Extension of CSRSM for the Parametric Study of the Face Stability of Pressurized Tunnels

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1 Extension of CSRSM for the Paraetric Study of the Face Stability of Pressurized Tunnels Guilhe Mollon 1, Daniel Dias 2, and Abdul-Haid Soubra 3, M.ASCE 1 LGCIE, INSA Lyon, Université de Lyon, Doaine scientifique de la Doua, Bât JCA Coulob, 20 Avenue Albert Einstein, Villeurbanne, France ; guilhe.ollon@insa-lyon.fr 2 LGCIE, INSA Lyon, Université de Lyon, Doaine scientifique de la Doua, Bât JCA Coulob, 20 Avenue Albert Einstein, Villeurbanne, France ; daniel.dias@insa-lyon.fr 3 GeM, University of Nantes, Boulevard de l'université, BP 152, Saint-Nazaire cedex, France ; abed.soubra@univ-nantes.fr ABSTRACT The Collocation-based Stochastic Response Surface Methodology (CSRSM is a powerful probabilistic ethod. It ais at replacing a coplex deterinistic odel by a siple analytical expression (called eta-odel to reduce the tie cost of the classical probabilistic ethods. The eta-odel is based on a Polynoial Chaos Expansion (PCE. The coefficients of the PCE are coputed in this paper by regression fro the response of the deterinistic odel at a liited nuber of collocation points. The conventional foralis of CSRSM requires perforing a new set of deterinistic coputations each tie the probabilistic paraeters of the input rando variables are slightly odified. An extension of CSRSM is therefore proposed in this paper. It allows the realization of a paraetric study at a liited tie cost without loss of accuracy. This is deonstrated by coparing the results obtained fro the original CSRSM and its extension when perforing a paraetric study concerning the stability analysis of a pressurized tunnel face. INTRODUCTION As is well known, the ost robust probabilistic approach is the Monte-Carlo siulation ethod. This ethod requires a large nuber of calls of the deterinistic odel especially for sall values of the failure probability (e.g. about 1,000,000 saples for a failure probability of Although the deterinistic nuerical ethods (such as those based on the Finite Eleent Method FEM or the Finite Difference Method FDM are appealing, these ethods are difficult to use in a probabilistic fraework particularly when eploying MC siulation ethod. This is because these ethods need a very long coputation tie. In order to overcoe the above shortcoing, the Collocationbased Stochastic Response Surface Methodology (CSRSM is applied herein. CSRSM akes possible the probabilistic study of a coplex deterinistic odel by replacing it by an analytical expression (called eta-odel with a reduced tie cost. This eta-odel is expressed by a Polynoial Chaos Expansion (PCE. The coefficients of the PCE are deterined in this paper by a regression approach using a liited nuber of calls of the deterinistic odel. The eta-odel can then be used in the classical probabilistic ethods such as the Monte Carlo siulation ethod. It should be entioned that the conventional foralis of CSRSM requires perforing a new set of deterinistic coputations each tie the probabilistic paraeters of the

2 input rando variables are slightly odified. An extension of CSRSM is therefore proposed in this paper. It allows the realization of a paraetric study at a liited tie cost without loss of accuracy. As an illustrative application of CSRSM and its extension, one considers in this paper the probabilistic stability analysis of a pressurized tunnel face. The paper is organised as follows: A brief description of the deterinistic odel concerning the collapse pressure of a pressurized tunnel face is first presented. It is followed by the presentation of CSRSM and its extension. Finally, a verification of CSRSM (by coparison with the results of Monte Carlo siulation ethod and a coparison between the results obtained fro both CSRSM and its extension are presented and discussed. DETERMINISTIC MODEL The face stability analysis of a circular tunnel driven by a pressurized shield is of ajor interest. It requires the deterination of a so-called critical collapse pressure of the tunnel face (denoted σ c, i.e. the greatest one under which the tunnel face collapses. The deterinistic odel chosen in this study is analytical. It is based on the kineatical theore of liit analysis. The collapse echanis is translational. It is coposed of five conical blocks (Fig. 1 with an opening angle equal to φ where φ is the soil friction angle. Figure 1. Five-block collapse echanis This echanis was extensively presented in Mollon et al. [2009]: it is able to provide a quite correct estiation of σ c despite the fact that it does not intersect the entire circular tunnel face as was observed experientally by Takano et al. (2006. The analytical five-block odel was chosen in this study because its extreely sall tie cost (saller than 0.1s akes it very convenient for the verification of CSRSM. This verification is perfored by coparison with Monte Carlo siulation ethod which requires a great nuber of calls of the deterinistic odel. In this study, the output rando variable is the critical collapse pressure, and the two input rando variables are the friction angle (φ and the cohesion (c of the soil.

3 BRIEF DESCRIPTION OF CSRSM In this section, the classical foralis of CSRSM (Isukapalli [1999], Sudret [2007], Phoon and Huang [2007] and Huang et al. [2009] is presented in the case of two input rando variables (φ and c. CSRSM ais at replacing the initial deterinistic odel by an approxiate analytical expression (eta-odel when perforing a probabilistic analysis. The eta-odel is a Polynoial Chaos Expansion (PCE. For a PCE of order n, the eta-odel is expressed in the basis of the ultidiensional Herite polynoials of order n. For a given set of the probabilistic paraeters of the input rando variables (i.e. type of distribution and statistical oents of the rando variables, and correlations between rando variables; the unknown coefficients of the PCE are obtained in this paper by regression using the response of the deterinistic odel at a given nuber of collocation points. The different steps of CSRSM ay be described as follows: The two rando variables (i.e. φ and c have first to be represented in the PCE by two standard variables ξ 1 and ξ 2, which are noral uncorrelated variables with zero ean and unit variance. For a PCE of order n, the available collocation points are classically deterined as follows: each standard variable is assued to take the values of the roots of the one-diensional Herite polynoial of order n+1, and the available collocation points result fro the cobinations of all these roots for the different rando variables. It should be noted that this ethod of choosing the collocation points is not andatory as will be seen later in this paper. For the present case of two input rando variables, the output rando variable ay be expressed by a PCE as follows: U = p a i i= 1 i ( ξ 1,ξ 2 (1 In this expression, the ters i are ultidiensional Herite polynoials (which constitute the basis of the PCE and the ters a i are unknown coefficients to be deterined. The nuber p of ters in this suation and the nuber M of the available collocation points are given by: ( n + n v! p = (2 n! n! v n ( 1 v M= n+ + 1 otherwise th 0 if one of the roots of the (n+1 univariate Herite polynoial is 0 (3 where n v is the nuber of rando variables. It should be entioned that when the nuber n v of the rando variables is iportant, the nuber M of the available collocation points becoes uch larger than the nuber p of the unknown coefficients. This leads to a linear syste of equations where the nuber of equations is greater than the nuber of unknown. It is therefore necessary to ake a judicious choice of the collocation points to be used in the regression process. Several ethods exist (Isukapalli [1999], Sudret [2007], but they are useless when using only two rando variables as is the case in the present paper. Hence, all the available collocation points are considered in the analysis. To be introduced in the deterinistic

4 odel, the collocation points represented by vectors ( ξ 1,,ξ 2, (with 1 M have to be expressed in the space of the physical variables corresponding to vectors ϕ,, using the following expressions: ( c ξ1 C, ξ1, = H ξ ξ 2 C, 2, ϕ = F c = Fc ϕ [ Φ( ξ1 ] [ Φ( ξ ] 2 (4 (5 In these equations, H is the Cholesky transfor of the correlation atrix of φ and c, F φ is the cuulative density function (CDF of φ, F c is the CDF of c, and Φ is the noral cuulative density function with zero ean and unit variance. When the response of the deterinistic odel (i.e. the value of σ c is obtained for each collocation point, the values of the PCE coefficients are deterined by regression using the following equation: t t N N a N f = (6 where a=colun vector of the unknown coefficients, f=colun vector of the deterinistic responses, and N is a atrix given by: N = 0,0 0,0 0,0 1,0 1,0 1,0 ( ξ1,1 ( ξ 1,2 ( ξ2,1 ( ξ ( ξ1,1 ( ξ ( ξ1,1, ξ2,1 ( ξ, ξ ( ξ2,1 ( ξ ( ξ ( ξ ( ξ ( ξ, ξ ( ξ 1, M 0,1 0,1 0,1 2,2 2, M 2,0 2,0 2,0 1,2 1, M Once the unknown coefficients are deterined, one obtains an analytical equation (i.e. a eta-odel that can be used in place of the deterinistic odel to perfor a probabilistic analysis using (for exaple Monte Carlo siulation ethod. VERIFICATION OF CSRSM A set of probabilistic paraeters is chosen as a reference case for the two input rando variables φ and c as follows: the two variables are considered as noral uncorrelated variables with μ φ =17 and COV(φ=10%, μ c =7 kpa and COV(c=20%. Characteristics of a soft clay are chosen because this kind of soil is likely to allow face instabilities. For this reference case, the coefficients of the PCEs of orders 2 to 5 are deterined using the process described above. Fig. 2 shows the response surfaces (represented by lines of equal value of σ c in a φ-c plane as provided by the original deterinistic odel and by its approxiations by the PCEs of orders 2 and 4. It clearly appears that both PCEs are able to provide a very good approxiation of the original deterinistic odel in the central part (i.e. in the region of axiu probability of occurrence, but that the order 4 is uch ore accurate in the surrounding areas corresponding to zones with lower probabilities of occurrence. 1,1 1,1 1,1 1,2 1, M 2,2 2, M 0,2 0,2 0,2 2,2 2, M (7

5 A Monte-Carlo (MC siulation with 10 6 saples is perfored on the original deterinistic odel and on the PCEs of orders 2 to 5. This siulation is only ade possible due to the very sall tie cost of the original deterinistic odel chosen in this paper, which allows a great nuber of saples. The results in ters of probability density function (PDF and in ters of failure probability are shown in Figs. 3a and 3b respectively. It appears fro Fig. 3a that the PDF obtained for the eta-odels are in good agreeent with the one of the original deterinistic odel, especially when the PCE order is larger than 2. Moreover, Fig. 3b shows that all the PCEs of order greater than 2 provide a good estiation of the failure probabilities at the distribution tail which is the zone of interest of the geotechnical engineer; the accuracy increases with the PCE order. Figure 2. Response surfaces of the critical collapse pressure (in kpa as provided by the original deterinistic odel and by the PCEs of orders 2 and 4 a. b. Figure 3. Results of the Monte-Carlo Siulation; a. PDF of σ c ; b. Failure probability As a conclusion, the PCE of order 2 provides good results only in ters of trends, but the PCEs of orders 3, 4 and 5 provide very close probabilities to the ones of the original deterinistic odel. There is also a good agreeent with FORM results obtained in a previous study (Mollon et al. [2009]. The PCE of order 4 sees

6 to be the best coproise between accuracy and coputational cost, but the order 3 ay reasonably be used for coputationally-expensive odels. Notice finally that the nuber of collocation points used for the fourth order PCE is equal to 25, which eans that 25 calls of the deterinistic odel were needed to obtain this PCE. EXTENSION OF CSRSM CSRSM is an efficient and accurate tool for the probabilistic study of a echanical odel, but it suffers fro the fact that its conventional foralis iplies a new set of deterinistic coputations each tie one of the probabilistic paraeters of the input variables (such as the COV or the correlation coefficient is changed. This is due to equations (4 and (5 which state that the position of the collocation points in the physical space is dependant on the correlation atrix and the CDF of the input variables. The tie cost of a paraetric study (when using a coputationallyexpensive odel is therefore very expensive, while such a study ay be interesting because these paraeters are usually not known with a great accuracy. This section presents an extension of CSRSM, which akes possible the use of existing deterinistic results to carry on a paraetric study with no additional tie cost. This extension is based on the fact that the position of the collocation points described earlier is not andatory and that it is possible to use other kinds of collocation schees, without loss of accuracy of the eta-odel. For exaple, Fig. 4a shows two response surfaces of σ c in the physical φ-c plane. The first one corresponds to a PCE of order 4 used for the reference case described above, and the second one corresponds to a PCE of order 4 for a different probabilistic case (lognoral variables with μ φ =17 and COV(φ=8%, μ c =7 kpa and COV(c=25% and with a correlation coefficient ρ φc =-0.3. Since the type of the probability distributions, the COVs and the correlation coefficient have been changed, the collocation points are different with respect to the reference case. However, the two response surfaces appear very siilar on the entire range of the paraeters. This illustrates the clai that there is no need for new calls of the deterinistic odel for a paraetric study. When a first set of deterinistic coputations has been perfored and a new set of probabilistic paraeters is chosen, one ay copute the positions of the collocation points in the standard space which would have led to the physical collocation points that were already coputed. These positions ( ξ 1,,ξ 2, are obtained by the following equations: ξ 1 ξ 2 ξ 1, ξ 2, = Φ = Φ = H ( Fϕ ( ϕ ( F ( c c ξ 1 ξ 2 1,,ξ 2, for which the corresponding deterinistic results are already available. Equation (6 can therefore be applied to copute the coefficients of a new PCE, with no additional deterinistic coputation. A MC siulation with 10 6 saples is carried out for the new probabilistic properties described above, using the three following odels: (i the original deterinistic odel, (ii the PCE of order 4 based on the conventional This process provides a new set of collocation points ( ξ (8 (9

7 CSRSM (which iplies 25 new calls to the deterinistic odel, and (iii the PCE of order 4 based on the proposed extension ethod (which uses only the existing 25 deterinistic results and do not require any new call to the deterinistic odel. a. b. Figure 4. a. coparison between the response surfaces provided by two PCEs with different probabilistic paraeters; b. Statistical oents of the conventional and proposed CSRSM (noralized to the ones of the original deterinistic odel a. b. Figure 5. Coparison of the classical and the proposed CSRSM with respect to the original deterinistic odel; a. PDF of σ c ; b. Failure probability Figs. 4b, 5a and 5b present a coparison between the three odels, respectively in ters of the statistical oents of σ c, in ters of the PDF of σ c, and in ters of the probability of failure at the distribution tail. A very good agreeent appears between the two eta-odels. It clearly eans that the proposed extension of CSRSM does not reduce the accuracy of the probabilistic results. Notice however that the proposed ethodology is only relevant if the area covered by the new collocation points in the standard space is not too different fro the one obtained using the original collocation schee. Fro a practical point of view, it eans that the changes ade to the probabilistic paraeters of the input variables should reain sall during the paraetric study.

8 CONCLUSION The Collocation-based Stochastic Response Surface Methodology (CSRSM is presented and applied to an analytical deterinistic odel (devoted to the deterination of the critical collapse pressure of a pressurized tunnel face. CSRSM is first verified by a coparison with the results obtained fro the Monte-Carlo (MC siulation ethod. In a second tie, an extension of the CSRSM is proposed, allowing a paraetric study for no additional call to the deterinistic odel. The proposed extension ethod is verified by a MC siulation approach. It shows that the proposed collocation schee does not reduce the accuracy of the eta-odel. However, the proposed ethodology is only relevant if the area covered by the new collocation points in the standard space is not too different fro the one obtained using the original collocation schee. Fro a practical point of view, it eans that the changes ade to the probabilistic paraeters of the input variables should reain sall during the paraetric study. REFERENCES Huang, S.P., Liang, B., Phoon, K.K. (2009. Geotechnical probabilistic analysis by collocation-based stochastic response surface ethod. Georisk, 3(2, Isukapalli, S.S. (1999. An uncertainty analysis of transport-transforation odels, Ph.D. thesis, The State University of New Jersey, New Brunswick, New Jersey. Mollon, G., Dias, D., and Soubra, A.-H. (2009. Probabilistic analysis and design of circular tunnels against face stability. Int. J. of Geoech., ASCE, 9(6, Phoon, K.K., and Huang, S.P. (2007. Geotechnical probabilistic analysis using collocation-based stochastic response surface ethod., Applications of Statistics and Probability in Civil Engineering, Kanda, Takada and Furada (eds., Taylor and Francis Group, London. Sudret, B. (2007. Global sensitivity analysis using polynoial chaos expansion., Reliability Engineering and Syste Safety, 93, Takano, D., Otani, J., Nagatani, H., Mukunoki, T. (2006. Application of X-ray CT on boundary value probles in geotechnical engineering Research on tunnel face failure. Proceedings of Geocongress, ASCE, Atlanta.