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FELSBAU 20 (2002) NO. 6

Tunnel Heading Stabilityin Drained Ground

By Pieter A. Vermeer, Nico Ruse and Thomas Marcher

When considering ground conditions fortunnelling one might distinguish between

ground with and without a macro structure due tostratification, schistosity and jointing. In thisstudy attention is focused on soils and very softrock without a significant macro structure. Thesematerials tend to be encountered for shallowtunnelling in urban areas, whereas macro-struc-tured ground is dominant in deep tunnelling. Forsoils and very soft rock, as considered in thisstudy, stability is governed by shear strengthparameters that can be measured in laboratorytests. For soft soils with little (effective) cohesion,it is necessary to drive the tunnel using a shield.For soils or rock with greater cohesion, it ispossible to use an open face tunnelling methodsuch as the NATM. Here, if necessary the stabilityof the tunnel face can be improved by inclining theface or by reducing the cross section of the exca-vation. In this study we will consider face stabilityboth for closed-face shield tunnelling and open-face NATM tunnelling.

When discussing previous research on tunnelheading stability, one has to distinguish betweendrained and undrained conditions. For un-drained conditions, as dominant in clays, practi-cal design curves have been derived on the basis

Standsicherheit der Ortsbrust von Tunnelnunter dränierten Baugrundbedingungen

Die Standsicherheit der Ortsbrust von Tunneln wird zu-nächst im Hinblick auf Schildvortrieb betrachtet. Anschlie-ßend wird die Spritzbetonbauweise einbezogen, indem ander Ortsbrust von keinem Stützdruck ausgegangen wird.Mit der Absicht, einfache Formeln zu entwickeln, werdenzunächst Tunnel mit einem Kreisquerschnitt in homoge-nem, dräniertem Mohr-Coulomb Material betrachtet. Er-gebnisse aus nichtlinearen Finite-Elemente-Berechnungenwerden verwendet, um zu zeigen, wie zumindest in Rei-bungsmaterial die Spannungsverteilung am Tunnel durchGewölbewirkung dominiert wird. Wenn der Reibungswin-kel größer als 20° ist, scheint die Standsicherheit der Orts-brust völlig unabhängig von der Überdeckung des Tunnelszu sein. Unter dränierten Bedingungen scheint die Spritz-betonbauweise möglich zu sein, wenn die effektive Kohäsi-on etwa 10 % von γ·D erreicht, wobei γ die Bodenwichte ist.D ist entweder der Durchmesser des Tunnels bei einemVollausbruch oder der Durchmesser der Ausbruchsflächebei Teilausbruch. Bei der Betrachtung typischer Quer-schnitte von NÖT-Tunneln wird gezeigt, daß ein äquivalen-ter Wert für D in den Standsicherheitsformeln verwendet

werden kann. In geschichtetem Baugrund sind die Glei-chungen schwierig anzuwenden, weswegen dafür eine nu-merische ϕ-c-Reduktion vorgeschlagen wird.

Tunnel heading stability is initially considered with a viewtowards closed face tunnelling. At the end open face tun-nelling is included by assuming face pressures to be equalto zero. In order to arrive at simple formulas, attention isinitially focused on circular tunnels in a homogenousMohr-Coulomb material. Data from non-linear finite ele-ment analyses are used to show that stress distributions indrained ground are dominated by arching. Once the fric-tion angle is larger than about 20°, stability appears to becompletely independent of the cover on top of the tunnel.For drained conditions, open face tunnelling appears to bepossible when the effective cohesion exceeds some 10 % ofγ·D, where γ is the unit soil weight. Here D may either bethe full tunnel diameter or a subsection diameter of a se-quential excavation. Considering typical shapes of NATMtunnels, it is shown that an equivalent D-value can be cal-culated for use in the stability formulas. For layeredground, the formulas are difficult to apply and it is pro-posed to use a numerical procedure named ϕ-c-reductionmethod.

of model tests (2) and these curves have beenlargely confirmed by theoretical studies (8). Thequestion whether a drained or undrained stabil-ity analysis should be carried out can be an-swered by considering the type of ground and theadvance rate of the tunnel face. According to aparametric study by Anagnostou and Kovári (1),drained conditions tend to apply when the groundpermeability is higher than 10-7 to 10-6m/s and thenet excavation advance rate is 0.1 to 1.0 m/hr orless. In a predominately sandy soil, therefore,drained stability conditions should be considered.In a clayey, low-permeability soil the undrainedanalysis is valid during excavation, but thedrained analysis applies in case of a standstill.Hence, even for excavations in clay it is importantto investigate drained soil conditions.

It would seem that Horn (25) was one of thefirst to propose a model for assessing the stabilityunder drained conditions. He considered the limitequilibrium of a sliding wedge at the tunnel face.Jancsecz and Steiner (11) applied this model toshield-tunnelling, whilst Sternath and Baumann(19) used it to analyse NATM-tunnelling.

The idea of considering drained stability on thebasis of a single equation was first suggested byAtkinson and Mair (2), as they proposed a formula

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of the form pf = qNq+γDNγ for shield tunnels in drycohesionless sand. Here pf is the minimum sup-port pressure at collapse, which will be referredto as failure pressure. The influence of a possibleground surface load is taken into account by auniformly distributed load q and a surchargestability number N

q. The diameter of the tunnel is

denoted as D, the unit soil weight as γ and Nγ is thesoil weight stability number. It should be notedthat Atkinson and Mair used the symbol T todenote stability numbers. The above formula forfailure pressure was extended by Anagnostouand Kovári (1) to cover cohesive-frictional mate-rials by proposing an equation of the form

p c N qN DNf c q= − ′ + + γ γ .............................. [1]

where c´ is the effective cohesion and Nc the cohe-

sion stability number. Anagnostou and Kováridenoted stability numbers by the symbol F insteadof N. The stability numbers are analogous to thebearing capacity factors of footings, in the sensethat they depend on the friction angle j´. Vermeerand Ruse (20) presented data from elastic-plasticfinite element analyses to show that the stabilitynumbers are independent of the depth of thetunnel, at least for friction angles beyond twentydegrees. Moreover, simple formulas were put for-ward for the stability numbers. Later Vermeer andRuse (21) extended this study by considering non-circular cross-sections to show that the shape ofthe excavation is not particular important.

The present paper summarises first of all pre-vious research on stability numbers for drainedsituations. Hereafter the influence of an unlinedwall near the tunnel face is considered, andattention is focused on NATM tunnelling withoutany supporting pressure. This paper will concen-trate mainly on results from numerical simula-tions, and for this reason a brief description of thefinite element procedure being used will be givenin the following section.

Finite element-analysesof failure pressures

Semprich (18) was one of the first to performthree-dimensional finite element calculations toanalyse the deformations near an open tunnelface. More recently Baumann et al. (3) studied theface stability of tunnels in soils and soft rocks by

using the finite element method in combinationwith the elastic-plastic Mohr-Coulomb constitu-tive model. Several authors (22, 24) have shownthat the elastic-plastic finite-element method iswell-suited to predict collapse loads of geotechni-cal structures. For limit load analyses, pre-failuredeformations are not of great importance and areassumed to be linearly elastic, as is usual withinthe elastic-plastic Mohr-Coulomb model beingused in this paper. Elastic strains are governed bythe elasticity modulus E and Poisson´s ratio ν. Theparticular values of these input parameters influ-ence load-displacement curves as shown inFigure 1a, but not the failure pressure pf. For thisreason they will not get any further attention inthis study. In addition to the elasticity modulusand Poisson´s ratio, there are three materialparameters for the plastic behaviour: the effec-tive cohesion c´, the effective angle of friction ϕ´and the angle of dilatancy ψ. Different dilatancyangles give different load-displacement curvesand different collapse mechanisms, but they havevery little influence on the failure load. For thisreason, nearly all our computations were per-formed for non-dilatant material.

As symmetrical tunnels are considered, thecollapse-load calculations are based on only halfa circular tunnel which is cut lengthwise alongthe central axis. Figure 1b shows a typical finiteelement mesh as used for the calculations. Theground is represented by 15-noded prismaticvolume elements and the tunnel lining is mod-elled with 8-noded shell elements. The boundaryconditions of the finite element mesh are asfollows: The ground surface is free to displace, theside surfaces have roller boundaries and the baseis fixed. It is assumed that the distribution of theinitial stresses is geostatic according to the ruleσ h́

= K0

σv́, where σ

h́ is the horizontal effective

stress and σv́ is the vertical effective stress. K

0 is

the coefficient of lateral earth pressure. Vermeerand Ruse (20) investigated the possible influenceof the initial state of stress, by varying the coeffi-cient of lateral earth pressure and found that theK0-value influences the magnitude of the dis-placements but not the pressure at failure.

The first stage of the calculations is to removethe volume elements inside the tunnel and toactivate the shell elements of the lining. This doesnot disturb the equilibrium as equivalent pres-

Fig. 1 Typical pres-sure-displacementcurve (a) and flowarea at collapse (b).Bild 1 TypischeDruck-Verschiebungs-kurve (a) und der Fließ-bereich im Bruch-zustand (b).

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FELSBAU 20 (2002) NO. 6

Fig. 2 Principalstresses (a - c) and

incremental displace-ments (d - f) at failure.

Close-up around theface for a tunnel with

H/D = 5.Bild 2 Hauptspan-

nungen (a - c) undinkrementelle Ver-

schiebungen (d - f)im Bruchzustand.

Ausschnitt aus demBereich der Ortsbrust

für einen Tunnel mitH/D = 5.

sures are applied on the inside of the entiretunnel. To get a full equivalence between theinitial supporting pressure and the initial geostat-ic stress field, the pressure distribution is notconstant but increases with depth. This is obvi-ously significant for very shallow tunnels, but anearly constant pressure occurs for deep tunnels.The minimum amount of pressure needed tosupport the tunnel is then determined by a step-wise reduction of the supporting pressure.

A typical pressure-displacement curve isshown in Figure 1a, where p is the supportingpressure at the level of the tunnel axis and u thedisplacement of the corresponding control pointat the tunnel face. The control point has to bechosen within the collapsing body; otherwise theload-displacement curve in Figure 1a will cometo an almost sudden end and the curve thencannot be used to conclude that failure has beenreached. Rather than selecting a single controlpoint, it is appropriate to select a few of suchpoints. With the reduction in supporting pres-sure, there is increasing displacement. Whenfailure occurs the curve has become horizontal.For shallow tunnels, a chimney-like collapsemechanism is obtained as indicated in Figure 1b,where incremental displacements at failure areshown as graded shades from blue to red.

Upon extending finite element procedures tolimit load computations, it appears that the entirenumerical procedure should be well designed inorder that an accurate assessment of the failureload can be made. For each decrement of sup-porting pressure, equilibrium iterations are per-formed and plastic stress redistribution is accom-plished by using a radial-return algorithm. Ageneral validation of the computer code is givenin the manual of the 3D-Plaxis program by Brink-greve and Vermeer (4) and the method of collapseload computations is fully described by Vermeerand Van Langen (22). In more recent papers the

authors have shown that such finite elementanalyses can also yield highly accurate data onfailure pressure of tunnel headings.

Arching at the face offully lined tunnels

Besides failure pressures, the finite element-meth-od produces insight into the stress distributionaround a tunnel face and the role of friction. Thiscan be seen from Figure 2 for a tunnel with arelative ground cover of H/D = 5. In this figureprincipal stresses are plotted in lengthwise sec-tions through circular tunnels; firstly for a tunnel innon-frictional soil, secondly for a friction angle ofonly 20° and finally for a highly frictional materialwith a friction angle of 35°. In all these differentcases the ground is non-cohesive and the support-ing pressure has been reduced down to the failurepressure by performing three-dimensional finite-element analyses. Moreover, fully lined tunnels areconsidered with a lining up to the very tunnel face.

Figure 2a shows a stress distribution withstress crosses that rotate around the tunnel face.All these crosses have about the same size, whichindicates a high supporting pressure. On theother hand, small stress crosses are seen aroundthe tunnel face of Figure 2b. Hence in frictionalmaterial the failure pressure is relatively low. Forthe highly frictional material of Figure 2c, thearching is extremely clear. Here the supportingpressure is nearly equal to zero and a strongstress arch is observed directly between the topand the bottom of the tunnel.

The influence of the angle of friction can also berecognised by the failure patterns in Figures 2d to2f. Here increasing displacements at failure areshown in graded shades from blue to red. Oneobserves in Figure 2d the extreme of a non-fric-tional material that flows more or less like a liquidinto the tunnel. For a moderate friction angle of

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20°, one observes in Figure 2e the development ofa tall cave. In case of a shallow tunnel this type offailure will extend to the surface to create a crater.For the highly frictional soil of Figure 2f, a relative-ly small body is dropping into the tunnel.

The consequences of friction dependent arch-ing are considerable. The stress arch carries theground cover independent of the magnitude of itsthickness. In an earlier study by Vermeer and Ruse(20) the ground cover has been varied systemati-cally to assess its influence on the failure pressure.It appeared that once the friction angle is largerthan about twenty degrees, stability is completelyindependent of the ground cover. Consideringnon-cohesive ground, it was found from a series of25 different calculations that p

f= γDNγ with

N γ ϕ=

′−1

90 05

tan. ..................................... [2]

under the conditions that ϕ´ > 20° and H/D > 1.Hence, the soil weight stability number is depend-ent on friction, but not dependent on groundcover.

Figure 3 shows that the finite element method(FEM) yields soil-weight stability numbers be-tween the theoretical bound solution by Léca andDormieux (14) and the results of a study byKrause (13). The latter assumed a shell-shapedfailure body at the tunnel face that can slide intothe tunnel. In finite element analyses no assump-tion at all is made about the failure mechanismand both dome-like failure bodies are found, asobserved in Figure 2e, and shell-shaped ones (seeFigure 2d) can be obtained. Anagnostou and Ko-vári (1) present curves for Nγ which lie well abovethe FE-results. Their sliding wedge model yieldsslightly different curves for different relativedepths. The lower boundary of the shaded area inFigure 3 corresponds to H/D = 1 and the upperone to situations with H/D > 5. Hence, the slidingwedge model would seem to be very conservativewhen cohesionless soils with friction angles lessthan about 30° are considered. The curve byAtkinson and Mair (2) is extremely conservativeas it is based on 2D-experiments, that showobviously less arching as 3D tunnel headings.

For high friction angles above forty degrees,most existing models give Nγ ≈ 0.1 and match theexperimental data by Chambon and Corté (6) rea-sonably well, as can be seen in Figure 3. Theexperimental data were obtained from 3D centri-fuge tests with nearly cohesionless sand and fric-tion angles in the range between 38° and 42°. Alltheir experimental data, i.e. assuming c´ = 0 as wellas c´ = 2.5 kPa, fit into the shaded bar in Figure 3.

The stability numbers forcohesion and surcharge

In contrast to the soil weight number, the cohesionnumber can be derived theoretically. In fact, Ver-meer and Ruse (20) derived the simple expressionN

c= cot ϕ´, which was also verified by use of the

Fig. 3 The soil weightstability number asdetermined by differ-ent methods.Bild 3 Die Stabilitäts-zahl Nγ für vollständigausgekleidete Tunnelsnach verschiedenenMethoden.

finite element method (FEM). Again this expres-sion can be compared to findings by other re-searchers, as also done in Figure 4. Once moreKrause´s results (13) are based on a shell-shapedfailure body, whereas the data by Anagnostou andKovári (1) are based on the sliding wedge model,as also described by Jancsecz and Steiner (11).

The theoretical derivation of Nc = cot ϕ´ isbased on the assumption that ground surfaceloads have no influence at all. In other words thestability number N

q in equation [1] is supposed to

be equal to zero. This has been checked by per-forming a series of 24 finite element analyses withdifferent uniformly distributed surface loads qand resulting data are presented in Figure 5. Forfriction angles above 25°, it is observed that asurface load has no influence on the failure pres-sure. In these cases arching is apparently sostrong that all surface loads can be carried,independent of the ground cover on top of thetunnel. For a low friction angle of only 20°, how-ever, the situation is slightly different. In this case

Fig. 4 The cohesionstability number ac-cording to differentmodels.Bild 4 Der Kohä-sionsbeiwert N

c auf

der Grundlage ver-schiedener Modelle.

Fig. 5 Influence ofuniformly distributedsurface load on failurepressure.Bild 5 Einfluß einergleichmäßig verteiltenAuflast an der Gelän-deoberfläche auf denBruchdruck.

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FELSBAU 20 (2002) NO. 6

Fig. 6 The soil weightstability number as afunction of the relativecutting length.Bild 6 Die Stabilitäts-zahl Nγ als Funktion derrelativen Abschlags-länge.

Fig. 7 The soil weightstability number as afunction of the relativecutting length.Bild 7 Die Stabilitäts-zahl Nγ als Funktion derrelativen Abschlags-länge.

one needs a ground cover of at least twice thetunnel diameter to eliminate the influence of asurface load completely. For a friction angle of20° and a cover of H = D, on the other hand, oneobserves a small increase of the failure pressureas a function of q such that N

q≈ 0.01. Compared

to the soil weight stability number of about 0.25(see Figure 3 for ϕ´ = 20°) and the cohesion stabil-ity number of 2.75 (see Figure 4), a N

q-value of

0.01 is very low and it can be disregarded. Hence

Nq ≈ 0 and Nc = cot ϕ´ ................................. [3]

at least under the conditions that ϕ´ > 20° andH > 2D. This relatively large cover is, however,not needed for high friction angles. For frictionangles beyond 25° the above equation appears tohold for smaller values of H, namely for H > D.

Influence of an unlined wallnear the tunnel face

In order to determine the influence of an unlinedwall with length d near the tunnel face, as indicat-ed in Figure 6b, additional finite element analyseswere carried out. In these analyses a supportingpressure was applied both at the tunnel face and atthe unlined part of the wall. This supporting pres-sure was then reduced until failure occurred.

Again circular tunnels are considered and againwe begin to consider purely frictional soil withoutany effective cohesion. Then it was found from aseries of 75 different calculations that pf = γDNγ

with the stability numbers as plotted in Figure 6. Itshows that the soil weight stability number in-creases as a function of the relative cutting lengthd/D. Different curves are obtained for differentfriction angles, but all curves show basically thesame shape. Indeed, up to d = D the curves areconcave and then one observes a convex shape.Finally they approach an asymptotic limiting valuethat depends on the friction angle. The asymptoticvalues correspond to completely unlined tunnels.

Considering the complex shape of the curves inFigure 6a, it would seem difficult to find an ana-lytical expression that matches these curves. Forsmall values of d/D, however, the curves have asimple concave shape, and the following fairlysimple analytical approximation can be used:

Nd D

γ

ϕ

ϕ=

+′

−′2 3

180 05

6( / )tan

.tan

....................... [4]

under the conditions that ϕ´ > 20° and d < 0.5D.This analytical approximation reduces to equa-tion [2] when the cutting length is equal to zero. Itmatches the computational results up to d = 0.5D,as can be seen in Figure 7 that shows a close up ofFigure 6a. It is expected that equation [4] is validfor a ground cover of at least 1.5D, but as yet thishas not been investigated. On considering Fig-ure 7, it would seem that the cutting length hasrelatively little influence on the tunnel headingstability. In particular up to d/D = 0.3, the soilweight stability number is found to be nearlyindependent of the cutting length.

The relatively small influence of the cuttinglength would seem to suggest that safety can hardlybe improved by a reduction of the cutting length,but distinction should be made between drainedand undrained conditions. Indeed, all results applyto drained situations with considerable archingand consequently a relatively small influence of thecutting length. For undrained conditions, on theother hand, the situation is different. Here modeltest results on tunnels in clay by Kimura and Mair

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(12) have shown a clear influence of the cuttinglength. This was even observed for small lengthswhere drained analyses show little or no influenceof d. It should also be realised that our “drained”formulas have been derived for circular tunnels innon-layered ground. In a later section it will beshown that the influence of the cutting length issomewhat more important when consideringNATM-shapes. Finally a significant influence of thecutting length will be reported later, when consid-ering NATM tunnels in layered ground.

Having shown failure mechanisms for fullylined tunnels in Figure 2, it is now of interest toconsider collapse patterns for partially lined tun-nels. To this end computational data are shown inthe three-dimensional perspective in Figure 8. Itshows close ups around the heading of deep tun-nels with different cutting lengths in ground witha friction angle of 30°. The lining is indicated byshades from white to black, the volume elementsof the ground by black lines. Similar to Figures 2eto 2f the rate of displacements are shown ingraded shades from blue to red. For small valuesof d, the failure mechanism remains at the tunnelface (Figure 8a), but for larger values one ob-serves a collapsing roof (Figures 8b and 8c).

Fully unlined tunnels

On increasing the cutting length up to extremelylarge values, one finally obtains the two-dimen-sional situation of an unlined tunnel. The data in

Figure 6 indicate that such a situation is virtuallyreached when d > 10D. Vermeer and Vogler(2002) performed two-dimensional analyses ofunlined tunnels and reported for the soil weightstability number the expression

N γ ϕ= ′ +0 6 2 0 182. cot . for d = ∞ .............. [5]

under the conditions that ϕ´ > 25° and H/D > 2. Oncomparing the extremes of a fully lined tunnel withd = 0 (equation [2]) and a fully unlined one withd = ∞ (equation [5]), one compares a face failure asindicated in Figure 8a with a plane strain rooffailure. The difference is tremendous and this isalso reflected in the failure pressure; Figure 6ashows that failure pressures differ at least a factortwo and for low friction angles even more than afactor three. This demonstrates that the three-dimensional arching for d = 0 is much strongerthan the two-dimensional arching for d = ∞. Thedata by Vermeer and Vogler (23) also show that fulltwo-dimensional arching, such that the failurepressure is independent of depth, requires a rela-tively large friction angle of at least 25° and arelatively large ground cover of at least 2D.

As the unlined tunnel involves a roof failure, itis logical to plot the supporting pressure as afunction of the roof settlements. In this case thecurve of Figure 1a reduces to the well-knownground response curve (15), which plays an im-portant conceptual role in NATM tunnelling. Inaddition there is the idea of a supporting groundring as indicated in Figure 9a. We will consider a

Fig. 8 Incrementaldisplacements at fail-ure for different cuttinglengths. Red colourindicates the zone withthe largest incrementaldisplacements.Bild 8 Inkremen-telle Verschiebungenim Bruchzustand beiunterschiedlichenAbschlagslängen. Rotgibt die Bereiche mitden größten inkremen-tellen Verschiebungenan.

Fig. 9 a) The ideaof a ground ring,b) deviatoric stresses|σ

1- σ

3|, c) mobilised

shear strength for adeep tunnel.Bild 9 a) Die Ideedes Gebirgstragrings,b) Deviatorspannungen|σ

1- σ

3|, c) mobilisierte

Scherfestigkeit füreinen tiefen Tunnel.

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Fig. 10 Possible se-quential excavations.

Bild 10 MöglicheTeilausbruchsflächen.

cross-section around a tunnel in order to investi-gate whether or not a so-called pressure ring inthe sense of Rabcewicz (17) will be formed. Withinsuch a ring one would expect tangential normalstresses (σ1) that are large with respect to theradial normal stresses (σ

3). The stress difference

σ1-σ

3 has thus been visualised to obtain Figure 9b.

The red colour in Figure 9b is used to indicateextreme stress differences and the blue colourmeans that there is either an isotropic state ofstress with σ

1= σ

3 or a state with very low stresses.

The large blue zone in Figure 9b relates to theassumption of an initial isotropic state withσ

1= σ

3. The smaller blue area just around the

tunnel indicates a zone with very small stresses.The red-green oval around the tunnel indicates anarching ground ring in the sense of Rabcewicz(17). The ring is characterised by large tangentialstresses (σ

1) and small radial stresses (σ

3). Excen-

tric ovals around the tunnel can also be observedin Figure 9c. This figure shows the mobilisation ofthe shear strength τ

f= c´+σ´tan ϕ´. The red colour

indicates full mobilisation and the blue colour isused for the area where there is no mobilisation atall, i.e. a zone with σ1 = σ3. This study thus con-firms the idea of a pressure ring around a tunnel.It is particularly observed when considering prin-cipal stress differences, as done in Figure 9b.

Maximum diameter inopen-face tunnelling

In usual open-face tunnelling, the face pressure isequal to zero and the failure pressure as comput-

ed from equation [1] must be negative; otherwisethe situation would not be stable. Open-face tun-nelling is thus subject to the criterion pf < 0. Inorder to consider this criterion in more detail,expressions [3] and [4] for the stability numbersare substituted into equation [1] to obtain:

p Dd D c

f =+ ( )

′−

− ′′

′

γϕ ϕ

ϕ2 3

180 05

6tan

tan.

tan ....... [6]

under the conditions that ϕ´ > 20° and d/D < 0.5.The stability criterion pf < 0 can now be reformu-lated to obtain an upper bound for the diameter ofa tunnel. It yields for d/D < 0.5

Dc

d D≤ ′

+ ( ) − ′′18

2 3 0 96

γ

ϕϕtan. tan ..................... [7]

This equation would seem to be implicit interms of the diameter, but this is not the case aslong as d/D has a given constant value. On theother hand, if the cutting length is taken as aconstant rather than the relative cutting length d/D, one has to solve the above equation iteratively.In many practical situations, however, d/D isrelatively small and its contribution to equa-tion [7] can be disregarded. On disregarding thisterm one obtains for d/D < 0.3

Dc≤ ′

− ′9 1

1 0 45γ ϕ. tan ...................................[8]

It can be observed from this equation that theangle of friction makes only a moderate contribu-tion to the safety of an open tunnel face. One canconsider for example a soil with a friction angle ofonly 20°, then equation [8] yields γD < 10 c´. For afriction angle of 30°, the situation is only slightlybetter as equation [8] then gives γD < 12 c´.Hence the friction angle is not of great impor-tance. It can thus be concluded that open facetunnelling under drained conditions is possiblewhen the effective cohesion is at least 10 % of γD.

If the cohesion is large enough a tunnel can bedriven at its full size. In case of relatively smallvalues of c´, on the other hand, the tunnel can bedriven in sections (Figure 10). In the latter case ofa sequential excavation the equivalent diameterof the top heading applies and equation [8] can beused to compute its maximum value. The abovecondition [8] on the tunnel diameter demon-strates the impact of the cohesion in open-facetunnelling. The maximum diameter is simplylinearly related to the cohesion.

Factor of safety foropen-face tunnelling

Instead of computing a maximum tunnel diame-ter, as done in the previous section, it is possibleto consider a tunnel with a given diameter and tocompute a factor of safety. In structural engineer-ing the safety factor is usually defined as the ratioof the collapse load to the working load, but fortunnel headings, this definition is not appropri-

Fig. 11 An entranceof the Rennsteigtunnel.Bild 11 Portalbereichdes Rennsteigtunnels.

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ate. Here one better adopts the definition that isused in the analysis of slope stability, i.e.η τ τ= f mob ..................................................... [9]

where τf represents the shear strength. This ratio

of the shear strength to the mobilised strength isa safety factor, which is to be considered in therest of this paper. By introducing the definitionτ

f= c´ + σ´ tan ϕ´, where σ´ is the effective normal

stress on a potential slip plane, the safety factor isfound to be

η σ ϕσ ϕ

= ′ + ′ ′+ ′

ccmob mob

tantan ................................. [10]

The parameters cmob and ϕmob are mobilisedshear strength parameters that are just largeenough to maintain equilibrium. Let us now re-turn to equation [6], where the failure pressure p

f

is related to the shear strength parameters. Thisrelationship does not only hold for the parame-ters p

f, c´ and ϕ´ but also for the mobilised ones

pmob

, cmob

and ϕmob

. It yields

p Dd D

c

mobmob

mob

mob

mob

=+ ( )

−

−

γϕ

ϕ

ϕ2 3

180 05

6 tan

tan.

tan

...... [11]

where pmob is the really applied face pressure.Within the concept of a single global factor ofsafety it is appropriate to define c

mob= c´/η and

ϕmob = ϕ´/η, as also suggested by equation [10]. Onsubstituting these expressions into equation [11]and on considering open-face tunnelling withpmob = 0, it follows that

η ϕ γϕ η= ′ + ′

+ ( ) ′0 9 18

2 36

. tantan

c D

d D ................................ [12]

For d = 0 the safety factor can be computedstraight forwardly, but an iterative procedure isneeded when the cutting length is not equal tozero. However, the influence of the cutting lengthon the safety factor is relatively small, as alreadydiscussed in a section above. For usual frictionangles beyond 20° and safety factors belowη = 1.5, it can be shown that

0 87 0 0. η η ηd d= =< ≤ for d< 0.2D ............. [13]

Hence, the influence of the relative cuttinglength on the safety factor is below 13 % and arelatively close estimate of the safety factor is thusobtained on using d = 0. For a more exact solutionof equation [12], the factor ηd=0 can be used as afirst iterate and a nearly exact solution will befound when performing two or three iterations.

Up to now open-face tunnelling has been con-sidered to be a special case of closed-face tunnel-ling, i.e. by setting pmob equal to zero. In non-linearfinite element analysis, however, the factor ofsafety can be computed directly by means of so-called ϕ-c-reduction. This procedure was basical-ly proposed by Zienkiewicz (24), improved byBrinkgreve and Bakker (4) and also published by

Dawson et al. (9). The procedure has been imple-mented in the Plaxis code (5) as well as in theFLAC code (10). In this ϕ-c-reduction procedurethe actual shear strength parameters are propor-tionally reduced until failure occurs for cmob andϕ

mob. Then the factor of safety is obtained from the

ratio of c´ and cmob

, or equivalently from the ratioof tan ϕ´ and tan ϕmob. Several such calculationswhere performed by Vermeer and Ruse (21) inorder to validate equation [12] for circular tun-nels in homogeneous ground.

NATM-tunnel in homogeneousground

As a first case study a cross section of theRennsteig tunnel in Thuringia is considered. Thisrelatively new tunnel with a length of nearly 8 kmis part of the German motorway A71 (Figure 11).The excavation of the double-tube tunnel wasdone by a sequential construction of a top headingfollowed by bench and invert. Figure 12 shows aparticular cross section with a relatively smallground cover of only 9 m. We have analysed thiscross section both for uniform ground as well asfor a layered ground profile. The layered groundprofile is to be considered in a subsequent section.In this section a non-layered ground with proper-ties as indicated in Figure 13 is to be considered.

Attention will be focused on the stability of thenon-circular top heading. Here it might be won-dered whether or not such an oval shape can beconsidered on the basis of equation [12], as thisrelation was derived for circular cross sections.However, it will be shown that equation [12] canhandle non-circular NATM tunnels. In order toapply this equation, an equivalent tunnel diameteris needed. The top heading of the Rennsteig tunnelhas a width of a = 11.5 m, a height of b = 5.3 m anda cross sectional area of 44 m2. The area can beused to calculate an equivalent diameter ofD = 7.5 m. The problem is now fully characterisedby the parameter set in Figure 12. On using equa-tion [12] this leads to a safety factor of η = 1.36.

In order to check the above result of equa-tion [12], a finite element analysis has been car-ried out for the real non-circular cross section. Inthis numerical analysis both c´ and ϕ´ were step-

Fig. 12 Top headingof Rennsteig tunnel inhomogeneous ground.Bild 12 Kalotten-vortrieb des Renn-steigtunnels in homo-genem Baugrund.

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Safety factors for Rennsteig tunnel in homogeneous ground.Sicherheitsfaktoren für den Rennsteigtunnel in homogenemBaugrund.

Method Geometry ηd=0 ηd=1.5

Equation 12 .... circle 1.40 1.36FEM ................ shallow top heading 1.48 1.35FEM ................ deep top heading 1.37

wise decreased down to failure values, i.e. downto c

mob and ϕ

mob, as explained at the end of the

previous section. In this manner a safety factor ofη = 1.35 was obtained, being practically equal tothe one from equation [12].

The considered cross section of the Rennsteigtunnel suffers from a relatively small ground coverof H/D = 1.2, where D is the equivalent diameter. Insuch a case one might wonder whether or not thestability of the tunnel heading is influenced by thissmall cover and for this reason we also analysed asituation with an extremely large ground cover.Again a factor of safety of about η = 1.35 was found,as also listed in the Table. This equivalence be-tween the very shallow and the very deep tunnelmay be more generally expected for friction anglesabove 25°. For lower angles of friction between 20°and 25°, depth independence requires relativeground covers of at least two, as also concludedfrom the data in Figure 5.

The nearly exact correspondence betweenpresent results for a circular shape and a topheading shape for d = 1.5 m are a coincidence. Onusing for instance d = 0 instead of d = 1.5 m, differ-ences between both shapes are found to be larger.For zero cutting length, the top heading shapehappens to yield η = 1.48, whereas a circular shapeleads to η = 1.40 as listed in the Table. Hence, itwould seem that the top heading shape tends to aslightly larger safety factor than the circular shape,but differences remain within a margin of 6 %. It isthus conducted that equation [12] applies both tocircular shapes and NATM shapes.

NATM-tunnel in layered ground

Instead of considering tunnel excavations in ho-mogeneous ground, as done in the previous sec-tion, attention will now be paid to excavations in

Fig. 13 Geologicalcross section of theBaecker Stream Val-ley.Bild 13 GeologischerSchnitt im BereichBäckerbachtal.

layered ground. Again the top heading excava-tion of the Rennsteig tunnel is considered and thecrossing of a valley with ground layers as indicat-ed in Figure 13. On the extreme left hand side oneobserves that the top heading and the groundcover are nearly completely in a weathered rocklayer. This case has already been considered inthe previous section on non-layered ground. Onmoving from the left hand side of Figure 13 to-wards the middle, the ground cover changes fromweathered rock into slope talus. The subsequentcross sections are clearly shown in Figure 14.Please note that the slope talus has a slightlysmaller friction angle and a considerably smallercohesion than the weathered rock underneath.The data for the weathered rock are as indicatedin Figure 12 and the properties of the slope talusare indicated in Figure 14.

All three different cases in Figure 14 have beenanalysed numerically. As equation [12] does nothold for layered ground, safety factors were com-puted by performing three-dimensional finite el-ement analyses. In all analyses the shape of thetop heading was exactly modelled and the ϕ-c-reduction procedure was applied. The computedη-values are indicated in Figure 14 and show avery logical trend; the thicker the soft top layer,the lower the factor of safety. On plotting resultsas a function of the effective ground cover

H H H top layer′ = − .......................................... [14]

one obtains Figure 15, in which the safety factor isplotted as a function of the effective ground cover.This figure illustrates the impact of an effectiveground cover. For relatively small effective groundcovers, the safety factor depends significantly onthis cover, but starting from H´≈ D, a more or lessconstant factor of safety is found. This is fully in theline with the data in Figure 5 on the influence of theground cover. It is also fully in line with ourfindings of arching around tunnel headings. ForH´ > D arching can fully develop and equation [12]applies. For smaller thicknesses of the effectivecover, the soft layer plays a role and equation [12]should not be applied. Instead it is recommendedto perform finite element analyses with ϕ-c-reduc-tion, as done to obtain Figure 15. On decreasing H´below the critical value of D, one observes first agradual decrease of the safety factor and a dra-matic decrease as soon as H´ has become negative.

For negative values of the effective groundcover, the tunnel face consists partly out of rela-tively soft ground and it influences the failuremechanism considerably. As long as the tunnelface is completely in the weathered rock, a full facefailure is obtained, as indicated in Figures 2e and2f. For the case of Figure 14c, on the other hand,a more local failure entirely inside the soft top layerwas observed. Figure 16 shows the local failure fora situation with zero cutting length; on the left across section and on the right a longitudinal sectionof the top heading. Increasing displacements areshown in graded shades from blue to red.

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Let us now consider the influence of the cuttinglength in somewhat more detail. To this end, itshould be realised that the very local face failurein Figure 16 corresponds to d = 0. For non-zerocutting lengths, however, the failure zone is larg-er due to the inclusion of the unsupported part ofthe tunnel roof. As long as the soft top layer is notfar below the tunnel roof and significant cuttinglengths are used, the failure mode is even a rooffailure instead of a face failure. As a consequenceof such different failure modes, the cutting lengthplays an important role in layered ground and thisis reflected by the two curves in Figure 15.

On performing non-linear finite element anal-yses in layered ground, it appears that the ele-ment mesh should be well designed in order thatan accurate assessment of the failure mechanismcan be made. For meshes with 15-noded prismat-ic elements as illustrated in Figure 1b and homo-geneous ground, fine meshes with an averageelement length of D/10 around the tunnel facewere used. In order to obtain accurate results fortunnel faces in layered ground, refined meshes inthe soft zones near the tunnel face were used, asfailure tends to localise in such zones.

Conclusions

Results of three-dimensional finite element calcu-lations have been considered for tunnel headingsin drained ground. Restriction has been made toan isotropic Mohr-Coulomb material, which ex-cludes materials with a highly anisotropic strengthsuch as jointed rocks and heavily bedded sedi-ments. Stress distributions around tunnel head-ings in soils and soft rocks were found to bedominated by arching, at least for friction anglesbeyond 20°. Once the friction angle is beyond thisvalue, stability appears to be independent of theground cover as well as possible surface loads. Forshield tunnelling, this leads to an extremely simpleexpression for the minimum support pressurerequired. For open face tunnelling, it results inrules for the maximum diameter of the excavationand it involves the cutting-length. Consideringsmall cutting lengths, the very simple approximatestability criteria of γD < 10 c´ was found to apply.The diameter D may either be the full tunneldiameter or a subsection diameter of a sequential

Fig. 14 Differentground profiles forRennsteig tunnel withcomputed safetyfactors for d = 1.5 m.Bild 14 Verschie-dene Bodenprofile fürden Rennsteigtunnelmit den jeweils be-rechneten Sicherheits-faktoren für d = 1,5 m.

excavation. Instead of considering a maximumdiameter, it is also possible to calculate a factor ofsafety for a fixed given value of the diameter.

In the last part of this study non-circular tun-nels have been considered. In such cases an equiv-alent diameter, for use in the stability criteria, canbe computed from the excavation area consid-ered. Unfortunately the stability equations do nothold when the tunnel face is dominated by differ-ent ground layers. In such cases it is advocated toapply a non-linear finite element analysis and touse the so-called ϕ-c-reduction method. Even inthe general case of layered ground the formulasfor non-layered ground remain of interest forunderstanding tendencies and for validating dif-ferent numerical models and computer codes.

In classical literature on the NATM one findsthe concept of the mobilisation of a supportingground ring around tunnels as a function ofdeformations. It would seem that we detectedsuch a ring numerically quite directly. It is nicelyvisualised by plotting contours of deviatoricstresses |σ

1- σ

3|. For circular tunnels such a

ground ring is found to be elliptical.It should be realised that not all aspects of

tunnel heading stability have been addressed,e.g. not the destabilising effect of pore waterpressure. When driving a shield tunnel under theground-water table and drained conditions ap-ply, the effective failure pressure naturally has tobe increased by the pore water pressure. Moreo-ver, one would have to use the submerged weight

Fig. 15 Factor ofsafety as a functionof the effective cover.Bild 15 Der Stand-sicherheitsfaktor alsFunktion der effektivenTunnelüberdeckung.

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γ´ in all previous formulas. When open face tun-nels are driven under the ground-water table anddrained conditions apply, the water has an addi-tional destabilising effect due to groundwaterflow towards the tunnel heading, which has notbeen considered in this paper.

The present equations do not always apply totunnelling in saturated clays, as they behaveinitially undrained with cu as the undrained shearstrength. For such situations, the equationp

f= -c

uT

c+ σ

0 applies, where σ

0 is the initial ver-

tical overburden stress at the tunnel axis. Atkin-son and Mair (2) used model test data to derivedesign curves that give Tc as a function of H/D andd/D. For a standstill, however, drained conditionswill occur and one may apply the formulas of thepresent study.

It should also be realised that perfect plasticityis a strong assumption, when considering highlycohesive clays and weak rocks that show a peakstrength and a much lower residual strength.This softening with its resulting tendency of pro-gressive failure is well-known for clay slopes, e.g.by the papers of Chen et al. (7) and Potts et al. (16),but as yet it was not thoroughly investigated fortunnel heading stability. For the two-dimensionalproblem of a fully unlined tunnel, consequencesof softening are studied in another paper in thesame issue of this journal. For the three-dimen-sional problem of a tunnel heading, it is a topic offurther research.

Finally it should be realised that attention hasbeen purely focused on stability and not on sur-face settlements, which are important for tunnel-ling in urban areas. There is no doubt that thesuccessful design of tunnel excavations in suchareas is based on both stability and deformationconsiderations.

References1. Anagnostou, G. ; Kovári, K.: Face Stability Conditions withEarth-Pressure-Balanced Shields. In: Tunnelling and Under-ground Space Technology 11 (1996), No. 2, pp. 165-173.2. Atkinson, J.H. ; Mair, R.J.: Soil mechanics aspects of softground tunnelling. In: Ground Engineering 14 (1981), No. 5, pp.20-38.3. Baumann, T. ; Sternath, R. ; Schwarz, J.: Face stability oftunnels in soft rock – Possibilities for the computational analysis.Proc. 14th Int. Conf. Soil Mech. Found. Engng, Hamburg, Vol. 3,pp. 1389-1392. Rotterdam: A.A. Balkema, 1997.4. Brinkgreve, R.B.J. ; Bakker, H.L. : Non-linear finite elementanalysis of safety factors. Proc. 7th Int. Conf. Computer Meth-ods and Advances in Geomechanics, Cairns, Vol. 2, pp. 1117-1122. Rotterdam: A.A. Balkema, 1991.

Fig. 16 Local failureinside the soft top

layer for d = 0.Bild 16 Ein lokaler

Verbruch bildet sich inder oberen Boden-

schicht aus (für d = 0).

5. Brinkgreve, R.B.J. ; Vermeer, P.A.: Manual of Plaxis 3DTunnel. Rotterdam: A.A. Balkema, 2001.6. Chambon, J.F. ; Corté, J.F.: Shallow tunnels in cohesionlesssoil: Stability of tunnel face. In: J. Geotech. Engng ASCE 120(1994), No. 7, pp. 1150-1163.7. Chen, Z. ; Morgenstern, N.R. ; Chan, D.H.: Progressivefailure of the Carsington Dam: a numerical study. In: Can.Geotech. J. 29 (1992), No. 6, pp. 971-988.8. Davis, E.H. ; Gunn, M.J. ; Mair, R.J. ; Seneviratne, H.N.: Thestability of shallow tunnels and underground openings in cohe-sive material. In: Géotechnique 30 (1980), No. 4, pp. 397-416.9. Dawson, E.M. ; Roth, W.H. ; Drescher, A.: Slope stabilityanalysis by strength reduction. In: Géotechnique 49 (1999),No. 6, pp. 835-840.10. Itasca Consulting Group Inc.: FLAC Version 4.0. User´sGuide. Itasca consulting Group Inc., Minneapolis, 2002.11. Jancsecz, S. ; Steiner, W.: Face support for large mix-shield in heterogeneous ground conditions. Proc. Tunnelling´94, London, pp. 531-550. London: Chapman & Hall, 1994.12. Kimura ; Mair (1981). Centrifugal Testing of Model Tunnelsin Soft Clay. Proc. 10th Int. Conf. Soil Mech. Found. Engng,Stockholm, Vol. 1, 319-322. Rotterdam: A.A. Balkema Publish-ers.13. Krause, T.: Schildvortrieb mit flüssigkeits- und erdgestütz-ter Ortsbrust. Report of the Inst. of Geotech. Engng of theUniversity of Braunschweig, Report No. 24, 1987.14. Léca, E. ; Dormieux, L.: Upper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterial. In: Géotechnique 40 (1990), No. 4, pp. 581-606.15. Pacher, F.: Deformationsmessungen im Versuchsstollenals Mittel zur Erforschung des Gebirgsverhaltens und zurBemessung des Ausbaus. Felsmechanik und Ingenieurgeolo-gie, Supplementum I. Wien: Springer, 1964.16. Potts, D.M. ; Kovacevic, N. ; Vaughan, P.R.: Delayedcollapse of cut slopes in stiff clay. In: Géotechnique 47 (1997),No. 5, pp. 953-982.17. Rabcewicz von, R.: Gebirgsdruck und Tunnelbau. Wien:Springer, 1944.18. Semprich, S.: Berechnung der Spannungen und Verfor-mungen im Bereich der Ortsbrust von Tunnelbauwerken inFels. Report of the Inst. of Geotech. Engng of the RWTHAachen, Report No. 8, 1980.19. Sternath, R. ; Baumann, T.: Face support for tunnels inloose ground. World Tunnel Congress Wien´97. Rotterdam:A.A. Balkema, 1997.20. Vermeer, P.A. ; Ruse, N.: Die Stabilität der Tunnelortsbrustin homogenem Baugrund. In: geotechnik 24 (2001), No. 3, pp.186-193.21. Vermeer, P.A. ; Ruse, N.: Neue Entwicklungen in derTunnelstatik. Proc. 3rd Kolloquium Bauen in Boden und Fels,Technische Akademie Esslingen (ed.: Schad, H.), pp. 3-14.Ostfildern: TAE, 2002.22. Vermeer, P.A. ; van Langen, H.: Soil collapse computationswith finite elements. In: Ingenieur-Archiv 59 (1989),pp. 221-236.23. Vermeer, P.A. ; Vogler, U.: On the stability of unlined tunnels.In Learned and Applied Soil Mechanics out of Delft (eds Barends& Steijger), pp. 127-134. Rotterdam: A.A. Balkema, 2002.24. Zienkiewicz, O.C. ; Humpheson, C. ; Lewis, R.W.: Associ-ated and non-associated visco-plasticity in soil mechanics. In:Géotechnique 25 (1975), No. 4, pp. 671-689.25. Horn, M.: Horizontaler Erddruck auf senkrechte Abschluß-flächen von Tunnelröhren. Landeskonferenz der UngarischenTiefbauindustrie, Budapest. Übersetzung ins Deutsche durchdie STUVA, 1961.

AcknowledgementsThe authors are indebted to Dr. ir. P. Bonnier from Plaxis for hissupport in the numerical calculations. The authors would alsolike to thank Dr.-Ing. habil. H. Schad for his very helpfulcomments and his thorough review of this paper.

AuthorsProfessor Dr.-Ing. Pieter A. Vermeer and Dipl.-Geol. NicoRuse, Institute of Geotechnical Engineering, University ofStuttgart, Pfaffenwaldring 35, D-70569 Stuttgart, Germany, E-Mail vermeer@igs.uni-stuttgart.de; ruse@igs.uni-stuttgart.de, Dipl.-Ing. Thomas Marcher, ILF Consulting Engineers ZTGmbH, Framsweg 16, A-6020 Innsbruck, Austria, E-Mailthomas.marcher@ibk.ilf.com.