squeezing loading of segmental lining and the effect of backfilling

Tunnelling and Underground Space Technology 26 (2011) 692–717

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Tunnelling and Underground Space Technology

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Squeezing loading of segmental linings and the effect of backfilling

M. Ramoni a,⇑, N. Lavdas b,1, G. Anagnostou a

a ETH Zurich, Switzerlandb Rothpletz, Lienhard + Cie AG, Olten, Switzerland

a r t i c l e i n f o

Article history:Received 22 November 2010Received in revised form 17 March 2011Accepted 6 May 2011Available online 14 June 2011

Keywords:Tunnel boring machineSqueezing groundSegmental liningSqueezing pressureOverstressingBackfillingNomogramsNumerical investigation

0886-7798/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.tust.2011.05.007

⇑ Corresponding author. Tel.: +41 44 633 32 71; faxE-mail address: [email protected] (MURL: http://www.tunnel.ethz.ch

1 Formerly: ETH Zurich, Switzerland.

a b s t r a c t

Overstressing of the segmental lining is one of the major hazard scenarios related to shielded TBM tun-nelling in squeezing ground. The present paper deals with this specific problem, addressing the key ques-tion of the ground pressure acting upon a segmental lining installed behind a single shielded TBM.Starting with a structured discussion of the influencing factors and their interactions, the paper investi-gates how the type, location and thickness of the backfilling play an important role with respect to theloading of a segmental lining. Secondly, it explains how to take due account of the actual thickness ofthe backfilling (which is not known a priori since it depends on the deformations of the bored profile)in a numerical simulation. Thirdly, the paper advances a number of theory-based decision aids whichcover the relevant range of ground parameters, initial stress, segmental lining and backfilling character-istics, thus supporting rapid initial assessments of the ground pressure acting upon a segmental liningand making a valuable contribution to the decision-making process.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The main hazard scenarios for shielded TBM tunnelling insqueezing ground are sticking of the cutter head, jamming of theshield or damage to the tunnel support. Furthermore, the occur-rence of significant deformations (ovalization) or even horizontalor vertical shifting of the segmental lining may lead to jammingof the back-up equipment or to violation of the clearance profile.

In a series of recent publications, the authors discussed the spe-cific problems of – and experience with – TBMs in squeezingground, reviewed the available countermeasures, commented onpossible technological improvements (including the developmentof deformable lining systems), analyzed the interaction betweenthe shield, ground and tunnel support quantitatively and provideddesign charts concerning the thrust force needed in order to avoidshield jamming (Ramoni and Anagnostou, 2010a, 2010b, 2010c,2010d). The present paper extends this research by addressingthe potential hazard of lining overstressing.

A realistic estimation of the loading of a segmental lining is onlypossible if due account is taken of the backfilling features. Section 2of the present paper shows – with a structured discussion of the

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: +41 44 633 10 97.. Ramoni).

influencing factors and their interactions – that the type, locationand thickness of the backfilling play an important role with respectto the ground pressure acting upon a segmental lining. Section 3explains how to take due account of these features in a numericalsimulation and, more specifically, how to deal with the non-linear-ity of the problem. The problem is demanding because the actualthickness of the backfilling is not known a priori, as it dependson the ground deformations that occur between the tunnel faceand the point at which the backfilling is completed. Section 4 pre-sents, in the form of dimensionless design nomograms, the resultsof a comprehensive parametric study into the ground pressure act-ing upon a segmental lining which exploits the numerical effi-ciency and reliability of the computational model introduced inSection 3. The nomograms cover the relevant range of groundparameters and initial stress, as well as different characteristicsof the TBM, the segmental lining and the backfilling (type and loca-tion), and allow a quick preliminary assessment to be made of theloading of a segmental lining. This is the first time that such a sys-tematic and thorough investigation has been presented.

An extended literature review on computational methods forTBM tunnelling in squeezing ground can be found in Ramoni andAnagnostou (2010b). Recent publications closely related to thetopic of the present paper include those of Simic (2005), Grazianiet al. (2007) and Schmitt (2009). Simic (2005) carried out numeri-cal investigations for the assessment of the loading of the segmen-tal lining in the ‘‘La Umbria’’ Fault of the Guadarrama Tunnel

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Nomenclature

D boring diameterdb thickness of the backfillingdl thickness of the segmental liningds thickness of the shieldE Young’s modulus of the groundEb Young’s modulus of the backfillingEl Young’s modulus of the segmental liningEs Young’s modulus of the shieldfc uniaxial compressive strength of the groundfc,l uniaxial compressive strength of the segmental liningG groundH depth of coverKb stiffness of the backfillingKc composite stiffness (segmental lining and backfilling)Kl stiffness of the segmental liningKs stiffness of the shieldL length of the shieldLf length of the front shield (double shielded TBM)Lr length of the rear shield (double shielded TBM)N number of entities of a N2 chartp ground pressurep� normalised ground pressurepmax bearing capacity of the segmental liningR tunnel radiusRl,o outer radius of the segmental liningRs,i inner radius of the shieldRs,o outer radius of the shield

SF safety factort difference between radius of the shield intrados and ra-

dius of the segmental lining extradosu radial displacement of the ground at the tunnel bound-

aryub radial displacement of the bored profile before comple-

tion of the backfillingvg gross advance ratex radial co-ordinate (distance from the tunnel axis)y axial co-ordinate (distance behind the tunnel face)DR difference between boring radius and radius of the

shield extradosDRl difference between boring radius and radius of the seg-

mental lining extradosDRr difference between boring radius and radius of the rear

shield extrados (double shielded TBM)c unit weight of the groundu angle of internal friction of the groundk location (distance behind the shield), where backfilling

is completedk� location (distance behind the shield), where the ground

closes the gap around the segmental liningm Poisson’s ratio of the groundr stressr0 initial stressw dilatancy angle of the ground

M. Ramoni et al. / Tunnelling and Underground Space Technology 26 (2011) 692–717 693

(Spain, double shielded TBM, D = 9.51 m), taking into account theeffect of creep. Graziani et al. (2007) investigated a double shieldedTBM drive (D = 11.00 m) for the planned Brenner Base Tunnel (Aus-tria/Italy) in the framework of the ‘‘TISROCK’’ research project,gaining a valuable insight into the effects of the length of a shearzone and of the stiffness of the backfilling on the sectional forcesin the segmental lining. The work of Schmitt (2009) is of a moregeneral nature and investigates the effects of non-uniform conver-gence and of non-hydrostatic shield and lining loading for singleshielded TBMs. All of these investigations are based upon fullythree-dimensional, step-by-step numerical simulations, assuminga priori the thickness of the backfilling and, consequently, the stiff-ness of the tunnel support. As will be shown later in the presentpaper, this simplification is unavoidable when using the commonlyavailable computational codes. Furthermore, it leads to a majorreduction in the computational effort (particularly when carryingout parametric studies).

2. Backfilling

2.1. Introduction

The factors influencing the ground pressure acting upon a seg-mental lining – particularly the properties of the backfilling –and their interactions can be mapped easily and efficiently usinga so-called ‘‘N2 chart’’ (Lano, 1990; NASA, 2007). Fig. 1 shows anN2 chart drawn up for the topic of the present paper. This is anN-by-N square matrix containing N = 13 entities on the main diag-onal and depicting their existing interactions in the non-blank off-diagonal cells. The interactions have to be read directionally be-tween the elements, i.e., first horizontally in the row and thenclockwise in the column. There are two further mapping rules con-cerning the shape and the colour of the off-diagonal cells. Concern-

ing the shapes, rhombuses indicate that an interaction exists onlyunder certain conditions, while circles denote unconditional inter-actions. As for the colours, green is used for interactions with a po-sitive effect (an increase in the first involved factor leads to anincrease in the second involved factor), red for a negative effectand black, where the effect may be either positive or negative.For a more detailed description of the applied diagramming tech-nique the reader is referred to Ramoni and Anagnostou (2010d),where an N2 chart was applied (using the same rules as in the pres-ent paper) for mapping the system behaviour of a gripper TBMdrive through squeezing ground.

Section 2.2 discusses – by making reference to the N2 chart ofFig. 1 – the usual case for rock TBM tunnelling, where backfillingof the segmental lining is carried out with pea gravel in the upperpart and with mortar in the bottom third of the cross-section at agiven distance behind the shield (Fig. 2a). Section 2.3 deals withthe rather rare case of grouting immediately behind the shieldvia the shield tail (Fig. 2b). For the sake of economy, details con-cerning backfilling technology are not given here, but can be found,e.g., in Thewes and Budach (2009).

For the sake of simplicity, pairs of numbers within curly brack-ets will be used for making reference to Fig. 1 and denoting theinteractions of the respective factors (e.g., {4-12} denotes the effectof the factor 4 on the factor 12), while a series of number in curlybrackets will denote a sequence of interactions (e.g., {7-9-10}abbreviates {7-9} and {9-10}).

As the shield slides along the tunnel floor, the gap aroundthe shield and the segmental lining is wider above the centrethan in the lower portion of the tunnel cross-section (Fig. 2).However, for the sake of simplicity, Sections 2.2 and 2.3 con-sider the theoretical case of axial symmetry, as this simplifica-tion can be made without loss of generality in the conclusionsdrawn.

Fig. 1. N2 chart concerning the ground pressure p acting upon a segmental lining.

694 M. Ramoni et al. / Tunnelling and Underground Space Technology 26 (2011) 692–717

2.2. Backfilling with pea gravel and mortar

2.2.1. Composite stiffnessThe ground pressure p acting upon a backfilled segmental lining

depends on the ground characteristics {1-13} and on thestiffness Kc of the tunnel support, which consists of the segmentsand the backfilling {12-13}. The composite stiffness Kc results fromthe stiffness of the segmental lining Kl {4-12} and the stiffness ofthe backfilling Kb {11-12}. In contrast to the lining stiffness Kl,the backfilling stiffness Kb is not known a priori. On the one hand,the properties of the backfilling material are known {5-11} but, onthe other hand, the characteristics of the ground encountered willinfluence the quality of the backfilling (e.g., blocky ground maylead to difficulties) {1-11}. Furthermore, the stiffness Kb dependson the thickness of the backfilling db (the thinner the backfillinglayer, the higher its stiffness {10-11}) and, therefore, on the defor-mations of the ground.

2.2.2. Thickness of the backfillingThe actual thickness of the backfilling db depends on the radial

displacement of the bored profile ub {9-10} occurring behind thetunnel face up to the point at which the backfilling is completed(Fig. 3a):

db ¼ DRl � ub

0 6 ub 6 DRl

�; ð1Þ

where DRl is the planned size of the radial gap of the segmental lin-ing {7-10}. DRl is equal to the difference between the boring radius Rand the radius of the segmental lining extrados Rl,o (cf. Fig. 3a) andrepresents an upper limit for the radial displacement of the boredprofile ub {7-9}. Eq. (1) disregards possible deformations of the seg-mental lining. This simplification is reasonable, as the deformations

of the segmental lining are small (due to its high stiffness) comparedto those of the ground.

With respect to the thickness of the backfilling db, two border-line cases can be distinguished. If the ground does not deform (i.e.,ub = 0, Fig. 3b), the backfilling is as thick as the planned size DRl ofthe radial gap between segments and ground. On the other hand, ifthe ground closes the radial gap before backfilling occurs (i.e.,ub = DRl, Fig. 3c), the thickness of the annular gap becomes equalto zero and backfilling is no longer possible.

2.2.3. Radial gap sizeThe radial gap size of the segmental lining DRl and the radial

gap size of the shield DR – i.e., the difference between boring ra-dius R and radius of the shield extrados Rs,o (cf. Fig. 3a) – are geo-metrically coupled with each other. An increase of the radial gapDR leads automatically to an increase of the radial gap DRl {3-7}:

DRl ¼ DRþ ds þ t; ð2Þ

where ds is the thickness of the shield and t the difference betweenthe radius of the shield intrados Rs,i {2-7} and the radius of the seg-mental lining extrados Rs,o (cf. Fig. 3a). With respect to the radialgap DR, it has also to be mentioned that if the shield has a ‘‘conical’’shape the radial gap size increases with increasing distance fromthe tunnel face, i.e., DR = DR(y).

2.2.4. Radial displacement of the bored profileThe radial displacement of the bored profile ub not only affects

the thickness of the backfilling db {9-10} but also has a direct effecton the ground pressure p {9-13}, as it is a part of the ‘‘pre-deforma-tion’’ that the ground experiences before the segmental lining be-comes loaded. It should be noted, that the convergence ub mayincrease (path {9-10-11-12-13}) or decrease (path {9-13}) the finalload p.

Fig. 2. (a) Backfilling with pea gravel and mortar, (b) annulus grouting via the shield tail.


The radial displacement of the bored profile ub depends, ofcourse, on the ground characteristics {1-9}. Two further influenc-ing factors are of a geometrical nature: the shield length L {2-9}and the distance behind the shield k {6-9}, where backfillingshould occur. The shield length L is a matter of TBM design, whilethe length k depends on the type of the backfilling material {5-6}and on operational decisions taken on the construction site. Thelonger the distance from the tunnel face L + k (cf. Fig. 3a), the big-ger will be the radial displacement of the bored profile ub {2-9, 6-9}and therefore the smaller the thickness of the backfilling db {9-10}.In the extreme case of Fig. 3c, where the convergence ub uses up

the available space DRl at a certain distance k� 6 k behind theshield, backfilling is no longer possible. It should be noted that adelayed backfilling might also be problematic with respect to thethrust force that the segmental lining can accommodate. In fact,an improper bedding of the segmental lining reduces its bearingcapacity in case of eccentric loading and this may limit the effec-tively available thrust force and slow down the TBM advance {6-8}.

2.2.5. Advance rateFirst of all, the gross advance rate vg plays a role with respect to

the radial displacement of the bored profile ub {8-9}. A slower TBM

Fig. 3. Thickness of the backfilling db when applying pea gravel and mortar:(a) general case, (b) borderline case of practically zero ground deformationsbetween face and location of backfilling, (c) rapidly converging ground closing thegap around the lining before completion of backfilling.

Fig. 4. Longitudinal arching action: (a) between shield and backfilled segmentallining (and between the shield and the core ahead of the tunnel face), (b) betweenthe core ahead of the tunnel face and the backfilled segmental lining.


advance results, as a rule, in bigger ground deformations in the ma-chine area (Ramoni and Anagnostou, 2010a).

In adverse ground conditions it is generally more difficult tomaintain a fast TBM advance {1-8}. For example, a high groundpressure p may lead to an overstressing of the segmental lining,thus necessitating repair works that may cause standstills and,consequently, reduce the gross advance rate vg {13-8}. Further-more, if the segmental lining near the TBM is subjected to a loadingin the proximity of its bearing capacity, it may be impossible to uti-lise the full installed thrust force of the TBM, if required, and theTBM may become jammed {13-8}. In this respect, a shorter shieldmay be advantageous {2-8}, as the required thrust force for over-coming shield skin friction decreases with the shield length L(Ramoni and Anagnostou, 2010b).

Another countermeasure that can be applied for reducing therequired thrust force and avoiding TBM jamming is an increasein the overcut DR {3-8}. This can be either a fixed countermeasure

realised a priori (by changing the layout of the TBM already in theplanning phase) or a temporary countermeasure realised duringconstruction through so-called ‘‘overboring’’. However, the appli-cation of overboring may lead to a reduction in the gross advancerate {3-8} and the feasibility of the increase in the boring radiuswill depend on the ground conditions encountered {1-3} (Ramoniand Anagnostou, 2010d).

2.2.6. Load transfer in longitudinal directionThe overcut DR may also have a direct effect on the radial dis-

placement of the bored profile ub {3-9}. A smaller gap betweenshield and ground is more likely to close than a larger gap. Afterclosing the gap, the ground starts to exert a load upon the shieldand, vice versa, the shield supports the ground (the stiffer theshield, the more pronounced will be its support action and the low-er will be the ground deformations {2-9}). This longitudinal archaction between the shield and the backfilled segmental lining(Fig. 4a) leads to a reduction of the ground deformations in thisarea {13-9}.

Such an effect is also possible between the core ahead of theface and the backfilled segmental lining (Fig. 4b). This effect ismore evident for short shields, and if the backfilling is carriedout near to the shield (i.e., if the distance L + k is short). Such a loadtransfer has a positive effect on the shield loading, but leads at thesame time to a higher loading of the segmental lining. It should benoted that the feedback effects {13-9-10-11-12-13} and {13-9-13}are conflictive with respect to lining loading.

2.3. Backfilling with grouting via the shield tail

Immediate backfilling with grouting via the shield tail is ratherrare in rock tunnelling. Basically, the same factors apply as in thelast section.

Fig. 5. Thickness of the backfilling db in the case of grouting via the shield tail.


One major difference concerns the parameter k, which becomesequal to zero (Fig. 5) {5-6}. Consequently, the thickness of thebackfilling db is governed by the radial displacement ub of thebored profile at the shield tail {9-10}. As this displacement is lim-ited by the overcut DR (rather than by the radial gap size DRl of thesegmental lining), in this respect interaction {3-9} substitutesinteraction {7-9}. In contrast to the case of ‘‘backfilling with peagravel and mortar’’, where rapidly converging ground may makeannulus backfilling impossible, the thickness of the backfillinglayer is here at least equal to the radius difference between the lin-ing extrados and the shield extrados:

db ¼ DRl � ub P ds þ t

0 6 ub 6 DR

�; ð3Þ

where as before, the deformations of the segmental lining and theshield were disregarded.

Another difference to the case of backfilling with pea graveland mortar concerns the longitudinal arching mentioned inSection 2.2.6. This effect is more pronounced in the case of groutingvia the shield tail (Ramoni and Anagnostou, 2010a, 2010b), as thesegmental lining begins to support the ground at a shorter distancebehind the tunnel face. This is particularly true if a rapidly harden-ing mortar (cf., e.g., Pelizza et al., 2010; Peila and Borio, 2011) isused, because the backfilling and the segmental lining then consti-tute a stiff system right from the start.

The considerations of Section 2.2 assumed that the backfillinglayer represents a compressible buffer and that its stiffness Kb

(and therefore the loading of the segmental lining p) increases ifthe thickness of the layer db decreases. However, this assumptiondoes not apply if the segmental lining is perfectly backfilled withgrout – a rather rare case considering the unavoidable imperfec-tions in construction. In this case, the backfilling layer forms a per-fect ring and, as shown later in Section 3.2.1, a reduction of itsthickness db leads to a decrease and not to an increase in thestiffness Kb. If a perfect ring is assumed, the effect of ub (paths{9-10-11-12-13} and {9-13}, cf. Section 2.2.4) will no longerbe ambiguous and the feedback effects {13-9-10-11-12-13} and{13-9-13} (cf. Section 2.2.6) will no longer be conflictive.

The N2 chart of Fig. 1 also assumes that backfilling with peagravel and mortar slow-downs the TBM advance to the samedegree as grouting via the shield tail. Otherwise, an interaction{5-8} should be added in the N2 chart.

Fig. 6. Problem layout.

3. Computational model

3.1. Introduction

The numerical investigations of the present paper are basedupon the axially symmetric model of Fig. 6. For a detailed discus-

sion of the assumptions underlying axial symmetry, the reader isreferred to Ramoni and Anagnostou (2010c). The condition of axialsymmetry presupposes that the tunnel is cylindrical, deep-seatedand crosses a homogeneous and isotropic ground in a hydrostaticand uniform stress field. Furthermore, the size of the gap betweenshield and ground, as well as between segmental lining andground, are taken as constant around the circumference, and theTBM weight is neglected. A further consequence of the assumptionof axial symmetry is that only normal forces are shown to developin the segmental lining, while in reality bending may also occur.The axially symmetric model is in general stiffer than in reality,and should lead to an overestimation of the ground pressure. How-ever, the stress peaks associated with bending moments may ex-ceed the compressive stresses developed in the axially symmetricmodel.

The occurrence of bending moments can be taken into accountin the structural assessment of the segmental lining in a simplifiedmanner by choosing a higher safety factor. For example, accordingto Kovári (1998), a non-uniformity of the loading of 1.2 (defined asthe ratio between the highest and the lowest radial pressure actingupon the lining) leads to a reduction of about 45% of the load whicha lining with a diameter of 10 m, a thickness of 50 cm and a rein-forcement content of 0.4% can accommodate. This can be coveredapplying a safety factor of about 2. Furthermore, tunnelling expe-rience indicates that in reality the non-uniformity of the loadingmust be quite limited and probably lower than 1.2 in most situa-tions (Kovári, 1998). Summarising, for taking account of possiblebending moments, choosing a higher safety factor is of course astrong simplification, which, however, can be applied at least fora pre-dimensioning.

The ground is modelled as a linearly elastic, perfectly plasticmaterial obeying the Mohr–Coulomb yield criterion and a non-associated flow rule. In order to reduce complexity, the time-dependency of the ground behaviour (due to creep or consolida-tion) is disregarded. Therefore, the gradual increase in groundpressure and deformations in the longitudinal direction is consid-ered to be due only to the spatial stress redistribution that is asso-ciated with the progressive advance of the working face (Lombardi,1973). The tunnel face is regarded as being unsupported.

As in the previous works by Ramoni and Anagnostou (2010a,2010b, 2010c), the numerical calculations have been performedwith the finite element code HYDMEC of the ETH Zurich (Anagnos-tou, 1992) applying the so called ‘‘steady state method’’ (NguyenMinh et al., 1991), which makes it possible to solve the advancingtunnel heading problem in one single computational step. For a de-tailed description of the computational method the reader is re-ferred to Anagnostou (2007), while validation examples,including a comparison between the steady state method and the


commonly used step-by-step method, can be found in Cantieni andAnagnostou (2009).

The shield is modelled by a mixed boundary condition accord-ing to Ramoni and Anagnostou (2010b, 2010c):

pðyÞ ¼0 if uðyÞ�uð0Þ6DR

KsðuðyÞ�uð0Þ�DRÞ if uðyÞ�uð0Þ>DR

�ðfor 06 y6 LÞ;

ð4Þ

where p is the ground pressure developing upon the shield, u the ra-dial displacement of the ground at the excavation boundary (and,consequently, u(0) denotes the pre-deformation of the groundahead of the tunnel face), L the shield length and DR the overcut.Ks denotes the stiffness of the shield and reads as follows:

Ks ¼Esds

R2 ; ð5Þ

where Es and ds denote the Young’s modulus and the thickness of theshield, respectively, and R is the tunnel radius. Eq. (4) takes accountof the fact that the ground starts to exert a load upon the shield onlyafter closing the radial gap DR. Assuming that the shield is able tobear the load without being overstressed, there is a linear depen-dency between the developing ground pressure p and the radialdisplacement of the ground u after gap closure (i.e., the stiffness Ks

is constant, cf. Eq. (5)).A mixed boundary condition also applies to the segmental lin-

ing, whereby, as explained in the next two sections, the featuresof the two backfilling types (pea gravel and mortar or groutingvia the shield tail) should be taken into account.

3.2. Backfilling with pea gravel and mortar

3.2.1. Boundary conditionIn this case, complete backfilling is achieved only after a certain

interval behind the shield (Fig. 2a, cross-section C–C). The segmen-tal lining is initially either not backfilled (cross-section A–A) orpartially backfilled (cross-section B–B). The length of the tunnelsection with incomplete backfilling depends on the steepness ofthe pea gravel ‘‘slope’’. Of course, with an axially symmetric com-putational model it is not possible to model the case of partialbackfilling. Disregarding the ‘‘intermediate’’ stage is, however, areasonable simplification with respect to the ground pressure act-ing upon the segmental lining because the stiffness of a partiallybackfilled segmental lining is negligible (Lavdas, 2010).

It is, nevertheless, important to distinguish between the back-filled and the non-backfilled sections of the lining (Fig. 6) and alsoto take into account the deformations that occur before backfilling(i.e., up to y = L + k, where k is the distance behind the shield, wherecompletion of backfilling is planned), because they influence thethickness of the backfilling layer and thus the stiffness of the sys-tem composed of the backfilling and the segmental lining. In thisrespect, a distinction must be made in the boundary condition ofthe lined tunnel section depending on whether the ground closesthe annular gap before backfilling or not. In the first case, i.e.:

for ub P DRl and y > L;

pðyÞ ¼0 if uðyÞ � uð0Þ 6 DRl

KlðuðyÞ � uð0Þ � DRlÞ if uðyÞ � uð0Þ > DRl

�; ð6Þ

while in the second case, i.e.:

for ub < DRl; pðyÞ ¼0 if L < y 6 Lþ k

KcðuðyÞ � uðLþ kÞÞ if y > Lþ k

�; ð7Þ

where ub and DRl denote the radial displacement of the bored pro-file before completion of the backfilling (ub = u(L + k) � u(0)) and the

theoretical size of the radial gap of the segmental lining,respectively.

In the first case, there is no backfilling layer at all and theground pressure p depends only on the stiffness of the segmentallining (cf. Eq. (6)):

Kl ¼Eldl

R2 ; ð8Þ

where El is the Young’s modulus of the concrete, dl the thickness of thesegments and R the tunnel radius. The assumption of a constant stiff-ness Kl presupposes that the segmental lining is not overstressed.

In general, the convergence ub, which occurs between the tun-nel face (y = 0, cf. Fig. 6) and the location of completion of backfill-ing (y = L + k), uses up only a part of the radial gap of the segmentallining DRl (cf. Fig. 3a). In this case, the stiffness Kc of the systemcomposed of the backfilling and the segmental lining shall beintroduced in the mixed boundary condition (cf. Eq. (7)). The com-posite stiffness Kc depends essentially on whether the backfillinglayer forms a perfect ring or represents rather a compressible buf-fer between the segmental lining and the ground. In the first case,the support consists of two concentric rings and the compositestiffness

Kc ¼ Kl þ Kb: ð9Þ

Kb is the stiffness of the outer ring:

Kb ¼Ebdb

R2 ; ð10Þ

where Eb and db denote the Young’s modulus and the thickness ofthe backfilling, respectively. According to Eq. (9), the backfillingwould in this case increase the stiffness of the support. In view ofthe common construction imperfections, the correctness of thisconclusion is rather questionable. It is much more plausible to as-sume that the backfilling rather represents a compressible layer –a layer that actually reduces the overall stiffness. In this case, thecomposite stiffness

Kc ¼1

1Klþ 1

Kb

; ð11Þ

where Kb is the stiffness of the backfilling layer under one-dimen-sional loading:

Kb ¼Eb

db: ð12Þ

If the backfill is highly compressible and highly deformed, a non-linear deformation behaviour (i.e., a non-constant stiffness Kb)should be considered.

As mentioned in Section 2.2.2, the thickness of the backfilling db

(which is important for the composite stiffness according toEqs. (10) and (12)) depends on the radial displacement of the boredprofile ub:

db ¼ DRl � ub ¼ DRl � uðLþ kÞ þ uð0Þ: ð13Þ

As the deformation ub of the bored profile is not known a priori anditself depends on the composite stiffness Kc, Eq. (7) introduces astrong non-linearity that cannot be dealt with by a standard finiteelement code. As shown in the next section, an iterative procedurewill be employed in order to find the numerical solution satisfyingthe highly non-linear boundary condition of Eqs. (6) and (7).

3.2.2. Iterative treatment of the boundary conditionFig. 7 shows the iterative procedure used for dealing with the

boundary condition according to Eqs. (6) and (7). The procedurestarts with the assumption that the ground closes the annulargap around the lining before backfilling. The computation for this

Fig. 7. Iterative procedure for the case of backfilling with pea gravel and mortar.

Fig. 8. First numerical example concerning the iterative procedure of Fig. 7:(a) convergence u � u(0) of the bored profile computed with Model A, (b) groundpressure p acting upon the shield and the segmental lining computed with Model A.


first iteration is carried out on the basis of Model A (upper part ofFig. 7). As there is no backfilling layer, the segmental lining is mod-elled by means of the following mixed boundary condition:

pðyÞ1 ¼0 if uðyÞ � uð0Þ 6 DRl

KlðuðyÞ1 � uð0Þ1 � DRlÞ if uðyÞ � uð0Þ > DRl

�ðfor y > LÞ:

ð14Þ

The first iteration provides the actual location of gap closure k�1. Ifthe gap is already closed at the planned location of backfilling

(i.e., if k�1 6 k), the assumption made (no backfilling) is correct andthe calculation is already finished after the first iteration. This hap-pens in the numerical example of Fig. 8, which concerns the hypo-thetical case of a 400 m deep tunnel crossing weak ground. Theboring diameter is equal to 10 m and the shield 10 m long. The seg-mental lining is 40 cm thick and has a radial gap size of 15 cm.Backfilling is planned 5 m behind the shield. Fig. 8a and b showthe convergence u � u(0) of the tunnel boundary and the distribu-tion of the ground pressure p, respectively, resulting from the firstiteration with Model A. The numerical results show that the groundestablishes contact with the segmental lining at a distance ofk�1 = 4.4 m behind the shield, i.e., before the planned backfilling,and confirm, therefore, the assumption underlying Model A, i.e.,that there is no backfilling layer. The ground pressure developingupon the lining far behind the shield amounts to p1 = 2.4 MPa.

Consider now the example of Fig. 9. The parameters of this exam-ple are the same as in Fig. 8, with the exception of the Young’smodulus E and the uniaxial compressive strength fc of the ground.(The ground is stiffer and stronger than in the previous example.)Fig. 9a shows the convergence along the tunnel based uponModel A, i.e., assuming that there is no backfilling. In this example,the ground does not close the gap at all. The numerical results arewrong because the model assumes that the support pressure is zerofor y > L + k = 15 m, while in reality a pressure will develop there dueto the backfilling. The same remark applies should the ground closethe gap at a distance k�1 > k, as occurs in the numerical example of

Fig. 9. Second numerical example concerning the iterative procedure of Fig. 7: (a and b) convergence u – u(0) of the bored profile computed with Models A and B,respectively, (c) ground pressure p acting upon the shield and the segmental lining computed with Model B.


Fig. 10 (k�1 = 19 m, point A), which concerns a ground with a slightlyhigher Young’s modulus E. In this case, Model A assumes that theground is unsupported in the interval L + k < y < L + k�1, while back-filling of the segmental lining already occurs at y = L + k. Addition-ally, Model A is erroneous with respect to the support stiffness inthe tunnel section y > L + k�1, because it does not consider the effectof backfilling on the stiffness of the support.

The results obtained from the first iteration are improved itera-tively by applying Model B, which takes into account the backfill-ing at y = L + k. In this case, the mixed boundary condition for thesimulation of the segmental lining in the second iteration readsas follows:

pðyÞ2 ¼0 if L < y 6 Lþ k

Kc;2ðuðyÞ2 � uðLþ kÞ2Þ if y > Lþ k:

�ð15Þ

The composite stiffness Kc,2 is calculated after Eqs. (8), (11)–(13),where the thickness of the backfilling db,1 is taken from the firstiteration (in the example of Fig. 9 db,1 = 68 mm and Kc,2 =362 MPa/m). The second iteration provides a new distribution ofthe convergence of the tunnel boundary u � u(0) and allows anew value to be determined in respect of the thickness of the back-filling db,2 (see, for example, Fig. 9b, where db,2 = 78 mm), which isused for the calculation of the composite stiffness in the next itera-tion step. The procedure is repeated as long as the value of db

changes significantly, i.e., until db,i � db,i�1.This iterative procedure converges very quickly and, in most

cases, three to four iterations suffice. This is true for both examplesof Figs. 9 and 10. In the example of Fig. 9, the third iteration, whichis carried out on the basis of a composite stiffness ofKc,3 = 349 MPa/m (calculated according to the thickness of the

backfilling after the second iteration, i.e., db,2 = 78 mm), leads tofollowing results: db,3 = 78.20 mm � db,2 = 78.25 mm andp3 = 0.71 MPa � p2 = 0.72 MPa. The example of Fig. 10 (where thebackfilling is weaker than in the example of Fig. 9) converges afterfour iterations, according to the following results (the results of thethird iteration are not plotted in Fig. 10 for the sake of clarity):db,4 = 54.71 mm � db,3 = 54.61 mm (db,2 = 55.49 mm) andp4 = 0.71 MPa � p3 = 0.70 MPa (p2 = 0.78 MPa) calculated withKc,4 = 133 MPa/m and Kc,3 = 131 MPa/m (Kc,2 = 170 MPa/m),respectively.

3.2.3. Simplifying assumptionThe computational effort can be reduced by assuming that the

backfilling is very stiff (i.e., Kb ?1). With this simplification,which is on the safe side with respect to the ground pressure p act-ing upon the segmental lining, the displacement-dependency ofthe composed stiffness Kc disappears, as Kc = Kl applies (cf.Eq. (11)). In this case, the procedure in Fig. 7 consists only of thefirst two iterations (i.e., one calculation with Model A and one withModel B), and since these iterations are no longer coupled to eachother, it has only to be evaluated whether Model A or Model Bdelivers the correct results for the given situation.

The assumption of Kc = Kl is correct in the particularly adverseconditions, where the ground closes the annular gap before back-filling is possible. Otherwise, it represents a simplification on thesafe side because it overestimates the stiffness of the tunnelsupport. Fig. 11a, which applies for the example introduced inSection 3.2.2 (and assumes that the backfilling layer is a compress-ible buffer, cf. Section 3.2.1), shows this overestimation by meansof the ratio of stiffness of the segmental lining Kl to composite

Fig. 10. Third numerical example concerning the iterative procedure of Fig. 7: (a) and (b) convergence u � u(0) of the bored profile computed with Models A and B,respectively, (c) ground pressure p acting upon the shield and the segmental lining computed with Model B.


stiffness Kc. As can be seen from Fig. 11a, the overestimation of thestiffness of the tunnel support decreases with decreasing thicknessof the backfilling db, i.e., for increasing ground deformations, andfor increasing value of the Young’s modulus of the backfilling Eb.In account of this, the conclusion can be drawn that the overesti-mation decreases with increasing squeezing potential of theground and increasing stiffness of the backfilling material. How-ever, as shown later (cf. Section 4.2.3), the ground pressure p actingupon the segmental lining does not depend linearly from the stiff-ness of the tunnel support used for the computations. Therefore,the overestimation of the support stiffness does not necessarilylead to a significant overestimation of the lining loading. This canbe observed, e.g., in Fig. 9c, where, for the sake of comparison,the dashed line plots the ground pressure p acting upon the seg-mental lining computed assuming Kc,2 = Kl = 480 MPa/m insteadof Kc,2 = 362 MPa/m. In this example, the overestimation of about11% in the loading of the segmental lining is not relevant from apractical point of view. On the contrary, there is a far more impor-tant overestimation (of 64%) in the example of Fig. 10c, which in-volves a much weaker backfilling material.

3.3. Backfilling with grouting via the shield tail

An iterative procedure is also necessary for the case of groutingvia the shield tail. As can be seen in Fig. 12, however, the procedureis simpler than in Fig. 7, and requires only one computational mod-el, which is practically identical to Model B of Fig. 7, the only dif-ference being that here k = 0. The boundary condition thereforebecomes simpler than in Eq. (15):

pðyÞi ¼ Kc;iðuðyÞi � uðLÞiÞ ðfor y > LÞ; ð16Þ

where Kc,i is the composite stiffness (segmental lining and annulusgrout), which has to be determined iteratively, because it dependson the thickness of the backfilling layer and thus on the ground con-vergence at the shield tail. The procedure begins (Fig. 12) with Kc,1

calculated according to an assumed initial value of backfilling thick-ness db,0 = DRl � DR. For the further iterations, the stiffness Kc,i iscalculated with Eqs. (8), (11)–(13) (or Eqs. (8)–(10), and (13) ifone assumes that the backfilling layer represents a perfect ring)using the improved value of db,i�1 resulting from the previous iter-ation. Analogously to Section 3.2.2, the procedure ends when theconvergence criterion db,i � db,i�1 is fulfilled. The main result ofthe procedure is the loading of the segmental lining p = pi.

The assumption of a very stiff backfilling (i.e., Kb ?1, analo-gously to Section 3.2.3) also leads in the present case (groutingvia the shield tail) to a considerable reduction in the computationaleffort. With this simplification, which is particularly appropriate ifa rapidly hardening mortar is used, the computation of the groundpressure acting upon the segmental lining requires only one itera-tion (i.e., p = p1). Depending on the static action of the backfillinglayer (compressible buffer or perfect ring, cf. Section 3.2.1) the sim-plification leads to an overestimation (Fig. 11a) or to an underesti-mation (Fig. 11b) of the stiffness of the tunnel support. For the firstcase, the considerations done at the end of the last section applyanalogously, while for the rather rare case of a ‘‘perfect ring’’,according to Fig. 11b the conclusion can be drawn that theunderestimation of the stiffness of the tunnel support becomesrelevant only for very stiff backfilling materials.

Fig. 11. Ratio of stiffness of the segmental lining Kl to composed stiffness Kc

(segmental lining and backfilling) for different values of the thickness of thebackfilling db as a function of the Young’s modulus of the backfilling Eb: (a) thebackfilling layer represents a compressible buffer and (b) the backfilling layer formsa perfect ring.

Fig. 12. Iterative procedure for the case of grouting via the shield tail.


4. Decision aids

4.1. Dimensionless parameters

The main result of the numerical computations is the final valueof the ground pressure far behind the tunnel face. Figs. 8b, 9c and10c show typical load distributions p(y) along the tunnel.

The ground pressure p acting upon the segmental lining dependson the material constants of the ground (Young’s modulus E,Poisson’s ratio m, uniaxial compressive strength fc, angle of internalfriction u and dilatancy angle w), on the initial stress r0, on thecharacteristics of the TBM (tunnel radius R, shield stiffness Ks, radialgap size DR and shield length L) and on the characteristics of thebackfilled segmental lining (composite stiffness Kc, radial gap sizeDRl and location of backfilling k):

p ¼ f ðE; m; fc;u;w;r0;R;Ks;DR; L;Kc;DRl; kÞ: ð17Þ

In order to reduce the computational effort, the calculations werecarried out on the basis of the simplifying assumption Kc = Kl (cf.Sections 3.2.3 and 3.3). The performance of a dimensional analysisleads to the following expression:

p� ¼ pr0¼ f

Er0; m;

fc

r0;u;w;

KsRE;DRR;LR;KlRE;DRl

DR;kR

� �: ð18Þ

After Anagnostou and Kovári (1993), the displacements u of an elas-to-plastic medium depend linearly on the reciprocal value of itsYoung’s modulus E (i.e., the product of u by E is constant). Thanksto this general property of elasto-plastic materials, it is possible toreduce further the number of parameters. More specifically, it isreasonable to expect that the product of E by DR (rather than theindividual parameters E and DR) will be significant. Consequently,the normalised ground pressure p� can be expressed as a functionof the product of the two normalised parameters E/r0 and DR/R:

pr0¼ f

Er0

DRR; m;

fc

r0;u;w;

KsRE;LR;KlRE;DRl

DR;kR

� �: ð19Þ

Ramoni and Anagnostou (2010c) tested and used this theoreticalhypothesis when working out nomograms for the required thrustforce. Lavdas (2010) carried out a series of numerical calculationscovering the entire range of material constants and initial condi-tions and showed that the hypothesis is also correct for the specificproblem of the present paper. Fig. 13 illustrates the approach bymeans of the results of numerical investigations concerning a singleshielded TBM with a normalised shield length of L/R = 2.0. Fig. 13aand b show the normalised ground pressure p� as function of E/r0

for the Model A and the Model B, respectively. Each curve appliesto another value of DR/R. The same results are plotted in Fig. 13cand d as a function of the product of E/r0 and DR/R. As all pointsin Fig. 13c and d fall on one single curve, the proposed normalisa-tion is correct. This is true, however, only for a constant ratioDRl/DR and not for a constant ratio DRl/R (Lavdas, 2010). Therefore,and with regard to the fact that the radial gap size of the segmentallining DRl depends geometrically on the radial gap size of theshield DR (cf. Eq. (2)), it was more reasonable to include in Eq. (18)the dimensionless parameter DRl/DR rather than the ratio DRl/R.

4.2. The parameter range covered

Although, as stated in Section 3.1, the applied computationalmethod is very efficient, a trade-off has had to be made betweenthe completeness of the parametric study and the cost of computa-tion and data processing. The numerical analyses have thereforebeen carried out only for selected parameters of the ground, TBMand segmental lining, as well as selected layouts of the backfilling.

Fig. 13. Results of numerical investigations: (a) and (b) normalised ground pressure p� as a function of E/r0 for different values of DR/R computed with Models A and B,respectively; (c) and (d) normalised ground pressure p� as a function of the dimensionless parameter EDR/r0R computed with Models A and B, respectively.


It should be noted that even with the limitations described inthe following sections, working out the nomograms has requiredalmost 8000 calculations.

4.2.1. Poisson’s ratio and dilatancy angle of the groundWith respect to the ground parameters, the first compromise

concerns the Poisson’s ratio, which has been kept constant tom = 0.25.

As a further simplification, analogously to Ramoni and Anag-nostou (2010c), the dilatancy angle of the ground w was taken asa function of the angle of internal friction u after Vermeer andde Borst (1984) and not as an independent parameter:

w ¼1� for u 6 20�

u� 20� for u > 20�

�: ð20Þ

However, the nomograms account for a wide variation in the otherthree material constants (E, u and fc) and cover the relevant rangeof ground strength and deformability.

4.2.2. Length and stiffness of the shieldAccording to the TBM technical data collected in Ramoni and

Anagnostou (2010c), the normalised shield length L/R of singleshielded TBMs is between 1 and 5 (the larger values applying tosmall diameter TBMs). The nomograms were worked out forL/R = 1. This assumption is safe with respect to the loading of thesegmental lining, as a shorter shield length facilitates load transferin the longitudinal direction (cf. Section 2.2.6 and Fig. 4b) and thisresults in a higher ground pressure acting upon the segmental lin-ing. Fig. 14a shows some of the results of a large series of numericalinvestigations carried out by Lavdas (2010) for testing this assump-tion. The solid lines apply to an overcut DR of 0.006R and differentground ‘‘qualities’’ (in terms of the normalised Young’s modulusE/r0 and the normalised uniaxial compressive strength fc/r0),while the dashed lines concern the case of a larger overcut(DR = 0.010R). As can be seen from Fig. 14a, the effect of the

normalised shield length L/R on the normalised ground pressurep� is negligible for L/R > 2 but may be important in the range1 6 L/R 6 2, particularly in the case of a higher quality groundand of a larger radial gap (compare, for example, points A and B).The load is higher if the shield is short (point A) because of the lon-gitudinal arching (Fig. 4b). This effect disappears if the gap is small,the ground quality low or the shield long, because the groundcloses the gap around the rear part of the shield and the arching oc-curs according to Fig. 4a.

The normalised shield stiffness KsR/E has been kept constantassuming the same value of KsR/E = 10 as in Ramoni and Anagnos-tou (2010c). As can be seen from Fig. 14b, which shows the effect ofthe normalised shield stiffness KsR/E on the normalised groundpressure p�, the assumed value of KsR/E is on the safe side with re-spect to the problem under consideration. Of course, the norma-lised shield stiffness KsR/E does not play a role if the shield is notloaded at all by the ground (this happens in the case of the lowestcurve in Fig. 14b), because a longitudinal arching action betweenshield and backfilled segmental lining (Fig. 4a) does not exist inthis case.

4.2.3. Stiffness and radial gap size of the segmental liningFig. 14c shows the effect of the normalised support stiffness

KlR/E on the normalised ground pressure p�. As expected, the stifferthe segmental lining, the higher will be the ground pressure actingupon it. In order to cover the relevant stiffness range, the nomo-grams were worked out for KlR/E = 1, 3 and 10.

In the most cases, the ratio of the radial gap size DRl of the seg-mental lining to the radial gap size DR of the shield is between 3and 10 (Ramoni and Anagnostou, 2010c). The nomograms applyfor a constant ratio of DRl/DR = 3 and are on the safe side for mostcases. The radial gap DRl affects the normalised ground pressure p�

only in the cases, where the radial gap DRl is closed before backfill-ing can occur (i.e., the cases, where Model A applies, cf.Section 3.2.2). Otherwise, if the ground does not close the radial

Fig. 14. Numerical results concerning the dependency of the normalised groundpressure p� on: (a) the normalised shield length L/R, (b) the normalised shieldstiffness KsR/E, (c) the normalised lining stiffness KlR/E, (d) the ratio of radial gapsize DRl of the lining to the radial gap size DR of the shield, (e) normalized locationof backfilling k/R.


gap DRl (i.e., if Model B applies), the ratio DRl/DR does not play anyrole. This is because of the simplifying assumption of a very stiff

backfilling done for working out the nomograms (cf. Sections3.2.3 and 3.3), which leads to the introduction of a constant stiff-ness Kc = Kl in the Model B (cf. Section 4.1). As the stiffness of thesimplified model does not depend on the actual thickness of thebackfilling, the normalised ground pressure p� does not dependon the radial gap DRl. This applies for the majority of the resultsshown in Fig. 14d, particularly in the practically relevant rangeDRl/DR = 3 � 10.

4.2.4. Location of backfillingThe nomograms were developed for the backfilling locations of

k = 0, R and 2R behind the shield (k = 0 applies to grouting via theshield tail). Fig. 14e shows the effect of the normalised backfillinglocation k/R on the normalised ground pressure p�. In general, theloading of the segmental lining decreases if backfilling is carriedout later. This happens, however, only up to a certain value ofthe distance k behind the shield (point C). If the planned locationof the backfilling is far behind the shield, the ground closes thegap earlier and the parameter k does not play a role anymore(points C and D show the same ground pressure).

4.3. The nomograms

Taking account of the dimensional analysis of Section 4.1 andbearing in mind that some of the model parameters are kept con-stant (cf. Section 4.2), the following expression applies for the nor-malised ground pressure:

pr0¼ f

Er0

DRR;

fc

r0;u;

KlRE;kR

� �: ð21Þ

Figs. 15–23 contain the graphical representation of this relation-ship. Each figure applies to a different pair of values (k/R, KlR/E).The diagrams of each figure apply to different values of the angleof internal friction u and show the normalised groundpressure p/r0 as a function of the dimensionless parameter(E/r0)(DR/R) and of the normalised uniaxial compressive strengthfc/r0. Table 1 contains an overview of the nomograms and of theconsidered parameter ranges.

4.3.1. Effect of the radial gap size of the shieldAs can be seen from the nomograms (Figs. 15–23), for given val-

ues of Young’s modulus E, uniaxial compressive strength of theground fc, initial stress r0 and tunnel radius R an increase of the ra-dial gap size of the shield DR leads to a reduction of the groundpressure p only up to a certain value of DR. For higher valuesof DR, the loading of the segmental lining p remains constant. Thisis because a bigger radial gap size DR of the shield means a higherradial gap size DRl of the segmental lining (cf. Eq. (2)) and thelatter, with the assumptions underlying the nomograms (cf.Section 4.2.3), affects the ground pressure p only if the groundcloses the gap before backfilling occurs.

4.3.2. Effect of the location of backfillingA closer examination of Figs. 15–23 reveals that the curves in

the left part of the nomograms, i.e., at small values of the dimen-sionless parameter (E/r0)(DR/R), are the same for all k/R values(all other parameters being constant). This is because the backfill-ing location k/R does not affect the ground pressure if the groundcloses the radial gap of the segmental lining before backfilling(cf. Section 4.2.4) and this happens when the Young’s modulus Eof the ground is low or the radial gap size of the shield DR small.

4.3.3. Counter-intuitive model behaviourAccording to the numerical results of Figs. 15–23, the effect of

the uniaxial compressive strength is not as straightforward as

Fig. 15. Backfilling with pea gravel and mortar: nomograms for k/R = 2 and KlR/E = 1.


Fig. 21. Backfilling with grouting via the shield tail: nomograms for k/R = 0 and KlR/E = 1.


Table 1Overview of the nomograms (a) and of the considered parameter ranges (b).

(a) Nomograms

Backfilling type k/R [-] KlR/E [-] Nomogram

Pea gravel and mortar 2 1 Fig. 153 Fig. 1610 Fig. 17

1 1 Fig. 183 Fig. 1910 Fig. 20

Grouting via shield tail 0 1 Fig. 213 Fig. 2210 Fig. 23

(b) Considered parameter ranges

Ground

Poisson’s ratio m [-] 0.25Angle of internal friction u [�] 15–35Dilatancy angle w [�] max(1, u– 20)Normalized compressive strength fc/r0 [-] 0.10–0.50

TBM

Normalized shield stiffness KsR/E [-] 10Normalized shield length L/R [-] 1

Backfilled segmental lining

Normalized stiffness KlR/E [-] 1, 3 or 10 (see above)Normalized radial gap size DRl/DR [-] 3Normalized location of backfilling k/R [-] 0, 1 or 2 (see above)

Table 2Application examples.

Application example 1 2 3

Ground1 E (MPa) 1000 570 h 10002 m (–) 0.253 fc (MPa) 3.0 3.0 1.0 j

4 u (�) 255 w (�) 56 c (kN/m3) 25

Initial stress7 H (m) 4008 r0 (MPa) 10 a

TBM9 R (m) 510 DR (cm) 5

Backfilled segmental lining11 El (MPa) 30,00012 dl (cm) 4013 Kl (MPa/m) 480 b

14 fc,l (MPa) 3015 DRl (cm) 1516 k (m) 5

Dimensionless products17 E/r0 (–) 100.00 57.00 100.0018 DR/R (–) 0.0119 (EDR)/(r0R) (–) 1.00 0.57 1.0020 fc/r0 (–) 0.30 0.30 0.1021 KlR/E (–) 2.40 4.21 2.4022 k/R (–) 1.00

Structural safety23 p� (–) 0.15 c 0.24 0.2424 p (MPa) 1.5 d,e 2.4 i 2.4 i

25 pmax (MPa) 2.4 f

26 SF (–) 1.6 g 1.0 1.0

a r0 = Hc.b Kl = Eldl/R2.c Linear interpolation between Figs. 18 and 19, design nomograms for u = 25�,

curve for fc/r0 = 0.30.d p = p�r0.e Bold value: main result of the calculation.


one might expect. At low values of (EDR)/(r0R), an increase in thenormalised uniaxial compressive strength fc/r0 leads to an increasein the normalised ground pressure p/r0, i.e., a better ground leadsto a higher loading of the segmental lining. This ‘‘paradox’’ is well-known in the literature (e.g., Boldini et al., 2000; Cantieni andAnagnostou, 2010; Graziani et al., 2005; Ramoni and Anagnostou,2010c) and is associated with the pre-deformations of the groundahead of the tunnel face (the bigger the pre-deformation, the morepronounced the stress relief).

f pmax = dl fc,l/R.g SF = pmax/p.h Linear interpolation between Figs. 19 and 20, design nomograms for u = 25�,

curve for fc/r0 = 0.30.i ‘‘Given’’, in this column the calculation proceeds from down to top.j Linear interpolation between Figs. 18 and 19, design nomograms for u = 25�,

curve for fc/r0 = 0.10.

4.4. Application examples

The nomograms enable an easy and quick computation to bemade of the ground pressure acting upon the segmental lining.The use of the nomograms will be illustrated by means of theapplication example introduced in Section 3.2.2. Table 2 showsthe input parameters as well as the calculations steps.

4.4.1. Assessment of the segmental lining for given conditionsColumn 1 of Table 2 shows how to calculate the ground pres-

sure acting upon the segmental lining for given ground conditionsand parameters of the TBM and of the segmental lining (i.e., thiscolumn deals with dimensioning).

With regard to the present normalised stiffness of the segmen-tal lining KlR/E = 2.4 (Row 21) and the normalised location of back-filling k/R = 1 (Row 22), the normalised ground pressure p�

(Row 23) is calculated by interpolating linearly between Figs. 18and 19. The appropriate curves have to be selected according tothe actual values of the angle of internal friction (u = 25�, Row 4)and normalised uniaxial compressive strength (fc/r0 = 0.3, Row20). The dimensionless parameter EDR/r0R (Row 19), which is en-tered in the nomograms in order to depict the value of the norma-lised ground pressure, is calculated on the basis of the parametersof Rows 1, 8, 9 and 10. In this example, p⁄ = 0.15 (Row 23), thusresults in a loading of the segmental lining of p = 1.5 MPa(Row 24). Taking into account the thickness and the uniaxialcompressive strength of the segmental lining (dl = 40 cm and

fc,l = 30 MPa according to Rows 12 and 14, respectively), a safetyfactor of SF = 1.6 (Row 26) results.

4.4.2. Estimation of critical ground conditions for a given segmentallining

Columns 2 and 3 of Table 2 show how to estimate the applica-bility range (in terms of ground quality) of a given segmental lin-ing. For this purpose (and for the sake of simplicity), the presentexample assumes that the ‘‘quality’’ of the ground can be describedby combinations of its Young’s modulus E and its uniaxial com-pressive strength fc (all other ground parameters considered asbeing constant).

The reference ground for the present investigation is that of Col-umn 1 of Table 2 (i.e., a Young’s modulus of E = 1000 MPa and auniaxial compressive strength of fc = 3 MPa). The question ad-dressed here is: how much worse would the ground have to bein order to endanger the structural safety of the segmental lining?In order to answer this question, the bearing capacity of thesegmental lining pmax = 2.4 MPa (Row 25) is made equal to theground pressure p (Row 24) acting upon it. Keeping the uniaxialcompressive strength equal to the reference value of fc = 3 MPa

Fig. 24. Histogram of the comparative calculations of Table 3.


(Row 3), it is possible to calculate back (Column 2, down to top) thecritical Young’s modulus E (Row 1) for which the structural safetyof the segmental lining becomes critical. In this example, this oc-curs if the Young’s modulus is E = 570 MPa. A similar calculation(Column 3) shows that (keeping the Young’s modulus equal to itsreference value of E = 1000 MPa) the critical uniaxial compressivestrength amounts to fc = 1 MPa.

Sensitivity analyses are important in order to identify the criti-cal geotechnical conditions with respect to a given set of designcriteria and problem parameters, as the intensity of squeezingmay vary significantly along the tunnel alignment (Cantieni andAnagnostou, 2007). The examples of this section of the paper dem-onstrate that such investigations can be carried out easily andquickly applying the proposed nomograms.

4.5. Applicability range of the nomograms

As mentioned above (cf. Sections 4.2.3 and 4.2.4), the nomo-grams were worked out only for specific values of the normalisedsupport stiffness KlR/E and normalised backfilling locations k/R. Ifthe actual values of KlR/E and k/R deviate from the values assumedfor the nomograms, the ground pressure can be obtained by inter-polating linearly between the nomograms (as done in the exam-ples of the last section).

The suitability of this procedure and, more general, the applica-bility range of the nomograms has been tested extensively by com-paring the ground pressures p obtained from the nomograms withthe pressures resulting from numerical analyses which take ac-count of the actual values of the model parameters (only the sim-plifying assumption of Sections 3.2.3 and 3.3 was retained). Table 3shows such a comparison for three sets of ground parameters andinitial conditions (Columns 1–6, 7–12 and 13–18, respectively). For

Table 3Comparison between the ground pressure p obtained from numerical analyses carried out

Applicationexample

1 2 3 4 5 6 7 8

Ground1 E (MPa) 1000 1000 1000 1000 1000 1000 2000 20002 m (–) 0.253 fc (MPa) 1 1 1 1 1 1 3 34 u (�) 20 20 20 20 20 20 25 255 w (�) 1 1 1 1 1 1 5 56 c (kN/m3) 25

Initial stress7 H (m) 250 250 250 250 250 250 500 5008 r0 (MPa) a 6.25 6.25 6.25 6.25 6.25 6.25 12.50 12.50

TBM9 R (m) 2.5 2.5 4 4 5.5 5.5 2.5 2.510 DR (cm) 4 4 5 5 6 6 4 411 L (m) 10.0 10.0 11.5 11.5 13.0 13.0 10.0 10.012 Es (MPa) 210,00013 ds (cm) 7.5 7.5 10.0 10.0 12.5 12.5 8.0 8.014 Ks (MPa/m) b 2520 2520 1313 1313 868 868 2520 2520

Backfilled segmental lining15 El (MPa) 30,00016 dl (cm) 30 50 30 50 30 50 30 5017 Kl (MPa/m) c 1440 2400 563 938 298 496 1440 240018 fc,l (MPa) 3019 DRl (cm) 14 14 17 17 20 20 14 1420 k (m) 3 3 0 0 8 8 3 3

Numerical analyses21 p (–) 1.2 1.3 1.5 1.6 1.2 1.3 1.1 1.3

Nomograms22 p (–) 1.2 1.3 1.6 1.8 1.1 1.2 1.3 1.6

a r0 = Hc.b Ks = Esds/R2.c Kl = Eldl/R2.

each set, three TBM and backfilling layouts (e.g., Columns 1–2, 3–4,5–6) and two different segmental linings (odd end even columns)have been considered. Fig. 24 shows that the results obtained byapplying the nomograms agree well with the results of the numer-ical investigations. As expected, the nomograms overestimate theloading of the segmental lining in the most cases. The overestima-tion is large only in the case of small diameter TBMs and of rela-tively good quality ground (Columns 13 to 16). This is becausethe assumption of a normalised shield length L/R = 1, which hasbeen made for working out the nomograms (cf. Section 4.2.2), isparticularly safe in these cases (cf. Fig. 14a).

Although the present paper focused on single shielded TBMs,the basic approach also applies to double shielded TBMs. In fact,there is no difference between the two machine types with respect

specifically for each example and by applying the nomograms.

9 10 11 12 13 14 15 16 17 18

2000 2000 2000 2000 4000 4000 4000 4000 4000 4000

3 3 3 3 10 10 10 10 10 1025 25 25 25 30 30 30 30 30 305 5 5 5 10 10 10 10 10 10

500 500 500 500 1000 1000 1000 1000 1000 100012.50 12.50 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00

4 4 5.5 5.5 2.5 2.5 4 4 5.5 5.55 5 6 6 4 4 5 5 6 611.5 11.5 13.0 13.0 10.0 10.0 11.5 11.5 13.0 13.0

10.0 10.0 12.5 12.5 8.0 8.0 10.0 10.0 12.5 12.51313 1313 868 868 2520 2520 1313 1313 868 868

30 50 30 50 30 50 30 50 30 50563 938 298 496 1440 2400 563 938 298 496

17 17 20 20 14 14 17 17 20 200 0 8 8 3 3 0 0 8 8

1.7 2.0 1.1 1.3 0.3 0.4 1.2 1.5 0.6 0.8

2.2 2.5 1.0 1.3 1.2 1.5 1.7 2.9 0.5 0.9


to the normalised location of the backfilling k/R. Furthermore, thenormalised shield length L/R (L = Lf + Lr) and the ratio DRl/DRr

(where DRr denotes the radial gap of the rear shield) are in thesame range as L/R and DRl/DR for single shielded TBMs (Ramoniand Anagnostou, 2010c). Consequently, the nomograms whichwere developed for single shielded TBMs can also be applied todouble shielded TBMs provided that the radial gap size of the rearshield DRr is taken into account in the computations (instead of theradial gap size of the single shield DR). This approach was con-firmed by a series of comparative computations (such as the onesin Table 3 and Fig. 24), which, for the sake of economy, are not re-ported here.

5. Closing remarks

When evaluating the feasibility of a shielded TBM drive insqueezing ground, one of the key questions to be addressed con-cerns the loading of the segmental lining. Therefore, as shown inSection 2, it is important to take due account of the type, locationand actual thickness of the backfilling of the segmental lining.

Computational investigations provide valuable indications as tothe magnitude of the ground pressure. In this respect, simplifiedinvestigations by means of the convergence-confinement methodare not satisfying, as plane strain analyses require an arbitrary esti-mation of the so-called ‘‘pre-deformation’’ of the ground and donot take due account of the stress history (Cantieni and Anagnos-tou, 2009). The correct simulation of a backfilled segmental liningis a highly non-linear problem, because the actual thickness of thebackfilling and, therefore, the actual stiffness of the tunnel supportare not known a priori but depend on the actual radial displace-ment of the bored profile at the point, where backfilling occurs.As proposed in Section 3, the simulation problem can be solvedby applying an iterative procedure.

A simplified iterative procedure made it possible to work out aseries of nomograms which cover the practically relevant range ofmaterial constants, initial stress, TBM, segmental lining and back-filling characteristics, and allow an easy and quick assessment tobe made of the loading of the segmental lining. The dimensionlessdesign charts of Section 4 represent a valuable contribution to thedecision-making process in the planning phase and – together withthe nomograms of Ramoni and Anagnostou (2010c) – are part of aset of decision aids for the planning of TBM drives in squeezingground worked out at the ETH Zurich.

In order to reduce the computational effort, the nomograms ofthe present paper disregard the fact that the stiffness Kc of the tun-nel support depends on the deformations of the bored profile, i.e.,they assume a constant stiffness Kc = Kl (cf. Sections 3.2.3 and 3.3),where Kl is the stiffness of the segmental lining. This assumption iseasy and reasonable (cf., e.g., Fig. 10), but not mandatory. Theintroduction into the computation of another tunnel support stiff-ness value is possible, but requires, of course, sufficient and reliableknowledge as to the actual value of the stiffness Kc of the systemcomposed of the backfilling and the segmental lining.

As for every geomechanical calculation, there are uncertaintiesassociated with the ground parameters and the ground behaviour.Of course, the proposed approach and the nomograms cannot elim-inate such uncertainties, but they do allow quick sensitivity analy-ses to be made. With respect to the ground behaviour, on the onehand it has to be said that disregarding its possible time-depen-dency is a major simplification which is, nevertheless, usuallymade due to a lack of knowledge concerning the behaviour ofthe ground over time. Neglecting time-dependency may be unsafewith respect to the lining loading. In fact, as shown in Ramoni andAnagnostou (2010a), in some cases a slower development of theground deformations (or a faster TBM advance) may lead to a high-

er lining load. On the other hand, it must also be said that the pro-posed nomograms assume homogeneous ground. This is anotherstandard assumption in tunnel analysis, but it is conservative forthe frequently encountered case of short weak zones alternatingwith stabilizing, competent rock (Graziani et al., 2007; Kováriand Anagnostou, 1995).

Besides the possible time-dependency of the ground behaviour,the present paper also disregards a time-dependency in the stiff-ness of the backfilling material. This is a reasonable assumptionfor pea gravel (the stiffness of which does not vary over the time)or rapidly hardening mortars (the stiffness of which achieves its fi-nal value very quickly). This does not apply, however, to backfillingmaterials with a pronounced time-dependent behaviour. For thesecases (and assuming that the backfilling layer represents a com-pressible buffer between the segmental lining and the ground),the approach proposed in this paper leads to an overestimationof the final value of the loading of the segmental lining. This over-estimation decreases with decreasing thickness of the backfillinglayer (cf. Section 3.2.3 and Fig. 11a), i.e., with increasing squeezingpotential of the ground. However, the quantification of these ef-fects needs further research work.

Acknowledgements

This paper evolved within the framework of the research pro-ject ‘‘Design aids for the planning of TBM drives in squeezingground’’, which is being carried out at the ETH Zurich, supportedby the Swiss Tunnelling Society (STS) and financed by the SwissFederal Roads Office (FEDRO).

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squeezing loading of segmental lining and the effect of backfilling

Documents

segmental liningand

ground pressure

present paper deals

present paper sho

tunnel support

backfilling features

single shielded tbm

paper investigates