UNDRAINED STABILITY OF SHALLOW SQUARE TUNNEL

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.