One of the major drawbacks of two-dimensional approaches in underground excavation design is the (mostly) inadequate modelling of the effects of the face advance. The amount of pre-displacements and the evolution of the displacements towards their final value are generally roughly estimated. On the other hand, 3D calculations pose a much greater modelling and evaluation challenge, and can be conducted only by the usage of numerical methods. The work presented in this publication is directed towards narrowing of this gap. It concerns the determination of the displacement evolution in an elasto-plastic, Mohr-Coulomb medium under hydrostatic primary stress state. 3D numerical simulations with FLAC 3D have been used to generate the displacement evolutions for various parameter combinations. The parameters have been chosen in such way that they form a regular grid in a co-ordinate system spanned by the friction angle and a unitless variable defined as the ratio between the depth of failure zone and the tunnel radius. Utilizing the closed-form solution by Feder and Arwanitakis, the equivalent fictitious support pressure was calculated from the from the displacement paths calculated with FLAC, thus allowing the establishment of three interpolation relationships. The development of the fictitious support pressure can be directly derived from these interpolation relationships, hence predicting both the displacement development and the amount of pre-relaxation of the ground. The effects of lining installation can be easily incorporated in this model by superposing the fictitious support pressure with the mobilized support pressure, while obeying the displacement compatibility between the liner and the rock mass. Both the solution scheme and its validation are presented in this paper, together with an empirical relationship depicting the rheological properties of shotcrete. The combination of these relationships enables an approximate estimation of the displacement development with installed shotcrete support and of the associated shotcrete utilisation ratio, hence allowing very fast assessment of the system behaviour in given tunnelling conditions. The results obtained with the application of the presented method are discussed, depicting selected tunnelling scenarios and their impact on the lining displacement development and associated utilisation ratio.


When tunnelling in difficult ground conditions, the importance of the ability to correctly predict the magnitude, spatial orientation and development of the displacements caused by the excavation and associated stress re-distribution cannot be overstressed. The entire scope of displacement information, as stated above, can be a priori predicted only by numerical methods. However, the prediction of radial displacement magnitude and its path towards final convergence are also possible with closed-form solutions and different empirical relationships [1,2,3]. Since the displacements in the plane perpendicular to the tunnel axis have the highest magnitude and result in the highest loading of the lining, this work omits the longitudinal displacements completely.

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