Thammasat
Int.J. Sc.Tech.,Vol. 10,No. 2, April-June
2005
Assessment
of Castigliano'sTheoremon the
Analysisof ClosingLoop for Canine
Retractionby Experiment and Finite
ElementMethod. Part I
Varintra Ungbhakorn
Departmentof Orthodontics,Faculty of Dentistry
ChulalongkornUniversity,Bangkok I 0330,Thailand
Tel: (662) 218-8154
E-mail: nchudhabuddhi
@hotmail.com
Variddhi Ungbhakorn
MahanakornUniversityof Technology,Nong Chok, Bangkok 10530,Thailand
Tel: (662) 218-6629
E-mail: [email protected]
Correspondingauthor
Paiboon Techalertpaisarn
Departmentof Orthodontics,Faculty of Dentistry
ChulalongkornUniversity,Bangkok I 0330,Thailand
Tel: (662)218-8951
E-mail: [email protected]
Abstract
In Part I, the Castigliano's theorem is applied to derive the equationsfor the ratios of M/F and
loop stiffnessfor four types of closing loop, namely, the vertical helical loop, T-loop, Opus9Oloop,
and helical T-loop. The last type of loop is a new designoffered in this study.The theoreticalM/F is
maximizedto yield the optimum closing loop configurationsfor canineretractionwheretheir heights
and widths are limited to 10x10millimeters.For closingloop applications,certainseriousassumptions
requiredby the Castigliano'stheoremare violated,therefore,accuracyin theoreticalpredictionis not
expected.However, trends indicating the influences of various variablesleading to the optimum
design of each loop type coincideswith the observationsof severalresearchersin the past. These
optimum configurationswill be used to study experimentallyin Part IL Since the facility for
determiningM/F by experimentsis not available,only experimentalresultson loop stiffnesswill be
comparedto thoseobtainedby Castigliano'stheoremand a finite elementmethod(FEM). If the FEM
yields good predictionof the loop stiffnesswithin allowablediscrepancy,then,the M/F resultingfrom
FEM may be usedfor actualestimates,eventhoughit requiresfuture experimentalverification.
Keywords: closingloop, canineretraction,Castigliano'stheorem,finite elementmethod
its size,shapeand supportingtissue,etc. [2,31.If
bodily movementcould be achieved,the level of
stress and strain of the supporting structure of
the tooth rootscan be minimized.Unfortunately,
in reality, the positionof the CRE is suchthat it
is impossibleto obtain a bodily movementby
l. Introduction
Bodily movement or translation of a
partially restraint body such as a tooth or a
group of teeth occurs only if the line of force
acts through its centerof resistanceor CRE [1].
The position of CRE for eachtooth dependson
28
ThammasatInt. J. Sc. Tech.,Vol. 10, No. 2, April-June2005
using a single force, hence, a system of force
consisting of forces and moments must be
applied to the brackets bonded to the crowns
instead. There are two methods in moving a
tooth or a segment of teeth [,4]. The first
method is called sliding mechanics and the
second, loop mechanics. The latter method is
our interestin this paper.The method involves
the application of a system of force in bending
arch wire into loops of various configurations in
continuous arch wires to deliver the appropriate
M/F to the tooth or a group of teeth when being
activated.These loops are called closing loops.
To designthe closing loop to deliver the desired
M/F with proper loop stiffness K (loop applied
force per unit deflection or the slope of the
graph of force vs. deflection) is the most
difficult task for a successful movement of a
tooth. If the value of the loop stiffnessis high,
then a large magnitude of force is needed to
activate one millimeter of the loop legs. For
example,if the loop stiffnessK is equal to 200
gm/mm (1.962 N/mm), then to activate 2
miflimeters,a force equivalentto 400 gm (3.92
N), is required at the loop legs. The high force
magnitude may hurt the patient becauseof high
stressat the periodontal ligament. Also the rate
of force decay will be rapid and hence more
frequent activation is needed than a loop with
lower stiffness. Some orthodontistschoose to
reduce loop stiffnessby using costly arch wire
with a lower value of Young's modulus [5-7]
suchas TMA arch wire.
As mentioned above, translation of a tooth
can occur only when the line of action of force
passingthroughCRE. But, in a real situation,we
can apply force to a tooth through the bracket
bonded on the crown only. If the distance
betweenthe bracket and CRE is equal to d, then
an undesirablemoment, M = F x d, arises.Tlre
concept of the closing loop is to generate a
moment with a magnitude equal and opposite to
the undesirable moment so that a pure
translationcan be realized.Hence, the ideal ratio
of M/F of the closing loop must be equal to d,
the distance between the bracket and CRE. In
the past many researchers[,2,8-12] tried to
determine this distance and came to the
conclusion that the distance d for the maxillary
is between 7.6 to 9.6 millimeters, and the
mandibular,7.6 to 10.3millimeters.
Several loop configurations believed to
yield high ratio of M/F and used by
29
orthodontists are: vertical helical loop, T-loop,
and L-loop. None of these loops is capableof
giving the required inherent ratio of M/F.
Therefore gable bends anterior and posterior to
the loop legs are neededto obtain a ratio of M/F
of about 8 to 10 millimeters. Siatkowski [1,8]
studied and designed new configurations of
loops, the Opus9O and Opus70, but still the
maximum inherent ratio of M/F is only about
5.5 millimeters when loops are centeredbetween
brackets.But if the Opus7Ois located off-center
with a distance from the anterior leg to the
bracket of 1.5 millimeters, he could obtain a
value of M/F close to the desired magnitude for
the tested wire size. Such an arrangement is
rather difficult in practice. Furthermore, once
the wire size changed, the M/F becomes an
unknownagain.
It can be seenthat if a reliable analytical or
numericalmethodof the closing loop analysisis
available, then any orthodontists can use this
tool to calculate the characteristic of their
closing loops theoretically without resorting to
costly and time-consumingexperiments.In this
first part of the paper, the authors employ the
existing Castigliano's theorem to derive the
expressionsfor the loop stiffnessand ratios of
M/F for various loop configurations. To
determineM/F experimentallyis costly, and it is
rather difficult to obtain accurateresults due to
its small size as comparedto the sensorsuch as
a strain gage. Therefore to check the validity of
the theoreticalresults,only experimentalresults
of the loop stiffness will be comparedin Part II.
Only the optimum design of each loop
using the
configuration obtained from
Castigliano'stheoremin Part I will be used to
study in part II of the paper employing the finite
element method. If FEM can predict the values
of the loop stiffness to within an acceptable
discrepancy, then we may assume that the
computed ratio of M/F by FEM can be used to
estimatethe actualvaluesalso.
2. Material and Methods
2.1 Castigliano'stheorem
ln 1879, Alberto Castigliano, an Italian
railroad engineer, outlined a method for
determiningthe displacementand rotation at a
point in a body. This method, which is referred
to as Castigliano'ssecondtheorem,appliesonly
to bodies that have constant temperature and
2005
ThammasatInt. J. Sc Tech.,Vol. 10,No. 2, April-June
material with linearly elastic behavior. The
theoremstatesthat [3]:
The first partial derivative of the strain
energy in the body with respect to a force (or
moment) acting at the point is equal to the linear
displacement (or angular displacement) of that
point in the direction of force (or moment).
In mathematicalform. it can be written as:
_ a u
O^- = -
aP"
where M , is the bending moment of each basic
element I of the loop, E, the Young's modulus
of the arch wire material, 1, the moment of
inertia of cross-sectionalarea of the arch wire
and ds, the differentiallengthofthe
arch wire.
(l)
where U is the strain energy of the elastic body
and if { representsa force, then { becomes
the linear displacement,but if { representsa
moment or couple, then d, becomesthe angular
i1
displacement.
In order to apply Castigliano's theorem
successfullythe following assumptionsmust be
met:
1. The material of the arch wire is linearly
elastic.
2. After activating the closing loop, the
material is still in the elastic range.
3. Displacementis small.
4. All arch wire forming the closing loop is
in the sameplane.
5. Distortion of the shape of the closing
loop after activation is small.
It is obvious that either some or all of the
assumptions 3 to 5 are violated in our
application. Hence, we are not expecting good
agreement with experimental results by
employing Castigliano's theorem in loop
analysis,as will be evident later. Nevertheless,
trends indicating how to optimize the loop
configuration are shown in the analytical
solutions.
In general, there are three types of strain
energy stored in the elastic body, namely, strain
energy due to axial load, torsion, and bending.
For the present types of closing loop
applications, the torsional strain energy due to
out-of-plane forces and strain energy due to
axial loads are small as comparedto its bending
strain energy, hence the total strain energy can
be approximately written as:
u=!ury
+-,
M
(2)
30
M
Fig. 1. Closingloop with gablebend
angle activatedby a force F
Considerthe closing loop model as shown
in Fig. 1 and from eqs.(l) and(2), the following
relationshipsare obtained:
' ^ T
M:dsf
6=o'=LLIF[""""1
aF
2AFl7r
'
^ l-
Er _l
,M2dsl
o=9Y-=l:-lFr
I
dM 2AMlar Er l
t:r
Gl
From eqs. (3) and (4), the expressionsfor
the loop stiffness, K = F I 6 andthe momentto-force ratio, M / F will be derived for each
type of closing loop consideredherein.
2.2 Y ertical helical loop
For a vertical helical loop as shown in Fig.
2, the bending moment for eachbasic elementof
the loop is as follows:
Mr=Fx-M
O<x<H
(5)
Mz = F(H + Rsin4,)-M
0 <Q, < r
(6)
Mt=F(H -Rsin4r)-M
0<dr<E
(1)
Int.J. Sc.Tech.,Vol. 10,No. 2, April-June2005
Thammasat
Eq. (13) coincideswith that given by Haack
[4] without the detail of derivation.
2.3 T-loop
For a T-loop as shown in Fig. 3, the
bending moment for eachbasic elementof the
loop is as follows:
{_
M
Fig. 2 Vertical helical loop
The strain energy for eachelementis:
t
H
(J,=' lG*-ula*
(s)
2EI i)
M
I\,7
Fig.3 T-loop
1
n
u ', = + f t n t n* R s i n )/ -, M l ' ( R d O )
zEri
(e)
1
r
u," = _- Irt u - Rsinfi,t- M l' (RdO,)
2Eri,
2
)
Mr=FH-M
0<!,<L
(1s)
r
Thereforethe total strainenergyof the Tloopis:
I
H
K =
t
"
P1U= l(Fx - M)'dx +
,
F
(17)
(ll)
where N is an odd number, the number of halfturns of the helices. For one helix N=3, with
Castigliano'stheoremaccordingto eqs. (3) and
(4), the following relationshipsare obtained:
M _ H 2 + r N R H + 2 R 2+ E I 7 / F
(16)
M^= F(n +2n)-M
Q<)z<@+atz)
f N-l\
\
(14)
0<d<n
Therefore the total strain energy of the vertical
helicalloop is:
+l-1U
0<x<H
M,= FIH +n(t- cos{)]-u
(10)
(N+t\
U=2U+l----lu
r \ 2 ) ,
M,:Fx-M
o
,
!r o,- M ) ' dy,
o
+ ltFH + FR(r- cos/)- Ml GdO)
(12)
2H + nNR
L+d/2
+ J trta +2R)-Ml'dy,
6EI
4 H 3 + 2 4 H R 2+ 3 n N R ( 2 H 2+ R 2 )
(l 3 )
(18)
Appling Castigliano'stheoremaccordingto
eqs. (3) and (4), the following relationships are
obtained:
3l
2005
Int.J. Sc.Tech.,Vol. 10,No. 2, April-June
Thammasat
M, = Fl(H * R)- *,1- M
L = W ' + 4 L ( H+ R )+ 2 n R ' + 2 n R H+
F
d ( H + 2 R ) +E I 0 l F l l [ 2 H+ 4 L + 2 n R + d )
(le)
(27)
0<x.<H+R
Therefore the strain energy of the
Opus9Oloop is:
K = 3EIl[2H' + 6H2L + 6nRH' +l2rHR2
1
+9nR'+3(2L+d)(H +2R)'l
EIU = ;
L
(:20)
I
H*!
Jrr*,
-M1' dx,+
o
E/.2
;t o J l n r a + R+ R s i on , \ -M f '
2.4Opus90loop
r
(Rdd,)
L
* : lrfrn +2R\M
- ]'dy+
,
L
o
"n.
|
t o
x. l
*-l
TI
il l l f
- ( r - c o s) /\ -, M l ' ( R d O , l
J [ r { r a+ 2 R t R
1 L .
+ : - ) r n n - M fd y , +
1 l
1"^e'.
;L oJ [ F { n + n r t
-cos4,)\-Ml'Gdd3l
Fig. 4 Opus9Oloop
+ IlF@+R-x,)-ttl'dx,
Since the Opus9O loop is unsymmetrical,
when it is centered within IBD (inter-bracket
distance). the inherent moments anterior and
posterior to both legs are not equal. For our
purpose of deriving the formulae for loop
optimization, we will assume that the Opus9O
loop is located off-center such that the inherent
moments at both legs are equal as shown in Fig.
4. The bending moment for each basic element
of the loop is as follows:
(28)
o
Applying Castigliano'stheorem to eq.(28)
yields the following relationships
:
M
=1u'12+3flRH +3(l+ n)R' +2HL
F
+2RL+ EIe I Flll2(H + R) +3rR +2Ll
(2e)
K = l 2 E I l [ 8 H 3+ 1 2 ( 2 + 3 n ) H ' R +
Mr=Fx,-M
0<t,(H+R(21)
72(1+r)HR' + (56 + 54r)R' +
M , = F l ( H + R ) + R s i n l . , 1 -M
0<Q,<rl2
(:22)
O<!,<L
(23)
24(2R' L+2RHL+n2r)l
(30)
M, = F(n +2n)- u
2.5 Helical TJoop
To obtain higher M/F ratio, in principle, we
should locate the wire material as far from the
base as possible. Hence, the authors are
proposing a new configuration of loop, called a
helical T-loop, as shown in Fig. 5. This
configuration has been modified from the simple
T-loop by adding helices at the upper wings.
During derivation of the formulae any number
of helices will be considered, but for practical
purpose only one helix on each wing will be
M, = Fl(n + 2R)-n(t - cos/,)] - n
Mr=FH-M
0<d, <ft
(24)
0<tr<L
(:25)
u
M, = FIH + n(t - cosQ,)]0 < Q,<3nl2 (26)
JZ
Int.J. Sc.Tech.,Vol. 10,No. 2, April-June2005
Thammasat
analyzedlater on. By increasing the length of
the arch wire loop in this manner,it will result in
greater flexibility, in other words, reduce the
stiffness of the equivalent simple T-loop. This is
a desirableloop property becausethe lower loop
stiffness implies that the rate of force decays as
the tooth moves to close the gap and will be
slower. Hence, larger activation distance per
patient visit can be achieved without causing
severe stress and strain to the supporting
structureof the tooth.
ul
=ri,
2EI i
(37)
u2
=rir
FH - M ) ' d y ,
2EI
(38)
Fx- M)'dx
i
1 " n
u', = - + l r { n + R ( l - c o s / , t \ - u l t n a 0 , t
2EI'
(3e)
1
E
u "^ : +
[ 1 n{ t n + 2 R )- R r l - c o s / , ) }
2Er t- Ml' Gdo)
(40)
t
u', = +
L+d/2
- l 'a y , @ t )
| l r W + 2 R )M
2EI i'
Observe from eq.(36) that for a simple Tloop N = l, one helix N = 3, and two helicesN =
5, applying Castigliano'stheorem to eq. (36)
and the following relationshipsare obtained:
M
::- =IH' + 4L(H + R)+2nNR' +2vNRH
F
+d(H + 2R)+ EI9 | Fl l[2H + 4L + 2n NR + d)
(42\
Fig.5 Helical T-looP
The bending moment for each basic
elementof the helical TJoop is as follows:
Mr=Fx-M
0<x<H
(31)
Mr=FH-M
0<!,<L
(32)
K = 3 E I / 1 2 n 3+ 6 n 2 t + 6 n R H 2 + 1 2 r H R 2
+3(2L+d)(H +zR)2 +gnNn3l
M 3= F I H + n ( t - c o s Q , ) l - u
(33)
0<A<,
M, = Fl(n +2R)-ft(l-cos/,)l-M
0<d,<E
(43).
3. Results and discussion
3.1 Theoretical findings
The aim in investigating the theoretical
results of each loop type is to find the optimum
loop configuration to yield the highest ratio of
M/F. The material of the arch wire is assumedto
be stainless steel of cross-sectional area
0.016x0.022 inch (0.40x0.55 mm) whose
Young's modulusis equal to 172,000MPa. The
size of any closing loop will be constrainedby
the anatomy of the oral cavity. For example,the
total height of the loop is limited by the
maximum height from the bracket on the tooth
crown to the vestibule.Sinceit is known that the
higher the loop, the higher the ratio of M/F will
be achieved [4,15], thereforein the following
investigation the total height and width (HrxW)
will be l0xl0 mm for upper canine retraction.
(34)
M, = F(n +2R)- M
0< )z < L+dl2
(35)
The total strain energy of the helical T-loop
is:
u=
(rrl-l\
f
/ A / -r l' \
I
z lu ,+ u ,* l ' " l u ,* 1: : - : ' l u^+ u ,I
\ 2 ) ' \ 2 ) - " 1
L
(36)
where N is an odd number, the number of halfturns of the helices, and the strain energy for
eachbasicelementis as follows:
JJ
Thammasat
Int.J. Sc.Tech.,Vol. 10,No. 2, April-June2005
stiffnessK can be calculatedby using eq. (13)
and the results are shown in Fig. 6. Notice that
the larger the values of H or R, the lower the
stiffness K will be. The optimum configuration
is summarizedas follows:
The obtained optimum configuration will be
usedin experimentsand analysisby FEM in part
II of this paper.
3.2 Vertical helical loop
To find the optimum configuration, we
assumethat there is no gable bend angle d in
eq. (12) and consideronly one helix in Fig.2.
The geometricalconstraintequationis:
H+R = l0
R = 1.5 mm, H = 8.5 mm, MJF = 6.32 mm, K =
33.8 sn/mm
(44)
81.5 m, FE8.5rr4 E=l7em tuF4 K:-338grnh
1m
1m
The calculated results for each helix radius
using eq. (12) yield the following:
R = 0.5 mm, H
R = 1.0mm, H
R = 1.5mm, H
R = 2.0 mm, H
1!m
e1m
Erm
5 m
= 9.5 mm, M/F = 5.71mm
= 9.0 mm, M/F = 6.1I mm
= 8.5 mm, M/F = 6.32 mm
= 8.0 mm, M/F = 6.39 mm
= 6 m
,m
m
0
Fig. 8 Moments of vertical helical loop with
gable bend
To see the effect of the gable bend angle,
eq. (12) is used to constructFig. 7 for various
magnitudes of activating forces F and their
correspondingmoments M are shown in Fig. 8.
In Part II, the more reliable FEM will be used to
construct the same graph as Fig. 7 to
demonstratethe right trend. This confirmation of
the correct trend will be done for all four types
of closingloop.
Fig. 6 Effect of helical radii on stiffnessof
vertical helical loop
12
10
8
6
l
t=r=?sF
+F=100
3.3 T-loop
To find the optimum configuration of the
T-loop, we assumethat there is no gable bend
angle 0 in eq. (19). The geometrical constraint
equationaccordingto Fig. 3 is:
l
B = 1 . 5 m m , H = 8 . 5 m m , E = 1 7 2 , 0 0 0W a , K = 3 3 . 8g m / m m
gm
L 1 F=15!91'r
H+2R= 10
2L+2R+d= 10
2
0
(45)
(46)
First we investigatethe influenceof d
by arbitrarily fixing R = I mm. Then eqs. (45)
and (46) become:
Fig. 7 M/F of vertical helical loop with gable
bend
H=8
2L+d= 8
It is seen that the larger the helix radius ,
the higher ratio of M/F is obtained. But for a
practical pulpose, we choose to limit the
maximum radius to R = 1.5 mm. The loop
(47)
(48)
Now the ratios of M/F are calculatedby eq.
( 1 9 )f o r d = 0 . l , a n d2 m m t o g i v e :
34
Int.J. Sc.Tech.,Vol. 10,No. 2, April-June2005
Thammasat
d = 0 mm. H = 8. L = 4 mm. M/F = 6.91 mm
d= 1 mm. H = 8, L = 3.5 mm, M/F = 6.88mm
d = 2 mm, H = 8, L = 3 mm, M/F = 6.85 mm
their corresponding moments M are shown in
Fig. 10. Sinceloop stiffnessis also an important
factor in determining the magnitude of
activating distance per patient visit, the
influences of the loop H and arch wire sizes on
the loop stiffnessare also shownin Fig. 11.
It is seen that d = 0 mm gives the highest
value of M/F. However, observe that the
influence of d on M/F is negligible. Also
constructinga closing loop with d = 0 mm will
facilitate the measurementof each activating of
distanceand the monitoring of the movement of
the tooth. Now the constraint equations are
reducedto:
d{ rm FE1rm FEgrm t=4 rrn K=238gnvrm
E=172mllPa
16m
14@
^ 12m
0
Fig. 10 Momentsof T-loop with gablebend
R = 0.5 mm, H = 9 mm, L = 4.5 mm, M/F =
7.20mm
R = 1.0mm, H = 8 mm, L = 4.0mm, M/F =
6 . 9 1m m
R = 1.5mm, H = 7 mm, L = 3.5 mm, M/F =
6.62mm
d=0 mm, FF1 mm, LF4 mm. E=172.000 MPa
_l
]
From the above results, the smallest R
yields the highest M/F. But for practical handbending of the closing loop, the smallestR is
chosen to be 1.0 mm, hence the optimum
configurationfor T-loop is:
7
4
+.016x.016
+.016x.022
9
l
H (mm)
Fig. 11 Effect of arch wire size on stiffnessof Tlooo
R = 1.0 mm, H = 8 mm, L = 4.0 mm, M/F =
6.91mm, K = 23.8gm/mm
3.4Opus90loop
loop asshownin Fig. 4, the
For the Opus9O
are:
constraintequations
seometrical
d=0 mm, R-1 mm, FE8 mm, b4 mm, K=23.8 grvmm,
E=172,M MPA
r=/c gm
+
+F=1009m
+E150
m
To search for optimum configuration from
eqs. (49) and (50), we try R = 0.5, 1.0, and 1.5
mm and the following results are obtained:
--:
+Elm
.99
E l m
(4e)
(s0)
H+2R = 10
L+R=5
+F-Egm
Erm
E m
H+2R= l0
L+2R= 10
I
(5l)
(s2)
+F=lsogTl
Following the similar procedure as the Tloop using eqs. (29) and (30), the optimum
configurationis derivedas follows:
R = 1.0mm, H = 8 mm, L = 8 mm, M/F = 7.18
mm,K=19.6gmlmm
Fig. 9 M/F of T-loop with gable bend
Here again the minimum radius R is limited
to 1.0 mm for practical hand-bending of the
loop. The effect of the gable bend angle on M/F
for various magnitudes of activating forces F
To seethe effect of the gable bend angle on
M/F, eq. (19) is used to construct Fig. 9 for
various magnitudes of activating forces F and
35
Thammasat
Int.J. Sc.Tech.,Vol. 10,No. 2, April-June2005
and their corresponding moments are shown in
Fig.12 and 13. The influencesofthe loop height
H and arch wire sizes on the loop stiffness are
demonstrated
in Fis. 14.
FE1 m,
FE8 m,
K=19.6 gn/m,
UB m,
Following a similar procedureas the T-loop
using eqs. (42) and (43) for one helix, the
optimum configurationis obtainedas follows:
d = 0 mm, R = 1.0 mm, H = 8 mm, L = 4 mm,
MIF =7.43 mm, K = 23.5 gm/mm
E=172,m lvPa
The effect of the gable bend angle on M/F
for various magnitudes of activating forces F
and their correspondingmoments are shown in
Fig. l5 and 16.The influencesofthe loop height
H and arch wire sizes on the loop stiffness are
demonstrated
in Fis. 17.
5
1
0
I
B (degrees)
d4trm &1 nm Fl€nmL4nm
l<gsgn'lml
E=l7ZmN,tu
13x
g
12&
F
1 1#
E
Fig. 12 M/F of Opus9Oloop with gablebend
+1 mm,l-t4mm,LJ mm,lc19.6gnvmm,812,m l,ft
1m
6€
10
5fs
1m
+F-759m
+E1mgm
+ ElsOgm
EgS
= 8 6s
A
' E&
5
:+E/5gm
:-
Etm
E
E m
1
0
1
5
d (degrees)
+E1mgm
Erg-l
= m
4{X)
Fig. 15 MiF of helicalT-loop with gablebend
200
0
5
1
0
1
5
d(degrees)
1m
Fig. 13 Momentsof Opus9Oloop with gable
bend
R=1 mm. L=8 mm. E=l72.m
't4@
^lm
tt t*
E m
966
Eno
MPa
n
50
0
40
5
1
0
1
5
d(degrees)
620
;
10
Fig. 16 Momentsof helical T-loop with gable
bend
0
d = Om m . B = l m m . L = 4 m m . E = 1 7 2 . m M P a
Fig. 14 Effect of arch wire sizeson stiffnessof
Opus9Oloop
m
50
c40
E
E30
3.5 Helical T-loop
For the helical T-loop as shown in Fig. 5,
the geometricalconstraintequationsare:
[+-!r6xt6
+.016x.022
I
Y29
10
0
H+2R= l0
2L+2R+d= l0
5
1 51 \
6
7
8
9
H (mm)
(54)
Fig. 17 Effect ofarch wire sizeson stiffnessof
helical T-loop
36
Int. J. Sc.Tech.,Vol. 10,No. 2, April-June2005
Thammasat
4. Conclusion
In Part I of the paper, the theoretical
derivations of the ratios of M/F and loop
stiffness of four types of loop configurations
have been shown in details. For maximum M/F,
the results indicate that if one is required of to
increasethe loop height (H), the loop width (L),
and use the smallest loop radius (R), except for
the vertical helical loop, the largest loop radius
is the best. This coincides with the observation
of
several researchers [1,4,6,15-19].
Experiments to compare the loop stiffness will
be done in Part II. The results of the more
reliable FEM will be presentedfor all optimum
loop configurationsobtainedin Part I.
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