In this master thesis we study the consistent semi-discretization of a plane linear elastic body. First we consider the theory of a three-dimensional linear elastic body by means of a primal formulation and a mixed formulation (Hellinger-Reissner-Principle). Assuming, that such a body is plane and its thickness is small compared to the planar dimensions, we separate the general three-dimensional problem into a so-called bending and a membrane problem. For both of these problems we perform a semi-discretization of the displacement by virtue of a series expansion. The obtained auxiliary problems are then solved on a two-dimensional computational domain. On this basis we analyse the consistency of the semi-discretized bending problem to the original three-dimensional problem for two different boundary conditions.