Approximate evaluation of stresses in degraded tunnel linings

Soil Dynamics and Earthquake Engineering 43 (2012) 45–57

Contents lists available at SciVerse ScienceDirect

Soil Dynamics and Earthquake Engineering

0267-72

http://d

n Corr

E-m

shinchb1 Te

journal homepage: www.elsevier.com/locate/soildyn

Approximate evaluation of stresses in degraded tunnel linings

Hany El Naggar a,n, Sean D. Hinchberger b,1

a Department of Civil Engineering, University of New Brunswick, Fredericton, Canada NB E3B 9P8b Hatch Renewable Power Geotechnical Lead, Senior Project Manager, 4342 Queen St. Suite 500, Niagara Falls, ON, Canada

a r t i c l e i n f o

Article history:

Received 21 December 2009

Received in revised form

27 February 2012

Accepted 14 July 2012Available online 10 August 2012

61/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.soildyn.2012.07.016

esponding author.

ail addresses: [email protected] (H. El Na

[email protected] (S.D. Hinchberger).

l.: þ905 374 5200; Direct: þ905 374 0701x

a b s t r a c t

This paper uses numerical and analytical methods to examine the static and seismic response of

tunnels with intact and degraded segmental concrete tunnel liners. Concrete degradation is simulated

using a non-linear finite element (FE) model that accounts for soil-structure interaction and the non-

linear stress–strain response of the soil and concrete. The non-linear FE model is used to calculate radial

stresses in tunnel linings with local concrete delaminations and that are subject to both static and

seismic loads. Then, the FE results are compared with an analytical solution for jointed tunnel linings

in order to assess the accuracy of the solution for predicting stresses in degraded liners. The analyses

and results presented in this paper illustrate a simple method for estimating and evaluating the effect

of concrete degradation on the distribution of thrust and moment in segmental tunnel linings subject to

either static or seismic loads.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Satisfactory long-term performance of buried reinforced con-crete structures depends in part on the durability of concrete inservice. In some cases, concrete tunnel liners may experiencedegradation when exposed to chloride-contaminated ground-water. For example, in Toronto, sections of the Toronto TransitCommission subway system that were constructed in the early1970s comprise precast segmental concrete lining rings. Sinceconstruction, significant portions of the segmental linings havebeen exposed to chloride contaminated groundwater and theynow exhibit significant concrete degradation in the form ofextensive concrete delaminations and steel deterioration [1].Concrete degradation, such as that described above, is prevalentin other cities which have significant snowfall and use deicingsalts on their roads.

Typically, estimating the effect of concrete degradation on theinternal loads in tunnel linings requires complicated analysessuch as those based on the finite element (FE) method, which canbe costly and time consuming. This paper uses a non-linear finiteelement (FE) model and linear elastic solution for jointed tunnellinings to examine the effect of concrete degradation on loads intunnel linings. The main objectives of the paper are: (i) toinvestigate the effect of local concrete degradation on the stress

ll rights reserved.

ggar),

6975.

distribution in degraded tunnel linings; (ii) to examine theaccuracy of a linear elastic solution (see [2] [3]) for static andpseudo-static analysis of degraded segmental tunnel linings and(iii) to study the effect of seismic events with different intensitieson intact and degraded segmental tunnel linings. To achieve theseobjectives, a typical tunnel case is examined using both non-linear FE model and an analytical solution for jointed tunnellinings in elastic ground. Internal lining loads are examined forthe static case corresponding to the end of construction and threedifferent seismic loading cases representing tunnels subject tolow, moderate and severe intensity earthquakes. For each loadingcase, the same soil profile and geometry are considered tofacilitate comparison of the seismic performance of the tunnellining. The results of this study provide insight into the effect ofconcrete degradation on the internal loads in tunnel linings anddemonstrate a simple method for developing upper and lowerbounds for the stresses in degraded linings.

2. Methodology

The finite element program AFENA [4] was used to performstatic and pseudo static analysis of tunnels with intact anddegraded linings. Seismic loading was estimated by performinga site response analysis (SRA) using bedrock input motionsrepresenting three levels of seismic intensity: low, moderateand high. The seismic shear strain deduced from the SRA corre-sponding to the tunnel spring line was used to develop anequivalent pure shear displacement field, which was applied asa boundary condition in the FE analysis similar to that done by

www.elsevier.com/locate/soildyn

www.elsevier.com/locate/soildyn

dx.doi.org/10.1016/j.soildyn.2012.07.016



mailto:[email protected]

mailto:[email protected]


H. El Naggar, S.D. Hinchberger / Soil Dynamics and Earthquake Engineering 43 (2012) 45–5746

Kontoe et al. [5]. This section describes the problem geometry,material models and FE discretization.

2.1. Problem geometry

As illustrated in Fig. 1, the problem geometry comprises acircular segmental concrete tunnel lining embedded in soil. TheI.D. (inner diameter) of the tunnel lining is 4.88 m and the O.D.(outer diameter) is 5.18 m and it is located at a depth of 29 mmeasured from the ground surface to the springline. As such, theC/D ratio of the tunnel is 5.5. The tunnel lining comprises eight150 mm thick concrete segments (1 m wide) with lining jointsevery 451. The first joint is situated at the crown. The tunnel isembedded in stiff soil; Table 1 summarizes the material para-meters for the soil and the lining.

2.2. Material models

2.2.1. Soil continuum

In this study, the soil continuum was modelled using anelastoplastic constitutive model based on the Mohr–Coulombfailure criterion. The input parameters for the model are: themodulus of elasticity, Es, Poisson’s ratio, n, effective cohesion, c0,and effective friction angle, f0. In addition to these parameters,the volumetric response of the soil at yield and failure is governedby the dilatancy angle, c, and a flow rule of the form proposed byDavis [6]. In the FE analyses, associated flow was assumed

29 m

5.18 m (O.D.)

G.L.

150 mm

θ

Fig. 1. Problem geometry.

Table 1Soil and tunnel liner properties used in the FE analysis.

Parameter Value

Soil elastic modulus, Es (MPa) 90

Soil Poisson’s ratio, ns 0.4

Effective shear strength, c0 0.2

Effective friction angle, f0 40

Coefficient of earth pressure at rest, K 0o 0.7

Soil Unit weight, g (kN/m3) 22

Initial elastic modulus of concrete (GPa) 37

Poisson’s ratio of concrete, nc 0.2

Joints rotational stiffness, ky (kN m/rad) 4500

Compressive strength of concrete, f 0c (MPa) 35

Ratio of flexural steel, rs 0.01

(c¼f0). As noted above, Table 1 summarizes the material proper-ties adopted for soil.

2.2.2. Concrete tunnel lining

The concrete lining was modeled using a nonlinear elastic,strain-softening plastic constitutive model. The failure criterionfor the model is (from Hinchberger [7]):

F ¼ J2 �ZðxpÞ 0:09358f 0:2

cu 0:3f cu�I1

� �1:80:81417�0:0327I1=f cu

�þ1:945J3=J2

3=2��0:345

¼ 0 ð1Þ

where I1 (¼trr), J2 (¼(s:s)/2) and J3 (¼det(s)) are the stressinvariants, Z(xp) is a strain-softening parameter, which is initiallyequal to one (see below) and fcu is the compressive strength ofconcrete. Fig. 2a compares compression and tension meridiansfrom (1) with strength test results and Fig. 2b shows thecorresponding failure envelop for conditions of plane stress(s3¼0). It is noted that (1) is accurate for mean stresses up to10 MPa and the plane stress strength, which governs in thisproblem, is very good.

Fig. 2. Failure criterion for concrete (from [7]). (a) Compression and tension

meridians, (b) failure criterion corresponding to plane stress.

H. El Naggar, S.D. Hinchberger / Soil Dynamics and Earthquake Engineering 43 (2012) 45–57 47

Prior to failure, the model response is assumed to be non-linear elastic using the following equation:

ET ¼ EoKh 1�RFJ2

0:09358f 0:2cu 0:3f cu�I1

� �1:80:81417�0:0327I1=f cu þ 1:945J3=J2

3=2� ��0:345

0B@

1CA

x

ð2Þ

Kh ¼ 1þrsf y

f cu

ð3Þ

where Eo is the initial tangent elastic modulus, Kh accounts for theinfluence of hoop reinforcement or stirrups [8], rs is the ratio ofhoop reinforcement to concrete confined inside the hoop reinfor-cement by volume, fy is the yield strength of the hoop reinforce-ment, and RF and x are 0.995 and 2, respectively. The resultantstress–strain response defined by (2) corresponding to uniaxialcompression is within 5% of that given by:

f c ¼ Khf cu 2ec

eo

� ��

ec

eo

� �2" #

, ð4Þ

which is commonly used for concrete (see [8]). In (4), eo is theaxial strain corresponding to fcu and fc is the compressive stresscorresponding to ec. Furthermore, if x¼1 is adopted in theconstitutive model, then (2) is essentially a hyperbolic elasticmodel [9] modified for use with (1). It is noted that Poisson’s ratiois assumed to be constant (n¼0.2) in the model even though nvaries with stress level.

After reaching failure governed by (1), the strain-softeningresponse of the concrete model has been calibrated to replicateconventional post-peak stress–strain relationships for concrete.First, for compression, the concrete is assumed to strain-softenand the softening parameter in (1) is:

Z xp� �¼ 1�Zxp

ð5Þ

where xp is the generalized plastic shear strain,

xp¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 ðe

p11�e

p22Þ

2þðep

22�ep33Þ

2þðep

33�ep11Þ

2þ2ep2

12þ2ep223þ2ep2

31

h irð6Þ

and

Z ¼0:5

ðð3þ0:29f cuÞ=ð145f cu�100ÞÞþ0:75rs

ffiffiffiffiffiffiffiffiffiffiffih0=Sh

q�0:002Kh

ð7Þ

-10

0

10

20

30

40

50

-0.004

ε

f (MPa)

ρs = 1%

Z

Extrados

Intrados

0.0060.0040.0020-0.002

Fig. 3. Stress–strain response—concrete model. (a) Elements with flexura

In (7), h0 is the width of the concrete core measured to theoutside of the transverse reinforcement, Sh is the centre-to-centre

spacing of the transverse reinforcement and all of the otherparameters have been previously defined. Eqs. (5) and (7) areadapted from [8] and are commonly used in North Americanconcrete design codes. Associated plastic flow has been assumed.

In tension, the concrete response is assumed to be linearelastic (ft¼Ece) until failure occurs as governed by (1). Afterfailure, the strain-softening is governed by two equations. First,in the presence of longitudinal rebar but without flexural steel,the tension stiffening model has been adapted from [10] viz:

Z xð Þ ¼1

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffi500xp

p ð8Þ

where xp is defined in (6). In the presence of flexural reinforce-ment, a smeared reinforcement model has been used and thestrain-softening parameter is thus:

Z xð Þ ¼1

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffi500xp

p þrsEseps ð9Þ

where eps is the normal plastic concrete strain parallel to the

flexural steel, rs is the ratio of flexural steel area to the gross areaof the element. Table 1 lists the material parameters assumed forthe concrete tunnel lining.

Fig. 3 shows the stress–strain response (compression andtension) of the concrete model used in this study. Such behaviouris typical of concrete (see [8], and [11]). In tension, the effect ofthe flexural steel has been smeared [10] and the flexural steel hasbeen neglected in compression. In general, the model is consid-ered to reasonably predict concrete behaviour up to failure and isadequate for some post-failure straining in both compression andtension. The insert in Fig. 3 illustrates where each of the tensionmodels was implemented in the FE model.

2.3. Finite element mesh

Fig. 4 shows the FE mesh used to study the problem illustratedin Fig. 1 and described above. The FE mesh comprised 61326-noded linear strain triangular elements. The following material

-10

0

10

20

30

40

50

ε

f (MPa)

No FlexuralReinforcement

Z

(a)(b)(a)

-0.004 0.0060.0040.0020-0.002

l steel, (b) elements confined by stirrups and without flexural steel.

t

tt

tt

t

t

t

Fig. 4. FE mesh (Note: t indicates the location of liner joints): (a) Entire mesh with inner soil, (b) Local details near the liner and (c) Excavated mesh.

Table 2Joint material properties.

Parameter Value

Joints rotational stiffness, ky (kN m/rad) 4500

Thickness of the joint zone, t (m) 0.02

Equivalent elastic modulus, Eequivalent (MPa) 320


zones were modelled: (1) soil, (2) reinforced concrete lining, and(3) joints as discussed below. As illustrated in Fig. 4, the meshdensity is very fine near the tunnel lining and is graded to becomecoarser with distance from the tunnel.

The soil continuum was modelled to a depth of 63 m and adistance of 25 m on either side of the tunnel axis. As such, thelateral boundaries were situated five times the tunnel diameterfrom the tunnel axis. Rough rigid boundary conditions wereadopted at the bottom and lateral mesh boundaries.

The liner was modelled as a reinforced concrete material withjoints starting at the crown and situated at 451 intervals. The linerjoints were modelled using thin zones of linear elastic elements;see Fig. 4b, at each joint. Thus, in total, 8 joints were consideredand the elastic properties where selected using on the followingequation derived from bending theory (see Appendix 1 forderivation details):

Eequivalent ¼kyt

Ið10Þ

where ky is the rotational stiffness, t is the thickness of the jointzone (Fig. 4b) and I is the moment of inertia of the liner section(I¼bd3/12). Table 2 summarizes the joint properties which give ajoint rotational stiffness of ky¼4500 kN m/rad/m and joint toliner stiffness ratio (Z¼ky/EI) of 0.45. There are no interfaceelements between the liner and soil and thus there is no slipbetween the liner and soil deposit.

2.4. FE solution sequence

Fig. 5 summarizes the FE solution sequence. First, the initialgeostatic stresses were established assuming increasing verticalstress with depth (sv¼gZ) and horizontal stresses based onsH¼Kosv, where Ko¼0.7. The unit weight of the liner wasassumed to be equal to that of the soil (Ko also) for the purposeof setting up the initial stresses and the ground water table isassumed to be very deep (e.g., well below the tunnel).

Next, the tunnel construction was simulated by incrementallyremoving elements representing the soil inside the tunnel and thetunnel lining in 100 steps. For this approach, the element stiffnessis removed instantly and the body forces caused by the elementunit weight are removed incrementally simulating stress relief inthe soil at the tunnel extrados. Typically there is some stress reliefduring tunnelling. Thus, some partial stress relief was accountedin the FE analysis. This was done by re-activating the elements


representing the tunnel liner after the 20th increment (seeabove); As such, there was soil-liner interaction from increment20 to 100 and the resultant solutions correspond to 20% stressrelief before installation of the lining.

Lastly, after establishing the post-construction stresses in thetunnel lining, displacements were incrementally (400 incre-ments) applied to the lateral mesh boundaries to induce pureshear in the soil resulting in in-plane stresses in the tunnel lining(see [5] and [3]). This will be discussed in more detail below.

2.5. Simulation of concrete delamination at the intrados

Three deferent liner degradation scenarios were consideredcorresponding to concrete delaminations at the liner intrados.Two of the scenarios represented local concrete loss (delamina-tion) at the liner intrados, whereas, the third scenario simulatedthe extreme case of complete intrados degradation. Such degra-dation may occur due to infiltration of chloride-contaminatedwater into the tunnel through one or more of the joints. In all

Geostatic(Gravity)

σh = Ko σv

Continuereducing Po

to zero

Fig. 5. Solution sequence for 2D plane strain simulations: (a) Initial stresses, (b) Remo

Prescribed displacements.

Infiltration

Scenario 1

Cl -1

2

3

4 5

8

7

6

1

5

8

7

6

Scenario

Fig. 6. Degradatio

cases, concrete delaminations were simulated by incrementallyremoving elements and their body forces from within thedegraded zone of the liner. The three scenarios considered in thisstudy and illustrated in Fig. 6 are:

Scenario 1: It comprised local concrete spalling at the crownzone. The elements of the inner layer of the liner betweeny¼7451 from the crown were removed gradually in 100 stepsto simulate concrete spalling to a depth of 50 mm (depth tothe bottom of the flexural steel).Scenario 2: This scenario comprised concrete spalling near they¼7451 locations or the shoulders. The elements of the innerlayer of the liner between y¼22.51 to 67.51 and betweeny¼�22.51 to �67.51 (from the crown) were removed gradu-ally in 100 steps to simulate local concrete spalling.Scenario 3: The last scenario comprised extensive spallingwere all elements of the inner layer (50 mm depth) of the linerwere removed gradually in 100 steps after establishing thepost construction stresses. This scenario was used to study theextreme case of complete intrados degradation.

Po

ve liner & inner soil (20 increments), (c) Reinstatement of liner elements and (d)

Cl - Cl -

2

3

4

1

2

3

45

8

7

6

Scenario 3 2

n scenarios.


2.6. Analytical solution

For all analyses, FE results were compared with the linearelastic jointed liner solution described in [2,3]. For comparisonpurposes, identical elastic properties were considered in both FEcalculations and the analytical solution (see Table 3 for the elasticproperties).

2.7. Pseudo static analysis of in-plane seismic shear stresses

Lastly, two approaches are commonly used for seismic designof tunnel linings. The first approach involves carrying out fullnonlinear dynamic soil-structure interaction analysis by FEmethod or finite difference method. In this approach, inertiaforces are accounted for and the earthquake is simulated byapplying a time history of horizontal displacement at the lowerboundary of the problem to induce vertically propagating shear

-0.20

-0.10

0.00

0.10

0.20

0

Acc

eler

atio

n (g

)

Time (s)

M 6.0 R50

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0

Acc

eler

atio

n (g

)

Time (s)

M 7.0 R70

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0

Acc

eler

atio

n (g

)

Time (s)

M 6.5 R30

5

2010

5

Fig. 7. Input motions and matched spect

Table 3Elastic properties used in the analytical solution (Refs. [2,3]).

Parameter Value

Soil elastic modulus, Es (MPa) 90

Soil Poisson’s ratio, ns 0.4

Initial elastic modulus of concrete (GPa) 37

Poisson’s ratio of concrete, nc 0.2

Joints rotational stiffness, ky (kN m/rad/m) 4500

waves. The second approach utilizes the pseudo-static techniquewhere the inertia forces are ignored and the earthquake loading issimulated as a static far-field shear stress (or strain) applied at theboundaries of the problem (see [3] and [5]). Current designpractice recommends using the pseudo-static approach (e.g.,[12], [13], [14] and [15]). The following is a summary of thepseudo static approach used in this study.

First, a site response analysis (SRA) was done to estimate theaverage far-field shear strain, gc, acting on the soil layer withembedded tunnel liner (e.g., [13]). The SRA was performed using(a) the program NERA [16] and (b) input ground motions obtainedfrom [17] and scaled according to Adams [18] for Type C soil (see[19]). NERA is a one-dimensional nonlinear earthquake siteresponse analysis code based on equivalent linear methodologyassuming stiffness degradation (G/Gmax) based on the magnitudeof the shear strain.

Fig. 7 shows the input ground motions. As illustrated in Fig. 7,three earthquake time histories were considered corresponding tolow, intermediate and high seismic events, which will be referredto as LSZ, MSZ and HSZ for the remainder of the paper. It is alsonoted that, although the time histories were taken from theuniform hazard spectra, UHS, used in Canada [19], the resultsshould be applicable elsewhere in the world.

Next, pseudo-static FE analyses were performed to estimatethe seismic induced in-plane shear stresses in the linings. The gc

estimated from the SRA and its effects on the tunnel lining wassimulated by incrementally applying prescribed boundary dis-placements as illustrated in Fig. 5d to achieve pure shear in the

0.00

0.10

0.20

0.30

0.40

0.50

Spec

tral

Acc

. (g)

Period (s)

M 6.0 R50

UHS Adams(1999)

00.10.20.30.40.50.60.70.8

0

Spec

tral

Acc

. (g)

Period (s)

M 7.0 R70

UHS Adams(1999)

00.20.40.60.8

11.21.4

Spec

tral

Acc

. (g)

Period (s)

M 6.5 R30

UHS Adams(1999)

10 0 105

42

0 4210

ra for (a) LSZ, (b) MSZ, and (c) HSZ.


soil layer. Such a displacement field induces racking of the liningembedded in the soil and approximately simulates the seismicinduced liner loads.

Finally, the gc estimated from the SRA was also converted to afar field shear stress viz.

tf f ¼ gcG ð11Þ

where gc is the shear strain deduced from the SRA, and G is theshear modulus of the soil. Then, the analytical solution in [3] wasused to estimate the seismic induced in-plane stresses caused byLSZ, MSZ and HSZ.

3. Results

This section summarizes the results of analyses for static andseismic loadings, respectively, and for scenarios involving intactand degraded tunnel liners. The static load results are presentedfirst followed by the evaluation of seismic effects from pseudo-static analysis.

3.1. Static loading

3.1.1. Intact liner

Figs. 8 and 9 compare numerical and analytical radial stresses(compression positive) at the liner extrados and intrados, respec-tively, after tunnel construction. Both of these figures show thatthere is good agreement between the stresses calculated by FEmethod and analytical solution. The good agreement is due to the

0

2

4

6

8

10

12

0θ°

Stre

ss (M

Pa)

AFENAClosed form sol.

18016014012010080604020

Fig. 8. Distribution of normal stresses at the liner extrados.

0

2

4

6

8

10

12

0θ°

Stre

ss (M

Pa)

AFENAClosed form sol.

18016014012010080604020

Fig. 9. Distribution of normal stresses at the liner intrados.

absence of plasticity in the FE calculations. Based on Figs. 8 and 9,it can be concluded that the analytical solution is adequate for thestatic case. In the following sections, the analytical solution iscompared with the results of nonlinear FE analysis for degradedlinings to explore its range of application.

3.1.2. Degraded liners

�

Stre

ss (M

Pa)

Stre

ss (M

Pa)

Scenario 1—Local delamination near the crownFigs. 10 and 11 compare calculated (analytical and FE) distribu-tions of radial stress at the extrados and intrados of the liner,respectively, corresponding to degradation Scenario 1 and staticloading. The results obtained using the analytical solution areshown for two extreme cases: (a) The lower dashed linecorresponds to a fully intact liner and (b) the upper dashed linecorresponds to a liner with the entire 50 mm of the intradosdelaminated (Scenario 3—Fig. 6).From Figs. 10 and 11, it can be seen that the analytical solutionprovides approximate upper and lower bounds for the stressescorresponding to degradation Scenario 1. In the degraded zone,the stress level obtained from the analytical solution assumingthe entire intrados is degraded matches that of Scenario 1 for theF.E. analysis. At a short distance from the degraded zone, resultsof the analytical solution for the intact liner match that ofScenario 1 in the undegraded zone. The fluctuation in stress atthe transition from intact to degraded zones (e.g., see abouty¼401) is due to the rotation of principle stresses at theselocations. For example, Fig. 12 shows the principle stresses nearthe transition from degraded to intact lining. On examination ofFig. 12, it can be seen that there is an inactive wedge of concretewith zero stress adjacent to the transition as expected. This zoneof zero stress is reflected in the FE results at 451 in Fig. 11. For

0

3

6

9

12

15

0θ°

F.E. Results(Scenario 1)

F.E. Results(Intact)

Closed form sol.(Full degradation)

Closed form sol.(Intact)

Scenario 1

18016014012010080604020

Fig. 10. Stresses at the liner extrados degradation Scenario 1.

0

4

8

12

16

20

θ°





Scenario 1

0 18016014012010080604020

Fig. 11. Stresses at the liner intrados degradation Scenario 1.

0

3

6

9

12

15

0θ°

Stre

ss (M

Pa)





Scenario 2

18016014012010080604020


0

4

8

12

16

20

Stre

ss (M

Pa)





Scenario 2

0θ°

18016014012010080604020


0

3

6

9

12

15

0θ°

Stre

ss (M

Pa)





Scenario 3

18016014012010080604020


0

4

8

12

16

20

Stre

ss (M

Pa)





Scenario 3

0θ°

18016014012010080604020


Fig. 12. Rotation of major principle stresses at the transition zone.


Scenario 1, the maximum calculated stress is 12 MPa (compres-sion) and there are no tensile stresses.
� Scenario 2—Local spalling at the tunnel shoulders
Figs. 13 and 14 compare the analytical and FE stress distribu-tions at the extrados and intrados of the liner due to staticloading corresponding to degradation Scenario 2. Scenario2 corresponds to local delaminations at the tunnel shoulders.Again as seen in the previous figures, the analytical solutionprovides upper and lower bounds for the stress states in thedegraded lining (Scenario 2). Similar to that observed forScenario 1, there are sudden variations in the FE calculatedstress at transitions from intact to degraded linings (thicknesschange). For this scenario, as one moves further from thedegraded zone (nearly one segment away) the stresses remainunchanged from that of the intact liner after construction. For

Scenario 2, the maximum stress is 17.9 MPa (compression) andthere are no tensile stresses.
� Scenario 3—Complete degradation of the intrados
This scenario represents the extreme case of intrados degrada-tion, where the entire inner layer of the lining is degraded.Figs. 15 and 16 show the stress distribution at the extradosand the intrados of the liner under static loading conditions.For this case; the maximum stress is 14.3 MPa (compression)and there are no tensile stresses in the lining. It can beobserved from the results that the value of maximum stressfor this degradation scenario is lower than the other twodegradation scenarios; even though the extent of degradationis greater. This occurs because there are no stress concentra-tions near transitions in the thickness of intact lining. Againthe analytical solution shows excellent ability to predict thestresses within the lining compared to the F.E. calculation. Thisis expected because both the FE and analytical solutionmethods are within the linear elastic range.

3.2. Evaluation of combined static and seismic loading

In this section the results of static plus seismic load cases arepresented for the scenarios considered in Section 4.1. As notedabove, low, medium and high intensity seismic events wereconsidered, which are hereafter referred to as LSZ, MSZ andHSZ. Fig. 17 shows the distribution of peak shear strain versusdepth for the three seismic intensities. From Fig. 18, it can be seenthat the peak shear strain at the tunnel level is 0.12%, 0.33% and1.25% for LSZ, MSZ and HSZ, respectively. The shear strain valueswere calculated using NERA and then used in the pseudo-staticanalyses presented in the following sections.

0

10

20

30

40

50

60

700.00

Peak shear strain (%)

Dep

th (m

)

Rock

0

10

20

30

40

50

60

700.00


Dep

th (m

)

Rock

0

10

20

30

40

50

60

700.00


Dep

th (m

)

Rock

LSZ HSZMSZ

0.200.10 0.600.400.20 4.002.00

Fig. 17. Results of the ground response analysis.

-5

0

5

10

15

20

25

0

θ°

Stre

ss (M

Pa)

HSZ

Static

MSZLSZ

LSZ

Static

MSZ

Tensile capacity

18016014012010080604020

Fig. 18. Stresses at the liner extrados for LSZ, MSZ and HSZ compared to the

static case.

θ°-5

0

5

10

15

20

25

Stre

ss (M

Pa)

HSZStatic MSZ

LSZ

LSZ

Static

MSZ

Tensile capacity

0 18016014012010080604020

Fig. 19. Stresses at the liner intrados for LSZ, MSZ and HSZ compared to the

static case.

0

3

6

9

12

0θ°

Stre

ss (M

Pa)

FEA LSZ Closed form solution LSZ

Intrados

Extrados

18016014012010080604020

Fig. 20. Comparison of FE and analytical radial stresses in the liner for LSZ.


3.2.1. The intact liner

Figs. 18 and 19 show the stress distribution at the extradosand the intrados of the intact liner for the three seismic eventsconsidered. In addition, the static loading is also presented tohighlight the impact of each seismic event. First, referring toFig. 18 corresponding to the liner extrados, the radial stress levelfrom the crown to the springline decreases as the seismicityincreases until in the worst case (HSZ) the tensile capacity of theconcrete is reached (Note: compression positive). On the otherhand, from the springline to the invert, the radial stresses increaseas the seismicity increases. Referring to Fig. 19, the reverse trend

is observed at the intrados. In general, the seismicity causesincreased bending in the jointed lining increasing the bendingstresses at the extrados and intrados. Although the lining con-sidered would be satisfactory for low and medium intensityearthquakes, the FE analyses indicate that it would developtensile cracking locally between 1201 and 1701 from the crown(counter clockwise) if exposed to high seismic intensity.

3.2.2. Influence of soil and concrete plasticity

Fig. 20 compares stresses calculated by both the FE methodand analytical solution at the extrados and intrados of the liner,respectively, for the LSZ. Figs. 21 and 22 show the same plots ofradial lining stress for MSZ and HSZ events. First, it can be seenfrom Figs. 20–22 that there is good agreement between theanalytical solution and the FE results for LSZ. However, for theMSZ and HSZ seismic cases, there is divergence in the order of 19%for MSZ and 28% for HSZ. The difference can be attributed toplasticity which develops in both the liner and soil, which is notaccounted for in the elastic analytical solution. Fig. 23 shows thelocation of plastic zones for MSZ and HSZ; there were no plasticzones for the LSZ case. As expected, the difference betweenanalytical and numerical solutions increases as the intensity ofthe seismic event increases and the extent of plasticity in the soiland lining increases. However, in spite of the inaccuracy intro-duced for MSZ and HSZ, the analytical solution is able to predictthe potential for cracking as it shows the tensile stresses locallyexceed the tensile capacity of the concrete (see Fig. 22).


3.2.3. The degraded liner during a seismic event

Figs. 24 and 25 present the radial stresses (compression positive)at the liner extrados and intrados, respectively, for LSZ, MSZ and HSZseismic events. Comparing Figs. 24 and 25 to Figs. 18 and 19,respectively, shows that the response of degraded linings duringseismic events of increasing intensity is similar to that seen for theintact linings. The observations from these analyses are: (a) For HSZ,there is an absence of tensile cracking of the extrados between 501and 801 in Fig. 24 compared with the intact case in Fig. 22. This is dueto the reduced flexural rigidity of the lining over the degraded zone,which in turn reduces the moments mobilized in the lining. (b) Theextent of tensile cracking at the intrados is identical for the degradedcase (Fig. 25) and intact case (Fig. 19) since the cracked zone is well

-15

-10

-5

0

5

10

15

20

25

30

θ°

Stre

ss (M

Pa)

FEA HSZ Closed form solution HSZ

Plasticity in the liner

Plasticity

Intrados

Extrados

0 18016014012010080604020

Fig. 22. Comparison of the stresses in the liner’s extrados for high seismicity.

Fig. 23. Location of plastic z

0

3

6

9

12

15

Stre

ss (M

Pa)

FEA MSZ Closed form solution MSZ

Intrados

Extrados

0θ°

18016014012010080604020

Fig. 21. Comparison of the stresses in the liner for medium seismicity.

away from the degraded zone and is isolated by joints. Similar trendsoccur for degradation Scenarios 2 and 3 as discussed below.

3.2.4. Influence of soil and concrete plasticity

In this section, only the results for Scenario 2 in MSZ arepresented due to the repeating trends in the analyses. Figs. 26and 27 compare radial stresses calculated by FE method andanalytical solution at the extrados and intrados of the liner,respectively, for the MSZ seismic event. It can be seen from thesefigures that the analytical solution is not able to predict the upperand lower bounds of the stresses in the degraded liner as observedearlier for the static cases. There is a 35% difference between the FE

ones for MSZ and HSZ.

-5

0

5

10

15

20

25

Stre

ss (M

Pa)

LSZStatic

MSZ

HSZ

Static

MSZLSZ

Absence of tensile cracking

Scenario 1

0

θ°

18016014012010080604020

Fig. 24. Stresses at the liner extrados in low, moderate and high seismic zones

(compared with the static case).

-10

-5

0

5

10

15

20

25

30

Stre

ss (M

Pa) LSZ

Static

MSZ

HSZ

Static

MSZ

LSZ

Tensile cracking

Scenario 1

θ°

0 18016014012010080604020

Fig. 25. Stresses at the liner intrados in low, moderate and high seismic zones

(compared with the static case).

Fig. 28. Stresses at the liner intrados and extrados under static conditions (both

the soil and the lining are elastic).


calculations and the analytical solutions at the location of degrada-tion and transitions from intact to degraded liner. This difference isattributed to plasticity that occurred in both the soil and the liner,which is not accounted for in the elastic solution. Figs. 26 and 27suggest that the analytical solution does not provide good esti-mates of the stresses in the liner for cases were the liner conditionhas degraded due to concrete delaminations at the intrados. Forsuch cases, a finite element analysis appears to be required.

3.3. Sensitivity analysis

To validate the analytical solution a finite element model withthe same elastic assumptions as that of the analytical solution wasdeveloped. The model adopted linear elastic material propertiesfor both the soil and the lining. Figs. 28 and 29 show comparisonsbetween the results of FE model and the predictions of the closedform solution for the radial stresses at the liner intrados andextrados under static and seismic loading conditions, respectively.It can be seen from Fig. 28 that under static loading conditions theresults agrees reasonably. For example, the difference between thepredictions of the FE analysis and the closed form solution is lessthan 10% at the springline (i.e., maximum less than 1 MPa in thiscase). For the seismic loading case the maximum differencebetween the two solutions at the extrados is only 1.22 MPa atthe location y¼67.51 and less than 1.73 MPa (about 15%) at thelocation y¼157.51 as shown in Fig. 29. These differences aremainly due to the assumptions of the analytical solution thataccounts for the joints by applying an overall reduction to thestiffness of an equivalent un-jointed tunnel lining [2].

This section presents the results of the sensitivity analysis thatwas conducted to determine a practical range of applicability for

0

4

8

12

16

20

Stre

ss (M

Pa) F.E. Results

(Scenario 2)



Scenario 2

0θ°

18016014012010080604020

Fig. 26. Stresses at the liner extrados degradation Scenario 3 and MSZ.

0

4

8

12

16

20

24

Stre

ss (M

Pa)




Scenario 2

0θ°

18016014012010080604020

Fig. 27. Stresses at the liner intrados degradation Scenario 3 and MSZ.

Fig. 29. Stresses at the liner intrados and extrados under seismic conditions (both

the soil and the lining are elastic).

the proposed analysis procedure. In this analysis the elasticmodulus of the soil was varied from 90 MPa to 180 MPa. Also,angles of internal friction of 341, 371, 401; typical for medium todense sand; were considered. Medium seismic intensity event(i.e., in MSZ) was assumed. The considered range of soil propertiesrepresent a practical rang for soils that remain predominatelyelastic after tunnel excavation as per the assumptions of theanalytical solution [2]. Tables 4–7 summarize the results of theconsidered cases. Under the static loading conditions the tablespresent the radial stresses at the crown, springline, and invertlocations which are the key locations for this loading conditions.Under the seismic loading condition the key locations are at thecrown, y¼67.51, y¼157.51, and the invert as it can be seen fromFig. 29. As it can be deduced from the tables the approximatepredictions of the closed form solution are close to that of themore sophisticated FE models. And consequently, the use of thisapproximate analytical solution does not introduce dramaticdifferences in the predicted stresses.

4. Summary and conclusions

The FE program AFENA [4] was used to perform static andpseudo static seismic response analyses of tunnels with intact anddegraded concrete liners. An advanced concrete model was usedto model the material behaviour of a segmental concrete tunnellining under both compression and tension loading. The effect oflocal concrete degradation on the stresses developed in degradedtunnel linings was investigated in addition to the effect of threeearthquake events with different intensities on the stresses inintact and degraded tunnel linings. Lastly, the accuracy of theresults obtained from analytical solutions in [2,3] were evaluated

Table 4Effect of the elastic modulus on the predicted stresses at key locations under static conditions.

Static case Crown Springline Invert

Es ðMPaÞ FE CFS Diff. FE CFS Diff. FE CFS Diff.

Intrados f0 ¼401 90 3.76 3.40 �0.36 10.59 11.17 0.58 3.82 3.40 �0.42

c0 ¼0.2 kPa 135 4.83 4.20 �0.63 9.04 10.18 1.14 4.88 4.2 �0.68

180 5.08 4.58 �0.50 8.61 9.63 1.02 5.24 4.59 �0.65

Extrados f0 ¼401 90 8.69 8.50 �0.19 5.37 5.23 �0.14 9.47 8.49 �0.98

c0 ¼0.2 kPa 135 6.95 7.53 0.58 6.52 6.02 �0.50 7.63 7.54 �0.09

180 6.56 7.00 0.44 6.77 6.37 �0.40 7.20 7.00 �0.20

Table 5Effect of the elastic modulus on the predicted stresses at key locations under seismic conditions.

Seismic case (MSZ) Crown At y¼67.51 At y¼157.51 Invert

Es ðMPaÞ FE CFS Diff. FE CFS Diff. FE CFS Diff. FE CFS Diff.

Intrados f0 ¼401 90 3.92 3.40 �0.52 13.67 13.91 0.24 0.50 1.23 0.73 4.02 3.42 �0.60

c0 ¼0.2 kPa 135 4.85 4.21 �0.64 13.55 13.87 0.32 0.7 1.50 0.80 4.91 4.22 �0.69

180 5.16 4.58 �0.58 13.69 14.08 0.39 1.2 1.42 0.22 5.26 4.59 �0.67

Extrados f0 ¼401 90 8.65 8.50 �0.15 2.62 4.08 1.46 11.60 9.41 �2.19 9.35 8.49 �0.86

c0 ¼0.2 kPa 135 6.91 7.53 0.62 3.78 5.08 1.30 10.12 8.21 �1.91 7.59 7.53 �0.06

180 6.53 7.00 0.47 4.56 5.82 1.26 9.05 7.39 �1.66 7.19 7.00 �0.19

Table 6Effect of the angle of internal friction on the predicted stresses at key locations under static conditions.

Static case Crown Springline Invert

c0 ¼0.2 kPa Es ðMPaÞ FE CFS Diff. FE CFS Diff. FE CFS Diff.

Intrados f0 ¼401 90 3.76 3.40 �0.36 10.59 11.17 0.58 3.82 3.40 �0.42

f0 ¼371 3.76 3.40 �0.36 10.59 11.17 0.58 3.82 3.40 �0.42

f0 ¼341 3.76 3.40 �0.36 10.59 11.17 0.58 3.82 3.40 �0.42

c0 ¼0.2 kPa

Extrados f0 ¼401 90 8.69 8.50 �0.19 5.37 5.23 �0.14 9.47 8.49 �0.98

f0 ¼371 8.69 8.50 �0.19 5.37 5.23 �0.14 9.47 8.49 �0.98

f0 ¼341 8.69 8.50 �0.19 5.37 5.23 �0.14 9.47 8.49 �0.98

Table 7Effect of the angle of internal friction on the predicted stresses at key locations under seismic conditions.

Seismic case (MSZ) Crown At y¼67.51 At y¼157.51 Invert

c0 ¼0.2 kPa Es ðMPaÞ FE CFS Diff. FE CFS Diff. FE CFS Diff. FE CFS Diff.

Intrados f0 ¼401 90 3.92 3.40 �0.52 13.67 13.91 0.24 0.50 1.23 0.73 4.02 3.42 �0.60

f0 ¼371 3.92 3.40 �0.52 13.61 13.91 0.30 0.48 1.23 0.75 4.02 3.42 �0.60

f0 ¼341 4.01 3.40 0.61 13.53 13.91 0.38 0.48 1.23 0.75 4.09 3.42 �0.67

Extrados c0 ¼0.2 kPa

f0 ¼401 90 8.65 8.50 �0.15 2.62 4.08 1.46 11.60 9.41 �2.19 9.35 8.49 �0.86

f0 ¼371 8.65 8.50 �0.15 2.36 4.08 1.72 11.58 9.41 �2.17 9.35 8.49 �0.86

f0 ¼341 8.61 8.50 �0.11 1.92 4.08 2.16 11.50 9.41 �2.09 9.32 8.49 �0.83


by comparing them to results obtained from a non-linear elasto-plastic finite element analysis.

The following conclusions can be made from the results anddiscussions presented above:

1)
There is good agreement between FE calculations and theanalytical solution for the static case of intact linings.
2)
The analytical solution for the intact and fully degraded liningsbracket the FE results for liners with local degradation andsubject to static loading, except at the section of discontinuity
where the cross-section of the lining changes suddenly. Thereason for this stress discontinuity is the rotation of theprinciple stresses. In the intact zones, the FE results arecomparable to those calculated by elastic closed-form solutionfor the intact case; whereas in the degraded zones the FEresults were comparable to those calculated by closed-formsolution for a fully degraded liner.

3)
Seismicity can induce significant stresses in tunnel linings. Forthe intact case, tensile cracking is possible in high seismiczones.


4)
There is good agreement between FE calculations and closed-form solutions for low seismicity, but progressively increasingerror (19% and 28%) was observed for medium and highseismic events due to soil and liner plasticity. However, theclosed-form solution is able to identify zones of potentialcracking.
5)
Liner degradation has a detrimental effect on stresses in thelining for the static loading case. However, during a seismicevent, the reduced liner stiffness in the degraded zone leads tolower moments [20] and stresses near the zones of degrada-tion. Thus, degradation does not appear to have a detrimentaleffect in this case due to soil-structure interaction, which iscontrary to that hypothesized in the introduction.
6)
Lastly, it is concluded that a detailed FE analysis should bedone to adequately assess the distribution of stresses indegraded concrete tunnel linings for medium and high seismicevents.
Appendix 1

This section presents the mathematical derivation of equiva-lent elastic properties to simulate the rotational stiffness of thejoints based on the bending theory. For joint b in depth and t inwidth and 1 unit in the third dimension and subjected to appliedmoment M as shown in the Figure below.

_ y¼ tan�1uy -for small strains y¼ u

y ð1Þ

u¼ ebt ð2Þ

From the bending theory,

sb ¼M

Iy and sb ¼ eb Eequivalent‘eb ¼

My

EI

From (2) we get:

‘u¼Myt

EequivalentI

By substituting from (1) in (2)

y¼Mt

EequivalentIð3Þ

Since the rotation, y, is related to the applied moment, M, viathe rotational stiffness, Ky, as:

y¼M

Kyð4Þ

By substituting from (3) in (4) we get,

Eequivalent ¼Kyt

I:

References

[1] Narduzzo L. Toronto’s tunnel solution. Civil Engineering, ASCE 2000;70:A10–6.

[2] El Naggar H, Hinchberger S. An analytical solution for jointed tunnel linings ina homogeneous infinite isotropic elastic medium. Canadian GeotechnicalJournal, Canada 2008;45(11):1572–93.

[3] El Naggar H, Hinchberger S, El Naggar MH. Simplified analysis of seismic in-plane stresses in composite and jointed tunnel linings. Soil Dynamics andEarthquake Engineering 2008;28:1063–77.

[4] Carter JP, Balaam, NP. 2005. AFENA v7.4. Finite Element Software—Usersmanual. Sydney, Australia.

[5] Kontoe S, Zdravkovic L, Potts DM, Menkiti O. Case study on seismic tunnelresponse. Canadian Geotechnical Journal 2000;45:1743–64.

[6] Davis EH. Theories of plasticity and the failure of soil masses. Soil mechanicsselected topics. Sydney: Butterworths; 1968.

[7] Hinchberger SD. Simple single-function failure criterion for concrete. ASCEJournal of Engineering Mechanics 2009;135(7):729–32.

[8] Scott BD, Park R, Priestley MJN. Stress–strain behaviour of concrete confined byoverlapping hoops at low and high strain-rates. ACI Journal 1982;79(1):13–27.

[9] Duncan JM, Chang CY. Nonlinear analysis of stress and strain in soils. Journalof Soil Mechanics and Foundation Division, ASCE 1970;96(SM5):1629–53.

[10] Vecchio, F. 1982. The response of reinforced concrete to in-plane shear andnormal stresses. PhD thesis, Department of Civil Engineering, University ofToronto, Canada.

[11] Youssef M, Moftah M. General stress–strain relationship for concrete at elevatedtemperatures. Journal of Engineering Structures 2007;29(10):2618–34.

[12] Wang JN. Seismic design of tunnels. Monograph, 7. Parsons BrinkerhoffQuade & Douglas; 1993.

[13] Penzien J, Wu CL. Stresses in linings of bored tunnels. Journal of EarthquakeEngineering and Structural Dynamics 1998;27:283–300.

[14] Penzien J. Seismically induced racking of tunnel linings. Journal of Earth-quake Engineering and Structural Dynamics 2000;29:683–91.

[15] Hashash YMA, Hook JJ, Schmidt B, Yao JI. Seismic design and analysis ofunderground structures. Journal of Tunneling and Underground SpaceTechnology 2001;16:247–93.

[16] Bardet JP, Tobita T. A computer program for nonlinear earthquake siteresponse analyses of layered soil deposits (NERA)April. University of South-ern California, Civil Engineering Department; 2001.

[17] Atkinson GM, Beresnev IA. Compatible ground-motion time histories for newnational seismic hazard maps. Canadian Journal of Civil 1998;25:305–18.

[18] Adams J, Weichert D, Halchuk S. 1999. Lowering the probability level—fourthgeneration seismic hazard results for Canada at the 2% in 50 year probabilitylevel. Proceedings of eighth Canadian conference on earthquake engineering,Vancouver, 6 p.

[19] NBCC. The National building code of Canada. Canada: National ResearchCouncil; 2005.

[20] El Naggar H, Hinchberger S, Lo KY. A closed-form solution for tunnel liningsin a homogenous infinite isotropic elastic medium. Canadian GeotechnicalJournal, Canada 2008;45(2):266–87.

Documents

Approximate evaluation of stresses in degraded tunnel linings