DAVIS,E. H., GUNN, M. J., MIUR, R. J. & SENEVIRATNE,H. N. (1980). G&technique
30, No. 4, 397416
The stability of shallow tunnels and underground
openings in cohesive material
E, H. DAVIS,* M. J. GUNN,_F R. J. MAIR$ and H. N. SENEVIRATNEg-
A basic engineering decision to be made in designing a
tunnel in soft clay is whether or not the tunnel can be
excavated without internal support. The safety of constructing a shallow heading in soft clay can be assessed in
terms of the fluid support pressure which may be required
to maintain stability. This can be estimated by means of
the lower and upper bound theorems of plasticity. The
Paper considers three different shapes of shallow underground opening relevant to tunnelling and upper and
lower bound stability solutions are derived for collapse
under undrained conditions.
Solutions are also derived
for assessment of the risk of blow-out failure caused by
excessively high fluid pressures.
Conditions
are considered under which local collapse can occur, independent
of the cover above the tunnel.
Une decision
fondamentale
doit Ctre prise lorsque
l’on
veut realiser un tunnel dans de l’argile molle; il s’agit, en
effet, de decider si un support
interne sera ou non
ntcessaire pour l’excavation du tunnel. La securitb de la
construction
dune galerie peu profonde dans de l’argile
molle peut itre evalute en fonction de la pression du fluide
support qui pourrait btre ntcessaire au maintien de la
stabilite. Cette estimation
peut se faire a l’aide des
thtoremes
a limites infirieure
et superieure.
L’article
envisage trois formes differentes d’ouverture souterraine
peu profonde interessant le creusement de tunnels, et des
solutions de stabilite a limites inferieure et superieure sont
trouvees pour l’affaissement
dans des conditions
non
drain&es. Des solutions sont egalement trouvees en ce qui
conceme l’evaluation du risque de rupture par eruption
due aux pressions excessives du fluide. Les conditions
dans lesquelles un affaissement local peut se produire sont
envisagees, independamment
de la couverture sit&e audessus du tunnel.
INTRODUCTION
Figure 1 shows schematically the normal method of constructing a shallow tunnel in soft
ground using a tunnel shield. Soil is removed from the tunnel face either by hand or by cutters on
a machine that forms a complete unit with the shield. The shield is jacked forward using
hydraulic rams which react against the tunnel lining. As the tunnel progresses, new rings of the
lining are installed at the rear of the shield. When the tunnel face is excavated by hand, the shield
protects the tunnellers from a roof collapse and the most obvious threat to the tunnellers arises
from possible instability of the face. When compressed air is used in the tunnel the stability of the
face is increased since the air pressure replaces to some extent the pre-existing in situ ground
stresses.
Compressed air is also often used in conjunction with tunnelling machines but here an
increasingly popular alternative is the use of bentonite or clay slurry under pressure. There is
usually a bead on the tunnelling machine which leads to overcutting, so that the initially
excavated diameter of the tunnel is greater than the diameter of the shield. If the fluid or air
pressure is great enough then the soil will not close up around the machine and an annular gap
will remain.
Under these circumstances it is possible to idealize the process of tunnelling as shown in Fig.
2, where a circular tunnel of diameter D is shown being constructed with a depth of cover C. The
tunnel lining is regarded as rigid and in front of it the tunnel heading is represented by a
Discussion on this Paper closes 1 March, 1981. For further details see inside hack cover.
*University
of Sydney.
t Cambridge University.
$ Scott Wilson Kirkpatrick
& Partners.
0 Ove Arup & Partners.
398
E. H. DAVIS,
M. J. GUNN,
R. J. MAIR
AND
H. N. SENEVIRATNE
Ground surface
~~~~_-~~~_~___
I
-----------
Bead
Tunnel shield
Fig. 1.
Shield tunnelling
Section
Fig. 2.
X-X
An idealization of shield tunnelling
cylindrical cavity of length P in which there acts a uniform fluid pressure cr.. The ground has a
unit weight y and a uniform pressure a, acts on the soil surface: this may be due to a large flexible
footing or an overburden of water (or very weak material). This Paper investigates what tunnel
pressures err are necessary to maintain the stability of the heading for different values of the
parameters that have been defined (D, C, P,y,a,)
and the strength of the ground. The collapse of
the tunnel heading will usually be a sudden event (caused, for example, by a sudden loss of tunnel
pressure) and hence it is appropriate to characterize the strength of the ground by its undrained
shear strength cU.In the following analysis it is assumed that c, is constant with depth, although
in practice c, will vary with depth depending on the history of the site. There are, however, many
situations where this assumption will be adequate and the methods of analysis that are used here
can be extended to cases where there is an arbitrary distribution
of c, with depth.
Broms & Bennermark
(1967) conducted experiments
in which they extruded clay under
pressure through vertical circular openings and they considered field observations
both where
failure had occurred and where stability had been maintained.
They defined a stability ratio N,
equal to the difference between the total overburden stress in the ground at the axis of the tunnel
STABILITY
OF SHALLOW
TUNNELS
IN COHESIVE
(before the tunnel is constructed)
strength c,.
MATERIALS
399
and the tunnel pressure divided by the undrained shear
N = [a,-a,+y(C
+D/~)]/c,
(1)
They concluded that if N is less than 6 then the opening will be stable. Their results are relevant
to the problem defined in this Paper for the particular case when P = 0.
In practice, the air pressure in a tunnel is often given a value to ensure that there is no flow of
water into the tunnel. This is achieved by applying a tunnel pressure greater than the pore-water
pressure at the tunnel invert. One possible approach to maintaining stability would be to set the
tunnel pressure equal to the overburden stress (i.e. N = 0). The problem with this approach is its
expense and also the health risks for tunnellers working at high air pressures. It is therefore
important to establish the minimum pressure necessary for stability.
It is appropriate to remark here that the Paper does not investigate whether the stability of a
tunnel heading improves or deteriorates in time. For a tunnel in heavily overconsolidated clay, a
reduction in tunnel pressure will be accompanied by the generation of negative excess pore
pressures which will dissipate as time passes. The clay around the tunnel will soften and after a
certain ‘stand-up’ time collapse is possible. Whether or not the heading collapses (and if it does,
the stand-up time) will depend on the geometry of the heading, the magnitude of the negative
pore pressures generated and the consolidation characteristics of the clay. In a tunnel in lightly
overconsolidated clay positive excess pore pressures will be generated and stability of the
heading will not deteriorate with time. Although an assessment of stand-up is of major concern
to tunnelling engineers (and is presently the subject of continued research at Cambridge
University) the Authors believe that the immediate undrained stability problem considered here
is a useful starting point when considering this phenomenon.
THE LIMIT
THEOREMS
AND
THEIR
APPLICATION
Stability solutions will be obtained using the limit theorems of plasticity. The soil is idealized
as an elastic, perfectly plastic material with a cohesion equal to c,. There is considerable
experimental evidence that this is a reasonable assumption for many clays. According to the
theory of plasticity, the collapse load for a particular configuration of loading on a perfectly
plastic body is unique, i.e. the load carrying capacity of the body cannot be changed by applying
the various loads in a different order. The lower bound theorem states that if any stress field can
be found which supports the loads, and is everywhere in equilibrium without yield being
exceeded, then the loads are lower than (or equal to) those for collapse. The upper bound
theorem states that if a work calculation is performed for a kinematically admissible collapse
mechanism then the loads thus deduced will be higher than (or equal to) those for collapse.
Since the tunnel pressure resists the collapse of soil into the tunnel it is a negative load in the
sense discussed above. The lower bound theorem will furnish a safe estimate of the tunnel
pressure required to maintain stability (i.e. higher or equal to that actually required) whereas the
upper bound theorem will provide an unsafe estimate. It is convenient to approach the solution
of the problem via a number of dimensionless parameters or groups which can be formally
derived by dimensional analysis, such as CID,P/D,cssjcu,
gT/c,,
and yDfc,.
For reasons which will
become clear later GJC, and or/c, can be replaced by the single parameter (a, - crr)/c,. Thus the
problem can be regarded as finding the value of (a, - r~r)/c, for limiting stability once the values
of the independent parameters CID,PJD and yDJc,
have been fixed.
Since the solution of the complete problem defined in Fig. 2 is not straightforward (in
particular, it is difficult to find a good lower bound) three simpler cases are considered in turn;
400
E. H. DAVIS, M. J. GUNN, R. J. MAIR AND H. N. SENEVIRATNE
ff.
1
!
1
4
4
4
Unit weighty
C
Undrained shear
strength c,
Undrained
cu
4. ,
II
1,
I--‘. . ”
k
CT
D
tf-.
1
Fig. 3.
shear strength
The plane strain unlined circular tunnel
Fig. 4.
‘7
D
. . . .
J.
The plane strain tunnel heading
each of which is relevant to the stability of tunnels or underground
openings and from which
some conclusions can be made as to the more general situation. The first and second cases are
shown in Figs 3 and 4: both are problems of plane strain. The stability of the long cylindrical
cavity of Fig. 3 (Case 1) will determine the radial pressure a cylindrical tunnel shield must resist.
This case is equivalent to the case illustrated in Fig. 2 when the ratio P/D is large. The case
shown in Fig. 4 (Case 2) is a ‘plane strain heading’; the excavated volume is not cylindrical but
instead is similar to a long wall mining excavation. The third case to be considered is the Broms
& Bennermark
problem which has the configuration
of Fig. 2 when P/D = 0. Of course the
results for this case are directly relevant to the stability of the hand excavated tunnel referred to
at the start of the introduction.
The three problems defined above will be referred to as Cases 1,2
and 3 in the remainder of this Paper.
THE PLANE
STRAIN
UNLINED
CIRCULAR
Lower bound. yD/c, = 0: weightless
The radially
symmetric
TUNNEL
(Case 1)
soil
stress field within
the annular
region shown in Fig. 5 is given by
c I =G T +2c ” In 2
0D
rJ@= fJ,+2c,
(2)
?I3 = 0
using the normal
the lower bound
notation.
solution
Outside
is
the annular
region there is an isotropic
(a, - or.)/cU = 2 In (2C/D + 1)
stress field a,. Thus
(3)
The influence of surface and tunnel pressures on the lower bound solutions appearing in this
Paper can be described by the single parameter o, -cr.. The reason for this is that the undrained
shear strength is independent
of the total mean normal stress; hence the addition of an arbitrary
isotropic stress produces an equally valid solution.
Lower bound, yDjc, > 0
For different values of yD/c, a computer program (Seneviratne,
1979) has been used to
generate equilibrium
stress fields around the tunnels which are everywhere in a state of plastic
STABILITY
OF SHALLOW
TUNNELS
IN COHESIVE
MATERIALS
401
yield. When the two plane equilibrium equations are combined with the yield condition a pair of
hyperbolic partial differential equations are obtained which can be numerically integrated
along two characteristic directions (e.g. Booker & Davis, 1977). Figure 6 shows the stress
characteristics generated for the case yD/c,
= 2. The characteristic lines are the directions of
maximum shear stress and are identical to the ‘slip-lines’ of metal plasticity. To obtain a
complete solution for a particular C/D ratio a strong discontinuity is constructed starting from
the point on the surface directly above the tunnel axis. This discontinuity completely surrounds
the tunnel as shown in Fig. 6. Outside the discontinuity, the above stress field is replaced by
Fig. 5.
A lower bound stress field for the plane strain circular tunnel (yD/c, = 0)
---
Fig. 6.
- --
Strong discontinuities
Stress characteristics
around a plane strain circular tunnel (yD/c, = 2)
402
E. H. DAVIS, M. .I. GUNN,
R. I. MAIR AND H. N. SENEVIRATNE
another stress field which satisfies the stress boundary condition at the surface and extends to
infinity without violating yield. This construction is illustrated in Fig. 7.
Lower bound solutions for values of yD/c, from 1 to 4 are shown in Fig. 8. For low values of
C/D when yD/c, = 3 or 4 it was not possible to complete the solution as described above
without violating yield and these solutions are not illustrated in Fig. 8. For values of yD/c,
greater than 4 the stress characteristics overlapped above the tunnel and no solution is
Fig. I.
Extension of the lower bound stress field for a plane strain circular tunnel (yD/c, > 0)
Fig. 8.
Lower bounds for the plane strain circular tunnel under gravity loading
STABILITY
OF SHALLOW
TUNNELS
IN COHESIVE
MATERIALS
403
presented. It can be shown (see the section entitled Local collapse) that as yD/c, is increased a
point is reached when it becomes impossible to maintain stability regardless of the magnitude of
the applied uniform tunnel pressure. The overlapping stress characteristics or the failure to
complete the extended stress field for low values of C/D when yD/c, = 3 or4 does not necessarily
mean that these cases are inherently unstable. No attempt has been made to find alternative
lower bounds since the generated solutions cover most of the range of parameters which is of
practical significance.
Upper bound
Four upper bound mechanisms are shown in Figs 9 to 12. Mechanisms A and B are simple
‘roof and ‘roof and sides’ mechanisms each containing one variable dimension (or angle) and
were deduced from model tunnel tests at Cambridge University (Cairncross, 1973; Mair, 1979).
The procedure for determining the critical collapse load is to derive an expression for (6, - (or)
(involving the variable dimension or angle) and then to minimize the value of (a, - oT) with
respect to the variation of the dimension or angle. This can be done either analytically or
numerically (e.g. by a digital computer program).
Fig. 9.
Fig. 10.
Upper bound mechanism A
Upper bound mechanism B
404
E. H. DAVIS, M. J. GUNN,
Fig. 11.
Upper bound mechanism C
Fig. 12.
Upper bound mechanism D
R. J. MAIR AND H. N. SENEVIRATNE
The reason that a, and rrr appear only in the form (oS- rrT)in the upper bound calculations is
that since a kinematically permissible mechanism for cohesive material involves no volume
change then the decrease in area of the tunnel must equal the area of ground loss at the surface.
Hence the work done by the pressures in the work calculation will be (0, -Q=) multiplied by that
area.
Mechanism C has four variable angles in its specification and includes mechanisms A and B
as special cases. Mechanism D is a ‘roof, sides and bottom’ mechanism with three variable
angles. Figures 13 and 14 show the results of a numerical optimization to discover the critical
mechanisms for yD/c, = 0 and 3. In both cases mechanism C is more critical for low values of
C/D and is superseded by mechanism D for high values of C/D. It can be seen that the value of
C/D at which this changeover takes place is lower for the greater value of yD/c,. Figures 13 and
14 also show the lower bounds which lie close to the best upper bounds indicating that the exact
collapse loads have been closely bracketed. In the neighbourhood of the optimum upper bound,
changes in the variable angles lead to small changes in the collapse load. There is not much
STABILITY OF SHALLOW
TUNNELS
Mechanism D
IN COHESIVE
405
MATERIALS
Mechanism C
Mechanism B
Mechanism A
Lower bound
1
1
1
2
3
8
4
5
CID
Fig. 13.
Stability solutions for the plane strain circular tunnel (yD/c, = 0)
2
t
CID
Lower bound
-12
t
Fig. 14.
Stability solutions for tbe plane strain circular tunnel (yD/c, = 3)
406
E. H. DAVIS, M. J. GUNN,
- - --
-
R. J. MAIR AND H. N. SENEVIRATNE
Lower bound
Upper bound
Fig. 15.
Upper and lower bound stability ratios for plane strain circular tunnels
Note: Stresses shown are normalized bye
Fig. 16.
”
A lower bound stress field for the plane heading (C/D = 4)
STABILITY OF SHALLOW TUNNELS IN COHESIVE
MATERIALS
407
Note: Stresses shown are normalized by cu.
Unlabelled stress components are mmor and are
two less than major principal stress components.
Fig. 17.
A lower bound stress field for the plane strain heading (C/D = 2.875)
difference between mechanisms B, C and D for practical purposes (i.e. in estimated collapse
load) although the mechanics of deformation are very different. In general it seems that a
mechanism with one variable will yield an adequate upper bound (i.e. close to a good lower
bound) providing an appropriate pattern of collapse is chosen.
Figure 15 shows the bounds for different values of yD/c, all plotted as stability ratio N against
CID. For values of CID greater than 3 the upper and lower bounds of N do not change
significantly with yD/c,. Below C/D = 3 there is a larger spread but adopting the lower bound
for yD/c, = 0 as a criterion for deciding the tunnel pressure should be a safe procedure. This is
because it always corresponds to a lower value of N, or a higher value of oT, than those
prescribed by the lower bounds for higher values of yD/c,. It is important, however, to consider
also the possibility either of local failure at high values of yD/c, (see the section entitled Local
collapse) or of failure caused by ‘blow-out’ for very shallow tunnels (see the section entitled
Blow-out).
THE PLANE
STRAIN
HEADING
(Case 2)
yD/c, = 0
Lower bound. Lower bound solutions for the plane strain heading can be constructed from three-
sided and four-sided areas of constant stress at or below yield. Two such typical fields of stress
are shown in Figs 16 and 17. These are systematically extended from two radial zones emanating
from the top and bottom of the heading (Gunn, 1980). These fields transmit a shear load from the
soil to the tunnel lining. Figure 16 shows 87% of the undrained shear strength being mobilized
on the soil-lining interface. This degree of mobilization would be reasonable for rough linings in
soft clay but solutions based on smooth linings would be more appropriate in other cases.
Solutions for a smooth lining can be obtained by adapting the analyses of Booker & Davis
(1973) or Ewing & Hill (1967) for the problems of bearing capacity near a vertical face and a Vnotched tension bar respectively. In particular the addition of an isotropic stress field to Ewing
& Hill’s slip line solution gives as a lower bound
(a, - a,)/~, = 2 + 2 In (C/D + 1)
(4)
408
E. H. DAVIS,
Fig. 18.
M. J. GUNN,
R. J. MAIR
AND
H. N. SENEVIRATNE
An upper bound mechanism for the plane strain heading
tar
a_&_---
6>
,,
q
. *.-.__,
Stress
field lower bound
Lower Douna (smoorn Immgj
(After Ewing & Hill)
N=2+21n(C/D+l)
01
0
Fig. 19.
1
2
CID
3
4
I
5
Stability solutions for tbe plane strain beading
Upper bound. When the upper bound mechanism illustrated in Fig. 18 is optimized with respect
to the three variable angles, the critical collapse load is found to be
(%--J/c,
= 4&P++)
(5)
with tan u = tan /I = 2 J(C/D+ 4)and 6 = 7~12.
The upper and lower bounds are plotted in Fig. 19 and again the exact collapse loads have
been reasonably closely bracketed. The upper bound is not affected by the degree of roughness
of the tunnel lining in contrast to the lower bounds where the collapse load for a rough lining is 0
to 20% higher (depending on C/D ratio) than that for a smooth lining.
STABILITY OF SHALLOW
TUNNELS IN COHESIVE MATERIALS
409
If the work done by the self weight of the soil is accounted for in the work calculation for the
mechanism shown in Fig. 18 then the same optimum is obtained and
(0s - ~r)/cu + YDMCID + )) = 4 &C/D +t)
(6)
Since the expression on the left is equal to the stability ratio N, it follows that this mechanism
predicts the same stability ratio regardless of the value of yD/c,.
Similarly it is possible to add a hydrostatic stress field to obtain new lower bounds for various
values of yD/c,. In these lower bounds, however, the tunnel pressures vary linearly with depth,
the pressures being yD higher at the bottom of the heading than at the top. If the average tunnel
pressure is adopted in the calculation of the stability ratio then the results for yD/c, = 0 apply
for any value of yD/c,. Figure 19 can therefore be regarded as a plot of stability ratio against C/D
for all values of yD/c,. The stability ratio for a safe tunnel pressure (determined as above) is not a
lower bound for the originally specified problem (constant or. with depth) but the results for the
circular tunnel suggest that this assumption is a safe one. As in the case of the circular tunnel the
possibilities of local collapse and blow-out must also be considered when determining a safe
tunnel pressure.
THE CIRCULAR
TUNNEL
HEADING
(Case 3)
yD/c, = 0
Lower bound. The first lower bound stress field is shown in Fig. 20(a). Within the cylindrical
volume of soil which is the continuation of the already excavated tunnel the axial stress is equal
to cr. In planes perpendicular to the tunnel axis, the two principal stresses are equal to a,+ 2c,.
Outside this cylinder there is a radially symmetric stress field similar to that adopted for the
plane strain circular tunnel with the axial stress being some intermediate value between the
radial and circumferential stresses. Outside the larger cylinder of diameter (C +(D/2)) there is an
isotropic stress field a,. The lower bound is
(Go- rrr)/c, = 2 + 2 In (2C/D + 1)
(7)
Figure 20(b) illustrates an alternative lower bound stress field. Within a hemispherical cap on
the end of the tunnel there is an isotropic stress field cr. Outside this hemisphere there is a
spherically symmetric stress field given by
0, = uT + 4c, In (2r/D)
(8)
1
Outside the sphere (to which the surface is a tangent plane) there is an isotropic stress field a,.
The lower bound is
00 = a, +2c,
(a, - G&C, = 4 In (2C/D + 1)
(9)
It can be seen from Fig. 21 that lower bound (a) allows higher loads to be supported for values
of CIDcO.86, whereas for values of C/D> 0.86 lower bound (b) is better.
Upper bound. The mechanism of Fig. 18 is adopted but the plane strain sliding blocks are
replaced by blocks with elliptical cross-sections. The length of the semi-axes perpendicular to
the plane of the diagram is equal to D/2. Optimization of the upper bound with respect to the
three variable angles leads to the line shown in Fig. 21.
E. H. DAVIS,
M. J. GUNN,
R. J. MAIR
AND
H. N. SENEWRATNE
Section X-X
(a) Thick cylinder
Section
X-X
(b) Thick sphere
Fig. 20.
Lower bound stress fields for the circular tunnel heading
The effect of the self weight of the soil can be added in exactly the same way as for case 2
leading to the interpretation of Fig. 21 in terms of stability ratio as well as (a, - c~)/c,, for the
weightless case. The relatively large gap between the upper and lower bounds for this case is a
reflection of the increased complexity of the three-dimensional problem.
STABILITY
OF SHALLOW
TUNNELS
IN COHESIVE
411
MATERIALS
24-
20 -
16Lower bound (thick sphere)
2 12-
Lower bound (thick cylinder)
N=2+2/0(2C/D+1)
0
1
2
3
4
5
CID
Fig. 21.
Stability solutions for the circular tunnel beading
(a) Plane strain
circular tunnel
Fig. 22.
Upper bound mechanisms
(b) Plane strain
tunnel heading
(c) 30 circular
tunnel heading
for local collapse
LOCAL COLLAPSE
It can easily be demonstrated
that if the ratio yD/c, is sufficiently large then collapse will
take place for any value of uniform tunnel pressure. For Cases 1, 2 and 3 which were defined
earlier optimization
of the mechanisms
shown in Figs 22(a), (b) and (c) results in upper
bounds on yD/c, for collapse of 8.71,8-28 and 1096 respectively. Even though these mechanisms
involve no immediate subsidence of the ground surface, it is likely that this would be the first
step of a progressive failure which would eventually propagate to the surface. Since for a given
site the value of y/c, would be predetermined,
the limiting value of yD/c, can be viewed as
specifying the maximum height of tunnel heading which can be constructed
under uniform
tunnel pressure. To calculate safe values of yD/c, for the various cases lower bound solutions are
required. For the plane strain circular tunnel the evidence of the stress characteristics
lower
bounds is that, for any value of C/D, stability is maintained provided that yD/c, < 2. Stability can
still be maintained for values of yD/c, between 2 and 4 when C/D exceeds certain minimal values
(see Fig. 8).
E. H. DAVIS, M. J. GUNN, R. J. MAIR AND H. N. SENEVIRATNE
412
Fig. 23.
A lower bound stress field for local collapse (Cases 2 and
3,62c,/y(Pastot’s
Fig. 24.
3)
solution)
A lower bound stress field for local collapse (Case 2) using Pastor’s solution
For both the plane strain tunnel heading and the circular tunnel heading, stress fields based
on Fig. 23 show that there will be no possibility of local collapse for tunnelling when yD/c, < 4
and uniform tunnel pressure equals y(C[D/2) (i.e. stability ratio N = 0). It is possible to improve
the solution for the plane strain heading by using Pastor’s (1978) lower bound for the critical
height of a vertical cut in cohesive material. Using a linear programming
approach, an allowable
stress field with 208 zones of linearly varying stresses and a safe height of 3.63 c,/y was obtained.
Adopting
this field as shown in Fig. 24 then the heading is stable for yD/c, = 5.63
when eT = y(C+O.3550).
In contrast to the plane strain circular tunnel these solutions are
valid for any value of C/D.
BLOW-OUT
The problem of causing a failure by having a tunnel pressure which is too large has already
been mentioned. For Cases 2 and 3 the solutions already obtained can be directly adapted as
follows.
STABILITY OF SHALLOW
TUNNELS
IN COHESIVE
413
MATERIALS
Upper bound
For both the plane strain and circular heading the result can be written in the form
(08- a,)/~, + yUc,(CP
+ 3) = N = f(V)
(10)
for collapse into the tunnel. It is possible to identify the first term in this equation with the net
work done by the pressures; the second term with work done by gravity; and the third term with
the plastic work dissipated on the sliding surfaces. When the direction of motion is reversed
or> as and
@Jr- %)/C, = YD/C”(CP + t) +f(cP)
or
(a, - D~)/c, + yD/c,(C/D + +) = -f(C/D)
= - N
(11)
Hence an unsafe (or high) estimate of tunnel pressure to cause blow-out can be obtained by
reversing the sign of the stability ratio for inward collapse.
Lower bound
If the directions of major and minor principal stress are reversed in the stress fields considered
for collapse-in and the gravity field added as before then equivalent results to the upper bounds
are obtained
(0, - cr)/c, + Y%(‘V
+ 3) = - N
This expression allows the calculation of safe tunnel pressures but as before the lower bound
stress field includes a tunnel pressure varying ‘linearly with depth and not the constant
distribution which is of real interest.
It is not possible to make use of the collapse-in results for case 1 in exactly the same way as
above. Figure 15 demonstrated that upper bound solutions do not lead to values of N that are
independent of yD/c,. This is because the work done by gravity would be equivalent to an
expression of the form yD/c,(C/D + x) where x has a value between 0 and f depending on the
optimum mechanism for a particular case. Hence to obtain results for blow-out the complete set
of upper bound mechanisms should be re-examined with the direction of movement reversed
and new lower bounds should be generated where the tunnel pressure is the major principal
stress.
However, it is likely that blow-out will only be a problem for very shallow tunnels when
failure will occur similarly to mechanism A (Fig. 9) and that as for collapse-in safe tunnel
pressures will be close to dT = a,.
DISCUSSION
AND CONCLUSIONS
Tunnel pressures necessary for the support of shallow tunnels and underground openings in
soft ground have been estimated using the limit theorems of plasticity. Two of the situations
considered have been special cases of the idealization suggested in the Introduction for the
process of constructing a shallow circular tunnel using a shield or tunnelling machine. The
results indicate (as might be expected) that higher tunnel pressures are required for stability of
the unlined circular tunnel (P/D + 00) than for the circular tunnel heading (P/D = 0).
The stability ratio N, originally proposed by Broms & Bennermark, has provided a
convenient framework for interpreting the results and can be used to estimate tunnel pressures
required for stability. In contrast to Broms & Bennermark, however, the results presented here
indicate that the critical values of N show a marked variation with the depth of burial (i.e. C/D)
414
E. H. DAVIS,
M. J. GUNN,
R. J. MAIR
AND
H. N. SENEVIRATNE
B Ii
+
Force
m73-y+7(C+qa
Fig. 25.
Moment
703/i 2
The equivalent stability problem in weightless medium
of the tunnel. Indeed, adoption of the Broms & Bennermark criterion (N < 6 for stability) would
indicate that some shallow headings would be stable with no support pressure whereas an upper
bound calculation
demonstrates
that collapse would be inevitable
without a supporting
pressure.
A possible objection to the lower bound or safe tunnel pressures suggested for the headings is
that they are based on stress fields where the tunnel pressures vary linearly with depth. Although
the adoption of the average pressure as safe seems reasonable, it is not satisfactory from the
point of view of the theory of plasticity. At first sight the adoption of the pressure at the bottom
of the heading as safe might seem to deal with this objection because now there is a greater
overall pressure supporting the heading than in the actual stress field. This process is equivalent
to a reduction in the safe value of N by &D/c,) but is again not satisfactory from the point of
view of plasticity theory because the greater constant support pressure is not proportionally
greater than the linearly varying support pressure. Indeed both these approaches could be used
to predict safe tunnel pressures when the results of the section entitled Local collapse indicate
that collapse would definitely occur. There is, however, a simple way in which the results already
presented can be adapted to predict safe tunnel pressures which are theoretically justified.
Figure 25 shows the loads acting when the effect of gravity and the surface pressure is
subtracted (according to plasticity theory the collapse load is not affected by this subtraction). A
linearly varying tunnel pressure is now the only load acting and it can be regarded as consisting
ofa tensile load equal to D[o, - a,+ y(C + D/2)] and a moment equal to yD3/12. The tensile load
is clearly responsible for failures involving the surface (as in Fig. 18) whereas the moment is
responsible for local collapse. These two loads can be regarded as independent and according to
plasticity theory the corresponding
failure locus on the interaction diagram must be convex.
Figure 26 shows (for the particular
cases of a plane strain heading with C/D = 2 and
C/D = 2.875) the three upper bound lines and lower bound points (A, B, C) on the appropriate
(nondimensionalized)
interaction
diagram. From the convexity property AB must represent
safe values of N, and this line can be regarded as a criterion for the reduction of N with
increasing yD/c,.
For the case of Fig. 26(a) with yD/c,
= 3 the average pressure criterion gives a
safe N of 4, i.e. o*/c,, > 1.5, whereas the above criterion gives a safe N of 2.3, i.e. GJC, > 3.2.
The practical tunnelling engineer will want to know what experimental evidence there is for
the solutions presented here. Tests on unlined circular model tunnels carried out at Cambridge
University (Mair, 1979; Seneviratne, 1979) have shown that the experimental collapse loads lie
between the lower and upper bound solutions that have been presented (see Fig. 27). A series of
STABILITY
OF SHALLOW
TUNNELS
IN COHESIVE
t
6
4
Upper bound (collapse in)
_------------_
A
Lower bound
415
MATERIALS
Upper bound (collapse in)
--------------
I
I
I
I Upper
1bound
I
Lower bound
f
I Upper
1bound
1(Local
1collapse)
I
I
I
I
(Local
1collapse)
i
I
B
I
20
6
‘1
9/c”
I
2
I
I
I
4
Upper bound (blowout)
_------_--e---s
I
,
I
1
6
Upper bound (blowout)
- - - --
(a) C/D = 413
-_---
---
i
(b) C/D = 2,875
Fig. 26.
Interaction diagrams for tbe plane strain beading
5-
4-
3-
4
2-
b"
Upper bound
D Experimental
CID
-1 -
Fig. 27. Comparison
yD/c, = 2.6)
of the stability solutions presented in this Paper with experimental
observations
(Case I:
416
E. H. DAVIS, M. J. GUNN, R. J. MAIR AND H. N. SENEVIRATNE
tests on model tunnel headings (Mair, 1979) has resulted in collapse loads which lie fairly close
to the lower bounds suggested for Case 3. For shallow tunnels (Case 1) the experimentally
observed collapse mechanism is usually very close to the optimum upper bound mechanism.
For deeper, i.e. (C/D)> 2, Case 1 tunnels and for most of the Case 3 tests, however, significant
differences are observed between the sliding block mechanisms presented here and the pattern of
deformation observed experimentally. The sliding block mechanisms suggest that a movement
of soil inwards near the tunnel is accompanied by an (approximately) equal settlement of the
surface. In practice, however, large movements near the tunnel are accompanied by much
smaller settlements of the surface, perhaps indicating a zone of plastically yielded soil near the
tunnel which is supported by a surrounding elastic region.
Nevertheless the Authors believe that the experimental evidence currently available indicates
that the results presented in this Paper can be used with confidence for the calculation of the
undrained stability of tunnels when C/D < 3.
ACKNOWLEDGEMENTS
The work presented in this Paper forms part of a wider programme of research into the
behaviour of shallow tunnels in soft ground at Cambridge University supported by the
Transport and Road Research Laboratory. The research has been conducted under the overall
direction of Professor A. N. Schofield and the Authors are grateful to him for several stimulating
discussions.
REFERENCES
Booker, J. R. & Davis, E. H. (1973). Some adaptations
of classical plasticity theory for soil stability problems.
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1973.
Booker, J. R. & Davis, E. H. (1977). ‘Stability analysis by plasticity theory’, in Numerical methods in geotechnical
engineering, eds C. S. Desai and J. T. Christian, Chapter 21. London: McGraw-Hill.
Broms, B. B. & Bennermark, H. (1967). Stability ofclay in vertical openings. J. Soil Mech. Fdns Div. Am. Sot. Cio. Engrs,
193, SMl, 71-94.
Cairncross, A. M. (1973). Deformations around model tunnels in stiff clay. PhD thesis, Cambridge University.
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Gunn, M. J. (1980). A note on the centred-fan stress field and its use in plasticity problems relevant to geotechnical
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University Engineering
Department
Internal Report.
Pastor, J. (1978). Analyse limite: determination
numirique
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au talus
vertical. J. Mtc. appl. 2, No. 2, 167-196.
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University.
Seneviratne, H. N. (1979). Deformations and pore pressure uariations around shallow tunnels in sqfi clay. PhD thesis,
Cambridge
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