ef
PAGE 118

LQ )LJXUH LV LQYHVWLJDWHG 7KLV FRQILJXUDWLRQ UHVXOWV LQ WKH IROORZLQJ ERXQGDU\ FRQGLWLRQV IRU WKH ERQH 1R GHIOHFWLRQV DU SUHVHQW DW SRLQWV $ DQG 1R PRPHQWV DUH WUDQVPLWWHG DW SRLQWV $ DQG 7KLV FRUUHVSRQGV WR WKH VLWXDWLRQ Df Ef Df Ef $V EHIRUH WKH QDLO FDQ PRYH UHODWLYH WR WKH ERQH DQG WKH ERXQGDU\ FRQGLWLRQV DUH WKDW 1R PRPHQWV DUH SUHVHQW DW SRLQWV $ DQG 1R VKHDUV DUH SUHVHQW DW SRLQWV $ DQG 7KHVH UHVXOW LQ Df Ef
PAGE 119

S S )LJXUH 6FKHPDWLF GLDJUDP RI D KHDOHG RVWHRV\QWKHVLV XQGHU HTXDOO\ VSDFHG IRXUSRLQW ORDGLQJ ZLWK WKH QDLO UHPDLQLQJ LQ VLWX

PAGE 120

DUH DSSOLHG WR VHYHUDO SRLQWV DORQJ WKH V\VWHP DQG UHVXOW LQ D VHULHV RI VLPXOWDQHRXV DOJHEUDLF HTXDWLRQV ZKLFK ZKHQ VROYHG \LHOG UHVXOWV ZKLFK VDWLVI\ WKH HTXDWLRQV DW WKHVH SDUWLFXODU SRLQWV :KLOH UHVXOWV DUH QRW REWDLQHG IRU HYHU\ SRVLWLRQ DORQJ WKH V\VWHP OHQJWK WKH XVH RI PDQ\ GLVFUHWH SRLQWV SURYLGHV D JRRG GHVFULSWLRQ RI WKH RYHUDOO V\VWHP UHVSRQVH &UDQGDOO f SUHVHQWV D GHWDLOHG GLVFXVVLRQ RI WKH XVH RI WKLV WHFKQLTXH DQG 7DEOH VKRZV WKH FRPSXWDWLRQDO PROHFXOHV IRU WKH IRUPDWLRQ RI ILQLWH GLIIHUHQFH UHSUHVHQWDWLRQV RI VHYHUDO GHULYDWLYHV %HFDXVH RI WKH SUHVHQFH RI GHULYDWLYHV LQ WKH FRHIILFLHQWV RI WKH JRYHUQLQJ HTXDWLRQV VHYHUDO DSSOLFDWLRQV RI WKH PROHFXOHV PXVW EH PDGH 'HVFULELQJ HTXDWLRQV LQ ILQLWH GLIIHUHQFH IRUP 7KH RVWHRn V\QWKHVLV RI OHQJWK / LV DVVXPHG WR EH GLYLGHG LQWR Q VHJPHQWV K LQ OHQJWK FHQWHUHG DERXW PHVK SRLQWV DORQJ LWV OHQJWK )LJXUH f 7KXV + K/ DQG ZH VHH WKDW + Q 7KH FRPSXWDWLRQDO PROHFXOH IRU WKH IRXUWK GHULYDWLYH LV FHQWHUHG RYHU D PHVK SRLQW L DQG HTXDWLRQ Df EHFRPHV >< < << < @ X OO O OO UL %n/ %+ >< < < @ U +A% N/ N/ < ‘ = % L % L \ % L R 4 R Df

PAGE 121

7DEOH )LQLWH GLIIHUHQFH FRPSXWDWLRQDO PROHFXOHV M M M M Kf Kf Kf Kf Kf

PAGE 122

ZKLOH HTXDWLRQ Ef UHGXFHV WR WKH IRUP >= = = = =B@ K [ L, L OO L N/ = 1 L 1 < O Ef 7KXV IRU D JHQHUDO ORDGLQJ FRQGLWLRQ WKH WHUPV LQ WKH DERYH HTXDWLRQV FDQ EH UHJURXSHG DQG VLPSOLILHG WR \LHOG 8 %Xf %OL %f
PAGE 123

$ \L T /+ R < ] N/+ 1 Gf Hf 1RWH WKDW A DQG GHSHQG RQ WKH JHRPHWULF DQG PDWHULDO SURSHUWLHV RI WKH V\VWHP FRPSRQHQWV LV D PRGLILHG FRXSOLQJ WHUP DQG LV QRW D IXQFWLRQ RI SRVLWLRQ GXH WR WKH RULJLQDO DVVXPSWLRQ RI FRQVWDQW PDWHULDO DQG JHRPHWULF SURSHUWLHV RI WKH QDLO DORQJ WKH OHQJWK RI WKH V\VWHP (TXDWLRQV Df DQG Ef DUH QRZ LQ WKHLU ILQLWH GLIIHUHQFH IRUP DV VKRZQ LQ HTXDWLRQV Df DQG Ef +RZHYHU WKH GHWHUPLQDWLRQ RI $ UHPDLQV \L EHIRUH WKH ILQLWH GLIIHUHQFH UHSUHVHQWDWLRQ RI WKH GHVFULELQJ HTXDWLRQV FDQ EH DSSOLHG :KHQ WKH EHDP RI OHQJWK / LV GLYLGHG LQWR Q VHJPHQWV IRU ILQLWH GLIIHUHQFH UHSUHVHQWDWLRQ WKH FRQFHQWUDWHG ORDGV 3 DUH UHSODFHG E\ ORDGV GLVWULEXWHG XQLIRUPO\ RYHU WKH VHJPHQW RI OHQJWK K ZKHUH K /Q )LJXUH f 7KLV UHSUHVHQWDWLRQ RI HTXLYDOHQW ORDGLQJ LV PRUH UHDOLVWLF LQ IDFW EHFDXVH WKH ORDGV DSSOLHG WR ERQHV E\ PXVFOH JURXSV DQG MRLQW UHDFWLRQV DUH WKHPVHOYHV GLVWULEXWHG RYHU DQ DUHDf 7KXV WKH ORDGLQJ LQWHQVLW\ EHFRPHV T[f 3K f ZKHUH T[f LV FRQVWDQW RYHU WKH VHJPHQW DQG H[LVWV RQO\ RQ WKRVH VHJPHQWV ZKHUH D FRQFHQWUDWHG ORDG DFWV $V ZDV VXJJHVWHG LQ WKH SUHYLRXV GLVFXVVLRQ RI WKH PDJQLWXGH RI WKH QRQGLPHQVLRQDO YDULDEOHV HDVH RI VROXWLRQ LV PDLQWDLQHG LI

PAGE 124

! L U ‘fL L IO } K e! )LJXUH )LQLWH GLIIHUHQFH UHSUHVHQWDWLRQ RI WKH RVWHRV\QWKHVLV ZLWK IUDFWXUH XQGHU IRXUSRLQW ORDGLQJ ZLWK FRQFHQWUDWHG ORDGV 3 UHVROYHG LQWR XQLIRUPO\ GLVWULEXWHG ORDGV RI PDJQLWXGH T 3K R

PAGE 125

,OO T4 DQG \4 DUH RI SURSHU PDJQLWXGH VR WKDW WKH QRQGLPHQVLRQDO YDULDEOHV DUH RI RUGHU ,I WKH FKDUDFWHULVWLF ORDGLQJ LQWHQVLW\ T4 LV FKRVHQ WR EH HTXDO WR T WKHQ 4 DW WKRVH SRLQWV ZKHUH ORDG LV DSSOLHG DQG ]HUR HOVHZKHUH )XUWKHUPRUH LI LW LV DVVXPHG WKDW WKH IUDFWXUHG RVWHRV\QWKHVLV XQGHU IRXUSRLQW ORDGLQJ GRHV QRW EHKDYH PXFK GLIIHUHQWO\ WKDQ DQ LQWDFW ERQH ZLWK VLPLODU IOH[XUDO ULJLGLW\ XQGHU VLPLODU ORDGLQJ WKHQ WKH FKDUDFWHULVWLFfGHIOHFWLRQ \ PD\ EH REWDLQHG IURP WKH VROXWLRQ RI WKH ORDG GHIOHFWLRQ HTXDWLRQ IRU WKH PD[LPXP GHIOHFWLRQ RI WKH LQWDFW V\VWHP 7KXV \A LV D IXQFWLRQ RI WKH ORDGLQJ OHQJWK DQG IOH[XUDO ULJLGLW\ RI WKH V\VWHP LH \A ITR/%f ,Q IDFW IRU D IRXUSRLQW ORDGHG V\VWHP ZLWK HTXDO ORDG VSDFLQJ DSSOLFDWLRQ RI WKH EHDP HTXDWLRQ WR DQ LQWDFW ERQH ZLWKRXW LQWHUQDO IL[DWLRQ \LHOGV WKH HTXDWLRQ 3/ \R % f DYJ ZKHUH %t LV WKH DYHUDJH IOH[XUDO ULJLGLW\ RI WKH ERQH 6XEn VWLWXWLQJ HTXDWLRQ f LQWR f DQG UHPHPEHULQJ WKDW T TA JLYHV T % f§ 69t f \R K/ 6LPSOLI\LQJ HTXDWLRQ Gf XVLQJ HTXDWLRQ f ZLWK WKH DVVXPSWLRQ WKDW WKH ORFDO IOH[XUDO ULJLGLW\ DW WKH ORDG SRLQWV LV QRW VLJQLILFDQWO\ GLIIHUHQW IURP WKH DYHUDJH IOH[XUDO ULJLGLW\ \LHOGV ; Q f \L

PAGE 126

6KRXOG WKH ORDGLQJ FRQILJXUDWLRQ GLIIHU IURP WKDW VKRZQ LQ )LJXUH WKH YDOXH RI $ ZLOO FKDQJH DV ZHOO \L 7KH ILQLWH GLIIHUHQFH UHSUHVHQWDWLRQV RI WKH JHQHUDO V\VWHP HTXDWLRQV Df DQG Ef DUH VKRZQ LQ WKHLU RZQ UHVXOWLQJ FRPSXWDWLRQDO PROHFXOH IRUP LQ )LJXUH 7KH PROHFXOHV ZLWK DOO FRHIILFLHQWV QRZ GHILQHG DUH DSSOLHG WR VXFFHVVLYH PHVK SRLQWV WR IRUP D VHULHV RI HTXDWLRQV ZKLFK FDQ EH PDQLSXODWHG WR \LHOG WKH GHVLUHG GHIOHFWLRQ UHVXOWV IRU WKH IRXUSRLQW ORDGHG V\VWHP &RQYHUVLRQ RI WKH SUHYLRXVO\ GHILQHG ERXQGDU\ FRQGLWLRQV WR WKHLU ILQLWH GLIIHUHQFH IRUP LV DOO WKDW UHPDLQV IRU WKH DSSOLFDn WLRQ RI WKH FRPSXWDWLRQDO PROHFXOHV WR DOO WKH QRGH SRLQWV RI WKH V\VWHP 2VWHRV\QWKHVLV ZLWK IUDFWXUH 8WLOL]LQJ WKH SUHYLRXVO\ GHILQHG ERXQGDU\ FRQGLWLRQV IRU WKH IUDFWXUHG ERQH V\VWHP HTXDWLRQV f DQG WKH ILQLWH GLIIHUHQFH DSSUR[LPDWLRQ PROHFXOHV LQ 7DEOH WR UHSUHVHQW WKH GHULYDWLYHV \LHOGV <$ <% < f < f $O $O < f < ‘ 'O 'O < f <
PAGE 127

)LJXUH 0RGLILHG FRPSXWDWLRQDO PROHFXOHV IRU WKH GHVFULELQJ QRQGLPHQVLRQDO GLIIHUHQWLDO HTXDWLRQV RI WKH FRXSOHG V\VWHP

PAGE 128

c L L L L

PAGE 129

6LPLODUO\ WKH QDLO ERXQGDU\ FRQGLWLRQV EHFRPH =$ n n =$ =$ r Df =' ‘ ]G ]GL r Ef =$ =$ =$ $$ r Df &0 4 1=' =' =' Ef :LWK WKHVH ERXQGDU\ FRQGLWLRQV IRU WKH RVWHRV\QWKHVLV ZLWK D IUDFWXUH SUHVHQW GHILQHG WKH DSSOLFDWLRQ RI WKH JHQHUDO PROHFXOH WR DOO WKH PHVK SRLQWV FDQ EH DFFRPSOLVKHG +HDOHG 2VWHRV\QWKHVLV ,Q D VLPLODU IDVKLRQ WKH ERXQGDU\ FRQGLWLRQV IRU WKH KHDOHG V\VWHP HTXDWLRQV f DUH UHFDVW LQWR ILQLWH GLIIHUHQFH IRUP )RU WKH ERQH \D r \G R < < < $O $ $O < < < 'O 'O 6LPLODUO\ IRU WKH QDLO = = = $O $ $O =' n = f =' =' r = f $ $O $O $ = f = = ] f 'O 'O Df Ef Df Ef Df Ef Df Ef

PAGE 130

ZKLFK SURYLGH WKH QHFHVVDU\ UHODWLRQVKLSV IRU WKH DSSOLFDWLRQ RI WKH FRPSXWDWLRQDO PROHFXOH IRU WKH V\VWHP WR DOO PHVK SRLQWV RI D KHDOHG RVWHRV\QWKHVLV 6ROXWLRQ RI 'LIIHUHQW (TXDWLRQV 8VLQJ WKH 5HOD[DWLRQ 7HFKQLTXH 7KH VROXWLRQ RI WKH Q VLPXOWDQHRXV HTXDWLRQV ZKLFK DUH GHYHORSHG E\ WKH VHTXHQWLDO DSSOLFDWLRQ RI WKH FRPSXWDWLRQDO PROHn FXOH WR HDFK RI WKH Q PHVK SRLQWV RI WKH V\VWHP LV FXPEHUVRPH DW EHVW +RZHYHU WKHUH DUH VHYHUDO QXPHULFDO WHFKQLTXHV ZKLFK SURn YLGH PHWKRGV IRU WKH GHWHUPLQDWLRQ RI WKH YDOXH RI WKH IXQFWLRQ DW HDFK PHVK SRLQW 7KH UHOD[DWLRQ WHFKQLTXH &UDQGDOO f ZDV XVHG LQ WKLV ZRUN DQG SURYLGHV DQ DOJRULWKP IRU GHWHUPLQLQJ WKH YDOXHV RI WKH IXQFWLRQV ZKLFK VDWLVI\ WKH HTXDWLRQV DQG WKHLU ERXQGDU\ FRQGLWLRQV $ VHW RI YDOXHV IRU < DQG = PXVW EH GHWHUPLQHG DW HDFK SRLQW DORQJ WKH RVWHRV\QWKHVLV VR WKDW ZKHQ WKHVH YDOXHV DUH DSSOLHG WR HTXDWLRQV Df DQG Ef WKH HTXDOLWLHV KROG IRU DOO SRLQWV DORQJ WKH EHDP ,Q JHQHUDO WKHVH YDOXHV DUH QRW NQRZQ +RZHYHU LI RQH DVVXPHV D VHW RI YDOXHV IRU WKH
PAGE 131

DQG 9 n ]L ]LL 9 ]L = =R < < ] L Ef ZKHUH WKH FRHIILFLHQWV DUH GHWHUPLQHG E\ WKH VSHFLILF JHRPHWU\ DQG PDWHULDO SURSHUWLHV RI WKH V\VWHP FRPSRQHQWV DV GHILQHG E\ HTXDWLRQV DHf 7KH PDJQLWXGH RI WKH UHVLGXDOV LQGLFDWH KRZ IDU WKH DVVXPHG YDOXHV DUH IURP VDWLVI\LQJ WKH GHVFULELQJ HTXDWLRQV DW WKDW PHVK SRLQW 7KH HVVHQFH RI WKH UHOD[DWLRQ WHFKQLTXH LV WR SURFHHG IURP PHVK SRLQW WR PHVK SRLQW XVLQJ WKH PROHFXOHV DQG ERXQGDU\ FRQn GLWLRQV SUHYLRXVO\ GHILQHG ,Q D V\VWHPDWLF IDVKLRQ WKH FXUUHQW YDOXHV RI < DQG = DUH DOWHUHG WR UHGXFH WKH UHVLGXDOV REWDLQHG L L 7KLV LV DFFRPSOLVKHG ZLWK WKH DLG RI WKH GLJLWDO FRPSXWHU 7KH SURJUDP GHYHORSHG WR SHUIRUP WKLV WDVN %21$,/ ILUVW SURFHVVHV WKH LQSXW YDOXHV RI V\VWHP FRPSRQHQW JHRPHWULF DQG PDWHULDO SURSHUWLHV WR JHQHUDWH WKH FRHIILFLHQWV HTXDWLRQV DHf XVLQJ WKHLU ILQLWH GLIIHUHQFH UHSUHVHQWDWLRQV %21$,/ DVVXPHV LQLWLDO YDOXHV RI WKH
PAGE 132

6LPLODUO\ LI LV DOWHUHG E\ $=BA ZKHUH 5f $= L \ Ef WKHQ WKH HTXDWLRQ IRU = ZRXOG DOVR EH VDWLVILHG +RZHYHU ZKHQ WKH SURJUDP PRYHV WR WKH QH[W SRLQW DQG DOWHUV WKH YDOXH RI
PAGE 133

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f 6WDUWLQJ ZLWK WKH GHIOHFWLRQV DW ]HUR YDULRXV YDOXHV IRU WKH UHOD[DWLRQ FRQVWDQWV ZHUH LQSXW WR WKH %21$,/ SURJUDP 7KHVH YDOXHV UDQJH IURP )RU 0RGHO YDOXHV RI IRU ERWK FRQVWDQWV FDXVHG D VOLJKW LQLWLDO RYHUVKRRW RI WKH DQWLFLSDWHG GHIOHFWLRQ EXW VHOIFRUUHFWHG WR JLYH D VLJQLILFDQW LQFUHDVH LQ WKH FRQYHUJHQFH UDWH RI WKH SURJUDP WR D FRUUHFW YDOXH WRWDO LWHUDWLRQV DW FRPSDUHG WR DW WR UHDFK b RI WKH WRWDO DQWLFLSDWHG GHIOHFWLRQf ,Q 0RGHO VLPLODU YDOXHV ZHUH DOVR IRXQG WR HQKDQFH

PAGE 134

WKH FRQYHUJHQFH DQG D YDOXH RI IRU ERWK FRQVWDQWV UHVXOWHG LQ VLJQLILFDQW DFFHOHUDWLRQ RI FRQYHUJHQFH WRWDO LWHUDWLRQV YHUVXV DW D YDOXH RI WR UHDFK b RI WKH DQWLFLSDWHG GHIOHFWLRQf 7KH WKLUG PRGHO SURGXFHG FRQVLGHUDEOH DFFHOHUDWLRQV XVLQJ WKH YDOXHV RI IRU WKH FRQVWDQWV DV ZHOO LWHUDWLRQV YHUVXV WR UHDFK b RI WKH DQWLFLSDWHG GHIOHFWLRQf $V WKH YDOXHV RI WKH FRQVWDQWV DSSURDFKHG RU ODUJHU WKH V\VWHP DFWXDOO\ GLYHUJHV DQG EHFRPHV XQVWDEOH 7KHUHIRUH WKH YDOXHV RI D D ZHUH FRQVLGHUHG VXLWDEOH IRU DOO V\VWHPV LQYROYHG ,W LV LPSRUWDQW WR QRWH WKDW DQ DVVXPSWLRQ PXVW EH PDGH DV WR WKH DFFHSWDEOH PDJQLWXGH RI WKH UHVLGXDOV RI WKH
PAGE 135

&+$37(5 9 (;3(5,0(17$/ 9(5,),&$7,21 2) 7+( 02'(/,1* 2) 7+( 267(26<17+(6,6 :HOO &KDUDFWHUL]HG 6\VWHPV 6HYHUDO H[SHULPHQWDO IRXUSRLQW ORDGLQJ WHVWV ZHUH SHUIRUPHG RQ ZHOO FKDUDFWHUL]HG V\VWHPV DQG DQDO\]HG ZLWK WKH %21$,/ FRPSXWHU SURJUDP LQ RUGHU WR HYDOXDWH WKH SURJUDPnV DFFXUDF\ DQG DSSOLFDELOLW\ LQ SUHGLFWLQJ RVWHRV\QWKHVLV ORDGGHIOHFWLRQ UHVSRQVH 0DWHULDOV DQG 0HWKRGV 8QLIRUP EHDP $ VWHHO EHDP RI FRQVWDQW FURVVVHFWLRQDO JHRPHWU\ PPKLJK [ PP ZLGH [ PP ORQJf ZDV SODFHG LQ D IRXUSRLQW ORDGLQJ DSSDUDWXV DV VKRZQ LQ WKH VFKHPDWLF UHSUHVHQn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

PAGE 136

n n Sf§/ Q LU n n Rf§ N)LJXUH 6FKHPDWLF UHSUHVHQWDWLRQ RI WKH IRXUSRLQW ORDGLQJ DSSDUDWXV XVHG IRU WKH LQ YLWUR YHULILFDWLRQ VWXGLHV

PAGE 137

WKH RXWHU EHDP GHIOHFWLRQV 7KH H[SHFWHG GHIOHFWLRQV ZHUH DOVR FDOFXODWHG XVLQJ DSSOLFDEOH HQJLQHHULQJ EHDP WKHRU\ VR WKDW FRPn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f DV GHILQHG OL L LQ HTXDWLRQV Ef DTG Ff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f 'HIOHFWLRQ PHDVXUHPHQWV ZHUH UHFRUGHG DW YDU\LQJ ORDGLQJ PDJQLWXGHV

PAGE 138

)LJXUH ,QWDFW KXPDQ IHPXU ZLWK VPDOO GLPSOHV DW FP LQWHUYDOV WR DLG LQ GHIOHFWLRQ PHDVXUHPHQWV

PAGE 139

)LJXUH ,QWDFW KXPDQ IHPXU PRXQWHG LQ WKH IRXUSRLQW ORDGLQJ IL[WXUH IRU LQ YLWUR YHULILFDWLRQ WHVWLQJ If§M 2 /Q

PAGE 140

)LJXUH ,QWDFW IHPXU ZLWK WKH GLDO LQGLFDWRU LQ SODFH LQ WKH IRXUSRLQW ORDGLQJ IL[WXUH UR 21

PAGE 141

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nV LQWUDn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

PAGE 142

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

PAGE 143

'HIOHFWLRQ PPf ? 3 1 3 / I )LJXUH *UDSKLFDO FRPSDUDWLYH GHIOHFWLRQ UHVXOWV LQFOXGLQJ EHDP HTXDWLRQ SUHGLFWLRQ %21$,/ SUHGLFWLRQ DQG PHDVXUHG GHIOHFWLRQ IRU WKH XQLIRUP VWHHO EHDP

PAGE 144

'HIOHFWLRQ PPf )LJXUH *UDSKLFDO FRPSDUDWLYH GHIOHFWLRQ UHVXOWV LQFOXGLQJ PHDVXUHG DQG %21$,/ SUHGLFWLRQV IRU WKH WDSHUHG VWHHO EHDP

PAGE 145

'HIOHFWLRQ PPf )LJXUH 'HIOHFWLRQ UHVXOWV REWDLQHG IURP EHDP HTXDWLRQ SUHGLFWLRQ %21$,/ SUHGLFWLRQ DQG DFWXDO PHDVXUHPHQW RI D IRXUSRLQW ORDGHG IHPXU L

PAGE 146

'HIOHFWLRQ PP )LJXUH &RPSDUDWLYH UHVXOWV REWDLQHG IURP WKH IRXUSRLQW ORDGLQJ RI DQ LQWDFW IHPXU ZLWK D PP )OXWHG QDLO LQ SODFH %21$,/ SUHGLFWHG DFWXDO PHDVXUHPHQW DQG SUHGLFWHG UHVXOWV IRU DQ XQQDLOHG IHPXUf

PAGE 147

'LVFXVVLRQ )RU WKH XQLIRUP VWHHO EHDP WHVWHG WKH %21$,/ SURJUDP \LHOGHG UHVXOWV ZKLFK ZHUH ZLWKLQ Obb RI WKRVH REWDLQHG E\ GLUHFW PHDVXUHn PHQW XVLQJ D GLDO LQGLFDWRU 7KHVH UHVXOWV ZHUH ZHOO ZLWKLQ WKH H[SHULPHQWDO DFFXUDF\ RI PHDVXUHPHQW RI DSSOLHG ORDG GHIOHFWLRQ JHRPHWULF DQG PDWHULDO SURSHUWLHV RI WKH VDPSOH WHVWHG )RU WKH WDSHUHG VWHHO VSHFLPHQ WKH PRGHOLQJ ZDV ZLWKLQ bb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f ZHUH XVHG LQ D WULDO DQG HUURU DSSURDFK WR DIIHFW VXLWDEOH %21$,/ GHIOHFWLRQ UHVSRQVH SUHGLFWLRQV

PAGE 148

)RU WKH XQQDLOHG IHPXU D ERQH PRGXOXV RI [ 1P \LHOGHG SUHGLFWHG GHIOHFWLRQV ZKLFK ZHUH ZLWKLQ b RI WKRVH DFWXDOO\ PHDVXUHG 7KH PRGHOLQJ RI WKH QDLOHG IHPXU \LHOGHG UHVXOWV ZLWKLQ bb RI WKRVH PHDVXUHG ZKHQ WKH ORZHU ERQH PRGXOXV RI O[ 1P ZDV XVHG LQ FRQMXQFWLRQ ZLWK WKH LQWHUIDFLDO VWLIIn QHVV RI [ 1PP DQG D PRGXOXV RI HODVWLFLW\ RI [ 1P IRU WKH VWDLQOHVV VWHHO QDLO 7KH PDLQWHQDQFH RI WKH LQWUDPHGXOODU\ QDLO LQ D KHDOHG RVWHRn V\QWKHVLV DV UHSUHVHQWHG E\ WKH DERYH LGHDOL]HG PRGHO RI DQ LQWDFW IHPXU ZLWK DQ LQWUDPHGXOODU\ QDLO LQ SODFH VHUYHV WR UHGXFH GHIOHFWLRQV WR b RI WKRVH SUHGLFWHG IRU WKH VDPH XQQDLOHG ERQH XQGHU VLPLODU ORDGLQJ FRQGLWLRQV +RZHYHU WKLV WKHRUHWLFDO SUHn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

PAGE 149

WKH DQWHULRU VXUIDFH RI WKH ERQH DV IRU WKH SUHYLRXVO\ WHVWHG LQWDFW IHPXUV )LJXUH f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

PAGE 150

)LJXUH &ORVHXS SKRWRJUDSK RI WKH IUDFWXUH VLWH RI D IHPXU ZLWK D )OXWHG QDLO LQ SODFH WKH IUDFWXUH VLWH KDV EHHQ GLVWUDFWHG WR VKRZ WKH QDLOf

PAGE 151

)LJXUH 2VWHRV\QWKHVLV LQ WKH WHVW IL[WXUH ZLWK WKH IUDFWXUH VXUIDFHV GLVWUDFWHG WR VKRZ WKH QDLO DQG IUDFWXUH VLWH ORFDWLRQ )LJXUH 2VWHRV\QWKHVLV LQ WKH WHVWLQJ IL[WXUH IRU LQ YLWUR YHULILFDWLRQ RI WKH %21$,/ SUHGLFWHG GHIOHFWLRQ UHVSRQVH

PAGE 152

ULIWrr}n ff

PAGE 153

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n V\QWKHVLV ZLWK D PP )OXWHG QDLO LQ SODFH +HUH WRR WKH FRUn UHODWLRQ LV TXLWH JRRG LQ WKH SUR[LPDO DQG PLGVKDIW UHJLRQ RI WKH RVWHRV\QWKHVLV EXW WKH LQDFFXUDF\ LQFUHDVHV LQ WKH GLVWDO SRUWLRQ 7KH SUHGLFWHG GHIOHFWLRQV IRU WKH LQWDFW ERQH ZLWK QR QDLO SUHVHQW DUH LQFOXGHG LQ ERWK H[DPSOHV WR VKRZ WKH VXEVWDQWLDO LQFUHDVH DV PXFK DV b RU PRUH LQ WKH GHIOHFWLRQV RI D IUDFWXUHG V\VWHP ZLWK LQWUDPHGXOODU\ IL[DWLRQ

PAGE 154

'HIOHFWLRQ PPf LQ SODFH 3UHGLFWHG GHIOHFWLRQ RI LQWDFW XQQDLOHG IHPXU LQFOXGHGf

PAGE 155

'HIOHFWLRQ PPf )LJXUH *UDSKLFDO UHSUHVHQWDWLRQ RI WKH DFWXDO GHIOHFWLRQ PHDVXUHPHQW DQG %21$,/ SUHGLFWLRQ RI D IHPXU ZLWK IUDFWXUH DQG PP )OXWHG QDLO LQ SODFH 3UHGLFWHG GHIOHFWLRQ RI LQWDFW XQQDLOHG IHPXU LQFOXGHGf

PAGE 156

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b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b RI WKRVH PHDVXUHG 7KXV LW DSSHDUV WKDW WKH FDQFHOOXV ERQH ZKLFK UHPDLQHG LQ WKH LQWUDPHGXOODU\ FDQDO ZDV VLJQLILFDQW HQRXJK WR UHWDLQ WKH LQWHUPHGLDWH LQWHUIDFLDO VWLIIQHVV YDOXH 7KH QHHG IRU DQ LQFUHDVHG PRGXOXV RI HODVWLFLW\ ZDV DWWULEXWHG WR WKH GU\LQJ RI WKH ERQH GXULQJ WKH ORQJHU SHULRG RI WLPH QHHGHG IRU UHDPLQJ DQG WHVWLQJ WKLV RVWHRV\QWKHVLV $V ZDV GLVFXVVHG LQ &KDSWHU ,, GU\LQJ UHVXOWV LQ VLJQLILFDQW LQFUHDVHV LQ WKH PRGXOXV RI HODVWLFLW\ RI ERQH

PAGE 157

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n GLFWHG GHIOHFWLRQV DQG LQWHUIDFLDO ORDGLQJ SURILOHV LQ WKH DUHD RI WKH IUDFWXUH VLWH LQ RUGHU WR H[SODLQ WKHVH UHODWLYH GHIRUPDWLRQV LQ D V\PPHWULFDOO\ ORDGHG RVWHRV\QWKHVLV 7KH QRQV\PPHWU\ RI WKH ERQH JHRPHWU\ UHVXOWHG LQ GHIOHFWLRQV DQG WKHLU ILUVW GHULYDWLYHV LH VORSHV ZKLFK ZHUH RQO\ VOLJKWO\ XQV\PPHWULFDO SUR[LPDO DQG GLVWDO WR WKH IUDFWXUH 7KHVH GLIIHUHQFHV ZHUH DSSUR[LPDWHO\ b RU OHVV +RZHYHU D PRUH VXEVWDQWLDO GLIIHUHQFH LQ ERQH FXUYDWXUH YDOXHV G
PAGE 158

,W ZDV QRWHG LQ &KDSWHU ,9 WKDW FRQFHQWUDWHG ORDGV DSSOLHG WR WKH H[WHUQDO VXUIDFH RI WKH ERQH EHFRPH GLVWULEXWHG RYHU DQ DUHD ZKHQ PRGHOHG XVLQJ ILQLWH GLIIHUHQFH UHSUHVHQWDWLRQV :KLOH WKLV LV FRQVLGHUHG WR EH PRUH FOLQLFDOO\ UHDOLVWLF LQ WKH FDVH RI DSSOLHG H[WHUQDO ORDGV LW PD\ SUHVHQW SUREOHPV LQ WKH PRGHOLQJ RI WKH LQWHUn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b OHVV WKDQ WKRVH PHDVXUHG 7KH PP V\VWHP ZLWK D PRUH ULJLG QDLO LQ SODFH KDV PRUH GLVWULEXWHG ORDGV LQ WKLV UHJLRQ DQG WKH FRPSXWHG GHIOHFWLRQV DUH IRXQG WR EH XQGHUHVWLPDWHG E\ RQO\ b ,W LV DOVR LPSRUWDQW WR QRWH WKDW %21$,/ FRPSXWHV GHIOHFWLRQV DQG VORSHV DW QRGH SRLQWV FP WR HLWKHU VLGH RI WKH IUDFWXUH VLWH :KLOH WKHVH SUHGLFWHG QRGDO GHIOHFWLRQV FRUUHODWH ZHOO ZLWK WKRVH DFWXDOO\ PHDVXUHG DW WKH QRGH SRLQWV WKH\ DUH QRW QHFHVVDULO\ LQGLFDWLYH RI WKH IUDFWXUH VLWH GHIOHFWLRQ )RU WKH WZR V\VWHPV DQDO\]HG DERYH WKH QRGDO GHIOHFWLRQV ZHUH b OHVV WKDQ DW WKH VLWH LWVHOI 7KLV GLIIHUHQFH ZDV SULPDULO\ GXH WR WKH VORSH RI WKH EHDP LQ WKH UHJLRQ ZLWK WKH FXUYDWXUH DIIHFWLQJ WKH UHVXOW E\

PAGE 159

OHVV WKDQ bf 7KH GLIIHUHQFHV LQ WKH VORSHV DQG FXUYDWXUHV SUR[LPDO DQG GLVWDO WR WKH IUDFWXUH VLWH DOVR HIIHFW WKH UHODWLYH WUDQVYHUVH GHIRUPDWLRQ DQG DQJXODWLRQ DW WKH IUDFWXUH LQWHUIDFH 7KH PDJQLWXGH RI WKH LQDFFXUDFLHV GHSHQG RQ WKH UHODWLYH GHIOHFWLRQV VORSHV DQG FXUYDWXUHV RI WKH SDUWLFXODU RVWHRV\QWKHVHV XQGHU LQYHVWLn JDWLRQ )RU WKH WZR V\VWHPV FLWHG DERYH WKH UHODWLYH GLIIHUHQFHV LQ WKHVH YDOXHV DIIHFWHG WKH UHVXOWV E\ OHVV WKDQ b &RQFOXVLRQV 7KH PRGHOLQJ RI V\VWHPV ZLWK ZHOO GHILQHG PDWHULDO DQG JHRPHWULF SURSHUWLHV E\ WKH %21$,/ FRPSXWHU SURJUDP \LHOGHG UHVXOWV ZKLFK ZHUH W\SLFDOO\ ZLWKLQ b RI WKRVH DFWXDOO\ PHDVXUHG 7KH KLJKO\ YDULDEOH SURSHUWLHV RI ERQH LQ YLWUR PDGH DFFXUDWH SUHGLFWLRQV RI WKH H[SHULPHQWDO RVWHRV\QWKHVHV GLIILFXOW +RZHYHU SUHGLFWHG UHVXOWV ZKLFK ZHUH ZLWKLQ b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b 7KH %21$,/ SURJUDP ZDV IRXQG WR XQGHUHVWLPDWH WKH GHIOHFWLRQV LQ WKH GLVWDO SRUWLRQ RI WKH RVWHRV\QWKHVLV E\ DSSUR[LPDWHO\ b

PAGE 160

7KLV ZDV GXH WR WKH RULJLQDO DVVXPSWLRQ RI FRQVWDQW QDLOERQH LQWHUn IDFLDO VWLIIQHVV DORQJ WKH OHQJWK RI WKH V\VWHP &DUHIXO REVHUYDWLRQ RI WKH IHPXUV WHVWHG LQGLFDWH WKDW HYHQ ZLWK UHDPLQJ RI WKH LQWUDn PHGXOODU\ FDQDO WR PP WKHUH ZDV FDQFHOOXV ERQH SUHVHQW WKURXJKRXW WKH GLVWDO IUDJPHQW DQG LQ SRUWLRQV RI WKH SUR[LPDO IUDJPHQW 7KH ILQLWH GLIIHUHQFH UHSUHVHQWDWLRQ RI WKH GHVFULELQJ HTXDWLRQV OHDGV WR LQDFFXUDFLHV ZKLFK DUH GHSHQGHQW RQ WKH UHODWLYH PDJQLWXGHV RI GHIOHFWLRQV DQG FXUYDWXUH ZLWKLQ WKH V\VWHP &RPSDULVRQ RI WKH GHIOHFWLRQV REWDLQHG LQ WKH LQ YLWUR WHVWLQJ RI DQ LQWDFW IHPXU ZLWK DQG ZLWKRXW DQ LQWUDPHGXOODU\ QDLO LQ SODFH LQGLFDWHV WKDW WKH VKDULQJ RI ORDG EHWZHHQ D QDLO DQG ERQH UHGXFHV WKH GHIOHFWLRQV LQ EHQGLQJ E\ DV PXFK DV b +RZHYHU LQ D VLPLODU LQWUDPHGXOODU\ RVWHRV\QWKHVLV ZLWK D IUDFWXUH SUHVHQW DW WKH PLGVKDIW GHIOHFWLRQV PD\ EH PRUH WKDQ WZR WLPHV WKRVH REVHUYHG IRU WKH LQWDFW ERQH 5HODWLYH WUDQVYHUVH GHIOHFWLRQV RFFXU DW WKH IUDFWXUH VLWH RI DQ RVWHRV\QWKHVLV XQGHU IRXUSRLQW EHQGLQJ ORDGV 7KHVH GHn IOHFWLRQV DUH GXH WR WKH GLIIHUHQFHV LQ QDLOERQH LQWHUIDFLDO ORDGLQJV ZKLFK DUH WKH UHVXOW RI WKH QRQFRQVWDQW JHRPHWULF SURSHUn WLHV RI WKH IHPXU ,W LV LPSRVVLEOH WR FRUUHODWH WKH DEVROXWH GHIOHFWLRQV LQ D SDUWLFXODU RVWHRV\QWKHVLV ZLWK WKH UHVXOWV RI WKH %21$,/ SURJUDP ZLWKRXW PDNLQJ PDWHULDO SURSHUW\ GHWHUPLQDWLRQV +RZHYHU VXFFHVV LQ SUHGLFWLQJ WKH DEVROXWH GHIOHFWLRQV XVLQJ WULDO DQG HUURU DSSOLFDWLRQ RI PDWHULDO SURSHUWLHV ZHOO ZLWKLQ WKH UDQJH RI WKRVH UHSRUWHG LQ WKH OLWHUDWXUH LQGLFDWH WKH HIIHFWLYHQHVV RI WKH

PAGE 161

%21$,/ SURJUDP LQ GHWHUPLQLQJ WKH UHODWLYH HIIHFW RI FKDQJHV RI WKH SURSHUWLHV RI WKH V\VWHP FRPSRQHQWV RQ WKH GHIOHFWLRQV WR EH H[SHFWHG

PAGE 162

&+$37(5 9, %21$,/ 35(',&7,21 2) 267(26<17+(6,6 '()/(&7,216 ,Q WKH SUHYLRXV FKDSWHU WKH %21$,/ PRGHOLQJ RI WKH GHIOHFn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f ZDV XVHG FI )LJXUH f 7KH ERQH PRGXOXV ZDV KHOG DW [ 1P DQG LWV JHRPHWU\ LH DUHD PRPHQW RI LQHUWLD DV D IXQFWLRQ RI OHQJWK ZDV WDNHQ IURP RQH RI WKH IHPXUV SURFHVVHG LQ WKH LQ YLWUR YHULILFDWLRQ VWXGLHV

PAGE 163

RI &KDSWHU 9 %\ KROGLQJ WKHVH SURSHUWLHV DQG WKH ORDGLQJ FRQILJXUDWLRQ DQG PDJQLWXGH FRQVWDQW GLUHFW FRPSDULVRQ RI QRQ GLPHQVLRQDO ERQH GHIOHFWLRQV < ZDV FDUULHG RXW DQG WKH GHIOHFWLRQ QRUPDOL]LQJ SDUDPHWHU \4 < \\4f UHPDLQHG IL[HG XQGHU WKHVH FRQGLWLRQV \ [ a Prf 7KH LQWUDPHGXOODU\ QDLO ZDV DVVXPHG WR KDYH DQ DUHD PRPHQW RI LQHUWLD RI [ P SDWWHUQHG DIWHU D PP )OXWHG QDLO 9DULDWLRQV LQ WKH UHODWLYH IOH[XUDO ULJLGLWLHV RI WKH FRPSRQHQWV VKRZQ LQ &KDSWHU ,9 WR KDYH D VLJQLn ILFDQW HIIHFW RQ WKH V\VWHP UHVSRQVH ZHUH FDUULHG RXW E\ YDU\LQJ WKH QDLO PRGXOXV RI HODVWLFLW\ RYHU WKH UDQJH WR [ 1P 7KH QDLOERQH LQWHUIDFLDO VWLIIQHVV ZDV DOVR YDULHG RYHU WKH UDQJH RI WR [ ,4 1PP DV SUHYLRXVO\ GHILQHG E\ LQ YLWUR WHVWLQJ 7KHVH FRPSRQHQW SURSHUWLHV ZHUH XVHG E\ WKH %21$,/ FRPSXWHU SURJUDP IRU ERWK KHDOHG DQG IUDFWXUHG RVWHRV\QWKHVHV 7KH UHVXOWn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f

PAGE 164

5HVXOWV +HDOHG RVWHRV\QWKHVHV )LJXUH LV SUHVHQWHG WR LOOXVWUDWH WKH QRQGLPHQVLRQDO GHIOHFWLRQ RI WKH ERQH DW WKH IUDFWXUH VLWH IRU D KHDOHG RVWHRV\QWKHVLV ZLWK DQ LQWUDPHGXOODU\ QDLO LQ SODFH 7KHVH UHVXOWV DUH SORWWHG DV D IXQFWLRQ RI WKH PRGXOXV UDWLRV IRU YDULRXV LQWHUIDFLDO VWLIIQHVVHV 7KH VKDGHG DUHD LQGLFDWHV WKH DSSUR[LPDWH UHJLRQ RI SURSHUW\ PDJQLWXGHV ZKLFK ZHUH XVHG LQ WKH LQn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f 1RWH WKH LQIOHFWLRQ LQ WKH PLGVSDQ ORDGLQJ LQWHQVLW\ DV LQWHUIDFLDO VWLIIQHVV LQFUHDVHV DQG RYHUDOO GHIOHFWLRQV GHFUHDVH )LJXUH VKRZV KRZ VLPLODU PLGVSDQ

PAGE 165

0LGVSDQ 'HIOHFWLRQ ?4f H$ )LJXUH 1RQGLPHQVLRQDO PLGVSDQ ERQH GHIOHFWLRQ IRU WKH KHDOHG RVWHRV\QWKHVLV ZLWK LQWUDPHGXOODU\ QDLO LQ VLWX WKH VKDGHG DUHD LQGLFDWHV DSSUR[LPDWH UHJLRQ RI LQ YLWUR YHULILFDWLRQf

PAGE 166

1RQGLPHQVLRQDO ,QWHU IDFLDO /RDGLQJ )LJXUH 1RQGLPHQVLRQDO LQWHUIDFLDO ORDGLQJ SURILOHV IRU D KHDOHG RVWHRV\QWKHVLV ZLWK LQWHUIDFLDO VWLIIQHVV RI [ 1PP DW YDULRXV ULJLGLW\ UDWLRV

PAGE 167

1RQGLPHQVLRQDO ,QWHUIDFLDO /RDGLQJ )LJXUH 1RQGLPHQVLRQDO LQWHUIDFLDO ORDGLQJ SURILOHV IRU D KHDOHG RVWHRV\QWKHVLV ZLWK LQWHUIDFLDO VWLIIQHVV RI [ a 1PP DW YDULRXV ULJLGLW\ UDWLRV

PAGE 168

1RQGLPHQVLRQDO ,QWHUIDFLDO /RDGLQJ 3RLQWV 2I /RDG $SSOLFDWLRQ )LJXUH 1RQGLPHQVLRQDO LQWHUIDFLDO ORDGLQJ SURILOHV IRU D KHDOHG RVWHRV\QWKHVLV ZLWK LQWHUIDFLDO VWLIIQHVV RI [ 1PP DW YDULRXV ULJLGLW\ UDWLRV

PAGE 169

1RQGLPHQVLRQDO ,QWHUIDFLDO /RDGLQJ )LJXUH 1RQGLPHQVLRQDO LQWHUIDFLDO ORDGLQJ SURILOHV IRU D KHDOHG RVWHRV\QWKHVLV DW D PRGXOXV UDWLR RI IRU YDULRXV LQWHUIDFLDO VWLIIQHVV

PAGE 170

1RQGLPHQVLRQDO ,QWHUIDFLDO /RDGLQJ )LJXUH 1RQGLPHQVLRQDO LQWHUIDFLDO ORDGLQJ SURILOHV ZKLFK ZHUH REWDLQHG XVLQJ GLIIHUHQW FRPSRQHQW SURSHUWLHV DQG LQWHUn IDFLDO VWLIIQHVVHV WR DFKLHYH HTXDO PLGVSDQ GHIOHFWLRQ

PAGE 171

GHIOHFWLRQV FDQ EH REWDLQHG E\ XVLQJ GLIIHUHQW FRPELQDWLRQV RI FRPSRQHQW SURSHUWLHV ZKLFK UHVXOW LQ GLVVLPLODU LQWHUIDFLDO ORDG GLVWULEXWLRQ 2VWHRV\QWKHVHV ZLWK IUDFWXUH )LJXUH VKRZV WKH UHVXOWV SUHGLFWHG E\ WKH %21$,/ SURJUDP IRU WKH QRQGLPHQVLRQDO PLGVSDQ ERQH GHIOHFWLRQV DW WKH IUDFWXUH VLWH 7KHVH UHVXOWV DUH SORWWHG DV D IXQFWLRQ RI PRGXOXV UDWLR IRU YDULRXV LQWHUIDFLDO VWLIIQHVVHV ZLWK WKH FXUYHV RYHUODSSLQJ DW ORZ PRGXOXV UDWLRV )RU KLJK PRGXOXV UDWLRV ( ( f WKH LQWHUIDFLDO VWLIIQHVVHV DERYH Q E [ 1PP KDYH OLWWOH HIIHFW RQ WKH GHIOHFWLRQV EXW IRU ORZHU PRGXOXV UDWLRV (A(A f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n SODFHPHQWV RI WKH IUDFWXUH VXUIDFHV UHVSHFWLYHO\ 7KHVH YDOXHV DUH DOVR SORWWHG DV D IXQFWLRQ RI PRGXOXV UDWLR IRU YDULRXV LQWHUn IDFLDO VWLIIQHVVHV 7KH FRQILJXUDWLRQ RI WKHVH JUDSKV LV VLPLODU WR )LJXUH

PAGE 172

n n f§ f f )LJXUH %21$,/ SUHGLFWHG UHVXOWV RI QRQGLPHQVLRQDO PLGVSDQ ERQH GHIOHFWLRQ IRU IUDFWXUHG RVWHR V\QWKHVHV DV D IXQFWLRQ RI PRGXOXV UDWLR IRU VHYHUDO LQWHUIDFLDO VWLIIQHVVHV

PAGE 173

)LJXUH %21$,/ SUHGLFWHG QRQGLPHQVLRQDO LQWHUIDFLDO UHDFWLRQ SURILOHV IRU IUDFWXUHG RVWHRV\QWKHVHV DW D PRGXOXV UDWLR RI IRU VHYHUDO LQWHUIDFLDO VWLIIQHVVHV

PAGE 174

/ 1RQGLPHQVLRQDO ,QWHU IDFLDO /RDGLQJ [ f

PAGE 175

1RQGLPHQVLRQDO ,QWHU IDFLDO /RDGLQJ [LRf N [ 1PP )LJXUH %21$,/ SUHGLFWHG QRQGLPHQVLRQDO LQWHUIDFLDO UHDFWLRQ SURILOHV IRU IUDFWXUHG RVWHRV\QWKHVHV DW D PRGXOXV UDWLR RI IRU YDULRXV LQWHUIDFLDO VWLIIQHVVHV

PAGE 176

1RQGLPHQVLRQDO ,QWHUIDFLDO /RDGLQJ [LRf )LJXUH %21$,/ SUHGLFWHG QRQGLPHQVLRQDO LQWHUIDFLDO UHDFWLRQ SURILOHV IRU IUDFWXUHG RVWHRV\QWKHVHV DW D PRGXOXV UDWLR RI IRU YDULRXV LQWHUIDFLDO VWLIIQHVVHV

PAGE 177

1RQGLPHQVLRQDO ,QWHUIDFLDO 0LVPDWFK / )LJXUH 1RQGLPHQVLRQDO IUDFWXUH VXUIDFH PLVPDWFK DV D IXQFWLRQ RI PRGXOXV UDWLR IRU YDULRXV LQWHUIDFLDO VWLIIQHVVHV ,QVHUW VKRZV UHODWLRQVKLS RI PLVPDWFK GLVWDO PRYHV IXUWKHU WKDQ SUR[LPDOf

PAGE 178

)LJXUH )UDFWXUH VXUIDFH DQJXODWLRQ DV D IXQFWLRQ RI PRGXOXV UDWLR IRU YDULRXV LQWHUIDFLDO VWLIIQHVVHV

PAGE 179

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n IDFLDO VWLIIQHVV EHWZHHQ WKH FRPSRQHQWV ZKLFK DOORZV UHODWLYH GLVn SODFHPHQW WR WDNH SODFH DQG LQWHUIDFLDO ORDGV WR EH GHYHORSHG 7KXV D KHDOHG V\VWHP ZLWK D VWLII LQWHUIDFH DQG KLJK QDLO PRGXOXV LH UHDPHG FRUWLFDO ERQH DQG ULJLG QDLOf ZLOO H[KLELW VPDOO GHIOHFWLRQV UHGXFWLRQV LQ HLWKHU QDLO PRGXOXV RU LQWHUIDFLDO VWLIIQHVV OHDG WR JUHDWHU GHIOHFWLRQV 7KH HIIHFW RI YDULDWLRQ LQ LQWHUIDFLDO VWLIIQHVV LV JUHDWHVW LQ WKH UHJLRQ RI ORZHU N YDOXHV 7KLV FRUUHVSRQGV WR WKH LQLWLDO FDQFHOOXV FUXVKLQJ UHJLRQ VKRZQ LQ )LJXUH 9DULDWLRQ RI VWLIIn QHVV LQ WKH LQWHUPHGLDWH UDQJH )LJXUHV DQG f KDYH D VPDOOHU HIIHFW DQG WKH LQFUHDVH RI LQWHUIDFLDO VWLIIQHVV DERYH [ 1PP ZHOO UHDPHG FRUWLFDO ERQHf KDV RQO\ D VPDOO HIIHFW RQ WKH PLGVSDQ GHIOHFWLRQV WKDW DUH SUHGLFWHG 7KLV GLPLQLVKLQJ HIIHFW RI LQWHUIDFLDO VWLIIQHVV RQ GHIOHFWLRQ OHQGV VXSSRUW WR WKH QHHG IRU IXUWKHU VWXG\ RI WKH QRQOLQHDU LQWHUIDFLDO VWLIIQHVV EHKDYLRU 7KH HIIHFWV RI PRGXOXV UDWLR DQG LQWHUIDFLDO VWLIIQHVV RQ LQWHUIDFLDO ORDGLQJ DUH VKRZQ LQ )LJXUHV )RU WKH ORZ

PAGE 180

LQWHUIDFLDO VWLIIQHVVHV [ 4 1PPf VKRZQ LQ )LJXUH WKH LQWHUIDFLDO ORDGLQJ LV ZHOO GLVWULEXWHG ZLWK QR ODUJH SHDNV DORQJ WKH V\VWHP 7KLV LV D FRQVHTXHQFH RI WKH IDFW WKDW ODUJH GHIOHFWLRQV DUH QHFHVVDU\ IRU ODUJH LQWHUIDFLDO VWUHVVHV 7KH PDJQLWXGHV DUH DIIHFWHG WR D VPDOO H[WHQW E\ LQFUHDVHV LQ QDLO PRGXOXV &KDQJLQJ WR WKH LQWHUPHGLDWH LQWHUIDFLDO VWLIIQHVVHV [ 1PPf )LJXUH OHDGV WR KLJKHU ORDG LQWHQVLWLHV 7KLV DGGLWLRQDO ORDG RQ WKH QDLO OHDGV WR GHFUHDVHV LQ RYHUDOO GHIOHFWLRQ EHFDXVH WKH QDLO ZKLFK LV VWLIIHU WKDQ WKH ERQH FDUULHV PRUH ORDG )LJXUH VKRZV WKH ORDGLQJ LQWHQVLW\ SURILOHV IRU WKH VWLIIHVW LQWHUIDFHV N [ 1Pf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

PAGE 181

WKH GLIIHUHQFHV LQ FXUYDWXUH FDXVHG E\ WKH GLIIHUHQW ORDGLQJ VHHQ E\ WKH WZR FRPSRQHQWV 7KHUH LV D WUDGH RII EHWZHHQ WKH GHFUHDVH LQ GHIOHFWLRQ DQG WKH ORFDOL]HG LQWHUIDFLDO ORDG LQWHQVLWLHV 7KLV LV H[HPSOLILHG LQ )LJXUH ZKHUH WKH ORDG LQWHQVLW\ SURILOHV DUH VKRZQ IRU V\VWHPV XQGHUJRLQJ HTXDO QRQGLPHQVLRQDO PLGVSDQ GHIOHFWLRQV < f 7KH QDWXUH RI WKH ORDGLQJ SURILOHV DUH DIIHFWHG E\ ERWK WKH FKRLFH RI QDLO ULJLGLW\ DQG LQWHUIDFLDO SURSHUWLHV 7KXV WZR VRPHZKDW GLIIHUHQW V\VWHPV FDQ \LHOG WKH VDPH PLGVKDIW GHIOHFWLRQV $ V\VWHP ZLWK LQWHUPHGLDWH LQWHUIDFLDO SURSHUWLHV DQG D PRUH ULJLG QDLO ZLOO \LHOG WKH VDPH PD[LPXP GHIOHFWLRQ DV D V\VWHP ZLWK JUHDWHU LQWHUn IDFLDO VWLIIQHVVHV DQG ORZHU QDLO ULJLGLW\ 7KH ODWWHU V\VWHP KRZHYHU LQYROYHV ORZHU LQWHUIDFLDO VWUHVVHV 7KXV IURP D FOLQLFDO VWDQGSRLQW VKRXOG WKH SK\VLFLDQ ZLVK WR UHGXFH WKH LQWHUIDFLDO ORDGV QHDU WKH VLWH RI WKH RULJLQDO IUDFWXUH D OHVV ULJLG QDLO SODFHG LQ ZHOO UHDPHG FRUWLFDO ERQH VHUYHV WKLV SXUSRVH $V H[SHFWHG WKH PRUH OLPEHU WKH QDLO WKH VPDOOHU WKH ORDGV RQ LW DQG WKH ORZHU WKH VWUHVV SURWHFWLRQ 7KH PRUH ORDG WUDQVPLWWHG WR WKH QDLO WKH OHVV RYHUDOO GHn IOHFWLRQ RI WKH RVWHRV\QWKHVLV DQG WKH EHQGLQJ VWUHVVHV LQ WKH ERQH %HQGLQJ VWUHVVHV DQG VWUDLQ LQ WKH ERQH RFFXU ZLWK PLGVSDQ GHIOHFWLRQ +RZHYHU WKH UHODWLRQVKLS EHWZHHQ EHQGLQJ VWUHVV DQG GHIOHFWLRQ LV QRW RQHWRRQH ,W LV DFWXDOO\ GHSHQGHQW RQ WKH FXUYDWXUH RI WKH EHDP DSSUR[LPDWHG E\ G\G[f 7RQLQR HW DO f :RR HW DO f DQG 8KWRII DQG 'XEXF f KDYH VKRZQ WKDW VWUHVV SURWHFWLRQ LQ SODWHG IUDFWXUHV OHDGV WR GLVXVH

PAGE 182

RVWHRSRURVLV DQG VXEVHTXHQW ZHDNQHVV RI WKH ERQH PDNLQJ LW OLDEOH WR IUDFWXUH RQFH WKH GHYLFH LV UHPRYHG :KLOH LQWUDPHGXOODU\ IL[DWLRQ LV GLIIHUHQW WKDQ SODWLQJ LQ WKDW LW DOORZV D[LDO ORDGV WR EH WUDQVPLWWHG DFURVV WKH IUDFWXUH VLWH WKH EHQGLQJ VWUHVVHV DUH XVXDOO\ DQ RUGHU RI PDJQLWXGH ODUJHU WKDQ WKH D[LDO VWUHVVHV GHYHORSHG 7KXV RQH PLJKW FRQMHFWXUH WKDW D VLPLODU VWUHVV SURn WHFWLRQ SKHQRPHQD LQ ERQHV IL[HG ZLWK LQWUDPHGXOODU\ QDLOVf§SHUKDSV WR D OHVVHU H[WHQWf§PD\ DOVR UHVXOW LQ KDPSHUHG VHFRQGDU\ UHPRGHOLQJ RI WKH IUDFWXUH DQG KHQFH D ZHDNHU ERQH $OO WKH DERYH GLVFXVVLRQV DUH SUHGLFDWHG RQ WKH DVVXPSWLRQ RI LQWLPDWH QDLOERQH LQWHUIDFLDO FRQWDFW +LJK ORFDO ORDGLQJ LQWHQVLn WLHV PD\ FDXVH WKH ERQH WR QHFURVH DQG WKH QDLO WR ORRVHQ 7KLV UHVXOWV LQ VLJQLILFDQWO\ ODUJHU GHIOHFWLRQV GXH WR WKH GLPLQLVKHG ORDGV EHLQJ FDUULHG E\ WKH QDLO ,W PD\ EH VSHFXODWHG WKDW LI WKLV ORRVHQLQJ FDQ EH IRUFHG WR RFFXU RYHU D SHULRG RI WLPH DV WKH ERQH KHDOV WKHQ WKH VWUHVV SURWHFWLRQ ZKLFK PD\ OHDG WR RVWHRSRURVLV ZLOO EH UHGXFHG ZLWK WLPH )LQDO UHPRYDO RI WKH ORRVHQHG LPSODQW WKHQ OHDYHV EHKLQG D UHODWLYHO\ QRUPDO ERQH WR FDUU\ WKH ORDGV 2VWHRV\QWKHVHV ZLWK IUDFWXUH )LJXUH LOOXVWUDWHV WKH PLGVSDQ QRQGLPHQVLRQDO GHIOHFWLRQV SUHGLFWHG IRU WKH LQWUDPHGXOODU\ RVWHRV\QWKHVHV ZLWK D IUDFWXUH SUHVHQW :KHQ FRPSDUHG WR WKH UHVSRQVH VHHQ LQ WKH KHDOHG V\VWHPV )LJXUH f LW EHFRPHV HYLGHQW WKDW WKH YDULDWLRQ RI FRPSRQHQW DQG LQWHUIDFLDO SURSHUWLHV KDYH D JUHDWHU HIIHFW RQ WKH V\VWHP SHUIRUPDQFH )RU H[DPSOH ZLWK DQ LQWHUIDFLDO VWLIIQHVV RI [ 1PP D FKDQJH LQ WKH PRGXOXV UDWLR IURP WR LH D VWLIIHQLQJ RI WKH QDLO UHVXOWV LQ

PAGE 183

D b GHFUHDVH LQ GHIOHFWLRQV IRU WKH KHDOHG V\VWHP ZKLOH WKH GHn IOHFWLRQ RI WKH IUDFWXUHG V\VWHP LV UHGXFHG E\ PRUH WKDQ b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n WULEXWHG DORQJ LWV OHQJWK DV LQ WKH KHDOHG V\VWHPV GLVFXVVHG LQ WKH SUHYLRXV VHFWLRQ 7KH QRQGLPHQVLRQDO GHIOHFWLRQ YDOXHV GHVFULEH D VXUIDFH ZKHQ FRQVLGHUHG DV D WKUHH GLPHQVLRQDO JUDSK ZLWK GHIOHFWLRQV SORWWHG YHUWLFDOO\ DV D IXQFWLRQ RI PRGXOXV UDWLR DQG LQWHUIDFLDO VWLIIQHVV 7KXV IRU D JLYHQ ULJLGLW\ UDWLR D ORFDO PLQLPXP LQ GHIOHFWLRQ LV VHHQ WR RFFXU DW LQWHUIDFLDO VWLIIQHVVHV EHWZHHQ [ 1PP DQG [ p 1PP ,QYHVWLJDWLRQ RI WKH LQWHUIDFLDO ORDGLQJ SURILOHV )LJXUHV f DW YDULRXV PRGXOXV UDWLRV DUH KHOSIXO LQ XQGHUVWDQGLQJ WKHVH GHIOHFWLRQV )LJXUH VKRZV WKH QRQGLPHQVLRQDO LQWHUIDFLDO ORDGLQJ SURILOHV IRU D PRGXOXV UDWLR RI DW YDULRXV LQWHUIDFLDO VWLIIQHVVHV )RU WKLV SDUWLFXODU VDPSOH WKH QDLOERQH ULJLGLW\

PAGE 184

UDWLR ( ( Af LV DSSUR[LPDWHO\ 7KXV WKH ERQH LV WLPHV DV ULJLG DV WKH QDLO DQG WKH ERLAH DSSHDUV DV D TXLWH ULJLG WXEH DURXQG D PRUH IOH[LEOH QDLO $SSOLFDWLRQ RI WKH H[WHUQDO ORDGV OHDGV WR GHIOHFWLRQ DQG LQWHUIDFLDO FRQWDFW VWUHVVHV :KHQ WKH LQWHUIDFH LV UHODWLYHO\ VRIW WKH LQWHUIDFLDO ORDGV DUH GLVn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

PAGE 185

LQ D PRGHO ZLWK D UDWKHU IOH[LEOH QDLO DQG YHU\ VWLII ZHOO UHDPHGf LQWHUIDFH LV [ 7KLV WUDQVODWHV LQWR D ORDG RI [ r 1 OEf RYHU WKH FP VSDQ DGMDFHQW WR WKH IUDFWXUH VLWH ,W LV GLIILFXOW WR HVWLPDWH WKH DUHD RYHU ZKLFK WKLV ORDG LV GLVWULEXWHG ,I RQH DVVXPHV KRZHYHU WKDW WKH FRQWDFW LV DORQJ D EDQG FP ZLGH DQG FP ORQJ WKHQ WKH VWUHVV LV DSSUR[LPDWHO\ 03D SVLf ZKLFK LV ZHOO DERYH WKH FRPSUHVVLYH VWUHQJWK RI FRUWLFDO ERQH 03Df DV UHSRUWHG E\ 5HLOO\ f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f 7KHVH UHVXOWV LQGLFDWH WKDW D PLQLPXP PLVPDWFK RFFXUV ZLWK KLJK LQWHUIDFLDO VWLIIQHVVHV DQG KLJK PRGXOXV UDWLRV ORFDOL]HG FUXVKLQJ LV QHJOHFWHGf 7KH PLVPDWFK

PAGE 186

LV LQFUHDVHG VLJQLILFDQWO\ LQ V\VWHPV ZKLFK KDYH FDQFHOORXV ERQH ORZ LQWHUIDFLDO VWLIIQHVVf SUHVHQW RU ZKHUH WKH ULJLGLW\ UDWLR LV ORZ ,W LV LQWHUHVWLQJ WR QRWH WKDW D QRQGLPHQVLRQDO GHIOHFWLRQ RI WKH PLQLPXP VHHQ LQ )LJXUH FRUUHVSRQGV WR DQ DFWXDO GHIRUPDWLRQ RI DSSUR[LPDWHO\ SP 7KH GLDPHWHU RI D FDSLOODU\ LV UHSRUWHG WR EH RQ WKH RUGHU RI SP .LWFKLQ DQG -XOLDQ f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n LQJ RU IUDFWXUH VXUIDFH LQWHUDFWLRQ LV DVVXPHG ,Q WKH FOLQLFDO VLWXDWLRQ KRZHYHU PXVFOH ORDGLQJ ZLWK RU ZLWKRXW ZHLJKW EHDULQJ ZLOO EH SUHVHQW DQG ZLOO WHQG WR UHGXFH WKH DQJXODWLRQ E\ FRPn SUHVVLQJ WKH IUDFWXUH VXUIDFHV

PAGE 187

6XPPDU\ 7KH %21$,/ FRPSXWHU SURJUDP KDV EHHQ XVHG WR PRGHO VHYHUDO RVWHRV\QWKHVHV XQGHU IRXUSRLQW EHQGLQJ ORDGLQJ 7KH SUHGLFWHG GHIOHFWLRQV KDYH EHHQ VKRZQ WR EH FRQVLVWHQW ZLWK H[SHULPHQWDO REVHUYDWLRQV DQG VHYHUDO FRQFOXVLRQV FDQ EH GUDZQ +HDOHG RVWHRV\QWKHVHV ,Q WKH RVWHRV\QWKHVLV ZLWK LQWUDn PHGXOODU\ QDLO UHPDLQLQJ LQ SODFH WKH ORDGV DSSOLHG WR WKH V\VWHP DUH VKDUHG E\ ERWK WKH ERXH DQG WKH QDLO 7KH DPRXQW RI ORDG WUDQVPLWWHG E\ WKH ERQH WR WKH QDLO LV GHWHUPLQHG E\ WKH UHODWLYH PDWHULDO DQG JHRPHWULF SURSHUWLHV RI ERWK WKH QDLO DQG ERQH DQG E\ WKH FKDUDFWHU RI WKH LQWHUIDFH EHWZHHQ WKHP 7KLV LQWHUn GHSHQGHQF\ RI WZR RU PRUH YDULDEOHV RQ WKH V\VWHP UHVSRQVH DOORZV VLPLODU GHIOHFWLRQV WR EH REWDLQHG ZLWK WKH XVH RI GLIIHUHQW QDLO DQG LQWHUIDFLDO SURSHUWLHV 7KLV LV D UHVXOW RI WKH UHGLVWULEXWLRQ RI WKH UQWHUIDFLDO ORDGLQJ SURILOHV WKH FRPSRQHQW LQWHUDFWLRQV ZKLFK RFFXUV ,Q JHQHUDO KRZHYHU WKH KHDOHG V\VWHPV XQGHUJR WKH OHDVW DPRXQW RI PLGVSDQ GHIOHFWLRQ D UHGXFWLRQ RI b ZKHQ FRPSDUHG WR WKH LQWDFW ERQH ZLWKRXW LQWHUQDO IL[DWLRQf ZLWK KLJK QDLOERQH PRGXOXV UDWLRV LH VWLIIHU QDLOV DQG KLJKHU LQWHUIDFLDO VWLIIn QHVVHV DQG WKH QDLOV FDUU\ D ODUJH SRUWLRQ RI WKH ORDG 7KH GDWD DOVR VKRZV WKDW WKH LQWHUIDFLDO ORDGLQJ PDJQLWXGHV DQG PLGVSDQ GHIOHFWLRQV RI V\VWHPV ZLWK VRIW LQWHUIDFLDO SURSHUWLHV DUH QRW DIIHFWHG DV PXFK E\ YDULDWLRQV LQ PRGXOXV UDWLR DV DUH WKRVH ZLWK VWLIIHU LQWHUIDFHV 7KH UHVXOWLQJ GHFUHDVH LQ GHIOHFWLRQ GXH WR WKH PDLQWHQDQFH RI WKH LQWUDPHGXOODU\ GHYLFH DIWHU LQLWLDO KHDOLQJ

PAGE 188

VHUYHV WR UHGXFH WKH VWUDLQV DSSHDULQJ LQ WKH ERQH XQGHU EHQGLQJ ORDG 7KLV PD\ OHDG WR D UHGXFWLRQ LQ VWUHVVHV DSSOLHG WR WKH ERQ\ PDWHULDO DQG KDPSHU WKH IXQFWLRQDO UHPRGHOLQJ RI WKH ERQH GXULQJ WKH ODWWHU VWDJHV RI WKH KHDOLQJ SURFHVV 7RQLQR HW DW f :RR HW DW f DQG 8KWRII DQG 'XEXF f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nV GHIOHFWLRQ 9DULDWLRQV LQ WKH UHODWLYH PDJQLWXGHV RI WKHVH SURSHUWLHV DOWHU WKH QDLOERQH LQWHUIDFLDO ORDGLQJ LQWHQVLW\ DQG GLVWULEXWLRQ )RU D JLYHQ PRGXOXV UDWLR WKH PLGVKDIW GHIOHFWLRQV DUH IRXQG WR EH D IXQFWLRQ RI WKH LQWHUIDFLDO SURSHUWLHV ZLWK ORFDO PLQLPD RFFXUULQJ DW YDULRXV PRGXOXV UDWLRV IRU GLIIHULQJ LQWHUIDFLDO VWLIIn QHVVHV 6\VWHPV ZLWK KLJKHU PRGXOXV UDWLRV PRUH ULJLG QDLOVf DQG LQWHUIDFLDO VWLIIQHVVHV ZHOO UHDPHG FRUWLFDO ERQHf H[KLELW WKH VPDOOHVW PLGVSDQ GHIOHFWLRQV 6LPLODU SURSHUWLHV DOVR PLQLPL]H WUDQVYHUVH GHIRUPDWLRQV DQG IUDFWXUH VLWH DQJXODWLRQ +RZHYHU

PAGE 189

WKH PLQLPXP WUDQVYHUVH PLVPDWFK IRU WKH PRGHO FRQVLGHUHG LV RQ WKH RUGHU RI \P DQG WKLV PD\ EH VXIILFLHQW WR GLVUXSW WKH FDSLOn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n LQJ DQG ORVV RI IL[DWLRQ $ VLPLODU SUREOHP RI ORRVHQLQJ ZDV REVHUYHG E\ $QGHUVHQ f LQ KLV H[SHULPHQWDO VWXGLHV RI LQWUDn PHGXOODU\ IL[DWLRQ LQ GRJV +RZHYHU WKH SURSHUWLHV RI WKH FRPn SRQHQWV XVHG DUH QRW VSHFLILHG 7KHVH UHVXOWV DSSHDU WR LPSO\ WKDW ZKHQ UHDPLQJ RI WKH ERQH LV QRW FOLQLFDOO\ IHDVLEOH D OHVV ULJLG QDLO VKRXOG EH XVHG WR PDLQWDLQ WKH IUDFWXUH IL[DWLRQ 7KH RYHUDOO UHVXOWV REWDLQHG IURP WKH %1$/ PRGHOLQJ RI WKH LQWUDPHGXOODU\ RVWHRV\QWKHVLV VXSSRUW WKH XVH RI PRUH IOH[XUDOO\ ULJLG LQWUDPHGXOODU\ GHYLFHV /LQGDKO f %UDGHQ f $QGHUVHQ f DQG 7RQLQR HW DO f KDYH DOVR IRXQG WKDW PRUH ULJLG IL[DWLRQ GHYLFHV OHDG WR FOLQLFDOO\ EHWWHU UHVXOWV 7KH DGGLWLRQDO UHDPLQJ RI WKH PHGXOODU\ FDQDO WR SHUPLW WKH SODFHPHQW

PAGE 190

RI WKHVH PRUH ULJLG GHYLFHV LQFUHDVHV WKH DPRXQW RI FRUWLFDO ERQH SUHVHQW DW WKH QDLOERQH LQWHUIDFH DQG UHVXOWV LQ D UHGXFWLRQ RI WKH EHQGLQJ GHIOHFWLRQV LQ WKH V\VWHP +RZHYHU RQFH LQLWLDO KHDOLQJ KDV EHHQ DIIHFWHG E\ WKHVH PRUH ULJLG V\VWHPV WKH VWUHVV SURWHFWLRQ DIIRUGHG WKH KHDOHG ERQH PD\ UHVXOW LQ KDPSHUHG IXQFWLRQDO UHPRGHOLQJ DQG GLPLQLVKHG ERQH VWUHQJWK 'LUHFWLRQV IRU )XWXUH :RUN 7KLV ZRUN KDV FRQFHQWUDWHG RQ WKH HIIHFW RI RQO\ WZR SDUDPn HWHUV RQ WKH UHVSRQVH RI WKH RVWHRV\QWKHVLV XQGHU IRXUSRLQW ORDGLQJ 7KHVH ZHUH WKH UHODWLYH IOH[XUDO ULJLGLWLHV RI WKH FRPSRQHQWV DQG WKH FKDUDFWHU RI WKH LQWHUIDFH VHSDUDWLQJ WKHP 7KH GDWD IURP WKLV PRGHO VXJJHVWV WKDW LQ VRPH V\VWHPV ODUJH LQWHUIDFLDO VWUHVVHV DQG PRWLRQV GHYHORS :KLOH LQ YLWUR VWXGLHV ZHUH XVHG WR YHULI\ WKH GHIOHFWLRQ SUHGLFWLRQV VDWLVIDFWRU\ PHWKRGV RI YHULI\LQJ WKH LQWHUIDFLDO ORDG GLVWULEXWLRQ DQG LQWHQVLW\ ZHUH QRW DYDLODEOH )XWXUH YHULILFDWLRQ RI WKH ORDGV GHYHORSHG EHWZHHQ LPSODQW DQG ERQH LV WKHUHIRUH QHFHVVDU\ DQG KDV PDQ\ H[SHULPHQWDO DQG FOLQLFDO DSSOLFDWLRQV LQ ELRPHFKDQLFDO VWXGLHV 7KH VXJJHVWHG VWUHVV SURn WHFWLRQ DIIRUGHG E\ LQWUDPHGXOODU\ GHYLFHV QHHGV IXUWKHU H[SHULn PHQWDO LQYHVWLJDWLRQ DQG YHULILFDWLRQ DV ZHOO 6HYHUDO RWKHU IDFWRUV ZHUH QRW ZHOO XQGHUVWRRG DQG LQGLFDWH WKH QHHG IRU IXUWKHU LQYHVWLJDWLRQ DQG LQFRUSRUDWLRQ LQ WKH JHQHUDO RVWHRV\QWKHVLV PRGHO 7KH ILUVW LV WKH QRQOLQHDU DQG QRQFRQVWDQW LQWHUIDFLDO VWLIIQHVV EHKDYLRU ZKLFK RFFXUV DORQJ WKH ERQH OHQJWK 7KH LQWHUIDFLDO VWLIIQHVV ZDV IRXQG WR EH QRQOLQHDU ZLWK WKH

PAGE 191

UHODWLYH GLVSODFHPHQW RI WKH FRPSRQHQWV DQG QRQFRQVWDQW DORQJ WKH OHQJWK RI WKH ERQH 7KH PRGHO VKRXOG EH PRGLILHG WR LQFRUSRUDWH WKLV H[SHULPHQWDO ILQGLQJ 7KH LQWHUDFWLRQ RI WKH IUDFWXUH VXUIDFHV ZDV DOVR DVVXPHG WR EH QRQH[LVWHQW 5RXJKQHVV RI WKH IUDFWXUH VXUIDFHV REOLTXLW\ RI WKH IUDFWXUH DQG IUDFWXUH FDOOXV EULGJLQJ WKH IUDFWXUH VLWH RFFXU LQ YLYR DQG VKRXOG EH DGGHG WR WKH PRGHO 9DULRXV ORDGLQJ FRQILJXUDWLRQV DQG IUDFWXUH VLWH SODFHPHQWV VKRXOG DOVR EH FRQVLGHUHG DQG WKH DGGLWLRQ RI D[LDO ORDGLQJ WR WKH EHQGLQJ ORDGV PRGHOHG VKRXOG EH LQFOXGHG DV ZHOO :LWK WKH LQFOXVLRQ RI WKH DERYH PRGLILFDWLRQV D PRUH JHQHUDO PRGHO ZLOO EH SURGXFHG DQG ZLOO DOORZ IXUWKHU XQGHUVWDQGLQJ RI WKH IDFWRUV DIIHFWLQJ WKH SHUIRUPDQFH RI WKH LQWUDPHGXOODU\ RVWHRn V\QWKHVHV DQG SURYLGH IXUWKHU LPSXW IRU WKH FOLQLFDO GHFLVLRQV PDNLQJ SURFHVV

PAGE 192

$33(1',; $ '(9(/230(17 2) 7+( ',))(5(17,$/ %($0 '()/(&7,21 (48$7,216 &RQVLGHU D EHDP LQ SXUH EHQGLQJ )LJXUH $O VKRZV D VHJPHQW RI WKH EHDP KDYLQJ EHHQ FXUYHG E\ WKH DSSOLHG PRPHQW 0A 3ODQH FURVVVHFWLRQV DUH DVVXPHG WR UHPDLQ SODQH DQG SHUSHQGLFXODU WR WKH ORQJLWXGLQDO D[LV $IWHU EHQGLQJ VRPH HOHPHQWV DUH OHQJWKHQHG RWKHUV VKRUWHQHG DQG DW RQH SRVLWLRQ WKH QHXWUDO VXUIDFH QR FKDQJH LQ OHQJWK RFFXUV /HW EH WKH FHQWHU RI FXUYDWXUH IRU WKH VHJPHQW $% Gf! LV WKH VPDOO LQFOXGHG DQJOH EHWZHHQ WKH WZR FXWWLQJ VXUIDFHV DQG U LV WKH UDGLXV RI FXUYDWXUH &RQVLGHU D KRUL]RQWDO HOHPHQW ORFDOL]HG DW VRPH GLVWDQFH Y IURP WKH QHXWUDO VXUIDFH :LWK %& SDUDOOHO WR 2$ HTXDO DQJOHV $2% DQG &%' DUH IRUPHG $SSO\LQJ WKH JHRPHWU\ RI VLPLODU WULDQJOHV DQG VPDOO DQJOHV DQG WDNLQJ FDUH WR FRQVLGHU WKH SRVLWLYH GLUHFWLRQV RI G-f DQG G[ DV VKRZQ LQ )LJXUH $O WKH IROORZLQJ UHODWLRQVKLSV DUH GHULYHG UG`! G[ DQG f§ f§\f§ H $Of U G[ ZKHUH H LV WKH VWUDLQ LQ WKH ILEHU LQ TXHVWLRQ H DSSHDUV WR EH D QHJDWLYH TXDQWLW\ +RZHYHU LW LV LPSRUWDQW WR QRWH WKDW GMfG[ LV LWVHOI D QHJDWLYH TXDQWLW\ LH VORSH GHFUHDVHV LQ WKH SRVLWLYH [ GLUHFWLRQ DQG WKHUHIRUH H LV LQGHHG D SRVLWLYH RU WHQVLOH VWUHVV LQ WKH HORQJDWHG ILEHU XQGHU FRQVLGHUDWLRQf

PAGE 193

)LJXUH $O $ VHJPHQW RI D EHDP KDYLQJ EHHQ FXUYHG E\ DQ DSSOLHG PRPHQW )LJXUH $ 6WUHVV GLVWULEXWLRQ SURILOHV GHYHORSHG LQ WKH FXUYHG EHDP

PAGE 194

:LWK WKH DSSOLFDWLRQ RI +RRNHnV /DZ OLQHDUO\ HODVWLF EHKDYLRUf ZKHUH 6 LV WKH ILEHU VWUHVV DQG ( LV WKH PRGXOXV RI HODVWLFLW\ 7KH VWUHVV DQG VWUDLQ DUH VHHQ WR YDU\ OLQHDUO\ ZLWK WKH GLVWDQFH IURP WKH QHXWUDO SODQH )LJXUH $ VKRZV WKLV VWUHVV GLVWULEXWLRQ $SSO\LQJ KRULn ]RQWDO IRUFH HTXLOLEULXP FRQGLWLRQV DQG XWLOL]LQJ HTXDWLRQ $f \LHOGV $f ,I WKH PDWHULDO LV LVRWURSLF LH ( I I G$f WKHQ HTXDWLRQ $f UHGXFHV WR $f n$ )RU WKH ILQLWH DUHD FRQVLGHUHG Y LH WKH QHXWUDO VXUIDFH SDVVHV WKURXJK WKH FHQWHU RI JUDYLW\ RI WKH FURVV VHFWLRQ $SSO\LQJ PRPHQW HTXLOLEULXP WR WKH HOHPHQW JLYHV 0 f$ VYG$ $f RU $f 7KH TXDQWLW\ I Y G$ LV UHFRJQL]HG DV WKH DUHD PRPHQW RI LQHUWLD 7KXV HTXDWLRQ efEHFRPHV B 0 U (O $f

PAGE 195

6XEVWLWXWLQJ HTXDWLRQ $Of UHVXOWV LQ GWS A 0 G[ (O $f 'LJUHVVLQJ D PRPHQW FRQVLGHU WKH VPDOO VHJPHQW RI WKH EHDP D VHFRQG WLPH )LJXUH $f :LWK WKH GLVWULEXWHG ORDG T[f DSSOLHG DQ HTXLOLEULXP IUHH ERG\ GLDJUDP FDQ EH FRQVWUXFWHG )LJXUH $f (TXLOLEULXP LPSOLHV 9 T[fG[ 9 G9f $f 9 G9f G[ 0E G9 ZKLFK \LHOGV $f G9 G[ T [f $OOf $f 'LIIHUHQWLDWLQJ HTXDWLRQ $f DQG VXEVWLWXWLQJ LQWR HTXDWLRQ $OOf \LHOGV G0 T[f $f G[ ,I IRU VPDOO GHIOHFWLRQV G\G[ WKHQ HTXDWLRQ $f EHFRPHV G! B B G\ B B0B G[ R (O G[A $f

PAGE 196

)LJXUH $ $ VHJPHQW RI D EHDP XQGHU GLVWULEXWLRQ ORDG T[f )LJXUH $ $ GLIIHUHQWLDO HOHPHQW RI WKH EHDP ZLWK GLVWULEXWHG ORDG T[f DQG UHDFWLRQ PRPHQWV DQG VKHDUV

PAGE 197

DQG FRPELQLQJ HTXDWLRQV $f DQG $f \LHOGV GA\ (O G[A T $f ZKLFK LV WKH GLIIHUHQWLDO HTXDWLRQ UHODWLQJ ORDG WR GHIOHFWLRQ LQ WKH EHDP

PAGE 198

/,67 2) 5()(5(1&(6 $OOHQ : & 3HUVRQDO FRPPXQLFDWLRQ $OOHQ : & 3LRWURZVNL %XUVWHLQ $ + DQG )UDQNHO 9 + %LRPHFKDQLFDO SULQFLSOHV RI LQWUDPHGXOODU\ IL[DWLRQ &OLQ 2UWKRS $PHULFDQ 6RFLHW\ IRU 7HVWLQJ DQG 0DWHULDOV $QQXDO %RRN RI $670 6WDQGDUGV 9RO (DVWRQ 0DU\ODQG $PHULFDQ 6RFLHW\ IRU 7HVWLQJ DQG 0DWHULDOV $QGHUVHQ / &RPSUHVVLRQ SODWH IL[DWLRQ DQG WKH HIIHFW RI GLIIHUHQW W\SHV RI LQWHUQDO IL[DWLRQ RQ IUDFWXUH KHDOn LQJ %RQH -RLQW 6XUJ $ $QGHUVHQ / *OLPHU : 6 -U DQG 7RRPV 5 ( ([SHULPHQWDO IUDFWXUHV WUHDWHG ZLWK ORRVH DQG WLJKW ILWWLQJ PHGXOODU\ QDLOV 6XUJ )RUXP %HHU ) 3 DQG -RKQVWRQ ( 5 9HFWRU 0HFKDQLFV IRU (QJLQHHUV 6WDWLFV 1HZ
PAGE 199

&XUUH\ 7KUHH DQDORJLHV WR H[SODLQ WKH PHFKDQLFDO SURSHUWLHV RI ERQH %LRUKHRORJ\ 'HQ +DUWRJ 3 $GYDQFHG 6WUHQJWK RI 0DWHULDOV 1HZ
PAGE 200

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f 3DXZHOV ) 8EHU GLH EHGHXWXQJ GHU EDXSULQ]LSLHQ GHV VWXW]XQG EHZHJXQJDSSDUDWHV IXU GLH EHDPVSUXFKXQJ GHU URKUHQNQRFKHQ $FWD $QDW 3HWHUV 5 0 %LRPHFKDQLFV DQG VXUJHU\ ,Q %LRPHFKDQLFV ,WV )RXQGDWLRQV DQG 2EMHFWLYHV (GLWHG E\ < & )XQJ 1 3HUURQH DQG 0 $QOLNHU (QJOHZRRG &OLIIV 1 3UHQWLFH +DOO

PAGE 201

3LRWURZVNL 6WUHVV DQDO\VLV RI ERQHV WHVWHG LQ WRUVLRQ $GYDQFHV LQ %LRHQJLQHHULQJ (GLWHG E\ $ & %HOO 1HZ
PAGE 202

6HGOLQ ( DQG +LUVFK )DFWRUV DIIHFWLQJ WKH GHWHUn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
PAGE 203

%,2*5$3+,&$/ 6.(7&+ *DU\ -RHO 0LOOHU ZDV ERUQ RQ 6HSWHPEHU LQ -DFNVRQYLOOH )ORULGD $IWHU DWWHQGLQJ HOHPHQWDU\ VFKRRO LQ ERWK )RUW /DXGHUGDOH DQG -DFNVRQYLOOH )ORULGD KH HQWHUHG WKH $OIUHG GX3RQW -XQLRU6HQLRU +LJK 6FKRRO LQ -DFNVRQYLOOH LQ 6HSWHPEHU +H UHFHLYHG D GHJUHH ZLWK KRQRUV LQ -XQH DQG HQWHUHG WKH 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH LQ 6HSWHPEHU RI WKH VDPH \HDU WR SXUVXH D GHJUHH LQ 0HFKDQLFDO (QJLQHHULQJ ,Q -XQH WKH GHJUHH RI %DFKHORU RI 6FLHQFH LQ 0HFKDQLFDO (QJLQHHULQJ ZDV FRQn IHUUHG ZLWK KRQRUV +H ZDV PDUULHG WR 6X]\ /LVV RI -DFNVRQ +HLJKWV 1HZ
PAGE 204

IL[DWLRQ 7KLV FXOPLQDWHG LQ WKH GLVVHUWDWLRQ $ %LRPHFKDQLFV $SSURDFK WR WKH ,QWHUDFWLRQ RI +RVW ,PSODQW 6\VWHPV DV SDUWLDO IXOILOOPHQW RI WKH GHJUHH UHTXLUHPHQWV DQG D 'RFWRU RI 3KLORVRSK\ 'HJUHH LQ 0HFKDQLFDO (QJLQHHULQJ ZDV H[SHFWHG WR EH FRQIHUUHG LQ -XQH

PAGE 205

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ &3 r Ws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

PAGE 206

7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH &RXQFLO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ -XQH 'HDQ &ROOHJH RI (QJLQHHULQJ 'HDQ *UDGXDWH 6FKRRO

PAGE 207

7 81,9(56,7< RI 8) c)/25,'$ 7KH )RXQGDWLRQ IRU 7KH &XWRU 1DWLRQ ,QWHUQHW 'LVWULEXWLRQ &RQVHQW $JUHHPHQW ,Q UHIHUHQFH WR WKH IROORZLQJ GLVVHUWDWLRQ $87+25 0LOOHU *DU\ 7,7/( $ ELRPHFKDQLFV DSSURDFK WR WKH LQWHUDFWLRQ RI KRVW LPSODQW V\VWHPV UHFRUG QXPEHU f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f DV ZHOO DV WR WKH PDLQWHQDQFH DQG SUHVHUYDWLRQ RI D GLJLWDO DUFKLYH FRS\ 'LJLWL]DWLRQ DOORZV WKH 8QLYHUVLW\ RI )ORULGD WR JHQHUDWH LPDJH DQG WH[WEDVHG YHUVLRQV DV DSSURSULDWH DQG WR SURYLGH DQG HQKDQFH DFFHVV XVLQJ VHDUFK VRIWZDUH 7KLV JUDULSRI SHUPLVVLRQV SURKLELWV XVH RI WKH GLJLWL]HG YHUVLRQV IRU FRPPHUFLDO XVH RU SURILW 6LJQDWXUH-R/&RS\ULJKW +ROGHU tN.\ PLOOHG 3ULQWHG RU 7\SHG 1DPH RI &RS\ULJKW +ROGHU/LFHQVHH 3HUVRQDO ,QIRUPDWLRQ %OXUUHG 3OHDVH SULQW VLJQ DQG UHWXUQ WR &DWKOHHQ 0DUW\QLDN 8) 'LVVHUWDWLRQ 3URMHFW 3UHVHUYDWLRQ 'HSDUWPHQW 8QLYHUVLW\ RI )ORULGD /LEUDULHV 32 %R[ *DLQHVYLOOH )/