Undrained Stability of Shallow Square Tunnel

Abstract

The upper‐ and lower‐bound theorems of classical plasticity are used to examine the undrained stability of a shallow square tunnel under conditions of plane‐strain loading. Rigorous bounds on the loads needed to support the tunnel against active or passive failure are derived using two numerical techniques that are based on a finite element type of discretization. Both techniques assume a perfectly plastic soil model with a linearized Tresca yield criterion and lead to large linear programming problems. The solution to the lower‐bound linear programming problem defines a statically admissible stress field, whereas the solution to the upper‐bound linear programming problem defines a kinematically admissible velocity field. For the range of tunnel geometries considered, the solutions obtained typically bracket the exact collapse load to within 15% or better. Where appropriate, the results for the upper‐bound formulation are verified using a variety of rigid block mechanisms.