  This paper focuses on the construction of periodic solutions of nonlinear
beam equations on the $d$-dimensional tori.
  For a large set of frequencies, we demonstrate that an equivalent form of the
nonlinear equations can be obtained by a para-differential conjugation. Given
the non-resonant conditions on each finite dimensional subspaces, it is shown
that the periodic solutions can be constructed for the block diagonal equation
by a classical iteration scheme.
