  In this paper, we analyze stability regions of a non-static restricted class
of axially symmetric spacetime with anisotropic matter distribution. We
consider $f(R)=R+{\epsilon}R^2$ model and assume hydrostatic equilibrium of the
axial self-gravitating system at large past time. Considering perturbation from
hydrostatic phase, we develop dynamical as well as collapse equations and
explore dynamical instabilities at Newtonian and post-Newtonian regimes. It is
concluded with the help of stiffness parameter, $\Gamma_1$, that radial profile
of physical parameters like pressure anisotropy, energy density and higher
curvature terms of the $f(R)$ model affect the instability ranges.
