EFFICIENCY CONSIDERATIONS FOR THE PURELY TAPERED INTERFERENCE FIT (TIF) ABUTMENTS USED IN DENTAL IMPLANTS

EFFICIENCY CONSIDERATIONS FOR THE PURELY TAPERED INTERFERENCE FIT (TIF) ABUTMENTS USED IN DENTAL IMPLANTS by Dinỗer Bozkaya, Graduate Student Sinan Müftü1, Ph.D Associate Professor Northeastern University Department of Mechanical Engineering Boston MA 02115 October 2003 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 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 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 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 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 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 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 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= π E ∆z Lc sin 2θ 3 ( b22 − rab2 ) − Lc sin θ ( 3rab + Lc sin θ )    6b22 (0) where Lc is the contact length, b2 is the outer radius of the implant, rab is the bottom radius of the abutment, θ 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 (0) The work done against friction Wf by sliding a tangential force µkN along the side s of the taper, by a distance ∆s, is found from, ∆s ∆z 0 W f = µ k ∫ Nds = µ k ∫ N dz cosθ (0) where μk is the kinetic coefficient of friction, and the geometric relation ∆s = ∆z/cosθ 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 = πµ k E ∆z Lc sin θ  2  3 ( b2 − rab ) − Lc sin θ ( 3rab + Lc sin θ )  6b22 (0) 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 cosθ b1 ∫ a a a a ∫ π r ( σ rrε rr + σ θθ εθθ ) drdz + Lc cosθ b2 ∫ ∫π r (σ b1 ε + σ θθi εθθi ) drdz , i i rr rr (0) 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 cosθ − z ) tan θ The stresses and strains for a TIF connection can be approximated as follows [7], σ rr = σ θθ a ε rr = εθθ a σ rr = i ε rr = i a a E ∆z tan θ   b1 ( z )   1 −  =− ÷ 2b1 ( z )   b2     (0) ∆z tan θ ( − ν )   b1 ( z )   1 −  =− ÷ 2b1 ( z )   b2   E ∆zb1 ( z ) tan θ   b2   E ∆zb1 ( z ) tan θ i −   ÷  ; σ θθ = 2b2 2b22   r   (0)   b2   1 +  ÷    r    b1 ( z ) ∆z tan θ  b ( z ) ∆z tan θ  b2  − + ν )  ÷ + ( − ν )  ; εθθ i =  ( 2b2 2b22  r       b2  + ν )  ÷ + ( −ν )  (  r    (0) (0) 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 Efficiency of a Tapered Interference Fit Abutment The efficiency ηc of a TIF type abutment system is defined here as the ratio of the pull-out force Fp to the insertion force Fi, ηc = Fp Fi (0) 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 mm implant-abutment system, using the parameters given in Table With this assumption the insertion force can be approximated by considering only the work done against friction ( Wi ≅ W f ) as, Fi = πµ k E ∆z Lc sin θ  2  3 ( b2 − rab ) − Lc sin θ [ 3rab + Lc sin θ ]  6b22 (0) The pull-out force of the tapered interference was given by Bozkaya and Müftü as [7], Fp = π E ∆z Lc  ( b22 − rab2 ) − Lc sin θ [ 3rab + Lc sin θ ]  ( µ s cos θ − sin θ ) cos2 θ  3b2  (0) 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 η% c formula for the TIF type abutment is obtained by using Eqns (11) and (12), η% c = Fp Fi = 2cosθ ( µ s cosθ − sin θ ) µk (0) The error ε involved in using Eqn (11) to find the insertion force is evaluated as, ε= (W f + U t ) ∆z − W f ∆z (W f + U t ) ∆z = Wf Ut = 1− Wf + Ut Wf + Ut (0) 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 σθθ and radial σrr 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 = Lccosθ , 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, σ= ( 2 ( σ1 − σ ) + ( σ1 − σ ) + ( σ − σ ) ) 1/ (0) where the principal stresses σ1, σ2 and σ3 are σθθ, and σrr respectively Then the following relation for the critical insertion depth ∆zp, which causes the onset of plastic deformation is obtained, −1/  r   b    ab ÷ +  ÷  (0)  b2   rab   where σY is the yield strength of the implant material obtained from uniaxial tension test, σ ∆z p = Rc−1  Y  E  rab ÷  tan θ 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 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 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 θ This figure demonstrates that if 10 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 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 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 12 Figure 7a shows the effect of taper angle θ on the insertion and pull-out forces, Fi and Fp given by Eqns (11) and (12), respectively In evaluating this figure, the interference δ = ∆z tanθ was kept constant at µm for θ = 1.5o and ∆z = 0.1524 mm Keeping the δ value constant implies that the insertion force is kept approximately constant as the taper angle varies in the range o - 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, < Lc ≤ 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 = mm to 700 N at Lc = 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 µs is greater than the kinetic coefficient of friction µk Second, the value of the 13 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 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 µk, and the pull-out force Fp on the static friction coefficient µs are shown to be linear in Eqns (11) and (12) Figure 7c demonstrates the effect of coefficient of friction when µk = µs This figure shows that the pull-out force is more adversely affected by the reduction of coefficient of friction For example, at µ = 0.1 the pull out force is equal to the insertion force (200 N), but at µ = 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 µk = µs, and for infinite friction (µ → ∞) the approximate efficiency of the system behaves as η% c → 2cos θ The actual and approximate efficiency formulas approach this limit in Figure 6b, which has the value of 1.997 for θ = 1.5o Figure 7d shows the effect of the kinetic coefficient of friction on the insertion force Fi by varying the ratio µk/µs in the range 0.7 − for µs = 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 14 Figure 7e for the papramater given in Table Note that the pull-out force was also found to be linearly dependent to insertion force in the experimental work presented in reference [8] DISCUSSION Investigating the effect of different design parameters on the efficiency of a TIF type implant-abutment interface, taper angle θ and friction, (μk, µs) were identified to be the most significant factors The efficiency of the interface was greater than 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 µ 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 µ 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 15 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 16 REFERENCES 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 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 Schwarz, M.S., (2000), "Mechanical complications of dental implants," Clin Oral Impl Res, 11, pp 156-158 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 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 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 Bozkaya, D and Müftü, S., (2003) "Mechanics of the tapered interference fit in dental implants," J Biomech, 36:11, pp 1649-1658 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 17 O'Callaghan, J., Goddard, T., Birichi, R., Jagodnik, J.J and Westbrook, S., "Abutment hammering tool for dental implants," American Society of Mechanical Engineers, IMECE-2002 Proceedings Vol 2, Nov 11-16, 2002, Paper No DE- 25112 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 18 List of Figures Figure Various implant-abutment attachment methods are used in commercially available dental implants Figure a) Definition of the design parameters of the tapered interface b) The free body diagram of the tapered abutment depicting the force balance Figure The distribution of the radial and circumferential stresses in the abutment and the implant at different axial (z) locations Figure 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 The critical insertion depth ∆zp as a function of b) bottom radius of the abutment rab for different taper angles θ, and c) implant outer radius b2 for different rab values Figure The variation of the efficiency of the attachment with respect to different parameters θ, μ and µ k / s are the significant parameters affecting the efficiency of the attachment The efficiency is larger than for θ < 6o The difference between kinetic and static friction coefficient causes high efficiency Figure 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 List of Tables Table 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 Implant Bicon Parameters θ μ µk / s Lc (mm) ∆z (mm) b2 (mm) E (GPa) rab (mm) Base Values 1.5 0.3, 0.5* 3.251 0.1524 1.372 113.8 0.762 Range - 10 0.1 - 0.9 0.7 - 1-5 – 0.2 1-4 50 - 200 N/A Table 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 20 Ankylos ITI Bicon Nobel Biocare Figure Various implant-abutment attachment methods are used in commercially available dental implants 21 Fi r Fi abutment r θ s z b1(z) δ z µN N θ Lc rab b2 implant a) b) Figure a) Definition of the design parameters of the tapered interface b) The free body diagram of the tapered abutment depicting the force balance 22 Non-dimensional stress σ/E 3.0x10 -03 2.0x10 -03 z=0 z=Lccosθ/2 z=Lccosθ σθθi 1.0x10-03 0.0x10+00 i σrr -1.0x10-03 -2.0x10-03 σθθa = σρρa = -Pc -3.0x10-03 0.0 0.5 1.0 Radial direction, r/rab 1.5 Figure 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 [µm] 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 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 23 0.30 0.35 0.25 b2 = 1.50 mm Lc = 3.25 mm Critical insertion depth, ∆z p (mm) Critical insertion depth, ∆z p (mm) o θ = 1.5 θ = 3.0 o θ = 4.5 o 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 θ = 1.5 o Lc = 3.25 mm 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 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 θ, and c) implant outer radius b2 for different rab values 24 5.0 2.8 2.8 0.4 2.2 Simplified Formula Simplified formula (Eqn Eqn 18) (13) 0.35 2.4 2.2 Error, ε 0.3 1.6 1.4 0.2 1.2 Error, ε 0.25 0.15 Efficiency, η 1.8 0.3 1.8 0.25 1.6 Actual 1.4 Simplified Formula Simplified formula(Eqn Eqn 18) (13) 1.2 Error, ε 0.15 0.8 0.8 0.1 0.6 0.4 0.4 0.05 0.2 0.2 0.1 0.6 0.05 0.1 10 0.2 0.3 0.4 0.5 0.6 0.7 Taper Angle, θ Coefficient of Friction, µ 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) 1.2 Error, ε Error, ε Efficiency, η 0.15 0.8 0.1 0.6 0.4 0.05 0.2 0.7 0.75 0.8 0.85 0.9 0.95 0.2 Kinetic Coefficient of Friction, µk,s c) Figure The variation of the efficiency of the attachment with respect to different parameters θ, μ and are the significant parameters affecting the efficiency of the attachment The efficiency is larger than for θ < 6o The difference between kinetic and static friction coefficient causes high efficiency 25 0.8 0.9 Error, ε Actual Efficiency, η 0.35 2.4 0.4 2.6 2.6 3000 3000 a) b) 2500 2500 Fp Fp Fi Fi 2000 Force (N) Force (N) 2000 1500 1500 1000 1000 500 500 10 1.5 Taper Angle, θ c) d) 3000 2500 2500 Fp 1500 1000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fi 1500 0.7 0.75 0.8 0.85 0.9 0.95 Kinetic Coefficient of Friction, (µk,s) Figure 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 700 600 500 400 300 200 100 500 Coefficient of Friction, µ 4.5 1000 500 Pull-out Force, Fp (N) 3.5 Fp 2000 Force (N) Force (N) 2000 e) 3000 Fi 0.1 2.5 Contact Length, Lc (mm) 250 500 750 1000 1250 Insertion Force, F i (N) 26 ... 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... 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... 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,