  Recently, a family of interesting analytical brane solutions were found in
$f(R)$ gravity with $f(R)=R+\alpha R^2$ in Ref. [Phys. Lett. B 729, 127
(2014)]. In these solutions, inner brane structure can be turned on by tuning
the value of the parameter $\alpha$. In this paper, we investigate how the
parameter $\alpha$ affects the localization and the quasilocalization of the
tensorial gravitons around these solutions. It is found that, in a range of
$\alpha$, despite the brane has an inner structure, there is no graviton
resonance. However, in some other regions of the parameter space, although the
brane has no internal structure, the effective potential for the graviton KK
modes has a singular structure, and there exists a series of graviton resonant
modes. The contribution of the massive graviton KK modes to the Newton's law of
gravity is discussed shortly.
