A simple equation for obtaining finite strain solutions from small strain analyses of tunnels with very large convergences

Abstract

This paper presents a novel, very simple, accurate, theoretically well-founded and widely applicable relationship expressing the tunnel convergences obtained from large strain elasto-plastic analyses as a hyperbolic function solely of the corresponding small strain convergences. It can thus be used for ‘self-correcting’ small strain solutions, removing the need for large strain elasto-plastic analyses at least at the preliminary design stage and quantifying a hitherto unknown error caused by disregarding the geometric non-linearity. The proposed relationship can be proved rigorously for the plane strain rotationally symmetric ground response problem with a general elasto-plastic constitutive law with or without dilatancy and hardening. Numerical analyses of characteristic two- and three-dimensional excavation problems show that this relationship is generally applicable, irrespective of the in situ stress state and the tunnel geometry. It is therefore very useful from an engineering point of view for the design of tunnels crossing heavily squeezing ground, where the convergences may be so large (sometimes well in excess of 10% of the tunnel radius) that the usual small strain elasto-plastic analyses are deficient.