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: ruigomes@civil.ist.utl.pt

0266-352X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.compgeo.2012.09.007

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.

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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 ½kPa

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 Ã Þ

Ã

1Àr

rÃ

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

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.

348

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 À 0:5ð1 À 2mm Þ2 þ 2

F ½ð3 À 2mm Þ þ ð1 À 2mm ÞC 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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Contents lists available at SciVerse ScienceDirect

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: ruigomes@civil.ist.utl.pt

0266-352X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.compgeo.2012.09.007

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

339

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 ½kPa

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 Ã Þ

Ã

1Àr

rÃ

with r Ã 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

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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.

344

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

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.

347

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.

348

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

350

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 À 0:5ð1 À 2mm Þ2 þ 2

F ½ð3 À 2mm Þ þ ð1 À 2mm ÞC 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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