17

Stability of the excavation face

In weak rock, the excavation face must be supported. It is important to esti-mate the necessary support pressure, in particular for slurry and EPB shields,where the pressure must be set by the operator. Several methods can be con-sulted 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-surfaceand the crown of a spherical cavity (Fig. 16.13, 17.1). We approximate thisdistribution by a quadratic parabola and assume that the material strengthis fully mobilised at the crown. As in equation 16.11, we obtain the necessarysupport pressure pc at the crown as

332 17 Stability of the excavation face

pc = h

γ − c

rf

cosϕ1 − sinϕ

1 +h

rf

2 sinϕ1 − sinϕ

. (17.1)

Thus, the crown of the unsupported excavation face is stable if:

c ≥ γrf1 − sinϕ

cosϕ.

17.2 Numerical results

From numerical results obtained with the FE-code PLAXIS, Vermeer andRuse

1 deduced the following approximation for the limit support pressureand the case ϕ > 20◦:

p ≈ − c

tanϕ+ 2γr

(1

9 tanϕ− 0.05

). (17.2)

The underlying results are obtained with an elastic-ideal plastic constitutivelaw assuming Mohr-Coulomb yield surface and associated plasticity. Theauthors 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. Forthis relation no analytic expression is given. Note that for ϕ = 0 Equ. 17.1reduces to

p = γh

(1 − c

γr

)

and for h→ ∞ Equ. 17.1 reduces to

p = − c

2 tanϕ+ 2γr

1 − sinϕ4 sinϕ

. (17.3)

Despite the striking similarity between (17.2) and (17.3), Equ. 17.1 providesmuch 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 thebound theorems

The lower-bound-theorem supplies a simple but very conservative estimationof the necessary support pressure p at the excavation face, for which the form

1P.A. Vermeer and N. Ruse, Die Stabilitat der Tunnelortsbrust im homogenenBaugrund, Geotechnik 24 (2001), Nr. 3, 186-193

17.3 Stability according to the bound theorems 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 weassume that the limit condition σθ − σr = 2c is fulfilled. For the sphericallysymmetric case regarded here, the equation of equilibrium in radial directionreads :

dσr

dr+

2r(σr − σθ) = 0 .

Using the limit condition and integration of the differential equation withconsideration of the boundary condition σr(r = r0) = p gives

p = σr − 4c lnr

r0.

Outside the spherical plastified zone we assume a constant hydrostatic stressσr = 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

p = q − 4c ln(

1 +h

r0

).

Fig. 17.2. Layout plan for the derivation of a lower bound for the support pressureat the excavation face. Case γ = 0

Now we consider the case γ > 0 by overlaying the hydrostatic stress σz =σx = σy = γz to the above mentioned stress field.2 We obtain, thus, a supportpressure that increases linearly with depth (Fig. 17.3):

p = γz + q − 4c ln(

1 +h

r0

).

2The limit condition is not violated by the overlay of a hydrostatic stress

334 17 Stability of the excavation face

Fig. 17.3. Support pressure increasing linearly with depth z

For plane deformation (’infinitely’ long tunnel with circular cross section) thenecessary support pressure can be estimated in a similar way3 as:

p = γz + q − 2c ln(

1 +h

r0

).

The necessary support pressure p at the excavation face can also be estimated(on the unsafe side) by the upper-bound-theorem, where we look at the slidingof two cylindrical rigid blocks made of rock (Fig. 17.4). By variation of thegeometry (i.e. of the angle shown in Fig. 17.4) the support pressure obtainedfrom the upper-bound-theorem is maximised. The results of the numericalcomputation of Davis et al. are plotted in Fig. 17.5. On the y-axis is plottedthe so-called stability ratio N :

N :=q − p+ γ(h+ r0)

c.

More complex collapse mechanisms for rocks with friction and cohesion areconsidered by Leca and Dormieux.4 From comparison with model testsit can be concluded that the kinematic solutions (upper bounds) are morerealistic than the ultra conservative static solutions (lower bounds).The assessment of excavation face stability is often accomplished followingthe collapse mechanism proposed by Horn (Section 16.1).

3A. Caquot: Equilibre des massifs a frottement interne. Gauthier-Villars, Paris,1934, p. 37

4E. Leca and L. Dormieux, Upper and lower bound solutions for the face stabilityof shallow circular tunnel in frictional material. Geotechnique 40, No. 4, 581–606(1990)

17.4 Stand-up time of the excavation face 335

Fig. 17.4. Failure mechanism at the excavation face

Fig. 17.5. Estimation of the support pressure at the excavation face with the upper-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 col-lapse is attributable partly to creep of the ground and partly to pore waterpressure.5,6 The latter effect can be explained as follows:According to Terzaghi’s consolidation theory, a load suddenly applied on awater-saturated cohesive soil acts, in the first instance, only upon the porewater. It is gradually transmitted to the grain skeleton, to the extent that thepore 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):

5P.R. Vaughan and H.J. Walbancke: Pore pressure changes and the delayed fail-ure of cutting slopes in overconsolidated clay. Geotechnique 23, 4, 1973, 531-539

6J.H. Atkinson and R.J. Mair: Soil mechanics aspects of soft ground tunnelling.Ground Engineering 1981

336 17 Stability of the excavation face

Initially, the grain skeleton ’does not feel’ the unloading, and the pressurein the pore water is reduced. The effective stresses are thus increased and,subsequently, reduced to the extent that water from the environment is suckedinto the voids. This reduction can finally lead to a cave-in. The so-calledconsolidation coefficient cv, which is proportional to the permeability of thematerial, controls the time necessary for this process. Consequently the lesspermeable the ground, the larger the delay of the cave-in is.