  There has been increasing evidence that at least some GRBs are emission
beamed. The beamed GRB afterglow evolution has been discussed by several
authors in the ultra-relativistic case.
  It has been shown that the dynamics of the blast wave will be significantly
modified by the sideways expansion, and there may be a sharp break in the
afterglow light curves under certain circumstances. However, this is true only
when the fireball is still relativistic. Here we present an analytical approach
to the evolution of the beamed GRB blast wave expanding in the surrounding
medium (density $n\propto r^{-s}$) in the non-relativistic case, our purpose is
to explore whether the sideways expansion will strongly affect the blast wave
evolution as in the relativistic case. We find that the blast wave evolution is
strongly dependent on the speed of the sideways expansion. If it expands with
the sound speed, then the jet angle $\theta $ increases with time as $\theta
\propto ln t$, which means that the sideways expansion has little effect on the
afterglow light curves, the flux $F\propto t^{-\frac{3(5\alpha -1)}{5}}$ for
$s=0$ and $F\propto t^{-\frac{7\alpha +1}{3}}$ for $s=2$.It is clear that the
light curve of $s=2$ is not always steeper than that of $s=0$, as in the
relativistic case.We also show that if the expansion speed is a constant, then
the jet angle $\theta \propto t$, and the radius $r \propto t^{0}$, in this
case the sideways expansion has the most significant effect on the blast wave
evolution, the flux $F\propto t^{-(5\alpha -1)}$ independent of $s$, and we
expect that there should be a smooth and gradual break in the light curve.
