In situ experimental study on high speed train induced ground vibrations with the ballast-less track

Soil Dynamics and Earthquake Engineering 102 (2017) 195–214

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In situ experimental study on high speed train induced ground vibrations with the ballast-less track

MARK

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Shi-Jin Fenga,b, , Xiao-Lei Zhanga,b, Lei Wanga, Qi-Teng Zhenga, Feng-Lei Dua, Zhi-Lu Wangc a b c

Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Shanghai 200092, China Department of Structural Engineering, Tongji University, Shanghai 200092, China

A R T I C L E I N F O

A B S T R A C T

Keywords: High speed train Ground vibrations Experimental study Embankment section Culvert section Transition section Viaduct section

Many conveniences and efficiencies have been brought to the passengers with the rapid growth and application of High Speed Train (HST), especially in China. However, the vibrations induced by HST can cause adverse influences on the surrounding environment, which arouses increasing concerns from researchers and governments. In this paper, in-situ HST generated ground vibrations were measured in the embankment, culvert, viaduct and transition sections of Beijing-Shanghai high speed railway (HSR) in China. The acceleration responses of the free field in three directions (x, y, z) were recorded with a train operation speed of around 250–350 km/h. The characteristics of the three directional free field responses in time and frequency domains are then be acquired. Moreover, the variations of vibration amplitude and its vibration level with the distance from the track centerline and train speed are investigated, respectively. Ground dynamic impact coefficient (GDIC) and ground remaining dominant frequency (GRDF) are first introduced and defined to identify the ground vibration performances. It is found that the vertical acceleration response is typically the largest in the near field, while in the far field the largest one is the transverse acceleration response. When a culvert or viaduct is included, the longitudinal vibration would be dominant in the near field. Typically, an obvious vibration amplification zone can be observed in the field around 20 m due to the wave interference. In all four measurement scenarios, the dominant frequencies of the free field are typically n times the characteristic frequencies, which have a strong relation to the train's geometry dimension and unevenness of the rail. The first dominant frequency of the free field is generally determined by the distance between wheelsets, bogies and the ground fundamental frequency. Some useful recommendations are also provided in this paper and the results are valuable for validating numerical prediction of HST induced vibration and ground-borne vibration mitigation.

1. Introduction

vibration issue because the vibration may cause unease to people, inaccuracy of precise instruments and even harm to surrounding buildings with the rise of train speed [1]. Because of the aforementioned negative effects, numerous prediction models have been proposed to evaluate the vibrations before the construction of new lines or the upgrade of existing lines [2,3]. Analytical study of Sneddon [4] was first applied to the vibration of elastic ground under moving load. Eason [5] then studied the three dimensional (3D) problem of an elastic half space subjected to moving loads with various distributions (e.g., point load, circular or rectangular load). And by introducing the model of a beam on Winkler springs, Kenny [6] clarified the track dynamics. Moreover, Sheng et al. [7] proposed a quite complete model of the track-ground system consisting of rail, sleeper, ballast and layered elastic ground, which was adopted

The high speed railway (HSR) has grown rapidly around the world. With great efforts from the Chinese government, the high speed train (HST) is now widely distributed in China and the running mileage has increased to 18,000 km. As one of the hottest means of transportation, HST is fast, punctual and convenient with high-quality experience. Its highest operation speed is typically between 300 and 350 km/h and will even surpass the sound velocity in future. Numerous new HSR lines are and will be under construction all over the world. Therefore, the 21st century will be an era for HST. Meanwhile, some unavoidable problems, such as safe operation, durability of bearing structures and impacts on the surrounding environment, have aroused strong attentions. Numerous scholars focus on the HST induced environmental

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Corresponding author at: Department of Geotechnical Engineering, Tongji University, Si Ping Road 1239, Shanghai 200092, China. E-mail addresses: [email protected] (S.-J. Feng), [email protected] (X.-L. Zhang), [email protected] (L. Wang), [email protected] (Q.-T. Zheng), [email protected] (F.-L. Du), [email protected] (Z.-L. Wang). http://dx.doi.org/10.1016/j.soildyn.2017.09.001 Received 25 February 2017; Received in revised form 29 August 2017; Accepted 2 September 2017 Available online 15 September 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved.

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(a) Embankment section

(b) Culvert section

(c) T Transition section

(d) Viaduct section

Fig. 1. Four scenarios of HSR test sites.

on soft soil ground. Lombaert and Degrande [30] proposed a solution strategy for evaluating the response to the dynamic excitation generated by track unevenness. Results of numerical predictions at different train speeds were then compared with field measurements. Galvín and Domínguez [31] analyzed the soil and structural vibration data obtained from the certification test of a HST line. And, Xia et al. [32] investigated the railway train-induced vibrations on the surrounding ground and a nearby multi-story building. The track and ground vibrations generated by HST were tested by [33,34] to obtain an appropriate mechanical characterization of the main elements involved in the process. The experimental study of train-induced environmental and nearby building vibrations at a bridge site were conducted by [35]. The viaduct behavior and nearby ground motion under the HST passage were studied by [36]. Generally, experimental investigations mainly concentrate on the vertical vibrations of the embankment itself with the ballasted track. Few of them [2,3,24,37–39] have focused on the ground vibrations. Vega et al. [40] presented a complete study of a culvert, including on-site measurement and numerical modeling. Another ground-borne vibration level measurement at a culvert section was included in a series of experimental investigations undertaken by [2]. To the authors’ knowledge, no other literatures have been published to experimentally investigate the vibration of railway culvert section. The in-situ measurements at the viaduct section and transition section are also rare [2,3]. In this paper, the vertical, transversal and longitudinal acceleration responses were thoroughly recorded during the in-situ tests of four scenarios, named as embankment, culvert, viaduct and transition sections based on the HSR's bearing structure. These in-situ tests were conducted on the Beijing-Shanghai HSR of China, along which the CRSII ballast-less track was adopted. Detailed experimental

by latter studies. On the basis of Sheng et al. [7], Cao et al. [8] proposed a vehicle-track-ground system and introduced a linear Hertz contact spring between wheelsets and the rail to consider the vertical track irregularities. Recently, Kahina Chahour et al. [9] proposed a novel approach for the plot of dispersion curves to analyze the resonant phenomenon of a track-ground system. All the models mentioned above are based on analytical methods without considering more complex scenarios. To overcome the disadvantages of analytical models, a mass of numerical studies focusing on the HST induced vibrations have also been presented. Two dimensional (2D) finite element method (FEM) models were first established [10,11] to simulate the vibrations caused by HSTs and subways, respectively. By assuming the un-variation of track structure, embankment and supporting foundation in the track direction, an efficient two-and-half dimensional (2.5D) finite element numerical modeling approach was developed [12–14] to simulate the traffic induced wave motions. Thus, a 3D issue can be turned into a 2D one with Fourier Transform. However, the applications of 2D or 2.5D models are limited and 3D models are then used for ground or in-door vibration prediction by FEM [15–18], boundary element method (BEM) [19,20] or the coupling method [21–23]. Although numerical studies are increasingly applied to practical projects, there are still some gaps with the reality [2,3] because of the complexity, inhomogeneity and uncertainty of the HSR system. Experimental studies of HST induced ground vibrations can innately avoid those drawbacks, provide verification [2] and help to improve scoping simulation accuracy [24–28]. Madshus and Kaynia [28] studied the dynamic behavior at critical speed of a HST based on in-situ tests on a soft soil site in Sweden. Auersch [29] presented an experimental study on the effect of moving load and the amplification of dynamic response 196

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investigations of HST induced ground vibrations in four scenarios are then presented with time history, frequency content, amplitude and vibration level analyses. Moreover, the primary vibration characteristics of four sections are summarized and compared in Section 5. The ground remaining dominant frequency (GRDF) and ground dynamic impact coefficient (GDIC) are first introduced and defined in this paper to identify the ground vibration performances. The above work is useful for the verification of analytical and numerical models, as well as the mitigation of HST induced ground vibrations. 2. In situ experiment arrangement of ground vibration During July 10 and 21, 2015, HST induced ground vibrations of four scenarios on the Beijing-Shanghai HSR were measured. The test sites were located in Fengyang Country (Anhui Province, China), which was close to Bengbu-Nan Station. Based on the HSR's bearing structure, the test sites are classified as embankment section, culvert section, viaduct section and transition section (Fig. 1). The vertical, transversal and longitudinal acceleration responses in the free field were simultaneously recorded in all the measurements. 2.1. High speed train in China

Fig. 2. Cross section view of the testing embankment section (unit: m).

The daily operating HSTs on the Beijing-Shanghai HSR can be classified into four types: CRH380A/B and CRH380AL/BL. The main difference between CRH380A and CRH380B is the distribution of drive motors. The letter “L” means that the train has 16 cars, otherwise, it is of 8-car formation. The dynamic parameters of CRH380 are summarized in the Appendix A. The side view of CRH380BL configuration can be found in [41]. The repetition of HST in geometry leads to a periodic excitation on railway structures. The geometrical features of railway structures could also result in a periodic acceptance of excitation. These periodic passages are then defined as “Geometrical Characteristic Frequency (GCF)” by Eq. (1).

fgi =

vt Li

i = 1, 2, 3, ⋯

direction), respectively. Additionally, some supplementary parameters, such as the embankment slope, spatial distribution and geometrical characteristics of each part, are shown in Fig. 2. Another typical bearing structure of HSR is viaduct, which includes simply-supported beams and continuous beams format. The simplysupported beam has proved effective in experimental tests and practical projects. It is the best choice for large-scale constructions [42] and is generally used for passenger-dedicated lines combined with a singlecell box. In the test site, 32 m long simply-supported beams were adopted. According to the previous studies [42,43], a 32 m prefabricated concrete simple beam with ballast-less track has a fundamental frequency of 4.66 Hz and a camber for long-term deflection of 5.9 mm. Its material quantities are estimated as follows: 323.2 m3 of concrete, 62.8 t of reinforcing bars and 10 t of pre-stressing tendons [42]. Twin rectangular type piers [42] were also adopted in the test site rather than those of round ended, round or rectangular type. Besides, CFG pile groups were adopted with four typical diameters (1.0, 1.25, 1.5 and 2.0 m) and with the pile spacing being two times the diameter. Other auxiliary facilities, such as cable conduits, noise barriers, overhead lines and emergency access, are generally installed in the viaduct structure [42,43].

(1)

where vt represents the HST's running speed; Li denotes the characteristic dimensions of HST and its railway structures. L1 and L2 are the distances between wheelsets of the same bogie and between bogies of the same car, respectively. L3 denotes the distance between adjacent bogies of the successive cars. L4 and L5 are the distances between neighboring cars and between rail-pads, respectively. L6 denotes the length of a culvert. When vt is equal to 300 km/h (maximum running speed of HST in China), the corresponding GCFs are listed in Table 1. 2.2. Track structure and its typical bearing structures in China

2.3. Characteristics of the test ground

The test sites are featured with a straight HSR of double lines and the ballast-less CRTSII slab track structure, as depicted in Fig. 2. The slab track consists of 60 kg/m rail, rail-pads, track slab, concrete asphalt (CA) mortar and concrete base. As one of the most widely used bearing structures of HSR, the composite embankment is typically composed of surface subgrade, bottom subgrade, subgrade bed and subfoundation with cement fly-ash gravel. The geometrical and physical characteristics of CRSTII slab track and high embankment structure are listed in Table 2. The rail-pad is composed of fastening parts and elastic substrate, and its vertical stiffness, damping and spacing are 2.50×107 N/m, 7.50×104 N·m/s, 0.65 m (along the longitudinal

The vibration measurements were performed at a soft soil region. Before the HSR construction, in-site characteristic tests were conducted to investigate the geometrical and physical parameters of the ground along the railway. Table 3 summarizes the estimated dynamic properties of the foundation soil. The distribution and dimensions of layered soils are shown in Fig. 2. The Q3al (Table 3) represents the alluvial pluvial layer in quaternary upper Pleistocene period. To aid the interpretation of the test results, the dispersion curves for the layered ground (without track), including several P-SV wave propagating modes and shear wave line of the bottom layer, are depicted in Fig. 3 based on the elastic wave theory [7,44,45]. The dispersion relationship is shown in terms of velocity (against frequency) rather than wavenumber to provide a more intuitive approach for the critical velocity calculation [46,47], which will be presented in Section 3.2. The intersected points between the curves are the so-called cut-off frequencies, or natural frequencies, of layered soils [24]. As indicated in Fig. 3, the first three natural frequencies of the test sites are 3.8, 25.8 and 54.1 Hz. These natural frequencies are then named as physical

Table 1 Geometrical characteristic frequencies (GCF) at HST speed of 300 km/h. Characteristic Dimensions (m)

L1 2.5

L2 17.5

L3 7.5

L4 25.0

L5 0.65

L6 5.0

GCF fgi (Hz)

33.3

4.8

11.1

3.3

128.2

16.7

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Table 2 Characteristic parameters of CRSTII slab track and high embankment structure. Components

Depth (m)

Density (kg/m3)

Elastic modulus (MPa)

Possion ratio

Damping ratio (%)

Rail Rail Slab CA Mortar Concrete Base High Embankment

— 0.2 0.05 0.3 0.4 2.3 2

7800 2500 2000 2500 2300 1950 2100

2.10×105 3.50×104 1.00×104 3.00×104 200.0 150.0 110.0

0.3 0.22 0.3 0.2 0.25 0.35 0.3

— 5 20 5 5 5 5

Surface Subgrade Button Subgrade Subgrade Bed

150

characteristic frequency (PCF). 2.4. Testing arrangement

120 112.2 m/s

Velocity (m/s)

The test process includes site selection, accelerometer planning, pretesting and acceleration responses recording. A low-pass filter with a cut-off frequency of 200 Hz was used for measurement in the free field (350 Hz for the culvert, 400 Hz for the viaduct pier). The background vibrations of the test site were also collected in the circumstances of no vehicles passing by. Thus, the corresponding frequency content can help to obtain more accurate data with a band-stop filter. As a train passes, its velocity was calculated using a laser velocity measuring instrument with a range of 5–500 km/h.

2rd mode

90

3rd mode

5rd mode

1st mode

60

Intersection points Free track Rayleigh wave Shear wave

30

0

2.4.1. Measurement instruments The adopted vibration signal acquisition instruments, SIRIUS-HD STGS, are made in Austria. The instruments have 4, 4, 8 and 16 channels (Fig. 4a), respectively. These instruments can be used in parallel during the test, thus providing totally 32 channels to collect the acceleration responses in the free field. The sampling frequency for embankment, culvert and transition sections were set to be 1000 Hz, which would be enough to record the high frequency components. For viaduct section, the sampling frequency was 1500 Hz. Two types of accelerometers were selected in the field experiment: three-component type (LC0161) and one-component type (LC0155). The three-component accelerometer, shown in Fig. 4b, can be initially used to measure three directional ground vibrations. Its measurement range is 0–5 g and its effective frequency ranges from 0.1 to 1000 Hz. Three-component accelerometers are commonly displayed at the first 3 measuring points. The one-component accelerometer (Fig. 4c) can be used to monitor three directional ground vibrations combined with another two ones and a designed steel angle. Its effective frequency range is 0.1–4000 Hz. Every three accelerometers were mounted on the steel angle by strong magnetic force and the steel angle would be stuck deep enough into the soil.

4rd mode

0

30

60

90

120

150

Frequency (Hz) Fig. 3. Dispersion curves of the test layered soil and critical speed.

The intervals of first 5 points are 2 m to acquire fast attenuation of ground vibrations. For the further ones, the intervals range from 4 to 10 m. As for the culvert section, the first two measuring points were located at 0 m and 8 m from the track centerline (in the culvert). The rest points are also shown in Fig. 5 (in red). Similarly, 10 measuring points in the transition (in red) and viaduct (in blue) sections were also arranged in a row, perpendicular to the track, with various intervals (Fig. 6). The first points of these two sections were placed at 3.0 m from the track centerline. The intervals of the viaduct section are relatively smaller than those of the transition section to acquire faster attenuation of acceleration responses in the free field.

3. Characteristics of three directional acceleration responses The three directional accelerations during 76 passages of HSTs (20 for the embankment section, 18 for the culvert section, 20 for the transition section and 20 for the viaduct section) were recorded with the train speed varying from 200 to 350 km/h. The passing HSTs are composed of 8 cars (CRH380A/B) or 16 cars (CRH380AL/BL). Except for the passing time, the main characteristics of CRH380A/B and CRH380AL/BL are similar. Thus, the time domain and frequency domain characteristics of CRH380A/B (8-car format) will not be listed in

2.4.2. Measuring points The first vibration measuring point of the embankment section was placed at 16.5 m from the track centerline due to the protection of guardrails. Totally, 10 measuring points were arranged in a row, perpendicular to the track, at various intervals, as shown in Fig. 5 (in blue). Table 3 Dynamic characteristics of the testing site. Soil layer

Soil type

Thickness (m)

Young's modulus (MPa)

Poisson ratio

Density (kg/ m3)

Shear wave velocity (m/s)

Rayleigh wave velocity (m/s)

Compression wave velocity (m/s)

(1) (2) (3)

Q3al clay Clay Ptlz completely-weathered amphibolite Ptlz highly-weathered amphibolite Weakly weathered amphibolite

2.4 13.1 10.5

42 83 122

0.3 0.36 0.32

1900 2010 2100

92.2 123.2 148.3

85.5 115.3 138.0

172.5 263.4 288.3

18.0

141

0.34

2300

151.2

141.2

307.2

∞

155

0.33

2320

158.5

147.7

314.6

(4) (5)

198

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(a) Signal acquisition instruments

(b) Three-component accelerometer

(c) One-component accelerometer

Fig. 4. The measuring instruments adopted in the vibration test.

section at 300 km/h, is presented in this part. For simplicity, only four out of ten measuring points, ranging from 16.5 to 38.5 m from the track centerline, are displayed. Their time history of vertical, transversal and longitudinal acceleration responses in the free field are shown in Fig. 7. The three directional accelerations of one measuring point were depicted in one figure to avoid repetition. The scale values of the vertical axis with brackets mean the end and start of two successive vibrations, such as, the bottom “(0.1)” in Fig. 7a means 0.1 m/s2 for the vertical vibration and −0.1 m/s2 for the transversal vibration. This is also applied for the Fourier spectra in the next sections. Moreover, the vertical scale is kept constant to better appreciate the effects of radiation and material damping in the soil. As expected, a succession of periodic vibration peak values, which indicate the specific geometry features of the passing train, can be found, especially in the field near the track (Fig. 7a and c). Moreover, the first peak value of transversal acceleration, which is perpendicular to the railway, is generally a little ahead of those in other two directions because the compression wave propagates faster than the shear wave. Within 28 m from the track centerline, the peak values of vertical acceleration (apz) are generally the largest and the peak values of longitudinal acceleration (apy) are the least. As for the transversal acceleration, its peak values (apx) are only a little weaker than apz (Fig. 7a, b and c). However, over 30 m away from the track centerline, apx (0.052 m/s2) becomes the largest, and apz (0.034 m/s2) and apy (0.025 m/s2) tend to be at the same level (Fig. 7d). Besides, the acceleration amplitudes of the third measuring point (at the distance of 24.5 m, Fig. 7c) are larger than those of the second one (at the distance of 20.5 m, Fig. 7b). This result is different from the previous research that a significant drop was expected [24].

Fig. 5. Distributions of measuring points in the embankment and culvert test sites.

3.1.2. Three directional time history of the other three scenarios As many fundamental variation rules of embankment, culvert, transition and viaduct sections are similar, this section mainly focuses on the differences between the embankment section and other three sections to avoid a repeat. The running speed of selected passing HST (16-cars format) for the next three measuring scenarios is also about 300 km/h.

Fig. 6. Distributions of measuring points in the transition and viaduct test sites.

this paper. All the vertical, transversal and longitudinal acceleration responses for the four scenarios (Fig. 1) of different measuring points (perpendicular to the track) are covered. It should be noted that the first two of the measuring points of the culvert section scenario were glued to the culvert (Fig. 5), the first measuring point of the viaduct section scenario was glued to the bottom of viaduct pier (Fig. 6) and all the other measuring points were placed at the free field (Figs. 5 and 6).

3.1.2.1. Time history of the culvert section. It is found that within 30 m from the track centerline (also called near field), the longitudinal acceleration is dominant rather than the vertical acceleration encountered in embankment section. The vertical acceleration in culvert section is typically a little weaker than the longitudinal one. Moreover, the vertical and longitudinal accelerations of the first two measuring point (which is in the culvert) arrive earlier and last longer due to the fact that wave propagate faster in stiff materials (Fig. 8a and b). Another two concentrated sustaining acceleration signals in vertical and longitudinal directions are also observed at 8.0 m from the centerline (besieged by dashed rectangles in Fig. 8b).

3.1. Time history characteristics of the four scenarios 3.1.1. Three directional time history of the embankment section A representative case of CRH380BL train, passing the embankment 199

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Fig. 7. Time histories of three directional ground vibration accelerations of 4 measuring points (embankment).

3.2. Frequency content characteristics of the four scenarios

3.1.2.2. Time history of the transition section. As for the transition section, despite the fact that apy and apz are of the same order of magnitude, apy gradually becomes the largest (Fig. 8c), which is parallel to the railway, beyond the distance of 20.0 m.

3.2.1. Results of the embankment section The acceleration amplitude spectra are achieved by Fourier Transformation (Fig. 9), corresponding to the time histories shown in Fig. 7. Fig. 9a emphasizes all the dominant frequencies of the first point with circles varying from about 3.5–137.5 Hz. The vertical, transversal and longitudinal accelerations almost have the same dominant frequencies. Moreover, the high frequencies are apparently attenuated by the radiation and material damping in the soil as the increase of the distance from the track centerline. The remarkable frequency components of accelerations in the free field are typically between 20 and 50 Hz with similar first three dominant frequencies for each measuring point (depicted in Fig. 9 in grey label). The first dominant frequency is

3.1.2.3. Time history of the viaduct section. As for the viaduct section, the time history of the first measuring point (on the viaduct pier) is much denser than those of others. The horizontal acceleration is dominant at the first two measuring points (Fig. 8d). Therefore, the horizontal vibration responses of the free field cannot be neglected, which is often neglected by most researchers.

Fig. 8. Time histories of three directional ground vibration accelerations of the other three scenarios (culvert, transition and viaduct).

200

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Fig. 9. Frequency spectra of three directional ground vibration accelerations of 4 measuring points (embankment).

3.2.2.2. Frequency content of the transition section. The three directional vibrations almost have the same frequency components and dominant frequencies, ranging from about 3.3–127.7 Hz. The remarkable frequency components of accelerations in the free field are between 10 and 40 Hz (Fig. 10c), which are slightly lower than those of the embankment and culvert sections (20–50 Hz).

generally around 34.4 Hz. The fact that some successive dominant frequencies, such as 31.0, 34.5 and 37.9 (Fig. 9a, c and d), typically have close differences may be the results of the frequency modulation provided by fg4. Besides, it is worth mentioning that the noticeable dominant frequency around 130 Hz (Fig. 9a), called as rail-pad passage frequency (caused by the distance between rail-pads at the speed of 300 km/h), is seldom pointed out in previous literature [24,34,38].

3.2.2.3. Frequency content of the viaduct section. Much wider frequency content distribution (10–350 Hz) can be found at the first measuring point (on the viaduct pier, Fig. 10d) due to the inclusion of viaduct structures. The existing high dominated frequencies (250–350 Hz) then explain the much denser acceleration signals in Fig. 8d.

3.2.2. Results of the other three scenarios 3.2.2.1. Frequency content of the culvert section. Similarly, the frequency contents of the measuring points of the free field in the culvert section are within 150 Hz. However, different from embankment section, much wider spectra distribution of 150–300 Hz can be found at the first two measuring points (in the middle and side of the culvert, Fig. 10a and b). There exist some new dominant frequencies, such as around 190 Hz (Fig. 10a) and 298 Hz (Fig. 10b). These new dominant frequencies are mainly products of characteristic frequencies of HST and free vibration frequencies of the culvert (first three: 56.6, 57.5 and 63.3 Hz through an additional finite element simulation). Moreover, the “two concentrated lasting vibrations” (besieged by dashed rectangles) shown in Fig. 8b corresponds to frequencies between about 250 and 300 Hz (Fig. 10b) and is mainly induced by torsional vibrations (third order natural frequency) at the edge of the culvert structure. Thus, for the first point (middle of the culvert), its spectra components are nearly absent in the range of 200–300 Hz due to the interplay between some axial and torsional vibrations at the symmetry axis of the culvert structure.

3.2.3. Analyses of frequency content of the four scenarios As the three directional accelerations almost have the same frequency contents and dominant frequencies, only vertical vibration will be analyzed here to avoid repetition. It is observed that the aforementioned first dominant frequency 34.4 Hz of the embankment section (Section 3.2.1) is close to the train characteristic frequency (also previously called as passage frequency [37,48,49]) fg1 induced by the distance between wheelsets. The GCFs (fgi) are determined by vt with known train's geometrical dimensions and Fig. 11 shows the relationship between GCFs and the frequency spectra of the 16.5 m (near field) and 32.5 m (far field) points (embankment section) at different speeds (260, 300 and 330 km/h). In all subfigures, the third and fourth red dotted lines correspondingly overlap with the spectra. As a single-frequency moving load would produce a transient with a broad band spectrum at a fixed point on the 201

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Fig. 10. Frequency spectra of three directional ground vibration accelerations of the other three scenarios (culvert, transition and viaduct).

ground due to the Doppler Effect, the fifth red dotted lines are regarded as overlapping with the spectra. Thus, it is believed that there exist multiple relationships among the dominant frequencies and GCFs (fgi) of the ground and the details for the embankment section are summarized in Table 4 (vt = 300 km/h). The quotients between [0.9 + Integer, 1.1 + Integer] can be treated as integers according to the Doppler Effect, and thus many of the dominant frequencies are respectively n times the system's characteristic frequencies fgi (in bold and grey highlight). For example, the dominant frequency of 35.7 Hz is 1.1 times fg1. As a result, it can be implied that the HST's characteristic length between wheelsets (fg1) and between adjacent bogies (fg3), which result in periodic quasi-static excitation, have more important roles in the ground vibration. However, the frequency contents at around 10–30 Hz and 38–60 Hz have fewer relationships with the GCFs. Fig. 11 also shows the relationship between the PCFs and the frequency spectra of the 16.5 m and 32.5 m points. The PCFs are represented by three blue dashed straight lines. Fewer overlaps between PCFs and the corresponding spectra can be found, except for fg2 (will be analyzed in the next paragraph). This is due to the fact that the quasistatic force applied by the HST may significantly contribute to the vibration under the track structure when HSTs travel at a speed below the ground wave velocity [50]. At this time, the quasi-static response of the free field would be rather small and no excitation of the propagating modes occurs. Thus, PCFs would not be the formation cause of the frequency contents of around 10–30 Hz and 38–60 Hz. In addition, as the commercial operation speed in China (83.3 m/s = 300 km/h) is smaller than the Rayleigh wave velocity of the surface ground (85.5 m/ s), no intersectional characteristic frequencies (ICF, fck) between vt and

dispersion curves (Fig. 3) are expected. That is the reason why ICFs are not analyzed in this paper. Due to the limited conditions of the field tests, the unevenness of the rail and the defect of the wheels were not measured. But, it has been established that the dominant frequencies of the ground vibration between 3 and 6 Hz [49], 20–40 Hz [50] and larger than 50 Hz [49] are typically induced by the unevenness in the ranges of long and short wavelengths. With a train speed below the ground wave velocity, this mechanism of excitation, which is the main contribution to dynamic vibration response, is much more essential in the response of free filed than under the track. Thus, it is inferred that the aforementioned peaks at around 10–30 Hz and 38–60 Hz are likely caused by the unevenness and rail defects. Typically, the vibrations observed at the track and in the free field are divided into two parts: 1) quasi-static vibration due to successive axles as the train passes, and 2) dynamic vibration caused by the train over the combined irregular profile of the wheels and rail. Based on the previous analysis and available publications [49–52], it is reasonable to deduce that the quasi-static vibration also has two qualities: periodicability (depend on the train's geometrical characteristics; GCF) and velocity-ability (depend on the relationship between train speed and field wave velocity; PCF and ICF). Finally, the general conclusion may be drawn that the periodic-ability and dynamic excitations are likely to significantly contribute to the vibration of the free field in different frequency bands, while the velocity-ability excitation makes a small contribution when the train speed is below the ground wave velocity. It has been checked that the dominated frequencies of the culvert section, transition section and viaduct section also have multiple 202

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relationships with fgi as in the embankment section. Moreover, the aforementioned conclusion is also applicable for these three measuring scenarios.

Table 4 The remaining vibration acceleration dominant frequencies of the 4 measuring points.

4. Characteristics of acceleration amplitudes in time domain 4.1. Variation of acceleration amplitudes with the distance 4.1.1. Results and analyses of the embankment section Another 9 sets of testing data at a train speed of around 300 km/h are added in this part. The variation of acceleration amplitudes of the free field (10 measuring points) with the distance from the track centerline are then shown in Fig. 12 with average values in blue. It can be found that the acceleration amplitudes in three directions are on a downward trend and with an obvious amplification zone around 24 m. The decrease of peak particle acceleration with the distance is due to the radiation and material damping in the soil, which can also be found in [2,35]. The amplification zone of vertical acceleration is relatively narrower than those of horizontal ones without a conventional explanation. The amplification zone is likely the product of wave interference between 1) the surface Rayleigh wave and reflected wave (Figs. 13a), and 2) scattering waves propagated from the neighboring fasteners (Fig. 13b). Similar results can also be found in some additional FE simulations by the authors and other researcher [53], especially when the distance between ground surface and bed rock is small. The relationships between acceleration amplitudes and the distance are fitted by polynomial equations in Fig. 12 (in red). The three directional fitting equations for vt = 300 km/h are expressed as follows:

Az = 0.0962 − 0.0023d + 1.15 × 10−5d 2

(2)

Ax = 0.0594 − 2.04 × 10−4d + 1.9 × 10−5d 2

(3)

Ay = 0.0326 − 1.84 × 10−5d + 6.98 × 10−6d 2

(4)

where Az, Ax and Ay denote the fitting vertical, transversal and

Fig. 11. Vertical frequency spectra of the 16.5 m and 32.5 m points at different train speeds (embankment section, GCF & PCF).

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Fig. 12. Time domain peak values of ground vibration accelerations with the distance (embankment).

longitudinal acceleration amplitudes of the free field, respectively; d is the distance of an observation point from the track centerline. It should be indicated that the fitting formulations in Figs. 12 and 16 can be used as a reference only for similar conditions of train, track, embankment/ culvert/ viaduct and foundation soil.

Totally, 10 sets of measuring data are included for the transition section scenario with the train speed around 300 km/h. Generally, the three directional acceleration amplitudes of the transition section are typically at the same level and smaller than those of the embankment section. Distinct amplifications of these amplitudes are observed in the zone around 25 m. In addition, an obvious rise can also be found at the last measuring point (Fig. 14c), which is similar to the observation in the culvert section (Fig. 14a and b). This may indicate the periodicity of the vibration amplification zone and matches with the theory of wave interference (Fig. 13). As for the viaduct section scenario, the acceleration amplitudes significantly attenuate within about 22.0 m (from the track centerline), beyond which the amplitudes then gently decline. The amplification

4.1.2. Results and analyses of the other three scenarios Totally, 8 series of measuring data at the speed of 300 km/h are included for the culvert section scenario. The expected amplification zone in vertical vibration is not obvious, and the transversal and longitudinal vibrations rise up at about 35 m from the centerline (Fig. 14a and b). These are different from the results in the embankment section.

(a) Surface wave and reflected wave

(b) Scattering waves from neighboring fasteners

Fig. 13. Diagrammatic sketch for the wave propagation path and wave interference.

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Fig. 14. Time domain peak values of ground vibration accelerations with the distance.

is linearly fitted using black (vertical), blue (transversal) and red lines (longitudinal). Generally, the slopes in three directions are very close to each other for each measuring point and become smooth as the increase of the distance from the track centerline. However, the upward tendency of vibration amplitudes with vt is weak, indicating a relatively independent relationship, which can also be found in [2,3,24,37]. It should be noted that the results of HST induced vibrations are sitespecific, thus the fitting lines are typically qualitatively similar with previous studies [2,3,24], yet quantitatively different due to the train type, track structure and ground. On the other hand, it is proved that critical train speed can be reached [28,29,46,54], especially in soft soil lands. As stated in [28], “the critical speed is controlled by the dispersion curve of the first Rayleigh mode of the soil and embankment profile at the site and the load distance of the train”, moreover, the embankment would help to improve the critical moving speed [9,28,46,52]. Thus, the critical train speed would be larger than the Rayleigh wave velocity of the half space. As the operating speed of HST in China (300 km/h = 83.3 m/s) is below the Rayleigh wave velocity (85.5 m/s, 115.3 m/s) and Shear wave velocity (92.2 m/s, 123.2 m/s) of the surface and second soil layers, the “dynamic amplification effect” (at critical speed) was not found in this in-situ test. Nevertheless, using a simplified approach proposed by Costa et al.[46,47], the critical speed of this track-ground system is estimated as 112.2 m/s. A similar critical speed range, between 380 km/h (105.6 m/s) and 400 km/h (111.2 m/s), was also found by Zhai et al. [24] in the site from Zaozhuang West station to Bengbu South station. This also explains why the PCF and ICF are not included in the frequency contents of the free field vibration in Section 3.2.3.

zone locates at around 20 m from the centerline. 4.1.3. Comparison of amplitudes with distance among the four scenarios Variations of acceleration amplitudes of the four sections with the distance are then depicted in Fig. 15 to achieve a comprehensive comparison. Generally, the ground vibrations of the embankment section are larger than those of the culvert, viaduct and transition sections in all three directions. It can be found that, the amplitudes of the culvert, transition and viaduct sections at 26 m are equal to those of the embankment section at 42 m, 52 m and 53 m, respectively. Moreover, the amplitudes of the culvert section are larger than those of the transition section and viaduct section. The acceleration amplitudes of the transition section keep stable and are generally the least among these four scenarios. However, as for the viaduct section, its acceleration amplitudes after the amplification zone gradually become the least. It can then be expected that the transition and viaduct sections would result in smaller ground vibrations than the embankment and culvert sections. In summary, the vertical vibration is typically dominant in the near field of four sections, while in the far field the horizontal vibrations would be larger. 4.2. Variation of acceleration amplitudes with train speed 4.2.1. Results and analyses of the embankment section A total of 20 sets of measuring data are included in this part. Fig. 16 shows the variation of ground vibration acceleration amplitudes (apz, apx and apy) with train speed varying from 250 to 350 km/h. apz, apx and apy are plotted in black, blue and red, respectively. It can be seen that the acceleration amplitudes (in three directions) generally display a linear uptrend with the increase of vt. As shown in Fig. 16, the uptrend 205

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Fig. 15. Vibration amplitudes of the four sections with the distance.

scenarios to achieve a comprehensive comparison. The relationships are also linearly fitted in black. It can be found that the relationships of the four sections are similar. Although a scattered distribution is non-ignorable, the expected positive trend between ground vibrations and train speed is rather weak, especially for the vertical and transversal vibrations.

4.2.2. Results and analyses of the other three scenarios Totally 18 series of measuring data are included for culvert section. The three directional acceleration amplitudes of the last three measuring points (on the ground) have linear uptrends with the increase of vt, the same as in the embankment section. However, adverse trends are found at the first two measuring points (Fig. 17a and b, in the culvert). It is then estimated that the maxima of acceleration amplitudes occur at a train speed of about 160 km/h, which is well below the maxima running speed of HST. It has been proved that, when a series of loads travel over a railway bridge/culvert, resonance condition would be expected when the loading frequency coincides with the natural frequency of the bridge/culvert [55]. Therefore, it is reasonable to take 160 km/h as one of the culvert's critical velocities, thus leading to a downward tendency of vibration amplitudes with vt at the measuring points in the culvert. Through an additional finite element simulation, it is found that the first 15-order natural frequencies of the culvert vary from 56 to 120 Hz. It is possible that the response at 56 Hz (medium range frequency) is due to the dynamic contribution of short rail unevenness, which has been discussed in Section 3.2.3. All the 20 sets and 18 series of measuring data are included for the transition section and viaduct section, respectively. Generally, the result shows that the acceleration amplitudes of the free field have a weak positive correlation with vt with a similar fitted slope for different measuring points. Little adverse trends (downtrend) are found at the third (Fig. 17c, longitudinal vibration) and fifth (Fig. 17d, transversal vibration) measuring points.

5. Characteristics of three directional vibration acceleration levels 5.1. Definition of the vibration acceleration level As an important index for evaluating the vibration duration, the vibration acceleration level of the free field is discussed in this section. As for the transient vibration situation, a commonly used metric, the running r.m.s. evaluation method recommended by ISO2631-1 (1997) and Chinese Standard of Environment Vibration in Urban Area is adopted as follows:

a w (t0) =

{ 1τ ∫

t0

t0 − τ

[a w (t )]2 dt

MTVV = max[a w (t0)]

VAL = 20 log

MTVV a0

}

1 2

(5) (6)

(7)

where aw(t) is the instantaneous frequency-weighted acceleration (m/ s2). τ is the integration time for running averaging. t and t0 are the time (integration variable) and the time of observation (instantaneous time), respectively. MTVV is the maximum transient vibration value. VAL represent the vibration acceleration level. It is recommended to use τ =

4.2.3. Comparison of amplitudes with train speed among the four scenarios Fig. 18 shows the relationships between acceleration amplitudes and train speed at the distance of around 20 m for the four measuring 206

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Fig. 16. Time domain peak values of three directional ground vibration accelerations with train speed (embankment).

1 s in measuring MTVV by the ISO2631-1 (1997). The reference acceleration a0 is set to 1.0×10−6 m/s in this paper.

4.1.2 and 4.2.2 except for smaller attenuation rates for vibration acceleration levels.

5.2. VAL of the embankment section

6. Summaries and discussion of the vibration measurements

The vertical (VALz), transversal (VALx) and longitudinal (VALy) variations of vibration acceleration level with the distance from the track centerline are illustrated in Fig. 19 with average values and errors in blue. Generally, with the increase of the distance, the vibration acceleration levels in three directions attenuate with amplification zones around 24 m. As mentioned in Part 4.1.1, the amplification zone in vertical direction is narrower than those in other two ones. Moreover, the average error, which is mainly induced by the diversity of HST operation condition and uncertainty of railway system, significantly decreases with the increased of the distance. This indicates that the influences of the diversity and uncertainty on the ground vibrations are limited or just confined to the field close to the railway. It is then possible for operators and environmentalists to distinguish the vibration sensitive zone using Fig. 19 and specific alarm values for vibration acceleration level. In addition, the variations of VAL with vt, which are not depicted here, agree with those of acceleration amplitudes in time domain (Section 4.2.1).

In case of repetition, some of the testing results of the culvert, viaduct and transition sections have not been plotted in this paper. So, the HST induced ground vibrations of this four-profiled sections are briefly summarized in this section. Based on these investigations, some recommendations are then provided, which may be useful for the analytical and numerical studies. 6.1. Summary of vibration measurements As the HST induced vibration characteristics of the four sections have been previously investigated, this section provides a further summary of their similarities and differences in Table 5. This table contains four columns of vibration characteristics, namely, Time Domain, Frequency Domain, Vibration Amplitude and Vibration Level. Every measuring section is divided into three parts (Before-AZ, AZ and After-AZ) to summarize their similarities and differences. AZ is short for amplification zone. In the “Time Domain” column, measuring points of the embankment section have primary similarities as follows: (1) time history is typically composed of a succession of periodic peak vibrations (termed as 1-1); (2) these peak vibrations indicate train geometrical characteristics (termed as 1–2); (3) periodicity is more distinct in vertical vibration acceleration than horizontal ones (termed as 1–3). These three similarities can also be found in the culvert section. Additionally, three

5.3. VAL of the other three scenarios To avoid repetition, the variations of vibration acceleration levels of the culvert, transition and viaduct sections with the distance and vt are not shown in this paper. Generally, their vibration acceleration levels have similar characteristics with the time history amplitudes in Sections 207

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Fig. 17. Time domain peak values of ground vibration accelerations with train speed.

are commonly close to each other (termed as 3-3). Similarities of 3-1, 32 and 3-3 also apply for the culvert, transition and viaduct sections. As for the differences, variations of vibration amplitude with the distance and speed are respectively listed. For the embankment section, the vibration at before-AZ decays faster with the distance than at AZ and after-AZ, and increases less gently with train speed than at after-AZ. AZ in the embankment section is approximately between 22–32 m and AZ of vertical vibration is a little narrower than those of horizontal ones. Details in other sections can be found in Table 5. In the “Vibration Level” column, variations are generally similar with those in the “Vibration Amplitude”. What is different is that its decrease rates (with the distance) and increase rates (with speed) of Vibration Level are typically smaller than those of Vibration Amplitude.

directional vibration amplitudes of the culvert section tend to be the same in the far field (termed by 1–4); and the transversal vibration peaks are generally a little ahead of those in other directions (termed by 1–5). Except for the culvert section, these 5 similarities also apply for the transition and viaduct sections. As for the differences, rankings of vertical (apz), transversal (apx) and longitudinal (apy) vibration amplitudes in three parts (Before-AZ, AZ and After-AZ) are compared in Table 5. For example, apx at after-AZ of the culvert section ranks first, and then apz and apy rank second. In the “Frequency Domain” column, dominating similarities of the embankment section are: (1) Fourier amplitudes have the same ranking with vibration amplitudes (“Time Domain” column) (termed as 2-1); (2) frequencies are abundant within 0–50 Hz (termed as 2-2), scarce within 50–100 Hz (termed as 2–3) and few within 100–150 Hz (termed as 2–4); (3) dominant frequencies (termed as 2–5) range from 3.5 to 137.5 Hz and remarkable frequencies (termed as 2–6) range between 20–50 Hz. Similarities of 2-1, 2-2, 2–3 and 2–4 also apply for the culvert, transition and viaduct sections (Table 5). Additionally, measuring points of the culvert and viaduct sections have other frequency range around 200 and 350 Hz, respectively. As for the differences, the first three dominant frequencies at three divided zones are presented. The multiple relationships between these dominant frequencies and fgi are also summarized in this column. In the “Vibration Amplitude” column, measuring points of the embankment section have elementary similarities as follows: (1) vibration amplitude generally decays with the distance from the track centerline (termed as 3-1); (2) ground vibration amplitude typically shows a weak linear uptrend with HST's running speed (termed as 3-2); (3) increase rates of vertical, transversal and longitudinal vibrations at each point

6.2. Dynamic impact coefficient for the ground vibration amplitudes In this section, results are further analyzed to discuss the application of experimental studies in practical ground vibration evaluation and prediction. Thus, the dynamic impact coefficient is defined to restrict a range rather than a specific value to estimate the ground vibrations. Another definition about the remaining dominant frequencies of the ground is also introduced in the next section. Dynamic impact coefficient for the ground vibration amplitudes: It is considered that the dispersion of measured data is mainly caused by the uncertainties of the HSR-bearing structure-ground system, such as the operation diversity of HST, uncertainty of manufacture error, variability of the track structure and complexity of inhomogeneous soil. Thus, the measured vibration peak values would underestimate the influence of vibration on the surroundings, especially when the testing 208

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Fig. 18. Vibration amplitudes of the four sections with train speed.

be regional. Thus, the interval ground dynamic impact coefficient (IGDIC) is further defined as:

data is not sufficient. It is also impossible for the analytical and numerical simulations to take these uncertainties into consideration. Therefore, the ground dynamic impact coefficient (GDIC, IFg) is introduced to define a vibration amplitude range rather than a tested or simulated specific value to estimate the ground vibration amplitudes. It is assumed that, there are N test data sets and K measuring points in every experiment case. The subscripts i and k represent the i-th test data and k-th measuring point. And, avk represents the average vibration amplitude of the N test data sets at k-th measuring point. The superscript Δ represents the relative error, e.g., aiΔk is the relative error of i-th test data and k-th point. Thus, these parameters as well as IFg are defined as:

K

IFgdis =

∑i = 1 aik N

a vΔk = aik − a vk

IFik =

aiΔk a vk

(8) (9)

The high frequencies (> 150 Hz) would be absorbed by the soil efficiently within the near field, the remarkable frequency components are typically within 50 Hz. It is then found that these dominant frequencies are mainly induced by the carriage periodicity, wheel/rail defects as presented in [56]. As indicated in Section 3.2 and Table 4, these remaining dominant frequencies of the ground are typically n times at least one of the geometrical characteristic frequencies (GCF, fgi) (of train, track or culvert) or wheel/rail defect characteristic

∑i = 1 IFik N

(11)

K

IFg =

∑k = 1 IFvk K

(13)

6.3. Remaining dominant frequencies of ground

(10)

N

IFvk =

K dis

One first divides the measuring range into several intervals, using dis to describe the range of the specific interval. Kdis represents the number of measuring points within that dis. Finally, both IFg and IFgdis of the embankment, culvert, transition and viaduct sections are calculated by Eqs. (12) and (13), which are listed in Tables 6–9. As indicated in Table 6, GDIC for vertical vibration is 0.14, which means that the alltime ground vibration acceleration amplitudes in the embankment section may lie within [0.86, 1.14]×Measured amplitude. Thus, 1.14ak can be used for the design of railway construction and vibration control. Additionally, it is necessary to mention that GDICs in tables would be more universal with more measuring data and the dependence of ground vibration on site characteristics should be further investigated.

N

a vk =

∑k =dis1 IFvk

(12)

As indicated in Figs. 12 and 19, the relative errors of adjacent measuring points change little, which means that the relative error may 209

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Fig. 19. Vibration level of ground vibration accelerations with the distance (embankment).

track interaction or ground vibration simulation at the first stage to minimize the calculation price.

frequencies (RCF, frj). Moreover, when the HST speed is approaching or larger than the critical track/soil velocity, multiple relationships between the dominant frequencies and the physical characteristic frequency of the ground (PCF, fpm) and the intersectional characteristic frequency (ICF, fcn) are also found in [24]. Thus, these characteristic frequencies would be included to define the ground remaining dominant frequencies (GRDF, fgrd):

fgrd = n (fgi *frj *fpm *fcn ) n = 1, 2, 3⋯

7. Conclusions Four scenarios of in situ measurements for HST induced ground vibrations were performed on the Beijng-Shanghai HSR close to Bengbu South high speed station in China. The acceleration responses in the three directions were recorded during the measurements. The vibration characteristics of the embankment, culvert, transition and viaduct sections in time history and frequency content are then analyzed. Also, the variations of acceleration amplitude and vibration acceleration level with distance (from the track centerline) and train speed are investigated, respectively. Based on these investigations, some recommendations are provided for practical design and further studies. These in-situ measurement data of these four scenarios are complementary to other experimental studies; which rarely focus on the culvert section and transition section scenarios. The measurements are useful to validate prediction models for HST induced vibrations. The key findings are:

(14)

The specific multiplication sign “*” is adopted here to remind that not all the characteristic frequencies, but at least one, is necessarily included. And n represents the integer multiple. Based on the prediction of GRDF, one can estimate its Rayleigh wave length [57–59], which is crucial for the surface ground vibration isolation. Thus, HSR operators and engineers can select and design the ground mitigation measurements with more efficiency and economic rationality. Moreover, GRDF is significant for validating ground vibration simulation in frequency domain. It may also contribute to the simplification of HST's impact on the railway [60,61], especially for the researchers focusing on the ground vibration. Therefore, the HSTrailway interaction force F(t), which is composed of cosine functions (based on GRDFs), can be used to investigate the HST induced ground vibrations with enough accuracies. The simplified interaction force is then described in Eq. (15).

F (t ) = F0 +

i t ⎞⎟ ∑ Fi cos ⎛⎜2πf grd

⎝

⎠

(1) Generally, the vertical vibration is the largest in the near field, but in the far field the transversal one dominates. When a culvert or a viaduct is included, the longitudinal vibration would also be dominant in the near field. The horizontal vibration cannot be neglected at most situations, which is often neglected by most researchers who only study vertical components. (2) The three directional vibrations of the free field have similar frequency contents at the same point. Its dominant frequencies

(15)

It should be noted that this is a suggestion for estimating the train210

211

Amplification Zone After-AZ

Before-AZ

After-AZ

Amplification Zone

Before-AZ

Amplification Zone After-AZ

Before-AZ

Amplification Zone After-AZ

Before-AZ

1–1 1–2 1–3 1–4 1–5

1–1 1–2 1–3 4. Same Vibrations at Far Field; 1–4 5. Transversal Vibration Peak Arrive Earlyer; 1–5 1–1 1–2 1–3 1–4 1–5

1st:apx; 2st:apy & apz 1st:apx & apy; 2st:apz

1st:apz; 2st:apx; 3st:apy

1st:apx; 2st:apz & apy

1st:apy; 2st:apz; 3st:apx

1st:apy; 2st:apz; 3st:apx 1st:apx; 2st:apz & apy 1st:apz; 2st:apx & apy

2–1 2–2 2–3 2–4 2–5: 9.92 − 131.8 Hz 2–6: 9.92 − 50.0 Hz 7. Around 350 Hz on Viaduct

1. Fourier Amplitude along with Vibrations'; 2–1 2. Abundent in 0–50 Hz; 2–2 3. Scarce in 50–100 Hz; 2–3 4. Little in 100–150 Hz; 2–45. Dominant Range: 3.5 − 137.5 Hz; 2–5 6. Remarkable Range: 20 − 50 Hz; 2–6 2–1 2–2 2–3 2–4 2–5: 13.4 − 133.8 Hz 2–6: 20.2 − 50.0 Hz 7. Around 200 Hz on Culvert 2–1 2–2 2–3 2–4 2–5: 6.7 − 127.7 Hz 2–6: 6.7 − 50 Hz

1st:Vertical(apz) 2st:Transversal(apx) 3st:Longitudinal(apy) 1st:apz; 2st:apx; 3st:apy 1st:apx; 2st:apz & apy

1st:apz; 2st:apy; 3st:apx

Similarity

Difference

Similarity

1. Periodic Vibration Peak Values; 1–1 2. Train Geometrical Characteristic; 1–2 3. Vertical Vibration Outline Better; 1–3

Frequency Domain

Time Domain

1.33.0=1.0·fg1; 2.9.92=2.0·fg2; 3.36.3=1.1·fg1 1.33.0=1.0·fg1; 2.36.3=1.1·fg1; 3.19.8 1.33.0=1.0·fg1; 2.19.8; 3.9.92=2.0·fg2

3–1 3–2 3–3

3–1 3–2 3–3

3–1 3–2 3–3

1. Decay with Distance; 3–1 2. Weak Linear Uptrend with Speed; 3–2 3. Same Point Similar Increase Rate for 3Direction; 3–3

1-order.34.4 Hz=1.0·fg1; 2-order.31.0 Hz=0.9·fg1; 3-order.37.9 Hz=1.1·fg1 1.34.5=1.0·fg1; 2.37.9=1.1·fg1; 3.24.0 1.37.8=1.1·fg1; 2.34.4=1.0·fg1; 3.24.1

1.123.5=1.0·fg5 2.33.5=1.0·fg1; 3.10.0=2.0·fg2 1.33.5=1.0·fg1; 2.36.8=1.0·fg1; 3.30.1 1.36.8=1.1·fg1; 2.33.5=1.0·fg1; 3.26.9 1.12.8=1.1·fg3; 2.32.0=1.0·fg1; 3.9.55=2.0·fg2 1.9.55=2.0·fg2; 2.12.7=1.1·fg3; 3.32.0=1.0·fg1 1.9.55=2.0·fg2; 2.6.37=2.0·fp1; 3.25.7

Similarity

Difference

Vibration Amplitude

Notes: 1-1 represent the first similarity of Time Domain; 2-1 represents the first similarity of Frequency Domian; fgi can be found in Tables 1 and 4; AZ represents Amplification Zone. Notes: 1-order represent the first order dominant frequency of the vibration; apx, apy and apz represent transversal, longitudinal and vertical vibration amplitude.

Viaduct Section

Transition Section

Culvert Section

Embank-ment Section

Measurement Situations

Table 5 Summary of HST induced vibration in four sections.

Distance:Decay Slower; Speed:Increase Almost flat Distance:Decay Fastest; Speed:Increase Rate Gentle AZ:15 − 25 m; Vertical is Narrower Distance:Decay Slower; Speed:Increase Rate More Gentle

Distance:Decay Fastest; Speed:Decrease with Speed on Culvert AZ:15 − 25 m; Vertical is Narrower Distance:Decay Slower; Speed:Almost flat; Distance:Decay Fastest; Speed:Increase Rate Gentle AZ:20 − 30 m; Vertical is Narrower

AZ:22 − 32 m; Vertical is Narrower Distance:Decay Slower; Speed:Increase Rate More Gentle

Distance:Decay Fastest; Speed:Week Positive

Difference

4–1 4–2 4–3

4–1 4–2 4–3

4–1 4–2 4–3

1. Vary along with Vibrations'; 4–1 2. Decay slower than Vibrations'; 4–2 3. Less Uptrend with Speed than Vibrations'; 4–3

Vibration Level

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Table 6 The dynamic impact coefficient IFg for the vibration amplitudes (embankment section). Test Points k Vertical avk Average Error IFvk IFgdis IFg Transversal avk Average Error IFvk IFgdis IFg Longitudinal avk Average Error IFvk IFgdis IFg

1

2

3

4

5

6

7

8

9

10

0.079 0.016 0.207 0.13 0.14

0.045 0.004 0.086

0.031 0.003 0.094

0.045 0.005 0.120 0.15

0.069 0.008 0.109

0.042 0.009 0.207

0.036 0.004 0.108 0.15

0.027 0.004 0.141

0.019 0.005 0.277

0.011 0.001 0.093

0.085 0.014 0.207 0.11 0.13

0.040 0.002 0.086

0.019 0.002 0.094

0.062 0.008 0.120 0.11

0.058 0.005 0.109

0.065 0.006 0.207

0.047 0.006 0.108 0.17

0.036 0.009 0.141

0.025 0.006 0.277

0.012 0.001 0.093

0.040 0.002 0.207 0.11 0.11

0.029 0.004 0.086

0.016 0.002 0.094

0.027 0.002 0.120 0.07

0.030 0.002 0.109

0.038 0.002 0.207

0.029 0.003 0.108 0.13

0.018 0.001 0.141

0.020 0.006 0.277

0.012 0.001 0.093

Table 7 The dynamic impact coefficient IFg for the vibration amplitudes (culvert section). Test Points k Vertical avk Average Error IFvk dis IFg IFg Transversal avk Average Error IFvk dis IFg IFg Longitudinal avk Average Error IFvk dis IFg IFg

1

2

3

4

5

6

7

8

9

0.618 0.284

0.368 0.219

0.036 0.003

0.027 0.003

0.023 0.003

0.025 0.002

0.018 0.002

0.018 0.002

0.037 0.041

0.459 0.38 0.31

0.596

0.084

0.104 0.11

0.128

0.096

0.088 0.43

0.087

1.108

0.880 0.456

0.507 0.309

0.043 0.006

0.038 0.006

0.034 0.007

0.025 0.002

0.033 0.005

0.030 0.004

0.165 0.033

0.518 0.42 0.25

0.609

0.143

0.156 0.14

0.206

0.071

0.148 0.17

0.151

0.202

0.288 0.154

0.392 0.255

0.056 0.006

0.086 0.046

0.040 0.007

0.026 0.003

0.018 0.002

0.022 0.003

0.159 0.028

0.535 0.43 0.28

0.650

0.108

0.528 0.27

0.165

0.124

0.137 0.14

0.120

0.173

Table 9 The dynamic impact coefficient IFg for the vibration amplitudes (viaduct section). Test Points k Vertical avk Average Error IFvk dis IFg IFg Transversal avk Average Error IFvk dis IFg IFg Longitudinal avk Average Error IFvk dis IFg IFg

1

2

3

4

5

6

7

8

9

0.204 0.038

0.195 0.021

0.110 0.009

0.046 0.004

0.027 0.003

0.022 0.003

0.046 0.008

0.075 0.004

0.012 0.002

0.185 0.13 0.17

0.106

0.083

0.097 0.12

0.103

0.155

0.179 0.24

0.054

0.175

0.117 0.019

0.162 0.029

0.116 0.008

0.088 0.018

0.029 0.006

0.022 0.004

0.101 0.013

0.079 0.004

0.017 0.002

0.162 0.14 0.16

0.178

0.070

0.209 0.20

0.198

0.199

0.124 0.14

0.045

0.108

0.292 0.070

0.101 0.024

0.106 0.018

0.061 0.008

0.018 0.004

0.019 0.004

0.098 0.014

0.084 0.002

0.016 0.002

0.239 0.22 0.18

0.236

0.170

0.129 0.20

0.243

0.232

0.139 0.14

0.028

0.139

Table 8 The dynamic impact coefficient IFg for the vibration amplitudes (transition section). Test Points k Vertical avk Average Error IFvk IFgdis IFg Transversal avk Average Error IFvk IFgdis IFg Longitudinal avk Average Error IFvk IFgdis IFg

1

2

3

4

5

6

7

8

9

10

0.026 0.004 0.172 0.18 0.22

0.016 0.002 0.134

0.016 0.004 0.236

0.012 0.002 0.127 0.16

0.013 0.002 0.155

0.023 0.005 0.204

0.011 0.002 0.134 0.29

0.011 0.002 0.196

0.011 0.004 0.325

0.032 0.016 0.514

0.028 0.003 0.119 0.16 0.18

0.019 0.004 0.218

0.014 0.002 0.145

0.010 0.001 0.096 0.15

0.017 0.004 0.229

0.023 0.003 0.116

0.014 0.004 0.248 0.21

0.010 0.001 0.113

0.020 0.005 0.243

0.021 0.005 0.225

0.027 0.004 0.130 0.14 0.19

0.022 0.004 0.183

0.013 0.001 0.115

0.015 0.003 0.182 0.21

0.017 0.003 0.199

0.025 0.006 0.248

0.019 0.005 0.249 0.21

0.013 0.003 0.203

0.017 0.003 0.168

0.021 0.005 0.225

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(3)

(4)

(5)

(6) (7)

speed, are consistent with those of time domain peak values.

typically have multiple relation with at least one of the GCF, RCF, PCF or ICF. The wheelsets and bogies passage frequency play a more important role in the ground vibration. The quasi-static vibration induced by the HST has two qualities: periodic-ability and velocity-ability. The periodic-ability and dynamic excitations are likely to significantly contribute to the vibration of free field in different frequency bands, while the velocityability excitation makes a small contribution when the train speed is below the critical velocity. The vibration amplitudes commonly decrease with the distance from the track centerline. However, obvious amplification zones can be found in three directions at around 20 m due to the wave interference between 1) surface Rayleigh wave and reflected wave and/or 2) scattering waves from the neighboring fasteners. Generally, the ground vibrations of the embankment section are larger than those of the culvert, viaduct and transition sections in all three directions. The viaduct and transition sections would result in the least ground vibrations, typically. The acceleration amplitudes typically exhibit a weak positive relationship with the train speed. The variations of the vibration level with the distance and train

In general, the high range frequencies (> 100 Hz) greatly attenuate within the track structure and the middle range frequencies (50–100 Hz) attenuate within the field close to the railway. The efforts for ground vibration isolation should focus on the low range frequencies (0–50 Hz). Moreover, in the far field, the frequencies around or within 20 Hz is the key for ground vibration isolation. The introduced GDIC and GRDF may be valuable for better evaluating the ground vibration and vibration mitigation.

Acknowledgments Much of the work described in this paper was supported by the National Basic Research Program of China (973 Program) under Grant No. 2014CB049101. The authors would like to greatly acknowledge this financial support and express the most sincere gratitude. And the authors would like to thank Zhang-Long Chen, Feng-Lei Du, Guang-Xin Zhou and Yi-Cheng Li for their assistance in site experiment and data processing.

Appendix A See Table A1.

Table A1 Three dimensional characteristic parameters of the HST. Notation Train mc Jcx Jcy Jcz k2x c2x k2y c2y k2z c2z Bogie mb Jbx Jby Jbz k1x c1x k1y c1y k1z c1z Wheelset mw Jwx Jwy Jwz

Item

Unit

Value

Car body mass Bogie moment of inertia to the x-axle Bogie moment of inertia to the y-axle Bogie moment of inertia to the z-axle Secondary suspension spring x-axle Secondary suspension damper x-axle Secondary suspension spring y-axle Secondary suspension damper y-axle Secondary suspension spring z-axle Secondary suspension damper z-axle

ton kg·m2 kg·m2 kg·m2 kN/m kN·s/m kN/m kN·s/m kN/m kN·s/m

40.0 2.7×105 1.15×105 2.7×105 240 30 240 500 400 60

Bogie mass Bogie moment of inertia to the x-axle Bogie moment of inertia to the y-axle Bogie moment of inertia to the z-axle Primary suspension spring x-axle Primary suspension damper x-axle Primary suspension spring y-axle Primary suspension damper y-axle Primary suspension spring z-axle Primary suspension damper z-axle

ton kg·m2 kg·m2 kg·m2 kN/m kN·s/m kN/m kN·s/m kN/m kN·s/m

3.2 7200 3200 6800 240 0 240 0 400 40

Wheelset Wheelset Wheelset Wheelset

ton kg·m2 kg·m2 kg·m2

2.4 – 1200 –

mass moment of inertia to the x-axle moment of inertia to the y-axle moment of inertia to the z-axle

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