Centrifuge and numerical modelling of ground-borne vibration from an underground tunnel

Centrifuge and numerical modelling of ground-borne vibration from an underground tunnel

Soil Dynamics and Earthquake Engineering 51 (2013) 23–34 Contents lists available at SciVerse ScienceDirect Soil Dynamics and Earthquake Engineering...

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Soil Dynamics and Earthquake Engineering 51 (2013) 23–34

Contents lists available at SciVerse ScienceDirect

Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Centrifuge and numerical modelling of ground-borne vibration from an underground tunnel W. Yang n,1, M.F.M. Hussein, A.M. Marshall Department of Civil Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK

art ic l e i nf o

a b s t r a c t

Article history: Received 27 April 2012 Received in revised form 28 March 2013 Accepted 6 April 2013 Available online 11 May 2013

Ground-borne vibration from underground tunnels is a major environmental concern in urban areas. Various studies, mostly based on numerical methods, have been conducted to investigate this problem. In the numerical models, soil stiffness and damping are commonly assumed to remain constant with depth (homogeneous assumption) in order to simplify the problem. However, various studies in the field of geotechnical engineering have shown that soil properties change with depth due to the effect of increased confining stress. This paper presents a study of the effects of variation of soil properties with depth (soil non-homogeneity) on ground-borne vibration from an underground tunnel. Both centrifuge and numerical modelling were used to perform the study. In the centrifuge model, a plastic pipe was buried within sand to model the underground tunnel. Vibration excitation was applied to the model tunnel invert by a small shaker. The tunnel and soil response was measured using accelerometers during the tests. Corresponding numerical models, based on FLAC 3D, were built to simulate the centrifuge tests. The potential boundary effects of the centrifuge tests were examined by the numerical models. The model responses for both homogenous and non-homogenous cases were also calculated by the numerical models. Comparison of the experimental and numerical results shows that a homogenous model can give acceptable estimates of the tunnel behaviour. However, a clear improvement of estimates of soil behaviour is observed when the variation of soil properties with depth is considered in the numerical model. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Centrifuge FLAC Ground-borne vibration Soil non-homogeneity Underground tunnel

1. Introduction 1.1. Vibration from underground tunnels Railways are often placed within underground tunnels for various safety, reliability, and aesthetic purposes. However, with the ongoing development of underground railways, there are certain environmental issues that need to be considered, including ground-borne vibrations. Vibrations from railway vehicles are generated at the rail–wheel interface and can propagate through the ground into a receiving building. This can cause a perceptible vibration as well as re-radiated noise which may have a significant impact on the comfort of residents of buildings. Various studies have been performed to model ground-borne vibration from underground railways using numerical and analytical methods. Krylov [1] reported one of the first analytical models (2D)

n

Corresponding author. Tel.: +44 115 956 6492; fax: +44 115 951 3898. E-mail addresses: [email protected] (W. Yang). [email protected] (M.F.M. Hussein), [email protected] (A.M. Marshall). 1 Postal address: Department of Civil Engineering, University Park, University of Nottingham NG7 2RD. 0267-7261/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.soildyn.2013.04.004

to calculate the ground response due to a point load applied at the base of a tunnel. This analytical model only considers excitation over a very low frequency range (1–4 Hz) as the model assumes that the wave lengths of generated waves in the soil are significantly larger than the tunnel diameter so that the influence of tunnel diameter can be ignored. Coupled finite element method–boundary element method (FEM–BEM) is widely used to study ground-borne vibration. For a tunnel-structure system, a tunnel is modelled by the FEM and the surrounding soil is modelled by the BEM. Jones et al. [2] presented a 2D model based on this method to analyse the effects of tunnel linings on vibration propagation. 2D underground railway induced vibration models are limited in that they are not able to account for both longitudinal and circumferential modes simultaneously [3]. The accuracy of 2D model estimates is therefore also limited. Andersen and Jones [4] confirmed this conclusion by comparing results of 2D and 3D modelling of vibration from two types of tunnels based on the coupled FEM–BEM method. A number of assumptions were made in order to reduce the computational processing time of the 3D model (e.g. 2.5D assumption, model uniformity along tunnel axis) which allowed for the use of the discrete wave number method and significantly reduced the degrees of freedom (also used by Sheng et al. [5]).

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Forrest and Hunt [6] developed a numerical model on the basis of cylinder theory in which concentric pipes are modelled using a 2.5D approximation (pipe-in-pipe, PiP). The tunnel lining is represented by the inner pipe and the surrounding soil is modelled as a full-space soil with an infinitely long cylindrical cavity. The tunnel and soil responses due to radial [6] and tangential [7] point harmonic loads on the tunnel invert are then calculated by applying an inverse Fourier transform. Hussein et al. [8] developed a model for analysing vibration from railway tunnels in a halfspace by using the PiP model. Clouteau et al. [9] assumed that the cross-section of the tunnel-soil system along the tunnel axis direction is periodic rather than invariant so that the Floquet transform could be applied. This assumption results in a significant reduction in computational time. Employing the 2.5D assumption, Muller et al. [10] developed two models based on coupling FE and analytical methods in order to calculate vibration from underground railway tunnels (both circular and non-circular) embedded in fullspace and half-space soil. All these numerical models assumed that the soil is a homogenous material in order to simplify the problem. An alternative way of studying underground railway induced vibration is through the use of physical modelling. Unlike the various numerical models which have been reported in the literature, only a limited number of physical modelling studies have been undertaken. The 1 g experimental work presented by Tamura et al. [11] and Asano and Kumagai [12], which are discussed by Tsuno et al. [13], can be considered the earliest physical models developed to study this problem. Various 1 g experiments have been developed to study ground-borne vibrations from tunnels (e.g. [12,14,15]). The effects of loading mechanism (e.g. impulsive or harmonic [12]) as well as tunnel lining material [15] have been studied. These experiments were conducted under normal gravity conditions and therefore the usefulness of their results is limited by the fact that full-scale prototype in-situ stresses were not realistically simulated. Soil behaviour is more accurately modelled using a geotechnical centrifuge which allows for the replication of prototype-scale ground stresses and soil behaviour within small-scale models. Very few cases of centrifuge modelling of ground-borne vibration from underground tunnels exist in the literature. Tsuno et al. [13] describe centrifuge modelling of underground railway induced vibration in which an acrylic pipe was used to represent an underground tunnel. The pipe was supported by a linear guide at each end to allow free motion of the model tunnel and a small dynamic shaker was placed inside the pipe to provide excitation. This paper presents the results of a programme of geotechnical centrifuge testing and numerical modelling undertaken to study the problem of ground-borne vibration from underground tunnels. The paper provides the relevant theory in Section 2, followed by experimental (Section 3) and numerical modelling (Section 4) results, a comparison and discussion of experimental and numerical results (Section 5), and finally conclusions (Section 6).

bender element tests are commonly used [16–18]. In the field, commonly used methods include the downhole test, crosshole test, seismic cone penetration test (SCPT) and spectral analysis of surface waves (SASW) [19,20]. According to the measured data, a number of empirical equations for G0 have been proposed [21–23]. In this study, the dynamic shear modulus of the soil layer used in the centrifuge was calculated by a widely used empirical relationship proposed by Hardin and Richart [24] which is given by Eq. (1) (for round grained sand). G0 ¼ 7000

ð2:17−eÞ2 ð1 þ 2K 0 Þ0:5 ′0:5 sv 3ð1 þ eÞ

ð1Þ

where e is the void ratio of the soil, K0 is the earth pressure coefficient at rest, and s′v is the vertical effective stress. The relationship between damping and confining stress has also been studied by many researchers [25–29,30]. Seed et al. [27] concluded that at low confining stresses (less than 25 kPa), the magnitude of soil damping can be very high, even at small strain levels. Large damping ratios within the top few meters of ground have also been observed in field measurements [31]. At high stress levels, and especially at low strains, soil damping tends to be low (generally less than 1%). Rollins et al. [28] observed only a slight reduction in soil damping ratio when confining stress was increased from 50 kPa to 400 kPa for all strain levels. This paper focuses on examining the effects of the variation of soil stiffness and damping with depth on ground-borne vibration from underground tunnels. The variation of soil properties with depth is referred to as soil non-homogeneity throughout the paper.

3. Centrifuge modelling of ground-borne vibration from underground tunnels Centrifuge tests were employed in this study to ensure that the variation of dynamic soil properties with depth (increasing confining stress) was effectively modelled in the experiments. The principle of centrifuge modelling is that by decreasing the dimensions of a prototype by a factor of N, whilst at the same time increasing the body force applied to the model (generated by centrifugal acceleration) by the same scale, the stresses within the model soil will be the same as those in the corresponding prototype. The fundamental scaling factors for centrifuge tests are summarized in Table 1. The centrifuge tests were performed at an acceleration of 60 times gravity using the Nottingham Centre for Geomechanics (NCG) beam centrifuge at the University of Nottingham. The equipment To simulate the three dimensional propagation of ground-borne vibration, a cylindrical steel container with a diameter of 490 mm and a depth of 500 mm was used. Undesirable reflection of vibration waves from the rigid boundaries of the container is one of the major problems of dynamic centrifuge testing. To reduce the boundary

2. Soil non-homogeneity due to the variation of soil properties with depth The dynamic shear modulus, G0, and damping ratio are two key parameters which define the dynamic soil behaviour for the study of underground railway induced ground-borne vibration. To simplify the problem, these two parameters are commonly assumed to be constant with depth [1,2,4–10]. However, it has been found that the dynamic shear modulus and damping are a function of soil depth due to the increase in confining stress [16]. Various studies, including both laboratory tests and field measurements, have been conducted to determine the dynamic shear modulus. In the laboratory, resonance column, torsion and

Table 1 Scaling factors for centrifuge tests. Parameters Length Velocity Acceleration Stress Strain Frequency Dynamic time

Model/prototype 1/N 1 N/1 1 1 N 1/N

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Table 2 Properties of the Leighton Buzzard Fraction C silica sand [35]. Property

Value

D50 dmin dmax Specific gravity, Gs emin emax

0.5 mm 0.3 mm 0.6 mm 2.65 0.552 0.802

Cross beam Accelerometer Cable

Shaker

Load cell

effects, an energy absorbing material, Duxseal (25 mm thick) [32–34] was installed on the bottom and side walls of the container to absorb incident waves. Dry fraction C silica sand supplied by David Ball Group Plc. was used in the centrifuge tests. The properties of the sand are given in Table 2. To achieve a uniform distribution of soil density, the sand was placed in a sand hopper and pluviated through a slotted plate at a constant height above the surface of the sample. The relative density and density of the soil models were 63% and 1610 kg/m3, respectively. To simulate the excitation from an underground tunnel, an LDS type V201 electromagnetic shaker was used in the tests. The shaker worked in conjunction with a TGA type 1230 wave generator and corresponding power amplifier. A Kistler type 9001 washer-shaped dynamic load-cell was used to measure the force from the shaker and eight Wilcoxon Research type 726 piezoelectric accelerometers were employed to measure the response of the model. The measured signals were sent to Kistler type 5134 signal conditioning units which filtered and amplified the signals before they were recorded by a National Instruments (NI) USB-6259 high-speed multifunction DAQ card. To ensure the accuracy of the measurement, the sampling frequency was chosen as 20 kHz which is at least four times the maximum frequency component of the measured signal. To eliminate aliasing errors, all the measured signals were filtered by an analogue filter in the signal conditioning unit to remove unwanted higher frequency components above 10 kHz before sampling. The experiment model The main components of the model included a tunnel and the surrounding soil layer. The model tunnel was a 160 mm diameter hollow PVC cylinder with a lining thickness of 4 mm, representing a tunnel prototype with 9.6 m diameter  0.24 m thickness. The shaker was placed vertically into the tunnel to apply a vertical dynamic excitation at the centre of the tunnel invert. The shaker was installed in conjunction with the load cell and a force transmission piece (Fig. 1). A steel cross beam was made to hold the shaker inside the centrifuge container. The cross beam was fixed to two wooden supports which were cut to fit the curvature of the container. The wooden supports were also used to control the height of the shaker and model tunnel. The two ends of the tunnel were sealed to prevent soil ingress (Fig. 2). The positioning of the accelerometers is shown in Fig. 3. Three accelerometers were bolted to the model tunnel to measure the response at the tunnel invert and apex. Three accelerometers were placed at the free surface and two accelerometers were placed within the interior of the soil layer to record the soil response. The centrifuge model was spun at a speed of 179 rpm, corresponding to an angular velocity of 18.7 rad/s, and an acceleration of 60 g at a radius of 1.67 m (at one third the model height). Based on previous experiments [35], the centrifuge was run for 20 min before the vibration tests were conducted to ensure the model reached a steady state condition.

Model tunnel

Fig. 1. Shaker in the tunnel.

Duxseal Wooden support Cross beam Model tunnel

Fig. 2. The model tunnel supported in the centrifuge container.

Centrifuge test results Typical model scale measurements from a centrifuge test are shown in Fig. 4. The dynamic force applied on the tunnel invert is shown in Fig. 4(a). The vibration signal generated from the wave generator was sinusoidal, the frequency of which was continuously varied from 1200 Hz to 4800 Hz (corresponding to 20 Hz to 80 Hz in prototype scale). At lower frequencies, signal noise can be significant in the centrifuge tests. 80 Hz is normally the highest frequency which can be perceived by people as mechanical vibration of the human body [36]. Due to the fact that the dynamic force from an underground tunnel is relatively small, the behaviour of soil will usually remain within the linear range. Therefore, a small amplitude excitation was applied to the model tunnel. According to Fig. 4(a), it was found that the applied force was not constant with time. The reason for this was that the force measured by the load cell equates to the force generated in the shaker minus the inertia force required to move the drive rod. The force generated in the shaker is constant however the inertia force to move the drive rod changes with time. Therefore, the force measured from the load cell also varies with time. The time histories of the tunnel and soil response are shown in Fig. 4(b) and (c), respectively. The results show that the tunnel response is larger than the soil response. This is due to the geometric

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2

Force [N]

1

Model tunnel

ac8

0 -1

ac7

ac6

-2 0

1

2

3

4

5

6

7

8

Time [s]

Acceleration [m/s2]

5

Container wall \ ac6

ac8

ac7

2.5 0 -2.5 -5

ac4

ac2

0

1

2

3

Duxseal

4

5

6

7

8

Time [s]

5

Shaker

Model tunnel Wooden support ac1 Load cell

Acceleration [ms/2]

ac3

ac5

0

-5 0

1

2

3

4

5

6

7

8

Time [s]

Fig. 3. Arrangement of the experiment for modelling vibration from an underground tunnel (a) top view (b) cross-section acceleration view (all dimensions in mm).

decay and material damping of the model. The tunnel and soil responses are also functions of time. The variation of model response with time is due to the fact that the applied force is not constant with time and soil response changes with frequency. The measured time domain data were transferred to frequency domain and the frequency response functions (FRFs) of the model responses were calculated. The coherence functions which were used to examine the quality of the FRF measurements were also calculated. The time domain data were treated as samples from random processes and analysed using a random vibration approach to reduce the random errors. The FRFs and corresponding coherence functions were calculated by Eqs. (2) and (3): HðωÞ ¼

SFA ðωÞ SFF ðωÞ

ð2Þ

γ 2 ðωÞ ¼

SFA ðωÞSAF ðωÞ SFF ðωÞSAA ðωÞ

ð3Þ

where ω is the frequency, H(ω) is the FRF, SFF(ω) is the auto-spectrum of the force and SAA(ω) is the auto-spectrum of the response. SFA(ω)is the cross-spectrum of the force and response and SAF(ω) is the crossspectrum of the response and force [37,38]. The FRFs of the tunnel and soil responses are shown in Figs. 5 and 6, respectively. The frequency domain results are presented in prototype scale because these can be directly compared with numerical modelling results. According to the results shown in Figs. 5 and 6, it can be seen that at all the measurement locations, the values of the coherence function are close to 1. A coherence value of 1 indicates that the measured response is 100% due to the measured excitation. At the

Fig. 4. Typical time domain results, (a) applied force, (b) response at a1 and (b) response at a6 (model scale).

frequencies where the troughs appear in the corresponding FRF curves, the coherence values are generally low. The model response is also small at the troughs of the FRF curves, suggesting that the measured signals were also small and that the noise (mainly due to the centrifuge drive system, associated on-board electronics, and heavy machinery located near the centrifuge) had large effects on the measured data at these points. It is also noted that the measurement of tunnel response in Fig. 5 is generally better than the measurement of soil response in Fig. 6 (i.e. the coherence of the tunnel response is generally higher than that of the soil). Since the effect of noise is proportionately higher when the magnitude of the measured signal is low, the coherence values will be lowest when the measured response is lowest. Because the force was applied within the tunnel, the magnitude of response within the tunnel was greater than within the soil and noise had less of an effect on the measured tunnel response (resulting in higher coherence values). The intention was to keep model behaviour within the linear small strain regime (to coincide with ground-borne vibrations from underground tunnels). To check the linearity of the model, vibration signals with different amplitudes were used to excite the centrifuge model. As shown in Fig. 7, although the amplitude of the excitation signal was increased twice, the FRFs of the tunnel and soil remained the same. It is suggested, therefore, that the FRFs were insensitive to the sweep amplitude and that the tunnel and soil behaviour remained within the linear range. The consistency of data was also checked by comparing FRFs measured at symmetric locations within the model and a reasonably good match of data was found. The experiment was also repeated on reprepared models and the average results of the FRFs at each

W. Yang et al. / Soil Dynamics and Earthquake Engineering 51 (2013) 23–34

ac1

-100 -120 -140 20

40

60

-60

Vertical acceleration [dBref (m/s2/N)]

-80

-80 -100 -120

20

40

60

-80 -100 -120 -140 20

-140

80

Frequency [Hz]

80

1

1

0.8

0.8

0.8

0.6 0.4

Coherence

1

0.6 0.4 0.2

0.2 0 20

40

60

40

0 20

80

80

0.6 0.4 0.2

40

60

0 20

80

40

60

80

Frequency [Hz]

Frequency [Hz]

Frequency [Hz]

60

Frequency [Hz]

Frequency [Hz]

Coherence

Coherence

ac3

ac2

-60

Vertical acceleration [dBref (m/s2/N)]

Vertical acceleration [dBref (m/s2/N)]

-60

27

Fig. 5. FRFs of the tunnel response and corresponding coherence (prototype scale).

ac4

-100 -120

40

60

-80

-100 -120 -140 20

80

60

-80 -100 -120 -140 20

80

Frequency [Hz] 1

0.8

0.8

0.8

0.4

0.6 0.4

40

60

80

0 20

Frequency[Hz]

60

80

0.6 0.4 0.2

0.2

0.2 0 20

Coherence

1

0.6

40

Frequency [Hz]

1

Coherence

Coherence

Frequency [Hz]

40

ac8

-60

Vertical acceleration [dBref (m/s2/N)]

-80

-140 20

ac6

-60

Vertical acceleration [dBref (m/s2/N)]

Vertical acceleration [dBref (m/s2/N)]

-60

40

60

80

0 20

Frequency [Hz]

40

60

80

Frequency [Hz]

Fig. 6. FRFs of the soil response and corresponding coherence (prototype scale).

measurement point were calculated. These averaged FRFs are used to compare against the numerical simulation results.

4. Numerical modelling of the experimental tests and comparison of results Description of the numerical model A numerical model based on FLAC 3D [39] was developed to simulate the centrifuge experiments. The mesh of the FLAC model is shown in Fig. 8 and was designed to replicate, as closely as possible, the prototype scale dimensions of the centrifuge model.

A vertical cylinder was used to model the soil medium (providing a cylindrical boundary condition) with a horizontal cylinder buried within it to model the tunnel. Due to symmetry along two axes, the FLAC model was built as a quarter of the physical model. A system of reference axes was selected, with orientation as indicated in Fig. 8 and with the origin located at the point where the dynamic excitation was applied. The tunnel within the numerical model was designed to coincide with the prototype dimensions of the experimental tunnel, with an outer diameter of 9.6 m, a lining thickness of 0.24 m, and a length of 12 m. At each end of the model tunnel, the displacement in the y direction was restrained to match conditions within the physical model. Different boundary conditions were applied at the bottom

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Vertical acceleration [dBref (m/s2/N)]

(z ¼25.8 m) and side boundary (R ¼13.5 m) of the mesh to assess the potential boundary effects on the model behaviour in the centrifuge tests. The displacement of the symmetry boundaries at x ¼0 and y¼0 was restricted in the x and y directions, respectively. A harmonic load at a single frequency was applied at the tunnel invert (x¼ 0, y¼0, z ¼0). The tunnel and soil responses at the locations corresponding to the centrifuge model measurements were calculated in the time domain by the FLAC model. The amplitudes of the model responses at the imposed loading frequency were recorded after the model reached a steady state. Tunnel response, ac1 -60

sweep amplitude a1 sweep amplitude 2*a1

-80 -100 -120 -140 20

30

40

50

60

70

80

Frequency [Hz]

Vertical acceleration [dBref (m/s2/N)]

Soil response, ac4 -60 -80 -100 -120 -140

20

30

50

40

60

70

80

Frequency [Hz] Fig. 7. Tunnel and soil response under different sweep amplitude (prototype scale).

planes of symmetry Soil

Soil

15.6 m

25.8 m

y

z

Boundary effects For analytical purposes, an underground tunnel is usually considered as embedded in a soil half space. In the experimental model, however, the tunnel was placed inside a steel container with walls that may cause reflection of waves which can propagate back into the model and interact with input waves, thereby affecting measurements. Duxseal was used in the experiments to try to reduce this effect. In the numerical model, four cases were simulated by applying different boundary conditions at the bottom and side walls to examine the boundary effects. The four cases are summarized in Table 3. Quiet (or absorbing) boundaries, which are used to absorb incident waves at the boundaries of the model, were applied to simulate an infinite medium. Fixed boundary implies restraint of displacements in all three coordinate directions (x, y, z). A homogenous model was used to investigate the boundary effects in order to reduce the computing cost for this study. The parameters used in the study are given in Table 4. The average shear modulus (calculated using Eq. (1)) of the soil layer in the centrifuge tests and a 1% damping ratio (due to the fact that low amplitude soil damping is usually very small) were assigned for the homogeneous model. The material parameters for the model tunnel are for PVC-U [40]. The response of the tunnel and soil under the four boundary conditions (Table 3) are shown in Figs. 9 and 10, respectively. Two rigid boundary effects can be found by comparing the results of the FLAC model with rigid (Case 1) and quiet (Case 2) boundaries. The first effect is that a fluctuation in FRFs is observed when the fixed boundaries are applied to the FLAC model (as seen in Fig. 9(a)). The tunnel response is smoother for the quiet boundary compared to the fixed boundary. The fluctuations in the FRFs are due to the interaction between the input waves and the reflected waves generated at the boundaries. At some frequencies, constructive interference of input and reflected waves occurs, leading to an increase in the model response. At other frequencies, destructive interference is observed and the model response is reduced. The second rigid boundary effect observed is that the tunnel crown response (Fig. 9(b) and (c)) and soil response (Fig. 10) are amplified for the rigid boundary case. The tunnel and soil response Table 4 Tunnel and soil model properties used to study the boundary effects.

x

Tunnel: the outer radius of the tunnel lining is 4.8 m and the thickness of the tunnel lining is 0.24 m 12 m

By varying the frequency of the harmonic load (from 20 Hz to 80 Hz), the FRFs were calculated at the same locations as in the centrifuge experiment.

13.5 m

13.5 m

Dynamic shear modulus (MPa) Density (kg/m3) Poisson's ratio Compression wave velocity (m/s) Shear wave velocity (m/s) Damping ratio

Tunnel

Soil

1111 1400 0.35 1855 890 5%

119 1610 0.2 444 272 1%

Fig. 8. FLAC model used to simulate the centrifuge tests, prototype scale.

Table 3 Boundary conditions applied on the numerical model.

Case Case Case Case

1 2 3 4

Bottom of the model

Sides of the model

Corresponding prototype

Fixed Quiet Fixed Quiet

Fixed Quiet Quiet Fixed

Soil Soil Soil Soil

cylinder with rigid boundaries half space layer resting on a rigid bed cylinder with infinite depth but rigid boundaries on the sides

W. Yang et al. / Soil Dynamics and Earthquake Engineering 51 (2013) 23–34

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Fig. 10. The soil response for different boundary conditions. Fig. 9. The tunnel response for different boundary conditions. (a) x=7.2m y=0 z=0, (b) x=0 y=0 z=9.36m and (c) x=7.2m y=0 z=9.36m.

from the model with fixed boundaries (Case 1) are larger than the corresponding results from the model with quiet boundaries (Case 2); the differences are more than 10 dB at many frequencies. This effect of amplifying the model response was not observed at the tunnel invert (Fig. 9(a)). The reason for this could be that the tunnel response at this point is relatively large (according to the FLAC model result with quiet boundary). This point is closest to the source (and relatively far from the boundaries), therefore the response is due mainly to the source and not significantly affected by reflected waves. By comparing the results from all the four cases shown in Figs. 9 and 10, it is also noted that the results of Cases 1 and 4 are closer compared to the results of Cases 1 and 3. This observation is clearer in the frequency range from 35 Hz to 60 Hz. Furthermore, the model response of Case 4 is larger than that of Case 3 at the majority of frequencies. These observations suggest that the reflection from the side walls more significantly influences the tunnel and soil response. The amplification of the model response is mainly due to the existence of the rigid side walls. This observation could be due to two reasons. The first is that most of the observation points are closer to the side walls than the bottom boundary. The second reason could be the geometry of the model. The effect of reflected waves from the rigid base could be reduced by the interruption of the transmission path due to the existence of the model tunnel. However, at a few frequencies both the results from Cases 3 and 4 are close but clearly smaller than the results of Case 1. This implies that the amplification of model results at these frequencies could be due to the combined effects from the rigid bottom and side walls. Therefore, the effects of a rigid base should not be ignored. These results show that the tunnel and soil response can be significantly affected by the boundary conditions. The two extreme

cases, the fixed boundary Case 1 and quiet boundary Case 2, are used to compare with experimental results in the following analysis.

5. Non-homogeneity effects In order to study the soil non-homogeneity effects on vibration from underground tunnels, the experimental results were compared with numerical results using homogeneous and nonhomogenous cases. The numerical model is shown in Fig. 8. For the homogenous case, constant shear modulus and damping were applied. For the non-homogenous case, the model was divided into 86 layers with a uniform depth of 0.3 m. The dynamic soil properties in each soil layer of the model were constant but varied from one layer to the next. 5.1. Homogenous model and centrifuge results Two sets of representative soil parameters were used for the homogenous model. Both sets of parameters assumed a small soil material damping ratio of 1% due to the fact that low amplitude soil damping is usually very small [27]. For the dynamic shear modulus, the first set of parameters, P1, used the average shear modulus of the whole soil layer in the centrifuge tests (prototype scale). The second set, P2, employed the average shear modulus of the soil layer between the surface and the tunnel invert, as shown in Fig. 11. The average dynamic shear modulus, G0, was calculated using Eq. (1). Poisson's ratio was taken as 0.2 for the numerical models which is a typical value for dry sand. The material properties of the model tunnel were for PVC, as shown in Table 4. The parameters used for the homogenous soil are summarized in Table 5. Both fixed and

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W. Yang et al. / Soil Dynamics and Earthquake Engineering 51 (2013) 23–34

G0

Distribution of G0with depth

G0

z

z

Average G0 from the surface to the tunnel invert, P2

Average G0 of the soil layer, P1

Fig. 11. The distribution of dynamic shear modulus for the homogenous model P1 and P2.

Table 5 Soil material properties for the homogeneous model.

Dynamic shear modulus (MPa) Density (kg/m3) Poisson's ratio Compression wave velocity (m/s) Shear wave velocity (m/s) damping ratio

Soil (P1)

Soil (P2)

119 1610 0.2 444 272 1%

106 1610 0.2 419 256 1%

quiet boundary conditions (Cases 1 and 2 from Table 3) were applied to the side and bottom boundaries of the numerical model. The comparison between experimental and numerical results using parameter sets P1 and P2 are shown in Figs. 12 and 13. To measure the difference of the FRFs calculated from the experimental data and numerical results, two commonly used distance measures, the RMS and COSH distances [41,42], were applied to the results. The RMS and COSH distances were calculated using Eqs. (4) and (5), respectively:   1 N H exp ðωÞ 2 ∑ log Nm¼1 H num ðωÞ

Drms ¼

   H exp ðωÞ H exp ðωÞ 1 N −log ∑ H num ðωÞ 2N m ¼ 1 H num ðωÞ    H num ðωÞ H num ðωÞ −log −2 þ H exp ðωÞ H exp ðωÞ

ð4Þ

Dcosh ¼

ð5Þ

where H exp ðωÞand Hnum(ω) are the experimental and numerical FRFs and N is the number of frequencies in the FRF. The results are given in Tables 6 and 7. According to the results, it was found that better estimates (smaller values of RMS and COSH distances) of the model response were obtained at most of the locations when the fixed boundary condition was applied in the FLAC model. This suggests that the boundary condition of the model in the centrifuge tests was closer to being fixed rather than quiet. This means that the rigid boundary effect was still observed even though Duxseal was used in the tests. Therefore, in the following discussion, the focus is placed on the comparison of experimental results with the fixed boundary numerical results. According to the results in Fig. 12, it was also observed that the tunnel response from the homogeneous model (with fixed boundary) is reasonably close to the experimental results. The distance measures clearly show that the estimates of the tunnel response

Fig. 12. Comparison of experiment and homogeneous numerical model results, tunnel response (prototype scale).

from the numerical model were better than the estimates of the soil response. The values of RMS and COSH distances of tunnel response are less than half of the values of soil response at most of the locations. This observation is due to two reasons. Firstly, the tunnel response depends on the properties of both the tunnel and the surrounding soil; therefore the effect of soil properties on the tunnel response is less significant than for the soil response. Secondly, the soil parameters used for the homogeneous model can be regarded as generally good estimates of the soil properties around the tunnel. These results, therefore, suggest that a homogeneous model can give a reasonably good prediction of tunnel response if appropriate tunnel and soil parameters are used. The results of Fig. 13 show that the soil response calculated from both the P1 and P2 homogeneous models have poor agreement with the centrifuge results. The numerical predictions (with fixed boundary) usually overestimate the soil response. The differences are more than 10 dB at many frequencies. This suggests that the homogeneous model with an average distribution of dynamic shear modulus and a low value of damping ratio (which is a commonly used method for practical problems), is incapable of providing a good estimate of the soil behaviour in the frequency range of interest. 5.2. Non-homogenous model and centrifuge results The assumed distributions of soil stiffness (in terms of compression wave and shear wave velocities) and material damping ratio with depth for the non-homogeneous model are shown in Fig. 14. This is the same distribution of soil properties used in [35] which gave good estimates of soil behaviour when studying

W. Yang et al. / Soil Dynamics and Earthquake Engineering 51 (2013) 23–34

ground-borne vibration from a surface source. The tunnel parameters are the same as the previous homogeneous model given in Table 5. Both fixed and quiet boundaries (Cases 1 and 2 in Table 3) are applied to the model. The comparison of centrifuge and non-homogeneous numerical model results for the tunnel and soil are shown in Figs. 15 and 16, respectively. The corresponding RMS and COSH distances of the FRFs of the non-homogenous and experimental results were calculated and are given in Tables 8 and 9. Similar to the comparison of experimental and numerical results from the homogenous models, it was found that the tunnel and soil responses from the numerical model with fixed boundaries agreed better with the experimental results than the numerical model with quiet boundaries. The RMS and COSH distances for the experiment versus non-homogenous model with fixed boundaries are significantly smaller than the distances for the experiment versus non-homogenous model with quiet boundaries. This again suggests that even with the use of Duxseal

31

in the experiment, the boundary condition in the centrifuge tests was closer to being fixed than quiet. According to the results shown in Fig. 15, the experimental and numerical tunnel response is reasonably close at most frequencies; the differences are smaller than 8 dB in the plots of FRF. Only at a few frequencies the differences are significant (more than 15 dB). The large differences observed could be due to the existence of the model tunnel. The distribution of the confining stress (and the assumed distribution of shear wave velocities in Fig. 14) does not consider the effect of the existence of the tunnel. Therefore, the parameters used in the FLAC model, which are based on the results from the previous soil layer tests with no tunnel [35] may not be accurate. By comparing the results of Figs. 12 and 15, it may be observed that the agreement between the tunnel response of the experimental data and non-homogeneous numerical model is similar to the agreement of the experimental data and homogeneous numerical model results. The distance values of tunnel response from experimental versus homogenous (fixed boundary) and experimental versus non-homogenous (fixed boundary) are similar. This observation is due to the fact that the tunnel response depends on both tunnel and soil properties. The soil nonhomogeneity effects have less of an impact on tunnel response. By comparing the results shown in Figs. 13 and 16, an improvement of the prediction of soil response can be observed compared to the non-homogeneous model with fixed boundary. This conclusion can also be drawn by examining the results of the distance measures of the FRFs. It can be seen that the RMS and COSH distances of the FRFs of the experimental versus non-homogeneous with fixed boundaries cases are about half of the distances of experimental versus homogeneous cases (Tables 8 and 9). This demonstrates that in order to improve the estimates of soil response from the numerical model, it is important to account for the variation of dynamic soil properties with depth.

6. Conclusions An experimental and numerical study of vibration from underground tunnels was performed. The intent of the investigations was to provide insight into the soil non-homogeneity effects on the propagation of vibrations from tunnels. The investigation successfully highlighted important issues related to the modelling of vibrations from underground tunnels, however, appropriate judgement should be taken when extending this work to fullscale prototype scenarios due to the effect of boundary conditions on results (which would not apply to the prototype). According to the results obtained from the centrifuge tests and FLAC simulations for modelling vibration from an underground tunnel, the following conclusions can be drawn:

 Centrifuge tests have been successfully conducted to study Fig. 13. Comparison of experiment and homogeneous numerical model results, soil (prototype scale).

ground-borne vibration induced from an underground tunnel. Due to limitations on the size of model container and the

Table 6 RMS distances of experimental and numerical results, homogenous model, at all the locations for the tunnel models (calculated from Eq. (4)). P1 fixed

P1 quiet

P2 fixed

P2 quiet

Tunnel response

x ¼7.2 m y¼ 0 z ¼ 0 x ¼0 y¼0 z¼ 9.36 m x ¼7.2 m y¼ 0 z ¼ 9.36 m

0.1879 0.1078 0.1563

0.1265 0.4334 0.3935

0.1498 0.1172 0.1997

0.1072 0.4970 0.4191

Soil response

x ¼7.2 m y¼ 0 z ¼ 15 m x ¼7.2 m y¼ 0 z ¼ 20.4 m x ¼0 y¼7 m z ¼ 20.4 m

0.3677 0.3128 0.2559

0.4227 0.4313 0.4341

0.1937 0.1975 0.2632

0.3367 0.2912 0.3944

32

W. Yang et al. / Soil Dynamics and Earthquake Engineering 51 (2013) 23–34

Table 7 COSH distances of experimental and numerical results, homogenous model, at all the locations for the tunnel models (calculated from Eq. (5)). P1 fixed

P1 quiet

P2 fixed

P2 quiet

Tunnel response

x ¼7.2 m y¼ 0 z ¼ 0 x ¼0 y¼ 0 z ¼9.36 m x ¼7.2 m y¼ 0 z ¼ 9.36 m

0.5821 0.3182 0.4921

0.3295 1.5460 1.4365

0.4498 0.3359 0.7124

0.2991 1.9862 1.7556

Soil response

x ¼7.2 m y¼ 0 z ¼ 15 m x ¼7.2 m y¼ 0 z ¼ 20.4 m x ¼0 y¼ 7.2 m z ¼ 20.4 m

1.3098 1.0110 0.8131

2.1689 1.8046 1.5995

0.6269 0.6366 0.8912

1.4040 1.0340 1.4127

Compression wave

Shear wave

Fig. 14. Distribution of dynamic soil properties with depth.

Fig. 15. Comparison of experiment and non-homogeneous numerical model results, tunnel response.

Fig. 16. Comparison of experiment and non-homogeneous numerical model results, soil responses.

W. Yang et al. / Soil Dynamics and Earthquake Engineering 51 (2013) 23–34

33

Table 8 RMS distances of experimental and numerical results, non-homogenous model, at all the locations for the tunnel models (calculated from Eq. (4)). P1 fixed

P2 fixed

Non-hom fixed

Non- hom quiet

Tunnel response

x ¼ 7.2 m y ¼0 z¼ 0 x ¼ 0 y¼ 0 z ¼9.36 m x ¼ 7.2 m y ¼0 z¼ 9.36 m

0.1879 0.1078 0.1563

0.1498 0.1172 0.1997

0.1390 0.1220 0.2105

0.1687 0.5531 0.4951

Soil response

x ¼ 7.2 m y ¼0 z¼ 15 m x ¼ 7.2 m y ¼0 z¼ 20.4 m x ¼ 0 y¼ 7 m z¼ 20.4 m

0.3677 0.3128 0.2559

0.1937 0.1975 0.2632

0.1115 0.1072 0.1583

0.3677 1.1592 0.4949

Table 9 COSH distances of experimental and numerical results, non-homogenous model, at all the locations for the tunnel models (calculated from Eq. (5)). P1 fixed

P2 fixed

Non-hom fixed

Non-hom quiet

Tunnel response

x ¼7.2 m y¼ 0 z ¼ 0 x ¼0 y¼ 0 z ¼9.36 m x ¼7.2 m y¼ 0 z ¼ 9.36 m

0.5821 0.3182 0.4921

0.4498 0.3359 0.7124

0.4323 0.4083 0.6513

0.5245 2.2068 1.9286

Soil response

x ¼7.2 m y¼ 0 z ¼ 15 m x ¼7.2 m y¼ 0 z ¼ 20.4 m x ¼0 y¼ 7.2 m z ¼ 20.4 m

1.3098 1.0110 0.8131

0.6269 0.6366 0.8912

0.3442 0.3292 0.4684

1.3085 6.5365 1.9953









dynamic shaker, the model boundaries had an effect on experimental results and a full half-space scenario was not replicated. The model did, however, provide consistent and reliable results. A FLAC 3D numerical model was built to simulate the centrifuge tests for modelling of ground-borne vibration from an underground tunnel. The numerical model was developed to replicate, as closely as possible, the prototype scale dimensions of the physical model. A vertical cylinder was used to model the soil medium with an associated cylindrical boundary condition. Different boundary conditions were applied to the numerical model to assess the potential boundary effects on the experimental results. The numerical simulation results showed that the rigid boundary had two main effects on the FRFs at the observation points. First, the rigid boundaries caused fluctuations of the FRFs in the frequency range considered. Second, the rigid boundary considerably increased the model response. The numerical results also demonstrated that the rigid side walls had more significant effects on the model response compared to the rigid bottom. The FLAC model with different boundary conditions was compared to the experimental results. The comparisons, using the RMS and COSH distances of the experimental and numerical FRFs, showed better agreement when the rigid boundary was used in the numerical model. This suggests that even though the standard material, Duxseal, was used in the centrifuge experiments, the rigid boundary of the centrifuge container still had clear effects on the experiment results. The numerical results from the homogeneous and nonhomogeneous models were compared to the experimental results in order to study the effect of soil non-homogeneity on the tunnel and soil response. The comparison between the homogeneous numerical results and experimental results suggests that the homogeneous model can give acceptable estimates of the tunnel behaviour if proper parameters of soil (average dynamic shear modulus and small material damping) are used in the model. However, the soil behaviour could not be accurately predicted by the homogeneous model. Improved estimates (based on a reduction of the RMS and COSH distances) of soil behaviour by the numerical model was observed when the non-homogeneous model was used. This illustrates the importance of accounting for the variation of

dynamic soil properties with depth when studying groundborne vibration from underground tunnels.

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