Interference fit and the Tresca limit

A shrink-fit hub on a shaft must transmit torque without yielding. Lamé’s equations give the stress state as a function of the contact pressure \(p_C\) and the geometry. One question remains: beyond what \(p_C\) does the hub yield?

Tresca’s criterion answers this by reducing a 3D stress state to a single scalar comparable to the uniaxial yield strength. The combination of Lamé and Tresca produces a closed-form expression for the maximum allowable contact pressure as a function of the hub geometry ratio \(Q_H\) and the yield strength \(\sigma_y\) alone.

Pipeline

(1) Stress state at the hub inner surface

Axial symmetry makes the stress tensor diagonal in cylindrical coordinates \((r, \theta, z)\) :

The diagonal components are principal stresses. Lamé’s equations for a hollow cylinder under internal pressure \(p_C\) , evaluated at the inner radius \(r = D_C/2\) :

with:

where \(D_{He}\) is the hub outer diameter and \(D_C\) the contact diameter. The axial stress \(\sigma_z\) lies between \(\sigma_r\) and \(\sigma_\theta\) for both plane-stress (\(\sigma_z = 0\) ) and plane-strain (\(\sigma_z = \nu(\sigma_r + \sigma_\theta)\) ) configurations.

(2) Ordering of principal stresses

With convention \(\sigma_1 \ge \sigma_2 \ge \sigma_3\) :

(3) Critical Mohr circle

The governing Mohr circle is \(C_{13}\) , built on the \((\sigma_1, \sigma_3)\) pair:

The intermediate stress \(\sigma_z\) does not appear: Tresca is insensitive to \(\sigma_2\) .

Move the slider to rotate the face normal in physical space. Watch the corresponding point trace an arc on the Mohr circle. The maximum shear stress is reached at $2\theta = 90°$ from the principal direction — that is, at $\theta = 45°$ in physical space.

(4) Tresca criterion

Uniaxial tension at yield: \(\sigma_1 = \sigma_y\) , \(\sigma_3 = 0\) , hence \(\tau_{max}^{(tension)} = \sigma_y/2\) . Equating to (6):

Tresca equivalent stress:

(5) Substitution and simplification

Inserting (3) into (9):

(6) Maximum allowable contact pressure

Imposing \(\sigma_{Tresca} \le \sigma_y\) and inverting:

Closing

Final formula. \(p_{C,\max} = \sigma_y (Q_H^2 - 1) / (2 Q_H^2)\) .

Assumptions (in order of appearance):

1. Axial symmetry (coaxial shaft and hub, \(\theta\) -independent loading) — eq. (1)
2. Lamé validity: homogeneous, isotropic, linear-elastic material, small strains — eqs. (2), (3)
3. \(\sigma_z\) intermediate between \(\sigma_r\) and \(\sigma_\theta\) — eq. (5)
4. Tresca criterion (ductile material, slip-dominated failure) — eq. (8)
5. Unity safety factor; substitute \(\sigma_y \to \sigma_y / n_s\) in design
6. No friction shear on coupling surface (\(\tau = 0\) ). The full formulation including friction \(\tau = \mu p_C + \tau_{ad}\) is in Croccolo et al. (2012), Eq. (12). Formula (11) is the \(\mu = 0\) , \(\tau_{ad} = 0\) limit, which overestimates \(p_{C,\max}\) by about 2–3% for typical friction coefficients.

Numerical validity limits:

• \(Q_H \to 1^+\) : \(p_{C,\max} \to 0\) (vanishingly thin hub)
• \(Q_H \to \infty\) : \(p_{C,\max} \to \sigma_y / 2\) (semi-infinite body)
• \(p_C > p_{C,\max}\) : incipient yielding at inner surface; progressive plastification not covered by elastic Lamé

Practical use. Equation (11) sets the upper bound on the interference \(Z\) : torque transmission requires \(Z \ge Z_{\min}\) ; hub strength requires \(p_C \le p_{C,\max}\) , hence \(Z \le Z_{\max}\) . If \(Z_{\min} > Z_{\max}\) , increase \(Q_H\) , choose a stronger hub material, or add adhesive.

Maximum contact pressure vs hub geometry ratio

import numpy as np
import matplotlib.pyplot as plt

Q = np.linspace(1.01, 5.0, 400)
ratio = (Q**2 - 1) / (2 * Q**2)

fig, ax = plt.subplots(figsize=(8, 5))
ax.plot(Q, ratio, linewidth=2, color='#1D9E75', label=r'$p_{C,max}/\sigma_y$')
ax.axhline(0.5, linestyle='--', color='gray', alpha=0.6, label=r'asymptote $\sigma_y/2$')
ax.scatter([1.5], [(1.5**2 - 1)/(2*1.5**2)], color='#D85A30', zorder=5, s=60)
ax.annotate(r'$Q_H = 1.5$ (Croccolo 2012, Sec. 4.1)',
            xy=(1.5, (1.5**2-1)/(2*1.5**2)), xytext=(2.2, 0.15),
            arrowprops=dict(arrowstyle='->', color='gray'))
ax.set_xlabel(r'$Q_H = D_{He} / D_C$')
ax.set_ylabel(r'$p_{C,max} / \sigma_y$')
ax.set_title('Maximum contact pressure vs hub geometry ratio')
ax.grid(True, alpha=0.3)
ax.legend()
plt.tight_layout()
plt.savefig('pc_max_vs_qh.png', dpi=150, bbox_inches='tight')

Numerical example — Croccolo, De Agostinis & Vincenzi (2012), Section 4.1

Data from the paper’s scenario 2: hollow steel shaft (39NiCrMo3, \(Q_S = 0.7\) ) press-fitted into an aluminium hub (EN-AW6082), no adhesive. The paper computes \(p_C = 82\) MPa as the Tresca-limited maximum pressure including friction (\(\mu = 0.4\) ).

Calculations.

From eq. (3):

From eq. (2): \(\sigma_r = -82\) MPa.

From eq. (10):

Cross-check: \(\sigma_\theta + p_C = 213.2 + 82 = 295.2\) MPa ✓

Safety factor: \(n_s = 304 / 295.2 = 1.03\) .

From eq. (11):

Interpretation. The simplified formula (11) gives \(p_{C,\max} = 84.4\) MPa. The paper reports 82 MPa from the full Tresca criterion including friction (\(\mu = 0.4\) ). The 2.9% difference confirms that friction has a modest effect on the yield limit. The safety factor \(n_s = 1.03\) indicates that 82 MPa is essentially at the Tresca limit — as expected, since the paper computed it as the maximum allowable pressure.

For a case with a more comfortable margin: at \(p_C = 67\) MPa (the paper’s hybrid scenario with adhesive), \(\sigma_{Tresca} = 67 \times 3.6 = 241.2\) MPa and \(n_s = 304/241.2 = 1.26\) .

References

1. Croccolo D., De Agostinis M., Vincenzi N. (2012), “Design and optimization of shaft–hub hybrid joints for lightweight structures: Analytical definition of normalizing parameters”, Int. J. Mechanical Sciences, 56, 77–85.
2. Croccolo D., Vincenzi N. (2009), “A generalized theory for shaft–hub couplings”, Proc. IMechE Part C, 223, 2231–2239.
3. Timoshenko S., Goodier J.N. (1970), Theory of Elasticity, McGraw-Hill, 3rd ed.

Related: Thick-Walled Cylinder Stress Analysis (Croccolo 2009)