Stress analysis of reinforced tunnel faces and comparison with the limit equilibrium method

Highlights

• We analysed the stability of a tunnel faces reinforced by bolts.

• We performed 3D numerical stress analyses modelling the bolts with tension elements.

• The behaviour of the adopted bolt model was investigated by numerical pull-out tests.

• We showed that this bolt model could depend on the fineness of the numerical mesh.

• We show that numerical analysis and limit equilibrium method give similar results.

Abstract

The present paper investigates the stability of reinforced tunnel faces in dry cohesive-frictional soils by means of 3D numerical stress analyses which take account of the individual bolts. The numerical calculations are performed assuming an elastic, perfectly plastic material obeying the Mohr–Coulomb yield criterion. As usual in this kind of problem, the bolts are modelled by one-dimensional tension elements, which have zero diameter and cannot take into account geometrically the diameter of the bolts or the borehole. The first part of the paper deals with the approximations induced by this simplification. More specifically, the paper shows by means of numerical pullout tests in respect of a single bolt in elasto-plastic soil that the behaviour of this model depends significantly on the fineness of the numerical mesh. The second part of the paper investigates the reinforcing effect of bolts on face stability assuming that the bond strength of the bolts depends on the confining stress with strength parameters equal to those of the soil. For several bolting patterns, the minimum cohesion $c_{\mathit{lim}}^{\prime }$ needed for face stability is calculated iteratively and compared with the predictions of the limit equilibrium method of Anagnostou and Serafeimidis (2007).

Introduction

Face reinforcement by fibreglass bolts is a very effective and flexible method for stabilizing the tunnel face in weak ground. It has been the subject of several papers over the past two decades, starting with Peila (1994), who performed spatial numerical stress analyses and showed that reinforcement reduces the deformations and the extent of the overstressed zone ahead of the face. Other related works are those of Ng and Lee (2002), which investigated the influence of the axial stiffness of the bolts with respect both to the stability of the tunnel heading and to the surface settlement in a stiff clay, and of Yoo and Shin (2003), which analysed the effect of bolt spacing, length and axial stiffness on face stability for different cover-to-diameter ratios. Dias and Kastner (2005) also modelled each bolt individually and investigated – by means of 3D finite difference analyses – the effects of bolt spacing and bond strength (i.e. the shear strength of the interface between grouted bolt and rock) on the face stability of a deep tunnel in soft rock. Furthermore, they compared the numerical results with the results of simplified analyses, which take into account the face reinforcement either by introducing an equivalent face support pressure or by considering a higher cohesion of the core ahead of the tunnel face. Kavvadas and Prountzopoulos (2009) performed spatial finite element calculations in order to find out the optimum bolt length and the overall face support pressure exerted by the bolts for different soil shear strength parameters and cover-to-diameter ratios.

The face reinforcement is tackled either by smearing the effect of the bolts and considering an equivalent higher strength ground (e.g., Indraratna and Kaiser, 1990, Grasso et al., 1991) or by taking account of individual bolts. The usual assumption concerning the bearing capacity of the bolts is that it is limited either by the tensile strength of the bar or by the shear strength of the soil–grout interface. Prete (2007) and later Oreste and Dias (2012) also took account of the bending failure of the bolt or failure of the soil due to the radial pressure exerted by the bolts (in a similar way to a soil nailing analysis).

In addition to the above-mentioned 3D numerical stress analyses, simpler approaches such as limit equilibrium methods (e.g., Cornejo, 1989, Mohkam and Wong, 1989, Anagnostou and Kovári, 1994) or methods based on plasticity theorems (Caquot and Kerisel, 1956, Mandel and Halphen, 1974, Leca and Dormieux, 1990) have also been proposed for assessing tunnel face stability. In fact, a 3D numerical stress analysis, besides being very time consuming and awkward to handle for practical engineering purposes, represents an unnecessarily complex approach (and actually a long way round) if it is only the stability of the face (rather than the deformation of the ground) that is concerned. On the other hand, numerical stress analysis represents the only computational possibility for checking the adequacy of a priori assumptions concerning the geometry of the failure mechanism and the horizontal stresses in the ground, which are needed in limit equilibrium analyses but are statically indeterminate (Anagnostou, 2012).

A simple, limit equilibrium based computational method for assessing the stability of a reinforced tunnel face was introduced by Anagnostou (1999) and refined by Anagnostou and Serafeimidis (2007). The present paper compares the results of this method with the results of spatial numerical stress analyses of the reinforced face. The underlying computational investigations include as a by-product the modelling of the bolts in numerical stress analyses which is of more general interest.

As in all above-mentioned stress analysis methods, the bolts are modelled here individually by one-dimensional structural elements. This reduces computational time considerably compared to more realistic models that use solid elements and two-dimensional interface elements to model the grouted bolts and their interfaces to the surrounding soil, respectively. On the other hand, however, the one-dimensional structural elements have zero diameter, i.e. they do not take account of the diameter of the boreholes geometrically. This has some important consequences for modelling, because the model behaviour proves to be mesh-sensitive, i.e. the structural behaviour of the reinforced core ahead of the face depends significantly on the fineness of the computational mesh.

The first part of the present paper deals with this issue. More specifically, Section 2 investigates aspects of bolt modelling in numerical stress analyses by considering the relatively simple problem layout of a bolt pullout test. The purpose of this section is to get a better understanding of the nature and effect of the approximations introduced by the simplified one-dimensional bolt model and to get some guidelines concerning the choice of computational mesh in large-scale numerical simulations involving bolts.

The second part of the paper presents a numerical solution to the reinforced tunnel face problem, determines the soil cohesion necessary in order for the face to be stable iteratively for different reinforcement layouts and compares the numerical results with those obtained by the limit equilibrium method (Section 3).