How to Select an Archwire










CHAPTER
OVERVIEW
Inventing is a combination of brains and materials. The more brains you
use, the less material you need.”
— Charles Kettering
“Make the workmanship surpass the materials.”
— Ovid
“Wisdom must be intuitive reason combined with scientic knowledge.”
— Aristotle
How to Select
an Archwire
21
Clinicians must select suitable archwires for their patients. This chapter serves as a simple
guide for traditional alloys such as stainless steel and beta-titanium with linear stress-strain
curves. Typical engineering formulas can be complicated and use units not readily familiar
to most clinicians, such as GPa. Wire stiffness can be determined by simply multiplying
material stiffness by cross-section stiffness. For cross-section stiffness, the base is 0.004-inch
round wire, with an arbitrary value of 1.0. For material stiffness, the base is stainless steel,
with a material number of 1.0. Convenient tables are included in this chapter that allow
the orthodontist to compare the stiffness and other properties by a simple ratio of the
cross-sectional and material stiffness numbers. In a similar manner, maximum moment and
maximum deection numbers can be compared by a ratio of their values.
491491

21
How to Select an Archwire
492
The typical orthodontic appliance is composed of a
wrought orthodontic wire; cast congurations and
more complicated shapes are rarely used. The ortho-
dontic appliance of the future may involve nonwire
shapes of both metals and polymers. For now, our
question is limited to a discussion of the factors that
should be considered in selecting an orthodontic
wire or archwire. Depending on application, the fol-
lowing discussion should be helpful; it is a mistake
to stereotype the sequence for alignment or general
treatment and not to take advantage of the mul-
titude of possibilities available to the modern or-
thodontist. The principles described in this chapter
should be applicable not only to xed bracket-wire
appliances but also to all present and future appli-
ances.
The Triad Characteristics
Three important characteristics during elastic de-
formation, forming a triad, are closely interrelated:
force-deection rate (F/Δ); maximum force (F
max
) or,
better stated, maximum moment (M
max
); and max-
imum elastic deection (Δ
max
). Here we consider
traditional materials such as stainless steel or beta-
titanium that have linear stress-strain plots in the
elastic range that follow or approach Hooke’s law
(Fig 21-1). Superelastic materials such as nickel-
titanium (Ni-Ti) wires are very important but are
nonlinear and do not follow Hooke’s law; therefore,
they are described elsewhere.
The formula for Δ
max
is as follows:
Δ
max
=
F
max
F
Wire properties in a simple tensile or exural
mode may follow Hooke’s law; however, a linear
relationship between force and deection may not
be produced in the overall appliance conguration
because the conguration changes as the spring ac-
tivates and because the force application point and
direction may change; nevertheless, the simple for-
mulas that are linear and describe small deections
are very useful.
Two fundamental beam formulas describe these
relationships:
F
= K
EI
Δ L
3
where E is the modulus of elasticity, I is the moment
of inertia, L is the wire length, and K is a constant
that changes with conguration and loading con-
ditions. When the loading condition is a cantilever
with a single force at the free end, K = 3, but this is
only one loading condition common in practice (Fig
21-2).
The other fundamental beam formula in exure
is the following:
F
max
=
σ
max
S
L
Fig 21-1 F/Δ rate, maximum force (F
max
) or maximum moment
(M
max
), and maximum (elastic) deection (Δ
max
) are closely inter-
related, and traditional materials such as stainless steel or beta-
titanium have linear stress-strain plots in the elastic range that fol-
low or approach Hooke’s law.
Fig 21-2 Flexure formula of a cantilever spring. F and Δ are propor-
tional within the range of small deection.
F
=
3EI
L
3

493
Force-Deection Rate
or
M
max
= σ
max
S
where σ
max
is the yield stress of the material, S is the
section modulus (I/c), c is the distance from the neu-
tral axis to the most outer surface of the wire, and
M
max
is the maximum moment that the wire can de-
liver in the elastic range. This simple exure formula
took nearly two centuries to be developed.
Clinicians have an interest in all three triad prop-
erties. A large Δ
max
is popular because wires that can
be deected during large activations without per-
manent deformation are desired. The magnitude of
F
max
or M
max
must be large enough to achieve the de-
livery of an optimum force level with a safety factor.
A wire with lower F
max
can also be easily deformed
by an intermittent heavy force from mastication.
The F/Δ rate determines the rate of force decay
or the force constancy. These triad wire characteris-
tics can be modied by altering wire material prop-
erties, the shape and size of the wire cross section,
and overall appliance conguration.
The F/Δ rate of
an appliance (spring) can be altered by complicated
congurations or more simple modications such as
increasing the interbracket distance.
Engineering formulas can predict the force sys-
tems from orthodontic appliance congurations for
both large and small deections. Formulas can be
long and complicated with quantities measured in
units not always familiar to the clinician. For that
reason, some of the overall design factors were
discussed in chapter 14, and this chapter considers
only the effect of wire cross section and the material
used on the force triad.
Force-Deection Rate
The clinician uses the F/Δ rate or the stiffness of a
wire in a number of ways. It tells us how much to
activate the wire to achieve a desired force. For ex-
ample, if 100 g of force is desired and the F/Δ is 20
g/mm, 5 mm of activation is required. Commonly,
leveling is accomplished by placing a series of wires
with increasingly larger cross sections. Sometimes
this sequence of wire changes is explained as in-
creasing the cross section to eliminate the play be-
tween the wire and the bracket. However, rarely is
wire play the biggest problem; the real reason wires
must be sequenced is to keep force levels more
constant.
When the teeth are grossly misaligned, a
low-stiffness wire is needed. As the teeth approach
nal alignment, a higher-stiffness wire must be used
with a smaller deection to give the same force.
Let us now consider the effect of the alloy or ma-
terial on stiffness or rigidity of an orthodontic wire.
F/Δ rates vary linearly with the modulus of elastic-
ity (E). It is convenient to use relative numbers to
describe the rigidity of a material’s contribution to
stiffness. Arbitrarily, let us compare all materials to
stainless steel, which will have a base number of 1.0.
Beta-titanium has an E that is 0.42 as great as that of
stainless steel, and hence its material stiffness num-
ber is 0.42. The material stiffness numbers for some
common orthodontic wire materials are given in Fig
21-3. One can compare the stiffness of two applianc-
es identical apart from their material by making a
ratio of their material stiffness numbers.
Both cross-sectional size and shape differences sig-
nicantly affect wire stiffness. For round wire, stiff-
ness varies as d
4
, where d is the wire diameter. For
Fig 21-3 Material stiffness numbers. The
numbers are relative values, with stainless
steel as the standard value of 1.0. Ni-Ti has
two material stiffness values depending on
crystal structure. The value for small deec-
tion is 0.26.
Alloy
Stainless steel
EIgiloy
Beta-titanium
Ni-Ti
1
1.2
0.42
0.26
Material stiffness number

21
How to Select an Archwire
494
edgewise wire, stiffness varies as bh
3
, where b is the
width and h is the height; the curved arrows in Fig
21-4 indicate the direction of bending moment.
In a similar manner to the material stiffness num-
ber, cross-section stiffness numbers use 0.004-inch
wire as the base, which has a value of 1.0. Figure
21-5 gives representative cross-section stiffness
numbers for both round (Fig 21-5a) and rectangu-
lar (Fig 21-5b) wires. A ratio of cross-section stiffness
numbers gives the relative stiffness of two identical
appliances that differ only in cross-sectional dimen-
sion. A more complete listing of cross-section stiff-
ness numbers is given in Table 21-1.
For example, two identical vertical loops (same
conguration and material) are fabricated. One is
0.014 inch and the other is 0.020 inch. Compare their
Fig 21-4 Cross-section stiffness. Both cross-
sectional size and shape differences signicantly
affect wire stiffness. For round wire, stiffness var-
ies as d
4
, where d is the wire diameter. For edge-
wise wire, stiffness varies as bh
3
, where b is the
width and h is the height.
Fig 21-5 Cross-section stiffness numbers. (a) Round
wires. The numbers are relative values based on
0.004-inch round wire as the standard value of
1.0. (b) Rectangular wires. The direction of bend-
ing affects the number.
a
Cross section
0.004
0.010
0.014
0.016
0.018
0.020
0.022
Cross-section stiffness number
1
39
150
256
410
625
915
b
Cross-section stiffness number
First-order Second-order
0.018 × 0.025
0.021 × 0.025
0.0215 × 0.028
1,865
2,176
3,130
967
1,535
1,845
F
=
3EI
L
3
I
=
bh
3
12
I
=
πd
4
64

495
Force-Deection Rate
Table 21-1 Relative force-deection rate, maximum moment, and maximum elastic deformation
Shape
Cross-section
dimensions (inch)
Relative force-
deection rate
Relative maximum
moment
Relative maximum elastic
deformation
Round 0.006 5.06 3.38 0.67
0.007 9.38 5.36 0.57
0.008 16.00 8.00 0.50
0.009 25.63 11.40 0.44
0.010 39.06 15.63 0.40
0.011 57.20 20.80 0.36
0.012 81.00 27.00 0.33
0.013 111.57 34.33 0.31
0.014 150.06 42.88 0.29
0.015 197.75 52.73 0.27
0.016 256.00 64.00 0.25
0.017 326.25 76.77 0.24
0.018 410.06 91.18 0.22
0.019 509.07 107.17 0.21
0.020 625.00 125.00 0.20
0.021 759.70 144.70 0.19
0.022 915.06 166.38 0.18
0.023 1,093.13 190.11 0.17
0.024 1,296.00 126.00 0.17
0.025 1,525.88 244.14 0.16
0.026 1,785.06 274.63 0.15
0.027 2,075.94 307.55 0.15
0.028 2,401.00 343.00 0.14
0.029 2,762.82 381.08 0.14
0.030 3,164.06 421.88 0.13
0.032 4,096.00 512.00 0.13
0.036 6,561.00 729.00 0.11
0.040 10,000.00 1,000.00 0.10
0.045 16,013.10 1,428.88 0.09
0.050 24,414.10 1,953.12 0.08
0.060 50,625.00 3,375.00 0.07
Square
0.016 × 0.016
434.60 108.65 0.25
0.017 × 0.017
3,524.23 522.11 0.15
0.018 × 0.018
969.14 154.70 0.22
0.019 × 0.019
864.22 181.94 0.21
0.021 × 0.021
1,289.69 245.66 0.19
Rectangular (rst-order)
0.006 × 0.020
318.31 63.66 0.20
0.008 × 0.020
424.41 84.88 0.20
0.010 × 0.020
530.53 106.10 0.20
0.0105 × 0.028
1,528.52 218.36 0.14
0.011 × 0.022
776.73 141.22 0.18
0.012 × 0.020
636.62 127.32 0.20
0.014 × 0.020
742.72 148.55 0.20
(cont)

21
How to Select an Archwire
496
Table 21-1
(cont)
Relative force-deection rate, maximum moment, and maximum elastic deformation
Shape
Cross-section
dimensions (inch)
Relative force-
deection rate
Relative maximum
moment
Relative maximum elastic
deformation
0.015 × 0.028
2,183.61 311.94 0.14
0.016 × 0.020
848.83 169.76 0.20
0.016 × 0.022
1,129.79 205.42 0.18
0.017 × 0.022
1,200.40 218.25 0.18
0.017 × 0.025
1,761.48 281.84 0.16
0.018 × 0.022
1,271.01 231.09 0.18
0.018 × 0.025
1,865.10 298.42 0.16
0.019 × 0.025
1,968.71 314.99 0.16
0.019 × 0.026
2,214.53 340.70 0.15
0.020 × 0.025
2,072.33 331.57 0.16
0.021 × 0.025
2,175.95 348.15 0.16
0.021 × 0.027
2,741.07 406.08 0.15
0.0215 × 0.028
3,129.83 447.12 0.14
Rectangular (second-order)
0.006 × 0.020
28.65 19.10 0.67
0.008 × 0.020
67.91 33.95 0.50
0.010 × 0.020
132.63 53.05 0.40
0.0105 × 0.028
214.95 81.89 0.38
0.011 × 0.022
194.18 70.61 0.36
0.012 × 0.020
229.18 76.40 0.33
0.014 × 0.020
363.93 103.98 0.29
0.015 × 0.028
626.67 167.11 0.27
0.016 × 0.020
543.25 143.25 0.25
0.016 × 0.022
597.57 149.40 0.25
0.017 × 0.022
716.77 168.65 0.24
0.017 × 0.025
814.51 191.65 0.24
0.018 × 0.022
850.84 189.08 0.22
0.018 × 0.025
966.87 214.86 0.22
0.019 × 0.025
1,137.13 239.40 0.21
0.019 × 0.026
1,182.61 248.97 0.21
0.020 × 0.025
1,326.29 265.26 0.20
0.021 × 0.025
1,535.18 292.45 0.19
0.021 × 0.027
1,658.18 315.84 0.19
0.0215 × 0.028
1,845.37 343.32 0.19

497
Maximum Force and Maximum Bending Moment
forces for the same activation. The cross-section stiff-
ness is 150.0 for the 0.014-inch loop and 625.0 for
the 0.020-inch loop. The 0.020-inch loop gives 4.3
times greater force (625/150).
There are three basic methods to reduce the F/Δ
rate: (1) using a different material with a lower E,
such as selecting a Ni-Ti or beta-titanium wire rather
than stainless steel; (2) adding wire to increase the
length (L), such as placing a helix or loop in the wire;
or (3) reducing the wire cross section. However, re-
ducing the cross-sectional size may be problematic
because the F
max
or M
max
is also dramatically reduced.
Maximum Force and Maximum
Bending Moment
The material property that determines the M
max
that
a wire can produce is the yield strength. The F
max
that an appliance delivers is partly determined by
design. The M
max
is more fundamental because it is
determined by material and cross section only. The
yield strength of a spring can vary considerably for
the same material. For example, consider two stain-
less steel vertical loops with the same cross-sectional
shape and size and the same overall conguration:
one made from a high-springback stainless steel and
the other from dead soft annealed stainless steel
ligature tying wire. The E of the two is about the
same, so within the elastic range, both would give
the same force for the same activation. The differ-
ence is that the ligature wire has much lower yield
strength and would permanently deform sooner,
giving potentially less F
max
. The dead soft wire is eas-
ily made in the clinic via annealing heat treatment
using a torch lamp. The wire is heated until a bright
yellow color is observed (Fig 21-6) and then slowly
cooled down. An annealed wire is very easy to bend.
The M
max
or σ
max
is reduced; however, the F/Δ rate
is the same because E is not changed. When a pas-
sive wire is needed with a large–cross-section wire,
an annealed wire is very easy to fabricate. Instanta-
neous excessive high force is avoided, and still high
stiffness is maintained for good stress distribution
(Fig 21-7). The clinician should not confuse stiffness
with F
max
in selecting a wire. The M
max
or F
max
varies
linearly with the yield strength.
Cross section of the wire also inuences the M
max
.
This is more signicant than the material because
M
max
varies as d
4
in round wire and bh
3
in rectan-
gular wire. Table 21-1 lists relative maximum bend-
ing moments for both round and rectangular cross
sections based on cross section. A ratio of the M
max
numbers can be used to compare appliances identi-
cal except for cross section in respect to F
max
or M
max
.
For example, compare the F
max
of 0.016-inch and
0.018-inch identical vertical loops. The M
max
is 64 for
the 0.016-inch loop and 91 for the 0.018-inch loop.
Therefore, the 0.018-inch loop can maximally deliv-
er 1.4 times the force of the 0.016-inch loop (91/64).
Fig 21-7 The maxillary left central incisor has a subgingival fracture line from trauma. The
fractured crown was temporarily bonded and extruded to expose the fracture line. When
a passive wire in bending and torsion are needed with a large stiff wire, an annealed steel
wire is very easy to fabricate; instantaneous excessive high force is avoided, and yet high
stiffness is still maintained. The bypass arch was fabricated of 0.017 × 0.025–inch stainless
steel wire. (a) Before extrusion. (b) After extrusion.
a b
Fig 21-6 A torch lamp is used to anneal
stainless steel.

21
How to Select an Archwire
498
Maximum Deection
The component properties that determine Δ
max
in
the cantilever-type loading condition are shown in
the following formula:
Δ
max
=
σ
max
L
2
KEc
where K is a constant that changes with congu-
ration and loading conditions and c is the distance
from the neutral axis to the most outer surface of
the wire (h/2 in rectangular wire and d/2 in round
wire). No table is given on yield strength because
there is great variation even in the same alloy, un-
like E, which is relatively constant for a given alloy.
The cross section inuences the Δ
max
in a linear
manner. It varies inversely with diameter (d) for
round wire and height (h) for rectangular wire.
Also,
it is affected by the direction of bending due to the
Bauschinger effect of residual stress. A relative Δ
max
number based on cross section is given in Table 21-1.
Compare the Δ
max
of 0.016-inch and 0.018-inch iden-
tical vertical loops. Δ
max
numbers are 0.25 and 0.22,
respectively. The 0.016-inch loop can be deected
1.1 times that of the 0.018-inch loop. The effect is
negligible. Interestingly, the labiolingual width for
the rectangular wire does not affect the maximum
range. A 0.017 × 0.022–inch or 0.017 × 0.025–inch
wire gives the same Δ
max
in a second- order bending
direction (see Table 21-1).
Clinicians like wires that deect large distances
without permanent deformation. Such appliances
require fewer adjustments and deliver more con-
stant forces. Another measurement is resilience,
which determines the amount of mechanical energy
the wire can store and release during deactivation
within the elastic range.
Other Wire Properties
Other than the triad of properties previously de-
scribed, many other wire properties need to be con-
sidered in choosing an orthodontic wire. In the plas-
tic range, the ultimate tensile strength denes the
stress levels at fracture. The percentage of elonga-
tion, cold bend tests, and strain difference between
the yield strength and the ultimate tensile strength
dene ductility. Fatigue involves cyclic loading usu-
ally dened in the elastic range.
The total area un-
der the stress-strain tensile plot is called toughness.
Depending on application, the clinician needs a wire
that can be shaped without fracture. Superelastic
Ni-Ti wire has less plastic range so that it lacks the
ability to bend into certain shapes. Brittle wires can
also fracture under occlusal loading by chewing or
during insertion.
Friction is a consideration during sliding mechan-
ics at all stages of treatment. The material property
that determines the frictional force between two
materials is the coefcient of friction (μ). A more
complete discussion is found in chapter 19.
Wire cross section selection is mainly related to
the stiffness wanted or the level of force or moment
to be delivered. A secondary consideration is the
amount of play desired between the bracket and
the wire. Some other characteristics to consider are
biocompatibility, resistance to corrosion and degra-
dation, and the ability to weld or solder.
Wire Stiffness Numbers
Chapter 14 discussed how wire conguration can af-
fect the F/Δ rate. We can compare orthodontic appli-
ances of identical design by isolating and evaluating
the effect of material and cross section of the wire.
In order to consider the wire component only and
not the overall design or conguration, the follow-
ing formula multiplies the material stiffness number
by the cross-section stiffness number. The product
is the wire stiffness number, a measure of the wire
rigidity independent of the appliance design.
W
s
= M
s
× C
s
where W
s
is the wire stiffness, M
s
is the material stiff-
ness, and C
s
is the cross-section stiffness.
For example, the cross-section stiffness of a
0.018-inch stainless steel wire is 410.0. The material
stiffness is 1.0. The wire stiffness is therefore 410
(410 × 1.0 = 410). The material stiffness of 0.016-inch
beta-titanium wire is 0.42, and the cross-sectional
stiffness is 256.0. The wire stiffness is therefore 107.5
(256 × 0.42 = 107.5). If we make two identical verti-
cal loops out of these materials and compare the
stiffness, we could compare the force levels for the
same activation in the elastic range. The ratio of the
wire stiffness numbers is 410 divided by 107.5, or 3.8.
The 0.018-inch stainless steel loop therefore gives
3.8 times the force of the 0.016-inch beta-titanium
loop.

499
Recommended Reading
Relative Torque (Torsion)
Numbers
Because different formulas are used for wires un-
dergoing torsion, a separate but similar table is
presented for relevant torque numbers (Table 21-2).
Numbers are given for relative torque/twist rates,
relative maximum torque, and maximum elastic
twist. Appliances can be compared in a similar man-
ner by making a ratio of the relevant numbers.
Recommended Reading
Burstone CJ. Application of bioengineering to clinical ortho-
dontics. In: Graber LW, Vanarsdall RL Jr, Vig KWL (eds). Ortho-
dontics: Current Principles and Techniques, ed 5. Philadelphia:
Elsevier Mosby, 2012:345–380.
Burstone CJ. Variable modulus orthodontics. Am J Orthod
Dentofacial Orthop 1981;80:1–16.
Burstone CJ, Goldberg AJ. Maximum forces and deections
from orthodontic appliances. Am J Orthod Dentofacial Orthop
1983;84:95–103.
Choy KC, Kim KH, Park YC, Kang CS. Torsional moment of
orthodontic wires. Korean J Orthod 2000;30:467–473.
Table 21-2 Relative torque/twist rate, maximum torque, and maximum elastic twist
Shape Cross-section dimensions (inch) Relative torque/twist rate Relative maximum torque Relative maximum elastic twist
Rectangular
0.006 × 0.020
46.47 16.19 0.35
0.008 × 0.020
101.66 27.38 0.27
0.010 × 0.020
182.14 40.81 0.22
0.0105 × 0.028
328.50 66.85 0.20
0.011 × 0.022
266.67 54.32 0.20
0.012 × 0.020
286.98 56.17 0.20
0.014 × 0.020
413.30 73.23 0.18
0.015 × 0.028
833.24 126.46 0.15
0.016 × 0.022
660.32 104.01 0.16
0.017 × 0.022
756.40 115.23 0.15
0.017 × 0.025
943.58 136.11 0.14
0.018 × 0.022
857.30 126.21 0.15
0.018 × 0.025
1,076.23 150.04 0.14
0.019 × 0.025
1,215.62 164.42 0.14
0.019 × 0.026
1,302.19 173.08 0.13
0.020 × 0.025
1,361.31 179.23 0.13
0.021 × 0.025
1,513.10 194.45 0.13
0.021 × 0.027
1,740.90 215.35 0.12
0.0215 × 0.028
1,957.06 235.04 0.12
Square
0.016 × 0.016
367.24 67.91 0.18
0.017 × 0.017
468.02 81.45 0.17
0.018 × 0.018
588.24 96.64 0.16
0.019 × 0.019
730.26 113.71 0.16
0.021 × 0.021
1,089.30 158.54 0.14

500
PROBLEMS
1. Compare 0.014-inch and 0.016-inch stainless steel wires
used for initial leveling. When 0.014-inch wire is used
instead of 0.016-inch wire, the ratio of the diameters is
reduced by 13%. How much reduction in the F/∆ rate is
anticipated? If 0.014-inch and 0.016-inch superelastic Ni-Ti
wires are compared, will the F/∆ difference be the same?
3. A helix is incorporated in the cantilever spring to reduce the
F/∆ rate. How would F/∆, F
max
, and ∆
max
be affected? Exact
numbers are not necessary. Compare A, B, and C.
2. In lingual orthodontics, the interbracket distance is small-
er than with a labial appliance. Suppose the interbracket
distance is reduced from 8 mm to 4 mm; how would this
affect the F/∆ rate?
4. 0.016 × 0.022–inch wires are inserted ribbon wise (A)
instead of edgewise (B). How would this change affect the
force system occlusogingivally and labiolingually? Ignore
the play between the wire and the brackets.
0.016-inch stainless steel
0.014-inch stainless steel
(13% reduction in diameter)
A
B
C
A B
For problems 1 to 7, assume a small deection and Hooke's law.

501
Problems
5. The size of a round wire is reduced by 50% to reduce the
F/∆ rate. How much reduction in the F/∆ rate is anticipat-
ed? How would it affect the F
max
?
6. Compare the stiffness of a 0.017 × 0.025–inch beta-titanium
spring and a 0.016 × 0.022–inch stainless steel spring with
the same conguration.
7. A 0.032 × 0.032–inch square stainless steel wire is used
for a lingual arch. Calculate the stiffness number, which is
not shown in the tables in the text.
The diameter of B is 50% of A.
A
B
0.017 × 0.025–inch
beta-titanium
0.016 × 0.022–inch
stainless steel
0.032"
0.032"

V
PART
Appendices
Hints for Developing
Useful Force Diagrams
Glossary
Solutions to Problems

505505
An important aspect of applying biomechanical
principles to clinical orthodontics is the use of val-
id force diagrams. A “good” diagram is needed to
explain any appliance or therapy. It is the  rst step
in solving the many problems presented in this text.
It is the fastest and most concise way for the author
or speaker to present new concepts or appliances.
A clinician can speak in generalities about his or
her treatment methodology, but evaluation of his
or her approach can be dif cult because of vague
language; however, a good force diagram can im-
mediately impart to the listener the presentation’s
validity. Double-talk is a form of speech in which in-
appropriate, invented, or nonsense words are used
to give the appearance of knowledge while actually
confusing an audience. It is hard to double-talk a
valid and concise force diagram.
In orthodontic books, journals, and lectures, it
is common to see incorrect force diagrams or dia-
grams that are dif cult to understand. The authors
have collected many examples and believe that they
are helpful in identifying and preventing common
errors. A major problem is that we do not want to
embarrass any orthodontist (many are our friends,
colleagues, and students) who may be guilty of a
mistaken force diagram. Our goal is to teach prop-
er force diagram construction and not to criticize.
Our solution is to take diagrams with actual errors
that we have observed and redraw the diagrams us-
ing different teeth and applications. Thus, none of
the diagrams that follow can be identi ed with any
particular individual. As is said in many  ction titles,
“Any similarity to any persons living or dead is pure-
ly coincidental.” Now let us consider some common
errors.
Floating Forces
Figure 1a shows an intrusion arch where the forces
on the incisors and posterior teeth are not precise-
ly placed at their line of action but are  oating in
space. Forces are vectors, and the force arrows must
be placed somewhere along the line of action of the
force. This is corrected in Fig 1b. The curved arrow
(moment) is correct in Fig 1a. Couples are free vectors
and can be placed anywhere on the diagram, unless
you want to emphasize where the appliance delivers
the moment. Figure 1b may be clearer, showing that
the moment is calculated around the center of re-
sistance (CR) of the posterior segment. Another ex-
ample of  oating forces is Fig 2a. An elastic to close
space acts at the level of the brackets in Fig 2b, not
occlusally as depicted in Fig 2a, away from the line
of action. Commonly, force diagrams in orthodon-
tic presentations and publications improperly show
forces not acting along their line of action.
Hints for
Developing Useful
Force Diagrams

Hints for Developing Useful Force Diagrams
506
Arrows That Are Not Forces
Straight and curved arrows are frequently used to
show the direction of movement (Fig 3). Because
arrows are used to represent forces, it is better to
use other methods to show movement. Additional
nonforce arrows may produce a confusing diagram.
Avoid curved arrows for labels, as shown in Fig 3
labeling the molar attachment. Style and color of
force arrows must be unique and not confused with
any other arrows. Some orthodontic publications
and slides show many arrows, and it is not always
clear whether they represent forces or movement
direction; hence, interpretation is difcult.
Equivalent Force Diagrams
Equivalent force systems are useful to include in an
orthodontic force diagram as discussed in this text.
Here, the force is placed at an arbitrary point not
necessarily along the line of action of the actual
force. Figure 4a shows that the intrusive force on
Fig 1a Fig 1b
Fig 2a
Fig 2b
Fig 3 Fig 4
a b

507
Equilibrium Diagrams
the maxillary incisor produces a moment of force
that tends to are the incisor. Figure 4b is better
because the force at the bracket is replaced with
an equivalent intrusive force at the CR and a coun-
terclockwise moment; this is a more complete de-
scription than the “moment of force” concept. The
equivalent force system should be in a separate col-
or or different format so that it is not identied as
additional applied forces.
Equilibrium Diagrams
Orthodontic appliances are in equilibrium, which
should be reected in force diagrams of all appli-
ances. Newton’s Third Law tells us that there are
equal and opposite forces (activation and deactiva-
tion); it is the activation forces that must be given
on the equilibrium diagram. The free-body diagram
that shows the forces acting on the teeth (deactiva-
tion forces) should also be in equilibrium because
it may be based on an appliance equilibrium condi-
tion. Figure 5a claims to show the forces acting from
accentuated curves and reverse curve of Spee arch-
wires (Fig 5b). Because the forces act on the teeth,
the diagram is a free-body diagram; nevertheless,
we can see that if the forces are reversed (Newton’s
Third Law), the archwire cannot be in equilibrium.
This cannot be a valid explanation of how these
arches work to correct a deep bite. What is shown is
impossible because the forces and moments do not
add to zero. A more detailed explanation of curve
of Spee arches is given in chapters 6 and 7.
Figure 6a has many errors that add to confusion
about its message. The Herbst-like appliance must
be in equilibrium; because there is a point attach-
ment on either end, only equal and opposite forc-
es are generated. In Fig 6a, the maxillary force and
mandibular force are not equal, nor are they along
the same line of action; impossible couples acting at
either end would be required for equilibrium. The
moment of force to the CR of the maxillary arch is
smaller than the moment of force to the mandib-
ular arch, yet the force is greater and the moment
arms the same. Overall, this a “nonsense” diagram
that does not enlighten the reader. Resolving the
force into horizontal and vertical components is re-
dundant because force direction is obvious. This is a
non-Newtonian appliance because it is claimed that
the maxillary distal force is greater than the man-
dibular mesial force. But is this possible? Figure 6b is
a correct diagram.
Fig 5a Fig 5b
Fig 6a Fig 6b

Hints for Developing Useful Force Diagrams
508
Confusing Activation and
Deactivation Forces
A free-body diagram showing all of the relevant
forces acting at the brackets (on the teeth) can be-
come complicated because more than one appli-
ance component can be present. This can be made
more difcult when two sets of forces are present:
activation and deactivation. Let us consider the sim-
ple mechanism for space closure in Fig 7. An elastic
is used to produce the horizontal forces. To add a
counterclockwise moment in the direction of lingual
apical root movement, a cantilever root spring is in-
serted into an auxiliary tube on the canine bracket.
To activate the spring, an occlusal force is required
(Fig 7a). This diagram is confusing because all forces
are acting on the teeth except one force that acts on
the wire hook, but the color is the same as the forces
on the teeth. For greater clarity, the diagram should
only show the deactivation forces in red, where all
red deactivation forces act on the teeth (Fig 7b). If a
cantilever is used, the clinician may prefer to show
the deactivation force system acting on the teeth
and not be concerned with an equilibrium diagram
of the cantilever spring itself. The downward force
on the hook should then be a different color to
show an equivalent diagram.
Fig 7a
Fig 7b
Fig 8a
Fig 8b
Fig 8c

509
Conclusion
It Is the Line of Force That Counts
It should be remembered that the line of force de-
termines the direction of the force and its point of
force application. In Fig 8a, a temporary anchorage
device (TAD) and a sliding jig were used to retract
the maxillary right molar. The red force parallel
to the occlusal plane apical to the brackets is the
correct line of action. The author of a publication
thought that the black diagonal force acted on the
molar or the maxilla (Fig 8b). But the line of action
is produced by the spring. What is the force system
on the archwire, assuming that friction prevents the
molar from sliding distally? The distal force and the
moment produced by the vertical forces (yellow) are
equivalent to the force (red) acting on the TAD api-
cal to the archwire (Fig 8c). If the TAD is at the CR,
the maxillary arch would tend to translate distally if
little play is present between the archwire and the
brackets. If the molar is free to slide, the force sys-
tem is more complicated, but still the force direction
is the same.
Equivalence: Replacing a Force
with a Force and a Couple
We have seen that it is useful to replace a force with
an equivalent force and a couple (moment). An ar-
bitrary point is chosen that helps our understand-
ing of the biomechanical system. Sometimes it is not
clear on the force diagram where that point is to be
found, or a useless point is selected. Consider Fig 9a,
where the four maxillary incisors require extrusion
to close an open bite. From rst molar forward to
the canine, the buccal segment is level and parallel
to the mandibular occlusal plane. An extrusion arch
from the molar auxiliary tube delivers an extrusive
force to the incisors. The deactivation force diagram
on the teeth correctly shows an intrusion force at
the molar and a counterclockwise tip- forward mo-
ment. What is the side effect that will be produced?
Because the buccal teeth from molar to canine are
joined together by a rigid wire, we are not interest-
ed in the force system at the molar tube but rather at
the CR of the entire buccal segment. The force and
moment at the rst molar are not useful depictions
unless we are studying molar tube deformation or
tube bonding strength. Figure 9b is more relevant;
the force system is replaced with an equivalent force
system at the CR of the buccal segment. A vertical
elastic is added at the canine so that the sum of all
moments on the buccal segment is zero. Because of
the varying distances, a net small extrusive force acts
on the buccal segments. Many orthodontic force di-
agrams show forces and moments, but the point at
which the moment is calculated is often a mystery.
Although couples are free vectors, we must know if
they are delivered by the appliance or are calculated
using a useful reference axis and where that axis is
located.
Conclusion
We have suggested some ways to simplify force
system diagrams to make communication among
orthodontists more transparent. There is still much
room for creativity in presentation without ignoring
Newtonian precepts. Force diagrams are important
because they are windows to the “soul” of the or-
thodontist. And you cannot double-talk a diagram.
Fig 9a Fig 9b

511
Glossary
511
Note: Some de nitions or terminology widely used in
physics are purposely narrowed to relate to clinical ortho-
dontics.
Activated shape The shape of a wire or a spring pro-
duced by an activation force system. See also Activation
force system, Deactivated shape, Passive shape, and Sim-
ulated shape.
Activation force system The force system applied to a
spring or archwire to insert it into a bracket. An activated
appliance by an activation force system is always in static
equilibrium. The deactivation force system acting on the
teeth is equal and opposite to the activation force system
(Newton’s Third Law). See also Deactivation force system
and Static equilibrium.
Activation moment The moment produced by the ac-
tivation of a spring such as a space closure loop. See also
Residual moment.
Active unit The part of an arch or appliance that is in-
volved with tooth movement. See also Reactive unit.
Alpha position The anterior component of a spring or
anterior point of attachment of a spring. See also Beta po-
sition.
Anatomical long axis of a tooth Arbitrary axis deter-
mined by the anatomy of the tooth. The longest dimen-
sion of the tooth measured determines the anatomical
long axis.
Anchorage unit See Reactive unit.
Anisotropic When a material shows different physical
properties depending on its direction. Wood is a typical
anisotropic material.
Annealing Heat treatment of the alloy to reduce strength
and increase ductility.
Axis of rotation An axis on the body around which all
points of the body rotate. In screw theory, the body may
translate along the axis. In two dimensions, the axis be-
comes a center of rotation.
Bauschinger effect The residual stress in a wire after per-
manent deformation that in uences the range of action.
Beam theory The science of explaining beam de ection
during loading. An orthodontic wire is a beam (ie, a struc-
ture with a large longitudinal dimension in relation to its
cross section).
Bending Produced when the wire structural axis chang-
es at right angles to the original structural axis. See also
Torsion.
Beta position The posterior components of a spring or
the posterior point of attachment of a spring. See also Al-
pha position.
Beta-titanium An alloy that contains titanium. Beta-
titanium is used in orthodontics because of its lower forces
and high springback. Its modulus of elasticity is 42% of
that of stainless steel. Another titanium alloy with a dif-
ferent crystalline structure, alpha-titanium, is the most
common titanium alloy used commercially, but it is not
suitable for orthodontic wire.
Biologic tooth movement The tooth movement that
involves bone resorption at compression sites and bone
apposition at tension sites by osteoclastic and osteoblastic
activity. Bone remodeling may be involved. See also Me-
chanical tooth displacement.
Bound vector A vector quantity that has a de nite point
of application. For example, force is a bound vector. See
also Free vector.
Cantilever A structure or an appliance that has only one
xed end. For example, a molar tip-back spring anchored
in a molar tube with an anterior hook is a cantilever.
Center of mass A point where the distribution of mass
of a body is concentrated. A nonconstrained body trans-
lates when the force is acting on the center of mass. See
also Center of resistance.
Center of resistance (CR) A point where force would
result in translation of a constrained body (tooth or group
of teeth). It may vary with the direction of force and is
commonly considered not a point but rather an area.
Center of rotation (CRot) A point on the body around
which all points of the body rotate. See also Axis of rota-
tion.

Glossary
512
Center of rotation constant (σ) A constant for a tooth
that determines tooth sensitivity to rotation or tipping. If
it is large, the tooth is less likely to tip.
Cobalt-chromium alloy (Elgiloy) Alloy of cobalt and
chromium. It is easy to bend because the yield strength is
low.
After bends are made, heat treatment will increase
the yield strength. Heat-treated cobalt-chromium alloy
shows similar physical properties to stainless steel alloy.
Coefcient of friction (μ) Dimensionless value that
represents the amount of friction between two materials.
There are static and kinetic coefcients of friction.
Controlled tipping A rotation of a tooth in the facial
view with a center of rotation near the apex or further
apically. No point of the tooth is displaced in the opposite
direction.
Couple A moment where the sum of force is zero,
achieved by two parallel equal and opposite forces not
in the same line of action acting on a body. The unit is
gram-millimeters (gmm). See also Moment.
Deactivated shape The shape of a wire or a spring be-
fore placement into the mouth or attachments. Arches can
be straight or bent wires. See also Activated shape, Passive
shape, and Simulated shape.
Deactivation force system The force system acting on
the tooth from an orthodontic appliance. It is equal and
opposite to the activation force system. See also Activa-
tion force system.
Derived (secondary) tooth movement Combined pri-
mary tooth movement (rotation and translation). Con-
trolled tipping and root movement are derived or second-
ary tooth movements. See also Primary tooth movement.
Edgewise A method of placing rectangular wire in which
there is a larger faciolingual dimension than occlusogingi-
val dimension. See also Ribbon wise.
Elastic limit The initial linear part of the force-deection
(F/Δ) curve or stress-strain (σ/ε) curve. Within the elastic
limit, the wire can be restored to its original shape by un-
loading the force.
Elgiloy See Cobalt-chromium alloy.
Energy A physical quantity that is transferable to work.
The unit is Nm. For example, elastics or coil springs are
energy-storing devices that slowly release the energy to
move a tooth. See also Resilience.
Equivalent force system A force system that has the
same effect as another force system.
Fatigue The weakening or fracturing of a wire under
repeated loading below the yield strength. Orthodontic
wires may show fatigue fracture by repeated masticatory
forces.
Force (F) A physical quantity that causes a change in shape
by Hooke’s law or an acceleration by Newton’s Second
Law. The unit is a Newton (N) or kg·m/sec
2
. 1 cN = 10
–2
N.
A gram (g) is the unit of mass. One gram has a gravitation-
al force of 0.98 cN on earth.
Force diagram A diagram of a body showing all the
forces acting on the body. Only the forces of interest are
depicted. Also known as a free-body diagram.
Force-deection (F/Δ) rate Amount of force need-
ed for unit displacement of a spring. The unit is gram/
millimeter (g/mm).
Force-driven appliance An appliance that produces the
correct force system for dentofacial modication. See also
Shape-driven appliance.
Free vector A vector quantity that has only magnitude
and direction, where the point of application is irrelevant.
For example, a couple is a free vector. See also Bound vec-
tor.
Free-body diagram See Force diagram.
Frictional force (F
f
) A force resisting the motion of an
object. F
f
= µN.
Functional axis of a tooth The axis of the tooth that is
determined by its behavior under loading. It is different
than the anatomical long axis of a tooth.
Gram (g) A unit of mass.
Isotropic When a material shows the same physical prop-
erties regardless of the direction. Most alloys are isotropic.
See also Anisotropic.
Lever arm An extension of wire attached to the bracket
or soldered to the arch. It replaces the point of force appli-
cation. Power arm is misused terminology in orthodontics
because power is a unit of work (or energy) per unit time.
Watt (W) is the unit of power.
Line of action An imaginary extended line of a vector.
For example, it is the line through the point at which force
is applied and along the direction in which force is ap-
plied. See also Transmissibility, law of.
Mass A physical property of a body that is determined by
the resistance to acceleration or gravitational force. The
unit is kilograms (kg). 1 kg = 10
3
g.
Mechanical tooth displacement The displacement of a
tooth within the periodontal ligament (PDL) space due to
mechanical compression, tension, or shearing of the PDL.
No biologic response is related. See also Biologic tooth
movement.

Glossary
513
M/F ratio A ratio between moment and force applied at
a dened point. Common ratios are measured to a point
at the bracket or center of resistance. The M/F ratio rep-
resents the distance from the dened point to an equiva-
lent single force. The unit is length measured in millime-
ters.
Modulus of elasticity (or Young modulus) The ratio
of stress to strain in the elastic range. It is the inherent
physical property of a material. The unit is Pascal (Pa) or
N/m
2
. One gigapascal = 10
9
Pa. For example, acrylic has a
modulus of elasticity of 3.2 GPa, and stainless steel has a
modulus of elasticity of 180 to 200 GPa.
Modulus of rigidity The ratio of shear stress to angu-
lar deformation in the elastic range. See also Torque and
Torsion.
Moment (M) A physical quantity to produce a turning
or rotation of the body. The magnitude of the moment is
measured by the product of the force times the perpendic-
ular distance from the line of action of that force to the
reference point; thus, M = F × D, where M is the moment,
F is the magnitude of force, and D is the perpendicular
distance from the line of action of the force to the point
being considered. The unit is Nm. In orthodontics, gmm
is commonly used. 1 gmm = 0.98 cNmm. See also Couple.
Moment of inertia (I) A geometric property of a body
that relates the moment needed for a desired amount of
bending. It depends on the cross-sectional size and shape
of the wire. I =
bh
3
12
for rectangular wire and I =
∏d
4
64
for
round wire.
Motion displacement diagram A diagram with a dot-
ted arrow in this book. It schematically represents the
direction of rotation and movement. It does not specify
rotation center or direction of forces or couples.
Neutral position If the residual moments are placed on
a spring, the neutral position is dened as the shape of the
spring when only the moment is applied and the horizon-
tal force is zero.
Newton (N) A unit of force. 1 N = 100 cN = 1 kg·m/sec
2
.
Nickel-titanium An alloy of nickel and titanium that en-
ables transformation of its crystalline structure from aus-
tenite to martensite and vice versa. It shows superelastici-
ty and shape-memory effects. Work-hardened martensite
has high springback but shows no superelasticity.
Normal force A component of force acting perpendicu-
lar to the contact surface.
Pascal (Pa) A unit of pressure. 1 Pa = 1 N/m
2
Passive shape The shape of a wire or a spring that pro-
duces no forces when placed in brackets. See also Activat-
ed shape, Deactivated shape, and Simulated shape.
Preactivation bends The nal or additional bends
placed into a wire or a spring conguration to produce a
residual moment or other forces.
Primary tooth movement Translation and pure ro-
tation of the tooth. See also Derived (secondary) tooth
movement.
Pure rotation A rotation of a tooth with center of rota-
tion at the center of resistance. It occurs when a couple is
applied.
Range of action The amount of wire deection without
permanent deection. See also Strength.
Reactive unit The part of an arch that is directly involved
with those teeth that are not to be displaced. Active and
reactive units are distinguished by clinical purpose only
and indistinguishable biomechanically by Newton’s First
Law. Synonym for anchorage unit. See also Active unit.
Residual moment The additional moment added to a
spring by preactivation bends. See also Activation moment.
Resilience A physical property of the material that rep-
resents its ability to store energy within the elastic range.
Resultant A single force that can replace multiple forces
and moments to produce the same external effect on the
body.
Ribbon wise A method of placing rectangular wire in
which there is a larger occlusogingival dimension than fa-
ciolingual dimension. See also Edgewise.
Rigid body A body that does not change in shape or size
under the action of forces.
Root movement A rotation of a tooth in the facial view
with the center of rotation near the crown.
Rowboat effect When a couple (in the facial view) is ap-
plied to a tooth, the crown moves in the opposite direc-
tion from the root. The root is analogous to the oar of the
boat. The oar moves back while the boat moves forward.
For example, during canine root movement, the posterior
segments move forward. This is not a scientic term relat-
ing to anchorage loss.
Section modulus A geometric property for a given cross
section used in the design of beams or wires undergoing
exure. It is calculated by dividing the moment of inertia
by the distance from the neutral axis to the most distant
outer surface of a wire (I/c). It is directly related to the
maximum bending moment of a spring or wire within the
elastic range.
Shape memory The phenomenon of remembering a
wire’s original shape. Nickel-titanium alloys have two
phases (martensite and austenite) with different physical

Glossary
514
properties. Phase transition occurs from one phase to an-
other by change of stress or temperature. In the marten-
sitic phase, nickel-titanium permanently deforms easily.
If temperature increases, the phase changes to austenite
and the original shape is restored. See also Superelasticity.
Shape-driven appliance An appliance made with a
shape that is passive after teeth move to that shape. The
force system is not always correct. See also Force-driven
appliance.
Simulated shape The elastically deformed shape of a
passive wire or a spring when a desired deactivation force
system is applied. See also Activated shape, Deactivated
shape, and Passive shape.
Stabilizing segment A group of teeth that is rigidly con-
nected by a high-stiffness wire or polymer.
Stainless steel A steel alloy with a minimum of 10.5%
chromium content. It is corrosion and rust resistant and
the stiffest alloy among orthodontic wires. Its modulus of
elasticity is 180 to 200 GPa.
Static equilibrium The condition of a body that exists
when all forces and moments acting on it sum to zero,
governed by Newton’s First Law. See also Activation force
system.
Statically determinate A structure or an appliance is
statically determinate when static equilibrium equations
are sufcient to solve for unknowns. For example, elastic
chain and cantilever springs are statically determinate if
the horizontal force is measured by a force gauge. See also
Statically indeterminate.
Statically indeterminate A structure or an appliance is
statically indeterminate when static equilibrium equations
are not sufcient to solve for unknowns. For example,
space-closing loops (springs) are statically indeterminate
if only horizontal forces are known. See also Statically de-
terminate.
Statics The branch of physics (mechanics) that deals with
the forces when bodies are in a state of rest or at a con-
stant velocity.
Stiffness Synonym for force-deection rate. Stiffness is the
strength per range of action.
Strain (ε) The ratio of the amount of elongation to the
original length. The unit is m/m or dimensionless.
Strength The maximum amount of force (or moment)
that the appliance can produce without permanent defor-
mation. See also Range of action.
Stress (σ) The amount of force per unit area. The unit is
Pascal (N/m
2
).
Superelasticity During unloading, the stress-strain curve is
not linear and hysteresis is present. During deactivation, a
plateau of constant force can be observed.
Torque Synonym for couple or moment. In orthodontics,
torque is produced by torsion of a wire. Incorrect deni-
tion: a change of axial inclination or third-order bracket
slot angle. Incorrect example: The prescription has –10 de-
grees of “torque” in the maxillary central incisor bracket.
Torsion The twisting of a wire produced by a torque op-
erating around a structural axis. See also Bending.
Toughness Total area under a force versus deection
curve up to fracture.
Translation A type of movement by which all points on
a body move the same amount in a parallel direction. Syn-
onym for bodily movement in clinical orthodontics.
Transmissibility, law of A force vector can be replaced
by any force as long as it is along the same line of action.
For example, distal force on a canine is applied either by
pulling from the distal or pushing from the mesial. See
also Line of action.
Trial activation The application outside the mouth of all
moments and forces used to activate a spring. It mimics
the activation of a spring during insertion in the mouth.
Twist Synonym for torsion.
Uncontrolled tipping A rotation of a tooth in the facial
view with the center of rotation located near the center
of the root. It literally means that tipping is not controlled
because some part of the tooth moved in a direction op-
posite to the applied force. Single forces acting at the
crowns of the teeth, such as the tongue, cheek force, or
a force from a simple nger spring of a removable appli-
ance, produce uncontrolled tipping.
Work A special form of energy. Also known as mechan-
ical work or mechanical energy. The unit is Nm. The area
under the force-deection (stress-strain) curve. The area
under the loading curve is the work done to the wire
during activation, and the area under the unloading curve
is the work done to the tooth. It is conserved in the elastic
range but partly dissipates as a result of the internal fric-
tion and heat in the plastic region. The unit is the same as
the unit for a moment, but work is a scalar and moment is
a vector. See also Energy.
Yield strength (or point) Theoretically, it is the stress at
which wire starts to deform plastically. The wire is elastic
under this point. Technically, this point is difcult to de-
termine; therefore, a line is drawn parallel to the straight-
line portion of the stress-strain curve, and an offset from it
in an amount equal to 0.1 or 0.2 or 1% unit strain is used.

515
Solutions to
Problems
515
As described in the text, an important property of a force or moment is direction. The
convention for moments in this solutions section is + for clockwise and – for coun-
terclockwise. For simpli cation, no sign is usually speci ed in the text or  gures for
force direction. Force direction is de ned by the force diagram alone unless required
for emphasis.

516
SOLUTIONS
2
2.
300 g – 100 g = 200 g (distal direction)
3.
300 g + 100 g = 400 g (on the same line of action)
1.
They are all the same because they share the same line of
action
(red dotted line)
.
4.
Horizontal components:
From lingual arch: 100 g (1)
From crisscross elastics: 300 g cos 60° = 150 g (2)
∑F
H
= (1) + (2) = 100 g + 150 g = 250 g
Vertical components:
From lingual arch: 0 g (3)
From crisscross elastic: 300 g sin 60° = 300 g × 0.87 =
260g (4)
F
R
= F
H
2
+ F
V
2
= 250
2
+ 260
2
= 360 g
θ = tan
–1
260
250
= 46°
∑F
V
= (3) + (4) = 0 g + 260 g = 260 g

517
2
SOLUTIONS
6a.
F
H
= 150 g cos 20° = 150 g × 0.94 = 141 g (horizontal
component)
F
V
= 150 g sin 20° = 150 g × 0.34 = 51 g (vertical
component)
6b.
F
H
= 150 g cos 45° = 150 g × 0.71 = 107 g (horizontal
component)
F
V
= 150 g sin 45° = 150 g × 0.71 = 107 g (vertical
component)
7.
F
buccolingual
= 100 g cos 20° = 100 g × 0.94 = 94 g (lingual
component)
F
mesiodistal
= 100 g sin 20° = 100 g × 0.34 = 34 g (mesial
component)
5a.
F
H
= 100 g sin 60° = 100 g × 0.87 = 87 g (lingual
component)
F
V
= 100 g cos 60° = 100 g × 0.5 = 50 g (intrusive
component)
5b.
F
H
= 100 g sin 45° = 100 g × 0.71 = 71 g (lingual
component)
F
V
= 100 g cos 45° = 100 g × 0.71 = 71 g (intrusive
component)
5c.
F
H
= 100 g cos 20° = 100 g × 0.94 = 94 g (lingual
component)
F
V
= 100 g sin 20° = 100 g × 0.34 = 34 g (extrusive
component)

518
SOLUTIONS
2
9.
To have the resultant force lie along the archwire, the sum of all
vertical components should be zero.
F
V
(headgear) = F (headgear) sin θ
F
V
(elastic) = 200 g sin 25° = 200 g × 0.42 = 84.5 g
F
V
(headgear) = F
V
(elastic)
Therefore,
F
V
(headgear)
sin θ = 84.5 g
θ = sin
–1
84.5
F
headgear
a. If F
headgear
= 200 g, θ = sin
–1
84.5
200
= 25°
b. If F
headgear
= 600 g, θ = sin
–1
84.5
600
=
c. If F
headgear
= 1,000 g, θ = sin
–1
84.5
1,000
= 4.8°
8.
F
H
(headgear) = 400 g cos 45° = 400 g × 0.71 = 284 g
F
V
(headgear) = 400 g sin 45° = 400 g × 0.71 = 284g
F
H
(elastic) = 200 g cos 35° = 200 g × 0.82 = 164 g
F
V
(elastic) = 200 g sin 35° = 200 g × 0.57 = 115 g
Therefore,
∑F
H
= F
H
(headgear) + F
H
(elastic) = 284 g + 164 g = 448 g
∑F
V
= F
V
(headgear) + F
V
(elastic) = 284 g – 115 g = 169g
F
R
= F
RH
2
+ F
RV
2
= 448
2
+ 169
2
g = 479 g
And
θ = tan
–1
169
448
= 21°

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CHAPTEROVERVIEW“ Inventing is a combination of brains and materials. The more brains you use, the less material you need.” — Charles Kettering“Make the workmanship surpass the materials.” — Ovid“Wisdom must be intuitive reason combined with scientic knowledge.” — AristotleHow to Select an Archwire 21Clinicians must select suitable archwires for their patients. This chapter serves as a simple guide for traditional alloys such as stainless steel and beta-titanium with linear stress-strain curves. Typical engineering formulas can be complicated and use units not readily familiar to most clinicians, such as GPa. Wire stiffness can be determined by simply multiplying material stiffness by cross-section stiffness. For cross-section stiffness, the base is 0.004-inch round wire, with an arbitrary value of 1.0. For material stiffness, the base is stainless steel, with a material number of 1.0. Convenient tables are included in this chapter that allow the orthodontist to compare the stiffness and other properties by a simple ratio of the cross-sectional and material stiffness numbers. In a similar manner, maximum moment and maximum deection numbers can be compared by a ratio of their values.491491 21How to Select an Archwire 492The typical orthodontic appliance is composed of a wrought orthodontic wire; cast congurations and more complicated shapes are rarely used. The ortho-dontic appliance of the future may involve nonwire shapes of both metals and polymers. For now, our question is limited to a discussion of the factors that should be considered in selecting an orthodontic wire or archwire. Depending on application, the fol-lowing discussion should be helpful; it is a mistake to stereotype the sequence for alignment or general treatment and not to take advantage of the mul-titude of possibilities available to the modern or-thodontist. The principles described in this chapter should be applicable not only to xed bracket-wire appliances but also to all present and future appli-ances.The Triad CharacteristicsThree important characteristics during elastic de-formation, forming a triad, are closely interrelated: force-deection rate (F/Δ); maximum force (Fmax) or, better stated, maximum moment (Mmax); and max-imum elastic deection (Δmax). Here we consider traditional materials such as stainless steel or beta- titanium that have linear stress-strain plots in the elastic range that follow or approach Hooke’s law (Fig 21-1). Superelastic materials such as nickel- titanium (Ni-Ti) wires are very important but are nonlinear and do not follow Hooke’s law; therefore, they are described elsewhere.The formula for Δmax is as follows: Δmax = Fmax F/ΔWire properties in a simple tensile or exural mode may follow Hooke’s law; however, a linear relationship between force and deection may not be produced in the overall appliance conguration because the conguration changes as the spring ac-tivates and because the force application point and direction may change; nevertheless, the simple for-mulas that are linear and describe small deections are very useful.Two fundamental beam formulas describe these relationships: F = K EI Δ L3where E is the modulus of elasticity, I is the moment of inertia, L is the wire length, and K is a constant that changes with conguration and loading con-ditions. When the loading condition is a cantilever with a single force at the free end, K = 3, but this is only one loading condition common in practice (Fig 21-2).The other fundamental beam formula in exure is the following: Fmax = σmax S LFig 21-1 F/Δ rate, maximum force (Fmax) or maximum moment (Mmax), and maximum (elastic) deection (Δmax) are closely inter-related, and traditional materials such as stainless steel or beta- titanium have linear stress-strain plots in the elastic range that fol-low or approach Hooke’s law.Fig 21-2 Flexure formula of a cantilever spring. F and Δ are propor-tional within the range of small deection.F = 3EI∆ L3 493Force-Deection RateorMmax = σmax Swhere σmax is the yield stress of the material, S is the section modulus (I/c), c is the distance from the neu-tral axis to the most outer surface of the wire, and Mmax is the maximum moment that the wire can de-liver in the elastic range. This simple exure formula took nearly two centuries to be developed.Clinicians have an interest in all three triad prop-erties. A large Δmax is popular because wires that can be deected during large activations without per-manent deformation are desired. The magnitude of Fmax or Mmax must be large enough to achieve the de-livery of an optimum force level with a safety factor. A wire with lower Fmax can also be easily deformed by an intermittent heavy force from mastication.The F/Δ rate determines the rate of force decay or the force constancy. These triad wire characteris-tics can be modied by altering wire material prop-erties, the shape and size of the wire cross section, and overall appliance conguration. The F/Δ rate of an appliance (spring) can be altered by complicated congurations or more simple modications such as increasing the interbracket distance. Engineering formulas can predict the force sys-tems from orthodontic appliance congurations for both large and small deections. Formulas can be long and complicated with quantities measured in units not always familiar to the clinician. For that reason, some of the overall design factors were discussed in chapter 14, and this chapter considers only the effect of wire cross section and the material used on the force triad.Force-Deection RateThe clinician uses the F/Δ rate or the stiffness of a wire in a number of ways. It tells us how much to activate the wire to achieve a desired force. For ex-ample, if 100 g of force is desired and the F/Δ is 20 g/mm, 5 mm of activation is required. Commonly, leveling is accomplished by placing a series of wires with increasingly larger cross sections. Sometimes this sequence of wire changes is explained as in-creasing the cross section to eliminate the play be-tween the wire and the bracket. However, rarely is wire play the biggest problem; the real reason wires must be sequenced is to keep force levels more constant. When the teeth are grossly misaligned, a low-stiffness wire is needed. As the teeth approach nal alignment, a higher-stiffness wire must be used with a smaller deection to give the same force.Let us now consider the effect of the alloy or ma-terial on stiffness or rigidity of an orthodontic wire. F/Δ rates vary linearly with the modulus of elastic-ity (E). It is convenient to use relative numbers to describe the rigidity of a material’s contribution to stiffness. Arbitrarily, let us compare all materials to stainless steel, which will have a base number of 1.0. Beta-titanium has an E that is 0.42 as great as that of stainless steel, and hence its material stiffness num-ber is 0.42. The material stiffness numbers for some common orthodontic wire materials are given in Fig 21-3. One can compare the stiffness of two applianc-es identical apart from their material by making a ratio of their material stiffness numbers. Both cross-sectional size and shape differences sig-nicantly affect wire stiffness. For round wire, stiff-ness varies as d4, where d is the wire diameter. For Fig 21-3 Material stiffness numbers. The numbers are relative values, with stainless steel as the standard value of 1.0. Ni-Ti has two material stiffness values depending on crystal structure. The value for small deec-tion is 0.26.AlloyStainless steelEIgiloyBeta-titaniumNi-Ti11.20.420.26Material stiffness number 21How to Select an Archwire 494edgewise wire, stiffness varies as bh3, where b is the width and h is the height; the curved arrows in Fig 21-4 indicate the direction of bending moment. In a similar manner to the material stiffness num-ber, cross-section stiffness numbers use 0.004-inch wire as the base, which has a value of 1.0. Figure 21-5 gives representative cross-section stiffness numbers for both round (Fig 21-5a) and rectangu-lar (Fig 21-5b) wires. A ratio of cross-section stiffness numbers gives the relative stiffness of two identical appliances that differ only in cross-sectional dimen-sion. A more complete listing of cross-section stiff-ness numbers is given in Table 21-1.For example, two identical vertical loops (same conguration and material) are fabricated. One is 0.014 inch and the other is 0.020 inch. Compare their Fig 21-4 Cross-section stiffness. Both cross- sectional size and shape differences signicantly affect wire stiffness. For round wire, stiffness var-ies as d4, where d is the wire diameter. For edge-wise wire, stiffness varies as bh3, where b is the width and h is the height.Fig 21-5 Cross-section stiffness numbers. (a) Round wires. The numbers are relative values based on 0.004-inch round wire as the standard value of 1.0. (b) Rectangular wires. The direction of bend-ing affects the number.aCross section0.0040.0100.0140.0160.0180.0200.022Cross-section stiffness number139150256410625915bCross-section stiffness numberFirst-order Second-order0.018 × 0.0250.021 × 0.0250.0215 × 0.0281,8652,1763,1309671,5351,845F = 3EI∆ L3I = bh3 12I = πd4 64 495Force-Deection RateTable 21-1 Relative force-deection rate, maximum moment, and maximum elastic deformationShapeCross-section dimensions (inch)Relative force- deection rateRelative maximum momentRelative maximum elastic deformationRound 0.006 5.06 3.38 0.670.007 9.38 5.36 0.570.008 16.00 8.00 0.500.009 25.63 11.40 0.440.010 39.06 15.63 0.400.011 57.20 20.80 0.360.012 81.00 27.00 0.330.013 111.57 34.33 0.310.014 150.06 42.88 0.290.015 197.75 52.73 0.270.016 256.00 64.00 0.250.017 326.25 76.77 0.240.018 410.06 91.18 0.220.019 509.07 107.17 0.210.020 625.00 125.00 0.200.021 759.70 144.70 0.190.022 915.06 166.38 0.180.023 1,093.13 190.11 0.170.024 1,296.00 126.00 0.170.025 1,525.88 244.14 0.160.026 1,785.06 274.63 0.150.027 2,075.94 307.55 0.150.028 2,401.00 343.00 0.140.029 2,762.82 381.08 0.140.030 3,164.06 421.88 0.130.032 4,096.00 512.00 0.130.036 6,561.00 729.00 0.110.040 10,000.00 1,000.00 0.100.045 16,013.10 1,428.88 0.090.050 24,414.10 1,953.12 0.080.060 50,625.00 3,375.00 0.07Square0.016 × 0.016434.60 108.65 0.250.017 × 0.0173,524.23 522.11 0.150.018 × 0.018969.14 154.70 0.220.019 × 0.019864.22 181.94 0.210.021 × 0.0211,289.69 245.66 0.19Rectangular (rst-order)0.006 × 0.020318.31 63.66 0.200.008 × 0.020424.41 84.88 0.200.010 × 0.020530.53 106.10 0.200.0105 × 0.0281,528.52 218.36 0.140.011 × 0.022776.73 141.22 0.180.012 × 0.020636.62 127.32 0.200.014 × 0.020742.72 148.55 0.20(cont) 21How to Select an Archwire 496Table 21-1 (cont) Relative force-deection rate, maximum moment, and maximum elastic deformationShapeCross-section dimensions (inch)Relative force- deection rateRelative maximum momentRelative maximum elastic deformation0.015 × 0.0282,183.61 311.94 0.140.016 × 0.020848.83 169.76 0.200.016 × 0.0221,129.79 205.42 0.180.017 × 0.0221,200.40 218.25 0.180.017 × 0.0251,761.48 281.84 0.160.018 × 0.0221,271.01 231.09 0.180.018 × 0.0251,865.10 298.42 0.160.019 × 0.0251,968.71 314.99 0.160.019 × 0.0262,214.53 340.70 0.150.020 × 0.0252,072.33 331.57 0.160.021 × 0.0252,175.95 348.15 0.160.021 × 0.0272,741.07 406.08 0.150.0215 × 0.0283,129.83 447.12 0.14Rectangular (second-order)0.006 × 0.02028.65 19.10 0.670.008 × 0.02067.91 33.95 0.500.010 × 0.020132.63 53.05 0.400.0105 × 0.028214.95 81.89 0.380.011 × 0.022194.18 70.61 0.360.012 × 0.020229.18 76.40 0.330.014 × 0.020363.93 103.98 0.290.015 × 0.028626.67 167.11 0.270.016 × 0.020543.25 143.25 0.250.016 × 0.022597.57 149.40 0.250.017 × 0.022716.77 168.65 0.240.017 × 0.025814.51 191.65 0.240.018 × 0.022850.84 189.08 0.220.018 × 0.025966.87 214.86 0.220.019 × 0.0251,137.13 239.40 0.210.019 × 0.0261,182.61 248.97 0.210.020 × 0.0251,326.29 265.26 0.200.021 × 0.0251,535.18 292.45 0.190.021 × 0.0271,658.18 315.84 0.190.0215 × 0.0281,845.37 343.32 0.19 497Maximum Force and Maximum Bending Momentforces for the same activation. The cross-section stiff-ness is 150.0 for the 0.014-inch loop and 625.0 for the 0.020-inch loop. The 0.020-inch loop gives 4.3 times greater force (625/150).There are three basic methods to reduce the F/Δ rate: (1) using a different material with a lower E, such as selecting a Ni-Ti or beta-titanium wire rather than stainless steel; (2) adding wire to increase the length (L), such as placing a helix or loop in the wire; or (3) reducing the wire cross section. However, re-ducing the cross-sectional size may be problematic because the Fmax or Mmax is also dramatically reduced.Maximum Force and Maximum Bending MomentThe material property that determines the Mmax that a wire can produce is the yield strength. The Fmax that an appliance delivers is partly determined by design. The Mmax is more fundamental because it is determined by material and cross section only. The yield strength of a spring can vary considerably for the same material. For example, consider two stain-less steel vertical loops with the same cross-sectional shape and size and the same overall conguration: one made from a high-springback stainless steel and the other from dead soft annealed stainless steel ligature tying wire. The E of the two is about the same, so within the elastic range, both would give the same force for the same activation. The differ-ence is that the ligature wire has much lower yield strength and would permanently deform sooner, giving potentially less Fmax. The dead soft wire is eas-ily made in the clinic via annealing heat treatment using a torch lamp. The wire is heated until a bright yellow color is observed (Fig 21-6) and then slowly cooled down. An annealed wire is very easy to bend. The Mmax or σmax is reduced; however, the F/Δ rate is the same because E is not changed. When a pas-sive wire is needed with a large–cross-section wire, an annealed wire is very easy to fabricate. Instanta-neous excessive high force is avoided, and still high stiffness is maintained for good stress distribution (Fig 21-7). The clinician should not confuse stiffness with Fmax in selecting a wire. The Mmax or Fmax varies linearly with the yield strength.Cross section of the wire also inuences the Mmax. This is more signicant than the material because Mmax varies as d4 in round wire and bh3 in rectan-gular wire. Table 21-1 lists relative maximum bend-ing moments for both round and rectangular cross sections based on cross section. A ratio of the Mmax numbers can be used to compare appliances identi-cal except for cross section in respect to Fmax or Mmax.For example, compare the Fmax of 0.016-inch and 0.018-inch identical vertical loops. The Mmax is 64 for the 0.016-inch loop and 91 for the 0.018-inch loop. Therefore, the 0.018-inch loop can maximally deliv-er 1.4 times the force of the 0.016-inch loop (91/64).Fig 21-7 The maxillary left central incisor has a subgingival fracture line from trauma. The fractured crown was temporarily bonded and extruded to expose the fracture line. When a passive wire in bending and torsion are needed with a large stiff wire, an annealed steel wire is very easy to fabricate; instantaneous excessive high force is avoided, and yet high stiffness is still maintained. The bypass arch was fabricated of 0.017 × 0.025–inch stainless steel wire. (a) Before extrusion. (b) After extrusion. a bFig 21-6 A torch lamp is used to anneal stainless steel. 21How to Select an Archwire 498Maximum DeectionThe component properties that determine Δmax in the cantilever-type loading condition are shown in the following formula: Δmax = σmax L2 KEcwhere K is a constant that changes with congu-ration and loading conditions and c is the distance from the neutral axis to the most outer surface of the wire (h/2 in rectangular wire and d/2 in round wire). No table is given on yield strength because there is great variation even in the same alloy, un-like E, which is relatively constant for a given alloy.The cross section inuences the Δmax in a linear manner. It varies inversely with diameter (d) for round wire and height (h) for rectangular wire. Also, it is affected by the direction of bending due to the Bauschinger effect of residual stress. A relative Δmax number based on cross section is given in Table 21-1. Compare the Δmax of 0.016-inch and 0.018-inch iden-tical vertical loops. Δmax numbers are 0.25 and 0.22, respectively. The 0.016-inch loop can be deected 1.1 times that of the 0.018-inch loop. The effect is negligible. Interestingly, the labiolingual width for the rectangular wire does not affect the maximum range. A 0.017 × 0.022–inch or 0.017 × 0.025–inch wire gives the same Δmax in a second- order bending direction (see Table 21-1).Clinicians like wires that deect large distances without permanent deformation. Such appliances require fewer adjustments and deliver more con-stant forces. Another measurement is resilience, which determines the amount of mechanical energy the wire can store and release during deactivation within the elastic range.Other Wire PropertiesOther than the triad of properties previously de-scribed, many other wire properties need to be con-sidered in choosing an orthodontic wire. In the plas-tic range, the ultimate tensile strength denes the stress levels at fracture. The percentage of elonga-tion, cold bend tests, and strain difference between the yield strength and the ultimate tensile strength dene ductility. Fatigue involves cyclic loading usu-ally dened in the elastic range. The total area un-der the stress-strain tensile plot is called toughness. Depending on application, the clinician needs a wire that can be shaped without fracture. Superelastic Ni-Ti wire has less plastic range so that it lacks the ability to bend into certain shapes. Brittle wires can also fracture under occlusal loading by chewing or during insertion.Friction is a consideration during sliding mechan-ics at all stages of treatment. The material property that determines the frictional force between two materials is the coefcient of friction (μ). A more complete discussion is found in chapter 19.Wire cross section selection is mainly related to the stiffness wanted or the level of force or moment to be delivered. A secondary consideration is the amount of play desired between the bracket and the wire. Some other characteristics to consider are biocompatibility, resistance to corrosion and degra-dation, and the ability to weld or solder.Wire Stiffness NumbersChapter 14 discussed how wire conguration can af-fect the F/Δ rate. We can compare orthodontic appli-ances of identical design by isolating and evaluating the effect of material and cross section of the wire. In order to consider the wire component only and not the overall design or conguration, the follow-ing formula multiplies the material stiffness number by the cross-section stiffness number. The product is the wire stiffness number, a measure of the wire rigidity independent of the appliance design.Ws = Ms × Cswhere Ws is the wire stiffness, Ms is the material stiff-ness, and Cs is the cross-section stiffness. For example, the cross-section stiffness of a 0.018-inch stainless steel wire is 410.0. The material stiffness is 1.0. The wire stiffness is therefore 410 (410 × 1.0 = 410). The material stiffness of 0.016-inch beta-titanium wire is 0.42, and the cross-sectional stiffness is 256.0. The wire stiffness is therefore 107.5 (256 × 0.42 = 107.5). If we make two identical verti-cal loops out of these materials and compare the stiffness, we could compare the force levels for the same activation in the elastic range. The ratio of the wire stiffness numbers is 410 divided by 107.5, or 3.8. The 0.018-inch stainless steel loop therefore gives 3.8 times the force of the 0.016-inch beta-titanium loop. 499Recommended ReadingRelative Torque (Torsion) NumbersBecause different formulas are used for wires un-dergoing torsion, a separate but similar table is presented for relevant torque numbers (Table 21-2). Numbers are given for relative torque/twist rates, relative maximum torque, and maximum elastic twist. Appliances can be compared in a similar man-ner by making a ratio of the relevant numbers.Recommended ReadingBurstone CJ. Application of bioengineering to clinical ortho-dontics. In: Graber LW, Vanarsdall RL Jr, Vig KWL (eds). Ortho-dontics: Current Principles and Techniques, ed 5. Philadelphia: Elsevier Mosby, 2012:345–380. Burstone CJ. Variable modulus orthodontics. Am J Orthod Dentofacial Orthop 1981;80:1–16.Burstone CJ, Goldberg AJ. Maximum forces and deections from orthodontic appliances. Am J Orthod Dentofacial Orthop 1983;84:95–103.Choy KC, Kim KH, Park YC, Kang CS. Torsional moment of orthodontic wires. Korean J Orthod 2000;30:467–473.Table 21-2 Relative torque/twist rate, maximum torque, and maximum elastic twistShape Cross-section dimensions (inch) Relative torque/twist rate Relative maximum torque Relative maximum elastic twistRectangular0.006 × 0.02046.47 16.19 0.350.008 × 0.020101.66 27.38 0.270.010 × 0.020182.14 40.81 0.220.0105 × 0.028328.50 66.85 0.200.011 × 0.022266.67 54.32 0.200.012 × 0.020286.98 56.17 0.200.014 × 0.020413.30 73.23 0.180.015 × 0.028833.24 126.46 0.150.016 × 0.022660.32 104.01 0.160.017 × 0.022756.40 115.23 0.150.017 × 0.025943.58 136.11 0.140.018 × 0.022857.30 126.21 0.150.018 × 0.0251,076.23 150.04 0.140.019 × 0.0251,215.62 164.42 0.140.019 × 0.0261,302.19 173.08 0.130.020 × 0.0251,361.31 179.23 0.130.021 × 0.0251,513.10 194.45 0.130.021 × 0.0271,740.90 215.35 0.120.0215 × 0.0281,957.06 235.04 0.12Square0.016 × 0.016367.24 67.91 0.180.017 × 0.017468.02 81.45 0.170.018 × 0.018588.24 96.64 0.160.019 × 0.019730.26 113.71 0.160.021 × 0.0211,089.30 158.54 0.14 500PROBLEMS1. Compare 0.014-inch and 0.016-inch stainless steel wires used for initial leveling. When 0.014-inch wire is used instead of 0.016-inch wire, the ratio of the diameters is reduced by 13%. How much reduction in the F/∆ rate is anticipated? If 0.014-inch and 0.016-inch superelastic Ni-Ti wires are compared, will the F/∆ difference be the same?3. A helix is incorporated in the cantilever spring to reduce the F/∆ rate. How would F/∆, Fmax, and ∆max be affected? Exact numbers are not necessary. Compare A, B, and C.2. In lingual orthodontics, the interbracket distance is small-er than with a labial appliance. Suppose the interbracket distance is reduced from 8 mm to 4 mm; how would this affect the F/∆ rate? 4. 0.016 × 0.022–inch wires are inserted ribbon wise (A) instead of edgewise (B). How would this change affect the force system occlusogingivally and labiolingually? Ignore the play between the wire and the brackets.0.016-inch stainless steel0.014-inch stainless steel(13% reduction in diameter)ABCA BFor problems 1 to 7, assume a small deection and Hooke's law. 501Problems5. The size of a round wire is reduced by 50% to reduce the F/∆ rate. How much reduction in the F/∆ rate is anticipat-ed? How would it affect the Fmax?6. Compare the stiffness of a 0.017 × 0.025–inch beta-titanium spring and a 0.016 × 0.022–inch stainless steel spring with the same conguration.7. A 0.032 × 0.032–inch square stainless steel wire is used for a lingual arch. Calculate the stiffness number, which is not shown in the tables in the text.The diameter of B is 50% of A.AB0.017 × 0.025–inchbeta-titanium0.016 × 0.022–inchstainless steel0.032"0.032" VPARTAppendices• Hints for Developing Useful Force Diagrams• Glossary• Solutions to Problems 505505An important aspect of applying biomechanical principles to clinical orthodontics is the use of val-id force diagrams. A “good” diagram is needed to explain any appliance or therapy. It is the  rst step in solving the many problems presented in this text. It is the fastest and most concise way for the author or speaker to present new concepts or appliances. A clinician can speak in generalities about his or her treatment methodology, but evaluation of his or her approach can be dif cult because of vague language; however, a good force diagram can im-mediately impart to the listener the presentation’s validity. Double-talk is a form of speech in which in-appropriate, invented, or nonsense words are used to give the appearance of knowledge while actually confusing an audience. It is hard to double-talk a valid and concise force diagram.In orthodontic books, journals, and lectures, it is common to see incorrect force diagrams or dia-grams that are dif cult to understand. The authors have collected many examples and believe that they are helpful in identifying and preventing common errors. A major problem is that we do not want to embarrass any orthodontist (many are our friends, colleagues, and students) who may be guilty of a mistaken force diagram. Our goal is to teach prop-er force diagram construction and not to criticize. Our solution is to take diagrams with actual errors that we have observed and redraw the diagrams us-ing different teeth and applications. Thus, none of the diagrams that follow can be identi ed with any particular individual. As is said in many  ction titles, “Any similarity to any persons living or dead is pure-ly coincidental.” Now let us consider some common errors.Floating ForcesFigure 1a shows an intrusion arch where the forces on the incisors and posterior teeth are not precise-ly placed at their line of action but are  oating in space. Forces are vectors, and the force arrows must be placed somewhere along the line of action of the force. This is corrected in Fig 1b. The curved arrow (moment) is correct in Fig 1a. Couples are free vectors and can be placed anywhere on the diagram, unless you want to emphasize where the appliance delivers the moment. Figure 1b may be clearer, showing that the moment is calculated around the center of re-sistance (CR) of the posterior segment. Another ex-ample of  oating forces is Fig 2a. An elastic to close space acts at the level of the brackets in Fig 2b, not occlusally as depicted in Fig 2a, away from the line of action. Commonly, force diagrams in orthodon-tic presentations and publications improperly show forces not acting along their line of action.Hints for Developing Useful Force Diagrams Hints for Developing Useful Force Diagrams506Arrows That Are Not ForcesStraight and curved arrows are frequently used to show the direction of movement (Fig 3). Because arrows are used to represent forces, it is better to use other methods to show movement. Additional nonforce arrows may produce a confusing diagram. Avoid curved arrows for labels, as shown in Fig 3 labeling the molar attachment. Style and color of force arrows must be unique and not confused with any other arrows. Some orthodontic publications and slides show many arrows, and it is not always clear whether they represent forces or movement direction; hence, interpretation is difcult. Equivalent Force DiagramsEquivalent force systems are useful to include in an orthodontic force diagram as discussed in this text. Here, the force is placed at an arbitrary point not necessarily along the line of action of the actual force. Figure 4a shows that the intrusive force on Fig 1a Fig 1bFig 2aFig 2bFig 3 Fig 4a b 507Equilibrium Diagramsthe maxillary incisor produces a moment of force that tends to are the incisor. Figure 4b is better because the force at the bracket is replaced with an equivalent intrusive force at the CR and a coun-terclockwise moment; this is a more complete de-scription than the “moment of force” concept. The equivalent force system should be in a separate col-or or different format so that it is not identied as additional applied forces.Equilibrium DiagramsOrthodontic appliances are in equilibrium, which should be reected in force diagrams of all appli-ances. Newton’s Third Law tells us that there are equal and opposite forces (activation and deactiva-tion); it is the activation forces that must be given on the equilibrium diagram. The free-body diagram that shows the forces acting on the teeth (deactiva-tion forces) should also be in equilibrium because it may be based on an appliance equilibrium condi-tion. Figure 5a claims to show the forces acting from accentuated curves and reverse curve of Spee arch-wires (Fig 5b). Because the forces act on the teeth, the diagram is a free-body diagram; nevertheless, we can see that if the forces are reversed (Newton’s Third Law), the archwire cannot be in equilibrium. This cannot be a valid explanation of how these arches work to correct a deep bite. What is shown is impossible because the forces and moments do not add to zero. A more detailed explanation of curve of Spee arches is given in chapters 6 and 7.Figure 6a has many errors that add to confusion about its message. The Herbst-like appliance must be in equilibrium; because there is a point attach-ment on either end, only equal and opposite forc-es are generated. In Fig 6a, the maxillary force and mandibular force are not equal, nor are they along the same line of action; impossible couples acting at either end would be required for equilibrium. The moment of force to the CR of the maxillary arch is smaller than the moment of force to the mandib-ular arch, yet the force is greater and the moment arms the same. Overall, this a “nonsense” diagram that does not enlighten the reader. Resolving the force into horizontal and vertical components is re-dundant because force direction is obvious. This is a non-Newtonian appliance because it is claimed that the maxillary distal force is greater than the man-dibular mesial force. But is this possible? Figure 6b is a correct diagram. Fig 5a Fig 5bFig 6a Fig 6b Hints for Developing Useful Force Diagrams508Confusing Activation and Deactivation ForcesA free-body diagram showing all of the relevant forces acting at the brackets (on the teeth) can be-come complicated because more than one appli-ance component can be present. This can be made more difcult when two sets of forces are present: activation and deactivation. Let us consider the sim-ple mechanism for space closure in Fig 7. An elastic is used to produce the horizontal forces. To add a counterclockwise moment in the direction of lingual apical root movement, a cantilever root spring is in-serted into an auxiliary tube on the canine bracket. To activate the spring, an occlusal force is required (Fig 7a). This diagram is confusing because all forces are acting on the teeth except one force that acts on the wire hook, but the color is the same as the forces on the teeth. For greater clarity, the diagram should only show the deactivation forces in red, where all red deactivation forces act on the teeth (Fig 7b). If a cantilever is used, the clinician may prefer to show the deactivation force system acting on the teeth and not be concerned with an equilibrium diagram of the cantilever spring itself. The downward force on the hook should then be a different color to show an equivalent diagram.Fig 7aFig 7bFig 8aFig 8bFig 8c 509ConclusionIt Is the Line of Force That CountsIt should be remembered that the line of force de-termines the direction of the force and its point of force application. In Fig 8a, a temporary anchorage device (TAD) and a sliding jig were used to retract the maxillary right molar. The red force parallel to the occlusal plane apical to the brackets is the correct line of action. The author of a publication thought that the black diagonal force acted on the molar or the maxilla (Fig 8b). But the line of action is produced by the spring. What is the force system on the archwire, assuming that friction prevents the molar from sliding distally? The distal force and the moment produced by the vertical forces (yellow) are equivalent to the force (red) acting on the TAD api-cal to the archwire (Fig 8c). If the TAD is at the CR, the maxillary arch would tend to translate distally if little play is present between the archwire and the brackets. If the molar is free to slide, the force sys-tem is more complicated, but still the force direction is the same.Equivalence: Replacing a Force with a Force and a CoupleWe have seen that it is useful to replace a force with an equivalent force and a couple (moment). An ar-bitrary point is chosen that helps our understand-ing of the biomechanical system. Sometimes it is not clear on the force diagram where that point is to be found, or a useless point is selected. Consider Fig 9a, where the four maxillary incisors require extrusion to close an open bite. From rst molar forward to the canine, the buccal segment is level and parallel to the mandibular occlusal plane. An extrusion arch from the molar auxiliary tube delivers an extrusive force to the incisors. The deactivation force diagram on the teeth correctly shows an intrusion force at the molar and a counterclockwise tip- forward mo-ment. What is the side effect that will be produced? Because the buccal teeth from molar to canine are joined together by a rigid wire, we are not interest-ed in the force system at the molar tube but rather at the CR of the entire buccal segment. The force and moment at the rst molar are not useful depictions unless we are studying molar tube deformation or tube bonding strength. Figure 9b is more relevant; the force system is replaced with an equivalent force system at the CR of the buccal segment. A vertical elastic is added at the canine so that the sum of all moments on the buccal segment is zero. Because of the varying distances, a net small extrusive force acts on the buccal segments. Many orthodontic force di-agrams show forces and moments, but the point at which the moment is calculated is often a mystery. Although couples are free vectors, we must know if they are delivered by the appliance or are calculated using a useful reference axis and where that axis is located.ConclusionWe have suggested some ways to simplify force system diagrams to make communication among orthodontists more transparent. There is still much room for creativity in presentation without ignoring Newtonian precepts. Force diagrams are important because they are windows to the “soul” of the or-thodontist. And you cannot double-talk a diagram.Fig 9a Fig 9b 511Glossary511Note: Some de nitions or terminology widely used in physics are purposely narrowed to relate to clinical ortho-dontics.Activated shape The shape of a wire or a spring pro-duced by an activation force system. See also Activation force system, Deactivated shape, Passive shape, and Sim-ulated shape.Activation force system The force system applied to a spring or archwire to insert it into a bracket. An activated appliance by an activation force system is always in static equilibrium. The deactivation force system acting on the teeth is equal and opposite to the activation force system (Newton’s Third Law). See also Deactivation force system and Static equilibrium.Activation moment The moment produced by the ac-tivation of a spring such as a space closure loop. See also Residual moment.Active unit The part of an arch or appliance that is in-volved with tooth movement. See also Reactive unit.Alpha position The anterior component of a spring or anterior point of attachment of a spring. See also Beta po-sition.Anatomical long axis of a tooth Arbitrary axis deter-mined by the anatomy of the tooth. The longest dimen-sion of the tooth measured determines the anatomical long axis.Anchorage unit See Reactive unit.Anisotropic When a material shows different physical properties depending on its direction. Wood is a typical anisotropic material.Annealing Heat treatment of the alloy to reduce strength and increase ductility.Axis of rotation An axis on the body around which all points of the body rotate. In screw theory, the body may translate along the axis. In two dimensions, the axis be-comes a center of rotation.Bauschinger effect The residual stress in a wire after per-manent deformation that in uences the range of action. Beam theory The science of explaining beam de ection during loading. An orthodontic wire is a beam (ie, a struc-ture with a large longitudinal dimension in relation to its cross section). Bending Produced when the wire structural axis chang-es at right angles to the original structural axis. See also Torsion.Beta position The posterior components of a spring or the posterior point of attachment of a spring. See also Al-pha position.Beta-titanium An alloy that contains titanium. Beta- titanium is used in orthodontics because of its lower forces and high springback. Its modulus of elasticity is 42% of that of stainless steel. Another titanium alloy with a dif-ferent crystalline structure, alpha-titanium, is the most common titanium alloy used commercially, but it is not suitable for orthodontic wire.Biologic tooth movement The tooth movement that involves bone resorption at compression sites and bone apposition at tension sites by osteoclastic and osteoblastic activity. Bone remodeling may be involved. See also Me-chanical tooth displacement.Bound vector A vector quantity that has a de nite point of application. For example, force is a bound vector. See also Free vector. Cantilever A structure or an appliance that has only one  xed end. For example, a molar tip-back spring anchored in a molar tube with an anterior hook is a cantilever. Center of mass A point where the distribution of mass of a body is concentrated. A nonconstrained body trans-lates when the force is acting on the center of mass. See also Center of resistance.Center of resistance (CR) A point where force would result in translation of a constrained body (tooth or group of teeth). It may vary with the direction of force and is commonly considered not a point but rather an area.Center of rotation (CRot) A point on the body around which all points of the body rotate. See also Axis of rota-tion. Glossary512Center of rotation constant (σ) A constant for a tooth that determines tooth sensitivity to rotation or tipping. If it is large, the tooth is less likely to tip.Cobalt-chromium alloy (Elgiloy) Alloy of cobalt and chromium. It is easy to bend because the yield strength is low. After bends are made, heat treatment will increase the yield strength. Heat-treated cobalt-chromium alloy shows similar physical properties to stainless steel alloy.Coefcient of friction (μ) Dimensionless value that represents the amount of friction between two materials. There are static and kinetic coefcients of friction. Controlled tipping A rotation of a tooth in the facial view with a center of rotation near the apex or further apically. No point of the tooth is displaced in the opposite direction.Couple A moment where the sum of force is zero, achieved by two parallel equal and opposite forces not in the same line of action acting on a body. The unit is gram-millimeters (gmm). See also Moment.Deactivated shape The shape of a wire or a spring be-fore placement into the mouth or attachments. Arches can be straight or bent wires. See also Activated shape, Passive shape, and Simulated shape.Deactivation force system The force system acting on the tooth from an orthodontic appliance. It is equal and opposite to the activation force system. See also Activa-tion force system.Derived (secondary) tooth movement Combined pri-mary tooth movement (rotation and translation). Con-trolled tipping and root movement are derived or second-ary tooth movements. See also Primary tooth movement.Edgewise A method of placing rectangular wire in which there is a larger faciolingual dimension than occlusogingi-val dimension. See also Ribbon wise.Elastic limit The initial linear part of the force-deection (F/Δ) curve or stress-strain (σ/ε) curve. Within the elastic limit, the wire can be restored to its original shape by un-loading the force.Elgiloy See Cobalt-chromium alloy.Energy A physical quantity that is transferable to work. The unit is Nm. For example, elastics or coil springs are energy-storing devices that slowly release the energy to move a tooth. See also Resilience.Equivalent force system A force system that has the same effect as another force system.Fatigue The weakening or fracturing of a wire under repeated loading below the yield strength. Orthodontic wires may show fatigue fracture by repeated masticatory forces.Force (F) A physical quantity that causes a change in shape by Hooke’s law or an acceleration by Newton’s Second Law. The unit is a Newton (N) or kg·m/sec2. 1 cN = 10–2 N. A gram (g) is the unit of mass. One gram has a gravitation-al force of 0.98 cN on earth.Force diagram A diagram of a body showing all the forces acting on the body. Only the forces of interest are depicted. Also known as a free-body diagram.Force-deection (F/Δ) rate Amount of force need-ed for unit displacement of a spring. The unit is gram/ millimeter (g/mm).Force-driven appliance An appliance that produces the correct force system for dentofacial modication. See also Shape-driven appliance.Free vector A vector quantity that has only magnitude and direction, where the point of application is irrelevant. For example, a couple is a free vector. See also Bound vec-tor.Free-body diagram See Force diagram.Frictional force (Ff) A force resisting the motion of an object. Ff = µN.Functional axis of a tooth The axis of the tooth that is determined by its behavior under loading. It is different than the anatomical long axis of a tooth.Gram (g) A unit of mass.Isotropic When a material shows the same physical prop-erties regardless of the direction. Most alloys are isotropic. See also Anisotropic.Lever arm An extension of wire attached to the bracket or soldered to the arch. It replaces the point of force appli-cation. Power arm is misused terminology in orthodontics because power is a unit of work (or energy) per unit time. Watt (W) is the unit of power.Line of action An imaginary extended line of a vector. For example, it is the line through the point at which force is applied and along the direction in which force is ap-plied. See also Transmissibility, law of. Mass A physical property of a body that is determined by the resistance to acceleration or gravitational force. The unit is kilograms (kg). 1 kg = 103 g.Mechanical tooth displacement The displacement of a tooth within the periodontal ligament (PDL) space due to mechanical compression, tension, or shearing of the PDL. No biologic response is related. See also Biologic tooth movement. Glossary513M/F ratio A ratio between moment and force applied at a dened point. Common ratios are measured to a point at the bracket or center of resistance. The M/F ratio rep-resents the distance from the dened point to an equiva-lent single force. The unit is length measured in millime-ters.Modulus of elasticity (or Young modulus) The ratio of stress to strain in the elastic range. It is the inherent physical property of a material. The unit is Pascal (Pa) or N/m2. One gigapascal = 109 Pa. For example, acrylic has a modulus of elasticity of 3.2 GPa, and stainless steel has a modulus of elasticity of 180 to 200 GPa.Modulus of rigidity The ratio of shear stress to angu-lar deformation in the elastic range. See also Torque and Torsion. Moment (M) A physical quantity to produce a turning or rotation of the body. The magnitude of the moment is measured by the product of the force times the perpendic-ular distance from the line of action of that force to the reference point; thus, M = F × D, where M is the moment, F is the magnitude of force, and D is the perpendicular distance from the line of action of the force to the point being considered. The unit is Nm. In orthodontics, gmm is commonly used. 1 gmm = 0.98 cNmm. See also Couple.Moment of inertia (I) A geometric property of a body that relates the moment needed for a desired amount of bending. It depends on the cross-sectional size and shape of the wire. I = bh312 for rectangular wire and I = ∏d4 64 for round wire.Motion displacement diagram A diagram with a dot-ted arrow in this book. It schematically represents the direction of rotation and movement. It does not specify rotation center or direction of forces or couples.Neutral position If the residual moments are placed on a spring, the neutral position is dened as the shape of the spring when only the moment is applied and the horizon-tal force is zero.Newton (N) A unit of force. 1 N = 100 cN = 1 kg·m/sec2.Nickel-titanium An alloy of nickel and titanium that en-ables transformation of its crystalline structure from aus-tenite to martensite and vice versa. It shows superelastici-ty and shape-memory effects. Work-hardened martensite has high springback but shows no superelasticity.Normal force A component of force acting perpendicu-lar to the contact surface.Pascal (Pa) A unit of pressure. 1 Pa = 1 N/m2Passive shape The shape of a wire or a spring that pro-duces no forces when placed in brackets. See also Activat-ed shape, Deactivated shape, and Simulated shape.Preactivation bends The nal or additional bends placed into a wire or a spring conguration to produce a residual moment or other forces.Primary tooth movement Translation and pure ro-tation of the tooth. See also Derived (secondary) tooth movement.Pure rotation A rotation of a tooth with center of rota-tion at the center of resistance. It occurs when a couple is applied.Range of action The amount of wire deection without permanent deection. See also Strength.Reactive unit The part of an arch that is directly involved with those teeth that are not to be displaced. Active and reactive units are distinguished by clinical purpose only and indistinguishable biomechanically by Newton’s First Law. Synonym for anchorage unit. See also Active unit.Residual moment The additional moment added to a spring by preactivation bends. See also Activation moment.Resilience A physical property of the material that rep-resents its ability to store energy within the elastic range.Resultant A single force that can replace multiple forces and moments to produce the same external effect on the body.Ribbon wise A method of placing rectangular wire in which there is a larger occlusogingival dimension than fa-ciolingual dimension. See also Edgewise.Rigid body A body that does not change in shape or size under the action of forces.Root movement A rotation of a tooth in the facial view with the center of rotation near the crown. Rowboat effect When a couple (in the facial view) is ap-plied to a tooth, the crown moves in the opposite direc-tion from the root. The root is analogous to the oar of the boat. The oar moves back while the boat moves forward. For example, during canine root movement, the posterior segments move forward. This is not a scientic term relat-ing to anchorage loss. Section modulus A geometric property for a given cross section used in the design of beams or wires undergoing exure. It is calculated by dividing the moment of inertia by the distance from the neutral axis to the most distant outer surface of a wire (I/c). It is directly related to the maximum bending moment of a spring or wire within the elastic range.Shape memory The phenomenon of remembering a wire’s original shape. Nickel-titanium alloys have two phases (martensite and austenite) with different physical Glossary514properties. Phase transition occurs from one phase to an-other by change of stress or temperature. In the marten-sitic phase, nickel-titanium permanently deforms easily. If temperature increases, the phase changes to austenite and the original shape is restored. See also Superelasticity.Shape-driven appliance An appliance made with a shape that is passive after teeth move to that shape. The force system is not always correct. See also Force-driven appliance.Simulated shape The elastically deformed shape of a passive wire or a spring when a desired deactivation force system is applied. See also Activated shape, Deactivated shape, and Passive shape.Stabilizing segment A group of teeth that is rigidly con-nected by a high-stiffness wire or polymer.Stainless steel A steel alloy with a minimum of 10.5% chromium content. It is corrosion and rust resistant and the stiffest alloy among orthodontic wires. Its modulus of elasticity is 180 to 200 GPa.Static equilibrium The condition of a body that exists when all forces and moments acting on it sum to zero, governed by Newton’s First Law. See also Activation force system.Statically determinate A structure or an appliance is statically determinate when static equilibrium equations are sufcient to solve for unknowns. For example, elastic chain and cantilever springs are statically determinate if the horizontal force is measured by a force gauge. See also Statically indeterminate.Statically indeterminate A structure or an appliance is statically indeterminate when static equilibrium equations are not sufcient to solve for unknowns. For example, space-closing loops (springs) are statically indeterminate if only horizontal forces are known. See also Statically de-terminate.Statics The branch of physics (mechanics) that deals with the forces when bodies are in a state of rest or at a con-stant velocity.Stiffness Synonym for force-deection rate. Stiffness is the strength per range of action.Strain (ε) The ratio of the amount of elongation to the original length. The unit is m/m or dimensionless.Strength The maximum amount of force (or moment) that the appliance can produce without permanent defor-mation. See also Range of action.Stress (σ) The amount of force per unit area. The unit is Pascal (N/m2).Superelasticity During unloading, the stress-strain curve is not linear and hysteresis is present. During deactivation, a plateau of constant force can be observed.Torque Synonym for couple or moment. In orthodontics, torque is produced by torsion of a wire. Incorrect deni-tion: a change of axial inclination or third-order bracket slot angle. Incorrect example: The prescription has –10 de-grees of “torque” in the maxillary central incisor bracket.Torsion The twisting of a wire produced by a torque op-erating around a structural axis. See also Bending.Toughness Total area under a force versus deection curve up to fracture.Translation A type of movement by which all points on a body move the same amount in a parallel direction. Syn-onym for bodily movement in clinical orthodontics.Transmissibility, law of A force vector can be replaced by any force as long as it is along the same line of action. For example, distal force on a canine is applied either by pulling from the distal or pushing from the mesial. See also Line of action.Trial activation The application outside the mouth of all moments and forces used to activate a spring. It mimics the activation of a spring during insertion in the mouth.Twist Synonym for torsion.Uncontrolled tipping A rotation of a tooth in the facial view with the center of rotation located near the center of the root. It literally means that tipping is not controlled because some part of the tooth moved in a direction op-posite to the applied force. Single forces acting at the crowns of the teeth, such as the tongue, cheek force, or a force from a simple nger spring of a removable appli-ance, produce uncontrolled tipping.Work A special form of energy. Also known as mechan-ical work or mechanical energy. The unit is Nm. The area under the force-deection (stress-strain) curve. The area under the loading curve is the work done to the wire during activation, and the area under the unloading curve is the work done to the tooth. It is conserved in the elastic range but partly dissipates as a result of the internal fric-tion and heat in the plastic region. The unit is the same as the unit for a moment, but work is a scalar and moment is a vector. See also Energy.Yield strength (or point) Theoretically, it is the stress at which wire starts to deform plastically. The wire is elastic under this point. Technically, this point is difcult to de-termine; therefore, a line is drawn parallel to the straight-line portion of the stress-strain curve, and an offset from it in an amount equal to 0.1 or 0.2 or 1% unit strain is used. 515Solutions toProblems515As described in the text, an important property of a force or moment is direction. The convention for moments in this solutions section is + for clockwise and – for coun-terclockwise. For simpli cation, no sign is usually speci ed in the text or  gures for force direction. Force direction is de ned by the force diagram alone unless required for emphasis. 516SOLUTIONS22.300 g – 100 g = 200 g (distal direction)3. 300 g + 100 g = 400 g (on the same line of action)1. They are all the same because they share the same line of action (red dotted line).4. Horizontal components:From lingual arch: 100 g (1)From crisscross elastics: 300 g cos 60° = 150 g (2)∑FH = (1) + (2) = 100 g + 150 g = 250 gVertical components:From lingual arch: 0 g (3)From crisscross elastic: 300 g sin 60° = 300 g × 0.87 = 260g (4)FR = FH2 + FV2 = 2502 + 2602 = 360 gθ = tan–1 260250 = 46°∑FV = (3) + (4) = 0 g + 260 g = 260 g 5172SOLUTIONS6a.FH = 150 g cos 20° = 150 g × 0.94 = 141 g (horizontal component)FV = 150 g sin 20° = 150 g × 0.34 = 51 g (vertical component)6b.FH = 150 g cos 45° = 150 g × 0.71 = 107 g (horizontal component)FV = 150 g sin 45° = 150 g × 0.71 = 107 g (vertical component)7.Fbuccolingual = 100 g cos 20° = 100 g × 0.94 = 94 g (lingual component)Fmesiodistal = 100 g sin 20° = 100 g × 0.34 = 34 g (mesial component)5a.FH = 100 g sin 60° = 100 g × 0.87 = 87 g (lingual component)FV = 100 g cos 60° = 100 g × 0.5 = 50 g (intrusive component)5b.FH = 100 g sin 45° = 100 g × 0.71 = 71 g (lingual component)FV = 100 g cos 45° = 100 g × 0.71 = 71 g (intrusive component)5c.FH = 100 g cos 20° = 100 g × 0.94 = 94 g (lingual component)FV = 100 g sin 20° = 100 g × 0.34 = 34 g (extrusive component) 518SOLUTIONS29. To have the resultant force lie along the archwire, the sum of all vertical components should be zero.FV (headgear) = F (headgear) sin θFV (elastic) = 200 g sin 25° = 200 g × 0.42 = 84.5 gFV (headgear) = FV (elastic)Therefore,FV (headgear) sin θ = 84.5 gθ = sin–1 84.5Fheadgeara. If Fheadgear = 200 g, θ = sin–184.5200= 25°b. If Fheadgear = 600 g, θ = sin–1 84.5600 = 8°c. If Fheadgear = 1,000 g, θ = sin–1 84.5 1,000= 4.8°8.FH (headgear) = 400 g cos 45° = 400 g × 0.71 = 284 gFV (headgear) = 400 g sin 45° = 400 g × 0.71 = 284gFH (elastic) = 200 g cos 35° = 200 g × 0.82 = 164 gFV (elastic) = 200 g sin 35° = 200 g × 0.57 = 115 gTherefore,∑FH = FH (headgear) + FH (elastic) = 284 g + 164 g = 448 g∑FV = FV (headgear) + FV (elastic) = 284 g – 115 g = 169gFR = FRH2 + FRV2 = 4482 + 1692 g = 479 gAnd θ = tan–1 169448 = 21° 519SOLUTIONS3In the following problems, the principle of equivalence is used for a solution. The green force system and red force system are equivalent. The arbitrary red dot is used for calculation of moments.1.∑Fgreen = ∑Fred (1)Fred = 100 g∑Mgreen = ∑Mred (2)100 g × 6 mm = 100 g × 0 mm + MredMred = 600 gmmLingual orthodontics requires the same force; however, an additional clockwise moment is needed.2.∑Fgreen = ∑Fred (1)Fred = 100 g∑Mgreen = ∑Mred (2)100 g × 7 mm + (–400 gmm) = 100 g × 0 mm + MredMred = 300 gmmNote that with lingual orthodontics, the direction of the moment is opposite to the moment applied on the buccal.3.∑Mgreen = ∑Mred (1)300 g × 4 mm = FA × 8 mm + FB × 0 mmFA = 150 g∑Fred = ∑Fgreen (2)300 g = FA + FB300 g = 150 g + FBFB = 150 gWhen a single force is difficult to apply, multiple forces can be used instead. 520SOLUTIONS34.∑Mgreen = ∑Mred (1)300 g × 2 mm = FA × 8 mm + FB × 0 mmFA = 75 g∑Fgreen = ∑Fred (2)300 g = FA + FB300 g = 75 g + FBFB = 225 g5.∑Mgreen = ∑Mred (1)200 g × 15 mm = FA × 0 mm + FB × 40 mmFB = 75 g∑Fgreen = ∑Fred (2)200 g = FA + 75 gFA = 125 g6.∑(F1 + F2) = R (1)R = 100 g + 100 gR = 200 g∑M = Rd (2)100 g × 30 mm + 0 = 200 g × d3,000 gmm / 200 g = dd = 15 mmd must be 15 mm distal and not mesial to the dot for the moment to be in the correct direction.Multiple forces can be replaced with a single force. 521SOLUTIONS37.Step 1. Resolve the forces into horizontal and vertical components and then sum them.∑Fgreen = ∑FredHorizontal components:FH1 = 100 g cos 70° = 34 gFH2 = 40 g cos 35° = 33 gFH1 + FH2 = FH = 67 gVertical components:100 g sin 70° + 40 g sin 35° = 117 gStep 2. Find the resultant of the summed horizontal and vertical forces.R = 67 g2 + 117 g2 = 135 gtan θ = 11767θ = 60°Step 3. Calculate the point of force application (d).∑Mgreen = ∑MredSum the moments from the vertical components only (horizontal components are ignored because no moments are produced at the red dot).100 g sin 70° × 0 mm + 40 g sin 35° × 30 mm = 117 g × d mmd = 5.9 mm8.∑Fgreen = ∑Fred (1)Fred = 100 g∑Mgreen = ∑Mred (2)–100 g × 4 mm + 1,000 gmm = 100 g × 0 mm + MredMred = 600 gmmLess lingual root torque is needed if the bracket is placed further apically, but the amount of reduction is small.9.∑Fgreen = ∑Fred (1)Fred = 300 g∑Mgreen = ∑Mred (2)300 g × 0 mm = –300 g × 5 mm + MredMred = 1,500 gmmPlacing the force and moment at the bracket is the same as placing the force apically on the lever; however, friction from the couple could change the M/F ratio at the bracket with sliding mechanics. 522SOLUTIONS310a.∑Fgreen = ∑Fred (1)Fred = 200 g∑Mgreen = ∑Mred (2)200 g × 0 mm = –200 g × 5 mm + MredMred = 1,000 gmm10b.∑Fgreen = ∑Fred (1)Fred = 200 g∑Mgreen = ∑Mred (2)–200 g × 10 mm = –200 g × 5 mm + MredMred = –1,000 gmmThe mesial force on the molar is the same in a and b. The rotation moment is opposite if the force is placed on thelingual.11a.∑Fgreen = ∑Fred (1)Fred = 100 g∑Mgreen = ∑Mred (2)100 g × 0 mm = –100 g × 10 mm + MredMred = 1,000 gmm11b.∑Fgreen = ∑Fred (1)Fred = 100 g∑Mgreen = ∑Mred (2)100 g × 0 mm = –100 g × 23 mm + MredMred = 2,300 gmm 523SOLUTIONS311c.∑Fgreen = ∑Fred (1)Fred = 100 g∑Mgreen = ∑Mred (2)100 g × 0 mm = –100 g × 33 mm + MredMred = 3,300 gmmThe extrusion force remains the same; however, the tip-back action increases as the lever arm becomes longer.12.∑Fgreen = ∑Fred (1)Fred = 80 g∑Mgreen = ∑Mred (2)80 g × 0 mm = 80 g × 4 mm + MredMred = –320 gmmA force placed at the labial bracket (A) translates the incisor to the left side with rotation in a counterclockwise direction. A force placed on the lingual attachment (B) at the CR translates the incisor to the left without rotation. 524SOLUTIONS41. No, there are no differences. Forces FA, FB, and FC are the same because they share the same line of action (law of transmissibility).2. Yes, they are different forces and create different effects. Equivalent force systems at the CR are shown.3. The molar will translate backward and rotate clockwise.5. The molar will translate backward and upward and rotate counterclockwise.4. The molar will translate backward and downward and rotateclockwise.6. The molar will translate backward and downward. 525SOLUTIONS413. The maxillary arch will translate backward and downward and rotate counterclockwise.7. The molar will translate backward and upward and rotate counterclockwise.8. The molar will translate backward and rotate counterclockwise.9.FR is the replaced with an equivalent single force by the graphic method. The molar will translate backward and upward. The line of action will coincide with the force vector FR. If θ1 and θ2 are equal and θ1 = 20°, the magnitude of the resultant will be 2× 500 g cos 20° = 940 g.11. The line of action of the force should pass through the CR. No, it is not possible to translate the maxillary dentition parallel to the occlusal plane.10.The effects are the same. Both forces FA and FB will translate the molar backward and rotate it counterclockwise.12. The maxillary arch will translate backward and upward and rotate counterclockwise. 526SOLUTIONS51.∑(F1 + F2) = R (1)R = 80 g + 100 gR = 180 g∑M = Rd (2)80 g × 0 mm + 100 g × 27 mm = 180 g × d2,700 gmm / 180 g = dd = 15 mm2a.Either maxillary or mandibular forces can be used. We will use the mandibular forces.∑(F1 + F2) = R (1)R = 100 g + 100 gR = 200 g∑M = Rd (2)100 g × 0 mm – 100 g × 22 mm = –200 g × d–2,200 gmm / –200 g = dd = 11 mmThe maxillary elastic force resultant lies on the same line of action as the mandibular resultant, with the same magnitude and opposite direction.2b.∑(F1 + F2) = R (1)R = 150 g + 100 gR = 250 g∑M = Rd (2)150 g × 0 mm – 100 g × 22 mm = –250 g × d–2,200 gmm / –250 g = dd = 8.8 mm 527SOLUTIONS54.In the lateral view:∑(F1 + F2) = R (1)R = 20 g + 80 gR = 100 g∑M = Rd (2)20 g × 0 mm + 80 g × 14 mm = 100 g × d1,120 gmm / 100 g = dd = 11.2 mmcontinuedIn the frontal view:∑(F1 + F2) = R (1)R = 80 g + 20 gR = 100 g∑M = Rd (2)80 g × 0 mm + 20 g × 40 mm = 100 g × d800 gmm / 100 g = dd = 8 mm3.∑(F1 + F2) = R (1)R = 100 g + 100 gR = 200 g∑M = Rd (2)100 g × 0 mm + 100 g × 36 mm = 200 g × d3,600 gmm / 200 g = dd = 18 mmThe resultant would be 200 g downward, located 18 mm from either of the green 100-g forces. Clinically, it is not possible to apply the single resultant force away from the arch, so two equivalent elastics must be used. 528SOLUTIONS55. 100-g force and 2,000-gmm moment (clockwise)6. 100-g force and –3,000-gmm moment (counterclockwise)7. 100-g force and –1,000-gmm moment (counterclockwise)As you can see from solutions 5 to 7, unilateral maxillo-mandibular Class II elastics provide not only mesiodistal force unilaterally but also various three-dimensional side effects. These side effects are undesirable most of the time, but not always.(4. cont)It is not necessary to identify the location of the CR to solve the problem. However, as seen in the occlusal view, the three-dimensional relationship of the location of resultant force (R) and the estimated CR (black circle) helps to predict the arch movement. The resultant force (red) is located anterior and to the right of the CR; therefore, the maxillary occlusal plane will steepen, and the right side will cant downward. 529SOLUTIONS58. First, analyze the occlusal view. There could be many possibilities; however, with the given dimensions, the two vertical forces may lie on the intersection of the archwire and a line connecting the left and right first premolars.Second, analyze the frontal view. Suppose the unknown forces at each first premolar are FA and FB.R = ∑F (1)100 g = FA + FBR × d = ∑M (2)The red dot was selected for convenience to sum the moments.100 g × 30 mm = FB × 50 mmFB = 60 gFA = 40 g9. At the CR of the maxillary arch, a 100-g distal occlusal force and 2,000-gmm clockwise moment are applied. At the CR of the mandibular arch, a 100-g mesial occlusal force and 1,500-gmm clockwise moment are applied.Both arches will rotate synchronously.10. At the CR of the maxillary arch, a 100-g distal occlusal force and 3,000-gmm clockwise moment are applied. At the CR of the mandibular arch, a 100-g mesial occlusal force is appliedonly.The maxillary and mandibular arches will rotate asynchronously. The maxillary arch will translate and rotate, while the mandibular arch will translate only. 530SOLUTIONS513. Any Class II elastic with a line of action passed in front of the maxillary and mandibular CRs will close the bite. One of the possible elastics shows that rotation of the mandibular arch can be prevented if the line of action (dotted line) passes through the CR of the arch.15. Synchronous occlusal plane canting of both arches is indicated. An equal distance from the line of action to the maxillary and mandibular CRs provides the same amount of moment.14. The elastic off-center to the right side produces only a moment to the maxillary arch. Rotation of the mandibular arch can be prevented if the line of action (dotted line) passes through the CR of thearch.11. At the CR of the maxillary arch, a 100-g leftward occlusal force and 3,000-gmm counterclockwise moment are applied. At the CR of the mandibular arch, a 100-g rightward occlusal force is applied only.The maxillary and mandibular arches will rotate asynchronously. The maxillary midline will move to the left side. The maxillary occlusal plane will cant right side downward and left side upward. An open bite will form laterally on the left side.12. At the CR of the maxillary arch, a 100-g leftward occlusal force and 2,000-gmm counterclockwise moment are applied. At the CR of mandibular arch, a 100-g rightward occlusal force and 2,000-gmm counterclockwise moment are applied.Both arches will rotate synchronously. 531SOLUTIONS6For solutions 1 to 4, only a single force is applied at the anterior end of the intrusion spring because it is a point contact (all incisor forces are known here). Therefore, only the laws of equivalence are needed to answer the questions. 1. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 7 mm = –210 gmm/sideThe incisor moment is –210 gmm/side. The incisors will tip to the labial.2. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 12 mm = –360 gmm/sideThe incisor moment is –360 gmm/side.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2The posterior moment is +900 gmm/side. This is conservative of anchorage; however, the posterior segment tends to rotate the crown backward and the roots forward.Determine the equivalent force system at the posterior CR. According to Newton’s Third Law, the reaction to the intrusive force (30 g, green arrow) is an equal and opposite 30-g extrusive force at the anterior segment. This can be replaced at the posterior CR with an extrusive 30-g force because of the laws of equivalence.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 30 mm = +900 gmm/sidecontinued continued 532SOLUTIONS63. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 7 mm = –210 gmm/sideThe incisor moment is –210 gmm/side.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 25 mm = +750 gmm/sideThe posterior moment is +750 gmm/side. The posterior segment will tend to rotate less than in problem 1 not only because the number of teeth in the posterior segment increased but also because the moment arm length was reduced to anteriorly relocate the posterior CR.(2. cont)Sum the moments around the arbitrary point at the posterior CR.30 g × 35 mm = +1,050 gmm/sideThe posterior moment is +1,050 gmm/side. The incisor and posterior segments will tend to rotate more than in problem 1. 533SOLUTIONS64. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 7 mm = –210 gmm/sideThe incisor moment is –210 gmm/side.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 20 mm = +600 gmm/sideThe posterior moment is +600 gmm/side. The number of teeth in the posterior segment is less than that in problem 3; although the posterior segment feels less moment, it does not necessarily mean it rotates less, because with fewer teeth stress can be increased in the periodontal ligament.5. First, determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 35 mm = +1,050 gmm/sideA posterior moment of +1,050 gmm/side is developed; therefore, the headgear should provide –1,050 gmm to prevent canting of the occlusal plane.Determine the sense and magnitude of the headgear force.The magnitude of FHG required is 210 g (= 1,050 gmm / 5 mm). The direction should be upward and backward so that the moment is counterclockwise.The magnitude of the headgear force is for exact balance. Clinically, if headgear wearing time is reduced, the posterior moment can be increased by increasing the headgear force and/or the distance. 534SOLUTIONS66. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 2 mm = –60 gmm/sideThe incisor moment is –60 gmm/side. The incisors will tip to the labial very little.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posterior CR.30 g × 22 mm = +660 gmm/sideThe posterior moment is +660 gmm/side.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 40 mm = +1,200 gmm/sideThe posterior moment is +1,200 gmm/side.The effect at the anterior segment will be same as in problem 1. The moment acting at the posterior segment is not only greater than the one in the problem 1, but it also acts on a single tooth. Clinically, a moment of 1,200 gmm is a recommended magnitude for tipping a single molar distally.7. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 7 mm = –210 gmm/sideThe incisor moment is –210 gmm/side. 535SOLUTIONS68. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 5 mm = –150 gmm/sideThe incisor moment is –150 gmm/side.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 28 mm = +840 gmm/sideThe posterior moment is +840 gmm/side.9. First, determine the equivalent force system at the anterior (incisor) CR.The intrusive force is 30 g/side. The incisors will translate apically at 90 degrees to the occlusal plane.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 10 mm = +300 gmm/sideThe posterior moment is +300 gmm/side.continued 536SOLUTIONS6Determine the equivalent force system at the anterior (incisor)CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 5 mm = –150 gmm/sideThe incisor moment is –150 gmm/side.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 15 mm = +450 gmm/sideThe posterior moment is +450 gmm/side.Determine the equivalent force system at the anterior (incisor)CR.∑F1 = ∑F2The intrusive force is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.30 g × 8 mm = –240 gmm/sideThe incisor moment is –240 gmm/side.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 30 g/side.∑M1 = ∑M2Sum the moments around the arbitrary point at the posteriorCR.30 g × 18 mm = +540 gmm/sideThe posterior moment is +540 gmm/side.(9. cont)continued 537SOLUTIONS610. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 60 g off-center.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.60 g × 10 mm = –600 gmmThe incisor moment is –600 gmm. The incisor segment will intrude and rotate counterclockwise. The left side will moveapically.Determine the equivalent force system at the posterior CR.∑F1 = ∑F2The extrusive force on the posterior segment is 60 g.∑M1 = ∑M2Sum the moments around the arbitrary point at the posterior CR.60 g × 10 mm = +600 gmmThe posterior moment is +600 gmm. The posterior segment tends to extrude and rotates clockwise (left side occlusally); however, its effect is minimal because the segments are rigidly connected by a lingual arch, and occlusal chewing forces mayhelp.11. First, determine the equivalent force system at the anterior (incisor) CR.∑F1 = ∑F2The intrusive force is 60 g off-center.∑M1 = ∑M2Sum the moments around the arbitrary point at the anterior CR.60 g × 10 mm = –600 gmmThe incisor moment is –600 gmm. The result is the same as in problem 10. The incisor segment will intrude and rotate counterclockwise.Determine the equivalent force system at the left posterior CR. Assume the intrusion spring is flexible enough that it allows complete freedom of individual movement of each posteriorsegment.∑F1 = ∑F2The extrusive force on the left posterior segment is 30 g.∑M1 = ∑M2Sum the moments around the arbitrary point at the left posterior CR.30 g × 10 mm = –300 gmmThe left posterior moment is –300 gmm.continued 538SOLUTIONS6Determine the equivalent force system at the right posterior CR.∑F1 = ∑F2The extrusive force on the right posterior segment is 30 g.∑M1 = M2Sum the moments around the arbitrary point at the right posterior CR.30 g × 30 mm = 900 gmmThe right posterior moment is 900 gmm. Note that buccal segments could tip to the buccal, particularly on the right side.(11. cont)12.FV = 30 g cos 60° = 30 g × 0.5 = 15 gFH = 30 g sin 60° = 30 g × 0.87 = 26 gAt the anterior segment:F = 30 gM = 150 gmmThe flared incisors will intrude along the long axis and tip slightly to the lingual.At the posterior segment:F = 30 gThe posterior segment will translate forward and downward; however, 30 g of force is probably too small to translate the whole posterior segment forward. 539SOLUTIONS71. A 100-g force and a –2,000-gmm couple at the posterier CR are equivalent to a 100-g extrusive force at the anterior hook. Note that the intrusive force on the incisors, which is not shown, is equal and opposite (Newton's Third Law).3a. FA × 20 mm = 2,000 gmm (counterclockwise).FA = 100 g3b.FB × 10 mm = 2,000 gmm (counterclockwise).FB = 200 g2. Equivalence between the red and green force systems.If the posterior moment is maintained at –2,000 gmm, the magnitude of the extrusive force at the posterior segment will be greater than in problem 1.The choice between two appliances is based on the magnitude of the vertical forces. If extrusion of the molar and intrusion of the anterior segment are not indicated, a is better. If extrusion of the molar or intrusion of the anterior segment is indicated, b is better. 540SOLUTIONS74a. The anterior segment will be intruded by translation because the force acts at the CR.4b. The anterior intrusive force of 50 g acting at the hook is replaced with an equivalent force system at the CR of the anterior segment. The anterior segment will be intruded by 50 g and will rotate clockwise from the 200-gmm moment. The anterior segment will be intruded less than in a because the number of teeth was increased in the anterior segment. However, the effect on the posterior segment is the same in a and b.5. The 150-g extrusive force acting on the mandibular arch is replaced with the equivalent force system at the CR of the mandibular arch. The mandibular arch extrudes and rotates clockwise. A steepened mandibular occlusal plane helps to correct the anterior deep bite.6. The equivalent force system at the CR is depicted in red arrows. The moment on the mandibular arch (2,000 gmm) is much greater than the moment on the maxillary posterior segment (–700 gmm). It is desirable that the maxillary right posterior segment will extrude more than the mandibular arch, but counterclockwise rotation of the segment is undesirable. It does not necessarily mean that the mandibular arch is displaced more, because more teeth comprise the stabilizingunit. 541SOLUTIONS77. The equivalent force system at the CR of the maxillary arch.The equivalent force system at the CR of the mandibular posterior segment by the Class II elastic.The resultant force system on the mandibular posterior segment is the sum of the force systems from the maxillomandibular elastic and tip-back spring. The moment is (–3,500 gmm) + 1,500 gmm = –2,000 gmm. The resultant force would be the vector sum of the two 100-g forces: 185 g in a 67.5° upward and forward direction.8. The resultant force system at each tooth is summed. The moments are not uniform on each tooth, even though uniform curvature is placed on the archwire. Moments at the terminal teeth—the second molar and the canine—are greater than at the teeth in between. There is not usually a clinical requirement for a force system produced by a wire shaped in this way.continuedThe equivalent force system at the CR of the mandibular posterior segment by the intrusion spring.The equivalent force system at the CR of the mandibular anterior segment by the tip-back spring. 542SOLUTIONS81.Step 1. The solution to the problem is recognized as an equilibrium problem where the spring is placed in equilibrium. The given 30-g force is a deactivation force, so the direction is reversed to obtain the blue activation force. The spring is placed in equilibrium with unknown moment (M) and force (F) in the correct (activation) direction to satisfy:∑F = 0 and ∑M = 0The equilibrium diagram shows the general solution to the problem without all force and moment magnitudes specified or yet calculated.2a.Step 1. The given 200-g force is the deactivation force, so the direction is reversed to obtain the blue activation force.Step 2. Calculate the magnitude of all unknowns. Assuming the reference point at the red dot to sum the moments,∑F = F – 200 g = 0 (1)∑M = F × 5 mm + 200 × 0 mm + M + M = 0 (2)From (1) and (2), F = 200 g and M = –500 gmmStep 3. The deactivation force system (forces acting on the lingual bracket) is in a reverse direction of the blue arrow activation force diagram, and the arrows are in red. The force system acting on each molar brackets includes a 500-gmm clockwise moment and 200-g expansive force.Step 2. Calculate the magnitude of all unknowns. ∑F = F – 30 g = 0 (1)∑M = F × 30 mm + M = 0 (2)From (1) and (2), F = 30 g and M = –900 gmmStep 3. The deactivation force system (forces acting on the tube or tooth) is in a reverse direction of the blue arrow activation force diagram, and the arrows are in red. The force system acting on the molar tube includes a 900-gmm clockwise moment and 30 g of extrusive force.In this chapter, blue arrows are the activation force system acting on the wire that is in equilibrium. The reversed-direction red arrows are the deactivation force system acting on the teeth. An arbitrary red dot is used to calculate the moments. 543SOLUTIONS82b.Step 1. The given 200-g force is the deactivation force, so the direction is reversed to obtain the blue activation force. The unknown moment is only acting on the left side.3.Step 1. The given –2,000-gmm moment is the deactivation moment, so the direction is reversed to obtain the blue activation moment. There are only mesial and distal forces on each side.Step 2. Calculate the magnitude of all unknowns. Assuming the reference point at the red dot to sum the moments,∑F = FA – FB = 0 (1)∑M = FA × 40 mm + FB × 0 mm + (–2,000 gmm) = 0 (2)From (1) and (2), FA = 50 g and FB = 50 gStep 2. Calculate the magnitude of all unknowns. Assuming the reference point at the red dot to sum the moments,∑F = F – 200 g = 0 (1)∑M = F × 5 mm + 200 × 0 mm + M = 0 (2)From (1) and (2), F = 200 g and M = –1,000 gmmStep 3. The deactivation force system (forces acting on the lingual bracket) is in a reverse direction of the blue arrow activation force diagram, and the arrows are in red. The force system acting on each molar brackets includes a 200-g expansive force and 1,000-gmm moment.continued 544SOLUTIONS8Step 3. The deactivation force system (forces acting on the lingual bracket) is in a reverse direction of the blue arrow activation force diagram, and the arrows are in red. The force system acting on the right molar bracket is a 50-g mesial force. The force system acting on the left molar bracket is a 50-g distal force and a 2,000-gmm clockwise moment.4.Step 1. The given –1,000 gmm, 2,000 gmm, and 400-g mesial force are the deactivation force system, so the directions are reversed to obtain the blue activation force system. The unknown force system includes FHA (anterior horizontal force), FVA (anterior vertical force), and FVB (posterior vertical force).Step 2. Calculate the magnitude of all unknowns. Assuming the reference point at the red dot to sum the moments,∑FH = 400-g mesial force + FHA = 0 (1)∑FV = FVA – FVB = 0 (2)∑M = 400 g × 0 mm + FHA × 0 mm + FVB × 0 mm – FVA × 10 mm + 2,000 gmm + (–1,000 gmm) = 0 (3)From (1), (2), and (3), FHA = 400 g, FVA = 200 g, and FVB = 200 gStep 3. The deactivation force system (forces acting on the brackets) is in a reverse direction of the blue arrow activation force diagram, and the arrows are in red. The force system acting on the canine bracket includes a 400-g distal force, a 200-g extrusive force, and a 2,000-gmm counterclockwisemoment.The force system acting on the premolar bracket includes a 400-g mesial force, a 200-g intrusive force, and a 1,000-gmm clockwise moment.(3. cont) 545SOLUTIONS86.Step 1. The given 2,000 gmm is the deactivation moment, so the direction is reversed to obtain the blue activation force. Because there is no couple on the left side, the unknowns are the vertical forces on each side.Step 2. Calculate the magnitude of all unknowns. Assuming the reference point at the red dot to sum the moments,∑F = FVA – FVB = 0 (1)∑M = FVA × 50 mm + FVB × 0 mm – 2,000 gmm = 0 (2)From (1) and (2), FVA = 40 g and FVB = 40 gStep 3. The deactivation force system (forces acting on the lingual bracket) is in a reverse direction of the blue arrow activation force diagram, and the arrows are in red. The force system acting on the right molar tube includes a 2,000-gmm clockwise moment and a 40-g extrusive force.The force system acting on the left molar tube is a 40-g intrusive force.The intrusive and extrusive forces are very low but may be considered side effects if unwanted.5.Step 1. The given 200-g force is the deactivation force, so the direction is reversed to obtain the blue activation force. Unknowns are the equal moments on each side with verticalforce.Step 3. The deactivation force system (forces acting on the bracket) is in a reverse direction of the blue arrow activation force diagram, and the arrows are in red. The force system acting on the canine bracket includes a 200-g extrusive force and a 1,000-gmm counterclockwise moment.The force system acting on the premolar bracket includes a 200-g intrusive force and a 1,000-gmm counterclockwisemoment.If only extrusion of the canine was planned, intrusion of the premolar and counterclockwise rotation of both canine and premolar are considered side effects.Step 2. Calculate the magnitude of all unknowns. Assuming the reference point at the red dot to sum the moments,∑F = –200 g + F = 0 (1)∑M = F × 0 mm – 200 g × 10 mm + M + M = 0 (2)From (1) and (2), F = 200 g and M = 1,000 gmm 546SOLUTIONS87. Step 1. The given 3,000 gmm, –2,400 gmm, and 200 g are the deactivation force system, so the directions are reversed to represent the activation force system in blue. The unknown force system includes FVA, FVB, and FHB.Step 2. Calculate the magnitude of all unknowns. Assuming the reference point at the red dot to sum the moments,∑FH = 200 g – FHB = 0 (1)∑FV = FVA – FVB = 0 (2)∑M = FHB × 0 mm + 200 g × 0 mm + FVB × 0 mm + (FVA × 30 mm) – 3,000 gmm + 2,400 gmm = 0 (3)From (1), (2), and (3), FVA = 20 g, FVB = 20 g, and FHB = 200g8.Step 1. The given 1,000-g force is acting on the mandible. The unknown force system includes Fcondyle and Fmastication.Step 2. Calculate the magnitude of all unknowns. Assuming the reference point at the red dot to sum the moments,∑F = Fcondyle – 1,000 g + Fmastication = 0 (1)∑M = –Fcondyle × 80 mm + 1,000 g × 30 mm + Fmastication × 0 mm = 0 (2)From (1) and (2), Fmastication = 625 g and Fcondyle = 375 gThe force system acting at the condyle is 375 g.This analysis suggests that the condyle is a weight-bearing joint. However, there are patients without condyles or a mandibular ramus that can still chew. The mandible must still be in equilibrium, so this simple analysis may be lacking.Step 3. The deactivation force system (forces acting on the bracket) is in a reverse direction of the blue arrow activation force diagram, and the arrows are in red. The force system acting on the canine bracket includes a 200-g distal force, a 20-g intrusive force, and a 2,400-gmm counterclockwisemoment.continuedThe force system acting on the molar tube includes a 200-g mesial force, a 20-g extrusive force, and a 3,000-gmm clockwise moment. 547SOLUTIONS8Step 5. Assuming the reference point at the red dot,∑FH = –300 g + FHA = 0 (1)∑FV = FVB – 50 g = 0 (2)∑M = –300 g × 0 mm + MB + 1,000 gmm + FVB × 20 mm + 50 g × 0 mm = 0 (3)From (1), (2), and (3), FHA = 300 g, FVB = 50 g, and MB = –2,000 gmmStep 6. The deactivation force system from the T-spring is depicted with red arrows.Step 7. The headgear force is acting on the molar tube.F = 500 g and M = 5,000 gmm9.Step 1. The maxillomandibular elastic is in equilibrium.Step 2.∑F = Felastic – 100 g = 0Felastic = 100 gStep 3. The deactivation force system is depicted in red (with reversed direction from the activation force system).Step 4. The T-spring is in equilibrium. The unknown force system includes MB, FVB, and FHA.continued continuedStep 8. The deactivation force system acting on the occipital bone and molar tube is depicted in red. 548SOLUTIONS8Step 10. The resultant force system acting on each attachment. The deactivation force system acting on the occipital bone by the harness is intentionally not depicted.Step 11. The replaced equivalent force system at the posterior CR. We are only interested in the force system acting at the posterior segment. The force system acting at the CR of the posterior segment includes a 1,000-gmm counterclockwise moment, a 200-g distal force, and a 50-g extrusive force.Note that the force from the maxillomandibular elastic is irrelevant to the posterior teeth because the T-spring is not a rigid body.10.Step 1. The appliance is in equilibrium by the activation force system in blue.F = 300 gStep 2. The deactivation force system is depicted in red (with reversed direction from the activation force system).Step 3. The force system is replaced at each CR. The force system acting on the CR of the maxillary molar includes a distal and intrusive 300-g force and a 2,400-gmm clockwisemoment.The force system acting on the CR of the mandibular molar includes a mesial and intrusive 300-g force and a 4,800-gmm clockwise moment.(9. cont)Step 9. The deactivation force system is depicted. 549SOLUTIONS92a. Lingual force with counterclockwise moment.2b. Labial force with clockwise moment only.2c. Clockwise moment only.1a. The replaced equivalent force system at the bracket is depicted in red (M/F = 5 mm). The CRot is at the apex, and the tooth will rotate clockwise (dotted curved arrow).1b. The replaced equivalent force system at the bracket is depicted in red (M/F = 12 mm). The CRot is at the incisal tip, and the tooth will rotate counterclockwise (dotted curved arrow).1c. The replaced equivalent force system at the bracket is depicted in red (M/F = –5 mm). The CRot is slightly above the CR, and the tooth will rotate clockwise (dotted curved arrow).1d. The replaced equivalent force system at the bracket is depicted in red (M/F = 10 mm). The CRot is at infinity, and the tooth does not rotate; only translation occurs.1e. The replaced equivalent force system at the bracket is depicted in red (M/F = 12 mm). The CRot is at the incisal tip, and the tooth will rotate clockwise (dotted curved arrow). 550SOLUTIONS13For equilibrium problems, it is necessary to solve first for forces on the lingual arch and then reverse the force direction for forces on the teeth. However, intermediate steps of these solutions are purposely omitted, and only forces on the teeth are given (red arrows) in the problem solutions.1. The force system acting at the maxillary right second molar includes a 500-gmm counterclockwise moment and a 100-g lingual force.The force system acting at the two–first molar anchorage unit includes a 1,700-gmm clockwise moment and a 100-g force in the opposite direction.2. Same as solution 1. The effect is the same regardless of stiffness of the wire as long as the line of action is the same.3.The moments given in a, b, and c are possible equilibrium solutions.a. 200 g of buccal force on the right molar with a 1,400-gmm clockwise moment at the right molar.b. 200 g of buccal force on the right molar with a 1,400-gmm clockwise moment at the left molar.c. 200 g of buccal force on the right molar with a 700-gmm clockwise moment at each molar. 551SOLUTIONS134. The relative force system is depicted in red. The vertical forces at each lingual bracket are considered side effects ifunnecessary.7.The only possible equilibrium diagram without any force is an equal and opposite 1,000-gmm tip-forward moment on the other side.8. Same as solution 7: 1,000 gmm of tip-forward moment. A couple is a free vector that is independent of location.9. A 1,000-gmm tip-forward moment at the right molar and additional 2,000-gmm clockwise moments at each bracket in the occlusal view are necessary for a transpalatal arch to be in equilibrium. A 100-g mesiodistal force and a 2,000-gmm clockwise moment at each molar are considered side effects, especially because the forces are acting in the wrong direction.10.a. One of the best deactivation force diagrams is depicted in red with equal moments acting at both molars with verticalforces.b. The horseshoe type is preferred because it exhibits dissociation for this application.5. Lingual forces with mesial-in rotation at both the molar CR and the bracket.6. The force system of solution 5 is replaced at the CR of the posterior segment. Therefore, the magnitudes and directions of the forces are the same. However, the magnitudes of the moments are decreased. They could even be reversed in direction depending on the M/F ratio at the bracket. 552SOLUTIONS141. The F/Δ rate decreases from 437 g/mm to 206 g/mm. Refer to Fig 14-25b at V = 8 mm.6. When the spring is activated, the activation moment is greater at the anterior end because more wire is distributed anteriorly at the critical section above the inflection point.7. In Fig 14-32d, 2 mm of activation will produce a M/F ratio of more than 10 mm. It will produce translation or root movement. As the spring deactivates, the M/F ratio will be increased, which could produce space opening.8. M/F ratios of approximately 7 mm at the posterior and 5 mm at the anterior tooth are expected with full activation (Δ = 6 mm). Therefore, both teeth will tip toward the extraction site with a CRot near the apex. With 3 mm of activation, M/F ratios of approximately 10 mm at the posterior and 7 mm at the anterior tooth are expected. Therefore, the posterior tooth will translate and the anterior tooth will undergo controlled tipping, and differential space closure will be produced.2. The F/Δ rate will be 874 (437 × 2) g/mm, and the M/F ratio will be the same (2.2 mm).3. The M/F ratio would be about the same because the amount of wire distribution is similar.4. The range of action is larger in spring B because the bend at the apex of the spring (critical section) is in the direction of the last permanent bend.5. Spring A will produce a greater M/F ratio and range of action and a lower F/Δ rate than spring B. 553SOLUTIONS1410. The posterior segment will tip while the anterior segment translates. This is not suitable for group A mechanics. It is suitable for group C mechanics. The red arrows are the replaced equivalent force system at the posterior CR.9. Differential space closure will not be produced because only controlled tipping is produced at both the posterior and anterior teeth. 554SOLUTIONS153. Equal and opposite vertical forces and moments are required from the bracket to hold the wire in equilibrium. The activation force system with unknown forces and moments is depicted.Conditions of static equilibrium:∑F = FA – FB = 0 (1)∑M = FB × 7 mm + MA + MB = 0 (2)The number of unknowns are too many to solve with only the conditions of equilibrium. A force gauge alone is not sufficient to measure the vertical force (FB) because the vertical force is associated with the moment. The force and the moment must be measured simultaneously with a special sensor.1.Step 1. The wire is in equilibrium. Therefore, the activation force and moment (blue arrows) are produced by the bracket of the maxillary right central incisor because the button on the left central incisor produces a single force only.Step 2. The directions of the force system are reversed to represent the deactivation force system acting on the tooth. The maxillary right central incisor will erupt and rotate clockwise.2.Step 1. The wire is in equilibrium by the activation force system. The unknown FA and MA are solved using two equilibrium conditions.Step 2. The directions of the force system are reversed to represent the deactivation force system acting on the tooth. The force system acting on the maxillary right central incisor bracket includes a 100-g extrusive force and a 700-gmm clockwise moment.The force system acting on the left central incisor bracket is a 100-g intrusive force. 555SOLUTIONS154. FB = 150 g is given.∑F = 150 g + FA = 0 (1)∑M = FB × 7 mm + MA + MB = 0 (2)From (1) and (2), FA = 150 g; however, MA and MB are still unknown. For full solution, see solutions 5 and 6.5. The Class II geometry is identified by reading the slot axis. In Class II geometry, MB/MA = 0.8. The relative activation force system is depicted.6.Step 1. As MB/MA = 0.8 in Class II geometry, the formula becomes:∑F = –150 g + FA = 0 (1)∑M = 150 g × 7 mm + MA + 0.8 MA = 0 (2)The activation force system includes FA = 150 g (extrusive), MA = –583 gmm (counterclockwise), and MB = –467 gmm (counterclockwise).Step 2. The directions are reversed for the deactivation force system at the brackets:On the right central incisor: 150-g extrusive force and 583-gmm clockwise momentOn the left central incisor: 150-g intrusive force and 467-gmm clockwise momentThe problem becomes statically determinate by identifying thegeometry. 556SOLUTIONS157.Step 1 (blue equilibrium diagram) is not used.In Class I geometry, MB = MA. The force system acting on the right tooth includes a 150-g extrusive force and a 525-gmm clockwise moment.The force system acting on the left tooth includes a 150-g intrusive force and a 525-gmm clockwise moment.9. The Z-bend produces a Class I geometry independent of the location of the bend. The force system on the left tooth includes a 150-g extrusive force and a 1,050-gmm counterclockwise moment.The force system on the right tooth includes a 150-g intrusive force and a 1,050-gmm counterclockwise moment.10. Two V-bends simulate virtual bracket repositioning. The geometry is the same as in solution 9, and therefore the force system is the same. The force system on the left tooth includes a 150-g extrusive force and a 1,050-gmm counterclockwise moment.The force system on the right tooth includes a 150-g intrusive force and a 1,050-gmm counterclockwise moment.8. The force system is the same because it is determined by the angled brackets. The length of the root and any tooth ankylosis are irrelevant for the force system. 557SOLUTIONS1511.Step 1. The geometry is Class III. The force system at the molar bracket includes a 150-g extrusive force and 4,050-gmm counterclockwise moment. The force system at the canine bracket is a 150-g intrusive force.Step 2. The replaced equivalent force system at each CR is calculated. The force system at the posterior segment CR includes a 150-g extrusive force and a 3,000-gmm counterclockwise moment.The force system at the anterior segment CR is a 150-g intrusive force.12a. The geometry is Class I because the slot axes intersect atinfinity.12b. The angles between the slots and the wires should be equal and opposite. Two bends should be placed for virtual bracket repositioning (θA = θB).13.Step 1. The force system is analyzed sequentially by two-tooth analysis. Each two-tooth relationship is a Class III geometry. The force system is depicted.Step 2. The resultant force system is calculated. The force system on the first premolar includes intrusive force and counterclockwise moment.The force system on the lateral incisor includes extrusive force and counterclockwise moment.The force system on the canine is counterclockwise moment (twice the magnitude) and no force. 558SOLUTIONS193. The magnitude of applied forces (FA) should be larger than the maximum static friction of 120 g to override the frictional force.1. The maximum static friction is 120 g (600 g × 0.2).4.FE = FA – FFEach tooth feels an effective force of 180 g (300 g – 120 g).5. 180 g (300 g – 120 g). Same as solution 4. Sliding occurs at the canine bracket only because the applied force at the molar does not override the maximum static friction force.2. An equal and opposite force of 100 g is applied at the molar. The applied force is the same as the frictional force because they are proportional until the maximum static friction force is reached. Therefore, an equal and opposite frictional force of 100 g is acting on each tooth. The canine and the molar do not feel any force.The applied force should overcome the maximum static friction to be effective.The differential frictional force does not produce a differential force system. 559SOLUTIONS196. Both the canine and the molar feel an equal and opposite force of 300 g. Sliding occurs at the molar tube only.7. The maximum static friction of the canine will be increased to 300 g (600 g × 0.5); however, both the canine and the molar feel an equal and opposite force of 300 g because sliding occurs at the molar tube only.8. The maximum static friction at the canine is 0.2 × 2 × 1,000 gmm / 4 mm = 100 g. The maximum static friction at the molar is 0.2 × 2 × 1,000 gmm / 6 mm = 66 g.166 g of applied force is needed to translate the teeth because sliding occurs at the bracket with the least friction, which is the molar tube.The sliding occurs only at the molar tube because the maximum static friction force is lower than at the canine.9. Each tooth feels an equal and opposite force of 100 g because the force is applied through the CR and no tipping occurs, so there are no normal forces on the wire. However, other normal forces can produce friction, such as the ligature tie and out-of-plane rotation. 560SOLUTIONS212. The stiffness is inversely proportional to L3. Therefore, the stiffness of the lingual to the labial is 83/43 = 8. Therefore, a 50% reduction in wire length leads to 800% the stiffness.1. According to Fig 21-5, the cross-section stiffness numbers of 0.014-inch and 0.016-inch stainless steel are 150 and 256, respectively. Therefore, 150/256 = 0.59. The 13%-reduced-diameter wire has 59% of the stiffness.This solution is based on the following formula:FΔ= K EIL3where E and L are the same because the same material (stainless steel) and interbracket distance are used. Therefore, only I, which is proportional to d4, affects the stiffness. The use of stiffness numbers simplifies the calculation for the clinician.0.0144/0.0164 = 0.59Ni-Ti wire does not follow the cross-section stiffness number ratios or the simple formulas. Force versus deflection curves are not linear, and heat treatment, composition, the amount of activation, and temperature can be important variables in forceprediction.3. The blue arrows are the activation force system for the given section. The stiffness is lowest in spring B because the helix is located at the critical section. There is little difference between springs A and C because the helix in spring C was added at the least-bending-moment area near the free end.The maximum force of the spring is directly related to the maximum bending moment of the spring. All springs have the same maximum bending moment at the fixed end. Therefore, the maximum force of all the springs is the same.The maximum deflection of the spring is highest in springB.4. The rectangular wire shows different stiffness in accordance with the direction of the bending because I is proportional to bh3 in rectangular wire.Stiffness of A/B = (0.22 × 0.163) / (0.16 × 0.223) = 0.53In ribbon wise application (A), the labiolingual stiffness is 53% that of the occlusogingival stiffness.In edgewise application (B), the occlusogingival stiffness is 53% that of the labiolingual stiffness. 561SOLUTIONS215. 50% reduction in diameter results in 6% of the stiffness (0.54 = 0.06).The maximum force is proportional to d3; therefore, the maximum force of 50%-reduced-diameter wire is 12.5% of the original (0.53 = 0.125).Reducing the diameter dramatically reduces stiffness; however, it also greatly reduces maximum force. This is why small-diameter stainless steel ligature wire is not suitable for leveling.7. The relative stiffness of a reference wire (0.004-inch) is 1. The stiffness is proportional to I ( πd464for round, bh312for rectangular wire). The relative stiffness number calculated for 0.032 × 0.032–inch square stainless steel wire is (0.0324/12) / (π 0.0044/64) = 6,954.The stiffness of 0.032 × 0.032–inch square stainless steel wire is therefore 695,400% that of 0.004-inch stainless steel wire.0.036-inch wire has a wire stiffness number of 6,561. Square 0.032 × 0.032–inch wire of the same material has similarstiffness.6. In the same configuration of the spring, the stiffness is affected by material stiffness (Ms) and cross-section stiffness (Cs) (see Fig 21-3 and Table 21-1).The relative stiffness of a 0.017 × 0.025–inch beta-titanium spring is Ms × Cs = 0.42 × 814.51 = 342.09.The relative stiffness of a 0.016 × 0.022–inch stainless steel spring is Ms × Cs = 1.0 × 597.57 = 597.57.The ratio is 342.09/597.57 = 0.57. Therefore, the stiffness of the 0.017 × 0.025–inch beta-titanium spring is 57% that of the 0.016 × 0.022–inch stainless steel spring. 563Index563Page numbers followed by “f” indicate figures; those followed by “t” indicate tables; those followed by “b” indicate boxes.AActivation force, 12f, 13, 136, 244, 341f, 508Activation force diagram, 136, 136f, 151Activation moment, 294, 295f, 302–306Active unit, 434–435, 435fAmontons-Coulomb Law, 455Anchoragebone quantity and quality effects on, 201clinical perception of, 200definition of, 5, 200fiber-reinforced composites as, 488ffriction effects on, 466–468, 467fgrowth-related changes effect on, 204–205for incisor intrusion, 110–114, 110f–114findirect, stainless steel ligatures for obtaining, 206, 206finflammatory response effects on, 200–201intraoraldevices, degrees of freedom and biomechanical basis of, 205–206differential moments used to attain differential stress, 202–203, 203fforce added to active unit, 204moment-to-force differential, 204number of teeth and segment units, 202occlusal interferences, 204occlusal interlocking, 204optimization of, 5overview of, 199parafunctional habits and, 204–205soft tissue load considerations, 204temporary devices for. See Temporary anchorage devices.in three-piece intrusion arch, 102ftip-forward, 271, 271ftotal tooth load at axis of resistance effects on, 200transpalatal arch effects on, 206variables that affectbiologic, 200–201mechanical, 200Angled bends, in spring, 296, 297fAngled elastics, resultants for, 26fAngular ramal positioning errors, 426, 427f–428fAnisotropic materials, 213–214Anisotropy, 158Anterior crisscross elastic, 72–74, 73fAnterior reverse articulation, 182fAnterior vertical elastics, 67, 68f, 83fAntirotation bends in spring, 308fAppliance ankylosis, 281Appliancesdesigning of, 5in equilibrium, 507evaluation of, 5force-deflection rates of, 95fselection of, 5shape-driven, 4simplification of, 19–20, 20fApplied force, 454, 455f, 467fArchwires. See also Wire.canine bypass, 310–312, 311fcharacteristics of, 492–493consistent force system, 345f–350f, 345–351continuous. See Continuous archwire.with curve of Spee, 473, 473fdeformation of, 473ductility of, 498effects produced by, 5in equilibrium, 140, 140fforce system delivered by, 28force-deflection rate of, 95f, 493–497, 495t–496tideal arch application of, 4inconsistent force system, 345f–350f, 345–351maximum bending moment of, 497maximum deflection of, 498maximum elastic twist of, 499tmaximum force of, 492, 497 Index564maximum torque of, 499tfor posterior intrusion, 117preformed, 241selection of, 5, 491–499stiffness of, 494f, 498, 499tthree-dimensional control using, 472torque/twist rate of, 499ttoughness of, 498triad characteristics of, 492–493types of, 471ultimate tensile strength of, 498Association, of moment and force, 250–252, 251f–252fAsymmetric applications, of lingual archunilateral expansion, 257–264, 257f–264funilateral rotation, 264–267, 264f–267funilateral tip-back and tip-forward mechanics, 267–271, 268f–271fAsymmetric headgear, 53, 53fAsynchronous Class II elastics, 66–69, 67f–69fAtherton’s patch, 416Austenitic nickel-titaniumdescription of, 483stress-strain curve of, 484fsuperelastic, 483–484transition temperature range of, 485–486Austenitic phase, 481, 485Auxiliary retraction spring, 468, 469fAuxiliary root spring, 8fAuxiliary spring attachment, to lingual arch, 235, 240fAxis of resistancedescription of, 196–198, 197ftotal tooth load at, 200BBalanced moments, 139, 139fBarycenter, 195, 195fBauschinger effect, 289–290, 290fBeam theory, 244Bendsangled, 296, 297fin continuous archwire, 355–358, 356f–358fV-, 302, 341f–344f, 341–345, 342t, 356, 357f, 471Z-, 339–341, 339f–341f, 342tBeta-titanium spring, 352, 353fBeta-titanium wirescoefficient of friction for, 455for composite cantilevers, 379force-deflection curve of, 478, 479fmechanical properties of, 482t, 482–483Bilateral constriction, 247, 248fBilateral expansion, 245–247, 246f–247f, 251fBilateral rotation, in shape-driven method, 251fBiologic variation, 6Biomechanicsdescription of, 4–5diagram of, 216fequilibrium and, 147f–152f, 147–151importance of, 9knowledge of, advantages of, 7–9Bodily movement, 167Body-centered cubic, 480, 481fBolton discrepancy, 410Bone modeling, 221–224Bone platesClass II malocclusion decompensation using, 399, 400fClass III malocclusion decompensation using, 403, 404fas temporary anchorage devices, 393fBone remodeling, 201, 221–224Bone resorption, 184, 200Bone stress field, 219Bone turnover rate, 201Boundary conditions, 140, 141f, 334–335Bracket(s)design of, 462–463force systems at, 176–180friction and, 462–463friction-free, 457–458geometries of. See Geometries.interbracket axis, 327, 327finterbracket distance, 335–336, 336fintrabracket forces, 473flingual precision, 231, 232flingual translation of, 164low-friction self-ligating, 457, 458fmisaligned, force from straight wire in, 324–326, 325f–326fnarrow, 463, 463fsingle force at, 181–182three-bracket segments, 351–355, 352f–355fthree-dimensional control using, 472tooth displacement at, 160–165virtual bracket repositioning, 356–358, 357f–358fBracket path, 183Bravais lattices, 481Buccal crown rotation, 206Buccolingual translation, 205Bypass arch, 355, 355fA Index565CCalibrated spring, 286Canine(s)activation forces on, 461axial inclination of, 352, 352fdistal movement of, 472fhigh, extrusion of, 353, 354fintrusion ofbypass arch with rectangular loop for, 112, 113fcantilever for, 112, 113fcontinuous archwire for, 358, 358fleveling of, with continuous archwire, 310, 310f, 312fretraction ofen masse, 279–281frictional force, 458–461, 459f–461f, 464fphases of, 460, 460fseparate, 307–309, 308f, 317rotation of, 472fCanine bypass archwire, 310–312, 311fCanine root spring, 310–312, 311fCanine-to-canine incisor bypass arch, 309, 309fCantilever(s)applications of, 385fbeta-titanium, 380f, 381biomechanical force generated by, 370, 371fbuccal, 377fcanine guidance using, 372fclinical uses of, 379fcombining activations on different planes, 381, 381fcomposite, 374, 375f, 379–381, 380fconfiguration of, 374–378, 375f–379fconsistent force system and, 347fcurved, 376f, 377definition of, 370design of, 371–374, 372f–373fforce system of, 370–371, 371fforce vector angulation and, 375, 377–378indications for, 370–371lingual arch used as, 377fload-deflection reduction using, 375, 375flong, 374loop added to, 377, 378fmajor configurations of, 377–378, 378f–379fmaxillary left canine rotation corrected using, 372ffrom maxillary second premolar, 266, 266fmechanics of, 378medium-length, 374minor configuration of, 375, 376f, 377molar tip-back generated by, 373fpalatal arch as, 377short, 374straight-wire, 250fsymmetric tip-back, 270, 270ftemporary anchorage device ligated to, 375fwire selection for, 374zigzag-shaped, 375, 375fCantilever springsbypass arch and, 355, 355fin continuous archwire, 282fstatically determinate retraction system, 284, 285fCartesian coordinate system, 14, 15fCenter of mass, 195Center of resistanceasymmetries and, 26definition of, 49determining of, 49force and, 49force system at, 179f, 334headgear force applied at, 46fintrusion of, 91flocation of, 169of molars, 257origin of concept of, 194–195positions of, 197rotation around, from couple, 323D asymmetric model of, 1963D symmetric model of, 1963D volumetric, 1982D projection model, 195–196Center of rotation. See also Pivot.for describing tooth movement, 165–167, 166f–167fside of rotation versus, 194–195Center of rotation constant, 179Centi-Newton, 14Centi-Newton-millimeter, 27Centroid, 195, 195f–196fCephalometric analyses, 390Cervical headgearcontinuous intrusion arch and, 125, 126fequivalence of, 42illustration of, 40flow, 42–43, 43foccipital headgear with, 47focclusal plane cant altered with, 50f, 51for translation, 43–44in type II posterior extrusion, 124–125, 125ftypical, 42, 42fChasles theorem, 195Class I geometry, 328–329, 329f, 336f, 354f, 359fClass II elasticsasynchronous, 66–69, 67f–69fforce with, 68fC Index566headgear and, 83, 83flong, 65f, 70fpurpose of, 64–65, 65fshort, 66f, 69ffor subdivision patients, 79, 79fsynchronous, 65–66, 66funilateral, 77, 77f, 79Class II geometry, 329–330, 336f, 341f, 342, 349f, 360fClass II malocclusioncase study of, 395fcasts of, 48fcharacteristics of, 399continuous intrusion arch for, 114decompensation of, using bone plates, 399, 400fdeep bite and, 114fheadgear application to, 52fincisor root movement for, 443f–444fleveling of, 108fmaxillomandibular surgery for, 429fproclination of mandibular incisors in, 399, 400fthree-piece intrusion arch for, 113, 113fClass II skeletal patterns, 391, 391f, 394fClass III geometry, 330f, 330–331, 336f, 342, 343f, 363, 364fClass III malocclusionappliances and jaw repositioning for, 406, 407f–408fcase studies of, 404f, 407f–408fdecompensation of, using bone plates, 403, 404fgrowth considerations in, 409f, 409–410management of, 409–410surgical correction of, biomechanics of, 410Class III skeletal patterns, 391Class IV geometry, 331, 331f, 335, 336f, 342, 360f, 361Class V geometry, 331f, 331–332, 333f, 336f, 342, 346, 361, 361fClass VI geometry, 331f, 331–332, 336f, 342, 356f, 362f–363f, 363Clinical studies, biomechanical approach to, 6Closed coil springs, 382, 382fClosed polygon method, 18, 19fCobalt-chromium alloys, 482, 482tCobalt-chromium wire, 478, 479fCoefficient of friction, 455, 464Coil springs, 12f, 13, 382Cold working, 485Compensation, for skeletal patterns, 391–392, 392fCompliance matrix, 213, 213fComposite cantilever, 374, 375f, 379–381, 380fComposites, in orthodontic wires, 487–488, 488fCondylar resorption, 428Condylar sag, 426, 426f–427fConsistent configuration, 254Consistent force systems, 345f–350f, 345–351Continuous archwire. See also Archwires.bends in, 355–358, 356f–358fcanine leveling with, 310f, 310–311, 312fcantilever intrusion spring in, 282fdeep bite leveling using, 124finterradicular miniscrew with, 441fnickel-titanium, 353, 354freverse curve of Spee with, 127–128, 128fvertical loop placement in, 304, 304fContinuous intrusion archactivation of, 96, 96fcase study application of, 106, 106fClass II malocclusion treated with, 114description of, 91f–92f, 91–92force system of, 93, 93fforce-deflection rates of, 95, 96fgingival tying of, 96, 97fheadgear used with, 110–111, 111f, 125incisor bracket placement of, 97f, 98reciprocal tip-back moment from, 113, 113fsecond-order side effects produced by, 96, 97fthird-order side effects produced by, 96–97, 97f–98fControlled tipping, 172f, 179, 179f, 200Coordinate systems, 14, 18Coupledefinition of, 28, 457force and, for tipping, 182–186in force diagrams, 505forces of, 28fas free vector, 28f, 169, 170fposterior crossbite correction using, 29, 30frotation produced by, 28unilateral expansion by, 260–264, 261f–264fCrisscross elastics. See Elastics, crisscross.Critical section, 249, 249f, 289Critical zone, 179, 180fCrossbitebilateral maxillary second molar buccal, 235, 236fbuccal, 338fcase study of, 29, 30fdefinition of, 6, 182, 387, 415lateral, 79, 235posterior, 29, 30fCross-section stiffness number, 480Crowding, canine retraction for, 279–280Curve of Speearchwires with, 473, 473fexaggerated, for posterior intrusion, 126–131illustration of, 127fC Index567leveling ofdescription of, 127f, 130, 131fmandibular incisor proclination for, 405fpostsurgical, 398fpresurgical, 393mandibular, excessive, 127, 127freduction of, 131reverse, 118type I posterior extrusion and, 118DDeactivated continuous intrusion arch, 91fDeactivation forcefor asymmetric unilateral expansion, 261fdescription of, 12f, 13, 136–137, 142on force diagram, 508Deactivation force diagrams, 137, 137f, 139, 145, 238, 257, 328, 509Deep biteClass II, 69f, 114fincisor intrusion for correction of. See Incisor intrusion.leveling of, 337, 338fposterior intrusion for correction of. See Posterior intrusion.pseudointrusion of, 91as symptom, 90Degrees of freedomdescription of, 160of intraoral anchorage devices, 205–206miniscrews used to modify, 206, 206fnumber of, 205–206Dental Movement Analysis, 371, 384Dentoalveolar assembly, 217Dentoalveolar complex, 214–216Derived tooth movement, 171–175, 172f–175fDifferential force, 279Differential moments, 202–203, 203fDifferential space closurecategories of, 276description of, 276force system for, 277–278group A mechanics, 276f, 276–277group B mechanics, 276f, 298f, 304group C mechanics, 276f, 306–307T-loop spring for, 299–307Directionof force, 14–16, 15f–16fof resultant, 26–27Dislocation, 479, 484, 485fDislocation-dependent metal alloys, 481–485Dissociation, of moment and force, 250–252, 251f–252fDistal rotation, 206Distalizationmolar, 445–446total-arch, 446Distraction osteogenesiscraniofacial deformities treated with, 422fdescription of, 419mandibular expansion through, 419–420, 420fmandibular lengthening through, 420–421, 421f–422fmaxillary advancement with, 423, 423fmaxillary expansion through, 415, 419Dual-wire system, 150fDynamic friction, 453EEdgewise arch, 182Edgewise bracketsarchwire tied to, 93ideal, 327fEffective force, 454Elastic chains, 382–383Elastic limit, 480Elastics. See also Maxillomandibular elastics.advantages of, 382anterior vertical, 67, 68f, 83fapplications of, 382Class II. See Class II elastics.crisscrossanterior, 72–74, 73fbilateral, 71, 72fbuccolingual, 70on continuous arch, 72description of, 14, 15fforce of, 72flingual hooks used with, 81, 81fpoint of force application of, 72fposterior, 70f–71f, 79, 80fon segments, 80, 80funilateral, 71f, 72definition of, 64disadvantages of, 382equilibrium principle applied to, 136intra-arch, 64lingual, 239, 240fredundancy of, 81–82segmental, 80f, 80–81subdivision, 77f–80f, 77–79up-and-down, 82, 82fvertical, 67, 68f, 76f, 76–78, 82E Index568woven up-and-down, 82, 82fEn masse canine retraction, 279Enamel hypoplasia, 286Enclosed polygon method, 18, 19f, 21fEnd-centered monoclinic, 481, 481fEquilibriumactivated intrusion arch in, 138applications of, 141–143archwires in, 140, 140fbiomechanics and, 147–151, 147f–152fboundary conditions, 140, 141fconcepts and formulas of, 138–139definition of, 136elastics example of, 136equivalence and, 145–147, 146fnonrigid deformed body application of, 140, 141foverview of, 135principle of, 136–139solving problems using, 141–145, 142f–145fEquilibrium diagrams, 137f, 139Equivalencecervical headgear, 42definition of, 32description of, 41equilibrium and, 145–147, 146fof force, 32–33, 33f–34f, 509force diagrams and, 509of force systems, 32, 73f, 506–507Newtonian, 32overview of, 25Equivalent force diagrams, 506–507Equivalent stress, 212Extraction therapiesindications for, 276overview of, 275posterior molar position, strategies for maintaining, 276–279space closure after. See Space closure.FFacial asymmetry. See Skeletal asymmetry.Failure of fixation, 428–430, 429f–430fFatigue, 480Fiber-reinforced composite ribbon, 123, 123fFiber-reinforced composite segment, 277Fiber-reinforced composites, 487, 488fFinger springs, 235, 236f, 240Finishing phase, excessive vertical overlap during, 130, 130fFinite element analysis, 197, 217, 378Finite element method, 215Finite element model, 217fFirst Law (Newton’s), 11–12, 12fForceactivation, 12f, 13, 136, 244, 341f, 508applications of, 19–20, 21fapplied, 454, 455f, 467fattributes of, 13f–16f, 13–16center of resistance and, 49changing position of, 173, 173fcharacteristics of, 13f–16f, 13–16components of, 16–18concurrent, 11–22continuity of, 188, 189bcouple and, for tipping, 182–186deactivation. See Deactivation force.definition of, 210description of, 4differential, 279effective, 454equivalence of, 32–33, 33f–34f, 509frictional. See Frictional forces.headgear. See Headgear.horizontal simulation of, 258f, 259intermittent, 188intrusive, 20, 20f–21flaw of transmissibility of, 16, 16f, 33mass and, 13maximum, 479, 492moment and, association and dissociation of, 250–252, 251f–252fnormal, 457f, 457–458occlusal, 465resultant, 17sense and direction of, 14–16, 15f–16fsource of, 457f, 457–458from straight wire in misaligned brackets, 324–326, 325f–326fstress and, differences between, 211tooth movement using, 5unilateral expansion by, 258–260, 258f–260fvertical simulation of, 259Force constancy, for incisor intrusion, 94–96, 95f–96fForce decay, 382Force diagramsactivation, 136, 136f, 151arrows used in, 506, 506fcouples on, 505deactivation, 137, 137f, 139, 145, 238, 257, 328, 509definition of, 136equivalent, 506–507floating forces, 505, 506fE Index569hints for developing, 505–509line of force, 509summary of, 152Force magnitudecalculation of, 142, 142fdescription of, 13–14, 13f–14f, 28for incisor intrusion, 94phenomena for reducing, 465Force systemsactive unit, 435–436at bracket, 176–180of cantilevers, 370–371, 371fat center of resistance, 334changes in, during deactivation of ideal shape, 242–243classification of, 28, 29fconsistent, 345f–350f, 345–351differential space closure, 277–278equivalence of, 32, 73f, 506–507estimation of, 436, 436fhorseshoe arch, 256, 256ffor incisor root movement, 313, 314finconsistent, 345f–350f, 345–351occipital headgear, 41foptimal, characteristics of, 186–1893D2D projections of, 35–36view of, 36ftooth movement and, 167, 171transpalatal arch, 256, 256fin unilateral tip-back and tip-forward lingual arch, 268–269, 269fForce-deflection curves, 478–479, 479fForce-deflection rate, 94, 95f, 188, 189f, 492–497, 495t–496tForce-drive orthodontics, 6, 96Force-driven methodbilateral constriction by, 248fbilateral expansion by, 247flingual arch, 243–246Four-incisor root movement, 315fFree bodies, 158–159Free vector, couple as, 28, 28f, 169, 170fFree-body diagram. See Force diagram.Frictionanatomical variation and, 465–466anchorage affected by, 466–468, 467fbenefits of, 463–464bracket design and, 462–463coefficient of, 455, 464dynamic, 453during initial alignment and finishing, 470–472, 470f–472fkinetic, 453during leveling, 470, 470fnegative aspects of, 464overriding, 464–465overview of, 453reduction of, during space closure, 468–469, 469frolling, 453static, 453summary of, 473–474torque and, 461, 461fFriction override, 464–465Frictional forcesanatomical variation and, 465–466anchorage affected by, 466–468, 467fbenefits of, 463–464calculation of, 463in canine retraction, 458–461, 459f–461f, 464fdescription of, 454–457, 454f–457fformula for, 463during initial alignment and finishing, 470–472, 470f–472freduction of, during space closure, 468–469, 469fsummary of, 473–474vibration effects on, 465Friction-free brackets, 457Frictionless springs, 469, 469fFrost’s mechanostat, 219Fulcrum, 194Full arch, headgear application to, 46–48, 47f–48fFunctional axis, 183, 184fGGeometriesboundary conditions and limitations, 334–335Class I, 328–329, 329f, 336f, 354f, 359fClass II, 329–330, 336f, 341f, 349f, 360fClass III, 330f, 330–331, 336f, 342, 343f, 363, 364fClass IV, 331, 331f, 335, 336f, 342, 360f, 361Class V, 331f, 331–332, 333f, 336f, 342, 346, 361, 361fClass VI, 331f, 331–332, 336f, 342, 356f, 362f–363f, 363classification of, 327–328as continuum, 332–335determination of, 327–328force system equivalence at center of resistance, 334straight wire used to correct, 359f, 359–360two-bracket, 359–363, 359f–363fvirtual bracket repositioning application to, 358visualization of, 335–339, 336f–338fG Index570Goshgarian-type transpalatal arch, 393fGram-millimeter, 27Grams, 13–14Gravitational acceleration, 14fGrowthin Class III patients, 409f, 409–410long-face problems and, 410–411HHeadgearasymmetric, 53, 53fcervical. See Cervical headgear.Class II elastics and, 83, 83fclassification of, 40clinical monitoring of, 51, 52fcombination, 47fcontinuous intrusion arch and, 110–111, 111f, 125, 126fdesigning of, 49f–50f, 49–51force from, 16, 16ffull arch application of, 46–48, 47f–48fincisor intrusion using, 110–111, 111finner and outer bowfrom frontal view, 54–55from lateral view, 40f–41f, 40–42, 50ffrom occlusal view, 52–54, 52f–54fintrusion arch and, for incisor intrusion, 110–111, 111fJ-hook, 40, 55, 55ffor molar translation along the occlusal plane, 45nonextraction Class II treatment use of, 46, 47f–48foccipital. See Occipital headgear.Oppenheim, 40overview of, 39points of force application for, 45fprotraction, 56f–59f, 56–57selection of, 6symmetric, 52f, 52–53Helices, 291–293, 292fHemifacial microsomia, 420, 421fHinge cap bracket, 231, 232fHooke’s law, 14, 94, 213, 213f, 334, 478, 492Horseshoe archconsistent configuration of, 254description of, 232, 250force systems delivered by, 256, 256fillustration of, 233finconsistent configuration of, 254maxillary posterior segments expanded using, 256f, 256–257transpalatal arch versus, 254Hydrostatic pressure, 212IIdeal arch, 4, 241, 241fIncisor(s). See also Maxillary incisors.diastema between, 185, 186feruption of, during sliding canine retraction, 282, 282flingual movement of, 163f–164fmandibular, proclined, 399, 400fprotrusive, 277f, 305root movement, 312–317, 313f–316f, 442, 442ftranslation of, using miniscrews, 442, 443f–444fIncisor brackets, continuous intrusion arch placement in, 97f, 98Incisor intrusionanchorage considerations for, 110f–114f, 110–114anterior, 90continuous intrusion arch. See Continuous intrusion arch.efficacy of, 90–91flaring of incisorsforce placement for avoidance of, 100illustration of, 99, 100f, 109fmandibular incisors, 99, 100fmaxillary incisors, 107fprevention of, 92–93three-piece intrusion arch for, 100–102, 101f–102ftying back the archwire for prevention of, 100, 100fforce forapplication at a point, 96f–99f, 96–99, 104fconstancy of, 94–96, 95f–96f, 104direction of, altering of, 102–107, 103f–107flingual application of, 99magnitude of, 94, 104posterior application of, 99predictability of, 98statically determinate, 99headgear for, 110–112, 111f–112fillustration of, 90finitial leveling arches, avoidance of, 108–109, 109fintrusion force system for, 92–99, 93f–98fJ-hook headgear for, 91, 112, 112fleveling and, 98, 98fmaxillary first premolars as anchorage for, 111, 112fminiscrews for, 444–445, 445fmisconceptions about, 90root resorption associated with, 90–91, 111–112three-piece intrusion arch foraltering force direction in, 102, 103f–104fanchorage control in, 102fcase study application of, 105, 113fdescription of, 100–102, 101f–102findications for, 105G Index571Incisor tippingcontrolled, 440–441, 441flingual, 437mesial, 348fminiscrews for, 440–441, 441fInconsistent configuration, 254Inconsistent force systems, 345f–350f, 345–351Indirect anchorage, using stainless steel ligatures, 206, 206fInflammatory response, 200–201Inner and outer bow headgearfrom frontal view, 54–55from lateral view, 40f–41f, 40–42, 50ffrom occlusal view, 52–54, 52f–54fInstantaneous center of rotation, 167Interbracket axis, 327, 327f, 337fInterbracket distance, 230, 293, 294f, 313, 326, 335–336, 336f, 360–361, 473Intermaxillary elastics. See Maxillomandibular elastics.Intermolar widthdescription of, 242, 243flingual arch for narrowing of, 250Interradicular miniscrews, 441f, 444Intra-arch elastics, 64Intra-arch space closure, 279–280Intrabracket forces, 473fIntrusiondefinition of, 91, 91fincisor. See Incisor intrusion.molar, 438, 438ftotal-arch, 446Intrusion archcontinuous. See Continuous intrusion arch.description of, 471, 472fthree-piece. See Three-piece intrusion arch.Intrusive force, 20, 20f–21f, 434Isotropic materials, 213JJ-hook headgeardescription of, 40, 55, 55fincisor intrusion using, 91, 112, 112fKKinetic friction, 453Kinetics, 11–12LLabial appliancelimitations of, 230–231lingual arch with, 230Labial archwire, 230Law of acceleration, 12Law of action and reaction, 12Law of inertia, 12Law of transmissibility of force, 16, 16f, 33Levelingcanine, 310ffriction during, 470, 470fincisor intrusion and, 98, 98fof occlusal planeafter surgery, 397f–398f, 397–399before surgery, 394–396, 395fat time of surgery, 396timing of, 396surgical, 396fLeveling archwires, force-deflection rates of, 95Line of force, 509Linear acceleration, 158, 158fLinear variable displacement transducer, 177, 183Linearity, 214Lingual archactive applications of, 240–246arch form preservation using, 469fassociated application of, 254–257asymmetric applications ofunilateral expansion, 257–264, 257f–264funilateral rotation, 264–267, 264f–267funilateral tip-back and tip-forward mechanics, 267–271, 268f–271fattachments, 231, 232fauxiliary springs attached using, 235, 240fbending and torsional moments at arbitrary sections along a wire, 248–250bilateral constriction, 247, 248f, 257bilateral expansion, 245–247, 246f–247f, 257bilateral rotation, 252–254, 253fas cantilever, 377fconfigurations of, 232–233, 233f, 244deactivated shape of, 242–243, 246, 248definition of, 230, 232description of, 92, 151design of, 230fdissociated application of, 254–257extraction case treated without, 234ffinger springs from, 235, 236f, 240force-driven, 243–246horseshoe arch. See Horseshoe arch.L Index572ideal shape, force system changes during deactivation of, 242–243intermolar widths narrowed using, 250with labial appliance, 230low-rigidity, 241, 242fmandibulardescription of, 232–233, 233funilateral rotation with, 264, 264funilateral tip-back and tip-forward with, 268fmaxillary, 232, 233f, 237fmolar rotation with, 252, 253focclusal planes equalized using, 269overview of, 229passive applications of, 234–240, 234f–240f, 276reciprocal anchorage with, 230, 231b, 246shape-driven, 241–243, 241f–243f, 255space closure-related side effects prevented using, 234as space maintainer, 234fstatically determinate use of, 377ftranspalatal arch. See Transpalatal arch.unilateral distal movement using, 268unilateral expansion or constrictionby couple, 260–264, 261f–264fdescription of, 257by force, 258–260, 258f–260fwire size and material for, 233, 234tLingual archwiresdescription of, 233relative stiffness of, 233, 234tLingual attachment, for separate canine retraction, 308fLingual bracket, 231, 232fLingual elastics, 239, 240fLingual force, 457Lingual hook, 81fLingual root torque, 178–179Lingual rotation, 206Lip protrusion, 305fLoadcouple, 211fdefinition of, 210Load-deflection reduction, cantilever configuration for, 375, 375fLong-faced patientsgrowth in, 410–411surgical management in, 411, 413fLoopBauschinger effect, 289–290, 290fcantilever with, 377, 378fhelices, 291–293, 292fhorizontal width of, 293, 293foff-center, 303ffor space closure, 286, 287fvertical height of, 290–291, 291fLoop space closure. See Space closure, loop.Loop spring, 289fLow cervical headgear, 42–43, 43fLow-friction self-ligating bracket, 457, 458fMMagnitudeof force. See Force magnitude.of resultant, 26–27MalocclusionClass II. See Class II malocclusion.Class III. See Class III malocclusion.segmental approach to, 129, 130fMandiblein Class III patients, 409f, 409–410expansion of, 419–420, 420flengthening of, 420–421, 421f–422fprognathism of, 404frepositioning of, 426–428, 427f–428fMandibular archarchwire with exaggerated reverse curve of Spee placed in, 128, 129fleveling of, 121type I posterior extrusion in, 118, 121Mandibular bite plate, 121, 121fMandibular lingual arches, 232–233, 233fMandibular second molarsillustration of, 356finclination of, 438Mandibular setback, 410Manipulating forcescomponents, 16f, 16–17overview of, 11resultants, 17–18, 19f, 21fMartensitic nickel-titanium, 483, 487Martensitic phase, 481, 485Mass, force and, 13Maxillaadvancement of, 423, 423frepositioning of, errors in, 424, 425fretrognathism of, 404fMaxillary arch, exaggerated curve of Spee placed in, 130Maxillary bite plate, 122fMaxillary expansionbiomechanics of, 416, 417fdistraction osteogenesis for, 415, 419osteotomy for, 415, 416f, 418, 418fsegmental, 415t, 418, 418f–419fL Index573sulcular epithelium, 415–416, 416fsurgically assisted, 415–418, 416f–417fMaxillary first premolars, as anchorage for incisor intrusion, 111, 112fMaxillary incisorscentral, 497flateral, 471, 471fMaxillary lingual arches, 232, 233f, 237fMaxillary molar translation, using cervical headgear, 43Maxillary second molar buccal crossbite, 235, 236f, 338fMaxillary second premolar, unilateral rotation with cantile-ver of, 266, 266fMaxillomandibular elastics. See also Elastics.definition of, 11, 13, 19, 20findications for, 64moment-to-force differential obtained using, 204overview of, 63posterior protraction using, 306, 307fside effects of, 82–83Maximum bending moment, 497Maximum deflection, 498Maximum elastic deflection, 492Maximum elastic deformation, 495t–496tMaximum elastic twist, 499tMaximum force, 479, 492, 497Maximum moment, 492, 497Maximum static friction force, 455Mechanicsdefinition of, 11–12kinetics, 11–12material science, 12statics, 11–12Mechanotransductionbone modeling, 221–224bone remodeling, 221–224description of, 216–217mechanical environment, 217–220mechanisms of, 217root resorption, 221–224Mesial crown rotation, 206Mesial-out rotation, 252Mesiodistal translation, 206Metal alloysbeta-titanium, 482t, 482–483body-centered cubic arrangement of, 480, 481fcobalt-chromium, 482, 482tcomposition of, 481–484crystal structure of, 480–481dislocation-dependent, 481–485end-centered monoclinic arrangement of, 481, 481fmechanical properties of, 481–484, 482tmicrostructural mechanisms of, 484–487nickel-titanium, 482t, 483phase transition–dependent, 483–487stainless steel, 481–482, 482tsuperelasticity of, 485transition temperature range of, 485–487M/F ratio. See Moment-to-force ratio.Midline deviation, 33, 34fMidline discrepancy, 29, 30f, 74, 74f–75fMiniscrews. See also Temporary anchorage devices.appliance design and application, 436characteristics of, 434description of, 206, 206fforce from, 434force vectors created by, 434fincisor intrusion using, 444–445, 445fincisor root movement using, 442, 442fincisor tipping using, 440–441, 441fincisor translation using, 442, 443f–444finterradicular, 441f, 444molar distalization using, 445–446for molar intrusion, 438, 438fmolar root movement, 440, 440f–441ffor molar uprighting, 438–439, 439foverview of, 433root spring assisted with, 441fsingle-tooth movement with, 436–437sites for, 434total-arch movement with, 436–437, 446uprighting spring with, 439, 440fMisaligned brackets, straight wire increative bends in, 355force from, 324–326, 325f–326f, 346results of, 336, 337fModel validation, 214Modulus of elasticity, 480Molar(s)bilateral rotation of, 252, 253fcenter of resistance of, 257distalization of, using miniscrews, 445–446intrusion of, 438, 438flabial archwire use of, 231frotation of, 252, 253ftipping of, using occipital headgear, 44, 44ftranslation of, using headgear, 45uprighting of, 438–439, 439fMolar root movementdistal, using occipital headgear, 44f, 44–45miniscrews for, 440, 440f–441fmoment-to-force ratios for, 440M Index574Momentactivation, 294, 295f, 302–306balanced, 139, 139fcalculation of, 27, 170definition of, 27differential, 202–203, 203fin first-order direction, 457force and, association and dissociation of, 250–252, 251f–252fformula for, 30in maxillary occlusal view, 72maximum, 492, 497maximum bending, 497overview of, 26pure, 27f, 28reciprocal tip-back, 113, 113f–114fresidual, 295, 296f, 298f, 299–302, 300f–301fMoment arm, 27Moment of force, 27, 27fMoment-to-force differential, 204Moment-to-force ratiodescription of, 169, 178, 186, 243, 293, 371for molar root movement, 440studies of, 6Moorrees mesh diagram, 390, 391fNNeutral position, 296–299, 297fNewtonian equivalence, 32Newton’s First Lawdescription of, 11–12, 12f, 40, 370equilibrium principle based on, 136Newton’s Second Law, 11–13Newton’s Third Lawdescription of, 12f, 12–13, 40, 135, 327, 455, 507illustration of, 136fNickel-titanium alloyscharacteristics of, 482t, 483martensitic, 483, 487shape memory of, 487Nickel-titanium continuous archwire, 353, 354f, 393Nickel-titanium wiressuperelastic, 486, 498transition temperature range of, 486OOccipital headgearcervical headgear with, 47fforce system from, 41ffrontal view of, 55ffor moving the molar root distally, 44f, 44–45occlusal plane cant altered with, 50f, 51, 111resultant force with, 83for tipping a molar distally, 44, 44fOcclusal forces, 465Occlusal interferences, 204Occlusal interlocking, 204Occlusal loading, 204Occlusal planecant of, 74occipital headgear used to alter, 50f, 51, 111extrusion of, 118interbracket axis and, 335leveling ofafter surgery, 397f–398f, 397–399before surgery, 394–396, 395fat time of surgery, 396timing of, 396stepped, leveling of, 396ftypes of, 396Occluso-apical translation, 205Off-center loop, 303fOffset yield strength, 480Omega stop, 467, 467fOpen bitein adolescents, 411case studies of, 386f, 412fClass I occlusion with, 446, 447f–448fClass II, 69flateral, 79occlusal planes with, 396orthodontic mechanotherapy and, 411, 412fposterior lateral, 82fprogress check for presurgical models in, 413b, 413fsequencing of treatment in, 411types of, 394fOppenheim headgear, 40O-rings, 457, 457f, 462, 465Orthodontic appliances. See Appliances.Orthodontic external root resorption, 223fOrthodontic forces. See Force.Orthodontic mechanotherapyClass III problems, 410in open bite patients, 411, 412fpresurgical orthodontic treatment. See Presurgical orthodontic treatment.Orthotropic material, 213Osteoclasts, 200–201, 222Osteotomymandibular-advancement, 420M Index575maxillary expansion, 415, 416f, 418, 418fsagittal split, 429Overbitecase study of, 397fdefinition of, 65, 183, 281sliding mechanics as cause of, 281, 281fstraight-wire appliance for, 183PPalatal arch, 416–418, 417fPeriodontal ligamentdescription of, 158, 187, 196stressanchorage values according to, 201–202description of, 220–222tooth movement initiated by, 200, 276in tooth movement modeling, 215Perpendicular distance, 27, 27fPhase transition–dependent alloys, 483–487Physiologic mobility, 186Pivot, 194Point of force applicationchanging of, for incisor intrusion, 99–100, 99f–100fof crisscross elastics, 72fdefinition of, 25for headgear, 45fof resultant, 25, 26f, 30–31, 31fPoisson’s effect, 217Polymer chains, 489f, 489–490Polymers, in orthodontic wires, 488–490, 489f–490fPolyphenylene, 488–489, 489fPosterior anchorage unit, 277, 277fPosterior crisscross elastics, 70f–71f, 79, 80fPosterior intrusionarchwires for, 117exaggerated curve of Spee for, 126–131exaggerated reverse curve of Spee for, 126–131overview of, 117three-piece intrusion arch for, 119–120, 119f–121ftype I, 118–121, 118f–121ftype II, 118, 118f, 121–125, 122f–126fPosterior segment, 351Postoperative complicationsmandibular repositioning, 426–428, 427f–428fmaxillary repositioning, 424, 425fPrecision fit lingual bracket, 231, 232fPreformed archwire, 241Premolarslabial archwire use of, 231fmaxillary first, 111, 112fmaxillary second, 266, 266fPrescriptions, 160, 164Pressure, 212Presurgical orthodontic treatmentalignment, 393decompensationbone plates for, 399–404, 400f–403fdescription of, 399goals of, 414leveling of occlusal planeafter surgery, 397f–398f, 397–399before surgery, 394–396, 395fat time of surgery, 396timing of, 396overview of, 392–393Primary tooth movement, 167–171, 168f–171f, 178Principal axes, 212, 212fPrincipal stress, 212, 212fPrincipal stress analysis, 202, 202fProportional limit, 480Protraction headgear, 56–57, 56f–59fPseudo-biomechanics, 4Pseudoelasticity, 483, 485Pseudointrusion, 91Pure moment, 27f, 28, 169Push spring, for molar uprighting, 439fPythagorean theorem, 18, 27RRamus angular errors, 426, 427f–428fRange, 478Rapid palatal expansion, 188Reciprocal anchorage, lingual arch with, 230, 231b, 246Reciprocal tip-back moment, 113, 113f–114fRectangular loop, 311fRegional acceleratory phenomena, 404, 406Residual moment, 295, 296f, 298f, 299–302, 300f–301fResilience, 498Restrained bodies, 158f, 158–159Resultant force, 17Resultantsfor angled elastics, 26fdescription of, 17–18, 19f, 21fdirection of, 26–27force used to replace, 35fmagnitude of, 26–27point of force application of, 25, 26f, 30–31, 31fReverse articulation, 387. See also Crossbite.Reverse curve of Speecontinuous archwire with, 127–128, 128fR Index576description of, 118exaggerated, for posterior intrusion, 126–131Rolling friction, 453Root movementfour-incisor, 315fincisor, 312–317, 313f–316f, 442, 442fmolardistal, using occipital headgear, 44f, 44–45miniscrews for, 440, 440f–441fmoment-to-force ratios for, 440Root resorptionhistologic features and events in, 224fillustration of, 185f, 313fincisor intrusion and, 90–91, 111–112mechanical environment for, 217–220orthodontic external, 223fRoot spring, miniscrew-assisted, 441fRotationaxes of, 197buccal crown, 206couple and, 28, 29fdegrees of freedom and, 205–206description of, 28distal, 206illustration of, 29f, 435flingual, 206mesial crown, 206posterior crossbite correction using, 29, 30fof teeth perpendicular to occlusal plane, 205unilateral, lingual arch for, 264–267, 264f–267fx-axis, 160, 161f, 162y-axis, 160, 161fz-axis, 160, 161f–162f, 198f“Round tripping,” 185, 186fRowboat effect, 310, 351SSagittal split osteotomy, 429“Sander spring,” 440Scientific biomechanics, 4–5Scientific terminology, 6–7Second Law (Newton’s), 11–13Second-order side effects, 96, 97fSecond-order tensor, 211Segmental elastics, 80f, 80–81Segmental maxillary expansion, 415t, 418, 418f–419fSelf-ligating systems, 462Sense and direction, of force, 14–16, 15f–16fSeparate canine retraction, 307–309, 308f, 317Shape memory, 483, 487Shape-drive orthodontics, 6, 8, 96Shape-driven appliances, 4Shape-driven methodbilateral expansion in, 247, 248f, 251fbilateral rotation in, 251flingual arch, 241–243, 241f–243fSharp bends, in spring, 299Shear stress, 211, 211f–212fSide effectssecond-order, 96, 97fspace closure-related, 234temporary anchorage devices for reduction of, 5, 7third-order, 96–97, 97f–98fSide of rotation, 194–195Simple tipping, 178Single-tooth movement, 436–437Skeletal asymmetryetiology of, 414imaging of, 414, 414fmaxillary expansion for, 415t, 415–418, 416f–418fSkeletal discrepancy, 404–408, 404f–408fSliding space closure. See Space closure, sliding.Slot-interbracket angle, 327Space closurecanine retraction, 279–281cantilevers for, 275continuous arches, 280differential. See Differential space closure.en masse, 469friction reduction during, 468–469, 469ffrictionless spring for, 469, 469fintra-arch, 279–280loop (frictionless) mechanicscanine retraction, 283, 283fdescription of, 279–280, 282–283illustration of, 279fincisor eruption during, 282, 282fspring design for, 275loops for, 286, 287fmechanics of, 279foverview of, 275segmented arches, 280sliding (friction) mechanicscontinuous arches, 280description of, 279f, 279–280phases of, 280–281, 281fproblem associated with, 281side effects of, 281fstatically determinate appliances for, 284–286, 285ftemporary anchorage devices used in, 277two-phase, 317, 318fR Index577Spring(s)angled bends in, 296, 297fantirotation bends in, 308fbeta-titanium, 352, 353fcalibrated, 286canine root, 310–312, 311fcantilever. See Cantilever springs.coil, 12f, 13, 382curvature in, 299force delivered by, 136frictionless, 469, 469fincisor root, 314, 314fneutral position of, 296–299, 297fproperties of, shape and dimension effect on, 290–293sharp bends in, 299statically determinate, 284, 306fstatically indeterminate, 286–287tip-back, 120f, 347, 349, 349fT-loop. See T-loop spring.types of, 136Spring wire, 287–289, 288fStainless steelannealing of, 497fligatures, 206, 206fproperties of, 481–482, 482tStainless steel wiredescription of, 478, 479fstiffness of, 498Static friction, 453Statically determinate appliancescantilevers. See Cantilever(s).mechanics of, 370foverview of, 369principles of, 370space closure, 284–286, 285fStatically determinate force, 99Statically determinate spring, 306fStatically indeterminate spring, 286–287Statics, 11–12Steel, 213Step bends. See Z-bends.Stiffness, 480, 493f–494f, 498, 499tStraight wiredescription of, 4, 241geometry corrections using, 359f, 359–360in misaligned bracketsforces from, 324–326, 325f–326f, 346results of, 336, 337fV-bend on, 302Straight-wire appliancesapplications of, 4for overbite, 183Straight-wire cantilevers, 250fStraindefinition of, 213description of, 216Hooke’s law and, 213, 213fStressbiologic response and, 222definition of, 210differential, 202–203, 203fequivalent, 212force and, differences between, 211Hooke’s law and, 213, 213fon periodontal ligament, 200polymer chain response to, 489f, 489–490principal, 212, 212fshear, 211, 211f–212ftipping-related, 218, 219f–220ffrom translation, 187types of, 211–212von Mises, 214Stress raisers, 290Stress relaxation, 489Stress tensordescription of, 211, 211fdeviatoric, 212Stress–biologic response relationship, 216Stress-strain, 6Stress-strain curves, 479–480, 484fSubdivision elastics, 77–79, 77f–80fSuperelastic austenitic nickel-titanium alloys, 483–484Superelastic nickel-titanium wire, 486, 498Superelasticity, 483, 485, 487Symmetric headgear, 52f, 52–53Synchronous Class II elastics, 65–66, 66fTTADs. See Temporary anchorage devices.T3D Occlusogram, 384Temporary anchorage devicesbone plates as, 393fcantilever ligation to, 375felastic chain attached to, 20, 21fforce-driven approach applied to, 435–436incisor root movement application of, 316f, 317miniscrews. See Miniscrews.molar intrusion using, 438, 438fmolar uprighting using, 438–439, 439frationale for, 434root spring side effects prevented using, 315T Index578side effects reduced using, 5, 7for space closure in maxilla, 277Tension-compression theory, 217Terminology, scientific, 6–7Thermoplastics, 488, 489fThird Law (Newton’s), 12f, 12–13Third-order side effects, 96–97, 97f–98fThree-bracket segments, 351–355, 352f–355f3D asymmetric model, 1963D force systems2D projections of, 35–36view of, 36f3D symmetric model, 1963D volume of resistance, 198Three-dimensional facial masks, 391, 391fThree-dimensional problems, 387, 387fThree-piece intrusion archBurstone, 205fcomponents of, 120f, 121incisor intrusion usingaltering force direction in, 102, 103f–104fanchorage control in, 102fcase study application of, 106, 107f, 113f, 205fdescription of, 100–102, 101f–102findications for, 105posterior extrusion using, 119f–120f, 119–121tip-back, 119f–120f, 119–121Tip effect, 96Tip-backfrom continuous intrusion arch, 113, 113f–114fof posterior teeth, 119, 119f, 131fthree-piece, 119ftip-back spring for creating, 349funilateral, using lingual arch, 267–271using cantilever, 373fTip-back spring, 120f, 347, 349, 349fTip-forward, using lingual arch, 267–271Tippingcontrolled, 172f, 179, 179f, 200force and couple needed for, 182–186illustration of, 29fincisor. See Incisor tipping.low cervical headgear for, 43simple, 178stresses associated with, 218, 219f–220fT-loop for prevention of, 293uncontrolled, 178, 183, 185, 185f–186f, 200, 218T-loop, 33, 34f, 203f, 283T-loop movementsactivation moment, 294, 295fangled bends in spring, 296–299neutral position, 296–299, 297fresidual moment, 295, 295fT-loop springantirotation bends with, 309fcase study of, 301, 301fdifferential space closure with, 299–307for group A mechanics, 300fresidual moment, 299–302, 300f–301fshape of, 293, 294ftemplate with, 304fTooth load at axis of resistance, 200Tooth movementasymmetric, 238fderived, 171–175, 172f–175ffinger springs for, 240force constancy for, 94force systemsat the bracket, 176–180description of, 158, 167, 171historical descriptions of, 194, 194fmechanical displacement stage of, 279mechanical environment for, 217–220modeling of, 215optimization of, 5overview of, 157–158periodontal ligament stress and, 200, 276predicting of, 32primary, 167–171, 168f–171f, 178rate of, 187fregional acceleratory phenomena, 404rotation. See Rotation.single-tooth, 436–437speed of, 2003D concepts in, 167ftranslation. See Translation.types of, 435fTooth positionbracket position change, 160–165, 161fcenter of rotation, 165–167, 166f–167fmethods for describing change of, 159–167, 160f–167fTorquedefinition of, 160friction and, 461, 461fillustration of, 29fimproper uses of, 182lingual root, 178–179in wire, 181Total-arch movement, 436–437, 446, 447f–448fToughness, 498Transition temperature range, 485–487T Index579Translationanchorage value of, 200of anterior segment, 436buccolingual, 205cervical headgear for, 43–44coordinate systems and, 162fdescription of, 161, 161fillustration of, 435fincisor, using miniscrews, 442, 443f–444fline of action of force delivered by, 285mesiodistal, 206occluso-apical, 205stress levels created by, 187Transpalatal archarch form preservation using, 469farch width affected by, 256fbilateral molar rotation with, 252, 253fdescription of, 232, 251fabrication of, 252, 253fforce systems delivered by, 256, 256fGoshgarian-type, 393fhorseshoe arch versus, 254ideal arch configuration using, 254, 255fillustration of, 233f, 238f, 252fmaxillary, 253fmaxillary molars connected with, 205mesial drift and, 206passive, 252, 253f, 255in type I posterior extrusion, 119unilateral constriction treated with, 262, 263funilateral rotation using, 264–265unilateral tip-back and tip-forward with, 268, 268fTreatmentbiomechanical approach to, 9. See also Biomechanics.presurgical. See Presurgical orthodontic treatment.results of, research and evaluation of, 6time required for, 5Treatment planalgorithm for, 390fdescription of, 390Moorrees mesh diagram, 390, 391fprioritized problem list as part of, 390three-dimensional facial masks, 391, 391fTrigonometric functions, resultant calculations using, 18, 19fTwo-bracket geometries, 359–363, 359f–363fTwo-bracket segments, 326–327, 327f2D projection model, 195–196Two-phase space closure, 317, 318fTwo-tooth root movement, 361, 363Two-vector mechanicsclinical applications of, 384–385, 385fdefinition of, 383mathematic procedure, 383–384Type I posterior intrusion, 118–121, 118f–121fType II posterior intrusion, 118, 118f, 121–125, 122f–126fUUltimate force, 479Ultimate tensile strength, 498Uncontrolled tipping, 178, 183, 185, 185f–186f, 200, 218Unilateral Class II elastics, 77, 77f, 79Unilateral rotation, 264–267, 264f–267fUnit cells, 481Up-and-down elastic, 82, 82fVV-bends, 302, 341f–344f, 341–345, 342t, 356, 357f, 471Vertical elastics, 67, 68f, 76f, 76–78, 82Vertical loopheight of, 290–291, 291fplacement of, in continuous archwire, 304, 304fVibration, 465Virtual bracket repositioning, 356–358, 357f–358fViscoelasticity, 214–215, 489fVolumetric body, 195von Mises stress, 214WWedging, 473, 473fWire. See also Archwires; Straight wire; specific type of wire.association and dissociation of, 250bending of, 249, 288ffor cantilever, 374characteristics of, 492–493coefficient of friction with, 455composites in, 487–488, 488fdislocations, 479flexure behavior of, 480force-deflection curves of, 478–479, 479fforce-deflection rate of, 493–497, 495t–496tforces acting on, 206fligation of, 462, 462fmaximum bending moment of, 480maximum elastic twist of, 499tmaximum torque of, 499tW Index580mechanical characteristics of, 478–480polymers in, 488–490, 489f–490fselection of, 491–499stiffness of, 494f, 498, 499tstress-strain curves, 479–480torque in, 181torque/twist rate of, 499ttriad characteristics of, 492–493V-bends in, 341f–344f, 341–345, 342tZ-bends in, 339–341, 339f–341f, 342tWork hardness, 485Woven up-and-down elastic, 82, 82fXx-axis rotation, 160, 161f, 162Yy-axis rotation, 160, 161fYield point, 480Yield stress, 480Young modulus, 480Zz-axis rotation, 160, 161f–162f, 198fZ-bends, 339f–341f, 339–341, 342tW

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