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: billy@civil.auth.gr (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.

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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