Consider a thin spherical shell of uniformly density of mass $M$ and radius $R$:

Say the shell has acquired a mass $m$ and further a mass $dM$ is be added

$dW = Vdm = \frac{Gm dm}{R}$ or $W = -{\int }_0^M\frac{Gmdm}{R}$

$=-\frac{G{M}^2}{2R }= self energy =U$

Say $F$is now the attractive force per unit area . If the shell expands from $R$ to $R +dR$ then work done by attractive force is $-F \times 4{\mathrm{\pi R}}^2\mathrm{dR}$ since this is the work done by gravitational field, this may be equal to reduction in gravitational potential energy.

or ,$4{\mathrm{\pi R}}^2\mathrm{FdR} =\mathrm{dU}$or $F = -\frac{1}{4{\mathrm{\pi R}}^2}\frac{dU}{dR} = \frac{G{M}^2}{8{\mathrm{\pi R}}^4}$

Now total force experienced by one hemisphere 

$= F \times {\mathrm{\pi R}}^2$(vector addition adds only components perpendicular to the equation plane)

$\Rightarrow$Net gravitational force $F_G = \frac{G{M}^2}{8R^2}$