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CALCULATION OF FACE STABILITY FOR EPB MACHINEMODEL OF ANAGNOSTOU & KOVARI (1996)

Analytical Calculation Scheme

Prof. Eng. Daniele PEILA

Course in Tunnelling and Tunnel Boring Machine

Kurs w zakresie drążenia tuneli oraz maszyny drążącej

Shape of the plastic zone around and ahead of the f ace –deep tunnel

N > 5

2 < N < 5

N < 2

c

02N

σσ=

Example of the results of an Axisimmetric numerical model

Typical properties for an average rock mass

Intact rock strenght σci 80 Mpa

Hoek – Brown constant mi 12

Geological Strenght Index GSI 50

Friction angle Φ’ 33°

Cohesive strenght C’ 3,5 Mpa

Rock mass compressive strenght σcm 13 Mpa

Rock mass tensile strenght σct -0,15

Deformation modulus Em 9000 Mpa

Poisson’s ratio ν 0,25

Dilation angle α Φ’/8 = 4°

Post – pick characteristics

Broken rock mass strenght σfcm 8 Mpa

Deformation modulus Efm 5000 MPa

N < 2

N > 5

2 < N < 5

PPP 0fc P

δ

The problem of face stability should be studied with a in 3D numerical method or with an axisimmetrical analysis.

Some simplified scheme can also be used if the following hypothesis are taken into account:- circular tunnel;- a rigid lining at p distance form the face;- an uniformly distributed pressure σt on the face.

σt

σs

Schéma de rupture du front de taille en terrain frottant

P. Chambon and J.F. Corté

Overall shape of the failure mechanism observed in sand and in clay

Clay Sand

Alternatively is possible to use the calulation scheme adopted for the evaluation of the optimal pressure at the tunnel face for shielded TBMs by Anagnostou & Kovari (1996).

Hypothesis:

• 3D rupture model;

• homogeneous and hysotropic ground;

• limite equilibrium computation following Horn model;

• Mohr – Coulomb yielding criteria on the sliding surfaces.

The following slides have been taken by the material given by Prof. Anagnostou at the post graduate master course in Tunnelling and TBMs

(2007-2008; 2009-2010)

HORN MODEL (1961)

Lateral shear force Ts

τ = c + σx tanφ

τ = c + σx tanφ

σx = λk σz (λk = coefficient of lateral stress)

σz

z

H

0

σv

γ H

σz = f (z, γ, σv)

Ts by integration of τ over lateral surface

τ = c + λk tanφ f (z, γ, σv)

3 Unknowns: S, N, T

3 Equations:

Equilibrium // Sliding (S, T, Ts, V, G)

Equilibrium ⊥ Sliding (S, N, V, G)

Coulomb Condition (T, N)

Solution:

Support Force S

S = f (ω, D, φ, c, G, V, Ts )

D

The support force S

D ω

S

ωcrit

Smax

The support force S

S = f (ω, D, φ, c, G, V, Ts )

The support force S

Consideration of a safety factor SF:

Exactly the same steps, but with reduced shear strength

parameters c/SF, tanφ/SF

Safety Factor of the unsupported face

• Total unit weight γtot

Short-term stability of a low-permeability ground

• Undrained shear strength su (φu = 0)

Total stress analysis

Long-term stability or high-permeability ground

Effective stress analysis

• Effective shear strength parameters φ’, c’

• Submerged unit weight γ’

• Seepage force fs (depending on the hydraulic conditions)

• Effective shear strength parameters φ’, c’

Long-term stability or high-permeability ground

• Submerged unit weight γ’

Effective stress analysis

• Seepage force fs (depending on the hydraulic conditions)

Working chamber closed & filled by water⇒ hydraulic equilibrium⇒ no seepage forces

• Effective shear strength parameters φ’, c’

Long-term stability or high-permeability ground

• Submerged unit weight γ’

Effective stress analysis

• Seepage force fs (depending on the hydraulic conditions)

Working chamber closed & filled by water⇒ hydraulic equilibrium⇒ no seepage forces

Support Force = S + W

Stabilityanalysis

• Effective shear strength parameters φ’, c’

Long-term stability or high-permeability ground

• Submerged unit weight γ’

Effective stress analysis

• Seepage force fs (depending on the hydraulic conditions)

Open face (under atmospheric pressure) ⇒ seepage towards the face⇒ seepage forces fs

• Effective shear strength parameters φ’, c’

Long-term stability or high-permeability ground

• Submerged unit weight γ’

Effective stress analysis

• Seepage force fs (depending on the hydraulic conditions)

Open face (under atmospheric pressure) ⇒ seepage towards the face⇒ seepage forces fs

Support Force: Design nomograms(lesson “Calculation of face stability for EPB machine model A&K”)

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The analysis is developed with a calculatio at the limit equilibrium, taking into account the following forces acting

on the wedge:

• Weight of the soil wedge (G);

• Vertical load due to the soil prism present upon the wedge (V);

• Tangential (T) and normal stresses (N) along the inclined sliding surface;

• Tangential (Ts) and normal stresses (Ns) along the lateral surfaces;

• Stabilization force (S), taking into account the presence of a pattern of grouted bars on the tunnel face;

• The friction along the contact surface between the wedge and the prism is not considered for safety reasons.

Anagnostou e Kovari, 2005

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( ) senωSTTcosωGV S ⋅++=+

Equilibrium equation in the sliding direction of the wedge:

tgωγBH2

1G 2=

( )senω GVcosωSN ++⋅=

Equilibrium equation in a the direction that is orthogonal to the sliding one:

cosω

HcBtgφNT +⋅=

Mohr-Coulomb strength criterion:

( ) ( )tgωtgφcosωcosωBH

cs

T

φωtg

GVS

+⋅

+−

++=

For drained conditions

For undrained conditions

+⋅⋅⋅⋅−

⋅⋅⋅+⋅⋅=ω

ωγσ2sin

)sin(2

2

1)( 2 BHHS

HBHBS ucv

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In a generic point (y,z), shear stresses can be evaluated using the Mohr-Coulomb criterion:

( ) ( ) φστ tgzyczy x ,, +=

( ) ( ) kzx zyzy λσσ ⋅= ,,

( ) ( ) vz H

ZzHzy σγσ +−=,

By the application of the silos theory, is possible to correlate the shear stress with the stresses acting in the verticel direction, defining an appropriate lateral thrust coefficient on the wedge, usually included between 0.4 and 0.5:

If is accepted the hypothesis that the vertical stress on the wedge depend linearly on the depth :

( ) ( )

+−⋅+= vk H

ZzHtgczy σγφλτ ,Consequently, the value of the tangential stress is:

( ) ( )∫ ⋅=H

S dzzbzT0

2 τIntegrating on the wedge height:

++⋅⋅=

3

22 γσφλω HtgctgHT v

kS

The shear stress acting on the lateral sides of the wedge is:

VALUTAZIONE DEL TERMINE TS

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VALUTAZIONE DEL TERMINE V

( )

−−⋅==

−R

Tλtgφ

v e1λtgφ

cRγBHtgωFσV ( )

−⋅==

Rγ

s1TγBHtgωFσV u

v

ωBHtgF =

U

FR =

( )ωHtgBU += 2 = perimeter of the soil prism

Drained conditions: Undrained conditions: