A technique for the solution of a large class of plane elastic-plastic problems is presented and applied to the bending of a simply supported singled crystal beam. The solution of elastic-perfectly plastic problems is accomplished by means of an iterative scheme for the location of the elastic-plastic interface at given load levels. This necessitates the solution of the plane elasticity problem in domains with irregular boundaries and numerical methods are thus dictated. An appropriate finite difference method and computer code are described. This method and code can be readily extended to include effects of elastic anisotropy and non-homogeneity. (Author).