Stress Concentration Factor Calculator (K_t & K_f)

From K_t to K_f

The theoretical (elastic) stress concentration factor relates the peak stress at the discontinuity to the nominal stress:

$$K_t = \frac{\sigma_{max}}{\sigma_{nom}}$$

For an elliptical hole in an infinite plate, Inglis derived the exact solution

$$K_t = 1 + 2\sqrt{\frac{a}{\rho}}$$

which shows why sharp notches (small tip radius $\rho$) are so damaging — and, in the limit $\rho \to 0$, why cracks require fracture mechanics rather than a stress concentration factor.

In fatigue, the full $K_t$ is usually not felt: small, sharp notches affect a volume of material too small to control crack initiation. Neuber's notch sensitivity captures this size effect:

$$q = \frac{1}{1 + \sqrt{a_N}/\sqrt{r}}, \qquad K_f = 1 + q\,(K_t - 1)$$

where $\sqrt{a_N}$ is the Neuber material constant, estimated here from $S_{ut}$ with the curve fits for steels given in Shigley (separate fits for bending/axial and for torsion). $K_f$ — not $K_t$ — is the factor to use in fatigue: applied to the endurance limit in stress-life, or as the concentration factor in Neuber's rule in the strain-life method.

A practical note for FEA users: a converged elastic finite element model of the notch returns $K_t$ directly — but it can never return $q$. Notch sensitivity is a material-scale effect that the mesh cannot see, and it must be applied separately.

Sources and validity

Shoulder fillet factors use Pilkey's polynomial fits to Peterson's charts (Peterson's Stress Concentration Factors), valid for $0.1 \le h/r \le 20$ in tension and bending and $0.25 \le h/r \le 4$ in torsion, with $h = (D-d)/2$. The two published $h/r$ fit ranges do not meet continuously at $h/r = 2$ for large step ratios (the printed $C_4$ polynomials diverge there, up to ~23% for a flat bar in bending at $D/d = 3$); this calculator blends the two fits linearly over $1.8 \le h/r \le 2.2$ to remove the artificial jump. In the $K_t$ chart, curve portions where the underlying fit validity is exceeded (and the value clamped) are drawn dashed. The plate with a central hole uses the Howland fit on the net-section nominal stress, valid for $d/W \le 0.65$. The Neuber constant fits apply to steels with $S_{ut}$ between roughly 345 and 1725 MPa (50–250 ksi). The calculator warns when an input leaves these ranges.

This calculator is provided for educational purposes and accompanies our course Introduction to Fatigue Calculations with FEA. Verify results against applicable standards before using them in design.