Abstract
The sparse polynomial chaos expansion is employed to perform a probabilistic analysis of the tunnel face stability in the spatially random soils. A shield tunnel under compressed air is considered which implies that the applied pressure is uniformly distributed on the tunnel face. Two sets of failure mechanisms in the context of the limit analysis theory with respect to the frictional and the purely cohesive soils are used to calculate the required face pressure. In the case of the frictional soils, the cohesion and the friction angle are modeled as two anisotropic cross-correlated lognormal random fields; for the purely cohesive soils, the cohesion and the unit weight are modeled as two anisotropic independent lognormal random fields. The influences of the spatial variability and of the cross-correlation between the cohesion and the friction angle on the probability density function of the required face pressure, on the sensitivity index and on the failure probability are discussed. The obtained results show that the spatial variability has an important influence on the probability density function as well as the failure probability, but it has a negligible impact on the Sobol’s index.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig6_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig7_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig8_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig9_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig10_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig11_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig12_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig13_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11440-017-0541-5/MediaObjects/11440_2017_541_Fig14_HTML.gif)
Similar content being viewed by others
References
Ahmed A (2012) Simplified and advanced approaches for the probabilistic analysis of shallow foundations. Doctoral dissertation, Nantes
Al-Bittar T, Soubra A-H (2013) Bearing capacity of strip footings on spatially random soils using sparse polynomial chaos expansion. Int J Numer Anal Methods Geomech 37(13):2039–2060
Al-Bittar T, Soubra A-H (2014) Efficient sparse polynomial chaos expansion methodology for the probabilistic analysis of computationally-expensive deterministic models. Int J Numer Anal Methods Geomech 38(12):1211–1230
Baecher GB, Christian JT (2005) Reliability and statistics in geotechnical engineering. Wiley, Hoboken
Blatman G, Sudret B (2010) An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probab Eng Mech 25(2):183–197
Blatman G, Sudret B (2010) Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab Eng Syst Saf 95(11):1216–1229
Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230(6):2345–2367
Chen RP, Tang LJ, Yin XS, Chen XM, Bian XC (2015) An improved 3D wedge-prism model for the face stability analysis of the shield tunnel in cohesionless soils. Acta Geotech 10(5):683–692
Ibrahim E, Soubra AH, Mollon G, Raphael W, Dias D, Reda A (2015) Three-dimensional face stability analysis of pressurized tunnels driven in a multilayered purely frictional medium. Tunn Undergr Sp Technol 49:18–34
Fenton GA, Griffiths D (2003) Bearing-capacity prediction of spatially random c φ soils. Can Geotech J 40(1):54–65
Fenton GA, Griffiths DV, Williams MB (2005) Reliability of traditional retaining wall design. Geotechnique 55(1):55–62
Fenton GA, Griffiths DV (2008) Risk assessment in geotechnical engineering. Wiley, Hoboken
Huang S, Quek ST, Phoon KK (2001) Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes. Int J Numer Methods Eng 52(9):1029–1043
Huang S, Liang B, Phoon KK (2009) Geotechnical probabilistic analysis by collocation-based stochastic response surface method: an Excel add-in implementation. Georisk 3(2):75–86
Jiang S, Li D, Cao Z, Zhou C, Phoon KK (2014) Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation. J Geotech Geoenviron Eng 141(2):04014096
Jiang S, Li D, Zhang L, Zhou C (2014) Slope reliability analysis considering spatially variable shear strength parameters using a non-intrusive stochastic finite element method. Eng Geol 168:120–128
Jin-Feng Z, Yu S (2015) Theoretical solutions of a circular tunnel with the influence of the out-of-plane stress based on the generalized Hoek–Brown failure criterion. Int J Geomech 16(3):06015006
Li D, Chen Y, Lu W, Zhou C (2011) Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables. Comput Geotech 38(1):58–68
Li D, Jiang S, Chen Y, Zhou C (2014) Reliability analysis of serviceability performance for an underground cavern using a non-intrusive stochastic method. Environ Earth Sci 71(3):1169–1182
Li D, Zheng D, Cao Z, Tang X, Phoon KK (2016) Response surface methods for slope reliability analysis: review and comparison. Eng Geol 203:3–14
Laloy E, Rogiers B, Vrugt JA, Mallants D, Jacques D (2013) Efficient posterior exploration of a high-dimensional groundwater model from two-stage Markov chain Monte Carlo simulation and polynomial chaos expansion. Water Resour Res 49(5):2664–2682
Mao N, Al-Bittar T, Soubra AH (2012) Probabilistic analysis and design of strip foundations resting on rocks obeying Hoek–Brown failure criterion. Int J Rock Mech Min Sci 49:45–58
Mollon G, Dias D, Soubra AH (2009) Probabilistic analysis and design of circular tunnels against face stability. Int J Geomech 9(6):237–249
Mollon G, Dias D, Soubra AH (2009) Probabilistic analysis of circular tunnels in homogeneous soil using response surface methodology. J Geotech Geoenviron Eng 135(9):1314–1325
Mollon G, Dias D, Soubra AH (2010) Probabilistic analysis of pressurized tunnels against face stability using collocation-based stochastic response surface method. J Geotech Geoenviron Eng 137(4):385–397
Mollon G, Dias D, Soubra AH (2011) Rotational failure mechanisms for the face stability analysis of tunnels driven by a pressurized shield. Int J Numer Anal Methods Geomech 35(12):1363–1388
Mollon G, Dias D, Soubra AH (2013) Range of the safe retaining pressures of a pressurized tunnel face by a probabilistic approach. J Geotech Geoenviron Eng 139(11):1954–1967
Mollon G, Dias D, Soubra AH (2013) Continuous velocity fields for collapse and blowout of a pressurized tunnel face in purely cohesive soil. Int J Numer Anal Methods Geomech 37(13):2061–2083
Pan Q, Dias D (2015) Face stability analysis for a shield-driven tunnel in anisotropic and nonhomogeneous soils by the kinematical approach. Int J Geomech 16(3):04015076
Pan Q, Dias D (2016) The effect of pore water pressure on tunnel face stability. Int J Numer Anal Methods Geomech 40(15):2123–2136
Pan Q, Dias D (2017) Upper-bound analysis on the face stability of a non-circular tunnel. Tunn Undergr Sp Technol 62:96–102
Phoon KK, Huang SP (2007) Uncertainty quantification using multi-dimensional hermite polynomials. In: Probabilistic applications in geotechnical engineering (GSP 170). Geotechnical Special Publication, ASCE, Reston, VA, pp 1–10. doi:10.1061/40914(233)12
Phoon KK (ed) (2008) Reliability-based design in geotechnical engineering: computations and applications. CRC Press, Boca Raton
Phoon KK, Ching J (eds) (2014) Risk and reliability in geotechnical engineering. CRC Press, Boca Raton
Senent S, Mollon G, Jimenez R (2013) Tunnel face stability in heavily fractured rock masses that follow the Hoek–Brown failure criterion. Int J Rock Mech Min Sci 60:440–451
Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979
Soubra AH, Mao N (2012) Probabilistic analysis of obliquely loaded strip foundations. Soils Found 52(3):524–538
Tang XW, Liu W, Albers B, Savidis S (2014) Upper bound analysis of tunnel face stability in layered soils. Acta Geotech 9(4):661–6718
Zeng P, Senent S, Jimenez R (2014) Reliability analysis of circular tunnel face stability obeying Hoek–Brown failure criterion considering different distribution types and correlation structures. J Comput Civ Eng 30(1):04014126
Acknowledgements
The first author thanks the China Scholarship Council for providing him with a Ph.D. Scholarship for his research work, and the financial support by the National Basic Research 973 Program of China (2013CB036004) is also greatly appreciated.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pan, Q., Dias, D. Probabilistic evaluation of tunnel face stability in spatially random soils using sparse polynomial chaos expansion with global sensitivity analysis. Acta Geotech. 12, 1415–1429 (2017). https://doi.org/10.1007/s11440-017-0541-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11440-017-0541-5