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VOLUME XLII NUMBER 10 563

The subject of wire-bracket angles is the basisfor the application of force systems in ortho-dontics. For many years, orthodontists have had tobend archwires to create their force systems ofchoice. This was time-consuming, so the profes-

sion welcomed the introduction of prescriptionbrackets, which eliminated the need for such bend-ing. In essence, archwire shape has been exchangedfor bracket design. At the time of this writing, therewere more than 1,100 different prescription brack-ets on the market.

If the ultimate in bracket design wereachieved, including features such as built-in tip andtorque, there would be no need for so many differ-ent designs. Regardless of the designer or manu-facturer, however, static equilibrium will alwaysrequire balanced force systems. If the orthodontistis interested primarily in applying torque at a giventime, the entire system must be balanced and, evenmore important, recognized. When a particulartype of bracket design is disappointing, the searchgoes on for others that are more likely to meet thedesired objectives.

Additional factors must be considered. For abracket to meet the objectives for which it has beendesigned, the slot sizes and cross-sectional dimen-sions of the archwire must be absolutely accurate,but this is difficult to achieve in manufacturing. In

spite of these weaknesses, prescription brackets dohave a place in orthodonticsat least toward theend of treatment, when major tooth movements areno longer required.

Regardless of whether a prescription bracket

or a standard bracket with a so-called neutral slotis used, force systems will be formed as a result ofthe wire-bracket angles created when the archwireis inserted into the bracket slots. In everyday prac-tice, these wire-bracket angles can easily be formedintraorally with a Tweed loop plier. The preferredangle to be formed in the archwire in any givenplane of spacefrontal, sagittal, or occlusalis45. This angle is easy to read and can be usedeffectively with round wire without causing thewire to twist and turn within the slots.

Round wire is the best choice for almost alltooth movements discussed here, in spite of thebad name it has achieved over the years becauseof its inability to produce torque within the slot.The biomechanics involved recognize the need fortorque, but given that a moment is the product offorce and distance, it is not necessary to producetorque within the slot. It is often assumed that if awire cannot produce a moment by twisting, it isincapable of producing torque. This is not true.Round wire can easily create moments at thebrackets because of the angles formed in the sagit-tal plane of space (Fig. 1). Certain moments mayappear to be in the wrong direction, but they are

correct because they result from archwire resil-ience. With a square or rectangular wire, themoment is also a product of force and distance,although it is not usually visualized in this mannerbecause the wire is first inserted into the molartubes, as a matter of convenience (Fig. 2). Onceinserted, the wire must be twisted to engage theanterior brackets.

Round wires may be incapable of producingtorque within anterior bracket slots, but they can

2008 JCO, Inc.

Understanding and ApplyingWire-Bracket Angles

THOMAS F. MULLIGAN, DDS, MSD

Dr. Mulligan is in the private practice oforthodontics at 6843 N. Eighth Ave.,Phoenix, AZ 85013; e-mail: [email protected]. This article is adapted by per-mission from his book, Common SenseMechanics in Everyday Orthodontics II(CSM Publishing, 1040 E. OsbornRoad, Phoenix, AZ 85014).

2008 JCO, Inc. May not be distributed without permission. www.jco-online.com

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564 JCO/OCTOBER 2008

Understanding and Applying Wire-Bracket Angles

create lingual root torque in the anterior segment.Intrusive forces acting through the incisor bracketsmay produce moments when exerted labially andperpendicular to the center of resistance in theincisor segment. The moments should actually bedescribed as labial crown moments, because inci-sors will flare unless they are prevented fromdoing so, at which time the moments will producelingual root movement. Before any of this can takeplace, it is essential to know the posterior moments.If the posterior moments in a partial appliance aregreater than the anterior moment, the incisors willrespond with lingual movement.

Wire-Bracket Relationships

The three most important wire-bracket rela-tionships are the center bend, the off-center bend,

Fig. 2 Lingual root torque produced by twistingrectangular wire for insertion into anterior bracketslots.

Fig. 1 Forces and moments produced by bends in three different locations.

Fig. 3 A. Center bend. Note wire-bracket anglescompared to straight wire and angulated brack-ets. B. Resilience of activated wire. C. Two off-center bends used to produce same force ascenter bend.

A

B

C

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and the parallel (step) bend. A fourth, the cantile-ver bend, has limited clinical application. Using aTweed loop plier, 45 bends can be placed at dif-ferent locations chosen by the orthodontist. Eachlocation will produce different wire-bracket anglesand a different force system. Unlike treatment withprescription brackets, a choice must be made as towhich of three force systems will best solve theproblem at hand. Obviously, the greater the inter-bracket distance, the more choices available forplacing these bends.

The Center Bend

Whether the slots are prescription or neutral,the wire-bracket relationship determines the forcesystem. The original malocclusion automaticallycreates wire-bracket angles because of malposi-tioned teeth.

Figure 3A shows the center bend, which isplaced equidistant from the adjacent brackets. Thewire-bracket angles seen with a center bend are

the same as those seen when the wire is straightand the brackets are angulated. (Each angle is

discussed here in relation to the brackets adjacentto the bend in question.) In this case, the forcesystem consists of equal and opposite moments.The resilience of the activated wire is one of themajor factors that can lead to a visual misinterpre-tation of force systems (Fig. 3B). Two off-centerbends can be used to produce the same force sys-tem as a center bend (Fig. 3C).

Figure 4 shows the paralleling of roots fol-lowing space closure in extraction treatmentinvolving all four first premolars. The applicationis simple, but the response is significant.

Figure 5 shows a diastema between centralincisors with converging roots. A center bendprovides the equal and opposite moments requiredfor root divergence and future stability followingspace closure. Crown movement always precedesroot movement, until the spaces are closed.

Figure 6 shows the simultaneous divergenceof all incisor roots with two sectional wires. Acontinuous wire cannot deliver four pure momentswithout creating vertical forces, but because acenter bend is equivalent to two off-center bends

Fig. 5 Center bends used for root divergence incase with midline diastema.

Fig. 4 Center bends used for root paralleling inextraction treatment.

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(Fig. 3C), a wire with a center bend between thetwo central incisors and a rectangular bypass seg-ment with 45 bends at the lateral incisor bracketseach provides the necessary moments for all fourteeth and creates root divergence without thedevelopment of vertical forces. This bypass seg-ment is one of the few instances where a rectan-

gular wire is preferred.Figure 7 shows the use of two off-center bends

to produce paralleling moments at an extractionsite. In this case, a military family did not knowwhether it would remain in town for a few weeks orseveral months. Therefore, the decision was madeto extract teeth and permit eruption of the maxillarycanines while retracting the mandibular canines.Because the mandibular second deciduous molarswere still in place, the canines were retracted withoff-center bends placed just mesial to the firstmolar tubes. (The bracket or tube closest to the

bend represents the location of the larger moment,making the molars the anchor units during canineretraction.) The canine roots then required upright-ing, however, so another off-center bend wasadded, allowing the roots to be paralleled beforethe eruption of the second premolars. Much canbe accomplished while a patients family is waitingfor a job transfer or a military assignment.

The Cantilever Bend

Whereas the center bend is placed equidis-

tant from two adjacent brackets, the cantileverbend is located one-third of the way between thetwo brackets, resulting in a long section and a shortsection. Again with level brackets, this bend posi-tion produces the same wire-bracket angles aswhen the wire is straight and the brackets areangulated (Fig. 8A). It is clear that determining thetotal force system by visually interpreting the

Fig. 6 Simultaneous divergence of all incisorroots with two sectional wires.

Fig. 8 A. Cantilever bend placed one-third of distance between brackets, resulting in long section and shortsection. B. Resilience of wire results in no angle at right bracket.

Fig. 7 Two off-center bends used to parallel rootsat extraction site.

A B

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brackets is much more difficult. The straight wirewith angulated brackets may be thought to containmoments at both brackets because of the anglesformed when reading an archwire before activa-tion. When the wire is activated by being insertedinto the bracket slots, however, the resilience of thewire results in no angle at one of the brackets (Fig.8B). This is true whether the slot size is .018" or.022" and regardless of interbracket distance.

The rules for the long section vs. the shortsection still apply: the long section points in thedirection of the force produced, while the short

section points in the direction opposite the forceproduced. The bracket located closer to the bendcontains the larger moment, and the bracket fartherfrom the bend contains the smaller moment, or nomoment. Thus, the rule itself remains valid, butthe information given is now more exact, as itidentifies the smaller moment as zero.

Minor tooth movements can alter the forcesystem. Figure 9 shows an anterior segment witha continuous overlay archwire. This can be appliedin various ways to provide a pure force actingthrough the center of resistance, if desired. Figure

10 illustrates the most commonly used cantileverapproach for canine intrusion. The archwire isplaced under the incisal wings of the canine brack-ets and within the slots of the incisor bracketsduring overbite correction.

Although the cantilever bend has limitedclinical application, it is one of the possible wire-bracket angles. When prescription brackets areused, this system may be encountered withoutbeing recognized, since the wire resilience is notvisualized when reading wire-bracket relation-ships. The locations of the bends when the slotsare level provide the most reliable determinationof the force systems.

The Off-Center Bend

Although the cantilever bend is technicallyan off-center bend, it has been described sepa-rately because of its limited use. Figure 11 showsa force system quite different from those discussedso far. Although the smaller moment might appearto be in the wrong direction, Figure 11B again

Fig. 9 Cantilever bend used in anterior segmentwith continuous overlay arch.

Fig. 10 Cantilever intrusion of canine, with arch-

wire placed under canine bracket wings.

Fig. 11 A. Off-center bend. B. Resilience of acti-vated wire responsible for counterclockwisemoment.

A

B

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shows that archwire resilience is responsible forthe counterclockwise moment, as the wire crossesthe bracket at a completely different angle whenactivated than when simply visualized before acti-vation (Fig. 11A). Forces and moments are pre-dicted by the rules regarding long section vs. shortsection, although the smaller moment is moreaccurately described when the force systems aredetermined by the locations of the bends. Insteadof referring to the moment simply as the smallermoment, the specific directionclockwise, coun-terclockwise, or neutralis indicated.

The clinical example shown in Figure 12illustrates the use of an off-center tipback bend atthe molar. As stated earlier, round wire cannotproduce torque within incisor bracket slots. Al-though the moments shown previously were in thesame direction when occurring in the sagittal planeof space, the anterior moment shown here is in theopposite direction because it is formed by an intru-sive force acting labial to the center of resistancein the anterior segment, rather than by torque pro-duced within the incisor bracket slots. The toe-inbend shown in Figure 13 is another form of off-

center bend that produces the force and momenton the molar predicted by the short section inFigure 12. Changing planes of space does not affectthe force system on the molar in either situation.

Figure 14 reveals an uncomplicated approach

Fig. 12 Off-center tipback bend at molar.

Fig. 14 Simultaneous rotation of molars and correction of crossbite.

Fig. 13 Off-center toe-in bend.

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to rotating molars and correcting crossbites simul-

taneously. The secret lies in avoiding engagementof the archwire into the second premolar brackets,as seen in the upper arch. If all brackets wereengaged, the off-center toe-in bends would becomecenter bends and would therefore not provide buc-cal forces. Here, the toe-in bend, representing theshort section, results in a force acting in the oppo-site direction. It is not uncommon to find molarsthat have drifted mesially before rotating and mov-ing into lingual crossbite. This is an easy andeffective method for simultaneous correction with-out the need for archwire removal.

In Figure 15A, the second molar has beenrotated with a center bend. It now requires buccalmovement, but there is no buccal force present. Bysimply removing the ligature on the first molar,the center bend is converted to an off-center bend,which introduces a buccal force in addition to therotational moment at the molar (Fig. 15B). This iscertainly a less time-consuming approach thanremoving the archwire to place a bend. In fact,these simplified procedures can make orthodonticsmore fun and less stressful.

The final clinical example of the off-centerbend is shown in Figure 16. Locating the bend justmesial to the premolar bracket creates an anchorside to the extraction site, since the tooth closestto the bend has the larger moment. Once the spaceis closed, the moments become equal and oppositebecause the bend becomes centered. This changein the force system takes place without removingthe archwire after the beginning of space closure.

In all these clinical examples, brackets werenot added or removed to create a particular forcesystem. For example, brackets may or may not be

Fig. 15 A. Center bend used to rotate second molar, which now requires buccal movement. B. Center bendconverted to off-center bend by removing first molar ligature, creating buccal force.

A B

Fig. 16 Off-center bend placed for canine retrac-tion becomes center bend after space closure.

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present on the premolars. The various wire-brack-et angles may be introduced at any time withoutbracket changes to create the force system ofchoice.

The Parallel (Step) Bend

The final wire-bracket relationship to be in-troduced is the parallel relationship, also referredto as the step bend. Again, the wire-bracket anglesare equal whether they involve activating bendsand level slots or a straight wire and angulatedslots (Fig. 17). Compared to the off-center bend,the step bend is nothing more than an additionalbend placed at the adjacent bracket in the oppositedirection, creating two parallel short sections. Ofall the wire-bracket angles discussed, this rela-tionship involves the greatest forces, since bothof the moments are equal in magnitude and actingin the same direction. Therefore, the balancingforces must be greater to create a balancing mo-ment, which must be equal and opposite to the

net moment.Because of the higher force magnitudes, the

step relationship is most useful in the occlusalplane of space when positioning molars that havebeen displaced, as in crossbites. It may also bedesirable in the treatment of older individuals,whose physiological response is usually not ashelpful as in younger patients. The step bend canbe applied in the sagittal plane of space whenvertical forces do not pose a threat to the verticaldimension of the patient, as in brachycephalicindividuals.

Figure 18 shows two examples of the stepbend, with and without canine brackets. Technicallyspeaking, the first example shows a genuine steprelationship; the second shows bends applied inthe same manner, but because a true step relation-ship does not exist, smaller forces are produced.

The higher force magnitudes from a steprelationship can be effective in restoring central-groove relationships between the first and secondmolars and thus increasing the posterior transversedimension (Fig. 19). On the other hand, the stepbend can have distinct disadvantages in the sagit-

tal plane of space when vertical problems arepresent. This is particularly true with full appli-ances; in many cases, the step bend will create theneed for posterior high-pull headgear to beemployed to overcome extrusive movements. Ifhigher force magnitudes are not desirable, the off-center bend should be used.

The step bend creates force magnitudes asmuch as four times greater than those produced bya cantilever bend acting at the same interbracketdistance with the same degree of activation. If astep bend is sectioned into two halves, each halfconstitutes a cantilever. In a cantilever bend, theload-deflection ratio is inversely proportional tothe cube of the wire length. If the length is dou-bled, the force becomes one-eighth per unit ofdeflection. Therefore, if the length is reduced byhalf, the force will increase eightfold, assumingthat the vertical deflection of the archwire remainsconstant. This helps explain why force levels areso high with step bends.

Of course, the archwire activation mayinvolve different bends. In the step-down arch, an

Fig. 17 Parallel (step) bend.

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Fig. 18 Two examples of step bends. A. Canines bonded. B. Canines not bonded.

Fig. 19 Step bend used to increase posterior trans-verse dimension.

A B

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increase in archwire length does not result in anincrease in vertical deflection (Fig. 20). As thelength of the wire doubles, it requires only one-eighth the force per unit of deflection. With thetipback bend, however, as the wire length doubles,vertical deflection also doubles. If the force isreduced to one-eighth per unit of deflection andthere are now two units of deflection, then the newforce is two times one-eighth, or one-fourth, of theoriginal force (Fig. 21). A half-length cantileverwill require four times as much force for activationas the same cantilever with twice the length and

only one unit of vertical deflection (Fig. 21C).

Discussion

The cantilever bend has created confusionamong many orthodontists because no moment ispresent at a bracket where one would appear to be,based on visual inspection of that bracket. If read-ing an archwire before it is inserted into thebracket slot indicates the presence of an angle, itis easy to assume that a moment must be presentat that bracket. To further clarify the matter beyond

the previous discussion of resilience, an additionalexplanation will be offered.In the discussion of the various wire-bracket

angles, the extremes involved the center bend, withequal and opposite moments, and the step bend,with equal moments in the same direction. Thestep bend also had balancing forces present, whilenone were required for the center bend. In Figure22A, the brackets on the right involve oppositeangular relationships. Figure 22B shows that as thebend moves from the centered position to the finalstep position, the new angle rotates counterclock-wise. In Figure 22C, the bracket slot has movedfrom its original angular position to its final angu-lar position, which is opposite the original. A wirecannot rotate from one angle to an opposite anglewithout passing through zero. At the zero point,shown in Figure 22D, only a pure force is presentat the bracket; the archwire forms no angle withthe slot.

Archwire resilience is responsible for thezero-angular relationship whenever a bend isplaced at the one-third position between two brack-

Fig. 20 In step-down arch, increased archwirelength does not increase vertical deflection.

Fig. 21 A. Increased vertical deflection from useof tipback bend. B. Load-deflection ratio is one-fourth of original value. C. Half-length cantileverrequires four times as much force for activationas cantilever twice as long with only one unit ofvertical deflection.

A

B

C

One Unit ofDeflection

Two Units ofDeflection

x

4x

18x

12x

14x

x

Force = x

Force = x

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ets. This zero angle must exist somewhere betweenthe two extremes of a clockwise moment in thecenter bend and a counterclockwise moment in thestep bend (Fig. 22A).

Conclusion

Various wire-bracket relationships are pos-sible in a loop-free wire, allowing the orthodontistto determine a force system by the location of anactivation bend. Several of these angular relation-ships are extremely useful, particularly duringtreatment with partial appliances.

With the cantilever bend, the archwire maybe placed under the wings of the brackets ratherthan in the slots, but this system has limited prac-tical application. The center bend is especiallyuseful for paralleling roots after space closure andfor root divergence after diastema correction. Theoff-center bend is extremely effective in tooth

retraction and protraction, as well as for buccal andlingual movements, rotational corrections, andvertical movements. The parallel or step bendproduces the greatest force of all the wire-bracketrelationships and is particularly useful in theocclusal plane of space. It can be disadvantageousin patients with vertical problems when used inthe sagittal plane of space. This bend is espe-ciallyeffective for increasing the posterior trans-verse dimension and is the bend of choice for theolder patient.

SUGGESTED READING

Burstone, C.J.: Rationale of the segmented arch, Am. J. Orthod.48:805-822, 1962.

Burstone, C.J. and Koenig, H.A.: Force systems from an ideal arch,Am. J. Orthod. 65:270-289, 1974.

Mulligan, T.F.: Common Sense Mechanics Office Course,Phoenix,AZ, 2008.

Fig. 22 A. Center bend and step bend. Brackets on right involve opposite angular relationships. B. Cantileverbend. As bend moves left, wire-bracket angle on right rotates counterclockwise. C. Bracket slot moves fromoriginal angular position to final angular position, opposite original. D. Zero point, with archwire forming no

angle with bracket slot.

A

C D

B

Center Bend

Step Bend

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