Experimental stress analysis of dental restorations. Part VII. Structural design and stress analysis of fixed partial dentures

Experimental

stress analysis

Part VII.

Structural

of fixed

partial

design

of dental and

restorations.*

stress analysis

dentures

M. Kamal El-Ebrashi, D.D.S., Ph.D.,** Robert G. Craig, Ph.D.,*** and Floyd A. Peyton, D.Sc.**** University of Michigan, School of Dentistry, Ann Arbor, Mich.

lh e principles of engineering design should be used in the structural design of fixed partial dentures as well as the biologic, esthetic, and mechanical restrictions of the oral cavity.’ The conventional structural design is primarily concerned with the analysis of given structures, using the conventional equations of strength of materials.2z 3 Bending moments in models representing fixed partial dentures were studied,’ and it was found that bending moments of semi-fixed partial dentures were higher than those estimated for fixed partial dentures attached at both ends. Load carrying capacities of dental beams were theoretically determined by Brumfield”g 6 who asserted that the most important factor was depth. The relationship of design to restorative materials was generally discussed in several papers.‘-g The first report concerning the measurement of stresses in fixed partial dentures using a brittle coating technique was published in 1965. lo Strain gages were also used to study the stress distribution on gold and chromium alloy bridges.ll Condensed from a dissertation in partial fulfillment of the requirements for the Ph.D. degree, University of Michigan, Horace H. Rackham Graduate School, 1968. Presented in part before The American Academy of Crown and Bridge Prosthodontics in Chicago, Ill., The Society of Experimental Stress Analysis, Chicago, Ill., and The International Association for Dental Research, San Francisco, Calif. This investigation was supported by United States Public DE-01817 from The National Institute of Dental Research, Bethesda, Md., and by United Arab Republic Graduate Grant, andria, School of Dentistry, Alexandria, Egypt, U.A.R. *Part I, J. PROS. DENT. 17: 277, 1967; Part II, 17: 292, Part IV, 22: 346, 1969; Part V, 22: 565, 1969 and Part VI, **Research

Health Service Research Grant National Institutes of Health, from The University of Alex1967; Part III, 22: 333, 1969; 22: 663, 1969.

Associate.

***Professor

and Chairman,

****Professor

of Dentistry.

Department

of Dental

Materials.

177

178

El-Ebrashi,

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J. Pros. Dent. February, 1970

/‘d

= 0*392

Fig. 1. All fixed partial denture models have a constant radius of curvature to depth ratio of the fixed joints.

The purpose of this study was to investigate stresses in the various components of fixed partial dentures restoring the posterior teeth of the lower jaw, and to measure quantitatively the effects of certain modifications in structural design on the stresses in the restorations using two-dimensional photoelasticity.

MATERIALS AND METHODS Two-dimensional photoelastic methods were used previously to investigate certain dental preparations,l”? I” and were used again in this study. Several models of fixed partial dentures were constructed (Fig. 1). Shoulderless margins and anatomic occlusal reduction were incorporated in Model 1. Rounded shoulders and flat occlusal reduction were incorporated in Model 2, while Model 3

“N”;lp?.? $3

Stress analysis

u

of

dental

restorations

179

was a cantilever fixed partial denture. All U notches, except when indicated, had radius-depth ratios of 0.392. Other similar fixed partial dentures were constructed with V and irregular U notches deliberately included in the region of the fixed joints for comparative reasons. The birefringent materials used in this study were polyester* and epoxy” resins in standard sheets. The stress optical coefficient of polyester was 40 p.s.i. per fringe per inch and that of epoxy was 60 p.s.i. per fringe per inch. Polyester was used for constructing the substructure, and epoxy was used in making the components of of the fixed partial dentures. I4 The two materials were used in the construction composite photoelastic models. Improved artificial stone? was used to represent dental cement in luting the composite photoelastic models. Static loading procedures were used at preplanned sites to represent occlusal loads in the mouth. Polaroid film (P/N No. 55) was used to record isochromatics in accordance with recData reduction was performed using the grid ognized photoelastic procedures. method, which helped in the mathematical integration procedure (shear difference method) to separate the principal stresses.” RESULTS AND

DISCUSSION

The interpretation was accomplished based on evaluation of cause and effect. The magnitude of maximum shear-stresses and principal stresses were calculated using conventional stress-analysis equations. lj Both full order and fractional isochromatic fringe orders were determined. Models 1 and 2 were loaded on the pontic using a 50 pound concentrated load using a dark field arrangement. The load(Fig. 2). Th e f rm ’ g es were photographed ing site was located to the mesial side of the pontic which explains the presence of more fringes at the mesial joint as compared to those at the distal fixed joint. Both models showed that even at low loads of 50 pounds, areas of concentration of stress appeared at the fixed joints and both areas were tensile in nature. Comparing the distal fixed joint (D) in both models, Model 2 had two isochromatic fringes at the center of the joint while Model 1 had three isochromatic fringes. The letter P denotes some parasitic birefringency due to the luting process of the retainer to the tooth structure. Comparing the mesial fixed joint (M) in both models, Model 2 had three isochromatic fringes while Model 1 had five isochromatic fringes at the center of the mesial joint. A fourth fringe was detected at the base of the U notch above the mesial joint in Model 2, while an isochromatic fringe of an order of 7 was detected at the same site in Model 1. Since in the mouth, the ratio of the radius of the groove to its depth (r/d) is not precisely duplicated, another model similar to Model 2 was used with V notches at the molar-pontic contact area. The U notches were not uniform at the bicuspidpontic contact area (Fig. 3). A “loader” was cut to represent upper teeth, and seven contact areas were established between the “loader” and the model. The areas of the fixed joints again showed high stress concentrations. Since the isochromatic fringes were highly complicated, a diagram was drawn to demonstrate the fringe *PSM-1

and PSM-5 sheets made by Photoelastic,

fDuroc,

Ransom and Randolph

Company,

Toledo,

Inc., Malvern, Ohio.

Pa.

180

El-Ebrashi,

Craig, and Peyton

Fig. 2. Dark-field isochromatics of Model 1 (top) and Model 2 (bottom) loaded on the pontic with a 50 pound load. In Model 1 (upper photograph), five fringes are shown at the me&l area (300 p.s.i. shear stress), while three fringes (180 p.s.i. shear stress) appeared at the distal area.

orders along the free boundaries of the model (Fig. 3). Alternate areas of tension and compression existed on both the upper and lower surfaces of the model. Cantilever Model 3 and a modified cantilever model with a sharp V notch above and below the fixed joint are shown in Fig. 4. A load of 10 pounds was used to produce a bending moment of 21 pound-inch. Ten and one-half fringes were counted at the tip of the V notch. The cantilever model with U notches resulted in a reduction of the number of isochromatic fringes, from 10.5 to 5.5 (Fig. 4). When the bending moment was increased to 40 pound-inch (Fig. 5) the number of the isochromatic fringes almost tripled, from 5.5 to 14. The letter H in Fig. 5 denotes an area of high concentration of stress in bending. By applying three as nearly equal moments as possible, but varying the site of the load, the absolute number of the fringes remained approximately the same (Fig. 6). The fringes were counted along the line AB. In general, the crowding of fringes at the fixed joint in the cantilever Model 3 was more than the corresponding crowding in Models 1 and 2. Modified Model 2 (with irregular U and V notches) was chosen for complete analysis of stress to compare it with Model 2 having only U notches (r/d ratio of 0.392). rry, the shear stress, was computed along the line AB and CD (Fig. 7). The multiple point loader was used for loading the model, because it represents a realistic loading device similar to the oral conditions of loading. Lines AB and CD passed from one proximal end of the model to the other end, and both passed

~~l:K‘2”

Stress analysis

+ TENSION - COMPRESSION

of dental

restorations

( LOAD.

181

180 lb8 )

Fig. 3. Modified Model 2 had irregular U and V notches at the fixed joints and was loaded using a multiple contact loader at a total load of 150 pounds. Light-field isochromatics (top) and fringe orders at the free boundaries (bottom) are shown. The numbers on the diagram indicate the fringe order, while + and .- denote tension or compression respectively.

through the two fixed joints. The beam model did not transmit a constant bending moment, and it could not be considered to be in pure bending. Therefore, the shear stress component did not vanish. rzy was computed from the photoelastic data, and it ranged from -200 p.s.i. to +410 p.s.i., i.e., shear stresses were not constant along the length of the beam because the unsymmetrical beam was loaded unsymmetrically. Modified Model 2 was loaded with a single loader on the pontic and the principal stresses were separated along a line midway between AB and CD (Fig. 8). (TV was compressive (-790 p.s.i.) near the distal contact area of the pontic (point 8 on the x-axis), while cI was compressive (-75 p.s.i.) at the same point. Both ml. and (TV were tensile near the mesial contact area of the pontic (point 21 on the x-axis). c2 had the largest algebraic value. The principle stresses were separated for Model 2 having only U notches along a line comparable to that selected for modified Model 2 (Fig. 8). The results are

182

El-Ebrashi,

J. Pros. Dent. February, 1970

Craig, and Peyton

Model 3 with a sharp V notch (top) is contrasted with the same model Both models were loaded with 10 pounds to produce a kbending with U notches (bottom). man lent of 21 pound-inches.

Fig. 4. Cantilever

Fig.

5. Cantilever

Model

3 was subjected

to a 40 pound-inch

bending

moment.

Stress analysis

I

of dental

Three

CL 4-v W

restorations

equal

183

momenfs.

B

O 3:

E

2-

I-

“A

“’

’

I.5

h

I

2.5

3

3.5 R INCH

DISTANCE ALONG LINE A-B Fig. 6. Fringe when

order it is subjected

is plotted as a function to three approximately

of position along line AB for cantilever equal bending moments.

Model

3

Q 400

300

200

mo

‘;; 0 P 100

200

C-D ;

MODIFIED Fig. 7. Shear stress, rev, along AB and CD in modified and V notches, at 150 pounds total load.

BRIDGE Model

2, which

‘2’

contains

irregular

U

104

El-Ebrashi,

Craig, and Peyton

900

800 700 600 500 400 300 200 100 ‘i 0 100 200 300 400 so0 600

(Bridge

2)

700 800

Fig. 8. The principal stresses are indicated along a line midway between AB and CD in modified Model 2 when loaded on the pontic with a 50 pound load. The difference between the principal stresses (CI - u2) is shown as the solid-line curve, u1 is denoted by the dashed line curve, and oz is denoted by the interrupted black line curve.

shown in Fig. 9. On comparing Figs. 8 and 9, there is a marked improvement of the design mainly due to the elimination of the V notches which were present in the fixed joint areas in the modified Model 2. The shear stress at point 8 in modified Model 2 with irregular U and V notches was about t850 p.s.i. (Fig. 8)) and at the same point in Model 2 with U notches (Fig. 9)) the shear stress was reduced to +420 p.s.i., or a fifty per cent reduction. The same was found for c1 and o2 values on comparing both models, e.g., at point 21, ul was +195 p.s.i. in the modified model, and it was reduced to +lOO psi. in Model 2 with only U notches. The data indicate a marked improvement in the structural design of the fixed joints in fixed partial dentures by the mere fact that concentration of stress in these critical areas was reduced by fifty per cent when U notches with r/d ratios of 0.392 replaced the irregular U and V notches.

Volume 23 Number 2

Stress analysis

of dental

restorations

185

C +

2

40(

3oc PO< lot 0

lot

20(

10(

BRIDGE 2 ( r/d = 0.392) ,50 lb I

Fig. 9. The principal stresses in Model 2 with only U notches are indicated line as in Fig. 8 when loaded with 50 pounds on the pontic.

along a comparable

CONCLUSIONS 1. Fixed

partial dentures do not function in bending as a symmetrical beam. of tension and compression were demonstrated when multiple contact loading was used. 2. The weakest part in posterior fixed partial dentures is the fixed joint (soldered area) since it exhibited high concentration of tensile and shear stresses. 3. V-grooves at the fixed joints should be avoided, and should be replaced by regular shaped U-grooves. The ratio of the radius of the groove to its depth (r/d) should be as large as possible to reduce the tensile and shear stresses at these critical locations. 4. Fixed partial dentures with chamfers and flat occlusal protection should be Alternate

areas

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structurally oc~lusal

Craig,

and Peyton

J. Pros. Dent. February, 1970

stronger than those with knife-edged proximal margins and anatomic protection as indicated by the number of isochromatics at the fixed joint

area. 5. Cantilever fixed partial dentures had much higher stresses at the fixed joint than fixed partial dentures that were attached at both ends. References 1. Ward, N. L., and Campbell, V. P.: Design and Construction of Bridges, Dent. Pratt. 4: 104-115, 1953. 2. Barnett, R. L.: Survey of Optimum Structural Design, Experimental Mechanics 6: 19A26A, 1966. 3. Brumfield, R. C.: Fundamental Mechanics of Dental Bridges, in Tylman, S. D., and Tylman, S. G., Theory and Practice of Crown and Bridge Prosthodontics, ed. 5, St. Louis, 1965, The C. V. Mosby Company, p. 1118-1196. 4. Smyd, E. S.: Mechanics of the Dental Structures: Guide to teaching dental engineering at undergraduate level, J. PROS. DENT. 2: 668-692, 1952. 5. Brumfield, R. C.: Load Capacities of Posterior Dental Bridges, J. PROS. DENT. 4: 530547, 1954. 6. Brumfield, R. C.: Dental Gold Structures, Ann Arbor, 1949, Edwards Bros., pp. 29-72. 7. Mahler, D. B., and Terkla, L. G.: Analysis of Stress in Dental Structures, Dent. Clin. N. Amer., Nov., 1958, pp. 789-798. 8. Mahler, D. B., and Terkla, L. G.: Relationship of Cavity Design to Restorative Materials, Dent. Clin. N. Amer., March, 1965, pp. 149-157. 9. Tylman, S. D.: Relationship of Structural Design of Dental Bridges to their Supporting Tissues, Int. Dent. J. 13: 303-317, 1963. 10. Craig, R. G., and Peyton, F. A.: Measurement of Stresses in Fixed-Bridge Restorations Using a Brittle-Coating Technique, J. Dent. Res. 44: 756-762, 1965. 11. Tillitson, E. W., Craig, R. G., and Peyton, F. A.: Experimental Stress Analysis of Gold and Chromium Alloy Bridges, I.A.D.R. Dental Materials Microfilm, Washington, D. C., March, 1967. 12. Craig, R. G., El-Ebrashi, M. K., LePeak, P. J., and Peyton, F. A.: Experimental Stress Analysis of Dental Restorations: Two-Dimensional Photoelastic Stress Analysis of Inlays, J. PROS. DENT. 17: 277-291, 1967. 13. Craig, R. G., El-Ebrashi, M. K., and Peyton, F. A.: Experimental Stress Analysis of Dental Restorations: Two-Dimensional Photoelastic Stress Analysis of Crowns, J. PROS. DENT. 17: 292-302, 1967. 14. El-Ebrashi, M. K., Craig, R. G., and Peyton, F. A.: Experimental Stress Analysis of Dental Restorations: The Concept of the Geometry of Proximal Margins, J. PROS. DENT. 22: 333-345, 1969. 15. Frocht, M. M.: The Shear-Difference Method. Proc. 13th Eastern Photoelasticity Conference, pp. 51-95, June, 1941. UNIVERSITY OF MICHIGAN SCHOOL OF DENTISTRY DEPARTMENT OF DENTAL MATERIALS ANN ARBOR, MICH. 48104