Upper bound limit analysis of collapse shape for circular tunnel subjected to pore pressure based on the Hoek–Brown failure criterion

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Abstract

Based on Hoek–Brown failure criterion and using the upper bound theorem of limit analysis, a numerical solution for the shape of collapsing block in circular tunnel subjected to pore pressure is derived. The effect of water pressure which is assumed to be a work rate of external force is included in the upper bound analysis. By employing variational calculation to minimize the objective function, the upper solution of collapsing block is obtained. In order to evaluate the validity of the method used in this paper, the result for pore pressure coefficient ru = 0, with no effect of pore pressure taken into account, is compared with previous work. The good agreement shows that the method of calculating the upper solution for the shape of collapsing block subjected to pore pressure is valid.

Highlights

► The shape of collapsing block in tunnel subjected to pore pressure is derived. ► The effect of water pressure is assumed to be a work rate of external force. ► The collapsing block decrease with the increase of pore pressure coefficient. ► The collapsing block increase with the increase of tunnel radius.

Introduction

The stability of tunnel face for a tunnel excavated in urban area is a major problem in tunnel engineering, which has been studied by many scholars since limit analysis theory is introduced to estimate the lower and upper bound stability solutions by Davis et al. (1980). Compared with the traditional stability analysis techniques, slice method and limit equilibrium method, solutions derived from limit analysis are more rigorous and no assumption with respect to forces is required (Yang and Yin, 2006). Due to these advantages, limit analysis method is used widely to analyze the stability of the front of tunnels driven in shallow strata. Leca and Dormieux (1990) derived upper and lower bound solutions of critical retaining pressure applied to the tunnel face by constructing a kinematically admissible failure mechanism in the framework of the limit analysis theory. In order to obtain a precise upper bound solution, Leca and Dormieux (1990) developed three types of three-dimensional failure mechanisms which represented collapse failure and blow-out failure respectively, and the failure mechanisms are cited frequently by other authors (Li et al., 2009, Lee et al., 2004, Lee and Nam, 2001, Buhan et al., 1999). For the purpose of improving the upper and lower bounds to describe the stability of a range of heading sizes, Augarde et al. (2003) employed finite element limit analysis methods to derive rigorous bounds on load parameters.

At present, the limit analysis theory in tunnel engineering is mainly applied to analyzing the stability of tunnel face excavated in shallow strata by using linear Mohr–Coulomb criterion. However, the collapse of deep tunnel is a complex nonlinear evolution process, thus, the difference of mechanical characteristics between deep tunnel and shallow tunnel is significant. Consequently, the linear failure criterion is not suited to solve the problem of roof collapsing for a deep-buried tunnel. Since Hoek–Brown failure criterion is developed for estimating the inherent characteristics and nonlinear failure feature of tightly hard rock mass, this nonlinear failure criterion has been widely applied in a variety of geotechnical engineering (Yang et al., 2004, Jimenez et al., 2008). Based on the generalized Hoek–Brown failure criterion, Merifield et al. (2006) used limit theorems to evaluate the ultimate bearing capacity of a surface footing on a rock mass, and the actual collapsing load is bracketed precisely by computing both the upper and lower bound solutions for the bearing capacity. To investigate the possible collapse of a rectangular cavern roof, Fraldi and Guarracino (2009) derived the exact solution of detaching profile by means of the upper bound theorem of limit analysis and Hoek–Brown failure criterion.

The effect of pore pressure should be included in the stability analysis for tunnels excavated in areas where the underground water table changes with season. When the limit analysis theory is used to analyze the stability of tunnels, with the effect of pore pressure taken into account, the main problem is how to calculate the rate of work done by pore pressure. The water pressure in the soil pores is regarded as an external force loaded on the soil skeleton by Michalowski (1995). In the light of this assumption, the work of water pressure can be expressed as a sum of pore pressure work on skeleton and the work of the water pressure on boundary. Due to its simplicity and validity, several authors (Viratjandr and Michalowski, 2006, Kim et al., 1997) used this assumption to analyze the stability of slope in the framework of limit analysis subjected to pore water pressure.

Located in high ground stress environment, the deformation, looseness and collapse of rock mass over the roof of deep-buried tunnel are confined in a certain region. Furthermore, the collapsing block over the roof of the tunnel forms a ‘collapsing arch’ which bears the whole load originated from the weight of rock mass. As a result, the study of collapsing shape for deep-buried tunnel will contribute to understanding the failure property of surrounding rock mass and providing theoretical basis for developing a supporting structure of deep-buried tunnels. The purpose of this paper is to make use of the upper bound theorem of limit analysis to derive the formula for constructing an effective shape of the collapsing block, taking the effect of pore pressure into account in the medium subjected to Hoek–Brown failure criterion.

Section snippets

Pore pressure in upper bound limit analysis

According to the upper bound theorem, when the velocity boundary condition is satisfied, the load derived by equating the external rate of work to the rate of the energy dissipation in any kinematically admissible velocity field is no less than the actual collapsing load (Michalowski, 1995). To achieve the effects of pore pressure in the framework of the upper bound theorem of limit analysis for slope stability, Viratjandr and Michalowski (2006) assumed that the work of water pressure is equal

Failure mode of rock mass for circular tunnel

Constructing a kinematically admissible failure mechanism is the key factor for upper bound theorem of limit analysis. According to the actual mechanical characteristics of rock mass over the roof of a deep-buried tunnel, an arched detaching curve f(x) is used to describe the velocity discontinuity surface, which can be seen from Fig. 1. (Fraldi and Guarracino, 2010). Due to the slide between the collapsing block and surrounding rock mass, the plastic flow occurs along the velocity

Limit analysis of collapsing shape with Hoek–Brown failure criterion

The widely used Hoek–Brown failure criterion is always represented in terms of the major and minor principal stresses. However, the relationship between normal and shear stresses can also be expressed by Hoek–Brown failure criterion, which is given by (Hoek and Brown, 1997)τ=Aσciσn+σtmσciBwhere σn is the normal stress, τ is the shear stress, A and B are material constants, and σci σtm are the uniaxial compressive strength and the tensile strength of the rock mass respectively.

According to the

Numerical result of collapsing shape for circular tunnel

In order to investigate the influence of the different rock parameters and pore pressure coefficients on the shape of collapsing block for circular tunnels, the collapsing blocks subjected to pore pressure using Hoek–Brown failure criterion are drawn in Fig. 2. With reference to the work of Fraldi and Guarracino (2010), the rock mass parameter B varies from 3/4 to 1 and pore pressure coefficient ranges from 0 to 0.75, which is clearly illustrated in Fig. 2. It can be seen from the figure that,

Conclusions

Based on Hoek–Brown failure criterion, a numerical solution for the shape of collapsing block in circular tunnel subjected to pore pressure is obtained in the framework of the upper bound theorem of limit analysis. The effect of water pressure which is regarded as a work rate of external force is included in the upper bound theorem. In order to derive the upper solution of collapsing block, variational calculation is employed to minimize the objective function. For the condition that the effect

Acknowledgments

This paper were supported by the Hunan Provincial Postgraduate Innovation Project (No. CX2009B043) and Doctoral Dissertation Innovation Project of Central South University (No. 2010bsxt07). The financial supports are greatly appreciated.

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