Keyway Stress Reduction⚠ unverified

Understanding the Parameters

• Shaft diameter $d$ and keyway width $w$ — a standard key width is roughly a quarter of the shaft diameter ($w/d \approx 0.25$), for which this heuristic returns $K_t \approx 1.05$ — noticeably lower than the ~1.6–2.0 that real keyways exhibit. The heuristic captures the direction (a wider slot relative to the shaft concentrates more stress) but not the true magnitude.

• The missing variable — fillet radius — the dominant driver of real keyway stress concentration is the radius at the bottom corners of the slot, not its width. A sharp corner ($r/d \to 0$) sends $K_t$ toward 3–4; a generous fillet relaxes it. Because this formula omits the fillet radius, it cannot reproduce the real behaviour.

Derivation (Approaching a Proof)

There is no first-principles derivation for this expression — it is an empirical linear fit intended to give a quick, monotonic estimate. The genuine physics of a keyway is a notch stress-concentration problem: the abrupt change in section and the small fillet radius force the stress trajectories to crowd together at the corner, raising the local stress to $K_t$ times the nominal $\sigma_{\text{nom}}$:

$$\sigma_{\max} = K_t\,\sigma_{\text{nom}}.$$

For real geometries $K_t$ is obtained from Peterson's stress-concentration charts (from photoelastic and finite-element studies) as a function of the fillet-radius-to-diameter ratio $r/d$ and the loading mode (bending vs torsion). Those charts show a strong nonlinear dependence on $r/d$ and values well above what the linear $1 + 0.2(w/d)$ heuristic gives. In fatigue, $K_t$ is further reduced to a fatigue stress-concentration factor $K_f = 1 + q(K_t - 1)$ via the notch-sensitivity $q$, and enters the endurance limit through the Marin miscellaneous factor (Marin Modification Factors).

Dimensional check. $w/d$ is dimensionless, so $K_t$ is dimensionless. ✓

History and Development

Stress-concentration factors were mapped experimentally (photoelasticity) and compiled by R. E. Peterson (Stress Concentration Factors, 1953/1974), the standard reference. Keyways receive special attention because they are the most common shaft fatigue initiator; standard profile and sled-runner keyways have tabulated $K_t$ values in Shigley and Peterson. This heuristic is a placeholder for those data.

Related Concepts: Torsional Shear Stress, Shaft Fatigue Factor, Marin Modification Factors, Combined Stress Shaft, Fatigue Endurance Limit

Notes: Crude heuristic — real keyway $K_t \approx 1.6$–$2.0$ (higher at sharp fillets) and depends on fillet radius, not $w/d$ linearly. Use Peterson charts/FEA for design and convert to $K_f$ via notch sensitivity for fatigue. Name is a misnomer ($K_t$ raises stress).