  We consider the Kirchhoff equation $$ \partial_{tt} u - \Delta u \Big( 1 +
\int_{\mathbb T^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus
$\mathbb T^d$, and its Cauchy problem with initial data $u(0,x)$, $\partial_t
u(0,x)$ of size $\varepsilon$ in Sobolev class. The effective equation for the
dynamics at the quintic order, obtained in previous papers by quasilinear
normal form, contains resonances corresponding to nontrivial terms in the
energy estimates. Such resonances cannot be avoided by tuning external
parameters (simply because the Kirchhoff equation does not contain parameters).
  In this paper we introduce nonresonance conditions on the initial data of the
Cauchy problem and prove a lower bound $\varepsilon^{-6}$ for the lifespan of
the corresponding solutions (the standard local theory gives
$\varepsilon^{-2}$, and the normal form for the cubic terms gives
$\varepsilon^{-4}$). The proof relies on the fact that, under these
nonresonance conditions, the growth rate of the "superactions" of the effective
equations on large time intervals is smaller (by a factor $\varepsilon^2$) than
its a priori estimate based on the normal form for the cubic terms. The set of
initial data satisfying such nonresonance conditions contains several
nontrivial examples that are discussed in the paper.
