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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 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 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 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 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 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 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) 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 > 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 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 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 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 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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