Abstract: Based on Bogy's characteristic equation, the geometrical conditions of eliminating stress singularities near bi-materials interface end are investigated, and contours of the combination of wedge angles are obtained for plane stress and plane strain problems. The results are verified using finite element method. The distributions of the stress along interface after the disappearance of stress singularity are analyzed and discussed. The results show that the stress distribution along the interface can become smoother with two critical wedge angles.