arching of soil

7/30/2019 Arching of Soil

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

Received 5 October 2004; received in revised form 2 April 2005; accepted 5 June 2005Available online 8 August 2005

Abstract

A series of centrifuge model tests and numerical simulations of these tests were carried out to investigate the surface settlementtroughs, excess pore water pressure generation, tunnel stability and arching effects 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 theoverload factors at collapse for both single and parallel tunneling. The arching ratio describes the evolution of the arching effects onthe soil mass surrounding tunnels and can be derived from the numerical analysis. The boundaries of the arching zones for bothsingle tunneling and parallel tunneling were determined. In addition, the boundaries of the positive and negative arching zones werealso proposed. 2005 Elsevier Ltd. All rights reserved.

Keywords: Arching effect; 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 transitsystems and sewage collection systems in densely pop-ulated cities. As the face of a tunnel is advanced, ameans of supporting the ground close to the facemay be needed; without such support, collapse mightoccur due to gross plastic deformation of the soil.Moreover, tunneling inevitably induces varying de-

grees of ground movement towards the tunnel openingand results in detrimental effects on nearby facilities,such as shallow foundations, piles, existing tunnelsand other pipeline systems. Taking appropriate mea-sures to protect nearby facilities before excavation isan important part of engineering practice. The predic-

tion of tunneling-induced ground movements duringexcavation of soft ground tunnel has been carriedout using various methods, including empirical meth-ods derived from field 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 andRowe, 1991).

Terzaghi (1943) explained how stress transfer fromyielding parts of a soil mass to adjacent non-yieldingparts leads to the formation of an arching zone. This

problem has two modes of displacement, dependingon whether the trap door is translated into the soil(passive mode) or away from it (active mode). Thepassive mode can be used for the evaluation of theuplift force of anchors, or of any buried structure thatcan be idealized as an anchor. The active mode can beused to study the gravitational flow of granular mate-rial between vertical walls (the silo problem) or theground pressure on tunnel liners. Ladanyi and Hoy-aux (1969) performed a series of model trap-door tests

0886-7798/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tust.2005.06.003

* Corresponding author. Tel.: +886 3 4227151x34135; fax: +886 34252960.

E-mail address: [email protected] (C.J. Lee).

www.elsevier.com/locate/tust

Tunnelling and Underground Space Technology 21 (2006) 119132

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Technologyincorporating Trenchless

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under 1 g conditions in order to check the validity ofthe classic bin theory. Handy (1983) analyzed soilarching action behind retaining walls, and Wang andYen (1973) carried out this analysis for slopes. Nakaiet al. (1997) performed a series of physical model testsunder 1 g conditions and carried out numerical analy-

sis of these tests to investigate the arching effect. Theyfound that the results obtained from the model testswere in good agreement with those obtained fromthe numerical analysis. Park and Adachi (2002) per-formed model tests under 1 g conditions to simulatetunneling events in unconsolidated ground with vari-ous levels of inclined layers. They found that remark-able non-symmetrical distributions of the earthpressure arose when a tunneling event took place ininclined layers with 60 of inclination. Stone and New-son (2002) presented the results of a series of centri-fuge tests designed to investigate the effects ofarching on soilstructure interaction. Koutsabeloulis

and Griffiths (1989) implemented a finite elementmethod 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 pilegroups subjected to lateral soil movement (Chen andMartin, 2002).

When tunneling is conducted in the vicinity of exist-ing pile foundations, the axial load transfer mechanismand failure mode on existing piles vary depending onthe distance between the existing piles and the new driv-ing tunnel and relative elevation of the piles with respectto 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 ofseveral tunnels stacked closely in an underground sta-tion needs to be established before appropriate protec-tion measures for nearby existing piles can beimplemented. By deepening our understanding of thearching effect in various geotechnical problems, we canimprove the design of the protection measures requiredfor existing underground structures nearby newtunneling.

Both centrifuge and numerical modeling were used inthe study. The stability of a tunnel, the movements ofsoil mass, the evolution of stress on the soil mass arounda tunnel, and the boundaries of the arching zone duringtunneling in clayey soils are investigated and discussed.Firstly, a series of centrifuge model tunnel tests was con-ducted. A finite difference program (FLAC) was thenchosen for numerical analysis of the system describedby the centrifuge model to provide insight into the arch-ing mechanism and the boundaries of the arching zoneduring tunneling. Finally, the results from the numericalmodeling and the measurements from the centrifugemodeling were compared in order to assess theirpredictions.

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. Thefull-scale system modeled with a centrifuge model (withdimensions Ntimes larger than those of the model if it istested in an acceleration that is Ntimes earth gravity) isreferred to as the prototype. It is intended that the pro-totype should include all the important characteristics ofthe field situation of interest. Centrifuge modeling pro-vides an opportunity to study for example the groundresponses due to tunneling before and after collapse; col-lapse is of course not permitted to occur in the field.

This experimental study was undertaken in the geo-technical centrifuge at the National Central University.The NCU centrifuge has a nominal radius of 3 m and

is capable of accelerating a 1 tonne model package to100 g and 0.55 tonne to 200 g. In the single-tunnel modeltests, one model tunnel, 60 mm in diameter, was embed-ded at various depths specified by the cover-to-diameterratio (C/D). In the parallel-tunnel model tests, two mod-el tunnels (60 mm in diameter) were separated by a spec-ified center-to-center distance (d) (as shown in Fig. 1),and buried at various depths specified by the cover-to-diameter ratio (C/D). All the model tests reported in thisstudy were carried out under a centrifugal accelerationof 100 g in order to model a prototype tunnel witha diameter of 6 m embedded at depths with the tested

C/D ratios.The soil used in the model tests had a plasticity index

of 18 and was classified as CL in the Unified Soil Clas-sification System. The soil slurry was remolded at abouttwice its liquid limit in a mixer and poured into the con-solidometer. Consolidation pressure was applied in fivestages, with a final pressure of 196 kPa. Further detailsof the soil bed preparation can be found in Wu and

d

PPT

Marked

spaghetti

LVDT

480mm

(Dimensions are in model scale)

820 mm

C

(60 mm)

Tunnel deformation gaugesTunnel

D

Fig. 1. Setup of test package for two parallel tunnels (model scale).

120 C.J. Lee et al. / Tunnelling and Underground Space Technology 21 (2006) 119132


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Lee (2003). The basic properties of the prepared soil bedare listed in Table 1. On completion of the consolida-tion, the soil bed, which had an undrained shearstrength profile of 3040 kPa, was lifted and placed ina 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 thepore water pressure. The pore pressure transducer(PPT) was inserted into a pre-drilled hole and then thehole was fully back-filled with thick slurry. EightLVDTs were fixed on top of the strong box to recordthe transverse surface settlements.

The test package was first spun at an acceleration of100 g for 5 min so that any voids generated during theinstallation of the PPTs and the assembly of the testpackage might be filled. After the centrifuge wasstopped, one or two 60 mm diameter model tunnels werecut manually, depending on the test conditions, andthen rubber bags were inserted into the tunnels. Tunnel

deformation gauges consisting of four thin cantileversmade from stainless steel were installed inside the rubberbags to measure the deformations at the crowns, inverts,and side-walls of the tunnels. The test package shown inFig. 1 was prepared for further tunnel collapse tests byconnecting 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 tunnelcenter during the reacceleration of the model up to acentrifuge acceleration of 100 g. The tunneling eventwas simulated by simultaneously reducing the air pres-sure inside the tunnels and eventually down to zero. This

air-pressure method of simulating tunnel excavation wasadapted from Mair (1979). The air-pressure method waschosen to support the tunnel during the acceleratingstages because measuring the tunnel deformation wasneeded and the waterproofing of this measuring devicewas difficult if an incompressible heavy fluid-pressuremethod was used. The method used in the study maycause smaller surface settlements and larger settlementtrough widths at the corresponding supporting pressuresbut no difference in the supporting pressure at collapsecompared to the incompressible heavy fluid-pressuremethod.

No more than 15 min were spent under 100 g prior tocollapse (including the accelerating stage). The dissipa-

tion of pore water pressure (or changes of effectivestress) during these two stages may be less than 1% inaverage. A torvane apparatus was used for determiningthe undrained shear strength at various depths on theside of the soil bed before and after the test. The test re-sults show that no obvious changes in the undrained

shear strengths (less than 2 kPa) at the same depths werefound. Therefore, we assumed that there was minimalmigration of pore water pressure and hence that the soilwas subjected to undrained shearing. The changes in thepore water pressures, the tunnel deformations, and thesurface settlements induced by tunneling were measuredcontinuously. After each model test, the soil bed wascautiously excavated to expose the implanted spaghetti.Further, six undisturbed samples were taken from thesoil bed at selected depths for use in unconfined com-pression tests. The undrained shear strength (su) andthe secant Youngs modulus (E50), for each soil bedwas determined from the average of the results for the

six samples, as listed in Table 2.The surrounding soil squeezes into the tunnel as the

supporting air pressure is gradually reduced, which fi-nally causes tunnel collapse. The overload factor (OF),as defined below, is a useful index for describing tunnelstability throughout the entire test process.

OF rvo pisu

; 1

where rvo is the overburden pressure at the tunnel cen-ter, and pi is the supporting pressure. Fig. 2 shows plotsof the surface settlements at the tunnel axis versus OFfor the single-tunnel test, Test7 (C/D = 3), and of thoseat the symmetrical axis of the settlement trough for theparallel-tunnel test, Twin3 (C/D = 3, d/D = 1.5). This

Table 1Basic properties of the prepared soil bed

Specific gravity, Gs 2.67Liquid limit, LL 40Plastic limit, PL 22Plasticity index, PI 18Unit weight, c (kN/m3) 18.1Compression index, Cc 0.28Swell index, Cr 0.0275Coefficient of consolidation, Cv (cm

2/s) 0.010524Permeability, k (m/s) 4.5 109

Table 2Test configurations

Test no.a C/D d/D su (kPa)

Test11 0.5 31.00Test12 0.5 35.12Test5 1 36.90Test8 1 37.90Twin4 1 1.5 33.00Twin5 1 1.5 39.10Test3 2 30.25Test9 2 35.79Twin1 2 3 48.70Twin2 2 1.5 41.00Twin9 2 1.5 39.52Twin12 2 2 35.10Test6 3 33.30Test7 3 34.00Twin3 3 1.5 36.10Twin6 3 1.5 32.90Test10 4 32.17Twin10 4 1.5 34.25Twin11 4 1.5 32.83

a Test3Test12: single tunnel; Twin1Twin11: two parallel tunnels.

C.J. Lee et al. / Tunnelling and Underground Space Technology 21 (2006) 119132 121


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figure shows that the surface settlements, S, increasedramatically once OFexceeds a critical value. Extendingthe straight-line portions of the first and second parts ofthe S vs OF curves to intersect at the points shown inFig. 2, the horizontal ordinate of each of these criticalpoints is defined as the overload factor at collapse,(OF)c. In engineering practice, the tunnel must be sup-ported against collapse during tunneling. Thus, the loadfactor (LF) is regarded as the reciprocal of the safetyfactor

LF rvo pirvo pic

OFOFc; 2

where (pi)c is the measured supporting pressure at col-lapse in the centrifuge model tests. The value ofLFvar-

ies from 0 to 1, which corresponds to variation of thetunnel stability from stable to critical.

A total of 19 model tests were performed in the study:nine single-tunnel tests and ten parallel-tunnel tests, aslisted in Table 2. The C/D ratio varied from 0.5 to 4for the single-tunnel tests, and from 1 to 4 for the paral-lel-tunnel tests. The d/D ratios of the parallel-tunneltests were 1.5, 2, and 3 for C/D = 2, and the d/D ratiowas 1.5 for the other C/D ratios.

2.2. Numerical modeling

The numerical experiments were carried out with atwo-dimensional explicit finite difference program,

FLAC2D (Cundall et al., 1993). A plane-strain modelwith large-strain formulation was used to simulate thedeformation behavior of the ground surrounding un-lined tunneling. In order to compare the results of thenumerical 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 amesh with a width of 82 m and a height of 48 m asshown in Fig. 3, which are the exact dimensions of thesoil bed used in the centrifuge model tests. The leftand right boundaries were fixed in the x-direction, andthe bottom boundary was fixed in the x- and y-directions. The grid size around the tunnels was0.5 m 0.5 m in the prototype and was enlarged by afactor of 1.08 as the distance to tunnel center increases.The soil bed was treated as an isotropic and elastic per-fectly plastic continuum following the MohrCoulombfailure criterion (/ = 0). A total of six numerical models

were analyzed. The model conditions and the mechani-cal properties of soil bed measured from the unconfinedcompression tests used in the numerical analysis areshown in Table 3.

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

Overload factor, (OF)

0 1 2 3 4 5 6 7 8 9 10 11 12

Settlementinproto

typescale,(m)

0.0

0.5

1.0

1.5

2.0

2.5

Twin3

Test7

(OF) c =3.96

(OF)c =3.2

Fig. 2. Definition of overload factor at collapse, (OF)c.

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

Table 3Geometric conditions and mechanical properties of the numerical models

Test no. C/D d/D su (kPa) E50 (kPa) c (kN/m3) m

Ntest8 1 37.90 3500 18.1 0.49Ntwin4 1 1.5 33.00 3500 18.1 0.49Ntest9 2 35.79 3500 18.1 0.49Ntwin2 2 1.5 41.00 4000 18.1 0.49Ntest7 3 34.00 3500 18.1 0.49Ntwin3 3 1.5 36.10 3500 18.1 0.49



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to tunneling. Secondly, the elements representing theexcavated soil in the tunnel were nulled and a uniformsupporting pressure equal to the overburden pressureat the center of tunnel was applied to the interior bound-ary of the circular opening to keep the tunnel stable. Thesupporting pressure was then reduced by a decrement of

10 kPa per step in order to simulate the decrease in thesupporting pressure that was applied in the centrifugemodel tests.

For FLAC, the value of maximum nodal unbalanceforce is used to determine if a simulation having reachedequilibrium. In the current study, an equilibrium statewas regarded as having converged when the maximumunbalance force of every node in the mesh was less than10 N (the ratio of the maximum unbalance force to theoverburden pressure on the element was about 0.005%)in the simulation of per step. The simulation was thenmoved to the next step (reducing the supporting pres-sure by a decrement of 10 kPa in the study). The sup-

porting pressure was reduced further until an errormessage 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 definedas the collapse supporting pressure, (pnum)c, which isdetermined from the numerical modeling. By substitut-ing (pnum)c into Eqs. (1) and (2), the overload factor,(OFnum)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 agreementbefore tunnel collapse, as shown in Fig. 4.

Numerical modeling can be used to examine defor-mation more precisely at small strain levels and at morelocations than can be achieved with centrifuge modeling,but is not as precise as the measurements of the failureand post-failure behavior that the centrifuge modeling

can provide. Therefore, integrating and comparing theresults derived from the numerical and centrifuge mod-els provides improved understanding of the deformationbehavior and the arching mechanism during tunneling.

3. Comparison of the results from centrifuge andnumerical modeling

3.1. Tunnel stability

The tunneling event was simulated by reducing thesupporting pressure inside the tunnels in both the centri-fuge modeling and the numerical experiments. Fig. 5shows the relations between the C/D ratio and (OF)cmeasured in the single-tunnel and the parallel-tunnelmodel tests (solid symbols), and the values of (OFnum)ccalculated 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. Thetest results from Mair (1979) are also included in this fig-ure. The increase in (OF)c with the C/D ratio illustratesthat the stability of the tunnel improves if the tunnel isembedded more deeply. In addition, Fig. 5 also showsthat the stability of parallel-tunnel is worse than thatof single tunnel; lower overload factors at collapse werefound 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 samevalues as those derived from the corresponding numeri-cal model (hollow symbols), as shown in Fig. 5. The val-ues of (OF)c and (OFnum)c for the single-tunnel modelare all confined by upper and lower bounds. The valuesof (OF)c and (OFnum)c for the parallel-tunnel model are


0 1 2 3 4 5 6 7 8 9 10 11 12

Settlementinprototype,(m

) 0.0

0.5

1.0

1.5

2.0

2.5

Test7 (C/D=3)

NTest7 (C/D=3)

Twin3 (C/D=3, d/D=1.5)

NTwin3 (C/D=3, d/D=1.5)

Fig. 4. Relations between the overload factor and maximum surfacesettlement derived from the numerical and centrifuge modeling.

Cover-to-diameter ratio, C/D

0 1 2 3 4

Overloadfactoratcolla

pse,(OF)c

0

1

2

3

4

5

6

Single - tunnel

Parallel -tunnel

centrifuge model of single - tunnel (Mair, 1979)

Numerical solution (single - tunnel)

Numerical solution (parallel- tunnel)

Upper bound(single - tunnel)(Wu and Lee, 2003)

Lower bound(single - tunnel)(Lee et al. , 1999)

Upper bound(parallel- tunnel)(Wu andLee, 2003)

Fig. 5. Comparison of the (OF)c values obtained from numerical andcentrifuge modeling.



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also confined by the upper bound, but no lower boundsolution has yet been derived. The centrifuge andnumerical modeling provide consistent evaluations ofthe tunnel stability.

3.2. Surface settlement troughs

Figs. 6 and 7 compare the computed (representedby lines) and measured surface settlement troughs(represented by symbols) at selected load factors forthe single-tunnel model (C/D = 2) and for the paral-lel-tunnel model (C/D = 1, d/D = 1.5), respectively.The distances, X, offset 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, areboth normalized with respect to the tunnel diameter,D. The computed settlement troughs compare reason-ably well both in shape and magnitude with thosemeasured in the centrifuge models at the correspond-

ing load factors. The shape of the settlement troughfor the single tunnel approximates closely to that ofthe error function. An empirical approach derivedfrom centrifuge modeling has been proposed for calcu-lating the surface and subsurface settlement troughsfor 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 theparameters obtained for single tunneling. Their meth-odology for predicting the settlement troughs based onthe ground loss was verified by comparison with 12sets of monitored field data (Wu and Lee, 2003).

3.3. Comparison of the excess pore water pressures

obtained from centrifuge and numerical modeling

In an undrained system, the stress changes

(Dr1,Dr2,Dr3) due to tunneling would generate excesspore water pressure within the soil mass. With the ap-proach suggested by Henkel, the changes in the porewater pressure, Du, can be determined with the equation

Du 13Dr1 Dr2 Dr3 aDr1 Dr22

Dr2 Dr32 Dr3 Dr121=2; 3where a is Henkels pore water pressure parameter, andDr1,Dr2,Dr3 are the changes in the major, intermediate,and minor principle stresses, respectively. A comparisonofa and Skemptons parameter, A, derived from triaxial

compression tests gives

a 1ffiffiffi2

p A 13

. 4

This definition is useful because it enables the predic-tion of the Du associated with loading conditions un-der plane strain conditions if we assume that the soilbed is an isotropic and elastic-plastic material. Byusing a value of a of 0.3 (determined from the triaxialcompression tests) and Dr2 = 1/2(Dr1 + Dr3) for theplane strain and undrained conditions, the developedexcess pore water pressure can be estimated after thechanges in total stress are calculated in the numericalsimulations.

In the centrifuge modeling, the stress changes on asoil element during tunneling cannot be measured,whereas the changes of pore water pressure can be di-rectly measured. The numerical analysis in terms of totalstress approach can give the changes of total stress but itcannot 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 wereobtained from the numerical analysis. In this study,the excess pore water pressure ratio, (Du/rvo), defined

Test9 (C/D=2)

Normalized distance from centre-line, (X/D)

-8 -6 -4 -2 0 2 4 6 8

Set

tlementratio,(S/D)

0.00

0.02

0.04

0.06

0.08

0.10

LF=0.567

LF=0.646

LF=0.730

LF=0.814

LF=0.893

LF=1.0

LF=0.570

LF=0.651

LF=0.733

LF=0.814

LF=0.896

LF=1.0

NTest9 (C/D=2)

Fig. 6. Comparison of surface settlement troughs at various loadfactors obtained from numerical and centrifuge modeling for a singletunnel.

NTwin4(C/D=1, d/D=1.5)

Normalized distance from centre-line of parallel tunnels, (X/D)

-8 -6 -4 -2 0 2 4 6 8

Settlementratio,(S/D

)

0.00

0.02

0.04

0.06

0.08

0.10

LF=0.369

LF=0.493

LF=0.616

LF=0.739

LF=0.862

LF=0.985

LF=0.374

LF=0.503

LF=0.615

LF=0.738

LF=0.880

LF=1.0

Twin4(C/D=1, d/D=1.5)

Fig. 7. Comparison of surface settlement troughs at various loadfactors obtained from numerical and centrifuge modeling for twoparallel tunnels.



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4. Mechanism of arching around tunnels

4.1. Evolution of arching effect

The effects of tunneling on the total stress within thesoil mass around a tunnel can be estimated from thecomputed responses at the grids of points shown inFig. 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 testsand from those shown in Figs. 9(a) (Twin3; C/D = 3,d/D = 1.5) and 9(b) (Twin2; C/D = 2, d/D = 1.5) forthe parallel-tunnel tests. A more detailed understandingof the stress transfer in a tunneling problem frommoving parts of the soil (settle more) to adjacent parts(settle less) can be achieved by considering the verticalstress redistributions in the soil mass above the springline. The arching ratio is defined as

AR% Drvrvo

100; 5

where Drv is the change in the vertical stress during tun-neling and rvo is the total overburden pressure. In thisstudy, we used the arching ratio to describe the archingbehavior quantitatively at various locations.

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

vo

(u

/

Test7 (C/D = 3)


)

vo

(u

/

)

vo

(u

/

)

vo

(u

/

)

vo

(u

/

)

0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

Calculated

B1

(OF)c


0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

Calculated

C3

(OF)c


0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

Calculated

A3

(OF)c


0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

Calculated


0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

Calculated

C1

(OF)c

A0

(OF)c

(a) (b)

(c) (d)

(e)

Fig. 10. Comparison of measured and calculated excess pore water pressures for Test7 (C/D = 3).



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on the spring line (B1E1) 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), themagnitude of the arching ratio is also related to the dis-tance offset from the tunnel center and to the overloadfactor (comparing the arching ratios on Lines A, B, C,D, and E). Fig. 13 summarizes the changes in thearching ratio for three overload factors (OF= 1,3, and(OFnum)c) for the elements on Lines A to E for NTest7(C/D = 3). The magnitude of the positive arching ratioincreases with increases in the overload factor for theelements along Line B but declines rapidly once the ele-ment is yielding (Fig. 13(b)). The element at C1 experi-ences the largest positive arching ratio (about a 9%rise). The elements located at a greater distance fromthe tunnel center experience smaller positive archingratios. In contrast, the arching ratio becomes more neg-

ative as the overload factor increases for the elements onLine A (Fig. 13(a)).

Similarly, Fig. 14 summarizes the arching ratio as afunction of the overload factor for the elements on thespring line (B1E1) and on the vertical center-line(A2A3) shown in Fig. 8(b) for NTest9 (C/D = 2).Fig. 15 summarizes the arching ratio as a functionof the overload factor for the elements on the springline (B1D1) 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 tunnelingshown in these two figures is similar to those pre-sented in Figs. 12 and 13 but smaller positive and neg-ative arching ratios are obtained at the correspondingpoints for deeper tunneling (NTest7, C/D = 3). Shal-lower tunneling imposes a larger arching effect onthe surrounding soil mass.

Twin3 (C/D=3, d/D=1.5)


) )

0 2 4 6 8 10 12

(u/ vo

)

(u/vo )

(u/vo

)

(u/vo

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

calculated

D1


0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

calculated

C1

(u/vo

D2


0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

calculated

B2


0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

calculated

A0


0 2 4 6 8 10 12

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Measured

calculated

(OF)c

(OF)c

(OF)c (OF)c

(OF)c

(a) (b)

(c) (d)

(e)

Fig. 11. Comparison of measured and calculated excess pore water pressures for Twin3 (C/D = 2, d/D = 1.5).



10/14

Fig. 16 summarizes the changes in the arching ratiofor the elements on Lines AF (Fig. 9(a)) for three over-load factors (OF= 1.5,2.5, and (OFnum)c) for NTwin3

(C/D = 3, d/D = 1.5). Fig. 17 shows the changes in thearching ratio for the elements on Lines AF (Fig. 9(b))for NTwin2 (C/D = 2, d/D = 1.5). A comparison ofthe changes in the arching ratios shown in Figs. 13and 16 indicates 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 Aand on the spring line. At this point, a larger positivearching 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 apillar. After examining in detail the changes in the arch-ing ratio shown in Figs. 1315 for the single-tunnel testsand in Figs. 16 and 17 for the parallel-tunnel tests, wereached the same conclusion for parallel tunneling asobtained for single tunneling, namely that shallower

tunneling imposes a larger arching effect on the sur-rounding soil mass because of the higher arching ratio.

In the centrifuge modeling, the excess pore waterpressure ratio will increase on an element in the soilbed that receives load transfers from adjoining yieldingor flexible elements. Conversely, the excess pore waterpressure ratio will decrease if an element shifts load tonon-yielding parts or to more rigid elements. Therefore,the changes in the measured excess pore water pressureratio can also be used to track the load transfers amongthe elements during tunneling simulation in the centri-fuge modeling. For example, the excess pore water pres-

sure ratios shown in Figs. 14 and 15 initially rise withincreases in the overload factor, but later fall as the ele-ments in non-yielding states progress to yielding statesduring 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 outwardfrom the excavated area as the overload factor increases.Knowledge of the boundaries of the arching and plasticzones in the tunnel collapse stage is thus crucial for engi-neering practice, and is now discussed.

Fig. 18 presents the variations in the shear stress ra-tio, q/su, at (OFnum)c for the soil elements on the selectedlines at the elevations of 3.5, 7.29, 11, and 16 m abovethe spring line (NTest7, C/D = 3). Here q =1/2(r1 r3). The arrows shown in Fig. 18 indicate thepositions of the outermost boundary of the plastic zoneat the elevations where the elements have stress states ofsu/q = 1 and are regarded as yielding. As can be seen inthe figure, the plastic boundaries extend from the tunnelcenter-line as far as 12 m along the spring line (abouttwice the tunnel diameter) and but only to 10 m at an

NTest7 (C/D=3)


0 1 2 3 4 5

Archingra

tio,(%)

-20

-15

-10

-5

0

5

10

A0

A3

A4

A5

B1

C1

D1

E1 (OF)c

NTest7 (C/D=3)

Overload factor, (OF)0 1 2 3 4 5

Archingratio,(%)

-2

0

2

4

6

8

10B0

B1

B3

B4

B5

C0

C1

C3

C4

C5

(OF)c

NTest7 (C/D=3)


0 1 2 3 4 5

Archingratio,

(%)

-2

0

2

4

6

8

10

D0

D1

D3

D4

D5

E1

E3

E4

(OF)c

(a)

(b)

(c)

Fig. 12. Arching ratio versus overload factor at various locations(NTest7 C/D = 3).



11/14

elevation 3.5 m above the spring line. Therefore, the out-er boundary of the plastic zone in the tunnel collapsestage can be inferred by examining all the elements re-garded as yielding (not just those considered inFig. 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. 19 presents the variations of the shearstress ratio, q/su, at (OFnum)c for the soil elements on the

NTest7 (C/D=3) Line B

Arching ratio, (%)

-20 -15 -10 -5 0 5 10

-10

-5

0

5

10

15

20

25

OF=1

OF=3

(OFnum)c

NTest7 (C/D=7) Line C

Arching ratio, (%)

-20 -15 -10 -5 0 5 10

-10

-5

0

5

10

15

20

25

OF=1

OF=3

(OFnum)c

NTest7 (C/D=3) Line E

Arching ratio, (%)

-20 -15 -10 -5 0 5 10

-10

-5

0

5

10

15

20

25

OF=1

OF=3

(OFnum

)c

NTest7 (C/D=3) Line D

Arching ratio, (%)

-20 -15 -10 -5 0 5 10

Distancefrom

springline,(m)

-10

-5

0

5

10

15

20

25

OF=1

OF=3

(OFnum

)c

NTest7 (C/D=3) Line A

Arching ratio, (%)

-20 -15 -10 -5 0 5 10

Distancefromspringline,(m)

-10

-5

0

5

10

15

20

25

OF=1

OF=3

(OFnum

)c

Spring line

Spring line

(a) (b) (c)

(d) (e)

Fig. 13. Changes in arching ratio in various overload factors along; (a) Line A; (b) Line B; (c) Line C; (d) Line D; (e) Line E (NTest7 C/D = 3).

NTest9 (C/D=2)


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Archingratio,(%)

-30

-25

-20

-15

-10

-5

0

5

10

15

20

B1

C1

D1

E1

A2

A3

(OF)c


Overload factor, (OF)0 1 2 3 4

Archingratio,(%)

-15

-10

-5

0

5

10

15

20

25

30

35

B1

C1

D1

A2

(OF)c




12/14

selected lines at elevations of 3.5, 7.29, 11 and 16 mabove the spring line (NTwin3, C/D = 3, d/D = 1.5).As can be seen in the figure, the boundaries of the plasticzone extend from the center-line of the two tunnels asfar as 15 m along the spring line and 8 m at an elevationof 3.5 m above the spring line. The outer boundary canalso be determined with the procedure described in theprevious paragraph, as shown in Figs. 9(a) (Twin3)and 9(b) (Twin2) (thick lines of dashes).

The non-yielding elements in the arching zone receivea load transfer from the elements in the plastic zone. Thearea lying between the outer boundary of the plasticzone and the boundary at which the elements have anarching ratio larger than 1% is regarded as the archingzone. Thus the outer boundaries of the single-tunnelingarching zones can easily be determined, and are shownas 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 sameprocedure can be used to infer the boundaries of thearching zone. They are depicted in Fig. 9(a) and (b)for Twin3 and Twin2.

Tunneling at different burial depths can result inarching and plastic zones of different extents. Deep sin-gle tunneling will result in a wider arching zone, but two

parallel tunnels only generate a slightly wider archingzone 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 inFig. 20(a) and (b), respectively. The curves representingthe outer bounds of the arching zones were obtained asfollows:

z 0.1 exp 0.305x 0.045 CD

x

for single tunnel; 6a

z

0.105 exp 0.27x

0.0375

C

D x

for two parallel tunnels d=D 1.5; 6bin which x is the distance offset from the tunnel centerfor the single-tunnel and the distance offset from thecenter-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 ratiosof the elements surrounding the tunnel (or tunnels), asshown in Figs. 13, 16, and 17, it was found that eacharching zone can be divided into two zones, the positive

NTwin3 Line A(C/D=3, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20


-10

-5

0

5

10

15

20

OF=1.5

OF=2.5

(OFnum

)c

NTwin3 Line B(C/D=3, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20

-10

-5

0

5

10

15

20

OF=1.5

OF=2.5

(OFnum)c

NTwin3 Line C(C/D=3, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20

-10

-5

0

5

10

15

20

OF=1.5

OF=2.5

(OFnum

)c

NTwin3 Line D(C/D=3, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20


-10

-5

0

5

10

15

20

OF=1.5

OF=2.5

(OFnum

)c

NTwin3 Line E(C/D=3, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20

-10

-5

0

5

10

15

20

OF=1.5

OF=2.5

(OFnum)c

NTwin3 Line F(C/D=3, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20

-10

-5

0

5

10

15

20

OF=1.5

OF=2.5

(OFnum)c

spring line spring line spring line


(a) (b) (c)

(d) (e) (f)

Fig. 16. Changes in arching ratio in various overload factors along; (a) Line A; (b) Line B; (c) Line C; (d) Line D; (e) Line E; (f) Line F (NTwin3

C/D = 3; d/D = 1.5).



13/14

arching zone (with positive AR) and the negative arch-ing zone (with negative AR). The boundaries (dashdot lines) of the two zones derived from the numericalanalysis are shown in Fig. 8(a)(c) for the single-tunnelwith various C/D ratios and in Fig. 9(a) and (b) fortwo parallel-tunnels with C/D = 2,3 and d/D = 1.5.The shaded rectangular regions in these figures corre-spond to the half-width of the sliding wedge, which

can be determined from the upper bound solution usingthe collapse mechanism proposed by Wu and Lee(2003). The sliding wedges are reasonably consistentwith the boundaries of the negative arching zones. A pileembedded in the negative arching zone would partiallylose both end bearing capacity and skin friction on thepile body due to the reduction of the vertical stresses

NTwin2 Line A(C/D=2, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20 30

Distancefrom

springline,(m)

-10

-5

0

5

10

15

OF=0.97

OF=1.95

(OFnum

)c

NTwin2 Line B(C/D=2, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20 30

-10

-5

0

5

10

15

(OF)=0.97

OF=1.95

(OFnum

)c

NTwin2 Line C(C/D=2, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20 30

-10

-5

0

5

10

15

OF=0.97

OF=1.95

(OFnum

)c

NTwin2 Line D(C/D=2, d/D=1.5)

Arching ratio, (%)

-30 -20 -10 0 10 20 30

Distancefromspr

ingline,(m)

-10

-5

0

5

10

15

OF=0.97

OF=1.95

(OFnum

)c

NTwin2 Line E(C/D=2, d/D=1.5)

Archinf ratio, (%)

-30 -20 -10 0 10 20 30

-10

-5

0

5

10

15

OF=0.97

OF=1.95

(OFnum

)c


spring line spring line

(a) (b) (c)

(d) (e)

Fig. 17. Changes in arching ratio in various overload factors along; (a) Line A; (b) Line B; (c) Line C; (d) Line D; (e) Line E (NTwin2 C/D = 2;d/D = 1.5).

NTest7 (C/D=3)

Distance from centerline of tunnel, (m)

0 10 20 30 40

Ratioofshearstress,(q/su)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

spring line

3.5m above spring line


11m above spring line


Fig. 18. Variations in the shear stress ratio at (OFnum)c for soilelements along lines at various elevations above the spring line for asingle tunnel (NTest7 C/D = 3).

NTwin3 (C/D=3, d/D=1.5)

Distance from center line of two parallel tunnels, (m)

0 10 20 30 40

RatioofShearstress,q

/su

0.0

0.2

0.4

0.6

0.8

1.0

1.2

spring line





Fig. 19. Variations in the shear stress ratio at (OFnum)c for soilelements along lines at various elevations above the spring line for twoparallel tunnels (NTwin3 C/D = 3 d/D = 1.5).



14/14

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 toinvestigate the surface settlement trough, excess porewater pressure generation, and tunnel stability of thetunnels with various C/D ratios (single-tunnel) and d/D ratios (two parallel-tunnels). Numerical analysis wasalso conducted to evaluate the tunnel stability and arch-ing effects that develop during tunneling in soft clayeysoil. The centrifuge and numerical modeling producedconsistent results in their predictions of the surface set-tlement trough, excess pore water generation, and the

overload factors at collapse for both single tunnelingand parallel tunneling. An arching ratio derived fromthe numerical analysis was defined to describe the evolu-tion of the arching effect in the soil mass surrounding thetunnels. The boundaries of the arching zones for bothsingle tunneling and parallel tunneling were determined.In addition, the boundaries of the positive and negativearching zones were also proposed. Construction engi-neers can easily locate the boundaries of the positiveand negative arching zones and then take appropriatemeasures to mitigate possible damages to undergroundstructures due to new nearby tunneling.

Acknowledgments

The financial support provided by the National ScienceCouncil, Taiwan, under Grants NSC 87-2211-E-008-024and NSC 91-2211-E-008-026 is gratefully acknowledged.

References

Clough, G.W., Schmidt, B., 1981. Design and performance ofexcavation and tunnels in soft clay. In: Soft Clay Engineering.Elsevier, Amsterdam, pp. 600634 (Chapter 8).

Chen, C.Y., Martin,G.R., 2002. Soilstructure interaction for landslidestabilizing pile. Computer and Geotechnics 29 (5), 363386.

Cundall, P.A., Coetzee, M.J., Hart, R.D., Varona, P.M., 1993. FLACUsers Manual. Itasca Consulting Group, USA.

Handy, R.L., 1983. The arch in soil arching. Journal of GeotechnicalEngineering, ASCE 111 (3), 302318.Koutsabeloulis, N.C., Griffiths, D.V., 1989. Numerical modeling of the

trap door problem. Geotechnique 39 (1), 7789.Ladanyi, B., Hoyaux, B., 1969. A study of the trap-door problem in a

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Lee, C.J., Chiang, K.H., 2004. Load transfer on single pile near newtunneling in sandy ground. In: Matsui, Tanaka, Mimura (Eds),Proceedings of the International Symposium on EngineeringPractice and Performance of Soft Deposits (IS-OSAKA 2004),pp. 495506.

Lee, C.J., Chiang, K.H., Kou, C.M., 2004. Ground movement and

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ground movements: the thunder bay tunnel. Canadian Geotechni-cal Journal 28 (1), 2541.

Mair, R.J., 1979. Centrifugal modeling of tunnel construction in softclay. Ph.D Thesis, University of Cambridge, UK.

Mair, R.J., Gunn, M.J., OReilly, M.P., 1981. Ground movementsaround shallow tunnels in soft clay. In: Proceedings of the 10thInternational Conference on Soil Mechanics and FoundationEngineering, pp. 323328.

Nakai, T., Xu, L., Yamazaki, H., 1997. 3D and 2D model tests andnumerical analyses of settlements and earth pressures due to tunnelexcavation. Soils and Foundations 37 (3), 3141.

Park, S.H., Adachi, T., 2002. Laboratory model tests and FE analyses

on tunneling in the unconsolidated ground with inclined layers.Tunneling and Underground Space Technology 17, 181193.

Peck, R.B., 1969. Deep excavation and tunneling in soft ground. In:Proceedings of 7th International Conference on Soil Mechanicsand Foundation Engineering, Mexico, State of the Art Volume, pp.225290.

Stone, K.J.L., Newson, T.A., 2002. Arching effects in soilstructureinteraction. In: Phillips, Guo, Popescu (Eds.), Physical Modeling inGeotechnics: ICPMG 02, pp. 935939.

Terzaghi, K., 1943. Theoretical Soil Mechanics. Wiley, New York.Wang, W.L., Yen, B.C., 1973. Soil arching in slopes. Journal of the

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nisms induced by tunneling in clayey soil. International Journal ofPhysical Modelling in Geotechnics 3 (4), 1327.

Distance from tunnel center, (m)

0 10 20 30

Depth

,(m)

0

5

10

15

20

25

C/D=1C/D=2

C/D=3

Fitting curvez=0.1exp{0.305x-0.045(C/D)x]

Paralle-tunnel

Distance from centerline of two parallel tunnels, (m)

0 10 20 30

Depth

,(m)

0

5

10

15

20

25

C/D=1, d/D=1.5

C/D=2, d/D=1.5

C/D=3, d/D=1.5

Fitting curvez=0.105exp[0.27x-0.0375(C/D)x]

Single-tunnel

(b)(a)

Fig. 20. Extents of the arching zones for single-tunnel and parallel-tunnel (d/D = 1.5).


arching of soil

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