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Creative wire bending-The force system from step and V bends

Charles J. Burstone, D.D.S., MS., and Herbert A. Koenig, Ph.Di Farmington, Corm.

The force system produced by wires with steps and V bends was sudied analytically by means of a small deflection mathematic analysis. Characteristic force relations ips were found in both the step and the V bend. Step bands centrally placed between adjacen brackets produce unidirectional couples that are equal in magnitude. Along with these couples, ve

!i

ical or horizontal forces are produced depending upon the plane of activation. Mesiodistal place ent of step bends is not critical because very little alteration in force system occurs if a step is cent red or positioned off center. V bends, on the other hand, are very sensitive to the positioning mesiodistally of the apex of the V. If the apex of the V bend is placed on center, equal and opposite couples are produced. As the V-bend apex is moved off center, predictable combinations of moments and forces are created. A method for determination of the relative force system is described that allows for simple interpretation and prediction of the force system from a V bend. The clinical applications of these data and a rational basis for wire bending are presented based on the producing of a desired force system. (AM J ORTHOO DENTOFAC ORTHOP 1988;93:59-67.)

I n a previous study, the force systems produced by a straight wire in malaligned brackets were determined. It was found that a predictable ratio of the moments produced between two adjacent brackets remained constant regardless of interbracket distance or the cross section of the wire used if the angles of the bracket remained constant to the interbracket axis. On the basis of this work, a classification of bracket-wire geometries was established that can serve as a guide to the force systems that would be expected from a straight wire appliance.

Although in recent years there has been a trend away from placing bends in wires by building more activation with bracket position and design, there is no question that comprehensive orthodontic treatment requires the placement of bends in wires. Bends are placed for a number of reasons-for example, (1) to correct tooth position if a discrepancy or error is present in bracket position; (2) as teeth approach their final position of alignment, it may be necessary to increase the magnitude of force, which can be accomplished by replac- ing the original wire with a wire of a larger cross section or by using the original wire and placing an exaggerated bend; (3) wires may be selected with very low load-

From the University of Connecticut Health Center Supported by NIHiNIDR Grant DE-03953.

deflection rates that require overbending beyond the position of an ideal arch to build up to more optimum force values for tooth movement; and (4) certain types of specialized tooth movement, such as space closure, tipback mechanics, intrusion, and root movement, require appropriate wire bending.

There are two approaches to the placement of bends in an arch wire. One could be described as the ideal shape-driven approach. The orthodontist forms a wire to the shape in which the desired occlusion would be if all of the brackets were to finally align themselves on the wire. This is the ideal arch concept. As has been previously shown, the ideal shape-driven approach to bending an orthodontic appliance can lead to undesirable side effects and may not operate in a manner predicted by a superficial reading of the relationship of a wire to a bracket. Even if eventually the teeth will align themselves on a straight wire (which may require increasing the stiffness of the wire by replacement with wires of a larger cross section), the overall orientation of the teeth in space may not be what is desired; furthermore, during this process of alignment, teeth do not move to the final straight wire position but undergo extraneous and undesirable movements because of the intervening force systems produced. A more rational approach to the bending of a wire is to make bends that deliver the required force system to produce wanted tooth movement. Although there is still much to be

59

60 But-stone and Koenig

STEP BEND FORCE SYSTEM

M,

n !

F,

Bracket 1

Fig. 1. Forces (F) and moments (M) produced by a step in a wire. Ail forces are acting on the wire. The forces to the teeth are equal and opposite (not shown).

100 ,-

F 14 ; - - - - -------A-

0; ILL _ L- -.~-----.-~_ --_-. - I 2 3 4 5 O/L

Fig. 2. Forces at bracket 1 and bracket 2 produced by a wire step. As the a/L ratio (mesiodistal position of the step) is changed, the forces remain relatively constant.

learned concerning an optimum force system, particularly in respect to the magnitude of moments and forces and moment to force ratios, it is important that the clinician make the bends so that at least the moments and forces are in a correct direction.

It is the thrust of this article to look at what might be called creative wire bending-not the ability to ma- nipulate wire and pliers, but how a wire should be bent to produce a desired force system. An infinite number of shapes or bends can be placed between two brackets. We will consider here only two commonly used basic bends-the step and the V bend. The force systems from these two types of bends will be studied, fundamental principles will be elicited, and the clinical applications shown.

METHOD

The wires used in this study were 0.016-inch stain less steel wires* with bends; they were placed between

*E = 0.2111104 x l@ g/mm; M,,, -= 1,860 g-mm The beam (or wire) is initially divided into separate

O/L

Fig. 3. Moments at bracket 1 and bracket 2 produced by a wire step. As the a/L ratio is changed, moments remain the same. A small negligible reduction in M, occurs if the bend is placed very close to bracket 1.

two aligned brackets with interbracket distances of 7 mm and 14 mm. Two sets of bends were studied- steps and V bends. As measured from the levels of the brackets vertically, the step and V bends were made so that their height was 0.35 mm (A). The position of the step or the V bend was changed mesiodistally so as to alter an a/L ratio (a = distance from V bend apex to bracket; L = interbracket distance). This ratio measured the centering or lack of centering of the bend within the interbracket distance. The step and V bend relationships are shown in Figs. 1 and 6, respectively. At each bracket the moments (M) and the forces (F) were calculated. In addition, the ratio between the moment produced at bracket 2 in comparison with bracket 1 was determined.

The solutions were carried out by means of an analytic model based upon small deflection theory. Large mesiodistal forces would be anticipated as deffections become great if wires are not free to slide in brackets. In previous work it has been shown that if wires are free to slide in brackets, the small deflection theory approximation is quite good for predicting force systems in which overall deflections are relatively small, such as in these applications.* It should be remembered that in this type of solution, we are purposely ignoring mesiodistal forces that could be present clinically so that the fundamental relationships can be more easily studied. Differences in boundary conditions from those assumed in the study could lead to differences in both the relative and absolute force systems.

The method used in this work to effect the analytic simulation of an activated wire is known as the transfer matrix method.

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Creative wire bending 61

A ( MM 1

Fig. 4. As step height (A) is increased, a linear relationship with the moment occurs.

geometric points along its length known as nodal points. The elastic beam relationships that exist between these nodes are stated in mathematic form and are related to each other.3.4 These relationships are known as transfer matrices. The wire is then constructed mathematically by superimposing these relationships until the force and deflection system at one end of the wire is directly related to the force and deflection system at the other end. The boundary conditions, which relate the forces and the activations at the termini of the wire, are then applied.

The solution occurs upon the inversion of a system matrix. Thus, the unknown activations and forces are calculated. The forces and the activations along the length of the entire wire are then calculated by means of the previously developed transfer matrices.

RESULTS Step bends

In a second-order direction, a step in a wire between two brackets produces equal and opposite vertical forces, and unidirectional couples whose moments are equal in magnitude (Fig. 1). Since the moments are in the same direction and of equal magnitude, the ratio is M,/M, = 1. This ratio defines the relative force system from a step in a wire that is identical to the Class I ratio of MJM, found with a straight wire with parallel and stepped brackets. If a/L is plotted against both force and moment, it can be seen in Figs. 2 and 3 that for a 14-mm interbracket distance, there is approximately no change in the magnitude of the force or the moment when the mesiodistal position of the step is changed. Basically, this is true of a 7-mm interbracket distance, although in this situation as the step approaches bracket 1, there is a very slight lowering in both force

e.

1210

31

347 gm 347 gm 1210 gm-mm

I lc> gm-mm

(t---------7 mm-1 1

Fig. 5. A, Force system produced by O.OlSinch stainless wire with 0.35mm step, 7-mm interbracket distance. Note large magnitude of forces and moments. All forces and moments are acting on the wire. B, Same force systems shown in A except directions of forces and moments have been reversed to show the force system acting on the teeth.

Fig. 6. Forces at bracket 1 and bracket 2 produced by V bend. As the V is placed off center, vertical forces increase. P and Ff4 denote 7-mm and 14-mm interbracket distances.

and moment. For all practical purposes then, the shift- ing of the step mesiodistally does not alter the absolute or relative force system. Furthermore, the ratio M,/M, = 1 is independent of the interbracket distance. When the height of the step is modified to reduce the deflection, a linear relationship of the moments is found (Fig. 4). This means that with smaller or larger steps, the relationship M,/M, = 1 remains constant.

The actual moments and forces over a typical 7-mm interbracket distance in an edgewise appliance using even a small 0.016-inch wire are very large: 347 g vertical force with 1210 g-mm moment (Fig. 5). It is obvious that both forces and moments are required for wire equilibrium; although the orthodontist may want either the force or the moment alone, he must accept both.

62 Burstone und Koenig

Fig. 7. Moments at bracket 1 and bracket 2 produced by a V bend. Crossover of a 0 g-mm at MZ occurs when bend is placed one third the distance between the brackets. Each a/L ratio gives a different M,/M, ratio.

The ratio M/F at each bracket is a measure of the relative amount of force and moment acting at each tooth. The M/F ratio is not a constant for a stepped wire because it is dependent upon the interbracket distance. The larger the interbracket distance, the greater M/F ratio. The significance of this ratio is that for any given magnitude of vertical force (in a second-order direction), the corresponding moments that would act would be greater with increasing interbracket distances. For example, if the interbracket distance is doubled from 7 mm to 14 mm, the force for a 0.35-mm step is reduced to 43 g. The M/F ratio for the 7-mm span (1210 g-mm) is 3.5 to 1. However, for the 14-mm span, the M/F ratio is 7.0 (301 g-mm/43 g). The M/F ratio is thus proportional to the interbracket distance.

V bends

Unlike a step, a V bend placed between two brackets radically alters the force system, depending upon the mesiodistal placement of the apex of the V (Figs. 6 and 7). A convenient measure of the mesiodistal placement is the ratio a/L. When the apex of the V bend is centered (a/L = 0.5), the moments at each bracket M,IM, are equal and opposite in direction. As the apex is moved to the left toward bracket 1, the ratio MJM, will change. If a/L is 0.4, M,/M, becomes -0.3, The moments at the two brackets are unequal, with the moment farthest away from the V-bend apex being one third the magnitude of the other moment. If one con- tinues to move the V bend farther toward bracket 1 so that a/L = 0.33, then no moment whatsoever is found at bracket 2, only a single force. Finally, if the V bend is moved farther toward M, (a/L = 0.2), the mo-

Table I. Force systems from step and V bends ---------T-- -- _-----.-- --. -_I

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Table II. MJM, ratios for V bends as a function of apex position from bracket 1

V-bend apex posirion (mm)

7 mm IBD*

Bracket 1 0.7 1.4 2.3 2.8 3.5

Bracket 2 _

*IBD = Interbracket distance.

I4 mm IBD* MJM,

1.4 0.4 2.8 0.3 4.6 0.0 5.6 -0.3 7.0 - 1.0

magnitude of the moment at M2 to 1720 g-mm. The vertical forces become 937 g. The large moments produced at M, in Fig. 8, B and C, could lead to permanent deformation of the wire. If one assumed a maximum moment at yield for the wire at 1860 g-mm, permanent deformation would occur at the point on the wire near bracket 1 since moments considerably more than 1860 g-mm are produced. When that happens, not only is there less force and moment but the entire wire bracket geometry changes, giving an entirely different M,/M, ratio and therefore a different type of tooth movement.

If the height of the apex of the V bend is increased or decreased, the relative force system M,/M, does not change. Fig. 9 shows the linear relationship between the height of the V bend and the moments produced. It should be remembered that if the height of the V bend becomes too great, (1) activations would be beyond the yield strength of the wire and permanent deformation would occur, and (2) these relationships are only true within the elastic limit.

Table II gives the actual positioning of the V bend in respect to bracket 1 in millimeters for both 7-mm and 14-mm interbracket distances. The high sensitivity of placing the apex of the V in short interbracket distances becomes evident. For a 7-mm interbracket distance, as would be expected, equal and opposite couples would be produced if the bend is placed 3.5 mm to the right of M,. If the bend is placed at 2.3 mm, the force system is radically changed so the moment at M, is completely eliminated. The difference in positioning between these two force systems is only 1.2 mm. Fur- thermore, if the bend is moved only an additional 0.9 mm toward M,, a moment will be produced at M, in the opposite direction-one third of the moment at M,. With a larger interbracket distance of 14 mm, considerably more leeway is present and small errors in

V BEND FORCE SYSTEMS

C.

4840 1720 gm-mm gm-mm

1

Fig. 8. Force systems from three a/L ratios. Interbracket distance is 7 mm; height of V bend is 0.35 mm. A, Centered V a/L = 0.5; equal and opposite couples. B, a/L = 0.29; a larger moment exists at M,. C, a/L = 0.14; forces and moments are much greater than in A and 8. In 6 and C, the wire would permanently deform.

Fig. 9. A linear relationship is shown between V-bend height (A) and M, and M, with varying a/L ratios.

the positioning of the V bend mesiodistally would not so significantly change the relative force system.

CLINICAL APPLICATIONS

The present study leads to important clinical applications in the use of two fundamental bends, a step and a V bend. Although the exact magnitudes of the forces and moments are significant and will become increasingly important as orthodontic appliances become more sophisticated, it is the relative force system that cur- rently is the most relevant. Even if the forces are not the exact magnitudes the orthodontist wishes or if the M/F ratios of a given tooth do not approach what we estimate would be the optimum, at a basic minimum it is necessary that the moments and forces at least be in

64 Burstone and Koenig

Fig. 10. Mesiodistal placement of V bend between molars and incisor has a large effect on the force system. A, Bend at a/L = 0.33 (a: measured from the molar tube; the incisor re- ceives only an intrusive force). B, Bend centered at a/L = 0.5. Incisors and molars are acted on by equal and opposite couples. If the arch is tied back, incisor root movement would occur. C, Bend placed anteriorly, a/L = 0.66. Lingual root torque and an eruptive force are acting on the incisors. All forces are shown acting on the teeth.

the correct direction, In this section on clinical applications, we will address the relative force systems. Since the relative magnitude of these forces and moments will be emphasized, no force magnitudes will be used in the diagrams. (Unlike previous equilibrium diagrams in the RESULTS section, the forces and moments are shown acting on the teeth.)

Fig. 10 shows a lateral view of an arch in which attachments have been placed only on the first molar and the four incisors. We are taking some liberties with the actual clinical situation, which might be more com- plicated since the arch curves around the corner and there may be friction and/or play between the attachments and the wire. Here we assume (1) forces acting in one plane, (2) full-bracket engagement from a rec- tangular wire, and (3) freedom to slide, eliminating mesiodistal forces.

Ffg. 11. Mesiodistal position of V bends placed between canine and molar alters force system on the molars. A, Lingual force only. B, A pure moment (couple) on the molar producing mesiobuccal rotation; no lateral forces. C, Moment producing mesiobuccal rotation. Buccal force is on the molar. All forces are shown acting on the teeth.

Let us place a V bend in a continuous arch with the apex of the V one third the distance from the molar tube to the incisor brackets (a/L = 0.33). A single intrusive force is produced on the incisor and on the molar both an eruptive force and a moment tending to move the roots forward and the crowns back. This could lead to incisor intrusion, molar tipback, and, if the arch wire is tied back, possible retraction of the upper incisors (Fig. 10, A). If we decide to place our V bend even farther posteriorly, perhaps next to the molar tube, the force system will change. This configuration would produce a moment on the central incisor tending to move the crown back and the root forward (labial root torque). This moment produced on the central incisor could be useful in preventing flaring of the upper incisor during intrusion; however, in other situations it could be an undesirable side effect, moving the root labially. When the V bend is placed off center posteriorly so that a/L is smaller than 0.33, the more the bend is increased, both the intrusive force and the labial root moment on the incisor will increase. If we place the V bend

Volume 93 Number I


one third of the distance from the molar tube to the bracket, we reach that crossover point where the moment is eliminated and any increased activation produces greater force without concern for spurious torque on the incisors.

In Fig. 10, B, the V bend is now placed in the center between the molar tube and the bracket; equal and opposite couples are produced. If the arch wire were tied back so anteroposterior forces were created, this wire could be used to move the roots lingually on the upper incisors. The roots would also move mesially on the molar, but this could be prevented by increasing the number of teeth in the posterior segment or backing up the posterior teeth with suitable headgear.

Fig. 10, C, shows the V bend now positioned closer to the central incisor bracket (a/L = 0.66 from the molar). The force system completely changes; instead of an intrusive force as in Fig. 10, A, there is now an eruptive force on the central incisor with lingual root torque. An intrusive force on the molar is also seen. By moving the V bend forward, we have a root spring for the incisors, provided the wire is tied back to prevent anterior flaring of the incisors. The eruptive force si- multaneously extrudes the incisors.

From an occlusal view, Fig. 11 shows the effects of altering the placement of a V bend between the canine and the first molar. The forces and the moments between the canines tend to cancel out across the anterior part of the arch (although side effects are still possible with highly flexible wires). For that reason we will primarily concentrate on the force system on the first molars. If the V bend is placed one third of the distance from the canine bracket to the molar tube as measured from the canine bracket, a lingual force would be produced on the molar, moving it lingually without any rotation (Fig. 11, A). If the V bend is placed close to the first molar, a force is produced moving the first molars buccally and a moment produces mesiobuccal rotation (Fig. 11, C). As would be expected, if the V bend is placed centrally between the canine and the first molar, equal and opposite couples would be produced. This allows for pure rotation of the first molar without the buccal force (Fig. 11, B). If we would like to rotate the molar with the mesial aspect out and also have a lingual force, the V bend must be placed somewhere between the a/L of 0.3 and 0.5 as measured from the canine bracket. In a situation in which mesiolingual rotation and lingual force on the first molar are required (this is sometimes indicated in uniarch extraction cases), the V bend should be placed as close as possible to the bracket on the canine (a/L greater than 0.66).

ESTHETIC BENDS - UPPER INCISORS

I I

A. n n 1 I

Fig. 12. Forces produced by anterior esthetic bends to move roots distally. A, V bend between central incisors gives equal and opposite couples. B, Step bends between central and lateral incisors give vertical forces and unidirectional couples. C, Summation of A and B. Moments are in correct direction for root movement but are not equal. Vertical forces will erupt the central incisors. All forces are shown acting on the teeth.

In the above two examples of force systems, one can see the principle of creative wire bending at work. The older, traditional approach in orthodontic treatment has been an ideal shape-drive appliance. The ideal shape thinking goes something like this: if you would like to move a molar lingually and rotate the tooth mesiobuccally, bend the wire so that it lies lingual and angled to the bracket. As we have seen, this may create many different force systems and may not produce the force system that is needed. A scientific and more creative approach is to (1) determine where the teeth should be moved and (2) establish the directions of both the force and the moment needed. Ideally, we should also establish the magnitudes of these forces and the ratio between the moment and force, but for now we will consider only the directions of the force and moment and (3) develop the proper shape in the wire to deliver that force system. This is what we mean by creative wire bending. A force-driven appliance is not fabricated by the shape of an ideal wire or by falla- ciously placing the wire in one bracket and looking to see where the wire is at the other bracket, imagining the teeth will move to that position on the wire. It is based on understanding some very fundamental relationships between the force system and what type of bend is used and where it is placed. The appliance thus becomes force driven and is shaped to deliver the desired forces rather than some arbitrary shape that some- how is related to the final position of the teeth or an exaggeration of the final position of the teeth.

The step bend, unlike the V bend, is not critical in its mesiodistal placement. For those orthodontists using

66 Bucvrone and Koenig

tipback mechanics with tipback bends such as the Tweed technique, this fact offers great convenience. Historically, bends have been placed that are called long ot short tipbacks; these bends are placed either off center or on center and may change in position during tipback mechanics. The favorable finding from this study shows that the placement of the bend does not appreciably influence the force system present.

The last clinical example that will be given is the placement of anterior esthetic bends. Fig. 12 shows an upper anterior segment in which esthetic bends have been placed between the four incisors. Although when play is present there may be other effects between brackets from one bracket down the line to nonadjacent brackets, our analysis will &ZOY~ this phenomenon and also the mesiodistal forces (in order) that the fundamental effects of this configuration can be studied.

Between the central incisors, a V bend is placed that produces equal and opposite couples. This is shown as the block A of the force system in Fig. 12, Between the central and the lateral incisors, a step bend is seen. The unidirectional couples and vertical forces of the step ate shown in block 3. If there is an addition of the force system from the V bend and the step, the total force system on each of the four brackets is shown in block C. The total force system acting at each of the incisors is very interesting, producing both desirable and undesirable effects in most patients. There ate moments tending to move the roots distally and this is desirable; unfortunately, the moments are not equal--- larger moments ate present on the central incisors in comparison to the lateral incisors. The actual moments that would be acting on the lateral incisor. of course, would be dependent upon what type of bend is placed between the lateral and the canine. which. for simplicity of analysis, we are ignoring. intrusive forces ate produced at the lateral brackets. These teeth will not be intruded provided the arch wire is connected to the posterior teeth. On the central incisors. an extrusive force is produced and in patients who previously have had intrusion of teeth or who have deep overbites, this extrusive force becomes an undesirable side effect Many orthodontists have noticed that when they place esthetic bends, the deep overbite increases. This could be anticipated by an understanding of the force system. Conversely, if one tried to correct a deep overbite by stepping the incisors apically, the moments produced would tend to move the roots mesially. For these reasons esthetic bends may not be the most sensible way to correct the axial inclinations of the incisors in many patients.

This study and the clinical applications ot results

have considered brackets in a straight hne with vana- tions of V and step bends. The usual clinical situation will have a combination of bends and malaligned brackets that ate not in a straight line. The actual force system is produced by adding together the two effects. Prcvi- ously, the force system from malaligned brackets in z; straight wire was discussed and certain principles elic ited on the basis of the relative angmation of adjacent brackets. To produce a desired force system, two approaches are possible. The first approach allows the orthodontist to use a straight wire, triggering the force system produced by the wire in malaligned brackets and adding the proper bends whose force systems are summated to the straight wire force system. Thi< IS called the super-position method. The other approach is to first contour the wire passively between two brackets and then to add the proper V or step bends. Since in this approach we remove the force system produced by a straight wire and maialigned brackets, it is referred to as the subtraction method. With small interbracket distances, it becomes increasingly difficult to use cithet the super-position or the subtraction method. Small CT- rots in wire bending can produce large errors in force systems. As the interbracket distance increases. out potential for delivering the desired force system 1s en hanced. Nevertheless. regatdless of the interbracket distance, knowing where to place the bend and knowing what type of bend to place at least heads the clinician in the right direction to achieve the proper tooth movement.

In this study we purposely have looked at the eltect of only two basic types of bends. V and step, on the force system using small deflection theory. .4tthough we have the capability of solving these problems in large deflection formulation, it is important to have n first approximation using small deflection theory tc: rh-- tablish some of the basic principles. Future studies are important to determine what differences in the actual force systems exist with large deflections. Particularly. the calculation of mesiodistal forces requires a consid- eration of both large deflection theory and bracket wire interface phenomena such as freedom to slide and play.

V and step bands ate not the only bends that can be placed to produce a given force system. Combma.. tions of V bends and steps can be used. 11-i particular. arcs or wire curvatures have many advantages in producing the desired forces and moments, more predict- ably, eliminating permanent deformation of wires and creating an activated wire shape that more closel:; ap- proximates a straight wire. A discussion of the u>e of wire curvatures instead of V and step bends IS beyond the scope of this article.

Vol1rme 91 Number 1

SUMMARY AND CONCLUSIONS

The effects of placing V bends and steps between brackets was studied by means of an analytic technique. It was found that the placement of a V or step bend and its positioning between two brackets will produce a characteristic force system.

1.

2.

3.

A step between two brackets produces unidirectional couples of equal magnitude and horizontal or vertical forces, depending upon the plane of

The mesiodistal placement of a step does not alter the force system in any appreciable amount. A V bend placed between brackets produces en-

activation.

tirely different force systems, depending upon the centering or lack of centering of the apex of the V.

4.

5.

The position of the apex of the V can be described conveniently as the ratio of a/L (the distance between the apex to one bracket over the total interbracket distance). Each a/L ratio has a characteristic force system. An orderly progression between incremental changes in the a/L ratio and the moments at the attached brackets is given.

6. With a V bend, the point of zero moment cross-


over occurs with an a/L of 0.33, where no moment is produced at the bracket farthest away from the V bend.

Creative wire bending is based upon first determin- ing a force system required for the desired tooth move-

The clinical application of understanding and using these basic relationships between wire shape and force systems is discussed.

ment and then placing the proper shape and bend to create that force system.

REFERENCES Burstone CJ, Koenig HA. Force systems from an ideal arch. AM J ORTHOD 1974;65:270-89. Burstone CJ, Koenig HA. Force systems from an ideal arch- large deflection considerations [in press]. Koenig HA, Burstone CJ. Analysis of generalized curved beams for orthodontic application. J Biomech 1974;7:429-35. DeFranco J, Koenig H, Burstone CJ. Three-dimensional large displacement analysis of orthodontic appliances. J Biomech 1976;9:793-801.

Reprint requests to: Dr. Charles J. Burstone University of Connecticut Health Center School of Dental Medicine Farmington, CT 06032-9984

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