Finite Element Analysis of the Relationship between Clasp

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					                Journal of Dental Research

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

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Finite Element Analysis of the Relationship
between Clasp Dimensions and Flexibility
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 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)

                                                                                                                                                            *     0.8JT   0.4J2
                                                                                                                                                            o     0.8J2   0.8J2
                                                                                                                                                            A     1.6J2 o.8j2
                                                                                                     0.                                                            t2/ tl =0.5
                                     ----      R                                                     a 0.6                                                 ---
                                                                                                                                                                   t2/tl = 1

                                                                                    I                C


   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
curvature radius of the clasp arm; C, the angle subtended by the c lasp
arm; F, load applied to the clasp tip.                                                               a)

                                                                                                                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)
                                              - W2 / Wl = 0.5                                                                     0    0.812 0.8I2
   0.                                         ---W2 / Wl = 1
   Z                                                                                                                              A    1.6F2 0.8[2
   an   0.3F
                                                                                                                                -       t2 / t =0.5
   a)                                                                                                                           ---     t2/tl = 1

        0.2   F                                                                                           z

   a)                                                                                                     IL
        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.
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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-
                                                                                                        CL      2.01-



                                           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
                                                                                                                                 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
                                          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.

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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    -------

   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
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1668          YUASA et al.                                                                                                                          J Dent Res October 1990

                                                                   TABLE 3
                    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
            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-
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  pp.                                                                                           tem for Cast Clasps, J Osaka Univ Dent Sch 27:175-187.
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  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-
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