A coupled model resulting from the boundary element method and eigen-analysis is proposed in this paper to analyse the stress field at crack tip. This new combine method can yield several terms of the non-singular stress in the Williams asymptotic expansion. Then the maximum circumferential stress (MCS) criterion taken the non-singular stress into account is introduced to predict the brittle fracture of cracked structures. Two earlier experiments are re-examined by the present numerical method and the role of the non-singular stress in the brittle fracture is investigated. Results show that if more terms of non-singular stress are taken into account, the predicted crack propagation direction and the critical loading by MCS criterion are much closer to the existing experimental results, especially for dominating mode II loading conditions. Moreover, numerical results manifest that Williams series expansion can describe the stress field further from the crack tip if more non-singular stress terms are adopted.