Rock Mech Rock Eng (2013) 46:1481–1498
DOI 10.1007/s00603-012-0362-7
ORIGINAL PAPER
Analytical Solutions for the Construction of Deeply Buried
Circular Tunnels with Two Liners in Rheological Rock
H. N. Wang • Y. Li • Q. Ni • S. Utili
M. J. Jiang • F. Liu
•
Received: 19 October 2012 / Accepted: 27 December 2012 / Published online: 21 February 2013
Springer-Verlag Wien 2013
Abstract The construction of underground tunnels is a
time-dependent process. The states of stress and strain in
the ground vary with time due to the construction process.
Stress and strain variations are heavily dependent on the
rheological behavior of the hosting rock mass. In this
paper, analytical closed-form solutions are developed for
the excavation of a circular tunnel supported by the construction of two elastic liners in a viscoelastic surrounding
rock under a hydrostatic stress field. In the solutions, the
stiffness and installation times of the liners are accounted
for. To simulate realistically the process of tunnel excavation, a time-dependent excavation process is considered
in the development of the solutions, assuming that the
radius of the tunnel grows from zero until its final value
according to a time-dependent function to be specified by
the designers. The integral equations for the supporting
pressures between rock and first liner are derived according
to the boundary conditions for linear viscoelastic rocks
(unified model). Then, explicit analytical expressions
are obtained by considering either the Maxwell or the
Boltzmann viscoelastic model for the rheology of the rock
mass. Applications of the obtained solutions are illustrated
using two examples, where the response in terms of
H. N. Wang (&) Y. Li
School of Aerospace Engineering and Applied Mechanics,
Tongji University, Shanghai 200092, People’s Republic of China
e-mail: wanghn@tongji.edu.cn
Q. Ni S. Utili
School of Engineering, University of Warwick,
Coventry CV4 7AL, UK
M. J. Jiang F. Liu
Department of Geotechnical Engineering, College of Civil
Engineering, Tongji University, Shanghai 200092,
People’s Republic of China
displacements and stresses caused by various combinations
of excavation rate, first and second liner installation times,
and the rheological properties of the rock is illustrated.
Keywords Tunnels Rheological rock Liner Analytical research
List of Symbols
A, B
a
D1, D2 … D12
EL1 (EL2 )
Ei ðGi Þ
fB1 ðfB2 Þ
fM1 ðfM2 Þ
GB1
GB
GL1 ðGL2 Þ
GM
GK
G
H
Laplace transform functions of s
Function of the excavation
process
Constant coefficients
Young’s modulus of the first
(second) liner
The ith Young’s (shear) modulus
in the general viscoelastic model
Free term of integral equation
Free term of integral equation
Shear elastic modulus of the
Hookean element in the
Boltzmann model
Shear elastic modulus of the
Kelvin element in the Boltzmann
model
Shear elastic modulus of the first
(second) liner
Shear elastic modulus of the
Hookean element in the Maxwell
model
Shear elastic modulus of the
Kelvin element in the Burgers
model
Relaxation shear modulus in the
rock viscoelastic model
Function defined in Eq. (13)
123
1482
H. N. Wang et al.
I
kM ðkM
Þ
kB ðkB Þ
kMi (i ¼ 1; 2; . . .; n)
KL1 ðKL2 Þ
KðtÞ
pðtÞ
p0
p1 ðp2 Þ
q
q1
R
R0
R1
R2 (R3 )
r; h; z
s
sij ðeij Þ
t
t0
t1 (t2 )
u0r
u1r
u2r
8
< u0r
ur ¼ u1r
: 2
ur
urL1 ðurL2 Þ
0 t\t1
t1 t\t2
t t2
v
WM and WB
Greek Symbols
mL1 ðmL2 Þ
dij
dðtÞ
Du1 ¼ u0r u1r
123
Function defined in Eq. (13)
Kernel of the integral equation
Kernel of the integral equation
Iterated kernel
Bulk elastic modulus of the first
(second) liner
Relaxation bulk modulus in the
rock viscoelastic model
Supporting pressure between first
liner and surrounding rock
Compressive stress at infinity
Supporting pressure pðtÞ during
the first (second) liner stage
Supporting pressure between first
and second liner
Supporting pressure qðtÞ during
the second liner stage
Tunnel radius
Initial radius of tunnel
Final radius of tunnel
Inner radius of the first (second)
liner
Cylindrical coordinates
Variable in the Laplace
transform
Deviator tensor of stress (strain)
tensors
Time
End time of excavation
Installation time of the first
(second) liner
Radial displacement in the
absence of liners (any t)
Radial displacement in the
presence of the first liner (t [ t1)
Radial displacement in the
presence of the second liner (t [ t2)
Radial displacement of the rock
Radial displacement of the first
(second) liner
Excavation rate
Kernel function of Eq. (35) and
Eq. (69)
Poisson’s ratio of the first
(second) liner
Unit tensor
Delta function
Reduction of radial
displacement in the rock due
Du2 ¼ u1r u2r
Du1þ ¼ ur ðr; tÞ ur ðr; t1 Þ
Du2þ ¼ ur ðr; tÞ ur ðr; t2 Þ
Du1tot ¼ ur ðr; t2 Þ ur ðr; t1 Þ
Du2tot ¼ ur ðr; 1Þ ur ðr; t2 Þ
k1 to k4
gM
gB
gi
rr ðrh Þ
rz
rrL1 ðrrL2 Þ
rhL1 ðrhL2 Þ
rij ðeij Þ
rmm ðemm Þ
to the presence of the first
liner (t [ t1)
Reduction of radial
displacement in the rock due
to the presence of the second
liner (t [ t2)
Incremental radial
displacement occurring
since the installation of the
first liner (t [ t1)
Incremental radial
displacement occurring
since the installation of the
second liner (t [ t2)
Total additional
displacement occurring
between the installation of
the two liners
Total additional displacement
occurring after the installation
of the second liner
Constant coefficients
Viscosity coefficient of the
dashpot element in the
Maxwell model
Viscosity coefficient of the
dashpot element in the
Boltzmann model
The ith viscosity coefficient
in the general viscoelastic
model
Radial (hoop) stress in the rock
z-Direction stress in the rock
Radial stress of the first
(second) liner
Hoop stress of the first
(second) liner
Stress (strain) tensor
Mean stress (strain)
1 Introduction
Most types of rocks exhibit time-dependent behavior.
During the excavation of a tunnel, the surrounding rock
deforms gradually due to the natural rock rheology. After
the excavation is complete, supports typically in the form
of concrete or shotcrete liners are put in place to seal the
rock and to reduce radial deformations. Displacements of
the surrounding rock and pressure between liner and rock
are critical parameters to be determined for tunnel design.
Therefore, proper simulation of the sequence of actions
involved in tunneling, including the whole process of
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels
excavation and support construction, is of great importance
to obtain a reliable tool for the determination of optimal
values of tunneling parameters to achieve optimal design.
This problem can be tackled either numerically or analytically. Numerical methods are widely used in many complex underground projects (Chehade and Shahrour 2008;
Prazeres et al. 2012). Although numerical methods such as
finite element, finite difference, discrete element, and
boundary element can provide useful answers, the insight into
the nature of the problem that can be gained from analytical
solutions is an important aspect that cannot be overlooked
(Carranza-Tores and Fairhurst 1999). In fact, analytical
solutions in closed form can be used to obtain a first estimation of the design parameters, providing guidance in the
conceptual stage of the design process. They make it possible
to run parametric analyses for a wide range of values of
the problem parameters so that a better understanding of the
physics of the problem can be gained by investigating the
sensitivity of the solution to the input parameters. In addition,
closed-form solutions provide a benchmark against which the
overall correctness of numerical analyses can be assessed.
Tunnel excavation is a long-term process which takes
place over a significant time, during which the tunnel face
advances at variable speed. If a cross-section of the tunnel is
considered, its radius grows over time according to the
excavation process employed. However, in the closed-form
analytical solutions presented in the literature to date, the
excavation process is assumed to be instantaneous. In this
paper, instead, the excavation time is accounted for so that
the influence of the excavation process on the final tunnel
convergence and pressure acting on the liners can be investigated by a parametric analysis. This allows optimization of
the excavation time, which thus becomes a design parameter.
Analytical solutions for stresses within a wedge-shaped
body subject to gravity were presented by Rashba (1953),
who carried out the earliest analytical studies on problems
with time-dependent geometry. The first analytical research
in civil engineering considering construction processes
considered stress analysis of filled culverts (Brown et al.
1968; Christiaon and Chuntranuluk 1974). Analytical
solutions for some problems with changing geometry were
also provided by Namov (1994). More recently, research
has been carried out to seek analytical solutions for viscoelastic problems involving time-dependent boundary
regions using the principle of correspondence (Cao 2000;
Wang and Nie 2010; Wang and Cao 2006). In tunnel construction, unlike the aforementioned research, the domain
of the problem decreases during the excavation stage, and
increases subsequently during the construction of the liners.
Furthermore, the surrounding ground material is rheological; i.e., its mechanical properties are time dependent.
These factors add to the complexity of the problem, making
it difficult to achieve analytical solutions in closed form.
1483
Unlike the case of linear elastic materials with constitutive equations in the form of algebraic equations, linear
viscoelastic materials have their constitutive relations
expressed by a set of operator equations. In general, it is
very difficult to obtain analytical solutions for most viscoelastic problems, especially in case of time-dependent
boundaries, although some closed-form or theoretical
solutions have been developed for excavations in rheological rock (Gnirk and Johnson 1964; Ladanyi and Gill 1984;
Brady and Brown 1985). However, in all these works,
excavation is assumed to take place instantaneously. Concerning the geomaterial–liner interaction, several studies
(Savin 1961; Peck et al. 1972; Einstein and Schwartz 1979)
have been carried out for excavations in elastic or viscoelastic rock. Sulem et al. (1987) presented an analytical
solution to determine the radial displacements and the
pressure applied on tunnel linings having assumed a Kelvin–Voigt model to describe the time-dependent behavior
of the surrounding rock. Using the complex variable method
and assuming the rock to be elastic, Wang and Li (2009)
achieved analytical solutions for the fields of stress and
displacement around a lined circular tunnel, accounting for
both misfit and the interaction between liner and the surrounding geomaterial. In previous research (Sulem et al.
1987; Ahmad et al. 2010), the effect of the progressive
advancement of the tunnel face along the longitudinal
direction was accounted for. Assuming an isotropic stress
state and a viscoelastic Burgers model for the rock, Nomikos et al. (2011) derived analytical solutions in closed form
and performed a parametric study on the effect of the liner
parameters. Different supports such as sprayed liners, twoliner systems, and anchor-grouting support were analyzed
by Mason and Stacey (2008) and Mason and Abelman
(2009). Liners were assumed to be instantaneously applied
at the end of the excavation. In tunnel practice, however,
liners may be installed at any time after excavation, which is
the case considered in this paper.
In summary, in the current literature on the construction
of lined tunnels in viscoelastic rock, the process of excavation is ignored and only the longitudinal advancement of the
tunnel face is considered by introducing a fictitious lining
pressure so that the problem can be mathematically cast as a
fixed boundary problem. Many problems of linear viscoelasticity can be solved using the principle of correspondence (Lee 1955; Christensen 1982; Gurtin and Sternberg
1962). However, the cross-section of a tunnel is excavated in
stages, which implies a time-dependent geometrical domain,
so the principle of correspondence cannot be employed.
In this paper, an axisymmetric tunnel excavated in rheological rock supported first by a primary liner, then by a
secondary one is analyzed. The rheological properties of
the rock mass are accounted for within a unified viscoelastic model. Also a time-dependent excavation process is
123
1484
H. N. Wang et al.
considered in the development of the solutions, assuming
that the radius of the tunnel grows from zero until its final
value according to a time-dependent function to be specified
by designers. Although the obtained analytical solutions are
rigorously applicable only to the axisymmetric case, i.e., a
single deeply buried tunnel, Schuerch and Anagnostou
(2012) demonstrated that solutions achieved for axisymmetric conditions are still valid for a wide range of different
ground conditions and for several cases of noncircular
tunnels with small error. In case of twin tunnels, the solution
is applicable only if the distance between the tunnels is such
that the influence of the presence of one tunnel on the stress
state of the rock around the other is negligible.
In this paper, first the proposed analytical solutions are
derived, then two examples are considered to illustrate the
influence of the excavation process and the installation
times of the liners on the obtained time-dependent stresses
and displacements.
2 Definition of the Problem
Excavation of an axisymmetric circular tunnel in a rheological
rock mass is considered. The following assumptions are made:
1.
2.
3.
The rock mass is homogeneous, isotropic, and linearly
viscoelastic.
The tunnel is first excavated and then supported by the
construction of two liners at different times. First the
radius of the tunnel grows from zero to R1, at the end
of the excavation; secondly support is provided by a
temporary concrete liner; third, after some time, a
second, permanent liner is installed. The problem is
cast as a two-dimensional (2D) infinite viscoelastic
plane subject to a hydrostatic uniform stress with a
circular opening whose radius varies with time.
The rate of excavation is small, so that it can be
assumed that it does not induce any dynamic stress.
In the analysis, the effect of the advancement of the
tunnel along the longitudinal direction is not accounted for.
This means that the cross-section considered in this analysis is at a sufficient distance from the tunnel face that
stresses and strains are unaffected by three-dimensional
effects. The tunneling process can be divided into three
stages. During the first stage (excavation stage) spanning
from time t ¼ 0 to t ¼ t1 , with t1 being the installation time
of the first liner, the radius of the tunnel varies as follows:
R0 þ aðtÞ 0 t t0
RðtÞ ¼
ð1Þ
R1
t [ t0
with RðtÞ being the time-dependent radius of the tunnel, R1
the final tunnel radius at the end of the excavation process,
R0 the initial radius ð0 R0 R1 Þ, and a ¼ aðtÞ being a
123
function accounting for the actual excavation process as
prescribed by the designers. t0 is the end time of the excavation. From t0 to t1, the rock is free to expand since no
liners are present. In this stage, no supporting pressure is
present. In order to release part of the pressure, the first liner
is often put in place after some time elapses from the end of
excavation. Hence, in this derivation, the first liner is built at
time t ¼ t1 , with p1 ðtÞ being the contact pressure between
rock and the first liner. The second stage (first liner stage)
spans from the installation of the first liner at t ¼ t1 to the
installation of the second liner at t ¼ t2 . The third stage
(second liner stage) spans from the installation of the second liner, at time t ¼ t2 , onwards. For t [ t2 , the contact
pressure between rock and the first liner is p2 ðtÞ with q1 ðtÞ
being the pressure between the first and the second liner.
3 Mechanical Analysis During the Construction
Process
3.1 Analysis of the Rock Mass
Cylindrical coordinates (r, h, z) are employed in the derivation of the analytical solution. As shown in Fig. 1, p0 is
the hydrostatic in situ stress. The boundary condition for
the rock mass is
rr ðRðtÞ; tÞ ¼ pðtÞ; rr ð1; tÞ ¼ p0 ðtÞ;
ð2Þ
8
0 t\t1
< 0
where pðtÞ ¼ p1 ðtÞ t1 t\t2 is an undetermined
:
p2 ðtÞ
t t2
function.
In rock mechanics, Hookean elastic springs and Newtonian viscous dashpots are used to model a variety of
rheological characteristics of the rock mass. Figure 2 provides a one-dimensional (1D) illustration of the general
Kelvin and Maxwell viscoelastic models with parameters
Ei and gi , to be experimentally determined (Flügge 1975).
To simulate more complex rock rheologies, additional
elastic springs or dashpots can be connected in parallel or
in series in the general Kelvin and Maxwell models. The
constitutive equations of a general viscoelastic model can
be expressed in the form of convolution integrals as
sij ðr; tÞ ¼ 2GðtÞ deij ðr; tÞ;
ð3aÞ
rmm ðr; tÞ ¼ 3KðtÞ demm ðr; tÞ
ð3bÞ
with sij and eij the deviator tensors of the stress and strain
tensors, rij and eij , respectively. By definition,
1
sij ¼ rij dij rmm ;
3
1
eij ¼ eij dij emm :
3
ð4aÞ
ð4bÞ
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels
1485
Fig. 1 Boundary conditions for
rock and liners
r
R (t )
p0 (t )
θ
q(t)
R1
R2
p1 (t )
p(t)
p0 (t )
first liner ( t > t1 )
rock
R3
R2
q(t)
second liner (t > t 2 )
Fig. 2 General viscoelastic
physical models
E1
En
η1
ηn
f1 ðtÞ df2 ðtÞ ¼ f1 ðtÞ f2 ð0Þ þ
η1
ηn
(b) General Maxwell viscoelastic physical
model
model
df2 ðsÞ
ds:
f1 ðt sÞ
ds
ð5Þ
For the case of axisymmetric deformations under planestrain conditions, the general solution for the radial
displacement and the three stress components of a rock mass
in the Laplace space can be written as (Wang and Nie 2010)
AðsÞ
þ rBðsÞ;
r
ð6Þ
and
1
1
rr ¼ 2GðtÞ s 2 AðsÞ þ BðsÞ þ 2KðtÞ sBðsÞ;
r
3
1
1
rh ¼ 2GðtÞ s 2 AðsÞ þ BðsÞ þ 2KðtÞ sBðsÞ;
r
3
2
rz ¼ 2 KðtÞ GðtÞ sBðsÞ:
3
f ðsÞ ¼
Z1
est f ðtÞdt;
0
the inverse transform of which is
1
L ½f ðsÞ ¼ f ðtÞ ¼
2pi
1
0
ur ¼
En
(a) General Kelvin viscoelastic physical
G(t) and K(t) are the so-called relaxation moduli, which
can be expressed by Ei (or Gi ) and gi in the viscoelastic
model. The asterisk (*) in Eq. (3) indicates the convolution
integral, the definition of which is
Zt
E1
ð7Þ
AðsÞ and BðsÞ are two undetermined functions of the
transform parameter s defined in the Laplace transform of a
time function, f ðtÞ, as
bþi1
Z
f ðsÞest dt:
bi1
The inverse transform of Eq. (7) is
2 1
1
1
rr ¼ 2 L ½GðtÞ sAðsÞ þ 2L
GðtÞ þ KðtÞ BðsÞs
r
3
2 1
1
1
rh ¼ 2 L ½GðtÞ sAðsÞ þ 2L
GðtÞ þ KðtÞ BðsÞs
r
3
h
i 4
h
i
rz ¼ 2L1 KðtÞ sBðsÞ L1 GðtÞ sBðsÞ :
3
ð8Þ
According to the boundary condition of Eq. (2), the
functions A(s) and B(s) can be determined as
i
1 h
AðsÞ ¼ p0 R2 ðtÞ pðtÞR2 ðtÞ ;
ð9Þ
2sGðtÞ
BðsÞ ¼ h
3p0
2s GðtÞ þ 3KðtÞ
i:
ð10Þ
123
1486
H. N. Wang et al.
Substituting Eqs. (9) and (10) into Eq. (8) provides the
explicit form for the radial and hoop stresses in the rock as
R2 ðtÞ
pðtÞR2 ðtÞ
rr ¼ p0 ðtÞ 1 2 ;
r
r2
ð11Þ
R2 ðtÞ
pðtÞR2 ðtÞ
:
rh ¼ p0 ðtÞ 1 þ 2 þ
r
r2
and
8
< Zt
1
1
p0 ðsÞR2 ðsÞHðt sÞds
ur ðr; tÞ ¼
2r :
0
9
Zt
=
for t1 \t t2 ; ð16bÞ
þR21 p1 ðsÞHðt sÞds
;
t1
The radial displacement in the time domain is
1
ur ¼ L1 ½AðsÞ þ rL1 ½BðsÞ:
r
Assuming
"
HðtÞ L
#
1
1
ð12Þ
"
1
sGðtÞ
IðtÞ L
#
1
1
;
s GðtÞ þ 3KðtÞ
and
8
< Zt
1
2
p0 ðsÞR2 ðsÞHðt sÞds
ur ðr; tÞ ¼
2r :
0
ð13Þ
þR21
then according to the properties of the convolution integral
of the Laplace transform, Eqs. (9) and (10) can be
simplified to
1
L ½AðsÞ ¼ HðtÞ ½p0 R2 ðtÞ pðtÞR2 ðtÞ
2
Zt
1
¼
p0 R2 ðsÞHðt sÞds
2
1
Zt2
t1
p1 ðsÞHðt sÞ ds þ R21
Zt
p2 ðsÞHðt sÞds
t2
for t [ t2 :
9
=
;
ð16cÞ
Displacements in the rock mass at a generic time t [ t1
can be calculated as long as the supporting pressures p1 ðtÞ
and p2 ðtÞ are known.
3.2 Mechanical Analysis of the Liners
0
1
þ
2
Zt
Liners are made of concrete. Here, the Young’s moduli EL1
and EL2 , and the Poisson’s ratios mL1 and mL2 are considered
for the first and second liner, respectively. According to
elasticity theory, the radial displacement of the first liner
subject to the stress boundary conditions outlined in Fig. 1 is
2
pðsÞR ðsÞHðt sÞds
0
3
3
L ½BðsÞ ¼ IðtÞ p0 ðtÞ ¼ 2
2
1
Zt
p0 ðsÞIðt sÞds:
0
ð14Þ
Substituting Eq. (14) into Eq. (12) yields
8
Zt
1<
p0 ðsÞR2 ðsÞHðt sÞds
ur ¼
2r :
0
9
Zt
= 3 Zt
2
þ pðsÞR ðsÞHðt sÞds r p0 ðsÞIðt sÞds:
; 2
0
0
ð15Þ
Equation (15) provides the general expression for the
radial displacements. If the rock mass is incompressible,
that is, KðtÞ ¼ 1, no displacements occur before
excavation. Considering now the installation of the first
liner at time t1 and the second liner at time t2, the
expression becomes
u0r ðr; tÞ ¼ 1
2r
Zt
with
ð17Þ
where R2 is the inner radius of the first liner, qðtÞ ¼
0
t1 t\t2
is the pressure between the liners, and
q1 ðtÞ
t t2
EL1
EL1
and KL1 ¼ 12m
are the shear and bulk
GL1 ¼ 2ð1þm
L1 Þ
L1
elastic moduli of the first liner, respectively. The
expression for the radial displacement undergone by the
second liner is
p0 ðsÞR2 ðsÞHðt sÞds
for 0 t t1 ;
ð16aÞ
1
R 2 R2
1 þ mL2 R22 q1 ðtÞ
2 2 3 2 q1 ðtÞ 2
r
2GL2 r R2 R3
KL2
R2 R23
t t2 ;
urL2 ðr; tÞ ¼ with
0
123
1
R 2 R2
1 þ mL1
2 1 2 2 ½pðtÞ qðtÞ 2GL1 r R1 R2
KL1
R21 pðtÞ R22 qðtÞ
r
R21 R22
t t1 ;
urL1 ðr; tÞ ¼ ð18Þ
where R3 is the inner radius of the second liner, and GL2
and KL2 are the shear and bulk elastic moduli of the
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels
second liner, respectively. The stresses acting on the first
liner are
R2
R22
R22
R21
rrL1 ¼ 1 21
qðtÞ
1
pðtÞ;
2
2
2
2
r R1 R2
r R1 R22
R21
R22
R22
R21
rhL1 ¼ 1 þ 2
qðtÞ 1 þ 2
pðtÞ;
2
2
2
r R1 R2
r R1 R22
1487
R21
p1 ðtÞ ¼ 2D1 þ 2D2 R21
p1 ðsÞHðt sÞds
t1
8 t
< Z1
1
p0 ðsÞR2 ðsÞHðt1 sÞds
þ
2D1 þ 2D2 R21 :
0
9
Zt
=
þ p0 ðsÞR2 ðsÞHðt sÞds :
ð24Þ
;
ð19Þ
whilst the stresses on the second liner are
R2
R22
q1 ðtÞ;
rrL2 ¼ 1 23
2
r R2 R23
R2
R22
q1 ðtÞ:
rhL2 ¼ 1 þ 23
2
r R2 R23
Zt
0
ð20Þ
Specifying the function H = H(t) according to the
viscoelastic model of interest, the analytical expression
for the supporting pressure acting after the construction of
the first liner can be obtained from Eq. (24).
3.3 Determination of the Supporting Pressure After
Construction of the First Liner
3.4 Determination of the Supporting Pressure After
Construction of the Second Liner
Having already imposed the boundary condition on the
stresses at the interface between the first liner and the rock,
the only boundary condition left concerns the displacements:
In the derivation of the solution, two boundary compatibility conditions will be imposed; the first one is at the
boundary between rock and the first liner:
ur ðR1 ; tÞ ur ðR1 ; t1 Þ ¼ urL1 ðR1 ; tÞ with t t1 :
ur ðR1 ; tÞ ur ðR1 ; t1 Þ ¼ urL1 ðR1 ; tÞ with
ð21Þ
According to Eq. (16), the increment of radial
displacement in the rock from time t1 until a generic
time t with t\t2 is
ur ðr; tÞ ur ðr; t1 Þ
8 t
Z1
1<
¼
p0 ðsÞR2 ðsÞHðt1 sÞds
2r :
p0 ðsÞR2 ðsÞHðt sÞds þ R21
Zt
t1
0
whilst the second one is at the boundary between the two
liners:
urL1 ðR2 ; tÞ urL1 ðR2 ; t2 Þ ¼ urL2 ðR2 ; tÞ with
9
=
p1 ðsÞHðt sÞds :
;
Substituting into Eq. (21) yields the following secondtype Volterra integral equation:
8 t
Z1
Zt
1 <
2
p0 ðsÞR ðsÞHðt1 sÞds p0 ðsÞR2 ðsÞHðt sÞds
2R1 :
0
0
9
Zt
=
þR21 p1 ðsÞHðt sÞds
;
t1
1
R1 R 2
1 þ mL1
R3
¼
2 2 2 p1 ðtÞ 2 1 2 p1 ðtÞ:
2GL1 R1 R2
KL1 R1 R2
ð23Þ
the
parameters
R2 R2
D1 ¼ 2G1L1 R21R22
1
L1
D2 ¼ 1þm
KL1 R21
R21 R22
ð26Þ
0
and
Zt
2
p0 ðsÞR ðsÞHðt sÞds þ
0
þR21
Zt
t2
R21
Zt2
p1 ðsÞHðt sÞds
t1
9
=
p2 ðsÞHðt sÞds :
;
ð27Þ
Substituting Eq. (27) into Eq. (25), the following is
obtained:
8 t
Z1
Zt
1 <
2
p0 ðsÞR ðsÞHðt1 sÞds p0 ðsÞR2 ðsÞHðt sÞds
2R1 :
0
0
9
Zt2
Zt
=
þR21 p1 ðsÞHðt sÞds þ R21 p2 ðsÞHðt sÞds
;
t1
2
, and rearranging Eq. (23) for p1 ðtÞ, the
following is obtained:
t t2 :
ur ðr; tÞ ur ðr; t1 Þ
8 t
Z1
1<
¼
p0 ðsÞR2 ðsÞHðt1 sÞds
2r :
ð22Þ
Defining
ð25Þ
The increment of radial displacement in the rock from
time t1 until a generic time t, with t [ t2 , is
0
Zt
t t2 ;
¼
t2
D1
½p2 ðtÞ q1 ðtÞ ½D2 p2 ðtÞ D3 q1 ðtÞ R1 ;
R1
ð28Þ
123
1488
H. N. Wang et al.
R2
1
unknown
functions.
2
R2
L2
2
D4 ¼ 1þm
KL2 R2 R2 ; D5 ¼
Defining
2
R22 R23
1
2GL2 R22 R23 ,
ηM
E M (G M )
L1
2
where D3 ¼ 1þm
KL1 R2 R2 , q1 ðtÞ and p2 ðtÞ being as yet
3
and substituting Eqs. (17) and (18) into
Eq. (26) yields
D1
D1
½p2 ðtÞ q1 ðtÞ ½D2 p2 ðtÞ D3 q1 ðtÞ R2 þ
p1 ðt2 Þ
R2
R2
D5
þ D2 p1 ðt2 Þ R2 ¼ q1 ðtÞ D4 q1 ðtÞ R2 :
ð29Þ
R2
D þD R2
1
2 2
Defining D6 ¼ D5 þD1 þD
2
2 , then rearranging, one
4 R þD3 R
2
2
Fig. 3 Maxwell viscoelastic physical model
Maxwell model and incompressible behavior, the constitutive parameters for the rock (see Eq. 3) become
G
GðtÞ ¼ GM e
ð30Þ
Substituting Eq. (30) into Eq. (28), the equation for p2 ðtÞ
is obtained as
M
;
KðtÞ ¼ 1:
ð32Þ
Substituting Eq. (32) into Eq. (13) yields
HðtÞ ¼
obtains
q1 ðtÞ ¼ D6 ½p2 ðtÞ p1 ðt2 Þ:
gMt
1
1
dðtÞ þ
;
GM
gM
IðtÞ ¼ 0:
ð33Þ
The radial displacement in the absence of liners is
2
3
Zt
2
1
p
ðtÞR
ðtÞ
1
0
þ
p0 ðsÞR2 ðsÞds5 for any t:
u0r ðr; tÞ ¼ 4
2r
GM
gM
0
R21
p2 ðtÞ ¼
2ðD1 D1 D6 D2 R21 þ D3 D6 R21 Þ
Zt
p2 ðsÞHðt sÞds
ð34Þ
4.1 Supporting Pressure and Displacements After
Construction of the First Liner
t2
Substituting Eq. (33) into Eq. (24) and defining k1 ¼
1
2ðD1 D1 D6 D2 R21 þ D3 D6 R21 Þ
8 t
<Z 1
p0 ðsÞR2 ðsÞHðt1 sÞds
:
Gg M D17 and D7 ¼ 1 þ 2GM DR21 þ 2GM D2 , the standard
þ
M
p1 ðtÞ ¼ k1
0
Zt
0
p0 ðsÞR
2
ðsÞHðt sÞds þ R21
Zt2
Zt
kM ðt; sÞ p1 ðsÞds þ fM1 ðtÞ:
ð35Þ
t1
p1 ðsÞHðt sÞds
t1
2D1 D6 p1 ðt2 Þ 2D3 D6 R21 p1 ðt2 Þ
1
integral equation becomes
)
:
ð31Þ
Hence, the equations for the supporting pressures p2 ðtÞ
and q1 ðtÞ after construction of the second liner are given in
Eqs. (30) and (31) by specifying the function H = H(t)
according to the viscoelastic model of interest.
The kernel of this integral equation is kM ðt; sÞ 1, and
the free term is fM1 ðtÞ k1 p0 ðt t1 Þ. According to the
theory of integral equations (Chambers 1976), the iterated
kernel can be determined by iteration as
kM1 ðt; sÞ ¼ kM ðt; sÞ ¼ 1;
Zt
kM2 ðt; sÞ ¼ kM ðt; uÞ kM1 ðu; sÞdu ¼ t s;
s
kM3 ðt; sÞ ¼
Zt
kM ðt; uÞ kM2 ðu; sÞdu ¼ ðt sÞ2 =2; . . .;
s
4 Analytical Solution for the Maxwell Model
Let us consider weak, soft or highly jointed rock masses,
and/or rock masses subject to high stresses, which are
prone to excavation-induced continuous viscous flows. The
Maxwell viscoelastic model (Fig. 3) is suitable to simulate
their rheology, since it accounts for both primary and
secondary rock creep. Now, assuming the validity of the
123
kMn ðt; sÞ ¼ ðt sÞn1 =ðn 1Þ!
ð36Þ
Then, the kernel function of Eq. (35) can be written
as
WM ðt; s; k1 Þ ¼
1
X
kn1
1 kMn ðt; sÞ ¼
n¼1
k1 ðtsÞ
¼e
1
X
n¼1
:
kn1
1
ðt sÞn1
ðn 1Þ!
ð37Þ
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels
Finally, the solution of Eq. (35) can be expressed as
p1 ðtÞ ¼ fM1 ðtÞ þ k1
Zt
WM ðt; s; k1 ÞfM1 ðsÞds
¼ k1 p0 ðt t1 Þ p0
Zt
ek1 ðtsÞ ðs t1 Þds:
p2 ðtÞ ¼ k2
t1
ð38Þ
Then, after integration and simplifying, the following
solution is obtained:
p1 ðtÞ ¼ p0 p0 e
k1 ðtt1 Þ
ð39Þ
:
Substituting Eq. (39) into Eq. (16), the radial
displacement of the rock mass at a time t with t1 t\t2
can be expressed as
p0
GM
GM
R21 ek1 ðtt1 Þ 1 þ
u1r ðr; tÞ ¼ R21
2GM r
gM k1
gM k1
Zt0
o
GM
GM 2
þ
R2 ðsÞds þ
R1 ðt1 t0 Þ :
ð40Þ
gM
gM
0
Hence, the obtained displacement varies with time
according to an exponential function. Substituting Eq. (39)
into Eq. (11) yields the following expressions for the stress
8
2
< p0 1 R ðtÞ
with 0 t t1
r2
rr ¼
;
:
R2
p0 þ p0 ek1 ðtt1 Þ r21 with t [ t1
8
ð41Þ
2
< p0 1 þ R ðtÞ
with 0 t t1
r2
rh ¼
:
R2
p0 p0 ek1 ðtt1 Þ r21 with t [ t1
Now considering the case of an instantaneous
excavation, that is, R(t) = R1 (with t [ 0) and t0 = 0, the
expressions for the displacements and stresses reduce to
p0 R21
GM
GM
GM
k1 ðtt1 Þ
e
1þ
þ
t1
ur ðr; tÞ ¼ 2GM r
gM k1
gM k1 gM
with t 0;
ð42Þ
rr ¼ p0 þ p0 ek1 ðtt1 Þ with
t 0:
R21
r2
4.2 Supporting Pressure and Displacements
After Construction of the Second Liner
Substituting the constitutive parameters into Eq. (31), the
integral equation for p2 ðtÞ can be obtained as
t1
k21
; rh ¼ p0 p0 ek1 ðtt1 Þ 1489
R21
r2
ð43Þ
The solution appearing in Eq. (42), which is valid only
for the particular case of an instantaneous excavation,
coincides with the solution provided by Nomikos et al.
(2011), who considered a Burgers viscoelastic model for
the rock mass, in case the spring of the Kelvin element is
infinitely stiff, GK ! 1.
Zt
kM
ðt; sÞ p2 ðsÞds þ fM2 ðtÞ;
ð44Þ
t2
where
kM
ðt; sÞ 1,
fM2 ðtÞ k2 p0 ðt t2 Þ þ kk21 D
p0 ½1 ek1 ðt2 t1 Þ þ D89 p1 ðt2 Þ, k2 ¼ Gg M D19 ;
M
D8 ¼ 2GM DR1 D2 6 2GM D3 D6 ; and D9 ¼ D7 þ D8 : By
1
using the same method outlined in Sect. 4.1, the following
is obtained:
p2 ðtÞ ¼ p0 p0 ek1 ðt2 t1 Þþk2 ðtt2 Þ :
ð45Þ
According to Eq. (30),
q1 ðtÞ ¼ D6 p0 ek1 ðt2 t1 Þ ½ek2 ðtt2 Þ þ 1;
ð46Þ
so that the displacement of the rock at a generic time t with
t t2 can be obtained as
p0
GM
2
2 k1 ðt2 t1 Þþk2 ðtt2 Þ
R1 e
ur ðr; tÞ ¼ 1þ
2GM r
gM k2
2
GM R1 k1 ðt2 t1 Þ 1
1
þ
e
k1 k2
gM
Zt0
GM
GM 2
2 GM
R1
þ
R2 ðsÞds þ
R1 ðt1 t0 Þ :
gM k1 g M
gM
0
ð47Þ
Now, for tunnel engineers it is of interest to know
the total additional displacement occurring after the
construction of the second liner, i.e.,
R2 p0 GM
Du2tot ¼ ur ðr; 1Þ ur ðr; t2 Þ ¼ 1
þ 1 ek1 ðt2 t1 Þ ;
2GM r gM k2
ð48Þ
and the total additional displacement occurring between the
installation of the two liners, i.e.,
Du1tot ¼ ur ðr; t2 Þ ur ðr; t1 Þ
h
i
R2 p0 GM
¼ 1
þ 1 ek1 ðt2 t1 Þ 1 :
2GM r gM k1
ð49Þ
Substituting Eq. (46) into Eq. (11), the stresses arising after
the construction of the second liner can be determined as
R21 k1 ðt2 t1 Þþk2 ðtt2 Þ
e
;;
r2
R2
rh ¼ p0 p0 21 ek1 ðt2 t1 Þþk2 ðtt2 Þ
r
with t t2 :
rr ¼ p0 þ p0 ð50Þ
123
1490
H. N. Wang et al.
0.18
4.3 Results and Discussion
v=1 m .day -1
0.16
An in situ stress at infinity of p0 ¼ 15 MPa is assumed.
In the example excavation process considered here, the
radius of the tunnel grows from R0 ¼ 1 m until R1 ¼ 6 m at
the end of the excavation process. The thicknesses of the
first and second liner are 100 and 200 mm, respectively, so
that R2 ¼ 5:9 m and R3 ¼ 5:7 m: For convenience of
notation, the absolute value of the radial displacement
will be presented in all the following figures, since radial
displacements are always opposite to the radial coordinate
r (i.e., the tunnel exhibits convergence).
4.3.1 Influence of the Excavation Rate
First, the displacement progression over time is investigated for three values of excavation rate: 0:625 m/day,
1 m/day, and 5 m/day, corresponding to total excavation
periods of 8, 5, and 1 day, respectively. In this example,
the first liner is installed immediately after completion of
the excavation process to reduce displacements as much
as possible. The second liner instead is installed after a
time interval of 7 days from the installation of the first
liner.
In Fig. 4, the radial displacements calculated at r ¼ R1
are plotted for the aforementioned values of excavation
rate. The asterisk and filled-triangle symbols indicate the
installation of the first and second liner, respectively. The
general trend is that the radial displacement increases over
time and reaches a constant value after around 30 days.
123
0.14
0.12
ur [m]
To illustrate the influence of the excavation process and the
installation times of the liners on the resulting stresses
and displacements, an example is presented herein. Backanalysis is often used to identify the most suitable rheological model for the rock and its related parameters. Yang
et al. (2001) derived rheological models and their constitutive parameters from in situ measurements during the
excavation of a long tunnel, whereas Feng et al. (2006)
derived them using genetic algorithms applied to the results
of rock creep tests. Considering these experimental results,
we have adopted the following values for the two parameters of the Maxwell model: GM ¼ 2; 000 MPa and gM ¼
4; 000 MPa day; which lie in the ranges of values identified
by Feng et al. (2006). Now, with regard to the elastic
parameters for the concrete liners, GL1 ¼ GL2 ¼
10; 000 MPa and mL1 ¼ mL2 ¼ 0:2 have been assumed;
hence, the elastic bulk moduli result to be KL1 = KL2 =
40; 000MPa: An excavation process with a linear increase
of radius is considered here, i.e.,
R0 þ vt 0 t t0
RðtÞ ¼
ð51Þ
R1
t [ t0
0.1
0.08
v=5 m .day-1
0.06
v=0.625 m .day-1
0.04
0.02
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
t [day]
Fig. 4 Radial displacement versus time for various excavation rates;
asterisks indicate the installation times of the first liner, whilst filled
triangles indicate the installation times of the second liner. In all
cases, the first liner is installed at t = t0 = t1
The results show that a lower excavation rate implies a
longer excavation time, which leads to a larger value of
displacement at the end of the excavation process at time
t1 = t0, since the tunnel is immediately supported by the
first liner after completion of the excavation. In the following two stages (t1 t\t2 and t [ t2 ), there is no significant difference among the three curves in terms of
additional displacement. Overall, faster excavation rates
imply lower total displacements at the end of every stage,
and also in the long term.
In Fig. 4 it can be observed that the three curves intersect at around the 17th day: at the beginning (before the
17th day), high excavation rates imply larger displacements, whereas later on (after the 17th day), the opposite is
true. So, it emerges that a fast excavation leads to high
rates of displacement early on. However, the final stable
state is reached earlier, with the final total displacement
being lower than the case of slow excavation, as can be
expected.
In Fig. 5, the stress responses of the rock calculated at
the interface between rock and the first liner, r = 6 m,
where the highest hoop stress is expected to develop, are
displayed for the analyzed excavation rates. In all three
cases considered, the stresses are the same at the end of
excavation and at the time of installation of the second
liner. However, for lower excavation rates, the change in
stress is more gradual. Absolute values of radial and circumferential stresses reach their minimum and maximum
at the end of the excavation phase, and would remain
unchanged if no liners were installed. After installation of
the first liner, the radial stress increases whereas the circumferential one decreases, so that the rock mass becomes
subject to a more isotropic state of stress, which is beneficial for the stability of the tunnel.
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels
0
-14
-2
-16
v=0.625 m .day -1
-4
-18
-6
v=1m .day -1
σθ [MPa]
σr [MPa]
1491
-8
-10
v=5 m .day-1
v= 0.625 m .day -1
-22
v=1 m.day -1
-24
-12
-26
-14
-28
-16
v=5 m . day -1
-20
-30
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
t [day]
t [day]
Fig. 5 Stresses calculated at the interface between rock and the first liner (r = 6 m) versus time for various excavation rates. Asterisks indicate
the installation times of the first liner, whilst filled triangles indicate the installation times of the second liner
0.25
t1=15 day
0.2
t1=12 day
t1=10 day
Du1þ ðr; tÞ u1r ðr; tÞ u1r ðr; t1 Þ
i
p0 R21
GM h k1 ðtt1 Þ
1þ
1
¼
e
2GM r
gM k1
ur [m]
with
0.15
t1=8 day
0.1
0.05
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
t [day]
ð53Þ
t1 t\t2 :
Here, the influence of the installation time of the first liner
was analyzed for an excavation rate of 0:625 m/day; which
implies an excavation period of 8 days ðt0 ¼ 8Þ: Four
options were considered, with the first liner installed on the
8th, 10th, 12th, and 15th day. In all the cases, the second liner
is installed on the 15th day. In Fig. 6, the resulting radial
displacements, calculated at the interface between rock and
the first liner, are displayed. It can be noted that earlier
Fig. 6 Radial displacement versus time for various installation times
of the first liner ðv ¼ 0:625 m/dayÞ: Asterisks indicate the installation
times of the first liner
0.08
0.07
4.3.2 Influence of the First Liner Installation Time
Du1 ðr; tÞ
u0r ðr; tÞ u1r ðr; tÞ
p0 R21
k1 ðtt1 Þ
¼
GM
1þ
gM k1
Δ u-2 [m]
The analytical expression for the reduction of the radial
displacements in the rock due to the presence of the first
liner results as
t1 =10 day
0.06
0.05
t1=6 day
0.04
t1=14 day
0.03
1e
2GM r
GM
GM
þ
þ
ðt t1 Þ ; with t1 t\t2 : ð52Þ
gM k1 gM
Du1 depends on the installation time of the first liner.
The earlier the first liner is put in place, the larger the
reduction of displacement will be. The analytical expression
for the incremental displacement occurring since the
installation of the first liner results as
0.02
0.01
0
6
8 10 12 14 16 18 20 22 24 26 28 30
t2 [day]
Fig. 7 Reduction of the radial displacement in the rock induced by
the second liner calculated at the interface between rock and the first
liner (r = 6 m) for various installation times of the first liner versus
the installation time of the second liner (t2)
123
1492
H. N. Wang et al.
installation of the liner reduces the rate of displacement, and
also leads to a shorter time to reach the stable state.
4.3.3 Influence of the Second Liner Installation Time
The analytical expression for the reduction of the radial
displacements in the rock due to the presence of the second
liner results as
Du2 ðr; tÞ u1r ðr; tÞ u2r ðr; tÞ
i
p0 R21 ek1 ðt2 t1 Þ h k2 ðtt2 Þ
e
ek1 ðtt2 Þ
¼
2GM r
GM ek1 ðtt2 Þ ek2 ðtt2 Þ
GM 1
1
þ
;
gM
k1
k2
gM k1 k2
ð54Þ
t t2 :
with
According to Eq. (54), it emerges that Du2 depends on
the time interval between the installation of the first and the
second liner, i.e., t2 t1 . Earlier installation of the first
liner leads to smaller reduction of displacements due to
the installation of the second liner. In Fig. 7, curves
representing the reduction of displacement versus the
installation time of the second liner, t2, are plotted for three
values of t1. From the figure, it emerges that, in order to
minimize deformations, the second liner has to be put in
place as soon as possible. Conversely, postponing the
installation of the second liner leads to more pressure
acting on the rock and the first liner, so that a more
economical design of the second liner can be achieved.
Now we consider the case of the first liner being built
immediately after completion of the excavation process on
the 8th day, with the second liner being installed at
0.2
t 2= 20 day
t 2= 15 day
ur [m]
0.15
different dates, namely the 8th, 15th, and 20th day. In
Fig. 8, the displacements calculated at the interface
between rock and the first liner (r ¼ 6 m) are displayed.
Early installation of the second liner leads to smaller displacements, with the stable state reached earlier.
In Fig. 9, the stresses in the rock at the interface between
rock and the first liner are plotted for the following time
intervals between the construction of the two liners: 2, 6, and
10 days, with v = 0.625 m/day and t0 = 8 days. It is worth
noting that early installation of the second liner leads to a
smaller radial stress and a larger circumferential one.
The incremental displacement occurring since the
installation of the second liner is
Du2þ ðr; tÞ u2r ðr; tÞ u2r ðr; t2 Þ
i
p0 R21 ek1 ðt2 t1 Þ
GM h k2 ðtt2 Þ
¼
1þ
1
e
2GM r
gM k2
with t t2 :
ð55Þ
This incremental displacement depends on the time
period t2 t1 and on the installation time of the second
liner. The larger t2 is, the larger the value of Du2þ will be.
5 Analytical Solution for the Boltzmann Model
For rock masses with good mechanical properties or subject to low stresses, the exhibited mechanical behavior
shows limited viscosity. For this type of behavior, the
Boltzmann viscoelastic model (Fig. 10) is commonly
employed. The material parameters adopted in the model
are the two elastic shear moduli GB1 and GB and the viscosity coefficient gB . Assuming that the rock is incompressible, the two relaxation moduli appearing in the
constitutive equations (see Eq. 3) are as follows:
GðtÞ ¼
G þG
G2B1
GB GB1
B1 B t
e gB þ
;
GB1 þ GB
GB1 þ GB
KðtÞ ¼ 1:
ð56Þ
t 2 =8 day
0.1
Substituting Eq. (56) into Eq. (13) yields
HðtÞ ¼
0.05
1
1 GB t
dðtÞ þ e gB ; IðtÞ ¼ 0:
GB1
gB
ð57Þ
Substituting Eq. (57) into Eq. (15), and according to the
properties of the Laplace transform, the analytical
expression for the radial displacement can be derived as
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
t [day]
Fig. 8 Radial displacement versus time for various installation times
of the second liner v ¼ 0:625 m/day). The asterisk indicates the
installation time of the first liner, whilst filled triangles indicate the
installation times of the second liner
123
ur ðr; tÞ ¼
1
R2 ðtÞ½pðtÞ p0 ðtÞ
2GB1 r
Zt
GB
1 Gg B t
s
B
e
R2 ðsÞ½pðsÞ p0 ðsÞe gB ds:
þ
2gB r
0
ð58Þ
1493
0
-14
-2
-16
-4
-18
t 2=10 day
-6
t 2=14 day
-8
t 2=18 day
-10
σ θ [MPa]
σr [MPa]
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels
-20
-22
t 2=14 day
t2=18 day
-24
-12
-26
-14
-28
-16
t 2=10 day
-30
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
t [day]
t [day]
Fig. 9 Stresses calculated at the interface between rock and the first liner (r = 6 m) versus time for various installation times of the second liner.
The asterisk indicates the installation time of the first liner, whilst filled triangles indicate the installation times of the second liner
Integrating and rearranging, the final solution is
obtained as
2 t
3
Z0
GB
GB
D11 p0 Gg B t1 4
g
s
t
e B
R2 ðsÞe gB ds B R21 e gB 0 5
u1 ðtÞ ¼
2D10 gB R1
GB
0
G
GB k tþ B t k t
t
ð61Þ
e 3 gB 1 3 1 e gB ;
EB (GB)
E B1(G B1 )
ηB
Fig. 10 Boltzmann viscoelastic physical model
GB
with D11 ¼ GB g
B k3
5.1 Supporting Pressure and Displacement After
Construction of the First Liner
Substituting
Eq.
u1 ðtÞ p1 ðtÞe
R1 R22
R21 R22
GB
gB t
(58)
,k3 ¼
into
2DR101g
B
;
(21),
and
D10 ¼
and
defining
R1
2GB1
þ 2G1L1 R3
u1 ðtÞ ¼ k3
Zt
2
kB ðt; sÞ u1 ðsÞds þ fB1 ðtÞ
ð59Þ
kB ðt; sÞ 1, fB1 ðtÞ 2g
GB
t
GB
t
p0
B R1 D10
G
e
g B t1
B
(
G
p0 R21
D11
Bt
¼
e gB 1
2GB1 r 2D10 gB R1
2 t
3
Z0
GB
GB
g
s
t
B
2
2
0
4 R ðsÞe gB ds R e gB 5
GB 1
u1r ðr; tÞ
t1
with
liner before the second liner is installed is obtained as
2 t
3
Z0
GB
GB
D11 p0 Gg B t1 4
g
s
t
p1 ðtÞ ¼
e B
R2 ðsÞe gB ds B R21 e gB 0 5
2D10 gB R1
GB
0
h
i
G
ðk3 g B Þðtt1 Þ
B
1 with t1 t\t2 :
ð62Þ
e
Substituting Eq. (62) into Eq. (58), the analytical
expression for the radial displacement of the rock mass
for t1 t\t2 is achieved as
L1
1
þ 1þm
KL1 R2 R2 , the integral equation becomes
1
. Thus, the pressure acting on the first
hR
t0
0
GB
s
R2 ðsÞe gB ds
0
GB
t
GgBB R21 e gB 0 ðe gB 1 e gB Þ. Following the methodology
shown in Sect. 4.1, solving the integral equations yields
2 t
3
Z0
G
GB
GB
p0
g
g B t1 4
s
t
u1 ðtÞ ¼
e B
R2 ðsÞe gB ds B R21 e gB 0 5
2D10 gB R1
GB
0
2
3
Zt
GB
GB
GB
GB
t
t
t
s
4e gB 1 e gB þ k3 ek3 ðtsÞ ðe gB 1 e gB Þds5:
t1
ð60Þ
h
G
ðk3 g B Þðtt1 Þ
e
B
i
1 1
)
p0 Gg B t
e B
2gB r
"Zt0
(Zt0
GB
s
R2 ðsÞe gB ds
0
GB GB
gB R21 Gg B t
D11 R1
t
s
e B e gB 0 R2 ðsÞe gB ds
GB
2D10 gB
0
# )
gB 2 Gg B t0
1 k3 ðtt1 Þ gB Gg B ðtt1 Þ gB
1
R eB e
eB
þ
:
GB 1
GB
G B k3
k3
þ
ð63Þ
123
1494
H. N. Wang et al.
When t1 t\t2 , the incremental displacement occurring
after the installation of the first liner is
Du1þ ðr;tÞ u1r ðr;tÞur ðr;t1 Þ
2 t
3
Z0
GB
GB
p0 R1
g
s
t
Gg t1 4
B
2
2
¼
e
R ðsÞe gB ds R1 e gB 0 5
4D10 gB GB1 r
GB
0
h
i p GB
G
G
0
ðk B Þðtt1 Þ
t
Bt
e gB e gB 1
1 e 3 gB
2gB r
2 t
3
0
Z
GB
GB
g
s
t
4 R2 ðsÞe gB ds B R21 e gB 0 5
GB
0
2 t
3
Z0
GB
GB
D11 p0 R1 Gg B t 4
g
s
t
R2 ðsÞe gB ds B R21 e gB 0 5
þ
e B
GB
4D10 g2B r
0
1 k3 ðtt1 Þ gB Gg B ðtt1 Þ gB 1
B
e
e
þ :
k3
GB
GB k3
ð64Þ
According to Eqs. (62) and (63), the supporting pressure
and displacements are related to the supporting time and
the excavation process, which was not the case for the
solution obtained for the Maxwell model. Substituting Eq.
(62) into Eq. (11) leads to the following expressions for the
stresses:
rr ¼
rh ¼
8
R2 ðtÞ
>
>
p0 1 2
>
>
r
>
<
GB
D1 D1 D6
R1
t
e gB p2 ðtÞ
þ D 2 R1 D 3 D 6 R1 þ
R1
R1
2GB1
Zt G
GB
B
R1
D1 D6
¼ e gB s p2 ðsÞds þ D3 D6 R1 e gB t p1 ðt2 Þ
2gB
R1
t2
GB
p0
ðtt Þ
½1 e gB 1 þ
2gB R1
GB
s
e gB ds þ
p 0 R1
2gB
Zt
GB
2
R ðsÞe
0
s
e gB ds t0
GB
gB s
1
ds R1
2gB
p0 R1 Gg B ðtt1 Þ
eB
2gB
Zt1
Zt2
p1 ðsÞ
t1
GB
s
e gB ds;
t0
ð66Þ
where p1 ðt2 Þ can be calculated by Eq. (62) as
2 t
3
Z0
GB
GB
D11 p0 Gg B t1 4
g
s
t
p1 ðt2 Þ ¼
e B
R2 ðsÞe gB ds B R21 e gB 0 5
2D10 gB R1
GB
0
h
i
G
ðk B Þðt t Þ
e 3 gB 2 1 1 :
ð67Þ
Assuming that
GB
t
u2 ðtÞ p2 ðtÞe gB
ð68Þ
0 t\t1
with
2 t
3
Z0
h
i p R2
2 GB
>
GB
GB
G
D
p
R
g
R
>
11 0 1
0
g t1 4
B 1 gB t0 5 ðk3 gBB Þðtt1 Þ
2
>
gB s
B
>
e
p
e
R
ðsÞe
ds
e
1
þ 21
>
: 0 2D10 gB r 2
GB
r
8
R2 ðtÞ
>
>
with
p0 1 þ 2
>
>
r
>
<
Zt0
with t1 t\t2 ;
0
ð65Þ
0 t\t1
2 t
3
Z0
h
i p R2
2 GB
>
GB
GB
G
D
p
R
g
R
>
11 0 1
0
g t1 4
B 1 gB t0 5 ðk3 gBB Þðtt1 Þ
2
>
gB s
B
>
e
p
þ
e
R
ðsÞe
ds
e
1
21
0
>
:
2D10 gB r 2
GB
r
with t1 t\t2 :
0
Hence, also the stresses turn out to depend on the
excavation procedure.
5.2 Supporting Pressure and Displacement
After Construction of the Second Liner
According to the analysis in Sect. 3.4, the integral equation
for the supporting pressure p2 = p2(t) is obtained by
substituting Eq. (57) into (31) as
123
R1
and D12 ¼ DR11 DR1 D1 6 þ D2 R1 D3 D6 R1 þ 2G
, the integral
1
equation for u2 ðtÞ can be obtained from Eq. (66) after some
rearrangements as
u2 ðtÞ ¼ k4
Zt
kB ðt; sÞ u2 ðsÞds þ fB2 ðtÞ;
t2
R1
where kB ðt; sÞ 1, k4 ¼ 2g
D112 , and
B
ð69Þ
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels
D1 D6
1 Gg B t
fB2 ðtÞ þ D 1 D 6 R1
e B p1 ðt2 Þ
D12
R1
Zt0
GB
GB
p0
ðtt
Þ
s
1
½1 e gB
R2 ðsÞe gB ds
þ
2gB R1 D12
8
G
Zt1
gBt <
2
GB
B
R
e
s
2
1
p0 R2 ðsÞe gB ds
½p2 ðtÞp0 þ
ur ðr;tÞ¼
2GB1 r
2gB r :
0
þR21
0
R1
2gB D12
Zt2
GB
s
p1 ðsÞ e gB ds þ
t1
p0 R1 Gg B ðtt1 Þ
eB
2gB D12
Zt1
e
GB
gB s
p 0 R1
2gB D12
Zt
GB
u2 ðtÞ ¼ fB2 ðtÞ þ k4
t0
ds:
ð70Þ
ð71Þ
The analytical expression for the radial displacement of
the rock mass for t t2 can be obtained by substituting
p1 ðtÞ (Eq. 62) and p2 ðtÞ into Eq. (58) and rearranging:
0.06
0.05
v=5 m .day-1
ur [m]
0.03
v=1 m .day-1
v=0.625 m .day-1
0
0
2
4
6
½p2 ðsÞp0 e
t2
9
=
ds :
;
R21
½p2 ðtÞ
Du2þ u2r ðr; tÞ ur ðr; t2 Þ ¼
2GB1 r
G
1 Gg B t
Bt
e B e gB 2
p2 ðt2 Þ 2gB r
2 t
3
0
Zt2
Z
GB
GB
s
s
4 p0 ðsÞR2 ðsÞe gB ds R21 p1 ðsÞ e gB ds5
t1
0
0
0.01
dsþR21
GB
gB s
The incremental displacement occurring in the second
liner stage is
So, the expression for p2 ðtÞ can be obtained from Eq.
(68). Since the integral expression in Eq. (70) is very long,
the explicit expression is here omitted. Substituting Eqs.
(67) and (70) into Eq. (30), we obtain
(
G
D11 p0
Bt
q1 ðtÞ ¼ D6 p2 ðtÞ e gB 1
2D10 gB R1
2 t
3
)
Z0
GB
GB
GB
g
s
t
ðk
Þðt
t
Þ
B
4 R2 ðsÞe gB ds R2 e gB 0 5½e 3 gB 2 1 1 :
GB 1
0.02
½p1 ðsÞp0 e
Zt
ð72Þ
t2
0.04
GB
gB s
t1
s
t0
WB ðt; sÞfB2 ðsÞds:
Zt2
e gB ds
The kernel function of Eq. (69) is written as
WB ðt; sÞ ¼ ek4 ðtsÞ . Thus, the solution for the integral
equation can be expressed analytically as
Zt
1495
8 10 12 14 16 18 20 22 24 26 28 30
t [day]
Fig. 11 Radial displacement versus time for various excavation
rates; asterisks indicate the installation times of the first liner, whilst
filled triangles indicate the installation times of the second liner
R21 Gg B t
e B
2gB r
Zt
t0
p0 ðsÞe
Zt
GB
s
p2 ðsÞ e gB ds þ
t2
GB
gB s
R2 G B t
ds 1 e gB 2
2gB r
R21 Gg B t
e B
2gB r
Zt2
GB
s
p0 ðsÞe gB ds:
t0
ð73Þ
5.3 Results and Discussion
According to the experimental data available (Feng et al.
2006), reasonable values for the constitutive parameters
of the viscoelastic rock are: GB ¼ 1; 000 MPa, GB1 ¼
2; 000 MPa, and gB ¼ 10; 000 MPa day. With regard to the
concrete employed for the liners, its properties are outlined
in Sect. 4.3. The same geometry (tunnel geometry and
thicknesses of the liners), in situ stress state, excavation
process, and excavation rates as in Sect. 4 were assumed.
The only difference from Sect. 4 is the rock model applied.
The displacement response calculated at the interface
between rock and the first liner (r ¼ 6 m) for various values
of excavation rate is illustrated in Fig. 11. Trends similar to
the response obtained for the Maxwell model, but with
faster convergence, are shown.
To assess the influence of the installation times of the
first and second liners, an excavation rate of 0.625 m/day
as in Sect. 4.3.2 was assumed. Four options were considered, with the first liner installed on the 8th, 10th, 12th, and
15th day. In all the cases, the second liner was installed on
the 15th day. In Fig. 12, the resulting radial displacements,
calculated at the interface between rock and the first liner
(r ¼ 6 m), are displayed. Again, it can be noted that early
installation of the liner reduces the rate of displacement,
and also leads to a shorter time to reach the stable state.
123
1496
H. N. Wang et al.
0.06
t1=12 day
t1=10 day
t1=15 day
0.05
ur [m]
0.04
t1=8 day
0.03
0.02
0.01
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
t [day]
Fig. 12 Radial displacement versus time for various installation
times of the first liner (v = 0.625 m/day). Asterisks indicate the
installation times of the first liner
0.06
t 2=15 day
t 2=12 day
t 2=10 day
0.05
ur [m]
0.04
t2=8 day
0.03
0.02
0.01
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
t [day]
Fig. 13 Radial displacement versus time for various installation
times of the second liner (v = 0.625 m/day). Filled triangles indicate
the installation time of the second liner
0
-15
v=0.625 m .day-1
v=1 m .day-1
-2.5
-17.5
v=5 m .day-1
-20
σθ [MPa]
-5
σr [MPa]
Considering now different times for the installation of the
second liner, assuming that the first liner was installed
immediately after completion of the excavation process,
the radial displacements calculated at the interface between
rock and the first liner are shown in Fig. 13. Comparing
Figs. 12 and 13 with Figs. 6 and 8, respectively, it emerges
that the trends are similar for both models; however, from a
quantitative point of view, the influence of the time of
installation of the liners on the radial displacements is more
significant in the Maxwell model.
With regard to the influence of the excavation rate on
the state of stress within the rock mass, in Fig. 14 the hoop
and radial stresses developed within the rock mass at the
interface with the first liner are plotted. As in Sect. 4, it was
assumed that the first liner was installed immediately after
excavation. In comparison with the Maxwell model (Sect.
4.3.4), the trends shown are similar, although the quantitative variation of the stress level over time is remarkably
less significant.
Finally, looking at the analytical expression derived for
the pressure acting on the first liner for the Maxwell model
(Eqs. 39, 45, 46), it can be observed that the supporting
pressure is independent of the excavation process. However, when applying the Boltzmann model instead, the
supporting pressures pðtÞ and qðtÞ depend on the excavation rate, as shown in Figs. 15 and 16. In Fig. 15, the
supporting pressure acting on the first liner is plotted
against the time difference Dt ¼ t t1 ; with t1 being the
installation time of the first liner, whilst in Fig. 16, the
supporting pressure acting on the second liner is plotted
against the time difference Dt0 ¼ t t2 ; with t2 being the
installation time of the second liner. From these figures it
emerges that the supporting pressure is larger for higher
excavation rates.
-7.5
-10
-22.5
-25
v=5 m .day-1
-12.5
-27.5
-15
-30
v=1 m.day-1
v=0.625 m .day-1
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
t [day]
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
t [day]
Fig. 14 Stresses calculated at the interface between rock and the first liner (r = 6 m) versus time for various excavation rates. Asterisks indicate
the installation times of the first liner, whilst filled triangles indicate the installation times of the second liner
123
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels
3.5
v=5 m .day -1
3
v=1 m .day -1
p [MPa]
2.5
v=0.625m .day -1
2
1.5
1
0.5
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Δ t [day]
Fig. 15 Pressure on the first liner versus time for various excavation
rates. Filled triangles indicate the installation times of the second
liner
1.5
v=5 m .day-1
1
v=1 m .day-1
q [MPa]
v=0.625 m.day -1
0.5
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
1497
supporting pressures, stresses, and displacements in the
rock and the liners could be derived.
The Maxwell viscoelastic model accounts for primary
and secondary creep commonly observed in rock masses
with weak mechanical properties or subject to high compressive stresses. In this case, radial displacements after
liner installation increase exponentially over time, and
reach a stable value asymptotically. Radial displacements
are a function of the excavation process and of the installation time of the liners. Assuming that the first liner is
installed immediately after excavation, a parametric analysis for various excavation rates showed that slow rates
lead to smaller displacements in the short term but to larger
ones in the long term, so that longer times are required for
the stresses to stabilize to a constant value. Then, a parametric analysis for various times of liner installation was
carried out, showing that early construction results in
smaller displacements in the long term.
The Boltzmann viscoelastic model is commonly
employed for rock masses exhibiting limited viscosity.
Unlike the Maxwell model, the closed-form solutions
obtained for the Boltzmann model showed that the development of pressure on the liners depends on the excavation
process and the times of liner installation.
Although the obtained analytical solutions are rigorously applicable only for axisymmetric plane-strain conditions, according to the recent work of Schuerch and
Anagnostou (2012), these solutions can be applied to a
much wider range of ground conditions and to several
cases of noncircular tunnels.
Δ t' [day]
Fig. 16 Pressure on the second liner versus time for various
excavation rates
6 Conclusions
Closed-form analytical expressions for stresses and displacements of deeply buried circular tunnels excavated in a
viscoelastic medium supported by two liners were derived
for any installation time of the liners. An initial hydrostatic
stress field was assumed, with the rock mass modeled as
linearly viscoelastic and the liners as purely elastic. To
simulate realistically the process of tunnel excavation,
solutions were developed for a time-dependent excavation
process with the radius of the tunnel growing from zero
until a final value according to a time-dependent function
to be specified by the designers. The integral equations for
the supporting pressures were established according to the
given boundary conditions. Solutions of the equations were
obtained assuming either the Maxwell or the Boltzmann
viscoelastic model for the rock mass so that explicit closedform analytical expressions for the time-dependent
Acknowledgments This work is supported by the National Natural
Science Foundation of Shanghai City (grant No. 11ZR1438700), the
Marie Curie Actions—International Research Staff Exchange Scheme
(IRSES): GEO—geohazards and geomechanics (grant No. 294976),
the National Natural Science Foundation of China (grants No.
10702052 and 51179128), and the China National Funds for Distinguished Young Scientists (grant No. 51025932). These supports are
greatly appreciated.
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