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Deterministic and probabilistic analysis of tunnel face stability

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## Deterministic and probabilistic analysis of tunnel face stability

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- 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

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

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

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

- 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

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

- 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

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

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

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

M3 Mechanism (blow-out) :

φ=30° ; c=0kPa

1. Deterministic analysis of the stability of a tunnel face

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

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

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

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

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

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

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

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

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

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

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

Validation by the response surfaces

Method is validated and Order 4 is considered as optimal

2. Probabilistic analysis

Statistical distribution of the critical pressures

Deterministic models: M3 (frictional soil) and M5 (purely cohesive soil)

2. Probabilistic analysis

Statistical distribution of the critical pressures

φ=25° ; c=0kPa φ=0° ; c=20kPa

Critical collapse pressure

Critical blow-out pressure

2. Probabilistic analysis

Failure probability of a tunnel face

Frictional soil:

φ=25° ; c=0kPa

Cohesive soil:

φ=0° ; cu=20kPa

2. Probabilistic analysis

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

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

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