the influence of pre-failure soil sfitness on the numerical analysis of tunnel construction

The in¯uence of pre-failure soil stiffness on the numerical analysisof tunnel construction

T. I . ADDENBROOKE, � D. M. POTTS � and A. M. PUZRIN{

The use of the ®nite-element method to analysetunnels is becoming more widespread, but anyprediction is dependent (among other things) onthe model adopted for the pre-failure soil be-haviour. This paper compares and contrastsplane strain predictions of ground movementfor both single- and twin-tunnel excavations instiff clay modelled as

(a) isotropic linear elastic perfectly plastic(b) anisotropic linear elastic perfectly plastic(c) isotropic non-linear elastic perfectly plastic

with shear stiffness dependent on deviatoricstrain and mean effective stress, and bulkmodulus dependent on volumetric strain andmean effective stress

(d) anisotropic non-linear elastic perfectly plas-tic employing the model in (c) above

(e) isotropic non-linear elastic perfectly plasticwith shear and bulk stiffness dependent ondeviatoric strain level, mean effective stress,and loading reversals.

The analyses model the geometry of the twinJubilee Line Extension Project tunnels beneathSt James's Park (London, UK), and ®eld dataare presented for comparison (Standing et al.,1996). By considering the predicted surfacesettlement, the study shows the importance ofmodelling non-linear elasticity, and the effect ofintroducing a soft independent shear modulus.The differences in subsurface displacements forisotropic and anisotropic models are highlighted.The subsequent modelling of an adjacent tunnelexcavation exposes more detailed features of allthe models. It is concluded (a) that anisotropicparameters appropriate to London Clay do notenhance the plane strain predictions of groundmovement as long as non-linear pre-failuredeformation behaviour is being modelled; (b)that a soft anisotropic shear modulus signi®-cantly improves green®eld predictions but nottwin-tunnel predictions; (c) and that accountingfor load reversal effects does in¯uence ananalysis of this problem (St James's Park twintunnels).

L'analyse des tunnels a de plus en plus souventrecours aÁ la meÂthode des eÂleÂments ®nis, maistoute preÂvision deÂpend (entre autres) de lamodeÂlisation du comportement des sols avantla rupture. L'article compare les mouvements desol de tunnels simples et doubles creuseÂs dansde l'argile rigide, sous l'effet de deÂformationsplanes, preÂdits dans les modeÁles suivants

(a) isotrope, lineÂaire, eÂlastique, parfaitementplastique

(b) anisotrope, lineÂaire, eÂlastique, parfaitementplastique

(c) isotrope, non lineÂaire, eÂlastique, parfaite-ment plastique, la rigiditeÂ au cisaillementvariant en fonction des contraintes deÂvia-trices et de la contrainte intergranulairemoyenne, et le module de compressionvariant en fonction de la contrainte hydro-statique et de la contrainte intergranulairemoyenne

(d) anisotrope, non lineÂaire, eÂlastique, parfaite-ment plastique, en utilisant le modeÁle ci-dessus (c)

(e) isotrope, non lineÂaire, eÂlastique, parfaite-ment plastique, la rigiditeÂ au cisaillement etla rigiditeÂ aÁ la compression variant enfonction des contraintes deÂviatrices, de lacontrainte intergranulaire moyenne et descontraintes subies reÂcemment.

Les analyses modeÂlisent la geÂomeÂtrie des tunnelsdoubles du prolongement de la ligne de meÂtroJubilee Line sous St James's Park, et preÂsententdes donneÂes releveÂes sur le terrain aÁ titre decomparaison (Standing et al., 1996). En exami-nant le tassement de surface preÂdit, l'eÂtudemontre l'importance de la modeÂlisation del'eÂlasticiteÂ non lineÂaire, et l'effet d'un modulede cisaillement tendre indeÂpendant. L'article faitressortir les diffeÂrences de deÂplacements souter-rains des modeÁles isotropes et anisotropes. LamodeÂlisation subseÂquente de l'excavation d'untunnel adjacent met en eÂvidence des aspects plusdeÂtailleÂs de tous les modeÁles. L'article conclut(a) que les parameÁtres anisotropes propres aÁl'argile londonnienne n'ameÂliorent pas les preÂ-visions des mouvements de sol sous l'effet dedeÂformations planes dans la modeÂlisation desdeÂformations non lineÂaires avant la rupture;

Addenbrooke, T. I., Potts, D. M. & Puzrin, A. M. (1997). GeÂotechnique 47, No. 3, 693±712

693

KEYWORDS: constitutive relations; ground move-ments; numerical modelling and analysis; settlement;tunnels.

(b) qu'un module de cisaillement anisotropetendre meÁne aÁ de meilleures preÂvisions dans lecas de nouveaux chantiers, mais pas celui detunnels doubles; et (c) que l'historique descontraintes reÂcentes compte peu dans l'analysede ce probleÁme (tunnels doubles de St James'sPark).

INTRODUCTION

It became evident during the 1970s that the stiff-ness of London Clay determined in the laboratorywas considerably lower than that determined fromback analysis of ®eld measurements (Burland,1989). Fig. 1 demonstrates this by plotting thevariation of undrained Young's modulus with meaneffective stress derived from a number of sitesaround London. Also plotted are the ranges ofresults from the careful testing of specimens fromblock samples of London Clay retrieved at AshfordCommon (Bishop et al., 1965). The values ofback-analysed stiffness are signi®cantly greater

than the laboratory stiffness. The stiffness deter-mined from deep excavations (5, 6 and 7 in Fig.1), with dominant horizontal stress changes, ishigher that that from vertical loading situations (1,2, 3 and 4 in Fig. 1), possibly indicating anisotropyof stiffness. At the end of the 1970s, local meas-urement of strains on samples during triaxial testsrevealed much larger stiffness at small strains thanpreviously determined with strain measurementacross the end platens. This explains the under-prediction of laboratory-determined stiffness seenin Fig. 1. Seismic and ultrasonic tests have recentlyshown the anisotropy of high stiffness at smallstrains (e.g. see Porovic, 1995). In 1993, Gunnreported the ®nite-element developments associatedwith the tunnelling research programme at Cam-bridge University in the late 1970s and early1980s. The ®nite-element calculations predictedmuch shallower and wider distributions of surfacesettlement than was seen in model tests or realtunnelling. The reason was quickly identi®ed asbeing the elastic part of any constitutive modelused to represent the stress±strain behaviour ofsoil. Gunn (1993) showed the improvement in pre-dicted surface settlement gained with non-linearelastic, rather than linear elastic, behaviour pre-failure.

Lee & Rowe (1989) carried out linear elasticperfectly plastic ®nite element analyses of tunnel-ling speci®cally to show the in¯uence of cross-anisotropic parameters pre-failure. They concludedthat particular attention should be given to the ratioof independent shear modulus to vertical modulus.Their study modelled a soil with an initial horizon-tal to vertical effective stress ratio (K0) , 1. Gunn(1993) in referring to Lee & Rowe (1989) statedthat, in his experience, the predictions of grounddeformations above tunnels in heavily overconsoli-dated London Clay were not improved by the useof appropriate anisotropic elastic moduli. Simpsonet al. (1996) concluded from analyses of excava-tion in London Clay that the predicted surfacesettlement trough is substantially in¯uenced by theanisotropic shear modulus, but little in¯uenced bynon-linearity. These ®ndings appear contradictory.

As numerical analysis of tunnelling has becomemore widely used, attempts have been made tomodel the characteristics of the pre-failure defor-

Manuscript received 31 October 1996; revised manuscriptaccepted 21 March 1997.Discussion on this paper closes 1 Septemebr 1997.� Imperial College of Science, Technology and Medicine,London.{ Technion ± Israel Institute of Technology, Haifa.

0 50 100 150Eu: MPa

0.1

0.2

0.3

0.4

0.5

Mea

n ef

fect

ive

stre

ss: M

Pa

5

6

7

153

42

Range from AshfordCommon triaxial tests

Fig. 1. Variation of undrained Young's modulus withmean normal effective stress derived from varioussites on London Clay: 1, Chelsea; 2, Hendon; 3, HydePark; 4, Commercial Union; 5, Brittanic House (twolines); 6, New Palace Yard; 7, YMCA. (After Burland,1989)

694 ADDENBROOKE, POTTS AND PUZRIN

mation behaviour of clays interpreted through backanalyses of ®eld data, and observed in the labora-tory. The analyses in this paper consider aspects ofall the models discussed above, by consideringisotropic and anisotropic, linear and non-linear pre-failure deformation behaviour, and also introducesthe use of a new non-linear elastic model whichaccounts for loading reversals (Puzrin & Burland,1997). The Imperial College Finite Element Pro-gramme (ICFEP) has been used to carry out all theanalyses presented. The tunnel analyses model thetwin tunnels of the Jubilee Line Extension at StJames's Park (London, UK), and ®eld data arepresented for comparison (Standing et al., 1996).

CONSTITUTIVE MODELS USED TO DEFINE PRE-

FAILURE DEFORMATION CHARACTERISTICS

A linear elastic model, if isotropic, requires thespeci®cation of two elastic constants. If the soil iscross-anisotropic, then ®ve parameters must bede®ned: Ev9, the vertical Young's modulus; Eh9,the horizontal Young's modulus; ì9vh, the Poisson'sratio for the in¯uence of increments of verticaleffective stress on horizontal strain; ì9hh, the Pois-son's ratio for the in¯uence of increments ofhorizontal effective stress on the horizontal strainin the orthogonal horizontal direction; and Gvh, theshear modulus in the vertical planes (the indepen-dent shear modulus). The ratios n9 � Ev9=Eh9 andm9 � Gvh=Ev9 are often quoted.

One of the models coded in ICFEP whichreproduces non-linear behaviour at small strains isoutlined by Jardine et al. (1986). This model iscalled model J4 for consistency with Puzrin &Burland (1996). The expressions that describe the

variation of shear and bulk moduli in the non-linear region are given in Appendix 1, as are theparameters used in model J4 throughout this paper.

Model J4 has been adapted to model non-linearanisotropy. Model J4 de®nes the variation of shearand bulk moduli G and K, which give a modulusE9 and a Poisson's ratio ì9. The anisotropic elasticstiffness matrix is obtained from constant ratios ofìvh9=ì9, ìhh9=ì9 Eh9=E9, n9 and m9. Thus theanisotropic moduli reduce by means of the trigono-metric functions de®ned in Appendix 1. This isreferred to as model AJ4.

Puzrin & Burland (1997) have proposed an al-ternative small strain model which has been codedin ICFEP. The stress space surrounding a stresspoint, or local origin, is divided into three regions(Fig. 2). The ®rst region is a linear elastic region(LER); in the second region, the small strainregion (SSR), the deviatoric and volumetric stress±strain behaviour is de®ned by a logarithmic curve(Puzrin & Burland, 1996). Each of these regions isbounded by an elliptical boundary surface. In the®nal region, between the SSR boundary and theplastic yield surface (which is de®ned independentof the pre-failure model), the soil behaviour isagain linear elastic, G and K being associated withthe SSR boundary. The model, called model L4by Puzrin & Burland (1996), is de®ned in Appen-dix 2.

Figure 3 shows the variation of secant Young'smodulus (Eu � 3G) normalized by mean effectivestress p9, when plotted against percentage axialstrain from triaxial test simulations. Both modelsJ4 and L4 compare extremely well with laboratorydata for London Clay undergoing undrained triaxialextension (CAU, `consolidated anisotropically un-

Bounding surfacefrom

plastic model(Mohr–Coulomb)

Jglobal

P ′global

J

P ′Local origin

LER

SSR

Fig. 2. Three stress regions surrounding a local origin in J± p9 space for modelL4 (tension positive)

NUMERICAL ANALYSIS OF TUNNEL CONSTRUCTION 695

drained test', and CIU, `consolidated isotropicallyundrained test'). Model L4 models a more rapidinitial reduction in stiffness in the small strainregion, and is softer than model J4 up to 0´01%axial strain; above this strain level, model J4 issofter than model L4.

There are a number of key differences betweenthe three non-linear pre-failure models, J4, L4 andAJ4.

· Models J4 and L4 are isotropic, while modelAJ4 is anisotropic.

· Models J4 and AJ4 use a trigonometric functionto de®ne the stress±strain behaviour in the non-linear region, while model L4 uses a logarithmicfunction.

· Models J4 and AJ4 allow only shear stiffnessreduction on shear straining with no volumetricstrain, and bulk stiffness reduction on volumetricstraining with no shear strain, while model L4forces shear and bulk moduli reduction togetherif there is only shear or only volumetric strain.

· Model L4 rede®nes the local origin of stress at aload reversal point, reinvoking the initial highelastic stiffness behaviour. Puzrin & Burland(1997) put forward a criterion for checkingwhether a change in direction of stress path isassociated with a load reversal in a soil element(see Appendix 2). Although acknowledged as asimpli®ed approach to modelling recent stresshistory effects, no such comparable feature existswithin model J4 or AJ4.

ANALYSIS DETAILS

Each of the analyses presented in this papermodelled excavation of the two diagonally orientedrunning tunnels of the Jubilee Line Extensionbeneath St James's Park. The geometry is presentedin Fig. 4, showing the soil pro®le, and the ®nite-element mesh in Fig. 5. The westbound tunnel wasexcavated ®rst: the ground response is not thereforein¯uenced by any existing surface or subsurface

0.0001 0.001 0.01 0.1 1

Axial strain: %

0

200

400

600

800

1000

1200

14003G

/p′ 5

Eu/

p′

CIU extension

CAU extension

Model L4

Model J4

Fig. 3. Secant stiffness±strain curves: comparison between ICFEP models and test data

4.5m

2.7m

34.3m

30.5m

21.5m

20m

Eastbound

Westbound

Sand

Thames Gravelc ′ 5 0 kPaφ′ 5 358

London Clayc ′ 5 5 kPaφ′ 5 258

Woolwich andReading BedClayc ′ 5 200 kPaφ′ 5 278

Tunnel diameter 5 4.75 m

Fig. 4. Diagonally oriented tunnels (St James's Park,London, UK)


structures, and is thus termed `green®eld'. Thesettlement above such an excavation is the `green-®eld settlement'. The eastbound tunnel is not agreen®eld excavation because of the existing west-bound tunnel.

A coupled consolidation formulation was em-ployed. Excavation of the tunnels over a simulatedtime period of 8 h was separated by a rest periodof 8´5 months. The sand and gravel were modelledas drained non-consolidating materials. The Lon-don Clay was attributed with an anisotropic per-

meability decreasing with depth. The reduction inhorizontal permeability was based on falling headtests carried out near this site at the Palace ofWestminster, reported by Burland & Hancock(1977) and shown in Fig. 6(a). The Palace ofWestminster site is 300 m from the St James's Parksite. The ratio of vertical to horizontal permeabilitywas taken as 1=10 at the top of the London Clay,increasing to 1=2 at the bottom. This pro®le wasadopted as it gave an excellent prediction of themeasured pore water pressure pro®le, as demon-

Fig. 5. Finite-element mesh

Fig. 6. (a) Permeability pro®le: (b) underdrained seepage pressure pro®le, Westminster, London, UK. (ICFEP and®eld data, Westminster, Higgins et al. (1996); St James's Park, Nyren (1995))

Horizontal permeability

Vertical permeability

Falling head data(Burland & Hancock, 1977)

Top of London Clay

Bottom of London Clay

Top of Woolwich and ReadingClay

1 3 10211 1 3 10210 1 3 1029 1 3 1028

Permeability: m/s(log scale)

(a)

240

230

220

210

0

Dep

th b

elow

wat

er ta

ble:

m

St James's Park field data

Westminster field data

ICFEP

Top of London Clay

HydrostaticBottom of London Clay

Top of Woolwich and Reading Clay

0 100 200 300

Pore pressure: kPa

(b)

240

230

220

210

0

Dep

th b

elow

wat

er ta

ble:

m


strated below. The Woolwich and Reading Claywas attributed with a constant anisotropic per-meability.

The initial stresses prescribed an underdrainedpore water pressure pro®le from a water tablelocated 4´5 m below the ground surface (hydrosta-tic in the Thames Gravel). Fig. 6(b) shows theagreement between the prescribed underdrainedpore water pressure pro®le and ®eld data from StJames's Park and Westminster. This prescribedpro®le is consistent with the anisotropic permeabil-ity de®ned above. An initial effective stress ratioK0 of 0´5 was prescribed in the sand and gravel,and 1´5 in the London Clay and Woolwich andReading Clay. In the absence of site measurementsof K0 the value adopted in the London Clay isfrom the `upper-bound' pro®le given by Hight &Higgins (1995), for London Clay between 10 and30 m below the ground surface. Their pro®le isbased on suction measurements in samples ofLondon Clay using the ®lter paper technique.

The displacement boundary conditions permittedno horizontal displacement along the two verticalmesh boundaries, and no displacement along thebottom horizontal boundary. Hydraulic boundaryconditions dictated that the pore water pressures inthe Thames Gravel remained hydrostatic. The ini-tial pore water pressures along the remote bound-aries to the clay strata were maintained throughoutthe analysis. The tunnels were treated as drainsusing a special boundary condition which preventswater being drawn across the boundary at nodeswhere suction exists in the adjacent soil, butpermits free ¯ow at nodes where the soil waterpressures are compressive. The excavation bound-ary condition simulates the removal of materialfrom the ®nite-element mesh; the stresses acting atthe excavation boundary as a result of the soilelements removed are divided by the number ofincrements of excavation. These linearly dividedstresses are applied in the reverse direction, at thatboundary, on each of the increments of excavation.Incremental excavation permits the monitoring ofthe volume of the developing surface settlementpro®le, this being a value frequently measured onsite. Volume loss is de®ned as the volume of soilexcavated in excess of the theoretical volume ofexcavation (per unit advance of tunnel), and isquoted as a percentage of the theoretical volume.Rapid excavation in a low-permeability clay resultsin no volume change, so the volume of the surfacesettlement pro®le is equal to the volume loss. Thetunnel lining (modelled as a continuous concretering: parameters listed in Appendix 5) is con-structed once the desired volume loss has beenobtained. A volume loss of 3´3% was prescribedfor the ®rst tunnel, and 2´9% for the second(calculated from the settlement data at St James'sPark; Standing et al., 1996).

SOIL PARAMETERS FOR THE ANALYSES

Made ground was isotropic linear elastic, E9 �3000 kPa and ì9 � 0:2. The Thames Gravel, Lon-don Clay and Woolwich and Reading Clay wereelastic perfectly plastic with Mohr±Coulomb yieldsurfaces and plastic potentials (de®nition and para-meters in Appendix 3). The Thames Gravel andthe Woolwich and Reading Clay were isotropiclinear elastic when the London Clay was linear(parameters in Appendix 4), and isotropic non-linear elastic using model J4 when the LondonClay was non-linear (parameters in Appendix 1).

Pre-failure, the London Clay was linear isotropicor anisotropic, or alternatively non-linear isotropic(model J4 or L4), or anisotropic (model AJ4). Thelinear elastic parameters modelled stiffness increas-ing with depth, using parameters based on theback-analysed data of Fig. 1 (see Appendix 4). Theisotropic non-linear elastic parameters were basedon curve ®tting to stiffness strain data from locallyinstrumented triaxial samples (see Appendix 1 formodel J4 and Appendix 2 for model L4). Twonon-linear anisotropic analyses are presented in thispaper, one de®nes n9 � 0:625 and m9 � 0:444(model AJ4i), the second reduces m9 to 0´2, mak-ing the clay very soft in shear (model AJ4ii).Published laboratory data do not give a relationshipfor the decay of Gvh with strain. Based on labora-tory and ®eld data for London Clay the initial Gvh

has been determined by Simpson et al. (1996) tobe 0´65 Ghh (Ghh, the shear modulus in a horizon-tal plane, is dependent on Eh9). With ì9 � 0:2 andn9 � 0:625, this gives m9 � 0:433. This is verysimilar to the value adopted in this paper, m9 �0:444 (assumed by Burland & Kalra, 1986). Lee &Rowe (1989) found an m9 value of between 0´2and 0´25 gave a good surface settlement matchwith centrifuge and ®eld data (data not for LondonClay). The value of m9 for model AJ4i is thereforesupported by laboratory and ®eld measurementsreported by others. The value for model AJ4ii,however, is adopted from the successful work ofLee & Rowe (1989), which is not justi®ed forLondon Clay by test data.

Model J4 does not automatically reinvoke highstiffness behaviour on loading reversal. For twoanalyses (isotropic and soft anisotropic), high stiff-ness behaviour was reinvoked across the entiremesh by zeroing the strains before starting thesecond excavation. The isotropic analysis is modelJ40, and the soft anisotropic analysis is modelAJ4ii0. This is an approach suggested by Jardineet al. (1991) for simulating the kinematic nature ofsoil stiffness in response to load reversals. Unlikemodel L4 which detects loading reversals and actsaccordingly, model J40 reinvokes high stiffnessbehaviour at the beginning of a user-de®ned incre-ment in all the elements in the analysis regardlessof the actual location of any reversals.


TRIAXIAL TESTS TO COMPARE THE DIFFERENT

MODELS

A single axisymmetric element was used tomodel ideal undrained extension triaxial tests onLondon Clay. The initial stresses represented theend of isotropic consolidation to 750 kPa meanstress. The tests were strain controlled by the incre-mental application of a prescribed displacement atthe top. Each non-linear model was used to obtaina comparison of the predicted pre-failure beha-viour. The soil was loaded to failure (Mohr±Cou-lomb yield surface). Fig. 7 shows the stress pathsfollowed, plotted as q against p9, where q is thedeviator stress and p9 the mean stress. Within theyield surface both the isotropic models follow avertical stress path, and the anisotropic model anegative gradient stress path with increasing p9accompanying the reduction in q.

The stress±strain behaviour and the developmentof excess pore water pressure with axial strain arecompared in Fig. 8. Shear stress is plotted as thechange in q. Fig. 8(a) shows the region up to0´01% axial strain, and Fig. 8(b) up to 0´4% (0´4%axial strain is the limit of non-linear behaviour).The laboratory data for the equivalent test is alsoplotted. In the very small strain region, the threenon-linear models show very similar stress±strainbehaviour, and agreement with the laboratory datais excellent (Fig. 8(a)). At larger strains the stress±strain curve for model L4 lies above the others(higher stiffness over the larger strain range shownin Fig. 4), and all the models plot slightly abovethe laboratory data (Fig. 8(b)). The anisotropic

model develops higher excess pore water pressuresthan the isotropic models.

These analyses show the similarity in thestress±strain behaviour for monotonic loading, andthe expected difference in pore water pressuredevelopment between isotropic and anisotropicmodels.

RESULTS FROM TUNNEL ANALYSES

The surface settlement pro®les are discussed®rst, drawing comparisons between the differentmodels and the ®eld data. Following this, the nu-merical predictions of subsurface ground move-ments from the different models are compared andcontrasted.

Surface settlementsThe ®rst series of results presented consider the

green®eld surface settlement pro®les above thewestbound tunnel.

Table 1 shows the consistency of the controlledvolume loss with the non-linear soil models, ran-ging from 3´2 to 3´3%. It is evident, however, thatsigni®cantly different degrees of unloading wererequired to achieve this volume loss for the differentmodels. Percentage unloading is obtained from theratio of the number of increments of excavation toachieve the desired volume loss to the total numberof increments of excavation. After 100% unloadingthe two linear pre-failure models failed to achievethe volume loss measured on site. The surface

300 600 900 1200 1500 1800p ′: kPa

0

200

400

2200

2400

2600

2800

21000

21200

q: k

Pa

Isotropic

Anisotropic

Fig. 7. Stress paths. Isotropically consolidated ( p09 � 750 kPa) extension test, comparison between isotropicand anisotropic models


0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Axial strain: %

(a)

10

20

30

40

50

q: k

Pa

u: k

Pa

CIU test

Model L4—q

Model J4—q

Model AJ4i—q

L4—u

J4—u

AJ4i—u

CIU test

Model L4—q

Model J4—q

Model AJ4i—q

L4—u

J4—u

AJ4i—u

0 0.1 0.2 0.3 0.4

Axial strain: %

(b)

100

200

300

400

500

q: k

Pa

u: k

Pa

Fig. 8. Stress±strain and excess water pore pressure±strain curves: comparison between ICFEP modelsand laboratory test data points. Isotropically consolidated ( p09 � 750 kPa) extension test


settlement pro®les are presented in Fig. 9(a) for theisotropic soil models. The linear pre-failure modelis shown for 100% unloading, and is clearly toowide and shallow, and of the wrong shape. The non-linear analyses predict deeper, narrower pro®les.There is little distinction between models J4 andL4. Fig. 9(b) presents the same comparison for theanisotropic soil models. The non-linear pre-failuremodel predicts deeper, narrower pro®les than thelinear model. The softer the independent shearmodulus (model AJ4ii), the deeper and narrowerstill the pro®le. It is clear that non-linear behaviourpre-failure is required to achieve deeper surfacesettlement pro®les closer to the ®eld measurements,

Table 1. Percentage unloading of excavation beforelining was constructed to achieve the measured volumeloss for the westbound excavation of 3´3%

Analysis(London Clay model)

Percentageunloading:

%

Volumeloss:

%

Isotropic linear elastic 100 1´89Anisotropic linear elastic 100 2´37Non-linear model J4 75 3´2Non-linear model L4 80 3´3Non-linear model AJ4i

(n9 � 0:625, m9 � 0:444)76 3´2

Non-linear model AJ4ii(n9 � 0:625, m9 � 0:2)

57´5 3´2

Fig. 9. Surface settlement pro®lesÐ®rst tunnel: (a) isotropic models; (b) anisotropicmodels. (Models, ICFEP data; ®eld data, Standing et al. (1996))

Linear elasticperfectly plastic

Model L4

Model J4

Field data

CL

0 10 20 30 40 50 60210

Offset from westbound tunnel axis: m

0

25

210

215

220

225

230

5

Ver

tical

dis

plac

emen

t: m

m(1

heav

e/2

settl

emen

t)

(a)

Linear elasticperfectly plastic

Model AJ4i

Model AJ4ii

Field data

CL

0 10 20 30 40 50 60210


230

225

220

215

210

25

0

5

Ver

tical

dis

plac

emen

t: m

m(1

heav

e/2

settl

emen

t)

(b)


and that the predicted shape and magnitude is im-proved with a soft anisotropic shear modulus (themaximum predicted settlement is compared withthe ®eld data in Table 2).

Having stiffness varying with strain level con-centrates the strains at the unloading boundary, soincreasing the movements close to the tunnel andreducing them further away to the sides. Adoptinga stiffness-strain relationship in model L4 withdouble the value of shear modulus in the LER (theinitial shear modulus, G0) may be thought to im-prove this still further. Fig. 10 shows that thestiffness±strain curve adopted still lies within thebounds indicated by the laboratory data for axialstrains greater than 0´001% (triaxial test). Carryingout the green®eld tunnel excavation with this stiffermodel, however, made no change to the distribu-

tion of strain level across the mesh, or to thesurface settlement pro®le at the desired volumeloss.

The linear pre-failure models are not consideredfurther as they do not accurately predict the strainsat the tunnel boundary in response to excavation.

Figure 11 plots the settlement as a result solelyof excavation of the second tunnel (eastboundtunnel), and Table 2 gives the magnitude of maxi-mum predicted settlement. At this stage two newanalyses are introduced, models J40 and AJ4ii0,for which high stiffness behaviour was reinvokedacross the entire mesh before the second excava-tion (zeroed strain analyses). Table 3 shows thepercentage unloading required to achieve the de-

CIU extension

CAU extension

Model L4

Model J4

Model L4—stiffer G0

0.0001 0.001 0.01 0.1 1

Axial strain: %

0

500

1000

1500

2000

2500

3G/p

′ 5 E

u/p

′

Fig. 10. Secant stiffness±strain curves: comparison between ICFEP models and test data

Table 2. Maximum surface settlement as a result ofexcavation of each tunnel

Pre-failure model Smax

westbound:mm

Smax

eastbound:mm

J4 11´1 10´8J40 ± 14´3L4 10´5 12´8

AJ4i 12´5 11´5AJ4ii 16´6 14´6AJ4ii0 ± 19´9

Field data 19´9 23´5(Standing et al., 1996)

Table 3. Percentage unloading of excavation beforelining was constructed to achieve the measured volumeloss for the eastbound excavation of 2´9%

Analysis(London Clay model)

Unloading:%

Volumeloss:%

Non-linear model J4 65 2´99Non-linear model J40 75 2´83Non-linear model L4 75 2´86Non-linear model AJ4i

(n9 � 0:625, m9 � 0:444)60´5 3´02

Non-linear model AJ4ii(n9 � 0:625, m9 � 0:2)

45 3´08

Non-linear model AJ4ii0(n9 � 0:625, m9 � 0:2)

57´5 2´94


sired volume loss, which is held in the range 2´8±3´1%. Fig. 11(a) shows the pro®les from the fournon-zeroed strain non-linear analyses. It is clearthat in each case the predicted surface settlementpro®le is far wider and shallower than the ®elddata. The similarity of prediction by models J4 andAJ4i implies that employing anisotropic parametersappropriate to London Clay is not a signi®cantfactor. The position of maximum surface settlementis pulled towards the ®rst tunnel in the cases ofmodels J4, L4 and AJ4i, but model AJ4ii predictsthe centre line settlement as the maximum. The®eld data exhibit a slight asymmetry, the pro®le

being wider on the side of the existing tunnel thanon the opposite side. This feature is displayed bythe numerical analyses. Model L4 predicts a dee-per, narrower settlement pro®le than model J4, thetwo models having predicted very similar pro®lesfor the green®eld tunnel (Fig. 9(a)). The differencehere must be attributed to the detection of loadreversals in some elements by model L4 and theconsequent reinvoking of high stiffness behaviour.Model AJ4ii still predicts the deepest, narrowestpro®le, but the difference between this model andthe others is less than in the green®eld instance(see Table 2).

Model L4

Model J4

Model AJ4i

Model AJ4iiField data

CLCL

1st tunnel2nd tunnel

0 10 20 30 40 50 60210


0

25

210

215

220

225

230

5

Ver

tical

dis

plac

emen

t: m

m(1

heav

e/2

settl

emen

t)

(a)

Model J4

Model J40 —high stiffness reinvoked

Model AJ4ii

Model AJ4ii0 —high stiffness reinvokedField data

CL

CL

1st tunnel2nd tunnel

0 10 20 30 40 50 60210


(b)

0

25

210

215

220

225

230

5

Ver

tical

dis

plac

emen

t: m

m(1

heav

e/2

settl

emen

t)

Fig. 11. Surface settlement pro®lesÐsecond tunnel: (a) non-linear models; (b) non-linearmodels with and without zeroed strains. (Models, ICFEP data; ®eld data, Standing et al.(1996))


Figure 11(b) shows the settlement predicted bythe two zeroed strain analyses compared with theirrespective non-zeroed equivalents. In both cases(comparison of model J4 with J40, and of modelAJ4ii with AJ4ii0) the magnitude of the predictedsettlement is greater, and the trough width nar-rower. The zeroed strain analyses do, however, losethe asymmetry of the predicted surface settlementpro®le.

Subsurface ground movements (®eld dataunpublished)

The surface settlement pro®les show differentresults for different pre-failure models for the be-haviour of the stiff clay being tunnelled through.

The plots of ground displacement towards thegreen®eld tunnel excavation, vertical down the axisline, and horizontal along the axis level (Figs 12(a)and 12(b), respectively) show that at the tunnel

20

40

60

Hor

izon

tal d

ispl

acem

ent:

mm

0 15 30 6045


Model L4

Model J4

Model AJ4i

Model AJ4ii

(b)

Model L4

Model J4

Model AJ4i

Model AJ4ii

0 20 40 60

Vertical displacement: mm

(a)

30

15

0

Dep

th b

elow

gro

und

surf

ace:

m

Fig. 12. (a) Vertical settlement down axis lineÐ®rst tunnel; (b) horizontal displacement along axis levelÐ®rst tunnel. (Non-linear models, ICFEP data)


boundary the vertical displacement at the crown isgreatest for model L4, reducing for model J4, thenagain for model AJ4i, and the least crown settle-ment is for model AJ4ii. The relative magnitudesof the horizontal displacement at the springline arein the opposite order (maintaining constant volumeloss). From the surface to 1 diameter above thecrown the vertical displacement down the axis lineis greatest with model AJ4ii, less for models AJ4iand J4 and least for model L4. There is lessconcentration of displacement within 1 diameter ofthe crown with model AJ4ii than the other models.The horizontal displacements for models J4, L4and AJ4i are almost indistinguishable from eachother. With model AJ4ii the horizontal displace-ment is increased at all points away from thetunnel. The deeper, narrower green®eld surface set-tlement pro®le is therefore achieved with a softindependent shear modulus by altering the move-ment at the tunnel boundary itself to give a moreuniform pattern of displacement (approximately45 mm inward displacement at the crown and thespringline, compared with nearly 60 mm inwarddisplacement at the crown and 35 mm at thespringline with model L4).

Figure 13 shows the development of volume losswith unloading into the second tunnel for modelsJ4 and L4, and plots the equivalent percentageunloading for a green®eld tunnel excavation at thesame depth. With model J4 the equivalent volumeloss is achieved at a lesser percentage unloadingwhen the tunnel is the second of a pair. This shows

the ground in the vicinity of the second tunnel tobe softened by excavation of the ®rst. In contrast,model L4 develops volume loss similarly in bothgreen®eld and twin-tunnel instances. The soilaround the second tunnel, although softened byexcavation of the ®rst tunnel, is clearly experien-cing load reversals on commencement of the sec-ond excavation. Interrogation of the analysis re-vealed that soil above and below the second tunnel,and soil directly between the ®rst and secondtunnel, experienced load reversals (as de®ned inmodel L4 using normalized stresses, see Appendix2). The evidence from the analyses is that with thesame tunnelling technique and standard of work-manship, a greater or equal volume loss is ex-pected in the second tunnel. The ®eld data, withless volume loss into the second tunnel than the®rst, implies some difference other than soil stiff-ness for west- and eastbound (for example, thestandard of workmanship, or construction sequenceat the tunnel face).

The ground movements into the second tunnelare shown in Fig. 14. Both vertical and horizontaldisplacement patterns are very similar for modelsJ4 and AJ4i, supporting the similarity in theirsurface settlement predictions. Model L4 (similarto model J4 in the green®eld situation) predictsgreater vertical and reduced horizontal displace-ments. Model AJ4ii, which still gives the leastcrown settlement, gives more settlement nearer theground surface. The asymmetry of the ground be-haviour, re¯ected in the surface settlement pro®les

40 50 60 70 80 90 100

Unloading: %

0

1

2

3

4

5

Vol

ume

loss

: %

J4 second

L4 second

J4 greenfield

L4 greenfield

Fig. 13. Development of volume loss with percentage unloading for the non-linear models J4 and L4,comparing the green®eld result with the second tunnel result


of Fig. 11(a), is clear in Fig. 14(b), which showsgreater horizontal displacement into the springlineon the side of the existing tunnel than on theopposite side for all the models. The differencebetween horizontal displacement predicted by mod-el AJ4ii and the other models is greater into the

second tunnel than into the ®rst (cf. Fig. 11(b)),while the difference in predicted vertical displace-ment above the crown with the different models isless into the second tunnel (cf. Fig. 11(a)), asre¯ected at the surface.

The zeroed strain analyses, which improved the

Fig. 14. (a) Vertical settlement down axis lineÐsecond tunnel; (b) horizontal displacement along axislevelÐsecond tunnel. (Non-linear models, ICFEP data)

Model L4

Model J4

Model AJ4i

Model AJ4ii

0 20 40 60


(a)

30

15

0

Dep

th b

elow

gro

und

surf

ace:

m

CL

Model L4

Model J4

Model AJ4i

Model AJ4ii

0 15 30 6045

20

40

60

Hor

izon

tal d

ispl

acem

ent:

mm

1st tunnel


(b)

CL

2nd tunnel


magnitude of the surface settlement prediction,show increased vertical displacements down theaxis line when compared with their non-zeroedcounterparts (Fig. 15(a)). The crown settlement isalso increased. The increased crown settlement iscountered by a reduction in horizontal movement

at the springline on the side of the existing tunnel(Fig. 15(b)). The horizontal displacements on theopposite side are comparatively similar with andwithout zeroed strains.

There are two contrasting patterns of subsurfaceground movement displayed here, both of which

Fig. 15. (a) Vertical settlement down axis lineÐsecond tunnel; (b) horizontal displacement along axislevelÐsecond tunnel. (Non-linear models with and without zeroed strains, ICFEP data)

Model J4

Model J40

Model AJ4ii

Model AJ4ii0

0 20 40 60


(a)

30

15

0

Dep

th b

elow

gro

und

surf

ace:

m

Model J4

Model J40

Model AJ4ii

Model AJ4ii0

0 15 30 45 60


20

40

60

Hor

izon

tal d

ispl

acem

ent:

mm

(b)


produce deeper, narrower surface settlement pro-®les, but which are achieved through differentvariations in the pre-failure deformation model.The green®eld surface settlement pro®le is im-proved by the use of non-linear stress±strain beha-viour at small strains. The settlement pro®le abovethe second adjacent excavation is similarly im-proved by the reintroduction of non-linear stress±strain behaviour at small strains through the auto-matic detection of load reversals or through themanual zeroing of the strains across the entiremesh. The in¯uence of such behaviour on theground movements adjacent to the tunnel is toincrease the crown settlement and reduce thespringline displacement. The settlement pro®lescan be further deepened and narrowed, in accor-dance with the ®eld data, by the use of anisotropicparameters within the non-linear model of stress±strain behaviour at small strains. This has beendemonstrated most clearly by the use of a very softindependent shear modulus (model AJ4ii) in thispaper. The in¯uence of softening the independentshear modulus within a non-linear anisotropic mod-el is, in contrast to the in¯uence of non-linearityitself, to reduce the crown settlement and increasethe springline displacement, producing a more uni-form pattern of deformation at the tunnel.

CONCLUSIONS

This paper has looked at the in¯uence of themodel adopted for pre-failure soil stiffness on thenumerical analysis of tunnel construction. The spe-ci®c geometry adopted was that of the Jubilee LineExtension tunnels beneath St James's Park in Lon-don, UK. The analyses were all plane strain, as theamount of computer resource required for sophisti-cated three-dimensional analysis is prohibitivelylarge. The in¯uence of three-dimensional effectson numerical predictions of settlement is beyondthe scope of this paper, but may contribute to thediscrepancy between measured ®eld settlementsand the predictions presented.

Green®eld tunnel excavationNon-linear pre-failure deformation models are

necessary to achieve volume loss of the magnitudemeasured at St James's Park. Isotropic and aniso-tropic linear pre-failure deformation models fail topredict suf®cient straining at the tunnel boundary.The non-linear models achieved the measured vo-lume loss at different increments of unloading.Models J4, L4 and AJ4i required the lining to beconstructed after 75±80% unloading. Making thesoil softer in shear (model AJ4ii) accelerated thedevelopment of volume loss with percentage un-loading, and the lining was constructed after 57´5%unloading. The effect of this is to reduce the extent

of any zones of yielding soil on completion ofexcavation, so reducing the magnitude of the dropin pore pressure and the increase in mean effectivestress during yielding in a dilatant soil. This in¯u-ences the stiffness of a stress-level-dependent soilmodel, and the degree of swelling or consolidationexpected in the intermediate term. Constructing thelining earlier in the analysis increases the predictedstresses carried in the lining in the short term.

Non-linear pre-failure models produce consider-ably deeper, narrower predictions of surface settle-ment than linear models. Isotropic non-linearmodels, however, still predict shallower and widerpro®les than that measured at St James's Park.Increasing the initial small strain stiffness in thelinear elastic region does not improve the settle-ment prediction at the desired volume loss.

Introducing anisotropic parameters appropriateto London Clay into a non-linear model gives littleimprovement on the isotropic results. A more rea-sonable prediction for settlement above a green®eldtunnel excavation can be achieved by employing avery soft independent shear modulus in the aniso-tropic stiffness matrix. The soft shear moduluscauses increased inward displacement at the spring-line and reduced settlement at the crown, leadingto a more uniform distribution round the tunnelthan the isotropic models, and increased horizontalsubsurface displacement over the full mesh width.Such a soft independent shear modulus cannot,however, be justi®ed from laboratory or ®eld testdata for London Clay.

It is apparent that unrealistic soil stiffness isrequired to achieve an improved prediction of sur-face settlement above a green®eld tunnel excava-tion when modelled in plane strain with K0 . 1. Ifonly surface settlement is of interest, then this is asatisfactory device for obtaining good plane strainpro®les. The usefulness of such an approach forobtaining information with respect to ground andlining stresses, and pore water pressure distribu-tions, and subsurface ground movements must,however, be questioned.

Adjacent tunnel excavation (non-linear pre-failuredeformation models)

If no account is taken of loading reversaleffects, the development of volume loss with per-centage unloading shows the soil to be softeraround the second tunnel than initially around the®rst. This is not the case when loading reversaleffects are accounted for in the fashion of modelL4 (a change in direction of the increment ofnormalized stress reinvoking high stiffness beha-viour), in which case the development of volumeloss is similar in both green®eld and twin tunnels.

All models predict shallower and wider surfacesettlement pro®les than the measured data. The


asymmetry of the measured pro®le is reproduced,but is overexaggerated above the ®rst tunnel. Anisotropic model which captures stress path reversaleffects by reinvoking small-strain high-stiffnessbehaviour (model L4) does improve the settlementprediction above the second excavation. Load re-versals were identi®ed above and below the sec-ond tunnel, and in the region between the two tun-nels. Manually introducing high-stiffness behaviouracross the entire mesh prior to excavation of thesecond tunnel improves the magnitude of the pre-dicted surface settlement by reducing the exagger-ated settlement above the existing tunnel. Thesubsurface displacements show this to be achievedby a reduction in the horizontal axis level move-ments.

The failure of a soft independent shear modulusto predict the settlement pro®le above the secondtunnel, and its lack of justi®cation from test data,leads further to the conclusion that this approach isa successful device for obtaining the short-termsurface settlement above a green®eld excavationalone (if restricted to plane strain analysis). Again,it cannot be justi®ed in general to model soilbehaviour accurately around tunnel excavations inhigh-K0 stiff clay. Modelling load reversal effectsby rede®ning the local stress origin at a point ofnormalized deviatoric or mean stress reversal hasbeen shown to be an in¯uencing factor in theseanalyses (model L4), and can improve predictions.The reinvoking of small-strain high-stiffness beha-viour across the entire mesh prior to excavation ofthe second tunnel proves to be a successful devicefor achieving a more accurate prediction of thesettlement above the second excavation. This ap-proach cannot, however, be justi®ed on the groundsof stress path reversal alone, as the entire soil massdoes not experience such reversals when modelledin plane strain.

ACKNOWLEDGEMENTS

The provision of triaxial test data by the Geo-technical Consulting Group is duly acknowledged.

The data were obtained from Jubilee Line Exten-sion Project site investigations which were used toestablish parameters appropriate to analyses ofWestminster Station.

APPENDIX 1. EQUATIONS DEFINING NON-LINEAR

ELASTIC MODEL J4

3G

p9� C1 � C2 cos á log10

Edp3C3

� �� ã( )

K

p9� C4 � C5 cos ä log10

(åv)

C6

� �ë" #

(1)

where G is the secant shear modulus, K is the secant bulkmodulus, p9 is the mean effective stress, Ed is thedeviatoric strain invariant used in ICFEP, åv is thevolumetric strain, and C1, C2, C3, C4, C5, C6, á, ã, äand ë are all constants. Ed is related to åa (the axial strainobserved in undrained triaxial tests) by the expression

Ed ��3p

åa (2)

where

Ed � 2

��16[(å1 ÿ å2)2 � (å1 ÿ å3)2 � (å2 ÿ å3)2]

q(3)

with å1, å2 and å3 being principal strains.The constants are obtained from a ®t to laboratory data

from stress path tests (Jardine et al., 1986). Throughout ananalysis the stiffness at a particular point is continuallychanging. It depends on both the current strain and thecurrent mean effective stress at that point. Until a speci®edminimum strain Ed min or åv min is exceeded, the stiffnessvaries only with the mean effective stress. This conditionalso applies once a speci®ed upper strain limit is ex-ceeded, Ed max or åv max. Ed min, åv min and Ed max, åv max arerequired `cut-offs' because of the trigonometric nature ofthe equations (see Fig. 3). The magnitude of the stiffnessis prevented from falling below speci®ed minimum values(Gmin or Kmin).

Table 4. Small-strain parameters for model J4

Strata C1 C2 C3: % á ã Ed min: % Ed max: % Gmin: kPa

Thames Gravel 1380´0 1248´0 5´0 3 10ÿ4 0´974 0´940 8´833 46 3 10ÿ4 0´346 41 2000´0London Clay 1120´0 1016´0 1´0 3 10ÿ4 1´335 0´617 8´660 25 3 10ÿ4 0´692 820 2333´3Woolwich and Reading

Clay1000´0 1045´0 5´0 3 10ÿ4 1´334 0´591 13´856 40 3 10ÿ4 0´381 050 2666´7

C4 C5 C6: % ä ë åv min: % åv max: % Kmin: kPa

Thames Gravel 275´0 225´0 2´0 3 10ÿ3 0´998 1´044 2´1 3 10ÿ3 0´20 5000´0London Clay 549´0 506´0 1´0 3 10ÿ3 2´069 0´420 5´0 3 10ÿ3 0´15 3000´0Woolwich and Reading

Clay530´0 460´0 5´0 3 10ÿ4 1´492 0´678 1´5 3 10ÿ3 0´16 5000´0


Anisotropic parameters for model AJ4 (LondonClay only).The parameters in Table A1 de®ne the variation of G andK, which de®ne the isotropic Young's modulus andPoisson's ratio, and the following relations form thecross-anisotropic elastic stiffness, matrix, and these re-lations remain constant throughout the analyses:

ìvh9 � ìhh9 � ìisotropic9 Eh9 � 1:28Eisotropic9

For model AJ4i, n9 � 0:625 and m9 � 0:444; for modelAJ4ii, n9 � 0:625 and m9 � 0:2. Note: with n9 � 0:625,the average of Eh9 and Ev9 is very nearly equal toEisotropic9 (0´5 (Eh9� Ev9) � 1:04Eisotropic9).

APPENDIX 2. EQUATIONS DEFINING NON-LINEAR

ELASTIC MODEL L4Note that p9, J (the mean and deviatoric stress

invariants), åv and Ed (the volumetric and shear straininvariants) are calculated from the local origin (see Fig. 2)

Elliptical boundary to the LER

FLER( p9, J ) � 1� n2 J

p9

� �2

ÿ aLER

p9

� �2

� 0 (4)

where aLER is a parameter de®ning the size of the LER. Itcan be determined from an undrained triaxial test asaLER � nJLER

u, where JLERu is the value of the deviatoric

stress at the LER boundary and n � p(KLER=GLER).Within this boundary, the moduli are de®ned by

KLER � K refLER

( p9ref )â( p9)â GLER � Gref

LER

( p9ref )ã( p9)ã (5)

where KLERref and GLER

ref are values of the bulk andshear elastic moduli, respectively, in the LER at the meaneffective stress p9ref .

Elliptical boundary to the SSR (same shape as LERboundary)

FSSR( p9, J ) � 1� n2 J

p9

� �2

ÿ aSSR

p9

� �2

� 0 (6)

where aSSR is a parameter de®ning the size of the SSR. Itcan be determined from an undrained triaxial test asaSSR � nJSSR

u, where JSSRu is the value of the deviatoric

stress at the SSR boundary, and n is de®ned above.Between the LER and the SSR the elastic moduli obey thefollowing logarithmic reductions

K � KLER 1ÿ á(xv ÿ xe)R[ln (1� xv ÿ xe)](Rÿ1)

(1� xv ÿ xe)

"

� [ln (1� xv ÿ xe)]R

��(7)

G � GLER 1ÿ á(xD ÿ xe)R[ln (1� xD ÿ xe)](Rÿ1)

(1� xD ÿ xe)

"

� [ln (1� xD ÿ xe)]R

��(8)

where

á � xu ÿ 1

(xu ÿ xe)[ln (1� xu ÿ xe)]R(9)

R � (1ÿ xe)

xu ÿ xe)ÿ b

� �(1� xu ÿ xe)[ln (1� xu ÿ xe)]

(xu ÿ 1)

� �� (10)

xu � 2KLER u

aSSR aSSR

xv � jåvjKLER

p9SSR

xD � EdGLER

JSSR

xe � aLER

aSSR

(11)

in which xu is the normalized strain at the SSR boundary,and xe the normalized strain at the LER boundary, and b isthe ratio of tangent modulus on the SSR boundary to thevalue within the LER

b � GSSR

GLER

� KSSR

KLER

(12)

and u is the incremental strain energy, which can bedetermined from an undrained triaxial test, being equal to0:5ESSR

u JSSRu, where ESSR

u and JSSRu are the deviatoric

strain and deviatoric stress on the SSR boundary, respec-tively.

Outside the SSR the moduli are bulk modulus,K � bKLER, and shear modulus, G � bGLER.

With this model, a stress path reversal results in therelocation of the local stress origin at the reversal point, soreinvoking the small-strain high-stiffness behaviour. In thisrespect, a stress path reversal is de®ned if the increment ofnormalized shear or mean stress is less than zero; thecurrent stress being normalized by its linear projection onthe SSR.

APPENDIX 3. MOHR±COULOMB YIELD SURFACE

AND PLASTIC POTENTIAL AND STRESS

INVARIANTSAll the analyses presented in this paper employed the

three-dimensional Mohr±Coulomb failure criterion tode®ne the yield of the stiff clay and the gravel. Thecriterion is speci®ed in terms of the effective stressinvariants p9, J and è

p9 � (ó 91 � ó 92 � ó 93)

3(13)

Table 5. Small-strain parameters for model L4

u=p9 5´851 064 3 10ÿ4

b 0´0643aLER=p9 2´524 468 3 10ÿ3

aSSR=p9 0´191 314 9KLER=p9 214.18GLER=p9 414.33

â 1.0ã 1.0

Kmin 3000.0Gmin 2333.3


J ��16[(ó 91 ÿ ó 92)2 � (ó 92 ÿ ó 93)2 � (ó 93 ÿ ó 91)2]

q(14)

è � ÿ tanÿ1 (2bÿ 1)��3p

� �(15)

where b � (ó 92 ÿ ó 93)=(ó 91 ÿ ó 93), such that in triaxialcompression (b � 0), è � 308, and in triaxial extension(b � 1), è � ÿ308. The yield surface, with a tensionpositive sign convention, is de®ned thus

F(ó ) � J

(ÿp9� a)g(è)ÿ 1 � 0 (16)

where a is the tensile limit of p9 (related to the cohesionc9 through g(è)), and g(è) is a function de®ning the shapeof the yield surface in the deviatoric plane and thegradient of the yield surface in the J, p9 plane

g(è) � sinè

cosè� (1=��3p

)(sinè sinö9)(17)

The plastic potential, G(ó ), de®ning plastic straining, hasthe same equation as the yield function, but with the angleof dilation, í9, de®ning the gradient of the function in theabove equation, rather than ö9.

NOTATION

Stiffness parametersEu undrained Young's modulusE9 drained Young's modulus

E9h drained Young's modulus in thehorizontal direction

E9v drained Young's modulus in the verticaldirection

G shear modulus

G0 initial value of GGvh value of G in the vertical planeGhh value of G in the horizontal plane

K bulk modulusm9 ratio of Gvh to E9v

n9 ratio of Ev9 to Eh9ì9 drained Poisson's ratio

ì9hh drained Poisson's ratio in the horizontalplane

ì9vh drained Poisson's ratio in the verticalplane

Stress parametersJ deviatoric stress invariant

K0 Ratio of initial vertical to horizontaleffective stress

p9 mean normal effective stressq deviatoric stress in triaxial spaceu pore water pressure

ó 91, ó 92, ó 93 principal effective stresses

Table 8. Anisotropic linear elastic parameters, varying with depth below ground surface, z (in metres)

Ev9 Eh9 ìvh9 ìhh9 Gvh n9 m9

London Clay 7500 � 3900z kPa 1´6Ev9 0´125 0´0 0´444Ev9 0´625 0´444

Burland & Kalra (1986).

APPENDIX 4. LINEAR ELASTIC SOIL PARAMETERSTable 7. Isotropic linear elastic parameters, varying with depth below ground surface, z (in metres)

Sand Gravel London Clay Woolwich and Reading Clay

Young's modulus, E9: kPa 5000 6000z 6000z 6000zPoisson's ratio, ì9 0´3 0´2 0´2 0´2

Table 6. Mohr±Coulomb yield surface parameters, plastic potential parameters, and unit weight

Sand Gravel London Clay Woolwich and ReadingClay

Strength parameters Linear elastic c9 � 0 kPaö9 � 35:08

c9 � 5:0 kPaö9 � 25:08

c9 � 200 kPaö9 � 27:08

Angle of dilation Linear elastic í9 � 17:58 í9 � 12:58 í9 � 13:58

Bulk unit weight: kN=m3 ãdry � 18ãsat � 20

ãsat � 20 ãsat � 20 ãsat � 20

APPENDIX 5. TUNNEL LINING PARAMETERSTable 9. Tunnel lining parameters

Young'smodulus

Poisson'sratio

Cross-sectional

area

Secondmoment of

area

28 3 106

kN=m20´15 0´168 m2=m 3´951 36 3

10ÿ4 m4=m

Unit weight � 24 kN=m3.


Strain parametersEd deviatoric strain invariantåv volumetric strainåa axial strain

å1, å2, å3 principal strains

Other parametersc9 cohesive strengthz depth below ground surfaceã unit weightí9 angle of dilationö9 angle of shearing resistance

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