the influence of soil anisotropy and ko on ground surface movements resulting form tunnel excavation

Franzius, J. N., Potts, D. M. & Burland, J. B. (2005). Geotechnique 55, No. 3, 189–199

189

The influence of soil anisotropy and K0 on ground surface movementsresulting from tunnel excavation

J. N. FRANZIUS*, D. M. POTTS† and J. B. BURLAND†

Finite element (FE) analysis is now often used in engi-neering practice to model tunnel-induced ground surfacesettlement. For initial stress regimes with a high coeffi-cient of lateral earth pressure at rest, K0, it has beenshown by several studies that the transverse settlementtrough predicted by two-dimensional (2D) FE analysis istoo wide when compared with field data. It has beensuggested that 3D effects and/or soil anisotropy couldaccount for this discrepancy. This paper presents a suiteof both 2D and 3D FE analyses of tunnel construction inLondon Clay. Both isotropic and anisotropic non-linearelastic pre-yield models are employed, and it is shownthat, even for a high degree of soil anisotropy, thetransverse settlement trough remains too shallow. Bycomparing longitudinal settlement profiles obtained from3D analyses with field data it is demonstrated that thelongitudinal trough extends too far in the longitudinaldirection, and that consequently it is difficult to establishsteady-state settlement conditions behind the tunnel face.Steady-state conditions were achieved only when applyingan unrealistically high degree of anisotropy combinedwith a low-K0 regime, leading to an unrealistically highvolume loss.

KEYWORDS: ground movements; numerical modelling andanalysis; settlement; tunnels

L’analyse d’elements finis (FE) est souvent utilisee apresent dans la pratique industrielle pour modeliser letassement de surface de sol provoque par les tunnels.Pour les regimes de contrainte initiaux, avec un fortcoefficient de pression terrestre laterale au repos K0, il aete montre par diverses etudes que l’auge de tassementtransversal predit par une analyse FE bidimensionnelle(2D) est trop large par rapport a ce qui se passe sur leterrain. Il a ete suggere que les effets 3D et /ou d’aniso-tropie du sol pourraient expliquer cet ecart. Cet exposepresente une suite d’analyses FE en 2D et 3D de laconstruction d’un tunnel dans de l’argile de Londres.Nous employons deux modeles elastiques non lineaires depre-ecoulement, un isotrope et un anisotrope, et nousmontrons que, meme pour un degre eleve d’anisotropiede sol, l’auge de tassement transversal reste trop peuprofonde. En comparant aux donnees de terrain lesprofils de tassement longitudinal obtenus a partir desanalyses 3D, nous demontrons que l’auge longitudinales’etend trop loin dans la direction longitudinale et que,par consequent, il est difficile d’etablir des conditions detassement de regime permanent derriere la face dutunnel. Des conditions de regime permanent ont eteobtenues uniquement en appliquant un degre anormale-ment eleve d’anisotropie combine a un regime de faibleK0, ce qui provoque une perte de volume anormalementelevee.

INTRODUCTIONFinite element (FE) analysis of tunnel construction in softsoil has become widely adopted over recent years. Most ofthe analyses are performed employing a plane strain model.It has been noted by several authors that the surface settle-ment trough in the transverse direction to the tunnel axisobtained from such analyses is too wide when comparedwith field data if the initial stress profile is described by ahigh value of the coefficient of lateral earth pressure at rest,K0.

Addenbrooke et al. (1997) presented a suite of two-dimensional (2D) FE analyses including both linear elasticand non-linear elastic pre-yield models, combined with aMohr–Coulomb yield surface. By modelling the constructionof the Jubilee Line Extension beneath St James’s Park,London, they concluded that for K0 ¼ 1.5 the predictedsurface settlement trough was too wide when soil parametersappropriate for London Clay were included in the soilmodels. Their study also showed that modelling soil aniso-tropy did not significantly improve the results when realisticsoil parameters were adopted. Similar conclusions were

drawn by Gunn (1993). In contrast, Simpson et al. (1996)presented results of a plane strain FE study modelling theHeathrow Express trial tunnel in which they concluded thatsoil anisotropy gives better surface settlement predictions foroverconsolidated clay. They compared results from a linearand a non-linear transversely anisotropic soil model withthose of a non-linear isotropic model. However, only limiteddetails about the applied soil model were given.

Tunnel construction clearly is a three-dimensional pro-blem, and one would expect that 3D FE analysis wouldimprove the surface settlement predictions, compared with2D modelling. Such a conclusion was drawn by Lee & Ng(2002) who compared results of a 3D study in which boththe degree of soil anisotropy and K0 were varied with theresults of Addenbrooke et al. (1997). The surface settlementtroughs from the 3D analyses by Lee & Ng (2002) weremuch more sensitive to changes in the ratio of horizontal tovertical Young’s modulus (defined as n9 ¼ E9h=E9v) thanobserved in the 2D study by Addenbrooke et al. (1997).However, Lee & Ng (2002) adopted a linear elastic perfectlyplastic soil model in contrast to the non-linear elasticperfectly plastic constitutive model adopted by Addenbrookeet al. (1997). Moreover, the tunnel diameter D and thetunnel depth z0 were different in the two studies.

The statement by Lee & Ng (2002) that 3D FE modellingleads to better surface settlement predictions than corre-sponding 2D analyses is in sharp contrast to the findings ofseveral other authors. Guedes & Santos Pereira (2000)presented a suite of FE studies (adopting an elastic soil

Manuscript received 13 June 2004; revised manuscript accepted 29November 2004.Discussion on this paper closes on 3 October 2005, for furtherdetails see p. ii.* Geotechnical Consulting Group, London; formerly ImperialCollege of Science, Technology and Medicine, London.† Imperial College of Science, Technology and Medicine, London.

model) that showed that, for both K0 ¼ 0.5 and 1.5, 3D and2D analyses give similar transverse surface settlementtroughs. Similar conclusions were drawn by Dolezalova(2002), who used both a linear elastic perfectly plastic and anon-linear elastic perfectly plastic constitutive model for thesoil. Vermeer et al. (2002) presented results from linearelastic perfectly plastic analyses that showed that the trans-verse settlement profile obtained from 3D analysis is similarto that from a corresponding 2D study. They also showedthat in the longitudinal direction the tunnel has to beconstructed over a certain length in order to achieve asteady-state settlement condition behind the tunnel face (i.e.settlement caused by the immediate undrained response). Fortheir particular analysis with K0 ¼ 0.66 the tunnel had to beconstructed over 80 m (10D) to achieve a sufficient distancefrom the vertical start boundary. Steady-state settlementdeveloped approximately 5D behind the tunnel face.

This paper investigates the differences between 3D and2D modelling on the prediction of tunnel-induced groundsurface settlement troughs when varying soil anisotropy andK0. The analyses presented here are similar to the planestrain study presented by Addenbrooke et al. (1997), whichcompared results from numerical analysis with field observa-tions from the Jubilee Line Extension at St James’s Park,London. For K0 ¼ 1.5 both isotropic and transversely aniso-tropic non-linear elastic perfectly plastic soil models havebeen adopted. The transverse surface settlement troughsfrom the 3D analyses are compared with corresponding 2Dresults and field data. Finally a K0 ¼ 0.5 initial stress regimeis incorporated in the study to highlight the significance ofthe magnitude of the coefficient of lateral earth pressure atrest on the predicted shape of the ground surface settlementtrough.

DETAILS OF ANALYSISAll FE analyses presented in this paper were carried out

using the Imperial College Finite Element Program (ICFEP).

Reduced integration was used, with an accelerated modifiedNewton–Raphson solution scheme with an error-controlledsubstepping stress point algorithm for solving the non-linearFE equations. All analyses were performed undrained,although the soil models applied are formulated in terms ofdrained parameters. Undrained conditions were enforced byusing a high bulk modulus for the pore water (Potts &Zdravkovic, 1999, 2001).

GeometryFigure 1 shows the 3D finite element mesh for the Jubilee

Line Extension westbound tunnel beneath St James’s Park,London. This was the first of the twin tunnels to beconstructed beneath the St James’s Park greenfield measure-ment site (Standing et al., 1996). The tunnel diameter D was4.75 m and the tunnel depth z0 was approximately 30.5 m.The subsequent construction of the eastbound tunnel was notincluded in the analyses described here.

Modelling of tunnel excavationFor the 3D analyses the tunnel was excavated in the

negative y-direction, starting from y ¼ 0 m (see Fig. 1). Onlyhalf of the problem was modelled, as the geometry issymmetrical. On all vertical sides of the mesh normalhorizontal movements were restrained, whereas for the baseof the mesh movements in all directions were restricted. Themesh shown in Fig. 1 consisted of 10 125 20-node hexadronelements which had 45 239 nodes. For the 2D analyses themesh in the x-z plane was adopted, and the soil wasmodelled by 8-node quadrilateral elements.

In the two-dimensional analyses tunnel excavation wasmodelled using the volume loss method (see Potts &Zdravkovic, 2001). The volume loss VL quantifies theamount of over-excavation and is defined as the ratio of thedifference between the volume of excavated soil and thetunnel volume (defined by the tunnel’s outer diameter)

Longitudinalprofile(x � 0 m)

30·5 m

55 m

Tunnel diameter � 4·75 m100 m

80 m

z

y

x

Transverseprofile(y � �50 m)

Fig. 1. FE mesh for 3D analyses of tunnel excavation beneath St James’s Park greenfield monitoring site

190 FRANZIUS, POTTS AND BURLAND

divided by the tunnel volume. Under undrained conditionsVL can also be obtained by dividing the volume (per runningmetre) of the surface settlement trough by the tunnel volume(per running metre). In London Clay values of VL between1% and 2% have been reported by several authors (Attewell& Farmer, 1974; O’Reilly & New, 1982). However, thevolume loss measured during construction of the JubileeLine Extension beneath St James’s Park was higher. Standinget al. (1996) reported a value of VL ¼ 3.3% for the construc-tion of the westbound tunnel, which is addressed in thispaper.

In the 2D analyses the tunnel was excavated over 15increments. This was done by evaluating the stresses that acton the tunnel boundary within the soil and applying them inthe reverse direction over the 15 increments. Elements with-in the tunnel boundary were not included in the analysesduring this procedure. After each increment the volume losswas calculated and results were taken from that increment inwhich the desired volume loss (i.e. VL � 3.3%) wasachieved.

The step-by-step approach (Katzenbach & Breth, 1981)adopted in the 3D analyses was not volume loss controlled.In this approach, excavation is modelled by successiveremoval of elements in front of the tunnel while successivelyinstalling lining elements behind the tunnel face. The tunnellining was modelled by elastic shell elements (Schroeder,2003) with a Young’s modulus of 28 3 106 kN/m2, a Pois-son’s ratio of 0.15, and a thickness of 0.168 m.

As noted above, in contrast to the 2D analyses there wasno volume loss control in the 3D analyses. Specification of acertain volume loss to be achieved during the tunnel excava-tion would require further assumptions, such as modellingthe tunnelling technique in more detail. In the excavationmodel presented in this paper the magnitude of VL dependson the excavation length Lexc, which is the length over whichsoil elements within the tunnel boundary are removed ineach excavation step. Over this length the soil around thetunnel boundary remains unsupported. Increasing Lexc leadsto higher values of VL. The choice of Lexc has seriousconsequences for the computational resources (storage andtime) needed. For a given tunnel length to be modelled areduction in Lexc not only increases the number of elementswithin the entire mesh but also requires more excavationsteps.

For the analyses presented here the excavation length wasset to Lexc ¼ 2.5 m, and therefore 40 excavation steps wererequired to model the tunnel construction over a length of100 m (21.0D), as shown in Fig. 1. The distance in thelongitudinal direction from the tunnel face to the remotevertical boundary after the last excavation step was 55 m(11.5D). The mesh dimensions and the value of Lexc werechosen after performing a number of analyses in which thesemeasures were varied (see Franzius, 2004). From this studyit was concluded that the dimensions used in this paperprovided the best compromise between mesh size and areasonable computational time (typical analyses took 15 daysto run on a Sun SF880 server). Furthermore, by varying Lexc

between 2.5 m and 2 m Franzius (2004) demonstrated thatsimilar surface settlement and horizontal surface strains wereobtained when normalising results against volume loss. Not-ing that a Lexc ¼ 2.0 m analysis took approximately 50%more calculation time than the 2.5 m analysis, it was decidedthat it was not possible to model a more realistic (i.e.smaller) value of Lexc, and it was therefore not attempted tosimulate the actual tunnelling technique applied during theconstruction of this part of the Jubilee Line Extension. Inpractice the value of Lexc is unlikely to be less than thewidth of the segmental lining (1000 mm), but its actualvalue is likely to be larger, and depends on workmanship, as

an open face excavation procedure was adopted. Nyren(1998) reports a maximum reach of the backhoe in advanceof the tunnel shield cutting edge of 1.9 m.

Initial stress profileThe ground profile consisted of London Clay with a

saturated bulk unit weight of ª ¼ 20 kN/m3. This is asimplification compared with the work of Addenbrooke etal. (1997), who modelled the layers of Thames Gravel andSand overlaying the clay. Comparison plane strain analyseswere performed to assess the influence of this change in soillayering on the surface settlement behaviour. It was foundthat modelling the gravel and the sand led to slightlynarrower transverse settlement troughs.

A constant value of the coefficient of lateral earth pres-sure at rest of K0 ¼ 1.5 was applied during the first analyses.Later, K0 ¼ 0.5 was also considered. The value of K0 ¼ 1.5was given as an upper bound value by Hight & Higgins(1995) for London Clay at a depth between 10 m and 30 mbelow the ground surface. At St James’s Park layers ofThames Gravel and Sand have been deposited on top of theLondon Clay in recent geological time. This has the effectof reducing K0 at the top of the soil profile. Addenbrooke etal. (1997) modelled a lower value of K0 ¼ 0.5 in the graveland the sand. Comparing the different initial stress profilesin a set of plane strain analyses showed only marginalinfluence in the choice of K0 in the top layer. It is the valueof K0 at tunnel depth that has the major influence on thesurface settlements. Consequently the simplification ofadopting a constant value of K0 is unlikely to have a majorinfluence on the results. A hydrostatic pore water pressuredistribution was prescribed, with a water table 2 m below theground surface. Above the water table pore water suctionswere specified. In all analyses the soil was modelled tobehave undrained, by specifying effective stress soil para-meters and a high value of the bulk stiffness of the porewater.

SOIL MODELIsotropic model

The London Clay was represented by a non-linear elasto-plastic model that was also included in the study presentedby Addenbrooke et al. (1997). The model described byJardine et al. (1986) was used to model the non-linearelastic pre-yield behaviour, and the yield surface and theplastic potential were described by a Mohr–Coulomb model.The non-linear elastic model accounts for the reduction ofsoil stiffness with strain. Trigonometric expressions describeG/p9 and K/p9 as a function of shear strain Ed and volu-metric strain �v respectively in the non-linear region. G isthe tangent shear modulus, K is the tangent bulk modulus,and p9 is the mean effective stress. The non-linear region isdefined by maximum and minimum values of shear andvolumetric strain. More details of this model and the soilparameters used are given in Appendix 1.

Anisotropic modelA new constitutive model was implemented into ICFEP to

combine the transversely anisotropic stiffness formulation ofGraham & Houlsby (1983) with the non-linear stiffnessbehaviour described above. It has been shown that transver-sely anisotropic material behaviour is fully described by fiveindependent material parameters (Pickering, 1970): Ev, thevertical Young’s modulus; Eh, the horizontal Young’s mod-ulus; �vh, the Poisson’s ratio for horizontal strain due tovertical strain; �hh, the Poisson’s ratio for horizontal strain

GROUND SURFACE MOVEMENTS FROM TUNNEL EXCAVATION 191

due to horizontal strain in the orthogonal direction; and Gvh,the shear modulus in the vertical plane.

Graham & Houlsby (1983) showed that only three ofthese material parameters can be obtained from conventionaltriaxial tests, as no shear stress can be applied to the sample.They introduced a material model that uses only threeparameters, namely Ev, �hh, and an anisotropic scale para-meter Æ that relates the remaining material properties as:

Æ ¼ffiffiffiffiffiffiEh

Ev

r¼ �hh

�vh

¼ Ghh

Gvh

(1)

where Ghh is the shear modulus in the horizontal plane.To implement this form of anisotropy into ICFEP, it was

combined with a small-strain formulation reducing the ver-tical Young’s modulus Ev with increasing deviatoric strain(within the strain limits Ed,min and Ed,max defining the non-linear range). As an additional option the value of Æ can bevaried linearly with strain level from its anisotropic value atEd,min to the isotropic case of Æ ¼ 1.0 at Ed,max. Appendix 2presents more details of this soil model. Although this modeldescribes stiffness anisotropy, the strength parameters werekept isotropic. Two parameter sets, referred to as ‘set 1’ and‘set 2’ and summarised in Table 1, were included in theanalyses presented here. A further set (‘set 2v’) adopted avariable Æ. The different parameter sets will be discussed inthe following section.

Anisotropic material parametersWhen using anisotropic soil parameters, the undrained

ratios

n ¼ Eh

Ev

and m ¼ Gvh

Ev

(2)

are often used. Similar expressions are defined for drainedstiffness properties. A relation between the drained andundrained ratios is given by Hooper (1975). Lee & Rowe(1989) showed that m influences the shape of the transversesettlement trough. For K0 ¼ 0.5 they concluded that a ratioof m ¼ 0.2–0.25 produced a reasonable match between FEresults and centrifuge tests. Field data for London Claysummarised by Gibson (1974), however, give a ratio ofapproximately m ¼ 0.38. In the same publication a ratio ofundrained Young’s moduli of n ¼ 1.84 is reported.Addenbrooke et al. (1997) included a transversely anisotro-pic model in their 2D study. The anisotropic parameters inthat study were chosen to match field data reported byBurland & Kalra (1986) giving drained ratios of n9 ¼ 1.6and m9 ¼ 0.44 with �9vh ¼ 0:125 and �9hh ¼ 0:0. Theequivalent undrained ratios can be calculated to be n ¼ 1.41and m ¼ 0.3 using the relations given by Hooper (1975).

Two parameter sets referred to as ‘set 1’ and ‘set 2’ wereapplied to the anisotropic model. Table 1 summarises theratios n9, m9, n and m for these sets. They were calculatedfor small strains (i.e. Ed , Ed,min), as they change with strainlevel. The material parameters (listed in Table 3) werederived from the isotropic model (Table 2) such that E9vreduces with increasing Æ, compared with the Young’smodulus used in the isotropic model. The parameters werechosen in order to obtain similar stress–strain curves whensimulating triaxial extension tests with the different materials(as outlined below). Of the two parameter sets, the first onerepresents a degree of anisotropy that is appropriate forLondon Clay. In contrast, the second set incorporates anextremely high degree of anisotropy, and is therefore moreof academic interest than for use in engineering practice.

In the first set, Æ ¼ 1.265 was chosen to match thedrained ratio of n9 adopted by Addenbrooke et al. (1997). APoisson’s ratio of �9hh ¼ 0:4 was adopted in order toachieve a ratio of m9 ¼ 0.46, which is close to that used byAddenbrooke et al. (1997) (0.44). Owing to the coupling ofthe anisotropic stiffness parameters through the scale factorÆ (see equation (1)) the ratios n and m derived for set 1differ slightly from those applied in the study ofAddenbrooke et al. (1997), where E9v, E9h, �9hh, �9vh and Gvh

were independent. The undrained ratio of n ¼ 1.18 is lower

Table 1. Stiffness ratios for the two sets of anisotropic soilparameters

Isotropic model Anisotropic model

Set 1 Set 2

n9 1.00 1.60 6.25m9 0.55 0.46 1.14n 1.00 1.18 1.66m 0.33 0.33 0.28

Table 2. Input parameters for isotropic pre-yield model (M1)

A B C: % � ª Ed,min: % Ed,max: % Gmin: kPa

373.3 338.7 1.0 3 10�4 1.335 0.617 8.66025 3 10�4 0.69282 2333.3

R S T: % � � �v,min: % �v,max: % Kmin: kPa

549.0 506.0 1.0 3 10�3 2.069 0.42 5 3 10�3 0.15 3000.0

Table 3. Input parameters for anisotropic pre-yield model (M2)

Parameter set 1

Aa Ba C: % � ª Ed,min: % Ed,max: % Ev,min: kPa Æ �9hh

373.3 338.7 1.0 3 10�4 1.335 0.617 8.66 3 10�4 0.69282 5558.8 1.265 0.4

Parameter set 2 and set 2v

Aa Ba C: % � ª Ed,min: % Ed,max: % Ev,min: kPa Æ �9hh

308.8 280.2 1.0 3 10�4 1.335 0.617 8.66 3 10�4 0.69282 5558.8 2.5 0.1


than n ¼ 1.41, which was calculated from their parameters.It is also below the value of n ¼ 1.84 given by Gibson(1974). In contrast the ratio m ¼ 0.33 is higher than in thework of Addenbrooke et al. (1997) and closer to the valueof 0.38 reported by Gibson (1974) for London Clay.

The second set was chosen in order to reduce the un-drained ratio m close to a value adopted by Lee & Rowe(1989) and bringing n close to the ratio reported by Gibson(1974). This was achieved by increasing the anisotropyfactor to Æ ¼ 2.5. Such a high value for London Clay is notsupported by any literature. Both Simpson et al. (1996) andJovicic & Coop (1998) reported ratios of approximately Ghh/Gvh ¼ Æ ¼ 1.5. The results of this parameter set can be seen,however, as an extreme example of how anisotropy affectstunnel-induced settlement predictions.

If, in the anisotropic model, the vertical Young’s modulusE9v was set to a similar magnitude as in the isotropic model(calculated from K9 and G for very small strains), E9hwould be increased by Æ2, hence leading to an overallstiffer behaviour. In order to obtain a similar overallstiffness response of the anisotropic model compared withthe isotropic one, E9v has to be reduced. The input para-meters Aa and Ba listed in Table 3 for the anisotropicmodel were chosen in order to give a similar response tothe isotropic model when used to simulate the behaviour ofan ideal undrained triaxial extension test. These tests weresimulated in a single-element FE analysis by prescribingvertical displacement at the top of the element (i.e. theywere strain controlled). The initial stress state within thesample was isotropic, with p9 ¼ 750 kPa. Fig. 2 presentsstress–strain curves from these tests. The deviatoric stress,defined as

q ¼ � ax � � r (3)

where �ax and �r are the axial and radial stress in the samplerespectively, is plotted against the axial strain �ax. Note thatthe values of q in the plots are negative, as triaxial extensionhas been considered. The results from the isotropic modelare referred to as ‘M1’, and ‘M2’ denotes the anisotropicresults. Laboratory data reported by Addenbrooke et al.(1997) are also shown.

Figure 2(a) shows the strain up to �ax ¼ 0.01%. For thisstrain range the results of the anisotropic parameter set 2coincide with the those from the isotropic model. Theresults of the anisotropic parameter set 1 are in slightlybetter agreement with the laboratory data. Fig. 2(b) presentsthe results for a strain range up to �ax ¼ 0.4%, which

corresponds to the upper limit of non-linear elastic behav-iour of Ed,max ¼ 0.69%. Different curves are given fordeviatoric stress q and excess pore water pressure u. It canbe seen that for �ax < 0.3% the isotropic analysis and thatbased on the anisotropic parameter set 1 are in very goodagreement with the laboratory data, whereas that based onparameter set 2 slightly over-predicts the data. The porewater pressure curves in this plot highlight the anisotropicbehaviour of parameter sets 1 and 2 (note: no laboratorydata were available). Higher excess pore pressure is gener-ated owing to the coupling of deviatoric and volumetricstrains, in contrast to isotropic behaviour, where volumetricstrain and therefore changes in pore water pressure areinduced only by changes in p9 whereas deviatoric strain iscaused only by changes in q.

This coupling is further illustrated in Fig. 3, which showsthe stress paths obtained during the single element analysesfor the different soil models. For the isotropic model thepath is vertical (showing no change in mean effective stressp9) until it reaches the Mohr–Coulomb yield surface. Forthe anisotropic model p9 changes during the tests, leading toinclined stress paths before the yield surface is reached. Thefigure also includes the stress path for an additional analysiswith the same input parameters as set 2 but with Æ beingvaried linearly with deviatoric strain from 2.5 at Ed,min to1.0 at Ed,max. It can be seen that the stress path from thisanalysis (referred to as set 2v) initially shows an anisotropicresponse but then becomes vertical.

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ISOTROPIC ANALYSESSettlement profiles from the 3D analyses are presented

along a longitudinal monitoring profile above the tunnelcentre line and along a transverse monitoring profile aty ¼ �50 m, as indicated in Fig. 1. Initially a 3D analysiswas performed with the isotropic soil model and K0 ¼ 1.5.The development of the longitudinal settlement profile as thetunnel heading is advancing is shown in Fig. 4. Differentcurves are given for every 10 m of tunnel progress. Theposition of the tunnel face for each curve is indicated by anarrow. This figure demonstrates that initially the settlementtrough has a similar shape to that of the cumulative errorcurve, which is often applied to describe the longitudinalsettlement behaviour (Attewell & Woodmann, 1982). How-ever, with further tunnel excavation, and in particular whenthe tunnel face reaches approximately y ¼ �80 m, the settle-ment does not continue to follow this anticipated trend.Instead the profile indicates that some reverse curvature isdeveloping at approximately y ¼ �60 m (as indicated in thefigure).

However, the main concern that is evident from Fig. 4 isthat for all stages of excavation additional settlement occursover the whole mesh length. One would expect that, oncethe tunnel heading had reached a certain y-position, therewould be no further settlement remote from the tunnel faceas a result of any additional tunnel excavation (note: onlythe short-term response is considered in the undrained FEanalysis). It can be seen in Fig. 4 that such a steady-statesettlement condition is not established during the analysis.There is still additional settlement at the y ¼ 0 m boundarywhen the tunnel is excavated from y ¼ �90 m to �100 m.Also, the remote boundary at y ¼ �155 m settles over thewhole analysis, although settlements for the first few incre-ments are negligible on an engineering scale. The additionalsettlement over the whole mesh length obtained in the lastincrements of the analysis indicates that the longitudinaldistance from the last excavation step to both the start andremote vertical boundaries is too small to obtain steady-stateconditions. An increase in the longitudinal dimension of themesh, however, would lead to excessive computational time,and therefore could have been achieved only in combinationwith a drastic increase of element size within the transversemesh plane, leading to a substantial loss in accuracy. Itshould be noted that the longitudinal dimension of the FEmesh used here and shown in Fig. 1 is considerably largerthan in most 3D studies recently published by other authors(e.g. Dolezalova, 2002; Lee & Ng, 2002; Vermeer et al.,2002).

Figure 5 presents transverse settlement profiles normalisedagainst maximum settlement, and compares the results with

field data from the St James’s Park monitoring site (Standinget al., 1996; Nyren, 1998). Field data are given for thetunnel face being just beneath the monitoring profile (re-ferred to as ‘set 22’) and for a distance between tunnel faceand monitoring section of 41 m (‘set 29’). Nyren (1998)reports no further short-term settlement after this survey.These results therefore represent the end of the immediatesettlement response. Comparing the two sets of field datashows that the shape of the settlement trough does notchange as the tunnel face moves away from the monitoringsection. The 3D FE results included in this figure are takenfrom the increment when the tunnel face was just beneaththe transverse monitoring section at y ¼ �50 m and for thelargest possible distance between tunnel face (y ¼ �100 m)and monitoring section of 50 m corresponding to the lastincrement of the analysis. The figure also includes resultsfrom a similar 2D analysis where the results were takenfrom increments where volume losses were close to thosecalculated from the field data (as listed in the figure). It canbe seen that all curves obtained from the FE analyses,regardless of whether 2D or 3D, are very similar, indicatingthat 3D effects cannot account for the discrepancy betweenplane strain results and field data. The results also show thatin the FE analysis the shape of the settlement trough doesnot change with VL (or face position) over a certain range ofVL. Similar conclusions were drawn by Potts & Addenbrooke(1997) when analysing tunnel-induced building deformation.

ANISOTROPIC ANALYSESInitially a set of 2D FE analyses were performed with the

anisotropic soil model. In all cases an initial stress regime

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Fig. 5. Transverse normalised settlement profiles for differentstages of isotropic 2D and 3D analyses together with field data


with K0 ¼ 1.5 was adopted. Fig. 6 shows the normalisedtransverse ground settlement troughs for these analyses takenfrom increments that achieved a volume loss of approxi-mately VL ¼ 3.3% (as listed in Fig. 6). This amount ofvolume loss was obtained from the field data for a tunnelface position approximately 41 m beyond the monitoringsection (which were previously included in Fig. 5 andlabelled ‘set 29’). The figure also lists the increments inwhich the desired volume loss was achieved. These incre-ment numbers show that, for the anisotropic analyses, vo-lume losses in the range 3.1–3.6% were obtained inincrement 10 (of 15 excavation steps) for constant Æ andincrement 9 for variable Æ. In contrast, results for theisotropic analysis are presented for increment 12. Thisindicates that, when comparing same stages in the analysis(i.e. same percentage of unloading), the anisotropic modelpredicts higher values of VL. Values of maximum settlementare also listed in the figure. The maximum settlementobtained for parameter set 1 is 10 mm and lies close to themaximum settlement (approx. 12 mm, VL ¼ 3.2%) obtainedby Addenbrooke et al. (1997) in their anisotropic analysiswith similar ratios of n9 and m9.

Comparing the settlement curve for the isotropic modelwith those for the anisotropic parameter sets 1 and 2 (with aconstant value of Æ) shows that the surface settlement troughbecomes narrower with increasing degree of anisotropy.Adopting a variable Æ (parameter set 2v) improves the settle-ment curve further. The reason for this behaviour is the factthat the anisotropic parameters were chosen such that E9vwas reduced (compared with the isotropic Young’s modulusfor very small strains) whereas E9h increased according to

the choice of Æ. This parameter choice also leads to a lowervalue of Gvh (compared with the isotropic shear modulus),whereas Ghh increases with the degree of anisotropy. Para-meter sets 2 and 2v show initially the same stiffness proper-ties. However, as the deviatoric strain increases from Ed,min

to Ed,max, the reduction of Æ leads to an additional decreaseof E9h and Ghh with deviatoric strain compared with thatobtained for the constant-Æ case. When the material withvariable Æ becomes isotropic at Ed,max it shows softerstiffness properties than the corresponding isotropic material(M1). During tunnel excavation the largest strains occurclose to the tunnel. The material with variable Æ thereforebehaves as softer in this region, which explains the narrowsettlement trough.

The figure demonstrates that even the narrowest settlementtrough obtained from the FE analysis is still too wide whencompared with field data. As parameter set 2v gave the bestresults of all plane strain analyses shown in Fig. 6 it wasadopted in a 3D greenfield analysis. The boundary condi-tions of the 3D mesh were the same as described previouslyfor the isotropic 3D model.

Figure 7 shows the development of longitudinal settlementprofiles during this 3D analysis (K0 ¼ 1.5). Comparing thisfigure with the isotropic results of Fig. 4 it can be seen thatthe anisotropic 3D analysis yielded settlement values thatare nearly one order of magnitude higher than those ob-tained with the isotropic model. Such behaviour is consistentwith the 2D results presented in Fig. 6, in which resultswere presented for earlier increments than in the isotropicanalyses in order to achieve similar volume losses (i.e. thevariation in percentage of unloading, indicated by the differ-ent increments in Fig. 6, is consistent with the observedtrend of the variation of volume loss in the 3D analyses).However, as discussed earlier, the higher volume losses areunlikely to affect the normalised shapes of the settlementtroughs.

The general shape of the longitudinal settlement troughsduring the initial stages of tunnel excavation is similar forboth isotropic and anisotropic analyses. The reverse curva-ture behind the tunnel face developing from a face positionof approximately y ¼ �60 m is magnified in the anisotropicanalysis.

Previously it was shown that a high degree of anisotropyleads to a narrower transverse settlement trough. A similareffect would be expected for the longitudinal curve, andcomparison of Figs 4 and 7 indicate that this is so. With anarrower and steeper trough steady-state conditions shoulddevelop earlier than for the wide trough obtained from theisotropic analysis. However, no steady-state conditions arereached in Fig. 7.

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The zone of reversed curvature behind the tunnel faceforms a settlement crater at the vertical starting boundary (aty ¼ 0 m). This arises because, owing to the symmetry condi-tion at the start boundary, tunnel construction essentiallycommences simultaneously in both the negative and positivey-directions (although only the negative y-part is modelled).A similar longitudinal settlement profile, including thissettlement crater, was presented by Vermeer et al. (2002),who performed 3D analyses for a K0 ¼ 0.65 situation.Applying an isotropic elasto-plastic soil model, they showedthat steady-state conditions were established approximately5D behind the face after the tunnel was constructed over alength of approximately 10D (with a tunnel diameterD ¼ 8 m and z0 ¼ 16 m). The present analysis indicates thata much greater length of tunnel is required to reach steady-state conditions in a high-K0 regime.

INFLUENCE OF K0

The anisotropic 3D analysis was repeated with K0 ¼ 0.5in order to investigate at which state of the analysis steady-state behaviour develops in a low-K0 regime. Fig. 8 showsthe development of longitudinal surface settlement profiles.It can be seen that points close to the start boundary settleduring the first excavation steps but then show small valuesof heave. It is interesting to note that no settlement crater asobserved for the K0 ¼ 1.5 case developed in this analysis.More importantly, a horizontal plane of vertical settlementemerges at about y ¼ �40 m after a face position ofy ¼ �80 m was reached. As tunnel construction continues,

the settlement at this point does not change: that is, steady-state settlement develops. However, comparing the magni-tude of settlement developing during this analysis with thesettlement values obtained in the anisotropic K0 ¼ 1.5 analy-sis reveals high values of settlement. At steady-state condi-tions the settlement above the tunnel centre line is 85.8 mm.This is high compared with field data, where a steady-statesettlement of approximately 20 mm developed. The volumeloss for the anisotropic K0 ¼ 0.5 is therefore significantlyhigher than that observed in field measurements:VL ¼ 18.1% compared with 3.3%. It is clear that such a highvolume loss is unacceptable engineering practice. However,the results highlight the two effects that a reduction ofK0 has on tunnel-induced surface settlement. It has beennoted by other authors that in a low-K0 regime the transversesettlement trough is narrower (Dolezalova, 2002) owing tosmaller lateral stress at tunnel depth. Such an effect is alsoevident when normalising the longitudinal settlement profilesof the different analyses and comparing them with normal-ised field data. Fig. 9 shows such a plot. The FE results arenormalised against the settlement at y ¼ �50 m, for whichFig. 8 indicated steady-state conditions for the anisotropicK0 ¼ 0.5 analysis. For consistency the same normalisationwas applied to the corresponding results from the other 3Danalyses (also including an isotropic K0 ¼ 0.5 case),although no steady-state settlement conditions were achievedthere. The field data from St James’s Park, London (Nyren,1998), are normalised against maximum settlement and areplotted such that the face position corresponds toy ¼ �100 m to compare them with the FE results. The figure

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Fig. 8. Longitudinal settlement profile from anisotropic 3D analysis with K0 0.5

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shows that the normalised anisotropic results for K0 ¼ 0.5have a shape similar to that of the normalised field data. Allother settlement troughs are too wide, with the isotropicK0 ¼ 1.5 analysis giving the widest trough.

A low value of K0, however, influences not only the shapebut also the magnitude of settlement, as listed in Fig. 9.Decreasing K0 reduces the mean effective stress p9 aroundthe tunnel. Both the isotropic and the anisotropic non-linearpre-yield models normalise the soil stiffness against p9. Alow K0 value therefore shows a reduction of soil stiffnesscompared with high-K0 situations. This reduction of stiffnessaround the tunnel results in an increase in volume loss.Combining K0 ¼ 0.5 with an extreme (and unrealistic) aniso-tropic scale factor of Æ ¼ 2.5 leads to a longitudinal settle-ment profile that shows steady-state settlement behind thetunnel face, but also results in an unsatisfactory degree ofvolume loss, which is further demonstrated in Fig. 10. Thisgraph compares the transverse settlement troughs at the endof the 3D analyses (from the anisotropic model M2, set 2v,K0 ¼ 0.5, the isotropic model M1 with K0 ¼ 1.5) with their2D counterparts and with the field data (set 29). The graphindicates that the 2D analysis with the anisotropic modeland with K0 ¼ 0.5 coincides well with the field measure-ments. However, this analysis is volume loss controlled, and,as pointed out above, this parameter set is not realistic forLondon Clay. Furthermore, this model is not applicable in3D analysis as it results in an unrealistic value of volumeloss, and hence exceeds the maximum measured settlementby more than four times. Applying an isotropic model withK0 ¼ 1.5, in contrast, predicts too small a maximum settle-ment that is only one third (3D analysis) or half (2D) of thefield measurements. The results of the other analyses are notshown in the diagram but lie within the range of settlementcurves presented in Fig. 10.

The wide range in volume losses observed in this studyshows that both 3D modelling and anisotropy do not resolvethe problem that numerical analysis predicts settlementtroughs that are too wide when compared with field data. Itfurthermore demonstrates the difficulty in modelling 3Dtunnelling. As pointed out earlier, the high volume lossesare also a result of the chosen excavation lengthLexc ¼ 2.5 m. Owing to computational resources at the timeof the analyses it was not possible to reduce this value ofLexc. With increasing computational power a refined 3Dtunnelling model should be addressed in future research.

CONCLUSIONSA suite of 2D and 3D FE analyses was performed to

investigate the influence of 3D effects, soil anisotropy and

K0 on the tunnel-induced ground settlement trough. The 3Dexcavation process was modelled by successive removal ofelements in front of the tunnel while successively installinglining elements behind the tunnel face.

In a first step the London Clay was modelled by a non-linear elasto-plastic isotropic soil model and a coefficient oflateral earth pressure at rest of K0 ¼ 1.5. Comparing 3Dwith 2D results showed that 3D modelling has a negligibleeffect on the shape of the transverse surface settlementtrough, which remained too wide when compared with fielddata. In the longitudinal direction the surface settlementtrough did not develop steady-state settlement conditions.The curve extended too far when compared with field data.Settlement was obtained on the vertical boundaries duringthe entire analysis, although the total length of tunnelconstruction was chosen as 21.0D. Such mesh dimensionsare considerably larger than most FE models used inrecently published studies. Soil anisotropy was included inthe study to investigate whether this additional soil charac-teristic could improve results. Plane strain results showedlittle improvement in the transverse trough when a level ofanisotropy appropriate for London Clay was adopted. Thetransverse settlement trough improved when an unrealisti-cally high degree of anisotropy was included. With this highlevel of anisotropy a 3D analysis was carried out. Thelongitudinal profile of this analysis was still too wide whencompared with field data, although it was steeper than thesettlement curve obtained from the isotropic analysis. Thisanalysis was then repeated with a low value of K0 ¼ 0.5.Only this combination of a high degree of anisotropy and alow K0, both unreasonable assumptions for London Clay,produced steady-state settlement conditions at the end of theanalysis. The magnitude of settlement, however, was toohigh, as unrealistically high values of volume loss developedduring the analysis.

The study demonstrates that incorporation of 3D model-ling and elastic soil anisotropy in the prediction of tunnel-induced ground surface settlement in London Clay does notsignificantly improve the settlement profile. Adopting realis-tic soil parameters brings only marginal improvements, andadopting an extreme case of soil anisotropy can lead toexcessive values of volume loss. This indicates that neither3D effects nor elastic soil anisotropy can account for theover-wide settlement curves obtained from FE tunnel analy-sis in a high-K0 regime.

APPENDIX 1: ISOTROPIC SOIL MODELThe non-linear elastic model (Jardine et al., 1986) describes the

secant soil stiffness as depending on strain level using atrigonometric expression. To use this model in a finite elementanalysis, the secant expressions are differentiated and then normal-ised against mean effective stress, giving the following tangentvalues (Potts & Zdravkovic, 1999):

G

p9¼ A þ B cos �X ªð Þ � B�ªX ª�1

2:303sin �X ªð Þ with X ¼ log10

Edffiffiffi3

pC

� �

(4)

K9

p9¼ R þ S cos �Y �ð Þ � S��Y ��1

2:303sin �Y �ð Þ with Y ¼ log10

j�vjT

� �

(5)

where G and K9 are the tangent shear and bulk moduli respectively,p9 is the mean effective stress, Ed is the deviatoric strain invariant,and �v is the volumetric strain. A, B, C, R, S, T, �, �, ª, and � areconstants, which are listed in Table 2. Ed,max, Ed,min, �v,max and �v,min

define strain limits above or below which the stiffness varies onlywith p9 and not with strain. Minimum values of tangent shear andbulk moduli are given by Gmin and K9min respectively.

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APPENDIX 2: ANISOTROPIC SOIL MODELThe transversely anisotropic formulation by Graham & Houlsby

(1983) was combined with a small-strain behaviour based on theisotropic model described in Appendix 1. The tangent verticalYoung’s modulus E9v is expressed as

E9v

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where p9 is the mean effective stress and Ed is the deviatoric straininvariant. The input parameters given in Table 3 are the same asused for the isotropic shear modulus in equation (4) apart from thevalues of Aa and Ba, which were chosen to be different from A andB. With the anisotropic scale parameter Æ and the Poisson’s ratio �9hh

being further input parameters the remaining stiffness properties canbe calculated from

E9h ¼ Æ2 E9v (7a)

�9vh ¼ �9hh

Æ(7b)

Gvh ¼ Æ2 E9v

2 1 þ �9vhð Þ (7c)

APPENDIX 3: MOHR–COULOMB MODELThe Mohr–Coulomb yield surface and plastic potential are

expressed in terms of effective stress invariants (Potts & Zdravkovic,1999). The input parameters are the angle of shearing resistance �9,the cohesion c9 and the angle of dilation ł9, which are summarisedin Table 4.

NOTATIONA, Aa, B, Ba, constants for non-linear soil models

C, R, S, Tc cohesion

D outer tunnel diameterEd deviatoric strain

Ed,min, Ed,max deviatoric strain range of non-linear behaviourEv vertical Young’s modulusEh horizontal Young’s modulusG shear modulus

Gvh shear modulus in vertical planeGhh shear modulus in the horizontal plane

K bulk modulusK0 coefficient of lateral earth pressure at rest

Lexc excavation length of step-by-step tunnel excavationm ¼ Gvh/Ev

n ¼ Eh/Ev

p9 mean effective stressq deviatoric stress

Sv vertical ground settlementVL volume loss

x, y, z coordinates of FE meshz0 tunnel depth (centre line to soil surface)Æ anisotropic scale factor

�, ª, �, � constants for non-linear soil modelsª soil bulk unit weight

�ax axial strain�v volumetric strain

�v,min, �v,max volumetric strain range of non-linear behaviour�vh Poisson’s ratio for horizontal strain due to vertical

strain�hh Poisson’s ratio for horizontal strain due to horizontal

strain in orthogonal direction

�ax axial stress�r radial stress� angle of shearing resistanceł angle of dilation9 index denoting effective parameter

REFERENCESAddenbrooke, T. I., Potts, D. M. & Puzrin, A. M. (1997). The

influence of pre-failure soil stiffness on the numerical analysisof tunnel construction. Geotechnique 47, No. 3, 693–712.

Attewell, P. B. & Farmer, I. W. (1974). Ground deformationsresulting from tunnelling in London Clay. Can. Geotech. J. 11,No. 3, 380–395.

Attewell, P. B. & Woodmann, J. P. (1982). Predicting the dynamicsof ground settlement and its derivatives cause by tunnelling insoil. Ground Engng 15, No. 7, 13–22, 36.

Burland, J. B. & Kalra, J. C. (1986). Queen Elizabeth II ConferenceCentre: geotechnical aspects. Proc. Instn Civ. Engrs 80, No. 1,1479–1503.

Dolezalova, M. (2002). Approaches to numerical modelling ofground movements due to shallow tunnelling. Proc. 2nd Int.Conf. on Soil Structure Interaction in Urban Civil Engineering,Zurich 2, 365–373.

Franzius, J. N. (2004). The behaviour of buildings due to tunnelinduced subsidence. PhD thesis, Imperial College, University ofLondon.

Gibson, R. E. (1974). The analytical method in soil mechanics.Geotechnique 24, No. 2, 115–140.

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Guedes, R. J. & Santos Pereira, C. (2000). The role of the soil K0

value in numerical analysis of shallow tunnels. Proc. Int. Symp.Geotechnical Aspects of Underground Construction in SoftGround, IS-Tokyo’99, Tokyo, 379–384.

Gunn, M. J. (1993). The prediction of surface settlement profilesdue to tunnelling. Predictive soil mechanics: Proceedingsof the Wroth Memorial Symposium (eds G. T. Houlsby andA. N. Schofield), pp. 304–316. London: Thomas Telford,London.

Hight, D. W. & Higgins, K. G. (1995). An approach to theprediction of ground movements in engineering practice: back-ground and application. Proc. 1st Int. Conf. Pre-failure Defor-mation of Geomaterials, Sapporo 2, 909–945.

Hooper, J. A. (1975). Elastic settlement of a circular raft inadhesive contact with a transversely isotropic medium. Geotech-nique 25, No. 4, 691–711.

Jardine, R. J., Potts, D. M., Fourie, A. B. & Burland, J. B. (1986).Studies of the influence of non-linear stress–strain character-istics in soil–structure interaction. Geotechnique 36, No. 3,377–396.

Jovicic, V. & Coop, M. R. (1998). The measurement of stiffnessanisotropy in clays with bender element tests in the triaxialapparatus. Geotech. Test. J. 21, No. 1, 3–10.

Katzenbach, R. & Breth, H. (1981). Nonlinear 3D analysis forNATM in Frankfurt Clay. In Proc. 10th Int. Conf. Soil Mech.Found. Engng, Stockholm 1, 315–318.

Lee, G. T. K. & Ng, C. W. W. (2002). Three-dimensional analysisof ground settlements due to tunnelling: role of K0 and stiffnessanisotropy. Proc. Int. Symp. on Geotechnical Aspects of Under-ground Construction in Soft Ground, Toulouse, 617–622.

Lee, K. M. & Rowe, R. K. (1989). Deformations caused by surfaceloading and tunnelling: the role of elastic anisotropy. Geotechni-que 39, No. 1, 125–140.

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O’Reilly, M. P. & New, B. M. (1982). Settlements above tunnels inthe United Kingdom: their magnitude and prediction. Proc. 3rdInt. Symp. Tunnelling ’82, London, 55–64.

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Potts, D. M. & Addenbrooke, T. I. (1997). A structure’s influenceon tunnelling-induced ground movements. Proc. Instn Civ. EngrsGeotech. Engineering 125, 109–125.

Table 4. Input parameter for Mohr–Coulomb model

�9: deg c9: kPa ł9: deg

25 5 12.5


Potts, D. M. & Zdravkovic, L. (1999). Finite element analysis ingeotechnical engineering. Vol. 1: Theory. London: ThomasTelford.

Potts, D. M. & Zdravkovic, L. (2001). Finite element analysis ingeotechnical engineering. Vol. 2: Application. London: ThomasTelford.

Schroeder, F. (2003). The influence of bored piles on existingtunnels. PhD thesis, Imperial College, University of London.

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Standing, J. R., Nyren, R. J., Burland, J. B. & Longworth, T. I.(1996). The measurement of ground movement due to tunnellingat two control sites along the Jubilee Line Extension. Proc. Int.Symp. Geotechnical Aspects of Underground Construction inSoft Ground, London, 751–756.

Vermeer, P. A., Bonnier, P. G. & Moller, S. C. (2002). On a smartuse of 3D-FEM in tunnelling. Proc.8th Int. Symp. on NumericalModels in Geomechanics – NUMOG VIII, Rome, 361–366.


the influence of soil anisotropy and ko on ground surface movements resulting form tunnel excavation

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