Tunnel roof is subjected to a complex tension-shear stress state after excavation. A tensile cut-off strength criterion is introduced in this study and combined with the upper bound limit analysis method to investigate the stability of a rectangular tunnel roof. First, the expression for the internal energy dissipation rate is derived for the circular cut-off segment of the failure criterion. Power functionals Φ are established for both two-dimensional and three-dimensional rotational collapse mechanisms. The analytical equations for the failure surface are obtained using the variational method. The strength reduction method that incorporates the cut-off criterion is proposed to quantify roof stability. The investigation into the morphology of the collapsing block indicates that the supporting pressure and the reduction coefficient ξ have a significant influence on the collapse shape of the tunnel, suggesting that attention should be paid to the suspension effect of the tunnel roof on stability. The range of the collapsing block under three-dimensional conditions is found to be larger than that under two-dimensional conditions. Parametric influences on the safety factor are examined. Finally, dimensionless design charts for the critical reinforcement pressure are provided for practical tunnel support design.
Cai et al. (2026) studied this question.