Analysis of Shield Tunnel

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2004; 28:57–91 (DOI: 10.1002/nag.327)

Analysis of shield tunnel

W. Q. Ding1, Z. Q. Yue2,n,y, L. G. Tham2, H. H. Zhu1, C. F. Lee2 and T. Hashimoto3

1Department of Geotechnical Engineering, Tongji University, Shanghai, China2Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

3Geo-Research Institute, Osaka, Japan

SUMMARY

This paper proposes a two-dimensional finite element model for the analysis of shield tunnels by takinginto account the construction process which is divided into four stages. The soil is assumed to behave as anelasto-plastic medium whereas the shield is simulated by beam–joint discontinuous model in which curvedbeam elements and joint elements are used to model the segments and joints, respectively. As grout isusually injected to fill the gap between the lining and the soil, the property parameters of the grout arechosen in such a way that they can reflect the state of the grout at each stage. Furthermore, the contactcondition between the soil and lining will change with the construction stage, and therefore, different stress-releasing coefficients are used to account for the changes. To assess the accuracy that can be attained by themethod in solving practical problems, the shield tunnelling in the No. 7 Subway Line Project in Osaka,Japan, is used as a case history for our study. The numerical results are compared with those measured inthe field. The results presented in the paper show that the proposed numerical procedure can be used toeffectively estimate the deformation, stresses and moments experienced by the surrounding soils and theconcrete lining segments. The analysis and method presented in this paper can be considered to be usefulfor other subway construction projects involving shield tunnelling in soft soils. Copyright # 2004 JohnWiley & Sons, Ltd.

KEY WORDS: subway construction; shield tunnelling; soil–structure interaction; numerical procedure;finite element method; ground settlement; lining; soft soils

1. INTRODUCTION

Since 1950s, urban developments in many cities have experienced continuous growth andexpansion. Because of the large populations and shortage of land resources, these cities havealways had a strong demand for efficient, economic and environmental friendly urban civilinfrastructure systems to accommodate the daily and routine travels of thousands and millionsof commuters. The subway system is an obvious solution to meet the demand. To minimize the

Contract/grant sponsor: Research Grant Council of Hong Kong SAR Government

Contract/grant sponsor: Hong Kong Jockey Club Charities Trust

Received 23 August 2002Revised 9 July 2003

Copyright # 2004 John Wiley & Sons, Ltd. Accepted 30 August 2003

yE-mail: [email protected]

nCorrespondence to: Dr. Q. Z. Q. Yue, Department of Civil Engineering, The University of Hong Kong, PokfulamRoad, Hong Kong, China.

impact on the existing traffic during construction, tunnelling is usually adopted for theconstruction of the subway. Because of its efficiency and safety, shield tunnelling is one of themost popular tunnelling methods for the construction of subway tunnels in soft soil ground.Over the last 30 years, the shield tunnelling method has experienced continuous improvementand development. New shield tunnelling methods including the earth pressure balanced shield,the slurry shield, the simultaneous backfill grouting as well as some improved grouting materialshave been introduced and developed in recent years [1].

Shield tunnelling will inevitably induce ground deformation and the surrounding soils willalso act on the shield lining segments. Quantitative and accurate prediction of such soil andstructure interaction will be of significant importance in many aspects including lining segmentdesign, construction safety, ground settlements, potential damage to existing structures andfacilities, and operation of the subway system.

Peck [2] developed an empirical method for the ground settlement associated with tunnellingin soft soils by assuming the settlement trough to be a Gaussian distribution curve. Theactual ground settlement is predicted based on the estimation of the ground loss ratio. Thismethod was used and improved by many engineers and researchers such as Clough andSchmidt [3] and Rowe et al. [4]. Recently, Schmidt [5] has proposed the semi-empiricalerror function method for the estimation and prediction of ground settlement. Though theseempirical or semi-empirical methods are useful in the evaluation of the ground settlementscaused by shield tunnelling construction, they must be applied with caution as they may not beapplicable to situations significantly different from the cases on which they are based onWang [6].

Another important design parameter is the load acting on the lining. Such load can betaken to be either the sum of the overburden pressure and water pressure or that determinedusing Terzaghi’s formula, Schulze–Duddeck method or other empirical methods. It is notedthat such load only considers the final state that is after the completion of the shield tunnelling.To provide a better simulation of the interaction between the lining and soil, beam–spring models, in which the lining and soil are modelled by beam and spring, respectively,can be adopted. As shield tunnel lining is composed of several concrete segments, a number ofresearchers and engineers had also taken into consideration the effects of the segment ringjoints [7, 8] in their studies. Lee and Ge [9] suggested an equivalent method to determinethe correction factor for approximating a jointed shield-driven tunnel lining as a continuousring structure under a plane strain condition. The present literature review has revealed thatthere are three widely accepted methods for modelling the effects of shield segment joints[10, 11].

Furthermore, shield tunnelling usually adopts staged construction and supporting techniques.Consequently, the responses such as soil displacements and lining forces induced by theconstruction will be different at different stages. It is believed that an optimal constructionprocess can improve the safety and reduce the disturbance to the surrounding soils. Hence, itbecomes very important to take into account the actual construction process in the shieldtunnelling design [12, 13]. Due to the complex nature of the problems, one may have to resort tonumerical approach for analysing the problems. It is well known that the finite element methodis a powerful tool for the analysis of soil–structure interaction in geomechanics and it has beenapplied to tunnel excavations by taking into account the construction process, the different soillayers, complex geometries, various loading conditions and soil–lining interfaces [14]. Thoughshield tunnelling, strictly speaking, is a three-dimensional and time-dependent soil–structure

Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2004; 28:57–91

W. Q. DING ET AL.58

interaction problem [15–18], many engineers and researchers have demonstrated that the two-dimensional finite element models can still give a fairly accurate prediction on the behaviour ofthe tunnel [12, 19, 20]. Benmebarek et al. [21] proposed two methods for taking into account thethree-dimensional effects in a two-dimensional model. Further assessment of two-dimensionalmodels in analysing shield tunnelling was reported in a recent publication by Negro and deQueiroz [22].

In this paper, we propose a two-dimensional finite element method for analysing shieldtunnels during construction. The construction of each lining segment of a shield tunnel isdivided into four stages. To model the discontinuous displacements between segments, a beam–joint discontinuous model is adopted for simulating the shield action. Moreover, the changesfrom fluid and solid states of the backfill grout at different stages are also considered.Furthermore, stress-releasing coefficients are used to account for the different contact conditionsbetween the soil and external lining surface. These coefficients are estimated based on fieldmeasured settlements and professional experiences. The numerical results obtained by thepresent model are compared with those measured during the construction of the No. 7 SubwayLine Project in Osaka, Japan. The comparison shows that the proposed model can be used toestimate the deformation, stresses and moments experienced by the surrounding soils and theconcrete lining segments during tunnelling.

2. BACKGROUND

During shield tunnelling, the shield will advance segment by segment with a balanced soilpressure that supports the soil. After cutting through a length of a segment, the lining willthen be installed. The lining is essentially an assembly of concrete segments linked by boltswhich joints will be covered with waterproof flexible plates. Before the shield tail isdetached, grout will be injected into the gap between the surrounding soil and the lining.After grouting, the shield will move forward. As the grout hardens and consolidates, thesurrounding soil will deform and induce pressures on the lining. Consequently, the ground willsettle gradually as well. All movements will cease only when a state of equilibrium state isreached.

Field observations have confirmed that the ground movements and the earth pressure on thelining segments have developed according to the construction process of the shield tunnelling[1, 21, 23–25]. Factors affecting ground movements due to shield tunnelling have beensummarized in recent publications by Nomoto et al. [25] and Hashimoto et al. [1, 26]. Theconstruction process is one of the most important factors. Closer study of the field observationshas revealed that the construction process for a segment can be divided into four stages below:

Stage 1}Balanced cutting and shield supporting. This stage includes face cutting andshield advancing. The existing soil pressure in the cutting face is balanced by pressurefrom the machine behind the cutter. Consequently, the stress changes due to the cutting andthe balancing pressure will not result in significant soil movement. Furthermore, thesurrounding soil outside the shield will not be able to release its stress due to the rigidsupport of the shield. However, one has to consider the possible disturbance on the surroundingsoil due to over-cutting, snaking and friction between the shield and the soil when the shieldadvances.


ANALYSIS OF SHIELD TUNNEL 59

Stage 2}Backfill grouting. The second stage is to install the lining segments and to backfill thegap at the shield tail with grout. The lining will form a circular ring within the shield tail coverplate. Once the shield moves forward, a gap between the soil and the lining segment will becreated. A grout must be simultaneously injected to backfill the gap space. The amount of groutis usually equal to 1.2–1.3 times of the gap volume and the grout pressure is between 0.2 and0:4 MPa: Such backfill grouting has three functions:

* preventing soil deformation immediately after the shield tail is detached,* stabilizing lining segments, and* improving tunnel water-proof performance.

The soil will start to interact with the lining although the grout is still in a fluid state. It isnoted that this backfill grouting serves a controlling measure to prevent soil deformation.Excessive grout volume and pressure may lead to soil heave around the shield tail and must beavoided.

Stage 3}Grout hardening. This stage is a transient stage. The grout will harden and consolidate,and the soil deformation will also increase with time though its rate decreases. The liningsegments and ground soil interaction will increase with time until an equilibrium state isreached. It is, therefore, necessary to assess the effects of the grout hardening and pressuredistribution on the response of the lining structure at this stage.Stage 4}Hardened grout. In the final stage, the grout becomes hardened and gains its fullstiffness and strength. The settlement will almost cease to increase.

3. NUMERICAL PROCEDURES FOR SHIELD TUNNELLING

In this paper, a two-dimensional finite element model is developed for modelling shieldtunnelling process. As such process is a very complicated one, exact simulation is almostimpossible. Therefore, the model only tries to take into account major factors, namely thecomplex properties of the soil materials, the snaking of the shield direction, the volume loss, thegrout properties and pressure, the joints of lining segment, and the construction process.Following the standard finite element procedures, one has to obtain the initial stress before theconstruction simulation can be carried out. The initial stresses at any point i is calculated by thefollowing equations:

sy0 ¼P

gjHj

sx0 ¼ K0ðsy0 � PwÞ þ Pwð1Þ

where sy0 and sx0 are the vertical and horizontal initial earth stresses, respectively, gj is the unitweight of the jth earth stratum above the point; Hj is the corresponding thickness; K0 is thelateral earth pressure coefficient, and Pw is the water pressure at the point.

As pointed out in Section 2, the construction of a tunnel segment can be divided intofour stages. As the soils behave non-linearly during the construction, an elasto-plasticmodel is adopted to model the soil. Similarly, the lining segment and the joints betweenlining segment are modelled by adopting appropriate curved members. Different modelsare adopted to simulate the grout as its behaviour is very much different at the various


W. Q. DING ET AL.60

stages. To take into account such non-linear complex problems, an incremental-iterated technique with constant stiffness within each construction stage is used in theanalyses. A flow chart illustrating the above FEM simulation for the soil–structure interactionanalysis of shield tunnelling is presented in Figure 1. Mathematically, the solution process

Input calculating data and controlling information

Calculate initial or current earth stress and stress releasing equivalent nodal forces { }reaF∆

Construction stage (stress releasing step): j=1

Yes

Form element and global stiffness matrix: [ ] [ ]∆+ ∑=

j

ini KK1ζ

ζ

Calculate newly added and stress releasing nodal forces at

current construction stage: { } { }addj

reaj FF ∆∆ ,

Solve FEM equation, gain: { }jkδ∆

Calculate incremental strain and stress:{ } { }jkjk σε ∆∆ , ,

then gain total displacement, strain and stress.

Soil and contact element nonlinear analysis

New over-excessive stress element exists?

Precision satisfied ?( )

00001.01

<∆−∆−∆

−pjkkj

pjk

ejk

σσσσ

Construction stage: j ≥ 4

No

Output results

j=j+1

Iteration step:

k ≥ N

k=k+1

Yes

Nonlinear equivalent

nodal forces:

{ }nonjkF∆

Yes

No

No

Iteration step: k=1

α

Figure 1. Flow chart of the finite element simulation for shield tunnelling.



can be expressed as

½K�ini þXjB¼1

½DKz�

!fDd jkg ¼ a jfDF grea þ fDF jgadd þ fDF jkgnon ð2Þ

where j ¼ 1–4 and k ¼ 1–N ; N is the number of non-linear iterative steps; ½K�ini is the initialstiffness matrix of soil ground and structure (if it exists) before excavation; ½DKz� is theincrement or decrement stiffness matrix of the soil ground excavated and the supportingstructure installed or removed at the construction stage z; fDF grea is the vector of the stressreleasing equivalent nodal forces due to the stress state of the previous construction processacting along the currently excavated boundary. For one construction (excavation) process, it isdue to the initial stress state; fDF jgadd is the vector of newly added nodal load at constructionstage j; fDF jkgnon is the vector of current incremental equivalent excessive nodal forces causedby non-linear stress. fDF jkgnon ¼

Pe

RVefBgTfDsag dV where fBg is strain matrix and fDsag is

the incremental non-linear stress vector; fDd jkg is the vector of incremental nodal displacementat the stage j and iterative step k; a j is the stress releasing coefficient at construction stage j:

The iteration for each stage will cease after 10 iterative steps, that is N ¼ 10; or the prescribediterative precision is achieved. In the analysis, the iterative precision is chosen to be

Dsejk � Dspjksjðk�1Þ � Dspjk

��50:00001 ð3aÞ

where Dsejk and Dspjk are the elements of the vectors of incremental elastic stress matrix fDsejkgand plastic stress matrix fDspjkg at the stage j and the iterative step k; respectively, and sjðk�1Þ isthe corresponding total stress at the stage j and the iterative step k � 1: Dsejk and Dspjk can beestimated using the following equations:

fDsejkg ¼ ½D�½B�fDd jkg ð3bÞ

fDspjkg ¼ ½Dp�½B�fDdjkg ð3cÞ

where ½D� and ½Dp� are the elastic and plastic material property matrices, respectively.After solving Equation (2), one can obtain the total displacement fdig; strain feig and stress

fsig at the construction stage i ði ¼ 1; 2; 3; 4Þ using the following equations:

fdig ¼ fd0g þXij¼1

XNk¼1

fDd jkg

feig ¼ fe0g þXij¼1

XNk¼1

fDe jkg

fsig ¼ fs0g þXij¼1

XNk¼1

fDs jkg ð4Þ

where fd0g and fe0g are the initial displacement and strain respectively, they are usually set tozero. fs0g is the initial earth stress; fDe jkg and fDs jkg are the vectors of the incremental strainand stress releasing step (construction stage) j and the iterative step k; respectively.Furthermore, the lining internal forces (hoop force, shear force and bending moment) canthen be determined from the displacements.


W. Q. DING ET AL.62

4. A CASE STUDY: SHIELD TUNNELLING FOR OSAKA NO. 7 SUBWAY LINE

Extensive monitoring was carried out during the construction of a tunnel of the Osaka SubwayLine. The results are readily available. This case history is used to assess the accuracy that can beachieved by the present approach. For completeness, a brief of the relevant information will begiven and more detailed information can be found in the report prepared by the JapaneseResearch Society of Construction Maintenance Techniques [27].

4.1. General

Figure 2 shows the general arrangement and longitudinal profile of the No. 7 Subway LineProject in Osaka, Japan. The subway is between the Morinomiya Station and the OsakaBusiness Park Station. Its east bound tunnel line is about 970:4 m long. Its west bound tunnelline is about 974:5 m long. The tunnels are about 16–30 m below the soil ground surface. Thelining is of reinforced concrete and each segment is 1:2 m wide and 280 mm thick, and has theouter diameter 5:3 m: Two 5:44 m in diameter earth pressure balanced shield machines withsynchronous grouting technique were used to construct the tunnels. In this paper, we will focuson Section A of the west bound tunnel.

4.2. Ground conditions

Figure 3 shows the soil stratum characteristics at Section A of the tunnels. There are threedifferent clayey and sandy soil strata, i.e. AC1, AC2 and AS1, above the proposed tunnel. Thetunnel is located at about 20 m below the ground. The tunnel passes through a sand layer, AS2,

Figure 2. General layout of the Osaka No. 7 Subway Line: (a) line plan; and (b) tunnel profile.



and a clayey layer, AC3. The thicknesses of these layers are about 11 and 5 m; respectively.Below the AC3 stratum, there are a sandy stratum, AS4, and a clayey stratum, AC4. Thepermanent ground water level is at about 3 m below the ground surface. The standardpenetration test results revealed that the three upper clayey soil strata (AC1–AC3) had a SPT N -values from 2.5 to 11.8 whereas the three sandy soil strata (AS1–AS3) had SPT N -values rangingfrom 34 to 43.05 (Figure 3 and Table I).

4.3. Instrumentation system for monitoring

In order to verify the shield tunnelling design and the settlement prediction, an instrumentationsystem was installed before and during the tunnel construction to monitor the ground settlement

Standard PenetrationTest (SPT) value

-40Depth (m)

Ac4

As3 -30

As2

Ac3

-20

-10As1

Ac2

Soil layer: Ac10

Tunnel

50

Figure 3. The soil layer of Section A.

Table I. The adopted ground soil parameters.

Soillayername

Thicknessof per layer

Hj ðmÞ

Averagevalue ofSPT N

ElasticmodulusE ðkPaÞ

Poisson’sratio m

Internalfriction

angle j ð8ÞCohesionC ðkPaÞ

Unitweight

gj ðkN=m3Þ

Lateralpressure

coefficient K0

AC1 2.0 2.50 8500 0.43 0.0 25.0 �16.0 0.75AC2 5.5 6.11 20774 0.43 0.0 61.1 �16.7 0.75AS1 3.0 34.00 39500 0.35 37.6 0.0 �16.6 0.55AS2 11.0 43.05 49003 0.35 40.4 0.0 �16.5 0.55AC3 5.0 11.80 40120 0.43 0.0 118.0 �17.0 0.75AS3 7.0 42.64 48572 0.35 40.3 0.0 �16.5 0.55AC4 26.0 13.80 46920 0.43 10.0 138.0 �17.4 0.75


W. Q. DING ET AL.64

and lining structural performance. Figure 4 shows the arrangement of the five settlementmonitoring points in the soil right above the west bound tunnel. Figure 5(a) shows thearrangement of eight earth pressure cells installed to measure the soil pressure acting on theouter surface of the concrete lining segments. Gauges are installed to measure moment as well ashoop forces. Their locations are shown in Figure 5(b) and 5(c), respectively.

4.4. Constitutive model for the surrounding soil

An elasto-plastic constitutive model was used in this paper for the soils surrounding the tunnel.The model employs the classical Drucker–Prager yield criterion and the Reyes elasto-plasticmatrix [28] after a careful examination of the soil properties [12]. The following empiricalequations are used to determine the Young’s modulus E ðkN=m2Þ the internal friction angle j,the lateral pressure coefficient K0 and Poisson’s ratio m for the sandy soil:

E ¼ 3800þ 1050N ð5aÞ

j ¼ 15þffiffiffiffiffiffiffiffiffi15N

pð5bÞ

K0 ¼ 1� sin j0 ð5cÞ

depth (m)

S5 -16.3S4 -15.8S3 -14.8

S2 -10.8

S1 -0.25

examed in this paper

West Boundbeing constructed andto be constructed

East Bound

S1 ~ S5 Measuring points

0

180°

90°270°

0°

o'

y

z

θ

Section A-A

Figure 4. The locations of the settlement gauges.



m ¼K0

1þ K0ð5dÞ

where j is the internal friction angle, j0 is effective internal friction angle, K0 is the lateralpressure coefficient, N is the number of standard penetration test (SPT) value.

EP2(131°)

EP3(180°)

EP8(58°)

EP1(90°)

EP4(224°)

EP5(272°)

EP6(310°)

EP7(355°)

90°

180°

270°

(a)

M01(75°)

M02(116°)

M03(151°)M04(188°)

M05(214°)

M06(261°)

M07(288°)

M08(323°)

270°

180°

90°

(b)

AF01(75˚)

AF02(116˚)

AF03(151˚)AF04(188˚)

AF05(214˚)

AF06(261˚)

AF07(288˚)

AF08(323˚)

AF09(5˚)

M09(5°)

M10(37°)

AF10(37˚)

270˚

180˚

90˚

0°

0°

(c)

0°

Figure 5. The locations of: (a) earth pressure cells; (b) moment gauges; and (c) hoop force gauges.


W. Q. DING ET AL.66

For the clayey soils, we used the following empirical relations:

E ¼ 170qu ð6aÞ

K0 ¼ OCR0:3 � 0:5 ð6bÞ

c ¼qu2

ð6cÞ

qu ¼N50

ð6dÞ

m ¼K0

1þ K0ð6eÞ

where c is the cohesion ðkN=m2Þ; qu is the unconfined compression strength (MPa), and OCR isthe over-consolidation ratio.

The parameters of the seven soil strata are given in Table I.

4.5. Modelling of the shield lining

Figure 6 shows a typical reinforced concrete segment with a circular-arc-shape. For suchsegment, one can adopt curved beam elements to simulate its action (Figure 7). As flexible plateswere used to ensure flexible joint contact between the concrete segments, the joints willsmoothen the stress concentration but they have some inherent discontinuous characteristics intransfering deformation. In this study, we have proposed a special joint element to represent theflexible contact between any two adjacent segments. A detailed description of the model is givenin Appendix A.

Based on the test results, the following lining parameters are adopted: the Young’s modulusE ¼ 3:5� 107 kN=m2; the rectangular cross-section area A ¼ 0:28 m2; the bending moment ofinertia I ¼ 1:83� 10�3 m4; the joint parameters: kW ¼ 2:5� 108 kN � m=rad=m2; kn ¼ 6:55�107 kN=m3; ks ¼ 2:5� 107 kN=m3:

4.6. Modelling of the grout

In the analysis, both the grout stiffness and grout pressure have to be modelled adequately andthe details of the models are outlined as follows:

(a) Grout stiffness. In this study, we adopted the Goodman joint model to represent such groutcontact property [28]. In this model, the element thickness is assumed to be zero and thestiffnesses are defined in terms of the normal and tangential stiffness. The normal stiffness Kn

joint

segment

Figure 6. Shield tunnel lining segment.



depends on the grout deformation modulus whereas the tangential stiffness Ks is related to thesliding feature of the contact.

If the normal contact stress sn is compressive, i.e. it has positive value, one can assume thatthe yield criterion of the contact element will follow the Mohr–Coulomb function, that is

f ¼ jtsj � ðcj þ sntgjjÞ ð7Þ

where cj and jj are, respectively, the cohesion and internal friction angle of the contact element,ts is the total shear stress on the contact element.

One can show readily that the elastic property matrix ½D� and plastic property matrix ½Dp� forthe ideal plastic and plane strain situation can be expressed as

½D� ¼Ks 0

0 Kn

" #ð8aÞ

½Dp� ¼1

S0

K2s KsS1

KsS1 S21

" #ð8bÞ

where S0 ¼ Ks þ Kntg2jj; S1 ¼ Kntgjj:The grout initially is in fluid state and it hardens gradually. Therefore, different parameters

should be used to simulate the behaviour of the grout. A brief description is given below toexplain how to adopt different grout parameters at the different construction stages.

In Stage 2, the grout is in the fluid state, and therefore, the tangential contact stiffness Ks isclose to zero. Although the liquid is incompressible, the trapped air voids can be compressed.Assuming 15% of the total grout volume is occupied by air voids, the relation between thegrouting pressure and volumetric strain can be expressed as

ev ¼ 0:15 1�Pa

Pa þ Pg

� �� ð9Þ

where Pa and Pg are the atmospheric pressure and the grouting pressure, respectively. In thenumerical calculations, we assumed Pa ¼ 0:1 MPa and Pg ¼ 0:15 MPa: The value of the normalcontact stiffness Kn can be estimated using the equation below

Kn ¼Pg

ev � t0ð10Þ

where t0 is the thickness of shield tail void and it equals to 0:07 m for the present shield tunnel.

n

s

2 1

Joint

Beam

Figure 7. Beam–joint discontinuous model.


W. Q. DING ET AL.68

In Stages 3 and 4, the grout hardens and its stiffness increases gradually. The tangentialcontact stiffness is given as:

Ks ¼Kn

2ð1þ nÞð11Þ

where n ¼ 0:3: The normal stiffness Kn can be estimated by using the following empiricalrelation [12]:

Kn ¼Et¼

150qut

ð12Þ

where qu is the unconfined compressive strength of grout, t is the thickness of contact element.In this study, qu are taken to be 0.2 and 3:0 MPa for Stages 3 and 4, respectively. The normal

and tangential stiffnesses adopted at various stages are given in Table II.(b) Distribution of grout pressure. Two models of grout pressure distribution (Figure 8) are

considered in the analysis. The first model (GP-A) assumes that the grout pressure is uniformand such model has the advantage of being simplicity, but, it may not be able to model thedecrease in pressure as the grout flows away from the grout hole. To overcome the above

Table II. The adopted parameters of the contact elements.

Shear stiffness Normal stiffness Internal friction CohesionConstruction stage Ks ðkN=m3Þ Kn ðkN=m3Þ angle j ð8Þ c ðkPaÞ

No. 1 } } } }No. 2 100 2:381� 104 0.0 0.0No. 3 1:648� 105 4:286� 105 20.0 100.0No. 4 2:473� 106 6:429� 106 50.0 3000.0

20406080

100120140160180

0 30 60 90 120 150 180 210 240 270 300 330 360θ (degree)

Gro

utin

g pr

essu

re (

kPa)

GP-A GP-BGrouting hole

Segment joint0°

270°

180°

90°95°

143°170°217°

232°242°

303°

19°θ

Figure 8. Grout pressure models.



limitation, a non-uniform pressure distribution is assumed in the second model (GP-B).Mathematically, the grout pressure at the hoop angle y is defined as

P ðyÞ ¼ P1 þ P2 cosy� b2

ð�p4y� b4pÞ ð13Þ

where b is the grout hole angle; P1 is the pressure at y ¼ b; P2 is the pressure at y ¼ pþ b:As two grout holes are used in the present case and they are symmetric about the vertical axis,

the non-uniform model of the grouting pressure distribution can be expressed as follows:

P ðyÞ ¼ 2P1 þ P2X2i¼1

cosy� bi

2ð�p4y� b1; y� b24pÞ ð14Þ

where b1 and b2 are the grout hole angles. In the present study, P1 ¼ 5:47 kPa; P2 ¼ 77:32 kPa;b1 ¼ 1438; b2 ¼ 2178:

4.7. Modelling of the stress release: stress releasing coefficient

The releasing coefficient for each construction stage is determined according to the fielddisplacements and experience.

Depending on the degree of snaking and volume loss, the stress release coefficient is taken tobe 0.0 to 0.1 in the first stage. In the second stage, the lining structure is installed and the grout isinjected at prescribed pressure to fill up the gap between the lining and the ground. The pressureapplied will not only restrain some of the ground movements but it may cause heaving, if thepressure is high. Consequently, the coefficient depends on the grout pressure. In the third stage,the stress releasing is mainly caused by the gap of the shield tail, and therefore, the coefficientwill depend on the gap. In the fourth stage, it depends on the stresses that have not yet released.The releasing coefficients for the first to the fourth construction stages are chosen to be 0.10,0.45, 0.3 and 0.15, respectively.

4.8. FEM mesh and boundary conditions

The initial finite element mesh adopted for the analysis is shown in Figure 9. In the finite elementsimulation, the soil is modelled by 512 elements containing 545 nodes. Lining segments and theirjoints are simulated with 30 beam elements and six joint elements, respectively. Thirty contactelements are used to model the interface between the ground and lining segments. The left andright boundaries of the mesh are laterally restricted. The bottom boundary is vertically restrictedwhereas the top surface is assumed to be a free boundary.

5. RESULTS AND ANALYSIS

Section A of the Osaka No. 7 subway line is analysed in this paper and the results are comparedwith the field measurement data documented in the report prepared by the Japanese ResearchSociety of Construction Maintenance Technique [27].

5.1. Soil displacements and settlements

Figure 10 shows the original and deformed meshes around the shield tunnel before grouting andafter completion of Stage 4. In Table III, the displacements at four points around the tunnel are


W. Q. DING ET AL.70

tabulated. They are the results before grouting and after the completion of the Stage 4 using theuniform grout pressure models. The settlement at the crown point A is about 7 mm: Thehorizontal displacement at the points C and D are about 2–3 mm: The heave at the bottompoint B is about 11 mm: These results have shown that the soil displacements can gainsubstantial increases with the progress of the construction stages.

The computed settlements above the tunnel after completion of Stage 4 are compared to themeasured values in Figure 11. The results show that the computed settlements agree fairly wellwith the measured ones.

5.2. Lining bending moment, hoop force and earth pressure

Figure 12 shows the measured and computed bending moments of shield lining at the Stages 2, 3and 4. In Figure 13, the measured and computed hoop forces of the shield lining segments arecompared for the Stages 2, 3 and 4. Furthermore, the earth pressure at the Stage 4 onlyis depicted in Figure 14. These results were obtained using the uniform grout pressure modelGP-A. In those figures, the corresponding predicted values using the conventional semi-analytical design method at the Stage 4 are also presented. The conventional method issummarized in Appendix B for an ease of reference. Details of this conventional method can befound in [10, 11, 29].

From Figure 12, it can be observed that the variations of the measured and the present FEMpredicted bending moments with respect to the hoop angle y and the construction stages arebasically similar although there are some significant discrepancies at the measurement locationsM10 ðy ¼ 758Þ and M08 ðy ¼ 3238Þ: At those two locations, the measured moments haveirregular tends with the construction stages 2–4, which indicates the measured moments mightnot be correct. The absolute values of the FEM predicated moments are generally greater thanthose of the measured moments. The moments predicted using the conventional method after

Figure 9. Initial finite element mesh.



the completion of Stage 4 are generally less than those measured values, which could cause anunsafe design.

From Figure 13, it can be observed that the variations of the measured and the present FEMpredicted hoop forces with respect to the hoop angle y and the construction stages are basicallysimilar although there are some significant discrepancies at the measurement locations AF01ðy ¼ 758Þ: Basically, the hoop forces increased as the construction stage progressed, whichcannot be predicted using the conventional method. Furthermore, the measured hoop forces arenot symmetric about the vertical centre (y ¼ 0 or 1808), while the predicted results are symmetricabout the vertical axis. The percentage relative differences between each set of the measured andFEM predicted hoop forces are between �113 and 35%. A majority of the relative differencesare within �20%: Furthermore, the predicted results using the conventional method at theStage 4 have the lowest variations with respect to the hoop angle y:

A

B

C D

20(mm)0Scale of total displacement from anoriginal point to its corresponding point:

(a)

A

C D

B

original point to its corresponding point:Scale of total displacement from an

20(mm)0

(b)

Figure 10. Deformed meshes of around the tunnel: (a) before grouting; and (b) after completion of Stage 4.


W. Q. DING ET AL.72

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7

Settlement (mm)

Dep

th (

m)

Measured value

GP-A

Figure 11. Variation of settlement with depth.

Table III. The displacements of the typical points around tunnel.

Point A Point B Point C Point DConstructionstage Horizontal Vertical Horizontal Vertical Horizontal Vertical Horizontal Vertical

1 0 �2.24 0 2.95 0.92 �0.12 �0.92 �0.122 0 �5.32 �0.03 6.06 �1.39 �0.38 1.45 �0.383 �0.08 �6.28 0.05 9.55 �1.86 0.93 2.34 0.694 �0.11 �6.67 0.11 11.21 �2.14 1.64 2.85 1.19

Note: The positive values are horizontal rightward and vertical upward; unit: mm.



From Figure 14, it is clear that the present FEM predicted earth pressure values are very closeto those measured results at the cell Nos. EP8 ðy ¼ 588Þ; EP1 ðy ¼ 908Þ and EP7 ðy ¼ 3558Þ andare about two times greater than those measured results at the other cell Nos. EP2 to EP6.Furthermore, the present FEM predicted result generally decreases as the hoop angle y increases

100

200

300

400

500

600

700

800

900

0 30 60 90 120 150 180 210 240 270 300 330 360

θ (degree)

Lin

ing

ring

hoo

p fo

rce

(kN

)

Stage 2 & measured Stage 2 & GP model AStage 3 & measured Stage 3 & GP model AStage 4 & measured Stage 4 & GP model AConventional design method Location of joints

Figure 13. Variation of measured and predicted hoop forces with hoop angle at thecompletion of construction stages 2, 3 or 4.

-70

-50

-30

-10

10

30

50

70

90

110

0 30 60 90 120 150 180 210 240 270 300 330 360θ (degree)

Ben

ding

mom

ent (

kN.m

)

Stage 2 & measured Stage 2 & GP model AStage 3 & measured Stage 3 & GP model AStage 4 & measured Stage 4 & GP model AConventional design method Location of joints

Figure 12. Variation of measured and predicted bending moments with hoop angle at thecompletion of construction stages 2, 3 or 4.


W. Q. DING ET AL.74

from 0 to 1808 and then increases as the hoop angle y increases from 180 to 3608; which isconsidered to be reasonable. However, the results predicted with the conventional method havefour cycles of increasing and decreasing. The four cycles correspond to the hoop angle intervals04y5908; 9084y51808; 18084y52708 and 27084y53608; which are evidently caused by thebasic assumptions associated with the semi-analytical formation of the conventional method(see Appendix B). Moreover, the predicted results using the conventional method are close tothose predicted using the present FEM method. The predicted results do not have significantdifferences.

From the above analysis and discussions, one can have the following general findings:

(a) The present method can be used to make a best prediction on the hoop forces and agood prediction on the earth pressures and a reasonable prediction on the bendingmoment. The predicted results can be used for a safe design for each of the threeconstruction stages.

(b) The in situ measurements have a best performance for the hoop forces, a goodperformance for the earth pressure and a reasonable performance for the bendingmoments. There seems a need to develop more reliable senses for in situmeasurement ofthe bending moment in the lining.

(c) The conventional method could give a reasonably good prediction on the hoop forces,an adequate estimation on the earth pressure and an under-estimation on the bendingmoment after the completion of the Stage 4. The use of this conventional method indesign estimation of the bending moment needs a great care.

(d) These results have also shown that the mechanical responses of the shield tunnelconstruction at the construction stages can be predicted by the numerical procedurepresented above. It is noted that it is very important to select correct soil parameters inthe numerical prediction.

40

70

100

130

160

190

220

250

280

310

340

0 30 60 90 120 150 180 210 240 270 300 330 360

θ (degree)

Nor

mal

ear

th p

ress

ure

(kPa

)

Measured Present methodConventional Location of joints

Figure 14. Variation of earth pressure after completion of Stage 4.



5.3. Joint displacement and rotation

Table IV gives the relative normal displacement, shearing displacements and rotational angulardisplacements between two nodes of the joint elements after the completion of Stage 4. FromTable IV, it can be observed that the relative normal displacements vary between 0.015 and0:022 mm: The relative shearing displacements vary between 0.000012 and 0:0014 mm: Therelative rotational angular displacements vary between 1:23� 10�7 and 1:23� 10�5: Such smallrelative displacements at the joints can be justified by the combined factors that the jointstiffness values (kW ¼ 2:8� 108 kN m=rad=m2; kn ¼ 6:55� 107 kN=m3; ks ¼ 2:5� 107 kN=m3)are much greater than those of the grout stiffness values between 100 and 6:429� 106 (seeTable II), these values are close to the lining concrete bending stiffness (say, Young’s modulus/thickness E=h ¼ 3:5� 107=0:28 ¼ 1:25� 108 kN=m3) in magnitude, as well as there is anarching effect along the circular lining.

Table IV. The relative displacements at joints of lining segment at the completionof construction stage No. 4.

y 198 958 1708 2328 2428 3038

GP-A Du ðmmÞ 1.91� 10�2 2.19� 10�2 1.60� 10�2 1.79� 10�2 1.91� 10�2 2.13� 10�2

Dv ðmmÞ �1.62� 10�4 �9.60� 10�4 �8.54� 10�4 �1.22� 10�5 6.15� 10�5 �1.35� 10�3

DW ð8Þ 8.83� 10�6 �1.12� 10�5 9.51� 10�6 3.05� 10�7 �3.87� 10�6 �5.08� 10�6

GP-B Du ðmmÞ 1.80� 10�2 2.12� 10�2 1.53� 10�2 1.72� 10�2 1.84� 10�2 2.03� 10�2

Dv ðmmÞ �1.46� 10�4 �9.01� 10�4 �8.55� 10�4 �1.66� 10�5 5.09� 10�5 �1.42� 10�3

DW ð8Þ 9.31� 10�6 �1.23� 10�5 1.06� 10�5 1.23� 10�7 �4.62� 10�6 �4.81� 10�6

Note: Du; Dv and DW are relative normal displacement, relative shearing displacement and relative rotational angulardisplacement between two nodes of joint element, respectively.

100

130

160

190

220

250

280

310

340

0 30 60 90 120 150 180 210 240 270 300 330 360

θ (degree)

Nor

mal

ear

th p

ress

ure

(kPa

) GP-A GP-B

Figure 15. Effect of grout pressure models: earth pressure.


W. Q. DING ET AL.76

5.4. Effects of grout pressure models

The effects of the distribution of the grout pressure were also studied by adopting both uniform(GP-A) and non-uniform (GP-B) models. The results are plotted in Figures 15–17. They haveshown the following phenomena:

(a) The GP-A model can result in an up to 24% higher earth pressure value than thoseassociated with the GP-B model since the grout pressure in GP-A model is assumed muchhigher than that in GP-B model (Figures 8 and 15). By comparing the correspondingresults in Figures 14 and 15, it is evident that the earth pressure values predicted from theconventional method is well covered by the earth pressure ranges predicted using the GP-A and GP-B over the entire hoop angles.

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 30 60 90 120 150 180 210 240 270 300 330 360

θ (degree)

Ben

ding

mom

ent (

kN.m

)

GP-A GP-B

Figure 16. Effect of grout pressure models: bending moment.

100

200

300

400

500

600

700

800

900

0 30 60 90 120 150 180 210 240 270 300 330 360

θ (degree)

Lin

ing

ring

hoo

p fo

rce

(kN

) GP-A GP-B

Figure 17. Effect of grout pressure models: hoop force.



(b) The two grout pressure models can produce very close results in the beam bendingmoments although it is interesting to note that the GP-B model has slightly higher earthpressure values around y ¼ 908; y ¼ 1808 or y ¼ 2708 (see Figure 16).

(c) The lining ring hoop forces predicted with the GP-A model is uniformly slightly (about7%) greater than those predicted with the GP-B model over the entire hoop angle ð04y53608Þ (see Figure 17).

6. CONCLUSIONS

Based on the main characteristics of the tunnel construction process, the properties of the groutand the stiffness of the lining and its joints, we have presented a two-dimensional finite elementmodel for shield tunnels. In the analysing, we pay special attentions to the contact factorsbetween the soil and the lining structure. These factors include the gap closing, the groutpressure distribution and hardening. Based on the analysis, we may have the followingconclusions:

* The results obtained by the present FEM method, in particular, the settlements, the earthpressures and the hoop forces, are very close to the values measured on site. The measuredvalues are well within the ranges of the numerical results. Furthermore, although thepresent predicted lining bending moment values are generally greater than those measuredon site, they can be used for a safe design. These findings have shown that the numericalprocedure presented in this paper is practical and reliable for evaluation, design anddisplacement prediction for shield tunnel construction.

* The hardening of the grout materials with the process of shield tunnelling can be simulatedby varying parameters such as the stiffness, the internal friction angle and cohesion atdifferent construction stages.

* The two grouting pressure models give similar results for the displacements, moments andforces. It may be concluded that the grouting pressure models do not have a significanteffect on the behaviour of the tunnelling in soil ground.

* Although the conventional method have been used for prediction of the lining bendingmoment, hoop forces and earth pressures after the completion of the construction stages,the predicted values could be lower and the method could not take into account manycomplex factors and construction stages that the present FEM method can accommodatein the design calculations.

* The above FEM analysis has further shown that numerical procedures can make adequatepredictions on the mechanical behaviour of complex soil–structure interaction system suchas shield tunnelling during and after construction. Such numerical predictions can enhancethe design and construction measures of geotechnical works in soft soils. It is noted that itis also critical to select correct values for the mechanical parameters of soils in thetheoretical predictions. Good knowledge and experience of the local soft soil conditionsand properties are always assets in selection of the parameter values for prediction.Additional experience on the selection of the soil parameters can be gained from thecomparisons between the predicted and measured mechanical responses that have beenpresented in this paper.


W. Q. DING ET AL.78

APPENDIX A: BEAM–JOINT DISCONTINUOUS MODEL

As shown in Figure 7, the beam–joint discontinuous model uses several curved beam elementsfor a segments and a joint element for a joint. Figure A1 depicts a curved beam element andFigure A2 shows a joint element in the forms of a double node.

By using the Castigliano’s second theorem [12] we can establish the relation of the nodalforces Fi ¼ fNi Qi Mig

T with the nodal displacements di ¼ fui vi WigT ði ¼ 1; 2Þ in a local co-

ordinate system ðn; sÞ: It can be shown readily that:

F1

F2

( )¼

K11 BK11

BK11 K22

" #d1

d2

( )¼ ½K�

d1

d2

( )ðA1Þ

where ½K� is the stiffness matrix and ½B� is the correlative matrix between the two nodal forces ofthe beam element.

ϕN1 u1

Q1 v11

2 P

M1 � 1

Q2 v2

M2 �2

N2 u2

x

z

o

o′

β

Figure A1. Nodal forces and displacements of curved beam model.

2

1

γ

s

n

21z

xo α

α

Figure A2. Double nodes of joint and nodal direction angles in global co-ordinates.



½K� and ½B� can be expressed as

Kii ¼ ½dik‘��13�3 ðA2Þ

½B� ¼

�cos b �sin b 0

sin b �cos b 0

Rðcos b� 1Þ R sin b 1

2664

3775 ðA3Þ

where R and b are the radius and central angle of the curved beam, respectively; dik‘ is element ofthe flexibility matrix, and

dikl ¼ dilk ðA4Þ

d111 ¼REA

ð0:5bþ 0:25 sin 2bÞ þR3

EIð1:5b� 2 sin bþ 0:25 sin 2bÞ ¼ d211 ðA5Þ

d112 ¼R

2EAsin2 bþ

R3

EIðcos bþ 0:5 sin2 b� 1Þ ¼ �d212 ðA6Þ

d113 ¼R2

EIðb� sin bÞ ¼ d213 ðA7Þ

d122 ¼REA

ð0:5b� 0:25 sin 2bÞ þR3

EIð0:5b� 0:25 sin 2bÞ ¼ d222 ðA8Þ

d123 ¼R2

EIðcos b� 1Þ ¼ �d223 ðA9Þ

d133 ¼REI

b ¼ d233 ðA10Þ

In Equations (A4)–(A10), E; A and I are the elastic modulus, cross-sectional area and momentof inertia of the curved beam, respectively.

The local stiffness ½K� can be assembled into the global stiffness ½KG� in a global co-ordinatesystem as follows:

½KG� ¼ ½T � ½K� ½T �T ðA11Þ

where ½T � is a transformation matrix between the local co-ordinate system and the globalco-ordinate system; and ½T �T is the transpose of the matrix ½T �:

The discontinuity of the two adjacent segments is described by the following relativedisplacements with three variables between the two nodes 1 and 2 in the local co-ordinatesystem ðn; sÞ

Du ¼ u1 � u2

Dv ¼ v1 � v2

DW ¼ W1 � W2

ðA12Þ

where s is defined as the direction of the bisecting angle between the two beams with a positivedirection inward to opening; n is the direction perpendicular to s with a positive direction ascounterclockwise. Accordingly, Du is the relative normal displacement of the joint along the n


W. Q. DING ET AL.80

direction, Dv is the relative shearing displacement along the s direction, and DW is the relativerotational angular displacement; ðui; vi; WiÞ; where i ¼ 1; 2; are the displacement components atthe node i along the local co-ordinate directions respectively.

Furthermore, the force–deformation relationship of the joint element can be expressed asfollows:

fF g ¼ ½K�fDdg ðA13Þ

where

fF g ¼ fN Q MgT; fDdg ¼ fDu Dv DWgT; ½K� ¼

kn 0 0

0 ks 0

0 0 kW

2664

3775

(Figure A3), and N ; Q; M are the two forces along the n and s directions and the moment of thejoint element, respectively.

Expressing ðui; vi;WiÞ as fdg ¼ fu1 v1 W1 u2 v2 W2gT; we can have the following equation:

fDdg ¼ ½C �fdg

½C � ¼

1 0 0 �1 0 0

0 1 0 0 �1 0

0 0 1 0 0 �1

2664

3775 ðA14Þ

As a result, we obtain the following equation:

½KJ� ¼ ½C�T ½K� ½C�

½KG� ¼ ½T � ½KJ� ½T �T

(ðA15Þ

where ½KJ� is the stiffness matrix of the joint element in the local co-ordinates, ½T � is a co-ordinatetransformation matrix, ½KG� is the stiffness matrix of the joint element in the global co-ordinates

k�

kn

ks

Figure A3. The stiffness kn; ks; kW in normal, shearing and rotational directions of joint.



ðx; zÞ; and

½KG� ¼

A1 A3 0 �A1 �A3 0

A2 0 �A3 �A2 0

kW 0 0 �kW

A1 A3 0

A2 0

sym: kW

2666666666664

3777777777775

ðA16Þ

where A1 ¼ kn sin2 gþ ks cos2 g; A2 ¼ kn cos2 gþ ks sin

2 g; A3 ¼ ðks � knÞ cos g sin g; g is intersec-tion angle between the opposite direction of local co-ordinate s and the direction of global co-ordinate x (Figure A2).

It is noted that the flexible plate has a large effect on the bending behaviour of the tunnellining [30]. When the bending moment exceeds a certain value at the joint, the joint will open.The rotational stiffness coefficient of the segment joint is, therefore, a non-linear function of therelative rotation angle DW as follows:

kW ¼ ðkW1 � kW2 Þ e�lDW þ kW2 ðA17Þ

where kW1 ; kW2 ; l are constants that can be determined from the bending test for the segmentjoint.

APPENDIX B: THE SEMI-ANALYTICAL METHOD FOR SHIELD TUNNELLING

For comparisons, we also present an analysis using the semi-analytical method [10, 11, 29] forshielding tunnelling. As the details are already document in the reference, we only provide ahighlight of the key points of the method.

The loads acting on the shield lining is shown in Figure B1. In the figure, P0 is overload, andusually equal to 10 kN=m2; R0 is external radius of shield lining; Rc is radius of middle line ofshield lining; g is gravity of lining; Pe1 and Pw1 are, respectively, the vertical earth pressure andwater pressure, respectively, acted on the up side of shield lining. The lateral earth pressure andwater pressure vary linearly and act on both sides of the shield lining. They are equal to qe1 andqw1 at the top of shield lining and qe2 and qw2 at the bottom of shield lining; Pe2 and Pw2 arerespectively the vertical earth pressure and water pressure respectively acted on the bottom sideof shield lining; Pg is the vertical resistance of lining weight acted on the bottom side of shieldlining.

For sandy clay, the earth pressure and water pressure are assumed to act on the liningseparately. If the overburden thickness is two times larger than the external diameter D of shieldlining, an effective overburden thickness h0 ðh052DÞ should be used and it is determined byfollowing Terzaghi’s formula (Figure B2) below:

h0 ¼B1ð1� ðc=B1gÞÞ

K0 tan fð1� e�K0 tan fðH=B1ÞÞ þ

P0gðe�K0 tan fðH=B1ÞÞ ðB1Þ


W. Q. DING ET AL.82

B1 ¼ R0 cotðp=4Þ þ ðf=2Þ

2

� �ðB2Þ

where c; f and g are cohesion, internal friction angle and unit weight of soil layer respectively;K0 is lateral pressure coefficient; H is overburden thickness. For many soil layers condition, theaverage parameter values with a weighting factor of the layer thickness will be used in Equations(B1) and (B2).

Figure B1. Loads acting on the shield.



The distribution of earth resistance is triangular in shape and its acting range is shown inFigure B1. The magnitude of the earth resistance qk is determined by following equation:

qk ¼ k � d ðB3Þ

where k is the soil resistance coefficient, d is the deformation at the horizontal quadrant point ofshield lining, and d can be calculated by following formula:

d ¼½2ðpe1 þ pw1Þ � ðqe1 þ qw1Þ � ðqe2 þ qw2Þ�R4

c

24ðEI þ 0:0454 � k � R4cÞ

ðB4Þ

Having these parameters determined, the internal forces of the shield lining can be easilycomputed. The formulae for the calculation of the beam bending moment and lining ring hoopforce are given in Table B1.

Figure B2. The overburden thickness h0 determined by Terzaghi’s formula.


W. Q. DING ET AL.84

Table B1. The internal forces caused by the different loads.

Loads Beam bending moment Lining ring hoop force

Vertical earth pressure andwater pressure: Pe1 þ Pw1

M ¼ 14ð1� 2 sin2 yÞ �

ðPe1 þ Pw1ÞR2c

N ¼ ðPe1 þ Pw1ÞRc sin2 y

Lateral earth pressure andwater pressure (rectangle part):qe1 þ qw1

M ¼ 14ð1� 2 cos2 yÞ�

ðqe1 þ qw1ÞR2c

N ¼ ðqe1 þ qw1ÞRc cos2 y

Lateral earth pressure andwater pressure (triangle part):qe2 þ qw2 � qe1 � qw1

M ¼ 148ð6þ 3 cos y�

12 cos2 y� 4 cos3 yÞ�ðqe2 þ qw2 � qel � qwlÞR2

c

N ¼ 116ð�cos yþ 8 cos2 yþ

4 cos3 yÞðqe2 þ qw2 � qel �qwlÞRc

Horizontal earth resistance: qk¼ kd

For 04y4p4;

M ¼ ð0:2346� 0:3536 cos yÞk �d � R2

c

For p44y4p

2;

M ¼ ð�0:3487þ 0:5 sin2 yþ0:2357 cos3 yÞ � k � d � R2

c

For 04y4p4;

N ¼ 0:3536 cos y � k � d � Rc

For p44y4p

2;

N ¼ ð�0:7071 cos yþ cos2 yþ0:7071 sin2 y cos yÞk � d � Rc

Lining unit weight: g For 04y4p2;

M ¼ ð�18pþ y sin yþ 5

6cos y�

12p sin2 yÞ � g � R2

c

For p24y4p;

M ¼ ð38p� ðp� yÞ sin yþ

56cos yÞg � R2

c

For 04y4p2;

N ¼ ð�p sin yþ ðp� yÞ sin yþp sin2 yþ 1

6cos yÞg � Rc

For p24y4p;

N ¼ ððp�yÞ sin yþ 16cos yÞg � R2

c

Note: The values beyond the expressed range can be easily obtained as the loads and lining are symmetric.

Table B2. Adopted values for the design parameters.

No. Parameters Design values

1 Lining segment: concrete plate2 External diameter of segment D ¼ 5:3 m3 External radius of shield lining R0 ¼ 2:65 m4 Radius of middle line of shield lining Rc ¼ 2:51 m5 Width of segment B ¼ 1:2 m (Calculating width:/m)6 Thickness of segment h ¼ 0:28 m7 Elastic modulus of segment E ¼ 3:5� 107 kN=m2

8 The bending moment of inertia I ¼ 1:83� 10�3 m4

9 Overburden thickness H ¼ 16:35 m10 Water head at the top of lining Hw ¼ 13:35 m11 Overload P0 ¼ 10 kN=m2

12 Soil resistance coefficient k ¼ 20000 kN=m3

Parameter of earth layer: mainlysandy clay, see Table I and Figure 3

13 The soil average unit weight g ¼ 16:52 kN=m3

14 The soil cohesion c ¼ 17:96 kPa15 The soil internal friction angle f ¼ 25:92816 The lateral pressure coefficient K0 ¼ 0:37817 The unit weight of water gw ¼ 10 kN=m3



Table B2 gives the values of the design parameters adopted in the present study andTable B3 tabulate the parameters computed by using these equations. These parameters arethen used to determine the beam bending moment and lining ring hoop force in the tunnellining.

APPENDIX C: NOMENCLATURE

Symbols in Main Text

sy0 vertical initial earth stresssx0 horizontal initial earth stressgj unit weight of the jth earth stratum above the pointHj thickness of the jth earth stratum above the pointK0 lateral earth pressure coefficientPw water pressure at the pointN number of non-linear iterative steps½K�ini initial stiffness matrix of soil ground and structure (if it exists) before excavation½DKz� increment or decrement stiffness matrix of the soil ground excavated and the

supporting structure installed or removed at the construction stage zDFrea vector of the stress releasing equivalent nodal forces due to the stress state of the

previous construction process acting along the currently excavated boundary

Table B3. Results of the semi-analytical design method.

No. Parameters Design values

1 Then B1 ¼ R0 cotp4þf2

2

!B1 ¼ 4:785 m

2 h0 ¼B1 1� c

B1g

� �K0 tan f 1� e

�K0 tan f�HB1

� �þ

P0g e

�K0 tan f�HB1

� �¼ 9:708 m52D

h0 ¼ 2D ¼ 10:6 m

3 Unit weight of lining g ¼ 25� h ¼ 0:7 ðkN=m2Þ4 Vertical earth pressure acted on the up side

of shield liningPe1 ¼ ðg� gwÞ � h0 ¼ 69:097 kPa

5 Water pressure acted on the up side ofshield lining

Pw1 ¼ gw � Hw ¼ 133:5 kPa

6 Lateral earth pressure and water pressureqe1

qe1 ¼ K0 � ðg� gwÞ � ðh0 þ h=2Þ ¼ 26:46 kPa

7 Lateral earth pressure and water pressureqw1

qw1 ¼ gw � ðh0 þ h=2Þ ¼ 134:9 kPa

8 Lateral earth pressure and water pressureqe2

qe2 ¼ K0 � ðg� gwÞ � ðh0 þ R0 þ RcÞ ¼ 38:82 kPa

9 Lateral earth pressure and water pressureqw2

qw2 ¼ gw � ðh0 þ R0 þ RcÞ ¼ 185:1 kPa

10 d ¼ ½2ðpe1þpw1Þ�ðqe1þqw1Þ�ðqe2þqw2Þ�R4c

24ðEIþ0:0454�k�R4c Þ

¼ 0:329 mm

11 qk ¼ k � d ¼ 6:582 kPa


W. Q. DING ET AL.86

DF jadd vector of newly added nodal load at construction stage j

DF jknon vector of current incremental equivalent excessive nodal forces caused by non-

linear stressB strain matrixDsa vector of incremental non-linear stressDdjk vector of incremental nodal displacement at the stage j and iterative step kaj stress releasing coefficient at construction stage jDsejk vector of incremental elastic stress at the stage j and iterative step kDspjk vector of incremental plastic stress at the stage j and iterative step ksjðk�1Þ total stress at the stage j and iterative step k � 1½D� elastic material property matrix½Dp� plastic material property matrixd0 vector of initial displacemente0 vector of initial strains0 vector of initial earth stressdi vector of total displacement at stage iei vector of total strain at stage isi vector of total earth stress at stage iDejk vector of incremental strain at stress releasing step j and iterative step kDsjk vector of incremental stress at stress releasing step j and iterative step kE Young’s modulusm Poisson’s ratioj internal friction anglej0 effective internal friction angleK0 lateral pressure coefficientN number of standard penetration test (SPT) valuec cohesionqu unconfined compression strengthOCR over-consolidation ratioA rectangular cross-sectional area of liningI bending moment of inertia of liningkW rotational stiffness of lining jointkn normal stiffness of lining jointks tangential stiffness of lining jointKn normal stiffness of the contact elementKs tangential stiffness of the contact elementf yield criterion value of the contact elementcj cohesion of contact elementjj internal friction angle of the contact elementsn normal stress of the contact elementts shear stress of the contact elementev volumetric strainPa atmospheric pressurePg grouting pressuret0 thickness of shield tail voidqu compressive strength



t thickness of contact elementb; b1; b2 grout hole angleP1 grouting pressure at y ¼ bP2 grouting pressure at y ¼ pþ b

Symbols in Appendix A

Fi nodal forces of nodal i; Fi ¼ Ni Qi MiT

Ni; Qi; Mi hoop force, shearing force and bending moment of nodal idi nodal displacements of nodal i; di ¼ ui vi Wi

T

ui; vi; Wi displacement components at the node i along the local co-ordinate directions½K� local stiffness matrix of the beam element and joint element[B] correlative matrix between the two nodal forces of the beam elementR radius of the curved beamb central angle of the curved beamKii block matrix of stiffness matrixdik‘ element of the flexibility matrixE elastic modulus of the curved beamA cross-sectional area of the curved beamI moment of inertia of the curved beam½KG� global stiffness matrix½T � transformation matrix between the local co-ordinate system and the global co-

ordinate system½T �T transpose of the matrix ½T �s direction of the bisecting angle between the two beams with a positive direction

inward to openingn direction perpendicular to s with a positive direction as counterclockwiseDu relative normal displacement of the joint along the n directionDv relative shearing displacement along the s directionDW relative rotational angular displacementF vector of joint nodal forces of hoop force, shearing force and bending moment,

F ¼ N Q MT

Dd vector of joint relative displacements in normal, shearing and rotational direction,Dd ¼ Du Dv DWT

½KJ� stiffness matrix of the joint element in the local co-ordinates½KG� stiffness matrix of the joint element in the global co-ordinates ðx; zÞg intersection angle between the opposite direction of local co-ordinate s and the

direction of global co-ordinate x (Figure A2)DW relative rotation anglekW1 constant that can be determined from the bending test for the segment jointkW2 constant that can be determined from the bending test for the segment jointl constant that can be determined from the bending test for the segment joint

Symbols in Appendix B

P0 overload


W. Q. DING ET AL.88

R0 external radius of shield liningRc radius of middle line of shield liningg unit weight of liningPe1; Pw1 vertical earth pressure and water pressure, respectively, acted on the up side of

shield lining, respectivelyqe1; qw1 lateral earth pressure and water pressure at the top of shield lining act on both

sides of the shield lining, respectivelyqe2; qw2 lateral earth pressure and water pressure at the bottom of shield lining act on both

sides of the shield lining, respectivelyPe2; Pw2 vertical earth pressure and water pressure, respectively, acted on the bottom side

of shield lining, respectivelyPg vertical resistance of lining weight acted on the bottom side of shield liningD external diameter of segmenth0 effective overburden thicknessc cohesion of soil layerf internal friction angle of soil layerg average unit weight of soil layerK0 lateral pressure coefficientH overburden thicknessqk earth resistancek soil resistance coefficientd deformation at the horizontal quadrant point of shield liningB width of segmenth thickness of segmentE elastic modulus of segmentI bending moment of inertia of segmentHw water head at the top of lininggw unit weight of water

ACKNOWLEDGEMENTS

The authors are indebted to Mr. Junichi Nagaya in Geo Research Institute, Osaka Soil Test Laboratory,Osaka, Japan for his kind assistance during this investigation. The work presented in this paper issupported by the Research Grants Council of Hong Kong SAR Government and the Hong Kong JockeyClub Charities Trust. The authors would also like to thank the three reviewers and Editor, Professor SteinSture, for their invaluable comments that have enhanced the paper presentation.

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Analysis of Shield Tunnel