  In this paper, we study the relaxed energy for biharmonic maps from a
$m$-dimensional domain into spheres. By an approximation method, we prove the
existence of a minimizer of the relaxed energy of the Hessian energy, and that
the minimizer is biharmonic and smooth outside a singular set $\Sigma$ of
finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we prove
that the singular set $\Sigma$ is 1-rectifiable.
