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Face stability analysis of shallow circular tunnels in cohesive–frictional soils Chengping Zhang , Kaihang Han, Dingli Zhang Key Laboratory for Urban Underground Engineering of the Education Ministry, Beijing Jiaotong University, Beijing 100044, China School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China article info Article history: Received 6 March 2015 Received in revised form 29 June 2015 Accepted 11 August 2015 Keywords: Shallow tunnel Face stability Failure mechanism Numerical simulation Limit analysis Face collapse abstract The face stability of a circular tunnel in cohesive–frictional soils was numerically and theoretically inves- tigated. Three-dimensional (3D) numerical simulations were first performed to analyze the face stability of a circular tunnel with a series of tunnel diameter-to-depth ratios and soil properties. The limit support pressure on the tunnel face and the failure zone in front of the tunnel face were both obtained from the numerical simulations. A simple and feasible criterion was suggested to outline the boundary strip of the failure zone at collapse in displacement clouds under different conditions. Based on the numerical sim- ulation results, a new 3D failure mechanism was proposed using the kinematic approach of limit analysis theory to determine the limit support pressure of the tunnel face. The new 3D failure mechanism was composed of four truncated cones on which a distributed force acts. Finally, the limit support pressures and failure zones obtained from the new failure mechanism and the numerical simulations were com- pared. In addition, comparisons between the results of this work and those of existing approaches were performed. Overall, the new failure mechanism is substantially more consistent with the shapes of the failure zones observed in numerical simulations and experimental tests than the existing multi-block failure mechanisms. The new failure mechanism is more effective and reasonable. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Because of the continuous expansion of cities and decrease in available land, new transportation and service networks must be placed underground (e.g., metro tunnels). Shield machines (earth, slurry and air) are widely used in urban shallow tunnel construc- tion in soft ground. However, if the support pressure in the cham- ber is not sufficient to balance the external earth and water pressure, the tunnel face may become unstable or collapse. Thus, the stability analysis of the tunnel face is essential for guaranteeing the safe construction of shallow shield tunnels. The primary con- cerns in stability analysis are the failure zone and the limit support pressure of the tunnel face. In purely cohesive soils, the so-called load factor N has been used to investigate the stability of the tunnel face. This load factor N was first defined by Broms and Bennermark (1967) as N =(r s + cH r t )/ c u , where r s is the possible surcharge loading acting on the ground surface, r t is the uniform pressure applied on the tunnel face, H is the depth of the tunnel axis, c is the soil unit weight, and c u is the undrained soil cohesion. From an experimental perspective, Broms and Bennermark (1967) derived the stability condition for N values between 6 and 7. Kimura and Mair (1981) found that the limit value depends on the tunnel cover when N is between 5 and 10 based on the centrifuge test results. Based on a limit equilibrium analytical approach, Ellstein (1986) gave an analytical expression of N for homogeneous cohesive soils, which agreed with the results of Kimura and Mair (1981). Davis et al. (1980) obtained upper and lower bound stability solutions for heading collapse under undrained conditions, which were based on three different shapes of shallow underground openings relevant to tunneling. More recently, a promising numerical approach (the finite element limit analysis) was proposed by Augarde et al. (2003) to investigate the stability of a plane strain heading in undrained soil conditions based on lower- and upper-bound theorems. This approach is cur- rently limited to a 2D analysis. In contrast with the rigid block fail- ure mechanisms, several kinematic approaches based on continuous velocity fields in limit analysis theory have been pro- posed. The difference between these approaches lies in the methods that are used to generate the continuous velocity fields. Klar et al. (2007) suggested a new kinematic approach in limit analysis theory for two-dimensional (2D) and three-dimensional (3D) stability analyses of circular tunnels in a purely cohesive soil based on an http://dx.doi.org/10.1016/j.tust.2015.08.007 0886-7798/Ó 2015 Elsevier Ltd. All rights reserved. Corresponding author at: School of Civil Engineering, Beijing Jiaotong Univer- sity, No. 3 Shangyuancun, Haidian District, Beijing 100044, China. E-mail addresses: [email protected], [email protected] (C. Zhang). Tunnelling and Underground Space Technology 50 (2015) 345–357 Contents lists available at ScienceDirect Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust

Tunnelling and Underground Space Technologyor.nsfc.gov.cn/bitstream/00001903-5/283913/1/1000014005291.pdf · Anagnostou and Kovári (1996) applied the wedge model to calculate the

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Tunnelling and Underground Space Technology 50 (2015) 345–357

Contents lists available at ScienceDirect

Tunnelling and Underground Space Technology

journal homepage: www.elsevier .com/ locate/ tust

Face stability analysis of shallow circular tunnelsin cohesive–frictional soils

http://dx.doi.org/10.1016/j.tust.2015.08.0070886-7798/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: School of Civil Engineering, Beijing Jiaotong Univer-sity, No. 3 Shangyuancun, Haidian District, Beijing 100044, China.

E-mail addresses: [email protected], [email protected] (C. Zhang).

Chengping Zhang ⇑, Kaihang Han, Dingli ZhangKey Laboratory for Urban Underground Engineering of the Education Ministry, Beijing Jiaotong University, Beijing 100044, ChinaSchool of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 March 2015Received in revised form 29 June 2015Accepted 11 August 2015

Keywords:Shallow tunnelFace stabilityFailure mechanismNumerical simulationLimit analysisFace collapse

The face stability of a circular tunnel in cohesive–frictional soils was numerically and theoretically inves-tigated. Three-dimensional (3D) numerical simulations were first performed to analyze the face stabilityof a circular tunnel with a series of tunnel diameter-to-depth ratios and soil properties. The limit supportpressure on the tunnel face and the failure zone in front of the tunnel face were both obtained from thenumerical simulations. A simple and feasible criterion was suggested to outline the boundary strip of thefailure zone at collapse in displacement clouds under different conditions. Based on the numerical sim-ulation results, a new 3D failure mechanism was proposed using the kinematic approach of limit analysistheory to determine the limit support pressure of the tunnel face. The new 3D failure mechanism wascomposed of four truncated cones on which a distributed force acts. Finally, the limit support pressuresand failure zones obtained from the new failure mechanism and the numerical simulations were com-pared. In addition, comparisons between the results of this work and those of existing approaches wereperformed. Overall, the new failure mechanism is substantially more consistent with the shapes of thefailure zones observed in numerical simulations and experimental tests than the existing multi-blockfailure mechanisms. The new failure mechanism is more effective and reasonable.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Because of the continuous expansion of cities and decrease inavailable land, new transportation and service networks must beplaced underground (e.g., metro tunnels). Shield machines (earth,slurry and air) are widely used in urban shallow tunnel construc-tion in soft ground. However, if the support pressure in the cham-ber is not sufficient to balance the external earth and waterpressure, the tunnel face may become unstable or collapse. Thus,the stability analysis of the tunnel face is essential for guaranteeingthe safe construction of shallow shield tunnels. The primary con-cerns in stability analysis are the failure zone and the limit supportpressure of the tunnel face.

In purely cohesive soils, the so-called load factorN has been usedto investigate the stability of the tunnel face. This load factor Nwasfirst defined by Broms and Bennermark (1967) as N = (rs + cH � rt)/cu, where rs is the possible surcharge loading acting on the groundsurface, rt is the uniform pressure applied on the tunnel face, H isthe depth of the tunnel axis, c is the soil unit weight, and cu is the

undrained soil cohesion. From an experimental perspective,Broms and Bennermark (1967) derived the stability condition forN values between 6 and 7. Kimura and Mair (1981) found that thelimit value depends on the tunnel cover when N is between 5 and10 based on the centrifuge test results. Based on a limit equilibriumanalytical approach, Ellstein (1986) gave an analytical expression ofN for homogeneous cohesive soils, which agreed with the results ofKimura and Mair (1981). Davis et al. (1980) obtained upper andlower bound stability solutions for heading collapse underundrained conditions, which were based on three different shapesof shallow underground openings relevant to tunneling. Morerecently, a promising numerical approach (the finite element limitanalysis) was proposed by Augarde et al. (2003) to investigate thestability of a plane strain heading in undrained soil conditionsbased on lower- and upper-bound theorems. This approach is cur-rently limited to a 2D analysis. In contrast with the rigid block fail-ure mechanisms, several kinematic approaches based oncontinuous velocity fields in limit analysis theory have been pro-posed. The difference between these approaches lies in themethodsthat are used to generate the continuous velocity fields. Klar et al.(2007) suggested a new kinematic approach in limit analysis theoryfor two-dimensional (2D) and three-dimensional (3D) stabilityanalyses of circular tunnels in a purely cohesive soil based on an

346 C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357

admissible continuous velocity field. The velocity field from the 2Dstability analysis was based on work performed by Verruijt andBooker (1996). The velocity field in the 3D stability analysis wasbased on work performed by Sagaseta (1987). More recently,Mollon et al. (2013) developed two continuous velocity fields forthe collapse and blowout of a pressurized tunnel face in purelycohesive soil. Those continuous velocity fields were based on thenormality condition, which states that any plastic deformation ina purely cohesive soil develops without any volume change. Thecontinuous velocity field results have shown significant improve-ments compared with the other approaches.

In cohesive–frictional soils, analytical approaches are primarilybased on limit equilibrium methods or limit analysis methods.Limit equilibrium methods are widely used in the theoretical anal-yses of tunnel face stability. Murayama et al. (1966) proposed a 2Dlogarithmic spiral model. Krause (1987) derived the limit supportpressure for tunnel face failure by assuming that the failure zonewas a half sphere, a half circle, or a quarter circle. In addition,Horn (1961) introduced a 3D wedge model that assumed a slidingwedge loaded by a soil silo. Anagnostou and Kovári (1996) appliedthe wedge model to calculate the limit support pressure in thehomogeneous stratum. Broere (2001) extended the wedge modelto a layered stratum. The other theoretical methods are the limitanalysis methods (based on the upper- and lower-bound theoremsof plasticity). Atkinson and Potts (1977) derived the limit supportpressure for an unlined cavity in a dry cohesionless material. Inaddition, Lyamin and Sloan (2000) investigated the stability of aplane strain circular tunnel in cohesive–frictional soils using finiteelement limit analysis methods. Leca and Dormieux (1990)assumed that the failure zone in front of the tunnel face consistedof a series of conical bodies and derived lower- and upper-boundsolutions for the limit support pressure in a dry Mohr–Coulombmaterial. Subsequently, by assuming different failure zone shapesin front of the tunnel face, Soubra (2000, 2002), Subrin andWong (2002) and Mollon et al. (2011) derived the upper-boundlimits for the limit support pressure in a dry Mohr–Coulomb mate-rial. Experimental tests can be used to study tunnel face stabilityproblems and the failure modes of the surrounding rock. Thesetests have played important roles in tunnel face stability studies.Chambon and Corté (1994) conducted a series of centrifuge modeltests to determine the tunnel face stability in dry sandy ground.Their results indicated that the relative depths of the tunnels andthe density of sand had little influence on the limit support pres-sure. In addition, their results indicated that the failure zone infront of the tunnel face was bulb-shaped. Takano et al. (2006) per-formed 1g experimental tests in which an X-ray computed tomog-raphy scanner was used to visualize the 3D shape of the failuremechanism. Kirsch (2010) performed small-scale model testsunder normal gravity (1g) to investigate the face stability of shal-low tunnels and to show that the necessary support pressure isindependent of the overburden and of the initial soil density.Numerical simulation is considered as an important method forinvestigating the stability of tunnel faces using both continuumand discrete approaches due to their good reproducibility. The con-tinuum numerical analysis can be performed using the FiniteElement Method (FEM) or the Finite Difference Method (FDM),whereas the discrete numerical analysis can be carried out usingthe Discrete Element Method (DEM). Vermeer et al. (2002) devel-oped a series of 3D FEM simulations for tunnel face stability anal-yses and demonstrated that the friction angle of the sand affectsthe failure zone in front of the tunnel face and that the limit sup-port pressure decreases as the friction angle of the sand increases.Based on 3D FEM simulations, Lu et al. (2014) considered the influ-ence of seepage on the face stability and analyzed the relationshipbetween the support pressure and displacement of the shield tun-nel face. Chen et al. (2013) conducted numerical simulations of the

tunnel face failures using the 3D FDM and compared the results ofthe limit support pressure with those obtained from their experi-mental tests. Senent et al. (2013) proposed a model with the 3DFDM to analyze the tunnel face failures in fractured rocks undertwo different distributions of normal stresses along the slip sur-face. Later on, Senent and Jimenez (2015) extended the model pro-posed by Senent et al. (2013) to layered soils and studied thepossibility for partial collapse of tunnel faces. Recently, severalresearchers have begun using DEM to analyze the stability of thetunnel face. Funatsu et al. (2008) investigated the stability of a sin-gle tunnel and two parallel tunnels by using a series of 2D DEMsimulations. Subsequently, Zhang et al. (2011) developed a 2DDEM simulation to study the behaviors of cohesive–frictional soilsduring the slurry shield tunneling process. Chen et al. (2011) con-structed a 3D DEM model to analyze the face stability of shallowshield tunnels in dry sand. The limit support pressure, failure zoneand soil arching were obtained and discussed in terms of the pro-cess of tunnel face failure.

This paper focuses on a face stability analysis of cohesive–fric-tional soils in the framework of the kinematic approach of limitanalysis theory. The rigid block failure mechanisms provide a sim-ple and intuitive approach and are either translational or rota-tional. Fig. 1 depicts various 3D rigid block failure mechanisms.These failure mechanisms only consider a portion of the tunnelface (an ellipse on a circular tunnel face) as the failure zone.Mollon et al. (2011) generated new failure mechanisms to extendthe failure zone to include the entire circular tunnel face using aspatial discretization technique. In those cases, the shapes of theblocks in the failure mechanisms are constrained by the normalitycondition (Chen, 1975), which implies that each velocity disconti-nuity should occur at an angle u from the corresponding velocitydiscontinuity surface, with u representing the internal frictionangle of the soil. In some sense, the normality condition causesthe 3D rigid block failure mechanism to consist of cones and/orlogarithmic-spiral-shaped rigid blocks. The upper shapes of thefailure zone are cusp-angle-shaped rather than arch-shaped.However, the results of the centrifuge tests proposed byChambon and Corté (1994) indicated that an arch effect occurs inthe upper part of the failure zone and that the failure soil massresembles a chimney for the cover-to-depth ratios C/D = 0.5, 1and 2, as shown in Fig. 2. There are some differences betweenthe assumptions in the limit analysis theory and the actual situa-tions in the upper parts of the failure mechanisms (cf. Fig. 1).

To address these issues, a numerical model is developed in thisstudy. The goal of the model is to accurately assess the limit sup-port pressure of tunnel face without any a priori assumptionregarding the shape of the failure mechanisms. Furthermore, a cri-terion is suggested to outline the boundary strip of the failure zoneat collapse in the displacement clouds. These results will serve as areference to develop a suitable collapse failure mechanism in thelimit analysis for the cohesive–frictional soils. Next, a new 3D fail-ure mechanism is proposed based on the results of numerical sim-ulations in the framework of the kinematic approach of limitanalysis theory. Finally, the limit support pressures and failurezones of a tunnel face obtained from the present failure mecha-nism and the numerical simulations are compared. In addition, acomparison between the results of the present study and existingapproaches is performed.

2. Numerical simulations with FLAC3D

2.1. FLAC3D numerical simulations

Among the numerical simulation methods, the FEM suffersfrom the shortcoming of the pathological mesh-dependency foranalyzing localization problems. The DEM is very time-

(a) (b) (c)Fig. 1. Some existing 3D rigid block failure mechanisms: perspective views. (a) The two-block failure mechanism defined by the rigid conical blocks (Leca and Dormieux,1990); (b) the multi-block failure mechanism defined by rigid conical blocks (Soubra, 2002); and (c) the rhinoceros horn failure mechanism defined by logarithmic spirals(Subrin and Wong, 2002).

Fig. 2. Failure pattern of a shallow tunnel in experimental tests (Chambon andCorté, 1994).

C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357 347

consuming for simulating the failure of a tunnel face by compar-ison with the computational efficiency of FEM and FDM (Chenet al., 2011). Moreover, the micro-parameters input into the DEMobtained from parameter calibration are difficult to represent thereal characteristics of the soils. The 3D finite difference codeFLAC3D (Itasca Consulting Group, 2006) used explicit Lagrangiancalculation scheme and a mixed discretization zoning technique.This is a forward scheme for nonlinear problem, which does notrequire iteration, unlike other techniques such as FEM that usesimplicit solution methods. Moreover, FLAC3D could deal with thelarge deformation problem very well. Therefore, FLAC3D is adopted

in this research aiming at outlining the failure zone and determin-ing the limit support pressure.

Because symmetrical tunnels are considered, the calculations ofthe limit support pressure are based on half of a cylindrical tunnelcut lengthwise along the central axis. Themodel is sufficiently largeto avoid boundary effects (cf. Fig. 3). A 3D non-uniform mesh isused. The present model is composed of approximately 45,823zones (‘zone’ is the FLAC3D terminology for each discretized ele-ment). A conventional elastic–plastic model based on the Mohr–Coulomb failure criterion is adopted to represent the soil. The shellsare simulated by a ‘‘liner” structural element. The boundary condi-tions in the model are as follows: the ground surface is free to dis-place, the side surfaces have roller boundaries, and the base is fixed.

The parameters of the soils adopted in the numerical simula-tions are given in Table 1. The four types of soils are loose sands,dense sands, soft clays and stiff clays, as suggested in Mollonet al. (2011). According to the studies proposed by Anagnostouet al. (2011) and Ibrahim et al. (2015), the Young’s Modulus hasno influence on the limit support pressure. Therefore, a Young’sModulus of 20 MPa is adopted for all the soils. The thickness,Young’s Modulus and Poisson’s ratio of the lining shells are35 mm, 33.5 GPa and 0.2, respectively.

In practical engineering, the excavation process during the con-struction of a shield tunnel is performed step by step. However, thisstudy focuses on the active failure zone and limit support pressureof a tunnel face. Therefore, the excavation process was simulatedusing a simplified single-step excavation scheme that assumes partof the tunnel (10 m in length) is excavated instantaneously.Simultaneously, lining shells are installed, and trapezoidally dis-tributed support pressures that are equal to the initial ground hor-izontal stress in the reversed direction are applied at the tunnelface. Then, the active limit support pressure is found by graduallydecreasing the support pressure until the tunnel face collapses. Ateach pressure, several cycles are performed until a steady state ofstatic equilibrium or plastic flow is developed in the soil.

2.2. Numerical simulation results

2.2.1. Limit support pressureFig. 4 shows the curves of the support pressure ratio (i.e., the

ratio of the specified face support pressure to the initial ground

Fig. 3. Numerical model for the analysis of tunnel face stability.

Table 1Soil parameters.

C/D c (kN/m3) K0 m c (kPa) u (�) Soil type

0.5 18 0.658 0.397 0 20 Loose sands18 0.357 0.263 0 40 Dense sands18 0.708 0.414 7 17 Soft clays18 0.577 0.366 10 25 Stiff clays

1 18 0.658 0.397 0 20 Loose sands18 0.357 0.263 0 40 Dense sands18 0.708 0.414 7 17 Soft clays18 0.577 0.366 10 25 Stiff clays

0 50 100 150 200 250 300 350 400 450 5000.0

0.2

0.4

0.6

0.8

1.0

0.200.14

0.24

Supp

ort p

ress

ure

ratio

Horizontal displacement of the central point of tunnel face / mm

C/D = 0.5, c = 0 kPa, = 20ο

ο

ο

ο

C/D = 0.5, c = 0 kPa, = 40C/D = 1.0, c = 0 kPa, = 20C/D = 1.0, c = 0 kPa, = 40

0.35

(a) Sands;

0 50 100 150 200 250 300 350 400 450 5000.0

0.2

0.4

0.6

0.8

1.0

0.08

0.17

0.11

0.24

Supp

ort p

ress

ure

ratio

Horizontal displacement of the central point of tunnel face / mm

C/D = 0.5, c = 7 kPa, = 17C/D = 0.5, c = 10 kPa, = 25C/D = 1.0, c = 7 kPa, = 17C/D = 1.0, c = 10 kPa, = 25

(b) Clays.

ϕϕϕϕ

οϕοϕοϕοϕ

Fig. 4. Relations between the horizontal displacement and the support pressureratio of a tunnel face. (a) Sands; (b) clays.

348 C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357

horizontal stress for the center of the tunnel) versus the horizontaldisplacement of the corresponding central point of the tunnel facefor various relative depths (C/D = 0.5 and 1.0) in different soils(sands or clays). As shown in Fig. 4, as the horizontal displacementof the tunnel face increases, the support pressure ratios graduallydecrease to a constant value. Specifically, the slopes of the curveswill reach zero. In Fig. 4, the dotted lines denote the horizontal tan-gents of the curves, and the y-intercepts indicate the limit supportpressure ratios. Smaller friction angles u, cohesion c and relativeratios C/D correspond with greater limit support pressure ratios.

2.2.2. The range of the failure zone on the limit conditionThe failure zone in front of the tunnel face is another main con-

cern. Displacement contours of the limit conditions are plotted inFigs. 5 and 6 for all cases. In these figures, the incremental displace-ments are shown as graded shades from blue to red. An ellipsoid-shape or truncated ellipsoid-shape is obtained in the upper part ofthe failure zone, as indicated in Figs. 5 and 6.

The specific boundaries of the failure zone obtained fromnumerical simulations based on the FEM would not be obtainedimmediately in displacement clouds. Therefore, it is very impor-tant to quickly set up criteria for outlining the boundary strip ofthe failure zone based on the displacement contours.

A simple and feasible criterion to outline the boundary strip ofthe failure zone at collapse in displacement clouds is proposed asfollows:

Fig. 5. Displacement contours of the limit conditions observed in the sands.

C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357 349

(i) For the condition with no outcropping of the failure zone atthe ground surface, most curves in the displacement cloudswill close (cf. Fig. 7a). In addition, the position within a sud-den increase gradient can be defined as the boundary strip ofthe failure zone.

(ii) For the outcropping condition of the failure zone at theground surface, most curves in the displacement clouds willnot close and will intersect the ground surface (cf. Fig. 7b).However, the ‘‘close” and ‘‘divergent” tendencies are stillobvious. The position within the perpendicularly intersectwith the ground surface can be defined as the boundary stripof the failure zone.

The criterion was used to outline the boundary strip of the fail-ure zone at collapse rather than to predict the accurate boundary ofthe failure zone. Therefore, even though the interval of the contourlines changed from 0.2 to 0.1, or to another value, the boundarystrip of the failure zone was consistent using the criterion.

3. Limit analysis of the face stability of shallow circular tunnels

3.1. The new failure mechanism

According to the results from the numerical simulation, thereare some differences between the model and the actual situationsin terms of the existing failure mechanisms (cf. Fig. 1), especially inthe upper parts of the failure mechanisms.

The shapes of the blocks in the failure mechanism are con-strained by the normality condition (Chen, 1975), which impliesthat each velocity discontinuity should form an angle u with thecorresponding velocity discontinuity surface, i.e., the internal

friction angle of the soil. The normality condition implies thatthe 3D rigid-block failure mechanism should consist of conesand/or logarithmic spiral-shaped rigid blocks.

Although the upper shape of the failure zone should be simu-lated with an ellipsoid, the ellipsoid does not fulfill the normalitycondition (Chen, 1975). Therefore, other improved treatmentsshould be considered. In this study, to propose an improved 3Dfailure mechanism, the failure causes are analyzed and someimproved treatments are proposed.

3.1.1. Failure causesAccording to the results of centrifuge model tests performed by

Idinger et al. (2011) (cf. Fig. 8), the lower failure plane is primarilyformed by shear failure, while the upper failure plane (arch-shaped) is more complicated and is likely affected by the combinedaction of tension and shear stresses. Hence, the upper and lowerfailure mechanisms are developed separately. Fig. 9 shows the fail-ure causes in longitudinal and crosswise profiles of the slip surface.

3.1.2. Improved treatmentsIn the limit equilibrium method, the upper part of the failure

zone, an ellipsoid block, is transformed into a distributed forcebased on the Terzaghi pressure-arch theory (cf. Fig. 10).Considering the failure causes and referring to the treatment ofthe upper part of failure zone in the limit equilibrium method,the new 3D failure mechanism is composed of four truncated conesand a distributed force acting on those truncated cones (cf. Fig. 11).The distributed force q is caused by the weight of the upper part ofthe failure zone, and the lower part of the failure zone is modeledas a multi-block failure mechanism [Blocks (1)–(4)].

Fig. 6. Displacement contours of the limit conditions observed in the clays.

350 C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357

This treatment, which transforms the effects of Blocks (5) and(6) into distributed forces, allows the failure mechanism to prop-erly satisfy the normality condition.

3.2. Geometric properties

The face stability analysis relevant to a circular rigid tunnel ofdiameter D driven under a depth of cover C could be idealized. Asurcharge rs is applied on the ground surface, and rt is the uniformsupport pressure on the tunnel face. Fig. 12 shows the two possiblecombinations of the improved 3D failure mechanism. When thestrength of the surrounding rock is low, or the cover depth of thetunnel is small, the tunnel face readily collapses and is likely tospread to the ground surface (cf. Fig. 12a). As the cover depthincreases, a collapsing arch can be formed before the tunnel face(cf. Fig. 12b). In general, the improved failure mechanism is com-posed of four rigid blocks with elliptical cross-sections and a dis-tributed force caused by oblique elliptical cylinders [Block (5)]and truncated ellipsoids or ellipsoids [Block (6)].

Moreover, the improvement (related to the ellipsoid shape) issimilar to the ellipsoid theory of particle flows in sublevel cavemining. Janelid and Kvapil (1966) developed the classical conceptof the gravity flow of ore in sublevel caving operations, whichstates that the loosening zone of a tunnel is an ellipse or a portionof an ellipse. To make the axis of the (truncated) ellipsoid vertical,the oblique elliptical cylinder [Block (5)] is added. The four trun-cated rigid cones have opening angles that are equal to 2u, whichare the same as the descriptions of the most popular multi-blockfailure mechanism proposed by Soubra (2000, 2002).

The four truncated rigid cones are translated with velocitieswith different directions, which are collinear with the cones’ axesand are at an angle u to the discontinuity surface (cf. Fig. 13).

The velocity of block i and the relative velocity between blocks iand i + 1 are described by the following equations:

v i ¼ v1

Yik¼2

cos Wk�1;k þu� �

cosðWk;kþ1 �uÞ ðfor i P 2Þ ð1Þ

v i;iþ1 ¼ v isinðWi;iþ1Þ

cosðWi;iþ1 �uÞ ðfor i P 1Þ ð2Þ

where

W0;1 ¼ aWi;iþ1 ¼ bi �Wi�1;i; ði P 1Þ

�ðfor i P 1Þ ð3Þ

The intersections of adjacent blocks are ellipses and are calledR1, Ri,i+1 [1 6 i 6 4], R5 and R6. The semi-axis lengths of theellipses are a1(b1), ai,i+1(bi,i+1) [1 6 i 6 4], a5(b5) and a6(b5).

The intersection of the first truncated cone (adjacent to the tun-nel face) with the circular tunnel face is an ellipse, with semi-axislengths of a1 and b1 that are calculated as follows (cf. Fig. 11):

a1 ¼ D2

ð4Þ

b1 ¼ D2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosða�uÞ cosðaþuÞp

cosuð5Þ

where u defines the opening angles of the four truncated rigid conesthat are equal to 2u and a is the angle between the axis of the firsttruncated rigid cone adjacent and the horizontal.

Therefore, the area A1 of the first truncated cone base is asfollows:

A1 ¼ pD2

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosða�uÞ cosðaþuÞp

cosuð6Þ

C/D =1.0c = 0 kPaφ = 20°

suggested boundarystrip of failure zone

ground surface

units: %

Tunnel2.0

2.0

2. 00.

4

0.4

0.60.8

1.01.

4

(a) C/D=1.0, φ = 20°, c=0 kPa

0.4

units: %

2.00.60.81.4

0.4

1.00.

20.

60.

4

ground surface

Tunnel

suggested boundarystrip of failure zone

C/D =0.5c = 0 kPaφ = 20°

(b) C/D=0.5, φ = 20°, c=0 kPa

Fig. 7. The suggested criteria for outlining the boundary strip of the failure zone forthe limit conditions in the displacement clouds.

C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357 351

In addition, the areas of the contact elliptical surfaces betweentwo successive truncated cones i and i + 1 are ellipses with semi-axis lengths of ai,i+1 and bi,i+1 (for 1 6 i 6 4) are described asfollows:

ai;iþ1 ¼ D2

Yik¼1

cosðWk�1;k þuÞcosðWk;kþ1 �uÞ ð7Þ

bi;iþ1 ¼ ai;iþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosðWi;iþ1 þuÞ cosðWi;iþ1 �uÞp

cosuð8Þ

In addition, the area Ai,i+1 (1 6 i 6 4) between two successiveblocks i and i + 1 is as follows:

Ai;iþ1 ¼ pD2

4

Yik¼1

cosðWk�1;k þuÞcosðWk;kþ1 �uÞ

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosðWi;iþ1 þuÞ cosðWi;iþ1 �uÞp

cosuð9Þ

The volumes V1 and Vi (for 2 6 i 6 4) of the truncated cones areas follows:

V1 ¼ A1h1 � A1;2h2

3ð10Þ

Vi ¼ Ai�1;ihi � Ai;iþ1hiþ1

3ð11Þ

where

h1 ¼ D cosðaþuÞ cosða�uÞsin 2u

h2 ¼ D cosðaþuÞ cosðb1�aþuÞsin 2u

hi ¼ h2Qi�1

k¼2cosðWk;kþ1þuÞcosðWk�1;k�uÞ ; ði P 3Þ

8>>><>>>:

ð12Þ

The intersection of a truncated ellipsoid with an oblique ellipticcylinder and the ground face forms an ellipse with semi-axislengths of a5, b5, a6 and b6 as follows:

a5 ¼ a4;5 sinðb1 þ b2 þ b3 þ b4Þ ð13Þ

b5 ¼ b4;5 ð14Þ

a6 ¼ a5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� C2

C20

sð15Þ

b6 ¼ b5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� C2

C20

sð16Þ

The upper rigid cone could intersect the ground surface,depending on C/D. Based on the theory of elasticity, the potentialcollapsing blocks are upright ellipsoids in the ground of the grav-itational stress field, i.e., if the coefficient of the horizontal pres-sure of the ground K0 is less than 1 (Li and Wu, 1996). The axialratio of the upright ellipsoid is equal to K0. Thus, the vertical lengthof the truncated ellipsoid is assumed to be C0, which is defined asfollows:

C0 ¼ a5K0

ð17Þ

K0 ¼ 1� sinu ðJaky’s empirical formulaÞ ð18ÞIn addition, the area Ai (i = 5, 6) of intersections of the truncated

ellipsoid with the oblique elliptic cylinder and the ground face is asfollows:

A5 ¼ pa5b5 ð19Þ

A6 ¼ pa5b5 1� C2

C20

!ð20Þ

To obtain the resultant force of the distributed force q, which iscaused by the weight of the ellipsoid and the oblique ellipticalcylinder, the volumes V5 of the truncated ellipsoid and V6 of theoblique elliptical cylinder are first obtained as follows:

V5 þ V6 ¼ pa5b5 C � C3

3C20

� a45 cosðb1 þ b2 þ b3 þ b4Þ2

" #ð21Þ

In general, the angles a, h1, h2, h3, and h4 are the five parametersthat define the specific geometry of the collapse mechanism.

3.3. Limit support pressure

To satisfy the stability conditions of the tunnel face according tothe upper bound theorem, the following relation is considered:

Pe 6 Pv ð22Þ

Fig. 8. The causes of failure zones: (a) C/D = 1.5; (b) C/D = 0.5 (Idinger et al., 2011).

Longitude profile Cross profile

vertical slidingzone

lateral slidingzone

slip surface withno friction

slip surface withfriction

Fig. 9. Longitudinal and crosswise profiles of the slip surface.

Fig. 10. Log-spiral shaped sliding wedge adopted in the limit equilibrium method(Murayama et al., 1966).

352 C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357

where Pe represents the power of the external loads and Pv denotesthe dissipation power. The power of the external loads, Pe, is thesum of three components, PT, the power of the support pressurerT, Ps, the power of the surcharge rs, and Pc, the power of the soilunit weight c.

Pe ¼ Pc þ Ps þ PT ð23ÞBy equating the total rate of external work to the total rate of

internal energy dissipation, as shown in Eq. (23), the pressure rT

at the face of the tunnel is obtained by Eq. (23) as follows:

rT ¼ NS � rS þ Nc � cDþ Nc � c ð24Þ

where Nc, Nc and Ns are non-dimensional coefficients that representthe effects of soil weight, cohesion and surcharge loading, respec-tively. For the collapse mechanism, the rate of external work ofthe surcharge loading should only be calculated when the mecha-nism outcrops on the ground surface, i.e., C0 P C.

The numerical results indicate that Nc and Ns are related by thefollowing classical formula when considering mass conservation:

Nc tanuþ 1� Ns ¼ 0 ð25Þ

In Eq. (24), rT, Nc, Nc and Ns depend on the mechanical and geomet-rical characteristics c, u, and C/D and on the angular parameters ofthe failure mechanism a and bi (for 1 6 i 6 4). These parameterswere obtained by maximizing rT in Eq. (24) with respect to theangles a and bi. An upper-bound solution can be found by numeri-cally optimizing Eq. (24) with respect to the five angles.

For the new failure mechanism, the limit support pressure wascalculated from Eqs. (24) and (25) with Eqs. (26) and (27) asfollows:

Nc ¼ P1 þ P2 þ P3 þ P4 þ P5

Dð26Þ

Fig. 11. The new 3D improved failure mechanism.

(a) Outcrop of the mechanism at the ground surface (b) No outcrop of the mechanism at the ground surface.

Fig. 12. Combinations of the improved 3D failure mechanism.

C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357 353

Ns ¼A6 sinð2b1 þ 2b3 � aÞ v4

v1

A1 cosað27Þ

where

P1 ¼ V1 sinaA1 cosa

P2 ¼ V2 sinð2b1�aÞv2v1

A1 cosa

P3 ¼ V3 sinð2b2þaÞv3v1

A1 cosa

P4 ¼ V4 sinð2b1þ2b3�aÞv4v1

A1 cosa

P5 ¼ ðV5þV6Þ sinð2b1þ2b3�aÞv4v1

A1 cosa

8>>>>>>>>>>>><>>>>>>>>>>>>:

Fig. 13. The velocity field of the new 3D failure mechanism.

The above results only apply when the ground surface isreached by the failure mechanism [C0 P C], i.e., when the followingis true:

Y4k¼1

cosðWk�1;k þuÞcosðWk;kþ1 �uÞ sinðb1 þ b2 þ b3 þ b4Þ P 2

CD

ð28Þ

b1 þ b2 þ b3 þ b4 6 p=2 ð29Þ

Fig. 14. Comparisons of the limit support pressures between the present mecha-nism and the numerical simulations.

Fig. 15. Comparisons of the failure zones between the present mechanism and the numerical simulations in sands.

354 C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357

When no outcropping of the mechanism occurs at the groundsurface, i.e., relation (28) reversed, Eqs. (24) and (25) are valid ifC is replaced by C0 in the above equations.

4. Comparisons

4.1. Comparisons of the present mechanisms with the numericalsimulation

The limit support pressures and failure zones obtained fromboth the present failure mechanism and the numerical simulationsare compared in this section.

4.1.1. Limit support pressureThe limit support pressures obtained from the numerical simu-

lations using FLAC3D were compared with those given by the

Fig. 16. Comparisons of the failure zones between the prese

present mechanism in the limit analysis. The results are shownin Fig. 14. In all cases, a maximum difference of approximately19.2% was obtained.

4.1.2. Failure zoneThe failure zones obtained from the numerical simulations

using FLAC3D were also compared with those given by the presentmechanism in the limit analysis. The results are shown in Figs. 15and 16. The failure zones given by the present mechanism agreewith those of the numerical simulation, especially for the high fric-tion angle case.

4.2. Comparisons with existing approaches

To further validate the results obtained from the numerical sim-ulations and the limit analysis developed in this paper,

nt mechanism and the numerical simulations in clays.

C/D=1.0

present mechanismσt =3.8 kPa

experimental mechanismσt =4.2 kPa

Tunnelface

1D

1D

0.6D0.46D

0.3D

0.33D

Fig. 17. Comparison of the limit support pressure and the failure zone between thepresent work and the centrifuge test (Chambon and Corté, 1994).

C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357 355

comparisons between the results of this work and existingapproaches were performed.

Fig. 18. Comparisons of the limit support pressure and the failure zo

4.2.1. Comparisons with a centrifuge testThe results from a centrifuge test (Chambon and Corté, 1994)

were compared with the present mechanism. As shown inFig. 17, the outlines of the failure zone obtained from the presentmechanism are similar to the results of the centrifuge test.Specifically, on the longitudinal profile, the boundary of the failurezone obtained from the present mechanism is approximately0.33D in front of the tunnel face and 0.46D above the tunnel crown.The centrifuge test results show that the boundary of the failurezone is approximately 0.3D in front of the tunnel face and 0.6Dabove the tunnel crown. In addition, the limit support pressureof the present mechanism is equal to 3.8 kPa, while that of the cen-trifuge test is 4.2 kPa. According to the above analysis, the failurezone and limit support pressure given by the present mechanismcorresponds with the results of the centrifuge test.

4.2.2. Comparisons with existing multi-block mechanismsThe outlines of the failure zone obtained from the numerical

simulations (adopting the proposed criteria) were compared withthose given by the present mechanism and by Mollon et al.(2009) using the multi-block mechanisms in the limit analysis.The results are shown in Fig. 18. For the upper-bound solutionsin the limit analysis, higher limit support pressure values indicatebetter solutions (Mollon et al., 2011). The limit support pressureobtained from this present mechanism is greater than thatobtained from the multi-block mechanism, which indicates thatthe present mechanism provides a better solution than themulti-block mechanisms. In addition, the results show that thepresent mechanism is more consistent with the outline of the fail-ure zones that were observed in the numerical simulations.

ne between the present work and the multi-block mechanism.

356 C. Zhang et al. / Tunnelling and Underground Space Technology 50 (2015) 345–357

5. Conclusions

To study the face stability of circular shield tunnels in cohesive–frictional soils, both numerical simulations and limit analyses wereperformed. The limit support pressure and failure zone of the tun-nel face for the limit conditions are the main concerns. A series of3D numerical simulations in different soils (clays or sands) for var-ious relative depths (C/D) were first performed. Considering theresults of the numerical simulation, a new 3D failure mechanismwas proposed in the framework of the kinematical approach oflimit analysis theory. Then, the limit support pressures and failurezones obtained from both the present mechanism and the numer-ical simulations were compared. Moreover, the comparisonsbetween the results of the present study and those of existingapproaches were provided. The main conclusions are presentedas follows:

(1) Numerical simulations were performed with FLAC3D toinvestigate the tunnel face stability as the support pressuregradually decreased. The limit support pressures of a tunnelface in different soils (clays or sands) for various relativedepths (C/D) were obtained. Based on the displacementclouds for the limit conditions, a simple and feasible crite-rion was suggested to outline the boundary strip of the fail-ure zone at collapse in displacement clouds under differentconditions. These results serve as a reference for exploringsuitable failure mechanisms in the limit analysis in cohe-sive–frictional soils.

(2) Based on the numerical simulation, a new 3D failure mech-anism was proposed in the framework of the kinematicalapproach of limit analysis theory. The new 3D failure mech-anism is composed of four truncated cones and a distributedforce acting on those truncated cones. The distributed forceis caused by the weight of the ellipsoid-shaped (or truncatedellipsoid-shaped) blocks.

(3) The support pressure and failure zone results of a tunnel facederived from the present mechanism using limit analysis arecompared with those obtained from the numerical simula-tions using FLAC3D. A maximum difference of approximately19.2% is observed for all cases. Moreover, the failure zonesare compared, and the results corresponded with oneanother.

(4) The failure zone predicted by the present mechanism corre-sponds with that of a centrifuge test (Chambon and Corté,1994). Specifically, in the longitudinal profile, the boundaryof the failure zone obtained from the present mechanism isapproximately 0.33D in front of the tunnel face and 0.46Dabove the tunnel crown. The centrifuge test shows that theboundary of the failure zone is approximately 0.3D in frontof the tunnel face and 0.6D above the tunnel crown. In addi-tion, the limit support pressure of the present mechanism isequal to 3.8 kPa, while that obtained from the centrifugetests is equal to 4.2 kPa.

(5) The outlines of the failure zone obtained from the numericalsimulations (adopting the proposed criteria) were comparedwith those given by the present mechanism and by theexisting multi-block mechanism in the limit analysis. Forthe upper bound solutions in the limit analysis, the highervalue of the limit support pressure indicates that the solu-tion is better. The limit support pressure obtained from thepresent mechanism is higher than that obtained from theexisting multi-block failure mechanisms, which shows thatthe present mechanism provides a better solution than theexisting multi-block failure mechanisms. In addition, theresults show that the present mechanism is substantially

more consistent with the shapes of the failure zones thatare observed in the numerical simulations than the existingmulti-block failure mechanisms, which shows that the pre-sent mechanism is more effective and reasonable.

Acknowledgments

The authors acknowledge the financial support provided by theNational Natural Science Foundation of China (Grant Nos.51008015, 51378002), and the Program for New CenturyExcellent Talents in University of China (Grant No. Ncet-12-0770).

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