Field experiment and numerical study on active vibration isolation by horizontal blocks in layered ground under vertical loading

Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

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

Field experiment and numerical study on active vibration isolation by horizontal blocks in layered ground under vertical loading Guangyun Gao a,b, Ning Li c, Xiaoqiang Gu a,b,n a

Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China c Shanghai Geotechnical Investigations and Design Institute Co., Limited, Shanghai 200032, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 15 July 2014 Received in revised form 8 November 2014 Accepted 12 November 2014

In this paper, a series of field experiments were carried out to investigate the active vibration isolation for a surface foundation using horizontal wave impedance block (WIB) in a multilayered ground under vertical excitations. The velocity amplitude of ground vibration was measured and the root-mean-square (RMS) velocity is used to evaluate the vibration mitigation effect of the WIB. The influences of the size, the embedded depth and the shear modulus of the WIB on the vibration mitigation were also systematically examined under different loading conditions. The experimental results convincingly indicate that WIB is effective to reduce the ground vibration, especially at high excitation frequencies. The vibration mitigation effect of the WIB would be improved when its size and shear modulus increase or the embedded depth decreases. The results also showed that the WIB may amplify rather than reduce the ground vibration when its shear modulus is smaller or the embedded depth is larger than a threshold value. Meanwhile, an improved 3D semi-analytical boundary element method (BEM) combined with a thin layer method (TLM) was proposed to account for the rectangular shape of the used WIB and the laminated characteristics of the actual ground condition in analyzing the vibration mitigation of machine foundations. Comparisons between the field experiments and the numerical analyses were also made to validate the proposed BEM. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Ground vibration Vibration isolation Layered soils Wave impedance block Field experiment Boundary element method Thin layer method

1. Introduction Man-made vibrations induced by machines, traffic, blasting, pile driving, etc. have become public nuisance annoying nearby structures, underground pipelines, sensitive electronic equipment and inhabitants [1–4]. To eliminate or reduce such disturbances, several kinds of countermeasures have been proposed in the past. The first kind of countermeasure is to reduce the vibration magnitude of the source, such as adjusting the frequency contents of the source, changing the location and direction of the source or installing dampers under the source. The second kind of countermeasure is to limit the magnitude of vibrations input into the structures, such as increasing the damping of the structure by installing additional dampers or other base-isolation systems. Beside the above two kinds, another possible kind of countermeasure is to modify the dynamic transmitting behavior of local subsoil via wave diffraction and scattering by installing wave barriers at suitable locations in the wave propagation path

n Corresponding author at: Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China. Tel.: þ86 21 65984551. E-mail address: [email protected] (X. Gu).

http://dx.doi.org/10.1016/j.soildyn.2014.11.006 0267-7261/& 2014 Elsevier Ltd. All rights reserved.

between the source and the structure. The commonly used wave barriers include open or in-filled trenches [5–11], rows of solid piles [12–16] or tubular piles [17], and gas mattresses [18,19]. For barriers installed near the vibration source, they are usually classified as active or near field vibration isolation. On the other hand, for barriers installed far from the source and close to the structure, they are classified as passive or far-field vibration isolation. One common characteristic of these wave barriers is that their dimension in the vertical direction (or depth) is significantly larger than those in the other directions. It is well recognized that the depth of the above barriers is the most crucial factor influencing the wave mitigation effect. Generally, depth of one Rayleigh wavelength at least is required in order to achieve a satisfactory vibration isolation effect. However, the Rayleigh wavelength generated by the man-made low frequency vibration can be very long. Therefore, in such cases, it is difficult to implement successfully in practical constructions due to soil instability, underground water, foundation requirements, cost and other issues. Except for the above vertically installed wave barriers, another promising horizontally installed wave barrier called wave impedance block (WIB) has drawn increasing attentions in vibration isolation of machine foundations, as shown in Fig. 1. The principle

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G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

P Surface foot h t

d

WIB

Layered ground

Fig. 1. Schematic of WIB in layered ground for vibration isolation.

of the WIB is to modify the wave propagation regime of the ground by introducing an artificial horizontal stiffened layer. In other words, the wave transmission regime in the ground depends on the relationship between the excitation frequency of the source and the cut-off frequency of the overlaying soil above the WIB. When the exciting frequency is higher than the cut-off frequency, wave will transmit. Otherwise, wave transmission will be prevented [20,21]. In the past, theoretical investigations on WIB have been carried out with numerical methods such as boundary element method (BEM), finite element method (FEM) as well as their combination, and concentrated on its impeding effect on the vibration exerted by foundations. Chouw et al. [22] first analyzed the active and passive vibration isolation using WIB with a 2D BEM in the frequency domain. Their findings indicated that increasing stiffness of WIB improves the screening effectiveness. Moreover, WIB does better in impeding low frequency vibration than in-filled trench. Following the pioneer work by Chouw et al. [22], Takemiya and Jiang [23] investigated the vibration isolation effects of WIB under loadings exerted by pile group embedded in a homogenous ground with FEM. Takemiya and Fujiwara [24] further explored the performance of WIB for vibrations due to sudden excitations using a 2D BEM in the time domain. Stamos et al. [25] investigated the vibration isolation effect of an open or filled trench and a stiff plate embedded in the soil between the road surface and the tunnel for a road-tunnel traffic system using BEM in frequency domain and FEM–BEM in time domain under plain strain conditions. Antes and von Estorff [26] investigated the isolation effects of WIB with varying stiffness using 3D BEM in the frequency domain and a combination of 2D BEM and FEM in the time domain. After that, several researchers began to introduce layered ground into WIB study, instead of homogenous ground. For example, Peplow et al. [27,28] employed the boundary integral equation method to study the active screening effectiveness of WIB in a double-layered ground. More recently, Xu et al. [29] invoked an integral equation and transfer-matrix formulation to study the wave mitigation effects of 2D WIB for the vibration due to a harmonic strip load on the stratified subgrade surface. With the rapid development of high-speed rail, subway and light rail, many researchers began to exploit WIB to screen the vibrations induced by tracks. Madshus et al. [30] and Kaynia et al. [31] studied the vibration induced by the high-speed railway and analyzed the vibration isolation efficiency of a WIB-like stiff plate buried beneath the ballast using FEM incorporated with a horizontal non-reflect boundary in the frequency domain and time domain. Andersen and Jones [32] adopted combined FEM–BEM to study the effect of grouting on the isolation of track vibrations in tunnels, in which the stiffened soil worked as a WIB. Peplow and Finnveden [33] investigated the isolation effects in the near field by WIB against vibrations below 200 Hz with a 2D spectral FEM.

More recently, Li et al. [34] compared the vibration isolation effects of entity WIB, honeycomb WIB, open trench and diaphragm wall against the low frequency vibrations induced by high-speed railway. However, in the previous theoretical studies, little attention has been paid to the influences of WIB geometrical, material properties and embedded depth on the wave mitigation effect in a multilayered ground. Therefore, Gao et al. [35–37] proposed a 2D plain-strain model and a 3D axisymmetric model with the aid of a semi-analytical BEM integrated with the fundamental solution derived by the thin layer method (TLM). Based on these two models, the influences of the WIB geometrical and material properties including diameter, shear modulus and embedded depth on the screening effectiveness in a double-layered ground and a Gibson-type ground were extensively investigated and several useful conclusions were drawn. Strictly, however, the 2D model and the 3D axisymmetric model are appropriated only for strip WIB and circular WIB, respectively. Nevertheless, the actual shape of WIB is usually rectangular and it is a general 3D problem, especially for the vibration isolation of machine foundations. Moreover, the actual ground was usually multilayered, instead of assumed homogeneous or double-layered. Most importantly, these theoretical studies usually were not verified by experiments. So far, though considerable theoretical achievements in WIB research have been made, very limited experimental works have been performed to investigate the vibration reduction effect of WIB and validate these theoretical findings. Kratzig and Niemann [38] reported the field tests of using horizontal WIB as an active or passive vibration isolation element for the surface foundation subjected to harmonic vertical excitations. The test results showed that the WIB can effectively reduce the vibration amplitude of the foundation, especially the vertical component. They also showed that WIB was effective to reduce the vibration in centrifuge model tests and holographic interferometry tests. Takemiya et al. [39] used the stiff honeycomb soil-columns formed by the soil improvement techniques under a road as a horizontal placed honeycomb WIB (HWIB) and studied its vibration isolation effect. Their results indicated that the WIB can reduce the traffic induced vibration. Besides the horizontal placed WIB, Takemiya [40] also successfully applied the HWIB which was vertically installed near the pile foundations of a high-speed train viaduct for vibration mitigation. Furthermore, he extended the HWIB by filling the precast honeycomb cells with tire shreds [41–44], which combines the wave scattering effect by stiff cell walls and the energy dissipation by fill-in tire shreds, and investigated the vibration mitigation efficiency under different load conditions by a series of field tests. Nevertheless, little attention was paid to the influences of WIB geometry, material properties (i.e. stiffness), embedded depth and the characteristic of the vibration (i.e. frequency) on the vibration isolation in these tests, although these factors will definitely affect the vibration isolation effect and the design of WIB in practice. In this paper, a series of field experiments were carried out to investigate the vibration isolation effect of a horizontal placed rectangular WIB in multilayered ground under vertical excitation induced by a vibrator. The amplitude of vibration at different distances to the source was measured to evaluate the degree of vibration reduction. The effects of the WIB geometry, the embedded depth and the shear modulus of the WIB on the vibration reduction were also systematically examined under different loading conditions. Meanwhile, an improved 3D semianalytical BEM combined with TLM was proposed to account for the rectangular shape of the used WIB and the laminated characteristics of the actual ground condition. Furthermore, comparisons between the field experiments and the numerical simulation were made to validate the proposed BEM and possible reasons were also given to explain the observed discrepancy.

G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

y

Table 1 Soil profile and properties.

4 5 6

Fill Clayey silt 2.5–9.0 Sandy silt 9.0–15.5 Silt 15.5–25.5 Sandy silt 25.5–30.0 Silty clay

Density (kg/m3)

Shear wave velocity (m/s)

Poisson's ratio

1.6 0.9

1830 1939

85 90

0.40 0.40

6.5

1888

100

0.40

6.5 10.0

1898 1837

160 130

0.45 0.45

4.5

1847

170

0.45

WIB

x d z

Electric vibrator Measuring point

Surface footing

200

3

0.0–1.6 1.6–2.5

Thickness (m)

b/2 b/2 200 200

1 2

Soil type

a/2 a/2 200 200

0.000

x

th

No. Depth (m)

253

WIB

500

500

500

2. Test site, equipments and program

500 500 4000

500

500

500

Fig. 2. Layout of the field test.

2.1. Test site The test site is located at the coastal plain of Shanghai city, China. According to the cone penetration test and other investigations, the top 30 m soils are classified as Quaternary Holocene sedimentary layers and consist of fill, clayey silt, sandy silt, silt and silty clay. The profile, unit weight and shear wave velocity of the soils are listed in Table 1. The background vibrations of the test site are measured for 24 h in the latitudinal, longitudinal and vertical directions. It is found that the vibration levels in the two horizontal directions are nearly the same as expected. It is also found that the vibration of the background is one or two orders of magnitude lower than that induced by the vibrator. Therefore, the background vibration is ignored in the following data analyses. 2.2. Test equipments and program A set of Continue Surface Wave System (CSWS) manufactured by GDS is employed for the field tests. The CSWS comprises an electric vibrator, a surface wave system control unit, a surface wave system drive unit, 6 geophones for in-situ shear wave velocity measurement and a power generator. The electric vibrator can generate harmonic waves from 10 Hz to 100 Hz with an interval of 5 Hz and supply a periodic vertical loading with a capacity of 489 N. Such harmonic waves are used to simulate the waves from common industrial machine operations. The horizontal and vertical ground vibrations are measured by two velocity transducers at each measuring point. Then the signals of the velocity transducer are recorded by a data logger and stored in a PC for further analyses. The sampling frequency of the data logger is 256 Hz. Fig. 2 shows the schematic diagram of the field test. The electric vibrator is fixed on a concrete block (i.e. footing) which is located on the ground surface. The length (x direction), width (y direction) and height (z direction) of the concrete block are 400 mm, 400 mm and 200 mm, respectively. The WIB is buried at a certain depth h under the block. The length, width and thickness of the WIB are denoted as l1, l2 and t, respectively. Note that the geometric centers of the vibrator, concrete block and WIB are coincided in the vertical direction. A total of 8 measuring points with 0.5 m interval are positioned along the measuring line (i.e. x axis). Table 2 lists the detail of the test program in this study. A total of 6 experimental cases were carried out in the field tests to investigate the influences of WIB size, shear modulus of the WIB material and embedded depth on the vibration isolation efficiency. The free field ground response (i.e. without WIB) was also measured for better comparisons.

Table 2 Experimental cases for the field test. WIB size a  b  t Depth Material Density (mm) h (mm) (kg/m3)

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Shear modulus (MPa)

Poisson's ratio

600  600  50

50

Concrete 2500

1.25  104

0.20

1000  700  50

50

Concrete 2500

1.25  104

0.20

710  710  25

50

Wood

654.8

0.26

710  710  25

50

Steel

7850

7.86  104

0.31

Concrete 2500

1.25  104

0.20

Concrete 2500

4

0.20

700  1000  50 200 700  1000  50 400

713

1.25  10

3. Test results and discussions Fig. 3 shows the vertical velocity-time histories at different measuring points in the tests with and without WIB. As seen in Fig. 3, in each case, the amplitude of the velocity decreases sharply as the distance between the source and the measurement increases due to the damping effect. Meanwhile, at the same measuring point, the amplitude of the velocity is considerately smaller in the test with WIB than that without WIB. It convincingly illustrates the effect of WIB on vibration mitigation. To quantify the vibration level, the root-mean-square (RMS) velocity is calculated and treated as the representative velocity response at the measuring point. To calculate the RMS velocity, first the velocity-time history sections for each exciting frequency are subtracted from the original data with band-pass filter. Then discrete Fourier transformation DFT is used to analyze the filtered velocity-time history and the RMS velocity is obtained according to the following equations: N1

XðkÞ ¼ ∑ xðnÞe  jð2n=NÞnk n¼0

Akms ¼

pffiffiffi  2 XðkÞ N

ð1Þ

ð2Þ

where N is the total number of sample points, x(n) refers to the velocity of the nth sample point, and Akrms means the RMS velocity in the frequency domain. The effect of WIB on the vibration mitigation can be reasonably described by the RMS velocity reduction ratio AR at each measuring

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G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

Vertical velocity (m/s)

-3

d = 0.5 m

0

-5

0

1

2

5

Vertical velocity (m/s)

x 10

5

-3

d = 0.5 m 0

-5

3

x 10

0

1

t (s)

Vertical velocity (m/s)

-3

5

d = 2.0 m 0

-5

0

1

2

Vertical velocity (m/s)

x 10

5

-3

0

0

1

-3

5

d = 3.5 m 0

-5

0

1

2

3

t (s)

2

Vertical velocity (m/s)

Vertical velocity (m/s)

x 10

3

d = 2.0 m

-5

3

x 10

t (s)

5

2

t (s)

-3

d = 3.5 m 0

-5

3

x 10

0

1

t (s)

2

3

t (s)

Fig. 3. Measured velocities at different locations (a) without WIB; (b) with WIB.

3.1. Effect of WIB size

point, which is given by [5] AR ¼

Amplitude with WIB Amplitude without WIB

ð3Þ

where the numerator and the denominator respectively refer to the RMS velocity before and after the WIB installation. The average amplitude reduction ratio AR [17] can be used to illustrate the overall vibration mitigation effect of the WIB and it can be obtained by AR ¼

1 l

Z AR dl

ð4Þ

where l is the length of measuring line. For the cases listed in Table 2, a considerable amount of the ground response data were recorded. For simplicity, only partial and represented results are analyzed and discussed in the following to illustrate the influences of WIB size, shear modulus and embedded depth on the mitigation of ground vibration. In order to enhance the universality of the conclusions, the word ‘longitudinal’ will be replaced by ‘radial’ in the following.

Herein, the plane size of the rectangular WIB is characterized by an equivalent diameter De as De ¼ 2ðab=π Þ0:5

ð5Þ

where a and b are the length and width of the rectangular WIB, respectively. The De values of the WIB are 677 mm, 944 mm and 801 mm in case 1, cases 2, 5 and 6, and cases 3 and 4, respectively. Fig. 4 compares the AR values of the radial and vertical velocities at each measuring point against its distance to the vibration source at a low excitation frequency (10 Hz) in case 1 and case 2, together with the AR value. As seen in Fig. 4, the AR value generally increases with increasing distance from the source, although some fluctuation exists. The AR value is about 0.2 for the measuring point at 0.5 m from the source, which means that about 80% vibrations are eliminated at this point by the installed WIB. The result convincingly illustrates that the WIB has the advantages to reduce the low frequency vibrations over an open or in-filled trench and a row(s) of piles in the near field. Meanwhile, the AR value is always less than 1.0 for all the measuring points. It indicates that the WIB plays a positive role in reducing the ground

G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

3

AR = 0.81 for case 1 1.0

0.8 0.6 0.4

AR = 0.63 for case 2 3

0.2

case 1: 600 × 600×50 mm

3

case 2: 1000×700×50 mm

AR for radial velocity

AR for radial velocity

1.0

0.6 AR = 0.66 for case 2 3

Case1: 600 × 600×50 mm

3

Case2: 1000×700×50 mm

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m) Fig. 4. Effect of WIB size on AR of (a) radial velocity and (b) vertical velocity for cases 1 and 2 at an excitation frequency of 10 Hz.

vibration, which agrees well with the experimental results in Kratzig and Niemann [38] and Takemiya et al. [39]. Fig. 5 shows the results at a medium excitation frequency (60 Hz) in case 1 and case 2. Compared with Fig. 4, it seems that the vibration mitigation effect of the WIB is more effective at the medium excitation frequency than that at the low excitation frequency. For example, regarding the radial vibration, the AR value is 0.24 at the excitation frequency of 60 Hz, much smaller than the value of 0.63 at the excitation frequency of 10 Hz. As seen in Figs. 4 and 5, the AR curves for the small WIB are generally higher than those for the large one, especially for the radial velocity. It is consistent with the theoretical finding that the WIB with larger diameter can impede more vibrations while the other WIB parameters keep constant [35,36]. Note that the difference between the two AR curves of the vertical velocity with different WIB sizes at the excitation of 10 Hz is minor, as shown in Fig. 4(b). This exceptional result may attribute to the reason that the difference of the WIB sizes cannot cause apparent difference in the AR curves because of the long Rayleigh wavelength (about 8.5 m) at the low frequency of 10 Hz. Moreover, it seems that the radial velocity is more sensitive to the WIB sizes than the vertical velocity, particularly at the excitation frequency of 60 Hz. Therefore, increasing the WIB size would be an effective way to reduce the radial vibration in practice. 3.2. Effect of material shear modulus Kratzig and Niemann [38] showed that resonant frequency of the surface foundation system increases as the WIB stiffness increases. Theoretical investigations [35,36] also indicated that the material shear modulus of the WIB is one of the most crucial factors affecting the vibration mitigation efficiency. Herein, two kinds of material

AR for vertical velocity

AR for vertical velocity

AR = 0.67 for case 1 0.8

AR = 0.46 for case 1

0.6 0.4 0.2

1.0

1.0

0.2

0.8

Case 1: 600 × 600×50 mm 3 Case 2: 1000×700×50 mm

AR = 0.24 for case 2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

0.4

255

0.8

3

Case 1: 600 × 600×50 mm 3 Case 2: 1000×700×50 mm AR = 0.47 for case 1

0.6 0.4 0.2 AR = 0.34 for case 2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

Fig. 5. Effect of WIB size on AR of (a) radial velocity and (b) vertical velocity for cases 1 and 2 at an excitation frequency of 60 Hz.

(i.e. wood in case 3 and steel in case 4) are used to experimentally investigate the effect of shear modulus. As seen in Table 2, the shear modulus of the steel is significantly larger than that of the wood. Fig. 6 compares the AR and AR values of the radial and vertical velocities in case 3 and case 4 at an excitation frequency of 30 Hz. As seen in Fig. 6, the AR values at all the measuring points in the test with steel WIB are much smaller than those with wood WIB, indicating WIB made of stiffer material may have better vibration mitigation effect. The AR values for the steel WIB are 0.57 and 0.47 in the radial and vertical directions, respectively, indicating 43% and 53% of the average ground vibrations in the radial and vertical directions are reduced by the steel WIB at the excitation frequency of 30 Hz. On the other hand, the AR values for the wood WIB are 1.46 in the radial direction and 0.97 in the vertical direction. Interestingly, due to its small shear modulus, the wood WIB amplifies the radial vibration by 140% at the measuring point with 1.0 m distance to the source and by 46% in average. These results indicate that improper design of the WIB may increase the vibration, rather than reduce the vibration. It agrees well with the theoretical findings that the ground vibration may be amplified rather than reduced when the shear modulus of WIB is below a threshold value [35,36]. Furthermore, the results in Fig. 6 also indicate that the WIB is more efficient in reducing the vertical vibration than the radial vibration at the excitation frequency of 30 Hz. From above analyses, a conclusion can be made that increasing the WIB shear modulus is an effective measure to improve the vibration mitigation effect, especially in the vertical direction. 3.3. Effect of embedded depth Fig. 7 compares the AR and AR values of radial and vertical velocities for WIB installed at different embedded depths (i.e. case 5 and case 6) at an excitation frequency of 35 Hz. As seen in Fig. 7,

G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

2.5 2.0

Case 5: embedded depth = 20 cm Case 6: embedded depth = 40 cm

Case 4: steel WIB

AR = 1.46 for case 3

1.5 1.0

2.0

Case 3: wood WIB

AR = 0.57 for case 4

0.5

1.5

AR = 1.05 for case 6

1.0

0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

AR = 0.53 for case 5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

1.4

1.0

1.2

AR = 0.97 for case 3

1.0 0.8 0.6

AR = 0.47 for case 4

0.4 0.2

Case 3: wood WIB Case 4: steel WIB

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m) Fig. 6. Effect of WIB material stiffness on AR of (a) radial velocity and (b) vertical velocity for cases 3 and 4 at an excitation frequency of 30 Hz.

AR for vertical velocity

AR for vertical velocity

AR for radial velocity

3.0

AR for radial velocity

256

0.8

AR = 0.67 for case 6

0.6 0.4 0.2

AR = 0.59 for case 5 Case 5: embedded depth = 20 cm Case 6: embedded depth = 40 cm

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m) Fig. 7. Effect of WIB embedded depth on AR of (a) radial velocity and (b) vertical velocity for cases 5 and 6 at an excitation frequency of 35 Hz.

the AR value increases from 0.53 to 1.05 in the radial direction and from 0.59 to 0.67 in the vertical direction when the embedded depth of the WIB increases from 20 cm to 40 cm. It can be seen that the radial velocity is more sensitive to the embedded depth than the vertical velocity. It also can be seen that the radial velocity at certain points (i.e. No. 1–3) even can be significantly amplified and the overall effect of the WIB is negative in reducing the radial vibration when the embedded depth is 40 cm. This phenomenon demonstrates that better vibration mitigation can be achieved by decreasing the embedded depth of the WIB, especially in the radial direction. This test result coincides with the theoretical findings that the WIB would enlarge the vibration instead of reducing the vibration when the embedded depth of the WIB increases to some extent [35,36]. Note that ideally no reduction can be observed that the embedded depth of WIB is larger than the critical depth [20,38]. Fig. 8. Elastic laminated half-space and the cylindrical coordinate system.

4. Numerical analysis by semi-analytical BEM 4.1. Green's function for 3D layered ground derived with the thin-layer method Boundary element method (BEM) is ideally suitable for converting many types of differential equations into boundary integration equations with the aid of fundamental solution, namely Green's function, which is the key of a BEM solving process. Analytical Green's function for homogeneous full-space or halfspace is always used in the previous soil–structure interaction problems (SSI). However, it is not proper for the dynamic problems in laminated ground since the interfaces between the soil layers sharply increase the freedoms of the dynamic system and therefore the computation effort. In comparison, the thin layer method (TLM) is universal and efficient for SSI problems in half-space.

Hence the Green's function derived with TLM can be incorporated into BEM to solve SSI problems in half-space [45,46]. TLM is a semi-analytical method used to solve wave propagation problems involving partially heterogeneous media. In this method, the media is discretized into a number of thin layers along the direction of material heterogeneity by the finite element method, while analytical methods are used in the other directions, as shown in Fig. 8. For each layer, the governing equation of motion in an isotropic and homogeneous linear elastic media with a cylindrical coordinate system can be expressed as follows:

i 8 h 2 εV ur 2 1 ∂uθ > μ ∇ u  ð2 þ Þ þ ðλ þ μÞ∂∂r þ f r ¼ ρ∂∂tu2r r > r r r∂θ > > < h i 2 ∂ur μ ∇2 uθ  1r ðurθ  2r∂ Þ þ ðλ þ μÞ∂r∂εθV þf θ ¼ ρ∂∂tu2θ θ > > > > : μ∇2 uz þ ðλ þ μÞ∂εV þ f ¼ ρ∂2 u2z z ∂z ∂t

ð6Þ

G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

where λ and μ are Lame's constants, ρ is the mass density of the material, ur, uθ and uz are the displacements in the radial, tangential and vertical directions, respectively, εv is the volumetric strain, and fr, fθ and fz are the body forces in the radial, tangential and vertical directions, respectively. When the thickness of the thin layer is small enough to some extent, the displacements of points in the vertical direction within a thin layer can be approximately obtained by interpolation techniques. In this study, quadratic interpolation is used and the soil displacements ur, uθ and uz in the ith thin layer can be given by 8 > < ur ¼ Nuri uθ ¼ Nuθi ð7Þ > : u ¼ Nu z zi n oT  T where uri ¼ uri ; urði þ 1=2Þ ; urðiþ 1Þ , uθi ¼ uθi ; uθði þ 1=2Þ ; uθði þ 1Þ T and uzi ¼ uzi ; uzði þ 1=2Þ ; uzði þ 1Þ are displacement vectors for the top, middle and bottom nodes of the ith thin layer in the radial, tangential and vertical directions, respectively, and  N ¼ ð2z2 =H 2 Þ  ðz=HÞ 1  ð4z2 =H 2 Þ ð2z2 =H 2 Þ þ ðz=HÞ  is the shape function in which H is the thickness of the layer and z is the vertical coordinate of any point in the ith thin layer. For harmonic vibration (e  iωt ), conducting the interpolation on the displacement u within each thin layer using Eq. (7) and then integrating over the z coordinate, Eq. (6) can be written as [47] o n2 n ðiÞ ð8Þ ∑ Pm  Ki umi ¼ 0 i¼1

where P is the external force vector, the subscript m means the mth term of the Fourier series decomposition, and 2

Ki ¼ k Ai þ kBi þ Ci

ð9Þ

in which k is the possible wave number and Ai, Bi and Ci are the matrices determined by the material properties and given by 2 3 2 3 " # A1i C1i 2 Bzi 6 7 6 7 7; B i ¼ 7 Ai ¼ 6 ; Ci ¼ 6 C2i A2i 4 5 4 5 B2T zi Czi Azi ð10Þ The relationship between forces and displacements in the frequency-wave number domain in a laminated media can be derived by solving the eigenvalue problem of Eq. (8). Then, performing the Fourier series decomposition on the given force P about the tangential coordinate and conducting the Hankel transformation on the obtained equations about the radial coordinate yield the displacements expressions in the frequency-wave number domain. Finally, fulfilling the inverse Hankel transformation and Fourier synthesis results in the displacement expressed in the Cartesian coordinates. It should be noted that there exists a conflict between the depth to be discretized and the calculation accuracy. On one hand, if the discretized soil is not deep enough, large calculation error may occur. On the other hand, with increasing discretized depth, the results may converge further towards the exact half-space solution, but the increasing nodes will significantly increase the computation effort. Hence, it is necessary to exploit certain kinds of techniques to simulate the bottom half-space accurately with limited amount of nodes. These techniques include setting springs, dampers or introducing paraxial boundary at the bottom nodes. The paraxial boundary, proposed by Kausel and Seal [46,48,49], is a kind of approximate solution suitable for TLM. It can be derived by conducting a quadratic Taylor series expansion of the stiff matrix of the homogeneous half-space or half-plane in the wave number. In a 3D cylindrical n 1 coordinate o system, the relation between the 2 traction vector P hm ; P hm ; P hzm and surface displacement vector

257

n

o u1hm ; u2hm ; uhzm corresponding to the mth order of Fourier series can be written as 2 8 1 9 6 > P > 6 > = 6 < hm > 6 2 ¼6 P 6 hm > > > > ; 6 : 6 P hzm 4

3 h i 9 2 78 2 2μh k k  α1 α2 þ μh kϑ2 7> u1 > > 7> = 7< hm 2 2 7 k  α1 α2 7> uhm > > 7> 7: uhzm ; μh ϑ2 α2 5 2 k  α1 α2

μh α2 μh ϑ2 α1 k  α1 α2 h i 2 2 2μh k k  α1 α2 þ μh kϑ2 2

k  α1 α2 2

ð11Þ where ϑ1 ¼ ω=vhp , ϑ2 ¼ ω=vhs , α21 ¼ k  ϑ1 , α22 ¼ k  ϑ2 . ω is the circular frequency, vhp and vhs are the P-wave and S-wave velocities in the bottom half-space, respectively, and μh is Lame's constant of the bottom half-space. Conducting the quadratic Taylor series expansion on the stiff matrix in Eq. (11) yields 8 i 2μ > > kh11m ¼ iρωvhs  k h vhs > > > 2 ω > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > i 2 > > k μh  2 μh ðλh þ 2μh Þ < kh22m ¼ iρωvhs þ 2ϑ2 ð12Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > i 2 > > ¼ i ρω v þ k λ þ 2 μ  2 μ ð λ þ2 μ Þ k > hzzm hp h h h h h > > 2 ϑ1 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > > > ðλ þ 2μh Þ=μh  2 : kh2zm ¼ khz2 ¼ μh k 2

2

2

2

Then, the general paraxial boundary can be employed by putting kh11m , kh22m , khzzm and kh2zm directly at the corresponding locations of the stiff matrices of A, B and C. For a 3D axisymmetric problem, the specific paraxial boundary can be derived with m ¼ 0. From above deducing processes, we can find that TLM itself has accounted for the contributions of each thin layer to the total stiff matrix of the ground. Thus, for a laminated ground, the ground stiff matrix could be easily obtained by substituting the material properties of each soil layer into the corresponding portion of thin layer stiff matrix and no extra work is needed. Therefore, TLM is especially suitable for analyzing the interaction between the layered ground and structures. To validate the proposed method, it is used to analyze the classic Lamb's problem which deals with the ground vibration induced by the wave source on the surface of an elastic homogenous half-space [50], and the result is compared with the closed-form solution by Wang [51]. In this study, a unit harmonic vertical excitation at a frequency of 16 Hz is applied on the surface of an elastic half-space. The density, shear modulus and Poisson's ratio of the half-space are 1800 kg/m3, 53 MPa and 0.25, respectively. In the calculation, the total depth is 60 m and it is evenly divided into 25 layers. Fig. 9 compares the vertical displacement of the ground surface in this study and by Wang [51]. The consistence of the results shown in Fig. 9 convincingly verifies the proposed method. 4.2. 3D semi-analytical BEM The presented formulation here for 3D linear elastodynamic problems using frequency domain BEM is an extension of the 2D formulation in Beskos et al. [8]. The equations of the motion in the frequency domain for a 3D layered and linear elastic body V with boundary S and zero body forces are of the form [52] ðc21 c22 Þui;ij þ c22 uj;ii þ ω2 uj ¼ 0;

i; j ¼ 1; 2; 3

ð13Þ

where ui ¼ ui ðx; ωÞ is the displacement of the point x, ω is the circular frequency, and c1 and c2 are the P-wave and S-wave velocities, respectively. Note that commas indicate the spatial differentiation and summation over repeated indices. The constitutive equation is of the form

σ ij ¼ ρ½ðc21  2c22 Þuk;k δij þ c22 ðui;j þ uj;i Þ

ð14Þ

258

G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

-9

Vertical displacement ( x 10 m)

2.5

Surface foot This study Wang's close-form solution[51]

2.0 1.5

Interface 1

l

Interface 2

l

1.0 0.5

b 0.0

0

10

20 30 Distance to the source (m)

40 Fig. 10. Semi-analytical BEM model (the soil is omitted).

Fig. 9. Comparison of the ground displacements under harmonic vertical vibration between the close-form solution and this study.

where σ ij ¼ σ ij ðx; ωÞ is the stress tensor and δij is Kronecker's delta. Zero initial conditions are assumed and the stress and displacement boundary conditions are

σ ij nj ¼ t i0 ðx; ωÞ; ui ¼ ui0 ðx; ωÞ;

x A Sσ

ð15Þ

x A Su

ð16Þ

where nj is the outward unit normal vector at the boundary S ¼ Sσ þ Sσ , and t i0 and ui0 are the tractions and the displacements on the boundary, respectively. According to reciprocal work theorem, the solution of Eq. (13) in the integrated form is given by Z Z cij ðxÞuj ðxÞ ¼ unij ðx; yÞt j ðyÞ dSðyÞ  t nij ðx; yÞuj ðyÞ dSðyÞ ð17Þ s

n

a

s

n

where uij and t ij are the fundamental solution of the displacement and traction tensors, respectively, and cij ðxÞ is a tensor depending on the position of x. If x is neither in the body V nor on the boundary S, then cij ðxÞ ¼ 0. If x is in the body V, then cij ðxÞ ¼ 1. If x is on the boundary S, then cij ðxÞ depends on the characteristics of the boundary near x, and the detailed determination can be found in [47].

with " g

H ¼

Hg11

Hg12

Hg21

Hg22

#

" ;

g

G ¼

Gg11

Gg12

Gg21

Gg22

#

" ;

g

U ¼

Ug1

#

UG2

;

T ¼

Tg1

#

TG2 ð19Þ

where the superscript g, 1 and 2 refer to the ground, the interface between the foundation and the ground (i.e. interface 1), and the interface between the WIB and the ground (i.e. interface 2), respectively, Ug1 and Tg1 , and Ug2 and Tg2 indicate the displacement and traction vectors of nodes on interface 1 and interface 2, respectively; and Hgij and Ggij (i; j ¼ 1; 2) are 3ðM þ NÞ  3ðM þNÞ matrices describing the influencing coefficients of the nodes located on the interface i on the displacement and traction of the nodes located on the interface j, where M and N are the numbers of nodes on the interface 1 and interface 2, respectively. Similarly, the BEM equation of the WIB is Hw Uw ¼ Gw Tw

ð20Þ w

w

where the superscript w refers to the WIB, U and T mean the displacement and traction vectors of nodes on the interface 2, respectively, and Hw and Gw are 3M  3M matrices describing interaction coefficients of the nodes located on the interface 2 regarding the displacement and traction. The force equilibrium equation for the surface foundation is P ¼ JΔ  ATr

4.3. Semi-analytical BEM model for WIB isolation system

" g

ð21Þ T

For simplicity and computational efficiency, a 3D elastic semianalytical BEM was established to calculate the vibration isolation by WIB in laminated ground based on the fundamental solution of thin layer method with paraxial boundary in the frequency domain. Fig. 10 shows the 3D BEM model of the WIB vibration isolation system and the material properties used in the analysis are listed in Tables 1 and 2. Since the boundary conditions for the interface between soil layers have been automatically met in the TLM fundamental solution, only the interface boundaries between soil, foundation and WIB are needed to be ramified. 8-noded semidiscontinuous boundary elements [53] are exploited to tackle the nodes located on the sharp edges and corners of the rectangular WIB. Meanwhile, the foundation and WIB are perfectly bonded with the soil. Note that the nodes at the vertical boundaries of WIB should be located on the surfaces of thin layer elements. For the sake of precision, the thickness of each thin layer and the element size of the interfaces should be less than 1/8 and 1/10 of the shear wavelength in the soil, respectively. In this study, the sixth soil layer of silty clay is treated as the bottom of the half-space and simulated by an elastic paraxial boundary. Regarding the model shown in Fig. 10, the BEM equation of the ground is H g Ug ¼ G g T g

ð18Þ

where P ¼ ½p1 ; p2 ; p3  is the harmonic force vector on the surface foundation, Δ ¼ ½Δ1 ; Δ2 ; Δ3 T means the displacement vector of the surface foundation, Tr is the force vector exerted to the foundation from the ground, and matrix J can be written as 2 2 3 6 J ¼  m4

ω1

ω22

ω

7 5

ð22Þ

2 3

where ωi (i¼ 1,2,3) are the three circular frequencies of the force P, m is the mass of the foundation, and A is given by 2 3 A1 A2 ⋯ Ai ⋯ AN 6 7 A1 A2 ⋯ Ai ⋯ AN A¼4 5 A1

A2

⋯

Ai

⋯

AN ð23Þ

where Ai (i¼ 1,2,…,N) is the representative area of the node i on the interface 1. The compatibility of the displacements at the interface of the rigid foundation and the soil can be expressed as Ug1 ¼ SΔ

ð24Þ n

where S refers to a 3N 3 geometric transformation matrix connecting the displacement vectors of nodes on the interface 1 and

takes the form 2 1 1 6 1 S¼4 1

⋯ ⋯

1 1

⋯

⋯ ⋯

1 1 1

⋯

3T

1

7 5

1

ð25Þ

1

Similarly, the compatibility of the displacements at the interface 2 can be written as Ug2 ¼ Uw

ð26Þ

The force equilibrium conditions at the interfaces 1 and 2 are ð27Þ

Tg2 ¼  Tw

where

9 ^ 11 ¼ AG11 S þ J > G > > > = ^ 12 ¼ AG12 G ^ 21 ¼ G21 S G ^ 22 ¼ G22 þ F G

"

G11

G12

G21

G22

ð30Þ

> > > > ;

# ¼ ðGg Þ  1 Hg

-6 Vertical displacement ( x 10 m)

Combining with Eqs. (20), (26)–(28), Eqs. (21) and (24) can be rewritten in a matrix form as " #( )

^ 12 ^ 11 G Δ P G ¼ ð29Þ g ^ 21 G ^ 22 U2 0 G

ð31Þ

F ¼ ðGw Þ  1 Hw

ð32Þ

Δ

By solving Eq. (29), we can get and Ug2 . Then Ug1 can be obtained based on Eq. (24). Finally, we can get Tg1 and Tg2 based on Eq. (21) with previously obtained Ug1 and Ug2 . Note that for the case without WIB, Ug2 ¼ 0. After obtaining the boundary displacements and tractions of the field with and without WIB, the displacement of any point p on the free surface of the half-space can be determined by ( ) ( ) h g i Tg h g i Ug g g 1 1 G ðpÞ G ðpÞ ðpÞ H ðpÞ H uðpÞ ¼  ð33Þ 1 2 1 2 Tg2 Ug2 uðpÞ ¼ Gg1 ðpÞTg1  Hg1 ðpÞUg1

ð34Þ

where uðpÞ is the displacement of point p, and Hgi ðpÞ and Ggi ðpÞ are two matrix describing the influencing coefficients of the point p on the displacement and traction of the nodes located on interface i (i¼1, 2), respectively. As seen above, the proposed method here has taken into account all the waves generated by the vibrating foundation and it treats the foundation–soil–WIB system as a whole. 4.4. Comparison of results in field tests and numerical analyses Figs. 11–13 compare the displacement amplitude attenuation curves obtained from the field tests and the 3D semi-analytical BEM. Generally, it can be seen that the vibration amplitude decreases with increasing distance from the source, especially for the vertical vibration. It also can be seen that there is a good agreement between the field measurements and the numerical analyses in general, although minor difference exists. The minor discrepancies may be due to the three reasons: (a) the actual inhomogeneous soil is assumed to be homogeneous for the numerical evaluation. In fact, stiff soil blocks and pieces of bricks were found during the excavation of the WIB pit, which may enhance the inhomogeneity of soil; (b) it is assumed that only the

0.8

259

Field test Numerical analysis

0.7

0.6

0.5

0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

ð28Þ

Field test Numerical analysis

1.5

1.0

0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m) Fig. 11. Comparison of (a) horizontal displacement and (b) vertical displacement in field test and numerical analysis in case 1 at an excitation frequency of 15 Hz.

-6 Horizontal displacement ( x 10 m)

¼ T

r

3.0 2.5

Field test Numerical analyses

2.0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

1.5

-6 Vertical displacement ( x 10 m)

Tg1

-6 Horizontal displacement ( x 10 m)

G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

Field test Numerical analysis

1.0

0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m) Fig. 12. Comparison of (a) horizontal displacement and (b) vertical displacement in field test and numerical analysis in case 2 at an excitation frequency of 60 Hz.

G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

-6 Horizontal displacement ( x 10 m)

260

1.5 Field test Numerical analysis

1.0

0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

-6 Vertical displacement ( x 10 m)

3.0 Field test Numerical analysis

2.5

embedded depth decreases. It should be emphasized that the installed WIB may amplify rather than reduce the ground vibration when the shear modulus is smaller than a threshold value or the embedded depth is larger than a threshold one. (c) A 3D semi-analytical BEM–TLM model was derived to account for the rectangular shape of the used WIB and the multilayered characteristics of the actual ground on the vibration isolation effect. The good agreement of the results between the field measurements and the numerical analyses indicates that this model can reasonably predict the vibration mitigation effect of the WIB.

Acknowledgments The research is partly supported by the National Natural Science Foundation of China (Grant nos. 50878155 and 51178342) and Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20130072110016).

2.0 1.5

References

1.0

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0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 d (m)

Fig. 13. Comparison of (a) horizontal displacement and (b) vertical displacement in field test and numerical analysis in case 3 at an excitation frequency of 85 Hz.

foundation and the WIB interact with the soil, which ignores the interaction between soil and other structures around. This assumption simplifies the vibration transmission mechanism of the SSI problem and may also introduce some inaccuracies; (c) the WIB is assumed to be fully bonded to the soil in the BEM model, which may not be the case in the field tests. In spite of the discrepancies, Figs. 11–13 convincingly illustrate that the semianalytical BEM solution can be used to reasonably predict the vibration isolation effect of the WIB for machine foundations.

5. Summary and conclusions In this study, a series of field experiments on active vibration isolation of machine foundations by WIB under vertical harmonic loading have been carried out. The influences of WIB size, shear modulus and embedded depth on the wave mitigation effectiveness were investigated. Meanwhile, a 3D semi-analytical BEM– TLM model was proposed to predict the vibration isolation effect of WIB and the results were compared with the field experiments. Based on the experiments and numerical analyses, the following conclusions can be made. (a) The experiments convincingly indicate that WIB is effective to reduce the ground vibration amplitude induced by a vertically loaded machine foundation, particularly at high excitation frequencies. The measured RMS velocity reduction ratio which quantifies the degree of the vibration mitigation by the WIB fluctuates with the distance from the measuring point to the vibration source. (b) The vibration mitigation effect of the WIB is improved as the plane size and shear modulus of the WIB increase or the

G. Gao et al. / Soil Dynamics and Earthquake Engineering 69 (2015) 251–261

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