Stress Concentration Factor Calculator

A stress concentration occurs wherever a structural discontinuity forces stress to flow around an obstacle. Holes, notches, fillets, grooves, and keyways all act as stress raisers. The peak stress at the discontinuity can be several times higher than the average (nominal) stress in the cross-section.

Stress concentration factor Kt:

sigma_max = Kt * sigma_nom

For a circular hole of diameter d in a flat bar of width W under uniaxial tension, Kt follows Peterson’s polynomial (Pilkey and Pilkey, 3rd ed.):

Kt = 3 - 3.13(d/W) + 3.66(d/W)^2 - 1.53(d/W)^3

For an infinitely wide plate (d/W approaching 0), Kt approaches 3.0, the classical result Ernst Gustav Kirsch published in Germany in 1898. Charles Inglis generalised it to elliptical holes in 1913, and his Kt = 1 + 2a/b collapses back to 3 for a circle. As d/W increases toward 1 the net section shrinks and the formula breaks down, so keep it to d/W below 0.8.

Which nominal stress does that Kt go with? This is the part that trips people up, and getting it backwards is a common source of unsafe answers. Peterson’s polynomial above is referenced to the net section stress, not the gross one:

sigma_net = sigma_gross * W / (W - d) sigma_max = Kt_net * sigma_net

The two conventions are related by that same width ratio:

Kt_gross = Kt_net * W / (W - d)

and both give the identical peak stress, which is the only number that matters. They agree at d/W = 0 and diverge fast: at d/W = 0.5 the net factor is 2.16 while the gross factor is 4.32.

Here is the sanity check that catches the mistake every time. The peak stress at the hole edge can never be lower than the average stress across the ligament that remains. If a calculation hands you a peak below the net section stress, the wrong nominal stress has been paired with the wrong Kt.

Why it matters for fatigue. Under cyclic loading, cracks initiate at stress concentrations. The stress concentration factor enters the fatigue life calculation directly. A Kt of 3 at a hole reduces the fatigue life of the structure far more than three times, because fatigue life scales roughly as sigma^(-3) to sigma^(-4) for metals.

Reducing stress concentrations. Larger fillet radii at notch roots, relief grooves adjacent to sharp features, and compressive residual stresses from shot-peening all reduce effective Kt in practice.