This document presents a closed-form elastic solution for stresses and displacements around tunnels using complex potential functions and conformal mapping. The solution models a homogeneous, isotropic, linear elastic geomaterial with an initially homogeneous stress state that is disturbed by the excavation of a tunnel. The tunnel can have a variety of cross-sectional shapes like semi-circular or double-arch. The analytical solution provides stress and displacement distributions that compare well with numerical modeling, providing insight into variable influences for preliminary tunnel design assessments.

1. International Journal of Rock Mechanics & Mining Sciences 39 (2002) 905–916
A closed-form elastic solution for stresses and displacements around
tunnels
G.E. Exadaktylosa,
*, M.C. Stavropouloub
a
Mining Engineering Design Laboratory, Department of Mineral Resources Engineering, Technical University of Crete, GR-73100 Chania, Greece
b
Department of Engineering Science, Section of Mechanics, National Technical University of Athens, 5 Heroes of Polytechnion Av,
GR-15773 Athens, Greece
Accepted 3 July 2002
Abstract
A closed-form plane strain solution is presented for stresses and displacements around tunnels based on the complex potential
functions and conformal mapping representation. The tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and
pre-stressed geomaterial. Further, the tunnel is considered to be deep enough such that the stress distribution before the excavation
is homogeneous. Needless to say that tunnels of semi-circular or ‘‘D’’ cross-section, double-arch cross-section, or tunnels with
arched roof and parabolic floor, have a great number of applications in soil/rock underground engineering practice. For the specific
type of semi-circular tunnel the distribution of stresses and displacements around the tunnel periphery predicted by the analytical
model are compared with those of the FLAC2D
numerical model, as well as, with Kirsch’s ‘‘circular’’ solution. Finally, a
methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of the tunnel.
r 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
Underground openings in soils and rocks are ex-
cavated for a variety of purposes and in a wide range of
sizes, ranging from boreholes through tunnels, drifts,
cross-cuts and shafts to large excavations such as
caverns, etc. A feature common to all these openings
is that the release of pre-existing stress upon excavation
of the opening will cause the soil or rock to deform
elastically at the very least. However, if the stresses
around the opening are not high enough then the rock
will not deform in an inelastic manner. This is possible
for shallow openings in relative competent geomaterials
where high tectonic stresses are absent. An under-
standing of the manner in which the soil or rock around
a tunnel deforms elastically due to changes in stress is
quite important for underground engineering problems.
In fact, the accurate prediction of the in situ stress field
and deformability moduli through back-analysis of
tunnel convergence measurements and of the ‘Ground
Reaction Curve’ is essential to the proper design of
support elements for tunnels [1,2].
The availability of many accurate and easy to use
finite element, finite difference, or boundary element
computer codes makes easy the stress-deformation
analysis of underground excavations. However, Carran-
za-Torres and Fairhurst note explicitly in their paper [3]:
‘‘yAlthough the complex geometries of many geotech-
nical design problems dictate the use of numerical
modeling to provide more realistic results than those
from classical analytical solutions, the insight into the
general nature of the solution (influence of the variables
involved etc.) that can be gained from the classical
solution is an important attribute that should not be
overlooked. Some degree of simplification is always
needed in formulating a design analysis and it is
essential that the design engineer be able to assess the
general correctness of a numerical analysis wherever
possible. The closed-form results provide a valuable
means of making this assessmenty’’.
One of the simplifying assumptions always made by
various investigators during studying—usually in a
preliminary design stage—analytically stresses and
*Corresponding author. Tel.: +30-8210-37450; fax: +30-8210-
69554.
E-mail addresses: exadakty@mred.tuc.gr (G.E. Exadaktylos),
m.stavropoulou@mechan.ntua.gr (M.C. Stavropoulou).
1365-1609/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 1 3 6 5 - 1 6 0 9 ( 0 2 ) 0 0 0 7 9 - 5

2. displacements around a tunnel is that it has a circular
cross-section [1,2,4]. This is due to the fact that the
celebrated Kirsch’s [5] analytical solution of the circular-
cylindrical opening in linear elastic medium is available
in the literature and it is rather simple for calculations
[6]. On the other hand, Gerc#
ek [7,8] was the first
investigator who presented a closed-form solution for
the stresses around tunnels with arched roofs and with
either flat or parabolic floor having an axis of symmetry
and excavated in elastic media subjected to an arbi-
trarily oriented in situ far-field biaxial stress state.
Gerc#
ek has used the method of conformal mapping and
Kolosov–Muskhelishvili complex potentials [9] along
with the ‘‘modified method of undetermined coeffi-
cients’’ of Chernykh [10]. However, Gerc#
ek did not
consider (a) the incremental release of stresses due to
excavation of the tunnel, (b) the solution for the
displacements,1
(c) the influence of support pressure on
tunnel walls on the stresses and displacements, and (d)
the methodology to derive the constant coefficients of
the series representation of the conformal mapping for
prescribed tunnel cross-sections.
In order to add the above essential elements for
appropriate tunnel and support design, we present here
the closed-form solution for the elastic stresses and
displacements around tunnels with rounded corners in
pre-stressed soil/rock masses. This solution is derived by
virtue of Muskhelishvili’s [9] complex potential repre-
sentation, the conformal mapping technique and the
properties of Cauchy integrals.2
The proposed closed-
form solution for the stresses and displacements that is
presented here is original, although Gerc#
ek following a
different methodology has derived the solution for a
different boundary value problem, that is appropriate
only for stress and not for deformation analysis of
underground excavations. The results of the analytical
solution pertaining to the stresses and displacements
around the tunnel with ‘‘D’’ cross-sectional shape are
compared with the predictions of the FLAC2D
numer-
ical code [11,13] for two far-field stress states. It is shown
that the numerical model predictions compare very well
with the analytical solution apart from the corner and
invert regions. Further, in Appendix A, we present a
simple methodology for the derivation of the coefficients
of the series representation of the complex conformal
mapping function that corresponds to a given tunnel
cross-section shape.
2. The closed-form full-field elastic solution for the tunnel
In this section, a plane strain elastic model is
considered for the influence of the excavation of an
underground opening on a homogeneous stress-defor-
mation state described by in situ principal stresses sxN
and syN referred to a Cartesian coordinate system Oxy.
That is, it is assumed that the tunnel-axis is aligned with
the direction of the third out-of-plane principal stress
szN: The direction of sxN forms an angle a with Ox-
axis. The cross-section of the tunnel has a vertical axis of
symmetry as it is illustrated in Fig. 1a. The wall of the
tunnel is subjected to uniform pressure P (with P to be
a positive number and tensile stresses are considered as
positive quantities in this work).
Before the tunnel is excavated the resultant vector of
forces acting along any contour in the soil/rock mass is
given by the relation
f ðx; yÞ ¼ 2Gz þ %
G0
%
z; ð1Þ
Fig. 1. (a) Schematic diagram of the tunnel and system of coordinates,
(b) unit circle and system of coordinates.
1
The displacement solution is of interest to the geotechnical engineer
for the design of tunnel linings and for the back-analysis of tunnel
closure measurements.
2
Jaeger and Cook [6] in their celebrated book of Rock Mechanics
state explicitly that ‘‘yBy far the most powerful method for the
solution of two-dimensional problems is the detailed use of complex
variable theory and conformal representation as developed in the
books of Muskhelishvili [9] and Savin [12]y’’.
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916
906

3. wherein the overbar denotes complex conjugate, z ¼
x þ iy; i ¼
ffiffiffiffiffiffiffi
1
p
is the imaginary unit and
G ¼ 1
4ðsxN þ syNÞ;
G0
¼ 1
2ðsxN syNÞe2ia
: ð2Þ
The methodology starts with the conformal mapping3
of
the boundary of the tunnel and its exterior (region S in
Fig. 1a) into the interior of the circle with unit radius
(region
P
in Fig. 1b). The position of every point in the
physical z-plane with z ¼ x þ iy ¼ reia
; where r; a denote
polar coordinates, is mapped into the unit circle in the z-
plane with z ¼ x þ iZ ¼ r eiy
by the complex function
z ¼ oðzÞ ¼ R
1
z
þ
X
3
k¼1
akzk
!
; z
j jp1; ð3Þ
where the constant term R is a real number and the
constant coefficients ak are in general complex numbers
with ak ¼ ak þ ibk ðk ¼ 1; 2; 3; yÞ: This relation of
conformal mapping with three terms in the series
representation is chosen because it is the simplest one
that may describe tunnels with conventional shapes and
rounded corners [8]. If excessive roundness is not wanted
then the number of terms in the series expression (3)
should be increased. This is demonstrated in Appendix
A. Note that the point z describes the contour C in the z-
plane in an anti-clockwise direction, as the point z
moves around the circle in the z-plane, likewise in a
clockwise direction (Fig. 1b). This is because the tunnel
exterior infinite region is mapped into an interior finite
region. Also, the boundary of the tunnel C is mapped
onto the circumference g of the unit circle (with z ¼ eiy
along g).
The parametric representation of the curves in the
Oxy-plane transformed by Eq. (3) has as follows
x ¼ R
cos y
r
þ
X
3
k¼1
rk
ðak cos ky bk sin kyÞ
( )
;
y ¼ R
sin y
r
þ
X
3
k¼1
rk
ðak sin ky þ bk cos kyÞ
( )
: ð4Þ
The above relations for a1 ¼ b3 ¼ 0; a2 ¼ b2 [8] give
tunnels with an axis of symmetry that forms an angle of
þp=4 with the Oy-axis, hence we apply the following
formula for rotation of the axis of symmetry of the
tunnel with respect to Oy-axis by p=4
x0
þ iy0
¼ ðx þ iyÞ exp ðip=4Þ: ð5Þ
The transformation of the Cartesian coordinates
through the parametric Eqs. (4), after their correction
according to Eq. (5), will result in a new orthogonal
system of coordinates (Fig. 2a) that corresponds to the
families of curves r ¼ constantðctÞ and y ¼ ðctÞ in the z-
plane (Fig. 2b). The parametric representation of the
tunnel boundary C is obtained by setting r ¼ 1:
The role played by the series real coefficients b1; a2; a3
may be realized by the examples illustrated in Figs. 3a–c.
The value of b1 controls the height-to-width ratio of the
tunnel and if only this term survives then the opening
takes the form of an ellipse (Fig. 3a). The value of a2
controls the triangularity of the tunnel cross-section
(Fig. 3b). Finally, the value of a3 depicts the resem-
blance of the tunnel with the square opening with
rounded corners (Fig. 3c).
Following Muskhelishvili’s [9] complex variable for-
mulation of plane elasticity problems, the stresses and
displacements may be fully described by two analytic
complex functions f0ðzÞ; c0ðzÞ inside the region pre-
scribed by unit circle. The excavation of tunnel can be
simulated by partially or totally relieving the surface
tractions at its periphery C due to the in situ stress field.
The integral of the surface tractions is represented here
by the function f ðx; yÞ; i.e.
f ðx; yÞ ¼ 2lðG þ PÞz l %
G0
%
z; zAC; ð6Þ
Fig. 2. Conformal mapping of (a) an infinite soil/rock mass surround-
ing a tunnel into (b) the interior of the unit circle (rp1).
3
A transformation of the form x ¼ xðx; ZÞ; y ¼ yðx; ZÞ is said to be
conformal if the angle between intersecting curves in the ðx; ZÞ-plane
remains the same for corresponding mapped curves in the ðx; yÞ-plane.
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916 907

4. where l is the in situ stress—relief factor. It is this relief
of stresses that causes the displacements. For l ¼ 0 no
excavation has been occurred, whereas for l ¼ 1 the
tunnel has been fully excavated (i.e. 0plp1).4
Also, in
the above relation we have set
G ¼
s
4
ð1 þ kÞ;
G0
¼
s
2
ð1 kÞe2iðaþp=4Þ
; ð7Þ
where syN ¼ s; sxN ¼ ks are the principal stress field
before the excavation of the tunnel. The solution of the
above boundary value problem may be found by the
method developed by Muskhelishvili [9] for regions
mapped on to the circle by the help of polynomials. The
details of the solution are not given herein. In the
transformed plane, the first complex function in Laurent
series form may be found as follows:
f0ðzÞ ¼
X
3
k¼0
ckzk
; z
j jp1 ð8Þ
with the constant coefficients to be given by the relations
Reðc1Þ ¼
lRð1 þ a3Þ
1 a2
3 b3
Ref %
G0
þ ðP þ 2GÞa1g
lRb3
1 a2
3 b3
Imf %
G0
þ ðP þ 2GÞa1g;
Imðc1Þ ¼
lRb3
1 a2
3 b3
Ref %
G0
þ ðP þ 2GÞa1g
lRð1 a3Þ
1 a2
3 b3
Imf %
G0
þ ðP þ 2GÞa1g;
c2 ¼ lð2G þ PÞRa2;
c3 ¼ lð2G þ PÞRa3;
c0 ¼ a2 %
c1 þ 2a3%
c2
: ð9Þ
In the above relations bars denote complex conjugates,
whereas Re( ) and Im( )denote the real and imaginary
value of what it enclose, respectively. The second
unknown complex function may be found by the
following relation:
c0ðzÞ ¼ lRð2G þ PÞz
1
z
z4
þ
P3
k¼1 %
akzkþ3
1 þ
P3
k¼1 kakzkþ1
!
X
3
k¼1
kckzk1
%
a3c1
z
lG0
R
X
3
k¼1
akzk
: ð10Þ
The final state of stress is found by adding to the
complex functions that have been found above, the
corresponding parts that account for the in situ stress
field, that is to say
fðzÞ ¼ GoðzÞ þ f0ðzÞ; cðzÞ ¼ G0
oðzÞ þ c0ðzÞ: ð11Þ
Fig. 3. Shapes of openings and corresponding curvilinear coordinates:
(a) elliptical opening (b1 ¼ 0:3 and a1 ¼ a2 ¼ b2 ¼ a3 ¼ b3 ¼ 0),
(b) hypotrochoidal-triangular opening (a2 ¼ b2 ¼ 0:3 and a1 ¼
b1 ¼ a3 ¼ b3 ¼ 0), (c) hypotrochoidal-square opening (a3 ¼ 0:3 and
a1 ¼ b1 ¼ a2 ¼ b2 ¼ b3 ¼ 0).
4
It must be noted that in the elastic case l ¼ uiðzÞ=uiðNÞ; ði ¼ x; yÞ;
where z is the distance behind the tunnel face and ui is the displacement
vector.
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916
908

5. Then, in polar coordinates ðr; yÞ—referring to the
conformal mapping plane—the radial, tangential and
shear stresses denoted as sr; sy; try; respectively, may be
computed by virtue of the following formulae:
sr þ sy ¼ 4Re
f0
ðzÞ
o0ðzÞ
;
sr itry ¼ 2Re
f0
ðzÞ
o0ðzÞ
z2
r2 %
o0ð%
zÞ
½ %
oð%
zÞff0
ðzÞ=o0
ðzÞg0
þ o0
ðzÞc0
ðzÞ ð12Þ
where primes denote differentiation (i.e. f0
df=dz).
Also, the incremental displacements due to stress relief
at the tunnel boundary, referred in the Cartesian
coordinate system Oxy, are given by
2Gðux þ iuyÞ ¼ kf0ðzÞ
oðzÞ %
f0
0ð%
zÞ
%
o0ð%
zÞ
%
c0ð%
zÞ ð13Þ
in which k denotes Muskhelishvili’s constant with k ¼
3 4n for plane strain conditions, G ¼ E=2ð1 þ nÞ is the
shear modulus, and E; n is the Young’s modulus and
Poisson’s ratio of the isotropic rock/soil mass, respec-
tively.
The final state of stress after the tunnel is fully
excavated (i.e. for l ¼ 1) is given by relationships (12),
as well as by relations (7)–(11). This stress solution is
exactly the same with that corresponding to the
boundary value problem in which the in situ stresses
are applied after the tunnel has been excavated. This
boundary value problem has been studied by Gerc#
ek
[7,8]. However, the problem of the tunnel that is
excavated in a pre-stressed rock/soil mass is another
type of boundary value problem that possess a different
displacement solution given by relation (13). This
solution for the incremental displacements tends to zero
far away from the tunnel.
The implementation of Eqs. (7)–(13) into a fast
computer code is quite easy with the computational
and software (e.g. Excel, Matlab, Maple, etc.) capabil-
ities of modern personal computers. Geotechnical
engineers should begin to exploit more the results of
applied elasticity theory in rock mechanics and rock
engineering applications. This is illustrated below with
some worked examples.
3. Verification of the proposed closed-form solution with
known solutions
The solution of the complex potentials fðzÞ and cðzÞ
that has been found above, is compared here with
existing solutions for the elliptical opening subjected to
internal pressure and to far-field uniaxial stress, and
with the square opening subjected to uniaxial stress s
along Ox-axis.
3.1. Case of the elliptical and circular openings
We consider first the elliptical opening subjected to
uniform internal pressure P and subjected to uniaxial
stress s in a direction that forms an angle a with the Ox-
axis. The relevant conformal transformation in the case
where the exterior of the ellipse with semi-axes a ¼
Rð1 þ mÞ; b ¼ Rð1 mÞ with 0pmp1 is mapped into
the interior of the unit circle is the following:
z ¼ oðzÞ ¼ R
1
z
þ mz ; z
j jp1; ð14Þ
hence, in this case a1 ¼ m; a2 ¼ a3 ¼ 0: Then from
formulae (7)–(11) it may be found
fðzÞ ¼ PRmz þ
sR
4
1
z
þ ð2e2ia
mÞz ;
cðzÞ ¼ PRz PRmz
m þ z2
1 mz2
sR
2
e2ia
z
þ
e2ia
z
m
ð1 þ m2
Þðe2ia
mÞ
m
z
1 mz2
; z
j jp1: ð15Þ
Fig. 4. Comparison of ur; ua for the cylindrical hole in an infinite
elastic medium characterized by Poisson’s ratio n ¼ 0:3 subjected to
uniaxial compression along Oy-axis s=E ¼ 0:01:
Fig. 5. Comparison of the prescribed and predicted semi-circular
tunnel.
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916 909

6. The above formulae are exactly the same with
that reported by Muskhelishvili [9] if z is sub-
stituted by 1=z (since he employed the con-
formal mapping on the exterior of the unit circle).
Further, the simpler case of the circular opening
subjected to the same stress field is obtained by setting
m ¼ 0:
Herein, a comparison is attempted between the
displacements around the circular tunnel wall given by
Kirsch’s solution and the proposed closed-form solu-
tion. Assuming conditions of plane strain the radial and
tangential displacements in polar coordinates (r; a) are
given by the formulae [6]
2Gur ¼
1
2
sð1 þ kÞ
R2
r
R2
2r
sð1 kÞ
4ð1 nÞ
R2
r2
cos 2a;
2Gua ¼
R2
2r
sð1 kÞ 2ð1 2nÞ þ
R2
r2
sin 2a: ð15aÞ
Fig. 4 displays the exact agreement between the two
displacement solutions for the case at hand.
Fig. 6. (a) Finite-difference mesh and (b) detail of the ‘‘D’’ tunnel model employed in FLAC2D
.
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916
910

7. 3.2. Case of the square opening
Next we consider the case of the square opening
subjected to uniaxial stress s along Ox-axis. The
conformal transformation in the case where the exterior
of the square with a length of its side ð5=3ÞR is mapped
into the interior of the unit circle is the following [10]:
z ¼ oðzÞ ¼ R
1
z
1
6
z3
; z
j jp1; ð16Þ
hence in this case a3 ¼ 1=6; a1 ¼ a2 ¼ 0: Then from
formulae (4)–(7) it may be found
fðzÞ ¼ sR
1
4z
þ
3
7
z þ
1
24
z3
;
cðzÞ ¼
sR
2
1
2z
þ
91z 78z3
84ð2 þ z4
Þ
; z
j jp1: ð17Þ
The above expressions are in full agreement with those
displayed in [10]. Hence, the proposed analytical
solution may be also employed for the stress–deforma-
tion analysis of caverns in rocks.
4. Stress–deformation analysis of the semi-circular tunnel
In order to demonstrate the potential applications of
the proposed solution in soil/rock engineering, a number
of examples have been worked out and they are
illustrated below. Namely, the comparability of analy-
tical model results concerning the distribution of stresses
and displacements around the semi-circular tunnel with
those predicted by FLAC2D
numerical code is demon-
strated. It may be argued that a boundary element code
would be more suitable for the comparison of boundary
stresses and displacements with the analytical solution.
However, we would like here to consider a numerical
code that is used extensively worldwide for the design of
tunnels and underground excavations.
First the unknown conformal mapping coefficients
b1; a2 ¼ b2; a3 are determined by an appropriate non-
linear constrained optimization algorithm presented in
Appendix A. As it is illustrated in Fig. 5 the Oy-axis is
an axis of symmetry of the tunnel whereas its floor is
located at y ¼ 0:53 and its width is 2.64. The values of
the constant conformal mapping coefficients have been
found by virtue of the methodology described in
Appendix A to be
R ¼ 0:9945; b1 ¼ 0:2836;
a2 ¼ b2 ¼ 0:092; a3 ¼ 0:0389: ð18Þ
The comparison of the predicted tunnel shape with the
actual one is illustrated in Fig. 5. As it may be seen from
this figure the truncated conformal mapping transfor-
mation with three terms in the series expansion (3) gives
corners with finite radius of curvature. However, for
b 101 and for xo0:72X (radius of the tunnel) the
predicted boundary almost coincides with the specified
tunnel boundary. It should be noted that more terms in
the conformal mapping series representation would
result in a better approximation. This is demonstrated
in Appendix A.
Furthermore, the geometrical model for the same
shape of tunnel that was prescribed into the FLAC2D
model is displayed in Fig. 6. The symmetry of the
problem with respect to Oy-axis has been exploited in
the numerical model by considering only the right-hand
part. A roller boundary is used to model zero displace-
ment along the line of the symmetry. The bottom of the
mesh and the right-hand boundary are pinned in both
Ox- and Oy-displacements.
Next, the distribution of tangential (hoop) stress at
the traction-free boundary of the semi-circular tunnel
predicted by the analytical model is compared with the
numerical code FLAC2D
, as well as with the Kirsch’s
‘‘circular’’ solution for the following two far-field stress
states:
Case I : sxN ¼ 0 MPa; syN ¼ 1 MPa;
Case II : sxN ¼ syN ¼ 1 MPa;
where compressive stresses are taken as negative
quantities.
Fig. 7. Plot of (a) hoop stress concentration sy=syN along tunnel
semi-circular boundary and (b) plot of the horizontal stress
concentration sx=syN along the floor of the tunnel (Case I loading).
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916 911

8. The comparison of the hoop stress around the tunnel
predicted by the three models are illustrated in Figs. 7
and 8, respectively, for the two far-field stress states at
hand. For both loading cases it is observed that the
numerical model is in close agreement with the
analytical solution with some difference of results close
to the ‘‘corner’’ of the tunnel (i.e. for bo151) that occurs
due to the following two reasons:
a. The conformal mapping representation introduces a
certain amount of rounding of the corner as it is
displayed in Fig. 5.
b. In contrast to the analytical solution, the FLAC
model predicts finite tractions at the corners of the
tunnel (e.g. Fig. 8a).
It is also interesting to note from Fig. 8b and
Table 1 that the greater discrepancy between the
analytical and numerical solutions occurs at the
invert of the tunnel for the isotropic loading case.
Further, the analytical model predicts that the hoop
stress concentration factor at the crown of the tunnel is
0.96 while the numerical model predicts the value of
0.8 (Fig. 8a). It is known that the stress concentration
factor for this stress state is always equal to 1
irrespective of the shape of the tunnel [1], hence the
analytical model leads to an improvement of prediction
of stresses compared to the numerical model.
It is also worth noting from Fig. 8a, as well as Table 1,
that the absolute values of the hoop stress concentration
predicted by both analytical and numerical solutions at
the crown of the tunnel that is subjected to isotropic
loading is appreciably smaller than that predicted by
Kirsch’s circular solution.
-2
-1
0
1
2
3
4
5
6
0 10 20 30 40 50 60 70 80 90
β [degrees]
Stress
concentration
Hoop stress (FLAC)
Radial stress (FLAC)
Shear stress (FLAC)
Hoop stress (Analytical)
Kirsch
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6
x [m ]
Horizontal
stress
concentration
FLAC
Analytical
(a)
(b)
Fig. 8. Plot of (a) hoop stress concentration sy=syN along tunnel
semi-circular boundary and (b) plot of horizontal stress concentration
sx=syN along the floor of the tunnel (Case II loading).
Table 1
Comparison of the hoop stress (in MPa) at the crown and invert of the
‘‘D’’ tunnel predicted by the three solutions for the two far-field stress
states at hand
Position Stress
state
Kirsch’s
solution
Analytical
model
FLAC2D
Relative
error (%)
Crown
(b ¼ 901)
Case I 1 0.96 0.80 16.6
Case II 2 1.50 1.55 3.3
Invert
(x ¼ 0)
Case I — 0.97 1.06 9.3
Case II — 0.43 0.33 23.2
-1.E-03
-8.E-04
-4.E-04
0.E+00
4.E-04
0 10 20 30 40 50 60 70 80 90
β [degrees]
Displacement
[m]
ux (FLAC)
ux (Analytical)
uy (FLAC)
uy (Analytical)
-4.E-04
0.E+00
4.E-04
8.E-04
1.E-03
0 1 2 3 4 5 6
x [m]
Displacement
[m]
ux (FLAC)
ux (Analytical)
uy (FLAC)
uy (Analytical)
(b)
(a)
Fig. 9. Plot of displacements ux; uy along (a) tunnel semi-circular
boundary with respect to the polar angle b and (b) along the floor of
the tunnel.
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916
912

9. Finally, a comparison between the FLAC model and
the analytical solution is attempted for the displace-
ments along the boundary of the tunnel for the isotropic
loading case. For this comparison the following values
of the elastic constants of the soil/rock mass were
assumed
E ¼ 10 GPa; n ¼ 0:3:
Fig. 10. Contour plots of the vertical displacement uy around the semi-circular tunnel that is subjected to isotropic far-field loading (Case II)
predicted (a) by FLAC and (b) by analytical solution.
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916 913

10. The comparison of the horizontal and vertical displace-
ments around the tunnel predicted by the two models
are illustrated in Fig. 9. Both these figures demonstrate
that the numerical model is in close agreement with the
analytical model except for some discrepancy of results
close to the ‘‘corner’’ of the tunnel (i.e. bo151). It may
also be noted that FLAC model predicts higher
horizontal displacements than the analytical solution
along the invert region of the tunnel boundary (Fig. 9b).
Fig. 10 also displays the contour plots of vertical
displacements that are predicted by the numerical
FLAC model and the analytical solution for the above
values of the elastic constants. The general agreement of
both predictions may be seen from these figures.
5. Conclusions
An exact solution has been presented for stresses and
displacements around tunnels with rounded corners. It
has been shown that the complex potential formulation
together with the conformal mapping representation can
be used successfully for the solution of plane elasticity
problems for any tunnel cross-sectional shape with an
axis of symmetry with prescribed surface tractions. The
solution method has been compared with the FLAC2D
numerical model for the particular case of the semi-
circular tunnel. It has been illustrated that both models
predict boundary stresses and displacements that are in
general agreement apart from the corner and invert
regions. Also, the formulation employed here is suitable
for the ground–support-interaction analysis of tunnels
constructed by the New Austrian Tunnelling Method.
Finally, a methodology is proposed for the estimation of
conformal mapping coefficients for a given cross-
sectional shape of a tunnel.
Appendix A
Herein, the procedure is described that is proposed for
the computation of the constant coefficients of poly-
nomial conformal mapping functions that map piece-
wise smooth opening contours onto the circular disc of
unit radius. First, it may be shown that along the
boundary C of the opening in the Oxy-plane the angle a
that is formed between ðrÞ and Ox axes measured from
the latter anti-clockwise is given by the relation [9]
(Fig. 11)
eia
¼
z
r
o0
ðzÞ
o0ðzÞ
j j
¼ eiy o0
ðzÞ
o0ðzÞ
j j
) a ¼ arg eiy o0
ðzÞ
o0ðzÞ
j j
; ðA:1Þ
where we have to set r ¼ 1 and z ¼ eiy
for the
corresponding contour of the opening g in the ðx; ZÞ
plane. Hence, from Eq. (3) the conformal mapping
function is given by relation:
z ¼ oðzÞ ¼ RðnÞ
eiy
þ
X
m
k¼1
aðnÞ
k þ ibðnÞ
k
eiky
!
; ðA:2Þ
where the superscript in parenthesis ðnÞ denotes the
iteration level and m is the degree of the polynomial
function ðm ¼ 0; 1; yÞ:
From Eq. (A.2) the parametric equations for the
coordinates xc; yc of the boundary of the opening in the
Oxy plane have as follows:
xc ¼ R cos y þ
X
m
k¼1
aðnÞ
k cos ky bðnÞ
k sin ky
( )
;
yc ¼ R sin y þ
X
m
k¼1
aðnÞ
k sin ky þ bðnÞ
k cos ky
( )
: ðA:3Þ
Also, from Eqs. (A.1) and (A.2) after some algebraic
manipulations the following formula for the angle a is
derived:
a ¼ tan1
sin y þ
Pm
k¼1 kaðnÞ
k sin ky þ
Pm
k¼1 kbðnÞ
k cos ky
cos y þ
Pm
k¼1 kaðnÞ
k cos ky
Pm
k¼1 kbðnÞ
k sin ky
!
:
ðA:4Þ
The piecewise smooth contour prescribing the opening is
divided into a number of smooth and simple curves or
arcs5
that are represented in the form
x ¼ xðsÞ; y ¼ yðsÞ; sapspsb; ðA:5Þ
Fig. 11. Definition of angle a that is formed between Ox and (r)
axes—with the latter being normal to the tangent axis (y) at some point
of the smooth curve—measured in an anti-clockwise sense.
5
Curves are called ‘‘smooth’’ when they have continuous first
derivatives, i.e. dxðsÞ=ds; dyðsÞ=ds; inside their interval of definition.
They are also called ‘‘simple’’ if xðs1Þ ¼ xðs2Þ; yðs1Þ ¼ yðs2Þ are
incompatible for saps1; s2psb; s1as2:
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916
914

11. in which sa; sb are finite constants defining the interval of
the curve, xðsÞ; yðsÞ are continuous functions in the
interval of definition and s denotes the arc coordinate.
Due to symmetry considerations only the one-half of the
contour may be considered. Next, the tangent line to
the arc which coincides with ðyÞ (Fig. 11) at any point of
the arc can be found by the formula
t
B
¼
dz
ds
¼
dxðsÞ
ds
þ i
dyðsÞ
ds
; ðA:6Þ
where the curly underline denotes that the quantity is a
vector. The normal line to tB denoted by the symbol nB;
which is also normal to the boundary of the hole and
coincides with the ðrÞ axis (Fig. 11) is then found from
the condition
n
B
t
B
¼ 0; ðA:7Þ
where the dot denotes operation of the inner (or scalar)
product. Having found the relation between the arc
coordinate s and the angle a we may easily correspond
its point ðx; yÞ to ðxc; ycÞ that is predicted by the
polynomial conformal mapping function.
Finally, the constant coefficients ak; bkðk ¼ 1; y; mÞ
are found by any of the available constrained nonlinear
minimization routines of the sum d of the distances
between the predicted coordinates ðxc; ycÞ and actual
coordinates ðx; yÞ of the contour C of the opening, i.e.
d ¼ min
X
N
j¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðxj xcjÞ2
þ ðyj ycjÞ2
q
pe; ðA:8Þ
where N is the total number of points along the contour
and e is the prescribed error tolerance.
As an example of the application of the above
methodology we present in Fig. 12 the predicted cross-
sectional shape of a tunnel with arched roof and invert.
As it can been seen in this figure by increasing the
number of terms in the series expansion (3) a smaller
rounding of the corner is achieved. Of course, in this
case the corresponding terms in the Laurent series
representation (8) for the potential function fðzÞ should
increase accordingly. This is a formidable task in the
frame of the proposed general complex variable
formulation [14].
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Fig. 12. Comparison of the prescribed tunnel with arched roof and floor and predicted tunnel shapes for the number of terms in the conformal
mapping series at hand.
G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics Mining Sciences 39 (2002) 905–916 915

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916