Document Sample

Journal of Dental Research http://jdr.sagepub.com/ Finite Element Analysis of the Relationship between Clasp Dimensions and Flexibility Y. Yuasa, Y. Sato, S. Ohkawa, T. Nagasawa and H. Tsuru J DENT RES 1990 69: 1664 DOI: 10.1177/00220345900690100701 The online version of this article can be found at: http://jdr.sagepub.com/content/69/10/1664 Published by: http://www.sagepublications.com On behalf of: International and American Associations for Dental Research Additional services and information for Journal of Dental Research can be found at: Email Alerts: http://jdr.sagepub.com/cgi/alerts Subscriptions: http://jdr.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations: http://jdr.sagepub.com/content/69/10/1664.refs.html Downloaded from jdr.sagepub.com by guest on May 6, 2011 For personal use only. No other uses without permission. Finite Element Analysis of the Relationship between Clasp Dimensions and Flexibility Y. YUASA, Y. SATO, S. OHKAWA, T. NAGASAWA, and H. TSURU Department of Removable Prosthodontics, Hiroshima University School of Dentistry, Kasumi 1-2-3, Minami-ku, Hiroshima 734, Japan A parameter study with use of the finite element method was of a "stiffness parameter" of the clasp. Morris et al. (1981) conducted for examination of the relationships between the proposed a stiffness index, which was the force required for shape parameters of a clasp (width and thickness at the base one unit of elastic deflection to be produced during stress- and tip of the clasp) and its displacement or stress. By synthesis relaxation testing. Johnson et al. (1983) used a stiffness pa- of these relationships, a simple formula defining the clasp tip rameter (force deflection) to determine the effect of single- displacement in terms of clasp dimensions ("displacement for- plane curvature on the flexibility of a half-round cast clasp. mula") was obtained. A stiffness parameter "Fd" (the load However, these results do not express quantitatively the rela- producing a 1-mm displacement of the clasp tip) was proposed, tionship between stiffness and clasp dimension. From the the- and a formula defining Fd in terms of shape parameters was ory of elasticity, Bates (1965) derived a formula for estimating derived from the displacement formula. Fd would be a prac- the deflection of a straight cantilever beam. Recently, Nokubi tical parameter for the definition of clasp retention, and the et al. (1987) used Po.1 (the force required to displace the tip present formulae appear to be useful tools for investigation of of the clasp by 0.1 mm) to estimate the bending rigidity of a the retention of removable partial dentures. circumferential clasp arm. They also derived another formula for estimating the deflection of a cast clasp. Although the for- J Dent Res 69(10):1664-1668, October, 1990 mula of Bates is simple, it is a formula for straight cantilever beams and thus cannot express the deflection of real clasps. Introduction. The equation developed by Nokubi et al. (1987) applies only to a clasp that has the same cross-sectional shape throughout, The retention force of a clasp is one of the most important and requires integration procedures. Approximate integration factors for successful removable partial dentures. Applegate with a personal computer is required for the calculation, and (1966) listed five factors that determine clasp retentiveness, hence, the clinical use of this formula is difficult. Therefore, and these can be condensed into the following three factors: a simple formula that can express the stiffness of each clasp (1) the accuracy with which the various parts of the clasp have used is necessary for clinical use. been adapted to the abutment tooth; (2) the condition of the The purpose of the present study was to investigate the abutment tooth; and (3) the flexibility of the retentive arm. relationship between clasp dimensions and stiffness and to ob- A poor fit may lead to plastic deformation of the clasp and, tain a more practical and simple relationship by the finite ele- consequently, to loss of retention. When the fit is passive, a ment method. poor fit is revealed as a decrease in the amount of undercut. The accuracy of fit is governed mainly by impression-making Materials and methods. and casting procedures. Since the fit depends on the devel- opment of these procedures, the regulation of retention on that Circumferential clasp arms for a mandibular second pre- basis is difficult. molar were analyzed by the two-dimensional finite element Abutment tooth conditions include the amount of undercut, method (FEM) (plane stress condition). The clasp arms were curvature and encirclement of abutment, and the friction coef- approximated by tapered, curved cantilever beams with a half- ficient. Although undercut is an effective parameter for reten- oval cross-section and subdivided into 440 triangular elements tion, a 0.25-mm undercut is used clinically for almost all Co- and 270 nodes (Fig. 1). The radius of curvature was set at 4 Cr clasps. The shallowest undercut is about 0.13 mm, because mm, according to the radius of a premolar reported by Wheeler errors in impression-taking, casting, and other procedures be- (1958), and the angle subtended by the clasp arm was 1200, gin to be significant below this limit (Warr, 1959). On the as obtained from clinical standards. The analysis was two- other hand, too deep an undercut causes permanent deforma- dimensional, since loading and displacements lie in the plane tion of the clasp arm. Other factors of the abutment condition of the clasp arm. However, the influence of out-of-plane width are governed by the tooth used as the abutment. Hence, the variation on the in-plane stiffness was taken into account, that ability to influence the factors involved in the abutment con- is, the thickness of each element was set equal to the width of dition is very limited, and the control of the retention force the clasp at the center of the element (Fig. 1). All nodes at the with these factors alone is difficult. bases of the clasps were restrained in all directions, and a On the other hand, flexibility is a factor that can be regu- concentrated load of 5 N was applied to the inner tip of the lated very easily for controlling the retention force of a clasp. clasp in a radial direction. The material properties of the clasp The flexibility of the clasp is affected by the clasp dimensions were set so as to be equivalent to those of a cobalt-chromium and the mechanical properties of the constituent alloy. For the alloy [Vitallium, which has a Young's modulus of 218 GPa, flexibility to be assessed, the relationship between the clasp from the value of Morris and Asgar (1975), and a Poisson's dimensions or the mechanical properties of the alloys and the ratio of 0.33]. Stress was expressed by the maximal equivalent resulting flexibility must be obtained. stress of von Mises (it was defined as ME stress), since the The flexibility of a clasp arm has been expressed in terms clasp material is a metal. The thickness of the clasp base (tj) and the tip (t2), and the width of the base (w1) and the tip (w2) were selected as the Received for publication September 5, 1989 four individual shape parameters that defined the dimensions Accepted for publication May 29, 1990 of the circumferential clasp arms. Table 1 shows the values of 1664 Downloaded from jdr.sagepub.com by guest on May 6, 2011 For personal use only. No other uses without permission. Vol. 69 No. 10 FINITE ELEMENT ANALYSIS OF CLASP FLEXIBILITY 1665 Wi W2 (mm) 0.81 * 0.8JT 0.4J2 o 0.8J2 0.8J2 E A 1.6J2 o.8j2 Q 0. t2/ tl =0.5 ---- R a 0.6 --- t2/tl = 1 co 0 a) I C 0 Fig. 1-The finite element model. The base of the clasp arm was fixed, E ° and load was applied to the tip of the clasp arm. t1, thickness of the bBase; a) t2, thickness of the tip; w1, width of the base; w2, width of the tip); R, a c curvature radius of the clasp arm; C, the angle subtended by the c lasp arm; F, load applied to the clasp tip. a) 6 0.21 0 0.4 0.4[2 0.8 0.8F2 0.5F Thickness of the clasp base(mm) ti t2 (f nm) Fig. 2-The displacement of the clasp tip as a function of the thickness of the clasp arm. * 0.4J2 0.2J2 o 0.4J 0.4J2 0.4F A 0.8J2 0.4J2 Wi W2 (mm) 0.4J2 E E 1- - W2 / Wl = 0.5 0 0.812 0.8I2 0. ---W2 / Wl = 1 Z A 1.6F2 0.8[2 a) an 0.3F 0 - t2 / t =0.5 a) --- t2/tl = 1 - E a) E 0.2 F z a) IL 0 Co ._ C) 5 0.1 1 0 0.8 0.8j9 1.6 1.6J2 Width of the clasp base(mm) 0o 0.4 0.4ft 0.8 Fig. 3-The displacement of the clasp tip as a function of the width of Thickness of the clasp base(mm) the clasp arm. Fig. 4-Fd (load for unit displacement of the clasp tip) as a function of the thickness of the clasp arm. Downloaded from jdr.sagepub.com by guest on May 6, 2011 For personal use only. No other uses without permission. 1666 YUASA et al. J Dent Res October 1990 ti t2 (mm) Wj W2 (mm) * 0.4JA 0.2P2 0 0.8J2- 0.4J2 1501- o 0.4JT 0 4JF 4.01- O 0.8J2 0.8J2 A 0.8J2 0.4[2 A 1.62 O.8J2 - W2 / W1 = 0.5 - t2 / tl - 0.5 ---W2/Wl = 1 ---t2/ tl = 1 E 3.01- 100o- E E a- z CL 2.01- IL - 50k 1.01 I_ I 0 0.8 0.8J 1.6 1.6J2 I 0.8 I0 Width of the clasp base(mm) 0 0.4 0.4j2- 0.8 0.8F2 Fig. 5-Fd as a function of the width of the clasp arm. Thickness of the clasp base(mm) Fig. 6-Sd (maximal equivalent stress at unit displacement of the clasp the four parameters. The types of clasp arm for all combina- tip) as a function of the thickness of the clasp arm. tions of these parameters were modeled. Each parameter was composed of a geometrical series with a common ratio of N2. sis, that is, all components for the functional parameter and The displacement of the tip and ME stress of these 576 kinds the underlined components in Table 1 for other parameters of models at a constant load (5 N) were calculated by the FEM. were combined, and their relationships were investigated. In the present study, an elastic analysis was undertaken. There- fore, the load and ME stress at a constant displacement of 1 The FEM program developed by the authors was used for mm, i.e., Fd (force per unit displacement) and Sd (stress per calculation on a personal computer (PC-9801RA, NEC Ltd., unit displacement), respectively, were also calculated. Tokyo). At first, the relationship between the thickness or width and Fd or Sd was investigated by varying tj and t2 or w, and w2 simultaneously (the ratios of t1 to t2 and w1 to w2 were con- Results. stant). Eight sets of combinations were chosen for each inves- For clarity in the description of the load vs. deformation tigation, that is, in the thickness investigation, the t2/t1 ratio behavior, four sets of combinations are shown in the following was 1 or 0.5, w1 and w2 were the underlined components in Figs. Figs. 2 and 3 show the displacement of the clasp tip. Table 1, and t1 (the functional parameter) was varied for all The parameter in Fig. 2 is the thickness of the clasp, and that the components in Table 1. On the other hand, in the width in Fig. 3 is the width. The displacement of the tip was inversely investigation, the wJw1 ratio was 1 or 0.5, tj and t2 were the proportional to the power 2.52-2.62 of the thickness and to the underlined components, and w1 (the functional parameter) was first power of the width. The ME stress was inversely propor- varied for all the components in Table 1. tional to the power 1.68-1.85 of the thickness and to the first Subsequently, the relationships among the individual shape power of the width. parameters (t1, t2, w1, and w2) and Fd or Sd were studied. Fd is the force in Newtons required for production of a Eight sets of these combinations were selected for each analy- displacement of 1 mm of the clasp tip, and Sd is the ME stress that occurs also due to 1-mm displacement. The relationships between the thickness or the width and Fd or Sd are shown in TABLE 1 Figs. 4-7. Fd was proportional to the 2.52th-2.61th power of FOUR INDIVIDUAL PARAMETERS OF THE CLASP Parameters TABLE 2 t1 (mm) 0.4 0.4 V2 0.8 0.8 2 RELATIONSHIP BETWEEN DISPLACEMENT AND PARAMETERS t2 (mm) 0.2 0.2 V2 0.4 0.4 \/2 0.8 0.8 2 Basic Parameters Thickness Width (mm) x value 2.52-2.62 1.00 0.8 0.8 \/2 w1 1.6 1.6 2- Individual Parameters t1 t2 wI W2 w2 (MM) x value 1.72-1.95 0.65-0.86 0.56-0.80 0.21-0.43 0.4 0.4 N/2 0.8 0.8 V/2 1.6 1.6 \/2 Displacement = K/(parameter)x, where K is a constant. Downloaded from jdr.sagepub.com by guest on May 6, 2011 For personal use only. No other uses without permission. Vol. 69 No. 10 FINITE ELEMENT ANALYSIS OF CLASP FLEXIBILITY 1667 the thickness and the first power of the width. On the other hand, Sd was proportional to the 0.77th-0.84th power of the thickness, but was independent of the width. For more detailed investigation, the effects of individual 4.01- A A A A shape parameters (t1, t2, wl, and w2) were studied. The dis- placement was inversely proportional to t1x, t2x, wjx, w2x within the practical range ("x" is a different value for each param- eter): t2 < t1 < 4*t2 and w2 < w1 < 4*w2. These relationships are shown in Table 2. The shape parameters and ME stress showed no clear inversely proportional relationship. E 3.0 F The relationships between Fd and individual parameters were E opposite those between clasp tip displacement and the same parameters, that is, Fd was approximately proportional to the *-*--0 "x"th power of these parameters. On the other hand, the re- -- lationships between Sd and these parameters were very com- CD 2.0F 0---0---- 0 ------- plex. When these results were synthesized, the following formula ti t2 (mm) for estimation of the tip displacement was obtained by the * 0.4J 0.2J2 least-squares methods: o 0.4Jf 0.4J[2 Displacement = {K*F/E}/{t1 1.87* t20 72*w1070 *w203°} (1) K = 2116 (mm2-59) 1.0 F A 0.8j2 0.4j where F is the load applied to the tip of the clasp, E is Young's - W2 / W1 - 0.5 modulus, t1 and t2 are the thicknesses of the clasp base and ---W2/W1 - 1 tip, and w1 and w2 are the respective widths. Although the effect of E was not investigated in this series of combinations, I I it was confirmed by FEM that the displacement of the clasp 0 0.8 0.8JF 1.6 1.6J2 tip was inversely proportional to E. Thus, this formula can be applied to all kinds of alloys that have different Young's mod- Width of the clasp base(mm) uli, such as Au alloys or Ni-Cr alloys. This formula can pro- Fig. 7-Sd as a function of the width of the clasp arm. vide the displacement of the tip within a 6.2% maximum error and a 3.0% mean error within the range of practical use. Fd (force per unit displacement of clasp tip) corresponds to and Sd (maximum equivalent stress per unit displacement of F/displacement in formula (1). Consequently, the formula that the clasp tip) was calculated. Assessment of fracture or per- defines the relationship between the four individual parameters manent deformation of a clasp is difficult. The effects of bend- and Fd for a premolar is as follows: ing, torsion, fatigue, and even accuracy of fit or porosity of Fd = {E/K} * {t11.87 * t20 72 *W10.70 * w20 30} (2) the casting should be considered. However, the focus of this study was clasp retention and not fracture. Thus, stress is not K = 2116 (mm259) discussed here, although Sd would be one of the factors for evaluation of clasp fracture or permanent deformation. Fd (force per unit displacement of the clasp tip) was used Discussion. as a stiffness parameter in this study. Fd is a parameter that In this study, force was applied to the clasp tip in a radial can be changed easily by practitioners and is very useful. From direction, and force and displacement lie in the same plane in the formula of clasp tip displacement [formula (1)], a formula this two-dimensional analysis. It is not known presently in for Fd [formula (2)] was derived. This formula is expressed what direction the functional force acts on the clasp. The func- as a function of clasp design and the mechanical properties of tional force on the clasp does not act exactly radially. The alloys and is able to express the flexibility of the clasp arm force has three components: the force in a radial direction, that quantitatively. in a tangential one (along the clasp arm), and that in an axial Henderson and Steffel (1977) listed length, diameter (in- one. The distributed force in a radial direction is the force that cluding its taper), cross-sectional form, and material used for moves the clasp tip outward by an amount equivalent to the the clasp arm as the four factors that influenced the flexibility undercut dimension and represents the most important force of a clasp arm. Three of these parameters are included in the when the stiffness of the clasp is considered. present formula, with the length factor being omitted. Al- It has been recognized for many decades that the displace- though the latter factor is an important one, it is governed by ment of a beam is inversely proportional to the "x"th power the abutment tooth used, and can be used only with difficulty of the thickness and width dimension. However, no study has for changing the clasp flexibility. This is because more than investigated the independent effects of the clasp tip and base 1800 encirclement must be provided to prevent the tooth from on the clasp tip displacement. In this study with the FEM, it moving away from the direct retainer and so that the direct became evident that the clasp tip displacement is inversely retainer will not slip off the tooth (Renner and Boucher, 1987). proportional to the "x"th power of four individual shape pa- Therefore, this formula is sufficiently useful for analysis of a rameters, as shown in Table 2. A simple and practical formula clasp arm for a premolar tooth, although it is accurate only for that defined the clasp tip deformation with four shape param- a clasp that has a curvature radius of 4 mm and an angle eters and Young's modulus was derived. This indicated that t1 subtended by the clasp arm of 1200. Table 3 shows Fd values is the most contributory factor and that w2 is the least contrib- for 144 kinds of premolar clasp arms calculated with use of utory for displacement of the clasp tip. this formula. Fd ranged from approximately 11 to 331 N/mm. The stress of the clasp was also investigated in this study, It is evident that a minor change in clasp thickness produces Downloaded from jdr.sagepub.com by guest on May 6, 2011 For personal use only. No other uses without permission. 1668 YUASA et al. J Dent Res October 1990 TABLE 3 EXAMPLES OF Fd VALUES IN VARIOUS DIMENSIONS OF A Co-Cr PREMOLAR CLASP ARM W_ 0.8 1.2 1.6 2.0 tj t2 W2 0.4 0.6 0.8 0.6 0.9 1.2 0.8 1.2 1.6 1.0 1.5 2.0 0.3 11 12 13 16 18 20 21 24 26 27 33 30 0.6 0.4 13 15 16 20 22 24 26 30 32 33 37 41 0.6 18 20 22 27 30 33 35 40 43 44 50 55 0.4 23 26 28 34 38 42 45 51 56 56 63 70 0.8 0.6 30 34 37 46 51 56 61 69 75 76 85 93 0.8 37 42 46 56 63 69 85 75 92 94 106 115 0.5 40 46 50 61 68 75 81 91 99 100 114 124 1.0 0.7 52 58 64 77 87 95 103 117 127 129 146 159 1.0 67 76 82 100 113 124 134 151 165 167 189 206 0.6 65 73 80 97 110 120 130 146 160 162 183 200 1.2 0.9 87 98 107 130 148 161 174 197 215 218 246 268 1.2 108 121 132 161 182 198 215 242 265 269 304 331 Units: shape parameters (mm); Fd (N/mm). Metal: Vitallium (Co-Cr alloy) (Young's modulus, 218 GPa). a considerable increase in Fd. From this Table, practitioners (1983): The Effect of Single Plane Curvature on Half-round Cast and technicians are able to estimate the approximate stiffness Clasps, J Dent Res 62:833-836. of clasps planned. Consequently, this Table and formula should MORRIS, H.F. and ASGAR, K. (1975): Physical Properties and Mi- make clasp work easier in the clinic, and they would be useful crostructure of Four New Commercial Partial Denture Alloys, J tools for the investigation of clasp retention. Prosthet Dent 33:36-46. MORRIS, H.F.; ASGAR, K.; and TILLITSON, E. (1981): Stress- relaxation Testing. Part I: A New Approach to the Testing of Removable Partial Denture Alloys, Wrought Wires and Clasp Be- REFERENCES havior, J Prosthet Dent 46:133-141. APPLEGATE, O.C. (1966): Essentials of Removable Partial Den- NOKUBI, T.; ONO, T.; MORIMITSU, T.; NAGASHIMA, T.; and ture Prosthesis, 3rd ed., Philadelphia: W.B. Saunders Co., 185 OKUNO, Y. (1987): Development of a Rational Fabricating Sys- pp. tem for Cast Clasps, J Osaka Univ Dent Sch 27:175-187. BATES, J.F. (1965): The Mechanical Properties of the Cobalt-chro- RENNER, R.P. and BOUCHER, L.J. (1987): Removable Partial mium Alloys and their Relation to Partial Denture Design, Br Dent Dentures, Chicago: Quintessence Pub. Co., pp. 83-84. J 119:389-396. WARR, J.A. (1959): An Analysis of Clasp Designs in Partial Den- HENDERSON, D. and STEFFEL, V.L. (1977): McCracken's Re- tures, Phys Med Biol 3:212-232. movable Partial Prosthodontics, 5th ed., St. Louis: C.V. Mosby WHEELER, R.C. (1958): A Textbook of Dental Anatomy and Co., pp. 59, 65-66. Physiology, 3rd ed., Philadelphia: W.B. Saunders Co., pp. 193- JOHNSON, D.L.; STRATTON, R.J.; and DUNCANSON, M.G. 200. Downloaded from jdr.sagepub.com by guest on May 6, 2011 For personal use only. No other uses without permission.

DOCUMENT INFO

Shared By:

Categories:

Tags:
Finite Element Analysis, Finite element, FEA analysis, Stress Analysis, the finite element method, structural analysis, FEA software, heat transfer, Finite elements, thermal analysis

Stats:

views: | 8 |

posted: | 5/9/2011 |

language: | English |

pages: | 6 |

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.