Abstract: The Sobolev embedding theorem implies correct formulation of the Dirichlet condition at isolated points or transmission conditions that simulate contact welding, bolt or screw fastening, etc. The problems on bending the Kirchhoff plate with periodically distributed point supports and the joint of two plates by rows of rivets are considered. The asymptotic analysis performed provides asymptotic expansions of solutions and error estimates, namely, the one-dimensional model of a narrow plate and the transmission conditions at the common boundary of two plates. The results of homogenization differ seriously in the cases of one or several rows of supports and rivets. In particular, one-row riveting provides only a hinge joint of plates (discontinuities in the rotation angle are allowed), but two-row riveting provides an almost complete clutch at which all elastic fields become, in main, continuous at the common edge.