Analysis of Convergence Confinement of Circular Tunnel by Supporting Elements Using Elasto-Plastic Solution of Ground
Authors: Nishimura T. , Kimura K. , Kawasaki T. , Kohno M. 
Published: August 2024

ABSTRACT:
The paper presents an analytical model for the interaction between the tunnel ground and the installed support system. Firstly, an analytical solution for the problem of stress-displacement around a circular tunnel in an elasto-perfectly-plastic medium is introduced. Shotcrete and steel sets are treated as the composite section of a straight beam as a homogenized section of equivalent mechanical properties of the two materials, and the contribution of the supporting elements, including rock bolt, is incorporated into the solution as uniformly distributed radial pressures. Then, the effect of the supporting elements is expressed with the line branching off from the non-supported relation in the fictitious radial pressure σra – radial displacement ua space. Secondly, for the case in which wall displacement is not evenly distributed, a procedure to obtain a representative value of wall displacement is presented. A series of FEM simulation is also presented, including the case for not fully / partially supported simulation. Observations of practical interest arising from the comparison of the analytical and numerical results are discussed.

1. INTRODUCTION
The stability of tunnel by the conventional excavation is treated based on a thought whereby the ground surrounding an underground opening becomes a load bearing structural component. The thought considers the activation of the support load of the ground and the interaction with the installed support system. The activation will result in a minimization of load supported by the artificial support elements, such as shotcrete (SC), steel sets (SS) and rockbolt (RB). The mechanical concept of this method is characterized by Ground Reaction Curve (GRC, or Ground Characteristic Curve, GCC) and the contribution of the support system. The tunnelling method has been used in the wide range of ground conditions, soft or hard, and shallow or deep. The problem of assessing the mechanical stability of tunnel ground itself and the interaction with the support elements is an important geotechnical problem with practical applications in the field of civil engineering. If the state of stress exceeds the strength of the ground due to the stress redistribution, yielding may occur in the vicinity of the tunnel opening. To prevent large deformation and the collapse in the ground during construction, support pressure must be applied to establish the load bearing zone within the ground in the vicinity of the opening.
Fig. 1 shows a schematic representation of ground reaction curve, and it is often said that the optimization postulates the curve turning upward (Rabcewicz, 1969, JSCE, 2016). The abscissa is the relative radial displacement and the ordinate the lining pressure. This figure also shows that the lining must be installed at the timing to meet at the minimum point of the GRC, not too early and not too late, to avoid the increase in the load to the support. The stress state around tunnel face can be expressed in the three-dimensions, representing major, intermediate, and minor principal stresses. Excavation-induced stresses shown in Fig. 1 has been primarily restricted to two-dimensions and the axial symmetry of both the geometry and the stresses is assumed. Numerical studies on GRC have been reported, e.g. Sezaki et. al. (1994), Nishimura (1991), and such solutions incorporate the effect of gravity and the strain softening behaviour of rocks, nevertheless, the solutions to explain the interaction between the ground and the supporting elements is never developed (Kovari, 1993).