A standard, reliable way of fastening substantial wheels, hubs and other fittings onto shafts is to make usage of a shrink fit.

When designing with shrink fits in mind, the estimation of the contact pressure induced in axisymmetric shrink fit problem is usually done via the classical well-known Lamé solution, which is normally used in plane strain and represents a very satisfactory solution at points away from the free edges of the shaft/hub assembly.

Dr Paynter and Prof Hills (2009) expanded this solution to an axisymmetric case where the frictional contact between the shaft and the hub is considered. Away from the free edges, Lamé's solution is still valid, but closer to the edge it was shown there is partial slip due to the frictional nature of the contact.


Their work considered only the partial slip solution when assembling the shrink fit system. In this project, we turned our attention at two other situations:

  1. What happens to the contact stresses when we try to pull out/push in the shaft?
  2. What happens to the contact stresses when we twist the shaft and let it go?

This implies applying to different loadings in addition to the assembly step performed by Paynter and Hills: a concentrated normal load in the first situation; and a concentrated torque in the second.

Results & Conclusions

While the first problem is very straightforward, the second one involved the solution of a plane/anti-plane partial slip problem, as the concentrated torque induces shear stresses that are perpendicular to the ones induced by the assembly and normal load.

For the normal loading problem, It was shown that, when the shaft is being ‘pulled out’ towards the exterior of the hub, the size of the slip zone increases with the load magnitude, but as we ‘push the shaft in’, towards the core of the hub, the slip zone remains locked at the assembly position.

For the second problem, It was shown that the problem exhibits considerable frictional coupling. In addition, it was shown that in monotonic loading the size of the slip zone is proportional to the magnitude of the applied torque. Finally, when unloading the system back to zero net torque it was shown that the residual stresses are history dependent. Although the final state represents no net torque, it still had locked-in anti-plane shear stress.

While developing a solution for the second problem, we also developed a novel algorithm for the integration of the stress fields history in a plane/anti-plane context.

The results for both models were verified against FE data.


The axisymmetric shrink fit problem subjected to axial force

The axisymmetric shrink fit problem subjected to torsion

An analysis of axisymmetric receding contacts