Analysis of face stability of circular tunnel in purely cohesive soils driven by shield

Analysis of face stability of circular tunnel in purely cohesive soils driven by shield Jinhui Liu, Wantao Ding, Mingbin Wang School of civil engineering, Shandong Jiaotong University, Jinan, China 2. Research Center of Geotechnical and Structural Engineering, Shandong University, Jinan, China. 3. School of Qilu Transportation, Shandong University, Jinan, China. 4. School of Civil Engineering, Ludong University, Yantai, China.


Introduction
The advanced shield techniques (earth, slurry or air shield) are widely used to construct a shallow tunnel in soft ground.Nevertheless, if the support pressure acting on the tunnel face is not sufficient to balance the external earth and water pressure, face collapse might occur.Hence, the failure mechanisms and the limit support pressures continue to be an important research issue in stability analysis of the tunnel face.
The face stability of shallow circular tunnels driven in cohesive soils has been investigated by several authors.Based on laboratory extrusion tests and field observations, Broms and Bennermark (1967) first defined a stability ratio N , which is expressed as , where s  is the possible surcharge loading on the ground surface, T  is the uniform pressure applied to the tunnel face,  is the soil unit weight, C is the depth of cover, D is the diameter of the circular tunnel, and u c is the soil undrained cohesion.
Subsequently, Davis et al.(1980) proposed a more theoretical approach derived from the limit state design concept.Schofield(1980) conducted a centrifuge model test and proposed a general expression for the stability ratio N that is dependent on tunnel depth.Kimura and Mair (1981)   performed a centrifuge test and noted that the limit support pressure depends on the tunnel cover for 5 10 N  .Idinger et al.(2011) investigated the face stability of shallow tunnel using the centrifuge model test and reported the arching effects of soil occurring at an overburden ratio of C/D=1.0.Based on a limit equilibrium analytical method, Ellstein(1986) presented an analytical solution for N for homogeneous cohesive soils, which verified the results of Kimura and Mair   (1981).Based on classical plasticity theory, Augarde et al.(2003) derived rigorous bounds on load parameters using the finite element limit analysis method and reported that the stability ratio N used to analyse the stability of an idealized heading in undrained soil conditions is not rigorous.Lee et al.(2006) carried out a series of centrifuge model tests and numerical simulations of these tests to study the tunnel stability and arching effects that develop during tunnelling in soft clayey soil and proposed the boundaries of the positive and negative arching zones.More recently, the kinematic methods in the field of limit analysis using continuous velocity fields have been proposed by researchers.Based on an admissible continuous velocity field obtained directly from elasticity theory, Klar et al.(2007) investigated the face stability analysis of circular tunnels in purely cohesive soils and proposed 2D and 3D upper-bound solutions for this problem.Mollon et al.(2013) proposed two new continuous velocity fields to analyse the collapse of tunnel face and noted that these velocity fields agreed with the actual failures observed in undrained clay.
For the face stability analysis of shallow tunnels, the rigid multi-blocks failure mechanisms were often adopted to solve the collapse pressure of the tunnel face.Davis et al.(1980) derived upper and lower bound stability solutions for collapse under undrained conditions.Leca et al.(1990) obtained three upper bound solutions from the consideration mechanisms based on the motion of rigid conical blocks in a frictional material.Mollon et al.(2009Mollon et al.( ,2010Mollon et al.( ,2011) ) improved these mechanisms and obtained solutions relatively close to those of real projects in cohesive and frictional soils.More recently, several authors adopted numerical and experimental methods to evaluate the collapse pressure and proposed results closer to those of actual situations.Chen et al.(2011Chen et al.( ,2013) ) analysed the face stability of shallow shield tunnels in dry sands using 3D DEM and 3D FDM and proposed a two-stage failure pattern based on the observation of earth pressure.Based on the results of smallscale model tests under normal gravity (1 g), Kirsch(2010) proposed that the overburden and the initial soil density do not influence the required support pressure.Considering the seepage force around the tunnel face, Lu et al.(2014) investigated the relationship between the support pressure and displacement of the shield tunnel face using a 3D FEM.For layered soils, Senent et al.(2015) improved the model proposed by Senent et al.(2013) to analyse the possibility of a partial collapse occurring on the tunnel face.Zhang et al.(2013Zhang et al.( ,2015) ) developed a 2D DEM model to analyse the behaviours of cohesive-frictional soils when tunnelling using a slurry shield and proposed a new 3D analytical model with four truncated cones on which a distributed force acts based on the results of numerical simulations.
This study aims at a face stability analysis of the purely cohesive soils in the framework of the kinematic approach of limit analysis theory.The rigid block failure mechanisms are a simple and intuitive approach and are either translational or rotational.Though the shapes of the blocks in the failure mechanisms well satisfy the normality condition proposed by Chen(1975), they may conflict with the yield criterion of the soils.Many three-dimensional analytical models and numerical simulations have also been constructed to analyse this problem using the finite element limit analysis method.However, the progressive failure of the tunnel face cannot be explained using these methods.
While the problem of tunnel face stability is inherently three-dimensional, much can be learned from the behaviour of a reasonable two-dimensional model.Therefore, considering the changes in the soil pressure state and the influence of the vertical soil arching effect in the failure zone, we proposed a new 2D failure mechanism and constructed an admissible continuous velocity field to analyse the critical collapse face pressure of purely cohesive soils using the slip-line and limit analysis theories.

Slip-line theory
The yield condition of the soils is often evaluated using the Coulomb criterion.Considering both the equations of equilibrium and the yield condition, a set of solutions of the plastic equilibrium in the yielded zone can be developed.Combined with the stress boundary conditions, the stresses in the yielded zone in front of the tunnel face can be studied using these solutions.For specific problems, the slip-line field can be established using the slip-line method, which means that the stress-strain relationship of the soil is ignored, and only the equilibrium and yield conditions are considered.Thus, only a partial stress field region is constructed, and occasionally, the solution can be invalid outside the partial stress field region.To solve this problem, a reasonable extended stress field is constructed in this study.Therefore, considering the theory of limit analysis, accurate solutions for the collapse face pressures of the tunnel can be obtained.

Limit analysis theory
In the limit analysis, the soil is idealized as a perfectly elastic-plastic material that obeys normality conditions (or the associated flow rules).Based on the limit analysis, the associated theorems are established.Much experimental evidence validates that this assumption is reasonable for many clays.
Thus, the critical collapse pressure of a tunnel face can be examined theoretically using the upperbound theorems for purely cohesive soils.The upper-bound theorem states that if a kinematically admissible velocity field can be found and if uncontained plastic flow previously occurred, then the deduced loads will be higher than or equal to those associated with collapse.
The aim of this study is to use the kinematic method of the limit analysis to study the face stability of purely cohesive soils.As the internal friction angle  of purely cohesive soils is zero, plastic deformation in a purely cohesive soil develops without any volume change and satisfies the normality condition.Thus, the failure surfaces are assumed to follow the stress characteristics, which are also the velocity characteristics for purely cohesive soils.The actual sliding surface is characterized by a velocity and a stress.

Admissible continuous translational velocity field
The yield condition of the soils is often evaluated using the Coulomb criterion.A purely cohesive soil can be regarded as a rigid plastic Tresca material, as its internal friction angle  is zero.
Thus, the Mohr circle of stress of a purely cohesive soil is shown in Figure 1.

Fig. 1 Mohr circle of stress of purely cohesive soil
Based on the slip-line theory, when the material is in the plastic state, two orthogonal shear planes exist at each point for plane strain problems, as shown in Figure 2.

Fig. 2 Two possible shear planes at a point
This study focuses on the reasonable failure mechanism and the collapse critical pressure of the shield tunnel under the plane strain heading (as shown in Figure 3) for purely cohesive soils.
Furthermore, the stress characteristics and the velocity characteristics are assumed to be identical.To construct a reasonable stress field, that is, an admissible continuous velocity field, the following assumptions are made.
(a) There is no shear force on the tunnel face; (b) The effect of the Terzaghi level arching effect on the upper soils of the lining in the rear of the tunnel face is ignored when 0.5 CD , where C is the depth of cover, and D is the tunnel diameter; (c) The soils at the top of the tunnel face is in the plastic state when tunnelling, that is,  + , where s  is the possible surcharge loading acting on the ground surface,  is the soil unit weight, C is the depth of cover, and u c is the soil undrained cohesion.
Based on these assumptions, a new 2D failure mechanism is proposed, as shown in Figure 4.The level of soil stress in the front of the tunnel face is analysed.

Fig. 4 A new 2D failure mechanism
The mechanism consists of four zones, i.e., Zone I, II, III and possibly IV.For failure zone I, when the support pressure T  acting on the tunnel face is not sufficient to ensure the stability of the tunnel face, the tunnel face can be regarded as the moveable vertical surface of failure zone I.The soil in zone I is in the limit state of the active earth pressure.Thus, zone I is a Rankine zone.For zone II, as the soils in zone II are constrained by the soils in zone I, zone III, and the elastic area in front of zone II, the stress principal axis in zone II should deflect to different degrees based on the associative flow rule(see.Chevalier,2013).Furthermore, the soils in zone II are in the plastic state and satisfy the Tresca yield criterion.Hence, zone II is a transition zone formed by a circular arc with radius 22 D and two adjacent boundaries (see Chen, 1975).Due to the assumption that the soils behind line EF are stable (i.e., the Terzaghi horizontal arc effect is ignored), deflection of the stress principal axis does not occur in zone III, in which the soils are in the plastic state.Therefore, zone III is also a Rankine zone.Zone IV is a possible failure zone.When the soils in zone IV are in a plastic state, zone IV can be regarded as a Rankine zone but is affected by the vertical soil arching effect occurring at the top of zone III.It must be determined whether a failure area appears in zone IV.
Based on the slip-line and limit analysis theories, together with considering the normality conditions and Tresca yield criterion, we construct an admissible continuous velocity field, as shown in Figure 5.In Figure 5, the line AB and the arc BE represent the maximum shear stress trajectory of the soils and are the stress boundaries of the plastic and elastic soil regions.The lines EJ and FK are the potential trajectories of the maximum shear stress when the soil in the yielding zone IV.
Moreover, the stability of soils in zone IV needs to be assessed.

Fig. 5 Admissible velocity field
In this study, the motion of the soils in failure zones I, II and III is assumed to be translational.
The fracture zone IV is assumed to respect Terzaghi's theory of relative soil pressure.Thus, a kinematically admissible velocity field is constructed, as shown in Figure 6; the admissible displaced pattern is shown in Figure 7.
The kinematic velocity magnitude at points B, E and F is given by the following: The kinematic velocity magnitude of the tunnel face can be written as follows: and III.In Figure 6, the velocity discontinuity surfaces consist of line AB , line OF and the circular arc BE .

Criterion of the limit collapse thickness
In this study, there are two possible failure patterns in front of the tunnel face, as shown in Figure 4.The shapes of the failure mechanism depend on the ratio CD .When CD is 0.5, the failure mechanism consists of three zones, i.e., zones I, II and III.When the values of CD are higher than 0.5, zone IV (i.e., the trapezoid EFJK with height H ) may be the possible failure zone, which is subjected to the influence of the vertical soil arching effect, as shown in Figure 8.The shape of the fracture mechanism is consistent with that of a test obtained by Schofield (1980).When the soil mass in trapezoid EFJK (i.e., zone IV) is unstable, zone IV becomes a possible fracture zone.In this case, the criterion of the limit collapse thickness needs to be defined to obtain the critical collapse pressure.Assuming that the collapse of the tunnel face just reaches the ground surface, the soils in zone IV can be regarded as loose earth, and the collapse pressure acting on line EF can be obtained using the Terzaghi's theory of relative soil pressure, which is modified as shown in Figure 9.
where C is the depth of cover.Equation ( 6) is the criterion of the limit collapse thickness.

Analysis of the limit collapse thickness
No uniform surcharge loading s

 on the ground surface
Neglecting the uniform surcharge loading, equation ( 6) is simplified as follows: , the soil mass in trapezoid EFJK is stable.Therefore, the fracture areas in front of the tunnel face involve zones I, II and III, and the limit collapse thickness is 0.5 D ; otherwise, it is the depth of cover, C ( H+D/2 ).An intuitive design chart to evaluate the limit collapse thickness is shown in Figure 10.EFJK is stable.Therefore, the fracture areas in front of the tunnel face involve zones I, II and III, and the limit collapse thickness is 0.5 D ; otherwise, it is the depth of cover, C ( H+D/2 ).A design chart is necessary to more intuitively evaluate the limit collapse thickness, as shown in Figure 11.


) is under or on the curve generated from equation (6).Therefore, the limit collapse thickness is equal to 0.5 D ; otherwise, it is equal to C ( H+D/2).

Critical collapse pressure
This paper aims at finding a more accurate critical collapse pressure of a circular tunnel driven by a shield in purely cohesive soils.Based on the kinematic approach of the limit analysis, together with the slip line theory a multi-zone failure mechanism is proposed.As described in the above section, the shape of the failure mechanism depends on the values of CD.As the critical collapse pressure depends on the failure mechanism, the critical collapse pressure of the tunnel face is calculated as follows.
C/D equal to 0.5 Figure 12 shows the fracture mechanism when C/D is 0.5.

CD=
This fracture mechanism consists of three zones.To obtain the critical collapse pressure using the limit analysis method, the rate of external work and that of the internal energy dissipation must be determined.The rate of external work is caused by the weight of the three zones, the possible uniform surcharge loading on the ground surface and the collapse pressure on the tunnel face.The rate of the internal energy dissipation is determined along the different velocity discontinuity surfaces.According to the analysis of the admissible continuous velocity field (see Figure 6), the rate of external work and that of the internal energy dissipation are calculated as follows.
The rate of work of the weight of the three zones, i.e., zone I, II and III, is shown below: The rate of work of the collapse pressure of the tunnel face is shown below: As undrained behaviour is assumed, the soil deforms at constant volume, as shown below.region i ( i =1,2,3), s n v is the downwards normal velocity at the ground surface, and T n v is the outward normal velocity on the tunnel face.

T A and
s A are the deforming areas on the tunnel face and at the ground surface, respectively.
The rate of internal energy dissipation along the different velocity discontinuity surfaces is shown below: where i S is the lateral surface along line AB , line OF and arc BE .The rate of the internal energy dissipation along the discontinuous surface is given as follows: The work equation consists of equating the rate of work of the external forces to the rate of internal energy dissipation and is given as follows: After simplification, the tunnel collapse pressure can be written as follows: where N  , c N , and s N are non-dimensional coefficients that represent, respectively, the effect of the soil weight, the cohesion, and the surcharge loading.The expressions of these coefficients are given as follows: Substituting equations ( 16), ( 17) and ( 18) into equation ( 15), the critical collapse pressure becomes as follows:

 += (20)
C/D greater than 0.5 Figure 13 shows the fracture mechanism when C/D is greater than 0.5.

CD
This fracture mechanism involves four zones.To obtain the critical collapse pressure of the tunnel face, the rate of work of the weight of the four regions, the possible uniform surcharge loading on the ground surface and the collapse pressure on the tunnel face must be determined.In addition, the rate of internal energy dissipation along the different velocity discontinuity surfaces must be obtained.According to the analysis of the admissible continuous velocity field (see Figure 6), the rate of external work and that of the internal energy dissipation are given as follows.
The equation for the rate of work of the weight of the three zones, i.e., zones I, II and III, is the same as in equation ( 8).The contributions of the weight of the soils in zone IV, the uniform surcharge  and the soil undrained cohesion u c can be expressed as follows: where q n v is the downwards normal velocity of the equivalent uniform distribution loads V q acting on line EF , and q A is equal to the length of line EF multiplied by unit thickness.
The equation for the rate of work of the collapse pressure of the tunnel face is the same as in equation ( 9).
The equation for the rate of internal energy dissipation along the different velocity discontinuity surfaces is the same as in equation ( 13).The work equation is obtained by equating the rate of work of the external forces to the rate of internal energy dissipation as follows: After simplification, it is found that the tunnel collapse pressure is given by where N  , c N and s N are non-dimensional coefficients that represent the effect of the soil weight, the cohesion, and the surcharge loading, respectively.The expressions of these coefficients are given as follows: zone IV; and 1 s N = is the effect of a possible uniform surcharge loading on the ground surface.
Note that in equations ( 24), ( 25) and ( 26  and CD .Thus, according to equation ( 6), the critical collapse pressure as a function of CD is calculated as follows: (a) N   and c N are both equal to zero.Thus, equations ( 24) and ( 25) are reduced, respectively, to the following: In this case, the equivalent uniform distribution load V q is less than or equal to zero.Thus, equation ( 26 24), ( 25) and ( 26) into equation ( 23), the critical collapse pressure is calculated as follows: When T  = 0, equation (32) becomes as follows: 4 (0.0138 ) 3.0708 ln( 1)

Analytical comparisons
In this study, it is assumed that the undrained shear strength u c of a purely cohesive soil is constant with depth.Thus, six variables model of the plane strain heading is formed by the following set: As the present model in this paper is 2D, the existing classical solutions of two-dimension models or some recent three-dimension models, including analytical and numerical solutions, are chosen to validate the results of the present model.For the cohesive soils, a traditional approach is to evaluate stability of a circular tunnel face in terms of a "load factor", usually denoted as N and defined as follows: The advantage of using the load factor to assess stability can reduce the complexity of the final results.However, several researchers demonstrated that the values of In this study, the load factor can be expressed as follows: 3+ 2 ln(4 1) The analytical comparisons of the load factor of the present model and the existing solutions are shown in Figure 14.

Fig. 14 Analytical comparisons of N of the present model and the existing solutions
Figure 14 shows that the reasonable solutions of the problem should be between the upper-bound solutions proposed by Davis et al.(1980) and the lower-bound solutions proposed by Davis et al.(1980).In Figure 14, the upper-bound solutions proposed by Sloan et al.(1994) and by Augarde et al.(2003) are numerical solutions obtained using finite element limit analysis methods based on the improved rigid-blocks mechanism proposed by Davis et al.(1980).The upper-bound solutions proposed by Augarde et al.(2003) were derived from updating the upper-bound solutions proposed by Sloan et al.(1994) using advanced algorithms of the finite element limit analysis.The upperbound solutions proposed by Augarde et al.(2003) are more accurate bounds on stability parameters.Therefore, the range of best solutions of the problem should be reduced to between the upper-bound solutions proposed by Augarde et al.(2003) and the lower-bound solutions proposed by Davis et al(1980).The comparisons of the upper-bound solutions proposed by Davis et al.(1980), Sloan et al.(1994) and Augarde et al.(2003) show that the best upper-bound solutions more closely approximate the lower-bound solutions.In the lower-bound solutions shown in Figure 14, the three lower-bound solutions proposed by Davis et al.(1980), Augarde et al.(2003), Ewing and Hill(1967) and the present model are calculated based on smooth linings.Only the lower-bound solution proposed by Gunn(1980) is computed based on rough linings.The lower-bound solutions based on rough linings would be appropriate in a purely cohesive soil.Thus, the reasonable upper-bound solutions more closely approximate the lower-bound solutions proposed by Gunn(1980).It can be seen from Figure 14  For the more general case of a cohesive soil, the traditional method used to assess the critical collapse pressure is using the following equation Since the critical coefficient s N is equal to 1.0 for purely cohesive soils under undrained conditions, equation (37) becomes The equation ( 38) can be transformed into  .This is in conflict with the real situation in the actual project.
Based on the research data proposed by Kimura et al.(1981), Ellstein et al.(1986), Augrade et al.(2003), Klar et al.(2007), Mollon et al.(2013) and the present model, the fitting equations of the critical coefficients Nc versus the ratio of the cover to the depth (C/D) can be expressed as ln( 41) Where A and B are constant coefficients which are showed in Table 1.18 shows that the size of the fracture zone obtained using the present model is much smaller than that obtained by Davis et al. (1980).This is because this study considers the influence of the vertical soil arching effect and the normality conditions, whereas the fracture mechanism proposed by Davis et al. (1980) neglects the normality condition.The fracture mechanism of the present model is consistent with the experimental result obtained by Schofield (1980), as shown in Figure 18(b).Based on the limit analysis and the slip-line theories and together with considering the normality condition and yield criterion, the reasonable size of the fracture zone in front of the tunnel face is obtained in this study.The proposed failure mechanism significantly improves the existing upper-bound solutions for the face stability of circular tunnels in purely cohesive soils.

Conclusions
Based on the kinematic approach of the limit analysis and slip-line theories, a new 2D analytical

Fig
Fig. 3 The plane strain heading

Fig. 7
Fig. 6 Kinematically admissible velocity field vertical velocity magnitude of the centroid of zones I, II, and III can be separately expressed as follows: are respectively the vertical velocity of the centroid of zones I, II, .preprints.org) | NOT PEER-REVIEWED | Posted: 26 November 2018 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 26 November 2018 doi:10.20944/preprints201811.0582.v1

PreprintsH
Fig. 9 Modified Terzaghi's theory of relative soil pressureThe contribution to the collapse pressure caused by the weight of the soils in trapezoid EFJK , 14

Fig
Fig. 10 Design chart for the relationship between


Fig. 11 Design chart for the relationship between Fig. 12 Failure mechanism for

V
rate of work of possible uniform surcharge loading on the ground surface is shown below: is the volume of region i , i v is the vertical velocity of centroid i in the volume of


loading s on the ground surface and the soil undrained cohesion u c to the critical collapse pressure are equivalent to that of the equivalent uniform distribution loads V q acting on line EF .Therefore, the rate of work of the weight of the soils in zone IV, the possible uniform surcharge π is the possible effect of cohesion of the soils in Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 26 November 2018 Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 26 November 2018 doi:10.20944/preprints201811.0582.v1 ), whether to consider the values of N   ,

N
must be considered.Substituting equations (

.
 .Dimensional analysis is used to solve the problem via the following dimensionless sets: As undrained behaviour is assumed, the first two in the dimensionless set are replaced by a single parameter,() that the upper-bound solutions proposed by the present model agree well with the lower-bound solutions proposed by Gunn(1980) when 4 CD .The upper-bound solutions obtained by the present model are better than other upper-bound solutions.The present upper bound mechanism more closely approximates the failure mechanism in the actual project and a safe estimate of the tunnel pressure.
self weight of the soil.

Fig. 15
Fig. 15 Analytical comparisons of N between the present model and the existing solutions (γD/cu>0) Figure 15 shows that the upper-bound solutions proposed by Davis et al.(1980), Augarde et al. (2003) and Klar et al.(2007) are than the experimental and analytical solutions obtained by Brom et al.(1967), Kimura et al.(1981) and Ellstein et al.(1986) for values of C/D from 0.5 to 3.0 when 0 u Dc  Fig. 16 Comparison of Nc,Nγ and Ns of the existing models and the present model: (a) Nc-Nγ-Ns;(b) Nγ

Figure 18
Figure18compares the failure zones of the present mechanism and that proposed byDavis et al.

Table 1
Constant coefficients and coefficients of determination of fitting equations