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Convergence-Confinement Method
in Shallow Tunnels

Z. Eisenstein and P. Branco

Abstract TheapplicationoftheConvergence-Confinementmethod
(C.-C. method) to design of shallow tunnels is investigated by
comparing results of the method with field measurements for two
tunnels in stiff clay in Edmonton, CanadcL Both tunnels were
excavated under very similar conditions. The only important
difference between them was the depth-to.diameter ratio. Because of
this ratio, the two tunnels exhibited different responses to analyses
by the C.-C. method. The deep tunnel showed a good agreement
between the analysis and field data; the shallow tunnel did not. This
discrepancy was attributed to the non-axisymmetric mode of
deformation developed around the shallow tunnel.

~ n examine ici l'utilisation de la rn~hode Conoer~nce-
Confinement (m~thode C-C) sur des modelss de tunmls en surface en
comparant lss r(,sultats de estte rm~thode aux mesures sur le terrain dans
ls oase de deux tunnels en argile dure a EMmonton au Canad~ Les deux
tunnels ont c~ cmus~s dane des conditions t~s g~Jaires. La seule
diff~rvwe notable entre lss deux oonommit le rapport profondeur-
~ . A eause de ce rapport, les deux tunnels ont pruebdt des ~ponees
d i f ~ v ~ aux analyses ~ , avec la m~thode C.-C Dane ls cas du
tunnel en profondeur, les analyses et les donn~ sur ls terrain ont bien
coneord~;maiscenefutpaslsoaspourletunnelensurfac~ Ced~alage
a ~ attribu~ au mode de ~ n n o n ~ & ~ o p #
autour du tunnel en surface.

A number of methods are cur-

rently used for design and
analysis of tunnels. Among

them, the convergence-confinement
method (C.-C. method) has played an
important role in provi~ng in~ght into
the interaction between lining support
and the surrounding ground m , ~ . The
method is relatively a~mgle , easy to use,
and can readily indicate the sensitivity of
the chosen solution through a range of
possible g~mndparametem, support ch ~-
acteristics and modes ofinstallatio~

However, to m- ln ta in simplicity, a
number of simplifying assumptions are
employed in its derivation. These as-
sumptious make the method applicable
only to deep tunnels in a hydrostatic
stress field; and, therefore, the use of
the C.-C. method in shallow tunnels is
open to question. The main purpose of
this paper is to discuss this problem.
The discussion is based on field data
obtained from two tllnnels constructed
in Edmonton, Alberta, Canada.

The Convergence-Confinement
Method (C.-C. Method)

The C.-C. method is based on a con-
cept tha t involves analysis of ground

Present address: Prof. Z. Eisenstein,
Department of Civil Engineering, University
of Alberta, Edmonton, Alberta, Canada; Dr.
P. Branco, Thurber Consultants Ltd.,
Edmonton, Alberta, Canada.
This paper is reprinted with permission
from Canadian Tunnelling Canadien 1990.

structure interaction by an indepen-
dent study of the behaviour of the
ground and of the t -nne l support. The
ground behaviour is represented by a
ground reaction curve (GRC); the lin-
ing is represented by a support reac-
tion curve (SRC). The former describes
the ground convergence in te rms of the
applied confining pressure; the latter
relates the confining pressure acting
on the lining to its deformation. The
solution for the ground support inter-
action is then given by the intersection
of these two curves, as illustrated in
Figure 1.

In the past, a number of approaches
for determining the GRC have been

F~ ELASI|C, Y|ELOING ~~/~/ /~ I

grounCl support


Figure 1. Concept of soil structure
interaction by the Convergence-
Confinement methovL

published. Brown et al. (1983) pre-
sented a summary of the characteristic
features of each of the main formula-
tions derived in the past 40 years.

Alarge n -mhe r of solutions for SRC
also have been published. SRCs are
determined, on the basis of the theory
of elasticity, from the ]inlng stiffness,
load capacity, and the displacement
that occurs before the lining activa-
tion, as indicated in Figure 1. The
support stiffness is defined as the
uniform all-around pressure required
to cause unit diametral swain on the
lining. Support stiff~esses and support
bearing capacity for different liners
have been presented by Lombardi
(1973) and Hoek and Brown (1981).

The idealization of the ground sup-
port interaction by the two reaction
curves, obtained from closed form solu-
tions, is only valid for the two-dlmen-
sional cylindrical model in which, ir-
respective of the lining and ground
mechanical properties, the soil and
support follow the same radial mode of
deformation. This is a major limitation
of the method with regard to shallow
tunnels, because the proximity of a
free surface above the tlmnel, a non-
hydrostatic stress field (K o ~ 1), and the
effects of gravity around the t~mnel
cannot be included in the analysis.

A review of other available lining
design methods, presented by Branco
(1981), has indicated tha t there is no
simple design method for shallow

The applicability ofthe C.-C. method
is investigated below, with special re-

Tunnelling and Underground Space Technology, Vol. 6, No. 3, pp. 343-346, 1991. 0886-7798/91 $3.00 + .00
Printed in Great Britain. (~) 1991 Pergamon Press plc 3 4 3

Page 2

gard to the influence of a free surface
above the tunnel.

C.-C. Method of Lining Design
of Shallow Tunnels in Stiff Soils

The construction of two t~mnels in
the city of Edmonton, Alberta, Canada,
permitted the use of the C.-C. method
to analyze the effect of the free surface
on the prediction of tunnel behaviour.
The tunnels were extensively instru-
mented for ground displacements and
lining loads. Both tunnels were exca-
vated in the Edmonton till, using very
similar construction methods.

The first tunnel, the experimental
t , nnel (EXP t-nnel), was comprehen-
sively studiedby E1-NAhhas (1980) and
Eisenstein et al. (1980 and 1981). It is
a small-diameter tunnel (D = 2.56 m)
driven by a full-face TBM at a depth of
27 m to the t -nnel centre line. The
primary lining of one of the sections of
the EXP t~mnel comprises segmented
steel ribs (WF100 x 19), 1.5 m from
centre to centre, and 5 x 20-cm timber
lagging placed between the webs of the
ribs. The rib and lagging system was
assembled within the shield ofthe TBM.
The drilllng machinewas then advanced
by jacking ags;n~t the ribs of the tempo-
rary lining. When the TBM advanced
sufficiently that the rib emerged from
the shield, the rib was expanded by
hydraulic jacks and 10-cm spacerswere
placed in the two upper joints of the
steelrib. The next rib-lagging assembly
was placed in the shield and the dr;lllng
operation continued.

The second bmnel, the LRT-South
Extension (LRT-SE) tu-nel, has been
extensively analyzed by Branco (1981).
It is a large-diameter t -nnel (D = 6.1
m) driven, like the experimental tun-
nel, by a shielded TBM, with the tun-
nel centre line at a depth of 11.8 m. The
primary lining of the LRT-SE tunnel is
composed of segmented steel ribs (W6
x 25), 1.22 m centre to centre, and 10 x
15-cm timber lagging between webs of
successive ribs. Installation of this
t~m hells slm;l ar to that described above
for the EXP tunnel.

Table 1 summarizes the lining and
ground parameters related to the two
tunnels that are used throughout the
calculations in this paper.

Both tunnels were excavated under
an approximately hydrostatic stress
field ( K =1.0). The difference in the
depth ratio (depth of the centre of the
tlmnel to the tunnel diameter) of the
LRT-SE tunnel and the EXP tunnel is
the most important difference between
the two t-nnels. Otherwise the tun-
nels are quite comparable, particularly
in terms of host ground and lining
method. The EXP tunnel has a depth
ratio of 10.56, and will be regarded as
a deep tunnel, whereas the LRT-SE
t -n nel has a depth ratio of 1.9 and will
be considered as a shallow t-nnel. The

Table 1. Lining and ground parameters for the LRT-SE and EXP Tunnels.


EXP Tunnel


i Cohesion (MPa)

" I ~} • ~" ¢ Before
"~ E o. I Expa
~" ~ ~"~m I
F , ~ Q I "

¢g t - ¢,'-
= • c ~ I After
o B = ~ i

LRT-SE Tunnel
(after Branco


Young's Modulus (MPa) 75 150

Polsson's Ratio 0.4 0.4

Coefficient of Dilatancy 1 1

Friction Angle (degrees) 30 40

0 0

Before Face 4 0

19 2.5

2.5 0.5

Young's Modulus (MPa) 207000 207000

Poisson's Ratio 0.25 0.25

Moment of Inertia (m ) 4.76 x 10 22.2 x 10

Cross-Section Area (m) 24.7 x 10 47.3 x 10

Rib Spacing (m) 1.5 1.2

Diameter (m) 2.56 6.1

Final Radial Load at the
Spdngline (Pi/Po) 3.02 to 0.12 0.18 to 0.24



e -

, m
e -

. m

. J

assumption of whether the t~mnels are
deep or shallow was based on the ex-
pected development of tangent ia l
stresses in the ground at the tunnel
wall, as proposed by Mindlin (1940).

Excluding three-d~rnensional and
gravity effects inherent in any t-nnel
construction, the EXP tunnel filiRll~
all of the boundary conditions associ-
ated with the C.-C. method. On the
ether hand, for the LRT-SE ~mnel, in
addition to three-dlmensional and grav-
ity effects, the proximity of the free
surface violates the imposed boundary
conditions. This indicates that the C.-
C method should better predict the
behaviour of the ground-support inter-

action for the EXP t , nnel than for the
LRT-SE t , nnel.

~igures 2 and 3 present the ground
reaction curves for the EXP tunnel and
for the LRT-SE tunnel, respectively.
These curves were derived according
to the formulation presented by Kaiser
(1980) for a circular opening excavated
in a material that is assumed to be
linearly elastic, brittle-perfectly plas-
tic, with yield surfaces described by the
Coulomb failure criterion.

Three different points of equilib-
rium for the ground-support interface
are shown in Figures 2 and 3 for both
t~mnels: they are Ea, Eb, and Ec.

344 TUm~.T.T.n~G AND UNDEROROUSD SPACE TECHNOLOGY Volume 6, Number 3, 1991

Page 3

a. 0.9



z o



O3 o

u ~ 0 2

~: O.t

E = 75 MPo
I Lake Edmonton

Sediments V = O. 4
Upper Till C =0

~p =30 °

Lower Till (3( = I (coefficient of dilotancy}
EX,~TUNNE m : ton 2 (45+(~ /2 ) =3.0

U ~e : (m-I)/(m-i-I) :0.5
Ko =l(Stress Field)

S R C - ~ J ~ -

0 1 t I I I I '
X J 2 3 4 5 6


Figure 2. Convergence-Confinement (C.-C.) method at the springline of the EXP

Point Ea

Ea is the point of equilibrium de-
fined by the intersection of the theo-
retical ground reaction and the sup-
port reaction curves. The plot of the
SRCs shown in Figures 2 and 3 re-
quires a knowledge of the compressive
stiffness of the support and of the
ground displacement close to the
ground-support interface, which takes
place before the l l n l n g expansion. In
this paper, the latter is obtained by the
sum of two ground displacements:

1. Ground displacements that take
place ahead of the face of the bmneh
assumed to be one-third of the elastic
wail displacement of the unlined tun-
nel (l~q-ken and Ghaboussi 1975).

2. Ground displacements that take
place along the length of the TBM
shield: assumed to be one-half of the
difference between the excavated di-
ameter and the diameter of the ex-
panded primary lining.

Branco (1981) presented a detailed
estimation of the displacements for both
the t|mnels.

Point Eb

Eb is another point of equih'brium
defined by the intersection ofthe ground
reaction and the support reaction
curves. The difference between Ea and
Eb is in the ground displacement that
is allowed to take place before the lin-
ing is expande& In order to find Ea,
the ground displacement that takes
place before the lln~ng expansion was
estimated without t~k~ng into account
any information derived from the tun-

nel instr~mentation. On the other
hand, the plot of the SRC that defines
the point of equilibrium Eb is based on
the measured ground displacements
that take place before the l l n l n g expan-
sion. These displacements, obtained
from field instrumentation, were pre-
sented by Branco (1981) and are sum-
mArized in Table 1.

Point Ec

Ec is the range of points of equilib-
rium obtained from the lln;n~ and


ground instrumentation, as presented
byE1-Nahhas (1980) and Braneo (1981).

The loads and displacement ratios
defining EC are related to t h e slprJng]ine
of the bmnele because a more complete
sot of field data was available at this
location. In addition, the gravity ef-
fects, which are not taken into account
in the C.-C method, are mlnim;,ed at
this location and thus can be neglected.

It is relevant to mention that for
both bl-nele, the displacements asso-
ciated with Ec were obtained at a dis-
tance of appro~mately one-quarter of
the bin-el diameter from the support
springline. This means that the true
Ec values at the ground-support should
be shifted to the right of the Ec shown
in Figures 2 and 3, because the closer
to the t-nnel walls, the larger are the
radial ground displacements.

The analysis ofthe points ofequilib-
rium, Ea and Ec, plotted for the ground-
support interface of the EXP t~mnel on
Figure 2, indicates that thrusts and
Hnlng displacements were reasonably
well predicted by the C.-C method.

On the other hand, the comparison
between Ea and Ec, shown in Figure 3
for the LRT-SE t~mnel, indicates that
the C.-C. method predicted loads and
displacements completely different
from those measure& After the mea-
sured ground displacements that took
place ahead of the lining expansion are
taken into account and when the SRC
is positioned along the horizontal axis

,t 0.9
--" r \ o i LRT South- k'~ Lake Edmonton

0 .8 I - \ 51 Bound Tunnel ~;;~ Sediment

z / \ lOl 6 ~/~ Upper T~II
~..~1'~0 0 .7 I " \ 151 r~ Lower Till

~ Saskatchewan
" ":'.- / \ 2° I ~O, , 0 . 6 F \ 25 M sands & Gravel

/ \
.--=""=== ° " tEB
• f___./_ i
• 03 i \


I I ; J ~._

X e l 2 3 4 5 6

I n t e r f a c e ( S o i l - l i n i n g ) D i s p a l c e m e n t / E l a s t i c W a l l
D i s p l a c e m e n t o f U n l i n e d T u n n e l --U/Uo e

Figure 3. Convergence-Confinement (C.-C.) method at the springline of the
LRT-SE Tunnel.

E = 150 MPa
v = 0.4
c = 0
~o = 40 °
a = 1 (coefficient of dilatancy)
m tan 2 (45 +0/2 = 4.6
Z e = (m-1)/(m + 1) = 0.64
k o = I(Stress Field)

Volume 6, Number 3, 1991 TUNN~T.T.T~a ~ U~'VF~m~OUND SPACE T~C~OLO~ 845

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