effect of root and bone morphology on the stress distribution in

98

To achieve predictable and physiologicorthodontic tooth movement, the axis of rotation andlevel and location of maximum stress should be identi-fied. Once force is applied to a tooth, stress is distrib-uted instantly along the periodontal ligament to achievea state of equilibrium. The axis of rotation and locationof maximum stress level can be revealed by investigat-ing this stress distribution.

Because direct investigation inside the periodon-tium is impossible in vivo, mathematical analyticalstudies,1-5 double exposure laser holographic interfer-ometry or reflection studies,6-8 displacement transduc-ers,9 and finite element methods10-12 have been used toreveal the axis of rotation and stress distribution.

Synge13 introduced the concept of the axis of rota-tion when a tooth is subjected to force. Burstone1 and

Christiansen and Burstone2 showed the relationshipbetween the axis of rotation and the location of forcerelative to root length in a 2-dimensional parabolic root.Nikolai3 has described the center of resistance and theaxis of rotation in a rhomboid root shape with a mathe-matical analytical model. Davidian14 and Halazonetis15used a computer model to study the center of resistanceand the axis of rotation. Despite the highly sophisticatedmethods used in each of these studies, variable resultswere obtained that showed that the mechanism of stressdistribution had not been clearly illustrated.

The purpose of this study is 2-fold: (1) to betterunderstand the location of the center of resistance, therelationship between the location of force, and the axisof rotation, and (2) to identify the ideal force magni-tude associated with various periodontal conditions,such as root resorption, alveolar bone loss, and varyingroot shape.

MATERIAL AND METHODSThe lateral view of an extracted upper canine was

scanned 2-dimensionally, and the scanned image wasdivided into 80 nodes (Fig 1A). The width of each nodebecame 0.2 mm along the long axis of the tooth. Fig 1shows detailed specifications of the model used in thisstudy. A canine was chosen because it is the tooth mostoften translated a long distance. A computer program,Visual Basic 4.0, was used to calculate the geometry of

ORIGINAL ARTICLE

Effect of root and bone morphology on the stress distribution inthe periodontal ligament

Kwangchul Choy, DDS, PhD,a Eung-Kwon Pae, DDS, MSc, PhD,b Youngchel Park, DDS, PhD,c Kyung-Ho Kim, DDS, PhD,d and Charles J. Burstone, DDS, MSeSeoul, Korea, and Farmington, Conn

To achieve predictable and physiologic orthodontic tooth movement, estimating the axis of rotation of a toothand the level and location of maximum stress distributed in the periodontal ligament is essential. An extractedupper canine was scanned into a computer 2-dimensionally and divided into 80 nodes along the long axis ofthe tooth. A mathematical formula was derived, and stress was calculated on each node. The purpose of thisstudy was to reveal the center of resistance, axis of rotation, and an ideal force magnitude associated withvarious periodontal conditions, such as potential root resorption, alveolar bone loss, and varying anatomic rootshape by analyzing the stress distribution in the periodontal ligament. The study demonstrates that the locationof center of resistance changes significantly with variation of shape and length of the root embedded in alveolarbone. In contrast, in response to alveolar bone loss, the relative location of the center of resistance to total rootlength remains constant. Analysis of the stress distribution pattern in our 2-dimensional model reveals that therelationship between location of force and axis of rotation is determined by s2 (that is) a constant depends onshape and length of a root in alveolar bone. Tapered and short roots that result from alveolar bone loss orapical root resorption are prone to tipping. The optimal orthodontic force may vary depending on the maximumstress in the periodontal ligament. (Am J Orthod Dentofacial Orthop 2000;117:98-105)

aAssistant Professor, Department of Orthodontics, College of Dentistry, YonseiUniversity.bAssistant Professor, Department of Orthodontics, School of Dental Medicine,University of Connecticut.cProfessor, Department of Orthodontics, College of Dentistry, Yonsei University.dAssistant Professor, Department of Orthodontics, College of Dentistry, YonseiUniversity.eProfessor, Department of Orthodontics, School of Dental Medicine, Universityof Connecticut.Reprint requests to: Eung-Kwon Pae, DDS, MSc, PhD, Assistant Professor,Department of Orthodontics, School of Dental Medicine, University of Con-necticut, Farmington, Conn 06030-1725; e-mail, [email protected] 2000 by the American Association of Orthodontists.0889-5406/2000/$12.00 + 0 8/1/102179

American Journal of Orthodontics and Dentofacial Orthopedics Choy et al 99Volume 117, Number 1

the scanned canine. All data were manipulated withMicrosoft Excel 97.

This standard model is not an exact parabola, butapproximates Burstones parabola.1,2 With this as astandard model, conditions were varied according toroot shape and alveolar bone quantity. First, the shape ofthe root was varied (triangular, tapered, blunt, and rec-tangular), while keeping the root length constant (Fig1B). Second, the amount of alveolar bone loss and rootresorption at the apex was gradually changed at 2-mmincrements. It was assumed that the tooth is a rigidbody, the periodontal ligament is homogenous isotropic,and stress and strain are linearly related. A computer

program was encoded to calculate area of root surface,the level of stress at each node, moment, and moment ofinertia. The center of resistance and the axis of rotationwere obtained in each case.

When a line of action of force perpendicular tothe long axis of a restrained object varies the appli-cation point at which stress is uniformly distributed,the center of resistance can be calculated similar toobtaining a centroid.16 When the line of action doesnot pass through the center of resistance, this forcecan be relocated to the center of resistance with theaddition of a moment; this is the concept of equiva-lent force systems. Therefore, the stress at an arbi-trary point in the periodontal ligament is the sum ofthe stress of the force applied at the center of resis-tance and its accompanying moment.

= MI +

AF

can be derived (M, moment; y, distance from center ofresistance to the node; I, moment of inertia; F, appliedforce; A, area of root).16

The node where stress is zero would be the axis ofrotation. As the stress is linearly increased from thispoint, maximum stress (max) should be found at themost distant node from the axis of rotation where y = c

Fig 1. A, Specifications of standard model used in thisstudy: a is distance between applied force and center ofresistance; b is distance between center of resistanceand axis of rotation. B, Schematic drawing of variousroot shapes and location of center of resistance.

Table I. Typical results on location of the center of resis-tance1,3,7,9,10

Model used in study (method) Percentage of root length*

2D parabola of Burstone (calculation) 403D parabola of Burstone (experiment) 332D Rhomboid of Nikolai (calculation) 55Tooth of Burstone (experiment) 27 - 423D of Tanne (calculation) 242D standard model in this study (calculation) 41.8*Measured from the alveolar crest

Table II. Changes in location of the center of resistancein relation to varying root shape, amount of alveolarbone loss, and amount of apical root resorption

Percentage of total Distance fromTooth length of the root bracket slot (mm)

Rectangular root 50.0 13.4Blunt root 43.9 12.4Standard root 41.8 12.1Tapered root 39.0 11.6Triangular root 33.0 10.8Amount of alveolar boneloss in standard root2 mm 41.2 13.24 mm 40.9 14.36 mm 40.7 15.58 mm 40.4 16.610 mm 40.1 17.812 mm 40.1 19.014 mm 41.0 20.2

Amount of apical rootresorption in standard root2 mm 44.8 11.74 mm 46.7 11.06 mm 47.9 10.28 mm 48.8 9.310 mm 49.7 8.412 mm 50.3 7.4

A

B

100 Choy et al American Journal of Orthodontics and Dentofacial OrthopedicsJanuary 2000

(c is distance from center of resistance to apex of theroot). Maximum stress was obtained using

max = MI +

AF

.

The relationship between the location of appliedforce and the axis of rotation was determined with a b = s2, where a is the distance between the appliedforce and the center of resistance, b is the distancebetween the center of resistance and the axis of rota-tion, and s2 indicates the variance of distribution ofstress (Fig 1A).

Based on these formulas, the location of the centerof resistance, s2, and a location and level of maximumstress was determined for that of varying root form,amount of loss of the alveolar bone, and amount of rootresorption. Physiologic optimal magnitude of ortho-dontic force was also estimated with respect to themaximum stress level.

RESULTSChange of the Center of Resistance in Accordancewith the Variation in Root Shape

The location of the center of resistance wasreported to be between 24% and 55% of root lengthmeasured from alveolar crest to root apex dependingon boundary conditions. Typical results of previousinvestigations are summarized in Table I. The center ofresistance of the standard model used in this study was41.8% of total root length measured from alveolarcrest. Assuming that the bracket is placed at the centerof the crown, the center of the resistance would be 12.1mm from the bracket slot (Table II). In other words, aM/F ratio of 12.1 is required at the bracket to translatethis standard model. This value was very similar to thatof the 2-dimensional model of Burstone1 and Chris-tiansen and Burstone2 with a parabolic-shaped root(40%). The evaluation of root shape on location of thecenter of resistance showed that as the shape of the root

Fig 2. Position of center of resistance (Cr) in responseto the amount of alveolar bone loss in relation to thebracket (A) and in relation to the root length (B).

Fig 3. Position of center of resistance (Cr) in responseto apical root resorption in relation to the bracket slot (A)and in relation to the root length (B).

B

A

B

A


was gradually tapered from a rectangular to a triangu-lar shape with the root length and maximum width keptconstant, the location of the center of resistance was50.0% in rectangular, 43.9% in blunt, 41.8% in stan-dard, 39.0% in tapered, and 33.0% in triangular shape,respectively (Fig 1B, Table II). It was found that thelocation of the center of resistance is not constant eventhough the width and length of the root is kept constant,but it was affected by the anatomic shape of the root.The more the root tapers, the more coronal the locationof the center of resistance moved.

Change of Center of Resistance in Relation to theAmount of Alveolar Bone Loss

As the amount of alveolar bone loss increased, thecenter of resistance gradually moved apically. Forinstance, the center of resistance moved 1.1 mm api-cally when 2 mm of alveolar bone was lost. In anextreme example, loss of 14 mm of alveolar bonewould cause the center of resistance to move apically8.1 mm. It was noted that the relationship betweenalveolar bone loss and changes in the center of resis-tance was almost linear when measured from thebracket. The relative location of the center of resistanceto the total root length is quite constant from 40.1% to41.8% (Table II, Fig 2). Change of Center of Resistance in Relation toRoot Resorption

When the apical part of the root is resorbed, the shapeand length of the root change simultaneously. As the rootwas shortened, the center of resistance moved coronally.The center of resistance moved 0.4 mm coronally whenroot was resorbed 2 mm; 12 mm of root resorptionresulted in the center of resistance moving 4.7 mm coro-nally. The relative location of the center of resistance tototal root length varied from 41.8% to 50.3% as theamount of root resorption increased (Table II, Fig 3B).Pattern of Stress Distribution in the PeriodontalLigament in Relation to Root Shape

Table III showed changes in the value of s2 accord-ing to various root shapes. The relationship betweenthe location of force and the axis of rotation based onthe formula is shown in Table IV. The value of s2decreases as the shape of the root tapered. Fig 4demonstrates that a minor change in positioning a forcearound the center of resistance results in a large changein the axis of rotation depending on the root shape. Forexample, in a triangular root that had a smaller s2, aclinically insignificant error such as 1 mm in posi-tioning of a horizontal force to the center of resistancemoves the axis of rotation from the root apex to the

bracket slot. With a rectangular root with a large s2,changes in the axis of rotation were much smaller thanthose in a triangular root (Table IV, Fig 4).Pattern of Stress Distribution in the PeriodontalLigament in Relation to the Amount of AlveolarBone Loss and Root Resorption

Length of the root embedded in alveolar bonedecreased as the amount of alveolar bone loss or rootresorption increased. The value of s2 was affectedgreatly by the amount of alveolar bone loss and rootresorption (Table III). Location of the axis of rotationin relation to amount of alveolar bone loss and rootresorption is shown in Fig 4. The data suggested that avery exact placement of force is required for the givenaxis of rotation as s2 decreases.

Optimal Magnitude of Orthodontic ForceThe optimal magnitude of orthodontic force was

determined based on the pattern of stress distribution inthe periodontal ligament. For practical reasons, we con-sidered maximum stress (max) in the periodontal liga-ment only. It was postulated that a stress greater thanblood pressure could make the capillary blood vessel inthe periodontal ligament collapse and can in turn causea blocking of the blood supply. Conversely, if maximumstress is less than the blood pressure, the capillary

Table III. The value of s2 in response to anatomic rootshape, amount of alveolar bone loss, and amount of api-cal root resorption; s2 decreases as the root becomestapered and as alveolar bone loss and root resorptionincrease. A decrease of s2 is associated with greater sen-sitivity to tooth tipping. Tooth s2

Rectangular root 27.929Blunt root 23.215Standard root 17.520Tapered root 16.349Triangular root 9.540Amount of alveolar bone loss in standard root

2 mm 13.5054 mm 9.9216 mm 7.4518 mm 4.39310 mm 2.49012 mm 1.13114 mm 0.292

Amount of apical root resorption in standard root2 mm 14.8734 mm 11.3976 mm 8.0558 mm 5.19110 mm 2.95812 mm 1.332


would not be collapsed in any part of the periodontalligament. If we assume that a rate of tooth movement isproportional to the magnitude of force, the optimumforce level would exist when the force producing themaximum stress was equal to the blood pressure.

Because normal systolic blood pressure is 120 mmHg (1.56 g/mm2) and the surface area of the root of ourstandard model is 94.1 mm2, more than 147 g of forcefor translation would result in a collapse of all capillar-ies at the compression site (147 g/94.1 mm2 = 1.56g/mm2); 74 g of force for tipping at the apex, 20 g foruncontrolled tipping, and 83 g for root movement withthe center of rotation at the tip of a bracket would ren-der 1.56 g/mm2 of stress in maximum.

DISCUSSIONWhen orthodontic force is applied to a tooth, stress

is distributed throughout the periodontal ligament andreaches a state of equilibrium within 2 minutes.8,17Biologic responses such as bone resorption and apposi-tion follow in response to strain in the periodontal lig-ament. An instant tooth movement within the peri-odontal ligament is called a primary displacement,whereas the movement followed by a biologic responseis called a secondary displacement. The exact relation-ship between these 2 types of tooth movement is com-pounded by biologic variation. For instance, tissueresponse to orthodontic force in older patients is muchslower than that in younger patients. Thus, the sec-ondary displacement would occur at a slower pace.18Nevertheless, secondary displacement can be predictedby the primary tooth movement that, in turn, can bedetermined by analyzing the stress distribution patternin the periodontal ligament.1,19,20 In our idealizedmodel, uniform strain should be found along the com-pression site to allow the teeth to be translated. Thelocation of force at which the tooth is translated isdefined as the center of resistance. The position of thecenter of resistance is influenced not only by alveolarbone loss or apical root resorption, but also by the vari-able shape of the root. When apical root resorptionoccurs, the root shape becomes more rectangular withthe center of resistance at 50% of total root length. In

contrast, during alveolar bone loss, the root anatomyremained unchanged. This may explain why the rela-tive location of the center of resistance remained con-stant (40.1% to 41.8%) during alveolar bone loss.

The results of this study demonstrate the importanceof s2 a concept developed by Ngerl et al.9 If we imag-ine a tooth as a stick and periodontal ligament fibers assmall springs, the center of resistance is a mean value ofthe distribution of the spring constant. The schematicdrawing (Fig 5) shows that continuous distribution ofthe springs can be substituted with 2 large and morerigid springs per side with the distance s apart from cen-ter of resistance (Fig 5A). The value s may be consid-ered as a standard deviation of the distribution of springconstant (Fig 5B). In the formula developed by Chris-tiansen and Burstone2 and Ngerl et al,9 y (M/F) =0.068h2, 0.068h2 is a special case of s2 when root shapeis exactly a parabola. Fig 5C shows zero s. Forceapplied to other than the center of resistance will rotatethe tooth around the center of resistance regardless ofthe location of the force and the magnitude of force.Translating the tooth is almost impossible in such acase. In Fig 5D, the force is applied at a = s. Althoughthe force application point is same as in Fig 5C, thetooth will rotate around the point b, and less tipping willbe developed. Therefore, s2 or s may be a constant thatrepresents the distribution of stress when translationoccurs. Clinically, the greater the s2, the less the toothwill tip. Thus, s2 can be considered as an index for sen-sitivity of tipping. The center of resistance and s appearto be analogous to the mean and standard deviation instatistics. As s becomes smaller, the location of forceshould be more precisely located for a given axis ofrotation. An s2 is influenced by shape and length of theroot embedded in alveolar bone. The axis of rotationwas influenced only by the constant s and the locationof applied force, yet it was independent from a modulusof elasticity (E) of the periodontal ligament or the mag-nitude of force. It may be a common misunderstandingthat less tipping will occur with a lighter force.

Clinicians are interested in an optimal magnitude oforthodontic force. The optimal force magnitude shouldbe modified depending on the type of tooth movement.

Table IV. Relationship between the axis of rotation and M/F ratio with varying root shapesLocation of the axis of rotation (Type of tooth movement)

Between Cr and apex Apex Infinity BracketM/F ratio at bracket (uncontrolled tipping) (controlled tipping) (bodily movement) (root movement)

Standard model 0 10.2 12.1 13.6Rectangular root 0 9.9 13.4 15.5Triangular root 0 9.8 10.7 11.6


Smith and Storey21 studied clinically optimal orthodonticforces. Unfortunately, a tipping occurred with an appli-ance designed to produce translation. Their results illus-trated that 150 to 200 g was optimal for single canine

movement. The actual force that the tooth felt shouldhave been much lower because friction was not consid-ered in the study. In the present study, an optimal ortho-dontic force was derived based on the assumption that thestress does not exceed the capillary blood pressure.Approximately 222 g (= 74 g 3) per side would be opti-mal for a controlled tipping (center of rotation at apex) of6 anterior teeth retraction. This approximates the valueBurstone et al22 suggested. Considering the result ofWeinstein23 (2 g) and this study (20 g), it is noted thateven a very light force could result in an uncontrolled tip-ping produced by a horizontal force on the crown of thetooth. A single force applied at the crown, such as retrac-tion of anterior teeth with round wires or by a removableappliance, can cause uncontrolled tipping. If heavierforces are used, there may be a higher risk of root resorp-tion. The force level should also be reduced in cases ofalveolar bone resorption or root resorption, not onlybecause the total root area is small, but also because, witha smaller s2, these teeth tip more easily.

Fig 4. A, Relationship between the location of force andthe axis of rotation (Arot) in response to alveolar boneloss. B, Relationship between the location of force andthe axis of rotation (Arot) in response to apical rootresorption. The value in parentheses of the graph insetindicates s2. Some lines in the graphs are omitted pur-posely for better visualization.

Fig 5. Stick diagrams of tooth; springs represent peri-odontal fibers. A, Continuous periodontal fibers are rep-resented by a series of small springs. B, Continuousperiodontal fibers are represented by 2 stiff springs perside. C, When s = 0, force acting on any point other thanCr will produce a rotation around Cr. D, When s = a, theaxis of location lies at distance b from Cr. Note that thestick shows less tendency to tip in D compared with C.

A

B

B

A

DC


The model used for the current study has been pur-posely oversimplified. It is 2-dimensional and assumesthe periodontal ligament to be homogenous andisotropic with a linear stress-strain relationship. It alsoignores variation in periodontal ligament thickness, aswell as irregularity in the bone and tooth. Nevertheless,it has been useful in delineating some fundamental rela-tionships between applied force and tooth movement.

CONCLUSIONSThe distribution of stress in the periodontal liga-

ment during tooth movement was investigated to findthe location of center of resistance and to elucidate therelationship between force and the axis of rotation. Inaddition, optimal magnitude of orthodontic forces invarious simulated conditions were examined. Signifi-cant findings are the following:1. The location of the center of resistance in the upper canine

is found at 42% of the root length measured from the alve-olar crest. The more a root tapers, the more the center ofresistance moves coronally.

2. As the amount of alveolar bone loss increases, the centerof resistance moves apically proportionally, but its per-centage of total root length remains unchanged.

3. As the amount of apical root resorption increases, the centerof resistance moves coronally with a nonlinear relationship.

4. The value s2, which explains the distribution of stress,decreases with increased root taper, increased alveolarbone loss, and increased apical root resorption. A toothwith a small s2 may therefore be more prone to tipping asit is difficult to manage its axis of rotation.

5. Optimal orthodontic force for tooth movement may bedefined by maximal limit of stress that capillary bloodvessels in the periodontal ligament can withstand. Theoptimal orthodontic force varies with the location of axisof rotation. Optimal forces increase in tooth movementapproaching translation.

REFERENCES

1. Burstone CJ. The biophysics of bone remodeling during orthodontics-optimal force con-sideration. In: Biology of tooth movement. Boca Raton, FL: CRC Press; 1989. p. 321-33.

2. Christiansen RL, Burstone CJ. Centers of rotation within the periodontal space. Am JOrthod Dentofacial Orthop 1969;55:353-69.

3. Nikolai RJ. Periodontal ligament reaction and displacements of a maxillary centralincisor loading. J Biomech 1974;7:93-9.

4. Steyn CL, Verwoerd WS, Merwe EJ, Fourie OL. Calculation of the position of the axisof rotation when single rooted teeth are orthodontically tipped. Br J Orthod1978;5:153-6.

5. Sutcliffe WJ, Atherton JD. The mechanics of tooth mobility. Br J Orthod 1980;7:171-8.6. Burstone CJ, Every TW, Pryputniewicz RJ. Holographic measurement of incisor

extrusion. Am J Orthod 1982;82:1-9. 7. Burstone CJ, Pryputniewicz RJ. Holographic determination of center of rotation pro-

duced by orthodontic forces. Am J Orthod Dentofacial Orthop 1980;77:396-409.8. Burstone CJ, Pryputniewicz RJ, Bowley WW. Holographic measurement of tooth

mobility in three-dimensions. J Periodont Res 1978;13:283-94.9. Ngerl H, Burstone CJ, Becher B, Messenburg DK. Center of rotation with transverse

forces: an experimental study. Am J Orthod Dentofacial Orthop 1991;99:337-45.10. Tanne K, Koenig HA, Burstone CJ. Moment to force ratios and center of rotation. Am

J Orthod Dentofacial Orthop 1988;94:426-31.

11. Tanne K, Nagataki T, Inoue Y, Sakuda M, Burstone CJ. Patterns of initial tooth dis-placements associated with various root length and alveolar bone height. Am J OrthodDentofacial Orthop 1991;100:66-71.

12. Yettram AL, Wright KWJ, Houston WJB. Center of rotation of a maxillary centralincisor under orthodontic loading. Br J Orthod 1977;4:23-7.

13. Synge JL. The tightness of teeth, considered as a problem concerning the equilibriumof a thin incompressible elastic membrane. Philos Trans R Soc Lond, Series A 231,1933;435-70.

14. Davidian EJ. Use of a computer model to study the force distribution on the root ofthe maxillary central incisor. Am J Orthod Dentofacial Orthop 1971;59:581-8.

15. Halazonetis DJ. Computer experiments using a two-dimensional model of tooth sup-port. Am J Orthod Dentofacial Orthop 1996;109:598-606.

16. Popov EP. Mechanics of materials. Englewood Cliffs, NJ: Prentice-Hall 2nd Ed., 1978.17. Pryputniewicz RJ, Burstone CJ. The effect of time and force magnitude on orthodon-

tic tooth movement. J Dent Res 1979;58:1754-64.18. Reitan K, Rygh P. Biomechanical principles and reactions. In: Graber TM, Vanarsdall

RL Jr, eds. Orthodontics, current principles and techniques. St Louis: CV Mosby;1994. p. 96-192.

19. Burstone CJ. The biomechanics of tooth movement. In: Kraus BS, Reidel RS, editors.Vistas in orthodontics. Philadelphia: Lea & Febiger; 1962. p. 197-213.

20. Burstone CJ. Application of bioengineering to clinical orthodontics. In: Graber TM,Swain BF, eds. Orthodontics, current principles and techniques. St Louis: CV Mosby;1985. p. 194-227.

21. Storey E, Smith R. Force in orthodontics and its relation to tooth movement. Aus JOrthod 1952;56:11-8.

22. Burstone CJ, Steenbergen EV, Hanley K. Modern edgewise mechanics and the seg-mented arch technique. Glendora, CA: Ormco Co; 1995.

23. Weinstein S. Minimal forces in tooth movement. Am J Orthod Dentofacial Orthop1967;53:881-903.

APPENDIX

1. Tooth translates when force passes through center ofresistance. Stress is evenly distributed along the peri-odontal ligament for a tooth to be translated (AppendixFig 1A). Sum of all forces should be zero to satisfy thefirst condition of state of equilibrium.

F = 0

Sum of all moments measured from the X axis should bezero to satisfy the second condition of state of equilibrium.

F d y = 0

d = Ay

if d = 0 then y = 02. Tooth rotates around center of resistance when a couple is

applied. When a couple is applied to a tooth, maximumstress(max) is found at the most distant node from theaxis of rotation (z axis) (Appendix Fig 1B). Stress at arbi-trary nodes are linearly increasing from the z axis. There-fore, stress at arbitrary node is

yc

max

To satisfy the first condition of equilibrium, the sum of allforces acting at each node along the periodontal ligamentshould be zero.


( yc

max) = 0

As c or max is not zero, y = 0In y = yA, as A is not zero, y = 0

3. The relationship between location of force and axis ofrotation is a b = s2 where a is the distance between theforce and the center of resistance, b is the distancebetween the center of resistance and the axis of rotation.As a tooth is in equilibrium,

Mz = 0

M = ( yc

max) = m

cax y2

max = (I = y2A)

= MIy

Mc

I

As force and moment is applied simultaneously at the cen-ter of resistance, a total stress is

= MyA +

FA

As b = y, and M/F = a

ab = IA

(Appendix Fig 1B)

As IA

=

y

2

resembles the variance

A= s2

ab = s2

As maximum stress (max) occurs where y = c,

max = Mc

I+

F

A

Appendix Fig 1

F ForceM Momenta Distance between center of resistance and forceb Distance between center of resistance and axis of rotationy Distance between node and center of resistancec Distance between center of resistance and apexA Total area of rootA Area of a nodeI Moment of inertia Stressmax Maximum stress

A B

effect of root and bone morphology on the stress distribution in

Documents