The design equations
Now put it together. You have an alternating von Mises stress σ′a, a steady one σ′m, a fatigue strength Se and a tensile strength Sut. A fatigue criterion is a line (or curve) in the σ′m–σ′a plane that separates safe from unsafe.
The four criteria
| Criterion | Equation | Use it when |
|---|---|---|
| modified Goodman | σ′aSe + σ′mSut = 1n | default; simple, always conservative |
| Gerber | n σ′aSe + (n σ′mSut)² = 1 | you want the best fit to test data |
| ASME elliptic | (n σ′aSe)² + (n σ′mSy)² = 1 | a good middle ground; used by the ASME shaft code |
| Soderberg | σ′aSe + σ′mSy = 1n | you want yielding covered by the same check |
They spread by roughly 20 % on a typical shaft, with Soderberg the most conservative and Gerber the least. Switch between them in panel 06 and watch n move — if the decision changes, your design is too close to the line anyway.
The shaft form of the equations
Substituting the round-shaft stresses gives the form you will find in every design handbook. For DE-Goodman:
and turning it round to size the shaft:
The cube root is worth internalising: the stress goes as 1/d³. Adding 10 % to a diameter buys 33 % more strength; adding 26 % doubles it.
Do not forget yielding
Goodman and Gerber say nothing about the first load cycle. A shaft can be perfectly safe in fatigue and still yield on start-up, so check it separately with the maximum von Mises stress:
ShaftLab reports both, side by side. If your drive can stall or has a big starting torque, run the yield check with the stall torque, not the running torque.
What safety factor?
n = 1.5 is a reasonable starting point for a well-understood machine with well-known loads. Push it to 2–3 when the loads are uncertain, when failure would be dangerous or expensive, or when the material data is a guess.
The simple single-gear shaft. Switch between the four criteria and watch the fatigue safety factor — then switch the gear from spur to helical (β = 15°) and see the axial force change the picture.