### Interference fits

Interference fits are machine elements that connect a shaft to a hub and are used for direct force-fit torque transmission.

The calculation of cylindrical and tapered interference fits is generally carried out according to DIN 7190-1 and 2 (2017) with extensions according to Kollmann. This calculation is based on the modified shear stress hypothesis according to Kollmann. As part of FVA 985, the option of selecting the shape modification hypothesis according to Von Mises as an alternative stress hypothesis was created. The calculation is then based on the approach from FVA 566.

The calculation of multiple interference fits goes beyond DIN 7190 and is also based on FVA 566.

The core results of the calculation are safety against (beginning) plasticity and sliding safety. Furthermore, a strength verification of the shaft in accordance with DIN 743 or the FKM guideline can be carried out.

#### Safety against plasticity

For the inner and outer parts of the interference fit (shaft and hub), the safety against plastic deformation and the safety against full plasticity in accordance with DIN 7190 is determined by comparing the joint pressure in the interference fit with the respective limit joint pressure.

The limit joint pressure for plastic deformation (start of the elastic-plastic range) of the outer part is calculated according to equation (15) of DIN 7190-1:

$p\_\{PA\}=\backslash frac\{1-Q\_A^2\}\{\backslash sqrt\{3\}\}R\_\{eLA\}$

The limit joint pressure for full plasticity of the outer part is calculated according to equations (27) and (28) of DIN 7190-1:

$p\_\{PA\}=2R\_\{eLA\}/\backslash sqrt\{3\}$ für $(Q\_A<e^\{-1\})$

$p\_\{PA\}=-2R\_\{eLA\}\backslash ln\; Q\_A$ für $(Q\_A\backslash ge\; e^\{-1\})$

The limiting joint pressure for plastic deformation (start of the elastic-plastic range) of the inner part is calculated according to equations (16) and (17) of DIN 7190-1:

$p\_\{PI\}=(1-Q\_I^2)R\_\{eLI\}/\backslash sqrt\{3\}$ für $(Q\_I>0)$

$p\_\{PI\}=2R\_\{eLI\}/\backslash sqrt\{3\}$ für $(Q\_I=0)$

The limiting joint pressure for full plastictiy of the inner part is calculated according to equation (2.42) in Kollmann (or for full inner parts according to equation (30) of DIN 7190-1):

$p\_\{PI\}=\backslash frac\{2(1-Q\_I)R\_\{eLI\}\}\{\backslash sqrt\{3\}\}$

If the calculation is performed on the basis of the shape modification hypothesis (Von Mises), the safety against beginning plasticity of each part is determined by comparing the equivalent strain on the inside with the limit strain (at which the yield point is reached):

The safety against full plasticity is determined by comparing the equivalent strain on the outside with the limiting strain and by comparing the radial stress on the inside with the yield point.

#### Sliding safety

According to DIN 7190-1 and DIN 7190-2, the following formulas are used to determine the maximum loads that the interference fit can transfer (assumption: joint pressure is completely utilized by the respective load):

Max. zulässiges Drehmoment (Formel (17) DIN 7190-2 bzw. (1) DIN 7190-1):

$T\_\{zul\}=\backslash frac\{\backslash pi\}\{2\backslash cos(\backslash beta)\}l\_FD\_\{Fm\}^2p\_m\backslash nu\_\{ru\}\backslash sqrt\{1-\backslash left(\backslash frac\{\backslash tan(\backslash beta)\}\{\backslash nu\_\{rl\}\}\backslash right)^2\}$

Max. permissible axial force (formula (18) DIN 7190-2 or (2) DIN 7190-1):

$F\_\{ax,zul\}=\backslash pi\; l\_FD\_\{Fm\}p\_m(\backslash nu\_\{rl\}-\backslash tan(\backslash beta))$

Max. permissible bending moment for tapered interference fits (formula (27) DIN 7190-2):

$M\_\{b,zul\}=f\_Qp\_mD\_\{Fm\}l\_F^2\backslash left[0.1+0.4\backslash frac\{\backslash nu\_\{ru\}\}\{\backslash cos(\backslash beta)\}+\backslash nu\_\{rl\}\backslash left(\backslash frac\{D\_\{Fm\}\}\{l\_F\}-0.05\backslash pi\backslash tan(\backslash beta)\backslash right)\backslash right]$

Since FVA-Workbench 8.1, the improved formula for the maximum permissible bending moment for tapered interference fits with formula (33) from FVA 787 I has been used:

$M\_\{b,zul\}=f\_Qp\_mD\_\{Fm\}l\_F^2\backslash left[0.12+0.3\backslash frac\{\backslash nu\_\{ru\}\}\{\backslash cos(\backslash beta)\}+\backslash nu\_\{rl\}\backslash left(\backslash frac\{D\_\{Fm\}\}\{l\_F\}-0.05\backslash pi\backslash tan(\backslash beta)\backslash right)\backslash right]$

The sliding safety for individual loads is determined by comparing the maximum permissible load with the transmitted loads.

In practice, there is usually a combined load on the compression joint, i.e. the joint pressure must be distributed. The validity of Coulomb's law of friction in the interference fit results in an ellipsoidal relationship for the transmission range of the interference fit in the case of combined loads (see also Final Report FVA 312-I Section 6.5). The combined sliding safety of a tapered interference fit results in

$S\_R=\backslash sqrt\{\backslash left(\backslash frac\{1\}\{S\_\{R,F\_\{ax\}\}\}\backslash right)^2+\backslash left(\backslash frac\{1\}\{S\_\{R,T\}\}\backslash right)^2+\backslash left(\backslash frac\{1\}\{S\_\{R,M\_b\}\}\backslash right)^2\}$

resp.

$\backslash frac\{1\}\{S\_R\}=\backslash sqrt\{\backslash left(\backslash frac\{F\_\{ax\}\}\{F\_\{ax,zul\}\}\backslash right)^\{2\}+\backslash left(\backslash frac\{T\}\{T\_\{zul\}\}\backslash right)^\{2\}+\backslash left(\backslash frac\{M\_b\}\{M\_\{b,zul\}\}\backslash right)^\{2\}\}$

The term with the bending moment is omitted for cylindrical interference fits.

#### Determination of joint pressure / oversize

There are two calculation procedures when designing interference fits:

Joint pressure p given, effective interference U_w or interference U sought.

Oversize U or effective oversize U_w given, joint pressure p sought.

Section 4.2 of DIN 7190-1 describes the two calculation procedures for interference fits subjected to purely elastic loads. Section 4.3 then describes the calculation of elastically-plastically stressed interference fits with certain restrictions. In the FVA-Workbench, the more general approach according to Kollmann is followed when using the modified shear stress hypothesis for elastically-plastically stressed interference fits; the formulas of DIN 7190 then result as a special case. When using the shape modification hypothesis, the calculation approach from FVA 566 is used.

In the case of tapered interference fits, the fitting distance and oversize are converted into each other.

#### Operating conditions

In the FVA-Workbench, the interference fit is first always calculated after joining (standing at room temperature). The user can optionally activate the consideration of the operating temperature and/or the rotational speed of the interference fit. In this case, the oversize change due to the operating temperature and the reduction in joint pressure due to the centrifugal force (see DIN 7190-1 section 10.2) are also taken into account. The centrifugal force also affects the stress curve in the components. The safeties against plasticity are given for all cases. The sliding safety is determined for the worst case of all variants considered.

#### Joining temperatures

If the user activates the consideration of the temperature, the required joining temperatures of the inner and outer part are determined for various common thermal joining processes of shrink fits while maintaining the given joining clearance. See also DIN 7190-1 section 8.2.

#### Interference fits with several joint sections

In the FVA-Workbench, it is possible to consider interference fits in which the outer hub diameter and/or the inner shaft diameter change over the joint length. The slice method according to DIN 7190-2 section 4.4 is used for this purpose. For this purpose, the interference fit is divided into slices with constant diameters. For each slice, the (average) joint pressure occurring in each case is then determined from the oversize and the corresponding safety against plasticity is determined. The maximum permissible loads of the individual slices are added together.

#### Input options

Cylindrical interference fit

The following calculation variants are available:

Joint pressure specification

Specification of oversize, option to calculate interference fits with multiple joint sections

Specification of tolerances or deviations, possibility of calculating interference fits with several joint sections

Joint pressure is determined based on the loads to be transferred while maintaining the required sliding safety

The joint diameter and the inner shaft diameter are determined automatically from the shaft contour.

Tapered interference fits

The following calculation variants are available:

Joint pressure specification

Specified utilization, i.e. the ratio of the joint pressure to the limit joint pressure for plastic deformation (corresponds to the reciprocal value of the safety against beginning plasticity)

Specification of min. and max. fitting distance, possibility to calculate interference fits with several joint sections

Joint pressure is determined on the basis of the loads to be transferred while maintaining the required sliding safety.

The orientation of the cone tip and the cone ratio C of the joint must be specified:

$C=1:\backslash frac\{l\_F\}\{D-d\}=\backslash frac\{D-d\}\{l\_F\}=2\backslash tan(\backslash beta)$

To determine the joint diameter, either the average joint diameter is determined from the shaft contour or the maximum joint diameter is specified.

If the cone angle error is specified, the resulting diameter clearance in the middle of the joint sections is determined, which reduces the effective interference.

It is possible to determine the axial shift from the tolerances of the shaft and hub.

Further geometry parameters

Up to 7 joint sections can be specified, each with length and corresponding hub outer diameter. In the case of tapered interference fits, the corresponding average joint diameter of each section is automatically determined from the average or maximum total joint diameter based on the cone ratio and the orientation of the cone tip.

Transmitted loads

The loads transmitted by the interference fit are automatically determined as part of the system analysis

In the case of a single calculation for the interference fit, the transmitted loads are specified by the user

In case of a tapered interference fit, only a releasing axial force (i.e. in the direction of the cone tip) is taken into account

Stress hypothesis

The calculation of cylindrical and tapered interference fits is generally carried out according to DIN 7190-1 and 2 (2017) with extensions according to Kollmann. This calculation is based on the modified shear stress hypothesis according to Kollmann. As part of FVA 985, the option of selecting the shape modification hypothesis according to Von Mises as an alternative stress hypothesis was created. The calculation is then based on the approach from FVA 566, which also allows the consideration of a stress-strain curve, otherwise an ideal-plastic material behavior is assumed.

Surface parameters

The smoothing of roughness peaks during joining affects the effective interference after joining. The average roughness depth Rz of the joining surfaces and the smoothing factor can be specified for this purpose.

The adhesion coefficients must be specified to determine the transmittable loads and the necessary (dis)assembly forces. See also section 5 of DIN 7190-1.

#### Design of interference fits

The following procedure is recommended when designing an interference fit

Determination of the minimum oversize based on the loads to be transferred and the required sliding safety

In the case of a tapered interference fit, the maximum interference (or the associated mounting distance) can now be determined directly by specifying the maximum utilization. In the case of cylindrical interference fits, this can be done by calculating the minimum interference with the associated safety against plasticity.

Finally, the determined limit values can be used to select a practical fit (cylindrical interference fit) or min. and max. mounting distance ( tapered interference fit) and the associated safety factors can be determined.

#### Multiple interference fits

The calculation of multiple interference fits goes beyond DIN 7190 and is based on FVA 566. Both elastically and elastically-plastically stressed multiple interference fits with up to 4 parts (i.e. 3 joints) can be calculated according to the shape modification energy hypothesis according to v. Mises. In the elastic-plastic range, the consideration of stress-strain curves is possible.

The following calculation variants are available:

Specification of the joint pressure in all joints

Specification of the oversize in all joints

Specification of tolerances or deviations in all joints

The outer and inner shaft diameters are automatically determined from the shaft contour. The outer diameter of all hub parts must also be specified. All hub parts have the same length (= joint length).

Transmitted loads

The loads transmitted by the interference fit are automatically determined as part of the system analysis

In the case of a single component calculatio for the interference fit, the transmitted loads are specified by the user

It is possible to apply additional pressure to the inside of the hollow shaft and the outside of the outermost hub.

Surface parameters

The smoothing of roughness peaks during joining affects the effective interference after joining. The average roughness depth Rz of the joining surfaces and the smoothing factor can be specified for this purpose.

The adhesion coefficients must be specified to determine the transmittable loads and the necessary (dis)assembly forces. See also section 5 of DIN 7190-1.

Joining process

The joining sequence (from inside to outside or from outside to inside) must be specified. The joining clearance must be specified to determine the joining temperatures. You can select whether joining is to take place using CO2 cooling, N2 cooling or user-defined temperatures.