The Dugdale model was initially developed in the case of a mode I loading. It was extended to other modes and to the mixed mode case. The exact solutions were given for all these modes in the case of an infinite medium with a straight crack. This work is an application of the Dugdale model to a crack in a semi infinite structure submitted to a mixed mode loading. The coupled system of singular integral equations of the first kind corresponding to the elastostatic problem is solved semi-analytically. Particular attention is needed in the resolution because of jump discontinuities in the loading of the crack faces. The criteria of propagation are deduced from the revisited Griffith theory (Francfort, Marigo, Journal of Mechanics and Physics of Solids, 1998). The presented results show the evolution of the applied load and critical stress with the crack length. The shape of the crack gap is also presented. A comparison with the problem of a crack in an infinite structure is performed.