ATKWSON,J. H. & Porrs, D. M. (1977).
Ge’otechnique 27, No. 2, 203-215
Stability of a shallow circular tunnel in cohesionless soil
J. H. ATKINSON*
This Paper investigates theoretically and experimentally the stability of a circular tunnel in a cohesionless
soil with support conditions similar to those found
during construction.
The experimental investigation
consisted of small-scale model tests in the laboratory
and on the Cambridge University large-diameter
centrifuge.
All the model tests were carried out in
plane strain using Leighton Buzzard sand and tests
with and without surcharge loading are reported.
The theoretical studies are based on the upper and
lower bound theorems and predictions of collapse
pressures from these theoretical solutions are shown
to bracket closely thevaluesobserved in themodel tests.
and D. M. PO’ITSt
Cette etude examine theoriquement et experimentalement la stabilite d’un tunnel circulaire dans un sol
pulverulent soutenu dans des conditions semblables a
celles trouvees pendant la construction.
L’etude
experimentale consista en essais sur modele a petite
6chelle en laboratoire et sur la centifugeuse a grand
diametre de l’Universit6 de Cambridge.
Tous les
essais sur modele furent mis a execution sous deformation plane en utilisant du sable de Leighton Buzzard,
et l’on rend compte des essais avec et sans surcharge
additionelle.
Les etudes theoriques sont basees sur
les theoremes limites suptrieures et inferieures et on
montre que des previsions de pressions de rupture a
partir de ces solutions sont encadrees Ctroitement par
les valeurs observees en essais sur modtles.
INTRODUCTION
As the face of a tunnel is advanced, a means of supporting the ground close to the face may
In competent rock, the tunnel will be stable without support, but in soft ground,
be needed.
with immediate support provided by such means as a shield, the use of compressed air or
clay slurry under pressure may be required; otherwise collapse may occur as a result of gross
plastic deformation
of the soil, possibly accompanied
by flooding.
Once a primary lining,
usually of concrete or cast-iron segments, is in place, instability is unlikely unless the fabric of
the lining deteriorates
or the ground in the vicinity of the tunnel is disturbed.
Ground movements during excavation of soft ground tunnels have been discussed recently by
The present Paper is concerned with the of radial pressures to be
Atkinson and Potts (1976).
supported by compressed air, bentonite slurry, a shield or by other means in order to achieve
One solution would be to allow for ground pressures close to the average total
stability.
stress in the soil before excavation.
This would maintain
stability and prevent excessive
ground movements and flooding, but the use of such large pressures at depth may be expensive
and, if air pressure is used as the support, may constitute a health hazard.
The support provided by air or bentonite pressure is equivalent to a uniform normal total
stress acting on the exposed periphery and face of the tunnel.
The state of stress in the soil
around a shield is rather more complicated but the support provided by a smooth circular
shield may be approximated
to a uniform normal total stress provided bending moments in the
Discussion on this Paper closes 1 September, 1977. For further details see inside back cover.
* Department of Civil and Structural Engineering, University College, Cardiff.
t Royal Dutch Shell Research Laboratories, Rijswijk, Netherlands.
204
J. H. ATKINSON
AND
D. M. POTTS
shield are not large.
Thus the state of stress in the soil around a newly excavated circular
tunnel supported by air or bentonite pressure or by a smooth shield, neglecting end effects, may
be approximated
to the state of stress associated with a uniform fluid pressure contained within
the heading.
The problem is to estimate the magnitude of the pressure required just to cause
the tunnel to collapse and an appropriate margin of safety may then be allowed.
The stability of a shallow circular tunnel in a weightless soil to which a uniform surface surcharge pressure was applied was investigated by Atkinson and Cairncross (1973).
The present
Paper extends this work for cohesionless soils to cover the more important problems of shallow
circular tunnels in soils whose self weight forces may themselves cause collapse.
The approach
to the problem will be through the bound theorems of plasticity theory together with the results
of tests on model tunnels in a sand soil.
MODEL
TUNNEL
TESTS
A number of model test to examine the behaviour of shallow tunnels in sand have been
carried out recently at Cambridge University.
Details of the apparatuses
and of the tests
have been given elsewhere (Atkinson et al., 1975; Atkinson et al., 1977; Potts, 1976) and only
the essential features will be described here.
It is important to appreciate that these model tests were not intended to reproduce precisely
to scale a real tunnel during construction
together with all the details of the methods of excavation and support.
Instead their purpose was to illustrate the way in which soil around a circular cavity deforms as a fluid pressure within the cavity is reduced, since this approximates to the
stress condition within a tunnel during construction.
The results of tests of this kind provide
a source of experimental data which may be used to examine the accuracy of stability calculations; examination
of the mechanisms of deformation
observed in such tests may lead to new
calculations for estimating the stability and deformations
of shallow soft ground tunnels.
In all the model tests the boundary conditions were the same and are illustrated in Fig. 1.
The block of soil containing
the tunnel was itself contained within an apparatus with rigid
boundaries;
the front and back faces imposed a condition of plane strain on the soil and the
bottom and side faces were sufficiently far removed from the tunnel not to influence the behaviour of the sand around and above the tunnel.
The sand surface may be unstressed with
us = 0 but in some tests a uniform surface surcharge us was applied by fluid pressure
contained
within a flexible rubber membrane.
Surface surcharge loadings may occur in
practice where tunnels are driven below structures on flexible foundations
or when a tunnel is
driven through dense sand overlain by a very soft deposit.
The tunnels were approximately
60 mm in diameter and were constructed completely through
the soil sample; consequently
the models were strictly examining the plane section of a tunnel
removed from its face. The actual diameters of the tunnels varied slightly according to the
The tunnels were lined with a thin
methods of construction
but each was measured directly.
cylindrical rubber membrane of negligible stiffness and strength and the tunnel pressure uT
applied a uniform radial total stress to the tunnel wall. The tunnel pressure a, is equivalent to
an excess air or bentonite pressure in a tunnel heading during construction
or to a circumferential thrust T = uT R per unit length in a smooth and moment-free tunnelling shield or tunnel
lining of radius R.
Most of the model tests were conducted on the laboratory floor under normal gravitational
accelerations but one series of tests was completed in which the models were accelerated in the
Cambridge large-diameter
centrifuge.
‘In a centrifuge the model is accelerated and stresses
due to body weight forces increase accordingly.
If a model is accelerated to ng (where g is
the acceleration due to the earth’s gravity) the stresses in the model are equal to those in a
STABILITY
Fig. 1.
TUNNEL
IN COHESIONLESS SOIL
Boundary conditions and dimensions
of model tunnel tests
OF A SHALLOW
CIRCULAR
205
In effect, so far as stress and strain are conprototype structure n times larger than the model.
cerned, a centrifugal mode1 becomes apparently n times larger when accelerated to ng. The
centrifugal
mode1 tests were conducted at constant accelerations
of 75g; consequently,
all
other things being equal, the behaviour of the 60 mm diameter model tunnels should be the
same as the behaviour of a 4.5 m diameter tunnel in Earth’s gravitational
field.
The sand used for these mode1 tests was the fraction of dry Leighton Buzzard sand passing a
No. 14 sieve and retained on a No. 25 sieve. This sand has been extensively examined in
laboratory tests at Cambridge in the past and its engineering behaviour is well documented.
Since the sand was dry, pore pressures were everywhere atmospheric, so that total and effective
For all tests, samples were prepared by pouring the dry sand
stresses were everywhere equal.
around a pre-formed tunnel; the sand was poured in the direction of the tunnel axis to produce
a homogeneous
sample which was isotropic in planes normal to the tunnel axis. A specified
All the results
sand density may be obtained by pouring the sand at a predetermined
rate.
discussed were taken from mode1 tests in which sand was dense with a voids ratio of approximately 0.52 and all were conducted in a similar fashion.
The tunnels were constructed with the tunnel pressure approximately
equal to the vertical
stress in the soil at the level of the tunnel axis; thus the initial conditions were
UT = y(C+R)+a,
where C is the cover above the tunnel crown as shown in Fig. 1 and y is the unit weight of the
soil. During a particular test the tunnel pressure was reduced in decrements until the tunnel
collapsed; this was, in all cases, a sudden and well defined event accompanied by very large soil
displacements
(Atkinson et al., 1975; Potts, 1976). In most tests lead shot were buried in the
sand or markers placed against a transparent
front face, and after each decrement of tunnel
pressure a radiograph or a photograph
was taken of the apparatus.
By observing the movements of the lead shot or of the markers on successive radiographs or photographs the complete
field of displacement and strain in the soil around the tunnel may be calculated (Potts, 1976).
Although soil deformations
and strains are not of specific concern here, such observations
may be of considerable
importance
when they suggest mechanisms
of collapse or fields of
stress necessary for stability calculations.
Three series of model tunnel tests have been completed in which such parameters as the
depth of cover C, the initial stress conditions, sand density, stress level and type of loading were
varied.
These tests are described in detail by Potts (1976).
Further tests on model tunnels in
J. H. ATKINSON
206
AND D. M. POTTS
0
0
9
0
0.5
IQ
1.5
CIZR
Fig. 2.
Variation
of tunnel pressure at collapse with depth of burial for model tunnels with crs=210
kN/m2
IO-
0
0
I
0
0
0
200
400
600
Qs : kN/m2
Fig. 3.
Variation
of tunnel pressure at collapse with surface pressure for model tunnels with C/2R=
0.5
sand have been carried out in which the tunnels were lined with thin metal tubes and strain
gauges used to measure the loads in the tubes (Potts, 1976) but, for the present, only those tests
in which tunnels contained thin rubber liners will be considered.
For all the model tests the tunnel pressures at collapse are shown plotted against depth of
burial in Figs 2 and 4, and against surcharge pressure ~a in Fig. 3.
THEORETICAL
ANALYSIS
The stability of a tunnel may be examined theoretically by making use of the upper and lower
bound theorems for perfectly plastic materials.
Strictly these. theorems are true only for
materials whose flow rule is associated and v=$ where Y is the angle of dilation (Hansen,
1958) and + is the maximum angle of shearing resistance.
In general the flow rule for real
soils is non-associated
and Y< $, and application
of the bound theorems to problems in soil
mechanics must be treated with some caution.
Nevertheless Davis (1968) has shown that
an upper bound calculated assuming Y= 4 is also an upper bound for the case where Y< 4.
Palmer (1966) has suggested that a lower bound solution based on the Coulomb failure criterion
may still be valid even if the flow rule is non-associated.
Application
of the bound theorems
involves, firstly, calculation of an admissible stress field, in which case the external loads cannot
cause collapse and, secondly, selection of a mechanism of collapse together with an appropriate
work rate calculation,
in which case the external loads must cause collapse.
Selection of a
realistic stress field and a realistic collapse mechanism may often be assisted by examination
of
soil deformations
observed during model tests.
Lower bound solution
If a statically admissible stress field can be found which nowhere violates the failure criterion
for the soil, the loading is a lower bound to the true collapse load; it may be noted that such a
solution will lead to a safe or high value for the tunnel pressure.
STABILITY
Fig. 4.
OF A SHALLOW
Variation
CIRCULAR
TUNNEL
IN COHESIONLESS
207
SOIL
of tunnel pressure at collapse with depth of burial for model tunnels without surface surcharge
z
0,
I.%*_l
e
(b)
Fig. 5.
States of stress in plane elements:
For a cohesionless
(a) close to the tunnel; (b) far from the tunnel
soil the failure criterion
may be written
.
al=/N3
.
.
.
.
.
.
.
.
.
(1)
where
l+sin+
C1=l_sin+
and 4 is the maximum angle of shearing resistance.
The state of stress in a small plane element in the vicinity
5(a) and the conditions of equilibrium
are
au, GJr- 4
ar+r
of the tunnel
1 aT,e
-rae+yCOSe=O
~~-!$!-$??-~sin~
=
0
is illustrated
in Fig.
.
.
.
.
.
.
.
.
.
.
.
.
.
*
(3)
where y is the unit of the soil. In general radial and tangential
stresses are not principal
stresses but, in order to obtain a solution, we assume that T,~ = 0, in which case the equilibrium
conditions become
.
z--yrsinB=O
and the failure criterion
.
.
.
.
.
.
.
.
(4)
.
.
.
.
.
.
.
*
(5)
.
.
.
.
.
.
.
.
(6)
becomes
0, = PO,
or
ue=-a,
1
P
J. H.
208
ATKINSON
AND
D. M. PO-ITS
Examination
of soil strains observed during model tests indicates that it is reasonable to assume
that, in the vicinity of the tunnel, principal strain rates are radial and tangential and so, accepting coaxiality of the directions of principal stress and principal strain rates, it appears
reasonable to assume that principal stresses are also radial and tangential,
and hence shear
stresses T,~ may be assumed to be zero.
The state of stress in a small plane element far from the tunnel is illustrated in Fig. 5(b) and
the conditions of equilibrium
are
$+2=-o
......
%+%=y
........
.
.
.
(7)
In general vertical and horizontal stresses are not principal stresses, but, in order to obtain
become
solution, we assume 7XQ=O, in which case the conditions of equilibrium
afJx
-=
3X
0
.
au
z,&-y
and the failure criterion
.
.
.
.
.
.
.
.
.
.
a
.
.
.
.
.
.
.
.
.
(10)
.
.
.
.
.
.
.
.
(11)
becomes
or
0% = P.?
1
OX=-a,
P
Consider the field of stress illustrated in Fig. 6. Within the region bounded by the circular
arc AFE principal stresses are radial and tangential, and beyond AFE they are vertical and
At the point A the failure criterion is just satisfied and the principal stresses
horizontal.
are tangential and radial; hence, if the tunnel pressure uT is reduced, the stresses at A are
(Jr = (Ta = ug
erg = 01 = jL*a,
At the point E the failure criterion is satisfied and the tangential stress may be found by integrating equation (5) around a circular arc just inside AFE; hence at E the stresses are
uB
=
u1
=
pa,+2yrs
UT
=
cr’3
=
as+-2yrs
CL
for (T@=~L(T~,
equation
The failure criterion is satisfied along EC and, substituting
integrated along EC with the limit uI = uT at r = R. The result may be expressed
UT =
(5) may be
in the form
~{(~)~-z[~+~)(~-2,+Y]
-Y} . . . .
Equation (12) relates the tunnel pressure
soil properties.
We have shown so far that when the
stress at A and along EC are statically
cular trajectories starting from EC and
to the surface pressure,
the tunnel
geometry
(12)
and the
tunnel pressure is given by equation (12) the states of
By integrating equation (5) around ciradmissible.
i.ptegrating equation (4) along radii it may be shown
STABILITY
Fig. 6.
OF A SHALLOW
CIRCULAR
TUNNEL
IN COHESIONLESS
209
SOIL
Possible states of stress in the soil around a shallow tunnel
(Potts, 1976) that the stress field is admissible everywhere within the region of soil contained
the circular arc AFE and that the soil just reaches limiting equilibrium
around AFE.
In the region beyond AFE horizontal and vertical stresses are related by
(JZ = Xa,
and the failure
criterion
is not violated
.
.
.
.
.
._.
.
.
.
.
.
.
.
.
(13)
.
(14)
when
.
A2lorl\<p
p
From equations
.
by
(10) and (13) the state of stress in an element
.
at a depth z is
a_. = a,+yz
a, = ; (0s + V)
Consider the change in the state of stress along a radius across the arc AFE at a point such
as F in Fig. 6. We imagine that the arc AFE represents a discontinuity
in the stress field of
finite width containing a large number of fan zones; the arc AFE is not itself a strong disconThe states
tinuity but the state of stress changes smoothly over a small distance across AFE.
of stress across such a discontinuity
are statically admissible when
S,
K‘
<
e26tane
.
.
.
.
.
.
.
.
.
.
(1%
where S=+(U~
+ u3) either side of the fan zones, S, > S_ and 6 is the change in the direction
of
the major principal stress.
Noting that S, 2 S_, we see that equation (15) is satisfied for all 6,
including
6 =O, when S, = S_ ; hence the states of stress across the discontinuity
are admissible
everywhere around AFE when the values of S=$(ul + u3) just inside and just outside the discontinuity are equal.
210
J. H. ATKINSON
AND
D. M. POTTS
CL?R
Fig. 7.
Form of the lower hound
solution of equation (16) for tunnels in a weightless soil
At the point F in Fig. 6 [z=r,(l
discontinuity
are given by
-cos
0)] the values of S just inside and just outside
(1 -cos
si = 3(p+ 1) Cus+?
0)
s, = 3
e)]
and the stress field is admissible
(
1 +i
1
[a,+yrs
(l-cos
the
I
when
s, = s,
To continue, it is convenient
to examine the two limiting cases when aa=0 and when ~a is
large and y may be assumed to be zero.
Firstly, when y=O, S,= S, when X= l/p and, from equation (14), the failure criterion is not
From equation (12) the lower bound or safe tunnel pressure is given by
violated.
cry. =
as
rs
R
0
1-p
-
or, since rs = C-k R
.
.
.
.
.
.
.
.
(16)
This result was obtained1 in slightly different form by Atkinson and Cairncross (1973).
However, they did not examine the stress field in the whole region of soil and the present analysis
confirms their result as a true lower bound within the limits of the assumptions
made here.
The form of equation (16) is shown in Fig. 7 as +/us plotted against depth of burial defined
When C= 0 and the crown and surface
by C/2R, where C is the cover above the tunnel crown.
coincide uT = us, which must be correct; as the tunnel gets deeper the safe tunnel pressure
reduces rapidly.
Secondly, when us = 0, Si = So when
r+l
-=
P
1 It should be noted that Atkinsonand
Cairncross(l973)
1,;
definep=(o,-uo,)/(o,+o,)and
not p= ol/oa as in this Paper.
OF A SHALLOW
STABKITY
Fig. 8.
CIRCULAR
TUNNEL
IN COHESIONLESS
211
SO11
Form of the lower hound solution from equations (18) and (22) for tunnels without surface surcharge loading
X=tL
i.e. when
and, from equation (14) the failure criterion
bound or safe tunnel pressure is given by
is not violated.
From
$=&)[($@-z(3-;)-l]
equation
(12) the lower
.
. ..
.
.
.
(17)
and, as before rs= C-t R.
Finally, we should note that we have not allowed the possibility that the radial stress in the
vicinity of the tunnel may be the major rather than the minor principal stress.
The critical
location will be at the tunnel crown, and we examine the state of stress in the soil adjacent to
the tunnel around BDC in Fig. 6. The tunnel pressure is a principal stress, hence for r= R
7T8= 0 and, from equation (3),
.
$--yRsinB=O
.
.
.
.
.
.
.
.
(18)
At the invert the failure criterion is satisfied,
stress; hence the stresses at C are
and the tunnel
pressure
is the minor
principal
At the crown the failure criterion
stress; hence the stresses at B are
and the tunnel
pressure
is the major
principal
is satisfied
(3, = a1 = 0.r.
u8B
Equation
(18) may be integrated
around
=
u3
=
+T/p
BDC giving
uBC-uBB = 2yR
Hence the failure criterion
and a lower bound
is satisfied simultaneously
or safe tunnel
pressure
.
.
.
.
.
.
.
.
.
(19)
.
.
(20)
at the crown and invert when
is given by
$=(p:_l)
. . . . .
.
.
212
Fig. 9.
J. H. ATKINSON
AND
D. M. POTTS
Collapse mechanism for an upper bound solution
For the case when y = 0, the solution uT > 0 is trivial, but for the case when us = 0 and y > 0 the
lower bound or safe tunnel pressure given by equation (20) may exceed that given by equation
(17).
The forms of equations (17) and (20) are shown in Fig. 8 as a,/2yR plotted against depth of
burial defined by C/2R; safe tunnel pressures must lie on or above the hatched line. The
relationship
between the two solutions illustrated in Fig. 8 suggests that stresses close to the
tunnel may be critical for deep tunnels.
Upper bound solution
An upper bound to the the true collapse load can be found by selecting any kinematically
possible collapse mechanism and performing an appropriate work rate calculation.
It may be
noted that the solution will lead to an unsafe or low value for the tunnel pressure.
The accuracy
of an upper bound calculation will depend, in part, on the closeness of the assumed mechanism
to the real one, and it is in the selection of a collapse mechanism that the model tests have
proved to be of value.
By observing the behaviour of the soil around the tunnels in a series of tests conducted in
the laboratory
without surcharge loading a collapse mechanism
similar to that shown in
Fig. 9 suggested itself (Atkinson et al., 1975). The wedge ABC in Fig. 9 moves downwards
but dilates at a rate sufficient to prevent separation along AB and BC. To satisfy compatibility
requirements
the planes AB and BC must make an angle 2~ at B, where v is the angle of
dilation.
Noting that the rate of work dissipated in a perfectly plastic frictional material with an
associated flow rule v = + is zero, the appropriate work rate calculation involves only the loads
due to the tunnel pressure and to the self weight of the soil. The upper bound or unsafe
tunnel pressure is then given by
f$=&&-&++;)
. . . . . . .
provided that C/R 2 1/sin 4 - 1. This limitation is necessary to ensure that the apex of the
sliding wedge at B is below the soil surface.
It may be noted that the upper bound, or unsafe,
collapse pressure given by equation (21) is independent
of the depth of burial and the magnitude of any surface pressure.
COMPARISON BETWEEN THE THEORETICAL AND EXPERIMENTAL
COLLAPSE PRESSURES
Equations (16), (17), (20) and (21) lead to values of the tunnel pressure at collapse which, in
theory, bound the true values and these may be compared with the collapse pressures observed
in the model tests and contained in Figs 2-4.
STABILITY
OF A SHALLOW
CIRCULAR
NOW : upper
bound ci=
TUNNEL
IN COHESIONLESS
SOIL
213
046 kN/ml
0
0
I.0
2.0
3.0
C/2R
(4
IO
f
Note : Upper
bound
UT=
OS:
046
kN/ml
kN/m’
6)
Fig. 10. Comparisons between model test results and theoretical predictions for the collapse of shallow circular
tunnels for +=SO”: (a) a,=210 kN/m’; (b) C/2R=05; (c) u,=O
The magnitude of the maximum angle of shearing resistance 4 depends on the magnitude of
the state of stress for a particular soil. The sand used in the model tests has been examined in
simple shear tests by Stroud (1971), and from his results a value of 4 of approximately
50” is
appropriate
for the stress levels in the soil around the model tunnels at collapse.
This relatively large value for + is a consequence of the very low stress levels in the model tests.
Equations (16), (17) (20) and (21) have been evaluated with d= 50” for the various series of mode1
tests and the bounds are shown together with the appropriate experimental
results in Fig. 10.
For +=50° the value of u,/2yR from equation (20) exceeds that given by equation (17) for all
depths of burial and hence, in Fig. 10(c) only the lower bound given by equation 20 is shown.
In Figs 10(a) and (b) the experimental
results seem to fall very close to the lower bound
solutions.
In some instances they lie above, or on the unsafe side of, the theoretical bound,
and this may be due to neglect of y in obtaining equation (16) or to an inappropriate
value of
+= 50” for mode1 tests in which the stress levels were raised owing to surface surcharge pressures; if a more conservative value of $= 45” is chosen all the model test results lie between the
214
J. H. ATKINSON
AND
D. M. POTTS
Nevertheless, the lower bound calculation of equation (16)
upper and lower bound solutions.
seems to predict the very small tunnel pressures at collapse with reasonable accuracy.
For the cases where u5=0, the experimental
points in Fig. 10(c) obtained from laboratory
and centrifuged models fall between the appropriate upper and lower bounds and, moreover,
It can be seen that for 4 = 50” the lower
the bounds solutions are relatively close to each other.
bound given by equation (17) applies only for tunnels at very small depths of burial and the
This is convenient since equation (20) was obbound given by equation (20) predominates.
tained without any assumptions
about the shear stresses in soil adjacent to the tunnel.
DISCUSSION
Figure 10 clearly indicates the ability of the simple upper and lower bound solutions to
predict the stability of the model tunnels with satisfying accuracy.
The work described in this Paper has been limited to plane strain and thus represents the
conditions in the plane tunnel section away from the tunnel heading.
However, recent model
tests conducted by Orme (1975), Argyle (I 976) and Aspden (1976) on headings in sand indicate
that collapse occurs in the cylindrical section of the tunnel away from the heading itself.
In
tests where the cylindrical section was lined collapse could only occur at the tunnel face, in
which case the collapse pressure was slightly lower than it was when collapse occurred in the
In both cases collapse pressures predicted by the
cylindrical
section of unlined headings.
bound calculations presented in this Paper were found to be in good agreement with the experimental results.
The calculations therefore seem to be applicable to the three-dimensional
as
well as the plane strain situation.
So far only dry sand has been considered, but the theoretical solutions, which are based on
effective stress, may be applied to saturated sands provided that the pore water is stationary
and the pore-water pressures around the tunnel are known.
The collapse pressure of a tunnel
in saturated sand is then simply the sum of the pressures predicted by either the upper or lower
The situation will, however, be rather more
bound solutions and the pore-water pressure.
complicated if there is a steady state, or transient, seepage.
Clearly the values of the angle of shearing resistance + applicable to the sand used in the
present investigations
may not be appropriate for granular soils in the field. To obtain satisfactory estimates of tunnel stability from the analyses given here a realistic value for 4 for the
field soil at the stress level in the tunnel at collapse should be found from laboratory tests.
It should be noted that while the Paper is entirely concerned with stress at failure, the
theory assumes that the tunnel support can readily adapt itself to the radial deformations
of
the ground for compliance with the mechanisms of the upper bound solution and the states
Such deformations cannot be predicted from the theory,
of stress of the lower bound solution.
but from the lead shot measurements
carried out during the model tests, it was found that
radial displacements
prior to collapse remained less than 5% of the original tunnel diameter
(Potts, 1976).
ACKNOWLEDGEMENTS
The research described in the Paper was carried out in the Cambridge University Engineering Department
and was supported by a Transport and Road Research Laboratory
research
contract for investigations
into the behaviour of tunnels and tunnel linings in soft ground; Dr
Potts was supported by an award from the Science Research Council.
The Authors are indebted to Dr E. T. Brown for contributions
made to the research, and to Professors A. N.
Schofield and J. A. Cheney and to Mr C. Collinson
for collaboration
in the use of the
Cambridge centrifuge.
STABILITY
OF A SHALLOW
CIRCULAR
TUNNEL
IN COHESIONLESS
SOIL
215
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