# Measurement of Partial Slip at the Interface of a Shrink Fit Assembly under Axial Load

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## Abstract

The slip at the interface of a shrink fit between a shaft and hub under axial load has been measured by a technique where a small cross hole is drilled through the assembly and a Talysurf profilometer is used to measure the profile of the hole. The measurements suggest that the technique is capable of measuring slip of the order of 1 to 2 μm. A finite element study has been carried out to predict the magnitude of slip with adequate agreement with the experimental results. The finite element study also shows that the extraction load for shrink fit assembly does not increase linearly with the coefficient of friction or the axial length of engagement of the shrink fit, as would be expected from a straightforward analysis.

### Keywords

Profilometer Shrink-fit Interfacial slip Residual stress Finite element method## Introduction

The shrink-fit is a routinely used method for attaching gearwheels and other components, collectively referred to here as hubs, onto shafts [1]. The standard calculation of the necessary interference for the shrink-fit to support given torsional and axial loads uses the Lamé thick walled cylinder equations combined with the assumption of Coulomb friction. However, detailed analysis shows a region of slip may develop between the hub and shaft close to the surface of the hub. Slip under cyclic loading may lead to fretting fatigue [2] but the standard approach to shrink-fit design is unable to determine whether such slip will occur.

Semi-analytical results are available for a rectangular cross section peg shrink-fitted into a cavity in a half-space where an extraction force, normal to the surface of the plane is applied to the peg [3]. The peg was assumed to be sufficiently long that the conditions at the end of the peg did not affect the behaviour. Results were presented for the opening and slip of the contact as the load is increased. This work accounts for the coupling between the extraction force and the interface pressure: the extraction force reduces the magnitude of the interface pressure up to the point where the surfaces separate. No such results have been obtained for the axisymmetric case where an extraction load is applied to a shaft shrink-fitted into a circular cavity in a half space. However, the case of torsion applied to a shaft shrink-fitted into a cavity has been addressed [4]. As the torsion is increased a slip region initiates at the surface of the half-space and then propagates along the interface between the shaft and circular cavity. Results are obtained for the depth to which the slip region propagates. In this work there is no coupling between the torsion and interface pressure such as would exist if an extraction load was applied to the shaft.

Finite element analysis may be combined with analytical solutions to aid the design of shrink-fits so that failure can be avoided in service [5, 6, 7]. These approaches can include complicating factors such as the geometry at the edge of the shrink-fit [8] and the influence of surface roughness at the interface [9]. These approaches however rely on the knowledge of the frictional conditions at the interface between the shrink-fitted components. For the high interface pressures typically encountered in shrink-fits, these frictional conditions are difficult to measure [10]. In cases where cyclic loading occurs, common in practice, the frictional behaviour becomes much more complex [11, 12].

There is limited previous experimental work where the interface conditions in a shrink-fit under applied load have been measured. An ultrasound technique has been used to determine the interface pressure by measuring the ratio of transmitted to reflected waves [13]. Photo-elastic measurements have also been made of the interface stress between a shaft and hub manufactured from an epoxy material [14]. Results were obtained for the interface pressure and the interface shear when the shaft was subjected to torsion. Measurements were also made of the residual shear stress after a cycle of torsion. Another study used neutron diffraction to measure the interface stress between a steel shaft and hub, although the low magnitude of the stress made accurate measurements difficult [15]. However, no work appears to have been carried out to measure the slip at the interface as increasing applied load is applied to the shrink-fit.

In the work described here experimental measurements are made of the slip between a hub and shaft in a shrink-fit assembly subjected to axial load. These measurements are then compared with the results of finite element analysis. The case of axial load is of particular interest because, just like the rectangular peg [3], Poisson contraction of the shaft resulting from the axial load reduces the interface pressure between the shaft and hub, increasing the likelihood of slip.

The technique for the measurement of slip that will be described here may be combined with measurements of the stress components at the interface to improve the understanding of the frictional conditions between shrink-fit components.

## Nominal Expressions

*r*

_{ i }while the hub an outer radius

*r*

_{ o }and length of engagement

*l*. The shaft and hub are assembled with a radial interference

*δ*. The nominal pressure

*p*developed at the interface between the shaft and hub is

*E*is Young’s modulus and

*ν*Poisson’s ratio for the hub and shaft, and

*k*=

*r*

_{ o }/

*r*

_{ i }. Equation (1) assumes that the shaft and hub remain elastic and that conditions of plane strain apply. It also ignores end effects and any influence of a tensile load applied to the end of the shaft. Such a tensile load will tend to reduce the interface pressure due to Poisson contraction of the shaft, hence the interface pressure calculated by equation (1) can only be considered to give a nominal value.

*τ*at the interface is

*μ*is the coefficient of friction. The nominal axial load to cause complete extraction of the shaft, called the nominal extraction load in the remainder of the paper, is

The actual extraction load will be lower than the nominal extraction load because axial load causes a Poisson contraction of the shaft and hence a reduction of the interface pressure leading to a reduced limiting shear stress. Slip initiates at the point where the shaft enters the hub and as the axial load increases a region of slip of axial length *d* forms between the shaft and hub, as shown in Fig. 1. With increasing load the slip region will eventually reach the end of the hub and extraction will occur.

## Experimental Measurements of Slip

Dimensions of test specimens

Hub | Hub bore (mm) | Shaft diameter (mm) | Interference (μm) | Extraction load (kN) |
---|---|---|---|---|

Hub A | 39.995 | 40.028 | 33 | 181.0 |

Hub B | 39.997 | 40.028 | 31 | 180.0 |

Hub C | 39.995 | 40.025 | 30 | 180.5 |

The shrink-fit assemblies were loaded in steps in a 250 kN servo-hydraulic test machine. The procedure was first to locate the upper and lower loading attachments of the shrink-fit assembly (Fig. 3) in the wedge grips of the test machine. Load was then applied to the assembly in steps. After each step in load had been applied the assembly was removed from the test machine so that the profiles of each of the holes could be measured. Removing the assembly from the test machine was achieved by tightening the locking nut and then slowly reducing the load. As the load was reduced the reading of the strain gauges attached to the shaft was monitored; the locking nut was tightened further to keep the strain gauge reading constant.

Experimental results of slip versus load for Hub A

Load (kN) | Slip at hole 1 (μm) |
---|---|

0 | 0.00 |

2.5 | −0.14 |

27.5 | 1.27 |

37.5 | 3.76 |

57.5 | 11.75 |

Experimental results of slip versus load for Hub B

Load (kN) | Slip at hole 1 (μm) | Slip at hole 2 (μm) |
---|---|---|

0 | 0.00 | 0.00 |

10 | 0.00 | 0.00 |

20 | 0.80 | 0.00 |

30 | 3.74 | 0.77 |

40 | 3.91 | −0.04 |

50 | 9.90 | 3.10 |

60 | 10.82 | 2.87 |

80 | 13.40 | 4.92 |

100 | 17.69 | 8.16 |

120 | 19.77 | 10.02 |

Experimental results of slip versus load for Hub C

Load (kN) | Slip at hole 1 (μm) | Slip at hole 2 (μm) | Slip at hole 3 (μm) |
---|---|---|---|

0 | 0.00 | 0.00 | 0.00 |

20 | −0.05 | 0.00 | 0.00 |

40 | 3.36 | 0.00 | 0.00 |

60 | 9.83 | 4.58 | 0.01 |

80 | 17.69 | 10.23 | 1.26 |

100 | 20.21 | 11.69 | 2.15 |

115 | 20.28 | – | – |

Slip was measured for loads up to 120 kN. All three hub assemblies were then loaded beyond this load until the extraction load was achieved. The extraction load for all three hubs was similar, about 180 kN. Table 1 lists the extraction load measured for the hubs.

## Finite Element Analysis

^{2}were used along the length of the contact interface. A detail of a small part of the mesh is shown in Fig. 9. The shrink fit was simulated by modelling the shaft and hub interference in the geometry and using the interference fit option in ABAQUS. An interference value of 31.5 μm was taken to represent the range of measured interferences provided in Table 1. The contact conditions at the interface were enforced using the penalty formulation. Load was applied to one node at the end of the shaft and the other nodes constrained so that the axial displacement was the same as the loaded node. The axial but not the radial displacements of nodes on the upper end of the hub were constrained to be zero, as shown in Fig. 8. Since the bolts attaching the hub to the lower loading attachment (Fig. 3) were only lightly preloaded and had a total axial stiffness much less than that of the hub, the constraints in the finite element model were chosen to represent the assumption that these bolts carried all the load applied to the shaft.

Before attempting to compare the finite element predictions with the experimental results, a series of analyses were conducted for a coefficient of friction equal to 0.15. This value is lower than that used to provide finite element predictions for the experimentally measured values.

*l*= 100 mm. Equation (3) suggests the extraction load increases linearly with the coefficient of friction but the finite element analysis demonstrates that this is only valid for small coefficients of friction.

## Discussion

Both experimental measurements and finite element predictions of slip versus axial load show slip occurring at small loads compared to the ultimate extraction load. For example, at an axial position 20 mm along the interface from the loaded end of the shaft, a slip of the order of 4 μm was measured at a load of 40 kN, much smaller that the extraction load of 180 kN. It would seem that to ensure fretting fatigue does not occur in axially loaded shrink fits, the level of the loading should be much smaller that the extraction load. However, no measurements of slip have been made under conditions of cyclic loading and it is possible that in practice shakedown would occur, reducing the likelihood of fretting fatigue.

The interaction of axial load and interface pressure ensures some nonlinearity in the behaviour of the shrink fit. As the length of engagement increases, the additional increase in the load carrying ability of the shrink fit reduces, as seen in Fig. 13. However, typical shrink fit designs have a length of engagement similar to the shaft diameter and for such designs the effect of the nonlinearity appears to be small. A similar nonlinearity is evident for variations of the coefficient of friction. Figure 14 shows that doubling the coefficient of friction does not quite double the extraction load. Again though, for typical levels of friction encountered between dry metallic surfaces, such nonlinearities are probably negligible.

The experimental measurements of slip in Fig. 15 show a maximum variation of the order of 5 μm between the three hub assemblies for the same load. This variation compares with a maximum measured slip of about 20 μm while the Talysurf data of Fig. 6 suggests the experimental error is less than 1 μm. Some of the variation may be attributed to dimensional differences between the three hubs, although the measurements in Table 1 show these differences are only of the order of 5%. Another potential cause of variation is due to axial stress: the shrink fit procedure generates axial stress as well as radial stress as indicated by the finite element results of Fig. 11 showing shear stress at the interface for the case of zero applied load. The precise distribution of these axial stresses will depend on the way the hub first comes into contact with the shaft as it cools down. These axial stresses will influence the behaviour of the hub assembly under superimposed axial load. We are not aware of any work that has been carried out to measure axial stress in a shrink fit assembly or assess its effect on the behaviour under load.

The finite element predictions of slip shown in Fig. 15 show differences compared to the experimental measurements of the order of the variability in these measurements. The finite element results assume an axial stress distribution that would exist if the contact between hub and shaft occurs uniformly along the length whereas in practice a different axial stress distribution may exist. In addition, the finite element analysis has assumed Coulomb friction using a range of coefficients of friction taken from existing experimental data [10]. Although these coefficients of friction led to adequate agreement between the finite element predictions and the experimental results for slip, the predictions of extraction load were significantly lower than the experimental results. A possible mechanism to explain this discrepancy is that the effective friction between two surface increases as slip accumulates and there is some experimental evidence that such an effect does occur [12].

Detailed measurements of the friction conditions between contacting surfaces at the microscopic scale are being made [17], allowing models of friction for such contacts to be generated. The work described in this paper to measure slip in a shrink fit component could be combined with measurement of the pressure and shear stress at the interface to enable friction models to be developed for accurate assessment of the behaviour of shrink-fit components.

## Conclusions

Experimental measurements have been made of the slip at the interface in a shrink fit assembly of a shaft and hub subjected to axial load. The measurements show that slip occurs near the loaded end of the shrink fit at levels of axial load much smaller that the ultimate extraction load. Finite element analysis has also been carried out to predict the slip under axial load with adequate agreement with the experimental results. The finite element analysis suggests that extraction loads for shrink fit components do not vary linearly with length of engagement or the coefficient of friction. However, for lengths of engagement similar to the diameter of the shrink fit and for typical levels of friction the estimates using the standard shrink fit analysis appears to be adequate.

The technique for measurement of slip that has been described offers the opportunity to carry out *in situ* measurements of frictional behaviour in a shrink-fit assembly when the measurement of slip is combined with a measurement of the pressure and shear stress at the interface.

## Notes

### Acknowledgements

We are grateful for the advice and assistance of Dr. Mahmoud Mostafavi with the finite element analysis.

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