= Solution
The uniform weight per unit length is $q=\rho abg$, where $\rho$ is the <mass density> and $g$ the <gravitational acceleration>. The total weight is $qL$, so each support provides an upward <force> $qL/2$. The weight on the portion from $0$ to $x$ is $qx$ acting at that portion's midpoint, a distance $x/2$ from the cut. In the same <bending moment> convention as part (c),
$$
B(x)=-\frac{qL}{2}x+(qx)\frac x2,
\qquad
\boxed{B(x)=-\frac12\rho abg\,x(L-x).}
$$
The units are $(\mathrm{N\,m^{-1}})\mathrm{m^2}=\mathrm{N\,m}$. Its greatest magnitude occurs at $x=L/2$, where $B=-qL^2/8$.
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