effect of anisotropy in ground movements caused by tunnelling

Zymnis, D. M. et al. Geotechnique [http://dx.doi.org/10.1680/geot.12.P.056]

1

Effect of anisotropy in ground movements caused by tunnelling

D. M. ZYMNIS�, I . CHATZIGIANNELIS† and A. J. WHITTLE�

This paper presents closed-form analytical solutions for estimating far-field ground deformationscaused by shallow tunnelling in a linear elastic soil mass with cross-anisotropic stiffness properties.The solutions describe two-dimensional ground deformations for uniform convergence (u�) andovalisation (u�) modes of a circular tunnel cavity, based on the complex formulation of planarelasticity and superposition of fundamental singularity solutions. The analyses are used to interpretmeasurements of ground deformations caused by open-face shield construction of a Jubilee LineExtension (JLE) tunnel in London Clay at a well-instrumented site in St James’s Park. Anisotropicstiffness parameters are estimated from hollow-cylinder tests on intact block samples of London Clay(from the Heathrow Airport Terminal 5 project), and the selection of the two input parameters isbased on a least-squares optimisation using measurements of ground deformations. The results showconsistent agreement with the measured distributions of surface and subsurface, vertical and horizontaldisplacement components, and anisotropic stiffness properties appear to have little effect on thepattern of ground movements. The results provide an interesting counterpoint to prior studies usingfinite-element analyses that have reported difficulties in predicting the distribution of ground move-ments for the instrumented section of the JLE tunnel.

KEYWORDS: anisotropy; elasticity; ground movements; settlement; theoretical analysis; tunnels

INTRODUCTIONAll tunnel operations cause movements in the surroundingsoil. Figs 1(a) and 1(b) illustrate the primary sources ofmovements for cases of closed-face shield tunnelling andopen-face sequential support and excavation (often referredto as NATM) respectively. For open-face shield tunnelling,the stress changes around the tunnel face and the unsupportedround length are primary sources of ground movements.Current geotechnical practice relies almost exclusively onempirical methods for estimating tunnel-induced ground de-formations. Following Peck (1969) and Schmidt (1969), thetransversal surface settlement trough can be fitted by aGaussian function as

uy x, yð Þ ¼ u0y exp

�x2

2x2i

!(1)

where x is the horizontal distance from the tunnel centreline,u0

y is the surface settlement at the tunnel centreline, and xi isthe location of the inflection point.

Mair & Taylor (1997) show that the width of the settle-ment trough is well correlated with the depth of the tunnel,H, and with the characteristics of the overlying soil (see Fig.2(a)). The same framework has been extended to subsurfacevertical movements by varying the trough width parameterto give

xi ¼ K H � yð Þ (2)

where K is the non-linear function shown in Fig. 2(b).There are very limited data for estimating the horizontal

components of ground deformations. The most commonlyused interpretation is to assume that the displacement vec-tors are directed to a point on or close to the centre of the

tunnel, as proposed by Attewell (1978) and O’Reilly & New(1982), such that

ux �x

H

� �uy (3)

There are also a variety of analytical solutions that havebeen proposed for estimating the two-dimensional distribu-tions of ground movements for shallow tunnels. Theseanalyses make simplifying assumptions regarding the consti-tutive behaviour of soil, and ignore details of the tunnelconstruction procedure, but otherwise fulfil the principles ofcontinuum mechanics. In principle, these analytical solutionsprovide a more consistent framework for interpreting hori-zontal and vertical components of ground deformations thanconventional empirical models, and use a small number ofinput parameters that can be readily calibrated to field data.They also provide a useful basis for evaluating the accuracyof more complex numerical analyses. However, the analyti-cal solutions do not purport to describe the processes oftunnel construction accurately, and hence are limited toestimation of far-field ground deformations. In contrast,more comprehensive finite-element (FE) analyses are alsoable to compute near-field behaviour (such as stresses intunnel lining systems), and hence offer a complete predictiveframework for simulating tunnel construction processes andtheir effects on adjacent structures.

The ‘far-field’ ground movements caused by shallowtunnelling processes (excavation and support) are solved as alinear combination of deformation modes occurring at thetunnel cavity (Fig. 3), with input parameters, u� and u�,corresponding to uniform convergence and ovalisationrespectively. Pinto & Whittle (2013) have shown that closed-form solutions obtained by superposition of singularity solu-tions (after Sagaseta, 1987) provide a good approximation tothe more complete (‘exact’) solutions obtained by represent-ing the finite dimensions of a shallow tunnel in a linearelastic soil (after Verruijt & Booker, 1996; Verruijt, 1997).

Pinto et al. (2013) have evaluated the analytical solutionsthrough a series of case studies involving tunnels excavated

Manuscript received 29 April 2012; revised manuscript accepted 22January 2013.Discussion on this paper is welcomed by the editor.� Massachusetts Institute of Technology, Cambridge, MA, USA.† Civil Engineer, Agia Paraskevi, Athens, Greece.

through different ground conditions using a variety ofclosed- and open-face construction methods. They generallyfound good agreement with measured data for tunnels con-structed in low-permeability clays, assuming isotropic elasticproperties. Although the analytical solutions do not simulatethe actual tunnel construction process, the effects of changesin control parameters (such as the face pressure in earthpressure balance (EPB) tunnelling) will affect far-fieldground movements, and will be reflected in changes in thecavity deformation parameters u� and u�: Pinto et al. (2013)noted significant limitations for the case of the HeathrowExpress trial tunnel (Deane & Bassett, 1995), and thediscrepancies between predicted and measured settlementswere attributed to anisotropic stiffness properties of theheavily overconsolidated London Clay.

More recently, Gasparre et al. (2007) have presentedresults from a comprehensive and definitive laboratoryinvestigation of the stiffness properties of natural LondonClay using block samples obtained during the excavationsfor Heathrow Terminal 5. Their test programme includedmeasurements of small-strain elastic properties (based prin-cipally on wave propagation data using triaxial devicesequipped with bender elements), limits on the reversibleelastic response (referred to as the Y1 yield condition)through drained and undrained triaxial stress probe tests,and measurements of the degradation of secant stiffnessparameters with strain level (using local strain measure-ments in triaxial and hollow-cylinder devices). They con-clude that the small-strain behaviour of the clay is welldescribed by the framework of cross-anisotropic elasticity,

and that ‘significant anisotropy was revealed at all scales ofdeformation’.

This paper extends the analytical solutions presented byPinto & Whittle (2013) to account for cross-anisotropicstiffness properties of the clay. The solutions are thenevaluated through comparisons with data from the JubileeLine Extension (JLE) project, involving open-face shieldtunnel construction beneath a well-instrumented site in StJames’s Park (Nyren, 1998). This is a very well-instrumentedand documented case site, with extensive supporting data oncross-anisotropic stiffness parameters for London Clay re-ported by Gasparre et al. (2007). The JLE test section hasbeen extensively analysed by others using FE analyses, manyhave reported problems in predicting far-field deformations,and hence it provides an interesting opportunity to assess thecapabilities of the proposed analytical solutions. Independentresearch by Puzrin et al. (2012) has attempted to model thesame case study using a related analytical approach.

ANALYTICAL SOLUTIONS FOR CROSS-ANISOTROPICELASTIC SOIL

The current analyses consider deformations in a verticalplane (x, y) through a cross-anisotropic, linear elastic soilwith isotropic properties in a plane with dip angle Æ to thehorizontal, as shown in Fig. 4. The stiffness parameters ofthe soil are given for a local (x9, y9, z9) coordinate system(Appendix 1 shows the transformation to the global frame(x, y, z)). The five independent anisotropic stiffness param-eters are defined in the local coordinate system as: E1, the

ShieldTunnel lining

Tunnel facewith support

Stress relief at tunnel face

Shield overcut & ploughingTail void

Deformation of lining

Consolidationof soil

(a)

(b)Round length

Anchors

Tunnel lining (shotcrete)

Deformation attunnel heading

Deformation of lining

Consolidationof soil

Fig. 1. Sources of ground movements associated with tunnelling (from Moller, 2006): (a) closed-facetunnel; (b) sequential excavation

2 ZYMNIS, CHATZIGIANNELIS AND WHITTLE

Young’s modulus of the soil in a direction parallel tothe isotropic plane; �1, the Poisson’s ratio of strains in theisotropic plane (x9–z9); E2, the Young’s modulus normal tothe isotropic plane; G2, the shear modulus for strain indirection y9; and �2, the Poisson’s ratio for strain in the y9direction due to strain in the x9 direction.

Following Milne-Thompson (1960) and Lekhnitskii (1963)the stress–strain relations for plane-strain geometry condi-tions can be written as

�x

�y

ªxy

8<:

9=; ¼

�11 �12 �16

�12 �22 �26

�16 �26 �66

24

35 � x

� y

�xy

8<:

9=; (4a)

where the �ij coefficients are related to the five independentstiffness parameters and the dip angle Æ, as shown in Table1. For Æ ¼ 08 (i.e. isotropic properties in the horizontalplane), E1 ¼ Eh, �1 ¼ �hh, E2 ¼ Ev, �2 ¼ �vh, G2 ¼ Gvh, andthe �ij coefficients are

40

35

30

25

20

15

10

5

00 5 10 15 20

Offset to inflection point, : mxi

Dep

th to

tunn

el a

xis,

: mH

0·50 � /x Hi

0·35

Compiled data

Clays

Sands and gravels

(a)

1·0

0·8

0·6

0·4

0·2

00 0·5 1·0 1·5 2·0

Trough width parameter, K

Dep

th r

atio

,/y H

Moh .(1996): silty sands

et al

Dyer .(1996): sands

et al

Mair & Taylor(1997): clays

Sym. Site Soil type R H/

Green Park

Regents Park

HEX

St James Park

Willington Qy

Centrifuge

London Clay

London Clay

London Clay

London Clay

Soft Clay

Kaolin

0·07

0·06, 0·10

0·26

0·08

0·16

0·23, 0·14

(b)

Fig. 2. Empirical estimation of inflection point (after Mair & Taylor, 1997): (a) width of surface settlement troughs;(b) width of subsurface settlement troughs

�uε

�uε

�

uδ

�Δuy

ΔV

VL

0

2u

Rε

� �Volume loss: Relative distortion:u

uδ

ερ � �

uδ

�Uniform

convergence OvalisationVertical

translation�

Finalshape

R

H

y u, y

x u, xGround surface

Net volume change No net volume change

(not to scale)

Fig. 3. Deformation modes around tunnel cavity (after Whittle & Sagaseta, 2003)

EFFECT OF ANISOTROPY IN GROUND MOVEMENTS CAUSED BY TUNNELLING 3

�11 ¼1� �2

hh

Eh

�12 ¼ ��vh

Ev

1þ �hhð Þ

�22 ¼1

Ev

1� n�2vh

� �

�66 ¼1

Gvh

�16 ¼ �26 ¼ 0

(4b)

where the stiffness ratios n ¼ Eh/Ev and m ¼ Gvh/Ev are usedlater in the paper.

Figure 5 shows non-linear secant stiffness measurementsof Ev, Eh and Gvh from drained, hollow-cylinder (HCA),uniaxial load tests on natural London Clay (unit B2) asfunctions of strain level (Gasparre et al., 2007). The data

show that London Clay is strongly anisotropic at very smallstrain levels (true elastic range). The stiffness ratio, n ¼ Eh/Ev, varies only slightly (n ¼ 1.72–2.30), while m ¼ Gvh/Ev

increases from 0.66 to 1.27 with increased strain level. Thesmall-strain stiffness ratios calculated from undrained testsare very similar to those from drained parameters, as shownin Fig. 5.

The elastic parameters are further constrained by thermo-dynamic considerations (e.g. Pickering, 1970), such that

Gvh, Ev, Eh . 0

0 , n , 4

�1 , nhh , 1

nhh þ 2nhvnvh < 1

(4c)

The conditions for incompressibility are given by Gibson(1974) as

( )xCorrective sheartractions, τ c

xy

H

x u, x

y u, y

Point source

H

α

Mirror imageof point sourceR

R

x �

y �

Local system

Planes of isotropic stiffness

Fig. 4. Superposition method to represent shallow tunnel in cross-anisotropic soil

Table 1. � coefficients used in analytical solution

�11 cos2 Æcos2 Æ

E1

� �2 sin2 Æ

E2

!þ sin2 Æ

sin2 Æ

E2

� �2 cos2 Æ

E2

!� E1

�1 cos2 Æ

E1

þ �2 sin2 Æ

E2

!2

þ sin2 2Æ

4G2

�12 sin2 Æcos2 Æ

E1

� �2 sin2 Æ

E2

!� E1

�1 cos2 Æ

E1

þ �2 sin2 Æ

E2

!�1 sin2 Æ

E1

þ �2 cos2 Æ

E2

!þ cos2 Æ

sin2 Æ

E2

� �2 cos2 Æ

E2

!� sin2 2Æ

4G2

�22 sin2 Æsin2 Æ

E1

� �2 cos2 Æ

E2

!þ cos2 Æ

cos2 Æ

E2

� �2 sin2 Æ

E2

!� E1

�1 sin2 Æ

E1

þ �2 cos2 Æ

E2

!2

þ sin2 2Æ

4G2

�16 sin 2Æsin2 Æ� �2 cos 2Æ

E2

� cos2 Æ

E1

!þ E1 sin 2Æ

�1 cos2 Æ

E1

þ �2 sin2 Æ

E2

!�1

E1

� �2

E2

� �þ sin 4Æ

4G2

�26 sin 2Æcos2 Æþ �2 cos 2Æ

E2

� sin2 Æ

E1

!þ E1 sin 2Æ

�1 sin2 Æ

E1

þ �2 cos2 Æ

E2

!�1

E1

� �2

E2

� �� sin 4Æ

4G2

�66 sin2 2Æ1

E1

þ 1þ 2�2

E2

� �� E1 sin2 2Æ

�1

E1

� �2

E2

� �2

þ cos2 2Æ

G2


�vh ¼ 0:5

�hh ¼ 1� 2n�2vh ¼ 1� n

2

(4d)

In the absence of body forces, the stresses can be solvedusing the Airy stress function, F(x, y), to give

�22

@4F

@x4� 2�26

@4F

@x3@yþ 2�12 þ �66ð Þ @4F

@x2@y2

� 2�16

@4F

@x@y3þ �11

@4F

@y4¼ 0

(5)

with

� x ¼@2F

@y2

� y ¼@2F

@x2

�xy ¼ �@2F

@x@y

Equation (5) is solved by means of the characteristicequation

�11º4 � 2�16º

3 þ 2�12 þ �66ð Þº2 � 2�26ºþ �22 ¼ 0 (6)

The roots of this equation are conjugate complex num-

bers, say º1, º1, º2, º2, and without loss of generalityº1 ¼ a1 þ ib1, º2 ¼ a2 þ ib2, and b1 . b2 . 0. Any analyticfunction g(x + ºy) satisfies equation (5) if º is a solution to

the characteristic equation. Since the resulting stress functionmust be real, the general solution is given by the expression

F x, yð Þ ¼ 2Re F1 z1ð Þ þ F2 z2ð Þ� �

¼ F1 z1ð Þ þ F1 z1ð Þ þ F2 z2ð Þ þ F2 z2ð Þ(7)

where z1 ¼ xþ º1y, z2 ¼ xþ º2y: Introducing the new func-tions �k(zk) ¼ F9k(zk), the stresses are found using thedefinition of complex variables z1, z2 as

� x ¼ 2Re º21�91 z1ð Þ þ º2

2�92 z2ð Þh i

(8a)

� y ¼ 2Re �91 z1ð Þ þ�92 z2ð Þ� �

(8b)

�xy ¼ �2Re º1�91 z1ð Þ þ º2�92 z2ð Þ� �

(8c)

and the displacements U(x, y), V(x, y) are found by integrat-ing the strains, to give

U ¼ 2Re p1�1 z1ð Þ þ p2�2 z2ð Þ� �

(9a)

V ¼ 2Re q1�1 z1ð Þ þ q2�2 z2ð Þ� �

(9b)

where the coefficients pk , qk are expressed as

pk ¼ �11º2k þ �12 � �16ºk

qk ¼ �12ºk þ�22

ºk

� �26

k ¼ 1, 2

(10)

Drained stiffness values

E �v

E �h

Gvh

n E /� �h E �v

m G E/� �vh v

(HC-DQ)

(IS-90-DZ)

(HC-DT)

–

–

Undrained stiffness ratios

n E /� uh E u

v

m G E/� vhuv

0

50

100

150

200

250

300S

tiffn

ess

,an

d: M

Pa

EE

Gh

vvh

0·001 0·01 0·10

0·5

1·0

1·5

2·0

2·5

3·0

Strain: %

Stif

fnes

s pa

ram

eter

s

and

n

m

Uniaxial HCA tests on block samples from 5·2 m depth

Fig. 5. Anisotropic stiffness ratios from drained HCA uniaxial loading tests on natural London Clay(after Gasparre et al., 2007)


Uniform convergence modeFor a cylindrical cavity of radius R in an infinite medium

undergoing uniform convergence, u�, the displacement com-ponents at the tunnel wall can be expressed by (Fig. 6(a))

uB Łð Þ ¼ u� cosŁ

¼ u�eiŁ þ e�iŁ

2

� �

¼ u�� þ ��1

2

� � (11a)

vB Łð Þ ¼ u� sinŁ

¼ u�eiŁ � e�iŁ

2i

� �

¼ u�� 1

2i

� � (11b)

where � ¼ eiŁ:The circular boundary of the tunnel cavity in the (x, y)

plane is transformed into an inclined ellipse in the plane ofthe complex variable z ¼ x + iy ¼ ReiŁ ¼ R� (Fig. 6(b)).

z1 ¼ xþ º1y

¼ xþ Re º1f gyþ iIm º1f gy

¼ x1 þ iy1

(12a)

z2 ¼ xþ º2y

¼ xþ Re º2f gyþ iIm º2f gy

¼ x2 þ iy2

(12b)

The boundary conditions can be solved by a furthermapping onto a circle of unit radius, as shown in Fig. 6(b).

zk ¼ R1� iºk

2�k þ

1þ iºk

2

1

�k

� �

, �k ¼zk � z2

k � R2 1þ º2k

� h i1=2

R 1� iºkð Þ

k ¼ 1, 2 �kj j . 1

(13)

The analytic functions �k(zk) can be expressed as aLaurent series of the conformed variable �k :

�1 z1ð Þ ¼ �1 z1 �1ð Þ½ �

¼ �1 �1ð Þ

¼X1n¼0

an��n1

(14a)

�2 z2ð Þ ¼ �2 z2 �2ð Þ½ �

¼ �2 �2ð Þ

¼X1n¼0

bn��n2

(14b)

�θ

R

uε

u

v u v� i

y

x

Uniform convergence

R

uδ

u vi�

y

x

Ovalisation

uδ

θθ

(a)

(b)

yk

Sk

xk

zk plane

ηk

Σk

�k plane

k

1

Fig. 6. (a) Prescribed displacement modes at tunnel cavity; (b) problem boundaries inzk-plane and in transformed plane


At the tunnel wall, |z| ¼ R and �1 ¼ �2 ¼ eiŁ ¼ �. Hence,from equation (9), the displacement components can befound from

p1

X1n¼0

an��n þ �pp1

X1n¼0

an�n þ p2

X1n¼0

bn��n

þ �pp2

X1n¼0

bn�n ¼ u�

� þ ��1

2

(15a)

q1

X1n¼0

an��n þ �qq1

X1n¼0

an�n þ q2

X1n¼0

bn��n

þ �qq2

X1n¼0

bn�n ¼ u�

� � ��1

2i

(15b)

Equating the coefficients for powers of �

n ¼ 1:

p1a1 þ p2b1 ¼u�

2

q1a1 þ q2b1 ¼iu�

2

(16a)

n 6¼ 1:

p1an þ p2bn ¼ 0

q1an þ q2bn ¼ 0(16b)

The series coefficients are then solved as

n ¼ 1:

a1 ¼u�

2

q2 � ip2

p1q2 � q1p2

� �

b1 ¼u�

2

�q1 þ ip1

p1q2 � q1p2

� � (17a)

n 6¼ 1:

an ¼ bn ¼ 0 (17b)

Ovalisation modeThe ovalisation mode involves no ground loss, and dis-

placements at the tunnel cavity can be represented asfollows.

uB Łð Þ ¼ u� cosŁ

¼ u�eiŁ þ e�iŁ

2

� �

¼ u�� þ ��1

2

� � (18a)

vB Łð Þ ¼ �u� sinŁ

¼ �u�eiŁ � e�iŁ

2i

� �

¼ �u�� 1

2i

� � (18b)

Applying the same methodology used above (for uniformconvergence) the series coefficients an, bn are found.

n ¼ 1:

a1 ¼u�

2

q2 þ ip2

p1q2 � q1p2

� �

b1 ¼u�

2

�q1 � ip1

p1q2 � q1p2

� � (19a)

n 6¼ 1:

an ¼ bn ¼ 0 (19b)

The displacements for uniform convergence and ovalisa-tion of a circular tunnel in an infinite cross-anisotropicelastic medium are then obtained by combining equations(17) or (19) with equations (14) and (9).

U x, yð Þ ¼ 2Re p1a1

1

�1 x, yð Þþ p2b1

1

�2 x, yð Þ

" #(20a)

V x, yð Þ ¼ 2Re q1a1

1

�1 x, yð Þþ q2b1

1

�2 x, yð Þ

" #(20b)

Effect of traction-free ground surfaceFollowing Sagaseta (1987), the ground movements asso-

ciated with a shallow tunnel located at a depth H below thetraction-free ground surface can be represented approxi-mately through a singularity superposition technique (Fig.4). The deformation field for the shallow tunnel is repre-sented by superimposing full-space solutions for a pointsource (0, �H) and mirror image sink (0, +H) (i.e. withequal and opposite cavity deformations) relative to thestress-free ground surface (y ¼ 0) respectively.

Contracting tunnel (�u� . 0, u� . 0):

uþx x, yð Þ ¼ U � x, yþ Hð Þ þ U� x, yþ Hð Þ

uþy x, yð Þ ¼ V � x, yþ Hð Þ þ V� x, yþ Hð Þ(21)

Mirror image (�u� , 0, u� , 0):

u�x x, yð Þ ¼ U � x, y� Hð Þ þ U� x, y� Hð Þ

u�y x, yð Þ ¼ V � x, y� Hð Þ þ V� x, y� Hð Þ(22)

The resulting normal and shear tractions at the surfacey ¼ 0 due to these mirror images are as follows.

N c xð Þ ¼ �þy x, 0ð Þ þ ��y x, 0ð Þ

¼ � y x, Hð Þ � � y x, �Hð Þ

¼ 0

(23a)

T c xð Þ ¼ �þxy x, 0ð Þ þ ��xy x, 0ð Þ

¼ �xy x, Hð Þ � �xy x, �Hð Þ

¼ �2�xy x, �Hð Þ

(23b)

A set of (equal and opposite) ‘corrective’ shear tractionsTc(x) must then be applied at the free surface (Fig. 4). Theresulting displacements on a half-plane due to these correc-tive stresses are

ucx ¼ 2Re p1�

c1 z1ð Þ þ p2�

c2 z2ð Þ

� �uc

y ¼ 2Re q1�c1 z1ð Þ þ q2�

c2 z2ð Þ

� � (24)

where the analytic functions �c1, �c

2 are obtained throughintegration (after Lekhnitskii, 1963),


�c1 z1ð Þ ¼

1

º1 � º2

1

2i

ð1�1

º2 f 1 ð Þ þ f 2 ð Þ� z1

d (25a)

�c2 z2ð Þ ¼ �

1

º1 � º2

1

2i

ð1�1

º1 f 1 ð Þ þ f 2 ð Þ� z2

d (25b)

and the integrals of the normal and shear tractions along theboundary are

f 1 sð Þ ¼ �ðs

1N xð Þdx ¼ 0 (25c)

f 2 sð Þ ¼ðs

1T xð Þdx ¼

ðs

1T c xð Þdx (25d)

Appendix 2 summarises the solution of the infinite inte-grals in equations 25(a), 25(b) and 25(d), from which thefollowing analytical functions are found.

�c1 z1ð Þ ¼

2

º1 � º2

º1�1 z1 � º1Hð Þ þ º2�2 z1 � º2Hð Þ½ �

(26a)

�c2 z2ð Þ ¼ �

2

º1 � º2

º1�1 z2 � º1Hð Þ þ º2�2 z2 � º2Hð Þ½ �

(26b)

The final field of ground deformations for a shallow

tunnel with uniform convergence at the tunnel cavity is thenobtained from equations (21), (22) and (24).

ux x, yð Þ ¼ uþx x, yð Þ þ u�x x, yð Þ þ ucx x, yð Þ (27a)

uy x, yð Þ ¼ uþy x, yð Þ þ u�y x, yð Þ þ ucy x, yð Þ (27b)

Typical resultsFigures 7 and 8 illustrate the effects of cross-anisotropic

stiffness properties on predictions of the shape of the surfacesettlement trough and lateral deflections for a ‘referenceinclinometer’ offset at a distance x/2R ¼ 1 from the tunnelcentreline. These results correspond to solutions for un-drained deformations (i.e. incompressible conditions withPoisson’s ratios defined in equation 4(d)) for a shallow tunnelin clay with R/H ¼ 0.22 (and typical cross-anisotropic stiff-ness ratios, n and m). Fig. 7 shows that for horizontal planesof isotropy (Æ ¼ 08), as the ovalisation ratio r ¼ �u�/u�

increases, the predicted settlement troughs become narrowerand the surface centreline settlement, u0

y , increases signifi-cantly. For r ¼ 0 the analyses predict inward horizontaldisplacements near the tunnel springline, while increases inr result in larger outward movements at this elevation (Fig.7(b)). Fig. 7 also illustrates results for cases where the planeof isotropy is dipping (Æ ¼ 08, 158, 308, 458 and r ¼ 1),representing (for example) conditions at the edge of a

�1·5 �1·0 �0·5 0 0·5 1·0 1·5�25

�20

�15

�10

�5

0

Normalised distance from tunnel centreline, /(a)

x H

Sur

face

set

tlem

ents

,: m

mu y

α 0� °

ρ 0�

ρ � 1

ρ � 2

ρ 1�

α 0°�

α °� 15

α °� 30

α °� 45

u Rx( 2 ): mm� u Rx(2 ): mm

(b)

R Hnm

/ 0·222·50·40·5

0·25

�

�

�

�

� �

νvh

hhν

�10 �5 0 5 10 15�15 �10 �5 0 5 10

�3·0

�2·5

�2·0

�1·5

�1·0

�0·5

0

Nor

mal

ised

dep

th,

/y H

α 0°� 1ρ �

Fig. 7. Effect of relative distortion and dip angle on predicted surface settlements andsubsurface lateral displacements for cross-anisotropic clay


sedimentary basin. As the dip angle decreases, the predictedsurface centreline settlement u0

y increases, while the effect onthe horizontal displacements is less pronounced.

Figure 8 shows the effects of the stiffness ratios n andm for the case where the soil has isotropic properties inthe horizontal plane (Æ ¼ 08). The results show a narrow-ing of the surface settlement trough for normal stiffnessratios, n . 1 (n ¼ Eh/Ev ¼ 1 is the isotropic case), whichis especially pronounced for n . 3. Increases in the shearstiffness ratio, m ¼ Gvh/Ev, have the opposite effect. Thesettlement trough for m ¼ 1 is much wider than theisotropic case (m ¼ 0.33). There is also a change in themode shape of the settlement trough shown for m ¼ 1.5,where the maximum settlement does not occur above thecentreline, but is instead offset at x/H � 0.5. This result isoften observed in two-dimensional FE analyses of shallowtunnel excavation for cases with high in situ K0 stressconditions (e.g. Addenbrooke et al., 1997; Franzius et al.,2005; Moller, 2006), but has not been reported in priortunnelling projects. The transition in mode shape is afunction of the anisotropic stiffness ratios (m and n) andthe ovalisation ratio, r, as shown in Fig. 9. The subse-quent applications of the analyses for the JLE tunnel usea constrained range of r to avoid the higher modesolutions.

PRIOR INTERPRETATION OF JLE TUNNEL IN STJAMES’S PARK

The JLE project (1994–1999) included 15 km of twinbored tunnels, 4.85 m in diameter, constructed using anopen-face shield and excavated by mechanical backhoe.Ground displacements were measured at a well-instrumentedgreenfield site in St James’s Park, and were described indetail by Nyren (1998). The westbound (WB) tunnel passedunder the instrumentation site in April 1995 with springlinedepth H ¼ 31 m and an advance rate of 45.5 m/day (i.e.1.9 m/h). The eastbound (EB) tunnel (not considered in thispaper) traversed the section in January 1996 at depthH ¼ 20.5 m (and offset at 21.5 m from the WB bore).

The instrumentation at the test section included an arrayof 24 surface monitoring points (SMP; surveyed by totalstations), and subsurface ground movements were recordedusing a set of: (a) nine electrolevel inclinometers, with tiltangles typically measured at vertical intervals of 2.5 m; and(b) 11 rod extensometers, each measuring vertical displace-ment components at up to eight elevations. Fig. 10 showseight locations (A–H) where two-dimensional vectors ofdisplacement can be interpreted from the inclinometer andextensometer data.

The soil profile comprises 12 m of fill, alluvium and terracegravels overlying a 40 m thick unit of low-permeability

R Hu u

/ 0·22/ 1

0°

��

�

ρ δ εα

�1·5 �1·0 �0·5 0 0·5 1·0 1·5�1·2

�1·0

�0·8

�0·6

�0·4

�0·2

0

Normalised distance from tunnel centreline, /(a)

x H

Nor

mal

ised

sur

face

set

tlem

ents

,/ |

|u

uy

y0

m 0·33� n 1·00�

3·993

2

0·01

� mn � 0·01

0·331

1

1·5

Higher mode

u R ux y(2 )/| |0

(b)

u R ux y(2 )/| |0

�0·5 0 0·5

�3·0

�2·5

�2·0

�1·5

�1·0

�0·5

0

Nor

mal

ised

dep

th,

/y H

m 0·33�

n 0·01�n 1·00�

n 2·00�n 3·00�n 3·99�

�0·5 0 0·5

n 1·00�

m 0·01�

m 0·33�m 1·00�

m 1·50�

Fig. 8. Effect of anisotropic stiffness ratios (n and m) on predicted surface settlements andsubsurface lateral displacements: (a) normalised surface settlement trough; (b) normalisedlateral displacements at offset, x/2R 1


London Clay (with four divisions shown in Fig. 10), abovethe Lambeth Group (lower aquifer system). The groundwatertable is located approximately 3 m below the ground surface,but pore pressures are 5–7 m below hydrostatic at the eleva-tion of the WB tunnel springline. Standing & Burland (2006)have carried out a detailed review of the physical andengineering properties of the four divisions of the LondonClay along this section of the JLE alignment. They report theundrained shear strength of London Clay, su, increasing from

215 � 80 kPa (unit A3) to 233 � 77 kPa (A2), and in situhydraulic conductivity values, k ¼ 0.15–2.0 3 10�10 m/s.

Surface displacementsFigures 11(a) and 11(b) show the vertical and horizontal

surface movements measured approximately 1 day after thepassage of the WB tunnel face, when it can reasonably beassumed that there is little consolidation within the London

nE

/�

� hE

� v

Mode I

Mode II

0 0·5 1·0 1·5 2·0 2·50

0·5

1·0

1·5

2·0

2·5

3·0

3·5

4·0

m G E/� �vh v

ρ 0�

ρ � 1

ρ � 2

Anisotropic stiffness ratios(after Gasparre ., 2007)et al

Fig. 9. Effect of anisotropic stiffness ratios and tunnel ovalisation ratio on surface settlementtrough mode shapes for shallow tunnels

HGFEDCBA

�4 0 4 9·6 16 22 26 32

�40

�31

�27

�22·5

�17

�13

�9

�5

0

Distance from tunnel centreline: m

Dep

th: m

R

Rp

Measurement pointsused in LSS analysis

Made ground/alluvium

Terrace gravels

London Clay(B)

London Clay(A3ii)

London Clay(A3i)

London Clay(A2)

Fig. 10. Cross-section and instrumentation of test section of JLE project in St James’s Park;shading indicates plastic zone around tunnel (Rp/R 4–13)


Clay. Standing & Burland (2006) fitted the transverse surfacesettlement trough using the empirical Gaussian relation(equation (1)) with a trough width, xi ¼ 13.3 m (i.e. K ¼xi/H ¼ 0.43) and maximum settlement above the crown,u0

y � 20 mm. Hence the volume loss at the ground surface,˜Vs (¼ 2:5u0

yxi) corresponds to an apparent ground loss atthe tunnel cavity, ˜VL/V0 ¼ 3.3%, caused by tunnel construc-tion. They attribute this unexpectedly high volume loss todetails of the construction method (the WB tunnel wasconstructed with up to 1.9 m of unsupported heading), andto a local ground zone above the WB tunnel crown with ahigher concentration of sand and silt partings in the LondonClay.

The horizontal surface displacements (Fig. 11(b)) are alsowell fitted by conventional empirical assumptions usingequation (3) with maximum surface horizontal movementux � 5.7 mm at x � 14 m east of the centreline. However, itshould be noted that the measured profile shows a loss ofanti-symmetry (e.g. ux 6¼ 0 mm at x ¼ 0 m) that Nyren(1998) attributes to a deviation in principal stresses acting inthe horizontal plane.

Prior numerical analysesSeveral researchers have attempted to compute the ground

movements reported by Nyren (1998) using non-linear FEmethods. For example, Franzius et al. (2005) compared two-dimensional and three-dimensional analyses using differentcoefficients of lateral earth pressure at rest, K0, and variousconstitutive models for simulating the construction of the

JLE WB tunnel. Their base-case scenario used a non-linear,isotropic elasto-plastic constitutive model with K0 ¼ 1.5. Intwo-dimensional analyses they assumed a volume loss ˜VL/V0 ¼ 3.3%, which resulted in a computed maximum surfacesettlement u0

y ¼ 10 mm and a transverse surface settlementtrough that was much wider than the measured behaviour(Fig. 12). Surprisingly, they also found similar results fromthree-dimensional analyses using a step-by-step procedurethat simulates the boundary conditions associated with open-face excavation and lining construction.

Franzius et al. (2005) then modified the constitutive modelto include non-linear cross-anisotropic stiffness properties(using a simplified three-parameter formulation proposed byGraham & Houlsby, 1983). They were able to achieve goodagreement with the measured settlement trough in the two-dimensional analyses only by using an unrealistically highelastic Young’s modulus ratio n ¼ Eh/Ev ¼ 6.5 (i.e. outsidethe theoretical elastic range of n; equation 4(c)) in combina-tion with a low value of K0 ¼ 0.5. However, when the samemodel parameters were used in a three-dimensional analysisof the open-face tunnel construction, much larger surfacesettlements were obtained (u0

y ¼ 85 mm with interpretedvolume loss, ˜VL/V0 ¼ 18%), as shown in Fig. 12.

Wongsaroj (2005) formulated a bespoke constitutivemodel to describe the non-linear, anisotropic behaviour ofLondon Clay, and used the model in three-dimensional FEsimulations for short- and long-term ground movementscaused by JLE tunnel construction. Fig. 13(a) compares themeasured surface settlements with computed results usingfour different input parameter sets. Models with both iso-

�10 0 10 20 30 40 50 60�25

�20

�15

�10

�5

0

5

10

Distance from tunnel centreline: m(a)

Ver

tical

set

tlem

ents

: mm Ground surface

Tunn

el c

entr

elin

e

Gaussian (after Standing &Burland, 2006)

ΔV VL 0/ : % xi: m

3·3 13·3

�10 0 10 20 30 40 50 60�25

�20

�15

�10

�5

0

5

10

Distance from tunnel centreline: m(b)

Hor

izon

tal d

ispl

acem

ents

: mm

Ground surface

Tunn

el c

entr

elin

e

Movements away from tunnel

Movements towards tunnel

Empirically fitted by Eq. (3)u x H ux y( / )�

Fig. 11. Empirical interpretation of surface displacements for WB JLE tunnel in St James’sPark: (a) surface settlement trough (after Standing & Burland, 2006); (b) empiricalinterpretation of surface displacements for WB JLE tunnel in St James’s Park


tropic and anisotropic small-strain stiffness (K0 ¼ 1.5; small-strain, drained elastic stiffness ratios, n ¼ 0.44, m ¼ 0.13,that are inconsistent with data shown in Fig. 5) resulted insettlement troughs that are wider than the field measure-ments for the WB JLE tunnel, and also significantly over-estimate the back-figured volume loss (˜VL/V0 ¼ 5.4–6.0%).Good agreement is achieved only by increasing the aniso-tropic stiffness ratio (Ghh/Gvh ¼ 5, corresponding tom ¼ 0.04) and reducing the assumed value of K0 ¼ 1.2. Fig.13(b) shows further comparisons with the subsurface hori-zontal displacements reported by Wongsaroj (2005). Theanalyses generally predict larger lateral deformations ofthe soil towards the tunnel centreline than are measured inthe field. The author attributed this discrepancy, in part, tosurveying errors in the field measurements. Subsurface hor-izontal displacements were not reported for the fourth model(Ghh/Gvh ¼ 5), and thus are not shown in Fig. 13(b).

APPLICATION OF ANALYTICAL SOLUTIONSIn contrast to the preceding analyses, which are based on

comprehensive three-dimensional FE analyses, the proposedanalytical solutions make simplifying constitutive assump-tions in order to solve the two-dimensional far-field grounddeformations as functions of the two cavity deformation

parameters, u� and u�. These parameters are back-fitted fromthe measured deformations of the WB JLE tunnel in StJames’s Park using a least-squares fitting approach. Thecurrent analyses assume linear elastic behaviour throughoutthe soil mass, and hence are likely to underestimate grounddeformations close to the tunnel lining, where plastic failureoccurs in the clay.

This near-field zone of plasticity can be estimated fromsolutions of a cylindrical cavity contraction in an elastic-perfectly plastic soil (e.g. Yu & Rowe, 1999). The radius ofthe plastic zone, Rp, can be found from

Rp

R¼ exp

N � 1

2

� �(28)

where N ¼ (p0 � pi)/su is the overload factor, and p0 and pi

are the pressures in the far field and within the tunnelcavity.

The radius of the plastic zone can then be estimated by

(a) equating p0 with the overburden pressure (�v0 � 600 kPa)at the springline

(b) assuming pi ¼ 0(c) considering a likely range of undrained shear strength for

Finite-element analysis results, after Franzius . (2005)et al

Line K0 ΔV VL 0/ : % m

Isotropic

Isotropic

Anisotropic

Anisotropic

Field measurements

x

DimensionsModel n

0·55

0·55

1·14

1·14

1

1

6·25

6·25

�3·3

�2·1

�3·5

�18·1

1·5

1·5

0·5

0·5

2D

3D

2D

3D

�10 0 10 20 30 40 50 60�90

�80

�70

�60

�50

�40

�30

�20

�10

0

10

Distance from tunnel centreline: m

Ver

tical

set

tlem

ents

: mm

Ground surface

Tunn

el c

entr

elin

e

Fig. 12. Surface settlement troughs as predicted by FE analysis undertaken by Franzius et al. (2005)


mnΔV VL 0/ : %Model

0·435

0·130

0·130

0·039

1·000

0·438

0·438

0·438

6·0

5·6

5·4

3·2

1·0

1·5

1·5

5·0

1·5

1·5

1·0

1·2

⎛⎜⎝ ⎛

⎜⎝v �vh

v �hvn �

⎛⎜⎝ ⎛

⎜⎝

m �n

v2(1 )� �hh

GG

vh

hh·

v � �hh v � �hv 0·15);Poisson’s ratios: isotropic (v � �vh

v � �hh 0·12, v � �hv 0·16)anisotropic model ( 0·07,v � �vh

HGFED

3D finite-element analysis results, after Wongsaroj (2005)

Line K0 G Ghh vh/

Isotropic

Anisotropic

Anisotropic

Anisotropic

Field measurements

�10 0 10 20 30 40 50 60�25

�20

�15

�10

�5

0

5

10


Ver

tical

set

tlem

ents

: mm Ground surface

Tunn

el c

entr

elin

e

�4 0 4 9·6 16 22 26 32

�31

�27

�22·5

�17

�13

�9

�5

0


Dep

th: m

�20 0 �20 0 �20 0 �20 0 �20 0 �20 0C

Horizontal displacements: mm

Fig. 13. Comparison between field measurements and FE analysis results undertaken by Wongsaroj(2005): (a) surface settlement trough; (b) subsurface lateral displacements


the London Clay, su ¼ 136–293 kPa (A3 unit; Standing& Burland, 2006).

Based on these assumptions, Rp � 4–13 m. The currentinterpretation excludes measured data within the estimatedplastic zone, but considers 49 subsurface deformations(along eight vertical lines, A–H, Fig. 10), together with 24locations where surface movements were surveyed. Fig. 14shows the derivation of the least-squares solution error(LSS) for the input parameter state space (u�, u�), where

LSS ¼ MinX

i

[(~uuxi � uxi)2 þ (~uuyi � uyi)

2] (29)

In most practical cases, engineers will expect to fit themeasured centreline surface settlement, u0

y , and hence it ispreferable to present a modified least-squares solution,LSS�, that includes this additional constraint.

Figures 14(a) and 14(b) compare results for two sets ofsoil stiffness properties: (a) the isotropic case (m ¼ 0.33,n ¼ 1, � ¼ �vh ¼ �hh ¼ 0.5); and (b) the cross-anisotropiccase (with Æ ¼ 08), based on the small-strain behaviourreported by Gasparre et al. (2007), and assuming incompres-sibility of the London Clay (m ¼ 0.66, n ¼ 2.07, �vh ¼ 0.5,�hh ¼ 1 � 0.5n ¼ �0.035). It should be noted that the small-strain elastic anisotropic stiffness ratio, n ¼ Eh/Ev, obtainedfrom undrained tests is very close to that obtained fromdrained tests, as shown in Gasparre et al. (2007).

There is little difference in the magnitude of the globalleast-squares error between the two sets of analyses, whilethe constrained LSS� solution for the isotropic case isslightly closer to the global minimum than the cross-aniso-tropic case. The derived cavity contraction parameter issmaller for the cross-anisotropic case (�u� ¼ 34 mm, com-pared with 36 mm for the isotropic case), with a higherrelative distortion, r ¼ �u�/u� ¼ 1.56 compared with 1.32.Both LSS� solutions imply slightly lower volume loss ratiosat the tunnel cavity (˜VL/V0 ¼ 3.0% and 2.8%; Figs 14(a)and 14(b) respectively) than were estimated by conventionalempirical solutions (3.3%; Fig. 11(a)).

Figure 15 compares analytical solutions of the distribu-tions of vertical and horizontal surface displacementcomponents for the WB JLE tunnel, using isotropic andcross-anisotropic soil properties (with LSS� tunnel modeinput parameters). The fields of vertical displacements arevery similar for both sets of analyses, while the cross-anisotropic case predicts slightly larger lateral ground move-ments around the tunnel springline than the isotropic case(Fig. 15(b)).

Figures 16(a) and 16(b) show that both sets of analysesproduce very reasonable agreement with the measured verti-cal and horizontal surface displacements. These results showthat reasonable predictions of surface displacements can beachieved using the analytical solutions with isotropic stiff-ness properties for the London Clay. This is a very surpris-ing result, which is due to the counteracting effects of thetwo key stiffness ratios, n and m (compare Figs 8(a) and8(b)).

Figures 17(a) and 17(b) compare the computed and meas-ured subsurface vertical and horizontal displacement compo-nents for the WB JLE tunnel. The computed deformationsare generally in very good agreement with both vertical andhorizontal components of movements measured in the farfield (i.e. outside the expected zone of plastic soil behav-iour). Very similar patterns of soil displacements are ob-tained using isotropic and anisotropic elastic stiffnessparameters. The analysis tends to overestimate measuredcentreline vertical settlements below 10 m, but produces veryaccurate predictions at the rest of the extensometer positions.The analytical solutions fit well the inclinometer readings atlocations from the ground surface up to a transition depthmarked by contour line ux ¼ 0 mm in Fig. 17(b), but predictoutward movement below this transition depth, while theinclinometers show zero ground movements.

CONCLUSIONSThis paper has presented new analytical solutions for

estimating two-dimensional ground deformations caused by

�100

�80

�60

�40

�20

0

20

40

60

80

100

�100 �80 �60 �40 �20 0 20 40 60 80 100�100

�80

�60

�40

�20

0

20

40

60

80

100

�100 �80 �60 �40 �20 0 20 40 60 80 100

Least-squares solution error analysis for isotropic analytical model

Method ΔV VL 0/ : % uε: mm ρ

LSS*

LSS

Square solution error: mm2

Centreline surface settlement fit

Least-squares solution error analysis for anisotropic analytical model

2000

0

40000

100000

60000

140000

80000

120000

20000

40000

10000

60000

10000080000

50003000

20000

5000

40000

10000

10000

60000

3000

5000

3000

20000

5000

40000

20000

10000

10000

5000

2000

0

40000

60000

80000

100000

140000120000

20000

40000

1000

0

60000

80000

100000

10000

5000

20000

3000

10000

40000

60000

5000

5000

3000

10000

3000

20000

10000

40000

5000

20000

10000

Higher settlement mode

Ova

lisa

tion,

: mm

u δ

Uniform convergence, : mm(a)

uε

Ova

lisa

tion,

: mm

u δ

Uniform convergence, : mm(b)

uε

Symbol

1·32

0·97

�36

�36

3·0

3·0

Method ΔV VL 0/ : % uε: mm ρ

LSS*

LSS

Square solution error: mm2

Centreline surface settlement fit

Symbol

1·56

1·08

�34

�34

2·8

2·8

Mode I

Mode II

Fig. 14. Least-squares error analysis undertaken for input parameter selection: (a) isotropic case; (b) cross-anisotropic case


�80

�70

�60

�50

�40

�30

�20

�10

0

�40 �30 �20 �10 0 10 20 30 40

Dep

th: m

Distance from centreline: m(a)

�80

�70

�60

�50

�40

�30

�20

�10

0

�40 �30 �20 �10 0 10 20 30 40

Dep

th: m

Distance from centreline: m(b)

0

�20

�20

�17·5

�15

�15

�12

·5

�12·5

�10

�10

�7·

5

�7·5

�5

�5

�5

�5

�2·5

�2·5

�2·5

�2·5

0

0

0

2·52·

5

5

5

5

�2·

5

�2·5

7·57·

5

�22·5

10�

25

12·5

�27·5

15

0

0

�80

�60

�40

�20

0

20

40

60

80

Anisotropic modelIsotropic model

�5

�2·

5

�2·5

�2·5

0

0

0

0

0

0

0

0

0

2·5

2·5

2·5

2·5

2·5

2·5

5

5

5

5

55 7·5

7·5

�2·5

�2·5

7·5

7·5

10

�5

�5

10

�5

�5

�80

�60

�40

�20

0

20

40

60

80

Anisotropic modelIsotropic model

Fig. 15. Analytical predictions of vertical and horizontal ground deformations for LSS� solutions withisotropic and cross-anisotropic stiffness properties for London Clay: (a) vertical displacements (mm);(b) horizontal displacements (mm)


shallow tunnelling in a cross-anisotropic soil. These analysesextend prior solutions derived by Pinto (1999), Whittle &Sagaseta (2003) and Pinto & Whittle (2013) in which thecomplete distribution of far-field ground movements can beinterpreted from two basic tunnel cavity deformation modeparameters (u� and u� or r), the dip angle of the isotopicstiffness plane, Æ, and two key anisotropic stiffness ratios,n ¼ Eh/Ev and m ¼ Gvh/Ev:

The analytical solutions have been applied to reinterpretground deformations associated with the open-face construc-tion of the WB tunnel for the Jubilee Line at a well-instrumented site in St James’s Park (Nyren, 1998). Thecurrent analyses benefit from high-quality measurements ofthe cross-anisotropic stiffness properties of intact LondonClay measured in an independent study for Heathrow Air-port T5 (Gasparre et al., 2007). These data show thatLondon Clay exhibits pronounced stiffness anisotropy atsmall strain levels.

The cavity deformation mode parameters are evaluatedusing a least-squares fit to surface and subsurface deforma-

tions at the instrumented test site. The results show that boththe isotropic and cross-anisotropic analytical solutions pro-duce very good fits to the measured ground displacements.Using the high-quality measurements undertaken by Gas-parre et al. (2007), it can indeed be concluded that cross-anisotropic stiffness parameters have only a small influenceon predictions of the far-field ground deformations causedby tunnelling in London Clay. The analytical solutionsachieve comparable levels of agreement with measurementsof the surface settlement trough that are conventionally fittedusing an empirical Gaussian distribution function. However,the current analytical solutions correspond to smaller volumelosses at the tunnel cavity than those estimated by conven-tional empirical assumptions (cf. Standing & Burland, 2006),while offering a more consistent framework for interpretingthe complete distribution of horizontal and vertical compo-nents of ground deformations. Although these results arevery encouraging, further case studies are needed to estab-lish how the cavity mode parameters are related to differentmethods of tunnel construction.

ΔV VL 0/ : %

1·32

1·56

3·0

2·8

�36·0

�34·0

Line Analytical model uε: mm ρ

Isotropic

Anisotropic

Field measurements

�10 0 10 20 30 40 50 60�25

�20

�15

�10

�5

0

5

10


Ver

tical

set

tlem

ents

: mm

Ground surface

Tunn

el c

entr

elin

e

�10 0 10 20 30 40 50 60�25

�20

�15

�10

�5

0

5

10


Hor

izon

tal d

ispl

acem

ents

: mm

Ground surface

Tunn

el c

entr

elin

e

Movements away from tunnel

Movements towards tunnel

Fig. 16. Comparison of computed and measured surface movements for WB JLE tunnel: (a)settlements; (b) horizontal displacements


�20 0 0 0 20 0 20 20 0 20 0 20 0 20HGFEDCB

�4 0 4 9·6 16 22 26 32

�31

�27

�22·5

�17

�13

�9

�5

0


Dep

th: m

Vertical displacements: mm

�40 0A

�4 0 4 9·6 16 22 26 32

�31

�27

�22·5

�17

�13

�9

�5

0


Dep

th: m

Horizontal displacements: mm

0A

ux 0 mm�

0�200�20�200�200�20�20 �40 �20�40�20HGFEDCB

ΔV VL 0/ : %

1·32

1·56

3·0

2·8

�36·0

�34·0

Line Analytical model uε: mm ρ

Isotropic

Anisotropic

Field measurements

Fig. 17. Comparison of computed and measured subsurface ground movements for WBJLE tunnel: (a) vertical displacements; (b) horizontal displacements


ACKNOWLEDGEMENTSThe lead author (DMZ) gratefully acknowledges support

provided by the George and Maria Vergottis and Goldberg-Zoino Fellowship programmes for her SM research at MIT.This work was initiated through a collaborative projectsupported by Tren Urbano GMAEC.

APPENDIX 1: ROTATION OF PLANES FROM LOCAL TOGLOBAL COORDINATE SYSTEM

Considering a cross-anisotropic, linear elastic soil with isotropicproperties in a general (x9, z9) plane with dip angle Æ to thehorizontal as shown in Fig. 4, the strains are related to the stresses inthe local (x9, y9, z9) coordinate system through the relation

�x9x9

�y9y9

�z9z9

ªx9y9

ªx9z9

ªy9z9

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

¼

1

E1

� �2

E2

� �1

E1

0 0 0

� �2

E2

1

E2

� �2

E2

0 0 0

� �1

E1

� �2

E2

1

E1

0 0 0

0 0 01

G2

0 0

0 0 0 02 1þ �1ð Þ

E1

0

0 0 0 0 01

G2

26666666666666666666666664

37777777777777777777777775

� x9x9

� y9y9

� z9z9

� x9y9

� x9z9

� y9z9

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

¼ Cx9y9z9� x9y9z9

(30)

The local material compliance matrix Cx9 y9z9 is transformed intothe global compliance matrix Cxyz as shown below

�xyz ¼ Cxyz�xyz

¼ RTCx9y9z9R�xyz

(31)

where R is the transformation matrix

R ¼

cos2 Æ sin2 Æ 0 0 0 � sin 2Æsin2 Æ cos2 Æ 0 0 0 sin 2Æ

0 0 1 0 0 0

0 0 0 cosÆ sinÆ 0

0 0 0 � sinÆ cosÆ 0

0:5 sin 2Æ �0:5 sin 2Æ 0 0 0 cos 2Æ

266666664

377777775(32)

APPENDIX 2: CALCULATION OF CORRECTIVESTRESSES INTEGRALS

The integral of the tractions along the free surface (equation25(d)) after some manipulation reduces to

f 2 sð Þ ¼ 2[º1�1 s� º1Hð Þ þ º1�1 s� º1Hð Þ þ º2�2 s� º2Hð Þ

þ º2�2 s� º2Hð Þ]

s 2 R

(33)

The calculation of the stress functions of the corrective stressesFc

1(z1), Fc2(z2) requires the calculation of the infinite integral

(equation (25a)),

�c1 z1ð Þ ¼

1

º1 � º2

1

2i

ð1�1

º2 f 1 ð Þ þ f 2 ð Þ� z1

d

¼ 1

º1 � º2

1

2i

ð1�1

f 2 ð Þ� z1

d

¼ 1

º1 � º2

1

i3

X2

k¼1

ð1�1

ºk�k � ºkHð Þ� z1

dþð1�1

ºk�k � ºkHð Þ� z1

d

" #

(34)

Consider the integrals of the complex functions �k(w)=(w� z),�k(w)=(w� z) along the path shown in Fig. 18. This path includesbranch points for function Fk

w1,2 ¼ ºkH � R

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ º2

k

q(35)

For small ratios R/H, and usual degrees of anisotropy, the twobranch points of Fk will lie in the upper plane (i.e. outside thechosen integration path), and therefore the integral of the analyticfunction Fk according to the Cauchy integral formula assumes thevalueþ

c

�k wð Þw� z

dw ¼ 2i�k zð Þ

þc

�k wð Þw� z

dw ¼ 0

(36)

Also

Þc

�k wð Þw� z

dw ¼ limR!1

ð�R

R

�k ð Þ� z

dþð

IR

�k wð Þw� z

dw

" #

¼ �Ð1�1

�k ð Þ� z

d

(37)

The final result is

�c1 z1ð Þ ¼

2

º1 � º2

º1�1 z1 � º1Hð Þ þ º2�2 z1 � º2Hð Þ½ � (38a)

Similarly,

�c2 z2ð Þ ¼ �

2

º1 � º2

º1�1 z2 � º1Hð Þ þ º2�2 z2 � º2Hð Þ½ � (38b)

NOTATIONan Laurent series coefficientsbn Laurent series coefficientsC integration path

Eh Young’s modulus in (any) horizontal direction (plane ofisotropy)

Ev Young’s modulus in vertical directionE1 Young’s modulus in direction parallel to isotropic planeE2 Young’s modulus in direction normal to isotropic planeF Airy’s stress function

w1

w2�R R

C

IR

z

w-plane

Fig. 18. Integration path


fk(x) integral of traction along boundaryGhh shear modulus for strain in horizontal planeGvh shear modulus for strain in (any) vertical plane (planes

of anisotropy)G2 shear modulus for strain in direction normal to isotropic

planeH depth to tunnel springlinei imaginary unitk hydraulic conductivityK empirical parameter related to settlement trough width

K0 coefficient of lateral earth pressure at restL radius of integration path

LSS� constrained least-squares solution that fits u0y

N overload factorN(x) normal traction on free surfacen, m stiffness ratios

pk , qk analytic coefficientsp0 pressure outside tunnel cavitypi pressure inside tunnel cavityR radius of tunnel

Rp radius of plastic zoneSk boundary in zk domain

s variablesu undrained shear strength

T(x) shear traction on free surfaceU, V full-space solution (horizontal and vertical

displacements)ux horizontal ground displacements~uuxi horizontal ground displacement measured at point iu y vertical ground displacements~uuyi vertical ground displacement measured at point iu0

y centreline surface settlementu� ovalisation parameteru� uniform convergence parameter

˜VL/V0 volume loss at tunnel cavity˜Vs volume loss at ground surfacewk branch points of �(w)

x distance from tunnel centreline(x, y, z) global coordinate system

(x9, y9, z9) local coordinate systemy depth measured from ground surface

z, zk complex parametersÆ dip angle of plane with isotropic properties�ij coefficients related to stiffness parametersªij shear strain�i normal strain�k transformed variableŁ angleºk roots of the characteristic equation (with positive

imaginary part)� Poisson’s ratio (isotropic case)

�hh Poisson’s ratio for effect of horizontal strain oncomplementary horizontal strain

�hv Poisson’s ratio for effect of horizontal strain on verticalstrain

�vh Poisson’s ratio for effect of vertical strain on horizontalstrain

�1 Poisson’s ratio for effect of strains in isotropic plane(x9–z9)

�2 Poisson’s ratio for effect of strain in y9 direction due tostrain in x9 direction

integration variabler ovalisation ratio� analytic coefficient�i normal stress�v0 overburden pressure�k boundary in �k domain�ij shear stress

�k (z) analytic function

Superscripts+ corresponding to cavity at (0, H)� corresponding to cavity at (0, �H)c ‘corrective’ solutions

SubscriptsB boundaryk integer (assumes values 1, 2)

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effect of anisotropy in ground movements caused by tunnelling

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