  Let $M$ be a hyperbolic fibered 3-manifold. We study properties of sequences
$(S_{\alpha_n}, \psi_{\alpha_n})$ of fibers and monodromies for primitive
integral classes in the fibered cone of $M$. The main tool is the asymptotic
translation length $\ell_{\mathcal{C}} (\psi_{\alpha_n})$ of the pseudo-Anosov
monodromy $ \psi_{\alpha_n}$ on the curve complex. We first show that there
exists a constant $C>0$ depending only on the fibered cone such that for any
primitive integral class $(S, \psi)$ in the fibered cone, $\ell_{\mathcal{C}}
(\psi)$ is bounded from above by $C/|\chi(S)|$. We also obtain a moral
connection between $\ell_{\mathcal{C}} (\psi)$ and the normal generating
property of $\psi$ in the mapping class group on $S$. We show that for all but
finitely many primitive integral classes $(S, \psi)$ in an arbitrary
2-dimensional slice of the fibered cone, $\psi$ normally generates the mapping
class group on $S$. In the second half of the paper, we study if it is possible
to obtain a continuous extension of normalized asymptotic translation lengths
on the curve complex as a function on the fibered face. An analogous question
for normalized entropy has been answered affirmatively by Fried and the
question for normalized asymptotic translation length on the arc complex in the
fully punctured case has been answered negatively by Strenner. We show that
such an extension in the case of the curve complex does not exist in general by
explicit computation for sequences in the fibered cone of the magic manifold.
