(This is a sample cover image for this issue. The actual cover is not yet available at this time.) This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright
Transcript
(This is a sample cover image for this issue. The actual cover is not yet available at this time.)
This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies areencouraged to visit:
Tunnel face stability in heavily fractured rock masses that followthe Hoek–Brown failure criterion
Salvador Senent a, Guilhem Mollon b, Rafael Jimenez a,n
a Technical University of Madrid, Spainb Hong Kong University of Science and Technology, Hong Kong
a r t i c l e i n f o
Article history:
Received 16 April 2012
Received in revised form
8 November 2012
Accepted 2 January 2013
Keywords:
Tunnel face stability
Critical pressure
Limit analysis
Collapse mechanism
Hoek–Brown
Non-linearity
a b s t r a c t
A tunnel face may collapse if the support pressure is lower than a limit value called the ‘critical’ or ‘collapse’
pressure. In this work, an advanced rotational failure mechanism is developed to compute, in the context of
limit analysis, the collapse pressure for tunnel faces in fractured rock masses characterized by the Hoek–
Brown non-linear failure criterion. The non-linearity introduces the need for additional assumptions about
the distribution of normal stresses along the slip surface, which translate into new parameters in the limit
analysis optimization problem. A numerical 3D finite difference code is employed to identify adequate
approximations of the distribution of normal stresses along the failure surface, with results showing that
linear stress distributions along the failure surface are needed to obtain improved results in the case of
weaker rock masses. Test-cases are employed to validate the new mechanism with the three-dimensional
numerical model. Results show that critical pressures computed with limit analysis are very similar to
those obtained with the numerical model, and that the failure mechanisms obtained in the limit analysis
approach are also very similar to those obtained in small scale model tests and with the numerical
simulations. The limit analysis approach based on the new failure mechanism is significantly more
computationally efficient than the 3D numerical approach, providing fast, yet accurate, estimates of critical
pressures for tunnel face stability in weak and fractured rock masses. The methodology has been further
employed to develop simple design charts that provide the face collapse pressure of tunnels within a wide
variety of practical situations.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
One of the most important issues when designing and con-structing a tunnel is to ensure the stability of the tunnel face, astunnelling experience indicates that most tunnel collapses havetheir origin in stability problems at the face [1]. Much researchhas been conducted to develop methods to assess the ‘critical’ or‘collapse’ pressure, or, in other words, the minimum pressure thatneeds to be applied at the tunnel face to avoid its instability.A wide variety of collapse mechanisms have been proposedwithin the framework of limit equilibrium analyses, such asHorn’s model [2] and its later variations (see e.g., Refs. [3,4]) orthose proposed by Vermeer et al. [5] and Melis [6]. Model tests atthe laboratory and centrifuge tests have also been conducted tostudy this problem (see e.g., Refs. [7–10]) and numerical modelssuch as the finite element method (FEM) [11–13] and the discreteelement method (DEM) [14,15] have been employed as well.
Another methodology to study the stability of the tunnel faceis limit analysis. In this context, Davis et al. [16] proposed an
initial solution for cohesive materials, whereas Leca and Panet[17] and Leca and Dormieux [18] proposed upper and lowerbound solutions for cohesive-frictional materials. Recent researchhas been focused on the development of improved failure geo-metries (see e.g., Refs. [19,20]), leading to failure surfaces that aredeveloped point-to-point and that allow rotational failure modesthat affect the whole excavation front [21,22].
These latter mechanisms have been shown to provide good resultswhen compared to limit equilibrium and numerical solutions. Inprevious works, geotechnical failure has been modelled using thelinear Mohr–Coulomb (MC) criterion that is traditionally applied tosoils. The actual strength of rock masses, however, is well known tobe a non-linear function of stress level. Although some methodologieshave been recently proposed to compute ‘equivalent’ MC parametersthat depend on the stress level for 2D tunnelling analyses (see e.g.,Refs. [23,24]) or, using limit analysis, to study the stability of 2Dtunnel sections excavated in materials with a non-linear failurecriterion (see e.g., Refs. [25,26]), only a few limited attempts havebeen made to consider the non-linearity of actual failure criteriawhen computing the critical pressure of the tunnel face.
One approach to consider non-linear failure criteria in the limitanalysis literature has been to employ a linear failure envelope thatis tangent to the original, and non-linear, failure criterion [27], sinceupper-bound solutions obtained using such envelope are also
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/ijrmms
International Journal ofRock Mechanics & Mining Sciences
1365-1609/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijrmms.2013.01.004
n Correspondence to: ETSI Caminos, C. y P., C/ Profesor Aranguren s/n, Madrid
International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451
Author's personal copy
upper-bound solutions to the original problem with the non-linearcriterion [28]. For instance, the ‘‘generalized tangential technique’’[29,30] is based on considering a linear envelope that is tangent atan ‘optimum’ point. Huang and Yang [31] have used this techniqueto study the stability of the tunnel face employing the passive ‘blow-out’ mechanism of Leca and Dormieux [18] and a non-linearstrength criterion previously proposed by Agar et al. [32]. Theirresults illustrate the influence of the degree of non-linearity of thecriterion on the collapse pressure, although they are not comparedwith other numerical or analytical methods.
In this work, we study the face stability of circular tunnelsexcavated in heavily fractured and ‘low quality’ rock masses thatfollow the non-linear Hoek–Brown (HB) failure criterion. To thatend, the rotational mechanism recently proposed by Mollon et al.[22] has been generalized to consider the non-linearity of the HBfailure criterion, so that we can employ ‘instantaneous’ values ofcohesion and of friction angle that depend on the stress level,at the same time that we fulfill the assumption of associated flow.To validate the new methodology, we compare our limit analysisresults with the results of a 3D finite difference numerical code.Finally, we employ the new improved failure mechanism todevelop design charts that can be used to estimate the tunnelface critical pressure, for a wide variety of practical cases, and as afunction of the tunnel diameter and of the rock mass parameters.
2. An improved rotational tunnel face failure mechanismfor non-linear materials
2.1. Principles of the collapse mechanism of Mollon et al.
The analytical collapse mechanism developed by Mollon et al.[22], in the framework of the kinematical theorem of limitanalysis applied to MC soils, relies on two main assumptions:(i) the collapse involves the rotational motion of a single rigid
block around an axis (Ox) (x being the horizontal direction perpen-dicular to the tunnel axis, see Fig. 1a), and (ii) the collapsing blockintersects the whole circular surface of the tunnel face. Thesekinematic assumptions were made after observations of numerical[12] and experimental [8] simulations of face collapses. Besides thekinematic aspects, the normality condition related to the assump-tions of the kinematical upper-bound theorem of limit analysishas to be fulfilled. In the case of a frictional soil, with or withoutcohesion, this condition states that the normal vector pointingoutward of the slip surface should make an angle p/2þj with thediscontinuity velocity vector in any point of the discontinuitysurface, with j being the internal friction angle of the material.Even in a homogeneous MC soil, there is no simple surface which isable to satisfy both the kinematic and the normality conditions. Itwas thus necessary to use a complex discretization scheme togenerate the external surface of the moving block from close toclose, using a collection of triangular facets respecting locally thenormality condition. The method and equations used for thisgeneration are described in detail in Ref. [22] and will be brieflyrecalled here to make understandable the changes needed to adaptthis formulation to the HB criterion, as described in Section 2.2.
Two levels of discretization are needed, as shown in Fig. 1.The first level consists in discretizing the circular tunnel face inny points Aj and A0j (for j¼1 to ny/2, see Fig. 1b). The second levelof discretization consists in defining a number of radial planes calledPj, which all meet at the centre O of the rotational motion andare thus all normal to the velocity field. As shown in Fig. 1c,the mechanism may be divided in two sections. The so-calledSection 1 corresponds to the planes Pj with j¼1 to ny/2, so thateach of these planes Pj contains the points O, Aj, and A0j,whereas the so-called Section 2 corresponds to the planes Pj withj4ny/2. In Section 2, the planes are generated with a constantuser-defined angular step db until Point F, which is the extremityof the mechanism and which is so far unspecified. Thus, thesurface generation process only depends on two discretization
Fig. 1. Discretizations used for the generation of the collapsing block. (a) Reference system; (b) cross section with points at the tunnel boundary and (c) longitudinal
section along tunnel axis of the failure mechanism.
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451 441
Author's personal copy
parameters: ny and db. It is based on an iterative scheme which isdescribed in Fig. 2. As shown in this figure, if two points Pi,j andPiþ1,j, belonging to Plane Pj, are known, it is possible to define athird point Pi,jþ1 belonging to Plane Pjþ1, in such a way that thetriangular facet (Pi,j, Piþ1,j, Pi,jþ1) respects the normality condition;i.e., its normal vector makes an angle p/2þj with the velocityvector. This triangular facet is called Fi,j. Moreover, if a similaroperation is performed for points Piþ1,j and Piþ2,j, it is possible tocompute the position of a point Piþ1,jþ1 and, in turn, to define the‘reversed’ facet (Piþ1,j, Pi,jþ1, Piþ1,jþ1), which is called F0i,j (Fig. 2).This process, described in detail in Ref. [22], starts from the pointsbelonging to the tunnel face and stops at the extremity of themechanism. At the end of the generation, the lateral surface of thecollapsing block is thus defined by a collection of triangular facets Fi,j
and F0 i,j, which makes it possible to compute its volume, weight, etc.When the entire external surface of the moving block has been
defined, the critical collapse pressure can be computed by applyingthe main equation of the kinematic theorem of limit analysis,which states that the rate of energy dissipation in the system isequal to the rate of work applied to the system by the externalforces. In the present case, assuming that the mechanism does notoutcrop at the ground surface, the forces applied to the movingblock of ground at collapse are its own weight and the ‘collapse’face pressure sc. The only possible energy dissipation in the systemarises from the slip surface and is proportional to the cohesion c.
After applying the work equation and performing a fewsimplifications, the collapse pressure is given by Eq. (1), whereSj, Rj, bj, Ri,j, Vi,j, bi,j, Si,j, R0i,j, V0i,j, b0i,j, and S0i,j are geometrical termsdefined in Ref. [22].
sc ¼
gPi,j
ðRi,jV i,j sin bi,jþR0i,jV0i,j sin b0i,jÞ�c cos j
Pi,j
ðRi,jSi,jþR0i,jS0i,jÞP
j
ðSjRj cos bjÞ
ð1Þ
This expression, however, is only valid for the velocity fieldobtained with a given position of the centre of rotation O. Thebest value of sc that the method can provide is then obtained bymaximization of this expression with respect to the two geo-metric parameters that define this point O.
2.2. Modification of the mechanism in the case of a spatial
variability of j
One of the key steps for generation of the mechanism dis-cussed above is the computation of the coordinates of Point F,since it allows (i) to know at which index j the generation shouldbe stopped when the extremity of the mechanism has beenreached, and (ii) to define the radius rf¼OF of a circle which in
turn provides the positions of the points Cj. As shown in Figs. 1cand 2, these points are located in each plane Pj and are neededbefore starting the generation process. The computation of thecoordinates of F is thus crucial. In the case of a homogeneous MCsoil, this computation is straightforward since F is the intersectionof two logarithmic spirals of parameter tan j emerging from thefoot and the crown of the tunnel. These two curves are indeed theexact analytical boundaries of the mechanism projected in thevertical plane of symmetry of the tunnel.
The use of a non-linear failure criterion, however, may lead toa spatial variability of the friction angle and of the cohesion thatneed to be considered for generation of the mechanism. This isdue to the non-linearity of the failure criterion, since the ‘equiva-lent’ c and j employed in the analysis are actually functions ofthe normal stress on the slip surface at failure, and such normalstress might not have a constant value on the entire slip surface.This observation has a crucial effect on the procedure for genera-tion of the mechanism presented in Ref. [22] and shortlydescribed above, as such procedure needs to be modified to allowfor non-constant friction angles along the slip surface. That can beachieved with the following modifications to the procedure forgeneration of the mechanism:
– The position of F is more complicated to determine since thespatial variability of j prevents from using the method basedon the intersection of two logarithmic spirals described above.In the modified mechanism, the position of F is obtained usinga process based on the one proposed in Ref. [33] in a 2Dversion of this mechanism: instead of defining the limit curvesof the mechanism projected in the plane of symmetry of thetunnel by two logarithmic spirals, we define these curves as asuccession of segments which locally respect the normalitycondition, as shown in Fig. 3. More precisely, the curve
Fig. 2. Principle of the point-by-point generation of the lateral surface of the
mechanism.
Fig. 3. Numerical determination of the position of point F.
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451442
Author's personal copy
emerging from A (respectively from B) is composed of asuccession of points Aj (respectively Bj) belonging to the planesPj. The determination of the position of the point Ajþ1 fromthe point Aj belonging to Pj is based on the fact that (i) Ajþ1
belongs to Pjþ1 and (ii) the segment AjAjþ1 makes an angle p/2�j(Aj) with the velocity vector, with j(Aj) being the localvalue of j at the coordinates of Aj. The same method is used tocompute the position of Bjþ1 from the one of Bj. This iterativeprocess starts at the points A and B and stops when the twocurves meet at the extremity of the mechanism. The point F isdefined as the intersection of the two curves. More detailsabout this process may be found in Ref. [33].
– During the generation of a new point Pi,jþ1 from two existingpoints Pi,j and Piþ1,j of the previous plane, the normalitycondition of the facet Fi,j should be verified with respect tothe local value of the friction angle j(x,y,z). In the presentstudy, this local value is taken at the middle point of thesegment Pi,jPiþ1,j.
– The expression of the energy dissipation has to be generalizedbecause the values of cohesion and of friction angle aredifferent for each facet Fi,j and F0i,j. The result is Eq. (2), whereo is the angular velocity of the moving block. For the samereason, the critical pressure given by Eq. (1) is no longer valid,and it now has to be computed using a newly developedexpression presented in Eq. (3).
In these expressions, ci,j and ji,j (respectively c0i,j and j0 i,j) are thevalues of c and j at the centroid of Fi,j (respectively F0i,j). As for Eq. (1),Eq. (3) is only valid for a given position of the centre O of the rotationalmotion. Moreover, since j and c are now a function of the stress field,it is also only valid for a given stress field; i.e., for a given spatialdistribution of the normal stress on the slip surface. Thus, finding thecritical collapse mechanism will require to maximize Eq. (3) not onlywith respect to the position of O (two geometric parameters), but alsowith respect to the stress field. To simplify such optimization, it isconvenient to make some assumptions on the ‘expression’ of thisstress field so that it can be expressed by means of a limited numberof parameters, which may then be introduced in the optimizationprocess. Section 3.2 analyzes this problem with more detail, and it alsodiscusses the types of stress distribution considered in this work.
3. Application to fractured rock masses with the Hoek–Brownfailure criterion
3.1. Introduction of the HB criterion in the modified collapse
mechanism
The HB failure criterion is typically applied to fractured rockmasses [34], since it accounts for the observation that rock massstrength is a non-linear function of stress level. The HB criterioncan be expressed as
s01 ¼ s03þsci mb
s03sciþs
� �a
, ð4Þ
where s01 and s03 are the major and minor effective principal stressesat failure; sci is the uniaxial compressive strength of the intact rock;
s and a are parameters that depend on rock mass quality as given byGSI and by the disturbance factor (D); and mb is a parameter thatdepends on rock mass quality and rock type. See Ref. [34] for furtherdetails of the latest available version of the HB criterion.
The introduction of the HB criterion in the context of limitanalysis with the newly proposed collapse mechanism requires tobe able to compute the ‘equivalent’ values of c and j correspond-ing to a given value of the normal stress s0n acting normal to thefailure surface. It is therefore necessary to define the HB failureenvelope in the Mohr-plane, since the ‘equivalent’ values of c andtan j that will be employed in the limit analysis computationcorrespond to the intercept and slope of the tangent line to thisenvelope at s0 ¼s0n (Fig. 4). In this way, we are substituting theoriginal failure criterion by its linear envelope at each stressvalue. As shown by Ref. [28], solutions obtained using suchenvelope are still valid upper-bound solutions that maintain therigour of the limit analysis approach.
The HB limit curve is the envelope of the Mohr circles definedby Eq. (4) for all the values of s03. More precisely, with thenotations of Fig. 4, we can write that
tmaxðs0nÞ ¼maxs0
3
½tðs03,s0nÞ� ð5Þ
In this expression, the function t(s03, s0n) is obtained from theMohr-circle indicating failure conditions for a stress state withminor principal stress s03 (see Fig. 4), and thus depends on theexpression of the HB criterion provided in Eq. (4). The equation ofthis Mohr-circle verifies
ðtðs03,s0nÞÞ2þ s0n�
s01þs032
� �� �2
¼s01�s03
2
� �2
ð6Þ
Introducing Eq. (4) into Eq. (6) and simplifying, we get
Solving Eq. (8) analytically would directly provide a closed-formexpression for the HB criterion limit curve. However, due to thecomplexity of Eq. (9), this analytical solution seems out of reach.
Fig. 4. Determination of the ‘equivalent’ Mohr–Coulomb parameters.
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451 443
Author's personal copy
The problem of deriving analytical expressions for the HB failureenvelope in t–s space has attracted significant attention in theliterature. Ucar [35] employed the general derivation procedurepreviously proposed by Balmer [36] to develop a first set ofnumerical equations to derive the value of t corresponding to thefailure envelope of the ‘original’ HB failure criterion; i.e., witha¼1/2. Such work was later generalized by Kumar [37], whodeveloped a numerical solution for the failure envelope of the‘generalized’ HB criterion. More recently, Shen et al. [38] havedeveloped approximate analytical expressions for the ‘instanta-neous’ c and j of the strength envelope which have beenvalidated within specific ranges of the material properties.
In this work, Eq. (5) is solved numerically using the optimizationtool of MATLAB, which directly provides the numerical value of tmax
for a given value of s0n, but does not provide any information aboutthe slope and intercept of the tangent line to the limit curve at thepoint (s0n, tmax). To obtain accurate estimates of c and j for any valueof s0n, the limit curve is approximated by a piecewise-linear function.In other words, the values of tmax are computed numerically at a largenumber of s0n values over the whole relevant range of normalpressures; in this case, a constant ‘step’ of 0.1 kPa between twosuccessive values of s0n was considered. Then, the values of c and jcorresponding to each segment are computed in the middle point ofeach of these small intervals and stored in a table. Values of c and jfor any arbitrary s0n are computed by linear interpolation betweentwo successive values in this table. Note that, thanks to the very smallstep chosen for s0n, two successive values of c or of j in this table arealways extremely close. Two examples are provided in Fig. 5 for twosets of mechanical parameters. It clearly appears that, as expected,the non-linearity of the criterion leads to a reduction of the‘equivalent’ friction angle and to an increase of the ‘equivalent’cohesion when the normal pressure increases. Thus, small values ofthe normal stress may lead to very high friction angles, up to 701.Such table of c�j values combined with the interpolation methodprovides two functions c(s0n) and j(s0n) which have the advantage ofbeing very fast. This is very important since the algorithm ofgeneration of the collapsing block very often calls these functions.Indeed, they are actually needed several times for each facet, there aremore than 10,000 facets in each mechanism, and the mechanism
generally has to be generated more than 100 times during the processof optimization of the centre of rotation and of the stress field neededfor limit analysis.
3.2. Hypothesis about the stress field
As indicated above, we need to define the distribution of normalstresses along the failure surface. Theoretically, we could aim for avery flexible, i.e., with many degrees of freedom, distribution thatwould introduce no constraints in the quality of the solutionobtained; this approach, however, has the shortcoming that itincreases the dimensionality and hence the difficulty of the optimi-zation problem.
We employed Fast Lagrangian Analysis of Continua in 3Dimensions (FLAC3D) [39] to conduct numerical simulations thathelp us identify reasonable, yet simple, stress distributions at thetunnel face and, in particular, to compute the distributions ofnormal stresses along the failure surface. The details of thenumerical model are discussed below. Using FLAC’s internalprogramming language (FISH) it is possible to compute and depictthe normal stress distribution along the failure surface. As anexample, Figs. 6 and 7 reproduce our computed results in twoparticular cases: Fig. 6 represents a case with extremely poormass rock properties, and Fig. 7 represents a case in whichslightly better properties have been used; this causes a smallerfailure mechanism that does not extend above the tunnel crown.
As it can be observed the normal stresses along the failuresurface tend to increase with depth. In the case with ‘worse’ rockmass properties this variation is pronounced, whereas, in the casewith ‘better’ properties, is small. Other cases computed, that, for thesake of space, are not reproduced herein, present an intermediatebehaviour between these two situations. Based on such results, andto assess the influence of the type of stress distribution consideredon the results computed, we choose to employ two differentdistributions of normal stresses along the slip surface in this work:
(1) A uniform distribution, defined by a constant stress in theentire failure surface. With this distribution, the optimizationof the failure mechanism needs to be performed with respect
0 10 20 30 40 50 60 70 80 90 1000
102030405060708090
100
Tang
entia
l stre
ss [k
Pa]
Normal stress [kPa]
Mohr Plane
GSI = 20mi = 9.6σci = 5 MPaD = 0
0 10 20 30 40 50 60 70 80 90 1000
102030405060708090
100
Tang
entia
l stre
ss [k
Pa]
Normal stress [kPa]
Mohr Plane
GSI = 10mi = 4.0σci = 3 MPa
D = 0
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
φ [º]
Normal stress [kPa]
Mohr−Coulomb equivalent
0 50 1000
5
10
15
20
25
30
35
c [k
Pa]
φ
c
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
φ [º]
Normal stress [kPa]
Mohr−Coulomb equivalent
0 50 1000
5
10
15
20
25
30
35
c [k
Pa]
φ
c
parameters
parameters
Fig. 5. Hoek–Brown limit curve and corresponding ‘equivalent’ values of c and j for two heavily fractured rock masses.
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451444
Author's personal copy
to three parameters: two of them are geometrical and definethe position of the centre of rotation, and the third onedefines the constant value of the stress distribution.
(2) A linear distribution, defined by two parameters that representthe stress value at the tunnel crown and the stress verticalgradient. Hence, the optimization is performed with respectto four parameters: two geometrical parameters to define thecoordinates of the centre of rotation and two additionalparameters relative to the stress distribution.
4. Validation of the model
4.1. Description of test-cases
To validate the proposed methodology, we have used 22 test-cases to compare results obtained using our improved failuremechanism with results obtained using numerical models inFLAC3D. Table 1 includes a list of parameters (mi, sci, GSI and g)employed for the analyses of the test-cases considered. In all casesa 10 m diameter tunnel has been modelled with a cover of 20 mthat is enough to eliminate the possibility that the mechanismoutcrops at the ground surface. Also, a damage parameter of D¼0has been employed to represent TBM-excavated tunnels with a
minimal disturbance of the rock mass surrounding the tunnel. Asshown in Table 1, the different cases have been selected so thatthey correspond to low-quality rock masses (GSIr25), since, inreal practice, problems associated to tunnel face instability couldbe mainly expected in such low-quality rock masses.
4.2. Results with limit analysis
The collapse pressures obtained by applying the new mechan-ism described in Sections 2 and 3 are shown in Table 1. Resultscomputed employing the assumed uniform and linear stressdistributions along the failure surface are included as well. Forcases of ‘weaker’ rocks, in which the collapse pressure is higherthan 8 kPa, the linear stress distribution improves the results, i.e.,the collapse pressure is higher than that computed with theuniform distribution. Note that, being an upper-bound limitanalysis approach, and considering that the face pressure actsagainst the movement of the mechanism, any solution thatincreases the collapse pressure represents an improvement of theresults. In such cases with ‘worse’ rock mass properties, the increasewith depth of normal stresses along the failure surface is similar tothat shown in Fig. 6b; hence, using a linear distribution improves thepredictions. Therefore, the newly proposed generalized failuremechanism, which allows considering non-constant stress
0 0.03 0.06 0.09 0.12−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
Y−A
xis
[m]
Normal Stress [MPa]
Fig. 6. Distribution of normal stresses along the failure surface (Case mi¼5;
and (b) at the cross section along the vertical plane of symmetry.
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451 445
Author's personal copy
distributions, is needed in this case to improve previous approachesthat only consider a constant stress field obtained from an optimumtangency point to the failure criterion. In other cases, correspondingto rock masses with ‘better’ properties that produce very smallvalues of the critical pressure, and taking into account thedistribution shown in Fig. 7b and the results presented in Table 1,there is a negligible advantage in considering the additional para-meter needed for the linear distribution and the results provided bythe constant stress field are probably acceptable.
Fig. 8 shows two examples, corresponding to test-cases 2and 7, of the optimal failure geometries obtained using the newcollapse mechanism. Fig. 8a corresponds to a case with ‘worse’properties, and shows an instability that extends upwards in thevertical direction, whereas Fig. 8b represents a case with betterproperties and shows a minor instability that only affects a smallvolume at the tunnel face.
4.3. Numerical results with a finite difference code
To validate our methodology, results have been compared withnumerical simulations using the finite difference code FLAC3D.Fig. 9 illustrates the FLAC3D model employed for numericalsimulation of tunnel face stability. The tunnel considered has adiameter of 10 m and a cover of 20 m. To minimize boundaryeffects and the needed computational effort, the tunnel has beenrepresented using a symmetric model with dimensions, in metres,of 25�30�35. A total of 163,260 elements have been employed,of which 819 are in the tunnel front, and the mesh has beendesigned to minimize element sizes where larger stress gradientsare expected. The boundary conditions of the model are given byfixed displacements at the ‘artificial’ boundaries of the model, i.e.,at its lateral perimeter and at its base. Similarly, and since we areonly concerned with face stability and not with tunnel conver-gences, the tunnel support has not been included and displace-ments at the tunnel excavation boundary have also been fixed.
The constitutive model employed for the fractured rock mass iselastic–perfectly plastic with the HB failure criterion as implementedin FLAC3D. The elastic properties employed, which have a negligible
effect on the collapse pressure, are given by a Young’s modulus ofE¼400 MPa and a Poisson’s ratio of n¼0.30. The rock density wastaken equal to 2.5 t/m3. As shown in Table 1, the remainingparameters to define the failure criterion are specific of each test-case.
In the framework of limit analysis, the assumption of anassociated flow rule with the dilatancy angle c in any point equalto the friction angle j, is necessary for the derivation of thefundamental theorems of this theory and, therefore, to providerigorous bounds of the critical load. This associated flow hypothesis,however, does not necessarily represent the real behaviour ofgeotechnical materials so that, in general, the dilatancy angle tendsto be lower than the friction angle (see eg. Ref. [40]). While it is notpossible to get rid of this assumption in our analytical mechanism, itis possible to evaluate its influence on the results by dealing withtwo limit cases in our numerical simulations: an associated flowrule (c¼j) and a zero-dilatancy behaviour (c¼0). As indicated byRef. [33], the key task is to evaluate the validity of this assumptionfor the problem studied. Numerical results provided in Table 1 showthat this assumption has actually a limited influence on the value ofthe critical collapse pressure, so that the theoretical background oflimit analysis is therefore acceptable in this case.
To find the collapse pressure of the tunnel face, the bisectionmethod proposed by [41] has been employed. Using this method,and for a given interval of pressure values, defined by one highervalue for which the face is stable and by one lower value for which itis unstable, the stability of the face is computed assuming the meanvalue; if the face is stable, the higher interval boundary is sub-stituted by such mean value (the lower interval value is substitutedif unstable), and the process is iteratively repeated for such newlydefined intervals until the required precision is achieved, which inthis case has been set to 0.1 kPa. To assess the stability of the tunnelface for each face pressure considered, we use a methodology that issimilar to that employed by FLAC3D for the determination of safetyfactors (refer to FLAC3D manual for details [39]) and that depends onthe ‘‘representative number of steps (N)’’ which characterizes themodel behaviour. To obtain N, we set an elastic behaviour in themodel; then, from an equilibrium state, and doubling the internalstresses in the elements, we obtain the number of steps needed to
Table 1Test-cases considered for validation.
Case Parameters Collapse pressure (kPa)
mi sci (MPa) GSI g (t/m3) Limit analysis Numerical model Difference limit analysis—numerical
model (associated flow)
Uniform Linear Associated flow Non-associated flow
1 5 1 10 2.5 49.5 52.0 61.9 62.2 9.9
2 5 1 15 2.5 34.9 36.8 38.6 39.3 1.8
3 5 1 20 2.5 25.6 26.8 26.3 27.7 �0.5
4 5 1 25 2.5 19.1 20.3 18.9 21.1 �1.4
5 5 5 10 2.5 16.4 17.1 15.3 15.4 �1.8
6 5 5 15 2.5 9.4 9.8 8.3 12.5 �1.5
7 5 5 20 2.5 5.2 5.2 5.3 6.1 0.1
8 5 5 25 2.5 2.3 2.3 2.4 2.3 0.1
9 5 10 10 2.5 8.4 9.1 7.7 10.9 �1.4
10 5 10 15 2.5 3.4 3.4 2.7 4.2 �0.7
11 5 10 20 2.5 0.0 0.0 0.0 0.0 0.0
12 5 10 25 2.5 0.0 0.0 0.0 0.0 0.0
13 5 15 10 2.5 5.2 5.2 4.8 6.9 �0.4
14 5 15 15 2.5 0.1 0.1 0.3 0.0 0.2
15 5 15 20 2.5 0.0 0.0 0.0 0.0 0.0
16 5 15 25 2.5 0.0 0.0 0.0 0.0 0.0
17 5 20 10 2.5 2.3 2.3 2.6 3.0 0.3
18 5 20 15 2.5 0.0 0.0 0.0 0.0 0.0
19 5 20 20 2.5 0.0 0.0 0.0 0.0 0.0
20 5 20 25 2.5 0.0 0.0 0.0 0.0 0.0
21 10 1 10 2.5 26.3 27.6 26.4 28.4 �1.2
22 15 1 10 2.5 18.1 18.9 17.1 20.1 �1.8
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451446
Author's personal copy
restore the model to equilibrium. For our model, this number isbetween 15,000 and 20,000 steps; because of this, we set N¼25,000steps in all cases. To assess stability we consider that the real tunnelmodel is unstable if it does not converge in these N steps. As aconvergence criterion, the FLAC3D default value of 1EXP-05 for theunbalanced mechanical-force ratio has been reduced to 1EXP-07.
To check the results of our proposed limit analysis approachwith the results of numerical simulations, we have compared(i) the computed values of the collapse pressure at the tunnel face;and (ii) the shapes of the failure mechanisms obtained by bothmethods. Failure mechanisms in the FLAC3D model have beenestimated considering the distribution of shear deformations.
Table 1 and Fig. 10 present a comparison of collapse facepressures computed using the FLAC3D numerical model and usingour limit analysis approach with the improved collapse mechan-ism. To compare results that correspond to equal conditions, inTable 1 the limit analysis results are only compared to thenumerical results for the case of associated flow. In most cases,
we obtain very similar critical pressures for the numerical modeland for the rotational failure mechanism. Except for test-case 1,all differences are lower than 2 kPa, a value which can beconsidered acceptable for practical purposes as a validation ofthe new methodology. However, note that small differences in thenumerical results arise from the definition of the convergencecriterion, which depends on the selected number of steps; andon the mesh size since, as shown by [33], the predictions areexpected to improve as the mesh size is decreased. This makes thenumerical model to be not completely accurate and it alsoexplains why, in some cases, collapse pressures in the numericalmodel are lower than in the framework of limit analysis. Asmentioned above, test-case 1 shows the only remarkable differ-ence, 9.9 kPa, from 52.0 to 61.9 kPa. The rock mass properties forthis case are: sci¼1 MPa, mi¼5 and GSI¼10, which are the worstproperties considered in this work, and that are probably morerepresentative of a ‘soil like’ material. It is likely that the solutioncould be improved using additional parameters in the distribution
Fig. 8. Collapse geometries obtained by the new mechanism. (a) Case 2 (mi¼5; sci¼1 MPa; GSI¼15; D¼0; g¼2.5 t/m3; Diameter¼10 m) and (b) Case 7 (mi¼5;
sci¼5 MPa; GSI¼20; D¼0; g¼2.5 t/m3; Diameter¼10 m).
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451 447
Author's personal copy
of normal stresses along the failure surface; however, since thesematerials seem to represent ‘marginal’ cases, we choose not togeneralize the shape of the stress distribution with additionalparameters to reduce difficulties during optimization.
Furthermore, the shapes of the failures mechanisms obtained withour limit analysis approach and with the FLAC3D numerical simula-tions are in good agreement. As an example, Fig. 11 presents acomparison of the failure mechanisms for the same test-cases,2 and 7, that were shown in Fig. 8. It has to be noted that test-case 7 illustrates an example in which the mesh size had aneffect on the shape of the failure mechanism obtained withFLAC3D: if the model described above, with 819 elements in thetunnel front, was employed, the slip surface obtained wasslightly different to the one obtained with the limit analysismechanism and, in fact, suggested the possibility of a partial face
failure. However, when the number of elements at the face wereincreased to 5846 using a much finer mesh, the failure surfacepresented in Fig. 11b was obtained. It can be clearly observedthat the failure mechanism affects the whole tunnel face, andalso that the shape of the failure surface agrees well with theshape provided by our limit analysis approach.
Finally, it is important to emphasize that the newly proposed limitanalysis approach is significantly more computationally efficient. Asan illustration, the times required to calculate test-cases 2 and 7, onan Intel Xeon CPU W3520 2.67-GHz PC are, respectively, 67 and23 min for the limit analysis approach coded in MATLAB, whereas 26and 20 h are needed for the numerical simulations with FLAC3D.Moreover, using the model with the much finer mesh, which could benecessary to predict slip surfaces accurately, the time required tocalculate test-case 7 is higher than 180 h.
Fig. 9. Geometry of the developed numerical model.
Fig. 11. Comparison of failure mechanisms computed with the limit analysis and
with the numerical simulation: (a) Case 2 (mi¼5; sci¼1 MPa; GSI¼15; D¼0;
g¼2.5 t/m3; Diameter¼10 m) and (b) Case 7 (mi¼5; sci¼5 MPa; GSI¼20; D¼0;
g¼2.5 t/m3; Diameter¼10 m).
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451448
Author's personal copy
5. Design charts
After its validation, the newly proposed mechanism is used toproduce design charts to compute the critical collapse pressure ofa tunnel face excavated in heavily fractured and ‘low quality’ rockmasses that follow the HB non-linear failure criterion. Fig. 12shows the collapse pressure of the tunnel face (sc), divided by thediameter of the tunnel (Dm) and by the unit weight of the rockmass (g), as a function of the uniaxial compressive strength (sci),also divided by Dm and g, for different values of the mi parameter
and of the Geological Strength Index (GSI). As mentioned above,we consider null disturbance (D¼0) to represent TBM-excavatedtunnels with a minimal damage of the rock mass surrounding thetunnel. Note that ranges of parameters have been selected torepresent poor quality rock masses, where face instability pro-blems are more likely in practice. Accordingly, sci/(gDm) is variedfrom 4 to 100 (equivalent in these charts to sci between 0.5 and30 MPa), which are typical of soft rocks to very soft rocks; and GSI
values are taken lower than 25, which are typical of very poorquality rock masses.
0 20 40 60 80 1000.00
0.05
0.10
0.15
0.20
0.25
σ c / γ
Dm
σci / γ Dm
mi = 5
GSI = 10GSI = 15GSI = 20GSI = 25
0 20 40 60 80 1000.00
0.05
0.10
0.15
σ c / γ
Dm
σci / γ Dm
mi = 10
GSI = 10GSI = 15GSI = 20GSI = 25
0 20 40 60 80 1000.00
0.05
0.10
0.15
σ c / γ
Dm
σci / γ Dm
mi = 15
GSI = 10GSI = 15GSI = 20GSI = 25
0 20 40 60 80 1000.00
0.05
0.10
0.15
σ c / γ
Dm
σci / γ Dm
mi = 20
GSI = 10GSI = 15GSI = 20GSI = 25
Fig. 12. Design charts of the critical collapse pressure for Hoek–Brown material.
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451 449
Author's personal copy
To produce the design charts presented in Fig. 12, a total of1632 cases were computed and included in the plots. Such casesconsidered tunnel diameters ranging between 6 and 12 m, andunit weights for the rock mass ranging between 2.0 and 2.5 t/m3.In all cases, it has been assumed that the collapse mechanismdoes not reach the surface, which, for the most unfavourable case,occurs when the tunnel cover is larger than 0.5D.
As expected, Fig. 12 shows a reduction of the collapse pressurewhen rock strength and rock mass quality increase, with a clearinfluence of sci and GSI. In most cases, the values of the criticalpressure are low and, in some cases, the collapse pressure is null;that means that the face is self-stable and that it is not necessaryto apply any pressure to support it. Note that, although our modelprovides negative values for the critical pressure when the face isself-stable (meaning that it would be necessary to ‘‘pull’’ the faceto trigger instability), this value of a null pressure was chosen as aconvention for such stable cases.
6. Conclusions
We present and validate a new analytical failure mechanismfor the determination, in the framework of limit analysis, of thecritical collapse pressure and of the geometry of the collapsemechanism, for the face of tunnels excavated in low quality rockmasses with the HB non-linear failure criterion. The use of a non-linear failure criterion introduces the need to consider thedistribution of normal stresses along the failure surface, so thatthe ‘local’ friction angle can be computed to fulfill the assumptionof associated flow that is inherent to limit analysis. As a conse-quence, there is a need to introduce new parameters in theoptimization problem that allow us to consider such stressdistribution. To be able to consider the non-linearity of the HBcriterion, we improve an advanced, and recently proposed, failuremechanism for the tunnel face [22]; the mechanism, that coversthe whole excavation front, is generated ‘‘point-by-point’’, and itprovides a rotational-type failure that is very similar to thatobserved in small-scale tunnel tests in the laboratory. Themechanism makes it possible to work with variable MC materialsproperties, and it represents the more advanced tunnel facefailure mechanism that has been proposed to this date. Numericalsimulations have been employed to identify adequate, yet simple,stress distributions at the tunnel face. The results of suchsimulations suggest that a linear distribution of stresses alongthe failure surface could be employed as an approximation to thereal stress distribution in many practical applications. Suchassumed linear distribution of normal stresses is further shownto capture better the normal stress variation in cases with ‘worse’rock mass properties, hence improving the prediction of thecritical pressure. The increased complexity of the stress distribu-tion, however, does not seem to improve the results in other caseswith ‘better’ rock mass properties, when the computed criticalpressures are almost equal to the uniform distribution case. Inaddition, to validate the new failure mechanism, 22 test-casescorresponding to rock masses with low quality, as indicated bytheir GSI value, have been employed to compare our limit analysisresults with results of three-dimensional simulations conductedwith FLAC3D. Two aspects have been compared: (i) the numericalvalue of the collapse pressure; and (ii) the shape of the failuremechanism. The obtained results suggest that the limit analysisapproach proposed herein successfully approximates the FLAC3D
numerical results but with a significantly reduced computationalcost, so that it could be applied for fast, and relatively reliable,estimations of the pressure needed for face support in shallowtunnels excavated in heavily fractured rock masses.
The new approach has been further employed to develop‘design charts’ to estimate the face collapse pressure of TBMtunnels excavated in weak or very weak HB rock masses with apoor quality. To limit the applicability of the approach, however, ithas be reminded that the HB failure criterion assumes an isotropicrock mass behaviour, and that it should only be applied when thestructure size, relatively to the spacing between discontinuities,makes it possible to consider the rock mass as a ‘continuum’instead of a blocky structure [42]. Therefore, the results presentedherein are only applicable to heavily fractured rock masses wherethese assumptions are valid and, for instance, they are not applic-able to unstable blocks or wedges defined by intersecting struc-tural discontinuities. Similarly, as solutions of the limit analysisproblem, solutions presented herein do not consider the deforma-tions at the tunnel face and, for instance, they do not account forsqueezing failures associated to high deformations of the materialwhen subjected to high stresses [43]. As a consequence, the use ofthe design charts presented in Fig. 12 should be made in a widerdesign context that considers other factors that could influence thesuccess of the project.
Acknowledgements
Salvador Senent holds a PhD Scholarship provided by Funda-cion Jose Entrecanales Ibarra. This research was funded, in part,by the Spanish Ministry of Economy and Competitiveness, undergrant BIA2012-34326. The support of both Institutions is grate-fully acknowledged.
Appendix A. Supporting information
Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.ijrmms.2013.01.004.
References
[1] Health and Safety Executive. Safety of new Austrian tunnelling method(NATM) tunnels. A review of sprayed concrete tunnels with particularreference to London Clay. Sudbury: (HSE) Books; 1996.
[2] Horn N. Horizontaler erddruck auf senkrechte abschlussflachen von tun-nelrohren. Landeskonferenz der ungarischen tiefbauindustrie. DeutscheUberarbeitung durch STUVA, Dusseldorf, 1961. p. 7–16.
[3] Anagnostou G, Kovari K. Face stability conditions with earth-pressure-balanced shields. Tunn Undergr Space Technol 1996;11:165–73.
[4] Broere W. Face stability calculation for a slurry shield in heterogeneous softsoils. In: Negro A, Ferreira AA, editors. Proceedings of the world tunnelcongress 98 on tunnels and metropolises, Sao Paulo, Brazil, 25–30 April 1998.Rotterdam: Balkema, 1998. p. 215–8.
[5] Vermeer PA, Ruse NM, Dong Z, Harle D. Ortsbruststabilitat von tunnelbau-werken am beispiel des Rennsteig tunnels. In: 2. Kolloquium—Bauen inBoden und Fels. Technische Akademie Esslinggen, Esslingen; January 2000.p. 195–202.
[6] Melis MJ. El colapso del tunel ferroviario por inestabilidad del frente ensuelos y rocas blandas o muy diaclasadas. Rev Obras Pub 2004;3450:33–64.
[7] Kimura T, Mair RJ. Centrifugal testing of model tunnels in soft clay. In:Proceedings of 10th International Conference on Soil Mechanics and Founda-tion Engineering, Stockholm, 15–19 June 1981. Rotterdam: Balkema, 1981.p. 319–22.
[8] Chambon P, Corte JF. Shallow tunnels in cohesionless soil: stability of tunnelface. J Geotech Eng 1994;120:1148–65.
[9] Takano D, Otani J, Nagatani H, Mukunoki T. Application of X-ray CT onboundary value problems in geotechnical engineering: research on tunnel facefailure. In: GeoCongress 2006: Geotechnical Engineering in the InformationTechnology Age. Proceedings of GeoCongress 2006, Atlanta, Georgia, USA.
[10] Kirsch A. Experimental investigation of the face stability of shallow tunnels insand. Acta Geotech 2010;5:43–62.
[11] Vermeer PA, Ruse N. Die stabilitat der tunnelortsbrust in homogenembaugrund. Geotechnik 2001;24:186–93.
[12] Dias D, Janin J, Soubra A, Kastner R. Three-dimensional face stability analysisof circular tunnels by numerical simulations. In: Alshawabkeh AN, Reddy KR,Khire MV, editors. GeoCongress 2008: Characterization, Monitoring, and
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451450
Author's personal copy
Modeling of GeoSystems. Proceedings of GeoCongress 2008, New Orleans,Louisiana, USA, 9–12 March 2008. p. 886–93.
[13] Kim SH, Tonon F. Face stability and required support pressure for TBM driventunnels with ideal face membrane—drained case. Tunn Undergr SpaceTechnol 2010;25:526–42.
[14] Melis MJ, Medina LE. Discrete numerical model for analysis of earth pressurebalance tunnel excavation. J Geotech Geoenviron Eng 2005;131:1234–42.
[15] Chen RP, Tang LJ, Ling DS, Chen YM. Face stability analysis of shallow shieldtunnels in dry sandy ground using the discrete element method. ComputGeotech 2011;38:187–95.
[16] Davis EH, Gunn MJ, Mair RJ, Seneviratine HN. The stability of shallow tunnels andunderground openings in cohesive material. Geotechnique 1980;30:397–416.
[17] Leca E, Panet M. Application du calcul �a la rupture �a la stabilite du front detaille d’un tunnel. Rev France Geotech 1988:5–19.
[18] Leca E, Dormieux L. Upper and lower bound solutions for the face stabilityof shallow circular tunnels in frictional material. Geotechnique 1990;40:581–606.
[19] Oberle S. Application de la methode cinematique �a l’etude de la stabilite d’unfront de taille de tunnel, Final project. ENSAIS, France. 1996.
[20] Soubra AH. Three-dimensional face stability analysis of shallow circulartunnels. In: GeoEng2000: An International Conference on Geotechnical andGeological Engineering. Proceedings of the International Conference onGeotechnical and Geological Engineering, 19–24 November 2000, Melbourne,Australia; 2000.
[21] Mollon G, Dias D, Soubra AH. Face stability analysis of circular tunnels drivenby a pressurized shield. J Geotech Geoenvir 2010;136:215–29.
[22] Mollon G, Dias D, Soubra AH. Rotational failure mechanisms for the facestability analysis of tunnels driven by a pressurized shield. Int J Numer AnalMethods Geomech 2011;35:1363–88.
[23] Sofianos AI, Nomikos PP. Equivalent Mohr–Coulomb and generalized Hoek–Brown strength parameters for supported axisymmetric tunnels in plastic orbrittle rock. Int J Rock Mech Min Sci 2006;43:683–704.
[24] Jimenez R, Serrano A, Olalla C. Linearization of the Hoek and Brown rockfailure criterion for tunnelling in elasto-plastic rock masses. Int J Rock MechMin Sci 2008;45:1153–63.
[25] Fraldi M, Guarracino F. Limit analysis of collapse mechanisms in cavities andtunnels according to the Hoek–Brown failure criterion. Int J Rock Mech MinSci 2009;46:665–73.
[26] Huang F, Yang XL. Upper bound limit analysis of collapse shape for circulartunnel subjected to pore pressure based on the Hoek–Brown failure criterion.Tunn Undergr Space Technol 2011;26:614–8.
[27] Yang XL, Yin JH. Slope stability analysis with nonlinear failure criterion. J EngMech 2004;130:267–73.
[28] Chen WF. Limit analysis and soil plasticity. Amsterdam: Elsevier; 1975.[29] Yang XL, Li L, Yin JH. Stability analysis of rock slopes with a modified Hoek–
Brown failure criterion. Int J Numer Anal Methods Geomech 2004;28:181–90.
[30] Yang XL, Li L, Yin JH. Seismic and static stability analyisis for rock slopes by akinematical approach. Geotechnique 2004;54:543–9.
[31] Huang F, Yang XL. Upper bound solutions for the face stability of shallowcircular tunnels subjected to nonlinear failure criterion. In: Tonon F, Liu X,Wu W, editors. Deep and underground excavations. Proceedings of the 2010GeoShangai international conference, Shangai, China; 2010. p. 251–6.
[32] Agar JG, Morgenstern NR, Scott JD. Shear strength and stress strain behaviourof Athabasca oil sand at elevated temperatures and pressures. Can Geotech J1987;24:1–10.
[33] Mollon G, Phoon KK, Dias D, Soubra AH. Validation of a new 2D failuremechanism for the stability analysis of a pressurized tunnel face in a spatiallyvarying sand. J Eng Mech 2011;137:8–21.
[34] Hoek E, Carranza-Torres C, Corkum B. Hoek–Brown failure criterion—2002edition. In: Hammah R, Bawden W, Curran J, Telesnicki M, editors. Proceed-ings of the North American rock mechanics symposium, University ofToronto; 2002, p. 267–73.
[35] Ucar R. Determination of shear failure envelope in rock masses. J Geotech Eng1986;112:303–15.
[36] Balmer G. A general analytical solution for Mohr’s envelope. Am Soc TestMater J 1952;52:1269–71.
[37] Kumar P. Shear failure envelope of Hoek–Brown criterion for rockmass. TunnUndergr Space Technol 1998;13:453–8.
[38] Shen J, Priest SD, Karakus M. Determination of Mohr–Coulomb shear strengthparameters from generalized Hoek–Brown criterion for slope stability ana-lysis. Rock Mech Rock Eng 2012;45:123–9.
[39] Itasca Consulting Group. FLAC3D 4.0 manual. Minneapolis; 2009.[40] Alejano LR, Alonso E. Considerations of the dilatancy angle in rocks and rock
masses. Int J Rock Mech Min Sci 2005;42:481–507.[41] Mollon G, Dias D, Soubra AH. Probabilistic analysis of circular tunnels in
homogeneous soil using response surface methodology. J Geotech Geoen-viron Eng 2009;135:1314–25.
[42] Hoek E, Brown ET. Practical estimates of rock mass strength. Int J Rock MechMin Sci 1997;34:1165–86.
[43] Hoek E. Big tunnels in bad rock. J Geotech Geoenviron Eng 2001;127:726–40.
S. Senent et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 440–451 451
