Face stability analysis of a shallow horseshoe-shaped shield tunnel in clay with a linearly increasing shear strength with depth

Tunnelling and Underground Space Technology 97 (2020) 103291

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Face stability analysis of a shallow horseshoe-shaped shield tunnel in clay with a linearly increasing shear strength with depth

T

⁎

Chengping Zhang , Wei Li, Wenjun Zhu, Zhibiao Tan Key Laboratory of Urban Underground Engineering of the Education Ministry, Beijing Jiaotong University, Beijing 100044, China School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Horseshoe-shaped shield tunnel Face stability Limit support pressure Continuous velocity field

The face stability of a horseshoe-shaped shield tunnel in undrained clays with a linearly increasing strength is numerically and analytically investigated in this paper. Based on the kinematic approach of limit analysis, a continuous velocity field for a horseshoe-shaped shield tunnel is proposed to evaluate the limit support pressure of both the collapse and blow-out of the tunnel face. Moreover, the analytical results are compared with those obtained from the numerical simulation and other existing literatures. The results show that the horseshoeshaped shield tunnel can improve the face stability compared to the circular shield tunnel. Finally, an improved velocity field is proposed, which provides a more realistic failure pattern and a better upper bound solution of the limit support pressure for the horseshoe-shaped shield tunnel.

1. Introduction The face stability analysis of tunneling projects becomes more and more popular due to the rapid development of urban subways, highway and railway tunnels. The shield tunneling machines are widely used in the constructions of urban metro, highway and railway tunnels, which makes it crucial to control supporting pressures distributed on the tunnel face. When the supporting pressure of the tunnel face is not sufficient, the shield tunnel face may become unstable or even collapse, which finally causes the subsidence of the ground surface. On the other hand, the tunnel face may blow out and push the soils towards the ground surface when the applied supporting pressure is too high. Thus it is crucial to perform the stability analysis on the practical shield tunneling projects. Because of its good reproducibility, many advanced numerical models were adopted to investigate the stability of the tunnel face using the Finite Element Method (FEM) (Ibrahim et al., 2015; Ukritchon et al., 2017; Vermeer et al., 2002), Finite Difference Method (FDM) (Senent and Jimenez, 2014; Senent et al., 2013; Zhang et al., 2015) and Discrete Element Method (DEM) (Chen et al., 2011; Zhang et al., 2011). But these numerical analyses were highly time-consuming and many input parameters were required, which implies that the analytical analysis may play a better role in the preliminary assessment on the stability of the tunnel face, especially when some design data about the operating details of the boring machine or soil properties are absent.

⁎

Moreover, for a better visualization of failure patterns, the experimental test was also an important approach to study the stability problem (Chambon and Corté, 1994; Chen et al., 2013; Idinger et al., 2011; Mair, 1969; Schofield, 1980; Takano et al., 2006). The stability of the tunnel face of circular tunnels has been widely investigated based on the limit analysis and the limit equilibrium approaches. The silo-wedge model was usually applied for the limit equilibrium method to deduce the limit supporting pressures, including different factors such as the horizontal arching, free span, seepage flow conditions and bolt reinforcements on the stability of the tunnel face (Anagnostou, 2012; Anagnostou and Kovari, 1996; Anagnostou and Perazzelli, 2013, 2015; Horn, 1961; Perazzelli et al., 2014). The main advantage lies in the simplicity of the calculation model. The limit analysis considered the stress-strain relationship of an ideally plastic soil mass and followed the associated flow rule, which provided a solution with a more rigorous theoretical basis compared with the limit equilibrium method. Compared with the silo-wedge model of the limit equilibrium method, the limit analysis method adopts the kinematically admissible velocity field and the statically admissible stress field to obtain the upper and lower bound solutions of the limit support pressure. And the failure pattern is obtained according to the normal flow condition associated with the yield condition rather than a vertical failure plane assumed by the silo-wedge model. Various kinematically admissible velocity fields were proposed to obtain critical upper bound solution of the tunnel face, both the translational rigid block

Corresponding author at: Key Laboratory of Urban Underground Engineering of the Education Ministry, Beijing Jiaotong University, Beijing 100044, China. E-mail address: [email protected] (C. Zhang).

https://doi.org/10.1016/j.tust.2020.103291 Received 9 May 2019; Received in revised form 6 October 2019; Accepted 3 January 2020 0886-7798/ © 2020 Elsevier Ltd. All rights reserved.

Tunnelling and Underground Space Technology 97 (2020) 103291

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Nomenclature

tangential velocity point E Rf distance between the origin O and the point E Rmax(θ, β)distance between the point E and the boundary of the cross-section Πβ L1, r2, r3 distances of three centres O1, O2, O3 to the point E vβ axial component of velocity vr radial component of velocity vθ orthoradial component of velocity vm(β) maximum value of the axial velocity ε̇ strain rate tensor Ẇγ rate of work of the soil mass rate of work of the surcharge Ẇs rate of work of the support pressure Wṫ ẆD rate of the energy dissipation Nγ, Ns, ND non-dimensional coefficients of soil unit weight, surface surcharge and cohesion

R1, R2, R3 three radii of the horseshoe-shaped tunnel face O1, O2, O3 three centres of the horseshoe-shaped tunnel face α1, α2, α3 ranges of three sections of circular arc W equivalent width of the tunnel face D equivalent height of the tunnel face C/D buried depth to diameter ratio γ soil unit weight cu undrained strength of the soils E Young’s modulus v Poisson’s ratio ρ variation gradient of soil undrained strength σs surface surcharge σt support pressure Πβ cross-section of the failure mechanism Ri distance between the tunnel crown and the maximal

(international tunneling association). Because of the first application of the horseshoe-shaped cross-section on the shield tunnels, the investigation on the face stability is very interesting and valuable. This paper focuses on the face stability analysis of a horseshoe-shaped shield tunnel. The study of Mollon et al. (2013) is extended to generate a more realistic continuous velocity field for a horseshoe-shaped shield tunnel in low-permeability clay. The closed-form formulas of the continuous velocity field are adopted to calculate the upper bound solution of the limit supporting pressure of both face collapse and blow-out. A series of numerical simulations for a horseshoe-shaped tunnel is performed to validate the analytical solution of the proposed failure mechanism. Finally, the results of the proposed failure mechanism are discussed by taking into account of the typical approaches available from the literatures.

mechanisms (Davis et al., 1980; Han et al., 2016a,b; Leca and Dormieux, 1990; Li et al., 2020; Li et al., 2019a,b; Mollon et al., 2009, 2010; Zhang et al., 2015; Soubra et al., 2008) and the rotational rigid block failure mechanisms (Li et al., 2018; Li and Zhang, 2020; Mollon et al., 2011; Pan and Dias, 2016, 2017; Subrin and Wong, 2002; Zou et al., 2019a,b; Zou and Qian, 2018) were proposed to be applied in the limit analysis for frictional soils. For purely cohesive soils, more realistic continuous velocity fields (Huang et al., 2018; Klar and Klein, 2014; Klar et al., 2007; Li et al., 2019a,b,c; Mollon et al., 2013; Osman et al., 2006; Zhang et al., 2018a,b) was applied to describe the continuous deformations of soils, which moved more like a ‘flow’ rather than a rigid block. According to the conclusions from the available literatures, it can be found that the face stability has been a key issue of the shield tunnel. Compared with the traditional mining method and NATM (New Austrian Tunneling method) in tunnel constructions, there are obvious advantages for the shield tunneling inchluding the security, quality, progress, environment and mechanization of construction. In order to adapt to different soil conditions, tunnel designs and operating requirements, various shapes of shield tunnels are adopted in the practical engineering projects such as circular, multi-circular and non-circular (including elliptical, rectangular and horseshoe-shaped crosssections) tunnel faces. And the non-circular tunnel face significantly can improve the space utilization and reduce the construction cost compared with the common circular shield tunnel. The first horseshoeshaped EPB (earth pressure balance) shield (as shown in Fig. 1) in the world has been put into construction in Baicheng tunnel of Menghua railway in China, which has attracted much attention of many engineers and won the 2018 annual technology innovation award of ITA

2. Limit analysis of the face stability of a horseshoe-shaped shield tunnel 2.1. Problem statement The new failure mechanism for the face stability of a horseshoeshaped shield tunnel is considered under the assumptions that the plastic deformation of soils is subjected to the undrained condition without any change of volume and the undrained strength of soils cu increases with depth y in a linear way (cu = cu0 + ρ(−y)). The soils are assigned with the Tresca yield criterion, which is a perfectly elasticplastic constitutive model (the Mohr-Coulomb criterion will be degenerated to the Tresca criterion when φ = 0). For the engineering problem, the specific geometrical face shape of the first horseshoe-

Fig. 1. The horseshoe-shaped shield used in Baicheng tunnel of Menghua railway in China. 2

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where L1 is the distance between O1 and Eon the tunnel face. The radius Rmax(θ, 0) of the tunnel face can be calculated as follows:

shaped EPB shield tunnel of Menghua railway is adopted herein as shown in Fig. 2. The following relations are about the geometrical properties of the horse-shaped tunnel. With the initial geometric parameters R1, R2, R3 and α1, α2, the following parameters are obtained:

α3 = π − α1 − α2 ⎧ d = R3 − (R3 − R2) sin α1·sin α2 ⎪ ⎪ D = R1 + d ⎨ W = 2·R1 ⎪ ⎪ Area = α1 R12 + α2 R22 + α3 R32 − (R1 − R2)·(R3 − d )·sin α1 ⎩

⎧ ⎪

L12cos2 θ − L12 + R12 0 ⩽ θ < θ1

Rmax (θ , 0) = r2 cos(θ + θ0 ) + r22cos2 (θ + θ0 ) − r22 + R22 θ1 ⩽ θ < θ2 ⎨ ⎪ r3 cos(θ ) + r32cos2 θ − r32 + R32 θ2 ⩽ θ < π ⎩ (4) where r2 and r3 are the distances from O2 and O3 to center E, respectively. θ0 is the angle between the EO1 and EO2. θ1 is the angle between EO1 and EP1, and θ2 is the angle between EO1 and EP2. The geometric relation of the horseshoe-shaped tunnel face is shown in Fig. 4. The Rmax of the tunnel face respectively follows different geometric relations in three arc segments. When θ varies from 0 to θ1, Rmax(θ,0) can be deduced by the triangular relation of O1A1E. Similarly, Rmax(θ,0) of the rest two segments can be obtained from the triangles O2A2E and O3A3E, respectively, when θ1 ≤ θ ≤ θ2 and θ2 ≤ θ ≤ π. A1, A2 and A3 are three arbitrary points in the corresponding arc segments. Substituting Eq. (4) into Eq. (2) yields, after some simplifications:

(1)

where α3 and d are shown in Fig. 2. D and W are the equivalent height and width of a horseshoe-shaped shield tunnel. The detailed geometric parameters of the tunnel face are calculated and summarized in Table 1. The parameters are designed for the first horseshoe-shaped EPB (earth pressure balance) shield of Baicheng tunnel of Menghua railway in China to provide a direct and generalized explanation for a typical horseshoe-shield tunnel. Moreover, since the face stability is closely related with the dimension of the tunnel face, the geometric parameters refer to a horseshoe-shaped face with equivalent height D and width L of approximately 9.6 m and 10.5 m, respectively, which corresponds to an equivalent size of the circular face with a diameter D of 10 m. The solutions obtained from this paper can be compared with the results of the existing researches (Davis et al., 1980; Mollon et al., 2009, 2010, 2011, 2013; Anagnostou and Kovari, 1996; Anagnostou, 2012) on the circular tunnel face with a diameter of 10 m. In addition, the new failure mechanism involves the continuous deformation of soil masses, which overcomes the limitation of common translational or rotational rigid block failure mechanism and builds a more realistic continuous velocity field for the collapse and blow-out of the tunnel face. The continuous velocity field represents the whole region of the failure mechanism. The velocity profile of the tunnel face completely covers the entire horseshoe-shaped face of the shield tunnel, which also overcomes the inherent shortcoming of incomplete contact between the excavation face and failure zone of the rigid block failure mechanism. Therefore, in order to estimate the face stability of the horseshoe-shaped shield tunnel, a rigorous analytical solution concerning the complexity of the geometry and the velocity distribution will be derived in this section.

Rmax (θ , β ) = ⎧ ⎪

πRi + 2β (Rf − Ri ) πRi L1 cos θ +

L12cos2 θ − L12 + R12 0 ⩽ θ < θ1

× r2 cos(θ + θ0 ) + r22cos2 (θ + θ0 ) − r22 + R22 θ1 ⩽ θ < θ2 ⎨ ⎪ r3 cos(θ ) + r32cos2 θ − r32 + R32 θ2 ⩽ θ < π ⎩

2.2.1. Velocity field The proposed 3D failure mechanism presents the plastic deformation space of the soil ground. As the face collapse in purely cohesive involves a continuous deformation of the soils, the velocity of the failure mechanism in this paper is divided into three components: the axial component vβ perpendicular to the rotational plane Πβ (velocity profile) and pointing towards the tunnel face, which is assumed to follow a parabolic distribution in the plane Πβ; the radial component vr located on the plane Πβ and pointing towards center Eβ; and the

As shown in Fig. 3, a tunnel with equivalent height D and width L is excavated under a cover depth of C in a purely cohesive soil with a linearly increasing undrained strength of cu = cu0 + ρ(−y). A surcharge σs is applied on the ground surface and no pressure is supported on the tunnel face. The failure mechanism is generated by rotating the horseshoe-shaped tunnel face of increasing area (shaded zone in Fig. 3) about the rotation center O on the symmetric vertical plane y-z. The whole 3D failure region is defined by the orthogonal curvilinear coordinate (r, θ, β). In an arbitrary rotational plane Πβ, the radial distance Rmax(θ, β) represents the radius from the maximal axial velocity point Eβ to the boundary of the rotational plane Πβ, which increases with the rotation angle β. The formula of Rmax(θ, β) is expressed as follows:

R3 Į3

O3

(R3-R2) D=(R1+d)

Į1 O 1 R2

Į2 O 2 R2

(2)

(R1-R2) d

where Rmax(θ, 0) is the radius of the tunnel face. R(β) is the corresponding length of Rmax(θ, β) of the rotational plane Πβ when θ = 0. And R(β) is assumed to linearly increase from Ri at the tunnel face to Rf at the ground surface as follows:

⎧ R (β ) = Ri + β (Rf − Ri ) (π 2) Ri = R1 + L1 ⎨ Rf = R1 + L1 + C ⎩

(5)

Any point in the space can be expressed in terms of the angle β (angle between the plane Πβ and the plane of the tunnel face) and the polar coordinates (r, θ) in the plane Πβ, which are defined with respect to the center Eβ of the plane Πβ.

2.2. New failure mechanism

R (β ) Rmax (θ , β ) = Rmax (θ , 0) × Ri

L1 cos θ +

W=(2R1)

Fig. 2. Image of the horseshoe-shaped shield tunnel.

(3) 3

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Table 1 Geometric parameters of the tunnel face. Geometric parameters

Values

R1 (m) R2 (m) R3 (m) α1 (°) α2 (°) α3 (°) d (m) Area (m2)

5.27 3.62 9.12 106 57 17 4.32 81.49

orthoradial component vθ located on the plane Πβ and orthogonal to the radial component, which is assumed to be zero. According to the results of FLAC3D, the maximal axial velocity point Eβ of every velocity profile Πβ is assumed to be located at a distance of R(β) below the vertex of the plane. The formulas of axial component vβ and the orthoradial component vθ of the velocity vector are defined as follows:

⎧ vβ (r , θ , β ) = vm (β ) × ⎡1 − ⎣ ⎨ vθ (r , θ , β ) = 0 ⎩

r2

Fig. 4. Calculation of Rmax of the tunnel face.

θ2 θ1

+ ∫

(6)

π

Rmax (θ, β )

⎡1 − ⎣

r2

⎤ rdrdθ

2 (θ, β ) Rmax ⎦

f (β ) ⎡ r2 ⎤ × 1− 2 ⎥ ⎢ A R max (θ , β ) ⎦ ⎣

(7)

̇ + εββ ̇ =0 div (ε )̇ = εrṙ + εθθ

}

(9)

The strain rate tensor in the orthogonal curvilinear coordinates (r, θ, β) can be expressed as follows: (8)

̇ εrβ ̇ ⎤ ⎡ εrṙ εrθ ̇ εθθ ̇ εθβ ̇ ⎥ ̇ε = ⎢ εθr ⎢ ⎥ ̇ ̇ ̇ ⎣ εβr εβθ εββ ⎦

where f(β) is a function of β. And A is a constant concerned with the geometry of the tunnel face.

f (β ) =

r32cos2 θ − r32 + R32 ]2 dθ

According to the normality condition of the limit analysis, the plastic deformation of a purely cohesive soil subjected to the undrained conditions occurs without any change of volume. The normality condition can be implemented in the equation of strain rate as follows:

Accomplishing the integration of vm(β) in Eq. (7), the formula of vβ(r, θ, β) is expressed as follows:

vβ (r , θ , β ) =

r22cos2 (θ + θ0) − r22 + R22 ]2 dθ

[r2 cos(θ + θ0) + π

1 2 ∫0 ∫0

L12cos2 θ − L12 + R12 ]2 dθ

[L1 cos θ +

+ ∫θ [r3 cos(θ) + 2

where vm(β) is a normalization function used to ensure that the velocity flux through any plane Πβ is constant and independent of angle β based on the principle of mass conservation, which is given by:

vm (β ) =

θ1

1

A = 2 {∫0

⎤

2 (θ, β ) Rmax ⎦

π 2Ri2 and [πRi + 2β (Rf − Ri )]2

where

Fig. 3. 3D failure mechanism for a horseshoe-shaped shield tunnel. 4

(10)

Tunnelling and Underground Space Technology 97 (2020) 103291

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⎧ ⎪ ⎪ ⎪ ⎪ ̇ = ⎪ εrβ

εrṙ = ̇ = εrθ

1 2r

(

∂vr ∂θ

+r

1 ⎡v cos θ 2(Rf − r cos θ ) ⎣ β

where vβ(r, θ, β = 0) and At represent the velocity and area of the tunnel face, respectively. And σt0 is the support pressure at the tunnel crown. γt is the unit weight of the material in the excavation chamber (γt = γ for the earth pressure shield; γt = γs for the slurry balance shield). The rate of the energy dissipation ẆD in the failure mechanism can be calculated based on the reference of Chen (1975):

∂vr ∂r

+

∂vθ ∂r

∂vr ∂β

− vθ

)

+ (Rf − r cos θ)

(

∂vβ ∂r

⎤ ⎦

)

∂v 1 ⎨ ̇ = r vr + ∂θθ εθθ ⎪ ∂vβ ⎪ ∂v 1 ̇ = 2r (R − r cos θ) ⎡r ∂βθ − vβ r sin θ + (Rf − r cos θ) ∂θ ⎤ ⎪ εθβ f ⎣ ⎦ ⎪ ∂vβ 1 ⎪ ̇ = R − r cos θ vθ sin θ − vr cos θ + ∂β εββ f ⎩

(

ẆD = 2

)

(11)

−K (θ , β ) r 3 − 2M (β ) r 4(Rf − r cos θ)

(12)

∂Rmax (θ, β ) 2 (θ, β ) df (β ) − Rmax ∂β dβ 4 (θ, β ) ARmax

and M (β ) =

ẆD ⩾ Wė = Ẇγ + Ẇs + Wṫ

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

(13)

π

Rmax (θ, β )

(vβ sin β + vr cos θ cos β )dr·r dθ ·

(Rf − r cos θ)dβ

(14)

∫A

s

π vβ ⎛r , θ , β = ⎞ dAs =2σs 2⎠ ⎝ r dθ

π

∫0 ∫0

Rmax θ, β = π 2

(

)v

β ⎛r ,

⎝

θ, β =

⎜

∫A

t

vβ (r , θ , β = 0)dAt =2

π

∫0 ∫0

Rmax (θ, β = 0)

+ γt (Ri − r cos θ)]dr·r dθ

Rmax θ, β = π 2

(

)

π vβ r , θ, β = dr·r dθ 2 π Rmax (θ, β = 0) vβ (r , θ, β = 0)dr·r dθ 2 ∫0 ∫0 2 π Rmax (θ, β = 0) vβ (r , θ, β = 0)·(Ri − r cos θ)dr·r dθ ∫0 ∫0 Nγt = D (θ, β = 0) π R vβ (r , θ, β = 0)dr·r dθ 2 ∫0 ∫0 max π π Rmax (θ, β ) 2 2∫ ∫ ∫ 2 max(| εi̇ |)dr·r dθ·(Rf − r cos θ)dβ Nc = 0 0 0π Rmax (θ, β = 0) vβ (r , θ, β = 0)dr·r dθ 2 ∫0 ∫0 π

Ns =

2 ∫0 ∫0

(

)

π 2 ∫ 2 ∫π ∫ Rmax (θ, β ) (Rf − r cos θ) cos β·2 max(| εi̇ |)dr·r dθ·(Rf − r cos θ)dβ D 0 0 0 (θ, β = 0) π R vβ (r , θ, β = 0)dr·r dθ 2 ∫0 ∫0 max

(21)

⎟

(22)

Thus it is easily obtained that Ns = 1 for purely cohesive soils. And Eq. (20) can be rearranged as a dimensionless form:

π ⎞ dr· 2⎠ (16)

γ ρD ⎞ γD ⎛ σt 0 − σs ·⎜Nγ − t Nγt ⎟⎞ − ⎛Nc + Nρ = cu 0 ⎠ γ ⎠ ⎝ cu 0 cu 0 ⎝ ⎜

⎟

(23)

When the applied supporting pressure is too high, a blow-out of the tunnel face will occur. The limit support pressure can be calculated by making the continuous velocity field in the opposite direction. σt0 in the case of blow-out is expressed as follows:

where vβ(r, θ, β = π/2) and As represent the velocity and area of the soil mass on the ground surface, respectively. The rate of work Wṫ of a linearly increasing support pressure σt distributed on the tunnel face is given by:

Wṫ = σt

π 2 ∫ 2 ∫π ∫ Rmax (θ, β ) (vβ sin β + vr cos θ cos β )dr·r dθ·(Rf − r cos θ)dβ D 0 0 0 (θ, β = 0) π R 2 ∫0 ∫0 max vβ (r , θ, β = 0)dr·r dθ

⎛Nc + ρD Nρ⎞ tan φ + 1 − Ns = 0 cu 0 ⎠ ⎝

(15)

The rate of work Ẇs of a possible uniform surcharge σs acting on the ground surface is given by:

Ẇs =

(20)

Considering the mass conservation of the failure mechanism, the numerical results indicate that the relation of Nc, Nρ and Ns is as follows (Leca and Dormieux, 1990; Mollon et al., 2009, 2010; Soubra et al., 2008; Zhang et al., 2015):

where VY is the vertical component of the velocity for a given soil element and dV is the volume of an elementary soil mass. And the formulas of VY and dV can be expressed as follows:

⎧ vY = vβ sin β + vr cos θ cos β dV = dr·r dθ ·(Rf − r cos θ)dβ ⎨ ⎩

Nγ =

⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Nρ = ⎩

∫V vY dV π 2

⎟

where Nγ, Nγt, Ns, Nc and Nρ are the non-dimensional coefficients that represent influences of soil unit weight γ, the gradient of support pressure on the tunnel face, the surface surcharge σs, the cohesion c and the soil inhomogeneity, respectively. The formulas are given as follows:

where Wė represents the total rate of work of the external forces and ẆD is the rate of the energy dissipation within the soil masses undergoing a plastic deformation. In the stability analysis of a shield tunnel, the external forces include the gravity of the soil unit weight γ, the support pressure σt and the surface surcharge σs. The rate of work of the soil mass Ẇγ is given by:

∫0 ∫0 ∫0

(18)

(19)

⎜

Based on the upper bound theorem of the limit analysis, a necessary stability condition for the tunnel face is considered according to Chen (1975):

= 2γ

[cu0

γ ρD ⎞ Nρ σt 0 = γD ·⎜⎛Nγ − t Nγt ⎟⎞ + σs·Ns − cu0·⎛Nc + cu 0 ⎠ γ ⎠ ⎝ ⎝

1 df (β ) · A dβ

2.3. Upper bound solution of the limit support pressure

Ẇγ = γ

Rmax (θ, β )

where cu0 is the undrained shear strength of the clay at the ground surface and ρ is the gradient of the undrained shear strength varying with the ground depth. Besides, soils are degenerated into homogeneous case when ρ = 0. By substituting Eqs. (14)–(18) into Eq. (13) and equating the total rate of the external work to the rate of the energy dissipation, the limit support pressure σt0 is obtained:

where 2f (β ) Rmax (θ, β )

π

cu (y ) = cu0 + ρ (−y )

∂vβ ∂vr =0 + (Rf − 2r cos θ) vr + r ∂β ∂r

K (θ , β ) =

π 2

∫0 ∫0 ∫0

where max(|εi̇ | ) denotes the maximal absolute value of the principal strain rate εi̇ , and cu is the undrained shear strength of the clay, which increases linearly with the ground depth y

Substituting Eq. (8) into Eq. (11), the radial component vr is obtained:

vr (r , θ , β ) =

=2

+ ρ (Rf − r cos θ) cos β ]·2 max(|εi̇ | )dr·r dθ ·(Rf − r cos θ)dβ

The radial component vr located on the plane Πβ can be derived according to the equilibrium equation of the strain rate tensor according to Eqs. (9) and (10):

r (Rf − r cos θ)

∫V 2cu·max(|εi̇ | )dV

γ σtb0 − σs ρD ⎞ γD ⎛ ·⎜Nγ − t Nγt ⎟⎞ + ⎛Nc + Nρ = cu 0 ⎠ γ ⎠ ⎝ cu 0 cu 0 ⎝

vβ (r , θ , β = 0)[σt 0

⎜

(17)

⎟

(24)

The optimal failure mechanism in both the collapse and blow-out of 5

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σtapplied (y ) = n·σh (y ) = n·K [γ (−y )] (C ⩽ y ⩽ C + D)

the tunnel face can be determined by the parameter L1. For a given horseshoe-shaped shield tunnel with buried depth C, unit weight of soil γ, undrained strength cu, inhomogeneous coefficient ρ, surface surcharge σs and the geometrical parameters of the tunnel face, the critical upper bound solution of the support pressure in the case of collapse and blow-out can be obtained with the help of the optimization tool implemented in the Matlab.

(25)

where σtapplied(y) represents the support pressure applied on the tunnel face in the numerical simulation. σh(y) is the initial ground horizontal stress at a depth of y. K is the lateral pressure coefficient of the soils and n is the support pressure ratio. As shown in Fig. 6, with the gradual decrease of the support pressure ratio, the collapse soils around the tunnel face move toward the shield tunnels and the horizontal displacement of the tunnel face will increase rapidly until the support pressure ratio decreases to a constant value. On the contrary, when the face support pressure is too high and the tunnel face blows out, the soils around the tunnel face move toward the soil ground. The horizontal displacement of the tunnel face will increase with the support pressure ratio. Specifically, when the slopes of the curves are approximate to zero, the corresponding support pressure ratios are defined as the limit support pressure ratios. In Fig. 6, the dotted lines show the horizontal tangents of the curves and denote the limit support pressure ratios. It is shown that a smaller relative ratio C/ D corresponds with a lower limit support pressure ratio for the collapse case. A greater C/D leads to a higher limit support pressure ratio for the blow-out case. Besides, difference between Fig. 6(a) and (b) shows that a higher undrained shear strength cu corresponds to a smaller limit support pressure ratio for collapse case (a larger limit support pressure ratio for blow-out case). This is because that a higher undrained shear strength can have a better resisting capacity. When the undrained shear strength cu is larger, a smaller limit support pressure ratio is required to maintain the face stability for the soil collapse. But when the face blows out, a larger limit support pressure ratio will be required to push the soils toward the ground surface.

3. Validation by FLAC3D 3.1. Numerical simulation To validate the proposed failure mechanism of a horseshoe-shaped shield tunnel, the specific geometrical face of the first horseshoe-shaped EPB shield tunnel of Menghua railway is adopted to perform the numerical simulation. The 3D finite difference code FLAC3D is adopted to investigate the stability of the tunnel face and outline the velocity field of the failure mechanism. Considering the symmetry of the tunnel on the vertical plane, a 3D numerical model is built based on half of the tunnel. The nodes at the bottom of mesh are fully fixed for the boundary conditions. Nodes on the vertical symmetric plane are only fixed in the X direction but to assure the vertical movement. Fig. 5 presents a full face excavation numerical model of a shield tunnel for C/D = 1.0 with a discretization of 58,410 zones (‘zone’ is the FLAC3D terminology for each discretized element) and a total of 62,186 nodes. The dimensions of the 3D numerical model are 20 × 30 × 50 m in the transversal, vertical, and longitudinal directions, which are large enough to avoid the influence of the boundary effects. To study a purely cohesive soil, an elastic perfectly plastic constitutive model based on Tresca failure criterion is assigned to the soils, which means that a purely clayey soil is in an undrained condition and the soil plastic deformation takes place without any volume change. The Young’s modulus E and Poisson’s ratio ν of the clay are taken as 500cu and 0.495, respectively (Huang et al., 2018; Ukritchon et al., 2017). It is worth noting that a higher value of the E/cu ratio is selected because it can increase the computation speed with no significant impact on the stability of the tunnel face. All the parameters used in the numerical model are summarized in Table 2. Moreover, the tunnel lining is modelled by the shell structural element with the Yong’s modulus of 33.5GPa, Poisson’s ratio of 0.2 and the thickness of 0.35 m. The full face excavation is performed in the FLAC3D. The excavation process is simulated using a simplified singlestep excavation scheme with the assumption that part of the tunnel (10 m in length) is excavated instantaneously. And the lining shells are installed simultaneously. The critical failure state is achieved by a stress-controlled method in both collapse and blow-out cases as follows: First, trapezoidal distributed support pressures that are equal to the initial ground horizontal stress are applied at the tunnel face in the reversed direction. Then, the active limit support pressure is found by gradually decreasing the support pressure until the tunnel face collapses (passive limit support pressure of blow-out is found by gradually increasing the support pressure). During each pressure, several cycles are performed until a steady state of static equilibrium or plastic flow is achieved in the soils. A large number of numerical simulations are conducted to obtain a satisfactory value of the critical collapse pressures, but it is an intuitive and simple approach to determine the critical failure of the tunnel face.

3.3. Failure zone Fig. 7 shows the displacement contours and velocity vectors of the critical state of a horseshoe-shaped shield tunnel for C/D = 0.5–3.0. It can be seen that the failure mechanism involves a continuous plastic deformation and the maximal velocity flow line locates at a distance below the center O1 of tunnel face for both the collapse and blow-out cases. Zhang et al. (2015) proposed a simple and feasible criterion to outline the boundary trip of the failure zone at collapse according to the displacement contour where a sudden increasing gradient happened. But this assessment is only suitable for the rigid-block mechanism of the frictional soils because a distinct shear band or failure plane can be found in the failure of the frictional soils. However, for the purely cohesive soils movement, there is no obvious shear band and the soil

3.2. Limit support pressure Fig. 6 shows the curves of the support pressure ratio (the ratio of the applied face support pressure to the initial ground horizontal stress for the tunnel face) versus the horizontal displacement of the corresponding monitoring point (O1 is chosen in the analysis) of the tunnel face for various relative depths C/D in different soils (cu0 = 20 kPa and 30 kPa).

Fig. 5. The numerical model of a horseshoe-shaped shield tunnel. 6

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horseshoe-shaped shield tunnel is reasonable, and the investigation on the continuous velocity filed of the horseshoe-shaped tunnel is necessary and significant. Moreover, the failure zones extend approximately 1.7D and 2.0D ahead of the tunnel face, respectively, when C/D = 0.5 and 1.0. When C/D = 2.0 and 3.0, the horizontal failure ranges are 2.2D and 3.1D, respectively. The ratios of the failure range to the buried depth are 3.4 and 2.0 when C/D = 0.5 and 1.0, and the ratios are only around 1.1 and 1.0 when C/D = 2.0 and 3.0. It is interesting to find that the increase on range of failure zone obviously slows down and the failure zone above the tunnel face is more prone to be vertical in the soil ground when C/D ≥ 2, although the influence of the failure zone on the ground surface basically increases with C/D. This may imply that the failure zone of the tunnel face will not constantly expand its range with C/D. It will be investigated and discussed in details in the next section.

Table 2 Parameters used in the numerical model. Parameters

Baicheng tunnel of Menghua railway

Geometrical parameters

Geotechnical parameters

R1 (m), R2 (m), R3 (m), d (m) α1 (°), α2 (°), α3 (°) Equivalent width B (m) Equivalent height H (m) Area (m2) Cohesion cu (kPa) Friction angle φ (°) Unit weight γ (kN/m3) Young’s modulus E (MPa) Poisson’s ratio v

5.27, 3.62, 9.12, 4.32 106, 57, 17 10.54 9.59 81.49 20/30 0 18 500cu 0.495

4. Results and discussions 1.8

1.71

1.65

Support pressure ratio

This section will discuss the solutions calculated by the proposed kinematic approach using continuous velocity field by taking into account of the results obtained by FLAC3D and other existing studies to validate the proposed failure mechanism of the horseshoe-shaped tunnels. In addition, the charts of the dimensionless parameters for calculating the limit support pressures are provided for a better evaluation on tunnel face stability and practical references for similar engineering projects.

Blow-out

1.6

1.58 1.41

1.4

1.49

1.35

C/D=0.5 C/D=0.75 C/D=1.0 C/D=1.5 C/D=2.0 C/D=2.5 C/D=3.0

1.32

1.2 1.0 0.8

0.69

0.65

0.6

4.1. Comparisons

0.58

0.4

0.42 0.37

0.50

4.1.1. Comparisons with the numerical simulation For a shield tunnel, a trapezoidal distributed support pressure is applied on the excavation face by the pressure chamber. The support pressure of the center of the circular tunnel face is generally adopted as the representative value of uniform support pressure. Thus, for a nonaxisymmetric tunnel face, the average values of the limit support pressure at the tunnel crown and tunnel invert are compared (σt = σt0 + γD/2). Figs. 8 and 9 present the limit support pressure σt of collapse and blow-out cases for cu0 = 20 kPa and 30 kPa with a specific geometrical face shape of the first horseshoe-shaped EPB shield tunnel of Menghua railway shown in Table 2 and a soil unit weight γ of 18kN/m3. The analytical solutions generally agree with the results of the numerical simulation when C/D varies from 0.5 to 2.0. But when C/D is higher than 2.0, a significant difference on the limit support pressure is found. This point will be discussed on the next section. Moreover, the analytical assumption of the position of the maximal axial velocity has an influence on the stability of the tunnel face. The critical value of L1 varies with the buried depth ratio C/D, which has been denoted nearby the symbol. It can be found that the critical position of the maximal axial velocity for the collapse case lie at around the tunnel invert when C/D = 0.5–1.5 (L1/d = 1.0, 0.8 or 0.6). And when C/D = 2.0–3.0, the critical L1/d is a bit smaller, which is equal to 0.6 or 0.4. Considering the computation time (a larger value of L1/d leads to a larger gradient in the radial direction on the bottom of the cross-section Πβ, which significantly increases the computation time), a compromising value of L1/d between the computation time and the optimality of the critical upper-bound solution is taken to be equal to 0.6 for face collapse case. For face blow-out case, L1/d = 0 is obviously the most critical value for the horseshoe-shaped tunnel face. Fig. 10 shows the comparison of the limit support pressure between the numerical simulaiton and limit analysis with dimensionless inhomogeneity parameter ρD/cu0. It is obvious that the limit support pressure decreases with ρD/cu0. Moreover, when C/D < 2.0, the results of the limit analysis correspond well with that of the numerical simulation. But the difference between the obtained upper solutions of the limit analysis and the results of the numerical simulation becomes more

Collapse

0.33

0.2 0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000

Horizontal displacement of the tunnel face (mm)

(a) 2.08 1.98 1.85 1.72 1.61

Support pressure ratio

2.0

Blow-out

1.52

1.5 1.45

C/D=0.5 C/D=0.75 C/D=1.0 C/D=1.5 C/D=2.0 C/D=2.5 C/D=3.0

1.0

0.54

0.5

0.0

0.0

0

500

0.15 0.06

0.49 0.38 0.28

Collapse

1000 1500 2000 2500 3000 3500 4000 4500 5000

Horizontal displacement of the tunnel face (mm)

(b) Fig. 6. Relations between the horizontal displacement and the support pressure ratio of the tunnel face: (a) cu0 = 20 kPa; (b) cu0 = 30 kPa.

failure involves a continuous deformation like a flow. The whole region of the displacement contour is considered to represent the failure zone better for the clayed soils. This failure pattern of numerical simulation coincides well with the proposed continuous deformation mechanism, but it is obviously different from the existing rigid block failure mechanisms. It is shown that the proposed failure mechanism for the 7

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Fig. 7. Displacement contour and velocity vector of the face collapse of the numerical simulation: (a) C/D = 0.5; (b) C/D = 1.0; (c) C/D = 2.0; (d) C/D = 3.0.

and more evident with ρD/cu0 for C/D ≥ 2.0, which implies that the continuous velocity field is more suitable for the horseshoe-shaped shield tunnel of C/D < 2.0.

4.1.2. Comparisons with the existing approaches For purely cohesive soils, a traditional method for estimating the stability of a tunnel face is based to the load factor N, which is a dimensionless parameter N (stability number) introduced by Davis et al. 8

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Fig. 7. (continued)

2013). For a given diameter of a shield tunnel, the horseshoe-shaped shield tunnel face not only obviously improves the face stability of a shield tunnel but also has the advantages of high efficiency of section utilization and low engineering cost in the practical engineering projects. The proposed failure mechanism provides the best upper-bound (safest) solution of the load factor N for C/D > 1.5, which implies that the continuous velocity field builds a more realistic and critical failure mechanism to describe the soil movement than the kinematic approaches (K.A.) of rigid blocks mechanism. The static approaches (S.A.) of rigid blocks mechanism by Davis et al. (1980) provide a lower bound solution for the stability number, thus an obvious difference is shown compared with those upper bound solution. Except the models of the limit analysis, two silo-wedge models (Anagnostou and Kovari, 1996; Anagnostou, 2012) of the limit equilibrium method are adopted to calculate the stability number N. For a circular shield tunnel of C/D = 0.5–3.0, the stability number N obtained from all the limit analysis methods is basically greater than the result deduced by the two silo-wedge models. The maximum difference of the stability number N between the proposed continuous velocity field for the horseshoe-shaped face and the silo-wedge model presented by Anagnostou and Kovari (1996); is about 20%, and the maximum difference can attain 77% with respect to the result of Anagnostou

(1980) as follows:

σ + γH − σt N = s cu

(26)

where H = C + D/2. σs is the surcharge of ground surface and σt is the critical supporting pressure applied on the tunnel face. By substituting Eqs. (23) and (24) into Eq. (26), the following expressions are obtained:

( (

⎧ Nc + ⎪ N= ⎨ ⎪− Nc + ⎩

)+ N)+

ρD N cu0 ρ

γD C ⎡ cu0 D

ρD cu0

γD C ⎡ cu0 D

ρ

⎣

+

1 2

+

1 2

⎣

( − (N

γ

) N ) ⎤ for blow - out ⎦

− Nγ − γt Nγt ⎤ for collapse ⎦ γ

−

γt γ

γt

(27) According to Eq. (27), the stability ratio N of the horseshoe-shaped shield tunnel can be deduced and compared with the existing mechanism as shown in Fig. 11. Since the existing studies on the circular tunnel face didn’t consider the gradient of the support pressure, γt is taken to be equal to 0 herein. The horseshoe-shaped tunnel face obviously improves the stability ratio N of the tunnel face by a maximum value of around 15% compared with that of the circular shield tunnel using continuous velocity field (M1 and M2 proposed by Mollon et al.,

FLAC3D L1/d=0

FLAC3D L1/d=0 L1/d=0.2

400

Limit support pressure σt (kPa)

Limit support pressure σt (kPa)

500

L1/d=0.4 L1/d=0.6 L1/d=0.8

300

L1/d=0.4

L1/d=1.0 L1/d=0.6

200 L1/d=0.6 L1/d=1.0

100

0

L1/d=1.0 L1/d=1.0 L1/d=1.0

0.5

1.0

L1/d=0.2

300

L1/d=0.4 L1/d=0.6 L1/d=0.8 L1/d=1.0

200

L1/d=0.4

100

L1/d=0.6

L1/d=0.8

1.5

2.0

2.5

0

3.0

L1/d=0.4

0.5

L1/d=0.6 L1/d=0.8

1.0

1.5

2.0

C/D

C/D

(a)

(b)

2.5

3.0

Fig. 8. Critical collapse pressure obtained from the analytical method and the numerical simulations: (a) cu0 = 20 kPa; (b) cu0 = 30 kPa. 9

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1000

L1/d=0.2

L1/d=0

L1/d=0.4

800

FLAC3D L1/d=0

L1/d=0

Limit support pressure σt (kPa)

Limit support pressure σt (kPa)

FLAC3D L1/d=0

L1/d=0.6

L1/d=0

L1/d=0.8 L1/d=1.0

L1/d=0

600 L1/d=0 L1/d=0

400 L1/d=0

L1/d=0.2

1000

0.5

1.0

1.5

2.0

2.5

L1/d=0

L1/d=0.4 L1/d=0.6

L1/d=0

L1/d=0.8

800

L1/d=1.0 L1/d=0

600

L1/d=0 L1/d=0 L1/d=0

400

200

L1/d=0

3.0

0.5

1.0

1.5

2.0

C/D

2.5

3.0

C/D

(a)

(b)

Fig. 9. Critical blow-out pressure obtained from the analytical method and the numerical simulations: (a) cu0 = 20 kPa; (b) cu0 = 30 kPa.

22

(2012). The stability number obtained from the former silo-wedge model is much larger than that of the later one. This is because the effect of the horizontal arch is taken into account by the former study. Moreover, the results of the former silo-wedge model are comparative with that of the proposed continuous velocity field. This difference is mainly influenced by the different limit states of the face failure defined in two methods. The silo-wedge method calculates the limit support pressure by the stress equilibrium of the assumed failure pattern (prismwedge model). But the limit analysis adopts the kinematically admissible velocity field to consider the balance of the internal and external energy.

20 18 16

N

14

C/D=2.0{ 300 C/D=1.5{ 200

100

C/D=1.0{ C/D=0.5{

0.0

12

6 4 2 0.5

1.0

1.5

2.0

2.5

3.0

C/D Fig. 11. Critical load factor obtained from the presented analytical method and other existing mechanisms.

1200

Limit support pressure σt (kPa)

Limit support pressure σt (kPa)

{

400 C/D=2.5

M1 Continuous velocity field (Mollon et al. 2013) M2 Continuous velocity field (Mollon et al. 2013) Rigid-block mechanism (Mollon et al. 2011) Rigid-block mechanism (Mollon et al. 2010) Rigid-block mechanism (Mollon et al. 2009) Rigid-block mechanism-K.A. (Davis et al. 1980) Rigid-block mechanism-S.A. (Davis et al. 1980) Silo-wedge model-L.E. (Anagnostou and Kovari 1996) Silo-wedge model-L.E. (Anagnostou 2012)

8

Dashed line=Numerical simulation Solid line=Limit analysis

{

Horseshoe-shaped L1/d=0.6

10

4.1.3. Comparisons with the circular face Fig. 12 shows the comparison of the limit support pressure between the horseshoe-shaped face and the circular face. The specific geometric parameters of R1 = 5.27 m, R2 = 3.62 m, R3 = 9.12 m and A = 81.49 m2 of the horseshoe-shaped shield tunnel of Menghua railway in China are adopted. Based on the geometric parameters of the horseshoe-shaped face, an equivalent circular cross section of D = 10.18 m and A = 81.49 m2 is obtained. The continuous velocity field is the most suitable kinematically admissible velocity field of the limit analysis for the purely cohesive soils, which has been shown in the

500 C/D=3.0

Horseshoe-shaped L1/d=0

}

Dashed line=Numerical simulation Solid line=Limit analysis

C/D=3.0

1000

}

C/D=2.5

}

800

C/D=2.0

} C/D=1.5

600

} C/D=1.0

400

} C/D=0.5 0.2

0.4

0.6

0.8

200 0.0

1.0

0.2

0.4

0.6

ρD/cu0

ρD/cu0

(a)

(b)

Fig. 10. Variation of face pressure σt with ρD/cu0: (a) Collapse; (b) blow-out. 10

0.8

1.0

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240

Equivalent circular face Horseshoe-shaped shield tunnel of Menghua railway

Limit support pressure σt (kPa)

220 200

previous section. The continuous velocity field of the limit analysis is used to calculate the limit support pressures of both the circular and horseshoe-shaped cross sections herein. It can be seen that horseshoeshaped face reduces the required face pressure by up to 21.5%, which obviously improves the face stability. Moreover, the silo-wedge model proposed by Anagnostou (2012) is adopted to compare the horseshoeshaped face with the circular face. Since the silo-wedge model of the limit equilibrium method needs to adopt an equivalent rectangular shape to represent a circular or horseshoe-shaped face, the height H and width B of a rectangle are equal to D and πD/4 respectively for a circular face. H and B are equal to R1 + d and A/H for a horseshoe-shaped face. A maximum difference of 9.6% between the horseshoe-shaped face and circular face is found for the silo-wedge model, which is less than that of the limit analysis. In general, there is an obvious difference between the horseshoe-shaped face and the circular cross section. The existing tunnel stability models don’t consider the horseshoe-shaped face well, which is not applicable to the horseshoe-shaped shield tunnel.

RD=Relative difference

RD=9.1%

RD=9.6%

180 160 140 120 100

RD=8.0% RD=13.3%

80 60 40

RD=4.8% RD=21.5%

20 0

C/D=0.5 C/D=1.0 C/D=2.0 C/D=0.5 C/D=1.0 C/D=2.0 Limit equilibrium Limit analysis

Fig. 12. Comparison of the limit support pressure between the horseshoeshaped face and circular face.

4.2. Design charts for the stability evaluation on a horseshoe-shape shield tunnel Using the results of the proposed failure mechanism, the design

4.0

1.0

Collapse Blow-out

3.5

Collapse Blow-out 0.8

3.0

ΝγCt=0.5764

C

Nγ =C/D+0.5764

0.6

Nγt

Nγ

2.5 2.0

ΝγBt=0.5223

0.4

NBγ =C/D+0.5223 1.5

0.2 1.0 0.5

0.5

1.0

1.5

2.0

2.5

0.0

3.0

0.5

1.0

1.5

C/D

(a) 50

N Cc=0.055×(C/D)3-0.760×(C/D)2+4.988×(C/D)+4.996

3.0

Collapse Blow-out Fitting by POLY3

45

R 2=0.9999

40 35

12

NBρ=-0.113×(C/D)3+1.998×(C/D)2+7.826×(C/D)+1.299 R2=1

Nc

Nρ

30 25 20

10 B c

3

2

15

N =0.0673×(C/D) -0.881×(C/D) +5.166×(C/D)+5.035 R 2=0.9998

8

6

2.5

(b)

16

14

2.0

C/D

10

Collapse Blow-out Fitting by POLY3 1.0

1.5

2.0

2.5

R2=1

5 0

0.5

NCρ =-0.107×(C/D)3+2.053×(C/D)2+6.915×(C/D)+1.776

3.0

0.5

1.0

1.5

2.0

C/D

C/D

(c)

(d)

Fig. 13. Stability charts of the coefficients: (a) Nγ; (b) Nγt; (c) Nc; (d) Nρ. 11

2.5

3.0

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C. Zhang, et al.

are provided as shown in Table 3.

charts for the stability evaluation on a horseshoe-shaped shield tunnel are provided. Nγ, Nγt, Nc and Nρ are the non-dimensional coefficients that represent the influences of soil unit weight γ, the gradient of support pressure, the cohesion c and the soil inhomogeneity on the stability of the tunnel face, respectively, as shown in Fig. 13. The coefficients Nγ, Nc and Nρ are dependent of the ratio C/D and given in Fig. 13. The coefficient Nγ of both collapse and blow-out cases can be expressed as a linear fitting curve. And the coefficients Nc and Nρ can be obtained by using a third order polynomial formula, which is in a good agreement with the solutions calculated by the analytical method. Therefore, the expressions of the limit support pressure of both the collapse and blow-out cases can be approximately expressed as follows: c

3

2

⎧ σt 0 − σs = γC − ⎡0.055 C − 0.760 C + 4.988 C + 4.996⎤ cu0 D D D ⎪ cu0 ⎣ ⎦ ⎪ ρD C 3 C 2 C − c ⎡−0.107 D + 2.053 D + 6.915 D + 1.776⎤ ⎪ ⎪ u0 ⎣ ⎦ b ⎨ σt 0 − σs γC C 3 C 2 C 0.067 D − 0.881 D + 5.166 D + 5.035⎤ ⎪ cu0 = cu0 − ⎡ ⎣ ⎦ ⎪ ρD C 3 C 2 C ⎪ ⎡ ⎤ 0.113 1.998 7.826 1.299 − − + + + ⎪ cu0 ⎣ D D D ⎦ ⎩

() () () ()

σt0c

() () () ()

() () () ()

4.3. Failure mechanism Fig. 16 plots two sketches of the failure mechanism for the horseshoe-shaped shield tunnels with different buried depths (C/D = 0.5 and 1.0) and different positions of maximal (L1/d = 0 and 0.6), and the distributions of the axial (vβ) and radial (vr) components are also included. The axial component vβ of velocity follows a parabola distribution in any cross-section passing through the origin O, which is consistent with the analytical assumption in the previous section. It is also worth noting that the derived radial velocity is zero at the point of maximal axial velocity and outside the envelope of the failure mechanism based on the normality condition of the limit analysis. The envelopes of vβ and vr well depict the continuous plastic deformation of soils influenced by the face collapse or blow-out of the shield tunnels. Fig. 17 shows the comparisons of the continuous velocity field with the results of FLAC3D, the rigid-block mechanism and the silo-wedge model. It can be found that the failure patterns of the presented continuous velocity field propagate from the tunnel face towards the ground surface, which are basically consistent with the displacement contours of FLAC3D. The failure zone extends in the soils ground with the buried depth C. When C/D = 0.5 and 1.0, the continuous velocity field is closer to the displacement contour of FLAC3D than the rigidblock mechanism and the silo-wedge model, which implies that the continuous velocity field is more suitable for the shallow buried tunnel. But when C/D = 2.0 and 3.0, it can be found that the continuous velocity field overestimates the failure zone and the assumed vertical failure plane of the silo-wedge model corresponds better with that of FLAC3D. It means that the failure zone will not infinitely expand its range with C/D and the boundary of the failure zone above the tunnel face is prone to be vertical in the soil ground. But it can be seen that there are still some differences between the result of FLAC3D and the silo-wedge model, the rigid-block mechanism when C/D = 2.0 and 3.0. The next section will focus on an improvement on the continuous velocity field for C/D ≥ 2.0.

(28)

where and σt0 represent the limit support pressures of collapse and blow-out, respectively. And the expressions of the critical load factor N are approximately given as follows: b

()

3

()

2

()

⎧ N Collapse = γD + ⎡0.055 C − 0.760 C + 4.988 C + 4.996⎤ 2cu0 D D D ⎪ ⎣ ⎦ ⎪ ρD C 3 C 2 C + c ⎡−0.107 D + 2.053 D + 6.915 D + 1.776⎤ ⎪ ⎪ u0 ⎣ ⎦ ⎨ Blow − out γD C 3 C 2 C = 2c − ⎡0.067 D − 0.881 D + 5.166 D + 5.035⎤ ⎪N u0 ⎣ ⎦ ⎪ ρD C 3 C 2 C ⎪ ⎡ ⎤ 0.113 1.998 7.826 1.299 − − + + + ⎪ cu0 ⎣ D D D ⎦ ⎩

()

()

()

()

()

()

()

()

()

(29) For soils with the undrained strength increasing linearly with the depth (ρ ≠ 0), the load factor N with different ρD/cu0 is shown in Fig. 14. It is obviously expected that the load factor N increases with ρD/cu0, which illustrates that the inhomogeneous soils will significantly reduce the stability of tunnel face compared with the homogeneous soils. From the view of the upper bound solutions, it is suggested that the neglect of the inhomogeneity of soils will overestimate the stability of the tunnel face. Thus it is necessary to take into account of the inhomogeneity of soils for the analytical method considering the economic benefit of the practical engineering. Fig. 15 presents the comparisons of the dimensionless coefficients Nγ, Nc and Nρ. Nγ generally follows a linear relationship of C/D. And non-linear increasing functions between Nc, Nρ and C/D are observed. Nγ in the present study agrees well with the design equation proposed by Ukritchon et al. (2017), the results obtained from Zhang et al. (2018a, 2018b) and Mollon et al. (2013), which are also calculated by the upper-bound method of the limit analysis. However, it is shown that the design equation proposed by Ukritchon et al. (2017) and the results of Mollon et al. (2013) provide smaller estimates on Nc and Nρ than the other upper-bound solutions. On one hand, the smaller Nc provided by Mollon et al. (2013) is because that an explicit finite-difference discretization scheme was numerically performed in the study, which finally resulted in the overestimation of the supporting collapse pressure (Klar and Klein, 2014; Zhang et al., 2017). On the other hand, the difference between the kinematical approach of the upper bound method and the finite element method of the numerical simulation also leads to smaller values of Nc and Nρ of the design equation proposed by Ukritchon et al. (2017). A higher σt for the collapse case and a lower σt for the blow-out case are obtained according to the numerical method. For the practical use in the geotechnical engineering, the dimensionless coefficients the coefficients Nγ, Nγt, Nc and Nρ versus C/D

5. Improvement on the continuous velocity field for a horseshoeshaped tunnel In the previous section, it can be found that when C/D is greater than and equal to 2.0, the obtained analytical solutions of the limit support pressures have a significant difference compared with the numerical results of FLAC3D and the failure zone of the analytical method 60 40

ρD/cu0=0 ρD/cu0=0.25 ρD/cu0=0.5 ρD/cu0=0.75

Collapse

ρD/cu0=1.0

N

20 0 -20

Blow-out

-40 -60 0.5

1.0

1.5

2.0

2.5

C/D Fig. 14. The load factor N for the nonhomogeneous soils. 12

3.0

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4.0 3.5

18

Present study (collapse) Present study (blow-out) Design equation by Ukritchon et al. (2017) (collapse) Analytical results by Zhang et al. (2018) (blow-out) Numerical results by Mollon et al. (2013) (collapse) Numerical results by Mollon et al. (2013) (blow-out)

Present study (collapse) Present sutdy (blow-out) Design equation by Ukritchon et al. (2017) (collapse) Fitting curve by Zhang et al. (2018) (collapse) Fitting curve by Zhang et al. (2018) (blow-out) Numerical results by Mollon et al. (2013) (collapse) Numerical results by Mollon et al. (2013) (blow-out)

16

3.0

14

Nc

Nγ

2.5 2.0

12

10

1.5 8

Design equation: Nc=7.8835×(C/D)0.3365

Design equation: Nγ=C/D+0.5

1.0

6

0.5

0.5

1.0

1.5

2.0

2.5

0.5

3.0

1.0

1.5

2.0

C/D

C/D

(a)

(b)

50

2.5

3.0

Present study (collapse) Present study (blow-out) Design equation by Ukritchon et al. (2017) (collapse) Fitting curve by Zhang et al. (2018) (collapse) Fitting curve by Zhang et al. (2018) (blow-out)

40

Nρ

30

20

10

Design equation: Nρ=7.6072×(C/D)1.2489 0

0.5

1.0

1.5

2.0

2.5

3.0

C/D

(c) Fig. 15. Comparisons of the non-dimensional coefficients: (a) Nγ; (b) Nc; (c) Nρ.

5.1. Improved velocity field

extends a larger range than the region of the displacement contour of FLAC3D. It is considered that the failure zone will not infinitely extend its region in the soil ground with C/D. Therefore, in order to improve the continuous velocity field of a horseshoe-shaped shield tunnel and make the velocity field closer to the realistic situation, this section will make some modifications to the original continuous velocity field.

Fig. 18 plots the improved velocity field for a horseshoe-shaped shield tunnel, which involves two zones of the velocity field: Zone Ⅰ is similar with the original continuous velocity field, but it is assumed that the axial component of the velocity turns vertical and the range of the velocity profile R(β) varies from Ri to Rf at a height of L0 above the

Table 3 Non-dimensional coefficients Nγ, Nγt, Nc and Nρ. C/D

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Collapse Nγ

Nγt

Nc

Nρ

Blow-out Nγ

Nγt

Nc

Nρ

1.0764 1.5764 2.0764 2.5764 3.0764 3.5764 4.0764 4.5764 5.0764 5.5764

0.5764 0.5764 0.5764 0.5764 0.5764 0.5764 0.5764 0.5764 0.5764 0.5764

7.2753 9.3232 10.9796 12.3610 13.5450 14.5810 15.5027 16.3323 17.0872 17.7799

5.7795 10.5716 16.3691 22.9767 30.2653 38.1419 46.5367 55.3961 64.6752 74.3387

1.0223 1.5233 2.0233 2.5233 3.0233 3.5233 4.0233 4.5233 5.0233 5.5233

0.5223 0.5223 0.5223 0.5223 0.5223 0.5223 0.5223 0.5223 0.5223 0.5223

7.3576 9.4546 11.0613 12.3589 13.4465 14.3830 15.2053 15.9385 16.6001 17.2030

5.7542 10.9308 17.1101 24.0606 31.6401 39.7515 48.3237 57.3027 66.6458 76.3184

13

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Fig. 16. Layout of the derived continuous velocity field for a horseshoe-shaped shield tunnel: (a) Collapse (C/D = 0.5, L1/d = 0.6); (b) blow-out (C/D = 1.0, L1/ d = 0).

tunnel crown rather than the buried depth C; Zone Ⅱ propagates vertically towards the ground surface and the range of the velocity profile Rf(y) increases with y in the vertical direction, varying from Rf to Rff. The formulas of R(β) and Rf(y) are expressed as follows:

R2max (θ , y ) = ⎧ ⎪

L1 cos θ +

Rf L12cos2 θ − L12 + R12 0 ⩽ θ < θ1

× r2 cos(θ − θ0 ) + r22cos2 (θ − θ0 ) − r22 + R22 θ1 ⩽ θ < θ2 ⎨ ⎪ r3 cos(θ ) + r32cos2 θ − r32 + R32 θ2 ⩽ θ < π ⎩

R (β ) = Ri + 2β (Rf − Ri ) π

⎧ ⎨ Rf (β ) = Rf + y (Rff − Rf ) (C − L0) ⎩

Rf + y (Rff − Rf ) (C − L0 )

(30)

(31)

And the vertical component v2y and the radial component v2r of velocity in the Zone Ⅱ are expressed as follows, respectively.

where L0 (0 ≤ L0 ≤ C), Rf (Ri ≤ Rf ≤ Ri + L0) and Rff (Rf ≤ Rff ≤ Ri + C)are considered as three variables herein. Thus the radius of the velocity profile R2max(θ, β) in the Zone Ⅱ is as follows:

14

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Fig. 17. Comparisons of the failure pattern of the tunnel face: (a) C/D = 0.5; (b) C/D = 1.0; (c) C/D = 2.0; (d) C/D = 3.0.

Fig. 18. Improved velocity field for a horseshoe-shaped shield tunnel. 2

Rf ⎧ f 2(y ) = [Rf + y·(Rff − Rf ) (C − L0 )]2 ⎪ ⎪ Rf θ1 ⎪ A2 = 2R {∫0 [L1 cos θ + L12cos2 θ − L12 + R12 ]2 dθ i ⎨ θ2 2 2 2 2 2 ⎪ + ∫θ1 [r2 cos(θ + θ0) + r2 cos (θ + θ0) − r2 + R2 ] dθ ⎪ π + ∫θ [r3 cos(θ) + r32cos2 θ − r32 + R32 ]2 dθ ⎪ 2 ⎩

2

f 2(y ) r ⎧ ⎪ v 2 y (r , θ , y ) = A2 × ⎡1 − R22 (θ, y) ⎤ max ⎣ ⎦ ⎨ K 2(θ, y ) M 2(y ) ⎪ v 2r (r , θ , y ) = − 4 r 3 − 2 r ⎩

(32)

where f2(y) is a function of y. And A2 is a constant. The expressions of f2(y) and A2 are given as follows:

}

The formulas of K2(θ, y) and M2(y) are expressed as follows: 15

(33)

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Fig. 21. Comparison of the stability number between the improved velocity field and the continuous velocity field with different ρD/cu0.

Fig. 19. The improvement of the limit support pressure σt.

∂v

⎧ ⎪ K 2(θ , y ) = ⎨ ⎪ ⎩

εrṙ = ∂rr ⎧ ⎪ ∂v 1 ∂v ⎪ εrθ ̇ = 2r ∂θr + r ∂rθ − vθ ⎪ ∂v y ⎪ 1 ∂v εrẏ = 2 ∂yr + ∂r ⎪

df 2(y ) ∂R2max (θ, y ) − R22max (θ, y ) dy ∂y 4 (θ, y ) A2·R2max

2·f 2(y ) R2max (θ, y )

M 2(y ) =

df 2(y ) dy

A2

(

(

(34)

⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

The corresponding strain rate tensor in the cylindrical coordinates (r, θ, y) of the Zone Ⅱ can be expressed as follows:

̇ εrẏ ⎤ ⎡ εrṙ εrθ ̇ εθθ ̇ εθy ̇ ⎥ ε ̇ = ⎢ εθr ⎢ ⎥ ̇yr εyθ ̇ εyy ̇ ε ⎣ ⎦

(35)

̇ = εθθ ̇ = εθy

1 2r

1 r

)

(v

(

r

+

∂v r ∂yθ

̇ = εyy

)

∂vθ ∂θ

+

)

∂v y ∂θ

)

∂v y ∂y

The Zone Ⅰ of the improved velocity field is the same as the original continuous veloctity field proposed in the previous section. The derivation process of the Zone Ⅰ can be conducted by Eqs. (14)–(18) only replacing C by a height of L0 in Eq. (3). Moreover, the work rates of the external forces and the energy dissipation of the Zone Ⅱ can be calculated as follows:

where

Fig. 20. Comparison of the limit support pressure between the improved velocity field and the existing literatures: (a) cu0 = 20 kPa; (b) cu0 = 30 kPa. 16

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Fig. 22. 3D improved failure mechanisms for the horseshoe-shaped shield tunnels: (a) C/D = 2.0; (b) C/D = 2.5; (c) C/D = 3.0.

⎧ ⎪ ⎪

Rff (Rf ≤ Rff ≤ Ri + C).

(θ, y ) C−L π R2 Ẇ 2 γ = 2γ ∫0 0 ∫0 ∫0 max v 2 y (r , θ , y )dr·r dθ ·dy (θ, y = C − L0) π R2 Ẇ 2s = 2σs ∫0 ∫0 max v 2 y (r , θ , y )dr·r dθ

(θ, y = C − L0) C − L0 π R2 ⎨ ̇ [cu0 + ρ (C − L0 − y ) cos β ]· ∫0 ∫0 max ⎪ W 2D = 2 ∫0 ⎪ ̇ ⎩ 2 max(|εi |)dr·r dθ ·dy

5.2. Results and discussion Fig. 19 presents the comparisons of the limit support pressure σt of C/D = 2.0 and 3.0. RD1 represents the relative differences between the original continuous velocity field and FLAC3D. And RD2 represents the relative differences between the improved velocity field and FLAC3D. It is shown that the improved velocity field provides a closer solution to the result of FLAC3D with the corresponding maximum difference less than around 16%. Moreover, due to the improvement on the continuous velocity field, the improved velocity field can improve the limit support pressure of the continuous velocity field by 7.8–19% for collapse case (lower bound) and decrease the limit support pressure of that by approximately 5% for blow-out case (upper bound). The improved velocity field provides a better solution of the limit support pressure than the continuous velocity field. Fig. 20 presents the comparison of the limit support pressure

(36)

The critical upper-bound solution of the support pressure for the collapse and blow-out cases can be calculated as follows: collapse = (Ẇγ + Ẇ 2 γ + Ẇ 2s − ẆD − Ẇ 2D) Wṫ ⎧ σt ⎨ σtblow − out = (Ẇγ + Ẇ 2 γ + Ẇ 2s + ẆD + Ẇ 2D) Wṫ ⎩

(37)

In the improved velocity field, L0, Rf and Rff are three variables to be determined to the critical support pressure σt. To determine the critical pressures for the collapse and blow-out cases, the maximization of σtcollapse and minimization of σtblow-out are performed under the help of the optimization tool implemented in the software MATLAB with respect to the three variables L0 (0 ≤ L0 ≤ C), Rf (Ri ≤ Rf ≤ Ri + L0) and 17

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Fig. 23. Comparison between the improved velocity field and the continuous velocity field for the horseshoe-shaped shield tunnels: (a) C/D = 2.0; (b) C/D = 2.5; (c) C/D = 3.0.

between the improved velocity field and the existing literatures. RD1 and RD2 represent the relative difference with the improved velocity field for cu = 20 kPa and 30 kPa, respectively. For a shield tunnel with an equivalent diameter, the horseshoe-shaped tunnel can reduce the required face pressure of the circular tunnel with a maximum difference of 19.8% for C/D = 2.0 and cu = 20 kPa, which improves the stability of the tunnel face. Compared with the result of the best rigid-block failure mechanism (Mollon et al., 2011), the improved velocity field can improve the limit support pressure of that with a maximum difference of 62.5% for C/D = 3.0 and cu = 30 kPa. The improved velocity field and continuous velocity field generally provide a better upper bound solution than the existing rigid-block mechanism. Fig. 21 shows the comparison of the stability number N between the the improved velocity field and the continuous velocity field. The stability ratio N increases linearly with the dimensionless parameter ρD/ cu0. The improved velocity field significantly improves the upper bound solutions of the original continuous velocity field. Compared with the result of the numerical simulation, the improved velocity field leads to a better result than that of the original velocity field. This comparison reveals that the improved velocity field can obtain a better upper solution when C/D ≥ 2.0. Fig. 22 shows the 3D improved velocity field for the horseshoeshaped Baicheng shield tunnel of Menghua railway of C/D = 2.0, 2.5 and 3.0. The specific geometric and gologic parameters are shown in Table 2. It can be seen that the improved velocity field obviously limits boundary of the continuous velocity field, and the improved failure mechanism overcomes the shortcoming that the failuer zone continuously expands its region with C/D. Fig. 23 shows the comparisons of the failure pattern between the improved velocity field and other

existing models. On one hand, it can be seen that the improved velocity field of the proposed model obviously limits boundary of the continuous velocity field, and overcomes the shortcoming that the failure zone constantly extends its region with C/D. On the other hand, the improved velocity field takes the vertical failure plane of the silo-wedge model and the rigid-block mechanism into account. The soil deformation zone is restricted in a narrower region and mainly concentrates in the vicinity of the tunnel face and the zone above the tunnel face by the parameters L0 and Rf, which is closer to the displacement contour of FLAC3D than other models. 6. Conclusions Based on the kinematic method of the limit analysis, this paper proposed a continuous velocity field to assess the face stability of a horseshoe-shaped tunnel in undrained clays. A series of numerical simulations are performed to validate the proposed analytical method. The comparisons between the proposed method and the numerical simulations, other available mechanisms are shown. Moreover, a fitting formula is provided for the limit support pressure and load factor. The design charts of the related non-dimensional parameters are presented for the stability evaluation on the horseshoe-shaped tunnel face. Finally, an improved velocity field of a horseshoe-shaped shield tunnel is proposed, which provides a closer result to that of the numerical simulations. The main conclusions are drawn as follows: (1) The continuous velocity field provided the critical upper-bound solution when the position of maximal axial velocity L1/d was equal to 0.6 (L1/d = 0 for blow-out case) by comparison with the results 18

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of FLAC3D, which represented a compromising value between the computation time and the most realistic deformation. (2) Compared with the circular shield tunnels, the horseshoe-shaped shield tunnels significantly improved the stability of the tunnel face by a maximum of 17%. The investigation on the critical velocity field of the horseshoe-shaped tunnel face was necessary and valuable. (3) The design charts of the non-dimensional coefficients Nγ, Nγt, Nc and Nρ were provided for estimating the limit support pressure and stability ratio. The coefficient Nγ with C/D conforms to a linear fitting curve. Nc and Nρ can be obtained by using a third order polynomial formula. Moreover, the inhomogeneity ρ of soil ground has a significant influence on the stability and the neglect of the inhomogeneity of soils will lead to an overestimate on the face stability. (4) The original continuous velocity field is modified to be closer to the realistic situation. The improved velocity field improves the critical upper bound solutions and provides a closer failure pattern to the result of FLAC3D.

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