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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 drenia tuneli oraz maszyny drcej

Shape of the plastic zone around and ahead of the face deep tunnel

N > 5

2 < N < 5

N < 2

c

02N=

Example of the results of an Axisimmetric numerical model

Typical properties for an average rock mass

Intact rock strenght ci 80 Mpa

Hoek Brown constant mi 12

Geological Strenght Index GSI 50

Friction angle 33

Cohesive strenght C 3,5 Mpa

Rock mass compressive strenght cm 13 Mpa

Rock mass tensile strenght ct -0,15

Deformation modulus Em 9000 Mpa

Poissons ratio 0,25

Dilation angle /8 = 4

Post pick characteristics

Broken rock mass strenght fcm 8 Mpa

Deformation modulus Efm 5000 MPa

N < 2

N > 5

2 < N < 5

PPP 0fc P

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 t on the face.

t

s

Schma de rupture du front de taille en terrain frottant

P. Chambon and J.F. Cort

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

= c + x tan

= c + x tan

x = k z (k = coefficient of lateral stress)

z

z

H

0

v

H

z = f (z, , v)

Ts by integration of over lateral surface

= c + k tan f (z, , v)

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 (, D, , c, G, V, Ts )

D

The support force S

D

S

crit

Smax

The support force S

S = f (, D, , 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, tan/SF

Safety Factor of the unsupported face

Total unit weight tot

Short-term stability of a low-permeability ground

Undrained shear strength su (u = 0)

Total stress analysis

Long-term stability or high-permeability ground

Effective stress analysis

Effective shear strength parameters , c Submerged unit weight Seepage force fs (depending on the hydraulic conditions)

Effective shear strength parameters , c

Long-term stability or high-permeability ground

Submerged unit weight

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 , c

Long-term stability or high-permeability ground

Submerged unit weight

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 , c

Long-term stability or high-permeability ground

Submerged unit weight

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 , c

Long-term stability or high-permeability ground

Submerged unit weight

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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( ) senSTTcosGV S ++=+

Equilibrium equation in the sliding direction of the wedge:

tgBH2

1G 2=

( )sen GVcosSN ++=

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

cos

HcBtgNT +=

Mohr-Coulomb strength criterion:

( ) ( )tgtgcoscosBH

cs

T

tg

GVS

+

+

++=

For drained conditions

For undrained conditions

+

+=

2sin

)sin(2

2

1)( 2

BHHSHBHBS ucv

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

( ) ( ) tgzyczy x ,, +=

( ) ( ) kzx zyzy = ,,

( ) ( ) vz HZ

zHzy +=,

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 HZ

zHtgczy ,Consequently, the value of the tangential stress is:

( ) ( ) =H

S dzzbzT0

2 Integrating on the wedge height:

++=3

22 HtgctgHT vkSThe shear stress acting on the lateral sides of the wedge is:

VALUTAZIONE DEL TERMINE TS

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VALUTAZIONE DEL TERMINE V

( )

==

R

Ttg

v e1tg

cRBHtgFV ( )

==

R

s1TBHtgFV uv

BHtgF =

U

FR =

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

Drained conditions: Undrained conditions: