  We develop and analyze an ultraweak variational formulation of the
Reissner-Mindlin plate bending model both for the clamped and the soft simply
supported cases. We prove well-posedness of the formulation, uniformly with
respect to the plate thickness $t$. We also prove weak convergence of the
Reissner-Mindlin solution to the solution of the corresponding Kirchhoff-Love
model when $t\to 0$.
  Based on the ultraweak formulation, we introduce a discretization of the
discontinuous Petrov-Galerkin type with optimal test functions (DPG) and prove
its uniform quasi-optimal convergence. Our theory covers the case of non-convex
polygonal plates.
  A numerical experiment for some smooth model solutions with fixed load
confirms that our scheme is locking free.
