Dynamic centrifuge tests on shallow tunnel models in ...

Physical Modelling in Geotechnics – Springman, Laue & Seward (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-59288-8

Dynamic centrifuge tests on shallow tunnel models in dry sand G. Lanzano University of Molise, StreGa Laboratory, Termoli (CB), Italy

E. Bilotta, G. Russo & F. Silvestri University of Naples “Federico II”, D.I.G.A. Department, Naples, Italy

S.P.G. Madabhushi Cambridge University, CUED Department, Cambridge, UK

ABSTRACT: In the framework of the Italian research project ReLUIS-DPC, a set of centrifuge tests were carried out at the Schofield Centre in Cambridge (UK) to investigate the seismic behaviour of tunnels. Four samples of dry sand were prepared at different densities, in which a small scale model of circular tunnel was inserted, instrumented with gauges measuring hoop and bending strains. Arrays of accelerometers in the soil and on the box allowed the amplification of ground motion to be evaluated; LVDTs measured the soil surface settlement. This paper describes the main results of this research, showing among others the evolution of the internal forces during the model earthquakes at significant locations along the tunnel lining.

1

2

INTRODUCTION

The behaviour of the tunnel lining under seismic loads is generally known by the observation of damage after the earthquake. Circular tunnels usually suffer ovalisation of the lining, producing the maximum diameter variation in the cross section at θ = 45°+ n90° (n = 0; 1; 2; 3) (Hashash et al. 2001). In common design practice, closed-form solutions are used in order to estimate the increments of internal forces during the seismic event, from synthetic parameters (Wang 1993; Penzien & Wu 1998; Hashash et al. 2001). On the other hand, measurements of seismic internal forces on real scale structure are difficult, because of the random occurrence of the earthquakes. Moreover, the instrumentation setups commonly used for the tunnel monitoring generally have too large a sampling interval to record seismic events, measuring only the internal forces values before and after (residual) the earthquake event. Small-scale physical modeling, like centrifuge tests, permits this limitation to be overcome by firing an artificial earthquake at the base of the model. A series of centrifuge tests were conducted in order to obtain artificial case-histories and calibrate the simplified analytical solutions. The paper shows some typical data records and processing, the whole results being reported by Lanzano (2009).

CENTRIFUGE TESTS

The tests were carried out at the “Schofield Centre” of the University of Cambridge, equipped with a large beam centrifuge called the Turner Beam Centrifuge. A dynamic earthquake actuator (Stored Angular Momentum—SAM, Madabhushi et al. 1998), and a special box (Laminar box), which allowed horizontal displacements during the dynamic phase, are also available at the Schofield Centre. In Table 1, the centrifuge test programme (Lanzano 2009) is shown. Four models of a circular tunnel in dry sand (Leighton Buzzard Sand fraction E) were tested at 80 g. Two different pluviation procedures were used to obtain different soil density: the loose sand models were prepared using a manual hopper; instead for the dense sand model, the material was poured into the box through an automatic procedure to control Table 1.

Tested models.

Model

Tunnel cover, C (mm)

Relative density, Dr (%)

T1 T2 T3 T4

75 75 150 150

∼75 ∼40 ∼75 ∼40

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the sand flow. The model tunnel lining was an aluminum tube 200 mm long, having external diameter D = 75 mm and thickness t = 0.5 mm. It was installed at two different depths. A typical layout of an instrumented model is shown in Figure 1 (T1). Each model was instrumented with miniature accelerometers, located in the soil and along the model container. Three vertical arrays in the models were instrumented by horizontal accelerometers: the first instrumented array was placed along the vertical passing through the tunnel axis (tunnel); the second array was located 125 mm far away from the tunnel axis (free-field); the third array was placed on the box at one side (reference). One of the external instruments was located on the base-plate, in order to measure the input motion at the rigid basement. The surface settlements were measured by LVDTs, placed in two gantries above the model. Eight pairs of strain gauges (Wheatstone bridges) were glued on the external and internal surface of the alloy tunnel, and wired to measure hoop forces and bending moments at four positions along the tunnel transverse section. Each of the tested models was spun up in steps, thereafter underwent four seismic events at increasing frequency and acceleration amplitudes. Table 2 shows the values of amplitude, nominal frequency and duration of each signal at both model and prototype scales (bracketed quantities refer to prototype).

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TYPICAL MEASUREMENTS

3.1

Acceleration time histories

A typical acceleration time history used as the input signal in the tests (model T1, signal EQ4) is shown in Figure 2, together with the corresponding Fourier spectrum. The signal was pseudo-harmonic, as its amplitude is not exactly constant, it is not symmetric about the time axis, and the frequency content is larger than the nominal frequency. According to Brennan et al. (2004), an effective energetic content is associated with such higher frequencies of the signal, therefore they were not filtered out. The effect of an over-filtering (close to the predominant frequency) would be a drastic reduction of the peak acceleration, which does not represent the true response of the soil at the wave passage (Lanzano 2009). In practice, all the acceleration time histories were centered, windowed (neglecting the noise before and after the significant signal), and filtered in the frequency domain: the filter was designed using a 4th order Butterworth type, which is an infinite-impulse-response filter (IIR). This digital filter was a typical “band pass” between the frequencies of 15 Hz and 250 Hz, in order to include all the meaningful frequency content of the Fourier spectrum of the input signal. The choice of the “band pass” filter was carried out in order to eliminate the low and the high frequency: the lowest frequency determined a drift of the signal during the integration of the acceleration time history; the highest frequencies were a recording noise, because all of these had almost zero spectral ordinates. Moreover a base-line correction with a linear law was applied to the input signal: this correction was useful to obtain a zero trend value at the end of the time history of velocity and displacement.

a [g]

20 10 0 -10 -20 0

Fourier amplitude

Figure 1. Schematic assembly of model T1.

Table 2. Earthquakes. Input signal

Frequency, f (Hz)

Duration (s)

Amplitude (g)

EQ1 EQ2 EQ3 EQ4

30 40 50 60

0.4 0.4 0.4 0.4

4.0 8.0 9.6 12.0

[0.375] [0.5] [0.625] [0.75]

[32] [32] [32] [32]

[0.05] [0.10] [0.12] [0.15]

0.4

t [s]

0.6

0.8

1

2 1.5 1 0.5 0 0

Figure 2. spectrum.

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0.2

100

200 f [Hz]

300

400

Acceleration time history and Fourier

The profiles of the peak accelerations at model scale during the EQ3 earthquake are shown in Figure 3 for each vertical array and each model. The models exhibited a slight amplification of the input acceleration (amax,s/amax,b < 1.5). The peak acceleration at the tunnel depth was generally lower than the peak value of the input signal. Moreover, the base accelerometer under the tunnel always measured a larger value of acceleration compared to the corresponding free-field accelerometer at the base. This is evidence of the influence of the cavity boundary on the wave propagation around the tunnel model. However, according to the observed amplification factors, the values of soil stiffness of both loose and dense models appear to be similar, despite the different values of relative density of the sand.

45°

3.5

EQ3

225° 315°

2.5 M [Nmm/mm]

135° EQ2

EQ1

1.5

EQ4

0.5 -0.5 -1.5 1.2

EQ4

N [N/mm]

1 EQ3

0.8 0.6

EQ2

0.4

EQ1

0.2 0 -0.2 0

3.2 Surface settlement The LVDT devices measured the surface settlement, w, at two symmetrical points during both the centrifuge flight and the shaking of models T1, T3 and T4. The values were recorded until the model swung down to 1 g. In Table 3, the average values of surface settlement are reported during these three important test steps at both model and prototype scale (square brackets). Sand densification occurred in all the tests for all the models: the measurements on the loose sand model (T4) showed settlements about twice larger than the dense models (T1 & T3). After the swing down

0 0

z [mm]

50

10

20 0

Ref EQ3

10

20 0

FF EQ3

10

20

Tun EQ3

250

4

3.3

Internal forces

The experimental values of internal forces in the lining were derived from the strain gauges records during each seismic event. Typical time histories of M and N in a test are shown in Figure 4 at model scale. Positive values represent inward curvature and compression, respectively. INTERPRETATION Transfer functions

The “similarity” between two time histories of horizontal accelerations measured in different points of the same array can be represented by the coherence function:

T1 T2 T3 T4 amax [g]

200

3

stages to 1 g, the cumulative settlements were only partially recovered.

4.1

150

2 t [s]

Figure 4. Typical time history of bending moment M and hoop force N in the dynamic phase (model T2).

4

100

1

Figure 3. Profiles of maximum acceleration for four different models (EQ3).

Table 3. Average measured settlements.

Model

End of spin up 80 g (cm)

After the eqs 80 g (cm)

End of the test 1 g (cm)

T1 T3 T4

0.26 [21.0] 0.21 [16.5] 0.42 [34.0]

0.42 [33.6] 0.41 [33.0] 0.8 [63.7]

0.35 [28.0] 0.35 [28.0] 0.56 [45.0]

Cohh jk (ω ) =

S jk (ω )

S jj (ω ) Skk (ω )

(1)

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

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ratio between the Fourier spectra of two different signals. The modulus of the transfer function represents the amplification function, which is defined as: A (ω )

H (ω ) =

X (ω ) Y (ω )

(2)

In Equation 2, X(ω) and Y(ω) are the Fourier spectra of two different signals. The combined interpretation of the transfer function and the coherency analysis was used to recognize which frequencies were amplified when the waves propagate through the soil layer from the first to the second accelerometers. In Figure 5 the transfer and coherence (dashed line) functions were calculated along the three instrumented verticals located in the model (reference, free-field and tunnel). Between 60 Hz and 175 Hz, the coherence function always had values higher than 0.6, which confirms that noise did not affect the soil response in this frequency range. The experimental amplification functions were plotted together with the theoretical function for a damped elastic layer on a rigid base. The value of the first natural frequency of the soil layer was computed by fitting the theoretical function to the experimental values. The experimental curves showed that the amplified frequencies of the reference and free-field verticals were very similar, while the tunnel vertical exhibited higher values. The presence of the tunnel modified the first natural frequency of the soil, increasing its stiffness. Figure 5 also shows that, along the tunnel vertical, the surface amplification appeared significantly reduced, especially around the resonant peak observed at the reference vertical. This is a clear evidence of the wave-screening effect of the tunnel structure. The mobilised transversal modulus of elasticity G and damping ratio D were evaluated from the first natural frequency f0 of the best fitted curve, using the simplified expression: Vs2 ρ = ( 4 Hff0 ) ρ 2

(3)

Amplification factor

1000

100

Reference Free-field Tunnel Best Fit (fo=120Hz; D=12,5%)

10

1 50

75

100

125 150 f [Hz]

175

Figure 5. Transfer function (T1 EQ1).

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 200

Coherence

G0

An average value of the shear strain mobilized during each earthquake can be calculated according to Newmark (1968):

γ =

vmax vs

(4)

In Equation 4, vmax is the peak ground velocity obtained from the integration of the acceleration time history recorded at the top of the free-field array; Vs is the shear wave velocity, obtained from the transfer function. The average values of the mobilized shear stiffness, damping ratio and shear strain are reported in Table 4. The values of the mobilized shear stiffness range between 8 MPa and 29 MPa, confirming that the stiffness of the soil was relatively low. The difference between the dense and the loose sand models was generally small. The assessment of the damping ratio was extremely affected by the false amplification peak, due to a very low coherence at the corresponding frequency, and the procedure could give an inaccurate estimation of the real value. Despite this limitation, the damping level was computed and it was generally very high, between 5.5% to 30%. The values of the maximum shear strain were relatively high, between 0.1% and 0.5%, without any apparent difference between dense and loose sand. 4.2

Shear stress-strain loops

Zeghal & Elgamal (1994) reported a procedure to evaluate the shear modulus and the damping ratio from the horizontal accelerometers data. Brennan et al. (2005) modified this procedure for Table 4. Mobilized shear stiffness, damping ratio and maximum shear strain. Model

EQ1

EQ2

EQ3

EQ4

G [MPa] T1 T2 T3 T4

29.2 24.4 26.8 29.3

26.5 18.3 27.9 25

26 22.5 24.8 19.7

17.1 8 17.1 8

D [%] T1 T2 T3 T4

12.5 13.2 7.6 18.6

15.2 14.8 5.5 17.8

13 32 10 16.4

9 31 9 31

γmax [%] T1 T2 T3 T4

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0.10 0.14 0.11 0.12

0.16 0.16 0.19 0.16

0.22 0.23 0.22 0.24

0.34 0.50 0.35 0.49

acceleration traces measured in dynamic centrifuge tests. They suggested a procedure to calculate the shear strain along the accelerometer arrays. From the time histories of acceleration, the displacements u (t) were obtained from double integration. In order to avoid integration errors like an unreal drift during the shaking, the signal was filtered twice, before each integration. The filter also eliminated the phase distortion due to the integration procedure. The shear strain could be evaluated using a first or second order approximation between two or more instruments positioned in the same vertical array:

) )

− −

(5)

⎡ ⎢(ui γ ( zi ) = ⎣

ui )

( zi ( zi

) + u − u ( zi (i i )z zi ) (i ( 1 − 1)

zi

zi ) ⎤ ⎥ zi −11 ) ⎦

) ∫ 0 ρ adz

Once the shear stresses and the shear strains were evaluated from the accelerometer outputs, the data were assembled in a graph of shear strain against shear stress, to evaluate the loops. Examples of stress-strain cycles are showed in Figure 7. As the signals were not single-frequency, the cycles were not regular. In spite of such irregular shapes, an overall estimation of shear stiffness G in the cycles was performed as: Gmob =

4.3 Remarks on stiffness and damping Brennan et al. (2005) suggested to carry out a ‘zero’ centrifuge test, in which the model (with or without the structure) is subjected to an input signal associated to a very low deformation level. In such a way,

τ [kPa]

15 -15 -30

0.3 0.2 0.1

-0.25% 0.00% 0.25% γ [%] 30 T4 - EQ3 15 0

-30

0

0.1 0.2 0.3 0.4 0.5 0.6 γ max [%] (Brennan et al.)

T4 - EQ2

0

-30

G=16MPa

-0.25% 0.00% 0.25% γ [%] 30 T4 - EQ4 15 0

-15

G=13MPa

-0.25% 0.00% 0.25% γ [%]

Figure 6. Maximum shear strains computed according Brennan et al. (2005) against Newmark (1968).

15

-15

G=15MPa

-15

0

30

T4 - EQ1

0

τ [kPa]

γ max [%] (Newmark)

T1 T2 T3 T4

0.4

(8)

The profiles of mobilized shear stiffness gave values always lower than 20 MPa and had a maximum during the EQ2 event, although this earthquake was not the weakest. The values of shear modulus G computed from the cycles were generally lower than those computed from the amplification functions.

30

0.5

τ max − τ min γ max − γ min

(6)

in which the index i was relative to the position of the central instruments and i − 1 and i + 1 to the top and bottom accelerometers. For each array, the shear strains were calculated using Equations. 5 and 6. They were generally high, larger than 0.05% and lower than 0.35%, which corresponded to a strain level where the mobilized shear stiffness was lower than the initial value. A comparison between the two different methods used to evaluate the average values of the maximum shear strain along the freefield accelerometer arrays is shown in Figure 6. Following Brennan et al. (2005), calculations were performed using a first order approximation. The graph shows a good agreement between the

0.6

(7)

τ [kPa]

( (

z

τ(

τ [kPa]

γ =

two methods, especially for the weaker earthquakes (EQ1, EQ2 and EQ3). The shear stresses τ were computed through the equilibrium of a deformable column, by integrating the acceleration time histories in the space domain (Zeghal & Elgamal 1994):

-30

G=12MPa

-0.25% 0.00% 0.25% γ [%]

Figure 7. Stress-strain cycles (reference array, 2nd order approximation, z = 0.165 m).

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the initial shear modulus can be estimated from the transfer functions. Unfortunately, even the weakest among the simulated earthquakes induced shear strains too high to be associated to the initial shear modulus. Therefore, the initial shear modulus G0 was estimated as a function of both the void ratio e and the mean effective stress p′ by the well-known empirical expression by Hardin & Richart (1963): ⎛ G0 =C⎜ p′ref ⎝

(

− 1+ e

)2 ⎞

⎛ p′ ⎞ ⎟ ⎜ ⎟ ⎠ ⎝ p′ref ⎠

n

(9)

in which p′ref = 1 kPa; C and n are empirical constants, which can be calibrated on a specific sand. A set of laboratory tests (Resonant Column and Torsional Shear) on Leighton Buzzard sand were performed at the University of Naples Federico II by Visone (2009), in order to obtain the initial properties of the sand and the curves G/G0 – γ and D0−γ. Starting from the results of the Resonant Column tests for different values of void ratio, the values of C = 22000 and n = 0.35 were obtained, performing a best fitting of the experimental data. The values of G/G0 and D, as back-calculated from the transfer functions in the different centrifuge tests, are plotted in Figures 8–9 with 1.2 1

0.0001

0.001

G/G0

0.8 RC tests T1 T2 T3 T4

0.6 0.4 0.2 0 0.01 γ [%]

0.1

1

different colors, showing the decay curves of shear modulus and the variation of damping ratio with the mobilized shear strain γ, as obtained from the Torsional Shear tests. The comparison between the two sets of experimental results showed a reasonable agreement for the damping ratio. The normalized shear stiffness decay obtained from the centrifuge tests appears lower than that from the laboratory tests. To date, the reasons of the discrepancies between normalised stiffness—strain curves are not well understood, since they may depend on both experimental factors and interpretation criteria. 4.4

Residual internal forces after earthquake

As shown in Figure 4, the values of the internal lining forces after each shaking are significantly different from the initial conditions. This behaviour was observed almost systematically for any event in all the models, and it seems to indicate that permanent deformations occurred around the tunnel during shaking. This is qualitatively consistent with the observed densification of the sand during shaking, shown by the surface settlements (see §3.2). The values of bending moments and hoop forces measured in the model T2, before and after each seismic event, are shown in Figure 10. The figure shows a general tendency of the hoop force to increase as the densification increases. Also the bending moment changed, indicating a redistribution of stresses around the tunnel. However, it must be observed that under static loads (i.e. before earthquakes), the value of bending moments at the four locations of the strain gauges are close to zero as they should be. Therefore the large change in bending moment values, in some cases even ten times larger than the static values after the last earthquake, can be misleading and needs to be evaluated with caution.

Figure 8. Mobilized shear stiffness against shear strain. 35

RC tests T1 T2 T3 T4

30 25 D [%]

20 15 10 5 0

0.0001

0.001

0.01

0.1

1

γ [%]

Figure 9. Damping ratio against shear strain rate.

Figure 10. Residual bending moments and hoop forces after each earthquake (T2).

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5

CONCLUSIONS

The features of four dynamic centrifuge tests were presented in order to study the seismic behavior of shallow tunnels. The following main remarks can be drawn: • The models exhibited a small amplification of the input signal on the soil surface and a reduction of the peak ground acceleration at tunnel depth. • Two procedures to evaluate the soil stiffness and maximum shear strains during the shaking were discussed using the acceleration time histories. The results are consistent in terms of maximum shear strains. • The normalized shear stiffness and damping evaluated in the centrifuge were compared to the laboratory data, showing a fair agreement for the damping. Some uncertainties still there exist for the shear modulus, since the small strain stiffness of the Leighton Buzzard sand was not directly measured in the centrifuge. • As long as sand densification occurred, residual forces accumulated in the lining after each seismic event. Although the measurement points were not sufficient to describe completely the evolution of the distribution of the internal forces along the tunnel transverse section, their values resulted significant. It therefore appears that such a behavior should be considered in design.

Hardin, B.O. & Richart, E.F. 1963. Elastic wave velocities in granular soils. Journal of the Soil Mechanics and Foundation Division ASCE 89 (SM 1): 33–65. Hashash, Y.M.A. Hook, J.J. Schmidt, B. & Yao, J.I.C. 2001. Seismic design of underground structures. Tunneling and Underground space technologies 16: 247–293. Lanzano, G. 2009. Physical and analytical modeling of tunnels under dynamic loadings. PhD thesis, University of Naples “Federico II”. Madabhushi, S.P.G. Schofield, A.N. & Lesley, S. 1998. A new Stored Angular Momentum (SAM) based earthquake actuator. Centrifuge 98: 111–116. Rotterdam: Balkema., Newmark, N.M. 1968. Problems in wave propagation in soil and rock. International Symposium on wave propagation and dynamic properties of earth materials: 7–26. Penzien, J. & Wu, C.L. 1998. Stresses in linings of bored tunnels, Earthquake Engineering and Structural Dynamics 27: 283–300. Visone, C. 2009. Performance based design of embedded retaining structures. PhD Thesis, University of Naples “Federico“. Wang, J.N. 1993. Seismic Design of Tunnels. Parson Brinckerhoff Inc. Zeghal, M. & Elgamal, A.W. 1994. Analysis of site liquefaction using earthquake records. Journal of Geotechnical and Geoenviromental Engineering ASCE 120: 71–85.

REFERENCES 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 Geoenv. Engineering ASCE 131 (12): 1488–1497.

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