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

Deterministic and probabilistic analysis of tunnel face stability

Guilhem MOLLON

Madrid, Sept. 2011

2Context:

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.

Introduction

Takano [2006]

Kirsh [2009]

Mashimo et al. [1999]

Schofield [1980]

3

Introduction

Probabilistic methods

Reliability methods

Deterministic modelDeterministic

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

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1. Deterministic analysis of the stability of a tunnel face

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

6


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


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Collapse (active case)Blow-out (passive case)

Purely cohesive soil

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 σc and a rigorous upper bound of σb.

Principles of the proposed models:8


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)


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


M3 Mechanism (frictional soils) :

Definition of a collection of points of the surface in the plane Πj+1, using the existing points in Πj

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M3 Mechanism (collapse) :

φ=30° ; c=0kPaφ=17° ; c=7kPa

12

Kirsh [2009]

φ=40° φ=25°φ=30°


M3 Mechanism (blow-out) :

φ=30° ; c=0kPa

13


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

0,,

1,,,2

2

2

2

rv

R

r

R

Rrfvrv i

m

vβ

vr vθ

14



-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

0div

15



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

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The point of maximum velocity is moved towards the foot or the crown of the tunnel face

17

Schofield [1980]


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


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


-> M3 (3 minutes) -> M5 (20 seconds)

2. Probabilistic analysis

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


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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

p

iiiaU

121 ,

22Collocation-based Stochastis Response Surface Methodology (CSRSM)


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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


1 , 1,

2 , 2,

C m m

C m m

H

mCcm

mCm

Fc

F

,21

,11

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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


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)

25Validation of CSRSM:


Validation by the response surfaces

Method is validated and Order 4 is considered as optimal

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Statistical distribution of the critical pressures

Type of soil

Type of failure

Scenario

Coefficients of variation

φ c γ C σt

Purely frictional

soils

Purely cohesive

soils

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

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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%

Optimistic 5% / 3% 1% 5%

Pessimistic 15% / 8% 5% 25%

Collapse(3 RV)

Neutral 10% / 5% / 15%

Optimistic 5% / 3% / 5%

Pessimistic 15% / 8% / 25%


Statistical distribution of the critical pressures

φ=25° ; c=0kPa φ=0° ; c=20kPaPDF

Critical collapse pressure

Critical blow-out pressure

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Failure probability of a tunnel face

Frictional soil:

φ=25° ; c=0kPa

Cohesive soil:

φ=0° ; cu=20kPa

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Comparison with a classical safety-factor approach


Test on 6 sands:

25°<φ<40° ; 150kPa<γD<250kPa

Test on 8 undrained clays:

20kPa<c<60kPa ; 150kPa<γD<250kPa

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Failure probability in a purely cohesive soil31


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

THANK YOU FOR YOUR ATTENTION

Guilhem MOLLON

Madrid, Sept. 2011

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

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