Stress Analysis of Reinforced Tunnel Faces and Comparison With the Limit Equilibrium Method 2013 Tunnelling and Underground Space Technology

Tunnelling and Underground Space Technology 38 (2013) 8798Contents lists available at SciVerse ScienceDirect

Tunnelling and Underground Space Technology

journal homepage: www.elsevier .com/ locate/ tustStress analysis of reinforced tunnel faces and comparison with the limitequilibrium method0886-7798/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.tust.2013.05.008

Corresponding author. Address: ETH Zurich, Wolfgang-Pauli-Strasse 15, 8093Zurich, Switzerland. Tel.: +41 44 633 07 29.

E-mail address: [email protected] (P. Perazzelli).P. Perazzelli , G. AnagnostouETH Zurich, Switzerland

a r t i c l e i n f oArticle history:Received 11 March 2013Accepted 20 May 2013Available online 13 June 2013

Keywords:TunnellingFace stabilityFace reinforcementBoltsNumerical stress analysisLimit equilibrium methoda b s t r a c t

The present paper investigates the stability of reinforced tunnel faces in dry cohesive-frictional soils bymeans of 3D numerical stress analyses which take account of the individual bolts. The numerical calcu-lations are performed assuming an elastic, perfectly plastic material obeying the MohrCoulomb yieldcriterion. As usual in this kind of problem, the bolts are modelled by one-dimensional tension elements,which have zero diameter and cannot take into account geometrically the diameter of the bolts or theborehole. The first part of the paper deals with the approximations induced by this simplification. Morespecifically, the paper shows by means of numerical pullout tests in respect of a single bolt in elasto-plas-tic soil that the behaviour of this model depends significantly on the fineness of the numerical mesh. Thesecond part of the paper investigates the reinforcing effect of bolts on face stability assuming that thebond strength of the bolts depends on the confining stress with strength parameters equal to those ofthe soil. For several bolting patterns, the minimum cohesion c0lim needed for face stability is calculatediteratively and compared with the predictions of the limit equilibriummethod of Anagnostou and Serafe-imidis (2007).

2013 Elsevier Ltd. All rights reserved.1. Introduction

Face reinforcement by fibreglass bolts is a very effective andflexible method for stabilizing the tunnel face in weak ground. Ithas been the subject of several papers over the past two decades,starting with Peila (1994), who performed spatial numerical stressanalyses and showed that reinforcement reduces the deformationsand the extent of the overstressed zone ahead of the face. Other re-lated works are those of Ng and Lee (2002), which investigated theinfluence of the axial stiffness of the bolts with respect both to thestability of the tunnel heading and to the surface settlement in astiff clay, and of Yoo and Shin (2003), which analysed the effectof bolt spacing, length and axial stiffness on face stability for differ-ent cover-to-diameter ratios. Dias and Kastner (2005) also mod-elled each bolt individually and investigated by means of 3Dfinite difference analyses the effects of bolt spacing and bondstrength (i.e. the shear strength of the interface between groutedbolt and rock) on the face stability of a deep tunnel in soft rock.Furthermore, they compared the numerical results with the resultsof simplified analyses, which take into account the face reinforce-ment either by introducing an equivalent face support pressure orby considering a higher cohesion of the core ahead of the tunnelface. Kavvadas and Prountzopoulos (2009) performed spatial finiteelement calculations in order to find out the optimum bolt lengthand the overall face support pressure exerted by the bolts for dif-ferent soil shear strength parameters and cover-to-diameter ratios.

The face reinforcement is tackled either by smearing the effectof the bolts and considering an equivalent higher strength ground(e.g., Indraratna and Kaiser, 1990; Grasso et al., 1991) or by takingaccount of individual bolts. The usual assumption concerning thebearing capacity of the bolts is that it is limited either by the ten-sile strength of the bar or by the shear strength of the soilgroutinterface. Prete (2007) and later Oreste and Dias (2012) also tookaccount of the bending failure of the bolt or failure of the soildue to the radial pressure exerted by the bolts (in a similar wayto a soil nailing analysis).

In addition to the above-mentioned 3D numerical stress analy-ses, simpler approaches such as limit equilibrium methods (e.g.,Cornejo, 1989; Mohkam and Wong, 1989; Anagnostou and Kovri,1994) or methods based on plasticity theorems (Caquot and Keri-sel, 1956; Mandel and Halphen, 1974; Leca and Dormieux, 1990)have also been proposed for assessing tunnel face stability. In fact,a 3D numerical stress analysis, besides being very time consumingand awkward to handle for practical engineering purposes, repre-sents an unnecessarily complex approach (and actually a longway round) if it is only the stability of the face (rather than thedeformation of the ground) that is concerned. On the other hand,numerical stress analysis represents the only computational

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Nomenclature

Ab cross section area of the grouted bolta bolt length inside the wedgeB width of the rectangular tunnel facec0 cohesion of the groundcI cohesion at the soilgrout interfaced grouted borehole diametere size of the grid elements adjacent to the boltE0 Youngs modulus of the soilEb Youngs modulus of the grouted boltFa1,y nodal forceFa2,y nodal forceFp tensile force at the head of the bolt (pullout analysis)h depth of coverH height of the rectangular tunnel faceI ratio of interface stiffness to axial bolt stiffnessKI shear stiffness of the interface between soil and cable

elementKnI normal stiffness of the interface elements between soil

and grouted boltKsI shear stiffness of the interface elements between soil

and grouted boltk0 coefficient of earth pressure at restl grid size in the axial directionL0 bolt lengthn reinforcement density (number of bolts per unit area)Nb axial force of the boltNby limit tensile force of the boltR maximum shear force at the soilgrouted bolt interface

per unit length of the boltri inner radiusro outer radiuss support pressureTb shear force at the soilgrouted bolt interface per unit

length of the bolt

ub displacement of the boltup displacement of the head of the boltuy longitudinal displacement at the centre of the facex horizontal co-ordinate perpendicular to tunnel axisy horizontal co-ordinate parallel to tunnel axisypl extent of bolt section with failed bondz vertical co-ordinate

Greek symbolsc unit weight of the soiley axial strain of the boltkp coefficient of lateral stress for the prism in Anagnostou

and Kovri (1994) and Anagnostou and Serafeimidis(2007) methods

kw coefficient of lateral stress for the wedge in Anagnostouand Serafeimidis (2007) and Anagnostou and Kovri(1994) methods

m Poissons ratio of the soilmb Poissons ratio of the boltr radial confining stress acting upon the grouted boltrby tensile strength of the boltrr radial stressrx horizontal stress perpendicular to the tunnel axisrz vertical stressu0 friction angle of the soiluI friction angle of the soilgrout interfacew0 dilatancy angle of the soilx angle between face and inclined sliding plane of the

wedgesI shear stress at the interface between soil and grouted

boltsm bond strength of the soilgrout interface

88 P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798possibility for checking the adequacy of a priori assumptions con-cerning the geometry of the failure mechanism and the horizontalstresses in the ground, which are needed in limit equilibrium anal-yses but are statically indeterminate (Anagnostou, 2012).

A simple, limit equilibrium based computational method forassessing the stability of a reinforced tunnel face was introducedby Anagnostou (1999) and refined by Anagnostou and Serafeimidis(2007). The present paper compares the results of this methodwith the results of spatial numerical stress analyses of the rein-forced face. The underlying computational investigations includeas a by-product the modelling of the bolts in numerical stress anal-yses which is of more general interest.

As in all above-mentioned stress analysis methods, the bolts aremodelled here individually by one-dimensional structural ele-ments. This reduces computational time considerably comparedto more realistic models that use solid elements and two-dimen-sional interface elements to model the grouted bolts and theirinterfaces to the surrounding soil, respectively. On the other hand,however, the one-dimensional structural elements have zerodiameter, i.e. they do not take account of the diameter of the bore-holes geometrically. This has some important consequences formodelling, because the model behaviour proves to be mesh-sensi-tive, i.e. the structural behaviour of the reinforced core ahead of theface depends significantly on the fineness of the computationalmesh.

The first part of the present paper deals with this issue. Morespecifically, Section 2 investigates aspects of bolt modelling innumerical stress analyses by considering the relatively simpleproblem layout of a bolt pullout test. The purpose of this sectionis to get a better understanding of the nature and effect of theapproximations introduced by the simplified one-dimensional boltmodel and to get some guidelines concerning the choice of compu-tational mesh in large-scale numerical simulations involving bolts.

The second part of the paper presents a numerical solution tothe reinforced tunnel face problem, determines the soil cohesionnecessary in order for the face to be stable iteratively for differentreinforcement layouts and compares the numerical results withthose obtained by the limit equilibrium method (Section 3).

2. Modelling of the bolts in the numerical stress analysis

2.1. Introduction

Pullout tests in the laboratory (Milligan and Tei, 1998; Yin et al.,2009) or in the field (Heymann et al., 1992; Franzn, 1998),so-called interface tests (Milligan and Tei, 1998) and numericalstress analyses (Di Fonzo et al., 2008; Su et al., 2008) show thatthe interaction between the bolts and the soil depends on severalfactors: the overburden pressure, the mechanical properties andsaturation degree of the soil, the roughness of the interface be-tween the grouted bolt and the soil, the properties of the grout,the grouting pressure and the borehole drilling procedure, whichmay cause greater or lesser disturbance to the soil in the vicinityof the borehole. In order to reproduce the effects of these factors,

m=160 kPa(a)

40

80

120

160

200

Fp [k

N]

FLAC3DFp

up

5103 kPa104 kPa107 kPa

109 kPa

P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798 89the numerical model should map the real geometry, i.e. the finitediameter, of the grouted bolt. Solid elements have to be used tomodel the bolt, while the possibility of shear failure at the interfacebetween the grouted bolt and the surrounding soil calls for the useof special interface elements. Such a bolt model would have a largenumber of static degrees of freedom and would be extremelydemanding in terms of computer time, if a large number of boltshas to be represented (as in the case of a reinforced tunnel face).

In such cases, a simplified model is usually adopted where thebolts are represented by one-dimensional elements with an idea-lised zero diameter. The interaction of these elements with the soilis dealt with by interface conditions that are incorporated into thenumerical formulation of the one-dimensional elements. Such asimplified model (the so-called cable element of the FLAC3Dcode, Fig. 1) has been used in the present work for the stress anal-ysis of the reinforced tunnel face. The cable element is a two-node,straight finite element with one axially oriented translational de-gree-of-freedom per node (no bending resistance). Its behaviouris linearly elasticperfectly plastic, taking account of the limitedtensile strength of the bolt. The numerical formulation of the cableelement incorporates an interface condition which accounts for theshear forces developing parallel to the bolt axes in response to therelative motion between the bolts and the surrounding soil. Theinterface model exhibits a linearly elasticperfectly plastic behav-iour, which is determined by two parameters: the shear stiffness KI,defined as the ratio between the bond force per unit length of cableto the elastic relative displacement between the cable and the sur-rounding soil; and the maximum bond force R per unit length. Thelatter is equal to pd sm, where d and sm denote the grouted bore-hole diameter and the bond strength, respectively. The bondstrength sm may be constant or may depend on the confining stressaccording to the MohrCoulomb criterion (cI + r tanuI). The con-fining stress r is computed at each nodal point along the cable axis,based on the stress-field in the soil zone to which the nodal point islinked.

In order to check the numerical formulation and to explainthe basic behaviour of this model, we studied the relativelysimple problem of the pullout test of a bolt in rigid soil andcompared the numerical results with analytical results (Sec-tion 2.2). Naturally, the rigid soil assumption of Section 2.2 doesnot account for the possibility of failure inside the soil awayfrom the interface. Soil failure in combination with the zerodiameter of the cable elements causes the numerical results tobe mesh-dependent. This issue is investigated by means ofnumerical pullout tests of Section 2.3 assuming an elasto-plasticsoil behaviour.Fig. 1. Representation of fully bonded reinforcement in FLAC3D (Itasca, 2009).2.2. Pullout test in rigid soil

The so-called pullout curve represents the relationship be-tween the tensile force Fp applied at the bolt head and the corre-sponding displacement up. The computational investigations inthe present section assume that, (i) the soil around the grouted boltis rigid; (ii) the bolt behaves as an elastic bar with axial stiffnessEbAb; and (iii) the behaviour of the bond between the grouted boltand the surrounding soil is linearly elasticperfectly plastic with ashear stiffness KI and a maximum bond shear force per unit lengthof bolt R. Under these conditions, a closed-form solution can be de-rived for the pullout curve (Appendix).

Fig. 2 shows pullout curves for different values of the two inter-face parameters, i.e. bond stiffness KI (Fig. 2a) and bond strength sm(Fig. 2b). The values of the other computational parameters are gi-ven in Table 1 (the rigid soil column). The tensile strength of thebolt was taken to be infinite, i.e. bond failure is the only failuremode considered. The curves were derived analytically. The resultsof one comparative numerical calculation by FLAC3D (marked bythe symbols in Fig. 2) are equal to the analytical results.

Due to the elastic behaviour of the interface model, the force Fpinitially increases linearly with the displacement up. At larger dis-placements up, the pullout line is curved because the shear stress atthe interface reaches the bond strength and can increase no more.The shear failure of the interface starts at the bolt head and prop-agates with increasing displacement up to the bolt end. Afterwardsthe force Fp remains constant.

If the bond stiffness KI is low, the interface also remains elasticfor relatively large displacements (Fig. 2a). On the other hand, if thestiffness is higher than a threshold value of about 10 GPa, the pull-out curves do not change significantly and the behaviour of theinterface is practically rigid-plastic (Fig. 2a). If the bond strengthis increased, the maximum pullout force increases and it is reachedat greater displacements (Fig. 2b).up [mm]

0

40

80

120

160

200

Fp [k

N]

FLAC3DKI=107 kPa

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7up [mm]

0

Fpup(b)

KI=103 kPa

40 kPa

m=160 kPa

120 kPa

80 kPa

Fig. 2. Pullout curves in rigid soil for different values (a) of the interface stiffness KIand (b) of the bond strength sm (based upon analytical solution except for the casewith Ki = 107 kPa and sm = 160 kPa, for which also numerical results are presented).

Table 1Parameter values assumed in the numerical pullout tests.

Computational case

Rigid soil Elasto-plastic soil

(i) (ii) (iii) (iv)

Parameters for the boltsInterface cohesion cI (kPa) 40160 1 10 10 0Interface angle of friction uI () 0 0 25 0 25Borehole diameter d (m) 0.1Bolt Youngs modulus Eb (GPa) 20

Additional parameters for the cable elements (Fig. 5)Interface shear stiffness KI (kPa) 103109 107

Bolt area Ab (m2) 0.00785

Additional parameters for the solid bolt elements (Fig. 4)Interface shear stiffness KsI (kPa/m) n/a 109

Interface normal stiffness KnI (kPa/m) n/a 109

Bolt Poissons number mb () n/a 0.25

Parameters for the soilYoungs modulus E0 (MPa) n/a 400Poissons number m () n/a 0.3Friction angle u0 () n/a 25Cohesion c0 (kPa) n/a 10Dilatancy angle w0 () n/a 0

Initial stress fieldInitial stress r0 (kPa) n/a 30

90 P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 87982.3. Pullout test in elasto-plastic soil

2.3.1. Numerical modelsAs mentioned above, the one-dimensional cable element does

not take account geometrically of the real diameter of the groutedbolt. The diameter is taken into account only indirectly, in thedetermination of the maximum shear force at the interface be-tween the grouted bolt and the soil. In order to show the effectub

ub

rr

r

r

rr

1 m

Soil

Soil

1 m

(a)

(b)

Fig. 3. Problem layout for the numerical pullout test in elasto-plastic soiof this simplification, we carried out comparative numerical pull-out tests in respect both of a model using elastic solid elementsfor the bolts (Fig. 3a) and a simplified model with one-dimensionalelastic cable elements (Fig. 3b). The two models are equivalent interms of the bolt stiffness and the maximum shear force at theinterface between the soil and the grouted bolt. The parametersare given in Table 1.

Fig. 3 shows the geometry and the boundary conditions of themodels under consideration. They are the simplest possible modelsfor investigating the interaction between a single bolt and the sur-rounding soil and they have some similarity to the situation at thetunnel face. The unsupported right vertical boundary is free tomove and corresponds in a very idealised way to a portion ofthe tunnel face (see inset in Fig. 3, top-right corner). The oppositevertical boundary, which is constrained with respect to axial dis-placement, corresponds ideally to a portion of the sliding surface.For the sake of simplicity (and contrary to what happens at thetunnel face, where the soil moves towards the excavated tunnel,while the bolts remain fixed if bond failure occurs in the activeanchorage zone, i.e. inside the sliding wedge), the models ofFig. 3 produce the relative motion between soil and bolt by fixingthe axial soil displacements and moving the bolt head in the oppo-site direction. In order to simplify the problem and to gain a betterunderstanding of the numerical results, a cylindrical computa-tional domain under a uniform radial confinement pressure wasconsidered. The axial stress rz was taken equal to zero.

The soil around the grouted bolt was modelled as a hollowthick-walled cylinder (in the case of solid bolt elements) or as a fullcylinder (in the case of one-dimensional cable elements). Thematerial behaviour was taken as linearly elastic, perfectly plasticwith the MohrCoulomb yield criterion. The soil parameters are gi-ven at the bottom of Table 1.

The interface between the grouted bolt and the soil wasmodelled by elements describing a cylindrical surface for the cased=0.1 m

3 m

Bolt - Cable elements

Bolt - Solid elementsInterface elements

l with bolts modelled (a) by solid elements or (b) by cable elements.

Fig. 5. Numerical discretisation in the spatial analysis with cable elements. (a)Finest grid and (b) coarsest grid.

P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798 91of solid bolt elements (Fig. 3a). For the other case (one-dimensionalbolt elements, Fig. 3b), the behaviour of the interface is mathemat-ically incorporated into the numerical formulation of the cable ele-ment. In both cases, the interface behaviour was taken to be rigid plastic with the MohrCoulomb yield criterion. The rigidity wasmaterialised by assuming a high value of interface stiffness (cf. Ta-ble 1). The computations were carried out for different values ofinterface shear strength parameters (cI, uI) in order to analyse bothof the failure modes that may occur during bolt pullout: the shearfailure in the soil around the bolt and the shear failure along theinterface between the grouted bolt and the soil. In order to enforcethe model for reproducing the first case, the interface shearstrength was taken to be equal to infinity. For the second case,we considered interface strength parameters lower than those ofthe soil. The particular case of interface shear strength parametersequal to those of the soil was analysed as well (see columns (i)(iv)of Table 1).

The numerical solution was carried out using the FLAC2D finitedifference code in the case of solid bolt elements (the axisymmet-ric model, Fig. 4) and FLAC3D in the case of cable elements (Fig. 5).Due to symmetry, the numerical discretisation under FLAC3D takesaccount of only one quarter of the cylinder and the grid points onthe two symmetry planes are fixed in the normal direction. Fur-thermore, the stiffness- and strength parameters of the cable ele-ments were taken to be equal to one quarter of the actual values.

In order to investigate mesh dependency effects, several numer-ical discretisations with more or less coarse grids (i.e. with differ-ent grid sizes e close to the bolt) were considered in bothanalyses. Figs. 4 and 5 show the finest and the coarsest grids underconsideration for the cases with solid bolt elements and cable ele-ments, respectively.

Every analysis starts with the initialization of the stress stateand proceeds with the numerical pullout test: gradual impositionof displacements up at the head of the bolt (i.e. to the grid-pointsor to the structural nodes depending on whether solid bolt or cableelements, respectively, are used) and calculation of the reaction ax-ial forces Fp.2.3.2. Results for bolt modelling by solid elementsFig. 6a shows the maximum pullout force Fpmax as a function of

the radial grid size e of the elements adjacent to the bolt. The upperline (1 in Fig. 6a) was derived assuming that the interface shearstrength is equal to infinity. Line 2 was calculated consideringinterface shear strength parameters equal to those of the soil,while the lower lines 3 and 4 assume lower interface strengthparameters.Fig. 4. Numerical discretisation in the axisymmetric analysis wiAccording to Fig. 6a, line 1, the maximum pullout force Fpmax in-creases linearly with the grid size e. This happens only in the caseof line 1, i.e. only if the limit state is associated with failure of thesoil (infinite interface shear strength). As explained below, the gridsize dependency of the maximum pullout force Fpmax is due to theuniformity of the stress field inside each element and to the rela-tionship between element stresses and nodal forces.

Consider (for the sake of simplicity) the case of a purely cohe-sive soil. If the interface strength is infinite, the pullout will causeshear failure of the first row of soil elements next to the bolt (seeFig. 7, elements a, b, c, . . .). At the limit state, the shear stresssry inside each element of the first row will be equal to the soilcohesion c. In axisymmetric, numerical analyses, the element con-tributions to the nodal forces are calculated considering the aver-age radius of every element. They depend, therefore, not only onthe inner element radius ri (=d/2) but also on the outer radius ro(=d/2 + e) and thus on the grid size e. In the present case (e.g., forelement a in Fig. 7),th solid bolt elements. (a) Finest grid and (b) coarsest grid.

(a)

0 0.05 0.1 0.15 0.2 0.25

e [m]

0

10

20

30

40

50

Fpm

ax [k

N]

(b)

0 0.04 0.08 0.12 0.16e [m]

0

10

20

30

40

50

Fpm

ax [k

N]

c'=10 kPa '=25

c'=10 kPa '=25

2 cI=c' I='

3 cI=0 I='4 cI=c' I=0

1 cI= I=0

1 cI= I=0

2 cI=c' I='

3 cI=0 I='4 cI=c' I=0

Fig. 6. Maximum pullout force as a function of the grid size of the soil elementsadjacent to the bolt for bolt modelling (a) by solid elements or (b) by cableelements.

92 P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798Fa1;y Fa2;y cl2pri ro

2 clpd e clpd 1 e

d

: 1

The pullout force increases linearly with the grid size e, becauseit is equal to the sum of the contributions Fa1,y, Fa2,y, Fb1,y, Fb2,y, . . . ofthe elements to the forces of the boundary nodes of the bolt. Notethat according to Eq. (1), the spatial discretisation of the problemincreases apparently the effective diameter of the grouted boltsfrom d to d + e, i.e. by the factor 1 + e/d. Consequently, the grid sizee has to be selected sufficiently small relatively to the bolt diame-ter in order to reduce the discretisation-induced error. Only theideal condition e = 0 would allow to reproduce the actual failureof the soil adjacent to the interface.

The erroneously high pullout force caused by the apparent in-crease in bolt diameter is practically irrelevant, if the interface failsbefore the soil. This is why the mesh-sensitivity disappears, if theinterface strength is equal or lower than the strength of the soil(curves 2, 3 and 4 in Fig. 6a).

For the smallest considered grid size (e = 0.001 m), the infiniteinterface strength model (line 1) leads expectedly to the sameFig. 7. Explanatory drawing on the mesh dependency of element nodal forces.maximum pullout force like the model with interface elementshaving the shear strength parameters of the ground. Consequently,there are practically two possibilities for modelling bond failure: (i)either selecting a very fine discretisation close to the bolt (andrefraining from modelling the interface explicitly) or (ii) selectinga coarse grid for the soil in combination with interface elementsaccounting for the shear strength of the soil.

The maximum pullout force calculated under the assumption ofa purely cohesive interface (line 4) is equal to the expected value(Fpmax = pdL0cI = p 0.1 3 10 kN = 9.4 kN). In the case of fric-tional interface (lines 2 and 3), however, the numerical maximumpullout force (about 18 and 10 kN for case 2 and 3, respectively) islower than the force that one might expect on the basis of the pre-scribed confining stress of 30 kPa (Fpmax = pdL0(cI + rrtanuI) = 22.6 kN and 13.2 kN). The reason is that the radial stressesalong the interface decrease during to the pullout.

2.3.3. Results for bolt modelling by one-dimensional cable elementsThe behaviour of the one-dimensional cable elements exhibits

some similarities to (but also some differences from) the behaviourof the solid bolt elements (Fig. 6b). Consider again line 1, which ap-plies to the case of infinite interface strength. The reason for theobservedmesh sensitivity is exactly the same like before: the nodalforces resulting from the internal element stresses are proportionalto the average radius, which in the present case is equal to e/2.Assuming for the sake of simplicity a purely cohesive material,the following equation applies for the example of Fig. 7 insteadof Eq. (1):

Fa1;y Fa2;y cl2pri ro

2 clpe clpd e

d

: 2

According to this equation, the effect of the spatial discretisation isequivalent to an apparent change of the bolt diameter from d to e(or by the factor e/d). Contrary to the solid bolt elements, wherethe discretisation always increases their apparent diameter, theapparent diameter in the case of cable elements may be bigger orsmaller than the actual diameter depending on whether e > d ore < d. This is why the cable element exhibits mesh sensitivity evenif assuming a low interface strength (lines 24 in Fig. 6b): there isalways such a small grid size, that the pullout force according toEq. (2) becomes lower than the actual pullout force. The resultsdo not depend on the grid size only if the shear failure occurs atthe interface rather than in the soil and this happens only if the gridis sufficiently coarse. The critical grid size ecr increases with theinterface strength (cf. lines 2, 3 and 4 in Fig. 6b). If the interfaceshear strength parameters are equal to those of the soil (curve 2),the critical grid size ecr is equal about to the bolt diameter d (cf.Eq. (2)).

It is, however, thoroughly possible that the interface exhibits aconsiderably higher strength sm than the soil due to the effects ofsoil dilatancy (Luo et al., 2000; Wang and Richwien, 2002) or ofgrouting pressure (Yin et al., 2012). In this case, the critical gridsize ecr may be much more bigger than the bolt diameter d. It iseasy to show, that the following condition must apply in order toavoid mesh sensitivity:

e >sm

c0 rr tan /0d: 3

This result is important with respect to the stress analysis of thereinforced face, because it means that the calculation grid shouldbe sufficiently coarse at the tunnel face in order to avoid mesh sen-sitivity and to map correctly the bolt behaviour. A very coarse gridis, nevertheless, in general computationally problematic with re-spect to the overall face stability problem, because it cannot mapthe displacement field and the localisation of the shear strain closeto the limit state.

Table 2Parameter values assumed in the stress analysis of the reinforced tunnel face.

Cable elementsLength L0 (m) 7Area Ab (m2) 0.00785Youngs modulus Eb (GPa) 20Borehole diameter d (m) 0.1Interface shear stiffness KI (kPa) 107

Interface cohesion cI (kPa) c0

Interface angle of friction uI() 25

SoilYoung modulus E0 (MPa) 400Poisson ratio m () 0.3Friction angle u0 () 25Cohesion c0(kPa) 015Dilatancy angle w0 () 0

Initial stress field (lithostatic)

P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798 93Another potential problem is associated with the frictional partof the interface shear resistance. In the case of finite interfacestrength and of a sufficiently coarse grid, the maximum pulloutforce of the cable element (Fig. 6b) is higher than the force of thesolid bolt element (Fig. 6a), if the interface has a frictional resis-tance (lines 2 and 3). This difference does not exist in the case ofpurely cohesive interface (line 4). The reason is that the radialstress acting upon the bolt decreases during pullout in the caseof the solid bolt elements (see Section 2.3.2) but remains equalto the far-field confining stress in the case of the cable elements.

In conclusion, mesh-sensitivity of the numerical results orinability to map adequately the failure mechanism as well as inac-curate consideration of the frictional component of the pulloutforce are potential problems of disregarding the actual geometryof the bolts in the numerical model.Coefficient of lateral stress k0 () 0.57Unit weight c (kN/m3) 173. Stress analysis of the reinforced tunnel face

3.1. Problem layout and computational model

The authors presented results of simplified numerical stressanalyses of the tunnel face, which rather than modelling the boltsand their interaction with the surrounding soil individually approximate the effect of face reinforcement by means of an equiv-alent face support pressure (Perazzelli and Anagnostou, 2011). Likein Perazzelli and Anagnostou (2011) we consider also here a tunnelhaving a square, 100 m2 big cross-section excavated throughhomogeneous soil at a depth of 23 m (Fig. 8). The reason for theunrealistic cross-section is that the numerical results shall be com-pared to those of the limit equilibrium method of Anagnostou andSerafeimidis (2007), which considers the simplified model of arectangular face. The authors investigated the effect of the shapeof the tunnel cross-section on the numerical face stability assess-ment and found that it is of rather secondary importance as longas the cross-sectional area is constant (Perazzelli and Anagnostou,2011).

A low-strength, cohesive-frictional ground and dry conditionsare considered. The soil is modelled as a linearly elastic, perfectlyplastic material obeying the MohrCoulomb yielding criterion witha non-associated flow rule. All calculations were carried out for afriction angle u0 of 25, a unit weight c of 17 kN/m2 and cohesionvalues c0 between 0 and 15 kPa. Table 2 shows all materialFig. 8. Computational domain and boundary conditions.constants as well as the assumptions concerning the initial stressfield. According to Perazzelli and Anagnostou (2011) the dilatancyangle w0: and the coefficient of lateral stress k0 do not affect facestability significantly.

The bolts were modelled individually using the cable elementsof the finite difference code FLAC3D. Table 2 includes also the as-sumed parameter values for the cable elements. The tensilestrength of the bolts was taken infinite in order to avoid tensilefailure and to focus to the interaction between grouted bolt andsoil. The behaviour of the interface was taken rigid-plastic byassuming a high value of the interface stiffness. Concerning thebond strength, the assumption was made that it depends on theconfining stress according to MohrCoulomb criterion withstrength parameters equal to those of the soil. This assumption isreasonable in the case of soft soil with non-dilatant behaviour(cf., e.g., Milligan and Tei, 1998) and bolts grouted at relativelylow pressure (a high grouting pressure would increase the radialstress in the soil around the bolt and thus also the frictional resis-tance of the bond).

The bolt lengths L0 were taken equal to 7 m, which corresponds,for example, to the minimum length of initially 12 m long boltsoverlapping by 5 m in the longitudinal direction. The assumedgrouted borehole diameter d of 0.1 m is typical for face reinforce-ments. The bolts are horizontal and uniformly distributed overthe face on a rectangular grid. Four different bolting patterns wereinvestigated, consisting of 3 6, 5 10, 6 12 or 7 14 bolts inthe modelled half face and corresponding to bolt spacings of0.71.7 m or to reinforcement densities n of 0.36, 1.00, 1.44 and1.96 bolts/m2, respectively.

Fig. 8 shows the dimensions of the computational domain andthe applied boundary conditions. Due to the vertical symmetryplane, only one half of the entire domain needs to be discretised.The displacements of the vertical model boundaries parallel andperpendicular to the tunnel axis are fixed in the direction x andy, respectively. All displacement components are fixed at the bot-tom model boundary. Square brick elements with size equal to0.125 m are used to discretise the face. This grid size was selectedon the basis of the conclusions of Section 2 concerning the possibil-ity of mesh dependent results using cable elements. The choice of astructured mesh ensures reproducibility of the numerical results.

In order to avoid rather secondary effects and to focus directlyto the tunnel face behaviour, the details of the excavation andsupport sequence were not simulated. Instead a simplified excava-tion scheme is adopted, where the position of the tunnel face re-mains fix in the model and the excavation process is simulatedby a gradual reduction of the longitudinal horizontal stress actingon the face. The tunnel support is considered rigid and extends up

Table 3Stress analysis results.

n (bolts/m2) c0equil (kPa) c0non-equil (kPa) c

0lim (kPa)

0.36 14 12 131.00 12 10 111.44 10 8 91.96 7 5 6

94 P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798to the face. The analysis consists of the following steps: (i) initial-ization of the in situ stress, considering a lithostatic distribution;(ii) removal of the brick elements representing the excavated tun-nel volume, fixing of the grid-points on the tunnel boundaries andcalculation of the reaction grid-point forces; (iii) replacement ofthe restraint conditions for the grid-points on the face by supportforces equal to the reactions; (iv) insertion of the cable elements;(v) gradual reduction of the support forces at the face grid-points,equivalent to a reduction of the original horizontal stress by a fac-tor, which is constant along the face.

For each reinforcement pattern the face stability was investi-gated for a series of closely spaced cohesion values in order todetermine an upper and a lower bound of the minimum cohesionneeded for stability for the given reinforcement pattern (Table 3).The limit cohesion c0lim was taken as the mean value of thesebounds (last column of Table 3).3.2. Model behaviour close to the limit equilibrium

The typical model behaviour will be discussed on the basis ofthe numerical results for a reinforcement by 1.44 bolts/m2 and acohesion c0 of 10 kPa, which is only slightly higher than the mini-mum cohesion required for stability (Table 3). Fig. 9 shows the ax-ial displacement and the minimum principal stress of the soily

-1.77E-02-1.2E-02-8.0E-03-2.0E-032.94E-03

Longitudinal soildisplacement uy(m)

lasticPlasticPlastic in the past

Interface condition

-8.7E+05-8.0E+05-6.5E+05

-5.0E+05-3.5E+05-2.0E+05-5.0E+04-5.3E+03

Minimum principalstress (Pa)

y

z

E

Fig. 9. (a) Contour lines of the minimum principal stress (compressive stresses are negatuy of the soil ahead of the face (yz-plane) and condition of the interface between bolt anof the longitudinal displacement uy of the soil at the face (zx-plane) (n = 1.44 bolts/m2ahead of the face as well as the interface condition along selectedbolts. For the same bolts, Fig. 10a and b show the distributions ofthe interface shear stress and of the axial tensile force, respectively.

According to Fig. 9b the boltsoil interface fails close to the face.The mobilised interface shear stress is, however, very low(Fig. 10a), because the confining stresses decreases strongly inthe vicinity of the face due to the soil plastification associated withthe axial stress release (Fig. 9a). As a consequence of the low bondstrength, the axial loads developing in the cable elements (Fig. 10b)are considerably lower than the limit loads of fibreglass bars,which are typically applied as a face reinforcement (200500 kN).3.3. Comparison with limit equilibrium method

The black circular markers in Fig. 11 show the relationship be-tween the limit cohesion c0lim and bolt density n according to thenumerical stress analyses. The other markers have been obtainedby the computational method of Anagnostou and Serafeimidis(2007). This method approximates the tunnel face by a rectangle(of height H and width B) and considers a failure mechanism thatconsists of a wedge at the face and of the overlying prism up tothe soil surface (Fig. 12). At limit equilibrium the prism load isequal to the bearing capacity of the wedge. The prism load is cal-culated on the basis of silo theory, while the bearing capacity ofthe wedge is calculated by considering the equilibrium of an infin-itesimal slice. Both the load of the prism and the bearing capacityof wedge depend on the inclination of the inclined slip plane. Thecritical inclination x of the inclined slip plane is determined itera-tively so that it maximises the support requirements. As explainedin detail in Anagnostou (2012), this method represents animprovement of the model of Anagnostou and Kovri (1994) inthat it eliminates the need for an a priori assumption of thedistribution of the vertical stress rz in the wedge and offers the0 0.0160

2

4

6

8

10

uy [m]

x

zz

(b)z [m]

(c) (d)

(a)

ive) ahead of the face (yz-plane); (b) contour lines of the longitudinal displacementd soil for selected bolts; (c) extrusion profile of the tunnel face; and (d) contour lines, c0 = 10 kPa).

y [m]

-160

-80

0

80

160

I[k

Pa]

z=1.3 mz=2.9 mz=4.6 mz=6.3 mz=7.9 mz=9.6 m

8 6 4 2 0

8 6 4 2 0 y [m]

0

20

40

60

80

100

Nb

[kN

]

(a)z

y

x=0.875 m

(b)

Fig. 10. Distribution (a) of the shear stress at the boltsoil interface and (b) of the axial bolt force along selected bolts (n = 1.44 bolts/m2, c0 = 10 kPa).

0 4 8 12 16 20c' [kPa]

0

0.5

1

1.5

2

2.5 Numerical stress analysisLimit equilibrium analysis with bondstrength according to Fig. 14 andconfining pressure for =45-'/2Limit equilibrium analysis with bondstrength according to Fig. 14 andconfining pressure for =0

'=25

n[b

olts

/m2 ]

Fig. 11. Necessary bolt density as a function of soil cohesion according to the stressanalysis of the reinforced face (Table 3) and to the model of Anagnostou andSerafeimidis (2007).

H=10 m

h=23 m

B=10 m

xy

zz

Fig. 12. Failure mechanism after Anagnostou and Serafeimidis (2007).

P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798 95possibility of analysing a layered ground with an arbitrary distribu-tion of reinforcement.

The stabilizing effect of the bolts is considered as a supportpressure s given by:s n MinNby;pdsma;pdsmL0 a; 4where Nby is the limit tensile force of the bolt, while a and (L0 a)denote the bond length inside and outside the wedge, respectively.Note that the bond lengths a and (L0 a) vary over the height of thewedge and, moreover, they depend also on the specific mechanismconsidered, i.e. on the angle x. Consequently, the support forceoffered by the reinforcement will not be uniformly distributed evenin the case of constant bolt spacing (Fig. 13).

The necessary reinforcement density n can be determined as afunction of the soil cohesion for fixed values of the parameters d,L0 and sm. The geometric parameters of the bolt diameter andlength d and L0 are the same as in the stress numerical analysis.The bond strength sm is determined taking account of the shearstrength parameters of the soil and the confining stress on thebolts. One can make a consistent assumption about the confiningstress (and thus also about the bond strength sm) by considering

Fig. 13. Example of support pressure distribution in the limit equilibrium methodof Anagnostou and Serafeimidis (2007).

0 4 8 12 16 20

c' [kPa]

0

20

40

60

m[k

Pa]

=45-'/2=0

z=H

z=silo()

x=wz

m=c'+[0.5(w+1)z]tan''=25

Fig. 14. Bond strength sm determined based upon the shear strength parameters ofthe soil and assuming a trapezoidal confining stress distribution according toAnagnostou and Kovri (1994).

(b)

0 1 2 3 4 5y [m]

0

50

100

150

200

250

Nb

[kN

]

(c)

0 1 2 3 4 5y [m]

0

10

20

30

40

50

60

T b [k

N/m

]

(a)

0 1 2 3 4 5y [m]

0

2

4

6

8

10

u b[m

m]

upy

y=0

L'

ub

NbTbdy Nb-Tbdy

dy

ypl

ub>R/KI

ub

P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798 97The predictions of the limit equilibrium method of Anagnostouand Serafeimidis (2007) are very close to those obtained by thenumerical stress analysis.

Appendix A. Analytical derivation of the pullout curve for rigidsoil

Under the rigid soil assumption, the relative displacement be-tween bolt and soil is equal to the displacement of the bolt ub.The bond shear force per unit length of the bolt reads then asfollows:

Tby KIuby; if ub < RKI ; andR; if ub P RKI ;

(A:1

where y denotes the spatial coordinate on the axis of the bolt(Fig. A.1). The maximum bond shear force

R pdsm; A:2

where d denotes the grouted bolt diameter and sm the maximumshear stress at the interface with the surrounding soil. The shearforce Tb is related to the axial force Nb of the bolt via the equilibriumcondition

Tb dNbdy

; A:3

while the axial force Nb is related to bolt axial strain ey and thus onthe displacement:

Nb eyEbAb dubdy

EbAb; A:4

where EbAb is the axial stiffness of the bolt.

A.1. Before bond failure

Let us consider first the case of elastic soilbolt interface(i.e. ub < R/KI for each y). From Eqs. (A.1), (A.3), and (A.4) followsthat

d2ubdy2

Iub; A:5

where

I KIEbAb

: A:6

This is a differential equation for the axial displacement of thebolt. Its solution for the boundary conditions

ub up at y 0 bolt head andNb 0 at y L0 bolt end

8>: ; A:7reads as follows:

uby upeL

0yffiffiI

p eL0y

ffiffiI

p

eL0ffiffiI

p eL0

ffiffiI

p : A:8

From Eqs. (A.4) and (A.8) we obtain the distribution of the axialforce along the bolt,

Nby EbAbupffiffiI

p eL0yffiffiI

p eL0y

ffiffiI

p

eL0ffiffiI

p eL0

ffiffiI

p ; A:9

and with y = 0 the initial linear part of the pullout curve:

Fp EbAbffiffiI

p eL0ffiffiI

p eL0

ffiffiI

p

eL0ffiffiI

p eL0

ffiffiI

p up: A:10A.2. Bond failure

The distribution of the interface shear force along the bolt be-fore failure is given by the following equation:

Tby KIuby KIupeL

0yffiffiI

p eL0y

ffiffiI

p

eL0ffiffiI

p eL0

ffiffiI

p : A:11

One can readily verify, that the maximum shear force occurs atthe bolt head (see also Fig. A.1). So, with increasing values of thehead displacement, bond failure will start at the bolt head (whenup = R/KI) and propagate afterwards towards the deepest point ofthe bolt. Let the symbol ypl denote the extent of the bolt sectionwith failed bond (Fig. A.1).

A.3. Elastic section of the bolt

In the deeper, elastic bolt section (i.e. for ypl 6 y 6 L0), Eq. (A.5) isstill valid. Its solutions for the boundary conditions

ub RKI at y ypl boundary of the section with failed bond andNb 0 at y L0 bolt end

(;

A:12

reads as follows:

uby RKI

eL0y

ffiffiI

p eL0y

ffiffiI

p

eL0ypl

ffiffiI

p eL0ypl

ffiffiI

p ypl 6 y 6 L0: A:13

From Eqs. (A.4) and (A.13) we obtain the distributions of the axialforce in the elastic section of the bolt as well as the axial force atthe boundary ypl:

Nby RffiffiI

p eL0y

ffiffiI

p eL0y

ffiffiI

p

eL0ypl

ffiffiI

p eL0ypl

ffiffiI

p ypl 6 y 6 L0; A:14

Nbypl RffiffiI

p eL0ypl

ffiffiI

p eL0ypl

ffiffiI

p

eL0ypl

ffiffiI

p eL0ypl

ffiffiI

p : A:15A.4. Section of the bolt with failed bond

In the bolt section with failed bond (0 6 y 6 ypl), the followingequation applies instead of Eq. (A.5):

d2ubdy2

REbAb

: A:16

The solution of this differential equation for the boundaryconditions

ub up at y 0 bolt head andNb Nbypl at y ypl boundary of the section with failed bond

(;

A:17

is

ub up R

EbAby y

2 ypl

1ffiffiI

p eL0ypl

ffiffiI

p eL0ypl

ffiffiI

p

eL0ypl

ffiffiI

p eL0ypl

ffiffiI

p

!0

6 y 6 ypl; A:18

while the axial force reads as follows:

Nby Rypl y Nbypl 0 6 y 6 ypl: A:19A.5. Extent of the failed bond section

The extent of the failed bond section ypl is derived from the con-dition of the continuity of the displacements at ypl. From Eqs. (A.13)

98 P. Perazzelli, G. Anagnostou / Tunnelling and Underground Space Technology 38 (2013) 8798and (A.18) with y = ypl, we obtain the following non-linear alge-braic equation for ypl:

up RKI

12Iy2pl

ffiffiI

p eL0yplffiffiI

p eL0ypl

ffiffiI

p

eL0ypl

ffiffiI

p eL0ypl

ffiffiI

p ypl 1 !

: A:20

This equation can be solved iteratively for given head displace-ment up. Once calculated ypl, the pullout force can be obtained fromEq. (A.19) with y = 0:

Fp Nb0 Rypl Nbypl: A:21

Alternatively, the pullout curve can be determined by consider-ing the extent of the failed bond zone ypl as the independentparameter and calculating the corresponding (up, Np) pairs withEqs. (A.20) and (A.21), respectively.

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Stress analysis of reinforced tunnel faces and comparison with the limit equilibrium method1 Introduction2 Modelling of the bolts in the numerical stress analysis2.1 Introduction2.2 Pullout test in rigid soil2.3 Pullout test in elasto-plastic soil2.3.1 Numerical models2.3.2 Results for bolt modelling by solid elements2.3.3 Results for bolt modelling by one-dimensional cable elements

3 Stress analysis of the reinforced tunnel face3.1 Problem layout and computational model3.2 Model behaviour close to the limit equilibrium3.3 Comparison with limit equilibrium method

4 ConclusionsAppendix A Analytical derivation of the pullout curve for rigid soilA.1 Before bond failureA.2 Bond failureA.3 Elastic section of the boltA.4 Section of the bolt with failed bondA.5 Extent of the failed bond section

References