Fatigue Notch Factor⚠ unverified

Understanding the Parameters

• Geometric factor $K_t$ — the elastic stress raiser from geometry (Stress Concentration); here it must be entered in the mislabelled Kf field. It sets the ceiling: $K_f \le K_t$.

• Notch sensitivity $q$ — the material-and-radius property that says how much of $K_t$ carries into fatigue. $q = 0$ (fully insensitive) gives $K_f = 1$; $q = 1$ (fully sensitive) gives $K_f = K_t$. High-strength steels approach $q = 1$; softer materials and sharper notches give lower $q$.

• Fatigue factor $K_f$ — bounded $1 \le K_f \le K_t$; it multiplies the alternating stress in fatigue checks (Goodman, shaft/bolt/weld fatigue), not the mean stress.

• Why $q < 1$ — the elastic peak is a point value; fatigue initiation needs a critically-stressed region, so the steep gradient at a sharp notch makes it less damaging than $K_t$ alone predicts.

Derivation (Approaching a Proof)

By definition, notch sensitivity interpolates $K_f$ between its two physical limits — no fatigue effect ($K_f = 1$) and full elastic effect ($K_f = K_t$):

$$q \equiv \frac{K_f - 1}{K_t - 1} \;\Longrightarrow\; K_f = 1 + q\,(K_t - 1).$$

The bounds are exact by construction. The physical content lives in $q$, predicted by the Neuber and Peterson material-length models from the notch root radius $r$ and a material constant $a$:

$$q_{\text{Neuber}} = \frac{1}{1 + \sqrt{a/r}}, \qquad q_{\text{Peterson}} = \frac{1}{1 + a/r}.$$

Both say sharper notches (small $r$) and tougher/softer materials (larger $a$) give smaller $q$, so $K_f < K_t$.

Dimensional check. $K_f = 1 + q\,(K_t - 1)$ is dimensionless — all three factors are pure ratios.

History and Development

The gap between the elastic $K_t$ and the fatigue-effective $K_f$ was a key 20th-century fatigue insight: real parts tolerate sharp notches better than elasticity warns. Neuber and Peterson supplied the notch-sensitivity models that make $K_f$ computable, and $K_f = 1 + q(K_t-1)$ became the standard bridge from a stress-concentration chart to a fatigue calculation (Shigley). That the handcalcs registry carries three copies of this one relation reflects overlapping calculator sets, not three different methods.

Related Concepts: Stress Concentration, Notch Sensitivity, Fatigue Stress Concentration, Stress Concentration and Notch Effects, Marin Endurance Limit, Modified Goodman Factor

Notes: Needs the geometric $K_t$ — enter it in the mislabelled Kf field ($K_t$ is not a separate input). Third duplicate of Notch Sensitivity / Fatigue Stress Concentration. $1 \le K_f \le K_t$; multiplies the alternating stress. Get $q$ from Neuber/Peterson.