Prediction of spatial displacement development / Vorhersage der räumlichen Verschiebungsentwicklung

One of the major shortcomings of two-dimensional approaches for underground excavation design is the (mostly) inadequate modelling of the effects of the face advance. Although they may roughly estimate the evolution of the displacements towards their final value, none of the 2D solutions available so far provide information about the extent of the pre-relaxation the ground has undergone. On the other hand, 3D calculations pose a much greater effort in modelling and evaluation, and can be conducted only by using numerical methods.
The method presented in this paper is intended to narrow this gap, representing an easy-to-implement tool allowing fast assessment of displacement development for a circular cavity in a linearly elastic-ideally plastic Mohr-Coulomb material under hydrostatic loading. A numerical parametric study has been performed on a 3D model in FLAC^3D. The parameters for the numerical simulations were chosen in such a way that they form a regular grid in a co-ordinate system spanned by the friction angle and a non-dimensional variable defined as the ratio between the depth of failure and the tunnel radius. Based on the displacement data obtained from numerical simulations, the respective fictitious support pressures have been back-calculated for every face position, using the closed-form solution developed by Feder & Arwanitakis. The back-calculated curves of the fictitious support pressure, its dimensionless form η (equivalent support pressure coefficient), defined as the ratio between the fictitious support pressure and the primary stress respectively, have been fitted with a slightly modified version of the function proposed by Sulem et al. The fitting process yields function shape parameters, allowing the establishment of three interpolation relationships. The knowledge of the shape parameters for an arbitrary set of mechanical parameters allows the calculation of the displacement path. The results have been verified on a set of ten 3D calculations with a random set of parameters and show very good agreement in all cases. In addition, the presented method is demonstrated through case histories.