# Face Stability Analysis of Shield Tunnels in Homogeneous Soil Overlaid by Multilayered Cohesive-Frictional Soils

Authors: Kaihang Han; Chengping Zhang; Wei Li; Caixia Guo
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

In order to better interpret failure features of the failure of soil in front of tunnel face, a new three-dimensional failure mechanism is proposed to analyze the limit support pressure of the tunnel face in multilayered cohesive-frictional soils. The new failure mechanism is composed of two truncated cones that represent the shear failure band and a distributed force acting on the truncated cones that represents the pressure arch effect. By introducing the concept of Terzaghi earth pressure theory, approximation of limit support pressures is calculated using the limit analysis methods. Then the limit support pressures obtained from the new failure mechanism and the existing approaches are compared, which show that the results obtained from the new mechanism in this paper provide relatively satisfactory results.


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## Body

## 1. Introduction

Nowadays, tunnel constructions in urban shallow soft ground are more frequently being carried out using closed shields, which are basically divided into compressed-air shields, slurry shields, and earth pressure balanced shields. By supporting the excavation face and preventing the seepage flow to the face, closed shields control the surface settlement and limit the risk of tunnel face failure through the continuous support of the face during excavation. However, lack of sufficient face support leads to the instability of tunnel face. In extreme cases, the collapse propagates up to the ground surface creating a surface depression. Researchers have developed experimental, numerical, and analytical approaches to investigate the tunnel face stability and to determine the limit support pressure of the tunnel face in cohesive-frictional soils.The stability of tunnel face in cohesive-frictional soils has been analyzed by using experimental tests (laboratory centrifuge tests [1–3] or 1 g model tests [4–6]) and the numerical simulations (the Finite Element Method [7–9], the Finite Difference Method [6, 10, 11], and the Discrete Element Method [12–14]). Both the limit equilibrium methods [15–23] and the limit analysis methods [9–11, 24–31] were used by other researchers to investigate the stability of tunnel face in cohesive-frictional soils. Actually, from the analytical approaches mentioned above, it can be concluded that either limit equilibrium methods or limit analysis methods have advantages and disadvantages. Sloan [32] presented detailed reviews and comparisons of the analytical approaches for assessing geotechnical stability.From the results obtained from experimental tests and numerical simulations, it is shown that the failure of soil in front of tunnel face shows two main features, that is, shear failure band in the lower part and pressure arch effect in the upper part. Referring to the remarks claimed by Sloan [32] and considering the specific problems of tunnel face stability analysis, the main characteristics of the limit equilibrium and limit analysis to assess the stability of tunnel face in cohesive-frictional soils are summarized as follows. As a tradition stability analysis method, the limit equilibrium methods are widely used in the practical engineering to analyze the stability of tunnel face. Due to its simplicity, a multitude of prediction models have been developed to solve the complex situations of tunnel face, for example, seepage flow conditions and bolt reinforcement. To reflect the feature of pressure arch effect, most of failure mechanisms based on limit equilibrium methods adopted Terzaghi earth pressure theory. However, limit equilibrium methods neglect the stress-strain relationship of the soil, which is a condition that must be satisfied for a complete solution stated by Chen [33]. Compared to the limit equilibrium methods, the limit analysis methods have a strictly theoretic basis.Based on the limit analysis methods, Zhang et al. [34] proposed a three-dimensional failure mechanism, which is composed of four truncated cones and a distributed force acting on those truncated cones. Their failure mechanism combines the generality of earth pressure theory with the rigour of limit analysis. And the results show that their improved treatments are effective. However, their works are limited to the case of homogeneous soils. Practical experience shows that the stability of the face is often a problem in heterogeneous soils (Ibrahim et al. [9]; Senent and Jimenez [11]). The shear strength parameters of soils in different strata are different from each other. In this paper, for multilayered cohesive-frictional soils, a new three-dimensional failure mechanism is proposed to determine the limit support pressure of the tunnel face using limit analysis methods. The new failure mechanism is composed of two truncated cones on which a distributed force acts. The distributed force could be calculated by using Terzaghi earth theory. Then, the limit support pressures obtained from the new failure mechanism and the existing approaches are compared.


## 2. Limit Analysis of the Tunnel Face Stability

### 2.1. The New Failure Mechanism

The tunnel face stability to be analyzed in this paper is idealized as shown in Figure1. By considering that the tunnel is a rigid circular cylinder of diameter D driven under a depth of cover C, a surcharge σs is applied on the ground surface, and σT is the uniform support pressure on the tunnel face.Figure 1
The improved failure mechanism.Referring to the concept proposed by Zhang et al. [34], a new prediction model is established to analyze face stability of shallow circular tunnels in multilayered cohesive-frictional soils. The new failure mechanism is composed of two truncated cones on which a distributed force acts. The two truncated cones were firstly proposed by Leca and Dormieux [26] to investigate the tunnel face stability in homogenous soil. Since this paper considers a multilayered cohesive-frictional soil (i.e., the crossed layer is assumed to be homogeneous whereas the cover soil is layered), two truncated cones proposed by Leca and Dormieux [26] are modified and only the part in crossed layer of the two truncated cones is preserved. The new failure mechanism considers the effect of the cover layers as a distributed force according to the Terzaghi earth theory. The same treatments were adopted in the works of Broere [18]; Kirsch and Kolymbas [19]; Anagnostou [20]; and Anagnostou and Perazzelli [21], which are based on the limit equilibrium method.


#### 2.1.1. Geometric Properties

The intersections of two truncated cones, tunnel face, and horizontal plane crossing the top of the tunnel face are two ellipses and a circle and are calledΣ1, Σ1,2, and Σ2. The intersection of the first truncated cone (adjacent to the tunnel face) with the circular tunnel face is an ellipse Σ1, with semiaxis lengths of a1 and b1 and with area of A1 that are calculated as follows (cf. Figure 1):(1)a1=D2,b1=D2cos⁡α-φcos⁡α+φcos⁡φ,A1=πD24cosα-φcos⁡α+φcos⁡φ,where φ defines the opening angles of the two truncated rigid cones that are equal to 2φ and α is the angle between the axis of the first truncated rigid cone which is adjacent and the horizontal.The area of the contact elliptical surfaceΣ1,2 between two truncated cones is ellipse with semiaxis lengths of a12 and b12 and with area of A12 is described as follows:(2)a12=D2cos⁡α+φsinβ+φ,b12=D2cos⁡α+φcos⁡φsin⁡β-φsin⁡β+φ1/2,A12=πD24cos2⁡α+φcos⁡φsin⁡β-φsin⁡β+φ3/2.The intersection of the second truncated cone with the horizontal plane across the top of the tunnel face is a circleΣ2 with radius of r2 and with area of A2 as follows:(3)r2=D2sin⁡βcos⁡αsin⁡φsin⁡β+φ-1tan⁡φ,A2=πD24sin⁡βcos⁡αsin⁡φsin⁡β+φ-12tan2⁡φ.In addition, the lateral surfaces and volumes of the two truncated cones are as follows:(4)Sb1=πD24cos⁡αsin⁡φRA-cos⁡α+φRCRD,Sb2=πD24cos⁡α+φRCRD-cos2⁡φsin⁡φcos2⁡α+φRC4,Vb1=πD312RARB-cos⁡φ2sin⁡φRC3,Vb2=πD312cos⁡φ2sin⁡φRC3-cos2⁡φ2cos2⁡α+φRC4,where(5)RA=cos⁡α-φcos⁡α+φcos⁡φ,RB=cos⁡α-φcos⁡α+φsin⁡2φ,RC=cos⁡α+φcos⁡φsin⁡β-φsin⁡β+φ1/2,RD=sin⁡βsin⁡φsin⁡β+φ.


#### 2.1.2. The Distributed Force Acting on the Two Truncated Cones

For cohesive-frictional soils, the results obtained from experimental tests and numerical simulations show that the failure of soil in front of tunnel face demonstrates two main features, that is, shear failure band in the lower part and pressure arch effect in the upper part. The existing researches [18–21] considered the effect of the cover layers as a distributed force and achieved satisfactory results, which is calculated using the Terzaghi earth pressure theory. In this paper, the distributed force acting on the two truncated cones is also calculated by Terzaghi earth pressure theory as shown in Figure 2.Figure 2
Distributed force calculated by Terzaghi earth pressure theory.Assuming that the cover layer above the tunnel is homogeneous and the soil in cover layer satisfies the Mohr-Coulomb failure criteria, the vertical equilibrium of a microthin layerdz at z depth and the boundary condition at the ground surface are as follows:(6)πr22σv+πr22γdz=πr22σv+dσv+2πrτdz,τ=c+K0tan⁡φ,z=0∣σv=σS,where the lateral pressure coefficient is defined as K0=(cos2⁡θ0+Kasin2⁡θ0)/(1/31-Kasin2⁡θ0+Ka),  θ0=π/4+φ/2,  Ka=tan2⁡π/4-φ/2 [35].Equation (6) is a differential equation for the vertical stress σv(z). To solve the differential equation and consider the boundary condition, the result of vertical stress at any depth is determined as follows:(7)σv=γr2K0tan⁡φ1-e-2K0tan⁡φ/rC-cK0tan⁡φ1-e-2K0tan⁡φ/rC+σSe-2K0tan⁡φ/rC.Since the ground above the tunnel is heterogeneous and consists ofn horizontal layers (cf. Figure 2), the vertical stress should be applied recursively for every layer from the top to down by treating the pressures of every overlying layer as a surface loading as follows: (8)Pi=γir2K0itan⁡φi1-e-2K0itan⁡φi/rCi-ciK0itan⁡φi1-e-2K0itan⁡φi/rCi+Pi-1e-2K0itan⁡φi/rCii=1,…,n,P0=σS,Pn=σv.Then the final result of vertical stress distribution acting on the two truncated cones is obtained as follows:(9)σv=γnr2K0ntan⁡φn1-e-2K0ntan⁡φn/rCn+∑k=1n-1γkr2K0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm-cnK0ntan⁡φn1-e-2K0ntan⁡φn/rCn-∑k=1n-1ckK0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm+σS∏k=1ne-2K0ktan⁡φk/rCk,where Ck is the thickness of layer k and ck, φk, K0k, and γk denote its unit weight and shear strength parameters.


#### 2.1.3. Velocity Field

The two truncated rigid cones are translated with velocities with different directions, which are collinear with the cones’ axes and are at an angleφ to the discontinuity surface. The velocity of first cone and the relative velocity between the two cones are described by the following equations:(10)V1=sin⁡β+φsin⁡β-φV2,V12=cos⁡αsin⁡β-φV2.


### 2.2. Limit Support Pressure

To satisfy the stability conditions of the tunnel face according to the upper bound theorem, the following relation is considered:(11)Pe≤Pv,where Pe represents the power of the external loads and Pv denotes the dissipation power. The power of the external loads, Pe, is the sum of three components: PT, the power of the support pressure σT, Ps, the power of the vertical stress σv, and Pγ, the power of the soil unit weight γ:(12)Pe=PT+Pσv+Pγ,PV=P1v+P2v+P3v,where(13)PT=πD24cos2⁡α+φcos2⁡φ-cos⁡αRC2RAσTV2,Pσv=πD24cos2⁡α+φcos2⁡φRE2cos2⁡α+φσvV2,Pγ=πD24cos2⁡α+φcos2⁡φsin⁡αRARBRC2+cos⁡αcos⁡φcos⁡β+φ2sin⁡φsin⁡β+φRC-RE32sin⁡φcos2⁡α+φγD3V2,P1v=πD24cos2⁡α+φcos2⁡φcos⁡αsin⁡φRARC2-cos⁡α+φRDRCccos⁡φV2,P2v=πD24cos2⁡α+φcos2⁡φcos2⁡φcos2⁡α+φRCRD-RE2sin⁡φcos2⁡α+φccos⁡φV2,P3v=πD24cos2⁡α+φcos2⁡φ2cos⁡βsin2φcos⁡α+φsin⁡βRD2RCccos⁡φV2.By equating the total rate of external work to the total rate of internal energy dissipation, as shown in (11), the pressure σT at the tunnel face is obtained by (14) with (15) as follows:(14)σT=NS·σS+Nγ·γD+Nc·c+∑k=1nNk·γkD+Nck·ck,(15)NS=1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RA∏k=1ne-2K0ktan⁡φk/rCk,Nγ=13tan⁡αRB+cos⁡φcos⁡β+φ2sin⁡φsin⁡β+φRC3RA-12sin⁡φcos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE3RA+1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RAr2K0tan⁡φD1-e-2K0tan⁡φ/rC,Nc=1tan⁡φ1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RA-1+1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RA-1K0tan⁡φ1-e-2K0tan⁡φ/rC,∑k=1nNγk·γkD+Nck·ck=1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RAγnr2K0ntan⁡φn1-e-2K0ntan⁡φn/rCn+∑k=1n-1γkr2K0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm-cnK0ntan⁡φn1-e-2K0ntan⁡φn/rCn-∑k=1n-1ckK0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm.In (14), σT, Nγ, Nc, and NS depend on the mechanical and geometrical characteristics c, φ, and C/D and on the angular parameters of the failure mechanism α. These parameters were obtained by maximizing σT in (14) with respect to the angles α. An upper bound solution can be found by numerically optimizing (14) with respect to the angle α.


## 2.1. The New Failure Mechanism

The tunnel face stability to be analyzed in this paper is idealized as shown in Figure1. By considering that the tunnel is a rigid circular cylinder of diameter D driven under a depth of cover C, a surcharge σs is applied on the ground surface, and σT is the uniform support pressure on the tunnel face.Figure 1
The improved failure mechanism.Referring to the concept proposed by Zhang et al. [34], a new prediction model is established to analyze face stability of shallow circular tunnels in multilayered cohesive-frictional soils. The new failure mechanism is composed of two truncated cones on which a distributed force acts. The two truncated cones were firstly proposed by Leca and Dormieux [26] to investigate the tunnel face stability in homogenous soil. Since this paper considers a multilayered cohesive-frictional soil (i.e., the crossed layer is assumed to be homogeneous whereas the cover soil is layered), two truncated cones proposed by Leca and Dormieux [26] are modified and only the part in crossed layer of the two truncated cones is preserved. The new failure mechanism considers the effect of the cover layers as a distributed force according to the Terzaghi earth theory. The same treatments were adopted in the works of Broere [18]; Kirsch and Kolymbas [19]; Anagnostou [20]; and Anagnostou and Perazzelli [21], which are based on the limit equilibrium method.


### 2.1.1. Geometric Properties

The intersections of two truncated cones, tunnel face, and horizontal plane crossing the top of the tunnel face are two ellipses and a circle and are calledΣ1, Σ1,2, and Σ2. The intersection of the first truncated cone (adjacent to the tunnel face) with the circular tunnel face is an ellipse Σ1, with semiaxis lengths of a1 and b1 and with area of A1 that are calculated as follows (cf. Figure 1):(1)a1=D2,b1=D2cos⁡α-φcos⁡α+φcos⁡φ,A1=πD24cosα-φcos⁡α+φcos⁡φ,where φ defines the opening angles of the two truncated rigid cones that are equal to 2φ and α is the angle between the axis of the first truncated rigid cone which is adjacent and the horizontal.The area of the contact elliptical surfaceΣ1,2 between two truncated cones is ellipse with semiaxis lengths of a12 and b12 and with area of A12 is described as follows:(2)a12=D2cos⁡α+φsinβ+φ,b12=D2cos⁡α+φcos⁡φsin⁡β-φsin⁡β+φ1/2,A12=πD24cos2⁡α+φcos⁡φsin⁡β-φsin⁡β+φ3/2.The intersection of the second truncated cone with the horizontal plane across the top of the tunnel face is a circleΣ2 with radius of r2 and with area of A2 as follows:(3)r2=D2sin⁡βcos⁡αsin⁡φsin⁡β+φ-1tan⁡φ,A2=πD24sin⁡βcos⁡αsin⁡φsin⁡β+φ-12tan2⁡φ.In addition, the lateral surfaces and volumes of the two truncated cones are as follows:(4)Sb1=πD24cos⁡αsin⁡φRA-cos⁡α+φRCRD,Sb2=πD24cos⁡α+φRCRD-cos2⁡φsin⁡φcos2⁡α+φRC4,Vb1=πD312RARB-cos⁡φ2sin⁡φRC3,Vb2=πD312cos⁡φ2sin⁡φRC3-cos2⁡φ2cos2⁡α+φRC4,where(5)RA=cos⁡α-φcos⁡α+φcos⁡φ,RB=cos⁡α-φcos⁡α+φsin⁡2φ,RC=cos⁡α+φcos⁡φsin⁡β-φsin⁡β+φ1/2,RD=sin⁡βsin⁡φsin⁡β+φ.


### 2.1.2. The Distributed Force Acting on the Two Truncated Cones

For cohesive-frictional soils, the results obtained from experimental tests and numerical simulations show that the failure of soil in front of tunnel face demonstrates two main features, that is, shear failure band in the lower part and pressure arch effect in the upper part. The existing researches [18–21] considered the effect of the cover layers as a distributed force and achieved satisfactory results, which is calculated using the Terzaghi earth pressure theory. In this paper, the distributed force acting on the two truncated cones is also calculated by Terzaghi earth pressure theory as shown in Figure 2.Figure 2
Distributed force calculated by Terzaghi earth pressure theory.Assuming that the cover layer above the tunnel is homogeneous and the soil in cover layer satisfies the Mohr-Coulomb failure criteria, the vertical equilibrium of a microthin layerdz at z depth and the boundary condition at the ground surface are as follows:(6)πr22σv+πr22γdz=πr22σv+dσv+2πrτdz,τ=c+K0tan⁡φ,z=0∣σv=σS,where the lateral pressure coefficient is defined as K0=(cos2⁡θ0+Kasin2⁡θ0)/(1/31-Kasin2⁡θ0+Ka),  θ0=π/4+φ/2,  Ka=tan2⁡π/4-φ/2 [35].Equation (6) is a differential equation for the vertical stress σv(z). To solve the differential equation and consider the boundary condition, the result of vertical stress at any depth is determined as follows:(7)σv=γr2K0tan⁡φ1-e-2K0tan⁡φ/rC-cK0tan⁡φ1-e-2K0tan⁡φ/rC+σSe-2K0tan⁡φ/rC.Since the ground above the tunnel is heterogeneous and consists ofn horizontal layers (cf. Figure 2), the vertical stress should be applied recursively for every layer from the top to down by treating the pressures of every overlying layer as a surface loading as follows: (8)Pi=γir2K0itan⁡φi1-e-2K0itan⁡φi/rCi-ciK0itan⁡φi1-e-2K0itan⁡φi/rCi+Pi-1e-2K0itan⁡φi/rCii=1,…,n,P0=σS,Pn=σv.Then the final result of vertical stress distribution acting on the two truncated cones is obtained as follows:(9)σv=γnr2K0ntan⁡φn1-e-2K0ntan⁡φn/rCn+∑k=1n-1γkr2K0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm-cnK0ntan⁡φn1-e-2K0ntan⁡φn/rCn-∑k=1n-1ckK0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm+σS∏k=1ne-2K0ktan⁡φk/rCk,where Ck is the thickness of layer k and ck, φk, K0k, and γk denote its unit weight and shear strength parameters.


### 2.1.3. Velocity Field

The two truncated rigid cones are translated with velocities with different directions, which are collinear with the cones’ axes and are at an angleφ to the discontinuity surface. The velocity of first cone and the relative velocity between the two cones are described by the following equations:(10)V1=sin⁡β+φsin⁡β-φV2,V12=cos⁡αsin⁡β-φV2.


## 2.1.1. Geometric Properties

The intersections of two truncated cones, tunnel face, and horizontal plane crossing the top of the tunnel face are two ellipses and a circle and are calledΣ1, Σ1,2, and Σ2. The intersection of the first truncated cone (adjacent to the tunnel face) with the circular tunnel face is an ellipse Σ1, with semiaxis lengths of a1 and b1 and with area of A1 that are calculated as follows (cf. Figure 1):(1)a1=D2,b1=D2cos⁡α-φcos⁡α+φcos⁡φ,A1=πD24cosα-φcos⁡α+φcos⁡φ,where φ defines the opening angles of the two truncated rigid cones that are equal to 2φ and α is the angle between the axis of the first truncated rigid cone which is adjacent and the horizontal.The area of the contact elliptical surfaceΣ1,2 between two truncated cones is ellipse with semiaxis lengths of a12 and b12 and with area of A12 is described as follows:(2)a12=D2cos⁡α+φsinβ+φ,b12=D2cos⁡α+φcos⁡φsin⁡β-φsin⁡β+φ1/2,A12=πD24cos2⁡α+φcos⁡φsin⁡β-φsin⁡β+φ3/2.The intersection of the second truncated cone with the horizontal plane across the top of the tunnel face is a circleΣ2 with radius of r2 and with area of A2 as follows:(3)r2=D2sin⁡βcos⁡αsin⁡φsin⁡β+φ-1tan⁡φ,A2=πD24sin⁡βcos⁡αsin⁡φsin⁡β+φ-12tan2⁡φ.In addition, the lateral surfaces and volumes of the two truncated cones are as follows:(4)Sb1=πD24cos⁡αsin⁡φRA-cos⁡α+φRCRD,Sb2=πD24cos⁡α+φRCRD-cos2⁡φsin⁡φcos2⁡α+φRC4,Vb1=πD312RARB-cos⁡φ2sin⁡φRC3,Vb2=πD312cos⁡φ2sin⁡φRC3-cos2⁡φ2cos2⁡α+φRC4,where(5)RA=cos⁡α-φcos⁡α+φcos⁡φ,RB=cos⁡α-φcos⁡α+φsin⁡2φ,RC=cos⁡α+φcos⁡φsin⁡β-φsin⁡β+φ1/2,RD=sin⁡βsin⁡φsin⁡β+φ.


## 2.1.2. The Distributed Force Acting on the Two Truncated Cones

For cohesive-frictional soils, the results obtained from experimental tests and numerical simulations show that the failure of soil in front of tunnel face demonstrates two main features, that is, shear failure band in the lower part and pressure arch effect in the upper part. The existing researches [18–21] considered the effect of the cover layers as a distributed force and achieved satisfactory results, which is calculated using the Terzaghi earth pressure theory. In this paper, the distributed force acting on the two truncated cones is also calculated by Terzaghi earth pressure theory as shown in Figure 2.Figure 2
Distributed force calculated by Terzaghi earth pressure theory.Assuming that the cover layer above the tunnel is homogeneous and the soil in cover layer satisfies the Mohr-Coulomb failure criteria, the vertical equilibrium of a microthin layerdz at z depth and the boundary condition at the ground surface are as follows:(6)πr22σv+πr22γdz=πr22σv+dσv+2πrτdz,τ=c+K0tan⁡φ,z=0∣σv=σS,where the lateral pressure coefficient is defined as K0=(cos2⁡θ0+Kasin2⁡θ0)/(1/31-Kasin2⁡θ0+Ka),  θ0=π/4+φ/2,  Ka=tan2⁡π/4-φ/2 [35].Equation (6) is a differential equation for the vertical stress σv(z). To solve the differential equation and consider the boundary condition, the result of vertical stress at any depth is determined as follows:(7)σv=γr2K0tan⁡φ1-e-2K0tan⁡φ/rC-cK0tan⁡φ1-e-2K0tan⁡φ/rC+σSe-2K0tan⁡φ/rC.Since the ground above the tunnel is heterogeneous and consists ofn horizontal layers (cf. Figure 2), the vertical stress should be applied recursively for every layer from the top to down by treating the pressures of every overlying layer as a surface loading as follows: (8)Pi=γir2K0itan⁡φi1-e-2K0itan⁡φi/rCi-ciK0itan⁡φi1-e-2K0itan⁡φi/rCi+Pi-1e-2K0itan⁡φi/rCii=1,…,n,P0=σS,Pn=σv.Then the final result of vertical stress distribution acting on the two truncated cones is obtained as follows:(9)σv=γnr2K0ntan⁡φn1-e-2K0ntan⁡φn/rCn+∑k=1n-1γkr2K0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm-cnK0ntan⁡φn1-e-2K0ntan⁡φn/rCn-∑k=1n-1ckK0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm+σS∏k=1ne-2K0ktan⁡φk/rCk,where Ck is the thickness of layer k and ck, φk, K0k, and γk denote its unit weight and shear strength parameters.


## 2.1.3. Velocity Field

The two truncated rigid cones are translated with velocities with different directions, which are collinear with the cones’ axes and are at an angleφ to the discontinuity surface. The velocity of first cone and the relative velocity between the two cones are described by the following equations:(10)V1=sin⁡β+φsin⁡β-φV2,V12=cos⁡αsin⁡β-φV2.


## 2.2. Limit Support Pressure

To satisfy the stability conditions of the tunnel face according to the upper bound theorem, the following relation is considered:(11)Pe≤Pv,where Pe represents the power of the external loads and Pv denotes the dissipation power. The power of the external loads, Pe, is the sum of three components: PT, the power of the support pressure σT, Ps, the power of the vertical stress σv, and Pγ, the power of the soil unit weight γ:(12)Pe=PT+Pσv+Pγ,PV=P1v+P2v+P3v,where(13)PT=πD24cos2⁡α+φcos2⁡φ-cos⁡αRC2RAσTV2,Pσv=πD24cos2⁡α+φcos2⁡φRE2cos2⁡α+φσvV2,Pγ=πD24cos2⁡α+φcos2⁡φsin⁡αRARBRC2+cos⁡αcos⁡φcos⁡β+φ2sin⁡φsin⁡β+φRC-RE32sin⁡φcos2⁡α+φγD3V2,P1v=πD24cos2⁡α+φcos2⁡φcos⁡αsin⁡φRARC2-cos⁡α+φRDRCccos⁡φV2,P2v=πD24cos2⁡α+φcos2⁡φcos2⁡φcos2⁡α+φRCRD-RE2sin⁡φcos2⁡α+φccos⁡φV2,P3v=πD24cos2⁡α+φcos2⁡φ2cos⁡βsin2φcos⁡α+φsin⁡βRD2RCccos⁡φV2.By equating the total rate of external work to the total rate of internal energy dissipation, as shown in (11), the pressure σT at the tunnel face is obtained by (14) with (15) as follows:(14)σT=NS·σS+Nγ·γD+Nc·c+∑k=1nNk·γkD+Nck·ck,(15)NS=1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RA∏k=1ne-2K0ktan⁡φk/rCk,Nγ=13tan⁡αRB+cos⁡φcos⁡β+φ2sin⁡φsin⁡β+φRC3RA-12sin⁡φcos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE3RA+1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RAr2K0tan⁡φD1-e-2K0tan⁡φ/rC,Nc=1tan⁡φ1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RA-1+1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RA-1K0tan⁡φ1-e-2K0tan⁡φ/rC,∑k=1nNγk·γkD+Nck·ck=1cos⁡αcos2⁡φsin⁡β-φsin⁡β+φRE2RAγnr2K0ntan⁡φn1-e-2K0ntan⁡φn/rCn+∑k=1n-1γkr2K0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm-cnK0ntan⁡φn1-e-2K0ntan⁡φn/rCn-∑k=1n-1ckK0ktan⁡φk1-e-2K0ktan⁡φk/rCk∏m=k+1ne-2K0mtan⁡φm/rCm.In (14), σT, Nγ, Nc, and NS depend on the mechanical and geometrical characteristics c, φ, and C/D and on the angular parameters of the failure mechanism α. These parameters were obtained by maximizing σT in (14) with respect to the angles α. An upper bound solution can be found by numerically optimizing (14) with respect to the angle α.


## 3. Comparisons

To validate the results obtained from the limit analysis developed in this paper, comparisons between the results of this work and existing approaches (Broere [18]; Tang et al. [30]; Senent and Jimenez [11]) were performed. Overall, the model proposed by Broere [18] is based on limit equilibrium methods while the models proposed by Tang et al. [30], Senent and Jimenez [11], and this paper are based on limit analysis methods, respectively.Three sets of analyses were carried out by referring to the researches recently reported by Tang et al. [30] and Senent and Jimenez [11]. And the detailed soil parameters of layers are shown in Table 1. The unit weight γ is equal to 18 kN/m3 for the layers in all cases (C/D=1.5).Table 1
Soil parameters of layers.
Sets’ number
Layers’ name
Layers’ number
c (kPa)
φ (°)
C (m)
1
Single crossed layer
Variable
(0)
c
0
=
2
.
5
φ
0
=
20
°
C
0
=
D
=
6
Δ
c
0
=
±
2
Δ
φ
0
=
-
5
°~15°
Single cover layer
Constant
(1)
c
1
=
2
.
5
φ
1
=
20
°
C
1
=
9
2
Single crossed layer
Constant
(0)
c
0
=
2
.
5
φ
0
=
20
°
C
0
=
D
=
6
Single cover layer
Variable
(1)
c
1
=
2
.
5
φ
1
=
20
°
C
1
=
9
Δ
c
1
=
±
2
Δ
φ
1
=
-
5
°~15°
3
Single crossed layer
Constant
(0)
c
0
=
2
.
5
φ
0
=
20
°
C
0
=
D
=
6
Two cover layers
Variable
(1)
c
1
=
2
.
5
φ
1
=
15
°
C
1 = 0~9
(2)
c
2
=
2
.
5
φ
2
=
1
0
°
C
2
=
9
-
C
1


### 3.1. Influence of the Variables of the Crossed Soil on Limit Support Pressures

The first set of analyses described the cases of a single cover layer with constant strength parameters (c1 and φ1) and a single crossed layer with different strength parameters (c0  and  φ0 and increments Δc0  and  Δφ0). Figures 3 and 4 show the curves of the limit support pressures with the variation of Δc0 and Δφ0, respectively.Figure 3
Influence ofΔφ0 on the limit support pressures (Δc0=0 kPa).Figure 4
Influence ofΔc0 on the limit support pressures (Δφ0=0°).In Figure3, the limit support pressures declined nonlinearly with the increase of Δφ0 from −5° to 15°. The decrease of the limit support pressures with Δφ0 from −5° and 0° was steeper than those with Δφ0 in the range between 0° and 15°, which shows that Δφ0 has a higher influence on the limit support pressures if the strength parameters of the crossed soil were weaker than strength parameters of the cover soil. Moreover, the results from this paper and Senent and Jimenez [11] were between the results from Broere [18] (the highest solutions) and from Tang et al. [30] (the lowest solutions). The approach proposed by Broere [18] overestimated the limit support pressure compared to the results from other approaches.Figure4 shows that the limit support pressures decreased linearly with the increase of Δc0. Similarly, the results from this paper and Senent and Jimenez [11] were almost equal and located between the results from Broere [18] (the highest solutions) and from Tang et al. [30] (the lowest solutions). It is noteworthy that rotational face collapse mechanism proposed by Senent and Jimenez [11] considers the whole face of the tunnel by employing a spatial discretization technique, which provides the results that outperform previous upper bound solutions. As mentioned above, the approximations of limit support pressures obtained from the new failure mechanism are not strictly an upper bound solution and the results from our paper are slightly less than the results from Senent and Jimenez [11] in most cases. The comparisons show that the results in this paper provide relatively satisfactory results. To compare Figures 3 and 4, it has also been found that the friction angle affected the support pressure effectively compared to the cohesion.


### 3.2. Influence of the Variables of the Cover Soil on Limit Support Pressures

#### 3.2.1. Single Cover Layer

The second set of analyses presented a reverse situation, which described the cases of a single cover layer with different strength parameters (c1  and  φ1 and increments Δc1  and  Δφ1) and a single crossed layer with constant strength parameters (c0 and φ0). Figures 5 and 6 compared change rules of the limit support pressures with the variation of Δc1 and Δφ1, respectively.Figure 5
Influence ofΔφ1 on the limit support pressures (Δc1=0 kPa).Figure 6
Influence ofΔc1 on the limit support pressures (Δφ1=0°).Figure5 shows that the limit support pressures decreased nearly linearly with the increase of Δφ1 from −5° to 15°. And the results from this paper were slightly less than the results from Senent and Jimenez [11] and were much higher than the results from Tang et al. [30].In Figure6, the limit support pressures dropped linearly as Δc1 increased. Similarly, the results from this paper were equal to the results from Senent and Jimenez [11] and were much higher than the results from Tang et al. [30].Since this section was a reverse situation with the last section, it is necessary to compare Figures3 and 5 (Figures 4 and 6) to analyze their influence degree on the limit support pressures. For the variation of friction angle (Δφ0 or Δφ1), the limit support pressures decreased from 40.09 kPa to 5.80 kPa (decrement 34.29 kPa) in Figure 3 while they decreased from 24.86 kPa to 19.44 kPa (decrement 5.42 kPa) in Figure 5. And for the variation of cohesion (Δc0 or Δc1), the limit support pressures decreased from 27.11 kPa to 18.11 kPa (decrement 9.00 kPa) in Figure 4 while they decreased from 23.46 kPa to 21.76 kPa (decrement 1.7 kPa) in Figure 6. Those data indicated that strength parameters’ variation in crossed layer plays bigger role in limit support pressures than in cover layers with the same variable range of strength parameters.


#### 3.2.2. Two Cover Layers

The third set of analyses described the cases of two cover layers with constant strength parameters (c1  and  φ1, c2  and  φ2, and relative thickness μ=C1/C2) and a single crossed layer with constant strength parameters (c0 and φ0). Figure 7 shows the curves of the limit support pressures with the relative thickness μ. It can be seen that the limit support pressures drop dramatically with the increase of μ from 0 to 0.5 while the limit support pressures remain constant when μ is bigger than 0.5. Moreover, the results obtained from this paper are slightly higher than the results obtained from Senent and Jimenez [11] and much higher than those of Tang et al. [30].Figure 7
Influence ofμ on the limit support pressures.


## 3.1. Influence of the Variables of the Crossed Soil on Limit Support Pressures

The first set of analyses described the cases of a single cover layer with constant strength parameters (c1 and φ1) and a single crossed layer with different strength parameters (c0  and  φ0 and increments Δc0  and  Δφ0). Figures 3 and 4 show the curves of the limit support pressures with the variation of Δc0 and Δφ0, respectively.Figure 3
Influence ofΔφ0 on the limit support pressures (Δc0=0 kPa).Figure 4
Influence ofΔc0 on the limit support pressures (Δφ0=0°).In Figure3, the limit support pressures declined nonlinearly with the increase of Δφ0 from −5° to 15°. The decrease of the limit support pressures with Δφ0 from −5° and 0° was steeper than those with Δφ0 in the range between 0° and 15°, which shows that Δφ0 has a higher influence on the limit support pressures if the strength parameters of the crossed soil were weaker than strength parameters of the cover soil. Moreover, the results from this paper and Senent and Jimenez [11] were between the results from Broere [18] (the highest solutions) and from Tang et al. [30] (the lowest solutions). The approach proposed by Broere [18] overestimated the limit support pressure compared to the results from other approaches.Figure4 shows that the limit support pressures decreased linearly with the increase of Δc0. Similarly, the results from this paper and Senent and Jimenez [11] were almost equal and located between the results from Broere [18] (the highest solutions) and from Tang et al. [30] (the lowest solutions). It is noteworthy that rotational face collapse mechanism proposed by Senent and Jimenez [11] considers the whole face of the tunnel by employing a spatial discretization technique, which provides the results that outperform previous upper bound solutions. As mentioned above, the approximations of limit support pressures obtained from the new failure mechanism are not strictly an upper bound solution and the results from our paper are slightly less than the results from Senent and Jimenez [11] in most cases. The comparisons show that the results in this paper provide relatively satisfactory results. To compare Figures 3 and 4, it has also been found that the friction angle affected the support pressure effectively compared to the cohesion.


## 3.2. Influence of the Variables of the Cover Soil on Limit Support Pressures

### 3.2.1. Single Cover Layer

The second set of analyses presented a reverse situation, which described the cases of a single cover layer with different strength parameters (c1  and  φ1 and increments Δc1  and  Δφ1) and a single crossed layer with constant strength parameters (c0 and φ0). Figures 5 and 6 compared change rules of the limit support pressures with the variation of Δc1 and Δφ1, respectively.Figure 5
Influence ofΔφ1 on the limit support pressures (Δc1=0 kPa).Figure 6
Influence ofΔc1 on the limit support pressures (Δφ1=0°).Figure5 shows that the limit support pressures decreased nearly linearly with the increase of Δφ1 from −5° to 15°. And the results from this paper were slightly less than the results from Senent and Jimenez [11] and were much higher than the results from Tang et al. [30].In Figure6, the limit support pressures dropped linearly as Δc1 increased. Similarly, the results from this paper were equal to the results from Senent and Jimenez [11] and were much higher than the results from Tang et al. [30].Since this section was a reverse situation with the last section, it is necessary to compare Figures3 and 5 (Figures 4 and 6) to analyze their influence degree on the limit support pressures. For the variation of friction angle (Δφ0 or Δφ1), the limit support pressures decreased from 40.09 kPa to 5.80 kPa (decrement 34.29 kPa) in Figure 3 while they decreased from 24.86 kPa to 19.44 kPa (decrement 5.42 kPa) in Figure 5. And for the variation of cohesion (Δc0 or Δc1), the limit support pressures decreased from 27.11 kPa to 18.11 kPa (decrement 9.00 kPa) in Figure 4 while they decreased from 23.46 kPa to 21.76 kPa (decrement 1.7 kPa) in Figure 6. Those data indicated that strength parameters’ variation in crossed layer plays bigger role in limit support pressures than in cover layers with the same variable range of strength parameters.


### 3.2.2. Two Cover Layers

The third set of analyses described the cases of two cover layers with constant strength parameters (c1  and  φ1, c2  and  φ2, and relative thickness μ=C1/C2) and a single crossed layer with constant strength parameters (c0 and φ0). Figure 7 shows the curves of the limit support pressures with the relative thickness μ. It can be seen that the limit support pressures drop dramatically with the increase of μ from 0 to 0.5 while the limit support pressures remain constant when μ is bigger than 0.5. Moreover, the results obtained from this paper are slightly higher than the results obtained from Senent and Jimenez [11] and much higher than those of Tang et al. [30].Figure 7
Influence ofμ on the limit support pressures.


## 3.2.1. Single Cover Layer

The second set of analyses presented a reverse situation, which described the cases of a single cover layer with different strength parameters (c1  and  φ1 and increments Δc1  and  Δφ1) and a single crossed layer with constant strength parameters (c0 and φ0). Figures 5 and 6 compared change rules of the limit support pressures with the variation of Δc1 and Δφ1, respectively.Figure 5
Influence ofΔφ1 on the limit support pressures (Δc1=0 kPa).Figure 6
Influence ofΔc1 on the limit support pressures (Δφ1=0°).Figure5 shows that the limit support pressures decreased nearly linearly with the increase of Δφ1 from −5° to 15°. And the results from this paper were slightly less than the results from Senent and Jimenez [11] and were much higher than the results from Tang et al. [30].In Figure6, the limit support pressures dropped linearly as Δc1 increased. Similarly, the results from this paper were equal to the results from Senent and Jimenez [11] and were much higher than the results from Tang et al. [30].Since this section was a reverse situation with the last section, it is necessary to compare Figures3 and 5 (Figures 4 and 6) to analyze their influence degree on the limit support pressures. For the variation of friction angle (Δφ0 or Δφ1), the limit support pressures decreased from 40.09 kPa to 5.80 kPa (decrement 34.29 kPa) in Figure 3 while they decreased from 24.86 kPa to 19.44 kPa (decrement 5.42 kPa) in Figure 5. And for the variation of cohesion (Δc0 or Δc1), the limit support pressures decreased from 27.11 kPa to 18.11 kPa (decrement 9.00 kPa) in Figure 4 while they decreased from 23.46 kPa to 21.76 kPa (decrement 1.7 kPa) in Figure 6. Those data indicated that strength parameters’ variation in crossed layer plays bigger role in limit support pressures than in cover layers with the same variable range of strength parameters.


## 3.2.2. Two Cover Layers

The third set of analyses described the cases of two cover layers with constant strength parameters (c1  and  φ1, c2  and  φ2, and relative thickness μ=C1/C2) and a single crossed layer with constant strength parameters (c0 and φ0). Figure 7 shows the curves of the limit support pressures with the relative thickness μ. It can be seen that the limit support pressures drop dramatically with the increase of μ from 0 to 0.5 while the limit support pressures remain constant when μ is bigger than 0.5. Moreover, the results obtained from this paper are slightly higher than the results obtained from Senent and Jimenez [11] and much higher than those of Tang et al. [30].Figure 7
Influence ofμ on the limit support pressures.


## 4. Conclusions

In order to better interpret the failure features, a new three-dimensional failure mechanism is proposed to analyze the limit support pressure of the tunnel face in multilayered cohesive-frictional soils based on limit analysis methods. The new failure mechanism is composed of two truncated cones that represent the shear failure band and a distributed force acting on the truncated cones that represents the pressure arch effect. The distributed force could be calculated by using Terzaghi earth pressure theory. Moreover, the comparisons between the results of the present study and those of existing approaches were provided. The main conclusions are presented as follows.(1) Results show that influence rules obtained from different models are similar. More importantly, our mechanism provides limit pressures that are almost equal to the results form Senent and Jimenez [11] and are higher than those computed by Tang et al. [30], which show that the results obtained from the new mechanism in this paper provide relatively satisfactory results.(2) For the strength parameters variations of soil in the crossed soil or in the cover soil, the results show thatΔφ0 has a higher influence on the limit support pressures if the strength parameters of the crossed soil were weaker than strength parameters of the cover soil (i.e., Δφ1≤0).(3) No matter strength parameters variations of soil in the crossed soil or in the cover soil, it has also been found that the friction angle affected the support pressure effectively compared to the cohesion.(4) The results indicated that strength parameters’ variation in crossed layer plays bigger role in limit support pressures than in cover layers with the same variable range of strength parameters.



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*Source: 1378274-2016-05-18.xml*