soil dynamics and earthquake...

Soil Dynamics and Earthquake Engineering 46 (2013) 41–51

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Soil Dynamics and Earthquake Engineering

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

Effect of interface friction on tunnel liner internal forces due to seismicS- and P-wave propagation

George P. Kouretzis n, Scott W. Sloan, John P. Carter

ARC Centre of Excellence for Geotechnical Science and Engineering, Faculty of Engineering and Built Environment, The University of Newcastle, Callaghan, NSW 2308, Australia

a r t i c l e i n f o

Article history:

Received 22 October 2012

Received in revised form

16 December 2012

Accepted 19 December 2012Available online 19 January 2013

Keywords:

Earthquake

Tunnel

Centrifuge test

Interface friction

Unreinforced final lining

61/$ - see front matter Crown Copyright & 2

x.doi.org/10.1016/j.soildyn.2012.12.010

esponding author. Tel.: þ61 2 4921 6449.

ail addresses: [email protected]

[email protected] (G.P. Kouretzis).

a b s t r a c t

The effects on the hoop force and bending moment developed in the lining of a circular tunnel of the

contact properties of the soil-lining interface are investigated numerically for both cases of S- and

P-seismic wave propagation. Development of a robust model for the dynamic simulation of this

problem includes: (i) the implementation of a hysteretic model of the non-linear soil response under

cyclic loads in the finite element code ABAQUS; and (ii) validation of the analyses results against

centrifuge tests from the literature and closed-from elasticity solutions. Accordingly, a parametric study

is conducted to quantify the effect of adopting different values of the friction coefficient of the tunnel

liner interface, while assuming that the relaxation load is transferred only to the temporary support

shell of the tunnel; a hypothesis applicable mainly to tunnels constructed with the NATM method

where an unreinforced concrete final lining is usually installed. Practical findings of this study suggest

that the full-slip assumption should be used in conjunction with closed-form expressions for

preliminary estimates of the tunnel response. On the contrary, for tunnels where the lining is designed

to bear the soil loads, numerical tools should be used for the rational assessment of their seismic

response. In the latter case, more experimental studies are needed to evaluate the friction properties at

the interface, since common expressions correlating the friction coefficient with the friction angle of the

surrounding soil do not appear compatible with the centrifuge test results examined herein.

Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Current state-of-practice in the seismic design of tunnels useseither closed-form linear elastic expressions for preliminaryestimates of the internal forces developing in the lining e.g.,[1–4], or comprehensive numerical analyses of the tunnel lining-soil system [5]. Recent research efforts also include the predic-tions from centrifuge tests [6–9]. These enrich our understandingof tunnel response under seismic wave propagation and alsoserve as a valuable tool for assessing the validity of variousanalysis methodologies and their assumptions.

One of the key issues regarding the simulation of tunnelresponse is the prediction of the contact behavior between thetunnel lining and surrounding medium. Many analytical solutionsfrom linear elasticity assume either zero friction (the full-slipcondition) or perfect bond between the tunnel and its surroundingsoil (the no-slip condition), and hence cannot simulate the interfaceresponse under cyclic loads reliably. Hashash et al. [10], referring tothe solution for shear or S-waves proposed by Wang [1], suggested

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that for most tunnels the interface condition is actually somewherebetween full-slip and no-slip. This implies that both cases should bemodeled to calculate the most critical internal forces induced byseismic wave propagation, since the solution for the full-slipcondition results in significantly lower hoop forces in the tunnellining. Indeed, the hoop forces calculated under the no-slip condi-tion may be up to 1000 times larger than those corresponding tothe full-slip assumption. Sedarat et al. [5] investigated the effect ofthe contact properties using extensive elastic-quasi static numericalanalyses which simulated S-wave propagation. Their results under-lined the influence of the interface properties on the internal forcesdeveloping in the tunnel liner, and they concluded that, sinceclosed-form solutions fail to adequately predict the tunnelresponse, numerical methods should be used instead.

Interestingly, the state-of-practice in tunnel lining design withnumerical methods, especially for tunnels constructed with theNATM method, is generally based on the rather crude assumptionof zero friction at the soil–tunnel final lining interface. This isjustified by the fact that a smooth waterproofing membrane isusually installed between the tunnel temporary support shell andthe final tunnel lining.

Following a brief overview of closed-form solutions for theestimation of tunnel liner forces due to seismic S- and P-wavepropagation, this paper describes a numerical model for investigating

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Nomenclature

M lining bending momentN lining axial hoop forceEm elasticity modulus of the surrounding mediumEl elasticity modulus of the tunnel linernm Poisson ratio of the surrounding mediumnl Poisson ratio of the tunnel linerr tunnel radiusD tunnel diametert liner thicknessI moment of inertia of the tunnel linergmax maximum free-field shear strainGm shear modulus of the surrounding mediumKm bulk modulus of the surrounding mediumGmax maximum soil shear modulusGt tangent soil shear modulus

rm density of the surrounding mediumVmax,S peak seismic velocity due to shear wave propagationVmax,P peak seismic velocity due to compressional wave

propagationF flexibility ratioC compressibility ratiogm unit weight of the surrounding mediumh tunnel overburden.Ko in-situ stress ratiosmax maximum free-field normal stresstmax maximum free-field shear stressCS shear wave velocityCP compressional wave velocityn interface friction coefficientd friction angle at the soil-liner interfacee sand void ratioj sand friction angle

G.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 46 (2013) 41–5142

the effect of friction at the soil–tunnel interface via non-lineardynamic finite element analysis. In order to simulate hysteretic soilresponse under cyclic loading, a user-defined model subroutine wasdeveloped for the finite element code ABAQUS/Standard [11]. Thenumerical results from this model are compared against the centri-fuge test measurements originally presented by Lanzano et al. [6] andLanzano [9]. This comparison shows the effect of the contact surfacebehaviour and permits the model to be calibrated and refined.

Accordingly, a simpler variant of the abovementioned model isused for comparison with the closed-form elasticity solutions, and aseries of parametric analyses is performed to quantify the effect ofinterface friction on the results. In contrast to the parametric study ofSedarat et al. [5], the current work examines the effect of compressionP-waves as well as S-waves, and the final lining of the tunnel isassumed not to bear any gravity loads resulting from soil massrelaxation. Thus all loads are assumed to be borne by the temporarytunnel shell created during tunnel excavation, and the final tunnellining is installed only after the soil mass has reached stressequilibrium, as is normal practice with the NATM method. Thisassumption applies chiefly to unreinforced concrete final liningsections, which are prone to earthquake effects (compared tostandard reinforced concrete), and is based on the following rationale:

(a)
The load from the surrounding soil mass acts beneficiallyunder seismic loading, as it induces a pre-stressing compres-sive axial force, which reduces the eccentricity of the hoopforce. However, the question of whether these loads from thetemporary excavation will have fully transferred to the finallining at the time of an earthquake event is uncertain.
(b)
Unreinforced concrete tunnel sections are constructed incompetent ground conditions, where long-term loads on thefinal lining from the surrounding medium are generally low, ifnot negligible.
The findings from this study can be used in conjunction withanalytical expressions to make preliminary predictions of theseismic response of tunnels. They can also be used as a guidelinefor the development of numerical models in the final design stages.

2. Closed-form expressions for the calculation of the internalforces in a circular tunnel due to seismic wave propagation

Of the closed-form solutions proposed for the determination oftunnel forces due to shear S-wave propagation, the most widely

used are due to Wang [1]. These provide different expressions forthe tunnel hoop force and the bending moment under full-slip,zero-friction conditions, as well as for the hoop force under no-slip conditions.

Under full-slip conditions at the surrounding medium-tunnelinterface, the following expressions are proposed for the designhoop force Nmax and the bending moment Mmax of the lining:

Nmax ¼ 71

6K1

Em

1þnmð Þrgmax ð1Þ

Mmax ¼ 71

6K1

Em

1þnmð Þr2gmax ð2Þ

where:

K1 ¼12 1�nmð Þ

2Fþ5�6nm

F ¼Em 1�n2

lð Þr3

6ElI 1þnmð Þthe flexibility ratio [12,13],

gmax ¼ Vmax,S=CS the maximum free-field seismic shear straindue to S-wave propagation correlated with the peakseismic velocity Vmax,S.

CS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGm=rm

pthe shear wave propagation velocity.

Under no-slip conditions at the soil-liner interface, Wang [1]proposed the following expression for the maximum hoop force:

Nmax ¼ 7K2Em

2 1þnmð Þrgmax ð3Þ

where:

K2 ¼ 1þ1�2nmð Þ � 1�Cð Þ F�0:5 1�2nmð ÞCþ2

½ 3�2nmð Þþ 1�2nmð ÞC�Fþ½0:5 5�6nmð Þ� 1�2nmð ÞCþ 6�8nmð Þ

and

C ¼Em 1�n2

l

� �r

Elt 1þnmð Þ 1�2nmð Þ

is the compressibility ratio [12,13].Using numerical analysis, the validity of this solution under

specific conditions has recently been verified and discussed byHashash et al. [10]. Note that Wang’s elastic solution, as well as allother similar solutions e.g., by Penzien and Wu [2], assumeinfinite bond strength at the soil–structure interface. As a result,tensile stresses are allowed to develop in the liner, and the

Fig. 1. Simulated centrifuge test T1 configuration (after Lanzano et al. [6]).

G.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 46 (2013) 41–51 43

maximum tensile hoop force in the liner is algebraically equal tothe maximum compressive axial force.

Considering S-wave propagation only is a reasonable assump-tion in free-field conditions, i.e., for tunnels under large over-burden stresses bored through homogeneous ground conditions,the shear waves will ‘‘carry’’ the highest portion of the seismicenergy. Nevertheless, for tunnels bored through areas of irregulartopographic relief or altering geotechnical conditions, the effect ofsecondary compressional P-waves resulting from reflection/refraction of S-waves at the ground surface (or at the interfaceof soil layers with different properties) cannot be ignored [14].

In order to cover this case, Kouretzis and Andrianopoulos [14]proposed a set of closed-form expressions (Table 1), based on thework of Ranken et al. [15,16]. These solutions follow a similarprocedure to the one proposed by Wang [1] for the S-wave case.

In the expressions reported in Table 1, smax is the maximumfree-field normal stress due to P-wave propagation, which is afunction of the amplitude of the strong motion as well as theelastic properties of the surrounding medium:

smax ¼7Vmax,P

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirm Kmþ

4

3Gm

� �s¼ 7Vmax,P

ffiffiffiffiffiffiffiffiffiffiffiffiffirmGm

qU

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1�nmð Þ

1�2nmð Þ

s

ð4Þ

In the above, the symbol (7) accounts for the cyclic nature ofthe loading.

Additionally, Kouretzis and Andrianopoulos [14] applied themethod of Ranken et al. [15] to supplement the set of expressionsproposed by Wang [1] and provide a solution for the maximumtunnel bending moment under no-slip conditions:

Mmax ¼ 7 2�K4�2K6½ � � tmax �r2

2ð5Þ

where tmax is the maximum free-field seismic shear stress:

tmax ¼ 7Vmax, S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirm � Gm

qð6Þ

with the factors K4 and K6 being reported in Table 1.As in the Wang [1] expressions, the linear elastic solution

proposed by Kouretzis and Andrianopoulos [14] was derived byassuming infinite normal bond strength at the soil–tunnel contactinterface. This assumption, studied numerically by Kouretzis andAndrianopoulos [14], leads to fictitious tensile hoop forces in thefinal lining but does not affect the compressive forces or bendingmoments. This effect is also reported for the S-wave case bySedarat et al. [5], who reach similar conclusions regarding thetensile liner stresses.

Table 1Internal forces of a circular liner due to P-wave propagation, under no-slip

Interface conditions Axial force, N

No slip 7 K3þK4½ � � s

Full slip 7 K3þK5½ � � s

where:

K3 ¼2 1�nmð Þ

1þ 1�2nmð ÞC

K4 ¼ 1þ1�2nmð Þ 1�Cð ÞF�0:5 1�2nmð ÞCþ2


K5 ¼4 1�nmð Þ

2Fþ5�6nm

K6 ¼1þ 1�2nmð ÞC½ �F�½0:5 1�2nmð ÞC��2


3. Numerical model and comparison of predictions withexperimental results

The numerical model to be used for the assessment of frictioneffects on the soil–structure interface under dynamic loads is nowpresented, together with a comparison of its predictions againstthe 80 g-centrifuge observations given by Lanzano et al. [6] andLanzano [9]. The latter describe a series of well-documented tests,simulating shallow tunnel response under seismic shear wavepropagation. Bilotta et al. [8] compared the test model T1 resultsunder earthquake excitation EQ2 [6,9] against the predictionsfrom linear dynamic numerical analyses, considering full-slipzero-friction conditions at the soil–structure interface. A sketchof the test model T1 is presented in Fig. 1, while details on thecentrifuge test configuration are reported in Table 2.

The numerical model presented in Fig. 2 was developed in thefinite element code ABAQUS/Standard to match the centrifugetest T1 by Lanzano et al. [6]. Unlike the elastic analyses presentedby Bilotta et al. [8], the non-linear hysteretic response of the soilis explicitly taken into account via the implementation of anappropriate constitutive model, which is based on the Rambergand Osgood [17] model. In the adopted hypoelastic formulation,the tangential shear modulus Gt of the soil gradually decreasesfrom its maximum value Gmax with increasing shear strain (blackline in Fig. 2c), while the opposite trend is followed by the viscousdamping x (grey line in Fig. 2c).

As described by Papadimitriou et al. [18], Papadimitriou andBouckovalas [19], Andrianopoulos et al. [20,21] and Karamitros[22], where the interested reader can find more details on thisformulation, the tangent shear modulus Gt is correlated to themaximum shear modulus Gmax according to

Gt ¼Gmax

Tð7Þ

and full-slip interface conditions.

max Bending Moment, Mmax

max �r

2 7C 1�2nmð Þ

6� FK3þ1�

K4

2�K6

� �� smax �

r2

2

max �r

2 7C 1�2nmð Þ

6� FK3þK5

� �� smax �

r2

2

Table 2Centrifuge test T1—EQ2 characteristics (after Lanzano et al. [6]).

Soil propertiesLeighton Buzzard sand Maximum void ratio Minimum void ratio Relative density Specific gravity Gs

1.014 0.613 75% 2.65

Aluminium alloy liner propertiesUnit weight Young’s modulus Poisson’s ratio Diameter Thickness

27 kN/m3 70 GPa 0.33 75 mm (6 m)a 0.5 mm (0.04 m)a

Excitation propertiesGravity level Frequency Duration Amplitude

80g 40 Hz (0.5 Hz)a 0.4 s(32 s)a 8.0 g (0.10 g)a

a Values in brackets correspond to prototype scale.

Fig. 2. Numerical model for comparison with centrifuge test results (a) finite element mesh, (b) input excitation and (c) non-linear soil response together with resonant

column data used for its calibration [24].


where TZ1.0 is a scalar quantifiying the reduction of the shearmodulus as the current deviatoric stress ratio

rij ¼sij

pð8Þ

diverts from a reference ratio rrefij , where sij is the deviatoric

component of stress

sij ¼ rij�pdij ð9Þ

and p is the isotropic stress component, i.e.,

p¼skk

3¼s11þs22þs33

3ð10Þ

The expression providing the scalar quantity T depends onwhether the soil is subjected to initial shearing (e LR), orsubsequent load reversals (A LR) according to [18,19]

T ¼1þ2 1

a1�1

9Xr92n1

ALR

1þ2 1a1�1

9Xr9n1

=2LR

8><>: ð11Þ

where Xr is a scalar quantifying the variation of the currentdeviatoric stress ratio rij from the reference ratio rref

ij

Xr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2rij�rref

ij

: rij�rref

ij

rð12Þ


and rrefij is the deviatoric stress ratio corresponding to the point of

load reversal. For the initinal shearing rrefij is the initial stress ratio.

The variable n1 in Eq. (11) is defined as

n1 ¼ a1GLR

max

pLR

!g1 ð13Þ

Fig. 3. Typical Ramberg–Osgood stress–strain curve, for the model parameters

used in the numerical analyses.

Fig. 4. Comparison of peak ground accelerations (a) at t

where GLRmax is the maximum shear modulus at the last load

reversal, pLR is the isotropic stress at the last load reversal, and a1,g1 are model parameters.

A shear reversal is defined when the scalar Xe:

Xe ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2eij�eref

ij

: eij�eref

ij

rð14Þ

changes sign between successive steps (i�1) and (i) i.e., when½XðiÞe �X i�1ð Þ

e �o0. In the above Eq. (14), eij is the deviatoric straintensor and eref

ij is the deviatoric strain tensor at the last shearreversal, while for the initial shearing it is the initial deviatoricstrain tensor. According to Papadimitriou and Bouckovalas [19],the elastic strain increments calculated with the above equationsare not fully recoverable, and the term ‘paraelastic’ [23] wouldbetter describe such a formulation.

For the case at hand involving dense Leighton Buzzard sand(Table 2), the maximum shear modulus is taken as Gmax¼45 Mpa,following Lanzano et al. [7] who calculated the average shearmodulus mobilized during each event from top-to-base transferfunctions. The model parameters a1¼0.80 and g1¼0.0002 werederived (Fig. 2c) on the basis of the experimental data from theresonant columns tests of Visone [24,6]. The calibration trial-and-error procedure is based on matching the analytical Ramberg–Osgood G/Gmax–g curve obtained for a certain set of (a1,g1)parameters, to the experimental G/Gmax data from the resonantcolumn tests for gog1, where g1 for sands ranges between0.0065% and 0.025% [18]. A typical stress–strain curve corre-sponding to the adopted parameters is presented in Fig. 3.

The Poisson’s ratio of the soil is taken as vm¼0.333, while itsdensity is equal to rm¼1.55 Mg/m3. No Rayleigh damping isintroduced in the model, with the only source of material dampingbeing energy dissipation from the hysteretic soil response.

The analyses were executed at model scale, with the alumi-num alloy tube modeling the tunnel liner having the properties

he free-field, and (b) along the tunnel vertical axis.

Fig. 5. Liner hoop forces and bending moment time histories at y¼3151, normal-

ized against their value at the end of the static step. Results for three different

friction coefficient assumptions at the liner-sand interface are presented.

Table 3aComparison of numerical against experimental results, in terms of dynamic axial

hoop forces.

Maximum axial hoop force dynamicincrement, Nmax (N/mm)a

Location along the cross-section

h¼1351 h¼2251 h¼3151

Centrifuge 0.0025 0.0050 0.0047

Numerical, n¼0 0.0065 0.0067 0.0075

Numerical, n¼0.1 0.0686 0.0683 0.0442

Numerical, n¼0.5 0.2206 0.2364 0.1520

a Values are presented as N per running mm of liner.

Table 3bComparison of numerical against experimental results, in terms of dynamic

bending moments.

Maximum bending moment dynamicincrement, Mmax (Nmm/mm)

Location along the cross-section

h¼451 h¼1351 h¼2251 h¼3151

Centrifuge 0.030 0.045 0.075 0.085

Numerical, n¼0 0.030 0.046 0.034 0.033

Numerical, n¼0.1 0.029 0.044 0.033 0.029

Numerical, n¼0.5 0.026 0.040 0.032 0.027

*Values are presented as Nmm per running mm of liner.


presented in Table 2. Beam elements were used for its simulation,while in order to account for plane strain conditions the Young’smodulus of the alloy was divided by 1�n2

l

� �. The tunnel embed-

ment depth in the model scale is 75 mm (Fig. 1).The numerical analyses consist of two steps: The first, a static

(swing-up) step, includes the application of a gravitational forceequal to 80g, which is held constant during the subsequentdynamic step, where the seismic excitation is applied. Duringthe first analysis step, a linear elastic model with shear modulusG¼Gmax was employed for the calculation of the gravity stressdistribution, while the bottom boundary of the model was fixed inthe vertical direction, and the lateral boundaries of the modelwere fixed in the horizontal direction.

In the subsequent dynamic analysis step, the input sinusoidalexcitation (shown in Fig. 2b) was applied at the fixed base of themodel, and features a frequency f¼40 Hz and a maximum

amplitude a¼5 g, as well as an initial ramp section of 8 periodsto eliminate the need for baseline correction. The excitationparameters were selected to match as accurately as possible theinput excitation presented by Bilotta et al. [8], since the latter wasnot available in digital format. The centrifuge input excitation iscompared with the applied excitation in Fig. 2b.

To account for deformation of the laminar box, the verticalboundaries of the mesh were connected via rigid links, while thebottom boundary was fixed. The contact between the sand andthe aluminium alloy tube allows for separation, while differentassumptions were adopted for the penalty friction formulationused to model the tangential behavior of the interface: In the firstanalysis, zero friction was considered following Bilotta et al. [8].In the second and third analyses, coefficients of friction n¼0.1and n¼0.5 were considered, which correspond to friction anglesat the soil–tube interface equal to d¼5.71 and d¼26.51, respec-tively. The linear penalty method was used to simulate thenormal surface behavior, together with an efficient iterativeprocedure that minimizes penetration of the surfaces (the Aug-mented Lagrangian method [11]).

Fig. 4 compares the peak ground accelerations from thenumerical analyses with those from the centrifuge test measure-ments, as well as the numerical results presented by Bilotta et al.[8]. The difference in the peak value of the input acceleration atthe bottom is due to the exclusion of the high-frequency compo-nent at the beginning of the experimental measurement of theacceleration. This feature does not affect the results higher in thesample since the high-frequency component is filtered by the soil.

Table 3a and b compare the peak axial (hoop) forces andbending moments predicted from the numerical analyses againstthe centrifuge test. These results are at the polar angles y alongthe tube periphery where strain gages (SG) were installed (seeFig. 1 and Fig. 2a). As in Bilotta et al. [8], the average values of thepeak-to-peak amplitude for axial forces and bending moments areused to compute the seismic increments DN and DM, while thesteady-state part of the seismic motion is considered when theacceleration has reached its peak value (Fig. 5).

The complete time-histories of the hoop force and bendingmoment at the polar angle y¼3151 are presented in Fig. 5 for thethree different interface friction coefficients. Forces and momentsare normalized against their value at the end of the static analysisstep, to obtain their dynamic component Ndynamic and Mdynamic,respectively, as the dynamic liner response is of interest here.

Notice that there is a clear shift in the internal liner forcesdeveloping throughout the duration of excitation, which is asso-ciated with non-recoverable deformation of the liner. This type ofbehaviour is attributed to the fact that as the shear modulus ofthe soil reduces with increasing shear strain, a re-distribution ofstresses takes place causing deformation of the tube, measured asheave of the liner crown (Fig. 5). This is due to the extremelyflexible liner used in the centrifuge tests to obtain accuratedynamic strain measurements [6], which features a diameter-to-thickness ratio D/t¼150. Such a flexible liner cannot contribute

Fig. 7. Finite element mesh to model seismic wave propagation in an infinite

elastic medium.

Table 4Input data of the basic set of analyses (Case 1).

Medium propertiesModulus of elasticity, Em 2 GPa

Poisson ratio, nm 0.2

Liner propertiesModulus of elasticity, El 19 GPa

Radius, r 4.0 m

Poisson ratio, vl 0.2

Thickness, t 0.15 m

Flexibility ratio 3193

Compressibility ratio 3.74

Fig. 6. Axial hoop force time histories at y¼3151, and crown displacements for a

very flexible (D/t¼150) and a stiffer (D/t¼15) liner. Analyses for a friction value

n¼0.1, with displacement results are presented at model scale.


singificant support to the surrounding sand, and the systemessentially behaves like a medium with a loosely supported void.

To further substantiate this aspect, an additional analysis wasperformed, considering a liner with a diameter-to-thickness ratioD/t¼15—a more realistic scenario, which corresponds to a tunnelwith diameter D¼6 m and liner thickness t¼0.4 m at prototypescale. As depicted in Fig. 6, this effect is then eliminated. A shift inaxial force and bending moment was also measured experimen-taly, and it was attributed by Lanzano et al. [6] to soil densifica-tion, as the void ratio of the sand is reduced from e¼0.71 toe¼0.68 after the completion of the test T1. This behaviour couldbe considered as an indication of insufficinent support providedby the liner. The hypoelastic model used in this study cannotcapture soil densification effects, and the shift measured duringcentrifuge tests cannot be addressed. However, such an effect isnot important for the problem examined herein, where we focuson the transient component of the internal force increments.

The best match to the centrifuge test results is achieved whena zero friction condition at the sand-tube interface is assumed, asin the numerical analyses presented by Bilotta et al. [8]. Theseauthors considered an elastic soil model, and incorporated Ray-leigh damping to simulate soil hysteretic behavior, yet theirresults are compatible with the results of the non-linear analysespresented here and the centrifuge test results. As the frictioncoefficient increases, a rapid increase in the dynamic portion ofthe axial forces is observed (Fig. 5, Table 3), while the bendingmoments are much less affected. This conclusion is consistent

with the findings of Sedarat et al. [5] for tunnels with similarflexibility factors, F. It is interesting to mention that even when avery low friction value is introduced at the interface, the axialhoop force values diverge grossly from the centrifuge test results(Fig. 5, Table 3). This divergence occurs for friction angles whichare much lower than tand¼tan(0.7j)—the recommended frictionangle for a cohesionless soil–smooth steel interface according toASCE-ALA [25] guidelines for axial static pipeline movementrelative to surrounding soil.

4. Parametrical investigation of friction effects on internalforces

In order to parametrically assess the effect of friction on thetunnel internal forces, a simple numerical model is developed.

Fig. 8. Comparison of analytical with numerical results from the basic analyses set (Case 1). Negative hoop force values correspond to compressional liner normal stresses.


This model, presented in Fig. 7, simulates shear and compres-sional wave propagation in an elastic space, i.e., it is compatiblewith the assumptions of the analytical solutions by Wang [1] andKouretzis and Andrianopoulos [14]. Infinite elements, functioningas absorbing boundaries, are incorporated at the top and bottomboundaries of the model to avoid wave reflection phenomena,while the lateral boundaries are not restrained. The input sinu-soidal motion, with the characteristics presented in Fig. 6, wasapplied at the bottom of the model as a stress time history. Thedirection of the input stress varies, depending on the wave typemodelled (shear or compressional).

First, a set of four basic analyses was performed in order tocompare the results of the numerical model with the analyticalclosed-form expressions presented in paragraph 2. The propertiesof the model are summarized in Table 4. The four analysessimulate the tunnel response under S- and P-wave propagation:first under full-slip conditions at the soil–structure interface,assuming zero-friction and allowing for separation under tensilecontact stresses, and second under no-slip conditions, where thebeam elements of the liner are tied to the nodes of the surround-ing soil. These two extreme cases correspond to the assumptionsunder which the analytical expressions were derived. For com-parison purposes, the hoop forces and bending moments arenormalized as N/(s � r) and M/(s � r2), respectively [1], with sbeing the maximum seismic normal stress for P-waves. This lattervalue is replaced by the maximum seismic shear stress, t, for thecase of shear S-waves.

The results presented in Fig. 8 show a fair agreement betweenthe analytical and numerical maximum values. Some numericalnoise is introduced in the P-wave bending moment time historyduring the separation phase, but this does not significantly affectthe solution (Fig. 8). Observe that the axial forces due to S-wavepropagation under full-slip conditions are practically zero for this‘flexible’ tunnel (with a flexibility factor of F¼3193) while, whenseparation is possible at the soil–structure interface, no tensileforces develop in the liner. The latter finding is consistent withthe results presented by Sedarat et al. [5] and Kouretzis andAndrianopoulos [14]. It is also compatible with the centrifugeobservations of Lanzano [9], who commented that ‘ythe hoop

forces are generally always positive, except some case(s) of null or

slightly negative value(s).’

To investigate the effect of friction at the soil–tunnel interface,a series of parametric analyses were performed for both theS- and the P-wave scenarios, considering three cases. Case1 corresponds to a ‘flexible’ tunnel (flexibility factor F¼3193,compressibility factor C¼3.74) with the properties presented inTable 4. Cases 2 and 3 are for stiffer tunnels, with F¼319/C¼0.37and F¼31.9/C¼0.037, respectively. These effectively cover thewhole range of flexibility and compressibility factors for typicaltunnels. This variation is achieved by reducing the surroundingsoil Young’s modulus in each analysis to attain the requiredfactors. No gravitational forces are acting on the model, thus theforces on the liner from the soil loads were equal to zero.

For each case five (5) different values of the friction coefficientare considered, ranging from zero up to a very high value of n¼5,while free separation at the soil–tunnel interface under tensilenormal stresses is allowed by default. An additional analysis wasperformed for each case, considering no-slip conditions andperfect bond at the interface. The results from the numericalanalyses are compared with the corresponding analytical solu-tions in Figs. 9 and 10.

As shown in Fig. 9, the effect of friction on the axial forces andbending moments for the S-wave case is trivial when the load ofthe soil mass is negligible. This is due to the fact that compressivenormal stresses at the contact interface are zero. This finding issignificant for the axial liner force, since the difference betweenthe cases where full-bond is assumed is dramatic. Notice thatthere is a slight divergence between the analytical and numericalaxial liner forces from the S-wave Case 3 analyses (Fig. 9). This isdue to the fact that the full-slip axial forces due to S-wavepropagation are very low, practically zero. A similar relativedivergence for cases 1 and 2 is not depicted in Fig. 9, due to thescale of the vertical axis.

On the other hand, due to the different mode of deformationimposed by P-wave loading, where normal stresses are applied atthe soil-lining interface as a result of propagation of the seismicwave train, the transition of the internal forces from the full-slipcase to the no-slip case is rather smooth. Indeed, as the frictioncoefficient increases, the numerical results tend to match the no-slip case results closely. However, the increase in the axial linerforces for realistic values of n (no0.5) is gradual, at least for‘flexible’ tunnels. The opposite effect occurs for the bending

Fig. 9. Results of the parametric analyses for the S-wave case. Absolute maximum values of normalized internal forces are presented.


moments, where the maximum values tend to decrease withincreasing friction at the interface.

5. Concluding remarks and practical implications

The effect of the soil-lining interface properties on the internalforces developed in the lining of a circular tunnel due to seismicloading was investigated numerically, for both (shear) S- and(compressional) P-wave propagation. The finite element results werevalidated against centrifuge test measurements and existing closed-from elastic solutions. The focus of the parametric part of this studywas on circular linings not carrying any loads from the soil mass, anassumption which is valid for unreinforced tunnel final linings wherethe relaxation load due to tunnel excavation is assumed to betransferred only to the temporary support of the tunnel. Practicalfindings of this study can be summarized as follows:

–
Comparison of non-linear dynamic numerical analyses withthe centrifuge test results presented by Lanzano et al. [6]suggests that test friction conditions at the dense sand-liner
interface are close to the zero-friction assumption. If loads aretransferred directly to the final tunnel lining from the sur-rounding soil, introducing even a very low friction value in thenumerical interface simulation results in grossly higher axialforces.

–
When the loads of the surrounding soil mass are assumed tobe borne by the temporary support shell, the effect of frictionangle on the structural response of the lining is negligible forthe S-wave case, since the normal stresses at the interface arezero. This is not the case when compressional P-waves areconsidered in the analysis, e.g., for tunnels bored throughirregular topography on through non-uniform geological for-mations, although for realistic friction values this effect is notsignificant.
–
For unreinforced tunnel final linings, when no soil mass loadsare generally expected to be transferred to the final lining andthe main failure mode in the final lining is the development ofintolerable cracks due to high eccentricity of the hoop force,the closed-form analytical solutions provide reasonable esti-mates of the liner forces under the full-slip condition. How-ever, for the no-slip, full-bond interface condition, the results

Fig. 10. Results of the parametric analyses for the P-wave case. Absolute maximum values of normalized internal forces are presented.


give unrealistically high axial hoop forces. These solutionswould also not yield realistic results when the final lining isdesigned to bear the surrounding soil loads (e.g., TBM tunnels,or even buried pipelines). The selection of the proper interfacefriction angle is an issue that deserves further experimentalinvestigation.

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