Probabilistic Analysis and Design of Circular Tunnels ...guilhem.mollon.free.fr/Telechargements/Mollon_Dias... · soil has been possible due to recent technological advances in-cluding

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Probabilistic Analysis and Design of Circular Tunnelsagainst Face Stability

Guilhem Mollon1; Daniel Dias2; and Abdul-Hamid Soubra, M.ASCE3

Abstract: This paper presents a reliability-based approach for the three-dimensional analysis and design of the face stability of a shallowcircular tunnel driven by a pressurized shield. Both the collapse and the blow-out failure modes of the ultimate limit state are studied. Thedeterministic models are based on the upper-bound method of the limit analysis theory. The collapse failure mode was found to give themost critical deterministic results against face stability and was adopted for the probabilistic analysis and design. The random variablesused are the soil shear strength parameters. The Hasofer-Lind reliability index and the failure probability were determined. A sensitivityanalysis was also performed. It was shown that �1� the assumption of negative correlation between the soil shear strength parameters givesa greater reliability of the tunnel face stability with respect to the one of uncorrelated variables; �2� FORM approximation gives accurateresults of the failure probability; and �3� the failure probability is much more influenced by the coefficient of variation of the angle ofinternal friction than that of the cohesion. Finally, a reliability-based design is performed to determine the required tunnel pressure for atarget collapse failure probability.

DOI: 10.1061/�ASCE�1532-3641�2009�9:6�237�

CE Database subject headings: Tunnels; Pressure; Passive pressure; Limit analysis; Probability; Design.

Introduction

Over the past 30 years, tunneling in a frictional and/or cohesivesoil has been possible due to recent technological advances in-cluding the pressurized shield. Face stability analysis of shallowcircular tunnels driven by the pressurized shield is of major im-portance. The tunnel face pressure must avoid both the collapse�active failure� and the blow-out �passive failure� of the soil massnearby the tunnel face. Active failure of the tunnel is triggered byapplication of surcharge and self-weight, with the tunnel facepressure providing resistance against collapse. Under passive con-ditions, these roles are reversed and the face pressure causesblow-out with resistance being provided by the surcharge andself-weight.

In this paper, the face stability analysis is conducted based ona probabilistic approach. The reliability-based analysis is morerational than the deterministic one since it takes into account theinherent uncertainty of the input variables. Nowadays, this is pos-sible because of the improvement of our knowledge on the statis-tical properties of the soil �Phoon and Kulhawy 1999; Baecherand Christian 2003�. Two performance functions may characterize

1Ph.D. Student, INSA Lyon, LGCIE Site Coulomb 3, Géotechnique,Bât. J.C.A. Coulomb, Domaine scientifique de la Doua, 69621 Villeur-banne cedex, France. E-mail: [email protected]

2Associate Professor, INSA Lyon, LGCIE Site Coulomb 3, Géotech-nique, Bât. J.C.A. Coulomb, Domaine scientifique de la Doua, 69621Villeurbanne cedex, France. Email: [email protected]

3Professor, Dept. of Civil Engineering, Univ. of Nantes, Bd. del’université, BP 152, 44603 Saint-Nazaire, France �corresponding author�.E-mail: [email protected]

Note. This manuscript was submitted on April 25, 2008; approved onJuly 1, 2009; published online on November 13, 2009. Discussion periodopen until May 1, 2010; separate discussions must be submitted for indi-vidual papers. This paper is part of the International Journal of Geome-chanics, Vol. 9, No. 6, December 1, 2009. ©ASCE, ISSN 1532-3641/
2009/6-237–249/$25.00.
INTERNATIONAL JOURNAL O

Int. J. Geomech. 200

the tunnel behavior: the serviceability limit state and the ultimatelimit state �ULS�. Only the collapse and the blow-out failuremodes of the ULS are analyzed herein. Two new rigorous deter-ministic limit analysis models are used. The soil shear strengthparameters are modeled as random variables. The main reliabilityconcepts are described next, followed by the two deterministicmodels and discussions of the deterministic and probabilistic nu-merical results based on these models.

Overview of Reliability Concepts

The reliability index is a measure of the safety that takes intoaccount the inherent uncertainties of the input variables. A widelyused reliability index is the Hasofer and Lind �1974� index. Itsmatrix formulation is �Ditlevsen 1981�

�HL = minx�F

��x − ��TC−1�x − �� 1�

in which x=vector representing the n random variables; �=vector of their mean values; C=covariance matrix; and F=failure region. The minimization of Eq. �1� is performed subjectto the constraint G�x��0 where the limit state surface G�x�=0,separates the n dimensional domain of random variables into tworegions: a failure region F represented by G�x��0 and a saferegion given by G�x��0.

The classical approach for computing �HL by Eq. �1� is basedon the transformation of the limit state surface into the rotatedspace of standard normal uncorrelated variates. The shortest dis-tance from the transformed failure surface to the origin of thereduced variates is the reliability index �HL.

An intuitive interpretation of the reliability index was sug-gested in Low and Tang �1997a, 2004� where the concept of anexpanding ellipsoid �or an ellipse in two-dimensions �2D� asshown in Fig. 1� led to a simple method of computing the
Hasofer-Lind reliability index in the original space of the random
F GEOMECHANICS © ASCE / NOVEMBER/DECEMBER 2009 / 237

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variables using an optimization tool available in most softwarepackages. Low and Tang �1997a,2004� reported that the Hasofer-Lind reliability index �HL may be regarded as the codirectionalaxis ratio of the smallest ellipsoid that just touches the limit statesurface to the unit dispersion ellipsoid �i.e., corresponding to�HL=1 in Eq. �1� without the minimum�. They also stated thatfinding the smallest ellipsoid �called hereafter critical ellipsoid�that is tangent to the limit state surface is equivalent to finding themost probable failure point. When the random variables are non-normal, the Rackwitz-Fiessler equivalent normal transformationwas used to compute the equivalent normal mean �N and theequivalent normal standard deviation �N. The iterative computa-tions of �N and �N for each trial design point are automatic duringthe constrained optimization search.

From the first-order reliability method FORM and the Hasofer-Lind reliability index �HL, one can approximate the failure prob-ability as follows:

Pf � ��− �HL� �2�

where �� · �=cumulative distribution function of a standard nor-mal variable. In this method, the limit state function is approxi-mated by a hyperplane tangent to the limit state surface at thedesign point.

Monte Carlo �MC� is another method of computing the failureprobability. It is the most robust simulation method in whichsamples are generated with respect to the probability density ofeach variable. For each sample, the response of the system iscalculated. An unbiased estimator of the failure probability isgiven by

P̃f =1

N�i=1

N

I�xi� �3�

where N=number of samples and I�x�=1 if G�x��0 and 0 else-where. The coefficient of variation of the estimator is given by

COV�P̃f� =��1 − Pf�PfN

�4�

Generally, for a given target of the coefficient of variation, thecrude MC simulation requires a large number of samples, i.e.,a large computation time. This is especially the case for smallvalues of the failure probability. The importance sampling �IS�simulation method is a more efficient approach; it requires fewersample points than the MC method. In this approach, the initial

Failure domain

Safe domain

Limit state surface

Design point

Criticaldispersionellipse

NHL 2.σβ

N2σ

N1σ

Unit dispersionellipse

N1μ

N2μ

x1

x2

NHL 1.σβ

Fig. 1. Design point and equivalent normal dispersion ellipses in thespace of two random variables

sampling density f� · � is shifted to the design point in order to

238 / INTERNATIONAL JOURNAL OF GEOMECHANICS © ASCE / NOVEMB


concentrate the samples in the region of greatest probability den-sity within the zone defined by G�x��0. The design point may bedetermined by using any of the classical methods such asRackwitz-Fiessler algorithm �Rackwitz and Fiessler 1978�, Lowand Tang’s ellipsoid approach �Low and Tang 1997a, 2004�, etc.An estimator of the failure probability Pf is obtained as follows�Melchers 1999�:

P̃f =1

N�i=1

N

I�vi�f�vi�h�vi�

�5�

where h� · �=new sampling density centered at the design pointand v=vector of sample values generated with the new probabil-ity density function �PDF�, i.e., h� · �. The coefficient of variationof the estimator is given by �Melchers 1999�

COV�P̃f� =1

Pf� 1

N� 1

N�i=1

N �I�vi�f�vi�h�vi�

�2

− �Pf�2� �6�

Reliability Analysis of a Circular Tunnel Face

The aim of this paper is to perform a reliability analysis of theface stability of a shallow circular tunnel driven by a pressurizedshield in a c-� soil. The problem can be idealized as shown inFigs. 2�a and b� by considering a circular rigid tunnel of diameterD driven under a depth of cover C. A surcharge �s is applied atthe ground surface and a uniform retaining pressure �t is appliedto the tunnel face to simulate tunneling under compressed air.The deterministic models are based on the upper-bound methodof the limit analysis theory. They are presented in the next sec-tion. Due to uncertainties in soil shear strength parameters, thecohesion c, and the angle of internal friction � are considered asrandom. They are modeled in the present analysis as random vari-ables. This means that the soil parameters are considered as ho-mogeneous in the whole soil mass. The randomness of the soil istaken into account from one simulation to another. The perfor-mance functions G1 and G2 used in the reliability analysis forboth the collapse and the blow-out cases are respectively definedas follows:

G1 = �t − �c �7�

G2 = �b − �t �8�

where �t=applied pressure on the tunnel face, and �c and�b=collapse and blow-out pressures, respectively.

Limit Analysis Models

Several theoretical models have been presented in literature forthe computation of the collapse and blow-out tunnel pressurescorresponding respectively to the active and passive modes offailure. The most recent and significant approach is the one pre-sented by Leca and Dormieux �1990� who considered three-dimensional �3D� failure mechanisms in the framework of theupper-bound method in limit analysis. In this paper, two newdeterministic models �3D multiblock failure mechanisms� basedon the upper-bound approach of limit analysis are proposed forthe probabilistic analysis. These mechanisms constitute an im-provement of the failure mechanisms by Leca and Dormieux�1990� since they allow the 3D slip surface to develop more freely
in comparison with the available one- and two-block mechanisms
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given by Leca and Dormieux �1990�. Notice that the use of alower-bound approach in limit analysis �using for instance finiteelements and linear programming� is another alternative ap-proach. It has the advantage of providing conservative solutions.However, this method leads, when dealing with the probabilisticanalysis, to complex and very expensive numerical computationswith a high computation time since the probabilistic analysis re-quires a significant number of calls of the deterministic model fora given soil variability. Thus, in order to optimize the computationtime and to get sufficiently accurate results, efforts were concen-trated in the present paper on the improvement of the best avail-able upper-bound solutions �i.e., those by Leca and Dormieux1990� by using multiblock failure mechanisms. Notice that amultiblock failure mechanism was also used by Soubra �1999�when dealing with the 2D analysis of the bearing capacity of stripfoundations. It was shown by Soubra �1999� that the multiblockmechanism significantly improves the solutions of the bearingcapacity as given by the available mechanisms �two-block andlog-sandwich mechanisms� and obtains smaller �i.e., better� upperbounds. This is due to the great freedom offered by this mecha-

c. Three-dimensional views of the M1 mechanism in the (x, y, z) space

a. M1 mechanism (collapse case) in the (y, z) plane

b. M2 mechanism (blow-out case) in the (y, z) plane

Fig. 2. Failure mechanisms M1 and M2 for the face stability

nism to move more freely with respect to traditional mechanisms.



The optimal radial shear zone found in the case of a ponderablesoil was not bounded by a log spiral as is the case of the tradi-tional Prandtl �i.e., log sandwich� mechanism but by a more criti-cal surface found by numerical optimization. Furthermore, themultiblock mechanism by Soubra �1999� led in some cases �forNq and Nc� to the exact solutions given by the log-sandwich fail-ure mechanism since both upper and lower bound solutions wereidentical in these cases. Notice finally that the two 3D multiblockfailure mechanisms presented in this paper for the stability analy-sis of circular tunnels make use of the idea of multiblock mecha-nisms suggested by Soubra �1999� in the 2D analysis of stripfootings in order to obtain better upper-bound solutions. A de-tailed description of these mechanisms is given in the followingsections.

Collapse Mechanism M1 (Active Case)M1 is an improvement of the two-block collapse mechanism pre-sented by Leca and Dormieux �1990�. This mechanism is a multi-block. It is composed of n truncated rigid cones with circularcross sections and with opening angles equal to 2�. Fig. 2�c�depicts three different 3D views of a five-block mechanism �i.e.,n=5�. The geometrical construction of this mechanism is similarto that of Leca and Dormieux �1990�, i.e., each cone is the mirrorimage of the adjacent cone with respect to the plane that is normalto the contact surface separating these cones �cf. Leca andDormieux 1990�. This is a necessary condition to assure the sameelliptical contact area between adjacent cones. In order to makeclearer the geometrical construction of the 3D failure mechanism,Fig. 3 shows how the first two truncated conical blocks adjacentto the tunnel face are constructed. The geometrical construction ofthe remaining truncated conical blocks is straightforward. InFig. 3, Block 1 is a truncated circular cone adjacent to the tunnelface. It has an opening angle equal to 2� �in order to respect thenormality condition in limit analysis� and an axis inclined at �with the horizontal direction. Thus, the intersection of this trun-cated cone with the tunnel face is an elliptical surface that doesnot cover the entire circular face of the tunnel �cf. Fig. 2�c�� Thisis a shortcoming not only of the present failure mechanism butalso of the one- and two-block mechanisms by Leca andDormieux �1990�. On the other hand, Block 1 is truncated withPlane 1, which is inclined at an angle �1 with the vertical direc-tion �cf. Fig. 3�. In order to obtain the same contact area with theadjacent truncated conical block, Block 2 is constructed in such amanner to be the mirror image of Block 1 with respect to theplane that is normal to the surface separating the two blocks �i.e.,Plane 2 as shown in Fig. 3�. The upper rigid cone will or willnot intersect the ground surface depending on the � and C /Dvalues. At first glance, the fact that the failure mechanism doesnot intersect the ground surface for some values of � and C /D is

Fig. 3. Detail of the construction of the M1 mechanism in the �y ,z�plane

striking. However, the same phenomenon was also observed


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when performing 3D numerical simulations using FLAC3D soft-
ware �Mollon et al. 2009�. Notice that while using numericalsimulations, no assumptions were made on the shape of the fail-ure mechanism and the optimal failure mechanism was explored.Thus, one may confirm that the present failure mechanism basedon limit analysis is acceptable even if the critical failure surfacedoes not outcrop.

M1 is a translational kinematically admissible failure mecha-nism. The different truncated conical blocks of this mechanismmove as rigid bodies. These truncated rigid cones translate withvelocities of different directions, which are collinear with thecones axes and make an angle � with the conical discontinuitysurface in order to respect the normality condition required by thelimit analysis theory. The velocity of each cone is determined bythe condition that the relative velocity between the cones in con-tact has the direction that makes an angle � with the contactsurface. The velocity hodograph is presented in Fig. 2�a�. Thepresent mechanism is completely defined by n angular parameters� and �i �i=1, . . . ,n−1� where n is the number of the truncatedconical blocks.

Blow-Out Mechanism M2 (Passive Case)Even though safety against collapse is a major concern duringtunneling, the blow-out mechanism may be of interest for veryshallow tunnels bored in weak soils, when the pressure �t

can become so great that soil is heaved in front of the shield. M2�Fig. 2�b�� is a blow-out mechanism. It represents the passive caseof the former mechanism. With reference to M1, the M2 mecha-nism presents an upward movement of the soil mass. Thus, thecones with an opening angle 2� are reversed. Contrary to M1, thepresent mechanism always outcrops.

Ellipsoid Approach via Spreadsheet

In the present paper, by the Low and Tang �1997a, 2004� method,one literally sets up a tilted ellipsoid in the Excel spreadsheet andminimizes the dispersion ellipsoid subject to the constraint that itbe tangent to the limit state surface using the Excel Solver withthe automatic scaling option. Eq. �1� may be rewritten as �Lowand Tang 1997b, 2004; Youssef Abdel Massih and Soubra 2008;Youssef Abdel Massih et al. 2008�

�HL = minx�F�� x − �N

�N T

R�−1� x − �N

�N �9�

in which R�−1inverse of the correlation matrix. This equationwill be used �instead of Eq. �1�� since the correlation matrix R�displays the correlation structure more explicitly than the covari-ance matrix C�.

Deterministic Numerical Results

For both M1 and M2 mechanisms, when the total rate of energydissipation and the total rate of external work are equated, theultimate tunnel pressure �u for the collapse and the blow-outmodes of failure can be expressed as follows:

�u = DN + cNc + �sNs �10�

where N, Nc, and Ns=nondimensional coefficients. They repre-sent, respectively, the effect of soil weight, cohesion, and sur-charge loading. The expressions of the different coefficients N,
Nc, and Ns are given in the Appendix. Notice that the external


forces involved in the present mechanisms are the weights of thedifferent truncated rigid cones, the surcharge loading acting onthe ground surface, and the pressure applied on the tunnel face.For the collapse mechanism, the rate of external work of thesurcharge loading should be calculated only in case of outcrop ofthe mechanism on the ground surface. The computation of therate of external work of the different external forces is straight-forward. The details are given in Oberlé �1996�. The rate of in-ternal energy dissipation takes place along the different velocitydiscontinuity surfaces. These are �1� the radial elliptical surfaceswhich are the contact areas between adjacent truncated cones and�2� the lateral surfaces of the different truncated cones. Noticethat the rate of internal energy dissipation along a unit velocitydiscontinuity surface is equal to c ·�u �Chen 2008� where c is thesoil cohesion and �u is the tangential component of the velocityalong the velocity discontinuity surface. Calculations of the rateof internal energy dissipation along the different velocity discon-tinuity surfaces are straightforward. The details are given inOberlé �1996�.

In Eq. �10�, �u, N, Nc, and Ns depend not only on themechanical and geometrical characteristics c, �, and C /D, butalso on the angular parameters of the failure mechanism � and �i

�i=1, . . . ,n−1�. In the following sections, the ultimate tunnelpressure of the collapse mode will be denoted �c, and that ofblow-out will be refereed to as �b. They were obtained respec-tively by maximization and minimization of �u in Eq. �10� withrespect to the � and �i angles. As for the ultimate tunnel pres-sures, the critical coefficients N

b, Ncb, and Ns

b �respectively Nc , Nc

c,and Ns

c� corresponding to the blow-out �respectively collapse�case, were obtained by minimization �respectively maximization�of these coefficients with respect to the � and �i angles. A com-puter program has been written in Microsoft Excel Visual Basic todefine the different coefficients N, Nc, and Ns and the tunnelpressure �u for the collapse and the blow-out modes of failure.The optimization was performed using the optimization tool“Solver” implemented in Microsoft Excel. It was shown that theincrease in the number of cones improves the solutions �i.e., in-creases the active coefficients, and reduces the passive ones�. Thenumerical results have shown that this improvement becomes in-significant �smaller than 1%� for a number of blocks greater thanfive. Therefore, only five blocks were used in this paper for thecollapse and the blow-out mechanisms.

Notice that in both the collapse and the blow-out cases, thenumerical results have shown that Nc and Ns are related by thefollowing classical formula:

Nc tan � + 1 − Ns = 0 �11�

This can be explained by the theorem of corresponding states�Soubra 1999�. Hence, in the following, only the N and Ns coef-ficients will be presented; the Nc coefficient can be obtained usingEq. �11�.

Comparison with Available Upper-Bound Solutions

Leca and Dormieux �1990� have considered a collapse failuremechanism composed of two rigid cones. Fig. 4 presents the N

c

and Nsc values given by the present analysis �M1 mechanism� and

the ones given by Leca and Dormieux �1990�. The Nc coefficient

increases with C /D; then, it becomes constant for large values ofC /D corresponding to the condition of no outcrop of the upperblock. However, coefficient Ns

c decreases with the C /D increaseand vanishes beyond a certain value of C /D corresponding to the
no-outcrop condition. In this case, the surcharge loading has no
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influence on the critical Nsc value. These conclusions conform

to those of Leca and Dormieux �1990�. It should be mentionedthat in the case of collapse, the present failure mechanism givesgreater upper-bound solutions than the available upper-boundsolutions proposed by Leca and Dormieux �1990�. The improve-ment of the solution is about 8% for the N

c coefficient when�=20° and C /D�0.55. For Ns

c, the improvement is equal to37.5% when �=20° and C /D=0.1.

For the blow-out case, Leca and Dormieux �1990� have con-sidered a mechanism composed of a single rigid cone movingupward. The upper-bound solutions given by these writers arecompared with the ones corresponding to the present M2 mecha-nism in Fig. 5. The M2 mechanism is better than the one pre-sented by Leca and Dormieux �1990� since the present upper-bound solutions are smaller. For the N

b coefficient, the reductionis very significant and it is of 41% when �=30° and C /D=1.4.For the Ns

b coefficient, significant reductions are also obtainedwith respect to the results presented by Leca and Dormieux. Forexample, when �=30° and C /D=1.4, the reduction attains 51%.

As a conclusion, the four design charts presented before �i.e.,Figs. 4 and 5� give the values of the coefficients N and Ns of theproposed mechanisms for both the collapse and blow-out casesfor different values of the governing parameters � and C /D. Theexisting values by Leca-Dormieux are also given in these charts,allowing one to appreciate the improvement with respect to priorsolutions. The determination of the critical collapse or blow-outpressure to be used in practice can then be made by using Eq. �10�where Nc in this equation is given by Eq. �11� and N and Ns aregiven in the design charts of Figs. 4 and 5.

It should be mentioned here that Fig. 2�c� presented earlier is a3D representation of the critical collapse mechanism obtainedafter optimization of the tunnel pressure with respect to the geo-

Fig. 4. Nc and Ns

c versus C/D as given by Leca and Dormieux andM1 mechanism

metrical parameters of the failure mechanism for a friction angle



equal to 17° and a cohesion equal to 7 kPa when C /D=2. It canbe seen that the 3D critical failure mechanism M1 involves aradial shear zone �composed of three small truncated rigid cones,i.e., Blocks 2–4� sandwiched between two greater rigid cones�i.e., Blocks 1 and 5�. Line BCDE as obtained by numerical op-timization is not a log spiral and this constitutes the major interestof the present multiblock mechanism with respect to the similar inshape mechanism �i.e., log sandwich� where line BCDE is re-placed by a log spiral.

For the currently encountered cases �i.e., 1�C /D�3; 10°��30°; 0�c�20 kPa�, Table 1 presents the results of thetunnel ultimate pressures �c and �b corresponding respectively tothe collapse and the blow-out modes of failure as given by thepresent failure mechanisms. The applied tunnel pressure �t shouldbe greater than the collapse pressure �c to avoid the active failure.If one adopts a safety factor against collapse �Fs=�t /�c� equal to2, the required applied pressure �t should be equal to twice thevalue of the collapse pressure as given in the fifth column ofTable 1. The comparison of these pressures with those of theblow-out case �i.e., �b� shows that the blow-out pressures aremuch higher than the practical applied tunnel pressures �t for allthe values of the governing parameters considered in this paper.Hence, the blow-out mode of failure does not occur for the casescurrently encountered in practice. In the following sections, onlythe collapse failure mode is considered in the probabilistic analy-sis and design of circular tunnels against face stability.

Comparison with Three-Dimensional NumericalSimulations

In order to assess the accuracy of the limit analysis collapse

Fig. 5. Nb and Ns

b versus C/D as given by Leca and Dormieux andM2 mechanism

model, complex 3D numerical simulations were performed using


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the finite difference commercial software FLAC3D. They allowone to determine the values of the critical collapse pressure �c.For a detailed description of these simulations, the reader mayrefer to Mollon et al. �2009� . Fig. 6�a� shows a comparison be-tween the collapse pressures as given by the FLAC3D model andby the M1 mechanism. Three cases of �c ,�� are considered toshow the effect of both c and � on the comparison between thelimit analysis and the numerical simulations. It appears that thelimit analysis results are not far from the ones obtained byFLAC3D. Even if they are upper-bounds �i.e., nonconservative� tothe exact collapse pressures, the present collapse pressures givenby limit analysis can be considered as sufficiently accurate for

Table 1. Values of the Collapse and Blow-Out Pressures �c and �b and tof C /D, �, and c

C /D��°�

c�kPa�

1 10 0

1 10 20

1 30 0

1 30 20

3 10 0

3 10 20

3 30 0

3 30 20

a. Critical collapse pressure

b. Failure pattern in the (y, z) plane

Fig. 6. Comparison between FLAC3D numerical simulations andlimit analysis



practical use. Notice also that the very short calculation time re-quired by the limit analysis model �smaller than 1 s� compared tothat required by the numerical simulations �90 min� is appealingfor the probabilistic analysis, which requires a significant numberof calls of the deterministic model for a given soil variability.Finally, Fig. 6�b� shows that the shape of the critical M1 collapsemechanism is very close to the one obtained by FLAC3D for thereference case �i.e., �=17°, c=7 kPa� that will be studied in theprobabilistic analysis.

Probabilistic Numerical Results

The present collapse failure mechanism M1 will be used in allsubsequent probabilistic analyses since �1� it gives better upper-bound solutions than the mechanism by Leca and Dormieux�1990� and �2� it gives results that are not very far from thesolutions given by complex 3D numerical simulations usingFLAC3D.

The probabilistic numerical results presented in this paper con-sider the case of a circular tunnel with a diameter D=10 m and acover C=10 m �i.e., C /D=1�. The soil has a unit weight of18 kN /m3. No surcharge loading ��s=0� is considered in theanalysis.

For the probability distribution of the random variables, twocases are studied. In the first case, referred to as normal variables,c and � are considered as normal variables. In the second case,referred to as nonnormal variables, c is assumed to be lognor-mally distributed while � is assumed to be bounded and a �distribution is used �Fenton and Griffiths 2003�. The parametersof the � distribution are determined from the mean value andstandard deviation of � �Haldar and Mahadevan 2000�. Forboth cases, correlated and uncorrelated variables are considered.In this paper, the illustrative values used for the statistical mo-ments of the shear strength parameters and their coefficient ofcorrelation �c,� are as follows: �c=7 kPa, ��=17°, COVc=20%,COV�=10%, and �c,�=−0.5.

A common approach to determine the reliability index of astability problem �slope stability, bearing capacity, etc.� is basedon the calculation of the reliability index corresponding to thedeterministic failure surface �i.e., the one corresponding to theminimum safety factor or the ultimate load� �Christian et al.1994�. In this paper, the reliability index is determined by mini-mizing the quadratic form of Eq. �9� not only with respect to therandom variables, but also with respect to the geometrical param-eters of the failure mechanism �� ,�i� i=1, .. ,n−1 �Bhattacharyaet al. 2003; Youssef Abdel Massih 2007; Youssef Abdel Massih

uired Face Pressure �t for a Safety Factor Fs=2 for the Common Values

c

a��t �kPa�for Fs=2

�b

�kPa�

0 200 660

, stable� 0 �i.e., stable� 94

2 44 3,800

, stable� 0 �i.e., stable� 4,700

6 212 2,570


2 44 20,900


he Req

��kP

10

0 �i.e.

2

0 �i.e.

10

0 �i.e.

2

0 �i.e.

et al. 2008�. Five rigid blocks �i.e., n=5 in Fig. 2� are considered.

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Therefore, the minimization is performed with respect to sevenparameters �� ,�i ,c ,��. The surface obtained corresponding tothe minimum reliability index is referred to here as the criticalprobabilistic surface.

For the configuration presented earlier, the tunnel collapsepressure corresponding to normal variables was found equal to�c=28.3 kPa. However, for nonnormal variables, the collapsepressure was computed using the equivalent mean values of therandom variables and was found equal to 28.8 kPa.

Reliability Index, Critical Dispersion Ellipses,and Partial Safety Factors

Fig. 7 presents the Hasofer-Lind reliability index versus the ap-plied pressure �t for four combinations of normal and nonnormal,uncorrelated and correlated shear strength parameters. For allcases, the reliability index increases with the increase of the tun-nel face pressure �t. The comparison of the results of correlatedvariables with those of uncorrelated variables shows that the re-liability index corresponding to uncorrelated variables is smallerthan the one of negatively correlated variables for both normaland nonnormal variables. One can conclude that assuming un-correlated shear strength parameters is conservative in compari-son to assuming negatively correlated parameters. For a targetreliability index of 3.8 as imposed by Eurocode 7, the requiredtunnel pressure is smaller for correlated and nonnormal variables.For instance, with respect to the reference case of normal anduncorrelated variables, �t decreases by 19% if the variables arecorrelated �54.3 kPa to be compared to 66.9 kPa� and by 7% ifthe variables are considered to follow nonnormal distributions�62.3 kPa to be compared to 66.9 kPa�.

The values �c� and �� of the design points corresponding todifferent values of the tunnel face pressure �t can give an ideaabout the partial safety factors of each of the strength parametersc and tan � as follows:

Fc =�c

c��12�

F� =tan��tan ��

�13�

Fig. 7. Reliability index versus �t for normal, nonnormal, uncorre-lated, and correlated variables

Table 2 gives the obtained partial safety factors Fc and F� and



the corresponding design point and reliability index for the fourcombinations of normal, nonnormal, uncorrelated, and correlatedvariables and for different values of the tunnel face pressure �t.This table also shows the empirical Fc and F� values suggestedby Eurocode 7. For a reliability index close to 3.8 as suggested byEurocode 7 in the ULS, the values obtained from the presentapproach for the four combinations of assumptions are between 1and 1.5 for Fc and about 1.5 for F�. The corresponding Eurocodevalues are 1.6 and 1.25. As can be seen, contrary to Eurocode 7,the present probabilistic approach attributes more safety to thecohesion parameter than Eurocode 7.

For �t=70 kPa �Fig. 7�, the collapse reliability index for un-correlated and correlated normal variables are respectively equalto 4.02 and 5.16. The corresponding most probable failure pointsobtained from the minimization procedure �Table 2� are found tobe at �c�=4.53 kPa, ��=10.87°� and �c�=7.11 kPa, ��=9.34°�.These are the points of tangency of the critical dispersion ellipseswith the limit state surface. Notice that the limit state surfacedivides the combinations of �c ,�� that would lead to failure from

Table 2. Reliability Index, Design Point, and Partial Safety Factors

�t �HL c� �� Fc F�

Normal uncorrelated variables

28.3 0.00 7.00 17.00 1.00 1.00

30 0.25 6.76 16.69 1.04 1.02

35 0.93 6.18 15.78 1.13 1.08

40 1.53 5.74 14.90 1.22 1.15

50 2.51 5.19 13.33 1.35 1.29

60 3.32 4.82 12.02 1.45 1.44

70 4.02 4.53 10.87 1.55 1.59

80 4.63 4.30 9.85 1.63 1.76

100 5.69 3.95 8.07 1.77 2.16

Normal correlated variables

28.3 0.00 7.00 17.00 1.00 1.00

30 0.35 6.79 16.66 1.03 1.02

35 1.30 6.38 15.57 1.10 1.10

40 2.11 6.32 14.40 1.11 1.19

50 3.35 6.54 12.37 1.07 1.39

60 4.34 6.82 10.73 1.03 1.61

70 5.16 7.11 9.34 0.98 1.86

80 5.87 7.40 8.12 0.95 2.14

100 7.08 7.94 6.05 0.88 2.88

Nonnormal uncorrelated variables

28.8 0.00 7.00 17.00 1.00 1.00

30 0.17 6.71 16.75 1.04 1.02

35 0.89 6.19 15.77 1.13 1.08

40 1.53 5.84 14.81 1.20 1.16

50 2.63 5.42 13.17 1.29 1.31

60 3.58 5.12 11.82 1.37 1.46

Nonnormal correlated variables

28.8 0.00 7.00 17.00 1.00 1.00

30 0.24 6.73 16.72 1.04 1.02

35 1.23 6.40 15.56 1.09 1.10

40 2.08 6.33 14.39 1.11 1.19

50 3.47 6.33 12.51 1.11 1.38

60 4.65 6.33 11.04 1.11 1.57

the combinations that would not. The �c ,�� values defining the


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limit state surface are obtained by searching c �or �� for a pre-scribed � �or c� that achieve both the conditions �1� G1=0 whereG1 is defined by Eq. �7� and �2� the collapse pressure �c in Eq. �7�is obtained by a maximization with respect to the geometricalparameters of the failure mechanism. For this purpose, a numeri-cal procedure was coded in Microsoft Excel Visual Basic. It callsthe Excel Solver iteratively in order to simultaneously satisfy thetwo preceding conditions. Fig. 8 provides graphical representationof the reliability analysis for both correlated and uncorrelatedshear strength parameters in the physical space of the randomvariables. One can easily see that negative correlation betweenshear strength parameters rotates the major axis of the ellipsefrom the vertical direction.

The critical probabilistic failure mechanisms obtained for bothuncorrelated and negatively correlated variables are plotted inFig. 9 using the values c� and �� of the design point �Table 2� andthe corresponding critical angular parameters of the failuremechanism. One can observe that the most probable failuremechanisms in the two cases are much more “extended” than thecritical failure mechanism obtained in the deterministic analysisby optimization of the tunnel pressure with respect to the geo-metrical parameters of the failure mechanism. This is due to the

Fig. 8. Unit and critical dispersion ellipses for correlated and uncor-related variables in the physical space of the random variables

Fig. 9. Critical collapse mechanisms in the �y ,z� plane



fact that the probabilistic failure mechanisms correspond to asmaller value of �. Thus, contrary to the critical failure mecha-nism obtained in the deterministic analysis, the probabilistic fail-ure mechanism outcrops the ground surface for both uncorrelatedand negatively correlated shear strength parameters.

Failure Probability

Both MC and IS simulations were performed for the computationof the failure probability. In this paper, these simulations werecarried out in the standardized space of uncorrelated variables.Hence, only uncorrelated normal random variables have beengenerated. The IS density function used in the standard uncorre-lated space is given as follows �e.g., Lemaire 2005�:

f�u� =1

�2�e−1/2�u − u��2

�14�

where u�=transformed value of the design point in the standarduncorrelated space of the random variables. When studying non-normal and/or correlated variables, the limit state surface, whichis determined point by point as explained in the previous section,was transformed to the standardized space of uncorrelated normalvariables using the equivalent normal transformation �i.e., theRackwitz-Fiessler equations� for each couple of �c ,��. The twoequations used for the transformation of each �c ,�� of the limitstate surface from the physical space to the standardized normaluncorrelated space �u1 ,u2� are �Lemaire 2005�

u1 = � c − �cN

�cN � �15�

u2 =1

�1 − �2�� − ��N

��N � − �� c − �c

N

�cN � �16�

where �=coefficient of correlation of c and �, and �cN, ��

N, �cN,

and ��N=respectively, the equivalent normal means and standard

deviations of the random variables c and �. They are determinedfrom the translation approach using the following equations:

c − �cN

�cN = �−1Fc�c�� 17�

� − ��N

��N = �−1F�� 18�

where Fc and F�=non-Gaussian cumulative distribution functions�CDFs� of c and �, and �−1� · �=inverse of the standard normalcumulative distribution. If desired, the original correlation matrix��ij� of the nonnormals can be modified to �ij� in line with theequivalent normal transformation, as suggested in Der Kiureghianand Liu �1986�. Some tables of the ratio �ij� /�ij are given inAppendix B2 of Melchers �1999�. For the cases illustrated herein,the correlation matrix, thus modified, differs only slightly fromthe original correlation matrix. Hence for simplicity, the examplesof this study retain the original unmodified correlation matrices�Low et al. 2007�.

Figs. 10 and 11 present, respectively, the failure probabilityand the corresponding coefficient of variation versus the numberof samples as given by MC and IS for normal and nonnormalcorrelated variables. The tunnel face pressure was equal to 50kPa. The expressions used for the computation of the failure prob-ability and the corresponding coefficient of variation in both MC
and IS simulations are given by Eqs. �3�–�6�. A computer program
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has been written in Microsoft Excel Visual Basic for these com-
putations. It should be mentioned that for the MC simulations, asingle set of samples was generated for the estimation of thefailure probability. This is because the difference between the twostudied cases was taken into account through the transformationof the limit state surface from the physical space to the rotatedstandard normal uncorrelated �u1, u2� space using Eqs. �15�–�18�.Also, the same set of samples can be used for uncorrelated vari-ables and for different values of the tunnel applied pressure �t.Notice however that in the IS method, a new set of samples wasgenerated for each probability distribution �normal and nonnor-mal� and correlation coefficient and for each value of the appliedpressure. This is because the design point changes with the prob-ability distribution and correlation of the random variables andwith the value of the tunnel applied pressure �t. Finally, noticethat the determination of the reliability index for use in the ISsimulations was determined using the dispersion ellipsoid ap-proach presented earlier.

Figs. 10 and 11 show that the convergence of the failure prob-ability calculated by IS is obtained for a sample size of 20,000with a coefficient of variation smaller than 1%. This value of thecoefficient of variation is much smaller than the commonlyadopted value used in the literature, i.e., 10%. In order to have aclear visualization of the convergence of the IS method, the maxi-mal number of samples represented on the x-axis of Figs. 10 and11 was limited to 200,000. For the MC simulation, a sample sizeof 5,000,000 was necessary to achieve an almost constant valueof the failure probability. The corresponding coefficient of varia-tion was smaller than 3%. Notice however that 35,000,000samples were necessary to achieve a coefficient of variationsmaller than 1%. Finally, notice that similar trends were obtainedin the case of uncorrelated normal and nonnormal variables �thefigures are not shown in the paper� and the same conclusions citedearlier remain valid in the case of uncorrelated variables �i.e., analmost constant value of Pf was obtained from MC simulationsbeyond 5,000,000 samples ; however, a smaller coefficient ofvariation of 1% was obtained in the present case�. In the fol-lowing, only IS simulation method will be used since it givesclose results with the MC simulations with a smaller sample size.All the subsequent results will be given for a maximal value of1% for the coefficient of variation of the estimator.

By varying the applied pressure on the tunnel face, the reli-ability index was calculated and the failure probability was plot-ted in Fig. 12 using FORM approximation and IS simulations fornormal, nonnormal, uncorrelated, and correlated variables. Fromthis figure, it is observed that the failure probability obtained fromFORM approximation are in good agreement with those obtainedfrom IS simulations for the commonly used values of the coeffi-cients of variation of the soil shear strength parameters �i.e.,COVc=20%, COV�=10%�. This means that FORM approxima-tion is an acceptable approach for estimating the failure probabil-ity for the commonly used values of the soil variability. It will beused in all subsequent computations.

In order to explain the good agreement between the two ap-proaches, the limit state surface is plotted. Fig. 13 shows the limitstate surface corresponding to a tunnel face pressure �t=50 kPafor normal and nonnormal uncorrelated random variables in thestandard space of normal uncorrelated variables. This figure alsoshows the linear FORM approximation, which is tangent to thelimit state surface at the design point. From this figure, it can beshown that the linear FORM approximation is very close to theexact limit state surface within the circle centered at the origin of

Fig. 10. Failure probability versus the number of samples for corre-lated variables as given by IS and MC

Fig. 11. Coefficient of variation of the failure probability versus thenumber of samples for correlated variables as given by IS and MC

Fig. 12. Comparison of the failure probability as given by FORMand IS
the rotated and transformed space and having a radius equal to 3.

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This explains why a good agreement between the two approachesis obtained especially for normal variables. In the case of uncor-related variables, the difference between the two failure probabili-ties given by FORM and IS is about 1.6% for normal variablesand becomes equal to 3.0% for nonnormal variables �Fig. 13�.

Sensitivity Analysis

Fig. 14 presents the CDFs of the tunnel face pressure for normal,nonnormal, correlated, and uncorrelated variables as given byFORM. When no correlation between shear strength parametersis considered, one can notice a more spread out CDF of the ap-plied pressure �i.e., a higher coefficient of variation of this pres-sure� with respect to the case of correlated shear strength. Thechosen probability distribution �i.e., normal, lognormal, and �distribution� does not significantly affect the values of the failureprobability.

Fig. 15 presents the effect of the coefficient of variation ofthe shear strength parameters on the failure probability. It can beseen that a small change in the coefficient of variation of � highlyaffects the failure probability. On the other hand, this failure prob-ability is less sensitive to changes in the uncertainty of the cohe-sion. Thus, the failure probability is highly influenced by thecoefficient of variation of �. The greater the scatter in �, the

a. Normal variables

b. Nonnormal variablesFig. 13. Limit state surface and FORM approximation in the uncor-related case

higher the failure probability. This means that accurate determi-



nation of the uncertainties of the angle of internal friction � isvery important in obtaining reliable probabilistic results.

Probability Density Function of the TunnelFace Pressure

Fig. 16 shows the PDFs corresponding to the CDFs given inFig. 14. The PDFs were determined by numerical derivation ofthe CDFs. It can be seen that the results of normal and nonnormalvariables are nearly similar. The correlation between the variableshas on the contrary an important influence, making the probabilitydensity more significant around the deterministic value of theapplied pressure. By fitting the PDF of the tunnel pressure to anempirical PDF �normal, lognormal, gamma� as shown in Fig. 17,it was found �after minimization of the sum of the relative errorsbetween the values of the computed PDF and those of the empiri-cal distribution� that the lognormal distribution is the one that bestfits the computed PDF especially in the distribution tail of interestto the engineering practice �i.e., where �t�2�c�. It is then easy touse the lognormal CDF to determine the failure probability for agiven applied tunnel pressure.

Fig. 14. CDFs of the tunnel face pressure

Fig. 15. Comparison of failure probabilities for different values ofthe coefficients of variation of c and �

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Reliability-Based Design

A reliability-based design �RBD� has been performed in this sec-tion. It consists of the calculation of the required tunnel facepressure for a target collapse reliability index of 3.8 as suggestedby Eurocode 7 for the ULS. This tunnel pressure is called here-after “probabilistic tunnel pressure.”

Fig. 18 presents the probabilistic tunnel pressure for differentvalues of the coefficients of variation of the shear strength param-eters and their coefficient of correlation when the random vari-ables follow normal distributions. This figure also presents thedeterministic tunnel face pressure �56.6 kPa� corresponding toa safety factor against collapse �Fs=�t /�c� equal to 2. Theprobabilistic tunnel face pressure decreases with the decrease ofthe coefficients of variation of the shear strength parametersand the increase of the negative correlation between these param-eters. It can become smaller than the deterministic tunnel facepressure for some values of the soil variability �i.e., COV�=5%,COVc=20%�. However, for high values of the coefficients ofvariation, the required tunnel face pressure is much higher than

Fig. 16. PDFs of the tunnel face pressure

Fig. 17. Fit of the PDF of the tunnel pressure in the normal uncor-related case



the deterministic value. As a conclusion, the deterministic tunnelface pressure may be higher or lower than the reliability-basedtunnel face pressure, depending on the uncertainties of the ran-dom variables and the correlation between these variables.

Conclusions

A reliability-based analysis and design of the face stability of ashallow circular tunnel driven by a pressurized shield was per-formed. Only the collapse and the blow-out failure modes of theULS were studied. Two rigorous deterministic models based onthe upper-bound method of limit analysis were used. It wasshown that• Although the results given by the upper-bound approach in

limit analysis are unsafe estimates of the collapse and blow-out loads, they are the best ones compared to the availablesolutions in both the active and passive cases. This is becausethe present mechanisms provide greater solutions than thosegiven by Leca and Dormieux �1990� in the collapse case andsmaller results than those of these writers in the case of blow-out.

• The blow-out mode of failure does not occur for the casescurrently encountered in practice. Hence, only the collapsefailure mode was considered in the probabilistic analysis anddesign against face stability.

• The present limit analysis results obtained from the M1 col-lapse mechanism are not very far from the solutions given bycomplex 3D numerical simulations using FLAC3D.

• The collapse reliability index increases with the increase of thetunnel face pressure.

• FORM approximation is an acceptable approach for estimatingthe failure probability against collapse for the commonly usedvalues of the soil variability.

• The assumption of uncorrelated shear strength parameters wasfound conservative �i.e., it gives a greater failure probability�in comparison to that of negatively correlated parameters;however, the type of the probability distribution does not sig-nificantly affect the values of the failure probability.

• The failure probability is more sensitive to � than to c. Thegreater the scatter in �, the higher the failure probability. Thismeans that the accurate determination of the uncertainties ofthe angle of internal friction � is important in obtaining reli-able probabilistic results.

• When no correlation between shear strength parameters is con-sidered, a more spread out CDF of the tunnel pressure wasobtained in comparison to the case of correlated shear strengthparameters.

• The distribution of the PDF of the tunnel pressure was foundvery close to a lognormal distribution. This allows one to eas-ily determine the failure probability against collapse for agiven face pressure.

• A RBD has shown that the tunnel pressure determined proba-bilistically decreases with the increase of the negative correla-tion between the shear strength parameters and the decrease oftheir coefficients of variation.

Appendix. Computation of the Tunnel CollapsePressure—M1 Mechanism

The formulation is given here for a five-block mechanism. The
formulation for another number of blocks is straightforward.

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Geometry

The distance from the extremity of each block i to the tunnelcrown �hi� and the height of the extremity of the last block �h5��are given by �Fig. 2�a��

h5� = h5 ·sin�2�4 + 2�2 + � − ��

cos��4,5 − ��− �H −

D

2� �19�

with � h2 = D ·cos�� + �� · cos��1 − � + ��

sin�2��

hi = h2 · k=2

i−1 � cos��k,k+1 + ��cos��k−1,k − �� for i � 3� �20�

The volumes of the extreme blocks �first and fifth blocks� are

Fig. 18. Comparison between determ

k=2



V1 =A1 · h1 − A1,2 · h2

3�21�

V5 =A4,5 · h5 − A5 · h5�

3�22�

The volume of an intermediate block i �for 2� i�4� is

Vi =Ai−1,i · hi − Ai,i+1 · hi+1

3�23�

If h5��0, the value of the outcropping surface of the fifthblock is

c and probabilistic design pressures

A5 =�

cos��· � h5� · sin�2��

2 sin�2�4 + 2�2 + � + �� · sin�2�4 + 2�2 + � − ��2

·�sin�2�4 + 2�2 + � + ��sin�2�4 + 2�2 + � − ��

�24�

Otherwise, the mechanism does not outcrop and A5=0.The area of the elliptical surface resulting from the intersection of the first cone �adjacent to the tunnel face� with the circular tunnel

face is

A1 =� · D2

4 cos��· �cos�� − �� · cos�� + �� 25�

The area of the contact elliptical surface between two successive blocks i and i+1 is given by

�A1,2 =� · D2

4 cos��· cos�� + ��2 ·

�cos��1 − � + ��cos��1 − � − ��1.5

Ai,i+1 =� · D2

4 cos��· cos�� + ��2 ·

�cos��i,i+1 + ��cos��i,i+1 − ��1.5 ·

k=2

i � cos��k−1,k + ��2

cos��k−1,k − �� for i � 2� �26�

Kinematics

The velocity of the block i and the relative velocity between theblocks i and i+1 are

vi = v1 · i

cos��k−1,k + ��cos��k−1,k − ��

for i � 2 �27�

vi,i+1 = vi ·sin�2�i,i+1�

cos��i,i+1 − ��for i � 1 �28�

where ��0,1 = � � �29�

inisti

�i,i+1 = �i − �i−1,i for i � 1

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Civ

il E

ngin

eers

. All

righ

ts r

eser

ved.
Expressions of N� and Ns
For a five-block collapse mechanism, the critical collapse pres-sure can be calculated from Eqs. �10� and �11�, with

N =P31 + P41 + P51 + P52

D�30�

Ns =

v5

v1· sin�2�2 + 2�4 + �� · A5

A1 · cos��31�

where P51 =

v4

v1· sin�2�1 + 2�3 − �� · V4

A1 · cos��32�

P52 =

v5

v1· sin�2�2 + 2�4 + �� · V5

A1 · cos��33�

P41 =

v2

v1· sin�2�1 − �� · V2 +

v3

v1· sin�2�2 + �� · V3

A1 · cos��34�

P31 =V1 · sin��A1 · cos��

�35�

References

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