  An $(\alpha,\beta)$-manifold $(M,F)$ is a Finsler manifold with the Finsler
metric $F$ being defined by a Riemannian metric $\alpha$ and $1$-form $\beta$
on the manifold $M$. In this paper, we classify $n$-dimensional
$(\alpha,\beta)$-manifolds (non-Randers type) which are positively complete and
locally projectively flat. We show that the non-trivial class is that $M$ is
homeomorphic to the $n$-sphere $S^n$ and $(S^n,F)$ is projectively related to a
standard spherical Riemannian manifold, and then we obtain some special
geometric properties on the geodesics and scalar flag curvature of $F$ on
$S^n$, especially when $F$ is a metric of general square type.
