Research ArticleUpper Bound Solution for the Face Stability ofShield Tunnel below the Water Table

Xilin Lu1 Haoran Wang2 and Maosong Huang1

1 Department of Geotechnical Engineering Tongji University 1239 Siping Road Shanghai 200092 China2 Shanghai Urban Construction Design amp Research Institute Shanghai 200125 China

Correspondence should be addressed to Xilin Lu xilinlutongjieducn

Received 16 May 2014 Revised 15 August 2014 Accepted 22 August 2014 Published 9 October 2014

Academic Editor Hang Xu

Copyright copy 2014 Xilin Lu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

By FE simulation with Mohr-Coulomb perfect elastoplasticity model the relationship between the support pressure anddisplacement of the shield tunnel face was obtained According to the plastic strain distribution at collapse state an appropriatefailure mechanism was proposed for upper bound limit analysis and the formula to calculate the limit support pressure wasdeduced The limit support pressure was rearranged to be the summation of soil cohesion c surcharge load q and soil gravity120574 multiplied by their corresponding coefficients119873

119888119873119902 and119873

120574 and parametric studies were carried out on these coefficients In

order to consider the influence of seepage on the face stability the pore water pressure distribution and the seepage force on thetunnel face were obtained by FE simulation After adding the power of seepage force into the equation of the upper bound limitanalysis the total limit support pressure for stabilizing the tunnel face under seepage conditionwas obtainedThe total limit supportpressure was shown to increase almost linearly with the water table

1 Introduction

The key issue during shield tunneling is to keep the stabilityof tunnel face and this generally depends on the supportpressure which was applied on the tunnel face after soilexcavation The pressure must be controlled at least no lessthan its limit value which corresponds to the active failurestate of the tunnel face So far various kinds of methods havebeen proposed to study this problem Experimental methodsincluding physical modeling and centrifuge modeling [1ndash4]were applied to study the failure mechanism of tunnel faceand the limit support pressure Some empirical method wasproposed to evaluate the stability of the tunnel face for exam-ple Broms and Bennermark [5] proposed stability numberto assess the stability of tunnel face in clay under undrainedcondition Cornejo [6] gave out a formula to determine thestability number of the tunnel face in clay based on the limitequilibrium method Based on the limit equilibrium of soilin front of the tunnel face Murayama proposed a formulato calculate the limit support pressure and the varyingcharacteristic of the formula with soil strength was studiedby Lu et al [7] Following Hornrsquos theory Jancsecz and Steiner

[8] studied the face stability of shield tunnel under drainedcondition by considering the equilibriumof the slidingwedgeat the tunnel face By employing limit analysis Davis et al[9] studied the stability of tunnel face and presented lowerand upper bound solutions of the limit support pressure tostabilize the tunnel face under undrained condition Augardeet al [10] employed finite element limit analysis to study theplane strain tunnel face under undrained condition Leca andDormieux [11] proposed a 3D failure mechanism to obtainthe upper bound solution of limit support pressure underdrained condition After the utilization of multiblock failuremechanism Soubra [12] and Mollon et al [13] obtainedbetter upper bound solutions which were closer to theexperimental results of Chambon and Corte [4] Mollon etal [14] proposed a 2D multiblock limit analysis method todetermine the critical collapse pressure of air pressurizedshield tunnel and Mollon et al [15] further studied theface stability by probabilistic analysis Numerical simulationmethod has been also adopted to study the stability of tunnelface Vermeer et al [16] studied the stability of the shieldtunnel face and the influences of cohesion surcharge loadand soil gravity on the limit support pressure by FEM

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 727964 11 pageshttpdxdoiorg1011552014727964

2 Mathematical Problems in Engineering

Li et al [17] studied the face stability of shield tunnel by FLAC3D analysis Discrete element method was also used for theface stability of shield tunnel [18ndash21] Although the failuremechanism and limit support pressure of shield tunnel facecould be obtained the complicated calculation in numericalmodelingmakes it too difficult to be used in real engineering

When shield tunnel locates under the water table linethe soil excavation often induces underground water seepageand apt to cause the collapse of shield tunnel face The facestability analysis under seepage condition was studied bynumerical simulation [22ndash24] or theoretical analysis whichwas based on the existing model Anagnostou and Kovari[25] studied the influence of seepage on the stability of tunnelface based on the wedge model de Buhan et al [26] analyzedthe face stability of shield tunnel by introducing the seepageforce into the model of Leca and Dormieux [11] Lee andNam [27] considered the influence of seepage by superposingthe results of seepage analysis on the mechanical analysisunder drained condition Lee et al [28] compared the resultsobtained from theoretical analysis by taking seepage forcesinto account with the results of the coupled finite elementanalysis Park et al [29] studied the stability of pressurizedshield tunnel by incorporating the results of Lee et al [28]into the upper bound analysis of Leca and Dormieux [11]The results from these works and recent study of Li et al [30]showed that the underground water seepage played crucialrole in the stability of tunnel face

In this paper the stability of shield tunnel facewas studiedby elastoplasticity FE simulation the collapsemechanism andlimit support pressure in active failure state were obtainedBased on the numerical results a failure mechanism wasproposed and a 2D upper-bound limit analysis model wasestablished and the formula for calculating the limit supportpressure was also deduced Following the Terzaghi superpo-sition method which has been commonly used in bearingcapacity analysis the limit support pressure was rearrangedas the summation of soil cohesion surcharge load andsoil gravity multiplied by their corresponding coefficientsand the varying characteristics of these coefficients with thedepth-to-diameter ratio of tunnel and the friction angle ofsoil were studied in detail The influence of seepage on thestability of shield tunnel under water table was also studiedThe pore water pressure distribution and seepage force on theshield tunnel face were obtained by FE numerical simulationAfter the calculation of seepage force on the failure area oftunnel face the proposed upper bound limit analysis modelwas extended into seepage condition

2 Finite Element Modeling of TunnelFace Stability

The relationship of deformation and support pressure ofshield tunnel face was obtained by FE analysis with PLAXISsoftware the constitutive model adopted is the widely usedMohr-Coulomb perfect elastoplasticity model The tunneldiameter is 119863 = 10m and the tunnel depth is 119862 = 10mthe finite element mesh constituted by 15-node triangularelement employed in numerical simulation was shown in

u = 0

u = 0

u = 0

u = = 0

u = = 0

120590t

C

D

q

Figure 1 The finite element mesh for the stability analysis of shieldtunnel

450

400

350

300

250

200

150

100

50

0

0 004 008 012 016 02

uD

120590t

(kPa

)

120593 = 5

120593 = 10

120593 = 15

120593 = 20

120593 = 25

120593 = 30

120593 = 35

120593 = 40

120593 = 45

Figure 2 The relationship between the support pressure anddisplacement at center-point of the tunnel face (119862119863 = 1)

Figure 1 The elastic modulus and Poisson ratio are 119864 =

20Mpa and ] = 03 Mohr-Coulomb model was adopted todescribe the constitutive relationship of soil in plastic stagethe cohesion 119888 = 2 kPa the friction angle 120593 ranges from 5∘to 45∘ and the soil gravity 120574 = 17 kNm3 Considering thatdilatancy angle has no influence on the limit support pressure[16] and in order to keep accordance with the assumption inupper bound limit analysis the dilatancy angle is assumed tobe equal to the friction angle

The initial stress field induced by the soil weight andsurcharge load was calculated After the excavation of the soilduring shield tunneling the initial condition was recoveredby applying lateral earth pressure with its value determinedby 1198700(119902 + 120574119862) In case of the lateral earth pressure 119870

0has

no influence on the support pressure at collapse [16] 1198700was

set as 1 minus sin(120593) After the initial pressure was applied thedisplacement of all nodes was set to zero The pressure wasreduced gradually from the initial value to obtain the curveof support pressure and displacement on the tunnel faceAs shown in Figure 2 the support pressure decreases withthe displacement after attaining its critical value it almostkeeps constant even when the displacement keeps increasing

Mathematical Problems in Engineering 3

(a) 119862119863 = 05 (120593 = 15∘ 30∘ 45∘)

(b) 119862119863 = 1 (120593 = 15∘ 30∘ 45∘)

(c) 119862119863 = 2 (120593 = 15∘ 30∘ 45∘)

Figure 3 The displacement increment around the tunnel face at collapse state

which indicates the collapse of the tunnel face It is also shownin Figure 2 that the limit support pressure obviously decreaseswith the friction angle

The increments of displacement and plastic strain distri-butions at collapse state are shown in Figures 3 and 4 Thefailure mode changes from global to local with the tunneldepth increases and the deformation area around the tunnelface reduces with the increase of the friction angle of soil

3 Upper Bound Limit Analysis of the ShieldTunnel Face Stability

31 Failure Mechanism In order to analyze the stability ofthe tunnel face an appropriate failuremechanism needs to beproposed According to the plastic strain distribution at col-lapse state obtained from numerical modeling in Section 2and referring to the Terzaghi failure mechanism for bearing

4 Mathematical Problems in Engineering

(a) 119862119863 = 05 (120593 = 15∘ 30∘ 45∘)

(b) 119862119863 = 1 (120593 = 15∘ 30∘ 45∘)

(c) 119862119863 = 2 (120593 = 15∘ 30∘ 45∘)

Figure 4 The equivalent plastic strain distribution around the tunnel face at collapse state

capacity analysis a failure mechanism which is composed ofa shearing zone 119887 and two rigid blocks 119886 and 119888 was proposedThe proposed failure mechanism could reflect the transitionfrom global failure mode to local mode with the increases ofthe tunnel depth which has been indicated by previous study[11 16] As shown in Figure 5 the upper isosceles triangle1198741015840

119874119861 which has an opening angle equal to 2120593 is block 119886and the axis of symmetry of the opening angle is verticalTheblock 119888 is isosceles triangle 1198741198601198601015840 the line 1198601198601015840 has an angleof 1205874 + 1205932 with the horizontal direction The shear zone 119887

is a log-spiral curve with the center is point119874 The geometricparameters of the failure mechanism are

1199030=

119863

2 sin (1205874 + 1205932) exp [(1205874 + 1205932) tan (120593)]

ℎ2=

1199030

(2 tan120593)

119897119861=

0 ℎ minus 119862 le 0

2 (ℎ minus 119862) tan120593 ℎ minus 119862 gt 0

(1)

Mathematical Problems in Engineering 5

2120593

O

O

998400

lB

aar0

120579OB

OE

B

bc

c

1205874 + 1205932

A

E

C

D

q

h

1205874 minus 1205932

Figure 5 The proposed failure mechanism

The comparison of the proposed failure mechanism and thenumerical results is shown in Figure 4 The block 119886 movesdownward with a velocity V

119886 the block 119888 moves left with an

angle of 1205874 + 1205932 with horizontal line and the velocity inshear zone increases from V

119886at 119874119861 line to V

119888at 1198741198601015840 line

32 The Limit Support Pressure Based on the failure mech-anisms proposed in previous section the upper boundsolution of the limit support pressure could be obtained byequaling the power of external force and plastic dissipationenergy [31] The power of the weight of block 119886 is

119875119882119886

=1

2120574V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (2)

The differential of the power of the weight of shearingzone 119887 is

119889119875119908119887

=V119886

2120574int

1205874+1205932

0

1199032

0exp (3120579 tan120593) cos 120579 119889120579 (3)

After integration the power of the weight of shearingzone 119887 is

119875119908119887

= int

1205874+1205932

0

119889119875119908119887

=120574V1198861199032

0

2 (1 + 9tan2120593)

sdot (sin(1205874+120593

2) exp [3 (120587

4+120593

2) tan120593]

+ 3 tan120593cos(1205874+120593

2)

sdot exp [3 (1205874+120593

2) tan120593] minus 1)

(4)

The power of the weight of block 119888 is

119875119882119888

=120574

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(5)

The power of the surcharge load is

119875119902= 119902119897119861V119886 (6)

The power of the support pressure on the tunnel face is

119875119905= 120590119905119863V119886exp [(120587

4+120593

2) tan120593] sin(120587

4+120593

2) (7)

The internal energy dissipation along line 1198741198741015840 and line1198741015840

1198611015840 is

119864119886= (2ℎ minus 119897

119861cot120593) 119888V

119886 (8)

The internal energy dissipation along the line 1198601198601015840 is

119864119888=119863 exp [(1205874 + 1205932) tan120593]

2 sin (1205874 + 1205932)119888V119886 (9)

The internal energy dissipation along line 1198611198601015840 which

equals the dissipation power rate in block 119887 is

119864119887= 1198641198611198601015840 =

1

2119888V1198861199030cot120593exp [(120587

2+ 120593) tan120593] minus 1 (10)

By equaling the power of external forces to the internalenergy dissipation we get

119875119882119886

+ 119875119882119887

+ 119875119882119888

+ 119875119902minus 119875119905= 119864119886+ 119864119887+ 119864119888+ 1198641198611198601015840 (11)

By combining from (2) to (11) the formula for calculat-ing the limit support pressure was obtained Analogous toTerzaghirsquos superposition method which has been commonlyused in bearing capacity analysis the obtained limit supportpressure could be rearranged to be

120590119905= 119888119873119888+ 119902119873119902+ 120574119863119873

120574 (12)

6 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

80

100Li

mit

supp

ort p

ress

ure (

kPa)

c (kPa)

120593 = 15

120593 = 30

120593 = 45

(a) The influence of cohesion

0 20 40 600

100

200

300

400

Lim

it su

ppor

t pre

ssur

e (kP

a)CD = 05CD = 1

CD = 2

120593 (∘)

(b) The influence of friction angle (119888 = 2 kPa)

Figure 6 The influence of the soil strength parameters on the limit support pressure (120574 = 17 kPa119863 = 10m)

where the influence coefficients of cohesion 119873119888 surcharge

load119873119902 and soil gravity119873

120574are

119873119888= minus

1

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

sdot ⟨(2ℎ minus 119897119861cot120593)

+119863 exp [(1205874 + 1205932) tan120593] cos120593

2 sin (1205874 + 1205932)

+ 1199030cot120593exp [(120587

2+ 120593) tan120593] minus 1⟩

119873119902=

119897119861

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

119873120574=

1

1198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot ⟨1199030ℎ minus 119897119861(ℎ minus 119862)

+ ((3 tan120601

sdot exp [3(1205874+120601

2) tan120601]

sdot cos(1205874+120601

2) minus 1

+ exp [3(1205874+120601

2) tan120601] sin(120587

4+120601

2))

sdot (1 + 9tan2120601)minus1

) 1199032

0

+1198632

2tan(120587

4minus120601

2)

sdot cos(1205874+120601

2) exp [(120587

4+120601

2) tan120601]⟩

(13)

As shown in Figure 6 the limit support pressuredecreases with the friction angle 120593 it increases with thedepth-to-diameter ratio of tunnel only when 120593 lt 20

∘ whileif 120593 gt 20

∘ the variation of depth-to-diameter ratio showsalmost no influence on the limit support pressure and theseresults agree well with those of Vermeer et al [16]

33 Parametric Studies on the Influence Coefficients Thecoefficients of cohesion surcharge load and soil gravity in(14) play important role in the calculation of limit supportpressure and deserve further study Different from the caseof bearing capacity analysis the coefficient of cohesion 119873

119888

here is negative As shown in Figure 7 119873119888increases with

the friction angle 120593 of soil and decreases with the depth-to-diameter ratio of shield tunnel Vermeer et al [16] statedthat119873

119888could be calculated by cot(120593) when the tunnel depth

119862 is more than twice of diameter 119863 or when friction angle120593 gt 35

∘ The upper bound solutions are close to the solutionsof Vermeer et al [16] obtained from FEM when 119862119863 = 2

and 120593 gt 10∘ When friction angle is less than a certain

value (120593 = 10∘) the 119873

119888-120593 curve changes which indicates

the intersection of the failure area on the ground surface

Mathematical Problems in Engineering 7

0 10 20 30 40 500

Vermeer et al (2002)

minus10

minus8

minus6

minus4

minus2

Nc

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

120593 (∘)

Figure 7 The relationship between119873119888and the friction angle of soil

(119863 = 10m)

Comparatively the formula119873119888= cot(120593) could not reflect this

property and is only suitable under the condition of smallsoil friction angles As shown in Figure 8 119873

119902is plotted as a

function of the friction angle for common values of the depthratio 119862119863 equal to 05 1 and 2 It is shown that the upperbound solutions decrease with the friction angle of soil andthe depth-to-diameter ratio of tunnel The value of 119873

119902turns

to be zerowhen friction angle reaches a certain value and thisvalue decreases with the depth-to-diameter ratio of tunnelThe relationship between the upper bound solution of119873

120574and

friction angle 120593 of soil is shown in Figure 9 The value of 119873120574

decreases with 120593 and it increases with the depth-to-diameterratio of tunnel only when friction angle is small otherwise itkeeps constant The upper bound solutions obtained in thispaper agree well with the formula 119873

120574= 1(9 tan120593) minus 005

proposed by Vermeer et al [16] when the friction angle of soilis large The figure also shows that the results in this paperare very close to the results of Mollon et al [13] and it isapplicable for wider range of the friction angle of soil

4 Influence of Seepage on the Stability ofTunnel Face

41 Seepage Analysis When shield tunnel locates underwater table line the soil excavation often induces seepageand causes the failure of tunnel face It is necessary toestablish a calculation model to estimate the influence ofseepage on the stability of tunnel face The key point of thestability analysis under seepage condition is the calculation of

0 10 20 30 40 500

02

04

06

08

1

120593 (∘)

LA (CD = 2)LA (CD = 1)LA (CD = 05)

FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nq

Figure 8The relationship between119873119902and the friction angle of soil

(119863 = 10m)

the seepage force By assuming the underground water seep-age to follow the Darcy law the seepage equation in steadystate is

120597

120597119909(119870119909

120597119867

120597119909) +

120597

120597119910(119870119910

120597119867

120597119910) = 0 (14)

where 119870119909 119870119910are the seepage coefficients in 119909 and 119910 direc-

tions in order to simplify the problem these two coefficientsare considered as the same 119867 is the water table from thebottom of the tunnel

FE simulation was employed to analyze the seepagecharacteristics of the tunnel face In the simulation two caseswith tunnel diameter of 5m and 10m were studied the watertable varies from the top of the tunnel to three times oftunnel diameters When tunnel diameter 119863 = 10m tunneldepth-to-diameter 119862119863 = 2 water table-to-tunnel diameter119867119863 = 2 and the seepage coefficient is 03 times 10minus5ms theporewater pressure distribution near the tunnel face obtainedfrom numerical simulation is shown in Figure 10 The porewater pressure on tunnel face is zero and the interval ofpressure line is 1m water level The failure area was dividedinto two parts the top part119860 is a triangle which is the same asblock 119886 in Figure 4 and the lower part 119861 is blocks 119887 and 119888 Asshown in Figure 10 the porewater pressure distributes almostuniformly in the failure area of tunnel face and it changessignificantly near the top and bottom of the tunnel face

From the pore water pressure distribution the water headdifference between the failure line and the tunnel face couldbe obtained and then the seepage force 119865 on the failure areacould be calculated the detailed derivation could be found atLee et al [28]The total seepage forcewas obtained and shown

8 Mathematical Problems in Engineering

0 10 20 30 40 500

1

2

3

4

5

Vermeer et al (2002)Mollon et al (2010)

120593 (∘)

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nr

Figure 9The relationship between119873120574and the friction angle of soil

(119863 = 10m)

A

B

Failure curve

Tunnel face

u = 0

u = 0

Figure 10 The distribution of the pore water pressure around thetunnel face

in Figure 11 the ratio of average seepage force over hydrostaticforce keeps almost constant and the value is slightly morethan that of Lee et al [28] obtained from 3D analysis

In order to study the seepage force in detail the horizontaland vertical components of the seepage forces in area119860 (area119886) and 119861 (areas 119887 and 119888) are studied separately The totalseepage force could be decomposed as

119865 = radic(119865119860119909

+ 119865119861119909)2

+ (119865119860119910

+ 119865119861119910)2

(15)

where 119865119860119909 119865119860119910 119865119861119909 and 119865

119861119910are the horizontal and vertical

components of seepage force in parts 119860 and 119861

05

04

03

02

01

0

1 15 2 25 3

HD

Ratio

D = 5mD = 10m

Lee et al (2003) (D = 5m)

Figure 11 The relationship between the average seepage force andthe water table

The ratios of each component of average seepage forceover hydrostatic force are shown in Figure 12 the biggestcomponent 119865

119861119909varies slightly with the water table 119865

119860119910and

119865119861119910

are smaller and 119865119860119909

is the smallest

42 Influence of Seepage on Limit Support Pressure By includ-ing the power rate of the seepage force in upper boundlimit analysis the influence of seepage on the limit supportpressure of tunnel face could be studied The power rate ofthe seepage force 119875

119865is

119875119865= 119875119865119909+ 119875119865119910 (16)

where 119875119865119909

is the power rate of the horizontal component ofseepage force 119875

119865119910is the power rate of the vertical component

of seepage force For the horizontal component velocity inpart119860 is zero so the power rates of the horizontal componentof seepage force component vanishes that is

119875119865119860119909

= 0 (17)

The power rate of the vertical component of seepage forcecomponent is

119875119865119860119910

=1

2119865119860119910V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (18)

In shear zone 119887 the power rates of the seepage force in soilelement are

119889119875119865119887119909

=V119886

21198651198611199091199032

0exp (3120579 tan120593) sin 120579 119889120579

119889119875119865119887119910

=V119886

21198651198611199101199032

0exp (3120579 tan120593) cos 120579 119889120579

(19)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Li et al [17] studied the face stability of shield tunnel by FLAC3D analysis Discrete element method was also used for theface stability of shield tunnel [18ndash21] Although the failuremechanism and limit support pressure of shield tunnel facecould be obtained the complicated calculation in numericalmodelingmakes it too difficult to be used in real engineering

When shield tunnel locates under the water table linethe soil excavation often induces underground water seepageand apt to cause the collapse of shield tunnel face The facestability analysis under seepage condition was studied bynumerical simulation [22ndash24] or theoretical analysis whichwas based on the existing model Anagnostou and Kovari[25] studied the influence of seepage on the stability of tunnelface based on the wedge model de Buhan et al [26] analyzedthe face stability of shield tunnel by introducing the seepageforce into the model of Leca and Dormieux [11] Lee andNam [27] considered the influence of seepage by superposingthe results of seepage analysis on the mechanical analysisunder drained condition Lee et al [28] compared the resultsobtained from theoretical analysis by taking seepage forcesinto account with the results of the coupled finite elementanalysis Park et al [29] studied the stability of pressurizedshield tunnel by incorporating the results of Lee et al [28]into the upper bound analysis of Leca and Dormieux [11]The results from these works and recent study of Li et al [30]showed that the underground water seepage played crucialrole in the stability of tunnel face

In this paper the stability of shield tunnel facewas studiedby elastoplasticity FE simulation the collapsemechanism andlimit support pressure in active failure state were obtainedBased on the numerical results a failure mechanism wasproposed and a 2D upper-bound limit analysis model wasestablished and the formula for calculating the limit supportpressure was also deduced Following the Terzaghi superpo-sition method which has been commonly used in bearingcapacity analysis the limit support pressure was rearrangedas the summation of soil cohesion surcharge load andsoil gravity multiplied by their corresponding coefficientsand the varying characteristics of these coefficients with thedepth-to-diameter ratio of tunnel and the friction angle ofsoil were studied in detail The influence of seepage on thestability of shield tunnel under water table was also studiedThe pore water pressure distribution and seepage force on theshield tunnel face were obtained by FE numerical simulationAfter the calculation of seepage force on the failure area oftunnel face the proposed upper bound limit analysis modelwas extended into seepage condition

2 Finite Element Modeling of TunnelFace Stability

The relationship of deformation and support pressure ofshield tunnel face was obtained by FE analysis with PLAXISsoftware the constitutive model adopted is the widely usedMohr-Coulomb perfect elastoplasticity model The tunneldiameter is 119863 = 10m and the tunnel depth is 119862 = 10mthe finite element mesh constituted by 15-node triangularelement employed in numerical simulation was shown in

u = 0

u = 0

u = 0

u = = 0

u = = 0

120590t

C

D

q

Figure 1 The finite element mesh for the stability analysis of shieldtunnel

450

400

350

300

250

200

150

100

50

0

0 004 008 012 016 02

uD

120590t

(kPa

)

120593 = 5

120593 = 10

120593 = 15

120593 = 20

120593 = 25

120593 = 30

120593 = 35

120593 = 40

120593 = 45

Figure 2 The relationship between the support pressure anddisplacement at center-point of the tunnel face (119862119863 = 1)

Figure 1 The elastic modulus and Poisson ratio are 119864 =

20Mpa and ] = 03 Mohr-Coulomb model was adopted todescribe the constitutive relationship of soil in plastic stagethe cohesion 119888 = 2 kPa the friction angle 120593 ranges from 5∘to 45∘ and the soil gravity 120574 = 17 kNm3 Considering thatdilatancy angle has no influence on the limit support pressure[16] and in order to keep accordance with the assumption inupper bound limit analysis the dilatancy angle is assumed tobe equal to the friction angle

The initial stress field induced by the soil weight andsurcharge load was calculated After the excavation of the soilduring shield tunneling the initial condition was recoveredby applying lateral earth pressure with its value determinedby 1198700(119902 + 120574119862) In case of the lateral earth pressure 119870

0has

no influence on the support pressure at collapse [16] 1198700was

set as 1 minus sin(120593) After the initial pressure was applied thedisplacement of all nodes was set to zero The pressure wasreduced gradually from the initial value to obtain the curveof support pressure and displacement on the tunnel faceAs shown in Figure 2 the support pressure decreases withthe displacement after attaining its critical value it almostkeeps constant even when the displacement keeps increasing

Mathematical Problems in Engineering 3

(a) 119862119863 = 05 (120593 = 15∘ 30∘ 45∘)

(b) 119862119863 = 1 (120593 = 15∘ 30∘ 45∘)

(c) 119862119863 = 2 (120593 = 15∘ 30∘ 45∘)

Figure 3 The displacement increment around the tunnel face at collapse state

which indicates the collapse of the tunnel face It is also shownin Figure 2 that the limit support pressure obviously decreaseswith the friction angle

The increments of displacement and plastic strain distri-butions at collapse state are shown in Figures 3 and 4 Thefailure mode changes from global to local with the tunneldepth increases and the deformation area around the tunnelface reduces with the increase of the friction angle of soil

3 Upper Bound Limit Analysis of the ShieldTunnel Face Stability

31 Failure Mechanism In order to analyze the stability ofthe tunnel face an appropriate failuremechanism needs to beproposed According to the plastic strain distribution at col-lapse state obtained from numerical modeling in Section 2and referring to the Terzaghi failure mechanism for bearing

4 Mathematical Problems in Engineering

(a) 119862119863 = 05 (120593 = 15∘ 30∘ 45∘)

(b) 119862119863 = 1 (120593 = 15∘ 30∘ 45∘)

(c) 119862119863 = 2 (120593 = 15∘ 30∘ 45∘)

Figure 4 The equivalent plastic strain distribution around the tunnel face at collapse state

capacity analysis a failure mechanism which is composed ofa shearing zone 119887 and two rigid blocks 119886 and 119888 was proposedThe proposed failure mechanism could reflect the transitionfrom global failure mode to local mode with the increases ofthe tunnel depth which has been indicated by previous study[11 16] As shown in Figure 5 the upper isosceles triangle1198741015840

119874119861 which has an opening angle equal to 2120593 is block 119886and the axis of symmetry of the opening angle is verticalTheblock 119888 is isosceles triangle 1198741198601198601015840 the line 1198601198601015840 has an angleof 1205874 + 1205932 with the horizontal direction The shear zone 119887

is a log-spiral curve with the center is point119874 The geometricparameters of the failure mechanism are

1199030=

119863

2 sin (1205874 + 1205932) exp [(1205874 + 1205932) tan (120593)]

ℎ2=

1199030

(2 tan120593)

119897119861=

0 ℎ minus 119862 le 0

2 (ℎ minus 119862) tan120593 ℎ minus 119862 gt 0

(1)

Mathematical Problems in Engineering 5

2120593

O

O

998400

lB

aar0

120579OB

OE

B

bc

c

1205874 + 1205932

A

E

C

D

q

h

1205874 minus 1205932

Figure 5 The proposed failure mechanism

The comparison of the proposed failure mechanism and thenumerical results is shown in Figure 4 The block 119886 movesdownward with a velocity V

119886 the block 119888 moves left with an

angle of 1205874 + 1205932 with horizontal line and the velocity inshear zone increases from V

119886at 119874119861 line to V

119888at 1198741198601015840 line

32 The Limit Support Pressure Based on the failure mech-anisms proposed in previous section the upper boundsolution of the limit support pressure could be obtained byequaling the power of external force and plastic dissipationenergy [31] The power of the weight of block 119886 is

119875119882119886

=1

2120574V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (2)

The differential of the power of the weight of shearingzone 119887 is

119889119875119908119887

=V119886

2120574int

1205874+1205932

0

1199032

0exp (3120579 tan120593) cos 120579 119889120579 (3)

After integration the power of the weight of shearingzone 119887 is

119875119908119887

= int

1205874+1205932

0

119889119875119908119887

=120574V1198861199032

0

2 (1 + 9tan2120593)

sdot (sin(1205874+120593

2) exp [3 (120587

4+120593

2) tan120593]

+ 3 tan120593cos(1205874+120593

2)

sdot exp [3 (1205874+120593

2) tan120593] minus 1)

(4)

The power of the weight of block 119888 is

119875119882119888

=120574

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(5)

The power of the surcharge load is

119875119902= 119902119897119861V119886 (6)

The power of the support pressure on the tunnel face is

119875119905= 120590119905119863V119886exp [(120587

4+120593

2) tan120593] sin(120587

4+120593

2) (7)

The internal energy dissipation along line 1198741198741015840 and line1198741015840

1198611015840 is

119864119886= (2ℎ minus 119897

119861cot120593) 119888V

119886 (8)

The internal energy dissipation along the line 1198601198601015840 is

119864119888=119863 exp [(1205874 + 1205932) tan120593]

2 sin (1205874 + 1205932)119888V119886 (9)

The internal energy dissipation along line 1198611198601015840 which

equals the dissipation power rate in block 119887 is

119864119887= 1198641198611198601015840 =

1

2119888V1198861199030cot120593exp [(120587

2+ 120593) tan120593] minus 1 (10)

By equaling the power of external forces to the internalenergy dissipation we get

119875119882119886

+ 119875119882119887

+ 119875119882119888

+ 119875119902minus 119875119905= 119864119886+ 119864119887+ 119864119888+ 1198641198611198601015840 (11)

By combining from (2) to (11) the formula for calculat-ing the limit support pressure was obtained Analogous toTerzaghirsquos superposition method which has been commonlyused in bearing capacity analysis the obtained limit supportpressure could be rearranged to be

120590119905= 119888119873119888+ 119902119873119902+ 120574119863119873

120574 (12)

6 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

80

100Li

mit

supp

ort p

ress

ure (

kPa)

c (kPa)

120593 = 15

120593 = 30

120593 = 45

(a) The influence of cohesion

0 20 40 600

100

200

300

400

Lim

it su

ppor

t pre

ssur

e (kP

a)CD = 05CD = 1

CD = 2

120593 (∘)

(b) The influence of friction angle (119888 = 2 kPa)

Figure 6 The influence of the soil strength parameters on the limit support pressure (120574 = 17 kPa119863 = 10m)

where the influence coefficients of cohesion 119873119888 surcharge

load119873119902 and soil gravity119873

120574are

119873119888= minus

1

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

sdot ⟨(2ℎ minus 119897119861cot120593)

+119863 exp [(1205874 + 1205932) tan120593] cos120593

2 sin (1205874 + 1205932)

+ 1199030cot120593exp [(120587

2+ 120593) tan120593] minus 1⟩

119873119902=

119897119861

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

119873120574=

1

1198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot ⟨1199030ℎ minus 119897119861(ℎ minus 119862)

+ ((3 tan120601

sdot exp [3(1205874+120601

2) tan120601]

sdot cos(1205874+120601

2) minus 1

+ exp [3(1205874+120601

2) tan120601] sin(120587

4+120601

2))

sdot (1 + 9tan2120601)minus1

) 1199032

0

+1198632

2tan(120587

4minus120601

2)

sdot cos(1205874+120601

2) exp [(120587

4+120601

2) tan120601]⟩

(13)

As shown in Figure 6 the limit support pressuredecreases with the friction angle 120593 it increases with thedepth-to-diameter ratio of tunnel only when 120593 lt 20

∘ whileif 120593 gt 20

∘ the variation of depth-to-diameter ratio showsalmost no influence on the limit support pressure and theseresults agree well with those of Vermeer et al [16]

33 Parametric Studies on the Influence Coefficients Thecoefficients of cohesion surcharge load and soil gravity in(14) play important role in the calculation of limit supportpressure and deserve further study Different from the caseof bearing capacity analysis the coefficient of cohesion 119873

119888

here is negative As shown in Figure 7 119873119888increases with

the friction angle 120593 of soil and decreases with the depth-to-diameter ratio of shield tunnel Vermeer et al [16] statedthat119873

119888could be calculated by cot(120593) when the tunnel depth

119862 is more than twice of diameter 119863 or when friction angle120593 gt 35

∘ The upper bound solutions are close to the solutionsof Vermeer et al [16] obtained from FEM when 119862119863 = 2

and 120593 gt 10∘ When friction angle is less than a certain

value (120593 = 10∘) the 119873

119888-120593 curve changes which indicates

the intersection of the failure area on the ground surface

Mathematical Problems in Engineering 7

0 10 20 30 40 500

Vermeer et al (2002)

minus10

minus8

minus6

minus4

minus2

Nc

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

120593 (∘)

Figure 7 The relationship between119873119888and the friction angle of soil

(119863 = 10m)

Comparatively the formula119873119888= cot(120593) could not reflect this

property and is only suitable under the condition of smallsoil friction angles As shown in Figure 8 119873

119902is plotted as a

function of the friction angle for common values of the depthratio 119862119863 equal to 05 1 and 2 It is shown that the upperbound solutions decrease with the friction angle of soil andthe depth-to-diameter ratio of tunnel The value of 119873

119902turns

to be zerowhen friction angle reaches a certain value and thisvalue decreases with the depth-to-diameter ratio of tunnelThe relationship between the upper bound solution of119873

120574and

friction angle 120593 of soil is shown in Figure 9 The value of 119873120574

decreases with 120593 and it increases with the depth-to-diameterratio of tunnel only when friction angle is small otherwise itkeeps constant The upper bound solutions obtained in thispaper agree well with the formula 119873

120574= 1(9 tan120593) minus 005

proposed by Vermeer et al [16] when the friction angle of soilis large The figure also shows that the results in this paperare very close to the results of Mollon et al [13] and it isapplicable for wider range of the friction angle of soil

4 Influence of Seepage on the Stability ofTunnel Face

41 Seepage Analysis When shield tunnel locates underwater table line the soil excavation often induces seepageand causes the failure of tunnel face It is necessary toestablish a calculation model to estimate the influence ofseepage on the stability of tunnel face The key point of thestability analysis under seepage condition is the calculation of

0 10 20 30 40 500

02

04

06

08

1

120593 (∘)

LA (CD = 2)LA (CD = 1)LA (CD = 05)

FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nq

Figure 8The relationship between119873119902and the friction angle of soil

(119863 = 10m)

the seepage force By assuming the underground water seep-age to follow the Darcy law the seepage equation in steadystate is

120597

120597119909(119870119909

120597119867

120597119909) +

120597

120597119910(119870119910

120597119867

120597119910) = 0 (14)

where 119870119909 119870119910are the seepage coefficients in 119909 and 119910 direc-

tions in order to simplify the problem these two coefficientsare considered as the same 119867 is the water table from thebottom of the tunnel

FE simulation was employed to analyze the seepagecharacteristics of the tunnel face In the simulation two caseswith tunnel diameter of 5m and 10m were studied the watertable varies from the top of the tunnel to three times oftunnel diameters When tunnel diameter 119863 = 10m tunneldepth-to-diameter 119862119863 = 2 water table-to-tunnel diameter119867119863 = 2 and the seepage coefficient is 03 times 10minus5ms theporewater pressure distribution near the tunnel face obtainedfrom numerical simulation is shown in Figure 10 The porewater pressure on tunnel face is zero and the interval ofpressure line is 1m water level The failure area was dividedinto two parts the top part119860 is a triangle which is the same asblock 119886 in Figure 4 and the lower part 119861 is blocks 119887 and 119888 Asshown in Figure 10 the porewater pressure distributes almostuniformly in the failure area of tunnel face and it changessignificantly near the top and bottom of the tunnel face

From the pore water pressure distribution the water headdifference between the failure line and the tunnel face couldbe obtained and then the seepage force 119865 on the failure areacould be calculated the detailed derivation could be found atLee et al [28]The total seepage forcewas obtained and shown

8 Mathematical Problems in Engineering

0 10 20 30 40 500

1

2

3

4

5

Vermeer et al (2002)Mollon et al (2010)

120593 (∘)

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nr

Figure 9The relationship between119873120574and the friction angle of soil

(119863 = 10m)

A

B

Failure curve

Tunnel face

u = 0

u = 0

Figure 10 The distribution of the pore water pressure around thetunnel face

in Figure 11 the ratio of average seepage force over hydrostaticforce keeps almost constant and the value is slightly morethan that of Lee et al [28] obtained from 3D analysis

In order to study the seepage force in detail the horizontaland vertical components of the seepage forces in area119860 (area119886) and 119861 (areas 119887 and 119888) are studied separately The totalseepage force could be decomposed as

119865 = radic(119865119860119909

+ 119865119861119909)2

+ (119865119860119910

+ 119865119861119910)2

(15)

where 119865119860119909 119865119860119910 119865119861119909 and 119865

119861119910are the horizontal and vertical

components of seepage force in parts 119860 and 119861

05

04

03

02

01

0

1 15 2 25 3

HD

Ratio

D = 5mD = 10m

Lee et al (2003) (D = 5m)

Figure 11 The relationship between the average seepage force andthe water table

The ratios of each component of average seepage forceover hydrostatic force are shown in Figure 12 the biggestcomponent 119865

119861119909varies slightly with the water table 119865

119860119910and

119865119861119910

are smaller and 119865119860119909

is the smallest

42 Influence of Seepage on Limit Support Pressure By includ-ing the power rate of the seepage force in upper boundlimit analysis the influence of seepage on the limit supportpressure of tunnel face could be studied The power rate ofthe seepage force 119875

119865is

119875119865= 119875119865119909+ 119875119865119910 (16)

where 119875119865119909

is the power rate of the horizontal component ofseepage force 119875

119865119910is the power rate of the vertical component

of seepage force For the horizontal component velocity inpart119860 is zero so the power rates of the horizontal componentof seepage force component vanishes that is

119875119865119860119909

= 0 (17)

The power rate of the vertical component of seepage forcecomponent is

119875119865119860119910

=1

2119865119860119910V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (18)

In shear zone 119887 the power rates of the seepage force in soilelement are

119889119875119865119887119909

=V119886

21198651198611199091199032

0exp (3120579 tan120593) sin 120579 119889120579

119889119875119865119887119910

=V119886

21198651198611199101199032

0exp (3120579 tan120593) cos 120579 119889120579

(19)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

(a) 119862119863 = 05 (120593 = 15∘ 30∘ 45∘)

(b) 119862119863 = 1 (120593 = 15∘ 30∘ 45∘)

(c) 119862119863 = 2 (120593 = 15∘ 30∘ 45∘)

Figure 3 The displacement increment around the tunnel face at collapse state

which indicates the collapse of the tunnel face It is also shownin Figure 2 that the limit support pressure obviously decreaseswith the friction angle

The increments of displacement and plastic strain distri-butions at collapse state are shown in Figures 3 and 4 Thefailure mode changes from global to local with the tunneldepth increases and the deformation area around the tunnelface reduces with the increase of the friction angle of soil

3 Upper Bound Limit Analysis of the ShieldTunnel Face Stability

31 Failure Mechanism In order to analyze the stability ofthe tunnel face an appropriate failuremechanism needs to beproposed According to the plastic strain distribution at col-lapse state obtained from numerical modeling in Section 2and referring to the Terzaghi failure mechanism for bearing

4 Mathematical Problems in Engineering

(a) 119862119863 = 05 (120593 = 15∘ 30∘ 45∘)

(b) 119862119863 = 1 (120593 = 15∘ 30∘ 45∘)

(c) 119862119863 = 2 (120593 = 15∘ 30∘ 45∘)

Figure 4 The equivalent plastic strain distribution around the tunnel face at collapse state

capacity analysis a failure mechanism which is composed ofa shearing zone 119887 and two rigid blocks 119886 and 119888 was proposedThe proposed failure mechanism could reflect the transitionfrom global failure mode to local mode with the increases ofthe tunnel depth which has been indicated by previous study[11 16] As shown in Figure 5 the upper isosceles triangle1198741015840

119874119861 which has an opening angle equal to 2120593 is block 119886and the axis of symmetry of the opening angle is verticalTheblock 119888 is isosceles triangle 1198741198601198601015840 the line 1198601198601015840 has an angleof 1205874 + 1205932 with the horizontal direction The shear zone 119887

is a log-spiral curve with the center is point119874 The geometricparameters of the failure mechanism are

1199030=

119863

2 sin (1205874 + 1205932) exp [(1205874 + 1205932) tan (120593)]

ℎ2=

1199030

(2 tan120593)

119897119861=

0 ℎ minus 119862 le 0

2 (ℎ minus 119862) tan120593 ℎ minus 119862 gt 0

(1)

Mathematical Problems in Engineering 5

2120593

O

O

998400

lB

aar0

120579OB

OE

B

bc

c

1205874 + 1205932

A

E

C

D

q

h

1205874 minus 1205932

Figure 5 The proposed failure mechanism

The comparison of the proposed failure mechanism and thenumerical results is shown in Figure 4 The block 119886 movesdownward with a velocity V

119886 the block 119888 moves left with an

angle of 1205874 + 1205932 with horizontal line and the velocity inshear zone increases from V

119886at 119874119861 line to V

119888at 1198741198601015840 line

32 The Limit Support Pressure Based on the failure mech-anisms proposed in previous section the upper boundsolution of the limit support pressure could be obtained byequaling the power of external force and plastic dissipationenergy [31] The power of the weight of block 119886 is

119875119882119886

=1

2120574V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (2)

The differential of the power of the weight of shearingzone 119887 is

119889119875119908119887

=V119886

2120574int

1205874+1205932

0

1199032

0exp (3120579 tan120593) cos 120579 119889120579 (3)

After integration the power of the weight of shearingzone 119887 is

119875119908119887

= int

1205874+1205932

0

119889119875119908119887

=120574V1198861199032

0

2 (1 + 9tan2120593)

sdot (sin(1205874+120593

2) exp [3 (120587

4+120593

2) tan120593]

+ 3 tan120593cos(1205874+120593

2)

sdot exp [3 (1205874+120593

2) tan120593] minus 1)

(4)

The power of the weight of block 119888 is

119875119882119888

=120574

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(5)

The power of the surcharge load is

119875119902= 119902119897119861V119886 (6)

The power of the support pressure on the tunnel face is

119875119905= 120590119905119863V119886exp [(120587

4+120593

2) tan120593] sin(120587

4+120593

2) (7)

The internal energy dissipation along line 1198741198741015840 and line1198741015840

1198611015840 is

119864119886= (2ℎ minus 119897

119861cot120593) 119888V

119886 (8)

The internal energy dissipation along the line 1198601198601015840 is

119864119888=119863 exp [(1205874 + 1205932) tan120593]

2 sin (1205874 + 1205932)119888V119886 (9)

The internal energy dissipation along line 1198611198601015840 which

equals the dissipation power rate in block 119887 is

119864119887= 1198641198611198601015840 =

1

2119888V1198861199030cot120593exp [(120587

2+ 120593) tan120593] minus 1 (10)

By equaling the power of external forces to the internalenergy dissipation we get

119875119882119886

+ 119875119882119887

+ 119875119882119888

+ 119875119902minus 119875119905= 119864119886+ 119864119887+ 119864119888+ 1198641198611198601015840 (11)

By combining from (2) to (11) the formula for calculat-ing the limit support pressure was obtained Analogous toTerzaghirsquos superposition method which has been commonlyused in bearing capacity analysis the obtained limit supportpressure could be rearranged to be

120590119905= 119888119873119888+ 119902119873119902+ 120574119863119873

120574 (12)

6 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

80

100Li

mit

supp

ort p

ress

ure (

kPa)

c (kPa)

120593 = 15

120593 = 30

120593 = 45

(a) The influence of cohesion

0 20 40 600

100

200

300

400

Lim

it su

ppor

t pre

ssur

e (kP

a)CD = 05CD = 1

CD = 2

120593 (∘)

(b) The influence of friction angle (119888 = 2 kPa)

Figure 6 The influence of the soil strength parameters on the limit support pressure (120574 = 17 kPa119863 = 10m)

where the influence coefficients of cohesion 119873119888 surcharge

load119873119902 and soil gravity119873

120574are

119873119888= minus

1

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

sdot ⟨(2ℎ minus 119897119861cot120593)

+119863 exp [(1205874 + 1205932) tan120593] cos120593

2 sin (1205874 + 1205932)

+ 1199030cot120593exp [(120587

2+ 120593) tan120593] minus 1⟩

119873119902=

119897119861

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

119873120574=

1

1198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot ⟨1199030ℎ minus 119897119861(ℎ minus 119862)

+ ((3 tan120601

sdot exp [3(1205874+120601

2) tan120601]

sdot cos(1205874+120601

2) minus 1

+ exp [3(1205874+120601

2) tan120601] sin(120587

4+120601

2))

sdot (1 + 9tan2120601)minus1

) 1199032

0

+1198632

2tan(120587

4minus120601

2)

sdot cos(1205874+120601

2) exp [(120587

4+120601

2) tan120601]⟩

(13)

As shown in Figure 6 the limit support pressuredecreases with the friction angle 120593 it increases with thedepth-to-diameter ratio of tunnel only when 120593 lt 20

∘ whileif 120593 gt 20

∘ the variation of depth-to-diameter ratio showsalmost no influence on the limit support pressure and theseresults agree well with those of Vermeer et al [16]

33 Parametric Studies on the Influence Coefficients Thecoefficients of cohesion surcharge load and soil gravity in(14) play important role in the calculation of limit supportpressure and deserve further study Different from the caseof bearing capacity analysis the coefficient of cohesion 119873

119888

here is negative As shown in Figure 7 119873119888increases with

the friction angle 120593 of soil and decreases with the depth-to-diameter ratio of shield tunnel Vermeer et al [16] statedthat119873

119888could be calculated by cot(120593) when the tunnel depth

119862 is more than twice of diameter 119863 or when friction angle120593 gt 35

∘ The upper bound solutions are close to the solutionsof Vermeer et al [16] obtained from FEM when 119862119863 = 2

and 120593 gt 10∘ When friction angle is less than a certain

value (120593 = 10∘) the 119873

119888-120593 curve changes which indicates

the intersection of the failure area on the ground surface

Mathematical Problems in Engineering 7

0 10 20 30 40 500

Vermeer et al (2002)

minus10

minus8

minus6

minus4

minus2

Nc

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

120593 (∘)

Figure 7 The relationship between119873119888and the friction angle of soil

(119863 = 10m)

Comparatively the formula119873119888= cot(120593) could not reflect this

property and is only suitable under the condition of smallsoil friction angles As shown in Figure 8 119873

119902is plotted as a

function of the friction angle for common values of the depthratio 119862119863 equal to 05 1 and 2 It is shown that the upperbound solutions decrease with the friction angle of soil andthe depth-to-diameter ratio of tunnel The value of 119873

119902turns

to be zerowhen friction angle reaches a certain value and thisvalue decreases with the depth-to-diameter ratio of tunnelThe relationship between the upper bound solution of119873

120574and

friction angle 120593 of soil is shown in Figure 9 The value of 119873120574

decreases with 120593 and it increases with the depth-to-diameterratio of tunnel only when friction angle is small otherwise itkeeps constant The upper bound solutions obtained in thispaper agree well with the formula 119873

120574= 1(9 tan120593) minus 005

proposed by Vermeer et al [16] when the friction angle of soilis large The figure also shows that the results in this paperare very close to the results of Mollon et al [13] and it isapplicable for wider range of the friction angle of soil

4 Influence of Seepage on the Stability ofTunnel Face

41 Seepage Analysis When shield tunnel locates underwater table line the soil excavation often induces seepageand causes the failure of tunnel face It is necessary toestablish a calculation model to estimate the influence ofseepage on the stability of tunnel face The key point of thestability analysis under seepage condition is the calculation of

0 10 20 30 40 500

02

04

06

08

1

120593 (∘)

LA (CD = 2)LA (CD = 1)LA (CD = 05)

FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nq

Figure 8The relationship between119873119902and the friction angle of soil

(119863 = 10m)

the seepage force By assuming the underground water seep-age to follow the Darcy law the seepage equation in steadystate is

120597

120597119909(119870119909

120597119867

120597119909) +

120597

120597119910(119870119910

120597119867

120597119910) = 0 (14)

where 119870119909 119870119910are the seepage coefficients in 119909 and 119910 direc-

tions in order to simplify the problem these two coefficientsare considered as the same 119867 is the water table from thebottom of the tunnel

FE simulation was employed to analyze the seepagecharacteristics of the tunnel face In the simulation two caseswith tunnel diameter of 5m and 10m were studied the watertable varies from the top of the tunnel to three times oftunnel diameters When tunnel diameter 119863 = 10m tunneldepth-to-diameter 119862119863 = 2 water table-to-tunnel diameter119867119863 = 2 and the seepage coefficient is 03 times 10minus5ms theporewater pressure distribution near the tunnel face obtainedfrom numerical simulation is shown in Figure 10 The porewater pressure on tunnel face is zero and the interval ofpressure line is 1m water level The failure area was dividedinto two parts the top part119860 is a triangle which is the same asblock 119886 in Figure 4 and the lower part 119861 is blocks 119887 and 119888 Asshown in Figure 10 the porewater pressure distributes almostuniformly in the failure area of tunnel face and it changessignificantly near the top and bottom of the tunnel face

From the pore water pressure distribution the water headdifference between the failure line and the tunnel face couldbe obtained and then the seepage force 119865 on the failure areacould be calculated the detailed derivation could be found atLee et al [28]The total seepage forcewas obtained and shown

8 Mathematical Problems in Engineering

0 10 20 30 40 500

1

2

3

4

5

Vermeer et al (2002)Mollon et al (2010)

120593 (∘)

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nr

Figure 9The relationship between119873120574and the friction angle of soil

(119863 = 10m)

A

B

Failure curve

Tunnel face

u = 0

u = 0

Figure 10 The distribution of the pore water pressure around thetunnel face

in Figure 11 the ratio of average seepage force over hydrostaticforce keeps almost constant and the value is slightly morethan that of Lee et al [28] obtained from 3D analysis

In order to study the seepage force in detail the horizontaland vertical components of the seepage forces in area119860 (area119886) and 119861 (areas 119887 and 119888) are studied separately The totalseepage force could be decomposed as

119865 = radic(119865119860119909

+ 119865119861119909)2

+ (119865119860119910

+ 119865119861119910)2

(15)

where 119865119860119909 119865119860119910 119865119861119909 and 119865

119861119910are the horizontal and vertical

components of seepage force in parts 119860 and 119861

05

04

03

02

01

0

1 15 2 25 3

HD

Ratio

D = 5mD = 10m

Lee et al (2003) (D = 5m)

Figure 11 The relationship between the average seepage force andthe water table

The ratios of each component of average seepage forceover hydrostatic force are shown in Figure 12 the biggestcomponent 119865

119861119909varies slightly with the water table 119865

119860119910and

119865119861119910

are smaller and 119865119860119909

is the smallest

42 Influence of Seepage on Limit Support Pressure By includ-ing the power rate of the seepage force in upper boundlimit analysis the influence of seepage on the limit supportpressure of tunnel face could be studied The power rate ofthe seepage force 119875

119865is

119875119865= 119875119865119909+ 119875119865119910 (16)

where 119875119865119909

is the power rate of the horizontal component ofseepage force 119875

119865119910is the power rate of the vertical component

of seepage force For the horizontal component velocity inpart119860 is zero so the power rates of the horizontal componentof seepage force component vanishes that is

119875119865119860119909

= 0 (17)

The power rate of the vertical component of seepage forcecomponent is

119875119865119860119910

=1

2119865119860119910V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (18)

In shear zone 119887 the power rates of the seepage force in soilelement are

119889119875119865119887119909

=V119886

21198651198611199091199032

0exp (3120579 tan120593) sin 120579 119889120579

119889119875119865119887119910

=V119886

21198651198611199101199032

0exp (3120579 tan120593) cos 120579 119889120579

(19)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

(a) 119862119863 = 05 (120593 = 15∘ 30∘ 45∘)

(b) 119862119863 = 1 (120593 = 15∘ 30∘ 45∘)

(c) 119862119863 = 2 (120593 = 15∘ 30∘ 45∘)

Figure 4 The equivalent plastic strain distribution around the tunnel face at collapse state

capacity analysis a failure mechanism which is composed ofa shearing zone 119887 and two rigid blocks 119886 and 119888 was proposedThe proposed failure mechanism could reflect the transitionfrom global failure mode to local mode with the increases ofthe tunnel depth which has been indicated by previous study[11 16] As shown in Figure 5 the upper isosceles triangle1198741015840

119874119861 which has an opening angle equal to 2120593 is block 119886and the axis of symmetry of the opening angle is verticalTheblock 119888 is isosceles triangle 1198741198601198601015840 the line 1198601198601015840 has an angleof 1205874 + 1205932 with the horizontal direction The shear zone 119887

is a log-spiral curve with the center is point119874 The geometricparameters of the failure mechanism are

1199030=

119863

2 sin (1205874 + 1205932) exp [(1205874 + 1205932) tan (120593)]

ℎ2=

1199030

(2 tan120593)

119897119861=

0 ℎ minus 119862 le 0

2 (ℎ minus 119862) tan120593 ℎ minus 119862 gt 0

(1)

Mathematical Problems in Engineering 5

2120593

O

O

998400

lB

aar0

120579OB

OE

B

bc

c

1205874 + 1205932

A

E

C

D

q

h

1205874 minus 1205932

Figure 5 The proposed failure mechanism

The comparison of the proposed failure mechanism and thenumerical results is shown in Figure 4 The block 119886 movesdownward with a velocity V

119886 the block 119888 moves left with an

angle of 1205874 + 1205932 with horizontal line and the velocity inshear zone increases from V

119886at 119874119861 line to V

119888at 1198741198601015840 line

32 The Limit Support Pressure Based on the failure mech-anisms proposed in previous section the upper boundsolution of the limit support pressure could be obtained byequaling the power of external force and plastic dissipationenergy [31] The power of the weight of block 119886 is

119875119882119886

=1

2120574V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (2)

The differential of the power of the weight of shearingzone 119887 is

119889119875119908119887

=V119886

2120574int

1205874+1205932

0

1199032

0exp (3120579 tan120593) cos 120579 119889120579 (3)

After integration the power of the weight of shearingzone 119887 is

119875119908119887

= int

1205874+1205932

0

119889119875119908119887

=120574V1198861199032

0

2 (1 + 9tan2120593)

sdot (sin(1205874+120593

2) exp [3 (120587

4+120593

2) tan120593]

+ 3 tan120593cos(1205874+120593

2)

sdot exp [3 (1205874+120593

2) tan120593] minus 1)

(4)

The power of the weight of block 119888 is

119875119882119888

=120574

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(5)

The power of the surcharge load is

119875119902= 119902119897119861V119886 (6)

The power of the support pressure on the tunnel face is

119875119905= 120590119905119863V119886exp [(120587

4+120593

2) tan120593] sin(120587

4+120593

2) (7)

The internal energy dissipation along line 1198741198741015840 and line1198741015840

1198611015840 is

119864119886= (2ℎ minus 119897

119861cot120593) 119888V

119886 (8)

The internal energy dissipation along the line 1198601198601015840 is

119864119888=119863 exp [(1205874 + 1205932) tan120593]

2 sin (1205874 + 1205932)119888V119886 (9)

The internal energy dissipation along line 1198611198601015840 which

equals the dissipation power rate in block 119887 is

119864119887= 1198641198611198601015840 =

1

2119888V1198861199030cot120593exp [(120587

2+ 120593) tan120593] minus 1 (10)

By equaling the power of external forces to the internalenergy dissipation we get

119875119882119886

+ 119875119882119887

+ 119875119882119888

+ 119875119902minus 119875119905= 119864119886+ 119864119887+ 119864119888+ 1198641198611198601015840 (11)

By combining from (2) to (11) the formula for calculat-ing the limit support pressure was obtained Analogous toTerzaghirsquos superposition method which has been commonlyused in bearing capacity analysis the obtained limit supportpressure could be rearranged to be

120590119905= 119888119873119888+ 119902119873119902+ 120574119863119873

120574 (12)

6 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

80

100Li

mit

supp

ort p

ress

ure (

kPa)

c (kPa)

120593 = 15

120593 = 30

120593 = 45

(a) The influence of cohesion

0 20 40 600

100

200

300

400

Lim

it su

ppor

t pre

ssur

e (kP

a)CD = 05CD = 1

CD = 2

120593 (∘)

(b) The influence of friction angle (119888 = 2 kPa)

Figure 6 The influence of the soil strength parameters on the limit support pressure (120574 = 17 kPa119863 = 10m)

where the influence coefficients of cohesion 119873119888 surcharge

load119873119902 and soil gravity119873

120574are

119873119888= minus

1

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

sdot ⟨(2ℎ minus 119897119861cot120593)

+119863 exp [(1205874 + 1205932) tan120593] cos120593

2 sin (1205874 + 1205932)

+ 1199030cot120593exp [(120587

2+ 120593) tan120593] minus 1⟩

119873119902=

119897119861

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

119873120574=

1

1198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot ⟨1199030ℎ minus 119897119861(ℎ minus 119862)

+ ((3 tan120601

sdot exp [3(1205874+120601

2) tan120601]

sdot cos(1205874+120601

2) minus 1

+ exp [3(1205874+120601

2) tan120601] sin(120587

4+120601

2))

sdot (1 + 9tan2120601)minus1

) 1199032

0

+1198632

2tan(120587

4minus120601

2)

sdot cos(1205874+120601

2) exp [(120587

4+120601

2) tan120601]⟩

(13)

As shown in Figure 6 the limit support pressuredecreases with the friction angle 120593 it increases with thedepth-to-diameter ratio of tunnel only when 120593 lt 20

∘ whileif 120593 gt 20

∘ the variation of depth-to-diameter ratio showsalmost no influence on the limit support pressure and theseresults agree well with those of Vermeer et al [16]

33 Parametric Studies on the Influence Coefficients Thecoefficients of cohesion surcharge load and soil gravity in(14) play important role in the calculation of limit supportpressure and deserve further study Different from the caseof bearing capacity analysis the coefficient of cohesion 119873

119888

here is negative As shown in Figure 7 119873119888increases with

the friction angle 120593 of soil and decreases with the depth-to-diameter ratio of shield tunnel Vermeer et al [16] statedthat119873

119888could be calculated by cot(120593) when the tunnel depth

119862 is more than twice of diameter 119863 or when friction angle120593 gt 35

∘ The upper bound solutions are close to the solutionsof Vermeer et al [16] obtained from FEM when 119862119863 = 2

and 120593 gt 10∘ When friction angle is less than a certain

value (120593 = 10∘) the 119873

119888-120593 curve changes which indicates

the intersection of the failure area on the ground surface

Mathematical Problems in Engineering 7

0 10 20 30 40 500

Vermeer et al (2002)

minus10

minus8

minus6

minus4

minus2

Nc

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

120593 (∘)

Figure 7 The relationship between119873119888and the friction angle of soil

(119863 = 10m)

Comparatively the formula119873119888= cot(120593) could not reflect this

property and is only suitable under the condition of smallsoil friction angles As shown in Figure 8 119873

119902is plotted as a

function of the friction angle for common values of the depthratio 119862119863 equal to 05 1 and 2 It is shown that the upperbound solutions decrease with the friction angle of soil andthe depth-to-diameter ratio of tunnel The value of 119873

119902turns

to be zerowhen friction angle reaches a certain value and thisvalue decreases with the depth-to-diameter ratio of tunnelThe relationship between the upper bound solution of119873

120574and

friction angle 120593 of soil is shown in Figure 9 The value of 119873120574

decreases with 120593 and it increases with the depth-to-diameterratio of tunnel only when friction angle is small otherwise itkeeps constant The upper bound solutions obtained in thispaper agree well with the formula 119873

120574= 1(9 tan120593) minus 005

proposed by Vermeer et al [16] when the friction angle of soilis large The figure also shows that the results in this paperare very close to the results of Mollon et al [13] and it isapplicable for wider range of the friction angle of soil

4 Influence of Seepage on the Stability ofTunnel Face

41 Seepage Analysis When shield tunnel locates underwater table line the soil excavation often induces seepageand causes the failure of tunnel face It is necessary toestablish a calculation model to estimate the influence ofseepage on the stability of tunnel face The key point of thestability analysis under seepage condition is the calculation of

0 10 20 30 40 500

02

04

06

08

1

120593 (∘)

LA (CD = 2)LA (CD = 1)LA (CD = 05)

FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nq

Figure 8The relationship between119873119902and the friction angle of soil

(119863 = 10m)

the seepage force By assuming the underground water seep-age to follow the Darcy law the seepage equation in steadystate is

120597

120597119909(119870119909

120597119867

120597119909) +

120597

120597119910(119870119910

120597119867

120597119910) = 0 (14)

where 119870119909 119870119910are the seepage coefficients in 119909 and 119910 direc-

tions in order to simplify the problem these two coefficientsare considered as the same 119867 is the water table from thebottom of the tunnel

FE simulation was employed to analyze the seepagecharacteristics of the tunnel face In the simulation two caseswith tunnel diameter of 5m and 10m were studied the watertable varies from the top of the tunnel to three times oftunnel diameters When tunnel diameter 119863 = 10m tunneldepth-to-diameter 119862119863 = 2 water table-to-tunnel diameter119867119863 = 2 and the seepage coefficient is 03 times 10minus5ms theporewater pressure distribution near the tunnel face obtainedfrom numerical simulation is shown in Figure 10 The porewater pressure on tunnel face is zero and the interval ofpressure line is 1m water level The failure area was dividedinto two parts the top part119860 is a triangle which is the same asblock 119886 in Figure 4 and the lower part 119861 is blocks 119887 and 119888 Asshown in Figure 10 the porewater pressure distributes almostuniformly in the failure area of tunnel face and it changessignificantly near the top and bottom of the tunnel face

From the pore water pressure distribution the water headdifference between the failure line and the tunnel face couldbe obtained and then the seepage force 119865 on the failure areacould be calculated the detailed derivation could be found atLee et al [28]The total seepage forcewas obtained and shown

8 Mathematical Problems in Engineering

0 10 20 30 40 500

1

2

3

4

5

Vermeer et al (2002)Mollon et al (2010)

120593 (∘)

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nr

Figure 9The relationship between119873120574and the friction angle of soil

(119863 = 10m)

A

B

Failure curve

Tunnel face

u = 0

u = 0

Figure 10 The distribution of the pore water pressure around thetunnel face

in Figure 11 the ratio of average seepage force over hydrostaticforce keeps almost constant and the value is slightly morethan that of Lee et al [28] obtained from 3D analysis

In order to study the seepage force in detail the horizontaland vertical components of the seepage forces in area119860 (area119886) and 119861 (areas 119887 and 119888) are studied separately The totalseepage force could be decomposed as

119865 = radic(119865119860119909

+ 119865119861119909)2

+ (119865119860119910

+ 119865119861119910)2

(15)

where 119865119860119909 119865119860119910 119865119861119909 and 119865

119861119910are the horizontal and vertical

components of seepage force in parts 119860 and 119861

05

04

03

02

01

0

1 15 2 25 3

HD

Ratio

D = 5mD = 10m

Lee et al (2003) (D = 5m)

Figure 11 The relationship between the average seepage force andthe water table

The ratios of each component of average seepage forceover hydrostatic force are shown in Figure 12 the biggestcomponent 119865

119861119909varies slightly with the water table 119865

119860119910and

119865119861119910

are smaller and 119865119860119909

is the smallest

42 Influence of Seepage on Limit Support Pressure By includ-ing the power rate of the seepage force in upper boundlimit analysis the influence of seepage on the limit supportpressure of tunnel face could be studied The power rate ofthe seepage force 119875

119865is

119875119865= 119875119865119909+ 119875119865119910 (16)

where 119875119865119909

is the power rate of the horizontal component ofseepage force 119875

119865119910is the power rate of the vertical component

of seepage force For the horizontal component velocity inpart119860 is zero so the power rates of the horizontal componentof seepage force component vanishes that is

119875119865119860119909

= 0 (17)

The power rate of the vertical component of seepage forcecomponent is

119875119865119860119910

=1

2119865119860119910V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (18)

In shear zone 119887 the power rates of the seepage force in soilelement are

119889119875119865119887119909

=V119886

21198651198611199091199032

0exp (3120579 tan120593) sin 120579 119889120579

119889119875119865119887119910

=V119886

21198651198611199101199032

0exp (3120579 tan120593) cos 120579 119889120579

(19)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

2120593

O

O

998400

lB

aar0

120579OB

OE

B

bc

c

1205874 + 1205932

A

E

C

D

q

h

1205874 minus 1205932

Figure 5 The proposed failure mechanism

The comparison of the proposed failure mechanism and thenumerical results is shown in Figure 4 The block 119886 movesdownward with a velocity V

119886 the block 119888 moves left with an

angle of 1205874 + 1205932 with horizontal line and the velocity inshear zone increases from V

119886at 119874119861 line to V

119888at 1198741198601015840 line

32 The Limit Support Pressure Based on the failure mech-anisms proposed in previous section the upper boundsolution of the limit support pressure could be obtained byequaling the power of external force and plastic dissipationenergy [31] The power of the weight of block 119886 is

119875119882119886

=1

2120574V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (2)

The differential of the power of the weight of shearingzone 119887 is

119889119875119908119887

=V119886

2120574int

1205874+1205932

0

1199032

0exp (3120579 tan120593) cos 120579 119889120579 (3)

After integration the power of the weight of shearingzone 119887 is

119875119908119887

= int

1205874+1205932

0

119889119875119908119887

=120574V1198861199032

0

2 (1 + 9tan2120593)

sdot (sin(1205874+120593

2) exp [3 (120587

4+120593

2) tan120593]

+ 3 tan120593cos(1205874+120593

2)

sdot exp [3 (1205874+120593

2) tan120593] minus 1)

(4)

The power of the weight of block 119888 is

119875119882119888

=120574

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(5)

The power of the surcharge load is

119875119902= 119902119897119861V119886 (6)

The power of the support pressure on the tunnel face is

119875119905= 120590119905119863V119886exp [(120587

4+120593

2) tan120593] sin(120587

4+120593

2) (7)

The internal energy dissipation along line 1198741198741015840 and line1198741015840

1198611015840 is

119864119886= (2ℎ minus 119897

119861cot120593) 119888V

119886 (8)

The internal energy dissipation along the line 1198601198601015840 is

119864119888=119863 exp [(1205874 + 1205932) tan120593]

2 sin (1205874 + 1205932)119888V119886 (9)

The internal energy dissipation along line 1198611198601015840 which

equals the dissipation power rate in block 119887 is

119864119887= 1198641198611198601015840 =

1

2119888V1198861199030cot120593exp [(120587

2+ 120593) tan120593] minus 1 (10)

By equaling the power of external forces to the internalenergy dissipation we get

119875119882119886

+ 119875119882119887

+ 119875119882119888

+ 119875119902minus 119875119905= 119864119886+ 119864119887+ 119864119888+ 1198641198611198601015840 (11)

By combining from (2) to (11) the formula for calculat-ing the limit support pressure was obtained Analogous toTerzaghirsquos superposition method which has been commonlyused in bearing capacity analysis the obtained limit supportpressure could be rearranged to be

120590119905= 119888119873119888+ 119902119873119902+ 120574119863119873

120574 (12)

6 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

80

100Li

mit

supp

ort p

ress

ure (

kPa)

c (kPa)

120593 = 15

120593 = 30

120593 = 45

(a) The influence of cohesion

0 20 40 600

100

200

300

400

Lim

it su

ppor

t pre

ssur

e (kP

a)CD = 05CD = 1

CD = 2

120593 (∘)

(b) The influence of friction angle (119888 = 2 kPa)

Figure 6 The influence of the soil strength parameters on the limit support pressure (120574 = 17 kPa119863 = 10m)

where the influence coefficients of cohesion 119873119888 surcharge

load119873119902 and soil gravity119873

120574are

119873119888= minus

1

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

sdot ⟨(2ℎ minus 119897119861cot120593)

+119863 exp [(1205874 + 1205932) tan120593] cos120593

2 sin (1205874 + 1205932)

+ 1199030cot120593exp [(120587

2+ 120593) tan120593] minus 1⟩

119873119902=

119897119861

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

119873120574=

1

1198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot ⟨1199030ℎ minus 119897119861(ℎ minus 119862)

+ ((3 tan120601

sdot exp [3(1205874+120601

2) tan120601]

sdot cos(1205874+120601

2) minus 1

+ exp [3(1205874+120601

2) tan120601] sin(120587

4+120601

2))

sdot (1 + 9tan2120601)minus1

) 1199032

0

+1198632

2tan(120587

4minus120601

2)

sdot cos(1205874+120601

2) exp [(120587

4+120601

2) tan120601]⟩

(13)

As shown in Figure 6 the limit support pressuredecreases with the friction angle 120593 it increases with thedepth-to-diameter ratio of tunnel only when 120593 lt 20

∘ whileif 120593 gt 20

∘ the variation of depth-to-diameter ratio showsalmost no influence on the limit support pressure and theseresults agree well with those of Vermeer et al [16]

33 Parametric Studies on the Influence Coefficients Thecoefficients of cohesion surcharge load and soil gravity in(14) play important role in the calculation of limit supportpressure and deserve further study Different from the caseof bearing capacity analysis the coefficient of cohesion 119873

119888

here is negative As shown in Figure 7 119873119888increases with

the friction angle 120593 of soil and decreases with the depth-to-diameter ratio of shield tunnel Vermeer et al [16] statedthat119873

119888could be calculated by cot(120593) when the tunnel depth

119862 is more than twice of diameter 119863 or when friction angle120593 gt 35

∘ The upper bound solutions are close to the solutionsof Vermeer et al [16] obtained from FEM when 119862119863 = 2

and 120593 gt 10∘ When friction angle is less than a certain

value (120593 = 10∘) the 119873

119888-120593 curve changes which indicates

the intersection of the failure area on the ground surface

Mathematical Problems in Engineering 7

0 10 20 30 40 500

Vermeer et al (2002)

minus10

minus8

minus6

minus4

minus2

Nc

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

120593 (∘)

Figure 7 The relationship between119873119888and the friction angle of soil

(119863 = 10m)

Comparatively the formula119873119888= cot(120593) could not reflect this

property and is only suitable under the condition of smallsoil friction angles As shown in Figure 8 119873

119902is plotted as a

function of the friction angle for common values of the depthratio 119862119863 equal to 05 1 and 2 It is shown that the upperbound solutions decrease with the friction angle of soil andthe depth-to-diameter ratio of tunnel The value of 119873

119902turns

to be zerowhen friction angle reaches a certain value and thisvalue decreases with the depth-to-diameter ratio of tunnelThe relationship between the upper bound solution of119873

120574and

friction angle 120593 of soil is shown in Figure 9 The value of 119873120574

decreases with 120593 and it increases with the depth-to-diameterratio of tunnel only when friction angle is small otherwise itkeeps constant The upper bound solutions obtained in thispaper agree well with the formula 119873

120574= 1(9 tan120593) minus 005

proposed by Vermeer et al [16] when the friction angle of soilis large The figure also shows that the results in this paperare very close to the results of Mollon et al [13] and it isapplicable for wider range of the friction angle of soil

4 Influence of Seepage on the Stability ofTunnel Face

41 Seepage Analysis When shield tunnel locates underwater table line the soil excavation often induces seepageand causes the failure of tunnel face It is necessary toestablish a calculation model to estimate the influence ofseepage on the stability of tunnel face The key point of thestability analysis under seepage condition is the calculation of

0 10 20 30 40 500

02

04

06

08

1

120593 (∘)

LA (CD = 2)LA (CD = 1)LA (CD = 05)

FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nq

Figure 8The relationship between119873119902and the friction angle of soil

(119863 = 10m)

the seepage force By assuming the underground water seep-age to follow the Darcy law the seepage equation in steadystate is

120597

120597119909(119870119909

120597119867

120597119909) +

120597

120597119910(119870119910

120597119867

120597119910) = 0 (14)

where 119870119909 119870119910are the seepage coefficients in 119909 and 119910 direc-

tions in order to simplify the problem these two coefficientsare considered as the same 119867 is the water table from thebottom of the tunnel

FE simulation was employed to analyze the seepagecharacteristics of the tunnel face In the simulation two caseswith tunnel diameter of 5m and 10m were studied the watertable varies from the top of the tunnel to three times oftunnel diameters When tunnel diameter 119863 = 10m tunneldepth-to-diameter 119862119863 = 2 water table-to-tunnel diameter119867119863 = 2 and the seepage coefficient is 03 times 10minus5ms theporewater pressure distribution near the tunnel face obtainedfrom numerical simulation is shown in Figure 10 The porewater pressure on tunnel face is zero and the interval ofpressure line is 1m water level The failure area was dividedinto two parts the top part119860 is a triangle which is the same asblock 119886 in Figure 4 and the lower part 119861 is blocks 119887 and 119888 Asshown in Figure 10 the porewater pressure distributes almostuniformly in the failure area of tunnel face and it changessignificantly near the top and bottom of the tunnel face

From the pore water pressure distribution the water headdifference between the failure line and the tunnel face couldbe obtained and then the seepage force 119865 on the failure areacould be calculated the detailed derivation could be found atLee et al [28]The total seepage forcewas obtained and shown

8 Mathematical Problems in Engineering

0 10 20 30 40 500

1

2

3

4

5

Vermeer et al (2002)Mollon et al (2010)

120593 (∘)

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nr

Figure 9The relationship between119873120574and the friction angle of soil

(119863 = 10m)

A

B

Failure curve

Tunnel face

u = 0

u = 0

Figure 10 The distribution of the pore water pressure around thetunnel face

in Figure 11 the ratio of average seepage force over hydrostaticforce keeps almost constant and the value is slightly morethan that of Lee et al [28] obtained from 3D analysis

In order to study the seepage force in detail the horizontaland vertical components of the seepage forces in area119860 (area119886) and 119861 (areas 119887 and 119888) are studied separately The totalseepage force could be decomposed as

119865 = radic(119865119860119909

+ 119865119861119909)2

+ (119865119860119910

+ 119865119861119910)2

(15)

where 119865119860119909 119865119860119910 119865119861119909 and 119865

119861119910are the horizontal and vertical

components of seepage force in parts 119860 and 119861

05

04

03

02

01

0

1 15 2 25 3

HD

Ratio

D = 5mD = 10m

Lee et al (2003) (D = 5m)

Figure 11 The relationship between the average seepage force andthe water table

The ratios of each component of average seepage forceover hydrostatic force are shown in Figure 12 the biggestcomponent 119865

119861119909varies slightly with the water table 119865

119860119910and

119865119861119910

are smaller and 119865119860119909

is the smallest

42 Influence of Seepage on Limit Support Pressure By includ-ing the power rate of the seepage force in upper boundlimit analysis the influence of seepage on the limit supportpressure of tunnel face could be studied The power rate ofthe seepage force 119875

119865is

119875119865= 119875119865119909+ 119875119865119910 (16)

where 119875119865119909

is the power rate of the horizontal component ofseepage force 119875

119865119910is the power rate of the vertical component

of seepage force For the horizontal component velocity inpart119860 is zero so the power rates of the horizontal componentof seepage force component vanishes that is

119875119865119860119909

= 0 (17)

The power rate of the vertical component of seepage forcecomponent is

119875119865119860119910

=1

2119865119860119910V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (18)

In shear zone 119887 the power rates of the seepage force in soilelement are

119889119875119865119887119909

=V119886

21198651198611199091199032

0exp (3120579 tan120593) sin 120579 119889120579

119889119875119865119887119910

=V119886

21198651198611199101199032

0exp (3120579 tan120593) cos 120579 119889120579

(19)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

80

100Li

mit

supp

ort p

ress

ure (

kPa)

c (kPa)

120593 = 15

120593 = 30

120593 = 45

(a) The influence of cohesion

0 20 40 600

100

200

300

400

Lim

it su

ppor

t pre

ssur

e (kP

a)CD = 05CD = 1

CD = 2

120593 (∘)

(b) The influence of friction angle (119888 = 2 kPa)

Figure 6 The influence of the soil strength parameters on the limit support pressure (120574 = 17 kPa119863 = 10m)

where the influence coefficients of cohesion 119873119888 surcharge

load119873119902 and soil gravity119873

120574are

119873119888= minus

1

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

sdot ⟨(2ℎ minus 119897119861cot120593)

+119863 exp [(1205874 + 1205932) tan120593] cos120593

2 sin (1205874 + 1205932)

+ 1199030cot120593exp [(120587

2+ 120593) tan120593] minus 1⟩

119873119902=

119897119861

119863 exp [(1205874 + 1205932) tan120593] sin (1205874 + 1205932)

119873120574=

1

1198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot ⟨1199030ℎ minus 119897119861(ℎ minus 119862)

+ ((3 tan120601

sdot exp [3(1205874+120601

2) tan120601]

sdot cos(1205874+120601

2) minus 1

+ exp [3(1205874+120601

2) tan120601] sin(120587

4+120601

2))

sdot (1 + 9tan2120601)minus1

) 1199032

0

+1198632

2tan(120587

4minus120601

2)

sdot cos(1205874+120601

2) exp [(120587

4+120601

2) tan120601]⟩

(13)

As shown in Figure 6 the limit support pressuredecreases with the friction angle 120593 it increases with thedepth-to-diameter ratio of tunnel only when 120593 lt 20

∘ whileif 120593 gt 20

∘ the variation of depth-to-diameter ratio showsalmost no influence on the limit support pressure and theseresults agree well with those of Vermeer et al [16]

33 Parametric Studies on the Influence Coefficients Thecoefficients of cohesion surcharge load and soil gravity in(14) play important role in the calculation of limit supportpressure and deserve further study Different from the caseof bearing capacity analysis the coefficient of cohesion 119873

119888

here is negative As shown in Figure 7 119873119888increases with

the friction angle 120593 of soil and decreases with the depth-to-diameter ratio of shield tunnel Vermeer et al [16] statedthat119873

119888could be calculated by cot(120593) when the tunnel depth

119862 is more than twice of diameter 119863 or when friction angle120593 gt 35

∘ The upper bound solutions are close to the solutionsof Vermeer et al [16] obtained from FEM when 119862119863 = 2

and 120593 gt 10∘ When friction angle is less than a certain

value (120593 = 10∘) the 119873

119888-120593 curve changes which indicates

the intersection of the failure area on the ground surface

Mathematical Problems in Engineering 7

0 10 20 30 40 500

Vermeer et al (2002)

minus10

minus8

minus6

minus4

minus2

Nc

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

120593 (∘)

Figure 7 The relationship between119873119888and the friction angle of soil

(119863 = 10m)

Comparatively the formula119873119888= cot(120593) could not reflect this

property and is only suitable under the condition of smallsoil friction angles As shown in Figure 8 119873

119902is plotted as a

function of the friction angle for common values of the depthratio 119862119863 equal to 05 1 and 2 It is shown that the upperbound solutions decrease with the friction angle of soil andthe depth-to-diameter ratio of tunnel The value of 119873

119902turns

to be zerowhen friction angle reaches a certain value and thisvalue decreases with the depth-to-diameter ratio of tunnelThe relationship between the upper bound solution of119873

120574and

friction angle 120593 of soil is shown in Figure 9 The value of 119873120574

decreases with 120593 and it increases with the depth-to-diameterratio of tunnel only when friction angle is small otherwise itkeeps constant The upper bound solutions obtained in thispaper agree well with the formula 119873

120574= 1(9 tan120593) minus 005

proposed by Vermeer et al [16] when the friction angle of soilis large The figure also shows that the results in this paperare very close to the results of Mollon et al [13] and it isapplicable for wider range of the friction angle of soil

4 Influence of Seepage on the Stability ofTunnel Face

41 Seepage Analysis When shield tunnel locates underwater table line the soil excavation often induces seepageand causes the failure of tunnel face It is necessary toestablish a calculation model to estimate the influence ofseepage on the stability of tunnel face The key point of thestability analysis under seepage condition is the calculation of

0 10 20 30 40 500

02

04

06

08

1

120593 (∘)

LA (CD = 2)LA (CD = 1)LA (CD = 05)

FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nq

Figure 8The relationship between119873119902and the friction angle of soil

(119863 = 10m)

the seepage force By assuming the underground water seep-age to follow the Darcy law the seepage equation in steadystate is

120597

120597119909(119870119909

120597119867

120597119909) +

120597

120597119910(119870119910

120597119867

120597119910) = 0 (14)

where 119870119909 119870119910are the seepage coefficients in 119909 and 119910 direc-

tions in order to simplify the problem these two coefficientsare considered as the same 119867 is the water table from thebottom of the tunnel

FE simulation was employed to analyze the seepagecharacteristics of the tunnel face In the simulation two caseswith tunnel diameter of 5m and 10m were studied the watertable varies from the top of the tunnel to three times oftunnel diameters When tunnel diameter 119863 = 10m tunneldepth-to-diameter 119862119863 = 2 water table-to-tunnel diameter119867119863 = 2 and the seepage coefficient is 03 times 10minus5ms theporewater pressure distribution near the tunnel face obtainedfrom numerical simulation is shown in Figure 10 The porewater pressure on tunnel face is zero and the interval ofpressure line is 1m water level The failure area was dividedinto two parts the top part119860 is a triangle which is the same asblock 119886 in Figure 4 and the lower part 119861 is blocks 119887 and 119888 Asshown in Figure 10 the porewater pressure distributes almostuniformly in the failure area of tunnel face and it changessignificantly near the top and bottom of the tunnel face

From the pore water pressure distribution the water headdifference between the failure line and the tunnel face couldbe obtained and then the seepage force 119865 on the failure areacould be calculated the detailed derivation could be found atLee et al [28]The total seepage forcewas obtained and shown

8 Mathematical Problems in Engineering

0 10 20 30 40 500

1

2

3

4

5

Vermeer et al (2002)Mollon et al (2010)

120593 (∘)

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nr

Figure 9The relationship between119873120574and the friction angle of soil

(119863 = 10m)

A

B

Failure curve

Tunnel face

u = 0

u = 0

Figure 10 The distribution of the pore water pressure around thetunnel face

in Figure 11 the ratio of average seepage force over hydrostaticforce keeps almost constant and the value is slightly morethan that of Lee et al [28] obtained from 3D analysis

In order to study the seepage force in detail the horizontaland vertical components of the seepage forces in area119860 (area119886) and 119861 (areas 119887 and 119888) are studied separately The totalseepage force could be decomposed as

119865 = radic(119865119860119909

+ 119865119861119909)2

+ (119865119860119910

+ 119865119861119910)2

(15)

where 119865119860119909 119865119860119910 119865119861119909 and 119865

119861119910are the horizontal and vertical

components of seepage force in parts 119860 and 119861

05

04

03

02

01

0

1 15 2 25 3

HD

Ratio

D = 5mD = 10m

Lee et al (2003) (D = 5m)

Figure 11 The relationship between the average seepage force andthe water table

The ratios of each component of average seepage forceover hydrostatic force are shown in Figure 12 the biggestcomponent 119865

119861119909varies slightly with the water table 119865

119860119910and

119865119861119910

are smaller and 119865119860119909

is the smallest

42 Influence of Seepage on Limit Support Pressure By includ-ing the power rate of the seepage force in upper boundlimit analysis the influence of seepage on the limit supportpressure of tunnel face could be studied The power rate ofthe seepage force 119875

119865is

119875119865= 119875119865119909+ 119875119865119910 (16)

where 119875119865119909

is the power rate of the horizontal component ofseepage force 119875

119865119910is the power rate of the vertical component

of seepage force For the horizontal component velocity inpart119860 is zero so the power rates of the horizontal componentof seepage force component vanishes that is

119875119865119860119909

= 0 (17)

The power rate of the vertical component of seepage forcecomponent is

119875119865119860119910

=1

2119865119860119910V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (18)

In shear zone 119887 the power rates of the seepage force in soilelement are

119889119875119865119887119909

=V119886

21198651198611199091199032

0exp (3120579 tan120593) sin 120579 119889120579

119889119875119865119887119910

=V119886

21198651198611199101199032

0exp (3120579 tan120593) cos 120579 119889120579

(19)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 10 20 30 40 500

Vermeer et al (2002)

minus10

minus8

minus6

minus4

minus2

Nc

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

120593 (∘)

Figure 7 The relationship between119873119888and the friction angle of soil

(119863 = 10m)

Comparatively the formula119873119888= cot(120593) could not reflect this

property and is only suitable under the condition of smallsoil friction angles As shown in Figure 8 119873

119902is plotted as a

function of the friction angle for common values of the depthratio 119862119863 equal to 05 1 and 2 It is shown that the upperbound solutions decrease with the friction angle of soil andthe depth-to-diameter ratio of tunnel The value of 119873

119902turns

to be zerowhen friction angle reaches a certain value and thisvalue decreases with the depth-to-diameter ratio of tunnelThe relationship between the upper bound solution of119873

120574and

friction angle 120593 of soil is shown in Figure 9 The value of 119873120574

decreases with 120593 and it increases with the depth-to-diameterratio of tunnel only when friction angle is small otherwise itkeeps constant The upper bound solutions obtained in thispaper agree well with the formula 119873

120574= 1(9 tan120593) minus 005

proposed by Vermeer et al [16] when the friction angle of soilis large The figure also shows that the results in this paperare very close to the results of Mollon et al [13] and it isapplicable for wider range of the friction angle of soil

4 Influence of Seepage on the Stability ofTunnel Face

41 Seepage Analysis When shield tunnel locates underwater table line the soil excavation often induces seepageand causes the failure of tunnel face It is necessary toestablish a calculation model to estimate the influence ofseepage on the stability of tunnel face The key point of thestability analysis under seepage condition is the calculation of

0 10 20 30 40 500

02

04

06

08

1

120593 (∘)

LA (CD = 2)LA (CD = 1)LA (CD = 05)

FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nq

Figure 8The relationship between119873119902and the friction angle of soil

(119863 = 10m)

the seepage force By assuming the underground water seep-age to follow the Darcy law the seepage equation in steadystate is

120597

120597119909(119870119909

120597119867

120597119909) +

120597

120597119910(119870119910

120597119867

120597119910) = 0 (14)

where 119870119909 119870119910are the seepage coefficients in 119909 and 119910 direc-

tions in order to simplify the problem these two coefficientsare considered as the same 119867 is the water table from thebottom of the tunnel

FE simulation was employed to analyze the seepagecharacteristics of the tunnel face In the simulation two caseswith tunnel diameter of 5m and 10m were studied the watertable varies from the top of the tunnel to three times oftunnel diameters When tunnel diameter 119863 = 10m tunneldepth-to-diameter 119862119863 = 2 water table-to-tunnel diameter119867119863 = 2 and the seepage coefficient is 03 times 10minus5ms theporewater pressure distribution near the tunnel face obtainedfrom numerical simulation is shown in Figure 10 The porewater pressure on tunnel face is zero and the interval ofpressure line is 1m water level The failure area was dividedinto two parts the top part119860 is a triangle which is the same asblock 119886 in Figure 4 and the lower part 119861 is blocks 119887 and 119888 Asshown in Figure 10 the porewater pressure distributes almostuniformly in the failure area of tunnel face and it changessignificantly near the top and bottom of the tunnel face

From the pore water pressure distribution the water headdifference between the failure line and the tunnel face couldbe obtained and then the seepage force 119865 on the failure areacould be calculated the detailed derivation could be found atLee et al [28]The total seepage forcewas obtained and shown

8 Mathematical Problems in Engineering

0 10 20 30 40 500

1

2

3

4

5

Vermeer et al (2002)Mollon et al (2010)

120593 (∘)

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nr

Figure 9The relationship between119873120574and the friction angle of soil

(119863 = 10m)

A

B

Failure curve

Tunnel face

u = 0

u = 0

Figure 10 The distribution of the pore water pressure around thetunnel face

in Figure 11 the ratio of average seepage force over hydrostaticforce keeps almost constant and the value is slightly morethan that of Lee et al [28] obtained from 3D analysis

In order to study the seepage force in detail the horizontaland vertical components of the seepage forces in area119860 (area119886) and 119861 (areas 119887 and 119888) are studied separately The totalseepage force could be decomposed as

119865 = radic(119865119860119909

+ 119865119861119909)2

+ (119865119860119910

+ 119865119861119910)2

(15)

where 119865119860119909 119865119860119910 119865119861119909 and 119865

119861119910are the horizontal and vertical

components of seepage force in parts 119860 and 119861

05

04

03

02

01

0

1 15 2 25 3

HD

Ratio

D = 5mD = 10m

Lee et al (2003) (D = 5m)

Figure 11 The relationship between the average seepage force andthe water table

The ratios of each component of average seepage forceover hydrostatic force are shown in Figure 12 the biggestcomponent 119865

119861119909varies slightly with the water table 119865

119860119910and

119865119861119910

are smaller and 119865119860119909

is the smallest

42 Influence of Seepage on Limit Support Pressure By includ-ing the power rate of the seepage force in upper boundlimit analysis the influence of seepage on the limit supportpressure of tunnel face could be studied The power rate ofthe seepage force 119875

119865is

119875119865= 119875119865119909+ 119875119865119910 (16)

where 119875119865119909

is the power rate of the horizontal component ofseepage force 119875

119865119910is the power rate of the vertical component

of seepage force For the horizontal component velocity inpart119860 is zero so the power rates of the horizontal componentof seepage force component vanishes that is

119875119865119860119909

= 0 (17)

The power rate of the vertical component of seepage forcecomponent is

119875119865119860119910

=1

2119865119860119910V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (18)

In shear zone 119887 the power rates of the seepage force in soilelement are

119889119875119865119887119909

=V119886

21198651198611199091199032

0exp (3120579 tan120593) sin 120579 119889120579

119889119875119865119887119910

=V119886

21198651198611199101199032

0exp (3120579 tan120593) cos 120579 119889120579

(19)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 10 20 30 40 500

1

2

3

4

5

Vermeer et al (2002)Mollon et al (2010)

120593 (∘)

LA (CD = 2)LA (CD = 1)

LA (CD = 05)FEM (CD = 2)FEM (CD = 1)FEM (CD = 05)

Nr

Figure 9The relationship between119873120574and the friction angle of soil

(119863 = 10m)

A

B

Failure curve

Tunnel face

u = 0

u = 0

Figure 10 The distribution of the pore water pressure around thetunnel face

in Figure 11 the ratio of average seepage force over hydrostaticforce keeps almost constant and the value is slightly morethan that of Lee et al [28] obtained from 3D analysis

In order to study the seepage force in detail the horizontaland vertical components of the seepage forces in area119860 (area119886) and 119861 (areas 119887 and 119888) are studied separately The totalseepage force could be decomposed as

119865 = radic(119865119860119909

+ 119865119861119909)2

+ (119865119860119910

+ 119865119861119910)2

(15)

where 119865119860119909 119865119860119910 119865119861119909 and 119865

119861119910are the horizontal and vertical

components of seepage force in parts 119860 and 119861

05

04

03

02

01

0

1 15 2 25 3

HD

Ratio

D = 5mD = 10m

Lee et al (2003) (D = 5m)

Figure 11 The relationship between the average seepage force andthe water table

The ratios of each component of average seepage forceover hydrostatic force are shown in Figure 12 the biggestcomponent 119865

119861119909varies slightly with the water table 119865

119860119910and

119865119861119910

are smaller and 119865119860119909

is the smallest

42 Influence of Seepage on Limit Support Pressure By includ-ing the power rate of the seepage force in upper boundlimit analysis the influence of seepage on the limit supportpressure of tunnel face could be studied The power rate ofthe seepage force 119875

119865is

119875119865= 119875119865119909+ 119875119865119910 (16)

where 119875119865119909

is the power rate of the horizontal component ofseepage force 119875

119865119910is the power rate of the vertical component

of seepage force For the horizontal component velocity inpart119860 is zero so the power rates of the horizontal componentof seepage force component vanishes that is

119875119865119860119909

= 0 (17)

The power rate of the vertical component of seepage forcecomponent is

119875119865119860119910

=1

2119865119860119910V119886[1199030ℎ minus 119897119861(ℎ minus 119862)] (18)

In shear zone 119887 the power rates of the seepage force in soilelement are

119889119875119865119887119909

=V119886

21198651198611199091199032

0exp (3120579 tan120593) sin 120579 119889120579

119889119875119865119887119910

=V119886

21198651198611199101199032

0exp (3120579 tan120593) cos 120579 119889120579

(19)

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

1

08

06

04

02

0

1 15 2 25 3

HD

FAx(D = 5m)FAy(D = 5m)FBx(D = 5m)FBy(D = 5m)

FAx(D = 10m)FAy(D = 10m)FBx(D = 10m)FBy(D = 10m)

Ratio

Figure 12The relationship between the seepage force and the watertable

After the integration of (21) and (22) we get

119875119865119887119909

=119865119861119909V1198861199032

0

2 (1 + 9tan2120593)

sdot [minus cos(1205874+120593

2) + 3 tan120593 sin(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] + 1

(20)

119875119865119887119910

=119865119861119910V1198861199032

0

2 (1 + 9tan2120593)

sdot [sin(1205874+120593

2) + 3 tan120593 cos(120587

4+120593

2)]

sdot exp [3 (1205874+120593

2) tan120593] minus 3 tan120593

(21)

The power rate of the seepage force in block 119888 is

119875119865119888119909

=119865119861119909

41198632 tan(120587

4minus120593

2) sin(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(22)

119875119865119888119910

=119865119861119910

41198632 tan(120587

4minus120593

2) cos(120587

4+120593

2) V119886

sdot exp [(1205874+120593

2) tan120593]

(23)

By summing up the power rates of the vertical and horizontalcomponents of seepage forces are

119875119865119909= 119875119865119886119909

+ 119875119865119887119909

+ 119875119865119888119909

119875119865119910= 119875119865119886119910

+ 119875119865119887119910

+ 119875119865119888119910

(24)

By noting that 120572 = 119865119860119910119865119861119909 120573 = 119865

119861119910119865119861119909

and adding (24)into the left-hand side of (12) the limit support pressure is

120590119905= 119888119873119888+ 119902119873119902+ 1205741015840

119863119873120574+ 119865119861119909119873119891 (25)

119873119891is the coefficient of seepage force and is

119873119865= 119873119865119861119909

+ 120572119873119865119860119910

+ 120573119873119865119861119910

(26)

where

119873119865119861119909

=1

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

sdot 1199032

0sdot (([3 tan120601 sin(120587

4+120601

2) minus cos(120587

4+120601

2)]

sdot exp [3(1205874+120601

2) tan120601] + 1)

sdot (1 + 9tan2120601)minus1

)

+1198632

2tan(120587

4minus120601

2) sin(120587

4+120601

2)

sdot exp [(1205874+120601

2) tan120601]

119873119865119860119910

=1199030ℎ minus 119897119861(ℎ minus 119862)

21198632 exp [(1205874 + 1206012) tan120601] sin (1205874 + 1206012)

119873119865119861119910

= 119873120574minus 119873119865119860119910

(27)

The formula of (26) was validated when the tunneldiameters are 5m and 10m and the tunnel depth is 20mThecohesion and friction angle of soil are 2 kPa and 30∘ the dryand saturated gravities of the soil are 17 kNm3 and 19 kNm3For simplicity the surcharge load is assumed to be zero Thecalculated limit support pressures in dry sand case are 119 kPa(119863 = 5m) and 272 kPa (119863 = 10m) By incorporatingthe seepage force in upper bound limit analysis the upperbound solution of total limit support pressure under seepagecondition can be calculated As shown in Figure 13 thecalculated total limit support pressure increases linearly withthe water table and agrees very well with the results of Lee etal [28]

5 Conclusions

The face stability of shield tunnel was studied by elastoplas-ticity FE simulation the equivalent plastic strain distributionand limit support pressure at collapse state were obtained

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

D = 5mD = 10m

Lee et al (2003) (D = 5m)

120

80

40

0

0 05 1 15 2 25 3

HD

Lim

it su

ppor

t pre

ssur

e (kP

a)

Figure 13The relationship between the total limit support pressureand the water table

Based on the numerical results a failure mechanism wasproposed to study the face stability of shield tunnel byupper bound limit analysis The calculating formula of thelimit support pressure was rearranged to be the summationof cohesion surcharge load and soil gravity multipliedby corresponding coefficients Parametric analysis showedthe coefficient of cohesion increases with friction angle ofthe soil and decreases with the depth-to-diameter ratio oftunnel The coefficients of surcharge load and soil gravitydecrease with the friction angle of soil Both coefficientsdecrease with the tunnel depth-to-diameter ratio only whenthe friction angle is less than an appropriate value otherwisethey are independent of the depth-to-diameter ratio andthe coefficient of surcharge load goes to zero The seepageanalysis was conducted by FE simulation the pore waterpressure distribution and seepage force on the tunnel facewere obtained By adding the power rate induced by seepageforce the proposed upper bound limit analysis was extendedto seepage condition The results showed that a large partof the limit support pressure was used to equilibrate theseepage force and the total limit support pressure variedalmost linearly with the water table

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The financial support by the National Science Foundationof China (NSFC through Grant no 50908171) and Shanghai

Municipal Science and Technology Commission (throughGrant no 13ZR1443800) is gratefully acknowledged

References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R-P Chen J Li L-G Kong and L-J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 pp 12ndash21 2013

[3] M A Meguid O Saada M A Nunes and J Mattar ldquoPhysicalmodeling of tunnels in soft ground a reviewrdquo Tunnelling andUnderground Space Technology vol 23 no 2 pp 185ndash198 2008

[4] P Chambon and J F Corte ldquoShallow tunnels in cohesionlesssoil stability of tunnel facerdquo Journal of Geotechnical Engineeringvol 120 no 7 pp 1148ndash1165 1994

[5] B B Broms and H Bennermark ldquoStability of clay at verticalopeningsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 96 no 1 pp 71ndash94 1967

[6] L Cornejo ldquoInstability at the face its repercussions for tun-nelling technologyrdquo Tunnels and Tunnelling International vol21 no 4 pp 69ndash74 1989

[7] X Lu M Huang and H Wang ldquoFace stability analysis ofplane strain tunnel in limit theoremrdquo in Recent Developmentsof Geotechnical Engineering pp 188ndash193 Japanese GeotechnicalSociety Okinawa Japan 2010

[8] S Jancsecz and W Steiner ldquoFace support for a large mix-shield in heterogeneous ground conditionsrdquo in Proceedings ofthe Tunnelling 94 pp 531ndash550 Springer New York NY USA1994

[9] E H Davis M J Gunn R J Mair and H N SeneviratneldquoThe stability of shallow tunnels and underground openingsin cohesive materialrdquo Geotechnique vol 30 no 4 pp 397ndash4161980

[10] C E Augarde A V Lyamin and S W Sloan ldquoStability ofan undrained plane strain heading revisitedrdquo Computers andGeotechnics vol 30 no 5 pp 419ndash430 2003

[11] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[12] A H Soubra ldquoThree-dimensional face stability analysis ofshallow circular tunnelrdquo in Proceedings of the International Con-ference on Geotechnical and Geological Engineering MelbourneAustralia 2000

[13] G Mollon D Dias and A-H Soubra ldquoFace stability analysisof circular tunnels driven by a pressurized shieldrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 1pp 215ndash229 2010

[14] GMollon K K Phoon D Dias and A-H Soubra ldquoValidationof a new 2D failure mechanism for the stability analysis of apressurized tunnel face in a spatially varying sandrdquo Journal ofEngineering Mechanics vol 137 no 1 pp 8ndash21 2010

[15] G Mollon D Dias and A Soubra ldquoRange of the safe retain-ing pressures of a pressurized tunnel face by a probabilisticapproachrdquo Journal of Geotechnical and Geoenvironmental Engi-neering vol 139 no 11 pp 1954ndash1967 2013

[16] P A Vermeer N Ruse and T Marcher ldquoTunnel headingstability in drained groundrdquo Felsbau vol 20 no 6 pp 8ndash182002

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[17] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[18] M J M Maynar and L E M Rodrıguez ldquoDiscrete numericalmodel for analysis of earth pressure balance tunnel excavationrdquoJournal of Geotechnical and Geoenvironmental Engineering vol131 no 10 pp 1234ndash1242 2005

[19] R P Chen L J Tang D S Ling and Y M Chen ldquoFace stabilityanalysis of shallow shield tunnels in dry sandy ground using thediscrete element methodrdquo Computers and Geotechnics vol 38no 2 pp 187ndash195 2011

[20] Z X Zhang X Y Hu and K D Scott ldquoA discrete numericalapproach for modeling face stability in slurry shield tunnellingin soft soilsrdquo Computers and Geotechnics vol 38 no 1 pp 94ndash104 2011

[21] Y Su G F Wang and Q H Zhou ldquoTunnel face stability andground settlement in pressurized shield tunnellingrdquo Journal ofCentral South University vol 21 no 4 pp 1600ndash1606 2014

[22] E-S Hong E-S Park H-S Shin and H-M Kim ldquoEffectof a front high hydraulic conductivity zone on hydrologicalbehavior of subsea tunnelsrdquo Journal of Civil Engineering vol 14no 5 pp 699ndash707 2010

[23] X L Lu and F D Li ldquoStudy on the stability of large cross-river shield tunnel face with seepagerdquo Applied Mechanics andMaterials vol 405-408 pp 1371ndash1374 2013

[24] X L Lu F D Li and M S Huang ldquoNumerical simulationof the face stability of shield tunnel under tidal conditionrdquo inGeo-Shanghai 2014 vol GSP242 pp 742ndash750 ASCE ShanghaiChina 2014

[25] G Anagnostou and K Kovari ldquoFace stability conditions withearth-pressure-balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[26] P de Buhan A Cuvillier L Dormieux and S Maghous ldquoFacestability of shallow circular tunnels driven under thewater tablea numerical analysisrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 23 pp 79ndash95 1999

[27] I-M Lee and S-W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[28] I M Lee S W Nam and J H Ahn ldquoEffect of seepage forceson tunnel face stabilityrdquo Canadian Geotechnical Journal vol 40no 2 pp 342ndash350 2003

[29] J K Park J T Blackburn and J-H Ahn ldquoUpper boundsolutions for tunnel face stability considering seepage andstrength increase with depthrdquo in Underground Space the 4thDimension of Metropolises pp 1217ndash1222 Taylor amp FrancisGroup Boca Raton Fla USA 2007

[30] C Li L Miao and W Lv ldquoNumerical analysis of face stabilityduring shield-driven tunneling under groundwater tablerdquo inProceedings of the Geo-Frontiers Advances in GeotechnicalEngineering Dallas Tex USA 2011

[31] W F Chen Limit Analysis and Soil Plasticity Elsevier ScientificNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of