The crack initiation equals to fracture for bonded joint with brittle adhesive. The criterion is formulated in terms of the quasi-stress intensity factor Kp, for the maximum principle stress, that is analogous to the stress intensity factor used to characterize the stress field in the vicinity of bond terminus. Kp is evaluated using a boundary element analysis. The crack initiation at the terminus of adhesive bonded joints is estimated with the critical quasi-stress intensity factor Kp. This method presented here hardly pays attention to the crack propagation. Since there is a large influence of crack propagation on the strength of adhesive joints and structures, crack propagation must be taken into account on strength prediction of bonded joints. The quasi-stress intensity factor Kp for the maximum principle stress can use as the criteria of the crack initiation at the terminus of adhesive bonded joints having various shapes.