-
ed
loni
logyngsentermseismalt we anand
dwayst con) concrunnelnot crnsideraelievesreinfoions a
schungsprojekte/e-laufende/e-fp-laufend-b3.html). Other
documentslike the French Recommendations for plain concrete in
tunnels(AFTES, 2000) adopt indirect criteria for crack control,
i.e. theyplace limits on the residual compression zone, by
requiring thatthe eccentricity of the axial load e =M/N (where M is
the bendingmoment and N the axial load) should not exceed 30% of
the liningthickness.
Noticeable differences exist among current codes with respectto
the assumptions made for the verication of unreinforced lin-
s still subsroom for improving/rening the existing procedures
for theof unreinforced concrete linings. Moreover, the
paramountparameters like the crack width, which are difcult to
esreliably using elastic methods, point to the need of using
sophisti-cated methods of analysis, namely nonlinear nite element
analy-sis, as part of the design process of plain concrete linings
and/or forcalibrating simpler methods for practical design.
The present study is a contribution in this direction, using
exist-ing regulations as a starting point for introducing
appropriate anal-ysis methods that allow proper checking of the
pertinent
Corresponding author. Tel.: +30 2310995662.E-mail addresses:
[email protected] (V.K. Papanikolaou), Andreas.Kappos.1@
Tunnelling and Underground Space Technology 40 (2014) 127140
Contents lists availab
ro
w.ecity.ac.uk (A.J. Kappos).allowable crack criteria for
unreinforced linings are the subjectof current research (see
http://www.bast.de/nn_74576/EN/E-For-
but are not required in others (DAUB).The above remarks make it
clear that there i0886-7798/$ - see front matter 2013 Elsevier Ltd.
All rights
reserved.http://dx.doi.org/10.1016/j.tust.2013.09.016tantialdesignrole
oftimateicted to waterproong membranes by the steel
bars.Unreinforced concrete linings are expected to crack and the
ex-
tent of cracking is the most critical design criterion in such
linings.It is notable that modern design specications for tunnels,
like theGerman ZTV-ING (BASt, 2007), or the American Technical
Manualfor Design and Construction of Road Tunnels (FHWA, 2009)
donot contain any specic requirements for this case; in fact,
(and also the compressive stresses) under the design M and
N.Other documents like the AFTES (2000) recommendations ignorethe
tensile strength of concrete and the basic design vericationis a
limitation of the eccentricity (see above); a similar procedureis
adopted in the German Recommendations for Unreinforced Lin-ings
(DAUB, 2007). Further discrepancies exist in shear verica-tions,
which are mandatory in some codes (FHWA, Eurocode 2)1.
Introduction
Although the lining of modern roastructed in reinforced
concrete, colead to the use of unreinforced (plainditions are met,
mainly when the trock and when dynamic loads arethe lining.
Notwithstanding cost coconcrete has the advantage that it rproblems
associated with the use ofpaction of concrete in congested
regtunnels is typically con-siderations sometimesete when the right
con-is constructed in soliditical for the design oftions, the use
of plainconstruction from thercement bars, i.e. com-nd possible
damage in-
ings against bending moment and axial load, in particular with
re-spect to the way the tensile strength of concrete is taken
intoaccount. The European code for concrete, Eurocode 2
(CEN,2004a), includes rather detailed provisions for plain
concrete(meant for static loading only) and species that tensile
strengthof concrete can be taken into account; however, the
pertinent de-sign equation adopted by Eurocode 2 ignores this
strength andonly involves the compression strength and the
eccentricity (e).The FHWA (2009) Manual requires a check of the
tensile stressesNonlinear analysisPractical nonlinear analysis of
unreinforc
Vassilis K. Papanikolaou a,, Andreas J. Kappos baCivil
Engineering Department, Aristotle University of Thessaloniki, P.O.
Box 482, ThessabDepartment of Civil Engineering, City University
London, London EC1V OHB, UK
a r t i c l e i n f o
Article history:Received 23 November 2012Received in revised
form 5 July 2013Accepted 27 September 2013Available online 22
October 2013
Keywords:TunnelsConcreteFinite elementsModelling
a b s t r a c t
A comprehensive methodoforced concrete tunnel liniplane nite
element represinterface conditions. Furthronmental, re, blast,
andidentied and properly deand the interpretation of thdifferent
analysis methods
Tunnelling and Underg
journal homepage: wwconcrete tunnel linings
ki 54124, Greece
for modelling, analyzing and assessing the structural response
of unrein-is presented. Various modelling techniques are described,
considering theation of the lining geometry, material constitutive
laws, and boundary andore, all relevant external loading cases are
studied, including gravity, envi-ic loading. Potential pitfalls in
the modelling and analysis procedures areith. The suggested
methodology is nally applied to actual tunnel liningsalysis results
leads to important conclusions regarding the applicability ofthe
performance of unreinforced concrete linings.
2013 Elsevier Ltd. All rights reserved.
le at ScienceDirect
und Space Technology
lsev ier .com/ locate/ tust
-
performance criteria, focussing on deformation quantities. It
has tobe pointed out here that use of advanced analysis tools
(nonlinearnite elements or nite differences) is already part of
actual designpractice (e.g. Corigliano et al., 2011), at least in
important tunnels,and are specically recognised by pertinent
documents like theAmerican FHWA(2009). The present study originated
in a very prac-tical context, i.e. the assessment of the capacity
of the unreinforcedconcrete linings proposed by the Constructors
team for parts ofthree major tunnels (up to 6 km long) currently
being built inGreece, and nonlinear nite element analysis was used
both bythe consultants of the Constructor and by the authors who
actedas reviewers of the design of the tunnel linings. The paper
attemptsto provide proper guidance for the reliable and efcient use
of theaforementioned advanced analysis tools by designers with
ade-quate experience in the eld. Furthermore, it addresses for the
rsttime the detailed analysis of plain concrete linings subjected
to seis-mic loading, an issue that is typically ignored in previous
studies.
(in reality the length of a tunnel segment is about 14 m), a
two-dimensional (2D) nite element formulation under planestrain
conditions (ez = 0, rz 0) is adopted. Four-node quadrilat-eral nite
elements with typical size of 7.5 cm and thickness of1.0 m are
utilized, leading to a dense mesh of a total of 2538 and4050
elements (6 elements across lining thickness) for the horse-shoe
and invert geometries, respectively (Fig. 2). The mesh densityis
selected with a view to striking a balance between (a)
adequateresolution in analysis results and (b) heavy computational
require-ments, considering the use of advanced nonlinear material
modelsand boundary conditions.
For the unreinforced concrete vault, a nonlinear
fractureplasticmaterial model (Cervenka et al., 1998; Cervenka and
Papanikolaou,2008) is assigned to the vault elements, capable of
capturingimportant aspects of concrete behaviour such as cracking,
crush-ing, and crack closure. On the contrary, the reinforced
concretefootings and invert are modelled using an elastic isotropic
material(concrete elastic properties), since cracking in these
regions is gen-
128 V.K. Papanikolaou, A.J. Kappos / Tunnelling and Underground
Space Technology 40 (2014) 1271402. Modelling procedures
In this section, various techniques for nite element modellingof
unreinforced tunnel linings are described, considering
geometry,material constitutive laws, and boundary and interface
conditions.Furthermore, the relevant load cases will be presented,
includinggravity, environmental, re, blast, and seismic loading.
Specicnumerical values together with corresponding analysis results
inan actual application will be presented in the next section. The
dis-cussion focuses on two typical cross-section types used in
roadwaytunnels, nevertheless the modelling approach used can be
appliedto other cross-sections as well.
Two typical lining cross-section prototypes are considered(Fig.
1), the rst of the horseshoe type with strip footings, andthe
second of the closed type with an invert (typically requiredin weak
rock conditions); the vault geometry is identical in bothsections.
The outer radius of the vault of the actual sections de-picted in
Fig. 1, from the centre of the trafc lane, is 7.85 m andthe lining
thickness is 0.45 m. For modelling and analysis, the niteelement
package ATENA (Cervenka et al., 2012), specically devel-oped for
plain and reinforced concrete structures, is employedthroughout the
present study.
2.1. Modelling of the lining section
Following the usual assumption of innite tunnel lengthFig. 1.
Typical lining sections: horseshoe (left) and closeerally not
expected due to the presence of reinforcement. An alter-native
approach would have been to model the above regionseither with
explicit or smeared reinforcement, however this wouldhave led to
increased computational demands without consider-able benet.
Boundary conditions between the tunnel lining and the
sur-rounding rock-mass are modelled following the familiar
Winklerspring approach (Dutta and Roy, 2002). Specically,
unilateralcompression-only, linearly distributed springs are
applied alongthe vault and the foundation (footings/invert) outer
boundary(Fig. 3). The spring compression stiffness (KV) is
calculated consid-ering plane strain conditions, as follows:
KV Es1 m2skN=m2
kN per meter of spring contraction per meter of line length1
where (Es) and (ms) are the subgrade reaction modulus and
Poissonsratio of the rock mass, respectively. Furthermore, the
horizontalfriction between footings and underlying rock mass is
modelledwith elastic bilateral springs of stiffness KH, typically
equal to 3050% of KV (Fig. 3, left).
Another important modelling aspect is the consideration of
theconstruction joints between the vault and the foundation.
Thesejoints are modelled using interface elements, connecting
theadjacent vault and foundation line boundaries. The interface
ele-ments are congured as unilateral contacts, incorporating
aconcrete-to-concrete friction coefcient (l), which depends ond
section with invert (right). Courtesy of EOAE SA.
-
oe s
tunn
Undesurface roughness (typical value 0.8). It is noted that
contact mod-elling is prone to introducing instabilities in the
numerical solutionand should be handled with caution. To improve
stability, it is rec-
Fig. 2. Finite element meshes: horsesh
Fig. 3. Boundary conditions between
V.K. Papanikolaou, A.J. Kappos / Tunnelling andommended to use a
low value of cohesion (e.g. 0.1 kN). In general, apreliminary
geometrically-nonlinear analysis with elastic materialproperties is
recommended, in order to identify and overcome anymodelling
pitfalls.
2.2. Modelling of various load types
In the following subsections, various load types and load
com-binations describing static, quasi-static, environmental,
accidental,re, and seismic loading on unreinforced concrete tunnel
liningsare described, with specic reference to the aforementioned
niteelement models.
2.2.1. Static, quasi-static, environmental, and accidental
loadingThe rst load case that develops after construction is the
lining
dead load, dened as a gravitational body load automatically
calcu-lated from nite element tributary areas, on the basis of the
densityof concrete (typically qc = 2400 kg/m3 for normal
unreinforced con-crete). This is followed by the development of
early creep effects,which can be simply represented by an
equivalent uniform temper-ature decrease (Dt) in concrete nite
elements. A similar load typeis subsequently applied to account for
environmental temperaturevariations, usually during winter
(typically Dt = 10 C in southernEurope). Furthermore, pavement and
trafc (on the inverts topboundary) live loads are applied in the
form of a constant distrib-uted force, as depicted in Fig. 4. The
last case in the above sequenceis the external rock mass load,
which gradually develops in a longterm fashion, after the
excavation and construction of the tunnel,depending on the physical
properties of the rock mass (higher val-ues correspond tomore
deteriorated rockmass). This may be repre-sented by a linearly
varying distributed force, normal to the tunnellining, with an
upper bound of p1 at the vault key and a lower boundof p2 at the
vault base (Fig. 5), further decreasing to zero at the in-vert oor.
The aforementioned sequential application of static, qua-si-static
and environmental loads is considered as the basiccombination (also
referred to as a load sequence) for the assessment
ection (left) and closed section (right).
el lining and surrounding rock mass.
rground Space Technology 40 (2014) 127140 129of the tunnel
structural response.A special situation that also has to be
assessed is the tunnel
structural integrity at the time of formwork demoulding. Thiscan
be conveniently modelled by modifying concrete materialparameters,
in order to reect the concrete strength at the timeof demoulding
(usually a small fraction of the 28-day strength).This updated dead
load case should be subsequently followed bythe application of
hydration heat, in the form of uniform tempera-ture increase (+Dt)
in concrete nite elements.
Accidental loading is another important case that should be
ad-dressed in design or assessment of tunnel linings. Three
accidentalload cases are described hereinafter, namely blast,
hydrostatic, andwedge loading, which should be properly combined
with the afore-mentioned static and environmental loads before
analysis:
(a) Accidental hydrostatic loading (due to possible blocking
ofthe drainage pipe next to the construction joint) is
modelledusing a horizontal triangularly distributed force on
theblocked side of the lining (Fig. 6, left). The load value atthe
base is p = qw g h, where qw is the density of water(1000 kg/m3)
and h is the vault height. This case succeedsthe [dead + creep +
pavement + rock mass] load sequencein the analysis.
(b) Blast (explosion) loading is modelled using a
constantinverted distributed force on the inner lining
boundary(Fig. 6, right). This case succeeds only the [dead + creep]
loadsequence in the analysis for safety, since it counteracts
bothenvironmental and rock mass loading.
(c) Wedge loading is a special case that emerges from the
acci-dental abruption of large rock volumes from the tunnel
sur-rounding rock mass, forming blocks (wedges) as shown inFig. 7
(left). The tendency of the wedge to detach from thesurrounding
rock mass will trigger an arching effect in the
-
nde130 V.K. Papanikolaou, A.J. Kappos / Tunnelling and Uinitial
stress eld above the wedge; in other words the void(the
discontinuity) created in the rock-mass due to thedetachment of the
wedge will magnify the stresses on eitherend, causing a local
overstressing in these areas. This can bemodelled by modifying the
aforementioned rock mass loadcase (Fig. 5) as follows: by operating
only on the half-lengthof the vault (from key to construction
joint), the rock mass
Fig. 4. Application of
Fig. 5. Application of
Fig. 6. Application of hydrostatic (
Fig. 7. Rock volume abruption (left) andrground Space Technology
40 (2014) 127140load is augmented in both half-length ends (p3) and
reducedin-between (p4). The augmentation length is expressed as
afraction (a) of the total half-length (l) and controls the
sever-ity of the wedging effect. Typical values for (a) are in
therange of 0.20.3. The wedge case succeeds the[dead + creep +
pavement] load sequence in the analysis.
pavement load.
rock mass load.
left) and blast loading (right).
application of wedge loading (right).
-
It is nally noted, that for carrying out the nonlinear analysis,
anadequate number of load steps (typically 20) is assigned for
eachload case participating in the load sequence. The solution
algo-rithm employed is a standard NewtonRaphson scheme with
linesearches, since snap-through and snap-back phenomena (i.e.
con-crete softening) are generally not expected herein. The
conver-gence criteria of the solution algorithm should be
carefullyselected, to maintain balance between numerical stability
andcomputational cost.
2.2.2. Fire loadingThe structural response of the unreinforced
concrete tunnel lin-
ings under re is a special accidental loading type that
requiresextensions and modications to the former conventional
nonlinearanalysis practices, mainly in the aspects of nite element
model-
V.K. Papanikolaou, A.J. Kappos / Tunnelling and Undeling,
material constitutive models and solution procedures(Cervenka et
al., 2004; Bergmeister, 2008). More specically, theanalysis
procedure is currently twofold; a heat transfer (transport)analysis
based on a specic temperature prole is rst performed,providing
concrete strain elds vs. re exposure time. These calcu-lated strain
elds are subsequently injected into a nonlinear static(mechanical)
analysis, which provides concrete stress and crackingevolution
during re exposure. It is assumed for simplicity that thetwo
aforementioned analysis procedures are uncoupled.
In order to conduct heat transfer analysis, it is advisable to
rstdensify the nite element mesh along the re front line to
providehigher resolution in the strain eld results that will be
introducedin the ensuing static analysis. Furthermore, to avoid
potentialnumerical instabilities during static analysis, the re
front is mar-ginally shifted away from the construction joint
interfaces (Fig. 8).The material heat transfer model employed,
assigned to concretenite elements (Cervenka et al., 2012) is an
advanced heat andmoisture diffusion model that requires the
following parameters:
(a) Concrete type (siliceous or calcareous).(b) Concrete
water-to-cement ratio (w/c).(c) Concrete thermal conductivity (kc)
vs. temperature (T).(d) Concrete volumetric specic heat (cV) vs.
temperature (T).
The evolution of the thermal conductivity (kc) and
volumetricspecic heat (cV) for concrete are dened according to
Eurocoderequirements for re design (EN1992-1-2, 2004b) and
depictedin Fig. 9. To be on the safe side, the lower limit of the
kcT curveis considered. Subsequently, the re boundary line (see
Fig. 8) isassociated with a temperature prole, i.e. the evolution
of temper-ature vs. re exposure time. Since the pertinent Eurocode,
EN1992-1-2 (CEN, 2004b) does not include explicit recommendations
forFig. 8. Vault mesh modication for heat transfer analysis.tunnel
re loading, the employed re scenarios are selected fromthe
literature (FIT, 2005; UPTUN, 2008; b, 2008, 2010). More
spe-cically, two proles are selected (Fig. 10), the standard
curve(Einheitstemperaturkurve ETK) of EN1991-1-2 (CEN, 2002) andthe
hydrocarbon re (modied hydrocarbon curve mHC), repre-senting the
most unfavorable scenario, both operating on a totalexposure period
of three hours (180 min). The coefcient of heattransfer by
convection (ac) is taken equal to 25 and 50 W/m2 Cfor ETK and mHC
proles, respectively, and the emissivity (er) asconstant, equal to
0.7 (EN1992-1-2, 2004b). The convection andemissivity heat ux is
calculated as follows:
qn ac Tg Tb er r T4g T4b 2
where qn is the heat ow at the re exposed boundary (W/m2), rthe
StefanBoltzmann constant (5.67 108 W/m2 C4), Tg theabsolute
temperature of radiation source (C), Tb is the structureboundary
temperature (C).
The strain elds calculated from the above heat transfer
analy-sis, are imposed on the original model, following the
application ofthe [dead + creep + pavement + rock mass] load
sequence. Sinceconcrete properties are temperature-dependent, they
are hereinintroduced as variables (rather than constants) in the
nonlinearstatic problem, following EN1992-1-2 (2004b) and specic
soft-ware recommendations (Cervenka et al., 2012). Variables
includeconcrete elastic modulus (Ec), compressive strength (fc),
tensilestrength (ft), plastic strain (ecp), fracture energy (GF)
and compres-sive failure displacement (wd), in ratio form with
respect to normal(15 C) temperature (Fig. 11). The nonlinear static
analysis is nallyperformed using a ne load step corresponding to 15
and 30 s ofre exposure (total 720 and 360 steps) for mHC and ETC.
curves,respectively. It is also possible to account for uneven step
size, tooptimize the computational cost.
2.2.3. Seismic loadingIn countries of high seismicity,
engineering experience has
shown that seismic excitation in the vicinity of a tunnel may
causeconsiderable structural damage (Dowding and Rozen, 1978;St.
John and Zahrah, 1987; Wang, 1993), especially to
unreinforcedconcrete linings. Consequently, the trend in
international codes ofpractice (Hashash et al., 2001) is to design
tunnel linings againstaxial strains and curvatures imposed by the
surrounding rock mass(where it can be assumed, for simplicity, that
the lining strains arepractically identical with those of the
adjacent rock mass, i.e. theseismic shear wave medium), as well as
against shape change(caused by shear deformation of the ground),
also referred to asovalization.
Consequently, the nal load type investigated in the presentstudy
is the seismic excitation of the tunnel lining, a situation thatis
often overlooked in similar studies due to the complexity in-volved
in its modelling and the associated high level of uncertainty.Two
modelling procedures regarding rock mass representation
aresuggested herein, one for the aforementioned rock mass
loadingleading to curvature of the longitudinal axis of the lining,
whichis based on the Winkler spring approach presented in previous
sec-tions, and the other for shape change loading, following the
moredemanding elastic continuum modelling approach.
The rst procedure is based on the same models already de-scribed
for nonlinear static analysis. In line with the plane
strainassumption, the seismic loading is applied to the lining
section(Fig. 12, left) as a horizontal body load (asymmetrical),
which isautomatically calculated from nite element tributary areas
usingthe specic weight of concrete, multiplied by the seismic
coef-
rground Space Technology 40 (2014) 127140 131cient (a). The
seismic coefcient is usually provided from NationalAnnexes to Code
regulations (e.g. EN1998-1, 2004c), correspondingto the seismic
zone where the structure is situated, while here it
-
into rock, but it is included herein for the sake of
completeness,since it can be a critical case in other tunnel types
(like cut-and-cover).
The second and computationally more demanding approach
ismodelling the ovalization effect due to the shape change
imposedby ground deformation caused by seismic excitation. The
ovaliza-tion load can be represented by a maximum shear distortion
(c)of the ground (Fig. 13, left). The shear distortion value can be
cal-
0.40
0.60
0.80
1.00
1.20
1.40c (W / moC)
(oC)
Lower limit
1500
2000
2500
3000
3500
4000
4500
5000
0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200
cv (kJ / m3oC)u = 3 %
(oC)
= 2400 kg/m3
Fig. 9. Evolution of thermal conductivity and volumetric specic
heat of concrete vs. temperature according to EN1992-1-2.
600
800
1000
1200
1400
mHC ETK
oC
c = 25 W / m2oC
c = 50 W / m2oC
Emissivity = 0.7
132 V.K. Papanikolaou, A.J. Kappos / Tunnelling and Underground
Space Technology 40 (2014) 127140was estimated from a site-specic
seismic hazard analysis (EAEE
0
200
400
0 30 60 90 120 150 180
min
Fig. 10. Fire temperature proles.SA internal report by Pavlidis
et al., Dec. 2010). Furthermore, asymmetrical seismic load, mainly
resulting from vertical groundshaking, is also considered, acting
simultaneously on both sidesof the vault. This is realized by
applying earth pressure loads atrest, suitably amplied to account
for seismic excitation (e.g.p1 = 1.5 a qc g h, p2 = 0.5 a qc g h,
where (h) is the vault height,Fig. 12, right). Similarly to re
loading, seismic loads are appliedsubsequent to the [dead + creep +
pavement + rock mass] load se-quence. This loading case is
generally not critical for tunnels bored
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000 1200
Ec / Ec (=15C)
(C)0.0
0.2
0.4
0.6
0.8
1.0ft / ft (=15C)
1.0
2.5
4.0
5.5
7.0
8.5
10.0cp / cp (=15C)
(C)1.0
1.2
1.4
1.6
1.8
2.0GF / GF (=15C)
0 200 400 600
0 200 400 600 800 1000 1200 0 200 400 600
Fig. 11. Temperature-dependculated as (St. John and Zahrah,
1987; Wang, 1993):
c VS=CS 3where (VS) is the maximum expected particle velocity
from shearwave and (CS) is the effective shear wave propagation
velocity. Thisrequirement inevitably requires the nite element
representation ofthe surrounding rock mass. To this purpose, an
adequately largerock mass prole (50 50 m), corresponding to 35
times the tun-nel gross dimensions is superimposed on the original
model, usingelastic isotropic material corresponding to the rock
mass elasticproperties (see Eq. (1)) as depicted in Fig. 13, right.
To optimizethe computational cost, the mesh topology of the elastic
continuumis described by a smooth transition from the (original)
very denselining mesh to a sparse discretisation at the model
boundary. Forthis reason, two different meshing zones are employed
(depictedwith different colour in Fig. 13).
As far as the new boundary conditions are concerned, the
origi-nal distributed springs along the vault outer boundary are
removedand replaced by unilateral contact interface conditions
(withoutfriction), corresponding to the waterproong membrane
placed
0.4
0.6
0.8
1.0fc / fc (=15C) (C)0.0
0.2 (C)
(C)1.0
1.5
2.0
2.5
3.0
3.5wd / wd (=15C)
(C)
800 1000 1200 0 200 400 600 800 1000 1200
800 1000 1200 0 200 400 600 800 1000 1200
ent concrete properties.
-
and loading parameters, followed by the presentation and
discus-
Fig. 12. Asymmetrical (left) and symmetrical (right) seismic
loading for Winkler spring models.
ma
V.K. Papanikolaou, A.J. Kappos / Tunnelling and Underground
Space Technology 40 (2014) 127140 133between concrete and rock mass
(or temporary lining, in othercases) during construction (Fig. 14).
However, the horizontal fric-tion between footings and underlying
rock mass is preserved. Dur-ing nonlinear analysis, the dead and
creep load are rst imposed
Fig. 13. Ovalization loading (left) and rockwith restrained
model boundaries, followed by the application ofshear distortion
(c) prole on the vertical model boundary (usingprescribed
displacements) and the restraint of the horizontal base(Fig. 13,
left). It is nally noted, that a simple and advisable validitycheck
for the integrity of the continuum model, is the comparisonof the
vault key vertical displacement due to (the same) gravityload,
between the continuum and the Winkler spring model. Inthe present
application, a small difference of 5% was found, whichis justied by
the fact that springs can contract independently ofeach other,
which does not apply in a continuum model (Duttaand Roy, 2002).
Nevertheless, this small difference implies thatthe application of
the spring approach on the other case studiespresented in this
paper is adequate, let alone the substantially low-er computational
cost required, which is convenient for practicalapplications.
In the next section, the aforementioned modelling techniquesare
applied to actually constructed tunnels in the region of
centralGreece with detailed reference to the pertinent constitutive
model
Fig. 14. Boundary conditions betwesion of selected nonlinear
analysis results.ss modelling as elastic continuum (right).3. Case
studies and discussion of analysis results
In this section, various case studies and representative
nonlin-ear analysis results, based on actually constructed tunnels
in cen-tral Greece (PATHE project, Maliakos-Kleidi section), will
bepresented and discussed. These case studies are based on two
dif-ferent lining geometries (see Fig. 1) and various load
cases/combi-nations, as described in the previous section. The
verications arecarried out with the aid of both inelastic
(nonlinear FEA) and mate-rial-elastic analysis methods, while
several criteria are checked, asprescribed by codes and/or
international recommendations. A de-tailed list of the considered
concrete and rock mass material prop-erties, as well as of load
magnitudes, for various load cases arepresented in Tables 1 and 2,
respectively. The basic, demouldingand blast combinations are
applied to both the horseshoe andclosed-section models, while the
hydrostatic, wedge, re, and seis-mic load combinations are applied
to the horseshoe model, corre-sponding to a lining constructed in a
different location (different
en tunnel lining and rock mass.
-
ground properties). Note that the seismic acceleration was
denedon the basis of the previously mentioned site-specic seismic
haz-ard analysis for the site of the tunnel analysed, while the
shearstrain was estimated from the acceleration and the properties
ofthe rock mass, using empirical relationships from the
literature.
As noted in the introduction, for unreinforced concrete
tunnellinings, the key performance criterion is crack width; this
can bechecked either indirectly (through simplied material-elastic
anal-ysis for bending moment and axial loading, followed by the
estima-tion of natural axis depth), or directly, through advanced
nonlinearanalysis that permits reliable estimation of crack width.
The hereinemployed analysis software (ATENA, Cervenka et al., 2012)
is capa-ble of providing both cracking propagation/orientation and
corre-sponding width. Consequently, the assessment of the
tunnellining response under various loading conditions can be
primarilyperformed in terms of (a) maximum crack width and (b)
remaininguncracked zone across the lining section thickness at the
end of theanalysis. The latter can be estimated either from
software crackingvisuals or from colour contours corresponding to
the initial (un-cracked) concrete tensile strength (fctd) (Fig. 15,
left). It is notedhowever, that a proper estimation of the
uncracked zone shouldnot only consider the above reduction of the
concrete tensilestrength itself but also the actual severity
(width) of cracking.For unreinforced concrete, tunnel designers
typically use the crite-rion that crack width should not exceed 1.0
mm, while the remain-ing uncracked zone should be at least half of
the total liningthickness (DIN 1045-1, 2000; AFTES, 2000).
(a) Flexural capacity ratio (kM =MEd/MRd) for a xed axial
load,where MEd is the moment derived from analysis (integratedfrom
stresses across the lining thickness) andMRd is the ex-ural
capacity, calculated from material properties and sec-tion
geometry.
(b) Axial capacity ratio (kN = NEd/NRd), where NEd is the
axialforce from analysis (compression negative) and NRd is theaxial
capacity of the section, calculated as follows(EN1992-1-1,
2004a):
NRd fcd b hw 1 2 e=hw 4
where e =MEd/NEd is the section eccentricity (Fig. 15, right),
applica-ble for hw/2 > e > 0, yet with an upper recommended
value of 0.3 hw(AFTES, 2000). In the present application, b = 1.0 m
and hw = 0.45 mare considered, whereas safety conditions correspond
to capacityratios k 6 1. It is noted, that if concrete cracking
does not occur inthe initial nonlinear analysis, it is not
necessary to repeat the anal-ysis considering elastic material
properties.
Other possible evaluation criteria are the vertical deection
atthe vault key, the overall section deformed shape and the
interfacebehaviour at the construction joints and concrete-rock
massboundaries (ovalization case). The application of all
aforemen-tioned evaluation procedures along with the respective
analysis re-sults are presented in the subsequent subsections.
0.3
pe
e*
134 V.K. Papanikolaou, A.J. Kappos / Tunnelling and Underground
Space Technology 40 (2014) 127140Since specic recommendations for
assessing the nonlinearstructural response of tunnel linings are
not generally providedin codes of practice, selection of
appropriate criteria is requiredto handle the present problem from
a conventional/code perspec-tive, as necessary for comparison and
validation reasons. More spe-cically, the nonlinear nite element
analysis is repeated here,considering elastic isotropic material
properties (analysis will stillbe geometrically nonlinear due to
boundary conditions), calculat-ing the section capacity factor (k)
for different action effects, i.e.:
Table 1Concrete and rock mass material properties.
Concrete Units Load case type
Demoulding
Ec GPa 32m 0.2acc, act 0.85cc 1.2fck MPa 2fcd = accfck/cc MPa
0.94fctm MPa 0.48fctk,0.05 MPa 0.33fctd = actfctk,0.05/cc MPa
0.18GF MN/m 1.191 105ec 2.20ecp = ecfctm/Ec 2.17w/c qc,unreinforced
kg/m3 2400qc,reinforced kg/m3 2500a 1/C 105
Rock mass Units Lining ty
Horsesho
Es MPa 1000ms 0.3KV (Eq. (1)) MN/m2 1098.9KH MN/m2 329.7q kg/m3
s
* For basic, demoulding and blast combinations.** For
hydrostatic, wedge, re and seismic combinations.Horseshoe** Closed
section*
800 3000.2 0.25833.3 320.0416.7 2600 Static Accidental/seismic
Fire
32 32 320.2 0.2 0.20.85 0.85 1.01.5 1.2 1.030 30 3017.00 21.25
30.002.90 2.90 2.902.00 2.00 2.001.13 1.42 2.007.241 105 7.241 105
7.241 1052.20 2.20 2.201.66 1.54 1.263.1. Static, quasi-static and
environmental loading
Fig. 16 shows the analysis results from the basic[dead + creep +
temperature + pavement + rock mass] load se-quence for both
horseshoe and closed section lining sections. Theresidual concrete
tensile strength (fctd) contours are displayed onthe model deformed
shape (50), overlaid by the correspondingcracking pattern.
Moreover, the vertical deection (d) history atthe vault key is
depicted, together with the maximum crack width,
-
(left
UndeTable 2Load values for various load cases.
Load case Units
Hydration CCreep CEnvironmental (winter) CPavement (Fig. 4)
kNRock mass (Fig. 5) kNBlast (Fig. 6, right) kNHydrostatic (Fig. 6,
left) kNWedge-a (Fig. 7) kNWedge-b (Fig. 7) kNSeismic, symmetric
(Fig. 12, left) Seismic, asymmetric (Fig. 12, right) kNSeismic,
ovalization*
* From Eq. (4)? c = VS/CS = 0.41/650 0.006.
Fig. 15. Approximation of the uncracked zone
V.K. Papanikolaou, A.J. Kappos / Tunnelling andremaining
uncracked zone and equivalent eccentricity, after theend of the
analysis.
From the analysis of the basic load combination, it is
observedthat crack widths and remaining uncracked zones remain well
be-low the acceptable limits (Fig. 16, right). This is also conrmed
bythe calculated equivalent section eccentricity from elastic
analysis,implying that simplied code recommendations may yield
reliableestimates in this case when advanced nonlinear analysis is
notavailable. Furthermore, from the vault key deection history, it
isobserved that mainly creep and temperature load stages
inducelarger displacements on the closed-section model due to its
geom-etry. Therefore, it is concluded that for the basic load
combination,the overall structural response of the unreinforced
tunnel lining issatisfactory.
As far as the demoulding [dead + hydration] load sequence
isconcerned, which incorporates very low values for
concretestrength (see Table 1), negligible cracking and practically
zerodeection is observed at the vault key after the end of the
analysis,for both the horseshoe and closed-section models. This is
justiedby the fact that the initial deection/cracking caused by
gravityload is counteracted by the subsequent concrete expansion
in-duced by hydration heat loading (Fig. 17). This favourable
responseis revealed due to the use of the advanced concrete
constitutivemodel, which can successfully handle crack closure.
Consequently,it is considered that the lining response during
demoulding re-mains well within safe limits.
3.2. Accidental loadings (static)
The rst accidental load combination analysed is
hydrostaticloading due to possible blocking of the drainage pipe.
Analysis isperformed on the horseshoe model and results indicate no
crackingalong the vault section, a small maximum vertical deection
ofValues of load parameters
Horseshoe model Closed-section model
Dt = +15.0 Dt = +15.0Dt = 7.7 Dt = 7.7Dt = 10.0 Dt = 10.0p =
10.0 p1 = 10.0, p2 = 33.3p1 = 180.0, p2 = 108.0 p1 = 200.0, p2 =
120.0p = 100.0 p = 100.0p = 90.0 a = 1/5, p3 = 270.0, p4 = 36.0 a =
1/3, p3 = 270.0, p4 = 36.0 a = 0.35 p1 = 129.6, p2 = 43.2 c =
0.006
) and denition of section eccentricity (right).
rground Space Technology 40 (2014) 127140 1356.6 mm (which is
1/2000 of the lining inner diameter, well belowany code limits that
vary from 1/500 to 1/250 of the span for hor-izontal straight
members), and an equivalent eccentricity of8.1 cm. This is justied
by the fact that hydrostatic loading is con-centrated at the vault
base and cannot cause important horizontaldeections (0.48 mm at
vault key). Since all evaluation criteria arefully satised, it is
considered that the hydrostatic combination isof minor importance
and can be safely ignored.
On the contrary, under wedge loading, which replaces the
con-ventional rock mass case, considerable concrete cracking is
esti-mated, especially when high (a) ratios are imposed.
Theconsidered [dead + creep + pavement + wedge] load sequence
isapplied on the horseshoe model for two different cases (a =
1/5and a = 1/3) and analysis results are summarized in Fig. 18. It
is ob-served that whereas crack widths are below allowable limits,
theuncracked zone is estimated to be smaller than the
minimumallowable value (hw/2). Moreover, cracking is not localized
at thevault key bottom bre, but is also signicant at the lateral
top -bres as well. Due to asymmetrical loading, an uneven
deformedshape is developed, with relatively high values of deection
atthe vault key. Therefore, wedge loading is found to be a
criticalaccidental situation, which should be handled with caution
in thedesign and assessment of unreinforced tunnel linings.
The nal accidental case considered is blast (explosion)
loading,following the [dead + creep] load sequence. Contrary to
intuition,the analysis results do not demonstrate any concrete
cracking atall, while vault key deection is negligible. However, a
closer obser-vation of the deformed shape explains this structural
response,since a large proportion of the blast energy is released
on the gapopening of the construction joints, which are modelled as
unilateralcontacts with friction (Fig. 19, see also Fig. 3). It is
noted that, if amonolithic modelling approach was followed instead
(using a per-fect connection between vault and footings/invert),
the blast energy
-
residual fctd (MPa)
-20
-16
-12
-8
-4
0
4
max 12.78 mm
dead creep temperature pavement rockmass
max crack width : 1.19103 mm < 1.0 mm uncracked zone : 34 cm
> hw/2 = 22.5 cm
eccentricity : 6.9 cm < 0.3hw = 13.5
(mm)
136 V.K. Papanikolaou, A.J. Kappos / Tunnelling and Underground
Space Technology 40 (2014) 127140would inevitably have been
absorbed by severe axial tensile failureof the vault sides, which
does not correspond to the actual behav-iour of a properly designed
lining under blast loading.
Fig. 16. Analysis results for th
-3
-2
-1
0
1 (mm) dead hydration
Fig. 17. Demoulding load sequence and corresponding
response.
Fig. 18. Analysis results for wresidual fctd (MPa)
-20
-16
-12
-8
-4
0
4
(mm) max 19.38 mm
max crack width : 4.26102 mm < 1.0 mm uncracked zone : 28 cm
> hw/2 = 22.5 cm
eccentricity : 7.9 cm < 0.3hw = 13.5
dead creep temperature pavement rockmass3.3. Fire loading
As already described in Section 2.2.2, re analysis is
performedusing an initial heat transfer analysis (thermal
response), followedby a nonlinear static analysis (mechanical
response). For thermalanalysis, two different re proles of
increasing severity (ETK,mHC) are applied. Fig. 20 shows the
evolution of heat transferusing the standard ETK re curve at the
vault key region (responseis identical along the entire re front),
for the considered exposuretime of three hours (max 1100 C). It is
observed that after the endof the exposure, at least half of the
lining thickness remains at ini-tial temperature (15 C). Similar
results are obtained also by apply-ing the mHC prole (max 1300
C).
Based on the strain elds resulting from the above heattransfer
analysis, a nonlinear static analysis with variable
(tem-perature-dependent) material properties is performed,
following
e basic load combination.
edge load combination.
-
-6
-4
-2
0
2
(mm)
dead creep blast
Fig. 19. Analysis results for blast loading combination.
V.K. Papanikolaou, A.J. Kappos / Tunnelling and Underground
Space Technology 40 (2014) 127140 137the initial application of the
[dead + creep + pavement + rock mass]load sequence. Fig. 21 shows
the residual concrete tensile strengthcontours during re exposure,
where it is clearly observed that afterthere event, the
remaininguncracked concrete regions correspondonly to a small
fraction of the lining section. This is also visible in
thecorresponding crack width contours on the same gure,
whereunacceptable cracking (over 1.0 mm width) is visible along the
refront boundary. Furthermore, in order to evaluate capacity
factorsaccording to code recommendations (EN1992-1-1, 2004a),
analysisis repeated using elastic concretematerial with constant
properties.However, exural (MRd) and axial (NRd, see Eq. (5))
capacities are cal-culated considering the variationof concrete
strength (fcd)with tem-perature. Fig. 22 shows the evolution of
exural capacity factor (kM),axial capacity factor (kN) and
eccentricity (e) during the ETK reexposure. It is observed that all
different evaluation factors reachunacceptable limits very early
(68 min of exposure) and thereforeit is conrmed that the analysed
unreinforced lining section is inad-equate to resist standard re
loading. As far as the more demandinghydrocarbonre curve (mHC) is
concerned, it is obvious that the lin-ing response is evenworse,
exhibiting severe cracking across the en-tire lining
thickness,while simplied assessment criteria exceed theallowable
limits after only two minutes of exposure. It is
thereforeconcluded, that the consideration of re loading is of
paramountimportance in the design and assessment of unreinforced
concretelinings.
3.4. Seismic loading
The rst two seismic load cases of asymmetric and symmetric
loading, used for checking against longitudinal strains
imposedby the surrounding rock mass, are applied following the
[dead +creep + pavement + rock mass] sequence. From the analysis
re-
0:30 h 1:00 h
2:00 h 2:30 h
Fig. 20. Evolution of heat transfer at the vsults, it is
observed that these cases do not induce any concretecracking on the
lining section, which is attributed to the stronghorizontal
resistance of the surrounding rock mass (also observedfor
hydrostatic loading). Furthermore, calculated eccentricities of8.5
and 7.3 cm, respectively, indicate that the lining response iswell
within safety limits. Therefore, it is conrmed that the abovecases
are not critical for the design and assessment of tunnel lin-ings,
especially in a rock terrain (Hashash et al., 2001).
On the contrary, the assessment of the lining response
againstshape change (ovalization analysis), has yielded interesting
results.Considering the rock mass as an elastic continuum
surrounding theconcrete liningmodel, the imposed shear distortion
(c) is applied ina cyclic fashion (c = + 0.006? 0? 0.006? 0),
following the deadandcreep load cases. Fig. 23 shows
thedeformedshape (100)of thecontinuum model during the application
of shear distortion. It isobserved that the lining undergoes a
considerable shape change,combined with local detachments from the
surrounding rockmass, a feature captured due to the direct
modelling of unilateralcontact interface conditions. Moreover, Fig.
24 shows the principalstress contours (00.5 MPa range), where the
bearing forces of therock mass continuum, due to the presence of
the lining, are visual-ised. As far as concrete cracking is
concerned, the maximum widthvalue (observed at c = 0.006) is equal
to 0.73 mm, which is closeto the adopted limit of 1.0 mm. Moreover,
the depth of the un-cracked zone is estimated to be about 5 cm,
which is considerablyless than the minimum allowable value of hw/2
(Fig. 23, bottom-right).
To further conrm the strong indication of liningsection
inadequacy due to shape change loading, the maximum
section eccentricities during cyclic analysis are calculated.
Sectioninadequacy was observed for all critical loading stages,
forinstance:
temperature (oC)
1:30 h
3:00 h
ault key region for the ETK re curve.
-
0:00 h 0:30 h 1:00 h 1:30 h 2:00 h 2:30 h
3:00 h
residual tensilestrength (MPa)
crack width (m)
Fig. 21. Evolution of concrete cracking during ETK re
exposure.
0.0
0.5
1.0
1.5
2.0
2.5
3.0M
min0.0
0.5
1.0
1.5
2.0
2.5
3.0
min0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
e
min
0.3hw
N
Fig. 22. Evolution of capacity factors (kM, kN) and eccentricity
(e) during ETK re exposure.
138 V.K. Papanikolaou, A.J. Kappos / Tunnelling and Underground
Space Technology 40 (2014) 127140
-
UndeV.K. Papanikolaou, A.J. Kappos / Tunnelling anddead creep!
emax M=N 8:76=16:93 51:7 cm > 22:5 cm! section inadequate
c 0:0005! emax M=N 64:87=27:29 233:7 cm> 22:5 cm! section
inadequateHowever, it is noted that consideration of the above
large
eccentricity values, which correspond to relatively low
momentsand axial loads (e.g. for the dead + creep case), may result
in mis-leading conclusions, regarding the adequacy of the section
if crack-ing has not actually taken place. This issue is also
identied insome guidance documents (e.g. AFTES, 2000), where the
tensilestrength of concrete is neglected. For this reason, the
above proce-dure may be rened by using elastic analysis of the
uncracked sec-tion (EN1992-1-1, 2004a) and checking the tensile
stress at thesection extreme bre, as follows:
rmax NA MIy y 5
Fig. 23. Model deformed shape and cra
Fig. 24. Principal stress contours of the rock mass
continurground Space Technology 40 (2014) 127140 139dead creep!
rmax 0:30 MPa < fctd 1:42 MPa! section adequate
c 0:0005! rmax 2:00 MPa > fctd 1:42 MPa! section
inadequateFrom the additional consideration of the above
conventional
checks, it is seen that the unreinforced lining section response
un-der shape-change effects induced by seismic loading, can be a
crit-ical design situation in earthquake-prone areas and should be
dulytaken into account.
4. Concluding remarks
The paper identied areaswherein existing codes and guidelinesfor
the design of unreinforced concrete linings need furtherimprovement
and/or harmonization. Since thebasic design criterionfor such
linings is the degree of cracking (depth, as well as width,
ofcritical cracks) it appearsnecessary toprovide to the tunnel
designer
cking during ovalization analysis.
um during ovalization analysis (undeformed shape).
-
the option of either using advanced analysis methods that
resultin reliable estimates of the cracking characteristics, or
using sim-plied methods, based on familiar elastic analysis that
lead to re-sults that are safe, without being
over-conservative.
The nite element analysis procedure described in detail hereinis
deemed to be useful to designers familiar with the basic con-cepts
of such an analysis, as it provides guidance both on settingup an
appropriate model of the lining and the different loads actingupon
it, and on interpreting the results of this analysis andexpressing
them in terms of quantities that allow proper safetyverications
(neutral axis depths, axial load eccentricities, crackwidths).
Comparisons between the key results of nonlinear nite
BASt [Bundesanstalt fr Straenwesen Federal Highway Research
Institute], 2007.Zustzliche Technische Vertragsbedingungen und
Richtlinien frIngenieurbauten ZTV-ING, Teil 5 Tunnelbau,
(Additional Technicalspecications and guidelines for Civil
Engineering Structures ZTV-ING, Part 5Tunnel Construction). Report.
No. S 1056.
Bergmeister, K., 2008. Innovative technologies to upgrade re
safety of existingtunnels. Beton- und Stahlbetonbau 103 (S1),
29.
CEN [Comit Europen de Normalisation], 2002. Eurocode 1: Actions
on Structures Part 12: General Actions Actions on Structures
Exposed to Fire (EN 1991-1-2). CEN, Brussels.
CEN Techn. Comm. 250, 2004a. Eurocode 2: Design of Concrete
Structures Part 11: General Rules and Rules for Buildings (EN
1992-1-1). CEN, Brussels.
CEN Techn. Comm. 250, 2004b. Eurocode 2: Design of Concrete
Structures Part 12: General Rules Structural Fire Design (EN
1992-1-2). CEN, Brussels.
CEN Techn. Comm. 250, 2004c. Eurocode 8: Design of Structures
for EarthquakeResistance Part 1: General Rules, Seismic Actions and
Rules for Buildings (EN19981). CEN, Brussels.
Cervenka, J., Cervenka, V., Eligehausen, R., 1998.
Fractureplastic material model forconcrete. Application to analysis
of powder actuated anchors. In: Proceedings ofthe 3rd International
Conference on Fracture Mechanics of Concrete Structures.
140 V.K. Papanikolaou, A.J. Kappos / Tunnelling and Underground
Space Technology 40 (2014) 127140element analysis and those of
simplied code-type relationshipsshowed that the latter are
generally safe, but often over-conserva-tive, especially when
tensile strength of concrete is ignored andlimitations on the
eccentricity are imposed.
The results of the analysis of the specic common cross-sec-tions
(horseshoe and closed) in Section 3 of the paper, althoughstrictly
applicable to the section geometries studied, are deemedto be of
broader interest. In addition to the above conclusionsregarding
sophisticated and simplied methods, it is interestingto note that
the most critical design situations were found to be reloading
(especially the hydrocarbon re case, which corresponds tothe rare,
but plausible, case that a lorry carrying fuel is set into rewhile
crossing a tunnel), and the ovalization of the lining due toseismic
ground deformations (note that the case studied here cor-responds
to a high seismicity area); formation of rock wedgesaround the
tunnel lining could also be a critical situation, albeitless
critical than the previous ones, and should be checked in tun-nels
bored into rock. Last and not least, it was found that amongthe two
typical cross sections studied, the horseshoe section is gen-erally
more appropriate than the closed (through an invert) sectionfor
unreinforced concrete linings in rock.
Acknowledgements
The authors would like to thank the Egnatia Motorway Com-pany
(EOAE SA) for making available to them all the constructiondrawings
of the linings of tunnels T1, T2, and T3 of the PATHE pro-ject, as
well as the results of the special seismic hazard study forthe area
(carried out by the team of Prof. S. Pavlidis, from the Aris-totle
University of Thessaloniki). They would also like to thankProf. T.
Tassios (from the National Technical University of Athens)and Mr S.
Raptopoulos (from EAAE SA) for the stimulating discus-sions of
several aspects pertaining to the performance of plain con-crete
linings, from which this study has benetted.
References
AFTES [Association Franaise des Tunnels et de l Espace
Souterrain], 2000. The useof plain concrete in tunnels. Tunnels et
Ouvrages Souterrains 158, 110118.Gifu, Japan.Cervenka, J.,
Papanikolaou, V.K., 2008. Three dimensional combined
fractureplastic
material for concrete. Int. J. Plast 24 (12), 21922220.Cervenka,
J., Surovec, J., Cervenka, V., 2004. Fractureplastic model for the
analysis
of reinforced concrete structures subjected to re. In:
Proceedings of theWorkshop: Fire Design of Concrete Structures:
What now? What next?. Milan,Italy.
Cervenka, V., Jendele, L., Cervenka, J., 2012. ATENA Program
Documentation. Part 1:Theory. Cervenka Consulting, Prague, Czech
Republic.
Corigliano, M., Scandella, L., Carlo G. Lai, C.G., Paolucci, R.,
2011. Seismic analysis ofdeep tunnels in near fault conditions: a
case study in Southern Italy. Bull.Earthquake Eng. 9 (4),
975995.
DAUB [Deutscher Ausschuss fr unterirdisches Bauen e.V. German
Committee forUnderground Construction], 2007. Recommendations for
Executing andApplication of unreinforced Tunnel Inner Linings.
Tunnel (int. journal ofDAUB, D-50827 Kln) 5, 1928.
DIN 10451, 2000. Tragwe rke aus Beton, Stahlbeton und
Spannbeton, Teil 1.Bemessung und Konstruktion. Berlin.
Dowding, C.H., Rozen, A., 1978. Damage to rock tunnels from
earthquake shaking. J.Geotech. Eng. Div., ASCE 104 (2), 175191.
Dutta, S.C., Roy, R., 2002. A critical review on idealization
and modeling forinteraction among soilfoundationstructure system.
Comput. Struct. 80 (2021), 15791594.
FHWA [Federal Highway Administration], 2009. Technical Manual
for Design andConstruction of Road Tunnels Civil Elements,
Publication No. FHWA-NHI-10-034, Washington, D.C.
b, 2008. Fire design of concrete structures structural behaviour
and assessment.b Bull. no. 46. Lausanne.
b, 2010. Structural concrete textbook. 2nd ed., vol. 4. b Bull.
54. Lausanne.FIT [Fire in Tunnels EU Thematic Network], 2005.
design re scenarios. Technical
Report (Rapporteur A. Haack). Brussels.Hashash, Y.M.A., Hook,
J.J., Schmidt, B., Yao, J.I.-C., 2001. Seismic design and
analysis
of underground structures. Tunn. Undergr. Space Technol. 16 (4),
247293.St. John, C.M., Zahrah, T.F., 1987. Aseismic design of
underground structures. Tunn.
Undergr. Space Technol. 2 (2), 165197.UPTUN [UPgrading Methods
for Fire Safety in existing TUNnels], 2008. WP 2 re
development and mitigation measures D211: re scenarios and
accidents inthe past. EU FP5 Contract G1RD-CT-2002-766. .
Wang, J.-N., 1993. Seismic Design of Tunnels: A State-of-the-Art
Approach.Monograph, monograph7. Parsons, Brinckerhoff, Quade and
Douglas Inc, NewYork.
Practical nonlinear analysis of unreinforced concrete tunnel
linings1 Introduction2 Modelling procedures2.1 Modelling of the
lining section2.2 Modelling of various load types2.2.1 Static,
quasi-static, environmental, and accidental loading2.2.2 Fire
loading2.2.3 Seismic loading
3 Case studies and discussion of analysis results3.1 Static,
quasi-static and environmental loading3.2 Accidental loadings
(static)3.3 Fire loading3.4 Seismic loading
4 Concluding remarksAcknowledgementsReferences