A simplified procedure for the interaction between fully-grouted bolts and rock mass for circular tunnels

doi:10.1016/j.compgeo.2018.10.008

Computers and Geotechnics

Volume 106, February 2019, Pages 177-192

https://doi.org/10.1016/j.compgeo.2018.10.008 Get rights and content

Abstract

This paper provides a simplified procedure for analysing the interaction between fully-grouted bolts and rock mass for circular tunnels under a hydrostatic stress field. In this procedure, the relative movement between the rock mass and the bolt is accounted for by considering the shearing stiffness at interface. Further, the elastic elongation of the bolt is considered. Two cases of elastic-brittle-plastic and elastic-perfectly-plastic rock mass behaviours are considered. The rock-bolt interaction is modelled at the initial and final states. At the initial state, immediately after the bolt is installed, a finite difference solution to obtain stresses and strains in the rock mass is proposed. At the final state, after the tunnel excavation advances and the fictitious support pressure becomes zero, another finite difference solution for computing stresses along the bolt and rock mass displacements is presented. To obtain the final converged results, an iterative process is performed by repeating the above solution at the final stage. The simplified procedure is validated by comparison with the commercial finite difference software, FLAC3D. A series of parametric studies is conducted to assess the sensitivity of the bolt and rock behaviours to variations in interface shear stiffness, the stiffness of the end plate and the assumed post-peak behaviour of the rock mass. The results indicate that the post-peak behaviour of the rock mass has an important effect on the bolt performance and, in turn, it should be a key design consideration. It is also shown that, under squeezing conditions, the failure mode of the bolt is influenced by the stiffness of the end plate.

Introduction

The fully-grouted bolt is one of the most widely used means to stabilise underground openings by strengthening the rock mass internally. The tension in grouted bolts is mobilised through the transfer of the shear load from the rock mass to the bolt, which reduces the deformation of the rock mass. If the bolt design is not carefully considered, rupture or shearing failures of the grouted bolts may occur. However, if the bolts are long and narrowly-spaced around the tunnels, the design may be overly conservative and uneconomical. In order to provide an appropriate design, understanding the interaction between the bolts and the rock mass is necessary.

For the preliminary design of fully-grouted bolts, installed around the perimeter of circular tunnels with a hydrostatic field, a series of analytical solutions have been previously presented to investigate the interaction mechanism between fully-grouted bolts and the rock mass. Bobet [1], [2] derived the tensile force in the bolt for elastic-perfectly-plastic rock in a hydrostatic field, and for elastic rock in a non-hydrostatic field. Stille et al. [3] presented an analytical solution for the elastic and plastic bolts in the elastic-brittle-plastic rock mass. Fahimifar [4], [5] proposed the improved ground reaction curves for elastic-perfectly-plastic rock and plastic softening rock by accounting for the supporting time of the bolt. Carranza-Torres [6] provided a closed-form solution for elastic rock reinforced by grouted bolts and quantified the decrease of the rock deformation by a dimensionless coefficient. Bernaud et al. [7], [8] proposed an approach for the design of the bolt at the horizontal and radial directions of the tunnel by regarding the bolted tunnel as a homogenized medium.

For simplicity, these solutions assume the grout material around the bolt as rigid, and thus no relative movement occurs between the bolt and rock, which results in an overly conservative design. Other analytical solutions account for the relative movement between the bolt and rock at the potential failure interface. Indraratna and Kaiser [9], [10] presented the analytical solutions of the equivalent properties of the rock that is based on the actual properties of rock and bolt geometry and spacing. Osgoui and Oreste [11] improved this solution by incorporating the Hoek-Brown failure criterion to represent the rock mass behaviour and using a more rigorous assessment of strain compatibility. The main limitation of these solutions is that the shearing behaviour between the rock and bolt is not defined explicitly. The bolt is artificially divided into a “pick-up” length and an “anchor” length. The transition point (i.e., neutral point) between the “pick-up” and “anchor” sections is determined using an empirical equation, which considers that the neutral point is merely correlated to the tunnel radius and bolt length. However, using a more robust equilibrium calculation, the location of the neutral point is determined to be affected by the mechanical properties of the in-situ rock and the bolt, and the shear behaviour at the potential failure interface [12], [13].

The behaviour of fully-grouted rock bolts is rather complex due to the interaction between bolt and the rock mass. Adequate description of the shearing behaviour between the bolt and rock mass requires more effort for the exact solution. To this end, a numerical calculation framework becomes necessary. Guan et al. [14] presented a numerical procedure to estimate the effect of passive bolt in a strain-softening rock mass based on a spring-slider shearing model. Tan et al. [15], [16] proposed a comprehensive numerical procedure that successfully implements a complex constitutive model. Their model considers the strain-softening behaviour of the rock mass and the trilinear bond-slip shearing relationship at the rock-bolt interface. However, Guan et al. ignores the increment of the elastic strain within the plastic zone, which tends to under-predict the rock deformation. Both Tan et al.’s and Guan et al.’s models assume a rigid bolt, which may under-predict the shear stress and tensile force for the bolts having fairly large shearing stiffness. Moreover, in their solutions, the fictitious support pressure, due to the tunnel face effect, gradually decreases from the initial state to the final state in finite stages. In each stage, an iterative process is required to obtain the stresses and strains in the rock and bolt. An excessive number of iterations may be required to obtain the final solution, which can be cumbersome and time-consuming.

In this paper, a simplified procedure for the interaction between the fully-grouted bolt and both elastic-perfectly-plastic and elastic-brittle-plastic rock masses is presented. The relative movement at the bolt-rock interface is accounted for by assuming a finite shearing stiffness between the rock and bolt. In comparison with existing procedures [14], [15], [16], the iteration process of the proposed procedure accounts for the elongation of the bolt, and is greatly simplified by considering only the initial and final states (i.e., no intermediate finite states as is the case for existing procedures) without compromising the accuracy of the predictions. In the procedure, at the initial state, immediately after the bolt is installed, a finite difference solution of the stresses and strains in the rock mass is proposed. At the final state, when the fictitious support pressure is negligible, another finite difference solution of the stresses along bolt as well as displacements in the rock mass is presented. To obtain the final results, a simplified iterative process is performed by repeating the finite difference solution at the final stage. The proposed procedure is validated by comparison with the commercially available finite difference software FLAC3D [17]. Parametric studies are conducted to assess the effects of the interface shearing stiffness, the installation of end plate and the post-peak behaviour of the rock mass.

Section snippets

Basic assumptions

The following assumptions were considered in the development of the proposed solution:

(1)
The opening is circular and excavated in a hydrostatic stress field, σ₀. The radius of the circular opening is R₀.
(2)
In the plane perpendicular to the tunnel axis, plane strain conditions are postulated, where σ_r and σ_θ represent the minor principal stress, σ₃, and the major principal stress, σ₁, respectively.
(3)
The rock mass is isotropic, continuous and initially elastic. In the vicinity of the tunnel face and

Equilibrium equations

For a rock mass without the reinforcement of bolt (see Fig. 3), the differential equation of equilibrium for axisymmetric geometry is expressed as: $\frac{d σ_{r}}{dr} + \frac{σ_{r} - σ_{θ}}{r} = 0$

Static equilibrium conditions for the rock mass with the reinforcement of bolt included at the infinitesimal element dr is also illustrated in Fig. 3. Relative movement is assumed to occur at the interface between the grout medium and the rock mass. Stress acting on the infinitesimal rock mass (with bolt) element dr are σ_r, σ_r + dσ_r, σ_θ

Evaluation of stresses and strains in rock and bolt

As mentioned above, the initial rock mass displacement occurs before the bolts are installed at a particular position. Immediately after installation, the rock bolts are not tensioned. As the tunnel excavation advances, further reduction of the fictitious support pressure occurs, causing an increase in the extent of the plastic zone and additional rock mass displacement, which induces tensile forces in the bolt. Therefore, prior to calculating stresses and displacements at the final stage of

Validation

The results of the proposed procedures were validated against these from numerical simulations for the cases with and without considering the reinforcement of the bolt using the commercially available finite difference software FLAC3D [17]. The properties of the severely soft rock mass and the bolt considered for the validation case are listed in Table 1. In the proposed procedure, Δr was given a small value of 0.005 to ensure the accuracy of the results. To reduce the simulation time, only a

Discussion

For the purpose of this discussion, two cases considering soft and hard rock masses are considered as shown in Table 1. The mechanical properties of the bolt are also presented in Table 1. c^peak and φ^peak represent the strength parameters in the plastic zone for the elastic-perfectly-plastic rock mass, and c^res and φ^res represent these for the elastic-brittle-plastic rock mass. It is noted that c^peak and φ^peak also denote the peak strength parameters before the plastic stage for both the

Remarks and conclusions

In this paper, a simplified procedure for the interaction between fully-grouted bolts and the rock mass was presented. Both elastic-perfectly-plastic and elastic-brittle-plastic rock mass behaviours were considered in the assessment. The rock-bolt interaction was investigated at the initial and final states. The initial state is when the bolt is initially installed near the tunnel face, while the final state is when the tunnel excavation is advanced to a location far from the bolts, and the

Acknowledgements

The authors acknowledge the financial support provided by National Key Research and Development Program of China (Grant No. 2016YFC0800200), National Natural Science Foundation of China (Grant No. 51678267), China Postdoctoral Science Foundation (Grant No. 2017M622450, Grant No. 2018T110768).

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Research PaperA simplified procedure for the interaction between fully-grouted bolts and rock mass for circular tunnels

Abstract

Introduction

Section snippets

Basic assumptions

Equilibrium equations

Evaluation of stresses and strains in rock and bolt

Validation

Discussion

Remarks and conclusions

Acknowledgements

Tunn Undergr Space Technol

Int J Rock Mech Min Sci Geomech Abstr

Tunn Undergr Space Technol

Tunn Undergr Space Technol

Tunnell Undergr Space Technol

Int J Rock Mech Min Sci Geomech Abstr

Int J Rock Mech Min Sci

Int J Rock Mech Min Sci

Int J Rock Mech Min Sci

Int J Rock Mech Min Sci

Int J Rock Mech Min Sci

Int J Rock Mech Min Sci

Research Paper
A simplified procedure for the interaction between fully-grouted bolts and rock mass for circular tunnels