Upper-bound limit analysis of passive failure of a 3D shallow tunnel face under the bidirectional inclined ground surfaces

Abstract A shallow tunnel entrance and exit zone generally includes two typical characteristics: a certain inclined angle of the ground surface and a shallow burial depth. Therefore, the accidents induced by tunnel excavation are common in this zone. However, the influence of the inclined ground surface, especially bidirectional inclined ground surfaces, is rarely considered in the stability analysis of shallow tunnel faces. When their influences are considered, the expression of volume for the last slider is not an integrable function, and the Gauss-Legendre quadrature method is used to solve it. Then, the analytical expression of the passive failure face supporting pressure for a 3D shallow tunnel face is derived. In addition, the influences of longitudinal and transverse (along and vertical the tunnel heading direction) inclined angles on the passive stability and failure modes of shallow tunnel face are examined. The results show that the longitudinal inclined angle has more significant influence on the passive stability than that of the transverse inclined angle due to the selection failure mode of the latter angle. Therefore, the effect of the transverse inclined angle cannot be neglected in research, especially in the tunnel entrance and exit zone.