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This paper investigates the undrained stability of a plane strain circular tunnel in clay, where the shear strength profile is assumed to increase linearly with depth. Stability solutions for a variety of geometries and soil conditions are found using rigid-block upper bound methods as well as finite element limit analysis (which gives both upper and lower bounds). The latter procedures employ a discrete form of the bound theorems of classical plasticity, use a bespoke conic programming scheme to solve the resulting optimization problems, and bracket the true collapse load to within 5% for all the cases considered. Results from the study are summarized in the form of stability charts as well as an approximate closed-form expression that can be used by practising engineers.
Wilson et al. (2011) studied this question.