2.1 Problem statement

A schematic diagram of the longitudinally inclined tunnel considering the effect of pore water pressure is shown in Figure 1. A longitudinally inclined circular tunnel with a diameter of D and inclined angle of δ is excavated under the water table. The water table elevation H_w is measured from the crown of the tunnel; C is the buried depth of the tunnel; and h_s refers to the piezometric head in the work chamber at the tunnel face. If h_s<(H_w+D), the groundwater will flow into the tunnel face, and cause the hydraulic gradients ahead of the tunnel face to vary rapidly. This phenomenon simultaneously produces the seepage force F_w acting on the soil in the direction of the groundwater flow, which results in surface settlement and collapse failure of the tunnel face [25, 26]. In this paper, only the problem of the collapse failure mode of longitudinally inclined tunnels is considered.

Figure 1全屏查看

Schematic diagram of a longitudinally inclined shield tunnel under the water table

Figure 1 shows that the collapse failure block ABF refers to a 3D rotational block that was first proposed by MOLLON et al [8]. Because the soil strength in the tensile regime is reduced, the effect of tensile strength cut-off was introduced by LI et al [21] to modify the traditional collapse failure mechanism. The modified failure mechanism proposed by LI et al [21] involves the formation of a rounded corner at the top of the failure surface rather than a sharp corner, which is more consistent with the actual failure mode [27].

2.2 Tensile strength cut-off

In the traditional Mohr-Coulomb failure criterion, the uniaxial compressive strength, uniaxial and triaxial tensile strengths are expressed by the following equations, respectively.

(1)

where f_c, f_m are the uniaxial compressive and tensile strengths, respectively.

The overall tensile strength of soil is much smaller than the one defined by the traditional Mohr-Coulomb failure criterion in practical situations, which is inconsistent with the calculation of tensile strength in Eq. (1). The classical Mohr-Coulomb failure criterion is considered to be not suitable to define the strength of soil in tensile regime. Thus, PAUL [28] introduced the tensile strength cut-off point into the principal stress space to define the local tensile strength f_t of soil, which can be seen in Figure 2. The local tensile stress in the soil can be expressed as

(2)

Figure 2全屏查看

The strength envelope of the tensile strength cut-off

where ξ is the reduced tensile strength coefficient, which ranges from 0 to 1. When ξ=0, the tensile strength corresponds to envelope C in Figure 2; when ξ=1, the tensile strength corresponds to envelope A; and when 0<ξ<1, the tensile strength is related to the dilatancy angle λ, which corresponds to envelope B.

It is assumed that the failure surface AF, BF in Figure 1 bears a traction vector T, and points located at AF, BF move at the velocity v in the kinematically admissible velocity field. The work rate done by internal energy for per unit length can be calculated as

(3)

Combined with Eqs. (1) and (2), the expression of w_D in tensile regime can be written as

(4)

2.3 Construction of modified failure mechanism

In this contribution, the effect of tensile strength cut-off was introduced to modify Mollon’s collapse failure mechanism. The collapse failure mechanism is composed of four log-spiral curves, AG, GFʹ, BH and HFʹ, as the two log-spiral curves, GFʹ and HFʹ, intersect at the top point Fʹ, which forms the tensile region, as shown in Figure 3(a). However, the buried depth of the tunnel decreases as the longitudinally inclined tunnel advances gradually, which leads to the collapse failure block extending to the ground surface. The two log-spiral curves intersect with the ground surface at points M and N, as shown in Figure 3(b). The collapse failure block rotates around the rotation centre O at angular velocity ω. The log-spiral curves of the 3D collapse failure mechanism in Figure 3 can be expressed in the form of polar coordinates:

(5)

Figure 3全屏查看

3D failure mechanism of the longitudinally inclined tunnel face: (a) The collapse failure mechanism with consideration of the tensile strength cut-off in the case of a deep burial depth; (b) The collapse failure mechanism with consideration of the tensile strength cut-off in the case of a shallow burial depth

where r_A, r_B, r_G, and r_H are the lengths of OA, OB, OG and OH, respectively; θ_A, θ_B and θ_G denote the inclinations of OA, OB and OG (OH), respectively; and φ is the friction angle, which is the angle between the velocity direction at any point on the log-spiral curves and its tangent direction. The dilatancy angle λ is a function of the geometric parameter θ, which reaches its maximum value λ_m at point Fʹ [29, 30]. The value range of λ is from φ to 90°. The expression of the dilatancy angle λ can be expressed as

(6)

When the equation of r₃(θ_F′)=r₄(θ_F′) is satisfied, the dilatancy angle λ and the geometric parameter θ reach their maximum values. The computational formula of λ_m can be written as

(7)

In the space rectangular coordinate system shown in Figure 3, the coordinates of the rotation centre O, point A at the vault of the tunnel and point B at the invert of the tunnel are (0, Y_O, Z_O), (0, 0, 0) and (0, -Dcosδ, Dsinδ), respectively. According to the law of tangents, the geometric parameters θ_A, θ_B, r_A and r_B can be expressed as

(8)

where δ is the inclined angle of the longitudinally inclined tunnel. When the value of δ is positive, the tunnel is inclined upwards; when the value of δ is negative, the tunnel is inclined downwards.

The generation of the collapse failure mechanism is determined by the location of the rotation centre O. To facilitate the optimal calculation, the geometric parameters θ_E and r_E/D of the tunnel centre E are introduced to calculate the coordinates of the rotation centre O. The coordinates of the rotation centre O can be expressed as

(9)

For the collapse failure mechanism at greater burial depths, the curves GFʹ and HFʹ intersect at point Fʹ. The generation of the collapse failure mechanism ends at point Fʹ. According to Eqs. (5) and (3), the computational formula of r_Fʹ and θ_Fʹ can be written as

(10)

If a shield tunnel excavates with an inclined angle in a shallow stratum, the failure mechanism of the tunnel face may approach to the ground surface gradually. When the shield machine excavates a specific position, the failure mechanism of the tunnel face will inevitably extend to the ground surface. The failure mechanism of the tunnel face in this case can be demonstrated in Figure 3(b). Based on the generation principle of the failure mechanism, the failure mechanism shown in Figure 3(b) is controlled by the termination points M and N. Therefore, to determine whether the failure mechanism has extended to the ground surface, an additional condition is imposed in the calculation program to distinguish whether the generated points are located on the ground surface or above the ground surface. The failure mechanism generation program can be written as the following idea: if the generated points are located below the ground surface, the generation of the failure mechanism continues; if the generated points happen to be located on the ground surface, the generation of the failure mechanism is completed; if the generated points are located above the ground surface, these points will be replaced by the points located on the ground surface which are calculated by using of interpolation.

2.4 The calculation of pore water pressure

To evaluate the influence of the pore water pressure on the face stability of longitudinally inclined shield tunnels, it is necessary to analyze the seepage field in front of the tunnel face during tunnel construction. VIRATJANDR et al [31] stated that the work produced by pore water pressure consists of two parts: the work produced by pore water pressure on the soil body and the work produced on the boundary of the velocity field. Due to the lack of an effective theoretical method for calculating the actual pore water pressure on a collapsed failure block, a numerical simulation is applied here to calculate the distribution of the pore water pressure.

In this work, it is assumed that the longitudinally inclined tunnel is located in a steady-state hydraulic field so that the soil permeability and porosity are equal anywhere in the numerical model. The numerical model is constructed by the standardized modelling method so that the pore water pressure on the nodes of the numerical model can correspond to that on the 3D collapse failure mechanism. A numerical model with approximately 389241 nodes and a total of 387000 zones for the case of D=10 m, C/D=2.0 and δ=10° is shown in Figure 4. Because the 3D numerical model is axisymmetric, only half of the longitudinally inclined tunnel is constructed to improve the operational speed. To ensure the elimination of the boundary effect, a numerical model is constructed with sizes of 35.00, 50.98 and 60.00 m in the transverse, longitudinal and vertical directions, respectively. The liner element is activated behind the tunnel face to simulate the shield lining segments, which are set to be impermeable so that the groundwater can only seep into the interior of the tunnel from the tunnel face. During the process of seepage simulation, a rapid change in the hydraulic head gradient appears near the tunnel face. Therefore, the mesh in the vicinity of the tunnel face is densified to increase the precision of the interpolation of the pore water pressure. In addition, it is assumed that there is continuous and sufficient groundwater recharge so that the water table elevation remains constant during the process of groundwater seepage.

Figure 4全屏查看

Numerical model of FLAC^3D for seepage simulation

Based on the constructed numerical model shown in Figure 4, the pore water pressure distribution contours of the tunnel face for D=10 m, C/D=2.0, δ=10° and H_w/C=1 are obtained, as shown in Figure 5. The tunnel face is assumed to be an impermeable interface so that the pore water pressure at the tunnel face is prescribed to be zero. It should be noted that the hydraulic elevation is measured from the crown of the tunnel face. The rapid variation in the hydraulic gradient at the tunnel face will result in large seepage forces acting on the collapse failure mechanism.

Figure 5全屏查看

Pore water pressure contours for longitudinally inclined tunnel face: (a) 3D view; (b) Horizontal symmetry plane; (c) Vertical symmetry plane (unit: kPa)

Based on the pore water pressure distribution obtained by numerical simulation, a FISH routine is used to extract the pore water pressure of each node around the tunnel. Then, the pore water pressure is introduced into the 3D rotational failure mechanism to establish the virtual work equation. However, the extracted pore water pressure of nodes in the numerical model cannot fully correspond to the points on the failure mechanism. Thus, based on the correspondence of the nodes between the numerical and theoretical models, the pore water pressure obtained from numerical simulation was interpolated at each discretization point of the constructed failure mechanism by using of linear interpolation method. The failure mechanism of the longitudinally inclined tunnel face with the consideration of the pore water effect was obtained. Using this failure mechanism, the work rate of the pore water pressure around the tunnel face can be calculated in the framework of upper bound theorem of limit analysis.

2.5 Upper bound limit analysis

To maintain the stability of the soil in front of a longitudinally inclined shield tunnel, it is necessary to calculate the critical chamber pressure. Based on the constructed failure mechanism of the longitudinally inclined tunnel face, the work rates of the external forces and internal energy dissipation can be calculated to obtain the objective function of the critical chamber pressure. Subsequently, the expressions of the external work rate and energy dissipation rate are derived separately in this paper.

On the basis of space discretization technology, a 3D rotational failure mechanism composed of several discretized points and triangular facets is constructed, as shown in Figure 6. By using these discretized points and triangular facets, the external work rate can be calculated, which includes the work generated by gravity, pore water pressure and chamber pressure.

Figure 6全屏查看

The element discretization for work rate calculations

According to MOLLON et al [8], there is no volumetric strain within the rotational failure block (assumption of rigid block), and the external work rate produced by gravity can be calculated as follows

(11)

where ω is the angular velocity of the whole failure mechanism and Vi,j (V′i,j) is the volume of the discretized element, which corresponds to the rotation radius Ri,j (R′i,j) and inclination angle θi,j(θ′i,j). It is assumed that the soil under the water table is completely saturated in this work. Therefore, the saturated unit weight γ_sat of the soil is used to calculate the work rate of gravity when the soil is under the water table; when the soil is above the water table, the saturated unit weight γ_sat in Eq. (11) is replaced by the dry unit weight γ_d. The conversion formula between the saturated unit weight and dry unit weight of soil is expressed as

(12)

where n is the porosity of the soil and γ_w is the water unit weight.

The calculation of the work rate of the pore water pressure should consider the influence of the seepage forces and buoyancy forces, which can be written as

(13)

where u refers to the pore water pressure; ni denotes the outwards unit vector normal to the surface boundary S of the failure mechanism; and vi is the velocity vector in the kinematically admissible velocity field. ɛi is the volumetric strain increment of the soil skeleton, which corresponds to the volume V of the failure mechanism.

According to the assumption of rigid blocks mentioned above, the parameter ɛi is equal to zero in Eq. (13). Therefore, Eq. (13) can be simplified to

(14)

Based on the discretized failure mechanism sketched in Figure 3, the work rate of the pore water pressure can be calculated as

(15)

where ui_,j and u′i_,j are the average pore water pressures of the triangular facets Fi_,j and F′i_,j, respectively; Si_,j and S′i_,j represent the areas of the triangular facets Fi_,j and F′i_,j, respectively; and Ri_,j and R′i_,j are the distances between the rotation centre O and the centre of gravity of the triangular facets Fi_,j and F′i_,j, respectively. ui₁, Si₁ and θi₁ are the parameters used to calculate the work rate of the pore water pressure at the intersection surface between the failure mechanism and the groundwater surface. The average pore water pressure is calculated by the pore water pressure at the corresponding discretized points, which can be expressed as

(16)

(17)

where pi_,j, pi_+1,j, pi_,j₊₁ (pi_+1,j, pi_,j₊₁, pi_+1,j₊₁) represent the pore water pressures on the three vertices of the triangular facet Fi_,j (F′i_,j), respectively.

Suppose that the chamber pressure σ_c is a uniform pressure that acts on the tunnel face vertically. Thus, the work rate of the chamber pressure can be calculated as

(18)

where S_Σj represents the area of the discretized element on the tunnel face; Rj and βj represent the rotation radius and inclination angle of the corresponding discretized element, respectively.

The internal energy dissipation rate occurs along the surfaces of the failure mechanism, which can be calculated as

(19)

where c and φ are the cohesion and friction angle of the soil, respectively.

Because the tensile strength cut-off was introduced in the construction of the collapse failure mechanism, the angle between the velocities and the tangential direction of the discretized points on the collapse surface in the tensile region is equal to dilatancy angle λ. According to Eq. (4) obtained in Section 2.2, the internal energy dissipation rate per unit area is defined by the reduced tensile strength, which can be expressed as

(20)

where ξ is the reduced tensile strength coefficient.

Thus, the internal energy dissipation rate in the tensile region is different from that in Eq. (19), and can be written as

In the framework of upper bound theorem of limit analysis, the objective function of the critical chamber pressure can be derived by equating the external work rates to the internal energy dissipation rate. Therefore, a strict upper bound solution of the chamber pressure can be expressed as

(22)

However, the derived objective function is the general solution of the critical chamber pressure. The MATLAB software is used here to optimize the position parameters θ_E and r_E/D of the rotation centre O, the geometric parameters θ_G of the tensile region and λ_m of the dilatancy angle. Finally, the optimal upper bound solution of the critical chamber pressure considering the effect of the pore water pressure is obtained. The optimal calculation of the objective functions of the critical chamber pressure should follows the relevant constraints:

(23)

To better understand the influence of the inclined angle and the pore water pressure on the face stability of a longitudinally inclined shield tunnel, the analytical solution of the critical chamber pressure is normalized. The general relation for the normalized critical chamber pressure can be written as

(24)

The whole process of the face stability analysis of a longitudinally inclined shield tunnel considering the effect of pore water pressure and tensile strength cut-off is shown in Figure 7.

Figure 7全屏查看

The flow diagram for face stability analysis of a longitudinally inclined shield tunnel

3.1 Normalized parametric analysis

Based on the constructed collapse failure mechanism and the derived objective function, the normalized critical chamber pressures are obtained by optimal computation using MATLAB software. To study the influence of the inclined angle and pore water pressure on the tunnel face stability, the normalized critical chamber pressures are plotted in Figures 8, 9 and 10 for the case of D=10 m, γ_d/γ_sat=0.85, γ_d=17 kN/m³, and ξ=0. The normalized critical chamber pressure σ_c/(γ_satD) decreases with increasing friction angle φ, cohesion c and reduced coefficient ξ, while increasing with increasing inclined angle δ. As shown in Figure 8, when the inclined angle δ changes from negative to positive, that is, when the inclined tunnel changes from downwards to upwards, the change in the value of σ_c/(γ_satD) markedly increases. Consequently, the inclined angle of the longitudinally inclined tunnel has a significant influence on the tunnel face stability, especially when the shield tunnel drills with an upwards inclined angle. The value of σ_c/(γ_satD) increases with increasing normalized groundwater elevation H_w/C, as shown in Figure 9. The differences in the critical chamber pressure range from 52.99% to 59.91% for H_w/C=0.5 and from 56.96% to 72.49% for H_w/C=1.0, which are comparable to the results for H_w/C=0.0, as shown in Figure 11. This shows that the change in groundwater elevation significantly influences the chamber pressure. However, when the inclined angle δ changes from -10° to 10°, the difference between H_w/C=0.5 (H_w/C=1.0) and H_w/C=0.0 slightly decreases. Figure 10 shows that the value of σ_c/(γ_satD) increases with increasing normalized cover depth C/D. The differences in the results between the cases of C/D=2.0 and C/D=1.0 range from 42.31% to 49.04%, and the differences in the results between the cases of C/D=3.0 and C/D=1.0 range from 59.28% to 64.46%, as shown in Figure 12. It is indicated that the influence of the cover depth on the face stability of longitudinally inclined tunnels cannot be ignored. The difference between C/D=2.0 (C/D=3.0) and C/D=1.0 slightly decreases as the inclined angle increases.

Figure 8全屏查看

The normalized critical chamber pressures σ_c/(γ_satD) versus inclined angle δ at H_w/C=1.0: (a) With different values of normalized cohesion c/(γ_satD); (b) With different values of friction angle φ; (c) With different values of reduced tensile strength parameter ξ

Figure 9全屏查看

The normalized critical chamber pressures σ_c/(γ_satD) versus the normalized groundwater elevation H_w/C with different values of inclined angle δ

Figure 10全屏查看

The normalized critical chamber pressures σ_c/(γ_satD) versus normalized cover depth C/D for different values of inclined angle δ

Figure 11全屏查看

(a) The value of the normalized critical chamber pressure σ_c/(γ_satD); (b) The difference in the results when H_w/C=0.5 and 1.0 for H_w/C=0.0

Figure 12全屏查看

(a) The value of the normalized critical chamber pressure σ_c/(γ_satD); (b) The difference in the results when C/D=2.0 and 3.0 for C/D=1.0

To visually study the influence of pore water pressure on the stability of longitudinally inclined tunnel faces, the critical chamber pressures considering the effect of pore water pressure are compared with those without considering the effect of pore water pressure. The critical chamber pressures of these two conditions are calculated under the same parameters to ensure the validity of the comparison. The critical chamber pressure considering the effect of the pore water pressure is much greater than that without considering the effect of the pore water pressure, which can be seen in Figure 13. Thus, the influence of pore water pressure on the face stability of longitudinally inclined shield tunnels cannot be ignored in actual engineering. Moreover, the critical chamber pressure considering the effect of the pore water pressure significantly increased with increasing buried depth C, while the change in the critical chamber pressure without considering the effect of the pore water pressure was minor or even unchanged, as shown in Figure 13(b). This rule indicates that the effect of pore water pressure will make the tunnel face collapse more easily with increasing buried depth C, as the groundwater level H_w/C remains constant.

Figure 13全屏查看

Comparisons of the critical chamber pressure with and without consideration of the effect of pore water pressure: (a) With different inclined angles δ; (b) With different buried depths C

3.2 Comparative analysis of collapse failure mode

Based on the upper bound solutions obtained from the theoretical analysis, the images of the 3D collapse failure block for the soil in front of the longitudinally inclined shield tunnel are plotted. Figures 14(a)-(c) show the critical collapse mechanisms after optimal computation for the cases of δ=-10°, δ=0° and δ=10° with D=10 m, C/D=2.0, H_w/C=1.0, φ=20° and c/(γ_satD)=0.05. Figure 14(d) presents the critical collapse mechanism for the case of H_w/C=0.5 with δ=10° and φ=20°. Figures 14(e) and (f) plot the critical collapse mechanisms for the cases of C/D=2.0 and C/D=0.5 with δ=10°, H_w/C=1.0, and φ=10°, respectively. The range of the collapse mechanism increases with increasing inclined angle δ and groundwater elevation H_w but decreases with increasing friction angle φ. It is indicated that the upwards inclined tunnel has a higher risk of collapse, which is consistent with the studies of CHENG et al [2]. As mentioned above, the effect of the pore water pressure strongly affects the stability of the tunnel face. Correspondingly, a comparison between Figures 14(c) and (d) indicates that the groundwater elevation also has a significant influence on the range of the critical collapse mechanism. A comparison between Figures 14(e) and (f) shows that the range of the collapse mechanism decreases with decreasing buried depth C as the longitudinally inclined tunnel drills upwards. Meanwhile, the collapse mechanism will extend to the ground surface when C/D is less than 0.5. Consequently, it is necessary to adjust the chamber pressure of the shield machine in real time during the construction of the longitudinally inclined shield tunnel under the groundwater table.

Figure 14全屏查看

Sketch of critical failure mechanisms for different parameters: When C/D=2.0, φ=20°, H_w/C=1.0 for (a) δ=-10°, (b) δ=0° and (c) δ=10°; (d) When C/D=2.0, φ=20°, δ=10°, H_w/C=0.5; When φ=10°, H_w/C=1.0, δ=10° for (e) C/D=2.0 and (f) C/D=0.5