EFFICIENCY CONSIDERATIONS FOR THE PURELY TAPERED
INTERFERENCE FIT (TIF) ABUTMENTS USED IN DENTAL IMPLANTS
by
Dinçer Bozkaya,
Sinan Müftü1, Ph.D.
Graduate Student
Associate Professor
Northeastern University
Department of Mechanical Engineering
Boston MA 02115
October 2003
1
Corresponding author: Northeastern University
Department of Mechanical Engineering, 334 SN
Boston, MA 02115
Tel: 617-373-4743, Fax: 617-373-2921
Email: smuftu@coe.neu.edu
1
ABSTRACT
A tapered interference fit provides a mechanically reliable retention mechanism
for the implant-abutment interface in a dental implant. Understanding the mechanical
properties of the tapered interface with or without a screw at the bottom has been the
subject of a considerable amount of studies involving experiments and finite element
(FE) analysis. In this paper approximate analytical formulas are presented to investigate
the effects of the parameters affecting the mechanical properties of a pure tapered
interference fit. It is shown that the connection strength of the tapered interference fit
interface, characterized by the pull-out force, is a function of the taper angle, the contact
length, the inner and outer radii of the implant, the static and the kinetic coefficients of
friction, and the elastic modulii of the implant/abutment materials. The efficiency of the
tapered interference fit abutment attachment method, which is defined as the ratio of the
pull-out force to insertion force, was found to be greater than one, for taper angles that
are less than 6o and when the friction coefficient is greater than 0.2. The magnitude of the
pull-out and insertion force depend significantly on the insertion depth, contact length,
radii of the implant and elastic modulus of the material.
Keywords: Dental implants; Taper lock; Morse taper; Conical interference fit; Tapered
interference fit; Connection mechanism; Pull-out force; Loosening torque
2
INTRODUCTION
The reliability and the stability of an implant-abutment connection mechanism is
an essential prerequisite for long-term success of dental implants [1]. High rate of screw
complications such as screw loosening has been encountered with screw-type implantabutment connection mechanism [2,3]. Inadequate preload, the misfit of the mating
components and rotational characteristics of the screws were considered to be the reasons
leading to screw loosening or fracture [3]. A tapered implant-abutment attachment with
or without a screw is an alternative method to the screw type attachment systems. In this
paper an abutment which uses only the tapered interference fit as the connection means is
called a tapered interference fit (TIF), whereas the term taper integrated screwed-in
(TIS) abutment is used to describe an abutment which uses a screw and a tapered fit
together. Four commercial implant systems are shown in Fig 1. The design by Nobel
Biocare (Nobel Biocare AB, Göteborg, Sweden) uses a screw, the designs by Ankylos
(Degussa Dental, Hanau-Wolfgang, Germany) and ITI (Institut Straumann AG,
Waldenburg, Switzerland) use TIS type abutments; and the design by Bicon (Bicon Inc.,
Boston, MA, USA) uses the TIF type abutment.
The main advantage of the TIS abutment is reducing screw-loosening incidents,
attributed to the increased interfacial strength between implant and abutment. A high
incidence of screw loosening, up to 40%, was found for systems using screw-only
implant-abutment connection; whereas, the failure rate for tapered interface implants was
lower, as much as 3.6% to 5.3% [4]. A retrospective study with 80 implants showed that
TIS connection provides a very low incidence of failure [5]. The lack of retrievability
could be considered as the disadvantage of a TIS system [6]. Clinical studies showing the
3
success of the TIS type implant-abutment interface encouraged the researchers and
implant companies to focus on understanding and evaluating the mechanical properties of
the tapered interface.
A considerable amount of experimental and finite element studies were performed
on understanding the mechanical properties of the tapered implant-abutment interface
with or without screw [2]. The mechanics of a TIF type implant was first explained
analytically by O’Callaghan et al. [9] and then by Bozkaya and Müftü [7]. Approximate
analytical solutions for the contact pressure, the pull-out force and loosening torque
acting in a tapered interference were developed by modeling the tapered interference as a
series of cylindrical interferences with variable radii. These formulas were verified by
non-linear finite element analyses for different design parameters [7]. TIS type implants
are investigated analytically in [10]; closed-form formulas are developed for estimating
the tightening and loosening torque values and to evaluate the efficiency of the implantabutment interface.
An elastic-plastic finite element analysis of a TIF implant-abutment interface,
with different insertion depths, showed that the stresses in the implant and abutment
locally exceeds the yield limit of the titanium alloy at the tips of the interface for an
insertion depth of 0.10 mm. The plastic deformation region spreads radially into implant,
for insertion depths greater than 0.1 mm. It was also found that the plastic deformation
decreases the increase in the pull-out force due to increasing insertion depth. The
optimum insertion depth is obtained when the implant starts to deform plastically [7].
A similar interface to the tapered implant-abutment exists between the sleeve and
the bone in total hip prosthesis where the tapered cone is press-fit to the sleeve drilled to
4
a mating tapered shape. Pennock et al. investigated the influence of the dryness of the
taper components, impaction force, number of impacts required to assemble the taper and
the taper angle on the pull-out force [8]. The pull-out force was found to be linearly
proportional to the insertion force. The experiments with successive impactions showed
that pull-out force gained from the multiple impactions is equal to the pull-out force
gained from the single largest impaction.
In this paper, approximate closed-form formulas are developed for a) estimating
the insertion force and b) evaluating the efficiency of the TIF abutments. The implant is
assumed to be a cylinder, and the taper of the abutment is modeled as a stack of
cylindrical interference fits with variable radius as in [7]. Commercially available
implants are not cylindrical; they typically have a variable outer radius profile. This issue
has been addressed in the authors' previous work [7]. The equations developed here,
provide a relatively simple way of assessing the interdependence of the geometric and
material properties of the system; and in one case, presented later, show a reasonably
good match with experimental measurements.
THEORY
Figure 2 describes the geometry of a TIF abutment system. The insertion force Fi
required to seat a taper lock abutment into the matching implant is typically applied by
tapping. The interference fit takes place, once the abutment is axially displaced by an
amount z by tapping. Interference gives rise to contact pressure pc ( z ) whose magnitude
changes along the axial direction z of the cone [7]. The resultant normal force N (Figure
5
2b), acting normal to the tapered face of the abutment, is obtained by integrating pc ( z )
along the length s of the interference, [7]
N
 E z Lc sin 2  2 2
3  b2  rab   Lc sin   3rab  Lc sin  
2

6b2
(1)

where Lc is the contact length, b2 is the outer radius of the implant, rab is the bottom
radius of the abutment,  is the taper angle as shown in Figure 2, and, E is the elastic
modulus of the implant and abutment, assumed to be made from the same material.
An average value for the insertion force Fi can be found from the energy balance,
where the work done by the insertion force Wi is equal to the sum of the work done
against friction Wf and the strain energy Ut stored in the abutment and the implant. This is
expressed as,
Wi  Fi z  W f  U t .
(2)
The work done against friction Wf by sliding a tangential force kN along the side
s of the taper, by a distance s, is found from,
s
z
0
0
W f   k  Nds   k  N
dz
cos
(3)
where μk is the kinetic coefficient of friction, and the geometric relation s = zcos is
used. Note that, in this equation the kinetic friction coefficient is used, as abutment
insertion is a dynamic process. The work done against friction is calculated from Eqns (1)
and (3) as,
Wf 
 k E z 2 Lc sin   2 2
3  b2  rab   Lc sin   3rab  Lc sin    .
2
6b2


(4)
6
During the insertion of the abutment, some portion of the work done by the
insertion force is stored in the abutment and the implant as strain energy. The total strain
energy Ut of the system is given by,
Ut 
Lc cos b1

0
a a
a a
  r  rr rr      drdz 
0
Lc cos b2
   r 
0
   i i  drdz ,
i i
rr rr
(5)
b1
where the radial and tangential stresses are σrr and σθθ and the radial and tangential strains
are εrr and εθθ, and the superscripts ‘a’ and ‘i’ refer to the abutment and the implant,
respectively. The radius of the abutment b1 varies along the axial direction z as
b1 ( z)  rab  (Lc cos  z)tan . The stresses and strains for a TIF connection can be
approximated as follows [7],
 rr   
a
 rr  
a
 rr
 rr
i
i
a
a
2
E z tan    b1  z   
1  

 
2b1  z    b2  


z tan  1      b1  z  
1  


2b1  z 
  b2 
(6)
2



2
E zb1  z  tan    b2  
E zb1  z  tan 
i

1     ;   
2
2b2
2b22
  r  
(7)
  b2 2 
1    
  r  
2

b1  z  z tan  
b1  z  z tan 
 b2 
i


1



1







 ;  


2
2b2
2b22
r 


2


 b2 
1       1   
r


(8)
(9)
The total strain energy Ut of the system is calculated by using Eqns (5)-(9). Once Ut is
evaluated, the insertion force Fi can be found in closed form, from Eqns (2) and (4). This
expression is not given here in order to conserve space. However, its results are presented
later in the paper.
7
Efficiency of a Tapered Interference Fit Abutment
The efficiency c of a TIF type abutment system is defined here as the ratio of the pullout force Fp to the insertion force Fi,
c 
Fp
Fi
.
(10)
An approximate relation for the efficiency can be obtained by noting that in Eqns (1)-(9)
the strain energy Ut of the system is small as compared to the work done against friction.
For example, the strain energy Ut of the system is approximately 6% of the total work
done Wi for a 5 mm implant-abutment system, using the parameters given in Table 1.
With this assumption the insertion force can be approximated by considering only the
work done against friction ( Wi  W f ) as,
Fi 
 k E z Lc sin   2 2
3  b2  rab   Lc sin  3rab  Lc sin   .
2

6b2

(11)
The pull-out force of the tapered interference was given by Bozkaya and Müftü as
[7],
Fp 
 E z Lc  2 2
2

2
3  b2  rab   Lc sin  3rab  Lc sin   (  s cos   sin  ) cos 
3b2
(12)
where the static coefficient of friction μs is used, as the pull-out force is applied on the
initially stationary implant. The following approximate efficiency c formula for the TIF
type abutment is obtained by using Eqns (11) and (12),
c 
Fp
Fi

2cos
k
  s cos  sin  .
(13)
The error  involved in using Eqn (11) to find the insertion force is evaluated as,

W
f
 U t  z  W f z
W
f
 U t  z

Wf
Ut
.
 1
W f  Ut
W f  Ut
(14)
8
Critical Insertion Depth
The interference fit results in a stress variation in the implant and the abutment as
predicted by equations (6)-(9). Typical circumferential  and radial rr stress variation
along the radial direction (r/rab) in the abutment and the implant, as predicted by these
formulas, is presented in Figure 3, for different locations (z) along the contact length Lc.
This figure shows that the maximum stresses occur in the implant at location z = Lccos ,
where the abutment radius is b1 = rab. It is clear, from equations (6)-(9), that both radial
and circumferential stresses are linearly proportional to the insertion depth z. Thus a
critical insertion depth value exists which causes plastic deformation of the implant
material. The von Mises stress yield criterion is used to determine the onset of yielding.
The equivalent von Mises stress is defined as,


1
2
2
2
 1   2    1   3    3   2 
2

1/ 2
(15)
where the principal stresses 1, 2 and 3 are , 0 and rr respectively. Then the
following relation for the critical insertion depth zp, which causes the onset of plastic
deformation is obtained,
1/ 2
 r  2  b  2 
1   Y  rab
 ab    2  
(16)
z p  Rc  
 E  tan   b2   rab  
where Y is the yield strength of the implant material obtained from uniaxial tension test,
and Rc is a stress concentration factor. It should be noted that the plain stress elasticity
approach used here provides only approximate answers. One drawback, of this approach
is that it does not capture the stress concentrations at the ends of the contact region [7].
The stress concentration factor Rc, which has a value greater than one, is an attempt to
take this effect into account.
9
RESULTS
The parameters of the implant-abutment system, given in Table 1, were taken as
base values to investigate the mechanics of the TIF type abutments.
Checking the Insertion Depth Formula
In Figure 4 the insertion depth z is plotted as a function of work done during
insertion Wi (= Fi z). The solid lines indicate the predictions based on the formulas
developed here, and the circles indicate the curve fit to the experimental results of
O’Callaghan et al. [9]. The curve fit, which is valid in the range 10-3  z  610-3
inches, is given as z = 1.910-3Wi0.579, where the units of z and Wi are “inch” and Wi
“oz.in,” respectively. On the other hand, by considering, for example, the simplified
insertion force formula Fi given in equation (11), the insertion depth z is found to be
proportional to Wi0.5. The error between the experimental curve fit formula and this work
is plotted as broken lines in Figure 4, and is seen to be less than 20%. The discrepancy is
largely due to the plastic deformation of the implant which is predicted to start around z
= 0.13 mm and occupy a wider area at deeper z values. Therefore, it is concluded that
equation (11) provides a fairly good estimate of the insertion force FI, when the material
remains elastic.
Critical Insertion Depth
Figure 5a shows the effect of the bottom radius of the abutment rab on the critical
insertion depth zp (Eqn (16)) for different taper angles . This figure demonstrates that if
a design has small radius rab and a large taper angle , then onset of plastic deformation
occurs at a lower insertion-depth value z. Figure 5b shows the variation of the critical
10
insertion depth zp with the outer radius b2 of the implant for different abutment radii rab.
This figure indicates that the critical insertion depth decreases with increasing implant
radius. This result may seem counter intuitive at first, but it can be explained by noting
that the contact pressure also increases with b2 at the tip of the abutment [7]. Therefore,
the stress levels rise with increasing (b2  rab) distance. On the other hand, for a fixed
value of implant radius b2, increasing the abutment radius rab has the effect of increasing
the value of the critical insertion depth.
Effects of System Parameters on Efficiency
The effect of the design parameters on the efficiency c of the system is
investigated for the TIF interface in Figure 6, using Eqns (10) and (13) with complete and
approximate insertion force Fi formulas. Investigation of Eqn (13) shows that the
efficiency of the interface c depends on the kinetic and static coefficients of friction μ,
and taper angle . In Figure 6, both the exact c and approximate c efficiency relations
are plotted for different taper angles θ in the range 1-10o, coefficient of friction μ
(  k  s ) in the range 0.1-0.9 and the kinetic coefficient of friction as a fraction of static
coefficient of friction  k / s in the range 0.7-1 for μs = 0.5. Figure 6a and Figure 6c show
that increasing θ and  k / s results in efficiency reduction, whereas Figure 6b shows that
increasing μ results in efficiency increase. For θ smaller than 5.8o, the efficiency of the
interface is larger than 1. For high taper angles such as 10o, the efficiency of the interface
is around 0.5. Increasing coefficient of friction from 0.1 to 0.2 increases the efficiency
from 1.24 to 1.56. A further increase in the coefficient of friction results in an increase in
the efficiency with decreasing slope as shown in Figure 6b. As the difference between
11
static and kinetic coefficient of friction is increased by taking the static friction
coefficient larger, the efficiency of the system increases. A difference of 30% of the static
friction coefficient results in an efficiency of 2.6.
The accuracy of the simplified insertion force Fi formula in Eqn (11), is also
investigated in Figure 6. In general, it is seen that Eqn (11) overestimates the efficiency
of the attachment. The error introduced by the use of this equation increases with
increasing θ and decreasing μ. The simplified formula can be used with less than 10%
error for the following ranges, 0.2  μ  0.9 and 1o  θ  2.4o.
Effects of System Parameters on Forces
In this work the implant is assumed to be a cylinder. Commercially available
implants are not cylindrical; they typically have a variable outer radius profile. This issue
has been addressed in the authors' previous work [7]. Eqns (11) and (12) provide a
relatively simple way of assessing the interdependence of the geometric and material
properties of the system. For example, the magnitudes of the pull-out Fp and insertion
forces Fi, found in Eqns (11) and (12), depend on the parameters ∆z, E, μk, μs linearly; on
the parameters b2, rab parabolically; on the parameter Lc in a cubic manner; and, on the
parameter θ trigonometrically. The details of these functional dependence are given next.
Effect of Taper Angle
Figure 7a shows the effect of taper angle  on the insertion and pull-out forces, Fi
and Fp given by Eqns (11) and (12), respectively. In evaluating this figure, the
interference  = z tan was kept constant at 4 m for  = 1.5o and z = 0.1524 mm.
12
Keeping the  value constant implies that the insertion force is kept approximately
constant as the taper angle varies in the range 1o - 10o. In fact Figure 7a shows this
assertion to be correct for the most part. The magnitude of the pull out force Fp, on the
other hand decreases from 1750 N to 500 N in the same range. The pull-out force
becomes less than the insertion force for taper angles greater than 5.8 o. This figure in
general shows that larger taper angles reduce the pull-out force; situation which should be
avoided for the long term stability of the interface.
Effect of the Contact Length
The pull-out and insertion forces increase with the cube of the contact length Lc as
shown in Eqns (11) and (12). However, in the region of interest for dental implants, 1 <
Lc  5 mm, this dependence appears linear, as shown in Figure 7b. Increasing the contact
length causes insertion force Fi to increase from 150 N at Lc = 1 mm to 700 N at Lc = 5
mm; In the same Lc range the pull-out force Fp varies between 290 N and 1250 N.
Effect of Friction
The coefficient of friction, despite its significant effects on the insertion and pullout processes, is difficult to determine exactly. First, a distinction must be made between
the static and kinetic coefficient of friction values; typically the static coefficient of
friction s is greater than the kinetic coefficient of friction k. Second, the value of the
coefficient of friction could be affected by the presence of saliva which acts as a lubricant
in the contact interface. The friction coefficient also depends on the surface roughness
and treatment. With many factors affecting its value, it is important to understand the
13
effect of a relatively wide range of friction coefficients, on the mechanics of the
connection.
The dependence of the insertion force Fi on the kinetic friction coefficient k, and
the pull-out force Fp on the static friction coefficient s are shown to be linear in Eqns
(11) and (12). Figure 7c demonstrates the effect of coefficient of friction when k = s.
This figure shows that the pull-out force is more adversely affected by the reduction of
coefficient of friction. For example, at  = 0.1 the pull out force is equal to the insertion
force (200 N), but at  = 0.7 the pull out force (2000 N) is nearly twice as much as the
insertion force (1000 N). This behavior is also evident in the approximate efficiency
formula, given in Eqn (13), and plotted in Figure 6b. Close inspection of this formula
shows that when k = s, and for infinite friction (  ) the approximate efficiency of
the system behaves as c  2cos2. The actual and approximate efficiency formulas
approach this limit in Figure 6b, which has the value of 1.997 for  = 1.5o. Figure 7d
shows the effect of the kinetic coefficient of friction on the insertion force Fi by varying
the ratio ks in the range 0.7  1 for s = 0.5. This figure shows that the insertion force
varies linearly in this range from 580 N to 800 N.
A relation between the insertion force and the pull-out force can be obtained from
equations (11) and (12). As both the insertion force Fi and the pull-out force Fp depend
on insertion depth z in a linear fashion, their interdependence is also linear. This is
shown Figure 7e for the papramater given in Table 1. Note that the pull-out force was
also found to be linearly dependent to insertion force in the experimental work presented
in reference [8].
14
DISCUSSION
Investigating the effect of different design parameters on the efficiency of a TIF
type implant-abutment interface, taper angle θ and friction, (μk, s) were identified to be
the most significant factors. The efficiency of the interface was greater than 1 for θ < 6o.
In order to maximize the pull-out force and increase the stability of the attachment, the
parameters should be maintained in this range. Additional efforts should be considered in
order to achieve the required amount of pull-out force by varying other parameters.
The amount of pull-out force necessary to achieve the stability of the interface can
be adjusted by varying Lc, ∆z, b2, E, μk, μs and θ. However there are some design
constraints on the parameters. Lc is constrained by the height of the implant and
abutment, ∆z by the stress state in the implant and abutment, b2 by the bone space
available and the stresses transferred to the bone, E by the elastic modulus of Titanium
alloy, μ and  k / s , by the condition of the implant-abutment interface (the existence of
lubrication) and θ by the efficiency of the system. From these parameters, b2 and E are
not varied since they are critical for implant-bone interface. Considering the effect of θ, μ
and  k / s on the efficiency of the interface, Lc and ∆z could be selected as the controlling
parameters to achieve the desired pull-out force.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the discussions they had with Mr. Fred Weekley
(United Titanium, Wooster, OH) and the partial support of Bicon Implants (Bicon Inc.,
Boston, MA, USA) for this work.
15
REFERENCES
1. Scacchi, M., Merz, B.R. and Schär, A.R. (2000), "The development of the ITI Dental
Implant System," Clin Oral Imp Res 11, pp. 22-32.
2. Geng, J., Tan, K. and Liu, G., (2001), "Application of finite element analysis in
implant dentistry: A review of the literature," J Prosthet Dent, 85, pp. 585-598.
3. Schwarz, M.S., (2000), "Mechanical complications of dental implants," Clin Oral
Impl Res, 11, pp. 156-158.
4. Merz, B.R., Hunenbart, S. and Belser, U.C., (2000), "Mechanics of the implantabutment connection: An 8-degree taper compared to a butt joint connection," Int J
Oral Maxillofac Implants, 15, pp. 519-526.
5. Mangano, C. and Bartolucci, E.G., (2001), "Single tooth replacement by morse taper
connection implants: A retrospective study of 80 implants," Int J Oral Maxillofac
Implants, 16, pp. 675-680.
6. Squier, R.S., Psoter, W.J. and Taylor, T.D., (2002), "Removal torques of conical,
tapered implant abutments: The effects of anodization and reduction of surface area,"
Int J Oral Maxillofac Implants, 17, pp. 24-27.
7. Bozkaya, D. and Müftü, S., (2003) "Mechanics of the tapered interference fit in
dental implants," J Biomech, 36:11, pp. 1649-1658.
8. Pennock, A.T., Schmidt, A.H. and Bourgeault, C.A., (2002), "Morse-type tapers:
Factors that may influence taper strength during total hip arthroplasty," The Journal
of Arthroplasty, 17, pp. 773-778.
9. O'Callaghan, J., Goddard, T., Birichi, R., Jagodnik, J.J. and Westbrook, S.,
"Abutment hammering tool for dental implants," American Society of Mechanical
16
Engineers, IMECE-2002 Proceedings Vol. 2, Nov. 11-16, 2002, Paper No. DE25112.
10. Bozkaya, D. and Müftü, S. (2003) "Mechanics of the Taper Integrated Screwed-In
(TIS) Abutments Used Dental Implants," submitted for review," J Biomech, October
2003.
17
List of Figures
Figure 1 Various implant-abutment attachment methods are used in commercially
available dental implants.
Figure 2 a) Definition of the design parameters of the tapered interface. b) The free body
diagram of the tapered abutment depicting the force balance.
Figure 3 The distribution of the radial and circumferential stresses in the abutment and
the implant at different axial (z) locations.
Figure 4 The insertion depth as a function of work of insertion. Experimental results of
O'Callaghan et al.9 is compared with the results of this work.
Figure 5 The critical insertion depth zp as a function of b) bottom radius of the abutment
rab for different taper angles , and c) implant outer radius b2 for different rab values.
Figure 6 The variation of the efficiency of the attachment with respect to different
parameters. θ, μ and  k / s are the significant parameters affecting the efficiency of the
attachment. The efficiency is larger than 1 for θ < 6o. The difference between kinetic and
static friction coefficient causes high efficiency.
Figure 7 Variation of pull-out and insertion force with a) taper angle, b) contact length, c)
coefficient of friction, d) ratio of kinetic to static friction coefficient, and e) pull-out force
vs. insertion force. The other parameters, which are held fixed, are reported in Table 1.
List of Tables
Table 1 Design parameters of the tapered interface used in a Bicon implant system
(implant: 260-750-308; abutment: 260-750-301).
* The static friction coefficient was fixed at 0.5 for analyzing the effect of different static and kinetic
friction offset values.
18
Implant
Bicon
Parameters
θ
μ
k / s
Lc
(mm)
∆z
(mm)
b2
(mm)
E
(GPa)
rab
(mm)
Base Values
1.5
0.3, 0.5*
1
3.251
0.1524
1.372
113.8
0.762
Range
1 - 10
0.1 - 0.9
0.7 - 1
1-5
0 – 0.2
1-4
50 - 200
N/A
Table 1 Design parameters of the tapered interface used in a Bicon implant system (implant:
260-750-308; abutment: 260-750-301).
* The static friction coefficient was fixed at 0.5 for analyzing the effect of different static and kinetic friction
offset values.
19
Ankylos
ITI
Bicon
Nobel Biocare
Figure 1 Various implant-abutment attachment methods are used
in commercially available dental implants.
20
Fi
r
Fi
abutment
r
 s
z
b1(z)

z
N

Lc
N
rab
b2
implant
a)
b)
Figure 2 a) Definition of the design parameters of the tapered interface. b) The free
body diagram of the tapered abutment depicting the force balance.
21
Non-dimensional stress /E
3.0x10-03
2.0x10
-03
1.0x10
-03
0.0x10
z=0
z=Lccos/2
z=Lccos
i
+00
-1.0x10
rri
-03
 =  = -Pc
a
-2.0x10-03
-3.0x10
a
-03
0.0
0.5
1.0
1.5
Radial direction, r/rab
Figure 3 The distribution of the radial and circumferential stresses in the abutment
and the implant at different axial (z) locations.
200
0.25
0.20
160
140
0.15
120
Error
Insertion depth, z [m]
180
100
0.10
80
60
This work
Experimental Data (O'Callaghan et al.)
Error
40
0.000
0.025
0.050
0.075
0.05
0.00
0.100
Insertion work, W i [N.m]
Figure 4 The insertion depth as a function of work of insertion. Experimental results
of O'Callaghan et al. [9] is compared with the results of this work.
22
0.30
0.35
Critical insertion depth, z p (mm)
o
0.25
b2 = 1.50 mm
Lc = 3.25 mm
Critical insertion depth, z p (mm)
 = 1.5
o
 = 3.0
o
 = 4.5
0.20
0.15
0.10
0.05
0.00
0.00
0.25
0.50
0.75
1.00
Abutment bottom radius, rab (mm)
a)
1.25
rab = 0.5 mm
rab = 1.0 mm
rab = 1.5 mm
0.30
0.25
 = 1.5
Lc = 3.25 mm
o
0.20
0.15
0.10
0.05
0.00
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Implant outer radius, b2 (mm)
b)
Figure 5 a) Stress distribution in the abutment and the implant at different axial (z)
locations. The critical insertion depth zp as a function of b) bottom radius of the
abutment rab for different taper angles , and c) implant outer radius b2 for different rab
values.
23
5.0
2.8
2.8
0.4
2.2
0.3
1.8
1.4
0.2
1.2
Error, 
0.25
1.6
0.15
1
0.3
2
1.8
0.25
1.6
Actual
1.4
0.2
Simplified Formula
Simplified
formula(Eqn
Eqn 18)
(13)
1.2
Error, 
0.15
1
0.8
0.8
0.1
0.6
0.4
0.1
0.6
0.4
0.05
0.05
0.2
0.2
1
2
3
4
5
6
7
8
9
0
0.1
0
10
0.2
0.3
0.4
0.5
0.6
0.7
Taper Angle, 
Coefficient of Friction, 
a)
b)
2.8
0.4
2.6
0.35
2.4
2.2
0.3
1.8
0.25
1.6
Actual
1.4
0.2
Simplified Formula
Simplified
formula (Eqn
Eqn 18)
(13)
Error, 
1.2
Error, 
Efficiency, 
2
0.15
1
0.8
0.1
0.6
0.4
0.05
0.2
0
0.7
0.75
0.8
0.85
0.9
0.95
1
0
Kinetic Coefficient of Friction, k,s
c)
Figure 6 The variation of the efficiency of the attachment with respect to different
parameters. θ, μ and  k / s are the significant parameters affecting the efficiency of the
attachment. The efficiency is larger than 1 for θ < 6o. The difference between kinetic
and static friction coefficient causes high efficiency.
24
0.8
0
0.9
Error, 
2
0.35
2.4
Efficiency, 
2.2
Efficiency, 
0.35
Actual
Simplified Formula
Simplified
formula(Eqn
Eqn 18)
(13)
Error, 
2.4
0
0.4
2.6
2.6
3000
a)
3000
b)
2500
2500
Fp
Fp
Fi
Fi
2000
Force (N)
Force (N)
2000
1500
1500
1000
1000
500
500
0
1
2
3
4
5
6
7
8
9
0
10
1
1.5
2
Taper Angle, 
3000
c)
d)
2500
1500
1000
500
500
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Coefficient of Friction, 
0
0.7
0.75
0.8
0.85
0.9
0.95
Kinetic Coefficient of Friction, (µk,s)
700
Figure 7 Variation of pull-out and insertion
force with a) taper angle, b) contact length, c)
coefficient of friction, d) ratio of kinetic to
static friction coefficient, and e) pull-out force
vs. insertion force. The other parameters, which
are held fixed, are reported in Table 1.
600
500
400
300
200
100
0
5
Fi
1500
0
4.5
2000
Force (N)
Force (N)
4
Fp
1000
Pull-out Force, Fp (N)
3.5
2500
Fp
2000
e)
3
3000
Fi
0
0.1
2.5
Contact Length, Lc (mm)
250
500
750
1000
1250
Insertion Force, Fi (N)
25
1