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Reinforcement optimization of fiber reinforced concrete liningsfor conventional tunnels

Giuseppe Tiberti, Fausto Minelli ⇑, Giovanni PlizzariDICATAM - Department of Civil Engineering, Architecture, Land, Environment and of Mathematics, University of Brescia, Italy

a r t i c l e i n f o

Article history:Received 26 September 2013Accepted 17 October 2013Available online 29 October 2013

Keywords:B. Fracture toughnessA. FibresC. Numerical analysisFinal concrete lining

a b s t r a c t

Tunnel linings are generally reinforced with conventional rebars that are placed to resist tensile stresses.As far as the service conditions are concerned, in recent years durability issues have became of para-mount importance, especially for underground infrastructures. Durability design generally requires rebarprotection against corrosion that can be achieved by reducing concrete porosity and crack width. The for-mer can be obtained by using a matrix with a low water/cement ratio while the latter can be achieved byusing a diffused reinforcement; to this aim, discrete fibrous reinforcement may represent an optimalsolution. In fact, the addition of fibers into concrete may provide noticeable residual tensile strength ata crack, linking the two adjacent faces of any crack due to the bridging effect provided by its enhancedtoughness. The latter also provides a significant resisting contribution against diffused tensile stressesacting in the structural element. However, when localized stresses (due to bending actions) occur, theyare more efficiently resisted by localized reinforcement (rebars).

Within this framework, this paper aims at investigating the behavior of FRC final tunnel linings, excavatedwith conventional method. Numerical non-linear analyses were carried out by considering different loadconditions in order to achieve a reinforcement optimization based on the combination of conventional(rebars) and fiber reinforcement (FRC). The procedure is applied to a real case of a road tunnel.

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1. Introduction

Fiber reinforced concrete (FRC) is a composite material with acementitious matrix and fibers as discontinuos reinforcement.FRC is gaining an increasing interest within the concrete commu-nity for the reduced construction time and labors costs. Besidescost issues, it is well recognized that fibrous reinforcement signif-icantly improves the mechanical performances of concrete asnumerous researches have demonstrated its effectiveness in manystructural applications, both with reference to Serviceability LimitStates (SLS) and Ultimate Limit States (ULS). For these reasons,many structural elements are now reinforced with fibers as partialor total substitution of conventional reinforcement (rebars orwelded mesh, [1]). The recent inclusion of FRC in the fib ModelCode 2010 [2], in national codes as well as the organization of con-ferences devoted to FRC [3–5] confirms this positive development.

FRC considerably improves the concrete post-cracking behaviordue to the remarkable residual post-cracking strength related to

the enhanced material toughness. The latter is a performancebased parameter useful for designers and depends on the fiber con-tent, material and geometry as well as on the concrete matrix [2].FRC toughness can be determined by means of fracture tests [6–8].However, fibrous reinforcement in combination with conventionalrebars is particularly effective in terms of crack control, since amore distributed crack pattern characterized by reduced crackwidths could be achieved [9]; this solution is also quite suitablein terms of structural strength since both diffused (resisted by fi-bers) and localized (resisted by rebars) stresses are generallypresent.

In the field of tunneling, the construction of tunnel lining in ashort time (to remain within the project schedule and budget)and in safe condition is a crucial issue in order to reduce the hin-drance to infrastructure above the ground and to the environmentas well as to guarantee durable and economic concrete structures.For these reasons, fibrous reinforcement can be a competitive solu-tion in linings for tunnels excavated with both conventional andmechanized methods [10–14].

Conventional tunneling is generally carried out in a cyclic exe-cution process of repeated steps of excavation followed by theapplication of relevant primary support, both of which dependon the existing ground conditions and ground behavior. The voidshould be fully stabilized with the first phase support (primarysupport system), which is designed to maintain a stable excavation

1359-8368/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compositesb.2013.10.012

⇑ Corresponding author. Address: DICATAM - Department of Civil Engineering,Architecture, Environment, Land Planning and of Mathematics, University ofBrescia, Via Branze, 43, 25123 Brescia, Italy. Tel.: +39 0303711223; fax: +390303711312.

E-mail addresses: giuseppe.tiberti@ing.unibs.it (G. Tiberti), fausto.minelli@in-g.unibs.it (F. Minelli), giovanni.plizzari@ing.unibs.it (G. Plizzari).

URL: http://dicata.ing.unibs.it/minelli/ (F. Minelli).

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so that a permanent lining can be placed [15,16]. The latter usuallyconsists of cast-in-place lightly reinforced massive concrete struc-ture subjected to bending and axial actions and is designed tostand for the whole life of the structure (final lining).

While FRC is already widely used in primary support system assprayed concrete (shotcrete), it is not commonly used in cast-in-place final linings and only just few experiences are reported intothe literature [17–19]. Some studies concern design provisions offinal lining cross-section to evaluate the benefits from steel fiberson crack control [20,21].

Actually, the presence of fibrous reinforcement allows for timereduction in fabrication, handling and placing of the curved rebarsthat has to be placed in cast-in-place and segmental linings. Fur-thermore, fibers represent a reinforcement spread out into thestructure, which could substitute the reinforcement placed alongthe longitudinal tunnel axis (secondary reinforcement) that is gen-erally used for stress redistribution. Since fibrous reinforcement isparticularly effective for diffused stresses [11,12] while rebars rep-resent an optimal reinforcement for localized stresses, it is of maininterest the evaluation of a proper combination of fibers and rebarsin order to achieve a reinforcement optimization, as proposed inother research works concerning precast tunnel linings [13,14,22].

To this aim, in the present paper several combinations of rein-forcement have been investigated by means of non-linear numer-ical simulations, which are suitable for taking into account theresidual strength provided by fibers after cracking. The numericalanalyses have been developed by using a global load progressiveincrement up to failure. In order to evaluate the bearing capacity,the development of crack patterns, the post-cracking stiffness aswell as the collapse mechanism, the structural analyses refer to areal case study of a cast-in-place final lining. Based on the initialdesign solution [23], the typical static loading conditions werenumerically simulated. It is shown that, by adopting a combinationof steel fibers and traditional reinforcement placed in appropriateregions, an optimized solution can be achieved, which guaranteesthe required structural performance at both SLS and ULS, by reduc-ing the amount of reinforcement globally used. Besides reinforce-ment savings (in terms of weight), this reduction also allowslabor savings and limited areas for the reinforcement storage.

2. Case study: Turina tunnel

The ‘‘Turina’’ road tunnel, which is part of the infrastructures ofa 3.5 km long road project, is taken as a reference case study here-in. The road section is 13 m wide and is made of two 3.75 m lanes,a hard shoulder (1.50 m wide) and a wide curb for each lane(1.25 m wide). The ‘‘Turina’’ tunnel is 665 m long, with an internal

diameter of 12.6 m (Fig. 1a) and it is excavated by means of con-ventional tunneling method.

The tunnel is located in a geo-mechanical group consisting ofpoor rock mass conditions and alluvial deposits. The most criticalsections investigated are positioned in layers of detritus with poormechanical properties and alluvial deposits having an unsure loca-tion with respect to the surrounding sub-layer of weak rock. Theaverage tunnel overburden for these sections is approximately40 m. Due to these uncertainties and the severe ground conditions,the injection-type fore-poling method (umbrella) was suggestedby designers in the preliminary tender project [23]. Fig. 1b showsthe longitudinal profile of the tunnel; it is evidenced that the injec-tion-type fore-poling method (umbrella) was adopted. Steel pipeswith a length of about 12 m, a diameter of 114.3 mm and a thick-ness of 10 mm were driven at the tunnel crown. Consequently thethickness of the arch (crown) cyclically varies along the tunnel lay-out as shown in Fig. 1a and b. On the contrary, the invert thicknessis constant and is equal to 900 mm (Fig. 1a).

More in details, the following construction sequence wasconsidered:

– Placement of sub-horizontal steel pipe umbrella and fiberglassbolts for providing temporary support and face reinforcement,respectively.

– Excavation of the heading of the tunnel.– Construction of the early stage lining (primary support) consist-

ing of fiber reinforced shotcrete and steel ribs.– Casting of the temporary invert.– Excavation of the invert of the tunnel and construction of the

corresponding first phase lining.– Casting of the invert of the final lining.– Casting of the benches and subsequently of the arch (crown) of

the final lining.

The present study focuses on the behavior of the final liningwith particular reference to the static load due to the surroundingground. In particular, both symmetrical and asymmetrical loadswere considered; the latter load distribution represents lining sec-tions along the track, which exhibit asymmetrical lateral groundcover, in proximity of the valley walls.

3. Materials

In the first final lining design [23], the minimum concrete char-acteristic cubic strength required (fck,cube) was 30 MPa. In order toquantify possible benefits from the use of FRC, an experimentalcampaign [24] was developed at the laboratory for testing materi-als of the University of Brescia on fibrous and non-fibrous

Minimum liningtransverse section

Maximum liningtransverse section

R6300

29.57°

109.

93°

19.9

3°

900

6365 6365 1510635

1360575

4220

5510

Invert

Bench

Crown

833511830

Steel ribs

Steel pipe umbrella

Temporary invert

Primary supportShotcrete-steel ribs

Final invert

(a) (b)

Fig. 1. Scheme of the transverse section of the final tunnel lining (a); simplified scheme of the longitudinal tunnel profile with evidenced the fore-poling method (b).

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reinforced samples with a concrete matrix designed to obtain aC 25/30 concrete, according to Eurocode 2 [25]. Fibrous reinforcedspecimens were cast by using steel macro fibers 50/1.0, having alength (Lf) of 50 mm, a diameter (uf) of 1.0 mm and an aspect ratio(Lf/uf) equal to 50. These fibers are cold drawn, have a hookedshape, a rounded shaft and a tensile strength higher than1100 MPa. Different fiber contents were considered: 20 kg/m3,30 kg/m3 and 60 kg/m3; the lowest dosage, corresponding to a vol-ume fraction (Vf) of 0.25%, was selected as a minimum amount. No-tice that numerical analyses were also performed on non-fibrousconcrete (plain concrete) for having a solution based on traditionalreinforced concrete (rebars only) [23].

A number of tests were conducted in order to determine thematerial properties. Standard tests on 150 mm cubes were carriedout for the determination of the concrete compressive strength.Young modulus (secant static modulus in compression accordingto UNI 6556 [26]) were measured from u 80 to 210 mm cylinders.All specimens were stored in a fog room (R.H. > 95%, T = 20 ± 2 �C)before testing.

Mechanical properties were measured at about 150 days aftercasting, in order to have results significant for the service life ofthe tunnel lining. Table 1 summarizes the mean values of the con-crete mechanical properties: the mean tensile and cylindrical com-pressive strength were estimated from the experimental cubicstrength by adopting the relationships proposed by Eurocode 2[25]. The concrete compressive strength was also measured at28 days in order to verify the target concrete class (Table 1). Thelatter, for the two batches considered in the present paper, isaround 40 MPa (mean value), which corresponds to the expectedone. It can be observed that the concrete compressive strength ina long-term condition (at 150 days) increased of about 30% withrespect to the standard condition (at 28 days).

Fracture properties of Steel Fiber reinforced concrete (SFRC) andof reference plain concrete were determined by using six notchedbeams for each batch (150 � 150 � 550 mm), tested under 3-pointbending according to EN 14651 [6]. These tests were carried outwith a closed-loop hydraulic testing machine by using the crackmouth opening displacement (CMOD) as control parameter, whichwas measured by means of a clip gauge positioned astride a notchat midspan, having a depth of 25 mm (Fig. 2a). The experimentalcurves, concerning the CMOD, are depicted in Fig. 2b. Based onthese curves, the residual post-cracking strengths fR,j (evaluatedat 4 different CMOD values, i.e. 0.5, 1.5, 2.5 and 3.5 mm [6]), andthe flexural tensile strength (limit of proportionality) fL were calcu-lated, as listed in Table 2.

The characteristic residual strengths fRk,j were determined fromthe mean values by assuming a T-student distribution, which takesinto account of the number of samples available for each batch.According to MC 2010 [2] and based on the characteristic fractureparameters reported in Table 2, the SFRC 0.25M can be classified as2c. Furthermore, it satisfies the minimum performance require-ments in terms of ratios fR1k/fLk (>0.4) and fR3k/fR1k (>0.5) introducedby MC 2010.

Regarding conventional reinforcement, rebars B450C (charac-teristic yielding strength, fsyk = 450 MPa, according to Eurocode 2[25]) were used in the preliminary project [23].

4. Numerical modeling

The determination of stress and strain fields in concrete finallining can be performed by using analytical [27] or numericalmethods [28,29].

Nowadays, several numerical approaches are available, sincethey allow to model the structure and its interaction with the sur-rounding ground. Several of these models are focused on a propersimulation of the ground including its non-linear behaviorwhereas, for the first phase and permanent linings, elastic materi-als are typically assumed.

The research work presented herein is focused on the lining as astructural element embedded in ground, with special reference toits behavior at SLS and, in addition, to its bearing capacity (ULS). Tothis aim, non linear fracture mechanics (NLFM, [30]) has to be usedin order to properly include the significant residual strength pro-vided by fibers after cracking and to evaluate the consequentimprovement of crack control, which plays a key role for struc-ture’s durability.

4.1. Numerical modeling of materials

The numerical analyses were performed by means of finite ele-ment code DIANA 9.4 [31], which implements several modelsallowing for the description of the concrete behavior after cracking[30]. The models developed by Rots [32], based on smeared crackapproach, were used. The smeared cracking models have orthotro-pic softening and are therefore suitable to model a multi-axialstress state. In order to develop the numerical simulations of tun-nel final linings (and concrete structures in general) with a reliablemodel, the functioning and numerical stability of these modelswere studied by simulating several experimental tests on FRCstructural members, available in literature [33]. Eventually, the to-tal strain rotating crack model was adopted in the numerical sim-ulations of tunnel linings based on smeared approach because itgives a reasonable approximation of the actual behavior of con-crete [33].

Concrete in tension was assumed to be linear up to the meantensile strength. After cracking, a softening post-cracking cohesivelaw was adopted according to the fictitious crack model (FCM,[30]). This law was approximated as bi-linear, where the first stee-per branch can be associated with (un-connected) micro crackingahead of the stress-free crack, while the second branch representsthe aggregate interlocking or the fiber bridging (Fig. 3a).

The post-peak behavior (after cracking) was evaluated bymeans of the inverse analysis method applied to the bending tests[34]; hence, numerical analyses based on NLFM, were performedfor determining the best-fitting softening law for the materialsadopted. In addition to the discrete crack approach, the numericalanalyses on the material were also carried out by assuming asmeared crack approach. Fig. 3b shows the FE mesh used for thediscrete approach; it is made by triangular plane stress elementsand interface elements for the crack. The material parametersidentified from the bending tests are reported in Table 3, accordingto the bi-linear law depicted in Fig. 3a. The numerical curves arecompared with the experimental ones in Fig. 4, for SFRC 0.25M

Table 1Mechanical properties of concrete and fiber contents.

Age (days) Batch ID fcm,cube (MPa) Ec (GPa) fcm* (MPa) fctm

* (MPa) Vf,tot (vol%)

28 0 Plain 40.0 – 33.2 2.58 –0.25M 40.9 – 33.9 2.63 0.25

150 0 Plain 59.9 37.5 49.7 3.61 –0.25M 52.5 37.5 43.6 3.25 0.25

* Estimated from fcm,cube.

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series; the mean experimental curve was also plotted in order tobetter evaluate the good agreement with the numerical curves(discrete and the smeared approach).

Concerning tunnel lining simulations, the uni-axial compressiveresponse of concrete was expressed by means of the relationshipproposed by Eurocode 2 [25] for non-linear analyses. The meanvalue of the cylindrical compressive strength (fcm), reported inTable 1, was used.

All the B450 C rebars in the model were defined as an elastic–plastic material with a linear strain hardening up to the ultimatestrength, according to Eurocode 2 [25]. An elastic modulus (Es) of200 GPa was assumed. The Von Mises yielding criterion was ap-plied and the following typical mean values of the main mechani-cal properties were adopted: fsym = 500 MPa, fsum = 575 MPa at aconventional ultimate strain of 7.5% [25].

3PBT - EN 14651

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5

CMOD [mm]

Nom

inal

Str

ess

σ N [

MP

a] Plain

SFRC 0.25M

(a) (b)

Fig. 2. Instrumentation of the notched specimen for beam test (a); experimental results of 3PBT notched beams according to EN 14651 (b).

Table 2Fracture parameters of the SFRC according to EN 14651.

Age (days) Batch ID Fracture parameters of the SFRCs according to EN-14651

fL (MPa) fR1 (MPa) fR2 (MPa) fR3 (MPa) fR4 (MPa)

150 0 Plain Mean values 5.74 0.33 – – –Characteristic values 5.15 0.19 – – –

0.25M Mean values 5.70 2.28 2.53 2.61 2.40Characteristic values 4.55 2.05 1.99 1.97 1.46

σ

w1 wc

w

fctm

σ1

Gf

σ

w1 wc

σGf

(a)(b)

Fig. 3. Bi-linear law adopted for the post-cracking behavior of SFRC (a); mesh adopted for the inverse analyses of the beam tests with a discrete crack approach (b).

Table 3Uni-axial tensile constitutive laws of materials.

Batch ID Ec (GPa) fctm (MPa) r1 (MPa) w1 (mm) wc (MPa)

0 Plain 37.5 3.61 0.28 0.050 0.350.25M 37.5 3.25 0.95 0.034 8.50

3PBT - EN 14651 - SFRC 0.25M

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5

CTODm [mm]

Nom

inal

Str

ess

σ N [

MP

a]

Experimental

Mean experimental

SFRC 0.25M Discrete approach

SFRC 0.25M Smeared approach

Fig. 4. Experimental and numerical results obtained from on beams with macrofiber 50/1.0, SFRC 0.25M.

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4.2. Final lining numerical model

The lining was studied by assuming a long-term condition withall the loads carried by the final lining (the resistant contribution ofthe first phase lining was neglected). This is a general acceptedassumption since the stiffness of the shotcrete outer lining, on di-rect contact with the ground, is often disregarded, because it mayhave undergone a chemical–physical reaction altering its initialmechanical properties [35].

As already mentioned, the final lining consists of a series of ‘‘ta-pered cylinders’’, because the fore-poling method was applied bymeans of steel pipe umbrella. Consequently, the transverse sectionof the final lining changes along the track: the numerical analyseswere accordingly carried out by referring to a transverse sectionhaving ‘‘average’’ geometrical dimensions, as shown in Fig. 5a.

It is worthwhile noticing that, due to the particular shape of thefinal lining, the boundary conditions are slightly different to theones typical of a plane strain situation. Hence, a 2D plane stressmodel was used because, for the final lining investigated, it pre-sents the same degree of reliability of a typical plane strain model.In addition, this numerical model enables a more precise descrip-tion of the rebars’ behavior in the transverse section, which is fun-damental, since the main aim of the research was the optimizationof the primary reinforcement (fibrous reinforcement in combina-tion with rebars).

Eight-nodes quadrilateral iso-parametric plane stress elementswith an average dimension of 110 mm were used, as shown inFig. 5b. The regions of the lining between the invert and thebenches are particularly crucial, since a change of thickness occurs.Therefore, a mesh refinement was adopted in these regions in or-der to better evaluate the corresponding local stress distributions(Fig. 5b).

During the construction process, firstly the invert is cast in placeand, subsequently, the benches and the arch of the final tunnel lin-ing are cast. As a result of this time-step procedure, the lining pre-sents cold joints that are weak regions (Fig. 6a). These weak jointsbetween the invert and the benches were modeled by consideringthat they are able to transmit compression stress only. In order tosimulate this behavior, as a first approximation, small layers of ele-ments were adopted along the cold joint having a very small ten-sile strength (0.01 MPa) for numerical stability (Fig. 6b).

Based on the material constitutive laws previously introduced,fibrous and non-fibrous concretes were simulated by means of asmeared crack approach. The conventional steel reinforcing bars(rebars) were implemented by means of embedded elements per-fectly bonded to the surrounding concrete.

Since the numerical analyses mainly focused on the permanentlining behavior, a simplified model was used for describing theinteraction between the structure and the surrounding ground,consisting of radial and tangential springs, the latter to simulate

the friction between the tunnel and the soil (Fig. 7). This type ofmodel is not proper for the evaluation of ground settlement but en-ables a good estimation of lining fields of stress and strains with anoticeable reduction of the computational time [27,36]. Conse-quently, the model was particularly adequate for the parametricstudy carried out.

Based on the geotechnical data available [23], the followingground mechanical properties were adopted for the critical tunnellining sections investigated: internal friction angle u = 35�, elasticground modulus Eground = 110 MPa and cohesion c = 5 kPa. The ra-dial spring stiffness (Kn; Fig. 7) was calculated by means of theBaguelin criterion, whereas the tangential spring stiffness (Kt)was estimated as Kn/3 [23]. Notice that, along the arch and thebenches, only radial springs were adopted, according to designerassumptions [23]. Despite the simplicity of the ground/lining inter-action simulated, it was possible, as a first approximation, to takeinto account the weak mechanical properties of the soil and to de-scribe a possible limit state governed by ground collapse. In fact,the spring behavior is elastic until a displacement limit of10 mm, after which it becomes perfectly plastic (Fig. 8).

5. Numerical results

A symmetrical load was simulated by using the previously de-scribed numerical model and by applying a suitable ground pres-sure, which was estimated according to designer assumptions[23]. In fact, by assuming that the primary support can distributethe actions of the surrounding ground, a radial pressure was ap-plied on the final tunnel lining, which varies from 507 kPa (onthe crown) to 302 kPa (on the benches), in relation to the tunneloverburden and in accordance with the ground properties(Fig. 9a). Moreover, lining sections along the track, which exhibitasymmetrical lateral ground cover, were simulating by means ofan asymmetrical applied radial pressure, as depicted in Fig. 9b.

Fig. 10a shows the conventional reinforcement (rebars) adoptedfor the lining which corresponds to a total amount of 48.9 kg/m3.

Non-linear analyses were carried out by increasing the radialpressure by means of load multiplier up to the ultimate load.Numerical results from lining with conventional reinforcementonly (RC) and with 20 kg/m3 of steel fiber reinforcement only (SFRC0.25M) are presented. Fig. 10b shows the numerical results interms of load multiplier (normalized to the service load) vs. the rel-ative (vertical) displacement between the crown and the invert. Itcan be noticed that the final lining reinforced with steel fibers only(SFRC 0.25M) exhibits slightly less ultimate load with respect tolining with rebars only (RC). However, both the reinforcement con-figurations are able to guarantee an ultimate load of about 1.9times the service load in presence of symmetrical loads, whereas,in case of asymmetrical load, a value of about 1.85 can be reached.

Final lining average transverse section

6365 126563651265

R6300

29.57°

6040

4220

Average R7450

109.

93°

19.9

3°

1150

900

Invert

Bench

Crown

Cold joint

Mesh refinement

(a) (b)

Fig. 5. Final lining average transverse section adopted (a); mesh adopted for the 2D plane stress finite element model (b).

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Therefore, by using fibrous reinforcement it is possible to achievethe same global response. Furthermore, by verifying the spring re-sponse, which simulate the lining-ground interaction, it was ob-served that the ultimate load reached in all the simulations ismainly governed by the weak ground properties taken into account

by means of the constitutive laws depicted in Fig. 8. In other words,even though toughness provided by SFRC generally enhances aredistribution of internal actions in statically indeterminate struc-tures as final tunnel linings, due to the poor ground mechanicalproperties, this benefit was not completely exploited at ULS, forthe lining investigated. Consequently, concerning the ULS, a furtherimprovement of FRC toughness, would not be exploited in terms ofglobal response.

Since the numerical analyses were carried out by progressivelyincreasing the applied pressure, it was possible to evaluate the cor-responding crack development in the final lining. For both the rein-forcement configurations and loading conditions considered, it wasevidenced that, at service load, no cracks occur in the center of thearch (crown). Micro-cracks appear in the center of the invert intra-dos as demonstrated by the very low cracking strain that can becaptured by means of the smeared crack approach. However,cracks occur in the region of connection between the invert andbenches, where the lining transverse section exhibits a change ofthickness. These local cracks govern the lining behavior at SLS.Cracking phenomena in the arch (crown) occur at about 1.4/1.6times the service load (respectively for asymmetrical and symmet-rical load). By analyzing the final lining crack pattern at the

Cold joint

Invert

Bench

(a) (b)

Fig. 6. Scheme of the lining with evidenced the cold joint between the invert and the bench (a); post cracking tensile law adopted for the cold joint and concrete lining (b).

Fig. 7. Distribution of radial and tangential springs along the final tunnel lining.

Fig. 8. Constitutive laws adopted for tangential (a) and radial springs (b), in order to simulate the ground-lining interaction.

302 kPa

507 kPa

302 kPa

(a)

318 kPa 211 kPa

545 kPa

(b)

Fig. 9. Radial pressure applied on the final lining for symmetrical (a) and asymmetrical (b) load conditions.

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maximum loads, a well distribute crack pattern can be observedalong the intrados of the invert and of the arch. The final estimatedmean crack width expected in the arch (crown) and invert of bothreinforcement configurations are quite low for both loading condi-tions. In fact, a mean cracking opening of about 0.3 mm was mea-sured in the SFRC 0.25M lining. The latter, as a first approximation,was estimated by multiplying the mean cracking strain by thecharacteristic length introduced in the model according to thesmeared approach. This length corresponds to the average elementdimension according to Rots [32]. These low crack openings, evenat ULS, confirm that the central part of the arch and of the invertare less relevant in terms of crack control than the joints betweenthe invert and benches.

Based on the crack development obtained from NLFEA, threesections of investigation (line 1, 2, 3) were considered in the crucial

region previously mentioned and evidenced in Fig. 11, in order tobetter estimate the crack development when increasing the ap-plied radial pressure. It should be observed that each base of mea-surement is about 300 mm long; hence, the relative displacementalong this length represents the effective crack width only in thecase of one crack occurring in this region. This hypothesis can bereliable since the local change of lining thickness probably tendsto localize cracking phenomena. The crack opening along line 1(which, as expected, is the most critical as it is the farthest fromthe neutral axis) is represented in Fig. 12a for SFRC 0.25M liningby comparing the symmetrical and asymmetrical loads.

The diagrams in Fig. 12a clearly demonstrated that, despite thesimilar load multipliers, the asymmetrical loading condition is themost severe situation in terms of cracking behavior at SLS. By refer-ring to this latter condition, a comparison between RC and SFRC

RC - Final lining configuration Relative vertical displacement: crown-invert

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 20 40 60 80 100

Displacement [mm]

Loa

d/Se

rvic

e L

oad

SFRC 0.25M: symm. l.c.

SFRC 0.25M: asymm. l.c.

RC: symm. l.c.

RC: asymm. l.c.

(a) (b)

Fig. 10. Scheme of lining with conventional reinforcement only (a); comparison between the load–displacement curves as obtained from RC and SFRC 0.25M (b).

Line 1

300

150

150

Line 2

Line 3

Fig. 11. Scheme with the three lines of investigation adopted to estimate the crack development between the bench and the invert.

Local behavior bench-invert, SFRC 0.25M, line 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.0 2.0 4.0 6.0 8.0 10.0

Displacement [mm]

Loa

d/Se

rvic

e L

oad

Symm. l.c.

Asymm. l.c.

Local behavior bench-invert, line 1, asymmetrical l.c.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.0 2.0 4.0 6.0 8.0 10.0

Estimated crack opening [mm]

Loa

d/Se

rvic

e L

oad

SFRC 0.25M

RC

(a) (b)

Fig. 12. Numerical curves of the load vs. crack opening along the lines of investigation: symmetrical vs. asymmetrical load condition (a), RC vs. SFRC 0.25 M (b).

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0.25M linings is reported in Fig. 12b, for line 1. It can be noticedthat steel fiber reinforcement is able to guarantee the same behav-ior of conventional rebars up to a load of 0.5 times the service load.At service load, the estimated crack opening of SFRC 0.25M config-uration is almost 3 times larger than that of RC one whereas at amultiplier of 1.5 is five times higher. Definitely, it emerges that, lo-cally, SFRC with 20 kg/m3 of steel fibers (SFRC 0.25M) is not able toguarantee the same performance level of the RC lining where, forasymmetrical loads, a maximum crack opening of 0.38 mm is ob-served at SLS. The latter can be considered, in absolute terms, anacceptable value in relation to the crucial region herein considered.

5.1. Optimized reinforcement

In order to better control the crack development in the criticalregions and to enhance the global behavior of the final lining, acombination of rebars and fibers (20 kg/m3, class 2c in concretehaving cubic compressive strength fcm = 52.5 MPa) is proposed(RCO-SFRC 0.25M, Fig. 13a). Notice that rebars are concentratedonly in the region connecting the bench and the invert (Fig. 13b).The proposed reinforcement combines the advantages of rebarsfor localized stresses and steel fibers for diffused stresses.

Fig. 14 shows the comparison between the three types of rein-forcement considered herein, in terms of local and global liningbehavior for the most severe loading condition (asymmetric) ana-lyzed. In particular, in Fig. 14a, the load vs. relative displacement ofthe crown/invert is presented, whereas in Fig. 14b the estimationof crack opening in the crucial region of the lining is shown; itcan be noticed that the proposed optimized reinforcement (RCO-SFRC 0.25M) is able to guarantee the same local and global

behavior of the reference configuration (RC). The global amountof reinforcement (fibers and conventional rebars) of the optimizedlining solution is 32 kg/m3, which corresponds to a reduction ofabout 30% with respect to the that used in the initial RC solutionproposed by designers.

6. Concluding remarks

The present paper reports the results from a non-linear numer-ical study of the final lining of a tunnel excavated with a conven-tional method.

The embedded in ground load condition was simulated bymeans of a numerical model which, despite a simplified descrip-tion of the interaction between lining and surrounding ground,can well capture the lining behavior in terms of crack develop-ment. Several reinforcement combinations were considered,including an optimized combination of SFRC with a rather lowtoughness and conventional rebars, localized only in the critical re-gions of connection between the invert and the benches wasproposed.

Based on the results and the discussion presented, the followingmain conclusions can be drawn:

– The optimized proposed solution enables a reduction of about30% of the total amount of steel reinforcement (fibers + rebars)with respect to the configuration usually adopted by designersrebars only (RC) and a reduction of about 75% of conventionalsteel reinforcing bars, which means savings in labor work andstorage areas.

Optimized local reinforcement RCO

1

7

43

5 1Ø16 @ 200 mm

61Ø12 @ 200 mm

2

1Ø16 @ 200 mm

1Ø20 @ 200 mm

1Ø20 @ 200 mm

1Ø20 @ 200 mm

1Ø20 @ 200 mm

RCO detail

(a) (b)RCO-SFRC 0.25M - Final lining configuration

Fig. 13. Optimized reinforcement proposed for the final tunnel lining based on rebars + steel fibers (a); detail of rebars placed in the critical region of the lining (b).

Asymmetrical load condition: relative vertical displacement crown-invert

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 20 40 60 80 100

Displacement [mm]

Loa

d/Se

rvic

e L

oad

RC

SFRC 0.25M

RCO-SFRC 0.25M

Local behavior bench-invert, line 1, asymmetrical l.c.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.0 2.0 4.0 6.0 8.0 10.0

Estimated crack opening [mm]

Loa

d/Se

rvic

e L

oad

SFRC 0.25M

RC

RCO-SFRC 0.25M

(a) (b)

Fig. 14. Comparison in terms of global behavior (a) and local behavior (b) of the final tunnel lining according to the reinforcement combinations considered: asymmetricalload condition.

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– Rebars are placed only in two critical regions of the lining: coldjoints (where the tensile strength is negligible), and connectionareas between the base of the bench and the invert (where athickness change occurs). In the first case, rebars guarantee abetter transmission of internal stresses, especially in tension.In the second case, high stress concentrations occur that coulddetermine local risks of large cracks, which can compromisethe structural durability and can be better controlled by rebarsin combination with fibrous reinforcement.

– In the arch (crown) no curved rebars should be placed. Conse-quently, a noticeable reduction in fabrication, handling andplacing of this type of reinforcement is expected for this partof the lining as well as for the invert.

– The proposed numerical procedure could be adopted also as adesign approach, based on non-linear simulations. It enablesto optimize the total amount of reinforcement by improvingthe structural behavior at both SLS and ULS.

Acknowledgments

The Authors wish to express their gratitude to the Italian Min-istry of University and Research (MIUR) for partially financing thisresearch work within the National Project ‘‘Optimisation of theStructural, Technological and Functional Performance of Construc-tion Methodologies and Materials in Tunnel Linings, (PRIN 2006).

The research project was also financed by Officine MaccaferriS.p.A (Bologna, Italy) whose support is gratefully acknowledged.

A special acknowledgment goes to GEODATA s.p.a (Turin, Italy)for the given technical documentation and technical consultancyprovided by engineers M. Pescara and N. Avagnina.

Finally, the authors are grateful to Mr. Eng. Giuseppe Barbierifor the assistance in performing non-linear FE analyses.

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