Analysis of the Limit Support Pressure of a Shallow Shield Tunnel in Sandy Soil Considering the Inﬂuence of Seepage

S S symmetry

Article Analysis of the Limit Support Pressure of a Shallow Shield Tunnel in Sandy Soil Considering the Inﬂuence of Seepage

Bo Mi 1,2 and Yanyong Xiang 1,2,*

1 Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China; [email protected] 2 School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China * Correspondence: [email protected]

Received: 20 May 2020; Accepted: 16 June 2020; Published: 17 June 2020

Abstract: The objective was to optimize the existing solution for the limit support pressure of a tunnel face. Firstly, based on the numerical simulation results, the existing three-dimensional analytical solution for pore water pressure distribution is expanded to a three-dimensional solution considering the pore water pressure distribution in the upper formation behind the tunnel face. Then, according to the results of physical model tests, a failure model considering the failure range in the upper formation behind the tunnel face is established, and the newly established three-dimensional solution for pore water pressure is introduced into the model, and then the limit effective support pressure of the tunnel face considering seepage is obtained by the method of soil–water joint calculation. Finally, the calculation results in this paper are compared with the experimental results, numerical simulation results and existing theoretical solutions. The major findings are as follows. The distribution of pore water pressure in the front and back strata above the tunnel face is basically symmetrical. The limit effective support pressure of the tunnel face will increase linearly with an increase in the hydraulic head difference between the tunnel face and the ground surface. The calculated results of the new limit equilibrium theory are obviously larger than those of the existing theory and numerical simulation and closer to the results of the physical model tests. Therefore, the new limit equilibrium model can better predict the limit effective support pressure of the tunnel face considering seepage and provide a reference for actual projects.

Keywords: seepage; limit eﬀective support pressure; three-dimensional analytical solution for pore water pressure distribution; limit equilibrium model; soil–water joint calculation

1. Introduction It is easy to form a seepage field near a tunnel face when the shield is excavated in a saturated sandy soil layer, which is not conducive to the stability of the tunnel face. The determination of the limit support pressure of the tunnel face under seepage conditions is very important to maintain the stability of the tunnel face in saturated sandy soil. Physical model tests [1–3] and numerical simulations [4–6] can be used to study the limit support pressure of the tunnel face under seepage conditions, but these methods are often limited to specific seepage fields and soil physical and mechanical parameters. However, theoretical calculation can often predict the limit support pressure of the tunnel face under different seepage conditions and soil physical and mechanical parameters. Theoretical calculation of the limit support pressure of the tunnel face under seepage conditions mainly includes limit analysis [7–9] and limit equilibrium analysis, and the limit equilibrium method is simple and easy to understand and more convenient to apply to actual projects.

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Anagnostou and Kovári [10] combined the seepage force calculated by numerical simulation and the classical wedge model, which gives the calculation expression of the limit support pressure of the tunnel face. Perazzelli et al. [11] and Cao et al. [12] fitted the pore water pressure distribution expression based on the numerical simulation results and combined the classical wedge model to study the effect of seepage on the limit support pressure of the tunnel face. Wang et al. [13] further optimized the pore water pressure distribution expression based on numerical simulation results, making the calculation result of the limit support pressure of the tunnel face more accurate. Based on engineering practice, Wei and He [14] proposed a trapezoidal wedge failure model, which is more in line with the actual situation than the classical wedge model. Based on the trapezoid wedge model, Qiao et al. [15] and Lü et al. [16], respectively, combined the seepage force and pore water pressure obtained from the numerical simulation results, and expressed the calculation of the limit support pressure of the tunnel face considering seepage. All of the above theories are based on numerical simulation to study the influence of seepage on the limit support pressure of the tunnel face, which is inconvenient for application in practical engineering. Cao et al. [17] extended the existing analytical solution for pore water pressure directly in front of the tunnel face [18] and the analytical solution for two-dimensional pore water pressure above the tunnel face [19] to a three-dimensional solution for pore water pressure distribution in front of the tunnel face. Combined with the limit instability failure model established by numerical simulation, the calculation of the limit support pressure of the tunnel face was obtained through the limit balance equation. In this paper, the limit equilibrium method is used to study the limit support pressure of the tunnel face in saturated sandy soil under seepage conditions. First of all, according to the numerical simulation results, the three-dimensional solution for pore water pressure distribution in front of the tunnel face from Cao et al. [17] is extended to a three-dimensional solution for pore water pressure considering the upper formation behind the tunnel face. Secondly, based on the results of physical model tests by Mi and Xiang [3], an instability failure model considering the failure range behind the tunnel face was established, and the newly established three-dimensional pore water pressure solution was introduced into the model to obtain the limit support pressure of the tunnel face. Finally, the calculation results in this paper are compared with experimental results, numerical simulation results and existing theoretical solutions.

2. Three-Dimensional Analytical Solution for Pore Water Pressure Distribution The distribution of pore water pressure is key to study of the limit support pressure of the tunnel face considering seepage by the limit equilibrium method. Figure1 shows the seepage model of the tunnel face, where, C is the cover thickness of the tunnel; D is the tunnel diameter; ht is the hydraulic head of the tunnel face; h0 is the hydraulic head of the ground surface; and ∆ is the hydraulic head diﬀerence between the tunnel face and the ground surface. Taking the bottom of the tunnel as the origin of the coordinates, the tunnel excavation direction is the positive x-axis direction, and the vertical upward direction is the positive z-axis direction. Cao et al. [17] obtained a three-dimensional solution for pore water pressure distribution in front of the tunnel face through Equations (1) and (2), but did not consider the distribution of pore water pressure behind the tunnel face. Here, u is the pore water pressure; γw is the unit weight of water; H is the equivalent height of the tunnel in this paper, H = D.

q " C+D # D x 2 u(x) = γw h + (ht h )e− CD (0 z < H) (1) 0 − 2 − 0 ≤

 r  C+D  x 2   (h h )e− CD2 y2+(z C D+ √C2+CD)   t 0 − −  u(x, y, z) = γw(h0 z) + − q ! ln 2   2 2 2   − 2 ln 2C+D ( 2C+D ) 1 y +(z C D √C +CD)  (2) D − D − − − − (H z H + C, x 0) ≤ ≤ ≥ Symmetry 2020, 12, 1023 3 of 13

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FigureFigure 1. 1.Seepage Seepage modelmodel of of the the tunnel tunnel face. face.

CD+ In order to expand the above three-dimensional solution−x without considering the pore water D 2 =−+−γ ()CD ()≤< (1 pressure behind the tunnel faceux() to thewt three-dimensional h00 h h e solution considering0 zH the pore water pressure 2 ) behind the tunnel face, a numerical model, as shown in Figure2, is established by FLAC 3D software, considering the symmetry of the tunnel. Here, the diameter of the tunnel D is 6 m, the thickness of  the covering soil C is 12 m, the top surface is free and the hydraulic head pressure is fixed at 1 m, CD+ 2 −x 2 22+−−+ + the bottom surface and the side wall of()hhe the− tunnelCD are fixed andyzCDCCD impermeable,( the surrounding) normal uxyz(,,)=−+γ () h z t 0 ln direction is fixed andw  impermeable,0 the tunnel face is fixed in the normal direction,2 the hydraulic(2 + + 2 22+−−− + 22CD−− CD yzCDCCD( ) head difference between the tunnel2ln face and the ground 1 surface is fixed at 0.5, 1.0 or 2.0 D, and) the DD friction angle of the soilϕ is 37◦, 31◦ or 25◦. Then, the distribution of pore water pressure is obtained by three-dimensional seepage calculation.()HzHCx≤≤ +,0 ≥ Symmetry 2020, 12, x FOR PEER REVIEW 4 of 15 In order to expand the above three-dimensional solution without considering the pore water pressure behind the tunnel face to the three-dimensional solution considering the pore water pressure behind the tunnel face, a numerical model, as shown in Figure 2, is established by FLAC3D software, considering the symmetry of the tunnel. Here, the diameter of the tunnel D is 6m, the thickness of the covering soil C is 12m, the top surface is free and the hydraulic head pressure is fixed at 1m, the bottom surface and the side wall of the tunnel are fixed and impermeable, the surrounding normal direction is fixed and impermeable, the tunnel face is fixed in the normal direction, the hydraulic head difference between the tunnel face and the ground surface is fixed at 0.5, 1.0 or 2.0 D , and the friction angle of the soil ϕ is 37°, 31° or 25°. Then, the distribution of pore water pressure is obtained by three-dimensional seepage calculation.

FigureFigure 2. Numerical 2. Numerical model model for for calculating calculating pore wa waterter pressure pressure distribution distribution (unit: (unit: m). m).

FigureFigure3 shows 3 shows the the cloud cloud image image of of the the pore pore water water pressurepressure distribution distribution obtained obtained by numerical by numerical simulation.simulation. According According to the to numericalthe numerical simulation simulation results, results, it isit foundis found that that the the pore pore water water pressure pressure in the in the front and back of the tunnel face is symmetrical. Therefore, this paper expands the front and back of the tunnel face is symmetrical. Therefore, this paper expands the three-dimensional three-dimensional solution for the pore water pressure distribution of Equations (1) and (2), which solution for the pore water pressure distribution of Equations (1) and (2), which did not consider the did not consider the pore water pressure distribution behind the tunnel face, into Equation (3), porewhich water comprehensively pressure distribution considers behind the thepore tunnel water face,pressure into distribution Equation (3), directly which in comprehensivelyfront of the tunnel face, the upper formation in front of the tunnel face and the upper formation behind the tunnel face.

(a)

(b)

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Figure 2. Numerical model for calculating pore water pressure distribution (unit: m).

Figure 3 shows the cloud image of the pore water pressure distribution obtained by numerical simulation. According to the numerical simulation results, it is found that the pore water pressure in the front and back of the tunnel face is symmetrical. Therefore, this paper expands the Symmetry 2020, 12, 1023 4 of 13 three-dimensional solution for the pore water pressure distribution of Equations (1) and (2), which did not consider the pore water pressure distribution behind the tunnel face, into Equation (3), which comprehensively considers the pore water pressure distribution directly in front of the considers the pore water pressure distribution directly in front of the tunnel face, the upper formation tunnel face, the upper formation in front of the tunnel face and the upper formation behind the in front of the tunnel face and the upper formation behind the tunnel face. tunnel face.

(a)

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(c)

(d)

(e)

Figure 3. Cloud image of the pore water pressure distribution. (a) The friction angle of the soilϕ = Figure 3. Cloud image of the pore water pressure distribution. (a) The friction angle of the soil ϕ = 37◦; 37°; the hydraulic head difference between the tunnel face and the ground surface Δ = 0.5 D ; (b) ϕ the hydraulic head diﬀerence between the tunnel face and the ground surface ∆ = 0.5 D;(b) ϕ = 37◦; = 37°; Δ = 1.0 D ; (c) ϕ = 37°; Δ = 2.0 D ; (d) ϕ = 31°; Δ = 2.0 D ; (e) ϕ = 25°; Δ = 2.0 D ∆ = 1.0 D;(c) ϕ = 37◦; ∆ = 2.0 D;(d) ϕ = 31◦; ∆ = 2.0 D;(e) ϕ = 25◦; ∆ = 2.0 D (unit: Pa). (unit: Pa).

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 " q #  x C+D  D + ( ) − CD2 ( )  γw h0 2 ht h0 e 0 z < H  − − ≤   r   C+D   x 2    (h h )e− CD2 y2+(z C D+ √C2+CD)    t 0 − −   γw(h0 z) + − q ! ln 2    2 2 2    − 2 ln 2C+D ( 2C+D ) 1 y +(z C D √C +CD)   D − D − − − − u(x, y, z) =  (3)  (H z H + C, x 0)   r   C+D ≤ ≤ ≥   x 2    (h h )e CD2 y2+(z C D+ √C2+CD)    t 0 − −   γw(h0 z) + − q ! ln 2    2 2 2    − 2 ln 2C+D ( 2C+D ) 1 y +(z C D √C +CD)   D − D − − − −   (H z H + C, x < 0) ≤ ≤ Figure4 shows the distribution of pore water pressure on the vertical symmetrical plane of the tunnel obtained by numerical simulation and theoretical calculation in this paper. It can be seen from Figure4 that the pore water pressure distribution law above the tunnel face obtained by the two methods is the same (the maximum relative error is less than 15%), both of which take the tunnel face as the symmetrical plane. Directly above the tunnel face, the pore water pressure gradient calculated by theory is obviously larger than that of the numerical simulation, which will make the seepage force in the theoretical calculation larger and make the limit support pressure of the tunnel face larger, and give the calculation result more safety. As the ground directly in front of the tunnel face is mainly aﬀected by the horizontal seepage force [18], the pore water pressure on the central axis of the tunnel represents the pore water pressure on the ground directly in front of the tunnel face. According to the distribution of pore water pressure on the central axis of the tunnel in Figure4, the theoretical calculation is basically consistent with the numerical simulation results. Based on the above analysis, it is safe to apply the pore water pressure calculated according to Equation (3) to calculation of the limitSymmetry support 2020, 12 pressure, x FOR PEER of the REVIEW tunnel face. 7 of 15

Figure 4. Pore water pressure distribution (ϕ = 37°; Δ = 1.0 D ; unit: kPa). Figure 4. Pore water pressure distribution (ϕ = 37◦; ∆ = 1.0 D; unit: kPa).

3. Limit Support Pressure of the Tunnel FaceFace ConsideringConsidering thethe InﬂuenceInfluence of of Seepage Seepage InIn thisthis section,section, ﬁrstly,firstly, accordingaccording toto thethe physicalphysical modelmodel teststests ofof MiMi andand XiangXiang [[3],3], aa failurefailure model consideringconsidering the the failure failure range range behind behind the tunnel the tunnel face is established,face is established, then the newlythen establishedthe newly established three-dimensional solution for the pore water pressure is introduced into the model, and, finally, the limit support pressure of the tunnel face is solved by the limit equilibrium method.

3.1. Establishment of the Failure Model Figure 5 shows the test model of excavation-seepage instability of the shield tunnel established by Mi and Xiang [3]. Different seepage fields are generated by regulating the pump switch. By rotating the runner of the tunnel model, the support plate moves backward gradually until the ground is unstable and damaged. In the process of the support plate moving backward, the ground settlement and the support pressure of the tunnel face are monitored. Based on analysis of the monitoring data, the ground collapse range and the limit support pressure of the tunnel face are obtained. Figure 6 shows the ground collapse range of the vertical symmetry plane in the center of the tunnel and the plane in front of the tunnel face under different seepage conditions based on the physical model tests of Mi and Xiang [3]. Here, Δ is the hydraulic head difference between the tunnel face and the ground surface and D is the tunnel diameter. It can be seen from Figure 6 that when the support pressure of the tunnel face is not enough, the ground in front and behind the tunnel face will lose stability within a certain range. The collapse range boundary in front of the tunnel face extends from the bottom of the tunnel to the upper part of the tunnel at a certain angle and reaches a certain height, and then extends vertically to the ground surface. The collapse range boundary behind the tunnel face extends from the top of the tunnel to the rear and up at a certain angle and reaches a certain height, then extends vertically upward to the ground surface. The horizontal collapse range boundary extends from the bottom of the tunnel to the upper side at a certain angle and reaches a certain height, then extends to the ground surface vertically.

Symmetry 2020, 12, 1023 6 of 13 three-dimensional solution for the pore water pressure is introduced into the model, and, ﬁnally, the limit support pressure of the tunnel face is solved by the limit equilibrium method.

3.1. Establishment of the Failure Model Figure5 shows the test model of excavation-seepage instability of the shield tunnel established by Mi and Xiang [3]. Diﬀerent seepage ﬁelds are generated by regulating the pump switch. By rotating the runner of the tunnel model, the support plate moves backward gradually until the ground is unstable and damaged. In the process of the support plate moving backward, the ground settlement and the support pressure of the tunnel face are monitored. Based on analysis of the monitoring data, theSymmetry ground 2020 collapse, 12, x FOR range PEER REVIEW and the limit support pressure of the tunnel face are obtained. 8 of 15

Figure 5. Test model of excavation-seepage instability ofof thethe shieldshield tunneltunnel [[3].3].

Figure6 shows the ground collapse range of the vertical symmetry plane in the center of the tunnel and the plane in front of the tunnel face under different seepage conditions based on the physical model tests of Mi and Xiang [3]. Here, ∆ is the hydraulic head difference between the tunnel face and the ground surface and D is the tunnel diameter. It can be seen from Figure6 that when the support pressure of the tunnel face is not enough, the ground in front and behind the tunnel face will lose stability within a certain range. The collapse range boundary in front of the tunnel face extends from the bottom of the tunnel to the upper part of the tunnel at a certain angle and reaches a certain height, and then extends vertically to the ground surface. The collapse range boundary behind the tunnel face extends from the top of the tunnel to the rear and up at a certain angle and reaches a certain height, then extends vertically upward to the ground surface. The horizontal collapse range boundary extends from the bottom of the tunnel to the upper side at a certain angle and reaches a certain height, then extends to the ground surface vertically. According to the ground collapse range in Figure6, considering the symmetry of the tunnel, a failure model considering the influence of seepage, as shown in Figure7, is established. This model consists of three parts: a pyramid, a prismoid and a prism from bottom to top. Here, H is the equivalent height of the tunnel (in this paper, H = D); C is the cover thickness of the tunnel; h0 is the hydraulic head of the ground surface; ht is the hydraulic head of the tunnel face; q is the additional pressure on

Figure 6. The ground collapse range of the vertical symmetry plane in the center of the tunnel and the plane in front of the tunnel face [3].

is the hydraulic head of the ground surface; ht is the hydraulic head of the tunnel face; q is the ω ω additional pressure on the ground surface; 1 is the break angle in front of the tunnel face; 2 is ω the horizontal break angle; 3 is the break angle behind the tunnel face.

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the ground surface; ω1 is the break angle in front of the tunnel face; ω2 is the horizontal break angle; ω3 is the break angle behind the tunnel face. Figure 5. Test model of excavation-seepage instability of the shield tunnel [3].

FigureFigure 6. 6. TheThe ground ground collapse range ofof thethe verticalvertical symmetrysymmetry plane plane in in the the center center of of the the tunnel tunnel and and the Symmetrytheplane plane 2020 in, front12in, frontx FOR of theof PEER the tunnel tunnelREVIEW face face [3]. [3]. 9 of 15

According to the ground collapse range in Figure 6, considering the symmetry of the tunnel, a failure model considering the influence of seepage, as shown in Figure 7, is established. This model consists of three parts: a pyramid, a prismoid and a prism from bottom to top. Here, H is the = equivalent height of the tunnel (in this paper, HD); C is the cover thickness of the tunnel; h0 is the hydraulic head of the ground surface; ht is the hydraulic head of the tunnel face; q is the ω ω additional pressure on the ground surface; 1 is the break angle in front of the tunnel face; 2 is ω the horizontal break angle; 3 is the break angle behind the tunnel face.

Figure 7. Pyramid–prismoid–prism failure model considering seepage.

Figure 8 shows the stress state of the prism micro-element, where, dpS1 and dpS2 are the

water pressure on the two sides of the prism micro-element, respectively; dN'1 and dN'2 are the normal effective pressure on the two sides of the prism micro-element, respectively. From the static equilibrium conditions of the prism micro-elements in the vertical direction, it can be obtained that ′ ++= + dV dpV dG22 dS12 dS (4)

where dV ' is the vertical effective pressure increment of the prism micro-element; ′′=+()()ωω ωσ dV2tan H13 H tan4tan H 2 d z (5) σ where 'z is the average vertical effective compressive stress on the z plane of the prism; dpV is the vertical water pressure increment of the prism micro-element; =+()()()ωω ω dpV 2 H tan13 H tan 4 H tan 2 du z (6)

where uz() is the average water pressure on the z plane of the prism; dG is the gravity of the prism micro-element, =+γωωω()() dGsat 2tan H13 H tan4tan H 2 dz (7) γ where sat is the saturated weight of the soil; dS1 is the friction resistance on the front and back sides of the prism micro-element,

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3.2. Calculation of the Limit Support Pressure of the Tunnel Face It is assumed that the soil in front of and behind the tunnel face is uniform and obeys the Mohr–Coulomb failure criterion. Considering that the divided calculation of water and soil needs to solve the seepage force, the process of solution is rather complicated, so this paper uses the method of soil–water joint calculation to analyze the stress of the failure model. Figure8 shows the stress state of the prism micro-element, where, dpS1 and dpS2 are the water pressure on the two sides of the prism micro-element, respectively; dN01 and dN02 are the normal eﬀective pressure on the two sides of the prism micro-element, respectively. From the static equilibrium conditions of the prism micro-elements in the vertical direction, it can be obtained that

dV0 + dpV + dG = 2dS1 + 2dS2 (4) where dV0 is the vertical eﬀective pressure increment of the prism micro-element;

dV0 = (2H tan ω1 + H tan ω3)(4H tan ω2)dσ0z (5) where σ0z is the average vertical eﬀective compressive stress on the z plane of the prism; dpV is the vertical water pressure increment of the prism micro-element;

dpV = (2H tan ω1 + H tan ω3)(4H tan ω2)du(z) (6) where u(z) is the average water pressure on the z plane of the prism; dG is the gravity of the prism micro-element, dG = γsat(2H tan ω1 + H tan ω3)(4H tan ω2)dz (7) where γsat is the saturated weight of the soil; dS1 is the friction resistance on the front and back sides of the prism micro-element, dS1 = (c + kσ0z tan ϕ)(4H tan ω2)dz (8) where c is the cohesion of the soil; k is the pressure coeﬃcient (in this paper, k = 1 sin ϕ); ϕ is the − friction angle of soil; dS2 is the friction resistance on the left and right sides of the prism micro-element:

dS2 = (c + kσ0z tan ϕ)(2H tan ω1 + H tan ω3)dz (9)

Substituting Equations (5)–(9) into Equation (4) and combining the surface boundary conditions σ z = + = q and u(H + C) = γw(h H C), the vertical eﬀective compressive stress of the prism 0 |z H C 0 − − σ0z can be obtained,

k tan ϕ k tan ϕ γsatR c (H+C z) (H+C z) σ z(z) = − 1 e− R − + [q + γw(h H C)]e− R − 0 k tan ϕ − 0 − − − k tan ϕ k tan ϕ k tan ϕ R z z z u(z) e− R dz e R u(z) R H+C − k tan ϕ k tan ϕ γsatR c (H+C z) (H+C z) = − 1 e− R − + [q + γw(h H C)]e− R − (10) k tan ϕ − 0 − − − k tan ϕ k tan ϕ k tan ϕ R z R 2H tan ω2 R 2H tan ω1 z z u(x y z) e R dxdydz e R R(2H tan ω +H tan ω )(4H tan ω ) H+C 2H tan ω H tan ω , , − 1 3 2 − 2 − 3 − R 2H tan ω R 2H tan ω 1 2 1 u(x y z) dxdy (2H tan ω +H tan ω )(4H tan ω ) 2H tan ω H tan ω , , 1 3 2 − 2 − 3 where R is the area-perimeter ratio of the cross-section of the prism,

(2H tan ω + H tan ω3)(4H tan ω2) R = 1 (11) [2(2H tan ω1 + H tan ω3) + 2(4H tan ω2)] Symmetry 2020, 12, x FOR PEER REVIEW 10 of 15

=+()()σϕ′ ω dS12 c kz tan 4 H tan dz (8) where c is the cohesion of the soil; k is the pressure coefficient (in this paper, k =−1sinϕ ); ϕ is the friction angle of soil; dS2 is the friction resistance on the left and right sides of the prism micro-element:

=+()()σϕ′ ω + ω dS213 c kz tan 2 H tan H tan dz (9)

Substituting Equations (5)–(9) into Equation (4) and combining the surface boundary conditions σ ′ = q and uH+C()=−−γ () h H C, the vertical effective compressive stress of z zHC=+ w 0 σ the prism 'z can be obtained,

kktanϕϕ tan γ Rc− −+−()HCz −+−() HCz σγ′()zeqhHCe=−sat 1 RR ++−−−() zwϕ 0 k tan  kktanϕϕ tan k tanϕ z − zz uz() eRR dze− uz ()  + R HC kktanϕϕ tan γ Rc− −+−()HCz −+−() HCz =esat 1−++−−−RR qhHCeγ () ϕ w 0 (10) k tan  kktanϕϕ tan ϕ zH2tanωω 2tan H − zz k tan 21 RR− +−ωω − u(,,) x y z e dxdydz e ()ωω+ ()ω HC2tan H23  H tan RH2tan13 H tan 4t Han 2  ωω 1 2tanHH21 2tan u(,,) x y z dxdy −−ωω ()()ωω+ ω2tanHH23 tan 2tanHH13 tan4tan H 2 where R is the area-perimeter ratio of the cross-section of the prism, ()()ωω+ ω 2tanHH13 tan4tan H 2 R = (11) Symmetry 2020, 12, 1023 ()()ωω++ ω 9 of 13 22HH tan13 tan 24 H tan 2

Figure 8. TheThe stress stress state state of of the the prism prism micro-element (soil–water joint calculation).

In order toto simplifysimplify thethe calculation,calculation, it it is is assumed assumed that that the the vertical vertical eff effectiveective compressive compressive stress stress in inthe the prismoid prismoid continues continues the changethe change rule inrule the in prism; the prism; that is, that within is, within the range theH rangez HzHCH ≤≤+ C, the + vertical, the ≤ ≤ effective compressive stress σ z(z) can be obtained by Equation (10). When z = H, the vertical effective 0 compressive stress of the interface between the prismoid and the pyramid can be obtained, that is, σ p = σ z(H). 0 0 Figure9 shows the stress state of the whole pyramid. According to the static equilibrium conditions in the x and z directions of the pyramid, it can be obtained that

T0 + pT + 2Ss sin α + S f sin ω1 = N0 f cos ω1 + p f cos ω1 (12)

V0 + pV + G = 2Ss cos α cos ω2 + S f cos ω1 + 2N0s sin ω2 + 2ps sin ω2 + N0 f sin ω1 + p f sin ω1 (13) where N0 f is the normal eﬀective pressure on the slope in front of the pyramid; S f is the friction resistance on the slope in front of the pyramid. The relationship between them is as follows: " !# 1 H S f = (2H tan ω2) c + N0 f tan ϕ (14) 2 cos ω1

Substitute Equation (14) into Equations (12) and (13), respectively, and eliminate N0 f simultaneously to obtain the eﬀective support pressure of the tunnel face T0,

1 2 T0 = V0 + pV + G 2Ss cos α cos ω2 H tan ω2c 2N0s sin ω2 2ps sin ω2 p f sin ω1 tan(ϕ+ω1) − − − − − − (15) 2 p 2Ss sin α H tan ω tan ω c + p cos ω T − − 1 2 f 1 where α is the angle between two side edges of the pyramid,

α = arctan(tan ω1 cos ω2) (16)

V0 is the vertical eﬀective pressure on the top surface of the pyramid,

V0 = (H tan ω1)(2H tan ω2)σ0z(H) (17) Symmetry 2020, 12, 1023 10 of 13

pV is the water pressure on the top surface of the pyramid,

Z H tan ω2 Z H tan ω1 pV = u(x, y, H) dxdy (18) H tan ω 0 − 2 G is the gravity of the pyramid, " # H(H tan ω1)(2H tan ω2) G = γ (19) sat 3

SS is the friction resistance on both sides of the pyramid,

1 H Ss = (H tan ω1) c + N0s tan ϕ (20) 2 cos ω2 where N0S is the normal pressure on both sides of the pyramid. Assuming that the pyramid adopts the vertical stress distribution form of gravity mode, as shown in Figure 10( γ0 in the ﬁgure is the eﬀective gravity of the soil), then,

2 3 ! k tan ω H σ0z(H) (γsat γw)H N = 1 + − (21) 0s 2 cos ω2 2 6 where γw is the water gravity. pS is the water pressure on both sides of the pyramid,

Z H tan ω1 H tan ω1 x ps = − u(x) dx (22) 0 tan ω1 cos ω2

p f is the water pressure on the slope in front of the pyramid,

Z H tan ω1 2x tan ω2 p f = u(x) dx (23) 0 tan ω1 sin ω1

pT is the water pressure on the front of the pyramid,

p = H2 u(x) T tan ω2 x=0 (24)

By substituting Equations (16)–(24) into Equation (15), the eﬀective support pressure of the tunnel face T0 can be obtained, then the eﬀective support pressure stress of the tunnel face σ0T can be obtained,

T σ = 0 (25) 0T 2 H tan ω2

It can be seen in Figure3 that the break angle behind the tunnel face, ω3, does not change much ϕ and can be approximately taken as 45 . Through the optimization calculation, it is determined that ◦ − 2 ω1 and ω2 make σ0T the maximum value σ0Tlim, and σ0Tlim is the limit eﬀective support pressure stress of the tunnel face considering the inﬂuence of seepage. Symmetry 2020, 12, x FOR PEER REVIEW 12 of 15

pS is the water pressure on both sides of the pyramid,

ω ω − H tan 1 Hxtan puxdx= 1 () (22) s 0 ωω tan12 cos

p f is the water pressure on the slope in front of the pyramid,

ω ω H tan 1 2tanx puxdx= 2 () (23) f 0 ωω tan11 sin

pT is the water pressure on the front of the pyramid,

pH= 2 tanω ux ( ) T 2 x=0 (24) By substituting Equations (16)–(24) into Equation (15), the effective support pressure of the σ tunnel face T ' can be obtained, then the effective support pressure stress of the tunnel face 'T can be obtained, T ' σ ' = T 2 ω (25) H tan 2 ω It can be seen in Figure 3 that the break angle behind the tunnel face, 3 , does not change ϕ much and can be approximately taken as 45° − . Through the optimization calculation, it is 2 ω ω σ σ σ Symmetrydetermined2020, 12that, 1023 1 and 2 make 'T the maximum value 'T lim , and 'T lim is the11 limit of 13 effective support pressure stress of the tunnel face considering the influence of seepage.

Symmetry 2020, 12, xFigure FOR PEER 9. The REVIEW stress state of the whole pyramid (soil–water joint calculation). 13 of 15

Figure 10. TheThe vertical vertical stress stress distribution form of gravity mode.

4. Comparative Comparative Analysis of the Limit EEffectiveffective SupportSupport PressurePressure ofof thethe TunnelTunnel Face Face Obtained Obtained by by DifferentDifferent Methods In orderorder toto verify verify the the correctness correctness of of the the theoretical theoretical solution solution for thefor limitthe limit equilibrium equilibrium in this in paper, this paper,the physical the physical model testmodel parameters test parameters of Mi and of XiangMi and [3] Xiang are used [3] forare calculation, used for calculation, and the calculation and the calculationresults are compared results are with compared the physical with model the physical test results model [3], thetest existing results theoretical[3], the existing calculation theoretical results calculationof the limit equilibriumresults of [11the] andlimit the equilibrium numerical simulation [11] and results the [numerical3], respectively. simulation The diameter results of [3], the respectively.tunnel D is 0.3 The m; thediameter covering of thicknessthe tunnelC is D 0.6 m;is 0.3m; the hydraulic the covering head of thickness ground surface C is h0.6m;0 is 1.0 the m; hydraulicthe hydraulic head head of ground of the tunnel surface face h h t isis 1.0m; 1.0, 0.9, the 0.8, hydraulic 0.7 or 0.5 head m; the of the saturated tunnel weight face h of is soil 1.0,γsat 0.9,is 3 0 t 21.2 kN/m ; the friction angle of the soil ϕ is 37◦; the cohesion of the soil c is 0; the additional pressure 0.8, 0.7 or 0.5m; the saturated weight of soil γ is 21.2 kN/ m3 ; the friction angle of the soil ϕ is on the ground surface q is 0. sat 37°; theFigure cohesion 11 shows of the the soil limit c eisff 0;ective the additional support pressure pressure of on the the tunnel ground face surface under diq ffiserent 0. seepage conditionsFigure obtained11 shows by the di fflimiterent effective methods, support where pre∆=ssureh0 hoft isthe the tunnel hydraulic face headunder di differentfference betweenseepage Δ−− conditionsthe tunnel faceobtained and the by ground different surface. methods, It can bewhere seen that=h0 there ht isis athe linear hydraulic relationship head between difference the betweenlimit effective the tunnel support face pressure and theσ0 Tgroundlim and surface. the hydraulic It can head be seen difference that there∆, that is is,a linearσ0Tlim willrelationship increase linearly with the increase in ∆. The theoretical calculationσ results of the limit equilibrium inΔ this paper between the limit effective support pressure 'T lim and the hydraulic head difference , that is, are obviously larger than those of Perazzelli et al. [11] and the numerical simulation results of Mi σ ' will increase linearly with the increase in Δ . The theoretical calculation results of the limit andT lim Xiang [3], which are closer to the physical model test results of Mi and Xiang [3]. The larger the equilibriumhydraulic head in this diff erencepaper are∆ is, obviously the closer larger the prediction than those result of Perazzelli calculated et in al. this [11] paper and isthe to numerical the actual simulationtest result. Theresults theoretical of Mi and model Xiang of [3], limit which equilibrium are closer established to the physical in this model paper test can results better predictof Mi and the Xiang [3]. The larger the hydraulic head difference Δ is, the closer the prediction result calculated in this paper is to the actual test result. The theoretical model of limit equilibrium established in this paper can better predict the limit effective support pressure of the tunnel face considering seepage and can provide a reference for practical engineering.

Symmetry 2020, 12, 1023 12 of 13 limit eﬀective support pressure of the tunnel face considering seepage and can provide a reference for practical engineering. Symmetry 2020, 12, x FOR PEER REVIEW 14 of 15

Figure 11. ComparisonComparison of of the the limit limit effective eﬀective support pressure of the tunnel face under seepage conditions obtained by didifferentﬀerent methods.methods.

5. Conclusions The objective objective was was to to study study the the limit limit support support pressure pressure of ofa tunnel a tunnel face face in sa inturated saturated sandy sandy soil soilunder under seepage seepage conditions. conditions. Firstly, Firstly, according according to to the the numerical numerical simulation simulation results, results, the the existing three-dimensionalthree-dimensional analytical solution for the pore water pressure distribution not consideringconsidering the distribution of of pore pore water water pressure pressure behind behind the tunnelthe tunnel face is face extended is extended to the three-dimensional to the three-dimensional solution consideringsolution considering the distribution the distribution behind the behind tunnel the face. tunnel Secondly, face. Secondly, based on based the results on the of results the physical of the modelphysical tests, model a failure tests, model a failure considering model consideringthe failure range the behindfailure therange tunnel behind face the was tunnel established, face was and theestablis newlyhed, established and the three-dimensional newly established pore three water-dimensional pressure solution pore water was introduced pressure into solution the model was tointroduced obtain the into limit the support model pressureto obtain ofthe the limit tunnel support face withpressure the methodof the tunnel of soil–water face with joint the calculation.method of Finally,soil–water the joint calculation calculation results. Finally, were compared the calculation with the results experimental were compa results,red with numerical the experimental simulation resultsresults, andnumerical existing simulation theoretical results solutions. and Theexisting main theoretical conclusions solutions. are as follows. The main conclusions are as follows.(1) The distribution of pore water pressure in the front and back strata above the tunnel face is basically(1) The symmetrical. distribution of pore water pressure in the front and back strata above the tunnel face is basically(2) When symmetrical the support. pressure of the tunnel face is not enough, both the ground in front and that behind(2)the When tunnel the face support will lose pressure stability of withinthe tunnel a certain face range.is not Therefore,enough, both when the establishing ground in thefront failure and model,that behind the failure the tunnel range offace the will ground lose directlystability in within front of a thecertain tunnel range. face, theTherefore, upper formation when establish in fronting of the tunnel failure face, model, and the the upper failure formation range of behindthe ground the tunnel directly face in should front be of considered the tunnel comprehensively. face, the upper formation(3) The in limit front eff ofective the support tunnel face pressure, and of the the upper tunnel formation face will increase behind linearly the tunnel with face the increaseshould be in theconsidered hydraulic comprehensively. head difference between the tunnel face and the ground surface. (4)(3) The The calculated limit effective results support in the new pressure limit equilibrium of the tunnel theory face are will obviously increase larger linearly than with those the in theincrease existing in the theory hydraulic and numerical head difference simulation between and arethe closertunnel to face the and results the ofground the physical surface. model tests. Therefore,(4) The the calculated new limit results equilibrium in the new model limit can equilibrium better predict theory the limitare obviously effective supportlarger than pressure those ofin the tunnel existing face theory considering and numerical seepage simulationand provide and a reference are closer for to practical the results engineering. of the physical model tests.It Therefore, is worth noting the new that the limit ground equilibrium studied in model this paper can better is sandy predict soil, and the the limit limit effective effective support support pressure of thethe tunneltunnel faceface forconsidering cohesive soilseepage needs and further provide study. a reference for practical engineering. It is worth noting that the ground studied in this paper is sandy soil, and the limit effective support pressure of the tunnel face for cohesive soil needs further study.

Author Contributions: B.M. and Y.X. contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

Symmetry 2020, 12, 1023 13 of 13

Author Contributions: B.M. and Y.X. contributed equally to this work. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Fundamental Research Funds for the Central Universities of China. (No. 2017YJS148). Acknowledgments: The authors are deeply thankful to the reviewers and editor for their valuable suggestions to improve the quality of the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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