Rotational Failure Mechanism for Face Stability of Circular Shield Tunnels in Frictional Soils

Ph.D, Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210098, China Master Student, Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210098, China Ph.D, East China Electric Power Design Institute of China Power Engineering Consulting Group, Shanghai 200063, China Ph.D, Senior Engineer, Nantong City Construction Group Co., Ltd., Nantong 226000, China


Introduction
Shield machines have been widely used for excavations of tunneling in loose grounds above or below the groundwater table.In the built-up areas, the tunnel face has to be adequately supported to prevent the excessive settlement at the ground.Different support methods are utilized in practice, such as compressed air, bentonite slurry, and earth pressure.Face stability analyses are often conducted to determine the required support pressure based on the limit equilibrium (LE), the limit analysis (LA), and the finite element (FE) or the finite difference (FD) methods.e face of most shielddriven tunnels is circular, and its stability should be analyzed in three-dimensional (3D) conditions.e numerical FE/FD method has advantage on the complex geometry of 3D problems, especially for modelling.
e analytical LE/LA methods need to develop a 3D failure mechanism of the tunnel face in advance.e criticality of their calculated support pressure depends on the developed mechanism.
erefore, the construction of the 3D mechanism is of importance in the stability analysis of tunnel faces.
Horn [1] earlier proposed a wedge-prism model based on the silo theory and then solved LE equations to yield the limit support pressure.
is model assumes the circular tunnel face as a square with slides corresponding to the tunnel diameter.It is further improved by Anagnostou and Kovári [2,3] and Chen et al. [4] to involve the influences of pore water pressures (PWP) and soil arching effects.Compared with the LE solutions, the LA method based on the theorem of soil plasticity is more rigorous to yield the upper or lower bound of the limit load.Numerous 3D stability analyses of tunnel faces have been conducted in the LA, especially for the upper-bound (UB) solutions.Unlike the lower-bound solutions, UB solutions are more critical for the determination of the support pressure on the tunnel face against the active collapse or passive blowout.To obtain the upper bound, a kinematically admissible velocity field needs to be proposed for the material with associated flow rule.For purely cohesive soils, Klar et al. [5] used the Tresca yield criterion to obtain the closed-form solution for the 3D deformation field.Mollon et al. [6] established continuous velocity fields, which can better estimate the upper bounds.However, it is difficult to present a velocity field for frictional soils obeying the Mohr-Coulomb yield criterion.Leca and Dormieux [7] proposed a 3D failure mechanism consisting of one or two conical blocks, and then Mollon et al. [8,9] modified it using a multiblock model to obtain more critical support pressures.Many researchers [10][11][12][13][14][15][16][17][18][19][20][21] further developed the mechanism and investigated some effects, such as seepage forces, reinforcement, nonhomogeneous soils, and longitudinal gradient of tunnels.However, these mechanisms are limited to translational failures in frictional soils.
With the help of the advanced measurements and image processing techniques, such as X-ray and CT scanners [22], digital image correlation (DIC), or particle image velocimetry (PIV) [23][24][25][26][27], the 3D failure behavior of the tunnel face can be clearly recorded in the experimental tests.e observed results are useful for improvements of the failure mechanism.
e measured results of Takano et al. [22] demonstrate the rotational movements for tunnel face failures in frictional soils.Subrin and Wong [28] considered the rotational movements of the tunnel face in cohesive frictional soils and proposed a rotational failure mechanism defined by two log spirals.e derived UB solutions of the support pressure have good agreements with the FE results of Wong and Subrin [29] and the experimental results measured by Berthoz et al. [30].Similar to the translational failure mechanisms of Leca and Dormieux [7] and Mollon et al. [8], the area inscribed to the circular tunnel face has an elliptical shape in the rotational mechanism, which cannot cover the entire face.e striking results on the failure surface are not consistent with the observed results in the experimental tests of Takano et al. [22].Recently, Mollon et al. [9] proposed a new rotational failure mechanism for circular faces of tunnels in frictional soils.A spatial discretization method is adopted to make the failure surface entirely passing through the circular tunnel face.e presented comparisons demonstrate that the presented mechanism can yield better upper bounds to the support pressure in the LA.
ese developments indicate the efficiency of the presented discretization method on the 3D geometrical construction.
e purpose of this study is to improve the rotational failure mechanism of Subrin and Wong [28] for circular faces of tunnels in frictional and/or cohesive soils.e cross section of the 3D mechanism is modified as an ellipse, to avoid the inscribed elliptical area to the circular tunnel face.Using the kinematic LA method, the least UB solutions of the support pressure are obtained through an optimization procedure.e PWP effects are also included in the stability analyses.Comprehensive comparisons are made to verify the criticality of the improved 3D mechanism.Finally, these calculated UB solutions are presented in some stability charts for convenient use in practice.

Three-Dimensional Rotational
Failure Mechanism e simple Mohr-Coulomb failure criterion is adopted for the shear strength of soils, and the cohesion and internal friction angle can be denoted as c and φ, respectively.ere is a support pressure (σ tc ) to prevent the tunnel face from active collapse.e support pressure is constant for compressed air.A rotational failure mechanism is proposed here to estimate the required support pressure against the tunnel face collapse.e face collapse only occurs below the ground surface.e case of the face blowout is not considered in this study.

2.2.
Geometry of the Failure Surface.In the framework of limit analysis, an upper bound of the support pressure on tunnel faces can be obtained from the kinematic approach.A kinematically admissible velocity field describing the possible failure must satisfy both of the flow rule associated with the yield condition and the velocity boundary conditions.e failure of the circular tunnel face is axisymmetric in 3D conditions.Drescher [36] presented the characteristic relations of the axisymmetric velocity field, in which the velocity jump across a strong discontinuity is inclined at an angle φ.
e velocity discontinuity satisfying the requirements is a straight line or a log spiral.e straight line is often used for translational failure mechanism in cohesionless soils, such as the classical wedge model generated by Horn [1] and the conical block model proposed by Leca and Dormieux [7].e log spiral is more suitable for the rotational failure mechanism in frictional soils, as presented by Subrin and Wong [28] and Mollon et al. [37].erefore, the log-spiral line is also employed here to construct the rotational failure mechanism in 3D conditions.e profile of the failure mechanism on the symmetry plane is bounded by two log-spiral lines, EF, and EF ′ : where the radius r 0 and angle θ 0 are illustrated in Figure 1. e cross section of the failure mechanism is assumed as an ellipse, which can make the cross section on the vertical tunnel face consistent with the entire circular tunnel face.
e ellipse can be expressed as where a local coordinate system (x, y) is introduced for each cross section and shown in Figure 1. e x-axis is perpendicular to the plane of the symmetry plane.
e parameters a and b are the semimajor and semiminor axes of the ellipse, respectively.e shape of the failure surface is generated by rotating an ellipse of increasing semiminor axis b about an axis passing through point O outside the ellipse.
e centerline of the curvilinear cone from the rotating axis, r m is e increasing semiminor axis b can be expressed as Using an eccentricity e, we can obtain the semimajor axis of the ellipse as In the zone I next to the tunnel face (Figure 1), the rotating angle θ ranges from θ TU to θ TD .e cross section of the failure mechanism on the tunnel face must be identical to the circle with tunnel diameter D. Based on the relationship, the eccentricity of the elliptical cross section can be obtained as where parameters d 1 and d 2 are Advances in Civil Engineering Angles θ TU and θ TD are shown in Figure 1.From the geometrical and trigonometric relations, the radius r 0 and the angle θ TD can be determined by the angles θ 0 and θ TU as e eccentricity of zone I is defined here, in order to make the interface of the tunnel face and the rotating plane conform to a whole circular face.For the remaining zone II of the failure mechanism (θ 0 ≤ θ < θ TU ), the ellipse in the cross section is assumed to gradually degenerate as a circle at θ � θ 0 , where the eccentricity e � 0.
e eccentricity is varying with the angle θ as where e 0 is the eccentricity of the elliptic cross section at θ � θ TU , which makes the surface smooth.Parameter η is the coefficient of the power function.When η � 1, the eccentricity decreases linearly with angle θ.

Upper-Bound Solutions of Support Pressure on Tunnel Face
3.1.Formula.Using the kinematically admissible velocity field, we can obtain an upper bound on the limit support pressure of tunnel faces by equating the rate of work done by the external forces to the internal rate of dissipation.In the failure mechanism, the rigid body with the shape of curvilinear cone rotates about the axis passing through point O. e linear velocity v can be expressed as where ω is the angular velocity and the infinitesimal volume element is en, the work rate of the soil weight can be written as e overall failure mechanism is divided into two parts, as the denoted zone I and zone II in Figure 1.Using equation ( 13) into the two parts, we can calculate the corresponding work rate of the soil weight, W c1 and W c2 , respectively.eir expressions are given as where angle θ TC can be obtained from the geometrical relation as Following the same procedure, the rate of work done by the support pressure can also be calculated over the circular tunnel face and expressed as e rate of internal energy dissipated over the velocity discontinuity surface S can be calculated as where the infinitesimal surface element of the failure surface is expressed as For the two parts of the failure surface, the rate of work dissipation D s1 and D s2 can be, respectively, written as Based on the energy balance, the limit support pressure can be determined and expressed in a dimensionless form: 4 Advances in Civil Engineering where the coefficients N c and N c can be written as When the PWP influences are included to obtain the support pressure, the rate of work due to the seepage and buoyancy forces within the soil volume should be included in the energy balance.As presented by Michalowski [38] and Michalowski and Nadukuru [39], the rate of work done by PWP can be reduced because of the assumption on the rigid body of the soil mass as where u is the magnitude of the PWP along the failure surface.During the excavation of the shield-driven tunnel, the distribution on PWP is complicated.To investigate the effects of the PWP on the support pressure, an approximate method is employed here to estimate the PWP using a coefficient r u , which is defined by Bishop and Morgenstern [40] as where h w is the depth of the failure surface below the groundwater table.Using equation (22), we can calculate the rate of work done by PWP for part I and II, respectively: Including the rate of work done by PWP into the energy balance, we can obtain the support pressure as where the coefficient N u is 3.2.Optimization Procedure.For a given problem (i.e., tunnel diameter D, cover depth C, soil density c, soil cohesion c, internal friction angle φ, and the possible PWP coefficient r u ), using the balance equation, we can find the maximum value of the support pressure against the face collapse through an optimization procedure.e independent variables describing the 3D failure surface are angles θ 0 and θ TU .In the search for the maximum value of σ tc , the optimization procedure is carried out until the band widths of the angles reduce to 0.01 °.In addition, the coefficient η defined in equation ( 10) is selected as η � 10 in the calculations.
rough numerous tests, when the coefficient η is more than 5, the difference is negligent, as shown in Figure 2.

Comparisons with the Existing Results
. Various mechanisms have been developed to determine the upper bounds of the support pressure.ese UB solutions are also given here for validating the presented failure mechanism of this study.For the case of the cohesive frictional soils shown in Figure 3(a), the result derived from this study is more critical than other UB solutions.
e required support pressure remains constant with the ratio of C/D.Because of the arching effects, the induced collapses are not reaching the ground surface, i.e., H e < C. e same examples are also used by numerical FD methods [14,41], and their results are given for comparisons, as shown in Table 1.e numerical results are smaller than those of this study.Based on the observed geometry of the numerically obtained failures, Zhang et al. [14] improved the existing multiblock mechanism of Mollon et al. [8] and then obtained more critical support pressure than that given by Mollon et al. [8].However, the improved multiblock mechanism is still less critical than the presented rotational mechanism of this study.
For the purely frictional soil, it can be seen from Figure 3(b) that the calculated pressures are larger than other results for φ � 20 °, but very closer to the results of Mollon et al. [37] for φ � 40 °.It indicates that the presented mechanism can better estimate the UB solution.e wedge model proposed by Horn [1] is often used to estimate the required support pressure of tunnel face in the cohesionless soil.Chen et al. [4] considered the soil arching effects and then improved the classic mechanism.
e limit support pressure is calculated by the LE method for different ratios of C/D, as shown in Figure 4(a).Because the present mechanism is not allowed to reach the ground surface, the obtained support pressure is independent on the cover depth.It can be seen that UB solutions of this study are close to the LE   solutions for C/D 4. Figure 4(b) illustrates the UB solutions derived from various mechanisms for purely frictional soils with di erent friction angles.e presented mechanism of this study yields the least upper bound on the support pressure.e di erences of the UB solutions between this study and Mollon et al. [37] decrease with the increasing friction angle.Pan and Dias [32] employed the rotational failure mechanism of Mollon et al. [37] to obtain the support pressure of tunnel face subjected to pore water pressure.e approximate method based on the PWP coe cient was also used to calculate the upper bounds.results are compared with the obtained UB solutions of this study, as shown in Figure 5. e presented mechanism can yield a greater value of the support pressure than the rotational mechanism of Mollon et al. [37].As the height of the groundwater table increases, greater support pressure is required on the tunnel face.In addition, the UB solutions derived from the rotational mechanism by Wong and Subrin [29] are given here for comparisons, as shown in Table 2. e presented rotational mechanism of this study yields more critical results because of the improvement on the entirely inscribed circular face of the failure mechanism.Wong and Subrin [29] also carried out FD analysis to calculate the limit support pressure, but it is still less than the pressure presented in this study.

Advances in Civil Engineering
Using the optimization procedure can not only obtain the least upper bound of the support pressure but also the corresponding critical failure surface.Figures 6 and 7 illustrate the 3D geometry of the obtained critical failure mechanisms for the cohesive frictional soil and purely frictional soil, respectively.It can be seen that the overall failure surface is smooth and passing through the entire circular tunnel face.e failure surface for the purely frictional soil with φ 40 °is very similar to the observed failure surface from the experimental tests by Kirsch [25] and Berthoz et al. [30].e 3D geometry of the proposed mechanism is close to the measured actual failure surface.rough numerous calculations, the obtained UB solutions are presented in a stability chart, as shown in Figure 8. e chart provides a simple way to predict the required support pressure on circular faces of tunnels in frictional soil.It should be noted that the chart is limited to the collapse occurring below the ground surface.Figure 9 illustrates the heights of the critical failure surface from the tunnel crown.e coe cients N c and N c in equation ( 20) are given for di erent friction angles, as shown in Figures 10 and 11, respectively.e friction angle has profound e ects on the coe cients, but the in uence of the soil cohesion is negligent.Compared with the values of N c given by Leca and Dormieux [7] and Subrin and Wong [28], this study can yield greater results.Subrin and Wong [28] obtained the coe cient N c 1/tan(φ) from their mechanism.As illustrated in Figure 11, the solutions of this study are close to their presented results.Reading the values of N c and N c illustrated in Figures 10 and 11, we can obtain the required    Advances in Civil Engineering support pressure through equation ( 20) for an arbitrary value of the friction angle.
Figures 12-14 show the stability charts for tunnel face subjected to pore water pressure with r u 0.1, 0.3, and 0.5, respectively.Several groundwater levels with H w /D 0.5, 1.0, 2.0, 3.0, and 4.0 are considered.As expected, the required support pressure increases with the increasing groundwater level.As the PWP coe cient increases to r u 0.5, the soil cohesion has a minor in uence on the support pressure for greater friction angle.In this situation, the support pressure depends on the groundwater level.It should be noted that the PWP coe cient cannot be determined in practice and can only be used for approximate estimation.As presented by Pan and Dias [32], the numerical FE/FD analyses can be conducted to obtain the PWP distribution during the excavation and then incorporating it into the UB solutions could yield an accurate result.

Concluding Remarks
is study presents an improved rotational failure mechanism for circular faces of shield-driven tunnels in the frictional and/or cohesive soils.
e mechanism has the shape of a curvilinear cone with an elliptical radial cross section and its lower and upper contours on the symmetry plane are de ned by two log-spiral lines.Unlike the mechanism proposed by Subrin and Wong [28], it avoids the inscribed ellipse to the circular tunnel face and can involve the entire circular face of the tunnels.e generating 3D failure surface is kinematically admissible in LA, and using 10 Advances in Civil Engineering the kinematic approach yields an upper bound to the support pressure on the tunnel face.Compared with the existing results derived from other mechanisms, the improved mechanism leads to the least UB solution in LA.It implies that the derived critical failure surface is closer to the actual one, as observed in some experimental tests.Numerous calculations are carried out, and then the obtained UB solutions are presented in a stability chart.It provides a tool for convenient estimation of the required support pressure on circular faces of tunnels in frictional soils.e e ects of the pore water pressure are also included in the stability analyses of tunnel faces.An approximate method based on the PWP coe cient is used to obtain the required support pressure on the face of shield-driven tunnels with a certain groundwater level.However, the value of the PWP coe cient cannot be determined in practice.When the PWP distribution during the excavations is derived from numerical FE/FD analyses, its involvement into the presented kinematical method can better estimate the limit support pressure.
e present mechanism can be further extended to tunnel face blowout, partial collapse on the tunnel face, and collapse reaching the ground surface in shallow tunnels.In fact, the modi cation on the elliptical cross section of the mechanism is not only suitable for the circular tunnel face but also used in other forms of the tunnel face, such as horseshoe and semiellipse.Although this study is limited to the frictional soil, the improvement of the rotational Advances in Civil Engineering mechanism could be useful in purely cohesive soil, as the continuous velocity eld proposed by Mollon et al. [6].
Besides the tunnel face stability, the improved mechanism can be extended into other 3D geotechnical problems, such as slope stability and bearing capacity of footings.

Notations a:
Semimajor axis of an ellipse b: Semiminor axis of an ellipse C: Tunnel cover depth c:

Soil cohesion D:
Tunnel diameter D s : Rate of work dissipation d 1 , d 2 : Parameters in equation (7) e: Eccentricity of an ellipse e 0 : Eccentricity of the elliptic cross section at θ θ TU H w : Height of the groundwater table H e : Height of the failure surface from the tunnel crown h w : Depth of the failure surface below the groundwater table N c , N c , N u : Dimensionless parameters r, r ′ : Radius of the lower and upper log-spiral lines r 0 : Log-spiral radius at θ θ 0 r m : Average radius of the two log spirals r u : Coe cient of pore water pressure u: Pore water pressure v: Radial velocity Work rate of pore water pressure c: Unit weight of soil η: Coefficient of the power function in equation ( 10) θ: Angular coordinate in polar system σ tc : Supporting pressure on the tunnel face φ: Internal friction angle of soils ω: Angular velocity.

Figure 1 :
Figure 1: Central cross section of 3D rotational failure mechanism for circular tunnel face.

Figure 3 :
Figure 3: Comparisons of the critical results derived from di erent postulated mechanisms (c 18 kN/m 3 and D 10 m).

Figure 2 :
Figure 2: In uence of the coe cient η on the critical results (D 10 m, C 10 m, and c 18 kN/m 3 ).

Figure 4 :Figure 5 :
Figure 4: Comparisons of the critical results for tunnel face in cohesionless soils.

Figure 6 :
Figure 6: View of the critical failure surfaces of tunnel face in cohesive frictional soil with c 7 kPa and φ 17 °.

Figure 7 :
Figure 7: View of the critical failure surfaces of tunnel face in purely frictional soil with c 0 kPa and φ 40 °.

Figure 8 :
Figure 8: Stability charts of the critical pressure supported on the tunnel face.

Figure 9 :
Figure 9: e critical heights of the tunnel face failures.

Figure 10 :
Figure 10: Dimensionless parameter N c for tunnel face stability.

Figure 11 :
Figure 11: Dimensionless parameter N c for tunnel face stability.

Figure 12 :
Figure 12: Stability charts of the critical pressure supported on the tunnel face under the pore water pressure r u 0.1.(a) φ 40 °.(b) φ 30 °.

Figure 13 :
Figure 13: Stability charts of the critical pressure supported on the tunnel face under the pore water pressure r u 0.3.(a) φ 40 °.(b) φ 30 °.

Figure 14 :
Figure 14: Stability charts of the critical pressure supported on the tunnel face under the pore water pressure r u 0.5.(a) φ 40 °.(b) φ 30 °.

Table 1 :
Comparisons of the obtained supported pressures (kPa) on the tunnel face for D 10 m, C 10 m, and c 18 kN/m 3 .

Table 2 :
[29]arison of the calculated values of σ tc /cD between this study and Wong and Subrin[29]for tunnels in purely frictional soils.