exadaktylos 2002-aclosed-form elastic solution for stresses and displacements around tunnel

7/30/2019 Exadaktylos 2002-Aclosed-Form Elastic Solution for Stresses and Displacements Around Tunnel

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International Journal of Rock Mechanics & Mining Sciences 39 (2002) 905916

A closed-form elastic solution for stresses and displacements around

tunnels

G.E. Exadaktylosa,*, M.C. Stavropouloub

aMining Engineering Design Laboratory, Department of Mineral Resources Engineering, Technical University of Crete, GR-73100 Chania, GreecebDepartment 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 Kirschs 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 studyingusually in a

preliminary design stageanalytically stresses and

*Corresponding author. Tel.: +30-8210-37450; fax: +30-8210-

69554.

E-mail addresses: [email protected] (G.E. Exadaktylos),

[email protected] (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


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displacements around a tunnel is that it has a circular

cross-section [1,2,4]. This is due to the fact that the

celebrated Kirschs [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 forthe 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

KolosovMuskhelishvili 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 Muskhelishvilis [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 adifferent 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 Pto 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

fx;y 2Gz %G0 %z; 1

Fig. 1. (a) Schematic diagram of the tunnel and system of coordinates,

(b) unit circle and system of coordinates.

1The displacement solution is of interest to the geotechnical engineer

for the design of tunnel linings and for the back-analysis of tunnel

closure measurements.2Jaeger 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) 905916906


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wherein the overbar denotes complex conjugate, z

x iy; i ffiffiffiffiffiffiffi

1p

is the imaginary unit and

G 14sxN syN;

G0 12sxN syNe

2ia: 2

The methodology starts with the conformal mapping3 of

the boundary of the tunnel and its exterior (region S inFig. 1a) into the interior of the circle with unit radius

(regionP

in Fig. 1b). The position of every point in the

physical z-plane with z x iy reia; where r; a denotepolar coordinates, is mapped into the unit circle in the z-

plane with z x iZ r eiy by the complex function

z oz R1

zX3k1

akzk

!; zj jp1; 3

where the constant term R is a real number and the

constant coefficients ak are in general complex numbers

with ak ak ibk k 1; 2; 3;y: This relation of

conformal mapping with three terms in the seriesrepresentation 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 Cin 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 eiy

along g).

The parametric representation of the curves in the

Oxy-plane transformed by Eq. (3) has as follows

x Rcos y

rX3k1

rkak cos ky bk sin ky

( );

y R sin y

rX3k1

rkak sin ky bk cos ky

( ): 4

The above relations for a1 b3 0; a2 b2 [8] givetunnels with an axis of symmetry that forms an angle of

p=4 with the Oy-axis, hence we apply the following

formula for rotation of the axis of symmetry of thetunnel with respect to Oy-axis by p=4

x0 iy0 x iy exp ip=4: 5

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 constantct and y ct in the z-

plane (Fig. 2b). The parametric representation of the

tunnel boundary C is obtained by setting r 1:The role played by the series real coefficients b1; a2; a3

may be realized by the examples illustrated in Figs. 3ac.

The value ofb1 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 a2controls 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 Muskhelishvilis [9] complex variable for-

mulation of plane elasticity problems, the stresses and

displacements may be fully described by two analytic

complex functions f0z; c0z 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 fx;y; i.e.

fx;y 2lG Pz l %G0 %z; zAC; 6

Fig. 2. Conformal mapping of (a) an infinite soil/rock mass surround-

ing a tunnel into (b) the interior of the unit circle (rp1).

3A transformation of the form x xx; Z; y yx; Z is said to beconformal if the angle between intersecting curves in the x; Z-planeremains the same for corresponding mapped curves in the x;y-plane.

G.E. Exadaktylos, M.C. Stavropoulou / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 905916 907


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where l is the in situ stressrelief factor. It is this relief

of stresses that causes the displacements. For l 0 no

excavation has been occurred, whereas for l 1 the

tunnel has been fully excavated (i.e. 0plp1).4 Also, in

the above relation we have set

G s

41 k;

G0 s

21 ke2iap=4; 7

where syN s; sxN ks are the principal stress fieldbefore 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:

f0z X3k0

ckzk

; zj jp1 8

with the constant coefficients to be given by the relations

Rec1 lR1 a3

1 a23 b3Ref %G0 P 2Ga1g

lRb3

1 a23 b3Imf %G0 P 2Ga1g;

Imc1 lRb3

1 a23 b3Ref %G0 P 2Ga1g

lR1 a3

1 a23 b3Imf %G0 P 2Ga1g;

c2 l2G PRa2;

c3 l2G PRa3;

c0 a2 %c1 2a3%c2 : 9

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:

c0z lR2G Pz 1

z

z4 P3

k1 %akzk3

1 P3k1 kakzk1 !

X3k1

kckzk1

%a3c1

z lG0R

X3k1

akzk: 10

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

fz Goz f0z; cz G0oz c0z: 11

Fig. 3. Shapes of openings and corresponding curvilinear coordinates:

(a) elliptical opening (b1 0:3 and a1 a2 b2 a3 b3 0),(b) hypotrochoidal-triangular opening (a2 b2 0:3 and a1 b1 a3 b3 0), (c) hypotrochoidal-square opening (a3 0:3 anda1 b1 a2 b2 b3 0).

4It must be noted that in the elastic case l uiz=uiN; i x;y;where z is the distance behind the tunnel face and ui is the displacement

vector.



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Then, in polar coordinates r; yreferring to theconformal mapping planethe radial, tangential and

shear stresses denoted as sr; sy; try; respectively, may becomputed by virtue of the following formulae:

sr sy 4Ref0z

o0z ;sr itry 2Re

f0z

o0z

z2

r2 %o0%z

%o%zff0z=o0zg0 o0zc0z 12

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

2Gux iuy kf0z oz %f00

%z

%o0%z %c0%z 13

in which k denotes Muskhelishvilis constant with k 3 4n for plane strain conditions, G E=21 n is theshear modulus, and E; n is the Youngs modulus andPoissons ratio of the isotropic rock/soil mass, respec-

tively.

The final state of stress after the tunnel is fully

excavated (i.e. for l 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 isexcavated 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 fz and cz

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 R1 m; b R1 m with 0pmp1 is mapped intothe interior of the unit circle is the following:

z oz R1

z mz

; zj jp1; 14

hence, in this case a1 m; a2 a3 0: Then fromformulae (7)(11) it may be found

fz PRmz sR

4

1

z 2e2ia mz

;

cz PRz PRmzm z2

1 mz2

sR

2

e2ia

z

e2iaz

m

1 m2e2ia m

m

z

1 mz2

; zj jp1: 15

Fig. 4. Comparison of ur; ua for the cylindrical hole in an infiniteelastic medium characterized by Poissons ratio n 0:3 subjected touniaxial compression along Oy-axis s=E 0:01:

Fig. 5. Comparison of the prescribed and predicted semi-circular

tunnel.



6/12

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 0:Herein, a comparison is attempted between the

displacements around the circular tunnel wall given by

Kirschs 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 1

2s1 k

R2

r

R2

2rs1 k

41 n R2

r2 cos 2a;2Gua

R2

2rs1 k 21 2n

R2

r2

sin 2a: 15a

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.



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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 5=3R is mapped

into the interior of the unit circle is the following [10]:

z oz R1

z

1

6z3

; zj jp1; 16

hence in this case a3 1=6; a1 a2 0: Then fromformulae (4)(7) it may be found

fz sR1

4z

3

7z

1

24z3

;

cz sR

2

1

2z

91z 78z3

842 z4

; zj jp1: 17

The above expressions are in full agreement with those

displayed in [10]. Hence, the proposed analytical

solution may be also employed for the stressdeforma-

tion analysis of caverns in rocks.

4. Stressdeformation 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 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 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 0:9945; b1 0:2836;

a2 b2 0:092; a3 0:0389: 18

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 Kirschs

circular solution for the following two far-field stress

states:

Case I : sxN 0 MPa;syN 1 MPa;

Case II : sxN syN 1 MPa;

where compressive stresses are taken as negative

quantities.

Fig. 7. Plot of (a) hoop stress concentration sy=syN along tunnelsemi-circular boundary and (b) plot of the horizontal stress

concentration sx=syN along the floor of the tunnel (Case I loading).



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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 occursdue 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

Kirschs circular solution.

-2

-1

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80 90

[degrees]

Stressconcen

tration

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 ]

Horizontalstressconcentration

FLAC

Analytical

(a)

(b)

Fig. 8. Plot of (a) hoop stress concentration sy=syN along tunnelsemi-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

Kirschs

solution

Analytical

model

FLAC2D Relative

error (%)

Crown(b 901)

Case I 1 0.96 0.80 16.6

Case II 2 1.50 1.55 3.3

Invert

(x 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

[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-circularboundary with respect to the polar angle b and (b) along the floor of

the tunnel.



9/12

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 10 GPa; n 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.



10/12

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 higherhorizontal 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 anddisplacements 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 groundsupport-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 ofunit 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 r and Ox axes measured from

the latter anti-clockwise is given by the relation [9]

(Fig. 11)

eia z

r

o0z

o0zj j eiy

o0z

o0zj j) a arg eiy

o0z

o0zj j

; A:1

where we have to set r 1 and z eiy for the

corresponding contour of the opening g in the x; Zplane. Hence, from Eq. (3) the conformal mapping

function is given by relation:

z oz Rn eiy Xmk1

ank ib

nk

eiky

!; A:2

where the superscript in parenthesis n denotes the

iteration level and m is the degree of the polynomial

function m 0; 1;y:From Eq. (A.2) the parametric equations for the

coordinates xc;yc of the boundary of the opening in theOxy plane have as follows:

xc R cos y Xmk1

ank cos ky b

nk sin ky

( );

yc R sin y Xmk1

ank sin ky b

nk cos ky

( ): A:3

Also, from Eqs. (A.1) and (A.2) after some algebraic

manipulations the following formula for the angle a is

derived:

a tan1

sin y Pmk1 kank sin ky Pmk1 kbnk cos kycos y

Pmk1 ka

nk cos ky

Pmk1 kb

nk sin ky

!:

A:4

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 xs; y ys; sapspsb; A:5

Fig. 11. Definition of angle a that is formed between Ox and (r)

axeswith the latter being normal to the tangent axis (y) at some point

of the smooth curvemeasured in an anti-clockwise sense.

5Curves are called smooth when they have continuous first

derivatives, i.e. dxs=ds; dys=ds; inside their interval of definition.They are also called simple if xs1 xs2;ys1 ys2 areincompatible for saps1; s2psb; s1as2:



11/12

in which sa; sb are finite constants defining the interval ofthe curve, xs;ys are continuous functions in theinterval 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 y (Fig. 11) at any point of

the arc can be found by the formula

tB

dz

ds

dxs

ds i

dys

ds; A:6

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 r axis (Fig. 11) is then found from

the condition

nB

tB

0; A:7

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 x;y to xc;yc that is predicted by thepolynomial conformal mapping function.

Finally, the constant coefficients ak; bkk 1;y; mare found by any of the available constrained nonlinear

minimization routines of the sum d of the distances

between the predicted coordinates xc;yc and actualcoordinates x;y of the contour C of the opening, i.e.

d minXNj1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixj xcj

2 yj ycj2

qpe; A:8

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 fz should

increase accordingly. This is a formidable task in the

frame of the proposed general complex variable

formulation [14].

References

[1] Hoek E, Brown ET. Underground excavations in rock. London:

Institute of Mining and Metallurgy, 1980.

[2] Brady BHG, Brown ET. Rock mechanics for underground

mining. London: Allen & Unwin, 1985.

[3] Carranza-Tores C, Fairhurst C. The elastoplastic response of

underground excavations in rock masses that satisfy the Hoek-

Brown failure criterion. Int J Rock Mech Min Geomech Abstr

1999;36:777809.

[4] Sulem J, Panet M, Guenot A. An analytical solution for time-

dependent displacements in a circular tunnel. Int J Rock MechMin Geomech Abstr 1987;24(3):15564.

[5] Kirsch G. Die theorie der Elastizit.at und die bed .urfnisse der

festigkeitslehre. Zeit Ver Deut Ing J 1898;42:797807.

[6] Jaeger JC, Cook NGW. Fundamentals of rock mechanics, 2nd ed.

London: Chapman & Hall, 1976.

[7] Gerc)ek H. Stresses around tunnels with arched roof. Proceedings

of the seventh International Congress on Rock Mechanics, vol. 2.

Balkema, Rotterdam, The Netherlands: ISRM, 1991. p. 129799.

[8] Gerc)ek H. An elastic solution for stresses around tunnels with

conventional shapes. Int J Rock Mech Min Sci 1997;34(34),

paper No. 096.

[9] Muskhelishvili NI. Some basic problems of the mathematical

theory of elasticity. Groningen, The Netherlands: Noordhoof Ltd,

1963.

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.



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[10] Novozhilov VV. Theory of elasticity. New York: Pergamon Press,

1961.

[11] ITASCA. Fast Lagrangian analysis of continua (FLACZD),

Minnesota: Itasca Consulting Group, Inc., 1995.

[12] Savin GN. Stress concentration around holes. Oxford: Pergamon

Press, 1961.

[13] Panet M. Understanding deformations in tunnels. In: Hudson JA,

editor. Comprehensive rock engineering, vol. 1 (Fundamentals),

Oxford: Pergamon Press, 1993. pp. 663690.

[14] Exadaktylos GE, Liolios PA, Stavropoulou MC. A closed-form

elastic stress-displacement solution for notched circular openings

in rocks. Int J Solids Struct, 2002, submitted for publication.


exadaktylos 2002-aclosed-form elastic solution for stresses and displacements around tunnel

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