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Deterministic and probabilistic analysis of tunnel face stability Guilhem MOLLON Madrid, Sept. 2011

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2 Context: Excavation of a circular shallow tunnel using a tunnel boring machine (TBM) with a pressurized shield Two main challenges: - Limit the ground displacements ->SLS - Ensure the tunnel face stability ->ULS Objectives of the study: - Improve the existing analytical models of assessment of the tunnel face stability - Implement and improve the probabilistic tools to evaluate the uncertainty propagation - Apply these tools to the improved analytical models Introduction

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Context: -Face failure by collapse has been observed in real tunneling projects and in small-scale experiments -To prevent collapse, a fluid pressure (air, slurry…) is applied to the tunnel face. If this pressure is too high, the tunnel face may blow-out towards the ground surface -It is desirable to assess the minimal pressure σ c (kPa) to prevent collapse, and the maximum pressure σ b (kPa) to prevent blow-out. -Many uncertainties exist for the assessment of these limit pressures -A rational consideration of these uncertainties is possible using the probabilistic methods. -The long-term goal is to develop reliability-based design methodologies for the tunnel face pressure. Introduction Takano [2006] Kirsh [2009] Mashimo et al. [1999] Schofield [1980] 3

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Introduction Probabilistic methods Reliability methods Deterministic model Deterministic input variables Deterministic output variables Random input variables Random output variables Failure probability Deterministic model Obstacle n°1 : Computational cost -Deterministic models are heavy -Large amount of calls are needed 4

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1. Deterministic analysis of the stability of a tunnel face

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Numerical model (FLAC 3D software) : -Application of a given pressure, and testing of the stability -Determination of the limit pressure by a bisection method -Average computation time : around 50 hours -Accuracy : 0.1kPa 6 1. Deterministic analysis of the stability of a tunnel face

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Observation of the failure shape: -The failure occurs in a different fashion if the soil is frictional or purely cohesive -Hence different failure mechanisms have to be developed for both cases Frictional soil 1. Deterministic analysis of the stability of a tunnel face 7 Collapse (active case)Blow-out (passive case) Purely cohesive soil

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Theory: -Models are developped in the framework of the kinematical theorem of the limit analysis theory -A kinematically admissible velocity field is defined a priori for the failure Assumptions: -Frictional and/or cohesive Mohr-Coulomb soil -Frictional soils: velocity vector should make an angle φ with the discontinuity (slip) surface -Purely cohesive soils: failure without volume change -Determination of the critical pressure of collapse or blow-out, by verifying the equality between the rate of work of the external forces (applied on the moving soil) and the rate of energydissipation (related to cohesion) Results: This method provides a rigorous lower bound of σ c and a rigorous upper bound of σ b. Principles of the proposed models: 8 1. Deterministic analysis of the stability of a tunnel face

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Existing mechanisms and first attempts: Blow-out : a.Leca and Dormieux (1990) b.Mollon et al. (2009) (M1 Mechanism) 9 Collapse: a.Leca and Dormieux (1990) b.Mollon et al. (2009) (M1 Mechanism) c. Mollon et al. (2010) (M2 Mechanism) 1. Deterministic analysis of the stability of a tunnel face

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M3 Mechanism (frictional soil): -We assume a failure by rotational motion of a single rigid block of soil -The external surface of the block has to be determined -No simple geometric shape is able to represent properly this 3D external surface -A spatial discretization has to be used 10 1. Deterministic analysis of the stability of a tunnel face

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M3 Mechanism (frictional soils) : Definition of a collection of points of the surface in the plane Π j+1, using the existing points in Π j 11 1. Deterministic analysis of the stability of a tunnel face

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M3 Mechanism (collapse) : φ=30° ; c=0kPaφ=17° ; c=7kPa 12 Kirsh [2009] φ=40°φ=25° φ=30° 1. Deterministic analysis of the stability of a tunnel face

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M3 Mechanism (blow-out) : φ=30° ; c=0kPa 13 1. Deterministic analysis of the stability of a tunnel face

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M4 Mechanism (purely cohesive soil): -Deformation with no velocity discontinuity and no volume change -All the deformation inside a tore of variable circular section -Parabolic velocity profile vβvβ vrvr vθvθ 14 1. Deterministic analysis of the stability of a tunnel face

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M4 Mechanism (purely cohesive soil): -The axial and orthoradial components are known by assumption -The remaining component (radial) is computed using -This computation is performed numerically by FDM in toric coordinates 15 1. Deterministic analysis of the stability of a tunnel face

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M4 Mechanism (purely cohesive soil): Layout of the axial and radial components at the tunnel face, at the ground surface, and on the tunnel symetry plane: The components are all null on the envelope: no discontinuity The tensor ot the rate of strain leads to the rate of dissipated energy and to the computation of the critical pressure 16 1. Deterministic analysis of the stability of a tunnel face

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M5 Mechanism (purely cohesive soil): The point of maximum velocity is moved towards the foot or the crown of the tunnel face 17 Schofield [1980] 1. Deterministic analysis of the stability of a tunnel face

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Numerical results (collapse): -M1 to M5 mechanisms are compared to the best existing mechanisms of the littérature, and to the results of the numerical model Frictional soil Purely cohesive soil -> M3 (3 minutes) -> M5 (20 seconds) 18 1. Deterministic analysis of the stability of a tunnel face

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19 1. Deterministic analysis of the stability of a tunnel face Numerical results (blow-out): -M1 to M5 mechanisms are compared to the best existing mechanisms of the littérature, and to the results of the numerical model Frictional soil Purely cohesive soil -> M3 (3 minutes) -> M5 (20 seconds)

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2. Probabilistic analysis

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Assessment of the failure probability: Random sampling methods Monte-Carlo Simulations: Random sampling around the mean point Sample size: 10 3 to 10 6 -> Unaffordable for most of the models Conclusion: -A less costly probabilistic methodology is needed : the CSRSM 2. Probabilistic analysis 21

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Simple case of study: 2 input RV: internal friction angle φ (°) cohesion c (kPa) 1 output RV: critical collapse pressure σ c (kPa) Principle: Substitute to the deterministic model a so-called meta- model with a negligible computational cost For two random variables, the meta model is expressed by a polynomial chaos expansion (or PCE) of order n: ξ 1 and ξ 2 are standard random variables (zero-mean, unit-variance), which represent φ et c in the PCE. The terms Γ i are multidimensional Hermite polynomials of degree n The terms a i are the unknown coefficients to determine 22 Collocation-based Stochastis Response Surface Methodology (CSRSM) 2. Probabilistic analysis

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23 Chosen model: Kinematic theorem of the limit analysis theory. -> Five-blocks translational collapse mechanism Shortcomings: -Geometrical imperfection of the model -Biased estimation of the collapse pressure Advantages: -Satisfying quantitative trends -Computation time < 0.1s 2. Probabilistic analysis

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24 Regression-based determination of the coefficients : -Consider the combinations of the roots of the Hermite polynomial of degree n+1 in the standard space -Express these points in the space of the physical variables (φ, c) : -Evaluate the response of the deterministic model at these collocation points, and determine the unknown coefficients a i by regression 2. Probabilistic analysis

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Set of reference probabilistic parameters -Gaussian uncorrelated random variables -Friction angle : μ φ =17° and COV(φ)=10% -Cohesion : μ c =7kPa and COV(c)=20% Validation by Monte-Carlo sampling (10 6 samples) 25 Validation of CSRSM: 2. Probabilistic analysis

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Validation by the response surfaces Method is validated and Order 4 is considered as optimal 26 2. Probabilistic analysis

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Statistical distribution of the critical pressures Type of soil Type of failure Scenario Coefficients of variation φcγCσtσt Purely frictional soils Purely cohesive soils Deterministic models: M3 (frictional soil) and M5 (purely cohesive soil) 27 Collapse (4 RV) Neutral /20%5%3%15% Optimistic/10%3%1%5% Pessimistic/30%8%5%25% Blow- out (4 RV) Neutral /20%5%3%15% Optimistic/10%3%1%5% Pessimistic/30%8%5%25% Blow- out (4 RV) Neutral 10%/5%3%15% Optimistic5%/3%1%5% Pessimistic15%/8%5%25% Collapse (3 RV) Neutral 10%/5%/15% Optimistic5%/3%/5% Pessimistic15%/8%/25% 2. Probabilistic analysis

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Statistical distribution of the critical pressures φ=25° ; c=0kPa φ=0° ; c=20kPa PDF Critical collapse pressure Critical blow- out pressure 28 2. Probabilistic analysis

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Failure probability of a tunnel face Frictional soil: φ=25° ; c=0kPa Cohesive soil: φ=0° ; c u =20kPa 29 2. Probabilistic analysis

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Comparison with a classical safety-factor approach Frictional soil Purely cohesive soil Test on 6 sands: 25°<φ<40° ; 150kPa<γD<250kPa Test on 8 undrained clays: 20kPa

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Failure probability in a purely cohesive soil 31 2. Probabilistic analysis

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Conclusions: -The continuous improvement of the computers velocities will make the probabilistic methods more and more affordable -The results of this work make possible to build up tools for the reliability-based design of tunnels in a close future -Most of the proposed methods and results may be transposed to other geotechnical fields, such as slopes or retaining walls -However, these methods are only acceptable if the probabilistic scenario is well-defined (dispersions, type of laws, correlations…). Efforts should be made to improve our knowledge on soil variability: What field/laboratory measurements methods are to be used to define properly the probabilistic scenario ? How could we investigate the physical reasons of the soil variability ? Conclusions - Perspectives 32

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THANK YOU FOR YOUR ATTENTION Guilhem MOLLON Madrid, Sept. 2011

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