Abstract: Rigid sliding blocks are used to construct two failure modes of the tunnel's circumferential excavation surface of circular tunnel. The compiled nonlinear programming program is used to solve the optimal upper bound solution of the support force coefficient σ_T /c (σ_T is uniformly distributed support load and c is the effective cohesion) and the stratum failure mode so as to reveal the influences of stratum parameters on tunnel stability. A simple and practical simplified formula for the ultimate support force of the tunnel's circumferential excavation surface is proposed. For undrained condition, the failure region is mainly concentrated in the upper part of the tunnel when the tunnel depth ratio H/D (H is the buried depth and D is the tunnel diameter) and the gravity coefficient γD/c (γ is unit weight) are small. With the increase of H/D and γD/c, the starting position of the slip line gradually expands to the bottom of the tunnel along the tunnel contour, and the failure region expands to the horizontal direction. For drainage conditions, there are three main stratum failure modes. When the internal friction angle ϕ and γD/c are large, with the decrease of the dilatancy coefficient, the ultimate supporting force increases significantly, and the range of failure region varies greatly, which may even cause a change in the failure mode. The ultimate supporting force of the tunnel circumferential excavation surface can be quickly obtained through the proposed simplified formulas.

Key words: Shield tunnel, Rigid block method, Failure mechanism, Ultimate support force, Non-associated flow rule