ts

Keywords:SqueezingYielding supportDesignAnalysisStress path

e ththathe

between the ground pressure and the radial displacement of the tunnel wall under plane strain condi-

e phening thr

structurally implemented in two main ways (Anagnostou and Can-tieni, 2007): either by arranging a compressible layer between theexcavation boundary and the extrados of a stiff lining (Fig. 1a) orthrough a suitable structural detailing of the lining that will allowa reduction in its circumference (Fig. 1b). In the rst case theground experiences convergences while the clearance prole re-mains practically constant. This solution has been proposed partic-ularly for shield tunnelling with practically rigid segmental linings

commonly, right from the start. In order to avoid overstressing ofthe shotcrete and, at the same time, to allow it to participate inthe structural system, special elements are inserted into longitudi-nal slots in the shotcrete shell (Fig. 1b, Section dd). Fig. 2 showsthe typical load deformation characteristics of such elements.As can be seen from the hoop force N vs. deformation e relation-ships, the elements yield at a specic load level, thereby limitingthe stress in the shotcrete shell. The so-called lining stress con-trollers (Schubert, 1996; Schubert et al., 1999) consist of co-axialsteel cylinders which are loaded in their axial direction, buckle instages and shorten up to 200 mm at a load of 150250 kN. Actual

* Corresponding author.

Tunnelling and Underground Space Technology 24 (2009) 309322

Contents lists availab

ro

.eE-mail address: linard.cantieni@igt.baug.ethz.ch (L. Cantieni).attempt is made to stop the deformations with the lining (so-calledresistance principle, Kovri, 1998), a so-called genuine rockpressure builds up, which may reach values beyond the structur-ally manageable range. The only feasible solution in heavilysqueezing ground is a tunnel support that is able to deform with-out becoming damaged, in combination with a certain amount ofover-excavation in order to accommodate the deformations. Sup-ports that are based on this so-called yielding principle can be

Steel sets applied in squeezing ground usually have a top hatcross-section. The hoop force in the steel sets is controlled by thenumber and by the pre-tensioning of the friction loops connectingthe steel segments. Up to four friction loops, each offering a slidingresistance of about 150 kN, may be used per connection. Shotcretemay be applied either after the occurrence of a pre-dened amountof convergence (as a recent example the lot Sedrun of the Gotthardbase tunnel can be mentioned, see Kovri et al. (2006)) or, more1. Introduction

The term squeezing refers to thmations that develop when tunnell0886-7798/$ - see front matter 2008 Elsevier Ltd. Adoi:10.1016/j.tust.2008.10.001tions. The computation of the characteristic line assumes a monotonic decrease of radial stress at theexcavation boundary, while the actual tunnel excavation and subsequent support installation involve atemporary complete radial unloading of the tunnel wall. This difference, in combination with the stresspath dependency of the ground behavior, is responsible for the fact that the results obtained by spatialanalysis are not only quantitatively, but also qualitatively different from those obtained by plane strainanalysis. More specically, the relationship between ground pressure and deformation at the nal stateprevailing far behind the face is not unique, but depends on the support characteristics, because theseaffect the stress history of the ground surrounding the tunnel. The yield pressure of the support, i.e. itsresistance during the deformation phase, therefore proves to be an extremely important parameter.The higher the yield pressure of the support, the lower will be the nal ground pressure. A targetedreduction in ground pressure can be achieved not only by installing a support that is able to accommo-date a larger deformation (which is a well-known principle), but also by selecting a support that yields ata higher pressure.

2008 Elsevier Ltd. All rights reserved.

omenon of large defor-ough weak rocks. If an

(Schneider et al., 2005; Billig et al., 2007). The second solution isthe one usually applied today. It involves steel sets having slidingconnections in combination with shotcrete (Fig. 1b, Sections ccand dd, respectively).Accepted 7 October 2008Available online 17 December 2008

that squeezing pressure will decrease by allowing the ground to deform. When estimating the amount ofdeformation required, one normally considers the characteristic line of the ground, i.e. the relationshipThe interaction between yielding suppor

L. Cantieni *, G. AnagnostouETH Zurich, Switzerland

a r t i c l e i n f o

Article history:Received 21 April 2008Received in revised form 16 September2008

a b s t r a c t

In this paper we investigatspatial numerical analysesnomograms which enable

Tunnelling and Underg

journal homepage: wwwll rights reserved.and squeezing ground

e interaction between yielding supports and squeezing ground by means oft take into account the stress history of the ground. We also present designrapid assessment of yielding supports. The idea behind yielding supports is

le at ScienceDirect

und Space Technology

l sevier .com/locate / tust

Nomenclature

a tunnel radiusb spacing of steel sets (Fig. 1)b0 clear distance between the steel sets, length of the

deformable elements (Fig. 1)c ground cohesiond lining thicknessdII thickness of the deformable elements and of the lining

during the deformation phasedIII lining thickness after the deformation phasee unsupported spanER Youngs modulus of the groundE Youngs modulus of the lining

r radial co-ordinate (distance from tunnel axis)s slot size in the circumferential direction (Figs. 1 and 2)u radial displacement of the groundu2D radial displacement of the ground assuming plane strain

conditionsu(0) radial displacement of the ground at the faceu(1) nal radial displacement of the grounduy maximum radial displacement of support in the defor-

mation phaseUy normalized maximum radial displacement of support in

the deformation phasex axial co-ordinate (distance behind the tunnel face)

310 L. Cantieni, G. Anagnostou / Tunnelling and Underground Space Technology 24 (2009) 309322L

fc uniaxial compressive strength of the groundfy yield strength of the highly-deformable concrete ele-

mentsH depth of coverk lining stiffnesskI support stiffness before the deformation phasekIII support stiffness after the deformation phasen number of deformable elements and sliding connections

in one cross-sectionnf number of friction loops per sliding connectionN lining hoop forceNy lining hoop force during the deformation phaseNf sliding resistance of one friction loopp radial pressure acting upon the liningp(1) nal radial pressure acting upon the liningpy yield pressure of supportpo initial stressexamples of this include the Galgenberg tunnel (Schubert, 1996),the Strenger tunnel (Budil et al., 2004) and the Semmering pilottunnel (Schubert et al., 2000). Additionally, there are the re-cently-developed highly-deformable concrete elements (Kovri,2005), which are composed of a mixture of cement, steel bersand hollow glass particles. These collapse at a pre-dened com-pressive stress which is dependent on the composition of the con-crete, thereby providing the desired deformability. These elementshave been applied in the Ltschberg Basetunnel and in the St. Mar-tin La Porte site access tunnel of the Lyon Turin Ferroviaire (Thutet al., 2006a,b).

The number n, the size s (in circumferential direction, Fig. 2)and the deformability of the compressible elements (or the slidingways of the steel set connections) will determine the possiblereduction of the circumference of the lining during the yieldingphase, limiting the amount of radial displacement that can occur

c - c

a1

2

b3

4

uu

s - ss

5

d - d

6

u 5

6

Fig. 1. Basic types of deformable supports: (a) compressible layer between lining and excinsets.c unit weight of the groundm Poissons ratio of the groundDs slot size reduction (Fig. 1)e slot deformation Ds/s (Fig. 2)ey slot deformation during the deformation phaseq2D radius of the plastic zone assuming plane strain condi-

tionsr1 maximum principal stressr3 minimum principal stressrs lining hoop stressrxx axial stressrrr radial stressrtt tangential stressrrx shear stress/ internal friction angle of the groundw dilatancy angle of the groundwithout damaging the lining. The elements selected must thereforebe compatible with the planned amount of over-excavation. Thelatter represents an important design parameter. If it is too low,costly and time-consuming re-proling works will be necessary.On the other hand, if it is too high, the over-prole will have tobe lled by the cast-in-situ concrete of the nal lining.

The idea behind all yielding support systems is that the groundpressure will decrease if the ground is allowed to deform. Duringconstruction, the support system and the amount of over-excava-tion can be adapted to changes in squeezing intensity throughthe use of advance probing, monitoring results and observationsmade in tunnel stretches excavated previously. In the planningphase, however, decision-making has to rely solely upon experi-ence and geomechanical calculations. When estimating the re-quired amount of over-excavation, the usual approach is toconsider a tunnel cross-section far behind the tunnel face and to

1 Compressible layer2 Lining3 Steel set4 Sliding connection with

three friction loops5 Shotcrete6 Compressible element

3

c d

c db'b

avation boundary; (b) yielding supports with steel sets, shotcrete and compressible

patan(kIII)

B

GRC

c

py

x

tunnel axisr

p( )

stiff support unsupported spandeformation phase

IIII

a

II core

dergassume plane strain conditions. Where there is rotational symme-

4

3

2

1

0

Hoop

forc

e N

[MN

]

0

0

10

15

5

0Sh

otc

rete

str

ess

s =

N/

d [M

Pa]

Radial displacement u = n s /2 [cm]

Deformation = s/s

hdc

lsc

5 1510

20% 40%0

200

400

600

Supp

ort

re

sista

nce

p

= N/

a [kP

a]d = 0.3 m

s = 0.4 m

N

a = 6 m

u

p

hdc

n = 6 connections

Fig. 2. Load deformation characteristics (N, Ds/s) of compressible elements andrespective convergence u vs. pressure p relationships for a circular lining with n = 6insets. lsc: lining stress controllers (four cylinders per linear meter) after Schubertet al. (1999); hdc: highly-deformable concrete elements after Thut et al. (2006a,b)having a yield stress of 47 MPa.

L. Cantieni, G. Anagnostou / Tunnelling and Untry, the plane strain problem is mathematically one-dimensional.The so-called characteristic line of the ground (also referred to asthe ground response curve, Panet and Guenot, 1982) expressesthe relationship between the radial stress p and the radial displace-ment u of the ground at the excavation boundary (Fig. 3a). Closed-form solutions exist for the ground response curve in a variety ofconstitutive models. The ground response curve can be employedfor estimating the radial convergence u(1) of the ground that mustoccur in order for the ground pressure to decrease to a chosen,structurally manageable value p(1) (Fig. 3a).

The rst problem with this approach is that the radial conver-gence u(1) represents only an upper limit in terms of the amountof over-excavation required, because it includes the ground dis-placement u(0) that has already occurred ahead of the tunnel face.The pre-deformation u(0) introduces an element of uncertaintyinto the estimation of the required amount of over-excavation. Thisuncertainty is particularly serious in the case of heavily squeezingground because its behavior is highly non-linear and, conse-quently, small variations in the deformation will have a large effecton the pressure.

A second, more fundamental problem is that all plane strainsolutions (whether closed-form solutions for the ground responsecurve or numerical simulations involving a partial stress releasebefore lining installation) assume that the radial stress at the exca-vation boundary decreases monotonically from its initial value (farahead of the face) to the support pressure (far behind the face),while the actual load history will include an intermediate stagewith a complete unloading of the excavation boundary in the radialdirection: the radial boundary stress is equal to zero over theunsupported span e > x > 0 between the tunnel face and the instal-lation point of the lining (Fig. 3b). Cantieni and Anagnostou (2008)have shown that the assumption of a monotonically decreasing ra-dial stress may lead (particularly under heavily squeezing condi-tions) to a more or less serious underestimation of grounde

uyu(e)

u(x)

u( )b

Necessary over-excavation

p

u

GRC

B

a

u( )

p( )

u(0) ?

C

D

round Space Technology 24 (2009) 309322 311pressure and deformation. The actual (u(1), p(1)) points pre-vailing at the equilibrium far behind the face are consistently lo-cated above the ground response curve (point B in Fig. 3a). Thismeans that, a plane strain analysis cannot reproduce at one andthe same time both the deformations and the pressures: in orderto determine the ground pressure through a plane strain solution,the deformations have to be underestimated (point C in Fig. 3a) or,vice versa, in order to determine the deformations, the groundpressure has to be underestimated (point D in Fig. 3a). This is par-ticularly relevant from the design standpoint for a yielding sup-port, because in this case one needs reliable estimates both ofthe deformations (for determining the amount of over-excavation)and of the pressures (for dimensioning the lining).

Moving on from these results, the present paper investigatesthe interaction between yielding supports and squeezing groundby means of numerical analyses that take into account the evolu-tion of the spatial stress eld around the advancing tunnelheading.

py

u

II III

atan(kI)

I

uy

py

B

A

Fig. 3. (a) Determination of amount of over-excavation based upon groundresponse curve; (b) development of ground pressure and deformation along thetunnel wall; (c) groundsupport interaction in the rotational symmetry modelunder plane strain conditions.

Anagnostou, 2007). As mentioned above, plane strain computa-tions are inadequate for the analysis of yielding supports. However,results of such calculations will also be presented in the next Sec-tions for the purpose of comparisons.

2. The inuence of the stress path on groundsupportinteraction

This Section provides some useful insights into the effect of thestress path by a comparative analysis of two hypothetical supportcases. Consider rstly a stiff, almost rigid lining (EL = 30 GPa,d = 35 cm), which is installed at a distance of e = 24 m behind thetunnel face (the other parameters are given in Table 1). Figs. 5aand 5b show on their left hand sides (support case 1) the support

dergIn view of the well-known uncertainties of all computationalmodels in tunnelling (in respect of the initial stress eld, the con-stitutive behavior and the material constants of the ground), it isreasonable to ask what is the value of such a rened model fromthe standpoint of practical design. In the present paper we see thatan approach that takes account of the stress history leads to re-sults, which are not only quantitatively but also qualitatively dif-ferent from the ones obtained by plane strain analyses. The mostsignicant and surprising result of our study is that the higherthe yield pressure of the support, the lower will be the nal pres-sure. In practical terms, the important conclusion here is that ayielding support should be designed so that it yields at the highestpossible pressure. An ideal yielding support is one that starts to de-form at a pressure just below the bearing capacity of the lining. Asexplained below, this result cannot be obtained through a planestrain model.

Fig. 3c illustrates the groundsupport interaction using thecharacteristic line method. In the case of rotational symmetry con-sidered here, the radial displacement u vs. pressure p relationshipfor the yielding support can be obtained from the hoop force N vs.deformation e curves of the compressible elements by means ofsimple algebraic operations (Fig. 2). The solid polygonal line inFig. 3c represents an idealized model of the characteristic line ofa yielding support. Phase I is governed by the stiffness kI of the sys-tem up to the onset of yielding. In Phase II the support system de-forms under a constant pressure py. In the example of Fig. 2, theyield pressure py amounts to 150400 kPa depending on the typeof the compressible elements. In the presence of steel sets(Fig. 1b), one should also take into account the resistance offeredby the sliding connections (i.e. (nfNf)/(ba), where nf, Nf and b denotethe number of friction loops per connection, the frictional resis-tance of one friction loop and the spacing of the steel sets, respec-tively). When the amount of over-excavation uy is used up, thesystem is made practically rigid (stiffness kIII), e.g. by applyingshotcrete, with the consequence that an additional pressure buildsup upon the lining (Phase III). Fig. 3b shows schematically thedevelopment of radial pressure along the tunnel. The amount ofover-excavation (uy) and the yield pressure (py) are the main de-sign parameters for a yielding support, while the stiffnesses kIand kIII are of secondary importance for the cases that are relevantin practical terms.

The intersection point of the ground response curve with thecharacteristic line of the support (Fig. 3c, point A) fullls the con-ditions of equilibrium and compatibility and shows the radialground convergence and the nal pressure acting upon the liningfar behind the face. In the case of a support having a lower yieldpressure p0y and the same deformation capacity uy (dashed polygo-nal line), one would obtain exactly the same intersection point, i.e.the same values of pressure and deformation. According to this ap-proach, the yield pressure has no bearing on the nal ground pres-sure developing upon the lining. We will see, however, that theactual equilibrium points (obtained by spatial, axisymmetric calcu-lations) are located above the ground response curve (obtained un-der plane strain conditions) and that the nal ground pressure ishigher in the case of supports involving a lower yield pressure(the points B and B0 in Fig. 3c apply to yield pressures of py andp0y, respectively). Section 2 of this paper explains why this is so,providing thereby a new insight into the problem of groundsup-port interaction, while Section 3 investigates in detail the inuenceof the main design parameters (yield pressure py, deformationcapacity uy).

The inuence of the yield pressure on the nal rock pressure hasnever before been investigated. The literature contains only a few

312 L. Cantieni, G. Anagnostou / Tunnelling and Unproject-specic three-dimensional numerical studies (e.g. Amberg,1999; Fellner and Amann, 2004). Furthermore, until now there hasbeen no reliable simplied method for estimating the requiredamount of over-excavation. Since a plane strain analysis cannotreproduce at one and the same time both the deformation andthe pressure, three-dimensional numerical computations areunavoidable at least for heavily squeezing conditions. We carriedout a comprehensive parametric analysis which involved morethan 1500 parameter combinations covering the relevant rangeof values and we developed dimensionless design nomograms thatcan be used for estimating the amount of over-excavation whenapplying yielding supports (Section 4).

All of the computations in the present paper apply to the case ofrotational symmetry. The underlying assumptions are: a cylindri-cal tunnel; uniform support pressure over the circumferentialdirection (but of course variable in the longitudinal direction); ahomogeneous and isotropic ground; a uniform and hydrostatic ini-tial stress eld. Fig. 4 shows the computational domain and theboundary conditions. The assumption concerning the initial stresseld does not allow determining bending moments, which woulddevelop in the case of a non-hydrostatic initial stress eld. How-ever, yielding support design is mostly based upon the widely usedmethod of characteristic lines, which assumes rotational symmetryas well and therefore agrees better with the chosen axisymmetricmodel.

Being aware of the great variety of the existing constitutivemodels, the mechanical behavior of the ground was modeled hereas linearly elastic and perfectly plastic according to the MohrCou-lomb yield criterion, with a non-associated ow rule. This constitu-tive model is widely used in the engineering practice. The liningwas modeled as a radial support having a deformation-dependentstiffness k = dp/du (Fig. 3c). Tunnel face support has not been takeninto account, and nor have any time dependencies of the behaviorof the ground or of the shotcrete lining. The numerical solutions ofthe axisymmetric tunnel problem have been obtained by means ofthe nite element method. The advancing tunnel heading was han-dled using the so-called steady state method (Corbetta, 1990;

po

40a

25 a75 ax

rp = k (u(x) - u(e))

e

a

Fig. 4. Model dimensions (not to scale) and boundary conditions.

round Space Technology 24 (2009) 309322pressure development in the longitudinal direction and the radialdisplacement of the ground at the excavation boundary, respec-tively. The nal ground pressure p(1) prevailing far behind the lin-

derging installation point amounts to 700 kPa, while the radial conver-gence u(1) u(0) of the opening is about 52 cm. As it is assumedthat the 24 m long span between the face and the lining has beenleft unsupported, this case is rather theoretical. It is equivalent,however, to a yielding support which is installed immediately atthe tunnel face and which is able to accommodate a radial conver-gence of at least 52 cm while offering only negligible resistance tothe ground in the deformation phase.

Consider now (support case 2) a yielding support that is in-stalled directly at the face and offers a resistance of py = 700 kPaduring the deformation phase (i.e. the yield pressure is assumedto be as high as the nal ground pressure p(1) in the rst case).Let us, additionally, assume that the support is able to accommo-date a sufciently large deformation so that the groundsupportsystem reaches equilibrium at the yield pressure py. The nalground pressure will therefore also amount to 700 kPa in this case.As the nal radial convergence of the opening amounts in this caseto about 24 cm (Fig. 5b, right hand side), the assumption made(that the system reaches equilibrium at the yield pressure) presup-poses that the support is able to accommodate a convergence of atleast 24 cm in the deformation phase.

The only difference between the two support cases mentionedabove lies in the stress history: in the rst case the pressure startsto develop at a distance e = 24 m from the face and reaches its nalvalue of 700 kPa far behind the face, while in the second case thesupport pressure of 700 kPa acts right from the start (see Fig. 5a).This difference leads to a considerable variation in the longitudinaldeformation proles, particularly in the region behind the tunnelface (Fig. 5b): the nal radial displacement of the ground at theexcavation boundary is twice as high in support case 1 as it is insupport case 2, while the displacements ahead of the tunnel faceare approximately equal. In view of the support pressure distribu-tions in Fig. 5a, this result makes sense intuitively, while clearlydiffering from what one might expect under the characteristic linemethod.

According to the latter, the radial displacement should be thesame in both support cases, because the relationship between

Table 1Model parameters (numerical examples of Sections 2 and 3).

Parameter Value

Initial stress p0 12.5 MPaDepth of cover H 500 mUnit weight of ground c 25 kN/m3

Tunnel radius a 4 mYoungs modulus (ground) ER 1000 MPaPoissons ratio (ground) m 0.3Angle of internal friction (ground) / 25Cohesion (ground) c 500 kPaDilatancy angle (ground) w 5

L. Cantieni, G. Anagnostou / Tunnelling and Unground pressure and deformation is unique (the ground responsecurve is one and the same for both cases as it does not dependon the support behavior) and both cases have the same nalground pressure p of 700 kPa. For this pressure, a plane strain anal-ysis would predict a radial displacement u2D of approximately35 cm. This value agrees well with the results of the axisymmetricanalysis obtained for support case 2, but underestimates consider-ably the nal displacement in support case 1 (Fig. 5b). For the lat-ter, the axisymmetric analysis leads to a radial displacement,which is closer to the plane strain displacement for an unsup-ported opening (marked by u2D(p = 0) in Fig. 5b).

Similar observations can be made regarding the extent of theplastic deformation zone (Fig. 5c): in support case 2 the nal radiusof the plastic zone amounts to 10.5 m, which is very close to theplane strain analysis result for a support pressure of 700 kPa(q2D = 11.0 m). In support case 1 the plastic zone extends up to aradius of 14.5 m. This value is closer to the plane strain analysisprediction for an unsupported opening (15.6 m).

Figs. 5d and 6 show the evolution of the stresses (rxx, rrr, rtt,rrx) at a point located at the tunnel boundary and the stress pathin the principal stress diagram, respectively. The stress state atthe tunnel boundary locations marked by a, b, c and d inFig. 5d is given by the respective points in the principal stress dia-gram of Fig. 6. According to Fig. 5d, the stress paths ahead of thetunnel face are very similar for the two support cases: with theapproaching excavation, the axial stress rxx decreases from its ini-tial value p0 (in the eld far ahead of the face) to zero (at the face).In the region far ahead of the tunnel face, a stress concentrationcan be observed (stress path portion ab in Fig. 6), while closeto the face the axial connement is lost to such a degree that thecore yields and, as a consequence of Coulombs yield criterion,the radial stress rrr and the tangential stress rtt decrease (stresspath portion bc in Fig. 6). Immediately after excavation the radialstress becomes equal to zero, while both the tangential and the ax-ial stresses become equal to the uniaxial compressive strength ofthe ground fc (stress state c in Fig. 6). This stress state persistsin support case 1 over the entire unsupported span. The radialstress increases to its nal value of 700 kPa after the installationof the practically rigid lining, because the latter hinders furtherground convergence (stress path portion cd in Fig. 6). In supportcase 2, the radial stress increases (due to the initial support stiff-ness kI, see Fig. 3c) practically immediately after support installa-tion to the yield pressure py and remains constant thereafter.

Plane strain analysis assumes a monotonic decrease in the ra-dial pressure from its initial value (that prevails in the natural statefar ahead the face) to the nal value (that prevails after excavationfar behind the face), while both support cases considered here in-volve an intermediate stage characterized by a complete unloadingof the excavation boundary in the radial direction. The differencebetween the two support cases lies in the length of this intermedi-ate stage: in support case 1, the biaxial stress state c persists overthe entire, 24 m long unsupported span, while in support case 2,the stress state remains biaxial only very briey, almost instanta-neously at the face. This is why the intermediate biaxial stress statec (zero radial stress) governs both the extent of the plastic zoneand the magnitude of ground deformations in support case 1 (theresults are closer to the plane strain predictions for an unsupportedopening), while the nal triaxial stress state d (700 kPa radialstress) governs the support case 2 results (the nal radius of theplastic zone and the radial displacement agree well with the planestrain results for an actual nal support pressure of 700 kPa).

The results of a parametric study with different values for theunsupported span e provide additional evidence for the signi-cance of the intermediate stage. Fig. 7a shows the ground pressurep(1) and the radial displacement of the ground u(1) at the nalequilibrium prevailing far behind the face for the two supportcases discussed above as well as, for the purpose of comparison,the ground response curve obtained under plane strain conditions.Note that each point under support case 1 applies to another valueof the unsupported length e (e values are reported besides theordinate axis), while each point under support case 2 applies to an-other value of the yield pressure py (py values are equal to the -nal pressures p(1) on the ordinate axis). The diagram illustratesquite plainly the non-uniqueness of the ground pressure vs. grounddisplacement relationship: the equilibrium points for support case1 are consistently located above the ground response curve, whileall of the support case 2 results agree well with the plane strainpredictions.

Fig. 7b is more useful from a practical point of view as it shows

round Space Technology 24 (2009) 309322 313the radial convergence u(1) u(0) of the opening instead of theradial displacement u(1) of the ground. This diagram includesonly the results of the axisymmetric computations because plane

dergSupport Case 1

p( ) = 700 kPaa

p

u

314 L. Cantieni, G. Anagnostou / Tunnelling and Unstrain analysis yields the total radial displacement u(1), but notthe pre-deformation u(0). Note that u(1) u(0) represents theminimum amount of over-excavation that is necessary in orderto preserve the clearance prole. The purpose of a yielding supportis to reduce ground pressure to a pre-dened, structurally manage-able level. Fig. 7b points to the interesting conclusion that theamount of over-excavation an essential design parameter doesnot depend only on the ground quality and on the desired designload level, but also on the characteristics of the support. In order

80

0

u[cm]

-16-808162432

-16-808162432

16

8

0

2D (p = 0)

u2D (p = 700 kPa)

u2D (p = 0)

x [m]

x [m]

b

c

40

60

20

d

-16-808162432x [m]

52 cm

plastic zone

zone with plastic deformations butstress state within elastic domain

rr

tt

xx

rx

p0

0fc

x [m]

c bd

-16-80832 1624

rrttxx

a

Fig. 5. Numerical results for a stiff support installed at e = 24 m behind the tunnel face (le(a) Ground pressure distribution along the lining; (b) radial displacement of the ground athe tunnel wall (r = 4.01 m).Support Case 2

p(x) = py = 700 kPa

py = 700 kPa

p

u

round Space Technology 24 (2009) 309322to reduce the ground pressure to 0.7 MPa, for example, the amountof over-excavation has to be 2452 cm depending on the supportsystem (see points A and B in Fig. 7b). On the other hand, for a gi-ven amount of over-excavation, the nal ground load will dependon the support characteristics as well. For an over-excavation of,e.g. 24 cm, the nal ground pressure in support case 1 will amountto 2.4 MPa, that is three to four times higher than the load in case 2(see points B and C in Fig. 7b). This difference is considerable froma design standpoint.

-16-808x [m]

162432

-16-808162432

16

8

0

2D (p = 700 kPa)

u2D (p = 700 kPa)

x [m]

80

0

u[cm]

40

60

20

-16-808162432x [m]

24 cm

plastic zone

rr

tt

xx

rx

-16-808x [m]

32

d b

1624

c

rr

tt, xx

p0

0

a

ft hand side) and for a yielding support installed at the tunnel face (right hand side).t the tunnel boundary; (c) extend of the plastic zone; (d) evolution of the stresses at

derg0

ps

0

yc

1

1

Support Case 1

a

b

c d

1/p0

1=3

1.5

0.5

0.5

L. Cantieni, G. Anagnostou / Tunnelling and UnNote that support case 1, which necessitates a larger amount ofover-excavation for a given design load level (or attracts a higherground load for a given amount of over-excavation), is equivalentto the case of a deformable support with negligible yield pressure.So Fig. 7b actually indicates that a high yield pressure is favorablein terms of the nal ground load and the amount of over-excava-tion. Moving on from this nding, we will examine in Section 3the inuence of yield pressure in more detail.

Before doing so, however, we will briey examine the theoret-ical case of a ground with elastic behavior. The response of an elas-tic material does not depend on stress history. Consequently, inboth of the support cases discussed above, the nal ground pres-sure and the ground displacement will fulll Kirschs solution, i.e.the nal equilibrium points prevailing far behind the tunnel facewill be located on the straight ground response line. The value ofthe nal ground pressure xes the value of the nal ground dis-placement. The nal ground displacement does not depend onhow the radial stress at the excavation boundary evolves before

3/p0

Fig. 6. Stress paths of the ground at the tunnel boundary in the principal stress diagram fodeformable support that reaches equilibrium at the yield pressure of 700 kPa (right hanFig. 5d. Straight lines yc and ps: yield condition and stress path under plane strain c

00

u( ) [cm]

0.7

6030

2

3

4

300

24

01

2

4

8

12

16

6448

32

e[m

]

B A

Support case 1Support case 2GRC

1

p()[

MPa

]

5

Fig. 7. (a) Ground response curve under plane strain conditions and nal equilibrium po1) and for a deformable support that is installed at the tunnel face and has such a large d2); (b) radial convergence of the tunnel wall and ground pressure at equilibrium for th0

yc

1

Support Case 2

py

a

b

dc

1/p0

1=3

0

1

1.5

0.5

0.5

ps

round Space Technology 24 (2009) 309322 315reaching its nal value. So, the ground displacement u(1) will bethe same for the two support cases in Fig. 5. The pre-deformationu(0), however, will be smaller in case 2, because the support exertsa pressure earlier than in case 1, which involves support installa-tion at a greater distance behind the face. Consequently, theconvergence of the opening u(1) u(0), i.e. the amount of over-excavation required, will be larger in case 2 for a given grounddisplacement u(1) or for a given ground pressure p(1). Also, fora given amount of over-excavation, the ground loading developingin case 2 will be higher than in case 1. This conclusion is diametri-cally opposite to the results obtained for elastoplastic ground. Thecase of elastic ground behavior is interesting only from thetheoretical point of view, because large convergences that necessi-tate a yielding support are always associated with an overstressingof the ground and cannot be reproduced by assuming elasticmaterial behavior. Our examination of elastic behavior shows,however, that the conclusions of the present paper are valid forheavily squeezing ground in particular.

3/p0

r a stiff support installed at e = 24 m behind the tunnel face (left hand side) and for ad side). Points a, b, c, d: stress state at the respective locations indicated inonditions, respectively.

90

a

00

p()[

MPa

]

u( ) - u(0) [cm]90

b

24 60

B A

C

30

2.4

0.7

52

ints for a stiff support installed at a distance e behind the tunnel face (support caseeformation capacity that it reaches equilibrium at the yield pressure (support casee two support cases.

3. The inuence of yield pressure and yield deformation

In this Sectionwe investigate numerically the effects of themainyielding support characteristics. All of the numerical analyses havebeen carried out for the parameters of Table 1 and an unsupportedspan of e = 1 m. For the sake of simplicity, the tunnel radiuswas keptxed, i.e. it was not increased by the amount of over-excavation thatis required in order to preserve the clearance prole when the sup-port deforms. Fig. 8a shows the characteristic lines of the tunnel sup-ports, while Table 2 summarizes the numerical values of their

parameters. The yielding supports are assumed to deform at a con-stant pressure py of 01.2 MPa up to a radial displacement ofuy = 0.15 m (cases O, A, B, C and D) or 0.30 m (cases O0, A0, B0, C0 andD0) and to be practically rigid after the deformation phase. Yieldingpressures like the ones assumed for cases A, A0, B and B0 are todayrealistic. Recently developed ductile concrete elements of particu-larlyhighyield strength (up to25 MPa, Solexperts, 2007)makehigh-er yield pressures (such as in cases C and C0) seem feasible at least inprinciple.CasesDandD0 areonlyof theoretical interest (theassumedyieldpressure is unrealisticallyhigh) andare consideredhereonly in

0 1 2py [MPa] 3

O'

u(

)-u

(e)[m

]p(

)[M

Pa]

O

0.1

0

0

1

2

3

4

5

B'

S

C, C'

D, D'

Bp(

) =py

A'

A uy = 0.15 m

Section 1:Deformation uyutilized,load increasesto p > py

Section 2:Deformation uynot utilized,equilibriumat p = py

Section 3:No yielding(py is too high),equilibriumat p < py

S

D, D'

uy = 0.30 m

0.30

2

0u [m]

00.15

1

p[M

Pa]

0.30

2

00

0.15

1 C D

C'D'

AO

O'

0.30

2

0u [m]

00.15

1

0.30

2

00

0.15

1

0.30

2

0u [m]

00.15

1

0.30

2

00

0.15

1

0.30

2

0u [m]

00.15

1

0.30

2

00

0.15

1

0.30

2

0u [m]

00.15

1

0.30

2

00

0.15

1

p[M

Pa]

a

b

c

d

B

B'A'

e

u( ) [m]0 0.3 0.60

1

2

3

4

5

OA

R

O'

B'

B

C, C'A'

GRC

uy = 0.30 m

uy = 0.15 mS

D, D'

p()[

MPa

]

3uy = 0.15 mpy [kPa] =

0OA

316 L. Cantieni, G. Anagnostou / Tunnelling and Underground Space Technology 24 (2009) 3093222

1

008162432

p[M

Pa]

x [m]

150

425

850

B

CFig. 8. (a) Characteristic lines of the analyzed yielding supports (cf. also Table 2) and nnal equilibrium points for the analyzed supports; (c) ground pressure p vs. distance x bC); (d) inuence of the yield pressure py on the nal ground pressure; (e) inuence of t0.4

0.3

0.2

0 1 2 3py [MPa]

C, C'

B'

uy = 0.30 m

uy = 0.15 m

B

O' A'

O Aal equilibrium points; (b) ground response curve under plane strain conditions andehind the face for different values of the yield pressure py (support cases O, A, B andhe yield pressure py on the radial displacement u(1) u(e) of the support.

rdininingat

20fy =yieler yihigh

po(cas

willa).sim

dergorder to showcompletemodelbehavior. The same is true for supportS,whoseyieldpressurehasbeenset sohigh that thegroundsupportsystem reaches equilibrium before the support yields.

The marked points in Fig. 8a represent the loading p(1) and theradial displacement u(1) u(e) of the support at the equilibriumstate far behind the tunnel face. Fig. 8b shows the correspondingequilibrium points (u(1), p(1)) of the ground and additionally,for the purposes of comparison, the ground response curve underplane strain conditions (solid line GRC) as well as the ground pres-sure and deformation for a practically rigid lining (point R).

As the effect of yield deformation uy is well known, attention ispaidhere to theeffectofyieldpressurepy.Weexaminehowtheequi-librium point changes in relation to the yield pressure py (cases O, A,B, C, . . .) starting with a support that can undergo a radial displace-ment of uy = 15 cm without offering any resistance to the ground(case O). The support O starts to develop pressure only after the uti-lization of the deformation margin uy. As can be seen from Fig. 8c(curve O), this happens at a distance of about 7 m behind the tunnelface. So, support O is structurally equivalent to the case of a practi-cally rigid liningwith a 7 m long unsupported span (cf. support case1 discussed in Section 2). Due to the stress path dependency of theground behavior (cf. Section 2), the equilibrium point for this caseis located above the plane strain response curve. When the yieldpressure increases, the nal ground pressure will decrease (Fig. 8c)and the equilibrium point will move towards the ground response

Table 2Support parameters (see also Fig. 3c).

Case kIa (MPa/m) py (kPa) uy (cm) kIII (MPa/m) Description

R n/a n/a 0 656 35 cm thick lining accoO (O) n/a 0 15(30) 656 35 cm thick shotcrete lA (A) 100 150 15(30) 656 Steel sets TH-44 spaced

resistance of 150 kNc

B (B) 100 425 15(30) 656 Like A, but additionallyinto the slots (ey = 50%,

C (C0) 100 850 15(30) 656 Like B, but with higherD (D0) 100 10200 15(30) 656 Like C, but with a highS 100 >20500 e e Like C, but with such a

a This parameter was kept constant in the numerical study as it is of subordinate imthe table are 74 MPa/m (cases A and A0), 97 MPa/m (cases B0 and C0) and 114 MPa/mnot affect the numerical results.

b This value assumes that, by the time the longitudinal slots close, the shotcretec After the yield phase, the support is set practically rigid (dIII = 35 cm, EL = 30 GPd The assumed yield pressure is only theoretically possible. This support has beene The parameter is irrelevant for this case.

L. Cantieni, G. Anagnostou / Tunnelling and Uncurve (O?A?B, see Fig. 8b), i.e. the deviation from the ground re-sponse curve will become smaller at higher yield pressure values.Note that the ground deformation remains approximately constantas it is governed by the yield deformation uy which is the same forthe support cases O, A and B. At a sufciently high yield pressure(850 kPa in this numerical example), the equilibrium point reachesthegroundresponsecurve (caseC, Fig. 8b). In caseC thegroundsup-port system reaches equilibrium just before reaching the nal risingbranch of the characteristic line of the support, i.e. the deformationmargin uy = 0.15 m is just used up. At yield pressures higher thanin case C, the amount of over-excavation is not utilized completely(Fig. 8a, caseD) and the systemreaches equilibriumat agroundpres-sure which is equal to the yield pressure. Consequently, the groundpressure, after reaching a minimum value (case C), increases withthe yield pressure and, as discussed in Section 2, the equilibriumpoints approximately follow the ground response curve (Fig. 8b,C?D). Similar conclusions can be drawn from an examination ofthe numerical results for the larger deformation margin ofuy = 0.30 m (cases O0, A0, B0, . . .). The deviation from the ground re-sponse curve is due to the increase in stress associatedwith the nalrising branch of the characteristic line of the support after the defor-mation margin is used up (see Fig. 8a and b, cases O, A, B, O0 and A0).Fig. 8d and e provide a complete picture of the effect of yieldpressure py on the nal ground loading p(1) and on the lining con-vergence u(1) u(e), respectively. There are three clear sections inthe p(1) vs. py and (u(1) u(e)) vs. py relationships: at low yieldpressures (Fig. 8d, Section 1), the amount of over-excavation iscompletely used up and an additional pressure builds up on thelining after the system is set rigid. The lining convergence remainspractically constant and equal to the deformation margin uy, whilethe ground load decreases with increasing yield pressure andreaches a minimum when the yield pressure is so high that theamount of over-excavation is just used up. At higher yield pres-sures (Fig. 8d, Section 2), the squeezing deformations take only afraction of the available deformation margin; the system comesto equilibrium under the yield pressure py and, therefore, theground loading increases with py. With further increasing yieldpressure, the deformation margin is utilized less and less, and atsome point the system reaches equilibrium just before the onsetof the yielding phase. From point S on (Fig. 8d, Section 3), the yieldpressure is no longer relevant (the loading and convergence of thesupport depend only on its initial stiffness).

It is remarkable that a similar reduction in the nal groundpressure can be achieved not only by installing a support that isable to accommodate a larger deformation (which is a well-knownprinciple), but also by selecting a support that yields at a higherpressure (see, e.g. cases A, A0 and C in Fig. 8d).n () s (cm)g to resistance principle (EL = 30 GPa) n/a n/awith open longitudinal slots (EL = 30 GPab) 6(6) 15(30)

1 m with sliding connections by 4 friction loops each offering a 6(6) 15(30)

cm shotcrete with highly-deformable concrete elements inserted7 MPa, cf. Solexperts, 2007)

6(9) 30(40)

d strength elements (fy = 17 MPa, Solexperts, 2007) 6(9) 30(40)eld pressured 6(9) 30(40)yield pressure that yielding does not occurd e e

rtance. The actual kI values of the support systems described in the last columns ofes B and C). A sensitivity analysis has shown that a variation of kI in this range does

have developed its nal stiffness.

ulated only in order to explain the model behavior.

round Space Technology 24 (2009) 309322 3174. Design nomograms

The relationship between the ground pressure p(1) and defor-mation u(1) at the nal equilibrium far behind the face is not un-ique, but depends on the support characteristics, as these willaffect the stress history of the ground surrounding the tunnel. Ingeneral, the groundsupport equilibrium points are located abovethe ground response curve. The ground response curve denes thelower limits for possible groundsupport equilibria and it thereforecontains the most favorable equilibrium points, because it showsthe minimum ground deformation that must take place in orderfor the ground pressure to decrease from its initial value to a givenvalue. It also shows, vice versa, the minimum ground pressure atequilibrium for a given ground deformation.

Under which conditions do the equilibrium points fall on theground response curve? According to the previous Section of thispaper, deviations from the ground response curve are associatedwith the nal rising branch of the characteristic line of thesupport and with the increase in radial stress, which follows theutilization of the deformation capacity uy of the support. When

the groundsupport system reaches equilibrium at the yield pres-sure py, the history of the radial stress at the excavation boundaryis similar to the stress history assumed by the plane strain modeland this leads to equilibrium points which are located very closeto theground response curve. Since reachingequilibriumat theyieldpressure py necessitates a sufciently large deformation capacity uyof the support, the yield pressure should be as high as possible inorder to limit the amount of over-excavation. So, the ideal supportsystem in squeezing ground is one that yields just before theground pressure reaches the bearing capacity of the support.

Todays realistic yield pressures, however, are in general lowerthan the feasible long-term bearing capacities of tunnel supports.Consequently, if a support were able to accommodate such a largedeformation uy that it reached equilibrium at the yield pressure py,then its bearing capacity would be under-utilized. In order tobridge the difference between yield pressure and the bearingcapacity of the support, the deformation margin uy has to be used

up completely and the ground pressure has to increase above theyield pressure. As explained above, plane strain computations arein this case inadequate as the actual equilibrium points are consis-tently located above the ground response curve. An analysis ofgroundsupport interaction must take account of the stress historyof the ground and this necessitates a spatial analysis. To facilitatethe design of yielding supports for tunnels through squeezingground, we carried out a large number of such analyses and weworked out dimensionless nomograms that cover the relevantparameter range.

The nal ground pressure p(1) developing upon the support farbehind the tunnel face depends in general on the material con-stants of the ground (ER, m, fc, /, w), on the initial stress p0, onthe problem geometry (tunnel radius a, unsupported span e) andon the support parameters taken from Fig. 3c (uy, py, kI, kIII)

p1 f ER; m; fc;/;w; p0; a; e; uy; py; kI; kIII: 1

0 4 8 12 16 20

0 4 8 12 16 20

20

20

0 4 8 12 16 20

0 4 8 12 16 20

= 20= 15py / p0 = 0.00

py / p0 = 0.02

8

py / p0 = 0.00

py / p0 = 0.02

py / p0 = 0.04

0

0.1

0.2

0.3

0

0.1

0.2

0.3

0.2

0.3

0.50.4 0.3 0.2

0.15

0.25

fc / p0 = 0.1

0.50.4 0.3

0.2

0.15

0.25

fc / p0 = 0.10

0.05 py / p0 = 0.04

fc / p0 = 0.10

0.10

fc / p0 = 0.05

0.150.20.5 0.3

0.10

fc / p0 = 0.05

0.150.20.5 0.3

fc / p0 = 0.05

p = pyp = py

p()/

p0p(

)/p0

)/p 0

318 L. Cantieni, G. Anagnostou / Tunnelling and Underground Space Technology 24 (2009) 3093220 4 8 12 16

0 4 8 12 16

py / p0 = 0.0

0

0.1

0

0.1

0.2

0.3

0.150.2

0.50.4

0.30.25

0.10

fc / p0 = 0.05

0.150.20.5 0.3 p = py

p = py

Uy = (uy ER ) / (a p0)

p(p(

)/p0Fig. 9. Normalized ground pressure p(1)/p0 as a function of the normalized yield deforfriction angles / = 15 or 20 and for normalized yield pressures py/p0 = 00.08 (other pa0 4 8 12 16 20

0 2 4 6 8 10

py / p0 = 0.08

0.100.150.20.5 0.3

fc / p0 =0.05

0.100.15

0.20.5 0.3 p = py

p = py

Uy = (uy ER ) / (a p0)

mation (uyER)/(ap0) and of the normalized uniaxial strength of the ground fc/p0 forrameters: m = 0.30, w = max (0, / 20), e/a = 0.25, akI/ER = akIII/ER = 4).

The number of parameters can be reduced from twelve to nine byperforming a dimensional analysis and by also taking account ofthe fact that the displacements of an elastoplastic medium dependlinearly on the reciprocal value of the Youngs modulus ER (Anag-nostou and Kovri, 1993)

p1p0

f uyERap0

;fcp0

;pyp0

;/; m;w;ea

;akIER

;akIIIER

: 2

In view of the still large number of variables, a trade-off has had to bemade between the completeness of the parametric study and thecost of computation and data processing. In order to limit the amountof numerical calculations and the number of nomograms, the Pois-sons number of the ground was kept constant to m = 0.3, while theangle of dilatancy w was taken equal to / 20 for / > 20 and to0 for / < 20 (Vermeer and Borst, 1984). Furthermore, the ratio e/awas kept constant to 0.25 in accordancewith common round lengths

of e = 11.5 mand typical cross-section sizes of trafc tunnels (a = 46 m). Finally, the supports have been assumed to be practically rigidup to the onset of the yielding phase as well as after the utilization ofthe deformation capacity (akI/ER = akIII/ER = 4). The initial supportstiffness akI/ER is of subordinate importance for the nal groundpressure, but the assumption of practically rigid behavior after thedeformation phase may lead to a considerable overestimation ofthe ground pressure if the actual stiffness akIII/ER is much lower thanthe value assumed for the numerical calculations. Comparativenumerical analyses have shown that the error involved in a rigid sup-port assumption is small (an overestimation of ground pressure byup to 1020%) if the normalized nal support stiffness akIII/ER is high-er than 12 or if the lining is thicker than a(ER/EL), where a, ER and ELdenote the tunnel radius and the YoungsModulus of the ground andof the lining, respectively. This condition is fullled in most cases oftunnelling through squeezing ground.

0 4 8 12 16 20

0 4 8 12 16 20

10

5

0 2 4 6 8 10

0 2 4 6 8 10

= 25 = 30py / p0 = 0.00

py / p0 = 0.02

py / p0 = 0.04

8

py / p0 = 0.00

py / p0 = 0.02

py / p0 = 0.04

0

0.1

0.2

0.3

0

0.1

0.2

0.3

0.2

0.3

0.10

0.150.20.5 0.3

fc / p0 = 0.05

0.100.15

0.20.5

fc / p0 =

fc / p0 =0.05

0.10

0.150.2

0.5

0.3

fc / p0 = 0.05

0.100.150.20.3

0.05

0.40.25

fc / p0 =

fc / p0 =

p = py0.5p = py

0.05

0.3

p() /

p0p(

) / p0

() /

p0

L. Cantieni, G. Anagnostou / Tunnelling and Underground Space Technology 24 (2009) 309322 3190 2 4 6 8

0 1 2 3 4

py / p0 = 0.0

0

0.1

0

0.1

0.2

0.3

0.100.150.20.5 0.3

0.100.20.50.05

fc / p0 =

0.3 p = py

p = py

Uy = (uy ER ) / (a p0)

pp(

) / p0Fig. 10. Normalized ground pressure p(1)/p0 as a function of the normalized yield defofriction angles / = 25 or 30 and for normalized yield pressures py/p0 = 00.08 (other pa0 1 2 3 4 5

0 0.4 0.8 1.2 1.6 2

py / p0 = 0.08

0.20.5fc / p0 =

0.30.4 0.10

0.100.150.20.5

0.3 0.050.4 0.25 p = py

p = py

Uy = (uy ER ) / (a p0)

rmation (uyER)/(ap0) and of the normalized uniaxial strength of the ground fc/p0 forrameters: m = 0.30, w = max (0, / 20), e/a = 0.25, akI/ER = akIII/ER = 4).

The nomograms in Figs. 911 apply to different values of fric-tion angle / and yield pressure py. They show the normalizedground pressure p(1)/p0 as a function of the normalized yielddeformation (uyER)/(ap0) and allow an estimation to be made ofthe amount of deformation that must take place in order for theground pressure to decrease to a technically manageable level.Note that the nomograms are based on the assumptions men-tioned at the end of Section 1. If the actual conditions deviate fromthese assumptions, the results should be considered with care. Forexample, a non-hydrostatic stress eld or a non-circular cross-sec-tion may lead to considerable asymmetric convergences or thedevelopment of high bending moments in the lining. The nomo-grams represent, nevertheless, a valuable decision-making aid atleast in the preliminary design phase.

0 1.6 3.2 4.8 6.4 8

0 1 2 3 4 5

= 35

py / p0 = 0.00

py / p0 = 0.02

py / p0 = 0.04

0

0.1

0.2

0.3

0

0.1

0.2

0.3

0

0.1

0.2

0.3

0.10

0.150.20.4 0.3

fc / p0 = 0.05

0.10

0.150.20.5 0.3

0.05

0.10.20.50.05

fc / p0 =

0.4 0.25

fc / p0 =

0.3

0.25

p = py

p = py

p() /

p0p(

) / p0

p() /

p0320 L. Cantieni, G. Anagnostou / Tunnelling and Underg0 0.4 0.8 1.2 1.6 2

py / p0 = 0.08

0

0.1

0.2

0.30 1 2 3 4 5

0.1fc / p0 = 0.5 0.3 0.2 p = py

Uy = (uy ER ) / (a p0)

p() /

p0

Fig. 11. Normalized ground pressure p(1)/p0 as a function of the normalized yield

deformation (uyER)/(ap0) and of the normalized uniaxial strength of the ground fc/p0for friction angle / = 35 and for normalized yield pressures py/p0 = 00.08 (otherparameters: m = 0.30, w = max (0, / 20), e/a = 0.25, akI/ER = akIII/ER = 4).5. Application examples

Table 3 shows, by means of two practical examples, how to ap-ply the nomograms step by step.

The rst example (columns 13) refers to a 250 m deep tunnel.Column 1 applies to a yielding support consisting of steel sets withsliding connections that deform at a ground pressure py of 60 kPa(rows 1522). For a lining thickness of 30 cm following the defor-mation phase and a design value of 15 MPa for the lining hoopstress rs, the nal pressure p(1) amounts to 0.9 MPa (rows 1113). In order to reduce the ground pressure to this value, the sup-port should be able to accommodate a deformation uy of 33 cm(rows 2325). This can be realized by arranging six 34 cm widelongitudinal slots in the shotcrete shell (rows 26 and 27).

Assume now that, in addition to the over-excavation and to thesupport measures according to column 1, deformable concrete ele-ments of 10 MPa yield strength are applied, thereby increasing theyield pressure py to 380 kPa (column 2, rows 2022). As thedeformable concrete elements become practically rigid at a strainey of about 50% (Fig. 2), the support will experience a convergenceuy of only 16 cm during the deformation phase (row 25). For thesevalues of yield pressure py and deformation uy, the ground pressurewill rise to p(1) = 0.68 MPa after the deformation phase (rows 13,14, 23 and 24). The nal hoop stress rs in the shotcrete shell is by33% lower than it would be without deformable concrete elements(cf. columns 1 and 2, row 12). So, for the same amount of over-excavation and for the same number n and size s of longitudinalslots, the application of the higher yield pressure support reducesthe risk both of a violation of the clearance prole during the defor-mation phase (since only 50% of the deformation margin of 33 cmis used up) and of an overstressing of the shotcrete shell after thedeformation phase (since the hoop stress rs is by 33% lower thanthe design value of 15 MPa).

Alternatively, for the same nal hoop stress (rs = 15 MPa), theminimum over-excavation and the size of the longitudinal slotscan be reduced to 12 and 25 cm, respectively, by applying the higheryield pressure support (column 3, rows 25 and 27). In addition, thehigher yield pressure will increase protection against crown insta-bilities during the deformation phase: Depending on the strengthand on the structure of the rock mass, structurally controlled blockdetachment or stress-induced loosening of an extended zonearound the openingmay occur (particularly in the case of a yieldingsupport, as it promotes loosening by allowing the occurrence of lar-ger deformations). If the resistance of the exible joints is not suf-ciently high, the support will experience settlement under theweight of the detached rock mass, with the result that both theover-excavation space and the deformation capacity of the supportwill be used up and therefore the behavior of the support duringsubsequent squeezing will resemble that of a support based onthe resistance principle. In order to safeguard against crown insta-bilities and to preserve the deformation capacity of the support,there has to be either a sufciently high yield pressure (so thatthe lining does not deform already under the loosening pressure)or additional support through sufciently long bolts.

Example (a) in Table 3 suggests the possibility of a trade-off be-tween, on the one hand, the greater structural effort associatedwith achieving a high yield pressure, and on the other hand a largeramount of over-excavation (possibly in combination with morebolting). It should be noted, however, that such a trade-off may of-ten not be possible. In mechanized tunnelling, for example, the op-tions for a trade-off are reduced dramatically as the boringdiameter largely pre-determines the space that is available for sup-port (dIII) and deformation (uy). At the same time, achieving a yield

round Space Technology 24 (2009) 309322pressure py that is higher than the loosening pressure may be dif-cult for trafc tunnels of large diameter, because the potentialsize and loading of the loosened zone increases with the size of

nel

2p

5

3011.30.68

dergTable 3Application examples.

(a) 250 m deep tun

1

1 Tunnel radius a (m) 52 Depth of cover H (m) 250

Ground3 Unit weight c (kN/m3) 204 Youngs modulus ER (MPa) 3005 Poissons ratio m () 0.36 Uniaxial compr. strength fc(kPa) 5007 Angle of internal friction / () 258 Dilatancy angle w () 59 Initial stressa p0 (MPa) 510 Normalized compr. strength fc/p0 () 0.1

Support after deformation phase11 Lining thickness dIII (cm) 3012 Concrete compress. stress rs (MPa) 1513 Final ground pressure p(1) (MPa) 0.9b

L. Cantieni, G. Anagnostou / Tunnelling and Unthe opening (Terzaghi, 1946), while the feasible yield pressure py ofthe support decreases with the tunnel radius a for given lininghoop force Ny (py = Ny/a), and structural detailing and materialtechnology impose limits on the hoop force Ny. For example,deformable concrete elements of high yield strength increase thehoop force Ny but may cause overstressing of the green shotcreteif the rock pressure develops quickly. This aspect may be importantparticularly for mechanized tunnelling, as the daily advance mayreach several meters even under adverse conditions.

Finally, the intensity of squeezing may also constrain the deci-sion space. columns 4 and 5 of Table 3 refer to an example withthe particularly heavy squeezing conditions that are associatedwith very high overburden and sheared rock of low strength. Inthe case of a support consisting of steel sets with six sliding con-nections in the deformation phase and a 50 cm thick concrete lin-ing afterwards, the required amount of over-excavation uy and thesliding way s of the connections amount to 67 and 70 cm, respec-tively (column 4). By applying a support of higher yield pressure(column 5, rows 19 and 20), the amount of over-excavation uycan be reduced to 47 cm (row 25). This convergence would neces-

14 Normalized ground pressure p(1)/p0 () 0.180 0.13Support during deformation phase

15 Steel set spacing b (m) 1.016 Steel set clear distance b0 (m) 0.817 Friction loop resistance Nf (kN) 15018 Lining thickness dII (cm) 2019 Number of friction loops nf () 2 220 Yield stress of deform. elem. fy (MPa) 0 1021 Max. slot deformation (%) ey () 100 5022 Yield pressure of supportc py (kPa) 60 38023 Normalized yield pressure py/p0 () 0.012 0.0724 Normalized yield deformation Uy () 3.9d 2l25 Yield deformation uy (cm) 33h 16k

26 Number of slots n () 6 627 Slot size s (cm) 34i 34j

a p0 = Hc.b p(1) = rs dIII/a.c py = Ny/a, where Ny = nf Nf /b + dIIb0 fy.d Based upon interpolation between nomograms for (/, py/p0) = (25, 0) and (25, 0.02e As Note (d) but from nomogram for py/p0 = 0.08.f As Note (d) but from nomogram for py/p0 = 0.00.g As Note (d) but from nomogram for py/p0 = 0.02.h uy = Uy a p0/ER.i s = 2 p uy/(n ey).j Value taken as in column 1.k uy = n s ey/(2p).l Uy = uy ER/(a p0).

m From Fig. 10, nomogram for / = 25 and py/p0 = 0.08 (curves for fc/p0 = 0.1, Uy acc. ton p(1) = p0 (p(1)/p0).o rs = p(1) a/dIII.p In this column, the calculation proceeds from bottom to top.800

2515000.32000255200.1

30 50 50o 15 15 15n 0.9b 1.5b 1.5bcrossing claystones (b) 800 m deep tunnel crossing sheared rock

3 4 5

round Space Technology 24 (2009) 309322 321sitate the application of deformable concrete elements in eight,74 cm wide longitudinal slots (note that the concrete elementscan shorten by a maximum of ey = 50%). Deformable concrete ele-ments of such a size have not been used in practice. The applicationof several smaller elements side by side would reduce the handlingdifculties associated with weight, but only at the expense ofreducing the exural stiffness of the lining. The latter is importantin the case of asymmetric squeezing. A larger number of longitudi-nal slots would involve a very great structural effort and wouldalso necessitate closely spaced bolts in order to control the riskof a snap-through in the case of non-uniform rock deformation.

6. Closing remarks

We have shown that an analysis of the ground-yielding supportinteraction that takes into account the stress history of the groundleads to conclusions which are qualitatively different from thoseobtained through plane strain analysis. The ground pressure devel-oping far behind the tunnel face in a heavily squeezing ground de-

5m 0.180 0.075 0.075

1.00.815020

2 2 410 0 1050 100 50380 60 440

6 0.076 0.003 0.0221.4e 10f 7g

12h 67h 47h

6 6 825i 70i 74i

) in Fig. 10 (curves for fc/p0 = 0.1, p(1)/p0 according to row 14).

row 24).

pends considerably on the amount of support resistance during theyielding phase. The higher the yield pressure of the support, thelower will be the nal load. A targeted reduction in ground pres-sure can be achieved not only by installing a support that is ableto accommodate a larger deformation (which is a well-knownprinciple), but also through selecting a support that yields at ahigher pressure. Furthermore, a high yield pressure reduces therisk of a violation of the clearance prole and increases safety levelagainst roof instabilities (loosening) during the deformation phase.

These results are important from the standpoint of conceptualdesign, even if the range of potential project conditions, design cri-teria and technological constraints does not allow us to make gen-eralizations about structural solutions for dealing with squeezingground. Some basic design considerations have been illustratedthrough the use of practical examples. Additionally, the nomo-grams presented in this paper contribute to the decision-makingprocess, as they allow for a quick assessment of different supportsand of their sensitivity with respect to variations in geology.

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The interaction between yielding supports and squeezing groundIntroductionThe influence of the stress path on groundsupport interactionThe influence of yield pressure and yield deformationDesign nomogramsApplication examplesClosing remarksReferences