We study deformations of a long (narrow after rescaling) Kirchhoff plate with periodic (rapidly oscillating) boundary. We deduce a limiting system of two ordinary differential equations of orders 4 and 2 which describe the deflection and torsion of a two-dimensional plate in the leading order. We also consider point supports (Sobolev conditions) whose configuration influences the result of homogenizing the biharmonic equation by decreasing the size of the limiting system of differential equations or completely eliminating it. The boundary-layer phenomenon near the end faces of the plate is studied for various ways of fastening as well as for angular junctions of two long plates, possibly by point clamps (Sobolev conjugation conditions). We discuss full asymptotic series for solutions of static problems and the spectral problems of plate oscillations.