Stability for 3D tunnel vaults in soils considering transition from shallow to deep burial conditions subjected to compound tension-shear failure

The stability of three-dimensional (3D) cylindrical tunnel vaults in soil strata subjected to a compound tension-shear failure is investigated based on the upper bound limit analysis. A failure mechanism for 3D cylindrical tunnel vaults reflecting the transition process from shallow to deep buried conditions, making the deep buried failure mechanisms a special case, is developed from the modified Mohr–Coulomb (MC) criterion with tension cut-off (TC). By deducing the energy balance equation, the expressions for three stability indices, i.e., the stability coefficient, the factor of safety (FoS) and critical required reinforcement strength, which can be used to quantify the tunnel vault stability, are derived. The optimal upper bound solution of stability indices is obtained using compiler optimization algorithms. The influences of 3D geometric characteristics and soil strength parameters with TC on the tunnel vault stability during the transition process from shallow to deep buried conditions of tunnel vault are revealed. Further, the stability charts for the 3D tunnel vault FoS under different parameters and for the critical reinforcement strength required to maintain the tunnel vault stability under different desired FoS are calculated and plotted. In practical engineering, the corresponding data can be acquired quickly by chart query, which provides a theoretical basis for the 3D preliminary design of cylindrical tunnels.