= Overhung shaft bending stress
\b[Overhung shaft bending stress] is the longitudinal normal <stress> caused by the <bending moment> from a transverse load beyond the shaft's supports. It differs from twisting due to shaft-axis <torque>. In the ideal two-support model, the largest <overhung shaft bending moment> has magnitude $M=Pe$, where $P$ is the end load and $e$ the overhang.<Shafts and Bearings by SDP-SI>[Section 4.0, case (b), overhung radial load; shaft-stress discussion of bending and torsion]{ref}
For a homogeneous linear <elastic beam>, plane sections in the beam model give longitudinal <strain> $\epsilon(y)=\kappa y$ relative to the <neutral axis>. With <Young's modulus> $E$,
$$
\sigma(y)=E\kappa y,\qquad
M=\int_A y\sigma(y)\,dA=E\kappa I,
$$
where $I$ is the <second moment of area>. Thus $\sigma(y)=My/I$. For a solid circular shaft of diameter $d$, $I=\pi d^4/64$ and the outermost distance is $d/2$, giving
$$
\sigma_{\max}=\frac{32Pe}{\pi d^3}.
$$
An original example uses $P=200\,\mathrm N$, $e=30\,\mathrm{mm}$ and $d=20\,\mathrm{mm}$. Then $M=6000\,\mathrm{N\,mm}$ and $\sigma_{\max}\simeq7.64\,\mathrm{MPa}$. Reducing diameter to $16\,\mathrm{mm}$ multiplies nominal stress by $(20/16)^3\simeq1.953$, raising it to about $14.9\,\mathrm{MPa}$. The cubic diameter dependence makes a diameter change different from an equal percentage change in overhang.
A measurement check can place opposed axial <strain gauges> at one shaft station. In a simple axial-plus-bending model, readings $\epsilon_+=44$ and $\epsilon_-=-32$ microstrain give axial strain $(\epsilon_++\epsilon_-)/2=6$ microstrain and bending strain $(\epsilon_+-\epsilon_-)/2=38$ microstrain. This separates two contributions that a single gauge would mix.
The nominal formula excludes local geometric concentrations, multiaxial loading and yielding. It is not a fatigue-life prediction. Under a stationary bending direction, a rotating material point can experience alternating <stress>, motivating <rotating shaft bending fatigue> even when the external load is steady. A permissible load requires the actual geometry, material and load history.
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