In this paper the stresses in a joint with a functionally graded material (FGM) are analyzed. In the middle of a joint with FGM the stresses have been calculated analytically by using the plate theory. The effect of the thickness of the FGM layer and the effect of the transition function form on the stresses in the joint is discussed. Near the free edges of the interface in a joint with FGM, the stresses are described analytically by using the Mellin transform method. Some examples are presented to show the good agreement of the stresses calculated from FEM and with the analytical description in a joint with graded material.