= Overhung shaft bending moment
{title2=$M=Pe$}
An \b[overhung shaft bending moment] is the internal <bending moment> produced when a transverse load acts beyond the shaft's support span. For a single end load $P$ at overhang distance $e$, its magnitude immediately next to the nearer support is
$$
M_{\max}=Pe
$$
in the ideal two-support shaft model. The relevant lever arm is the distance beyond that support, rather than the distance between supports. Manufacturer bearing-load guidance distinguishes this arrangement from a load located between bearings.<Shafts and Bearings by SDP-SI>[Section 4.0, case (b), Overhung Radial Load]{ref}
Place ideal supports A and B at axial positions $z=0$ and $z=L$, and apply an upward load $P$ at $z=L+e$. With upward reactions positive, <static equilibrium> gives
$$
R_A=P\frac eL,\qquad R_B=-P\left(1+\frac eL\right).
$$
Choosing the internal moment sign consistently with the left-hand force sums, the <bending moment> along the shaft is
$$
M(z)=\begin{cases}
Pe\,z/L,&0\le z\le L,\\
P(L+e-z),&L\le z\le L+e.
\end{cases}
$$
It is continuous at B, reaches $Pe$ there and vanishes at the free end. An ideal bearing need not apply a concentrated couple for the shaft's internal moment at its location to be nonzero: the shaft continues beyond that support.
For original values $P=200\,\mathrm N$, $L=100\,\mathrm{mm}$ and $e=30\,\mathrm{mm}$, the reactions are $+60$ and $-260\,\mathrm N$ and the peak moment is $6\,\mathrm{N\,m}$. Doubling $e$ at fixed $P$ doubles that moment. Changing $L$ while keeping $P,e$ fixed changes the reactions but leaves $Pe$ unchanged.
\Image[/overhung-shaft-moment-diagram.png]
{title=Original overhung-shaft equilibrium and <bending moment> diagram for a 200 N upward end load}
For a uniform solid circular shaft in simple elastic bending, <overhung shaft bending stress> follows from $\sigma_{\max}=Mc/I$ and $I=\pi d^4/64$:
$$
\sigma_{\max}=\frac{32Pe}{\pi d^3}.
$$
At diameter $d=20\,\mathrm{mm}$, the example gives about $7.64\,\mathrm{MPa}$. This excludes stress concentrations and other loads; a shaft's allowable load requires more than this nominal estimate. Bending can remain even when opposing gear <torques> give zero net shaft-axis <torque>, because transverse-force moments and axial twisting are different components of the mechanical loading.
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