Page 15th International Itasca Symposium

Accounting for long term effects for the structural design of deep tunnels in

claystones

Victor Augusto RIBEIRO LIMA, Jean-François BRUCHON and Sébastien BURLON

Setec Terrasol

Page 2

1. Introduction

2. Accounting for long term effects

3. Calibration with measurements in terms of displacements

4. Comparison in terms of structural forces

5. Conclusion

Summary

Page 3

Introduction

- Complex behaviour of claystones or shales: multiscale material, thermo-hydro-mechanical couplings,brittle/ductile, anisotropy, short and long term behaviour …

- Large bibliography with complex models: hard to choice and certainly to use

Page 4

Introduction

- Complex behaviour of claystones or shales: multiscale material, thermo-hydro-mechanical couplings,brittle/ductile, anisotropy, short and long term behaviour …

- Large bibliography with complex models: hard to choice and certainly to use

Seyedi et al. (2017)

Different models tested

In situ measurements

Page 5

Introduction

From an engineer point of view, where structures have to be designed:

- “simple” models are preferred but able to catch evolution of forces in structure

- especially when long term effects play an important part in final loadings = extrapolation

Page 6

Introduction

Example of “simple” model in the context of the Meuse/Haute-Marne laboratory - Saitta et al. (2017) :

- Mohr-Coulomb’s criterion and Norton’s law

- 2 domains define from in-situ observations (intact and disturbed zones) -> anisotropy forced

- In consistency with principal results in terms of displacement of the

soil in a quasi-isotropic stress state and with isotropic laws

Saitta et al. (2017)

Saitta et al. (2017)

Intact zone

Fractured zone

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Accounting for long term effects

Norton’s law, initially developed for steels at high temperatures (1929)

- visco-plastic strain rate:

ሶ𝜖𝑣𝑝 depends only of deviatoric stress q

independent of mean stress p

ሶ𝜖𝑖𝑗𝑣𝑝=

3

2( ሶ𝜖𝑣𝑝)

sij

𝑞where ሶ𝜖𝑣𝑝 = 𝛾 < 𝑞 − 𝑞0 >

𝑛

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Accounting for long term effects

Modified Norton’s law

- ሶ𝜖𝑣𝑝 depends on the strength mobilization (p,q and Lode angle θ):

with

𝑓 𝑋 =1

1+𝑎

𝑋

𝛼 𝑓 𝑋 ∈ 0,1

𝑋 =𝑑1

𝑑2𝑋 ∈ 0,+∞

ሶ𝜖𝑖𝑗𝑣𝑝=

3

2( ሶ𝜖𝑣𝑝)

sij

𝑞where ሶ𝜖𝑣𝑝 = 𝐴 × 𝑓 𝑋

Page 9

Accounting for long term effects

Modified Norton’s law

- ሶ𝜖𝑣𝑝 depends on the strength mobilization (p,q and Lode angle θ):

with

𝑓 𝑋 =1

1+𝑎

𝑋

𝛼 𝑓 𝑋 ∈ 0,1

𝑋 =𝑑1

𝑑2𝑋 ∈ 0,+∞

ሶ𝜖𝑖𝑗𝑣𝑝=

3

2( ሶ𝜖𝑣𝑝)

sij

𝑞where ሶ𝜖𝑣𝑝 = 𝐴 × 𝑓 𝑋

cos 3𝜃 =3 3

2×

𝐽3

𝐽2Τ3 2

𝑑1 𝑞 =< 𝑞 − 𝑞0 >

𝑑2 𝑝, 𝑞, 𝜃 =𝑀 𝜃 × 𝑝 + 𝑞 − 𝑁 𝜃

𝑀 𝜃 2 + 1 2

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Accounting for long term effects

Modified Norton’s law

- q= 𝑞0 : 𝑋 =𝑑1

𝑑2= 0→ 𝑓 𝑋 = 0 → ሶ𝜖𝑣𝑝 = 0

- q= 𝑞𝑚𝑎𝑥 : 𝑋 =𝑑1

𝑑2= ∞→ 𝑓 𝑋 = 1 → ሶ𝜖𝑣𝑝 = 𝐴

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Titr

e d

e l'

axe

q (MPa)

Calcul du paramètre (d1^n) / (d1^n + d2^n) p = 13.8 MPa

α = 2 α = 3 α = 4 α = 8 α = 30

𝑓 𝑋 =1

1+1𝑋

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Titr

e d

e l'

axe

q (MPa)

Calcul du paramètre (d1^n) / (d1^n + d2^n) p = 13.8 MPa

a=1 a=2 a=3 a=4 a=5

𝑓 𝑋 =1

1+ 𝑋

2

θ = π/3, p = 13.8 MPa, q0 = 0, c = 6 MPa and ϕ = 20

≠ α ≠ a

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Calibration with measurements in terms of displacements

Application to the Meuse/Haute-Marne laboratory

Parameters: 0

40

80

120

0 30 60

Clo

sure

(mm

)

Time (day)

GRD4Horizontal closure

Vertical closure

Semi-empirical law

𝜖ሶ𝑣𝑝 = 𝐴 × < 𝑞 − >𝑛 n [-] A [ 1 . ]

[Pa]

GCS – GRD4 6.8 2. × 10 𝑞0 = 3 𝐽2,0 (initial invariant deviatoric stress)

ሶ = ×

+

Τ

A [ ] a [-] [-]

GCS – GRD4 × 10 11 2 3

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Comparison in terms of structural forces

Forces in rigid structure

- Normal force : quite similar

- Bending moment : quite different and more logical with in-situ observations (to be confirm)

- Bending moment more sensitive with variations of normal and shear stresses.

-12

-8

-4

0

-0.4 -0.2 0 0.2 0.4

No

rma

l fo

rce

(MN

/m)

Bending moment (MN.m/m)

Norton's law

New creep law

Page 13

Conclusion

- Models (simple and complex) have to be compared to measurements in terms of displacements andforces (only displacement not sufficient for structure design)

- Modified Norton’s law proposed here:

- is still a simple law based on a simple approach

- seems more reliable

- Others comparisons have to be done to confirm these results

- Test new idea is quite simple with Flac thanks to fish functions !

Acknowledgements

Andra for experimental data used in this study