Undrained stability of a square tunnel where the shear strength increases linearly with depth

This paper investigates the stability of a plane strain square tunnel in undrained clay, where the shear strength profile increases linearly with depth. The stability for a range of tunnel geometries and soil conditions are found using rigid-block upper bound methods as well as finite element limit analysis. The latter procedures employ a discrete form of the bound theorems of classical plasticity, use a bespoke conic programming scheme to solve the resulting optimisation problems, and bracket the true collapse load with upper and lower bound solutions to within 5% for all the cases considered. Results from the parametric study are summarised in the form of stability charts. An approximate closed-form expression is developed for use by practising engineers.