analysis of structure using ccm

Tunnelling and Underground Space Technology 18(2003) 347–363

0886-7798/03/$ - see front matter� 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0886-7798(03)00004-X

Analysis of structural interaction in tunnels using the covergence–confinement approach

P.P. Oreste*

Department of Georesources and Land, Technical University of Turin, Corso Duca Degli Abruzzi 24, Turin 10129, Italy

Received 8 June 2002; received in revised form 16 December 2002; accepted 4 January 2003

Abstract

The rock-support interaction in tunnels is studied through the use of the convergence–confinement method. The equations thatcharacterize the behaviour of the most important support types are given together with a set of conceptual interaction schemes.As far as the behaviour of the support is concerned, reference is made to the ultimate limit state concept, which is widely usedin civil engineering. This approach is linked to the classical convergence–confinement method. The interaction between thetemporary support system and the final lining is dealt with, and the noteworthy case of presupport ahead of the face, followed bya further internal support(usually steel sets and shotcrete) is also included. Finally, the ‘ground reaction curve of the reinforcedtunnel’, which allows one to analyse the interaction between the reinforcement around the tunnel and supports, is introduced.� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Structural interaction; Tunnels; Convergence–confinement approach

1. Introduction

During the excavation of tunnels in unstable rockmasses using conventional methods, several types oftemporary supports interact with the purpose of stabilis-ing the opening before the final lining is completed.With the tendency, in some cases, to advance full faceeven in large tunnels and under difficult conditions, thesupport structures and the rock reinforcement can bevery complex, requiring a rational understanding of theinteraction with the rock mass.

The ever increasing attention into the costs of workhas led to research for the economic optimisation of thesupport structures, that is, for the choice of the differentcombinations of types, which, with an equality oftechnical results, would permit lower costs. If, forexample, rock reinforcements and support structures areplanned to stabilise a large tunnel, it is necessary toknow exactly how they interact and to define the bestcombination of all the possible technical combinationsfrom the economic point of view.

The study of the interaction between support struc-tures can easily be carried out using the convergence–

*Tel.: q39-1-1564-7608; fax:q39-1-1564-7699.E-mail address: [email protected](P.P. Oreste).

confinement method. This method allows one to have aqualitative understanding of the interaction phenomenonand helps operative choices. In more complex cases, itis necessary to use numerical schemes; however, thecomputational effort required to analyse the resultswould need to be intensive to produce a parametricanalysis that is able to improve the final design.

2. The convergence–confinement method

The convergence–confinement method for a circulartunnel in a hydrostatic stress field(for example: Lom-bardi, 1975; Hoek and Brown, 1980; Bouvard-Lecoanetet al., 1988; Brown et al., 1983; A.F.T.E.S., 1993; Panet,1995; Peila and Oreste, 1995) gives the displacementand the load acting on the support through the intersec-tion of the ground reaction curve of the tunnel and thesupport reaction line(Fig. 1): the reaction line can bedefined on the basis of the following four parameters:

– stiffness k: ratio, in elastic conditions, between thepressure applied by the ground and the displacementof the support(at the extrados);

– displacement of the wallu , which has alreadyin

developed on installation of the supports;

348 P.P. Oreste / Tunnelling and Underground Space Technology 18 (2003) 347–363

Fig. 1. The convergence–confinement method. Key:p: internal tunnelpressure;u: radial displacement of the wall(positive towards the tun-nel axis); p : in situ hydrostatic stress;p : pressure acting on the0 eq

support structure;p : pressure that induces the plastic failure of themax

structure(support capacity); k: support stiffnesswforceylength x; u :3in

displacement of the wall before support installation;u : displacementeq

at equilibrium;u : displacement of the wall on reaching the elasticel

limit in the support;u : displacement of the wall on collapse of themax

support; andA: equilibrium point of the tunnel-support system.

Fig. 2. Elastic-perfectly plastic behaviour assumed for the supportmaterial: once the yield strengths is reached the material contin-max

ues to develop plastic strains under constant load, until the strain´ is reached, corresponding to collapse of the support.max

– pressurep which induces the yielding of themax

support; and– displacement of the tunnel wallu which causesmax

the support to collapse.

A support defined with the reaction line of Fig. 1 hasan ideal elastic-perfectly plastic behaviour: havingreached yielding(elastic limit), it continues to deformwith the same load until collapse occurs(Fig. 2). Thestiffness of the support in plastic conditions is zero.

There are often some uncertainties in the evaluationof u . As a first approximation,u is given by thein in

distance from the excavation face where the support isinstalled, by the diameter of the tunnel and by themechanical characteristics of the ground(Panet andGuenot, 1982; Panet, 1995). The stiffnessk of thestructure and the equilibrium loadp also influence theeq

value ofu . These uncertainties influence the evaluationin

of the equilibrium point on the convergence–confine-ment curve. It is advisable to use a parametric analysisfor this calculation, varyingu in the interval of valuesin

that is considered to be correct.The behaviour of steel sets, shotcrete lining and

rockbolts, the most frequently used support systems, hasbeen analysed. The interaction between the supports andthe final lining is dealt with later in the paper. The waysof taking into account the interaction between rockreinforcement(through bolting) and ground improve-ment,(e.g. through injection) around the tunnel and thetraditional support systems are also analysed.

3. Typical reaction line of a support structure

3.1. Steel sets

The k stiffness of a closed circular steel set is givenset

by the following simplified expression(Eq. (1)) (Hoekand Brown, 1980):

1k ( (1)set 2B EhsetC FdØ Ryt yblock

D G2 2dqØtblockq ØR2E Øbwood blockE ØAst set

where: k is the stiffness of the steel setwforceyset

length x; E is the elastic modulus for the steel;R is the3st

radius of the tunnel;d is the steel set spacing along thetunnel axis;A is the cross-sectional area of the steelset

section; 2q is the angle between the connection points;t is the thickness of the connection blocks(in theblock

radial direction); b is the width of the connectionblock

block (in the circumferential direction); andE is thewood

elastic modulus of the wood that the blocks are madeof.

In the case where wood blocks are not used, Eq.(1)can be further simplified as follows:

E ØAst setk s (2)set 2B EhsetC FdØ RyD G2

The equilibrium condition gives(Eq. (3)) the maxi-mum pressurep sustainable from the steel set:max

s ØAst,y setp ( (3)max,set B EhsetC FRy ØdD G2

349P.P. Oreste / Tunnelling and Underground Space Technology 18 (2003) 347–363

where:s is the yield strength of the steel andh isst,y set

the cross-sectional height of the steel section.The collapse of the steel set occurs when the´br,st

failure strain is reached in the steel, that is, when the´ circumferential strain in the steel set equals´ .q br,st

Given that (where u is the radialur

´ sq rB EhsetC FRyD G2

displacement of the steel set), one obtains failure for:

u u yur max,set in,sets s´ (4)br,sth hset setRy Ry2 2

and it results that:

B EhsetC Fu yu s´ Ø Ry (5)max,set in,set br,stD G2

where the term within the round brackets represents themean radius of the steel set.

3.2. Shotcrete lining

The structural behaviour of the shotcrete lining isevaluated on the basis of the general equation of theradial displacement(Eq. (6)) which is obtained byresolving the differential equation system that governsthe stress and strain behaviour of the elastic media inaxialsymmetrical conditions:

Bu sAØrq (6)r r

where:u is the radial displacement in the lining at ther

distancer andA andB are integration constants.From this equation, using also the definition of the

axialsymmetric strain and the constitutive law of theelastic material, taking in consideration the correctboundary conditions(see Appendix 1), one can obtainthe stiffness of the shotcrete ringk , the maximumshot

pressure that can be applied on its extradosp andmax,shot

the radial displacement of the wall related to the shot-crete ring failureu .max,shot

22w zx |R y RytŽ .shoty ~E 1conk s Ø Ø (7)shot w z22

x |1qn RŽ . 1y2n R q RytŽ . Ž .con con shoty ~

2w zRytŽ .shot1 x |p s Øs Ø 1y (8)max,shot c 22 Ry ~

u (u q´ Ø RytŽ .max,shot el,shot br,con shot

2Ø 1yn ØRØ RytŽ . Ž .con shot pmax,shoty Ø (9)2 2Ryt q 1y2Øn ØR kŽ . Ž .shot con shot

where: E and n are the concrete elastic moduluscon con

and Poisson’s ratio, respectively;t is the liningshot

thickness; ands is the uniaxial strength of thec

shotcrete.

3.3. Radial anchored bolts (active bolts)

The study of a pattern of radial anchored bolts canconveniently be carried out according to Hoek andBrown (1980) (when the bolt length is such as toconsider the stress perturbation caused by anchoringnegligible). A bolt can be represented by a series oftwo springs: the first one refers to the stiffness of thebar between the anchorage and the head(free boltlength), while the second one refers to a set of defor-mational effects due to non- perfect anchorage, inflectionand yielding of the washer plate on the tunnel wallduring loading. The bolting stiffness is given by thefollowing expression:

1k sbol w z4ØLbolS S Ø qQx |t l 2pØF ØEy ~st

where:Q is the load-deformation constant for the anchorand headwforce =length x (Hoek and Brown, 1980);y1 1

S andS are the circumferential spacing and longitudinalt l

spacing;L is the bolt length;F is the bolt diameter;bol

andE is the elastic modulus for the steel.st

In this case,p is given by Eq.(10), whereT ismax max

the force that induces yielding of the steel.

Tmaxp s (10)max,bol S ØSt l

The collapse of the bolting system occurs when therupture deformation in the steel is reached(Eq.br,st

(11)):

u yu (L Ø´ (11)max,bol in,bol bol br,st

Eq. (11) is based on the hypothesis that the boltanchorage does not undergo any displacement afterbolting. This assumption produces a safer bolting design.

In the case where the force that causes collapse ofthe anchorage is lower than the force that inducesyielding of the steel, it is advisable to makeT equalmax

to the first of the two forces andu su .max,bol el,bol

If the bolts are prestressed with a forceT , the reaction0

line would assume the form shown in Fig. 3.Stiffness k and p do not change;u isbol max,bol el,bol

defined by the other parameters, as can be seen in Fig.3: it decreases whenDp increases, ifu is consid-bol in,bol

ered constant. The radial displacement of the wall thatcauses collapse of the bolting system, is given by Eq.


Fig. 3. Reaction line for pre-stressed radial bolts. Key:Dp : internalbol

pressure due to the stressing: .T0

Dp sbol S ØSl t

(12) which takes the pre-stressing deformation intoaccount:

w z4ØT0u yu (L Ø ´ y (12)x |max,bol in,bol bol br,st 2pØF Ey ~st

4. The evaluation of support efficiency

A support can be considered suitable when:– its safety factor concerning the collapse(or sometimes

concerning only yielding) is greater than an accepta-ble minimum value:F GF ; ands s,min

– the wall displacements in equilibrium conditionsresult to be lower than a given limit in relation to thedesign criteria of the tunnel:u Fu .eq lim

In the presence of radial bolting, it is acceptable tointroduce a third evaluation criterion which prevents theplastic radius in the rock mass from exceeding theanchorage point of the bolts:R F(RqaØL ), wherepl bol

as0.5y0.75.The safety factor that refers to the support collapse is

based on an examination of the maximum principalstrain induced at the intrados, for an elastic-perfectlyplastic behaviour of the material(Fig. 2):

´brF s (13)s´max

where: ´ is the failure strain of the support materialbr

and´ is the maximum strain induced in the support.max

For the supports examined in Sections 3.1, 3.2 and

3.3, the safety factor is, therefore, expressed by Eq.(14)–Eq. (16):steel sets:

B EhsetC F´ Ø Rybr,stD G2

F s (14)s,set u yueq in,set

shotcrete lining:

u Gu : Feq el s,shot

´ Ø RytŽ .br,con shots

2Ø 1yn ØRØ Ryt pŽ . Ž .con shot max,shotu yu q Øeq el,shot2 2 kshotRyt q 1y2Øn ØRŽ . Ž .shot con

(15)

u -u : Feq el,shot s,shot

w z2 2x |´ Ø Ryt q 1y2Øn ØRŽ . Ž .br,con shot cony ~

s2Ø u yu ØRØ 1ynŽ . Ž .eq in,shot con

radial bolting with anchoring:

´ ØLbr,st bolF s (16)s,bol 4ØT ØL0 bolu yu qeq in,bol 2pØF ØEst

It is possible to evaluate a maximum allowable walldisplacementu that defines the range of existenceamm

for each single type of analysed support(u Fu ) byeq amm

placing the minimum allowable value of the safetyfactor (F sF ) into Eq. (14)–Eq. (16).s s,min

Frequently, however, the long term performance ofsupports is not considered acceptable when they over-come elastic limits. In the case of the concrete being inplastic field, in fact, fissuring can arise and decrease themechanical characteristics of the concrete. It is, there-fore, justifiable to impose the elastic behaviour of thelining through the definition of the safety factor thatrefers to the applied loadsp and not to the displacementsu:

pmaxF s (17)s peq

5. Compound support

When a compound support that behaves like a linearelastic model is considered, the stiffness of the systemis simply given by the sum of the stiffness of eachsingle element in the system:

k s k (18)tot i8i


Fig. 4. Parallel scheme of the support interaction: the total stiffnessof the support system in the elastic field is given by simply summingthe single stiffnesses.

Fig. 5. Example of the reaction line of a support system composed ofthree different types of structures installed at the same time and at thesame distance from the face. Key:u : radial displacement of the wallin

when the supports are installed;u , u , u : radial displacementel,1 el,2 el,3

of the wall when the elastic limit is reached in the three supports;u : min wu x; .p s pmax,tot i max,i max,tot max,i8

i

Fig. 6. Example of a reaction line of a support system composed ofthree different types of structures installed in the tunnel at differenttimes and at different distances from the excavation face.

where: k is the total stiffness of the support systemtot

andk is the single support stiffness.i

The calculation scheme is that of several stiffnessesplaced in parallel(Fig. 4).

If the supports are assumed to be installed at thesame time and at the same distance from the excavationface (u su ), the load p , which is applied to thein,i in i

generic supporti (if it has not reached its elastic limitu ) results to be a function of the radial displacementel,i

(uyu ) and of its own stiffnessk :in i

psk Ø uyu (19)Ž .i i in

For u greater than its own elastic limit, the load bornby the generic support is equal top , according tomax,i

the specific support type.As u increases, the elastic limit can be reached in

some supports and its contribution, in terms of stiffness,result to be zero:

¯k s k (20)tot i8i

where: sk for u-u ; and s0 for uGu .¯ ¯k ki i el,i i el,i

The reaction line of the support system assumes theshape given in Fig. 5, with a progressive reduction ofstiffness.

If one assumes that the collapse of the support systemcoincides with the collapse of the weakest element(collapse of the first support), then:

w xu smin u (21)max,tot i max,i

The u value, therefore, indicates the end of themax,tot

reaction line of the support system.In the more usual case, when different supports are

installed at different times or at different distances fromthe excavation face(u /u , for i/j), the reactionin,i in,j

line of the whole system can assume the form of Fig.

6. The stiffness varies asu increases, always on thebasis of Eq.(20), but the values of are expressed by:ki

ksk for u Fu-u ;i i in,i el,i

ks0 for u-u anduGu .i in,i el,i

The load acting on the single generic supporti isgiven by Eq.(22) whenu Fu-u :in,i el,i

psk Ø uyu (22)Ž .i i in,i

Eq. (21) is still valid in the case shown in Fig. 6.


Table 1Results of the numerical analysis in the studied example

Case a Case b

Final vertical displacement in the crown 4.8 mm 6.1 mmTotal vertical load on the support system 0.20 MPa 0.14 MPaRadial displ. at the installation of shotcrete 2.1 mm 2.1 mmRadial displ. at the installation of steel sets 2.5 mm 3.2 mm

The definition for the safety factor of the singlesupport is the same as that given in Section 4, as afunction of the following:

● tunnel wall displacement at equilibrium,u ;eq

● displacement on the installation of each single supportu ; andin,i

● geometrical and mechanical characteristics of thestructure.

A maximum admissible wall displacementu ofadm,tot

the support system can be defined as the minimumvalue of the maximum admissible wall displacementsu for each single support(u smin wu x);adm,i adm,tot i adm,i

u is the maximum displacement which guaranteesadm,i

the minimum required safety factor for the genericsupport:

u yuŽ .max,i in,iu s qu (23)adm,i in,iFs,min,i

where:F is the minimum admissible safety factors,min,i

for the generic support.As an alternative, when it is necessary to guarantee

the permanency of each support in the elastic field,u is defined by the following equation:adm,i

u yuŽ .el,i in,iu s qu (24)adm,i in,iFs,min,i

A correct design of a support system should be ableto specify the single supports and to choose the instal-lation times in such a way that the values ofu areamm,i

as similar and coinciding as possible or slightly greaterthanu :eq

u (u Gu for i/j (25)amm,i amm,j eq

The mathematical procedure presented in this sectionmeets the target of an economic design, i.e. preventingsome supports from working under safety conditionsthat are unjustifiably high in comparison to the others.

The study developed in this section using the conver-gence–confinement method allows one to quantify con-cepts that engineers encounter in the monitoring data ofstresses and displacements in the support structures andin the numerical modelling of tunnels:

● the loads applied on the support are a function of thestiffness of the support, of the radial displacement ofthe wall when the support is installed and of the finalradial displacement of the wall; and

● when more than one support are used,(i.e. shotcretelining and steel sets) the load on each of them dependon the radial displacement of the wall when the singlesupport is installed; for a constant value of the

stiffness, the load is, therefore, a function of thedistance from the face of the point of installation.

Heredown are illustrated, for example, the results ofa numerical analysis, using FLAC, of a tunnel 12-mwidth, 10-m height, 140-m depth, excavated in a weakrock mass (friction angle: 308, cohesion: 0.1 MPa,deformation modulus: 5000 MPa) in which the in situcoefficient of horizontal stressK is equal to 0.5.

The mesh is constituted by 12 000 quadrilateral ele-ments and extend 180=140 m.

Two support structures were considered: a shotcretelining 20-cm thick realised close to the face, and doubleINP 160 steel sets spaced 75 cm, installed at a certaindistance from the face. When the shotcrete lining isinstalled, the radial displacement of the wall is lowerthan that one with reference to the installation of steelsets.

Two cases were considered: firstly, steel sets installedat a short distance(1.5 m) from the face(case a);secondly, steel sets installed at great distance from theface(7 m) (case b).

The numerical calculation has confirmed the impor-tance of the distance at which the supports are installed.Table 1 summarises the results for case a and case b.Fig. 7 shows the detail of the mesh of the numericalmodel adopted to analyse the illustrated problem. Fig. 8reports the final vertical displacements around the tunnelfor case a.

From these results it is possible to note that theefficiency of the second support(steel sets) decreasewith the distance of the point of installation from theface: the final displacement of the tunnel increase; andfinal vertical load on the support system decrease. Boththe results are quickly estimated through the conver-gence–confinement method and the mathematical pro-cedure presented in this paper.

6. A particular case: supports placed inside a shot-crete lining

When other supports(for example steel sets oranchored radial bolts) are placed after and inside ashotcrete lining, it is no longer possible to simply sumthe individual stiffnesses(as shown in Section 5) tofind the global stiffness of the support system. Theparallel scheme is in fact no longer suitable. Thestiffness of the shotcrete lining, due to the changed


Fig. 7. Detail of the mesh of the numerical method developed to study the illustrated example.

Fig. 8. Final vertical displacements for case a of the illustrated example.

boundary conditions at the intrados, can no longer becalculated using Eq.(7).

This situation occurs, for example, using the Mechan-ical Pre-cutting Method(Bougard, 1988; van Walsum,

1991; Puglisi, 1991), where a shotcrete lining pre-support is placed ahead of the excavation face and onlylater, with the advancement of the excavation face, aresteel sets installed. It is not unusual for steel sets and


radial bolts to be installed after and inside a first layerof shotcrete during traditional advancement.

The radial pressure at the lining intrados is no longerzero and it is applied by the internal support. Nowsro

is function of its stiffnessk and of the radial displace-*

ment d that develop at the intrados of the lining afterthe installation of the internal support.

*s sk Ød (26)r0

After the definition of the new integration constantsdue to the new boundary condition(see Appendix 2),one can obtain the stiffness of the support systemktot

(liningqinternal support):

k stot

w zEcon *2Ø 1yn ØE ØRØ q Ryt Økx |Ž . Ž .con con shot1qny ~Ž .con Econy (27)w zB E 1qn ØRR Ž .con22 * C FE Ø 1y2Øn ØR q Ryt Ø E q 1y2Øn Ø 1qn Øk Øt Ø 1qx |Ž . Ž . Ž . Ž .con con shot con con con shot

D GRyty ~Ž .shot

Fig. 9. Typical trends of the reaction line of a concrete lining with internal confinement produced by another support. Key:(a): initial yieldingof the shotcrete lining(u -u ); (b): initial yielding of the internal support(u )u ); u : displacement of the tunnel wall when the lining is* *

el el el el in

installed;u : displacement when the internal support is installed;u : displacement when the yielding of the lining occurs;u : displacement when* *in el el

the yielding of the internal support occurs;u : displacement when the support system collapses(minimum value betweenu andu ); k:*max,tot max max

stiffness of only the lining;k : stiffness of the whole support system;k : stiffness of the internal support;p : maximum load that can be*tot max,tot

reached by the support system.

Having reached the internal support capacity( *k Ødsfor usu ), the boundary condition at the intrados* *pmax el

of the lining is no longer represented by Eq.(26) butby Eq. (28):

*s sp (28)r0 max

where: is the yield strength of the internal support,*pmax

calculated according to one of the methods found inSections 3.1, 3.2 and 3.3, taking its actual geometry intoaccount.

6.1. A calculation example

In order to verify the importance of a correct inter-pretation of the structural interaction in the case inwhich an auxiliary support is installed inside the shot-crete lining, a comparison is made between the typicalapproach(simplified) which makes use of the parallelstiffness scheme(Section 5) and the more completeapproach proposed in Section 6. The analysed exampleconcerns two circular tunnels of 3 and 8.5 m in diameter;the mechanical and geometrical characteristics of thesupports are given in Table 2.

The stiffness of the total system is the same stiffnessthat the lining has in the absence of internal supports(Eq. (7)), in that it has reached the elastic limit(k s*

0).The lining yielding and collapse verification must be

carried out at its intrados where, even in this case, acritical stress condition is reached.

On the basis of the contents of this section, it ispossible to identify two typical situations according towhether the concrete lining reaches plastic conditionsfirst (u -u ) or the internal support reaches plastic*

el el

conditions first (u )u *). These two different situa-el el

tions are schematically illustrated in Fig. 9.


Table 2Mechanical and geometrical characteristics of the supports used in thecalculation example

Shotcrete lining

Elastic modulus E : 8000 MPacon

Poisson ratio n : 0.15con

Uniaxial compressive strength s : 16 MPac

Internal friction angle f: 398Maximum principal rupture strain ´ : 0.0045br,con

Thickness t : 0.25 mshot

Minimum admissible safety factor F : 1.3s,min

Internal support (HEB 240 steel sets)

Elastic modulus E : 210 000 MPast

Yielding strength of the steel s : 450 MPast,y

Maximum principal rupture strain ´ : 0.0050br,st

Cross-sectional area of the steel section A : 106 cm2set

Cross-sectional height of the steel section h : 0.24 mset

Steel set spacing along tunnel axis d: 1.2 mMinimum admissible safety factor F s1.1s,min

Table 3Results of the comparison of the two different approaches

Tunnel diameter: 3 m

Simplified calculation Proposed approach

k (MPaym)tot 2469 2218u (mm)el 12.4 15.0u (mm)*

el 14.4 14.5u (mm)max 15.6 16.0u (mm)amm 14.3 14.5

Tunnel diameter: 8.5 m

Simplified calculation Proposed approachk (MPaym)tot 241 235u (mm)el 17.8 18.5u (mm)*

el 24.3 24.5u (mm)max 27.9 28.0u (mm)amm 23.8 24.0

Fig. 10. Results of the comparison between a complete evaluation of the support system reaction line according to the proposed approach and tothe simplified calculation, on the basis of Section 5. The example refers to the mechanical and geometrical conditions summarised in Table 2.Key: (a) tunnel diameter: 3 m;(b) tunnel diameter: 8.5 m; A: installation of the shotcrete lining; B: installation of the internal support; C: yieldingof one of the two supports; D: yielding of both supports; and E: collapse of the system.

The following values have also been hypothesised:

● Ds3 m: u s10 mm;u s12 mm*in in

● Ds8.5 m: u s10 mm;u s16 mm.*in in

The reaction lines of the support systems obtainedusing the two different approaches are shown in Fig.10.

More details on the results of the comparison aregiven in Table 3.

From an analysis of Fig. 10 and Table 3 one canobserve that:– the total stiffness of the support system results to be

slightly lower for the proposed procedure than for the

simplified calculation: the reduction is greater(10.2%) for the 3-m diameter tunnel, but less impor-tant (2.5%) for the 8.5-m diameter tunnel;

– the displacement at collapse(u ) and those consid-max

ered to be admissible(u ) do not significantly varyadm

with the different calculation methods;– for the 8.5-m diameter tunnel, the reaction line

obtained for the two methods, qualitively speaking,have a similar trend;

– for the 3-m diameter tunnel, the reaction lines insteadhave very different trends: when considering thestiffness in parallel one can notice the yielding of theshotcrete lining immediately after the installation ofthe internal support; the proposed procedure permitsone to verify how the concrete lining is in elastic


Fig. 11. Definition of the maximum admissible pressure on the system when it is necessary to guarantee that both supports remain in the elasticfield (the example refers to the 3 m diameter tunnel). Key: (a): proposed approach; and(b): supports considered to be active in the parallelscheme.

conditions for a larger deformation range, due to thebeneficial confinement produced by the internal sup-port; and

– the evaluation of the extension of the elastic field ofthe support structures, carried out with the proposedprocedure for the 3-m diameter tunnel, shows, as aconsequence, that the reaction line reaches thep pressure which is obviously greater than thatmax,tot

which can be deduced from the simplified calculation(q35.6%).

It is, therefore, possible to draw the conclusions thatusing the proposed procedure for small diameter tunnels,that is, when the boundary conditions of the lining areof greater importance, can favourably influence thedesign of the support structure. This is even moreobvious when, as often happens, it is necessary toguarantee the elastic behaviour of the foreseen supports:the difference between the two reaction curves of Fig.10a consists not only of different qualitative trends butalso of different values ofp and above all ofmax,tot

pressurep in points C(points corresponding to yieldingof the first support).

The dimensioning should however be based, in thesecases, on the maximum admissible pressurep actingadm

on the support system, calculated starting from thepressure relative to point C:

p C( )p s (29)adm FS

The two curves of Fig. 10a are shown with moredetail in Fig. 11(the reaction line calculated with theproposed procedure in Fig. 11a and that calculated byconsidering the supports with the simple parallel scheme

in Fig. 11b). It is evident thatp is very different inadm

the two cases.

7. The interaction between preliminary supports andthe final lining

The final lining has the purpose of making a tunnelstable in the long term, when, because of viscosity andweathering of the rock the mechanical characteristics ofthe rock mass appear to be lower than in the assumedshort-term conditions.

The final lining, from the moment it is placed,constitutes a further support on the inside of the tem-porary ones. The global reaction line(temporaryqfinalsupports) can be calculated according to the formulationobtained in Section 6.

Fig. 12 shows the global reaction line of the supportswhich is necessary for the evaluation of the loadsproduced on the final lining and for the study of theinteraction between the temporary supports and the finallining. The reaction line of the temporary supportsintercepts the short-term ground reaction curve: B rep-resents the equilibrium that occurs before the final liningis cast. The ground reaction curve then moves until itreaches the long-term situation; a further deformativeprocess of the tunnel is now contrasted not only by thetemporary supports but by these supports together withthe final lining. Point C represents the final equilibriumbetween the tunnel and the support system when onesupposes that the effectiveness of the temporary supportsremains unaltered in time.

In reality, it is often preferred to neglect the effect ofthe temporary supports in the long term as it is consid-ered that these may have undergone a chemical–physicalreaction, on direct contact with the rock, so that the


Fig. 12. Interaction between the temporary supports and the final lin-ing. Key: A: installation of the temporary supports; B: equilibriumpoint for the temporary supports; C: equilibrium point of the supportsystem(temporary supportsqfinal lining) in the long term; D: equi-librium point for only the final lining in the long term;u : radial wallin

displacement on the installment of the temporary supports;u : radialeq,1

wall displacement in correspondence to the short term equilibriumpoint; u : radial wall displacement in correspondence to the longeq,2

term equilibrium point;p : final load on the final lining;k : stiffnesseq,2 1

of the temporary supports;k : stiffness of the whole system(tempo-2

rary supportsqfinal lining); andk : stiffness of only the final lining.3

Fig. 13. Improvement of the strength parameters(generally cohesion)of the rock mass in the plastic field, due to the presence of radialbolts.

Fig. 14. Calculation scheme for the calculation of the convergence–confinement curve in presence of reinforced rock around the tunnel(Oreste, 1994).

original mechanical properties are altered. On the basisof this second hypothesis, the final equilibrium pointresults to be D, which is the intersection between thesingle reaction line of the final lining and the long-termground reaction curve. It is worthwhile to observe how,even when point C refers to a higher pressurep, thepart of the load that weighs directly on the final liningis usually lower than the pressure that refers to point D,as pressurep, in the first case, is distributed over thetemporary supports and the final lining. The minimumadmissible safety factor required for the final liningvaries in function of the hypotheses assumed on theefficiency of the temporary supports in time: it willobviously be lower when the temporary support presenceis neglected.

8. The interaction between the ground improvementinterventions and the supports

When ground improvement techniques are usedaround the tunnel(passive bolts such as dowels andcemented cables, grouting and freezing of the ground,radial jet-grouting, etc.) together with temporary sup-ports or final lining, the study of the interaction becomescomplex and, as a consequence, the designing of thedifferent structural elements becomes difficult.

It is possible to effectively face this problem byimagining that the mechanical properties of the rock areimproved after the interventions in the reinforced zone.Obviously, the evaluation of the magnitude of such anincrease could be difficult but there are some usefulindications in literature(Grasso et al., 1989, 1991;Indraratna and Kaiser, 1987): some authors suggest toincrease the cohesion value of the rock in the plastic

field (residual cohesionc*) (Fig. 13) on the basis ofthe Eq.(30).

1qsenw*c scq ØDs (30)32Øcosw

where:Ds is the confinement produced by the action3

of the grouted bolts:

TmedDs s3 S ØSt l

where T : is the mean force along each bolt;S andmed t

S : transversal and longitudinal spacing;c andw: cohe-l

sion and friction angle of the rock mass.Furthermore, thanks to specific calculation methods

(Oreste, 1994; Oreste and Peila, 1996; Pelizza et al.,1994) (Fig. 14), it is possibly to easily obtain theground reaction curve(Fig. 15) when two differentmaterials co-exist at the boundary of the tunnel: the firstmaterial represents the natural rock while the second isthe reinforced rock in a concentric zone around thetunnel. The ground reaction curve thus obtained is called


Fig. 15. Typical ground reaction curve in the presence of rock rein-forcing. Key: A: rock improvement interventions; B: yielding of thenatural or reinforced rock(loss of linearity); and C: radial wall dis-placement in the absence of supports.

Table 4Main characteristics of the Serena tunnel section studied in the cal-culation example

Depth(m) 120Tunnel radius(m) 6.5Thickness of the reinforced zone(m) 8In situ hydrostatic stress(MPa) 4

Fig. 16. Trend of the plastic radii in the natural rock and in the rein-forced zone as a function of the internal pressure applied by the sup-ports (p ). Key: A: yielding in the reinforced zone; B: yielding ineq

the natural rock; and C: complete yielding of the reinforced zone.

the ‘ground reaction curve in the presence of rockreinforcing’.

Once the convergence–confinement curve has beenobtained in the presence of rock reinforcing it is thenpossible to proceed according to the methods given inthe previous sections to evaluate the stress and strainconditions of the support structures. The work conditionsof the reinforcement elements or of the improved zonesare then evaluated on the basis of the equilibriumpressurep (Fig. 1) which influences the stress andeq

strain fields of the natural rock and of the reinforcedzone. In this way it is also possible to verify, forexample, that the plastic radius in the improved zonedoes not exceed a certain limit in order to preventcompromising the functioning of the completely cement-ed radial bolts or cables(they should, in fact, have anadequate anchorage length in the elastic rock). The

plastic radii in the two different materials result to bedirect functions of pressurep applied by the supportseq

onto the tunnel walls(Fig. 16).An example of the application of the convergence–

confinement concept in the presence of rock reinforcinghas been carried out for a real case for which conver-gence measurements were available: the Serena tunnelnear La Spezia(Grasso et al., 1989, 1990, 1991). TheSerena tunnel, which is part of the Nuovo ItinerarioFerroviario Pontremolese(The New Pontremolese Rail-way Itinerary) which connects La Spezia to Parma, is adouble track tunnel(excavation section of approx. 110m , 13-m span) of a total length of 7 km and a maximum2

depth of 500 m. The tunnel progresses along geologicalformations that are distinguished by poor geomechanicalcharacteristics and which can be defined as ‘structuralycomplex’ due to the marked lithological and structuralinhomogeneity. From a geomechanical point of view,the ground under examination prevalently belongs toBieniawski’s IV and V technical classes (Bieniawski,1984). Tunnelling was carried out using hydraulic ham-mers or simply with excavators.

A reinforcement intervention of the rock was carriedout in a section of approximately 1 km, in a clay andcalcarous formation(passive cables cemented along thefull length in 51-mm diameter bore-holes with a finalstrength equal to 45 t, quincunx placed with a 1=1 mspacing) as soon as a high convergence velocity wasencountered(3 cmyday), this problem being difficult tosolve in any other manner. The intervention entailed theplacing of radial reinforcing elements of a length similarto the hypothesised plastic radius(approx. 12 m) at thetunnel border from a pilot tunnel and from the largeradius section. These elements were able to exercise a‘coaction’ effect on the rock and, therefore, to mobilisea remarkable additional strength.

The ground reaction curve was determined in thepresence of rock reinforcing for a particular section(progressive length: 6600 m) with a depth of 120 m,according to the criterion shown in this section. Themain characteristics of the studied section are shown inTable 4. The mechanical parameters of the natural rockand of the reinforced zone used in the calculation aresummarised in Table 5.

The ground reaction curves, in the absence and in thepresence of rock reinforcements, and the mean value ofthe convergence measured in concomitance with thesupport structures that apply an internal pressure of


Table 5Mechanical and strength parameters(according to Hoek and Brown’s strength criterion) for the natural rock and for the reinforced zone of theSerena tunnel

Mechanical and strength parameters Natural Reinforcedrock zone

Elastic modulusE (MPa) 800 800Peak strength parameterm 0.4 0.4Peak strength parameters 0.001 0.001Residual strength parameterm 0.35 0.502Residual strength parameters 0.00025 0.0305Uniaxial compressive strengths (MPa)c 3.48 3.48Deformation parameterf in plastic conditions 1 1

Fig. 17. Comparison between the ground reaction curve of the tunnelin the presence and absence of rock reinforcements and the meanvalue of the in situ measured radial wall displacements.

approximately 0.13 MPa to the walls are given in Fig.17. An analysis of the graph permits one to notice agood agreement between the calculation results and thein situ measured values.

9. Conclusions

The interaction of support structures in tunnels hasbeen studied in this work through the use of theconvergence–confinement method. After having recalledthe equations that characterize the behaviour of the mostimportant support types usually used, the equations thatpermit one to obtain the loads which are applied to thedifferent supports simultaneously present in a tunnel,have been obtained. Two different criteria for the eval-uation of the safety factors of each single support havebeen proposed: one is based on the strain level and isof general validity; the other is based on the analysis ofthe stresses and should be used in all those cases inwhich one wishes to avoid the plastic conditions in oneor all the supports. In both cases, it is possible tocalculate the maximum admissible radial wall displace-

ments connected to the minimum admissible safetyfactor for each single support. The maximum admissibledisplacement for each support results to be very usefulfor a correct design, from the economic point of view,and is able to avoid injustifiable high values of thesafety factors of some structures in comparison to others.

A case, which does not fit into the usual rock-supportinteraction schemes, was also analysed: an auxiliarysupport is placed inside a concrete lining(for example,a presupport ahead of the faceqsteel sets). The completeanalysis has shown how the global stiffness of thesystem results to be slightly lower than that obtainedusing the traditional approach(stiffnesses in a parallelscheme), while the interval of elastic behaviour of theconcrete lining is considerably more extended for smalldiameter tunnels, producing a trend of the global reactionline that is basically different from the traditionalapproach.

Finally, the interaction between temporary supportsand a final lining was studied in depth with the usuallyused calculation hypotheses and reference was made tothe interaction between the support structures and rockimproving works around the tunnel excavation throughan approach concerning the ground reaction curve in thepresence of rock reinforcing.

Appendix 1: Definition of the mechanical parametersof the shotcrete lining

The structural behaviour of the shotcrete lining isevaluated on the basis of the general equation of theradial displacement Eq.(A1.1) which is obtained byresolving the differential equation system that governsthe stress and strain behaviour of the elastic media inaxialsymmetrical conditions:

Bu sAØrq (A1.1)r r

where:u is the radial displacement in the lining at ther

distancer andA andB are integration constants.By using Eq.(A1.1) and from the definition of the

axialsymmetric strain one can calculate the radial and


Fig. 18. Geometric loading scheme of a shotcrete lining. Key:r:generic radial coordinate;R: tunnel radius;p: pressure acting on thelining extrados;s : radial stress at the intrados.ro

circumferential strains Eq.(A1.2) and Eq.(A1.3) (pos-itive if it is a compression strain) and the stresses onthe inside of the lining vs. the distancer Eq. (A1.6)and Eq.(A1.7) from the constitutive law of the elasticmaterial Eq.(A1.4) and Eq.(A1.5):

w zdu Br´ s s Ay (A1.2)x |r 2dr ry ~

w zu Br´ s s Aq (A1.3)x |q 2r ry ~

1 w z2 2x |´ s Ø 1yn Øs y n qn Øs (A1.4)Ž . Ž .r con r con con qy ~Econ

1 w z2 2x |´ s Ø n yn Øs y n qn Øs (A1.5)Ž . Ž .q con con q con con ry ~Econ

s sCØ´ qDØ´ (A1.6)q q r

s sCØ´ qDØ´ (A1.7)r r q

where:

;1ynŽ .con

Cs ØEcon1y2Øn Ø n q1Ž . Ž .con con

; and E and n arenconDs ØEcon con con1y2Øn Ø n q1Ž . Ž .con con

the concrete elastic modulus and Poisson’s ratio,respectively.

The boundary conditions around the lining ring(Fig.18) are:

1. rsR: u s(uyu ): the displacement of the extra-R in,shot

dos is equal to the displacement of the tunnel wallafter the installation of the lining; and

2. rs(Ryt ): s s0: the internal pressure acting onshot ro

the intrados is zero.One obtains Eq.(A1.8) and Eq. (A1.9) from Eq.

(A1.1) and Eq.(A1.7), which give theA andB constantsEq. (A1.10).

B w zx |uyu sAØRq ´ BsR uyu yAØRŽ . Ž .in,shot in,shoty ~R

(A1.8)

w zw zx |RØ uyu yAØRŽ .in,shoty ~x |0sCØ Ay qD2RytŽ .y ~shot

w zw zx |RØ uyu yAØRŽ .in,shoty ~x |Ø Aq (A1.9)2RytŽ .y ~shot

w z1y2Øn ØRŽ .conx |As Ø uyu ;Ž .in,shot2 2Ryt q 1y2Øn ØRy ~Ž . Ž .shot con

2w zRØ RytŽ .shotx |Bs Ø uyuŽ .in,shot2 2Ryt q 1y2Øn ØRy ~Ž . Ž .shot con

(A1.10)

By substituting the constantsA and B obtained fromEq. (A1.10) in Eq. (A1.2) and Eq.(A1.3) and theselast two equations in Eq.(A1.7), one obtains theexpression of the radial stress forrsR (p) vs. the radialwall displacement after installation of the lining(uyu ):in,shot

psk Ø uyu where:Ž .shot in,shot

22w zx |R y RytŽ .shoty ~E 1conk s Ø Ø (A1.11)shot w z22

x |1qn RŽ . 1y2n R q RytŽ . Ž .con con shoty ~

where:k is the stiffness of the shotcrete ring.shot

P is obtained, in this case, by taking thesmax q,max

maximum circumferential stress at the intrados equal tos (the uniaxial compression strength of the shotcrete);c

s Eq. (A1.12) is calculated as follows:q,max

– determining expression(uyu ) from Eq.(A1.11)in,shot

in function of p and substituting this in Eq.(A1.10);and

– evaluating Eq.(A1.2) and Eq.(A1.3) at the liningintrados(for rsRyt ) and substituting these in Eq.shot

(A1.6).

ps ss sVØ where:q,max q rsRyt( )shot kshot

E RconVs2Ø Ø w z22

x |1qnŽ . 1y2Øn ØR q RytŽ . Ž .con con shoty ~

(A1.12)

By making s Eq. (A1.12) equal tos and byq,max c

simplifying one obtains:


2w zRytŽ .shot1 x |p s Øs Ø 1y (A1.13)max,shot c 22 Ry ~

Failure will occur when the circumferential compres-sive strain at the intrados attains the failure value´ :br,con

w z2Ø 1yn ØRŽ .conx |´ s´ (q,max q rsRyt( ) 2shot 2Ryt q 1y2Øn ØRy ~Ž . Ž .shot con

Ø u yuŽ .el,shot in,shot

u yuŽ .max,shot el,shotq s´br,conRytŽ .shot

from which:

u (u q´ Ø RytŽ .max,shot el,shot br,con shot

2Ø 1yn ØRØ RytŽ . Ž .con shoty 2 2Ryt q 1y2Øn ØRŽ . Ž .shot con

pmax,shotØ (A1.14)

kshot

Appendix 2: Evaluation of the mechanical parame-ters of the system composed by shotcrete lining andanother internal support (steel sets)

Pressures (Fig. 18) is, in this case, no longer zeroro

and it is applied by the internal support in function ofits stiffnessk and of the radial displacementd that*

develop at the intrados of the lining after the installationof the internal support.

*s sk Ød (A2.1)r0

d is obtained by the following equation:

*ds uyu q D yD (A2.2)*Ž . Ž .in u uin

where:D is the increase of the thickness of the liningu

corresponding to displacementu of the tunnel wall;is the increase of the thickness of the lining onD *uin

installation of the internal support(displacement of thetunnel wall equal tou ).*

in

The increase of the thickness of the lining with theincrease of the displacement(uyu ) at the extradosin

(u being the displacement of the wall on installationin

of the lining) is given by the following equationobtained from Eq.(6), as the difference between theradial intrados and extrados displacements:

* *w z w zB B* *D s A Ø Ryt q y A ØRq (A2.3)x | x |Ž .u shot Ryt Ry ~ y ~Ž .shot

where: A and B are integration constants due to the* *

changed boundary conditions and are still unknown.At the moment of installing the internal support(us

u on the tunnel wall), the increase of the thickness of*in

the lining is given by the following equation:

w zB *usu( )inD s A Ø Ryt qx |* * Ž .u usu shot( )in in Ryty ~Ž .shot

w zB *usu( )iny A ØRq (A2.4)x |*usu( )in Ry ~

where: A and B are integration constants of the liningalone Eq.(A1.10), for usu .*

in

Finally, by substituting Eq.(A2.3) and Eq.(A2.4) inEq. (A2.2) and by simplifying one obtains the requiredd expression, which is a function of theA and B* *

integration constants and of the radial wall displacementu:

* * *ds uyu q yt ØA qlØBŽ . Ž .in shot

y yt ØA qlØB (A2.5)* *Ž .shot usu usu( ) ( )in in

where:

tshotls (A2.6)

RØ RytŽ .shot

The boundary condition at the intrados is now repre-sented by the following new relations:

rs(Ryt ): s sk* Ødshot r0

The two boundary conditions(at the intrados and atthe extrados) lead one to obtain Eq.(A2.7) and Eq.(A2.8) which take the place of Eq.(A1.8) and Eq.(A1.9).

*B w z* * *x |uyu sA ØRq ´ B sRØ uyu yA ØRŽ . Ž .in iny ~R

(A2.7)

w zw z*x |RØ uyu yA ØRŽ .iny ~

* *x |s sk ØdsCØ A y qDr0 2RytŽ .y ~shot


*x |Ø A q (A2.8)2RytŽ .y ~shot

By substituting Eq.(A2.5) with Eq. (A2.6) in Eq.(A2.8), one obtains the following expression ofA (B* *

is given by Eq.(A2.7), A being known):*

* *A sbØ uyu qgØ u yu (A2.9)Ž . Ž .in in in

where:


w z*x |1y2Øn ØRØ 1qn Ø Ryt Øk qEŽ . Ž . Ž .con con shot cony ~

bsw zB ER22 * C FE Ø 1y2Øn ØR q Ryt Ø E q 1y2Øn Ø 1qn Øk Øt Ø 1qx |Ž . Ž . Ž . Ž .con con shot con con con shot

D GRyty ~Ž .shot

w z1y2Øn Øt Ø 2RytŽ . Ž .con shot shot* x |1y2Øn Ø 1qn Ø Ryt Øk ØRØ 1yŽ . Ž . Ž .con con shot 2 2Ryt q 1y2Øn ØRy ~Ž . Ž .shot con

gsyw zB ER22 * C FE Ø 1y2Øn ØR q Ryt Ø E q 1y2Øn Ø 1qn Øk Øt Ø 1qx |Ž . Ž . Ž . Ž .con con shot con con con shot

D GRyty ~Ž .shot

In the same way as in Section 3.2, the new integrationconstantsA andB being known, it is now possible to* *

obtain the expression of pressurep (radial stress at theextrados) in function of displacementu:

w zCyD *w xps 2ØCØby Ø uyu q 2ØCØg Ø u yux | Ž . Ž .in in inRy ~

(A2.10)

By deriving Eq.(A2.10), with respect to displacementu, and by developing the known terms, one finallyobtains the stiffness of the support systemk (liningqtot

internal support):

w zEcon *2Ø 1yn ØE ØRØ q Ryt Økx |Ž . Ž .con con shot1qny ~Ž .con Econk s y (A2.11)tot w zB E 1qn ØRR Ž .con22 * C FE Ø 1y2Øn ØR q Ryt Ø E q 1y2Øn Ø 1qn Øk Øt Ø 1qx |Ž . Ž . Ž . Ž .con con shot con con con shotD GRyty ~Ž .shot

Having reached the internal support capacity( *k Ødsfor usu ), the boundary condition at the intrados* *pmax el

of the lining is no longer represented by Eq.(A2.1) butby Eq. (A2.12):

*s sp (A2.12)r0 max

where: is the yield strength of the internal support,*pmax

calculated according to one of the methods found inSections 3.1 and 3.2 and Section 3.3, taking its actualgeometry into account.

While Eq. (A2.7) is still valid, Eq. (A2.8) is nowsubstituted by Eq.(A2.13):

where:

1y2Øn ØRŽ .conhs 2 2Ryt q 1y2Øn ØRŽ . Ž .shot con

21y2Øn Ø 1qn Ø RytŽ . Ž . Ž .con con shot 1ms Ø2 2Ryt q 1y2Øn ØR EŽ . Ž .shot con con

The newA and B parameters permit one to obtain* *

the p expression in function of the displacementu atthe extrados; deriving expressionp Eq. (A2.10) withrespect tou, it is possible to obtain the same stiffnessthat the lining has in the absence of internal supports(Eq. (7)), in that it has reached the elastic limit(k s*

0). The new parameters are also necessary for theevaluation of the stresses and strains in the lining forthe u greater thanu .*

el


* *x |s sp sCØ A y qDr0 max 2RytŽ .y ~shot


*x |Ø A q (A2.13)2RytŽ .y ~shot

By resolving Eq.(A2.13) one obtains the new expres-sion for A (B is also now given by Eq.(A2.7), as* *

A is known):*

* *A shØ uyu qmØp (A2.14)Ž .in max


The displacement of the tunnel wall which causes*umax

collapse of the internal support is evaluated startingfrom thed displacement at the lining intrados(frommax

Eq. (A2.5)) which induces a principal strain in thesupport that is equal to the failure strain(paragraph 4).

Is the maximum displacement that guarantees the*uammminimum required safety factor for the internal support.

The verification of the yielding and collapse of thelining must be carried out at the intrados where, evenin this case, a critical stress condition is reached. Thecircumferential stress and strain at the intrados arecalculated makingrsRyt , with Eq. (A1.6) and Eq.shot

(A1.3) (with A and B or A and B calculated on the* *

basis of Eq.(A1.10) for an internal support that is notyet active, or with Eq.(A2.9) and Eq.(A2.7) for aninternal support in elastic conditions or finally with Eq.(A2.14) and Eq.(A2.7) for internal support in plasticconditions); the radial stress, when different from zero,is calculated with Eq.(A2.1), Eq.(A2.5) and Eq.(A2.6)(internal support in elastic conditions) or with Eq.(A2.12) (internal support in plastic conditions).

If one hypothesises that the Mohr–Coulomb strengthcriterion is valid for concrete, the maximum principalstress (circumferential stress) at the intradoss ,q,max

which induces yielding in the lining, is a function ofthe minimum principal stresss (radial stress), of ther0

uniaxial compression strengths and of the internalc

friction angle F, according to the well-known Eq.(A2.15).

B E1qsinfC Fs ss q Øs (A2.15)q,max c r0D G1ysinf

By making Eq.(A2.15) equal to Eq.(A1.6) (adoptingthe opportune integration constants, as already seen) itis possible to calculate the displacement of the tunnelwall u for which yielding of the lining occurs while,el

if one makes Eq.(A1.3) equal to the principal rupturestrain ´ , one obtains the wall displacementubr,con max

which produces collapse of the lining.u is theamm

maximum displacement that guarantees the minimumrequired safety factor of the lining.

As, in this case, the concrete is usually shotcrete, itis necessary to carefully evaluate both the value of theelastic modulus and the strength parameters(c and F)as these vary in time after installation: it is necessary toadopt mean parameters during loading of the structure(Oreste, 2003). It is, however, useful to proceed with aparametric analysis in order to be able to evaluate theimportance of the uncertainty on these parameters.

The displacement at the intrados foru)u is alwaysel

calculated with Eq.(A2.5) but, in this case, the para-

metersA andB are evaluated forusu . The stiffness* *el

of the support system, once the lining is in a yieldingcondition, is given only by the stiffness of the internalsupport.

References

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analysis of structure using ccm

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