tunnel stability and arching effects during tunnelling in soft clayey soil

Tunnel stability and arching e?ects during tunneling in soft clayey soil

C.J.Lee

a,*

,B.R.Wu b ,H.T.Chen a ,K.H.Chiang

a

a Department of Civil Engineering,National Central University,No.300,Jung-da Rd.,Chungli,Taoyuan 32054,Taiwan b

National Science and Technology Center for Disaster Reduction,3F.,No.106,Sec.2,HoPing E.Rd.,Taipei 106,Taiwan

Received 5October 2004;received in revised form 2April 2005;accepted 5June 2005

Available online 8August 2005

Abstract

A series of centrifuge model tests and numerical simulations of these tests were carried out to investigate the surface settlement troughs,excess pore water pressure generation,tunnel stability and arching e?ects that develop during tunneling in soft clayey soil.The two methods were found to provide consistent results of the surface settlement troughs,excess pore water generation,and the overload factors at collapse for both single and parallel tunneling.The arching ratio describes the evolution of the arching e?ects on the soil mass surrounding tunnels and can be derived from the numerical analysis.The boundaries of the arching zones for both single tunneling and parallel tunneling were determined.In addition,the boundaries of the positive and negative arching zones were also proposed.

ó2005Elsevier Ltd.All rights reserved.

Keywords:Arching e?ect;Tunnel stability;Centrifuge modeling;Numerical modeling

1.Introduction

Tunneling in soft clayey soils has become very pop-ular in recent years because it is one of the best con-struction methods for building mass rapid transit systems and sewage collection systems in densely pop-ulated cities.As the face of a tunnel is advanced,a means of supporting the ground close to the face may be needed;without such support,collapse might occur due to gross plastic deformation of the soil.Moreover,tunneling inevitably induces varying de-grees of ground movement towards the tunnel opening and results in detrimental e?ects on nearby facilities,such as shallow foundations,piles,existing tunnels and other pipeline systems.Taking appropriate mea-sures to protect nearby facilities before excavation is an important part of engineering practice.The predic-tion of tunneling-induced ground movements during excavation of soft ground tunnel has been carried out using various methods,including empirical meth-ods derived from ?eld observations (Peck,1969;Clough and Schmidt,1981)and centrifuge modeling (Mair et al.,1981;Wu and Lee,2003;Lee et al.,2004),or numerical and analytical methods (Lee and Rowe,1991).

Terzaghi (1943)explained how stress transfer from yielding parts of a soil mass to adjacent non-yielding parts leads to the formation of an arching zone.This problem has two modes of displacement,depending on whether the trap door is translated into the soil (passive mode)or away from it (active mode).The passive mode can be used for the evaluation of the uplift force of anchors,or of any buried structure that can be idealized as an anchor.The active mode can be used to study the gravitational ?ow of granular mate-rial between vertical walls (the silo problem)or the ground pressure on tunnel https://www.360docs.net/doc/7013361648.html,danyi and Hoy-aux (1969)performed a series of model trap-door tests

0886-7798/$-see front matter ó2005Elsevier Ltd.All rights reserved.doi:10.1016/j.tust.2005.06.003

*

Corresponding author.Tel.:+88634227151x34135;fax:+88634252960.

E-mail address:cjleeciv@https://www.360docs.net/doc/7013361648.html,.tw (C.J.Lee).

https://www.360docs.net/doc/7013361648.html,/locate/tust

Tunnelling and Underground Space Technology 21(2006)

119–132

Tunnelling and

Underground Space Technology

incorporating Trenchless Technology Research

under1g conditions in order to check the validity of the classic bin theory.Handy(1983)analyzed soil arching action behind retaining walls,and Wang and Yen(1973)carried out this analysis for slopes.Nakai et al.(1997)performed a series of physical model tests under1g conditions and carried out numerical analy-sis of these tests to investigate the arching e?ect.They found that the results obtained from the model tests were in good agreement with those obtained from the numerical analysis.Park and Adachi(2002)per-formed model tests under1g conditions to simulate tunneling events in unconsolidated ground with vari-ous levels of inclined layers.They found that remark-able non-symmetrical distributions of the earth pressure arose when a tunneling event took place in inclined layers with60°of inclination.Stone and New-son(2002)presented the results of a series of centri-fuge tests designed to investigate the e?ects of arching on soil–structure interaction.Koutsabeloulis and Gri?ths(1989)implemented a?nite element method to investigate the trap-door problem.The con-cept of soil arching was recently adopted in the anal-ysis of the mobilization of resistance from passive pile groups subjected to lateral soil movement(Chen and Martin,2002).

When tunneling is conducted in the vicinity of exist-ing pile foundations,the axial load transfer mechanism and failure mode on existing piles vary depending on the distance between the existing piles and the new driv-ing tunnel and relative elevation of the piles with respect to the centerline of the tunnel(Lee and Chiang,2004). These behaviors result from the complicated redistribu-tions of stress around tunnels during tunneling.Hence, the stress distribution in the vicinity of a tunnel or of several tunnels stacked closely in an underground sta-tion needs to be established before appropriate protec-tion measures for nearby existing piles can be implemented.By deepening our understanding of the arching e?ect in various geotechnical problems,we can improve the design of the protection measures required for existing underground structures nearby new tunneling.

Both centrifuge and numerical modeling were used in the study.The stability of a tunnel,the movements of soil mass,the evolution of stress on the soil mass around a tunnel,and the boundaries of the arching zone during tunneling in clayey soils are investigated and discussed. Firstly,a series of centrifuge model tunnel tests was con-ducted.A?nite di?erence program(FLAC)was then chosen for numerical analysis of the system described by the centrifuge model to provide insight into the arch-ing mechanism and the boundaries of the arching zone during tunneling.Finally,the results from the numerical modeling and the measurements from the centrifuge modeling were compared in order to assess their predictions.2.Centrifuge and numerical modeling

2.1.Centrifuge modeling

The basic principle of centrifuge modeling is to recre-ate the stress conditions that are present in full-scale constructions in models of greatly reduced scale.The full-scale system modeled with a centrifuge model(with dimensions N times larger than those of the model if it is tested in an acceleration that is N times earth gravity)is referred to as the prototype.It is intended that the pro-totype should include all the important characteristics of the?eld situation of interest.Centrifuge modeling pro-vides an opportunity to study for example the ground responses due to tunneling before and after collapse;col-lapse is of course not permitted to occur in the?eld.

This experimental study was undertaken in the geo-technical centrifuge at the National Central University. The NCU centrifuge has a nominal radius of3m and is capable of accelerating a1tonne model package to 100g and0.55tonne to200g.In the single-tunnel model tests,one model tunnel,60mm in diameter,was embed-ded at various depths speci?ed by the cover-to-diameter ratio(C/D).In the parallel-tunnel model tests,two mod-el tunnels(60mm in diameter)were separated by a spec-i?ed center-to-center distance(d)(as shown in Fig.1), and buried at various depths speci?ed by the cover-to-diameter ratio(C/D).All the model tests reported in this study were carried out under a centrifugal acceleration of100g in order to model a prototype tunnel with a diameter of6m embedded at depths with the tested C/D ratios.

The soil used in the model tests had a plasticity index of18and was classi?ed as CL in the Uni?ed Soil Clas-si?cation System.The soil slurry was remolded at about twice its liquid limit in a mixer and poured into the con-solidometer.Consolidation pressure was applied in?ve stages,with a?nal pressure of196kPa.Further details of the soil bed preparation can be found in Wu and

120 C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132

Lee(2003).The basic properties of the prepared soil bed are listed in Table1.On completion of the consolida-tion,the soil bed,which had an undrained shear strength pro?le of30–40kPa,was lifted and placed in a strong box.The set-up of the test package for the par-allel-tunnel model is shown in Fig.1.Five PPTs were in-serted at selected positions to monitor changes in the pore water pressure.The pore pressure transducer (PPT)was inserted into a pre-drilled hole and then the hole was fully back-?lled with thick slurry.Eight LVDTs were?xed on top of the strong box to record the transverse surface settlements.

The test package was?rst spun at an acceleration of 100g for5min so that any voids generated during the installation of the PPTs and the assembly of the test package might be?lled.After the centrifuge was stopped,one or two60mm diameter model tunnels were cut manually,depending on the test conditions,and then rubber bags were inserted into the tunnels.Tunnel deformation gauges consisting of four thin cantilevers made from stainless steel were installed inside the rubber bags to measure the deformations at the crowns,inverts, and side-walls of the tunnels.The test package shown in Fig.1was prepared for further tunnel collapse tests by connecting air pressure lines to the rubber bags.

The air pressure in the rubber bags was carefully reg-ulated to balance the overburden pressure at the tunnel center during the reacceleration of the model up to a centrifuge acceleration of100g.The tunneling event was simulated by simultaneously reducing the air pres-sure inside the tunnels and eventually down to zero.This air-pressure method of simulating tunnel excavation was adapted from Mair(1979).The air-pressure method was chosen to support the tunnel during the accelerating stages because measuring the tunnel deformation was needed and the waterproo?ng of this measuring device was di?cult if an incompressible heavy?uid-pressure method was used.The method used in the study may cause smaller surface settlements and larger settlement trough widths at the corresponding supporting pressures but no di?erence in the supporting pressure at collapse compared to the incompressible heavy?uid-pressure method.

No more than15min were spent under100g prior to collapse(including the accelerating stage).The dissipa-tion of pore water pressure(or changes of e?ective stress)during these two stages may be less than1%in average.A torvane apparatus was used for determining the undrained shear strength at various depths on the side of the soil bed before and after the test.The test re-sults show that no obvious changes in the undrained shear strengths(less than2kPa)at the same depths were found.Therefore,we assumed that there was minimal migration of pore water pressure and hence that the soil was subjected to undrained shearing.The changes in the pore water pressures,the tunnel deformations,and the surface settlements induced by tunneling were measured continuously.After each model test,the soil bed was cautiously excavated to expose the implanted spaghetti. Further,six undisturbed samples were taken from the soil bed at selected depths for use in uncon?ned com-pression tests.The undrained shear strength(s u)and the secant Young?s modulus(E50),for each soil bed was determined from the average of the results for the six samples,as listed in Table2.

The surrounding soil squeezes into the tunnel as the supporting air pressure is gradually reduced,which?-nally causes tunnel collapse.The overload factor(OF), as de?ned below,is a useful index for describing tunnel stability throughout the entire test process.

OF?

r voàp i

s u

;e1Twhere r vo is the overburden pressure at the tunnel cen-ter,and p i is the supporting pressure.Fig.2shows plots of the surface settlements at the tunnel axis versus OF for the single-tunnel test,Test7(C/D=3),and of those at the symmetrical axis of the settlement trough for the parallel-tunnel test,Twin3(C/D=3,d/D=1.5).This

Table1

Basic properties of the prepared soil bed

Speci?c gravity,G s 2.67 Liquid limit,LL40

Plastic limit,PL22 Plasticity index,PI18

Unit weight,c(kN/m3)18.1 Compression index,C c0.28 Swell index,C r0.0275 Coe?cient of consolidation,C v(cm2/s)0.010524 Permeability,k(m/s) 4.5·10à9Table2

Test con?gurations

Test no.a C/D d/D s u(kPa) Test110.5–31.00 Test120.5–35.12 Test51–36.90 Test81–37.90 Twin41 1.533.00 Twin51 1.539.10 Test32–30.25 Test92–35.79 Twin12348.70 Twin22 1.541.00 Twin92 1.539.52 Twin122235.10 Test63–33.30 Test73–34.00 Twin33 1.536.10 Twin63 1.532.90 Test104–32.17 Twin104 1.534.25 Twin114 1.532.83

a Test3–Test12:single tunnel;Twin1–Twin11:two parallel tunnels.

C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132121

?gure shows that the surface settlements,S,increase dramatically once OF exceeds a critical value.Extending the straight-line portions of the?rst and second parts of the S vs OF curves to intersect at the points shown in Fig.2,the horizontal ordinate of each of these critical points is de?ned as the overload factor at collapse, (OF)c.In engineering practice,the tunnel must be sup-ported against collapse during tunneling.Thus,the load factor(LF)is regarded as the reciprocal of the safety factor

LF?

r voàp i

r voàep iT

c

?

OF

eOFT

c

;e2T

where(p i)c is the measured supporting pressure at col-lapse in the centrifuge model tests.The value of LF var-ies from0to1,which corresponds to variation of the tunnel stability from stable to critical.

A total of19model tests were performed in the study: nine single-tunnel tests and ten parallel-tunnel tests,as listed in Table2.The C/D ratio varied from0.5to4 for the single-tunnel tests,and from1to4for the paral-lel-tunnel tests.The d/D ratios of the parallel-tunnel tests were1.5,2,and3for C/D=2,and the d/D ratio was1.5for the other C/D ratios.

2.2.Numerical modeling

The numerical‘‘experiments’’were carried out with a two-dimensional explicit?nite di?erence program,FLAC2D(Cundall et al.,1993).A plane-strain model with large-strain formulation was used to simulate the deformation behavior of the ground surrounding un-lined tunneling.In order to compare the results of the numerical and centrifuge modeling,the boundary condi-tions and soil properties used in the numerical model were chosen to be the same as those studied in the cen-trifuge model tests.The numerical analysis considered a mesh with a width of82m and a height of48m as shown in Fig.3,which are the exact dimensions of the soil bed used in the centrifuge model tests.The left and right boundaries were?xed in the x-direction,and the bottom boundary was?xed in the x-and y-directions.The grid size around the tunnels was 0.5m·0.5m in the prototype and was enlarged by a factor of1.08as the distance to tunnel center increases. The soil bed was treated as an isotropic and elastic per-fectly plastic continuum following the Mohr–Coulomb failure criterion(/=0).A total of six numerical models were analyzed.The model conditions and the mechani-cal properties of soil bed measured from the uncon?ned compression tests used in the numerical analysis are shown in Table3.

The numerical modeling was commenced at a state of geostatic equilibrium,and allowed to come to numerical equilibrium under the force of gravity.This step pro-vides an estimate of the in situ stress in the soil prior

Fig.3.Geometry used in the numerical analysis(C/D=3,d/D=1.5). Table3

Geometric conditions and mechanical properties of the numerical models

Test no.C/D d/D s u(kPa)E50(kPa)c(kN/m3)m Ntest81–37.90350018.10.49 Ntwin41 1.533.00350018.10.49 Ntest92–35.79350018.10.49 Ntwin22 1.541.00400018.10.49 Ntest73–34.00350018.10.49 Ntwin33 1.536.10350018.10.49 122 C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132

to tunneling.Secondly,the elements representing the excavated soil in the tunnel were nulled and a uniform supporting pressure equal to the overburden pressure at the center of tunnel was applied to the interior bound-ary of the circular opening to keep the tunnel stable.The supporting pressure was then reduced by a decrement of 10kPa per step in order to simulate the decrease in the supporting pressure that was applied in the centrifuge model tests.

For FLAC,the value of maximum nodal unbalance force is used to determine if a simulation having reached equilibrium.In the current study,an equilibrium state was regarded as having converged when the maximum unbalance force of every node in the mesh was less than 10N(the ratio of the maximum unbalance force to the overburden pressure on the element was about0.005%) in the simulation of per step.The simulation was then moved to the next step(reducing the supporting pres-sure by a decrement of10kPa in the study).The sup-porting pressure was reduced further until an error message indicating bad geometry of mesh appeared. The message of the bad geometry of mesh implies a dra-matic increase in the displacement within the mesh, therefore,the supporting pressure at this step is de?ned as the collapse supporting pressure,(p num)c,which is determined from the numerical modeling.By substitut-ing(p num)c into Eqs.(1)and(2),the overload factor, (OF num)c,at collapse and the load factor(LF)num, respectively,can be determined.The relationship be-tween the overload factor and the maximum surface set-tlement obtained from the centrifuge modeling and that computed with the numerical modeling for the single-tunnel and parallel-tunnel models are in good agreement before tunnel collapse,as shown in Fig.4.

Numerical modeling can be used to examine defor-mation more precisely at small strain levels and at more locations than can be achieved with centrifuge modeling, but is not as precise as the measurements of the failure and post-failure behavior that the centrifuge modeling can provide.Therefore,integrating and comparing the results derived from the numerical and centrifuge mod-els provides improved understanding of the deformation behavior and the arching mechanism during tunneling.

https://www.360docs.net/doc/7013361648.html,parison of the results from centrifuge and numerical modeling

3.1.Tunnel stability

The tunneling event was simulated by reducing the supporting pressure inside the tunnels in both the centri-fuge modeling and the numerical experiments.Fig.5 shows the relations between the C/D ratio and(OF)c measured in the single-tunnel and the parallel-tunnel model tests(solid symbols),and the values of(OF num)c calculated from the numerical modeling(hollow sym-bols).The relationships between the C/D ratio and the lower bound of the overload factor,(OF)L,for single-tunnel(Lee et al.,1999),and the upper bound of over-load factor,(OF)L for single-tunnel and parallel tunnels (Wu and Lee,2003),are also displayed in Fig.5.The test results from Mair(1979)are also included in this?g-ure.The increase in(OF)c with the C/D ratio illustrates that the stability of the tunnel improves if the tunnel is embedded more deeply.In addition,Fig.5also shows that the stability of parallel-tunnel is worse than that of single tunnel;lower overload factors at collapse were found in the parallel-tunnel model.

The overload factors at collapse(solid symbols)ob-tained from the centrifuge model for both the single-tun-nel and parallel-tunnel models have nearly the same values as those derived from the corresponding numeri-cal model(hollow symbols),as shown in Fig.5.The val-ues of(OF)c and(OF num)c for the single-tunnel model are all con?ned by upper and lower bounds.The values of(OF)c and(OF num)c for the parallel-tunnel model are

C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132123

also con?ned by the upper bound,but no lower bound solution has yet been derived.The centrifuge and numerical modeling provide consistent evaluations of the tunnel stability.

3.2.Surface settlement troughs

Figs.6and7compare the computed(represented by lines)and measured surface settlement troughs (represented by symbols)at selected load factors for the single-tunnel model(C/D=2)and for the paral-lel-tunnel model(C/D=1,d/D=1.5),respectively. The distances,X,o?set from the tunnel center-line (or from the center-line of the two tunnels in the par-allel-tunnel tests)and the surface settlements,S,are both normalized with respect to the tunnel diameter, D.The computed settlement troughs compare reason-ably well both in shape and magnitude with those measured in the centrifuge models at the correspond-ing load factors.The shape of the settlement trough for the single tunnel approximates closely to that of the error function.An empirical approach derived from centrifuge modeling has been proposed for calcu-lating the surface and subsurface settlement troughs for various ground losses due to tunneling in soft clay and in sandy soils(Wu and Lee,2003;Lee et al., 2004).In addition,these researchers proposed a super-imposition method for estimating the surface settle-ment troughs caused by parallel tunneling from the parameters obtained for single tunneling.Their meth-odology for predicting the settlement troughs based on the ground loss was veri?ed by comparison with12 sets of monitored?eld data(Wu and Lee,2003).

https://www.360docs.net/doc/7013361648.html,parison of the excess pore water pressures obtained from centrifuge and numerical modeling In an undrained system,the stress changes (D r1,D r2,D r3)due to tunneling would generate excess pore water pressure within the soil mass.With the ap-proach suggested by Henkel,the changes in the pore water pressure,D u,can be determined with the equation

D u?

1

3

eD r1tD r2tD r3Tta?eD r1àD r2T2

teD r2àD r3T2teD r3àD r1T2 1=2;e3Twhere a is Henkel?s pore water pressure parameter,and D r1,D r2,D r3are the changes in the major,intermediate, and minor principle stresses,respectively.A comparison of a and Skempton?s parameter,A,derived from triaxial compression tests gives

a?

1

???

2

p Aà

1

3

.e4T

This de?nition is useful because it enables the predic-tion of the D u associated with loading conditions un-der plane strain conditions if we assume that the soil bed is an isotropic and elastic-plastic material.By using a value of a of0.3(determined from the triaxial compression tests)and D r2=1/2(D r1+D r3)for the plane strain and undrained conditions,the developed excess pore water pressure can be estimated after the changes in total stress are calculated in the numerical simulations.

In the centrifuge modeling,the stress changes on a soil element during tunneling cannot be measured, whereas the changes of pore water pressure can be di-rectly measured.The numerical analysis in terms of total stress approach can give the changes of total stress but it cannot give the changes of pore water pressure.How-ever,the changes of pore water pressure can be esti-mated using Eq.(3)after the total stress changes were obtained from the numerical analysis.In this study, the excess pore water pressure ratio,(D u/r vo),de?ned

124 C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132

as the excess pore water pressure normalized with the to-tal overburden pressure of the measured point,is used to follow the variations of the excess pore water pressure during the tunneling simulations.The method for com-parison of the measured and calculated(D u/r vo)is de-scribed as follows.

The e?ects of tunneling on the total stress within the soil mass around a tunnel can be estimated from the computed responses at the grids of points shown in Fig.8(a)(Test7,C/D=3),Fig.8(b)(Test9,C/D=2), and Fig.8(c)(Test8,C/D=1)for the single-tunnel tests and from those shown in Figs.9(a)(Twin3;C/D=3, d/D=1.5)and9(b)(Twin2;C/D=2,d/D=1.5)for the parallel-tunnel tests.In the centrifuge model,the pore water pressures were measured at the points repre-sented by solid circle symbols in Figs.8and9(i.e.,A0, A3,B1,C1,and C3in Fig.8(a);A0,B2,C1,D1,and D2 in Fig.8(a)).

Figs.10and11compare the variations in the mea-sured(represented by lines)and computed(repre-sented by hollow circle symbols)D u/r vo with the overload factor at the corresponding locations for Test7(C/D=3)and for Twin3(C/D=3.d/D=1.5), respectively.There is very reasonable agreement be-tween the measured and computed excess pore water pressures before tunnel collapse.The small discrep-ancy between them may result from pore water pres-sure dissipation caused by partial drainage due to the appearance of micro-cracks around the small holes into which the pore water pressure transducers were inserted.Results of similar consistency were also found in the tests with di?erent C/D and d/D ratios. Thus the stress states computed from the numerical experiments can be and are used to investigate the arching behavior of the soil mass around the tunnel in the next section,although the stress measurements in the centrifuge model tests were impossible.

C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132125

4.Mechanism of arching around tunnels

4.1.Evolution of arching e?ect

The e?ects of tunneling on the total stress within the soil mass around a tunnel can be estimated from the computed responses at the grids of points shown in Fig.8(a)(Test7,C/D=3),Fig.8(b)(Test9,C/D=2), and Fig.8(c)(Test8,C/D=1)for the single-tunnel tests and from those shown in Figs.9(a)(Twin3;C/D=3, d/D=1.5)and9(b)(Twin2;C/D=2,d/D=1.5)for the parallel-tunnel tests.A more detailed understanding of the stress transfer in a tunneling problem from moving parts of the soil(settle more)to adjacent parts (settle less)can be achieved by considering the vertical stress redistributions in the soil mass above the spring line.The arching ratio is de?ned as ARe%T?

D r v

r vo

?100;e5T

where D r v is the change in the vertical stress during tun-neling and r vo is the total overburden pressure.In this study,we used the arching ratio to describe the arching behavior quantitatively at various locations.

An element that receives higher load transfers from adjoining yielding or?exible elements will generate a larger positive arching ratio.Conversely,a negative arching ratio will arise if an element shifts load to non-yielding parts or to more rigid elements.Fig.12 shows a plot of the arching ratio versus overload factor for NTest7(C/D=3)at the grid of points shown in Fig.8(a).As the overload factor increases,all of the ele-ments on the vertical center-line(Line A)experience a decrease in vertical stress(negative AR)but the elements

126 C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132

on the spring line(B1–E1)experience an increase in ver-tical stress(positive AR),as shown in Fig.12(a).Simi-larly,as can be seen in Fig.12(b)and(c),the magnitude of the arching ratio is also related to the dis-tance o?set from the tunnel center and to the overload factor(comparing the arching ratios on Lines A,B,C, D,and E).Fig.13summarizes the changes in the arching ratio for three overload factors(OF=1,3,and (OF num)c)for the elements on Lines A to E for NTest7 (C/D=3).The magnitude of the positive arching ratio increases with increases in the overload factor for the elements along Line B but declines rapidly once the ele-ment is yielding(Fig.13(b)).The element at C1experi-ences the largest positive arching ratio(about a9% rise).The elements located at a greater distance from the tunnel center experience smaller positive arching ratios.In contrast,the arching ratio becomes more neg-ative as the overload factor increases for the elements on Line A(Fig.13(a)).

Similarly,Fig.14summarizes the arching ratio as a function of the overload factor for the elements on the spring line(B1–E1)and on the vertical center-line (A2–A3)shown in Fig.8(b)for NTest9(C/D=2). Fig.15summarizes the arching ratio as a function of the overload factor for the elements on the spring line(B1–D1)and on the vertical center-line(A2) shown in Fig.8(c)for NTest8(C/D=1).The evolu-tion of the arching ratio in the shallower tunneling shown in these two?gures is similar to those pre-sented in Figs.12and13but smaller positive and neg-ative arching ratios are obtained at the corresponding points for deeper tunneling(NTest7,C/D=3).Shal-lower tunneling imposes a larger arching e?ect on the surrounding soil mass.

C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132127

Fig.16summarizes the changes in the arching ratio for the elements on Lines A–F(Fig.9(a))for three over-load factors(OF=1.5,2.5,and(OF num)c)for NTwin3(C/D=3,d/D=1.5).Fig.17shows the changes in the arching ratio for the elements on Lines A–F(Fig.9(b)) for NTwin2(C/D=2,d/D=1.5).A comparison of the changes in the arching ratios shown in Figs.13 and16indicates that the arching ratios in parallel tun-neling evolve in nearly the same way as in single tunnel-ing,except at Element A1,which is located on Line A and on the spring line.At this point,a larger positive arching ratio develops initially,but its value rapidly de-clines to near zero once the stress state becomes yielding, in the same manner as the other elements(Fig.16(a)). This part of the soil mass takes the load that is trans-ferred from the two compressive arches above the tun-nels due to parallel tunneling,and behaves like a pillar.After examining in detail the changes in the arch-ing ratio shown in Figs.13–15for the single-tunnel tests and in Figs.16and17for the parallel-tunnel tests,we reached the same conclusion for parallel tunneling as obtained for single tunneling,namely that shallower tunneling imposes a larger arching e?ect on the sur-rounding soil mass because of the higher arching ratio.

In the centrifuge modeling,the excess pore water pressure ratio will increase on an element in the soil bed that receives load transfers from adjoining yielding or?exible elements.Conversely,the excess pore water pressure ratio will decrease if an element shifts load to non-yielding parts or to more rigid elements.Therefore, the changes in the measured excess pore water pressure ratio can also be used to track the load transfers among the elements during tunneling simulation in the centri-fuge modeling.For example,the excess pore water pres-sure ratios shown in Figs.14and15initially rise with increases in the overload factor,but later fall as the ele-ments in non-yielding states progress to yielding states during increases in the overload factor.

4.2.Boundaries of the arching and plastic zones in the tunnel collapse stage

As discussed in the previous section,the outer bound-aries of the arching and plastic zones expand outward from the excavated area as the overload factor increases. Knowledge of the boundaries of the arching and plastic zones in the tunnel collapse stage is thus crucial for engi-neering practice,and is now discussed.

Fig.18presents the variations in the shear stress ra-tio,q/s u,at(OF num)c for the soil elements on the selected lines at the elevations of3.5,7.29,11,and16m above the spring line(NTest7,C/D=3).Here q= 1/2(r1àr3).The arrows shown in Fig.18indicate the positions of the outermost boundary of the plastic zone at the elevations where the elements have stress states of s u/q=1and are regarded as yielding.As can be seen in the?gure,the plastic boundaries extend from the tunnel center-line as far as12m along the spring line(about twice the tunnel diameter)and but only to10m at an

128 C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132

elevation3.5m above the spring line.Therefore,the out-er boundary of the plastic zone in the tunnel collapse stage can be inferred by examining all the elements re-garded as yielding(not just those considered in Fig.18),and is depicted in Fig.8(a)as a thick line of dashes.Fig.8(b)and(c)also display the outer bound-aries of the plastic zone for the single-tunnel models (C/D=1,2).

Similarly,Fig.19presents the variations of the shear stress ratio,q/s u,at(OF num)c for the soil elements on the

C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132129

selected lines at elevations of3.5,7.29,11and16m above the spring line(NTwin3,C/D=3,d/D=1.5). As can be seen in the?gure,the boundaries of the plastic zone extend from the center-line of the two tunnels as far as15m along the spring line and8m at an elevation of3.5m above the spring line.The outer boundary can also be determined with the procedure described in the previous paragraph,as shown in Figs.9(a)(Twin3) and9(b)(Twin2)(thick lines of dashes).

The non-yielding elements in the arching zone receive a load transfer from the elements in the plastic zone.The area lying between the outer boundary of the plastic zone and the boundary at which the elements have an arching ratio larger than1%is regarded as the arching zone.Thus the outer boundaries of the single-tunneling arching zones can easily be determined,and are shown as thick lines in Fig.8(a)–(c)for various C/D ratios.

The arching mechanism in the case of two parallel-tunnels is similar to that for a single-tunnel,so the same procedure can be used to infer the boundaries of the arching zone.They are depicted in Fig.9(a)and(b) for Twin3and Twin2.

Tunneling at di?erent burial depths can result in arching and plastic zones of di?erent extents.Deep sin-gle tunneling will result in a wider arching zone,but two parallel tunnels only generate a slightly wider arching zone for the same burial depth.The extents of the arch-ing zones for a single-tunnel and parallel-tunnels embed-ded at various depths(C/D=1,2,3)are shown in Fig.20(a)and(b),respectively.The curves representing the outer bounds of the arching zones were obtained as follows:

z?0.1exp0.305xà0.045

C

D

x

for singleàtunnel;e6aTz?0.105exp0.27xà0.0375

C

D

x

for two parallelàtunnelsed=D?1.5T;e6bTin which x is the distance o?set from the tunnel center for the single-tunnel and the distance o?set from the center-line of the two parallel-tunnels,and z is the depth.

4.3.Positive and negative arching zones

After examining the variations in the arching ratios of the elements surrounding the tunnel(or tunnels),as shown in Figs.13,16,and17,it was found that each arching zone can be divided into two zones,the positive

130 C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132

arching zone(with positive AR)and the negative arch-

ing zone(with negative AR).The boundaries(dash–dot lines)of the two zones derived from the numerical analysis are shown in Fig.8(a)–(c)for the single-tunnel with various C/D ratios and in Fig.9(a)and(b)for two parallel-tunnels with C/D=2,3and d/D=1.5. The shaded rectangular regions in these?gures corre-spond to the half-width of the sliding wedge,which can be determined from the upper bound solution using the collapse mechanism proposed by Wu and Lee (2003).The sliding wedges are reasonably consistent with the boundaries of the negative arching zones.A pile embedded in the negative arching zone would partially lose both end bearing capacity and skin friction on the pile body due to the reduction of the vertical stresses

C.J.Lee et al./Tunnelling and Underground Space Technology21(2006)119–132131

and would experience a large amount of settlement dur-ing new nearby tunneling.

5.Summaries and conclusions

A series of centrifuge model tests were carried out to investigate the surface settlement trough,excess pore water pressure generation,and tunnel stability of the tunnels with various C/D ratios(single-tunnel)and d/ D ratios(two parallel-tunnels).Numerical analysis was also conducted to evaluate the tunnel stability and arch-ing e?ects that develop during tunneling in soft clayey soil.The centrifuge and numerical modeling produced consistent results in their predictions of the surface set-tlement trough,excess pore water generation,and the overload factors at collapse for both single tunneling and parallel tunneling.An arching ratio derived from the numerical analysis was de?ned to describe the evolu-tion of the arching e?ect in the soil mass surrounding the tunnels.The boundaries of the arching zones for both single tunneling and parallel tunneling were determined. In addition,the boundaries of the positive and negative arching zones were also proposed.Construction engi-neers can easily locate the boundaries of the positive and negative arching zones and then take appropriate measures to mitigate possible damages to underground structures due to new nearby tunneling. Acknowledgments

The?nancial support provided by the National Science Council,Taiwan,under Grants NSC87-2211-E-008-024 and NSC91-2211-E-008-026is gratefully acknowledged. References

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