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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”)
25
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
26
( ) 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
27
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
28
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: