Predicting the dynamics of groundsettlement and its derivitivescaused by tunnelling in soilby P. B.ATTEWELL, DEng, PhD, MICE, IIt J. P. WOODMAN, MSc

Equations are developed for the predic-tion of the three-dimensional ground move-ments and strains caused by tunnelling insoil. Predicted settlements are comparedwith some reported case histories and aworked example for the calculation of de-formations and strains is given.

1. IrltfocIUcbcÃlAT THE PLANNING STAGE of a tunnellingscheme, and in the light of factual andinterpretive ground investigation evidence,the project promoter or his representativewill be keen to ensure, by choice of ap-propriate construction techniques and aids,that structures adjacent to the tunnel suf-fer no damage or disruption as a result ofthe tunnelling operations. Any attempt topredict the effects of ground movements,caused by excavation, on buildings andin-ground structures such as water and gasmains encounters difficulties not only of atheoretical nature but also arising from therequirement to have sufficient informationabout practical circumstances. However,recognition of these problems does notsignificantly reduce the need for predic-tive information on induced movementsin the soil. The analyses outlined in ahisPaper were developed as a result of par-ticular environmental problems, and it issuggested that any predictions based onthe analyses are likely to be more relevantto overall planning and design than tospecific practical cases.

During recent years numerous case his-tories of ground movement measurementshave been reported in the literature. Mostof the measurements relate to ground sur-face since it is less easy to instrument forand actually measure movements belowground. Furthermore, the surface measure-ments are usually restricted to verticalmovements, or settlements, since the mea-surement of lateral movement again pres-

eEngineering Geology Laboratories, University ofDurham, England

ents some difficulty. But a more severelimitation arises with respect to the datainterpretation. Measured movements aheadof the tunnel face have effectively beeriignored in the interpretation, attention hav-ing been concentrated on the permanent-set movements that have developed in

planes transverse (normal) tio the tunnelcentre-line and behind the face. However,above-ground and in-ground structureswill respond transiently to the forward (ofthe tunnel face) elements of the settle-ment trough, and before any more complexstructural analysis can be contemplatedit is necessary to provide an interpretivethree-dimensional setting for the expectedground movements.

A useful starting point for the analysisis the form of the transverse settlementtrough that has developed after passage ofthe tunnel face. It is assumed that mostof the displaced volume of the trough perunit distance of face advance can be at-tributed to face- and radial-take losses.These have been discussed in some detailby Attewell (1978). As time passes, andthe face advances well beyond the trans-verse vertical plane of interest, additionaltime-dependent settlements caused bysuch mechanisms as ground consolidationare superimposed on the ground-loss set-tlements. These longer-term processes ofspecial importance in clay soils, and whichwill increase the final volume of the sur-face settlement depression, contribute re-latively little to the early transient de-

formation picture, and so are not consider-ed in the Paper.

There seems to be reasonable case his-tory evidence from tunnelling in differentgenerally-cohesive soils ranging from clays(see, for example, Schmidt, 1969; Peck,1969; Attewell 8t Farmer, 1974; Attewellet a/., 1978; Glossop fit Farmer, 1977; Glos-sop, 1980) to fill (Dobson et al., 1979;McDeitnott, 1979) that the permanenttransverse settlement profile can be des-cribed in terms of a normal probability, orGaussian, equation (Fig. 1). The applica-tion of such an equation to settlementtroughs in granular soi'Is is less secure.For the more cohesive soils the method ofmeasurement data fits to a normal proba-bility equation has been noted in Attewell(1978) and Dobson et al. (1979), and it isfound —as might be expected —that thequality of fit is variable. In spite of attem-pts to formulate other functions for parti-cular cases it has been concluded that thenormal probability curve remains the mostappropriate for fitting to transverse set-tlement profiles in general, and for use inthe prediction of permanent settlementdistribution (Norgrove et al., 1979). Stoch-astic analysis also provides analytical jus-tification for its adoption (Litwiniszyn,1956; Sweet 8t Bogdanoff, 1965; Glossop,1977; Attewell, 1978). Having achieved abest-fit curve to transverse settlementdata, the particular profile may then becharacterised by its maximum settlement(equivalent to the mean point of a statis-

Transverse distance from tunnel centre line. Iy axis)

2I I 0 i ~Bi 2i

aximumty=+3i

0 445 w mex /I

Seulement at point of inflexion (y =i)w=O.BOB wmsx

Fig. 1. Normal probability form of transverse settlement profile

FILL

70 ~&

g x7 Sandy

clay 7 Building wasteand brick rubble

Grey silty

Ash v material

Boulder clayFill boundary

EFill b

Silty clay

nd and

0 5 10 15 20

Scale —Metres—————10—————Settlement contours. mm

V GWL when first struckHighest recorded GWL

10 30 500 20 40

0''cales —metres

Interpolated Sl boreholesshort at this level

Ider clay

Fig. 2(a). Measured distribution of ground settlement above atunnel (face entered clay; upper overburden fill) Fig. 2(b). Cross-section of tunnel for Fig. 2(a)

November, 1982 13

tical normal distribution), and the trans-verse distance from t'hat maximum to thepoint of inflexion or maximum gradient(equivalent to the standard deviation of astatistical normal distribution). These twoparameters, classified with respect to thetype of ground and tunnel geometry (ex-cavated face area and depth below groundsurface), may then be used for the predic-tion of settlement and settlement deriva-tives (displacement, strain, gradient, cur-vature) across the transverse profile (Nor-grove et al., 1979).

Having accepted the normal probabilitycurve in respect of the transverse settle-ment trough, it seems logical to assign acumulative probability function to the set-tlement trough on the tunnel centre line

projecting forward from the face, notingthe general form of settlement distribu-tion about a tunnel face as typified in Fig.2. The aim of this present Paper is toinvestigate such a procedure by analysisand by reference to some case historydata, and also to derive equations for set-tlement, lateral displacement, and strain,that might be used predictively for anypoint on the ground surface radially distantfrom the tunnel face.

It is emphasised that the present Paperis concerned largely with predicted grounddeformation parame:ers. The amplitudesand amplitude-locations of each parametershould not be interpreted directly in termsof structural deformations. It is of interestto note that Geddes (1978, 1981) has sug-gested one method of approaching theground-structure interaction problem in thecontext of coal mining subsidence.

2. AnalysisFig. 3 shows the coordinate system to

be adopted: x, or x,—parallel to the tunnel centre line; y, or x,—transverse (nor-mal) to the tunnel centre line; z, or xs-vertical. The respective displacements andstrains are u, v, w, e, ev, e . Thus, thesettlement in this notation is w rather thans or Rv, which have been used previouslyin the literature. The tunnel is at depth z0.

In conformity with earlier Papers (see,for example, Norgrove et al., 1979, Appen-dix 2), the following symbols apply:

V is the volume (m') displaced by thesettlement per unit face advance (m)for a linear loss source, and is simplythe volume displaced for a point losssource;

i is the transverse horizontal distancebetween the points of maximum set-tlement and inRexion at depth z given

I 2 —Z fl0by —= K„—,where

a 2an is usually an empirical parameter but

may sometimes be derived theoretic-ally,

a is a material parameter, andK, is chosen in accordance with the defi-

nition of a.A discussion on the justification of equa-tions of this type is given in Attewell(1978), and in Appendix A.2.1 Point source of loss

Consider, first, a point source of groundloss at x, = 0, y, = 0, z,. Let r be the

horizontal radial coordinate (= VX2 + y'),and let p be the radial displacement (p =0 when x = y = r = 0). The tangentialdisplacement q equals 0 as a consequenceof circular symmetry.

Then, on the basis of experimental evi-dence or stochastic theoiy, the following

14 Ground Engineering

assumption is made for ground that is flator nearly flat relative to source depth:

V —I'2

w = —exp —,z(z,... (1),2r//2 2/2

2,+2I +22=0 (3),with 2> written as the hoop strain. Theabove equality is clearly approximate andarguable, but under the envisaged condi-tions of low confinement it is thought tobe not unreasonable.Since

Bp

Brand

P2// = —+

I'hen

ry + fg

(4).

1Bq p——= — (because q =0) (5),r B/ii r

Bp p+ —= —2

Br r

w —-2 ... (6).

The solution of eqn. 6 with boundary con-dition p = 0 at r = 0 is given by

I

p = — ————2r X

0

exp —dr'/2

-nI'W

2 —Z0Now

(7).

u = —pI'nd

yV = —P

rtherefore

(8).

(9).

u = —xwZ —Z0

and

( 10),

V = —yWZ —Z0

It follows that

Bu

BX

andBv

vBy

n X2—w —-1 ... (12),Z —Z0

/2

n—w —-1 ... (13).Z —Z0 /2

I z —z n0and so, given that —= K,

a 2a

Bw r2 n2. = —= -w —-2 —... (2).

Bz I2 Z —Z0

For non-flat ground this assumption canalso be made, but only at levels belowthose of neighbouring ground troughs, orat least close to them in comparison withsource depth.

It is further assumed that the soil atground surface undergoes no volumetricstrain as settlement develops, that is,

Fig. 3. Tunnel coordinate system

2.2 Linear source of lossThe foregoing equations are of limited

value to the practical tunnelling case. Itis necessary to consider a point sourceof loss at depth z,, on the line y = 0, moving forwards along the x-coordinate axisfrom x,. to xl. (Subscript 'i's used to de-note 'initial'r tunnel start point, and sub-script 'f'enotes 'face'r 'final'. The con-dition is shown in Fig. 3. Time will be sub-sumed under the parameter xl, and so itwill not be taken into account specific-ally. In effect this means that the rate oftunnel advance is assumed to be slowcompared to the dynamics of groundmovement, or that tunnel advance hasstopped temporarily at xf, and time is al-lowed for dynamic effects to fade awaybefore measurements of ground movementare taken. By using an effective value ofxl one may approximate the situation whenradial loss occurs over a short length oftunnel as well as face loss, or when thetunnel moves at a moderate but constantspeed.

It is supposed that ground loss settle-ments consequent to increments of tun-nel advance are additive. In this case

XlV -( (x-x~) '+y'-)

w = —exp2/ri 2 2/2

x;

dX0

exp —G

( 14),

where

1 —P2G(x) = —exp —dP ... (15)

can be found from standard probabilitytables (see Table I). In particular,

G(0) = ~ ~

G(<x:) = 1.

In practice, given that the tunnel is suffi-ciently deep to be regarded as a linearsource, then the prediction of settlementusing eqn. 14 may be resolved into thefollowing steps:(a) Estimate the volume ground loss V perunit distance of advance. Do this by mea-

surement on a transverse profile —on-sitelevelling when the tunnel face has advan-ced a distance of approximately z, be-yond the measurement line, followed byscale section planimeter integration or by

the use of the equation V = ~2~ i w ax—or from a consensus of case history evi-dence of tunnelling in similar soils (seeAttewell, 1978). The existence of consoli-dation will complicate this calculation, butthere are two approaches under which theuse of eqn. 14 is acceptable. The consoli-dation component of a composite groundloss-consolidation V may be extracted inorder to predict the corresponding con-solidation-free component of settlement.Alternatively, it may be retained if con-solidation does not alter the source dis-tribution in such a way as to significantlychange the form of w.(b) Estimate the transverse profile inflex-ion distance i. (Again, do this by directmeasurement on a fully-established trans-verse profile, or from a consensus of casehistory evidence).(c) Note the y (tranverse) coordinatedistance from the tunnel centre line forthe point of interest on ground surface.(d) Take the differences between the x-coordinate of this point and the x-coordin-ates of starting (x,.) and current (x<)points, with the proviso that xf mayhave to be replaced by an effective value—see later discussion. Normalise each ofthe two values by i, and determine ineach case the cumulative probabilityfrom Table I.

To illustrate the procedure, considerthe following example, which relates tothe Fig. 4 case history:

(x —xI) /i

0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.22.32.42.52.62.72.82.93.0

0 1 2 3.500 .504 508 .512.540 .544 .548 .552.579 .583 .587 .591.618 .622 .626 .629.655,659,663 .666.691 .695 .698 .702.726 .729 .732 .736758 .761 .764 .767.788 .791 .794 .797.816 .819 .821 .824.841 .844,846 .848.864 .867 .869 .871.885 .887,889 .891.903,905 .907 .908.919 .921 .922 .924.933 .934 .936 .937.945 .946 .947 .948.955 .956 .957 .958.964 .965 .966 .966.971 .972 .973 .973.977 .978 .978 .979.982 .983 .983 .983.986 .986 .987 .987.989 .990 .990 .990.992 .992 .992 .992.994 .994 .994 .994.995 .995 .996 .996.997 .997 .997 .997.997 .998 .998 .998.998 .998 .998 .998.999 .999 .999 .999

4 5

.516 .520

.556 .560

.595 .599

.633 .637

.670 .674

.705 .709

.739 .742

.770 .773

.800 .802

.826 .829

.851 .853

.873 .875

.893 .894

.910 .911:925 .926.938 .939.949 .951.959 .960.967 .968.974 .974.979 .980.984 .984.987 .988,990 .991.993 .993.994 .995.996 .996.997 .997.998 .998.998 .998.999 .999

6 7

.524 .528

.564 .567

.603 .606.641,644

.677 .681

.712 .716

.'I45 .749

.776 .779

.805 .808

.831 .834

.855 .858

.877 .879

.896 .898,913 .915.928 .929,941 .942.952 .953.961 .962.969 .969.975 .976.980 .981.985 .985.988 .988.991 .991.993 .993.995 .995.996 996.997 .997.998 .998.998 .999.999 .999

8 9.532 .536.571 .575.610 .614.648 .652,684 .688.719 .722.752 .755.782 .785.811 .813.836 .839.860 .862.881 .883.900 .901.916 .918.931 .932.943 .944.954 .954.962 .963.970 .971.976 .977.981 .982985 .986.989 .989.991 .992.993 .994.995 .995.996 .996.997 .997.998 .998.999 .999.999 .999

X.XII

TABLE I. NUMERICAL INTEGRATION OF THE NORMAL PROBABILITY CURVE

X-XITable of G

I

w „=7.86mm;i = 3.9m.

Thus, V = 0.08m'/m.

Let (x-xI) = 4m; (x-x,.) ~ac; y = 1.5m.

Therefore, G —= 1; G

= G(1.02) = 0.846 (from Table I).

exp + —-1 x

G G — ... (18).

The transverse, y-coordinate displacementis given by

The x-coordinate strain is

W — X

exp — —X

It follows that w = 1.12mm.

Eqn. 14 can be expressed in the form

v =—ywz —z0

(19),—(X-XI)2

exp2I2

(22) .

whereV

w„=—expy2

2I2

w=w„G ——G

(16),

(17)

where w is given by eqn. 14.

The y-coordinate strain is

Bv ny'9= —= —w ——1 ... (20).

y zo —z

Since ground gradient has been usedas one criterion for predicting the stabilityof a surface building (Norgrove et al.,1979), and since it can also affect the in-

tegrity of, say, a linear in-ground structuresuch as a pipe, it is useful to obtain anexpression for that gradient fI.

The x-coordinate displacement is given by Along the y-coordinate axis

is the asymptotic form of w as x, m —oc,xI ~ oc. w„ is the established 2-dimen-sional 'permanent set'ormal probabilitysettlement form used by the several auth-ors quoted above.

I z —zn0Assuming that —= K, —,other re-

a 2asuits follow from eqn. 14. For the verticalstrain:

U

z —z0

XIV

(x-x,) X2III2

x,

—( (x —x0) ' y2)

exp dxo2I2

II, = —= —w= —G

— (',")Iand along the x-coordinate axis

(23),

2 = —=—wz —z0

Iw„—ex'p—(x-x,) 2-

2I 2

Bw wII, = —=—exp

Bx

-(x-x,)2

2I2

exp — —X

—(X —XI) 2

-exp2I2

(21)—(X —XI) 2

—exp2I2

(24),

November, 1982 15

where x'=x+0) and y'=y+v) are thefinal position coordinates.

Excavations of finite width and length atdepth z, are immediate generalisations ofthe above results. In particular, the verticaldisplacement is still given by eqn. 16, but

with w„= V G ——G

and V defined as the volume displaced perunit areal face advance. Excavations of fin-ite volume are best treated by noting thatground loss only occurs at the boundarysurface of the excavation, but there isanother complication. The existence of afinite hole violates the assumption of mat-erial continuity, so that, even within theframework of this particular theory, resultsare approximate only, unless this specificproblem is treated.

2.3 Structural deformationWhen considering the integrity of a

grounded structure, the direct connectionof the above results to structural disp'lace-ment and strains is problematic. Indeed,this is true for any theoretical approachthat does not specifically include the geo-metry, and the material constitutive rela-tions of the structure itself. Accordingly,the remaining displacement derivatives areincluded in Appendix B. If ground integrityis pf interest, or in the unlikely event thatstructure and ground are indistinguishablein terms of material behaviour (except pos-sibly in failure point), the shear strainsand rotations are given in terms of the de-rivatives by the following formulae:

Bu Bu.

yj = —+-Bx. Bx.

(25),

(26)

Note that ej = 0 for both linear and pointBv Bu

sources, since —= —.Bx By

Given that, at the levels in the groundof structural significance, the assumption

+ e„+ e, = 0 is reasonable, the resultsin this Paper can be used as boundaryconditions for conventional stress analy-sis of grounded structures. However, thereis one other proviso. Under the "egis ofstochastic theory used in the calculationof w, basic assumptions are violated bythe structure. Although it may conceivablysatisfy the same material constitutive re-lations as the ground, it cannot respondin the same way to random displacements(almost by definition). This will alwayshave a local effect, but if the structureis sufficiently extensive the global volumeloss may also be infiuenced to producelarge errors (large variations in groundsurface produce similar results). The em-pirical approach to the form of w does notexplicitly, or implicitly, make the assump-tions underlying stochastic theory, but it islikely that experimental observation woulddemonstrate analogous effects. This pointwill not be considered further. It will alsobe assumed that material non-linear be-haviour can be catered for.

Once slipping occurs at the ground-structure interface, displacement ratherthan strain becomes the material con-stitutive parameter over areas of slippage

16 Ground Engineering

(Geddes, 1978, 1981). It is sometimes ap-propriate to resolve the slip displacementfunction on the interface into a mean trans-lation, and a differential displacement func-tion. 'If the structure is smail, a Taylor ser-ies approximation to the displacementfunction may be adequate, but althoughs.rain reappears as a parameter its use isfundamentally different from that beforeslipping.

By way of example, two contrasting andlimiting cases follow. In the first place,consider a pipe, which is materially indis-tinguishable from the ground, buried paral-lel to the x-axis. Competence of the pipecan be assessed in terms of the groundstrain components, but it is more usefulto consider parameters that are more ap-propriate to the pipe geometry. Assumingthat the pipe is not subject to crushing orshear failure, then, under conditions ofsmall strain, the most important paramet-ers are

axial strain, e ,Bcjj

the material curvatures, —,Bx

BQj

and —,Bx

BCBj»

and the torsional strain, —.Bx

The general case also requires e> e» y ~y„, and y,.

Now suppose that the same pipe isfurnished with an almost perfectly smoothsurface such that the small frictional forcegenerated on the surface is proportional toslip. There will be a loss of shear acrossthe ground-pipe interface but, for simplic-ity, the effect on the displacement field will

be neglected. Let u be the mean grounddisplacement along the line of the pipe.(If there are external constraints on theaxial movement of the pipe, then put u =0). Then the displacement field in thepipe (structure S) is given approximatelyby

08 0v„= v(x, y,, z,)w,= w(x,y,, v)

where x is the apparent initial position ofthe point at x, relative to the ground. xis found by equating position coordinatesafter displacement, in ground, and S.

x =X '(x„+0) (29)

Alternatively, from eqn. 28

Bux» + 0 = x + 0 (x,) + —(x-x,)

Bx

x, ... (30).Therefore,

0 —0(x )x x +

Bu1+—Bx

XB

(31)

x + 0 —0(x) (32),

x' X(x) = x+0(x) = x', x +0... (28)

Therefore,

Buif — (x- x,) is small (small strain

Bx

XB

and/or small slip, x —x, = u —0(x,) ).From eqn. 27,

v, = v(x,y,,z,)Bv

= v (x„, y„, z,) + — (u-u(x,))Bx

x(33)

= v(x,, y„, z,) for small strain and/or slip (34) .

Similar arguments apply for w, .

BuThe axial strain e,

Bxethere is no shear or torsional strain

The parameters that remain are

B»v

and —.Bx»e

0, and

either.Bewe

Bx»e

Another common structure is a horizon-tal planar slab foundation perpendicularto the z-axis. If it fo'Ilows ground dis-placement exactly, the relevant parameter

BCjj BCjj»j BCjj»

liSt iS e, e», y»<, —,—,—Bx By Bx

Bcjj—.parameters e,, y„», y, may be ad-By

ded if needed. When the slab is almostperfectly smooth, only the bending terms

Bew Bew Bew»and are re-

Bx,e 'y,e Bx, By,quired, and the calculation of 0,, v,, w,is analogous to that for the smooth pipe.

By paying excessive attention to thedetailed analysis, and in regarding theabove examples simply as limiting cases,there is a risk of ignoring what could turnout to be good first-order approximationsin many cases. Whenever the stress-bear-ing part of the structure is sulciently thin,on strain-energy considerations the totalforce it can exert is relatively small what-ever its stiffness, ground-structure interac-tion will be highly localised, and providedthat the ground does not shear and flownormally to the structure by this interac-tion then structural displacement will beclose to ground displacement. In this situ-ation, the 'rough'nd 'smooth'odelsmay be considered reasonable approxima-tions to structures with and without pro-tuberances (flanges, for example, in thecase of in-ground pipes), respectively.

2.4 Stationary points on the linear-sourcedisplacement profile

There are obvious difficulties of accurate-ly relating structural movement to groundmovement. One method of tackling theproblem would be to create a 3-D finiteelement model of the structure in-ground,to apply the theoretical ground displace-ments 0, v, w to the boundaries of thesoil-structure system, and to observe theinduced strains in the structure. Since thelocation of any structural failures will bedetermined by the position of certain maxi-ma on the displacement profile, and sincethe values of these maxima could be help-ful in assessing the risk of failure, it is

reasonable also to examine the analyticalexpressions for the maxima and their lo-cations.

—n V—expZo —Z I- I

3X

2exp

—(s —I) 2

2J2(46),

Vw =—G'aty=0

/z(35).

nIvl .„=—

Z —Z0

Vexp

1

G2

X —XfLet G 'enote the factor G

I

X —Xr X —X/G —,which becomes 1 —G

I I

as xt m —0'or a semi infinite tunnel. Ontransverse profiles, x is constant, and thefollowing equations are obtained simply byinspection or differentiation of earlierresults:

-(x-x,)'(x-x/)'xp

—exp2I2 2I2

Most of these equations replicate the ear-lier-published equations for the transversetrough case except for the multiplying fac-tor G'.

For the off-centre line profile y is con-stant, and a related set of results can beobtained. Let x denote the mid-point x-coordinate of the tunnel, and 2I = x/-xf,tunnel length. Arrows ' 're used toindicate limiting values for the semi-infinitetunnel when x,. m —ltd.

w „=w„2G ——1 atx=x

(47) .

—(t —I)'xp

2I2

atx=x ~t... (48),

-+n

Z —Z0

Wm 1—exp2

at x = x/ ~ i ... (49),

n w„ I (t+I) (t-I)+i'Xss'ttt

z, —z ~2 I(t+I)'-i't

y (36) . (4f). where t is a positive or zero root of

n V2ytmax

z,-z

32exp —— G'

aty = ~ /3i (38) .

Vl(i„,l „=—exp

By eqn. 23

2G'sty = ~i

(39).

—n V—G't y = 0 ... (37).Zo —Z 21r I

—> w„at x ~ —00 aS x,. —> —<x

(any x if x/ ~ rx: as well) ... (42).

n i 2IIul „,, =—w„——x

zo z ~2fr s+ I

—(S—I) 2

exp atx = x ~s ... (43),2I 2

n I~—w —at x = x/ ... (44),Z,-Z

where s is the positive root of

-(t+I)'t

+ I)' i'xp2I'2

—(t —I) 2

(t —I) ' i'xp2I 2

(50)

w I I2

There are three non-trivial roots (0, ~ t,)to this equation if I ( lt3 i, and five roots

(0, ~ t,, ~ t,) if I ) +3i . The solutiont = 0 yields a compressive maximum orminimum according to whether I is less

than or greater than +3i, respectively.

(40) . —(s + l)2(s+I) exp = (s-I) x

2I2

e/

Z —Z0

(51).Hence, maximum bending in the verticalplane of a horizontal transverse pipe is

exp—(s-I)'45) .

Eqn. 50 also locates the following station-ary values:

2V 3exp —— G', and occurs at

$ 2fr Is 2

2I 2

This equation also gives an approximationto the stationary points of 8,. (

a w w„ I (t+I) (t-I)+isX

t)x2 +2;2; (t+ I) ' is

y = m +3i. Maximum bending in the hori

zontal plane also occurs at y = ~ )J3 i,and has the value

I8'I -=i)w

t)xmax

s+ I

—(t —I) 2

exp (52),

TABLE II. CASE HISTORY TUNNEL AND GROUND INFOR'MATION

Fig.No.

TunnelLocation

Ground Exes v.die,2R

(m)

Axis i K„ndepth (m)

to(m)

Max. Troughsettlement vol.

(mm) (% or ms/m)

CohesionC

(kN/mr)

Liquidlimit(%)

Plastic Bulk Moisturelimit density content(%) (kN/ms) m

(%)

Permeabilityk

(m/s)

4 N.W.A. Contract 31 Stony/laminatedTyneside Sewerage clayScheme (Hebburn)

2.014 7.5 3.9 1 0.98 7 86 0.077m'/m 73to

275

53 23 19.0 28 2x10'

London Transport London ClayJubilee Line(Green Park)

4.146 29.3 14.5 61 0.196m"/m 60to90

24to30

19.0to

201

23to29

6 Sewerage tunnel, SleechSydenham, (soft, saturated

Belfast alluvial silt)41 kN/ms c.a.

2.7 5 75 2.5 16 2.1% 8.7 47 14.8 75 157 x10ito

382 x 10"

Anglian WaterAuthority, Haycroft

Relief Sewer,Grimsby

Silty clay(Grimsby

marinewarp)

41 kN/mr c.a

3.1 5.7 3.2 16%( 1.08m*/mwithout c.ax0.65m"/mwith c.a.)

12 27 180 10 l0

8 N.W.A. Contract 276 Urban andTyneside Sewerage industrial

Scheme fill

(Ousebum)

3 471 13 7.29 1.69 0.69 81 0.355m'/m 145 21

9 Newcastle upon Tyne Urban fill

City Sewer, and stiffdark clay

2.15 113 35 1 08 17.2 0.21ms/m 171 35 17 21 0 12

18 Ground Engineering

82w

ifx2

xaI

exp —— ... (53).~21—w„ I -I2—exp —... (54) .

Distance ahead of tunnel face. x axis. m

12 8 4 0 -4 -8 .12 .16I

y=0

-3E's

t,E -i4

+

01

++

Distance'ahead of tunnel face. x axis, m

8 4 0 -4 -8 -12I I I I

4

9-6E 1E2-8

e --1E

--1IO

-16I

(t+I) (t —I)+i't

+ l)2 —i2

~n w y

ze —z

—(t-I) 2

exp2f2

(55),1

exp —— ... (56).2

$2v

8X2 Zo —Z

"m

w y I I2—exp

(57) .

Ground

Sand above watertable

Stiff clays

Sand below watertable

Silts and soft clays

Face of Tail ofshield shield

30- 50% 60- 80%30 —60% 50- 75%

0- 25%0 —25%

50- 75%30 —50%

Even accepting that some delay on set-tlement transmission to ground surfacemust occur (any delay depending, amongother factors, upon the type of groundmaterial through which the movements

3. Case histories: x-coordinateground settlement

Six settlement case histories are shownin Figs. 4 to 9 inclusive, with some addi-tional data being given in Table II. In mostcases they provide some qualified con-firmation, via the form of the curves, thatthe extension of the normal probab'.Iitycurve-fitting concept from the transversey-axis two-dimensional case to the fullthree-dimensional profile is reasonable forthe important zone of ground movementprojecting transiently ahead of the tunnelface. The x-axis offset between the pre-dicted and measured curves arises simplybecause the 50%w „point (and thepoint where the rate of change of settle-ment maximises) on the former curve hasbeen drawn to coincide with the passageof the tunnel face beneath the point, withthe implied assumption that the surfacesettlement occurs at the same time asthe ground loss which causes it. Withthis last assumption, the separation ineach case between the two curves maybe taken to indicate that in the case ofclay soils a greater percentage of the over-all settlement can be attributed to face-and early radial —take as compared withthe weaker, less cohesive soils. It is gen-erally understood that a higher percentageof the overall ground loss at a tunnel inmore granular soil occurs towards the tailof a shield, and that in such material thetotal losses are much more dependent onexcavation and ground support technique.A suggestion by Craig (1975) as to thepartition of the total settlement to theground loss at a shield-driven tunnel isgiven below:

y = 1.5m

ce --6E

--7+

6I

EE --3~

--4

y = 4.5m

Fig. 4. Forward x-coordinate settlementdevelopment profiles for NorthumbrianWater Authority's Contract 31 (Hebburn)South Bank Interceptor Sewerin stony/laminated c/ay (Attewell 8 Farmer, 1973).Full lineis measured profile; broken lineis theoretical profile (50% w,„assumedcoincident with plane of tunnel face)

Distance ahead of tunnel face. x axis, m

40 30 20 10 0 -10 .20 -30 -40

y=O

c --400 EE E--5III

~ h--6--7

y= Sm

EE

I I

E-"3y= 10m

I I I

4-5

I

y= 15m002

3-4

Fig. 5. Forward x-coordinate settlementdevelopment profiles for a Jubilee Linerunning tunnel at Green Park Station, Lon-don in London Clay (see Attewell 8 Far-mer, 1974). Full line is measured profile;broken lineis theoretical profile (50%wassumed coincident with plane of tunnelface)

are taking place) it would seem, from anexamination of the several predicted andmeasured curves, that a better matchcould be achieved by assigning localground loss sub-maxima to different posi-tions at and back from the tunnel face. Asa general design guide for tunnelling in,for example, firm-to-stiff clay, the ratiow/w „at x = 0 (the face position) would

EE-1

+IU

E

IllIO

EE-

3.Om

Distance ahead of tunnel face.x axis, m

8 4 0 -4 -8I I I I

-10

II —-20

E sI-30E

y --40

ce --50a --60Io 70

-12 -16I

y=o

E»0~E

1c

III 5li

I

EE ~ 010

-20

y = 4m

I I I I I I

y=6m

Fig. 7. Forward x-coordinate settlement de-velopment profiles for a tunnel at Grimsbyin alluvial silt (Glossop, 1980). Full lineis measured profile; broken line is theo-retical profile (50% w „assumed coinci-dent with plane of tunnel face)

seem to lie between 30 and 50% with anaverage value at about 40%. Contributingto this percentage will be a portion of thelosses at the face plus a decreasing por-tion of the losses back down the shieldand the.jirsed tunnel. It will also be notedthat the losses at the face contribute sub-stantially to the settlement projected be-yond the face (x ) 0). In such a clay,

November, 1982 19

Fig. 6. Forward x-coordinate settlementdevelopment profiles for a tunnel in Belfastsleech (Glossop 8 Farmer, 1977). Full lineis measured profile; broken line is theore-tical profile (50% w „assumed coinci-dent with plane of tunnel face)

Distance ahead of tunnel face. x axis. m

25 20 15 10 5 0 -10 -20 -30 -40 -50

--10 y= 0--20

1--30

E )-40E

--60cE"--70 ~

R~ --80<Il gp

--10E --20

W %30E1

- -%40cE

--50y --60(I( 70

I I I I

g~E 10 y= lpm

1- - 20

u -F30E"--40J) --50

Fig. 8. Forward x-coordinate settlementdevelopment profiles for NorthumbrianWater Authority's Contract 276 (Ouse-burn) North Bank Interceptor Sewer infilled ground (see Dobson et al., 1979).Full line is measured profile; broken lineis theoretical profile (50% w,„assumedcoincident with plane of tunnel face)

y= 5m

Distance ahead of tunnel face. x axis, m

12 8 4 0 -4 -8 -12 -16 -20

2

-4 y=p

.6E.I(8Eg--ip

c "12XeE- 14Ol

a 16i)I ..--18

Fig. 9. Forward x-coordinate centre linesettlement development profile from asewer tunnelin Newcastle upon Tyne. Con-struction at the point of measurement waspartly in clay and partly in fill, with filloverburden. Full lineis measured profile;broken line is theoretical profile (50% w „assumed coincident with plane of tunnelface)

practical observation suggests that dis-cernib'le settlement development would ex-tend over a distance of approximately 1z,ahead of, and 2z, behind the tunnel face.This attenuated settlement behind the tun-nel face can be modelled by assigning,say, a ground loss of 80% to the face,and the remaining 20% to the point x =—z, . If the face advanced at a uniform rate,then, given the means to perform calcula-tions rapidly, the total loss can be distri-buted along the tunnel to match an x-axissettlement profile. It is pointed out that theform of the x-axis profile over a distancex/z, > -0.25 x/zc (that is, the forward profile and the advance portion of the laggingprofile) is relatively insensitive to suchdistribution of ground loss.

An underlying problem in such predic-tive analyses is the assignment of a set-tlement volume figure, and a value forthe n-parameter. The latter is determinedfrom the variation of the transverse settle-

20 Ground Engineering

ment trough with depth (see AppendixA(iii)). The former depends strongly uponthe geotechnical properties of the groundat and above tunnel level. Both parametersare interdependent, and rely, for their ap-plication, upon a corpus of case historydata. Reference may be made to the datatables in Attewell (1978), and to Norgroveet al. (1979), for the selection of n-values,and it is noted that more case historydata has become available since thosedates. It should also be noted in the equa-tions given earlier that the predicted set-tlements, displacements, and strains, aredirectly proportional to the chosen valuesof settlement volume or maximum settle-ment, but not to n, and so a sensitivityanalysis on these parameters is notstraightforward.

4. ExampleAssume that it is proposed to tunnel

through filled ground, and the groundmovements in the vicinity of an in-groundstructure 1.5m below ground level are tobe examined. Relevant data are listed be-low:(a) Soil: cohesive fill.

(b) Tunnel depth from ground surface:9.2m.

(c) Tunnel depth from in-ground struc-ture: 7.7m.

(d) Tunnel excavated diameter: 2.44m.(e) Predicted settlement volume: 5% of

excavated face area.(f) Predicted K = 1; predicted n = 1.

(K, is independent of a when n = 1,and so Ko = K).

(g) Inflexion distance i: 3.85m (from 0and f —see equation in Section 2introduction).

From eqn. 14, Fig. 10a may be construc-ted. Additionally or alternatively, plan set-tlement contours may be constructedaround the tunnel face. The predictedmaximum settlement (y = 0) ) is 24.23mm(Fig. 10a), and the 0.5w, point is loca-ted, for convenience, on the graph at x

0. For practical assessment it wouldbe recommended that the x = 0 ordinateaxis be re-set to intersect the cumula-tive probability curve (y = 0) at w =9.7mm (i.e. 0.4w,„), this re-set applyingfor all y.

Predicted x-coordinate strains and dis-placements are shown in Figs. 10b and10c, respectively and, as with the settle-ments, could be presented as plan contourplots. As above, an x-axis re-set forwardfor x = 0 at x = 1.1m would be appro-priate.

Predicted y-coordinate strains and dis-placements are shown in 'Figs. 10d and 10e,respectively, and again these could bepresented as plan contour plots. A re-setof the x = 0 axis would require each curveto be relabelled for its specified x distanceminus 1.1m.

5. CondusionsBy assuming that the soil at ground sur-

face undergoes no volume change, andthat ground movement is the sum total ofelementary movements of assumed formresulting from increments of tunnel ad-vance, it is 'possible to derive expressionswhich quantify the soil movement aheadand at the sides of a tunnel face. With em-p'irical support and some theoretical justi-fication, these expressions are based on anormal probability form of transverse set-tlement trough, and a cumulative probabil-ity form of centre line trough. The primaryexpressions for ground loss settlements

seem to approximate fairly well to mea-sured settlement profiles on and parallelto the tunnel centre line. The greatest de-parture from the predicted form occursat the fully-developed end of the centre-line profile, and an explanation for this hasbeen suggested in terms of a distributionof ground loss along the tunne'I, and aconsolidation settlement contribution.

Although ground surface settlement canbe measured relatively easily by preciselevelling, lateral movements and strains,being smaller in magnitude, can be mea-sured only with great difficulty, if at all.Analytically, because of the relative insen-sitivity of the integration operation to theassumed form of the transverse settle-ment curve, a higher degree of reliancemay be placed on the equations for grounddisplacement than on those for groundstrain, gradient and curvature which invol-ve differentiations of the profile equation.Furthermore, the results should be seen aspertaining to an approximate generalmodel, which may not apply very well toany particular case, both because of theassumptions made and the local variationson site.

More three-dimensional deformation da-ta are needed from field measurements in

order to provide a basis for modifying andextending the above approach, which canperhaps be justified at the present timemainly on the grounds of convenience. In

the longer term a physical understandingof the ground deformation mechanisms,perhaps on the lines discussed by Atkin-son & Mair (1981), should prove to bemore rewarding, and may go some waytowards answering the cr'ticisms of deMello (1981). In both the short and longterms, a major objective must be to inter-pret the predicted ground movements in

terms of above-ground and in-groundstructural response.

AcknowledgementsMr. R. McMillan and Mr. J. Yea:es pro-

vided much of the encouragement for pur-suing the analyses in the Paper and offeredvalued comments on an earlier draft. Theauthors are grateful for the interest shown.

Dr. I. W. Farmer end Dr. N. H. Glossopof Newcastle upon Tyne University kindlyprovided the case history information onthe Belfast and Grimsby tunnels.

Some of the preliminary work was per-formed by Mr. R. I. G. Gordon as part ofhis MSc. Engineering Geology AdvancedCourse at Durham University.

ReferencesArkinson, J. H. si Mair, R. J. (1981): "Soil msc-hsnics aspects of soft ground tunnelling", GroundEngineering, v, 14, No. 5, pp. 20-26.Attswell, P. B. (1978): "Ground movements csu-sed by tunnelling in soil", Proc. Conf. on LsrgeGround Movemenrs snd Srrucrures, Csrdlff, July1977, Ed. Gsddss, J. D., Publ. Peotsch Press, Lon-don, pp. 812-948.Anewell, P. B., Farmer, I. W. si G/ossop, N. H.(1978): "Ground dsformstlon caused by tunnellingin s silty slluvisl clay", Ground Engineering, v. 11,No. 8, pp. 23-41.Attswell, P. B. 8i Fermer, I. W. (1974): "Grounddisturbance caused by shisfd tunnelling in s stifoverconsolidstsd clay', Engineering Geology, v. 8,PP. 361-381.Anewell, P. B., Fermsr, I. W., G/ossop, N. H. 8Kusznir, N. J. (1975): "A csss history of grounddeformation caused by tunnslllng in laminatedclay", Proc. Conf. on Subway Construction, Buds-psst-BslstonfQrsd, pp. 165-178.Anewell, P. B. 8i Fermer I. W. (1973): "Measure-ment snd interpretation of ground movementsduring construction of s tunnel in lsmlnstsd clsyst Hebburn, County Durham", Report to Transportsod Road Research Laboratory, DoE, RssssrchContrscf No. ES/GW/842/68.Craig, R. (1975): Discussion st meeting of BritishTunnelling Society, Tunnels snd Tunnslling, v. 7,pp. 61-65.