soil mechanics aspects of soft ground tunnelling

Soil mechanics aspects ofsoft ground tunnellingby J. H. ATKINSON*, BSc, MA, MSc, DIC, PhD, MICE, FGS at R. J. MAIR~, MA, PhD, MICE

SynopsisTHE BASIC PRINCIPLES of soil mechanicsapplicable to the behaviour of slopes,foundations and retaining walls apply eq-ually to the stability of tunnels in softground and to the settlements caused bytunnelling. Tunnel engineers, however,have a separate terminology to describecertain aspects of tunnelling and they em-ploy terms such as squeeze, stand up timeand ground loss. In addition, tunnel en-gineers often attribute to creep thosetime-dependent phenomena which founda-tion engineers rightly associate with pri-mary consolidation.

Recently a number of calculations havebeen developed, mostly as a result ofresearch carried out at Cambridge Univer-

sity, which deal with the stability of tun-nels and tunnel headings and with settle-ments caused by tunnelling, These calcu-lations always consider quite separatelydrained cases and undrained cases in ac-cordance with the basic principles of soilmechanics but it is not always obviouswhich calculation is applicable for a parti-cular practical tunnelling problem.

This Paper considers the stresses andpore pressures in soft ground due to tun-nelling and it discusses the calculationsappropriate for estimating the stability ofthe tunnel and its heading and the settle-ments caused by tunnelling.

IntroductionGeotechnical engineers accept that the

principle of effective stress provides thefundamental basis for understanding thebehaviour of saturated soils and that, asa consequence, it is necessary to distin-guish clearly between stability and set-tlement calculations for drained loading,for undrained loading and for consolida-tion. Thus, calculations for the stabilityof a quickly excavated trench aro differentto those for the stability of an an-cient hillside; similarly calculations forthe stability and settlement of a foun-dation on a fine grained clayey soil aredifferent to those for the same foundationon a coarse grained sandy soil. In bothcases further calculation may be requiredto estimate the rate at which a trenchwill become unstable and the rate at whicha foundation settles due to consolidation.

It is always necessary to distinguishbetween drained, undrained and consoli-dation calculations for soils for which theprinciple of effective stress applies. Thus,for calculations for the stability of tun-nels and for settlements due to tunnellingoperations it is important to determine atthe outset whether the construction should

*Reader and Head of the Geotechnical Engineer-ing Division, The City University, London.

IIAssIstant Principal Engineer, Scott Wilson Kirk-patrick fk Partners

This Paper will be the subject of a British Geo-technical Society Informal Discussion to be heldon Wednesday, September 30, 19B1, at the in-stitution of Civil Engineers

20 Ground Engineering

be taken as drained or as undrained sothat the appropriate calculations may bedone. This Paper considers the factorswhich determine whether a particular tun-nelling operation may be taken as drainedor as undrained and discusses some cal-culations for the stability of the headingand the settlements caused by construc-tion of the tunnel.

The principle of effective stress andfundamental soil behaviour

The principle of effective stress wasstated by Terzaghi (1936) and it requ'resthat all measurable effects of a changeof stress, such as straining and changesof strength, occur as a consequence of achange of effective stress. Thus, if soil isloaded or unloaded by changing total stres-ses and pore pressure together in such away that the effective stresses remainconstant there will be no measurable ef-fect on the soil, there will be no strainsand its strength will be unaltered. Con-versely, if soil is loaded or unloaded with-out strains occurring the effective stressesand the strength will remain unaltered.If the total normal and shear stresses are<r and -, and the pore pressure is u, theeffective stresses tr'nd -,'re

If a soil structure, such as a slope or afoundation or a tunnel, remains undistur-bed for a length of time the pore pres-sure will reach steady state values in equi-librium with the boundary conditions andthe steady state pore pressure is denotedu,. If now, due to some loading, the porepressures are altered so that they are nolonger in equilibrium with the boundaryconditions the out of balance porepressure is known as the excess porepressure and is denoted as u which maybe positive or negative. Thus, the porepressure u is given by

u = u + ue (2)

Normally u„will be simply found fromhydrostatic conditions or from a steadystate seepage flownet but u will not us-ually be easily determined.

For the present we will examine soilbehaviour in terms of stress parameters tand s and strain parameters e~ and e„given by

= —,'<,—irs) S = T (<, + ~a) (3)

ey —iti e« i!i. —ei + es (4)

and in terms of the specific volume v isgiven by

v =1+ wG,

where w is the water content andG, is the specific gravity of the soil

grains,

The derivation and use of these para-meters was discussed by AtkinsonBransby (1978),

From eqn. 1 we have

t'rand s' s-u (6)

The essential features of soil behav;'ourare illustrated in Fig. 1. The line AB isthe normal consolidation line and the lineBD is a swelling line, and these representthe states of soil in the ground before anyconstruction has taken place; the sampleat B is normally consolidated, that atC is lightly overconsolidated and that atD is heavily overconsolidated.

In practice few, if any, natural soilsare truly normally consolidated; the statesof soft clays and loose sands are at pointssuch as C and these soils are taken to belightly overconsolidated; the states of stiffclays and dense sands are at points suchas D and these soils are taken to be hea-vily overconsolidated. The double line isthe critical state line which representsthe states of all samples at ultimate fail-ure, and the projection of the critical stateline on the t': s'lane is given by

t = s . Sintii..'8)where yea's the critical state friction an-gle. Alternatively, the ultimate failure ofsoil is given by

-,„' tr„'an tjiea (9)

where -,„' rn and o.„' <rn —u are theshear and normal stresses on a particularslip plane through the soil on which fail-ure is taking place,

Eqns, 8 and 9 are the same as the Mohr-Coulomb criterion of failure with zero co-hesion appropriate for ultimate, or criticalstate failure of remoulded soil, For over-consolidated remoulded soils the peakstrength is greater than the critical statestrength but the peak strength is onlymobilised for very small strains. For nat-ural soils, the peak strength may be grea-ter than the peak strength of the samesoil remoulded to the same state but,again, the peak strength will be mobilisedonly for very small strains and, after mod-est strains, the strength of natural soilwill approach the ultimate, or critical statestrength of the same soil remoulded tothe same state.

In practice it is quite possible that soilstrains which occur during construction

The parameter t is a deviator or shearstress and the parameter s is an averageor normal stress neglecting the influenceof the intermediate principal stress trs .

Similarly e> is a deviator or shear strainand ee is a volumetric strain and is givenalso by

5vhe'V

v

(CriticalStateLine)

0

U,

(a) S =S —U

I ~Undrained(a) (b)

drained

lb)

= cII ( 10)

where c„, the undrained shear strength,decreases exponentially with increasingspecific volume.

will be large enough to take the soil closeto the ultimate, or critical state strengthand, as argued by Atkinson & Bransby(1978), it is logical to base designs on theultimate strength given by eqns. 8 and 9.This approach is logical and conservativeand it would be reasonable to use lowerfactors of safety in the design than wouldcommonly be used if the peak strengthwas chosen.

As normally consolidated soils areloaded towards ultimate failure the statesmap out a surface which is known as thestate boundary surface, discussed in somedetail by Atkinson & Bransby (1978).The essential features of a state boundarysurface for soil are that it limits all pos-sible states for a particular soil; if the statelies on the state boundary surface thesoil is normally consolidated and its be-haviour is inelastic and if the state liesinside the state boundary surface the soilis overconsolidated and its behaviour iselastic.

An underlying assumption within theframework of critical state soil mechanicsis that all samples with the same specificvolume and loaded undrained with nochange of specific volume will ultimatelyreach the critical state line at the samepoint and they will all have the same un-drained shear strength, irrespective of thetotal stress loading path. For undrainedloading we may write

Fig. 1 (left). Con-solidati on, swellingand failure lines forsoil

CSL

S =S —U

Fig. 2 (above).Stresses, pore pres-sures and volume-tric strains duringdrained and un-drained loading andconsolidati on(a) Undrained load-

ing and consoli-dation

(b) Drained loading

The important feature for design pur-poses is that if the loading is such thatthere is no volume change the soil strengthis given by eqn. 10 and we need not con-sider effective stresses; for all cases whenthere are volume changes the strengthis given by eqns. 8 or 9 and it is neces-sary to determine the pore pressures andhence the effective stresses.

For overconsolidated soil for which thestates are inside the state boundary sur-face the behaviour is taken to be elasticand increments of stress are related toincrements of strain by an appropriateform of Hooke's law. For drained loadingof isotropic elastic soil strains are relatedto effective stresses by the effective stressYoung's modulus E'nd the effectivestress Poisson's ratio I 'ut for undrainedloading it is possible to relate strains tototal stresses by the undrained Young'smodulus E„ together with v„——-',. For an-isotropic elastic soil relationships be-tween stresses and strains are morecomplex but it is still necessary to dis-tinguish between drained and undrainedloading.

For normally consolidated soil for whichthe states are on the state boundary sur-face the behaviour is inelastic and stressesand strains must be related through someelasto-plastic stress-strain law. The criti-cal state model provides one possibleelasto-plastic stress-strain law for soiland stress-strain equations are given bySchofield & Wroth (1968) and by Atkin-son & Bransby (1978). Even with theseinelastic laws however, it is still essen-tial to distinguish between drained andundrained loading.

During drained loading there are no ex-cess pore pressures and the steady statepore pressures can always be determinedfrom hydrostatic conditions or from asteady state seepage flow-net. During un-drained loading, however, there will us-ually be excess pore pressures and asthese change with time they are not easilycalculated. Thus, for drained loadingstrains are related to effective stresseswhile for undrained loading strains may bedetermined from total stress changes eventhough they depend on changes of effec-tive stress.

Excess pore pressures dissipate withtime and, even if total stresses remainconstant, the changes of pore pressurecause changes of effective stress whichin turn lead to strains. These deformationsare due to consolidation and they occurwithout any change of external loading;thus a foundation may continue to settlelong after the loading is complete and set-tlements above tunnels may occur longafter construction, For undrained loadingof isotropic elastic soil ))s' 0 (e.g. At-kinson & Bransby, 1978) and, from eqn. 6we have

)(u = ))s

Thus, knowing the steady state porepressures before and after construction(these may or may not be the same) andthe changes of total stress due to con-struction we may calculate the changesof pore pressure, and hence effectivestress, during consolidation and so, forelastic soil, calculate the consolidationsettlements. For inelastic or anisotropicsoil similar calculations for consolidationsettlements are possible but they are moredifficult to perform.

Thus there are a number of differentcases to be considered. Firstly, for drainedloading for which excess pore pressuresare zero and effective stresses can bedetermined the behaviour of soil shouldbe analysed in terms of effective stresses.The strength is given by eqns. 8 or 9 and,for elastic soil, strains are given byHooke's law in terms of effective stresses.

Secondly, for undrained loading of satu-rated soil for which there are no volumechanges, analyses may be carried out interms of total stresses and there is noneed to determine the pore pressures. Theundrained shear strength given by eqn, 10,

July, 1981 21

h

WRY LY/

t f

C

.zvz <

I

D

(b) lc)

Fig. 3. Stages in the construction of a soft ground tunnel: (a) before construction;(b) tunnel heading; (c) lined tunnel

than those applied by the temporary sup-port system but for the present, and forsimplicity, we will assume that the valueof <rr at the heading is the same as thevalue of <rr due to the permanent liningWe will assume, also, that shields andlinings are smooth and relatively flexibleand hence <r, is uniform, radial andconstant.

At (a) in Fig. 3, ahead of the advancingtunnel, the vertical and horizontal totalstresses at any depth z are o, andrespectively and the groundwater condi-tions are steady state and hydrostatic. If,for simplicity, we take Ke = 1 appropriatefor a soil which is moderately overcon-solidated, vertical and horizontal stressesare equalt and

= <rh——<r,. + ./z (13)

depends only on the water content andis independent of the loading and, for elas-tic soil, strains are given by Hooke'slaw in terms of total stresses with t „——2

Thirdly, excess pore pressures producedas a result of undrained loading dissipatewith time and, for elastic soil, the addi-tional strains may be calculated fromelastic theory. These calculations arefamiliar to geotechnical engineers con-cerned with the stability and settlementof foundations and the same principlesapply equally to calculations for the sta-bility of tunnels and for settlements dueto tunnelling.

Rates of loading and consolidationAny changes of total stress or boundary

conditions give rise to excess pore pres-sures which in turn give rise to migrationof the pore water, volume changes andconsolidation. The rate of change of totalstress due to loading or excavation maybe very fast compared to the rate at whichexcess pore pressures diminish, so thatit may be assumed that there is no con-solidation during loading or construction.In this case, the loading is known asundrained and, for saturated soil, therewill be no volumetric strains but usuallythere will be undrained shear strains. Al-

ternatively, the rate of change of totalstress may be very slow compared tothe rate at which excess pore pressuresdiminish, so that it may be assumed thatall consolidation takes place during load-ing. In this case, the loading is known asdrained, there are no excess pore pres-sures but there will probably be volumetricand shear strains.

Fig, 2 (a) shows a case of undrainedloading where the total stress <r is appliedrelatively quickly so that at a time t,. whenthe loading is complete there are no vol-umetric strains e, but the excess porepressure is u,.; thereafter volumetric strainsincrease as excess pore pressures dimin-ish. Fig, 2(b) shows a case of drainedloading where the total stress o. is ap-plied relatively slowly so that excess porepressures are always zero and volumetricstrains follow the loading. The essentialdistinction between drained and undrainedloading rests with the rate of loading com-pared to the rate of consolidation; it isnot the absolute rate of loading which isimportant.

The rate of loading is determined sim-ply by the construction. Thus, a riverwill erode a natural slope over severaldecades, a foundation will be constructedand loaded over a year or so, a cuttingmade in several months, several metres


of tunnel driven in a few days, a trenchwill be dug in a few hours and earthquakeloads may last less than a minute. Therate of erosion of a natural slope will beof the order of 10" times slower than therate at which loads are imposed by anearthquake.

Consolidation takes place as waterseeps from places where the excess porepressures are large toward the boundar-ies where the excess pore pressures arezero and the rate of consolidation depends,among other things, on the rate of seepage.The velocity V of water seeping throughsoil is related to the hydraulic gradient iby Darcy's law,

V =ki ( 12)

Total stress changes in the grounddue to tunnelling

Without complex analysis it is impos-sible to determine accurately the stressesin the ground due to tunnelling but sim-ple estimates can be made which illus-trate important features.

Fig. 3. shows a section of a long cir-cular tunnel during construction. At thesoil surface there is a uniform surcharge<rs which may arise due to buildings, traf-fic or the presence of a layer of very weaksoil, Inside the tunnel there is assumedto be a uniform total stress or appliedto the tunnel wall and the face of theheading. The tunnel pressure o-r may arisefrom air or bentonite slurry pressure, fromthe support of a shield and face plate orfrom the support of a permanent lining.In practice the total stresses applied by apermanent lining may be greater or smaller

where k is known as the coefficient ofpermeability and is taken to be a constantfor a given soil.

Values for k for coarse grained sandysoils are of the order of 10" times greaterthan values of k for fine grained clayeysoils and hence, all other things beingequal, consolidation of sandy soils willtake place of the order of 10e times fas-ter than consolidation of clayey soils.Thus earthquakes may cause undrainedloading of sandy soils while natural slopesin clayey soils should be considered asdrained.

The first task of a geotechnical engin-eer is to determine whether a particularcase may be taken to be drained or un-drained and, in the latter case, there willbe subsequent settlements due to consoli-dation, These general principles applyequally to all soil structures, to founda-tions, to slopes, to retaining walls as wellas to tunnels.

where ./ is the unit weight of the soil,At (b) in Fig. 3, at the heading, the

horizontal total stress in an element in theface and the vertical total stress in anelement above the crown are <rz asshown. For a drained case the groundwater conditions are steady state seepagetowards, or away from, the heading andfor an undrained case there will be ex-cess pore pressures induced by the chan-ges of total stress. If compressed air orbentonite slurry is used to maintain sta-bility or to prevent excessive seepageof ground water into the heading the valueof <r, is simply the air or slurry pressurewhich may be controlled by the engineer.Small values of o-r are desirable for eco-nomic and health reasons and a value onlyslightly greater than the original steadystate pore pressure is usually chosen.The value of o-r due to a shield dependson a number of factors including the sizeof any overcutting bead, the use of faceplates, the stiffness of the soil and thestresses applied as the shield is jackedahead. It is unusual for o-r greatly to ex-ceed the original overburden pressuregiven by eqn. 13, and clearly in an un-lined tunnel with no air or bentonite pres-sure <rr = 0. In many cases it would bea reasonable estimate that o-r during con-struction falls in the range 0 <(o-r (~ zzy

(C + —,'D), the lower values applying forheavily overconsolidated stiff soils and thehigher values applying for shallow tun-nels in lightly overconsolidated soft soils.Thus, since o-r is likely to be appreciablyless than the original total stress in theground, tunnelling will in general cause adecrease in the mean total stress s and anincrease in the total shear stress t, Forthe special case of tunnelling in isotropicelastic soil it turns out that the changesof radial and tangential total stesses closeto the tunnel and close to the headingare equal and opposite with the resultthat hs = 0 but for anisotropic and in-elastic soil the value of s decreases (Mair,1979; Seneviratne, 1979) .

At (c) in Fig. 3, where the tunnel iscomplete with its permanent lining andgroundwater conditions are steady state,the value of <Tr depends on a number offactors including the construction proced-ure, the stiffness of the soil, the stiffnessof the lining and any stresses jacked intoan expanded lining. For the present, how-ever, we will assume that the total stress

Strictly K —~ '/~ 'elates horizontal andevertical effect<'ve stresses in the ground but if

~ ' <r„'hen ~ —~ also.h h

which will usually be atmospheric, butif the lining is sealed the pore pressuresdepend on the conditions in the regionof disturbed ground immediately behindthe lining.

For the present it is assumed that thefinal pore pressures u are a little lessthan the initial pore pressures u, but in

any case, at (c) in Fig. 3, there is steadystate two-dimensional flow and pore pres-sures everywhere may be found from asimple two-dimensional flow-net (e.g. Tay-lor, 1979)

At (b) in Fig. 3, the pore pressuresare u,. and water rises in standpipes toheights h,. as shown, but the values ofu,. depend on whether the construction isdrained or undrained. If the tunnelling op-erations can be taken as drained the porepressures u,. correspond to conditions ofsteady state seepage towards, or awayfrom, the heading and they are in equili-brium with the tunnel pressure o-r in theheading due to air or bentonite slurrypressure. For the drained case pore pres-sures u,. everywhere may be found from athree-dimensional flow-net and these porepressures u,. will usually be different fromthe final steady state pore pressures u„corresponding to two-dimensional seep-age. If the tunnelling operations can betaken as undrained so there are no vol-ume changes during construction, the porepressures u,. will be determined by theinitial pore pressures u, and the changesof total stress due to the tunnelling, Thesepore pressures are not in equilibrium withthe boundary conditions at the tunnel andthe excess pore pressures dissipate asconsolidation occurs.

We must be very careful here to notethat for conditions of steady state see-page there will be additional effectivestresses due solely to the drag of theflowing water on the soil. These seepagestresses u-,.'ct in the direction of theflowlines and are given by

nI

S

Fig. 4. Total stress path for a typical soilelement near a tunnel during construction

applied by a permanent lining is thesame as that applied by a temporary sup-port system, and hence total stresses at(b) in Fig. 3 are taken to be the same asthose at (c). When soil deformations oc-cur around the tunnel, o-T applied by asupport system will be less than the origi-nal total stress in the ground, due to thecontribution from the mobilised strengthof the ground itself. The stresses imposedupon the permanent lining would thereforealmost certainly be less than the originaltotal stress in the ground since soil defor-mations inevitably occur during construc-tion.

We may now sketch the total stresspath for a typical element near the tun-nel as ABC in Fig. 4, where the points A,B and C correspond to the sectiors (a)(b) and (c) in Fig. 3. Since the total stres-ses at (b) and (c) are assumed to be thesame the points B and C coincide. In

practice the loads on a permanent liningwill probably be a little greater than thetotal stresses applied by a temporary sup-port system and hence the total stressesat (b) and at (c) in Fig. 3 will differ, andthe points B and C in Fig. 4 will not coin-cide but they will probably not be signi-ficantly different. The precise shape of thestress path ABC in Fig, 4 depends onthe nature of the soil and for the pres-ent illustration we have simply sketchedan arbitrary curve, For the special case ofisotropic elastic soil fjs = 0 and the lineABC is straight and vertical, and so thecurve shown in Fig, 4 is appropriate forinelastic or anisotropic soil.

(15)rr,. = 7«. I

where i is the hydraulic gradient (AtkinsonIk Bransby, 1978).

It is these seepage stresses which giverise to piping at the toe of a sheet pilewall and, for seepage towards a tunnel,they reduce the stability of the tunnel andmay cause ravelling in granular soils well-known to tunnelling engineers,

Pore pressures in the ground due totunnelling

At any instant the pore pressure u in

the ground is given by

(14)U = 7 . h

Effective stress paths for tunnellingWe have already seen that, in accor-

dance with the principle of effective stress,it is the effective, not the total, stresseswhich determine soil behaviour and so wemust consider the changes of pore pres-sure as well as the changes of total stress.In considering pore pressures we must bevery careful to consider the case for

where 7„, is the unit weight of water andh is the rise of water in a stand-

pipe as shown in Fig. 3.

Fig. 5. Total and ef-fective stress pathsfor a typical soil ele-ment near a tunnelduring drained ex-cavation

At (a) in Fig. 3, ahead of the advanc-ing tunnel the initial steady state porepressures are u„and water rises in stand-pipes to heights h, corresponding to thenatural water table. Thus it is usually asimple matter to determine the initial porepressure u, everywhere in the soil.

At (c) in Fig. 3, the final steady statepore pressures are u„and water rises in

standpipes to heights h„as shown. Thesefinal pore pressures correspond to con-ditions of steady state seepage towards,or away from, the tunnel and they arein equilibrium with the pore pressuresjust behind the tunnel lining.

The precise values of u„just behindthe lining will depend on whether or notthe lining is impermeable and whether thetunnel contains fluid pressures. If thelining leaks then u„just behind the lin-

ing is simply the pressure in the tunnel

u,.=u, +flu ( 16)

and f)u are the changes of pore pressuredue to the changes of total stress duringexcavation. In most cases pore pressuresdecrease during undrained tunnelling so buhas a negative value. At (c) in Fig. 3,steady state pore pressures u„ in equili-brium with the pressures just behind thelining are exactly the same at those fol-lowing drained loading discussed in theprevious section. Thus, immediately after

A

July, 1981 23

drained tunnelling separately from thecase for undrained tunnelling followed byconsolidation.

We will begin by considering the drain-ed case for which the rate of excavationis relatively slow compared with the rateof consolidation so that excess pore pres-sures are zero and all changes of effectivestress and all strains take place duringexcavation.

Fig. 5 shows total and effective stresspaths sketched for typical elements ofsoil near a tunnel for drained excavation.The total stress path ABC is the sameas that shown in Fig. 4, and the porepressures have been sketched foru„) u,. ) u„where u„are pore pres-sures before construction, u,. are steadystate pore pressures near the heading andu„are steady state pore pressures aroundthe lined tunnel. In Fig. 5, the effectivestress state B'or excavation approachesthe critical state line and, if the effectivestress path A''eaches the critical stateline, the soil fails. The effective stress pathB''orresponding to the reduction in

pore pressure from ur to u at constant total stress moves away from the criticalstate line and hence the factor of safety at(c) is greater than that at (b). The pathA''nvolves a modest reduction of

s'nd

a relatively large increase in t'ndwill give rise to settlements due largelyto shear straining. The path B''nvolvesan increase of s'nd will give rise to set-tlements due to volumetric compressivestrains. It must be emphasised that, fordrained loading, all strains and settlementsoccur immediately the loads or pore pres-sures are changed. Since the pore pres-sures are known from the appropriateflow-nets all calculations for stability andsettlement may be carried out in termsof effective stresses.

For undrained tunnelling the rate of ex-cavation is relatively quick compared tothe rate of consolidation so that there isno drainage during construction and vol-ume changes are zero. Thus, any set-tlements that occur during undrained tun-nelling are due to shear strains only. At(a) in Fig. 3 the steady state pore pres-sure u, before construction is hydrostatic.At (b) in Fig. 3, the pore pressures areu,. where

SLFig. 6. Total and effecti ve stress pathsfor a typical soil ele-ment near a tunnelduring undrainedexcavation followedby consolidation

required to maintain stability is given by

( 19)

and for the case when the surface pres-sure is large and the weight of the soilmay be neglected the required tunnel pressure is given by

(20)

construction the excess pore pressure u,.IS

0;=0,—Lj =u +jILI —ij ... (17)

noting that Iiu is usually negative for tun-nelling. Consolidation occurs as the excesspore pressures change from u,. to u„= 0and strains and settlements are due to thecorresponding changes of effective stress.

Fig. 6 shows total and effective stresspaths sketched for typical elements ofsoil near a tunnel for undrained tunnellingfollowed by consolidation, The total stresspath is the same as that shown in Fig. 4and the pore pressures have been sketchedfor u, ) u„. The path A''orrespond-ing to undrained tunnelling involves a re-latively large increase of t'hich will giverise to settlements due solely to shearstraining and, of course, there are no vol-umetric strains. The path B''orres-ponding to consolidation as the pore pres-sures change from u,. to 0 involves anincrease of s'hich gives rise to consoli-dation settlements due solely to volume-etric straining.

In Fig. 6, the effective stress pathA''or

excavation approaches the criticalstate line and if the effective stress stateB'eaches the critical state line the soilfails. The effective stress path B''or-responding to the increase of pore pres-sure from u,. to u during consolidationalso moves towards the critical state lineand hence the factor of safety at (c) isless than that at (b) and, if either pointB'r point C'n Fig. 6 reach the criticalstate line the soil fails. The time for theeffective stress to move from B'o a pointon the critical state line when failure oc-curs is often known as the stand-up timeand it depends on the rate of consolida-tion.

The rate at which the consolidation set-tlements occur depends largely on thepermeability of the soil and, for clayey soilsfor which undrained loading applies, theconsolidation process may be lengthy. Forthe undrained case there are excess porepressures which are not easily determinedbut, provided there are no volume chan-ges, calculations for stability and settle-ment may be carried out in terms of totalstresses.

is whether or not the tunnel can be safelyexcavated without compressed air, pres-surised slurry or other temporary supportbeing required before erection of the lining,In addition, as in most geotechnical engin-eering problems, it is necessary to esti-mate and minimise deformations. But tun-nel engineers must firstly assess the over-all stability of their tunnels so that theyhave an appreciation of the factor of safe-ty under working conditions.

For fine-grained clayey soils, for whichtunnelling may be considered to take placeunder undrained conditions, stability ana-lyses should be carried out in terms oftotal stresses using the undrained shearstrength c„. Care must be taken, how-ever, to determine whether there are anylayers of coarse-grained sandy soil pre-sent which may accelerate the rate of consolidation so that the conditions are nolonger fully undrained. For cases whereundrained loading applies the tunnel pres-sure Irr required to maintain stability fora factor of safety F, is given byc„2C

<rr = rr,. ——Tr+ 2»D (1+—) (18)F, D

where T, is a dimensionless tunnel stabilitynumber and the other terms are as de-fined in Fig. 3.

For coarse grained sandy soils for whichtunnelling may often be taken as drained,stability analyses should be carried out interms of effective stresses and, for a logi-cal and conservative design, it is appro-priate to take the critical state soil strengthwith y„'nd c' 0. For dry soil and forthe case when Ir, = 0 the tunnel pressure

D

where T» and T, are dimensionless stabil-ity numbers.

In order to estimate the tunnel pressureat collapse the soil strength is given byq,.,'ut, to include a factor of safety thesoil strength is given by an allowableangle of shearing resistance y„'iven by

1

tan y„' —tanIfr,'x

(21)

The tunnel stability numbers T,, T» andT,. are analogous to the bearing capacityfactors N, N» and N„and their valuesdepend on such things as the geometryof the tunnel and soil strength. Values fortunnel stability numbers have been ob-tained using the limit theorems of plas-ticity theory in conjunction with modeltests.

Values of Tr based on theoretical analy-ses for headings and for plane sectionsof circular tunnels were given by Daviset al (1980) and results of centrifugalmodel tests reported by Kimura & Mair(1981) are in good agreement with thesetheoretical analyses. It was found thattunnel stability in clay soils is stronglyinfluenced by the length of unlined head-ing P, the cover C and the tunnel diameterD. Values of T, obtained from theoreticalanalyses and centrifugal model tests fordifferent ratios P/D and C/D are given inFig. 7. The results shown are for tunnelsin clay with constant undrained shearstrength c„.Where c„varies with depth,appropriate calculations may be doneusing the average strength between thesurface and the tunnel axis.

Tunnel stability numbers T» and T,based on theoretical analyses and modeltest results were discussed by Atkinson& Potts (1977a). The values of 7> dependonly on the soil strength and are indepen-dent of the depth of the tunnel unless thetunnel is extremely shallow, but values

Calculations for stability of tunnelsTunnels in soft ground are usually rela-

tively shallow and the total stresses ap-plied by the soil and the pore-water aretoo small to cause the failure of a shieldor a permanent lining. Thus, possible in-stability is likely to occur at the headingand at sections which are unsupported orsupported by insufficient air or bentoniteslurry pressure.

A basic engineering decision to be madewhen designing a tunnel in soft ground


1 2 3Fig. 7. Influence of heading geometry and depth on the tunnel stability number To

C/D

2.0— 10

1.0— 0.5

I

10' 20' 30'I40'0'0 30'0

Fig. 8. Influence of soil strength on the tunnel stability number T> Fig. 9. Influence of soil strength and depth on the tunnel stabilitynumber T,

for T, given by Atkinson & Potts (1977a)depend both on the soil strength and onthe depth. Values for T> and T, shown in

Figs. 8 and 9 are those obtained as theo-retical safe bounds and centrifugal andlaboratory models collapsed when the sup-port pressures fell below those given bythese values of T~ and T,. Model testson tunnel headings in sand by Argyle(1976) and by Aspden (1976) indicatedthat the three-dimensional influence on sta-bility in cohesionless soils is much lesssignificant than observed for clay soils(Casarin, 1977; Mair, 1979).

The solutions for drained loading wereobtained for dry soil for which total andeffective stresses are the same and forsoil which is not dry allowance must bemade for pore pressures and seepagestresses. If the lining is impermeable andthere is no seepage the lining must sup-port hydrostatic water pressures in ad-dition to the effective stresses from thesoil while if there is seepage the effectivestresses must be modified to account forthe seepage stresses. If there is excessiveseepage towards an unlined tunnel fail-ure may occur due to piping. Loads ap-pl.'ed to impermeable linings due to waterpressures may exceed the loads appliedby the soil.

Calculations of settlemertts causedby tunnelling

Settlements due to tunnelling may beestimated by empirical methods (Atkin-son & Potts, 1977b) or by analysis usingthe finite element method (Atkinson, Orr& Wroth, 1978), the associated fields method (Atkinson & Potts, 1978) or similarcomputer based calculations. For soilwhich is overconsolidated and for tunnelsfor which the factor of safety is relative-ly large the state of the soil will remaininside the state boundary surface and itsbehaviour will be elastic.

Settlements of elastic soil may be cal-culated using the finite element methodtaking the effective stress elastic para-meters E'nd >'or drained loading andthe total stress elastic parameters E„and~ „———,'or undrained loading. For elasticso'.I the behaviour is path independent andthis means that the settlements dependon the initial and final states and not onthe stress path. If the final condit.'ons fordrained loading are the same as those forundrained loading followed by consolida-tion (i.e, the points C're the same in

Figs. 5 and 6) settlements due to drainedloading are the same as those due to un-drained loading followed by consolidation.


Thus consolidation settlements may befound as the difference between drainedand undrained settlements. For these elas-tic calculations it will be necessary, in

most cases, to make allowance for aniso-tropy and inhomogeneity of the soil and,for drained loading, the value of E'illnot be constant.

If the soil is only very lightly over-consolidated, or if the tunnel is close to astate of collapse the state of the soil mayreach the state boundary surface and thebehaviour will no longer be purely elastic.In this case the calculations must be modi-fied to take account of the plastic compo-nents of strain. This may be done by in-cluding an elasto-plastic soil model, suchas that based on theories of criticalstate soil mechanics, into the finite ele-ment program or by making use of aplastic method of deformation analysissuch as the associated fields method. In

any case however it is necessary, as al-ways, to distinguish between drained andundrained loading,

Due to inward displacement of the sur-rounding soil, the volume of soil excavatedduring tunnelling is always somewhatgreater than the volume correspondingto the tunnel cross-section and the addi-tional volume is known as ground loss.Under undrained conditions, when thesoil must deform at constant volume, theground loss into the tunnel per unit len-gth of tunnel advance is equal to the areaof the surface settlement trough, Groundlosses for shallow tunnels in soft clayscan be predicted by finite element met-hods using critical state soil models (Mair,Gunn & O'Reilly, 1981) and hence, bycombining these methods with the empiri-cal patterns of surface settlement sug-gested by Peck (1969) settlements abovetunnels can be predicted.

The rate at which consolidat:on set-tlements occur depends on the depth ofthe tunnel, on its geometry and on thecoefficient of consolidation of the soilwhich depends, in turn, on its permea-bility and compressibility. The problem issimilar to that of estimating the rate ofsettlement of a foundation due to con-sol:dation and was examined by Senevi-ratne (1979). While it is of interest tocalculate the rate at which consolidationsettlements occur the most important as-pect of calculations for the rate of con-solidation is for estimating the stand-uptime of a tunnel for which the construc-t':on was undrained and for which effec-tive stresses are moving towards the cri-tical state line as shown by the path

B''n

Fig, 6. We have already seen that theprincipal factor controlling stand-up timeof a tunnel is the rate at which excesspore pressures dissipate due to consoli-dation and the rate of consolidation de-pends on the permeability and compres-sibility of the soil and on lengths of drain-age paths.

Calculations for settlements caused bytunnelling are more complex and, at pres-ent, seem to be less accurate than thecalculations for stability discussed in theprevious section. Nevertheless, the samebasic soil mechanics principles apply toeach analysis; in particular it is essentialto distinguish between drained and un-drained loading and all time-dependentphenomena are taken as being due toconsolidation.

DiscussionTunnelling in soft ground is basically a

problem of soil mechanics and so all theprinciples of soil mechanics apply. A sim-ple and unified view of the behaviour ofsoils is given by the ideas of critical statesoil mechanics and these ideas have beenused to examine the behaviour of soilduring tunnelling construction. In thisPaper a number of broad simplifying as-sumptions have been made concerning thechanges of stress and pore pressure in

order to expose the basic principles. Inpractice changes of stress and pore pres-sure may be different to those assumedbut the basic principles will remain thesame. Particular care has been taken todistinguish between drained loading ap-plicable for tunnelling in coarse grainedsoils from undrained loading followed byconsolidation applicable for tunnelling infine grained soils. Different calculations arerequired for stability and settlements foreach case.

In the ideas of critical state soil mec-hanics and, indeed, in the classical theoriesof soil mechanics, strains due to creepare neglected and all time-dependent ef-fects are regarded as being due to primaryconsolidation. Thus, in the present analy-sis, settlements occurring after construc-tion and the stand-up time are shown tobe dependent on the rate of consolidation.

Tunnel construction in soft ground canbe achieved by a variety of techniquesand many factors affect the developmentof ground movements: these include thegeometry characterising the unlined tun-nel heading and its depth from the sur-face, the properties of the soil strata andtheir stress histories, the type of lining

(concluded on page 38j

l- iltiltilllII

5m3m

C LAY

A=I+ ik

TB

D 3 y

Sm r4 1L

GEOMETRY

Fig. 17. Example 2

ELATERAL EARTH PRESSURES

Soil mechanicsand tunnelling

(continued from page 26)

used, and the method and rate of exca-vation. It is recognised that it is oftendifficult to assess the influence of each ofthese factors in isolation, but it has beenthe aim of this Paper to show how it ispossible to separate some of the moreimportant parameters and interpret thebehaviour of a tunnel in soft ground interms of well-established principles of soilmechanics. With this approach to softground tunnelling, a more rational under-standing of tunnel deformation behaviourcan be developed and a framework provi-ded for tunnel design and interpretationof field data.

Moments about tie position

Moment due to net available passive re-sistance

Zone Moment (kNm/m run)

Earth pressure 5 0.5 x 223.9 x6 x (4 + 6) = 6 717.0

Moments activated by retained material

Zone

Earth pressure 1

Earth pressure 2

Earth pressure 3

Earth pressure 4

Water pressure 6

Water pressure 7

Water pressure 8

Water pressure 9

Moment (kNm/m run)

0.5 x 15.7x 2.67x (0.67 x 2.67 —2)

—4.415.7 x 5.33 x (0.67+ 2.67) = 2790.5 x (33.8—15.7)x 5.33x (0.67 +0.67 x 5.33) = 20533.8x6x(6 + 3) =18250.5 x 22.1 x 2.67x (0.67 + 0.67 x2.67) = 72

18 x 2.67 x (0.67+ 1.5x 2.67) = 224

0.5 (22.1 —18)x 2.67 x (0.67+ 2.67 + 0.33x 2.67) = 2305x18x6x(6 + 2) = 432

Tot a I 3 056.0

Factor of safety F„:

moment of net availablepassive resistance 6 717

F„ =—= 2.2moment activated by 3 056retained material

For comparison the factors of safety cor-responding to Methods 1, 2 and 4 respec-tively are: F = 1.5

F „=4.7F, = 1.4

Example 2: Short-term analysis of apropped wall retainingclay overlain by a gran-ular fill and a surcharge

(Fig. 17)Soil properties:

(i) Fill: i)i' 35, )I' 0,. Kx = 0.27, 7»x = 20kN/ms


(ii) Clay: c„=35kN/m', /', ——15kN/m'et

lateral earth pressures:

ACTIVE SIDE

Position

A

—D

+D

Earth Pressure (kN/m')10 x 0.27 = 2.70.27 (10 + 20x 2) 13.510+ 2x 20+3 x 15—2 x 35 = 2510+ 2x20+3x15 95

PASSIVE SIDE

D 4x35

Moments about tie position:

25Length y = —= 1.67m

15

Moment due to net available passive re-sistance


Clay 5 140 x 5 x 5.5 =3850

Moments activated by retained material


1 2x2.7x (-1) = -5,4Fill 2 0.5 x (13.5—2.7) x

2 x (-0.67) = -7.2Clay 3 0.5 x 25 x 1.67x (1.33

+ 0.67 x 1.67) = 51.12Clay 4 95 x 5 x 5.5 =2 612.5

Total 2 651

Factor of safety F„:

moment of net availablepassive resistance 3 850

F„= =—= 1.45moment activated by 2 651retained material

For comparison the factors of safety cor-responding to Methods 1, 2 and 4 res-pectively are:—F, = 1.63

F „=32.1F, = 1.41

Acknowledge merttsMuch of the work referred to in this

Paper was carried out at Cambridge Uni-versity as part of a programme of inves-tigations into the behaviour of tunnels insoft ground, supported by the Transportand Road Research Laboratory (TRRL).The authors are grateful to Mr. M. P.O'Reilly, Head of the Tunnels Division atthe TRRL, for his continual interest andconstructive criticisms. The authors areindebted to Professors A. N. Schofield andC. P. Wroth for their overall direction ofthe research work.

ReferencesArgyle, D. N. (1976): "An investigation into thecollapse of tunnel headings in dense sand".Cambridge University Engineering Tripos Part IIResearch Report

Aspden, R. (1976): "Co'lapse of unlined tunnelswith headings in dense sand". Cambridge Uni-versity Engineering Tripos Part II Research ReportAtkinson, J. H. & Bransby, P. L. (1978): The Mec-hanics of Soils. McGraw-Hill, London

Atkinson, J. H„orr, T. L. L, & Wroth, C, P.(1978): "Finite element calculations for the de-formations around model tunnels". In ComputerMethods in Tunnel Design. ICE, pp, 121-144.Atkinson, J. H. & Potts, D, M. (1979): "The sta-bility of a shallow circular tunnel in cohesionlesssoil". Geotechnique, Vol 27: 2, pp. 203-215.Atkinson, J. H & Potts, D. M. (1977); "Subsi-dence above shallow tunnels in soft ground".Jnl, Geot. Eng. Div., ASCE., Vol. 103, GT4, pp.307-325.Atkinson, J, H. & Potts, D M. (1978): "Calcula-tion of stresses and deformations around shallowcircular tunne's in soft ground by the methodof associated fields" In Computer methods inTunnel Design. ICE, pp. 61-84.Casarin, C. (1977): "Soil deformations aroundtunnel headings in c'ay". MSc. Thesis, Universityof Cambridge

Davis, E. H., Gunn, M, J., Mair, R. J. & Sene-viratne, H, N. (1980): "The stability of shallowtunnels and underground openings in cohesivematerial". Geotechnique 30, No 4, 397-416.Kimura, T. & Mair, R. J (1981): "Centrifuqaltesting of model tunnels in soft clay". Proc, 10thInternational Conference on Soil Mechanics andFoundation Engineering, Stockholm.Mair, R. J. (1979): "Centrifugal modelling of tun-nel construction in soft clay" PhD Thesis, Uni-versity of Cambridge.Mair, R. J., Gunn, M, J. & O'Reilly, M. P. (1981):"Ground movements around shallow tunnels insoft c'ay" Proc. 10th International Conferenceon Soil Mechanics and Foundation Engineering,Stockholm.Peck. R. B. (1969): "Deep excavations and tun-nelling in soft ground". Proc. 7th Int. Conf, onSoil Mechanics and Foundation Engineering, State-of-the-Art Vo'ume, pp. 226-290,Schofield, A. N. & Wroth C P, (1968): Crit'calState Soil Mechanics. McGraw-Hill, London.Seneviratne, H. N. (1979): "Deformations and porepressures around model tunnels in soft clay".PhD Thesis, Cambridge UniversityTaylor, R. N. (1979): "Stand-up of a model tun-nel in silt". MPhil Thesis, University of Cam-bridge.

Terzaghi, K. (1936): "The shearing resistance ofsaturated soil and the angle between the planesof shear", Proc. 1st Int. Conf, Soil Mech andFoundn. Enging., Vol. 1, pp. 54-56,

soil mechanics aspects of soft ground tunnelling

Documents