Deterministic and probabilistic analysis of tunnel face stability

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

2. 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

3. 3 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 σc (kPa) to prevent collapse, and the maximum pressure σb (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. Mashimo et al. [1999] Schofield [1980] Takano [2006] Kirsh [2009] Introduction

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

5. 1. Deterministic analysis of the stability of a tunnel face

6. 6 Numerical model (FLAC3D 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 1. Deterministic analysis of the stability of a tunnel face

7. 7 • 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 Collapse (active case) Blow-out (passive case) Frictional soil Purely cohesive soil 1. Deterministic analysis of the stability of a tunnel face

8. 8 Principles of the proposed models: 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 ofenergy dissipation (related to cohesion) Results: This method provides a rigorous lower bound of σcand a rigorous upper bound ofσb. 1. Deterministic analysis of the stability of a tunnel face

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

10. 10 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 1. Deterministic analysis of the stability of a tunnel face

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

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

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

14. 14 vθ vr vβ 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 1. Deterministic analysis of the stability of a tunnel face

15. 15 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 1. Deterministic analysis of the stability of a tunnel face

16. 16 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 1. Deterministic analysis of the stability of a tunnel face

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

18. 18 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) 1. Deterministic analysis of the stability of a tunnel face

19. 19 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) 1. Deterministic analysis of the stability of a tunnel face

20. 2. Probabilistic analysis

21. 21 Assessment of the failure probability: Random sampling methods Monte-Carlo Simulations: Random sampling around the mean point Sample size: 103 to 106 -> Unaffordable for most of the models Conclusion: -A less costly probabilistic methodology is needed : the CSRSM 2. Probabilistic analysis

22. 22 Collocation-based Stochastis Response Surface Methodology (CSRSM) Simple case of study: 2 input RV: internal friction angle φ (°) cohesion c (kPa) 1 output RV: critical collapse pressure σc (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: ξ1 and ξ2 are standard random variables (zero-mean, unit-variance), which represent φ et c in the PCE. The terms Γi are multidimensional Hermite polynomials of degree ≤ n The terms ai are the unknown coefficients to determine 2. Probabilistic analysis

23. 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

24. 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 ai by regression 2. Probabilistic analysis

25. 25 Validation of CSRSM: Set of reference probabilistic parameters -Gaussian uncorrelated random variables -Friction angle : μφ=17° and COV(φ)=10% -Cohesion : μc=7kPa and COV(c)=20% Validation by Monte-Carlo sampling (106 samples) 2. Probabilistic analysis

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

27. 27 Statistical distribution of the critical pressures Deterministic models: M3 (frictional soil) and M5 (purely cohesive soil) 2. Probabilistic analysis

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

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

30. 30 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<c<60kPa ; 150kPa<γD<250kPa 2. Probabilistic analysis

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

32. 32 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

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