Stability analysis of unlined elliptical tunnel using finite element upper-bound method with rigid translatory moving elements

The stability of an unlined elliptical tunnel in cohesive-frictional soils is determined. The analysis has been performed with two methods: finite element upper-bound method with plastic deformation elements (UP-PDE) and finite element upper-bound method with rigid translatory moving elements (UP-RTME). UP-PDE has been used to study tunnel stability by many scholars. The UP-RTME in combination with a finite element approach and triangular rigid translator moving elements is presented in detail. In the proposed method, the node coordinates and velocities of rigid elements are treated as unknowns without considering the rotating freedom. A specific plane strain formulation is proposed using nonlinear programming, and the optimal slip lines are determined by automatically adjusting the velocity discontinuities. Solutions for the influence of a range of soil parameters, dimensionless depths C/D and dimensionless spans B/D on the stability numbers γD/c and collapse mechanisms are solved using this method. The γD/c values increase with ϕ and decrease with C/D and B/D. γD/c is less sensitive to C/D as ϕ increases. The collapse mechanisms of unlined elliptical tunnels comprising two groups of slip lines are also presented, and they explicitly reflect the relative movement of blocks. The results show that these two methods are in agreement with each other.