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Three dimensional face stability of a tunnel in weak rock masses subjected to seepage forces Qiujing Pan, Daniel Dias

MARK

⁎

Laboratory 3SR, Grenoble Alpes University, CNRS UMR 5521, Grenoble, France

A R T I C L E I N F O

A B S T R A C T

Keywords: Tunnel face Stability Limit analysis Kinematical approach Seepage forces The Hoek-Brown failure criterion

Reliable prediction of tunnel face stability is a key challenge for tunnel engineering, especially when drilling in highly fractured rock masses under the water table. This work aims to study face stability of a circular tunnel in weak rock masses under the water table based on an advanced three-dimensional (3D) rotational collapse mechanism in the context of the kinematical approach of limit analysis. The fractured rock masses are characterized by the Hoek-Brown failure criterion. A 3D steady-state seepage ﬁeld obtained numerically is used to interpolate hydraulic heads of the 3D collapse mechanism and seepage force is incorporated into this predicting model by directly considering it as a body force. The results provided by the presented approach are compared with those of numerical calculations, showing a good agreement. The proposed work also provides an improvement with respect to other existing solutions. Thanks to the high computational eﬃciency of the presented method, four sets of normalized charts are obtained for a tunnel driven in weak rock masses under the water table.

1. Introduction Tunnel excavation mainly refers to the closed-face tunneling by means of the tunneling boring machine (TBM) and the open-face conventional tunneling. The closed-face TBM technique, including the slurry-shield tunnelling and the earth pressure balance (EPB)-shield tunneling, is widely used for tunnel constructions in recent decades, especially in urban region, due to the advantages of rapid tunnelling and eﬀective controlling of ground movements. One of the most important issues for safe excavation is the face stability in the tunnel engineering. For TBM tunnelling, shield machines can provide continuous supports on the tunnel face to compensate earth pressures and underground water pressures with freshly excavated soil or pressurized mixture of bentonite and water. At the preliminary design stage, it is of highly practical value to predetermine a reasonable range of face pressures to avoid both the ground subsidence (if the face pressure is not enough) and the ground uplift (if the face pressure is too high). This stability issue has been investigated by several researchers by means of numerical approaches (Vermeer et al., 2002; Lambrughi et al., 2012; Mollon et al., 2013a), experimental tests (Kamata and Mashimo, 2003; Kirsch, 2010). Besides, many theoretical models, e.g. the silo wedge model (Anagnostou and Kovari, 1996; Anagnostou and Perazzelli, 2015) and the triangular base prism model (Oreste and Dias, 2012) within the framework of limit

⁎

equilibrium method, the classical cone models (Leca and Dormieux, 1990), the horn model (Subrin and Wong, 2002), the multi-blocks models (Mollon et al., 2009), the 3D rotational models (Mollon et al., 2011; Pan and Dias, 2017), the continuous velocity models (Mollon et al., 2013b) in the context of limit analysis theory were also developed to predict face pressures. The comparisons with numerical simulations or experimental measurements show that these analytical models work well to estimate required face pressures. Particularly, the advanced 3D rotational collapse mechanism generated by the spatial discretization technique improves existing solutions signiﬁcantly with respect to translational failure mechanisms (Mollon et al., 2011). In practical engineerings, tunnels are often constructed in aquifers, like subsea tunnels, cross-river tunnels and many urban tunnels. Therefore the destabilizing eﬀect induced by the underground water (seepage forces or excessive pore-water pressures) should be taken into account. Many contributions have been devoted to study the face stability when tunnelling in water bearing zone. In addition to experimental investigation (Pellet et al., 1993) and numerical researches (Anagnostou, 1995; De Buhan et al., 1999), simpliﬁed approximate approaches, mainly including the limit equilibrium method and the kinematical approach, are common techniques to deal with this problem. Based on the silo wedge model, Anagnostou and Kovari (1996) and Perazzelli et al. (2014) studied the face stability under seepage ﬂow, while Bezuijen et al. (2005), Broere (2015) investigated the

Corresponding author. E-mail address: [email protected] (D. Dias).

http://dx.doi.org/10.1016/j.tust.2017.11.003 Received 20 December 2016; Received in revised form 11 June 2017; Accepted 1 November 2017 0886-7798/ © 2017 Elsevier Ltd. All rights reserved.

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Notations

Nw

a, m, mi, δβ, n ct C D Di Fy, Fz GSI h hk hF Hw

Ri,j, βi,j

i ny,k, nz,k Nγ Nc Ns

sHoek-Brown constants discretization parameters equivalent cohesion tunnel overburden tunnel diameter disturbance factor components of the resultant seepage force geological strength index hydraulic head average hydraulic head at each element surface piezometric head on the tunnel face undisturbed hydraulic elevation measured from the tunnel crown hydraulic head gradient direction cosines of the unit normal vector of each element surface non-dimensional coeﬃcient representing the contribution of soil weight non-dimensional coeﬃcient representing the contribution of cohesion non-dimensional coeﬃcient representing the contribution of surcharge loading

Ri , βi Ri,0 , βi,0 sk Si Si,0 Si, j Vi,j vx , vy, vz Wseepage γd γw γ′ σ1 σ3 σc σs σt ω φt

non-dimensional coeﬃcient representing the contribution of seepage forces polar coordinates of the barycenter of the triangular facets Fi,j Polar coordinates of the point on the possible outcropping surface polar coordinates of point on the tunnel face area of the element surface area of the element at the outcropping surface area of the element at the discretized face area of the triangular facet Fi,j volume of the element corresponding to facets Fi,j velocity components work rate of seepage force dry unit weight water unit weight submerged unit weight maximum eﬀective principal stresses minimum eﬀective principal stresses uniaxial compressive strength of the intact rock. ground surface surcharge eﬀective face pressure velocity angle of the failure mechanism equivalent friction angle

approach of limit analysis. Based on the advanced 3D collapse mechanism proposed by Mollon et al. (2011), Senent et al. (2013) computed necessary face pressures of a tunnel face in weak rock masses characterized by the modiﬁed Hoek-Brown yield criterion. The distribution of normal stresses along the failure surface, which is identiﬁed by 3D numerical ﬁnite diﬀerence simulations, is required in an attempt to ﬁnding the lower-bound solutions of face pressure. However, the distribution of normal stresses is generally intractable. Yang and Huang (2013) investigated the three-dimensional collapsing shape and collapsing range for a deep cavity roof in the Hoek-Brown medium. Nevertheless, the Hoek-Brown failure criterion has never been taken to examine the tunnel face stability in the presence of seepage forces. This work will ﬁll this gap. This paper aims to study the face stability of a circular tunnel in highly degraded rock masses characterized by the modiﬁed HoekBrown failure criterion in the context of the kinematical approach of limit analysis. The advanced 3D collapse mechanism proposed by Mollon et al. (2011) is extended to analyze the eﬀect of seepage forces, which represent a destabilizing factor that contributes to the failure of a tunnel face. Numerical calculations are employed to compute hydraulic head distributions under a steady state ﬂow due to tunnel excavations. Then interpolated calculations of hydraulic heads on each point of the 3D collapse mechanism are done using numerical results. In order to validate the presented method, the obtained results are compared with those of hydro-mechanical numerical simulations as well as with previously published solutions when the modiﬁed Hoek-Brown failure criterion is reduced to a Mohr-Coulomb (MC) one. This paper ends up with providing several sets of charts, respectively corresponding to diﬀerent water table elevations and rock strength parameters.

inﬂuence of excessive pore-water pressure, for slurry-shield tunnels, induced by the slurry inﬁltration. With the application of the corn failure mechanism, Lee and Nam (2001); Lee et al. (2003) and Park et al. (2007) investigated tunnel face stability subjected to seepage forces. Pan and Dias (2016) studied the eﬀect of pore-water pressure on the tunnel face stability with application of the 3D rotational failure mechanism. All these researches highlight that both seepage forces and the excessive pore water pressures exert a greatly adverse eﬀect on the face stability. In these studies, the critical issues are how to calculate seepage forces or pore-water pressures at the vicinity of a tunnel face and how to incorporate the contribution of seepage forces into the predicting models. Several approximated and simpliﬁed expressions for the hydraulic head distribution according to numerical simulations (Perazzelli et al., 2014) and ﬁeld measurements (Broere and Van Tol, 2000; Bezuijen et al., 2005) were presented to address this problem. Lee et al. (2003) directly calculated the seepage forces using ﬁnite element simulations, but the seepage forces were not regarded as a body force acting on the soil particles when it is incorporated into the computational model. Pan and Dias (2016) interpolated the pore-water pressure in the failure mechanism by using numerical results, in which the porewater pressure is used to account for the contribution of seepage forces and buoyancy forces. Besides, all these aforementioned theoretical models related to the face stability under the water table are based on the linear MohrCoulomb failure criterion. However, the non-linear character of the failure envelopes of geo-materials has been observed in many experimental works, among which the non-linear Hoek-Brown criterion, e.g. the original Hoek-Brown criterion (Hoek and Brown, 1980, 1997) and its modiﬁed version (Hoek et al., 2002), has been widely applied to characterize the strength of isotropic rock masses. Several attempts have been made to evaluate the tunnel face stability (Saada et al., 2013; Senent et al., 2013) and the tunnel roof stability (Yang and Huang, 2013) using the modiﬁed Hoek-Brown failure criterion in the light of the kinematical approach. Saada et al. (2013) studied the tunnel face stability in the context of the modiﬁed Hoek-Brown strength criterion, in which both the corn failure mechanism and the horn failure mechanism were employed to assess the lower-bound solutions of face pressure, and the classical tangential line technique is adopted to incorporate the Hoek-Brown strength criterion into the kinematical

2. The problem statement 2.1. Schematic diagram for a circular tunnel advancing under the water table The schematic diagram of the problem under consideration is sketched in Fig. 1. A circular tunnel with diameter of D at a buried depth of C is excavated under the water table. The water table elevation Hw is measured from the tunnel crown; hF refers to the piezometric head 556

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m GSI −100 ⎞ = exp ⎛ mi ⎝ 28−14Di ⎠

(2)

GSI −100 ⎞ s = exp ⎛ ⎝ 9−3Di ⎠

(3)

⎜

⎜

a=

in the shield chamber or in the tunnel face. If hF < (Hw + D), the ground water ﬂows into the tunnel face associated with the variation of hydraulic head gradients i ahead of the tunnel face. This process simultaneously induces the seepage force iγw, a body force, acting on rock masses in the direction of the groundwater ﬂow. This phenomenon, as reported by Anagnostou and Kovari (1996) in a EPB shield tunnel and by Lee and Nam (2001) and Lee et al. (2003) in a NATM tunnel, further results in surface settlement and causes face instability during the process of excavation. This work is only devoted to the face stability problem. In Fig. 1, the region ABF refers to a 3D collapse body which rotates around a horizontal axis with an angular velocity. The velocity in the failure mechanism is equal to the product of the angular velocity and the distance between the rotating center and the point under consideration. The discretization mainly includes the tunnel face discretized by 2n points and the collapse body by a series of radial planes separated by an angle of δβ. The precision of the generation of the failure mechanism is controlled by these two discretization parameters, which are taken to be n = 200 and δβ = 0.5° in the following calculations. The geometry of the failure mechanism is determined by two dimensionless parameters. The process of the generation of the rotational collapse mechanism is presented in Mollon et al. (2011).

1 1 GSI ⎞ 20 + ⎡exp ⎛− −exp ⎛− ⎞ ⎤ ⎥ ⎢ 2 6⎣ ⎝ 3 ⎠⎦ ⎝ 15 ⎠

(4)

a

1

cosφt ⎡ ma (1−sinφt ) ⎤1 − a tanφt ⎛ sinφt ⎞ ⎡ ma (1−sinφt ) ⎤1 − a ct − 1+ = ⎥ ⎢ ⎥ 2 ⎣ 2sinφt 2sinφt σc m ⎝ a ⎠⎢ ⎣ ⎦ ⎦ s + tanφt (5) m ⎜

⎟

In the kinematical approach of the limit analysis, the equivalent friction angle φt is optimized together with other variables that determine the geometry of the failure mechanism during the search of the best upper-bound solution.

2.2. The modiﬁed Hoek-Brown criterion Most previously published works mentioned above employed the linear Mohr-Coulomb criterion to derive critical face pressures on the basis of the upper-bound limit analysis or the limit equilibrium method. However, most geo-materials present a non-linear failure envelope. The Hoek-Brown criterion is non-linear and is extensively used for rocks with varying degrees of fracture. The modiﬁed Hoek-Brown criterion reported by Hoek et al. (2002) can be written as the following equation:

⎜

⎟

The magnitude of constant mi can be determined by compression testes; the geological strength index GSI characterizes the rock mass quality. The disturbance factor Di reﬂects the disturbance degree to the rock mass in situ, varying from 0.0 for the undisturbed rock masses to 1.0 for highly disturbed rock masses. For example it is suggested to be 0.0 to represent a TBM-excavated tunnel with a minimal disturbance to the rock masses around the tunnel face (Hoek et al., 2002). The main approaches for the implementation of a non-linear failure criterion into the kinematic approach include the variational method (Baker and Frydman, 1983; Zhang and Chen, 1987), the tangential line technique (Drescher and Christopoulos, 1988; Collins et al., 1988; Yang et al., 2004) and the limit analysis ﬁnite element method (Meriﬁeld et al., 2006; Li et al., 2008). The tangential line technique is an eﬀective and convenient approach to deal with this problem. The main idea of the tangential line technique is to replace the non-linear failure criterion by a tangential line, in which the slope and the intercept of the tangential line correspond respectively to the equivalent friction angle φt as well as the equivalent cohesion ct with respect to the non-linear envelope The tangential line technique yields an upper-bound solution to the original problem since it exceeds the original non-linear envelope. This eﬀective tangential technique was widely adopted by researchers to study the stability of slopes, foundations and tunnels by means of the modiﬁed Hoek-Brown yield criterion (Saada et al., 2013; Yang and Xu, 2017; Qin et al., 2017; Pan et al., 2017). The equivalent friction angle φt as well as the equivalent cohesion ct in the context of modiﬁed Hoek-Brown criterion follows the following relationship (Yang et al., 2004):

Fig. 1. The schematic diagram for a circular shield-driven tunnel advancing under the water table.

σ σ1 = σ3 + σc ⎛m 3 + s ⎞ ⎝ σc ⎠

⎟

2.3. Application of the kinematical approach to the tunnel face stability The external loads acting on this system include face pressures provided by the shield machine or by the face reinforcement system, the gravity, the seepage force, and the surface surcharge if the failure mechanism outcrops. As many authors did (Anagnostou and Perazzelli, 2015; Lambrughi et al., 2012; Mollon et al., 2011; Zou and Zuo, 2017; among others), the eﬀective face pressure σt is assumed to be uniformly distributed at the tunnel face. For simpliﬁcation, the ground surface surcharge σs is set to be zero in this work. Based on the kinematical approach of limit analysis, an upper-bound estimate of the ultimate load can be derived when the rate of work of external forces applied to the system is equal to or less than the rate of energy dissipation within the system (Chen, 1975). If the ultimate load to be determined oﬀers a resistance against the failure for a system, the kinematical approach of limit analysis yields a lower-bound estimate. Therefore, a lower-bound

a

⎟

(1)

where σ1 and σ3 represents respectively the maximum and minimum eﬀective principal stresses; σc denotes the uniaxial compressive strength of the intact rock. The Hoek-Brown parameters, m, s and a are determined by the constant mi, the geological strength index GSI and the disturbance factor Di,

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In order to obtain hydraulic head distributions in the failure mechanism, a ﬁnite diﬀerence software FLAC3D is employed to conduct seepage calculations. The seepage ﬁeld obtained by the numerical modellings is extracted using the FISH language and then used to compute hydraulic head of each point in the 3D failure mechanism. The process of seepage ﬂow calculations will be given in a later section. The ﬂow diagram for performing the stability analysis of the tunnel face under water-table in this paper is illustrated at Fig. 3.

estimate of the required eﬀective face pressure can be formulated by:

σt = γ ′DN γ ′ + σs Ns + γw Hw Nw−ct Nc

(6)

where Nγ′, Ns, Nw, and Nc are non-dimensional coeﬃcients, representing, respectively, the contribution of rock submerged weight, ground surface surcharge, seepage force and equivalent cohesion, see the Appendix A for their expressions; ct is the equivalent cohesion which is related to equivalent friction angle φt given by Eq. (5), γ′ the rock submerged unit weight (the dry unit weight γd should be used for calculation above the water table), γw the water unit weight. In order to obtain critical eﬀective face pressure, σt is minimized in Eq. (6) with respect to φt and other two variables that deﬁne the shape of the failure mechanism. The optimal calculations are achieved by application of the optimization tool (the fminsearch function) of the commercial software Matlab.

3. Comparisons 3.1. Comparisons with numerical modellings In order to verify the presented method, the results obtained by the presented method are compared with those provided by the numerical analysis using the ﬁnite diﬀerence software FLAC3D. Because of the symmetry of this three-dimensional problem at the vertical plane, only half of the domain is modelled. Fig. 4 provides a sketch of a used numerical model of a circular tunnel with a diameter of 10 m under a buried depth of 20 m. The model contains approximately 95,000 zones and 99,000 nodes. The dimensions of the numerical model are taken as 40 m in the X direction, 55 m in the Y direction and 40 m in the Z direction. These dimensions are large enough to eliminate the boundary eﬀects. A ﬂuid-mechanical analysis with FLAC3D is employed to model this problem. The simulation proceeds in two steps: (1) ﬂuid ﬂow-only calculation during which stage the model is cycled to equilibrium to establish a steady-ﬂow state for the changed hydraulic boundary conditions (this steady-state seepage ﬁeld is also used to interpolate hydraulic heads of the 3D rotational failure mechanism in the kinematical approach); (2) the mechanical-only calculation with supporting pressures acting on the tunnel face. A similar sequential seepage and stress analysis was adopted by Lee and Nam (2001) to account for the eﬀect of seepage forces on the tunnel face stability, and by FLAC3D to simulate staged constructions of a vertical excavation supported by an impermeable concrete caisson below the water level as an application example (FLAC3D 4.0 manual). In the ﬂuid-mechanical simulation, both mechanical boundary conditions and hydraulic boundary conditions must be speciﬁed. For the mechanical boundary conditions, the displacements at the bottom of the model are ﬁxed, and only the normal directions are ﬁxed on the vertical boundaries. For the hydraulic boundary conditions, the porewater pressure in the work chamber is presumed to be zero, and the

2.4. Work rate of seepage force The seepage force is a body force, so its work rate can be obtained by integrals over the whole failure mechanism volume:

Wseepage = γw

∭V ⎛ ∂∂hx vx + ∂∂hy vy + ∂∂hz vz ⎞ dV ⎜

⎟

⎝

(7)

⎠

where h is the hydraulic head and

∂h ∂h , ∂x ∂y

and

∂h ∂z

are hydraulic gradient

components; vx, vy and vz are velocity components. Please note that vx = 0 in the rotational velocity ﬁeld, only y and z directions are involved. As seen in Fig. 2, the triangular facet Pi, j, Pi+1, j, Pi, j+1 constitutes the boundary of the failure mechanism; P′i, j, P′i+1, j, P′i, j+1 are projections of points Pi, j, Pi+1, j, Pi, j+1 on the symmetry plane. The entity Pi,jPi+1,jPi,j+1 − P′i, jP′i+1,jP′i,j+1 is used for the calculation of the work rate of gravity which is obtained by a simple summation of the work rate on each element volume. Similarly, this element is taken to calculate the work rate of seepage forces, which can be formulated as:

Wseepage = ω ∑ i

∑

[Ri,j (Fy sinβi,j + Fz cosβi,j )]

j

(8)

where Fy and Fz denote components of the resultant seepage force acting on this element, which can be calculated by:

Fy = −γw

∫V

∂h dV ∂x

(9)

Fz = −γw

∫V

∂h dV ∂z

(10)

By applying the divergence theorem to the integrals over the volume of the element Pi, jPi+1, jPi, j+1 − P′i, jP′i+1, jP′i, j+1, the components of the resultant seepage force can be further expressed as:

Fy = −γw

∫S hn y ds = γw ∑ hk n y,k sk

(11)

∫S hnz ds = γw ∑ hk nz,k sk

(12)

k

Fz = −γw

k

where the summation index k indicates the ﬁve boundary surfaces of the element Pi, jPi+1,jPi, j+1 − P′i, jP′i+1,jP′i, j+1, and ny,k and nz,k denotes the direction cosines of the unit normal vector of each surface; sk is the area; hk is the average hydraulic head at each surface. Therefore, the term Nw representing the seepage force contribution can be formulated as:

Nw =

∑i ∑j [Ri,j (sinβi,j ∑k hk n y,k sk + cosβi,j ∑k hk nz,k sk )] Hw ∑i (Si,0 Ri,0 cosβi,0)

Fig. 2. The element discretization for seepage work calculations.

(13)

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Fig. 3. The ﬂow diagram to perform the tunnel face stability analysis.

analysis, the horizontal displacements of the middle point at the tunnel face versus diﬀerent face pressures are plotted in Fig. 5 for the case of mi = 7, GSI = 10, σc = 1.0 MPa (very poor rock masses) and the case of mi = 10, GSI = 20, σc = 5.0 MPa (relatively better rock masses) under C/D = 2.0 and Hw/D = 2.0. The minimum required face pressure may be deﬁned as the value at which the horizontal displacement suddenly begins to increase signiﬁcantly (Lee and Nam, 2001; Lambrughi et al., 2012; Ibrahim et al., 2015). The critical eﬀective face pressures calculated by the presented kinematical approach are also given in the ﬁgure (153.7 kPa and 57.5 kPa, respectively). According to Fig. 5, the critical eﬀective face pressures obtained by the presented kinematical approach are close to those provided by the hydro-mechanical analysis, showing that the presented method is valid to deal with the considered problem in this paper. The proposed kinematical approach provides a critical pressure that is higher (more conservative) than the numerical value obtained (Fig. 5). This discrepancy between these two methods can be explained by two reasons:

40m

Z 55m

40m

Y

X

Fig. 4. A sketch of the numerical model.

– The tunnel lining is considered using shell structural elements in the numerical model, and small displacements due to the soil load can be produced on the tunnel periphery. The tunnel concrete lining is elastic linear simulated with shell structural elements. In the LA model, the soil mass behind the tunnel face is considered as ﬁxed in all directions. – The mesh grid size: assuming that the LA and the ﬁnite diﬀerence

groundwater level measured from the crown of tunnel is equal to the water table elevation Hw which is assumed to stay constant for the steady-state ﬂuid ﬂow analysis. Since the shield tunnel has a waterproof lining, it is assumed that the tunnel walls are impermeable, which means that underground water only ﬂows into the tunnel face. This ﬂuid-mechanical analysis which is not a coupled analysis permits to consider the seepage in the numerical analysis in the same way as in the LA model (the eﬀect of the pore pressure change on deformation due to the seepage is not taken into account). In numerical models, a linear perfectly elastic-plastic constitutive model based on the modiﬁed Hoek-Brown failure criterion (with an associated ﬂow rule) is assigned to the rock masses. The corresponding rock Young’s modulus is obtained using the empirical formula according to GSI and Di (Hoek and Diederichs, 2006). The tunnel concrete lining is simulated with shell structural elements with the Young’s modulus of 20 GPa, the Poisson’s ratio of 0.2 and thickness of 0.22 m. The other mechanical parameters used in the numerical analysis are listed in Table 1. In order to determine the critical face pressures by the numerical

Table 1 The other mechanical parameters for the hydro-mechanical analysis.

559

Parameters

Case 1

Case 2

Rock submerged unit weight Rock dry unit weight Permeability coeﬃcient Young’s modulus Poisson’s ratio

20 kN/m3 25 kN/m3 1.0 × 10−1 cm/s 270 MPa 0.3

1.0 × 10−4 cm/s 670 MPa 0.2

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Fig. 5. The horizontal displacements at the center of tunnel face versus face pressures for two cases. Fig. 6. Critical collapse mechanisms provided by limit analysis for case of (a) mi = 7, GSI = 10, σc = 1.0 MPa and (b) mi = 10, GSI = 20, σc =5.0 MPa.

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Fig. 6. (continued)

models use the same boundary conditions, the same loadings, the same (associated) Mohr-Coulomb model (tangent approximation to Hoek-Brown model) and material properties. Considering a stable state calculated by FLAC3D, for example points at the upper right corner in Fig. 5, the ﬁnite diﬀerence model provides an approximate exact solution to the problem in the sense that local equilibrium may not be satisﬁed everywhere at the boundary between zones; but if the zone size is reduced to zero, local equilibrium will be satisﬁed to the limit. In particular, the limit stress ﬁeld satisﬁes the lower-bound theorem. The face pressure in the calculation provides an upper bound for the critical face pressure. Also, the deformation ﬁeld at the “failure state” calculated by FLAC3D (e.g. last point recorded towards the bottom of the Fig. 5) is kinematically admissible (it fulﬁlls all criteria of the upper bound theorem). Thus, the “critical pressure” tends to a limit as the grid size is reduced; this limit may be considered to be very close to the exact critical pressure for the problem. Besides, it is also worthy to note that the hydro-mechanical analysis in FLAC3D requires a very long computational time, even though it is uncoupled analysis. For instance, the consuming time to obtain one horizontal displacement (from the stage 1 to stage 2) on an Intel Xeon

Fig. 7. Comparisons with previously published solutions.

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Fig. 8. The normalized critical face pressures versus normalized uniaxial compressive strength at Hw/D = 1.0.

degenerated to the linear Mohr-Coulomb criterion using a = 1 (Yang and Pan, 2015). Fig. 7 plots a comparison of critical eﬀective face pressures between the proposed method and other four solutions for the case of φ = 35°, c = 0.0 kPa, γw/γ′ = 1.81, D = 5 m and C/D = 2.0. It is observed that the critical face pressure increases almost linearly with the water table elevation, which means that underground water seriously impacts the tunnel face stability. In addition, the critical effective face pressures according to the presented method are higher than those provided by other approaches in most cases. It should be noted that the presented method yields a rigorous lower-bound solution, which means that the presented method improve the critical effective face pressures compared with the other approaches. For example, the maximum improvement reaches 31% with respect to the solutions by Lee et al. (2003). The presented results also increase the existing best solutions by Pan and Dias (2016) by 3.2%. This slight discrepancy may be due to diﬀerent approaches introducing the seepage force contribution into the predicting model between these two works, for instance the method presented by Pan and Dias (2016) neglects the pore-water pressure work on the volumetric strain of the

CPU E5-1620 3.5 GHz PC is about 70 h for the case of mi = 7, GSI = 10, σc = 1.0 MPa. However, the computational time is around 30 min with the kinematical approach of limit analysis coded in Matlab. This indicates that the presented approach is highly eﬃcient in comparison to the numerical analysis. The numerical hydro-mechanical analysis is practically not applicable to perform parametric analysis, for example to obtain the design charts as given in a later section. Fig. 6(a) and (b), respectively, present the critical collapse mechanism provided by the upper-bound limit analysis for these two cases with Hw/D = 2.0. A comparison between the Fig. 6(a) and (b) indicates that the critical mechanism extends to a wider range ahead of the tunnel face for the weaker rock masses. 3.2. Comparisons with existing solutions This section devotes to a comparison between the presented work and those reported by Anagnostou and Kovari (1996), Lee et al. (2003), Perazzelli et al. (2014), Pan and Dias (2016). In order to perform the comparisons, the non-linear Hoek-Brown failure criterion is 562

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Fig. 9. The normalized critical face pressures versus normalized uniaxial compressive strength at Hw/D = 2.0.

to zero in all calculations. The diameter of the tunnel is taken to be 10 m; the ratio of submerged unit weight γ′ to water unit weight γw is set to 2.0. The charts in Figs. 8–11 are useful for direct computations of critical face pressures at the preliminary stage. Making reference to Figs. 8–11, not surprisingly, the normalized critical face pressure rises with the increase of Hw/D ratio, but decreases with the rock mass quality. It is observed that the σt/γ′D shows an obvious non-linear relationship with σc/γ′D, especially for σc/γ′D smaller than 20.0. When σc/γ′D > 20, the variation of σt/γ′D with respect to σc/γ′D becomes gradually smaller. A similar non-linear relationship between these two parameters was also reported by Senent et al. (2013) who considered the tunnel face stability in heavily fractured rock masses in the absence of seepage forces.

skeleton. Besides, the approach reported by Pan and Dias (2016) is invalid for a mechanism whose velocity at the boundary is null, for example the velocity continuity failure mechanism for purely cohesive soils (Mollon et al., 2013a). 4. Design charts Thanks to the eﬃcient computational speed of the presented approach, it is possible to perform calculations for the purpose of parametric analysis. Several design charts are provided in Figs. 8-11, which respectively corresponds to Hw/D = 1.0, 2.0, 3.0 and 4.0. In these charts, the normalized critical face pressure σt/γ′D is plotted as a function of the normalized uniaxial compressive strength σc/γ′D. In each set of ﬁgure, GSI changes from 10 to 25, and mi from 5 to 20; the overburden ratio C/D is taken as 2.0 and the disturbance factor Di is set

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Fig. 10. The normalized critical face pressures versus normalized uniaxial compressive strength at Hw/D = 3.0.

5. Conclusions

numerical simulation of hydro-mechanical analysis, but requires a much shorter computational time. When the Hoek-Brown failure criterion degenerates to linear Mohr-Coulomb criterion with a = 1, the presented work is compared with other existing solutions, which shows that the presented method provides an improvement for critical eﬀective face pressures. Thanks to the fast computational speed of the presented approach, several useful charts are obtained for parametric analysis, in which the normalized critical face pressure is plotted as a function of the normalized uniaxial compressive strength corresponding to diﬀerent water table elevations and for several Hoek-Brown sets of parameters. Numerical results show that the normalized critical face pressure rises with the increase of water-table elevation, but decreases as the rock mass quality.

For underground structures, one of the most important destabilizing factors is the underground water. This work aims to study the face stability of a tunnel excavated in weak rock masses under the water table in the context of the kinematical approach of limit analysis. An advanced 3D rotational collapse mechanism is used to calculate critical eﬀective face pressures. The seepage force is incorporated into this predicting model by considering it as a body force acting on rock masses. A 3D steady-state seepage ﬁeld obtained numerically by FLAC3D is used to interpolate the hydraulic head of each point on the 3D collapse mechanism. The computed results of the presented approach are compared with other approaches. The presented approach agrees well with the

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Fig. 11. The normalized critical face pressures versus normalized uniaxial compressive strength at Hw/D = 4.0.

him with a PhD Scholarship for his research work and the ﬁnancial support by the National Basic Research 973 Program of China (2013CB036004) is also greatly appreciated.

Acknowledgements The ﬁrst author thanks the China Scholarship Council for providing Appendix A The expressions of coeﬃcients Nγ′, Ns and Nc are given as follows:

Nγ′ =

Ns =

Nc =

∑i ∑j (Ri,j Vi,j sinβi,j ) D ∑i (Σi Ri,0cosβi,0)

∑i (Si Risinβi ) ∑i (Si,0 Ri,0cosβi,0)

cosφt ∑i ∑j (Ri,j Si,j ) ∑i (Σi Ri,0cosβi,0)

where φt is the equivalent friction angle; Ri,j, Vi,j, βi,j, Ri,0, βi,0, Si,j, and Si, Ri, βi are explained in details in Mollon et al. (2011).

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