= Rotating shaft bending fatigue
\b[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 is
$$
\sigma_a=\frac{M(d/2)}I=\frac{32M}{\pi d^3}.
$$
A rotating surface point has $\sigma(t)=\sigma_a\cos(\omega t+\phi)$ 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, $M=20\,\mathrm{N\,m}$ and $d=20\,\mathrm{mm}$ give $\sigma_a\approx25.5\,\mathrm{MPa}$. 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.<AMB2025-03 High-Cycle Rotating Bending Fatigue Tests of PBF-L Ti-6Al-4V>[Challenge Summary and Fatigue Testing]{ref} A lifetime claim needs appropriate experimental fatigue data and treatment of scatter; the elastic stress formula does not supply one.
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