Effect of stress disturbance induced by construction on the seismic response of shallow bored tunnels

Computers and Geotechnics 49 (2013) 338–351

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Computers and Geotechnics
journal homepage: www.elsevier.com/locate/compgeo

Effect of stress disturbance induced by construction on the seismic response
of shallow bored tunnels
Rui Carrilho Gomes ⇑
Civil Engineering and Architectural Dept., Technical Univ. of Lisbon, IST, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal

a r t i c l e

i n f o

Article history:
Received 28 July 2011
Received in revised form 26 June 2012
Accepted 13 September 2012
Available online 13 October 2012
Keywords:
Tunnels
Earthquake
Stress disturbance
Lining forces

a b s t r a c t
This paper examines the effect of the stress disturbance induced by tunnel construction on the completed
tunnel’s seismic response. The convergence-confinement method is used to simulate the tunnel construction prior to the dynamic analysis. The analysis is performed using the finite element method and drained
soil behaviour is simulated with an advanced multi-mechanism elastoplastic model, utilising parameters

derived from laboratory testing of Toyoura sand. The response of the soil and of the lining during dynamic
loading is studied. It is shown that stress disturbance due to tunnel construction may significantly
increase lining forces induced by earthquake loading, and Wang’s elastic solution appears to underestimate the increase.
Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction
Historically, the general conviction has been that the effect of
earthquakes on tunnels was not very important. Nevertheless,
some underground structures have experienced significant damage in recent earthquakes, including the 1995 Kobe Japan earthquake [1], the 1999 Chi-Chi Taiwan earthquake [2] and the 1999
Kocaeli Turkey earthquake [3].
In recent decades, the number of large tunnels and underground spaces constructed has grown significantly. In addition,
the high cost of real estate has increased the demand for tunnels
in large urban centres. This work studies large-diameter tunnels
at relatively shallow depth, commonly used in urban areas for metro structures, highway tunnels and large water and sewage transportation ducts. In urban areas, tunnel excavation by boring maybe
preferable to cut-and-cover excavation due to the existence of
overlying structures. Thus, excavation by boring is considered in
this work.
Tunnels in earthquake prone areas are subjected to both static
and seismic loading. The most important static loads acting on
underground structures are ground pressures and water pressure;
in general, live loads can be safely neglected. It is well known that
tunnel excavation and the application of support measures induces
three-dimensional (3D) deformation and stress redistribution during tunnel face advance. The convergence-confinement method [4]
is one of the most common assumptions used for considering the
⇑ Tel.: +351 218 418 420; fax: +351 218 418 427.
E-mail address:
0266-352X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
/>
3D face effect in two-dimensional (2D) plane-strain analysis, as it
approximates the stress relaxation of the ground due to the

delayed installation of the lining [5,6].
Before earthquake loading is applied to the tunnel, the initial
stress field has already been disturbed by tunnel construction.
There are studies that simulate the tunnel construction prior to
the seismic analysis of underground structures (e.g. [7,8]), while
other studies do not simulate tunnel construction (e.g. [9]). The
main focus of this paper is to assess the influence of stress disturbance induced by construction on the seismic response of shallow
tunnels.
The approaches used to quantify the seismic effect on an underground structure are summarized by Hashash et al. [10] and
include (i) closed-form elastic solutions to compute deformations
and forces in tunnels for a given free-field deformation [11], (ii)
numerical analysis to estimate the free-field shear deformations
using one-dimensional (1D) wave propagation analysis (e.g. Shake
[12]), and (iii) 2D or 3D finite element or finite difference codes to
simulate soil-structure response (e.g. Flac [13], Flush [14], Gefdyn
[15], CESAR-LCPC [16]).
In this work, the finite element method is used, because an
advanced constitutive model is required to evaluate accurately
the effect of stress disturbance due to tunnel construction on the
tunnel seismic response. The advanced multi-mechanism elastoplastic model developed at École Centrale de Paris, ECP [17,18] is
used, since this model can take into account important factors that
affect soil behaviour, such as the strain level and the stress
conditions, while the influence of other factors which control the
stiffness degradation, such as the plasticity index and the initial


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R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

state (OCR, void ratio, stress state, etc.) are considered via the model parameters. The ECP model has been widely used to simulate
soil behaviour under static and cyclic loading (e.g. [19,20]) and is
implemented in the general purpose finite element code Gefdyn
[15]. This code is particularly suitable for modeling the cyclic
behaviour of soils and soil-structure interaction, and it has been
successfully used in the past to study the behaviour of geotechnical
structures [21,22]. In this work, the behaviour of Toyoura sand is
simulated.
A 2D plane-strain finite element simulation of bored tunnel
construction was performed using the convergence-confinement
method [4]. It was assumed that the stress disturbance induced
by tunnel construction is controlled by a single parameter, namely
the decompression level. Three values of decompression level
within the range of values usually adopted in practice are considered, to assess their effect on lining seismic response.
Subsequent to the simulation of tunnel construction, a set of
eight dynamic analyses were performed in the time domain. Ovaling deformation of the cross-section of circular tunnel due to
ground shaking is studied, as it refers to the deformation of the
ground produced by seismic wave propagation through the Earth’s
crust.

–

–
–

–
–

–

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r0 Àr0 2
ii
jj
qk ¼
þ ðr0ij Þ2 is the radius of the Mohr circle in the
2
plane of the generic deviatoric mechanism of normal ~
ek . Here
i, j, k e {1, 2, 3}; i = 1 + mod(k, 3), and j = 1 + mod(k + 1, 3), with
mod(k,0 i)0 representing the residue of the division of k by i;
r þr
p0k ¼ ii 2 jj is the centre of the Mohr circle in the plane of the
deviatoric mechanism of normal ~
ek ;
p0c is the critical pressure that is linked to the volumetric plastic
strain epv by the relation p0c ¼ p0co expðbepv Þ, where p0co represents
the initial critical pressure, which is the critical mean effective
stress that corresponds to the initial state defined by the initial
void ratio, and b the plasticity compression modulus of the
material in the isotropic plane (ln p0 , epv );
/0pp is the friction angle at the critical state;
b is a numerical parameter which controls the shape of the yield
surface in the ðp0k ; qk Þ plane (Fig. 1) and varies from b = 0–1,
passing from a Coulomb surface to Cam-Clay surface type;
rk is an internal variable that defines the degree of mobilised
friction of the mechanism k and introduces the effect of shear
hardening of the soil.

The last variable is linked to the plastic deviatoric strain,

according to the following hyperbolic function:

epd;k ,

R

r elk

depd;k dt
R
a þ depd;k dt

2. Constitutive model for soil

rk ¼

2.1. Model description

where a is a parameter which regulates the deviatoric hardening of
the material. It varies between a1 and a2, such that:

The soil behaviour is simulated over a large range of strains
with ECP’s elastoplastic multi-mechanism model developed by
Aubry et al. [17] and extended to cyclic behaviour by Hujeux [18].
The model is written in the framework of the incremental plasticity and is characterised by both isotropic and kinematic hardening. It decomposes the total strain increment into elastic and
plastic parts. Whilst the elastic response is assumed to be isotropic,
the plastic behaviour is considered to be anisotropic by superposing the response of three plane-strain deviatoric mechanisms
(k = 1, 2, 3) and one purely isotropic (k = 4).
With these assumptions, the total plastic strain increment deP is
written as:

deP ¼ depd þ depv ¼
p
d

3
X

4
X

k¼1

k¼1

epd;k þ

epv ;k

ð1Þ

p

where de and dev represent, respectively, the total deviatoric and
volumetric plastic strain increments. The former is given by the
contributions, depd;k , of the three deviatoric mechanisms, the latter
by the contributions of all four mechanisms.
The elastic response is assumed to be isotropic and non-linear
with the bulk, K, and shear moduli, G, functions of the mean effective stress according to the relations:

K ¼ K ref

p0
p0ref

!n e
;

G ¼ Gref

p0
p0ref

þ

ð4Þ

a ¼ a1 þ ða2 À a1 Þak ðrk Þ

ð5Þ

where the intermediate of the parameter ak(rk) (Fig. 2), integrates
the decomposition of the behaviour domain into pseudo-elastic,
hysteretic and mobilised domains, where:

ak ðrk Þ ¼ 0
ak ðrk Þ ¼



Àr hys

rk
r mob Àr hys

ak ðrk Þ ¼ 1

m

if

rel < r k < r hys

if

rhys < r k < rmob

if

r

mob

ð6Þ

< rk < 1

el

r defines the extent of the elastic domain and is the minimum

value that rk can take, while rhys and rmob designate the extent of
the domain where hysteresis degradation occurs.
The evolution of the volumetric plastic strains is controlled by a
flow rule based on Roscoe-type dilatancy rule:



q
@ epv ;k ¼ @ epd;k ak ðr k Þ Á sin w À k0 Á aw
pk

ð7Þ

with aw a constant parameter and w representing the characteristic
angle [23] defining the limit between dilating ð@ epv ;k < 0Þ and contracting ð@ epv ;k > 0Þ of the soil (Fig. 3).
The primary yield function defined in Eq. (3) can also be written
in the form:

!n e
ð2Þ
b=0

where Kref and Gref are respectively the bulk and shear modulus at
the reference mean effective stress p0ref . The degree of the non-linearity is controlled by the exponent ne.
During monotonic loading, the primary yield function associated to the generic mechanism k has the following expression:


 0 
À
Á

p
Á rk
fk qk ; p0k ; epv ; rk ¼ qk À p0k Á sin /0pp Á 1 À b Á ln k0
pc
where the following variables have been introduced:

b=0.25

qk

b=0.50
b=0.75
b=1

ð3Þ
p'k
Fig. 1. Influence of parameter b on the yield surface shape.


340

R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

Whenever a stress reversal occurs, the primary yield function
(3) and (8) is abandoned and the cyclic surface becomes active.
The latter is defined by the function:

À
Á  n À h
Á

cyc h 
p
h
h

fkc p0k ; sk ; r cyc
À rcyc
k ; ev ; dk ; nk ¼ sk À dk À r k nk
k

ð11Þ

The surfaces associated to deviatoric yield functions are the
circles of radius r cyc
interior to circles of primary loading, both
k
tangents at the point dhk of exterior normal nhk , where the point dhk
corresponds to the last load reversal h of the mechanism k
(Fig. 5):

dhk ¼

Fig. 2. Evolution of ak(rk).

qhk
1

p0  ;
0
k

p0h
k sin /pp 1 À ln p0
c

rcyc
k
p'k
Fig. 3. Critical state and characteristic state lines.

fk ðqk ; p0k ; epv ; r k Þ ¼ jsnk j À r k

ð8Þ

where the following variables have been introduced:
– sk is the
deviator stress vector in the k-plane of components
r0 Àr0
sk1 ¼ ii 2 jj and sk2 ¼ r0ij and norm jsk2 j ¼ qk . The yield surfaces
of each mechanism k can be interpreted in the normalised deviatoric plane ðsnk1 ; snk2 Þ:

sk1

gk

;

snk2 ¼

sk2

ð9Þ

gk

Á

gk p0k ; epv ¼ p0k sin /0pp



ð13Þ

in which ep;h
d;k is the plastic deviatoric strain of the mechanism k at
the last load reversal h. The variable a(rk) obeys to the same relation
as in monotonic loading (5).
Finally, the constitutive equation set is completed with the
equation describing the isotropic yield function which defines
the last mechanism (k = 4) of the model. The isotropic yield function is assumed to be:

À
Á
f p0 ; epv ¼ p0 À p0c Á d Á r 4
0

ð14Þ

p0c

where p and

are respectively the current and the critical state
mean effective pressure. The parameter d represents the distance
between the isotropic consolidation line and the critical state line
in the (ln p0 , e) plane. The internal variable r4 depends on model
parameter c that controls the volumetric hardening of the soil as:

r4 ¼

 0 
p
1 À ln k0
pc

ð10Þ

In this plane, the deviatoric yield surfaces are circles of radius rk
(Fig. 4).

(a)

R


p
p;h 
 ded;k dt À ed;k 
R

¼ r elk þ



a þ  depd;k dt À ep;h
d;k 

R

with the normalisation factor gk given by:

À

ð12Þ

h
The vector ðdhk À rcyc
k nk Þ corresponds to the vector going from the
origin of the normalised deviatoric plane to the centre of the cyclic
circle. The vectors dhk and nhk are discontinuous parameters introducing kinematic hardening to the model [18].
The hardening variable r cyc
k can be expressed in terms of the position of the current stress state with respect to the position of the
last load reversal with initial value equal to r elk :

qk

snk1 ¼

snh
nhk ¼  knh 
sk

r el4

þ

depd;4 dt
R
c Á pc =pref þ depd;4 dt

The model parameter c is equal to c1 during monotonic loading,
and equal to c2 during cyclic loading.
All the mechanisms are coupled through the total volumetric
plastic strain given by:

(b)

τk

ð15Þ

s

n
k2

1

σ'ij
s

n
k2

s

sk

n
k

σ'jj

s
p'k σ'ii

σk

n
s k1

n
k1

1

Fig. 4. Stress state representations for deviatoric mechanism k: (a) Mohr’s representation (normal stress, rk, vs. shear stress, sk) in the i–j plane; (b) stress state in the
normalised deviatoric plane ðsnk1 ; snk2 Þ.


341

R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

(a)

s

(b)

n
k2

s

1

n
k2

h

nk

1
h

dk
el

rk
cyc
rk

s
el

rk

n
k1

s

1

n
k1

1

m

rk

Fig. 5. Evolution of deviatoric yield surface in the normalised deviatoric plane of the mechanism k for: (a) monotonic loading; (b) cyclic loading.

depv ¼ depv ;1 þ depv ;2 þ depv ;3 þ depv ;4

ð16Þ

Further analytical details concerning the model can be found in
specific Refs. [17,18].

2.2. Model’s parameters and laboratory tests simulations
2.2.1. Introduction
Toyoura sand is a clean, uniform, fine sand with zero fines content, commercially available washed and sieved, which has been
widely used for liquefaction and other studies, in Japan and worldwide [24,25]. The behaviour of Toyoura sand has been characterised under a number of different stress paths, the results of
which have been used to provide constitutive parameters for this
study.
The soil is assumed to be homogeneous in each analysis performed. The model’s parameters were evaluated for the sand with
relative density equal to 40% (initial void ratio e0 = 0.833).
The model’s parameters can be divided in those than can be
directly measured:
– Elasticity: Kref, Gref, ne and p0ref .
– Critical state and plasticity: /0pp , b, d and p0co .
– Yield function and hardening: w.
and those that are non-directly measured:
– Critical state and plasticity: b.
– Yield function and hardening: a1, a2, c1, c2, aw and m.
– Threshold domains: rel, rhys, rmob and r el4 .

ð2:17 À e0 Þ2 0 0:4
ðp Þ ½kPaŠ
1 þ e0

bffi

1 þ e0
¼ 21:5
k

ð19Þ

2.2.3. Determination of non-directly measurable parameters
As the paper is devoted to seismic applications, in the numerical
simulations made for identification of the non-directly measurable
parameters, simple shear loading was assumed. The strategy for
model parameter identification detailed in Santos et al. [19] and
Gomes [28] has already been explored and verified. The strategy
to derive model parameters related to shear hardening relies directly on experimental data represented by a strain-dependent
stiffness degradation curve.
The calibration of the parameters is derived from the reference
‘‘threshold’’ shear strain, c0.7, that defines the shear strain for a
stiffness degradation factor of G/G0 = 0.7. In effect, the reference
threshold shear strain defines the beginning of significant stiffness
degradation. The ‘‘standard’’ shape for the ak = f(rk) curve proposed
by Santos et al. [19] is shown in Fig. 6. This curve can be determined by means of the point (r0.7; a0.7) obtained from the experimental strain-dependent stiffness curve, according to the
following equations:

r0:7 ¼

2.2.2. Determination of directly measurable parameters
The initial shear modulus, G0, is estimated according to the following equation from Iwasaki et al. [26] for Toyoura sand:

G0 ¼ 14100

For e0 = 0.833, the critical state line, CSL, presented by Ishihara
[24] has abscissa p0co ¼ 1200 kPa and slope, k, equal to 0.0852.
The parameter d (horizontal distance between the isotropic consolidation line and the critical state line) is equal to 5.8.
Hajal [27] proposed the following relationship to calculate the
plastic compressibility modulus, b:

0:7G0 c0:7


;
p0
p0k sin /0pp 1 À b ln pk0
c

0:3c0:7
aðr 0:7 Þ ¼ 1
1
þ
lnð1 À r à Þ
Ã
1Àr
rÃ

with r à ¼ r 0:7 À r elas

ð17Þ

For e0 = 0.833 and p0ref ¼ 1 MPa, Gref defined by Eq. (2) becomes
equal to 218 MPa and ne = 0.4.
The initial bulk modulus, k0, was derived from the following
relationship, valid for homogeneous isotropic linear elastic materials, assuming the Poisson’s ratio equal to m = 0.2:

K0 ¼

2G0 ð1 þ mÞ
¼ 1:333 Á G0
3ð1 À 2mÞ

ð18Þ

According to Eq. (2), Kref becomes equal to 291 MPa.
Based on drained and undrained monotonic triaxial tests,
Ishihara [24] determined that /0pp ¼ w ¼ 31 .

Fig. 6. Standard shape for the relationship rk À ak.

ð20Þ


342

R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

Table 1
Model parameters for Toyoura sand (Dr = 40%).
Layer

0–5 m

el

5–10 m
À3

r
a1
a2

10–15 m
À3

2.18 Â 10
1.50 Â 10À5
2.06 Â 10À3

15–20 m
À3

1.13 Â 10
2.50 Â 10À5
3.13 Â 10À3

1.01 Â 10
2.50 Â 10À5
2.60 Â 10À3

20–25 m
À4

25–30 m
À4

8.18 Â 10
2.50 Â 10À5
3.13 Â 10À3

7.61 Â 10
2.50 Â 10À5

4.44 Â 10À3

6.81 Â 10À4
2.50 Â 10À5
5.33 Â 10À3

À3
Gref = 218 MPa; Kref = 291 MPa; p0c0 ¼ 1200 kPa; p0ref ¼ 1 MPa, m = 1; ne = 0.4; /0pp ¼ 31 ; w = 31°; b = 21.5; aw = 1; rmob = 0.8; b = 0.2; d = 5.8; c1 = 0.06; c2 = 0.03; r el
;
4 ¼ 10
Coefficient of earth pressure at rest: k0 = 0.5. Volumetric unit mass: q = 1900 kg/m3.

The parameters rel and a2 can be determined using the following
relationship:

r el ¼

p0k

G0 cet

;
p0
1 À b ln pk0

sin /0pp

a2 ¼

a0:7 ðr mob À rel Þ

ðr 0:7 À rel Þ

ð21Þ

c

where cet is the elastic threshold shear strain. For Toyoura sand,
cet ffi 10À6. For sands, b is small (b = 0.2). The parameter a1 assumes
small values from matching simulated and experimental straindependent shear modulus curves. The parameter rmob is usually taken equal to 0.8, and rel = rhys.
In addition, it was noticed that varying the parameters rel, a1
and a2 with mean effective stress improve the match between
experimental and simulated results.
At last, the parameters c1, c2, rel4 and aw were determined to obtain best fitting of the experimental undrained triaxial tests [24].
According to the proposed strategy and the available experimental data [24,26,29] the parameters summarized in Table 1
were evaluated for the sand with relative density equal to 40%
(initial void ratio e0 = 0.833).
Fig. 7 shows experimental data from resonant column (RC) and
cyclic torsional shear (CTS) tests [26,29] and the model response
for a single element under stress controlled cyclic simple shear
loading.
The model response is in good agreement with the experimental stiffness degradation curve, while damping tends to be underpredicted by the model.
3. Simulation of construction
3.1. Model
The finite element mesh (Fig. 8) simulates a soil mass 30 m
thick in plane-strain conditions with 1254 isoparametric 4-node
rectangular elements, overlying an impervious isotropic linear
elastic half-space (G = 500 MPa, q = 2000 kg/m3).

0.5

0.8

0.4
increase p'
increase p'

0.6

0.3

0.4

0.2

0.2

0.1

Damping ratio,

Shear modulus ratio, G/G0

1.0

During the different stages of the static analysis that simulated
tunnel construction, the nodes along lower boundary of the mesh
were fixed in both the horizontal and vertical direction, the nodes
along the lateral boundaries were fixed in the horizontal direction,
while all other nodes were free in both the horizontal and vertical
direction. During the dynamic stage, the nodes along the lower

boundary were freed in the horizontal direction in order to apply
the seismic input motion and to activate the absorbing elements.
These are linear 2-node elements developed to simulate radiation
conditions at the base of the finite element model by eliminating
the elastic waves that would otherwise be reflected back into the
interior of the finite domain by the artificial boundaries of the
model [30]. This objective is achieved by imposing additional actions reproducing the dynamic impedance at the nodes of the model boundary which characterises the interface between the finite
and infinite domain [31]. The latter is regarded as a 1-phase elastic
medium.
Kuhlemeyer and Lysmer [32] suggested that the maximum element size, hmax, in the direction of wave propagation should be less
than approximately one-tenth to one-eighth of the lowest wavelength of interest in the simulation. The latter may be evaluated
by the ratio between the minimum wave velocity, Vmin, and the
highest frequency of the input wave, fmax. In the model studied,
Vmin is equal to 100 m sÀ1 near the surface, and fmax can be taken
as equal to 10 Hz, thus, hmax should not exceed 1.25 m in the direction of wave propagation. In this work, the vertical element size
adopted is 1.0 m.
The lining is modelled as continuous and impervious circular
ring with linear elastic behaviour using 40 beam elements. No relative movement (no-slip) was allowed on the tunnel lining-soil
interface. Interface elements were not employed as there was no
basis to determine their properties and they could potentially
dominate the model response.
The potential separation of the two materials was examined by
monitoring the normal stresses acting at the interface. In all computations this potential separation never occurred.
The maximum mobilized strength ratio (ratio between mobilized strength to the available strength) was examined along the
perimeter of the tunnel. The maximum mobilized strength ratio
reaches a value of 1 only for the two strongest input motions. This
occurs in limited regions of the soil along the perimeter of the
tunnel.

100 m

0.0
1E-06

1E-05

1E-04

1E-03

1E-02

0.0
1E-01

Sand (30 m)

Shear strain, γ
Experimental

15 m

Simulation

Fig. 7. Toyoura sand (Dr = 40%, p0 = 25, 50, 100 and 200 kPa): strain-dependent
shear modulus and damping curves from RC and CTS tests and model response for
single element under stress controlled drained cyclic simple shear loading.

10 m

Elastic half-space

Fig. 8. Finite element mesh.


R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

3.2. Lining properties
A concrete lining subjected to bending and axial load often
cracks and behaves in a non-linear fashion, and/or may have joints.
According to Wang [11], the ratio between lining effective stiffness
and full-section stiffness is within the range 0.30–0.95.
Lining material non-linearity effects were taken into account in
an approximate manner by adopting an effective thickness of
0.35 m in the numerical simulations, which corresponds to fullsection thickness of 0.45 m. So, a ratio between effective and fullstiffness of 0.48 was adopted for bending stiffness and 0.78 for
axial stiffness, reflecting that the effect of cracking is greater for
the bending stiffness than the axial stiffness.
The mechanical parameters are taken as the typical properties
of a reinforced concrete lining: Young modulus, Ec = 32 GPa, Poisson ratio m = 0.2, volumetric unit mass q = 2550 kg/m3, compressive strength of concrete, fc = 38 MPa, and yield strength of the
rebar, fsy = 500 MPa.

343

as consequence of the tunnel construction. The higher the
k-parameter, the lower the vertical effective stress around the tunnel, as higher decompression of the soil is allowed.
The stress field at the end of the construction phase is the initial
stress field for the subsequent seismic analysis.
To evaluate if the maximum strength of the concrete lining is
exceeded, bending moment-axial load interaction diagrams for
the lining were computed. According to Eurocode 2 [33], the maximum reinforcement area, As,max, should be 0.04Ac, where Ac is the
cross sectional area of concrete. Fig. 11 plots the evolution of lining
forces during the construction stages against the maximum

strength of the concrete lining for As,max = 0.04Ac.
It is verified that maximum lining forces are about 5% of the
maximum strength of the concrete.
The maximum compressive stress in the concrete lining from all
analyses performed is about 4.8 MPa. According to Eurocode 2 [33],
as the maximum compressive strength does not exceed
0.4fcm = 15.2 MPa, the hypothesis of linear elastic behaviour is
valid.

3.3. Simulation of construction
Tunnel construction has been simulated using a procedure
based on the convergence-confinement method [4].
The first calculation phase of the procedure involves switching
off both lining and ground elements inside the tunnel and applying
an initial pressure p0 inside the tunnel to balance the initial geostatic stress field. Afterward, the pressure p0 are reduced to
(1 À k) Á p0, where k is the proportion of unloading before the lining
is installed and is called the decompression level. In the second calculation phase, the lining elements are activated, and the pressure
applied inside the tunnel reduced to zero. For stiff linings, the
remaining ground stresses will largely go into the lining. A large
k-parameter corresponds to large unsupported lengths and/or late
installation of the tunnel lining. In this case, ground deformations
will be relatively large, whilst structural forces in the lining will be
relatively low. Conversely, a smaller k-parameter leads to reduced
ground deformations and larger structural forces in the lining.
In 2D numerical analyses of open face tunnels, a decompression
level k of around 50% is commonly used. Closed shield tunnelling,
however, is typically represented by a reduced decompression level of around 20–30% [6].
To cover a large range of tunnelling methods and to evaluate the
influence of the k-parameter on the seismic response, three values
were used in the simulations presented in this work: 5%, 20% and

50%.
3.4. Tunnel construction results
The convergence curve at the tunnel crown due to tunnel construction is shown in Fig. 9a. The dashed line represents the vertical stress evolution at the tunnel crown without lining placement,
while the continuous line represents the vertical stress after lining
placement. Lining axial load, N, and bending moment, M, at the end
of tunnel construction are shown in Fig. 9b. The curves in Fig. 9b
show that the less the decompression level is prior to lining installation, the higher the forces induced in the lining. The sections with
higher forces are in the crown (h % 0), floor (h % 180°) and sidewalls (h % 90° and 270°). The deformed shapes of the lining are
shown in Fig. 9c. As expected, higher lining deformation of the soil
occurs for larger values of k-parameter.
The profile of effective vertical stress, r0v , at the end of construction phase in the free-field and in the tunnel centre (Fig. 10), are
compared with the initial effective stress, r0v 0 . In the free-field r0v
is equal to the initial vertical effective stress, r0v 0 , as it is not affected by tunnel construction. In the profile crossing the tunnel
centre, r0v , diverges from r0v 0 for depths greater than about 7 m,

4. Seismic response
4.1. Input earthquake motions
The near source seismic scenario established by the Portuguese
National Annex to Eurocode 8 [34] was used as reference to select
input earthquake motions from the European Strong-Motion Database [35]. The following criteria were adopted: local magnitude
between 6 and 7, source-to-site distance from 15 to 35 km. Eight
seismic records from measurement sites on rock were available
and the horizontal component with higher peak ground acceleration was used.
The properties of the selected records are presented in Table 2,
namely the surface wave magnitude, Ms, the epicentral distance, R,
the horizontal peak ground acceleration, PGA, the Arias Intensity,
AI, and the mean period, Tm. Fig. 12 shows the pseudo-spectral
acceleration (PSA) of all the time histories.

4.2. Seismic analysis

As tunnels are completely embedded in the ground and the
inertial force induced by seismic wave propagation on the surrounding soil is large relative to the inertia of the structure, the
model must be able to simulate the free-field deformation of the
ground and its interaction with the structure.
In the analysis, where only vertically incident shear waves are
introduced into the domain and the lateral limits of the problem
are considered to be sufficient far not to influence the predicted
response, the ground response is assumed to be the free-field
response. Thus, the width of the model plays an important role
in ensuring the development of free-field deformation far away
from the tunnel. A sensitivity study based on similar conditions
and using modal analysis, found that the soil-tunnel interaction region can extend up to three diameters from the tunnel centre [36].
In this work, the lateral boundaries of the mesh were placed five
diameters from the tunnel centre and the equivalent node condition was imposed at the nodes of the lateral boundaries, i.e. the
displacements of nodes at the same depth on the lateral boundaries are equal in all directions.
The incident waves defined at the outcropping bedrock (elastic
half-space of the soil profile defined in Fig. 8) are introduced into
the base of the model after deconvolution performed in the linear
range. Thus, the obtained movement at the top of the elastic halfspace is composed of the incident waves and the reflected signal.


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R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

(b)

300

N:

50%
50%

M:

20%
20%

5%
5%

0

250

60
40

200
λ=5%

150

-400
λ=20%
λ=50%

100

20

N (kN/m)

Vertical stress (kPa)

Sand - t=0.45m

-800

0
-20

-1200

50

After lining placement
Without lining

0
0.000

-40

-1600
0.005

0.010

0.015

0.020

0.025

-60

0
Crown

Vertical displacement (m)

(c)

M (kNm/m)

(a)

90

180
Floor

270

360
Crown

θ (º)

6.0

Distance from tunnel centre (m)

Displacement amplification factor = 100
2 cm

4.0

2.0

θ

0.0

-2.0

-4.0

-6.0
-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

Distance from tunnel centre (m)
Undeformed shape
20%

50%
5%

Fig. 9. Effect of the decompression level on (a) the convergence curve at the tunnel crown, (b) lining forces and (c) lining deformed shape at the end of construction.

σ'v
0

200

σ'v
400

600

0

0.0

200

400

600

0.0

5.0

5.0

10.0

10.0

Depth (m)

Depth (m)

Free-field

15.0

20.0

15.0
20.0

Tunnel

25.0

25.0

σ'v0

σ'v0
30.0

30.0

50%

20%

5%

50%

20%

5%

Fig. 10. Effect of the decompression level on the vertical effective stress at the end of construction: profile in the free-field (lateral boundary), and profile in the centre of the
tunnel.


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R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

Bending moment (kNm/m)

Maximum strength
1000

Evolution of
construction stage

500
0

Start

End

-500

20

-2

Spectral acceleration (ms )

1500

15

10

5

0
-1000

0.0

0.5

Maximum strength
-1500
-2500

-2000

-1500

-1000

-500

1.0

1.5

2.0

2.5

Period (s)
0

Fig. 12. Pseudo-acceleration response spectra of the selected time-histories.

Axial force (kN/m)
50%

20%

5%
6

Fig. 11. Interaction diagrams: evolution of lining forces during the construction
stages and the maximum strength of the concrete lining.
-2

Acc. (ms )

4

In all the analyses, a time step of Dt = 0.005 s was used and an
implicit Newmark numerical integration scheme with c = 0.625
and b = 0.375 is used in the dynamic analysis [37].

2
0
-2
-4

4.3. Seismic response of the soil

-6

4.3.1. Single earthquake time-history
This section highlights the modifying effects of the stress disturbance induced by tunnel construction on the seismic response of
the soil. First, the details of the seismic response computed from
a single time-history are presented, then the results from all eight
records are collated and discussed.
The selected time-history is the Avej earthquake (number 8
from Table 2, and Fig. 13). This time-history has the highest Arias
Intensity and the second highest PGA, thus it induces large deformations on the tunnel and in the ground.
Fig. 14 presents the p0k À qk stress path of the activated deviatoric mechanism of two integration points at the depth of the tunnel
centre (depth = 20 m): one point at the free-field (lateral boundary
of the mesh, 50 m from the tunnel centre) and the other point is
near the tunnel (1 m from the tunnel sidewall).
In this figure the peak yield surface (Eq. (3) with rk = 1), the
mobilized yield surface, the critical state line, CSL, and the initial
stress line in the free-field computed with coefficient of earth pressure at rest, k0, equal to 0.5 are presented.
The stress path starts from point 1. The integration points in the
free-field coincide with the initial stress line. The integration points
near the tunnel are affected by the stress disturbance induced by
tunnel construction. The position of point 1 is consistent with the
deformation mechanism shown in Fig. 9c. For low k-value, the
tunnel deforms and pushes into soil at sidewalls, and thus the
normal stress increases leading to reduced qk and increased p0k .

0

2

4

6

8

10

t (s)
Fig. 13. Acceleration time-history number 8 (Table 2): Avej earthquake.

The maximum extent of the mobilized yield surface is identified
as Point 2, in some cases this coincides with the peak yield surface.
Point 2 is reached simultaneously at t = 4.510 s in all the analyses.
From Point 2 onwards the behaviour is highly non-linear near the
tunnel, as it produces significant hysteresis loops. The dynamic
analysis ends at Point 3.
In the free-field, the integration points exhibit relatively narrow
stress paths, with a single large loop preceded and followed by several small loops. Near the tunnel, the integration points have more
large loops and bigger variation of mean effective pressure.
The acceleration time histories at surface above the tunnel (1 m
from the sidewall) and at the free-field are shown in Fig. 15. The
accelerations in the free-field and near the tunnel are similar.
The effect of the decompression level on the acceleration is relatively small.
The time that the stress path reaches Point 2 (t = 4.51 s) is
marked in Fig. 15. This instant coincides with a negative peak
acceleration that appears after a set of cycles of high peak acceleration. So, the maximum extent of the mobilized yield surface
(Point 2) is consequence of these cycles of high peak acceleration.

Table 2
Properties of the selected records.

No.

Earthquake

Country/year

Station

Ms

R (km)

PGA (m sÀ2)

AI (m sÀ1)

Tm (s)

1
2
3
4
5
6
7
8

Friuli
Montenegro
Campano Lucano

Campano Lucano
Kozani
Umbria Marche
South Iceland
Avej

Italy, 1976
Serbia and Montenegro, 1979
Italy, 1980
Italy, 1980
Greece, 1995
Italy, 1997
Iceland, 2000
Iran, 2002

Tolmezzo-Diga Ambiesta
Ulcinj-Hotel Albatros
Bagnoli-Irpino
Sturno
Kozani-Prefecture
Assisi-Stallone
Thjorsarbru
Avaj Bakhshdari

6.5
7.0
6.9
6.9
6.5
5.9

6.6
6.4

23
21
23
32
17
21
15
28

3.50
2.20
1.78
3.17
2.04
1.83
5.08
4.37

0.80
0.74
0.45
1.51
0.23
0.27
1.35
1.74

0.39
0.72
0.94
0.82
0.33
0.28
0.36
0.28


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300

300

Initial stress line at the free-field
Peak yield surface
2
Mobilized yield surface
CSL

Initial stress line at the free-field
Peak yield surface
Mobilized yield surface
CSL

200

Free-field
5%

100

3

qk (kPa)

qk (kPa)

200

2
1

Tunnel
5%

100

3

1
0

0
0

100

200

300

400

500

0

100

200

p'k (kPa)
300

qk (kPa)

qk (kPa)

Free-field
20%

500

Initial stress line at the free-field
Peak yield surface

Mobilized yield surface
CSL

2

200

2

100

400

300

Initial stress line at the free-field
Peak yield surface
Mobilized yield surface
CSL

200

300

p'k (kPa)

3

Tunnel
100 20%

3
1

1

0

0
0

100

200

300

400

0

500

100

200

300

300

Initial stress line at the free-field
Peak yield surface
Mobilized yield surface
CSL

200

500

400

500

2

qk (kPa)

qk (kPa)

Free-Field
50%

400

Initial stress line at the free-field
Peak yield surface
Mobilized yield surface
CSL

200

2

100

300

p'k (kPa)

p'k (kPa)

1

Tunnel
50%

100

3

1

3

0

0
0

100

200

300

400

500

0

100

200

p'k (kPa)

300

p'k (kPa)

Fig. 14. p0 –q stress paths of integration points at depth of the tunnel centre (z = 20 m) for various decompression levels (time-history 8).

8

8

Free-field
(surface)

6

Point 2 (t = 4.51 s)

6

Point 2 (t = 4.51 s)

4
-2

Acc. (ms )

-2

Acc. (ms )

4

Near the tunnel
(surface)

2
0
-2
-4
-8
0

2

4

6

8

0
-2
-4

50%
20%
5%

-6

2

50%
20%
5%

-6
-8
10

t (s)

0

2

4

6

8

10

t (s)

Fig. 15. Acceleration time histories for various decompression levels (time-history 8): free-field vs. near the tunnel.

Fig. 16 shows the acceleration response spectra in the free-field
(lateral boundary) and above the tunnel at ground surface. The
response spectra in the free-field are similar for the three values
of decompression level. Above the tunnel, the response spectra

are also similar in shape, but the effect of stress disturbance can
be observed in the ordinates axe. In general, a lower decompression level leads to slightly higher spectral acceleration, compared
to the cases with larger decompression level.


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R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

40

Free-field (surface)

35
30

Spectral acceleration (ms-2)

Spectral acceleration (ms-2)

40

50%
20%
5%
Input signal (earthquake 8)

25
20
15
10
5
0
0.0

Tunnel (surface)

35
30

50%
20%
5%
Input signal (earthquake 8)

25
20
15
10
5
0

0.5

1.0

1.5

2.0

2.5

0.0

0.5

1.0

Period (s)

1.5

2.0

2.5

Period (s)

Fig. 16. Pseudo-response spectrum at ground surface for various decompression levels (time-history 8): free-field vs. tunnel.

4.3.2. All time-histories
The influence of site effects on the seismic response of the
ground is analysed in this section.
The PGA of the input motion is compared with the PGA at soil
surface in the free-field and above the tunnel for all the selected
time histories and for various decompression levels (Fig. 17).
In the free-field, the peak acceleration is nearly independent of
the decompression level, no clear trend can be observed. Above the
tunnel, the influence of the decompression level can be seen, particularly at higher input PGA and, in general, the lower decompression level lead to slightly higher PGA at the soil surface.
4.4. Seismic response of the lining
4.4.1. Single earthquake time-history
This section highlights the modifying effects of stress disturbance induced by tunnel construction on the seismic response of
the lining. Again, the results based on time-history number 8 are
presented in detail.
Fig. 18 shows the variation of the lining forces in the sidewall of
the tunnel (h = 90°, depth = 20 m) during the seismic loading. The
axial force increases during the seismic action, because progressive
plastification of the soil increases the vertical load transmitted to
the lining. This effect is greatest for the higher value of k. The bending moment varies significantly during the seismic event, due to

the ovalization of the tunnel lining. The post-event bending
moment increases in absolute value, approximately doubling the
values prior to seismic loading. The larger variation of the lining
forces occurs between 3 and 5 s approximately, which correspond
to the most intense part of the earthquake. For t > 5 s, the lining
forces remain relatively constant.

Fig. 19 plots the lining forces along the lining section (angle h) at
the end of construction, at the end of the seismic analysis and the
maximum forces envelop, for the analysis with decompression
level equal to 50% and time-history number 8. Just occasionally
the envelop curves coincide with the end of construction or the
end of the seismic analysis curves, which indicates that in general
the maximum forces occur during earthquake loading.
The absolute maximum increment in lining forces along the lining section developed during the seismic loading are presented in
Fig. 20.
The absolute maximum increment in bending moment, |DM|,
and axial force, |DN|, due to seismic loading has peaks near the
45° diagonals (h % 45°, 135°, 225° and 315°). This is consistent with
the ovalisation of a ring due to shear loading (e.g. [10]). The maximum increment in axial force occurs near the floor (h % 135° and
225°), while the maximum increment in bending moment occurs
near the crown (h % 45° and 315°). Higher k-parameter leads to
higher increment in axial force for all lining sections, while for
increment in bending moment no clear trend is noticeable.
4.4.2. All time-histories
The lining forces computed with all selected time-histories (see
Section 4.1) are analysed in this section. To assess the importance
of taking into account the seismic loading in the lining design,
Fig. 21 shows the ratio between total maximum lining forces (construction + seismic loading) and the maximum lining forces
induced by construction simulation.

The lining forces ratio grows proportionally to the PGA of the input motion and to the decompression level. The bending moment
ratio varies from 48% to 766%, while the axial force ratio varies
from 7% to 126%. When the decompression level changes from

1.0

1.0
0.8

50%

Free-field (surface)

PGA - soil surface (g)

PGA - soil surface (g)

50%
20%
5%

0.6
1:1

0.4
0.2

Above the tunnel (surface)

20%

5%

0.8
0.6

1:1

0.4
0.2
0.0

0.0
0.0

0.1

0.2

0.3

0.4

PGA - input motion (g)

0.5

0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

PGA - input motion (g)

Fig. 17. Peak ground acceleration (PGA) at the soil surface for various decompression levels and with all time-histories: free-field vs. above the tunnel.


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R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

0

50%

20%

100

5%

M (kNm/m)

N (kN/m)

-500
-1000
-1500

50
0
-50
50%

20%

5%

-100

-2000
0

2

4

6

8

0

10

2

4

6

8

10

t (s)

t (s)

Fig. 18. Lining forces during seismic loading (sidewall h = 90°, z = 20 m) for various decompression levels (time-history 8).

0

200

λ = 50%

λ = 50%

-500

N (kN/m)

M (kNm/m)

100
0
-100

-1500

End construction
End seismic analysis
Envelop

-200
0

90

180

-1000

End construction
End seismic analysis
Envelop

-2000

270

360

0

90

180

θ (º)
Crown

270

360

θ (º)

Floor

Crown

Crown

Floor

Crown

Fig. 19. Lining forces at end of construction phase, end of the seismic analysis and envelop for k = 50% and time-history 8.

1000

200
20%

50%

5%

150

| N| (kN/m)

| M| (kNm/m)

50%

100

20%

5%

750
500
250

50
0

0
90

0

180

270

360

90

0

Crown

270

180

θ (º)

360

θ (º)

Floor

Crown

Crown

Floor

Crown

Fig. 20. Maximum increment in lining forces during earthquake loading for various decompression levels (time-history 8).

150%
50%

800%

20%

0.97

5%

600%

0.97
400%

0.98

200%
R2

|Ntotal|/|Nconstruction|-1

|Mtotal|/|Mconstruction|-1

1000%

50%
125%

0.79

20%

100%

5%

75%

0.74

50%

0.76

25%

R2

0%

0%
0.0

0.1

0.2

0.3

0.4

PGA input motion (g)

0.5

0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

PGA input motion (g)

Fig. 21. Ratio between maximum total force and maximum forces during construction phase for various decompression levels and for all time-histories.


349

R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

5% to 20%, the axial force ratio increases in average 47% and the
bending moment ratio increases 31%. When the decompression level changes from 5% to 50%, the axial force ratio increases in average 184% and the bending moment ratio increases 126%. Thus,
larger construction induced stress disturbance lead to greater lining forces ratio.
The R-square coefficients are high (above 0.97 for bending moment ratio, and around 0.75 for axial force ratio), indicating strong
correlation between the lining forces ratio and the PGA of the input
motion.
Fig. 22 plots the overall maximum lining forces computed for all
time-histories, and compares them with the maximum strength of
the concrete lining.
The maximum lining forces do not exceed about 35% of the
maximum strength of the concrete, indicating that a concrete lining can be designed to withstand the combined static and seismic
loading. The maximum compressive stress in the concrete lining in
all analyses performed is about 12.6 MPa. According to Eurocode 2
[33], as the maximum compressive strength does not exceed
0.4fcm = 15.2 MPa, the hypothesis of linear elastic behaviour is
valid.

– the ground is an infinite, elastic, homogeneous and isotropic

medium;
– the tunnel and the lining are circular and the lining thickness is
small in comparison to the tunnel diameter.
Seismic actions are considered as external static forces acting
on the tunnel lining, induced by the ground distortion related to
a vertically propagating shear wave. The resulting ovalisation of
the tunnel lining is assumed to occur under plane strain conditions.
The detailed solution for the no-slip condition is summarised in
Appendix A.
The maximum free-field shear strain at the tunnel depth, cff,
introduced in the analytical solution (Table 3) is the average of
the maximum shear strain on the lateral boundary of the numerical model, at the depth of the tunnel. Fig. 23 shows a strong rela-

2.E-03

R2 = 0.91

4.5. Analytical solutions
1.E-03

The closed-form solution to predict the transverse seismic response of the tunnel, summarised in [11], was adopted for comparison with the numerical analyses. This solution takes into account
explicitly the soil-structure interaction effect under both no-slip
and full-slip conditions. This method is based on the following
assumptions:

0.E+00
0.0

0.1

Maximum strength

0.4

0.5

0.6

Fig. 23. Relation between maximum free-field shear strain at the tunnel depth, cff
with the peak acceleration of all time-histories.

1000
500

Mobilized strength ratio

Bending moment (kNm/m)

0.3

PGA input motion (g)

1500

0
-500
-1000

Maximum strength
-1500

-2500

0.2

1.00
0.75
0.50
0.25

t=4.510 s

0.00
-2000

-1500

-1000

-500

0

0

180

90

Axial force (kN/m)

50%

20%

270

360

θ (º)

5%

50%

Fig. 22. Interaction diagrams for various decompression levels: overall maximum
lining forces and the maximum strength of the concrete lining.

20%

5%

Fig. 24. Mobilized strength ratio along the lining at t = 4.510 s for various
decompression levels (time-history 8).

Table 3
Maximum increment in lining forces due to seismic loading computed by the numerical model and Wang solution (no-slip) for all time-histories.
TH

1
2

3
4
5
6
7
8

PGA (m sÀ2)

3.50
2.20
1.78
3.17
2.04
1.83
5.08
4.37

cff
9.5 Â 10À4
6.6 Â 10À4
5.1 Â 10À4
9.0 Â 10À4
5.9 Â 10À4
2.9 Â 10À4
1.4 Â 10À3
1.0 Â 10À3

Gm (kPa)

56.7
67.3
74.8
58.4
70.3
91.2
46.0
54.6

|DMmax| (kN m/m)

|DNmax| (kN/m)

Wang

50%

20%

5%

Wang

50%

20%

5%

83

58
45
78
53
26
119
89

120
66
43
112
64
27
189
152

138
72
45
116
65
27
206
178

134
71
45
121

71
31
233
188

363
296
252
352
278
172
432
377

837
764
473
781
500
258
1062
872

736
742
428
752
435
221
905

827

710
580
407
636
395
210
875
649


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R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

tion between the PGA of the input motion and the maximum freefield shear strain at the tunnel depth.
The shear modulus of the ground, Gm, (Table 3) was estimated
using the stiffness degradation curve (G/G0 À c, Fig. 7). The G0 at
the depth of the centre of the tunnel (%125 MPa) was multiplied
by the shear modulus ratio corresponding to the value of cff to
compute Gm.
According to Wang [11], the flexibility ratio F is the most important parameter to quantify the ability of the lining to resist the distortion imposed by the ground. For the cases considered, F is
between 17 and 34 with an average value of 24. Thus, according
to this parameter, the lining deflects more than the soil being excavated. Within this range of values of F, no relevant slippage between the soil and the tunnel is expected. In fact, this feature
becomes crucial only for F < 1, as, for example, in the case of the
tunnel built in very soft ground.
Fig. 24 presents the mobilized shear strength ratio, defined as
the ratio between the mobilized strength to the available strength,
distribution in the soil along the perimeter of the tunnel for

t = 4.510 s, the instant where the maximum extent of mobilized
yield surface occurs (Fig. 14).
Although the maximum mobilized strength ratio at t = 4.510 s
reaches a value of 1.0, it occurs in confined regions of the soil.
No-slip assumption remains adequate, because slip in the interface
soil-tunnel may occur only in these limited zones.
Since the effects of tunnel construction are not taken into account in analytical solutions, the comparison gives an indication
of the significance of modelling the tunnel construction for practical applications.
Table 3 summarises the increments in the axial force and bending moment in the tunnel lining, computed for no-slip conditions
using Wang’s method and those obtained with the numerical model for various decompression levels. Fig. 25 shows the deviation between these two approaches, defined as:

DeviationjDM max j ¼

jDM max jNumerical model À jDM max jWang
jDMmax jWang

are small, because the degree of non-linearity at tunnel depth is
relatively small (e.g. for cff % 5 Â 10À4 the G/G0 % 0.7). The regression lines clearly diverge and the values of the deviation are larger
for higher intensity motion, indicating that the decompression level and a larger degree of non-linearity (for cff % 10À3 the G/
G0 % 0.5) have an increasing influence.
The regression lines of the deviation of axial force are nearly
parallel for the different decompression levels, indicating that the
effect of decompression level has an important role for all input
motions. The progressive plastification of the soil above the tunnel
induced by the seismic loading increases the axial force in the lining. The deviation of axial force grows for higher input motion because this effect is not caught by the analytical solution.
5. Conclusions
The effect of stress disturbance induced by tunnel construction
on the seismic response of shallow bored tunnels was evaluated
using numerical simulations.
The presence of tunnel and associated stress disturbance does

not significantly affect the seismic response at the ground surface.
Some reduction in the peak acceleration occurs with the increasing
k-value.
During seismic loading, stress paths in the soil close to the tunnel exhibit wider perturbations in terms of both qk and p0k than in
the free-field. The stress path perturbation is also wider for
increasing k-value.
Seismic loading causes significant fluctuation in tunnel lining
forces during the event, and higher permanent lining forces in
the post-event state. This is attributed to the progressive plastification of the soil that increases the vertical load transmitted to the
lining and that increases the distortion of the tunnel.
Comparison of numerical predictions with an analytical solution highlights that the founding assumptions in the latter may result in the underestimate of tunnel lining forces resulting from
seismic loading, particularly for higher intensity motions.

ð22Þ
Appendix A

In general, Wang’s solution underestimates the lining seismic
forces in comparison with the numerical model. The deviation varies up to 110% for increment in bending moment, and from 22% to
158% for increment in axial force. The higher deviation in the incremental axial force occurs for k = 50%, while for the increment in
bending moment it occurs for k = 5%.
The deviation grows with the intensity of the input motion and,
thus, with the maximum free-field shear strain at the tunnel depth.
For low intensity motions, the regression lines of the deviation
of the bending moment are close and the values of the deviation

For no-slip conditions the maximum increments in the axial
force and bending moment in the transverse direction of the tunnel
are given by:

DNmax ¼ Æ1:15

DMmax ¼ Æ

200%

ðA1Þ

K l Em
R2 cff
6 1 þ mm

ðA2Þ

200%

50%

20%

150%

0.90
0.85
0.87

5%
100%
50%

2

R

0%
-50%

0.0

0.1

0.2

0.3

0.4

PGA input motion (g)

0.5

0.6

Deviation | N max |

50%

Deviation | Mmax |

K 2 Em
R Á cff

2 1 þ mm

150%

0.40

20%

0.25

5%

0.40

100%
50%

R2

0%
-50%
0.0

0.1

0.2

0.3

0.4

0.5

0.6

PGA input motion (g)

Fig. 25. Deviation between the maximum increment in lining forces due to seismic loading computed with Wang solution and the numerical model for all time-histories and
for various decompression levels.


R.C. Gomes / Computers and Geotechnics 49 (2013) 338–351

where Em is the mobilised soil Young’s modulus (evaluated with reference to the previously calculated shear modulus Gm), mm indicates
the corresponding Poisson’s ratio (here assumed equal to 0.3), R is
the tunnel radius and cff the maximum free-field shear strain at
the tunnel depth. The lining response coefficients are given by the
following expression:

12ð1 À mm Þ
Kl ¼
2F þ 5 À 6mm
K2 ¼ 1 þ

ðA3Þ

F ½ð1 À 2mm Þ À ð1 À 2mm ÞC Š À 0:5ð1 À 2mm Þ2 þ 2
F ½ð3 À 2mm Þ þ ð1 À 2mm ÞC Š þ Cð2:5 À 8mm þ 6m2m Þ þ 6 À 8mm

[13]

[14]

[15]
[16]

[17]

[18]

ðA4Þ
where F is the flexibility ratio:

F¼

3
2
t ÞR

Em ð1 À m
6Et Ið1 þ mm Þ

[19]

ðA5Þ
[20]

with I corresponding to the moment of inertia of the tunnel lining in
the transverse direction, Et is the lining Young’s modulus, mt the lining Poisson’s ratio and C the compressibility ratio:

C¼

Em ð1 À m2t ÞR
2Gm ð1 À m2t ÞR
¼
Et tð1 þ mm Þð1 À 2mm Þ
Et tð1 À 2mm Þ

[21]
[22]

ðA6Þ
[23]

with t corresponding to the thickness of the tunnel lining in the
transverse direction.

[24]
[25]

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