Probabilistic analysis of the face stability of circular tunnels

Transcription

1 Probabilisti analysis of the fae stability of irular tunnels Guilhem Mollon 1, Daniel Dias 2 and Abdul-Hamid Soubra 3, M. ASCE 1 PhD Student, INSA Lyon, LGCIE Site Coulomb 3, Géotehnique, Bât. J.C.A. Coulomb, Domaine sientifique de la Doua, Villeurbanne edex, Frane. Guilhem.Mollon@insa-lyon.fr 2 Assoiate Professor, INSA Lyon, LGCIE Site Coulomb 3, Géotehnique, Bât. J.C.A. Coulomb, Domaine sientifique de la Doua, Villeurbanne edex, Frane. Daniel.Dias@insalyon.fr 3 Professor, University of Nantes, GeM, UMR CNRS 6183, Bd. de l université, BP 152, Saint- Nazaire edex, Frane ABSTRACT: This paper presents a reliability-based approah for the threedimensional analysis of the fae stability of a shallow irular tunnel driven by a pressurized shield. Only the ollapse failure mode of the ultimate limit state is studied. The deterministi model is based on the upper-bound method of the limit analysis theory. The random variables used are the soil shear strength parameters. The Hasofer-Lind reliability index and the failure probability were determined. A sensitivity analysis was also performed. It was shown that (i) the assumption of negative orrelation between the soil shear strength parameters gives a greater reliability of the tunnel fae against ollapse, with respet to the hypothesis of unorrelated variables, (ii) the failure probability is muh more influened by the oeffiient of variation of the angle of internal frition than that of the ohesion and (iii) when no orrelation between shear strength parameters is onsidered, a more spread out CDF of the tunnel pressure was obtained in omparison to the ase of orrelated shear strength parameters. INTRODUCTION Over the past thirty years, tunnelling in a fritional and/or ohesive soil has been possible due to reent tehnologial advanes inluding the pressurized shield. Fae stability analysis of shallow irular tunnels driven by the pressurized shield is of major importane. The tunnel fae pressure must avoid both the ollapse (ative failure) and the blow-out (passive failure) of the soil mass nearby the tunnel fae. In this paper, only the ollapse mode of the ultimate limit state (ULS) is onsidered in the analysis. 1

2 The fae stability analysis is onduted based on a probabilisti approah. The soil shear strength parameters are modeled as random variables. The reliability-based analysis is more rational than the deterministi one sine it takes into aount the inherent unertainty of the input variables. Nowadays, this is possible beause of the improvement of our knowledge on the statistial properties of the soil (Phoon and Kulhawy 1999; Baeher and Christian 2003). OVERVIEW OF RELIABILITY CONCEPTS The reliability index is a measure of the safety that takes into aount the inherent unertainties of the input variables. A widely used reliability index is the Hasofer and Lind (1974) index. Its matrix formulation is (Ditlevsen 1981) β HL T 1 ( x µ ) C ( x µ ) = min (1) x F in whih x =vetor representing the n random variables; µ =vetor of their mean values; C=ovariane matrix; and F=failure region. The minimization of Eq. 1 is performed subjet to the onstraint G ( x) 0 where the limit state surfae G ( x) = 0, separates the n dimensional domain of random variables into two regions: A failure region F represented by G ( x) 0 and a safe region given by G ( x) > 0. Low and Tang (1997) have shown that the minimization of the reliability index is equivalent to find the smallest dispersion ellipsoid (or ellipse in the two-dimensional ase) that is tangent to the limit state surfae and whose entre is at the equivalent normal mean point of the random variables. This ellipsoid is alled 'ritial dispersion ellipsoid' and the ellipsoid orresponding to β HL = 1 in Eq. 1 without the min is named 'unit dispersion ellipsoid'. The ratio between the ritial dispersion ellipsoid and the unit dispersion one is the Hasofer-Lind reliability index. When the random variables are non-normal and orrelated, the optimisation approah uses the Rakwitz-Fiessler equivalent normal transformation without the need to diagonalize the orrelation matrix as shown in Youssef Abdel Massih et al. (2008). The N omputations of the equivalent normal mean µ and equivalent normal standard N deviation σ for eah trial design point are automatially found during the onstrained optimization searh. The method of omputation of the reliability index using the onept of an expanding ellipse suggested by Low and Tang (1997) is used in this paper. RELIABILITY ANALYSIS OF A CIRCULAR TUNNEL AGAINST FACE COLLAPSE The aim of this paper is to perform a reliability analysis of a shallow irular tunnel against fae ollapse. The irular tunnel is driven by a pressurized shield in a -ϕ soil. The problem an be idealized as shown in Fig. 1 by onsidering a irular rigid tunnel of diameter D driven under a depth of over C. A surharge σ is applied at s 2

3 the ground surfae and a onstant retaining pressure σ t is applied to the tunnel fae to simulate tunnelling under ompressed air. Due to unertainties in soil shear strength parameters, the ohesion, and the angle of internal frition ϕ are onsidered as random variables. The performane funtion G used in the reliability analysis is defined as follows: G = σ σ (2) t where σ t is the applied pressure on the tunnel fae, and σ is the ollapse pressure. For the omputation of the ollapse pressure σ, a multiblok failure mehanism is proposed in the framework of the kinematial approah of limit analysis. The proposed M1 mehanism allows the slip surfae to develop more freely in omparison with the available mehanism given by Lea and Dormieux (1990). It is omposed of several trunated rigid ones with irular ross-setions and with opening angles equal to 2 ϕ. Eah one is the mirror image of the adjaent one with respet to the normal plane to the surfae separating these ones. This is a neessary ondition to assure the same elliptial ontat area between adjaent ones. The upper rigid one will or will not interset the ground surfae depending on the C/D value. It should be mentioned that M1 is a translational kinematially admissible failure mehanism. The different bloks of this mehanism move as rigid bodies. These rigid ones translate with veloities of different diretions, whih are ollinear with the ones axes and make an angle ϕ with the disontinuity surfae. The veloity of eah one is determined by the ondition that the relative veloity between the ones in ontat has the diretion that makes an angle ϕ with the ontat surfae. The veloity hodograph is presented in Fig. 1. The present mehanism is ompletely defined by n angular parameters α and i = 1,..., n 1 where n is the number of rigid bloks. β ( ) i FIG. 1. Failure mehanism M1 In the present paper, by the Low and Tang (1997) method, one literally sets up a tilted ellipsoid in the Exel spreadsheet and minimizes the dispersion ellipsoid subjet to the onstraint that it be tangent to the limit state surfae. 3

4 NUMERICAL RESULTS For the M1 mehanism, when the total rate of energy dissipation and the total rate of external work are equated, the ollapse tunnel pressure σ an be expressed as follows: σ = γ DN + N + σ N (3) γ s s where N γ, N, and N s are non-dimensional oeffiients. They represent respetively the effet of soil weight, ohesion, and surharge loading. The expressions of the different oeffiients are given in Oberlé (1996). In Eq. 3, σ depends not only on the mehanial and geometrial harateristis, ϕ, and C / D, but also on the geometrial parameters of the failure mehanism α and β i i = 1,..., n 1. The deterministi ollapse pressure σ was obtained by maximization ( ) of the Eq. 3 with respet to the α and β i angles. It was shown that the inrease in the number of ones improves the solution. This improvement beomes insignifiant (smaller than 1%) for a number of bloks greater than five. Therefore, only five bloks were used in this paper. The present failure mehanism was shown to give a better (i.e. greater) solution than the available upper-bound solution given by Lea and Dormieux (1990). For more details on this omparison, the reader may refer to Soubra et al. (2008). FIG. 2. Reliability index versus The probabilisti numerial results presented in this paper onsider the ase of a irular tunnel with a diameter D=10 m and a over C=10 m (i.e. C/D=1). The soil σ = 0 is onsidered in the has a unit weight of 18 kn/m 3. No surharge loading ( ) analysis. For the probability distribution of the random variables, two ases are σ t s 4

5 studied. In the first ase, referred to as normal variables, and ϕ are onsidered as normal variables. In the seond ase, referred to as nonnormal variables, is assumed to be lognormally distributed while ϕ is assumed to be bounded and a beta distribution is used. The parameters of the beta distribution are determined from the mean value and standard deviation of ϕ (Haldar and Mahadevan 2000). For both ases, orrelated and unorrelated variables are onsidered. In this paper, the illustrative values used for the statistial moments of the shear strength parameters and their oeffiient of orrelation ρ, ϕ are as follows: µ = 7 kpa, µ ϕ = 17 o, COV = 20%, COV ϕ = 10% and ρ, ϕ = The values of the COV and the negative orrelation are taken from Phoon and Kulhawy (1999). The reliability index is determined by minimizing the quadrati form of Eq. 1 not only with respet to the random variables, but also with respet to the geometrial parameters of the failure α, (f. Youssef Abdel Massih et al. 2008). Thus, the minimization is mehanism ( β i ) performed with respet to 7 parameters ( α, β,, ϕ) i. Table 1. Reliability index and design point Unorrelated σ t Correlated variables variables (kpa) β HL * φ* β HL * φ* Normal variables Nonnormal variables β HL * φ* β HL * φ* The ollapse pressure was found equal to σ = 28.3 kpa for normal variables and to 28.8 kpa for nonnormal variables, using respetively the mean and equivalent mean values of the random variables. The orresponding reliability index is equal to zero. Fig. 2 and Table 1 present the Hasofer-Lind reliability index versus σ t for four ombinations of normal, nonnormal, unorrelated and orrelated shear strength parameters. The reliability index orresponding to unorrelated variables is smaller than the one of negatively orrelated variables for both normal and nonnormal variables. One an onlude that assuming unorrelated shear strength parameters is onservative in omparison to assuming negatively orrelated parameters. 5

6 FIG. 3. Unit and ritial dispersion ellipses for orrelated and unorrelated variables, and ritial ollapse mehanisms For σ t = 70 kpa (Table 1), the reliability index for unorrelated and orrelated normal variables are respetively equal to 4.02 and The orresponding most probable failure points obtained from the minimization proedure are found to be at * 4.53, * = kpa ϕ = o * * o = 7.11 kpa, ϕ = These are the points of ( ) and ( ) tangeny of the ritial dispersion ellipses with the limit state surfae. Fig. 3 provides graphial representation of the reliability analysis for both ases in the physial spae of the random variables. One an easily see that negative orrelation between shear strength parameters rotates the major axis of the ellipse from the vertial diretion. The ritial probabilisti failure mehanisms obtained for both unorrelated and negatively orrelated variables are also plotted in Fig. 3 using the values * and φ* of the design points and the orresponding ritial angular parameters of the failure mehanism. One an observe that the most probable failure mehanisms (i.e. the mehanisms plotted with the optimized angles at the design point) in the two ases are muh more extended than the ritial failure mehanism obtained in the deterministi analysis by optimization of the tunnel pressurethis is due to the fat that the probabilisti failure mehanisms orrespond to a smaller value of φ. Fig. 4 presents the CDFs of the tunnel fae pressure for normal, nonnormal, orrelated and unorrelated variables as given by FORM. When no orrelation between shear strength parameters is onsidered, one an notie a more spread out CDF of the tunnel pressure (i.e., a higher oeffiient of variation of this pressure) with respet to the ase of orrelated shear strength. Notie however that the hosen probability distribution (i.e., normal, lognormal, and beta distribution) does not signifiantly affet the values of the failure probability. Fig. 5 presents the effet of the oeffiient of variation of and φ on the failure probability. A small hange in the oeffiient of variation of φ highly affets the failure probability, but this failure probability is less sensitive to hanges in the unertainty of. Thus, the failure probability is highly influened by the oeffiient of variation of φ. 6

7 FIG. 4. CDFs of the tunnel fae pressure FIG. 5. Influene of the oeffiients of variation of and φ FIG. 6. PDFs of the tunnel fae pressure Fig. 6 shows the PDFs orresponding to the CDFs given in Fig. 4. The PDFs were determined by numerial derivation of the CDFs. It an be seen that the results of normal and nonnormal variables are nearly similar. The orrelation between the variables has on the ontrary an important influene, making the probability density more signifiant around the deterministi value of the applied pressure. By fitting the PDF of the tunnel pressure to an empirial probability density funtion (Normal, Lognormal, Gamma) as shown in Fig. 7, it was found (after minimization of the sum of the relative errors between the values of the omputed PDF and those of the empirial distribution) that the lognormal distribution is the one that best fits the omputed PDF espeially in the distribution tail (i.e. where σ t >2σ ). It is then easy to use this lognormal distribution funtion to determine the failure probability for a given applied tunnel pressure. CONCLUSION FIG. 7. Fit of the PDF of the tunnel pressure A reliability-based analysis of the fae stability of a shallow irular tunnel driven by a pressurized shield was performed. Only the ollapse failure mode of the ultimate limit state was studied. A deterministi model based on the upper-bound method of limit analysis was used. The main onlusions of the paper are: 7

8 The assumption of unorrelated shear strength parameters was found onservative (i.e., it gives a greater failure probability) in omparison to that of negatively orrelated parameters; however, the type of the probability distribution does not signifiantly affet the values of the failure probability; The failure probability is more sensitive to ϕ than to. The greater the satter in ϕ, the higher the failure probability. This means that the aurate determination of the unertainties of ϕ is important in obtaining reliable probabilisti results; When no orrelation between shear strength parameters is onsidered, a more spread out CDF of the tunnel pressure was obtained in omparison to the ase of orrelated shear strength parameters; The distribution of the probability density funtion of the tunnel pressure was found very lose to a lognormal distribution. This allows one to easily determine the failure probability against ollapse for a given fae pressure. In this study, the soil is onsidered as homogeneous in eah simulation, and the variability is taken into aount from one simulation to another. A next step would be to study the influene of a spatially varying field of the strength parameters, to be as lose as possible to a real soil. REFERENCES Baeher, G.B., and Christian, J.T. (2003). Reliability and statistis in geotehnial engineering, Wiley. Ditlevsen, O. (1981). Unertainty modelling: With appliations to multidimensional ivil engineering systems, MGraw-Hill, New York. Haldar, A., and Mahadevan, S. (2000). Probability, reliability and statistial methods in engineering design, Wiley, New York. Hasofer, A. M., and Lind, N. C. (1974). "Exat and invariant seond-moment ode format." J. of Engrg. Meh. Div., 100 (1): Lea, E., and Dormieux, L. (1990). "Upper and lower bound solutions for the fae stability of shallow irular tunnels in fritional material." Géotehnique, 40 (4): Low, B. K., and Tang, W. H. (1997). "Effiient reliability evaluation using spreadsheet." J. Eng. Meh., 123 (7): Oberlé, S. (1996). Appliation de la méthode inématique à l étude de la stabilité d un front de taille de tunnel, Final Projet, ENSAIS, Frane, in Frenh. Phoon, K.-K., and Kulhawy, F. H. (1999). "Evaluation of geotehnial property variability." Can. Geoteh. J., 36: Soubra, A.-H., Dias, D., Emeriault, F. and Kastner, R. (2008). "Three-dimensional fae stability analysis of irular tunnels by a kinematial approah." Geoongress 2008, GSP N 179, ASCE, New Orleans/Louisiana: Youssef Abdel Massih, D.S., Soubra, A.-H., and Low, B.K. (2008). "Reliabilitybased analysis and design of strip footings against bearing apaity failure." J. of Geoteh. & Geoenvironmental Engrg., 134 (7):