The bonded interface between dissimilar materials is often the site of premature failures, especially when there is a stress singularity at the intersection of the interface and the free edge or surface. However, the singularities, which depend upon the elastic constants and the interior angles of the two materials, may vanish for certain ranges of these angles. Thus, proper selection of the interior angles offers the possibility of enhancing joint strength. In this work, the focus is on the shear strength of bonded aluminum-epoxy cylinders of uniform diameter under torsional loading and the effect of selecting angles for which the stress singularity at the edge disappears. A new boundary element framework is developed for pure shear loading to determine the power of the singularity for general interface angles and the value of the associated generalized stress intensity factor (GSIF), using a weighted traction axisymmetric torsional mode formulation. Physical experiments are conducted to examine whether changes in the interior angles produce differences in measured joint strength and failure patterns. Three different geometries are considered: a “concave” angle of 37° for the aluminum cylinder, a butt joint with 90° angles for both cylinders and a “convex” aluminum cylinder with an interior angle of 143°. Based upon elastic theory, the first two cases are associated with singularities, while the third is singularity-free. The experiments demonstrate that the joint shear strength can be greatly improved by avoiding stress singularities through the selection of a proper edge angle. From a physical perspective, this strength improvement is due to crack tortuosity, extensive adherence of epoxy to aluminum, and epoxy matrix spalling. Low strength specimens show rapid debonding and minimal adherence.