  This paper focuses on the simultaneous homogenization and dimension reduction
of periodic composite plates within the framework of non-linear elasticity. The
composite plate in its reference (undeformed) configuration consists of a
periodic perforated plate made of stiff material with holes filled by soft
matrix material. The structure is clamped on a cylindrical part. Two cases of
asymptotic analysis are considered: one without pre-strain and the other with
matrix pre-strain. In both cases, the total elastic energy is in the
von-K\'arm\'an (vK) regime ($\varepsilon^5$).
  A new splitting of the displacements is introduced to analyze the asymptotic
behavior. The displacements are decomposed using the Kirchhoff-Love (KL) plate
displacement decomposition. The use of a re-scaling unfolding operator allows
for deriving the asymptotic behavior of the Green St. Venant's strain tensor in
terms of displacements. The limit homogenized energy is shown to be of vK type
with linear elastic cell problems, established using the $\Gamma$-convergence.
  Additionally, it is shown that for isotropic homogenized material, our limit
vK plate is orthotropic. The derived results have practical applications in the
design and analysis of composite structures.
