Tunnelling in soft ground - The Mechanics of Soils and Foundations - page 402

Values for T c depend on the values of C/D and P/D. Some values have been obtained
from upper and lower bound and limit equilibrium solutions of the kind described in
Chapters 19 and 20 but these are in three dimensions and are quite complicated.
The values of T c commonly used in design are given by Atkinson and Mair (1981)
and these are shown in Fig. 25.5(b). They were obtained from centrifuge model tests
of the kind described in Chapter 27. This is a very good example of an instance where
information obtained from centrifuge model tests has been applied with an equation
derived from soil mechanics theory to give a design method.
To apply a single factor of safety, s u can be replaced with a safe undrained strength
s us , as described in Sec. 19.5. Alternatively, partial factors can be applied to all the
design parameters in Eq. (25.1.) Notice that the tunnel and its heading become safer
as the support pressure

t is increased but may become less safe if the tunnel pressure
becomes so large that it is close to causing a blow-out failure.
From Eq. (25.1) with

σ

σ = 0 the face is self-supporting and can be safely excavated

tc

with an open shield if

1
T c (γ

s u ≥ z + q ) (25.2)

with a value of T c obtained from Fig. 25.5(b). Notice that the stability of an unsup-
ported tunnel face is similar to the stability of an unsupported vertical cut discussed
in Sec. 21.8. In each case the undrained strength of the soil arises from negative pore
pressures which are developed by the excavation. With time the pore pressures rise
towards their steady state values, so the soil swells and weakens and the face becomes
less stable and sooner or later both must collapse. The question is not whether an
unsupported face or heading collapses but how long will it be before it does. This is a
problem of consolidation time, discussed in Chapter 15.

(b) Stability of tunnel headings for drained loading
Model tests on tunnels in dry sand show that the tunnel pressure at the collapse state

σ tc
is always relatively small. It depends strongly on the tunnel diameter D and is almost
independent of the depth of cover C . Figure 25.6(a) shows a circular tunnel section in
dry soil. The stability is rather like that of an arch in a building: it is necessary only to
maintain a ring of stable grains round the circumference.
Figure 25.6(b) shows a soil wedge behind the face which is like that on the active
side of a retaining wall shown in Fig. 24.6. Due to arching the vertical stress at the top
of the wedge is very small so the vertical stress in the dry soil near the invert is D .

The horizontal stress in the soil in the wedge corresponds to the active pressures so the
collapse pressure on the face

γ

σ tc is approximately

D tan 2 45 ◦ −

2 φ

1

σ = K a γ D = γ (25.3)

tc

φ is the friction angle. As before, because we are considering an ultimate limit
state of collapse this should be taken as the critical state strength friction angle

where

φ c ,as

φ c can be replaced with a discussed in Chapter 18. To apply a single factor of safety