17 Stability of the excavation face
In weak rock, the excavation face must be supported. It is important to estimate the necessary support pressure, in particular for slurry and EPB shields, where the pressure must be set by the operator. Several methods can be consulted for the estimation:
Fig. 17.1. Excavation face as hemisphere
17.1 Approximate solution for ground with own weight We regard the distribution of the vertical stress between the ground-surface and the crown of a spherical cavity (Fig. 16.13, 17.1). We approximate this distribution by a quadratic parabola and assume that the material strength is fully mobilised at the crown. As in equation 16.11, we obtain the necessary support pressure pc at the crown as
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17 Stab Stabil ilit ity y of the the exca excav vatio ation n face face
c cos ϕ rf 1 − sin ϕ pc = h h 2sin ϕ 1+ rf 1 − sin ϕ γ −
(17.1)
.
Thus, the crown of the unsupported excavation face is stable if: c ≥ γ rf
1 − sin ϕ cos ϕ
.
17.2 Numerical results From numerical results obtained with the FE-code PLAXIS, Vermeer and Ruse1 deduced the following approximation for the limit support pressure and the case ϕ > 20 : ◦
p ≈ −
c
tan ϕ
+ 2γ r
1 9tan ϕ
− 0.05
.
(17.2)
The underlying results are obtained with an elastic-ideal plastic constitutive law assuming Mohr-Coulomb yield surface and associated plasticity. The authors point to the fact that p is independent of the overburden height h. For the case ϕ = 0, instead, they obtain a linear relation between p and h. For this relation no analytic expression is given. Note that for ϕ = 0 Equ. 17.1 reduces to
c p = γ h 1 − γr
and for h → ∞ Equ. 17.1 reduces to p = −
c
2tan ϕ
+ 2γ r
1 − sin ϕ 4sin ϕ
.
(17.3)
Despite the striking similarity between (17.2) and (17.3), Equ. 17.1 provides much higher p-values than (17.2). Also the bound theorems (Section 17.3) provide higher p-values than Equ. 17.2.
17.3 Stability of the excavation face according to the bound theorems The lower-bound-theorem supplies a simple but very conservative estimation of the necessary support pressure p at the excavation face, for which the form 1
P.A. Vermeer and N. Ruse, Die Stabilit¨ at der Tunnelortsbrust im homogenen at Baugrund, Geotechnik 24 (2001), Nr. 3, 186-193
17.3 17.3 Stabil Stabilit ity y acco accord rding ing to the bound bound theor theorem emss
333 333
of a hemisphere is assumed (Fig. 17.2). We first consider the case γ = 0. Within a spherical zone (radius r = r0 + h) around the excavation face we assume that the limit condition σθ − σr = 2c is fulfilled. For the spherically symmetric case regarded here, the equation of equilibrium in radial direction reads : 2 dσr + (σr − σθ ) = 0 . dr r Using the limit condition and integration of the differential equation with consideration of the boundary condition σr (r = r0 ) = p gives p = σr − 4c ln
r r0
.
Outside the spherical plastified zone we assume a constant hydrostatic stress σr = σθ = q . Equilibrium at the boundary of the two zones ( r = r0 + h) requires σr = q . Thus we obtain the necessary support pressure p as
h p = q − 4c ln 1 + r0
.
Fig. 17.2. Layout plan for the derivation of a lower bound for the support pressure
at the excavation face. Case
γ =
0
Now we consider the case γ > 0 by overlaying the hydrostatic stress σz = σx = σy = γ z to the above mentioned stress field. 2 We obtain, thus, a support pressure that increases linearly with depth (Fig. 17.3):
h p = γ z + q − 4c ln 1 + r0 2
.
The limit condition is not violated by the overlay of a hydrostatic stress
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17 Stab Stabil ilit ity y of the the exca excav vatio ation n face face
Fig. 17.3. Support pressure increasing linearly with depth
z
For plane deformation (’infinitely’ (’infinitely’ long tunnel tunnel with circular circular cross section) section) the necessary support pressure can be estimated in a similar way 3 as:
h p = γ z + q − 2c ln 1 + r0
.
The necessary support pressure pressure p at the excavation face can also be estimated (on the unsafe side) by the upper-bound-theorem, upper-bound-theorem, where we look at the sliding of two cylindrical rigid blocks made of rock (Fig. 17.4). By variation of the geometry (i.e. of the angle shown in Fig. 17.4) the support pressure obtained from the upper-bound-theorem is maximised. The results of the numerical computation computation of Davis et al. are plotted in Fig. 17.5. On the y -axis is plotted the so-called stability ratio N : N :=
q − p + γ (h + r0 ) c
.
More complex collapse mechanisms for rocks with friction and cohesion are considered considered by Leca and Dormieux.4 From com compari parison son with with model model tests tests it can be concluded that the kinematic solutions (upper bounds) are more realistic than the ultra conservative static solutions (lower bounds). The assessment of excavation face stability is often accomplished following the collapse mechanism proposed by Horn (Section (Section 16.1).
´ A. Caquot: Equilibre des massifs a ` frottement interne. Gauthier-Villars, Paris, 1934, p. 37 4 E. Leca and L. Dormieux, Upper and lower lower bound b ound solutions for the face stability stability of shallow circular tunnel in frictional material. G´eotec eo techn hniq ique ue 40, No. 4, 581–606 (1990) 3
17.4 17.4 Stan Standd-up up time time of the the exca excav vatio ation n face face
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Fig. 17.4. Failure mechanism at the excavation face
Estimation of the support pressure at the excav excavation face with the upp erFig. 17.5. Estimation bound-theorem (according to
Davis
et al.)
17.4 Stand-up time of the excavation face The excavation face is stable for a certain stand-up time. The delay of collapse is attributable partly to creep of the ground and partly to pore water pressure.5 , 6 The latter effect can be explained as follows: According to Terzaghi’s consolidation theory, a load suddenly applied on a water-saturated cohesive soil acts, in the first instance, only upon the pore water. It is gradually transmitted to the grain skeleton, to the extent that the pore water is squeezed out. Exactly the same procedure occurs at unloading (for instance due to the construction of a cut or the excavation of a tunnel): 5
P.R. Vaughan and H.J. Walbancke: Pore pressure changes and the delayed failure of cutting slopes in overconsolidated clay. G´eotec eo techn hniq ique ue 23, 4, 1973, 531-539 6 J.H. Atkinson and R.J. Mair: Soil mechanics aspects of soft ground tunnelling. Ground Engineering 1981
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17 Stab Stabil ilit ity y of the the exca excav vatio ation n face face
Initially, the grain skeleton ’does not feel’ the unloading, and the pressure in the pore water is reduced. The effective stresses are thus increased and, subsequently, reduced to the extent that water from the environment is sucked into the voids. This reduction can finally lead to a cave-in. The so-called consolidation coefficient cv , which is proportional to the permeability of the material, controls the time necessary for this process. Consequently the less permeable the ground, the larger the delay of the cave-in is.
