Lea,
E. & Domieux,
L. (1990). Giotechnique
40, No. 4, 58146
Upper and lower bound solutions for the face stability of
shallow circular tunnels in frictional material
E. LECA*
and L. DORMIEUX*
With the recent increase in underground urban
development as well as for transportation, tunnels
need to he driven in increasingly dif&ult soil conditions. In most cases the ground itself is not stable
and face stability is achieved by applying fluid
pressure to the tunnel front. The question of determining the retaining pressure to he used has
received special consideration in the past hecause
of the concern for safety during construction, and
also hecause of the damage that could he caused to
surface structures by the failure of a shallow
tunnel. The problem is three-dimensional and can
he studied by using the limit state design method.
Solutions are available for the case of a circular
tunnel in purely cohesive ground, but very little is
known of the face stability of a tunnel driven in
sandy soils. This Paper approaches this latter
problem from the point of view of limit analysis.
Both safety against face collapse and blow-out are
considered. Three upper hound solutions are
derived from the consideration of mechanisms
based on the motion of rigid conical blocks. The
results are compared with lower hound solutions
published in a previous article. A failure criterion is
proposed for the tunnel face in the general case of
a cohesive and frictional soil, and charts are provided to allow a bracketed estimate of the required
retaining pressure. A comparison with centrifuge
laboratory tests shows close agreement between the
theoretical upper hound solutions and the face
pressures at collapse measured experimentally.
KEYWORDS: limit state design/analysis;
city; research; sands; stability; tunnels.
plasti-
INTRODUCTION
The stability of the front of a tunnel driven in
cohesive material has been studied by several
authors since the paper by Broms & Bennermark
(1967) on a stability criterion based on laboratory
extrusion
tests and field observations.
Similar
Discussion on this Paper closes 5 April 1991; for further
details see p. ii.
* Ecole Nationale des Ponts et Chaussees, 93167,
Noisy-laGrand Cedex, Paris, France.
L’utilisation croissante du sous-sol en site urbain
ainsi que le dtveloppement des riseaux de transports enter& conduit i construire des tunnels dans
des conditions toujours plus difficiles. Dans la
plupart des cas le terrain est instable et ii est n&
cessaire d’appliquer une pression de souttinement
au niveau du front de taille. Le cboix de la pression
I utiliser a fait l’objet de qombreuses 6tudes au
tours des derniLres an&es en raison des problemes
po&s pour de tels projets du point de vue de la
&curiti en tours de construction et des cons&
quences d’une rupture sur les structures situ&s en
surface. 11 s’agit d’un problbme tridimensionnel qui
peut 6tre notamment btudii! par la mbtbode de
I’analyse limite. Des solutions ont dbjji &ti! propos&s pour le cas d’un tunnel circulaire en terrain
cob&rent, mais les connaissances sont encore limit&es quant ai la stabilite du front de taille I’un
tunnel creu& dans des sols sableux. Ce dernier
problkme est examin dans le pr&sent article du
point de vue de I’analyse limite. On s’intkresse h la
fois aux risques de rupture par effondrement et par
explosion. L’examen de trois m6canismes ha.6 sur
le mouvement rigide de blocs coniques permet
d’ahoutir 1 une approche par l’extbrieur des conditions de stabilitb. Les rbultats sont comparb i
I’approcbe par I’int&ieur p&sent&e dans un article
p&&dent. L’6tude ahoutit P la mise en C?vidence
d’un critere de rupture pour le front de taille dans
le cas g&n&al d’un sol frottant-cob&ent.
On
propose igalement une drie d’abaques permettant
d’obtenir un encadrement de la pression de soutb
nement P utiliser. L’application I des essais en centrifuge
montre que les homes sup&ieures
obtenues analytiquement sont t&s proches des
P la
pressions
mesuri?es exp&imentalement
rupture.
conclusions
were presented by Peck (1969) and
more recently a more theoretical
approach was
proposed by Davis et al. (1980), which is derived
from the limit state design concept. Their results
are
supported
by
centrifuge
model
tests
(Schofield.
1980) and nrovide a more general
expression
of the stability
criterion, -which
accounts for the effects of tunnel depth.
Yet very little is known about the face stability
of tunnels when the ground is characterized
by a
Mohr-Coulomb
yield criterion.
This question
581
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LECA AND DORMIEUX
582
has, however, received more attention
recently
because of the construction
of tunnels in increasingly difficult ground conditions
and even in
sandy materials.
Some lower bound solutions
were proposed by Muelhaus (1984); they essentially refer to the stability of the tunnel lining, but
an estimate of the critical unsupported
length
behind the face is also provided. A more general
solution for the tunnel front is given by Leca &
Panet (1988).
This Paper addresses the problem of the face
stability of a tunnel driven in a Mohr-Coulomb
material from the point of view of limit analysis.
The major difficulty of such a problem comes
from the fact that it is fully three-dimensional.
An
upper
bound
solution
is derived
from the
analyses of three failure mechanisms. The results
obtained for the lower bound (Leca & Panet,
1988) are re-examined
and compared
with the
upper bound solutions. This allows a bracketed
estimate to be made of stability conditions for the
tunnel front. The critical pressures obtained from
this method
are compared
with experimental
results from centrifuge tests.
The construction
procedure for shallow tunnels
in soils is described in the Paper by Davis et al.
(1980). The problem can be idealized, as shown in
Fig. 1, by considering
a circular rigid tunnel of
diameter D driven under a depth of cover C, i.e.
at a depth
FM+;
The unit weight of the soil is y and a surcharge us
is applied at ground surface. The unsupported
length behind the face P is taken as zero (which is
a reasonable assumption for shield driven tunnels)
and a retaining pressure rrr is applied to the front.
Such support can be achieved by using com-
pressed air, bentonite
slurry or earth pressure
(EPB shield). In the case of this study err is
assumed to be constant, which best represents the
case of compressed air, but also gives some information for slurry or EPB shields.
The ground conditions are also assumed to be
uniform around the tunnel. The soil is modelled
as a Mohr-Coulomb
material, characterized
by
its cohesion c’ and its friction angle 4’. For such
materials it is helpful to introduce the unconfined
compression strength
Uc = 2
c’ cos #J’
1 - sin 4’
and the Rankine
K, =
earth pressure
coefficients
1 - sin 4’
1 + sin f#~’
for active failure and
K = 1 + sin 4
P
1 - sin 4
for passive failure. As an alternative set of parameters uc and K, (or KP) can be used to characterize the soil’s resistance.
For a cohesionless
material K, or K, can be used alternatively with
4’. Dimensional analysis shows that this problem
can be analysed in terms of five dimensionless
parameters : C/D, a&,,
4~,
Y%,
and K, (or
KA).
LIMIT ANALYSIS
The purpose of limit analysis is to provide an
estimate of stability conditions
for a mechanical
system regardless of the behaviour of the material
it is made of (Salencon, 1983, 1990). Stability con-
Sectm
A-A
Fig. 1. Simplified geometry for tbe front stability of a shallow tunnel
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FACE STABILITY
OF TUNNELS
IN FRICTIONAL
SOILS
583
Fig. 2. Bracketed estimate of the domain of supported loads
ditions for this system are derived in terms of the
loads that can be applied to the system without
causing its failure. An upper bound estimate of
such loads is found by considering
a kinematically admissible failure mechanism for which
the power 8, of the loads applied to the system is
larger than the power P, that can be dissipated
inside the system during its movement (the upper
bound theorem). On the other hand, any set of
loads for which a stress field can be found that
satisfies equilibrium
and the material yield criterion, is a lower bound solution (lower bound
theorem). Stability can be characterized
by the
domain .X of supported loads; .X is bracketed by
a lower bound domain Xx- and an upper bound
domain X’. This is shown in Fig. 2 for the case
of a system subjected to a set of two loads Qi and
Q,.Any set of loads located inside X- is potentially supported
by the system while any set of
loads outside X’ is unstable. The domain X is
convex, which means that if the two sets of loads
Q’ and Q” are supportable,
any linear combination
of these loads @’ + (1 - A@” where
0 < 1 < 1 is also supportable.
In Fig. 2 any point
M located on the straight line between A and A’
is inside &‘- as long as A and A’ both belong to
X‘-. This convexity can be used to obtain an
improved lower bound solution.
For the tunnel problem analysed in this Paper
three loading parameters
need to be considered:
%lo, t %/UC7 and yDJa,.Since soil yield is controlled by the Mohr-Coulomb
criterion,
the
failure mechanisms
must be chosen such that
along any failure surface X,, the angle ed between
the discontinuity
velocity
V,, and & (Fig. 3)
satisfies the condition
&<8*<rC-4
(5)
Otherwise, the dissipated power P, is equal to
infinity and no upper bound solution can be
derived from the mechanisms analysed.
Fig. 3. Discontinuity velocity along the failure surface
UPPER
BOUND
SOLUTIONS
Three failure mechanisms
have been considered. They all involve the movement of solid
conical blocks with circular cross-sections.
The
opening of each cone is equal to 24’ and its velocity Y is parallel to its axis (Fig. 4). Therefore condition (5) is satisfied along the failure surfaces
between the moving blocks and the rest of the
ground. The three mechanisms
MI, MI1 and
MI11 are shown in Fig. 5. MI and MI1 are collapse mechanisms,
whereas MI11 refers to blowout failure. Even though safety against collapse is
a major concern during tunnelling,
case MI11
may be of interest for very shallow tunnels bored
in weak soils, when the pressure (TVcan become
so great that soil is heaved in front of the shield.
\ --_
\ \\
//fi\
TR
\\F--- \
\\
V
Fig. 4. Conical blocks and kinematic conditions used in
MI, MI1 and MID
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LECA AND DORMIEUX
/ A’
(4
/
I
(?I’)
/
Fig. 5. Mechanisms (a) MI, (b) MI1 and (c) MI11
Such a phenomenon
has been observed during
tunnelling projects (Clough et al., 1983).
Failure is due to the collapse of one conical
block in MI and two blocks in MII. The
geometry is a little more complex in the latter
case: the first cone (block 1) is truncated
by a
plane K perpendicular
to the plane of symmetry of
the tunnel, which projects as A on Fig. 5(b). The
second cone (block 2) is a mirror image of the
first with respect to plane rr’ (perpendicular
to rr
and going through the centre of the intersection
Xi2 between n and block 1; plane ri projects as
A on Fig. 5(b)). This ensures that plane x intersects both blocks along the same elliptical surface
Zr2. Plane x is chosen such that the axis of the
second cone is vertical. Therefore, both MI and
MI1 are characterized
by only one parameter, the
angle a between the axis of the cone adjacent to
the tunnel and the horizontal. MI11 is also characterized by a (Fig. 5(c)); the geometry is similar
to that of MI except that the cone is inverted and
the velocity reversed. It should be noted that for
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FACE STABILITY
OF TUNNELS
Fig. 6. Area of failure at the tunnel face
all three mechanisms
the intersection
of the
tunnel with its adjacent cone is an ellipse &,i the
major semi-axis length of which is equal to D/2
(Fig. 6). This implies that only part of the tunnel
face is failing. However, limit analysis theory
remains valid for such a geometry and upper
bound solutions can still be derived from these
three mechanisms.
The derivations related to MI, MI1 and MI11
are given in Appendices I, II and III respectively.
In all three cases the power 8, of external loads
Wffc > a&,, yD/u,}and the dissipation power
P, are first calculated
separately.
An upper
bound solution is then found, given that in order
for the set of loads {as/u,, a&,-,
yD/u,}to be
stable, 8, and Pv must satisfy
PC < Pv
(6)
To interpret the results obtained in Appendices I,
II and III it is convenient
to rewrite the three
loading parameters
Qs = (K, - 1) z + 1
QT = (KP - 1)
2 +1
Q, = WP- 1)
$
(7a)
(7c)
Then, relation (6) leads to upper bound
that can be put in the form
(8)
NsQs+N,Q,GQ,
for collapse mechanisms
solutions
MI and MI1 and
NsQs+N,Q,aQ,
for blow-out mechanism
MIII.
and (9) Ns and N, are weighting
depend on the angle a between
cone adjacent to the tunnel and
(9)
In relations (8)
coefficients that
the axis of the
the horizontal.
IN FRICTIONAL
SOILS
585
The two collapse mechanisms are optimized when
u is chosen such that N, and N, are at maximum.
MI11 is optimized
when Ns and N, are at
minimum. For all three mechanisms
the coefficients N, and N, of the best upper bound results
will be respectively referred to as Ns’+, N,‘+ for
collapse and Nsb+, NYb+ for blow-out. The results
of these
optimizations
are summarized
on
Fig. 7(a) for collapse and Fig. 7(b) for blow-out
failure.
In Fig. 7, optimal values Ns’+, N,” and Nsb+,
N b+ are plotted as functions of the depth ratio
CID for common values of the friction angle 4’
(20”45”). It was found that MIX provides the best
upper bound for collapse in most cases except for
very shallow tunnels (C/D< 0.25) or friction
angle values smaller than 30”. MI and MI1 lead
to similar results when C/D is greater than 1.0.
Fig. 7(a) shows that N,‘+ is almost
always
smaller than N,‘+ and is equal to zero for any
value of 4’ when C/D 2 0.6. This suggests that if
the actual failure conditions are similar to those
predicted by MI and MII, the surcharge us will
have very little influence on face collapse, unless
the tunnel is very shallow. For most conditions,
failure would not reach the ground surface and
MI as well as MI1 could be considered as local
collapse mechanisms. However, such mechanisms
could lead to some larger scale failure with the
formation of sink holes, since a large amount of
soil would be left unsupported
once initial collapse of the tunnel face had occurred.
Another conclusion
from the analysis of collapse mechanisms
was that the optimal values
N Et NC+ are always obtained for essentially the
sa”,e’vaLe a’+ of parameter a
a’+
N 49” - $
This means that in the plane of symmetry of the
tunnel, the angle of the critical failure surface to
the horizontal P+ = 49” + r&/2 is larger than the
angle of active failure in plane strain conditions
6, = 45” + #/2 (Fig. 8(a)). Therefore the area in
front of the tunnel which is influenced by the collapse is more limited than in the case of a long
open cut. This could be seen as a stabilizing effect
due to three-dimensional
equilibrium conditions
around the tunnel face and would again need to
be compared
to actual failures observed in the
field.
The values of NSb+ and NYb+ computed for the
blow-out case, however, are large and increase
sharply with the depth ratio CID,which is consistent with the fact that such a failure would only
occur with very shallow tunnels. The critical
geometry is obtained when
ab+ N 49”
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(11)
586
LECA AND
e
I N”,’ ’
DORMIEUX
___
Iv,+
- ---
NC,+
022 /
\
g’ = 20
0.16-
\
\
\
0.16-
\
\
\
0.14-
I$’ = 25
\
\
\
0.12-\
\
\
O.lO-
\
‘\
0.06 -
lp’ = 30
\
\
\
\
\
\
\
\
$’ = 35’
\
fp’ = 40”
f$’ = 45”
_ I’ = 20”
-L__
0.2
0.4
0.6
(a1
Fig. 7. Upper bound values of Ns and N, (a) for collapse and (b) for blow-out
Downloaded by [ Middle East Technical University] on [19/06/17]. Copyright © ICE Publishing, all rights reserved.
CID
FACE STABILITY
OF TUNNELS
IN FRICTIONAL
SOILS
587
(W
Fig. 8. Critical geometries for (a) collapse and (b) blow-out
This means that the angle of the failure surface
to the horizontal
Sb+ = 49” - 4’ is smaller than
the angle of passive failure in plane strain conditions 6, = 45” - @/2 for common values of 4’
(Fig. 8(b)).
LOWER BOUND SOLUTIONS
Some lower bound solutions for the case of a
Mohr-Coulomb
material have been published
previously (Leca & Panet, 1988). They are based
on three stress fields similar to those used by
Davis et al. (1980). These three stress fields SI, SII,
311 are shown on Fig. 9.
SI is a geostatic stress field and it can actually
be used in the general case of a soil with weight
(y > 0). In the geometry shown on Fig. 9(a), the
ground is divided in three layers: above and
below the tunnel (y > D/2 or y < -D/2)
the
stress field is also isotropic; in the layer located at
tunnel depth (-D/2 < y < D/2) the horizontal
component
of stress along Oz is equal to a,. SII
and SIII both apply to a weightless soil (y = 0).
Even though this assumption may not be realistic
in the case of a shallow tunnel the lower bound
derived from SII and SIII will be used to improve
the general solution obtained from the consideration of stress field SI. SII is axisymmetric around
the tunnel axis. Within the cylinder C, which
extends the tunnel in the Oz direction, the axial
stress is equal to (or and the radial and tangential
stresses to a constant value go, which is chosen
such that the ground is at yield everywhere in C,.
Outside the cylinder C,, which is tangential to
the ground surface, the stress field is isotropic and
equal to 0s. Between cylinders C, and C, a solution for radial stress or and tangential stress bg is
derived from solving the equations of equilibrium
with the assumption that the soil is at yield. SIII
is spherically symmetrical around the centre 0 of
the tunnel face. The stresses are isotropic inside
sphere S, and outside sphere S,, where they are
equal to rrT and us respectively. Between S, and
S, the radial stress ur and both tangential stresses
bg, B, are determined
as for SII by solving the
equations
of equilibrium
with the assumption
that the soil is at yield.
SI, SII and SIII all satisfy the equations
of
equilibrium
and boundary
conditions
of the
present problem. Therefore a lower bound solution can be found from these three stress fields by
assuming that the yield criterion is not exceeded
in the soil mass. The derivations
of these lower
bounds can be found in the paper by Leca &
Panet (1988). The result can be put in each case in
the form of a double inequality. For stress field SI
and
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588
LECA AND DORMIEUX
os+yv-Y) ~s+Y(+Y)
+!I
a__________
I
-+
Section
A
A-A
(b)
Fig. 9. Stress fields (a) SI, (b) SII and (c) SIII
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FACE STABILITY
OF TUNNELS
IN FRICTIONAL
Table 1. Values of (C/D)* for
collapse and blow-out.
For stress field SII
l'y
+2;+1
P (
>
(Kp- l)a&, + 1
WP
-
lb,/%
+
*(i’Kp- l) ~ (Kr - l)a$a,
( >
zw-
<
( >
2;+1
+ 1
+ 1
1)
(14)
(15)
and for safety against blow-out
N,Q,+N,Q,~QT
(16)
In relations (15) and (16) Qs, Q= and Q, are
defined as in equations (7). The values of N, and
N, related to the best lower bound will be
referred to respectively as Nsc-, NYC- for collapse
and Nab-, NYb- for blow-out. A closed form solution can be obtained for these coefficients;
for
stress field SI, we can find for collapse
= K,
(174
NYC- = K,
and for blow-out
Nsbm = K,
NbY
(17c)
=K$
D
(174
For both SII and SIII, Q, is not present in the
solution and NYC-, NYb- are thus equal to zero.
The values of NSC-, Nsb- associated to SII are,
for collapse
N,‘-
= K,
The coefficients Nsb- and Nsb- associated
are, for collapse
to SIII
2(1-KP)
It can easily be shown that the three sets of
inequalities can be written with the same general
form as for upper bound solutions. i.e. for safety
against collapse
&Qs+N,Q,GQ,
(C/D)*
blow-out
20”
25”
30”
35”
40”
45”
2;+1
(KP - l)or/ck
(C/D)*
collapse
1
1
For stress field SIII
Ns’-
589
SOlLS
(lga)
N,‘-=
2;+1
(
(19a)
>
and for blow-out
Z(1-W
Nsb- =
2;
(
+ 1
>
Relations (17) to (19) show that the estimated
lower bounds all depend on the depth ratio C/D
and the friction angle 4’ of the soil. The values of
Nsc-, N,‘- and Nsb-, NYb- have been computed
for 4’ = 20”, 25”, 30”, 35”, 40” and 45” and
plotted as a function of C/D (for C/D < 3.0).
These plots are shown in Fig. 10 for the general
case (y > 0) and Fig. 11 for the case of a weightless soil. Figs 10(a) and 11(a) refer to collapse
and Figs 10(b) and 11(b) to blow-out. The results
obtained from stress fields SII and SIII can be
compared on Fig. 11. For these conditions, NYCand NYb- are equal to zero, and the best lower
bound is obtained when N,‘- is at minimum for
collapse and when Nsb- is at maximum for blowout. It is found, as in the case of cohesive material
(Davis et al., 1980), that this best estimate of the
lower bound is provided
by SII for shallow
tunnels and SIII for deeper tunnels. The values
(C/D)*of the depth ratio, for which both stress
fields lead to the same value of Nsc- or Nsb- are
given in Table 1.
DISCUSSION
From an engineering point of view, the parameters Qs and Q, are imposed by geometric and
loading conditions,
and the supporting
pressure
oT (i.e. the parameter QT) should be chosen such
that failure of the tunnel during construction
is
prevented. The upper and lower bound solutions
derived can be written
N,Qs+N,Q,GQ,
and for blow-out
(19b)
(20)
for collapse and
(lgb)
NsQs + N,Q, 2 QT
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(21)
LECA AND DORMIEUX
590
N”,-, N”,- i
-
w,
----
N”,-
2
-,N:“Nb,
i
-
Iv,-
--__
e-
20
(b)
Fig. 10. Lower bouud values of N, and N, (7 > 0) (a) for collapse and (b) for blow-out
Downloaded by [ Middle East Technical University] on [19/06/17]. Copyright © ICE Publishing, all rights reserved.
FACE STABILITY
OF TUNNELS
IN FRICTIONAL
N,C~/:’ 0,
---30
-
SII
-----
Sill
591
SOILS
Sill
,
01
0
I
,
,
,
,
,
1
’
I
I
I
I
2
I
I
I
I
I
I
3
(b)
Fig. 11. Lower bound values of N, in a weightless soil (a) for collapse and (b) for blow-out
Downloaded by [ Middle East Technical University] on [19/06/17]. Copyright © ICE Publishing, all rights reserved.
C/D
LECA AND DORMIEUX
592
for blow-out. This means that the value of Qr at
failure can be written
(2-a
QT=NsQs+N~Q~
where Qr, Qs and Q, are given by equations (7)
and Ns, N, are weighting coefficients for loads Qs
and Q, that can be bracketed from the values of
NsC- , Nsc+ , N,C- and N ‘+, or Nsb-, Nsb+, NYband N, bt derived previoYusly. In other words the
problem of the stability of the tunnel front can be
analysed by the same methods used for determining the bearing capacity of a foundation.
The
analogy applies better to blow-out
since the
ground is then failing when Qr becomes too large.
Equation (22) actually provides a lower estimate Qr- and an upper estimate Qr+ of the ultimate load Qr*. Qr- is found by using the lower
bound values NsC- and N,,for collapse and
N b- and N b- for blow-out;
Qr+ is found by
using the up& bound values Ns’+ and N,,‘+ for
collapse and Nsb+ and Nrb+ for blow-out.
In
order to reduce the uncertainty
on this estimate
of Qr*, the general lower bound solution (y > 0)
can then be improved by making use of the solutions obtained in the case of a weightless soil.
Relations (20) and (21) can also be written
N,G+N
Qr
N,s+N
QT
&l
’ QT
’
&l
(24)
QT.
A Q,./ QT
5-
I
Upper
I
Therefore
only two loading parameters
Qs/Qr
and Q,/Qr need to be considered and the stability
of the tunnel front can be investigated
in the
loading plane (Qs/Qr. Q,/Qr) by braketing
the
domain Xx’ of the load combinations
(QJQr,
Q,/Qr) that are stable. The procedure is shown in
Fig. 12 for the case 4 = 20”, C/D = 0.5.
Lines A,E, and A,& in Fig. 12 represent the
lower bound solution provided by the general
stress field SI (y > 0). Since Xx’ is convex and contains A,, B, and B,’ (best lower bound solution
for y = 0, i.e. Q,/Q, = 0) all points between lines
A,B, and A,B,’ belong to X’. For the same
reason, all points between lines A, B, and A, B,’
also belong to X’. The lower bound estimate
Xx’- of X’ can in this way be extended from the
domain AlBIB A, to the domain A,B,‘B2’A,.
From a practical point of view, this improved
lower bound solution can be obtained for collapse, by rewriting relation (20) with NYC- taken
equal to the value found in case SI and with Nscequal to the best solution found in the case of a
weightless soil (i.e. in case SII or in case SIII). For
blow-out, it can be obtained by rewriting relation
(21) with NYb- equal to the solution found for SI,
and N,“- equal to the solution for a weightless
soil. This improved lower bound solution is compared with the upper bound solution on Figs
13(a) and 14(a) for Ns values, and Figs 13(b) and
14(b) for N, values.
For any value of the friction angle 4’ between
20” and 45” and a depth ratio C/D between 0.0
and 3.0, the failure load Qr* can be bracketed by
I
bound
I
I
I
for collapse
I
I
4---
Lower
.
3.
bound
Best lower
and Sill
from
bound
SI
from
Fig. 12. Improved lower bound solution for the case I$’ = 20”, C/D = 05
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SII
FACE STABILITY
OF TUNNELS
IN FRICTIONAL
4 1
1
/
/’
I’
/’
f’
/
/’
/
/
/’
/’
,,’
,I’
/
I’
,-’
I
/’
/
/’
/ /
/’
_,’
,/’
I’
0
+“-
./’
.’
/’
.’
/-
I-
,’
0,’
,.’
rp’ = 20”
Y-
OI
,I’
r’
,,_.
/’
./’
/
,Q.-
//’
’
/’
,-
/’
,’
,/’
TX’
ho”,’
,G
Q, x’
,’
,,/
/’
/ /’
,,I’
,/’
/’
/
/
I
Q/’
,I’
,/’
,/’
/’
/’
-,~‘,,,~‘//rl
,,’
,/
/
/’
,,’
,’
,’
0.5~,/’/’
//
/
& ,/’
,/I
6!/ /
,
/’
Y
4, /’
,/’
1’
’
/’
I’
I
/’
NC,+
,/‘,
if
I’
/’
/
-
r$ I’
1/,/
0,
I
w-
I
o /
8,
i
--__
593
SOILS
fp’ = 25”
1
(b)
2
, $5’ =
_ @’ =
-rp’ =
‘4)’ =
30”
35”
40
45
CID
Fig. 13. Bracketing values of (a) N, and (b) N, for collapse
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594
LECA AND DORMIEUX
NS
_____
t
-
N,
Nb+
5
-_--
&it-
-
Nb'
Y
40
30
20
io_ q’ = 25”
_ @’ = 20”
O(
1
(W
2
Fig. 14. Bracketing values of (a) Ns and (b) N, for blow-out
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FACE
STABILITY
OF TUNNELS
using equation (22) together with Figs 13 and 14.
Using the values of NSc- , NYC- from Fig. 13
provides a lower estimate QT of QT* for collapse.
Using NSc+, NYC+ provides an upper estimate
QT+ . The weighting coefficients for blow-out
failure are given in Fig. 14. QT- is obtained by
substituting
Nsb- and NYb- for N, and N, in
equation
(21), and QT+ by substituting
Nsb+,
N Yb+ for N,, N,in this equation.
The case of a tunnel driven in a cohesionless
material is of special interest. In this situation ~c
is equal to zero and it is not possible to consider
the dimensionless
coefficients a Ja,,
U&T,- and
yD/a,. As a result, coefficients Q,, & and G, are
not defined. However. it can easilv be seen that
the pressure at failure CJ~ can be estimated by
writing
oT= N,a,+N,yD
(25)
and by choosing N, and N, from Figs 13 and 14
as described previously.
COMPARISON WITH EXPERIMENTAL
RESULTS
Having provided a simple method to estimate
the value of the face pressure bT at failure, we
may now apply it to typical conditions
for the
experimental
study of face stability of tunnels in
sands. In this way one can quantify the differences between the predicted
upper and lower
bound values and establish the validity of the
method. In this application,
it must be remembered that oT acts as a retaining load for collapse.
Therefore upper bound estimates of the pressure
at failure (iT* should be expected to be smaller
than both oT* and the lower bounds.
Centrifuge
tests have been carried
out in
Nantes, France, to study the face stability of
tunnels in sands (Chambon
& Corte, 1989). In
these tests, the tunnel was modelled as a rigid
cylinder, a soft membrane covered the front part
of the cylinder and allowed a supporting pressure
+ to be applied to the face. The centrifuge was
operated at 50 g; at this acceleration
level, the
80 mm cylinder modelled a tunnel with a 4 m
Table 2.
CID
1.0
1.0
2.0
2.0
IN FRICTIONAL
SOILS
internal diameter. Failure at the face was induced
by decreasing the face pressure Go. The soil used
for the experiments
was a dry fine sand
(Fontainebleau
sand). The pressure
gT. was
obtained by filling the cylinder with air (uniform
pressure) or with water (hydrostatic
pressure). A
surcharge os could also be applied on top of the
model.
Test results reported
by Chambon
& Corte
(1989) show that (1) failure is sudden; (2) it occurs
when the face pressure is decreased to a small
value r+/ (a few kPa); (3) for the range of values
C/D has little influence on the limit
considered,
pressure; (4) the failed area is bulb-shaped with its
largest dimensions at face level; (5) this geometry
is not affected much by C/D or soil density; and
(6) failure does not reach the ground surface for
C/D > 1.0. The tests with compressed air support
were run at two depth ratios, C/D = I.0 and
C/D = 2.0. Two soil conditions were examined. A
loose sand (7 = 15.3 kN/m3, D, = 62%), and a
dense sand (y = 16.1 kN/m3, D, = 86%). Shear
strength
tests on these soils had shown that
c‘ = 2.3 kPa, 4’ = 35.2” for the loose sand, and
c’ = 1.1 kPa, 4’ = 38.3” for the dense sand. The
results obtained
from the four tests are summarized in Table 2. The limit analysis estimates
of the critical face pressures for these four tests
are also recorded in Table 2. It is apparent that
lower bound solutions
are significantly
higher
than the upper bound values, as well as measured
pressures at failure. On the other hand, the upper
bound estimates gT+ are in close agreement with
test results, with uTt values slightly lower and
almost identical to the pressures c+/ measured at
failure in the centrifuge.
Other similarities between upper bound solutions and experimental
results are shown in Fig.
15, in which the failure zone observed in the centrifuge along the tunnel centreplane is represented
for the case of a loose sand for C/D = 1.0. The
critical geometry associated with the best upper
bound solution is shown with dashed lines. Even
though it does not extend in the vertical direction
as much as the actual failure area, it coincides
almost perfectly with the observed surface in front
Comparison between predicted and measured pressures at fake
y: kN/m3
15.3
16.1
15.3
16.1
595
Critical pressures predicted
from limit analysis: kPa
QT-
+
UT
29
29
46
44
2
3
2
3
Measured pressures at failure
in the centrifuge: kPa
OI
6
3
4
4
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LECA AND DORMIEUX
596
1
\\\\\\\\\\\\\\\<
----
Failure
atea observed
Crltlcal
failure
surface
in the centrifuge
from
llmlt
analysrs
Fig. 15. Comparison between theoretical critical surface and observed failure area
of the tunnel. In particular the extent of failure
ahead of the tunnel face is the same as that
observed. The large amount of failed material in
the centrifuge over the tunnel crown, however,
could have resulted
from the progression
of
failure in unsupported
ground once face collapse
had occurred.
CONCLUSION
The limit analysis concept has been used to
examine the stability conditions of the face of a
shallow tunnel driven in a frictional material.
Safety against both collapse and blow-out have
been analysed. Upper bound solutions have been
derived
from consideration
of three
failure
mechanisms
based on the movement
of rigid
blocks with conical
shapes. The amount
of
material involved in these mechanisms is limited,
but such geometries
could be representative
of
initial ground movements
that could lead to
larger scale failures. In particular
the results
suggest that if the predicted failure conditions are
close to the actual ones, the surcharge us has little
effect on face stability (except for very shallow
tunnels), and the extent of the failure zone in front
of the tunnel is smaller than in the case of a long
open cut.
The upper bounds have been compared with
lower bounds
published
previously
(Leca &
Panet, 1988). In both cases the problem reduces
to one of two loading parameters
Qs/Q= and
QYIQT,with Qs, Q, and Qr defined above. It is
found that it can be treated similar to the method
used for determining the bearing capacity of foundations, i.e. the critical supporting load is equal to
QT=bQ,+N,Q,
(26)
with N, and N, estimated from Fig. 13 (collapse)
or Fig. 14 (blow-out). These figures actually allow
one to bracket N, and N, between lower bound
values N,‘- (or N,“-) and N,,- (or N,b-) and
upper bound values NQ+ (or Nsb+) and N,‘+ (or
NYb+). This way a lower estimate and an upper
estimate can be found for the collapse load QT.
All conclusions remain valid in the particular case
of a cohesionless soil as long as Qs, Qr and Q, are
assumed to be equal to es, eT and yD respectively.
The method has been applied to centrifuge
tests for the face stability of shallow circular
tunnels in sands. Reasonable agreement has been
found between the theoretical upper bound estimates and the face pressures measured at failure
in the tests. Other similarities are evident between
the critical failure mechanisms derived from limit
analysis, and observed failure areas in the centrifuge. These conclusions support the idea that the
upper bound solutions are closer to the actual
pressures at failure than the lower bound values,
and can provide reasonable estimates of critical
face pressures.
APPENDIX
I. DERIVATION
OF THE
UPPER BOUND SOLUTIONS
ASSOCIATED
WITH MECHANISM
MI
Geometric properties
First, some geometric
quantities
need to be
determined, prior to the derivation of the external
power B, and the dissipation power P, associated
with mechanism
MI (see Fig. 16). For this
purpose it is more convenient
to consider two
cones W and ‘8’. Both cones have the same apex R
and the same axis A, but the base C of V is in the
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FACE STABILITY
Fig. 16.
OF TUNNELS
SOILS
597
Geometry of mechanism MI
plane of the tunnel face (xOy), whereas the base
r of W is at the ground surface. The moving
block ~8 corresponds
to cone V minus cone r. Of
course only one cone, V, needs to be considered
when the ground surface is not reached.
Two axis systems are used: (0, x, y, z), associated with the tunnel and (a, X, Y, Z), associated
with the cones. The equations below allow transformation between both co-ordinate systems.
x=x
(27a)
Y = 4 sin a tg@ - y cos a + 2 sin o!
(27b)
D cos a
- y sin a - 2 cos a
(27~)
Z=-
IN FRICTIONAL
2 tsdf
Both cones are characterized
by the same equation with respect to co-ordinate
system (Cl, X, Y,
Z)
x2 + Y* = tg%#J’z*
Therefore,
the area J& of the cone base is
~ _ ~0' J[cos (a - 4’) cos (a + &)]
4
The same parameters
v’. Its height h’ is
sin 2ci - F
can be determined
for cone
sin 2f#/
h’+
sin 2&
c’ is an ellipse, the semi-axis
D
sin 2a - y
lengths of which are
sin 24’
(33a)
” = Z 2 sin (a - 4’) sin (a + 4’)
D
(28)
sin 2~ - g
” = Z 2 cos #&sin
The height h of cone V is
(31)
cos l#J’
sin 24’
(a - 4’) sin (a + #)I
(34b)
h = D cos (a - 4’) cos (a + 4’)
sin ~C#J
(29)
Its area ~4’ is
2
Its intersection with the tunnel face (Fig. 16) is an
ellipse for which semi-axis lenths a and b are
given by
KD=
d’
= 4
4 sin 2a - $!
(
cos #[sin
sin 24’
>
(a - 9’) sin (a + @)1312
D
(304
a=-
2
b
=
D JCcos
(a - 4’) ~0s (a + 4’11
2
cos 4’
(35)
The volume Y, of the block is equal to
(30b)
Vs=Y-V-’
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(36)
LECA AND DORMIEUX
598
where $‘” is the volume of cone W, and Y’ is the
volume of cone v’.
y _
nD3[cos(a - 4’)
12
qs’)]3’2
cos (u +
(37)
cos 4 sin 24’
sin 24’
>
’ cos 4’ sin 2&[sin (a - 4’) sin (a + #)13”
The lateral area 9,
(38)
of the block is equal to
Y,=Y-Y
(39)
where Y is the lateral area of cone %‘, and Y’ is
the lateral area of cone v’. An expression for Y
as well as Y’ is found by using complex number
integration
x-cos
a [cos(a -
4’) cos (a + $‘)I”’
(40)
ml u.
4
2
The expressions for Yb can be simplified
the following parameters are introduced
JCcos(c(- 4’) cos(a + 401
cos fp’
R = cos (a - &) cos (a + #I
B
dpv
~=C’Y-ncotc$’
(46)
where n is the unit vector normal to the discontinuity surface at the point where dP,/dC is computed (Fig. 3). Since the angle Od between the
velocity Y and the discontinuity
surface is chosen
equal to $‘, equation (46) can be written
dP
--I = C’V cos Cp’
dZ
9, = 8,
loads 9. has three com-
+ 8, + Yr
(48)
sin 2a - E sin 24’
>
(
’ [sin (u - 4’) sin (a + #‘)13”
R, =
(45)
where 8, is the power of the retaining pressure
trr , Ps is the power of the surcharge rrs, and S’r is
the power of the soil weight y.
sin #J’ cos 4’
f
V= Ve,
Power of external loads
The power of external
ponents
sin 4’ cos 4’
y’ = c
all points
Plastic energy can only be dissipated along discontinuities
and the dissipation
energy per unit
area dP,JdZ is
3
sin 2a - i
f
Velocity field
The moving block is rigid. Therefore
of the block have the same velocity
sin 24’
(41)
once
B, =
(are3
ss I:
WW
(49)
= -o,vcosad
After substituting
(424
. (Ve,) dC
pT=
-$cos
expression
(31) for & we find:
aR,a,V
The same method is used together with equations
(35) and (43) for determining 8, and qY.
cos ci
R, = -
(42~)
w
R, = sin 2a - F
R, = &sin
Equations
(51)
sin 24’
(42d)
t--q)
2(a - 4’) sin 2(a + &)]
(Vez)dV
(42e)
=yVsina*Y,
(36) and (39) then become
(52)
that is to say
(43)
$/,=--
R,= R,’
1 - 2 co3 f#hga cos Cp’
RE3
nD= R,Rc
4
1
8, = $
2 sin a sin 24’
Rn2 R., R,
0, V
(53)
Rn3
%=4
nD2 sin a
- 3 R,R.[I
- ($+V
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(54)
FACE STABILITY
Finally equation
OF TUNNELS
2 sin a sin 24’ -%?-J
Dissipation power
The dissipation
nism MI is
N, =
os
D,V(55)
RB(, -;
power
y
associated
with mecha-
P, =
(56)
where dP,/dx
is given by equation
(47). Therefore
P, = C’V cos 4’9,
in equation (44). If we
for 9, in the above
Ro2 R,*
1 - 2 cos* #tgci ~
RE3
1
c’V
(58)
Upper bound theorem
The upper bound solutions
associated
with
mechanism MI are found by writing relation (6)
with 8, and P, computed
as in equations (55)
and (58) respectively. This substitution
leads to
the inequation
RD*R, ~
RE3
+ +tgzR,
599
thus
1
Rn2
cos (24’) - cos (2a) R, tgcl
N, = $R,
(61)
tga
The best upper bound solution for MI is obtained
by choosing
n such that N, and N, are at
maximum. These expressions only apply to cases
where failure reaches the ground surface; that is
to say for depth ratios C/D such that
2$+147
sin 2a
sin 24’
If C/D is larger than [(sin Zu/sin 24’) - 1]/2 relation (60) remains valid provided R, is set equal to
zero in equations (61) and (62).
(57)
Y, has been determined
substitute
this expression
equation we find
2tga sin 24’ -
SOILS
where Ns and N, are obtained
(48) can be rewritten
+ +x
IN FRICTIONAL
[
0,
c’ cot gr$’
1 - R,3
RE3
cr
- ~
c’ cot g#
1
YD
x,=x
(5%
If we introduce the soil unconfined
compression strength uf and the Rankine earth pressure
coefficient K, (equations (2) and (4)) relation (59)
can be rearranged in the form
E
1
+N,(K,-l)$
< (&, - 1) aT + 1 (60)
n,
(64a)
Y, = i sin atg# - y cos a + z sin a
c’ cot g@
R,* R,*
< 1 - 2tga cos2 I$’ ~
RE3
APPENDIX II. DERIVATION OF THE
UPPER BOUND SOLUTION ASSOCIATED
WITH MECHANISM MI1
Geometric properties
Two cones are considered, vi (apex a,, axis
A,) of which the base x1 is in the same plane as
the tunnel face; and V, (apex Q2, axis A2) of
which the base El2 is in plane 7~(U, and %?*have
the same geometric properties) (see Fig. 17). The
first moving block .%3icorresponds
to the portion
of %?i located below plane x and g2 is the portion
of %?zlocated below the ground surface. Four axis
systems will be referred to: (0, x, y, z) associated
with the tunnel front; (Q,, Xi, Yi, Z,) associated
with vi; (a,, X2, Y,, Z,) associated with %:,;
and (B, X’, Y’, Z’) associated with planes II and R’.
The following co-ordinate
transformations
are
used
D cos a
- y sin a - z cos a
2 tg4’
z,=X,=x’
(64b)
0%
(654
y _ -Dcos(a+@)sin&cos/?
1
2
sin (B + 4’) cos 4’
- Y’ sin p - z’ cos jl
-Dcos(a+#~‘)
”
= 2
sin (@+ 4’)
Wb)
cos 4
sin p 7
sin 4’
+ Y’ cos /I - z’ sin /3
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(65~)
LECA AND DORMIEUX
600
Fig. 17. Geometry of mechanism MI1
Since plane rc is chosen such that A2 is vertical,
we have between a and p the relation
The area &,,
d
28 - a = n/2
Wb)
12 -
of X:,, is
0
cos’ (a + 4’) &sin (B - $‘)I
cos fp’
[sin (B + @)1312
4
The heights h, and h, of cones V, and V, respectively are
h = D cos (a - 4’) cos (a + 4’)
1
(67)
sin 24’
(72)
The intersection
C, of Wz with ground surface
is a circle of which the radius r2 is equal to
D
h = D sin (/I - 4’) cos (a + 4’)
2
sin 24’
(68)
‘* = 2
W, intersects with the tunnel face along an ellipse
C, (Fig. 17) of which the semi-axis lengths are
D
a, = -
(@a)
2
b
1
= g JCcos (a -
4’) cos (a +
441
cos Cp’
2
sin /? cos a
sin 4’ sin @ + 4’)
-
2;+1
tg@
(
>I
Its area CZI, is
sin b cos a
4 sin (j? + 4’)
(69b)
The area zZI of X1 is
~
1
_ aDZ J[COS (a - 4’) cos (a + @)]
cos 4’
4
(74)
(70)
The intersection Z,, of V, with plane n is also an
ellipse (Fig. 17) with semi-axial lengths
a ,2=D2
(73)
cos (a + 4’)
sin (.jI + 4’)
P’lb)
Since .G’#ris obtained by removing a cone identical
to V$ to cone V,, its volume Vt,, and its lateral
area Y,,, can be put in the form
“Vbr = W”, - Vz
(75)
Y,,
(76)
= 9,
- ,4v,
where V, and V2, and 9, and 9, are, respectively, the volumes and lateral areas of cones %‘r
and Wz. Y,, ^v,, 9, and 9, are determined by
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FACE STABILITY
using the same methods
= c
as in Appendix
OF TUNNELS
I, i.e.
IN FRICTIONAL
SOILS
601
These coefllcients allow one to write Yb,,
Y,, and Y,, in a simplified form
[cos (cz- 4’) cos (a + @)]3’2
Y,,
=
$ (R/,
R, -
s
R,‘)
(90)
R, - cos (a + qY)Rc R,
2
Y,
[cos
(a -
4’) cos (a + #)]“2
(79)
sin 4’ cos 4’
Y,=-_-
nD2 sin p cos’(a + 4’)
4
sin 4’
(80)
The volume Vb2 and the lateral
block .??I2are determined thus
area Y,,
of
sin 4’ cos (a + 4’)
*ybz=V2-ly3
(81)
.4pb,=c4p2-Ys
(82)
where Y2, 9, are given by equations (78) and
(80), and V”, and Y, are, respectively, the volume
and the lateral area of the portion of V2 located
above the ground
sin 4’ sin (B + 4’)
(83)
nD2 sin 4’
y,=--
4
cos2 4’
sin /? cos a
x
2
B
= JCcos (a - 4’) cos (a +
cos l$’
4’11
= cos (a - 4’) cos (a + 4’)
sin (24’)
R c = cos;o:+/)
[;z
;;
VelocityJeld
Both BI and a2 are
respective velocities are
rigid
blocks
;;]I”
co2 4’
cos (a + 4’)
R, - $
the
VI = be,,
(94)
V2 = bez2
(95)
Since W, and g2 are not moved at the same
speed, a discontinuity
in velocities is created
along their intersection YZ12,and a relative velocity V,, needs to be considered between gI and
B2 (Fig. 18).
In order for relation (5) to be satisfied, the
surface
angle between V,, and the discontinuity
XI2 will again be chosen equal to 4’. This implies
the following relations between VI, V, and VI,
the fol-
v,=
sin (D + 4’) v
sin (j3 - 4’)
(96)
’
(85)
(86)
(87)
sin j3
(88)
RD = sin 4’ sin (j3 + 4’)
R, =
and
(84)
sin 4’ sin (B + 4’)
It is convenient at this point to introduce
lowing coefficients
A
(92)
sin 4’ cos (a + 4’)
cos f#l’
&sin (B - &‘)I
x [sin @ + @)13/*
R
1
+cos.
x
R
Yob,,
sin 4’
(89)
Fig. 18. Relative velocities of a,
and 9,
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602
LECA AND DORMIEUX
cos
&:2=
CL
sin @ - I$‘)
(97)
V2
p,=--
aDZ 1 cos’ (CL+ 4’)
4
3
cos2 4’
R,R, + cos 0:
X sin a ~,2
Plastic energy can be dissipated along the lateral
area of Y?r and Wz and along Z,,. In all three
cases the dissipation
energy per unit area is (cf.
Appendix I)
cos f#J’cos (B + 4’) R
x 2 sin 4 sin (B + 4’)
dpv
E
C’V cos (fi
=
’
cos2 4’
(98)
cos (a + 4’)
withV=Vion%?,,V=I/,onV,,andV=I/,,
on Zi2.
3
Rc2 - 2 g sin 4’
2 sin f$’ cos2 (cr + 4’)
rDv,
'I
(105)
Equation
Power of external loads
The power of external
9, =
(99) can then be written
nD2 cos2 (a + 4’)
4
loads 8, is
9, = Yr + 8s + Yy
co? f#l’
RE2
x cos2(a + 4’)
(99)
where Yr, 9, and S’r are defined as in Appendix
I.
+
cos a
0
s-R,~RA~T
RA4,
sin a -
Rc2
8,=
(Vlez,) dC
(-~TeJ
+ cos a cos 4’ cos @ + 4’) R
2 sin C#J’
sin (J + 4’)
ufw
= -crTVl cos ad,
’
22
1v,
Rn3
_ 2 sin 4’ cos’
(a + c#J’)
> 3
9s =
(V2ez,) dC
(-u,e,J
= a,V*d*
9, =
(101)
(-re,Wd
Dissipation power
The dissipation energy associated
nism MI1 can be written
d-lr
( - re,XV,ez,) dv
+
= yV, sin czYb, + yV2’zy,,
P” = p,, + p,, + P,,,
(102)
After substituting
equations (70), (74), (90), (92)
and (96) for &‘,, d,, Yb,, Y,, and VI we find
=
9
_
T
e
COS
c( CO2
RA UT v,
RC2
zz
co?
x
I$’
sin #
P2”
Rc2
cos (a + c#/)
P 12v
(108)
=
= CT, cos 4’ ,40b,
sin’ 4
D2 cos2 4’
;
= C’V, cos f$’ P,,
(103)
(107)
contri-
PI” =
(a + +)
x -
8,
where PI,, P,, and P,,, are the respective
butions of gi, SJ2 and X,2
cos2cp
4
with mecha-
(109)
=
= C’V12cos I#/ d,,
Equations
(91), (93) and (72) together
equations
(96) and (97) allow one to
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(110)
with
write
FACE STABILITY
equations
p
cosz (a + 4’)
4
IN FRICTIONAL
cos2 Cp’
xcr
cos
c#l’V, 1 (111)
(a
+4’)
p=gcos2
cos a R,
X
2”
-
1-
cos
sin 6’ R,
4
cos2
x
(a: +
2
N, (K, - 1) ;
C
RcRD
(a + 4’)
sin f#~’~0s’ (a + 4’)
1
N, =
x c’ cos r#J’V,
--
9
12” -
(112)
4
cos2
sin /3
x cl cos 4’
RC
Y
v,
X
cos24’
RE2
cos u R,
--_
sin 4’ RC2
sin 4’ cos2 (a + @)
x cl cos l#l’V,
1
RE2
+
sin tl-
a
-=
R,aT
RARB
RC2
2 sin 4’ sin @I+ 4’)
<
RE3
’
YD
2 sin 4’ cos’ (0: + 4’) > 3
R,
RE2
cos lx RC2 cos2 (a + 4’) >
x c’ cot g&
cos 4’ cos (B + 4’) Rc3
2 sin 4’ sin (B + #)
R,
1
sin (j3 - 4’) RE3
’ sin (B + #)
R,
(118)
Relation (116) provides the best upper bound
associated with MI1 when a is chosen such that
N, and N, are at maximum. As in Appendix I,
the above results only apply when the ground
surface is reached by the failure mechanism, that
is to say when
5 < cos (a + #) sin (B - 4’)
D’
2 sin c$’
sin (B + 4’)
(119)
For deeper
tunnels
(relation
(119) reversed)
expression (116) is still valid provided R, is set
equal to zero in relations (117) and (118).
APPENDIX III. DERIVATION OF THE
UPPER BOUND SOLUTION ASSOCIATED
WITH MECHANISM MI11
Geometric properties
Two cones, $? and v’ are considered (see Fig.
19). Both have the same appex n and the same
axis A. The moving block SI corresponds
to cone
W minus cone v’. %’ intersects the tunnel face
along an ellipse z’ (base of cone W) and the
ground surface along another ellipse x (base of
cone U).
Two axis systems are used: (0, x, y, z) associated with the tunnel, and (a, X, Y, Z) associated
with the cones. The co-ordinate
transformation
between both systems is given by the following
equations.
s RC2
+ cos a cos 4’ cos (B + I$‘) R
-
(117)
(114)
Upper bound theorem
The upper bound solutions
associated
with
mechanism
MI1 are obtained
by substituting
equations (106) and (114) for B, and P, in relation (6)
cos2(a + 4’)
[
(113)
nD2cos’ (a + 4’)
4
N, and N, are in the
cos a 60s’ 4’ sin (B + 4’) R,
tgaR, +
(116)
- 2 sin 4’ cos a cos’ 4
1
Substituting
equations (1 ll), (112) and (113) for
pi,, p2, and pi,, in equation (107) leads to
p _
1
1
f#J’
2 cos /I sin 24 cos (a + 4’) Ro2
X
l)%+
ec
c
sin @ - 4’) RE2
1
N, = f
cos’ (a + 4’)
nD2
<(K,-
The weighting coeflients
present case equal to
cos2 4’
cos
1
+ 1 + N,(K, - 1) F
4
RE2
-
4’)
603
SOILS
Relation (115) can be rearranged
in the same
form as in Appendix I once the unconfined compression strength cr, and the passive earth pressure coefficient K, are introduced.
(108) (109) and (110) in the form
= e
I”
OF TUNNELS
(115)
x=x
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(120)
604
LECA AND DORMIEUX
Fig. 19. Geometry of Mechanism MI11
Y = t sin atgf#/ + y cos a - 2 sin a
D cos w.
z=--
2 44’
The semi-axis lengths of ellipse c’ are
D
a’ = -
+ y sin u + 2 cos a
The height h of cone V and the height h’ of cone
v’ are
sin 2a + g
D
e JCcos(a -
=
2
44 ~0s (a + &‘)I
(129)
cos 4’
of
(123)
sin 24’
=
b’
Equations
(126H129) allow determination
the area S’ of x and the area &’ of r
sin 24’
h=;
h’
(128)
2
cm (a + 4’) cm (a - 4’)
sin 2q5’
(124)
&cjf=E!c
4
The equation of cones V and v’ with respect
the co-ordinate system (a, X, Y, Z) is
x2 + Y2 = rgSj5’22
2
(
sin 2cr + T
sin 24’
>
4 cos @[sin (a - 4’) sin (a + &)13j2
to
(125)
(130)
&,
=
9
JCCOS
(a + 4’) cm
4
(a - f$‘)]
cos 4’
(131)
The semi-axis lengths of ellipse I: are
D
sin 2a + $
a = Z 2 sin (a - 4’) sin (a + 4’)
sin 2a + F
(a - 4’) sin (a + #)I
of block
(126)
sin 24’
b=D
2 2 cos $‘J[sin
The volume Y, and the lateral area 9,
g are found by writing
sin 24’
(127)
V,=Y-v-’
(132)
9,
(133)
= Y - 9’
where V” and y, and V’ and Y’ are the volume
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FACE STABILITY
and lateral areas respectively
W
>I
3
[(
y-=e
of cone V and cone
sin 2a + g
f
OF TUNNELS
sin 2 4’
12 sin 24’ cos 4’
(134)
sin a
(a + 4’) sin (a - C/J’)]
X
l
^I” =
e [cos(a -
(135)
I
dP
-=c’vcos~’
dZ
(144)
with 8,, 8, and 8, defined as in Appendix I.
After substitution
of equations (139) to (141) for
&, &’ and Y, we find
(136)
(146)
2
.V’=+cosa
g?,=
---
nD2 R, Rc2
4
x
YY = - G
As in Appendices I and II, Y and Y’ are determined by using complex integration.
Equation
(130) to (133) can be put in a simplified form by
introducing three coefficients
R, = cos aJ[cos
(a + 4’) cos (a - c#J’)] (138a)
R, = sin a&sin
(a + 4’) sin (a - @)]
R = sin 2a + (2CID + 1) sin 24’
E
cos 24’ - cos 2a
The expressions
(138b)
(EF)‘-(SJDv
sin a
sin 24’ cos $J’
(148)
and equation
(145) can be written
I
RB Rc2
-------a,+-0
cos 4’
R.4
cos 4’
T
_isin-a(E)3-(2Y%
sin 24’ cos I#J’
I
V
(149)
4
sm a cos f$’
R.4
(139)
1
x[(zz>‘-(ZJ]
that
can be dissipated
is given by equation
(144)
P, = C’V cos qb’Yb
(141)
(142)
in
(150)
where dP,/dX
~0~ R, Rc2 - R,
12 sin 4’ cos 4
Dissipation power
The plastic energy
mechanism MI11 is
(140)
12 sin 24’ cos 4’
9,=-
2
(138~)
RBRc2
sin a cos 4’
y,x,
(147)
05 v
for &, zZ’, Vb and Y, are then
4
.d’=aD2
cos qs
JCcos(a + 4’) cos(a - 4’11
sin 4’ cos f$’
&I!!?,
(145)
4’) cos (a + &)13”
sin 24’ cos f#i
12
(143)
Y’, = YT + Ys + YY
sin 24’
cos 24’ - cos 2a
Velocityjeld
The velocity of moving block _6gis
Power of external loads
The power of external loads Y’, can be written
sin f$’ cos 4’
sin 2a + T
605
Since this is a rigid body motion, plastic energy
can only be dissipated along the lateral surface of
&?; the dissipation energy per unit area dP,/dX is
2
x &sin
SOILS
V= Ve,
x [sin (a - 4’) sin (a + @)I””
Y = $
IN FRICTIONAL
(151)
After substitution
of equation
equation (151) we find
p, = q
[R, Rc2 -
(142) for 9,
C’V
&I 7
sm 4’
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in
(152)
606
LECA AND DORMIEUX
Upper bound theorem
The upper bound solutions
associated
with
MI11 are found by writing relation (6) with 8,
and P, as computed in equations (149) and (152)
K,
K,
y
0,
or
D
-R,Rc2s-+RR,~
cos I$’
cos 4’
H
C
P
Ye
P,
(RBRc2 - RA) -T----
(153)
can also be rearranged
into the
<
This inequality
form
N, (K,, - 1) ;
sin 4’
1
+ 1 + N,(K,
>(K,-
- 1) F
l)z+
1
(154)
where IS=and K, refer to the unconfined compression strength and the passive earth pressure coefficient of the soil respectively, and N, and N, are
N,=-
4, Rc2
RA
N,=
3R;;J2,‘[(g-($J] (156)
(155)
The best upper bound associated with MI11 is
found by choosing a such that N, and N, are at
minimum.
Since failure
always
reaches
the
ground surface, expression (154) is valid for all
values of C/D.
NOTATION
’ soil cohesion
soil friction angle
;I
oc unconfined compression
strength
Rankine earth pressure coefficient (active)
Rankine earth pressure coefficient (passive)
soil unit weight
surcharge pressure
tunnel pressure
tunnel diameter
tunnel depth
tunnel depth of cover
unsupported
length behind tunnel front
power of external loads
dissipation power
REFERENCES
H. (1967). Stability of clay
at vertical openings. J. Soi/ Me& Fndn Div. Am.
Sot. Civ. Engrs 93, No. SMl, 71-94.
Chambon, P. & Corte, J. F. (1989). Stabiliti: du front de
taille d’un tunnel faiblement enterrt: modtlisation
en centrifugeuse. Proc. Int. Co@ Tunneling and
Broms, B. B. & Bennermark,
Microtunneling in Soft
Ground: From Field to
Theory, Paris, pp. 307-315.
Clough, G. W., Sweeney, S. P. & Finno, R. J. (1983).
Measured soil response to EPB shield tunnelling. J.
Geotech. Engng Div., Am. Sot. Ciu. Engrs 109, No. 2,
131-149.
Davis, E. H., Gunn, M. J., Mair, R. J. & Seneviratne,
H. N. (1980). The stability
of shallow
tunnels
and underground
openings
in cohesive material.
GCotechnique 30, No. 4,397-416.
Leca, E. & Panet, M. (1988). Application du Calcul a la
Rupture a la stabilitb du front de taille dun tunnel.
Revue FranGaise de Giotechnique, No. 43, 5-19.
Muelhaus, H. B. (1985). Lower bound solutions for circular tunnels in two or three dimensions.
Rock
Mech. & Rock Engng l&37-52.
Peck, R. B. (1969). Deep excavations
and tunneling in
soft ground. Proc. 7th Int. Con& Soil Mech. and
Fndn Engng, Mexico, Balkema 3,225290.
Salenqon, J. (1983). Calcul b la Rupture et Analyse
Limite.
Salenqon, J. (1990). An introduction
to the yield design
theory and its application
to soil mechanics. European J. of Mech., A/solids 9, No. 5. 477-500.
Schofield, A. N. (1980). Cambridge
geotechnical
centrifuge operations. Gbotechnique 30, No. 3, 227-288.
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