 Rotating shaft bending fatigue

Rotating shaft bending fatigue occurs when a shaft rotates while carrying a bending moment whose direction is approximately fixed in space. A material point on the outer surface moves alternately through the tensile and compressive sides of the bent section. A constant external bending load can therefore produce cyclic local stress.

For an ideal circular shaft in elastic pure bending, with diameter $d$, second moment of area $I=\pi d^4/64$ and steady moment magnitude $M$, the nominal surface stress amplitude isA rotating surface point has $\sigma (t)={\sigma }_a\cos (\omega t+\varphi )$ in this model. The mean is zero, ${\sigma }_{\max }={\sigma }_a$, ${\sigma }_{\min }=-{\sigma }_a$ and stress ratio $R={\sigma }_{\min }/{\sigma }_{\max }=-1$. One shaft revolution produces one full stress cycle at that point. A bending moment rotating with the shaft would require a different analysis; shaft rotation alone does not guarantee this cycle.

For an original example, $M=20\mathrm{N}\mathrm{m}$ and $d=20\mathrm{mm}$ give ${\sigma }_a\approx 25.5\mathrm{MPa}$. At $1800$ revolutions per minute, one hour supplies $108000$ cycles. These values specify a nominal loading example, not a predicted life or a safe design stress. Stress concentration, surface finish, material condition and other loads affect the local response and mechanical fatigue behavior.

A motivating apparatus is four-point rotating-bending testing. A NIST/NASA benchmark used machined and polished titanium-alloy specimens, a $100\mathrm{Hz}$ test frequency and fully reversed loading with $R=-1$. Comparing differently treated specimens under a controlled arrangement helps connect material state to fatigue response.^[ref] A lifetime claim needs appropriate experimental fatigue data and treatment of scatter; the elastic stress formula does not supply one.