Experimental and numerical study on circular tunnels

This article was downloaded by: [Giovanni Lanzano] On: 10 March 2014, At: 08:14 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

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Experimental and numerical study on circular tunnels under seismic loading a

b

b

Giovanni Lanzano , Emilio Bilotta , Gianpiero Russo & Francesco b

Silvestri a

DiBiT, University of Molise, Campobasso, Italy

b

DICEA, University of Napoli Federico II, Naples, Italy Published online: 06 Mar 2014.

To cite this article: Giovanni Lanzano, Emilio Bilotta, Gianpiero Russo & Francesco Silvestri (2014): Experimental and numerical study on circular tunnels under seismic loading, European Journal of Environmental and Civil Engineering, DOI: 10.1080/19648189.2014.893211 To link to this article: http://dx.doi.org/10.1080/19648189.2014.893211

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European Journal of Environmental and Civil Engineering, 2014 http://dx.doi.org/10.1080/19648189.2014.893211

Experimental and numerical study on circular tunnels under seismic loading Giovanni Lanzanoa, Emilio Bilottab*, Gianpiero Russob and Francesco Silvestrib a DiBiT, University of Molise, Campobasso, Italy; bDICEA, University of Napoli Federico II, Naples, Italy

Downloaded by [Giovanni Lanzano] at 08:14 10 March 2014

(Received 9 May 2013; accepted 21 January 2014) This paper compares the experimental results of a set of centrifuge models of tunnels in sand under seismic loadings with the predictions of finite element dynamic analyses and of simplified methods. In order to characterise the soil behaviour, mobilised shear stiffness and damping ratio of the sand model have been back-calculated from the experimental results according to two different procedures. Starting from the accelerometer measurements, one was based on the transfer functions from surface to base and the other one on the average shear stress–strain cycles along the sand layer. A series of viscoelastic 2D dynamic analyses were performed to simulate the model tests by a linear equivalent approach. The equivalent shear stiffness and damping ratio determined from stress–strain cycles were used as input values for the analyses. The shear stress transfer at the ground-lining interface was back-analysed to calibrate the interface elements used in the numerical code, in order to improve the assessment of the transient changes of hoop force. Finally, the numerical results have been compared to analytical solutions, widely adopted in the design, and to the experimental data in terms of transient increments of internal forces in the lining. Such a comparison indicates that the analytical formulations give a good estimation of the seismic increment of bending moment in the lining and a reasonable lower bound for the transient changes of hoop forces, provided that cyclic shear strains are correctly evaluated. Keywords: centrifuge; tunnel; numerical analysis; back-analysis

1. Introduction Shallow circular tunnels in soft ground are largely used for every kind of transportation systems. The seismic response of these structures is generally safer compared to above ground structures; nevertheless, several tunnels suffered strong damage in recent earthquakes (Bäckblom & Munier, 2002; Wang et al., 2001; Yoshida, 2009), which may be associated with the onset of loading conditions incompatible with the lining resistance. The increments of internal forces induced in a tunnel lining during earthquakes can be predicted with several procedures at different levels of complexity (e.g. Bilotta et al., 2007; Hashash, Hook, Schmidt, & Yao, 2001). For preliminary and intermediate design stages, pseudo-static and simplified dynamic analyses are suggested by guidelines (AGI, 2005; ISO 3010:2001, 2001). These methods permit to calculate the transient changes of internal forces in the lining once the seismic increment of shear strain at the tunnel depth is estimated (Penzien & Wu, 1998; Wang, 1993). *Corresponding author. Email: [email protected] © 2014 Taylor & Francis


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On the other hand, experimental and numerical evidences of permanent changes of internal loads in the tunnel lining (e.g. Amorosi & Boldini, 2009; Lanzano, Bilotta, Russo, Silvestri, & Madabhushi, 2012; O’Rourke, Goh, Menkiti, & Mair, 2001) suggest that soil-structure interaction (SSI) should be ideally modelled through full dynamic analyses accounting for soil plastic straining under cyclic loads to capture such effects. However, these approaches require advanced soil characterisation and numerical tools which are not conventionally used in the design practice. The calibration of all the above methods should require validation against experimental data, which are seldom available since measurements of seismic internal forces on real-scale structures during earthquakes are very difficult. Complication arises not only due to the random occurrence of earthquakes, but also because the routine monitoring instrumentation of the existing tunnels has too large time sampling. Hence, it is not able to catch the transient nature of dynamic soil-tunnel kinematic interaction. Due to the substantial lack of well-documented full-scale case histories, any of the proposed design methods is difficult to be validated. To bridge this gap, centrifuge seismic tests on a model tunnel (Lanzano et al., 2012) were carried out at the Schofield Centre of Cambridge University (UK) in the framework of an Italian research project (ReLUIS). The final aim of that work was to calibrate and compare numerical predictions of different complexity on the basis of reliable experimental data (Bilotta & Silvestri, 2012). A primary scope of the study presented in this paper was to interpret the experimental data from the centrifuge tests (Section 2), in order to complement the mechanical characterisation of the sand from the laboratory with the soil properties measured at the model scale (Section 3). The second part of the paper (Section 4) compares the experimental results with the predictions of simplified approaches (i.e. those suggested by Wang, 1993) and SSI analyses. These latter were targeted to capture only the transient behaviour during shaking, poorly affected by other well-known mechanical phenomena, such as sand densification during shaking, which is expected to primarily induce permanent changes of internal forces. A simple homogeneous viscoelastic model, calibrated on the basis of equivalent linear soil parameters back-figured from the centrifuge tests, was therefore adopted. 2. Reference centrifuge tests In this section, a brief summary of the centrifuge tests on tunnel models is given. More explanatory details about the equipment, measuring instruments, materials, test preparation, procedures and results are reported by Lanzano et al. (2012). The models were made by using dry Leighton Buzzard sand (fraction E); the sandy layers were deposited at two different relative densities (Dr = 40 and 75%). The static and dynamic properties of the sand were already measured during a comprehensive laboratory investigation performed on specimens prepared at comparable densities (Visone & Santucci de Magistris, 2009). Nevertheless, a back-calculation of the actual dynamic properties in the centrifuge model has been preferred in this study, as discussed in Section 3. The tunnel lining was physically modelled by an aluminium–copper alloy tube (density, ρ = 2770 kg/m3; Young’s modulus, E = 70 GPa; Poisson’s ratio ν = .33) having an external diameter D = 75 mm and a thickness t = .5 mm (Figure 1). The basic features of the four series of centrifuge tests are summarised in Table 1. The tube was located at two different depths; the layouts of the tests are shown in Figure 2. Further details can be found in Lanzano et al. (2012).

European Journal of Environmental and Civil Engineering


Figure 1. Table 1. Model T1 T2 T3 T4

Figure 2.

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Aluminium alloy tube and drawing of strain gauges layout. Tested models. Tunnel cover, C [mm] 75 75 150 150

Relative density, Dr [%] ~75 ~40 ~75 ~40

Instrumentation layout of models (Lanzano et al., 2012).

The tests were performed after accelerating the models at 80 g. According to the centrifuge scale factors, each model tunnel reproduces the behaviour of a prototype concrete tunnel with a diameter of 6 m and a thickness of .06 m embedded in a soil layer about 24 m thick. Accelerometers were installed along three vertical arrays (i.e. “tunnel”, “free-field” and “reference” in Figure 2) to measure the horizontal and the vertical components of

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the acceleration at the base of the model, on the container and in the ground. One accelerometer of the reference array was located on the base plate, in order to record the applied motion at the rigid basement. Strain gauges along the tube were arranged in order to measure bending moments and hoop stresses at four locations along two transverse sections (see Figure 1). The input motions were pseudo-harmonic and they had similar features in the four models. Table 2 shows the values of amplitude, nominal frequency and duration of each signal both at model and prototype scale (bracketed figures). As an example, the time histories of the acceleration applied to the model T3 and the corresponding Fourier spectra are shown in Figure 3. 3. Evaluation of soil properties The fabric of sand in the centrifuge models might have been different from the laboratory specimens, since the procedure of preparation of the centrifuge models could not reproduce exactly that adopted for the laboratory element tests. On the other hand, the laboratory stress paths could not reproduce all the possible loading paths experienced by the model during shaking. Therefore, the dynamic properties of the sand to be used in the numerical analyses were back-figured from the results of the centrifuge tests as shown in the following. The reference shear strain, γ, the mobilised shear stiffness, G, and the damping ratio, D, of the soil during each shaking were estimated following two different approaches starting from the acceleration time histories. The two procedures are hereafter described and the results are discussed and compared. 3.1. Coherence and transfer functions The “similarity” between two time histories of horizontal accelerations measured in different points ( j, k) of the same array can be represented by the coherence function, Cohjk(ω): Sjk ðxÞ Cohjk ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sjj ðxÞSkk ðxÞ

(1)

where Sjk is the Fourier transform of the cross-covariance and Sjj and Skk are the autospectra of the Fourier transform of the auto-covariance. For each frequency, the coherence has a value included between 0 and 1: the higher the coherence, the higher the correlation between the two signals. The transfer function, instead, is a representation of the ground motion variability due to the wave propagation inside a medium. This function is defined as the ratio

Table 2.

Model earthquakes.

Input signal EQ1 EQ2 EQ3 EQ4

Gravity level [g] 80 80 80 80

Frequency [Hz] 30 40 50 60

(.375) (.5) (.625) (.75)

Duration [s] .4 .4 .4 .4

(32) (32) (32) (32)

Amplitude [g] 4.0 8.0 9.6 12.0

(.05) (.10) (.12) (.15)



5

Figure 3. Input signal and smoothed Fourier spectra at model scale recorded on the base-plate (T3; 80 g).

between the Fourier spectra, X(ω) and Y(ω), of two different signals. The modulus of the transfer function represents the amplification function, A(ω), which is defined as: AðxÞ ¼ jHðxÞj ¼

X ðxÞ Y ðxÞ

(2)

The combined interpretation of the transfer and the coherency functions was used in this case to recognise which frequencies were amplified in the waves propagating through a soil layer between two accelerometers. Figure 4 shows all the transfer and coherence functions which were calculated along the vertical reference arrays located in the four models, between the base, Y(ω), and the top accelerometers, X(ω). In the range of frequencies where the coherence was higher than .9, it was assumed that the intensity of the recorded signal was about one order of magnitude higher than any possible noise. Hence, experimental values in this range were best fitted with the theoretical amplification function of a viscoelastic layer, i.e: 1 AðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 F þ ðDFÞ2

(3)

where D is the soil damping ratio and F is the frequency ratio, defined as F ¼ x H=VS , being H the thickness of the sand layer and VS the shear wave velocity. In most cases, the two signals, X(ω) and Y(ω), met the requirement on the minimum coherence value through more than 50% of the frequency range, and the coefficient of determination of the best fitting, R2, was higher than .9. Similar curves were fitted also on the basis of experimental data along free-field and tunnel verticals (Bilotta, Lanzano, Russo, Silvestri, & Madabhushi, 2009; Lanzano, Bilotta, Russo, Silvestri, & Madabhushi, 2009).


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Figure 4.

Coherence and transfer functions relative to reference vertical.

In this way, the fundamental frequency of the layer: f1 ¼ x1 =2p ¼ VS =4H

(4)

and the mobilised damping ratio, D, was obtained from the peaks of the back-figured amplification functions. The mobilised transversal modulus of elasticity, G, was evaluated from f1 using the expression: GTF ¼ VS2 q ¼ ð4Hf1 Þ2 q

(5)

being ρ the soil mass density. An average value of the shear strain mobilised during each earthquake can be calculated according to Newmark (1967): vmax cTF ¼ (6) VS In Equations (5) and (6), VS is the equivalent shear wave velocity obtained from the transfer function through Equation (4), and vmax is the average peak particle velocity in the soil layer. The latter was back-figured from the integration of the acceleration time history recorded by the central accelerometer of the reference array (Conti & Viggiani, 2012).


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The mobilised values of the dynamic shear modulus, the damping ratio and the maximum shear strain are reported in Table 3. The average values of the dynamic shear modulus range between 28 and 10 MPa, from EQ1 to EQ4, confirming that the mobilised stiffness of the soil was relatively low. Some values in the table are reported in brackets, since in these cases, either the best fitting was only possible on a partial span of the data points (lower than 25%) or the determination coefficient R2 was lower than .85. These values are considered as not reliable. The average back-figured values of the damping ratio are rather high (14–35%); however, they could well be overestimated due to low coherence of the signals in the frequency range around the resonance peak. The average values of the maximum shear strain range between .07 and .23%, without any clear difference between dense and loose sand. 3.2. Shear stress–strain loops Zeghal and Elgamal (1994) suggested a procedure to evaluate the shear modulus, the damping ratio and the shear strain from the data recorded by arrays of accelerometers. Brennan, Thusyanthan, and Madabhushi (2005) adapted the method to back-figure the same parameters in dynamic centrifuge tests; the procedure has been more recently validated also by Li, Escoffier, and Kotronis (2013). Following the same approach, the time histories of the displacements, u(t), were obtained from double integration of the accelerograms, a(t). In order to avoid errors like an unrealistic drift during the shaking, the signal was band-pass filtered twice (between 15 and 250 Hz), before each integration. The filter also eliminated the phase distortion due to the integration procedure. The time histories of shear strain could be calculated by differentiating the displacements, u(t), with respect to depth, z, using a second-order approximation between two or more instruments positioned along the same vertical array: Table 3. Mobilised shear stiffness, damping ratio and maximum shear strain from transfer functions. Model GTF [MPa] T1 T2 T3 T4 Avg DTF [%] T1 T2 T3 T4 Avg γTF [%] T1 T2 T3 T4 Avg

EQ1

EQ2

EQ3

EQ4

31 27 32 23 28

37 45 29 23 33

39 (21) (25) (24) 27

34 (10) (10) (11) 16

14 8 17 17 14

20 14 19 14 17

28 (28) (16) (14) 22

21 (40) (39) (40) 35

.07 .07 .07 .08 .07

.08 .07 .08 .09 .08

.11 (.15) (.13) (.14) .13

.15 (.26) (.27) (.25) .23

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cðzi Þ ¼

h i ðzi zi1 Þ ðziþ1 zi Þ ðuiþ1 ui Þ ðz þ ðui ui1 Þ ðz iþ1 zi Þ i zi1 Þ ðziþ1 zi1 Þ

(7)

in which the index i was relative to the position of the central instrument and i – 1 and i + 1 to the upper and lower accelerometer, respectively. The shear stress τ was computed through the dynamic equilibrium of a soil column, by integrating the acceleration time histories with respect to depth: Z z sðzÞ ¼ qadz (8)


0

As the signals were not of single frequency, the shape of the cycles was not regular (cf. also Brennan et al., 2005; Conti & Viggiani, 2012; Li et al., 2013). An equivalent shear stiffness, Gcyc, for each cycle was generally estimated as: smax smin Gcyc ¼ (9) cmax cmin The equivalent damping ratio, Dcyc, was calculated by integrating the area of each stress–strain loop as follows: H 2 sdc (10) Dcyc ¼ p Ds Dc where Δτ and Δγ are the peak-to-peak amplitudes of the shear stress and strain in each loop. Table 4 shows the values of Gcyc and Dcyc averaged over the stress–strain cycles. The single amplitudes of cyclic shear strain were rather high, varying between .02 and .3%, which correspond to a strain range in which the stress–strain behaviour of sand is clearly non-linear. As a consequence, the equivalent shear stiffness decreases from values as high as 41 MPa (T1, EQ1) to 11 MPa (T2, EQ4); accordingly, the equivalent damping increases, ranging between 11 and 38%. 3.3. Comparison between the two procedures Figure 5 compares the peak shear strains and the equivalent parameters calculated according to the two procedures above described. The subscript “TF” refers to the values computed using the transfer functions for G and D, and Equation (6) for the shear strain γ; the subscript “cyc” is used for the values obtained from the τ – γ cycles. The graphs show a reasonable good agreement between the two methods. The values computed for dense (T1, T3) and loose (T2, T4) soil are plotted with full and open symbols, respectively. The interpretation based on the transfer functions produces higher values for both the shear strain, γTF, and stiffness, GTF. It may be also observed that the interpretation of the cycles shows slightly larger strains, γcyc, for the looser models than for the denser ones (Figure 5(a)). Correspondingly, the shear stiffness values, Gcyc, of the looser models are lower (Figure 5(b)). It is worth noting that the quality of the results based on the transfer function is affected by the use of an almost single-frequency shaking event. In fact, in most cases, the frequency of the input signal was far from the fundamental frequency of the soil layer, hence the most amplified harmonics of the signal were associated to a very low energy content. The use of a shaking signal with a wider range of frequencies, as natural earthquakes are, would likely improve the reliability of the transfer function method.

European Journal of Environmental and Civil Engineering Table 4. cycles.

Mobilised shear stiffness, damping ratio and maximum shear strain from stress–strain

Model


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EQ1

EQ2

EQ3

EQ4

Gcyc [MPa] T1 T2 T3 T4 Avg

41 22 21 22 30

30 22 20 16 22

26 14 20 14 19

21 11 14 13 15

Dcyc [%] T1 T2 T3 T4 Avg

17 14 12 16 14

20 27 19 11 19

20 32 22 28 25

28 38 28 31 31

γcyc [%] T1 T2 T3 T4 Avg

.02 .04 .04 .04 .03

.05 .07 .06 .09 .07

.09 .15 .09 .13 .11

.13 .26 .18 .21 .19

Figure 5.

Comparison between the two different estimations of (a) γ, (b) G and (c) D.

The comparison in Figure 5(c) between the two methods in terms of the damping ratio, D, is affected by the larger scatter in the results, as also observed in similar works (cf. also Brennan et al., 2005; Conti & Viggiani, 2012; Li et al., 2013). 3.4. Strain dependence of shear stiffness and damping The small strain shear modulus, G0, of the LB sand was measured at varying values of confining pressure and relative density in the low-amplitude resonant column (RC) and torsional shear (TS) tests on laboratory specimens (Visone & Santucci de Magistris, 2009). However, for the reasons mentioned at the beginning of this section, a direct assessment of G0 from the back-analysis of the centrifuge tests was preferred in this work. Brennan et al. (2005) suggested to evaluate the initial soil stiffness of the centrifuge model by a low-energy dynamic shaking test, associated to a very low deformation

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level. More recently, Lee, Wang, Wei, and Hung (2012) have further demonstrated that the values of shear wave velocity, VS, obtained from such low-intensity shaking tests agree with the measurements in bender element tests during the centrifuge flight. Unfortunately, even the weakest model earthquake (EQ1) induced shear strains too high to be associated to the initial value of the shear modulus, G0. Therefore, this parameter was obtained by extrapolation from the experimental data according to the following procedure: (1) Preliminarily, a mean “laboratory” G(γ)/G0 decay curve (solid line in Figure 6) was fitted on the data of G(γ)/G0 from RC and TS tests (Lanzano, Bilotta, Russo, Silvestri, & Madabhushi, 2010); the solid line was fitted on both RC and TS laboratory tests data carried out at 100 and 200 kPa of confining pressure (Visone & Santucci de Magistris, 2009), being such stress range representative of the stress level in the sand model at 80 g; (2) centrifuge experimental data points (G:γ) were obtained for both loose and dense sand centrifuge models from the (τ, γ) cycles; (3) reference values of shear modulus G(γmin) were determined (see Figure 7) corresponding to the lowest measured shear strain γmin in centrifuge (about .02% for the denser and .04% for the looser soil models); (4) G(γmin) values were hence used to plot dimensionless “centrifuge” decay curves, G/G(γmin); (5) in order to extrapolate the value of G0 in the centrifuge model, these curves were subsequently scaled down to match the “laboratory” G(γ)/G0 decay curve; (6) since the necessary scaling factor is equal to G(γmin)/G0, the “centrifuge” values of G0 could be back-figured. On average, G0 resulted equal to 60 MPa for the denser models and to 30 MPa for the looser ones. They result lower than the corresponding values from laboratory element tests, which induced to infer that the overall fabric of sand in centrifuge models was different from the laboratory specimens. At low strain level, the calculated shear moduli were affected by large scatter, therefore for the lowest amplitude events (i.e. EQ1 of all models), an average value of G/G0 over all the cycles was plotted in Figure 6(a). In the case of the strongest events (e.g. EQ4), a decay of the shear modulus was observed during the event and data from each single cycle were plotted in the same figure. A similar procedure was followed by using the experimental transfer functions shown in Figure 4; the corresponding data are shown in Figure 6(b). In this case, values of G0 equal to about 50 MPa for the densest models and 43 MPa for the loosest ones were obtained. In the figure, the points corresponding to the events EQ4 of models T2– T4 have been highlighted; since the procedure was applied to a very small portion of the transfer function, the results should be considered as less reliable (cf. §3.1). In Figure 8, a comparison between laboratory and centrifuge tests is shown in terms of damping ratio D. Overall, the values of damping ratio D obtained from the centrifuge (τ, γ) cycles are relatively higher than the mean D – γ curve measured in the laboratory. The observation of lower stiffness and higher damping in the centrifuge models concurrently suggest that the model preparation and stress-induced anisotropy considerably affected the fabric of the sand layer making it weaker than the laboratory specimens.



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Figure 6. G(γ)/G0 laboratory curve compared with the values back-figured from stress–strain cycles (a) and transfer functions (b) in centrifuge tests.

Figure 7. Procedure to compare the decay of stiffness with strain as obtained from laboratory and centrifuge tests.


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Figure 8. Damping ratio laboratory curve compared with the values back-figured from stress–strain cycles (a) and transfer functions (b) in centrifuge tests.

4. Experimental vs. numerical and analytical modelling 4.1. Equivalent linear viscoelastic analyses Equivalent linear viscoelastic dynamic analyses of the coupled ground-tunnel system undergoing shaking were performed by the FE code PLAXIS 2D (Brinkgreve, Swolfs, & Engin, 2011). The geometry of the centrifuge models was reproduced by the finite element meshes shown in Figure 9, for both shallow and deep tunnels. The PLAXIS mesh consisted of triangular 15-node elements. The elements were small enough to ensure the frequency content of the input signal not to be artificially filtered (Kuhlemeyer & Lysmer, 1973). The lining was modelled by using beam elements. The two vertical boundaries were linked by rigid “node-to-node anchors”, forcing them to have identical displacements as in the rigid laminar box in the centrifuge tests. The vertical spacing of the anchors is .01 m; hence, 29 constraints were used in the meshes. A rigid bottom boundary was assumed in the FE models and the signal recorded at the base of the reference array was assumed as input signal for the analyses. In the numerical simulations shown hereafter, the soil was intentionally modelled as linear viscoelastic and homogeneous, by assuming the shear stiffness G and the damping ratio D constant with depth, and corresponding to the values back-figured from the τ – γ cycles (Table 4). Local changes of density were not modelled in the analyses. This mimics the assumptions at the base of the simplified analytical methods such as that proposed by Wang (1993). In PLAXIS, the viscous damping is frequency dependent



Figure 9.

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Analysed numerical models.

and traditionally modelled through the well-known Rayleigh formulation, for which the damping matrix is evaluated as the sum of mass and stiffness matrix, multiplied for the coefficients αR and βR, respectively. These coefficients were estimated using the “double frequency approach” suggested by Park and Hashash (2004): the Rayleigh damping was assumed coincident with the assigned damping ratio, Dcyc, at the predominant frequency of the input signal (cf. Table 2) and at the first natural frequency of the soil layer. The relevant Rayleigh functions in the frequency domain are plotted in Figure 10 for all the analysed models, together with the Fourier spectrum of the input signals (grey shadow). Note that the input frequency increases with the model earthquake intensity from EQ1 to EQ4 (cf. Table 2) while, correspondingly, the natural frequency decreases due to the

Figure 10.

Adopted Rayleigh damping functions vs. experimental values of damping ratio.

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non-linear behaviour of sand (cf. Section 3.1); since the two frequencies approach each other, the Rayleigh formulation produces larger overdamping at high frequencies when passing from weaker (EQ1) to stronger (EQ4) input motions. 4.2. Dynamic response of soil layer The time histories of accelerations were calculated at the same locations of the accelerometers in the models (cf. Figure 2). In Figure 11, a comparison is shown between measured and calculated accelerations at the upper sensor along the reference array. The curves were truncated at .23 s to allow a better insight. The calculated time histories of acceleration for EQ3 and EQ4 are generally smoother than the corresponding experimental ones. Compared to EQ1 and EQ2, such a larger filtering of the higher frequencies is systematic in all models and it is consistent with the numerical overdamping shown in Figure 10. The effect of overdamping at higher frequencies can be appreciated also in Figure 12, where a similar comparison between experimental and numerical pseudo-acceleration

Figure 11. Measured and calculated time histories of acceleration at the “reference” top accelerometer (window between .03 and .23 s).



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response spectra is shown. In all cases, the pseudo-acceleration is well computed around the predominant period of the input signal. At lower periods, on the contrary, the numerical predictions generally tend to underestimate the experimental spectral acceleration. This is particularly evident for EQ3 and EQ4 in all models. The most accurate description of the spectral response was achieved for EQ1 of model T1: in this case, the Rayleigh function reproduced a damping value close to the measured damping ratio over a wider range of frequencies of the input signal (cf. Figure 10). The maximum values of measured acceleration, amax, are plotted against depth in Figure 13, together with the corresponding calculated values. The agreement between experimental data and numerical calculation is generally fine along both the “reference” and the “tunnel” vertical arrays. Better predictions were achieved for weaker shakings. This has been previously observed and discussed in terms of time histories of acceleration and response spectra of the accelerometer at the top of the reference array (Figures 11 and 12). The quality of prediction is somehow confirmed also in depth, although the agreement with the experimental data seems higher for the denser models (T1 and T3). A possible reason is that the experimental loose models are

Figure 12. Numerical vs. experimental response spectra of pseudo-acceleration (“reference” top accelerometer).

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Figure 13.

Profiles of computed and measured amax with depth.


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less homogeneous than the dense ones due to the different sand deposition technique which has been adopted in the two cases (cf. Lanzano et al., 2012). The bottom accelerometer of the tunnel array was not connected to the box, hence it did not record the input signal. This is the reason why the maximum values of acceleration at that level differ from the corresponding ones in the reference array. Along the tunnel array, the numerical analyses systematically underpredict these values, being closer to the maximum values of the input signals (i.e. measured at the bottom of the reference array): such a difference can be expected since the experimental detail of the actual contact between the sand and the bottom of the box could not be modelled in the numerical analyses. Figure 14 illustrates the overall cyclic stress–strain behaviour of the sand model considered as subjected to simple shear. “Experimental” and “numerical” plots are

Figure 14. Shear stress–strain cycles calculated from the “reference” accelerometer array (experimental vs. numerical).

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compared in the figure. The “numerical” curves were obtained according to the same procedure used for the experimental data (cf. §3.2) along the “reference” array. On the average, the predicted shear strains result higher than experimental ones, since the computed accelerations sometimes overestimate measurements (see for instance response spectra in Figure 12). The figure also shows the difficulties which may arise in back-figuring such values, due to the irregular shape of most cycles, particularly for the weaker motions (EQ1), as a consequence of the particular pseudo-harmonic shape of the input signal. Moreover, the comparison between numerical and experimental cycles shows a systematic higher damping in the numerical results, as it would be expected due to the numerical overdamping related to the Rayleigh formulation adopted. 4.3. Internal forces in the lining during shaking The centrifuge tests have shown that both transient (reversible) and accumulated (permanent) changes of internal forces arise in the lining, the latter due to sand densification (cf. Lanzano et al., 2012). The simplified constitutive model adopted in both the numerical and analytical predictions only allowed for the calculation of the reversible changes of hoop forces and bending moments, due to ovalisation of the tunnel lining under cyclic ground shear straining.

Figure 15. Non-dimensional increments of hoop forces calculated in the lining by varying the factor Rinter.


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According to the analytical formulations in literature (e.g. Penzien & Wu, 1998; Wang, 1993), the changes of hoop forces in the tunnel lining during shaking are largely influenced by the contact conditions at the interface between the soil and the lining. A parametric study was therefore performed on the numerical models in order to analyse the sensitivity of the predicted internal forces to the degree of roughness of the soil-lining interface.


4.3.1. Back-analysis of the interface conditions The interface elements used in the analysis were elastic, with a reduced stiffness compared to the soil. In the Plaxis code, the degree of such reduction is defined by an interface factor, Rinter, which is equal to 1 if no reduction is considered. In this case, no relative slippage is allowed between the lining and the soil. As long as the factor Rinter is reduced, a larger amount of relative slippage is allowed, tending to full-slip conditions for very low values of Rinter. Four values of Rinter were used in the analyses (Rinter = .01; .05; .1; and 1). In Figure 15, the non-dimensional increments of hoop forces, ΔN/τR, calculated in the lining by varying the factor Rinter are shown for all the models and all the shaking events (τ is the reference shear stress, calculated as Gcyc · γcyc, and R is the tunnel radius). In the same figure, the values analytically computed with the formulas by Wang (1993) in both full-slip and no-slip conditions (see Appendix) are shown for comparison with dashed lines. It is worth noting that, according to Wang (1993), the value of the normalised increment of hoop force depends on the mobilised shear stiffness.

Figure 16.

Measured vs. calculated changes of hoop force: influence of the choice of Rinter.

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It can be observed that all the calculations performed with Rinter = 1 predict an increment of hoop force which practically coincides with the analytical predictions for noslip conditions. On the other side, the calculations with Rinter = .01 are very close to the analytical predictions for full-slip conditions. In order to choose a reasonable value of Rinter to reproduce the fairly smooth lining– ground interface in the centrifuge model, the experimental variations of hoop force, ΔNexp, were compared to the corresponding numerical computations, ΔNnum, for three different assumptions of Rinter (.01; .05; .1). In Figure 16, the numerical changes of hoop forces, ΔNnum, are plotted against the experimental ones, ΔNexp. The black solid line represents the condition DNnum ¼ DNexp ; the black dashed and the grey solid lines bound a difference of ±20 and ±50%, respectively, from that condition. The comparison shows that the best prediction of hoop forces is obtained by assuming Rinter = .05, although for model T3 only this choice overestimates the experimental measurements.

Figure 17.

Measured and calculated values of changes of hoop force.


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4.3.2. Prediction of transient changes of internal forces A comparison among the reversible increments of hoop forces, ΔN, and bending moment, ΔM, is proposed in Figures 17 and 18. In Figure 17, the experimental increments of hoop force are shown with black dots, together with the numerical prediction for Rinter = .05 (black line) and the analytical solution (grey line). This latter was computed by assuming in Equation (A2) (full slip, see Appendix) γ = γcyc, i.e. the back-figured experimental values of average shear strain in the model. The analytical solution, corresponding to full-slip conditions, is obviously a lower bound of the experimental values. The FE analyses predict values which are in most cases in good agreement with the experimental ones. Figure 18 shows similar plots for bending moments, with those relevant to the analytical solutions computed again with reference to the experimental strains. In this case, the Wang’s analytical solution (Equation (A1)) does not depend on the slippage at the interface. The differences between analytical and numerical calculations are consistent with the differences between experimental and numerical shear strain previously observed. They are however very small for the weaker events.

Figure 18.

Measured and calculated values of changes of bending moment.

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5. Conclusions In this paper, the results of centrifuge tests on reduced scale models of a circular tunnel in sand have been interpreted and back-analysed. The aim was to compare the experimental results with the predictions of simplified methods and finite element analyses. The dynamic response of the physical models has been modelled by a simple equivalent linear approach in a series of finite element analyses. This is the common procedure adopted for site seismic response analysis in design, where the soil is usually modelled as an equivalent linear viscoelastic medium. Although both experimental and numerical evidences of permanent changes of internal loads in the tunnel lining are reported in literature, this aspect was intentionally not examined in this work. The mobilised shear stiffness and damping ratio were back-calculated from the experimental results. The centrifuge values of small strain shear stiffness were on average lower than those measured in the laboratory element tests, due to unexpected weaker fabric of the sand and differences in the followed loading paths. Two procedures were used to back-figure the mobilised shear stiffness and damping ratio from the centrifuge tests: one was based on the interpretation of the spectral ratio between surface and bottom accelerometric signals and the other on the shear stress– strain cycles during each event. The degree of uncertainty of both procedures is increasingly high at the lowest strain levels, where the input signal was most affected by high frequency noise, although time histories were filtered for high frequencies before each integration step. The scatter of the parameters derived from the stress–strain loops is related to the back-calculation procedure, which determines the shear strain by numerical manipulation (time integration and spatial derivation) of the recorded accelerations. Due to the characteristics of the input signals, which were pseudo-harmonic (i.e. nominally characterised by a single frequency), the procedure based on the spectral ratio allowed for results of lower reliability than the interpretation of cycles. Therefore, only the latter was used to back-figure the mobilised stiffness and damping for the numerical and analytical predictions. The stiffness reduction factor of the ground-lining interface (Rinter) was back-calculated to be used in the numerical analyses in order to provide a reliable estimate of the transient changes of hoop force. The comparison of experimental and numerical results in terms of transient changes of internal forces in the lining was satisfactory both in terms of bending moment and of hoop force. The results indicate that the analytical formulations proposed by Wang (1993), widely adopted in the design practice, can give a good estimation of the seismic increment of bending moment in the lining and a reasonable lower bound for the transient changes of hoop forces, provided that cyclic shear strain are correctly evaluated. Although such a simplified approach may be sufficient for a preliminary design, concern arises when the sand layer is interested by intense densification during shaking. In this case, permanent changes of internal forces may originate, which can be predicted only with more sophisticated constitutive models. A careful investigation of this issue of concern was however out of the scope of this work.

Acknowledgements The experimental data at the basis of this paper were obtained in the framework of an agreement between Universities of Napoli and Cambridge, as a part of a Research Project funded by ReLUIS


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(University Network of Seismic Engineering Laboratories) Consortium. The writers wish to thank Dr Gopal Madabhushi, who coordinated the experimental campaign in centrifuge, Dr Filippo Santucci de Magistris, for making available the characterization of LB sand by laboratory tests, and the coordinator of the research project, Prof. Stefano Aversa, for his continuous support.


References AGI. (2005). Guidelines for geotechnical design in seismic zones (Patron ed.). Bologna: Special volume of Italian Geotechnical Association (in Italian). Amorosi, A., & Boldini, D. (2009). Numerical modelling of the transverse dynamic behaviour of circular tunnels in clayey soils. Soil Dynamics and Earthquake Engineering, 29, 1059–1072. Bäckblom, G., & Munier, R. (2002). Effects of earthquakes on the deep repository for spent fuel in Sweden based on case studies and preliminary model results (Technical Report TR-02-24). Stockholm: Swedish Nuclear Fuel and Waste Management. Bilotta, E., Lanzano, G., Russo, G., Santucci de Magistris, F., Aiello, V., Conte, E., … Valentino, M. (2007). Pseudo-static and dynamic analyses of tunnels in transversal and longitudinal direction. In Proceedings of the 4th international conference on Earthquake Geotechnical Engineering. Thessaloniki: Springer. Bilotta, E., Lanzano, G., Russo, G., Silvestri, F., & Madabhushi, S. P. G. (2009). Seismic analyses of shallow tunnels by dynamic centrifuge tests and finite elements. In Proceedings of the 17th international conference on Soil Mechanics and Geotechnical Engineering. Alexandria, Egypt: Balkema. Bilotta, E., & Silvestri, F. (2012). A predictive exercise on the behaviour of tunnels under seismic actions. In Geotechnical aspects of underground construction in soft ground – Proceedings of the 7th international symposium on Geotechnical Aspects of Underground Construction in Soft Ground (pp. 1071–1077). Rome: Balkema. Brennan, A. J., Thusyanthan, N. I., & Madabhushi, S. P. G. (2005). Evaluation of shear modulus and damping in dynamic centrifuge tests. Journal of Geotechnical and Geoenvironmental Engineering, 131, 1488–1497. Brinkgreve, R. B. J., Swolfs, W. N., Engin, E. (2011). Plaxis 2D 2011 manuals. Delft: Plaxis bv. Conti, R., & Viggiani, G. M. B. (2012). Evaluation of soil dynamic properties in centrifuge tests. Journal of Geotechnical and Geoenvironmental Engineering, 138, 850–859. Hashash, Y. M. A., Hook, J. J., Schmidt, B., & I-Chiang Yao, J. (2001). Seismic design and analysis of underground structures. Tunnelling and Underground Space Technology, 16, 247–293. Hoeg, K. (1968). Stresses against underground structural cylinders. Journal of the Soil Mechanics and Foundation Division, ASCE, 94, 833–858. ISO 3010:2001. (2001). Basis for design of structures – Seismic actions on structures. Kuhlemeyer, R. L., & Lysmer, J. (1973). Finite element method accuracy for wave propagation problems. Journal of the Soil Mechanics and Foundation Division, ASCE, 99, 421–427. Lanzano, G., Bilotta, E., Russo, G., Silvestri, F., & Madabhushi, S. P. G. (2010). Dynamic centrifuge tests on shallow tunnel models in dry sand. In Proceedings of the ICPMG2010 Physical Modeling in Geotechnics (pp. 561–567). Zurich, Switzerland: Balkema. Lanzano, G., Bilotta, E., Russo, G., Silvestri, F., & Madabhushi, S. P. G. (2012). Centrifuge modeling of seismic loading on tunnels in sand. Geotechnical Testing Journal, 35, 854–869. Lanzano, G., Bilotta, E., Russo, G., Silvestri, F., & Madabhushi, S. P. G. (2009). Experimental assessment of performance-based methods for the seismic design of circular tunnels. In Proceedings of the 1st international conference on Performance Based Design in Earthquake Geotechnical Engineering. IS-Tokyo, Tokyo (Japan). Lee, C.-J., Wang, C.-R., Wei, Y.-C., & Hung, W.-Y. (2012). Evolution of the shear wave velocity during shaking modeled in centrifuge shaking table tests. Bulletin of Earthquake Engineering, 10, 401–420. Li, Z., Escoffier, S., & Kotronis, P. (2013). Using centrifuge tests data to identify the dynamic soil properties: Application to Fontainebleau sand. Soil Dynamics and Earthquake Engineering, 52, 77–87. Newmark, N. M. (1967). Problems in wave propagation in soil and rocks. In Proceedings of the international symposium on Wave Propagation and Dynamic Properties of Earth Materials (pp. 7–26). Albuquerque, NM: University of New Mexico Press.


24

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O’Rourke, T. D., Goh, S. H., Menkiti, C. O., & Mair, R. J. (2001). Highway tunnel performance during the 1999 Duzce earthquake. In Proceedings of the 15th international conference on Soil Mechanics and Geotechnical Engineering (pp. 1365–1368). Istanbul, Turkey. Park, D., & Hashash, Y. M. A. (2004). Soil damping formulation in nonlinear time domain site response analysis. Journal of Earthquake Engineering, 8, 49–274. Penzien, J., & Wu, C. L. (1998). Stresses in linings of bored tunnels. Earthquake Engineering & Structural Dynamics, 27, 283–300. Visone, C., & Santucci de Magistris, F. (2009). Mechanical behaviour of the Leighton Buzzard Sand 100/170 under monotonic, cyclic and dynamic loading conditions. In Proceedings of the XIII conference L’Ingegneria Sismica in Italia, ANIDIS. Bologna, Italy. Wang, J.-N. (1993). Seismic design of tunnels: A state-of-the-art approach. In Monograph 7 (147 p). New York, NY: Parsons Brinckeroff. Wang, W. L., Wang, T. T., Su, J. J., Lin, C. H., Seng, C. R., & Huang, T. H. (2001). Assessment of damage in mountain tunnels due to the Taiwan Chi-Chi earthquake. Tunnelling and Underground Space Technology, 16, 133–150. Yoshida, N. (2009). Damage to subway station during the 1995 Hyogoken-Nambu (Kobe) earthquake. In T. Kokusho (Ed.), Earthquake geotechnical case histories for performance-based design (pp. 373–389). Tokyo: CRC Press. Zeghal, M., & Elgamal, A.-W. (1994). Analysis of site liquefaction using earthquake records. Journal of Geotechnical and Geoenvironmental Engineering ASCE, 120, 71–85.

Appendix Wang (1993) proposed expressions to calculate the increments of bending moments and hoop forces due to the ovalisation of a circular elastic lining (Es, νs) of thickness ts around a cavity of radius R in an elastic ground (E, ν or G). Assuming an average ground shear strain γ, they read as follows: ð1 þ 3a2 4a3 Þ p cos 2 h þ (A1) DM ¼ G c R2 3 4 DN ¼

ð1 þ 3a2 4a3 Þ p cos 2 h þ GcR 3 4

(full slip allowed at the interface) (A2)

p DN ¼ ð1 þ a2 Þ cos 2 h þ G c R (no slip allowed at the interface) (A3) 4 where θ is the angular coordinate of the lining section; a1, a2 and a3 are functions (see Table A1) of the soil Poisson’s ratio, ν, and of two compressive and flexural relative stiffness ratios, C and F, defined as: C¼

Eð1 m2s ÞR Es ts ð1 þ mÞð1 2mÞ

(A4)

Eð1 m2s ÞR3 6Es Ið1 þ mÞ

(A5)

F¼

being I the moment of inertia per unit length of lining.

European Journal of Environmental and Civil Engineering Table A1.

Relative stiffness parameters (Hoeg, 1968).

No slip at the interface

Full slip at the interface a1 ¼

a2 ¼


a3 ¼

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ð12mÞð1CÞF12ð12mÞ2 Cþ2 ½ð32mÞþð12mÞC Fþð528mþ6m2 ÞCþ68m ½1þð12mÞC F12ð12mÞC2 ½ð32mÞþð12mÞC Fþð528mþ6m2 ÞCþ68m

ð12mÞðC1Þ ð12mÞCþ1

a2 ¼ 2Fþ12m 2Fþ56m 2F1 a2 ¼ 2Fþ56m